User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain - Revision history

Tohline: /* P-V Diagram */

2015-01-21T19:18:30Z

P-V Diagram

← Older revision		Revision as of 19:18, 21 January 2015
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	====P-V Diagram====		====P-V Diagram====

	For an arbitrary value of the adiabatic exponent, <math>~\gamma_g</math>, it isn't possible to invert this virial relation to obtain an analytic expression for <math>~\chi_\mathrm{ad}</math> as a function of <math>~\Pi_\mathrm{ad}</math>. But, as written, the virial relation dictates the behavior of <math>~\Pi_\mathrm{ad}</math> as a function of <math>~\~~chi_~~\mathrm{ad}</math>. Figure 4 displays this <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math> behavior for a number of different values of <math>~\gamma_g</math>.		For an arbitrary value of the adiabatic exponent, <math>~\gamma_g</math>, it isn't possible to invert this virial relation to obtain an analytic expression for <math>~\chi_\mathrm{ad}</math> as a function of <math>~\Pi_\mathrm{ad}</math>. But, as written, the virial relation dictates the behavior of <math>~\Pi_\mathrm{ad}</math> as a function of <math>~\Chi_\mathrm{ad}</math>. Figure 4 displays this <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math> behavior for a number of different values of <math>~\gamma_g</math>.

	<div align="center">		<div align="center">
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	<div align="center">		<div align="center">
	<math>		<math>
	~\Pi_\mathrm{ad} = (\~~chi_~~\mathrm{ad}^{4-3\gamma_g} - 1)/\~~chi_~~\mathrm{ad}^4 \, ,		~\Pi_\mathrm{ad} = (\Chi_\mathrm{ad}^{4-3\gamma_g} - 1)/\Chi_\mathrm{ad}^4 \, ,
	</math>		</math>
	</div>		</div>
	for six different values of <math>~\gamma_g</math> — specifically, for <math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled — and show the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, that is required to construct a nonrotating, self-gravitating, pressure-truncated adiabatic sphere with an equilibrium radius <math>~\~~chi_~~\mathrm{ad}</math>. The solid red curve identifies the behavior of an isothermal <math>~(\gamma_g=1)</math> system. The mathematical solution becomes unphysical wherever the pressure becomes negative.		for six different values of <math>~\gamma_g</math> — specifically, for <math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled — and show the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, that is required to construct a nonrotating, self-gravitating, pressure-truncated adiabatic sphere with an equilibrium radius <math>~\Chi_\mathrm{ad}</math>. The solid red curve identifies the behavior of an isothermal <math>~(\gamma_g=1)</math> system. The mathematical solution becomes unphysical wherever the pressure becomes negative.
	</td>		</td>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
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	For physically relevant solutions, both <math>~\~~chi_~~\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> must be nonnegative. Hence, as is illustrated by the curves in Figure 4, the physically allowable range of equilibrium radii is,		For physically relevant solutions, both <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> must be nonnegative. Hence, as is illustrated by the curves in Figure 4, the physically allowable range of equilibrium radii is,
	<div align="center">		<div align="center">
	<math>		<math>
	1 \le \~~chi_~~\mathrm{ad} \le \infty \, ~~~~~\mathrm{for}~ \gamma_g < 4/3 \, ;		1 \le \Chi_\mathrm{ad} \le \infty \, ~~~~~\mathrm{for}~ \gamma_g < 4/3 \, ;
	</math>		</math>

	<math>		<math>
	0 < \~~chi_~~\mathrm{ad} \le 1 \, ~~~~~~\mathrm{for}~ \gamma_g > 4/3 \, .		0 < \Chi_\mathrm{ad} \le 1 \, ~~~~~~\mathrm{for}~ \gamma_g > 4/3 \, .
	</math>		</math>
	</div>		</div>

	Each of the <math>~\Pi_\mathrm{ad}(\~~chi_~~\mathrm{ad})</math> curves drawn in Figure 4 exhibits an extremum. In each case this extremum occurs at a configuration radius, <math>\~~chi_~~\mathrm{extreme}</math>, given by,		Each of the <math>~\Pi_\mathrm{ad}(\Chi_\mathrm{ad})</math> curves drawn in Figure 4 exhibits an extremum. In each case this extremum occurs at a configuration radius, <math>~\Chi_\mathrm{extreme}</math>, given by,
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{\partial\Pi_\mathrm{ad}}{\partial\~~chi_~~\mathrm{ad}} = 0 \, ,		\frac{\partial\Pi_\mathrm{ad}}{\partial\Chi_\mathrm{ad}} = 0 \, ,
	</math>		</math>
	</div>		</div>
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	<div align="center">		<div align="center">
	<math>		<math>
	4 - 3\gamma_g \~~chi_~~\mathrm{ad}^{4-3\gamma_g} = 0 ~~~~\Rightarrow ~~~~~ \~~chi_~~\mathrm{extreme} = \biggl[ \frac{4}{3\gamma_g} \biggr]^{1/(4-3\gamma_g)} \, .		4 - 3\gamma_g \Chi_\mathrm{ad}^{4-3\gamma_g} = 0 ~~~~\Rightarrow ~~~~~ \Chi_\mathrm{extreme} = \biggl[ \frac{4}{3\gamma_g} \biggr]^{1/(4-3\gamma_g)} \, .
	</math>		</math>
	</div>		</div>
	For each value of <math>\gamma_g</math>, the corresponding dimensionless pressure is,		For each value of <math>~\gamma_g</math>, the corresponding dimensionless pressure is,
	<div align="center">		<div align="center">
	<math>		<math>
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	<tr>		<tr>
	<td align="center">		<td align="center">
	<math>~\~~chi_~~\mathrm{extreme} = \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}</math>		<math>~\Chi_\mathrm{extreme} = \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</table>		</table>
	</div>		</div>
			(In a [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Stability\|separate, related discussion of the free-energy function]], we demonstrate that this "extremum" also serves as a dividing line between dynamically stable and unstable models along a given curve.)

	~~Note~~, ~~first~~, ~~that~~ for <math>~\gamma_g > 4/3</math>, an equilibrium configuration with a positive radius can be constructed for all physically realistic — that is, for all positive — values of <math>~\Pi_\mathrm{ad}</math>. Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant — ''i.e.,'' positive — pressures only for values of <math>~\gamma_g < 4/3</math>; and in each case it represents a ''maximum'' limiting pressure.
			In examining the group of plotted curves, notice that, for <math>~\gamma_g > 4/3</math>, an equilibrium configuration with a positive radius can be constructed for all physically realistic — that is, for all positive — values of <math>~\Pi_\mathrm{ad}</math>. Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant — ''i.e.,'' positive — pressures only for values of <math>~\gamma_g < 4/3</math>; and in each case it represents a ''maximum'' limiting pressure.

Tohline: /* Virial Equilibrium of Adiabatic Spheres */

2015-01-20T20:54:28Z

Virial Equilibrium of Adiabatic Spheres

← Older revision		Revision as of 20:54, 20 January 2015
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	=Virial Equilibrium of Adiabatic Spheres=		=Virial Equilibrium of Adiabatic Spheres=
	Highlights of the rather detailed discussion presented below have been [[User:Tohline/SSC/Virial/~~PolytropesSummary~~\|summarized in an accompanying chapter]] of this H_Book.		Highlights of the rather detailed discussion presented below have been [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29\|summarized in an accompanying chapter]] of this H_Book.
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

Tohline: /* Renormalization */ Various changes to make this chapter more compatible with the next (second effort) chapter

2015-01-20T19:32:26Z

Renormalization: Various changes to make this chapter more compatible with the next (second effort) chapter

← Older revision		Revision as of 19:32, 20 January 2015
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~chi_~~\mathrm{ad}^4 </math>		<math>~\Chi_\mathrm{ad}^4 </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	\biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}		\biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}
	\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{3\gamma_g}		\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{3\gamma_g}
	\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}}\cdot \frac{\~~chi_~~\mathrm{ad}^{4-3\gamma_g} }{\Pi_\mathrm{ad}} -		\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}}\cdot \frac{\Chi_\mathrm{ad}^{4-3\gamma_g} }{\Pi_\mathrm{ad}} -
	\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2}		\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2}
	\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{4}		\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{4}
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~chi_~~\mathrm{ad}</math>		<math>~\Chi_\mathrm{ad}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Pi_\mathrm{ad} = \~~chi_~~\mathrm{ad}^{-3\gamma_g} - \~~chi_~~\mathrm{ad}^{-4}</math>		<math>~\Pi_\mathrm{ad} = \Chi_\mathrm{ad}^{-3\gamma_g} - \Chi_\mathrm{ad}^{-4}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ \Pi_\mathrm{ad} = \~~chi_~~\mathrm{ad}^{-3(n+1)/n} - \~~chi_~~\mathrm{ad}^{-4}</math>		<math>~ \Pi_\mathrm{ad} = \Chi_\mathrm{ad}^{-3(n+1)/n} - \Chi_\mathrm{ad}^{-4}</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~\~~chi_~~\mathrm{ad}^{4-3\gamma_g} - \Pi_\mathrm{ad} \~~chi_~~\mathrm{ad}^4 = 1</math>		<math>~~\Chi_\mathrm{ad}^{4-3\gamma_g} - \Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = 1</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\~~chi_~~\mathrm{ad}^{(n-3)/n} - \Pi_\mathrm{ad} \~~chi_~~\mathrm{ad}^4 = 1 \, .</math>		<math>~\Chi_\mathrm{ad}^{(n-3)/n} - \Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = 1 \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</div>		</div>

	~~For later use we should develop expressions for both <math>~\chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> that are in terms~~ of ~~our structural form factors and Hoerdt's two dimensionless functions, <math>~r_a</math> and <math>~p_a</math>. (Adopting a unified notation, we will set <math>~\gamma_g \rightarrow (n+1)/n</math>.)~~		=====In Terms of Free-Energy Coefficients=====

			Referring back to relations between our free-energy coefficients, as [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#ABratios\|presented earlier]], we note that,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\chi_\mathrm{ad} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{ad}}</math>		<math>~\frac{R_\mathrm{ad}}{R_\mathrm{norm}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{\mathcal{A}}{\mathcal{B}}\biggr)^{n/(n-3)} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{P_\mathrm{ad}}{P_\mathrm{norm}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{3}{4\pi} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Hence, we can write,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Chi_\mathrm{ad}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\chi_\mathrm{eq} ~\biggl( \frac{\mathcal{B}}{\mathcal{A}}\biggr)^{n/(n-3)} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Pi_\mathrm{ad}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\mathcal{D} ~\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr]^{1/(n-3)} \, .</math>
			</td>
			</tr>
			</table>
			</div>



			=====In Terms of Horedt's Equilibrium Parameters=====
			For later use it is also worth developing expressions for both <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> that are in terms of our structural form factors and Horedt's two dimensionless functions, <math>~r_a</math> and <math>~p_a</math>. (Adopting a unified notation, we will set <math>~\gamma_g \rightarrow (n+1)/n</math>.)

			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Chi_\mathrm{ad} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{ad}}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~\~~chi_~~\mathrm{ad}^{n-3} </math>		<math>~\Rightarrow~~~\Chi_\mathrm{ad}^{n-3} </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~chi_~~\mathrm{ad}^{n-3} </math>		<math>~\Chi_\mathrm{ad}^{n-3} </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~chi_~~\mathrm{ad}</math>		<math>~\Chi_\mathrm{ad}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\~~chi_~~\mathrm{ad}^{-3(n+1)/n} - \~~chi_~~\mathrm{ad}^{-4} \, .</math>		<math>~\Chi_\mathrm{ad}^{-3(n+1)/n} - \Chi_\mathrm{ad}^{-4} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Virial Equilibrium */ Correct exponent on f_M in expression for (A/B)^n

2015-01-20T17:16:08Z

Virial Equilibrium: Correct exponent on f_M in expression for (A/B)^n

← Older revision		Revision as of 17:16, 20 January 2015
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	<div align="center">		<div align="center" id="ABratios">
	<table border="1" cellpadding="10" align="center">		<table border="1" cellpadding="10" align="center">
	<tr><td align="left">		<tr><td align="left">
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	<td align="left">		<td align="left">
	<math>~\frac{1}{5^n} \biggl( \frac{4\pi}{3} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1}		<math>~\frac{1}{5^n} \biggl( \frac{4\pi}{3} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1}
	\cdot \frac{\mathfrak{f}_W^n}{\mathfrak{f}_A^n \mathfrak{f}_M^{1-n}} \, ;		\cdot \frac{\mathfrak{f}_W^n}{\mathfrak{f}_A^n \mathfrak{f}_M^{n-1}} \, ;
	</math>		</math>
	</td>		</td>

Tohline: /* Virial Equilibrium */ Copy and paste from earlier derivations the boxed-in equations showing various ratios of A/B

2015-01-20T17:04:05Z

Virial Equilibrium: Copy and paste from earlier derivations the boxed-in equations showing various ratios of A/B

← Older revision		Revision as of 17:04, 20 January 2015
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	<div align="center">		<div align="center">
	<math>		<math>
	2C \chi_\mathrm{eq}^{-2} + ~3 B\chi_\mathrm{eq}^{3 -3\gamma_g} -~3A\chi_\mathrm{eq}^{-1} -~ 3D \chi_\mathrm{eq}^3 = 0 \, ,		2\mathcal{C} \chi_\mathrm{eq}^{-2} + ~3 \mathcal{B}\chi_\mathrm{eq}^{3 -3\gamma_g} -~3\mathcal{A}\chi_\mathrm{eq}^{-1}
			-~ 3\mathcal{D} \chi_\mathrm{eq}^3 = 0 \, ,
	</math>		</math>
	</div>		</div>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~A</math>		<math>~\mathcal{A}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~B</math>		<math>~\mathcal{B}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~C</math>		<math>~\mathcal{C}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~D</math>		<math>~\mathcal{D}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	(The [[User:Tohline/SphericallySymmetricConfigurations/Virial#Structural_Form_Factors\|dimensionless structural form factors]], <math>~\mathfrak{f}_i,</math> that appear in these expressions are defined ''for isolated polytropes'' in our accompanying introductory discussion and are [[User:Tohline/SSC/Virial/Polytropes#Role_of_Structural_Form_Factors\|discussed further, below]].) Once the pressure exerted by the external medium (<math>~P_e</math>), and the configuration's mass (<math>~M_\mathrm{tot}</math>), angular momentum (<math>~J</math>), and specific entropy (via <math>~K</math>) have been specified, the values of all of the coefficients are known and <math>~\chi_\mathrm{eq}</math> can be determined.		(The [[User:Tohline/SphericallySymmetricConfigurations/Virial#Structural_Form_Factors\|dimensionless structural form factors]], <math>~\mathfrak{f}_i,</math> that appear in these expressions are defined ''for isolated polytropes'' in our accompanying introductory discussion and are [[User:Tohline/SSC/Virial/Polytropes#Role_of_Structural_Form_Factors\|discussed further, below]].) Once the pressure exerted by the external medium (<math>~P_e</math>), and the configuration's mass (<math>~M_\mathrm{tot}</math>), angular momentum (<math>~J</math>), and specific entropy (via <math>~K</math>) have been specified, the values of all of the coefficients are known and <math>~\chi_\mathrm{eq}</math> can be determined.


			<div align="center">
			<table border="1" cellpadding="10" align="center">
			<tr><td align="left">

			For later use, we note that after making the substitution, <math>~\gamma_g \rightarrow (n+1)/n</math>,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl( \frac{\mathcal{A}}{\mathcal{B}} \biggr)^n</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl\{
			\frac{3}{20\pi} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W
			\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{-(n+1)/n}
			\cdot \mathfrak{f}_A^{-1}
			\biggr\}^n</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~5^{-n}
			\biggl( \frac{3}{4\pi} \biggr)^n \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{2n}
			\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{-(n+1)}
			\cdot \biggl( \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr)^{n}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{5^n} \biggl( \frac{4\pi}{3} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1}
			\cdot \frac{\mathfrak{f}_W^n}{\mathfrak{f}_A^n \mathfrak{f}_M^{1-n}} \, ;
			</math>
			</td>
			</tr>
			</table>
			</div>
			and,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl\{
			\frac{1}{5} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \biggr\}^{-3(n+1)}
			\biggl\{ \frac{4\pi}{3}
			\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{(n+1)/n}
			\cdot \mathfrak{f}_A
			\biggr\}^{4n}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			5^{3(n+1)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{-6(n+1)}
			\biggl( \frac{4\pi}{3} \biggr)^{4n-4(n+1)}
			\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{4(n+1)} \cdot\frac{\mathfrak{f}_A^{4n}}{\mathfrak{f}_W^{3(n+1)}}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			5^{3(n+1)} \biggl( \frac{3}{4\pi} \biggr)^{4} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2(n+1)}
			\cdot\frac{\mathfrak{f}_A^{4n} \mathfrak{f}_M^{2(n+1)} }{\mathfrak{f}_W^{3(n+1)}}
			</math>
			</td>
			</tr>
			</table>
			</div>

			</td></tr>
			</table>
			</div>

	==Isolated Nonrotating Adiabatic Configuration==		==Isolated Nonrotating Adiabatic Configuration==

Tohline: /* P-V Diagram */

2015-01-19T20:51:50Z

P-V Diagram

← Older revision		Revision as of 20:51, 19 January 2015
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	<div align="center">		<div align="center">
	<math>		<math>
	~\Pi_~~\mathrm{ad} \biggr\|_~~\mathrm{extreme} = \biggl(\frac{4}{3\gamma} - 1 \biggr) \biggl[ \frac{3\gamma_g}{4} \biggr]^{4/(4-3\gamma_g)} \, .		~\Pi_\mathrm{extreme} = \biggl(\frac{4}{3\gamma} - 1 \biggr) \biggl[ \frac{3\gamma_g}{4} \biggr]^{4/(4-3\gamma_g)} \, .
	</math>		</math>
			</div>
			In terms of the polytropic index, the equivalent limiting expressions are,
			<div align="center">
			<table border="0" cellpadding="5">
			<tr>
			<td align="center">
			<math>~\chi_\mathrm{extreme} = \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}</math>
			</td>
			<td align="center">
			and
			</td>
			<td align="center">
			<math>~\Pi_\mathrm{extreme}^{(n-3)} = (4n)^{-4n} (n-3)^{n-3} [3(n+1)]^{3(n+1)} \, .</math>
			</td>
			</tr>
			</table>
	</div>		</div>

Tohline at 17:32, 19 January 2015

2015-01-19T17:32:49Z

Show changes

Tohline: /* P-V Diagram for Unity Form Factors */ Rework final subsection with "Figure 4" plot

2015-01-19T17:27:25Z

P-V Diagram for Unity Form Factors: Rework final subsection with "Figure 4" plot

← Older revision		Revision as of 17:27, 19 January 2015
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	{{LSU_WorkInProgress}}		{{LSU_WorkInProgress}}

	====P-V Diagram ~~for Unity Form Factors~~====		====P-V Diagram====

	~~Writing the coefficient, <math>B</math>, in terms of the average sound speed and setting the radial scale factor equal to the equilibrium radius of an isolated adiabatic sphere, that is, setting,~~		For an arbitrary value of the adiabatic exponent, <math>~\gamma_g</math>, it isn't possible to invert this virial relation to obtain an analytic expression for <math>~\chi_\mathrm{ad}</math> as a function of <math>~\Pi_\mathrm{ad}</math>. But, as written, the virial relation dictates the behavior of <math>~\Pi_\mathrm{ad}</math> as a function of <math>~\chi_\mathrm{ad}</math>. Figure 4 displays this <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math> behavior for a number of different values of <math>~\gamma_g</math>.
	~~<div align="center">~~
	~~<math>~~
	~~R_0 = \frac{GM}{5\bar{c_s}^2} \, ,~~
	~~</math>~~
	~~</div>~~
	~~the equation governing the radii of ''adiabatic'' equilibrium states becomes,~~
	~~<div align="center">~~
	~~<math>~~
	~~\chi^4 - \frac{1}{\Pi_a} \chi^{(4-3\gamma_g)} + \frac{1}{\Pi_a} = 0 \, ,~~
	~~</math>~~
	~~</div>~~
	~~where,~~
	~~<div align="center">~~
	~~<math>~~
	~~\Pi_a \equiv \frac{4\pi P_e G^3 M^2}{3\cdot 5^3 \bar{c_s}^8} \, .~~
	~~</math>~~
	~~</div>~~
	As in the isothermal case, for a given choice of configuration mass and sound speed, this parameter, <math>\Pi_a</math>, can be viewed as a dimensionless external pressure. Alternatively, for a given choice of <math>P_e</math> and <math>\bar{c_s}</math>, <math>\Pi_a^{1/2}</math> can represent a dimensionless mass; or, for a given choice of <math>M</math> and <math>P_e</math>, <math>\Pi_a^{-1/8}</math> can represent a dimensionless sound speed. Here we will view it as a dimensionless external pressure.

	~~Unlike the isothermal case, for~~ an arbitrary value of the adiabatic exponent, <math>\gamma_g</math>, it isn't possible to invert this ~~equation~~ to obtain an analytic expression for <math>\~~chi~~</math> as a function of <math>\~~Pi_a~~</math>. But ~~we can straightforwardly solve for~~ <math>\~~Pi_a~~</math> as a function of <math>\~~chi</math>. The solution is,~~
	~~<div align="center">~~
	~~<math>~~
	~~\Pi_a =~~ \~~frac{\chi^~~{~~(4- 3\gamma_g)} - 1}{\chi^4~~} ~~\, .~~
	~~</math>~~
	~~</div>~~
	~~For physically relevant solutions, both <math>\chi</math> and <math>\Pi_a~~</math> ~~must be nonnegative~~. ~~Hence, as is illustrated by the curves in~~ Figure 4~~, the physically allowable range of equilibrium radii is,~~
	~~<div align="center">~~
	<math>
	1 \le \~~chi~~ \~~le \infty \, ~~~~~~~\mathrm{~~for~~}~~~ \gamma_g < 4/3 \, ;~~
	</math>

	<math>
	~~0 < \chi \le 1 \, ~~~~~~\mathrm{for}~~~ \gamma_g ~~> 4/3 \, .~~
	</math>
	~~</div>~~

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td valign="top" width=450 rowspan="1">		<td valign="top" width=450 rowspan="1">
	The curves trace out the function,		The curves shown here, on the right, trace out the function,
	<div align="center">		<div align="center">
	<math>		<math>
	\~~Pi_a~~ = (\~~chi~~^{4-3\gamma_g} - 1)/\~~chi~~^4 \, ,		~\Pi_\mathrm{ad} = (\chi_\mathrm{ad}^{4-3\gamma_g} - 1)/\chi_\mathrm{ad}^4 \, ,
	</math>		</math>
	</div>		</div>
	for six different values of <math>\gamma_g</math> (<math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled) and show the dimensionless external pressure, <math>\~~Pi_a~~</math>, that is required to construct a nonrotating, self-gravitating, ~~uniform density,~~ adiabatic sphere with an equilibrium radius <math>\~~chi~~</math>. ~~The mathematical solution becomes unphysical wherever the pressure becomes negative.~~		for six different values of <math>~\gamma_g</math> — specifically, for <math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled — and show the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, that is required to construct a nonrotating, self-gravitating, pressure-truncated adiabatic sphere with an equilibrium radius <math>~\chi_\mathrm{ad}</math>. The solid red curve identifies the behavior of an isothermal <math>~(\gamma_g=1)</math> system. The mathematical solution becomes unphysical wherever the pressure becomes negative.

	The solid red curve~~, drawn for~~ the ~~case~~ <math>\gamma_g = 1</math>~~, is identical to~~ the ~~solid black (isothermal) curve displayed above in Figure 1~~.
	</td>		</td>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
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	</div>		</div>

	Each of the <math>\~~Pi_a~~(\~~chi~~)</math> curves drawn in Figure 4 exhibits an extremum. In each case this extremum occurs at a configuration radius, <math>\chi_\mathrm{extreme}</math>, given by,
			For physically relevant solutions, both <math>~\chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> must be nonnegative. Hence, as is illustrated by the curves in Figure 4, the physically allowable range of equilibrium radii is,
			<div align="center">
			<math>
			1 \le \chi_\mathrm{ad} \le \infty \, ~~~~~\mathrm{for}~ \gamma_g < 4/3 \, ;
			</math>

			<math>
			0 < \chi_\mathrm{ad} \le 1 \, ~~~~~~\mathrm{for}~ \gamma_g > 4/3 \, .
			</math>
			</div>

			Each of the <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math> curves drawn in Figure 4 exhibits an extremum. In each case this extremum occurs at a configuration radius, <math>\chi_\mathrm{extreme}</math>, given by,
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{\partial\~~Pi_a~~}{\partial\~~chi~~} = 0 \, ,		\frac{\partial\Pi_\mathrm{ad}}{\partial\chi_\mathrm{ad}} = 0 \, ,
	</math>		</math>
	</div>		</div>
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	<div align="center">		<div align="center">
	<math>		<math>
	4 - 3\gamma_g \~~chi~~^{4-3\gamma_g} = 0 ~~~~\Rightarrow ~~~~~ \chi_\mathrm{extreme} = \biggl[ \frac{4}{3\gamma_g} \biggr]^{1/(4-3\gamma_g)} \, .		4 - 3\gamma_g \chi_\mathrm{ad}^{4-3\gamma_g} = 0 ~~~~\Rightarrow ~~~~~ \chi_\mathrm{extreme} = \biggl[ \frac{4}{3\gamma_g} \biggr]^{1/(4-3\gamma_g)} \, .
	</math>		</math>
	</div>		</div>
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	<div align="center">		<div align="center">
	<math>		<math>
	\~~Pi_a~~ \biggr\|_\mathrm{extreme} = \biggl(\frac{4}{3\gamma} - 1 \biggr) \biggl[ \frac{3\gamma_g}{4} \biggr]^{4/(4-3\gamma_g)} \, .		~\Pi_\mathrm{ad} \biggr\|_\mathrm{extreme} = \biggl(\frac{4}{3\gamma} - 1 \biggr) \biggl[ \frac{3\gamma_g}{4} \biggr]^{4/(4-3\gamma_g)} \, .
	</math>		</math>
	</div>		</div>

	Note, first, that for <math>\gamma_g > 4/3</math>, an equilibrium configuration with a positive radius can be constructed for all physically realistic — that is, for all positive — values of <math>\~~Pi_a~~</math>. Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant — ''i.e.,'' positive — pressures only for values of <math>\gamma_g < 4/3</math>; and in each case it represents a ''maximum'' limiting pressure.		Note, first, that for <math>~\gamma_g > 4/3</math>, an equilibrium configuration with a positive radius can be constructed for all physically realistic — that is, for all positive — values of <math>~\Pi_\mathrm{ad}</math>. Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant — ''i.e.,'' positive — pressures only for values of <math>~\gamma_g < 4/3</math>; and in each case it represents a ''maximum'' limiting pressure.

	====Maximum Mass====		====Maximum Mass====

Tohline: /* Summary */

2015-01-19T16:59:51Z

Summary

← Older revision		Revision as of 16:59, 19 January 2015
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~a_\mathrm{ad} + b_\mathrm{ad} \, ,</math>		<math>~a_\mathrm{ad} + b_\mathrm{ad} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	the expressions for the dimensionless equilibrium radius and the dimensionless external pressure, as just derived, may be written as, respectively,		In addition, the expressions for the dimensionless equilibrium radius and the dimensionless external pressure, as just derived, may be written as, respectively,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Renormalization */ Delete unnecessary "work in progress" sign

2015-01-19T01:19:17Z

Renormalization: Delete unnecessary "work in progress" sign

← Older revision		Revision as of 01:19, 19 January 2015
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	</table>		</table>
	</div>		</div>

	~~{{LSU_WorkInProgress}}~~


	For later use we should develop expressions for both <math>~\chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> that are in terms of our structural form factors and Hoerdt's two dimensionless functions, <math>~r_a</math> and <math>~p_a</math>. (Adopting a unified notation, we will set <math>~\gamma_g \rightarrow (n+1)/n</math>.)		For later use we should develop expressions for both <math>~\chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> that are in terms of our structural form factors and Hoerdt's two dimensionless functions, <math>~r_a</math> and <math>~p_a</math>. (Adopting a unified notation, we will set <math>~\gamma_g \rightarrow (n+1)/n</math>.)