Steady flows, nonlinear gravitostatic waves, and Zeldovich pancakes in a Newtonian gas

We show that equations of Newtonian hydrodynamics and gravity describing one-dimensional steady gas flow possess nonlinear periodic solutions. In the case of a zero-pressure gas, the solution exhibits hydrodynamic similarity and is universal: it is a lattice of integrable density singularities coinciding with maxima of the gravitational potential. With finite-pressure effects included, there exists critical matter density that separates two regimes of behavior. If the average density is below the critical, the solution is a density wave which is in phase with the wave of the gravitational potential. If the average density is above the critical, the waves of the density and potential are out of phase. Traveling plane gravitostatic waves are also predicted and their properties elucidated. Specifically, a subsonic wave is made of two out-of-phase oscillations of matter density and gravitational potential. If the wave is supersonic, the density-potential oscillations are in phase.

Figure 1

Sketches of solutions $\rho (\varphi )$ to Eq. (9). In the generic case (black), there are two branches, ${\rho }_{+}(\varphi )$ and ${\rho }_{-}(\varphi )$, separated by the critical density ${\rho }_c$ [Eq. (10)], indicated by the straight dotted red line. For flowing zero-pressure matter, $w=0$, the solution ${\rho }_{-}(\varphi )$ [Eq. (15)] is shown in blue. For static, $j=0$, finite-pressure matter, the corresponding solution ${\rho }_{+}(\varphi )$ is sketched in orange.

Figure 2

Sketch of potential well (for fictitious particle of unit mass and position $\varphi$) formed by the monotonically decreasing potential energy function $U(\varphi )$ [Eq. (11)] and the constraint $\varphi \le 0$. For fixed energy $g^2/2$, the motion is oscillatory: at the leftmost turning point $\varphi ={\varphi }_1$ the particle temporarily stops and turns around, while at the rightmost point $\varphi =0$ it bounces off the wall and just reverses its direction of motion. Grayscale regions are not accessible for motion.

Figure 3

One period of spatial variation of the gravitational potential ${\varphi }_{-}$ (black) [Eq. (17)] and density ${\rho }_{-}=1/\sqrt{-{\varphi }_{-}}$ (blue) in reduced units [Eq. (16)] for a zero-pressure matter. Density singularity at the origin is the Zeldovich pancake.