On the Gravitational Collapse of a Gas Cloud in Presence of Bulk Viscosity

We analyze the effects induced by the bulk (or second) viscosity on the dynamics associated to the extreme gravitational collapse. Aim of the work is to investigate whether the presence of viscous corrections to the evolution of a collapsing gas cloud influence the top-down fragmentation process. To this end, we generalize the approach presented in the Hunter work [1] to include in the dynamics of the (uniform and spherically symmetric) cloud the negative pressure contribution associated to the bulk viscosity phenomenology. Within the framework of a Newtonian approach (whose range of validity is outlined), we extend to the viscous case either the Lagrangian, either the Eulerian motion of the system addressed in [1] and we treat the asymptotic evolution in correspondence to a viscosity coefficient of the form ζ = ζ0 ρ 5/6 (ρ being the cloud density and ζ0 = const.). We show how the adiabatic-like behavior of the gas is deeply influenced by viscous correction when its collapse reaches the extreme regime toward the singularity. In fact the density contrast associated to a given scale of the fragmentation process acquires, asymptotically, a vanishing behavior which prevents the formation of sub-structures. Since in the non-viscous case the density contrasts remain constant, we can conclude that in the adiabatic-like collapse the top down mechanism of structures formation is suppressed as soon as viscous effects are taken into account. Such a feature is not present in the isothermal-like collapse because the sub-structures formation is yet present and outlines the same behavior as in the non-viscous case. We emphasize that in the adiabatic-like collapse the bulk viscosity is also responsible for the appearance of a threshold scale beyond which perturbations begin to increase; this issue, absent in the non-viscous case, is equivalent to deal with a Jeans length. A discussion of the physical character that the choice ν = 5/6 takes place in the present case is provided.