At best, galaxy-galaxy lensing and satellite dynamics have the potential to constrain the dependence of the line of sight velocity dispersion on the projected radius, v(rp). Determining v(rp) has proven to be quite a challenge to galaxy-galaxy lensing studies, in large part because the shear profiles of NFW lenses and isothermal sphere lenses are not dramatically different, except on the very smallest (r < rs) and very largest (r > rvir) scales . To date, only one tentative measurement of v(rp) has been made from observations of galaxy-galaxy lensing . Kleinheinrich et al.  modeled the lens galaxies in the COMBO-17 survey as singular isothermal spheres with velocity dispersions that scaled with luminosity as
where v* is the line of sight velocity dispersion of the halo of an L* galaxy. Kleinheinrich et al.  fixed to be 0.35 and determined best-fitting values of v* for projected radii in the range 20 h-1 kpc < rp < rmax. When they considered all lenses in their sample, Kleinheinrich et al.  found v* ~ 139 km sec-1for rmax = 50 h-1 kpc, v* ~ 164 km sec-1for rmax = 150 h-1 kpc, and v* ~ 123 km sec-1for rmax = 500 h-1 kpc. This suggests a velocity dispersion profile that rises at small radii, reaches a maximum, then decreases at large radii. However, the formal error bars on these measurements show that all of these values of v* agree to within one to two standard deviations. In addition, it should be kept in mind that each of these measurements of v* is not independent (as they would be if a differential measurement of v*(rp) were made), so the data points and their error bars are all correlated with one another.
Considerably stronger constraints on the dependence of the halo velocity dispersion with projected radius have come from the most recent investigations of the motions of satellites about host galaxies. In particular, both Prada et al.  and Brainerd  have measured decreasing velocity dispersion profiles for the satellites of host galaxies in the SDSS and 2dFGRS, respectively. Although they used different data sets and different host-satellite selection criteria, both Prada et al.  and Brainerd  used the same technique to make measurements of the velocity dispersion profiles. That is, the distribution of velocity differences, N(|dv|), for satellites found within projected radii of rmin < rp < rmax was modeled as a combination of a Gaussian and an offset due to interlopers. In both studies, the interloper fraction was determined separately for each of the independent radial bins.
Prior to correcting for the contamination of interlopers, Prada et al.  found a velocity dispersion profile, v(rp), that increased with projected radius. After the removal of the interlopers, however, Prada et al.  found decreasing velocity dispersion profiles in both cases. The corresponding velocity dispersion profiles are shown in Figure 5. Moreover, their corrected velocity dispersion profiles were fitted well by the velocity dispersion profiles of NFW halos with virial masses of 1.5 × 1012 M (hosts with absolute magnitudes -19.5 < MB < - 20.5) and 6 × 1012 M (hosts with absolute magnitudes -20.5 < MB < - 21.5). Since Prada et al.  adopted a value of h = 0.7 and since the absolute magnitude of an L* galaxy is MB* ~ - 19.5, these results suggest that the virial mass of the halo of an L* galaxy is 10 × 1011 h-1 M.
Figure 5. Velocity dispersion profiles for satellites of SDSS host galaxies . Circles: host galaxies with -20.5 < MB < - 21.5, Squares: host galaxies with -19.5 < MB < - 20.5. Left panel: "raw" velocity dispersion profiles prior to correction for contamination by interlopers. Right panel: velocity dispersion profiles after correction for contamination by interlopers. After correction for interlopers, v(rp) for the satellites of the fainter hosts is consistent with the expectations for an NFW halo with M200 = 1.5 × 1012 M, and v(rp) for the satellites of the brighter hosts is consistent with the expectations for an NFW halo with M200 = 6 × 1012 M. Here h = 0.7 has been adopted.
Brainerd  selected hosts and satellites from the final data release of the 2dFGRS using criteria identical to those of Sample 3 in Prada et al. . In addition, she used these same criteria to select hosts and satellites from the present epoch galaxy catalogs of the flat, -dominated the GIF simulation . This is a publicly-available simulation which includes semi-analytic galaxy formation in a CDM universe. Brainerd  restricted her analysis to hosts with luminosities in the range 0.5 L* L 5.5 L*, and found a roughly similar number of hosts and satellites in both the 2dFGRS (1345 hosts, 2475 satellites) and the GIF simulation (~ 1200 hosts, ~ 4100 satellites, depending upon the viewing angle). Like Prada et al. , Brainerd  obtained a decreasing velocity dispersion profile for the satellites of the 2dFGRS galaxies once the effects of interlopers were removed. In addition, excellent agreement between v(rp) for the 2dFGRS galaxies and v(rp) for the GIF galaxies was found, showing consistency between the motions of satellites in the 2dFGRS and the expectations of a -dominated CDM universe. See Figure 6.
Figure 6. Velocity dispersion profiles for satellites in the final data release of the 2dFGRS and the flat, -dominated GIF simulation . Here h = 0.7 has been adopted.
Further, Brainerd  divided her sample of 2dFGRS host galaxies into thirds based upon the spectral index parameter, , and computed the dependence of the velocity dispersion profile on host spectral type. The subsamples corresponded to hosts which are expected to have morphologies that are approximately: (i) E/S0, (ii) Sa, and (iii) Sb/Scd. The median luminosities of the hosts in the subsamples were all fairly similar: (i) 2.64 LbJ*, (ii) 2.25 LbJ*, and (iii) 2.11 LbJ*. The velocity dispersion profiles of all three samples decreased with radius and, moreover, v(rp) was found to have a much higher amplitude and steeper decline for the satellites of early-type hosts than it did for the satellites of late-type hosts. See Figure 7. Although there is some difference in the median luminosities of the hosts in the subsamples, the difference is too small to have a significant effect on the velocity dispersion profiles. Therefore, the results of Brainerd  seem to indicate that early-type galaxies have deeper potential wells (and hence more massive halos) than late-type galaxies.
Figure 7. Velocity dispersion profiles for satellites in the final data release of the 2dFGRS as a function of the host spectral parameter, . The morphology of the hosts is expected to be roughly E/S0 in the left panel, Sa in the middle panel, and Sb/Scd in the right panel. The median luminosities of the subsamples in each of the panels is somewhat different, but the difference is too small to account for the differences in the velocity dispersion profiles. Here h = 0.7 has been adopted.
Previous work on the dependence of v with projected radius using SDSS galaxies  and 2dFGRS galaxies  concluded that v(rp) was consistent with an isothermal profile; i.e., v(rp) = constant. In both of these investigations, the hosts and satellites were selected in a manner that was identical to that of Sample 3 in Prada et al. . In both previous analyses, however, the number of hosts and satellites was significantly smaller than the more recent studies, and the formal error bars were correspondingly larger. In addition, the original analysis of SDSS host-satellite systems  neglected to account for the fact that the interloper fraction increases with radius, which would have biased measurements of v at large rp towards values which are higher than the actual satellite velocity dispersion at those radii.
Even more recently, Conroy et al.  used satellites of z ~ 0.8 host galaxies in the DEEP2 survey to investigate v(rp). DEEP2 (Deep Extragalactic Evolutionary Probe 2) is being carried out with the DEIMOS spectrograph at the Keck-II telescope, and will ultimately collect spectra of ~ 60, 000 galaxies with redshifts of 0.7 z 1.4 to a limiting magnitude of RAB = 24.1 . Unfortunately, the survey is still far from complete and only 61 isolated host galaxies (having a total of 75 satellites) were found in the current DEEP2 data. Because of this, the errors on v(rp) are large, and formally v(rp) for the DEEP2 galaxies is fitted well by a constant value: v(110 h-1 kpc) = 162+44-30 km sec-1, v(230 h-1 kpc) = 136+26-20 km sec-1, v(320 h-1 kpc) = 155+55-38 km sec-1. Therefore, isothermal halos for the DEEP2 galaxies cannot be ruled out at the moment. Conroy et al.  show, however, that their velocity dispersion measurements are consistent with expectations for NFW halos with virial masses in the range 3.5 × 1012 h-1 M M200 8.0 × 1012 h-1 M. This is in good general agreement with the results of Prada et al. , especially considering that the DEEP2 hosts are of order one magnitude brighter than the SDSS hosts (i.e., the virial mass implied for the halos of the brightest galaxies in the SDSS sample is ~ 4 × 1012 h-1 M). At the moment, however, the DEEP2 data are too severely limited by small number statistics to place strong constraints on the nature of the dark matter halos of galaxies with redshifts of order unity.