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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 ~n_c=1 polytropic index and the envelope has an ~n_e=5 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 ~(n_c, n_e) = (1, 5) 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 ~n_c = 1 core of Murphy's bipolytrope is,


\theta(\xi) = \frac{\sin\xi}{\xi} \, ,

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


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

Also as is detailed in our accompanying discussion, the Lane-Emden function governing the structure of the ~n_e = 5 envelope of Murphy's bipolytrope is,

~\phi(\xi)

~=

~\frac{B^{-1}\sin[\ln(A\xi)^{1/2})]}{\xi^{1/2}\{3-2\sin^2[\ln(A\xi)^{1/2}]\}^{1/2}}

 

~=

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

and the first derivative of this function is,

~\frac{d\phi}{d\xi}

~=

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

where we have adopted the shorthand notation,

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

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 ~n_c=1 core, this pair of referenced functions is:

~U_{1E} \equiv \xi \theta \biggl(- \frac{d\theta}{d\xi}\biggr)^{-1}

~=

~\sin\xi\biggl[\frac{1}{\xi^{2}} (\sin\xi - \xi\cos\xi)\biggr]^{-1}

 

~=

~\frac{\xi^2}{(1 - \xi\cot\xi)} \, ;

~(n_c+1) V_{1E} \equiv  (n_c+1)\frac{\xi}{ \theta} \biggl(- \frac{d\theta}{d\xi}\biggr)

~=

~\frac{2\xi^2}{\sin\xi} \biggl[ \frac{1}{\xi^{2}} (\sin\xi - \xi\cos\xi) \biggr]

 

~=

~ 2(1 - \xi\cot\xi) \, .

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

~U_{5F} \equiv \xi \phi^5 \biggl(- \frac{d\phi}{d\xi}\biggr)^{-1}

~=

~
\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]

 

~=

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

 

~=

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

~(n_e+1) V_{5F} \equiv (n_e+1) \frac{\xi}{ \phi} \biggl(- \frac{d\phi}{d\xi}\biggr)

~=

~\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}}

 

~=

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

 

~=

~\frac{-6[3\cos\Delta - \frac{3}{2}\sin\Delta - \frac{1}{2}\sin(3\Delta)  ] }{2\sin\Delta [2+\cos(2\Delta)]} \, .

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)

U and V Functions from Murphy (1983)

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

~\Delta

~=

~\ln\sqrt{A\xi}

~\Rightarrow ~~~~ 2\Delta

~=

~2 \ln\sqrt{A\xi} = \ln(A\xi)

and,       ~3\Delta

~=

~3\ln[\sqrt{A\xi}] \, ;

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

CAUTION: Presented in this fashion — that is, by using ~\xi 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, ~\eta, 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, ~(A_0\eta)_\mathrm{root} — see the next-to-last column of the table — from ten different interface-coordinate locations, ~\xi_i = \zeta_J, as specified in the first (left-most) column of the table. The interface parameter, ~\kappa_i, is determined directly from the specified value of ~\xi_i; the parameter, ~p=(1-\kappa_i)^{-1}; then, given ~p, the root to the cubic equation and, in turn, the value of ~\Delta_i, are determined; and, finally, a choice of the integer phase-shift parameter, ~m, leads to the calculation of ~(A_0\eta)_\mathrm{root}. For comparison, the last (right-most) column of the table lists the corresponding coordinate, ~\xi_J, as originally derived and published by Murphy (1983). Coordinate pairs shown in the orange-colored panels of our table correspond to the six models whose characteristics are provided in Murphy's Table 3; coordinate pairs shown in the green-colored 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 blue-colored panels have been drawn from Murphy's Table 2.


Murphy (1983)

Tables 1, 2, & 3

Here … setting ξi = ζJ … and assuming μe / μc = 1

Murphy (1983)

Tables 1, 2, & 3

1st Constraint

~\zeta_J

~\kappa_i

~p

~y_\mathrm{root}

~(\Delta_i-m\pi)

~= \tan^{-1}(y_\mathrm{root})

~m

~(A_0\eta)_\mathrm{root} = e^{2\Delta_i}

~\xi_J

3.2678E-02

2.3732E-04

1.00023737

2.26083

1.15435

-3

6.55237E-08

6.55E-08

-2

3.50874E-05

3.508E-05

-1

1.87890E-02

1.8789E-02

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

99.997π

100π

3.1415

22604.6

-4.42408E-05

-4.42408E-05

-4.42408E-05

2

91268π

91275π

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 log-log 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, ~\xi_J = 10.0164, that has been displayed using a red font in the right-most 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 ~\xi_J = 10.0614. 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, ~\zeta_J = 2.6914, that has been displayed using a red font in the left-most 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 ~(A_0\eta)_\mathrm{root} = 290.943, which is almost a factor of ten larger than Murphy's published root coordinate value, ~\xi_J = 33.0249. Working backwards via an iterative procedure, we have determined that Murphy's listed root of 33.0249 can be obtained if the originating value of ~\zeta_J \leftrightarrow \xi_i is set to 2.10961 (see the black-on-white panel entry in the left-most 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 ~\zeta_J = 2.0914, 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

ξ = ζJ

(Aξ) = ξJ

A = ξJ / ζJ

Δ = ln(Aξ)1 / 2

V1E

V5F

\frac{(n_c+1)V_{1E}}{(n_e+1)V_{5F}}

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.


Some Model Characteristics (assuming μe / μc = 1)

Model

Specified

Analytically Determined Here

… and Constraint Implications

~\xi_i

~m

~\Delta_i

~(A_0\eta)_\mathrm{root}

~\frac{A_0}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr)

~B_0

~\frac{B_0}{3^{1/4}} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{1/2} \theta_i^{-1}

~\xi_s

~V_{1E}

~V_{5F}

\frac{(n_c+1)V_{1E}}{(n_e+1)V_{5F}}

~U_{1E}

\frac{U_{1E}}{U_{5F}}

1 3.2678E-02 0 1.1544 10.0614 307.894 5.77929 4.39209 1.7392 3.5598E-04 1.1866E-4 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 1.8966E-05 0.48862 3.141778 3.3907E+04 1.1302E+04 1.000 2.9106E-04 1.000

Corresponding Values Extracted Directly from Murphy's (1983) Table 3

 

Column 1 from Table 3 of Murphy (1983)
Column 3 from Table 3 of Murphy (1983)
 
Column 3 from Table 3 of Murphy (1983)
Column 4 from Table 3 of Murphy (1983)
 
Column 5 from Table 3 of Murphy (1983)
Column 6 from Table 3 of Murphy (1983)
Comment by J. E. Tohline on 11 May 2015:  The heading on this last column of numbers extracted from Murphy's (1983) paper has a subscript "J" whereas the subscript should be "R" to indicate the configuration's total radius.

See an accompanying discussion for definitions of the functions, ~U_{1E}, ~V_{1E}, ~U_{5F}, and ~V_{5F} \, .

Related Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

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