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>
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Ramblings
Generic Setup
Dividing the above, 2nd-order ODE through by the quantity, <math>~[R^2 (P_0/P_c)]</math>, gives,
<math>
\frac{d^2x}{dr_0^2}
+ \biggl[\frac{4}{r_0} - \frac{1}{R}\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggl(\frac{P_c}{P_0}\biggr)\biggr] \frac{dx}{dr_0}
- \biggl[\frac{1}{R}\biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{\alpha}{r_0} \biggr] x
= - \frac{1}{R^2}\biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl[ \biggl( \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} \biggr) \biggr] x \, ,
</math>
which matches Prasad's (1949) equation (1), namely,
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<math>~x^{' '} + \biggl[\frac{4}{r_0} - \frac{\mu(r_0) }{r_0}\biggr] x^{'} - \biggl[ \frac{\alpha \mu(r_0)}{r_0^2} \biggr] x</math> |
<math>~=</math> |
<math>~- \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl[ \frac{n^2\rho_c}{\gamma_g P_c} \biggr] x \, ,</math> |
where, primes indicate differentiation with respect to <math>~r_0</math>, and,
<math>~\mu(r_0) \equiv \frac{r_0}{R} \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \, .</math>
(Note that Prasad's equation has the awkward units of inverse length-squared.) Regrouping terms in Prasad's governing equation, multiplying through by <math>~R^2</math> (to make the equation dimensionless), and now letting primes denote differentiation with respect to the dimensionless radial coordinate, <math>~\chi_0</math>, we quite generally can write the linear adiabatic wave equation as,
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<math>~- \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) \sigma^2 x </math> |
<math>~=</math> |
<math>~\biggl[x^{' '} + \frac{4 x^'}{\chi_0}\biggr] - \frac{\mu(\chi_0)}{\chi_0} \biggl[ x^{'} + \frac{\alpha x}{\chi_0} \biggr]</math> |
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<math>~=</math> |
<math>~\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) - \frac{\mu(\chi_0)}{\chi_0} \biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \biggr] \, . </math> |
Defining,
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<math>~A</math> |
<math>~\equiv</math> |
<math>~\biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_c}{\rho_0}\biggr) \, ,</math> |
|
<math>~B</math> |
<math>~\equiv</math> |
<math>~\frac{A\mu(\chi_0)}{\chi_0} = \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \, ,</math> |
the governing equation becomes,
|
Notice that, because,
<math>~g_0 = - \frac{1}{\rho_0} ~\frac{dP_0}{dr_0} \, ,</math>
at every radial location throughout the configuration, it must also be true that, for any equilibrium configuration,
|
<math>~B</math> |
<math>~=</math> |
<math>~- \biggl(\frac{\rho_0}{\rho_c}\biggr)^{-1} \frac{d(P_0/P_c)}{d\chi_0}</math> |
|
<math>~\Rightarrow ~~~~ \frac{B}{A}</math> |
<math>~=</math> |
<math>~- \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \, .</math> |
The following table shows that this relationship holds for a collection of analytically described equilibrium structures.
| Table 1a: Properties of Analytically Defined Equilibrium Structures | |||
|---|---|---|---|
| Model | <math>~\frac{\rho_0}{\rho_c}</math> | <math>~\frac{P_0}{P_c}</math> | <math>~\frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)</math> |
| Uniform-density | <math>~1</math> | <math>~1 - \chi_0^2</math> | <math>~-2\chi_0</math> |
| Linear | <math>~1-\chi_0</math> | <math>~\tfrac{1}{5} (5 -24 \chi_0^2 + 28\chi_0^3 - 9\chi_0^4)</math> | <math>~\tfrac{1}{5}[- 48\chi_0 + 84\chi_0^2 - 36\chi_0^3]</math> |
| Parabolic | <math>~1-\chi_0^2</math> | <math>~\tfrac{1}{2} (2 - 5\chi_0^2 + 4\chi_0^4 - \chi_0^6)</math> | <math>~- 5\chi_0 + 8\chi_0^3 - 3\chi_0^5</math> |
| <math>~n=1</math> Polytrope | <math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math> | <math>~\biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}\biggr]^2</math> | <math>~\frac{2\sin(\pi\chi_0)}{(\pi^2\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]</math> |
| Table 1b: Properties of Analytically Defined Equilibrium Structures | ||||
|---|---|---|---|---|
| Model | <math>~\frac{\rho_0}{\rho_c}</math> | <math>~B\equiv \frac{g_0}{g_\mathrm{SSC}}</math> | <math>~A \equiv \biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr)^{-1}</math> | <math>~\biggl(\frac{P_0}{P_c}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr)^{-2}</math> |
| Uniform-density | <math>~1</math> | <math>~2\chi_0</math> | <math>~1 - \chi_0^2</math> | <math>~1 - \chi_0^2</math> |
| Linear | <math>~1-\chi_0</math> | <math>~\tfrac{48}{5}(\chi_0 - \tfrac{3}{4}\chi_0^2)</math> | <math>~\tfrac{1}{5} (1-\chi_0) (5 + 10\chi_0 - 9\chi_0^2)</math> | <math>~(1 + 2\chi_0 - \tfrac{9}{5}\chi_0^2)</math> |
| Parabolic | <math>~1-\chi_0^2</math> | <math>~5\chi_0 - 3\chi_0^3</math> | <math>~\tfrac{1}{2} (1-\chi_0^2) (2 - \chi_0^2)</math> | <math>~(1 - \tfrac{1}{2} \chi_0^2)</math> |
| <math>~n=1</math> Polytrope | <math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math> | <math>~\frac{2}{\pi\chi_0^2}\biggl[ \sin(\pi\chi_0) - \pi\chi_0 \cos(\pi\chi_0) \biggr]</math> | <math>~\frac{\sin(\pi\chi_0)}{\pi\chi_0}</math> | <math>~1</math> |
Leaning on this new expression for the ratio, <math>~B/A</math>, let's play with the form of the governing equation.
|
<math>~- \sigma^2 x </math> |
<math>~=</math> |
<math>~A \biggl\{ \frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) + \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \biggr\} </math> |
|
|
<math>~=</math> |
<math>~A\biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl\{ \biggl(\frac{P_0}{P_c}\biggr)\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) + \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \biggr\} </math> |
Polytropic Configurations
Let's compare this presentation of the LAWE to the form of the LAWE that has been derived specifically for polytropic equilibrium configurations, namely,
|
<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> |
<math>~=</math> |
<math>0 \, ,</math> |
where,
|
<math>~V(\xi)</math> |
<math>~\equiv</math> |
<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> |
[Note that <math>~\theta_c = 1</math> and, therefore for all practical purposes, it can be dropped. This notation was introduced in our separate discussion of the polytropic LAWE in order to make it clear how our derivations have overlapped earlier published work.] Regrouping terms, we have,
|
<math>~-\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta}\biggr]x</math> |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
Next we note that, written in terms of the traditional polytropic radial coordinate, <math>~\xi</math>, the fractional radius,
<math>~\chi_0 \equiv \frac{r_0}{R} = \frac{\xi}{\xi_1} = \frac{a_n \xi}{R} \, .</math>
Hence, multiplying the polytropic LAWE through by the quantity, <math>~(R/a_n)^2</math>, gives,
|
<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> |
<math>~=</math> |
<math>~-\biggl[\omega^2 \biggl(\frac{R^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta}\biggr]x</math> |
|
|
<math>~=</math> |
<math>~-\biggl(\frac{\theta_c}{\theta}\biggr)\sigma^2 x \, .</math> |
Finally, noting that, for polytropic configurations,
|
<math>~\frac{\theta}{\theta_c}</math> |
<math>~=</math> |
<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> |
we can rewrite the polytropic LAWE in the form,
|
<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> |
<math>~=</math> |
<math>~-\biggl( \frac{P_0}{P_c} \biggr)^{-1}\biggl( \frac{\rho_0}{\rho_c} \biggr)\sigma^2 x \, ,</math> |
which precisely matches the general expression for the LAWE presented at the end of our 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 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 dictated by,
Hydrostatic Balance
<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} = - \frac{GM_r}{r^2} </math> ,
As has been detailed in the last column of 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,
|
<math>~\frac{\rho}{\rho_c}</math> |
<math>~=</math> |
<math>~1 - a z - b z^2 \, ,</math> |
where both coefficients, <math>~a</math> and <math>~b</math>, are positive.
| ASIDE: Surface Location | |||||||||
|
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,
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,
We acknowledge, as well, that the density profile can now be written in terms of these roots; specifically,
|
This specified density profile implies a mass distribution,
|
<math>~M_r</math> |
<math>~=</math> |
<math>~4\pi R^3 \rho_c \int_0^z (1 - a z - b z^2)z^2 dz</math> |
|
|
<math>~=</math> |
<math>~4\pi R^3 \rho_c \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) \, .</math> |
The hydrostatic balance condition therefore implies,
|
<math>~\frac{1}{R\rho_c} \biggl( \frac{\rho}{\rho_c} \biggr)^{-1} \frac{dP}{dz}</math> |
<math>~=</math> |
<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> |
|
<math>~\Rightarrow ~~~~\biggl[ \frac{1}{4\pi G R^2 \rho_c^2 } \biggr] \frac{dP}{dz}</math> |
<math>~=</math> |
<math>~-\biggl( \frac{z}{3} - \frac{a z^2}{4} - \frac{b z^3}{5} \biggr)\biggl( \frac{\rho}{\rho_c} \biggr) </math> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
<math>\Rightarrow ~~~~\biggl( \frac{P}{P_n} \biggr)</math> |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<math>~ \biggl[ -\biggl(\frac{1}{6}\biggr)z^2 + z^3 \biggl( \frac{7a}{36} \biggr) + z^4 \biggl( \frac{8b}{60} - \frac{a^2}{16}\biggr) - z^5 \biggl( \frac{9ab}{100} \biggr) - z^6 \biggl(\frac{b^2}{30} \biggr) \biggr] + C \, . </math> |
where, <math>~C</math>, is an integration constant, and,
<math>~P_n \equiv 4\pi G \rho_c^2 R^2 \, .</math>
The integration constant — which also proves to be the normalized central pressure — is determined by ensuring that the pressure goes to zero at the surface of the configuration, <math>~z_s</math>. That is,
|
<math>~C = \frac{P_c}{P_n}</math> |
<math>~=</math> |
<math>~ - \biggl[ -\biggl(\frac{1}{6}\biggr)z_s^2 + z_s^3 \biggl( \frac{7a}{36} \biggr) + z_s^4 \biggl( \frac{8b}{60} - \frac{a^2}{16}\biggr) - z_s^5 \biggl( \frac{9ab}{100} \biggr) - z_s^6 \biggl(\frac{b^2}{30} \biggr) \biggr] \, . </math> |
Hence, the pressure profile is,
|
<math>\frac{P}{P_n} </math> |
<math>~=</math> |
<math>~ \biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{7a}{36} \biggr)(z_s^3 - z^3) - \biggl( \frac{8b}{60} - \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> |
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 above ASIDE, this means that <math>~z_s=a^{-1}</math>. So the pressure distribution is,
|
<math>\frac{P}{P_n} </math> |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<math>~a^{-2} \biggl\{ \frac{1}{6}[1 - (az)^2] - \frac{7}{36} [1 - (az)^3] + \frac{1}{16} [1 - (az)^4] \biggr\} \, . </math> |
And from the expression for the integration constant, we have,
|
<math>~\frac{P_c}{P_n}</math> |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<math>~a^{-2} \biggl[\frac{1}{6} - \frac{7}{36} + \frac{1}{16} \biggr] </math> |
|
|
<math>~=</math> |
<math>~\frac{5}{2^4\cdot 3^2 a^2} \, . </math> |
Hence, dividing one expression by the other, we obtain,
|
<math>\frac{P}{P_c} </math> |
<math>~=</math> |
<math>~\frac{1}{5}\biggl\{ 24[1 - (az)^2] - 28 [1 - (az)^3] + 9 [1 - (az)^4] \biggr\} \, . </math> |
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,
|
<math>~\frac{\rho}{\rho_c}</math> |
<math>~=</math> |
<math>~1 - a z \, .</math> |
Strategically rewriting the expression for the pressure distribution gives,
|
<math>\frac{P}{P_c} </math> |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{5+ 5(az) - 19 (az)^2 + 9 (az)^3 \biggr\} \, . </math> |
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,
|
<math>\frac{P}{P_c} </math> |
<math>~=</math> |
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr)^2 [5+ 10(az) - 9 (az)^2 ] \, . </math> |
Uniform Density
In the case of a uniform-density configuration, the governing equation is,
|
<math>~\sigma^2 x </math> |
<math>~=</math> |
<math>~2\chi_0 \biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - (1-\chi_0^2) \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, , </math> |
where,
<math>~\sigma^2 \equiv \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} = \frac{6}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr] \, .</math>
The following individual mode analyses should be compared with the results found in our discussion of Sterne's general solution.
Mode 0
Try an eigenfunction of the form,
<math>x = a_0\, ,</math>
in which case,
<math>x^' = x^{' '} = 0 \, .</math>
In order for this to be a solution, we must have,
|
<math>~\sigma^2 a_0 </math> |
<math>~=</math> |
<math>~2\chi_0 \biggl[ \frac{\alpha a_0}{\chi_0} \biggr] </math> |
|
<math>~\Rightarrow ~~~~ \frac{6}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr]</math> |
<math>~=</math> |
<math>~2\alpha = 2\biggl(3 - \frac{4}{\gamma_g}\biggr) </math> |
|
<math>~\Rightarrow ~~~~ \frac{\omega^2}{4\pi G\bar\rho}</math> |
<math>~=</math> |
<math>~\gamma_g - \frac{4}{3}\, . </math> |
Mode 2
Try an eigenfunction of the form,
<math>x = a_0 + a_2\chi_0^2 \, ,</math>
in which case,
<math>~x^' = 2 a_2\chi_0 </math> and <math>~x^{' '} = 2 a_2 \, . </math>
In order for this to be a solution, we must have,
|
<math>~\sigma^2 \biggl(a_0 + a_2\chi_0^2\biggr) </math> |
<math>~=</math> |
<math>~2 \biggl[ \alpha (a_0 + a_2\chi_0^2) + \chi_0 (2 a_2\chi_0 )\biggr] - (1-\chi_0^2) \biggl[ \frac{4(2 a_2\chi_0 )}{\chi_0}+ 2a_2 \biggr] </math> |
|
|
<math>~=</math> |
<math>~2\alpha a_0 + \chi_0^2[2a_2 (2+\alpha)] - 10a_2(1-\chi_0^2) </math> |
|
<math>~\Rightarrow ~~~~ \sigma^2 a_0 - 2\alpha a_0 + 10a_2</math> |
<math>~=</math> |
<math>~\chi_0^2 [-\sigma^2 + 2 (2+\alpha) + 10 ]a_2 \, . </math> |
Given that the coefficients on both sides of this expression must independently be zero, we have:
|
<math>~\sigma^2 </math> |
<math>~=</math> |
<math>~2 (2+\alpha) + 10</math> |
|
<math>~\Rightarrow ~~~~ \frac{\omega^2}{4\pi G\bar\rho} </math> |
<math>~=</math> |
<math>~\frac{\gamma_g}{6}\biggl[14 +2\biggl(3-\frac{4}{\gamma_g} \biggr)\biggr]</math> |
|
|
<math>~=</math> |
<math>~\frac{\gamma_g}{6}\biggl[20 -\frac{8}{\gamma_g}\biggr] = \frac{1}{3}\biggl(10\gamma_g - 4\biggr) \, ,</math> |
and
|
<math>~\frac{a_2}{a_0}</math> |
<math>~=</math> |
<math>~\frac{1}{10} \biggl[ 2\alpha - \sigma^2 \biggr] </math> |
|
|
<math>~=</math> |
<math>~\frac{1}{10} \biggl\{ 2\alpha - [14+2\alpha) ] \biggr\} = - \frac{7}{5} \, .</math> |
Parabolic Density Distribution
In the case of a parabolic density distribution, the governing equation is,
|
<math>~\sigma^2 x </math> |
<math>~=</math> |
<math>~(5\chi_0 - 3\chi_0^3)\biggl[ \frac{\alpha x}{\chi_0} + x^'\biggr] - \tfrac{1}{2} (1-\chi_0^2) (2 - \chi_0^2) \biggl[ \frac{4x^'}{\chi_0}+ x^{' '} \biggr] \, , </math> |
where,
<math>~\sigma^2 \equiv \frac{\tau_\mathrm{SSC}^2 \omega^2}{\gamma_g} = \frac{15}{\gamma_g}\biggl[\frac{\omega^2}{4\pi G\bar\rho}\biggr] \, .</math>
First Trial
Try an eigenfunction of the form,
<math>x = (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) \, ,</math>
in which case,
|
<math>~x^' </math> |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<math>~ 2\chi_0 (2-\chi_0^2)^{-2}[ (a + 2b)+ 4c\chi_0^2 -c \chi_0^4 ] \, ;</math> |
and,
|
<math>~x^{' '} </math> |
<math>~=</math> |
<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> |
|
|
|
<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> |
|
|
<math>~=</math> |
<math>~(2-\chi_0^2)^{-3} [ 2(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> |
|
|
|
<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> |
|
|
<math>~=</math> |
<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> |
|
|
|
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<math>~2 (2-\chi_0^2)^{-3} \{(2a+4b) + \chi_0^2(3a + 6b + 24c) + \chi_0^4(-6c) + \chi_0^6(c)\} </math> |
In order for this to be a solution, we must have,
|
<math>~\sigma^2 x </math> |
<math>~=</math> |
<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> |
|
<math>~\Rightarrow~~~~\sigma^2 (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) </math> |
<math>~=</math> |
<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> |
|
|
|
<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> |
Multiplying through by <math>~(2-\chi_0^2)^2</math> gives,
|
<math>~\sigma^2 (2-\chi_0^2) (a + b\chi_0^2 + c\chi_0^4) </math> |
<math>~=</math> |
<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> |
|
|
|
<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> |
|
<math>~\Rightarrow~~~~\sigma^2 [2a + \chi_0^2(2b-a) + \chi_0^4(2c -b) - c\chi_0^6]</math> |
<math>~=</math> |
<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> |
|
|
|
<math>~ + (5 - 3\chi_0^2)\biggl\{ (2a + 4b)\chi_0^2 + 8c\chi_0^4 - 2c \chi_0^6 \biggr\} </math> |
|
|
|
<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> |
|
|
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<math>~ - (1-\chi_0^2) \biggl\{ (2a+4b) + \chi_0^2(3a + 6b + 24c) -6c \chi_0^4 + c\chi_0^6 \biggr\} </math> |
|
|
<math>~=</math> |
<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> |
|
|
|
<math>~ - (1-\chi_0^2) \biggl\{ (10a + 20b)+ \chi_0^2[56c - a - 2b] +\chi_0^4 [-30c ]+ 5c \chi_0^6 \biggr\} </math> |
So, the coefficients of each even power of <math>~\chi_0^n</math> are:
| <math>~\chi_0^0</math> | : |
<math>~2a\sigma^2 - 10a\alpha +10a +20b =~a(2\sigma^2 - 10\alpha+10) + 20b</math> |
| <math>~\chi_0^2</math> | : |
<math>~(2b-a)\sigma^2 -5[(2b-a)\alpha + (2a + 4b) ] +6a\alpha+[56c - a - 2b] - (10a + 20b)</math> <math>=~a[-\sigma^2 -5(-\alpha + 2) +6\alpha-1-10] + b[2\sigma^2-10\alpha-20-2-20 ] + c[ 56]</math> <math>=~a[-\sigma^2 + 11\alpha - 21] + b[2\sigma^2-10\alpha-42 ] + 56 c</math> |
| <math>~\chi_0^4</math> | : |
<math>~(2c-b)\sigma^2 - 5[ (2c -b)\alpha + 8c] - 30c + 3[(2b-a)\alpha + (2a + 4b)] - [56c - a - 2b ] </math> <math>=~a[-3\alpha+7] + b[-\sigma^2+5\alpha+6\alpha+12+2] + c[2\sigma^2-10\alpha-40-30-56]</math> <math>=~a[7-3\alpha] + b[11\alpha-\sigma^2+14] + c[2\sigma^2-10\alpha-126]</math> |
| <math>~\chi_0^6</math> | : |
<math>~- c\sigma^2 +(10+5\alpha)c +3[ (2c -b)\alpha + 8c] +5c +30c</math> <math>=~b[-3\alpha] + c[-\sigma^2 + 10 + 5\alpha+6\alpha+24+35]</math> <math>=~b[-3\alpha] + c[11\alpha -\sigma^2 + 69]</math> |
| <math>~\chi_0^8</math> | : |
<math>~- 3(2+\alpha)c -5c =~c[-11-3\alpha]</math> |
Independent Investigation of Parabolic Distribution
In the specific case of a parabolic density distribution, the leading factor on the LHS is,
|
<math>~\frac{1}{A_\mathrm{parab}} \equiv \biggl(\frac{P_c}{P_0}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) </math> |
<math>~=</math> |
<math>~\frac{(1-\chi_0^2)}{(1-\chi_0^2)^2 (1-\tfrac{1}{2}\chi_0^2)} = \frac{1}{(1-\chi_0^2) (1-\tfrac{1}{2}\chi_0^2)} = \frac{2}{2 - 3\chi_0^2 + \chi_0^4} \, ,</math> |
and the function appearing on the RHS is,
|
<math>~\mu(\chi_0)</math> |
<math>~=</math> |
<math>~\frac{\chi_0^2(1-\chi_0^2)(5-3\chi_0^2)}{(1-\chi_0^2)^2 (1-\tfrac{1}{2}\chi_0^2)} = \frac{\chi_0^2 (5-3\chi_0^2)}{A_\mathrm{parab}} \, .</math> |
Multiplying the linear adiabatic wave equation through by <math>~A_\mathrm{parab}</math>, gives,
|
<math>~- \sigma^2 x </math> |
<math>~=</math> |
<math>~\frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) - \frac{B_\mathrm{parab} }{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, , </math> |
where,
<math> ~B_\mathrm{parab} \equiv \chi_0 (5-3\chi_0^2) \, . </math>
Now we note that,
|
<math>~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr]</math> |
<math>~=</math> |
<math>~ \frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl[ \chi_0^4 x^' \biggr] + \chi_0^4 x^' \frac{d}{d\chi_0}\biggl[ \frac{A_\mathrm{parab}}{\chi_0^4} \biggr] </math> |
|
<math>~\Rightarrow ~~~~\frac{A_\mathrm{parab}}{\chi_0^4} \frac{d}{d\chi_0}\biggl[ \chi_0^4 x^' \biggr] </math> |
<math>~=</math> |
<math>~ \frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \chi_0^4 x^' \frac{d}{d\chi_0}\biggl[ \frac{A_\mathrm{parab}}{\chi_0^4} \biggr] </math> |
|
|
<math>~=</math> |
<math>~ \frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \frac{\chi_0^4 x^'}{2} \frac{d}{d\chi_0}\biggl[ 1 - \frac{3}{\chi_0^2} + \frac{2}{\chi_0^4}\biggr] </math> |
|
|
<math>~=</math> |
<math>~ \frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] - \chi_0^4 x^' \biggl[ \frac{3}{\chi_0^3} - \frac{4}{\chi_0^5}\biggr] </math> |
|
|
<math>~=</math> |
<math>~ \frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] + \frac{x^'}{\chi_0} (4 - 3\chi_0^2 ) \, . </math> |
Similarly we note that,
|
<math>~\frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr]</math> |
<math>~=</math> |
<math>~ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \frac{d}{d\chi_0}\biggl[ \chi_0^\alpha x \biggr] + \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \biggr] </math> |
|
<math>~\Rightarrow ~~~~ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \frac{d}{d\chi_0}\biggl[ \chi_0^\alpha x \biggr] </math> |
<math>~=</math> |
<math>~ \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ \frac{B_\mathrm{parab}}{\chi_0^\alpha} \biggr] </math> |
|
|
<math>~=</math> |
<math>~ \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \frac{d}{d\chi_0}\biggl[ 5\chi_0^{1-\alpha} -3 \chi_0^{3-\alpha}\biggr] </math> |
|
|
<math>~=</math> |
<math>~ \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - \chi_0^\alpha x \biggl[ 5(1-\alpha)\chi_0^{-\alpha} -3 (3-\alpha)\chi_0^{2-\alpha}\biggr] </math> |
|
|
<math>~=</math> |
<math>~ \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] - x \biggl[ 5(1-\alpha) -3 (3-\alpha)\chi_0^{2}\biggr] \, . </math> |
Hence, the LAWE can be rewritten as,
|
<math>~- \sigma^2 x </math> |
<math>~=</math> |
<math>~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' \biggr] + \frac{x^'}{\chi_0} (4 - 3\chi_0^2 ) - \frac{d}{d\chi_0}\biggl[ B_\mathrm{parab}~x \biggr] + x \biggl[ 5(1-\alpha) -3 (3-\alpha)\chi_0^{2}\biggr] \, ; </math> |
then multiplying through by <math>~\chi_0</math>, and rearranging terms gives,
|
<math>~- x \biggl\{ [ 5(1-\alpha)+ \sigma^2]\chi_0 -3 (3-\alpha)\chi_0^{3} \biggr\}</math> |
<math>~=</math> |
<math>~\chi_0 ~\frac{d}{d\chi_0}\biggl[ A_\mathrm{parab}~x^' - B_\mathrm{parab}~x\biggr] + x^'(4 - 3\chi_0^2 ) \, . </math> |
Next, we note that,
|
<math>~ A_\mathrm{parab}~x^' </math> |
<math>~=</math> |
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - x \biggl[\frac{d}{d\chi_0}\biggl(A_\mathrm{parab}\biggr) \biggr]</math> |
|
|
<math>~=</math> |
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - \frac{x}{2} \biggl[\frac{d}{d\chi_0}\biggl(2 - 3\chi_0^2 + \chi_0^4\biggr) \biggr]</math> |
|
|
<math>~=</math> |
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) + x (3\chi_0 - 2\chi_0^3 )</math> |
|
<math>~\Rightarrow ~~~~A_\mathrm{parab}~x^' - B_\mathrm{parab}~x</math> |
<math>~=</math> |
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) + \biggl[(3\chi_0 - 2\chi_0^3 ) - ( 5\chi_0 - 3\chi_0^3 )\biggr] x</math> |
|
|
<math>~=</math> |
<math>~\frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - (2\chi_0 - \chi_0^3 ) x \, .</math> |
So, the LAWE becomes,
|
<math>~- x \biggl\{ [ 5(1-\alpha)+ \sigma^2]\chi_0 -3 (3-\alpha)\chi_0^{3} \biggr\}</math> |
<math>~=</math> |
<math>~\chi_0 ~\frac{d}{d\chi_0}\biggl[ \frac{d}{d\chi_0} \biggl( A_\mathrm{parab}~x \biggr) - (2\chi_0 - \chi_0^3 ) x\biggr] + x^'(4 - 3\chi_0^2 ) </math> |
|
|
<math>~=</math> |
<math>~ \chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) - \chi_0 ~\frac{d}{d\chi_0}\biggl[ (2\chi_0 - \chi_0^3 ) x\biggr] + x^'(4 - 3\chi_0^2 ) </math> |
|
|
<math>~=</math> |
<math>~ \chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) - \biggl\{ \frac{d}{d\chi_0}\biggl[ (2\chi_0^2 - \chi_0^4)x\biggr] - (2\chi_0 - \chi_0^3)x \biggr\} + \biggl\{ \frac{d}{d\chi_0}\biggl[ (4-3\chi_0^2)x\biggr] + 6\chi_0 x \biggr\} </math> |
|
|
<math>~=</math> |
<math>~ \chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) + \frac{d}{d\chi_0}\biggl[(4-3\chi_0^2)x -(2\chi_0^2 - \chi_0^4)x\biggr] + (2\chi_0 - \chi_0^3)x + 6\chi_0 x </math> |
|
|
<math>~=</math> |
<math>~ \chi_0 \frac{d^2}{d\chi_0^2} \biggl( A_\mathrm{parab}~x \biggr) + \frac{d}{d\chi_0}\biggl[(4-5\chi_0^2 + \chi_0^4)x \biggr] + (8\chi_0 - \chi_0^3)x \, . </math> |
Moving the last term on the RHS of this expression to the LHS, and factoring the polynomial coefficients of the terms inside of the first and second derivatives gives,
|
<math>~x \biggl\{ (5\alpha - 13 - \sigma^2)\chi_0 + (10 - 3\alpha)\chi_0^{3} \biggr\}</math> |
<math>~=</math> |
<math>~ \frac{\chi_0}{2} \frac{d^2}{d\chi_0^2} \biggl[ (1-\chi_0^2)(2-\chi_0^2)x \biggr] + \frac{d}{d\chi_0}\biggl[(1-\chi_0^2)(4-\chi_0^2)x \biggr] \, . </math> |
First Trial (Same as Above)
Try an eigenfunction of the form,
<math>x = (2-\chi_0^2)^{-1} (a + b\chi_0^2 + c\chi_0^4) \, ,</math>
in which case,
|
LHS |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
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<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> |
|
RHS (1st term) |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<math>~ \frac{\chi_0}{2} [ 2(b-a) + 12(c-b)\chi_0^2 - 30c\chi_0^4] </math> |
|
|
<math>~=</math> |
<math>~ (b-a)\chi_0 + 6(c-b)\chi_0^3 - 15c\chi_0^5 </math> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
RHS (2nd term) |
<math>~=</math> |
<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> |
|
|
<math>~=</math> |
<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> |
|
|
|
<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> |
|
|
<math>~=</math> |
<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> |
|
|
|
<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> |
|
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<math>~=</math> |
<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> |
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<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> |
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<math>~=</math> |
<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> |
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<math>~ - \biggl[\chi_0^2 (8b-10a) + \chi_0^4 (16c-20b+4a) + \chi_0^6 (6b-30c) + 8c\chi_0^8 \biggr] </math> |
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<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> |
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<math>~=</math> |
<math>~ \chi_0 (2-\chi_0^2)^{-2}\biggl\{[(16b-20a) + 8a] + \chi_0^2 [(32c-40b+8a) + (10a-8b) + (8b-10a)] </math> |
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<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> |
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<math>~=</math> |
<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> |
So, the coefficients of each even power of <math>~\chi_0^n</math> are:
| <math>~\chi_0^0</math> | : |
<math>~a(10\alpha - 26 - 2\sigma^2) - (4b-4a) - [16b-12a] ~=a(10\alpha - 10 - 2\sigma^2) - 20b</math> |
| <math>~\chi_0^2</math> | : |
<math>~[a(\sigma^2 -11\alpha + 33 ) + b(10\alpha - 26 - 2\sigma^2)] - [ 24c- 28b + 4a]- [(32c-40b+8a) ]</math> <math>= ~[a(\sigma^2 -11\alpha + 21 ) + b(10\alpha + 42 - 2\sigma^2)] - 56c</math> |
| <math>~\chi_0^4</math> | : |
<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> <math>~=~[ a (3\alpha -7) + b(\sigma^2 -11\alpha -14 ) + c(10\alpha + 126 - 2\sigma^2)]</math> |
| <math>~\chi_0^6</math> | : |
<math>~[ b (3\alpha -10) + c(\sigma^2 -11\alpha + 33 )] - [66c -6b ]-[36c -4b ]</math> <math>~=~[ b (3\alpha ) + c(\sigma^2 -11\alpha - 69 )] </math> |
| <math>~\chi_0^8</math> | : |
<math>~c(3\alpha -10) + 15c + 6c = c[3\alpha+11]</math> |
The expressions for the coefficients presented in this table exactly match the entire set of expressions derived earlier, except for the adopted sign convention — every term in this second derivation has the opposite sign to the corresponding term in the earlier derivation.
Conjecture
Returning to the generic formulation derived earlier, we have,
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<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) x </math> |
<math>~=</math> |
<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> |
Now, suppose that the expression on the RHS is of the form,
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<math>~\mathrm{RHS}</math> |
<math>~=</math> |
<math>~u dv + v du \, ,</math> |
where <math>~u \equiv (P_0/P_c)</math>? Then the function,
<math>~v \equiv \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, ,</math>
and the eigenvector, <math>~x</math>, must satisfy both of the relations:
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Relation I |
<math>~:</math> |
<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> |
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Relation II |
<math>~:</math> |
<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> |
The validity (or not) of this conjecture can be tested against both configurations whose
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© 2014 - 2021 by Joel E. Tohline |