SSC/BipolytropeGeneralization/Pt2

Bipolytrope Generalization (Pt 2)

Part I:  Bipolytrope Generalization

Part II:  Derivations

Part III:  Examples

Part IV:  Best of the Best

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Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Detailed Derivations

Dividing the free-energy expression through by $E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ generates,

${\mathfrak{𝔊}}^{*}\equiv \frac{\mathfrak{𝔊}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$	$=$	$-\frac{3}{5}\left(\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\right)\left(\frac{1}{R}\right)\cdot {\mathfrak{𝔣}}_{WM}+\left[\frac{\nu s_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{({\gamma }_c-1)}\right]\left[\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}{K}_c{\rho }_{ic}^{{\gamma }_c-1}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\right]$
		$+\left[\frac{(1-\nu )s_{\mathrm{e}\mathrm{n}\mathrm{v}}}{({\gamma }_e-1)}\right]\left[\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}{K}_e{\rho }_{ie}^{{\gamma }_e-1}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}\right]$
	$=$	$-\frac{3\cdot {\mathfrak{𝔣}}_{WM}}{5}\left[\frac{K_c{G}^{(3{\gamma }_c-4)}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2(3{\gamma }_c-4)}}{G^{3{\gamma }_c-3}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{5{\gamma }_c-6}}{]}^{1/(3{\gamma }_c-4)}\right(\frac{1}{R})$
		$+\left[\frac{\nu s_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{({\gamma }_c-1)}\right]\left[\frac{K_c{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3{\gamma }_c-4}{K}_c^{3{\gamma }_c-4}}{G^{3{\gamma }_c-3}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{5{\gamma }_c-6}}{]}^{1/(3{\gamma }_c-4)}\right(\frac{{\rho }_{ic}}{\overline{\rho }}{)}^{{\gamma }_c-1}[\frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi R^3}{]}^{{\gamma }_c-1}$
		$+\left[\frac{(1-\nu )s_{\mathrm{e}\mathrm{n}\mathrm{v}}}{({\gamma }_e-1)}\right]\left[\frac{K_c{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3{\gamma }_c-4}{K}_e^{3{\gamma }_c-4}}{G^{3{\gamma }_c-3}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{5{\gamma }_c-6}}{]}^{1/(3{\gamma }_c-4)}\right(\frac{{\rho }_{ie}}{\overline{\rho }}{)}^{{\gamma }_e-1}[\frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi R^3}{]}^{{\gamma }_e-1}$
	$=$	$-\frac{3\cdot {\mathfrak{𝔣}}_{WM}}{5}\left(\frac{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{R}\right)$
		$+\left[\frac{\nu s_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{({\gamma }_c-1)}\right]\left(\frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi }{)}^{{\gamma }_c-1}\right[\frac{K_c^{3{\gamma }_c-3}}{G^{3{\gamma }_c-3}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2{\gamma }_c-2}}{]}^{1/(3{\gamma }_c-4)}\left[\frac{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{R}{]}^{3({\gamma }_c-1)}\right[\left(\frac{K_c}{G}\right)M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{{\gamma }_c-2}{]}^{-3({\gamma }_c-1)/(3{\gamma }_c-4)}(\frac{{\rho }_{ic}}{\overline{\rho }}{)}^{{\gamma }_c-1}$
		$+\left[\frac{(1-\nu )s_{\mathrm{e}\mathrm{n}\mathrm{v}}}{({\gamma }_e-1)}\right]\left(\frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi }{)}^{{\gamma }_e-1}\right[\frac{K_c^{3{\gamma }_c-3}(K_e/K_c{)}^{3{\gamma }_c-4}}{G^{3{\gamma }_c-3}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2{\gamma }_c-2}}{]}^{1/(3{\gamma }_c-4)}\left[\frac{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{R}{]}^{3({\gamma }_e-1)}\right[\left(\frac{K_c}{G}\right)M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{{\gamma }_c-2}{]}^{-3({\gamma }_e-1)/(3{\gamma }_c-4)}(\frac{{\rho }_{ie}}{\overline{\rho }}{)}^{{\gamma }_e-1}$
	$=$	$-\frac{3\cdot {\mathfrak{𝔣}}_{WM}}{5}\left(\frac{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{R}\right)$
		$+\left[\frac{\nu s_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{({\gamma }_c-1)}\right]\left(\frac{3}{4\pi }{)}^{{\gamma }_c-1}\right[M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3{\gamma }_c-4}{]}^{({\gamma }_c-1)/(3{\gamma }_c-4)}\left[M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-2}{]}^{({\gamma }_c-1)/(3{\gamma }_c-4)}\right[M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-3{\gamma }_c+6}{]}^{({\gamma }_c-1)/(3{\gamma }_c-4)}\left[\frac{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{R}{]}^{3({\gamma }_c-1)}\right(\frac{{\rho }_{ic}}{\overline{\rho }}{)}^{{\gamma }_c-1}$
		$+\left[\frac{(1-\nu )s_{\mathrm{e}\mathrm{n}\mathrm{v}}}{({\gamma }_e-1)}\right]\left(\frac{3}{4\pi }{)}^{{\gamma }_e-1}\right(\frac{K_e}{K_c}\left)\right[M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2({\gamma }_e-1)-2({\gamma }_c-1)}{]}^{1/(3{\gamma }_c-4)}\left[\frac{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{R}{]}^{3({\gamma }_e-1)}\right[\left(\frac{K_c}{G}{)}^{({\gamma }_c-1)-({\gamma }_e-1)}{]}^{3/(3{\gamma }_c-4)}\right(\frac{{\rho }_{ie}}{\overline{\rho }}{)}^{{\gamma }_e-1}$
	$=$	$-\frac{3\cdot {\mathfrak{𝔣}}_{WM}}{5}(\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^{-1}+\frac{\nu s_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{({\gamma }_c-1)}[\left(\frac{3}{4\pi }\right)\frac{{\rho }_{ic}}{\overline{\rho }}{]}^{{\gamma }_c-1}[\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{-3({\gamma }_c-1)}$
		$+\frac{(1-\nu )s_{\mathrm{e}\mathrm{n}\mathrm{v}}}{({\gamma }_e-1)}\left(\frac{K_e}{K_c}\right)\left[\frac{K_c^3}{G^3{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{]}^{({\gamma }_c-{\gamma }_e)/(3{\gamma }_c-4)}\right[\left(\frac{3}{4\pi }\right)\frac{{\rho }_{ie}}{\overline{\rho }}{]}^{{\gamma }_e-1}[\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{-3({\gamma }_e-1)}.$

We also want to ensure that envelope pressure matches the core pressure at the interface. This means,

$K_e{\rho }_{ie}^{{\gamma }_e}$	$=$	$K_c{\rho }_{ic}^{{\gamma }_c}$
$\Rightarrow \frac{K_e}{K_c}$	$=$	${\rho }_{ic}^{{\gamma }_c}{\rho }_{ie}^{-{\gamma }_e}$
	$=$	$\left[\frac{{\rho }_{ic}}{{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{{\gamma }_c}\right[\frac{{\rho }_{ie}}{{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{-{\gamma }_e}{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^{{\gamma }_c-{\gamma }_e}$
	$=$	$\left[\frac{{\rho }_{ic}}{{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{{\gamma }_c}\right[\frac{{\rho }_{ie}}{{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{-{\gamma }_e}\left\{\frac{3}{4\pi }\right[\frac{G^3{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{K_c^3}{]}^{1/(3{\gamma }_c-4)}{\}}^{{\gamma }_c-{\gamma }_e}$
$\Rightarrow \frac{K_e}{K_c}\left[\frac{K_c^3}{G^3{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{]}^{({\gamma }_c-{\gamma }_e)/(3{\gamma }_c-4)}\right[\left(\frac{3}{4\pi }\right)\frac{{\rho }_{ie}}{\overline{\rho }}{]}^{{\gamma }_e-1}$	$=$	$\left[\right(\frac{3}{4\pi }\left)\frac{{\rho }_{ic}}{{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{{\gamma }_c}\right[\left(\frac{3}{4\pi }\right)\frac{{\rho }_{ie}}{{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{-{\gamma }_e}\left[\right(\frac{3}{4\pi })\frac{{\rho }_{ie}}{\overline{\rho }}{]}^{{\gamma }_e-1}$
	$=$	$\left[\right(\frac{3}{4\pi }\left)\frac{{\rho }_{ic}}{\overline{\rho }}{]}^{{\gamma }_c-1}\right(\frac{{\rho }_{ic}}{{\rho }_{ie}}\left)\right(\frac{{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{\overline{\rho }}{)}^{{\gamma }_e-{\gamma }_c}$
	$=$	$\left[\right(\frac{3}{4\pi }\left)\frac{{\rho }_{ic}}{\overline{\rho }}{]}^{{\gamma }_c-1}\right(\frac{{\rho }_{ic}}{{\rho }_{ie}}\left)\right(\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^{3({\gamma }_e-{\gamma }_c)}$

Hence, we can write the normalized (and dimensionless) free energy as,

${\mathfrak{𝔊}}^{*}$

$=$

$-\frac{3\cdot {\mathfrak{𝔣}}_{WM}}{5}(\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}^{-1}+\{\frac{\nu s_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{({\gamma }_c-1)}+\frac{(1-\nu )s_{\mathrm{e}\mathrm{n}\mathrm{v}}}{({\gamma }_e-1)}\left(\frac{{\rho }_{ic}}{{\rho }_{ie}}\right)\left\}\right[\left(\frac{3}{4\pi }\right)\frac{{\rho }_{ic}}{\overline{\rho }}{]}^{{\gamma }_c-1}[\frac{R}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{-3({\gamma }_c-1)}.$

Keep in mind that, if the envelope and core both have uniform (but different) densities, then ${\rho }_{ic}={\rho }_c$, ${\rho }_{ie}={\rho }_e$, and

$\frac{{\rho }_c}{\overline{\rho }}=\frac{\nu }{q^3};\frac{{\rho }_e}{\overline{\rho }}=\frac{1-\nu }{1-q^3};\frac{{\rho }_e}{{\rho }_c}=\frac{q^3(1-\nu )}{\nu (1-q^3)}.$

Free Energy and Its Derivatives

Now, the free energy can be written as,

$\mathfrak{𝔊}$	$=$	$U_{\mathrm{t}\mathrm{o}\mathrm{t}}+W$
	$=$	$\left[\frac{2}{3({\gamma }_c-1)}\right]S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+\left[\frac{2}{3({\gamma }_e-1)}\right]S_{\mathrm{e}\mathrm{n}\mathrm{v}}+W$
	$=$	$\left[\frac{2}{3({\gamma }_c-1)}\right]C_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{R}^{3-3{\gamma }_c}+\left[\frac{2}{3({\gamma }_e-1)}\right]C_{\mathrm{e}\mathrm{n}\mathrm{v}}{R}^{3-3{\gamma }_e}-A{R}^{-1}.$

The first derivative of the free energy with respect to radius is, then,

$\frac{d\mathfrak{𝔊}}{dR}$

$=$

$-2C_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{R}^{2-3{\gamma }_c}-2C_{\mathrm{e}\mathrm{n}\mathrm{v}}{R}^{2-3{\gamma }_e}+A{R}^{-2}.$

And the second derivative is,

$\frac{d^2\mathfrak{𝔊}}{d{R}^2}$	$=$	$-2(2-3{\gamma }_c)C_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{R}^{1-3{\gamma }_c}-2(2-3{\gamma }_e)C_{\mathrm{e}\mathrm{n}\mathrm{v}}{R}^{1-3{\gamma }_e}-2A{R}^{-3}.$
	$=$	$\frac{2}{R^2}[(3{\gamma }_c-2)C_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{R}^{3-3{\gamma }_c}+(3{\gamma }_e-2)C_{\mathrm{e}\mathrm{n}\mathrm{v}}{R}^{3-3{\gamma }_e}-A{R}^{-1}]$
	$=$	$\frac{2}{R^2}[(3{\gamma }_c-2)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(3{\gamma }_e-2)S_{\mathrm{e}\mathrm{n}\mathrm{v}}+W].$

Equilibrium

The radius, $R_{\mathrm{e}\mathrm{q}}$, of the equilibrium configuration(s) is determined by setting the first derivative of the free energy to zero. Hence,

$0$	$=$	$2C_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{R}_{\mathrm{e}\mathrm{q}}^{2-3{\gamma }_c}+2C_{\mathrm{e}\mathrm{n}\mathrm{v}}{R}_{\mathrm{e}\mathrm{q}}^{2-3{\gamma }_e}-A{R}_{\mathrm{e}\mathrm{q}}^{-2}$
	$=$	$R_{\mathrm{e}\mathrm{q}}^{-1}[2C_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{R}_{\mathrm{e}\mathrm{q}}^{3-3{\gamma }_c}+2C_{\mathrm{e}\mathrm{n}\mathrm{v}}{R}_{\mathrm{e}\mathrm{q}}^{3-3{\gamma }_e}-A{R}_{\mathrm{e}\mathrm{q}}^{-1}]$
	$=$	$R_{\mathrm{e}\mathrm{q}}^{-1}[2S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+2S_{\mathrm{e}\mathrm{n}\mathrm{v}}+W]$
$\Rightarrow 2S_{\mathrm{t}\mathrm{o}\mathrm{t}}+W$	$=$	$0.$

This is the familiar statement of virial equilibrium. From it we should always be able to derive the radius of equilibrium configurations.

Stability

To assess the relative stability of an equilibrium configuration, we need to determine the sign of the second derivative of the free energy, evaluated at the equilibrium radius. If the sign of the second derivative is positive, the system is dynamically stable; if the sign is negative, he system is dynamically unstable. Using the above statement of virial equilibrium, that is, setting,

$2S_{\mathrm{t}\mathrm{o}\mathrm{t}}+W$	$=$	$0,$
$\Rightarrow S_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$=$	$-S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}-\frac{W}{2},$

we obtain,

$\frac{d^2\mathfrak{𝔊}}{d{R}^2}{|}_{\mathrm{e}\mathrm{q}}$	$=$	$\frac{2}{R_{\mathrm{e}\mathrm{q}}^2}[(3{\gamma }_c-2)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W-(3{\gamma }_e-2)(S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+\frac{W}{2}){]}_{\mathrm{e}\mathrm{q}}$
	$=$	$\frac{2}{R_{\mathrm{e}\mathrm{q}}^2}[3({\gamma }_c-{\gamma }_e)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(2-\frac{3}{2}{\gamma }_e)W{]}_{\mathrm{e}\mathrm{q}}$
	$=$	$\frac{6}{R_{\mathrm{e}\mathrm{q}}^2}[({\gamma }_c-{\gamma }_e)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+\frac{1}{2}(\frac{4}{3}-{\gamma }_e)W{]}_{\mathrm{e}\mathrm{q}}$
	$=$	$\frac{6}{R_{\mathrm{e}\mathrm{q}}^2}[-\frac{W}{2}({\gamma }_e-\frac{4}{3})-({\gamma }_e-{\gamma }_c)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{]}_{\mathrm{e}\mathrm{q}}.$

So, if when evaluated at the equilibrium state, the expression inside of the square brackets of this last expression is negative, the equilibrium configuration will be dynamically unstable. We have chosen to write the expression in this particular final form because we generally will be interested in bipolytropes for which the adiabatic exponent of the envelope is greater than $4/3$ and the adiabatic exponent of the core is less than or equal to $4/3$ — that is, ${\gamma }_e>4/3\ge {\gamma }_c$. Hence, because the gravitational potential energy, $W$, is intrinsically negative, the system will be dynamically unstable only if the second term (involving $S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$) is greater in magnitude than the first term (involving $W$).

It is worth noting that, instead of drawing upon $S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ and $W$ to define the stability condition, we could have used an appropriate combination of $S_{\mathrm{e}\mathrm{n}\mathrm{v}}$ and $W$, or the $S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ and $S_{\mathrm{e}\mathrm{n}\mathrm{v}}$ pair. Also, for example, because the virial equilibrium condition is $S_{\mathrm{t}\mathrm{o}\mathrm{t}}=-W/2$, it is easy to see that the following inequality also equivalently defines stability:

$S_{\mathrm{t}\mathrm{o}\mathrm{t}}({\gamma }_e-\frac{4}{3})-({\gamma }_e-{\gamma }_c)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$>$

$0.$

Related Discussions

• Newer, Version2 of Bipolytrope Generalization derivations.
• Analytic solution with $n_c=0$ and $n_e=0$.
• Analytic solution with $n_c=5$ and $n_e=1$.