User:Tohline/SSC/Structure/Other Analytic Models
Other Analytically Definable, Spherical Equilibrium Models
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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,
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<math>~M_r(r)</math> |
<math>~=</math> |
<math>~\int_0^r 4\pi r^2 \rho(r) dr</math> |
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<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,
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<math>~g_0(r) \equiv \frac{G M_r(r) }{r^2} </math> |
<math>~=</math> |
<math>~\frac{4\pi G \rho_c r}{3} \biggl[1 - \frac{3}{4} \biggl( \frac{r}{R} \biggr)\biggr] \, .</math> |
Hence, proceeding via what we have labeled as "Technique 1", and enforcing the surface boundary condition, <math>~P(R) = 0</math>, Stein (1966) determines that (see his equation 3.5),
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<math>~P(r)</math> |
<math>~=</math> |
<math>~- \int_0^r g_0(r) \rho(r) dr</math> |
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<math>~=</math> |
<math>~\frac{\pi G\rho_c^2 R^2}{36} \biggl[5 - 24 \biggl( \frac{r}{R} \biggr)^2 + 28 \biggl( \frac{r}{R} \biggr)^3 - 9 \biggl( \frac{r}{R} \biggr)^4 \biggr] \, ,</math> |
where, it can readily be deduced, as well, that the central pressure is,
<math>~P_c = \frac{5\pi}{36} G\rho_c^2 R^2 \, .</math>
Stabililty
Lagrangian Approach
As has been derived in an accompanying discussion, the second-order ODE that defines the relevant Eigenvalue problem is,
<math>
\biggl(\frac{P_0}{P_c}\biggr)\frac{d^2x}{d\chi_0^2}
+ \biggl[\biggl(\frac{P_0}{P_c}\biggr)\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0}
+ \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2
+ (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr] x = 0 .
</math>
where the dimensionless radius,
<math>
\chi_0 \equiv \frac{r_0}{R} \, ,
</math>
<math> g_\mathrm{SSC} \equiv \frac{P_c}{R\rho_c}</math> and <math>\tau_\mathrm{SSC} \equiv \biggl( \frac{R^2\rho_c}{P_c}\biggr)^{1/2} \, . </math>
For Stein's configuration with a linear density distribution,
<math> g_\mathrm{SSC} = \frac{5\pi G\rho_c R}{36}</math> and <math>\tau_\mathrm{SSC} \equiv \biggl( \frac{36}{5\pi G \rho_c }\biggr)^{1/2} = \biggl( \frac{12}{5}\cdot \frac{R^3}{GM_\mathrm{tot} }\biggr)^{1/2} \, . </math>
Hence,
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<math>~\frac{g_0}{g_\mathrm{SSC}} </math> |
<math>~=</math> |
<math>~\frac{48}{5}\cdot \chi_0\biggl(1 - \frac{3}{4} \chi_0 \biggr) \, .</math> |
and the governing adiabatic wave equation takes the form,
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<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),
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<math>~\frac{\partial}{\partial r} \biggl( \gamma P_0 \nabla\cdot \vec{\xi}\biggr) + \biggl( \omega^2 + \frac{4g_0}{r}\biggr) \rho_0 \xi</math> |
<math>~=</math> |
<math>~0 \, ,</math> |
where,
<math>~\vec\xi = \mathbf{\hat{e}}_r \xi(r) \, .</math>
Realizing that, for a spherically symmetric system,
<math>\nabla\cdot \vec\xi = \frac{1}{r^2}\frac{\partial}{\partial r}\biggl(r^2 \xi\biggr) = \frac{\partial \xi}{\partial r} + \frac{2\xi}{r} \, ,</math>
and remembering that,
<math>~\frac{\partial P_0}{\partial r} = -g_0 \rho_0 \, ,</math>
we can rewrite this relation in the more familiar form of a 2nd-order ODE, namely,
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<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> |
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<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> |
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<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> |
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<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,
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<math>~0</math> |
<math>~=</math> |
<math>~\biggl(\frac{P_0}{P_c}\biggr) \frac{\partial^2 \xi}{\partial \chi_0^2} + \biggl[ \frac{2}{\chi_0}\biggl(\frac{P_0}{P_c}\biggr) - \frac{g_0 }{g_\mathrm{SSC}}\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggr] \frac{\partial \xi}{\partial \chi_0} + \biggl\{ \biggl[\frac{\omega^2\tau_\mathrm{SSC}^2}{\gamma} + \frac{2}{\chi_0 } \biggl(\frac{2}{\gamma } - 1\biggr)\frac{g_0}{g_\mathrm{SSC}}\biggr] \biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2}{\chi_0^2} \biggl(\frac{P_0}{P_c}\biggr) \biggr\} \xi \, . </math> |
Now, plugging in the functional expressions that specifically apply to the linear model gives,
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<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> |
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<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> |
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<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,
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<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,
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<math>~M_r(r)</math> |
<math>~=</math> |
<math>~\int_0^r 4\pi r^2 \rho(r) dr</math> |
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<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,
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<math>~g_0(r) \equiv \frac{G M_r(r) }{r^2} </math> |
<math>~=</math> |
<math>~\frac{4\pi G \rho_c r}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r}{R} \biggr)^2\biggr] \, .</math> |
Hence, proceeding via what we have labeled as "Technique 1", and enforcing the surface boundary condition, <math>~P(R) = 0</math>, Prasad (1949) determines that,
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<math>~P(r)</math> |
<math>~=</math> |
<math>~- \int_0^r g_0(r) \rho(r) dr</math> |
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<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> |
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<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> |
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<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> |
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<math>~=</math> |
<math>~\frac{4\pi G\rho_c^2 R^2}{15} \biggl[1-\biggl(\frac{r}{R}\biggr)^2\biggr]^2 \biggl[1-\frac{1}{2}\biggl(\frac{r}{R}\biggr)^2\biggr] \, ,</math> |
where, it can readily be deduced, as well, that the central pressure is,
<math>~P_c = \frac{4\pi}{15} G\rho_c^2 R^2 \, .</math>
Stabililty
As has been derived in an accompanying discussion, the second-order ODE that defines the relevant Eigenvalue problem is,
<math>
\biggl(\frac{P_0}{P_c}\biggr)\frac{d^2x}{d\chi_0^2}
+ \biggl[\biggl(\frac{P_0}{P_c}\biggr)\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0}
+ \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2
+ (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr] x = 0 \, ,
</math>
where the dimensionless radius,
<math>
\chi_0 \equiv \frac{r_0}{R} \, ,
</math>
<math> g_\mathrm{SSC} \equiv \frac{P_c}{R\rho_c}</math> and <math>\tau_\mathrm{SSC} \equiv \biggl( \frac{R^2\rho_c}{P_c}\biggr)^{1/2} \, . </math>
For Prasad's configuration with a parabolic density distribution,
<math> g_\mathrm{SSC} = \frac{4\pi G\rho_c R}{15}</math> and <math>\tau_\mathrm{SSC} \equiv \biggl( \frac{15}{4\pi G \rho_c }\biggr)^{1/2} = \biggl( \frac{2R^3}{GM_\mathrm{tot} }\biggr)^{1/2} = \biggl( \frac{3}{2\pi G\bar\rho}\biggr)^{1/2}\, . </math>
Hence,
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<math>~\frac{g_0}{g_\mathrm{SSC}} </math> |
<math>~=</math> |
<math>~(5 - 3 \chi_0^2)\chi_0 \, ,</math> |
and the governing adiabatic wave equation takes the form,
<math>
(1-\chi_0^2) \biggl( 1 - \frac{1}{2}\chi_0^2 \biggr)\frac{d^2x}{d\chi_0^2}
+ \frac{1}{\chi_0}\biggl[4 (1-\chi_0^2) \biggl( 1 - \frac{1}{2}\chi_0^2 \biggr) - (5 - 3 \chi_0^2)\chi_0^2\biggr] \frac{dx}{d\chi_0}
+ \biggl[\frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_\mathrm{g}}
-\alpha (5 - 3 \chi_0^2)\biggr] x = 0 \, ,
</math>
where,
<math>~\alpha \equiv 3 - \frac{4}{\gamma_\mathrm{g}} \, .</math>
In keeping with Prasad's presentation — see, specifically, his equations (2) & (3) — this wave equation can also be written as,
<math>
(1-\chi_0^2) \biggl( 1 - \frac{1}{2}\chi_0^2 \biggr)\frac{d^2x}{d\chi_0^2}
+ \frac{1}{\chi_0}\biggl[4 - 11\chi_0^2 + 5\chi_0^4\biggr] \frac{dx}{d\chi_0}
+ \biggl[\mathfrak{J}+3\alpha \chi_0^2 \biggr] x = 0 \, ,
</math>
where,
<math>~\mathfrak{J} \equiv \frac{3\omega^2}{2\pi G \gamma_\mathrm{g} \bar\rho} - 5\alpha \, .</math>
For what it's worth, we have also deduced that this expression can be written as,
<math> (1-\chi_0^2) \biggl( 1 - \frac{1}{2}\chi_0^2 \biggr)\chi_0^{-4} \frac{d}{d\chi_0} \biggl[\chi_0^4 \frac{dx}{d\chi_0} \biggr] -(5-3\chi_0^2)\chi_0^{1+\alpha} \frac{d}{d\chi_0} \biggl[ \chi_0^{-\alpha} x \biggr] + \biggl(\frac{\tau_\mathrm{SSC}^2~ \omega^2}{\gamma_\mathrm{g}}\biggr) x = 0 \, , </math>
Ramblings
The material originally contained in this "Ramblings" subsection has been moved to generate a separate chapter that stands on its own.
Promising Avenue of Exploration
What follows is a direct extension of what is referred to in our "Ramblings" chapter as the third guess under "Exploration2". We pursue this line of reasoning, here, because it appears to be a particularly promising avenue of exploration.
In the case of a parabolic density distribution, the LAWE becomes,
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<math>~\frac{2}{(1-x^2)(2-x^2)} \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math> |
<math>~=</math> |
<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} \, . </math> |
We have chosen to examine the suitability of an eigenfunction of the form,
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<math>~\mathcal{G}_\sigma</math> |
<math>~=</math> |
<math>~(a_0 + a_2x^2)^n \cdot (2 - x^2)^m \, ,</math> |
where, for a given value of <math>~\alpha</math>, the four parameters, <math>~a_0</math>, <math>~a_2</math>, <math>~n</math> and <math>~m</math> are to be determined in concert with a value of the square of the eigenfrequency, <math>~\sigma^2</math>. From the accompanying discussion we have determined that the following five coefficient expressions must independently be zero in order for this trial eigenfunction to satisfy the LAWE:
| <math>~x^0</math> | : |
<math>~ \alpha(10a_0^2) + \sigma^2(- 2a_0^2) -20n a_0a_2 + 10ma_0^2 </math> |
| <math>~x^2</math> | : |
<math>~\alpha(- 11a_0^2 +20 a_0a_2) + \sigma^2(a_0^2 - 4 a_0a_2) + 60n a_0a_2 -20na_2^2- 25m a_0^2 + 20m a_0a_2 + 8n m a_0a_2 </math><math> - [ 8n(n-1)a_2^2 + 2m(m-1)a_0^2 ] </math> |
| <math>~x^4</math> | : |
<math>~ \alpha(10a_2^2 - 22 a_0a_2 +3a_0^2) - \sigma^2(2a_2^2- 2a_0a_2 ) - 47n a_0a_2+ 60n a_2^2 - 50m a_0a_2 +11m a_0^2+ 10ma_2^2-12 n m a_0a_2 + 8n m a_2^2 </math><math> ~ + 16n(n-1)a_2^2 - 4m(m-1)a_0 a_2 + 2m(m-1)a_0^2 </math> |
| <math>~x^6</math> | : |
<math>~ \alpha(6a_0a_2 - 11a_2^2) + \sigma^2(a_2^2) + 11 n a_0a_2 +22 m a_0a_2 - 47n a_2^2 - 25m a_2^2 -12 n m a_2^2 + 4n m a_0a_2 </math><math>~ - 10 n(n-1)a_2^2 -2m(m-1)a_2^2 +4m(m-1)a_0 a_2 </math> |
| <math>~x^8</math> | : |
<math>~ \{ 3\alpha + [ 4n m + 11n + 11m ] + [ 2n(n-1) + 2m(m-1) ]\}a_2^2 </math> |
First Constraint
We begin by manipulating the last expression — that is, the coefficient expression for the <math>~x^8</math> term. Rejecting the trivial option of setting <math>~a_2 = 0</math>, in order for this expression to be zero the terms inside the curly braces must sum to zero. Rewriting this expression in terms of the sum of the exponents,
<math>~s_{nm} \equiv n + m\, ,</math>
we obtain the quadratic expression,
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<math>~0</math> |
<math>~=</math> |
<math>~3\alpha + [ 4n m + 11n + 11m ] + [ 2n(n-1) + 2m(m-1) ]</math> |
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<math>~=</math> |
<math>~3\alpha + 4n m + 9n + 9m + 2n^2 + 3m^2</math> |
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<math>~=</math> |
<math>~3\alpha + 9s_{nm} + 2s_{nm}^2 \, .</math> |
This means that, once the physical parameter, <math>~\alpha = (3 - 4/\gamma_g)</math>, has been specified, the sum of the exponents must be,
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<math>~s_{nm}</math> |
<math>~=</math> |
<math>~\frac{1}{4}\biggl[ -9 \pm (81 - 24\alpha)^{1/2} \biggr]</math> |
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<math>~=</math> |
<math>~\frac{3^2}{2^2}\biggl[ -1 \pm \biggl(1 - \frac{2^3\alpha}{3^3} \biggr)^{1/2} \biggr] \, .</math> |
Second Constraint
Next we examine the expression that serves as the coefficient of <math>~x^0</math>. Setting that coefficient expression to zero while replacing <math>~m</math> in favor of <math>~s_{nm}</math> — via the relation, <math>~m = (s_{nm}-n)</math> — gives,
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<math>~0</math> |
<math>~=</math> |
<math>~\alpha(10a_0^2) + \sigma^2(- 2a_0^2) -20n a_0a_2 + 10ma_0^2</math> |
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<math>~=</math> |
<math>~2a_0^2 \biggl[5\alpha -\sigma^2 + 5(s_{nm}-n) - 10n \biggl(\frac{a_2}{a_0} \biggr)\biggr]</math> |
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<math>~=</math> |
<math>~2a_0^2 \biggl[5\alpha -\sigma^2 + 5s_{nm} -5n\biggl(1 - \frac{2a_2}{a_0} \biggr)\biggr]</math> |
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<math>~\Rightarrow ~~~~ \frac{\sigma^2}{5}</math> |
<math>~=</math> |
<math>~(\alpha + s_{nm}) -n(1 - 2\lambda) \, ,</math> |
where, we have set,
<math>~\lambda \equiv \frac{a_2}{a_0} \, .</math>
So, once <math>~\alpha</math> is specified and <math>~s_{nm}</math> is known from the first constraint, we can use this expression to replace <math>~\sigma^2</math> in the other three coefficient expressions.
Intermediate Summary
Written in terms of the three remaining unknowns, <math>~n</math>, <math>~a_0</math>, and <math>~\lambda</math>, the three remaining constraints are:
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<math>~x^2:</math> |
<math>~0</math> |
<math>~=</math> |
<math>~ \alpha(- 11 +20 \lambda) + \sigma^2(1 - 4 \lambda) + 60n \lambda -20n \lambda^2- 25m + 20m \lambda + 8n m \lambda - [ 8n(n-1)\lambda^2 + 2m(m-1) ] </math> |
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<math>~=</math> |
<math>~ \alpha(- 11 +20 \lambda) + 5(1 - 4 \lambda)[ (\alpha + s_{nm}) -n(1 - 2\lambda) ] + 60n \lambda -20n \lambda^2 - 8n(n-1)\lambda^2 </math> |
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<math>~ + (s_{nm}-n)[- 23 + 20 \lambda + 8n \lambda] - 2(s_{nm}-n)^2 </math> |
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<math>~x^4:</math> |
<math>~0</math> |
<math>~=</math> |
<math>~ \alpha(10\lambda^2 - 22 \lambda +3) - 10[ (\alpha + s_{nm}) -n(1 - 2\lambda) ](\lambda^2- \lambda ) - 47n \lambda+ 60n \lambda^2 + 16n(n-1)\lambda^2 </math> |
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<math>~ + (s_{nm} - n)[ 9 - 46 \lambda + 10\lambda^2-12 n \lambda + 8n \lambda^2] + (2-4\lambda)(s_{nm} - n)^2 </math> |
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<math>~x^6:</math> |
<math>~0</math> |
<math>~=</math> |
<math>~ \alpha </math> |
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<math>~ + </math> |
| <math>~x^2</math> | : |
<math>~\alpha(- 11a_0^2 +20 a_0a_2) + \sigma^2(a_0^2 - 4 a_0a_2) + 60n a_0a_2 -20na_2^2- 25m a_0^2 + 20m a_0a_2 + 8n m a_0a_2 </math><math> - [ 8n(n-1)a_2^2 + 2m(m-1)a_0^2 ] </math> |
| <math>~x^4</math> | : |
<math>~ \alpha(10a_2^2 - 22 a_0a_2 +3a_0^2) - \sigma^2(2a_2^2- 2a_0a_2 ) - 47n a_0a_2+ 60n a_2^2 - 50m a_0a_2 +11m a_0^2+ 10ma_2^2-12 n m a_0a_2 + 8n m a_2^2 </math><math> ~ + 16n(n-1)a_2^2 - 4m(m-1)a_0 a_2 + 2m(m-1)a_0^2 </math> |
| <math>~x^6</math> | : |
<math>~ \alpha(6a_0a_2 - 11a_2^2) + \sigma^2(a_2^2) + 11 n a_0a_2 +22 m a_0a_2 - 47n a_2^2 - 25m a_2^2 -12 n m a_2^2 + 4n m a_0a_2 </math><math>~ - 10 n(n-1)a_2^2 -2m(m-1)a_2^2 +4m(m-1)a_0 a_2 </math> |
Third Constraint
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© 2014 - 2021 by Joel E. Tohline |