SSC/Structure/BiPolytropes - Revision history

Jet53man: /* Solution Steps */

2023-07-06T17:13:37Z

Solution Steps

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Jet53man: /* The Schönberg-Chandrasekhar Condition */

2023-07-05T22:42:03Z

The Schönberg-Chandrasekhar Condition

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Jet53man: /* The Tohline Generalization */

2023-05-27T18:47:37Z

The Tohline Generalization

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	So a discontinuity across the interface will arise if the ratio of the molecular weights, <math>\mu_c/\mu_e</math>, is not unity, or if there is a discontinuity in the temperature across the interface — that is, if <math>T(r_i)\|_e \ne T(r_i)\|_c</math>, or both. Because they were using bipolytropes to model optically thick stellar interiors, ~~[http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]~~ argued that the temperature should also be continuous across the interface and, hence, that a discontinuity in the density would be introduced at the interface from a discontinuity in the molecular weight. [[SSC/Structure/LimitingMasses#Relationship_Between_the_Bonnor-Ebert_and_Sch.C3.B6nberg-Chandrasekhar_Critical_Masses\|In a separate chapter where we discuss the relationship between the Schönberg-Chandrasekhar critical mass and the Bonnor-Ebert critical mass]], we will argue that a discontinuous drop in the density is introduced by a substantial jump in the gas temperature at the interface. In making this alternate assumption, the structural equations describing the bipolytropic model will remain unchanged; we will only need to replace the ratio <math>\mu_c/\mu_e</math> by the ratio <math>T(r_i)\|_e/T(r_i)\|_c</math>.		So a discontinuity across the interface will arise if the ratio of the molecular weights, <math>\mu_c/\mu_e</math>, is not unity, or if there is a discontinuity in the temperature across the interface — that is, if <math>T(r_i)\|_e \ne T(r_i)\|_c</math>, or both. Because they were using bipolytropes to model optically thick stellar interiors, {{ SC42 }} argued that the temperature should also be continuous across the interface and, hence, that a discontinuity in the density would be introduced at the interface from a discontinuity in the molecular weight. [[SSC/Structure/LimitingMasses#Relationship_Between_the_Bonnor-Ebert_and_Sch.C3.B6nberg-Chandrasekhar_Critical_Masses\|In a separate chapter where we discuss the relationship between the Schönberg-Chandrasekhar critical mass and the Bonnor-Ebert critical mass]], we will argue that a discontinuous drop in the density is introduced by a substantial jump in the gas temperature at the interface. In making this alternate assumption, the structural equations describing the bipolytropic model will remain unchanged; we will only need to replace the ratio <math>\mu_c/\mu_e</math> by the ratio <math>T(r_i)\|_e/T(r_i)\|_c</math>.

	==Solution Steps==		==Solution Steps==

Jet53man: /* Solution Steps */

2023-05-27T18:41:56Z

Solution Steps

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	* Step 1: Choose <math>n_c</math> and <math>n_e</math>.		* Step 1: Choose <math>n_c</math> and <math>n_e</math>.
	* Step 2: Adopt boundary conditions at the center of the core (<math>\theta = 1</math> and <math>d\theta/d\xi = 0</math> at <math>\xi=0</math>), then solve the Lane-Emden equation to obtain the solution, <math>\theta(\xi)</math>, and its first derivative, <math>d\theta/d\xi</math> throughout the core; the radial location, <math>\xi = \xi_s</math>, at which <math>\theta(\xi)</math> first goes to zero identifies the natural surface of an [[~~User:Tohline/~~SSC/Structure/Polytropes#Lane-Emden_Equation\|isolated polytrope]] that has a polytropic index <math>n_c</math>.		* Step 2: Adopt boundary conditions at the center of the core (<math>\theta = 1</math> and <math>d\theta/d\xi = 0</math> at <math>\xi=0</math>), then solve the Lane-Emden equation to obtain the solution, <math>\theta(\xi)</math>, and its first derivative, <math>d\theta/d\xi</math> throughout the core; the radial location, <math>\xi = \xi_s</math>, at which <math>\theta(\xi)</math> first goes to zero identifies the natural surface of an [[SSC/Structure/Polytropes#Lane-Emden_Equation\|isolated polytrope]] that has a polytropic index <math>n_c</math>.
	* Step 3 Choose the desired location, <math>0 < \xi_i < \xi_s</math>, of the outer edge of the core.		* Step 3 Choose the desired location, <math>0 < \xi_i < \xi_s</math>, of the outer edge of the core.
	* Step 4: Specify <math>K_c</math> and <math>\rho_0</math>; the structural profile of, for example, <math>\rho(r)</math>, <math>P(r)</math>, and <math>M_r(r)</math> is then obtained throughout the core — over the radial range, <math>0 \le \xi \le \xi_i</math> and <math>0 \le r \le r_i</math> — via the relations shown in the <math>2^\mathrm{nd}</math> column of Table 1.		* Step 4: Specify <math>K_c</math> and <math>\rho_0</math>; the structural profile of, for example, <math>\rho(r)</math>, <math>P(r)</math>, and <math>M_r(r)</math> is then obtained throughout the core — over the radial range, <math>0 \le \xi \le \xi_i</math> and <math>0 \le r \le r_i</math> — via the relations shown in the <math>2^\mathrm{nd}</math> column of Table 1.

Jet53man: /* The Schönberg-Chandrasekhar Condition */

2023-05-27T18:41:10Z

The Schönberg-Chandrasekhar Condition

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	==Interface Conditions==		==Interface Conditions==
	===The Schönberg-Chandrasekhar Condition===		===The Schönberg-Chandrasekhar Condition===
	The <math>2^\mathrm{nd}</math> and <math>3^\mathrm{rd}</math> columns of Table 1 provide the mathematical relations that are needed to implement the interface matching conditions specified by equation (1) of ~~[http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]~~ (see the above article excerpt) when the core obeys a polytropic equation of state. For example, in order to ensure that,		The <math>2^\mathrm{nd}</math> and <math>3^\mathrm{rd}</math> columns of Table 1 provide the mathematical relations that are needed to implement the interface matching conditions specified by equation (1) of {{ SC42 }} (see the above article excerpt) when the core obeys a polytropic equation of state. For example, in order to ensure that,
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	if, as assumed by ~~[http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]~~ (again, see the above article excerpt), <math>T</math> is related to <math>P</math> and <math>\rho</math> through the ideal-gas equation of state. More specifically, according to the ideal-gas equation of state,		if, as assumed by {{ SC42 }} (again, see the above article excerpt), <math>T</math> is related to <math>P</math> and <math>\rho</math> through the ideal-gas equation of state. More specifically, according to the ideal-gas equation of state,
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	After setting <math>n_e = 1</math>, these four relations are essentially identical to, respectively, equations (13), (10), (12), & (11) of ~~[http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Beech 1988b]~~; adopting <math>n_e=3</math> instead, they serve as the foundation of earlier work by ~~[http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)]~~.		After setting <math>n_e = 1</math>, these four relations are essentially identical to, respectively, equations (13), (10), (12), & (11) of {{ Beech88full }}; adopting <math>n_e=3</math> instead, they serve as the foundation of earlier work by {{ HC41full }}.
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	<span id="TohlineGeneralization">		<span id="TohlineGeneralization">

	===The Tohline Generalization===		===The Tohline Generalization===
	</span>		</span>

Jet53man: /* BiPolytropes */

2023-05-27T18:35:04Z

BiPolytropes

← Older revision		Revision as of 14:35, 27 May 2023
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	\|-		\|-
	! style="height: 125px; width: 125px; background-color:#ffff99;" \|		! style="height: 125px; width: 125px; background-color:#ffff99;" \|
	<font size="-1">[[H_BookTiledMenu#MoreModels\|<b>Composite<br />~~Polytropies~~</b>]]<br />(Bipolytropes)</font>		<font size="-1">[[H_BookTiledMenu#MoreModels\|<b>Composite<br />Polytropes</b>]]<br />(Bipolytropes)</font>
	\|}		\|}

Jet53man: /* Background */

2023-05-27T18:33:52Z

Background

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	==Background==		==Background==
	While [[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 [[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)], {{ Milne30full }} 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>.

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	~~[http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)]~~ envisioned that the temperature, {{Math/VAR_Temperature01}}, and pressure, {{Math/VAR_Pressure01}}, should be "continuous across the surface of fit." Given that {{Math/VAR_Temperature01}} and {{Math/VAR_Pressure01}} are continuous, he envisioned that the gas density, {{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 {{Math/VAR_Density01}}/{{Math/MP_MeanMolecularWeight}} that should be continuous across the interface. A discontinuous jump in {{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 {{Math/VAR_Density01}} at the interface.		{{ Milne30 }} envisioned that the temperature, {{Math/VAR_Temperature01}}, and pressure, {{Math/VAR_Pressure01}}, should be "continuous across the surface of fit." Given that {{Math/VAR_Temperature01}} and {{Math/VAR_Pressure01}} are continuous, he envisioned that the gas density, {{Math/VAR_Density01}}, should be continuous across the surface of fit as well. {{ SC42full }} later pointed out — see the following journal article excerpt — that, if the core and envelope have different molecular weights, it is the ratio {{Math/VAR_Density01}}/{{Math/MP_MeanMolecularWeight}} that should be continuous across the interface. A discontinuous jump in {{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 {{Math/VAR_Density01}} at the interface.

	<div align="center">		<div align="center">
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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>		Text excerpts<sup>&dagger;</sup> from §2 (pp. 161-162) of <br />{{ SC42figure }}<br />© American Astronomical Society
	~~"''On the Evolution of the Main-Sequence Stars''"<p~~></p>
	~~ApJ, 96, 161-172~~ © American ~~Astronomicall~~ Society
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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>.		{{ SC42 }} 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>.

	==Setup==		==Setup==

Jet53man: /* Related Discussions */

2021-08-12T17:37:32Z

Related Discussions

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	=Related Discussions=		=Related Discussions=
	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]
	* [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]		* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
	* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]		* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
	** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia		** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia

Jet53man: /* Example Solutions */

2021-08-12T17:37:07Z

Example Solutions

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	=Example Solutions=		=Example Solutions=

	* [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic solution]] for <math>n_c = 5, ~n_e = 1</math>.		* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic solution]] for <math>n_c = 5, ~n_e = 1</math>.
	* [[SSC/Structure/BiPolytropes/~~Analytic0_0~~\|Analytic solution]] for <math>n_c = 0, ~n_e = 0</math>.		* [[SSC/Structure/BiPolytropes/Analytic00\|Analytic solution]] for <math>n_c = 0, ~n_e = 0</math>.

	=Related Discussions=		=Related Discussions=

Jet53man: /* BiPolytropes */

2021-07-29T21:45:39Z

BiPolytropes

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	__FORCETOC__		__FORCETOC__
	=BiPolytropes=		=BiPolytropes=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:#ffff99;" \|
			<font size="-1">[[H_BookTiledMenu#MoreModels\|<b>Composite<br />Polytropies</b>]]<br />(Bipolytropes)</font>
			\|}

	==Background==		==Background==