6 Cluster mass

In this section we present mass estimates for Abell 970. We first consider the mass derived from its X-ray emission and then we present masses computed from the velocity and galaxy distributions.

6.1 X-ray mass

Supposing that the gas is isothermal, in hydrostatic equilibrium and distributed with spherical symmetry, the dynamical mass inside r is given by:

$$M(r) = - \frac{k \, T_{\rm X} \, r}{G \, \mu \, m_{\rm p}} \frac{{\rm d} \log \rho(r)}{{\rm d} \log r},$$ (1)

where $\mu$ is the mean molecular weight ($\mu=0.59$ for a fully ionized primordial gas) and $m_{\rm p}$ is the proton mass. If we assume that the gas is described by a $\beta$-model,

$$\rho(r) = \rho_{0} (1 + (r/r_{\rm c})^2)^{-3\beta/2},$$ (2)

the dynamical mass becomes:

$$M(r) = 6.68\times 10^{10} \frac{\beta\, T_{\rm X}}{\mu} \frac{r^3}{r^2 + r_{\rm c}^2}~M_{\odot},$$ (3)

with r and $r_{\rm c}$ measured in kpc and $T_{\rm X}$ in keV.

Furthermore, if we have the density contrast $\delta = \bar{\rho}(r)/\rho_{\rm c}$ (where $\bar{\rho}(r)$ is the mean density inside the radius r and $\rho_{\rm c}$ is the critical density at the redshift of the cluster) then we can define $r_{\delta}$ as

$$\left(\frac{r_{\delta}}{r_{\rm c}}\right)^2 = \frac{2.3\time... ...mega_{\Lambda})} \frac{\beta\, T_{\rm X}}{\mu \, r_{\rm c}^2},$$ (4)

where $f(z, \Omega_{0}, \Omega_{\Lambda})$ depends on the cosmological parameters:

$$f^2(z, \Omega_0, \Omega_{\Lambda})\!= \!\Omega_{\Lambda} + \Omega_0 (1+z)^3 - (\Omega_0 + \Omega_{\Lambda} - 1) (1 + z)^2.$$ (5)

When $\delta=200$ we have the usual r₂₀₀ radius, which is about the virial radius, $r_{\rm vir}$ (e.g. Lacey & Cole 1993). The parameters of the $\beta$-model may be estimated from the fitting of the observed X-ray brightness profile, since $\Sigma_{\rm X} \propto 1/(1 + (R/r_{\rm c})^2)^{3\beta-1/2}$. The profile was obtained with the STSDAS/IRAF task ellipse; it extends only to $\sim$ 900 h₅₀^-1 kpc. A least-squares fitting of this profile (see Fig. 9) gives: $\beta = 0.66 \pm 0.04$ and $r_{\rm c} = (260 \pm 20)~h_{50}^{-1}$ kpc. The quality of the fitting is good: $\chi^2 = 8.4$ for 37 degrees of freedom. Taking $T_{\rm X}=3.3$ keV (cf. Sect. 4), we obtain r₂₀₀ = 2.00_-0.28^+0.38 Mpc and $M(r_{200}) = 4.8_{-0.8}^{+1.5} \times 10^{14}~M_\odot$. The total mass in the region where the velocities have been measured is $M(r=1200\, \mbox{kpc}) = 2.79_{-0.74}^{+1.18} \times 10^{14}~M_\odot$. White et al. (1997) obtained $M(r = 675~ {\rm kpc}) = 2.6 \times 10^{14}~M_\odot$. This value is about 1.4 times larger than ours at the same radius (cf. Fig. 10 below), probably as a consequence of the high value of $T_{\rm X}$ assumed by these authors; anyway, within the errors both results are in agreement.

Figure 9: X-ray brightness profile (IPC data with error bars), obtained with the STSDAS/IRAF task ellipse; the continuous line is the best-fit $\beta$-model.

6.2 Optical virial mass

In Fig. 10 we show the cluster mass profile computed with the virial mass estimator (VME) which, as discussed by Aceves & Perea (1999), gives less biased mass estimates when the system is not completely sampled. These authors also show that the VME overestimates the real mass by no more than 20% at small radii, being more reliable at larger apertures. The error bars in Fig. 10 are 1-$\sigma$ standard deviations computed using the bootstrap method. The VME assumes, of course, that the system is virialized. In general, the presence of substructures or large-scale flows tend to increase the velocity dispersion of the galaxies, leading to an overestimation of the mass of the system.

The VME of Abell 970, within 1.2 h₅₀^-1 Mpc, is $M = (6.80 \pm 1.34) \times 10^{14}~M_{\odot}$, where the error, as before, was computed with the bootstrap method. Note that, for a virialized cluster, these are lower limits for the mass, since we have velocities only for the central region of the system. Indeed, assuming a relation between virial radius and velocity dispersion similar to that adopted by Girardi et al. (1998), we estimate that $r_{\rm vir} \sim 3.4~h_{50}^{-1}$ Mpc, while the velocities have been measured within a region of radius $\sim$ 1.2 h₅₀^-1 Mpc.

Figure 10 also displays the run of the total b_j luminosity of the cluster (up to b_j = 19.75). Considering the VME masses, we find that the mass-luminosity ratio ranges from 1360  $M_\odot/L_\odot$at the cluster central region, to $\sim$450  $M_\odot/L_\odot$ at the largest aperture.

The mass profile derived from the X-ray emission is also presented in Fig. 10. The VME masses are in excess of the X-ray mass estimates by large factors, ranging from $\sim$16 for the central apertures, to about 4 at $\sim$1.3  h₅₀^-1 Mpc aperture, which encompasses the whole velocity sample. These factors are well above the uncertainties discussed above for virialized clusters. In fact, the dynamical mass determined by the X-ray emission at radius $r \gg r_{\rm c}$depends essentially on the temperature and the asymptotic slope of the gas density. Both are poorly determined with the available data; it is then possible that one of them (or both) are under-estimated, which implies that we under-estimate the dynamical X-ray mass. For instance, if $\beta$ is as high as 0.70 and $T_{\rm X} = 4.4$ (cf. the error bars in Table 2), then the dynamical X-ray mass would be twice the estimated value, i.e., M(r=1.2 h₅₀^-1 Mpc $)=4.1\times10^{14}~M_\odot$. On the other hand, the presence of a substructure associated with the cluster brightest galaxy, as well as the mean velocity gradient, may be an indication of non-virialization and, consequently, the VME may be largely overestimated.

Figure 10: Cluster optical masses (circles and dotted lines), X-ray masses (squares and dot-dashed lines) and luminosities (triangles and dashed lines).