Difference between revisions of "User:Tohline/SSC/Perspective Reconciliation"

From VistrailsWiki
Jump to navigation Jump to search
(→‎Eulerian Reformulation: Add reference link to Stothers & Frogel (1967))
(Tidy up references, etc.)
Line 14: Line 14:
</div>
</div>


and, hence, as established via a [[User:Tohline/SSC/Perturbations#Consistent_Lagrangian_Formulation|Lagrangian formulation of the problem]].  After finishing that review, we became aware that a separate study of radial pulsation modes in the homogeneous sphere has been published by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf S. Rosseland (1969)]
and, hence, as established via a [[User:Tohline/SSC/Perturbations#Consistent_Lagrangian_Formulation|Lagrangian formulation of the problem]].  The eigenvectors and eigenvalues that Sterne derived for the first two or three radial modes have also appeared &#8212; usually in the context of separate, re-derivations &#8212; in other publications:  See, for example, &sect;38.2 (pp. 402-403) of [[User:Tohline/Appendix/References#KW94|[<font color="red">KW94</font>]]].
 
After finishing that review, we became aware that a separate study of radial pulsation modes in the homogeneous sphere has been published by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf S. Rosseland (1969)]
In his book titled, ''The Pulsation Theory of Variable Stars'' (see, specifically his &sect; 3.2, beginning on p. 27).  Rosseland solved an eigenvalue problem as defined by the relation (see his equation 2.23 on p. 20, with the adiabatic condition being enforced by setting the right-hand-side equal to zero),
In his book titled, ''The Pulsation Theory of Variable Stars'' (see, specifically his &sect; 3.2, beginning on p. 27).  Rosseland solved an eigenvalue problem as defined by the relation (see his equation 2.23 on p. 20, with the adiabatic condition being enforced by setting the right-hand-side equal to zero),
<div align="center">
<div align="center">
Line 37: Line 39:
</div>
</div>


which he derived in an earlier section of his book via an Eulerian formulation of the problem.   
Rosseland derived this expression in an earlier section of his book via an Eulerian formulation of the problem.   
Realizing that, for a spherically symmetric system,
Realizing that, for a spherically symmetric system,
<div align="center">
<div align="center">
<math>\nabla\cdot \vec\xi = \frac{1}{r^2}\frac{\partial}{\partial r}\biggl(r^2 \xi\biggr) = \frac{\partial \xi}{\partial r} + \frac{2\xi}{r} \, ,</math>
<math>\nabla\cdot \vec\xi = \frac{1}{r^2}\frac{\partial}{\partial r}\biggl(r^2 \xi\biggr) = \frac{\partial \xi}{\partial r} + \frac{2\xi}{r} \, ,</math>
</div>
</div>
as is demonstrated in [[User:Tohline/SSC/Structure/Other_Analytic_Models#Eulerian_Approach|and accompanying discussion]], we can rewrite this relation in the more familiar form of a 2<sup>nd</sup>-order ODE, namely,
as is demonstrated in [[User:Tohline/SSC/Structure/Other_Analytic_Models#Eulerian_Approach|and accompanying discussion]], this relation can be rewritten in the more familiar form of a 2<sup>nd</sup>-order ODE, namely,


<div align="center">
<div align="center">
Line 86: Line 88:
</math><br />
</math><br />
</div>
</div>
Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, <math>~n=1</math> polytrope are plugged into it, we obtain exactly the same 2<sup>nd</sup>-order ODE as published by Murphy.
Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, <math>~n=1</math> polytrope are plugged into it, as is demonstrated in an [[User:Tohline/SSC/Stability/Polytropes#MurphyFiedler1985b|accompanying discussion]], we obtain exactly the same 2<sup>nd</sup>-order ODE as published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985)].


==Eulerian Reformulation==
==Eulerian Reformulation==
Line 114: Line 116:
Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, [[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution|"linear stellar model"]] are plugged into it, we obtain exactly the same 2<sup>nd</sup>-order ODE as published by [http://adsabs.harvard.edu/abs/1967ApJ...148..305S R. Stothers &amp; J. A. Frogel (1967, ApJ, 148, 305)] &#8212; see their equation (2).
Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, [[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution|"linear stellar model"]] are plugged into it, we obtain exactly the same 2<sup>nd</sup>-order ODE as published by [http://adsabs.harvard.edu/abs/1967ApJ...148..305S R. Stothers &amp; J. A. Frogel (1967, ApJ, 148, 305)] &#8212; see their equation (2).


==Linearizing the Key Relations==
 
{{LSU_WorkInProgress}}
 
=Linearizing the Key Relations=


<table border="3" align="center" cellpadding="10">
<table border="3" align="center" cellpadding="10">

Revision as of 20:36, 22 June 2015

Reconciling Eulerian versus Lagrangian Perspectives

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |


Comment by J. E. Tohline on 22 June 2015: Last night I realized that a key to understanding how to reconcile the Eulerian and Lagrangian perspectives was an analysis of the eigenvalue problem for the homogeneous sphere.

In an accompanying discussion, we have reviewed T. E. Sterne's (1937, MNRAS, 97, 582) study of radial pulsation modes in the homogeneous sphere. He solved the eigenvalue problem as defined by the

Adiabatic Wave (or Radial Pulsation) Equation

LSU Key.png

<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>

and, hence, as established via a Lagrangian formulation of the problem. The eigenvectors and eigenvalues that Sterne derived for the first two or three radial modes have also appeared — usually in the context of separate, re-derivations — in other publications: See, for example, §38.2 (pp. 402-403) of [KW94].

After finishing that review, we became aware that a separate study of radial pulsation modes in the homogeneous sphere has been published by S. Rosseland (1969) In his book titled, The Pulsation Theory of Variable Stars (see, specifically his § 3.2, beginning on p. 27). Rosseland solved an eigenvalue problem as defined by the relation (see his equation 2.23 on p. 20, with the adiabatic condition being enforced by setting the right-hand-side equal to zero),

<math>~\frac{\partial}{\partial r} \biggl( \gamma P_0 \nabla\cdot \vec{\xi}\biggr) + \biggl( \omega^2 + \frac{4g_0}{r}\biggr) \rho_0 \xi</math>

<math>~=</math>

<math>~0 \, ,</math>

where,

<math>~\vec\xi = \mathbf{\hat{e}}_r \xi(r) \, ,</math>

Rosseland derived this expression in an earlier section of his book via an Eulerian formulation of the problem. Realizing that, for a spherically symmetric system,

<math>\nabla\cdot \vec\xi = \frac{1}{r^2}\frac{\partial}{\partial r}\biggl(r^2 \xi\biggr) = \frac{\partial \xi}{\partial r} + \frac{2\xi}{r} \, ,</math>

as is demonstrated in and accompanying discussion, this relation can be rewritten in the more familiar form of a 2nd-order ODE, namely,

<math>~P_0 \frac{\partial^2 \xi}{\partial r^2} + \biggl[ \frac{2P_0}{r}- \rho_0 g_0 \biggr] \frac{\partial \xi}{\partial r} + \biggl[ \biggl( \frac{\omega^2\rho_c}{\gamma} + \frac{4\rho_c g_0}{\gamma r}\biggr) \biggl(\frac{\rho_0}{\rho_c}\biggr) - \biggl(\frac{2\rho_c g_0 }{r}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2P_0}{r^2} \biggr] \xi </math>

<math>~=</math>

<math>~0 \, .</math>

Lagrangian Reformulation

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

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

and the characteristic gravitational acceleration as,

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

we can rewrite the Lagrangian-formulated wave equation as,

<math> \biggl(\frac{P_0}{P_c}\biggr)\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0}\biggl(\frac{P_0}{P_c}\biggr) - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \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>

Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, <math>~n=1</math> polytrope are plugged into it, as is demonstrated in an accompanying discussion, we obtain exactly the same 2nd-order ODE as published by Murphy & Fiedler (1985).

Eulerian Reformulation

Using the same characteristic time scale and gravitational acceleration, we can similarly rewrite the Eulerian-formulated expression as,

<math>~\biggl(\frac{P_0}{P_c}\biggr) \frac{\partial^2 \xi}{\partial \chi_0^2} + \biggl[ \frac{2}{\chi_0}\biggl(\frac{P_0}{P_c}\biggr) - \frac{g_0 }{g_\mathrm{SSC}}\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggr] \frac{\partial \xi}{\partial \chi_0} + \biggl\{ \biggl[\frac{\omega^2\tau_\mathrm{SSC}^2}{\gamma} + \frac{2}{\chi_0 } \biggl(\frac{2}{\gamma } - 1\biggr)\frac{g_0}{g_\mathrm{SSC}}\biggr] \biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2}{\chi_0^2} \biggl(\frac{P_0}{P_c}\biggr) \biggr\} \xi </math>

<math>~=</math>

<math>~0 \, .</math>

Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, "linear stellar model" are plugged into it, we obtain exactly the same 2nd-order ODE as published by R. Stothers & J. A. Frogel (1967, ApJ, 148, 305) — see their equation (2).



Work-in-progress.png

Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
|   Go Home   |


Linearizing the Key Relations

Continuity Equation
Lagrangian Perspective Eulerian Perspective

<math>~\frac{d\rho}{dt}</math>

<math>~=</math>

<math>~- \rho \nabla\cdot\vec{v}</math>

<math>~\frac{\partial\rho}{\partial t}</math>

<math>~=</math>

<math>~- \rho \nabla\cdot\vec{v} - \vec{v}\cdot \nabla\rho</math>

Spherically Symmetric Initial Configurations & Purely Radial Perturbations

<math>~\frac{d\rho}{dt}</math>

<math>~=</math>

<math>~- \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr)</math>

<math>~\frac{\partial\rho}{\partial t}</math>

<math>~=</math>

<math>~- \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr) - v_r \frac{\partial \rho}{\partial r}</math>

In an interval of time, <math>~dt = \partial t</math>, a fluid element initially at position <math>~r_0</math> moves to position, <math>~r = r_0 + r_1 = r_0(1 + \xi)</math>. [For later reference, note that <math>~\xi</math> can be a function of <math>~r_0</math> as well as of <math>~t</math>.] On the righthand side of the expression, the radial coordinate will be handled as follows: From the Lagrangian perspective, <math>~r \rightarrow r_0 (1+ \xi)</math>, while from the Eulerian perspective, we want to stay at the original coordinate location, so <math>~r \rightarrow r_0</math>. From both perspectives,

<math>~v_r = \frac{\partial ( r_0 \xi )}{\partial t} = r_0 \frac{\partial \xi}{\partial t} \, .</math>

Riding with the fluid element (Lagrangian perspective), <math>~\rho \rightarrow (\rho_0 + \rho_L) = \rho_0(1+s_L)</math>, while at a fixed coordinate location (Eulerian perspective), <math>~\rho \rightarrow (\rho_0 + \rho_E) = \rho_0(1 + s_E)</math>. Finally, in maintaining a Lagrangian perspective, we will need to ensure that the same element of mass is being tracked as we "ride along" with the fluid element to its new position. For radial perturbations associated with a spherically symmetric configuration, this means that the differential mass in each spherical shell, <math>~dm = 4\pi r^2 \rho dr</math>, must remain constant; that is,

<math>~4\pi r_0^2 \rho_0 dr_0</math>

<math>~=</math>

<math>~4\pi r^2 \rho dr</math>

 

<math>~=</math>

<math>~4\pi [r_0(1+\xi)]^2 \rho_0(1+s_L) dr</math>

 

<math>~=</math>

<math>~4\pi r_0^2 \rho_0 \biggl(1+2\xi + \cancelto{\scriptstyle\text{small}}{\xi^2} + \cdots \biggr) (1+s_L) dr</math>

 

<math>~\approx</math>

<math>~4\pi r_0^2 \rho_0 \biggl(1+2\xi + s_L + 2\cancelto{\scriptstyle\mathrm{small}}{\xi s_L} \biggr) dr</math>

<math>~\Rightarrow~~~ \frac{d}{dr}</math>

<math>~\approx</math>

<math>~(1+2\xi + s_L ) \frac{d}{dr_0} \, .</math>

<math>~\frac{d}{dt}\biggl[\rho_0(1+s_L)\biggr]</math>

<math>~=</math>

<math>~- \biggl\{ \frac{\rho_0(1+\cancelto{}{s_L})}{[r_0(1+\cancelto{}{\xi})]^2} \biggr\}(1+2\cancelto{}{\xi s_L}) \frac{\partial}{\partial r_0} \biggl\{ [r_0(1+\cancelto{}{\xi})]^2 v_r \biggr\}</math>

<math>~\Rightarrow~~~ \frac{d s_L}{dt}</math>

<math>~=</math>

<math>~- \biggl[\frac{2v_r}{r_0} + \frac{\partial v_r}{\partial r_0} \biggr] - \frac{1}{\rho_0} \frac{d\rho_0}{dt}</math>

<math>~\frac{\partial}{\partial t}\biggl[\rho_0(1+s_E)\biggr]</math>

<math>~=</math>

<math>~- \frac{\rho_0(1+\cancelto{\mathrm{small}}{s_E})}{r_0^2} \frac{\partial}{\partial r_0} \biggl( r_0^2 v_r \biggr) - v_r \frac{\partial [\rho_0(1+\cancelto{\mathrm{small}}{s_E})] }{\partial r_0}</math>

<math>~\Rightarrow~~~ \frac{\partial s_E}{\partial t}</math>

<math>~=</math>

<math>~- \biggl[\frac{2v_r}{r_0} + \frac{\partial v_r}{\partial r_0} \biggr] - \frac{v_r}{\rho_0} \frac{\partial \rho_0 }{\partial r_0}</math>

Note: The last term that appears on the righthand side of the two expressions appears to be different. But if, as we are assuming here, <math>~\rho_0</math> has no explicit time dependence but may be considered to be a function of the radial coordinate, <math>~r_0</math>, then the two terms are the same. This is because, quite generically for any scalar function <math>~q</math>, the total time-derivative (Lagrangian perspective) differs from the partial time-derivative (Eulerian perspective) via the expression, <math>dq/dt - \partial q /\partial t = \vec{v}\cdot \nabla q</math>. In our case, <math>~\partial \ln \rho_0/\partial t = 0</math>, so <math>~d\ln\rho_0/dt = \vec{v}\cdot \nabla \ln \rho_0</math>.

<math>~s_L ~~\rightarrow~~ \Delta_L(r_0) e^{i\omega t}</math>             … and …             <math>~s_E ~~\rightarrow~~ \Delta_E(r_0) e^{i\omega t}</math>

<math>~\xi ~~\rightarrow~~ x(r_0) e^{i\omega t}</math>             <math>\Rightarrow</math>             <math>~v_r ~~\rightarrow~~ (i\omega)r_0 x(r_0) e^{i\omega t}</math>

<math>~e^{i\omega t} \biggl[ (i\omega)\Delta_L + \frac{d\Delta_L}{dt} \biggr]</math>

<math>~=</math>

<math>~- e^{i\omega t} \biggl[ 2(i\omega)x + (i\omega)x + (i\omega)r_0 \frac{\partial x}{\partial r_0} \biggr]</math>

<math>~\Rightarrow ~~~r_0 \frac{\partial x}{\partial r_0} </math>

<math>~=</math>

<math>~- \Delta_L -3 x - \frac{1}{(i\omega)} \biggl[ v_r \frac{\partial\Delta_L}{\partial r_0} \biggr]</math>

 

<math>~=</math>

<math>~- \Delta_L - x \biggl[3 + \cancelto{\scriptstyle\mathrm{small}}{\frac{\partial\Delta_L}{\partial \ln r_0} }\biggr]</math>

<math>~e^{i\omega t} (i\omega)\Delta_E </math>

<math>~=</math>

<math>~- e^{i\omega t} \biggl[ 2(i\omega)x + (i\omega)x + (i\omega)r_0 \frac{\partial x}{\partial r_0} \biggr] - \biggl[\frac{1}{\rho_0} \frac{\partial \rho_0}{\partial r_0} \biggr](i\omega)r_0 x e^{i\omega t}</math>

<math>~\Rightarrow ~~~r_0 \frac{\partial x}{\partial r_0} </math>

<math>~=</math>

<math>~- \Delta_E - x \biggl[3 + \frac{\partial \ln\rho_0}{\partial \ln r_0} \biggr] </math>

See Also