SSC/StabilityConjecture/Bipolytrope51: Difference between revisions
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=Examine B- | =Examine B-KB74 Conjecture in the Context of <math>(n_c, n_e) = (5 , 1)</math> Bipolytropes= | ||
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! style="height: 125px; width: 125px; background-color:white;" |<font size="-1">[[H_BookTiledMenu#MoreModels|<b>B-KB74<br />Conjecture<br /> RE: Bipolytrope</b>]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)</font> | ! style="height: 125px; width: 125px; background-color:white;" |<font size="-1">[[H_BookTiledMenu#MoreModels|<b>B-KB74<br />Conjecture<br /> RE: Bipolytrope</b>]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)</font> | ||
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In §6 of their paper, [http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974; hereafter, B-KB74)] have suggested that <font color="darkgreen">"… a static configuration close to an extremum of the</font> [mass-radius equilibrium] <font color="darkgreen">curve may be considered as a perturbed state of a model of the same mass situated on the other side of the extremum. The difference of the two models approximately represents the eigenfunction of the neutral mode."</font> In an [[Appendix/Ramblings/NonlinarOscillation|accompanying discussion]] we have demonstrated that this "B-KB74 conjecture" applies exactly in the context of an analysis of the stability of pressure-truncated, n = 5 polytropes. We know that it applies exactly in this case because, along the n = 5 mass-radius sequence, the eigenfunction of the fundamental mode of radial oscillation is known analytically. | |||
Here we turn to the B-KB74 conjecture to assist us in examining the stability of models that lie along the sequence of bipolytropes with <math>(n_c, n_e) = (5 , 1)</math>. The internal structure of these bipolytropic structures can be defined analytically. But, as far as we have been able to determine, nothing is known about the eigenvectors describing their natural modes of radial oscillation. We hope to be able to use the B-KB74 conjecture to determine the eigenfunction of the fundamental mode of radial oscillation for the model along the sequence that is marginally [dynamically] unstable. | |||
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Revision as of 19:40, 2 August 2021
Examine B-KB74 Conjecture in the Context of Bipolytropes
| B-KB74 Conjecture RE: Bipolytrope (nc, ne) = (5, 1) |
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In §6 of their paper, G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974; hereafter, B-KB74) have suggested that "… a static configuration close to an extremum of the [mass-radius equilibrium] curve may be considered as a perturbed state of a model of the same mass situated on the other side of the extremum. The difference of the two models approximately represents the eigenfunction of the neutral mode." In an accompanying discussion we have demonstrated that this "B-KB74 conjecture" applies exactly in the context of an analysis of the stability of pressure-truncated, n = 5 polytropes. We know that it applies exactly in this case because, along the n = 5 mass-radius sequence, the eigenfunction of the fundamental mode of radial oscillation is known analytically.
Here we turn to the B-KB74 conjecture to assist us in examining the stability of models that lie along the sequence of bipolytropes with . The internal structure of these bipolytropic structures can be defined analytically. But, as far as we have been able to determine, nothing is known about the eigenvectors describing their natural modes of radial oscillation. We hope to be able to use the B-KB74 conjecture to determine the eigenfunction of the fundamental mode of radial oscillation for the model along the sequence that is marginally [dynamically] unstable.
See Also
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