SSC/Structure/BiPolytropes/Analytic15 - Revision history

Xxsrm: /* Step 6: Envelope Solution */

2025-01-24T00:17:31Z

Step 6: Envelope Solution

← Older revision		Revision as of 20:17, 23 January 2025
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	</table>		</table>
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	[[File:CommentButton02.png\|right\|100px\|Note that our homology factor and scaling coefficient serve virtually the same roles as the homology factor, A, and scaling coefficient, B, used by Murphy (1983), but they are not mathematically identical so we have added a subscript "0" to highlight the distinctions.]]where <math>~A_0</math> is a "homology factor" and <math>~B_0</math> is an overall scaling coefficient — the values of both will be determined presently from the interface conditions — and we have introduced the notation,		[[File:CommentButton02.png\|right\|100px\|Note that our homology factor and scaling coefficient serve virtually the same roles as the homology factor, A, and scaling coefficient, B, used by Murphy (1983), but they are not mathematically identical so we have added a subscript "0" to highlight the distinctions.]]
			where <math>~A_0</math> is a "homology factor" and <math>~B_0</math> is an overall scaling coefficient — the values of both will be determined presently from the interface conditions — and we have introduced the notation,
	<div align="center">		<div align="center">
	<math>~\Delta \equiv \ln(A_0\eta)^{1/2} = \frac{1}{2} (\ln A_0 + \ln\eta) \, .</math>		<math>~\Delta \equiv \ln(A_0\eta)^{1/2} = \frac{1}{2} (\ln A_0 + \ln\eta) \, .</math>

Xxsrm: /* Step 5: Interface Conditions */

2025-01-24T00:14:25Z

Step 5: Interface Conditions

← Older revision		Revision as of 20:14, 23 January 2025
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	<span id="Alternative"><font color="red">Alternative:</font> In our</span> [[SSC/Structure/BiPolytropes/#UVplane\|introductory description of how to build a bipolytropic structure]], we pointed out that, instead of employing these last two fitting conditions, Chandrasekhar [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]] found it useful to employ, instead, the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions, which in the present case produces,		<span id="Alternative"><font color="red">Alternative:</font> In our</span> [[SSC/Structure/BiPolytropes#UVplane\|introductory description of how to build a bipolytropic structure]], we pointed out that, instead of employing these last two fitting conditions, Chandrasekhar [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]] found it useful to employ, instead, the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions, which in the present case produces,
	<div align="center">		<div align="center">
	<math>		<math>

Joel2: /* Comment on Murphy's Scalings */

2024-01-03T22:45:08Z

Comment on Murphy's Scalings

← Older revision		Revision as of 18:45, 3 January 2024
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	===Comment on Murphy's Scalings===		===Comment on Murphy's Scalings===

	~~[http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy's (1983)]~~ derivations also include an homology factor, <math>~A</math>, and an overall scaling factor, <math>~B</math>, but they are calculated differently from our <math>~A_0</math> and <math>~B_0</math>. In the righthand column of the third page of his paper, Murphy states that,		The {{ Murphy83a }} derivations also include an homology factor, <math>~A</math>, and an overall scaling factor, <math>~B</math>, but they are calculated differently from our <math>~A_0</math> and <math>~B_0</math>. In the righthand column of the third page of his paper, Murphy states that,
	<div align="center">		<div align="center">
	<math>~A = \frac{\xi_J}{\zeta_J} \, ,</math>		<math>~A = \frac{\xi_J}{\zeta_J} \, ,</math>
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	<math>~A = \frac{A_0}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>		<math>~A = \frac{A_0}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>
	</div>		</div>
	It is usually the value of this quantity, rather than simply our derived value of <math>~A_0</math>, that is tabulated below — both [[SSC/Structure/BiPolytropes/Analytic15#Murphy.27s_Example_Model_Characteristics\|here]] and [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29\|here]] — as we make quantitative comparisons between the characteristics of our derived models and those published by ~~[http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)]~~ and by ~~[http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy & Fiedler (1985a)]~~.		It is usually the value of this quantity, rather than simply our derived value of <math>~A_0</math>, that is tabulated below — both [[SSC/Structure/BiPolytropes/Analytic15#Murphy.27s_Example_Model_Characteristics\|here]] and [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29\|here]] — as we make quantitative comparisons between the characteristics of our derived models and those published by {{ Murphy83a }} and by {{ MF85a }}.


	In the lefthand column of the fourth page of his paper, ~~[http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)]~~ defines the coefficient <math>~B</math> in such a way that the ''value'' of the envelope function, <math>~\phi_{5F}</math>, equals the ''value'' of the core function, <math>~\theta_{1E}</math>, at the interface. Specifically, he sets,		In the lefthand column of the fourth page of his paper, {{ Murphy83a }} defines the coefficient <math>B</math> in such a way that the ''value'' of the envelope function, <math>\phi_{5F}</math>, equals the ''value'' of the core function, <math>\theta_{1E}</math>, at the interface. Specifically, he sets,

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Joel2: /* Obtained from Chandrasekhar's U-constraint */

2024-01-03T22:42:04Z

Obtained from Chandrasekhar's U-constraint

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	which, in the particular case being examined here, becomes — again, see [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar.27s_U_and_V_Functions\|our accompanying discussion]] for elaboration on ~~[http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy's (1983)]~~ "U<sub>5F</sub>" and "U<sub>1E</sub>" function notations —		which, in the particular case being examined here, becomes — again, see [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar.27s_U_and_V_Functions\|our accompanying discussion]] for elaboration on the "U<sub>5F</sub>" and "U<sub>1E</sub>" function notations used by {{ Murphy83a }} —
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Joel2: /* First Constraint */

2024-01-03T22:38:27Z

First Constraint

← Older revision		Revision as of 18:38, 3 January 2024
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	</div>		</div>

	In a [[SSC/Structure/BiPolytropes/Analytic15/Pt2#Analytic_Solution_of_Key_Interface_Relation\|separate subsection of this chapter~~, below~~]], we present a closed-form analytic solution, <math>\Delta_i(\kappa_i)</math>, to this nonlinear equation.		In a [[SSC/Structure/BiPolytropes/Analytic15/Pt2#Analytic_Solution_of_Key_Interface_Relation\|separate subsection (Part II) of this chapter]], we present a closed-form analytic solution, <math>\Delta_i(\kappa_i)</math>, to this nonlinear equation.

Joel2: /* First Constraint */

2024-01-03T22:36:58Z

First Constraint

← Older revision		Revision as of 18:36, 3 January 2024
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	<math>~\kappa_i</math>		<math>\kappa_i</math>
	</td>		</td>
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	<math>~=</math>		<math>=</math>
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	<td align="left">		<td align="left">
	<math>~\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, .</math>		<math>\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, .</math>
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	</tr>		</tr>
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	</div>		</div>

	In a [[SSC/Structure/BiPolytropes/Analytic15#Analytic_Solution_of_Key_Interface_Relation\|separate subsection of this chapter, below]], we present a closed-form analytic solution, <math>~\Delta_i(\kappa_i)</math>, to this nonlinear equation.		In a [[SSC/Structure/BiPolytropes/Analytic15/Pt2#Analytic_Solution_of_Key_Interface_Relation\|separate subsection of this chapter, below]], we present a closed-form analytic solution, <math>\Delta_i(\kappa_i)</math>, to this nonlinear equation.

Joel2: /* Step 6: Envelope Solution */

2024-01-03T22:18:21Z

Step 6: Envelope Solution

← Older revision		Revision as of 18:18, 3 January 2024
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	</div>		</div>

	As has been explained in the context of our [[SSC/Structure/Polytropes#Srivastava~~.27s_F~~-Type_Solution\|more general discussion of Srivastava's function]], if we ignore, for the moment, the additional "<math>m\pi</math>" phase shift that can be attached to a determination of the angle, <math>~\Delta</math>, the physically viable interval for the dimensionless radial coordinate is, <math>~e^{2\pi} \ge A_0\eta \ge \eta_\mathrm{crit} \equiv e^{2\tan^{-1}(1+2^{1/3})} \, .</math>		As has been explained in the context of our [[SSC/Structure/Polytropes/Analytic#Srivastava's_F-Type_Solution\|more general discussion of Srivastava's function]], if we ignore, for the moment, the additional "<math>m\pi</math>" phase shift that can be attached to a determination of the angle, <math>~\Delta</math>, the physically viable interval for the dimensionless radial coordinate is, <math>~e^{2\pi} \ge A_0\eta \ge \eta_\mathrm{crit} \equiv e^{2\tan^{-1}(1+2^{1/3})} \, .</math>

Joel2: /* Step 6: Envelope Solution */

2024-01-03T22:16:06Z

Step 6: Envelope Solution

← Older revision		Revision as of 18:16, 3 January 2024
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	==Step 6: Envelope Solution==		==Step 6: Envelope Solution==

	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 20 April 2015: There is a type-setting error in this function expression as published in the upper left-hand column of the second page of the article by Murphy (1983); the sine function in the denominator should be sine-squared, as presented here.]]Following the work of {{ Murphy83a }} and of {{ MF85a }}, we will adopt for the envelope's structure the F-Type solution of the n = 5 Lane-Emden function discovered by {{ Srivastava62full }} and described in an [[SSC/Structure/Polytropes#Srivastava~~.27s_F~~-Type_Solution\|accompanying discussion]], namely,		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 20 April 2015: There is a type-setting error in this function expression as published in the upper left-hand column of the second page of the article by Murphy (1983); the sine function in the denominator should be sine-squared, as presented here.]]Following the work of {{ Murphy83a }} and of {{ MF85a }}, we will adopt for the envelope's structure the F-Type solution of the n = 5 Lane-Emden function discovered by {{ Srivastava62full }} and described in an [[SSC/Structure/Polytropes/Analytic#Srivastava's_F-Type_Solution\|accompanying discussion]], namely,

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

Joel2: /* Step 6: Envelope Solution */

2024-01-03T22:12:51Z

Step 6: Envelope Solution

← Older revision		Revision as of 18:12, 3 January 2024
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	==Step 6: Envelope Solution==		==Step 6: Envelope Solution==

	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 20 April 2015: There is a type-setting error in this function expression as published in the upper left-hand column of the second page of the article by Murphy (1983); the sine function in the denominator should be sine-squared, as presented here.]]Following the work of {{ Murphy83a }} and of {{ MF85a }}, we will adopt for the envelope's structure the F-Type solution of the ~~<math>~~~n=5~~</math>~~ Lane-Emden function discovered by ~~[http://adsabs.harvard.edu/abs/1962ApJ...136..680S S. Srivastava (1968, ApJ, 136, 680)]~~ and described in an [[SSC/Structure/Polytropes#Srivastava.27s_F-Type_Solution\|accompanying discussion]], namely,		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 20 April 2015: There is a type-setting error in this function expression as published in the upper left-hand column of the second page of the article by Murphy (1983); the sine function in the denominator should be sine-squared, as presented here.]]Following the work of {{ Murphy83a }} and of {{ MF85a }}, we will adopt for the envelope's structure the F-Type solution of the n = 5 Lane-Emden function discovered by {{ Srivastava62full }} and described in an [[SSC/Structure/Polytropes#Srivastava.27s_F-Type_Solution\|accompanying discussion]], namely,

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

Joel2: /* Step 5: Interface Conditions */

2024-01-03T21:50:00Z

Step 5: Interface Conditions

← Older revision		Revision as of 17:50, 3 January 2024
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	<span id="Alternative"><font color="red">Alternative:</font> In our</span> [[SSC/Structure/BiPolytropes/~~MurphyUVplane~~#~~Chandrasekhar's_U_and_V_Functions~~\|introductory description of how to build a bipolytropic structure]], we pointed out that, instead of employing these last two fitting conditions, Chandrasekhar [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]] found it useful to employ, instead, the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions, which in the present case produces,		<span id="Alternative"><font color="red">Alternative:</font> In our</span> [[SSC/Structure/BiPolytropes/#UVplane\|introductory description of how to build a bipolytropic structure]], we pointed out that, instead of employing these last two fitting conditions, Chandrasekhar [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]] found it useful to employ, instead, the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions, which in the present case produces,
	<div align="center">		<div align="center">
	<math>		<math>