User:Tohline/SSC/Stability/Isothermal
Radial Oscillations of Pressure-Truncated Isothermal Spheres
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Groundwork
Equilibrium Model
Review
In an accompanying discussion, while reviewing the original derivations of Ebert (1955) and Bonnor (1956), we have detailed the equilibrium properties of pressure-truncated isothermal spheres. These properties have been expressed in terms of the isothermal Lane-Emden function, <math>~\psi(\xi)</math>, which provides a solution to the governing,
Isothermal Lane-Emden Equation
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A parallel presentation of these details can be found in §2 — specifically, equations (2.4) through (2.10) — of Yabushita (1968). Each of Yabushita's key mathematical expressions can be mapped to ours via the variable substitutions presented here in Table 1.
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Table 1: Mapping from Yabushita's (1968) Notation to Ours |
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| Yabushita's (1968) Notation: | <math>~x</math> | <math>~\psi</math> | <math>~\mu</math> | <math>~M</math> | <math>~x_0</math> | <math>~p_0</math> |
| Our Notation: | <math>~\xi</math> | <math>~-\psi</math> | <math>~\bar\mu</math> | <math>~M_{\xi_e}</math> | <math>~\xi_e</math> | <math>~P_e</math> |
For example, given the system's sound speed, <math>~c_s</math>, and total mass, <math>~M_{\xi_e}</math>, the expression from our presentation that shows how the bounding external pressure, <math>~P_e</math>, depends on the dimensionless Lane-Emden function, <math>~\psi</math>, is,
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<math>~P_e</math> |
<math>~=</math> |
<math>~\biggl( \frac{c_s^8}{4\pi G^3 M_{\xi_e}^2} \biggr) ~\xi_e^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2_e e^{-\psi_e}</math> |
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<math>~\Rightarrow ~~~ \xi_e^2 \biggl(\frac{d\psi}{d\xi}\biggr)_e e^{-(1/2)\psi_e}</math> |
<math>~=</math> |
<math>~\frac{1}{c_s^4}\biggl[ G^3 M_{\xi_e}^2 ~(4\pi P_e)\biggr]^{1 / 2} \, ,</math> |
which — see the boxed-in excerpt that follows — exactly matches Yabushita's (1968) equation (2.9), after recalling that the system's sound speed is related to its temperature via the relation,
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} \, .</math>
And, our expression for the truncated configuration's equilibrium radius is,
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<math>~R</math> |
<math>~=</math> |
<math>~\frac{GM_{\xi_e}}{c_s^2} \biggl[ \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-1}</math> |
which — see the boxed-in excerpt that follows — matches Yabushita's (1968) equation (2.10).
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Equations extracted† from p. 110 of S. Yabushita (1968, MNRAS, 140, 109)
"Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres"
MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society |
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†Mathematical expressions displayed here with layout modified from the original publication. |
P-V Diagram
As we have discussed in a separate chapter that focuses on the structural properties of pressure-truncated Isothermal spheres, Bonnor (1956) examined the sequence of equilibrium models that is generated by varying the truncation radius over the range, <math>~0 < \xi_e < \infty</math>. In a diagram that shows how <math>~P_e(\xi_e)</math> varies with equilibrium volume, <math>~V(\xi_e) \propto R^3</math>, Bonnor noticed that there is a pressure above which no equilibrium configurations exist. The original P-V diagram published by Bonnor (1956) has been reproduced here, in the left-hand panel of our Figure 1. The right-hand panel of Figure 1 shows the same equilibrium sequence, as generated from our numerical integration of the isothermal Lane-Emden equation; in our plot, we have adopted Whitworth's normalizations, <math>~P_\mathrm{rf}</math> and <math>~V_\mathrm{rf}</math>.
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Figure 1: Bonnor's P-V Diagram |
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As is discussed below, in separate studies, Yabushita (1968) and Taff & Van Horn (1974) examined the lowest-order modes of radial oscillations that arise in pressure-truncated isothermal spheres. In both studies, numerical techniques were used to solve the eigenvalue problem associated with the isothermal LAWE. The nine individual equilibrium models that were studied by Taff & Van Horn are identified by the nine small, filled circular markers along the sequence that has been displayed in the right-hand panel of our Figure 1; as labeled, they correspond to models with <math>~\xi_e</math> = 2, 3, 4, 5, 6, 7, 8, 9, and 10. Earlier, Yabushita studied the oscillation modes of three of these same configurations; specifically, the models with <math>~\xi_e</math> = 6, 7, and 8, which straddle the location along the equilibrium sequence of the model associated with the pressure maximum (the turning point labeled "A" in Bonnor's P-V diagram).
Other Properties
Also, as has been summarized in our accompanying discussion of the equilibrium properties of pressure-truncated isothermal spheres, we have,
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<math>~r_0 </math> |
<math>~=</math> |
<math>~\biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi \, ;</math> |
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<math>~P_0 = c_s^2 \rho_0 </math> |
<math>~=</math> |
<math>~(c_s^2 \rho_c) e^{-\psi} \, ;</math> |
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<math>~M_r </math> |
<math>~=</math> |
<math>~\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr] \, .</math> |
Hence, for isothermal configurations,
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<math>~g_0 \equiv \frac{GM_r}{r_0^2}</math> |
<math>~=</math> |
<math>~G\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr] \biggl[ \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi\biggr]^{-2}</math> |
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<math>~=</math> |
<math>~c_s^2 \biggl( \frac{4\pi G \rho_c}{c_s^2} \biggr)^{1 / 2} \biggl( \frac{d\psi}{d\xi} \biggr) \, . </math> |
Linearized Wave Equation
In our introductory discussion of techniques that facilitate linear stability analyses, we derived what we now repeatedly refer to as the "key" form of the
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> |
Here we review two published articles that have presented a partial analysis of radial modes of oscillation in pressure-truncated isothermal spheres. The analyses presented in both of these papers, effectively, employ this key wave equation, but the authors of these articles present it in different forms.
Yabushita (1968)
The linearized wave equation that Yabushita (1968) used to examine the radial pulsation modes of pressure-truncated isothermal spheres is displayed in the following, boxed-in image:
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Equation extracted from p. 111 of S. Yabushita (1968, MNRAS, 140, 109)
"Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres"
MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society |
 |
This equation can be obtained straightforwardly through a strategic combination of three of the following four linearized principal governing equations that we have derived in our accompanying, broad introductory discussion of linear stability analyses, namely,
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Linearized Linearized Linearized <math> P_1 = \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\, , </math> Linearized <math> \nabla^2 \Phi_1 = 4\pi G \rho_1\, . </math> |
Taking the partial time-derivative of the linearized equation of continuity gives,
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<math>~- \nabla\cdot \frac{\partial \vec{v}}{\partial t} </math> |
<math>~=</math> |
<math>~\frac{1}{\rho_0}\frac{\partial^2 \rho_1}{\partial t^2} + \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial\vec{v}}{\partial t} \, ;</math> |
and, taking the divergence of the linearized Euler equation gives,
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<math>~-\nabla\cdot \frac{\partial \vec{v}}{\partial t} </math> |
<math>~=</math> |
<math>~\nabla^2 \Phi_1 + \nabla\cdot \biggl[\frac{1}{\rho_0} \nabla P_1\biggr] - \nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] \, .</math> |
Combining the two, then making two substitutions using (1) the linearized Poisson equation and (2) the linearized Euler equation, we have,
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<math>~\frac{\partial^2 \rho_1}{\partial t^2} + \nabla\rho_0 \cdot \frac{\partial\vec{v}}{\partial t} </math> |
<math>~=</math> |
<math>~\rho_0 \nabla^2 \Phi_1 + \rho_0 \nabla\cdot \biggl[\frac{1}{\rho_0} \nabla P_1\biggr] - \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] </math> |
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<math>~\Rightarrow ~~~ \frac{\partial^2 \rho_1}{\partial t^2} + \nabla\rho_0 \cdot \biggl[ - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] </math> |
<math>~=</math> |
<math>~4\pi G \rho_0 \rho_1 + \nabla^2 P_1 + \rho_0 \nabla P_1 \cdot \nabla \biggl(\frac{1}{\rho_0} \biggr) - \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] \, .</math> |
Rearranging terms, and using the replacement equilibrium relation, <math>~\nabla P_0 = - \rho_0\nabla\Phi_0</math>, gives,
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<math>~ \frac{\partial^2 \rho_1}{\partial t^2} - \nabla^2 P_1 - 4\pi G \rho_0 \rho_1 - \nabla\rho_0\cdot\nabla\Phi_1 </math> |
<math>~=</math> |
<math>~ \frac{\nabla\rho_0}{\rho_0} \cdot \biggl[ \nabla P_1 + \rho_1 \nabla \Phi_0 \biggr] + \rho_0 \nabla P_1 \cdot \nabla \biggl(\frac{1}{\rho_0} \biggr) + \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0} \nabla \Phi_0 \biggr] </math> |
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<math>~=</math> |
<math>~ \frac{\nabla\rho_0}{\rho_0} \cdot \biggl[ \nabla P_1 \biggr] + \frac{\rho_1}{\rho_0} \biggl[ \nabla\rho_0\cdot \nabla \Phi_0 \biggr] - \frac{1}{\rho_0} \nabla P_1 \cdot \nabla \rho_0 + \rho_0 \nabla \Phi_0 \cdot \nabla \biggl[ \frac{\rho_1}{\rho_0} \biggr] + \rho_1\nabla^2 \Phi_0 </math> |
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<math>~=</math> |
<math>~ \frac{\rho_1}{\rho_0} \biggl[ \nabla\rho_0\cdot \nabla \Phi_0 \biggr] - \frac{\rho_1}{\rho_0} \biggl[ \nabla \Phi_0 \cdot \nabla\rho_0\biggr] + \nabla \Phi_0 \cdot \nabla \rho_1 + 4\pi G \rho_0 \rho_1 </math> |
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<math>~\Rightarrow ~~~ \frac{\partial^2 \rho_1}{\partial t^2} - \nabla^2 P_1 - 8\pi G \rho_0 \rho_1 - \nabla\rho_0\cdot\nabla\Phi_1 - \nabla \Phi_0 \cdot \nabla \rho_1 </math> |
<math>~=</math> |
<math>~0 \, .</math> |
This is identical to Yabushita's (1968) equation (2.12).
Taff and Van Horn (1974)
Drawing on the expressions for the radial profiles of various physical variables in equilibrium isothermal spheres, as provided above, our more familiar, "key" form of the wave equation can be rewritten as,
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<math>~0</math> |
<math>~=</math> |
<math>~\frac{\xi^2}{r_0^2}\biggl\{ \frac{d^2x}{d\xi^2} + \biggl[4 - \biggl(\frac{r_0 g_0 \rho_0}{P_0}\biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)\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\biggr\} </math> |
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<math>~=</math> |
<math>~\frac{\xi^2}{r_0^2}\biggl\{ \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} + \frac{1}{4\pi G \rho_c \gamma_\mathrm{g}} \biggl[\omega^2 + (4 - 3\gamma_\mathrm{g}) \frac{4\pi G \rho_c }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] x\biggr\} </math> |
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<math>~=</math> |
<math>~\frac{4\pi G \rho_c}{\gamma_\mathrm{g} c_s^2} \biggl\{ \gamma_\mathrm{g}\frac{d^2x}{d\xi^2} + \gamma_\mathrm{g}\biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[\frac{\sigma_c^2}{6} - (3\gamma_\mathrm{g} - 4)~ \frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] x\biggr\} </math> |
Aside from the leading (constant) coefficient, this expression is identical to the linearized wave equation that Taff & Van Horn (1974) used to examine the radial pulsation modes of pressure-truncated isothermal spheres; their governing relation is displayed in the following, boxed-in image:
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Equation extracted from p. 427 of L. G. Taff & H. M. Van Horn (1974, MNRAS, 168, 427-432)
"Radial Pulsations of Finite Isothermal Gas Spheres"
MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society |
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A mapping between our expression and the one copied directly from Taff & Van Horn (1974) is facilitated by the variable mapping provided here in Table 2; note, in particular, that the roles of the two variables, <math>~x</math> and <math>~\xi</math> are swapped.
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Table 2: Mapping from Taff & Van Horn's (1974) Notation to Ours |
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| Taff & Van Horn's (1974) Notation: | <math>~x</math> | <math>~\xi</math> | <math>~\psi</math> | <math>~\Gamma_1</math> | <math>~\lambda^2</math> |
| Our Notation: | <math>~\xi</math> | <math>~x</math> | <math>~\psi</math> | <math>~\gamma_\mathrm{g}</math> | <math>~\sigma_c^2/6</math> |
Previously Published Eigenvalues and Eigenfunctions
From the Analysis of Taff and Van Horn (1974)
The left-hand column of the following composite figure presents a pair of images that present properties of the eigenvectors that resulted from the Taff & Van Horn (1974) analysis of radial oscillations in pressure-truncated isothermal spheres, assuming that the configurations remain isothermal — that is, adopting an adiabatic exponent of <math>~\Gamma_1 = 1</math> — during the oscillations. In the image extracted from their paper that is titled, "Table I", the first column of numbers identifies values of nine adopted truncation radii in the range, <math>~2 \le x_0 \le 10</math>, while the second column lists the corresponding value of <math>~\lambda_0^2</math> that were determined via their numerical analysis. For three values of the truncation radius — <math>~x_0 = 6, 7, \And 8</math> — the third column lists the values of <math>~\lambda_0^2</math> that had been previously reported by Yabushita (1968).
In the right-hand column of this composite figure, we have detailed some results from our own numerical analysis of the same set of nine configurations that were studied by Taff & Van Horn (1974) . In the second column of our table, whose heading is "Fundamental Mode," we have listed the value of <math>~\mathfrak{F}</math> that was required in our analysis in order to generate an eigenfunction whose logarithmic derivative at each configuration's surface was precisely negative three (to five significant digits). In order to facilitate quantitative comparison with the work of Taff & Van Horn, the third column of our table lists for each model the corresponding value of <math>~\lambda_0^2</math>; recalling that <math>~\gamma = 1</math> and <math>~\alpha = -1</math>, each of these values was determined via the relation,
<math>~\lambda_0^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6} = \frac{(\mathfrak{F}-2)}{6} \, .</math>
The agreement between our numerically determined fundamental-mode eigenvalues (highlighted by pink, rectangular boxes) and the ones reported by Taff & Van Horn is excellent, across the board.
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Data extracted from Tables I & II (p. 429) of L. G. Taff & H. M. Van Horn (1974, MNRAS, 168, 427-432)
"Radial Pulsations of Finite Isothermal Gas Spheres"
MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society |
From Our Analysis<math>~\alpha = -1</math> <math>~(\gamma=1)</math>
B.C.: <math>~\frac{d\ln x}{d\ln\xi} = -3.00000</math> |
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Fundamental Mode | |||||||||||||||||||||||||||||
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First Harmonic | |||||||||||||||||||||||||||||
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For three of their models — the ones having <math>~x_0 = 3, 6, \And 9</math> — Taff & Van Horn (1974) also determined an eigenvalue for the first harmonic mode of oscillation; see the seventh column of numbers, labeled <math>~\lambda_1^2</math>, that appears in the digital image of their "Table II" that we have extracted from their paper an presented in the above composite figure. Via our own numerical analysis, we have determined an eigenvalue for the first harmonic mode of oscillation in all nine of the models. These results are presented in the right-hand panel of the above composite figure, under the heading, "First Harmonic." Again, the agreement between our numerically determined eigenvalues (see the three highlighted by pink, rectangular boxes) and the ones reported by Taff & Van Horn is excellent.
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Figure extracted from p. 430 of L. G. Taff & H. M. Van Horn (1974, MNRAS, 168, 427-432)
"Radial Pulsations of Finite Isothermal Gas Spheres"
MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society | |
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From Yabushita's (1992) Analysis
In the portion (§5) of his analysis that is focused on the stability of pressure-truncated polytropic spheres, S. Yabushita (1992) examined the eigenvalue problem governed by the following wave equation:
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Radial Pulsation Equation Extracted† from p. 182 of S. Yabushita (1992)
"Similarity Between the Structure and Stability of Isothermal and Polytropic Gas Spheres"
Astrophysics and Space Science, vol. 193, pp. 173-183 © Springer |
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†Equations and text displayed here exactly as it appears in the original publication. |
Let's examine the overlap between this pair of governing relations and the ones employed by HRW66. If we replace the variable <math>~X</math> with <math>~h</math>, set <math>~\gamma = (n+1)/n</math>, and set the dimensionless eigenfrequency, <math>~s</math>, to zero in the radial pulsation equation employed by HRW66, we have,
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<math>~0 </math> |
<math>~=</math> |
<math>~ \frac{d^2 h}{dx^2} + \biggl[\frac{4}{x} + (n+1) \frac{\theta^'}{\theta} \biggr] \frac{dh}{dx} + (n+1)\biggl[ 3 - \frac{4n}{(n+1)} \biggr] \biggl[ \frac{\theta^' h}{\theta x} \biggr] </math> |
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<math>~=</math> |
<math>~ \frac{d^2 h}{dx^2} + \biggl[\frac{4}{x} + (n+1) \frac{\theta^'}{\theta} \biggr] \frac{dh}{dx} + (3-n) \biggl[ \frac{\theta^' h}{\theta x} \biggr] \, . </math> |
This matches equation (5.3) of Yabushita (1992) — see the above boxed-in image — except the <math>~(4/x)</math> term appears as <math>~(2/x)</math> in Yabushita's article; giving the benefit of the doubt, this is most likely a typographical error in Yabushita (1992). According to HRW66, the corresponding central boundary condition is,
<math>\frac{dh}{dx} = 0</math> at <math>x=0 \, .</math>
While — after changing the sign on the right-hand side of HRW66's equation (58) as argued in our accompanying discussion in order to align with the separate derivations presented by Christy (1965) and Cox (1967) — the corresponding boundary condition at the surface is,
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<math>~\frac{dh}{dx}</math> |
<math>~=</math> |
<math>~- \frac{h}{x} \biggr[ 3 - \frac{4}{\gamma} + \cancelto{0}{\frac{x s^2}{\gamma q}} \biggr]</math> |
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<math>~=</math> |
<math>~\frac{n-3}{n+1} \biggl(\frac{h}{x} \biggr) \, .</math> |
at |
<math>~x = x_0 \, .</math> |
This surface boundary condition, which has been used by the astrophysics community in the context of isolated polytropic configurations, is different from the one displayed as equation (5.4) of Yabushita (1992). The surface boundary condition chosen by Yabushita — effectively,
<math>~\frac{d \ln h}{d\ln x} = -3 \, ,</math>
— does seem to be more appropriate in the context of a study of the stability of pressure-truncated polytropes because, as argued by Ledoux & Pekeris (1941) and as reviewed in our accompanying discussion, it ensures that the pressure fluctuation at the surface is zero. It is worth noting that Yabushita's surface boundary condition matches the surface boundary condition chosen by Taff & Van Horn (1974) in their study of pressure-truncated isothermal spheres; in their words (see p. 428 of their article): [Setting the surface logarithmic derivative to negative 3] expresses the condition that the pressure at the perturbed surface always remain[s] equal to the confining pressure exerted by the external medium in which the [pressure-truncated] sphere must be embedded.
Our Numerical Integration
Algorithm
Here we show how a relatively simple, finite-difference algorithm can be developed to numerically integrate the governing LAWE from the center of the isothermal configuration, outward to its surface, which we will mark by the radial location, <math>~\xi_\mathrm{surf}</math>.
Following a parallel discussion, in preparation for discretization on a discrete radial grid, <math>~0 \le \xi_i \le \xi_\mathrm{surf}</math>, the above-defined LAWE for an isothermal may be written as,
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<math>~x_i</math> |
<math>~=</math> |
<math>~ - \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{x_i'}{\xi_i} - \biggl[ \frac{\sigma_c^2}{6\gamma} - \frac{\alpha}{\xi} \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] x_i \, . </math> |
Now, using the general finite-difference approach described separately, we make the substitutions,
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<math>~x_i'</math> |
<math>~\approx</math> |
<math>~ \frac{x_+ - x_-}{2 \Delta_\xi} \, ; </math> |
and,
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<math>~ x_i </math> |
<math>~\approx</math> |
<math>~\frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} \, ,</math> |
which will provide an approximate expression for <math>~x_+ \equiv x_{i+1}</math>, given the values of <math>~x_- \equiv x_{i-1}</math> and <math>~x_i</math>. Specifically, if the center of the configuration is denoted by the grid index, <math>~i=1</math>, then for zones, <math>~i = 3 \rightarrow N</math>,
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<math>~\frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} </math> |
<math>~=</math> |
<math>~ - \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{1}{\xi_i} \biggl[ \frac{x_+ - x_-}{2 \Delta_\xi}\biggr] - \biggl[ \frac{\sigma_c^2}{6\gamma} - \frac{\alpha}{\xi} \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] x_i </math> |
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<math>~\Rightarrow ~~~ x_+\biggl\{ \frac{1 }{\Delta_\xi^2} + \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{1}{\xi_i} \biggl[ \frac{1}{2 \Delta_\xi}\biggr] \biggr\} </math> |
<math>~=</math> |
<math>~ x_- \biggl\{ \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{1}{\xi_i} \biggl[ \frac{1}{2 \Delta_\xi}\biggr] - \frac{1 }{\Delta_\xi^2} \biggr\} + x_i \biggl\{ \frac{2}{\Delta_\xi^2} - \biggl[ \frac{\sigma_c^2}{6\gamma} - \frac{\alpha}{\xi} \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \biggr\} </math> |
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<math>~\Rightarrow ~~~ x_+\biggl\{ 2 + \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{\Delta_\xi}{\xi_i} \biggr\} </math> |
<math>~=</math> |
<math>~ x_- \biggl\{ \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{\Delta_\xi}{\xi_i} - 2 \biggr\} + x_i \biggl\{ 4 - 2\Delta_\xi^2 \biggl[ \frac{\sigma_c^2}{6\gamma} - \frac{\alpha}{\xi} \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \biggr\} </math> |
At the very center <math>~(\xi_1=0)</math>, we will set <math>~x_1=1</math>; then at the first radial grid line off the center <math>~(\xi_2 = \Delta_\xi)</math>, we will call upon the derived series expansion to set,
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<math>~x_2</math> |
<math>~=</math> |
<math>~1-\frac{\mathfrak{F}\Delta_\xi^2}{60} \, .</math> |
Some Results
When examining isothermal <math>~(\gamma_\mathrm{g} = 1)</math> radial oscillations in pressure-truncated isothermal spheres, the marginally unstable equilibrium configuration should be the one along the equilibrium sequence for which the fundamental-mode eigenfrequency is precisely zero — that is, <math>~\mathfrak{F} = -2\alpha = 2</math> — and for which the fundamental-mode eigenfunction displays <math>~d\ln x/d\ln\xi = -3</math> at the surface of the truncated sphere. Via the numerical modeling, just described, we have determined that this marginally unstable configuration has the following properties.
| Numerically Determined Properties of Marginally Unstable Model |
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| <math>~\xi_\mathrm{surf}</math> | 6.4510534 |
| <math>~\psi_\mathrm{surf}</math> | 2.642199 |
| <math>~\frac{d\psi}{d\xi}\biggr|_\mathrm{surf}</math> | 0.3773673 |
| <math>~e^{-\psi_\mathrm{surf}}</math> | 0.0712045 |
| <math>~\biggl[ e^\psi \biggl( \frac{d\psi}{d\xi} \biggr)^2\biggr]_\mathrm{surf}</math> | 1.9999588 |
| <math>~\mathfrak{F}</math> | 2 |
| <math>~\frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{surf}</math> | -3.0000000 |
| <math>~\frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2}\biggr|_\mathrm{surf}</math> | 2.262333 |
Implications
Our own analysis of this problem decidedly validates the results published by earlier authors. In particular, it appears as through Yabushita's (1968) conjecture is correct; specifically, a careful LAWE analysis seems to indicate that — if we consider only isothermal fluctuations, that is, <math>~\gamma_\mathrm{g}=1</math> — the dynamical instability sets in at the point along the equilibrium sequence where the pressure maximum arises. Let's see if we can put this conjecture on an even firmer foundation by manipulating the LAWE analytically.
When <math>~\gamma_\mathrm{g} = 1</math>, the LAWE for pressure-truncated isothermal spheres is,
|
<math>~0</math> |
<math>~=</math> |
<math>~ \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[\frac{\sigma_c^2}{6} + \frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] x </math> |
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|
<math>~=</math> |
<math>~ \frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{d\ln x}{d\ln\xi} + \xi^2\biggl[\frac{\sigma_c^2}{6} + \frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] </math> |
|
|
<math>~=</math> |
<math>~ \frac{d}{d\ln \xi}\biggl[ \frac{d\ln x}{d\ln\xi}\biggr] + \biggl[\frac{d\ln x}{d\ln\xi}-1 \biggr] \frac{d\ln x}{d\ln\xi} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{d\ln x}{d\ln\xi} + \xi^2\biggl[\frac{\sigma_c^2}{6} + \frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] </math> |
|
|
<math>~=</math> |
<math>~ \frac{dy_\xi}{d\ln \xi} + \biggl[y_\xi-1 \biggr] y_\xi + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] y_\xi + \xi^2\biggl[\frac{\sigma_c^2}{6} + \frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \, , </math> |
where, we have introduced the shorthand notation,
<math>y_\xi \equiv \frac{d\ln x}{d\ln\xi} \, .</math>
Now, precisely at the onset of dynamical instability, we should find that, <math>~\sigma_c^2 = 0</math>. We also know, from Bonnor's (1956) work, that the equilibrium configuration at the pressure maximum is identified by the model whose Lane-Emden function at its surface exhibits the property,
|
<math>~\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_\mathrm{surf}</math> |
<math>~=</math> |
<math>~2 </math> |
|
<math>~\Rightarrow ~~~ \biggl( \frac{d\psi}{d\xi}\biggr)_\mathrm{surf}</math> |
<math>~=</math> |
<math>~\biggl[ 2^{1 / 2} e^{-\psi/2} \biggr]_\mathrm{surf} \, .</math> |
Therefore, at the surface the condition expected from the LAWE is,
|
<math>~0</math> |
<math>~=</math> |
<math>~\biggl\{ \xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi-1 ) y_\xi + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] y_\xi + \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr\}_\mathrm{surf} </math> |
|
|
<math>~=</math> |
<math>~\biggl\{ \xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi + 3 ) y_\xi + \xi (1-y_\xi)\biggl( \frac{d\psi}{d\xi} \biggr) \biggr\}_\mathrm{surf} </math> |
|
|
<math>~=</math> |
<math>~\biggl\{ \xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi + 3 ) y_\xi + 2^{1 / 2}\xi (1-y_\xi)e^{-\psi/2} \biggr\}_\mathrm{surf} \, . </math> |
Let's see what this implies if, at the surface of the configuration, the logarithmic derivative of the fundamental-mode eigenfunction has the value,
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<math>~[y_\xi]_\mathrm{surf} = \frac{d\ln x}{d\ln\xi} \biggr|_\mathrm{surf}</math> |
<math>~=</math> |
<math>~-3 \, .</math> |
In this case, we have,
|
<math>~0</math> |
<math>~=</math> |
<math>~\biggl\{ \frac{dy_\xi}{d\xi} + 2^{5 / 2}e^{-\psi/2} \biggr\}_\mathrm{surf} \, . </math> |
Not sure what to make of this!
References
Here we have drawn primarily from the following three sources:
- §5.3.8 (p. 372) of Horedt's (2004) treatise on Polytropes: Applications in Astrophysics and Related Fields
- S. Yabushita (1968, MNRAS, 140, 109) — Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres
- L. G. Taff & H. M. Horn (1974, MNRAS, 168, 427-432) — Radial Pulsations of Finite Isothermal Gas Spheres
See also:
- S. Yabushita (1975, MNRAS, 171, 85) — Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres — II
- S. Yabushita (1992, Astrophys. & Space Sciences, 193, 173 - 183) — Similarity between the structure and stability of isothermal and polytropic gas spheres
- J. P. Chieze (1987, A&A, 171, 225 - 232) — The fragmentation of molecular clouds. I - The mass-radius-velocity dispersion relations
- Radial Oscillations of Polytropic Spheres
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© 2014 - 2021 by Joel E. Tohline |