Self-gravitating adiabatic displacement equations

The fluid displacement $\mathbf{\xi }$ determines Eulerian density through $\delta \rho =-\nabla \cdot (\rho \mathbf{\xi })$. The pressure law is $\delta p+\mathbf{\xi }\cdot \nabla p=-\gamma p\nabla \cdot \mathbf{\xi }$. The linear momentum equation must include both $-\delta \rho \nabla \Phi$ and $-\rho \nabla \delta \Phi$, with ${\nabla }^2\delta \Phi =4\pi G\delta \rho$. Retaining the last term incorporates perturbed self-gravity.

A constant-density gas sphere in hydrostatic equilibrium has $g={\omega }_d^{2}r$ and $p=\rho {\omega }_d^2(R^2-r^2)/2$. Its stellar oscillations can be expanded using regular solid harmonics and radial displacement amplitudes. Uniform density does not imply uniform specific entropy, so this equilibrium can be radially stable and still exhibit convective instability.

For the even polynomial family of degree $n\ge 2$, the physical radial squared frequency is $[\gamma n(n+1)/2-4]{\omega }_d^2$. The fundamental $n=2$ mode is stable when $\gamma >4/3$ and marginal at $\gamma =4/3$. At angular degree zero the gradient of the regular solid harmonic vanishes, so the auxiliary displacement amplitude multiplying it is redundant.

When the amplitudes of uniform-density stellar oscillation are finite even polynomials, their highest coefficients close algebraically. With $L=n(2l+n+1)$ and $x={\omega }^2/{\omega }_d^2$, the nonradial coefficient determinant is $x^2+(4-\gamma L/2)x-l(l+1)$. A degree balance gives a necessary frequency relation without solving every lower coefficient or imposing surface conditions.

The frequency ${\omega }_d=(GM/R^3{)}^{1/2}=(4\pi G\rho /3{)}^{1/2}$ measures the inverse gravitational timescale of a uniform-density sphere. It is the natural normalization for uniform-density stellar oscillation and the linear restoring or destabilizing accelerations.