The Stability of Uniform-Density Spheres

The Eigenvalue Problem

Our Approach

As has been derived in an accompanying discussion, the second-order ODE that defines the relevant Eigenvalue problem is,

<math> \frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr] x = 0 . </math>

where the dimensionless radius,

<math> \chi_0 \equiv \frac{r_0}{R} , </math>

the characteristic time for dynamical oscillations in spherically symmetric configurations (SSC) is,

<math> \tau_\mathrm{SSC} \equiv \biggl[ \frac{R^2 \rho_c}{P_c} \biggr]^{1/2} , </math>

and the characteristic gravitational acceleration is,

<math> g_\mathrm{SSC} \equiv \frac{P_c}{R \rho_c} . </math>

The Approach Taken by Sterne (1937)

T. E. Sterne (1937) begins his analysis by deriving the

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>

in a manner explicitly designed to reproduce Eddington's pulsation equation — it appears as equation (1.8) in Sterne's paper — and, along with it, the surface boundary condition,

<math>~\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math> at <math>~r_0 = R \, ,</math>

which appears in Sterne's paper as equation (1.9). Then, as shown in the following paragraph extracted directly from his paper, Sterne (1937) rewrites both of these expressions in, what he considers to be, "more convenient forms."

Paragraph extracted from T. E. Sterne (1937)

"Modes of Radial Oscillation"

Monthly Notices of the Royal Astronomical Society, vol. 97, pp. 582 - © Royal Astronomical Society

Notation:
Sterne's		Ours
<math>~\xi_0 = Rx</math>		<math>~r_0</math>
<math>~\xi_1</math>		<math>~x</math>
<math>~n^2</math>		<math>~\omega^2</math>
<math>~\alpha</math>		<math>~3-4/\gamma_g</math>
<math>~\mu</math>		<math>~g_0 \rho_0 r_0/P_0</math>
<math>~g_0 R^2</math>		<math>~GM_\mathrm{tot}</math>

Properties of the Equilibrium Configuration

Our Setup

From our derived structure of a uniform-density sphere, in terms of the configuration's radius <math>R</math> and mass <math>M</math>, the central pressure and density are, respectively,

<math>P_c = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) </math> ,

and

<math>\rho_c = \frac{3M}{4\pi R^3} </math> .

Hence the characteristic time and acceleration are, respectively,

<math> \tau_\mathrm{SSC} = \biggl[ \frac{R^2 \rho_c}{P_c} \biggr]^{1/2} = \biggl[ \frac{2R^3 }{GM} \biggr]^{1/2} = \biggl[ \frac{3}{2\pi G\rho_c} \biggr]^{1/2}, </math>

and,

<math> g_\mathrm{SSC} = \frac{P_c}{R \rho_c} = \biggl( \frac{GM}{2R^2} \biggr) . </math>

The required functions are,

Density:

<math>\frac{\rho_0(r_0)}{\rho_c} = 1 </math> ;

Pressure:

<math>\frac{P_0(r_0)}{P_c} = 1 - \chi_0^2 </math> ;

Gravitational acceleration:

<math> \frac{g_0(r_0)}{g_\mathrm{SSC}} = 2\chi_0 . </math>

So our desired Eigenvalues and Eigenvectors will be solutions to the following ODE:

<math> \frac{1}{(1 - \chi_0^2)} \biggl\{ (1 - \chi_0^2) \frac{d^2x}{d\chi_0^2} + \frac{4}{\chi_0}\biggl[1 - \frac{3}{2}\chi_0^2 \biggr] \frac{dx}{d\chi_0} + \frac{1}{\gamma_\mathrm{g}} \biggl[\tau_\mathrm{SSC}^2 \omega^2 + 2 (4 - 3\gamma_\mathrm{g}) \biggr] x \biggr\} = 0 . </math>

Setup as Presented by Sterne (1937)

In §2 of his paper, Sterne (1937) details the structural properties of an equilibrium, uniform-density sphere as follows. (Text taken verbatim from Sterne's paper are presented here in green.) Given that the undisturbed density is constant and equal to the mean density, <math>~\bar\rho</math>, the mass within any radius is,

<math>M_r = \biggl( \frac{4\pi}{3} \biggr) \bar\rho \xi_0^3 \, ;</math>

the undisturbed values of gravity and the pressure are, respectively,

<math>g_0 \equiv \frac{GM_r}{\xi_0^2} = \biggl( \frac{4\pi}{3} \biggr) G\bar\rho R x \, </math>

and

<math>P_0 = \biggl( \frac{2\pi}{3} \biggr) G R^2 \bar\rho^2(1 - x^2) \, ;</math>

and the quantity,

<math>\mu \equiv \frac{g_0 \bar\rho \xi_0}{P_0} = \frac{2x^2}{(1-x^2)} \, .</math>

Hence, for this particular equilibrium model, Sterne's derived wave equation — his equation (1.91), as displayed above — becomes,

<math>~0</math>	<math>~=</math>	<math>~\xi_1^{ ' ' } + \biggl[\frac{4-\mu}{x} \biggr]\xi_1^' + \frac{R\bar\rho}{P_0} \biggl( \frac{n^2 R}{\gamma} - \frac{\alpha g_0}{x} \biggr) \xi_1</math>
	<math>~=</math>	<math>~\xi_1^{ ' ' } + \frac{1}{x}\biggl[4 -\frac{2x^2}{(1-x^2)} \biggr]\xi_1^' + \frac{3}{2\pi G R \bar\rho (1 - x^2)}\biggl[ \frac{n^2 R}{\gamma} - \biggl( \frac{4\pi}{3} \biggr) \alpha G\bar\rho R \biggr] \xi_1</math>
	<math>~=</math>	<math>~(1-x^2) \xi_1^{ ' ' } + \frac{1}{x}\biggl[4(1-x^2) - 2x^2 \biggr]\xi_1^' + \biggl[ \frac{3n^2 }{2\pi \gamma G \bar\rho} - 2 \alpha \biggr] \xi_1</math>
	<math>~=</math>	<math>~(1-x^2) \xi_1^{ ' ' } + \frac{1}{x}\biggl[4 - 6x^2 \biggr]\xi_1^' + \mathfrak{J} \xi_1 \, ,</math>

where,

<math>~\mathfrak{J} \equiv \frac{3n^2 }{2\pi \gamma G \bar\rho} - 2 \alpha \, .</math>

Analytic Solution

First few lowest-order modes

Mode 0:

<math>x_0 = \mathrm{constant}</math>, in which case,

<math> \omega_0^2 = - 2(4 - 3\gamma_\mathrm{g})\biggl[ \frac{2\pi G\rho_c}{3} \biggr] = 4\pi G \rho_c \biggl[ \gamma_\mathrm{g}- \frac{4}{3} \biggr] </math>

Mode 1:

<math>x_1 = a + b\chi_0^2</math>, in which case,

<math> \frac{1}{(1 - \chi_0^2)} \biggl\{ 2b (1 - \chi_0^2) + 8b \biggl[1 - \frac{3}{2}\chi_0^2 \biggr] + A_1 \biggl(1 + \frac{b}{a}\chi_0^2 \biggr) \biggr\} = 0 , </math>

where,

<math> A_1 \equiv \frac{a}{\gamma_\mathrm{g}}\biggl[ \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2+ 2(4 - 3\gamma_\mathrm{g}) \biggr] . </math>

Therefore,

<math> (A_1 + 10b) + \biggl[ \biggl(\frac{b}{a}\biggr) A_1 - 14b \biggr] \chi_0^2 = 0 , </math>

<math> \Rightarrow ~~~~~ A_1 = - 10b ~~~~~\mathrm{and} ~~~~~ A_1 = 14a </math>

<math> \Rightarrow ~~~~~ \frac{b}{a} = -\frac{7}{5} ~~~~~\mathrm{and} ~~~~~ \frac{A_1}{a} = 14 = \frac{1}{\gamma_\mathrm{g}}\biggl[ \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2+ 2(4 - 3\gamma_\mathrm{g}) \biggr] .

</math>

Hence,

<math> \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2 = 20\gamma_\mathrm{g} -8 </math>

<math> \Rightarrow ~~~~~ \omega_1^2 = \frac{2}{3}\biggl( 4\pi G\rho_c \biggr) (5\gamma_\mathrm{g} -2) </math>

and, to within an arbitrary normalization factor,

Sterne's General Solution

n=1 Polytrope

This discussion has been moved to another chapter.

User:Tohline/SSC/UniformDensity

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