SR - Revision history

Jet53man: /* Barotropic Structure */

2022-10-17T16:28:51Z

Barotropic Structure

← Older revision		Revision as of 12:28, 17 October 2022
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	{{ Math/EQ_ZTFG01 }}		{{ Math/EQ_ZTFG01 }}
	<font size="-1">		<font size="-1">
	Reference (original): {{ ~~Chandrasekhar35~~ }}		Reference (original): {{ Chandrasekhar35full }}
	</font>		</font>

Jet53man: /* Barotropic Structure */

2022-10-17T16:18:07Z

Barotropic Structure

← Older revision		Revision as of 12:18, 17 October 2022
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	<tr>		<tr>
	<td align="center">		<td align="center">
	{{~~Template:~~Math/EQ_Polytrope01}}		{{ Math/EQ_Polytrope01 }}
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~Template:~~Math/EQ_ZTFG01}}		{{ Math/EQ_ZTFG01 }}
	<font size="-1">		<font size="-1">
	Reference (original): ~~[http://adsabs.harvard.edu/abs/1935MNRAS..95..207C S. Chandraskehar (1935)]~~		Reference (original): {{ Chandrasekhar35 }}
	</font>		</font>

Jet53man: /* ZTF Gas in Relativistic Limit */

2022-05-01T18:08:30Z

ZTF Gas in Relativistic Limit

← Older revision		Revision as of 14:08, 1 May 2022
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	This agrees with, for example, the asymptotic form presented as equation (25) in §X.1 (p. 361) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]. Keeping only the leading term leads to the expression,		This agrees with, for example, the asymptotic form presented as equation (25) in §X.1 (p. 361) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]. Keeping only the leading term leads to the expression,
	~~<div align="center">~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<td align="left">		<td align="left">
	<math>\frac{1}{2^3}\biggl(\frac{3}{\pi}\biggr)^{1/3} (hc) \biggl(\frac{\rho}{m_p \mu_e}\biggr)^{4/3} \, .</math>		<math>\frac{1}{2^3}\biggl(\frac{3}{\pi}\biggr)^{1/3} (hc) \biggl(\frac{\rho}{m_p \mu_e}\biggr)^{4/3} \, .</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], §X.1, Eq. (28)<br />
			<!-- [<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>], Eq. (2-32)<br /> -->
			[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], Eq. (11.43)<br />
			[<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], Eq. (15.26)<br />
			[<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], lower expression in Eq. (5.163)<br />
			[<b>[[Appendix/References#Choudhuri10\|<font color="red">Choudhuri10</font>]]</b>], Eqs. (5.11)-(5.12)
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	<~~/div>~~		<!--

	This matches equation (28) in §X.1 (p. 362) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]; see also, equation (11.43) of [<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], equation (15.26) of [<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], the lower expression labeled as equation (5.163) in [<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], and equations (5.11)-(5.12) of [<b>[[Appendix/References#Choudhuri10\|<font color="red">Choudhuri10</font>]]</b>].		This matches equation (28) in §X.1 (p. 362) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]; see also, equation (11.43) of [<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], equation (15.26) of [<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], the lower expression labeled as equation (5.163) in [<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], and equations (5.11)-(5.12) of [<b>[[Appendix/References#Choudhuri10\|<font color="red">Choudhuri10</font>]]</b>].
			-->

	===Steady-state Flow-Field Specification===		===Steady-state Flow-Field Specification===

Jet53man: /* Nonrelativistic ZTF Gas */

2022-05-01T18:05:23Z

Nonrelativistic ZTF Gas

← Older revision		Revision as of 14:05, 1 May 2022
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	This agrees with, for example, the asymptotic form presented as equation (24) in §X.1 (p. 361) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]. Keeping only the leading term leads to the expression,		This agrees with, for example, the asymptotic form presented as equation (24) in §X.1 (p. 361) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]. Keeping only the leading term leads to the expression,
	~~<div align="center">~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<td align="left">		<td align="left">
	<math>\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl( \frac{\rho}{m_p \mu_e}\biggr)^{5/3} \, .</math>		<math>\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl( \frac{\rho}{m_p \mu_e}\biggr)^{5/3} \, .</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], §X.1, Eq. (27)<br />
			[<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>], Eq. (2-32)<br />
			[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], Eq. (11.42)<br />
			[<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], Eq. (15.23)<br />
			[<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], upper expression in Eq. (5.163)<br />
			[<b>[[Appendix/References#Choudhuri10\|<font color="red">Choudhuri10</font>]]</b>], Eqs. (5.9)-(5.10)
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	<~~/div>~~		<!--

	This matches equation (27) in §X.1 (p. 362) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]; see also, equation (2-32) of [<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>], equation (11.42) of [<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], equation (15.23) of [<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], the upper expression labeled as equation (5.163) in [<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], and equations (5.9)-(5.10) of [<b>[[Appendix/References#Choudhuri10\|<font color="red">Choudhuri10</font>]]</b>].		This matches equation (27) in §X.1 (p. 362) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]; see also, equation (2-32) of [<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>], equation (11.42) of [<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], equation (15.23) of [<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], the upper expression labeled as equation (5.163) in [<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], and equations (5.9)-(5.10) of [<b>[[Appendix/References#Choudhuri10\|<font color="red">Choudhuri10</font>]]</b>].
			-->

	====ZTF Gas in Relativistic Limit====		====ZTF Gas in Relativistic Limit====

Jet53man: /* The Parameter, β */

2021-10-08T16:23:20Z

The Parameter, β

← Older revision		Revision as of 12:23, 8 October 2021
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	<math>\beta = \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, .</math>		<math>\beta = \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, .</math>
	</div>		</div>
	Examples include our discussion of [[SSC/Structure/BiPolytropes/Analytic1.53\|bipolytropic configurations with]] <math>(n_c, n_e) = (\tfrac{3}{2}, 3)</math>, as introduced by ~~[http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]~~; the study of [[Apps/SMS#Equation_of_State\|rotating, supermassive stars]], especially as introduced by ~~[http://adsabs.harvard.edu/abs/1984ApJ...280..825B J. R. Bond, W. D. Arnett, & B. J. Carr (1984)]~~; and our reference to a [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|derivation of the linear adiabatic wave equation]] by ~~[http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux & C. L. Pekeris (1941)]~~.		Examples include our discussion of [[SSC/Structure/BiPolytropes/Analytic1.53\|bipolytropic configurations with]] <math>(n_c, n_e) = (\tfrac{3}{2}, 3)</math>, as introduced by {{ Milne30full }}; the study of [[Apps/SMS#Equation_of_State\|rotating, supermassive stars]], especially as introduced by {{ BAC84full }}; and our reference to a [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|derivation of the linear adiabatic wave equation]] by {{ LP41full }}.

	====Adiabatic Exponent====		====Adiabatic Exponent====

Jet53man at 16:19, 8 October 2021

2021-10-08T16:19:26Z

Jet53man: /* ZTF Gas in Relativistic Limit */

2021-10-08T16:13:30Z

ZTF Gas in Relativistic Limit

← Older revision		Revision as of 12:13, 8 October 2021
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	</div>		</div>

	This matches equation (28) in §X.1 (p. 362) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]; see also, equation (11.43) of [<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], equation (15.26) of [<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], the lower expression labeled as equation (5.163) in [<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], and equations (5.11)-(5.12) of [~~http:~~//~~adsabs.harvard.edu~~/~~abs/2010asph.book.....C A. R. Choudhuri (2010)~~].		This matches equation (28) in §X.1 (p. 362) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]; see also, equation (11.43) of [<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], equation (15.26) of [<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], the lower expression labeled as equation (5.163) in [<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], and equations (5.11)-(5.12) of [<b>[[Appendix/References#Choudhuri10\|<font color="red">Choudhuri10</font>]]</b>].

	===Steady-state Flow-Field Specification===		===Steady-state Flow-Field Specification===

Jet53man: /* Nonrelativistic ZTF Gas */

2021-10-08T16:11:14Z

Nonrelativistic ZTF Gas

← Older revision		Revision as of 12:11, 8 October 2021
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	</div>		</div>

	This matches equation (27) in §X.1 (p. 362) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]; see also, equation (2-32) of [~~http:~~//~~adsabs.harvard.edu~~/~~abs/1968psen.book.....C D. D. Clayton (1968)~~], equation (11.42) of [<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], equation (15.23) of [<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], the upper expression labeled as equation (5.163) in [<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], and equations (5.9)-(5.10) of [~~http:~~//~~adsabs.harvard.edu~~/~~abs/2010asph.book.....C A. R. Choudhuri (2010)~~].		This matches equation (27) in §X.1 (p. 362) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]; see also, equation (2-32) of [<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>], equation (11.42) of [<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], equation (15.23) of [<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], the upper expression labeled as equation (5.163) in [<b>[[Appendix/References#P00\|<font color="red">P00</font>]]</b>], and equations (5.9)-(5.10) of [<b>[[Appendix/References#Choudhuri10\|<font color="red">Choudhuri10</font>]]</b>].

	====ZTF Gas in Relativistic Limit====		====ZTF Gas in Relativistic Limit====

Jet53man: /* Entropy Tracer */

2021-10-06T23:58:55Z

Entropy Tracer

← Older revision		Revision as of 19:58, 6 October 2021
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	</tr>		</tr>
	</table>		</table>
	See §4.1, Eq. (33) of ~~[https://ui.adsabs.harvard.edu/abs/2002ApJS..138..121M/abstract Motl et al. (2002)]~~ or §2.2, Eq. (31) of ~~[http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello & Tohline (2012)]~~.		See §4.1, Eq. (33) of {{ MTF2002 }} or §2.2, Eq. (31) of {{ MT2012 }}.

	===Initial Conditions===		===Initial Conditions===

Jet53man: /* The Parameter, β */

2021-09-20T16:38:34Z

The Parameter, β

← Older revision		Revision as of 12:38, 20 September 2021
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	<math>\beta = \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, .</math>		<math>\beta = \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, .</math>
	</div>		</div>
	Examples include our discussion of [[SSC/Structure/BiPolytropes/Analytic1.~~5_3~~\|bipolytropic configurations with]] <math>(n_c, n_e) = (\tfrac{3}{2}, 3)</math>, as introduced by [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]; the study of [[Apps/SMS#Equation_of_State\|rotating, supermassive stars]], especially as introduced by [http://adsabs.harvard.edu/abs/1984ApJ...280..825B J. R. Bond, W. D. Arnett, & B. J. Carr (1984)]; and our reference to a [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|derivation of the linear adiabatic wave equation]] by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux & C. L. Pekeris (1941)].		Examples include our discussion of [[SSC/Structure/BiPolytropes/Analytic1.53\|bipolytropic configurations with]] <math>(n_c, n_e) = (\tfrac{3}{2}, 3)</math>, as introduced by [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]; the study of [[Apps/SMS#Equation_of_State\|rotating, supermassive stars]], especially as introduced by [http://adsabs.harvard.edu/abs/1984ApJ...280..825B J. R. Bond, W. D. Arnett, & B. J. Carr (1984)]; and our reference to a [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|derivation of the linear adiabatic wave equation]] by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux & C. L. Pekeris (1941)].

	====Adiabatic Exponent====		====Adiabatic Exponent====

← Older revision		Revision as of 12:19, 8 October 2021
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	<td align="center" colspan="3">		<td align="center" colspan="3">
	[[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], Chapter II, §12, Eqs. (123 - 125)<br />		[[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], Chapter II, §12, Eqs. (123 - 125)<br />
	[~~https:~~//~~www.amazon.com~~/~~Principles-Stellar-Evolution-Nucleosynthesis-Clayton/dp/0226109534/ref=sr_1_1?dchild=1&keywords=Clayton%2C+Donald+nucleosynthesis&qid=1625416068&s=books&sr=1-1 Clayton (1968)~~], §2-2, Eqs. (2-121)<br />		[<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>], §2-2, Eqs. (2-121)<br />
	[[Appendix/References#KW94\|[<b><font color="red">KW94</font></b>]]], §13.2, Eqs. (13.27 - 13.29)<br />		[[Appendix/References#KW94\|[<b><font color="red">KW94</font></b>]]], §13.2, Eqs. (13.27 - 13.29)<br />
	[[Appendix/References#HK94\|[<b><font color="red">HK94</font></b>]]], §3.7.1, Eqs. (3.87 - 3.98)<br />		[[Appendix/References#HK94\|[<b><font color="red">HK94</font></b>]]], §3.7.1, Eqs. (3.87 - 3.98)<br />
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	For derivations and additional discussions of the triplet of adiabatic exponents, <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, see:		For derivations and additional discussions of the triplet of adiabatic exponents, <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, see:
	* [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]] — especially pp. 56-58; note that <math>~d\ln V = - d\ln \rho</math>;		* [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]] — especially pp. 56-58; note that <math>~d\ln V = - d\ln \rho</math>;
	* [~~http:~~//~~adsabs.harvard.edu~~/~~abs/1968psen.book.....C Donald D. Clayton (1968)~~]; especially p. 118; note that <math>~d\ln V = - d\ln \rho</math>;		* [<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>]; especially p. 118; note that <math>~d\ln V = - d\ln \rho</math>;
	* [[Appendix/References#KW94\|[<b><font color="red">KW94</font></b>]]] — especially p. 106;		* [[Appendix/References#KW94\|[<b><font color="red">KW94</font></b>]]] — especially p. 106;
	* [~~http:~~//~~adsabs.harvard.edu~~/~~abs/1994sipp.book.....H Carl J. Hansen & Steven D. Kawaler (1994)~~] — especially pp. 136-138;		* [<b>[[Appendix/References#HK94\|<font color="red">HK94</font>]]</b>] — especially pp. 136-138;
	* Our own brief presentation of [[Apps/SMS#Expression_for_Adiabatic_Exponent\|several equivalent expressions for <math>~\Gamma_1</math>]].		* Our own brief presentation of [[Apps/SMS#Expression_for_Adiabatic_Exponent\|several equivalent expressions for <math>~\Gamma_1</math>]].


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