SSC/Structure/BiPolytropes/Analytic51/Pt4

BiPolytrope with n_c = 5 and n_e = 1 (Pt 4)

Part I:  (nc,ne) = (5,1) BiPolytrope

Part II:  Example Models

Part III:  Limiting Mass

Part IV:  Free Energy

Free Energy

Here we use this bipolytrope's free energy function to probe the relative dynamical stability of various equilibrium models. This derivation for $(n_c,n_e)=(5,1)$ bipolytropes is similar to the one that has been presented elsewhere in the context of (n_c, n_e) = (0, 0) bipolytropes and follows the analysis outline provided in our discussion of the stability of generalized bipolytropes.

Expression for Free Energy

In order to construct the free energy function, we need mathematical expressions for the gravitational potential energy, $W$, and for the thermal energy content, $S$, of the models; and it will be natural to break both energy expressions into separate components derived for the $n_c=5$ core and for the $n_e=1$ envelope. Consistent with the above equilibrium model derivations, we will work with dimensionless variables. Specifically, we define,

$W^{*}$

$\equiv$

$\frac{W}{[K_c^5/G^3{]}^{1/2}}$

;

$S^{*}$

$\equiv$

$\frac{S}{[K_c^5/G^3{]}^{1/2}}.$

Drawing on the various functional expressions that are provided in the above derivations, including the Table of Parameters, integrals over the material in the core give us,

$S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$	$=$	$\frac{3}{2}\underset{0}{\overset{r_i}{\int }}(\frac{P^{*}}{{\rho }^{*}}{)}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}(4\pi {\rho }^{*}{)}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}(r^{*}{)}^{2}d{r}^{*}$		Mathematica Integral
	$=$	$6\pi \left(\frac{3}{2\pi }{)}^{3/2}\underset{0}{\overset{{\xi }_i}{\int }}\right(1+\frac{1}{3}{\xi }^2{)}^{-3}{\xi }^{2}d\xi$
	$=$	$6\pi \left(\frac{3^2}{2\pi }{)}^{3/2}\underset{0}{\overset{x_i}{\int }}\right(1+x^2{)}^{-3}{x}^{2}dx$
	$=$	$\left(\frac{3^8}{2^7\pi }{)}^{1/2}\right[\frac{x_i}{(1+x_i^2)}-\frac{2x_i}{(1+x_i^2{)}^2}+{\tan }^{-1}(x_i)]$
	$=$	$\frac{1}{2}\left(\frac{3^8}{2^5\pi }{)}^{1/2}\right[x_i(x_i^4-1)(1+x_i^2{)}^{-3}+{\tan }^{-1}(x_i)],$

where, in order to streamline the integral for Mathematica, we have used the substitution, $x\equiv \xi /\sqrt{3}$; and,

$W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$	$=$	$-\underset{0}{\overset{r_i}{\int }}(4\pi M_r^{*}{\rho }^{*}{)}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}(r^{*})d{r}^{*}$		Mathematica Integral
	$=$	$-4\pi \underset{0}{\overset{{\xi }_i}{\int }}\left(\frac{2\cdot 3}{\pi }{)}^{1/2}\right[{\xi }^3(1+\frac{1}{3}{\xi }^2{)}^{-3/2}](1+\frac{1}{3}{\xi }^2{)}^{-5/2}(\frac{3}{2\pi })\xi d\xi$
	$=$	$-\left(\frac{2^3\cdot 3^8}{\pi }{)}^{1/2}\underset{0}{\overset{x_i}{\int }}\right(1+x^2{)}^{-4}{x}^{4}dx$
	$=$	$-\left(\frac{2^3\cdot 3^8}{\pi }{)}^{1/2}\right[3{\tan }^{-1}(x_i)+\frac{x_i(3x_i^4-8x_i^2-3)}{(1+x_i^2{)}^3}\left]\right(\frac{1}{2^4\cdot 3})$
	$=$	$-\left(\frac{3^8}{2^5\pi }{)}^{1/2}\right[x_i(x_i^4-\frac{8}{3}{x}_i^2-1)(1+x_i^2{)}^{-3}+{\tan }^{-1}(x_i)].$

(Apology: The parameter $x_i$ introduced here is identical to the parameter ${\ell }_i$ that was introduced earlier in the context of our discussion of the "Limiting Mass" of these models. Sorry for the unnecessary duplication of parameters and possible confusion!)

While our aim, here, has been to determine an expression for the gravitational potential energy of a truncated $n=5$ polytropic sphere, our derived expression can also give the gravitational potential of an isolated $n=5$ polytrope by evaluating the expression in the limit $x_i\to \mathrm{\infty }$. In this limit, the first term inside the square brackets goes to zero, while the second term, $\underset{x_i\to \mathrm{\infty }}{\lim }{\tan }^{-1}(x_i)=\frac{\pi }{2}.$ We see, therefore, that, $W^{*}{|}_{\mathrm{t}\mathrm{o}\mathrm{t}}=\underset{x_i\to \mathrm{\infty }}{\lim }{W}^{*}=-(\frac{3^8}{2^5\pi }{)}^{1/2}\frac{\pi }{2}=-(\frac{3^8\pi }{2^7}{)}^{1/2}.$ Taking into account our adopted energy normalization, this can be rewritten with the dimensions of energy as, $W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}{|}_{\mathrm{t}\mathrm{o}\mathrm{t}}$ $=$ $-\left(\frac{3^8\pi }{2^7}{)}^{1/2}\right(\frac{K_c^5}{G^3}{)}^{1/2}=-\left(\frac{3^8\pi }{2^7}{)}^{1/2}\right[\frac{\pi }{2^3\cdot 3^7}{]}^{1/2}\frac{G{M}^2}{a_5}$   $=$ $-(\frac{3{\pi }^2}{2^{10}}{)}^{1/2}\frac{G{M}^2}{a_5},$ where we have elected to write the total gravitational potential energy in terms of the natural scale length for $n=5$ polytropes, which, as documented elsewhere, is, $a_5$ $=$ $[\frac{3K}{2\pi G}{]}^{1/2}{\rho }_c^{-2/5}=[\frac{3K}{2\pi G}{]}^{1/2}\left[\frac{\pi M^2{G}^3}{2\cdot 3^4}\right]K^{-3}=G{M}^2[\frac{\pi G^3}{2^3\cdot 3^7{K}^5}{]}^{1/2}.$ As can be seen from the following, boxed-in equation excerpt, our derived expression for the total gravitational potential energy of an isolated $n=5$ polytrope exactly matches the result derived by 📚 H. A. Buchdahl (1978, Aust. J. Phys., Vol. 31, pp. 115 - 116). The primary purpose of Buchdahl's short communication was to point out that, despite the fact that its radius extends to infinity, "the gravitational potential energy of [an isolated] polytrope of index 5 is finite." Equation excerpt from p. 116 of H. A. Buchdahl (1978) Remark on the polytrope of index 5 Australian Journal of Physics, Vol. 31, pp. 115 - 116 $\mathrm{\Omega }=-(\pi \sqrt{3}/32)G{M}^2/\alpha .$ Note that a comparison between Buchdahl's derived expression and our expression in the limit $x_i\to \mathrm{\infty }$ requires the parameter substitutions, $\mathrm{\Omega }\to W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}{|}_{\mathrm{t}\mathrm{o}\mathrm{t}}$       and       $\alpha \to a_5$

Notice that these two terms combine to give, for the core,

$(2S+W{)}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$=$

$(\frac{2\cdot 3^6}{\pi }{)}^{1/2}\frac{x_i^3}{(1+x_i^2{)}^3}=(\frac{2}{\pi }{)}^{1/2}{3}^{3/2}{\xi }_i^3(1+\frac{1}{3}{\xi }_i^2{)}^{-3}.$

Similarly, integrals over the material in the envelope give us,

$S_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}$	$=$	$\frac{3}{2}\underset{r_i}{\overset{R}{\int }}(\frac{P^{*}}{{\rho }^{*}}{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}(4\pi {\rho }^{*}{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}(r^{*}{)}^{2}d{r}^{*}$		Mathematica Integral
	$=$	$6\pi \underset{{\eta }_i}{\overset{{\eta }_s}{\int }}[{\theta }_i^6{\varphi }^2]\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}{]}^3{\eta }^{2}d\eta$
	$=$	$\left(\frac{3^2}{2\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\underset{{\eta }_i}{\overset{{\eta }_s}{\int }}[\sin (\eta -B){]}^{2}d\eta$
	$=$	$\left(\frac{3^2}{2^5\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\{2(\eta -B)-\sin [2(\eta -B)]{\}}_{{\eta }_i}^{{\eta }_s}$
	$=$	$\left(\frac{1}{2^5\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\{6(\eta -B)-3\sin [2(\eta -B)]{\}}_{{\eta }_i}^{{\eta }_s};$

and,

$W_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}$	$=$	$-\underset{r_i}{\overset{R}{\int }}(4\pi M_r^{*}{\rho }^{*}{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}(r^{*})d{r}^{*}$		Mathematica Integral
	$=$	$-4\pi \underset{{\eta }_i}{\overset{{\eta }_s}{\int }}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5\varphi \left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}{]}^2\eta d\eta$
	$=$	$-\left(\frac{2^3}{\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}\underset{{\eta }_i}{\overset{{\eta }_s}{\int }}(-{\eta }^2\frac{d\varphi }{d\eta })\varphi \eta d\eta$
	$=$	$-\left(\frac{2^3}{\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\underset{{\eta }_i}{\overset{{\eta }_s}{\int }}[\sin (\eta -B)-\eta \cos (\eta -B)]\sin (\eta -B)d\eta$
	$=$	$-\left(\frac{1}{2^3\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\{-3\sin [2(\eta -B)]+2\eta \cos [2(\eta -B)]+4(\eta -B)+2B{\}}_{{\eta }_i}^{{\eta }_s}$
	$=$	$-\left(\frac{1}{2^3\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\{6(\eta -B)-3\sin [2(\eta -B)]-4\eta {\sin }^{}(\eta -B)+4B{\}}_{{\eta }_i}^{{\eta }_s}.$

In this case, the two terms combine to give, for the envelope,

$(2S+W{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$=$	$\left(\frac{1}{2^3\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2[4\eta {\sin }^2(\eta -B)+4B{]}_{{\eta }_i}^{{\eta }_s}$
	$=$	$\left(\frac{2}{\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2[{\eta }_s{\sin }^2({\eta }_s-B)-{\eta }_i{\sin }^2({\eta }_i-B)].$

Equilibrium Condition

Global

Recognizing from the above Table of Parameters that,

$A$	$=$	$\frac{{\eta }_i}{\sin ({\eta }_i-B)},$	[because ${\varphi }_i=1$]
$({\eta }_s-B)$	$=$	$\pi ,$	[hence, ${\sin }^2({\eta }_s-B)=0$]
${\eta }_i$	$=$	$3^{1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right){\xi }_i(1+\frac{1}{3}{\xi }_i^2{)}^{-1},$

we can rewrite this last "envelope virial" expression as,

$(2S+W{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$=$	$-\left(\frac{2}{\pi }{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{\eta }_i^3$
	$=$	$-\left(\frac{2}{\pi }{)}^{1/2}{3}^{3/2}{\xi }_i^3\right(1+\frac{1}{3}{\xi }_i^2{)}^{-3}.$

This expression is equal in magnitude, but opposite in sign to the "core virial" expression derived earlier. Hence, putting the core and envelope contributions together, we find,

$(2S+W{)}_{\mathrm{t}\mathrm{o}\mathrm{t}}=2(S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+S_{\mathrm{e}\mathrm{n}\mathrm{v}})+(W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{e}\mathrm{n}\mathrm{v}})$

$=$

$0.$

This demonstrates that the detailed force-balanced models of $(n_c,n_e)=(5,1)$ bipolytropes derived above are also all in virial equilibrium, as should be the case. More importantly, showing that these four separate energy integrals sum to zero helps provide confirmation that the four energy integrals have been derived correctly. This allows us to confidently proceed to an evaluation of the relative dynamical stability of the models.

In Parts

In section ⑩ of our Tabular Overview, we speculated that, in bipolytropic equilibrium structures, the statements

$2S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

and

$2S_{\mathrm{e}\mathrm{n}\mathrm{v}}+W_{\mathrm{e}\mathrm{n}\mathrm{v}}=-3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}},$

hold separately. Let's evaluate the "PV" term. We find that,

$3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=4\pi P_i{r}_i^3$	$=$	$4\pi (1+\frac{{\xi }_i^2}{3}{)}^{-3}(\frac{3}{2\pi }{)}^{3/2}{\xi }_i^3$
	$=$	$(1+\frac{{\xi }_i^2}{3}{)}^{-3}(\frac{2\cdot 3^3}{\pi }{)}^{1/2}{\xi }_i^3.$

This is precisely the "extra term" that shows up (with opposite signs) in the above-derived expressions for the separate quantities, $(2S+W{)}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ and $(2S+W{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}$. Hence our speculation has been shown to be correct, at least for the case of bipolytropes with, $({\gamma }_c,{\gamma }_e)=(\frac{6}{5},2)$.

Stability Condition

According to the accompanying free-energy based, generalized formulation of stability in bipolytropes, our above derived $(n_c,n_e)=(5,1)$ bipolytropes — or, equivalently, $({\gamma }_c,{\gamma }_e)=(6/5,2)$ bipolytropes — will be dynamically stable only if,

$-(W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{e}\mathrm{n}\mathrm{v}})({\gamma }_e-\frac{4}{3})$

$>$

$2({\gamma }_e-{\gamma }_c)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}.$

Otherwise, they will be dynamically unstable toward radial perturbations. For various values of the ${\mu }_e/{\mu }_c$ ratio, Table 3 identifies the value of ${\xi }_i$ — and the corresponding values of $q$ and $\nu$ — at which the left-hand side of this stability relation equals the right-hand side. The locus of points provided by Table 3 defines the curve that separates stable from unstable regions of the $q-\nu$ parameter space. The red-dashed curve drawn in Figure 3 graphically depicts this demarcation: the region below the curve identifies bipolytrope models that are dynamically stable while the region above the curve identifies unstable models.

Table 3: Points Defining Stability Curve ${\mu }_e/{\mu }_c$ ${\xi }_i$ $q$ $\nu$   ${\mu }_e/{\mu }_c$ ${\xi }_i$ $q$ $\nu$ 1 2.416 0.5952 0.6830 0.375 6.259 0.1695 0.6054 0.95 2.500 0.5805 0.6884 0.350 7.341 0.1284 0.5439 0.90 2.594 0.5642 0.6937 0.340 7.991 0.1109 0.5081 0.80 2.816 0.5255 0.7031 ⅓ 8.548 0.0990 0.4790 0.70 3.109 0.4775 0.7104 0.32 10.2 0.0744 0.4038 0.65 3.296 0.4481 0.7124 0.31 12.4 0.05536 0.3264 0.60 3.523 0.4142 0.7125 0.305 14.4 0.04494 0.2772 0.55 3.809 0.3748 0.7096 0.3 17.733 0.03412 0.2186 ½ 4.186 0.3284 0.7014 0.295 25.737 0.02165 0.14347 0.45 4.719 0.2733 0.6830 0.291 75.510 0.00666 0.0450 0.40 5.574 0.2073 0.6429

Figure 3: Largely the same as Figure 1, above, but a red-dashed curve has been added that separates the $q-\nu$ domain into regions that contain stable models (lying below the curve) from dynamically unstable models (lying above the curve), as determined by the virial stability analysis presented here.

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