User:Tohline/SSC/VariationalPrinciple
Ledoux's Variational Principle
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All of the discussion in this chapter will build upon our derivation elsewhere of the,
LAWE: Linear 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> |
We will draw heavily from the paper published by Ledoux & Pekeris (1941), and from pp. 458-474 of the review by P. Ledoux & Th. Walraven (1958) in explaining how the variational principle can be used to identify the eigenvector of the fundamental mode of radial oscillation in spherically symmetric configurations.
Ledoux and Pekeris (1941)
Historically, by the 1940s, the LAWE was a relatively familiar one to astrophysicists. For example, the opening paragraph of a 1941 paper by Ledoux & Pekeris (1941, ApJ, 94, 124), reads:
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Paragraph extracted from P. Ledoux & C. L. Pekeris (1941)
"Radial Pulsations of Stars"
ApJ, vol. 94, pp. 124-135 © American Astronomical Society |
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If we divide their equation (1) through by <math>~Xr = \Gamma_1 P r</math> and recognize that,
<math> \frac{dX}{dr} = \frac{dX}{dm}\frac{dm}{dr} = - \Gamma_1 g_0 \rho \, , </math>
we obtain,
<math> \frac{d^2\xi}{dr^2} + \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr} +\frac{\rho}{\Gamma_1 P} \biggl[ \sigma^2 + (4 - 3\Gamma_1) \frac{g_0}{r} \biggr] \xi = 0 \, . </math>
Clearly, this 2nd-order, ordinary differential equation is the same as our derived LAWE, but with a more general definition of the adiabatic exponent that allows consideration of a situation where the total pressure is a sum of both gas and radiation pressure.
Multiplying this last equation through by <math>~\Gamma_1 P r^4</math>, and recognizing that,
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<math>~(r^4 \Gamma_1 P)\frac{d^2\xi}{dr^2} </math> |
<math>~=</math> |
<math>~ \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr] </math> |
we can write,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr] + ( \Gamma_1 P r^4 ) \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr} +\rho \biggl[ \sigma^2 r^4 + (4 - 3\Gamma_1) g_0 r^3\biggr] \xi </math> |
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<math>~=</math> |
<math>~ \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \biggl[4r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr} \biggr] \frac{d\xi}{dr} + \biggl[ 4 r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr}\biggr] \frac{d\xi}{dr} +\biggl[ \sigma^2 \rho r^4 - (4 - 3\Gamma_1) r^3 \frac{dP}{dr} \biggr] \xi </math> |
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<math>~=</math> |
<math>~ \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] +\biggl[ \sigma^2 \rho r^4 + (3\Gamma_1 - 4) r^3 \frac{dP}{dr} \biggr] \xi </math> |
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<math>~=</math> |
<math>~ \frac{d}{dr}\biggl[ \Gamma_1 P r^4 ~\frac{d\xi}{dr} \biggr] +\biggl[ \sigma^2 r^4 \rho + 4 Gm (r ) r \rho + 3\Gamma_1 r^3 \frac{dP}{dr} \biggr] \xi \, . </math> |
Assuming that <math>~\Gamma_1</math> is uniform throughout the configuration, this last expression is the same as equation (3) of Ledoux & Pekeris (1941), while the next-to-last expression is identical to equation (58.1) of Ledoux & Walraven (1958).
Review by Ledoux and Walraven (1958)
Here we are especially interested in understanding the origin of equation (59.10) of P. Ledoux & Th. Walraven (1958), which appears in §59 (pp. 464 - 466) of their Handbuch der Physik article.
From our accompanying summary of the set of nonlinear governing relations, we highlight the
Euler + Poisson Equations
<math>\frac{d^2 r}{dt^2} = - 4\pi r^2 \frac{dP}{dm} - \frac{Gm}{r^2} </math>
Repeating a result from our separate derivation, linearization of the two terms on the righthand side of this equation gives,
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<math> r^2 \frac{dP}{dm} </math> |
<math> \rightarrow </math> |
<math> r_0^2 \biggl[1 + x~ e^{i\omega t} \biggr]^2 \biggl\{\frac{dP_0}{dm} \biggl[1 + p~ e^{i\omega t} \biggr] + P_0~e^{i\omega t} \frac{dp}{dm} \biggr\} \approx r_0^2 \frac{dP_0}{dm} \biggl[1 + (2x+p)~ e^{i\omega t} \biggr] + P_0 r_0^2~e^{i\omega t} \frac{dp}{dm} </math> |
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<math> \frac{Gm}{r^2} </math> |
<math> \rightarrow </math> |
<math> \frac{Gm}{ r_0^2} \biggl[1 + x~ e^{i\omega t} \biggr]^{-2} \approx \frac{Gm}{ r_0^2} \biggl[1 -2 x~ e^{i\omega t} \biggr] \, . </math> |
Adopting the terminology of Ledoux & Walraven (1958), the "variation" of each of these terms is obtained by subtracting off the leading order pieces — which presumably cancel in equilibrium. In particular, drawing a parallel with their equation (59.1), we can write,
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<math> ~\delta \biggl( - \frac{Gm}{r^2} \biggr) </math> |
<math> ~\approx </math> |
<math> \frac{Gm}{ r_0^2} \biggl[2 x~ e^{i\omega t} \biggr] \, . </math> |
And, drawing a parallel with their equation (59.2), we have,
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<math> ~\delta \biggl( - \frac{1}{\rho} \cdot \frac{dP}{dr} \biggr) = \delta \biggl( - 4\pi r^2 \frac{dP}{dm} \biggr) </math> |
<math> ~\approx </math> |
<math> -4\pi r_0^2 \frac{dP_0}{dm} \biggl[(2x+p)~ e^{i\omega t} \biggr] - 4\pi P_0 r_0^2~e^{i\omega t} \frac{dp}{dm} </math> |
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<math> ~= </math> |
<math>~\biggl\{ \frac{Gm}{r_0^2}\biggl[(2x)\biggr] -4\pi r_0^2 \frac{dP_0}{dm} \biggl[(p) \biggr] - 4\pi P_0 r_0^2 \frac{dp}{dm} \biggr\} e^{i\omega t} </math> |
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<math> ~= </math> |
<math>~\biggl\{ \biggl( \frac{2 Gm}{r_0^2}\biggr) x -\frac{1}{\rho_0} \cdot \frac{d}{dr_0} \biggl[ P_0 p \biggr] \biggr\} e^{i\omega t} \, . </math> |
Now, if we combined the linearized continuity equation and the linearized (adiabatic form of the) first law of thermodynamics, as derived elsewhere, we can write,
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<math>~p = \gamma_\mathrm{g} d</math> |
<math>~=</math> |
<math>~- \gamma_\mathrm{g} \biggl[ 3x + r_0 \frac{dx}{dr_0} \biggr] </math> |
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<math>~=</math> |
<math>~- \frac{\gamma_\mathrm{g}}{r_0^2} \frac{d}{dr_0} \biggl( r_0^3 x \biggr) \, . </math> |
Hence,
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<math> ~\delta \biggl( - \frac{1}{\rho} \cdot \frac{dP}{dr} \biggr) </math> |
<math> ~\approx </math> |
<math>~\biggl\{ \biggl( \frac{2 Gm}{r_0^2}\biggr) x +\frac{1}{\rho_0} \cdot \frac{d}{dr_0} \biggl[ \frac{\gamma_\mathrm{g} P_0}{r_0^2} \frac{d}{dr_0} \biggl( r_0^3 x \biggr) \biggr] \biggr\} e^{i\omega t} \, . </math> |
So, given that a mapping from our notation to that used by Ledoux & Walraven (1958) requires <math>~xe^{i\omega t} \rightarrow \zeta/r_0</math>, I understand the origins of their equations (59.1) and (59.2). But I do not yet understand how … "Accordingly, the acting forces per unit volume can be considered as deriving from a potential density"
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<math>~\rho_0 \mathcal{V}</math> |
<math>~=</math> |
<math>~ - \biggl( \frac{2Gm}{r_0^3}\biggr) \rho_0 \zeta^2 + \frac{\gamma_\mathrm{g} P_0}{2} \biggl[ \frac{1}{r_0^2} \frac{\partial(r_0^2 \zeta)}{\partial r_0} \biggr]^2 \, . </math> |
It is clear that, once I understand the origin of this expression for the potential density, I will understand how the "Lagrangian density" as defined by their equation (47.8), viz.,
<math>~\mathcal{L} = \rho_0 [\mathcal{K} - \mathcal{V}] \, ,</math>
becomes (see their equation 59.5),
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<math>~\mathcal{L}</math> |
<math>~=</math> |
<math>~\frac{\rho_0}{2} {\dot\zeta}^2 + \biggl( \frac{2Gm}{r_0^3}\biggr) \rho_0 \zeta^2 - \frac{\gamma_\mathrm{g} P_0}{2} \biggl[ \frac{1}{r_0^2} \frac{\partial(r_0^2 \zeta)}{\partial r_0} \biggr]^2 \, . </math> |
Noting that, <math>~\dot\zeta = i\omega r_0 x e^{i\omega t}</math>, this in turn gives,
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<math>~L \equiv \int_0^R 4\pi r_0^2 \mathcal{L} dr_0</math> |
<math>~=</math> |
<math>~ 4\pi \int_0^R \biggl\{ \frac{\rho_0}{2} {\dot\zeta}^2 + \biggl( \frac{2Gm}{r_0^3}\biggr) \rho_0 \zeta^2 - \frac{\gamma_\mathrm{g} P_0}{2} \biggl[ \frac{1}{r_0^2} \frac{\partial(r_0^2 \zeta)}{\partial r_0} \biggr]^2 \biggr\}r_0^2 dr_0 </math> |
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<math>~=</math> |
<math>~ 4\pi e^{2i\omega t} \int_0^R \biggl\{ \frac{\rho_0}{2} (i \omega r_0 x)^2 + \biggl( \frac{2Gm}{r_0^3}\biggr) \rho_0 (r_0 x)^2 - \frac{\gamma_\mathrm{g} P_0}{2} \biggl[ \frac{1}{r_0^2} \frac{\partial(r_0^3 x)}{\partial r_0} \biggr]^2 \biggr\}r_0^2 dr_0 </math> |
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<math>~=</math> |
<math>~ 2\pi e^{2i\omega t} \int_0^R \biggl\{ -\rho_0 \omega^2 r_0^2 x^2 - 4 r_0 x^2 \frac{dP_0}{dr_0} - \frac{\gamma_\mathrm{g} P_0}{r_0^4} \biggl[ 3r_0^2 x + r_0^3 \frac{\partial x}{\partial r_0} \biggr]^2 \biggr\}r_0^2 dr_0 </math> |
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<math>~=</math> |
<math>~ 2\pi e^{2i\omega t} \int_0^R \biggl\{ -\rho_0 \omega^2 r_0^4 x^2 - \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 - 4 r_0^3 x^2 \frac{dP_0}{dr_0} - 3\gamma_\mathrm{g} P_0 \biggl[ 3r_0^2 x^2 + 2r_0^3 x \frac{\partial x}{\partial r_0} \biggr] \biggr\}dr_0 </math> |
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<math>~=</math> |
<math>~ 2\pi e^{2i\omega t} \int_0^R \biggl\{ -\rho_0 \omega^2 r_0^4 x^2 - \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 - 4 r_0^3 x^2 \frac{dP_0}{dr_0} +r_0^3 x^2 \frac{d}{dr_0}\biggl(3\gamma_\mathrm{g}P_0\biggr) -\frac{d}{dr_0}\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr] \biggr\}dr_0 </math> |
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<math>~=</math> |
<math>~ 2\pi e^{2i\omega t} \biggl\{ - \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0 - \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0 + \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0 -\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R} \biggr\} \, . </math> |
The group of terms inside the curly braces, here, matches the group of terms inside the curly braces of Ledoux & Walraven's equation (59.8) if we acknowledge that:
- Our <math>~\omega^2</math> has the same meaning as, but the opposite sign of, their <math>~\sigma^2</math>.
- Our last term goes to zero because, <math>~r_0 = 0</math> at the center, while <math>~P_0 = 0</math> at the surface.
LP41 Again
After setting the last term to zero, this last expression can be rewritten as,
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<math>~2 e^{-2i\omega t} L </math> |
<math>~=</math> |
<math>~ - \omega^2 \int_0^R 4\pi\rho_0 r_0^4 x^2 dr_0 - \int_0^R 4\pi \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0 - (3\gamma_\mathrm{g} - 4)\int_0^R 4\pi\rho_0 r_0^3 x^2 \biggl( -\frac{1}{\rho_0}\frac{dP_0}{dr_0} \biggr)dr_0 </math> |
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<math>~=</math> |
<math>~ - \omega^2 \int_0^R x^2 r_0^2 dm - \gamma_\mathrm{g} \int_0^R \biggl[ r_0 \biggl( \frac{\partial x}{\partial r_0}\biggr) \biggr]^2 P_0 dV - (3\gamma_\mathrm{g} - 4) \int_0^R r_0 x^2 \biggl( \frac{Gm}{r_0^2} \biggr)dm </math> |
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<math>~- \biggl[ \frac{2 e^{-2i\omega t}}{\int_0^R x^2 r_0^2 dm } \biggr] L </math> |
<math>~=</math> |
<math>~ \omega^2 + \biggl\{ \frac{\gamma_\mathrm{g} \int_0^R \bigl[ r_0 \bigl( \frac{\partial x}{\partial r_0}\bigr) \bigr]^2 P_0 dV + (3\gamma_\mathrm{g} - 4) \int_0^R x^2 \bigl( \frac{Gm}{r_0} \bigr)dm}{\int_0^R x^2 r_0^2 dm} \biggr\} \, . </math> |
As is explained in detail in §59 (pp. 464 - 465) of Ledoux & Walraven (1958), and summarized in §1 of Ledoux & Pekeris (1941), the function inside the curly braces of this last expression will be minimized if the radially dependent displacement function, <math>~x</math>, is set equal to the eigenfunction of the fundamental mode of radial oscillation, <math>~x_0</math>; and, after evaluation, the minimum value of this expression will be equal to (the negative of) the square of the fundamental-mode oscillation frequency, <math>~\omega^2</math>. This explicit mathematical statement is contained within equation (8) of Ledoux & Pekeris and within equation (59.10) of Ledoux & Walraven.
Now, as we have discussed separately — see, also, p. 64, Equation (12) of [C67] — the gravitational potential energy of the unperturbed configuration is given by the integral,
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<math>~W_\mathrm{grav}</math> |
<math>~=</math> |
<math>~ - \int_0^{M} \biggl( \frac{Gm}{r_0} \biggr) dm \, ;</math> |
for adiabatic systems, the internal energy is,
<math> U_\mathrm{int} = \frac{1}{({\gamma_g}-1)} \int_0^R P_0 dV
\, ;</math>
and — see the text at the top of p. 126 of Ledoux & Pekeris (1941) — the moment of inertia of the configuration about its center is,
<math> I = \int_0^M r_0^2 dm
\, .</math>
Hence, the function to be minimized may be written as,
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<math>~ \biggl\{ \frac{\gamma_\mathrm{g} (\gamma_\mathrm{g}-1) \int_0^R \bigl[ r_0 \bigl( \frac{\partial x}{\partial r_0}\bigr) \bigr]^2 dU_\mathrm{int} - (3\gamma_\mathrm{g} - 4) \int_0^R x^2 dW_\mathrm{grav}}{\int_0^R x^2 dI} \biggr\} \, . </math> |
This expression appears in equation (9) of Ledoux & Pekeris (1941).
Chandrasekhar (1964)
In a paper titled, A General Variational Principle Governing the Radial and the Non-Radial Oscillations of Gaseous Masses, S. Chandrasekhar (1964, ApJ, 139, 664) independently derived the Ledoux-Pekeris Lagrangian. Returning to the second line of our effort to simplify the above definition of the Lagrangian, and making the substitution, <math>~\psi \equiv r_0^3 x</math>, we have,
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<math>~2L </math> |
<math>~=</math> |
<math>~ 4\pi e^{2i\omega t} \int_0^R \biggl\{ \rho_0 \biggl( \frac{i \omega \psi}{r_0^2} \biggr)^2 + \biggl( \frac{4Gm}{r_0^3}\biggr) \rho_0 \biggl( \frac{\psi}{r_0^2} \biggr)^2 - \gamma_\mathrm{g} P_0 \biggl[ \frac{1}{r_0^2} \frac{\partial(\psi)}{\partial r_0} \biggr]^2 \biggr\}r_0^2 dr_0 </math> |
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<math>~=</math> |
<math>~ 4\pi e^{2i\omega t} \int_0^R \biggl\{ -\omega^2 \rho_0 \biggl( \frac{\psi}{r_0} \biggr)^2 - 4\biggl( \frac{dP_0}{dr_0}\biggr) \biggl( \frac{\psi^2}{r_0^3} \biggr) - \gamma_\mathrm{g} P_0 \biggl[ \frac{1}{r_0} \frac{\partial(\psi)}{\partial r_0} \biggr]^2 \biggr\}dr_0 </math> |
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<math>~=</math> |
<math>~- 4\pi e^{2i\omega t} \int_0^R \biggl\{ \omega^2 \rho_0 \psi^2 + 4\biggl( \frac{dP_0}{dr_0}\biggr) \biggl( \frac{\psi^2}{r_0} \biggr) + \gamma_\mathrm{g} P_0 \biggl[ \frac{\partial\psi}{\partial r_0} \biggr]^2 \biggr\} \frac{dr_0}{r_0^2} . </math> |
This integral expression matches the integral expression that appears in equation (49) of Chandrasekhar (1964), if we accept that our squared frequency, <math>~\omega^2</math>, has the opposite sign to Chandrasekhar's <math>~\sigma^2</math>. Chandrasekhar acknowledged that, for radial modes of oscillation, his result was the same as that derived earlier by Ledoux and his collaborators.
Examples
Returning to the last line of our above definition of the Lagrangian, that is,
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<math>~L </math> |
<math>~=</math> |
<math>~ 2\pi e^{2i\omega t} \biggl\{ - \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0 - \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0 + \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0 -\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R} \biggr\} \, , </math> |
let's attempt to evaluate the terms inside the curly braces for the case of pressure-truncated polytropic configurations because, as has been discussed separately, we have an analytic expression for the eigenvector of the fundamental-mode of radial oscillation. Dividing through by <math>~P_c R_\mathrm{eq}^3</math> gives,
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<math>~\frac{L_{\{\}} }{P_c R_\mathrm{eq}^3}</math> |
<math>~=</math> |
<math>~ - \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0 - \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0 + \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0 -\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R} </math> |
where, we have set the pressure at the (truncated) surface to the value, <math>~P_0|_\mathrm{surface} = P_e</math>.
We will adopt the normalizations used in the Stahler presentation. Specifically,
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<math> ~M </math> |
<math>~=~</math> |
<math> M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\tilde\xi} </math> |
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<math> ~R_\mathrm{eq} </math> |
<math>~=~</math> |
<math> R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\tilde\xi} </math> |
where,
<math>M_\mathrm{SWS} = \biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
<math> R_\mathrm{SWS} = \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, . </math>
In addition, from our introductory discussion of polytropes, we have,
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<math>~\rho_0</math> |
<math>~\equiv</math> |
<math>~\rho_c \theta^n </math> |
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<math>~\Rightarrow ~~~ P_0</math> |
<math>~=</math> |
<math>~K_n \biggl[ \rho_c \theta^n \biggr]^{(n+1)/n} </math> |
and,
<math> r_0 \equiv a_\mathrm{n} \xi , </math>
where,
<math>~ a^2_\mathrm{n} = \frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \, . </math>
Also, from our definition of the structural form factors and our evaluation of them in the context of pressure-truncated configurations, we know that,
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<math>~\rho_c</math> |
<math>~=</math> |
<math>~\frac{\bar\rho}{ {\tilde{f}}_M}</math> |
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<math>~=</math> |
<math>~\biggl[\frac{3M}{4\pi R_\mathrm{eq}^3}\biggr] \biggl[ \frac{3}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr) \biggr]_{\tilde\xi}^{-1}</math> |
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<math>~=</math> |
<math>~\frac{1}{4\pi} \biggl[\frac{M/M_\mathrm{SWS}}{(R/R_\mathrm{SWS})^3}\biggr] \biggl[ \frac{M_\mathrm{SWS}}{ R_\mathrm{SWS}^3} \biggr]\biggl(- \frac{d\theta}{d\xi}\biggr)_{\tilde\xi}^{-1} \tilde\xi</math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |