The Jeans criterion for hydrostatic and infalling gas

Aims. We study the local gravitational instability of non-rotating astrophysical fluids allowing for the presence of an external gravitational potential in addition to the fluid self-gravity. Methods. We present a self-consistent linear-perturbation analysis, taking into account pressure and density gradients in the background medium. We explore two different steady-state configurations for the unperturbed gas: hydrostatic equilibrium and infall into a gravitational potential well. Results. We show that in both cases the instability criterion is the classical Jeans criterion, which, contrary to previous claims, is not modified by the presence of the external gravitational field. While in the case of hydrostatic equilibrium linear local perturbations are always gravitationally stable, the conditions for gravitational instability can be met in the case of infalling gas, also in the presence of additional non-gravitational forces such as those due to winds. Conclusions. We conclude that the Jeans criterion can have a role in regulating the formation of clumps and star clusters in streams or shells of gas infalling into galactic gravitational potential wells, as well as, on smaller scales, the fragmentation of gas in collapsing molecular clouds.