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Radial Oscillations of a Zero-Zero Bipolytrope
This is a chapter that summarizes an accompanying, detailed derivation.
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In a separate chapter on astrophysical interesting equilibrium structures, we have derived analytical expressions that define the equilibrium properties of bipolytropic structures having <math>~(n_c, n_e) = (0, 0)</math>, that is, bipolytropes in which both the core and the envelope are uniform in density, but the densities in the two regions are different from one another. These configurations are fully defined (in a dimensionless sense) once any two of the following three key parameters have been specified: The envelope-to-core density ratio, <math>~\rho_e/\rho_c</math>; the radial location of the envelope/core interface, <math>~q \equiv r_i/R</math>; and, the fractional mass that is contained within the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. These three parameters are related to one another via the expression,
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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> |
Equilibrium configurations can be constructed that have a wide range of parameter values; specifically,
<math>~0 \le q \le 1 \, ;</math> <math>~0 \le \nu \le 1 \, ;</math> and, <math>~0 \le \frac{\rho_e}{\rho_c} \le 1 \, .</math>
(We recognize from buoyancy arguments that any configuration in which the envelope density is larger than the core density will be Rayleigh-Taylor unstable, so we restrict our astrophysical discussion to structures for which <math>~\rho_e < \rho_c</math>.)
By employing the linear stability analysis techniques described in an accompanying chapter, we should, in principle, be able to identify a wide range of eigenvectors — that is, radial eigenfunctions and accompanying eigenfrequencies — that are associated with adiabatic radial oscillation modes in any one of these equilibrium, bipolytropic configurations. Using numerical techniques, Murphy & Fiedler (1985), for example, have carried out such an analysis of bipolytropic structures having <math>~(n_c, n_e) = (1,5)</math>. A pair of linear adiabatic wave equations (LAWEs) must be solved — one tuned to accommodate the properties of the core and another tuned to accommodate the properties of the envelope — then the pair of eigenfunctions must be matched smoothly at the radial location of the interface; the identified core- and envelope-eigenfrequencies must simultaneously match.
After identifying the precise form of the LAWEs that apply to the case of <math>~(n_c, n_e) = (0,0)</math> bipolytropes, we discovered that the pair of equations can both be solved analytically, for a limited range of key parameters.
Two Separate Eigenvectors
Core
<math>~\alpha_c \equiv 3-\frac{4}{\gamma_c}</math>
<math>~g = \frac{1}{1+2q^3}</math>
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Mode |
Core Eigenvector |
<math>~\frac{3\omega_\mathrm{core}^2}{2\pi \gamma_c G \rho_c} = 2\alpha_c + 2j(2j+5)</math> |
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<math>~j=0 </math> |
<math>~x_\mathrm{core} = a_0 </math> |
<math>~6-8/\gamma_c</math> |
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<math>~j=1 </math> |
<math>~x_\mathrm{core} = a_0 \biggl[ 1 - \frac{7}{5}\biggr(\frac{\xi^2}{g^2}\biggr) \biggr]</math> |
<math>~20-8/\gamma_c</math> |
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<math>~j=2 </math> |
<math>~x_\mathrm{core} = a_0 \biggl[ 1 - \frac{18}{5}\biggr(\frac{\xi^2}{g^2}\biggr) + \frac{99}{35}\biggr(\frac{\xi^2}{g^2}\biggr)^2 \biggr]</math> |
<math>~42-8/\gamma_c</math> |
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