User:Tohline/SSC/Structure/PolytropesASIDE1
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ASIDE: Whitworth's Scaling
Bounded Adiabatic
For adiabatic configurations <math>(\delta_{1\gamma_g} = 0)</math>, equilibrium states exist at radii given by the roots of the following expression:
<math> 3(\gamma_g-1) B\chi^{3 -3\gamma_g} ~-~A\chi^{-1} -~ 3D\chi^3 = 0 \, . </math>
This is precisely the same condition that derives from setting equation (3) to zero in Whitworth's (1981, MNRAS, 195, 967) discussion of the "global gravitational stability for one-dimensional polyropes." The overlap with Whitworth's narative is perhaps clearer after introducing the algebraic expressions for the coefficients <math>A</math>, <math>B</math>, and <math>D</math>, dividing the equation through by <math>(3\chi^3 V_0) = (4\pi R^3)</math>, and rewriting <math>R</math> as <math>R_\mathrm{eq}</math> to obtain,
<math> P_e = K \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} - \biggl( \frac{3GM^2}{20\pi R_\mathrm{eq}^4} \biggr) \, . </math>
This exactly matches equation (5) of Whitworth, which reads:
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Ideally we would like to invert this equation to obtain an analytic expression for the configuration's equilibrium radius in terms of the physical parameters, <math>M</math>, <math>K</math>, and <math>P_e</math>. However, this cannot be accomplished for an arbitrary value of the adiabatic exponent, <math>\gamma_g</math>.
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