= Solution
{c}
Work in the rest frame of the hole and neglect stellar velocities compared with $v$. In addition to $v\gg\sigma_*$, assume the galactic encounter speed is nonrelativistic, $v\ll c$. Treat a star as a test particle with specific <angular momentum> $\ell=bv$. At its pericentre $r_{\min}$, its radial velocity vanishes. <Conservation of energy> and <conservation of angular momentum> give
$$
\frac{v^2}{2}=\frac{\ell^2}{2r_{\min}^2}-\frac{GM_{\rm BH}}{r_{\min}},
\qquad
b^2=r_{\min}^2+\frac{2GM_{\rm BH}r_{\min}}{v^2}.
$$
The second term is <gravitational focusing>. For the rough Newtonian estimate in the question, assume capture when the pericentre reaches the <Schwarzschild radius>, $r_S=2GM_{\rm BH}/c^2$. It follows that
$$
b_{\rm cap}^2=r_S^2\left(1+\frac{c^2}{v^2}\right),
\qquad
\boxed{b_{\rm cap}\simeq\frac{2GM_{\rm BH}}{cv}.}
$$
The approximation keeps the focusing term because $v\ll c$.
Let $\rho_*=mn_*$ be the stellar mass density. Multiplying the <geometric collision cross-section> by the incoming mass flux gives the <collisionless black-hole capture> rate
$$
\dot M_*\simeq\pi b_{\rm cap}^2\rho_*v
=\frac{4\pi G^2M_{\rm BH}^2\rho_*}{c^2v}.
$$
For a supersonic gas with negligible <sound speed>, <Bondi--Hoyle--Lyttleton accretion> has capture radius $r_a=2GM_{\rm BH}/v^2$ and rate
$$
\dot M_{\rm BHL}\simeq\pi r_a^2\rho_gv
=\frac{4\pi G^2M_{\rm BH}^2\rho_g}{v^3}.
$$
Consequently
$$
\boxed{\frac{\dot M_*}{\dot M_{\rm BHL}}
\simeq\frac{\rho_*}{\rho_g}\left(\frac vc\right)^2.}
$$
For comparable densities and $v=200\,\mathrm{km\,s^{-1}}$, the suppression is approximately $4\times10^{-7}$. Gas can collide, dissipate orbital energy, and be captured from a region of size $GM/v^2$; collisionless particles generally leave on an unbound orbit unless their <angular momentum> is small enough for direct capture. <Chandrasekhar dynamical friction> can slow the hole by transferring energy to a stellar wake, but does not mean that the stars producing that wake are swallowed.
There is an important accuracy limitation in the requested numerical coefficient. Newtonian orbital mechanics evaluated at a relativistic <event horizon> cannot give the exact capture boundary. In <Schwarzschild spacetime>, a slowly incoming massive particle has dimensionless energy $\mathcal E\simeq1$ and <timelike geodesic effective potential>
$$
V(r)=\left(1-\frac{2r_g}{r}\right)
\left(1+\frac{\ell^2}{c^2r^2}\right),\qquad r_g=\frac{GM_{\rm BH}}{c^2}.
$$
The double turning point satisfying $V=1$ and $dV/dr=0$ occurs at $r=4r_g$ and $\ell=4GM_{\rm BH}/c$. The exact low-speed boundary is therefore $b_{\rm cap}\simeq4GM_{\rm BH}/(cv)$, twice the rough value; it makes the rate ratio $4(\rho_*/\rho_g)(v/c)^2$. The key suppression scaling is unchanged.
Direct capture of ordinary stars or <dark matter> is therefore an inefficient route to rapid growth of a <supermassive black hole> under these conditions. <Tidal disruption events> can capture stellar debris without swallowing an intact point-like star, and special dense environments can modify the supply. These qualifications do not make collisionless capture equivalent to dissipative gas accretion.
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