= Solution
<Hydrostatic equilibrium> means that the local <pressure> gradient balances <gravity>, with no large <acceleration> or strength-supported departure from that balance. In a spherical <planet>,
$$
\boxed{\frac{dP}{dr}=-\rho(r)\frac{GM(r)}{r^2}.}
$$
Here $M(r)$ is the enclosed <mass>, not a constant total <mass> at every interior <radius>. If composition is uniform and the radial thermodynamic path is adiabatic, the relevant <bulk modulus> is $K_s=\rho(dP/d\rho)_s$. The <chain rule> then gives the Adams-Williamson relation
$$
\boxed{\frac{d\rho}{dr}=\frac{\rho}{K_s}\frac{dP}{dr}
=-\frac{GM(r)\rho^2}{K_s r^2}.}
$$
A <phase> transition or composition jump must be treated separately; its <density> change is not obtained by integrating one smooth homogeneous-adiabatic equation through it.
The <seismic parameter> is
$$
\boxed{\Phi=V_P^2-\frac43V_S^2=\frac{K_s}{\rho},\qquad
\frac{d\rho}{dr}=-\frac{GM(r)\rho}{r^2\Phi}.}
$$
The <shear modulus> $\mu$ measures resistance to shear deformation. Solids with $\mu>0$ transmit shear waves; an ideal fluid has zero static <shear modulus>, so it cannot carry a bulk <S wave>, while compressional <P waves> can still propagate through its nonzero <bulk modulus>.
Limitations include uncertain seismic coverage, <temperature> and composition tradeoffs, nonadiabatic boundary layers, <phase> changes, lateral heterogeneity, <rotation> and finite material strength. Small asteroids and comet nuclei commonly retain irregular shapes because their self-gravity is too weak to overcome strength and impact history; a few larger bodies, such as Ceres, are much closer to global hydrostatic figures. A spherical shape alone is not a complete proof of present <hydrostatic equilibrium>.
Two complementary remote constraints are precise spacecraft tracking, which measures gravitational harmonics and constrains <mass> distribution when combined with shape and <rotation>; and tidal or rotational measurements, such as <Love numbers>, <librations> and <spin> <precession>, which constrain deformability, <moment of inertia> and possible fluid layers. Magnetic induction measurements in a time-varying external field can provide an additional conductivity constraint, but are not a unique <density> inversion.
For the \b[June 2018] lunar picture, start with a <regolith> and fractured <crust>, a largely anorthositic <crust> tens of kilometres thick, a silicate <mantle>, a deep partially molten region and a small iron-rich core. GRAIL and Apollo-based models gave a global mean crustal thickness around $34$-$43$ km.<The Crust of the Moon as Seen by GRAIL by Mark A. Wieczorek and colleagues>[abstract: mean crustal thickness and <gravity>/seismic constraints]{ref} A representative seismic model has a partial-melt boundary near <radius> $480$ km, a liquid outer-core boundary near $330$ km and a solid inner-core <radius> near $240$ km; these are model-dependent \b[<radii> from the centre], not depths below the surface.<Seismic detection of the layers of the lunar core by Renee C. Weber and colleagues>[core-reflection analysis and inferred layer <radii>]{ref}
\Image[/past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018-lunar-interior-model.png]
{title=A representative pre-June-2018 lunar interior model}
{description=An original schematic cross-section labels the crust, silicate mantle, deep partial-melt region, fluid outer core and solid inner core. Approximate central layer radii are 480, 330 and 240 kilometres; the thin crust is shown with a visually enlarged width and all boundaries are model dependent.}
The long-running Apollo network had only four principal stations on the near side, with poor global coverage and strong scattering in the fractured near-surface material. Far-side structure and weak deep reflected <phases> were therefore difficult to constrain. The historical date matters: the answer does not import later Mars seismology or later lunar-core refinements into the 2018 premise.
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