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>~\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,

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>~\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>

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>~\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>~\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,

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>

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>

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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Ramblings

Generic Setup

Dividing the above, 2^nd-order ODE through by the quantity, <math>~[R^2 (P_0/P_c)]</math>, gives,

<math> \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \frac{1}{R}\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggl(\frac{P_c}{P_0}\biggr)\biggr] \frac{dx}{dr_0} - \biggl[\frac{1}{R}\biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{\alpha}{r_0} \biggr] x = - \frac{1}{R^2}\biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl[ \biggl( \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} \biggr) \biggr] x \, , </math>

which matches Prasad's (1949) equation (1), namely,

<math>~x^{' '} + \biggl[\frac{4}{r_0} - \frac{\mu(r_0) }{r_0}\biggr] x^{'} - \biggl[ \frac{\alpha \mu(r_0)}{r_0^2} \biggr] x</math>

<math>~- \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl[ \frac{n^2\rho_c}{\gamma_g P_c} \biggr] x \, ,</math>

where, primes indicate differentiation with respect to <math>~r_0</math>, and,

<math>~\mu(r_0) \equiv \frac{r_0}{R} \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \, .</math>

(Note that Prasad's equation has the awkward units of inverse length-squared.) Regrouping terms in Prasad's governing equation, multiplying through by <math>~R^2</math> (to make the equation dimensionless), and now letting primes denote differentiation with respect to the dimensionless radial coordinate, <math>~\chi_0</math>, we quite generally can write the linear adiabatic wave equation as,

<math>~- \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \sigma^2 x </math>	<math>~=</math>	<math>~\biggl[x^{' '} + \frac{4 x^'}{\chi_0}\biggr] - \frac{\mu(\chi_0)}{\chi_0} \biggl[ x^{'} + \frac{\alpha x}{\chi_0} \biggr]</math>
	<math>~=</math>	<math>~\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) - \frac{\mu(\chi_0)}{\chi_0} \biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \biggr] \, . </math>

Defining,

<math>~A</math>	<math>~\equiv</math>	<math>~\biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_c}{\rho_0}\biggr) \, ,</math>
<math>~B</math>	<math>~\equiv</math>	<math>~\frac{A\mu(\chi_0)}{\chi_0} = \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \, ,</math>

the governing equation becomes,

<math>~\sigma^2 x </math>	<math>~=</math>	<math>~\frac{B}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) -\frac{A}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) </math>
	<math>~=</math>	<math>~B \biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - A \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, . </math>

Notice that, because,

at every radial location throughout the configuration, it must also be true that, for any equilibrium configuration,

<math>~B</math>	<math>~=</math>	<math>~- \biggl(\frac{\rho_0}{\rho_c}\biggr)^{-1} \frac{d(P_0/P_c)}{d\chi_0}</math>
<math>~\Rightarrow ~~~~ \frac{B}{A}</math>	<math>~=</math>	<math>~- \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \, .</math>

The following table shows that this relationship holds for a collection of analytically described equilibrium structures.

Properties of Analytically Defined Equilibrium Structures
Model	<math>~\frac{\rho_0}{\rho_c}</math>	<math>~B\equiv \frac{g_0}{g_\mathrm{SSC}}</math>	<math>~A \equiv \biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr)^{-1}</math>	<math>~\frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)</math>
Uniform-density	<math>~1</math>	<math>~2\chi_0</math>	<math>~1 - \chi_0^2</math>	<math>~-2\chi_0</math>
Linear	<math>~1-\chi_0</math>	<math>~\tfrac{48}{5}(\chi_0 - \tfrac{3}{4}\chi_0^2)</math>	<math>~\tfrac{1}{5} (1-\chi_0) (5 + 10\chi_0 - 9\chi_0^2)</math>	<math>~\tfrac{1}{5}[- 48\chi_0 + 84\chi_0^2 - 36\chi_0^3]</math>
Parabolic	<math>~1-\chi_0^2</math>	<math>~5\chi_0 - 3\chi_0^3</math>	<math>~\tfrac{1}{2} (1-\chi_0^2) (2 - \chi_0^2)</math>	<math>~- 5\chi_0 + 8\chi_0^3 - 3\chi_0^5</math>
<math>~n=1</math> Polytrope	<math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>	<math>~\frac{2}{\pi\chi_0^2}\biggl[ \sin(\pi\chi_0) - \pi\chi_0 \cos(\pi\chi_0) \biggr]</math>	<math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>	<math>~\frac{2\sin(\pi\chi_0)}{(\pi^2\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]</math>

Leaning on this new expression for the ratio, <math>~B/A</math>, let's play with the form of the governing equation.

<math>~- \sigma^2 x </math>	<math>~=</math>	<math>~A \biggl\{ \frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) + \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \biggr\} </math>
	<math>~=</math>	<math>~A\biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl\{ \biggl(\frac{P_0}{P_c}\biggr)\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) + \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \biggr\} </math>

Uniform Density

In the case of a uniform-density configuration, the governing equation is,

<math>~\sigma^2 x </math>

<math>~2\chi_0 \biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - (1-\chi_0^2) \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, , </math>

where,

<math>~\sigma^2 \equiv \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} = \frac{6}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr] \, .</math>

The following individual mode analyses should be compared with the results found in our discussion of Sterne's general solution.

Mode 0

Try an eigenfunction of the form,

in which case,

In order for this to be a solution, we must have,

<math>~\sigma^2 a_0 </math>	<math>~=</math>	<math>~2\chi_0 \biggl[ \frac{\alpha a_0}{\chi_0} \biggr] </math>
<math>~\Rightarrow ~~~~ \frac{6}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr]</math>	<math>~=</math>	<math>~2\alpha = 2\biggl(3 - \frac{4}{\gamma_g}\biggr) </math>
<math>~\Rightarrow ~~~~ \frac{\omega^2}{4\pi G\bar\rho}</math>	<math>~=</math>	<math>~\gamma_g - \frac{4}{3}\, . </math>

Mode 2

Try an eigenfunction of the form,

in which case,

In order for this to be a solution, we must have,

<math>~\sigma^2 \biggl(a_0 + a_2\chi_0^2\biggr) </math>	<math>~=</math>	<math>~2 \biggl[ \alpha (a_0 + a_2\chi_0^2) + \chi_0 (2 a_2\chi_0 )\biggr] - (1-\chi_0^2) \biggl[ \frac{4(2 a_2\chi_0 )}{\chi_0}+ 2a_2 \biggr] </math>
	<math>~=</math>	<math>~2\alpha a_0 + \chi_0^2[2a_2 (2+\alpha)] - 10a_2(1-\chi_0^2) </math>
<math>~\Rightarrow ~~~~ \sigma^2 a_0 - 2\alpha a_0 + 10a_2</math>	<math>~=</math>	<math>~\chi_0^2 [-\sigma^2 + 2 (2+\alpha) + 10 ]a_2 \, . </math>

Given that the coefficients on both sides of this expression must independently be zero, we have:

<math>~\sigma^2 </math>	<math>~=</math>	<math>~2 (2+\alpha) + 10</math>
<math>~\Rightarrow ~~~~ \frac{\omega^2}{4\pi G\bar\rho} </math>	<math>~=</math>	<math>~\frac{\gamma_g}{6}\biggl[14 +2\biggl(3-\frac{4}{\gamma_g} \biggr)\biggr]</math>
	<math>~=</math>	<math>~\frac{\gamma_g}{6}\biggl[20 -\frac{8}{\gamma_g}\biggr] = \frac{1}{3}\biggl(10\gamma_g - 4\biggr) \, ,</math>

and

<math>~\frac{a_2}{a_0}</math>	<math>~=</math>	<math>~\frac{1}{10} \biggl[ 2\alpha - \sigma^2 \biggr] </math>
	<math>~=</math>	<math>~\frac{1}{10} \biggl\{ 2\alpha - [14+2\alpha) ] \biggr\} = - \frac{7}{5} \, .</math>

Parabolic Density Distribution

In the case of a parabolic density distribution, the governing equation is,

<math>~\sigma^2 x </math>

<math>~(5\chi_0 - 3\chi_0^3)\biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - \tfrac{1}{2} (1-\chi_0^2) (2 - \chi_0^2) \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, , </math>

where,

<math>~\sigma^2 \equiv \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} = \frac{15}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr] \, .</math>

First Trial

Try an eigenfunction of the form,

in which case,

In order for this to be a solution, we must have,

<math>~\sigma^2 x </math>

<math>~(5 - 3\chi_0^2)\biggl[ \alpha x + \chi_0 x^'\biggr] - \tfrac{1}{2} (1-\chi_0^2) (2 - \chi_0^2) \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] </math>

Independent Investigation of Parabolic Distribution

In the specific case of a parabolic density distribution, the leading factor on the LHS is,

<math>~\frac{1}{A_\mathrm{parab}} \equiv \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) </math>

<math>~\frac{(1-\chi_0^2)}{(1-\chi_0^2)^2 (1-\tfrac{1}{2}\chi_0^2)} = \frac{1}{(1-\chi_0^2) (1-\tfrac{1}{2}\chi_0^2)} = \frac{2}{2 - 3\chi_0^2 + \chi_0^4} \, ,</math>

and the function appearing on the RHS is,

<math>~\frac{\chi_0^2(1-\chi_0^2)(5-3\chi_0^2)}{(1-\chi_0^2)^2 (1-\tfrac{1}{2}\chi_0^2)} = \frac{\chi_0^2 (5-3\chi_0^2)}{A_\mathrm{parab}} \, .</math>

Multiplying the linear adiabatic wave equation through by <math>~A_\mathrm{parab}</math>, gives,

<math>~- \sigma^2 x </math>

<math>~\frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) - \frac{B_\mathrm{parab} }{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, , </math>

where,

<math> ~B_\mathrm{parab} \equiv \chi_0 (5-3\chi_0^2) \, . </math>

Now we note that,

<math>~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr]</math>	<math>~=</math>	<math>~ \frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl[ \chi_0^4 x^' \biggr] + \chi_0^4 x^' \frac{d}{d\chi_0}\biggl[ \frac{A_\mathrm{parab}}{\chi_0^4} \biggr] </math>
<math>~\Rightarrow ~~~~\frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl[ \chi_0^4 x^' \biggr] </math>	<math>~=</math>	<math>~ \frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \chi_0^4 x^' \frac{d}{d\chi_0}\biggl[ \frac{A_\mathrm{parab}}{\chi_0^4} \biggr] </math>
	<math>~=</math>	<math>~ \frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \frac{\chi_0^4 x^'}{2} \frac{d}{d\chi_0}\biggl[ 1 - \frac{3}{\chi_0^2} + \frac{2}{\chi_0^4}\biggr] </math>
	<math>~=</math>	<math>~ \frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \chi_0^4 x^' \biggl[ \frac{3}{\chi_0^3} - \frac{4}{\chi_0^5}\biggr] </math>
	<math>~=</math>	<math>~ \frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] + \frac{x^'}{\chi_0} (4 - 3\chi_0^2 ) \, . </math>

Similarly we note that,

<math>~\frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr]</math>	<math>~=</math>	<math>~ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \frac{d}{d\chi_0}\biggl[ \chi_0^\alpha x \biggr] + \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \biggr] </math>
<math>~\Rightarrow ~~~~ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \frac{d}{d\chi_0}\biggl[ \chi_0^\alpha x \biggr] </math>	<math>~=</math>	<math>~ \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \biggr] </math>
	<math>~=</math>	<math>~ \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ 5\chi_0^{1-\alpha} -3 \chi_0^{3-\alpha}\biggr] </math>
	<math>~=</math>	<math>~ \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \biggl[ 5(1-\alpha)\chi_0^{-\alpha} -3 (3-\alpha)\chi_0^{2-\alpha}\biggr] </math>
	<math>~=</math>	<math>~ \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - x \biggl[ 5(1-\alpha) -3 (3-\alpha)\chi_0^{2}\biggr] \, . </math>

Hence, the LAWE can be rewritten as,

<math>~- \sigma^2 x </math>

<math>~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] + \frac{x^'}{\chi_0} (4 - 3\chi_0^2 ) - \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] + x \biggl[ 5(1-\alpha) -3 (3-\alpha)\chi_0^{2}\biggr] \, ; </math>

then multiplying through by <math>~\chi_0</math>, and rearranging terms gives,

<math>~- x \biggl\{ [ 5(1-\alpha)+ \sigma^2]\chi_0 -3 (3-\alpha)\chi_0^{3} \biggr\}</math>

<math>~\chi_0 ~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' - B_\mathrm{parab}~x\biggr] + x^'(4 - 3\chi_0^2 ) \, . </math>

Next, we note that,

<math>~ A_\mathrm{parab}~x^' </math>	<math>~=</math>	<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - x \biggl[\frac{d}{d\chi_0}\biggl(A_\mathrm{parab}\biggr) \biggr]</math>
	<math>~=</math>	<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - \frac{x}{2} \biggl[\frac{d}{d\chi_0}\biggl(2 - 3\chi_0^2 + \chi_0^4\biggr) \biggr]</math>
	<math>~=</math>	<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) + x (3\chi_0 - 2\chi_0^3 )</math>
<math>~\Rightarrow ~~~~A_\mathrm{parab}~x^' - B_\mathrm{parab}~x</math>	<math>~=</math>	<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) + \biggl[(3\chi_0 - 2\chi_0^3 ) - ( 5\chi_0 - 3\chi_0^3 )\biggr] x</math>
	<math>~=</math>	<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - (2\chi_0 - \chi_0^3 ) x \, .</math>

So, the LAWE becomes,

<math>~- x \biggl\{ [ 5(1-\alpha)+ \sigma^2]\chi_0 -3 (3-\alpha)\chi_0^{3} \biggr\}</math>	<math>~=</math>	<math>~\chi_0 ~\frac{d}{d\chi_0}\biggl[ \frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - (2\chi_0 - \chi_0^3 ) x\biggr] + x^'(4 - 3\chi_0^2 ) </math>
	<math>~=</math>	<math>~ \chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) - \chi_0 ~\frac{d}{d\chi_0}\biggl[ (2\chi_0 - \chi_0^3 ) x\biggr] + x^'(4 - 3\chi_0^2 ) </math>
	<math>~=</math>	<math>~ \chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) - \biggl\{ \frac{d}{d\chi_0}\biggl[ (2\chi_0^2 - \chi_0^4)x\biggr] - (2\chi_0 - \chi_0^3)x \biggr\} + \biggl\{ \frac{d}{d\chi_0}\biggl[ (4-3\chi_0^2)x\biggr] + 6\chi_0 x \biggr\} </math>
	<math>~=</math>	<math>~ \chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) + \frac{d}{d\chi_0}\biggl[(4-3\chi_0^2)x -(2\chi_0^2 - \chi_0^4)x\biggr] + (2\chi_0 - \chi_0^3)x + 6\chi_0 x </math>
	<math>~=</math>	<math>~ \chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) + \frac{d}{d\chi_0}\biggl[(4-5\chi_0^2 + \chi_0^4)x \biggr] + (8\chi_0 - \chi_0^3)x \, . </math>

Moving the last term on the RHS of this expression to the LHS, and factoring the polynomial coefficients of the terms inside of the first and second derivatives gives,

<math>~x \biggl\{ (5\alpha - 13 - \sigma^2)\chi_0 + (10 - 3\alpha)\chi_0^{3} \biggr\}</math>

<math>~ \frac{\chi_0}{2} \frac{d^2}{d\chi_0^2} \biggl[ (1-\chi_0^2)(2-\chi_0^2)x \biggr] + \frac{d}{d\chi_0}\biggl[(1-\chi_0^2)(4-\chi_0^2)x \biggr] \, . </math>

User:Tohline/SSC/Structure/Other Analytic Models

Contents

Other Analytically Definable, Spherical Equilibrium Models

Linear Density Distribution

Stabililty

Lagrangian Approach

Eulerian Approach

Parabolic Density Distribution

Equilibrium Structure

Stabililty

Ramblings

Generic Setup

Uniform Density

Mode 0

Mode 2

Parabolic Density Distribution

First Trial

Independent Investigation of Parabolic Distribution

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