Scaling Symmetry and Integrable Spherical Hydrostatics

Sidney Bludman, Dallas C. Kennedy

Departamento de Astronomía, Universidad de Chile, Santiago, Chile.
Natick, USA.

DOI: 10.4236/jmp.2013.44069                

Abstract

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 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 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.

Keywords

Lagrangian Mechanics; Symmetry; Hydrodynamics; Astrophysics; Stellar Structure

Conflicts of Interest

The authors declare no conflicts of interest.

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