§19 Gas within a dark matter halo

Let us consider next the case of a galaxy dominated by a dark matter halo for which its density is given by the relation (Binney & Tremaine, 1997):

$$\rho_d = \frac{ \rho_{d_\star} }{ 1 + \left( r / \mathsf{a} \right)^2 },$$ (19.1)

in which $\mathsf{a}$ is the core radius and quantities with a star refer to the value at the centre of the galaxy.

The potential resulting from such a density profile can be calculated by means of eq.(17.10),

$$\phi_{\mathrm{d}} -\phi_{\mathrm{d}_\star} = 4 \pi G \rho_{\mathr... ...+ \frac{\mathsf{a}}{r} \arctan \left( \frac{r}{\mathsf{a}} \right) -1 \right\},$$ (19.2)

in which the value of the gravitational potential $\phi_{\mathrm{d}_\star}$ has been evaluated at the centre of the galaxy $r_\star \! = \! 0$. If the gas in the galaxy is in hydrostatic equilibrium with the dark matter halo, then $\boldsymbol{\mathrm{grad}} p \! = -\rho { \boldsymbol{\mathrm{grad}} } \: \phi_\mathrm{d}$. In this case, the enthalpy of the isothermal gas is given by:

$$w - w_\star = -\phi_\mathrm{d} + \phi_{\mathrm{d}_\star} = c^2_\star \ln \left( \frac{p}{p_\star} \right),$$ (19.3)
and so the pressure takes the value:

$$p = p_\star \exp \left\{ \left( -\phi_\mathrm{d} + \phi_{\mathrm{d}_\star} \right) / c^2_0 \right\}.$$ (19.4)

It is possible to simplify the above expressions by using the fact that for astronomical cases $r_0 \! \gg \! a$. In other words:

$$\frac{p}{p_0} = \eta^{-\mathit{k}},$$ (19.5)

where the dimensionless parameter $\mathit{k}$ is given by:

$$\mathit{k} \! \equiv \! - 4 \pi G \frac{ \rho_{\mathrm{d}_\star} \mathsf{a}^2 }{ c_\star^2 }.$$ (19.6)

Adopting these approximations, the required analytic solutions can be found:

$$\alpha = \frac{1}{2} \left( \mathit{k} \Gamma + 1 \right) \left( \mathit{k} \Gamma + 2 \right) \sin^{ \mathit{k} \Gamma } \! \varphi_0,$$

$$\beta = - \mathit{k} \Gamma \left( 2 + \mathit{k} \Gamma \right) \sin^{ \mathit{k} \Gamma} \! \varphi_0,$$ (19.7)

$$\zeta = \frac{1}{2} \mathit{k} \Gamma \left( \mathit{k} \Gamma + 1 \right) \sin^{ \mathit{k} \Gamma} \! \varphi_0,$$

where for simplicity it was assumed that $\tilde\alpha \negmedspace = \negmedspace \tilde\beta \negmedspace = \negmedspace \tilde\zeta \negmedspace = \negmedspace 0$. In other words, the gravitational field induced by the mass of the cloud has been neglected. Using typical values (Binney & Tremaine, 1997) for galaxies then $\rho_\star \sim \! \unit{ 0.1 }{ M_\odot \usk pc\rpcubed }$, $\mathsf{a} \sim \! \unit{ 1 }{ \kilo pc }$ . Taking central values for the gas in the galaxy as $n_\star \! \sim \! \unit{ 1 }{ \centi\meter\rpcubed }$ and $T_\star \! \sim \! \unit{ \power{10}{5} }{ \kelvin }$ then $\mathit{k} \! \sim \! -10$. Fig.(III.4) shows plots for different values of $M_0$ and $\mathit{k}$.

Figure III.4: Deflection produced in a jet as it travels across a galaxy, for which its gravitational potential is dominated by a dark matter halo. The jet penetrates the galaxy parallel to the $x/r_0$ axis. Various trajectories are shown in each diagram for different initial heights $y/r_0 \! = \! 0.5$, $0.15$,...,$0.95$ measured from the $x/r_0$ axis. The top and bottom diagrams were calculated for the case of a non relativistic jet with an initial Mach number of $M_0 \! = \! 10$ and $M_0 \! = \! 20$ respectively. For each of this diagrams a value of $\mathit{k} \! = \! -1$, $-2$, $-3$ was used for the top, middle and bottom panels respectively (see text). The dashed lines in the figures represent the direct numerical integration of eq.(17.14) with the pressure given by eq.(19.4) for the case in which the ratio of the core radius $\mathsf{a}$ to the initial radius $r_0$ is given by $\mathsf{a} / r_0 \! = \! 10^{-3} \ll 1$. The continuous lines are analytic approximations found under this conditions.

The number $\mathit{k}$ can be rewritten as $\mathit{k} \! = - \left( 4 / ( 4 - \pi ) \right) ( G { \mathsf{M} } / \mathsf{a} ) ( 1 / c_\star^2 )$, where ${ \mathsf{M} }$ is the mass of a sphere with radius $\mathsf{a}$. This quantity is proportional to the gravitational energy of the cloud evaluated at the core radius divided by the sonic kinetic energy that a fluid element in the jet has. In other words, in the same way as in Section §18, the dimensionless number $\mathit{k}$ is an indicator as to how big deflections produced by gravity are.

Footnotes

... by

In exactly the same form as it was done in the footnote of p., the dimensionless parameter $\mathit{k}$ (apart from an unimportant proportionality factor of $-4 \pi$ ) can be calculated by standard dimensional analysis. In this case the important parameters in the problem are the gravitational constant $G$, the characteristic length $a$ and the sound speed $c_\star$ together with the density $\rho_\star$ evaluated at the centre of the cloud.