Self-gravitating fluid dynamics, unstabilities and solitons
This work studies the hydrodynamics of self-gravitating compressible isothermal fluids. We show that the hydrodynamic evolution equations in absence of viscosity are scale covariant. We study the evolution of the time dependent fluctuations around singular and regular isothermal spheres. We linearize the fluid equations around such stationary solutions and apply Laplace transform to solve them. We find that the system is stable below a critical size (X ~ 9.0 in dimensionless variables) and unstable above; this size is the same critical size found in the study of the thermodynamical stability in the canonical ensemble and associated to a center-to-border density ratio of 32.1 . We prove that the value of this critical size is independent of the Reynolds number of the system. Furthermore, we give a detailed description of the series of successive dynamical instabilities that appear at higher and higher sizes following the geometric progression X_n ~ 10.7^n. We turn then to study exact solutions of the hydrodynamic equations without viscosity and we provide analytic and numerical axisymmetric soliton-type solutions. The stability of exact solutions corresponding to a collapsing filament is studied by computing linear fluctuations. Radial fluctuations growing faster than the background are found for all sizes of the system. However, a critical size (X ~ 4.5) appears, separating a weakly from a strongly unstable regime.