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

Tohline: /* Setup */

2019-05-22T22:54:49Z

Setup

← Older revision		Revision as of 22:54, 22 May 2019
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	<font color="red">'''NOTE typed on 13 May 2019:'''</font>  ~~Previous~~ to this date, in both the middle and right-hand colums, the RHS of the ''polytropic'' version of the Lane-Emden equation incorrectly had a ''positive'' sign. This was a type-setting error that has now been corrected.		<font color="red">'''NOTE typed on 13 May 2019:'''</font>  Prior to this date, in both the middle and right-hand colums, the RHS of the ''polytropic'' version of the Lane-Emden equation incorrectly had a ''positive'' sign. This was a type-setting error that has now been corrected.
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Tohline: /* Setup */

2019-05-13T23:18:51Z

Setup

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	\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\theta}{d\xi} \biggr) = \theta^{n_c}		\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\theta}{d\xi} \biggr) = - \theta^{n_c}
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	\frac{1}{\eta^2} \frac{d}{d\eta} \biggl( \eta^2 \frac{d\phi}{d\eta} \biggr) = \phi^{n_e}		\frac{1}{\eta^2} \frac{d}{d\eta} \biggl( \eta^2 \frac{d\phi}{d\eta} \biggr) = - \phi^{n_e}
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			<font color="red">'''NOTE typed on 13 May 2019:'''</font>  Previous to this date, in both the middle and right-hand colums, the RHS of the ''polytropic'' version of the Lane-Emden equation incorrectly had a ''positive'' sign. This was a type-setting error that has now been corrected.
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Tohline: /* Related Discussions */

2017-03-17T22:05:21Z

Related Discussions

← Older revision		Revision as of 22:05, 17 March 2017
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	* [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Schoenberg-Chandrasekhar Limit: A BiPolytropic Approximation (Beech 1988b)]		* [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Schoenberg-Chandrasekhar Limit: A BiPolytropic Approximation (Beech 1988b)]
	* [http://adsabs.harvard.edu/abs/1988Ap%26SS.146..299B BiPolytropic Model for Low-Mass Stars (Beech 1988a)]		* [http://adsabs.harvard.edu/abs/1988Ap%26SS.146..299B BiPolytropic Model for Low-Mass Stars (Beech 1988a)]
			* An analysis (published in French) of the secular stability of configurations containing an isothermal core, with relevance to the Schönberg-Chandrasekhar mass limit; [http://adsabs.harvard.edu/abs/1967AnAp...30..975G M. Gabriel and P. Ledoux (1967, Annals d'Astrophysique, 30, 975)]

	=See Also=		=See Also=

Tohline: /* Background */

2016-11-22T22:27:53Z

Background

← Older revision		Revision as of 22:27, 22 November 2016
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	==Background==		==Background==
	While [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres\|polytropic spheres]] can give us useful ''general'' insight into the internal structure of stars, they do not faithfully represent the detailed structure of most stars, in part, because the equation of state that is most relevant to the densest regions of a star usually is different from the equation of state that is relevant to the star's envelope. As is highlighted in the following excerpt from [http://adsabs.harvard.edu/abs/1931MNRAS..91..472C Cowling (1931)], [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] was among the first to suggest that more realism could be achieved by constructing what is now commonly referred to as a bipolytrope: a composite model in which the envelope, obeying one equation of state — described by polytropic index <math>n_e</math> — is fitted to a core obeying a different equation of state — described by polytropic index <math>n_c</math>.		While [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres\|polytropic spheres]] can give us useful ''general'' insight into the internal structure of stars, they do not faithfully represent the detailed structure of most stars, in part, because the equation of state that is most relevant to the densest regions of a star usually is different from the equation of state that is relevant to the star's envelope. As is highlighted in the following excerpt from [http://adsabs.harvard.edu/abs/1931MNRAS..91..472C Cowling (1931)], [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] was among the first to suggest that more realism could be achieved by constructing what is now commonly referred to as a bipolytrope: a composite model in which the envelope, obeying one equation of state — described by polytropic index <math>n_e</math> — is fitted to a core obeying a different equation of state — described by polytropic index <math>n_c</math>.
	~~<div align="center">~~
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	~~Excerpt from the introductory paragraphs of [http://adsabs.harvard.edu/abs/1931MNRAS..91..472C Cowling (1931)]~~
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			Text excerpts<sup>&dagger;</sup> from §2 (pp. 161-162) of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]<p></p>
			"''On the Evolution of the Main-Sequence Stars''"<p></p>
			ApJ, 96, 161-172 © American Astronomicall Society
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	~~Excerpt from §2 of~~ [~~http~~:~~//adsabs~~.~~harvard.edu/abs/1942ApJ....96..161S~~ Schönberg & Chandrasekhar (1942)]		[[Image:SchonbergChandra1942.jpg\|600px\|center\|Schönberg & Chandrasekhar (1942)]]
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	[[Image:~~SchonbergChandra1942~~.~~jpg~~\|~~600px~~\|~~References~~]]
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	[http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] furthermore pointed out — again, see the relevant article excerpt — that, when constructing a bipolytropic model, the mathematical function that specifies the mass enclosed inside a spherical radius <math>r</math> for the ''core'', <math>M(r)\|_\mathrm{c}</math>, must give the same value as the mathematical function that specifies the mass enclosed inside a spherical radius <math>r</math> for the ''envelope'', <math>M(r)\|_\mathrm{e}</math> ''at the radial location of the interface'', <math>r_i</math>.		[http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] furthermore pointed out — again, see the relevant article excerpt — that, when constructing a bipolytropic model, the mathematical function that specifies the mass enclosed inside a spherical radius <math>r</math> for the ''core'', <math>M(r)\|_\mathrm{c}</math>, must give the same value as the mathematical function that specifies the mass enclosed inside a spherical radius <math>r</math> for the ''envelope'', <math>M(r)\|_\mathrm{e}</math> ''at the radial location of the interface'', <math>r_i</math>.

Tohline: /* Background */

2016-11-22T22:07:11Z

Background

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			Text extracted<sup>&dagger;</sup> from p. 472 of [http://adsabs.harvard.edu/abs/1931MNRAS..91..472C T. G. Cowling (1931)]<p></p>
			"''Note on the Fitting of Polytropic Models in the Theory of Stellar Structure''"<p></p>
			MNRAS, 91, 472-478 © Royal Astronomical Society
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			[[Image:CowlingRE_Milne1931.jpg\|500px\|center\|Cowling (1931)]]
			<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->
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	[http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] envisioned that the temperature, {{User:Tohline/Math/VAR_Temperature01}}, and pressure, {{User:Tohline/Math/VAR_Pressure01}}, should be "continuous across the surface of fit." Given that {{User:Tohline/Math/VAR_Temperature01}} and {{User:Tohline/Math/VAR_Pressure01}} are continuous, he envisioned that the gas density, {{User:Tohline/Math/VAR_Density01}}, should be continuous across the surface of fit as well. [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] later pointed out — see the following journal article excerpt — that, if the core and envelope have different molecular weights, it is the ratio {{User:Tohline/Math/VAR_Density01}}/{{User:Tohline/Math/MP_MeanMolecularWeight}} that should be continuous across the interface. A discontinuous jump in {{User:Tohline/Math/MP_MeanMolecularWeight}} at the interface — for example, switching from a pure helium core to an envelope whose dominant element is hydrogen — will therefore lead to a discontinuous jump in {{User:Tohline/Math/VAR_Density01}} at the interface.		[http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] envisioned that the temperature, {{User:Tohline/Math/VAR_Temperature01}}, and pressure, {{User:Tohline/Math/VAR_Pressure01}}, should be "continuous across the surface of fit." Given that {{User:Tohline/Math/VAR_Temperature01}} and {{User:Tohline/Math/VAR_Pressure01}} are continuous, he envisioned that the gas density, {{User:Tohline/Math/VAR_Density01}}, should be continuous across the surface of fit as well. [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] later pointed out — see the following journal article excerpt — that, if the core and envelope have different molecular weights, it is the ratio {{User:Tohline/Math/VAR_Density01}}/{{User:Tohline/Math/MP_MeanMolecularWeight}} that should be continuous across the interface. A discontinuous jump in {{User:Tohline/Math/MP_MeanMolecularWeight}} at the interface — for example, switching from a pure helium core to an envelope whose dominant element is hydrogen — will therefore lead to a discontinuous jump in {{User:Tohline/Math/VAR_Density01}} at the interface.

Tohline: /* Background */

2016-09-25T15:12:00Z

Background

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	~~<!--~~ Image:SchonbergChandra1942.jpg\|600px\|References -->		[[Image:SchonbergChandra1942.jpg\|600px\|References]]
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Tohline: Add "See Also" section along with link to paper by Rappaport, Verbunt, and Joss (1983)

2015-05-26T03:27:40Z

Add "See Also" section along with link to paper by Rappaport, Verbunt, and Joss (1983)

← Older revision		Revision as of 03:27, 26 May 2015
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	* [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Schoenberg-Chandrasekhar Limit: A BiPolytropic Approximation (Beech 1988b)]		* [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Schoenberg-Chandrasekhar Limit: A BiPolytropic Approximation (Beech 1988b)]
	* [http://adsabs.harvard.edu/abs/1988Ap%26SS.146..299B BiPolytropic Model for Low-Mass Stars (Beech 1988a)]		* [http://adsabs.harvard.edu/abs/1988Ap%26SS.146..299B BiPolytropic Model for Low-Mass Stars (Beech 1988a)]


			=See Also=

			* [http://adsabs.harvard.edu/abs/1983ApJ...275..713R Rappaport, Verbunt, & Joss (1983, ApJ, 275, 713)] — ''A New Technique for Calculations of Binary Stellar Evolution, with Application to Magnetic Braking''.


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Background */ Comment-out image from Cowling and another from SchonbergChandrasekhar

2015-05-24T18:01:26Z

Background: Comment-out image from Cowling and another from SchonbergChandrasekhar

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Tohline: /* Setup */

2015-05-04T18:55:46Z

Setup

← Older revision		Revision as of 18:55, 4 May 2015
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	The <math>1^\mathrm{st}</math> column of Table 1 provides the relevant set of associated structural relations for an isothermal core.		The <math>1^\mathrm{st}</math> column of Table 1 provides the relevant set of associated structural relations for an isothermal core.

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Tohline: /* Solution Steps */ Add "UVplane" span tag

2015-04-23T20:31:11Z

Solution Steps: Add "UVplane" span tag

← Older revision		Revision as of 20:31, 23 April 2015
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	Taking the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions on the left-hand side of Table 3 produces,		<span id="UVplane">
			Taking the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions on the left-hand side of Table 3 produces,</span>
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	<math>		<math>