= Solution
The <Hill radius> compares planetary <gravity> with the differential stellar tide. Expanding the stellar tide near a circular orbit gives $GM_p/r_H^2\simeq3\Omega^2r_H$, so
$$
\boxed{r_H=a\left(\frac{M_p}{3M_\star}\right)^{1/3}.}
$$
It sets the size of the <planet>'s tidal neighbourhood, controls encounter and <feeding zone> scales, and limits the region within which material can remain bound against stellar shear. It is not itself a solid surface or a guarantee of capture.
The <Bondi radius> compares planetary potential energy per unit <mass> with the <gas> thermal scale,
$$
\boxed{r_B=\frac{GM_p}{c_s^2}.}
$$
This convention uses the ambient <sound speed>; definitions based on escape speed can differ by a factor of two. Inside $r_B$, <gravity> can substantially perturb thermal <gas>. <Gas> still needs a way to dissipate energy and <angular momentum> to accrete; a geometric boundary alone does not supply a <gas>-capture rate.
For an embedded hydrostatic envelope a useful outer scale is $r_{\rm out}\sim\min(r_B,r_H)$, subject to disk thickness and <gas> recycling. Let $h=H/a=c_s/(a\Omega)$ and $q=M_p/M_\star$. The <Bondi-Hill crossover in a protoplanetary disk> obeys
$$
\frac{r_B}{a}=\frac q{h^2},\qquad\frac{r_H}{a}=\left(\frac q3\right)^{1/3},\qquad\boxed{r_B=r_H\ \Longrightarrow\ q=\frac{h^3}{\sqrt3}.}
$$
Small cores are usually thermally limited, whereas larger ones become tidally limited. These radii also organize <pebble accretion>: <gas> <drag> can turn an encounter into capture, with drift-dominated and shear-dominated regimes depending on particle stopping and encounter times. There is no universal accretion efficiency fixed by $r_H$ or $r_B$ alone.<Lecture notes on the formation and early evolution of planetary systems by Philip J. Armitage>[Version1, sectionIII.C.1, pages32–34: <core accretion> and envelope boundaries]{ref}
\Image[/past-exam-of-the-exoplanets-and-planetary-systems-course-of-the-university-of-cambridge/2015/paper-1-bondi-hill-scales.png]
{title=Bondi and Hill radii exchange their roles as the smaller planetary-envelope scale}
{source=paper-1-bondi-hill-scales.py}
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