Evidence for dark matter from X-rays

An important application of X-ray observations is to probe the gravitational potential of the galaxy, and hence infer its density distribution. Consider a thin shell of hot gas. The gravitational force due to mass enclosed by the shell (consisting of stars, gas and dark matter) will pull the shell inward. At the same time, if the pressure decreases outward, then the pressure gradient will try to push the shell outward. In hydrostatic equilibrium, these forces balance. By estimating the pressure gradient from X-rays, we can infer the gravitational force and hence look for evidence for dark matter. Let's assume for simplicity that the elliptical galaxy is spherically symmetric.

Let $M(<r)$ be the mass inside the shell of radius $r$ (consisting of stars, gas and dark matter: $M=M_\star+M_{\rm gas}+M_{\rm dm}$), and $dr$ its thickness The gravitational force on the shell is

$$F_g(r)= {GM(<r)M_s\over r^2}\,,$$ (7.7)

where $M_s =4\pi r^2\rho_{\rm gas}(r) dr$ is the mass of the shell.

Let $p(r)$ be the pressure in the hot gas. The pressure force on a surface is $p(r)$ times the surface area, $4\pi r^2$. The net outward force is the difference between the outward and inward forces:

$$F_p(r) = 4\pi r^2 (p(r)-p(r+dr)) = -4\pi r^2 {dp\over dr}\,dr\,$$ (7.8)

where the pressure was expanded in a Taylor series. Combining the last three equations gives

$${GM(<r)\over r^2} 4\pi r^2 \rho_{\rm gas}(r) dr = -4\pi r^2 {dp\over dr} dr\,,$$ (7.9)

hence

$${GM(<r)\over r^2} = -{1\over \rho_{\rm gas}(r)} {dp\over dr}\,,$$ (7.10)

or alternatively in terms of the gravitational potential $\Phi$

$$\nabla\Phi=-\nabla p/\rho_{\rm gas}\,.$$ (7.11)

This should look familiar from hydrostatic equilibrium in stars!

By modelling the spectrum as function of position we can determine $T(r)$, and since the X-ray intensity $\propto \rho_{\rm gas}^2 T^{1/2}$, we can also determine $\rho_{\rm gas}(r)$. Hence $p\propto rho_{\rm gas}T$, and so we have determined the rhs of Eq.(7.11). From that, one can reconstruct the gravitational potential, and hence infer the mass distribution.

The temperature $T$ is observed to remain approximately constant, in which case we can obtain an approximate expression for the density profile. The pressure is

$$p={\hbox{${\rm k}_{\sc\rm B}$}\,T\over \mu\,\hbox{$m_{\sc\rm p}$}}\,\rho_{\rm gas}\,.$$ (7.12)

If we take $T$ constant then the rhs of Eq. (7.10) becomes $-{\hbox{${\rm k}_{\sc\rm B}$}\,T\over \mu\,\hbox{$m_{\sc\rm p}$}}{d\ln(\rho_{\rm gas})/dr}$, and hence

$$G M(<r) = -{\hbox{${\rm k}_{\sc\rm B}$}T\over \mu\,\hbox{$m_{\sc\rm p}$}} r^2 {d\ln\rho_{\rm gas}\over dr}\,.$$ (7.13)

Taking the derivative of both sides with respect to $r$, gives

$$G 4\pi r^2 \rho_{\rm tot} = -{\hbox{${\rm k}_{\sc\rm B}$}T\over \mu\,\hbox{$m_{\sc\rm p}$}} {d\over dr} (r^2 {d\ln\rho_{\rm gas}\over dr})\,,$$ (7.14)

since $dM(<r)/dr=4\pi\,r^2\,\rho_{\rm tot}$. $\rho_{\rm tot}$ is the total mass density, i.e. due to stars, gas and dark matter.

Now assume that the gas density and total mass density are proportional, $\rho_{\rm gas}\propto \rho_{\rm total}$, and try a solution of the form $\rho_{\rm totcl}=\rho_0\,(r_0/r)^\beta$. Substitution shows that this is indeed a solution when $\beta=2$, in which case

$$\rho_{\rm tot}(r) = {\hbox{${\rm k}_{\sc\rm B}$}\,T\over 2\pi\,G\mu\hbox{$m_{\sc\rm p}$}}\,r^{-2}\,.$$ (7.15)

So a measurement of $T$ can constrain the total mass density $\rho_{\rm tot}$. Estimating the stellar mass density from the luminosity, the gas density from the X-ray emissivity, we find $\rho_{\rm dm}=\rho_{\rm total}-\rho_\star-\rho_{\rm gas}$. More detailed modelling of this type confirms that also elliptical galaxies have most of their mass in the form of dark matter.