2011/12/31 by Sidney Bludman, Dallas C. Kennedy
Physics and Astronomy · Mathematics · #math-ph #astro-ph.SR #math.MP #physics.class-ph #physics.flu-dyn
published as Journal of Modern Physics, Vol. 4 No. 4 (April 2013) 486-494 · 10 pages, 4 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1106.1222
arxiv created 2012/11/07 · arxiv updated 2013/04/29
Any symmetry reduces a second-order differential equation to a first integral: variational symmetries of the action (exemplified by central field dynamics) lead to conservation laws, but symmetries of only the equations of motion (exemplified by scale-invariant hydrostatics) yield first-order \em non-conservation laws between invariants. We obtain these non-conservation laws by extending Noether's Theorem to non-variational symmetries and present an innovative variational formulation of spherical adiabatic hydrostatics. For the scale-invariant case, this novel synthesis of group theory, hydrostatics, and astrophysics allows us to recover all the known properties of polytropes and define a \em core radius, inside which polytropes of index n share a common core mass density structure, and outside of which their envelopes differ. The Emden solutions (regular solutions of the Lane-Emden equation) are obtained, along with useful approximations. An appendix discusses the n=3 polytrope in order to emphasize how the same mechanical structure allows different thermal structures in relativistic degenerate white dwarfs and zero age main sequence stars.