SSC/Structure/Lane1870 - Revision history

Jet53man: /* SEGMENT II */

2021-08-27T21:11:54Z

SEGMENT II

← Older revision		Revision as of 17:11, 27 August 2021
Line 472:		Line 472:

	Drawing upon ''[[SSCpt2/SolutionStrategies#Technique_2\|Technique 2]]'' from our discussion of various solution strategies, in a [[SSC/Structure/Polytropes#Lane-Emden_Equation\|separate chapter]] we have derived what is now commonly referred to as the,		Drawing upon ''[[SSCpt2/SolutionStrategies#Technique_2\|Technique 2]]'' from our discussion of various solution strategies, in a [[SSC/Structure/Polytropes#Lane-Emden_Equation\|separate chapter]] we have derived what is now commonly referred to as the,
	<div align="center">		<div align="center" id="LaneEmden">
	<font color="maroon">'''Polytropic Lane-Emden Equation'''</font>		<font color="maroon">'''Polytropic Lane-Emden Equation'''</font>

Jet53man: /* Lane's 1870 Work */

2021-07-29T03:38:37Z

Lane's 1870 Work

← Older revision		Revision as of 23:38, 28 July 2021
Line 3:		Line 3:
	=Lane's 1870 Work=		=Lane's 1870 Work=

			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:#ffeeee;" \|
			<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>Lane</b>]]<br />(1870)</font>
			\|}
	In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been copied ''verbatim'' from a scientifically impactful 1870 article by [https://en.wikipedia.org/wiki/Jonathan_Homer_Lane J. Homer Lane]. Note that we have broken the article into separate '''segments''' in order to facilitate the insertion of interspersed comments. Our accompanying discussion of ''[[SSC/Structure/Polytropes#Polytropic_Spheres\|Polytropic Spheres]]'' shows how this groundbreaking physical/mathematical model developed by Lane to probe the interior structure of the Sun — encapsulated in what is now commonly referred to as the ''Lane-Emden'' equation — provides a tool for understanding stars that have a wide range of structural properties.		In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been copied ''verbatim'' from a scientifically impactful 1870 article by [https://en.wikipedia.org/wiki/Jonathan_Homer_Lane J. Homer Lane]. Note that we have broken the article into separate '''segments''' in order to facilitate the insertion of interspersed comments. Our accompanying discussion of ''[[SSC/Structure/Polytropes#Polytropic_Spheres\|Polytropic Spheres]]'' shows how this groundbreaking physical/mathematical model developed by Lane to probe the interior structure of the Sun — encapsulated in what is now commonly referred to as the ''Lane-Emden'' equation — provides a tool for understanding stars that have a wide range of structural properties.
			<br />
			<br />

	==SEGMENT I==		==SEGMENT I==

Jet53man: /* SEGMENT III */

2021-07-27T16:57:05Z

SEGMENT III

← Older revision		Revision as of 12:57, 27 July 2021
Line 772:		Line 772:
	</table>		</table>

	According to our [[~~User:Tohline/SSC~~/Virial/FormFactors#PTtable\|accompanying summary of structural form factors]], the central-to-mean density of an isolated polytropic configuration is,		According to our [[SSCpt1/Virial/FormFactors#PTtable\|accompanying summary of structural form factors]], the central-to-mean density of an isolated polytropic configuration is,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* SEGMENT II */

2021-07-27T16:55:27Z

SEGMENT II

← Older revision		Revision as of 12:55, 27 July 2021
Line 275:		Line 275:
	----		----
	<p></p>		<p></p>
	In this last expression, the parameter <math>~k</math> that appears in the exponent <font color="darkgreen">represents the [[~~User:Tohline/~~SR/IdealGas#gamma_g\|ratio of specific heat]] of a gas under constant pressure to its specific heat under constant volume.</font> Instead of <math>~k</math>, it is more common in modern notation to use, <math>~\gamma_g\equiv c_P/c_V</math>.		In this last expression, the parameter <math>~k</math> that appears in the exponent <font color="darkgreen">represents the [[SR/IdealGas#gamma_g\|ratio of specific heat]] of a gas under constant pressure to its specific heat under constant volume.</font> Instead of <math>~k</math>, it is more common in modern notation to use, <math>~\gamma_g\equiv c_P/c_V</math>.
	<p></p>		<p></p>
	----		----
Line 464:		Line 464:


	Drawing upon ''[[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Technique_2\|Technique 2]]'' from our discussion of various solution strategies, in a [[~~User:Tohline/~~SSC/Structure/Polytropes#Lane-Emden_Equation\|separate chapter]] we have derived what is now commonly referred to as the,		Drawing upon ''[[SSCpt2/SolutionStrategies#Technique_2\|Technique 2]]'' from our discussion of various solution strategies, in a [[SSC/Structure/Polytropes#Lane-Emden_Equation\|separate chapter]] we have derived what is now commonly referred to as the,
	<div align="center">		<div align="center">
	<font color="maroon">'''Polytropic Lane-Emden Equation'''</font>		<font color="maroon">'''Polytropic Lane-Emden Equation'''</font>

	{{ ~~User:Tohline/~~Math/EQ_SSLaneEmden01 }}		{{ Math/EQ_SSLaneEmden01 }}
	</div>		</div>
	where, given that <math>~n = (\gamma_\mathrm{g}-1)^{-1}</math> and <math>~K_n = P_c \rho_c^{-(n+1)/n}</math>, we have,		where, given that <math>~n = (\gamma_\mathrm{g}-1)^{-1}</math> and <math>~K_n = P_c \rho_c^{-(n+1)/n}</math>, we have,

Jet53man: /* SEGMENT I */

2021-07-27T16:53:39Z

SEGMENT I

← Older revision		Revision as of 12:53, 27 July 2021
Line 59:		Line 59:
	the temperature of the element,		the temperature of the element,
	</td>		</td>
	<td align="center">{{ ~~User:Tohline/~~Math/VAR_Temperature01 }}</td>		<td align="center">{{ Math/VAR_Temperature01 }}</td>
	</tr>		</tr>

Line 111:		Line 111:
	its elasticity, or elastic force per unit surface,		its elasticity, or elastic force per unit surface,
	</td>		</td>
	<td align="center"><math>\biggl( \frac{R_\oplus^2}{GM_\oplus} \biggr)</math> {{ ~~User:Tohline/~~Math/VAR_Pressure01 }}</td>		<td align="center"><math>\biggl( \frac{R_\oplus^2}{GM_\oplus} \biggr)</math> {{ Math/VAR_Pressure01 }}</td>
	</tr>		</tr>

Line 210:		Line 210:
	<font color="maroon">'''Hydrostatic Balance'''</font>		<font color="maroon">'''Hydrostatic Balance'''</font>

	{{ ~~User:Tohline/~~Math/EQ_SShydrostaticBalance01 }}		{{ Math/EQ_SShydrostaticBalance01 }}
	</div>		</div>
	where,		where,
	<div align="center">		<div align="center">
	{{ ~~User:Tohline/~~Math/EQ_SSmassConservation01 }}		{{ Math/EQ_SSmassConservation01 }}
	</div>		</div>

	Our derivation of this mathematical statement of hydrostatic balance has been tagged, ''[[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Technique_1\|Technique 1]]'', in our accompanying discussion of various [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Solution_Strategies\|solution strategies]].		Our derivation of this mathematical statement of hydrostatic balance has been tagged, ''[[SSCpt2/SolutionStrategies#Technique_1\|Technique 1]]'', in our accompanying discussion of various [[SSCpt2/SolutionStrategies#Solution_Strategies\|solution strategies]].

	==SEGMENT II==		==SEGMENT II==

Jet53man: /* Lane's 1870 Work */

2021-07-27T16:50:35Z

Lane's 1870 Work

← Older revision		Revision as of 12:50, 27 July 2021
Line 3:		Line 3:
	=Lane's 1870 Work=		=Lane's 1870 Work=

	In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been copied ''verbatim'' from a scientifically impactful 1870 article by [https://en.wikipedia.org/wiki/Jonathan_Homer_Lane J. Homer Lane]. Note that we have broken the article into separate '''segments''' in order to facilitate the insertion of interspersed comments. Our accompanying discussion of ''[[~~User:Tohline/~~SSC/Structure/Polytropes#Polytropic_Spheres\|Polytropic Spheres]]'' shows how this groundbreaking physical/mathematical model developed by Lane to probe the interior structure of the Sun — encapsulated in what is now commonly referred to as the ''Lane-Emden'' equation — provides a tool for understanding stars that have a wide range of structural properties.		In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been copied ''verbatim'' from a scientifically impactful 1870 article by [https://en.wikipedia.org/wiki/Jonathan_Homer_Lane J. Homer Lane]. Note that we have broken the article into separate '''segments''' in order to facilitate the insertion of interspersed comments. Our accompanying discussion of ''[[SSC/Structure/Polytropes#Polytropic_Spheres\|Polytropic Spheres]]'' shows how this groundbreaking physical/mathematical model developed by Lane to probe the interior structure of the Sun — encapsulated in what is now commonly referred to as the ''Lane-Emden'' equation — provides a tool for understanding stars that have a wide range of structural properties.

	==SEGMENT I==		==SEGMENT I==

Jet53man: Created page with "FORCETOC =Lane's 1870 Work= In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been cop..."

2021-07-27T16:49:59Z

Created page with "__FORCETOC__  =Lane's 1870 Work= In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been cop..."

Show changes

SSC/Structure/Lane1870 - Revision history

Jet53man: /* SEGMENT II */

Jet53man: /* Lane's 1870 Work */

Jet53man: /* SEGMENT III */

Jet53man: /* SEGMENT II */

Jet53man: /* SEGMENT I */

Jet53man: /* Lane's 1870 Work */

Jet53man: Created page with "__FORCETOC__ =Lane's 1870 Work= In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been cop..."

Jet53man: Created page with "FORCETOC =Lane's 1870 Work= In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been cop..."