The non-homogenous 1D Euler equations with self-gravity are given by

\frac{\partial }{\partial t} \begin{pmatrix} \rho \\ \rho u \\ E\end{pmatrix} + \frac{\partial }{\partial x} \begin{pmatrix} \rho u \\ \rho u^2 + p \\ u(E+p) \end{pmatrix} = \begin{pmatrix} 0 \\ - \rho \triangledown \phi_g \\ -\rho u \cdot \triangledown \phi_g \end{pmatrix}

which in short hand notation is

\vec{u}_t + \vec{F}(\vec{u})_x = \vec{S}(\vec{u})

where \vec{u} is vector of fluid variables, \vec{F} is their fluxes, and \vec{S} is the source-term vector.