Write $w=1/r$ and $h=r^2\dot\theta=bv_\infty$. Since $\dot r=-h\,dw/d\theta$, the radial equation becomes the <Binet equation>
$$
\frac{d^2w}{d\theta^2}+w=\frac{GM}{h^2},
$$
whose solution is
$$
w=c_1\cos\theta+c_2\sin\theta+\frac{GM}{b^2v_\infty^2}.
$$
Choose the incoming asymptote at $\theta=\pi$ and the downstream axis at $\theta=0$. Then $w\to0$ as $\theta\to\pi$, while $r\sin\theta\to b$. These two conditions give
$$
c_1=\frac{GM}{b^2v_\infty^2},
\qquad c_2=\frac1b.
$$
The mirror-image streamlines meet on the downstream axis at
$$
\boxed{r_{\rm coll}=\frac{b^2v_\infty^2}{2GM}.}
$$
At 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 is
$$
E_{\rm after}=\frac{v_\infty^2}{2}-\frac{GM}{r_{\rm coll}}
=\frac{v_\infty^2}{2}-\frac{2G^2M^2}{b^2v_\infty^2}.
$$
The gas is bound when $E_{\rm after}<0$, or
$$
\boxed{b<b_{\rm crit}=\frac{2GM}{v_\infty^2}.}
$$
Sweeping the corresponding capture cylinder through gas of density $\rho_\infty$ gives the <Bondi--Hoyle--Lyttleton accretion rate>
$$
\boxed{\dot M=\pi b_{\rm crit}^2\rho_\infty v_\infty
=\frac{4\pi G^2M^2\rho_\infty}{v_\infty^3}.}
$$
Unlike 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.

Solution

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

{c}
{title2=$\dot M_{\rm BHL}$}

In the ballistic supersonic limit, streamline collisions bind gas with impact parameter $b<2GM/v_\infty^2$, giving
$$
\dot M_{\rm BHL}=\frac{4\pi G^2M^2\rho_\infty}{v_\infty^3}.
$$

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 Incoming links: Bondi--Hoyle--Lyttleton accretion rate