Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 57 / 1 / c / Solution

Use the positive mass accretion rate $\dot{M}=-\mathcal{J}$, and retain $\delta =\gamma -1$, $q=2/\beta -1/2$. Define $h=1-q\delta$, which is positive under the condition in (b). Matching the Bernoulli function to the reservoir and using the polytropic equation of state givesThe transonic spherical accretion rate in a power-law potential is thereforeThe combination $A/c_{\infty }^2$ has dimensions of length to the power $\beta$, so this expression has dimensions of mass per time. The sonic point selects the flux that connects the subsonic reservoir to the inward supersonic transonic branch.

For the endpoint limits of power-law spherical accretion, hold $A,\beta ,{\rho }_{\infty },c_{\infty }$ fixed. As $\delta \to 0$, $\ln h=-q\delta +O({\delta }^2)$, so $(q-1/\delta )\ln h\to q$. HenceThis is also obtained directly from an isothermal equation of state: the Bernoulli function becomes $u^2/2+c_{\infty }^2\ln (\rho /{\rho }_{\infty })-A/r^{\beta }=0$, so ${\rho }_{\ast }/{\rho }_{\infty }=e^q$. For $\beta =1$, $A=GM$, it recovers the isothermal Bondi accretion rate.

For $0<\beta <4$, $q>0$ and $\gamma \uparrow f(\beta )$ means $h\downarrow 0$. Since $\delta =(1-h)/q$,ThusThe finite limiting flux accompanies $r_{\ast }\to 0$ and $c_{\ast },{\rho }_{\ast }\to \infty$; it does not assert a finite-radius sonic point at the endpoint. For $\beta =1$ this is the familiar $\gamma \uparrow 5/3$ limit $\pi G^2{M}^2{\rho }_{\infty }/c_{\infty }^3$.

For completeness, the printed range $\beta \ge 4$ has the endpoint $f=+\infty$. At $\beta =4$, $q=0$ and the flux is independent of $\gamma$, namely $4\pi {\rho }_{\infty }(2A{)}^{1/2}$. For $\beta >4$, $q<0$ and $h\sim |q|\delta$, so $\dot{M}\to 0$ as $\gamma \to \infty$. These are the corresponding extended endpoint limits.