User:Tohline/Appendix/Ramblings/Radiation/InitialTemperatures - Revision history

Tohline: /* Other Physical Variables */ Delete extraneous paragraph and equation

2010-08-16T18:57:42Z

Other Physical Variables: Delete extraneous paragraph and equation

← Older revision		Revision as of 18:57, 16 August 2010
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	</math>		</math>
	</div>		</div>


	~~— set by values at the center of our accretor (<math>\beta_c = 0.99352</math>; <math>\Gamma_1 = 5/3</math>) — is,~~
	~~<div align="center">~~
	~~<math>~~
	~~\gamma_\mathrm{crit} = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2} \biggr]_c = 1.67765 .~~
	~~</math>~~
	~~</div>~~


	Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into these expressions lets us tabulate various properties at the center of both stars.		Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into these expressions lets us tabulate various properties at the center of both stars.

Tohline: /* Other Physical Variables */ Clean up table and related discussion

2010-08-16T18:41:41Z

Other Physical Variables: Clean up table and related discussion

← Older revision		Revision as of 18:41, 16 August 2010
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	===Other Physical Variables===		===Other Physical Variables===

	The ratio of radiation pressure to gas pressure ~~can be~~ calculated via the relation,		The ratio of radiation pressure to gas pressure (see the last column of the above table) is calculated via the relation,
	<div align="center">		<div align="center">
	<math>		<math>
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	</div>		</div>

	In order to avoid establishing stellar structures that are convectively unstable, Dominic also needs to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value, <math>\gamma_c</math>, established at the center of the accretor. From Equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|Chandrasekhar (1967)]] we see that ~~the critical value of~~ <math>\~~gamma~~</math> depends on each star's central value of <math>\beta</math> — that is, it depends on <math>\beta_c</math> — and on each star's ''structural'' <math>\Gamma_1 \equiv d\ln P/d\ln \rho</math> (which is <math>5/3</math> for our two {{User:Tohline/Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars) in the following way:		In order to avoid establishing stellar structures that are convectively unstable, Dominic also needs to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value, <math>\gamma_c</math>, established at the center of the accretor. From Equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|Chandrasekhar (1967)]] we see that <math>\gamma_c</math> depends on each star's central value of <math>\beta</math> — that is, it depends on <math>\beta_c</math> — and on each star's ''structural'' <math>\Gamma_1 \equiv d\ln P/d\ln \rho</math> (which is <math>5/3</math> for our two {{User:Tohline/Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars) in the following way:
	<div align="center">		<div align="center">
	<math>		<math>
	\~~gamma_\mathrm{crit}~~ = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2} \biggr]_c.		\gamma_c = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2} \biggr].
	</math>		</math>
	</div>		</div>
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	Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into ~~the~~ these expressions lets us ~~identify~~ properties at the center of both stars~~; note that,~~		Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into these expressions lets us tabulate various properties at the center of both stars.

	<table align="center" border="1" cellpadding="5" width="75%">		<table align="center" border="1" cellpadding="5" width="75%">
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	</td>		</td>
	<td align="center" colspan="4">		<td align="center" colspan="3">
	Approximations		Approximations
	</td>		</td>
	<td align="center" colspan="4">		<td align="center" colspan="5">
	From Quartic Solution		From Quartic Solution
	</td>		</td>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>T_\mathrm{code}~~\|_c</math>~~		<math>T_\mathrm{code}^\mathrm{max}</math>
	~~</td>~~
	~~<td align="center">~~
	~~<math>T_~~\mathrm{~~cgs~~}~~\|_c~~</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="center">		<td align="center">
	<math>T_\mathrm{code}\|_c</math>		<math>T_\mathrm{code}^\mathrm{max}</math>
			</td>
			<td align="center">
			<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr\|_c</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>T_\~~mathrm{cgs}\|_c~~</math>		<math>\beta_c</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\~~frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr\|_c~~</math>		<math>\gamma_c</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\~~beta_c~~</math>		<math>T_\mathrm{cgs}^\mathrm{max}</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="center">		<td align="center">
	<math>0.5843</math>		<math>0.5843</math>
	~~</td>~~
	~~<td align="center">~~
	~~<math>9.45\times 10^{7}~\mathrm{K}</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="center" colspan="1">		<td align="center" colspan="1">
	~~ ~~		<math>6.522\times 10^{-3}</math>
			</td>
			<td align="center" colspan="1">
			0.99352
	</td>		</td>
	<td align="center" colspan="1">		<td align="center" colspan="1">
	~~<math>6~~.~~522\times 10^{-3}</math>~~		1.67765
	</td>		</td>
	<td align="center" colspan="1">		<td align="center" colspan="1">
	0.~~99352~~		<math>9.39\times 10^{7}~\mathrm{K}</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="center">		<td align="center">
	<math>0.3854</math>		<math>0.3854</math>
	~~</td>~~
	~~<td align="center">~~
	~~<math>6.24\times 10^{7}~\mathrm{K}</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>

	<td align="center" colspan="4">		<td align="center" colspan="1">
	~~ ~~		0.3842
			</td>
			<td align="center" colspan="1">
			<math>3.111\times 10^{-3}</math>
			</td>
			<td align="center" colspan="1">
			0.99690
			</td>
			<td align="center" colspan="1">
			1.67188
			</td>
			<td align="center" colspan="1">
			<math>6.22\times 10^{7}~\mathrm{K}</math>
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>

	The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> match exactly the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.)		The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> obtained via the quartic solution match exactly the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.) In this email, Dominic also stated that he chose <math>\gamma = 1.67114094</math>; I'm not quite sure how he derived this value.
	In ~~the~~ email ~~I received from~~ Dominic~~, he~~ stated that he chose <math>\gamma = 1.67114094</math>. I'm not quite sure how he derived this value.


	<br />		<br />
	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Application to Dominic Marcello's Rad-Hydro Models */ Rearrange discussion paragraphs

2010-08-16T17:07:32Z

Application to Dominic Marcello's Rad-Hydro Models: Rearrange discussion paragraphs

← Older revision		Revision as of 17:07, 16 August 2010
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	==Application to Dominic Marcello's Rad-Hydro Models==		==Application to Dominic Marcello's Rad-Hydro Models==
	Currently Dominic is ignoring the effects of electron degeneracy pressure, so in applying ~~this~~ <math>T(\rho)</math> solution to his models we can set <math>F(\chi) = 0</math> in the definition of <math>a_0</math>. Doing this, we find that,		===Temperature===
			Currently Dominic is ignoring the effects of electron degeneracy pressure, so in applying the above <math>T(\rho)</math> solution to his models we can set <math>F(\chi) = 0</math> in the definition of <math>a_0</math>. Doing this, we find that,

	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">
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	</tr>		</tr>
	</table>		</table>

	~~In either case, the ratio of radiation pressure to gas pressure can be calculated via the relation,~~
	~~<div align="center">~~
	~~<math>~~
	~~\frac{1}{\Gamma} = \frac{P_\mathrm{rad}}{P_\mathrm{gas}} = \biggl( \frac{\tilde{a}}{3\tilde{r}} \biggr) \frac{T_\mathrm{code}^3}{\rho_\mathrm{code}} .~~
	~~</math>~~
	~~</div>~~

	Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].		Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].
Line 637:		Line 631:
	</table>		</table>

	~~Plugging~~ <math>\~~rho_~~\mathrm{code}^\mathrm{~~max~~}</math> ~~into the these expressions lets us identify properties at the center of both stars;~~ note that,
			===Other Physical Variables===

			The ratio of radiation pressure to gas pressure can be calculated via the relation,
			<div align="center">
			<math>
			\frac{1}{\Gamma} = \frac{P_\mathrm{rad}}{P_\mathrm{gas}} = \biggl( \frac{\tilde{a}}{3\tilde{r}} \biggr) \frac{T_\mathrm{code}^3}{\rho_\mathrm{code}} .
			</math>
			</div>
			Also note that,
	<div align="center">		<div align="center">
	<math>		<math>
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	</math>		</math>
	</div>		</div>

			In order to avoid establishing stellar structures that are convectively unstable, Dominic also needs to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value, <math>\gamma_c</math>, established at the center of the accretor. From Equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|Chandrasekhar (1967)]] we see that the critical value of <math>\gamma</math> depends on each star's central value of <math>\beta</math> — that is, it depends on <math>\beta_c</math> — and on each star's ''structural'' <math>\Gamma_1 \equiv d\ln P/d\ln \rho</math> (which is <math>5/3</math> for our two {{User:Tohline/Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars) in the following way:
			<div align="center">
			<math>
			\gamma_\mathrm{crit} = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2} \biggr]_c.
			</math>
			</div>


			— set by values at the center of our accretor (<math>\beta_c = 0.99352</math>; <math>\Gamma_1 = 5/3</math>) — is,
			<div align="center">
			<math>
			\gamma_\mathrm{crit} = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2} \biggr]_c = 1.67765 .
			</math>
			</div>


			Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into the these expressions lets us identify properties at the center of both stars; note that,

	<table align="center" border="1" cellpadding="5" width="75%">		<table align="center" border="1" cellpadding="5" width="75%">
Line 758:		Line 779:

	The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> match exactly the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.)		The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> match exactly the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.)

	In order to avoid establishing stellar structures that are convectively unstable, we also need to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value established at the center of the accretor, keeping in mind that the ''structural'' gamma, <math>\Gamma_1 \equiv d\ln P/d\ln \rho = 5/3</math> for our two {{User:Tohline/Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars. From Equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|Chandrasekhar (1967)]] we see that the critical value of <math>\gamma</math> — set by values at the center of our accretor (<math>\beta_c = 0.99352</math>; <math>\Gamma_1 = 5/3</math>) — is,
	~~<div align="center">~~
	~~<math>~~
	~~\gamma_\mathrm{crit} = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2} \biggr]_c = 1.67765 .~~
	~~</math>~~
	~~</div>~~
	In the email I received from Dominic, he stated that he chose <math>\gamma = 1.67114094</math>. I'm not quite sure how he derived this value.		In the email I received from Dominic, he stated that he chose <math>\gamma = 1.67114094</math>. I'm not quite sure how he derived this value.

Tohline: /* Application to Dominic Marcello's Rad-Hydro Models */ Embellish tables

2010-08-16T14:27:38Z

Application to Dominic Marcello's Rad-Hydro Models: Embellish tables

← Older revision		Revision as of 14:27, 16 August 2010
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	<table align="center" border="1" cellpadding="5" width="60%">		<table align="center" border="1" cellpadding="5" width="60%">
	<tr>		<tr>
	<td align="center" colspan="5">		<td align="center" colspan="7">
	<font color="red">Case A:</font>   <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;		<font color="red">Case A:</font>   <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;
	</td>		</td>
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	<td align="center">		<td align="center">
	<math>K_\mathrm{code}</math>		<math>K_\mathrm{code}</math>
			</td>
			<td align="center">
			<math>\theta</math>
			</td>

			<td align="center">
			<math>\lambda</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="center">		<td align="center">
	<math>0.2571</math>		<math>0.2571</math>
			</td>
			<td align="center">
			<math>1.957 \rho_\mathrm{code}^{1/3}</math>
			</td>

			<td align="center">
			<math>0.3980\rho_\mathrm{code}^{1/3}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="center">		<td align="center">
	<math>0.2364</math>		<math>0.2364</math>
			</td>
			<td align="center">
			<math>1.957 \rho_\mathrm{code}^{1/3}</math>
			</td>

			<td align="center">
			<math>0.3660\rho_\mathrm{code}^{1/3}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</table>		</table>

	Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into the these expressions lets us identify properties at the center of both stars.		Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into the these expressions lets us identify properties at the center of both stars; note that,
			<div align="center">
			<math>
			\beta \equiv \frac{P_\mathrm{gas}}{P_\mathrm{total}} = \frac{1}{1+P_\mathrm{rad}/P_\mathrm{gas}} .
			</math>
			</div>

	<table align="center" border="1" cellpadding="5" width="75%">		<table align="center" border="1" cellpadding="5" width="75%">
			<tr>
			<td align="center" colspan="10">
			<b><font color="darkblue">Central Stellar Values</font></b>
			</td>
			</tr>
			<tr>
			<td colspan="2">

			</td>
			<td align="center" colspan="4">
			Approximations
			</td>
			<td align="center" colspan="4">
			From Quartic Solution
			</td>
			</tr>

	<tr>		<tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>\beta_c \~~equiv~~ \~~biggl[~~\frac{~~1}{1 +~~ P_\mathrm{rad}/P_\mathrm{gas}}\biggr]_c</math>		<math>\beta_c</math>
			</td>

			<td align="center">
			<math>T_\mathrm{code}\|_c</math>
			</td>
			<td align="center">
			<math>T_\mathrm{cgs}\|_c</math>
			</td>
			<td align="center">
			<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr\|_c</math>
			</td>
			<td align="center">
			<math>\beta_c</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="center">		<td align="center">
	<math>0.99339</math>		<math>0.99339</math>
			</td>

			<td align="center" colspan="1">
			0.5805
			</td>
			<td align="center" colspan="1">

			</td>
			<td align="center" colspan="1">
			<math>6.522\times 10^{-3}</math>
			</td>
			<td align="center" colspan="1">
			0.99352
	</td>		</td>
	</tr>		</tr>
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	<td align="center">		<td align="center">
	<math>0.99687</math>		<math>0.99687</math>
			</td>

			<td align="center" colspan="4">

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

	The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> match ~~pretty well~~ the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.) An even better match to the accretor is obtained if we use the exact solution to the quartic equation: For <math>\rho_\mathrm{code} = 1.000</math>, we find <math>T_\mathrm{code} = 0.5805</math> and <math>P_\mathrm{rad}/P_\mathrm{gas} = 6.522\times 10^{-3}</math> (and <math>\beta_c = 0.99352</math>).		The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> match exactly the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.)

	In order to avoid establishing stellar structures that are convectively unstable, we also need to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value established at the center of the accretor, keeping in mind that the ''structural'' gamma, <math>\Gamma_1 \equiv d\ln P/d\ln \rho = 5/3</math> for our two {{User:Tohline/Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars. From Equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|Chandrasekhar (1967)]] we see that the critical value of <math>\gamma</math> — set by values at the center of our accretor (<math>\beta_c = 0.99352</math>; <math>\Gamma_1 = 5/3</math>) — is,		In order to avoid establishing stellar structures that are convectively unstable, we also need to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value established at the center of the accretor, keeping in mind that the ''structural'' gamma, <math>\Gamma_1 \equiv d\ln P/d\ln \rho = 5/3</math> for our two {{User:Tohline/Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars. From Equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|Chandrasekhar (1967)]] we see that the critical value of <math>\gamma</math> — set by values at the center of our accretor (<math>\beta_c = 0.99352</math>; <math>\Gamma_1 = 5/3</math>) — is,

Tohline: /* Application to Dominic Marcello's Rad-Hydro Models */ Add information on central beta and limiting adiabatic exponent

2010-08-16T03:41:40Z

Application to Dominic Marcello's Rad-Hydro Models: Add information on central beta and limiting adiabatic exponent

← Older revision		Revision as of 03:41, 16 August 2010
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	<td align="center">		<td align="center">
	<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr\|_c</math>		<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr\|_c</math>
			</td>
			<td align="center">
			<math>\beta_c \equiv \biggl[\frac{1}{1 + P_\mathrm{rad}/P_\mathrm{gas}}\biggr]_c</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="center">		<td align="center">
	<math>6.65 \times 10^{-3}</math>		<math>6.65 \times 10^{-3}</math>
			</td>
			<td align="center">
			<math>0.99339</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="center">		<td align="center">
	<math>3.14\times 10^{-3}</math>		<math>3.14\times 10^{-3}</math>
			</td>
			<td align="center">
			<math>0.99687</math>
	</td>		</td>
	</tr>		</tr>
Line 676:		Line 685:
	</table>		</table>

	The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> match pretty well the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.) An even better match to the accretor is obtained if we use the exact solution to the quartic equation: For <math>\rho_\mathrm{code} = 1.000</math>, we find <math>T_\mathrm{code} = 0.5805</math> and <math>P_\mathrm{rad}/P_\mathrm{gas} = 6.522\times 10^{-3}</math>.		The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> match pretty well the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.) An even better match to the accretor is obtained if we use the exact solution to the quartic equation: For <math>\rho_\mathrm{code} = 1.000</math>, we find <math>T_\mathrm{code} = 0.5805</math> and <math>P_\mathrm{rad}/P_\mathrm{gas} = 6.522\times 10^{-3}</math> (and <math>\beta_c = 0.99352</math>).

			In order to avoid establishing stellar structures that are convectively unstable, we also need to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value established at the center of the accretor, keeping in mind that the ''structural'' gamma, <math>\Gamma_1 \equiv d\ln P/d\ln \rho = 5/3</math> for our two {{User:Tohline/Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars. From Equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|Chandrasekhar (1967)]] we see that the critical value of <math>\gamma</math> — set by values at the center of our accretor (<math>\beta_c = 0.99352</math>; <math>\Gamma_1 = 5/3</math>) — is,
			<div align="center">
			<math>
			\gamma_\mathrm{crit} = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2} \biggr]_c = 1.67765 .
			</math>
			</div>
			In the email I received from Dominic, he stated that he chose <math>\gamma = 1.67114094</math>. I'm not quite sure how he derived this value.


	<br />		<br />
	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Application to Dominic Marcello's Rad-Hydro Models */ Central values

2010-08-16T02:18:17Z

Application to Dominic Marcello's Rad-Hydro Models: Central values

← Older revision		Revision as of 02:18, 16 August 2010
Line 553:		Line 553:
	Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].		Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].

	<table align="center" border="1" cellpadding="5" width="75%">		<table align="center" border="1" cellpadding="5" width="60%">
	<tr>		<tr>
	<td align="center" colspan="6">		<td align="center" colspan="5">
	<font color="red">Case A:</font>   <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;		<font color="red">Case A:</font>   <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;
	</td>		</td>
Line 566:		Line 566:
	<td align="center">		<td align="center">
	<math>K_\mathrm{code}</math>		<math>K_\mathrm{code}</math>
	~~</td>~~
	~~<td align="center">~~
	~~<math>\rho^\mathrm{max}_\mathrm{code}</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
Line 587:		Line 584:
	<td align="center">		<td align="center">
	<math>0.2571</math>		<math>0.2571</math>
	~~</td>~~
	~~<td align="center">~~
	~~<math>1.0000</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
Line 608:		Line 602:
	<td align="center">		<td align="center">
	<math>0.2364</math>		<math>0.2364</math>
			</td>
			<td align="center">
			<math>20.4</math>
			</td>
			<td align="center">
			<math>0.5373~\rho_\mathrm{code}^{2/3}</math>
			</td>
			<td align="center">
			<math>5.17\times 10^{-3}~\rho_\mathrm{code}</math>
			</td>
			</tr>

			</table>

			Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into the these expressions lets us identify properties at the center of both stars.

			<table align="center" border="1" cellpadding="5" width="75%">

			<tr>
			<td align="center">
			'''Star'''
			</td>
			<td align="center">
			<math>\rho^\mathrm{max}_\mathrm{code}</math>
			</td>
			<td align="center">
			<math>T_\mathrm{code}\|_c</math>
			</td>
			<td align="center">
			<math>T_\mathrm{cgs}\|_c</math>
			</td>
			<td align="center">
			<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr\|_c</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			Accretor
			</td>
			<td align="center">
			<math>1.0000</math>
			</td>
			<td align="center">
			<math>0.5843</math>
			</td>
			<td align="center">
			<math>9.45\times 10^{7}~\mathrm{K}</math>
			</td>
			<td align="center">
			<math>6.65 \times 10^{-3}</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			Donor
	</td>		</td>
	<td align="center">		<td align="center">
Line 613:		Line 664:
	</td>		</td>
	<td align="center">		<td align="center">
	<math>20.4</math>		<math>0.3854</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>0.~~5373~~~~~\rho_~~\mathrm{~~code}^{2/3~~}</math>		<math>6.24\times 10^{7}~\mathrm{K}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>5.17\times 10^{-3~~}~\rho_\mathrm{code~~}</math>		<math>3.14\times 10^{-3}</math>
	</td>		</td>
	</tr>		</tr>
Line 625:		Line 676:
	</table>		</table>

			The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> match pretty well the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor). (See email from Dominic to Joel dated 8/4/2010.) An even better match to the accretor is obtained if we use the exact solution to the quartic equation: For <math>\rho_\mathrm{code} = 1.000</math>, we find <math>T_\mathrm{code} = 0.5805</math> and <math>P_\mathrm{rad}/P_\mathrm{gas} = 6.522\times 10^{-3}</math>.

	<br />		<br />
	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Application to Dominic Marcello's Rad-Hydro Models */ Enter corrected value of polytropic constant into table!

2010-08-15T19:37:30Z

Application to Dominic Marcello's Rad-Hydro Models: Enter corrected value of polytropic constant into table!

← Older revision		Revision as of 19:37, 15 August 2010
Line 586:		Line 586:
	</td>		</td>
	<td align="center">		<td align="center">
	<math>0.~~02732~~</math>		<math>0.2571</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 607:		Line 607:
	</td>		</td>
	<td align="center">		<td align="center">
	<math>0.~~02512~~</math>		<math>0.2364</math>
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Application to Dominic Marcello's Rad-Hydro Models */ Plug in corrected values of polytropic constant

2010-08-15T19:34:00Z

Application to Dominic Marcello's Rad-Hydro Models: Plug in corrected values of polytropic constant

← Older revision		Revision as of 19:34, 15 August 2010
Line 592:		Line 592:
	</td>		</td>
	<td align="center">		<td align="center">
	<math>1.~~322\times 10^{4}~~</math>		<math>15.86</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>0.~~06209~~~\rho_\mathrm{code}^{2/3}</math>		<math>0.5843~\rho_\mathrm{code}^{2/3}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>7.98\times 10^{-6}~\rho_\mathrm{code}</math>		<math>6.65\times 10^{-3}~\rho_\mathrm{code}</math>
	</td>		</td>
	</tr>		</tr>
Line 613:		Line 613:
	</td>		</td>
	<td align="center">		<td align="center">
	<math>1.~~700\times 10^~~4</math>		<math>20.4</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>0.~~05709~~~\rho_\mathrm{code}^{2/3}</math>		<math>0.5373~\rho_\mathrm{code}^{2/3}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>6.~~202~~\times 10^{-6}~\rho_\mathrm{code}</math>		<math>5.17\times 10^{-3}~\rho_\mathrm{code}</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Application to Dominic Marcello's Rad-Hydro Models */ Add more columns of information to table

2010-08-15T03:42:18Z

Application to Dominic Marcello's Rad-Hydro Models: Add more columns of information to table

← Older revision		Revision as of 03:42, 15 August 2010
Line 543:		Line 543:
	</tr>		</tr>
	</table>		</table>

			In either case, the ratio of radiation pressure to gas pressure can be calculated via the relation,
			<div align="center">
			<math>
			\frac{1}{\Gamma} = \frac{P_\mathrm{rad}}{P_\mathrm{gas}} = \biggl( \frac{\tilde{a}}{3\tilde{r}} \biggr) \frac{T_\mathrm{code}^3}{\rho_\mathrm{code}} .
			</math>
			</div>

	Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].		Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].

	<table align="center" border="1" cellpadding="5" width="50%">		<table align="center" border="1" cellpadding="5" width="75%">
	<tr>		<tr>
	<td align="center" colspan="5">		<td align="center" colspan="6">
	<font color="red">Case A:</font>   <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;		<font color="red">Case A:</font>   <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;
	</td>		</td>
Line 568:		Line 575:
	<td align="center">		<td align="center">
	<math>T_\mathrm{code}</math> (for <math>\rho_\mathrm{code} \ll \rho_1</math>)		<math>T_\mathrm{code}</math> (for <math>\rho_\mathrm{code} \ll \rho_1</math>)
			</td>
			<td align="center">
			<math>P_\mathrm{rad}/P_\mathrm{gas}</math> (for <math>\rho_\mathrm{code} \ll \rho_1</math>)
	</td>		</td>
	</tr>		</tr>
Line 586:		Line 596:
	<td align="center">		<td align="center">
	<math>0.06209~\rho_\mathrm{code}^{2/3}</math>		<math>0.06209~\rho_\mathrm{code}^{2/3}</math>
			</td>
			<td align="center">
			<math>7.98\times 10^{-6}~\rho_\mathrm{code}</math>
	</td>		</td>
	</tr>		</tr>
Line 604:		Line 617:
	<td align="center">		<td align="center">
	<math>0.05709~\rho_\mathrm{code}^{2/3}</math>		<math>0.05709~\rho_\mathrm{code}^{2/3}</math>
			</td>
			<td align="center">
			<math>6.202\times 10^{-6}~\rho_\mathrm{code}</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Application to Dominic Marcello's Rad-Hydro Models */ Plug in numerical values

2010-08-15T02:58:42Z

Application to Dominic Marcello's Rad-Hydro Models: Plug in numerical values

← Older revision		Revision as of 02:58, 15 August 2010
Line 542:		Line 542:
	</td>		</td>
	</tr>		</tr>
			</table>

			Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings#TableProperties\|accompanying ''properties'' table]].

			<table align="center" border="1" cellpadding="5" width="50%">
			<tr>
			<td align="center" colspan="5">
			<font color="red">Case A:</font>   <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;
			</td>
			</tr>

			<tr>
			<td align="center">
			'''Star'''
			</td>
			<td align="center">
			<math>K_\mathrm{code}</math>
			</td>
			<td align="center">
			<math>\rho^\mathrm{max}_\mathrm{code}</math>
			</td>
			<td align="center">
			<math>\rho_1</math>
			</td>
			<td align="center">
			<math>T_\mathrm{code}</math> (for <math>\rho_\mathrm{code} \ll \rho_1</math>)
			</td>
			</tr>

			<tr>
			<td align="center">
			Accretor
			</td>
			<td align="center">
			<math>0.02732</math>
			</td>
			<td align="center">
			<math>1.0000</math>
			</td>
			<td align="center">
			<math>1.322\times 10^{4}</math>
			</td>
			<td align="center">
			<math>0.06209~\rho_\mathrm{code}^{2/3}</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			Donor
			</td>
			<td align="center">
			<math>0.02512</math>
			</td>
			<td align="center">
			<math>0.6077</math>
			</td>
			<td align="center">
			<math>1.700\times 10^4</math>
			</td>
			<td align="center">
			<math>0.05709~\rho_\mathrm{code}^{2/3}</math>
			</td>
			</tr>

	</table>		</table>