ReadingTimeMachine/rtm-sgt-ocr-v1

1source,target2 We have kept GRB090423 in our analysis. despite the debate regarding its classification as along burst (see Salvaterra et al.," We have kept GRB090423 in our analysis, despite the debate regarding its classification as a long burst (see Salvaterra et al."3" 2009) since its duration does satisfy Too>2 s. For bursts reported in Table 2 with spectra fitted by a ""Band law"". dN,/dE«E"" for E«E, and dN,/dExE! for E>E,. we find «a>=-0.8. «B>=-23. and «E,>=200 keV. Using these average parameters. we estimate. basec on Band (2003). a GBM sensitivity in the 50-300 keV band of M~ 0.6 ph em s7!."," 2009) since its duration does satisfy $T_{90}>2$ s. For bursts reported in Table 2 with spectra fitted by a ""Band law"", $dN_\gamma/dE\propto E^{\alpha}$ for $E<E_p$ and $dN_\gamma/dE\propto E^{\beta}$ for $E>E_p$, we find $<\alpha>\simeq -0.8$, $<\beta>\simeq -2.3$, and $<E_p>\simeq 200$ keV. Using these average parameters, we estimate, based on Band (2003), a GBM sensitivity in the 50-300 keV band of $P^{(50-300)\rm keV}_{\rm lim, GBM}\sim$ 0.6 ph $^{-2}$ $^{-1}$."4 For our LF analysis we use only the 144 long (Too>2 s) GBM bursts with peak flux higher thar that threshold., For our LF analysis we use only the 144 long $T_{90} > 2$ s) GBM bursts with peak flux higher than that threshold.5 In the BATSE sample we have included all LGRBs detectec while the BATSE onboard trigger was set to a significance of, In the BATSE sample we have included all LGRBs detected while the BATSE onboard trigger was set to a significance of6neutrals.,neutrals.7 In the above definitions. 2=|D|sqn[D]. so the sign of the Hall parameter is sensitive to the magnetic field polarity.," In the above definitions, $B \equiv8|\mathbf{B}|\, sgn\{B_z\}$, so the sign of the Hall parameter is sensitive to the magnetic field polarity."9" Lt is further assumed that the ions are ""heavy and the electrons are light. so that qiI."" Another kev parameter employed. in Paper L (see also WI93) is (where ij ds the ion mass densitv). the ratio of the Keplerian rotation time to the neutralion momentum exchange time."," It is further assumed that the ions are `heavy' and the electrons are `light', so that $q \ll101$ Another key parameter employed in Paper I (see also WK93) is (where $\rho_{\rm i}$ is the ion mass density), the ratio of the Keplerian rotation time to the neutral–ion momentum exchange time."11" In the ambipolar dilfusivitv limit. (eoo,»low] or. equivalently. [4|>> 1) the Elsasser number A (Section 2.2)) reduces to Y. whereas in the Hall cülfusivity limit (ap«Joy)σο or. equivalently. |]& 3p A=Ys.p! "," In the ambipolar diffusivity limit $\sigma_{\rm O} \gg \sigma_{\rm P} \gg12|\sigma_{\rm H}|$ or, equivalently, $|\beta_{\rm i}| \gg 1$ ) the Elsasser number $\Lambda$ (Section \ref{subsec:param}) ) reduces to $\Upsilon$ , whereas in the Hall diffusivity limit $\sigma_{\rm P} \ll13|\sigma_{\rm H}| \ll \sigma_{\rm O}$ or, equivalently, $|\beta_{\rm i}|14\ll 1 \ll |\beta_{\rm e}|$ ) $\Lambda = \Upsilon |\beta_{\rm i}|$."15The parameter constraints for the four Hall sub-regimes are presented in Table Al. which reproduces Table L1.," The parameter constraints for the four Hall sub-regimes are presented in Table \ref{table:constraints}, which reproduces Table I.1."16 The physical origin of the imposed constraints is summarized in the caption of this table., The physical origin of the imposed constraints is summarized in the caption of this table.17 Phe key predicted properties of the solutions in each sub-reeime are listed in Table A2.. which reprocduces Table L2.," The key predicted properties of the solutions in each sub-regime are listed in Table \ref{table:constraints1}, which reproduces Table I.2."18 We also reproduce below some of the expressions used in the derivation of these results that are relevant to the analysis presented in this paper., We also reproduce below some of the expressions used in the derivation of these results that are relevant to the analysis presented in this paper.19"N The ratio |db,db,|, is given by (equation L115). where 3=1/3ig."," The ratio $|db_r/db_\phi|_0$ is given by (equation I.115), where $\beta \equiv 1/\betio$."20 This expression can be used to approximate [b../bo.]. the ratio of the corresponding magnetic field components at the base of the wind.," This expression can be used to approximate $|b_{r{\rm b}}/b_{\phi{\rm b}}|$, the ratio of the corresponding magnetic field components at the base of the wind."21 Anotherrelationship between the field components at Zyds provided by A2 Al. 2.1)). 62. 63)). 4.3..," Anotherrelationship between the field components at $\zt_{\rm b}$ is provided by \ref{table:constraints1} \ref{table:constraints}, \ref{subsec:gov}) \ref{eq:kappa} \ref{eq:ele}) \ref{subsec:global}."22"galaxies, with no sign of ongoing interaction or disturbed morphology, in all five u,g,r,i, and z-bands from SDSS DR6 images.","galaxies, with no sign of ongoing interaction or disturbed morphology, in all five $u,g,r,i,$ and $z$ -bands from SDSS DR6 images."23 Cross-correlation of the SDSS sample with the LEDA catalogue has enabled us to investigate the variation of the scale lengths for different types of disk/spiral galaxies., Cross-correlation of the SDSS sample with the LEDA catalogue has enabled us to investigate the variation of the scale lengths for different types of disk/spiral galaxies.24" Although the typical scale length in u-band is larger than that in the r-band, the scale lengths in the g,r,7, and z-bands are similar and only become smaller on the average for late morphological types."," Although the typical scale length in $u$ -band is larger than that in the $r$ -band, the scale lengths in the $g,r,i,$ and $z$ -bands are similar and only become smaller on the average for late morphological types."25" This result remains consistent when using by-eye morphological classification or when using asymmetry parameter, concentration parameter, or velocity dispersion as an indicator for galaxy morphological type."," This result remains consistent when using by-eye morphological classification or when using asymmetry parameter, concentration parameter, or velocity dispersion as an indicator for galaxy morphological type."26" Our sample spans a range of total stellar masses between 1099and 10!?Mo with a typical galaxy mass of 101952931Mo, and shows that the while scale length increases for more massive galaxies, the scale length spread also increases with galaxy mass."," Our sample spans a range of total stellar masses between $10^{6.6}$and $10^{12.2} \rm{~M}_\odot$ with a typical galaxy mass of $10^{10.8\pm0.54} \rm{~M}_\odot$, and shows that the while scale length increases for more massive galaxies, the scale length spread also increases with galaxy mass."27" Overall, these results are in full agreement with the recent work by Courteauetal. (2007).."," Overall, these results are in full agreement with the recent work by \citet{Courteauetal07}."28" Scale length variations between bands are commonly studied to better understand the content and distribution of different stellar populations, metals, and/or dust."," Scale length variations between bands are commonly studied to better understand the content and distribution of different stellar populations, metals, and/or dust."29" A colour gradient is expected to increase from early-type spirals to late-type spirals, mainly due to extinction, which increases to later types."," A colour gradient is expected to increase from early-type spirals to late-type spirals, mainly due to extinction, which increases to later types."30" However, for Scd galaxies or later the colour gradient becomes smaller, because of decreasing amount of extinction (see,e.g.,Peletier&Balcells1996;deGrijs 1998)."," However, for Scd galaxies or later the colour gradient becomes smaller, because of decreasing amount of extinction \citep[see, e.g., ][]{PB96,deGrijs98}."31". Changes in galaxy scale length in different wavelengths can be attributed to extinction by moderate amounts of dust, with radial metallicity and age gradients as other contributing factors (Elmegreen&Elmegreen1984;Peletieretal.1994;Beckman 1996)."," Changes in galaxy scale length in different wavelengths can be attributed to extinction by moderate amounts of dust, with radial metallicity and age gradients as other contributing factors \citep[][]{EE84,Petal94, Betal96}."32". All these parameters wil probably change as a function of redshift, enabling us to measure the variation of intrinsic scale length with cosmological epoch."," All these parameters will probably change as a function of redshift, enabling us to measure the variation of intrinsic scale length with cosmological epoch."33" It is expected that the opacity of disk galaxies is expected to have been systematically higher in the past (e.g.,Dwek1998;Peietal.1999)."," It is expected that the opacity of disk galaxies is expected to have been systematically higher in the past \citep[e.g., ][]{Dwek98, Pei99}."34. The fact that the observed radial colour vary little suggests strongly that stellar population effects are not important here., The fact that the observed radial colour vary little suggests strongly that stellar population effects are not important here.35" Dust effects are studied by using radiative transfer models which take into account scattering as well as absorption by dust, and observationally by investigating the scale length ratio in different bands as a function of inclination, i.e., optical depth."," Dust effects are studied by using radiative transfer models which take into account scattering as well as absorption by dust, and observationally by investigating the scale length ratio in different bands as a function of inclination, i.e., optical depth."36" However, there are some degeneracies."," However, there are some degeneracies."37 Any tendency of the stars in the outer parts of disks to be bluer would tend to result in underestimated dust content., Any tendency of the stars in the outer parts of disks to be bluer would tend to result in underestimated dust content.38" Any tendency of the dust to concentrate towards the centre would result in an overestimate of the bluer scale lengths, and would not be distinguishable photometrically from a tendency of the stars in the outer disc to be bluer (c.f.Pohlen&Trujillo2006;Er- 2008).."," Any tendency of the dust to concentrate towards the centre would result in an overestimate of the bluer scale lengths, and would not be distinguishable photometrically from a tendency of the stars in the outer disc to be bluer \citep[c.f.][]{PT06,Erwinetal08,Azzolinietal08}. ."39" Peletieretal.(1994, found that scale length ratios could change, due to"," \citet{Petal94,Petal95} found that scale length ratios could change, due to"40To convert our V—4 colors to metallicities we use tlie relations derived by Couture aic Ixissler-Patigetal.(1998b).,To convert our $V-I$ colors to metallicities we use the relations derived by \citet{cou91} and \citet{kis98b}.41. These relatious provide the best calibrations for the low and the ugh metallicity rauges. respectively.," These relations provide the best calibrations for the low and the high metallicity ranges, respectively."42 Usiug the V.—/ values for our brighter iuagnitude iuterval. i.e. 0.91 and 1.09. the conversion relations vield metallicities of /H)~—1.50 aud —0.59 dex and [Fe/H]»—1.52 and —0.91 dex. for the blue and red peaks respectively.," Using the $V-I$ values for our brighter magnitude interval, i.e. $0.91$ and $1.09$ , the conversion relations yield metallicities of $\sim -1.50$ and $-0.59$ dex and $\sim -1.52$ and $-0.94$ dex, for the blue and red peaks respectively."43 The errors introduced by he internal uucertainties of the conversion relatious are around Q.:) dex., The errors introduced by the internal uncertainties of the conversion relations are around 0.3 dex.44 Note that if we used the 'eddeuing value adopted by Seckeretal.(1995) the peak colors would reddeu to (V—£)~¢).96 auc 1.11. correspoucdiug to metallicity shifts of ouly +0.2 dex.," Note that if we used the reddening value adopted by \citet{sec95} the peak colors would redden to $(V-I) \sim 0.96$ and 1.14, corresponding to metallicity shifts of only $\sim +0.2$ dex."45 These values we derive are typical or bright elliptical galaxies (Forbes. Brodie Crillimair 1997).," These values we derive are typical for bright elliptical galaxies (Forbes, Brodie Grillmair 1997)."46 In particular. the peaks in M87 are ound at V— £=0.95 and 1.20 mag.," In particular, the peaks in M87 are found at $V-I$ =0.95 and 1.20 mag."47 (Ixunduetal.1999) corresponding to [Fe/H]~—1.1a xd —0.6 dex (using the Ixissler-Patig et al., \citep{kun99} corresponding to $\sim -1.4$ and $-0.6$ dex (using the Kissler-Patig et al.48 relation)., relation).49 The peaks in NGC 1399 occur a V—1=0.99 aud 1.18 Csissler-Patigetal.1998b).. corresponding to [Fe/H]~—1.3 aud —0.6 dex.," The peaks in NGC 1399 occur at $V-I=$ 0.99 and 1.18 \citep{kis98b}, corresponding to $\sim -1.3$ and $-0.6$ dex."50 If auything. the uetal-rich peak in NGC 3311 may bave somewhat lower metallicity than the metal-rich peaks in he central cDs of the Virgo and Foruax clusters.," If anything, the metal–rich peak in NGC 3311 may have somewhat lower metallicity than the metal–rich peaks in the central cDs of the Virgo and Fornax clusters."51 The data of Seckeretal.(1995) were obtained uncer nonu-photometric conditions., The data of \citet{sec95} were obtained under non–photometric conditions.52 As described in their text. these authors clearly mace every attempt to correctly calibrate their frames using conventional aud appropriate techulques.," As described in their text, these authors clearly made every attempt to correctly calibrate their frames using conventional and appropriate techniques."53 However. given that our HST colors agree well with those or other central cD galaxies. globular cluster systems. it sees likely that tle Secker et al.," However, given that our HST colors agree well with those for other central cD galaxies' globular cluster systems, it seems likely that the Secker et al."54 results suller [rom a zero-point error., results suffer from a zero–point error.55 Zepletal.(1995). studied the globular clusters in NGC 3923 aud reported au exceptionally red color distribution for this galaxys luminosity., \citet{zep95} studied the globular clusters in NGC 3923 and reported an exceptionally red color distribution for this galaxy's luminosity.56 We note. however. that these observatious were mace on the same CTIO lin. uou-photometric. observing run as NGC 3311 auc were calibraed with the same data.," We note, however, that these observations were made on the same CTIO 4m, non–photometric, observing run as NGC 3311 and were calibrated with the same data."57 We suggest that these observatious be revisited iu light of he differences found for NGC 3311., We suggest that these observations be revisited in light of the differences found for NGC 3311.58 In the Coma cluster. WoodworthaudHarris(2000) [ouud. frou HST data. that IC 1051 las a high mean globular cluster metallicity (Fe/H]~—0.3 dex).," In the Coma cluster, \citet{woo00} found, from HST data, that IC 4051 has a high mean globular cluster metallicity $\sim -0.3$ dex)."59 The color distribution appears to be uuimocdal with a narrow dispersion. although Woodworth aud Harris were not able to exclude he possibility of a bi-modal distribution with two close peaks and a uetal-poor population arotud /H]~—1.0 dex.," The color distribution appears to be unimodal with a narrow dispersion, although Woodworth and Harris were not able to exclude the possibility of a bi–modal distribution with two close peaks and a metal–poor population around $\sim -1.0$ dex."60 The specific frequency of IC. 1051 is high (Sa= 12353). comparable to those of cluster central cDs. although IC 1051 is neitherthe central galaxy 1i tliis rich galaxy. cluster nor is it exceptionally luminous (My)— 21.9).," The specific frequency of IC 4051 is high $_N$ $\pm$ 3), comparable to those of cluster central cDs, although IC 4051 is neitherthe central galaxy in this rich galaxy cluster nor is it exceptionally luminous $_V -21.9$ )."61 Iu their survey of archival HST data. GebhardtandIExissler-Patis(1999). fouud evidence for," In their survey of archival HST data, \citet{geb99}62 found evidence for"63"After bars, spirals are the most prominent features of galaxy discs.","After bars, spirals are the most prominent features of galaxy discs."64 The question of whether the spirals are short- or long-lived features is of major importance to the internal evolution of disc galaxies., The question of whether the spirals are short- or long-lived features is of major importance to the internal evolution of disc galaxies.65" In this paper, I evaluate the evidence that spirals are transient features that change within a few dynamical times, but are superseded by fresh patterns in a recurrent manner."," In this paper, I evaluate the evidence that spirals are transient features that change within a few dynamical times, but are superseded by fresh patterns in a recurrent manner."66" We still lack a complete, and widely accepted, theory for the origin of spiral patterns in galaxies."," We still lack a complete, and widely accepted, theory for the origin of spiral patterns in galaxies."67" There is good evidence (e.g. that many prominent spiral patterns are found in barred galaxies, or are driven by tidal interactions, or perhaps even DM halo substructure (e.g.[Dubinskietal.|/2008)."," There is good evidence \citep[\eg][]{KN79} that many prominent spiral patterns are found in barred galaxies, or are driven by tidal interactions, or perhaps even DM halo substructure \citep[\eg][]{Dubi08}."68. There seems little doubt that a tidally-driven spiral pattern evolves rapidly (eg.2010). but repeated excitations may be possible (Byrd&Howard)," There seems little doubt that a tidally-driven spiral pattern evolves rapidly \citep[\eg][]{SL93,Dobb10}, but repeated excitations may be possible \citep{BH92}."69" If bars are long-lived features, as is generally believed ∙∙(foradissentingviewseeBournaud,Combes&Semelin spirals driven by bars could also be long-lived 2005),patterns."," If bars are long-lived features, as is generally believed \citep[for a dissenting view see][]{BCS05}, spirals driven by bars could also be long-lived patterns."70" However, simulations suggest that spirals and bars could have differing pattern speeds, which seemed to be supported by observational data ci although the latter group recently changed their minds Buiaal]2009"," However, simulations \citep{SS88} suggest that spirals and bars could have differing pattern speeds, which seemed to be supported by observational data \citep{SCJ03,Buta09}, although the latter group recently changed their minds \citep{Salo10}!"71"),, ∥Although many spirals in galaxies could be driven responses, the ubiquity of the spiral phenomenon suggests others are likely to be self-excited features of discs."," Although many spirals in galaxies could be driven responses, the ubiquity of the spiral phenomenon suggests others are likely to be self-excited features of discs."72" In particular, all driving agents can be excluded from N- body simulations of isolated stellar discs, which continue to manifest spiral patterns that must be self-excited."," In particular, all driving agents can be excluded from $N$ -body simulations of isolated stellar discs, which continue to manifest spiral patterns that must be self-excited."73" A satisfactory theory to account for self-excited spirals has yet to emerge,1977; despite decades of effort (seeinlreviewsby(Toomra[wood] 2010).."," A satisfactory theory to account for self-excited spirals has yet to emerge, despite decades of effort \citep[see reviews74 by][]{Toom77,Atha84,BL96,Sell10a}."75". While all agree spirals are& gravitationally- variations in the surface density of the old stellar disc, there is no consensus even on the expected lifetimes of the patterns."," While all agree spirals are gravitationally-driven variations in the surface density of the old stellar disc, there is no consensus even on the expected lifetimes of the patterns."76" Here I first explain why the duration of spiral features is an important issue, and then elaborate on the brief discussion of the lifetimes of spirals presented by hereafterBT08,p. 526).."," Here I first explain why the duration of spiral features is an important issue, and then elaborate on the brief discussion of the lifetimes of spirals presented by \citet[][hereafter BT08,77 p.~526]{BT08}."78 I evaluate four different types of evidence that bear on the question whether spirals are short- or long-lived patterns., I evaluate four different types of evidence that bear on the question whether spirals are short- or long-lived patterns.79 It is not obvious that the rate of evolution of galaxy discs is affected by the lifetimes of the patterns., It is not obvious that the rate of evolution of galaxy discs is affected by the lifetimes of the patterns.80" It may seem that if the time-averaged amplitude and pitch angle of transient spirals does not differ much from those of a steady long-lived pattern, the rate of change in the distribution of angular momentum and other quantities would be similar."," It may seem that if the time-averaged amplitude and pitch angle of transient spirals does not differ much from those of a steady long-lived pattern, the rate of change in the distribution of angular momentum and other quantities would be similar."81" However, the rate of angular momentum transport by spiral waves is given by the rate at which wave action (Lynden-Bell&Kalnajs|1972) is transported at the"," However, the rate of angular momentum transport by spiral waves is given by the rate at which wave action \citep{LBK72} is transported at the"82results from complex ω and real &. to real «o and complex Κ..,results from complex $\omega$ and real $k_z$ to real $\omega$ and complex $k_z$.83 The conversion is done following the method explained in TGV (see their Equation (40)., The conversion is done following the method explained in TGV (see their Equation (40)).84 First. we consider the same parameters as in Figure 2 and compute the numerically determined Lp/R versus vin/e.," First, we consider the same parameters as in Figure \ref{fig:ld} and compute the numerically determined $\ld/R$ versus $\nu_{\rm in}/\omega$."85 To compare with the analytical approximation. the numerical result is plotted using symbols © in Figure 2..," To compare with the analytical approximation, the numerical result is plotted using symbols $\Diamond$ in Figure \ref{fig:ld}."86 Α very good agreement between approximate and numerical results is found., A very good agreement between approximate and numerical results is found.87" This means that for wR/v,«1 the approximate analytical theory provides accurate results.", This means that for $\omega R / \vk \ll 1$ the approximate analytical theory provides accurate results.88 Next we numerically explore the effect of increasing the wave frequency., Next we numerically explore the effect of increasing the wave frequency.89 We take the same parameters as in Figure 3.., We take the same parameters as in Figure \ref{fig:ldw}.90 Again. we use symbols © to represent the eigenvalue result in Figure 3..," Again, we use symbols $\Diamond$ to represent the eigenvalue result in Figure \ref{fig:ldw}."91 We find that in the eigenvalue computations the transition between the regime dominated by resonant damping and that dominated by collisional damping occurs. around oRf/vy~d. while in the analytical approximation collisions start to become important for coR/vy~10.," We find that in the eigenvalue computations the transition between the regime dominated by resonant damping and that dominated by collisional damping occurs around $\omega R / \vk \sim 1$, while in the analytical approximation collisions start to become important for $\omega R / \vk \sim 10$."92 This discrepancy is an effect of the TT approximation., This discrepancy is an effect of the TT approximation.93 For realistic values of the wave frequency (ωδή~107— 1074) both numerical and analytic results are in excellent agreement., For realistic values of the wave frequency $\omega R / \vk \sim 10^{-2} - 10^{-1}$ ) both numerical and analytic results are in excellent agreement.94 The results of Section + have direct implications for MHD waves propagating in partially tonized plasmas of the solar atmosphere., The results of Section \ref{sec:tube} have direct implications for MHD waves propagating in partially ionized plasmas of the solar atmosphere.95 For kink waves studied in this paper. both resonant absorption and ion-neutral collisions decrease the amplitude of the waves.," For kink waves studied in this paper, both resonant absorption and ion-neutral collisions decrease the amplitude of the waves."96 However. the two processes represent very different physical mechanisms.," However, the two processes represent very different physical mechanisms."97 On the one hand. resonant absorption is an ideal process that transfers wave energy from global kink motions to localized azimuthal motions within. the. transversely inhomogeneous part of the flux tube.," On the one hand, resonant absorption is an ideal process that transfers wave energy from global kink motions to localized azimuthal motions within the transversely inhomogeneous part of the flux tube."98 These azimuthal motions keep propagating along magnetic field lines (seethenumeri-calsimulationsby.e.g..Pascoeetal.2010. 2011).," These azimuthal motions keep propagating along magnetic field lines \citep[see the numerical simulations by, e.g.,][]{pascoe1,pascoe2}."99. A detailed investigation of the energy transfer in the case of standing waves was done in Arreguietal.(2011). by analyzing the Poynting flux in à two-dimensional configuration., A detailed investigation of the energy transfer in the case of standing waves was done in \citet{arregui2d} by analyzing the Poynting flux in a two-dimensional configuration.100 However. resonant absorption itself does not dissipate wave energy in the plasma.," However, resonant absorption itself does not dissipate wave energy in the plasma."101 The energy fed into the inhomogeneous layer will be dissipated by another mechanism later (see.e.g..there-sultsofPoedtsetal.1989a.b.1990a.b.c.inresistive MHD).," The energy fed into the inhomogeneous layer will be dissipated by another mechanism later \citep[see, e.g., the results of][ in resistive MHD]{poedtskerner, poedtskerner2,poedts,poedts2,poedts3}."102. Hence. the damping length due to resonant absorption. Lpga. represents the length scale for the kink motions to be converted into azimuthal motions.," Hence, the damping length due to resonant absorption, $L_{\rm D, RA}$, represents the length scale for the kink motions to be converted into azimuthal motions."103 On the other hand. ion-neutral collisions is a true dissipative process which deposits wave energy and so it contributes to plasma heating.," On the other hand, ion-neutral collisions is a true dissipative process which deposits wave energy and so it contributes to plasma heating."104 Hence. the damping length due to ton-neutral collisions. Epin. represents the length scale for the kink wave energy to be dissipated by 10n-neutral collisions.," Hence, the damping length due to ion-neutral collisions, $L_{\rm D, IN}$, represents the length scale for the kink wave energy to be dissipated by ion-neutral collisions."105 By comparing the values of Epa and Lpin we can estimate the fraction of energy converted to Alfvénnic. azimuthal motions and the fraction of energy dissipated by collisions.," By comparing the values of $L_{\rm D, RA}$ and $L_{\rm D, IN}$ we can estimate the fraction of energy converted to Alfvénnic, azimuthal motions and the fraction of energy dissipated by collisions."106 Let us apply this theory to kink waves propagating along chromospheric waveguides (spicules)., Let us apply this theory to kink waves propagating along chromospheric waveguides (spicules).107 We assume that the driver of the waves is located at the photosperic level and the waves propagate through the chromosphere to the corona., We assume that the driver of the waves is located at the photosperic level and the waves propagate through the chromosphere to the corona.108 We take the variation of physical parameters with height (e.g.. density. temperature. ionization degree. ete.)," We take the variation of physical parameters with height (e.g., density, temperature, ionization degree, etc.)"109 from the VALC model (Vernazzaetal.1981)., from the VALC model \citep{valc}.110. For the chromospheric magnetic field we consider the model used by Leake&Arber(2006)., For the chromospheric magnetic field we consider the model used by \citet{leakearber}.111. Then we use Equations (64)) and (65)) to compute the values of Epa and Lpin.," Then we use Equations \ref{eq:ldrasf}) ) and \ref{eq:ldinsf}) ) to compute the values of $L_{\rm D, RA}$ and $L_{\rm D, IN}$."112 As the physical parameters change along the spicule. both {ρα and Lpjy are functions of height in the chromosphere.," As the physical parameters change along the spicule, both $L_{\rm D, RA}$ and $L_{\rm D, IN}$ are functions of height in the chromosphere."113 We plot in Figure 4((a) the values of the damping lengths as functions of height for a wave period of 45 s (Okamoto&DePontieu2011)., We plot in Figure \ref{fig:averagedld}( (a) the values of the damping lengths as functions of height for a wave period of 45 s \citep{okamotodepontieu}.114. The damping length due to resonant absorption increases with height., The damping length due to resonant absorption increases with height.115 This is an effect of the increase of the kink velocity. νι. with height (Soleretal.2011c).," This is an effect of the increase of the kink velocity, $\vk$, with height \citep{stratified}."116. At low heights. {ρα is comparable to the thickness of the whole chromosphere. meaning that à large fraction of wave energy is in the form of azimuthal motions when the wave reaches the coronal level.," At low heights, $L_{\rm D, RA}$ is comparable to the thickness of the whole chromosphere, meaning that a large fraction of wave energy is in the form of azimuthal motions when the wave reaches the coronal level."117 On the contrary. Liv is several orders of magnitude longer.," On the contrary, $L_{\rm D, IN}$ is several orders of magnitude longer."118 As Lpjn decreases with height. the effect of collisions is more important in the upper chromosphere.," As $L_{\rm D, IN}$ decreases with height, the effect of collisions is more important in the upper chromosphere."119 Next. we calculate the damping length averaged along the spicule. Lp. as where s represents the direction along the spicule and H is the height of the chromosphere above the photosphere.," Next, we calculate the damping length averaged along the spicule, $\bar{L}_{\rm D}$, as where $s$ represents the direction along the spicule and $H$ is the height of the chromosphere above the photosphere."120 We take H= 3.000 km.," We take $H=$ 3,000 km."121 Equation (66)) is used to calculate the averaged values of both {ρα and Lpin.," Equation \ref{eq:averagedld}) ) is used to calculate the averaged values of both $L_{\rm D, RA}$ and $L_{\rm D, IN}$."122 We plot in Figure 4((b) the averaged values of the damping lengths as functions of the wave period., We plot in Figure \ref{fig:averagedld}( (b) the averaged values of the damping lengths as functions of the wave period.123 First. we obtain that the averaged Lpin is several orders of magnitude longer than the averaged Lpga in the range of periods taken into account in Figure 4((b).," First, we obtain that the averaged $L_{\rm D, IN}$ is several orders of magnitude longer than the averaged $L_{\rm D, RA}$ in the range of periods taken into account in Figure \ref{fig:averagedld}( (b)."124 This means that ion-neutral collisions have little impact on wave propagation., This means that ion-neutral collisions have little impact on wave propagation.125 On the contrary. the averaged Lp for periods less than 1O s is smaller than or of the same order as the height of the chromosphere.," On the contrary, the averaged $L_{\rm D, RA}$ for periods less than 10 s is smaller than or of the same order as the height of the chromosphere."126 This result points out that only waves with periods longer than. 10 s are able to reach the coronal level in the form of kink motions., This result points out that only waves with periods longer than 10 s are able to reach the coronal level in the form of kink motions.127 Waves with shorter periods reach the corona as small-scale azimuthal motions. which are unobservable with present day instruments.," Waves with shorter periods reach the corona as small-scale azimuthal motions, which are unobservable with present day instruments."128 This effectively imposes a lower limit for the period of kink waves observable in the corona., This effectively imposes a lower limit for the period of kink waves observable in the corona.129 This 1s consistent with the observed periods of coronal waves (e.g..Tomezyketal.2007;&Meln-tosh2009:McIntoshetal.201 1).," This is consistent with the observed periods of coronal waves \citep[e.g.,][]{tomczyk07,tomczyk09,mcintosh2011}."130. In this paper we have investigated resonant Alfvénn waves in partially ionized plasmas., In this paper we have investigated resonant Alfvénn waves in partially ionized plasmas.131 We find that the conserved quantity at the resonance and the jump of the perturbations across the resonant layer are the same as in fully tonized. ideal plasmas.," We find that the conserved quantity at the resonance and the jump of the perturbations across the resonant layer are the same as in fully ionized, ideal plasmas."132 We have derived expressions for the damping lengths due to resonant absorption and due to ton-neutral collisions for the case of propagating kink waves in straight magnetic tubes., We have derived expressions for the damping lengths due to resonant absorption and due to ion-neutral collisions for the case of propagating kink waves in straight magnetic tubes.133 In the limit of large collision frequencies. the damping length," In the limit of large collision frequencies, the damping length"134(2008a).,.135. Backerouud stars reddened by interstellar extinction and unresolved galaxies could also populate he optical photometric sequence of the cluster., Background stars reddened by interstellar extinction and unresolved galaxies could also populate the optical photometric sequence of the cluster.136 Iu his case additional selection criteria are necessurv o distinguish bona fide cluster menibers from these contanunauts., In this case additional selection criteria are necessary to distinguish bona fide cluster members from these contaminants.137 The combination of optical and infrared data has proved to be a fiducial techuique to distiuguisli 1a fide cool cluster mcuibers from background objects (ZapateroOsorioetal.1997:.b:Martin2000:: DNZO).," The combination of optical and infrared data has proved to be a fiducial technique to distinguish bona fide cool cluster members from background objects \citealt{osorio97a,osorio97b,martin00}; BMZO)."138 The membership of most of the low mass stars and brown dwirfs (€ photometric sequeuces. m low-extinction clusters like he Pleiades and 6 Orionis. was later confined frou ΧΟΡΟΥ ΠΟΙΟΣ. radial velocity or the preseuce of lithiuu (ZapateroOsorioetal 1997b: Morauxetal. 2001: Ienvouetal. 2005: Bihainetal.2006: al. 2007)).," The membership of most of the low mass stars and brown dwarfs $>$ photometric sequences, in low-extinction clusters like the Pleiades and $\sigma$ Orionis, was later confirmed from proper motions, radial velocity or the presence of lithium \citealt{osorio97b}; ; \citealt{moraux01}; ; \citealt{kenyon05}; ; \citealt{bihain06}; \citealt{cab07a}) )."139 Figure 3. represents a J. [7 diagrn with cluster uenmber candidates from previous survevs (BZOR. Béjaretal 2001a)). indicated by open stars. aud the 113 objects with available J-baud photometry from the ποσο survey. represented by solid circles aud. open riangles.," Figure \ref{fig3} represents a $I$, $I$ $J$ diagram with cluster member candidates from previous surveys (BZOR, \citealt{bejar04a}) ), indicated by open stars, and the 143 objects with available $J$ -band photometry from the present survey, represented by solid circles and open triangles."140 All of them are within the /conpleteuess naenitude of the survey (J=16 uae)., All of them are within the completeness magnitude of the survey $I$ mag).141 According to evolutionary theoretical models. this corresponds to a uass juterval from 0.1 down to 0.013 ML...," According to evolutionary theoretical models, this corresponds to a mass interval from 0.1 down to 0.013 $_{\odot}$."142 It can be seen frou Figure 3. that 1214 candidates (solid. circles) of the 151 selected. objects iu the optical diagrams withinLB. the completeness maecuitude show redder colors and magnitudes brighter than the lower euvelope of the photometric sequence of previously coufizmied members. which roughly corresponds to the LO Myv isochrone: we will thus consider these objects as the likely photometric cluster member caudidates of the preseut survey.," It can be seen from Figure \ref{fig3} that 124 candidates (solid circles) of the 151 selected objects in the optical diagrams within the completeness magnitude show redder colors and magnitudes brighter than the lower envelope of the photometric sequence of previously confirmed members, which roughly corresponds to the 10 Myr isochrone; we will thus consider these objects as the likely photometric cluster member candidates of the present survey."143 There are 27 objects. within the completeucss of the survey. hat preseut bluer J colors than expected for the photometyic sequence of the cluster in Figure 23:5 these will be cousidered as probable nonauenmboers in the rest of the paper.," There are 27 objects, within the completeness of the survey, that present bluer $I-J$ colors than expected for the photometric sequence of the cluster in Figure \ref{fig3}; these will be considered as probable non-members in the rest of the paper."144 The full list of objects preseuted im this oper is eiven iu Table 2.., The full list of objects presented in this paper is given in Table \ref{tab2}.145 Their membership status is also indicated in the last columium of Table 2.., Their membership status is also indicated in the last column of Table \ref{tab2}.146 There are also two objects with 7> nunag for which there are 10. available infrared data deep cnough to restrict their o»longiue to the infrared photometric sequence., There are also two objects with $I\ge$ mag for which there are no available infrared data deep enough to restrict their belonging to the infrared photometric sequence.147 Since hey are not within the completeness magnitude of our survey. we will not consider them for the analysis of the very low ass stars and brown dwarfs of the cluster iu jext sections.," Since they are not within the completeness magnitude of our survey, we will not consider them for the analysis of the very low mass stars and brown dwarfs of the cluster in next sections."148 Iu this section we analyze the spatial distributious of the verv low παπα stars and brown dwarf caucliclates selected iu the present survey., In this section we analyze the spatial distributions of the very low mass stars and brown dwarf candidates selected in the present survey.149 We have selected 121 eood cluster member candidates witlin the completeness magnitude that follow both the optical and infrared photometric sequence of previously kuown low mass iienibers of σ Orionis., We have selected 124 good cluster member candidates within the completeness magnitude that follow both the optical and infrared photometric sequence of previously known low mass members of $\sigma$ Orionis.150 The first question arisen in our study is whether there is clear evidence of the existence of a clustering of substelhu: objects around the inultiple star σ Oriouis., The first question arisen in our study is whether there is clear evidence of the existence of a clustering of substellar objects around the multiple star $\sigma$ Orionis.151 The representation of the spatial distribution in Figure shows that there isa conceutratiou of substellar objects around σ Oriouis., The representation of the spatial distribution in Figure shows that there isa concentration of substellar objects around $\sigma$ Orionis.152 To test this we have calculated the distributions of the object density per arcmin? iu the oesent survey along the à and ó axis centered on tle σ Orvionis AB coordinates., To test this we have calculated the distributions of the object density per $^2$ in the present survey along the $\alpha$ and $\delta$ axis centered on the $\sigma$ Orionis AB coordinates.153 A representation of these distributions can be seen in the top aud bottom panels of Figure L.., A representation of these distributions can be seen in the top and bottom panels of Figure \ref{fig4}.154 It can be seen that the distribution decreases roni the central star in both the à and the à axis. indicating the existence of a greater concentration of objects around σ Orionis.," It can be seen that the distribution decreases from the central star in both the $\alpha$ and the $\delta$ axis, indicating the existence of a greater concentration of objects around $\sigma$ Orionis."155 It is interesting to note that roni both figures we can see that there is also au increase oeithe last bins (at separations larger than 30 arciiu) to he north and west of σ Orionis., It is interesting to note that from both figures we can see that there is also an increase in the last bins (at separations larger than 30 arcmin) to the north and west of $\sigma$ Orionis.156 Some of these objects are located at distances closer to the star ¢ Orlouis and could be related to the existence of a substellar o»pulation around this star., Some of these objects are located at distances closer to the star $\zeta$ Orionis and could be related to the existence of a substellar population around this star.157 The rest of these objects are located closer to σ Orionis than to any other OD star. but this population can be related to the sparse. wide clustering around the € Oriouis cluster (Sherry2003:Bricenoctal.2005:Caballero&Solano 2008).," The rest of these objects are located closer to $\sigma$ Orionis than to any other OB star, but this population can be related to the sparse, wide clustering around the $\epsilon$ Orionis cluster \citep{sherry03,briceno05,cabysol08}."158. It might be that the population ideuti&ed bv Jeffriesetal.(2006).. which is kinematically cüfferent from the σ Oviouis cluster aud sec to be concentrated to the west of the star. is also related with the excess of sources observed.," It might be that the population identified by \cite{jeffries06}, which is kinematically different from the $\sigma$ Orionis cluster and seem to be concentrated to the north-west of the star, is also related with the excess of sources observed."159 Objects in this region are identified in the last column of Table 2.., Objects in this region are identified in the last column of Table \ref{tab2}.160 After discarding this population of possible Oriou background. objects a total of 102 bona fide member candidates remains concentrated around the σ Orionis star., After discarding this population of possible Orion background objects a total of 102 bona fide member candidates remains concentrated around the $\sigma$ Orionis star.161 We determine that the central coordinates of the cluster. according to the substellar distribution. are within the ceutral bius of 5 arcimin around. the massive star.," We determine that the central coordinates of the cluster, according to the substellar distribution, are within the central bins of 5 arcmin around the massive star."162 This estimate is not τον precise because of the limits imposed bv the ecometry of our survey ancl the radial dependence of the object density ina cluster., This estimate is not very precise because of the limits imposed by the geometry of our survey and the radial dependence of the object density in a cluster.163 The center of mass of the umltiple star σ Ovionis is located a few arcseconds from the ceutral star o Oriouis AB (see Caballero2007. 2008b)).," The center of mass of the multiple star $\sigma$ Orionis is located a few arcseconds from the central star $\sigma$ Orionis AB (see \citealt{cab07,cab08b}) )."164 We can estimate the ceuter of mass of the substellar population of the cluster aud fiud that this is located within 2 arcuünu of the more niüassive stars: consideriue the uncertainties. we can argue that they are the same.," We can estimate the center of mass of the substellar population of the cluster and find that this is located within 2 arcmin of the more massive stars; considering the uncertainties, we can argue that they are the same."165 Iu the remaining discussion of the spatial distribution we consider the coordinates of 7 Orionis AB as the ceutral coordinates of the cluster., In the remaining discussion of the spatial distribution we consider the coordinates of $\sigma$ Orionis AB as the central coordinates of the cluster.166 The uext objective iu our study is to characterize the radial distribution of the surface deusitv of the substellar objects in the cluster., The next objective in our study is to characterize the radial distribution of the surface density of the substellar objects in the cluster.167 Di order to calculate this we have estimated the surface deusitv of the umuber of objects in concentric coronas at different distances to the ceuter of the cluster., In order to calculate this we have estimated the surface density of the number of objects in concentric coronas at different distances to the center of the cluster.168 This is a decreasing function with distance to the center and is shown in Fieure 5.., This is a decreasing function with distance to the center and is shown in Figure \ref{fig5}.169 The increase in this distribution at distances larger than 30 arcinin can be explained by the presence of other population of Orion (see previous subsection)., The increase in this distribution at distances larger than 30 arcmin can be explained by the presence of other population of Orion (see previous subsection).170 We can try to adjust this distribution with different clupizical functions obtained for starclusters such as an exponential (VandenBergh&Sher1960) or a Ίνπιο distribution (sine 1962).., We can try to adjust this distribution with different empirical functions obtained for starclusters such as an exponential \citep{van60} or a King distribution \citep{king62}. .171 In the case of the σ Orionis cluster. duc to the sharpness of the decay audthe contamination of a backeround population at larec distances. the former seenis to be more suitable.," In the case of the $\sigma$ Orionis cluster, due to the sharpness of the decay andthe contamination of a background population at large distances, the former seems to be more suitable."172 Iu addition.the Kine function was developed to reproduce," In addition,the King function was developed to reproduce"173to the SDSSO2 criteria: These basic selection criteria result in 861 hosts and 2310 satellites.,to the SDSS02 criteria: These basic selection criteria result in 864 hosts and 2340 satellites.174 As noted bv SDSS02. however. may of the hosts have a large nuniber of satellites. around them Gn one case. a potential host iun our sample has 605 satellites}.," As noted by SDSS02, however, many of the hosts have a large number of satellites around them (in one case, a potential host in our sample has 605 satellites)."175 These are. therefore. most likely to be associated with cluster svstenis. rather than απο trulv isolated.," These are, therefore, most likely to be associated with cluster systems, rather than being truly isolated."176 To eliminate these objects. we impose a further restriction that the luminosity of the host be ereater than the stm total of the ΗχοςΊος of the satellites.," To eliminate these objects, we impose a further restriction that the luminosity of the host be greater than the sum total of the luminosities of the satellites."177 This. too. was done by SDSS02. aud reduces our 2dF souple to 859 hosts and 1693 satellites.," This, too, was done by SDSS02, and reduces our 2dF sample to 859 hosts and 1693 satellites."178" Finally, we impose two additional cuts on the lost ealaxies."," Finally, we impose two additional cuts on the host galaxies."179" First. eveball inorphiologies are available for the 2dF ealaxies with b;z18. aud 29 of the above hosts are classified as galaxvgalaxy οσο»,"," First, eyeball morphologies are available for the 2dF galaxies with $b_J \ls 18$, and 29 of the above hosts are classified as galaxy–galaxy mergers."180 We delete these hosts from the sample ou the basis that they are unlikely to be fully relaxed systems., We delete these hosts from the sample on the basis that they are unlikely to be fully relaxed systems.181 Secoud. we delete all hosts with £>6£° because the velocity dispersions of their satellites are poorly fit by the technique we adopt (see below). aud the uuuber of iuterloper galaxies (as opposed to ecnuine satellites) appears to be both large (2 15) and have a laree dispersion (~20% )).," Second, we delete all hosts with $L > 6 L^\ast$ because the velocity dispersions of their satellites are poorly fit by the technique we adopt (see below), and the number of interloper galaxies (as opposed to genuine satellites) appears to be both large $\gs 45$ ) and have a large dispersion $\sim 20$ )."182 These additional cuts leave us with a final sample of 809 host ealaxies aud 1556 satellites., These additional cuts leave us with a final sample of 809 host galaxies and 1556 satellites.183 Of these. 75 are classified as ellipticals. 81 are classified as SO. aud 213 are classified as spirals.," Of these, 75 are classified as ellipticals, 84 are classified as S0, and 243 are classified as spirals."184 The sunuple of spirals is uniforiülv distributed in inclination auele. aud there is no correlation between host Iunuinositv and median inclination angle.," The sample of spirals is uniformly distributed in inclination angle, and there is no correlation between host luminosity and median inclination angle."185 The ellipticals have a total of 171 satellites. the SOs have a total of 303 satellites. and the spirals have a total of 178 satellites.," The ellipticals have a total of 171 satellites, the S0's have a total of 303 satellites, and the spirals have a total of 478 satellites."186 The median redshift of the S09 host galaxies in the full sample is ted=0.073.while for the spiral hosts μα=0.055. and for the elliptical aud SO hosts τμ=0.062.," The median redshift of the 809 host galaxies in the full sample is $z_{\rm med}= 0.073$,while for the spiral hosts $z_{\rm med} = 0.055$, and for the elliptical and S0 hosts $z_{\rm med} = 0.062$."187 The probability distribution of the luminosities of the host galaxies. the probability distribution of the nuuber of satellites around. individual hosts. and the probability distribution of the difference iun apparent b; magnitude between the hosts and their satellites are shown iu paucls ἂν b. aud c of 1. 2. and 3.," The probability distribution of the luminosities of the host galaxies, the probability distribution of the number of satellites around individual hosts, and the probability distribution of the difference in apparent $b_J$ magnitude between the hosts and their satellites are shown in panels a, b, and c of 1, 2, and 3."188 11 shows results for the cutive sample of 5089 hosts. while 22 shows the results for the 159 hosts classified as elliptical or SO. aud 33 shows the results for the 213 hosts classified as spirals.," 1 shows results for the entire sample of 809 hosts, while 2 shows the results for the 159 hosts classified as elliptical or S0, and 3 shows the results for the 243 hosts classified as spirals."189 Iu order to compare with SDSS02. we adopt an analysis technique that is ideutical to theirs.," In order to compare with SDSS02, we adopt an analysis technique that is identical to theirs."190 The radial velocity dispersions of the host ealaxy halos. σοι are computed by fitting a combination of a Gaussian and a coustaut offset to histograms of the velocity differences between the hosts and satellites.," The radial velocity dispersions of the host galaxy halos, $\sigma_v$, are computed by fitting a combination of a Gaussian and a constant offset to histograms of the velocity differences between the hosts and satellites."191 The width of the bestfitting Gaussian is a nueasure of σε. while the offset accounts for the fact that there will. necessarily. be some fraction of iuterloper ealaxies that are selected as satellites when. im fact. they are nof dynamically associated with the host galaxy.," The width of the best–fitting Gaussian is a measure of $\sigma_v$, while the offset accounts for the fact that there will, necessarily, be some fraction of interloper galaxies that are selected as satellites when, in fact, they are not dynamically associated with the host galaxy."192" Like SDSS02. we fiud that this techuique provides very good fits to the velocity difference histoerams. vielding values of 4? per deeree of freedom. \7/v. that are z1 for hosts with L<62""."," Like SDSS02, we find that this technique provides very good fits to the velocity difference histograms, yielding values of $\chi^2$ per degree of freedom, $\chi^2/\nu$, that are $\ls 1$ for hosts with $L \le 6L^\ast$."193 Tu the case of hosts with L> 6L. ον22.5 anc. hence. we do not consider these objects further.," In the case of hosts with $L > 6L^\ast$ $\chi^2/\nu \gs 2.5$ and, hence, we do not consider these objects further."194μα simlessbox-DhboxzDhtDdp22—0pt Many tvpes of astrophysical objects are powered by black hole (BID) accretion.,1=0pt 2=0pt Many types of astrophysical objects are powered by black hole (BH) accretion.195 Racliation enerey produced by accretion can be very hieh and can explain such dramatic phenomena as quasars. powerful radio galaxies. X-ray binaries. and gamma rav. bursts (GRBs).," Radiation energy produced by accretion can be very high and can explain such dramatic phenomena as quasars, powerful radio galaxies, X-ray binaries, and gamma ray bursts (GRBs)."196 However. DII accretion does not always result in high radiative output.," However, BH accretion does not always result in high radiative output."197 This is true for both stellar DII and super massive black holes (SMBID)., This is true for both stellar BH and super massive black holes (SMBH).198 In particular. objects with SAIBIL appear to spend statistically most of their time in an inactive phase.," In particular, objects with SMBH appear to spend statistically most of their time in an inactive phase."199 Inactive SMDIIS are not something that one would expect. because (hese black holes are embedded in the relatively dense environments of galactic nuclei.," Inactive SMBHs are not something that one would expect, because these black holes are embedded in the relatively dense environments of galactic nuclei."200 Therefore it is natural to suppose that the gravity due to an SMDII will draw in matter at high rates. leading to a high svstem luminosity.," Therefore it is natural to suppose that the gravity due to an SMBH will draw in matter at high rates, leading to a high system luminosity."201 Monitoring of X-ray. binaries with stellar DIHIs reveals (hat these objects often exhibit large time variability in (he total energy output in the spectral energy. distribution., Monitoring of X-ray binaries with stellar BHs reveals that these objects often exhibit large time variability in the total energy output in the spectral energy distribution.202 Then one of the main goals of anv theory of DII accretion is (wo explain why accretion proceeds through very different mocles., Then one of the main goals of any theory of BH accretion is two explain why accretion proceeds through very different modes.203" Generally. the radiative output. [rom accretion depends on the mass accretion rate AL, and an efficiency factor. η."," Generally, the radiative output from accretion depends on the mass accretion rate $\MDOT_a$ and an efficiency factor, $\eta$."204" In. previous papers of this series (Proga Begelman 2003a.b. hereafter PDO03a. ancl PBO3b. respectively). we studied how plvsical conditions at large distances from SMDIT affect. A, in the so-called radiatively inefficient accretion flows (RIAF)."," In previous papers of this series (Proga Begelman 2003a,b, hereafter PB03a and PB03b, respectively), we studied how physical conditions at large distances from SMBH affect $\MDOT_a$ in the so-called radiatively inefficient accretion flows (RIAF)."205" RIAF with very low η and also low M, have been proposed to explain very low radiative Iuminosities in svstems as such Ser À* (e.g.. Ichimaru 1977: Rees οἱ al."," RIAF with very low $\eta$ and also low $\MDOT_a$ have been proposed to explain very low radiative luminosities in systems as such Sgr A* (e.g., Ichimaru 1977; Rees et al."206 1982: Naravan Yi 1994. 1995: Abramowiez et al.," 1982; Narayan Yi 1994, 1995; Abramowicz et al."207 1995: Blandford Begelman 1999: Sharma et al., 1995; Blandford Begelman 1999; Sharma et al.208 2007 and reference therein)., 2007 and reference therein).209" In PBO3a. we addressed the issue of how M, depends on the distribution of specilic angular momentum. / at large radii assuming that the adiabatic index. 5=5/3."," In PB03a, we addressed the issue of how $\MDOT_a$ depends on the distribution of specific angular momentum, $l$ at large radii assuming that the adiabatic index, $\gamma=5/3$."210 For high /. corresponding to the circularization radius larger than the last stable orbit. eas cannot directly accrete onto a BIL. unless some physical mechanism like e.g.. viscosity or magnetic fields. or both. (rausports angular momentum towards.," For high $l$, corresponding to the circularization radius larger than the last stable orbit, gas cannot directly accrete onto a BH, unless some physical mechanism like e.g., viscosity or magnetic fields, or both, transports angular momentum towards."211 For inviscid accretion flow (he matter with too high /. either flows outward due to gas pressure ancl centrifugal forces or," For inviscid accretion flow the matter with too high $l$, either flows outward due to gas pressure and centrifugal forces or"212Application to the Crab pulsar vields that the primary beam produces an IC luminosity by upseattering the X-ray photons with keV energies. luminosity of Ly~10° erg s|. tha 1s. xy —D This is much lower then the IC: luminosity produced by upscattering the UV photons wilh eV energies. Li:c1011 erg sf:,"Application to the Crab pulsar yields that the primary beam produces an IC luminosity by upscattering the X-ray photons with keV energies, luminosity of $L_{X} \sim 10^{36} $ erg $s^{-1}$, that is, =5 This is much lower then the IC luminosity produced by upscattering the UV photons with eV energies, $L_{UV} \sim 10^{34} $ erg $s^{-1}$:."213 The above is an estimate of the peak power., The above is an estimate of the peak power.214 The average Iuminositv is lower bv at least one order of magnitude., The average luminosity is lower by at least one order of magnitude.215 Thus. we conclude that the IC scattering by the primary beam is unlikelv to be the origin of the VERITAS signal.," Thus, we conclude that the IC scattering by the primary beam is unlikely to be the origin of the VERITAS signal."216 In the previous section we discussed (he gamma-ray emission produced by (he primary bean., In the previous section we discussed the gamma-ray emission produced by the primary beam.217 In this section we discuss (he gamma-ray emission bv (he particles that are produced in pair cascades of the particles in the primary beam. the secondary plasma.," In this section we discuss the gamma-ray emission by the particles that are produced in pair cascades of the particles in the primary beam, the secondary plasma."218 We recall that the primary beam has a density 26;; and a Lorentz factor 55 (5))., We recall that the primary beam has a density $n_{GJ}$ and a Lorentz factor $\gamma_b$ \ref{1}) ).219" As nomenclature [or the secondaryplasma we use n, for its density and 5, lov its Lorentz factor.", As nomenclature for the secondaryplasma we use $n_p$ for its density and $\gamma_p$ for its Lorentz factor.220 We assume energv equiparliGon between (he primary beam ancl the secondary plasma here).., We assume energy equipartition between the primary beam and the secondary plasma \citep[the assumption of equipartition between the primary beam and secondary plasma is justified in the polar cap models;][ we assume a similar parametrization here]{1996ApJ...458..278D}.221" From equipartition it follows that np5,= Πεν.", From equipartition it follows that $n_p \gamma_p = n_{GJ} \gamma_b$ .222 The two particle populations are connected through (he pair cascading process. np=Aptos. Where A=LOOAs is the multiplicity factor of the secondary particles.," The two particle populations are connected through the pair cascading process, $n_p = \lambda_p n_{GJ}$, where $\lambda = 100 \lambda_2 $ is the multiplicity factor of the secondary particles."223 Multiplicities of the order A~10? are typical in ouler gap models (e.g..Wang&IHirotani 2011).. but canalso reach much higher values. Àc104—109 (Takataοἱal. 2010)..," Multiplicities of the order $\lambda \sim 10^2$ are typical in outer gap models \citep[\eg][]{2011ApJ...736..127W}, , but canalso reach much higher values, $\lambda \sim 10^4-10^6$ \citep{2010ApJ...715.1318T}. ."224ssuch that we get ad Integrating equation (53) wwhere ©) is an arbitrarv constant of integration.,such that we get and Integrating equation (58) where $\phi _{0}$ is an arbitrary constant of integration.225 It is very difficult to integrate the equation (60) in a closed form., It is very difficult to integrate the equation (60) in a closed form.226 However if at this stage we take the m=—1£56... 3i ihen we are able (ο integrate and get the following solution.," However if at this stage we take the $m = -1$, $\frac{\gamma k}{M} = 1$ then we are able to integrate and get the following solution."227 and, and228(<0.1 mag).,$\leq 0.1$ mag).229 Also in accordance with the results obtained in Section 3.1.. the light curve is brightest and rises to maximum fastest when viewed downm.," Also in accordance with the results obtained in Section \ref{sect:toy-models}, the light curve is brightest and rises to maximum fastest when viewed down."230 In contrast to the toy-model results. however. the anti-parallel direction does not show the dimmest light curve for the 3D model — this is a consequence of the more complex distribution of ash material which determines both the location of energy generation (Ni) and the distribution of opacity (via the dependence on Xi in equation 5).," In contrast to the toy-model results, however, the anti-parallel direction does not show the dimmest light curve for the 3D model – this is a consequence of the more complex distribution of ash material which determines both the location of energy generation $^{56}$ Ni) and the distribution of opacity (via the dependence on $X_{\mbox{\scriptsize Fe-grp}}$ in equation 5)."231 As might be anticipated from Figure 5 (lower panels). the dimmest light curves are actually seen by observers whose line-of-sight is nearly perpendicular to (see below).," As might be anticipated from Figure \ref{fig:td} (lower panels), the dimmest light curves are actually seen by observers whose line-of-sight is nearly perpendicular to (see below)."232 While the light curves shown in Figure 6. indicate that it is possible to see significant differences in the peak magnitude as a function of direction. it is important to consider how probable a randomly aligned observer is to see a given range of light curve parameters.," While the light curves shown in Figure \ref{fig:3d-lcs} indicate that it is possible to see significant differences in the peak magnitude as a function of direction, it is important to consider how probable a randomly aligned observer is to see a given range of light curve parameters."233 To this end. a grid of 100 viewing directions (evenly spaced in solid angle). has been employed to map out the angular distribution of the light curve properties.," To this end, a grid of 100 viewing directions (evenly spaced in solid angle), has been employed to map out the angular distribution of the light curve properties."234 In Figure 7. the peak magnitudes are shown for this grid of directions as a function of the polar angle. 9. (6 being the angle between the z-axis of the model and the observer's line-of-sight).," In Figure \ref{fig:3d-mpeak} the peak magnitudes are shown for this grid of directions as a function of the polar angle, $\theta$, $\theta$ being the angle between the $z$ -axis of the model and the observer's line-of-sight)."235" There is clearly a net trend in the variation of AZ, with 6: for large values of cos@ the mean trend is very similar to that found in the toy models.", There is clearly a net trend in the variation of $M_{\rm p}$ with $\theta$; for large values of $\cos \theta$ the mean trend is very similar to that found in the toy models.236 But. as noted above. owing to the complex distribution of ash material. a much weaker trend is revealed in the region where cos8.«0.," But, as noted above, owing to the complex distribution of ash material, a much weaker trend is revealed in the region where $\cos \theta < 0$."237 The grid of light curves as a function of viewing directions can be used to determine the probability distribution for the peak magnitude., The grid of light curves as a function of viewing directions can be used to determine the probability distribution for the peak magnitude.238 This has been done and the resulting cumulative probability distribution is shown in Figure 8.., This has been done and the resulting cumulative probability distribution is shown in Figure \ref{fig:p-dist}.239 Although more complex in detail. the probability distribution function (pdf) obtained shares most of the important properties suggested by the toy models: namely. it is fairly wide (spanning ~ 0.5 mag) and the probability is not strongly concentrated around the median (for example. fully one quarter of the directions give peak magnitudes that are brighter than the median by at least 0.1 mag).," Although more complex in detail, the probability distribution function (pdf) obtained shares most of the important properties suggested by the toy models; namely, it is fairly wide (spanning $\sim$ 0.5 mag) and the probability is not strongly concentrated around the median (for example, fully one quarter of the directions give peak magnitudes that are brighter than the median by at least 0.1 mag)."240 An important difference from the pdf implied by the toy models is a distinct asymmetry (as could be anticipated from Figure 7)) — there is a probability tail extending to brighter magnitudes meaning that one may see larger differences on the bright side than on the dim side of the median., An important difference from the pdf implied by the toy models is a distinct asymmetry (as could be anticipated from Figure \ref{fig:3d-mpeak}) ) – there is a probability tail extending to brighter magnitudes meaning that one may see larger differences on the bright side than on the dim side of the median.241" The probability distribution for A, will be used in Section 5.2) to quantify the relevance of this work to the interpretation of the local Hubble diagram.", The probability distribution for $M_{\rm p}$ will be used in Section \ref{sect:hubble} to quantify the relevance of this work to the interpretation of the local Hubble diagram.242 For completeness. Figure 9. shows the range of light-curve rise times with viewing direction.," For completeness, Figure \ref{fig:3d-tpeak} shows the range of light-curve rise times with viewing direction."243 As with the toys models. it can be seen that there is significant diversity in the rise time and that it tends to be shortest when viewing down (i.e. close to #= 0) and longest when viewing closer to—7 (i.e. €.— s.," As with the toys models, it can be seen that there is significant diversity in the rise time and that it tends to be shortest when viewing down (i.e. close to $\theta = 0$ ) and longest when viewing closer to (i.e. $\theta \sim \pi$ )."244 In the next section. we examine the relationship between rise time and peak magnitude in some detail.," In the next section, we examine the relationship between rise time and peak magnitude in some detail."245 We now consider some ramitications of the results obtained from the 3D model that has been discussed in Section 4.., We now consider some ramifications of the results obtained from the 3D model that has been discussed in Section \ref{sect:real_model}.246 In Section 3.?. we used the toy models to investigate the possible systematic effects of an off-centre explosion on the inference of Ni masses., In Section \ref{sect:toy-discuss} we used the toy models to investigate the possible systematic effects of an off-centre explosion on the inference of Ni masses.247 Here. we extend that discussion using the results obtained with the 3T2d200 explosion model.," Here, we extend that discussion using the results obtained with the 3T2d200 explosion model."248where the dusty L dwarf atmospheres begin to clear. and the objeet's atmosphere becomes dominated by methane absorption. the defining characteristic of T dwarfs.,"where the dusty L dwarf atmospheres begin to clear, and the object's atmosphere becomes dominated by methane absorption, the defining characteristic of T dwarfs."249" The processes involved are not vet well understood. and the majority of evolutionary models do not cover this region. instead having a “dusty” set of models (e.g. DUSTY, ?: CI00. 23) and a clear atmosphere set of models (e.g. COND. ?:: Clr. 23) for the later T dwarfs."," The processes involved are not yet well understood, and the majority of evolutionary models do not cover this region, instead having a “dusty” set of models (e.g. DUSTY, \citealt{chabrier00}; C100, \citealt{burrows06}) ) and a clear atmosphere set of models (e.g. COND, \citealt{baraffe03}; Clr, \citealt{burrows06}) ) for the later T dwarfs."250 ? attempted to reproduce this region using a hybrid model for the colour evolution and cooling of the brown dwarfs., \citet*{saumon08} attempted to reproduce this region using a hybrid model for the colour evolution and cooling of the brown dwarfs.251 They had some success in reproducing the photometric characteristics of the field population. but made less progress with the younger members of the Pleiades cluster.," They had some success in reproducing the photometric characteristics of the field population, but made less progress with the younger members of the Pleiades cluster."252 These currently known low-gravity objects have been identitied as such thanks to fiducial age constraints demonstrating the youth of the systems., These currently known low-gravity objects have been identified as such thanks to fiducial age constraints demonstrating the youth of the systems.253 Such objects represent benchmarks as with a known age (either from their being a cluster member or in a binary with a star/ white dwarf of a known age). they can be used to critically examine the predictions made by evolutionary models.," Such objects represent benchmarks as with a known age (either from their being a cluster member or in a binary with a star/ white dwarf of a known age), they can be used to critically examine the predictions made by evolutionary models."254 Brown dwarfs are classified as either L or T dwarfs based on the strengths of lines within their spectra. and the general spectral shape in comparison to standards (see ο for a review). although estimates based on broadband photometry are often made.," Brown dwarfs are classified as either L or T dwarfs based on the strengths of lines within their spectra, and the general spectral shape in comparison to standards (see \citealt{kirkpatrick05} for a review), although estimates based on broadband photometry are often made."255 For T dwarfs this classification is usually based on the infrared spectra (22)... while the hotter L dwarfs are classified using optical spectra (2)..," For T dwarfs this classification is usually based on the near-infrared spectra \citep{burgasser02, geballe02}, while the hotter L dwarfs are classified using optical spectra \citep{kirkpatrick99}."256 Obviously. to achieve any sort of classification. “standard” objects must be selected. and so objects with unusual colours such as low gravity objects were often excluded as outliers.," Obviously, to achieve any sort of classification, “standard” objects must be selected, and so objects with unusual colours such as low gravity objects were often excluded as outliers."257 ? notes that. for example. any LO dwarf will fall between a higher mass (285 Maa old C a few Gyr). stellar object to à very young (< 20 Myr) low mass 620 Mii.) brown dwarf.," \citet{kirkpatrick05} notes that, for example, any L0 dwarf will fall between a higher mass $\approx$ 85 $_{\rm Jup}$ ), old $>$ a few Gyr), stellar object to a very young $<$ 20 Myr) low mass $<$ 20 $_{\rm Jup}$ ) brown dwarf."258 Understandably. this is a large range in mass and age. and it is therefore perhaps naive to assume every L dwarf of a defined spectral type to have identical photometry and indeed spectra. particularly while so many absorption lines are sensitive to gravity (e.g. K1... Nab. Rb I. Cs I.," Understandably, this is a large range in mass and age, and it is therefore perhaps naive to assume every L dwarf of a defined spectral type to have identical photometry and indeed spectra, particularly while so many absorption lines are sensitive to gravity (e.g. K, Na, Rb , Cs )."259 Tt was suggested in? that brown dwarf classification have a suffix added where o. 2. 5 and 9 could be used to denote decreasing gravity.," It was suggested in \citet{kirkpatrick05} that brown dwarf classification have a suffix added where $\alpha$, $\beta$, $\gamma$ and $\delta$ could be used to denote decreasing gravity."260 ? have recently used this notation while creating a low gravity sequence for early L dwarfs., \citet{cruz09} have recently used this notation while creating a low gravity sequence for early L dwarfs.261 This follows the work of ? who studied 20 low gravity dwarfs including 8 M dwarfs and 11 L dwarfs., This follows the work of \citet{kirkpatrick08} who studied 20 low gravity dwarfs including 8 M dwarfs and 11 L dwarfs.262 ? identified 23 low gravity L dwarfs from LO-L5. and beginning to detine a low gravity sequence based on that presented in ? using gravity sensitive features in the spectra.," \citet{cruz09} identified 23 low gravity L dwarfs from L0-L5, and beginning to define a low gravity sequence based on that presented in \citet{kirkpatrick99} using gravity sensitive features in the spectra."263 They use .7 to describe an object of the age of the Pleiades. and show that typically hese objects are red for their spectral class in JAvs.," They use $\beta$ to describe an object of the age of the Pleiades, and show that typically these objects are red for their spectral class in $J-K_{S}$."264 Tt will. 10wever. take time before a similar feat can be accomplished based on late L dwarfs and T dwarfs. as so few are currently known.," It will, however, take time before a similar feat can be accomplished based on late L dwarfs and T dwarfs, as so few are currently known."265" As discussed earlier. open star clusters have long been the ""uvourite hunting ground for these benchmark objects as they are coeval."," As discussed earlier, open star clusters have long been the favourite hunting ground for these benchmark objects as they are coeval."266 However. as we search further into the T dwarf regime to cooler temperatures. it is becoming evident that the majority of surveys do not probe deep enough to discover T dwarfs.," However, as we search further into the T dwarf regime to cooler temperatures, it is becoming evident that the majority of surveys do not probe deep enough to discover T dwarfs."267 To date. there are Τ dwarfs known in the Hyades (2)... although at an age of 625 Myr this means that they have a gravity similar to that of field T dwarfs.," To date, there are 2 T dwarfs known in the Hyades \citep{bouvier08}, although at an age of 625 Myr this means that they have a gravity similar to that of field T dwarfs."268 SOri70 (2). was discovered in σ Orionis. and i has been recently suggested that it may be surrounded by a dus disk accounting for its redness in the [3.6] and [4.5] micron bands (2)..," SOri70 \citep{zapatero02} was discovered in $\sigma$ Orionis, and it has been recently suggested that it may be surrounded by a dust disk accounting for its redness in the [3.6] and [4.5] micron bands \citep{scholz08}."269 One object was reported by (2). in IC348 with an estimatec spectral type of T6., One object was reported by \citep{burgess09} in IC348 with an estimated spectral type of T6.270 However at the voung age of this population (3 Myr). the reliability of evolutionary models is questionable due to the rather ad-hoe nature of the starting point of these caleulations (2).," However at the young age of this population (3 Myr), the reliability of evolutionary models is questionable due to the rather ad-hoc nature of the starting point of these calculations \citep{baraffe02}."271 There have been many studies of the Pleiades to search for young brown dwarfs. although until our previous work (?) only L dwarfs were discovered (e.g. 2222222).," There have been many studies of the Pleiades to search for young brown dwarfs, although until our previous work \citep{casewell07} only L dwarfs were discovered (e.g. \citealt{bouvier98, pinfield00, moraux01, dobbie02, bihain06, stauffer07, lodieu07}) )."272 In ?. we presented the results of a 2.5 square degree survey of the Pleiades in near-IR and optical wavebands., In \citet{casewell07} we presented the results of a 2.5 square degree survey of the Pleiades in near-IR and optical wavebands.273 After selecting candidate cluster members using the £Z. / and ZJ. Z colour magnitude diagrams. we measured proper motions using the 5 year baseline between the near-IR and optical datasets.," After selecting candidate cluster members using the $I-Z$, $I$ and $Z-J$, $Z$ colour magnitude diagrams, we measured proper motions using the 5 year baseline between the near-IR and optical datasets."274 After our analysis. 6 candidates were consistent with being late L or early T dwarfs.," After our analysis, 6 candidates were consistent with being late L or early T dwarfs."275 These objects are far too faint to obtain a spectrum with which to confirm their nature. and so we present here additional photometry in the form of Spilzer IRAC [3.6] and [4.5] micron bands. methane imaging and near-IR ἐν band imaging of the faintest candidates.," These objects are far too faint to obtain a spectrum with which to confirm their nature, and so we present here additional photometry in the form of $Spitzer$ IRAC [3.6] and [4.5] micron bands, methane imaging and near-IR $K$ band imaging of the faintest candidates."276 This additional information will allow us to place constraints on atmospheric models appropriate to a young cluster and hence a low gravity environment., This additional information will allow us to place constraints on atmospheric models appropriate to a young cluster and hence a low gravity environment.277 Methane imaging is a very useful tool for determining whether a brown dwarf is a bona fide T dwarf when a spectrum is not available as shown by ?.., Methane imaging is a very useful tool for determining whether a brown dwarf is a bona fide T dwarf when a spectrum is not available as shown by \citet{tinney05}.278 As T dwarfs get cooler. methane absorption becomes more dominant in the // band as seen in Figure |..," As T dwarfs get cooler, methane absorption becomes more dominant in the $H$ band as seen in Figure \ref{meth_flt}."279 The magenta curve (long dashed line) is a TO.5 dwarf. SDSSJOIS141.694124429.6 (2). which shows very little methane absorption.," The magenta curve (long dashed line) is a T0.5 dwarf, SDSSJ015141.69+124429.6 \citep{geballe02} which shows very little methane absorption."280 The flux in the // band forms a broad hump., The flux in the $H$ band forms a broad hump.281 However. as we progress to the T4+.5 dwarf. 2MASSJOSS919]4-1]404488 (green. short dashed line: 2)) this region. and in particular the red side of the flux. appears to have been suppressed by the presence of methane in the atmosphere.," However, as we progress to the T4.5 dwarf, 2MASSJ05591914-1404488 (green, short dashed line; \citealt{burgasser00}) ) this region, and in particular the red side of the flux, appears to have been suppressed by the presence of methane in the atmosphere."282 This is further shown in the TS dwarf. 2MASSJIO415195-093506 (blue. dotted line: 21) which shows sharply peaked flux in this region.," This is further shown in the T8 dwarf, 2MASSJ0415195-093506 (blue, dotted line; \citealt{burgasser02}) ) which shows sharply peaked flux in this region."283" We can further characterise the degree of methane absorption by using the methane filters. C//, ""long"" (grey line in Figure 19) and ""short"" (black line in Figure 13)."," We can further characterise the degree of methane absorption by using the methane filters, $CH_{4}$ “long” (grey line in Figure \ref{meth_flt}) ) and “short” (black line in Figure \ref{meth_flt}) )."284" We can calculate a methane colour index C'//,,- 47. which will be negative for objects with a late T spectral type as all the flux is in the short filter and not the long (where the majority of the methane absorption is occurring)."," We can calculate a methane colour index $CH_{4s}$ $_{4l}$, which will be negative for objects with a late T spectral type as all the flux is in the short filter and not the long (where the majority of the methane absorption is occurring)."285 We imaged each of the S T dwarf candidates from ? (PLZI93. PLZI721. PLZI235. PLZHI2 and PLZJ100) using NIRT on Gemini North in both the methane short and methane long filters on the nights of 02/09/2007 and 04/09/2007.," We imaged each of the 5 T dwarf candidates from \citet{casewell07} (PLZJ93, PLZJ721, PLZJ235, PLZJ112 and PLZJ100) using NIRI on Gemini North in both the methane short and methane long filters on the nights of 02/09/2007 and 04/09/2007."286 We requested that the seeing be better than 1.27 and that the cloud cover was at the 70th percentile or better (corresponding to at worst. thin cirrus).," We requested that the seeing be better than 1.2” and that the cloud cover was at the 70th percentile or better (corresponding to at worst, thin cirrus)."287 It was noted in the observing log there was some passing cloud on the observation of PLZIIOO. but this does not appear to have affected the images.," It was noted in the observing log there was some passing cloud on the observation of PLZJ100, but this does not appear to have affected the images."288 The remaining candidate PLZJ23. has an estimated photometrie spectral type of L8. and therefore was not observed as part of the methane imaging programme.," The remaining candidate PLZJ23, has an estimated photometric spectral type of L8, and therefore was not observed as part of the methane imaging programme."289 IRI consists of a 1024. 1024 pixel ALADDIN InSb array and when combined with the f/6 camera. provides a plate scale of 0.117 areseconds per pixel and a field of view of 120 areseconds.," NIRI consists of a $\times$ 1024 pixel ALADDIN InSb array and when combined with the f/6 camera, provides a plate scale of 0.117 arcseconds per pixel and a field of view of $\times$ 120 arcseconds."290" The data were obtained using a 9 point dither pattern and total exposure times of 30 mins for the C'//,, filter and Ihr for the C'77,; filter.", The data were obtained using a 9 point dither pattern and total exposure times of 30 mins for the $CH_{4s}$ filter and 1hr for the $CH_{4l}$ filter.291 We also observed aknown TS dwarf 9MASSIJO407088541514565 (2). for | min in both the ἐς and C'ILj filters to act as a standard object with which to check our results., We also observed aknown T5 dwarf 2MASSJ04070885+1514565 \citep{burgasser04} for 1 min in both the $CH_{4s}$ and $CH_{4l}$ filters to act as a standard object with which to check our results.292 The images were reduced using IRAT.. the package v1.9 and the NIRI specific tasks.," The images were reduced using , the package v1.9 and the NIRI specific tasks."293 Firstly the images were prepared using the task which adds keywords to the image, Firstly the images were prepared using the task which adds keywords to the image294(e.g.2011).,"\citep[e.g.][]{Franx1997on1358highz, Frye2002z4outflow,295 Bouwens2009behindclusters, Zheng2009, Bradley2011}."296". ol=7.6 the field of Δυο 1689 (Bradleyetal.2008).. but amo18o the highest-redshift ealaxies known are a ealaxv at ;=603 in the field of Abell 383 (couprisiis two images: Ricluudetal.2001:Zitrin2011c ). a —6.5 galaxy in the field of Abell 2218 (coniprisi18o three images: IKribetal.200[:Egi 2005)). ancl alc Tocandida ein the ""bullet cluster (comprising two nuages: Tallctal. 2011))."," $z=7.6$ the field of Abell 1689 \citep{Bradley2008}, but among the highest-redshift galaxies known are a galaxy at $z=6.03$ in the field of Abell 383 (comprising two images; \citealt{Richard2011,Zitrin2011c}) ), a $z\sim6.5$ galaxy in the field of Abell 2218 (comprising three images; \citealt{Kneib2004on2218highz,Egami2005on2218highz}) ), and a $z\sim7$ candidate in the “bullet cluster” (comprising two images; \citealt{Hall2011bullet}) )."297 Iu addition. the leus moel can be used to map the leused images back iuto fie source plane. wile the high magnification euables thi11 spatially resolve«d internal structural properties to TE micasured.," In addition, the lens model can be used to map the lensed images back into the source plane, while the high magnification enables their spatially resolved internal structural properties to be measured."298 Such measurements are not possible without the aid of the chsing power of the cluster (c.g.Zitrin 201110)., Such measurements are not possible without the aid of the lensing power of the cluster \citep[e.g.][]{Zitrin2011b}.299 The Lyian-apha forest produces a sharp drop in fUs. below 12]6A., The Lyman-alpha forest produces a sharp drop in flux below $1216$.300. The expansion of the universe moves this spectral break to longer waveleueths. allowing ligh-redshift ealaxics to be ideutified as “dropouts” 1d ifthe observed-friuae optical aud nem-IR (seeMadan19€111Frauxcta].1997 )..," The expansion of the universe moves this spectral break to longer wavelengths, allowing high-redshift galaxies to be identified as “dropouts” in the observed-frame optical and near-IR \citep[see][]{Madau1995dropout,Franx1997on1358highz}."301 As these voune. high-redshift eaANICS are usualv actively forming stux. their rest-frame [0Nu spectra should be relatively due.," As these young, high-redshift galaxies are usually actively forming stars, their rest-frame UV spectra should be relatively blue."302 This combinatiOl of properties allows the redshift of he galaxy Q be estimated accurately. evel without spectroscopy: as low as 1( uncertzntv on he recshift is obtained here. for confidence leveIx.," This combination of properties allows the redshift of the galaxy to be estimated accurately, even without spectroscopy: as low as $\sim1\%$ uncertainty on the redshift is obtained here, for confidence levels."303 The study of hiehl-redshift galaxies enables important ¢Onstroadnts to be placed on galaxy evolution axd strcture forinatknl., The study of high-redshift galaxies enables important constraints to be placed on galaxy evolution and structure formation.304 Particularly. observing high-edshift sealaxies provides direct iieasureiments of the Le: πμi. early SER. aud the epoch of tje interealactic imiecdiunm," Particularly, observing high-redshift galaxies provides direct measurements of the $z$ luminosity-function, early SFR, and the epoch of the intergalactic medium"305Clearly. luminosity on the main sequence. relative to the solar isochrone. is very tightly correlated with metallicity.,"Clearly, luminosity on the main sequence, relative to the solar isochrone, is very tightly correlated with metallicity."306 ancl (d) shows the relationship between spectroscopic metallicity ιν and the metallicity computed. via [τς, Panel (d) shows the relationship between spectroscopic metallicity $_{\mathrm spec}$ and the metallicity computed via Eq.307 ," 1, $_{\mathrm KF}$."308The line shows a 1:1 relation., The line shows a 1:1 relation.309 Clearly. Eq.," Clearly, Eq."310 1 oroduces a à very accurate metallicity estimate for the stars: he scatter in the residuals is a mere 0.08 dex in. Fe/L]., 1 produces a a very accurate metallicity estimate for the stars: the scatter in the residuals is a mere 0.08 dex in [Fe/H].311 Since he spectroscopic metallicities are accurate to approximately 1.05 dex. the metallicity Ην. (derived. directly. from he position of the star relative to the ΓΙΟ isochrone in he CMD) appears to be as accurate as a spectroscopically measured metallicity.," Since the spectroscopic metallicities are accurate to approximately 0.05 dex, the metallicity $_{\mathrm KF}$ (derived directly from the position of the star relative to the JFK0 isochrone in the CMD) appears to be as accurate as a spectroscopically measured metallicity."312 Eq., Eq.313 1 turned out to vield a far more accurate metallicity than we had anticipatect., 1 turned out to yield a far more accurate metallicity than we had anticipated.314 We have performed a preliminary check on the metallicity-luminosity relation by using i0 to compute expected apparent magnitudes of [x cdwarfs in some well. studied open and globular clusters. for which sullicientlv/ deep DV. photometry exists.," We have performed a preliminary check on the metallicity-luminosity relation by using it to compute expected apparent magnitudes of K dwarfs in some well studied open and globular clusters, for which sufficiently deep $BV$ photometry exists."315 A suitable colour to compare the observed. main sequence with the JEIXO. isochrone is at DVξ0.9. Let the main sequence magnitude at this colour »j denoted. Ady(0.9)., A suitable colour to compare the observed main sequence with the JFK0 isochrone is at $B-V = 0.9.$ Let the main sequence magnitude at this colour be denoted $M_V(0.9)$.316 For solar metallicity. A44(0.9)=6.17 (rom Eq.," For solar metallicity, $M_V(0.9) = 6.17$ (from Eq."317 1 and Table 3)., 1 and Table 3).318 The above calculations arc summarised in Table. 5., The above calculations are summarised in Table 5.319 Note that as far as possible we have used. distance moduli which are independent of main sequence fitting. in order to avoid. adopting moduli which are ultimately based on the same (or similar) subcwarls to those studied: here.," Note that as far as possible we have used distance moduli which are independent of main sequence fitting, in order to avoid adopting moduli which are ultimately based on the same (or similar) subdwarfs to those studied here."320 Overall the agreement between the observed luminosity of the main sequences at (D139)=0.9 and the value obtained from Ίσα., Overall the agreement between the observed luminosity of the main sequences at $(B-V)_0 = 0.9$ and the value obtained from Eq.321 1 is satisfactory: a more detailed study would. be interesting., 1 is satisfactory; a more detailed study would be interesting.322 One should note that at. very. low metallicities. the metallicitv-Iuminositv relation may not continue in the linear fashion shown in panel (b) of Figure 15..," One should note that at very low metallicities, the metallicity-luminosity relation may not continue in the linear fashion shown in panel (b) of Figure \ref{CMDkdwarfs}."323 More Ix charts with high (super-solar) and intermediate ασ Μο < 0.5) and Uipparcos parallaxes would be of great interest. in this regard. to check. the lincarity of the relation.," More K dwarfs with high (super-solar) and intermediate $-1.2 < $ [Fe/H] $< -0.5$ ) and Hipparcos parallaxes would be of great interest in this regard, to check the linearity of the relation."324 There are presently no Ix subcwarfs with good parallaxes and metallicities below Fe/H] =—2 with which to check the relation at low metallicitv., There are presently no K subdwarfs with good parallaxes and metallicities below [Fe/H] $ \approx -2$ with which to check the relation at low metallicity.325 Figure 10 shows the observed. absolute magnitudes Mq: at (D.Vy=0.90 versus the value Ao(0.9) computed. using Iq.," Figure \ref{VversusV} shows the observed absolute magnitudes $M_V$ at $(B-V)_0326= 0.90$ versus the value $M_V(0.9)$ computed using Eq."327 1., 1.328 The tightness of the metallicitv-ITuminosity relation [or lx cdwarfs is a remarkable result. for which there would be a wide range of applications., The tightness of the metallicity-luminosity relation for K dwarfs is a remarkable result for which there would be a wide range of applications.329 For example:, For example:330The integrated inclination curve of the flow channel component yields a slightly elevated arched loop.,The integrated inclination curve of the flow channel component yields a slightly elevated arched loop.331 The apex height above the 7—1 level is around 300-400 km., The apex height above the $\tau = 1$ level is around 300-400 km.332" With the height fixed to 0 km at the spot radius. the curve intersects the Tr=| level at around r/r;,4,=0.6."," With the height fixed to 0 km at the spot radius, the curve intersects the $\tau = 1$ level at around $r/r_{spot} =0.6$."333 Outside the sunspot. the field lines point downwards.," Outside the sunspot, the field lines point downwards."334 As the background component inclination never reaches horizontal fields. the integration yields a much steeper curve than for the flow channels.," As the background component inclination never reaches horizontal fields, the integration yields a much steeper curve than for the flow channels."335" If taken at face value. the curve reaches a depth of 2 (4) Mm at Fray,= 0.8(0.6). implyimg a thick penumbra. not a shallow surface layer."," If taken at face value, the curve reaches a depth of 2 (4) Mm at $r/r_{spot} =0.8 $ (0.6), implying a thick penumbra, not a shallow surface layer."336 For the backgrounc component fortunately a direct comparison with a theoretical model is possible., For the background component fortunately a direct comparison with a theoretical model is possible.337 I overplotted the boundary layers between umbra and penumbra and between penumbra and surroundings from the sunspot model of ?.1994 in Fig. 10., I overplotted the boundary layers between umbra and penumbra and between penumbra and surroundings from the magneto-static sunspot model of \citet[][JS94]{jahn+schmidt1994} in Fig. \ref{integ1}.338". E reduced the radius in their original calculation to 93 and 95%.. respectively. of its value. to fit with the dimensions of the sunspot in the observations. and shifted the curves in height to be at zzO km at r/ro,=| like the integrated inclination curves."," I reduced the radius in their original calculation to 93 and 95, respectively, of its value, to fit with the dimensions of the sunspot in the observations, and shifted the curves in height to be at z=0 km at $r/r_{spot} = 1$ like the integrated inclination curves."339" For r/?xpo, between 0.6 and I. the agreement between the integrated bg curve and the magneto-statical model is astonishingly good. whereas for smaller radii the JS94 model is than the integrated curve."," For $r/r_{spot}$ between 0.6 and 1, the agreement between the integrated bg curve and the magneto-statical model is astonishingly good, whereas for smaller radii the JS94 model is than the integrated curve."340 The good agreement in the mid to outer penumbra comes a bit as a surprise: the JS94 model gives the location (and thus also the field inclination) of the boundary layer between spot and surroundings at depths up to some Mm. whereas the integrated curve uses the inclination observed close to the 7=1 level.," The good agreement in the mid to outer penumbra comes a bit as a surprise: the JS94 model gives the location (and thus also the field inclination) of the boundary layer between spot and surroundings at depths up to some Mm, whereas the integrated curve uses the inclination observed close to the $\tau = 1$ level."341 The observed surface inclination thus seems to be identical to the inclination in the deeper layers — which was simply an assumption in the derivation of the integrated curves., The observed surface inclination thus seems to be identical to the inclination in the deeper layers – which was simply an assumption in the derivation of the integrated curves.342 The deviation between the curves also points in the correct direction: if the inclination changes with depth. the fields should get more vertical in the deeper layers. às the expansion of flux concentrations happens near the surface layer.," The deviation between the curves also points in the correct direction: if the inclination changes with depth, the fields should get more vertical in the deeper layers, as the expansion of flux concentrations happens near the surface layer."343 Thus. the surface inclination should be larger than in the deep layers. leading exactly to the less steep integrated curve as obtained.," Thus, the surface inclination should be larger than in the deep layers, leading exactly to the less steep integrated curve as obtained."344 Despite the surprisingly good agreement with a theoretical model — which also indicates that the main assumption of a small depth dependence of the inclination could be valid — the integration of the surface inclination of course can not be fully correct., Despite the surprisingly good agreement with a theoretical model – which also indicates that the main assumption of a small depth dependence of the inclination could be valid – the integration of the surface inclination of course can not be fully correct.345 In the left panel of Fig. 10," In the left panel of Fig. \ref{integ1},"346.. Loverplotted the approximate formation height(s) of the observed spectral lines throughout the penumbra., I overplotted the approximate formation height(s) of the observed spectral lines throughout the penumbra.347 All information retrieved from the observed spectra thus only refers to this small layer of the atmosphere., All information retrieved from the observed spectra thus only refers to this small layer of the atmosphere.348 There is no guarantee that fields do not for example bend strongly as soon as they leave the height range. in which the spectral lines are sensitive.," There is no guarantee that fields do not for example bend strongly as soon as they leave the height range, in which the spectral lines are sensitive."349 In fact. one expects exactly this behavior for the field lines that pass the upper boundary of the formation height. because of the exponential decrease of density with height.," In fact, one expects exactly this behavior for the field lines that pass the upper boundary of the formation height, because of the exponential decrease of density with height."350 ? found no height dependence of the field inclination using spectral lines similar to the nm pair. but this would be valid only inside approximately the same formation height range as plotted in Fig. 10.," \citet{sanchezcuberes+etal2005} found no height dependence of the field inclination using spectral lines similar to the nm pair, but this would be valid only inside approximately the same formation height range as plotted in Fig. \ref{integ1}."351. To investigate the influence of inclination changes with height (or depth) on the integration of inclination. | repeated the integration with the assumptions that the inclination may vary with height by +20 for the background component. respectively. 415 for the flow channels.," To investigate the influence of inclination changes with height (or depth) on the integration of inclination, I repeated the integration with the assumptions that the inclination may vary with height by $\pm$ 20 for the background component, respectively, $\pm$ 15 for the flow channels."352 The upper and lower limits of the retrievec curves are given by shaded areas in Fig. 10., The upper and lower limits of the retrieved curves are given by shaded areas in Fig. \ref{integ1}.353. As the expected change should be a deviation towards more inclined fields in higher and less inclined fields in lower layers. any curve that would not leave the shaded areas could be in agreement with the observed surface inclinations. within the 15-20 range of variation allowed for.," As the expected change should be a deviation towards more inclined fields in higher and less inclined fields in lower layers, any curve that would not leave the shaded areas could be in agreement with the observed surface inclinations, within the 15-20 range of variation allowed for."354 Even if one takes the extreme case of a field line starting at the lower and ending at the upper boundary of the shaded area. the curves would not change enough to contradict the description above.," Even if one takes the extreme case of a field line starting at the lower and ending at the upper boundary of the shaded area, the curves would not change enough to contradict the description above."355 The background component still would imply a thick penumbra: the flow channels could change from non-elevated to strongly elevated loops. but without giving rise to a new topology of the flow channels relative to the background component.," The background component still would imply a thick penumbra; the flow channels could change from non-elevated to strongly elevated loops, but without giving rise to a new topology of the flow channels relative to the background component."356 In the previous section. the integration was performed for azimuthal averages. where the azimuthal fine-structure of the penumbra is lost.," In the previous section, the integration was performed for azimuthal averages, where the azimuthal fine-structure of the penumbra is lost."357 The integration can however also be performed for individual radial cuts., The integration can however also be performed for individual radial cuts.358 To simplify the calculation and to smooth out slightly the pixel-to-pixel variations in the inversion results. I decided to use 92 of about 4 deg angular extend for the construction of à 3-D model that takes the azimuthal fine-structure into account.," To simplify the calculation and to smooth out slightly the pixel-to-pixel variations in the inversion results, I decided to use 92 of about 4 deg angular extend for the construction of a 3-D model that takes the azimuthal fine-structure into account."359 The model itself is represented in the following way: the integration of the background component gives a mesh of height with radial and azimuthal position. 7/77.;).," The model itself is represented in the following way: the integration of the background component gives a mesh of height with radial and azimuthal position, $h(r_i, \phi_i)$."360 This mesh is interpolated to a smooth surface. whose color is set to represent the field strength. of the background component.," This mesh is interpolated to a smooth surface, whose color is set to represent the field strength of the background component."361 The flow channels are overplotted as thin lines. with a color code corresponding to their temperature.," The flow channels are overplotted as thin lines, with a color code corresponding to their temperature."362 The temperature ts scaled individually between the respective minimum and maximum for each of the 92 bins: thus. the color bar only gives an average range of temperature.," The temperature is scaled individually between the respective minimum and maximum for each of the 92 bins; thus, the color bar only gives an average range of temperature."363 This description applies to Figs., This description applies to Figs.364 11. and B2.., \ref{penumbralgrains} and \ref{integ2}.365 As an intermediate step to the full 3-D model. Fig.," As an intermediate step to the full 3-D model, Fig."366 shows a top view of the model.," \ref{penumbralgrains}367 shows a top view of the model."368 This view corresponds to the commonly used 2-D maps of physical quantities., This view corresponds to the commonly used 2-D maps of physical quantities.369 For comparison. [ also show a speckle-reconstructed image of NOAA 10425 in the G-band taken with the Dutch Open Telescope (DOT) on 9th of August 2003 about half an hour later than the observation analyzed here.," For comparison, I also show a speckle-reconstructed image of NOAA 10425 in the G-band taken with the Dutch Open Telescope (DOT) on 9th of August 2003 about half an hour later than the observation analyzed here."370 The figure serves as a comparison of the spatial resolution of the polarimetric to speckle-reconstructed data., The figure serves as a comparison of the spatial resolution of the polarimetric to speckle-reconstructed data.371 The boundary between umbra and penumbra has the same shape in both observations (cf., The boundary between umbra and penumbra has the same shape in both observations (cf.372 inside the white rectangles). even with the time difference of half an hour.," inside the white rectangles), even with the time difference of half an hour."373 Bright penumbral grains can be seen in the DOT map near the umbral boundary. but only on the limb side.," Bright penumbral grains can be seen in the DOT map near the umbral boundary, but only on the limb side."374to2~3 (eg.Steideletal.1999)...,"to$z\sim3$ \citep[e.g.,][]{99steidel}."375 However. here are still some discrepancies in the interpretation of lis evolution. in terius of density. slope. Iuninositv. or a combination ofthese.," However, there are still some discrepancies in the interpretation of this evolution, in terms of density, slope, luminosity, or a combination of these."376 Dunkeretal.(2001). undertake a photometric analysis of the UDF aud xopose that the deusity increases six-fold frou to Do 3. in aerecmuent with Beckwithetal.(2006)...," \citet{04bunker} undertake a photometric analysis of the HUDF and propose that the density increases six-fold from to $z\sim3$ , in agreement with \citet{06beckwith}."377 Yan&Windhorst(2001). push the detection Wut deeper o lnaenitude 30. finding a steeper faint slope at compared to z~3 by 0.2-0.3.," \citet{04yw} push the detection limit deeper to magnitude 30, finding a steeper faint slope at compared to $z\sim3$ by 0.2-0.3."378 Furthermore. Bowweusetal.(2006) estimate corrections to the measured quantities to account for various observational effects andconclude that the intriusic hDunuinositv is ~0.8 mag fainter at +—6.," Furthermore, \citet{06bouwens} estimate corrections to the measured quantities to account for various observational effects andconclude that the intrinsic luminosity is $\sim$ 0.8 mag fainter at $z\sim6$."379 Their conclusious remain qualitatively unchauged after Reddy&Steidel(2009) receutly revisit the LF parameters at 2~3., Their conclusions remain qualitatively unchanged after \citet{09reddy} recently revisit the LF parameters at $z\sim3$.380 Ou the other haud. erouud-based observations. e.g. MeLbureetal.(2009).. find au even stronger Iuuinositv evolution.," On the other hand, ground-based observations, e.g., \citet{09mclure}, find an even stronger luminosity evolution."381 Different nicasureineuts of the hunimositv density (LD) or star formation rate (SER) also eive somewhat differeut results (0.9..Bunkeretal.2001:Dowwenus2006).," Different measurements of the luminosity density (LD) or star formation rate (SFR) also give somewhat different results \citep[e.g.,][]{04bunker,06bouwens}."382. It is nuportaut to establish whether these observed differences are due to intrinsic differences in the evolution ofdiffereut galaxy populations or due to issues with the derivation of the LF., It is important to establish whether these observed differences are due to intrinsic differences in the evolution of different galaxy populations or due to issues with the derivation of the LF.383 Spectroscopic confirmations of galaxies. e.g. Malhotraetal.(2005)... Dow-IIveecluudetal. (2007).. IIathietal.(2008) aud Vauzellaetal. (2009)... have already. proven theeffectiveness and robustuess of the dropout technique iu selecting LBCs.," Spectroscopic confirmations of galaxies, e.g., \citet{05malhotra}, \citet{07dow}, , \citet{08hathi} and \citet{09vanzella}, , have already proven theeffectiveness and robustness of the dropout technique in selecting LBGs."384 However. the faint," However, the faint"385binaritv. which has since been confirmed in 2 cases bv Ixoóhhler (2001).,"binarity, which has since been confirmed in 2 cases by Köhhler (2001)."386 These characteristics can be usec to select candidate cluster members with similar photometric properties to the known cluster stars., These characteristics can be used to select candidate cluster members with similar photometric properties to the known cluster stars.387 Spectroscopic study can then confirm signs of stellar vouth. in particular enhanced lithium.," Spectroscopic study can then confirm signs of stellar youth, in particular enhanced lithium."388 Thus new members of the cluster can be found. irrespective of their X-ray properties and without prior knowledge of their distances or proper motions., Thus new members of the cluster can be found irrespective of their X-ray properties and without prior knowledge of their distances or proper motions.389 However. such a sample might suller contamination from backgrouncd giants and foreground stars.," However, such a sample might suffer contamination from background giants and foreground stars."390 Mlamajek et al. (, Mamajek et al. (3912000: see their fig.,2000; see their fig.392" 4) showed a colour-magnitude diagram for > 21.000 stars within a 1"" radius of the cluster based upon United States Naval Observatory (USNO) A2.0 photographie photometry (Monet et αἱ."," 4) showed a colour-magnitude diagram for $>$ 21,000 stars within a $^{\rm o}$ radius of the cluster based upon United States Naval Observatory (USNO) A2.0 photographic photometry (Monet et al."393 1998)., 1998).394 The RECN stars were elevated above the vast majority of stars. but a broad giant branch overlapped the region of the diagram occupied by the Ix-tvpe RIECK stars.," The RECX stars were elevated above the vast majority of stars, but a broad giant branch overlapped the region of the diagram occupied by the K-type RECX stars."395gs Lawson et al. (, Lawson et al. (3962001) used the 1-m telescope ancl Lk lk SILC charge-coupled. device. (CCD) at. the South AfricanAstronomical Observatory (SAAQ) to obtain multi-epoch dillerential. V -band. observations and. Cousins ποιοιεν of the late-twpe members of the cluster.,2001) used the 1-m telescope and 1k $\times$ 1k SITe charge-coupled device (CCD) at the South AfricanAstronomical Observatory (SAAO) to obtain multi-epoch differential $V$ -band observations and Cousins photometry of the late-type members of the cluster.397 1n addition to these data. fields neighbouring those containing he known members were observed. to search for new members.," In addition to these data, fields neighbouring those containing the known members were observed to search for new members."398 frames were obtained with exposure times ranging rom | 300s to encompass the magnitude range of the late-ype members. and to obtain photometry to fainter limits.," frames were obtained with exposure times ranging from $1-300$ s to encompass the magnitude range of the late-type members, and to obtain photometry to fainter limits."399 The deepest frames obtained: useful (few percent accuracy) ohotometry. of stars of VzIS. or stars of spectral type AI5 ancl mass AL& O.LAL. at the distance and age of the cluster according to the models of Siess. Dufour Forestini (2000).," The deepest frames obtained useful (few percent accuracy) photometry of stars of $V \approx 18$, or stars of spectral type M5 and mass $M \approx 0.1$ $_{\odot}$ at the distance and age of the cluster according to the models of Siess, Dufour Forestini (2000)."400 These data were calibrated against equatorial and southern photometric standard. stars., These data were calibrated against equatorial and southern photometric standard stars.401 “Phe SITe CCD has a field of view of 26 arcmin at the f//16 Cassegrain focus of the lem telescope., The SITe CCD has a field of view of 26 $^{2}$ at the $f$ /16 Cassegrain focus of the 1-m telescope.402 23 fields were observed. many over-lapping. for a total area of zz500 aremin or z of the extent of the cluster as defined by Mamajek et al. (," 23 fields were observed, many over-lapping, for a total area of $\approx 500$ $^{2}$ or $\approx$ of the extent of the cluster as defined by Mamajek et al. ("4032000).,2000).404 Stars in cach field were compared to the sequence of RECN stars in the (Vo 2) versus V colour-magnitude diagram., Stars in each field were compared to the sequence of RECX stars in the $V-I$ ) versus $V$ colour-magnitude diagram.405 Candidates were selected as those objects with V mags that Fell. for their AO 2) colour. within the range LO<V<L0 mag of the sequence of KECK stars (Pig.," Candidates were selected as those objects with $V$ mags that fell, for their $V-I$ ) colour, within the range $-1.0 < V < 1.0$ mag of the sequence of RECX stars (Fig."406 1)., 1).407 This criterion accounted. for the possibility of a greater. spread: of ages within the cluster than is apparent in the RECN stars. uehly reddened stars or binary stars with elevated 1 mags. and non-linearities in the colour-magnitude relationship for he cluster (Lawson Feigelson 2001).," This criterion accounted for the possibility of a greater spread of ages within the cluster than is apparent in the RECX stars, highly reddened stars or binary stars with elevated $V$ mags, and non-linearities in the colour-magnitude relationship for the cluster (Lawson Feigelson 2001)."408 Of z2000 stars measured. only 6met the criterion (Vable 1).," Of $\approx 2000$ stars measured, only 6met the criterion (Table 1)."409 Most. stars in the fields. were 2 mag fainter than the sequence: in xwticular the 3 M-type candidates were the onlystars with (V. 1) 2. and with V magnitudes that dilfered by <2.5 mag from the extrapolated linear sequence.," Most stars in the fields were $> 2$ mag fainter than the sequence; in particular the 3 M-type candidates were the onlystars with $V-I$ ) $> 2$, and with $V$ magnitudes that differed by $< 2.5$ mag from the extrapolated linear sequence."410 The differential V-bancl cata were obtained with a 30 s exposure time. with =40 observations of cach Ποιά obtained during 1999 and z25 during 2000.," The differential $V$ -band data were obtained with a 30 s exposure time, with $\approx 40$ observations of each field obtained during 1999 and $\approx 25$ during 2000."411 Phe methocls used to reduce ane analyse the calibrated and. cülferential photometric data are discussed by Lawson et al. (, The methods used to reduce and analyse the calibrated and differential photometric data are discussed by Lawson et al. (4122001).,2001).413 Optical spectroscopy of the 6 candidates was obtained. on 2000 April 21 and 22 using the 2.3-mi telescope ancl dual-beam spectrograph at Mount. Stromlo ancl Siding Springs Observatories (AISSSO)., Optical spectroscopy of the 6 candidates was obtained on 2000 April 21 and 22 using the 2.3-m telescope and dual-beam spectrograph at Mount Stromlo and Siding Springs Observatories (MSSSO).414 In the red beam. the 120011 (1200 +) grating gave a 2-pixel resolution of 1.1 with coverage from. AAG2007160.A.," In the red beam, the 1200R (1200 $^{-1}$ ) grating gave a 2-pixel resolution of 1.1 with coverage from $\lambda\lambda 6200-7160$."415 In the bluebeam. the 600B (600 1) grating gave a 2-pixel resolution of 2.2 with coverage from AAZSL05400A.," In the bluebeam, the 600B (600 $^{-1}$ ) grating gave a 2-pixel resolution of 2.2 with coverage from $\lambda\lambda 3840-5400$."416 Exposure times ranged from 720 s to 3000 s. ancl vielded continuum signal-to-noise (S/N) ratios of 15130 in the τοῦ and. for the 4 brightest stars. 1040 in the blue.," Exposure times ranged from 720 s to 3000 s, and yielded continuum signal-to-noise (S/N) ratios of $15-130$ in the red and, for the 4 brightest stars, $10-40$ in the blue."417 Phe spectra. were calibrated using come-llats. bias frames and le-Ar are frames. making use of standard. library routines such as withinIRAF.," The spectra were calibrated using dome-flats, bias frames and Fe-Ar arc frames, making use of standard library routines such as within."418 Analysis of the spectra showed 2 of these stars were active. lithium-rich late-type objects.," Analysis of the spectra showed 2 of these stars were active, lithium-rich late-type objects."419 One of these stars (listed. as USNO Anon 1 in Table 1) has only a USNO- catalogue entry., One of these stars (listed as USNO Anon 1 in Table 1) has only a USNO-A2.0 catalogue entry.420 The other star (Listed as USNO Anon 2 in Table. 1) appears to be the optical counterpart: of the Faint Source. Catalogue. FSC: Moshir et al., The other star (listed as USNO Anon 2 in Table 1) appears to be the optical counterpart of the Faint Source Catalogue FSC; Moshir et al.421 1989) object FOS450τοῦ., 1989) object F08450–7854.422 Phe FSC position dillers by 5 aresee from the USNO position. with," The FSC position differs by 5 arcsec from the USNO position, with"423"An accurate knowledge of the physical properties of young stellar objects (like their mass, age and luminosity) is important to constrain theoretical pre-main sequence evolutionary models.","An accurate knowledge of the physical properties of young stellar objects (like their mass, age and luminosity) is important to constrain theoretical pre–main sequence evolutionary models."424" The determination ol these properties, however, depends criucally on the availability of accurate distances."," The determination of these properties, however, depends critically on the availability of accurate distances."425" Unfortunately, since distances to regions of star formation are often uncertain by more than 20 or30%.,, errors on the luminosity and age of young stars are typically about70%."," Unfortunately, since distances to regions of star formation are often uncertain by more than 20 or, errors on the luminosity and age of young stars are typically about."426". Significant progress has been possible in recent years thanks to Very Long Baseline Interferometry (VLBI) observations, particularly with the Very Long Base Array (VLBA —Loinard et 22005. 2007. 2008; Torres et 22007, 2009; Menten et 2007; Xu et 22006)."," Significant progress has been possible in recent years thanks to Very Long Baseline Interferometry (VLBI) observations, particularly with the Very Long Base Array (VLBA –Loinard et 2005, 2007, 2008; Torres et 2007, 2009; Menten et 2007; Xu et 2006)."427" Owing to the very accurate astrometry delivered by such instruments, trigonometrie parallaxes (and therefore distances) can be measured very precisely if multi-epoch observations spread over a [ew years are obtained."," Owing to the very accurate astrometry delivered by such instruments, trigonometric parallaxes (and therefore distances) can be measured very precisely if multi-epoch observations spread over a few years are obtained."428" VLBI instruments are only sensitive to high surface brightness emission, and can only detect objects where non-thermal processes are at work."," VLBI instruments are only sensitive to high surface brightness emission, and can only detect objects where non-thermal processes are at work."429" Such non-thermal sources must, therefore, be identified in the regions of interest before their distance can be measured using mulu-epoch VLBI observations."," Such non-thermal sources must, therefore, be identified in the regions of interest before their distance can be measured using multi-epoch VLBI observations."430" Fortunately, many young stars are magnetically active, and do exhibit detectable levels of non-thermal radio This type of emission is typically characterized by strong variability, some level of circular polarization, and a negative spectral index."," Fortunately, many young stars are magnetically active, and do exhibit detectable levels of non-thermal radio This type of emission is typically characterized by strong variability, some level of circular polarization, and a negative spectral index."431" Also, for magnetically active stars, there is a good correlation between X-ray and non-thermal radio emission (Benz Güddel 2004) so voung stars with detectable levels of non-thermal radio emission are associated with bright X-ray sources."," Also, for magnetically active stars, there is a good correlation between X-ray and non-thermal radio emission (Benz Güddel 2004) so young stars with detectable levels of non-thermal radio emission are associated with bright X-ray sources."432" In this work, we will focus on the star-forming region associated with the Serpens molecular cloud (Strom et 11974; see Eiroa 1992 and Eiroa et 22008 [or two recent reviews)."," In this work, we will focus on the star-forming region associated with the Serpens molecular cloud (Strom et 1974; see Eiroa 1992 and Eiroa et 2008 for two recent reviews)."433" More specifically, we will concentrate on the SVS 4 region (Strom et 11976), an infrared cluster of at least 1 pre-main sequence sources deeply embedded within the Serpens core, and one of the densest young stellar sub-clusters known, with a stellar mass density of ~10?M. ? (Eiroa Casali 1989)."," More specifically, we will concentrate on the SVS 4 region (Strom et 1976), an infrared cluster of at least 11 pre-main sequence sources deeply embedded within the Serpens core, and one of the densest young stellar sub-clusters known, with a stellar mass density of $\sim\, 10^5\, {\rm M}_\odot$ $^{-3}$ (Eiroa Casali 1989)."434" In the direction of SVS 4, Preibisch (1998) detected a bright X-ray source (Ly ~ 4 x 10?! erg !: Preibisch 20034) now known to be associated"," In the direction of SVS 4, Preibisch (1998) detected a bright X-ray source $L_X$ $\sim$ 4 $\times$ $^{31}$ erg $^{-1}$; Preibisch 2003a) now known to be associated"435]t is straighforwarel to calculate the proper motions for jets of a given 3 and 06. and compare them to the values. we would. derive using the method outlined above. for varving estimates of the distance to the source.,"It is straighforward to calculate the proper motions for jets of a given $\beta$ and $\theta$, and compare them to the values we would derive using the method outlined above, for varying estimates of the distance to the source."436 In Figs 2(ad) we plot the inferred. Lorentz factor as a fraction of the intrinsic Lorentz factor of the jet. as à function of the distance estimated to the source expressed as à fraction of the true distance.," In Figs 2(a–d) we plot the inferred Lorentz factor as a fraction of the intrinsic Lorentz factor of the jet, as a function of the distance estimated to the source expressed as a fraction of the true distance."437 In each figure the dillerent curves indicate dillerent intrinsic angles to the line of sight. and cach of the four panels represents a dilferent. intrinsic Lorentz factor (2. 5. 10. 50).," In each figure the different curves indicate different intrinsic angles to the line of sight, and each of the four panels represents a different intrinsic Lorentz factor (2, 5, 10, 50)."438 Ehe points at which the curves intersect with the upper abscissa corresponds to dias [or 1e particular combination of proper motions observed.," The points at which the curves intersect with the upper abscissa corresponds to $d_{\rm439max}$ for the particular combination of proper motions observed."440 Apart from the smallest angles and lowest. velocities. rere is for all the curves an extremely rapic variation in 1e inferred. Lorentz factor close to the true| distance to 10 source.," Apart from the smallest angles and lowest velocities, there is for all the curves an extremely rapid variation in the inferred Lorentz factor close to the true distance to the source."441 The figures demonstrate clearly tjab it will be cHeetively impossible to measure the distance accurately enough to constrain the Lorentz factor., The figures demonstrate clearly that it will be effectively impossible to measure the distance accurately enough to constrain the Lorentz factor.442 A related point is wt all significantly relativistic jets will bv necessity Πο very ‘lose to dis (rie 3)., A related point is that all significantly relativistic jets will by necessity lie very close to $d_{\rm max}$ (Fig 3).443 Phis leads to one useful conclusion — i£ je jets we observe are intrinsically significantly relativistic. which seems to be the case. then measurements of two-sicec oper motions will give us an accurate distance estimate.," This leads to one useful conclusion – if the jets we observe are intrinsically significantly relativistic, which seems to be the case, then measurements of two-sided proper motions will give us an accurate distance estimate."444 As a result. this means that observations of two-sided je oper motions in AGN. were they ever το be observed. would be extremely useful lor calibrating the cosmologica distance scale.," As a result, this means that observations of two-sided jet proper motions in AGN, were they ever to be observed, would be extremely useful for calibrating the cosmological distance scale."445 Unfortunately. to. date most. well-stuclicc AGN are significantly. Doppler-boosted (so-called “Doppler [avouritism). implving small angles of the jets to the line of sight. and resulting in no measurements of two-siclec relativistic proper motions so far.," Unfortunately, to date most well-studied AGN are significantly Doppler-boosted (so-called `Doppler favouritism'), implying small angles of the jets to the line of sight, and resulting in no measurements of two-sided relativistic proper motions so far."446 As noted above. there is a glimmer of hope for the owest velocities and smallest angles. where the swing in the Curves around the true distance is not too dramatic.," As noted above, there is a glimmer of hope for the lowest velocities and smallest angles, where the swing in the curves around the true distance is not too dramatic."447 However (a) this discussion is really concentrating on significantlv relativistic jets. and (b) the smallest. angles to the line of sight will have the largest ratios of proper motions and luxes between the approaching and receding sides of the jet. making the measurements increasingly hare to make.," However (a) this discussion is really concentrating on significantly relativistic jets, and (b) the smallest angles to the line of sight will have the largest ratios of proper motions and fluxes between the approaching and receding sides of the jet, making the measurements increasingly hard to make."448" This is illustrated in Fig 4. in which the pro»er motions (scaled. to a distance of 1 kpe) and resultard apparent velocity as a fraction of the speed. of. light are plotted or dillerent intrinsic Lorentz factors as a ""unction. of angle to the line of sight."," This is illustrated in Fig 4, in which the proper motions (scaled to a distance of 1 kpc) and resultant apparent velocity as a fraction of the speed of light are plotted for different intrinsic Lorentz factors as a function of angle to the line of sight."449 The receding proper motions are very similar for all intrinsic Lorentz factors. but. the approaching proper motions are dilfering functions. which peak at progressively smaller angles (the peaks occur at ϱ~L/U radians).," The receding proper motions are very similar for all intrinsic Lorentz factors, but the approaching proper motions are differing functions which peak at progressively smaller angles (the peaks occur at $\theta \sim4501/\Gamma$ radians)."451 Note that for both GRS 1915|105 and GRO J1655-40 the ratio of approaching to receding proper motions has been «3. which. as this figure. illustrates. indicates immediatelv that whatever the Lorentz [actor. they must be at [large angles to the line of sight. (and therefore. consulting Fig 3. unless the jets are only mildly relativistic. means that they must both lie at d7 du).," Note that for both GRS 1915+105 and GRO J1655-40 the ratio of approaching to receding proper motions has been $<3$, which, as this figure illustrates, indicates immediately that whatever the Lorentz factor, they must be at large angles to the line of sight (and therefore, consulting Fig 3, unless the jets are only mildly relativistic, means that they must both lie at $d \sim d_{\rm max}$ )."452 Uf we are hoping to measure the Lorentz factor from the proper motions of a jet close to the line of sight then the ratio of oper motions becomes increasingly large ancl therefore increasingly hard to measure accurately., If we are hoping to measure the Lorentz factor from the proper motions of a jet close to the line of sight then the ratio of proper motions becomes increasingly large – and therefore increasingly hard to measure accurately.453 Ehe ratio of Duxes is even ereater. being the ratio of proper motions raisec ο SOME power & (at the same angular separation). ancl so compared to the approaching component the receding je will appear o be extremely faint and. slow moving.," The ratio of fluxes is even greater, being the ratio of proper motions raised to some power $k$ (at the same angular separation), and so compared to the approaching component the receding jet will appear to be extremely faint and slow moving."454 Mos ikelv we wil observe only the approaching jet. or jet. plus core if activi vds still ongoing (as has been the case to date or AGN).," Most likely we will observe only the approaching jet, or jet plus core if activity is still ongoing (as has been the case to date for AGN)."455 This leads us to consider an alternative approach to at least limiting the Lorentz factor., This leads us to consider an alternative approach to at least limiting the Lorentz factor.456 For a jet of apparent velocity Xs. the intrinsic Lorentz factor is at least as large as Sapp. Corresponding to the solution for €=1/L.," For a jet of apparent velocity $\beta_{\rm app}$, the intrinsic Lorentz factor is at least as large as $\beta_{\rm457app}$, corresponding to the solution for $\theta = 1 / \Gamma$."458 In this way observations of one-sided. proper motions can allow us to place a lower limit on the Lorentz factor., In this way observations of one-sided proper motions can allow us to place a lower limit on the Lorentz factor.459 How accurate is this method compared to two-sided proper motions?, How accurate is this method compared to two-sided proper motions?460 In [act it can never place a more constraining lower limit on E than can be obtained by measurement of two-sided: proper motions., In fact it can never place a more constraining lower limit on $\Gamma$ than can be obtained by measurement of two-sided proper motions.461 This is natural. since the lower limits to the Lorentz factors measured [rom one-sided proper motions assume the," This is natural, since the lower limits to the Lorentz factors measured from one-sided proper motions assume the"462The IL I toward PISS 2155-301 appears to represent gas with the lowest detected metallicity.,The H I toward PKS 2155-304 appears to represent gas with the lowest detected metallicity.463 Was this eas was once inside the galaxies at liis +. or is it pristine?," Was this gas was once inside the galaxies at $cz = 17,000 \pm 1000$ km $^{-1}$, or is it pristine?"464 We can perhaps answer this question by deeper spectral searches for traces of metals., We can perhaps answer this question by deeper spectral searches for traces of metals.465 The origin of the lower-columu Lyra systenis would seei to be more diverse. possibly arising in extended halos or debris disks of dwarf galaxies. large galaxies. aud small eroups (Morris viui den Bereh 1991).," The origin of the lower-column $\alpha$ systems would seem to be more diverse, possibly arising in extended halos or debris disks of dwarf galaxies, large galaxies, and small groups (Morris van den Bergh 1994)."466 A primary theoretical issue is whether low-: clouds have anv relation to the evolution of the birvous in the hielh-: forest., A primary theoretical issue is whether $z$ clouds have any relation to the evolution of the baryons in the $z$ forest.467 A quick estimate sugecsts that the low-: absorbers could contain a substantial (25%)) fraction of the total ↴⋝⋜∐⋅∙↖⇁∪∐↴∖↴↸∖↴∖↴↑∐⊔⋜↧↑↸∖≼↧↕↥⋅∪⋯↕≧↕∶↴∙⊾↕≧⋜⋯∶↴∙⊾∐⋯⊳↕↸∖≺≻↴∖↴⋅↖⇁∐↑∐↸∖↴∖↴↕↴∖↴∙≤≥⊥⋝⊥⋝⋋↽∶⋖∩∙∩⊔≩∩∶∶∩⋅∪∩∣⋝∕∣⊤⊽↴. . ⋅ ⋅ ∙⋅↼⊐ for↴ κ.α. .," A quick estimate suggests that the $z$ absorbers could contain a substantial ) fraction of the total baryons estimated from Big Bang nucleosynthesis, $\Omega_{\rm BBN} = (0.036 \pm 0.007) h_{75}^{-2}$ (Burles Tytler 1997)."468 : ⋅≽ ≺↕≧↿∐⋅↕↸∖↴∖↴∙∖↽⊺⋅↖↽↑↕↸∖↥⋅⊥≝⋟⋂∣⋟∙≼⊲∪∐↴∖↴↕≼∐∖↥⋅↑∐∪↴∖↴↸∖∫⇀⋅↖⇁∩↴∖↴⋅↖↽↴∖↴↑↸∖⋯↴∖↴↖↖⇁↕↑∐⋀∖↕∐≧↓∩↓⇝↸⊳↕⊔−∙ which one can derive the space density op. dA major uncertainty in deriving absorber masses is the ionization correction. which depends on the profile of eas density around the cloud centers.," Consider those $\alpha$ systems with $_{\rm HI} \geq 10^{13}$ $^{-2}$, for which one can derive the space density $\phi_0$, The major uncertainty in deriving absorber masses is the ionization correction, which depends on the profile of gas density around the cloud centers."469" Asse. for simplicity. that ο) —notr/rg).aud adopt photoionization equilibria at rate Dy at 20.000 IX. The ionizing radiation field is J, =Judy/my)""with qoo mLs+and Jg =(107223.erescin2lsiτνaoi:os."," Assume, for simplicity, that $n_H(r) = n_0 (r/r_0)^{-2}$, and adopt photoionization equilibrium at rate $\Gamma_{\rm HI}$ at 20,000 K. The ionizing radiation field is $J_{\nu} = J_0 (\nu / \nu_0)^{-\alpha_s}$ with $\alpha_s \approx 1.8$ and $J_0 = (10^{-23}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}~{\rm Hz}^{-1}~470{\rm sr}^{-1}) J_{-23}$."471The IIau I column density iuteerated through the cloud at iupact parameter b is. οὐ. We can solve for nya aud fud the total eas mass within 5=(100kpc)/499 for a fiducial colum deusity Nyy=(103emP23N44. which vields a cloud closure parameter in barvons. Note that ο is insensitive to the scaling parameters: For the spherical-cloud model. the radiation field. cloud size. and columu- distribution probably each contribute to the nucertaimty in," The H I column density integrated through the cloud at impact parameter $b$ is, We can solve for $n_0r_0^2$ and find the total gas mass within $b = (100~{\rm kpc}) b_{100}$ for a fiducial column density $_{\rm HI} = (10^{14}~{\rm cm}^{-2}) N_{14}$, which yields a cloud closure parameter in baryons, Note that $\Omega_b$ is insensitive to the scaling parameters: For the spherical-cloud model, the radiation field, cloud size, and column-density distribution probably each contribute to the uncertainty in"472applications of the wave coupling discussed iu this paper.,applications of the wave coupling discussed in this paper.473 The variation of the solar radius duriug the li-vear period activity. cycle is a loug time problem., The variation of the solar radius during the 11-year period activity cycle is a long time problem.474 The characteristics of the variatiou lav reveal some basic mechanisms of activity. therefore many efforts have heen mace i order to πια observational evideuce of a correlation between radius variation and activity parameters.," The characteristics of the variation may reveal some basic mechanisms of activity, therefore many efforts have been made in order to find observational evidence of a correlation between radius variation and activity parameters."475 The observational results are coutroversial aud solmewhat inconsistent., The observational results are controversial and somewhat inconsistent.476 Some observations indicate a negative correlation (Laclarectal.1996).. some results show uo systematic variation in time (Brown&Christenscn-Dalseaard1998).," Some observations indicate a negative correlation \citep{lac}, some results show no systematic variation in time \citep{bro}."477 Most observations reveal a positive correlation (Ulrich& 2001).," Most observations reveal a positive correlation \citep{ul,ba,em,no}."478 The observed amplitude of the variation is also controversial., The observed amplitude of the variation is also controversial.479" Based ou Mount. Wilson data. Ulrich&Bertello(1995). found ~0.2%: Noel sueeested a similar amplitude ~0.2%, while in Laclareetal.(1996) the amplitude is ~0.08""."," Based on Mount Wilson data, \citet{ul} found $\sim 0.2^{\prime\prime}$; \citet{no} suggested a similar amplitude $\sim 0.2^{\prime\prime}$, while in \citet{lac} the amplitude is $\sim 0.08^{\prime\prime}$."480" Probably. the most realistic value of the amplitude ~0.02"" was obtained by Eiulioetal.(2000). by analyzing SOIIO-MDI data."," Probably, the most realistic value of the amplitude $\sim4810.02^{\prime\prime}$ was obtained by \citet{em} by analyzing SOHO-MDI data."482 The variation of the solar radius during the activity evele naturally leads to the idea of its commection to he magnetic fell pressure variation iu the interior due to the large-scale turbulent cya., The variation of the solar radius during the activity cycle naturally leads to the idea of its connection to the magnetic field pressure variation in the interior due to the large-scale turbulent dynamo.483 ILlowever. even a very πα variation of the radius requires a luee Lniagsneic CLOYSV to conipensate the corresponding eravitational energv chanec (see the above estimate).," However, even a very small variation of the radius requires a huge magnetic energy to compensate the corresponding gravitational energy change (see the above estimate)."484 Then it secius dlausible that the radius variation is due to the globa slow maguctoacoustic m0de., Then it seems plausible that the radius variation is due to the global slow magnetoacoustic mode.485 Slow lnagnetoacoustic waves propagating along the poloidal naenetic field have both R aud Z-couiponeuts of velocity (see Eqs.(9)-(1U)}. therefore the global slow uode will lead to the periodical variation of he solar oblateness. which may cause interesting consequencOR for solar total iiyadiauce.," Slow magnetoacoustic waves propagating along the poloidal magnetic field have both $R$ and $Z$ -components of velocity (see Eqs.(9)-(14)), therefore the global slow mode will lead to the periodical variation of the solar oblateness, which may cause interesting consequences for solar total irradiance."486" The fundamental period of pulsatious in t1e elobal slow magucetoacoustic mode will be of order 2R8,/e4. which for the li-vear period implies the mean magnetic field streueth iu the solar iuterior to be ~107 C. Another possible candidate for the global mode coupling can be tιο observed 1.3 vr variation at the base of the solar convection zone2000)."," The fundamental period of pulsations in the global slow magnetoacoustic mode will be of order $2R_0/c_A$, which for the 11-year period implies the mean magnetic field strength in the solar interior to be $\sim 10^3$ G. Another possible candidate for the global mode coupling can be the observed 1.3 yr variation at the base of the solar convection zone."487. I£ the elobal slow mode has the period of ~ 0.65 vr. then it may drive the torsional oscillations with the period of 1.3 vx in the solar tachocline.," If the global slow mode has the period of $\sim$ 0.65 yr, then it may drive the torsional oscillations with the period of 1.3 yr in the solar tachocline."488 Long-term observations of close binary stars show a variation of the orbital period over long ine-sceales with relative amplitude AP/P~10. where P is the orbital period.," Long-term observations of close binary stars show a variation of the orbital period over long time-scales with relative amplitude ${\Delta P}/P \sim 10^{-5}$ , where $P$ is the orbital period."489 A variety of weorctical models have been proposed to explain us phenomenon., A variety of theoretical models have been proposed to explain this phenomenon.490 The first models included ie observed period changes due to the apsidal notion. or the influence of a distant. unseen companion.," The first models included the observed period changes due to the apsidal motion, or the influence of a distant, unseen companion."491 Then Applegate(1992)— sugeested jii the variation can be explained bv the eravitational coupling of the orbit to the variation in shape of a imaeuetically active companion., Then \citet{ap} suggested that the variation can be explained by the gravitational coupling of the orbit to the variation in shape of a magnetically active companion.492 The variable deformation of the active star is produced by the variation in the distrinition of angular momentum durine the activity evcle., The variable deformation of the active star is produced by the variation in the distribution of angular momentum during the activity cycle.493 Ou another haud. Lanzactal.(1998) sugeested that a torsional oscillation driven by fje. stellar dynamo nav account for the observed behaviour of the binary svstem.," On another hand, \citet{lan} suggested that a torsional oscillation driven by the stellar dynamo may account for the observed behaviour of the binary system."494 IHowewer the prodo still reniadns. because the magnetic energv can not conrpeusate the eravitational energv change. as has been shown for the Sun.," However the problem still remains, because the magnetic energy can not compensate the gravitational energy change, as has been shown for the Sun."495 lustead. were we propose that the long term radial pulsation due to the oelobal slow 1uocde considered iun our paoer (which causes a periodic variatiou in oblateuess) may be respouside for the variation of the orbital period wi la gravitationa coupling similar to Applegate’s.," Instead, here we propose that the long term radial pulsation due to the global slow mode considered in our paper (which causes a periodic variation in oblateness) may be responsible for the variation of the orbital period with a gravitational coupling similar to Applegate's."496 However coutrary o Applegaew nmechanin dere stellar radia mulsation cai be the reason for both orbital perio nodulation and magnetic activity iuthe primary star.," However contrary to Applegate's mechanism, here stellar radial pulsation can be the reason for both orbital period modulation and magnetic activity inthe primary star."497 The relative amplitude of the orbital perio nodulation is proportional to the relative variation of the eravitational acceleration of the primary AP/P~Αι (Applegate1992)., The relative amplitude of the orbital period modulation is proportional to the relative variation of the gravitational acceleration of the primary ${\Delta P}/P\sim {\Delta g}/g$ \citep{ap}.498. Ou the other rand. the relative variation of the eravitational acceleration is of the order of Ag/g~ AR/R. where Rods radius of the primary.," On the other hand, the relative variation of the gravitational acceleration is of the order of ${\Delta g}/g\sim499{\Delta R}/R$ , where $R$ is radius of the primary."500" Then iu order o ect the observed relative period modulation ΔΡ/Ρ~10”, "," Then in order to get the observed relative period modulation ${\Delta P}/P\sim50110^{-5}$ "502from a number of potential fields primarily for their observability during the first few allocated INT runs: their centres are shown in table |..,from a number of potential fields primarily for their observability during the first few allocated INT runs; their centres are shown in table \ref{tab:fields}.503 The Virgo field was added at a later date to optimise year round observability and provide a field visible from the Southern hemisphere., The Virgo field was added at a later date to optimise year round observability and provide a field visible from the Southern hemisphere.504 In addition. the selected fields had to have low extinction. overlap with existing (multi-wavelength) datasets and have a lack of bright stars. nearby galaxies and bright. large clusters.," In addition, the selected fields had to have low extinction, overlap with existing (multi-wavelength) datasets and have a lack of bright stars, nearby galaxies and bright, large clusters."505 The mean extinction. (71). in each field. was determined from the DIRBE corrected TIRAS 1005701 maps of and was found to be < 0.06 in Andromeda and < 0.02 in the remaining fields.," The mean extinction, $E(B-V)$, in each field was determined from the DIRBE corrected IRAS $100 \mu m$ maps of \scite{schlegel} and was found to be $<$ 0.06 in Andromeda and $<$ 0.02 in the remaining fields."506" The large (4) pixel scale of the IRAS maps provides little information about the small scale distribution of the Galactic dust. and therefore making extinction. corrections using 4"" cells. as opposed to individual galaxy corrections. could imprint a low level spurious clustering pattern."," The large $4^{\prime}$ ) pixel scale of the IRAS maps provides little information about the small scale distribution of the Galactic dust, and therefore making extinction corrections using $4^{\prime}$ cells, as opposed to individual galaxy corrections, could imprint a low level spurious clustering pattern."507 This potential bias combined with the low galactic extinction values meant no extinction corrections for galactic dust were applied., This potential bias combined with the low galactic extinction values meant no extinction corrections for galactic dust were applied.508 Observations were made over 63 nights between August 1998 and March 2003. during which time several nights were lost due in part to instrument problems. but mostly due to bad weather conditions.," Observations were made over 63 nights between August 1998 and March 2003, during which time several nights were lost due in part to instrument problems, but mostly due to bad weather conditions."509 The data were obtained using the WFC which is a mosaic of four 4096 x 2048 pixel CCD chips. each chip covering of DIS.11A (zz0.29 degs? per WFC pointing) with a pixel scale of 0.33” per pixel.," The data were obtained using the WFC which is a mosaic of four 4096 x 2048 pixel CCD chips, each chip covering of $22.8^{\prime} \times 11.4^{\prime}$ $\approx0.29$ $^{2}$ per WFC pointing) with a pixel scale of $0.33^{\prime\prime} $ per pixel."510 At the beginning and end of each night bias. dark and twilight flat-tield frames were acquired.," At the beginning and end of each night bias, dark and twilight flat-field frames were acquired."511 Landolt standard star frames were also observed several times throughout each night (see section 4.1., Landolt standard star frames were also observed several times throughout each night (see section \ref{sec:photo}) ).512 On the whole. data were taken when conditions were either photometric or light cirrus was present. with variable seeing across the fields.," On the whole, data were taken when conditions were either photometric or light cirrus was present, with variable seeing across the fields."513 Assuming the median INT seeing of the 5¢ depths shown in table 2. implied a total of 3.7 hours per pointing were required to obtain the multi band data (BV97Z y. resulting in a survey speed of ~1.2 deg? per night (not including the ( data).," Assuming the median INT seeing of the $5\sigma$ depths shown in table \ref{tab:filters} implied a total of 3.7 hours per pointing were required to obtain the multi band data $BVRi^{\prime}Z$ ), resulting in a survey speed of $\sim 1.2$ $^{2}$ per night (not including the $U$ data)."514 The median seeing values actually obtained for each field in each band are presented in table 3. alongside the median depth reached in each band in each field., The median seeing values actually obtained for each field in each band are presented in table \ref{tab:seeing} alongside the median depth reached in each band in each field.515 There are no values for the Z band data at present because the fringing proved too severe (see section 3.1)., There are no values for the $Z$ band data at present because the fringing proved too severe (see section \ref{sec:datared}) ).516 Figure 2. illustrates how the /? band depths vary across the Andromeda field (fully reduced data only) and figure 3 shows the corresponding cumulative distribution of fraction of total area versus depth in 7., Figure \ref{fig:greyscale} illustrates how the $R$ band depths vary across the Andromeda field (fully reduced data only) and figure \ref{fig:cumulative} shows the corresponding cumulative distribution of fraction of total area versus depth in $R$.517 In order to cover each of the ODTS fields efficiently with the WFC focal plane geometry. a tiling pattern involving using the camera in two rotational positions. 180 apart. was employed.," In order to cover each of the ODTS fields efficiently with the WFC focal plane geometry, a tiling pattern involving using the camera in two rotational positions, $180^\circ$ apart, was employed."518 The fields were covered by a diagonal grid. with the rotation alternating with each row.," The fields were covered by a diagonal grid, with the rotation alternating with each row."519 Each grid element. representing one pointing of the INT WFC. was assigned a unique ODTS identification number.," Each grid element, representing one pointing of the INT WFC, was assigned a unique ODTS identification number."520 Figure + depicts the sub-fields (i.e. the WFC camera pointings). where the coloured bands within the grid elements indicate which filters each of the sub-tields have been observed in thus far.," Figure \ref{fig:fieldplots} depicts the sub-fields (i.e. the WFC camera pointings), where the coloured bands within the grid elements indicate which filters each of the sub-fields have been observed in thus far."521 A couple of WFC pointings have been highlighted for clarity in tizure + (0) and the WFC chips have been labelled., A couple of WFC pointings have been highlighted for clarity in figure \ref{fig:fieldplots} (c) and the WFC chips have been labelled.522 In this paper. he terminology refers to one entire WFC image and refers to one chip of the Observations for the optical INT) portion of the ODTS were completed in March 2003 with approximately 23ddeg of the survey observed in VRI. although the data reduction is still ongoing.," In this paper, the terminology refers to one entire WFC image and refers to one chip of the Observations for the optical (INT) portion of the ODTS were completed in March 2003 with approximately $23$ $^2$ of the survey observed in $BVRi^{\prime}$, although the data reduction is still ongoing."523 The (/ data. taken in the best observing conditions. currently cover | deg? in the Andromeda field.," The $U$ data, taken in the best observing conditions, currently cover $\sim$ 1 $^2$ in the Andromeda field."524 The total coverage of the fully reduced data is summarised in table 4.., The total coverage of the fully reduced data is summarised in table \ref{tab:area}. .525than in quiescence).,than in quiescence).526 Due to the Linitecl statistics of the data. the exact degree of spectral variability cannot be deteriuued nor can the exact cause of these possible variations be established.," Due to the limited statistics of the data, the exact degree of spectral variability cannot be determined nor can the exact cause of these possible variations be established."527 Dining the 2000 outburst. the source f£uctuated in Imuinosity by over 3 orders of magnitude on timescales of days (Wijnandsetal.2001).," During the 2000 outburst, the source fluctuated in luminosity by over 3 orders of magnitude on timescales of days \citep{wijnandsetal2001_rxte}."528 However. theBeppoSAN observatious reporte by Wijnandsetal.(2002) provided only. rough of estimatesthe fux ofSAN ΤΟΝΙ3658 between 2000 March 5 and 85.," However, the observations reported by \citet{wijnandsetal2002_bepposax} provided only rough estimates of the flux of SAX J1808.4–3658 between 2000 March 5 and 8."529 With the detection of the source on 2000 [απο 6. we now have a clearer picture of how diii the source could become during certain phases of its 2000 outburst.," With the detection of the source on 2000 March 6, we now have a clearer picture of how dim the source could become during certain phases of its 2000 outburst."530 As stated by Wijnandsetal.(2001).. the large luninositv variations observed during the 2000 outburst are difficult to understand as due to siuilulv dramatic variations in the mass accretion rate.," As stated by \citet{wijnandsetal2001_rxte}, the large luminosity variations observed during the 2000 outburst are difficult to understand as due to similarly dramatic variations in the mass accretion rate."531 It is more likely that oulv modest variations in the accretion rate can trigger transitious between two significautly different huuinositv states., It is more likely that only modest variations in the accretion rate can trigger transitions between two significantly different luminosity states.532 For example. ceutrifugal imbibition of accretion by the neutron stars magnetic field is expected below a certain critical accretion rate (the‘propellerregine:Mlarionov&Suvaevo1975) aud small but erratic viudations iu the accretion rate around this critical rate could iu principle give rise to the enormous Wuinosity swings observed dunus the 2000 outburst.," For example, centrifugal inhibition of accretion by the neutron star's magnetic field is expected below a certain critical accretion rate \citep[the 'propeller533regime';][]{is1975} and small but erratic variations in the accretion rate around this critical rate could in principle give rise to the enormous luminosity swings observed during the 2000 outburst."534" When the source is in this propeller τοσο, accretion is iubibited. but it is evident roni the large brightuess fluctuations during the 2000 outburst. that a considerable amount of natter was still available in the accretion disk."," When the source is in this propeller regime, accretion is inhibited, but it is evident from the large brightness fluctuations during the 2000 outburst, that a considerable amount of matter was still available in the accretion disk."535 According to Campanactal.(2002). a pure xopeller contribution is ruled out in quiescenuce siuce this mechanisin is expected to stop operating at luminosities below 10°? because the source should turu on then as a radio pulsar.," According to \citet{campanaetal2002}, a pure propeller contribution is ruled out in quiescence since this mechanism is expected to stop operating at luminosities below $10^{33}$ because the source should turn on then as a radio pulsar."536 Stellaetal.(2000). and Campanactal.(2002) sugecsted that a possible explanation for the quiescent flix is the cussion from the shock front between the relativistic wind of the radio pulsar and the matter out-flowing from the companion star., \citet{stellaetal2000} and \citet{campanaetal2002} suggested that a possible explanation for the quiescent flux is the emission from the shock front between the relativistic wind of the radio pulsar and the matter out-flowing from the companion star.537 Therefore. the flux during the 2000 outburst observation could be higher than that observed in quiescence because of the large amount of matter still present close to the neutron star.," Therefore, the flux during the 2000 outburst observation could be higher than that observed in quiescence because of the large amount of matter still present close to the neutron star."538 DiSalvo&Burderi(2003) sugsested an alternative explanation for the quicscent eiission of SAN Jlsos.b3658 iu which this enüssion is produced by direct dipole radiation from the radio pulsar., \citet{db2003} suggested an alternative explanation for the quiescent emission of SAX J1808.4–3658 in which this emission is produced by direct dipole radiation from the radio pulsar.539" The quiesceut X-ray huuinosity aud spectral shape of SAX J1808.13658 are indeed cousistent with those observed. frou several field wullisecond radio pulsars (seeBecker&Pavlov2002.forarecentreview). but Caindlayetal.(2002) found that the millisecond pulsars iu the elobular cluster £7 Tuc have a predominautly soft spectral shape suggestive of a thermal origin in contrast fo what is observed for SAN JIsUs.|3658,"," The quiescent X-ray luminosity and spectral shape of SAX J1808.4–3658 are indeed consistent with those observed from several field millisecond radio pulsars \citep[see][for540a recent review]{bp2002}, but \citet{grindlay2002} found that the millisecond pulsars in the globular cluster 47 Tuc have a predominantly soft spectral shape suggestive of a thermal origin in contrast to what is observed for SAX J1808.4--3658."541" Couclusive proof for an active radio pulsar iu SAN 505.£3658 would come from the detection of radio pulsations during quiescence, although such a detection iighlt be inhibited by the ambicut uatter still present in the system (seealsothedis-cussioninDiSalvo&Burd"," Conclusive proof for an active radio pulsar in SAX J1808.4–3658 would come from the detection of radio pulsations during quiescence, although such a detection might be inhibited by the ambient matter still present in the system \citep[see also the discussion in][]{db2003}."542er 2003).. Campanaetal.(2002) noted that the quiesceut xoperties of SAN Jlsds.l3658 are remarkably different from those observed for other quiesceut jeutrou star systems: (ajyits0.5 10 keV Iunimosity is the lowest observed so far for anv neutron star svsteni and (b) its spectrun is dominated * a power-law coniponent instead of a thermal conrponeut.," \citet{campanaetal2002} noted that the quiescent properties of SAX J1808.4–3658 are remarkably different from those observed for other quiescent neutron star systems: (a) its 0.5–10 keV luminosity is the lowest observed so far for any neutron star system, and (b) its spectrum is dominated by a power-law component instead of a thermal component."543 The quiesceunt enuüssion of neutron star Norav trausicuts is most often explained bv henual cussion frou the neutron star surface releasing the heat deposited iu the crust and core of the neutron star during outburst (seeBrown. herein).., The quiescent emission of neutron star X-ray transients is most often explained by thermal emission from the neutron star surface releasing the heat deposited in the crust and core of the neutron star during outburst \citep[see][ and references therein]{bbr1998}.544 In this model. the exact luuinosities of he svstenis should depend on their time-averaged accretion rates (Campanaetal.1998:Brown.Bild-steu.&Rautledee 1998).," In this model, the exact luminosities of the systems should depend on their time-averaged accretion rates \citep{campanaetal1998,bbr1998}."545. Due to the low peak uninositv of SAN J1s08.13658. Brown.Bild-sten.&Rutledge—(1998) predicted that this source should be rather faint im quiesceuce.," Due to the low peak luminosity of SAX J1808.4–3658, \citet{bbr1998}546 predicted that this source should be rather faint in quiescence."547 At first sieht the low detected quiescent πιοαν is consistent with this prediction. but no strong evidence could be found for a thermal coiiponeut inthe quiescent spectrum obtained.," At first sight the low detected quiescent luminosity is consistent with this prediction, but no strong evidence could be found for a thermal component in the quiescent spectrum obtained."548 This indicates that the thermal luuinosity of this source is very low. implying a rapidly cooling neutron star which would require culauced core cooling processes to explain (Campanaetal.2002).," This indicates that the thermal luminosity of this source is very low, implying a rapidly cooling neutron star which would require enhanced core cooling processes to explain \citep{campanaetal2002}."549. The combination of the low time-averaged aceretion rate aud the possibility of rapid core cooling müght also be able to explain why the quiesceut spectraofSAN J1808.|3658 is dominated by the power-law componcut., The combination of the low time-averaged accretion rate and the possibility of rapid core cooling might also be able to explain why the quiescent spectra of SAX J1808.4–3658 is dominated by the power-law component.550 If iu the “ordinary, If in the 'ordinary'551higher if compared for example to the good fit of Figure 12..,higher if compared for example to the good fit of Figure \ref{bfrc}.552 Similarly. for the opposite case of à slowly rising rotation curve. it is the parametric model which is quite unable to account for such a shape of the rotation velocity. with a similar high value for the best-fit residuals (see Fig. B4)).," Similarly, for the opposite case of a slowly rising rotation curve, it is the parametric model which is quite unable to account for such a shape of the rotation velocity, with a similar high value for the best-fit residuals (see Fig. \ref{pl}) )."553dependence of the background radiation but also a further exponential drop. and thus Q=0 Therefore we conclude that we can approximate the total energy-density with that of the interacting dark-matter density (o= Pips).,"dependence of the background radiation but also a further exponential drop, and thus $Q\simeq 0$ Therefore we conclude that we can approximate the total energy-density with that of the interacting dark-matter density $\rho \simeq \rho_{\rm IDM}$ )."554 Note. that L/e can be viewed as the mean life time of the residual radiation particles.," Note, that $1/\epsilon$ can be viewed as the mean life time of the residual radiation particles."555 With the aid of the ditterential equation theory we present solutions that are relevant to our eq.(7)., With the aid of the differential equation theory we present solutions that are relevant to our \ref{bol2}) ).556 In general a Riccati differential equation Is given by and it is fully solvable only when a particular solution is known., In general a Riccati differential equation is given by and it is fully solvable only when a particular solution is known.557 Below we present two cases in which analytical solutions are possible., Below we present two cases in which analytical solutions are possible.558 Note that the solution of is the only one providing a «a? dependence of the scale factor (see eqs. 12... 19.," Note that the solution of is the only one providing a $\propto \alpha^{-3}$ dependence of the scale factor (see eqs. \ref{sol1},"559 and 209)., \ref{sol4a} and \ref{eq:lam}) ).560 Suppose that we have a non-perfect cosmic fluid in a disequilibrium phase with energy density p., Suppose that we have a non-perfect cosmic fluid in a disequilibrium phase with energy density $\rho$.561 Then from the collisional Boltzmann equation. we have: Furthermore. we assumeitl that for a convenient period of time the cosmic fluid. in an expanding Universe. is slowly diluted according to poCia nx 0).," Then from the collisional Boltzmann equation, we have: Furthermore, we assume that for a convenient period of time the cosmic fluid, in an expanding Universe, is slowly diluted according to $\rho \sim C_{1}\alpha^{m}$ $m\le 0$ )."562 From a mathematical point of view. the latter assumption simply means that a solution of the form «a is a particular solution of the Boltzmann equation.," From a mathematical point of view, the latter assumption simply means that a solution of the form $\propto \alpha^{m}$ is a particular solution of the Boltzmann equation."563" Therefore. we have finally that: We remind the reader that for homogeneous and isotropic. flat cosmologies (Q,,+5=1). driven by non relativistic DM and a DE with a constant equation of state parameter (0). the density evolution of this cosmic fluid can be written as: where poy and pos are the present-day DM and DE densities. respectively."," Therefore, we have finally that: We remind the reader that for homogeneous and isotropic flat cosmologies $\Omega_{\rm m}+\Omega_{Q}=1$ ), driven by non relativistic DM and a DE with a constant equation of state parameter $w$ ), the density evolution of this cosmic fluid can be written as: where $\rho_{\rm m,0}$ and $\rho_{Q,0}$ are the present-day DM and DE densities, respectively."564 Thenecessary criteria to have cosmic acceleration and an inflection point in our past (;«ο). are: (ay P.«0 and (b) &=0. which leads to the conditions:," Thenecessary criteria to have cosmic acceleration and an inflection point in our past $t_{i}<t_{0}$ ), are: (a) $P<0$ and (b) $\ddot{\alpha}=0$ , which leads to the conditions:"565"produces synchrotron radiation, visible mainly at radio frequencies (e.g. Jonesetal. 1974)).","produces synchrotron radiation, visible mainly at radio frequencies (e.g. \citealt{1974ApJ...192..261J}) )."566" Therefore, the radio luminosity due to jets is inherently related to the central engine that powers the AGN."," Therefore, the radio luminosity due to jets is inherently related to the central engine that powers the AGN."567 One of the most prominent features in the optical spectrum of an AGN is the [OII] emission line., One of the most prominent features in the optical spectrum of an AGN is the [OII] emission line.568" It is produced in a very low density ionized medium, such as the NLR, when thermal photoelectrons collide with oxygen atoms and excite their lower energy levels."," It is produced in a very low density ionized medium, such as the NLR, when thermal photoelectrons collide with oxygen atoms and excite their lower energy levels."569" Even though de-excitation through downward radiation has a very low probability, in such a low density medium, collisions are much more sparse, so the downward radiation results in the emission of this forbidden-line."," Even though de-excitation through downward radiation has a very low probability, in such a low density medium, collisions are much more sparse, so the downward radiation results in the emission of this forbidden-line."570" Since the BLR can be obscured by the dusty torus, depending on the galaxy’s orientation, and the NLR is always visible, [ΟΠ] is a good tracer of the underlying continuum emission in an AGN (e.g. Hesetal. 1993)), although other narrow lines, e.g. [OIII], may be better (Simpson1998).."," Since the BLR can be obscured by the dusty torus, depending on the galaxy's orientation, and the NLR is always visible, [OII] is a good tracer of the underlying continuum emission in an AGN (e.g. \citealt{1993Natur.362..326H}) ), although other narrow lines, e.g. [OIII], may be better \citep{1998MNRAS.297L..39S}."571" It is now well established that there is a correlation between the luminosity of the NLR and the radio luminosity for radio-loud objects and that this relation is independent of redshift (e.g. Rawlings&Saunders1991,, Willottetal. 1999))."," It is now well established that there is a correlation between the luminosity of the NLR and the radio luminosity for radio-loud objects and that this relation is independent of redshift (e.g. \citealt{1991Natur.349..138R}, \citealt{1999MNRAS.309.1017W}) )."572 This supports the idea that the physical process behind radio jets is intrinsically linked to the source of the narrow lines via a common central engine (an accreting SMBH)., This supports the idea that the physical process behind radio jets is intrinsically linked to the source of the narrow lines via a common central engine (an accreting SMBH).573" In this paper we explore these ideas further by showing how rest-frame 124m luminosity, vLvi2,m, as a probe of warm dust emission, correlates with Dvisimuz, a tracer of jet power, and with which we assume tracks the underlying continuum Ljorj,emission of the AGN."," In this paper we explore these ideas further by showing how rest-frame $\rm 12\,\mu m$ luminosity, $\nu L_{\nu \rm 12\mu m}$, as a probe of warm dust emission, correlates with $L_{\nu \rm 151MHz}$, a tracer of jet power, and with $L_{\rm [OII]}$, which we assume tracks the underlying continuum emission of the AGN."574" Throughout this paper we adopt the following values for the cosmological parameters: Ho=70kms!Mpc!, 0.7 and Q4=0.3."," Throughout this paper we adopt the following values for the cosmological parameters: $\rm H_0=70\,km\,s^{-1}\,Mpc^{-1}$, $\rm575\Omega_M=0.7$ and $\rm \Omega_\Lambda=0.3$."576" We use the convention S,cv.", We use the convention $S_{\nu}\propto\rm\nu^{-\alpha}$.577" We select all of the narrow-line radio galaxies from the complete low-frequency selected radio samples 3CRR (Laingetal. 1983),, 6CE (Ealesetal.1997;Rawlings2001),, 6C* ( Blundelletal.1998;Jarvis 2001aa,b), 7CRS (Lacyetal.1999;Willott2003),, and TOOTOO (Hill&Rawlings2003;Vardoulakietal.2009) surveys by requiring a narrow redshift span of 0.9«z1.1."," We select all of the narrow-line radio galaxies from the complete low-frequency selected radio samples 3CRR \citep{1983MNRAS.204..151L}, , 6CE \citep{1997MNRAS.291..593E,2001MNRAS.322..523R}, 6C* ( \citealt{1998MNRAS.295..265B,2001MNRAS.326.1563J}a a,b), 7CRS \citep{1999MNRAS.308.1096L,2003MNRAS.339..173W}, and TOOT00 \citep{2003NewAR..47..373H,Vardoulaki} surveys by requiring a narrow redshift span of $0.9<z<1.1$."578 The low selection frequency of these samples (either 178 or 151 MHz) ensures that they are selected based on their optically thin lobe emission and thus in an orientation independent way., The low selection frequency of these samples (either 178 or 151 MHz) ensures that they are selected based on their optically thin lobe emission and thus in an orientation independent way.579" The narrow-line selection ensures that we are not including Blazars or type-I AGN, although 3C343 has since been classified as a QSO (Cleary et al."," The narrow-line selection ensures that we are not including Blazars or type-I AGN, although 3C343 has since been classified as a QSO (Cleary et al."580 2007)., 2007).581" Assembling together galaxies from the different radio surveys, allows a wide range of radio luminosities (25<logy)(Lv15imnz/W 29) to be covered."," Assembling together galaxies from the different radio surveys, allows a wide range of radio luminosities $25 <\log_{10}(L_{\nu \rm 151MHz}/\rm582\,W\,Hz^{-1}\,sr^{-1})<29$ ) to be covered."583" At the same time, such a narrow| width in redshift allows us to study correlations independent of possible evolutionary effects and any issues with assumed k-corrections."," At the same time, such a narrow width in redshift allows us to study correlations independent of possible evolutionary effects and any issues with assumed k-corrections."584" In the local Universe there is not enough cosmic volume to gather a reasonable number of luminous sources, whereas at high redshifts there are no statistically complete samples for the low flux limits that distant luminous sources require to be observed, therefore 0.9<z1.1 was chosen as a reasonable compromise."," In the local Universe there is not enough cosmic volume to gather a reasonable number of luminous sources, whereas at high redshifts there are no statistically complete samples for the low flux limits that distant luminous sources require to be observed, therefore $0.9 < z < 1.1$ was chosen as a reasonable compromise."585 The span of our sample in the redshift-radio luminosity plane is shown in Figure 1.., The span of our sample in the redshift–radio luminosity plane is shown in Figure \ref{fig:l151_z}.586" Most of the radio galaxies in our sample are classified as FRIIs, i.e. they are brightest at the edges of the radio structure."," Most of the radio galaxies in our sample are classified as FRIIs, i.e. they are brightest at the edges of the radio structure."587" The only exception is 3C343, which is core dominated and has since been identified as a quasar (Clearyetal. 2007),, and 6CE1217--37 and 5C7.242, both of which could be classified as either an FRI or FRII."," The only exception is 3C343, which is core dominated and has since been identified as a quasar \citep{2007ApJ...660..117C}, and 6CE1217+37 and 5C7.242, both of which could be classified as either an FRI or FRII."588" 'The observations presented in this paper were made with the Multiband Imaging Photometer for Spitzer (MIPS) camera on the Spitzer Space Telescope which provides long-wavelength capability, under the programme ID30344 (PI."," The observations presented in this paper were made with the Multiband Imaging Photometer for Spitzer (MIPS) camera on the Spitzer Space Telescope which provides long-wavelength capability, under the programme ID30344 (PI."589 Jarvis)., Jarvis).590 Our observations were carried out between August 2006 and August 2007., Our observations were carried out between August 2006 and August 2007.591 Our objects are relatively bright and short exposures were sufficient., Our objects are relatively bright and short exposures were sufficient.592" For the quasars we observed for one cycle using 7 jitters of 10 seconds, resulting in a total exposure time of 70 seconds on source."," For the quasars we observed for one cycle using 7 jitters of 10 seconds, resulting in a total exposure time of 70 seconds on source."593" For the radio galaxies we adopted a slightly different strategy; the 3CRR sources were observed with the same strategy as the quasars (i.e. 70 second total exposure time), while the less radio luminous objects from the 6CE, 7CRS and TOOT samples were observed for two 7-jitter cycles of 10 second per jitter, resulting in 140 seconds on source, to ensure significant detections, as previous observations have shown there to be a correlation between host galaxy mass/luminosity and the radio luminosity for radio galaxies (e.g. Willott et al."," For the radio galaxies we adopted a slightly different strategy; the 3CRR sources were observed with the same strategy as the quasars (i.e. 70 second total exposure time), while the less radio luminous objects from the 6CE, 7CRS and TOOT samples were observed for two 7-jitter cycles of 10 second per jitter, resulting in 140 seconds on source, to ensure significant detections, as previous observations have shown there to be a correlation between host galaxy mass/luminosity and the radio luminosity for radio galaxies (e.g. Willott et al."594 2003; McLure et al., 2003; McLure et al.595 2004)., 2004).596" Some of the 3C radio galaxies already had adequate data in the archive, for MIPS these include 3C22, 3C184, 3C280, 3C268.1, 3C289 and 3C343 (ID74; PI Houck) and 3C356 (ID3329; PI Stern)."," Some of the 3C radio galaxies already had adequate data in the archive, for MIPS these include 3C22, 3C184, 3C280, 3C268.1, 3C289 and 3C343 (ID74; PI Houck) and 3C356 (ID3329; PI Stern)."597 The data reduction was performed using the standard pipeline version S15.0.5., The data reduction was performed using the standard pipeline version S15.0.5.598" We measured the 24m flux densities by using an aperturediameter of 6.12 pixels and a 1.61 aperture correction, to match the procedure used by the MIPS instrument team to derive calibration factors from standard star observations."," We measured the $\mu$ m flux densities by using an aperturediameter of 6.12 pixels and a 1.61 aperture correction, to match the procedure used by the MIPS instrument team to derive calibration factors from standard star observations."599This section presents the sampο of red giauts that has been used for our now motaicity calibration.,This section presents the sample of red giants that has been used for our new metallicity calibration.600 T10 PAectroscopicallv deteruiue iroi abundances of f10 different authors are all cousisteut with the Ziun Wewf 1981)) abuudauce scale., The spectroscopically determined iron abundances of the different authors are all consistent with the Zinn West \cite{zinn}) ) abundance scale.601 All Stróunmeren colors refer tO t1¢ photometric svstem defired ]x Olsen (1993))., All Strömmgren colors refer to the photometric system defined by Olsen \cite{olse93}) ).602 T10 photometric colors of the previous calibration (Carelol Richtler 1992)) are based O1 hne svsteiia of primary y.zudards by Doud (1980)) axl Olsen (1983.. 1981)) axd have been corrected to the Olseu (1993) system accordi18o to the transformations eiven by Olsen (1995)).," The photometric colors of the previous calibration (Grebel Richtler \cite{greb92}) ) are based on the system of primary standards by Bond \cite{bond}) ) and Olsen \cite{olse83}, \cite{olse84}) ) and have been corrected to the Olsen (1993) system according to the transformations given by Olsen \cite{olse95}) )."603 Tn the same way the im color of the Held stars sample of Authousv-Twarog Twaroeg (1998) ) has been corrected according to Olseu’s raustormationus., In the same way the $m_1$ color of the field stars sample of Anthony-Twarog Twarog \cite{anth98}) ) has been corrected according to Olsen's transformations.604 The V and (®g) colors in this sample are on the svsein of Olsen (1993)., The $V$ and $(b-y)$ colors in this sample are on the system of Olsen (1993).605" For our calibration. wie uxed 12 E region stars from Joucli-Sorewen (1993)) and 5 stars from his 1991 list (199 L)). namely E3-33. E-37, EL-108. E5-32. E5-Is. 56. L6-Is. EG-9s. τσι. E8-39. Es-17. ES8-I8. aud FI-2. F5-2. F5-3. F6-1. F6-Mi3."," For our calibration, we used 12 E region stars from rensen \cite{jonc93}) ) and 5 stars from his 1994 list \cite{jonc94}) ), namely E3-33, E4-37, E4-108, E5-32, E5-48, E5-56, E6-48, E6-98, E7-64, E8-39, E8-47, E8-48, and F4-2, F5-2, F5-3, F6-1, F6-3."606 Their colors mre cousistent with the Olsen (1993) photometric svstem., Their colors are consistent with the Olsen (1993) photometric system.607 The standard stars are unifoniulv distributed over the color range 0.2< mag which has been used for our inetallicity calibration., The standard stars are uniformly distributed over the color range $0.2 < (b-y) < 1.3$ mag which has been used for our metallicity calibration.608The target galaxw in all our simulations is similar to NGC 1251 (the one counected to VirgollI21 by a tidal bridge) which is a typical spiral ealaxy with a circular velocity of 190 lan 1.,"The target galaxy in all our simulations is similar to NGC 4254 (the one connected to VirgoHI21 by a tidal bridge), which is a typical spiral galaxy with a circular velocity of 190 km $^{-1}$."609 Unless specified otherwise. the physical parameters for this galaxy were taken in the extragalacticdatabaset.," Unless specified otherwise, the physical parameters for this galaxy were taken in the extragalactic."610. The stellar disk is truncated at a radius 13 kpc (compatible with the Hos optical radius) with initially a Toomre(1963) profile of scale-leneth ο=L kpc., The stellar disk is truncated at a radius 13 kpc (compatible with the $R_{25}$ optical radius) with initially a \citet{toomre63} profile of scale-length $a=4$ kpc.611 The eas disk has an initial radius of 30 kpc. with a flat distribution from 7=0 to r=25 kpe aud a linear decrease from 25 to 30 kpc.," The gas disk has an initial radius of 30 kpc, with a flat distribution from $r=0$ to $=25$ kpc and a linear decrease from 25 to 30 kpc."612 This extent is compatible with the presently observed IT disk. taking iuto account the fact that the outer parts are stripped during the interaction.," This extent is compatible with the presently observed HI disk, taking into account the fact that the outer parts are stripped during the interaction."613 The final HIE radius im our model is about 25 kpc. while Phookunctal.(1993). detect III up to radii of 20 kpc aud the deeper observations by Tavucsetal.(2007) show III detection up to 25 kpc.," The final HI radius in our model is about 25 kpc, while \citet{phookun93} detect HI up to radii of 20 kpc and the deeper observations by \cite{haynes07} show HI detection up to 25 kpc."614 Note that an UT disk trumcated at 25 kpe already forms the main III bridge and the VirgollI21-like cloud. whereas a shehltly larger one is required to account for the most recent III data showing au extension North of VirgoIII21.," Note that an HI disk truncated at 25 kpc already forms the main HI bridge and the VirgoHI21-like cloud, whereas a slightly larger one is required to account for the most recent HI data showing an extension North of VirgoHI21."615 Both the stellar and gaseous disks are initially set up as 'Toonure(1963) disks with a radial scale leneth of 1 ipc for stars and 8 kpe for gas., Both the stellar and gaseous disks are initially set up as \citet{toomre63} disks with a radial scale length of 4 kpc for stars and 8 kpc for gas.616 A central bulee (PIlununer profile of scale-lenegth 1 kpe) is implemented. with a bulge:disk mass oratio of 0.25.," A central bulge (Plummer profile of scale-length 1 kpc) is implemented, with a bulge:disk mass ratio of 0.25."617 The dark matter halo is modeled with a softened isothermal sphere with a mass density profile given by: The sphere is truncated at r=100 kpe aud we use a core radius s2=Ll kpc., The dark matter halo is modeled with a softened isothermal sphere with a mass density profile given by: The sphere is truncated at $r=100$ kpc and we use a core radius $r_c=4$ kpc.618 This profile provides an extended flat rotation curve aud was fouud by Ducetal.(2001). to fairly reproduce the distribution of gas along tidal tails when compared to real systems., This profile provides an extended flat rotation curve and was found by \citet{DBM04} to fairly reproduce the distribution of gas along tidal tails when compared to real systems.619" The dark halo mass within the optical radius (13 kpc) is SB,«10? AL. (dark-to-wisible mass ratio 0.6).", The dark halo mass within the optical radius (13 kpc) is $8.5 \times 10^9$ $_{\sun}$ (dark-to-visible mass ratio 0.6).620 Within the outer edge of the initial III disk at 30 kpe. the halo nass is 2.2<1011 AL. (darketo visible ratio 1.6).," Within the outer edge of the initial HI disk at 30 kpc, the halo mass is $2.2 \times 10^{11}$ $_{\sun}$ (dark-to visible ratio 1.6)."621 The total mass of the dark halo up to its truucation radius at 100 kpe is 9&LOM ALL., The total mass of the dark halo up to its truncation radius at 100 kpc is $9 \times 10^{11}$ $_{\sun}$.622 The total dark-to-visible ratio. 75:1. is compatible with those favored by Dubiuskictal.(1996) to form long tidal tails.," The total dark-to-visible ratio, 7.5:1, is compatible with those favored by \citet{dubinski1996} to form long tidal tails."623 The initial gas mass fraction in the disk is7%... compatible with a prescut-day eas fraction of about 5 percent after star formation.," The initial gas mass fraction in the disk is, compatible with a present-day gas fraction of about 5 percent after star formation."624 The mass of each componcut is computed to eet a circular velocity Των 190 aus bat kr—15 kpe. as observed in NGC 51: this give a stellar mass of 1.2«Lott AL. aud initial eas mass of 8.5«10? ML...," The mass of each component is computed to get a circular velocity $V_\mathrm{circ} \simeq $ 190 km $^{-1}$ at $r=15$ kpc, as observed in NGC 4254; this give a stellar mass of $1.2 \times 10^{11}$ $_\sun$ and initial gas mass of $8.5 \times 10^9$ $_\sun$."625 The modeled interloper ealaxy is gas-free., The modeled interloper galaxy is gas-free.626 This choice is nade for simplicity aud. CPU-time efficiency. because the eas disk of the iuterloper can disturb the target disk ouly for close encounters and/or low-velocity encounters with coplanar disks. when the two eas disks overlap aud shock.," This choice is made for simplicity and CPU-time efficiency, because the gas disk of the interloper can disturb the target disk only for close encounters and/or low-velocity encounters with coplanar disks, when the two gas disks overlap and shock."627 This is not the case for the high-velocity fiv-bys studied here., This is not the case for the high-velocity fly-bys studied here.628 In particular the periceuter distance m our uodel for VirgoIlI21 (see Sect. Ly) , In particular the pericenter distance in our model for VirgoHI21 (see Sect. \ref{sect:virgohi}) )629is 59 kpc with au orbital plane differeut from the target disk plane. walking any overlap of the eas disks uulikelv.," is 59 kpc with an orbital plane different from the target disk plane, making any overlap of the gas disks unlikely."630 We chose to model he interloper as a spiral with the same mass xoportious and relative sizes for the bulge. disk. aud dark matter halo.," We chose to model the interloper as a spiral with the same mass proportions and relative sizes for the bulge, disk, and dark matter halo."631 All sizes are scaled as the square root of the mass relatively to the target galaxy. which keeps he central deusitv coustaut.," All sizes are scaled as the square root of the mass relatively to the target galaxy, which keeps the central density constant."632 We performed a nunber of simulations aimed at coluparing the formation of tidal tails in hieh aud low velocity encounters., We performed a number of simulations aimed at comparing the formation of tidal tails in high and low velocity encounters.633 In all the seven rus we have carried out. the orbit is direct. ie. the orbital momenta of the interloper w.r.f.," In all the seven runs we have carried out, the orbit is direct, i.e. the orbital momentum of the interloper w.r.t."634 the target has the same direction as the spin of the target disk., the target has the same direction as the spin of the target disk.635" The orbital parameters. which are given iu Table 1. for all απλα», are: the iuitial velocity V4. at an iufiuite distance: the velocity at the beeiuniug of the simulations is computed with the assumption that the dynamical fraction is uceheible before the beeimnuiug of the he mnipact parameter 5. which is the distance at which the two galaxies would cross if they had linear trajectories along their initial he actual periceuter distance Rp aud velocity at the pericenter Vp."," The orbital parameters, which are given in Table \ref{tab:param} for all runs, are: – the initial velocity $V_\infty$ at an infinite distance; the velocity at the beginning of the simulations is computed with the assumption that the dynamical fraction is negligible before the beginning of the – the impact parameter $b$, which is the distance at which the two galaxies would cross if they had linear trajectories along their initial – the actual pericenter distance $R_\mathrm{P}$ and velocity at the pericenter $V_\mathrm{P}$."636 These quautites are measured frou the he inclination of the orbit plane wrt., These quantites are measured from the – the inclination of the orbit plane w.r.t.637 the target ealaxy disk plane 4, the target galaxy disk plane $i$.638 A coplanar cucouuter corresponda tor=U. the ratio of the interloper-to-tarect total masses Mj. We compare here the properties of tidal tails formed in low and Ligh velocity encounters., A coplanar encounter corresponds to $i=0$ – the ratio of the interloper-to-target total masses $M_i$ We compare here the properties of tidal tails formed in low and high velocity encounters.639 All these simmlatious are for direct orbits. with lieh velocity ln s 1) for runs 1-3-5. and low velocity (V4=230 lau +) for runs 2-6. the other paramcters being unchanged.," All these simulations are for direct orbits, with high velocity $V_\infty =900$ km $^{-1}$ ) for runs 1-3-5, and low velocity $V_\infty =230$ km $^{-1}$ ) for runs 2-4-6, the other parameters being unchanged."640 The impact parameters b were chosen to provide comparable periceuter distances Rp for cach siuulation pairs 1-2. 3-1. 5-6 (although the exact," The impact parameters $b$ were chosen to provide comparable pericenter distances $R_\mathrm{P}$ for each simulation pairs 1-2, 3-4, 5-6 (although the exact"641Haussleretal.(2007) catalogs.,\citet{haussler2007} catalogs.642 Our analysis hinges on the consistent measurement of the axis-ratios of galaxies at very different redshifts and from very different imaging data sets., Our analysis hinges on the consistent measurement of the axis-ratios of galaxies at very different redshifts and from very different imaging data sets.643" In Holdenetal. (2009),, we tested our our axis-ratio measurements fromGALFIT with simulations of observations of high redshift galaxies using real low redshift galaxies as templates."," In \citet{holden2009}, we tested our our axis-ratio measurements from with simulations of observations of high redshift galaxies using real low redshift galaxies as templates."644" We found these measurements to be robust, with a negligible shift in the axis-ratio from z=0 to z=1 of ddprojc—0.01 with a scatter of σᾳ~0.01—0.03 depending on galaxy magnitude."," We found these measurements to be robust, with a negligible shift in the axis-ratio from $z=0$ to $z=1$ of $\delta q_{proj}645\simeq -0.01$ with a scatter of $\sigma_q \simeq 0.01 - 0.03$ depending on galaxy magnitude."646" In addition to data-related differences, the fitting algorithms also differ between the low- and high-redshift galaxy samples."," In addition to data-related differences, the fitting algorithms also differ between the low- and high-redshift galaxy samples."647 We use the adaption of for SDSS imaging from Guoetal.(2009) to measure the axis-ratios of a sub-sample of our SDSS galaxies in a manner that is fully consistent with the treatment of the high-redshift galaxies., We use the adaption of for SDSS imaging from \citet{guo2009} to measure the axis-ratios of a sub-sample of our SDSS galaxies in a manner that is fully consistent with the treatment of the high-redshift galaxies.648" For small axis-ratios, systematic differences are expected to be largest."," For small axis-ratios, systematic differences are expected to be largest."649" Therefore, we select 412 SDSS galaxies from our sample with axis-ratios 0.3«Gproj0.305 as determined by the SDSS pipeline."," Therefore, we select 412 SDSS galaxies from our sample with axis-ratios $0.3 < q_{proj} < 0.305$ as determined by the SDSS pipeline."650" These represent the extreme end of the distribution, where systematic differences in fitting"," These represent the extreme end of the distribution, where systematic differences in fitting"651Why study the B-mode polarization of the cosmic microwave background (CMB)?,Why study the $B$ -mode polarization of the cosmic microwave background (CMB)?652" Detection of the primordial gravitational waves generated during inflation would give us a direct insight into the physical condition of the universe when the energy scale was close to the grand unification scale, ~1016 GeV (seeLiddle&Lyth2009,forarecentreviewandreferences"," Detection of the primordial gravitational waves generated during inflation would give us a direct insight into the physical condition of the universe when the energy scale was close to the grand unification scale, $\sim 10^{16}$ GeV \citep[see][for a recent review and references653therein]{liddle/lyth:PDP}."654" While a direct detection of the primordial gravitationaltherein).. waves using, e.g., laser interferometers, seems not possible with the present-day technology, an indirect detection using the B-mode polarization of the CMB (Seljak&Zaldar-riaga1997;Kamionkowskietal.1997) may be possible in the near future (most optimistically, within a few years), provided that the energy scale of inflation at which the observed gravitational waves were generated was indeed as high as the grand unification scale."," While a direct detection of the primordial gravitational waves using, e.g., laser interferometers, seems not possible with the present-day technology, an detection using the $B$ -mode polarization of the CMB \citep{seljak/zaldarriaga:1997,kamionkowski/kosowsky/stebbins:1997} may be possible in the near future (most optimistically, within a few years), provided that the energy scale of inflation at which the observed gravitational waves were generated was indeed as high as the grand unification scale."655" We often characterize the amplitude of gravitational waves known as tensor using the so- (also“tensor-to-scalar ratio,” which is perturbations)conventionally defined as and are the Fourier transform of the whereamplitudeshy of hytwo linear polarization states of gravitational waves, and y is the primordial curvature perturbation, which is a scalar perturbation (hence the name, “tensor-to-scalar "," We often characterize the amplitude of gravitational waves (also known as tensor perturbations) using the so-called “tensor-to-scalar ratio,” which is conventionally defined as where $h^{+}_{\mathbf k}$ and $h^{\times}_{\mathbf k}$ are the Fourier transform of the amplitudes of two linear polarization states of gravitational waves, and ${\cal R}_{\mathbf k}$ is the primordial curvature perturbation, which is a scalar perturbation (hence the name, “tensor-to-scalar ratio”)."656"It is Ry that seeded the observed structure in the ratio"").universe, as well as the dominant component of the observed CMB temperature anisotropy (seeWeinberg2008,forarecentreviewandreferences therein).."," It is ${\cal R}_{\mathbf k}$ that seeded the observed structure in the universe, as well as the dominant component of the observed CMB temperature anisotropy \citep[see][for a657recent review and references therein]{weinberg:COS}."658" The dominant, scalar part of the temperature anisotropy generates radial and tangential polarization patterns around hot and cold spots (Coulsonetal."," The dominant, scalar part of the temperature anisotropy generates radial and tangential polarization patterns around hot and cold spots \citep{coulson/crittenden/turok:1994}."659" This is called the E-mode polarization, and has been 1994)..detected with high statistical significance (Brownmatsuetal.2010;QUIET 2010)."," This is called the $E$ -mode polarization, and has been detected with high statistical significance \citep{brown/etal:2009,chiang/etal:2010,larson/etal:prep,komatsu/etal:prep,quiet:prep}."660" However, the B- polarization, which cannot be generated by the scalar perturbations but can be generated by the tensor perturbations, has not been found yet."," However, the $B$ -mode polarization, which cannot be generated by the scalar perturbations but can be generated by the tensor perturbations, has not been found yet."661" The current upper limit on the tensor-to-scalar ratio is r«0.24, which mainly comes from the upper limit on the tensor contribution to the temperature anisotropy on large angular scales (Komatsuetal.2010)."," The current upper limit on the tensor-to-scalar ratio is $r<0.24$, which mainly comes from the upper limit on the tensor contribution to the temperature anisotropy on large angular scales \citep{komatsu/etal:prep}."662". Given the upper limit on r, one can calculate the expected level of the B-mode power spectrum (see Figure 1))."," Given the upper limit on $r$, one can calculate the expected level of the $B$ -mode power spectrum (see Figure \ref{fig:clnow}) )."663" For r—0.24, the B-mode power spectrum is smaller than the E-mode power spectrum by a factor of 10 at the first bump (created by electrons at z 10)."," For $r=0.24$, the $B$ -mode power spectrum is smaller than the $E$ -mode power spectrum by a factor of 10 at the first bump (created by electrons at $z\lesssim66410$ )."665" At the second bump (created by electrons at z~ the B-mode power spectrum is smaller than the E-mode1090), power spectrum by factor of 50."," At the second bump (created by electrons at $z\simeq 1090$ ), the $B$ -mode power spectrum is smaller than the $E$ -mode power spectrum by a factor of 50."666 It is the smallness of the B-mode power spectruma that makes the detection of this signal challenging., It is the smallness of the $B$ -mode power spectrum that makes the detection of this signal challenging.667 'There are three sources of noise for B-mode detection: (1) Detector noise; (2) Galactic foregroundemission; and (3) Gravitational lensing., There are three sources of noise for $B$ -mode detection: (1) Detector noise; (2) Galactic foregroundemission; and (3) Gravitational lensing.668" In this paper, we shall focus on the Galactic foreground."," In this paper, we shall focus on the Galactic foreground."669" We use a map-based method for reducing the Galactic foreground, and study how the residual foreground limits à measurement of the primordial B-mode polarization."," We use a map-based method for reducing the Galactic foreground, and study how the residual foreground limits a measurement of the primordial $B$ -mode polarization."670" The foreground reduction technique we use is motivated by the “template cleaning method"" used bythe WMAP team (Pageetal."," The foreground reduction technique we use is motivated by the “template cleaning method” used bythe team \citep{page/etal:2007,gold/etal:2009,gold/etal:prep}."671 This method was further investigated by Efstathiou2010)..etal.(2009) in the context of the Planck mission., This method was further investigated by \citet{efstathiou/gratton/paci:2009} in the context of the mission.672" We shall study this technique in the context of a next-generation, low-noise, low-resolution (0.5 degree FWHM) space-borne experiment."," We shall study this technique in the context of a next-generation, low-noise, low-resolution (0.5 degree FWHM) space-borne experiment."673frequency derivative) cau be explained as a sequence of : ↺∱≺∪⊼∕−⋝↖↖↽∐↸⊳∐⋯⊳↸⊳↿∐⋅⋜↧↑⋜↧↸⊳∪∐↴∖↴↑⋜⋯⊓⋅⋜↧↑↸∖↕⊰∙↽∕∏∐∖∐↑∐↸∖↕⊰⋀∖↕⊱↖↽⋜∐⋅↕⋜↧↑↕∪∐ : Jj ⋅⋅ ∪↕≯↑∐∖↻∏↕↴∖↴↸∖↕⋟↥⋅↸∖≺∣⋯∖∐↸⊳⋅↖↽↴∖↴↸⊳⋜↧↕↸∖↴∖↴↖↖↽↕↕⊔∖↕⋜⋯∖↴↸∖≼⊔↕⋯↸∖∣−⋜↧↴∖↴ ⋖∆↗∕⋝⇉∶∶∫↿⋟≺∪∖⊼∕⋝⇉∣−≺∐∑⋟∙ ↖↖↽∐↸∖↥⋅↸∖∆∡↰⊲∶∫↿⋟ÓpT> ds defined as noise streneth.,"frequency derivative) can be explained as a sequence of which occur at a constant rate R. Then the RMS variation of the pulse frequency scales with elapsed time $\tau$ as $<(\Delta \nu) ^{2}> = R <(\delta \nu) ^{2}> \tau$ (Hz), where $S= R <\delta \nu ^{2}>$ is defined as noise strength."674" Then. RAIS scaling for the pulse frequency derivaives. can be obtained. as <CAP)?4DοDτς(Sírzj?Izs""1|."," Then, RMS scaling for the pulse frequency derivatives can be obtained as $<(\Delta \dot \nu) ^{2}>^{1/2} = (S/\tau )^{1/2} {\rm{Hz.s}}^{-1}$."675 As soon from. Table 1. iu our fits. upper lits on intrinsic pulse frequency derivatives are 7-10 times higher than the oue-term spin up rates.," As seen from Table 1, in our fits, upper limits on intrinsic pulse frequency derivatives are 7-10 times higher than the long-term spin up rates."676 If white noise in the pulse frequency derivative can be interpolaed to a few days.of frequency derivative. obtaiied froma ~ Loweek observation should typically have a maenitucde that cau be estiuated from <(Ar)DwreckPESE60difs«15? Previous mareiual measurement of change in the orbital period. was (-2.9 £ «10.© yr! (Clark 2000). and our new value for the orbital period change.," If white noise in the pulse frequency derivative can be interpolated to a few days, frequency derivative obtained from a $\sim 1$ week observation should typically have a magnitude that can be estimated from $<(\Delta \dot \nu) ^{2}_{week}>=677 <(\Delta \dot \nu) ^{2}_{1600 days} \times 15^2 $ Previous marginal measurement of change in the orbital period, was (-2.9 $\pm$ $\times 10^{-6}$ $^{-1}$ (Clark 2000), and our new value for the orbital period change,"678Microquasars are binary svsclus hosting a star and au accreting black hole or neuron star.,Microquasars are binary systems hosting a star and an accreting black hole or neutron star.679 Matter from tlhe star ds ransterred to the compact object. part of it being launched through maguetoceitifugal forces (e.e..Blase&Whaiculvan 2011).," Matter from the star is transferred to the compact object, part of it being launched through magnetocentrifugal forces \citep[e.g.,][]{bla77,bla82,bar11}."680. This triggers the formation of bi- jets. which generate ron-therimal radio enission (e.g...Miraveleta.1999:Ribó2005).. and are thought to be tre Kkcation from where the eamuna ravs observed 1l SOC S01YCCR ALC! emitted (ee.Albertetal.2007:Ta-vaietal.2009;Axlo2009:Sabatijioct 2010).," This triggers the formation of bi-polar jets, which generate non-thermal radio emission \citep[e.g.,][]{mir99,rib05}, and are thought to be the location from where the gamma rays observed in some sources are emitted \citep[e.g.,][]{alb07,tav09,abd09,sab10}."681. Jets could © Inaenetically dominated at their base. but naegnctoly¢lrodyuaniücal processes occurring at Ligher jet reight would accecrate the fQW. cfficietly converting naegnetie ewrev iuto kiuetic oie (e.g.[xennssarovctal. 2007).," Jets could be magnetically dominated at their base, but magnetohydrodynamical processes occurring at higher jet height would accelerate the flow, efficiently converting magnetic energy into kinetic one \citep[e.g.][]{kom07}."682". At the scales of the bilary svsteu {ο109Haas. where Rea, is the Sclwwarzsclid radius). the jet is lisely ο be alreac va lydrodyvuuuica (IID) flow."," At the scales of the binary system $\sim 10^6\,R_{\rm Sch}$, where $R_{\rm Sch}$ is the Schwarzschild radius), the jet is likely to be already a hydrodynamical (HD) flow."683" We focus were outje persistent jets thought to be present during the QW-jud state ¢of unicroquasars. altrough some considerations OL ransicut ejecta. associatcc to low-hard to high-soft Sate traIS]1015, are done below (sec.e.g..FenderOL.50ἱ9,OYreviewsonnücroquasar states)."," We focus here on the persistent jets thought to be present during the low-hard state of microquasars, although some considerations for transient ejecta, associated to low-hard to high-soft state transitions, are done below \citep[see, e.g.,][for reviews on microquasar states]{fen04,fen09}."684 Part « Etio energv carried by the jet can be dissipated iu he OYu of magnetic reconnection. recollimation aid interua shocks. shear lavers in fιο jet walls. aud tiwhulence.," Part of the energy carried by the jet can be dissipated in the form of magnetic reconnection, recollimation and internal shocks, shear layers in the jet walls, and turbulence."685" Part of the dissipated elOYSv cali So to non-therlia particles. generating low- and high-cucrey (nission via different maechanisiis. svuchrotron from radio to N-ravs. aud inverse Compon (IC) aid hacdrou-related processes UL» to Calla Lars (sec.Ca,Bosch-Ramon&Ikhaugulvau2009.andreferc1οhere 1j."," Part of the dissipated energy can go to non-thermal particles, generating low- and high-energy emission via different mechanisms, synchrotron from radio to X-rays, and inverse Compton (IC) and hadron-related processes up to gamma rays \citep[see, e.g.,][and references therein]{bos09}."686 As shown bw Perucho&Bosch-Ramoun(2008) (PBOs hereafter) and Peruchoeal.(2010a) (PBIS10 hereafter). in nücroquasars nsting au OB star (high-mass mucroquasars: hereafter IINAICn he jet may be strongly influenced 1w the stellar wixl.," As shown by \cite{per08} (PB08 hereafter) and \cite{per10} (PBK10 hereafter), in microquasars hosting an OB star (high-mass microquasars; hereafter HMMQ) the jet may be strongly influenced by the stellar wind."687 The one-side iupact of the wind on the (presuniabY] aready IID jet leads to strong aud asviunietric recollimation shocks. bending aud differeit types of instabilities: the recolliamation shocks seen suitable candidates for particle acceeration aud nou-hermal enusson: bending iav be noticeable iu radio at 111larcsecond scales: iustabilitics may destroy the jet flow even within ε1 binary svstei.," The one-side impact of the wind on the (presumably) already HD jet leads to strong and asymmetric recollimation shocks, bending and different types of instabilities: the recollimation shocks seem suitable candidates for particle acceleration and non-thermal emission; bending may be noticeable in radio at milliarcsecond scales; instabilities may destroy the jet flow even within the binary system."688" For ypical wiud aud jet velocities. sav Ce~2s105 aud cPp""1079 ons, respectively, stidies show that for comac‘ft Dinaries aud jet-to-wineL moneitu flux ratios s0.) he jet can be already. disrupted. (PDüs. PBILO)."," For typical wind and jet velocities, say $v_{\rm w}\sim 2\times 10^8$ and $v_{\rm j}\sim 10^{10}$ cm/s, respectively, studies show that for compact binaries and jet-to-wind momentum flux ratios $\la 0.1$ the jet can be already disrupted (PB08, PBK10)."689 This tuber is rather coustraiueg. siicc| only a few ΠΛΑΤΟ iieht be aOVE his threshold.," This number is rather constraining, since only a few HMMQ might be above this threshold."690 Tus could be the reasoi for the low miber o ΠΙΟ deteced. as sueeested in PBIN1U)," This could be the reason for the low number of HMMQ detected, as suggested in PBK10."691 Iu ALY case. even I: nof destroved within the MALY SVSena. jets cal πιufer srong pcrurbatious with both dvuaimical ancl raciaive COlLISCQUOCIHCCS.," In any case, even if not destroyed within the binary system, jets can suffer strong perturbations with both dynamical and radiative consequences."692 Previous work in ΠΑΙΑΤΟ wind-jet iucractions was done unuder the assuiptiou hat the winel is homogencous. nmt iu fact stelax winds are thought o be chuupx (e.g.Owocki&Cohen2006:Moffat.2008).," Previous work in HMMQ wind-jet interactions was done under the assumption that the wind is homogeneous, but in fact stellar winds are thought to be clumpy \citep[e.g.,][]{owo06,mof08}."693. For this reason. it has beeu propose| that wind chuupiness shouk ο taken iuto account whe1 stidying IEMMO (Owocetal.2009:ArarQot2008 -ADRUO hereafter: Romeroetal.2tM:Araudoeta]. 20011)).," For this reason, it has been proposed that wind clumpiness should be taken into account when studying HMMQ \citealt{owo09,ara09} -ABR09 hereafter-; \citealt{rom10,ara11}) )."694" Au importa xuneter tha determines na1onioeeneity of the wind is the wind Ἡhue actor f. which determines the wind volume fracticn with hielkY density,"," An important parameter that determines the inhomogeneity of the wind is the wind filling factor $f$, which determines the wind volume fraction with higher density."695 For à siguificau departure from homogencity. tie dntrachnunp medi Lass ali moleutiii fluxes will e negligible aud onY chuups will have a dynamical iiet on the jet.," For a significant departure from homogeneity, the intraclump medium mass and momentum fluxes will be negligible and only clumps will have a dynamical impact on the jet."696 Siuce he interactioi between a TD je and a chuupy wixd las nof jen studied iu detail. we have earrk«d out Bendiimensioial (3D) simulations oftus scenario.," Since the interaction between a HD jet and a clumpy wind has not been studied in detail, we have carried out 3-dimensional (3D) simulations of this scenario."697 Sinilatious je been done for two differen jet powers aid jet-o-wind momenun ratio. Lj2&1001QU ore!os and ozOU.05üAOS. respectively. Q CXlore wla could 0 the ransitkn between jet «lestiiction ai loic-ena collatio1," Simulations have been done for two different jet powers and jet-to-wind momentum ratio, $L_{\rm j}=3\times69810^{36}-10^{37}$ erg/s and $\approx 0.03-0.08$, respectively, to explore what could be the transition between jet destruction and long-term collimation."699 Another siuulatiou has focused on he evohtion of individual chua5 111]ected in the jet at different heights., Another simulation has focused on the evolution of individual clumps injected in the jet at different heights.700 Unlike in PDKI1h in which he sinulation start with the jet being injected at its base. rere the jet is coical aud crossing he wrole erid. aud the chuupy wind is injected from oue of the jet sides.," Unlike in PBK10, in which the simulation started with the jet being injected at its base, here the jet is conical and crossing the whole grid, and the clumpy wind is injected from one of the jet sides."701By using these Hermitian products we not only dispense with the unnecessary burden of Majorana phases but we are also able to remove one additional phase from the lepton mixing matrix. (hus. simplihing the subsequent analvsis.,"By using these Hermitian products we not only dispense with the unnecessary burden of Majorana phases but we are also able to remove one additional phase from the lepton mixing matrix, thus, simplifying the subsequent analysis."702" The lepton mass matrices can be diagonalized as Thus. the product M,MJ becomes which can be written as using Eq.(7)."," The lepton mass matrices can be diagonalized as Thus, the product $M_l M_l^{\dagger}$ becomes which can be written as using Eq.(7)."703" Similarly. for the product M,M, we obtain We can absorb two phases [rom £7 ancl one phase [rom P, in the left handed lepton fields aud (he resulting lepton mixing matrix is given bv where U, and U, contain three real parameters aud one phase each while P, contains one phase 2, = diag(1.1.€"")."," Similarly, for the product $M_\nu M_\nu^{\dagger}$ we obtain We can absorb two phases from $P_l$ and one phase from $P_\nu$ in the left handed lepton fields and the resulting lepton mixing matrix is given by where $\tilde{U_l}$ and $\tilde{U_\nu}$ contain three real parameters and one phase each while $P_\nu$ contains one phase $P_\nu$ = $(1,1,e^{i \phi})$."704 Thus. in this formalism Ü is expressed in terms of six real parameters and three In the present work. we discuss a parametrization of the lepton mixing matrix which allows for the large deviations from 0644=0 and has the form of Eq.(4) at zeroth order.," Thus, in this formalism $U$ is expressed in terms of six real parameters and three In the present work, we discuss a parametrization of the lepton mixing matrix which allows for the large deviations from $\theta_{13}=0$ and has the form of Eq.(4) at zeroth order."705 Here 815 can have the values sinειν) for TBM mixing. tan1/4) for GRI mixing. cos'(2/2) for GR2 mixing where y = (1+V/5)/2. 7/6 for hexagonal mixing and 7/4 for bimasximal mixing.," Here $\theta_{12}'$ can have the values $\sin^{-1}(1/\sqrt{3})$ for TBM mixing, $\tan^{-1}(1/\varphi)$ for GR1 mixing, $\cos^{-1}(\varphi/2)$ for GR2 mixing where $\varphi$ = $(1+\sqrt{5})/2$, $\pi/6$ for hexagonal mixing and $\pi/4$ for bimaximal mixing."706 Deviations [from the above mentioned scenarios are parametrized in terms of charged lepton corrections represented by small parameters, Deviations from the above mentioned scenarios are parametrized in terms of charged lepton corrections represented by small parameters707llaving the magnetic field for a simulated AISP. we can obtain its initial period with a spin-up relation.,"Having the magnetic field for a simulated MSP, we can obtain its initial period with a spin-up relation."708 Alparetal.(1982) and similarly Bhattacharya&vandenHetvel(1991) found that for reevelecl pulsars spun up by accretion at the Eddington limit from a binary companion. (he initial period is estimated by the expression which ean be rewritten as a birth line in the 2—P diagram bv the form where the period is in seconds and the period derivative is ss.I.," \citet{Alpar82} and similarly \citet{Bhat91} found that for recycled pulsars spun up by accretion at the Eddington limit from a binary companion, the initial period is estimated by the expression which can be rewritten as a birth line in the $\dot P - P$ diagram by the form where the period is in seconds and the period derivative is ${\rm709s\cdot{s^{-1}}}$."710 Having the magnetic liekl. the initial period. and the age. we can determine the present period and period derivative of the MSP assuming a pure dipole spin down.," Having the magnetic field, the initial period, and the age, we can determine the present period and period derivative of the MSP assuming a pure dipole spin down."711 However. recent studies of LAINBs where (he spin periods of accreting neutron stars were measured have allowed (he estimation of the surface magnetic fields.," However, recent studies of LMXBs where the spin periods of accreting neutron stars were measured have allowed the estimation of the surface magnetic fields."712 Lamb&Yu(2005). conclude that the general properties n LAIXNBs can be understood if the accretion rates range from the Ecddiugton critical accretion rate to 5xLO! times that rate.," \citet{Lamb05}713 conclude that the general properties of LMXBs can be understood if the accretion rates range from the Eddington critical accretion rate to $5\times 10^{-4}$ times that rate."714 These different accretion rates lead to different birth lines in (he P?—P diagram., These different accretion rates lead to different birth lines in the $\dot P-P$ diagram.715 To incorporate a distribution of accretion rates. we have included an approximate procedure in which we dither the intercept of the following birth line where the dithering parameter 9 varies from 0 to 2.8.," To incorporate a distribution of accretion rates, we have included an approximate procedure in which we dither the intercept of the following birth line where the dithering parameter $\delta$ varies from 0 to 2.8."716 The birth line can be reformulated bv the expression The extremes ofthe dithering parameter 9 represent (he approximate Edclington accretion critical rate and 5xLO! times that rate (see figure 4 in Lamb&Yu (2005)))., The birth line can be reformulated by the expression The extremes of the dithering parameter $\delta$ represent the approximate Eddington accretion critical rate and $5\times 10^{-4}$ times that rate (see figure 4 in \citet{Lamb05}) ).717 While we explored a Gaussian distribution of the dithering parameter. we obtain better agreement wilh a ramp cdistribution (hat increases by a [actor of 4 between 0 and 2.8.," While we explored a Gaussian distribution of the dithering parameter, we obtain better agreement with a ramp distribution that increases by a factor of 4 between 0 and 2.8."718 With (he magnetic field randomly selected from the distribution of Equation 2. we then use Equation 6 to obtain the initial period of (he simulated. MSDP.," With the magnetic field randomly selected from the distribution of Equation 2, we then use Equation 6 to obtain the initial period of the simulated MSP."719 However. we impose a minimum initial period οἱ Da 7 123 ms in accord with the RATE studies of LMXDs by Chakrabarty(2005).," However, we impose a minimum initial period of $P_{o_{\rm min}}$ = 1.3 ms in accord with the RXTE studies of LMXBs by \citet{Chak05}."720. Having randomlv determined (he pulsars age assuming a uniform birth rate. we spin (he pulsar down to obtain the present period and period derivative assuming constant magnetic surface field.," Having randomly determined the pulsar's age assuming a uniform birth rate, we spin the pulsar down to obtain the present period and period derivative assuming constant magnetic surface field."721 A constant magnetic field used in this study of MSPs is incontrast to the assumption ol magnetic field decay in our simulations of NPs (Gonthierοἱal.2002.2004. 2007)..," A constant magnetic field used in this study of MSPs is incontrast to the assumption of magnetic field decay in our simulations of NPs \citep{Gon02, Gon04, Gon07}. ."722nunake it an ideal source to carry oul pliase-resolved spectroscopy and probe changes in the spectral οποιον distribution between different phases.,make it an ideal source to carry out phase-resolved spectroscopy and probe changes in the spectral energy distribution between different phases.723 The origin of the soft excess in iis not clear. and its (nme-averaged spectral energy distribution can be well modeled by a vide variety of models including reflection from a partially ionized accretion disc 2006).. Comptonized disc emission [rom a low temperature disc. ionized partial covering. or a smeared disc wind seen in absorption (seee.g.Middletonetal.2009).," The origin of the soft excess in is not clear, and its time-averaged spectral energy distribution can be well modeled by a wide variety of models including reflection from a partially ionized accretion disc \citep{crummy2006}, Comptonized disc emission from a low temperature disc, ionized partial covering, or a smeared disc wind seen in absorption \citep[see e.g.][]{middletonetal2009}."724". Phase-resolved spectra could give important insights for breaking this degeneracy,", Phase-resolved spectra could give important insights for breaking this degeneracy.725 Therelore one of our major goals in (his work is (ο search for signatures of variable reflection (in the continuum and reflection lines) and/or variable absorption features (e.g. changes in the properties of ο VII. ο VIII edees which are the hallmark of the presence of warm absorber in active galactic nuclei).," Therefore one of our major goals in this work is to search for signatures of variable reflection (in the continuum and reflection lines) and/or variable absorption features (e.g. changes in the properties of O VII, O VIII edges which are the hallmark of the presence of warm absorber in active galactic nuclei)."726 Middletonetal.(2009) analvzed the enerev dependence of the observed variability in (he EPIC data and showed that the variability primarily originated in the hieh-enerey photons., \citet{middletonetal2009} analyzed the energy dependence of the observed variability in the EPIC data and showed that the variability primarily originated in the high-energy photons.727 They also presented a spectral decomposition of the EPIC MOS spectrum. averaged over the entire observation.," They also presented a spectral decomposition of the EPIC MOS spectrum, averaged over the entire observation."728 Here we analvze the EPIC PN data taken during this observation to test if Chere is any. signature of spectral variations between (he different phases., Here we analyze the EPIC PN data taken during this observation to test if there is any signature of spectral variations between the different phases.729" Due to limitations in photon statistics (largely caused by the high pileup on the CCD as described in relpreparegata: i alsoseeMiddletonetal.2009) wecouldertractspeclrawilhgoodsignal noiseraliofromlwophasesonly : lhevhigh""phasespectrumbyaccumulalingspectranearthecrests. andthe cL re"," Due to limitations in photon statistics (largely caused by the high pileup on the CCD as described in \\ref{prepare_data}; also see \citealt{middletonetal2009}) ) we could extract spectra with good signal-to-noise ratio from two phases only: the “high” phase spectrum by accumulating spectra near the crests, and the “low” phase spectrum by accumulating spectra near the troughs."730"fpreparegalawedescribeindelailthealgorithmweusedlocrealelhephase—resolved high""and low"" spectr", In \\ref{prepare_data} we describe in detail the algorithm we used to create the phase-resolved “high” and “low” spectra.731e refanalysis..andconelusionsaresummearizedint refconclusion..," The spectral analysis is described in \\ref{analysis}, and conclusions are summarized in \\ref{conclusion}."732 The QPO in wwas noticed in an oobservation starting on 2007 May 31 (observation ID., The QPO in was noticed in an observation starting on 2007 May 31 (observation ID.733 0506440101)., 0506440101).734 The EPIC-PN detector was operated in conjunction with the{ή filler in mode durime tliis observation., The EPIC-PN detector was operated in conjunction with the filter in mode during this observation.735 We extracted the lieht curves ancl spectral information of this data set from the ddetector using SScience Analvsis Software (SAS: v.9.0.0) and following the, We extracted the light curves and spectral information of this data set from the detector using Science Analysis Software (SAS; v.9.0.0) and following the736is the solution of the following ODE: We will see below that this solution is unique.,is the solution of the following ODE: We will see below that this solution is unique.737" For the sake of simplicity of notations. we will omit the depeudeuce of X"" on (f..c) aud we will simply write From (6.3)) and (1.9)). we can easilv check that NYT)=ayιτ. hence using (6.1)). we ect where XY satisfies: Iu oxder to show that XT(0) is uniquely defined. we solve (6.1)) backwards. iu other words. we let Iu this case: where XT satisfies: ∙∙↽ ⊡⋅∪⋯↕⋟↥⋅∪↻∪↴∖↴↕⊓∪∐↓∙↓∙∙↑∐↸∖↴∖↴∪↕∏⊓∪∐⊸∖⊥⊂∊⊽⊥∏∩∙∖⋝∶⇀↴⇀ ⋅≓↽⋃↽) is⋅ uuique⋅ aud hence X-10)=XQ(f) Lv.is uniquely. determined."," For the sake of simplicity of notations, we will omit the dependence of $X^{0}$ on $(t,x)$ and we will simply write From \ref{eq2_sec6}) ) and \ref{att_sign}) ), we can easily check that $X^{0}_{2}(\t)=x_{2}-t+\tau$, hence using \ref{eq1_sec6}) ), we get where $X_{1}^{0}$ satisfies: In order to show that $X_{1}^{0}(0)$ is uniquely defined, we solve \ref{eq3_sec6}) ) backwards, in other words, we let In this case: where $\o{X}_{1}^{0}$ satisfies: From Proposition \ref{ganaza}, the solution $\o{X}_{1}^{0}\in738C^{1}([0,\infty);\R)$ is unique and hence $X_{1}^{0}(0)=\o{X}_{1}^{0}(t)$ is uniquely determined."739". ≼⊲∪∐↴∖↴↸∖≺∣⋯∖∐↑↕⋅↖⇁↑∐↸∖↕≯∏∐↸⊳↑↕∪↕∩⊽∪↕↴∖↴↖↖⇁↸∖∐≼∐∖∐∐↸∖≼↧∙ Frou Step 1. we know that withoy V, os,given by (6.6)) also depends on cj aud ey."," Consequently the function $V^{0}$ is well From Step 1, we know that with $\o{X}_{1}^{0}$ given by \ref{eq4_sec6}) ) also depends on $x_{1}$ and $x_{2}$."740 Let the fuuction. Y2-0:Z.>EX be defined as follows. (with. simplified notation showing the dependence on the variables (ανο} ): Iu order to show that V? is Lipschitz. it suffices (see (6.7))) to show that Y. is Lipschitz.," Let the function $Y:\R^{3}\rightarrow \R$ be defined as follows (with simplified notation showing the dependence on the variables $(t,x_{1},x_{2})$ ): In order to show that $V^{0}$ is Lipschitz, it suffices (see \ref{Ya1}) )) to show that $Y$ is Lipschitz."741 First. it is easily secu from (6.6)) that Y is Lipschitz iu time f£.," First, it is easily seen from \ref{eq4_sec6}) ) that $Y$ is Lipschitz in time $t$."742 The Lipschitz coutimuty with respect to the variable wy directly follows from the monotonicity of f (see (1.1))). and the comparison principle.," The Lipschitz continuity with respect to the variable $x_{1}$ directly follows from the monotonicity of $\o{f}$ (see \ref{Ymala3}) )), and the comparison principle."743 In order to show the Lipschitz continuity with respect to ay. we first eive a formal proof by assuming that f is smooth. aud then we present the main idea that permut to make the proof rigorous.," In order to show the Lipschitz continuity with respect to $x_{2}$, we first give a formal proof by assuming that $\o{f}$ is smooth, and then we present the main idea that permit to make the proof rigorous."744 Suppose that Take then the above two fuuctious satisfv:, Suppose that Take then the above two functions satisfy:745ideal for selecting quasar candidates for spectroscopic follow-up. independent of quasar colour.,"ideal for selecting quasar candidates for spectroscopic follow-up, independent of quasar colour."746 The kev to this result is that the lines of constant 6;A in Figure 14 cut through the stellar locus almost perpendicularly: therefore. quasar bydy colour does not alfect the likelihood of quasar selection using our KA method variant.," The key to this result is that the lines of constant $b_J - K$ in Figure \ref{UonlyKX} cut through the stellar locus almost perpendicularly; therefore, quasar $b_J - K$ colour does not affect the likelihood of quasar selection using our KX method variant."747" In this paper. we constructed. a sample of GO quasars with measured 6,A colours. using a combination of APAICAT. IRIS2 imagingoe and FCSS spectroscopic identifications."," In this paper, we constructed a sample of 69 quasars with measured $b_J - K$ colours, using a combination of APMCAT, IRIS2 imaging and FCSS spectroscopic identifications."748 11 of these quasars are red. satisfying οAz 3.5.2 ancl all of these τοῦ quasars are raclio-quict according to the All Sky Optical Catalogue of RaciofX-Ray Sources.," 11 of these quasars are red, satisfying $ b_J - K \geq$ 3.5, and all of these red quasars are radio-quiet according to the All Sky Optical Catalogue of Radio/X-Ray Sources."749 In accordance with Barkhouse&Llall(2001) and Maddox&Llewett (2006).. as the 1 red quasars founcl here are unresolved. luminous anc racio-quict sources predominantly located at 1. this strongly indicates that the main cause of the red b;—IN colours is dust at the quasar redshift.," In accordance with \citet{2001AJ....121.2843B} and \citet{2006MNRAS.367..717M}, as the 11 red quasars found here are unresolved, luminous and radio-quiet sources predominantly located at $z > 1$ , this strongly indicates that the main cause of the red $b_J - K$ colours is dust at the quasar redshift."750 Comparing our quasar sample to LBQS quasars. we found. that the two b;—Ix colour distributions are cüllerent at the confidence level. with our sample having a significantlv broader. redder distribution.," Comparing our quasar sample to LBQS quasars, we found that the two $b_J - K$ colour distributions are different at the confidence level, with our sample having a significantly broader, redder distribution."751 Analysis of the uncertainties in the b;—dy colours demonstrated that neither our red quasar detections or our comparison to the LBQS quasars is allected by them., Analysis of the uncertainties in the $b_J - K$ colours demonstrated that neither our red quasar detections or our comparison to the LBQS quasars is affected by them.752" A second analysis. on the elfects of our datasets magnitude ancl completeness limits. revealed. that they limited our measured 6,A distribution tob;dy« "," A second analysis, on the effects of our datasets magnitude and completeness limits, revealed that they limited our measured $b_J - K$ distribution to $ b_J - K <$ 5."753"As red. quasars are observed: up to this colour limit. we concluded that the trucb,A distribution of quasars is even broader and. redder than observed here."," As red quasars are observed up to this colour limit, we concluded that the true $b_J - K$ distribution of quasars is even broader and redder than observed here."754 From the observed by—dx colour distribution. we robustly constrained the red quasar fraction of the Ax:18.4 quasar population to be ereater than2274... and. from. a model estimated it to be A1'A.," From the observed $b_J - K$ colour distribution, we robustly constrained the red quasar fraction of the $K \leq 18.4$ quasar population to be greater than, and from a model estimated it to be ."755. Using the quasar saniple constructed here. the viability ofa KX method variant was tested.," Using the quasar sample constructed here, the viability of a KX method variant was tested."756 Using ab; Rvs RoW plot. the KN method variant was capable of separating the quasar sample out from the other objects in the APAICAT catalogue.," Using a $b_J - R$ vs $R - K$ plot, the KX method variant was capable of separating the quasar sample out from the other objects in the APMCAT catalogue."757 Comparing the KN method variant to the UVX and 2PQZ multicolour methods. the IXN method variant was found to be as elective in the number of quasars it selected: however. it was superior at. selecting quasars indepencdoent of colour.," Comparing the KX method variant to the UVX and 2QZ multicolour methods, the KX method variant was found to be as effective in the number of quasars it selected; however, it was superior at selecting quasars independent of colour."758 For those reasons. the WN method. variant. used here is an ideal technique for future large surveys to use in selecting potential quasars. whether red. or blue. for spectroscopic follow-up.," For those reasons, the KX method variant used here is an ideal technique for future large surveys to use in selecting potential quasars, whether red or blue, for spectroscopic follow-up."759 We thank the AAO for use of the LRIS2instrument. and Stuart Racer of the AAO for hishelp in reducing the LUS2 imaging., We thank the AAO for use of the IRIS2instrument and Stuart Ryder of the AAO for hishelp in reducing the IRIS2 imaging.760probability of finding a bubble at large. enough distance from the cluster centre (and would also possibly modify the heating rates). thus more sophisticated studies ancl more stringent observational constrains are needed to pin-down the relevance of CR. diffusion out of the bubbles.,"probability of finding a bubble at large enough distance from the cluster centre (and would also possibly modify the heating rates), thus more sophisticated studies and more stringent observational constrains are needed to pin-down the relevance of CR diffusion out of the bubbles."761 The second. factor which may bias the sizes of our simulated CR. bubbles high is the fact that obscrvationally only relativistic electrons have been detected so far in racio obes. while we have conjectured in our simulations that here should. be a spatially extended: distribution of CT orotons as well.," The second factor which may bias the sizes of our simulated CR bubbles high is the fact that observationally only relativistic electrons have been detected so far in radio lobes, while we have conjectured in our simulations that there should be a spatially extended distribution of CR protons as well."762 Many of our CR. bubbles. especially the ones hat are further away from the central AGN. are expected to rave a rather aged population of CR electrons. which would not have detectable radio emission.," Many of our CR bubbles, especially the ones that are further away from the central AGN, are expected to have a rather aged population of CR electrons, which would not have detectable radio emission."763 In. our picture these oibbles. correspond to the so-called. ghost cavities which ave identified as clepressions in N.rav emission without any notable racio emission. or only radio emission at lower requencies.," In our picture these bubbles correspond to the so-called ghost cavities which are identified as depressions in X–ray emission without any notable radio emission, or only radio emission at lower frequencies."764 Because of the very low contrast in. projected surface brightness maps. many of these ghost cavities are missed both in observations and simulations.," Because of the very low contrast in projected surface brightness maps, many of these ghost cavities are missed both in observations and simulations."765 Le will be an important. task for future work to better quantify. this bias. and to find powerful observational constraints for the relative content of relativistic electrons and protons in ACN-inllated bubbles.," It will be an important task for future work to better quantify this bias, and to find powerful observational constraints for the relative content of relativistic electrons and protons in AGN-inflated bubbles."766 Our simulations certainly suggest that CRs may play a crucial role in shaping the central ICM properties and in regulating AGN activity in clusters of galaxies., Our simulations certainly suggest that CRs may play a crucial role in shaping the central ICM properties and in regulating AGN activity in clusters of galaxies.767for the irradiated Class IV. albedo function.,for the irradiated Class IV albedo function.768 For an orbital inclination of/=60°. the 99.9 confidence upper limit to the planet-to-star flux ratio for the grey albedo model is already e-254.x107?. while it is €=5.7x107? when adopting the irradiated Class IV albedo function.," For an orbital inclination of $i=60^{\circ}$, the 99.9 confidence upper limit to the planet-to-star flux ratio for the grey albedo model is already $\epsilon=5.1\times10^{-5}$, while it is $\epsilon=5.7\times10^{-5}$ when adopting the irradiated Class IV albedo function."769 Similar to our results for the hot Jupiter HD 75289Ab (Rodler et al., Similar to our results for the hot Jupiter HD 75289Ab (Rodler et al.770 2008). Figure 4 shows that the upper limits established by adopting the grey albedo model are deeper than the ones found with the irradiated Class IV model.," 2008), Figure \ref{f6:upp} shows that the upper limits established by adopting the grey albedo model are deeper than the ones found with the irradiated Class IV model."771 Assuming a planetary radius Ry=1.2Ay and an orbital inclination /=46° (cf., Assuming a planetary radius $R_{\rm p}=1.2~R_{\rm Jup}$ and an orbital inclination $i\approx46^{\circ}$ (cf.772 Section 3)). we find the upper limit to the geometric albedo to be p.«0.40 for the grey albedo model and p<0.44 for the irradiated Class IV nodel at a wavelength of ty=462nm. which corresponds to the centre of gravity of the irradiated Class IV. albedo function in the observed wavelength range.," Section \ref{S:6:2}) ), we find the upper limit to the geometric albedo to be $p<0.40$ for the grey albedo model and $p<0.44$ for the irradiated Class IV model at a wavelength of $\lambda_0=462~{\rm nm}$, which corresponds to the centre of gravity of the irradiated Class IV albedo function in the observed wavelength range."773 For comparison. at this wavelength Jupiter's geometric albedo is p=0.44 (Karkoschka 1994).," For comparison, at this wavelength Jupiter's geometric albedo is $p=0.44$ (Karkoschka 1994)."774 We have observed the hot Jupiter orbiting 7 Boo for two half nights with UVES. mounted at the VLT/UT2. in an attempt to measure starlight reflected from the planetary companion.," We have observed the hot Jupiter orbiting $\tau$ Boo for two half nights with UVES, mounted at the VLT/UT2, in an attempt to measure starlight reflected from the planetary companion."775ines of constant height above/below the plaue (Z linear coordinate).,lines of constant height above/below the plane $Z$ linear coordinate).776 The color map has been normalized so that he peak density alone cach line of sight ects the naxinunun vellow intensity., The color map has been normalized so that the peak density along each line of sight gets the maximum yellow intensity.777 Again. this allows us o trace the bulee shape even far away from the aue. but of course the density of bulec stars at b|= aud |b]=8° is not the same.," Again, this allows us to trace the bulge shape even far away from the plane, but of course the density of bulge stars at $|b|=4^{\circ}$ and $|b|=8^{\circ}$ is not the same."778 Looking at HxY. the bulge splits in two components both above and below the plauc.," Looking at $|l|\leq1^{\circ}$, the bulge splits in two components both above and below the plane."779 However. when moving ο sections at positive longitudes (top panels) oulv he one closer to the Sun is visible. while at reeative longitudes (bottom panels) ouly the far one remains.," However, when moving to sections at positive longitudes (top panels) only the one closer to the Sun is visible, while at negative longitudes (bottom panels) only the far one remains."780 Although we do not have data in the region close to the plane. it is reasonable to assume that the clongated structures secu at |b|c1 would moree in the ceuter. completing an XN-shaped structure.," Although we do not have data in the region close to the plane, it is reasonable to assume that the elongated structures seen at $|b|\geq7814^\circ$ would merge in the center, completing an X-shaped structure."782 If we make the exercise of joiuiug the of the N above and below the plane. we fud that the whole structure is approximatcly centered at a distance of 7.1c0.6 kpc from the Sun. which is consistent with the most recent determination of the distance to the Calactic ceuter.," If we make the exercise of joining the of the X above and below the plane, we find that the whole structure is approximately centered at a distance of $7.4\pm0.6$ kpc from the Sun, which is consistent with the most recent determination of the distance to the Galactic center."783 The latter was found to be 7.52£0.10 kpc by Nishivamia ct al. (, The latter was found to be $7.52\pm0.10$ kpc by Nishiyama et al. (7842005). 8.040.6 kpe by Chez et al. (,"2005), $8.0\pm0.6$ kpc by Ghez et al. ("7852008). and 8.3340.35 kpe by Callessen et al. (,"2008), and $8.33\pm0.35$ kpc by Gillessen et al. ("7862009).,2009).787 The accuracy of the distauce from the Sun to the center of the N-shaped structure found here depends ou how appropriate is the adopted value for the absolute maguitucde of the bulge RC., The accuracy of the distance from the Sun to the center of the X-shaped structure found here depends on how appropriate is the adopted value for the absolute magnitude of the bulge RC.788 Cneertaintics on this parameter may come from the broad bulge metallicity distribution function. and the unknown precise age and helimm couteut of the bulge.," Uncertainties on this parameter may come from the broad bulge metallicity distribution function, and the unknown precise age and helium content of the bulge."789 However. this is bevoud the purpose of the present paper. which addresses the of the outer ([b|> 17) Galactic bulge.," However, this is beyond the purpose of the present paper, which addresses the of the outer $|b|>4^\circ$ ) Galactic bulge."790 Tn closing this section let us comment on the effect of incompleteness., In closing this section let us comment on the effect of incompleteness.791 Tncompletcucss affects the present analysis in two wavs., Incompleteness affects the present analysis in two ways.792 First. it artificially reduces the star density. especially in the faint RC (Fig. 2)).," First, it artificially reduces the star density, especially in the faint RC (Fig. \ref{cmds}) )."793 For a qualitative analysis of the bulge Sipe such as the preseut ouc. this might not be a serious bias.," For a qualitative analysis of the bulge shape such as the present one, this might not be a serious bias."794 Second. severe imconipleteuess produces a cutoff iu the middle of the RC. so that the peak in the RC Iuninosity fiction looks brighter than it really is.," Second, severe incompleteness produces a cutoff in the middle of the RC, so that the peak in the RC luminosity function looks brighter than it really is."795 Iudeed. when fields at 0<35° were tentatively iucluded d the analysis. the map would show the bulk of bulge stars ectting closer and closer to the Sun. for ower and lower |b].," Indeed, when fields at $|b|<3.5^{\circ}$ were tentatively included in the analysis, the map would show the bulk of bulge stars getting closer and closer to the Sun, for lower and lower $|b|$."796 This effect was uot truste as real. aud this is where our decision to exclude fields at |b]«3.5° comes from.," This effect was not trusted as real, and this is where our decision to exclude fields at $|b|<3.5^{\circ}$ comes from."797 Even for high [b]. rowever. the extinction is lighly variable on a fiek o field basis. and so it is the 2MÀSS degree of incompleteness.," Even for high $|b|$, however, the extinction is highly variable on a field to field basis, and so it is the 2MASS degree of incompleteness."798 The wigelesof the arms of the X in Fie., The wigglesof the arms of the $X$ in Fig.799 L (sec. e.g... the map at positive latitudes. or/ =|5°) nieht iu fact be due to differeutia extinction aud thus incompleteness.," \ref{xshape} (see, e.g., the map at positive latitudes, for $l=+5^{\circ}$ ) might in fact be due to differential extinction and thus incompleteness."800 ere we investigate the deusitv of RC stars iu the faint and bright RCs., Here we investigate the density of RC stars in the faint and bright RCs.801 The RC deusitics not ouly reveal the structure aud sviuuetrv of the X-shaped bulec. but also provide a consistency check to address any lingeriug doubts that the brielituess differeuce between the two RCs might be due to something other than distance.," The RC densities not only reveal the structure and symmetry of the X-shaped bulge, but also provide a consistency check to address any lingering doubts that the brightness difference between the two RCs might be due to something other than distance."802" We aeasured the density of RC stars as a ,uction of vertical height below the Galactic plane. at longitude 7~[5° for the bright red chunp. aud at /~3° for the faint red clump: at these latitudes the bright and faint RCs. respectively. are maxim (seeMcWilliam&Zoc-cali 2010)."," We measured the density of RC stars as a function of vertical height below the Galactic plane, at longitude $l\sim+5^{\circ}$ for the bright red clump, and at $l\sim-3^{\circ}$ for the faint red clump; at these latitudes the bright and faint RCs, respectively, are maximum \citep[see][]{2010ApJ...724.1491M}."803. The RC densities were measured from ) arcuinute radius circular 2\LASS fields. iu A. frequency histoeraims simular to MeWillima&Zoccali(2010) aud included a linear iuterpolation o remove the backerouud counts," The RC densities were measured from 30 arc-minute radius circular 2MASS fields, in $K_{\rm s}$ frequency histograms similar to \cite{2010ApJ...724.1491M} and included a linear interpolation to remove the background counts."804 Dackerouud subtraction would be significantly improved with he use of isochrouc fitting techniques. but this is bevoud the scope of the curent paper.," Background subtraction would be significantly improved with the use of isochrone fitting techniques, but this is beyond the scope of the current paper."805 To compute the space densities of the RC stars Toni the nunuber counts it was necessary to asstuue distances for the bright aud fait RC opulatious: we adopted 6.1 and 9.0 pe distances or these populations. respectively (see MeWilliamn Zoccali 2010).," To compute the space densities of the RC stars from the number counts it was necessary to assume distances for the bright and faint RC populations; we adopted 6.4 and 9.0 kpc distances for these populations, respectively (see McWilliam Zoccali 2010)."806" The vertical extent of our density ficlds ranges ron bo=6.5"" to 12.6"" in latitude. and eenerallv corresponds to vertical distances. Z. roni 1.0 to Leb kpc."," The vertical extent of our density fields ranges from $b=-6.5^{\circ}$ to $-12.6^{\circ}$ in latitude, and generally corresponds to vertical distances, $Z$, from 1.0 to 1.4 kpc."807 All projection effect have oen takine iuto account when computing the stellar surface densities for the bright aud faint RC populations., All projection effect have been taking into account when computing the stellar surface densities for the bright and faint RC populations.808 Qur results for the two regious are plotted iu Fig. 5.. ," Our results for the two regions are plotted in Fig. \ref{density}, ,"809which shows that the faint aud bright RCs possess sinilar declining trends iu space deusity, which shows that the faint and bright RCs possess similar declining trends in space density810ate times.,late times.811 Close to the peak of the luminosity. the lightcurve is very sensitive to the structure of the star. being shallower for stars with polytropic index close to 4/3. expected for solar type stars.," Close to the peak of the luminosity, the lightcurve is very sensitive to the structure of the star, being shallower for stars with polytropic index close to 4/3, expected for solar type stars."812 In this case. the /δολ oofile is reached only after the luminosity jas dropped by at least two magnitudes.," In this case, the $t^{-5/3}$ profile is reached only after the luminosity has dropped by at least two magnitudes."813 For stars with a relatively Hat density profile. such as red giants and low mass stars. the /72 orofiles is reached earlier.," For stars with a relatively flat density profile, such as red giants and low mass stars, the $t^{-5/3}$ profiles is reached earlier."814 In this paper we have only investigated a very simple setup. with a given mass ratio between the star and the supermassive black hole. and one given set of orbital parameters.," In this paper we have only investigated a very simple setup, with a given mass ratio between the star and the supermassive black hole, and one given set of orbital parameters."815 It is expected hat the results would be further dependent on the such additional »arameters. such as the ratio of tidal radius to pericentre distance and the eccentricity of the orbit.," It is expected that the results would be further dependent on the such additional parameters, such as the ratio of tidal radius to pericentre distance and the eccentricity of the orbit."816 We plan to consider these ettects in subsequent investigations., We plan to consider these effects in subsequent investigations.817 Finally. it should be further emphasised that all these results refer essentially to the return time of the disrupted debris. and only correspond to an actual luminosity under the further assumption that the subsequent accretion is perfectly ethcient and occurs on a much shorter timescale. which may not be the case (see 2).," Finally, it should be further emphasised that all these results refer essentially to the return time of the disrupted debris, and only correspond to an actual luminosity under the further assumption that the subsequent accretion is perfectly efficient and occurs on a much shorter timescale, which may not be the case (see \citealt{ayal2000}) )."818 We thank Walter Dehnen for providing us with his setup routine for polytropic spheres., We thank Walter Dehnen for providing us with his setup routine for polytropic spheres.819 We acknowledge several interesting discussions with Walter Dehnen. Mark Wilkinson. Sergei Nayakshin and Paul O'Brien.," We acknowledge several interesting discussions with Walter Dehnen, Mark Wilkinson, Sergei Nayakshin and Paul O'Brien."820 We also thank the Referee. Stephan Rosswog. for an insightful report.," We also thank the Referee, Stephan Rosswog, for an insightful report."821 All the visualization of SPH simulations have been obtained using the SPLASH visualization tool by Dan Price (2) ," All the visualization of SPH simulations have been obtained using the SPLASH visualization tool by Dan Price \citep{splash}822 "823"Doppler imaging has shown that rapidly rotating RS CVn binary systems. such as SV Cam ΕΚΚΑVV. P,,,20.59dd) frequently show spots at high and polar latitudes.","Doppler imaging has shown that rapidly rotating RS CVn binary systems, such as SV Cam V, $_{rot}$ d) frequently show spots at high and polar latitudes."824 A successful theoretical model of the formation of polar spots assumes that magnetic flux from decaying active regions is swept toward the poles by meridional flows (?).., A successful theoretical model of the formation of polar spots assumes that magnetic flux from decaying active regions is swept toward the poles by meridional flows \citep{schrijver01polar}.825 However. in order to produce polar spots bipolar acive regions have to emerge at a rate approximately 30 times faster than in the case of the Sun. implying that the photospheres of active stars should be peppered with a large number of small spots.," However, in order to produce polar spots bipolar active regions have to emerge at a rate approximately 30 times faster than in the case of the Sun, implying that the photospheres of active stars should be peppered with a large number of small spots."826 Jeffers et al.(2005). used spectrophotometric data from. the on board the Hubble Space Telescope to eclipse map the inner face of the primary of the RS CVn SV Cam., Jeffers et al.(2005) used spectrophotometric data from the on board the Hubble Space Telescope to eclipse map the inner face of the primary of the RS CVn SV Cam.827 These observations and the parallax showed that the surface flux in the eclipsed low latitude region is approximately lower than computed from the bestfitting ? model atmosphere.," These observations and the parallax showed that the surface flux in the eclipsed low latitude region is approximately lower than computed from the bestfitting \citet{allard00}828 model atmosphere."829 This flux deficit can only be accounted for if approximately of the primary’s surface is peppered with dark spots too small to be resolved through eclipse-mapping techniques where the resolution limit is approximately 1.5 degrees., This flux deficit can only be accounted for if approximately of the primary's surface is peppered with dark spots too small to be resolved through eclipse-mapping techniques where the resolution limit is approximately 1.5 degrees.830 This paper extends the work of Jeffers et al. (, This paper extends the work of Jeffers et al. (8312005) by applying the extrapolated solar spot size distribution of ?. to a hypothetically immaculate primary star of SV Cam.,2005) by applying the extrapolated solar spot size distribution of \citet{solanki99} to a hypothetically immaculate primary star of SV Cam.832 We determine how different degrees of spot coverage influence the shape of the binary system's lighteurve. and how accurately these lightcurves are reconstructed into surface brightness distributions using the Maximum Entropy(Max Ent) eclipse mapping technique (2:: 29).," We determine how different degrees of spot coverage influence the shape of the binary system's lightcurve, and how accurately these lightcurves are reconstructed into surface brightness distributions using the Maximum Entropy(Max Ent) eclipse mapping technique \citealt{cameron97dots}; \citealt{cameron97xyuma}) )."833 The variable nature of the spot coverage of active stars makes the quantification of their spot size distribution an intriguing problem., The variable nature of the spot coverage of active stars makes the quantification of their spot size distribution an intriguing problem.834 In the case of the Sun. the spot size distribution has been determined by ?. from direct observations taken from the Mount Wilson white-light plate collection covering the period 1917-1982.," In the case of the Sun, the spot size distribution has been determined by \citet{bogdan88} from direct observations taken from the Mount Wilson white-light plate collection covering the period 1917-1982."835 When plotted on a log-log scale the size distribution of sunspots is. parabolic. implying a two-parameter log-normal distribution. which canno be represented by a single parameter distribution such as a power law.," When plotted on a log-log scale the size distribution of sunspots is parabolic, implying a two-parameter log-normal distribution, which cannot be represented by a single parameter distribution such as a power law."836 Following ὁ the number of sunspots. N. as a function of the solar surface area. A. is equated as. where the constants and σι are the mean and geometric standard deviation of the log-normal distribution. and (4X) the maximum value reached by the distribution.," Following \citet{bogdan88} the number of sunspots, N, as a function of the solar surface area, A, is equated as, where the constants $\langle$ $\rangle$ and $\sigma_A$ are the mean and geometric standard deviation of the log-normal distribution, and $\left(\frac{dN}{dA}\right)_{max}$ the maximum value reached by the distribution."837 For the case of the Sun. these values are tabulated in Table | where set | is for an inactive Sun and set 2 is for an active Sun.," For the case of the Sun, these values are tabulated in Table \ref{input} where set 1 is for an inactive Sun and set 2 is for an active Sun."838 The extrapolation ofthe solar spot size distribution to active stars can be represented by different extrapolations e.g. ? and ?..," The extrapolation ofthe solar spot size distribution to active stars can be represented by different extrapolations e.g. \citet{solanki99}839 and \citet{solanki04}."840 As long as the size distribution shows many sub-resolution spot that are not clumped together. the exact representation of the extrapolated size distribution is of minor importance.," As long as the size distribution shows many sub-resolution spot that are not clumped together, the exact representation of the extrapolated size distribution is of minor importance."841 The extrapolation of ? is used. which is determined by analysing how the observable parameters of the log normal distribution. (Ν/Α a. GAD. and σι. change with increasing magnetic," The extrapolation of \citet{solanki99} is used, which is determined by analysing how the observable parameters of the log normal distribution, $_{max}$ , $\langle$ $\rangle$ , and $\sigma_A$ , change with increasing magnetic"842ον PandeyanclαναShankar(2005).,by \cite{ursiga:pandey05}.843.. A suite of programs developed: in-house was used. to. reduce ~5000 hours of he survey data (a quarter of the total 20.000) hours observed over a span of ~5 vears).," A suite of programs developed in-house was used to reduce $\sim 5000$ hours of the survey data (a quarter of the total $\sim 20,000$ hours observed over a span of $\sim 5$ years)."844 The deconvolved images and a source catalogue of ~2.800 sources were published wv Pandey(2006).," The deconvolved images and a source catalogue of $\sim 2,800$ sources were published by \cite{thesis:pandey06}."845 Systematics in positional errors were found. when the »ositions of sources common to MICE. catalogue. anc the Alolonelo Relerenee Catalogue (AIRC) (Largeetal.1981) were compared., Systematics in positional errors were found when the positions of sources common to MRT catalogue and the Molonglo Reference Catalogue (MRC) \citep{mnras:large81} were compared.846 Pandey(2006) treated: the svstematies in errors in o and sinze independently., \cite{thesis:pandey06} treated the systematics in errors in $\alpha$ and $\sin za$ independently.847 Dy estimating two separate 1-D least-squares fits for errors in à and sina the systematics were corrected only in the source catalogue., By estimating two separate 1-D least-squares fits for errors in $\alpha$ and $\sin za$ the systematics were corrected only in the source catalogue.848 However. errors remained in the images which impede usefulness of MICE. images for. multi-wavelength analvsis of sources.," However, errors remained in the images which impede usefulness of MRT images for multi-wavelength analysis of sources."849 In addition. the source. of. errors was not investigated.," In addition, the source of errors was not investigated."850 At MICE. the. visibility data ds processed through several complex stages of data reduction specific to the array. especially. arising due to its non-coplanarity (UdavaShankaretal.2002).," At MRT, the visibility data is processed through several complex stages of data reduction specific to the array, especially, arising due to its non-coplanarity \citep{apss:uday02}."851.. It was therefore decided to correct for errors in the image domain and avoid re-processing the visibility data., It was therefore decided to correct for errors in the image domain and avoid re-processing the visibility data.852 This paper describes the application of |.2-D lomography. a technique ubiquitous in the computer vision and graphics community. to correct the errors. in he image domain.," This paper describes the application of 2-D homography, a technique ubiquitous in the computer vision and graphics community, to correct the errors in the image domain."853 Hlomography is used to estimate a transformation matrix (which includes rotation. translation and. non-isotropic scaling) that accounts for positional errors in the linearly evideed 2-D images.," Homography is used to estimate a transformation matrix (which includes rotation, translation and non-isotropic scaling) that accounts for positional errors in the linearly gridded 2-D images."854 In our view. this echnique will be of relevance to the new generation racio elescopes where. owing to huge data rates. only images after a certain integration would be recorded. as opposed to raw visibilities (Lonsdaleetal.2009).," In our view, this technique will be of relevance to the new generation radio telescopes where, owing to huge data rates, only images after a certain integration would be recorded as opposed to raw visibilities \citep{ieee:lonsdale09}."855. Phis paper also describes our investigations tracing the positional errors to errors in the array geometry. used. for imaging., This paper also describes our investigations tracing the positional errors to errors in the array geometry used for imaging.856 Our hypothesis on the array geometry. its subsequent confirmation endorsed by re-estimation of the array geometry. and its ellect on the images are also described.," Our hypothesis on the array geometry, its subsequent confirmation endorsed by re-estimation of the array geometry and its effect on the images are also described."857 The rest of the paper is organised as follows., The rest of the paper is organised as follows.858 Section 2 compares positions of sources common to MICI catalogue and AIRC., Section \ref{s:poserror} compares positions of sources common to MRT catalogue and MRC.859 The 2-D homography estimation is brielly described in Section 3.., The 2-D homography estimation is briefly described in Section \ref{s:homography}. .860 Section 4 presents the correction scheme and typical results., Section \ref{s:scheme} presents the correction scheme and typical results.861 The re-estimation of MICE array geometry is described in Section 5.., The re-estimation of MRT array geometry is described in Section \ref{s:arraygeometry}.862 Finally. we summarise ane present our conclusions in Section 6..," Finally, we summarise and present our conclusions in Section \ref{s:conclusions}."863 The positions of sources common to MICE. catalogue and AIRC were compared., The positions of sources common to MRT catalogue and MRC were compared.864 We used \IRC because of its overlap with MICE survey. its proximity in frequeney compared to other reliable catalogues available ancl. comparable resolution (27.62.2°86see(8|35°.5)).," We used MRC because of its overlap with MRT survey, its proximity in frequency compared to other reliable catalogues available and, comparable resolution $(2\arcmin.62\times2\arcmin.86\sec(\delta+35^\circ.5))$."865 Aloreover. lor sources of listed [ux density :100 Jy (at 408 AllIz) the catalogue is reported to be substantially complete and. the reliability is reported to be (Largeetal.1981).," Moreover, for sources of listed flux density $\geq 1.00$ Jy (at 408 MHz) the catalogue is reported to be substantially complete and, the reliability is reported to be \citep{mnras:large81}."866. For our further discussions. errors in. MIC source positions are considered random. without any systeniatics.," For our further discussions, errors in MRC source positions are considered random, without any systematics."867 About 400 bright sources common to the two catalogues and with Εαν density at 151.5. MlIz greater than 5 Jw (> 15-0) were identified and their positions were compared., About 400 bright sources common to the two catalogues and with flux density at 151.5 MHz greater than 5 Jy $> 15$ $\sigma$ ) were identified and their positions were compared.868 'l'he sources were labellec as common if they lie within 4 of cach other., The sources were labelled as common if they lie within $4\arcmin$ of each other.869 Since MIRC has a source densitv of ~ source 27. the chances of considering two unrelated SOULCES às Conimon are extremely low.," Since MRC has a source density of $\sim0.5$ source $^{-2}$, the chances of considering two unrelated sources as common are extremely low."870 A Dux threshold of 70 ensures a source population abundant to reliably estimate homography (explained in next section)., A flux threshold of $\sigma$ ensures a source population abundant to reliably estimate homography (explained in next section).871 The positional errors in à and ὁ show no systematics as à [function of à (referof Fie., The positional errors in $\alpha$ and $\delta$ show no systematics as a function of $\alpha$ (referof Fig.872 laa and Ibb)., \ref{f:sourcecomparison}a a and \ref{f:sourcecomparison}b b).873 For visualisation. the errors are shown in percentages of MICE beamwidths.," For visualisation, the errors are shown in percentages of MRT beamwidths."874The errors in à and 9 show a linear gradient as a function of sinze.,The errors in $\alpha$ and $\delta$ show a linear gradient as a function of $\sin za$.875 The errors in a. plotted against sinze. reach ~410% ofthe AIRT beaniwidth (refer of lig.," The errors in $\alpha$, plotted against $\sin za$, reach $\sim \pm10\%$ of the MRT beamwidth (refer of Fig."876 laa)., \ref{f:sourcecomparison}a a).877 Whereas. the errors in 0. plotted against sinτα. are significant and reach ~250% of NICE beamwicdth. (," Whereas, the errors in $\delta $ , plotted against $\sin za$, are significant and reach $\sim \pm50\%$ of MRT beamwidth. ("878refer of Fie.,refer of Fig.879 HIbb)., \ref{f:sourcecomparison}b b).880 Histograms in Fig., Histograms in Fig.881 lee and Fig., \ref{f:sourcecomparison}c c and Fig.882 Lele show the distribution of errors in a. and 9. respectively.," \ref{f:sourcecomparison}d d show the distribution of errors in $\alpha$ and $\delta$, respectively."883 The histogram of errors in do shows a broader spread compare lo errors in à., The histogram of errors in $\delta$ shows a broader spread compared to errors in $\alpha$.884 lte-imaging. to correct [or errors in the images. woul involve re-reducing the ~5.000 hours of observed. data.," Re-imaging, to correct for errors in the images, would involve re-reducing the $\sim 5,000$ hours of observed data."885 Owing to the complexity involved it was decided to correc [or the positional errors in the images. thus avoiding re-processing.," Owing to the complexity involved it was decided to correct for the positional errors in the images, thus avoiding re-processing."886 Phe 2-D homography estimation technique was emplovecl for correcting positional errors in images and is discussed in detail in the following section., The 2-D homography estimation technique was employed for correcting positional errors in images and is discussed in detail in the following section.887 The 2-D planar homography is a non-singular linear relationship between points on planes., The 2-D planar homography is a non-singular linear relationship between points on planes.888" Civen two sets of ÁN corresponding5 image5 points in projective coordinates. (p,andpj.)ore homography maps p, to the corresponding pj (lartleyand.Zisserman2000)."," Given two sets of $K$ corresponding image points in projective coordinates, $({\mathbf p}^{}_{k} \mbox{ and } {\mathbf p}'_{k}) \in \mathbb{P}^{2}$, homography maps ${\mathbf p}^{}_{k}$ to the corresponding ${\mathbf p}'_{k}$ \citep{book:hartley}."889. Where. k-l...A.," Where, $k=1,\ldots,K$."890 The homography sought here is a non-singular 3.3 matrix H such that: ∖∖⊽↓↕⋖⊾↓⋅∢⋅⋡↿∖⇁≀⋅∕↴⋡⋮∣∣∕↔↴⋜⋯∠↿∖⇁≀⋅≦↴⊳∕∣∕↙↔↴↓⋅∢⊾↓≻↓⋅∢⋅⊳∖⋖⊾⊔↿↿∖⋂⊳⊳∖⊲↓⊔∶↙∩∪⇂⋅∫∖⋡ ≼↛∪↓⋅↓⋅∢⊾⊳∖↓≻∪↓∐⇂⊀↓⊔⋏∙≟↳∖∐⊰↾↓∖⋜⋯∠∟∖∐⊰≺⊲≱∖⋯⊔," The homography sought here is a non-singular $3\times 3$ matrix $\tt H$ such that: Where, $(x^{}_{k},y^{}_{k})$ and $(x'_{k},y'_{k})$ represent $(\alpha,\sin za)$ of $K$ corresponding MRT and MRC sources, respectively."891⋅⊓⋅≱∖⋡↓⋅⋖⋅≱∖↓≻∢⋅≼∙↥⋠↓∖⇁⋖⋅⇂∙∖⇁⋡ ↓⊔∣⊲⇀⊲⊏⇂⊔⋜⊔⊲↓∪⊔↓↦↿∖⇁≀⋅∕↴⊳⋮∣∣∕↴⊳↓⊐⋜⋃⊔⊔⊽≀⋅≦↴⊳∕∣∕↙⋰↓∃⋜⊔⋅⋖⋅↓⋅∢⋅⇂∎∢⋅↓⋅↓⋅∢⋅∠⊓∪ one dimension higher than the dimension of the problem space.," In Equation \ref{eq:inhomog}, $\left(x^{}_{k},y^{}_{k},1\right)$ and $\left(x'_{k},y'_{k},1\right)$ are referred to as the and are always represented one dimension higher than the dimension of the problem space."892 This is à commonly. used representation in computer graphics., This is a commonly used representation in computer graphics.893 The simple reason is that with a 2 «2matrix one can only rofafe set of 2-D points around the origin and «σας towards or away [rom the origin., The simple reason is that with a $2\times2$ matrix one can only a set of 2-D points around the origin and them towards or away from the origin.894 A 2. 2matrix is incapable of franslating set of 2-D points., A $2\times2$ matrix is incapable of a set of 2-D points.895 The homogeneous coordinates allow one to express a translation as a multiplication., The homogeneous coordinates allow one to express a translation as a multiplication.896 A single 3. 3matrix. with homogeneous coordinates. can account for rotation. scaling and translation of 2-D coordinates.," A single $3\times3$ matrix, with homogeneous coordinates, can account for rotation, scaling and translation of 2-D coordinates."897" For example. from Equation Ἐν. οτε =hyper Notice. fy, (representing translation in a-dimension) is simply being added to the normal clot product (ων |P459)that together represents rotation and scaling."," For example, from Equation \ref{eq:inhomog}, , $x'_{k} = h^{}_{11}x^{}_{k} + h^{}_{12}y^{}_{k} + h^{}_{13}$ Notice, $h^{}_{13}$ (representing translation in$\alpha$ -dimension) is simply being added to the normal dot product $(h^{}_{11}x^{}_{k} + h^{}_{12}y^{}_{k})$ that together represents rotation and scaling."898 In homogeneous coordinates. the 2-D problemspace is a plane hovering in the third. dimension at a unit distance.," In homogeneous coordinates, the 2-D problemspace is a plane hovering in the third dimension at a unit distance."899 A general homography matrix. for projective transformation.— has 8 cleerees-ol-freedom (DOE).," A general homography matrix, for projective transformation, has 8 degrees-of-freedom (DOF)."900— For our, For our901In the temperature-gravitv plane. the wCCen HB stars match (the expectations οἱ lle-enriched models rather than canonical ones.,"In the temperature-gravity plane, the $\omega$ Cen HB stars match the expectations of He-enriched models rather than canonical ones."902 However. the resulting underestimate of (heir mass prevents us from straight(forwardly concluding that this is evidence of helium enrichment.," However, the resulting underestimate of their mass prevents us from straightforwardly concluding that this is evidence of helium enrichment."903 In fact. (he progenies of He-rich stars in the HD phase should not be noticeably less massive (han stars of canonical composition (DAntona&Caloi2004).. and the difference al any temperature is expected (ο be (ny (0.03M...\loehleretal.2003).," In fact, the progenies of He-rich stars in the HB phase should not be noticeably less massive than stars of canonical composition \citep{DAntona04}, and the difference at any temperature is expected to be tiny \citep[$\leq$0.03~M$_\sun$,][]{Moehler03}."904. Moreover. the derived masses are on average well below the value required to ignite helium in the core (00.45 ALL).," Moreover, the derived masses are on average well below the value required to ignite helium in the core $\sim$ 0.45 $_\sun$ )."905 The easiest interpretation of our observations is (he presence of a svstenialtic error. biasing the results (toward lower eravilies ancl. as a consequence. lower masses.," The easiest interpretation of our observations is the presence of a systematic error, biasing the results toward lower gravities and, as a consequence, lower masses."906 However. in this work we used the same instrument. software. and models as 2009).. [indiug a clear dillerence between wCCen and the other clusters. while our measurements well agree with Moehleretal.(2011).. who also investigated uw CCen but with a different instrument. higher resolution. different. models for stars hotter than 200000 Ix. and only a subset of Balmer lines.," However, in this work we used the same instrument, software, and models as \citet{Moni07,Moni09}, finding a clear difference between $\omega$ Cen and the other clusters, while our measurements well agree with \citet{Moehler11}, who also investigated $\omega$ Cen but with a different instrument, higher resolution, different models for stars hotter than 000 K, and only a subset of Balmer lines."907 Therefore. even if we cannot completely exclude. an observational bias with respect to theoretical expectations. the difference between w CCen and (he three comparison clusters must be real:G," Therefore, even if we cannot completely exclude an observational bias with respect to theoretical expectations, the difference between $\omega$ Cen and the three comparison clusters must be real:."908"Cs, The same conclusion can be drawn even if the offset is a product of (he inaclequacy of the emploved models: in (his case. (hey would be reproducing sullicientlv well the atmospheric structure of the WB objects in the comparison clusters. but nol in iw CCen. hence a physical difference would be present."," The same conclusion can be drawn even if the offset is a product of the inadequacy of the employed models: in this case, they would be reproducing sufficiently well the atmospheric structure of the HB objects in the comparison clusters, but not in $\omega$ Cen, hence a physical difference would be present."909 The observed trends cannot be due to a wrong (hotter) temperature scale., The observed trends cannot be due to a wrong (hotter) temperature scale.910 In fact. for each star we (translated (he measured temperature in a reddening estimate. comparing the observed (5— V) color to the theoretical value obtained interpolating (he erid. lor the same metallicities as the model spectra used in (he fits.," In fact, for each star we translated the measured temperature in a reddening estimate, comparing the observed $B-V$ ) color to the theoretical value obtained interpolating the \citet{Kurucz93} grid, for the same metallicities as the model spectra used in the fits."911 The average value is E(B—V )-0.114. in perlect agreement with the literature. and will no significant trend along the HB.," The average value is $B-V$ )=0.114, in perfect agreement with the literature, and with no significant trend along the HB."912 A temperature scale hotter by (5%)) would have caused an overestimate ol reddening by 0.04 (0.02) mag at 100000 Ix. In Figure 3. we compare the IL; line profiles of (vo stars with similar temperature in 330 and oí CCen: the line core and depth are very similar. indicating no noliceably difference in temperature. but the star in 880. whose measured eravityv is 0.40 dex higher. shows wider wings.," A temperature scale hotter by ) would have caused an overestimate of reddening by 0.04 (0.02) mag at 000 K. In Figure \ref{f_spec} we compare the $_\beta$ line profiles of two stars with similar temperature in 80 and $\omega$ Cen: the line core and depth are very similar, indicating no noticeably difference in temperature, but the star in 80, whose measured gravity is 0.40 dex higher, shows wider wings."913 This comparison indicates that the peculiar stellar gravities rellect a real difference in the spectra of the target stars., This comparison indicates that the peculiar stellar gravities reflect a real difference in the spectra of the target stars.914 We are aware that some stellar parameters unaccounted for in our study. such as stellar wind and rotation. can cause wider line wings mimicking a difference in gravity. and this degeneracy cannot be avoided at our resolution.," We are aware that some stellar parameters unaccounted for in our study, such as stellar wind and rotation, can cause wider line wings mimicking a difference in gravity, and this degeneracy cannot be avoided at our resolution."915 The underestimate of eravilies can also be caused by, The underestimate of gravities can also be caused by916plotted as a fundamental plane relation in ,plotted as a fundamental plane relation in 2.917The crosses show the luminosity vs. the virial quantity a7μέ: where σι is the intensity weighted mean line-ol-sight velocity dispersion within an ellective radius., The crosses show the luminosity vs. the virial quantity ${\sigma_e}^2 \mathit{R_{eff}}/G$ where $\sigma_e$ is the intensity weighted mean line-of-sight velocity dispersion within an effective radius.918 Units are 101 L... and M..., Units are $10^{11}$ $_\odot$ and $_\odot$.919 The expectation from the virial theorem is where e; is the structure. constant which depencds upon the MOND parameter η. or. alternatively. upon the contribution of dark matter to the total mass within an cllective radius.," The expectation from the virial theorem is where ${c_1}$ is the structure constant which depends upon the MOND parameter $\eta$, or, alternatively, upon the contribution of dark matter to the total mass within an effective radius."920 The two parallel lines in 22 show this relation for two cifferent values of ej For the upper curve. labelled No the structure constant. e;=4.54. is that »propriate For the Newtonian Jalfe model (jj;= 10) or an equivalent dark matter fraction within an ellective radius of about 0.1.," The two parallel lines in 2 show this relation for two different values of $c_1$ For the upper curve, labelled $N$, the structure constant, $c_1=4.54$, is that appropriate for the Newtonian Jaffe model $\eta =10$ ) or an equivalent dark matter fraction within an effective radius of about 0.1."921 Η all systems were represented perfectly. by isotropic Newtonian Jalle models with AZ/L=1 then they would lie on this line., If all systems were represented perfectly by isotropic Newtonian Jaffe models with $M/L=1$ then they would lie on this line.922" For the lower line labelled AZ. the structure constant. e,=0.74. corresponds to a deep MOND Jalle moclel (7= 0.1) or an equivalent dark matter fraction within A, of 0.9."," For the lower line labelled $M$, the structure constant, $c_1 = 0.74$, corresponds to a deep MOND Jaffe model $\eta = 0.1$ ) or an equivalent dark matter fraction within $\mathit{R_{eff}}$ of 0.9."923 Note that deep MOND Jalfe models with a fixed value of 7g will also define a virial FP relation because they correspond to homologous objects with a fixed fraction of dark matter within the elfective radius., Note that deep MOND Jaffe models with a fixed value of $\eta$ will also define a virial FP relation because they correspond to homologous objects with a fixed fraction of dark matter within the effective radius.924 Because lower accelerations (lower 5g) corresponds to more dark matter (higher elfective M/L) the MOND line lies to the right of the Newtonian line in this figure., Because lower accelerations (lower $\eta$ ) corresponds to more dark matter (higher effective M/L) the MOND line lies to the right of the Newtonian line in this figure.925 The solid points in 22 are the same systemsbut with stellar plotted against the virial quantity., The solid points in 2 are the same systemsbut with stellar plotted against the virial quantity.926 The masses are estimated from the luminosity multiplied. by ALL derived from population synthesis models (Cappellari et al., The masses are estimated from the luminosity multiplied by $M/L$ derived from population synthesis models (Cappellari et al.927 2006)., 2006).928 Now we see that the points lic very. near the virial relation for pure Newtonian Jalfe mocels (small discrepaney or little dark matter within an elective radius), Now we see that the points lie very near the virial relation for pure Newtonian Jaffe models (small discrepancy or little dark matter within an effective radius).929 With MOND. this is the expected. result for high surface brightness galaxies.," With MOND, this is the expected result for high surface brightness galaxies."930 There are other classes of high surface. brightness objects with a raclially declining velocity dispersion which. apart from small shifts due to variations in homology. shoulc lic on the same ET: specifically. chwarl elliptical galaxies. the ultra-compact dwarls. and the globular star clusters.," There are other classes of high surface brightness objects with a radially declining velocity dispersion which, apart from small shifts due to variations in homology, should lie on the same FP: specifically, dwarf elliptical galaxies, the ultra-compact dwarfs, and the globular star clusters."931 In Fig., In Fig.932 3 a wide range of objects is plotted on a viria representation of the FP., 3 a wide range of objects is plotted on a virial representation of the FP.933" Here again the logarithm of the uminosity (107 2. joa proxy for the mass. is plotted agains he logarithm of the virial quantity ah,ayie (lott 41. 1."," Here again the logarithm of the luminosity $10^{11}$ $L_\odot$ ), a proxy for the mass, is plotted against the logarithm of the virial quantity $\sigma^2 \mathit{R_{eff}}/G$ $10^{11}$ $M_\odot$ )."934 AL he objects are Newtonian (satisfving the Newtonian viria heorem) and homologous with constant M/L they shoulc ie on the FP defined by the elliptical galaxics., If the objects are Newtonian (satisfying the Newtonian virial theorem) and homologous with constant M/L they should lie on the FP defined by the elliptical galaxies.935 On this plot he cllipticals drawn from the larger sample of Jorgensen et al. (, On this plot the ellipticals drawn from the larger sample of rgensen et al. (9361996) are shown as crosses: the globular clusters are solic »oints. the triangles are dwarlelliptical: the open points are ultra-compact dwarfs: and the stars are the dwarf spheroida companions of the Milkv Way.,"1996) are shown as crosses; the globular clusters are solid points, the triangles are dwarf elliptical; the open points are ultra-compact dwarfs; and the stars are the dwarf spheroidal companions of the Milky Way."937 The open squares are X-ray emitting clusters of galaxies but with gas mass plotted rather than the luminosity., The open squares are X-ray emitting clusters of galaxies but with gas mass plotted rather than the luminosity.938 The references for these observations are given in the figure caption., The references for these observations are given in the figure caption.939 Points significantly below the parallel lines have a Large Newtonian dynamical M/L. the conventional explanation being the presence of dark matter.," Points significantly below the parallel lines have a large Newtonian dynamical M/L, the conventional explanation being the presence of dark matter."940 For example. the galaxy clusters are well below the Newtonian line rellecting the classical virial cliscrepaney in these objects.," For example, the galaxy clusters are well below the Newtonian line reflecting the classical virial discrepancy in these objects."941 The essential feature of this plot is that the high surface brightness objects. ranging from. ellipticals το globular clusters define a fairly narrow fundamental plane which is consistent with the virial theorem., The essential feature of this plot is that the high surface brightness objects ranging from ellipticals to globular clusters define a fairly narrow fundamental plane which is consistent with the virial theorem.942 There are. of course. deviations due to non-homology and. more seriously. to the jeteroseeneity of the data samples.," There are, of course, deviations due to non-homology and, more seriously, to the heterogeneity of the data samples."943 But the point is that the same EP is delineated by all high. surface brightness (i.6.. ugh internal acceleration) objects.," But the point is that the same FP is delineated by all high surface brightness (i.e., high internal acceleration) objects."944 The low-surlace-brightness dwarf spheroidals. as expected. lic significantly below this EP defined. by the ugh surface brightness elliptical galaxies.," The low-surface-brightness dwarf spheroidals, as expected, lie significantly below this FP defined by the high surface brightness elliptical galaxies."945 Lhe conventional explanation is that. these object are dominated: by. dark matter in the inner regions., The conventional explanation is that these object are dominated by dark matter in the inner regions.946 With MOND:. this is precisely he expectation for low surface brightness objects where a arge discrepancy is predicted and the size has disappeared as à parameter.," With MOND, this is precisely the expectation for low surface brightness objects where a large discrepancy is predicted and the size has disappeared as a parameter."947 44 is the FJ relation for the earlv-tvpe galaxies shown in 22 (from Cappellari et al., 4 is the FJ relation for the early-type galaxies shown in 2 (from Cappellari et al.948 2006)., 2006).949 Llere the luminosity is plotted: against loglat (Cay): the parallel lines are the relation, Here the luminosity is plotted against $\log(\sigma^4/Ga_0)$ ; the parallel lines are the relation950A few ultra deep (R of at least - 25) luminosity functions (LFs) have become available in the literature for the Coma cluster (Bernstein et al.,A few ultra deep (R of at least $\sim$ 25) luminosity functions (LFs) have become available in the literature for the Coma cluster (Bernstein et al.951 1995; B95 or Milne et al., 1995: B95 or Milne et al.952 2007: MO7) but only in small fields., 2007: M07) but only in small fields.953 Recent large scale surveys for Coma are limited to much brighter magnitudes (e.g. Lobo et al., Recent large scale surveys for Coma are limited to much brighter magnitudes (e.g. Lobo et al.954 1997. Trentham 1998. Terlevich et al.," 1997, Trentham 1998, Terlevich et al."955 2001. Andreon & Cuillandre 2002. Beijersbergen et al.," 2001, Andreon $\&$ Cuillandre 2002, Beijersbergen et al."956 2002. Iglesias-Párramo et al.," 2002, Iglesias-Párramo et al."957 2003)., 2003).958 Ultradeep LFs sample the Coma cluster population down to the limit of what can be called galaxies and then include large populations of globular clusters., Ultradeep LFs sample the Coma cluster population down to the limit of what can be called galaxies and then include large populations of globular clusters.959 This is also shown by MOT. who considered a catalog of objects detected along the Coma cluster line of sight down to R~25.75.," This is also shown by M07, who considered a catalog of objects detected along the Coma cluster line of sight down to $\sim$ 25.75."960 Among this population. the objects belonging to the Coma cluster are My~—-9 systems.," Among this population, the objects belonging to the Coma cluster are $_R \sim -9$ systems."961 This corresponds to galaxies with a mass similar to that of globular clusters. and sometimes even smaller.," This corresponds to galaxies with a mass similar to that of globular clusters, and sometimes even smaller."962 The formation of such systems (that would not be globular clusters) remains quite puzzling and could e.g. be related to formation processes of tidal dwarf galaxies (e.g. Bournaud et al., The formation of such systems (that would not be globular clusters) remains quite puzzling and could e.g. be related to formation processes of tidal dwarf galaxies (e.g. Bournaud et al.963 2003)., 2003).964 This raises the question of galaxy formation in clusters from material coming from other already existing galaxies and the question of the globular cluster removal from their parent galaxies., This raises the question of galaxy formation in clusters from material coming from other already existing galaxies and the question of the globular cluster removal from their parent galaxies.965 It also relates to the cluster ability to influence the different giant galaxy types (e.g. Boselli et al., It also relates to the cluster ability to influence the different giant galaxy types (e.g. Boselli et al.966 2006) that will contribute in different ways to the formation of these very faint dwarf galaxies., 2006) that will contribute in different ways to the formation of these very faint dwarf galaxies.967 A nice example of such very faint objects torn off from galaxies infalling into massive clusters can be found 1 Cortese et al. (, A nice example of such very faint objects torn off from galaxies infalling into massive clusters can be found in Cortese et al. (9682007).,2007).969 To initiate the formation processes of such faint systems. material needs to be expelled from existing galaxies by several nternal (e.g. gas expulsion via supernova winds) or external (tidal disruptions. harrassment. ...) processes.," To initiate the formation processes of such faint systems, material needs to be expelled from existing galaxies by several internal (e.g. gas expulsion via supernova winds) or external (tidal disruptions, harrassment, ...) processes."970 At least external processes are driven by environmental effects that act at the cluster scale., At least external processes are driven by environmental effects that act at the cluster scale.971 In order to test these scenarios. we therefore need surveys with such large characteristic scales.," In order to test these scenarios, we therefore need surveys with such large characteristic scales."972 However. up to now these ultra-deep surveys were limited to very small areas (a few to a few tens of aremin- for MO7 and B95 respectively). which is a limitation to study possible environmental effects in the Coma cluster.," However, up to now these ultra-deep surveys were limited to very small areas (a few to a few tens of $^2$ for M07 and B95 respectively), which is a limitation to study possible environmental effects in the Coma cluster."973 In order to fill this lack of a large ultradeep field on Coma. we added several pointings from our previous deep multiband survey made at CFHT (Adami et al.," In order to fill this lack of a large ultradeep field on Coma, we added several pointings from our previous deep multiband survey made at CFHT (Adami et al."974 2005 (A05). 2006 (a and b: A06a and b). and 2007 (A07)) which have a subarea I common.," 2005 (A05), 2006 (a and b: A06a and b), and 2007 (A07)) which have a subarea in common."975 The resulting field of view of this new dataset combination (~32x7 aremin* or ~300 aremin) is large enough to study possible environmental effects and deep enough - 25.5) to reach the very faint object populations in the globular cluster regime., The resulting field of view of this new dataset combination $\sim 42\times$ 7 $^2$ or $\sim$ 300 $^2$ ) is large enough to study possible environmental effects and deep enough $\sim$ 25.5) to reach the very faint object populations in the globular cluster regime.976 Section 2 presents our data and methods., Section 2 presents our data and methods.977 Luminosity functions are computed in Section 3., Luminosity functions are computed in Section 3.978 We discuss the nature of the faintest objects statistically in the Coma cluster in Section 4 and give conclusions in Section 5., We discuss the nature of the faintest objects statistically in the Coma cluster in Section 4 and give conclusions in Section 5.979" All along that paper we use the following cosmological parameters: Hy = 71 km s! Mpe7!. O4=0.73 and O,,=0.27."," All along that paper we use the following cosmological parameters: $_0$ = 71 km $^{-1}$ $^{-1}$, $\Omega _\Lambda =0.73$ and $\Omega _m=0.27$."980 The resulting Coma distance modulus is 34.98., The resulting Coma distance modulus is 34.98.981 The Coma cluster images are described in detail in. AO6a., The Coma cluster images are described in detail in A06a.982 The set of data was made of two deep 42x30 arcmin? fields observed at the CFHT 3.6 m telescope with the CFHI2K camera: one covering the north and the other the south part of Coma., The set of data was made of two deep $\times$ 30 $^2$ fields observed at the CFHT 3.6 m telescope with the CFH12K camera: one covering the north and the other the south part of Coma.983 Each field was observed in the B. V. R. and I CFHT filters (u* band data is presently being acquired with the Megacam camera) and magnitudes were derived in the Vega system.," Each field was observed in the B, V, R, and I CFHT filters (u* band data is presently being acquired with the Megacam camera) and magnitudes were derived in the Vega system."984 These observations had an overlap of ~42x7 arcmin? in the form of a horizontal central. strip (CS) crossing the Coma cluster from East to West., These observations had an overlap of $\sim 42 \times 7$ $^2$ in the form of a horizontal central strip (CS) crossing the Coma cluster from East to West.985 The CS covers the qz[193.51.195.32] and 6=|27.93.28.06] rectangle.," The CS covers the $\alpha$ =[194.51,195.32] and $\delta$ =[27.93,28.06] rectangle."986 In the CS the exposure time was therefore at least doubled compared to, In the CS the exposure time was therefore at least doubled compared to987exponential nor a truncated exponential represent then well.,exponential nor a truncated exponential represent them well.988 Furthermore. there are subtle dillerences between the undisturbed and warped composite profiles: The latter are gaighthy Hatter overall and exhibit a particularly strong drop ονομα ~1.des. which may be à signature of the onset of je warp.," Furthermore, there are subtle differences between the undisturbed and warped composite profiles: The latter are slightly flatter overall and exhibit a particularly strong drop beyond $\sim 1.1 R_{25}$, which may be a signature of the onset of the warp."989 For these reasons. we decide to fit all composite profiles with broken exponential laws according to Iq. 1..," For these reasons, we decide to fit all composite profiles with broken exponential laws according to Eq. \ref{eq_brokenexp}."990 —- is possible that even a broken exponential law provides inly. an insullicient representation of the true profile shape. uad that additional parameters would be required to model un accurately. (e.g... a three-component broken exponential model for the warped objects): however. given the small size of the sample. we decide not to investigate more complex models in this paper.," It is possible that even a broken exponential law provides only an insufficient representation of the true profile shape, and that additional parameters would be required to model it accurately (e.g., a three-component broken exponential model for the warped objects); however, given the small size of the sample, we decide not to investigate more complex models in this paper."991 Given that the constituent galaxics may represent a range of profile shapes. the uncertainties in these fitting parameters for the composite profile. are. likely to. be domünated by scatter between the galaxies. rather than the measurement uncertainties.," Given that the constituent galaxies may represent a range of profile shapes, the uncertainties in these fitting parameters for the composite profile are likely to be dominated by scatter between the galaxies, rather than the measurement uncertainties."992 We therefore determine the uncertainties on these parameters by bootstrapping. 1.0. by randomly resampling the set of profiles (two profiles. per galaxy. representing the two sides) that we superpose to construct the composite. and fitting cach of the resulting realizations of the composite spectrum separately.," We therefore determine the uncertainties on these parameters by bootstrapping, i..e, by randomly resampling the set of profiles (two profiles per galaxy, representing the two sides) that we superpose to construct the composite, and fitting each of the resulting realizations of the composite spectrum separately."993 Because of the small size of the sample. we cannot guarantee that his procedure adequately samples the truc variance in the went population. but it provides a representation of the variance within the sample itself.," Because of the small size of the sample, we cannot guarantee that this procedure adequately samples the true variance in the parent population, but it provides a representation of the variance within the sample itself."994 We begin our analysis and comparison of the composite xofile shapes by examining the break radius. ry. for both the Leand stellar components.," We begin our analysis and comparison of the composite profile shapes by examining the break radius, $r_0$, for both the $\alpha$and stellar components."995 Fig., Fig.996 2 shows the distribution of »eak radii recovered. from our bootstrapping procedure for he uncisturbed (solid contours). warped (dashed contours). and full (grevscale) samples.," \ref{fig_rr} shows the distribution of break radii recovered from our bootstrapping procedure for the undisturbed (solid contours), warped (dashed contours), and full (greyscale) samples."997 In the full sample. represented » grevscales. the probability distribution. for the La component exhibits a strong peak around. ro70.7 and an extended: tail towards larger radii. as well as a secondary »ealk for the stellar continuum break radius at ~1.1405.," In the full sample, represented by greyscales, the probability distribution for the $\alpha$ component exhibits a strong peak around $r_0\approx0.7$ and an extended tail towards larger radii, as well as a secondary peak for the stellar continuum break radius at $\sim 1.1 R_{25}$."998 The contours encompass the locus rgaΞMoonta! herefore. we cannot rule out that the break raclii or the stellar continuum and Ilo. components are the same.," The contours encompass the locus $r_{H\alpha}=r_{continuum}$; therefore, we cannot rule out that the break radii for the stellar continuum and $\alpha$ components are the same."999 The uncisturbed and warped subsamples exhibit a qualitatively similar behaviour. with the dillerence that the secondary maximum for an Ha break radius around 1.1. feo; is much more pronounced in the warped than in the uncdisturbed or [ull samples.," The undisturbed and warped subsamples exhibit a qualitatively similar behaviour, with the difference that the secondary maximum for an $\alpha$ break radius around 1.1 $R_{25}$ is much more pronounced in the warped than in the undisturbed or full samples."1000 To further explore these results. we show the projection of this probability distribution onto the r axis in lig. 3..," To further explore these results, we show the projection of this probability distribution onto the $r_0^{H\alpha}$ axis in Fig. \ref{fig_r}."1001 All samples exhibit a strong peak in the probability distribution. at. ry.0.7. as well as a secondary peak around ry10l.l.," All samples exhibit a strong peak in the probability distribution at $r_0\sim 0.7$, as well as a secondary peak around $r_0\sim1.0 - 1.1$."1002 However. this secondary peak is much stronger in the warped than in the undisturbed sample: we therefore believe that it is related to the onset of the warp. where a drop in the surface. brightness. is expected.," However, this secondary peak is much stronger in the warped than in the undisturbed sample; we therefore believe that it is related to the onset of the warp, where a drop in the surface brightness is expected."1003 Nevertheless. the evidence for the 0.7/6; peak is clear even among the warped galaxies. thus justifving the decision to include both undisturbed and. warped: galaxies in our analysis.," Nevertheless, the evidence for the $R_{25}$ peak is clear even among the warped galaxies, thus justifying the decision to include both undisturbed and warped galaxies in our analysis."1004 These results. modulo slight. quantitative cilferences. are also obtained when weighting the individual spectra inversely by their integrated: surface. brightness in the composite (not shown in the figure. as the differences are small).," These results, modulo slight quantitative differences, are also obtained when weighting the individual spectra inversely by their integrated surface brightness in the composite (not shown in the figure, as the differences are small)."1005" We conclude 1) that the Lla break radius of 0.7 appears across both types of galaxies and so is not a result of warps. 2) that this Lla break occurs in unwarped galaxies at (d=0.6500"" (0.68208 for the full sample). and 3) that the break radii in the stellar continuum and Πα are consistent."," We conclude 1) that the $\alpha$ break radius of 0.7 appears across both types of galaxies and so is not a result of warps, 2) that this $\alpha$ break occurs in unwarped galaxies at $r_0^{H\alpha}=0.65^{+0.05}_{-0.06}$ $0.68^{+0.08}_{-0.02}$ for the full sample), and 3) that the break radii in the stellar continuum and $\alpha$ are consistent."1006 The other important characteristic of the. profiles that we consider in our comparison of Ila and stellar continuum surface brightness profiles are the inner and outer slopes., The other important characteristic of the profiles that we consider in our comparison of $\alpha$ and stellar continuum surface brightness profiles are the inner and outer slopes.1007 Fig., Fig.1008 4. shows the distribution of inner and. outer slopes recovered from the bootstrapping procedure for both components (filled ancl empty. circles for the 446. crosses for the stellar continuum).," \ref{fig_mm} shows the distribution of inner and outer slopes recovered from the bootstrapping procedure for both components (filled and empty circles for the $H\alpha$, crosses for the stellar continuum)."1009 Our discussion will focus on the full sample. but notable dilferences between the undisturbed and warped samples will be discussed.," Our discussion will focus on the full sample, but notable differences between the undisturbed and warped samples will be discussed."1010 Our first conclusion from this Figure is that the locus Ay=Az. indicated. by a solid line in the bottom right corner of cach panel. is inconsistent with both distributions: instead. Ao>At.," Our first conclusion from this Figure is that the locus $\lambda_1=\lambda_2$, indicated by a solid line in the bottom right corner of each panel, is inconsistent with both distributions; instead, $\lambda_2>\lambda_1$."1011 This implies that the composite surface brightness profiles decline more steeply in the outer disk than in the inner one. i.e. xh the stellar continuum and lla profiles in the composite spectrum are of the Freeman Type HE (the sub-exponential case). and hypothesis Lf (Freeman Type L the exponential xolile) is ruled out.," This implies that the composite surface brightness profiles decline more steeply in the outer disk than in the inner one, i.e., both the stellar continuum and $\alpha$ profiles in the composite spectrum are of the Freeman Type II (the sub-exponential case), and hypothesis $H_3$ (Freeman Type I, the exponential profile) is ruled out."1012 While the stellar. continuum fits populate a fairly compact region in the AjAz parameter space. [its to he lla composite profile exhibit an extended: tail," While the stellar continuum fits populate a fairly compact region in the $\lambda_1-\lambda_2$ parameter space, fits to the $\alpha$ composite profile exhibit an extended tail."1013 The maximum of the probability distribution for the full sample ies around Ayz1.5 and Aoz Y. but a significant fraction," The maximum of the probability distribution for the full sample lies around $\lambda_1\approx1.5$ and $\lambda_2\approx7$ , but a significant fraction"1014ealaxies that are members of the Local Cvoup.,galaxies that are members of the Local Group.1015 This leaves us with 7£8.080 ealaxies covering 90% of the sky.," This leaves us with $748,080$ galaxies covering $90\%$ of the sky."1016 Note that applying no dust correction reduces the uunuber of galaxies to 701.035.," Note that applying no dust correction reduces the number of galaxies to $704,035$."1017 We display the most comprehensive represeutatiou of the local Universe in Figure 2.., We display the most comprehensive representation of the local Universe in Figure \ref{fig:pic}.1018 We see a complex web of galaxies stretched out along fibuneuts between dense uodes often corresponding to the location of giaut superclusters., We see a complex web of galaxies stretched out along filaments between dense nodes often corresponding to the location of giant superclusters.1019 The most obvious structure is the “Creat Attractor” found within the region bounded by 300<7!«3607 aud Ibcbox|L, The most obvious structure is the “Great Attractor” found within the region bounded by $300<l<360\arcdeg$ and $-45<b<+45\arcdeg$.1020"h This structure is roughly divided iuto north aud south Calactic hemisphere complexes the overlapping Uvdra-Ceutamrus aud ""Shaplev supercluster complex (0—5<| 15°) and the Pavo-Indus wall 15<b« 0) (27).."," This structure is roughly divided into north and south Galactic hemisphere complexes – the overlapping Hydra-Centaurus and “Shapley” supercluster complex $0<b<+45\arcdeg$ ) and the Pavo-Indus wall $-45<b<0\arcdeg$ ) \citep{dres:88,fisher:95}. ."1021 The Perseus-Pisces “chain” extends from the Perseus supercluster (7— 1507.55= 157) to (1= 1107.5= 35747). aud appears to reach up to b=[30° along latitude 160° (2)..," The Perseus-Pisces “chain” extends from the Perseus supercluster $l=150\arcdeg$ $b=-15\arcdeg$ ) to $l=110\arcdeg$ $b=-35\arcdeg$ \citep{hg:86}, and appears to reach up to $b=+30\arcdeg$ along latitude $160\arcdeg$ \citep{fisher:95}."1022 It las been demonstrated that there is no physical link between the Perscus-Pisces chain and the Pavo-Iundus wall(?).., It has been demonstrated that there is no physical link between the Perseus-Pisces chain and the Pavo-Indus \citep{dinel:96}.1023" Additional features include the Torologimm-Reticulum supercluster (Fo= 2657.b= 557) (T)... Virgo supercluster (Fo= 2s5b.b=| 75""). the Ursa Major cloud (= 1157.5= (65°). and the NGC 1600 exoup (|= 3107.05= 30°) (?).."," Additional features include the Horologium-Reticulum supercluster $l=265\arcdeg$ $b=-55\arcdeg$ ) \citep{lucey:83}, , Virgo supercluster $l=285\arcdeg$ $b=+75\arcdeg$ ), the Ursa Major cloud $l=145\arcdeg$ $b=+65\arcdeg$ ), and the NGC 1600 group $l=210\arcdeg$ $b=-30\arcdeg$ ) \citep{fisher:95}."1024" Finally. the ""Coat Wall at e:~δ000 kin/s (7) is the faint feature running from the Hercules superchister (P= 308.6=|157) to (= Is07.5—| 30°),"," Finally, the “Great Wall” at $cz\sim8000$ km/s \citep{gh:89} is the faint feature running from the Hercules supercluster $l=30\arcdeg$ $b=+45\arcdeg$ ) to $l=180\arcdeg$ $b=+30\arcdeg$ )."1025 The inhomogeneity of these structures causes a eyxavitational acceleration on the Local Group of ealaxies., The inhomogeneity of these structures causes a gravitational acceleration on the Local Group of galaxies.1026 The CAB velocity dipole (?) results from the motion of the Sun relative to the CMD standard of rest.," The CMB velocity dipole \citep{line:96}1027 results from the motion of the Sun relative to the CMB standard of rest."1028 This motion cau be broken down iuto two parts: the motion of the sun relative to the Local Coup and the motion of the Local Croup with respect to the CAIB., This motion can be broken down into two parts; the motion of the sun relative to the Local Group and the motion of the Local Group with respect to the CMB.1029" Using the most recent values (?) we fud that the Local Group velocity relative to the CAIB is 622kins in thedirection Lian=212.5,44—28."," Using the most recent values \citep{cv:99} we find that the Local Group velocity relative to the CMB is $622 \kms$ in thedirection $l_{cmb} = 272, b_{cmb} = 28$."1030 Since both eravitational force aud fux fall off as distance squared. the net acceleration of the Local Group is proportional to the dipole of the light distribution for a constant mass-to-light ratio Y.," Since both gravitational force and flux fall off as distance squared, the net acceleration of the Local Group is proportional to the dipole of the light distribution for a constant mass-to-light ratio $\Upsilon$."1031 Thus. where 9; is the flux received from cach galaxy and sc; is a directional unit vector.," Thus, where $S_i$ is the flux received from each galaxy and $\hat{r}_i$ is a directional unit vector."1032 However. the light mav be a biased tracer of mass (?): if there is Huear biasing then the true acceleration would be a factor 1/5 times the measured value.," However, the light may be a biased tracer of mass \citep{kais:84}; if there is linear biasing then the true acceleration would be a factor $1/b$ times the measured value."1033" Linear theory can be used to estimate the resultant velocity. (7). which depends on the matter deusitv of the universe O,,."," Linear theory can be used to estimate the resultant velocity \citep{peeb:80}, which depends on the matter density of the universe $\Omega_{m}$."1034 The expected velocity of the Local Cvoup is therefore. where FO)zQUO (7) aud fj is the Ilubble coustaut.," The expected velocity of the Local Group is therefore, where $f(\Omega_m) \simeq \Omega_m^{0.6}$ \citep{peeb:80} and $H_0$ is the Hubble constant."1035 Of course. this is only true in linear theory.," Of course, this is only true in linear theory."1036" Nonlinear N-body sinulations show that the velocity of a region like the local eroup is typically within 7"" of the computed linear acceleration aud that the magnitude agrees to within 20% (?)..", Nonlinear N-body simulations show that the velocity of a region like the local group is typically within $7\arcdeg$ of the computed linear acceleration and that the magnitude agrees to within $20\%$ \citep{dsy:91}.1037" Measuring the lieht dipole therefore achieves two aus: (1) It verifies that the CXMB dipole is truly caused by he Suus motion and (2) it can cetermine a combination of Ty. Q,, aud b."," Measuring the light dipole therefore achieves two aims: (1) It verifies that the CMB dipole is truly caused by the Sun's motion and (2) it can determine a combination of $\Upsilon_K$, $\Omega_{m}$ and $b$."1038" Since Q,, aud Yy can be measured by other means we can therefore determine the value of b.", Since $\Omega_{m}$ and $\Upsilon_K$ can be measured by other means we can therefore determine the value of $b$.1039 To accurately calculate a clustering dipole we reed to consider the effect of the masked region., To accurately calculate a clustering dipole we need to consider the effect of the masked region.1040 We address this in two wavs: by cloniug the sky above aud below the masked regious (2).. aud by filling the masked region with randomly choseu ealaxies such that it has the same surface density as the mumasked area.," We address this in two ways: by cloning the sky above and below the masked regions \citep{llb:89}, and by filling the masked region with randomly chosen galaxies such that it has the same surface density as the unmasked area."1041 We apply both methods o estimate the uncertaüntv coutributed bv the nasked region., We apply both methods to estimate the uncertainty contributed by the masked region.1042 Figure 3 shows the convergence ofthe maguitude of the clustering dipole as a function of the flux nuit of the survey., Figure \ref{fig:conv} shows the convergence of the magnitude of the clustering dipole as a function of the flux limit of the survey.1043" The dipole iosth arises roni galaxies with A,«12 mae. the brightest 50.000. galaxies."," The dipole mostly arises from galaxies with $K_s < 12$ mag, the brightest $50,000$ galaxies."1044 Thedipole converges at fainter naguitudes and the addition of 300.000. galaxics with AW.>13.2changes the dipole by less then 5%.," Thedipole converges at fainter magnitudes and the addition of $300,000$ galaxies with $K_s > 13.2$changes the dipole by less then $5\%$ ."1045 For comparison. the dashed line shows the growth in the total flux as a function of the nuuber of ealaxies used. which coutiuues to grow for faint imaenitudes.," For comparison, the dashed line shows the growth in the total flux as a function of the number of galaxies used, which continues to grow for faint magnitudes."1046 This is compelling evidence that the, This is compelling evidence that the1047where P? is the neutron star spin period.,where $P$ is the neutron star spin period.1048 Failure of pulsar emission may also partly result from the fact that coherent radiowaves with a wavelength longer than 75 em can be absorbed effectively in the wind plasma (Ilarionov&Sunvave1975)., Failure of pulsar emission may also partly result from the fact that coherent radiowaves with a wavelength longer than 75 cm can be absorbed effectively in the wind plasma \citep{ill75}.1049. The flow in the inner part of the accretion disk is expected to have density fluctuations (“chimps) produced by a variety of mechanisms. such as thermal instability. INelvin-Hehlmholtz instability. ancl magnetoturbulence (seeLambetal.1935:Shibazaki&1987).," The flow in the inner part of the accretion disk is expected to have density fluctuations (“clumps"") produced by a variety of mechanisms, such as thermal instability, Kelvin-Helmholtz instability, and magnetoturbulence \citep[see][]{lam85,shi87}."1050. The clumpy wind density would be much higher than the averaged value estimated above., The clumpy wind density would be much higher than the averaged value estimated above.1051" Thev nav also leave short. sporadic ""transparent"" time for the development of particle acceleration in the gap aud generation of pulsar emission."," They may also leave short, sporadic “transparent"" time for the development of particle acceleration in the gap and generation of pulsar emission."1052" Η we assume that the tvpical clump separation is less than the disk height. //;, at the inner edge {τμ of the disk. the duration of successful pulsar emission should be less than Where Coe is the escape velocity at At."," If we assume that the typical clump separation is less than the disk height $H_{\rm1053in}$ at the inner edge $R_{\rm in}$ of the disk, the duration of successful pulsar emission should be less than where $v_{\rm esc}$ is the escape velocity at $R_{\rm in}$ ."1054" IE. <HyRy and H7/R<0.1. we have LLP ms <7<LTD? s, Here Re=(GAP?/A4x)? is the corotation radius and Ij.=cP/2s is the light evlinder radius. respectively."," If $R_{\rm c}<R_{\rm in}<R_{\rm lc}$ and $H/R\la 0.1$, we have $11P$ ms $<\tau<1.7P^{3/2}$ s. Here $R_{\rm c}\equiv(GMP^2/4\pi)^2$ is the corotation radius and $R_{\rm lc}\equiv cP/2\pi$ is the light cylinder radius, respectively."1055 As pointed out by Zhang et al. (, As pointed out by Zhang et al. (10562006). the dynamical time scale of the inner gap (745/67105—10.! s where ~Pu is the height of the gap) is much smaller (han (he rotation period P.,"2006), the dynamical time scale of the inner gap $\sim h_{\rm1057gap}/c\sim 10^{-6}-10^{-4}$ s, where $\sim h_{\rm gap}$ is the height of the gap) is much smaller than the rotation period $P$."1058 So the time scale to develop a pair cascade is much shorter (han 7., So the time scale to develop a pair cascade is much shorter than $\tau$.1059" [ts magnitude seems compatible with the burst durations measured so far,", Its magnitude seems compatible with the burst durations measured so far.1060 The debris disk may be popular in relatively voung neutron stars., The debris disk may be popular in relatively young neutron stars.1061 Jiang Li (2005) performed Monte-Carlo simulation of pulsar evolution. assuming (hat all neutron stars are born with a surrounding supernova fallback disk with (he initial masses of the disk ranging from 10°AL. to 107AL..," Jiang Li (2005) performed Monte-Carlo simulation of pulsar evolution, assuming that all neutron stars are born with a surrounding supernova fallback disk with the initial masses of the disk ranging from $10^{-6}\,M_{\sun}$ to $10^{-2}\,M_{\sun}$."1062 They found that the emereing proportion of disk-fed neutron stars (ie. with the disk extending inside the light evlinder) is ~20%—50% at age of 10° vears. and ~LOW—25% at age of 10! vears.," They found that the emerging proportion of disk-fed neutron stars (i.e. with the disk extending inside the light cylinder) is $\sim106320\%-50\%$ at age of $10^3$ years, and $\sim 10\%-25\%$ at age of $10^4$ years."1064 Obviously these numbers are sensitive to the assumptions for the initial parameters. e.g.. the distributions of the initial disk masses. of (he neutron star spin periods. magnetic fields. and most importantly. (he mechanisms of the propeller spin-down.," Obviously these numbers are sensitive to the assumptions for the initial parameters, e.g., the distributions of the initial disk masses, of the neutron star spin periods, magnetic fields, and most importantly, the mechanisms of the propeller spin-down."1065 However. it clearly demonstrates (hat a considerable [raction of isolated neutron stars could harbor a debris disk with sufficiently long time (Popovetal.(2000) suggested that a fraction of 0.1%—0.2% of all isolated neutron stars may be presently in the propeller stage cue interaction with the interstellar medium).," However, it clearly demonstrates that a considerable fraction of isolated neutron stars could harbor a debris disk with sufficiently long time \citet{pop00} suggested that a fraction of $0.1\%-0.2\%$ of all isolated neutron stars may be presently in the propeller stage due interaction with the interstellar medium)."1066" It was also Found that the ratio of the characteristic age /.=P/2P and the true age / distributes within a relatively wide range [rom ~0.1 to ~10. indicating that 7, and the magnetic field strength estimated from magnetic dipole radiation may considerably deviate from (he actual values lor theseneutron stars."," It was also found that the ratio of the characteristic age $t_{\rm c}=P/2\dot{P}$ and the true age $t$ distributes within a relatively wide range from $\sim 0.1$ to $\sim 10$, indicating that $t_{\rm c}$ and the magnetic field strength estimated from magnetic dipole radiation may considerably deviate from the actual values for theseneutron stars."1067 The disk-assisted spin-down may also explain why RRATs have relatively long spin periods compared with normal isolated radio pulsars., The disk-assisted spin-down may also explain why RRATs have relatively long spin periods compared with normal isolated radio pulsars.1068This means that ina time comparable to the precursor jet breakout time. the channel closes up again.,"This means that in a time comparable to the precursor jet breakout time, the channel closes up again."1069 From Fig.5 of Lazzati (2005). it is known that the precursor duration is always shorter than the gap period of tune.," From Fig.5 of Lazzati (2005), it is known that the precursor duration is always shorter than the gap period of time."1070 Therefore. we conclude that the chamnel opened by the precursor jet is indeed closed again before the main jet starts to propagate out.," Therefore, we conclude that the channel opened by the precursor jet is indeed closed again before the main jet starts to propagate out."1071 The free-fall time of the material along the rotation axis is For the C/O core material of the collapsed progenitor. whose density is pSO0ecm7. its free-fall time is comparable to the ~100 eap period and therefore this :ὲ will uaterialbe essentially↔ὲ evacuated.," The free-fall time of the material along the rotation axis is For the C/O core material of the collapsed progenitor, whose density is $\rho\sim 500 {\rm g cm^{-3}}$, its free-fall time is comparable to the $\sim 100 \,{\rm s}$ gap period and therefore this material will be essentially evacuated."1072ac. However.. for Tle core material with p—1eem3 the free-fall time is nmcli longer than the eap period. aud the fuunel in this The chauuel made bv the initial⋅⋅⋅ precursor jet. cau. while it remains open. produce an iuteresting Cclission signature.," However, for He core material with $\rho\sim 1~{\rm g cm^{-3}}$, the free-fall time is much longer than the gap period, and the funnel in this The channel made by the initial precursor jet can, while it remains open, produce an interesting emission signature."1073∙ As the cocoon expands transversally iuto⋅ the stellar material. it ⋅⋅⋅will drive⋅ a shock.," As the cocoon expands transversally into the stellar material, it will drive a shock."1074 Since⊲⋅ this ⋅⋅shock takes place in⋅ the highly⋅ optically. thick ⋅⋅interior⋅ of⋟ the progenitor. star. itf ⋅⋅will become a radiatiou-doninatedD.. shock. Similarly⋅⋅ case of the supernova⋅ shock propagating inside the iu⋅⋅↔⋅≽≽ ↽ 1976).," Since this shock takes place in the highly optically thick interior of the progenitor star, it will become a radiation-dominated shock, similarly to the case of the supernova shock propagating inside the star (Weaver 1976)."1075" The. velocity :of he‘ shockὋ driven by thematter ""NS : He coreteal lerivedmateria £m )e derivedei⋅ 5from e; ↕∖= satished2 ↽of,LeHowhereνι5).is""T given byThis ;Ex"," The velocity of the shock driven by the cocoon into the He core material can be derived from $v_s=(p_c/\rho)^{1/2}=v_\perp$ , where $p_c$ is given by Eq.(5)."1076" lis Givesa .loonsbeing DP./D,Dell l= aud/ Vp op .""n:⋅⋅asstuuptionshock-heatede.of=a"," This gives a shock velocity $v_s\simeq10^9{\rm cm1077s^{-1}}L_{j,49}^{3/8}\theta_{j,-1}^{-1/4}r_{11}^{-3/4}\rho^{-3/8}\alpha^{-1/4}$."1078shock|radiation-dominatedvelocityshock.For to a The walls of the jet again. can radiate density constant of2r s Loe Lin 6)(esl? Bong “ergsp» radiation.," For a radiation-dominated shock, this corresponds to a radiation temperature of where $a$ is the Boltzmann energy density constant of radiation."1079" To check the selfbcousisteney of having assumed a radiation-doniuated shock. we need to compare the the radiation diffusion leneth A.= ef(3e,;0;0T) (where DS is the electron uuuber deusitv of the plasima) with the stopping distance of the ious due to Coulomb collisios 7."," To check the self-consistency of having assumed a radiation-dominated shock, we need to compare the the radiation diffusion length $\Delta_\gamma=c/(3n_e v_s \sigma_{\rm T})$ (where $n_e$ is the electron number density of the plasma) with the stopping distance of the ions due to Coulomb collisios $l$."1080 According to Weaver Chapline (1971). the Coulomb friction is the dominant dissipatio- mechanisn coutrolling the shock width (ie. the shock is collisional). provided that the stopping distance of the ious / is larger than the raciation diffusiou lenet[um A..," According to Weaver Chapline (1974), the Coulomb friction is the dominant dissipation mechanism controlling the shock width (i.e. the shock is collisional), provided that the stopping distance of the ions $l$ is larger than the radiation diffusion length $\Delta_\gamma$."1081. Otherwise. the shock is racdiation-dominated.," Otherwise, the shock is radiation-dominated."1082" For an electron temperature T;—T,. the ion-clectron collision rate is rn=2s|TTLecBYT./10KeV)LEN1l while the iondon collision rate. for au jon temperature of T;=Ορος-— 200KeV. is rg—dosLOM(pflecmn2)07;/200K06V)72s1 (Spitzer 1962: Waxnanu Loch 2001)."," For an electron temperature $T_e=T_r$, the ion-electron collision rate is $\nu_{ie}=2\times10^{10} (\rho/11083{\rm g cm^{-3}}) (T_e/{\rm 10KeV})^{-3/2} {\rm s^{-1}}$, while the ion-ion collision rate, for an ion temperature of $T_i=(3/16) m_p1084v_s^2=200{\rm KeV}$, is $\nu_{ii}=1\times10^{10} (\rho/1 {\rm g1085cm^{-3}}) (T_i/200{\rm KeV})^{-3/2} {\rm s^{-1}}$ (Spitzer 1962; Waxman Loeb 2001)."1086 Thus. the stopping of the ious is donüuated by iuteraction with the electrons. with a mean free path eiven by 1," Thus, the stopping of the ions is dominated by interaction with the electrons, with a mean free path given by ."1087) The radiation diffusion leneth is A.=20ci(p/lecii7)tCe./LO?cms|," The radiation diffusion length is $\Delta_{\gamma}=20{\rm cm}1088(\rho/1 {\rm g cm^{-3}})^{-1} (v_s/10^9{\rm cm s^{-1}})^{-1}$."1089 Since A.o>{ων the seale (width) of the shock is determined by A..," Since $\Delta_{\gamma}\gg l_{ie}$, the scale (width) of the shock is determined by $\Delta_{\gamma}$."1090 Except for a thin laver at the leading οσο of the shock front. the Coulomb friction is uniiportant. and the shock is effectively radiatiou-domunated.," Except for a thin layer at the leading edge of the shock front, the Coulomb friction is unimportant, and the shock is effectively radiation-dominated."1091" For a radiation temperature of T,xLokeV. Compton scattering gives a cooling rate of coup(Nor/Ansc)T)=wo.Q0(T,νοyisd. while the Breisstrabhme cooling rate is plos=3o.H(p/lecanCQ(T,/10KeV)Ves 1, "," For a radiation temperature of $T_r\simeq 10{\rm KeV}$ , Compton scattering gives a cooling rate of $t_{\rm1092Comp}^{-1}=(8\sigma_T/3m_e c) a T_r^4= 8\times10^{10} (T_r/10{\rm1093KeV})^4 {\rm s^{-1}}$, while the Bremsstrahlung cooling rate is $t_{Brem}^{-1}=3\times10^8 (\rho/1 {\rm g cm^{-3}}) (T_e/{\rm109410KeV})^{-1/2} {\rm s^{-1}}$ ."1095Thus Compton scattering also provides the donmunant imechanisin for cooling the electrons. aud this cau reduce the ion telmperature significantlySUM from: those that would occur in à Coulomb viscositv-dominated shock.," Thus, Compton scattering also provides the dominant mechanism for cooling the electrons, and this can reduce the ion temperature significantly from those that would occur in a Coulomb viscosity-dominated shock."1096 These phlotous diffuse⋅ alea. creating∙ sufficiout∙∙ pressure to decelerate the electrons (as viewed in the frame of the shock front) aud the fous are then by. these. electrons through (oulouib friction.," These photons diffuse ahead, creating sufficient pressure to decelerate the electrons (as viewed in the frame of the shock front) and the ions are then decelerated by these electrons through Coulomb friction."1097decelerated We note that the above rough quautitative argument is only au approximate treatment of thisB issue., We note that the above rough quantitative argument is only an approximate treatment of this issue.1098B A amore detailedB umuericalB treatiueut takingB8 into. account the euergy aud momentum balance in the shock transition as well as relativisticeffects. as performed by Weaver (1976). is beyond the scope of the present- work.," A more detailed numerical treatment taking into account the energy and momentum balance in the shock transition as well as relativistic, as performed by Weaver (1976), is beyond the scope of the present work."1099"Loa.M. However.⋅⋅↴⋅. therem is, €a usefuln⋅⋅ muuerical""PA€ criterion for ↸⊳∐↸∖↸⊳↨↘∐↕∶↴∙↑∐↸∖↴∖↸∖∐⊣⊳∪∐↴∖↕↴∖↑↸∖∐↸⊳⋅↖∪↕⋜↧⋯≼∐⋜↧⊓≺≻∐⊣↧∪⋯∐↕⋜↧↑↸∖≼↧""M m⋅∙∙ ∙ shock."," However, there is a useful numerical criterion for checking the self-consistency of a radiation-dominated shock."1100"- According⋅⋅↜ to Weaver⇁↽⋅↽∢↼⋅ (1976). the radiative. heat ⊓⋅⋜⋯↴∖↴⋯↥⋅↑↴∖↴∏↕−⊔↸⊳↸∖↴∖↴↕⋟∪↥⋅∐⋅∪≼⊔∐⊳↕∐∩⊾↑∐↸∖∐↸∖↸⊳↸∖↴∖∷∖↴⋜∐⋅↖↽≼∐↴∖↴↴∖↴↕≻⋜↧↑↕∪∐ oO strong: shock wave↴⋅↜↴ if ⋅∙∙the ratio of the⋅↔⋅ raciation pressure to. the⋅⋅⋅ pressure P./P, ↴exceeds L 15."," According to Weaver (1976), the radiative heat transport suffices for producing the necessary dissipation in a strong shock wave if the ratio of the radiation pressure to matter pressure $P_r/P_m$ exceeds 4.45."1101"↱↴∙↴∙∙ Thiscondition⇁ star (WeaverBM our .case. the⋅∙ ratio⊽ of the two∙↴↴∙↴ pressures⋅ cocoon intothe theII KI. =pelínsgkT,~100foryf]pyr‘ οι.oe T2=T,i MükeV. justifvinethe↽≝≻↴3 ήν""n . a0, radiation-donminated22 dock."," This condition is satisfied in our case, the ratio of the two pressures being $P_r/P_m=a T_r^4/n_e kT_e=\rho v_s^2/ n_e k1102T_r\simeq 100$ for $v_s=10^9 {\rm cm s^{-1}}$ and $T_e=T_r=10 {\rm1103KeV}$, justifying the assumption of a radiation-dominated shock."1104 this107008δω corresponds chaunel. before the radiation withtemperature of latter closes up a transicut thermal ! pulse characteristic photon energv ~ 10IeV aud SN ! .Lipen ο I~ |luninosityJ escapes from the channel. the cocoon wressure decreases significantly. as does the radiatiou energv density of the photou field.," The shock-heated walls of the jet channel, before the latter closes up again, can radiate a transient thermal pulse with characteristic photon energy $kT\sim {\rm 10KeV}$ and a luminosity escapes from the channel, the cocoon pressure decreases significantly, as does the radiation energy density of the photon field."1105 Finally. as the pressure wanes. the channel closes wp again under the pressure of he external stellar material.," Finally, as the pressure wanes, the channel closes up again under the pressure of the external stellar material."1106 of ~10 seconds duration. besides the nou-thermal enüssion conrponeut roni the cocoon itself.," of $\sim10$ seconds duration, besides the non-thermal emission component from the cocoon itself."1107 Towdoes the weak precursorjet form during the initial core collapse?, How does the weak precursor jet form during the initial core collapse?1108 There is a long-standing speculation that when rotation aud imagnuetie fields are taken iuto account. the core collapse of massive stars can lead to some formof MIID outflows. although it is unknown whether they can power supernovaeor eanuni-ray bursts (0.8. LeBlauce," There is a long-standing speculation that when rotation and magnetic fields are taken into account, the core collapse of massive stars can lead to some formof MHD outflows, although it is unknown whether they can power supernovae or gamma-ray bursts (e.g. LeBlance"1109center row as a function of frequency at cillerent fiber iunubers.,center row as a function of frequency at different fiber numbers.1110 Phase fiting residual (shown οἱ the bottom panel of Fie. L.," Phase fitting residual (shown on the bottom panel of Fig. \ref{fig:PH_Fre_FiberNum},"1111 RMSQ.9 rad) is cousisterjowl bh photou-uoise limited meastuement error (see 82.| for details)., RMS=0.9 rad) is consistent with photon-noise limited measurement error (see \ref{sec:Err_Ana} for details).1112 GD for a pa‘ticular fiber uuiiber is obtained by averagiug the results of CD measurenents for those rows associated wilh the fiber., GD for a particular fiber number is obtained by averaging the results of GD measurements for those rows associated with the fiber.1113" ""nο,", Fig.1114 Ὁ shows the resuts at v=950 THz as a function of fiber number., \ref{fig:GD_SpecNum} shows the results at $\nu$ =550 THz as a function of fiber number.1115 Note tha the twe) nllus ο ‘the interferometer are intentionally tiltec to each other aid the 60 fibers are evenly mounted akng the slit clirection., Note that the two arms of the interferometer are intentionally tilted to each other and the 60 fibers are evenly mounted along the slit direction.1116" The measured CDs should gradually vary with fiber junbe""s", The measured GDs should gradually vary with fiber number.1117 We use a secotd-orcer polynomial to fi the GD variation with the fiber number., We use a second-order polynomial to fit the GD variation with the fiber number.1118 The [ittiὃν resiual has an IMS of 0.0016 ps., The fitting residual has an RMS of 0.0046 ps.1119 Fig., Fig.1120 6 shews fitted GD as a [uuction of frequency Lor differeu fibers., \ref{fig:GD_Fre_FiberNum} shows fitted GD as a function of frequency for different fibers.1121 CD varies 0.15 ps (0.6%)) across rueast'ement range from 510 to 565 THz., GD varies 0.15 ps ) across measurement range from 540 to 565 THz.1122 lenoring GD dependence of frequency would result in 180 m51 1neasurernment offset between two ends of measwement range (assuming a true RV of 30.000 mi«s.+. which is a typical stellar RV value due to tve Earth's barycentrie inotion).," Ignoring GD dependence of frequency would result in 180 $\rm{m}\cdot\rm{s}^{-1}$ measurement offset between two ends of measurement range (assuming a true RV of 30,000 $\rm{m}\cdot\rm{s}^{-1}$, which is a typical stellar RV value due to the Earth's barycentric motion)."1123 Table 1. provides the polynomial fitting coefficieuts of GD vs. fibe: number at dillerent frequencies within measurement rauge., Table \ref{tab:GD_err_fre} provides the polynomial fitting coefficients of GD vs. fiber number at different frequencies within measurement range.1124 Two johwsical parameters. © aud v. are measured in the experiment.," Two physical parameters, $\phi$ and $\nu$ , are measured in the experiment."1125 The uncertainty of the © measurenient is 0.5 rad under phoon-noise lini(ου conclition assuming a S/N of 120 and a ypleal friige visibility of1., The uncertainty of the $\phi$ measurement is $\sim$ 0.8 rad under photon-noise limited condition assuming a S/N of 120 and a typical fringe visibility of.1126556.. The uicertainty due to the waveleneth calibration is ~0.002 THz (0.02A))., The uncertainty due to the wavelength calibration is $\sim$ 0.002 THz ).1127 We conduct a bootstrapping process to ivestigate 1ο uncertainty of GD caused by he measurement uucertaluties of © ai dav., We conduct a bootstrapping process to investigate the uncertainty of GD caused by the measurement uncertainties of $\phi$ and $\nu$.1128 We ad eaussiall loises with standard deviation of heastlremel| errors to both © and v anc calculate ile group celay., We add gaussian noises with standard deviation of measurement errors to both $\phi$ and $\nu$ and calculate the group delay.1129 We run 1000 iterations [or ο 1 order to estimate the lice‘tainty of iD., We run 1000 iterations for bootstrapping in order to estimate the uncertainty of GD.1130 The nedian of the relative error of GD leasnents. 5GD/GD. is LIx107.," The median of the relative error of GD measurements, $\delta \rm{GD}/\rm{GD}$, is $4.4\times10^{-5}$."1131 In compariso he meclia 10[ the relative CD meastuement error | Lax:LO+ after smoothing by fittii QaOi polylal [9] CD variation with the fiber number.," In comparison, the median of the relative GD measurement error is $\sim1.8\times10^{-4}$ after smoothing by fitting a polynomial to GD variation with the fiber number."1132 This 1e' does not agree with the relative GD err 291cted wv the bootstrapping experiment., This number does not agree with the relative GD error predicted by the bootstrapping experiment.1133 WestSE lia the uncounted error in the 300siraplug siinlation comes [rom image distortion dte 10 optics wuch the data pipeline has uo hly corrected for. e.g.. spectrum curvature. spectla ine slant. etc.," We suspect that the uncounted error in the bootstrapping simulation comes from image distortion due to optics which the data pipeline has not fully corrected for, e.g., spectrum curvature, spectral line slant, etc."1134 Iun a 2-D s»ectruin as illustraed in Fig. Ἐν," In a 2-D spectrum as illustrated in Fig. \ref{fig:DFDI_setup},"1135 1le dhase shift betwee adjacent pixels ong sli jrection Is  0.6 ‘acl. aud he phase slift between each wavelength cliaue is 72.5 rac.," the phase shift between adjacent pixels along slit direction is $\sim$ 0.6 rad, and the phase shift between each wavelength chanel is $\sim$ 2.5 rad."1136 Αι iperfect spectrum curvature tracing ten stoslift pixel in tl esit cirection while au iniperfect 5at uyrection can affect. pixe slifine i1 bot1 slit:1ud dispersiou directions., An imperfect spectrum curvature tracing tends to shift pixel in the slit direction while an imperfect slant correction can affect pixel shifting in both slit and dispersion directions.1137 The range of unwrappec 9vase is ~ 1000. rad., The range of unwrapped phase is $\sim$ 4000 rad.1138 For o fiber. ilae[n]παv-chaugiug plase error is tutroduced by the ¢ala jxpeliue when correcting f he optical «Istorti¢ii. for exampe. 0.1 rad deviation from true veiue al oje end. while no deviatio rat tle otler end. a relative er‘or of GD would be caused witl all estiration of 0.1£/000=1x10.1.," For one fiber, if a gradually-changing phase error is introduced by the data pipeline when correcting for the optical distortion, for example, 0.4 rad deviation from true value at one end while no deviation at the other end, a relative error of GD would be caused with an estimation of $0.4/4000=1\times10^{-4}$."1139 If dil‘erent fibers are t|aed indepeuclently. which is the «ease for the MARVELS data rediction. plpeliie. then. this grallaly-changing phase error. introdiced by iuperlect optical distortiou correctiou. mayexplain the saucdard deviation error we see after the »olyuoumial fitting for CDs as a fuuction of fiber uumber.," If different fibers are treated independently, which is the case for the MARVELS data reduction pipeline, then this gradually-changing phase error, introduced by imperfect optical distortion correction, mayexplain the standard deviation error we see after the polynomial fitting for GDs as a function of fiber number."1140"other hand. equation (S4)) implies. given that 4,=ofer; (equation LST) and &o are fixed. that. [/Ai as u|/&i goes up (and σι=(σιTH)1/2TL goes down).","other hand, equation \ref{eq:betio}) ) implies, given that $\tilde\sigma_\perp =1141\lao/a_0^2$ (equation I.87) and $\sigpar$ are fixed, that $\betiabs$ as $|\sigH|/\tilde\sigma_\perp$ goes up (and $\sigP /\tilde\sigma_\perp = (\tilde\sigma_\perp^2 -1142\sigH^2)^{1/2}/\tilde\sigma_\perp$ goes down)."1143 Taken together. these results explain the manifested dependence of the solutions in Figs.," Taken together, these results explain the manifested dependence of the solutions in Figs."1144 4. and 5 on Ιση]έσι., \ref{fig:10_1} and \ref{fig:5_10} on $|\sigH|/\tilde\sigma_\perp$.1145" As a further check on the applicability of the analytical framework developed. in Paper Lo we note that equations (AS)) and. (A9)) imply a similar dependence of ty and 3, on |&g|/e&, in the limit syσ1 (Le. in the Ohm cilfusion-mocdified. Hall regime: Cases in and iv)."," As a further check on the applicability of the analytical framework developed in Paper I, we note that equations \ref{eq:ht}) ) and \ref{eq:zb}) ) imply a similar dependence of $\zt_{\rm h}$ and $\zt_{\rm b}$ on $|\sigH|/\tilde\sigma_\perp$ in the limit $s_0 \ll 1$ (i.e. in the Ohm diffusion-modified Hall regime; Cases iii and iv)."1146 In fact. fixing the values of the same parameters as before. these equations imply that. to leading order in small ratios. twohx1ο and. Zox[teu] in this case. while equation (85)) shows that [A45] as Ἰση]ήσι goes up (assuming again that σι and 6o remain unchanged).," In fact, fixing the values of the same parameters as before, these two equations imply that, to leading order in small ratios, $\tilde h \propto 1/|\beteo|$ and $\zt_{\rm b} \propto |\beteo|$ in this case, while equation \ref{eq:beteo}) ) shows that $|\beteo|$ as $|\sigH|/\tilde\sigma_\perp$ goes up (assuming again that $\tilde\sigma_\perp$ and $\sigpar$ remain unchanged)."1147" We have verified that solutions in these regimes indeed exhibit the expected dependence on [e&g|/&,.", We have verified that solutions in these regimes indeed exhibit the expected dependence on $|\sigH|/\tilde\sigma_\perp$ .1148 Yet another test of the predictions of the hydrostatic analysis regarding the dependence of the solutions on σµ]/σι can be constructed using equation (83)) for the parameter Yo., Yet another test of the predictions of the hydrostatic analysis regarding the dependence of the solutions on $|\sigH|/\tilde\sigma_\perp$ can be constructed using equation \ref{eq:upo}) ) for the parameter $\upo$.1149 As ciseussecl in Section L6 ancl illustrated in Fig., As discussed in Section I.6 and illustrated in Fig.1150 L2. this analvsis indicates that the requirement Yol (ie. that the mid-plane neutralion momentum exchange time be shorter than the local orbital time) is a fundamental constraint on viable wind-driving disc models of the type that we consider. and applies to solutions in all the diffusivity regimes (ambipolar. Hall and Ohm).," I.2, this analysis indicates that the requirement $\upo \gtrsim 1$ (i.e. that the mid-plane neutral–ion momentum exchange time be shorter than the local orbital time) is a fundamental constraint on viable wind-driving disc models of the type that we consider, and applies to solutions in all the diffusivity regimes (ambipolar, Hall and Ohm)."1151 This condition follows directly from the inequalities οιπο)71 and Jin)<1 that characterize the Hall Cases (i) and. (iii). but. as can be seen by combining the first. two parameter constraints reproduced in the first row of Table 1.. it can be formally inferred (within the framework of the hivydrostatic analvsis) to apply also in the other Hall sub-regimes.," This condition follows directly from the inequalities $\upo \betiabs > 1/2$ and $\betiabs < 1$ that characterize the Hall Cases (i) and (iii), but, as can be seen by combining the first two parameter constraints reproduced in the first row of Table \ref{table:boundary}, it can be formally inferred (within the framework of the hydrostatic analysis) to apply also in the other Hall sub-regimes."1152 The illustrative solutions for the Cases (ii) and (iv) in Fig., The illustrative solutions for the Cases (ii) and (iv) in Fig.1153 3 salisfv this inequality. as verified. explicitly. in. Table ," \ref{fig:iii-iv} satisfy this inequality, as verified explicitly in Table \ref{table:boundary}."1154One can further test. the foregoing result. in these regimes by holding Ag. σι and σω fixed (as was done above) and decreasing [eg|/e (or. equivalently. increasing σι σι} until (i£ Ag is sulliciently small) Yu declines to a value <]l where. according to the hycrostatic analysis. viable solutions cease to exist.," One can further test the foregoing result in these sub-regimes by holding $\lao$, $\tilde\sigma_\perp$ and $\sigpar$ fixed (as was done above) and decreasing $|\sigH|/\tilde\sigma_\perp$ (or, equivalently, increasing $\sigP/\tilde\sigma_\perp$ ) until (if $\lao$ is sufficiently small) $\upo$ declines to a value $<1$ where, according to the hydrostatic analysis, viable solutions cease to exist."1155 Such calculations are presented in lig. G..," Such calculations are presented in Fig. \ref{fig:eta_sh},"1156 where we plot Ἰση[ίσι as a function of Yo for representative solutions in these two sub-regimes., where we plot $|\sigH|/\tilde\sigma_\perp$ as a function of $\upo$ for representative solutions in these two sub-regimes.1157 IH is seen that all viable solutions are indeed. restricted to the region Yoc1. validating the above prediction.," It is seen that all viable solutions are indeed restricted to the region $\upo > 1$, validating the above prediction."1158 As was noted in Section. L.2.1. the Obm ancl Pedersen conductivities are always positive. even under a global reversal of the field. polarity. since 8o is not a function of the magnetic field strength and a. only contains magnetic terms that scale as Bo. Don," As was noted in Section I.2.1, the Ohm and Pedersen conductivities are always positive, even under a global reversal of the field polarity, since $\sigpar$ is not a function of the magnetic field strength and $\sigP$ only contains magnetic terms that scale as $B^{2}$."1159sThe Hall conductivity.MEN however. has an overall lincar dependence. on DIB]sgnD] and can thus assume both positive and negative. values depending on the direction of the vertical field component.," The Hall conductivity, however, has an overall linear dependence on $B \equiv1160|\mathbf{B}|\, sgn\{B_z\}$ and can thus assume both positive and negative values depending on the direction of the vertical field component."1161 The dependence of ση on the magnetic field. polarity was shown in Paper LE t0 alfect both the extent of the parameter ranges where viable winc-clriving disc solutions can exist in the Hall domain and the properties of these solutions., The dependence of $\sigH$ on the magnetic field polarity was shown in Paper I to affect both the extent of the parameter ranges where viable wind-driving disc solutions can exist in the Hall domain and the properties of these solutions.1162 We now brielle summarize these results. which were obtained in the hyclrostatic approximation assuming an ionelectron plasma.," We now briefly summarize these results, which were obtained in the hydrostatic approximation assuming an ion–electron plasma."1163 The range of values of the parameter 3—ανω for viable solutions was found to be restricted by the following two conditions (see Section L6): The above two constraints together imply ὃν combining this inequality with the classification criterion Yo/|3|>1/2 that distinguishes the Hall. regimes (1) ancl (ii) (see Section 4.1)). one obtains Equation (87)) prediets that. although there could. be both positive- and negative-polaritv solutions in the Llall sub-regimes (1) and (iii). no viable solutions should. exist in these cases when 23 decreases below Τον.," The range of values of the parameter $\beta \equiv 1/\beta_{\rm i0}$ for viable solutions was found to be restricted by the following two conditions (see Section I.6): The above two constraints together imply By combining this inequality with the classification criterion $\upo/|\beta| > 1/2$ that distinguishes the Hall sub-regimes (i) and (iii) (see Section \ref{subsec:illus}) ), one obtains Equation \ref{eq:beta_constraint1}) ) predicts that, although there could be both positive- and negative-polarity solutions in the Hall sub-regimes (i) and (iii), no viable solutions should exist in these cases when $\beta$ decreases below $-\upo/2$."1164" This prediction of the hyelrostatic analysis is examined in Fig. Ἐν,"," This prediction of the hydrostatic analysis is examined in Fig. \ref{fig:5_26},"1165 which plots f«.. the normalized density at the sonic point. as a function of the coupling parameter Yu for C'ase-(i) solutions derived using the indicated model. parameters and. corresponding to both positive anc negative values of 3xay (see equation S4)).," which plots $\tilde \rho_{\rm s}$, the normalized density at the sonic point, as a function of the coupling parameter $\upo$ for Case-(i) solutions derived using the indicated model parameters and corresponding to both positive and negative values of $\beta \propto \sigH$ (see equation \ref{eq:betio}) )."1166 Since the magnitudes of all the conductivity tensor components are held. constant. all the solutions are characterized by the same absolutevalue of the ion Hall parameter (||= 1/145).," Since the magnitudes of all the conductivity tensor components are held constant, all the solutions are characterized by the same absolutevalue of the ion Hall parameter $\betiabs =11671/1.45$ )."1168 The figure verifies that viable, The figure verifies that viable1169All three observing runs terminated. close to the telescope elevation limit but the source exhibits significant. variability which is not apparent for the plotted comparison star or for others not shown here.,All three observing runs terminated close to the telescope elevation limit but the source exhibits significant variability which is not apparent for the plotted comparison star or for others not shown here.1170 The light curve for the last four hours of the night of Mav 2 (11JD. 397) shows clear indications of modulation at or near the orbital period. together with slow changes in the average magnitude., The light curve for the last four hours of the night of May 2 (HJD 397) shows clear indications of modulation at or near the orbital period together with slow changes in the average magnitude.1171 We corrected the data commencing from ILJD 396.92 (— 22h UT) for these slow changes using a second order polynomial and. used the OQ method (Warner Robinson. 1972) to search the cle-trenclecl data for periodicitv in the range 0.01 to 0.05 days.," We corrected the data commencing from HJD 396.92 $\sim 22$ h UT) for these slow changes using a second order polynomial and used the Q method (Warner Robinson, 1972) to search the de-trended data for periodicity in the range 0.01 to 0.05 days."1172 There is à strong single peak corresponding to a period of 0.030320.001 davs., There is a strong single peak corresponding to a period of $0.030 \pm 0.001$ days.1173 This is consistent with the orbital period. of 0.030263 clays discovered by Galloway et al. (, This is consistent with the orbital period of 0.030263 days discovered by Galloway et al. (117420025).,2002b).1175 In Fig., In Fig.1176 5., 5.1177 we plot the corrected. light curve folded at the X-ray period and ephemeris where phase zero is defined as the time when the companion is at its greatest. distance from the observer Le. it lies bevond the neutron star., we plot the corrected light curve folded at the X-ray period and ephemeris where phase zero is defined as the time when the companion is at its greatest distance from the observer i.e. it lies beyond the neutron star.1178 The vertical error bars are those generated by the DoPIIOTT photometry which appear to be slightly over-estimated., The vertical error bars are those generated by the DoPHOT photometry which appear to be slightly over-estimated.1179 To clarify this we repeated. the phase folding process for the nearby comparison star. using the same X-ray 'phemeris. so that the real scatter for a ‘constant’ source can be seen.," To clarify this we repeated the phase folding process for the nearby comparison star, using the same X-ray ephemeris, so that the real scatter for a 'constant' source can be seen."1180 These points are plotted in the lower part of Fig., These points are plotted in the lower part of Fig.1181 5 and have a slightly cilferent offset to that used in Fig., 5 and have a slightly different offset to that used in Fig.1182 4., 4.1183 The formal DoPIIO'T average error for these 22 points is 0.021 mag but they have a clo scatter of only 0.017 mag so the NTE J0929314 error bars in Eys, The formal DoPHOT average error for these 22 points is 0.021 mag but they have a $\pm 1 \sigma$ scatter of only 0.017 mag so the XTE J0929–314 error bars in Figs.1184 4 5 are probably ~ 20 per cent too large., 4 5 are probably $\sim$ 20 per cent too large.1185 The modulation is approximately sinusoidal with amplitude 0.09 magnitudes peak to peak and maximum at phase 0.19d:0.05., The modulation is approximately sinusoidal with amplitude $\sim 0.09$ magnitudes peak to peak and maximum at phase $0.19 \pm 0.05$.1186 Hence the modulation is unlikely to be due to X-ray heating of the companion as this would have maximum light at phase zero., Hence the modulation is unlikely to be due to X-ray heating of the companion as this would have maximum light at phase zero.1187 In this respect NTL 0029314 dillers from SAN 1505.3658 in which the orbital »eriod. optical modulation had a maximum at phase zero (Ciles et ab.," In this respect XTE J0929--314 differs from SAX J1808.4–3658 in which the orbital period optical modulation had a maximum at phase zero (Giles et al.,"1188 1999)., 1999).1189 Nor is it likely that the modulation is due to emission from a hot spot on the disc since this would lave a maximum at a phase between 0.31.5., Nor is it likely that the modulation is due to emission from a hot spot on the disc since this would have a maximum at a phase between $\sim 0.3 - 0.5$.1190 Perhaps th. processes contribute., Perhaps both processes contribute.1191 Phe first PCA X-ray observation (obs., The first PCA X-ray observation (obs.1192 01-00) occurred during the time we detected an orbital period modulation on he night of 2002 May 2 (11D 397) and its duration is shown in the top panel of Fie., 01-00) occurred during the time we detected an orbital period modulation on the night of 2002 May 2 (HJD 397) and its duration is shown in the top panel of Fig.1193 4., 4.1194 However. this observation consists of crossed slows to determine the X-ray source position and is thus not suitable for modulation analysis.," However, this observation consists of crossed slews to determine the X-ray source position and is thus not suitable for modulation analysis."1195 As noted earlier. no X-ray amplitude modulation was reported in any of the many following PCA observations (Juett et al.," As noted earlier, no X-ray amplitude modulation was reported in any of the many following PCA observations (Juett et al."1196 2003)., 2003).1197 Vhere are many occasions within Tables 2 3 where we have values on the same night but some caution is required. in combining these into broadband spectra due to the variability detected on several nights., There are many occasions within Tables 2 3 where we have values on the same night but some caution is required in combining these into broadband spectra due to the variability detected on several nights.1198 In most instances the dillerent colour averages for each night are derived from. one or more measurements taken in a time interval of — 1 hour., In most instances the different colour averages for each night are derived from one or more measurements taken in a time interval of $\sim$ 1 hour.1199 It should be clearly noted that the central wavelength ancl bandwidth for the filters dillers between the Cousins and older Johnston svstems and this can allect the, It should be clearly noted that the central wavelength and bandwidth for the filters differs between the Cousins and older Johnston systems and this can affect the1200of masses. making possible to generate a M—Z relation.,"of masses, making possible to generate a $M-Z$ relation."

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