User:Tohline/SSC/Structure/BiPolytropes/Analytic1 5
BiPolytrope with <math>n_c = 1</math> and <math>n_e=5</math>
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Here we construct a bipolytrope in which the core has an <math>~n_c=1</math> polytropic index and the envelope has an <math>~n_e=5</math> polytropic index. As in the case of our separately discussed, "mirror image" bipolytropes having <math>~(n_c, n_e) = (5, 1)</math>, this system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. [On 12 April 2015, J. E. Tohline wrote: I became aware of the published discussions of this system by Murphy — and especially the work of Murphy & Fiedler (1985) — (see itemization of additional key references, below) in March of 2015 after searching the internet for previous analyses of radial oscillations in polytropes and, then, reading through Horedt's (2004) §2.8.1 discussion of composite polytropes.]
Key References
- J. O. Murphy (1980a, Proc. Astr. Soc. of Australia, 4, 37): A Finite Radius Solution for the Polytrope Index 5
- J. O. Murphy (1980b, Proc. Astr. Soc. of Australia, 4, 41): On the F-Type and M-Type Solutions of the Lane-Emden Equation
- J. O. Murphy (1981, Proc. Astr. Soc. of Australia, 4, 205): Physical Characteristics of a Polytrope Index 5 with Finite Radius
- J. O. Murphy (1982, Proc. Astr. Soc. of Australia, 4, 376): A Sequence of E-Type Composite Analytical Solutions of the Lane-Emden Equation
- J. O. Murphy (1983, Australian Journal of Physics, 36, 453): Structure of a Sequence of Two-Zone Polytropic Stellar Models with Indices 0 and 1
- J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175): Composite and Analytical Solutions of the Lane-Emden Equation with Polytropic Indices n = 1 and n = 5
- J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219): Physical Structure of a Sequence of Two-Zone Polytropic Stellar Models
- J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222): Radial Pulsations and Vibrational Stability of a Sequence of Two-Zone Polytropic Stellar Models
Steps 2 & 3
Based on the discussion presented elsewhere of the structure of an isolated <math>n=5</math> polytrope, the core of this bipolytrope will have the following properties:
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