Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2015 / Paper 1 / 1 / a / Solution

For a steady, spherically symmetric radial inflow, conservation of mass makes the mass accretion rate independent of radius:Atomic hydrogen gives $\rho \simeq m_{H}n$. The Newtonian free-fall speed from rest at infinity is $|v_r|=\sqrt{2GM/r}$, obtained by setting the specific mechanical energy $v_r^2/2-GM/r$ to zero. Hence the spherical free-fall accretion rate isUsing the given values gives $\rho \simeq 1.67\times 10^{-21}\mathrm{kg}{\mathrm{m}}^{-3}$, $|v_r|\simeq 2.98\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$, andThe area factor assumes a full spherical inflow with negligible angular momentum. With only a local number density and speed, an arbitrary nonspherical inflow's total rate is not fixed; one instead integrates $-\rho \mathbf{v}\cdot d\mathbf{S}$ over the inflow surface. This is a flux estimate, rather than the pressure-selected Bondi accretion solution.