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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. Letting <math>~R</math> be the radius and <math>~M_\mathrm{tot}</math> be the total mass of the bipolytrope, these configurations are fully defined 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, for a restricted range of key parameters, the pair of equations can both be solved analytically.
Two Separate LAWEs
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> |
For both regions of the bipolytrope, we define the dimensionless (Lagrangian) radial coordinate,
<math>~\xi \equiv \frac{r_0}{r_i} \, .</math>
So, the interface is, by definition, located at <math>~\xi = 1</math>; and, the surface is necessarily at <math>~\xi = q^{-1}</math>. As the material in the bipolytrope's core (envelope) is compressed/de-compressed during a radial oscillation, we will assume that heating/cooling occurs in a manner prescribed by an adiabat of index <math>~\gamma_c ~(\gamma_e)</math>; in general, <math>~\gamma_e \ne \gamma_c</math>. For convenience, we will also adopt the frequently used shorthand "alpha" notation,
<math>~\alpha_c \equiv 3 - \frac{4}{\gamma_c} \, ,</math> and <math>~\alpha_e \equiv 3 - \frac{4}{\gamma_e} \, .</math>
After adopting, for convenience, the function notation,
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<math>~g^2</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> |
we have deduced that, for the core, the LAWE may be written in the form,
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<math>~0</math> |
<math>~=</math> |
<math>~ (1 - \eta^2)\frac{d^2x}{d\eta^2} + ( 4 - 6\eta^2 ) \frac{1}{\eta} \cdot \frac{dx}{d\eta} + \mathfrak{F}_\mathrm{core} x \, . </math> |
where,
<math>~\eta \equiv \frac{\xi}{g} \, ,</math> and <math>~\mathfrak{F}_\mathrm{core} \equiv \frac{3\omega_\mathrm{core}^2}{2\pi G\gamma_c \rho_c} - 2\alpha_c\, .</math>
Eigenvector
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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