User:Tohline/SSC/Structure/BiPolytropes/MurphyUVplane
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 <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 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 <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 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 <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 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 <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 LaneEmden 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}\{32\sin^2[\ln(A\xi)^{1/2}]\}^{1/2}} </math> 

<math>~=</math> 
<math>~\frac{B^{1}\sin\Delta}{\xi^{1/2}(32\sin^2\Delta)^{1/2}} \, ,</math> 
and the first derivative of this function is,
<math>~\frac{d\phi}{d\xi}</math> 
<math>~=</math> 
<math>~ \frac{B^{1}(3\sin\Delta  2\sin^3\Delta 3\cos\Delta) }{2\xi^{3/2}(32\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 "UV" 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}\{32\sin^2\Delta\}^{5/2}} \biggl[ \frac{2\xi^{3/2}(32\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}{[32\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}\{32\sin^2\Delta\}^{1/2}} {\sin\Delta} \frac{[3\sin\Delta  2\sin^3\Delta 3\cos\Delta ] }{2\xi^{3/2}(32\sin^2\Delta)^{3/2}} </math> 

<math>~=</math> 
<math>~\frac{3[3\sin\Delta  2\sin^3\Delta 3\cos\Delta ] }{\sin\Delta (32\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 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,
<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
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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As a supplement to the preceding two figures, the following table shows in quantitative detail how we are able to analytically derive individual coordinate roots, <math>~(A_0\eta)_\mathrm{root}</math> — see the nexttolast column of the table — from ten different interfacecoordinate locations, <math>~\xi_i = \zeta_J</math>, as specified in the first (leftmost) column of the table. The interface parameter, <math>~\kappa_i</math>, is determined directly from the specified value of <math>~\xi_i</math>; the parameter, <math>~p=(1\kappa_i)^{1}</math>; then, given <math>~p</math>, the root to the cubic equation and, in turn, the value of <math>~\Delta_i</math>, are determined; and, finally, a choice of the integer phaseshift parameter, <math>~m</math>, leads to the calculation of <math>~(A_0\eta)_\mathrm{root}</math>. For comparison, the last (rightmost) column of the table lists the corresponding coordinate, <math>~\xi_J</math>, as originally derived and published by Murphy (1983). Coordinate pairs shown in the orangecolored panels of our table correspond to the six models whose characteristics are provided in Murphy's Table 3; coordinate pairs shown in the greencolored panels correspond to two of the ten "F2 = 0" roots identified in Murphy's Table 1; and the additional three coordinate pairs shown in the bluecolored panels have been drawn from Murphy's Table 2.
Murphy (1983) Tables 1, 2, & 3 
Here … setting <math>\xi_i = \zeta_J</math> … and assuming <math>\mu_e/\mu_c = 1</math> 
Murphy (1983) Tables 1, 2, & 3 

1^{st} Constraint 

<math>~\zeta_J</math> 
<math>~\kappa_i</math> 
<math>~p</math> 
<math>~y_\mathrm{root}</math> 
<math>~(\Delta_im\pi)</math><math>~= \tan^{1}(y_\mathrm{root})</math> 
<math>~m </math> 
<math>~(A_0\eta)_\mathrm{root} = e^{2\Delta_i}</math> 
<math>~\xi_J</math> 
3.2678E02 
2.3732E04 
1.00023737 
2.26083 
1.15435 
3 
6.55237E08 
6.55E08 
2 
3.50874E05 
3.508E05 

1 
1.87890E02 
1.8789E02 

0 
10.0614 
10.0164 

0.8154 
0.15474 
1.18307 
2.93939 
1.24287 
0 
12.0101 
12.0083 
1.6598 
0.76541 
4.26281 
12.6311 
1.49179 
0 
19.7585 
20.4312 
[ 2.0914 ] 
1.46609 
2.14552 
6.11816 
1.40878 
1 
31.9964 
33.0249 
[ 2.10961 ] 
1.50744 
1.97067 
5.56379 
1.39296 
1 
33.0249 

2.6914 
4.37925 
0.29592 
0.31486 
0.30503 
1 
290.943 

2.7302 
4.83853 
0.26052 
0.27316 
0.26666 
1 
99.997<math>\pi</math> 
100<math>\pi</math> 
3.1415 
22604.6 
4.42408E05 
4.42408E05 
4.42408E05 
2 
91268<math>\pi</math> 
91275<math>\pi</math> 
4.2993 
0.58961 
0.62908 
0.89355 
0.72924 
0 
4.2994 
4.2993 
7.6882 
0.19091 
0.83969 
1.6268 
1.01964 
0 
7.6851 
7.6882 
As this table and the loglog plots presented in two preceding figures illustrate, by and large, the agreement between our analytically determined coordinate roots and those derived & published by Murphy (1983) is excellent. There are a couple of mismatches, but we attribute both to typesetting errors in the Murphy publication:
 The coordinate, <math>~\xi_J = 10.0164</math>, that has been displayed using a red font in the rightmost column of our table is the value drawn directly from Murphy's Table 3. It appears to us that two digits of this number were inadvertently transposed during publication and that the coordinate value should have appeared as <math>~\xi_J = 10.0614</math>. We are convinced that, as published, this is a typographical error in Murphy's Table 3 because: (a) the corrected value matches our derived value of that coordinate to more significant figures, and (b) the same model is also listed in Murphy's Table 2 with this corrected value.
 The coordinate, <math>~\zeta_J = 2.6914</math>, that has been displayed using a red font in the leftmost column of our table is the value drawn directly from Murphy's Table 3. As shown in our table, from this coordinate value we derive a root coordinate of <math>~(A_0\eta)_\mathrm{root} = 290.943</math>, which is almost a factor of ten larger than Murphy's published root coordinate value, <math>~\xi_J = 33.0249</math>. Working backwards via an iterative procedure, we have determined that Murphy's listed root of <math>33.0249</math> can be obtained if the originating value of <math>~\zeta_J \leftrightarrow \xi_i</math> is set to <math>2.10961</math> (see the blackonwhite panel entry in the leftmost column of our table). By replacing the "6" by a zero in Murphy's published coordinate parameter — that is, by assuming that there was a typographical error and Murphy's interface coordinate was actually <math>~\zeta_J = 2.0914</math>, we are able to derive a root coordinate value (31.9964) that is quite close to Murphy's published value (33.0249).
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 
3.5598E04 
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 <math>\mu_e/\mu_c</math> = 1) 

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

<math>~\xi_i</math> 
<math>~m</math> 
<math>~\Delta_i</math> 
<math>~(A_0\eta)_\mathrm{root}</math> 
<math>~\frac{A_0}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) </math> 
<math>~B_0</math> 
<math>~\frac{B_0}{3^{1/4}} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{1/2} \theta_i^{1} </math> 
<math>~\xi_s</math> 
<math>~V_{1E}</math> 
<math>~V_{5F}</math> 
<math>\frac{(n_c+1)V_{1E}}{(n_e+1)V_{5F}}</math> 
<math>~U_{1E}</math> 
<math>\frac{U_{1E}}{U_{5F}}</math> 

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  100.00<math>\pi</math>  115.065  0.12408  0.64370  4.6539  7.25779  2.41926  1.000  1.02703  1.000 
6  3.1415  2  6.283141  91268<math>\pi</math>  170.44223<math>e^{2\pi}</math>  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, <math>~U_{1E}</math>, <math>~V_{1E}</math>, <math>~U_{5F}</math>, and <math>~V_{5F} \, .</math> 
Related Discussions
 Bipolytrope with n_{c} = 1 and n_{e}=5
 Polytropes emdeded in an external medium
 Constructing BiPolytropes
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