SSC/Structure/PolytropesASIDE1

ASIDE: Whitworth's Scaling

This provides details in support of our associated discussion of embedded polytropic spheres.

In his study of the "global gravitational stability [of] one-dimensional polytropes," Whitworth (1981, MNRAS, 195, 967) normalizes (or "references") various derived mathematical expressions for configuration radii, $R$, and for the pressure exerted by an external bounding medium, $P_{\mathrm{e}\mathrm{x}}$, to quantities he refers to as, respectively, $R_{\mathrm{r}\mathrm{f}}$ and $P_{\mathrm{r}\mathrm{f}}$. The paragraph from his paper in which these two reference quantities are defined is shown here:

Whitworth (1981, MNRAS, 195, 967)

In order to map Whitworth's terminology to ours, we note, first, that he uses $M_0$ to represent the spherical configuration's total mass, which we refer to simply as $M$; and his parameter $\eta$ is related to our $n$ via the relation,

$\eta =1+\frac{1}{n}.$

Hence, Whitworth writes the polytropic equation of state as,

$P=K_{\eta }{\rho }^{\eta },$

whereas, using our standard notation, this same key relation is written as,

$P=K_{\mathrm{n}}{\rho }^{1+1/n}$ ;

and his parameter $K_{\eta }$ is identical to our $K_{\mathrm{n}}$.

According to the second (bottom) expression identified by the red outlined box drawn above,

$P_{\mathrm{r}\mathrm{f}}=\frac{3^4{5}^3}{2^{10}\pi }\left(\frac{K_1^4}{G^3{M}^2}\right),$

and inverting the expression inside the green outlined box gives,

$K_1=[K_n(4P_{\mathrm{r}\mathrm{f}}{)}^{\eta -1}{]}^{1/\eta }.$

Hence,

$P_{\mathrm{r}\mathrm{f}}=\frac{3^4{5}^3}{2^{10}\pi }\left(\frac{1}{G^3{M}^2}\right)[K_n(4P_{\mathrm{r}\mathrm{f}}{)}^{\eta -1}{]}^{4/\eta },$

or, gathering all factors of $P_{\mathrm{r}\mathrm{f}}$ to the left-hand side,

$P_{\mathrm{r}\mathrm{f}}^{(4-3\eta )}=2^{-2(4+\eta )}\left(\frac{3^4{5}^3}{\pi }{)}^{\eta }\right[\frac{K_n^4}{G^{3\eta }{M}^{2\eta }}].$

Analogously, according to the first (top) expression identified inside the red outlined box,

$R_{\mathrm{r}\mathrm{f}}=\frac{2^{2}GM}{3\cdot 5K_1}=2^{2/\eta }\left(\frac{GM}{3\cdot 5}\right)K_n^{-1/\eta }{P}_{\mathrm{r}\mathrm{f}}^{(1-\eta )/\eta }\Rightarrow R_{\mathrm{r}\mathrm{f}}^{\eta }=\frac{2^2}{K_n}(\frac{GM}{3\cdot 5}{)}^{\eta }{P}_{\mathrm{r}\mathrm{f}}^{(1-\eta )}.$

Examples
$n$	$\eta =1+1/n$	$P_{\mathrm{r}\mathrm{f}}$	$R_{\mathrm{r}\mathrm{f}}$
1	2	$\frac{2^6\pi }{3^4{5}^3}\left[\frac{G^3{M}^2}{K^2}\right]$	$\left[\frac{3^{2}5}{2^4\pi }\right(\frac{K}{G}){]}^{1/2}$
5	6/5	$\frac{3^{12}{5}^9}{2^{26}{\pi }^3}\left[\frac{K^{10}}{G^9{M}^6}\right]$	$\left[\frac{2^{12}\pi }{3^6{5}^5}\right(\frac{G^5{M}^4}{K^5}){]}^{1/2}$
$\mathrm{\infty }$	1	$\frac{3^4{5}^3}{2^{10}\pi }\left[\frac{K^4}{G^3{M}^2}\right]$	$\frac{2^{2}GM}{3\cdot 5K}$