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Homologously Collapsing Polytropic Spheres
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Review of Goldreich and Weber (1980)
In an accompanying discussion, we have reviewed the selfsimilar, dynamical model that Peter Goldreich & Stephen Weber (1980, ApJ, 238, 991) developed to describe the nearhomologous collapse of stellar cores that obey an equation of state having an adiabatic index, — and polytropic index, . Here we examine whether or not this selfsimilar collapse model can be extended to configurations with arbitrary polytropic index.
Governing Equations
We begin with the set of principal governing equations that are already written in terms of a stream function and a timevarying radial coordinate, , as developed in the accompanying discussion of Goldreich & Weber's (1980) work. The continuity equation, the Euler equation, and the Poisson equation are, respectively,

Following Goldreich & Weber (1980), the normalization length scale, , that appears in this set of equations is the same length scale that is used in deriving the LaneEmden equation, namely,
where the subscript, "c", denotes central values and, as presented in our introductory discussion of barotropic supplemental relations,
But, unlike Goldreich & Weber, we will leave the polytropic index unspecified. Inserting this equation of state expression into the definition of the normalization length scale leads to,
Following Goldreich & Weber, we allow the normalizing scale length to vary with time and adopt an accelerating coordinate system with a timedependent dimensionless radial coordinate,
(The spatial operators in the above set of principal governing equations exhibit a subscript, , indicating the adoption of this accelerating coordinate system.) This, in turn, reflects a timevarying central density; specifically,
Next, we normalize the density by the central density, defining a dimensionless function,
which is in line with the formulation and evaluation of the LaneEmden equation, where the primary dependent structural variable is the dimensionless polytropic enthalpy,
And we normalize both the gravitational potential and the enthalpy to the square of the central sound speed,









giving,



and,



With these additional scalings, the continuity equation becomes,



the Euler equation becomes,



and the Poisson equation becomes,
Homologous Solution
Goldreich & Weber (1980) discovered that the governing equations admit to an homologous, selfsimilar solution if they adopted a stream function of the form,



which, when acted upon by the various relevant operators, gives,









This streamfunction appears to be a reasonable choice, as well, in the more general case that we are considering, in which case the radial velocity profile is, as in the Goldreich & Weber derivation,



Also, in our more general case, the continuity equation gives,





constant, 
independent of time. But the Euler equation becomes,









Because everything on the lefthand side of this scaled Euler equation depends only on the dimensionless spatial coordinate, , while everything on the righthand side depends only on time — via the parameter, — both expressions must equal the same (dimensionless) constant. We will follow the lead of Goldreich & Weber (1980) and call this constant, . From the terms on the lefthand side, we conclude that the dimensionless gravitational potential is,



Inserting this expression into the Poisson equation gives,









which becomes the familiar LaneEmden equation for arbitrary n when
From the terms on the righthand side we conclude, furthermore, that the nonlinear differential equation governing the timedependent variation of the scale length, , is,



© 2014  2020 by Joel E. Tohline 