The stability of an isothermal gaseous sphere with gravitation

A dispersion relation has been obtained for waves propagating in an inhomogeneous gas with viscosity and gravity in the direction of the density gradient. It has been shown that a stationary distribution of isothermal gas is unstable under gravitational—convective perturbations. It is suggested that the initialand boundary-value problems of the non-stationary dynamics of an isothermal gas with gravity but without viscosity are incorrect and require regularization, for example, by adding viscosity. The presence of viscosity does not prevent the development of the Jeans long-wave instability. Numerical experiments based on a fully three-dimensional, non-stationary numerical code have demonstrated the development of a collapse inside an isothermal cloud.