SSCpt1/Virial/FormFactors

Structural Form Factors (Pt 1)

Part I:  Synopsis

Part II:  n = 5 Polytrope

Part III:  n = 1 Polytrope

Structural Form Factors

As has been defined in a companion, introductory discussion, three key dimensionless structural form factors are:

${\mathfrak{𝔣}}_M$	$\equiv$	$\underset{0}{\overset{1}{\int }}3\left[\frac{\rho (x)}{{\rho }_0}\right]x^{2}dx,$
${\mathfrak{𝔣}}_W$	$\equiv$	$3\cdot 5\underset{0}{\overset{1}{\int }}\left\{\underset{0}{\overset{x}{\int }}\right[\frac{\rho (x)}{{\rho }_0}\left]x^{2}dx\right\}\left[\frac{\rho (x)}{{\rho }_0}\right]xdx,$
${\mathfrak{𝔣}}_A$	$\equiv$	$\underset{0}{\overset{1}{\int }}3\left[\frac{P(x)}{P_0}\right]x^{2}dx,$

where, $x\equiv r/R_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}$, and the subscript "0" denotes central values. The principal purpose of this chapter is to carry out the integrations that are required to obtain expressions for these structural form factors, at least in the few cases where they can be determined analytically. These form-factor expressions will then be used to provide expressions for the two constants, ${𝒜}$ and ${ℬ}$, that appear in the free-energy function and in the virial theorem, and to provide corresponding expressions for the normalized energies, $W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}/E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ and $S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}/E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$.  

Synopsis

Summary of Derived Structural Form-Factors

Isolated Polytropes $(n\ne 5)$	Pressure-Truncated Polytropes $(n\ne 5)$
${\mathfrak{𝔣}}_M$ $=$ $[-\frac{3{\theta }^{\text{'}}}{\xi }{]}_{{\xi }_1}$ ${\mathfrak{𝔣}}_W$ $=$ $\frac{3^2\cdot 5}{5-n}[\frac{{\theta }^{\text{'}}}{\xi }{]}_{{\xi }_1}^2$ ${\mathfrak{𝔣}}_A$ $=$ $\frac{3(n+1)}{(5-n)}[{\theta }^{\text{'}}{]}_{{\xi }_1}^2$	${\widetilde{\mathfrak{𝔣}}}_M$ $=$ $(-\frac{3{\tilde{\theta }}^{\text{'}}}{\tilde{\xi }})$ ${\widetilde{\mathfrak{𝔣}}}_W$ $=$ $\frac{3\cdot 5}{(5-n){\tilde{\xi }}^2}[{\tilde{\theta }}^{n+1}+3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]$ ${\widetilde{\mathfrak{𝔣}}}_A$ $=$ $\frac{1}{(5-n)}\{6{\tilde{\theta }}^{n+1}+(n+1)[3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]\}$
Isolated n = 1 Polytrope $\tilde{\xi }\to {\xi }_1=\pi$	Pressure-Truncated n = 1 Polytropes $0<\tilde{\xi }<\pi$
${\mathfrak{𝔣}}_M$ $=$ $\frac{3}{{\pi }^2}$ ${\mathfrak{𝔣}}_W$ $=$ $\frac{3^2\cdot 5}{2^2{\pi }^4}$ ${\mathfrak{𝔣}}_A$ $=$ $\frac{3}{2{\pi }^2}$	${\widetilde{\mathfrak{𝔣}}}_M$ $=$ $\frac{3}{{\tilde{\xi }}^3}[\sin \tilde{\xi }-\tilde{\xi }\cos \tilde{\xi }]$ ${\widetilde{\mathfrak{𝔣}}}_W$ $=$ $\frac{3\cdot 5}{2^3{\tilde{\xi }}^6}[4{\tilde{\xi }}^2-3\tilde{\xi }\sin (2\tilde{\xi })+2{\tilde{\xi }}^2\cos (2\tilde{\xi })]$ ${\widetilde{\mathfrak{𝔣}}}_A$ $=$ $\frac{3}{2^2{\tilde{\xi }}^3}[2\tilde{\xi }-\sin (2\tilde{\xi })]$
Isolated n = 5 Polytrope	Pressure-Truncated n = 5 Polytropes
	${\widetilde{\mathfrak{𝔣}}}_M$ $=$ $(1+{\ell }^2{)}^{-3/2}$ ${\widetilde{\mathfrak{𝔣}}}_W$ $=$ $\frac{5}{2^4}\cdot {\ell }^{-5}\left[\ell \right({\ell }^4-\frac{8}{3}{\ell }^2-1)(1+{\ell }^2{)}^{-3}+{\tan }^{-1}(\ell )]$ ${\widetilde{\mathfrak{𝔣}}}_A$ $=$ $\frac{3}{2^3}{\ell }^{-3}[{\tan }^{-1}(\ell )+\ell ({\ell }^2-1)(1+{\ell }^2{)}^{-2}]$ where,     $\ell \equiv \frac{\tilde{\xi }}{\sqrt{3}}$

Expectation in Context of Pressure-Truncated Polytropes

For pressure-truncated polytropic configurations, the normalized virial theorem states that,

$2\left(\frac{S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\right)+\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$

$=$

$\frac{4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^3}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}.$

This provides one mechanism by which the correctness of our form-factor expressions can be checked. Specifically, having determined $S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}$ and $W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$ from the derived form factors, we can see whether the sum of these energies as specified on the lefthand-side of this virial theorem expression indeed match the normalized energy term involving the external pressure, as specified on the righthand side. In order to facilitate this "reality check" at the end of each example, below, we will use Stahler's detailed force-balanced solution of the equilibrium structure of embedded polytropes to provide an expression for the term on the righthand side of the virial theorem expression.

We begin by plugging our general expression for $E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ into this righthand-side term and grouping factors to facilitate insertion of Stahler's expressions.

$\frac{4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^3}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$	$=$	$4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^3[K^{-n}{G}^3{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{(5-n)}{]}^{1/(n-3)}$
	$=$	$4\pi \left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{(n-5)/(n-3)}{P}_e{R}_{\mathrm{e}\mathrm{q}}^3\right[K^{-n}{G}^3{M}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}^{(5-n)}{]}^{1/(n-3)}.$

From Stahler's equilibrium solution, we have,

$R_{\mathrm{e}\mathrm{q}}$	$=$	$R_{\mathrm{S}\mathrm{W}\mathrm{S}}\left(\frac{n}{4\pi }{)}^{1/2}\right\{\xi {\theta }_n^{(n-1)/2}{\}}_{\tilde{\xi }}$
	$=$	$\left[\xi {\theta }_n^{(n-1)/2}{]}_{\tilde{\xi }}\right(\frac{n+1}{4\pi }{)}^{1/2}{G}^{-1/2}{K}_n^{n/(n+1)}{P}_{\mathrm{e}}^{(1-n)/[2(n+1)]}$
$\Rightarrow P_e{R}_{\mathrm{e}\mathrm{q}}^3$	$=$	$\left[\xi {\theta }_n^{(n-1)/2}{]}_{\tilde{\xi }}^3\right(\frac{n+1}{4\pi }{)}^{3/2}{G}^{-3/2}{K}_n^{3n/(n+1)}{P}_{\mathrm{e}}^{1+3(1-n)/[2(n+1)]}$
	$=$	$\left[\xi {\theta }_n^{(n-1)/2}{]}_{\tilde{\xi }}^3\right(\frac{n+1}{4\pi }{)}^{3/2}{G}^{-3/2}{K}_n^{3n/(n+1)}{P}_{\mathrm{e}}^{(5-n)/[2(n+1)]};$
$M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}$	$=$	$M_{\mathrm{S}\mathrm{W}\mathrm{S}}\left(\frac{n^3}{4\pi }{)}^{1/2}\right\{{\theta }_n^{(n-3)/2}{\xi }^2\left|\frac{d{\theta }_n}{d\xi }\right|{\}}_{\tilde{\xi }}$
	$=$	$\left[{\theta }_n^{(n-3)/2}{\xi }^2\right|\frac{d{\theta }_n}{d\xi }\left|{]}_{\tilde{\xi }}\right[\frac{(n+1{)}^3}{4\pi }{]}^{1/2}{G}^{-3/2}{K}_n^{2n/(n+1)}{P}_{\mathrm{e}}^{(3-n)/[2(n+1)]}$
$\Rightarrow K^{-n}{G}^3{M}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}^{(5-n)}$	$=$	$\left[{\theta }_n^{(n-3)/2}{\xi }^2\right|\frac{d{\theta }_n}{d\xi }\left|{]}_{\tilde{\xi }}^{(5-n)}\right[\frac{(n+1{)}^3}{4\pi }{]}^{(5-n)/2}{G}^{3-3(5-n)/2}{K}_n^{-n+2n(5-n)/(n+1)}{P}_{\mathrm{e}}^{(3-n)(5-n)/[2(n+1)]}$
	$=$	$\left[{\theta }_n^{(n-3)/2}{\xi }^2\right|\frac{d{\theta }_n}{d\xi }\left|{]}_{\tilde{\xi }}^{(5-n)}\right[\frac{(n+1{)}^3}{4\pi }{]}^{(5-n)/2}{G}^{3(n-3)/2}{K}_n^{3n(3-n)/(n+1)}{P}_{\mathrm{e}}^{(3-n)(5-n)/[2(n+1)]};$
$\Rightarrow P_e{R}_{\mathrm{e}\mathrm{q}}^3[K^{-n}{G}^3{M}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}^{(5-n)}{]}^{1/(n-3)}$	$=$	$\left\{\right[{\theta }_n^{(n-3)/2}{\xi }^2\left|\frac{d{\theta }_n}{d\xi }\right|{]}_{\tilde{\xi }}[\frac{(n+1{)}^3}{4\pi }{]}^{1/2}{\}}^{(5-n)/(n-3)}{G}^{3/2}{K}_n^{-3n/(n+1)}{P}_{\mathrm{e}}^{(n-5)/[2(n+1)]}$
		$\times \left[\xi {\theta }_n^{(n-1)/2}{]}_{\tilde{\xi }}^3\right(\frac{n+1}{4\pi }{)}^{3/2}{G}^{-3/2}{K}_n^{3n/(n+1)}{P}_{\mathrm{e}}^{(5-n)/[2(n+1)]}$
	$=$	$\left\{\right[{\theta }_n^{(n-3)/2}{\xi }^2\left|\frac{d{\theta }_n}{d\xi }\right|{]}_{\tilde{\xi }}^{(5-n)}\left[\frac{(n+1{)}^3}{4\pi }{]}^{(5-n)/2}\right[\xi {\theta }_n^{(n-1)/2}{]}_{\tilde{\xi }}^{3(n-3)}(\frac{n+1}{4\pi }{)}^{3(n-3)/2}{\}}^{1/(n-3)}$
	$=$	$\{(n+1{)}^{3[(5-n)+(n-3)]/2}(4\pi {)}^{[(n-5)+(9-3n)]/2}|\frac{d{\theta }_n}{d\xi }{|}_{\tilde{\xi }}^{(5-n)}({\theta }_n{)}_{\tilde{\xi }}^{[(n-3)(5-n)+3(n-1)(n-3)]/2}{\tilde{\xi }}^{[2(5-n)+3(n-3)]}{\}}^{1/(n-3)}$
	$=$	$\{(n+1{)}^3(4\pi {)}^{(2-n)}|\frac{d{\theta }_n}{d\xi }{|}_{\tilde{\xi }}^{(5-n)}({\theta }_n{)}_{\tilde{\xi }}^{(n+1)(n-3)}{\tilde{\xi }}^{(n+1)}{\}}^{1/(n-3)}.$

Hence, the expectation based on Stahler's equilibrium models is that,

$\frac{4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^3}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$

$=$

$\left[\frac{(n+1{)}^3}{4\pi }{]}^{1/(n-3)}\right[\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right)\frac{1}{(-\theta {\text{'}}_n{)}_{\tilde{\xi }}}{]}^{(n-5)/(n-3)}({\theta }_n{)}_{\tilde{\xi }}^{(n+1)}{\tilde{\xi }}^{(n+1)/(n-3)}.$

As a cross-check, multiplying this expression through by $[(R_{\mathrm{e}\mathrm{q}}/R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}})(M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}/M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}{)}^2]$ — where the expression for $R_{\mathrm{e}\mathrm{q}}/R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ can be obtained from our discussions of detailed force-balanced models — gives a related result that can be obtained directly from Horedt's expressions, namely,

$[\frac{4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^4}{G{M}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}^2}{]}_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}$

$=$

$\frac{{\tilde{\theta }}^{n+1}}{(n+1)(-{\tilde{\theta }}^{\prime }{)}^2}.$

Viala and Horedt (1974) Expressions

Presentation

📚 Y. P. Viala & Gp. Horedt (1974, Astron. & Ap., Vol. 33, pp. 195 - 202) have provided analytic expressions for the gravitational potential energy and the internal energy — which they tag with the variable names, $\mathrm{\Omega }$ and $U$, respectively — that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres. [The same expression for $\mathrm{\Omega }$ is also effectively provided in §1 of Horedt (1970) through the definition of his coefficient, "A" (polytropic case).]

Y. P. Viala & Gp. Horedt (1974) Polytropic Sheets, Cylinders and Spheres with Negative Index Astronomy & Astrophysics, Vol. 33, pp. 195 - 202 $\mathrm{\Omega }$ $=$ $-G\underset{0}{\overset{M}{\int }}\frac{MdM}{r}=\frac{16{\pi }^{2}G{\rho }_0^2{\alpha }^5}{(5-n)}[\mp {\xi }^3{\theta }^{n+1}-3{\xi }^3({\theta }^{\prime }{)}^2-3{\xi }^2\theta ({\theta }^{\prime })],$ $U$ $=$ $\frac{1}{\gamma -1}\underset{V}{\int }pdV=\frac{\alpha K{\rho }_0^{1+1/n}}{\gamma -1}\underset{0}{\overset{\xi }{\int }}{\theta }^{n+1}4\pi {\alpha }^2{\xi }^{2}d\xi$   $=$ $\frac{\alpha K{\rho }_0^{1+1/n}}{\gamma -1}\cdot \frac{4\pi {\alpha }^2(n+1)}{(5-n)}[\frac{2{\xi }^3{\theta }^{n+1}}{n+1}\pm {\xi }^3({\theta }^{\prime }{)}^2\pm {\xi }^2\theta ({\theta }^{\prime }){]}_0^{\xi }.$ (the superior sign holds if $-1<n<\mathrm{\infty }$, the inferior if $-\mathrm{\infty }<n<-1$)

A couple of key equations drawn directly from 📚 Viala & Horedt (1974) have been shown here. As its title indicates, the paper includes discussion of — and accompanying equation derivations for — equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries: planar sheets, axisymmetric cylinders, and spheres. We have extracted derived expressions for the gravitational potential energy, $\mathrm{\Omega }$, and the internal energy, $U$, that apply to spherically symmetric configurations only. These authors also consider negative polytropic indexes; we are considering only values in the range, $0\le n\le \mathrm{\infty }$, so, as the accompanying parenthetical note indicates, when either $\pm$ or $\mp$ appears in an expression, we will pay attention only to the superior sign.

Rewriting these two expressions to accommodate our parameter notations — recognizing, specifically, that $\alpha$ is the familiar polytropic length scale ($a_n$; expression provided below), ${\rho }_0$ is the central density $({\rho }_c)$, and $(\gamma -1)=1/n$ — we have from 📚 Viala & Horedt (1974),

$[W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}{]}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$	$=$	$-\frac{(4\pi {)}^2}{(5-n)}\cdot G{\rho }_c^2{a}_n^5[{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }],$
$[{\mathfrak{𝔖}}_{\mathrm{A}}{]}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$	$=$	$\frac{n(4\pi {)}^2}{3(5-n)}\cdot G{\rho }_c^2{a}_n^5[\frac{6}{(n+1)}\cdot {\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }].$

First Reality Check

As a quick reality check, let's see whether, when appropriately added together, these two energies satisfy the scalar virial theorem for isolated polytropes.

$[W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+2S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}{]}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$	$=$	$W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+\frac{3}{n}{\mathfrak{𝔖}}_A$
	$=$	$-\frac{(4\pi {)}^2}{(5-n)}\cdot G{\rho }_c^2{a}_n^5[{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }]$
		$+\frac{(4\pi {)}^2}{(5-n)}\cdot G{\rho }_c^2{a}_n^5[\frac{6}{(n+1)}\cdot {\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }]$
	$=$	$\frac{(4\pi {)}^2}{(5-n)}\cdot G{\rho }_c^2{a}_n^5[\frac{6}{(n+1)}-1]{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}$
	$=$	$\frac{(4\pi {)}^2}{(n+1)}\cdot G{\rho }_c^2{a}_n^5{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}.$

For isolated polytropes, $\tilde{\theta }\to 0$, so this sum of terms goes to zero, as it should if the system is in virial equilibrium.

Renormalization

Both of the energy-term expressions derived by 📚 Viala & Horedt (1974) are written in terms of ${\rho }_c$ and

$a_{\mathrm{n}}$

$=$

$[\frac{(n+1)K_n}{4\pi G}\cdot {\rho }_c^{(1-n)/n}{]}^{1/2}$

— that is, effectively in terms of ${\rho }_c$ and $K_{\mathrm{n}}$ — whereas, in the context of our discussions, we would prefer to express them in terms of our generally adopted energy normalization,

$E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$

$=$

$[K_n^n{G}^{-3}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{n-5}{]}^{1/(n-3)}.$

In order to accomplish this, we need to replace the central density with the total mass of an isolated polytrope, $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$, whose generic expression is (see, for example, equation 69 of Chandrasekhar),

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$=$

$(4\pi {)}^{-1/2}[\frac{(n+1)K_n}{G}{]}^{3/2}{\rho }_c^{(3-n)/2n}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}.$

Hence, we have,

$E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^{n-3}$	$=$	$K_n^n{G}^{-3}\{(4\pi {)}^{-1/2}[\frac{(n+1)K_n}{G}{]}^{3/2}{\rho }_c^{(3-n)/2n}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}{\}}^{n-5}$
	$=$	$[(4\pi {)}^{-1/2}(n+1{)}^{3/2}{\rho }_c^{(3-n)/2n}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}{]}^{n-5}{K}_n^{[2n+3(n-5)]/2}{G}^{[-6-3(n-5)]/2}$
	$=$	$[(4\pi {)}^{-1/2}(n+1{)}^{3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}{]}^{n-5}{\rho }_c^{(n-3)(5-n)/2n}{K}_n^{5(n-3)/2}{G}^{-3(n-3)/2}$
$\Rightarrow E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$	$=$	$[(4\pi {)}^{-1/2}(n+1{)}^{3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}{]}^{(n-5)/(n-3)}{\rho }_c^{(5-n)/2n}{K}_n^{5/2}{G}^{-3/2}$
	$=$	$[(4\pi {)}^{-1/2}(n+1{)}^{3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}{]}^{(n-5)/(n-3)}G{\rho }_c^2{\rho }_c^{[-4n+(5-n)]/2n}(\frac{K_n}{G}{)}^{5/2}$
	$=$	$[(4\pi {)}^{-1/2}(n+1{)}^{3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}{]}^{(n-5)/(n-3)}G{\rho }_c^2[\frac{K_n}{G}\cdot {\rho }_c^{(1-n)/n}{]}^{5/2}$
	$=$	$[(4\pi {)}^{-1/2}(n+1{)}^{3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}{]}^{(n-5)/(n-3)}G{\rho }_c^2[\frac{4\pi }{(n+1)}\cdot a_n^2{]}^{5/2}$
$\Rightarrow (4\pi {)}^{2}G{\rho }_c^2{a}_n^5$	$=$	$E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}(4\pi {)}^2[(4\pi {)}^{-1/2}(n+1{)}^{3/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}{]}^{(5-n)/(n-3)}[\frac{(n+1)}{4\pi }{]}^{5/2}$
	$=$	$E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}^{(5-n)/(n-3)}(4\pi {)}^{[-(n-3)-(5-n)]/2(n-3)}(n+1{)}^{[3(5-n)+5(n-3)]/2(n-3)}$
	$=$	$E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}[(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}^{(5-n)}\cdot \frac{(n+1{)}^n}{4\pi }{]}^{1/(n-3)}.$

So, employing our preferred normalization, the 📚 Viala & Horedt (1974) expressions become,

$[\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$	$=$	$-\frac{1}{(5-n)}[{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }\left]\right[(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}^{(5-n)}\cdot \frac{(n+1{)}^n}{4\pi }{]}^{1/(n-3)},$
$[\frac{{\mathfrak{𝔖}}_{\mathrm{A}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$	$=$	$\frac{n}{3(5-n)}[\frac{6}{(n+1)}\cdot {\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }\left]\right[(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}^{(5-n)}\cdot \frac{(n+1{)}^n}{4\pi }{]}^{1/(n-3)}.$

Second Reality Check

If we now renormalize the sum of energy terms discussed in our first reality check, above, we have,

$\frac{1}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}[W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+2S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}{]}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}=\frac{4\pi P_e{R}_{\mathrm{e}\mathrm{q}}^3}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$

$=$

$(n+1{)}^{-1}{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}[(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}^{(5-n)}\cdot \frac{(n+1{)}^n}{4\pi }{]}^{1/(n-3)}.$

(This may or may not be useful!)

Implication for Structural Form Factors

On the other hand, our expressions for these two normalized energy components written in terms of the structural form factors are,

$\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$	$=$	$-\frac{3}{5}{\chi }^{-1}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^2\cdot \frac{{\widetilde{\mathfrak{𝔣}}}_W}{{\widetilde{\mathfrak{𝔣}}}_M^2},$
$\frac{{\mathfrak{𝔖}}_A}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$	$=$	$\frac{4\pi n}{3}\cdot {\chi }^{-3/n}\left[\frac{3}{4\pi }\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}})\frac{1}{{\widetilde{\mathfrak{𝔣}}}_M}{]}_{\mathrm{e}\mathrm{q}}^{(n+1)/n}\cdot {\widetilde{\mathfrak{𝔣}}}_A,$

where, in equilibrium (see here and here for details),

${\chi }_{\mathrm{e}\mathrm{q}}\equiv \frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$	$=$	$\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}}\left\{\frac{R_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{t}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\right\}$
	$=$	$\tilde{\xi }(-{\tilde{\xi }}^2{\tilde{\theta }}^{\prime }{)}^{(1-n)/(n-3)}\{\left[\frac{4\pi }{(n+1{)}^n}{]}^{1/(n-3)}\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{(n-1)/(n-3)}\},$
$\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$=$	$\left(\frac{{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}}{{\xi }_1^2{\theta }_1^{\text{'}}}\right),$
${\widetilde{\mathfrak{𝔣}}}_M$	$=$	$(-\frac{3{\tilde{\theta }}^{\text{'}}}{\tilde{\xi }}).$

Hence, we deduce that,

${\widetilde{\mathfrak{𝔣}}}_W$	$=$	$-\frac{5}{3}\left[\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\right]{\chi }_{\mathrm{e}\mathrm{q}}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{-2}\cdot {\widetilde{\mathfrak{𝔣}}}_M^2$
	$=$	$\frac{5}{3}\left\{\right[-\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\left]\right[\frac{4\pi }{(n+1{)}^n}{]}^{1/(n-3)}\}\tilde{\xi }(-{\tilde{\xi }}^2{\tilde{\theta }}^{\prime }{)}^{(1-n)/(n-3)}(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{[(n-1)-2(n-3)]/(n-3)}\cdot (-\frac{3{\tilde{\theta }}^{\text{'}}}{\tilde{\xi }}{)}^2$
	$=$	$3\cdot 5\left\{\right[-\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\left]\right[\frac{4\pi }{(n+1{)}^n}{]}^{1/(n-3)}\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{(5-n)/(n-3)}\right\}(-{\tilde{\theta }}^{\text{'}}{)}^{[(1-n)+2(n-3)]/(n-3)}{\tilde{\xi }}^{[-(n-3)+2(1-n)]/(n-3)}$
	$=$	$3\cdot 5\left\{\right[-\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\left]\right[\frac{4\pi }{(n+1{)}^n}{]}^{1/(n-3)}\left(\frac{{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}}{{\xi }_1^2{\theta }_1^{\text{'}}}{)}^{(5-n)/(n-3)}\right\}(-{\tilde{\theta }}^{\text{'}}{)}^{(n-5)/(n-3)}{\tilde{\xi }}^{(5-3n)/(n-3)}.$

If we now adopt the VH74 expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,

$\{{\}}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$	$=$	$\frac{1}{(5-n)}[{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }\left]\right[(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}_{{\xi }_1}^{(5-n)}\cdot \frac{(n+1{)}^n}{4\pi }{]}^{1/(n-3)}[\frac{4\pi }{(n+1{)}^n}{]}^{1/(n-3)}(\frac{{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}}{{\xi }_1^2{\theta }_1^{\text{'}}}{)}^{(5-n)/(n-3)}$
	$=$	$\frac{1}{(5-n)}[{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }](-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}{)}^{(5-n)/(n-3)}.$

Therefore,

$[{\widetilde{\mathfrak{𝔣}}}_W{]}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$	$=$	$\frac{3\cdot 5}{(5-n)}[{\tilde{\xi }}^3{\tilde{\theta }}^{n+1}+3{\tilde{\xi }}^3({\tilde{\theta }}^{\text{'}}{)}^2-3(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}})\tilde{\theta }]{\tilde{\xi }}^{-5}$
	$=$	$\frac{3\cdot 5}{(5-n){\tilde{\xi }}^2}[{\tilde{\theta }}^{n+1}+3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }].$

Now, from our earlier work we deduced that ${\widetilde{\mathfrak{𝔣}}}_A$ is related to ${\widetilde{\mathfrak{𝔣}}}_W$ via the relation,

${\widetilde{\mathfrak{𝔣}}}_A$

$=$

${\tilde{\theta }}^{n+1}+{\widetilde{\mathfrak{𝔣}}}_W\left[\frac{(n+1)}{3\cdot 5}\right]{\tilde{\xi }}^2.$

Hence, we now have,

$[{\widetilde{\mathfrak{𝔣}}}_A{]}_{\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$	$=$	${\tilde{\theta }}^{n+1}+\frac{(n+1)}{(5-n)}[{\tilde{\theta }}^{n+1}+3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]$
	$=$	$\frac{1}{(5-n)}\{6{\tilde{\theta }}^{n+1}+(n+1)[3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]\}.$

Building on the work of VH74, we have, quite generally,

Structural Form Factors for Isolated Polytropes	Structural Form Factors for Pressure-Truncated Polytropes
${\mathfrak{𝔣}}_M$ $=$ $[-\frac{3{\theta }^{\text{'}}}{\xi }{]}_{{\xi }_1}$ ${\mathfrak{𝔣}}_W$ $=$ $\frac{3^2\cdot 5}{5-n}[\frac{{\theta }^{\text{'}}}{\xi }{]}_{{\xi }_1}^2$ ${\mathfrak{𝔣}}_A$ $=$ $\frac{3(n+1)}{(5-n)}[{\theta }^{\text{'}}{]}_{{\xi }_1}^2$	${\widetilde{\mathfrak{𝔣}}}_M$ $=$ $(-\frac{3{\tilde{\theta }}^{\text{'}}}{\tilde{\xi }})$ ${\widetilde{\mathfrak{𝔣}}}_W$ $=$ $\frac{3\cdot 5}{(5-n){\tilde{\xi }}^2}[{\tilde{\theta }}^{n+1}+3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]$ ${\widetilde{\mathfrak{𝔣}}}_A$ $=$ $\frac{1}{(5-n)}\{6{\tilde{\theta }}^{n+1}+(n+1)[3({\tilde{\theta }}^{\text{'}}{)}^2-{\widetilde{\mathfrak{𝔣}}}_M\tilde{\theta }]\}$
We should point out that 📚 D. Lai, F. A. Rasio, & S. L. Shapiro (1993b, ApJ Suppl., Vol. 88, pp. 205 - 252) define a different set of dimensionless structure factors for isolated polytropic spheres — $k_1$ (their equation 2.9) is used in the determination of the internal energy; and $k_2$ (their equation 2.10) is used in the determination of the gravitational potential energy. $k_1$ $\equiv$ $\left[\frac{n(n+1)}{5-n}\right]{\xi }_1|{\theta }_1^{\text{'}}|$ $k_2$ $\equiv$ $\frac{3}{5-n}[\frac{4\pi |{\theta }_1^{\text{'}}|}{{\xi }_1}{]}^{1/3}$ Note that these are defined in the context of energy expressions wherein the central density, rather than the configuration's radius, serves as the principal parameter. We note, as well, that for rotating configurations they define two additional dimensionless structure factors — $k_3$ (their equation 3.17) is used in the determination of the rotational kinetic energy; and ${\kappa }_n$ (their equation 3.14; also equation 7.4.9 of [ST83]) is used in the determination of the moment of inertia.

The singularity that arises when $n=5$ leads us to suspect that these general expressions fail in that one specific case. Fortunately, as we have shown in an accompanying discussion, ${\mathfrak{𝔣}}_W$ and ${\mathfrak{𝔣}}_A$, as well as ${\mathfrak{𝔣}}_M$, can be determined by direct integration in this single case.

Related Discussions

• See our plot of, what Kimura (1981b) would refer to as, several $M_1$ sequences

See Also

• Index to a Variety of Free-Energy and/or Virial Analyses
• Spherically Symmetric Configurations (SSC) Index