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Stability Analyses of PP Tori (Part 2)
[Comment by J. E. Tohline on 24 May 2016] This chapter contains a set of technical notes and accompanying discussion that I put together several months ago as I was trying to gain a foundational understanding of the results of a large study of instabilities in selfgravitating tori published by the Imamura & Hadley collaboration. I have come to appreciate that some of the logic and interpretation of published results that are presented, below, has serious flaws. Therefore, anyone reading this should be quite cautious in deciding what subsections provide useful insight. I have written a separate chapter titled, "Characteristics of Unstable Eigenvectors in SelfGravitating Tori," that contains a much more trustworthy analysis of this very interesting problem.
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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This is a direct extension of our Part 1 discussion. Here we continue our effort to check the validity of the Blaes85 eigenvector. The relevant reference is:
 Blaes (1985), MNRAS, 216, 553 (aka Blaes85) — Oscillations of slender tori.
Start From Scratch
Basic Equations from Blaes85
Blaes85
Eq. No. 





(2.6) 



(2.6) 



(2.5) 
Blaes85
Eq. No. 









(4.2) 



(4.1) 



(4.2) 



(4.13) 



(4.14) 
Our Manipulation of These Equations
Analytic



























Also,






Putting the two together implies,
Definition of Eigenvalue Problem Associated with the Stability of Slim, PapaloizouPringle Tori  


The first line of this governing, twoline expression contains the function, , as a leading factor, while the leading factor in the second line is the ratio, . Presumably the three terms (hereafter, TERM1, TERM2, & TERM3, respectively) inside the curly brackets on the first line must cancel — to a sufficiently high order in — and, independently, the two terms (hereafter, TERM4 & Term5, respectively) inside the curly brackets on the second line must cancel. Furthermore, these cancellations must occur separately for the real parts and the imaginary parts of each bracketed expression.
Example Evaluation
Evaluating various terms using the parameter set, as begun in our "Part 1" analysis, we have:
TERM1 








TERM2 





TERM3 





The sum of these three terms gives,
TERM1 + TERM2 + TERM3 














Moving on to the last pair of terms …
TERM4 








TERM5 (Case B) 














Evaluating this TERM5 expression for the case of , we have,
TERM5 (Case B) 














Testing for Expected Cancellations
Note first that, adopting the shorthand notation,












Real Parts
TERM1





































































TERM2






















































Sum of TERM1 and TERM2


















TERM3












Sum of TERM1 + TERM2 + TERM3
Therefore,



























TERM4





















Or, continuing to develop the analytic powerlaw expression,














\, . 
TERM5
Now, let's examine the TERM5 expressions.



Case B: 
























































Sum of TERM$ and TERM5
When added together, we obtain,

























































So we see that the coefficients of the lowestorder terms are zero, and the coefficient of the term is almost zero! My analysis the second time around gives,
























Exactly the same as the first time around.
Imaginary Parts
TERM1












TERM2









TERM3






TERM4















Alternatively we can write,







































TERM5



Case B: 











Let's rewrite both of these expressions in terms of a power series in .




































Dropping all terms on the righthandside that are or higher, we have,




































Together
Together, then, we have:





















Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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When added together, we obtain,




































Summary
As stated above, the eigenvalue problem that must be solved in order to identify the eigenfunction, , and eigenfrequency, , of unstable (as well as stable) nonaxisymmetric modes in slim , polytropic PP tori with uniform specific angular momentum is defined by the following twodimensional , 2^{nd}order PDE:



where, is the enthalpy distribution in the unperturbed, axisymmetric torus, and















We also should appreciate that,












If an exact solution, , to this eigenvalue problem were plugged into this governing PDE, we would expect that both of the following summations would be exactly zero at all meridionalplane locations throughout the torus:






While an exact analytic solution to this eigenvalue problem is not (yet) known, Blaes (1985) has determined that a good approximate solution is an eigenvector defined by the complex eigenfrequency,



and, simultaneously, the complex eigenfunction,



where,



Real Components of Various Terms  

Order  
        
        

The loglog plot shown here, on the right, illustrates the behavior of the "TERM4 + TERM5" sum for the example parameter set, . As the blue diamonds illustrate, the real part of this sum drops by approximately two orders of magnitude for every factor of ten drop in . The total drop is roughly eight orders of magnitude over the displayed range, . As the salmoncolored squares in the same plot indicate, the imaginary part of the sum, "TERM4 + TERM5," is even closer to zero, dropping roughly 12 orders of magnitude over the same range of . This indicates that, with the Blaes85 eigenvector, the real part of the sum of this pair of terms differs from zero by a residual whose leadingorder term varies as while the corresponding imaginary part of the sum differs from zero by a residual whose leadingorder term varies as .
As our above analytic analysis shows, when each of the expressions for TERM4 and TERM5 is rewritten as a power series in , a sum of the two analytically specified TERMs results in precise cancellation of leadingorder terms. For the imaginary component of this sum, our derived expression for the residual is,






The dotted, salmoncolored line of slope 3 that has been drawn in our accompanying loglog plot was generated using this analytic expression for the residual term. It appears to precisely thread through the points (the salmoncolored squares) whose plot locations have been determined via our numerical spreadsheet evaluation of the imaginary component of the "TERM4 + TERM5" sum. Additional confirmation that we have derived the correct analytic expression for comes from subtracting this analytically defined residual from the numerically determined sum: The result is the greendashed curve in the accompanying loglog plot, which appears to be a line of slope 4.
Analogously, for the real component of this sum, the precise expression for the residual is,






The dotted, light blue line of slope 2 that has been drawn in our accompanying loglog plot was generated using this analytic expression for the residual term. It appears to precisely thread through the points (the light blue diamonds) whose plot locations have been determined via our numerical spreadsheet evaluation of the real part of the "TERM4 + TERM5" sum. Notice that at the surface of the torus — that is, when — this residual goes to zero, in which case the leading order term in the "real" component residual will be drop to .
See Also
 Imamura & Hadley collaboration:
 Paper I: K. Hadley & J. N. Imamura (2011, Astrophysics and Space Science, 334, 126), "Nonaxisymmetric instabilities in selfgravitating disks. I. Toroids" — In this paper, Hadley & Imamura perform linear stability analyses on fully selfgravitating toroids; that is, there is no central pointlike stellar object and, hence, .
 Paper II: K. Z. Hadley, P. Fernandez, J. N. Imamura, E. Keever, R. Tumblin, & W. Dumas (2014, Astrophysics and Space Science, 353, 191222), "Nonaxisymmetric instabilities in selfgravitating disks. II. Linear and quasilinear analyses" — In this paper, the Imamura & Hadley collaboration performs "an extensive study of nonaxisymmetric global instabilities in thick, selfgravitating stardisk systems creating a large catalog of star/disk systems … for star masses of and inner to outer edge aspect ratios of ."
 Paper III: K. Z. Hadley, W. Dumas, J. N. Imamura, E. Keever, & R. Tumblin (2015, Astrophysics and Space Science, 359, article id. 10, 23 pp.), "Nonaxisymmetric instabilities in selfgravitating disks. III. Angular momentum transport" — In this paper, the Imamura & Hadley collaboration carries out nonlinear simulations of nonaxisymmetric instabilities found in selfgravitating star/disk systems and compares these results with the linear and quasilinear modeling results presented in Papers I and II.
© 2014  2021 by Joel E. Tohline 