Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 314 / 2

Let $u(r)>0$ denote the inward radial velocity. Steady spherical symmetry and mass conservation giveThe radial Euler momentum equation for the globally isothermal equation of state $p=c_s^2\rho$ isThe logarithmic derivative of mass conservation is ${\rho }^{\prime }/\rho =-u^{\prime }/u-2/r$. Substitution gives the Isothermal Bondi equationAt a smooth sonic point, $u=c_s$ makes the coefficient of $u^{\prime }$ vanish, so the right-hand side must vanish too. ThereforeThe solution that crosses this critical point of the isothermal Bondi equation continuously is the transonic accretion solution.

The radial equation can be writtenThus the Bernoulli function is constant on each streamline:The boundary conditions $u\to 0$ and $\rho \to {\rho }_0$ as $r\to \infty$ fix $\mathcal{B}=c_s^2\ln {\rho }_0$.

At the Bondi sonic point, $u_s=c_s$ and $GM/r_s=2c_s^2$. Evaluating the Bernoulli function there givesand henceThe conserved mass accretion rate is thereforeThus

With the dimensionless variables $x=r/r_s$ and $y=u/c_s$, mass conservation and part (c) giveSubstitution into the Bernoulli function, together with $GM/r=2c_s^2/x$, eliminates every dimensional parameter and gives the transcendental equationEquivalently,orThe transonic branch passes through $(x,y)=(1,1)$.

Put $Y=y^2$. On the inner supersonic branch, the equation from part (d) isIts asymptotic expansion as $x\to 0^{+}$ beginsTaking the positive square root givesSince $x=r/r_s$ and $y=u/c_s$,The leading term is the free-fall speed $\sqrt{2GM/r}$; the logarithmic term is the first pressure correction to the inner Isothermal Bondi accretion flow.