 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.