User:Tohline/SSC/Structure/Other Analytic Models
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Other Analytically Definable, Spherical Equilibrium Models
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Linear Density Distribution
In an article titled, "Stellar Evolution: A Survey with Analytic Models," R. F. Stein (1966, in Stellar Evolution, Proceedings of an International Conference held at the Goddard Space Flight Center, Greenbelt, MD, U.S.A., edited by R. F. Stein & A. G. W. Cameron, pp. 1105) defines the "Linear Stellar Model" as a star whose density "varies linearly from the center to the surface," that is (see his equation 3.1),
where, is the central density and, is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our general solution strategy for determining the equilibrium structure of spherically symmetric, selfgravitating configurations,






in which case we have,
and we can write,



Hence, proceeding via what we have labeled as "Technique 1", and enforcing the surface boundary condition, , Stein (1966) determines that (see his equation 3.5),






where, it can readily be deduced, as well, that the central pressure is,
Stabililty
Lagrangian Approach
As has been derived in an accompanying discussion, the secondorder ODE that defines the relevant Eigenvalue problem is,
where the dimensionless radius,
and
For Stein's configuration with a linear density distribution,
and
Hence,



and the governing adiabatic wave equation takes the form,
























where, following R. Stothers & J. A. Frogel (1967, ApJ, 148, 305),
Eulerian Approach
In his book titled, The Pulsation Theory of Variable Stars, S. Rosseland (1969) defines the relevant eigenvalue problem for adiabatic, radial pulsations in terms of the governing relation (see his equation 2.23 on p. 20, with the adiabatic condition being enforced by setting the righthandside equal to zero),



where,
Realizing that, for a spherically symmetric system,
and remembering that,
we can rewrite this relation in the more familiar form of a 2^{nd}order ODE, namely,












Multiplying through by and, again, letting , we have,



Now, plugging in the functional expressions that specifically apply to the linear model gives,









and, multiplying through by gives,





















where, following R. Stothers & J. A. Frogel (1967, ApJ, 148, 305),
Parabolic Density Distribution
Equilibrium Structure
In an article titled, "Radial Oscillations of a Stellar Model," C. Prasad (1949, MNRAS, 109, 103) investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,
where, is the central density and, is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our general solution strategy for determining the equilibrium structure of spherically symmetric, selfgravitating configurations,






in which case we can write,



Hence, proceeding via what we have labeled as "Technique 1", and enforcing the surface boundary condition, , Prasad (1949) determines that,















where, it can readily be deduced, as well, that the central pressure is,
Stabililty
As has been derived in an accompanying discussion, the secondorder ODE that defines the relevant Eigenvalue problem is,
where the dimensionless radius,
and
For Prasad's configuration with a parabolic density distribution,
and
Hence,



and the governing adiabatic wave equation takes the form,
where,
In keeping with Prasad's presentation — see, specifically, his equations (2) & (3) — this wave equation can also be written as,
where,
For what it's worth, we have also deduced that this expression can be written as,
Ramblings
The material originally contained in this "Ramblings" subsection has been moved to generate a separate chapter that stands on its own.
Promising Avenue of Exploration
What follows is a direct extension of what is referred to in our "Ramblings" chapter as the third guess under "Exploration2". We pursue this line of reasoning, here, because it appears to be a particularly promising avenue of exploration.
In the case of a parabolic density distribution, the LAWE becomes,



We have chosen to examine the suitability of an eigenfunction of the form,



where, for a given value of , the four parameters, , , and are to be determined in concert with a value of the square of the eigenfrequency, . From the accompanying discussion we have determined that the following five coefficient expressions must independently be zero in order for this trial eigenfunction to satisfy the LAWE:
: 


:  
: 


: 


: 

First Constraint
We begin by manipulating the last expression — that is, the coefficient expression for the term. Rejecting the trivial option of setting , in order for this expression to be zero the terms inside the curly braces must sum to zero. Rewriting this expression in terms of the sum of the exponents,
we obtain the quadratic expression,









This means that, once the physical parameter, , has been specified, the sum of the exponents must be,






Second Constraint
Next we examine the expression that serves as the coefficient of . Setting that coefficient expression to zero while replacing in favor of — via the relation, — gives,












where, we have set,
So, once is specified and is known from the first constraint, we can use this expression to replace in the other three coefficient expressions.
Intermediate Summary
The three remaining constraints emerge from the remaining three coefficient expressions, namely,
:  
: 


: 

Written in terms of the three remaining unknowns, , , and , the three constraints are:




















































At first glance, this is still not as promising as I had hoped. In practice there are only two unknowns — because the parameter, , has divided out — while there are three constraints. So the problem remains over constrained.
Remaining Group of Three Constraints
Let's adopt another approach. Let's assume that the parameter, , is also initially unspecified and replace it in all three remaining constraint expressions, in favor of , using the abovespecified, first constraint, namely,



This gives,
























The three unknowns are: , , and .
Prasad's Work
Overview
C. Prasad (1949, MNRAS, 109, 103) performed a semianalytic analysis of the radial oscillations and stability of structures having a parabolic density distribution. Let's examine his tabulated results to see if they help us understand more fully whether or not our analysis is on the right track. For example, from his Table I, we see that when , where, according to his equation (3),
This means that, also, . Now, from our derived second constraint, we deduce that,



Hence, since , we conclude that,
Also, since by definition , we conclude that,
Next, given that , we conclude from our derived first constraint, that


and 
The Minus Root
Combining these two results for the "minus" solution, we furthermore conclude that, for this specific mode, the relationship between the two exponents and are,
and
The Plus Root
Next, let's examine the "plus" solution. Because , this solution implies that,
.
In this case, then, we deduce that,
.
So, even though these first two constraints have not revealed the value of either of the exponents, and , we see that the resulting trial eigenfunction must be,









Interesting!
Third Constraint
The Minus Root
Let's insert all of these relations into the algebraic expression that we have derived from the coefficient:

RHS 


































Let's repeat this step, but start from an earlier expression for the coefficient, namely,












The first two terms on the RHS immediately go to zero because, for this specific eigenfunction, both and are zero. Plugging in our determined expressions for and gives,

RHS 














which exactly matches the previous, but messier, derivation. Now, the two roots of the quadratic expression inside the square brackets are,






Both roots are imaginary numbers and therefore not of interest in the context of this astrophysical problem.
The Plus Root
Next, in addition to setting , we'll plug and into the third constraint expression as follows:

RHS 


















So the nontrivial solution is — and, hence, — in which case the trial eigenfunction is,



Fifth Constraint
The Minus Root
In a similar vein, let's insert all of the deduced relations into the algebraic expression that we have derived from the coefficient:

RHS 






















As above, the first two terms on the RHS immediately go to zero because, for this specific eigenfunction, both and are zero. Plugging in our determined expressions for and gives,

RHS 


















The Plus Root
Next, in addition to setting , we'll plug and into the fifth constraint expression as follows:

RHS 






















From this constraint, it appears that the nontrivial result is, .
Fourth Constraint

RHS 















The Minus Root
In addition to setting , here we plug and into the fourth constraint expression as follows:

RHS 


















The Plus Root
Next, in addition to setting , we'll plug and into the fourth constraint expression as follows:

RHS 






From this constraint, it appears that the nontrivial result is, .
More General Approach
The specific trial eigenfunction that we have just examined does not appear to simultaneously satisfy all constraints prescribed by the LAWE. So, in a separate chapter, we will examine an even more general trial eigenfunction. It is the one that also has previously been introduced in our "Ramblings" chapter under the subheading, "Consider Parabolic Case", having the form,



In this accompanying chapter, we will be examining whether or not it satisfies the (same) version of the LAWE that describes stability in structures having a parabolic density profile, namely,



© 2014  2020 by Joel E. Tohline 