User:Tohline/SSC/Structure/WhiteDwarfs - Revision history

Tohline: /* The Nauenberg Mass-Radius Relationship */

2019-08-09T15:34:30Z

The Nauenberg Mass-Radius Relationship

← Older revision		Revision as of 15:34, 9 August 2019
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	</math>		</math>
	</div>		</div>
	where,		<span id="ChandrasekharMass">where,</span>
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: /* See Also */

2019-08-04T01:09:21Z

Tohline: /* See Also */

2019-08-04T01:03:07Z

Tohline at 02:07, 20 June 2019

2019-06-20T02:07:29Z

← Older revision		Revision as of 02:07, 20 June 2019
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	which matches the [[#Chandrasekhar_mass\|expression presented above]] for the Chandrasekhar mass if we set <math>~m_u = m_p</math>.		which matches the [[#Chandrasekhar_mass\|expression presented above]] for the Chandrasekhar mass if we set <math>~m_u = m_p</math>.

			=See Also=

			* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer & D. Lynden-Bell (1966)], Phys. Rev. Letters, 17, 816: ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''
			<table border="0" align="center" width="100%" cellpadding="1"><tr>
			<td align="center" width="5%"> </td><td align="left">
			<font color="green">… work by Roxburgh (1965, Z. Astrophys., 62, 134), Anand (1965, Proc. Natl. Acad. Sci. U.S., 54, 23), and James (1964, ApJ, 140, 552) shows that the</font> [Chandrasekhar (1931, ApJ, 74, 81)] <font color="green">mass limit <math>~M_3</math> is increased by only a few percent when uniform rotation is included in the models, …</font>

			<font color="green">In this Letter we demonstrate that white-dwarf models with masses considerably greater than <math>~M_3</math> are possible if differential rotation is allowed … models are based on the physical assumption of an axially symmetric, completely degenerate, self-gravitating fluid, in which the effects of viscosity, magnetic fields, meridional circulation, and relativistic terms in the hydrodynamical equations have been neglected.</font>
			</td></tr></table>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Highlights from Discussion by Shapiro & Teukolsky (1983) */

2019-03-13T22:41:11Z

Highlights from Discussion by Shapiro & Teukolsky (1983)

← Older revision		Revision as of 22:41, 13 March 2019
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	</div>		</div>

	As is shown by the following boxed-in ~~image~~, this expression matches equation (3.3.11) from [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], except for the sign of the exponent on <math>~\xi_1</math>, which is demonstratively correct in our expression.		As is shown by the following boxed-in equation table, this expression matches equation (3.3.11) from [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], except for the sign of the exponent on <math>~\xi_1</math>, which is demonstratively correct in our expression.

	<!-- <table border="1" cellpadding="5" align="center">		<!-- <table border="1" cellpadding="5" align="center">
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	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">
	~~Equation~~ extracted<sup>&dagger;</sup> from §3.3 (p. 63) and §2.3 (p. 27) of [http://adsabs.harvard.edu/abs/1983bhwd.book.....S Shapiro & Teukolsky (1983)]<p></p>		Equations extracted<sup>&dagger;</sup> from §3.3 (p. 63) and §2.3 (p. 27) of [http://adsabs.harvard.edu/abs/1983bhwd.book.....S Shapiro & Teukolsky (1983)]<p></p>
	"''Black Holes, White Dwarfs, and Neutron Stars:   The Physics of Compact Objects''"<p></p>		"''Black Holes, White Dwarfs, and Neutron Stars:   The Physics of Compact Objects''"<p></p>
	(New York: John Wiley & Sons)		(New York: John Wiley & Sons)
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:ST83Eq2_3_23.png\|600px\|Equation 2.3.23 from ST83]]		<!-- [[File:ST83Eq2_3_23.png\|600px\|Equation 2.3.23 from ST83]]<br />-->
	<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->		<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->
			<math>~\Gamma = \frac{4}{3} \, ,</math>
			<math>~K = \frac{3^{1 / 3} \pi^{2 /3}}{4} \frac{\hbar c}{m_u^{4 / 3}\mu_e^{4 / 3}} = \frac{1.2435 \times 10^{15}}{\mu_e^{4 /3}} ~\mathrm{cgs} \, .</math>
			(Eq. 2.3.23)
	</td>		</td>
	</tr>		</tr>
	<tr><td align="left"><sup>&dagger;</sup>~~Equations displayed~~ here~~, as individual digital images,~~ exactly as ~~they appear~~ in the original publication.</td></tr>		<tr><td align="left"><sup>&dagger;</sup>Each equation has been retyped here exactly as it appears in the original publication.</td></tr>
	</table>		</table>
	</div>		</div>


	Given that (see equation 3.3.12 of [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>]~~, reprinted above~~) in the relativistic limit, <math>~\Gamma = \gamma_g = 4/3</math> — that is, <math>~n=3</math> — and acknowledging as [[#m3\|we have above]] that, for isolated <math>~n = 3</math> polytropes,		Given that (see equation 3.3.12 of [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>]; see the boxed-in equation table) in the relativistic limit, <math>~\Gamma = \gamma_g = 4/3</math> — that is, <math>~n=3</math> — and acknowledging as [[#m3\|we have above]] that, for isolated <math>~n = 3</math> polytropes,
	<div align="center">		<div align="center">
	<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,		<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,
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	</div>		</div>

	Separately, [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] show that the effective polytropic constant for a relativistic electron gas is (see their equation 2.3.23, reprinted above),		Separately, [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] show that the effective polytropic constant for a relativistic electron gas is (see their equation 2.3.23, reprinted above in the boxed-in equation table),

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

Tohline: /* Highlights from Discussion by Shapiro & Teukolsky (1983) */

2019-03-12T17:06:02Z

Highlights from Discussion by Shapiro & Teukolsky (1983)

← Older revision		Revision as of 17:06, 12 March 2019
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:ST83Eq3_3_11.png\|600px\|Equation 3.3.11 from ST83]]		<!-- [[File:ST83Eq3_3_11.png\|600px\|Equation 3.3.11 from ST83]] -->
	<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->		<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~M</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			4\pi R^{(3-n)/(1-n)} \biggl[ \frac{(n+1)K}{4\pi G} \biggr]^{n/(n-1)} \xi_1^{(3-n)/(1-n)} \xi_1^2 \|\theta^'(\xi_1)\| \, .
			</math>
			</td>
			<td align="right">
			(Eq. 3.3.11)
			</td>
			</tr>
			</table>

	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:ST83Eq3_3_12.png\|600px\|Equation 3.3.12 from ST83]]		<!-- [[File:ST83Eq3_3_12.png\|600px\|Equation 3.3.12 from ST83]] -->
	<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->		<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->
			<math>~\Gamma = \tfrac{4}{3} \, ,</math>
			<math>~n = 3 \, ,</math>
			<math>~\xi_1 = 6.89685 \, ,</math>
			<math>~\xi_1^2\|\theta^'(\xi_1)\| = 2.01824 \, .</math>
			(Eq. 3.3.12)
	</td>		</td>
	</tr>		</tr>

Tohline: /* Highlights from Discussion by Shapiro & Teukolsky (1983) */ Rearrange presentation of 3 equations from ST83

2016-11-21T21:53:33Z

Highlights from Discussion by Shapiro & Teukolsky (1983): Rearrange presentation of 3 equations from ST83

← Older revision		Revision as of 21:53, 21 November 2016
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	As is shown by the following boxed-in image, this expression matches equation (3.3.11) from [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], except for the sign of the exponent on <math>~\xi_1</math>, which is demonstratively correct in our expression.		As is shown by the following boxed-in image, this expression matches equation (3.3.11) from [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], except for the sign of the exponent on <math>~\xi_1</math>, which is demonstratively correct in our expression.

	<table border="1" cellpadding="5" align="center">		<!-- <table border="1" cellpadding="5" align="center">
	<tr>		<tr>
	<td align="center">		<td align="center">
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	</tr>		</tr>
	</table>		</table>
			-->

	Given that (see equation 3.3.12 of [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>]) in the relativistic limit, <math>~\Gamma = \gamma_g = 4/3</math> — that is, <math>~n=3</math> — and acknowledging as [[#m3\|we have above]] that, for isolated <math>~n = 3</math> polytropes,		<div align="center">
			<table border="1" cellpadding="5" width="80%">
			<tr><td align="center">
			Equation extracted<sup>&dagger;</sup> from §3.3 (p. 63) and §2.3 (p. 27) of [http://adsabs.harvard.edu/abs/1983bhwd.book.....S Shapiro & Teukolsky (1983)]<p></p>
			"''Black Holes, White Dwarfs, and Neutron Stars:   The Physics of Compact Objects''"<p></p>
			(New York: John Wiley & Sons)
			</td></tr>
			<tr>
			<td align="center">
			[[File:ST83Eq3_3_11.png\|600px\|Equation 3.3.11 from ST83]]
			<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->
			</td>
			</tr>
			<tr>
			<td align="center">
			[[File:ST83Eq3_3_12.png\|600px\|Equation 3.3.12 from ST83]]
			<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->
			</td>
			</tr>
			<tr>
			<td align="center">
			[[File:ST83Eq2_3_23.png\|600px\|Equation 2.3.23 from ST83]]
			<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->
			</td>
			</tr>
			<tr><td align="left"><sup>&dagger;</sup>Equations displayed here, as individual digital images, exactly as they appear in the original publication.</td></tr>
			</table>
			</div>


			Given that (see equation 3.3.12 of [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], reprinted above) in the relativistic limit, <math>~\Gamma = \gamma_g = 4/3</math> — that is, <math>~n=3</math> — and acknowledging as [[#m3\|we have above]] that, for isolated <math>~n = 3</math> polytropes,
	<div align="center">		<div align="center">
	<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,		<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,
	</div>		</div>

	<table border="1" cellpadding="5" align="center">		<!-- <table border="1" cellpadding="5" align="center">
	<tr>		<tr>
	<td align="center">		<td align="center">
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	</tr>		</tr>
	</table>		</table>
			-->

	this polytropic expression for the mass becomes,		this polytropic expression for the mass becomes,
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	</div>		</div>

	Separately, [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] show that the effective polytropic constant for a relativistic electron gas is (see their equation 2.3.23),		Separately, [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] show that the effective polytropic constant for a relativistic electron gas is (see their equation 2.3.23, reprinted above),

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

	<table border="1" cellpadding="5" align="center">		<!-- <table border="1" cellpadding="5" align="center">
	<tr>		<tr>
	<td align="center">		<td align="center">
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	</tr>		</tr>
	</table>		</table>
			-->

	Together, then, the [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] analysis gives,		Together, then, the [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] analysis gives,

Tohline: /* Highlights from Discussion by Shapiro & Teukolsky (1983) */ Finished showing parallel derivation of M_Ch by ST83

2016-10-04T16:12:37Z

Highlights from Discussion by Shapiro & Teukolsky (1983): Finished showing parallel derivation of M_Ch by ST83

← Older revision		Revision as of 16:12, 4 October 2016
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	<td align="left">		<td align="left">
	<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2} \frac{\mu_e^2}{B_F^2}		<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2} \frac{\mu_e^2}{B_F^2}
	\biggl[ \frac{ m_p }{m_u \mu_e } \biggr]^{2} \, .		\biggl[ \frac{ m_p }{m_u \mu_e } \biggr]^{2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \mu_e^2 M_\mathrm{tot}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2} \frac{\mu_e^2}{B_F^2}
			\biggl[ \frac{ m_p }{m_u } \biggr]^{2} \, ,
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>

			which matches the [[#Chandrasekhar_mass\|expression presented above]] for the Chandrasekhar mass if we set <math>~m_u = m_p</math>.


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Highlights from Discussion by Shapiro & Teukolsky (1983) */

2016-10-04T00:49:50Z

Highlights from Discussion by Shapiro & Teukolsky (1983)

← Older revision		Revision as of 00:49, 4 October 2016
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\biggl( \frac{3\pi^2}{2^6}\biggr)^{1/ 3} \biggl[ \frac{ \hbar^3 c^3 }{ m_u^4 \~~mu_3~~^4 } \biggr]^{ 1 / 3}		\biggl( \frac{3\pi^2}{2^6}\biggr)^{1/ 3} \biggl[ \frac{ \hbar^3 c^3 }{ m_u^4 \mu_e^4 } \biggr]^{ 1 / 3}
	= \biggl[ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \~~mu_3~~^4 } \biggr]^{ 1 / 3} \, .		= \biggl[ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \mu_e^4 } \biggr]^{ 1 / 3} \, .
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~4\pi m_3 \biggl[\frac{ 1 }{\pi G}\biggr]^{3/2} \biggl[ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \~~mu_3~~^4 } \biggr]^{ 1 / 2}		<math>~4\pi m_3 \biggl[\frac{ 1 }{\pi G}\biggr]^{3/2} \biggl[ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \mu_e^4 } \biggr]^{ 1 / 2} \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
	~~And~~, ~~given that~~,		Given that the definitions of the characteristic Fermi pressure, {{ User:Tohline/Math/C_FermiPressure }}, and the characteristic Fermi density, {{ User:Tohline/Math/C_FermiDensity }}, are,

			<div align="center">
			<math>A_\mathrm{F} \equiv \frac{\pi m_e^4 c^5}{3h^3} </math><br /><p></p>

			<math>\frac{B_\mathrm{F}}{\mu_e} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \, ,</math>
			</div>

			we have,
	<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>~</math>		<math>~ M_\mathrm{tot}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~</math>		<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2} \frac{\mu_e^2}{B_F^2} \biggl[ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \mu_e^4 } \biggr]^{ 1 / 2}
			\biggl[ \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \biggr]^2 \biggl[\frac{3h^3}{2\pi m_e^4 c^5} \biggr]^{3 / 2}
			</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~
	~~</div>~~

	<~~div~~ align="center">		<tr>
	<math>A_\~~mathrm~~{F} \~~equiv~~ \frac{\~~pi m_e~~^~~4 c^5~~}{3h^3} ~~</math><br />~~		<td align="right">
	~~<math>~~\~~frac~~{B_\~~mathrm~~{F}}{\mu_e} \~~equiv~~ \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \~~, ,~~</math>
	</~~div~~>		</td>
			<td align="center">
	~~we have,~~		<math>~=</math>
	<~~div align="center">~~		</td>
	~~<table border="0" cellpadding="5" align="center"~~>		<td align="left">
			<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2} \frac{\mu_e^2}{B_F^2}
			\biggl\{ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \mu_e^4 } \cdot
			\frac{2^{12} \pi^4 m_p^4}{3^4} \biggl( \frac{m_e c}{h} \biggr)^{12} \cdot \frac{3^3h^9}{2^3\pi^3 m_e^{12} c^{15}}
			\biggr\}^{1 / 2}
			</math>
			</td>
			</tr>

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~ M_\mathrm{tot}</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2} \frac{\mu_e^2}{B_F^2} \biggl[ \frac{~~3 h^3 c^3~~ }{~~2^9\pi~~ m_u^4 \~~mu_3^4 }~~ ~~\biggr]^{ 1 / 2}~~		<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2} \frac{\mu_e^2}{B_F^2}
	~~\biggl[ \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h~~} ~~\biggr)^3~~ \biggr]^~~2 \biggl[\frac{3h^3}~~{2~~\pi m_e^4 c^5~~} \~~biggr]^{3 / 2}~~		\biggl[ \frac{ m_p }{m_u \mu_e } \biggr]^{2} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* White Dwarfs */

2016-10-02T03:34:29Z

White Dwarfs

← Older revision		Revision as of 03:34, 2 October 2016
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~4\pi \biggl[\frac{ K_3 }{\pi G}\biggr]^{3/2} ~~m_3~~		<math>~4\pi m_3 \biggl[\frac{ K_3 }{\pi G}\biggr]^{3/2} \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			</table>
			</div>

			Separately, [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] show that the effective polytropic constant for a relativistic electron gas is (see their equation 2.3.23),

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

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~ ~~		<math>~K_3</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[\frac{2^4 ~~K_3~~^3 }{\pi G^3}\biggr]^{1/2} ~~m_3~~ \, .		<math>~
			\biggl( \frac{3\pi^2}{2^6}\biggr)^{1/ 3} \biggl[ \frac{ \hbar^3 c^3 }{ m_u^4 \mu_3^4 } \biggr]^{ 1 / 3}
			= \biggl[ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \mu_3^4 } \biggr]^{ 1 / 3} \, .
	</math>		</math>
	</td>		</td>
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	</div>		</div>

	~~Separately,~~ [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] ~~show that the effective polytropic constant for a relativistic electron gas is (see their equation~~ 2.3.23),		<table border="1" cellpadding="5" align="center">
			<tr>
			<td align="center">
			Equation extracted from §2.3 (p.27) of [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>]
			</td>
			</tr>
			<tr>
			<td align="center">
			[[File:ST83Eq2_3_23.png\|600px\|Equation 2.3.23 from ST83]]
			</td>
			</tr>
			</table>

			Together, then, the [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>] analysis gives,

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~K_3~~</math>		<math>~ M_\mathrm{tot}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 285:		Line 307:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~4\pi m_3 \biggl[\frac{ 1 }{\pi G}\biggr]^{3/2} \biggl[ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \mu_3^4 } \biggr]^{ 1 / 2}
	\biggl[\biggl( \frac{3~~\pi~~^2}{2^6}\biggr)^{1/ 3} ~~\biggr]~~
	</math>		</math>
	</td>		</td>
Line 292:		Line 313:
	</table>		</table>
	</div>		</div>
			And, given that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

	~~<table border="1" cellpadding="5" align="center">~~
	<tr>		<tr>
			<td align="right">
			<math>~</math>
			</td>
	<td align="center">		<td align="center">
	~~Equation extracted from §2.3 (p.27) of [~~<b>~~[[User:Tohline~~/~~Appendix~~/~~References#ST83\|~~<~~font color~~="~~red~~">~~ST83~~<~~/font~~>]]</b>]		<math>~=</math>
			</td>
			<td align="left">
			<math>~</math>
	</td>		</td>
	</tr>		</tr>
			</table>
			</div>

			<div align="center">
			<math>A_\mathrm{F} \equiv \frac{\pi m_e^4 c^5}{3h^3} </math><br />
			<math>\frac{B_\mathrm{F}}{\mu_e} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \, ,</math>
			</div>

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

	<tr>		<tr>
			<td align="right">
			<math>~ M_\mathrm{tot}</math>
			</td>
	<td align="center">		<td align="center">
	[[~~File:ST83Eq2_3_23.png\|600px\|Equation~~ 2.3~~.23 from ST83~~]]		<math>~=</math>
			</td>
			<td align="left">
			<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2} \frac{\mu_e^2}{B_F^2} \biggl[ \frac{3 h^3 c^3 }{2^9\pi m_u^4 \mu_3^4 } \biggr]^{ 1 / 2}
			\biggl[ \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \biggr]^2 \biggl[\frac{3h^3}{2\pi m_e^4 c^5} \biggr]^{3 / 2}
			</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			</div>



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

← Older revision		Revision as of 01:09, 4 August 2019
Line 450:		Line 450:
	=See Also=		=See Also=

	* [https://ui.adsabs.harvard.edu/abs/1930LEDPM...9..944S/abstract E. Stoner (1930)], The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 7, Volume 9, Issue 60, p. 944-963: ''The Equilibrium of Dense Stars''		* [https://ui.adsabs.harvard.edu/abs/1930LEDPM...9..944S/abstract Edmund C. Stoner (1930)], The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 7, Volume 9, Issue 60, p. 944-963: ''The Equilibrium of Dense Stars''

	* [https://ui.adsabs.harvard.edu/abs/1931ApJ....74...81C/abstract S. Chandrasekhar (1931)], ApJ, 74, p. 81: ''The Maximum Mass of Ideal White Dwarfs''		* [https://ui.adsabs.harvard.edu/abs/1931ApJ....74...81C/abstract S. Chandrasekhar (1931)], ApJ, 74, p. 81: ''The Maximum Mass of Ideal White Dwarfs''

← Older revision		Revision as of 01:03, 4 August 2019
Line 449:		Line 449:

	=See Also=		=See Also=

			* [https://ui.adsabs.harvard.edu/abs/1930LEDPM...9..944S/abstract E. Stoner (1930)], The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 7, Volume 9, Issue 60, p. 944-963: ''The Equilibrium of Dense Stars''

			* [https://ui.adsabs.harvard.edu/abs/1931ApJ....74...81C/abstract S. Chandrasekhar (1931)], ApJ, 74, p. 81: ''The Maximum Mass of Ideal White Dwarfs''

	* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer & D. Lynden-Bell (1966)], Phys. Rev. Letters, 17, 816: ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''		* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer & D. Lynden-Bell (1966)], Phys. Rev. Letters, 17, 816: ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''