= Angular-degree bound on perturbed stellar self-gravity
{title2=$|E_g|\leq\frac{4\pi G}{\ell(\ell+1)}\int r^4|\rho_{1,\ell}(r)|^2\,dr$}
For a normalized <spherical harmonic> of degree $\ell\geq1$, write $\rho_1=s(r)Y$ and $\phi_1=F(r)Y$. Integration of the radial <Poisson equation> gives $A=-4\pi G\int r^2F^*s\,dr$, where $A=\int[r^2|F'|^2+\ell(\ell+1)|F|^2]dr$. <Cauchy-Schwarz inequality> and $\int|F|^2\leq A/[\ell(\ell+1)]$ give the displayed bound on $|E_g|=A/(4\pi G)$. For bounded density amplitudes with fixed radial support this decays as $\ell^{-2}$, giving a quantitative angular short-scale justification of the <Cowling approximation>.
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