User:Tohline/SSC/Structure/BiPolytropes/Analytic1.5 3 - Revision history

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

2017-07-06T20:12:07Z

Milne's (1930) Choice of Equations of State

← Older revision		Revision as of 20:12, 6 July 2017
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	<math>\frac{P_\mathrm{gas} + P_\mathrm{deg}}{P_\mathrm{rad}} = \frac{\beta}{1-\beta} \, .</math>		<math>\frac{P_\mathrm{gas} + P_\mathrm{deg}}{P_\mathrm{rad}} = \frac{\beta}{1-\beta} \, .</math>
	</div>		</div>


			(We also have referenced this [[User:Tohline/SR#The_Parameter.2C_.CE.B2\|parameter, β]], in the context of a broader discussion of equations of state and modeling [[User:Tohline/SR#Time-Dependent_Problems\|time-dependent flows]].)

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

2015-06-05T20:15:10Z

Milne's (1930) Choice of Equations of State

← Older revision		Revision as of 20:15, 5 June 2015
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	</table>		</table>
	</div>		</div>
	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>.		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,
			<div align="center">
			<math>~P = \frac{K}{\beta} ~\rho^{5/3} \, ,</math>
			</div>
			with Milne's coefficient, <math>~K</math>, having the same definition as our coefficient, <math>~K_c</math>

	==Our Derivation==		==Our Derivation==

Tohline: /* Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s) */ Insert footnote to equation image from Milne (1930)

2015-06-05T19:50:09Z

Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s): Insert footnote to equation image from Milne (1930)

← Older revision		Revision as of 19:50, 5 June 2015
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	<tr>		<tr>
	<th align="center">		<th align="center">
	Equations extracted ~~(with presentation order rearranged)~~ from [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]<p></p>		Equations extracted<sup>&dagger;</sup> from [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]<p></p>
	"''The Analysis of Stellar Structure''"<p></p>		"''The Analysis of Stellar Structure''"<p></p>
	MNRAS, vol. 91, pp. 4 - 55 © Royal Astronomical Society		MNRAS, vol. 91, pp. 4 - 55 © Royal Astronomical Society
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	</td>		</td>
	</tr>		</tr>
			<tr><td align="left">
			<sup>&dagger;</sup>Equations displayed here, as a single digital image, with presentation order & layout modified from the original publication.
			</td></tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Step 4: Throughout the core ~(0 \le \xi \le \xi_i) */ Modified description of equations extracted from Milne (1930)

2015-06-05T19:46:56Z

Step 4: Throughout the core ~(0 \le \xi \le \xi_i): Modified description of equations extracted from Milne (1930)

← Older revision		Revision as of 19:46, 5 June 2015
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	<tr>		<tr>
	<th align="center">		<th align="center">
	Equations extracted ~~(with presentation order rearranged)~~ from [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]<p></p>		Equations extracted<sup>&dagger;</sup> from [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]<p></p>
	"''The Analysis of Stellar Structure''"<p></p>		"''The Analysis of Stellar Structure''"<p></p>
	MNRAS, vol. 91, pp. 4 - 55 © Royal Astronomical Society		MNRAS, vol. 91, pp. 4 - 55 © Royal Astronomical Society
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	</td>		</td>
	</tr>		</tr>
			<tr><td align="left"><sup>&dagger;</sup>Equations displayed here, as a single digital image, with presentation order & layout modified from the original publication.</td></tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Core / Clarify how betaP relates to 5/3 gas law

2015-06-05T18:55:13Z

Core: Clarify how beta*P relates to 5/3 gas law

← Older revision		Revision as of 18:55, 5 June 2015
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	which can be associated with a polytropic relation of the form,		which can be associated with a polytropic relation of the form,
	<div align="center">		<div align="center">
	<math>~P = K_c \rho^{1 + 1/n_c} \, ,</math>		<math>~P_\mathrm{deg}\biggr\|_\mathrm{NR} = K_c \rho^{1 + 1/n_c} \, ,</math>
			</div>
			that is, a total pressure of the form,
			<div align="center">
			<math>~\beta P = K_c \rho^{1 + 1/n_c} \, ,</math>
	</div>		</div>
	with,		with,

Tohline: /* Milne's (1930) Choice of Equations of State */ Reword discussion of Milne's equation of state to conform with his definition of beta

2015-06-05T18:49:45Z

Milne's (1930) Choice of Equations of State: Reword discussion of Milne's equation of state to conform with his definition of beta

← Older revision		Revision as of 18:49, 5 June 2015
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	With this construction in mind, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] also introduced the parameter, <math>~\beta</math>, to define the ratio of gas pressure (meaning, ideal-gas plus degeneracy pressure) to total pressure ~~in the envelope~~, that is,		With this construction in mind, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] 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>
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	===Core===		===Core===

	In contrast to the envelope, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] assumed that the (non-relativistic) electron degeneracy pressure dominates over the ideal-gas pressure in the core. That is, he assumed that, throughout the core of his composite polytropic ~~configuraion~~,		In contrast to the envelope, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] assumed that the (non-relativistic; "NR") electron degeneracy pressure dominates over the ideal-gas pressure in the core. That is, he assumed that, throughout the core of his composite polytropic configuration,
	<div align="center">		<div align="center">
	<math>~\beta P = \cancelto{0}{P_\mathrm{gas}} + P_\mathrm{deg} \,.</math>		<math>~\beta P = \cancelto{0}{P_\mathrm{gas}} + P_\mathrm{deg}\biggr\|_\mathrm{NR} \,.</math>
	</div>		</div>

	As we have [[User:Tohline/SR#Nonrelativistic_ZTF_Gas\|demonstrated elsewhere]], the non-relativistic ~~(NR)~~ expression for the degeneracy pressure is,		As we have [[User:Tohline/SR#Nonrelativistic_ZTF_Gas\|demonstrated elsewhere]], the non-relativistic expression for the degeneracy pressure is,

	<div align="center">		<div align="center">
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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 core structure that will be incorporated into our derivation of the bipolytrope's properties, below.

	This is precisely the approach taken by [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)]. Just before his equation (43), Milne states that the equation of state ~~that is relevant for~~ the ~~core is~~,		This is precisely the approach taken by [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)]. Just before his equation (43), Milne states that, "the equation of state when the electrons alone are degenerate can be shown" to be,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	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 ~~of the core~~ exactly matches our expression for <math>~P_\mathrm{deg}\|_\mathrm{NR}</math>.		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 Derivation==		==Our Derivation==

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

2015-06-05T16:56:58Z

Milne's (1930) Choice of Equations of State

← Older revision		Revision as of 16:56, 5 June 2015
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	==Milne's (1930) Choice of Equations of State==		==Milne's (1930) Choice of Equations of State==

	As has been detailed in our [[User:Tohline/SR#Equation_of_State\|introductory discussion of analytically expressible equations of state]] and as is summarized in the following table, often the total gas pressure can be expressed as the sum of three separate components: a component of ideal gas pressure, a component of radiation pressure, and a component due to a degenerate electron gas.		As has been detailed in our [[User:Tohline/SR#Equation_of_State\|introductory discussion of analytically expressible equations of state]] and as is summarized in the following table, often the total gas pressure can be expressed as the sum of three separate components: a component of ideal gas pressure, a component of radiation pressure, and a component due to a degenerate electron gas. As a result, the total pressure is given by the expression,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~P</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} \, .</math>
			</td>
			</tr>
			</table>
			</div>

	<table width="95%" align="center" border=1 cellpadding=5>		<table width="95%" align="center" border=1 cellpadding=5>
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	</table>		</table>

	~~===Envelope===~~

	[http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] considered that the effects of electron degeneracy pressure could be ignored in the envelope of his composite polytrope and, accordingly, employed an expression for the total pressure of the form,
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~P</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~P_\mathrm{gas} + P_\mathrm{rad} \, .</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~

	Milne also introduced the parameter, <math>~\beta</math>, to define the ratio of gas pressure to total pressure in the envelope, that is,		With this construction in mind, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] also introduced the parameter, <math>~\beta</math>, to define the ratio of gas pressure (meaning, ideal-gas plus degeneracy pressure) to total pressure in the envelope, that is,
	<div align="center">		<div align="center">
	<math>\beta \equiv \frac{P_\mathrm{gas}}{P} ~~\biggr\|_\mathrm{env}~~ \, ,</math>		<math>\beta \equiv \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, ,</math>
	</div>		</div>
	in which case, also,		in which case, also,
	<div align="center">		<div align="center">
	<math>\frac{P_\mathrm{rad}}{P} ~~\biggr\|_\mathrm{env}~~ = 1-\beta </math>		<math>\frac{P_\mathrm{rad}}{P} = 1-\beta </math>
	and		and
			<math>\frac{P_\mathrm{gas} + P_\mathrm{deg}}{P_\mathrm{rad}} = \frac{\beta}{1-\beta} \, .</math>
			</div>


			===Envelope===

			Now, inside the ''envelope'' of his composite polytrope, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] considered that the effects of electron degeneracy pressure could be ignored and, accordingly, employed throughout the envelope the expression,
			<div align="center">
	<math>\frac{P_\mathrm{gas}}{P_\mathrm{rad}} \biggr\|_\mathrm{env} = \frac{\beta}{1-\beta} \, , </math>		<math>\frac{P_\mathrm{gas}}{P_\mathrm{rad}} \biggr\|_\mathrm{env} = \frac{\beta}{1-\beta} \, , </math>
	</div>		</div>
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	<math>T = \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \rho^{1/3} \, .</math>		<math>T = \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \rho^{1/3} \, .</math>
	</div>		</div>
	Now, returning to the definition of <math>~\beta</math>, we recognize that the total pressure in the envelope can be written in the form of a ''modified'' ideal gas relation, namely,		Now, returning to the definition of <math>~\beta</math> while ignoring the effects of degeneracy pressure, we recognize that the total pressure in the envelope can be written in the form of a ''modified'' ideal gas relation, namely,
	<div align="center">		<div align="center">
	<math>~P = \~~frac~~{1}{\~~beta~~}\biggl(\frac{\Re}{\mu_e}\biggr) \rho T \, ,</math>		<math>~\beta P = P_\mathrm{gas} + \cancelto{0}{P_\mathrm{deg}} = \biggl(\frac{\Re}{\mu_e}\biggr) \rho T \, ,</math>
	</div>		</div>
	with the ''specific'' <math>~T(\rho)</math> behavior just derived. This allows us to write the envelope's total pressure as,		with the ''specific'' <math>~T(\rho)</math> behavior just derived. This allows us to write the envelope's total pressure as,
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	===Core===		===Core===

	In contrast to the envelope, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] assumed that the (non-relativistic) electron degeneracy pressure dominates over the ~~other two~~ pressure ~~contributions~~ in the core. As we have [[User:Tohline/SR#Nonrelativistic_ZTF_Gas\|demonstrated elsewhere]], the non-relativistic (NR) expression for the degeneracy pressure is,		In contrast to the envelope, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] assumed that the (non-relativistic) electron degeneracy pressure dominates over the ideal-gas pressure in the core. That is, he assumed that, throughout the core of his composite polytropic configuraion,
			<div align="center">
			<math>~\beta P = \cancelto{0}{P_\mathrm{gas}} + P_\mathrm{deg} \,.</math>
			</div>

			As we have [[User:Tohline/SR#Nonrelativistic_ZTF_Gas\|demonstrated elsewhere]], the non-relativistic (NR) expression for the degeneracy pressure is,

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

Tohline: /* Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s) */ Associate copyright acknowledgement with equations extracted from Milne

2015-06-03T16:05:57Z

Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s): Associate copyright acknowledgement with equations extracted from Milne

← Older revision		Revision as of 16:05, 3 June 2015
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	<tr>		<tr>
	<th align="center">		<th align="center">
	~~Structure of the Bipolytrope's Envelope Presented by~~ [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]		Equations extracted (with presentation order rearranged) from [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]<p></p>
			"''The Analysis of Stellar Structure''"<p></p>
			MNRAS, vol. 91, pp. 4 - 55 © Royal Astronomical Society
	</th>		</th>
			</tr>
	<tr>		<tr>
	<td>		<td>

Tohline: /* Step 4: Throughout the core ~(0 \le \xi \le \xi_i) */ Restructure citation of Milne's equations

2015-06-03T15:59:46Z

Step 4: Throughout the core ~(0 \le \xi \le \xi_i): Restructure citation of Milne's equations

← Older revision		Revision as of 15:59, 3 June 2015
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	<tr>		<tr>
	<th align="center">		<th align="center">
	~~Structure of the Bipolytrope's Core Presented by~~ [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]		Equations extracted (with presentation order rearranged) from [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]<p></p>
			"''The Analysis of Stellar Structure''"<p></p>
			MNRAS, vol. 91, pp. 4 - 55 © Royal Astronomical Society
	</th>		</th>
			</tr>
	<tr>		<tr>
	<td>		<td>

Tohline: /* Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s) */ Show correspondence between our envelope structure and Milne's

2015-06-02T01:55:04Z

Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s): Show correspondence between our envelope structure and Milne's

← Older revision		Revision as of 01:55, 2 June 2015
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	</div>		</div>



	Instead of working completely across this table in order to relate the envelope's density, radial coordinate, and mass to properties of the core, it is worth pausing to insert into the leftmost set of relations the expressions for <math>~\rho_e</math> and <math>~K_e</math> that were [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1.5_3#HighlightedExpressions\|derived above]]. In doing this, we obtain,		Instead of working completely across this table in order to relate the envelope's density, radial coordinate, and mass to properties of the core, it is worth pausing to insert into the leftmost set of relations the expressions for <math>~\rho_e</math> and <math>~K_e</math> that were [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1.5_3#HighlightedExpressions\|derived above]]. In doing this, we obtain,
	<div align="center">		<div align="center">
	<table border="1" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~\lambda^3~~ \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^{-1} \frac{\beta}{(1-\beta)} \cdot \frac{1}{3} a_\mathrm{rad} \biggr] \phi^3</math>		<math>~\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^{-1} \frac{\beta}{(1-\beta)} \cdot \frac{1}{3} a_\mathrm{rad} \biggr] \lambda^3 \phi^3 \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	<math>~\frac{1}{(\pi G)^{1/2}} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/6}		<math>~\frac{1}{(\pi G)^{1/2}} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/6}
	\biggl\{ \lambda^3 \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^{-1} \frac{\beta}{(1-\beta)} \cdot \frac{1}{3} a_\mathrm{rad} \biggr] \biggr\}^{-1/3} \eta</math>		\biggl\{ \lambda^3 \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^{-1} \frac{\beta}{(1-\beta)} \cdot \frac{1}{3} a_\mathrm{rad} \biggr] \biggr\}^{-1/3} \eta</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{(\pi G)^{1/2}} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/6}
			\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^2\frac{(1-\beta)^2}{\beta^2} \biggl( \frac{3}{a_\mathrm{rad} }\biggr)^2 \biggr] ^{1/6} \frac{\eta}{\lambda} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl(\frac{3}{\pi a_\mathrm{rad} G } \biggr)^{1/2} \biggl( \frac{\Re}{\mu_e}\biggr) \frac{(1-\beta)^{1/2}}{\beta} \cdot \frac{\eta}{\lambda} \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	</tr>		</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~4 \biggl( \frac{3}{\pi a_\mathrm{rad}G^3} \biggr)^{1/2} \biggl(\frac{\Re}{\mu_e}\biggr)^2 \frac{(1-\beta)^{1/2}}{\beta^2}
			\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \, .</math>
			</td>
			</tr>
			</table>
			</div>


			By comparison, the expressions that [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] 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.

			<div align="center">
			<table border="2" cellpadding="10">
			<tr>
			<th align="center">
			Structure of the Bipolytrope's Envelope Presented by [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]
			</th>
			<tr>
			<td>
			[[File:EnvelopeRelations01.png\|500px\|center\|Milne (1930)]]
			</td>
			</tr>
	</table>		</table>
	</div>		</div>

			The agreement between our derivation and Milne's is exact, once it is realized that Milne has used <math>~\theta(\xi)</math> to represent the Lane_Emden function for the <math>~n_e = 3</math> envelope, whereas we have represented this function by <math>~\phi(\eta)</math>; and in place of Milne's coefficient, <math>~\lambda_1</math>, we have simply written, <math>~\lambda</math>.



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

User:Tohline/SSC/Structure/BiPolytropes/Analytic1.5 3 - Revision history

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

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

Tohline: /* Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s) */ Insert footnote to equation image from Milne (1930)

Tohline: /* Step 4: Throughout the core ~(0 \le \xi \le \xi_i) */ Modified description of equations extracted from Milne (1930)

Tohline: /* Core */ Clarify how beta*P relates to 5/3 gas law

Tohline: /* Milne's (1930) Choice of Equations of State */ Reword discussion of Milne's equation of state to conform with his definition of beta

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

Tohline: /* Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s) */ Associate copyright acknowledgement with equations extracted from Milne

Tohline: /* Step 4: Throughout the core ~(0 \le \xi \le \xi_i) */ Restructure citation of Milne's equations

Tohline: /* Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s) */ Show correspondence between our envelope structure and Milne's

Tohline: /* Core / Clarify how betaP relates to 5/3 gas law