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Continue Search for Marginally Unstable (5,1) Bipolytropes

This Ramblings Appendix chapter — see also, various trials — provides some detailed trial derivations in support of the accompanying, thorough discussion of this topic.

Key Differential Equation

In an accompanying discussion, we derived the so-called,

Linear Adiabatic Wave (or Radial Pulsation) Equation

<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. After adopting an appropriate set of variable normalizations — as detailed here — this becomes,

<math>~0</math>

<math>~=</math>

<math>~ \frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} + \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\} x \, , </math>

where, <math>~\alpha_g \equiv (3 - 4/\gamma_g)</math>.

Applied to the Core

As we have already summarized in an accompanying discussion, throughout the core we have,

<math>~r^*</math>	<math>~=</math>	<math>~\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi \, ;</math>
<math>~\frac{\rho^*}{P^*}</math>	<math>~=</math>	<math>~\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2} \, ;</math>
<math>~\frac{M_r^*}{r^*}</math>	<math>~=</math>	<math>~ \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]\biggl( \frac{2\pi}{3}\biggr)^{1 / 2} \frac{1}{\xi} = 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \, . </math>

So the relevant core LAWE becomes,

<math>~0</math>

<math>~=</math>

<math>~ \frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} + \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\} x \, , </math>

Applied to the Envelope

And as we have also summarized in the same accompanying discussion, throughout the envelope we have,

<math>~r^*</math>	<math>~=</math>	<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \, ;</math>
<math>~\frac{\rho^*}{P^*}</math>	<math>~=</math>	<math>~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1} \, ; </math>
<math>~\frac{M_r^*}{r^*}</math>	<math>~=</math>	<math>~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \biggr]^{-1} = 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i \eta \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \, . </math>

So the relevant envelope LAWE becomes,

<math>~0</math>

<math>~=</math>

<math>~ \frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} + \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\} x \, , </math>

See Also

• K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206), A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation