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Astron. Astrophys. 363, 440-450 (2000)

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4. Analysis

While 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:

[EQUATION]

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:

[EQUATION]

Hence, the gravitational potential is given by

[EQUATION]

where [FORMULA] and the total mass is given by

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where [FORMULA] is the spatial galaxy number density, [FORMULA] is the anisotropy index and [FORMULA] 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

[EQUATION]

[FORMULA] and [FORMULA] are related to the observed line-of-sight velocity dispersion [FORMULA] through

[EQUATION]

In the simple case, where the galaxy orbits are isotropic, Eq. 7 is equivalent to Eq. 2 with [FORMULA] replaced by [FORMULA].

If we make a further simplification by assuming that not only the gas but also the galaxies are isothermal, i.e. [FORMULA] is a constant, then we have

[EQUATION]

where [FORMULA]. Given the above parametrisation for the X-ray surface brightness and the resultant expression for [FORMULA] given by Eq. 3, we deduce the spatial galaxy density distribution as

[EQUATION]

where [FORMULA]. The observed line-of-sight velocity dispersion is trivially given by [FORMULA] and [FORMULA].

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

[EQUATION]

where again [FORMULA]. The line-of-sight velocity dispersion [FORMULA] can be deduced from Eq. 9. However, the measured velocity dispersion is an average of [FORMULA] within a certain radius:

[EQUATION]

In the case of Abell 2104, we have the observables [FORMULA], [FORMULA], [FORMULA], [FORMULA]. Since the ASCA PSF was too poor to deduce a meaningful temperature profile, we will assume that the gas is isothermal for the time being. In the following section we will study the cluster total mass deduced from the various methods and examine their consistency using the simple parametrised [FORMULA]-model given above.

4.1. Mass estimate from optical data

The projected galaxy density distribution is consistent with a wide range of models. The following family of parametrised functions

[EQUATION]

were fitted to the projected galaxy density distribution after background subtraction using the density of galaxies in the annulus [FORMULA] to [FORMULA] as background. If we fix the core radius to the X-ray determined value of [FORMULA], then we found the best fit to be [FORMULA] ([FORMULA] with 10 degrees of freedom), though [FORMULA] to [FORMULA] were also statistically consistent with the observed data. Note that [FORMULA] corresponds to spatial galaxy distributions of the form given in Eq. 11 with [FORMULA] respectively. The projected total mass density distribution given by Eq. 6 was also statistically consistent with the projected galaxy density distribution ([FORMULA] with 10 degrees of freedom), which means mass-follows-light is not excluded. The 2D projection of the functional form [FORMULA] (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 [FORMULA] 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.

[FIGURE] Fig. 9. The radially averaged galaxy number density distribution (with background). The curves show a number of statistically consistent model fits to the data points. The short-dashed, solid and long-dashed curves corresponds to the [FORMULA] cases of the family of curves given by Eq. 14. The dotted curve shows the 2D projection of the Navarro model (Navarro et al. 1996).

If we estimate the total mass distribution from the galaxy density distribution and velocity dispersion assuming that the galaxies are isothermal, then [FORMULA] where the observed data give [FORMULA] to [FORMULA] and [FORMULA] km s-1, implying that [FORMULA] to [FORMULA] km s-1. Thus from Eq. 5 the total mass is between [FORMULA] and [FORMULA] within a radius of [FORMULA] (or 0.76 Mpc), and between [FORMULA] and [FORMULA] extrapolating to [FORMULA] (or 1.46 Mpc). Note that optical data alone does not constrain the mass very well, even under assumptions such as isothermality of the galaxy distribution and isotropy of the orbits.

On the other hand, if the galaxy distribution is not isothermal but follows that of the mass then the measured velocity dispersion implies that [FORMULA] km s-1 from Eq. 13 and the total mass is [FORMULA] within a radius of [FORMULA] (or 1.46 Mpc).

4.2. Mass estimate from X-ray data

The values of [FORMULA], [FORMULA] and [FORMULA] keV have been determined from spatial analysis of the HRI data and the spectro-analysis of the ASCA data respectively. Thus from the X-ray data, [FORMULA] implies a [FORMULA] km s-1 and a X-ray deduced total mass of [FORMULA] out to a radius of [FORMULA] (or 1.46 Mpc).

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 [FORMULA]. The X-ray and optical data are also marginally consistent if mass-follows-light.

The total gas mass within [FORMULA] was found to be [FORMULA] which gives a gas fraction of [FORMULA]% compared to the X-ray deduced mass, but 5-10% compared to the dynamically deduced mass. The gas fraction within a radius of [FORMULA] Mpc (where the over-density is 500 times the critical density of the Universe) is [FORMULA]%, which is lower than the average gas fraction of [FORMULA]% for nearby hot ([FORMULA] 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 [FORMULA] (Walker et al. 1991) from the measured light element abundance, the lower limit of the baryonic fraction of this cluster is thus consistent with [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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