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

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.

Let all velocity amplitudes refer to the same positive normal direction, from medium 1 into medium 2. For a right-going linear sound wave, its pressure amplitude is $p^{\prime }=Z{u}^{\prime }$ with acoustic impedance $Z=\rho c_s$; for a left-going wave it is $p^{\prime }=-Z{u}^{\prime }$. Continuity of normal velocity and pressure at the interface givesEliminating $u_t$ yields the normal-incidence acoustic velocity reflection coefficientThe common adiabatic index and mean particle mass $\bar{m}$ give $c_{s,j}=\sqrt{\gamma k_B{T}_j/\bar{m}}$, soA negative coefficient reverses the reflected velocity perturbation's phase in this fixed coordinate system. The reflected pressure coefficient has the opposite sign. Pressure balance also gives ${\rho }_1{T}_1={\rho }_2{T}_2$, so the impedance ratio can equivalently be written $Z_2/Z_1=\sqrt{{\rho }_2/{\rho }_1}$. These are normal-incidence, small-amplitude results without a surface mass or tension.

Work in an inertial centre of mass frame and write $I={\sum }_i{m}_i|{\mathbf{r}}_i{|}^2$. Differentiation and Newton's second law giveThe pairwise Newtonian gravitational potential energy isFor each pair, combining the two force contributions gives $-G{m}_i{m}_j/|{\mathbf{r}}_i-{\mathbf{r}}_j|$. Therefore ${\sum }_i{\mathbf{r}}_i\cdot {\mathbf{F}}_i=\Omega$, andFor a stationary configuration, or after a long-time average of a bounded system whose endpoint term vanishes, the virial theorem givesThe brackets may be suppressed when $T,\Omega$ denote equilibrium averages. A collapsing cloud or an eccentric Kepler orbit need not obey $2T+\Omega =0$ instantaneously. Confining surface pressure, magnetic stress or other external forces add their own virial terms; the displayed result concerns an isolated gravitational particle system.