UV Plane Functions as Analyzed by Murphy (1983)

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

S. Srivastava (1968, ApJ, 136, 680) A New Solution of the Lane-Emden Equation of Index n = 5
H. A. Buchdahl (1978, Australian Journal of Physics, 31, 115): Remark on the Polytrope of Index 5 — the result of this work by Buchdahl has been highlighted inside our discussion of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>.
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

Relevant Lane-Emden Functions

As is detailed in our accompanying discussion, the Lane-Emden function governing the structure of the <math>~n_c = 1</math> 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 <math>~n_e = 5</math> envelope of Murphy's bipolytrope is,

<math>~\phi(\xi)</math>	<math>~=</math>	<math>~\frac{B^{-1}\sin[\ln(A\xi)^{1/2})]}{\xi^{1/2}\{3-2\sin^2[\ln(A\xi)^{1/2}]\}^{1/2}} </math>
	<math>~=</math>	<math>~\frac{B^{-1}\sin\Delta}{\xi^{1/2}(3-2\sin^2\Delta)^{1/2}} \, ,</math>

and the first derivative of this function is,

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

where we have adopted the shorthand notation,

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

Chandrasekhar's U and V Functions

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_c=1</math> core, this pair of referenced functions is:

<math>~U_{1E} \equiv \xi \theta \biggl(- \frac{d\theta}{d\xi}\biggr)^{-1}</math>	<math>~=</math>	<math>~\sin\xi\biggl[\frac{1}{\xi^{2}} (\sin\xi - \xi\cos\xi)\biggr]^{-1}</math>
	<math>~=</math>	<math>~\frac{\xi^2}{(1 - \xi\cot\xi)} \, ;</math>
<math>~(n_c+1) V_{1E} \equiv (n_c+1)\frac{\xi}{ \theta} \biggl(- \frac{d\theta}{d\xi}\biggr)</math>	<math>~=</math>	<math>~\frac{2\xi^2}{\sin\xi} \biggl[ \frac{1}{\xi^{2}} (\sin\xi - \xi\cos\xi) \biggr]</math>
	<math>~=</math>	<math>~ 2(1 - \xi\cot\xi) \, .</math>

Correspondingly, in the context of the model's <math>~n_e=5</math> envelope, the pair of referenced functions is:

<math>~U_{5F} \equiv \xi \phi^5 \biggl(- \frac{d\phi}{d\xi}\biggr)^{-1}</math>	<math>~=</math>	<math>~ \frac{B^{-4}\xi \sin^5\Delta}{\xi^{5/2}\{3-2\sin^2\Delta\}^{5/2}} \biggl[ \frac{2\xi^{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>~(n_e+1) V_{5F} \equiv (n_e+1) \frac{\xi}{ \phi} \biggl(- \frac{d\phi}{d\xi}\biggr)</math>	<math>~=</math>	<math>~\frac{6\xi^{3/2}\{3-2\sin^2\Delta\}^{1/2}} {\sin\Delta} \frac{[3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ] }{2\xi^{3/2}(3-2\sin^2\Delta)^{3/2}} </math>
	<math>~=</math>	<math>~\frac{3[3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ] }{\sin\Delta (3-2\sin^2\Delta)} </math>
	<math>~=</math>	<math>~\frac{-6[3\cos\Delta - \frac{3}{2}\sin\Delta - \frac{1}{2}\sin(3\Delta) ] }{2\sin\Delta [2+\cos(2\Delta)]} \, . </math>

In an effort to demonstrate correspondence with the published work of Murphy (1983), we have reproduced his expressions for these governing U-V functions in the following boxed-in image.

U-V Functions Extracted from Murphy (1983)

(slightly edited before reproduction as an image, here)

The match between our expressions and those presented by Murphy becomes clear upon recognizing that, in our notation,

<math>~\Delta </math>	<math>~=</math>	<math>~\ln\sqrt{A\xi} </math>
<math>~\Rightarrow ~~~~ 2\Delta </math>	<math>~=</math>	<math>~2 \ln\sqrt{A\xi} = \ln(A\xi) </math>
and, <math>~3\Delta </math>	<math>~=</math>	<math>~3\ln[\sqrt{A\xi}] \, ;</math>

and, in laying out these function definitions, Murphy has implicitly assumed that the two scaling coefficients, <math>~A</math> and <math>~B</math>, are unity.

CAUTION: Presented in this fashion — that is, by using <math>~\xi</math> to represent the dimensionless radial coordinate in all four expressions — Murphy's expressions seem to imply that the independent variable defining the radial coordinate in the bipolytrope's core is the same as the one that defines the radial coordinate in the structure's envelope. In general, this will not be the case, so we have explicitly used a different independent variable, <math>~\eta</math>, to mark the envelope's radial coordinate in our expressions. It is clear from other elements of his published derivation that Murphy understood this distinction but, as is explained more fully below, errors in his final model specifications may have resulted from not explicitly differentiating between this variable notation.

Critique of Murphy's Model Characteristics

Model Characteristics from Murphy's Table 3				Implications
Model	<math>\xi = \zeta_J</math>	<math>(A\xi) = \xi_J</math>	<math>A = \xi_J/\zeta_J</math>	<math>\Delta = \ln(A\xi)^{1/2}</math>	<math>V_{1E}</math>	<math>V_{5F}</math>	<math>\frac{(n_c+1)V_{1E}}{(n_e+1)V_{5F}}</math>
1	0.032678	10.0614	307.895	1.1544	3.5598E-04	1.1871E-04	1.000
2	0.8154	12.0083	14.727	1.2428	2.3212E-01	7.7302E-02	1.001
3	1.6598	20.4312	12.309	1.5085	1.1481	0.40720	0.940
4	2.6914	33.0249	12.2704	1.7486	6.5689	0.75371	2.905
5	2.7302	314.159	115.068	2.8750	7.2578	2.4193	1.000
6	3.1415	2.8675E+05	9.1278E+04	6.2832	3.3907E+04	2.9313E+04	0.386
NOTE: As is explained in an accompanying discussion, we suspect that the two numbers drawn from Murphy's Table 3 that are displayed here in a red font contain typographical errors.

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