Bipolytrope Generalization

Setup

Before working with the free energy expression, we will need mathematical expressions for the gravitational potential energy, <math>~W</math>, and for the thermal energy content of the core and envelope, <math>~S_\mathrm{core}</math> and <math>~S_\mathrm{env}</math>, respectively. Generally for spherically symmetric polytropic configurations, we should be able to write these three energy quantities in the following forms:

<math>~W</math>	<math>~=~</math>	<math>~ - \frac{A}{R} \, ;</math>
<math>~S_\mathrm{core}</math>	<math>~=~</math>	<math>~ B_\mathrm{core} R^3 P_{ic} = C_\mathrm{core} R^{3-3\gamma_c} \, ;</math>
<math>~S_\mathrm{env}</math>	<math>~=~</math>	<math>~ B_\mathrm{env} R^3 P_{ie} = C_\mathrm{env} R^{3-3\gamma_e} \, ;</math>

where,

<math>~C_\mathrm{core}</math>	<math>~=~</math>	<math>~ B_\mathrm{core} [K_c \rho_{ic}^{\gamma_c} ]R^{3\gamma_c} </math>
	<math>~=~</math>	<math>~ B_\mathrm{core} K_c \biggl[ \frac{3M_\mathrm{core}}{4\pi (r_i/R)^3} \biggl( \frac{\rho_{i}}{\bar\rho}\biggr)_c \biggr]^{\gamma_c} </math>
	<math>~=~</math>	<math>~ B_\mathrm{core} K_c \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \frac{\nu}{q^3} \biggl( \frac{\rho_i}{\bar\rho}\biggr)_c \biggr]^{\gamma_c} \, ,</math>
<math>~C_\mathrm{env}</math>	<math>~=~</math>	<math>~ B_\mathrm{env} [K_e \rho_{ie}^{\gamma_e} ]R^{3\gamma_e} </math>
	<math>~=~</math>	<math>~ B_\mathrm{env} K_e \biggl[ \frac{3M_\mathrm{tot}(1-\nu)}{4\pi (1 - q^3)} \biggl( \frac{\rho_{i}}{\bar\rho}\biggr)_e \biggr]^{\gamma_e} \, .</math>

For a given structure, the three coefficients, <math>~A, C_\mathrm{core}</math>, and <math>~C_\mathrm{env}</math>, should remain constant during a radial perturbation of the configuration.

Later it will also be useful to recognize that, in equilibrium <math>~(R = R_\mathrm{eq})</math>, we will demand that <math>~P_{ie} = P_{ic}</math>. As a result, we can choose to write the total thermal energy either entirely in terms of the exponent, <math>~\gamma_c</math> or the exponent, <math>~\gamma_e</math>. Letting the properties of the core take the lead, we can write,

<math>~S_\mathrm{tot}</math>

<math>~R^3 P_{ic} (B_\mathrm{core} + B_\mathrm{env}) = C_\mathrm{core} \biggl( 1 + \frac{B_\mathrm{env}}{B_\mathrm{core}} \biggr) R_\mathrm{eq}^{3-3\gamma_c} \, .</math>

Letting the properties of the envelope take the lead, we obtain,

<math>~S_\mathrm{tot}</math>

<math>~R^3 P_{ie} (B_\mathrm{core} + B_\mathrm{env}) = C_\mathrm{env} \biggl( 1 + \frac{B_\mathrm{core}}{B_\mathrm{env}} \biggr) R_\mathrm{eq}^{3-3\gamma_e} \, .</math>

Free Energy and Its Derivatives

Now, the free energy can be written as,

<math>~\mathfrak{G}</math>	<math>~=~</math>	<math>~U_\mathrm{tot} + W</math>
	<math>~=~</math>	<math>~\biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] S_\mathrm{core} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] S_\mathrm{env} + W</math>
	<math>~=~</math>	<math>~\biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] C_\mathrm{core} R^{3-3\gamma_c} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] C_\mathrm{env} R^{3-3\gamma_e} - A R^{-1} \, .</math>

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

<math>~\frac{d\mathfrak{G}}{dR}</math>

<math>~ -2 C_\mathrm{core} R^{2-3\gamma_c} -2 C_\mathrm{env} R^{2-3\gamma_e} + A R^{-2} \, .</math>

And the second derivative is,

<math>~\frac{d^2\mathfrak{G}}{dR^2}</math>	<math>~=~</math>	<math>~ -2 (2-3\gamma_c) C_\mathrm{core} R^{1-3\gamma_c} -2 (2-3\gamma_e) C_\mathrm{env} R^{1-3\gamma_e} - 2A R^{-3} \, .</math>
	<math>~=~</math>	<math>~ \frac{2}{R^2} \biggl[(3\gamma_c-2) C_\mathrm{core} R^{3-3\gamma_c} + (3\gamma_e-2) C_\mathrm{env} R^{3-3\gamma_e} - A R^{-1} \biggr]</math>
	<math>~=~</math>	<math>~ \frac{2}{R^2} \biggl[(3\gamma_c-2) S_\mathrm{core} + (3\gamma_e-2) S_\mathrm{env} +W \biggr] \, .</math>

Equilibrium

The radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration(s) is determined by setting the first derivative of the free energy to zero. Hence,

<math>~0 </math>	<math>~=~</math>	<math>~ 2 C_\mathrm{core} R_\mathrm{eq}^{2-3\gamma_c} + 2 C_\mathrm{env} R_\mathrm{eq}^{2-3\gamma_e} - A R_\mathrm{eq}^{-2} </math>
	<math>~=~</math>	<math>~ R_\mathrm{eq}^{-1} \biggl[ 2 C_\mathrm{core} R_\mathrm{eq}^{3-3\gamma_c} + 2 C_\mathrm{env} R_\mathrm{eq}^{3-3\gamma_e} - A R_\mathrm{eq}^{-1} \biggr]</math>
	<math>~=~</math>	<math>~ R_\mathrm{eq}^{-1} \biggl[ 2 S_\mathrm{core} + 2 S_\mathrm{env} +W \biggr]</math>
<math>\Rightarrow ~~~~ 2 S_\mathrm{tot} + W </math>	<math>~=~</math>	<math>~0 \, .</math>

This is the familiar statement of virial equilibrium. From it we can derive the following, very general expression for the equilibrium radius:

<math>~\frac{A}{2R_\mathrm{eq}} = S_\mathrm{tot}</math>	<math>~=~</math>	<math>~R_\mathrm{eq}^3 P_i (B_\mathrm{core} + B_\mathrm{env}) </math>
<math>\Rightarrow ~~~~ \biggl[ \frac{P_i}{GM_\mathrm{tot}^2} \biggr] R^4_\mathrm{eq}</math>	<math>~=~</math>	<math>~\biggl[ \frac{A/(GM_\mathrm{tot}^2)}{2(B_\mathrm{core} + B_\mathrm{env})} \biggr] \, .</math>

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,

<math>~2 S_\mathrm{tot} + W</math>	<math>~=~</math>	<math>~0 \, ,</math>
<math>\Rightarrow ~~~~ S_\mathrm{env} </math>	<math>~=~</math>	<math>~- S_\mathrm{core} - \frac{W}{2} \, ,</math>

we obtain,

<math>~\frac{d^2\mathfrak{G}}{dR^2}\biggr\|_\mathrm{eq}</math>	<math>~=~</math>	<math>~ \frac{2}{R_\mathrm{eq}^2} \biggl[ (3\gamma_c-2) S_\mathrm{core} +W - (3\gamma_e-2)\biggl( S_\mathrm{core} + \frac{W}{2}\biggr) \biggr]_\mathrm{eq} </math>
	<math>~=~</math>	<math>~ \frac{2}{R_\mathrm{eq}^2} \biggl[ 3(\gamma_c-\gamma_e) S_\mathrm{core} + \biggl(2 - \frac{3}{2}\gamma_e\biggr)W \biggr]_\mathrm{eq} </math>
	<math>~=~</math>	<math>~ \frac{6}{R_\mathrm{eq}^2} \biggl[ (\gamma_c-\gamma_e) S_\mathrm{core} + \frac{1}{2}\biggl(\frac{4}{3} - \gamma_e\biggr)W \biggr]_\mathrm{eq} </math>
	<math>~=~</math>	<math>~ \frac{6}{R_\mathrm{eq}^2} \biggl[ -\frac{W}{2}\biggl( \gamma_e - \frac{4}{3}\biggr) - (\gamma_e-\gamma_c) S_\mathrm{core} \biggr]_\mathrm{eq} \, .</math>

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 <math>~4/3</math> and the adiabatic exponent of the core is less than or equal to <math>~4/3</math> — that is, <math>~\gamma_e > 4/3 \ge \gamma_c</math>. Hence, because the gravitational potential energy, <math>~W</math>, is intrinsically negative, the system will be dynamically unstable only if the second term (involving <math>~S_\mathrm{core}</math>) is greater in magnitude than the first term (involving <math>~W</math>).

It is worth noting that, instead of drawing upon <math>~S_\mathrm{core}</math> and <math>~W</math> to define the stability condition, we could have used an appropriate combination of <math>~S_\mathrm{env}</math> and <math>~W</math>, or the <math>~S_\mathrm{core}</math> and <math>~S_\mathrm{env}</math> pair. Also, for example, because the virial equilibrium condition is <math>~S_\mathrm{tot} = -W/2</math>, it is easy to see that the following inequality also equivalently defines stability:

<math>~ S_\mathrm{tot}\biggl( \gamma_e - \frac{4}{3}\biggr) - (\gamma_e-\gamma_c) S_\mathrm{core} </math>

Examples

(0, 0) Bipolytropes

In an accompanying discussion we have derived analytic expressions describing the equilibrium structure and the stability of bipolytropes in which both the core and the envelope have uniform densities, that is, bipolytropes with <math>~(n_c, n_e) = (0, 0)</math>. From this work, we find that integrals over the mass and pressure distributions give:

<math>~ \frac{W}{R_\mathrm{eq}^3 P_i} = - \frac{A}{R_\mathrm{eq}^4 P_i} </math>	<math>~=~</math>	<math>- ~ \frac{3}{5} \biggl[ \frac{GM_\mathrm{tot}^2}{R^4P_i} \biggr] \biggl( \frac{\nu^2}{q} \biggr) f </math>
	<math>~=~</math>	<math>- ~4\pi q^3 \Lambda f \, ,</math>
<math>~\frac{S_\mathrm{core}}{R_\mathrm{eq}^3 P_i} = B_\mathrm{core}</math>	<math>~=~</math>	<math> ~2\pi q^3 (1 + \Lambda) \, ,</math>
<math>~\frac{S_\mathrm{env}}{R_\mathrm{eq}^3 P_i} = B_\mathrm{env}</math>	<math>~=~</math>	<math> 2\pi \biggl[ (1-q^3) + \frac{5}{2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] \, ,</math>

where,

<math>~\Lambda</math>	<math>~\equiv~</math>	<math> \frac{3}{2^2 \cdot 5} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4 P_i} \biggr) \frac{\nu^2}{q^4} \, ,</math>
<math>~f(q,\rho_e/\rho_c)</math>	<math>~\equiv~</math>	<math>1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2} - 1 \biggr) + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \biggl(\frac{1}{q^5} - 1 \biggr) - \frac{5}{2}\biggl(\frac{1}{q^2} - 1 \biggr) \biggr] </math>
	<math>~=~</math>	<math>1 + \frac{5}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q^2) + \frac{1}{2q^5} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5) \, ,</math>
<math>~g^2(q,\rho_e/\rho_c)</math>	<math>~\equiv~</math>	<math>1 + \biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) + \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] </math>
	<math>~\equiv~</math>	<math>1 + \biggl[ 2\biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q) + \frac{1}{q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (1 - 3q^2 + 2q^3 ) \biggr] \, , </math>

With these expressions in hand, we can deduce the equilibrium radius and relativity stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes using the generalized expressions provided above. For example, from the statement of virial equilibrium <math>~(2S_\mathrm{tot} = - W )</math> we obtain,

<math>~q^3 (1 + \Lambda) + (1-q^3) + \frac{5}{2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) </math>

<math>~q^3 \Lambda \biggl[ 1 + \frac{5}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q^2) + \frac{1}{2q^5} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5) \biggr] </math>

<math>\Rightarrow ~~~~ \frac{1}{\Lambda}</math>	<math>~=~</math>	<math>\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (q-q^3) + \frac{1}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5) - \biggl[ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) + \frac{3}{2q^2} \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] </math>
	<math>~=~</math>	<math>\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (q-q^3 + 2 -3q +q^3) + \frac{1}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5 +3 - 15q^2+15q^3 -3q^5) </math>
	<math>~=~</math>	<math>\frac{5}{2}\biggl[ 2\biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q) + \frac{1}{q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (1 - 3q^2 + 2q^3 ) \biggr] </math>
	<math>~=~</math>	<math>\frac{5}{2}(g^2-1) </math>
<math>\Rightarrow ~~~~ \biggl[ \frac{P_i}{GM_\mathrm{tot}^2} \biggr] R_\mathrm{eq}^4</math>	<math>~=~</math>	<math>\biggl( \frac{3}{2^3 \pi } \biggr) \frac{\nu^2}{q^4} (g^2-1) \, . </math>

And the condition for dynamical stability is,

<math>-\frac{W}{2}\biggl( \gamma_e - \frac{4}{3}\biggr) - (\gamma_e-\gamma_c) S_\mathrm{core} </math>	<math>~>~</math>	<math>~0 \, .</math>
<math>\Rightarrow ~~~~ 2\pi q^3 \Lambda \biggl[ \biggl( \gamma_e - \frac{4}{3}\biggr) f - (\gamma_e-\gamma_c) \biggl( 1 + \frac{1}{\Lambda}\biggr) \biggr] </math>	<math>~>~</math>	<math>~0 \, .</math>
<math>~\biggl( \gamma_e - \frac{4}{3} \biggr)f - (\gamma_e - \gamma_c) \biggl[1 + \frac{5}{2}(g^2-1) \biggr]</math>	<math>~>~</math>	<math>~0 \, .</math>

(5, 1) Bipolytropes

In another accompanying discussion we have derived analytic expressions describing the equilibrium structure of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>. Can we perform a similar stability analysis of these configurations? Work in progress!

Related Discussions

Analytic solution with <math>~n_c = 0</math> and <math>~n_e=0</math>.
Analytic solution with <math>~n_c = 5</math> and <math>~n_e=1</math>.

User:Tohline/SSC/BipolytropeGeneralization

Contents