Self-gravitating adiabatic displacement equations
= Self-gravitating adiabatic displacement equations
The <fluid displacement> $\boldsymbol\xi$ determines Eulerian density through $\delta\rho=-\nabla\cdot(\rho\boldsymbol\xi)$. The pressure law is $\delta p+\boldsymbol\xi\cdot\nabla p=-\gamma p\nabla\cdot\boldsymbol\xi$. 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.