User:Tohline/SSC/Stability/MurphyFiedler85
Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b)
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Overview
In the stability analysis presented by Murphy & Fiedler (1985b), the relevant polytropic indexes are, <math>~(n_c, n_e) = (1,5)</math>. Structural properties of the underlying equilibrium models have been reviewed in our accompanying discussion.
The Linear Adiabatic Wave Equation (LAWE) that is relevant to polytropic spheres may be written as,
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<math>~0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr) \frac{\xi^2}{\theta} - \alpha Q\biggr] \frac{x}{\xi^2} </math> |
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where: <math>~Q(\xi) \equiv - \frac{d\ln\theta}{d\ln\xi} \, ,</math> <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, ,</math> and, <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math> |
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See also …
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As we have detailed separately, the boundary condition at the center of a polytropic configuration is,
<math>~\frac{dx}{d\xi} \biggr|_{\xi=0} = 0 \, ;</math>
and the boundary condition at the surface of an isolated polytropic configuration is,
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<math>~\frac{d\ln x}{d\ln\xi}</math> |
<math>~=</math> |
<math>~- \alpha + \frac{\omega^2}{\gamma_g } \biggl( \frac{1}{4\pi G \rho_c } \biggr) \frac{\xi}{(-\theta^')} </math> at <math>~\xi = \xi_s \, .</math> |
But this surface condition is not applicable to bipolytropes. Instead, let's return to the original, more general expression of the surface boundary condition:
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<math>~ \frac{d\ln x}{d\ln\xi}\biggr|_s</math> |
<math>~=</math> |
<math>~- \alpha + \frac{\omega^2 R^3}{\gamma_g GM_\mathrm{tot}} \, .</math> |
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Utilizing an accompanying discussion, let's examine the frequency normalization used by Murphy & Fiedler (1985b) (see the top of the left-hand column on p. 223):
For a given radial quantum number, <math>~k</math>, the factor inside the square brackets in this last expression is what Murphy & Fiedler (1985b) refer to as <math>~\omega^2_k \theta_c</math>. Keep in mind, as well, that, in the notation we are using,
This also means that the surface boundary condition may be rewritten as,
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Murphy & Fiedler (1985b) realized that they could not simply integrate the above-presented polytropic LAWE from the center of the configuration to its surface because the underlying bipolytropic equilibrium structure of the envelope and the core are defined by two different polytropic indexes. Instead, they separated the problem into two pieces — integrating the relevant core LAWE from the center to the interface, then integrating the relevant envelope LAWE from the interface to the surface — being careful to properly match the two solutions at the interface.
More Detailed Setup
Core Layers With n = 1
For n = 1 structures the LAWE is,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{d^2x}{d\xi^2} + \biggl[ 4 - 2 Q_1 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr) \frac{\xi^2}{\theta} - \alpha_\mathrm{core} Q_1\biggr] \frac{x}{\xi^2} </math> |
where,
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<math>~Q_1</math> |
<math>~\equiv</math> |
<math>~- \frac{d\ln\theta}{d\ln\xi} \, .</math> |
Given that, for <math>~n = 1</math> polytropic structures,
<math> \theta(\xi) = \frac{\sin\xi}{\xi} </math> and <math> \frac{d\theta}{d\xi} = \biggl[ \frac{\cos\xi}{\xi}- \frac{\sin\xi}{\xi^2}\biggr] </math>
we have,
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<math>~Q_1</math> |
<math>~=</math> |
<math>~ - \frac{\xi^2}{\sin\xi} \biggl[ \frac{\cos\xi}{\xi}- \frac{\sin\xi}{\xi^2}\biggr] </math> |
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<math>~=</math> |
<math>~ 1 - \xi\cot\xi \, . </math> |
Hence, the governing LAWE for the core is,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{d^2x}{d\xi^2} + \biggl[ 4 - 2 ( 1 - \xi\cot\xi ) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr) \frac{\xi^3}{\sin\xi} - \alpha_\mathrm{core} ( 1 - \xi\cot\xi )\biggr] \frac{x}{\xi^2} </math> |
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<math>~=</math> |
<math>~ \frac{d^2x}{d\xi^2} + \biggl[ 1 + \xi\cot\xi \biggr] \frac{2}{\xi} \cdot \frac{dx}{d\xi} + 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr) \frac{\xi^3}{\sin\xi} - \alpha_\mathrm{core} ( 1 - \xi\cot\xi )\biggr] \frac{x}{\xi^2} \, . </math> |
This can be rewritten as,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{d^2x}{d\xi^2} + \frac{2}{\xi} \biggl[ 1 + \xi\cot\xi \biggr]\frac{dx}{d\xi} + \biggl[ \biggl( \frac{\sigma_c^2}{3\gamma_\mathrm{core} } \biggr) \frac{\xi}{\sin\xi} + \frac{2 \alpha_\mathrm{core} ( \xi\cos\xi - \sin\xi) }{\xi^2 \sin\xi} \biggr] x </math> |
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<math>~=</math> |
<math>~ \frac{d^2x}{d\xi^2} + \frac{2}{\xi} \biggl[ 1 + \xi\cot\xi \biggr]\frac{dx}{d\xi} + \biggl[ \frac{\gamma_g}{\gamma_\mathrm{core}}\biggl( \omega_k^2 \theta_c \biggr) \frac{\xi}{\sin\xi} + \frac{2 \alpha_\mathrm{core} ( \xi\cos\xi - \sin\xi) }{\xi^2 \sin\xi} \biggr] x \, , </math> |
which matches the expression presented by Murphy & Fiedler (1985b) (see middle of the left column on p. 223 of their article) if we set <math>~\theta_c = 1</math> and <math>~\gamma_g/\gamma_\mathrm{core} = 1</math>. This LAWE also appears in our separate discussion of radial oscillations in n = 1 polytropic spheres.
Envelope Layers With n = 5
The LAWE for n = 5 structures is,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{d^2x}{d\eta^2} + \biggl[ 4 - 6Q_5 \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta} + 6 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{env} } \biggr) \frac{\eta^2}{\phi} - \alpha_\mathrm{env} Q_5\biggr] \frac{x}{\eta^2} </math> |
where,
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<math>~Q_5</math> |
<math>~\equiv</math> |
<math>~- \frac{d\ln\phi}{d\ln\eta} \, .</math> |
From our accompanying discussion of the underlying equilibrium structure of <math>~(n_c, n_e) = (1, 5)</math> bipolytropes, we know that,
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<math>~\phi</math> |
<math>~=</math> |
<math>~\frac{B_0^{-1}\sin\Delta}{\eta^{1/2}(3-2\sin^2\Delta)^{1/2}} \, ,</math> |
and,
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<math>~\frac{d\phi}{d\eta}</math> |
<math>~=</math> |
<math>~ \frac{B_0^{-1}[3\cos\Delta-3\sin\Delta + 2\sin^3\Delta] }{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}} \, . </math> |
where <math>~A_0</math> is a "homology factor," <math>~B_0</math> is an overall scaling coefficient, and we have introduced the notation,
<math>~\Delta \equiv \ln(A_0\eta)^{1/2} = \frac{1}{2} (\ln A_0 + \ln\eta) \, .</math>
Hence,
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<math>~Q_5</math> |
<math>~=</math> |
<math>~ - \eta \biggl[ \frac{\eta^{1/2}(3-2\sin^2\Delta)^{1/2}}{B_0^{-1}\sin\Delta} \biggr] \frac{B_0^{-1}[3\cos\Delta-3\sin\Delta + 2\sin^3\Delta] }{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}} </math> |
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<math>~=</math> |
<math>~ \frac{ 3\sin\Delta - 3\cos\Delta - 2\sin^3\Delta }{2 \sin\Delta (3-2\sin^2\Delta)} \, . </math> |
And,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{d^2x}{d\eta^2} ~+~ \biggl[ 4 + \frac{ 3(3\cos\Delta - 3\sin\Delta + 2\sin^3\Delta) }{ \sin\Delta (3-2\sin^2\Delta)} \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta} ~+~ \biggl[ \biggl( \frac{\sigma_c^2}{\gamma_\mathrm{env} } \biggr) \frac{B_0 \eta^{1/2}(3-2\sin^2\Delta)^{1/2}}{\sin\Delta} ~+~ \frac{ 3\alpha_\mathrm{env} (3\cos\Delta -3\sin\Delta + 2\sin^3\Delta )}{\eta^2 \sin\Delta (3-2\sin^2\Delta)}\biggr] x </math> |
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<math>~=</math> |
<math>~ \frac{d^2x}{d\eta^2} ~+~ \biggl[ 4 ~+~ \frac{ 3(3\cos\Delta - \tfrac{3}{2}\sin\Delta - \tfrac{1}{2}\sin3\Delta) }{ \sin\Delta (2 + \cos2\Delta)} \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta} ~+~ \biggl[\omega^2_k \theta_c \biggl( \frac{\gamma_g}{\gamma_\mathrm{env} } \biggr) \frac{B_0 \eta^{1/2}(2 + \cos2\Delta)^{1/2}}{\sin\Delta} ~+~ \frac{ 3\alpha_\mathrm{env} (3\cos\Delta -\tfrac{3}{2}\sin\Delta - \tfrac{1}{2}\sin3\Delta )}{\eta^2 \sin\Delta (2 + \cos2\Delta)}\biggr] x \, , </math> |
which matches the expression presented by Murphy & Fiedler (1985b) (see middle of the left column on p. 223 of their article) if we set <math>~\theta_c = 1</math> and <math>~\gamma_g/\gamma_\mathrm{env} = 1</math>.
Surface Boundary Condition
Next, pulling from our accompanying discussion of the stability of polytropes and an accompanying table that details the properties of <math>~(n_c, n_e) = (1, 5)</math> bipolytropes, the surface boundary condition is,
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<math>~ \frac{d\ln x}{d\ln\eta}\biggr|_s</math> |
<math>~=</math> |
<math>~- \biggl(\frac{\gamma_g}{\gamma_\mathrm{env}}\biggr) \alpha + \frac{\omega^2 R^3}{\gamma_\mathrm{env} GM_\mathrm{tot}} </math> |
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<math>~\Rightarrow ~~~ \frac{d\ln x}{d\ln\eta}\biggr|_s + \biggl(\frac{\gamma_g}{\gamma_\mathrm{env}}\biggr) \alpha </math> |
<math>~=</math> |
<math>~ \frac{\omega^2 (R_s^*)^3}{\gamma_\mathrm{env} GM^*_\mathrm{tot}} \biggl( \frac{K_c}{G}\biggr)^{3 / 2}\biggl( \frac{K_c}{G}\biggr)^{-3 / 2} \frac{1}{\rho_0}</math> |
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<math>~=</math> |
<math>~ \frac{\omega^2 }{\gamma_\mathrm{env} G\rho_0 } \biggl[ (2\pi)^{-1/2} \xi_i e^{2(\pi - \Delta_i)} \biggr]^3 \biggl[ \biggl( \frac{3}{2\pi} \biggr)^{1/2} \sin\xi_i \biggl( \frac{3}{\sin^2\Delta_i} - 2 \biggr)^{1/2} e^{(\pi - \Delta_i)} \biggr]^{-1} \biggl( \frac{\mu_e}{\mu_c}\biggr)</math> |
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<math>~=</math> |
<math>~ \frac{\omega^2 }{\gamma_\mathrm{env}(2\pi G\rho_0)} \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{1}{\sqrt{3}} \biggl[ \frac{\xi_i^2}{\theta_i} \biggr] \biggl( \frac{3}{\sin^2\Delta_i} - 2 \biggr)^{-1 / 2} e^{5(\pi - \Delta_i)}</math> |
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<math>~=</math> |
<math>~ \frac{\omega^2 }{\gamma_\mathrm{env}(2\pi G\rho_0)} \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{e^{5\pi}}{\sqrt{3}} \biggl[ \frac{\xi_i^2}{\theta_i} \biggr] \xi_i^{1 / 2}B\theta_i (\xi_i A)^{-5/2}</math> |
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<math>~=</math> |
<math>~ \frac{\omega^2 }{\gamma_\mathrm{env}(2\pi G\rho_0)} \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{B e^{5\pi}}{\sqrt{3} ~A^{5 / 2}} </math> |
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<math>~=</math> |
<math>~ \frac{2\omega_k^2 \theta_c}{(n_c+1)} \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{B e^{5\pi}}{\sqrt{3} ~A^{5 / 2}} \, . </math> |
After acknowledging that, in their specific stability analysis, <math>~\theta_c = 1</math>, <math>~n_c = 1</math>, and <math>~\mu_e/\mu_c = 1</math>, this right-hand-side expression matches the equivalent term published by Murphy & Fiedler (1985b) (see the bottom of the left-hand column on p. 223).
Interface Conditions
Here, we will simply copy the discussion already provided in the context of our attempt to analyze the stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes; specifically, we will draw from STEP 4: in the Piecing Together subsection. Following the discussion in §§57 & 58 of P. Ledoux & Th. Walraven (1958), the proper treatment is to ensure that fractional perturbation in the gas pressure (see their equation 57.31),
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<math>~\frac{\delta P}{P}</math> |
<math>~=</math> |
<math>~- \gamma x \biggl( 3 + \frac{d\ln x}{d\ln \xi} \biggr) \, ,</math> |
is continuous across the interface. That is to say, at the interface <math>~(\xi = \xi_i)</math>, we need to enforce the relation,
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<math>~0</math> |
<math>~=</math> |
<math>~\biggl[ \gamma_c x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr) - \gamma_e x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr)\biggr]_{\xi=\xi_i}</math> |
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<math>~=</math> |
<math>~\gamma_e \biggl[ \frac{\gamma_c}{\gamma_e} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr) - \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr)\biggr]_{\xi=\xi_i}</math> |
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<math>~\Rightarrow~~~ \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=\xi_i}</math> |
<math>~=</math> |
<math>~3\biggl(\frac{\gamma_c}{\gamma_e} -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} \, .</math> |
In the context of this interface-matching constraint (see their equation 62.1), P. Ledoux & Th. Walraven (1958) state the following: In the static (i.e., unperturbed equilibrium) model … discontinuities in <math>~\rho</math> or in <math>~\gamma</math> might occur at some [radius]. In the first case — that is, a discontinuity only in density, while <math>~\gamma_e = \gamma_c</math> — the interface conditions imply the continuity of <math>~\tfrac{1}{x} \cdot \tfrac{dx}{d\xi}</math> at that [radius]. In the second case — that is, a discontinuity in the adiabatic exponent — the dynamical condition may be written as above. This implies a discontinuity of the first derivative at any discontinuity of <math>~\gamma</math>.
The algorithm that Murphy & Fiedler (1985b) used to "… [integrate] through each zone …" was designed "… with continuity in <math>~x</math> and <math>~dx/d\xi</math> being imposed at the interface …" Given that they set <math>~\gamma_c = \gamma_e = 5/3</math>, their interface matching condition is consistent with the one prescribed by P. Ledoux & Th. Walraven (1958).
Our Confession
When we tried to integrate the governing LAWEs in the piecemeal fashion described by Murphy & Fiedler (1985b) — as we have just detailed — we initially failed to match their published eigenvector solutions. In retrospect, it appears as though we did not correctly implement the interface-matching conditions. In an effort to diagnose this problem, we backed up to a more generalized prescription of the LAWE that allowed us to smoothly integrate a single equation from the center to the surface of the configuration without having to mess with interface-matching conditions. In what follows, we describe this alternate approach. We show how this approach has allowed us to derive radial-oscillation eigenvectors that match in detail the results published by Murphy & Fiedler (1985b).
See Also
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© 2014 - 2021 by Joel E. Tohline |