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On the gravitational collapse of a gas cloud in the presence of bulk viscosity

We analyse the effects induced by the bulk (or second) viscosity on the dynamics associated with the extreme gravitational collapse. The aim of the work is to investigate whether the presence of viscous corrections to the evolution of a collapsing gas cloud influences the top-down fragmentation process. To this end, we generalize the approach presented by Hunter (1962 Astrophys. J. 136 594) to include in the dynamics of the (uniform and spherically symmetric) cloud the negative pressure contribution associated with 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 or the Eulerian motion of the system addressed in Hunter (1962 Astrophys. J. 136 594) and we treat the asymptotic evolution in correspondence with a viscosity coefficient of the form ζ = ζ0 ρ5/6(ρ being the cloud density and ζ0= const). We show how the adiabatic-like behaviour of the gas (i.e. when the polytropic index γ takes values 4/3 < γ ⩽ 5/3) is deeply influenced by viscous correction when its collapse reaches the extreme regime toward the singularity. In fact, for sufficiently large viscous contributions, density contrasts associated with a given scale of the fragmentation process acquire, asymptotically, a vanishing behaviour which prevents the formation of sub-structures. Since in the non-dissipative case density contrasts diverge (except for the purely adiabatic behaviour γ = 5/3 in which they remain constant), we can conclude that in the adiabatic-like collapse the top-down mechanism of structure formation is suppressed as soon as enough strong viscous effects are taken into account. Such a feature is not present in the isothermal-like (i.e. 1 ⩽ γ < 4/3) collapse because the sub-structure formation is yet present and outlines the same behaviour 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 (dependent on the polytropic index) beyond which perturbations begin to increase; this issue, absent in the non-viscous case, is equivalent to dealing with a Jeans length. A discussion of the physical character that the choice ν = 5/6 takes place in the present case is provided.