= Surface gravity perturbation of a uniform-density sphere
For a uniform-density sphere with <incompressible flow> of radius $R$ with one surface <spherical harmonic> displacement $\xi_r(R)=lA R^{l-1}Y_l^m$, the bulk density perturbation vanishes but the surface contributes $\delta\rho=\rho\xi_r(R)\delta(r-R)$. Continuity of the <Newtonian gravitational potential> and the <Poisson equation> give $[\partial_r\delta\Phi]_{\mathrm{out}-\mathrm{in}}=4\pi G\rho\xi_r(R)$. The regular interior and decaying exterior solutions are
$$
\delta\Phi_{\mathrm{in}}=C r^lY_l^m,\qquad
\delta\Phi_{\mathrm{out}}=C R^{2l+1}r^{-l-1}Y_l^m,\qquad
C=-\frac{4\pi G\rho l A}{2l+1}.
$$
The coefficient follows because the derivative jump is $-(2l+1)C R^{l-1}Y_l^m$. This surface source is why neglecting the bulk density perturbation does not justify neglecting the perturbed gravity.
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