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<math>~ | <math>~q^3 = \frac{(\rho_e/\rho_c)}{2[1-(\rho_e/\rho_c) ]} \, ;</math> | ||
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<math>~\frac{ | <math>~\sigma^2 = 3\biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[ (\alpha-1) \pm \sqrt{\alpha+1} \biggr] \, .</math> | ||
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Revision as of 03:56, 28 November 2016
Radial Oscillations of a Zero-Zero Bipolytrope
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Groundwork
In an accompanying discussion, we derived the so-called,
Adiabatic Wave (or Radial Pulsation) Equation
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<math>~ \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x = 0 </math> |
whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. According to our accompanying derivation, if the initial, unperturbed equilibrium configuration is an <math>~(n_c, n_e) = (0,0)</math> bipolytrope, then we know that the relevant functional profiles are as follows for the core and envelope, separately. Note that, throughout, we will preferentially adopt as the dimensionless radial coordinate, the parameter,
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<math>~\xi</math> |
<math>~\equiv</math> |
<math>~\frac{r}{r_i} \, ,</math> |
in which case,
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<math>~\chi</math> |
<math>~=</math> |
<math>~ \chi_i \xi = q \biggl( \frac{G\rho_c^2 R^2}{P_c} \biggr)^{1 /2 }\xi \, .</math> |
The corresponding radial coordinate range is,
<math>~0 \le \xi \le 1 </math> for the core, and
<math>~1 \le \xi \le \frac{1}{q} </math> for the envelope.
Core
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<math>~r_0</math> |
<math>~=</math> |
<math>~\biggl( \frac{P_c}{G\rho_c^2}\biggr)^{1 / 2} \chi = (qR) \xi \, ,</math> |
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<math>~\rho_0</math> |
<math>~=</math> |
<math>~\rho_c \, ,</math> |
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<math>~\frac{P_0}{P_c}</math> |
<math>~=</math> |
<math>~1 - \frac{2\pi}{3} \chi^2 = 1 - \frac{2\pi}{3} \biggl[ \frac{G\rho_c^2 R^2}{P_c} \biggr] q^2 \xi^2 = 1 - \frac{\xi^2}{g^2} \, ,</math> |
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<math>~M_r</math> |
<math>~=</math> |
<math>~\frac{4\pi}{3} \biggl( \frac{P_c^3}{G^3 \rho_c^4} \biggr)^{1 / 2}\chi^3 = \frac{4\pi}{3} \biggl( \frac{P_c^3}{G^3 \rho_c^4} \biggr)^{1 / 2} \biggl( \frac{G\rho_c^2 R^2}{P_c} \biggr)^{3 /2 } (q\xi)^3 </math> |
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<math>~=</math> |
<math>~ \frac{4\pi}{3} ( \rho_c R^3 ) (q\xi)^3 = \frac{4\pi}{3} (q\xi)^3 \rho_c \biggl[ \biggl( \frac{P_c}{G\rho_c^2} \biggr)^{1 / 2} \biggl( \frac{3}{2\pi} \biggr)^{1 / 2} \frac{1}{qg}\biggr]^3 </math> |
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<math>~=</math> |
<math>~ \frac{4\pi}{3} (q\xi)^3 \biggl[ \biggl( \frac{P_c^3}{G^3\rho_c^4} \biggr)^{1 / 2} \biggl( \frac{3}{2\pi} \biggr)^{3 / 2} \frac{1}{q^3g^3}\biggr] = \frac{4\pi}{3} \biggl[ \biggl(\frac{\pi}{6}\biggr)^{1 / 2} \nu g^3 M_\mathrm{tot} \biggl( \frac{3}{2\pi} \biggr)^{3 / 2} \frac{1}{g^3}\biggr]\xi^3 </math> |
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<math>~=</math> |
<math>~ M_\mathrm{tot} \nu \xi^3 \, , </math> |
where,
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<math>~g^2(\nu,q)</math> |
<math>~\equiv</math> |
<math> \biggl\{ 1 + \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( 1-q \biggr) + \frac{\rho_e}{\rho_c} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \biggr\} \, , </math> |
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<math>~\frac{\rho_e}{\rho_c}</math> |
<math>~=</math> |
<math> \frac{q^3}{\nu} \biggl( \frac{1-\nu}{1-q^3}\biggr) \, . </math> |
Hence,
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<math>~g_0</math> |
<math>~=</math> |
<math>~\frac{G(M_\mathrm{tot} \nu \xi^3)}{(qR\xi)^2} = \biggl( \frac{GM_\mathrm{tot} }{R^2 } \biggr) \frac{\nu \xi}{q^2} </math> |
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<math>~=</math> |
<math>~ G \biggl[\biggl( \frac{P_c^3}{G^3\rho_c^4} \biggr)^{1 / 2} \biggl(\frac{6}{\pi}\biggr)^{1 / 2} \frac{1}{\nu g^3} \biggr] \biggl[\biggl(\frac{G\rho_c^2}{P_c} \biggr)^{ 1 / 2} \biggl(\frac{2\pi}{3} \biggr)^{1 / 2} qg \biggr]^2 \frac{\nu \xi}{q^2} </math> |
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<math>~=</math> |
<math>~ (P_c G)^{1 / 2} \biggl(\frac{2^3\pi}{3} \biggr)^{1 / 2} \frac{\xi}{g} </math> |
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<math>~\frac{\rho_0}{P_0}</math> |
<math>~=</math> |
<math>~ \frac{\rho_c}{P_c} \biggl[ 1 - \frac{\xi^2}{g^2} \biggr]^{-1} = \frac{\rho_c}{P_c} \biggl( \frac{g^2}{g^2 - \xi^2} \biggr) \, ;</math> |
and the wave equation for the core becomes,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{1}{(qR)^2} \cdot \frac{d^2x}{d\xi^2} + \biggl[\frac{4qR}{r_0} - \biggl(\frac{qR g_0 \rho_0}{P_0}\biggr) \biggr] \frac{1}{(qR)^2} \cdot \frac{dx}{d\xi} + \biggl(\frac{\rho_0}{P_0} \biggr)\biggl[ \frac{\omega^2}{\gamma_\mathrm{g} } + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g} } \biggr)\frac{g_0}{r_0} \biggr] x </math> |
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<math>~=</math> |
<math>~ \frac{1}{(qR)^2} \biggl\{ \frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - q\biggl(\frac{P_c}{G\rho_c^2} \biggr)^{1 / 2}\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \frac{1}{qg} (P_c G)^{1 / 2} \biggl(\frac{2^3\pi}{3} \biggr)^{1 / 2} \frac{\xi}{g} \frac{\rho_c}{P_c} \biggl( \frac{g^2}{g^2 - \xi^2} \biggr) \biggr] \frac{dx}{d\xi} \biggr\} </math> |
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<math>~ + \frac{\rho_c}{P_c} \biggl( \frac{g^2}{g^2 - \xi^2} \biggr) \biggl[ \frac{\omega^2}{\gamma_\mathrm{g} } + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g} } \biggr)(P_c G)^{1 / 2} \biggl(\frac{2^3\pi}{3} \biggr)^{1 / 2} \frac{\xi}{g} \cdot \frac{1}{qR\xi}\biggr] x </math> |
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<math>~=</math> |
<math>~ \frac{1}{(qR)^2} \biggl\{ \frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \biggl( \frac{2\xi}{g^2 - \xi^2} \biggr) \biggr] \frac{dx}{d\xi} \biggr\} </math> |
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<math>~ + \frac{\rho_c}{P_c} \biggl( \frac{g^2}{g^2 - \xi^2} \biggr) \biggl[ \frac{\omega^2}{\gamma_\mathrm{g} } + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g} } \biggr)(P_c G)^{1 / 2} \biggl(\frac{2^3\pi}{3} \biggr)^{1 / 2} \frac{1}{qg} \biggl(\frac{G\rho_c^2}{P_c} \biggr)^{1 / 2} \biggl( \frac{2\pi}{3} \biggr)^{1 / 2} qg \biggr] x </math> |
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<math>~=</math> |
<math>~ \frac{1}{(qR)^2(g^2 - \xi^2)} \biggl\{ (g^2 - \xi^2)\frac{d^2x}{d\xi^2} + ( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \frac{q^2 g^2 R^2 \rho_c}{P_c} \biggl[ \frac{\omega^2}{\gamma_\mathrm{g} } + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g} } \biggr) \frac{4\pi G\rho_c}{3} \biggr] x \biggr\} </math> |
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<math>~=</math> |
<math>~ \frac{1}{(qR)^2(g^2 - \xi^2)} \biggl\{ (g^2 - \xi^2)\frac{d^2x}{d\xi^2} + ( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 2\biggl[ \frac{3\omega^2}{\gamma_\mathrm{g}4\pi G\rho_c} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g} } \biggr) \biggr] x \biggr\} \, . </math> |
Envelope
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<math>~r_0</math> |
<math>~=</math> |
<math>~ (qR) \xi \, ,</math> |
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<math>~\rho_0</math> |
<math>~=</math> |
<math>~\rho_e \, ,</math> |
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<math>~\frac{P_0}{P_c}</math> |
<math>~=</math> |
<math> 1 - \frac{2\pi}{3}\chi_i^2 + \frac{2\pi}{3} \biggl(\frac{\rho_e}{\rho_c}\biggr) \chi_i^2 \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{1}{\xi} - 1\biggr) - \frac{\rho_e}{\rho_c} (\xi^2 - 1) \biggr] </math> |
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|
<math>~=</math> |
<math> 1 - \frac{1}{g^2}\biggl\{ 1 - \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{1}{\xi} - 1\biggr) - \frac{\rho_e}{\rho_c} (\xi^2 - 1) \biggr] \biggr\} </math> |
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<math>~\Rightarrow ~~~ \frac{g^2 P_0}{P_c}</math> |
<math>~=</math> |
<math> g^2 - 1 + \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{1}{\xi} - 1\biggr) - \frac{\rho_e}{\rho_c} (\xi^2 - 1) \biggr] \, , </math> |
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<math>~M_r</math> |
<math>~=</math> |
<math>\frac{4\pi}{3} \biggl[ \frac{P_c^3}{G^3 \rho_c^4} \biggr]^{1/2} \chi_i^3\biggl[1 +\frac{\rho_e}{\rho_c} \biggl( \xi^3 - 1\biggr) \biggr]</math> |
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<math>~=</math> |
<math>M_\mathrm{tot} \frac{4\pi}{3} \biggl[\biggl( \frac{\pi}{6}\biggr)^{1 / 2}\nu g^3 \biggr] \biggl[ \biggr(\frac{3}{2\pi}\biggr)\frac{1}{g^2} \biggr]^{3 /2} \biggl[1 +\frac{\rho_e}{\rho_c} \biggl( \xi^3 - 1\biggr) \biggr] </math> |
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<math>~=</math> |
<math> \nu M_\mathrm{tot} \biggl[1 +\frac{\rho_e}{\rho_c} \biggl( \xi^3 - 1\biggr) \biggr] \, . </math> |
Hence,
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<math>~g_0</math> |
<math>~=</math> |
<math>~ \frac{G M_\mathrm{tot}\nu }{ R^2 q^2\xi^2} \biggl[1 +\frac{\rho_e}{\rho_c} \biggl( \xi^3 - 1\biggr) \biggr] \, , </math> |
and, after multiplying through by <math>~(q^2 R^2 g^2P_0/P_c)</math>, the wave equation for the envelope becomes,
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<math>~0</math> |
<math>~=</math> |
<math>~\frac{q^2 g^2 R^2 P_0}{P_c} \biggl\{ \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} \biggr\} + \frac{q^2 g^2 R^2 \rho_0}{P_c} \biggl[ \frac{\omega^2 }{\gamma_\mathrm{g}} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g}} \biggr)\frac{g_0}{r_0} \biggr] x </math> |
|
|
<math>~=</math> |
<math>~\frac{g^2 P_0}{P_c} \biggl\{ \frac{d^2x}{d\xi^2} + \biggl[4 - \biggl(\frac{qRg_0 \rho_e}{P_0}\biggr) \xi\biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} \biggr\} + \frac{q^2 g^2 R^2 \rho_e}{P_c} \biggl[ \frac{\omega^2 }{\gamma_\mathrm{g}} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g}} \biggr)\frac{g_0}{r_0} \biggr] x </math> |
|
|
<math>~=</math> |
<math>~ \frac{g^2 P_0}{P_c} \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - \biggl(\frac{qg^2Rg_0 \rho_e}{P_c}\biggr) \frac{dx}{d\xi} </math> |
|
|
|
<math>~ + 2\biggl(\frac{\rho_e}{\rho_c}\biggr) \frac{3}{4\pi G \rho_c} \biggl\{ \frac{\omega^2 }{\gamma_\mathrm{g}} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g}} \biggr)\biggl(\frac{4\pi G \rho_c}{3}\biggr) \biggl[ \frac{1}{\xi^3} + \frac{\rho_e}{\rho_c}\biggl(1-\frac{1}{\xi^3}\biggr) \biggr] \biggr\} x </math> |
|
|
<math>~=</math> |
<math>~ \frac{g^2 P_0}{P_c} \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - 2 \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[1 +\frac{\rho_e}{\rho_c} \biggl( \xi^3 - 1\biggr) \biggr] \frac{1}{\xi^2} \cdot \frac{dx}{d\xi} </math> |
|
|
|
<math>~ + 2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl\{ \frac{3\omega^2 }{4\pi G\rho_c \gamma_\mathrm{g}} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g}} \biggr) \biggl[ \frac{1}{\xi^3} + \frac{\rho_e}{\rho_c}\biggl(1-\frac{1}{\xi^3}\biggr) \biggr] \biggr\} x </math> |
Check1
If <math>~\rho_e/\rho_c = 1</math>, this envelope wave equation should match seamlessly into the core wave equation. Let's see if it does. First,
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<math>~g^2(\nu,q)|_{\rho_e=\rho_c}</math> |
<math>~=</math> |
<math> 1 + \biggl(\frac{1}{q^2} - 1\biggr) =\frac{1}{q^2} \, , </math> |
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<math>~\frac{g^2 P_0}{P_c} \biggr|_{\rho_e = \rho_c}</math> |
<math>~=</math> |
<math>~ g^2 - \xi^2 = \frac{1}{q^2} - \xi^2 \, . </math> |
Hence, for the envelope,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{g^2 P_0}{P_c} \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - 2 \biggl[1 + \biggl( \xi^3 - 1\biggr) \biggr] \frac{1}{\xi^2} \cdot \frac{dx}{d\xi} </math> |
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<math>~ + 2 \biggl\{ \frac{3\omega^2 }{4\pi G\rho_c \gamma_\mathrm{g}} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g}} \biggr) \biggl[ \frac{1}{\xi^3} + \biggl(1-\frac{1}{\xi^3}\biggr) \biggr] \biggr\} x </math> |
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<math>~=</math> |
<math>~ \biggl( \frac{1}{q^2} - \xi^2 \biggr) \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - 2\xi \cdot \frac{dx}{d\xi} + 2 \biggl\{ \frac{3\omega^2 }{4\pi G\rho_c \gamma_\mathrm{g}} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g}} \biggr) \biggr\} x </math> |
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<math>~=</math> |
<math>~ \biggl( \frac{1}{q^2} - \xi^2 \biggr) \frac{d^2x}{d\xi^2} + \biggl\{ 4\biggl( \frac{1}{q^2} - \xi^2 \biggr) - 2\xi^2 \biggr\} \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 2 \biggl[ \frac{3\omega^2 }{4\pi G\rho_c \gamma_\mathrm{g}} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g}} \biggr) \biggr] x </math> |
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<math>~=</math> |
<math>~ \biggl( \frac{1}{q^2} - \xi^2 \biggr) \frac{d^2x}{d\xi^2} + \biggl( \frac{4}{q^2} - 6\xi^2 \biggr) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 2 \biggl[ \frac{3\omega^2 }{4\pi G\rho_c \gamma_\mathrm{g}} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g}} \biggr) \biggr] x \, . </math> |
Whereas, for the core,
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<math>~0</math> |
<math>~=</math> |
<math>~ \biggl(\frac{1}{q^2} - \xi^2 \biggr)\frac{d^2x}{d\xi^2} + \biggl( \frac{4}{q^2} - 6\xi^2 \biggr) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 2\biggl[ \frac{3\omega^2}{\gamma_\mathrm{g}4\pi G\rho_c} + \biggl( \frac{4 - 3\gamma_\mathrm{g}}{\gamma_\mathrm{g} } \biggr) \biggr] x \, , </math> |
which matches exactly.
Attempt to Solve
Adopting some of the notation used by T. E. Sterne (1937) and enunciated in our accompanying discussion of the uniform-density sphere, we'll define,
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<math>~\alpha</math> |
<math>~\equiv</math> |
<math>~3 - 4/\gamma_\mathrm{g} \, ,</math> |
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<math>~\mathfrak{F}</math> |
<math>~\equiv</math> |
<math>~\frac{3\omega^2 }{2\pi \gamma_\mathrm{g} G \rho_c} - 2 \alpha \, ,</math> |
in which case the wave equation for the core becomes,
|
<math>~0</math> |
<math>~=</math> |
<math>~ \frac{1}{(qR)^2(g^2 - \xi^2)} \biggl\{ (g^2 - \xi^2)\frac{d^2x}{d\xi^2} + ( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \mathfrak{F} x \biggr\} \, , </math> |
and the wave equation for the envelope becomes,
|
<math>~0</math> |
<math>~=</math> |
<math>~ \frac{g^2 P_0}{P_c} \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - 2 \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[1 +\frac{\rho_e}{\rho_c} \biggl( \xi^3 - 1\biggr) \biggr] \frac{1}{\xi^2} \cdot \frac{dx}{d\xi} </math> |
|
|
|
<math>~ + \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl\{ \mathfrak{F} + 2\alpha \biggl[1 - \frac{1}{\xi^3} - \frac{\rho_e}{\rho_c}\biggl(1-\frac{1}{\xi^3}\biggr) \biggr] \biggr\} x \, . </math> |
A Specific Choice of the Density Ratio
Now, let's focus on the specific model for which <math>~\rho_e/\rho_c = 1/2</math>. In this case,
|
<math>~g^2(\nu,q) \biggr|_{\rho_e/\rho_c=1/2}</math> |
<math>~=</math> |
<math> 1 + \frac{1}{2} \biggl[ 1-q + \frac{1}{2} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] </math> |
|
|
<math>~=</math> |
<math>\frac{1}{4q^2}\biggl\{ 4q^2 + \biggl[ 2q^2 - 2q^3 + 1-q^2 \biggr] \biggr\} </math> |
|
|
<math>~=</math> |
<math>~ \biggl[ \frac{1+5q^2 - 2q^3 }{4q^2} \biggr] \, ; </math> |
|
<math>~\frac{g^2 P_0}{P_c}\biggr|_{\rho_e/\rho_c=1/2}</math> |
<math>~=</math> |
<math> g^2 - 1 + \frac{1}{2} \biggl[ \biggl( \frac{1}{\xi} - 1\biggr) - \frac{1}{2} \biggl(\xi^2 - 1 \biggr) \biggr] </math> |
|
|
<math>~=</math> |
<math> g^2 - 1 - \frac{1}{4} \biggl[ \xi^2 + 1 - \frac{2}{\xi} \biggr] </math> |
|
|
<math>~=</math> |
<math> g^2 - \frac{\xi^2}{4} \biggl[ 1 + \frac{5}{\xi^2} - \frac{2}{\xi^3} \biggr] \, . </math> |
Note that this last expression goes to zero at the surface of the bipolytrope, that is, at <math>~\xi = 1/q</math>. For this specific case, the wave equation for the envelope becomes,
|
<math>~0</math> |
<math>~=</math> |
<math>~ \frac{g^2 P_0}{P_c} \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - \biggl[1 +\frac{1}{2} \biggl( \xi^3 - 1\biggr) \biggr] \frac{1}{\xi^2} \cdot \frac{dx}{d\xi} + \frac{1}{2} \biggl\{ \mathfrak{F} + 2\alpha \biggl[1 - \frac{1}{\xi^3} + \frac{1}{2}\biggl(-1 + \frac{1}{\xi^3}\biggr) \biggr] \biggr\} x </math> |
|
|
<math>~=</math> |
<math>~ \biggl\{ g^2 - \frac{\xi^2}{4} \biggl[ 1 + \frac{5}{\xi^2} - \frac{2}{\xi^3} \biggr] \biggr\} \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - \frac{1}{2}\biggl[1 + \xi^3 \biggr] \frac{1}{\xi^2} \cdot \frac{dx}{d\xi} + \frac{1}{2} \biggl\{ \mathfrak{F} + \alpha \biggl[1 - \frac{1}{\xi^3} \biggr] \biggr\} x </math> |
|
|
<math>~=</math> |
<math>~\frac{1}{4\xi^3} \biggl\{ \biggl[ 4g^2\xi^3 - \xi^5 \biggl( 1 + \frac{5}{\xi^2} - \frac{2}{\xi^3} \biggr) \biggr] \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - 2\xi (1 + \xi^3 ) \frac{dx}{d\xi} + 2 \xi^3 \biggl[ \mathfrak{F} + \alpha \biggl(1 - \frac{1}{\xi^3} \biggr) \biggr] x \biggr\} </math> |
|
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<math>~=</math> |
<math>~\frac{1}{4\xi^3} \biggl\{ \biggl[ 4g^2\xi^3 - \xi^5 - 5\xi^3 + 2\xi^2 \biggr] \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - 2\xi (1 + \xi^3 ) \frac{dx}{d\xi} + \biggl[ 2 \xi^3 (\mathfrak{F} + \alpha) - 2\alpha \biggr] x \biggr\} </math> |
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<math>~=</math> |
<math>~\frac{1}{4\xi^3} \biggl\{ \biggl[ 2 + (4g^2 - 5)\xi - \xi^3 \biggr] \biggl[ \xi^2 \cdot \frac{d^2x}{d\xi^2} + 4\xi \cdot \frac{dx}{d\xi} \biggr] - 2(1 + \xi^3 ) \biggl[ \xi \cdot \frac{dx}{d\xi} \biggr] - \biggl[ 2\alpha - 2 \xi^3 (\mathfrak{F} + \alpha) \biggr] x \biggr\} </math> |
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<math>~=</math> |
<math>~\frac{1}{4\xi^3} \biggl\{ \biggl[ 2 + (4g^2 - 5)\xi - \xi^3 \biggr] \biggl[ \xi^2 \cdot \frac{d^2x}{d\xi^2} \biggr] +\biggl[ 3 + (8g^2 - 10)\xi - 3\xi^3 \biggr] \biggl[ 2\xi \cdot \frac{dx}{d\xi} \biggr] - \biggl[ 2\alpha - 2 \xi^3 (\mathfrak{F} + \alpha) \biggr] x \biggr\} </math> |
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<math>~=</math> |
<math>~\frac{x}{4\xi^3} \biggl\{ \biggl[ 2 + (4g^2 - 5)\xi - \xi^3 \biggr] \biggl[ \frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} \biggr] +\biggl[ 6 + 4(4g^2 - 5)\xi - 6\xi^3 \biggr] \biggl[ \frac{\xi}{x} \cdot \frac{dx}{d\xi} \biggr] - \biggl[ 2\alpha - 2 \xi^3 (\mathfrak{F} + \alpha) \biggr] \biggr\} \, . </math> |
Idea Involving Logarithmic Derivatives
Notice that the term involving the first derivative of <math>~x</math> can be written as a logarithmic derivative; specifically,
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<math>~\frac{\xi}{x} \cdot \frac{dx}{d\xi} </math> |
<math>~=</math> |
<math>~\frac{d\ln x}{d\ln \xi} \, .</math> |
Let's look at the second derivative of this quantity.
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<math>~\frac{d}{d\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr]</math> |
<math>~=</math> |
<math>~ \frac{\xi}{x} \cdot \frac{d^2x}{d\xi^2} + \frac{dx}{d\xi} \cdot \biggl[ \frac{1}{x} - \frac{\xi}{x^2} \cdot \frac{dx}{d\xi}\biggr] </math> |
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<math>~=</math> |
<math>~ \frac{\xi}{x} \cdot \frac{d^2x}{d\xi^2} + \frac{1}{x} \biggl[ 1 - \frac{d\ln x}{d\ln \xi} \biggr]\cdot \frac{dx}{d\xi} </math> |
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<math>~\Rightarrow ~~~ \frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} </math> |
<math>~=</math> |
<math>~ \frac{d}{d\ln\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr] - \biggl[ 1 - \frac{d\ln x}{d\ln \xi} \biggr]\cdot \frac{d\ln x}{d\ln \xi} \, . </math> |
Now, if we assume that the envelope's eigenfunction is a power-law of <math>~\xi</math>, that is, assume that,
<math>~x = a_0 \xi^{c_0} \, ,</math>
then the logarithmic derivative of <math>~x</math> is a constant, namely,
<math>~\frac{d\ln x}{d\ln\xi} = c_0 \, ,</math>
and the two key derivative terms will be,
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<math>~\frac{\xi}{x} \cdot \frac{dx}{d\xi} = c_0 \, ,</math> |
and |
<math>~\frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} = c_0(c_0-1) \, .</math> |
Hence, in order for the wave equation for the envelope for the specific density ratio being considered here to be satisfied, we need,
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<math>~0</math> |
<math>~=</math> |
<math>~ \biggl[ 2 + (4g^2 - 5)\xi - \xi^3 \biggr] \biggl[ c_0(c_0-1) \biggr] +\biggl[ 6 + 4(4g^2 - 5)\xi - 6\xi^3 \biggr] \biggl[ c_0 \biggr] - \biggl[ 2\alpha - 2 \xi^3 (\mathfrak{F} + \alpha) \biggr] </math> |
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<math>~=</math> |
<math>~ \biggl[ 2 + (4g^2 - 5)\xi - \xi^3 \biggr] c_0(c_0-1) +c_0\biggl[ 6 + 4(4g^2 - 5)\xi - 6\xi^3 \biggr] - \biggl[ 2\alpha - 2 \xi^3 (\mathfrak{F} + \alpha) \biggr] </math> |
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<math>~=</math> |
<math>~ \biggl[2c_0(c_0-1) + 6c_0 - 2\alpha \biggr] + \biggl[(4g^2-5)(c_0^2 - c_0 + 4c_0 ) \biggr]\xi + \biggl[ -c_0(c_0-1) -6c_0 + 2(\mathfrak{F}+\alpha) \biggr]\xi^3 </math> |
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<math>~=</math> |
<math>~ 2\biggl[c_0^2 + 2c_0 - \alpha \biggr] + \biggl[(4g^2-5)(c_0^2 + 3c_0 ) \biggr]\xi + \biggl[ 2(\mathfrak{F}+\alpha) - c_0(c_0+5) \biggr]\xi^3 \, . </math> |
This means that three algebraic relations must simultaneously be satisfied, namely:
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<math>~\xi^{0}:</math> |
<math>~c_0^2 + 2c_0 - \alpha =0</math> |
<math>~\Rightarrow~</math> |
<math>~c_0 = -1 \pm (1+\alpha)^{1 / 2} \, ;</math> |
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<math>~\xi^{1}:</math> |
<math>~g^2 = \frac{5}{4}</math> |
<math>~\Rightarrow~</math> |
<math>~q=\biggl(\frac{1}{2}\biggr)^{1 / 3} </math> and, hence, <math>~\nu = \frac{2}{3} \, ;</math> |
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<math>~\xi^{3}:</math> |
<math>~2(\mathfrak{F}+\alpha) = c_0(c_0+5)</math> |
<math>~\Rightarrow~</math> |
<math>~\frac{2}{3}\cdot \sigma^2 = (\alpha-1) \pm \sqrt{\alpha+1} \, .</math> |
More General Solution
Leaving the density ratio unspecified, let's try to write the wave equation for the envelope in the same form, and see if the logarithmic derivatives can be manipulated in a similar fashion.
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{g^2 P_0}{P_c} \biggl[ \frac{d^2x}{d\xi^2} + \frac{4}{\xi} \cdot \frac{dx}{d\xi} \biggr] - 2 \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[1 +\frac{\rho_e}{\rho_c} \biggl( \xi^3 - 1\biggr) \biggr] \frac{1}{\xi^2} \cdot \frac{dx}{d\xi} </math> |
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<math>~ + \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl\{ \mathfrak{F} + 2\alpha \biggl[1 - \frac{1}{\xi^3} - \frac{\rho_e}{\rho_c}\biggl(1-\frac{1}{\xi^3}\biggr) \biggr] \biggr\} x </math> |
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<math>~\Rightarrow ~~~\frac{\xi^3}{x} \cdot 0</math> |
<math>~=</math> |
<math>~ \frac{g^2\xi P_0}{P_c} \biggl[ \frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} \biggr] + \biggl\{ \frac{4g^2 \xi P_0}{P_c} - 2 \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[\biggl( 1-\frac{\rho_e}{\rho_c} \biggr) +\biggl(\frac{\rho_e}{\rho_c} \biggr) \xi^3 \biggr] \biggr\} \frac{\xi}{x} \cdot \frac{dx}{d\xi} </math> |
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<math>~ +\xi^3 \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl\{ \biggl[ \mathfrak{F} + 2\alpha -2\alpha\frac{\rho_e}{\rho_c} \biggr] + 2\alpha\biggl( \frac{\rho_e}{\rho_c} -1\biggr) \cdot \frac{1}{\xi^3} \biggr\} </math> |
where,
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<math>~\frac{g^2\xi P_0}{P_c}</math> |
<math>~=</math> |
<math> \xi \biggl\{ g^2 - 1 + \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{1}{\xi} - 1\biggr) - \frac{\rho_e}{\rho_c} (\xi^2 - 1) \biggr] \biggr\} </math> |
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<math>~=</math> |
<math> \xi \biggl\{ \biggl[ 2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggr] \frac{1}{\xi} +\biggl[ g^2 - 1 - 2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) + \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \biggr] - \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^2 \biggr\} </math> |
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<math>~=</math> |
<math> \biggl[ 2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggr] +\biggl[ g^2 - 1 - 2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) + \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \biggr]\xi - \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^3 </math> |
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<math>~=</math> |
<math> \biggl[ 2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggr] +\biggl[ g^2 - 1 - 2\biggl(\frac{\rho_e}{\rho_c}\biggr) + 3\biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \biggr]\xi - \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^3 \, . </math> |
Hence, the wave equation becomes,
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<math>~0</math> |
<math>~=</math> |
<math>~ \biggl[ \mathcal{A} + (g^2-\mathcal{B}) \xi - \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^3\biggr] \biggl[ \frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} \biggr] + \biggl\{ 4\biggl[ \mathcal{A} + (g^2-\mathcal{B}) \xi - \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^3\biggr] - \mathcal{A} - 2\biggl(\frac{\rho_e}{\rho_c} \biggr)^2 \xi^3 \biggr\} \frac{\xi}{x} \cdot \frac{dx}{d\xi} </math> |
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<math>~ + \biggl[ \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl( \mathfrak{F} + 2\alpha -2\alpha\frac{\rho_e}{\rho_c} \biggr)\xi^3 + 2\alpha\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl( \frac{\rho_e}{\rho_c} -1\biggr) \biggr] </math> |
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<math>~=</math> |
<math>~ \biggl[ \mathcal{A} + (g^2-\mathcal{B}) \xi - \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^3\biggr] \biggl[ \frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} \biggr] + \biggl\{ 3\mathcal{A} + 4(g^2-\mathcal{B}) \xi - 6\biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^3 \biggr\} \frac{\xi}{x} \cdot \frac{dx}{d\xi} </math> |
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<math>~ + \biggl[ \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl( \mathfrak{F} + 2\alpha -2\alpha\frac{\rho_e}{\rho_c} \biggr)\xi^3 -\alpha \mathcal{A} \biggr] \, , </math> |
where,
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<math>~\mathcal{A}</math> |
<math>~\equiv</math> |
<math>~2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \, ; </math> |
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<math>~\mathcal{B}</math> |
<math>~\equiv</math> |
<math>~1 + 2\biggl(\frac{\rho_e}{\rho_c}\biggr) - 3\biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \, . </math> |
As before, if we assume a power-law solution, the wave equation for the envelope becomes,
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<math>~0</math> |
<math>~=</math> |
<math>~ \biggl[ \mathcal{A} + (g^2-\mathcal{B}) \xi - \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^3\biggr] \biggl[ c_0(c_0-1) \biggr] + \biggl\{ 3\mathcal{A} + 4(g^2-\mathcal{B}) \xi - 6\biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \xi^3 \biggr\} c_0 </math> |
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<math>~ + \biggl[ \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl( \mathfrak{F} + 2\alpha -2\alpha\frac{\rho_e}{\rho_c} \biggr)\xi^3 -\alpha \mathcal{A} \biggr] </math> |
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<math>~=</math> |
<math>~\xi^0 \biggl[ \mathcal{A}c_0(c_0-1) + 3\mathcal{A}c_0 -\alpha\mathcal{A} \biggr] + \xi^1 \biggl[ (g^2-\mathcal{B})c_0(c_0-1) +4(g^2-\mathcal{B})c_0 \biggr] + \xi^3 \biggl[\biggl(\frac{\rho_e}{\rho_c}\biggr)^2c_0(1-c_0) - 6\biggl(\frac{\rho_e}{\rho_c}\biggr)^2c_0 +\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl( \mathfrak{F} + 2\alpha -2\alpha\frac{\rho_e}{\rho_c} \biggr) \biggr] </math> |
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<math>~=</math> |
<math>~\xi^0 \biggl[ c_0(c_0-1) + 3c_0 -\alpha \biggr]\mathcal{A} + \xi^1 \biggl[ (g^2-\mathcal{B})(c_0^2+3c_0)\biggr] + \xi^3 \biggl[( \mathfrak{F} + 2\alpha ) - \biggl(\frac{\rho_e}{\rho_c}\biggr)(5c_0 +c_0^2 + 2\alpha) \biggr]\biggl(\frac{\rho_e}{\rho_c}\biggr) \, .
</math> |
This means that three algebraic relations must simultaneously be satisfied, namely:
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<math>~\xi^{0}:</math> |
<math>~c_0^2 + 2c_0 - \alpha =0</math> |
<math>~\Rightarrow~</math> |
<math>~c_0 = -1 \pm (1+\alpha)^{1 / 2} \, ;</math> |
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<math>~\xi^{1}:</math> |
<math>~g^2 = 1 + 2\biggl(\frac{\rho_e}{\rho_c}\biggr) - 3\biggl(\frac{\rho_e}{\rho_c}\biggr)^2</math> |
<math>~\Rightarrow~</math> |
<math>~q^3 = \frac{(\rho_e/\rho_c)}{2[1-(\rho_e/\rho_c) ]} \, ;</math> |
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<math>~\xi^{3}:</math> |
<math>~(\mathfrak{F}+2\alpha) = \biggl(\frac{\rho_e}{\rho_c}\biggr)(5c_0 +c_0^2 + 2\alpha)</math> |
<math>~\Rightarrow~</math> |
<math>~\sigma^2 = 3\biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[ (\alpha-1) \pm \sqrt{\alpha+1} \biggr] \, .</math> |
Related Discussions
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