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2000: Lou 2002: Lou Fan 2002: Lou Shen 2003: Lou Zou 2004: Lou Wu 2004). | 2000; Lou 2002; Lou Fan 2002; Lou Shen 2003; Lou Zou 2004; Lou Wu 2004). |
Specifically. we write | Specifically, we write |
criteria have exclucled all sources with radio spectral index between 151. Alllz and 1.4 11 flatter than 1. or with radio angular size larger than 13 aresec. | criteria have excluded all sources with radio spectral index between 151 MHz and 1.4 GHz flatter than 1, or with radio angular size larger than 13 arcsec. |
The final sample consists of GS objects over an area of skv of 0.421 sr. and is statistically complete at an angular size limit. of 6<ll aresec. | The final sample consists of 68 objects over an area of sky of 0.421 sr, and is statistically complete at an angular size limit of $\theta < 11$ arcsec. |
Full details of how the 6C**. sample was selected can be found in Cruz et al. ( | Full details of how the 6C** sample was selected can be found in Cruz et al. ( |
2006. hereafter Paper 1). | 2006, hereafter Paper I). |
The selection eriteria just described are similar to those of the 6C* sample (Blundell et al. | The selection criteria just described are similar to those of the 6C* sample (Blundell et al. |
1998: Jarvis οἱ al. | 1998; Jarvis et al. |
2001a.b). which was one of the samples usec by Jarvis et al. ( | 2001a,b), which was one of the samples used by Jarvis et al. ( |
200160) to constrain the co-moving space density of Iow-frequeney. selected radio sources. | 2001c) to constrain the co-moving space density of low-frequency selected radio sources. |
The 6C* sample was crucial in that study in sampling to high redshift (2— 4.4). | The 6C* sample was crucial in that study in sampling to high redshift $z \simeq
4.4$ ). |
The οςἘν sample. being larger (cf. | The 6C** sample, being larger (cf. |
0.13 sr) and deeper (cf. | 0.13 sr) and deeper (cf. |
0.06<Sys)2.0 Jv) than 6C. aims to improve on the small-numboer statistics limitation of this previous work. anc ultimately to extend it to higher redshifts (22 5). | $0.96 \leq S_{151} \leq 2.0$ Jy) than 6C*, aims to improve on the small-number statistics limitation of this previous work, and ultimately to extend it to higher redshifts $z \,\,\gtsim\,\, 5$ ). |
Deep imaging follow-up with UETL/UIST on ΕΙΗΕ. NIRD on Gemini. anc NIRC on Weck provide photometry for all members of the ος* sample (Paper I). | Deep imaging follow-up with UFTI/UIST on UKIRT, NIRI on Gemini and NIRC on Keck provided photometry for all members of the 6C** sample (Paper I). |
Optical spectroscopy provided: redshifts for 32 per cent of the sources (Paper Land references therein). | Optical spectroscopy provided redshifts for 32 per cent of the sources (Paper I and references therein). |
A summary of key observational information is given in Table | A summary of key observational information is given in Table \ref{tab:6cssummary1_median}. |
In this paper we describe a. method. of. redshif estimation based on the Az diagram of radio galaxies. | In this paper we describe a method of redshift estimation based on the $K-z$ diagram of radio galaxies. |
‘This is presented in Section ?77.. | This is presented in Section \ref{sec:estimation}. |
In Section 3.. we use the complete set of magnitudes of the 6€ sample to estimate redshifts for all its optically identified members. | In Section \ref{sec:estimates}, we use the complete set of magnitudes of the 6C** sample to estimate redshifts for all its optically identified members. |
These are compared to spectroscopic redshifts in Section 4.. in order to assess the robustness of the method. | These are compared to spectroscopic redshifts in Section \ref{sec:comparison}, in order to assess the robustness of the method. |
The resulting estimated redshift cüstribution is discussed in Section 5.. | The resulting estimated redshift distribution is discussed in Section \ref{sec:est-zdist}. |
In Section 6 we summarize the model radio luminosity function (IRLE) of Jarvis et al. ( | In Section \ref{sec:JarvisRLF} we summarize the model radio luminosity function (RLF) of Jarvis et al. ( |
20010). | 2001c). |
This is the most relevant model to compare our data to. because ib. takes into account the selection ellects. of the 6C7. sample. | This is the most relevant model to compare our data to, because it takes into account the selection effects of the 6C* sample. |
In. Section. 77. we compare the redshift distribution (including spectroscopic and discuss the evolution of the co-moving space density of the most radio luminous. low-frequencey selected: sources. | In Section \ref{sec:RLF} we compare the redshift distribution (including spectroscopic redshifts) of the 6C** sample with the model predictions, and discuss the evolution of the co-moving space density of the most radio luminous, low-frequency selected sources. |
Unless otherwise stated. we assume throughout that ff=τοkms*\Ipe 1 On,=03 and O4—0.7. | Unless otherwise stated, we assume throughout that $H_{0}=70~ {\rm km~s^{-1}Mpc^{-1}}$ , $\Omega_ {\mathrm
M} = 0.3$ and $\Omega_ {\Lambda} = 0.7$. |
The convention used. for radio spectral index is S,xν"o where S, is the IIux-density at frequency v. | The convention used for radio spectral index is $S_{\nu} \propto \nu^{-\alpha}$, where $S_{\nu}$ is the flux-density at frequency $\nu$. |
Infrarecd-photometry provides a method of. recdshif estimation by utilising the tightness of the relation between magnitude and recshift. which is characteristic of he near-infrared. Hubble. clagran of radio galaxies (Lilly Loneair 1984: Eales et al. | Infrared-photometry provides a method of redshift estimation by utilising the tightness of the relation between magnitude and redshift, which is characteristic of the near-infrared Hubble diagram of radio galaxies (Lilly Longair 1984; Eales et al. |
1997: Jarvis ct al. | 1997; Jarvis et al. |
2001a: De Breuck et al. | 2001a; De Breuck et al. |
2002. Willott et al. | 2002, Willott et al. |
2003). | 2003). |
Phe physica xls for the A> relation is not well understood. | The physical basis for the $K-z$ relation is not well understood. |
A ow redshifts. the emission is dominated by the ol stellar population in the host galaxy: at high redshifts. samples rest-[ramoe optical wavelengths. where the star ormation history can have a significant. οσοι. | At low redshifts, the emission is dominated by the old stellar population in the host galaxy; at high redshifts, samples rest-frame optical wavelengths, where the star formation history can have a significant effect. |
Non-stellar contamination to the light. in the form of reddened quasar light and/or narrow emission lines. also contributes to the difficulty of interpreting the dyz diagram of radio ealaxies. particularly at high redshifts (223). | Non-stellar contamination to the light, in the form of reddened quasar light and/or narrow emission lines, also contributes to the difficulty of interpreting the $K-z$ diagram of radio galaxies, particularly at high redshifts $z >
3$ ). |
Despite these caveats. the ἐνz diagram is still of interest as a tool for recshift estimation. | Despite these caveats, the $K-z$ diagram is still of interest as a tool for redshift estimation. |
ltedshift estimates based on the Az diagram have seenerally been obtained. by simple application of the empirical A> relation (e.g. Dunlop Peacock 1990). | Redshift estimates based on the $K-z$ diagram have generally been obtained by simple application of the empirical $K-z$ relation (e.g. Dunlop Peacock 1990). |
llowever. the significant amount of scatter around this relation requires the use of à more sophisticated: method one which takes into account all the available information in the clagram. and also which allows us to characterise the uncertainty on the output redshift’ estimates. | However, the significant amount of scatter around this relation requires the use of a more sophisticated method – one which takes into account all the available information in the diagram, and also which allows us to characterise the uncertainty on the output redshift estimates. |
With these requirements in mind the following approach is adopted: (i) we use Monte Carlo simulations to generate a statistical universe of svnthetic realisations of the Ας diagram. based on à model of its underlving galaxy distribution. and (ii) we extract individual photometric redshift probability density functions from this simulated. population. | With these requirements in mind the following approach is adopted: (i) we use Monte Carlo simulations to generate a statistical universe of synthetic realisations of the $K-z $ diagram, based on a model of its underlying galaxy distribution, and (ii) we extract individual photometric redshift probability density functions from this simulated population. |
The most well defined. A2 diagram [or radio galaxies currently available is the one obtained by Willott et. al. ( | The most well defined $K-z$ diagram for radio galaxies currently available is the one obtained by Willott et al. ( |
2003) from a combined dataset of the radio galaxies [roni the ὃςπι (Laine. Riley Longair 1983). GCE (Eales οἱ al. | 2003) from a combined dataset of the radio galaxies from the 3CRR (Laing, Riley Longair 1983), 6CE (Eales et al. |
1997: Rawlines. Eales Lacy 2001). 6C (Jarvis et al. | 1997; Rawlings, Eales Lacy 2001), 6C* (Jarvis et al. |
2001a.b) and TORS (Lacy ct al. | 2001a,b) and 7CRS (Lacy et al. |
2000. Willott ct al. | 2000, Willott et al. |
2003) Ilux-lipited. samples. | 2003) flux-limited samples. |
LO is based. on a total of 204 racio ealaxies with redshifts ranging from 0.05 to 4.4. and its Az relation is well fitted by a second-order polynomial between dy-magnitude anc log),2 (Willott et al. | It is based on a total of 204 radio galaxies with redshifts ranging from 0.05 to 4.4, and its $K-z$ relation is well fitted by a second-order polynomial between $K$ -magnitude and $\log_{10} z$ (Willott et al. |
2003): The main advantage of using this fv2 diagram is that it has been obtained from completely identified samples with close to complete. or complete redshift) information. | 2003): The main advantage of using this $K-z$ diagram is that it has been obtained from completely identified samples with close to complete, or complete redshift information. |
This ensures the absence of significant biases in terms of sources with the weakest lines being missed. because their redshifts are cillicult to obtain. | This ensures the absence of significant biases in terms of sources with the weakest lines being missed because their redshifts are difficult to obtain. |
Another advantage is that these samples have been selected at a similar radio-frequency to 6C**, with progressively fainter Dux-density limits. | Another advantage is that these samples have been selected at a similar radio-frequency to 6C**, with progressively fainter flux-density limits. |
The brightest sample is 83CRAR selected at MMIETZ. with a flux-density limit of στ> 10.9.J.]v (8124c 12.4]v. assuming a spectral index of 0.8): the faintest sample is TORS selected at MMLbIz. with a Ilux-densitv limit of Sy.)2 0.5.JJv. | The brightest sample is 3CRR selected at MHz, with a flux-density limit of $S_{178}
\geq 10.9$ Jy $S_{151} \geq 12.4$ Jy, assuming a spectral index of 0.8); the faintest sample is 7CRS selected at MHz, with a flux-density limit of $S_{151} \geq 0.5$ Jy. |
The intermediate samples are GCE and 6€7 selected. at MAIIz. with llux-density limits of 2.0<Sys)x3.93 JJv and O.OG<<Sys)2.00 JJ. respectively. | The intermediate samples are 6CE and 6C* selected at MHz, with flux-density limits of $2.0
\leq S_{151} \leq 3.93$ Jy and $0.96 \leq S_{151} \leq 2.00$ Jy, respectively. |
This. results in a wide range in radio luminosity. which has made the investigation of the radio-Iuminosity dependence of the Az relation possible in an unprecedented. way. | This results in a wide range in radio luminosity, which has made the investigation of the radio-luminosity dependence of the $K-z$ relation possible in an unprecedented way. |
The correlation between luminosity and racio luminosity has been one of the major worries with redshift estimates based on the A zorelation. | The correlation between luminosity and radio luminosity has been one of the major worries with redshift estimates based on the $K-z$ relation. |
Willott ct al. ( | Willott et al. ( |
2003). found a statistically significant mean luminosity dillerence between the ὃςRR. and. TORS radio galaxies of 0.55 mae in A-band. over all redshilts. | 2003) found a statistically significant mean luminosity difference between the 3CRR and 7CRS radio galaxies of 0.55 mag in $K$ -band, over all redshifts. |
Llowever. the 6C radio galaxies were found to cdiller on average from the 3€ ones by only 2:0.3 mag. which is much smaller than the value (20.6 mag) reported previously | However, the 6C radio galaxies were found to differ on average from the 3C ones by only $\simeq 0.3$ mag, which is much smaller than the value $\simeq
0.6$ mag) reported previously |
derived. | derived. |
These equations (38)) surpass in consistency the usual Fokker-Planck equations (49)) — (52)) — (532). | These equations \ref{LAequation}) ) surpass in consistency the usual Fokker-Planck equations \ref{formeFP}) ) – \ref{Asanseffetscoll}) ) – \ref{Bsanseffetscoll}) ). |
The latter are unsatisfactory [rom a principle point of view. being local and. non-collective. | The latter are unsatisfactory from a principle point of view, being local and non-collective. |
ὃν contrast. the proposed. equations Lully account for the system's inhomogencity ancl for the collective gravitational dressing of the colliding particles. | By contrast, the proposed equations fully account for the system's inhomogeneity and for the collective gravitational dressing of the colliding particles. |
Equations (38)) describe the evolution of distribution functions in action and angle space. which is possible when the hamiltonian associated with the average potential is integrable. | Equations \ref{LAequation}) ) describe the evolution of distribution functions in action and angle space, which is possible when the hamiltonian associated with the average potential is integrable. |
Physically. these equations describe the evolution of the distribution functions in action space as a result of the weak eravitational noise caused. by. the cliscreteness of the particles. dressed. with the polarization clouds that their own gravity induces in the system. | Physically, these equations describe the evolution of the distribution functions in action space as a result of the weak gravitational noise caused by the discreteness of the particles, dressed with the polarization clouds that their own gravity induces in the system. |
This gravitational polarization is accounted for in equation (38)) in à manner that is fully consistent with the distribution functions. as they are at the moment. | This gravitational polarization is accounted for in equation \ref{LAequation}) ) in a manner that is fully consistent with the distribution functions, as they are at the moment. |
Equation (38)) is the sum. of a second. order derivative term with respect to actions and of a first order one. | Equation \ref{LAequation}) ) is the sum of a second order derivative term with respect to actions and of a first order one. |
It therefore basically is of the Fokker-Planck type. although it is definitely simpler in the form of expression (38)). | It therefore basically is of the Fokker-Planck type, although it is definitely simpler in the form of expression \ref{LAequation}) ). |
The dillusion coelIicient involved depends on the l-bocly distributions theniselves. in particular through the factor |D|> which represents the elect of the dressing of the colliding particles hy the gravitational polarization induced around them by their own inlluence. | The diffusion coefficient involved depends on the 1-body distributions themselves, in particular through the factor $\mid \!{\cal{D}}\!\mid^{-2}$ which represents the effect of the dressing of the colliding particles by the gravitational polarization induced around them by their own influence. |
Unlike in electrical plasmas. the polarization dressing in sell-gravitational systems does not cause any screening of the interaction. which remains elfective even between distant. particles. | Unlike in electrical plasmas, the polarization dressing in self-gravitational systems does not cause any screening of the interaction, which remains effective even between distant particles. |
The mutual distance of such particles is limited only. by the finite size of the system. | The mutual distance of such particles is limited only by the finite size of the system. |
Were the gravitational inlluence of particles on their surrounding to be neglected. the response matrix 5 (equation. (34))) would. reduce to unity and the coellicients of the corresponding Fokker-Planck kinetic. equation would simply be averages by the distribution functions of functions of velocity. as in equations (52))— (53)). | Were the gravitational influence of particles on their surrounding to be neglected, the response matrix $\varepsilon$ (equation \ref{epsilonalphabeta}) )) would reduce to unity and the coefficients of the corresponding Fokker-Planck kinetic equation would simply be averages by the distribution functions of functions of velocity, as in equations \ref{Asanseffetscoll}) ) – \ref{Bsanseffetscoll}) ). |
]t is apparent from the developments of appendix A. which lead to equation (38)). that the k component in angle Fourier space of the gravitational polarization response given to a particle has frequency w=k-Q. | It is apparent from the developments of appendix \ref{grossesmagouilles}, which lead to equation \ref{LAequation}) ), that the ${\mathbf{k}}$ component in angle Fourier space of the gravitational polarization response given to a particle has frequency $\omega = {\mathbf{k}}\! \cdot \! {\mathbf{\Omega}}$. |
This means that the polarization cloud. which accompanies a particle forms a structure in angle space which vary as wo£6: it corotates in angle with that particle. | This means that the polarization cloud which accompanies a particle forms a structure in angle space which vary as ${\mathbf{w}} - {\mathbf{\Omega}}t$: it corotates in angle with that particle. |
The presence of the Dirac function 0(k;:€,Κυ£22) in equation (38)) indicates that particles interact. resonantly. | The presence of the Dirac function $\delta({\mathbf{k}}_1\!\cdot {\mathbf{\Omega}}_1 - {\mathbf{k}}_2\!\cdot {\mathbf{\Omega}}_2)$ in equation \ref{LAequation}) ) indicates that particles interact resonantly. |
This certainly is an important physical property of remote interactions. for which the components of the angle wave vectors Κι and k» must be small. | This certainly is an important physical property of remote interactions, for which the components of the angle wave vectors ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ must be small. |
For closer encounters. the modulus of these wave vectors is larger and the resonance condition ky,-Q)=ko:Qe becomes less selective. being more easily. satisficc. | For closer encounters, the modulus of these wave vectors is larger and the resonance condition ${\mathbf{k}}_1\cdot {\mathbf{\Omega}}_1 = {\mathbf{k}}_2\cdot {\mathbf{\Omega}}_2$ becomes less selective, being more easily satisfied. |
The correlation function has been calculated on the basis of a linearized theory. which is justified by the weakness of the average interactions in this many-body system. | The correlation function has been calculated on the basis of a linearized theory, which is justified by the weakness of the average interactions in this many-body system. |
This means that the trajectories of the particles during the collision are regarded as being the unperturbed trajectories. | This means that the trajectories of the particles during the collision are regarded as being the unperturbed trajectories. |
Similarly. the gravitational polarization cloud around any one of the colliding particles is caleulated as if the partner in the collision were not present: equation (38)) is still à weak collision approximation. | Similarly, the gravitational polarization cloud around any one of the colliding particles is calculated as if the partner in the collision were not present: equation \ref{LAequation}) ) is still a weak collision approximation. |
A cutolf at small impact. parameters is therefore needed to account for the rare strong collisions. | A cutoff at small impact parameters is therefore needed to account for the rare strong collisions. |
Equation (38)) takes full account of the inhomogeneity of the system. which is embodied. in the dependence of the distribution functions on the actions J's. Lt requires no artificial cutoff at large impact parameters. | Equation \ref{LAequation}) ) takes full account of the inhomogeneity of the system, which is embodied in the dependence of the distribution functions on the actions $\mathbf{J}$ 's. It requires no artificial cutoff at large impact parameters. |
The details of the trajectories followed by the particles in the present gravitational potential are also fully accounted for. being implicit in the relations which link the angle anc action variables to the position and momentum ones. | The details of the trajectories followed by the particles in the present gravitational potential are also fully accounted for, being implicit in the relations which link the angle and action variables to the position and momentum ones. |
These relations depend on the actual global gravitational potential of the svstem. which slowly evolves in time together with the distribution functions. | These relations depend on the actual global gravitational potential of the system, which slowly evolves in time together with the distribution functions. |
The clensity-potential basis functions (C7(r) are choosen at the beginningὃνe of the calculation once and for all. but their angleo Fourier transforms csk(J). which depend on the actual trajectories of the particles. changeD with time because the trajectoryJ ofa particle of given actions slowly evolves with the general potential of the svstem as the relaxation proceeds. | The density-potential basis functions $\psi^{\alpha}({\mathbf{r}})$ are choosen at the beginning of the calculation once and for all, but their angle Fourier transforms $\psi^{\alpha}_{\mathbf{k}}({\mathbf{J}})$, which depend on the actual trajectories of the particles, change with time because the trajectory of a particle of given actions slowly evolves with the general potential of the system as the relaxation proceeds. |
As long as it sullers no collision. a given particle keeps its vector J fixed because the actions are acliabatic invariants. | As long as it suffers no collision, a given particle keeps its vector ${\mathbf{J}}$ fixed because the actions are adiabatic invariants. |
Collisions. however. cause a secular evolution of the functions f£"(J). which is exactly what equation (38)) describes. | Collisions, however, cause a secular evolution of the functions $f^a({\mathbf{J}})$, which is exactly what equation \ref{LAequation}) ) describes. |
The description of particle motions is mace simple by the use of action and angle variables. | The description of particle motions is made simple by the use of action and angle variables. |
Their complexity is embocied in the supposedly known relation between position and momentum variables and action and angle variables. | Their complexity is embodied in the supposedly known relation between position and momentum variables and action and angle variables. |
The usefulness | The usefulness |
positions. | positions. |
The back-traced initial conditions are listed in the third column of table | The back-traced initial conditions are listed in the third column of table (3). |
Incidentally, these initial conditions are close to a (3).multi-resonant configuration where Saturn Uranus and Uranus Neptune are both in 4:3 MMR’s. | Incidentally, these initial conditions are close to a multi-resonant configuration where Saturn Uranus and Uranus Neptune are both in 4:3 MMR's. |
Recall that this initial condition is indeed one of the setups that consistently exhibit scattering. | Recall that this initial condition is indeed one of the setups that consistently exhibit scattering. |
However, given the similarities in dynamical evolutions among the successful initial conditions of this family, at this level of accuracy, it is probably safe to say that all four of them are compatible with the classical Nice model results. | However, given the similarities in dynamical evolutions among the successful initial conditions of this family, at this level of accuracy, it is probably safe to say that all four of them are compatible with the classical Nice model results. |
Let us now consider the final family of initial conditions, listed in table where Jupiter and Saturn are initially in a 2:1 MMR. | Let us now consider the final family of initial conditions, listed in table (1), where Jupiter and Saturn are initially in a 2:1 MMR. |
(1),Unlike the scenario of the classical Nice model (Tsiaganis et al. | Unlike the scenario of the classical Nice model (Tsiaganis et al. |
2005), there are no major resonances to cross for Jupiter and Saturn between the 2:1 and the 5:2 MMR’s. | 2005), there are no major resonances to cross for Jupiter and Saturn between the 2:1 and the 5:2 MMR's. |
Consequently, a different mechanism, involving different resonances, is needed to create the instability. | Consequently, a different mechanism, involving different resonances, is needed to create the instability. |
Thommes et al. ( | Thommes et al. ( |
2008) considered the dynamical evolution of a system where Jupiter Saturn are in a 2:1 MMR, Saturn Uranus are in a 3:2 MMR, and Uranus Neptune are in a 4:3 MMR. | 2008) considered the dynamical evolution of a system where Jupiter Saturn are in a 2:1 MMR, Saturn Uranus are in a 3:2 MMR, and Uranus Neptune are in a 4:3 MMR. |
In such a system, the instability is triggered by Uranus and Neptune crossing a 7:5 MMR. | In such a system, the instability is triggered by Uranus and Neptune crossing a 7:5 MMR. |
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