SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt1

Virial Equilibrium of Adiabatic Spheres (1^st Effort)

Part I:   Isolated Configurations

Part II:  Configurations Embedded in an External Medium

Highlights of the rather detailed discussion presented below have been summarized in an accompanying chapter of this H_Book.

Review

Adopted Normalizations

In an introductory discussion of the virial equilibrium structure of spherically symmetric configurations, we adopted the following physical parameter normalizations for adiabatic systems.

Adopted Normalizations for Adiabatic Systems
In Terms of ${\gamma }_g$	In Terms of $n$
$R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $\left[\right(\frac{G}{K})M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2-{\gamma }_g}{]}^{1/(4-3{\gamma }_g)}$ $P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $[\frac{K^4}{G^{3{\gamma }_g}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2{\gamma }_g}}{]}^{1/(4-3{\gamma }_g)}$ $E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}{R}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^3=[K{G}^{3(1-{\gamma }_g)}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{6-5{\gamma }_g}{]}^{1/(4-3{\gamma }_g)}$ ${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $\frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^3}=\frac{3}{4\pi }[\frac{K^3}{G^3{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{]}^{1/(4-3{\gamma }_g)}$ $c_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^2$ $\equiv$ $\frac{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}=\frac{4\pi }{3}[\frac{K}{(G^3{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2{)}^{{\gamma }_g-1}}{]}^{1/(4-3{\gamma }_g)}$	$R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $=$ $\left[\right(\frac{G}{K}{)}^n{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{n-1}{]}^{1/(n-3)}$ $P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $=$ $[\frac{K^{4n}}{G^{3(n+1)}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2(n+1)}}{]}^{1/(n-3)}$ $E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $=$ $[K^n{G}^{-3}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{n-5}{]}^{1/(n-3)}$ ${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $=$ $\frac{3}{4\pi }[\frac{K^3}{G^3{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{]}^{n/(n-3)}$ $c_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^2$ $=$ $\frac{4\pi }{3}[\frac{K^n}{G^3{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{]}^{1/(n-3)}$

Virial Equilibrium

Also in our introductory discussion — see especially the section titled, Energy Extrema — we deduced that an adiabatic system's dimensionless equilibrium radius,

${\chi }_{\mathrm{e}\mathrm{q}}\equiv \frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}},$

is given by the root(s) of the following equation:

$2{𝒞}{\chi }_{\mathrm{e}\mathrm{q}}^{-2}+3{ℬ}{\chi }_{\mathrm{e}\mathrm{q}}^{3-3{\gamma }_g}-3{𝒜}{\chi }_{\mathrm{e}\mathrm{q}}^{-1}-3{𝒟}{\chi }_{\mathrm{e}\mathrm{q}}^3=0,$

where the definitions of the various coefficients are,

${𝒜}$	$\equiv$	$\frac{1}{5}\cdot \left[\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}})\frac{1}{{\mathfrak{𝔣}}_M}{]}^2\cdot {\mathfrak{𝔣}}_W,$
${ℬ}$	$\equiv$	$\frac{4\pi }{3}\left[\frac{3}{4\pi }\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}})\frac{1}{{\mathfrak{𝔣}}_M}{]}_{\mathrm{e}\mathrm{q}}^{\gamma }\cdot {\mathfrak{𝔣}}_A$
	$=$	$\frac{4\pi }{3}\left[\right(\frac{P_c}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}){\chi }^{3\gamma }{]}_{\mathrm{e}\mathrm{q}}\cdot {\mathfrak{𝔣}}_A,$
${𝒞}$	$\equiv$	$\frac{3\cdot 5}{2^4\pi }\left[\frac{J^2{c}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^2}{G^2{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^4}\right]\cdot \frac{{\mathfrak{𝔣}}_T}{{\mathfrak{𝔣}}_M},$
${𝒟}$	$\equiv$	$\left(\frac{4\pi }{3}\right)\frac{P_e}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}.$

(The dimensionless structural form factors, ${\mathfrak{𝔣}}_i,$ that appear in these expressions are defined for isolated polytropes in our accompanying introductory discussion and are discussed further, below.) Once the pressure exerted by the external medium ($P_e$), and the configuration's mass ($M_{\mathrm{t}\mathrm{o}\mathrm{t}}$), angular momentum ($J$), and specific entropy (via $K$) have been specified, the values of all of the coefficients are known and ${\chi }_{\mathrm{e}\mathrm{q}}$ can be determined.

For later use, we note that after making the substitution, ${\gamma }_g\to (n+1)/n$, $(\frac{{𝒜}}{{ℬ}}{)}^n$ $=$ $\left\{\frac{3}{20\pi }\right[\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right)\frac{1}{{\mathfrak{𝔣}}_M}{]}^2\cdot {\mathfrak{𝔣}}_W\left[\frac{3}{4\pi }\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}})\frac{1}{{\mathfrak{𝔣}}_M}{]}^{-(n+1)/n}\cdot {\mathfrak{𝔣}}_A^{-1}{\}}^n$   $=$ $5^{-n}\left(\frac{3}{4\pi }{)}^n\right[\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right)\frac{1}{{\mathfrak{𝔣}}_M}{]}^{2n}\left[\frac{3}{4\pi }\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}})\frac{1}{{\mathfrak{𝔣}}_M}{]}^{-(n+1)}\cdot (\frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_A}{)}^n$   $=$ $\frac{1}{5^n}\left(\frac{4\pi }{3}\right)(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{n-1}\cdot \frac{{\mathfrak{𝔣}}_W^n}{{\mathfrak{𝔣}}_A^n{\mathfrak{𝔣}}_M^{n-1}};$ and, $\frac{{{ℬ}}^{4n}}{{{𝒜}}^{3(n+1)}}$ $=$ $\left\{\frac{1}{5}\right[\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right)\frac{1}{{\mathfrak{𝔣}}_M}{]}^2\cdot {\mathfrak{𝔣}}_W{\}}^{-3(n+1)}\left\{\frac{4\pi }{3}\right[\frac{3}{4\pi }\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right)\frac{1}{{\mathfrak{𝔣}}_M}{]}^{(n+1)/n}\cdot {\mathfrak{𝔣}}_A{\}}^{4n}$   $=$ $5^{3(n+1)}\left[\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\left)\frac{1}{{\mathfrak{𝔣}}_M}{]}^{-6(n+1)}\right(\frac{4\pi }{3}{)}^{4n-4(n+1)}\left[\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}})\frac{1}{{\mathfrak{𝔣}}_M}{]}^{4(n+1)}\cdot \frac{{\mathfrak{𝔣}}_A^{4n}}{{\mathfrak{𝔣}}_W^{3(n+1)}}$   $=$ $5^{3(n+1)}\left(\frac{3}{4\pi }{)}^4\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^{-2(n+1)}\cdot \frac{{\mathfrak{𝔣}}_A^{4n}{\mathfrak{𝔣}}_M^{2(n+1)}}{{\mathfrak{𝔣}}_W^{3(n+1)}}$

Isolated Nonrotating Adiabatic Configuration

For a nonrotating configuration $(C=J=0)$ that is not influenced by the effects of a bounding external medium $(D=P_e=0)$, the statement of virial equilibrium is,

$3B{\chi }_{\mathrm{e}\mathrm{q}}^{3-3{\gamma }_g}-3A{\chi }_{\mathrm{e}\mathrm{q}}^{-1}=0.$

Hence, one equilibrium state exists for each value of ${\gamma }_g$ and it occurs where,

${\chi }_{\mathrm{e}\mathrm{q}}^{4-3{\gamma }_g}=(\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^{4-3{\gamma }_g}$

$=$

$\frac{A}{B}.$

Two Points of View
In terms of $K$ and $M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}(=M_{\mathrm{t}\mathrm{o}\mathrm{t}})$	In terms of $P_c$ and $M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}(=M_{\mathrm{t}\mathrm{o}\mathrm{t}})$
${\chi }_{\mathrm{e}\mathrm{q}}^{4-3{\gamma }_g}$ $=$ $\{\frac{1}{5}\cdot [\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right)\frac{1}{{\mathfrak{𝔣}}_M}{]}^2\cdot {\mathfrak{𝔣}}_W\}$     $\times \left\{\frac{3}{4\pi }\right[\frac{3}{4\pi }\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right)\frac{1}{{\mathfrak{𝔣}}_M}{]}_{\mathrm{e}\mathrm{q}}^{-{\gamma }_g}\cdot \frac{1}{{\mathfrak{𝔣}}_A}\}$ $\Rightarrow R_{\mathrm{e}\mathrm{q}}^{4-3{\gamma }_g}$ $=$ $\frac{4\pi }{3\cdot 5}\left[\frac{3}{4\pi }\right(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}})\frac{1}{{\mathfrak{𝔣}}_M}{]}_{\mathrm{e}\mathrm{q}}^{2-{\gamma }_g}\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_A}[\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2-{\gamma }_g}}{K}]$ $\Rightarrow \frac{K{R}_{\mathrm{e}\mathrm{q}}^{4-3{\gamma }_g}}{G{M}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}^{2-{\gamma }_g}}$ $=$ $\frac{1}{5}(\frac{4\pi }{3}{)}^{{\gamma }_g-1}\frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_A{\mathfrak{𝔣}}_M^{2-{\gamma }_g}}$ — — — — — —     or, inverted and setting ${\gamma }_g=1+1/n$     — — — — — — $4\pi (\frac{G}{K}{)}^n{M}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}^{n-1}{R}_{\mathrm{e}\mathrm{q}}^{3-n}=(\frac{5{\mathfrak{𝔣}}_A{\mathfrak{𝔣}}_M}{{\mathfrak{𝔣}}_W}{)}^n\frac{3}{{\mathfrak{𝔣}}_M}$	${\chi }_{\mathrm{e}\mathrm{q}}^{4-3{\gamma }_g}$ $=$ $\{\frac{1}{5}\cdot [\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right)\frac{1}{{\mathfrak{𝔣}}_M}{]}^2\cdot {\mathfrak{𝔣}}_W\}$     $\times \left\{\frac{3}{4\pi }\right[\left(\frac{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{P_c}\right){\chi }^{-3\gamma }{]}_{\mathrm{e}\mathrm{q}}\cdot \frac{1}{{\mathfrak{𝔣}}_A}\}$ $\Rightarrow {\chi }_{\mathrm{e}\mathrm{q}}^4$ $=$ $\frac{3}{20\pi }\left(\frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^2\right(\frac{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{P_c})\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_A\cdot {\mathfrak{𝔣}}_M^2}$ $\Rightarrow \frac{P_c{R}_{\mathrm{e}\mathrm{q}}^4}{P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}{R}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^4}(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{)}^2$ $=$ $\frac{3}{20\pi }\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_A\cdot {\mathfrak{𝔣}}_M^2}$ $\Rightarrow \frac{P_c{R}_{\mathrm{e}\mathrm{q}}^4}{G{M}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}^2}$ $=$ $\frac{3}{20\pi }\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_A\cdot {\mathfrak{𝔣}}_M^2}$

According to the solution shown in the left-hand column, for fluid with a given specify entropy content, the equilibrium mass-radius relationship for adiabatic configurations is,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{({\gamma }_g-2)}\propto R_{\mathrm{e}\mathrm{q}}^{(3{\gamma }_g-4)}.$

We see that, for ${\gamma }_g=2$, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass. Conversely, for ${\gamma }_g=4/3$, the mass of the configuration is independent of the radius. For ${\gamma }_g>2$ or ${\gamma }_g<4/3$, configurations with larger mass (but the same specific entropy) have larger equilibrium radii. However, for ${\gamma }_g$ in the range, $2>{\gamma }_g>4/3$, configurations with larger mass have smaller equilibrium radii. (Note that the related result for isothermal configurations can be obtained by setting ${\gamma }_g=1$ in this adiabatic solution, because $K=c_s^2$ when ${\gamma }_g=1$.)

Role of Structural Form Factors

When employing a virial analysis to determine the radius of an equilibrium configuration, it is customary to set the structural form factors, ${\mathfrak{𝔣}}_M$, ${\mathfrak{𝔣}}_W$ and ${\mathfrak{𝔣}}_A$, to unity and accept that the expression derived for $R_{\mathrm{e}\mathrm{q}}$ is an estimate of the configuration's radius that is good to within a factor of order unity. As has been demonstrated in our related discussion of the equilibrium of uniform-density spheres, these form factors can be evaluated if/when the internal structural profile of an equilibrium configuration is known from a complementary detailed force-balance analysis. In the case being discussed here of isolated, spherical polytropes, solutions to the,

Lane-Emden Equation

$\frac{1}{{\xi }^2}\frac{d}{d\xi }\left({\xi }^2\frac{d{\mathrm{\Theta }}_H}{d\xi }\right)=-{\mathrm{\Theta }}_H^n$

can provide the desired internal structural information. Here we draw on Chandrasekhar's [C67] discussion of the structure of spherical polytropes to show precisely how our structural form factors can be expressed in terms of the Lane-Emden function, ${\mathrm{\Theta }}_H$, dimensionless radial coordinate, $\xi$, and the function derivative, ${\mathrm{\Theta }}^{\text{'}}=d{\mathrm{\Theta }}_H/d\xi$.

Mass

We note, first, that Chandrasekhar [C67] — see his Equation (78) on p. 99 — presents the following expression for the mean-to-central density ratio:

$\frac{\overline{\rho }}{{\rho }_c}$

$=$

$[-\frac{3{\mathrm{\Theta }}^{\text{'}}}{\xi }{]}_{{\xi }_1},$

where the notation at the bottom of the closing square bracket means that everything inside the square brackets should be, "evaluated at the surface of the configuration," that is, at the radial location, ${\xi }_1$, where the Lane-Emden function, ${\mathrm{\Theta }}_H(\xi )$, first goes to zero. But, as we pointed out when defining the structural form factors, the form factor associated with the configuration mass, ${\mathfrak{𝔣}}_M$, is equivalent to the mean-to-central density ratio. We conclude, therefore, that,

${\mathfrak{𝔣}}_M$

$=$

$[-\frac{3{\mathrm{\Theta }}^{\text{'}}}{\xi }{]}_{{\xi }_1}.$

Gravitational Potential Energy

Second, we note that Chandrasekhar's [C67] expression for the gravitational potential energy — see his Equation (90), p. 101 — is,

$-W$

$=$

$\frac{3}{5-n}\left(\frac{G{M}^2}{R}\right),$

whereas our analogous expression is,

$-W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$

$=$

$\frac{3}{5}\left(\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{R_{\mathrm{e}\mathrm{q}}}\right)\frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2}.$

We conclude, therefore, that,

$\frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2}$	$=$	$\frac{5}{5-n}$
$\Rightarrow {\mathfrak{𝔣}}_W$	$=$	$\frac{3^2\cdot 5}{5-n}[\frac{{\mathrm{\Theta }}^{\text{'}}}{\xi }{]}_{{\xi }_1}^2.$

Mass-Radius Relationship

Third, Chandrasekhar [C67] shows — see his Equation (72), p. 98 — that the general mass-radius relationship for isolated spherical polytropes is,

$G{M}^{(n-1)/n}{R}^{(3-n)/n}$

$=$

$\frac{(n+1)K}{(4\pi {)}^{1/n}}[-{\xi }^{(n+1)/(n-1)}\frac{d{\mathrm{\Theta }}_H}{d\xi }{]}_{\xi ={\xi }_1}^{(n-1)/n},$

which we choose to rewrite as,

$4\pi (\frac{G}{K}{)}^n{M}^{(n-1)}{R}^{(3-n)}$	$=$	$(n+1{)}^n[{\xi }^{(n+1)}(-{\mathrm{\Theta }}^{\text{'}}{)}^{(n-1)}{]}_{\xi ={\xi }_1}$
	$=$	$-\left(\frac{\xi }{{\mathrm{\Theta }}^{\text{'}}}{)}_{\xi ={\xi }_1}\right[(n+1)\xi (-{\mathrm{\Theta }}^{\text{'}}){]}_{\xi ={\xi }_1}^n.$

By comparison, the expression for the equilibrium radius that has been derived, above, from an analysis of extrema in the free energy function — specifically, see the last expression in the left-hand column of the table titled "Two Points of View" — we obtain,

$4\pi (\frac{G}{K}{)}^n{M}^{(n-1)}{R}_{\mathrm{e}\mathrm{q}}^{(3-n)}$

$=$

$\frac{3}{{\mathfrak{𝔣}}_M}(\frac{5{\mathfrak{𝔣}}_A{\mathfrak{𝔣}}_M}{{\mathfrak{𝔣}}_W}{)}^n.$

Hence, it appears as though, quite generally,

$\frac{1}{{\mathfrak{𝔣}}_M}(\frac{5{\mathfrak{𝔣}}_A{\mathfrak{𝔣}}_M}{{\mathfrak{𝔣}}_W}{)}^n$

$=$

$-\left(\frac{\xi }{3{\mathrm{\Theta }}^{\text{'}}}{)}_{\xi ={\xi }_1}\right[(n+1)\xi (-{\mathrm{\Theta }}^{\text{'}}){]}_{\xi ={\xi }_1}^n.$

Or, taking into account the expressions for ${\mathfrak{𝔣}}_M$ and ${\mathfrak{𝔣}}_W$ that have just been uncovered, we conclude that,

$\frac{5{\mathfrak{𝔣}}_A{\mathfrak{𝔣}}_M}{{\mathfrak{𝔣}}_W}$	$=$	$[(n+1)\xi (-{\mathrm{\Theta }}^{\text{'}}){]}_{\xi ={\xi }_1}$
$\Rightarrow \frac{{\mathfrak{𝔣}}_A}{{\mathfrak{𝔣}}_W}$	$=$	$\frac{(n+1)}{3\cdot 5}{\xi }_1^2.$
$\Rightarrow {\mathfrak{𝔣}}_A$	$=$	$\frac{(n+1)}{3\cdot 5}{\xi }_1^2\left\{\frac{3^2\cdot 5}{5-n}\right[\frac{{\mathrm{\Theta }}^{\text{'}}}{\xi }{]}_{{\xi }_1}^2\}.$
	$=$	$\frac{3(n+1)}{(5-n)}[{\mathrm{\Theta }}^{\text{'}}{]}_{{\xi }_1}^2.$

Central and Mean Pressure

It is also worth pointing out that Chandrasekhar [C67] — see his Equations (80) & (81), p. 99 — introduces a dimensionless structural form factor, $W_n$, for the central pressure via the expression,

$P_c$

$=$

$W_n\left(\frac{G{M}^2}{R^4}\right),$

and demonstrates that,

$\frac{1}{W_n}$

$\equiv$

$4\pi (n+1)[{\mathrm{\Theta }}^{\text{'}}{]}_{{\xi }_1}^2.$

It is therefore clear that a spherical polytrope's central pressure is expressible in terms of our structural form factor, ${\mathfrak{𝔣}}_A$, as,

$P_c$

$=$

$\frac{3}{4\pi (5-n)}\left(\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{R_{\mathrm{e}\mathrm{q}}^4}\right)\frac{1}{{\mathfrak{𝔣}}_A}.$

Looking back at our original definition of the structural form factors, we note that,

${\mathfrak{𝔣}}_A=(\frac{\overline{P}}{P_c}{)}_{\mathrm{e}\mathrm{q}}.$

Hence, this last equilibrium relation can be rewritten as,

$\frac{\overline{P}{R}_{\mathrm{e}\mathrm{q}}^4}{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}=\frac{3}{4\pi (5-n)}.$

Alternate Derivation of Gravitational Potential Energy

As has been discussed elsewhere, we have learned from Chandrasekhar's discussion of polytropic spheres [C67] — see his Equation (16), p. 64 — that if a spherically symmetric system is in hydrostatic balance, the total gravitational potential energy can be obtained from the following integral:

$W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$

$=$

$+\frac{1}{2}\underset{0}{\overset{R}{\int }}\mathrm{\Phi }(r)dm.$

Using "technique #3" to solve the differential equation that governs the statement of hydrostatic balance, we know that in any polytropic sphere, $\mathrm{\Phi }(r)$ is related to the configuration's radial enthalpy profile, $H(r)$, via the algebraic expression,

$\mathrm{\Phi }(r)+H(r)$

$=$

$C_B,$

where, $C_B$, is an integration constant. At the surface of the equilibrium configuration, $H=0$ and $\mathrm{\Phi }=-G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}/R_{\mathrm{e}\mathrm{q}}$, so the integration constant is,

$C_B$

$=$

$-\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{R_{\mathrm{e}\mathrm{q}}},$

which implies,

$\mathrm{\Phi }(r)$

$=$

$-H(r)-\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{R_{\mathrm{e}\mathrm{q}}}.$

Now, from our general discussion of barotropic relations, we can write,

$H(r)$

$=$

$(n+1)\frac{P(r)}{\rho (r)}.$

Hence,

$-\mathrm{\Phi }(r)$

$=$

$(n+1)\frac{P(r)}{\rho (r)}+\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{R_{\mathrm{e}\mathrm{q}}},$

and,

$W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$	$=$	$-\frac{1}{2}\underset{0}{\overset{R}{\int }}[(n+1)\frac{P(r)}{\rho (r)}+\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{R_{\mathrm{e}\mathrm{q}}}]4\pi \rho (r)r^{2}dr$
	$=$	$-2\pi \{(n+1)\underset{0}{\overset{R}{\int }}P(r)r^{2}dr+\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{R_{\mathrm{e}\mathrm{q}}}\underset{0}{\overset{R}{\int }}\rho (r)r^{2}dr\}$
	$=$	$-2\pi \{\frac{1}{3}(n+1)P_c{R}_{\mathrm{e}\mathrm{q}}^3\underset{0}{\overset{1}{\int }}3[\frac{P(x)}{P_c}]x^{2}dx+\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{3R_{\mathrm{e}\mathrm{q}}}({\rho }_c{R}_{\mathrm{e}\mathrm{q}}^3\left)\underset{0}{\overset{1}{\int }}3\right[\frac{\rho (x)}{{\rho }_c}\left]x^{2}dx\right\}$
	$=$	$-2\pi \{\frac{1}{3}(n+1)P_c{R}_{\mathrm{e}\mathrm{q}}^3{\mathfrak{𝔣}}_A+\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{3R_{\mathrm{e}\mathrm{q}}}({\rho }_c{R}_{\mathrm{e}\mathrm{q}}^3\left){\mathfrak{𝔣}}_M\right\}$
	$=$	$-\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{R_{\mathrm{e}\mathrm{q}}}\{\frac{2\pi }{3}(n+1)[\frac{P_c{R}_{\mathrm{e}\mathrm{q}}^4}{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}]{\mathfrak{𝔣}}_A+\frac{1}{2}[\frac{4\pi {\rho }_c{R}_{\mathrm{e}\mathrm{q}}^3}{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}\left]{\mathfrak{𝔣}}_M\right\}$
	$=$	$-\frac{1}{2}\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{R_{\mathrm{e}\mathrm{q}}}\{\frac{4\pi }{3}(n+1)[\frac{P_c{R}_{\mathrm{e}\mathrm{q}}^4}{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}]{\mathfrak{𝔣}}_A+1\}.$

We now recall two earlier expressions that show the role that our structural form factors play in the evaluation of $W$ and $P_c$, namely,

$W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$

$=$

$-\frac{3}{5}\left(\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{R_{\mathrm{e}\mathrm{q}}}\right)\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2}$

and,

$P_c$

$=$

$\frac{3}{20\pi }\left(\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{R_{\mathrm{e}\mathrm{q}}^4}\right)\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_A{\mathfrak{𝔣}}_M^2}.$

Plugging these into our newly derived expression for the gravitational potential energy gives,

$-\frac{3}{5}\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2}$	$=$	$-\frac{1}{2}\{\frac{4\pi }{3}(n+1)[\frac{3}{20\pi }\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_A{\mathfrak{𝔣}}_M^2}]{\mathfrak{𝔣}}_A+1\}$
$\Rightarrow (2\cdot 3)\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2}$	$=$	$(n+1)\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2}+5$
$\Rightarrow (5-n)\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2}$	$=$	$5$
$\Rightarrow \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2}$	$=$	$\frac{5}{5-n}.$

As it should, this agrees with the expression for the ratio, ${\mathfrak{𝔣}}_W/{\mathfrak{𝔣}}_M^2$, that was derived in our above discussion of the gravitational potential energy.

Summary

In summary, expressions for the three structural form factors associated with isolated, spherically symmetric polytropes are as follows:

Structural Form Factors for Isolated Polytropes
${\mathfrak{𝔣}}_M$ $=$ $[-\frac{3{\mathrm{\Theta }}^{\text{'}}}{\xi }{]}_{{\xi }_1}$ ${\mathfrak{𝔣}}_W$ $=$ $\frac{3^2\cdot 5}{5-n}[\frac{{\mathrm{\Theta }}^{\text{'}}}{\xi }{]}_{{\xi }_1}^2$ ${\mathfrak{𝔣}}_A$ $=$ $\frac{3(n+1)}{(5-n)}[{\mathrm{\Theta }}^{\text{'}}{]}_{{\xi }_1}^2$

See Also