SSC/Structure/PolytropesEmbedded

Embedded Polytropic Spheres

Part I:   General Properties

Part II:  Truncated Configurations with n = 1

Part III:  Truncated Configurations with n = 5

Part IV:  Other Considerations

Embedded Polytropes

In a separate discussion we have shown how to determine the structure of isolated polytropic spheres. These are rather idealized stellar structures in which the pressure and density both drop to zero at the surface of the configuration (for 0 ≤ $n$ < 5) or in which the equilibrium configuration extends to infinity (for 5 ≤ $n$ ≤ ∞). Here we consider how the equilibrium radius of a polytropic configuration of a given $M$ and $K_{\mathrm{n}}$ is modified when it is embedded in an external medium of pressure $P_e$. We will begin by reviewing the general properties of embedded (and truncated) polytropes for a wide range of polytropic indexes, principally summarizing the published descriptions provided by 📚 Gp. Horedt (1970, MNRAS, Vol. 151, pp. 81 - 86), by 📚 A. Whitworth (1981, MNRAS, Vol. 195, pp. 967 - 977), by Kimura (1981a), and by 📚 S. W. Stahler (1983, ApJ, Vol. 268, pp. 165 - 184). Then we will focus in more detail on polytropes of index $n$ = 1 and $n$ = 5 because their structures can be described by closed-form analytic expressions.  
 

General Properties

Horedt's Presentation

It appears as though 📚 Horedt (1970) — hereafter, Horedt70 — was the first to draw an analogy between the mass limit that is associated with bounded isothermal spheres — the so-called Bonnor-Ebert spheres — and the limiting mass that can be found in association with equilibrium sequences of embedded polytropes that have polytropic indexes $n>3$. Using a tilde to denote values of parameters at the (truncated) edge of a pressure-bounded polytropic sphere, Horedt70 (see the bottom of his p. 83) derives the following set of parametric equations relating the configuration's dimensionless radius, $r_a$, to a specified dimensionless bounding pressure, $p_a$:

$r_a\equiv \frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}}$	$=$	$\tilde{\xi }(-{\tilde{\xi }}^2{\tilde{\theta }}^{\prime }{)}^{(1-n)/(n-3)},$
$p_a\equiv \frac{P_{\mathrm{e}}}{P_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}}$	$=$	${\tilde{\theta }}_n^{n+1}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\prime }{)}^{2(n+1)/(n-3)},$

where it is understood that, as discussed elsewhere, ${\theta }_n(\xi )$ is the solution to the Lane-Emden equation for a polytrope of index $n$,

${\tilde{\theta }}^{\prime }$	$\equiv$	$\frac{d{\theta }_n}{d\xi }\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{a}\mathrm{t}\tilde{\xi },$
$R_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}$	$\equiv$	${\alpha }_r(\frac{{\alpha }_M}{M}{)}^{(1-n)/(n-3)}=[\frac{4\pi }{(n+1{)}^n}(\frac{G}{K_n}{)}^n{M}^{n-1}{]}^{1/(n-3)},$
$P_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}$	$\equiv$	$K_n(\frac{{\alpha }_M}{M}{)}^{2(n+1)/(n-3)}=K_n^{4n/(n-3)}[\frac{(n+1{)}^3}{4\pi G^3{M}^2}{]}^{(n+1)/(n-3)}.$

Notice that, via these normalizations, Horedt chose to express $R_{\mathrm{e}\mathrm{q}}$ and $P_{\mathrm{e}}$ in terms of $K_{\mathrm{n}}$ and the system's total mass, $M$.

Whitworth's Presentation

In §5 of his paper, 📚 Whitworth (1981) — hereafter, Whitworth81 — also presents the set of parametric equations that define what the equilibrium radius, $R_{\mathrm{e}\mathrm{q}}$, is of an embedded polytrope for a certain imposed external pressure, $P_{\mathrm{e}}$, namely,

$R_{\mathrm{e}\mathrm{q}}$	$=$	$R_{\mathrm{r}\mathrm{f}}\left\{\frac{4\eta }{5|\eta -1|}\right(\frac{\xi }{3}{)}^{\eta }|\frac{d{\theta }_n}{d\xi }{|}^{(2-\eta )}{\}}_{{\xi }_e}^{1/(3\eta -4)}$
$\Rightarrow (\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{r}\mathrm{f}}}{)}^{(3-n)}$	$=$	$\left[\frac{4(n+1)}{5}{]}^n\right(\frac{{\xi }_e}{3}{)}^{(n+1)}|\frac{d{\theta }_n}{d\xi }{|}_{{\xi }_e}^{(n-1)},$
$P_{\mathrm{e}}$	$=$	$P_{\mathrm{r}\mathrm{f}}\left\{2^{-8/\eta }\right(\frac{5|\eta -1|}{\eta }{)}^3\left(\frac{3}{\xi }{)}^4\right|\frac{d{\theta }_n}{d\xi }{|}^{-2}{\}}_{{\xi }_e}^{\eta /(3\eta -4)}{\theta }_n^{\eta /(\eta -1)}$
$\Rightarrow (\frac{P_{\mathrm{e}}}{P_{\mathrm{r}\mathrm{f}}}{)}^{(3-n)}$	$=$	$2^{-8n}\left\{\right(\frac{5}{n+1}{)}^3\left(\frac{3}{\xi }{)}^4{\theta }_n^{(3-n)}\right|\frac{d{\theta }_n}{d\xi }{|}^{-2}{\}}_{{\xi }_e}^{(n+1)},$

where, in order to obtain the second line of the two relations we have used the substitution, $\eta \to (1+1/n)$, and, as is detailed in an accompanying ASIDE, Whitworth81 "referenced" $P_{\mathrm{e}}$ and $R_{\mathrm{e}\mathrm{q}}$ to, respectively,

$P_{\mathrm{r}\mathrm{f}}^{(4-3\eta )}$	$=$	$2^{-2(4+\eta )}\left(\frac{3^4\cdot 5^3}{\pi }{)}^{\eta }\right[\frac{K_n^4}{G^{3\eta }{M}^{2\eta }}]$
$\Rightarrow P_{\mathrm{r}\mathrm{f}}^{(n-3)}$	$=$	$2^{-2(5n+1)}\left(\frac{3^4\cdot 5^3}{\pi }{)}^{(n+1)}\right[\frac{K_n^{4n}}{G^{3(n+1)}{M}^{2(n+1)}}],$
$R_{\mathrm{r}\mathrm{f}}^{\eta }$	$=$	$\frac{2^2}{K_n}(\frac{GM}{3\cdot 5}{)}^{\eta }{P}_{\mathrm{r}\mathrm{f}}^{(1-\eta )}$
$\Rightarrow R_{\mathrm{r}\mathrm{f}}^{(n+1)}$	$=$	$\left(\frac{2^2}{K_n}{)}^n\right(\frac{GM}{3\cdot 5}{)}^{(n+1)}{P}_{\mathrm{r}\mathrm{f}}^{-1}$
$\Rightarrow R_{\mathrm{r}\mathrm{f}}^{(3-n)}$	$=$	$\left(\frac{2^2}{K_n}{)}^{n(3-n)/(n+1)}\right(\frac{GM}{3\cdot 5}{)}^{(3-n)}{P}_{\mathrm{r}\mathrm{f}}^{(n-3)/(n+1)}$
	$=$	$\left(\frac{2^2}{K_n}{)}^{n(3-n)/(n+1)}\right(\frac{GM}{3\cdot 5}{)}^{(3-n)}\left\{2^{-2(5n+1)}\right(\frac{3^4\cdot 5^3}{\pi }{)}^{(n+1)}\left[\frac{K_n^{4n}}{G^{3(n+1)}{M}^{2(n+1)}}\right]{\}}^{1/(n+1)}$
	$=$	$K_n^n(2^2{)}^{-(n+1)}(\frac{GM}{3\cdot 5}{)}^{(3-n)}\left(\frac{3^4\cdot 5^3}{\pi }\right)\left[\frac{1}{G^3{M}^2}\right]$
	$=$	$2^{-2(n+1)}{\pi }^{-1}{3}^{n+1}\cdot 5^n{K}_n^n{G}^{-n}{M}^{1-n}$

Via these normalizations, Whitworth81 — as did Horedt70 — chose to express $R_{\mathrm{e}\mathrm{q}}$ and $P_{\mathrm{e}}$ in terms of $K_{\mathrm{n}}$ and the system's total mass, $M$.

To convert from Whitworth's expressions, which use one set of normalization parameters $(R_{\mathrm{r}\mathrm{f}},P_{\mathrm{r}\mathrm{f}})$, to Horedt's expressions, which use a somewhat different set of normalization parameters — identified here as $(R_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}},P_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}})$ — one simply needs to make use of the relations,

$(\frac{R_{\mathrm{r}\mathrm{f}}}{R_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}}{)}^{(3-n)}$	$=$	$3^{(n+1)}[\frac{5}{2^2(n+1)}{]}^n.$
$(\frac{P_{\mathrm{r}\mathrm{f}}}{P_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}}{)}^{(3-n)}$	$=$	$2^{8n}[\frac{(n+1{)}^3}{3^4\cdot 5^3}{]}^{(n+1)},$

Kimura's Presentation

At the same time Whitworth's work was being published, Kimura (1981a) also published a derivation of the equations that define the equilibrium properties of embedded, pressure-truncated polytropic configurations. (Note that an erratum has been published correcting typographical errors that appear in a few equations of the original paper.) When compared with, for example, Horedt's published work — which Kimura references — Kimura's set of structural equations are a bit more difficult to digest because they include (a) an equation-of-state index that is different from the traditional polytropic index — specifically (see his equation 6),

$\sigma \equiv (n+1{)}^{-1}$

— which was Kimura's effort to more gracefully accommodate discussions of isothermal $(n=\mathrm{\infty })$ configurations; and (b) an additional integer index, $m$, so that a single set of equations can be used to specify the structure of planar $(m=1)$ and cylindrical $(m=2)$ as well as spherical $(m=3)$ configurations. In the present context, we will fix the value to $m=3$. Kimura also chose to express his structural solutions in terms of a dimensionless radius, $\zeta$, instead of the traditional variable, $\xi$ — note that the two are related via the expression,

$\zeta =(n+1{)}^{1/2}\xi ;$

and in terms of a dimensionless gravitational potential, $\varphi$, instead of the traditional dimensionless enthalpy variable, ${\theta }_n$ — note that the two are related via the expression (see Kimura's equation 12),

$\varphi ={\sigma }^{-1}(1-{\theta }_n).$

Given this relationship, we note as well that,

${\varphi }^{\text{'}}\equiv \frac{d\varphi }{d\zeta }=-\frac{d{\theta }_n}{d\xi }\cdot \left[{\sigma }^{-1}\frac{d\xi }{d\zeta }\right]=-\frac{d{\theta }_n}{d\xi }(n+1{)}^{1/2}.$

The set of equilibrium equations derived by Kimura (1981a) in what he identifies as "Paper I" — see especially his equations number (16) and (23) — are summarized most succinctly in Table 1 of his "Paper II" (Kimura 1981b). The equations he presents for "radial distance," "pressure," and "fractional mass within $\tilde{\zeta }$" are, respectively,

$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{K}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}}}$	$=$	$\tilde{\zeta }=(n+1{)}^{1/2}\tilde{\xi },$
$\frac{P_{\mathrm{e}}}{P_{\mathrm{K}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}}}$	$=$	${\tilde{\theta }}_n^{n+1},$
$\frac{M}{M_{\mathrm{K}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}}}$	$=$	${\tilde{\zeta }}^2{\tilde{\varphi }}^{\text{'}}=(n+1{)}^{3/2}[-{\xi }^2\frac{d{\theta }_n}{d\xi }{]}_{\tilde{\xi }},$

where, expressed in terms of the central pressure, $p_{*}$, and the polytropic constant, $K_n,[$note that, in Kimura's paper, $H=K_n^{n/(n+1)}]$, the relevant normalization parameters are,

$R_{\mathrm{K}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}}$	$\equiv$	$(4\pi G{)}^{-1/2}H{p}_{*}^{\sigma -1/2}=(4\pi G{)}^{-1/2}{K}_n^{n/(n+1)}{p}_{*}^{(1-n)/[2(n+1)]},$
$P_{\mathrm{K}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}}$	$=$	$p_{*},$
$M_{\mathrm{K}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}}$	$=$	$(4\pi G{)}^{-3/2}(4\pi )H^2{p}_{*}^{2\sigma -1/2}=(4\pi G^3{)}^{-1/2}{K}_n^{2n/(n+1)}{p}_{*}^{(3-n)/[2(n+1)]}.$

In order to compare Kimura's equilibrium expressions for $R_{\mathrm{e}\mathrm{q}}$ and $P_e$ with the corresponding expressions presented by Horedt and by Whitworth, we need to replace $p_{*}$ by $M$ in both expressions. Inverting Kimura's expression for $M$, we have,

$p_{*}^{(3-n)/[2(n+1)]}$

$=$

$M(n+1{)}^{-3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{-1}(4\pi G^3{)}^{1/2}{K}_n^{-2n/(n+1)}.$

Hence,

$P_{\mathrm{K}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}}$	$=$	$[M(n+1{)}^{-3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{-1}(4\pi G^3{)}^{1/2}{K}_n^{-2n/(n+1)}{]}^{2(n+1)/(3-n)},$
	$=$	$[M^{-2}(n+1{)}^3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^2(4\pi G^3{)}^{-1}{K}_n^{4n/(n+1)}{]}^{(n+1)/(n-3)},$
	$=$	$P_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}[(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^2{]}^{(n+1)/(n-3)}$
$\Rightarrow P_e$	$=$	$P_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}{\tilde{\theta }}^{n+1}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{2(n+1)/(n-3)},$

which matches Horedt's expression for $P_e$. Also after replacement we obtain,

$R_{\mathrm{K}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}}$	$=$	$(4\pi G{)}^{-1/2}{K}^{n/(n+1)}[M(n+1{)}^{-3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{-1}(4\pi G^3{)}^{1/2}{K}_n^{-2n/(n+1)}{]}^{(1-n)/(3-n)}$
	$=$	$M^{(n-1)/(n-3)}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{(1-n)/(n-3)}(n+1{)}^{3(1-n)/2(n-3)}(4\pi {)}^{[(1-n)-(3-n)]/[2(3-n)]}{G}^{[3(1-n)-(3-n)]/[2(3-n)]}[K_n^{n(3-n)-2n(1-n)}{]}^{1/[(n+1)(3-n)]}$
	$=$	$M^{(n-1)/(n-3)}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{(1-n)/(n-3)}(n+1{)}^{3(1-n)/2(n-3)}(4\pi {)}^{1/(n-3)}{G}^{n/(n-3)}{K}_n^{-n/(n-3)}$
$\Rightarrow R_{\mathrm{e}\mathrm{q}}$	$=$	$\tilde{\xi }(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{(1-n)/(n-3)}(n+1{)}^{[3(1-n)+(n-3)]/2(n-3)}\left[4\pi \right(\frac{G}{K_n}{)}^n{M}^{(n-1)}{]}^{1/(n-3)}$
	$=$	$R_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}\tilde{\xi }(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{(1-n)/(n-3)},$

which exactly matches Horedt's expression for $R_{\mathrm{e}\mathrm{q}}$.

Stahler's Presentation

Similarly, in Appendix B of his work, 📚 Stahler (1983) — hereafter, SWS — states that the mass, $M$, associated with the equilibrium radius, $R_{\mathrm{e}\mathrm{q}}$, of embedded polytropic spheres is,

$M$	$=$	$M_{\mathrm{S}\mathrm{W}\mathrm{S}}\left(\frac{n^3}{4\pi }{)}^{1/2}\right\{{\theta }_n^{(n-3)/2}{\xi }^2\left|\frac{d{\theta }_n}{d\xi }\right|{\}}_{{\xi }_e}$
$R_{\mathrm{e}\mathrm{q}}$	$=$	$R_{\mathrm{S}\mathrm{W}\mathrm{S}}\left(\frac{n}{4\pi }{)}^{1/2}\right\{\xi {\theta }_n^{(n-1)/2}{\}}_{{\xi }_e}$

where, from his equations (7) and (B3) we deduce,

$M_{\mathrm{S}\mathrm{W}\mathrm{S}}=(\frac{n+1}{nG}{)}^{3/2}{K}_n^{2n/(n+1)}{P}_{\mathrm{e}}^{(3-n)/[2(n+1)]},$

$R_{\mathrm{S}\mathrm{W}\mathrm{S}}=(\frac{n+1}{nG}{)}^{1/2}{K}_n^{n/(n+1)}{P}_{\mathrm{e}}^{(1-n)/[2(n+1)]}.$

Notice that, via these two normalizations, SWS chose to express $R_{\mathrm{e}\mathrm{q}}$ and $M$ in terms of $K_{\mathrm{n}}$ and the applied external pressure, $P_{\mathrm{e}}$.

NOTE:   An accompanying chapter presents a much more detailed description of the sequences of truncated polytropic spheres that are derived and discussed by SWS.

Reconciliation

Here we demonstrate that Whitworth's and Stahler's presentations are equivalent to one another. We begin by plugging Stahler's definition of $M_{\mathrm{S}\mathrm{W}\mathrm{S}}$ into his expression for $M$, then inverting it to obtain an expression for $P_{\mathrm{e}}$ in terms of $M$ and $K_{\mathrm{n}}$.

$M$	$=$	$\left[\frac{(n+1{)}^3}{4\pi G^3}{]}^{1/2}{K}_n^{2n/(n+1)}{P}_{\mathrm{e}}^{(3-n)/[2(n+1)]}\right\{{\theta }_n^{(n-3)/2}{\xi }^2\left|\frac{d{\theta }_n}{d\xi }\right|{\}}_{{\xi }_e}$
$\Rightarrow P_{\mathrm{e}}^{(3-n)}$	$=$	$\left[\frac{4\pi G^3}{(n+1{)}^3}{]}^{(n+1)}{K}_n^{-4n}{M}^{2(n+1)}\right\{{\theta }_n^{(n-3)/2}{\xi }^2\left|\frac{d{\theta }_n}{d\xi }\right|{\}}_{{\xi }_e}^{-2(n+1)}$
	$=$	$\left[\frac{4\pi G^3{M}^2}{(n+1{)}^3}{]}^{(n+1)}{K}_n^{-4n}\right\{{\theta }_n^{(3-n)}{\xi }^{-4}|\frac{d{\theta }_n}{d\xi }{|}^{-2}{\}}_{{\xi }_e}^{(n+1)}$

Alternatively, plugging Whitworth's definition of $P_{\mathrm{r}\mathrm{f}}$ into his expression for $P_{\mathrm{e}}$ gives,

$P_{\mathrm{e}}^{(3-n)}$	$=$	$2^{2(5n+1)}(\frac{\pi }{3^4\cdot 5^3}{)}^{(n+1)}{2}^{-8n}\cdot 3^{4(n+1)}(\frac{5}{n+1}{)}^{3(n+1)}[G^3{M}^2{]}^{(n+1)}{K}_n^{-4n}\{{\theta }_n^{(3-n)}{\xi }^{-4}|\frac{d{\theta }_n}{d\xi }{|}^{-2}{\}}_{{\xi }_e}^{(n+1)}$
	$=$	$2^{2(n+1)}[\frac{\pi }{(n+1{)}^3}{]}^{(n+1)}[G^3{M}^2{]}^{(n+1)}{K}_n^{-4n}\left\{{\theta }_n^{(3-n)}{\xi }^{-4}\right|\frac{d{\theta }_n}{d\xi }{|}^{-2}{\}}_{{\xi }_e}^{(n+1)}.$

So Whitworth's and Stahler's relations for $P_{\mathrm{e}}(M,K_n)$ are, indeed, identical. Similarly examining Stahler's expression for the equilibrium radius, we find,

$R_{\mathrm{e}\mathrm{q}}$	$=$	$\left(\frac{n+1}{4\pi G}{)}^{1/2}{K}_n^{n/(n+1)}\right[\xi {\theta }_n^{(n-1)/2}{]}_{{\xi }_e}\{P_{\mathrm{e}}^{1/(n+1)}{\}}^{(1-n)/2}$
	$=$	$\left(\frac{n+1}{4\pi G}{)}^{1/2}{K}_n^{n/(n+1)}\right[\xi {\theta }_n^{(n-1)/2}{]}_{{\xi }_e}\left\{\right[\frac{4\pi G^3{M}^2}{(n+1{)}^3}\left]K_n^{-4n/(n+1)}\right[{\theta }_n^{(3-n)}{\xi }^{-4}|\frac{d{\theta }_n}{d\xi }{|}^{-2}{]}_{{\xi }_e}{\}}^{(1-n)/[2(3-n)]}$
$\Rightarrow R_{\mathrm{e}\mathrm{q}}^{(3-n)}$	$=$	$\left(\frac{n+1}{4\pi G}{)}^{(3-n)/2}{K}_n^{n(3-n)/(n+1)}{\xi }_e^{3-n}\right\{\left[\frac{4\pi G^3{M}^2}{(n+1{)}^3}{]}^{1/2}{K}_n^{-2n/(n+1)}\right[{\xi }^{-2}|\frac{d{\theta }_n}{d\xi }{|}^{-1}{]}_{{\xi }_e}{\}}^{(1-n)}$
	$=$	$(n+1{)}^{[(3-n)-3(1-n)]/2}(4\pi {)}^{[(n-3)+(1-n)]/2}{G}^{[(n-3)+3(1-n)]/2}[K_n^{(3-n)+2(n-1)}{]}^{n/(n+1)}{\xi }_e^{(3-n)+2(n-1)}{M}^{(1-n)}|\frac{d\theta }{d\xi }{|}_{{\xi }_e}^{(n-1)}$
	$=$	$(n+1{)}^n(4\pi {)}^{-1}{G}^{-n}{K}_n^n{M}^{(1-n)}\left[{\xi }^{(n+1)}\right|\frac{d{\theta }_n}{d\xi }{|}^{(n-1)}{]}_{{\xi }_e}.$

And Whitworth's expression becomes,

$R_{\mathrm{e}\mathrm{q}}^{(3-n)}$	$=$	$2^{-2(n+1)}{\pi }^{-1}{3}^{n+1}\cdot 5^n{K}_n^n{G}^{-n}{M}^{1-n}\left[\frac{4(n+1)}{5}{]}^n\right(\frac{{\xi }_e}{3}{)}^{(n+1)}|\frac{d{\theta }_n}{d\xi }{|}_{{\xi }_e}^{(n-1)}$
	$=$	$(n+1{)}^n(4\pi {)}^{-1}{K}_n^n{G}^{-n}{M}^{1-n}{\xi }_e^{(n+1)}|\frac{d{\theta }_n}{d\xi }{|}_{{\xi }_e}^{(n-1)}.$

Hence, Stahler's equilibrium radius, $R_{\mathrm{e}\mathrm{q}}$, exactly matches Whitworth's $R_{\mathrm{e}\mathrm{q}}$.

Summary

Once the function, ${\theta }_n(\xi )$, and its first derivative with respect to the dimensionless radial coordinate, $d{\theta }_n/d\xi$, are obtained via a solution of the Lane-Emden equation, the equilibrium radius, $R_{\mathrm{e}\mathrm{q}}$, and total mass, $M$, of a pressure-bounded polytrope can be expressed in terms of Stahler's normalizations as follows: $\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$ $=$ $\left(\frac{n}{4\pi }{)}^{1/2}\right[\xi {\theta }_n^{(n-1)/2}{]}_{{\xi }_e},$ $\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$ $=$ $(\frac{n^3}{4\pi }{)}^{1/2}{p}_a^{(n-3)/[2(n+1)]},$ where, $p_a$ $\equiv$ $\left[{\theta }_n^{(n-3)/2}{\xi }^2\right|\frac{d{\theta }_n}{d\xi }|{]}_{{\xi }_e}^{2(n+1)/(n-3)}={\theta }_n^{(n+1)}({\xi }^2\left|\frac{d{\theta }_n}{d\xi }\right|{)}_{{\xi }_e}^{2(n+1)/(n-3)}.$ Then, the external pressure, expressed in terms of Whitworth's normalization, is, $\frac{P_{\mathrm{e}}}{P_{\mathrm{r}\mathrm{f}}}$ $=$ $2^{8n/(n-3)}[\frac{(n+1{)}^3}{3^4\cdot 5^3}{]}^{(n+1)/(n-3)}{p}_a;$ and the conversion from Stahler's normalization to Whitworth's normalization of the radius is achieved via the expression, $\frac{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}{R_{\mathrm{r}\mathrm{f}}}$ $=$ $\left[\frac{3^{(n+1)}}{2^{(n+3)}}\right(\frac{5}{n+1}{)}^n{]}^{1/(n-3)}(\frac{\pi }{n}{)}^{1/2}{p}_a^{(1-n)/[2(n+1)]}.$

Chieze's Presentation

From equations (8), (10), and (68) in Chapter IV of [C67], we can immediately formulate the following expressions for, respectively, $P_e(\tilde{\xi }),R_{\mathrm{e}\mathrm{q}}(\tilde{\xi })$, and $M_{\mathrm{t}\mathrm{o}\mathrm{t}}(\tilde{\xi })$:

$\frac{P_e}{P_{\mathrm{C}\mathrm{h}}}$	$=$	${\tilde{\theta }}^{n+1},$
$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{C}\mathrm{h}}}$	$=$	$[\frac{n+1}{4\pi }{]}^{1/2}\tilde{\xi }$
$\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{C}\mathrm{h}}}$	$=$	$[\frac{(n+1{)}^3}{4\pi }{]}^{1/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}),$

where,

$P_{\mathrm{C}\mathrm{h}}$	$\equiv$	$K{\rho }_c^{(n+1)/n},$
$R_{\mathrm{C}\mathrm{h}}$	$\equiv$	$\left[\right(\frac{K}{G}){\rho }_c^{1/n-1}{]}^{1/2},$
$M_{\mathrm{C}\mathrm{h}}$	$\equiv$	$\left[\right(\frac{K}{G}{)}^3{\rho }_c^{(3-n)/n}{]}^{1/2}.$

In this case, the expressions for the physical variable normalizations have been defined in terms of — in addition to $G$ and/or $K$ — the equilibrium configuration's central density, ${\rho }_c$, instead of in terms of $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ or $P_e$. These are precisely the expressions for, respectively, $P_s({\xi }_s)$, $R_s({\xi }_s)$, and $M_s({\xi }_s)$ that appear in the appendix of J. P. Chieze (1987, A&A, 171, 225-232) — see, respectively, his equations (A7), (A5), and (A6). [Note that, for the polytropic systems of interest to us, here — that is, systems having $0\le n<\mathrm{\infty }$ — Chieze's parameter $ϵ\equiv \mathrm{sgn}(n+1)=1$.]

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