Kelvin stellar mode (${\omega }_l$)

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 $U=A{r}^l{Y}_l^m$, the free-surface Lagrangian pressure perturbation condition and the surface gravity perturbation of a uniform-density sphere giveIn detail $\delta p/\rho ={\omega }^{2}U-\delta \Phi$ 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: $\mathbf{\xi }=\mathbf{\Omega }\times \mathbf{r}$ is tangent to the sphere and has zero restoring force but nonzero curl.