User:Tohline/SSC/Structure/BiPolytropes/MurphyUVplane - Revision history

Tohline: /* Critique of Murphy's Model Characteristics */ Insert long multi-colored table from separate chapter

2015-05-14T21:38:18Z

Critique of Murphy's Model Characteristics: Insert long multi-colored table from separate chapter

Show changes

Tohline: /* Critique of Murphy's Model Characteristics */ Insert comparison table from another chapter

2015-05-14T18:59:14Z

Critique of Murphy's Model Characteristics: Insert comparison table from another chapter

Show changes

Tohline: /* Critique of Murphy's Model Characteristics */ Add footnote to table of numbers

2015-05-02T15:28:39Z

Critique of Murphy's Model Characteristics: Add footnote to table of numbers

← Older revision		Revision as of 15:28, 2 May 2015
Line 293:		Line 293:

	<div align="center">		<div align="center">
	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center" width="90%">
	<tr>		<tr>
	<td align="center" colspan="4">		<td align="center" colspan="4">
Line 486:		Line 486:
	<td align="center">		<td align="center">
	0.386		0.386
			</td>
			</tr>

			<tr>
			<td align="left" colspan="8">
			<font color="red">NOTE:</font> As is [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Murphy.27s_Example_Model_Characteristics\|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.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Critique of Murphy's Model Characteristics */ Clean up numerical values listed in table

2015-05-02T15:21:12Z

Critique of Murphy's Model Characteristics: Clean up numerical values listed in table

← Older revision		Revision as of 15:21, 2 May 2015
Line 348:		Line 348:
	</td>		</td>
	<td align="right">		<td align="right">
	1.~~1963E~~-04		1.1871E-04
	</td>		</td>
	<td align="center">		<td align="center">
	0.~~9919~~		1.000
	</td>		</td>
	</tr>		</tr>
Line 378:		Line 378:
	</td>		</td>
	<td align="center">		<td align="center">
	1.~~0009~~		1.001
	</td>		</td>
	</tr>		</tr>
Line 404:		Line 404:
	</td>		</td>
	<td align="center">		<td align="center">
	0.~~9398~~		0.940
	</td>		</td>
	</tr>		</tr>
Line 425:		Line 425:
	</td>		</td>
	<td align="right">		<td align="right">
	6.~~5693~~		6.5689
	</td>		</td>
	<td align="right">		<td align="right">
	0.~~75372~~		0.75371
	</td>		</td>
	<td align="center">		<td align="center">
	2.~~9053~~		2.905
	</td>		</td>
	</tr>		</tr>
Line 455:		Line 455:
	</td>		</td>
	<td align="right">		<td align="right">
	2.~~4315~~		2.4193
	</td>		</td>
	<td align="center">		<td align="center">
	0.~~9950~~		1.000
	</td>		</td>
	</tr>		</tr>
Line 482:		Line 482:
	</td>		</td>
	<td align="right">		<td align="right">
	1.~~1734E~~+05		2.9313E+04
	</td>		</td>
	<td align="center">		<td align="center">
	~~---~~		0.386
	</td>		</td>
	</tr>		</tr>

Tohline: /* Critique of Murphy's Model Characteristics */

2015-05-02T03:34:26Z

Critique of Murphy's Model Characteristics

← Older revision		Revision as of 03:34, 2 May 2015
Line 298:		Line 298:
	Model Characteristics from Murphy's Table 3		Model Characteristics from Murphy's Table 3
	</td>		</td>
	<td align="center" colspan="3">		<td align="center" colspan="4">
	Implications		Implications
	</td>		</td>
Line 314:		Line 314:
	<td align="center">		<td align="center">
	<math>A = \xi_J/\zeta_J</math>		<math>A = \xi_J/\zeta_J</math>
			</td>
			<td align="center">
			<math>\Delta = \ln(A\xi)^{1/2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 337:		Line 340:
	<td align="right">		<td align="right">
	307.895		307.895
			</td>
			<td align="right">
			1.1544
	</td>		</td>
	<td align="right">		<td align="right">
Line 361:		Line 367:
	<td align="right">		<td align="right">
	14.727		14.727
			</td>
			<td align="right">
			1.2428
	</td>		</td>
	<td align="right">		<td align="right">
Line 384:		Line 393:
	<td align="right">		<td align="right">
	12.309		12.309
			</td>
			<td align="right">
			1.5085
	</td>		</td>
	<td align="right">		<td align="right">
Line 395:		Line 407:
	</td>		</td>
	</tr>		</tr>

			<tr>
			<td align="center">
			4
			</td>
			<td align="right">
			<font color="red">2.6914</font>
			</td>
			<td align="right">
			33.0249
			</td>
			<td align="right">
			12.2704
			</td>
			<td align="right">
			1.7486
			</td>
			<td align="right">
			6.5693
			</td>
			<td align="right">
			0.75372
			</td>
			<td align="center">
			2.9053
			</td>
			</tr>

			<tr>
			<td align="center">
			5
			</td>
			<td align="right">
			2.7302
			</td>
			<td align="right">
			314.159
			</td>
			<td align="right">
			115.068
			</td>
			<td align="right">
			2.8750
			</td>
			<td align="right">
			7.2578
			</td>
			<td align="right">
			2.4315
			</td>
			<td align="center">
			0.9950
			</td>
			</tr>

			<tr>
			<td align="center">
			6
			</td>
			<td align="right">
			3.1415
			</td>
			<td align="right">
			2.8675E+05
			</td>
			<td align="right">
			9.1278E+04
			</td>
			<td align="right">
			6.2832
			</td>
			<td align="right">
			3.3907E+04
			</td>
			<td align="right">
			1.1734E+05
			</td>
			<td align="center">
			---
			</td>
			</tr>

	</table>		</table>
	</div>		</div>

Tohline: /* Critique of Murphy's Model Characteristics */ Add more lines to "critique" table

2015-05-02T02:10:54Z

Critique of Murphy's Model Characteristics: Add more lines to "critique" table

← Older revision		Revision as of 02:10, 2 May 2015
Line 297:		Line 297:
	<td align="center" colspan="4">		<td align="center" colspan="4">
	Model Characteristics from Murphy's Table 3		Model Characteristics from Murphy's Table 3
			</td>
			<td align="center" colspan="3">
			Implications
	</td>		</td>
	</tr>		</tr>
Line 311:		Line 314:
	<td align="center">		<td align="center">
	<math>A = \xi_J/\zeta_J</math>		<math>A = \xi_J/\zeta_J</math>
			</td>
			<td align="center">
			<math>V_{1E}</math>
			</td>
			<td align="center">
			<math>V_{5F}</math>
			</td>
			<td align="center">
			<math>\frac{(n_c+1)V_{1E}}{(n_e+1)V_{5F}}</math>
	</td>		</td>
	</tr>		</tr>
Line 325:		Line 337:
	<td align="right">		<td align="right">
	307.895		307.895
			</td>
			<td align="right">
			3.5598E-04
			</td>
			<td align="right">
			1.1963E-04
			</td>
			<td align="center">
			0.9919
			</td>
			</tr>

			<tr>
			<td align="center">
			2
			</td>
			<td align="right">
			0.8154
			</td>
			<td align="right">
			12.0083
			</td>
			<td align="right">
			14.727
			</td>
			<td align="right">
			2.3212E-01
			</td>
			<td align="right">
			7.7302E-02
			</td>
			<td align="center">
			1.0009
			</td>
			</tr>
			<tr>
			<td align="center">
			3
			</td>
			<td align="right">
			1.6598
			</td>
			<td align="right">
			20.4312
			</td>
			<td align="right">
			12.309
			</td>
			<td align="right">
			1.1481
			</td>
			<td align="right">
			0.40720
			</td>
			<td align="center">
			0.9398
	</td>		</td>
	</tr>		</tr>

Tohline at 01:32, 2 May 2015

2015-05-02T01:32:29Z

← Older revision		Revision as of 01:32, 2 May 2015
Line 290:		Line 290:
	<font color="red">'''CAUTION:'''</font> 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.		<font color="red">'''CAUTION:'''</font> 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===		==Critique of Murphy's Model Characteristics==

	<div align="center">		<div align="center">

Tohline: Begin constructing table to critique Murphy's analysis

2015-05-02T01:29:42Z

Begin constructing table to critique Murphy's analysis

← Older revision		Revision as of 01:29, 2 May 2015
Line 289:		Line 289:

	<font color="red">'''CAUTION:'''</font> 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.		<font color="red">'''CAUTION:'''</font> 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===

			<div align="center">
			<table border="1" cellpadding="5" align="center">
			<tr>
			<td align="center" colspan="4">
			Model Characteristics from Murphy's Table 3
			</td>
			</tr>
			<tr>
			<td align="center">
			Model
			</td>
			<td align="center">
			<math>\xi = \zeta_J</math>
			</td>
			<td align="center">
			<math>(A\xi) = \xi_J</math>
			</td>
			<td align="center">
			<math>A = \xi_J/\zeta_J</math>
			</td>
			</tr>
			<tr>
			<td align="center">
			1
			</td>
			<td align="right">
			0.032678
			</td>
			<td align="right">
			<font color="red">10.0614</font>
			</td>
			<td align="right">
			307.895
			</td>
			</tr>
			</table>
			</div>

	=Related Discussions=		=Related Discussions=

Tohline: /* Chandrasekhar's U and V Functions */Change notation from eta to xi

2015-05-02T01:09:37Z

Chandrasekhar's U and V Functions: Change notation from eta to xi

← Older revision		Revision as of 01:09, 2 May 2015
Line 148:		Line 148:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~U_{5F} \equiv \~~eta~~ \phi^5 \biggl(- \frac{d\phi}{d\~~eta~~}\biggr)^{-1}</math>		<math>~U_{5F} \equiv \xi \phi^5 \biggl(- \frac{d\phi}{d\xi}\biggr)^{-1}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 155:		Line 155:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{B^{-4}\~~eta~~ \sin^5\Delta}{\~~eta~~^{5/2}\{3-2\sin^2\Delta\}^{5/2}}		\frac{B^{-4}\xi \sin^5\Delta}{\xi^{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]		\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>
	</td>		</td>
Line 191:		Line 191:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~(n_e+1) V_{5F} \equiv (n_e+1) \frac{\~~eta~~}{ \phi} \biggl(- \frac{d\phi}{d\~~eta~~}\biggr)</math>		<math>~(n_e+1) V_{5F} \equiv (n_e+1) \frac{\xi}{ \phi} \biggl(- \frac{d\phi}{d\xi}\biggr)</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 197:		Line 197:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{6\~~eta~~^{3/2}\{3-2\sin^2\Delta\}^{1/2}} {\sin\Delta}		<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\~~eta~~^{3/2}(3-2\sin^2\Delta)^{3/2}}		\frac{[3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ] }{2\xi^{3/2}(3-2\sin^2\Delta)^{3/2}}
	</math>		</math>
	</td>		</td>
Line 255:		Line 255:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\ln\sqrt{A\~~eta~~} </math>		<math>~\ln\sqrt{A\xi} </math>
	</td>		</td>
	</tr>		</tr>
Line 267:		Line 267:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~2 \ln\sqrt{A\~~eta~~} = \ln(A\~~eta~~) </math>		<math>~2 \ln\sqrt{A\xi} = \ln(A\xi) </math>
	</td>		</td>
	</tr>		</tr>
Line 280:		Line 280:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~3\ln[\sqrt{A\~~eta~~}] \, ;</math>		<math>~3\ln[\sqrt{A\xi}] \, ;</math>
	</td>		</td>
	</tr>		</tr>
Line 289:		Line 289:

	<font color="red">'''CAUTION:'''</font> 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.		<font color="red">'''CAUTION:'''</font> 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.


	=Related Discussions=		=Related Discussions=

Tohline: /* Relevant Lane-Emden Functions */ Change "eta" to "xi"

2015-05-02T01:01:25Z

Relevant Lane-Emden Functions: Change "eta" to "xi"

← Older revision		Revision as of 01:01, 2 May 2015
Line 19:		Line 19:
	==Relevant Lane-Emden Functions==		==Relevant Lane-Emden Functions==

	As is detailed in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Steps_2_.26_3\|our accompanying discussion]], the Lane-Emden function governing the structure of the ''core'' of Murphy's bipolytrope is,		As is detailed in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Steps_2_.26_3\|our accompanying discussion]], the Lane-Emden function governing the structure of the <math>~n_c = 1</math> ''core'' of Murphy's bipolytrope is,
	<div align="center">		<div align="center">
	<math>		<math>
Line 31:		Line 31:
	</math>		</math>
	</div>		</div>
	Also as is detailed in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Step_6:__Envelope_Solution\|our accompanying discussion]], the Lane-Emden function governing the structure of the ''envelope'' of Murphy's bipolytrope is,		Also as is detailed in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Step_6:__Envelope_Solution\|our accompanying discussion]], the Lane-Emden function governing the structure of the <math>~n_e = 5</math> ''envelope'' of Murphy's bipolytrope is,

	<div align="center">		<div align="center">
Line 37:		Line 37:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\phi(\~~eta~~)</math>		<math>~\phi(\xi)</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 43:		Line 43:
	</td>		</td>
	<td align="left">		<td align="left">
	<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>~\frac{B^{-1}\sin[\ln(A\xi)^{1/2})]}{\xi^{1/2}\{3-2\sin^2[\ln(A\xi)^{1/2}]\}^{1/2}} </math>
	</td>		</td>
	</tr>		</tr>
Line 55:		Line 55:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{B^{-1}\sin\Delta}{\~~eta~~^{1/2}(3-2\sin^2\Delta)^{1/2}} \, ,</math>		<math>~\frac{B^{-1}\sin\Delta}{\xi^{1/2}(3-2\sin^2\Delta)^{1/2}} \, ,</math>
	</td>		</td>
	</tr>		</tr>
Line 68:		Line 68:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{d\phi}{d\~~eta~~}</math>		<math>~\frac{d\phi}{d\xi}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 75:		Line 75:
	<td align="left">		<td align="left">
	<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}} \, ,		\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>		</math>
	</td>		</td>
Line 83:		Line 83:
	where we have adopted the shorthand notation,		where we have adopted the shorthand notation,
	<div align="center">		<div align="center">
	<math>~\Delta \equiv \ln(A\~~eta~~)^{1/2} \, .</math>		<math>~\Delta \equiv \ln(A\xi)^{1/2} \, .</math>
	</div>		</div>