SSC/Structure/BiPolytropes/Analytic1.53 - Revision history

Joel2: /* Milne's (1930) Choice of Equations of State */

2024-01-16T21:43:44Z

Milne's (1930) Choice of Equations of State

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	With this construction in mind, {{ Milne30 }} also introduced the parameter, <math>~\beta</math>, to define the ratio of gas pressure (meaning, ideal-gas plus degeneracy pressure) to total pressure, that is,		<span id="BetaDefinition">With this construction</span> in mind, {{ Milne30 }} also introduced the parameter, <math>~\beta</math>, to define the ratio of gas pressure (meaning, ideal-gas plus degeneracy pressure) to total pressure, that is,
	<div align="center">		<div align="center">
	<math>\beta \equiv \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, ,</math>		<math>\beta \equiv \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, ,</math>

Joel2: /* BiPolytrope with (nc, ne) = (3/2, 3) */

2024-01-15T23:09:07Z

BiPolytrope with (nc, ne) = (3/2, 3)

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	In deriving the properties of this model, we will follow the [[SSC/Structure/BiPolytropes#Solution_Steps\|general solution steps for constructing a bipolytrope]] that are outlined in a separate chapter of this H_Book. That group of general solution steps was drawn largely from chapter IV, §28 of Chandrasekhar's book titled, "An Introduction to the Study of Stellar Structure" [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]]. At the end of that chapter (specifically, p. 182), Chandrasekhar acknowledges that "[Milne's] method is largely used in § 28." It seems fitting, therefore, that we highlight the features of the ''specific'' bipolytropic configuration that {{ Milne30 }} chose to build.		In deriving the properties of this model, we will follow the [[SSC/Structure/BiPolytropes#Solution_Steps\|general solution steps for constructing a bipolytrope]] that are outlined in a separate chapter of this H_Book. That group of general solution steps was drawn largely from chapter IV, §28 of Chandrasekhar's book titled, "An Introduction to the Study of Stellar Structure" [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]]. At the end of that chapter (specifically, p. 182), Chandrasekhar acknowledges that "[Milne's] method is largely used in § 28." It seems fitting, therefore, that we highlight the features of the ''specific'' bipolytropic configuration that {{ Milne30 }} chose to build.

	~~{{ SGFworkInProgress }}~~


	==Milne's (1930) Choice of Equations of State==		==Milne's (1930) Choice of Equations of State==

Joel2: /* Core */

2024-01-15T22:56:30Z

Core

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	Recognizing that Milne set <math>~q_e = 2</math>, as "the statistical weight of an electron," and that he adopted a molecular weight of the electrons, <math>~\mu_{e^-}=2</math>, this expression for the equation of state exactly matches our expression for <math>~P_\mathrm{deg}\|_\mathrm{NR}</math>. Our enlistment of an <math>~n_c = \tfrac{3}{2}</math> polytropic equation of state for the core is therefore also perfectly aligned with Milne's treatment of the core; in particular, according to Milne, at each radial location throughout the core the total pressure can be obtained from the expression,		Recognizing that Milne set <math>~q_e = 2</math>, as "the statistical weight of an electron," and that he adopted a molecular weight of the electrons, <math>\mu_{e^-}=2</math>, this expression for the equation of state exactly matches our expression for <math>P_\mathrm{deg}\|_\mathrm{NR}</math>. Our enlistment of an <math>n_c = \tfrac{3}{2}</math> polytropic equation of state for the core is therefore also perfectly aligned with Milne's treatment of the core; in particular, according to Milne, at each radial location throughout the core the total pressure can be obtained from the expression,
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	<math>~P = \frac{K}{\beta} ~\rho^{5/3} \, ,</math>		<math>P = \frac{K}{\beta} ~\rho^{5/3} \, ,</math>
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	with Milne's coefficient, <math>~K</math>, having the same definition as our coefficient, <math>~K_c</math>		with Milne's coefficient, <math>K</math>, having the same definition as our coefficient, <math>K_c</math>.

	=See Also=		=See Also=

Joel2: /* Core */

2024-01-15T22:54:45Z

Core

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	This is the core structure that will be incorporated into our derivation of the bipolytrope's properties~~, below~~.		This is the [[SSC/Structure/BiPolytropes/Analytic1.53/Pt3#Step_4:_Throughout_the_core_(0_≤_ξ_≤_ξi)\|core structure that will be incorporated into our derivation]] of the bipolytrope's properties.

	This is precisely the approach taken by {{ Milne30 }}. Just before his equation (43), Milne states that, "the equation of state when the electrons alone are degenerate can be shown" to be,		This is precisely the approach taken by {{ Milne30 }}. Just before his equation (43), Milne states that, "the equation of state when the electrons alone are degenerate can be shown" to be,

Joel2: /* Core */

2024-01-15T22:51:33Z

Core

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	(Note that, here only, we have used the parameter, <math>~\mu_{e^-}</math>, to denote the molecular weight of electrons, instead of just <math>~\mu_e</math>, in order not to confuse it with the mean molecular weight assigned~~, below,~~ to the envelope material.) So, from the solution, <math>~\theta(\xi)</math>, to the Lane-Emden equation of index <math>~n=\tfrac{3}{2}</math>, we will be able to determine that,		(Note that, here only, we have used the parameter, <math>\mu_{e^-}</math>, to denote the molecular weight of electrons, instead of just <math>\mu_e</math>, in order not to confuse it with the mean molecular weight assigned to the envelope material.) So, from the solution, <math>\theta(\xi)</math>, to the Lane-Emden equation of index <math>n=\tfrac{3}{2}</math>, we will be able to determine that,
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Joel2: /* Envelope */

2024-01-15T22:47:02Z

Envelope

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	It is clear, therefore, that the two radial scale-factors are the same. In preparation for our [[SSC/Structure/BiPolytropes/Analytic1.53/Pt3#Step_8:~~__Throughout_the_envelope~~\|further discussion of the structure of this bipolytrope's envelope, below]], it is useful to highlight the following two expressions that have been developed here in the process of showing the correspondence between our work and that of Milne:		It is clear, therefore, that the two radial scale-factors are the same. In preparation for our [[SSC/Structure/BiPolytropes/Analytic1.53/Pt3#Step_8:_Throughout_the_envelope_(ηi_≤_η_≤_ξs)\|further discussion of the structure of this bipolytrope's envelope, below]], it is useful to highlight the following two expressions that have been developed here in the process of showing the correspondence between our work and that of Milne:

	<div align="center" id="HighlightedExpressions">		<div align="center" id="HighlightedExpressions">

Joel2: /* Envelope */

2024-01-15T22:44:35Z

Envelope

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	It is clear, therefore, that the two radial scale-factors are the same. In preparation for our [[#Step_8:__Throughout_the_envelope\|further discussion of the structure of this bipolytrope's envelope, below]], it is useful to highlight the following two expressions that have been developed here in the process of showing the correspondence between our work and that of Milne:		It is clear, therefore, that the two radial scale-factors are the same. In preparation for our [[SSC/Structure/BiPolytropes/Analytic1.53/Pt3#Step_8:__Throughout_the_envelope\|further discussion of the structure of this bipolytrope's envelope, below]], it is useful to highlight the following two expressions that have been developed here in the process of showing the correspondence between our work and that of Milne:

	<div align="center" id="HighlightedExpressions">		<div align="center" id="HighlightedExpressions">

Joel2: /* BiPolytrope with (nc, ne) = (3/2, 3) */

2024-01-15T22:29:40Z

BiPolytrope with (nc, ne) = (3/2, 3)

Show changes

Joel2: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ξs) */

2023-12-27T20:11:33Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ξs)

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	By comparison, the expressions that {{ Milne30 }} derived for the run of <math>~\rho</math>, <math>~r</math>, and <math>~M_r</math> throughout the envelope are presented in his paper as, respectively, equations (89), (86), and (85). In an effort to facilitate this comparison, Milne's expressions — which also specifically apply to the base of the envelope, whose identity is associated with primed variable names in Milne's notation — are reprinted as extracted ~~images~~ in the following boxed-in table.		By comparison, the expressions that {{ Milne30 }} derived for the run of <math>~\rho</math>, <math>~r</math>, and <math>~M_r</math> throughout the envelope are presented in his paper as, respectively, equations (89), (86), and (85). In an effort to facilitate this comparison, Milne's expressions — which also specifically apply to the base of the envelope, whose identity is associated with primed variable names in Milne's notation — are reprinted as extracted equations in the following boxed-in table.

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

Joel2: /* Step 4: Throughout the core (0 ≤ ξ ≤ ξi) */

2023-12-27T20:10:22Z

Step 4: Throughout the core (0 ≤ ξ ≤ ξi)

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	By comparison, the expressions that {{ Milne30 }} derived for the run of <math>~\rho</math>, <math>~r</math>, and <math>~M_r</math> throughout the core are presented in his paper as, respectively, equations (90), (88), and (87). In an effort to facilitate this comparison, Milne's expressions — which also specifically apply to the outer edge of the core, whose identity is associated with primed variable names in Milne's notation — are reprinted as extracted ~~images~~ in the following boxed-in table.		By comparison, the expressions that {{ Milne30 }} derived for the run of <math>~\rho</math>, <math>~r</math>, and <math>~M_r</math> throughout the core are presented in his paper as, respectively, equations (90), (88), and (87). In an effort to facilitate this comparison, Milne's expressions — which also specifically apply to the outer edge of the core, whose identity is associated with primed variable names in Milne's notation — are reprinted as extracted equations in the following boxed-in table.

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