User:Tohline/Apps/SMS - Revision history

Tohline: /* Polytropic Index Slightly Less Than 3 */ Tag the approximation to n_p

2015-12-21T19:59:10Z

Polytropic Index Slightly Less Than 3: Tag the approximation to n_p

← Older revision		Revision as of 19:59, 21 December 2015
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	Now, from above we have,		Now, from [[#GammaApprox\|above]] we have,
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	This matches exactly expression (7) in BAC84. Again from above — and continuing to assume <math>~\sigma \gg 1</math> — we have,		This matches exactly expression (7) in BAC84. Again from [[#GammaApprox\|above]] — and continuing to assume <math>~\sigma \gg 1</math> — we have,
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Used by BAC84 */

2015-12-21T19:57:26Z

Used by BAC84

← Older revision		Revision as of 19:57, 21 December 2015
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	</table>		</table>
	</div>		</div>
	They also point out that, in the case where a system is dominated by radiation pressure <math>~(\sigma \gg 1)</math>, this expression becomes,		<span id="GammaApprox">They also point out that, in the case where a system is dominated by radiation pressure <math>~(\sigma \gg 1)</math>, this expression becomes,</span>
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Expression for Adiabatic Exponent */ Tidy up

2015-12-21T19:52:56Z

Expression for Adiabatic Exponent: Tidy up

← Older revision		Revision as of 19:52, 21 December 2015
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	====For any value of the ratio of specific heats====		====For any value of the ratio of specific heats====
	From equation (2) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124) — see, for example, [[User:Tohline/SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,		From equation (2) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124) — see, for example, [[User:Tohline/SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — or, equally well, from equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|<b>[<font color="red">C67</font>]</b>]], we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,
	<div align="center">		<div align="center">
	<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>		<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>
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	</div>		</div>
	They also point out that, in the case where a system is dominated by radiation pressure (<math>~\sigma \gg 1</math>), this expression becomes,		They also point out that, in the case where a system is dominated by radiation pressure <math>~(\sigma \gg 1)</math>, this expression becomes,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Expression for Adiabatic Exponent */ Table of 3 equivalent expressions

2015-12-20T17:57:20Z

Expression for Adiabatic Exponent: Table of 3 equivalent expressions

← Older revision		Revision as of 17:57, 20 December 2015
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	===Expression for Adiabatic Exponent===		===Expression for Adiabatic Exponent===
			<div align="center">
			<table border="1" cellpadding="8" align="center">
			<tr><th align="center">Three Equivalent Expressions</th>
			</tr>
			<tr><td align="center">

			<!-- BEGIN TABLE LAYOUT -->
			<table border="0" cellpadding="5" align="center">

			<tr><td align="center" colspan="3">
			Equation (24) from [http://adsabs.harvard.edu/abs/1918MNRAS..79R...2E A. S. Eddington (1918, MNRAS, 79, 2)]
			</td></tr>
			<tr>
			<td align="right">
			<math>~\Gamma_1 - \frac{4}{3}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{(\gamma-\tfrac{4}{3})(4-3\beta)}{1 + 12(\gamma-1) (1 - \beta)/\beta} </math>
			</td>
			</tr>

			<tr><td align="center" colspan="3">
			----
			</td></tr>
			<tr><td align="center" colspan="3">
			Equation (131) from Chapter II of [[User:Tohline/Appendix/References#C67\|<b>[<font color="red">C67</font>]</b>]]
			</td></tr>
			<tr>
			<td align="right">
			<math>~\Gamma_1</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\beta + \frac{(\gamma-1)(4-3\beta)^2}{\beta + 12(\gamma-1) (1 - \beta)} </math>
			</td>
			</tr>
			<tr><td align="center" colspan="3">
			----
			</td></tr>
			<tr><td align="center" colspan="3">
			[[#GeneralizingBAC84\|Derived below]] … noting that <math>~\beta = (1+\sigma)^{-1}</math>
			</td></tr>

			<tr>
			<td align="right">
			<math>~\Gamma_1 - \frac{4}{3}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{(3\gamma -4)(1+4\sigma)}{3(1+\sigma)[1 + 12(\gamma-1)\sigma]} </math>
			</td>
			</tr>
			</table>

			<!-- END TABLE LAYOUT -->
			</td></tr>
			</table>
			</div>


	====For any value of the ratio of specific heats====		====For any value of the ratio of specific heats====
	From equation (2) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124) — see, for example, [[User:Tohline/SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,		From equation (2) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124) — see, for example, [[User:Tohline/SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,
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	</div>		</div>

	Or, even more generally, we can show that,		<span id="GeneralizingBAC84">Or, even more generally, we can show that,</span>
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Used by BAC84 */ Even more general expression in terms of sigma and gamma

2015-12-20T00:10:23Z

Used by BAC84: Even more general expression in terms of sigma and gamma

← Older revision		Revision as of 00:10, 20 December 2015
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			Or, even more generally, we can show that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1 - \frac{4}{3}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{(3\gamma -4)(1+4\sigma)}{3(1+\sigma)[1 + 12(\gamma-1)\sigma]} \, .</math>
			</td>
			</tr>
			</table>
			</div>

Tohline: /* Expression for Adiabatic Exponent */ show how Ledoux and Pekeris matches BAC84

2015-12-19T23:49:03Z

Expression for Adiabatic Exponent: show how Ledoux and Pekeris matches BAC84

Show changes

Tohline: /* Equation of State */

2015-12-19T19:29:50Z

Equation of State

← Older revision		Revision as of 19:29, 19 December 2015
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	===Expression for Adiabatic Exponent===		===Expression for Adiabatic Exponent===
			====For any value of the ratio of specific heats====
	From equation (2) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124) — see, for example, [[User:Tohline/SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,		From equation (2) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124) — see, for example, [[User:Tohline/SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,
	<div align="center">		<div align="center">
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	is,		is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\beta + \frac{(\gamma-1)(4-3\beta)^2}{\beta + 12(\gamma-1) (1 - \beta)} \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where, <math>~\gamma</math> is the ratio of specific heats associated with the ideal-gas component of the equation of state. Notice that <math>~\beta = 1 </math> represents a situation where there is no radiation pressure. In this limit the expression simplifies to,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1\biggr\|_{\beta=1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\gamma \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			which makes sense. On the other hand, setting <math>~\beta = 0</math> represents the other extreme, where there is no ideal-gas contribution to the pressure. In this case, we have,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1\biggr\|_{\beta=0}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{16(\gamma-1)}{12(\gamma-1) } = \frac{4}{3} \, .</math>
			</td>
			</tr>
			</table>
			</div>

			====Used by BAC84====
			On the other hand, without derivation BAC84 state (see their equation 4) that the adiabatic exponent is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{4}{3} + \frac{4\sigma + 1}{3(\sigma+1)(8\sigma + 1)} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			They also point out that, in the case where a system is dominated by radiation pressure, <math>~\sigma \gg 1</math>, and this expression becomes,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1\biggr\|_{\sigma \gg 1}</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~\frac{4}{3} + \frac{1}{6\sigma} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Clearly, in the limit <math>~\sigma \rightarrow \infty</math>, this gives <math>~\Gamma_1 \rightarrow 4/3</math>, which, as it should, matches the limiting value obtained from the Ledoux & Pekeris expression when <math>~\beta = 0</math>.

			BAC84 do not explicitly state what value of the ratio of specific heats they used in deriving their expression for the adiabatic exponent. But this can be deduced by examining how their expression behaves in the limit of no radiation pressure, that is, for <math>~\sigma = 0</math>.


			<!-- COMMENT OUT BRIEF SECTION
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	</table>		</table>
	</div>		</div>
			END OF COMMENTED OUT SECTION -->

Tohline: /* Equation of State */

2015-12-19T00:09:35Z

Equation of State

← Older revision		Revision as of 00:09, 19 December 2015
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\beta + \frac{(4-3\beta)^2 (\~~gamma~~-1)}{\beta ~~+ 12~~(\gamma - 1)(1 - \~~beta~~)} \~~, .~~</math>		<math>~\beta + \frac{g(4-3\beta)^2}{\beta + 12g (1 - \beta)} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\beta + \frac{g}{\beta}\biggl[4-3\beta\biggr]^2\biggl[1 + \frac{12g (1 - \beta)}{\beta}\biggr]^{-1} \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where we are using the temporary shorthand notation, <math>g \equiv (\gamma - 1)</math>. Replacing <math>~\beta</math> with its expression in terms of <math>~\sigma</math> gives us,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{(1+\sigma)} + g(1+\sigma)\biggl[4-\frac{3}{(1+\sigma)}\biggr]^2\biggl[1 + 12g \sigma\biggr]^{-1} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{(1+\sigma)} + \frac{g[4(1+\sigma)-3]^2 }{(1 + 12g \sigma)} </math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Equation of State */ Begin subsection that presents expression for adiabatic exponent

2015-12-18T21:14:45Z

Equation of State: Begin subsection that presents expression for adiabatic exponent

← Older revision		Revision as of 21:14, 18 December 2015
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	which is precisely the definition provided in equation (5) of BAC84.		which is precisely the definition provided in equation (5) of BAC84.

			===Expression for Adiabatic Exponent===
			From equation (2) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124) — see, for example, [[User:Tohline/SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,
			<div align="center">
			<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>
			</div>
			is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\beta + \frac{(4-3\beta)^2 (\gamma-1)}{\beta + 12(\gamma - 1)(1 - \beta)} \, .</math>
			</td>
			</tr>
			</table>
			</div>


	===Mass Normalization===		===Mass Normalization===

Tohline: /* Polytropic Index Slightly Less Than 3 */

2015-12-18T19:41:23Z

Polytropic Index Slightly Less Than 3

← Older revision		Revision as of 19:41, 18 December 2015
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	\approx \biggl( 1 - \frac{3}{4\sigma^2}\biggr) (n m_B)^{1/8\sigma^2}\, ,</math>		\approx \biggl( 1 - \frac{3}{4\sigma^2}\biggr) (n m_B)^{1/8\sigma^2}\, ,</math>
	</div>		</div>
	which certainly is close to unity when <math>~\sigma \gg 1</math>. After ~~settomg~~ <math>~f=1</math>, this last expression for <math>~M_u</math> exactly matches the expression presented as equation (9) in BAC84.		which certainly is close to unity when <math>~\sigma \gg 1</math>. After setting <math>~f=1</math>, this last expression for <math>~M_u</math> exactly matches the expression presented as equation (9) in BAC84.



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