## 4. AnalysisWhile the X-ray image show significant substructure in the cluster indicating deviations from hydrostatic equilibrium, the cluster total mass deduced from assumptions of dynamical equilibrium are still reliable, as is shown by numerical simulations (Evrard et al 1996; Schindler 1996). Under the assumption of hydrostatic equilibrium and spherical symmetry, the cluster total mass is directly related to the intracluster gas properties as: In general, a good fit can be found for the X-ray surface brightness distribution using the parametrisation given in Eq. 1, which in turn gives the gas density as follows if the gas is isothermal: Hence, the gravitational potential is given by where and the total mass is given by The lensing effects of the background galaxies by the cluster gravitational field is directly related to the 2-D projection of the total mass density. In this case, the projected total mass density is given by If we consider galaxies as test particles in the cluster potential well, then Jean's equation for a collisionless, steady state, non-rotating spherically symmetric system gives where is the spatial galaxy number density, is the anisotropy index and is the radial velocity dispersion. The spatial galaxy number density is related to the observed 2-D projection of the galaxy number density through the Abel inversion given by and are related to the observed line-of-sight velocity dispersion through In the simple case, where the galaxy orbits are isotropic, Eq. 7 is equivalent to Eq. 2 with replaced by . If we make a further simplification by assuming that not only the gas but also the galaxies are isothermal, i.e. is a constant, then we have where . Given the above parametrisation for the X-ray surface brightness and the resultant expression for given by Eq. 3, we deduce the spatial galaxy density distribution as where . The observed line-of-sight velocity dispersion is trivially given by and . Alternatively, if we simplify the case by assuming that the galaxy density distribution follows that of the total mass, i.e. mass-follows-light, then from Jean's equation (Eq. 7) we see that the galaxies cannot be isothermal if the gas is isothermal and the X-ray surface brightness is parametrised as in Eq. 1. The radial velocity dispersion is given by where again . The line-of-sight velocity dispersion can be deduced from Eq. 9. However, the measured velocity dispersion is an average of within a certain radius: In the case of Abell 2104, we have the observables
, ,
,
. Since the ## 4.1. Mass estimate from optical dataThe projected galaxy density distribution is consistent with a wide range of models. The following family of parametrised functions were fitted to the projected galaxy density distribution after background subtraction using the density of galaxies in the annulus to as background. If we fix the core radius to the X-ray determined value of , then we found the best fit to be ( with 10 degrees of freedom), though to were also statistically consistent with the observed data. Note that corresponds to spatial galaxy distributions of the form given in Eq. 11 with respectively. The projected total mass density distribution given by Eq. 6 was also statistically consistent with the projected galaxy density distribution ( with 10 degrees of freedom), which means mass-follows-light is not excluded. The 2D projection of the functional form (Navarro et al. 1996) was also found to be statistically consistent with the observed galaxy distribution. The projected galaxy distribution is shown in Fig. 9 along with the various model fits. The observed galaxy density distribution is still declining towards the edge of the image indicating a wider field is needed to reach the true "edge" of the cluster. The X-ray data show that the cluster extends at least out to a radius of which is beyond the optical field of view for the current observation. A wider field of view would help to reject some of the above models.
If we estimate the total mass distribution from the galaxy density
distribution and velocity dispersion assuming that the galaxies are
isothermal, then where the observed
data give to
and
km s On the other hand, if the galaxy distribution is not isothermal but
follows that of the mass then the measured velocity dispersion implies
that km s ## 4.2. Mass estimate from X-ray dataThe values of ,
and
keV have been determined from
spatial analysis of the HRI data and the spectro-analysis of the
Note that if the galaxies are isothermal, then the X-ray deduced mass is consistent with the optically deduced mass (or generalised "Virial" mass) if . The X-ray and optical data are also marginally consistent if mass-follows-light. The total gas mass within was found to be which gives a gas fraction of % compared to the X-ray deduced mass, but 5-10% compared to the dynamically deduced mass. The gas fraction within a radius of Mpc (where the over-density is 500 times the critical density of the Universe) is %, which is lower than the average gas fraction of % for nearby hot ( keV) non-cooling flow clusters (Arnaud & Evrard 1999). The gas fraction within a radius of 1.46 Mpc gives a lower limit to the baryonic fraction. Since the baryonic matter density predicted from the Big Bang nucleosynthesis gives (Walker et al. 1991) from the measured light element abundance, the lower limit of the baryonic fraction of this cluster is thus consistent with . © European Southern Observatory (ESO) 2000 Online publication: December 11, 2000 |