User:Tohline/Apps/HomologousCollapse
Homologously Collapsing Polytropic Spheres
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Review of Goldreich and Weber (1980)
This is principally a review of the dynamical model that Peter Goldreich & Stephen Weber (1980, ApJ, 238, 991) developed to describe the near-homologous collapse of stellar cores. As we began to study the Goldreich & Weber paper, it wasn't immediately obvious how the set of differential governing equations should be modified in order to accommodate a radially contracting (accelerating) coordinate system. I did not understand the transformed set of equations presented by Goldreich & Weber as equations (7) and (8), for example. At first, I turned to Poludnenko & Khokhlov (2007, Journal of Computational Physics, 220, 678) — hereafter, PK07 — for guidance. PK07 develop a very general set of governing equations that allows for coordinate rotation as well as expansion or contraction. Ultimately, the most helpful additional reference proved to be §19.11 (pp. 187 - 190) of Kippenhahn & Weigert [ KW94 ].
Governing Equations
Length
Following Goldreich & Weber (1980), we choose the same length scale for normalization that is used in deriving the Lane-Emden equation, which governs the hydrostatic structure of a polytrope of index <math>~n</math>, that is,
<math> a_\mathrm{n} \equiv \biggl[\frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr)\biggr]^{1/2} \, , </math>
where the subscript, "c", denotes central values and, as presented in our introductory discussion of barotropic supplemental relations,
<math>~H = (n+1) \kappa \rho^{1/n} \, .</math>
Substitution of this equation of state expression leads to,
<math> a = \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \rho_c^{-(n-1)/(2n)} \, . </math>
Most significantly, Goldreich & Weber (see their equation 6) allow the normalizing scale length to vary with time in order for the governing equations to accommodate a self-similar dynamical solution. In doing this, they effectively adopted an accelerating coordinate system with a time-dependent dimensionless radial coordinate,
<math>~\vec\mathfrak{x} \equiv \frac{1}{a(t)} \vec{r} \, .</math>
This, in turn, will mean that either the central density varies with time, or the specific entropy of all fluid elements (captured by the value of <math>~\kappa</math>) varies with time, or both. In practice, Goldreich & Weber assume that <math>~\kappa</math> is held fixed, so the time-variation in the scale length, <math>~a</math>, reflects a time-varying central density; specifically,
<math> \rho_c = \biggl\{ \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \frac{1}{a(t)} \biggr\}^{2n/(n-1)} \, . </math>
Given the newly adopted dimensionless radial coordinate, the following replacements for the spatial operators should be made, as appropriate, throughout the set of governing equations:
<math>~\nabla_r ~\rightarrow~ a^{-1} \nabla_\mathfrak{x}</math> and <math>~\nabla_r^2 ~\rightarrow~ a^{-2} \nabla_\mathfrak{x}^2 \, .</math>
Specifically, the continuity equation, the Euler equation, and the Poisson equation become, respectively,
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Mass-Density and Speed
Next, Goldreich & Weber (1980) (see their equation 10) choose to normalize the density by the central density, specifically defining a dimensionless function,
<math>f \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} \, .</math>
Keeping in mind that <math>~n = 3</math>, this is also in line with the formulation and evaluation of the Lane-Emden equation, where the primary dependent structural variable is the dimensionless polytropic enthalpy,
<math>\Theta_H \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
Also, Goldreich & Weber (1980) (see their equation 11) normalize the gravitational potential to the square of the central sound speed,
<math>c_s^2 = \frac{\gamma P_c}{\rho_c} = \frac{4}{3} \kappa \rho_c^{1/3} = \frac{4}{3}\biggl(\frac{\kappa^3}{\pi G}\biggr)^{1/2} [a(t)]^{-1} \, .</math>
Specifically, their dimensionless gravitational potential is,
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<math>~\sigma</math> |
<math>~\equiv</math> |
<math>~\frac{\Phi}{c_s^2} = \biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr] \Phi \, ,</math> |
and the similarly normalized enthalpy may be written as,
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<math>~\frac{H}{c_s^2} </math> |
<math>~=</math> |
<math>~\biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr] 4\kappa \rho^{1/3} </math> |
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<math>~=</math> |
<math>~3 \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} </math> |
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<math>~=</math> |
<math>~3f \, .</math> |
With these additional scalings, the continuity equation becomes,
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<math>~\frac{d\ln f^3}{dt} </math> |
<math>~=</math> |
<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi \, ,</math> |
the Euler equation becomes,
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<math>~ \biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr] \biggl[ \frac{d\psi}{dt} - \frac{1}{2a^2} ( \nabla_\mathfrak{x} \psi )^2 \biggr] </math> |
<math>~=</math> |
<math>~ - 3 f - \sigma \, ;</math> |
and the Poisson equation becomes,
<math>\nabla_\mathfrak{x}^2 \sigma = 3f^3 \, .</math>
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© 2014 - 2021 by Joel E. Tohline |