User:Tohline/SSC/Structure/Other Analytic Models

Other Analytically Definable, Spherical Equilibrium Models

Linear Density Distribution

In an article titled, "Stellar Evolution: A Survey with Analytic Models," R. F. Stein (1966, in Stellar Evolution, Proceedings of an International Conference held at the Goddard Space Flight Center, Greenbelt, MD, U.S.A., edited by R. F. Stein & A. G. W. Cameron, pp. 1-105) defines the "Linear Stellar Model" as a star whose density "varies linearly from the center to the surface," that is (see his equation 3.1),

<math>\rho(r) = \rho_c\biggl( 1 - \frac{r}{R} \biggr) \, ,</math>

where, <math>~\rho_c</math> is the central density and, <math>~R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our general solution strategy for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,

<math>~M_r(r)</math>	<math>~=</math>	<math>~\int_0^r 4\pi r^2 \rho(r) dr</math>
	<math>~=</math>	<math>~\frac{4\pi\rho_c r^3}{3} \biggl[1 - \frac{3}{4} \biggl( \frac{r}{R} \biggr)\biggr] \, ,</math>

in which case we have,

<math>M_\mathrm{tot} \equiv M_r(R) = \frac{\pi\rho_c R^3}{3} \, ,</math>

and we can write,

<math>~g_0(r) \equiv \frac{G M_r(r) }{r^2} </math>

<math>~=</math>

<math>~\frac{4\pi G \rho_c r}{3} \biggl[1 - \frac{3}{4} \biggl( \frac{r}{R} \biggr)\biggr] \, .</math>

Hence, proceeding via what we have labeled as "Technique 1", and enforcing the surface boundary condition, <math>~P(R) = 0</math>, Stein (1966) determines that (see his equation 3.5),

<math>~P(r)</math>	<math>~=</math>	<math>~- \int_0^r g_0(r) \rho(r) dr</math>
	<math>~=</math>	<math>~\frac{\pi G\rho_c^2 R^2}{36} \biggl[5 - 24 \biggl( \frac{r}{R} \biggr)^2 + 28 \biggl( \frac{r}{R} \biggr)^3 - 9 \biggl( \frac{r}{R} \biggr)^4 \biggr] \, ,</math>

where, it can readily be deduced, as well, that the central pressure is,

<math>~P_c = \frac{5\pi}{36} G\rho_c^2 R^2 \, .</math>

Stabililty

Lagrangian Approach

As has been derived in an accompanying discussion, the second-order ODE that defines the relevant Eigenvalue problem is,

<math> \biggl(\frac{P_0}{P_c}\biggr)\frac{d^2x}{d\chi_0^2} + \biggl[\biggl(\frac{P_0}{P_c}\biggr)\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr] x = 0 . </math>

where the dimensionless radius,

<math> \chi_0 \equiv \frac{r_0}{R} \, , </math>

<math> g_\mathrm{SSC} \equiv \frac{P_c}{R\rho_c}</math>           and           <math>\tau_\mathrm{SSC} \equiv \biggl( \frac{R^2\rho_c}{P_c}\biggr)^{1/2} \, . </math>

For Stein's configuration with a linear density distribution,

<math> g_\mathrm{SSC} = \frac{5\pi G\rho_c R}{36}</math>           and           <math>\tau_\mathrm{SSC} \equiv \biggl( \frac{36}{5\pi G \rho_c }\biggr)^{1/2} = \biggl( \frac{12}{5}\cdot \frac{R^3}{GM_\mathrm{tot} }\biggr)^{1/2} \, . </math>

Hence,

<math>~\frac{g_0}{g_\mathrm{SSC}} </math>

<math>~=</math>

<math>~\frac{48}{5}\cdot \chi_0\biggl(1 - \frac{3}{4} \chi_0 \biggr) \, .</math>

and the governing adiabatic wave equation takes the form,

<math>~0</math>	<math>~=</math>	<math>~ \frac{1}{5}\biggl(5 - 24 \chi_0^2 + 28 \chi_0^3 - 9 \chi_0^4 \biggr)\frac{d^2x}{d\chi_0^2} + \biggl[\frac{1}{5}\biggl(5 - 24 \chi_0^2 + 28 \chi_0^3 - 9 \chi_0^4 \biggr)\frac{4}{\chi_0} - \biggl(1-\chi_0\biggr) \frac{48}{5}\cdot \chi_0\biggl(1 - \frac{3}{4} \chi_0 \biggr)\biggr] \frac{dx}{d\chi_0} </math>
		<math>~ + \biggl(1-\chi_0\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\frac{12}{5} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) + (4 - 3\gamma_\mathrm{g})\frac{48}{5}\cdot \chi_0\biggl(1 - \frac{3}{4} \chi_0 \biggr)\frac{1}{\chi_0} \biggr] x </math>
<math>~0</math>	<math>~=</math>	<math>~ \biggl(5 - 24 \chi_0^2 + 28 \chi_0^3 - 9 \chi_0^4 \biggr)\frac{d^2x}{d\chi_0^2} + \frac{4}{\chi_0}\biggl[\biggl(5 - 24 \chi_0^2 + 28 \chi_0^3 - 9 \chi_0^4 \biggr)- 12\biggl(1-\chi_0\biggr) \chi_0^2\biggl(1 - \frac{3}{4} \chi_0 \biggr)\biggr] \frac{dx}{d\chi_0} </math>
		<math>~ + 12\biggl(1-\chi_0\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) + 4(4 - 3\gamma_\mathrm{g})\biggl(1 - \frac{3}{4} \chi_0 \biggr)\biggr] x </math>
<math>~0</math>	<math>~=</math>	<math>~ \biggl(5 - 24 \chi_0^2 + 28 \chi_0^3 - 9 \chi_0^4 \biggr)\frac{d^2x}{d\chi_0^2} + \frac{4}{\chi_0}\biggl[\biggl(5 - 24 \chi_0^2 + 28 \chi_0^3 - 9 \chi_0^4 \biggr)- \biggl(12\chi_0^2 - 21\chi_0^3 + 9\chi_0^4 \biggr)\biggr] \frac{dx}{d\chi_0} </math>
		<math>~ + \biggl(1-\chi_0\biggr) \biggl[\biggl(\frac{12}{\gamma_\mathrm{g}} \biggr)\biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) + \biggl(\frac{12}{\gamma_\mathrm{g}} \biggr)(4 - 3\gamma_\mathrm{g})\biggl(4 - 3 \chi_0 \biggr)\biggr] x </math>
<math>~0</math>	<math>~=</math>	<math>~ \biggl(5 - 24 \chi_0^2 + 28 \chi_0^3 - 9 \chi_0^4 \biggr)\frac{d^2x}{d\chi_0^2} + \frac{4}{\chi_0}\biggl[5 - 36 \chi_0^2 + 7 \chi_0^3 \biggr] \frac{dx}{d\chi_0} </math>
		<math>~ + \biggl(1-\chi_0\biggr) \biggl[\Omega^2 + \biggl(\frac{12}{\gamma_\mathrm{g}} \biggr)(4 - 3\gamma_\mathrm{g})\biggl(4 - 3 \chi_0 \biggr)\biggr] x \, , </math>

where, following R. Stothers & J. A. Frogel (1967, ApJ, 148, 305),

<math>~\Omega^2 \equiv \frac{12}{\gamma_\mathrm{g}} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \, .</math>

Eulerian Approach

In his book titled, The Pulsation Theory of Variable Stars, S. Rosseland (1969) defines the relevant eigenvalue problem for adiabatic, radial pulsations in terms of the governing relation (see his equation 2.23 on p. 20, with the adiabatic condition being enforced by setting the right-hand-side equal to zero),

<math>~\frac{\partial}{\partial r} \biggl( \gamma P_0 \nabla\cdot \vec{\xi}\biggr) + \biggl( \omega^2 + \frac{4g_0}{r}\biggr) \rho_0 \xi</math>

<math>~=</math>

<math>~0 \, ,</math>

where,

<math>~\vec\xi = \mathbf{\hat{e}}_r \xi(r) \, .</math>

Realizing that, for a spherically symmetric system,

<math>\nabla\cdot \vec\xi = \frac{1}{r^2}\frac{\partial}{\partial r}\biggl(r^2 \xi\biggr) = \frac{\partial \xi}{\partial r} + \frac{2\xi}{r} \, ,</math>

and remembering that,

<math>~\frac{\partial P_0}{\partial r} = -g_0 \rho_0 \, ,</math>

we can rewrite this relation in the more familiar form of a 2^nd-order ODE, namely,

<math>~0</math>	<math>~=</math>	<math>~ \frac{1}{\gamma} \biggl( \omega^2 + \frac{4g_0}{r}\biggr) \rho_0 \xi + \nabla\cdot \vec{\xi} ~\biggl(\frac{\partial P_0}{\partial r}\biggr) + P_0 \frac{\partial}{\partial r} \biggl( \nabla\cdot \vec{\xi} \biggr) </math>
	<math>~=</math>	<math>~ \frac{\xi \rho_c}{\gamma} \biggl( \omega^2 + \frac{4g_0}{r}\biggr) \biggl(\frac{\rho_0}{\rho_c}\biggr) - \rho_0 g_0 \biggl[\frac{\partial \xi}{\partial r} + \frac{2\xi}{r}\biggr] + P_0 \frac{\partial}{\partial r} \biggl[\frac{\partial \xi}{\partial r} + \frac{2\xi}{r}\biggr] </math>
	<math>~=</math>	<math>~ \frac{\xi \rho_c}{\gamma} \biggl( \omega^2 + \frac{4g_0}{r}\biggr) \biggl(\frac{\rho_0}{\rho_c}\biggr) + \biggl[ - \rho_0 g_0 + \frac{2P_0}{r}\biggr] \frac{\partial \xi}{\partial r} + P_0 \frac{\partial^2 \xi}{\partial r^2} + \xi \biggl[ - \biggl(\frac{2\rho_0 g_0 }{r}\biggr) - \frac{2P_0}{r^2}\biggr] </math>
	<math>~=</math>	<math>~P_0 \frac{\partial^2 \xi}{\partial r^2} + \biggl[ \frac{2P_0}{r}- \rho_0 g_0 \biggr] \frac{\partial \xi}{\partial r} + \biggl[ \biggl( \frac{\omega^2\rho_c}{\gamma} + \frac{4\rho_c g_0}{\gamma r}\biggr) \biggl(\frac{\rho_0}{\rho_c}\biggr) - \biggl(\frac{2\rho_c g_0 }{r}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2P_0}{r^2} \biggr] \xi \, . </math>

Multiplying through by <math>~(R^2/P_c)</math> and, again, letting <math>~\chi_0 \equiv r/R</math>, we have,

<math>~0</math>

<math>~=</math>

<math>~\biggl(\frac{P_0}{P_c}\biggr) \frac{\partial^2 \xi}{\partial \chi_0^2} + \biggl[ \frac{2}{\chi_0}\biggl(\frac{P_0}{P_c}\biggr) - \frac{g_0 }{g_\mathrm{SSC}}\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggr] \frac{\partial \xi}{\partial \chi_0} + \biggl\{ \biggl[\frac{\omega^2\tau_\mathrm{SSC}^2}{\gamma} + \frac{2}{\chi_0 } \biggl(\frac{2}{\gamma } - 1\biggr)\frac{g_0}{g_\mathrm{SSC}}\biggr] \biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2}{\chi_0^2} \biggl(\frac{P_0}{P_c}\biggr) \biggr\} \xi \, . </math>

Now, plugging in the functional expressions that specifically apply to the linear model gives,

<math>~0</math>	<math>~=</math>	<math>~\frac{1}{5}\biggl[5 - 24 \chi_0^2 + 28 \chi_0^3 - 9 \chi_0^4 \biggr]\frac{\partial^2 \xi}{\partial \chi_0^2} </math>
		<math>~ + \biggl\{ \frac{2}{5\chi_0}\biggl[5 - 24 \chi_0^2+ 28 \chi_0^3 - 9 \chi_0^4 \biggr] - \frac{48}{5}\chi_0\biggl(1 - \frac{3}{4} \chi_0 \biggr)\biggl(1-\chi_0\biggr) \biggr\} \frac{\partial \xi}{\partial \chi_0} </math>
		<math>~ + \biggl\{ \biggl[ \frac{\Omega^2}{5} + \frac{96}{5} \biggl(\frac{2}{\gamma } - 1\biggr)\biggl(1 - \frac{3}{4} \chi_0 \biggr)\biggr] \biggl(1-\chi_0\biggr)- \frac{2}{5\chi_0^2} \biggl[5 - 24 \chi_0^2+ 28 \chi_0^3 - 9 \chi_0^4 \biggr] \biggr\} \xi \, , </math>

and, multiplying through by <math>~(5\chi_0^2)</math> gives,

<math>~0</math>	<math>~=</math>	<math>~\biggl(5\chi_0^2 - 24 \chi_0^4+ 28 \chi_0^5 - 9 \chi_0^6 \biggr) \frac{\partial^2 \xi}{\partial \chi_0^2} </math>
		<math>~ + \biggl[ 2\chi_0\biggl(5 - 24 \chi_0^2+ 28 \chi_0^3 - 9 \chi_0^4 \biggr) - 12\chi_0^3 \biggl(4-7\chi_0 +3\chi_0^2\biggr) \biggr] \frac{\partial \xi}{\partial \chi_0} </math>
		<math>~ + \biggl[ \Omega^2 \chi_0^2 \biggl(1-\chi_0\biggr) + 24 \chi_0^2\biggl(\frac{2}{\gamma } - 1\biggr)\biggl(4-7 \chi_0 +3\chi_0^2\biggr) - 2\biggl(5 - 24 \chi_0^2+ 28 \chi_0^3 - 9 \chi_0^4 \biggr) \biggr] \xi </math>
	<math>~=</math>	<math>~\biggl(5\chi_0^2 - 24 \chi_0^4+ 28 \chi_0^5 - 9 \chi_0^6 \biggr) \frac{\partial^2 \xi}{\partial \chi_0^2} + \biggl(10\chi_0 - 96 \chi_0^3+ 140 \chi_0^4 - 54 \chi_0^5 \biggr) \frac{\partial \xi}{\partial \chi_0} </math>
		<math>~ + \biggl[ \Omega^2 \biggl(\chi_0^2-\chi_0^3\biggr) + \biggl(\frac{2}{\gamma } - 1\biggr)\biggl(96 \chi_0^2 - 168 \chi_0^3 +72\chi_0^4\biggr) + \biggl(-10 + 48 \chi_0^2 - 56 \chi_0^3 + 18 \chi_0^4 \biggr) \biggr] \xi </math>
	<math>~=</math>	<math>~\biggl(5\chi_0^2 - 24 \chi_0^4+ 28 \chi_0^5 - 9 \chi_0^6 \biggr) \frac{\partial^2 \xi}{\partial \chi_0^2} + \biggl(10\chi_0 - 96 \chi_0^3+ 140 \chi_0^4 - 54 \chi_0^5 \biggr) \frac{\partial \xi}{\partial \chi_0} </math>
		<math>~ + \biggl[ -10 + \chi_0^2 \biggl( \Omega^2 + \frac{192}{\gamma} - 48 \biggr) - \chi_0^3 \biggl(\Omega^2 + \frac{336}{\gamma} - 112 \biggr) + \chi_0^4\biggl(\frac{144}{\gamma} - 54 \biggr) \biggr] \xi \, , </math>

where, following R. Stothers & J. A. Frogel (1967, ApJ, 148, 305),

<math>~\Omega^2 \equiv \frac{12}{\gamma_\mathrm{g}} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \, .</math>

Parabolic Density Distribution

Equilibrium Structure

In an article titled, "Radial Oscillations of a Stellar Model," C. Prasad (1949, MNRAS, 109, 103) investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,

<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>

where, <math>~\rho_c</math> is the central density and, <math>~R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our general solution strategy for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,

<math>~M_r(r)</math>	<math>~=</math>	<math>~\int_0^r 4\pi r^2 \rho(r) dr</math>
	<math>~=</math>	<math>~\frac{4\pi\rho_c r^3}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r}{R} \biggr)^2 \biggr] \, ,</math>

in which case we can write,

<math>~g_0(r) \equiv \frac{G M_r(r) }{r^2} </math>

<math>~=</math>

<math>~\frac{4\pi G \rho_c r}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r}{R} \biggr)^2\biggr] \, .</math>

Hence, proceeding via what we have labeled as "Technique 1", and enforcing the surface boundary condition, <math>~P(R) = 0</math>, Prasad (1949) determines that,

<math>~P(r)</math>	<math>~=</math>	<math>~- \int_0^r g_0(r) \rho(r) dr</math>
	<math>~=</math>	<math>~- \frac{4\pi G \rho_c^2 R^2}{15} \int_0^r \biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr]\biggl[5 - 3\biggl( \frac{r}{R} \biggr)^2\biggr] \biggl( \frac{r}{R} \biggr) \frac{dr}{R}</math>
	<math>~=</math>	<math>~- \frac{4\pi G \rho_c^2 R^2}{15} \int_0^r \biggl[ 5\biggl(\frac{r}{R} \biggr) - 8\biggl(\frac{r}{R} \biggr)^3 + 3\biggl(\frac{r}{R} \biggr)^5\biggr] \frac{dr}{R}</math>
	<math>~=</math>	<math>~\frac{2\pi G\rho_c^2 R^2}{15} \biggl[2 - 5 \biggl( \frac{r}{R} \biggr)^2 + 4 \biggl( \frac{r}{R} \biggr)^4 - \biggl( \frac{r}{R} \biggr)^6 \biggr] </math>
	<math>~=</math>	<math>~\frac{4\pi G\rho_c^2 R^2}{15} \biggl[1-\biggl(\frac{r}{R}\biggr)^2\biggr]^2 \biggl[1-\frac{1}{2}\biggl(\frac{r}{R}\biggr)^2\biggr] \, ,</math>

where, it can readily be deduced, as well, that the central pressure is,

<math>~P_c = \frac{4\pi}{15} G\rho_c^2 R^2 \, .</math>

Stabililty

As has been derived in an accompanying discussion, the second-order ODE that defines the relevant Eigenvalue problem is,

<math> \biggl(\frac{P_0}{P_c}\biggr)\frac{d^2x}{d\chi_0^2} + \biggl[\biggl(\frac{P_0}{P_c}\biggr)\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr] x = 0 \, , </math>

where the dimensionless radius,

<math> \chi_0 \equiv \frac{r_0}{R} \, , </math>

<math> g_\mathrm{SSC} \equiv \frac{P_c}{R\rho_c}</math>           and           <math>\tau_\mathrm{SSC} \equiv \biggl( \frac{R^2\rho_c}{P_c}\biggr)^{1/2} \, . </math>

For Prasad's configuration with a parabolic density distribution,

<math> g_\mathrm{SSC} = \frac{4\pi G\rho_c R}{15}</math>           and           <math>\tau_\mathrm{SSC} \equiv \biggl( \frac{15}{4\pi G \rho_c }\biggr)^{1/2} = \biggl( \frac{2R^3}{GM_\mathrm{tot} }\biggr)^{1/2} = \biggl( \frac{3}{2\pi G\bar\rho}\biggr)^{1/2}\, . </math>

Hence,

<math>~\frac{g_0}{g_\mathrm{SSC}} </math>

<math>~=</math>

<math>~(5 - 3 \chi_0^2)\chi_0 \, ,</math>

and the governing adiabatic wave equation takes the form,

<math> (1-\chi_0^2) \biggl( 1 - \frac{1}{2}\chi_0^2 \biggr)\frac{d^2x}{d\chi_0^2} + \frac{1}{\chi_0}\biggl[4 (1-\chi_0^2) \biggl( 1 - \frac{1}{2}\chi_0^2 \biggr) - (5 - 3 \chi_0^2)\chi_0^2\biggr] \frac{dx}{d\chi_0} + \biggl[\frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_\mathrm{g}} -\alpha (5 - 3 \chi_0^2)\biggr] x = 0 \, , </math>

where,

<math>~\alpha \equiv 3 - \frac{4}{\gamma_\mathrm{g}} \, .</math>

In keeping with Prasad's presentation — see, specifically, his equations (2) & (3) — this wave equation can also be written as,

<math> (1-\chi_0^2) \biggl( 1 - \frac{1}{2}\chi_0^2 \biggr)\frac{d^2x}{d\chi_0^2} + \frac{1}{\chi_0}\biggl[4 - 11\chi_0^2 + 5\chi_0^4\biggr] \frac{dx}{d\chi_0} + \biggl[\mathfrak{J}+3\alpha \chi_0^2 \biggr] x = 0 \, , </math>

where,

<math>~\mathfrak{J} \equiv \frac{3\omega^2}{2\pi G \gamma_\mathrm{g} \bar\rho} - 5\alpha \, .</math>

For what it's worth, we have also deduced that this expression can be written as,

<math> (1-\chi_0^2) \biggl( 1 - \frac{1}{2}\chi_0^2 \biggr)\chi_0^{-4} \frac{d}{d\chi_0} \biggl[\chi_0^4 \frac{dx}{d\chi_0} \biggr] -(5-3\chi_0^2)\chi_0^{1+\alpha} \frac{d}{d\chi_0} \biggl[ \chi_0^{-\alpha} x \biggr] + \biggl(\frac{\tau_\mathrm{SSC}^2~ \omega^2}{\gamma_\mathrm{g}}\biggr) x = 0 \, , </math>

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation