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Continue Search for Marginally Unstable (5,1) Bipolytropes
This Ramblings Appendix chapter — see also, various trials — provides some detailed trial derivations in support of the accompanying, thorough discussion of this topic.
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Key Differential Equation
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. After adopting an appropriate set of variable normalizations — as detailed here — this becomes,



where, . Alternatively — see, for example, our introductory discussion — for polytropic configurations we may write,



Applied to the Core
As we have already summarized in an accompanying discussion, throughout the core we have,









So the relevant core LAWE becomes,






Now, following our separate discussion of an analytic solution to this LAWE, we try,









Plugging this trial function into the relevant LAWE gives,
LAWE 





Now, if we set and , we find that the terms on the RHS sum to zero. It therefore appears that we have identified a dimensionless displacement function that satisfies the core LAWE.
Applied to the Envelope
And as we have also summarized in the same accompanying discussion, throughout the envelope we have,









So the relevant envelope LAWE becomes,





















where,



and 



If we set and , the envelope LAWE simplifies to the form,



In yet another Ramblings Appendix derivation we have explored a trial dimensionless displacement for the envelope of the form,


In this case,


















and it can be shown that the simplified envelope LAWE is perfectly satisfied. Notice that, with this adopted segment of the eigenfunction for the envelope, we have,












Interface Matching
According to our accompanying discussion of the interface matching condition — as we presently understand it — the proper eigenfunction will exhibit a discontinuity in the slope of the dimensionless displacement function such that,






See Also
 K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201  206), A Hot and Fast Ultrastripped Supernova that likely formed a Compact Neutron Star Binary.
© 2014  2020 by Joel E. Tohline 