Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 347 / 1

Let $u=-v_r>0$ denote the inward speed. Steady spherical mass conservation and the radial momentum equation for an isothermal gas giveEliminating $d\log \rho /dr$ produces the Isothermal Bondi equationA smooth flow can cross Mach number one only where both sides vanish. Its Bondi sonic point is thereforeThe integrated Bernoulli equation which approaches rest and density ${\rho }_{\infty }$ at infinity isAt the sonic point this gives ${\rho }_s=e^{3/2}{\rho }_{\infty }$, and hence the unique regular transonic branch has

This mode is most plausible for a nearly stationary supermassive black hole immersed in the hot, pressure-supported, X-ray-emitting atmosphere of a massive elliptical galaxy. Such gas is comparatively smooth, its random and rotational speeds can be smaller than its sound speed, and radiative cooling can be slow on the inflow scale.

The ideal Bondi accretion assumptions can fail in several independent ways. The gas may carry enough specific angular momentum to circularize into an accretion disk; a galaxy's stars and dark matter can dominate the gravitational potential outside the black hole's sphere of influence; the hole can move relative to the gas; cooling, conduction, or external heating can destroy adiabaticity or isothermality; a multiphase or clumpy medium is not a uniform reservoir; magnetic fields, turbulence, and viscosity add stresses absent from the spherical model; and AGN feedback, winds, or jets can expel or recirculate the inflow. Time dependence and self-gravity provide further failures. Thus a Bondi estimate is best interpreted as an idealized supply rate, not automatically as the rate crossing the event horizon.

If $c_{s,\infty }\gg \sigma$ and $M_{\mathrm{enc}}\to M$, expansion of the denominator givesthe stated model's pressure-dominated Bondi accretion limit. Its order-unity coefficient differs from the $\pi e^{3/2}$ found for the exactly isothermal critical solution because such coefficients depend on the adopted equation of state and interpolation.

If instead $\sigma \gg c_{s,\infty }$,The gas's thermal motion is then negligible beside the galactic velocity dispersion, and the enclosed galactic mass, rather than the black hole alone, focuses the gas. For a singular isothermal sphere, $M_{\mathrm{enc}}(R)=2{\sigma }^{2}R/G$, so this becomes $\dot{M}\simeq 4\pi {\rho }_{\infty }\sigma R^2$. This second limit describes capture controlled by the host potential and is therefore not a genuinely spherical black-hole Bondi solution.

Write $w=1/r$ and $h=r^2\dot{\theta }=b{v}_{\infty }$. Since $\dot{r}=-hdw/d\theta$, the radial equation becomes the Binet equationwhose solution isChoose the incoming asymptote at $\theta =\pi$ and the downstream axis at $\theta =0$. Then $w\to 0$ as $\theta \to \pi$, while $r\sin \theta \to b$. These two conditions giveThe mirror-image streamlines meet on the downstream axis atAt that point each streamline has radial velocity $-v_{\infty }$ and equal and opposite azimuthal velocity. An inelastic collision cancels the latter, so the specific energy afterwards isThe gas is bound when $E_{\mathrm{after}}<0$, orSweeping the corresponding capture cylinder through gas of density ${\rho }_{\infty }$ gives the Bondi--Hoyle--Lyttleton accretion rateUnlike stationary spherical Bondi accretion, this is a directed, supersonic flow with a downstream focusing wake; bulk speed replaces sound speed as the main resistance to capture.