# Stahler's Mass-Radius Relationship for Embedded Polytropes

## Review

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,

 $\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr) + \frac{20\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4$ $~=~$ $~0 \, .$

In what was intended to be a complementary discussion, our free-energy analysis of embedded polytropes produced a virial equilibrium expression of the general form,

$\mathcal{A} - \mathcal{B}\chi_\mathrm{eq}^{4 -3\gamma_g} +~ \mathcal{D}\chi_\mathrm{eq}^4 = 0 \, ,$

where,

 $~\chi_\mathrm{eq}$ $~\equiv$ $\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, ,$ $~\mathcal{A}$ $~\equiv$ $\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2 \cdot \tilde\mathfrak{f}_W \, ,$ $~\mathcal{B}$ $~\equiv$ $\frac{4\pi}{3} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{\gamma} \cdot \tilde\mathfrak{f}_A \, ,$ $~\mathcal{D}$ $~\equiv$ $\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, ,$

and,

 $~R_\mathrm{norm}$ $~\equiv$ $~ \biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma_g} \biggr]^{1/(4-3\gamma_g)} =\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{(n-1)} \biggr]^{1/(n-3)} \, ,$ $~P_\mathrm{norm}$ $~\equiv$ $~ \biggl[ \frac{K^4}{G^{3\gamma_g} M_\mathrm{tot}^{2\gamma_g}} \biggr]^{n/(n-3)} = \biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, ,$

and,

Structural Form Factors for Pressure-Truncated Polytropes

 $~\tilde\mathfrak{f}_M$ $~=$ $~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi}$ $\tilde\mathfrak{f}_W$ $~=$ $~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi}$ $\tilde\mathfrak{f}_A$ $~=$ $\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} + \tilde\Theta^{n+1}$

When we went back to compare the mass-radius relationship that results from our very general virial equilibrium expression to the one published by Stahler for pressure-truncated $~n = 5$ polytropes, they did not appear to agree. In what follows, we methodically plow through this comparison in considerable detail to uncover whatever discrepancies might exist.

## Comparison

First, let's insert the definitions of the coefficients $~\mathcal{A}$, $~\mathcal{B}$, and $~\mathcal{C}$ into the virial equilibrium expression, replacing, where necessary, the adiabatic exponent in favor of the polytropic index, using the relation, $~\gamma_g = (n+1)/n$.

 $~0$ $~=$ $~\mathcal{A} - \mathcal{B}\chi_\mathrm{eq}^{(n-3)/n} +~ \mathcal{D}\chi_\mathrm{eq}^4$ $~=$ $~ \frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2 \cdot \tilde\mathfrak{f}_W ~-~ \frac{4\pi}{3} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n} \cdot \tilde\mathfrak{f}_A \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr)^{(n-3)/n} +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr) ^4$

Next, explicitly spelling out as well the definitions of our adopted normalization radius and normalization pressure — recognizing that $~P_\mathrm{norm} R_\mathrm{norm}^4 = GM^2_\mathrm{tot}$ — and multiply the expression through by $[3GM_\mathrm{tot}^2/(4\pi)]$.

 $~0$ $~=$ $~ \frac{3}{20\pi} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) GM_\mathrm{limit}^2 ~-~ GM_\mathrm{tot}^2 \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \biggr]_\mathrm{eq}^{(n+1)/n} \biggl[ \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \biggr] \biggl[ \biggl( \frac{K}{G} \biggr) M_\mathrm{tot}^{(1-n)/n} \biggr] R_\mathrm{eq}^{(n-3)/n} +~ P_e R_\mathrm{eq}^4$ $~=$ $~ \frac{3}{20\pi} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) GM_\mathrm{limit}^2 ~-~ K \biggl[ \frac{3}{4\pi} \cdot \frac{M_\mathrm{limit}}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n} \mathfrak{f}_A R_\mathrm{eq}^{(n-3)/n} +~ P_e R_\mathrm{eq}^4 \, .$

As has been pointed out in our separate, more general discussion of the virial equilibrium of polytropes, if we multiply this expression through by $~R_\mathrm{eq}^{-4}$, set all three structural form factors, $~\mathfrak{f}_i$, equal to unity, and replace $~M_\mathrm{limit}$ with the notation, $~M_0$, the expression exactly matches the one presented as equation (5) of Whitworth, which reads:

But I like this last version of our derived expression as well because it shows some resemblance to the mass-radius relationship presented by Stahler and highlighted above: The first term on the left-hand-side is a constant times the square of the mass; the third term is a constant times the fourth power of the equilibrium radius; and the middle term shows a cross-product of the mass and radius (in our case, each is raised to a power other than unity). In an effort to make the comparison with Stahler even clearer, let's rewrite our expression in terms of the mass and equilibrium radius, normalized respectively to $~M_\mathrm{SWS}$ and $~R_\mathrm{SWS}$.

 $~0$ $~=$ $~ \frac{3G}{20\pi} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2 \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]^2 ~+~ P_e \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr]^4$ $~ ~-~ K \mathfrak{f}_A \biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]^{(n+1)/n} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr]^{(n-3)/n}$ $~=$ $~ \frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr)^{3} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2 \biggl[ K_n^{4n} P_\mathrm{e}^{(3-n)} \biggr]^{1/(n+1)} G^{-2} ~+~ \biggl( \frac{n+1}{n} \biggr)^{2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \biggl[ K_n^{4n} P_\mathrm{e}^{[(n+1)+2(1-n)]} \biggr]^{1/(n+1)} G^{-2}$ $~ ~-~\mathfrak{f}_A \biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n} \biggl( \frac{n+1}{nG} \biggr)^{[3(n+1)+(n-3)]/2n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} \biggl[ K^{1 + 2 + (n-3)/(n+1)} \biggr] \biggl[ P_\mathrm{e}^{(3-n) + (1-n)(n-3)/(n+1) } \biggr]^{1/(2n)}$ $~=$ $~ \frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr)^{3} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2 \biggl[ K_n^{4n} P_\mathrm{e}^{(3-n)} \biggr]^{1/(n+1)} G^{-2} ~+~ \biggl( \frac{n+1}{n} \biggr)^{2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \biggl[ K_n^{4n} P_\mathrm{e}^{(3-n)} \biggr]^{1/(n+1)} G^{-2}$ $~ ~-~\mathfrak{f}_A \biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n} \biggl( \frac{n+1}{n} \biggr)^{2} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} \biggl[ K^{4n/(n+1)} \biggr] \biggl[ P_\mathrm{e}^{(3-n)/(n+1) } \biggr] G^{-2}$ $~=$ $~ \frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2 ~-~\mathfrak{f}_A \biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} ~+~ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4$

 © 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation