User:Tohline/SSC/Structure/BiPolytropes/MurphyUVplane

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UV Plane Functions as Analyzed by Murphy (1983)

Whitworth's (1981) Isothermal Free-Energy Surface
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This chapter supports and expands upon an accompanying discussion of the construction of 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. This system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. Here we provide an in-depth analysis of the work published by J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175) in which the derivation of this particular bipolytropic configuration was first attempted. As can be seen from the following list of "key references," however, this publication was only one of a series of interrelated works by Murphy. We will henceforth refer to this <math>~(n_c, n_e) = (1, 5)</math> system as "Murphy's bipolytrope."

Key References

Relevant Lane-Emden Functions

As is detailed in our accompanying discussion, the Lane-Emden function governing the structure of the core of Murphy's bipolytrope is,

<math> \theta(\xi) = \frac{\sin\xi}{\xi} \, , </math>

and the first derivative of this function with respect to the dimensionless radial coordinate, <math>~\xi</math>, is,

<math> \frac{d\theta}{d\xi} = -\frac{1}{\xi^{2}} (\sin\xi - \xi\cos\xi) \, . </math>

Also as is detailed in our accompanying discussion, the Lane-Emden function governing the structure of the envelope of Murphy's bipolytrope is,

<math>~\phi(\eta)</math>

<math>~=</math>

<math>~\frac{B^{-1}\sin[\ln(A\eta)^{1/2})]}{\eta^{1/2}\{3-2\sin^2[\ln(A\eta)^{1/2}]\}^{1/2}} </math>

 

<math>~=</math>

<math>~\frac{B^{-1}\sin\Delta}{\eta^{1/2}(3-2\sin^2\Delta)^{1/2}} \, ,</math>

and the first derivative of this function is,

<math>~\frac{d\phi}{d\eta}</math>

<math>~=</math>

<math>~- \frac{B^{-1}(3\sin\Delta - 2\sin^3\Delta -3\cos\Delta) }{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}} \, , </math>

where we have adopted the shorthand notation,

<math>~\Delta \equiv \ln(A\eta)^{1/2} \, .</math>

Chandrasekhar's U and V Functions

ASIDE: Comments on Presentation by Murphy (1983)

As presented by Murphy (1983), most of the development and analysis of this model was conducted within the framework of what is commonly referred to in the astrophysics community as the "U-V" plane. Specifically in the context of the model's <math>~n=5</math> envelope, this pair of referenced functions are:

<math>~U_{5F} \equiv \eta \phi^5 \biggl(- \frac{d\phi}{d\eta}\biggr)^{-1}</math>

<math>~=</math>

<math>~ \frac{B^{-4}\eta \sin^5\Delta}{\eta^{5/2}\{3-2\sin^2\Delta\}^{5/2}} \biggl[ \frac{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}}{3\sin\Delta - 2\sin^3\Delta -3\cos\Delta } \biggr] </math>

 

<math>~=</math>

<math>~ \frac{2B^{-4}\sin^5\Delta}{[3-2\sin^2\Delta][3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ]} </math>

 

<math>~=</math>

<math>~ \frac{-2B^{-4}\sin^5\Delta}{[2+\cos(2\Delta)][3\cos\Delta - \frac{3}{2}\sin\Delta - \frac{1}{2}\sin(3\Delta) ]} \, ; </math>

<math>~V_{5F} \equiv \frac{\eta}{ \phi} \biggl(- \frac{d\phi}{d\eta}\biggr)</math>

<math>~=</math>

<math>~\frac{\eta^{3/2}\{3-2\sin^2\Delta\}^{1/2}} {\sin\Delta} \frac{[3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ] }{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}} </math>

 

<math>~=</math>

<math>~\frac{[3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ] }{2\sin\Delta (3-2\sin^2\Delta)} </math>

 

<math>~=</math>

<math>~\frac{-[3\cos\Delta - \frac{3}{2}\sin\Delta - \frac{1}{2}\sin(3\Delta) ] }{2\sin\Delta [2+\cos(2\Delta)]} \, . </math>

After recognizing that <math>~\cos(2\Delta) = \cos(2\ln\eta^{1/2}) = \cos(\ln\eta)\, ,</math> we see that these expressions for the functions, <math>~U_{5F}</math> and <math>~V_{5F}</math>, match the expressions used by Murphy (1983) and reproduced (slightly edited) here as an image, for ease of comparison:

U_5F and V_5F Functions by Murphy (1983)


Related Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

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