User:Tohline/ProjectsUnderway/Core Collapse Supernovae
A Template for Gravitational Wave Signals from Core-Collapse Supernovae
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29 August 2014: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.
Free-Fall Collapse
I want to begin this section by paying tribute to Donald Lynden-Bell (1962, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 58, pp. 709-711), who was the first to appreciate the relatively simple behavior that should be exhibited by the free-fall collapse of a uniformly rotating, uniform-density spheroid. In an article less than two pages in length, Lynden-Bell first noted that the governing dynamical equations (written in cylindrical coordinates) take the form,
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<math>~\ddot{R}</math> |
<math>~=</math> |
<math>~-~ 2A_L(t) R + \frac{h_{LB}^2}{R^3} \, ,</math> |
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<math>~\ddot{Z}</math> |
<math>~=</math> |
<math>~-~ 2C_L(t) Z \, ,</math> |
where the dots denote differentiation with respect to time, <math>~h_{LB}</math> is a constant and the two time-dependent coefficients, <math>~A(t)</math> and <math>~C(t)</math>, come from the gravitational potential that, according to Lyttleton (1953), has the form,
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<math>~\Phi</math> |
<math>~=</math> |
<math>~-~ 2A_L(t) R^2 ~-~ C_L(t) Z^2 \, .</math> |
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