Scaling Symmetry and Integrable Spherical Hydrostatics

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.

Cite this paper

S. Bludman and D. Kennedy, "Scaling Symmetry and Integrable Spherical Hydrostatics,"*Journal of Modern Physics*, Vol. 4 No. 4, 2013, pp. 486-494. doi: 10.4236/jmp.2013.44069.

S. Bludman and D. Kennedy, "Scaling Symmetry and Integrable Spherical Hydrostatics,"

References

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[1] G. W. Bluman and S. C. Anco, “Symmetry and Integration Methods for Differential Equations,” Springer-Verlag, Berlin, 2010.

[2] S. Bludman and D. C. Kennedy, “Invariant Relationships Deriving from Classical Scaling Transformations,” Journal of Mathematical Physics, Vol. 52, 2011, Article ID: 042092.

[3] S. Chandrasekhar, “An Introduction to the Study of Stellar Structure, Chapters III, IV,” University of Chicago, 1939.

[4] M. Schwarzschild, “Structure and Evolution of the Stars,” Princeton University Press, Princeton, 1958.

[5] R. Kippenhahn and A. Weigert, “Stellar Structure And Evolution,” Springer-Verlag, Berlin, 1990.

[6] C. J. Hansen and S. D. Kawaler, “Stellar Interiors: Physical Principles, Structure, and Evolution,” Springer-Verlag, Berlin, 1994.

[7] G. P. Horedt, “Polytropes: Applications in Astrophysics and Related Fields,” Kluwer, Dordrecht, 2004.

[8] F. K. Liu, “Polytropic Gas Spheres: An Approximate Analytic Solution of the Lane-Emden Equation,” Monthly Notices of the Royal Astronomical Society, Vol. 281, No. 4, 1996, pp. 1197-1205.

[9] S. A. Bludman and D. C. Kennedy, “Analytic Models for the Mechanical Structure of the Solar Core,” The Astrophysical Journal, Vol. 525, No. 2, 1999, pp. 1024-1031.

[10] P. Pascual, “Lane-Emden Equation and Padé’s Approximants,” Astronomy & Astrophysics, Vol. 60, 1977, pp. 161-163.

[11] Z. F. Seidov, “Lane-Emden Equation: Picard vs Pade,” arXiv:astro-ph/0107395.

[12] W. E. Boyce and R. C. DiPrima, “Elementary Differential Equations and Boundary Value Problems,” 7th Edition, John Wiley and Sons, Hoboken, 2001.

[13] D. W. Jordon and P. Smith, “Nonlinear Ordinary Differential Equations,” 3rd Edition, Oxford University Press, Oxford, 1999.