Revision as of 22:55, 6 August 2015

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>

Ramblings

The material originally contained in this "Ramblings" subsection has been moved to generate a separate chapter that stands on its own.

Difference between revisions of "User:Tohline/SSC/Structure/Other Analytic Models"

Revision as of 22:55, 6 August 2015

Contents

Other Analytically Definable, Spherical Equilibrium Models

Linear Density Distribution

Stabililty

Lagrangian Approach

Eulerian Approach

Parabolic Density Distribution

Equilibrium Structure

Stabililty

Ramblings

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@@ Line 773: / Line 773: @@
 </math>
 </div>
-{{LSU_WorkInProgress}}
 ==Ramblings==
-===Generic Setup===
+The material originally contained in this "Ramblings" subsection has been moved to generate [[User:Tohline/SSC/Structure/Other_Analytic_Ramblings|a separate chapter that stands on its own.]]
-Dividing the [[User:Tohline/SSC/Structure/Other_Analytic_Models#Stabililty_2|above, 2<sup>nd</sup>-order ODE]] through by the quantity, <math>~[R^2 (P_0/P_c)]</math>, gives,
-<div align="center">
-<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><br />
-</div>
-which matches [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P Prasad's (1949)] equation (1), namely,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<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>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-where, primes indicate differentiation with respect to <math>~r_0</math>, and,
-<div align="center">
-<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>
-</div>
-(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,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \sigma^2  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl[x^{' '} + \frac{4 x^'}{\chi_0}\biggr] - \frac{\mu(\chi_0)}{\chi_0} \biggl[ x^{'}   + \frac{\alpha x}{\chi_0} \biggr]</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-Defining,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~A</math>
-  </td>
-  <td align="center">
-<math>~\equiv</math>
-  </td>
-  <td align="left">
-<math>~\biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_c}{\rho_0}\biggr) \, ,</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~B</math>
-  </td>
-  <td align="center">
-<math>~\equiv</math>
-  </td>
-  <td align="left">
-<math>~\frac{A\mu(\chi_0)}{\chi_0}  = \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \, ,</math>
-  </td>
-</tr>
-</table>
-</div>
-the governing equation becomes,
-<div align="center" id="LAWE">
-<table border="1" cellpadding="5"><tr><td align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\sigma^2  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~B \biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - A \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, .
-</math>
-  </td>
-</tr>
-</table>
-</td></tr></table>
-</div>
-Notice that, because,
-<div align="center">
-<math>~g_0 = - \frac{1}{\rho_0} ~\frac{dP_0}{dr_0} \, ,</math>
-</div>
-at every radial location throughout the configuration, it must also be true that, for any equilibrium configuration,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~B</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~- \biggl(\frac{\rho_0}{\rho_c}\biggr)^{-1} \frac{d(P_0/P_c)}{d\chi_0}</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ \frac{B}{A}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~- \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-<span id="Examples">The following table shows that this relationship holds for a collection of analytically described equilibrium structures.</span>
-<table border="1" cellpadding="5" align="center" width="90%">
-<tr>
-  <th align="center" colspan="4"><font size="+1">Table 1a:  Properties of Analytically Defined Equilibrium Structures</font></th>
-</tr>
-<tr>
-  <td align="center" width="10%">Model</td>
-  <td align="center"><math>~\frac{\rho_0}{\rho_c}</math>
-  <td align="center"><math>~\frac{P_0}{P_c}</math>
-  <td align="center"><math>~\frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)</math>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
-  <td align="center"><math>~1</math>
-  <td align="center"><math>~1 - \chi_0^2</math>
-  <td align="center"><math>~-2\chi_0</math>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution|Linear]]</td>
-  <td align="center"><math>~1-\chi_0</math>
-  <td align="center"><math>~\tfrac{1}{5} (5 -24 \chi_0^2 + 28\chi_0^3 - 9\chi_0^4)</math>
-  <td align="center"><math>~\tfrac{1}{5}[- 48\chi_0 + 84\chi_0^2 - 36\chi_0^3]</math>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Parabolic_Density_Distribution|Parabolic]]</td>
-  <td align="center"><math>~1-\chi_0^2</math>
-  <td align="center"><math>~\tfrac{1}{2} (2  - 5\chi_0^2 + 4\chi_0^4 - \chi_0^6)</math>
-  <td align="center"><math>~- 5\chi_0 + 8\chi_0^3 - 3\chi_0^5</math>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/Stability/Polytropes#n_.3D_1_Polytrope|<math>~n=1</math> Polytrope]]</td>
-  <td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
-  <td align="center"><math>~\biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}\biggr]^2</math>
-  <td align="center"><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>
-</tr>
-</table>
-<div align="center" id="Table1b">
-<table border="1" cellpadding="5" align="center" width="90%">
-<tr>
-  <th align="center" colspan="5"><font size="+1">Table 1b:  Properties of Analytically Defined Equilibrium Structures</font></th>
-</tr>
-<tr>
-  <td align="center" width="10%">Model</td>
-  <td align="center"><math>~\frac{\rho_0}{\rho_c}</math>
-  <td align="center"><math>~B\equiv \frac{g_0}{g_\mathrm{SSC}}</math>
-  <td align="center"><math>~A \equiv \biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr)^{-1}</math>
-  <td align="center"><math>~\biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr)^{-2}</math>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
-  <td align="center"><math>~1</math>
-  <td align="center"><math>~2\chi_0</math>
-  <td align="center"><math>~1 - \chi_0^2</math>
-  <td align="center"><math>~1 - \chi_0^2</math>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution|Linear]]</td>
-  <td align="center"><math>~1-\chi_0</math>
-  <td align="center"><math>~\tfrac{48}{5}(\chi_0 - \tfrac{3}{4}\chi_0^2)</math>
-  <td align="center"><math>~\tfrac{1}{5} (1-\chi_0) (5 + 10\chi_0 - 9\chi_0^2)</math>
-  <td align="center"><math>~(1 + 2\chi_0 - \tfrac{9}{5}\chi_0^2)</math>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Parabolic_Density_Distribution|Parabolic]]</td>
-  <td align="center"><math>~1-\chi_0^2</math>
-  <td align="center"><math>~5\chi_0 - 3\chi_0^3</math>
-  <td align="center"><math>~\tfrac{1}{2} (1-\chi_0^2) (2  - \chi_0^2)</math>
-  <td align="center"><math>~(1  - \tfrac{1}{2} \chi_0^2)</math>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/Stability/Polytropes#n_.3D_1_Polytrope|<math>~n=1</math> Polytrope]]</td>
-  <td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
-  <td align="center"><math>~\frac{2}{\pi\chi_0^2}\biggl[ \sin(\pi\chi_0) - \pi\chi_0 \cos(\pi\chi_0) \biggr]</math>
-  <td align="center"><math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
-  <td align="center"><math>~1</math>
-</tr>
-</table>
-</div>
-<span id="Generic">Leaning on this new expression for the ratio, <math>~B/A</math>, let's play with the form of the governing equation.</span>
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- \sigma^2  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-===Polytropic Configurations===
-Let's compare this presentation of the LAWE to the form of the LAWE that has been derived [[User:Tohline/SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation|specifically for polytropic equilibrium configurations]], namely,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} +
-\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta} -
-\biggl(3-\frac{4}{\gamma_g}\biggr)  \cdot \frac{(n+1)V(x)}{\xi^2} \biggr]  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>0 \, ,</math>
-  </td>
-</tr>
-</table>
-</div>
-where,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~V(\xi)</math>
-  </td>
-  <td align="center">
-<math>~\equiv</math>
-  </td>
-  <td align="left">
-<math>~- \frac{\xi}{(\theta/\theta_c)} \frac{d (\theta/\theta_c)}{d\xi} = \frac{g_0}{a_n}\biggl(\frac{a_n^2\rho_0}{P_0}\biggr)\frac{\xi}{(n+1)} \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-[Note that <math>~\theta_c = 1</math> and, therefore for all practical purposes, it can be dropped.  This notation was introduced in our [[User:Tohline/SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation|separate discussion of the polytropic LAWE]] in order to make it clear how our derivations have overlapped earlier published work.]  Regrouping terms, we have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~-\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta}\biggr]x</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} -
-\biggl[ \frac{\alpha (n+1)V(x)}{\xi^2} \biggr]  x</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl[\frac{d^2x}{d\xi^2} + \biggl(\frac{4}{\xi}\biggr)\frac{dx}{d\xi} \biggr] -\biggl[\frac{(n+1)V(\xi)}{\xi} \biggr]
-\biggl[\frac{dx}{d\xi} + \frac{\alpha x}{\xi} \biggr] </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{\xi^4}\frac{d}{d\xi}\biggl(\xi^4 \frac{dx}{d\xi}\biggr) -\biggl[\frac{(n+1)V(\xi)}{\xi} \biggr]
-\biggl[\frac{1}{\xi^\alpha}\frac{d}{d\xi} \biggl(\xi^\alpha x\biggr)\biggr] </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{\xi^4}\frac{d}{d\xi}\biggl(\xi^4 \frac{dx}{d\xi}\biggr) +\biggl[(n+1)\frac{d\ln(\theta/\theta_c)}{d\xi} \biggr]
-\biggl[\frac{1}{\xi^\alpha}\frac{d}{d\xi} \biggl(\xi^\alpha x\biggr)\biggr] \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-Next we note that, written in terms of the traditional polytropic radial coordinate, <math>~\xi</math>, the fractional radius,
-<div align="center">
-<math>~\chi_0 \equiv \frac{r_0}{R} = \frac{\xi}{\xi_1} = \frac{a_n \xi}{R} \, .</math>
-</div>
-Hence, multiplying the polytropic LAWE through by the quantity, <math>~(R/a_n)^2</math>, gives,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{1}{\chi_0^4}\frac{d}{d\chi_0}\biggl(\chi_0^4 \frac{dx}{d\chi_0}\biggr) +\biggl[(n+1)\frac{d\ln(\theta/\theta_c)}{d\chi_0} \biggr]
-\biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0} \biggl(\chi_0^\alpha x\biggr)\biggr] </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~-\biggl[\omega^2 \biggl(\frac{R^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta}\biggr]x</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~-\biggl(\frac{\theta_c}{\theta}\biggr)\sigma^2 x \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-Finally, noting that, for polytropic configurations,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{\theta}{\theta_c}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl( \frac{P_0}{P_c} \biggr)\biggl( \frac{\rho_0}{\rho_c} \biggr)^{-1}
-= \biggl( \frac{P_0}{P_c} \biggr)^{1/(n+1)} \, ,
-</math>
-  </td>
-</tr>
-</table>
-</div>
-we can rewrite the polytropic LAWE in the form,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{1}{\chi_0^4}\frac{d}{d\chi_0}\biggl(\chi_0^4 \frac{dx}{d\chi_0}\biggr) +\biggl[\frac{d\ln(P_0/P_c)}{d\chi_0} \biggr]
-\biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0} \biggl(\chi_0^\alpha x\biggr)\biggr] </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~-\biggl( \frac{P_0}{P_c} \biggr)^{-1}\biggl( \frac{\rho_0}{\rho_c} \biggr)\sigma^2 x \, ,</math>
-  </td>
-</tr>
-</table>
-</div>
-which precisely matches the general expression for the LAWE presented at the end of our [[User:Tohline/SSC/Structure/Other_Analytic_Models#Generic_Setup|generic setup, directly above]].
-This seems to be a particularly insightful way to write the LAWE, as the only structural functions that appear explicitly are <math>~P_0(\chi_0)</math> and <math>~\rho_0(\chi_0)</math>.  It appears as though the eigenfunctions that describe ''adiabatic'' radial pulsations do not explicitly depend ''a priori'' on the radial dependence of the equilibrium gravitational acceleration.
-===Examine Structural Pressure-Density Relation===
-====Derivation====
-One striking property exhibited by the [[User:Tohline/SSC/Structure/Other_Analytic_Models#Examples|example configurations tabulated above]] is the ''structural'' relationship between the chosen function, <math>~\rho_0(\chi_0)</math>, and the corresponding radial pressure distribution, <math>~P_0(\chi_0)</math>, that is  [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|dictated by]],
-<div align="center">
-<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />
-<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} = - \frac{GM_r}{r^2} </math> ,
-</div>
-As has been detailed in the last column of [[User:Tohline/SSC/Structure/Other_Analytic_Models#Table1b|Table 1b]], in all four cases, the ratio, <math>~(P_0/P_c)(\rho_0/\rho_c)^{-2}</math>, is an analytically prescribed polynomial expression.  That is, the pressure is "evenly divisible" by the square of the density.  Let's examine how broadly reliable this behavior is.  [Note that, for simplicity in typing, hereafter throughout this subsection we will drop the subscript zero and, rather than <math>~\chi_0</math>, we will use <math>~z \equiv r/R</math> to denote the dimensionless radial coordinate.]
-Assume a mass-distribution given by the general quadratic function,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{\rho}{\rho_c}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~1 - a z - b z^2 \, ,</math>
-  </td>
-</tr>
-</table>
-</div>
-where both coefficients, <math>~a</math> and <math>~b</math>, are positive.
-<div align="center" id="Surface">
-<table border="1" width="75%" align="center" cellpadding="5">
-<tr>
-<td align="center">ASIDE:  Surface Location</td>
-</tr>
-<tr><td align="left">
-The surface of the configuration will be defined by the radial location, <math>~z_s</math>, at which the density first goes to zero.  If <math>~b = 0</math>, then the surface will be located at <math>~z_s = a^{-1}</math>; and if <math>~a = 0</math>, it will be located at <math>~z_s = b^{-1/2}</math>.  More generally, however, the roots of the quadratic equation that results from setting <math>~\rho/\rho_c</math> to zero are,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~z_\pm</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~- \frac{a}{2b}\biggl[1 \mp \biggl(1+\frac{4b}{a^2}\biggr)^{1/2}  \biggr] \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-Because only the <math>~z_+</math> solution provides positive roots, we conclude that, when both <math>~a</math> and <math>~b</math> are nonzero, the radial coordinate of the surface is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~z_s = z_+</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{a}{2b}\biggl[\biggl(1+\frac{4b}{a^2}\biggr)^{1/2}  - 1\biggr] \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-We acknowledge, as well, that the density profile can now be written in terms of these roots; specifically,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{\rho}{\rho_c}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~b(z_+ - z)(z - z_-) \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-In the discussion, below, it may be advantageous to adopt the following notation:
-<div align="center">
-<math>~\ell^2 \equiv \frac{4b}{a^2} ~~~~~\Rightarrow ~~~~~ \ell = \frac{2b^{1/2}}{a} \, ,</math>
-</div>
-in which case,
-<div align="center">
-<math>~az_s = \frac{2}{\ell^2}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr]</math>
-&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
-<math>~b^{1/2}z_s = \frac{1}{\ell}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr] \, .</math>
-</div>
-</td></tr>
-</table>
-</div>
-This specified density profile implies a mass distribution,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~M_r</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~4\pi R^3 \rho_c \int_0^z (1 - a z - b z^2)z^2 dz</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~4\pi R^3 \rho_c \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-The hydrostatic balance condition therefore implies,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{1}{R\rho_c} \biggl( \frac{\rho}{\rho_c} \biggr)^{-1} \frac{dP}{dz}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~-\frac{G}{R^2 z^2} \biggl[ 4\pi R^3 \rho_c \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) \biggr]</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~\biggl[ \frac{1}{4\pi G R^2 \rho_c^2 } \biggr] \frac{dP}{dz}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~-\biggl( \frac{z}{3} - \frac{a z^2}{4} - \frac{b z^3}{5} \biggr)\biggl( \frac{\rho}{\rho_c} \biggr) </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-- \biggl( \frac{z}{3} - \frac{a z^2}{4} - \frac{b z^3}{5} \biggr)
-+ a \biggl( \frac{z^2}{3} - \frac{a z^3}{4} - \frac{b z^4}{5} \biggr)
-+ b \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{a}{4} + \frac{a}{3}  \biggr) + z^3 \biggl( \frac{b}{5} - \frac{a^2}{4} + \frac{b}{3}  \biggr)
-+ z^4 \biggl( - \frac{ab}{5} - \frac{ab}{4}  \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{7a}{12} \biggr) + z^3 \biggl( \frac{8b}{15} - \frac{a^2}{4}\biggr)
-- z^4 \biggl( \frac{9ab}{20} \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>\Rightarrow ~~~~\biggl( \frac{P}{P_n} \biggr)</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\int \biggl[ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{7a}{12} \biggr) + z^3 \biggl( \frac{8b}{15} - \frac{a^2}{4}\biggr)
-- z^4 \biggl( \frac{9ab}{20} \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr) \biggr] dz
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl[ -\biggl(\frac{1}{6}\biggr)z^2 + z^3 \biggl( \frac{7a}{36} \biggr) + z^4 \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr)
-- z^5 \biggl( \frac{9ab}{100} \biggr) - z^6 \biggl(\frac{b^2}{30}   \biggr) \biggr] + C \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-where, <math>~C</math>, is an integration constant, and,
-<div align="center">
-<math>~P_n \equiv 4\pi G \rho_c^2 R^2 \, .</math>
-</div>
-The integration constant &#8212; which also proves to be the normalized central pressure &#8212; is determined by ensuring that the pressure goes to zero at [[User:Tohline/SSC/Structure/Other_Analytic_Models#Surface|the surface of the configuration]], <math>~z_s</math>.  That is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~C = \frac{P_c}{P_n}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ -
-\biggl[ -\biggl(\frac{1}{6}\biggr)z_s^2 + z_s^3 \biggl( \frac{7a}{36} \biggr) + z_s^4 \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr)
-- z_s^5 \biggl( \frac{9ab}{100} \biggr) - z_s^6 \biggl(\frac{b^2}{30}   \biggr) \biggr] \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Hence, the pressure profile is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_n} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{7a}{36} \biggr)(z_s^3 - z^3) - \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr) (z_s^4 - z^4)
-+ \biggl( \frac{9ab}{100} \biggr)(z_s^5 - z^5) + \biggl(\frac{b^2}{30}   \biggr)(z_s^6 - z^6) \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Let's check this general expression against the specific cases described above.
-====Example1====
-First, let's set <math>~b=0</math>, but leave <math>~a</math> general.  As described in the [[User:Tohline/SSC/Structure/Other_Analytic_Models#Surface|above ASIDE]], this means that <math>~z_s=a^{-1}</math>.  So the pressure distribution is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_n} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{7a}{36} \biggr)(z_s^3 - z^3) + \biggl( \frac{a^2}{16}\biggr) (z_s^4 - z^4)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl(\frac{1}{6}\biggr)(a^{-2} - z^2) - \biggl( \frac{7a}{36} \biggr)(a^{-3} - z^3) + \biggl( \frac{a^2}{16}\biggr) (a^{-4} - z^4)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~a^{-2} \biggl\{
-\frac{1}{6}[1 - (az)^2] - \frac{7}{36} [1 - (az)^3] + \frac{1}{16} [1 - (az)^4]
-\biggr\} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-And from the expression for the integration constant, we have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{P_c}{P_n}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl[
-\biggl(\frac{1}{6}\biggr)a^{-2}  - a^{-3} \biggl( \frac{7a}{36} \biggr) + a^{-4} \biggl( \frac{a^2}{16}\biggr) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~a^{-2} \biggl[\frac{1}{6} - \frac{7}{36}  + \frac{1}{16} \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{5}{2^4\cdot 3^2 a^2} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Hence, dividing one expression by the other, we obtain,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_c} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{5}\biggl\{ 24[1 - (az)^2] - 28 [1 - (az)^3] + 9 [1 - (az)^4] \biggr\} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Let's check to see if this "general linear" pressure distribution is evenly divisible by the square of the density distribution which, in this case, is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{\rho}{\rho_c}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~1 - a z  \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-Strategically rewriting the expression for the pressure distribution gives,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_c} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{5}\biggl\{ 24[1 - (az)][1 + (az)] - 28 [1 - (az)][1 + (az) + (az)^2] + 9 [1 - (az)][1 + (az)][1 + (az)^2] \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{ 24[1 + (az)] - 28 [1 + (az) + (az)^2] + 9 [1 + (az)][1 + (az)^2] \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{5+ 24(az) - 28 [(az) + (az)^2] + 9 [(az) + (az)^2 + (az)^3] \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{5+ 5(az) - 19 (az)^2 + 9 (az)^3 \biggr\} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-And, as luck would have it, the expression inside the curly braces can be "divided evenly" by the quantity, <math>~[1-(az)]</math>, one more time.  Specifically, the expression becomes,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_c} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr)^2 [5+ 10(az) - 9 (az)^2 ] \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-====Example2====
-First, let's set <math>~a=0</math>, but leave <math>~b</math> general.  As described in the [[User:Tohline/SSC/Structure/Other_Analytic_Models#Surface|above ASIDE]], this means that <math>~z_s=b^{-1/2}</math>.  So the pressure distribution is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_n} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{2b}{15} \biggr) (z_s^4 - z^4) + \biggl(\frac{b^2}{30}   \biggr)(z_s^6 - z^6)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl(\frac{1}{6}\biggr)(b^{-1} - z^2) - \biggl( \frac{2b}{15} \biggr) (b^{-2} - z^4) + \biggl(\frac{b^2}{30}   \biggr)(b^{-3} - z^6)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{b}\biggl\{
-\biggl(\frac{1}{6}\biggr)[1 - bz^2] - \biggl( \frac{2}{15} \biggr) [1 - (bz^2)^2] + \biggl(\frac{1}{30}   \biggr)[1 - (bz^2)^3] \biggr\} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-And from the expression for the integration constant, we have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{P_c}{P_n}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl[ \biggl(\frac{1}{6}\biggr)z_s^2 - z_s^4 \biggl( \frac{2b}{15}\biggr) + z_s^6 \biggl(\frac{b^2}{30}   \biggr) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ \frac{1}{b}
-\biggl[ \frac{1}{6} -  \frac{2}{15} + \frac{1}{30} \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ \frac{1}{3 \cdot 5b} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Hence, dividing one expression by the other, we obtain,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_c} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{2}\biggl\{
-[1 - bz^2] - 4  [1 - (bz^2)^2] +  [1 - (bz^2)^3] \biggr\} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Let's check to see if this "general parabolic" pressure distribution is evenly divisible by the square of the density distribution which, in this case, is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{\rho}{\rho_c}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~1 - b z^2  \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-Strategically rewriting the expression for the pressure distribution gives,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_c} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{2}\biggl\{ 5[1 - (bz^2)] - 4[1 - (bz^2)][1 + (bz^2)] + [1 - (bz^2)][1 + (bz^2) + (bz^2)^2] \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{2} \biggl( \frac{\rho}{\rho_c}\biggr) \biggl\{ 5 - 4 [1 + (bz^2)] + [1 + (bz^2) + (bz^2)^2] \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{2} \biggl( \frac{\rho}{\rho_c}\biggr) \biggl[ 2 - 3 (bz^2) + (bz^2)^2 \biggr]
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Again, as luck would have it, the expression inside the square brackets can be "divided evenly" by the quantity, <math>~[1-(bz^2)]</math>, one more time.  Specifically, the expression becomes,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_c} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{2} \biggl(\frac{\rho}{\rho_c}\biggr)^2 [2-(bz^2) ] \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-====Example3====
-In the most general quadratic case, we should rewrite the pressure distribution as,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\frac{P}{P_n} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ \frac{1}{2^4\cdot 3^2\cdot 5^2}\biggl\{
-^3\cdot 3\cdot 5^2 (z_s^2 - z^2) - 2^2\cdot 5^2 \cdot 7 a(z_s^3 - z^3) - [2^5\cdot 3\cdot 5 b - 3^2\cdot 5^2 a^2] (z_s^4 - z^4)
-+ 2^2\cdot 3^4 ab (z_s^5 - z^5) + 2^3\cdot 3\cdot 5b^2 (z_s^6 - z^6)
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ \frac{z_s^2}{2^4\cdot 3^2\cdot 5^2}\biggl\{
-(1 - \zeta^2) - 700 (a z_s) (1 - \zeta^3) - [480 (b z_s^2) - 225 (a z_s)^2] (1 - \zeta^4)
-+ 324 (az_s)(b z_s^2) (1 - \zeta^5) + 120(b z_s^2)^2 (1 - \zeta^6)
-\biggr\} \, ,
-</math>
-  </td>
-</tr>
-</table>
-</div>
-where, <math>~\zeta \equiv z/z_s</math>.  Similarly, let's rewrite the integration constant as,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{P_c}{P_n}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ \frac{z_s^2}{2^4\cdot 3^2\cdot 5^2}\biggl\{
-- 700 (a z_s)  - [480 (b z_s^2) - 225 (a z_s)^2]
-+ 324 (az_s)(b z_s^2)  + 120(b z_s^2)^2
-\biggr\} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-So the pressure can be written as,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>\biggl[ \frac{2^4\cdot 3^2\cdot 5^2}{z_s^2} \biggr] \biggl[ \frac{P}{P_n} \biggr]</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
- <td align="left">
-<math>~ \biggl[ \frac{2^4\cdot 3^2\cdot 5^2}{z_s^2} \biggr] \biggl[ \frac{P_c}{P_n}\biggr]
-- 600 \zeta^2 + 700 (a z_s) \zeta^3 + [480 (b z_s^2) - 225 (a z_s)^2] \zeta^4
-- 324 (az_s)(b z_s^2) \zeta^5 - 120(b z_s^2)^2 \zeta^6  \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-The question that remains to be answered is:  Is this expression for the pressure distribution "evenly divisible" by the square (or even the ''first'' power) of the normalized density distribution which, [[User:Tohline/SSC/Structure/Other_Analytic_Models#Derivation|as defined above]] for the general quadratic case, is,
-<div align="center">
-<math>\frac{\rho}{\rho_c} = 1 - az - bz^2 = 1 - (az_s)\zeta - (bz_s^2)\zeta^2 \, .</math>
-</div>
-In attempting to answer this question, it may prove advantageous to refer back to the [[User:Tohline/SSC/Structure/Other_Analytic_Models#Surface|above ASIDE discussion]] of the roots of this quadratic function and, in particular, that,
-<div align="center">
-<math>~az_s = \frac{2}{\ell^2}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr]</math>
-&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
-<math>~b^{1/2}z_s = \frac{1}{\ell}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr] \, ,</math>
-</div>
-where, <math>~\ell^2 \equiv 4b/a^2</math>.
-<!--  HIDE TABLE 1C
-<table border="1" cellpadding="5" align="center" width="90%">
-<tr>
-  <th align="center" colspan="5"><font size="+1">Table 1c:  Properties of Analytically Defined Equilibrium Structures</font></th>
-</tr>
-<tr>
-  <td align="center" colspan="5">
-<math>~\frac{\rho}{\rho_c} = 1 - a\chi_0 - b\chi_0^2</math>
-  </td>
-</tr>
-<tr>
-  <td align="center" width="10%">Model</td>
-  <td align="center"><math>~a</math></td>
-  <td align="center"><math>~b</math></td>
-  <td align="center"><math>~C = \frac{P_c}{P_n}</math></td>
-  <td align="center"><math>~\frac{P_0}{P_n}</math></td>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
-  <td align="center"><math>~0</math></td>
-  <td align="center"><math>~0</math></td>
-  <td align="center"><math>~\frac{1}{6}</math>
-  </td>
-  <td align="center">
-<math>C - \frac{1}{6}\chi_0^2 </math>
-  </td>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution|Linear]]</td>
-  <td align="center"><math>~1</math></td>
-  <td align="center"><math>~0</math></td>
-  <td align="center"><math>~\frac{1}{6}\biggl[1 - \biggl( \frac{7}{6} \biggr) + \biggl( \frac{15}{40} \biggr) \biggr] = \frac{5}{2^4\cdot 3^2}</math></td>
-  <td align="center">
-<math>~
-C + \biggl[ -\biggl(\frac{1}{6}\biggr)\chi_0^2 + \chi_0^3 \biggl( \frac{7}{36} \biggr) - \chi_0^4 \biggl( \frac{1}{16}\biggr) \biggr]
-= C + \frac{1}{2^4\cdot 3^2}\biggl[ -24\chi_0^2 +  28\chi_0^3  - 9 \chi_0^4\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Parabolic_Density_Distribution|Parabolic]]</td>
-  <td align="center"><math>~0</math></td>
-  <td align="center"><math>~1</math></td>
-  <td align="center"><math>~\frac{1}{6}\biggl[1 - \frac{32}{40}  + \frac{1}{5}  \biggr] = \frac{1}{15}</math></td>
-  <td align="center">
-<math>~
-C + \biggl[ -\biggl(\frac{1}{6}\biggr)\chi_0^2 + \chi_0^4 \biggl( \frac{8}{60} \biggr) - \chi_0^6 \biggl(\frac{1}{30}   \biggr) \biggr]
-= C + \frac{1}{30}\biggl[ -5\chi_0^2 + 4 \chi_0^4  - \chi_0^6 \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="center">General Linear</td>
-  <td align="center"><math>~a</math></td>
-  <td align="center"><math>~0</math></td>
-  <td align="center"><math>~\frac{1}{2^4\cdot 3^2}\biggl[
-- 28a  + 9a^2  \biggr]
-</math>
-  </td>
-  <td align="center">
-<math>~
-C + \frac{1}{2^4\cdot 3^2}\biggl[ -24 \chi_0^2 + 28a \chi_0^3 -9a^2 \chi_0^4  \biggr]
-</math>
-  </td>
-</tr>
-</table>
-END HIDING of TABLE 1C -->
-===Uniform Density===
-In the case of a uniform-density configuration, the governing equation is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\sigma^2  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2\chi_0 \biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - (1-\chi_0^2) \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, ,
-</math>
-  </td>
-</tr>
-</table>
-</div>
-where,
-<div align="center">
-<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>
-</div>
-The following individual mode analyses should be compared with the results found in [[User:Tohline/SSC/UniformDensity#Sterne.27s_General_Solution|our discussion of Sterne's general solution]].
-====Mode 0====
-Try an eigenfunction of the form,
-<div align="center">
-<math>x = a_0\, ,</math>
-</div>
-in which case,
-<div align="center">
-<math>x^' = x^{' '} = 0 \, .</math>
-</div>
-In order for this to be a solution, we must have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\sigma^2  a_0 </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2\chi_0 \biggl[ \frac{\alpha a_0}{\chi_0} \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ \frac{6}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr]</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2\alpha = 2\biggl(3 - \frac{4}{\gamma_g}\biggr)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ \frac{\omega^2}{4\pi G\bar\rho}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\gamma_g - \frac{4}{3}\, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-====Mode 2====
-Try an eigenfunction of the form,
-<div align="center">
-<math>x = a_0 + a_2\chi_0^2 \, ,</math>
-</div>
-in which case,
-<div align="center">
-<math>~x^' = 2 a_2\chi_0 </math>&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; <math>~x^{' '} = 2 a_2 \, . </math>
-</div>
-In order for this to be a solution, we must have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\sigma^2  \biggl(a_0 + a_2\chi_0^2\biggr) </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2\alpha a_0 + \chi_0^2[2a_2 (2+\alpha)] - 10a_2(1-\chi_0^2)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ \sigma^2 a_0 - 2\alpha a_0 + 10a_2</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\chi_0^2 [-\sigma^2  + 2 (2+\alpha) + 10   ]a_2 \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Given that the coefficients on both sides of this expression must independently be zero, we have:
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\sigma^2 </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2 (2+\alpha) + 10</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ \frac{\omega^2}{4\pi G\bar\rho} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{\gamma_g}{6}\biggl[14 +2\biggl(3-\frac{4}{\gamma_g} \biggr)\biggr]</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{\gamma_g}{6}\biggl[20 -\frac{8}{\gamma_g}\biggr] = \frac{1}{3}\biggl(10\gamma_g - 4\biggr) \, ,</math>
-  </td>
-</tr>
-</table>
-</div>
-and
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{a_2}{a_0}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{10} \biggl[ 2\alpha - \sigma^2 \biggr] </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{10} \biggl\{ 2\alpha - [14+2\alpha) ] \biggr\} = - \frac{7}{5} \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-===Parabolic Density Distribution===
-In the case of a [[User:Tohline/SSC/Structure/Other_Analytic_Models#Parabolic_Density_Distribution|parabolic density distribution]], the governing equation is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\sigma^2  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-where,
-<div align="center">
-<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>
-</div>
-====First Trial====
-Try an eigenfunction of the form,
-<div align="center">
-<math>x = (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) \, ,</math>
-</div>
-in which case,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~x^' </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ (2-\chi_0^2)^{-1} (2 b\chi_0 + 4c\chi_0^3) + 2\chi_0 (2-\chi_0^2)^{-2} (a + b\chi_0^2 + c\chi_0^4) </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ (2-\chi_0^2)^{-2}[(2-\chi_0^2) (2 b\chi_0 + 4c\chi_0^3) + 2\chi_0 (a + b\chi_0^2 + c\chi_0^4) ]</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ 2\chi_0 (2-\chi_0^2)^{-2}[(2-\chi_0^2) (b + 2c\chi_0^2) +  (a + b\chi_0^2 + c\chi_0^4) ]</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ 2\chi_0 (2-\chi_0^2)^{-2}[  (2b+4c\chi_0^2-b\chi_0^2 -2c\chi_0^4) +  (a + b\chi_0^2 + c\chi_0^4) ]</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ 2\chi_0 (2-\chi_0^2)^{-2}[  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ] \, ;</math>
-  </td>
-</tr>
-</table>
-</div>
-and,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~x^{' '} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~(2-\chi_0^2)^{-1} (2 b + 12 c\chi_0^2) + 2\chi_0 (2-\chi_0^2)^{-2} (2 b\chi_0 + 4c\chi_0^3)
-+ 2(2-\chi_0^2)^{-2} (a + b\chi_0^2 + c\chi_0^4) </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~+ 2\chi_0 (a + b\chi_0^2 + c\chi_0^4) [-2(2-\chi_0^2)^{-3}(-2\chi_0)] + 2\chi_0 (2-\chi_0^2)^{-2} (2 b\chi_0 + 4c\chi_0^3) </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~(2-\chi_0^2)^{-3} [
-(4-4\chi_0^2 + \chi_0^4) (b + 6 c\chi_0^2) + 4\chi_0^2 (2-\chi_0^2) (b + 2c\chi_0^2)
-+ 2(2-\chi_0^2) (a + b\chi_0^2 + c\chi_0^4) </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~+ 8\chi_0^2 (a + b\chi_0^2 + c\chi_0^4)  + 4\chi_0^2 (2-\chi_0^2) (b + 2c\chi_0^2) ]</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2(2-\chi_0^2)^{-3} \{
-( 4b+24c\chi_0^2 - 4b\chi_0^2-24c\chi_0^4 + b\chi_0^4 + 6c\chi_0^6 ) +(8b \chi_0^2+16c\chi_0^4 -4b\chi_0^4 - 8c\chi_0^6)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-+(2a+2b\chi_0^2 + 2c\chi_0^4 -a\chi_0^2 - b\chi_0^4 - c\chi_0^6)
-+  (4a\chi_0^2 + 4b\chi_0^4 + 4c\chi_0^6)  \}</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2(2-\chi_0^2)^{-3} \{
-(4b+2a) + \chi_0^2(24c-4b+8b+2b-a+4a) + \chi_0^4(-24c+b+16c-4b+2c-b+4b) + \chi_0^6(6c-8c-c+4c)
-\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2 (2-\chi_0^2)^{-3} \{(2a+4b) + \chi_0^2(3a + 6b + 24c) + \chi_0^4(-6c) + \chi_0^6(c)\}
-</math>
-  </td>
-</tr>
-</table>
-</div>
-In order for this to be a solution, we must have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\sigma^2  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~(5 - 3\chi_0^2)\biggl[ \alpha x + \chi_0 x^'\biggr] - (1-\chi_0^2) (2  - \chi_0^2) \biggl[ \frac{2x^'}{\chi_0}+ \frac{x^{' '}}{2} \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow~~~~\sigma^2  (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(5 - 3\chi_0^2)\biggl[ \alpha  (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) + 2\chi_0^2 (2-\chi_0^2)^{-2}[  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ]\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-- (1-\chi_0^2) \biggl[ 4 (2-\chi_0^2)^{-1}[  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ]+ (2-\chi_0^2)^{-2} \{(2a+4b) + \chi_0^2(3a + 6b + 24c) -6c \chi_0^4 + c\chi_0^6\} \biggr] \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Multiplying through by <math>~(2-\chi_0^2)^2</math> gives,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\sigma^2  (2-\chi_0^2) (a + b\chi_0^2 + c\chi_0^4) </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(5 - 3\chi_0^2)\biggl\{ \alpha  (2-\chi_0^2) (a + b\chi_0^2 + c\chi_0^4) + 2\chi_0^2 [  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ]\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-- (1-\chi_0^2) \biggl\{ 4 (2-\chi_0^2)[  (a + 2b)+ 4c\chi_0^2 -c \chi_0^4  ]+ [(2a+4b) + \chi_0^2(3a + 6b + 24c) -6c \chi_0^4 + c\chi_0^6] \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow~~~~\sigma^2  [2a + \chi_0^2(2b-a) + \chi_0^4(2c -b) - c\chi_0^6]</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(5 - 3\chi_0^2)\biggl\{ \alpha  [2a + \chi_0^2(2b-a) + \chi_0^4(2c -b) - c\chi_0^6]\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-+ (5 - 3\chi_0^2)\biggl\{   (2a + 4b)\chi_0^2 + 8c\chi_0^4 - 2c \chi_0^6  \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-- (1-\chi_0^2) \biggl\{ [  8(a + 2b)+ 32c\chi_0^2 - 8c \chi_0^4  ] - [  4\chi_0^2(a + 2b)+ 16c\chi_0^4 -4c \chi_0^6  ]\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-- (1-\chi_0^2) \biggl\{ (2a+4b) + \chi_0^2(3a + 6b + 24c) -6c \chi_0^4 + c\chi_0^6 \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(5 - 3\chi_0^2)\biggl\{ 2a\alpha + \chi_0^2 [(2b-a)\alpha +  (2a + 4b) ] + \chi_0^4[ (2c -b)\alpha + 8c] - (2+\alpha)c\chi_0^6 \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-- (1-\chi_0^2) \biggl\{  (10a + 20b)+ \chi_0^2[56c - a - 2b] +\chi_0^4 [-30c  ]+ 5c \chi_0^6  \biggr\}
-</math>
-  </td>
-</tr>
-</table>
-</div>
-So, the coefficients of each even power of <math>~\chi_0^n</math> are:
-<div align="center" id="FirstTable">
-<table border="1" cellpadding="8" align="center">
-<tr>
-  <td align="right"><math>~\chi_0^0</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~2a\sigma^2 - 10a\alpha +10a +20b =~a(2\sigma^2 - 10\alpha+10) + 20b</math>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^2</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~(2b-a)\sigma^2 -5[(2b-a)\alpha +  (2a + 4b) ] +6a\alpha+[56c - a - 2b] - (10a + 20b)</math><p>
-<math>=~a[-\sigma^2 -5(-\alpha + 2) +6\alpha-1-10] + b[2\sigma^2-10\alpha-20-2-20 ] + c[ 56]</math></p><p>
-<math>=~a[-\sigma^2 + 11\alpha - 21] + b[2\sigma^2-10\alpha-42 ] + 56 c</math></p>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^4</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~(2c-b)\sigma^2 - 5[ (2c -b)\alpha + 8c] - 30c + 3[(2b-a)\alpha +  (2a + 4b)] - [56c - a - 2b ] </math><p>
-<math>=~a[-3\alpha+7] + b[-\sigma^2+5\alpha+6\alpha+12+2] + c[2\sigma^2-10\alpha-40-30-56]</math></p><p>
-<math>=~a[7-3\alpha] + b[11\alpha-\sigma^2+14] + c[2\sigma^2-10\alpha-126]</math></p>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^6</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~- c\sigma^2 +(10+5\alpha)c +3[ (2c -b)\alpha + 8c] +5c +30c</math><p>
-<math>=~b[-3\alpha] + c[-\sigma^2 + 10 + 5\alpha+6\alpha+24+35]</math></p><p>
-<math>=~b[-3\alpha] + c[11\alpha -\sigma^2 + 69]</math></p>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^8</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~- 3(2+\alpha)c -5c =~c[-11-3\alpha]</math>
-  </td>
-</tr>
-</table>
-</div>
-===Independent Investigation of Parabolic Distribution===
-In the specific case of a parabolic density distribution, the leading factor on the LHS is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{1}{A_\mathrm{parab}} \equiv \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-and the function appearing on the RHS is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\mu(\chi_0)</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-Multiplying the linear adiabatic wave equation through by <math>~A_\mathrm{parab}</math>, gives,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- \sigma^2  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-where,
-<div align="center">
-<math>
-~B_\mathrm{parab} \equiv \chi_0 (5-3\chi_0^2) \, .
-</math>
-</div>
-Now we note that,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr]</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~\frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl[ \chi_0^4 x^' \biggr]
-</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] + \frac{x^'}{\chi_0} (4 -  3\chi_0^2 ) \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Similarly we note that,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr]</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~
-\frac{B_\mathrm{parab}}{\chi_0^\alpha} \frac{d}{d\chi_0}\biggl[ \chi_0^\alpha x \biggr]
-</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-Hence, the LAWE can be rewritten as,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- \sigma^2  x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-then multiplying through by <math>~\chi_0</math>, and rearranging terms gives,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- x \biggl\{ [ 5(1-\alpha)+ \sigma^2]\chi_0 -3 (3-\alpha)\chi_0^{3} \biggr\}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-Next, we note that,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~
-A_\mathrm{parab}~x^'
-</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) + x (3\chi_0 - 2\chi_0^3 )</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~A_\mathrm{parab}~x^' - B_\mathrm{parab}~x</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - (2\chi_0 - \chi_0^3 ) x \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-So, the LAWE becomes,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- x \biggl\{ [ 5(1-\alpha)+ \sigma^2]\chi_0 -3 (3-\alpha)\chi_0^{3} \biggr\}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-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,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~x \biggl\{ (5\alpha - 13 - \sigma^2)\chi_0 + (10 - 3\alpha)\chi_0^{3} \biggr\}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<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>
-  </td>
-</tr>
-</table>
-</div>
-====First Trial (Same as Above)====
-Try an eigenfunction of the form,
-<div align="center">
-<math>x = (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) \, ,</math>
-</div>
-in which case,
-<!--  LHS -->
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-LHS
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\chi_0 (2-\chi_0^2)^{-2}\biggl\{ (a + b\chi_0^2 + c\chi_0^4) (2-\chi_0^2) [ (5\alpha - 13 - \sigma^2) + (10 - 3\alpha)\chi_0^{2}] \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\chi_0 (2-\chi_0^2)^{-2} (a + b\chi_0^2 + c\chi_0^4) \biggl\{
-(10\alpha - 26 - 2\sigma^2) + (\sigma^2 -11\alpha + 33  )\chi_0^2 + (3\alpha -10)\chi_0^{4}
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\chi_0 (2-\chi_0^2)^{-2}\biggl\{
-a(10\alpha - 26 - 2\sigma^2) + \chi_0^2[a(\sigma^2 -11\alpha + 33  ) + b(10\alpha - 26 - 2\sigma^2)]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-+\chi_0^{4}[ a (3\alpha -10) + b(\sigma^2 -11\alpha + 33  ) + c(10\alpha - 26 - 2\sigma^2)]
-+ \chi_0^{6}[ b (3\alpha -10) + c(\sigma^2 -11\alpha + 33  )] + c(3\alpha -10)\chi_0^{8}
-\biggr\}
-</math>
-  </td>
-</tr>
-</table>
-</div>
-<!--  RHS (1st part) -->
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-RHS (1<sup>st</sup> term)
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{\chi_0}{2} \frac{d^2}{d\chi_0^2} \biggl[ (1-\chi_0^2)(a + b\chi_0^2 + c\chi_0^4) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{\chi_0}{2} \frac{d^2}{d\chi_0^2} \biggl[ a + (b-a)\chi_0^2 + (c-b)\chi_0^4 - c\chi_0^6\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{\chi_0}{2} \frac{d}{d\chi_0} \biggl[ 2(b-a)\chi_0 + 4(c-b)\chi_0^3 - 6c\chi_0^5\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{\chi_0}{2} [ 2(b-a) + 12(c-b)\chi_0^2 - 30c\chi_0^4]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(b-a)\chi_0 + 6(c-b)\chi_0^3 - 15c\chi_0^5
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\chi_0 (2-\chi_0^2)^{-2}\biggl\{ (4-4\chi_0^2 + \chi_0^4)
-[(b-a) + 6(c-b)\chi_0^2 - 15c\chi_0^4 ] \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\chi_0 (2-\chi_0^2)^{-2}\biggl\{
-[(4b-4a) + (24c- 24b)\chi_0^2 - 60c\chi_0^4 ]
-+ [(-4b + 4a)\chi_0^2 + (-24c + 24b)\chi_0^4 + 60c\chi_0^6 ]
-+ [(b-a)\chi_0^4 + 6(c-b)\chi_0^6 - 15c\chi_0^8 ]
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\chi_0 (2-\chi_0^2)^{-2}\biggl\{
-(4b-4a) + [(24c- 24b) + (-4b + 4a)]\chi_0^2 +[  - 60c  + (-24c + 24b) + (b-a)]\chi_0^4
-+ [60c  + 6(c-b)]\chi_0^6 - 15c\chi_0^8
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\chi_0 (2-\chi_0^2)^{-2}\biggl\{
-(4b-4a) + [ 24c- 28b + 4a]\chi_0^2 +[  - 84c  + 25b  -a ]\chi_0^4
-+ [66c  -6b ]\chi_0^6 - 15c\chi_0^8
-\biggr\}
-</math>
-  </td>
-</tr>
-</table>
-</div>
-<!--  RHS (2nd part) -->
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-RHS (2<sup>nd</sup> term)
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{d}{d\chi_0}\biggl[(1-\chi_0^2)(4-\chi_0^2)(2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(2-\chi_0^2)^{-1} \frac{d}{d\chi_0}\biggl[(4-5\chi_0^2+\chi_0^4)(a + b\chi_0^2 + c\chi_0^4) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~+
-(4-5\chi_0^2+\chi_0^4)(a + b\chi_0^2 + c\chi_0^4) \frac{d}{d\chi_0}\biggl[(2-\chi_0^2)^{-1} \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(2-\chi_0^2)^{-1} \frac{d}{d\chi_0}\biggl[4a + \chi_0^2(4b-5a) + \chi_0^4(4c-5b+a) + \chi_0^6(b-5c) + c\chi_0^8 \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~+
-\biggl[4a + \chi_0^2(4b-5a) + \chi_0^4(4c-5b+a) + \chi_0^6(b-5c) + c\chi_0^8 \biggr]  \frac{d}{d\chi_0}\biggl[(2-\chi_0^2)^{-1} \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(2-\chi_0^2)^{-2}\biggl\{(2-\chi_0^2) \biggl[\chi_0 (8b-10a) + \chi_0^3 (16c-20b+4a) + \chi_0^5 (6b-30c) + 8c\chi_0^7 \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~+2\chi_0
-\biggl[4a + \chi_0^2(4b-5a) + \chi_0^4(4c-5b+a) + \chi_0^6(b-5c) + c\chi_0^8 \biggr]
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\chi_0 (2-\chi_0^2)^{-2}\biggl\{\biggl[(16b-20a) + \chi_0^2 (32c-40b+8a) + \chi_0^4 (12b-60c) + 16c\chi_0^6 \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-- \biggl[\chi_0^2 (8b-10a) + \chi_0^4 (16c-20b+4a) + \chi_0^6 (6b-30c) + 8c\chi_0^8 \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~+
-\biggl[8a + \chi_0^2 (8b-10a) + \chi_0^4 (8c-10b+2a) + \chi_0^6 (2b-10c) + 2c\chi_0^8 \biggr]
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\chi_0 (2-\chi_0^2)^{-2}\biggl\{[(16b-20a) + 8a] + \chi_0^2 [(32c-40b+8a) + (10a-8b) + (8b-10a)]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-+ \chi_0^4 [(12b-60c)+ (20b-4a-16c) + (8c-10b+2a)] + \chi_0^6[16c + (30c-6b) + (2b-10c) ] - 6c\chi_0^8
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\chi_0 (2-\chi_0^2)^{-2}\biggl\{[16b-12a] + \chi_0^2 [(32c-40b+8a) ] + \chi_0^4 [-2a + 22b -68c] + \chi_0^6[36c -4b ] - 6c\chi_0^8
-\biggr\}
-</math>
-  </td>
-</tr>
-</table>
-</div>
-So, the coefficients of each even power of <math>~\chi_0^n</math> are:
-<div align="center" id="SecondTable">
-<table border="1" cellpadding="8" align="center">
-<tr>
-  <td align="right"><math>~\chi_0^0</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~a(10\alpha - 26 - 2\sigma^2) - (4b-4a) - [16b-12a] ~=a(10\alpha - 10 - 2\sigma^2) - 20b</math>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^2</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~[a(\sigma^2 -11\alpha + 33  ) + b(10\alpha - 26 - 2\sigma^2)] - [ 24c- 28b + 4a]- [(32c-40b+8a) ]</math><p>
-<math>= ~[a(\sigma^2 -11\alpha + 21  ) + b(10\alpha + 42 - 2\sigma^2)] -  56c</math></p>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^4</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~[ a (3\alpha -10) + b(\sigma^2 -11\alpha + 33  ) + c(10\alpha - 26 - 2\sigma^2)] - [  - 84c  + 25b  -a ] - [-2a + 22b -68c]</math><p>
-<math>~=~[ a (3\alpha -7) + b(\sigma^2 -11\alpha -14  ) + c(10\alpha + 126 - 2\sigma^2)]</math></p>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^6</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~[ b (3\alpha -10) + c(\sigma^2 -11\alpha + 33  )] - [66c  -6b ]-[36c -4b ]</math><p>
-<math>~=~[ b (3\alpha ) + c(\sigma^2 -11\alpha - 69  )] </math></p>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^8</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~c(3\alpha -10) + 15c + 6c = c[3\alpha+11]</math>
-  </td>
-</tr>
-</table>
-</div>
-The expressions for the coefficients presented in [[User:Tohline/SSC/Structure/Other_Analytic_Models#SecondTable|this table]] exactly match the entire set of expressions [[User:Tohline/SSC/Structure/Other_Analytic_Models#FirstTable|derived earlier]], except for the adopted sign convention &#8212; every term in this second derivation has the opposite sign to the corresponding term in the earlier derivation.
-===Conjecture===
-Returning to the [[User:Tohline/SSC/Structure/Other_Analytic_Models#Generic|generic formulation derived earlier]], we have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\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) \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Now, ''suppose'' that the expression on the RHS is of the form,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\mathrm{RHS}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~u dv + v du \, ,</math>
-  </td>
-</tr>
-</table>
-</div>
-where <math>~u \equiv (P_0/P_c)</math>?  Then the function,
-<div align="center">
-<math>~v \equiv \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, ,</math>
-</div>
-and the eigenvector, <math>~x</math>, must satisfy ''both'' of the relations:
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<font size="+1">Relation I</font>
-  </td>
-  <td align="center">
-<math>~:</math>
-  </td>
-  <td align="left">
-<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) x
-= \frac{d}{d\chi_0}\biggl[ \biggl(\frac{P_0}{P_c}\biggr)\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr)\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<font size="+1">Relation II</font>
-  </td>
-  <td align="center">
-<math>~:</math>
-  </td>
-  <td align="left">
-<math>~\frac{d}{d\chi_0}\biggl[ \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr)\biggr]
-= \frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr)
-</math>
-  </td>
-</tr>
-</table>
-</div>
-The validity (or not) of this conjecture can be tested against both configurations whose  --- ABANDON!
-===Exploration===
-====Compare LAWE to Hydrostatic Balance Condition====
-Returning to the [[User:Tohline/SSC/Structure/Other_Analytic_Models#Generic|generic formulation derived earlier]], we have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\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) \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Dividing this entire expression through by <math>~(P_0/P_c)x</math> gives,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{x \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}{x\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, .
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl( \frac{x^'}{x} \biggr) \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \chi_0^4 x^' \biggr) \biggr]
-+ \frac{d}{d\chi_0}\biggl[ \ln \biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl(\chi_0^\alpha x\biggr)\biggr] \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-Now, let's step aside from the LAWE and look directly at the differential relationship between the mass-density and the pressure, as dictated by combining the [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|two principal governing relations]], the
-<div align="center">
-<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />
-<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} </math> ,<br />
-</div>
-and,
-<div align="center">
-<span id="Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />
-<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr)  = 4\pi G \rho </math> .<br />
-</div>
-In combination, we have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~-4\pi G \rho_0 </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{r_0^2}\frac{d}{dr_0}\biggl[ \frac{r_0^2}{\rho_0} \frac{dP_0}{dr_0}\biggr] </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ -4\pi G \rho_0 \biggl(\frac{R^2\rho_c}{P_c}\biggr)</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{1}{\chi_0^2}\frac{d}{d\chi_0}\biggl[ \frac{\chi_0^2}{(\rho_0/\rho_c)}\cdot \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr] </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ -[4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl( \frac{\rho_0}{\rho_c} \biggr)^{-1}\frac{1}{\chi_0^2}\frac{d}{d\chi_0}\biggl[ \chi_0^2 \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr]
-+ \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{d}{d\chi_0}\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ -[4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)^2\biggl(\frac{P_0}{P_c}\biggr)^{-1}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{1}{\chi_0^2 (P_0/P_c)}\frac{d}{d\chi_0}\biggl[ \chi_0^2 \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr]
-+ \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl(\frac{p^'}{p}\biggr)
-\frac{1}{\chi_0^2 p^'}\frac{d}{d\chi_0}\biggl[ \chi_0^2 p^'\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{d\ln(p)}{d\chi_0}\cdot
-\frac{d}{d\chi_0}\biggl[ \ln(\chi_0^2 p^')\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr] \, ,
-</math>
-  </td>
-</tr>
-</table>
-</div>
-where,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~p^'</math>
-  </td>
-  <td align="center">
-<math>~\equiv</math>
-  </td>
-  <td align="left">
-<math>~\frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-Let's compare the form of this "equilibrium" relation with the form of the LAWE just constructed:
-<div align="center" id="Compare">
-<table border="1" align="center" width="75%">
-<tr><td align="center">
-<table border="0" cellpadding="5">
-<tr>
-  <td align="right">
-<math>~- [4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)^2\biggl(\frac{P_0}{P_c}\biggr)^{-1}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl( \frac{p^'}{p} \biggr) \cdot
-\frac{d}{d\chi_0}\biggl[ \ln(\chi_0^2 p^')\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<font size="+1">''versus''</font>
-  </td>
-  <td align="left">
-&nbsp;
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl( \frac{x^'}{x} \biggr) \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \chi_0^4 x^' \biggr) \biggr]
-+ \frac{d\ln(p)}{d\chi_0}\cdot  \frac{d}{d\chi_0}\biggl[\ln\biggl(\chi_0^\alpha x\biggr)\biggr]
-</math>
-  </td>
-</tr>
-</table>
-</td></tr>
-</table>
-</div>
-I like this layout because it unveils similarities in the way the differential operators interact with the functions that describe the radial profiles of variables &#8212; specifically, the mass-density, the pressure, and the fractional radial displacement, <math>~x</math>, during pulsations.  However, it is not yet obvious how best to translate between the two differential equations in order to aid in solving for the unknown variable, <math>~x(\chi_0)</math>.
-====Dabbling with Equilibrium Condition====
-In the meantime, I've found it instructive to play with the first of these two expressions to see how it might be restructured in order to most directly confirm that it is satisfied by the expressions presented in [[User:Tohline/SSC/Structure/Other_Analytic_Models#Examples|Table 1]].  Adopting the shorthand notation,
-<div align="center">
-<math>~\Gamma \equiv 4\pi G\rho_c \tau_\mathrm{SSC}^2</math>
-&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
-<math>~\varpi \equiv \frac{\rho_0}{\rho_c} \, ,</math>
-</div>
-and multiplying the "equilibrium" relation through by <math>~(-\varpi p)</math>, we have,
-<div align="center">
-<table border="0" cellpadding="5">
-<tr>
-  <td align="right">
-<math>~\Gamma \varpi^3</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-- \varpi p^'\biggl\{
-\frac{1}{\chi_0^2 p^'} \frac{d}{d\chi_0} (\chi_0^2 p^') - \frac{1}{\varpi}\frac{d\varpi}{d\chi_0}
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-p^' \varpi^' - \varpi \frac{dp^'}{d\chi_0}  -\frac{2\varpi p^'}{\chi_0}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{p^' }{\chi_0}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \varpi \frac{dp^'}{d\chi_0}  \, ;
-</math>
-  </td>
-</tr>
-</table>
-</div>
-or,
-<div align="center">
-<table border="0" cellpadding="5">
-<tr>
-  <td align="right">
-<math>~\Gamma \varpi^2</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{p^' }{\chi_0 \varpi}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \frac{dp^'}{d\chi_0}  \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-<font color="red">'''Case 1''' (Parabolic)</font>:
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\varpi = 1 -\chi_0^2</math>
-  </td>
-  <td align="center">
-&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp;
-  </td>
-  <td align="left">
-<math>~\varpi^' = -2\chi_0 </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~p^' = -5\chi_0 + 8\chi_0^3 - 3\chi_0^5 = \chi_0(1-\chi_0^2)(-5+3\chi_0^2)</math>
-  </td>
-  <td align="center">
-&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp;
-  </td>
-  <td align="left">
-<math>~\frac{p^'}{\chi_0 \varpi} = -5 +3\chi_0^2  \, .</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-Also, note:
-  </td>
-  <td align="left">
-<math>~\frac{d(p^')}{d\chi_0} = -5 +24\chi_0^2 -15\chi_0^4 \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-For the parabolic case, therefore, the RHS of the "equilibrium" expression is,
-<div align="center">
-<table border="0" cellpadding="5">
-<tr>
-  <td align="right">
-<font size="+1">RHS</font>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(-5+3\chi_0^2)\biggl[ -2\chi_0^2 - 2(1-\chi_0^2) \biggr] - (-5 +24\chi_0^2 -15\chi_0^4)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(10 - 6\chi_0^2) + (5 -24\chi_0^2 +15\chi_0^4)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~15(1-2\chi_0^2+\chi_0^4) \, ,
-</math>
-  </td>
-</tr>
-</table>
-</div>
-which, indeed, matches the LHS of the "equilibrium" relation, if,
-<div align="center">
-<math>~\Gamma = 15</math>
-&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
-<math>~\tau_\mathrm{SSC}^2 = \frac{15}{4\pi G \rho_c} \, .</math>
-</div>
-This has all worked satisfactorily because, [[User:Tohline/SSC/Structure/Other_Analytic_Models#Stabililty_2|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the parabolic density distribution.
-<font color="red">'''Case 2''' (Linear)</font>:
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\varpi = 1 -\chi_0</math>
-  </td>
-  <td align="center">
-&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp;
-  </td>
-  <td align="left">
-<math>~\varpi^' = -1 </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~p^' = \tfrac{12}{5}[- 4\chi_0 + 7\chi_0^2 - 3\chi_0^3]</math>
-  </td>
-  <td align="center">
-&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp;
-  </td>
-  <td align="left">
-<math>~\frac{p^'}{\chi_0 \varpi} = \tfrac{12}{5}(-4 +3\chi_0)  \, .</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-Also, note:
-  </td>
-  <td align="left">
-<math>~\frac{d(p^')}{d\chi_0} = \tfrac{12}{5}[- 4 + 14\chi_0 - 9\chi_0^2] \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-For the linear case, therefore, the RHS of the "equilibrium" expression is,
-<div align="center">
-<table border="0" cellpadding="5">
-<tr>
-  <td align="right">
-<font size="+1">RHS</font>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\tfrac{12}{5}(-4 +3\chi_0)\biggl[ -\chi_0  - 2(1-\chi_0) \biggr] - \tfrac{12}{5}(- 4 + 14\chi_0 - 9\chi_0^2)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\tfrac{12}{5}\biggl[ (4 -3\chi_0)( 2-\chi_0 ) + (4 - 14\chi_0 + 9\chi_0^2) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\tfrac{12}{5}(12-24\chi_0 + 12\chi_0^2)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\tfrac{2^4\cdot 3^2}{5}(1-2\chi_0 + \chi_0^2) \, ,
-</math>
-  </td>
-</tr>
-</table>
-</div>
-which, indeed, matches the LHS of the "equilibrium" relation, if,
-<div align="center">
-<math>~\Gamma = \frac{2^4\cdot 3^2}{5}</math>
-&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
-<math>~\tau_\mathrm{SSC}^2 = \frac{2^2\cdot 3^2}{5\pi G \rho_c} \, .</math>
-</div>
-This has all worked satisfactorily because, [[User:Tohline/SSC/Structure/Other_Analytic_Models#Lagrangian_Approach|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the linear density distribution.
-<font color="red">'''Case 3''' (n = 1 polytrope)</font>:
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\varpi = \frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
-  </td>
-  <td align="center">
-&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp;
-  </td>
-  <td align="left">
-<math>~\varpi^' = \frac{\cos(\pi\chi_0)}{\chi_0} -  \frac{\sin(\pi\chi_0)}{\pi\chi_0^2}</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~p^' = \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>
-  </td>
-  <td align="center">
-&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp;
-  </td>
-  <td align="left">
-<math>~\frac{p^'}{\chi_0 \varpi} = \frac{2}{(\pi\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]  \, .</math>
-  </td>
-</tr>
-<tr>
-  <td align="center" colspan="3">
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-Also, note: &nbsp; &nbsp; &nbsp;<math>~\frac{d(p^')}{d\chi_0} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl[ \frac{2\pi \cdot \cos(\pi\chi_0)}{(\pi^2\chi_0^3)} - \frac{6\sin(\pi\chi_0)}{(\pi^2\chi_0^4)}  \biggr] \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]
-+\frac{2\sin(\pi\chi_0)}{(\pi^2\chi_0^3)} \biggl[ \pi\cos(\pi\chi_0) - \pi^2\chi_0 \sin(\pi\chi_0) - \pi \cos(\pi\chi_0) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ \frac{1}{\pi^2 \chi_0^4} \biggl\{
-\biggl[ 2\pi \chi_0\cdot \cos(\pi\chi_0) - 6\sin(\pi\chi_0)  \biggr] \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]
-+2 \chi_0 \sin(\pi\chi_0) \biggl[ - \pi^2\chi_0 \sin(\pi\chi_0) \biggr]
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~ \frac{1}{\pi^2 \chi_0^4} \biggl\{
-\pi^2\chi_0^2\cos^2(\pi\chi_0) -8\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+6\sin^2(\pi\chi_0)
-- 2 \pi^2 \chi_0^2 \sin^2(\pi\chi_0)
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{2}{\pi^2 \chi_0^4} \biggl\{3\sin^2(\pi\chi_0)
--4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+\pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
-\biggr\} \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-  </td>
-</tr>
-</table>
-</div>
-For the case of an n = 1 polytropic configuration, therefore, the equilibrium requirement is,
-<div align="center">
-<table border="0" cellpadding="5">
-<tr>
-  <td align="right">
-<math>~\Gamma \varpi^2</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{p^' }{\chi_0 \varpi}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \frac{dp^'}{d\chi_0}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{2}{(\pi\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]\biggl[ \cos(\pi\chi_0) -  \frac{\sin(\pi\chi_0)}{\pi\chi_0} - \frac{2\sin(\pi\chi_0)}{\pi\chi_0} \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
-- \frac{2}{\pi^2 \chi_0^4} \biggl\{3\sin^2(\pi\chi_0)
--4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+\pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{2}{(\pi^2\chi_0^4)} \biggl\{ \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]\biggl[\pi\chi_0 \cos(\pi\chi_0) -  3\sin(\pi\chi_0) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
--3\sin^2(\pi\chi_0) + 4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0) - \pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{2}{(\pi^2\chi_0^4)} \biggl\{3\sin^2(\pi\chi_0) - 4\pi\chi_0\sin(\pi\chi_0)\cos(\pi\chi_0) + (\pi\chi_0)^2 \biggl[1-\sin^2(\pi\chi_0) \biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~
--3\sin^2(\pi\chi_0) + 4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0) - \pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{2}{(\pi^2\chi_0^4)} \biggl\{ (\pi\chi_0)^2 \sin^2(\pi\chi_0)  \biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~2\pi^2 \biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}  \biggr]^2
-</math>
-  </td>
-</tr>
-</table>
-</div>
-So, the equilibrium condition is satisfied if,
-<div align="center">
-<math>~\Gamma = 2\pi^2</math>
-&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
-<math>~\tau_\mathrm{SSC}^2 = \frac{2\pi^2}{4\pi G \rho_c} = \frac{\pi}{2G \rho_c}  \, .</math>
-</div>
-This has all worked satisfactorily because, [[User:Tohline/SSC/Stability/Polytropes#Setup|as presented in a separate chapter discussion]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of an n = 1 polytropic configuration.
-====Dabbling with LAWE====
-Now, let's experiment with the [[User:Tohline/SSC/Structure/Other_Analytic_Models#Compare|LAWE as presented above]], that is,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\biggl( \frac{x^'}{x} \biggr) \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \chi_0^4 x^' \biggr) \biggr]
-+ \frac{d\ln(p)}{d\chi_0}\cdot  \frac{d}{d\chi_0}\biggl[\ln\biggl(\chi_0^\alpha x\biggr)\biggr] \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-After multiplying though by <math>~(-p)</math>, this expression may be written as,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~-\sigma^2 \varpi</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{p}{x}\biggl[ \frac{dx^'}{d\chi_0} + \frac{4x^'}{\chi_0}\biggr]
-+ \frac{\alpha p^'}{\chi_0}\biggl[1 + \frac{\chi_0 x^'}{\alpha x}\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\biggl(\frac{p}{x}\biggr) x^{' '} + p\biggl[ \frac{4}{\chi_0}\biggr]\frac{x^'}{x}
-+ p^'\biggr[\frac{x^'}{x}\biggr] + \frac{\alpha p^'}{\chi_0}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ -\biggl[\sigma^2 \varpi + \frac{\alpha p^'}{\chi_0}  \biggr]x</math>
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-px^{' '} + \biggl[ \frac{4p}{\chi_0} + p^'\biggr]x^'
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow ~~~~ 0</math>
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-px^{' '} + \biggl[ 4p + \chi_0 p^'\biggr]\frac{x^'}{\chi_0} + \biggl[\sigma^2 \varpi + \frac{\alpha p^'}{\chi_0}  \biggr]x \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-(We could have, perhaps, obtained this expression in a more direct fashion had we started directly from the form of the [[User:Tohline/SSC/Structure/Other_Analytic_Models#LAWE|LAWE derived earlier]].)
-<font color="red">'''Case 0''' (Uniform density)</font>:
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\varpi</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~1 \, ;</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~p</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~1-\chi_0^2 \, ;</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\frac{\alpha p^'}{\chi_0}</math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~- 2\alpha \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-For the uniform-density case, therefore, the the LAWE becomes,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~-\sigma^2 </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\frac{(1-\chi_0^2)}{x}\biggl[ \frac{dx^'}{d\chi_0} + \frac{4x^'}{\chi_0}\biggr]
--2\alpha \biggl[1 + \frac{\chi_0 x^'}{\alpha x}\biggr]
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~\Rightarrow~~~~ (2\alpha -\sigma^2)x </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(1-\chi_0^2)\biggl[ \frac{dx^'}{d\chi_0} + \frac{4x^'}{\chi_0}\biggr] - 2\chi_0 x^'
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(1-\chi_0^2)\frac{dx^'}{d\chi_0}  + (1-\chi_0^2)\biggl[ \frac{4x^'}{\chi_0}\biggr] - 2\chi_0 x^'
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(1-\chi_0^2)\frac{dx^'}{d\chi_0}  + \frac{2x^'}{\chi_0}\biggl( 2 - 3\chi_0^2 \biggr) \, ,
-</math>
-  </td>
-</tr>
-</table>
-</div>
-where, [[User:Tohline/SSC/Structure/Other_Analytic_Models#Stabililty_2|as defined above]],
-<div align="center">
-<math>~\alpha \equiv 3 - \frac{4}{\gamma_g} \, .</math>
-</div>
-=====Mode 3=====
-Try an eigenfunction of the form,
-<div align="center">
-<math>x = a + b\chi_0^2 + c\chi_0^4 \, ,</math>
-</div>
-in which case,
-<div align="center">
-<math>~\frac{2x^'}{\chi_0} = \frac{2}{\chi_0}(2 b\chi_0 +4c\chi_0^3) = 4b+8c\chi_0^2</math>
-&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
-<math>~x^{' '} = 2 b + 12c\chi_0^2 \, . </math>
-</div>
-In order for this to be a solution, we must have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~(2\alpha -\sigma^2)( a + b\chi_0^2 + c\chi_0^4) </math>
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(1-\chi_0^2)(2 b + 12c\chi_0^2 )  + ( 2 - 3\chi_0^2 )(4b+8c\chi_0^2 )
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-(2 b + 12c\chi_0^2)  - \chi_0^2(2 b + 12c\chi_0^2 )  + 2(4b+8c\chi_0^2 ) - 3\chi_0^2 (4b+8c\chi_0^2 )
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~10b + \chi_0^2(12c-2b+16c-12b) - \chi_0^4(12c + 24c)
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-  </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~10b + \chi_0^2(28c-14b) - \chi_0^4(36c) \, .
-</math>
-  </td>
-</tr>
-</table>
-</div>
-So, the coefficients of each even power of <math>~\chi_0^n</math> are:
-<div align="center" id="FirstTable">
-<table border="1" cellpadding="8" align="center">
-<tr>
-  <td align="right"><math>~\chi_0^0</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~a\mathfrak{F} +10b</math>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^2</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~b\mathfrak{F} -14b + 28c</math>
- </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^4</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~c[\mathfrak{F} -36]</math>
-  </td>
-</tr>
-</table>
-</div>
-where, following [[User:Tohline/SSC/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29|Sterne's (1937) presentation]],
-<div align="center">
-<math>~\mathfrak{F}  \equiv \sigma^2 - 2 \alpha \, .</math>
-</div>
-In order for all three of the coefficients to be zero, we must have:
-First: &nbsp;&nbsp;&nbsp; <math>~\mathfrak{F} = 36 \, ;</math>
-Second: &nbsp;&nbsp;&nbsp; <math>~22b = -28c ~~~~~\Rightarrow ~~~~~ c = - (11/14)b \, ;</math>
-Third: &nbsp;&nbsp;&nbsp; <math>~36a = -10b ~~~~~\Rightarrow ~~~~~ b = -(18/5)a \, .</math>
-Hence, choosing <math>~a = 1</math> implies:  <math>~b = -18/5</math> &nbsp; &nbsp; &nbsp; &nbsp;and  &nbsp; &nbsp; &nbsp; &nbsp;
-<math>~ c = (11/7)(9/5) = +99/35 \, .</math>  This precisely matches the "j = 2" mode identified by Sterne.
-<font color="red">'''Case 1''' (Parabolic)</font>:
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~\varpi = 1 -\chi_0^2</math>
-  </td>
-  <td align="center">
-&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp;
-  </td>
-  <td align="left">
-<math>~\varpi^' = -2\chi_0 </math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~p = \tfrac{1}{2}(1 -\chi_0^2)^2 (2-\chi_0^2)</math>
-  </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-&nbsp;
-  </td>
-</tr>
-<tr>
-  <td align="right">
-<math>~p^' = -5\chi_0 + 8\chi_0^3 - 3\chi_0^5 = \chi_0(1-\chi_0^2)(-5+3\chi_0^2)</math>
-  </td>
-  <td align="center">
-&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp;
-  </td>
-  <td align="left">
-<math>~\frac{\alpha p^'}{\chi_0} = \alpha (1-\chi_0^2)(-5+3\chi_0^2)  \, .</math>
-  </td>
-</tr>
-</table>
-</div>
-For the parabolic case, therefore, the the LAWE becomes,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~0</math>
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-px^{' '} + \biggl[ 4p + \chi_0 p^'\biggr]\frac{x^'}{\chi_0} + \biggl[\sigma^2 \varpi + \frac{\alpha p^'}{\chi_0}  \biggr]x
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~
-\tfrac{1}{2}\varpi^2 (2-\chi_0^2)x^{' '}
-+ \biggl[ 2\varpi^2 (2-\chi_0^2) + \chi_0^2\varpi(-5+3\chi_0^2)\biggr]\frac{x^'}{\chi_0}
-+ \biggl[\sigma^2 \varpi + \alpha \varpi(-5+3\chi_0^2)  \biggr]x
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{\varpi}{2}\biggl\{
-\varpi (2-\chi_0^2)x^{' '}
-+ \biggl[ 4\varpi (2-\chi_0^2) + \chi_0^2(-10+6\chi_0^2)\biggr]\frac{x^'}{\chi_0}
-+ \biggl[\mathfrak{K}+6\alpha\chi_0^2  \biggr]x
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{\varpi}{2}\biggl\{
-(2-3\chi_0^2 + \chi_0^4)x^{' '}
-+ \biggl[ 8-22\chi_0^2 + 10\chi_0^4\biggr]\frac{x^'}{\chi_0}
-+ \biggl[\mathfrak{K}+6\alpha\chi_0^2  \biggr]x
-\biggr\}
-</math>
-  </td>
-</tr>
-</table>
-</div>
-where,
-<div align="center">
-<math>~\mathfrak{K} \equiv 2(\sigma^2 - 5\alpha) \, .</math>
-</div>
-=====Mode 3P=====
-Try an eigenfunction of the form,
-<div align="center">
-<math>x = a + b\chi_0^2 + c\chi_0^4 \, ,</math>
-</div>
-in which case,
-<div align="center">
-<math>~\frac{x^'}{\chi_0} = \frac{1}{\chi_0}(2 b\chi_0 +4c\chi_0^3) = 2b+4c\chi_0^2</math>
-&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
-<math>~x^{' '} = 2 b + 12c\chi_0^2 \, . </math>
-</div>
-In order for this to be a solution, we must have,
-<div align="center">
-<table border="0" cellpadding="5" align="center">
-<tr>
-  <td align="right">
-<math>~0</math>
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{\varpi}{2}\biggl\{
-(2-3\chi_0^2 + \chi_0^4)(2 b + 12c\chi_0^2 )
-+ (8-22\chi_0^2 + 10\chi_0^4 )(2b+4c\chi_0^2 )
-+ (\mathfrak{K}+6\alpha\chi_0^2 )( a + b\chi_0^2 + c\chi_0^4 )
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{\varpi}{2}\biggl\{
-(4b+24c\chi_0^2 - 6b\chi_0^2 -36c\chi_0^4 + 2b\chi_0^4 + 12c\chi_0^6)
-+ (16b + 32c\chi_0^2 - 44b\chi_0^2 - 88c\chi_0^4 + 20b\chi_0^4 + 40c\chi_0^6 )
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-   </td>
-  <td align="center">
-&nbsp;
-  </td>
-  <td align="left">
-<math>~+ (a\mathfrak{K} + b\mathfrak{K}\chi_0^2 + c\mathfrak{K}\chi_0^4 + 6a\alpha\chi_0^2 + 6b\alpha\chi_0^4 + 6c\alpha\chi_0^6 )
-\biggr\}
-</math>
-  </td>
-</tr>
-<tr>
-  <td align="right">
-&nbsp;
-   </td>
-  <td align="center">
-<math>~=</math>
-  </td>
-  <td align="left">
-<math>~\frac{\varpi}{2}\biggl[ (20b + a\mathfrak{K}) \chi_0^0 +(24c - 6b + 32c - 44b + b\mathfrak{K} + 6a\alpha)\chi_0^2
-+ (-36c+2b -88c+20b +c\mathfrak{K} + 6b\alpha) \chi_0^4 + (12c + 40c + 6c\alpha )\chi_0^6 \biggr]
-</math>
-  </td>
-</tr>
-</table>
-</div>
-So, the coefficients of each even power of <math>~\chi_0^n</math> are:
-<div align="center" id="FirstTable">
-<table border="1" cellpadding="8" align="center">
-<tr>
-  <td align="right"><math>~\chi_0^0</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~20b + a\mathfrak{K}</math>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^2</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~56c + (\mathfrak{K}- 50)b + 6a\alpha</math>
- </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^4</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~b(22+6\alpha) + c(\mathfrak{K}-124)</math>
-  </td>
-</tr>
-<tr>
-  <td align="right"><math>~\chi_0^6</math></td>
-  <td align="center">&nbsp; : &nbsp;</td>
-  <td align="left">
-<math>~c(52 + 6\alpha)</math>
-  </td>
-</tr>
-</table>
-</div>
-This is disappointing, as it does not result in nonzero coefficient values.
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