= Kelvin stellar mode
{c}
{title2=$\omega_l$}
= Kelvin f mode
{c}
{synonym}
The <Kelvin stellar modes> are the potential <incompressible stellar surface modes> of a nonrotating self-gravitating uniform-density sphere with <incompressible flow>. For a regular displacement potential $U=A r^lY_l^m$, the free-surface <Lagrangian pressure perturbation> condition and the <surface gravity perturbation of a uniform-density sphere> give
$$
\omega_l^2=\frac{2l(l-1)}{2l+1}\frac{GM}{R^3},\qquad l\geq1.
$$
In detail $\delta p/\rho=\omega^2U-\delta\Phi$ and $\delta p/\rho=(GM/R^2)\xi_r$ at $r=R$, so $\omega_l^2=lGM/R^3-4\pi G\rho l/(2l+1)$. The $l=1$ branch is rigid translation with zero restoring force. This potential family does not exhaust vortical zero-frequency displacements: $\boldsymbol\xi=\boldsymbol\Omega\times\mathbf r$ is tangent to the sphere and has zero restoring force but nonzero <curl>.
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