= Transonic accretion in a steep dark-matter cusp
{title2=Transonic accretion in an $r^{-5/2}$ density cusp}
Consider steady spherical <accretion> of a <gas> obeying a <polytropic equation of state> in <isentropic flow>, with fixed background potential $\Phi=-A r^{-1/2}$, where $A=16\pi G\Psi_0$, with reservoir <mass density> $\rho_0$ and <sound speed> $c_0$ at infinity. At a regular <sonic point>, radial momentum balance requires $c_c^2=A/(4\sqrt{r_c})$. The <Bernoulli function> is
$$
\frac{u^2}{2}+\frac{c_s^2}{\gamma-1}+\Phi=\frac{c_0^2}{\gamma-1}.
$$
Setting $u=c_s=c_c$ at the <sonic point> gives
$$
c_c^2\left[\frac1{\gamma-1}-\frac72\right]=\frac{c_0^2}{\gamma-1},\qquad \frac{c_c^2}{c_0^2}=\frac2{9-7\gamma}.
$$
A finite positive critical <sound speed> therefore requires $1<\gamma<9/7$. <Mass conservation> gives $\dot M=4\pi r_c^2\rho_c c_c$, while the constant-entropy gas law gives $\rho_c/\rho_0=(c_c^2/c_0^2)^{1/(\gamma-1)}$. Since $r_c\propto A^2c_0^{-4}$ at fixed $\gamma$, these equations yield $\dot M\propto\rho_0G^4\Psi_0^4c_0^{-7}$. The steep sound-speed dependence comes from both the selected capture scale and the critical flow speed. The background potential is prescribed; gas self-gravity and finite outer boundaries require a different model.
For a background gravitating density $\rho_{\rm dm}=\Psi_0r^{-5/2}$, the potential is $\Phi=-16\pi G\Psi_0r^{-1/2}$. A steady polytropic inflow has a regular sonic point only for $1<\gamma<9/7$, and its accretion rate scales as $\dot M\propto\rho_0G^4\Psi_0^4c_0^{-7}$.
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