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Radial Oscillations of a ZeroZero Bipolytrope
IMPORTANT NOTE! While the overall development in this chapter is correct and, in particular, the chronology of discovery is properly reflected, the final quantitative results — for example, the recorded root(s) of the governing quartic equation — are incorrect because, beginning in the Example21 subsection, below, we started using an incorrect expression for the function . In an accompanying summary presentation, we have corrected this mistake.
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Groundwork
In an accompanying discussion, we derived the socalled,
whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, selfgravitating fluid configurations. According to our accompanying derivation, if the initial, unperturbed equilibrium configuration is an bipolytrope, then we know that the relevant functional profiles are as follows for the core and envelope, separately. Note that, throughout, we will preferentially adopt as the dimensionless radial coordinate, the parameter,



in which case,



The corresponding radial coordinate range is,
for the core, and
for the envelope.
Core





















where,






Hence,












and the wave equation for the core becomes,





















Envelope
























Hence,



and, after multiplying through by , the wave equation for the envelope becomes,


















Check1
If , this envelope wave equation should match seamlessly into the core wave equation. Let's see if it does. First,






Hence, for the envelope,















Whereas, for the core,



which matches exactly.
Boundary Condition
In order to ensure finite pressure fluctuations at the surface of this bipolytropic configuration, we need the logarithmic derivative of to obey the following relation:



Now, according to our accompanying discussion of the equilibrium mass and radius of a zerozero polytrope, we know that,






Hence, a reasonable surface boundary condition is,



Attempt to Find Eigenfunction for the Envelope
Adopting some of the notation used by T. E. Sterne (1937) and enunciated in our accompanying discussion of the uniformdensity sphere, we'll define,






in which case the wave equation for the core becomes,



and the wave equation for the envelope becomes,






A Specific Choice of the Density Ratio
Now, let's focus on the specific model for which . In this case,


















Note that this last expression goes to zero at the surface of the bipolytrope, that is, at . For this specific case, the wave equation for the envelope becomes,





















Idea Involving Logarithmic Derivatives
Notice that the term involving the first derivative of can be written as a logarithmic derivative; specifically,



Let's look at the second derivative of this quantity.









Now, if we assume that the envelope's eigenfunction is a powerlaw of , that is, assume that,
then the logarithmic derivative of is a constant, namely,
and the two key derivative terms will be,

and 

Hence, in order for the wave equation for the envelope for the specific density ratio being considered here to be satisfied, we need,












This means that three algebraic relations must simultaneously be satisfied, namely:







and, hence, 




More General Solution
Leaving the density ratio unspecified, let's try to write the wave equation for the envelope in the same form, and see if the logarithmic derivatives can be manipulated in a similar fashion.












where,












Hence, the wave equation becomes,












where,






As before, if we assume a powerlaw solution, the wave equation for the envelope becomes,












This means that three algebraic relations must simultaneously be satisfied, namely:














and, hence, 

Surface Boundary Condition
Given that, with this solution, the ratio,



we see that the desired surface boundary condition is,









But, for our identified solution, this is the logarithmic derivative of throughout the envelope as well as at the surface. So the boundary condition is automatically satisfied.
Match to a Core Eigenfunction (First Blundering)
If we define,
the above wave equation for the core becomes,



Not surprisingly, this is identical in form to the eigenvalue problem first presented by Sterne (1937) in connection with an examination of radial oscillations in uniformdensity spheres. For the core of our zerozero bipolytrope, we can therefore adopt any one of the polynomial eigenfunctions and corresponding eigenfrequencies derived by Sterne. We will insist that the eigenfrequency of the envelope match the eigenfrequency of the core; and, following J. O. Murphy & R. Fiedler (1985b) (see the top paragraph of the righthand column on p. 223 of their article), we seek solutions for which there is continuity in both the eigenfunction and its first derivative at the interface .
Try Quadratic Core Eigenfunction
Let's begin with Sterne's quadratic function and see if we can match it to the envelope's powerlaw eigenfunction. Keeping in mind that the overall normalization is arbitrary, from Sterne's presentation, we have,






and the associated eigenfrequency is obtained by setting,
In this case, then, the eigenfrequency for the envelope will match the eigenfrequency of the core if,






Now, the eigenfunction for the envelope is,
where,



The value of this function will match the value of its core counterpart at the interface if,






Finally, the slope (first derivative) of the core eigenfunction will match the slope of the envelope eigenfunction at the interface if,















The solution to this quadratic equation gives,






In order for this condition to hold while also meeting the demands of the eigenfrequency, we need to satisfy the relation,












where, keep in mind,



RESULT: After examining a range of physically reasonable values of , we do not find any values for which the lefthandside of this condition matches the righthandside.
Try Quartic Core Eigenfunction
Let's begin with Sterne's quartic function and see if we can match it to the envelope's powerlaw eigenfunction. From Sterne's presentation, we have,



and the associated eigenfrequency is obtained by setting,
In this case, then, the eigenfrequency for the envelope will match the eigenfrequency of the core if,






The eigenfunction for the envelope is, as before. The value of this envelope function will match the value of its core counterpart at the interface if,






Finally, the slope (first derivative) of the core eigenfunction will match the slope of the envelope eigenfunction at the interface if,









Eureka Regarding Prasad's 1948 Paper
Envelope Solution Outline
C. Prasad (1948, MNRAS, 108, 414416) has examined a closely related problem and, as it turns out, the mathematical approach that he used to solve that problem analytically is gratifyingly useful to me here. If, as above, we restrict our investigation to configurations for which,
and if we multiply through by , our governing wave equation becomes,






where,
This wave equation is very similar to equation (2) of Prasad (1948). If, following Prasad's guidance, we then assume a series solution of the form,



the indicial equation gives,
This is precisely the value of the exponent, , that we derived — in a more stumbling fashion — above and, as is shown by the following framed image, it is identical to the exponent derived by Prasad (1948).
Equation and accompanying text extracted^{†} from C. Prasad (1948)
"Radial Oscillations of a Particular Stellar Model"
Monthly Notices of the Royal Astronomical Society, vol. 108, pp. 414416 © Royal Astronomical Society 

^{†}Displayed here exactly as presented in the original publication. 
Using equation (7) from Prasad (1948) as a guide, we hypothesize that the eigenfrequency of the j^{th} mode in the envelope is given by the relation,






where, the last expression results from recognizing that and we have adopted the notation,
And guided by equation (6) from Prasad (1948), we hypothesize that successive coefficients in the (truncated) series that defines the radial structure of each mode is governed by the recurrence relation,



Example Envelope Eigenvectors
Mode j = 0
Here we assume that the series defining the eigenfunction has only one term. This should match our earlier restricted solution. Specifically,









In this case, the wave equation becomes,



The coefficients of the terms will sum to zero if the abovedefined indicial exponent condition is satisfied; that is, by setting,



In order for the coefficients of the terms to sum to zero, we need,






Mode j = 1
Here we assume that the series defining the eigenfunction has two terms: and . Specifically,









In this case, after factoring out , the wave equation becomes,






Again, the coefficients of the terms will sum to zero if the abovedefined indicial exponent condition is satisfied; that is, by setting,



In order for the coefficients of the terms to sum to zero, we need,












In addition, we must also examine what condition is required for the terms to sum to zero. We have,






Mode j = 2
Here we assume that the series defining the eigenfunction has three terms: , , and . Specifically,









In this case, after factoring out , the wave equation becomes,






Again, the coefficients of the terms will sum to zero if,



In order for the coefficients of the terms to sum to zero, we need,









In order for the coefficients of the terms to sum to zero, we need,









And the frequency determined from setting to zero the sum of coefficients of the terms is,









Match Prasadlike Envelope Eigenvector to the Core Eigenvector
If we define,
the above wave equation for the core becomes,



Not surprisingly, this is identical in form to the eigenvalue problem first presented by Sterne (1937) in connection with an examination of radial oscillations in uniformdensity spheres. For the core of our zerozero bipolytrope, we can therefore adopt any one of the polynomial eigenfunctions and corresponding eigenfrequencies derived by Sterne. We will insist that the eigenfrequency of the envelope match the eigenfrequency of the core; and, following J. O. Murphy & R. Fiedler (1985b) (see the top paragraph of the righthand column on p. 223 of their article), we seek solutions for which there is continuity in both the eigenfunction and its first derivative at the interface .
Eigenfrequencies
We must note that, heretofore, we have used the following dimensionless frequency notations:

and 

This means that, demanding that the two dimensional frequencies be the same requires that the ratio of the dimensionless frequencies be,



Now, according to Sterne's derivation, the dimensionless eigenfrequency associated with the mode in the core is,



And, as we have just discussed, the dimensionless eigenfrequency associated with the Prasadlike mode in the envelope is,



where,



Hence, in order for any specific pair of modes to have the same dimensional eigenfrequencies, we must have an envelopetocore density ratio given by the expression,



Put another way, the ratio of the dimensional eigenfrequencies is,



We also should keep in mind that, in our particular case, the envelope density must not be greater than the core density. So, demanding that the (dimensional) eigenfrequencies be equal and, simultaneously, that , implies the following constraint on the integer index, for each choice of the index, :









The following table lists values of for various values of the companion index, , and an assumed value of the parameter,
Limiting Index, , assuming 

0  0  0   
1  2  1  1 
2  3  3  2 
3  5  4  4 
4  6  6  5 
5  8  7  7 
6  9  9  8 
7  11  10  10 
8  12  12  11 
9  14  13  13 
10  15  15  14 
Limiting Index, , assuming 

0       
1  1  0  0 
2  2  2  1 
3  4  3  3 
4  5  5  4 
5  7  6  6 
6  8  8  7 
7  10  9  9 
8  11  11  10 
9  13  12  12 
10  14  14  13 
Implications
Keeping in mind that,



and that, in order for the Prasadlike modes to be relevant in the envelope, we must have,






we recognize that once the eigenfrequency match is used to define the relevant value of the density ratio, , the relevant values of both and are set as well. Specifically, as derived above, in the context of our "more general" envelope solution,

and 

This also means that the parameter,



Eigenfunctions
The eigenfunction associated with the Sternelike mode of the core is,



where, for the specified, mode, the value of the leading coefficient, , is arbitrary, but for all other coefficients,






The eigenfunction associated with the Prasadlike mode of the envelope is,



where, for the specified, mode, the value of the leading coefficient, , is arbitrary, but for all other coefficients,






Example11
Let's try

































Let's define both and such that the values of both eigenfunctions is unity at the interface . This means that,









Hence, in order for the first derivative of both eigenfunctions to be equal at the interface, we need,















Example12
Let's try


















In order for the core's eigenfunction to have the value of unity at , we need,









Hence, in order for the first derivative of both eigenfunctions to be equal at the interface, we need,


















where,



The solution to this quadratic equation gives,






Material that appears after this sign is under development and therefore may contain incorrect mathematical equations and/or physical misinterpretations.  Go Home  
Example21
Let's try

































Let's define both and such that the values of both eigenfunctions is unity at the interface . This means that,






Hence, in order for the first derivative of both eigenfunctions to be equal at the interface, we need,


















But, we can show that,
WRONG! This relation should read,

Hence,







































Allow Different Adiabatic Exponents
Let's allow the core to have a different adiabatic exponent, , from the envelope, ; this also means that will be correspondingly different in the two regions. First, note that the ratio of the eigenfrequencies is,



where,



So, if we want the dimensional eigenfrequencies to be identical, this means we need,






No Physical Solution11
Let's try Skimming through the earlier Example11 discussion, above, it looks like the only thing we need to do in order to allow the core and envelope to have different adiabatic exponents is to explicitly add the "envelope" subscript to on the righthand side of the last expression for . That is, in order to match both the value and the first derivative of the two eigenfunctions at the core/envelope interface, we need,






But, as before, the requirement that means that,






Notice that the righthand side of this expression only depends on the choice of the adiabatic exponent in the envelope. When combined with the condition imposed by setting the dimensional frequency ratio to unity, we have,









In this last expression, the lefthand side only depends on the adiabatic exponent in the core, while the righthand side only depends on the adiabatic exponent in the envelope.
Evaluation (Tohline's Excel spreadsheet AdExp inside workbook AnalyticEigenvector.xlsx): Over the range, , we plotted and for the example case of . There was a fairly wide range of pairings, for which the LHS = RHS; for example, at , both sides give , and at , both sides give . But in all cases, the inferred density ratio was greater than unity. Hence, this example index pairing does not seem to result in physically relevant coreenvelope eigenvectors.
Solution21
Setup
Let's try . Examining the above, Example21 discussion, we deduce that, in order for the first derivative of both eigenfunctions to be equal at the interface, we need,















Notice that the core's adiabatic exponent does not explicitly enter into this condition. Given that,

and 

WRONG! This relation should read,

we can view this expression as a conditional relationship between and . Once a pair has been found that satisfies this condition, we can use the matchingfrequency condition to give the corresponding, required value of ; specifically, for ,















First Few Numerically Determined Model Parameters
Example Solutions  

Envelope  Interface  Core  
(plus)  
55.118  15.508  
58.136  15.923  
62.247  16.479  
66  16.978 
Note that, in all cases, we find,



thereby demonstrating that the ratio of the dimensional frequencies is unity.
Special Case of 4/3 Envelope
It is worth examining in more detail the specific case of because, in this case, and (plus) are both zero, so the constraint equations become simpler. For this specific case, in order for the two eigenfunctions and their first derivatives to match at the interface, we have,


















In the physically relevant range of the parameter, , the parameter value that satisfies this constraint is,
From the frequencyratio constraint, therefore, we have,



Roots of Quartic Equation
Let,
for  



… 




… 

Then the interface constraint equation takes the form,




































To solve this equation analytically, we follow the Summary of Ferrari's method that is presented in Wikipedia's discussion of the Quartic Function to identify the roots of an arbitrary quartic equation.
First, we adopt the shorthand notation:
where, in our particular case,
Now, define,
Then the four roots of the quartic equation are,
The root is the physically relevant one, and it matches the interface value of associated with in our above table of example solutions. Excellent! 
In summary then, for any choice of the envelope's adiabatic exponent, , the physically relevant root of the quartic equation gives us the interface location of the model for which our analytically specified eigenvector applies; specifically,
From this value, we also know that,
and
WRONG! The first of these three expressions should read,

Combined Eigenfunction
From above, we know that the eigenfunction for the core is,



And the matching eigenfunction for the envelope is,



Related Discussions
 Summary of Above, Detailed Derivation
 Searching for Additional, Analytically Specified Eigenvectors of ZeroZero Bipolytropes
© 2014  2020 by Joel E. Tohline 