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Stahler's Mass-Radius Relationship for Embedded Polytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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In an accompanying chapter that discusses detailed force-balanced models of embedded (and pressure-truncated) polytropes, we review S. W. Stahler's (1983) pair of parametric relations for the equilibrium mass and equilibrium radius for such systems, namely,


~M

~=~


M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2 
\biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\xi_e}


~R_\mathrm{eq}

~=~


R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\xi_e}

where,

M_\mathrm{SWS} = 
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,


R_\mathrm{SWS} = \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,

and point out that Stahler (1983) (see his equation B13) explicitly states that the relevant mass-radius relationship for ~n = 5 embedded polytropes is,


\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
+ \frac{20\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4

~=~


~0 \, .


Whitworth's (1981) Isothermal Free-Energy Surface

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