Astron. Astrophys. 360, L43-L46 (2000)

2. Theoretical expectations

We briefly summarize the mathematical treatment of the intracluster gas tracing the underlying dark matter distribution of clusters. First, if the X-ray surface brightness profile of a cluster can be well approximated by the conventional model (Cavaliere & Fusco-Femiano 1976)

this would indicate (Cowie, Henriksen & Mushotzky 1987),

for an optically-thin, thermal bremsstrahlung emission, where and T are the electron number density and temperature, respectively. The central electron number density , temperature and X-ray surface brightness are connected by

where , with X being the primordial hydrogen mass fraction, is the average Gaunt factor, and z is the cluster redshift. The total mass in gas within r is simply

Secondly, if the intracluster gas is in hydrostatic equilibrium with the underlying dark matter distribution, we have

where is the average molecular weight. For NFW profile

Here we have neglected the self-gravity of the gas. Using the normalized gas temperature and the volume-averaged baryon fraction as the two variables, we obtain the following two first-order differential equations

where , , and . The first equation can be straightforwardly solved with :

In order to solve the second equation and determine the free parameters a, b and , we use the following boundary conditions

Namely, we demand that the baryon fraction should asymptotically match the universal value of at the virial radius defined by

where represents the overdensity of dark matter with respect to the average background value , for which we take and . We now come to the free parameters involved in Eqs. (8) and (9). With the X-ray imaging observation, we can obtain the best-fit values of , and . If, on the other hand, the X-ray spectroscopic measurement can set a useful constraint on the central temperature , we will be able to derive the central electron density from Eq. (3). As a result, there are only two free parameters in the above equations: (or equivalently ) and . These two parameters can be fixed during the numerical searches for the solution of Eqs. (8) and (9) using the boundary conditions Eqs. (10) and (11). This will allow us to work out simultaneously the radial profiles of gas density and temperature, and fix the dark matter (NFW) profile of the cluster characterized by and .

© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000
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