Radial Oscillations of a Zero-Zero Bipolytrope

This is a chapter that summarizes an accompanying, detailed derivation.

In a separate chapter on astrophysical interesting equilibrium structures, we have derived analytical expressions that define the equilibrium properties of bipolytropic configurations 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,

<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,

(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

<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>

The Core's LAWE

After adopting, for convenience, the function notation,

<math>~\equiv</math>

<math> 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] \, , </math>

we have deduced that, for the core, the LAWE may be written in the form,

<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>

Not surprisingly, this is identical in form to the eigenvalue problem that was first presented — and solved analytically — by Sterne (1937) in connection with his examination of radial oscillations in isolated uniform-density spheres. As is demonstrated below, for the core of our zero-zero bipolytrope, we can in principle adopt any one of the polynomial eigenfunctions and corresponding eigenfrequencies derived by Sterne.

The Envelope's LAWE

Subsequently, we also have deduced that, for the envelope, the governing LAWE becomes,

<math>~ \biggl[ 1 + \frac{(g^2-\mathcal{B}) \xi}{\mathcal{A}} - \mathcal{D} \xi^3\biggr] \frac{d^2x}{d\xi^2} + \biggl\{ 3 + \frac{4(g^2-\mathcal{B}) \xi}{\mathcal{A}} - 6\mathcal{D} \xi^3 \biggr\} \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[ \mathcal{D} \biggl(\frac{\rho_c}{\rho_e}\biggr) \biggl( \mathfrak{F}_\mathrm{env} + 2\alpha_e -2\alpha_e\frac{\rho_e}{\rho_c} \biggr)\xi^3 -\alpha_e \biggr]\frac{x}{\xi^2} \, , </math>

where,

<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>
<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>
<math>~\mathcal{D}</math>	<math>~\equiv</math>	<math>~\frac{1}{\mathcal{A}}\biggl( \frac{\rho_e}{\rho_c}\biggr)^2 = \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr] \, , </math>
<math>~\mathfrak{F}_\mathrm{env}</math>	<math>~\equiv</math>	<math>~\frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e} - 2\alpha_e \, . </math>

We have been unable to demonstrate that this governing equation can be solved analytically for arbitrary pairs of the key model parameters, <math>~q</math> and <math>~\rho_e/\rho_c</math>. But, if we choose parameter value pairs that satisfy the constraint,

<math>~g^2 = \mathcal{B} </math> <math>~\Rightarrow</math> <math>~g = \frac{1}{1+2q^3} \, ,</math> and, <math>~q^3 = \mathcal{D} = \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr] \, ,</math>

then the LAWE that is relevant to the envelope simplifies. Specifically, it takes the form,

<math>~0</math>	<math>~=</math>	<math>~ ( 1 - q^3 \xi^3 ) \frac{d^2x}{d\xi^2} + ( 3 - 6q^3 \xi^3 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[ q^3 \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e \biggr]\frac{x}{\xi^2} </math>
	<math>~=</math>	<math>~\frac{x}{\xi^2}\biggl\{ ( 1 - q^3 \xi^3 ) \biggl[ \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}\biggr] + ( 3 - 6q^3 \xi^3 ) \frac{d\ln x}{d\ln \xi} + \biggl[ q^3 \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e \biggr] \biggr\} \, . </math>

Shortly after deriving this last expression, we realized that one possible solution is a simple power-law eigenfunction of the form,

where the (constant) exponent is one of the roots of the quadratic equation,

<math>~c_0^2 + 2c_0 - \alpha_e = 0 \, ,</math> <math>~\Rightarrow</math> <math>~c_0 = -1 \pm \sqrt{1+\alpha_e} \, .</math>

This power-law eigenfunction must be paired with the associated eigenfrequency,

<math>~\mathfrak{F}_\mathrm{env}</math>	<math>~=</math>	<math>~c_0(c_0+5) = 3c_0 + \alpha_e</math>
<math>~\Rightarrow ~~~ \frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e} </math>	<math>~=</math>	<math>~ 3(c_0 + \alpha_e) = 3[\alpha_e -1 \pm \sqrt{1+\alpha_e}] \, .</math>

Next, we noticed the strong similarities between the mathematical properties of this eigenvalue problem and the one that was studied by C. Prasad (1948, MNRAS, 108, 414-416) in connection with, what we now recognize to be, a closely related problem. Drawing heavily from Prasad's analysis, we discovered an infinite number of eigenfunctions (each, a truncated polynomial expression) and associated eigenfrequencies that satisfy this governing envelope LAWE. The eigenvectors associated with the lowest few modes are tabulated, below.

Eigenvector

Core

<math>~\alpha_c \equiv 3-\frac{4}{\gamma_c}</math>
<math>~g = \frac{1}{1+2q^3}</math>

Mode	Core Eigenvector	<math>~\frac{3\omega_\mathrm{core}^2}{2\pi \gamma_c G \rho_c} = 2\alpha_c + 2j(2j+5)</math>
<math>~j=0 </math>	<math>~x_\mathrm{core} = a_0 </math>	<math>~6-8/\gamma_c</math>
<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>
<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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