User:Tohline/SSC/Structure/BiPolytropes/MurphyUVplane
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UV Plane Functions as Analyzed by Murphy (1983)
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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 polytropic index and the envelope has an polytropic index. This system is particularly interesting because the entire structure can be described by closedform, analytic expressions. Here we provide an indepth 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 system as "Murphy's bipolytrope."
Key References
 S. Srivastava (1968, ApJ, 136, 680) A New Solution of the LaneEmden 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 .
 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 FType and MType Solutions of the LaneEmden 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 EType Composite Analytical Solutions of the LaneEmden Equation
 J. O. Murphy (1983, Australian Journal of Physics, 36, 453): Structure of a Sequence of TwoZone 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 LaneEmden 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 TwoZone 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 TwoZone Polytropic Stellar Models
Relevant LaneEmden Functions
As is detailed in our accompanying discussion, the LaneEmden function governing the structure of the core of Murphy's bipolytrope is,
and the first derivative of this function with respect to the dimensionless radial coordinate, , is,
Also as is detailed in our accompanying discussion, the LaneEmden function governing the structure of the envelope of Murphy's bipolytrope is,






and the first derivative of this function is,



where we have adopted the shorthand notation,
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 "UV" plane. Specifically in the context of the model's core, this pair of referenced functions is:












Correspondingly, in the context of the model's envelope, the pair of referenced functions is:


















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






and, 


and, in laying out these function definitions, Murphy has implicitly assumed that the two scaling coefficients, and , are unity.
CAUTION: Presented in this fashion — that is, by using 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, , 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 
ξ = ζ_{J} 
(Aξ) = ξ_{J} 
A = ξ_{J} / ζ_{J} 
Δ = ln(Aξ)^{1 / 2} 
V_{1E} 
V_{5F} 

1 
0.032678 
10.0614 
307.895 
1.1544 
3.5598E04 
1.1871E04 
1.000 
2 
0.8154 
12.0083 
14.727 
1.2428 
2.3212E01 
7.7302E02 
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. 
Some Model Characteristics (assuming μ_{e} / μ_{c} = 1) 

Model 
Specified 
Analytically Determined Here 
… and Constraint Implications^{†} 















1  3.2678E02  0  1.1544  10.0614  307.894  5.77929  4.39209  1.7392  3.5598E04  1.1866E4  1.000  2.9998  1.000 
2  0.8154  0  1.2429  12.0101  14.729  1.25567  1.06865  36.356  0.23212  0.077372  1.000  2.8644  1.000 
3  1.6598  0  1.4918  19.7585  11.9041  1.01206  1.28146  44.984  1.14812  0.38271  1.000  2.3995  1.000 
4  2.0914  1  1.7328  31.9964  15.2990  0.87563  1.60398  35.0016  2.19913  0.73304  1.000  1.98894  1.000 
5  2.7302  1  2.8749  100.00π  115.065  0.12408  0.64370  4.6539  7.25779  2.41926  1.000  1.02703  1.000 
6  3.1415  2  6.283141  91268π  170.44223e^{2π}  1.8966E05  0.48862  3.141778  3.3907E+04  1.1302E+04  1.000  2.9106E04  1.000 
Corresponding Values Extracted Directly from Murphy's (1983) Table 3 


^{†}See an accompanying discussion for definitions of the functions, , , , and 
Related Discussions
 Bipolytrope with n_{c} = 1 and n_{e}=5
 Polytropes emdeded in an external medium
 Constructing BiPolytropes
© 2014  2021 by Joel E. Tohline 