SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

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 [FORMULA] model (Cavaliere & Fusco-Femiano 1976)

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where [FORMULA] is the average molecular weight. For NFW profile

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where [FORMULA] represents the overdensity of dark matter with respect to the average background value [FORMULA], for which we take [FORMULA] and [FORMULA]. 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 [FORMULA], [FORMULA] and [FORMULA]. If, on the other hand, the X-ray spectroscopic measurement can set a useful constraint on the central temperature [FORMULA], 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: [FORMULA] (or equivalently [FORMULA]) and [FORMULA]. 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 [FORMULA] and [FORMULA].

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000
helpdesk.link@springer.de