Mass accretion rate ($\dot{M}$)

The mass accretion rate is the mass entering an accretor per unit time. In a steady spherical flow, $\dot{M}=4\pi r^2\rho u$ is independent of radius.

A steady full-sphere radial inflow has mass flux $\dot{M}=4\pi r^2\rho |v_r|$ by conservation of mass. With Newtonian free-fall speed from rest at infinity, $|v_r|=\sqrt{2GM/r}$, this becomes the displayed expression. The assumed velocity then requires $\rho \propto r^{-3/2}$ along the steady flow. A local density and speed do not determine the total mass accretion rate of a general nonspherical inflow, and this estimate does not impose the sonic boundary condition of Bondi accretion.

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 isSetting $u=c_s=c_c$ at the sonic point givesA 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 $r_c\propto A^2{c}_0^{-4}$ at fixed $\gamma$, these equations yield $\dot{M}\propto {\rho }_0{G}^4{\Psi }_0^4{c}_0^{-7}$. 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 ${\rho }_{\mathrm{dm}}={\Psi }_0{r}^{-5/2}$, the potential is $\Phi =-16\pi G{\Psi }_0{r}^{-1/2}$. A steady polytropic inflow has a regular sonic point only for $1<\gamma <9/7$, and its accretion rate scales as $\dot{M}\propto {\rho }_0{G}^4{\Psi }_0^4{c}_0^{-7}$.