Overview: Marginally Unstable Pressure-Truncated Configurations

Additional details may be found here.

Isothermal

Once a central density, <math>~\rho_c</math>, and constituent fluid sound speed, <math>~c_s</math>, have been specified, the internal structure of an equilibrium, isothermal sphere can be completely described in terms of the function, <math>~\psi(\xi) \equiv \ln(\rho_c/\rho)</math>, which is a solution of the,

Isothermal Lane-Emden Equation

<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr) = e^{-\psi}</math>

subject to the boundary conditions, <math>~ \psi = 0</math> and <math>~d\psi/d\xi = 0</math> at <math>~\xi = 0</math>. In isolation, the isothermal sphere extends to infinity. But configurations of finite extent can be constructed by truncating the function, <math>~\psi</math>, at some radius, <math>~0 < \xi_e < \infty</math> — such that the surface density is finite and set by the value of <math>~\psi_e \equiv \psi(\xi_e)</math> — and embedding the configuration in a hot, tenuous medium that exerts an "external" pressure, <math>~P_e = c_s^2 \rho_c e^{-\psi_e}</math>, uniformly across the surface of the — now, truncated — sphere. The internal structure of such a "pressure-truncated" isothermal sphere is completely describable in terms of the same function, <math>~\psi(\xi)</math>, that describes the structure of the isolated isothermal sphere, except that beyond <math>~\xi_e</math> the function becomes irrelevant.

A sequence of equilibrium, pressure-truncated isothermal spheres is readily defined by varying the value of <math>~\xi_e</math>. Figure 1 displays the behavior of such an equilibrium sequence, as viewed from three different astrophysical perspectives (in all cases, <math>~c_s</math> is held fixed while <math>~\xi_e</math> is varied monotonically along the sequence): Left panel — A diagram showing how the truncated configuration's equilibrium volume varies with the externally applied pressure, if the configuration's mass is held fixed. Center panel — A diagram showing how the truncated configuration's mass varies with the equilibrium radius, if the external pressure is held fixed. Right panel — A diagram that shows how the configuration's mass varies with central density, if the external pressure is held fixed.

Figure 1: Equilibrium Sequences of Pressure-Truncated Isothermal Spheres (viewed from three different astrophysical perspectives)
●	<math>~\xi_e</math>	^†External Pressure vs. Volume (Fixed Mass)	Mass vs. Radius (Fixed External Pressure)	^‡Mass vs. Central Density (Fixed External Pressure)
●	4.05
●	6.45
●	9.00
●	67.00
●	98.50
●	735.00
●	1060.00
^†This is the classic P-V diagram that shows up in most discussions of Bonnor-Ebert spheres. ^‡In a similar diagram in which the radius (rather than external pressure) is held fixed, the purple (rather than green) markers identify mass extrema.

This equation — in the following, slightly rewritten form — can be found among our selected set of key equations associated with the study of radial pulsation, and will henceforth be referred to as the,

Isothermal LAWE

	<math>~0 = \frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{g}}\biggr)\xi^2 - \alpha \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{x}{\xi^2} </math>
where: <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>

Yabushita (1974, 1975) showed that the following eigenvector specification provides a

Precise Solution to the Isothermal LAWE
<math>~\sigma_c^2 = 0</math>	and	<math>~x = 1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} \, ,</math>

if the adiabatic exponent is assigned the value, <math>~\gamma_g = 1</math>, in which case the parameter, <math>~\alpha = -1</math>. When viewed in concert with the surface boundary condition,

the relevant configuration is precisely defined by the surface condition, xxx, which is identical to the configuration at the turning point.

Polytropic

Given a value of the polytropic index, <math>~n</math>, the internal structure of a detailed force-balance model is provided via the function, <math>~\theta(\xi)</math>, which is a solution of the,

Polytropic Lane-Emden Equation

<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math>

subject to the boundary conditions, <math>~\Theta_H = 1</math> and <math>~d\Theta_H/d\xi = 0</math> at <math>~\xi = 0</math>.

To identify pure radial oscillation modes, we seek solutions to the,

Polytropic LAWE

	<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>
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>

We have discovered that, for any value of the polytropic index in the range, <math>~3 \le n < \infty</math>, the following eigenvector specification provides a

Precise Solution to the Polytropic LAWE
<math>~\sigma_c^2 = 0</math>	and	<math>~x = 1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi} </math>

if the adiabatic exponent is assigned the value, <math>~\gamma_g = (n+1)/n</math>, in which case the parameter, <math>~\alpha = (3-n)/(n+1)</math>.

References

User:Tohline/SSC/Stability/InstabilityOnsetOverview

Contents

Overview: Marginally Unstable Pressure-Truncated Configurations

Isothermal

Polytropic

References

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