Mechanics of the affine star model

Abstract

In a recent pioneering study of the phenomena occurring when a star is disrupted by passage through the tidal field of a large black hole, it was found convenient to make use of a simplified affine star model (described in terms of a 3 × 3 deformation matrix q) as a preliminary approximation pending more accurate hydrodynamical investigations. The present work consists of a general study of the intrinsic mechanical properties of a wide class of such affine star models (including allowance for non-adiabatic effects such as thermonuclear energy release and viscous dissipation) independently of any particular physical context. It is first shown explicitly that the canonical equations of internal motion, as derived directly from the appropriate internal Lagrangian function L_I(q, $$\rm {\dot q}$$), are the same as those obtainable by applying the tensor virial theorem to the first terms in a Taylor expansion about the centre of a corresponding Newtonian hydrodynamic model. An extension to the general compressible case of the classical Dedekind theorem for incompressible fluid models is shown to arise from the discrete symmetry of the Lagrangian L_I under the operation of replacing q by its transpose. The Lagrangian L_I is also invariant under a continuous 0(4) symmetry group action which is the direct product of the usual 0(3) action associated with angular momentum conservation and an independent 0(3) action which is shown to be associated with vorticity conservation. After a general discussion of the mechanical properties arising from these symmetries, it is shown in particular that any (pseudo-stationary) equilibrium state of a general compressible affine star model is conformally related to one of the (well known) equilibrium states of the corresponding incompressible fluid model.