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

Tohline: /* Ratio of Gas Pressure to Radiation Pressure */ Central temperature of polytrope

2010-08-12T20:25:13Z

Ratio of Gas Pressure to Radiation Pressure: Central temperature of polytrope

← Older revision		Revision as of 20:25, 12 August 2010
Line 1,084:		Line 1,084:
	</math>		</math>
	</div>		</div>

			We might, in addition, ask what the central temperature is in an <math>n=3/2</math> polytrope. Well, if the gas pressure dominates (''i.e.,'' if <math>\Gamma \gg 1</math>),
			<div align="center">
			<math>
			P_\mathrm{cgs} \approx \frac{\Re}{\bar\mu} \rho_\mathrm{cgs} T_\mathrm{cgs};
			</math>
			</div>
			and in code units,
			<div align="center">
			<math>
			P_\mathrm{code} = \kappa_\mathrm{code} \rho_\mathrm{code}^{5/3} .
			</math>
			</div>

			Hence,

			<table align="center" border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>
			T_\mathrm{code}
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} T_\mathrm{cgs} \approx \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \frac{1}{(\Re/\bar\mu)} \biggl(\frac{P_\mathrm{cgs}}{\rho_\mathrm{cgs}}\biggr)

			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\frac{1}{(\Re/\bar\mu)} \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \biggl(\frac{P_\mathrm{cgs}/P_\mathrm{code}}{\rho_\mathrm{cgs}/\rho_\mathrm{code}}\biggr) \biggl(\frac{P_\mathrm{code}}{\rho_\mathrm{code}}\biggr)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\frac{1}{(\Re/\bar\mu)} \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{2}\biggl( \frac{t_\mathrm{cgs}}{t_\mathrm{code}} \biggr)^{-2} \kappa_\mathrm{code}\rho_\mathrm{code}^{2/3}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\frac{1}{\tilde{r}} ~\kappa_\mathrm{code}\rho_\mathrm{code}^{2/3} .
			</math>
			</td>
			</tr>

			</table>

			This, in turn, tells us that at the center of a polytropic star,
			<div align="center">
			<math>
			\Gamma \approx \frac{3\tilde{r}^4}{\tilde{a}}~\kappa_\mathrm{code}^{-3} \rho_\mathrm{code}^{-1} .
			</math>
			</div>

			This derivation will need to be modified to handle the more general case when <math>\Gamma</math> is not necessarily large.

	==Super-Eddington Accretion==		==Super-Eddington Accretion==

Tohline: /* Ratio of Gas Pressure to Radiation Pressure */ Correct and expand discussion of the ratio of radiation pressure to gas pressure

2010-08-12T19:30:08Z

Ratio of Gas Pressure to Radiation Pressure: Correct and expand discussion of the ratio of radiation pressure to gas pressure

← Older revision		Revision as of 19:30, 12 August 2010
Line 1,065:		Line 1,065:
	<div align="center">		<div align="center">
	<math>		<math>
	\Gamma \equiv \frac{P_\mathrm{gas}}{P_\mathrm{rad}} = \frac{3 \Re}{a_\mathrm{rad~~} \bar{\mu~~} } \biggl[ \frac{\rho}{T^3} \biggr] ,		\Gamma \equiv \frac{P_\mathrm{gas}}{P_\mathrm{rad}} = \frac{3 (\Re/\bar\mu)}{a_\mathrm{rad}} \biggl[ \frac{\rho}{T^3} \biggr]_\mathrm{cgs} ,
	</math>		</math>
	</div>		</div>
Line 1,075:		Line 1,075:
	</div>		</div>

	Following Dominic's definition of code units, above, <math>T^3</math> should be normalized by <math>( c^6/\Re^3 )</math> ~~and we see that~~ the mass density should be normalized by the quantity <math>(a_\mathrm{rad} c^6/\Re^4)</math>. Hence, the ratio of gas pressure to radiation pressure can be written as,		Following Dominic's definition of code units, above: <math>T^3</math> should be normalized by <math>[ c^6/(\Re/\bar\mu )^3]</math>; the mass density should be normalized by the quantity <math>[a_\mathrm{rad} c^6/(\Re/\bar\mu)^4]</math>; and <math>(\Re/\bar\mu)</math> should be replaced by <math>\tilde{r}</math>. Hence, the ratio of gas pressure to radiation pressure can be written as,
	<div align="center">		<div align="center">
	<math>		<math>
	\Gamma = \frac{3}{\bar{\mu} } \biggl[ \frac{\rho_\mathrm{code}}{T_\mathrm{code}^3} \biggr] \frac{\tilde{r}}{\tilde{a}} .		\Gamma = \frac{3(\Re/\bar\mu)}{a_\mathrm{rad}} \biggl[\frac{\rho}{T^3}\biggr]_\mathrm{code} \biggl[\frac{c^6 a_\mathrm{rad}}{(\Re/\bar\mu)^4}
			\biggl( \frac{\tilde{r}^4}{\tilde{c}^6 \tilde{a}} \biggr)\biggr] \biggl[\frac{(\Re/\bar\mu)^3}{c^6}
			\biggl( \frac{\tilde{c}^6 }{\tilde{r}^3} \biggr)\biggr]
			= 3\biggl[ \frac{\rho_\mathrm{code}}{T_\mathrm{code}^3} \biggr] \frac{\tilde{r}}{\tilde{a}} .
	</math>		</math>
	</div>		</div>


	==Super-Eddington Accretion==		==Super-Eddington Accretion==

Tohline: /* Chandrasekhar Mass and Radius */ Minor modifications to discussion

2010-08-03T13:03:39Z

Chandrasekhar Mass and Radius: Minor modifications to discussion

← Older revision		Revision as of 13:03, 3 August 2010
Line 850:		Line 850:

	===Chandrasekhar Mass and Radius===		===Chandrasekhar Mass and Radius===
	The characteristic mass and ~~length~~ scales that are associated with a self-gravitating, degenerate-electron gas are identified [[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass\|in an accompanying Wiki page]] in the context of our discussion of the structure of spherically symmetric white dwarfs and the Chandrasekhar mass. ~~Both~~ scales depend on the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, and characteristic Fermi density, {{User:Tohline/Math/C_FermiDensity}}, that are familiar to the condensed-matter community. As recorded in our [[User:Tohline/Appendix/Variables_templates\|accompanying variables appendix]], the definition of these two ''condensed-matter relevant'' quantities is, respectively,		====Review====
			The characteristic mass, length, and time scales that are associated with a self-gravitating, degenerate-electron gas are identified [[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass\|in an accompanying Wiki page]] in the context of our discussion of the structure of spherically symmetric white dwarfs and the Chandrasekhar mass. All three of these scales depend on the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, and characteristic Fermi density, {{User:Tohline/Math/C_FermiDensity}}, that are familiar to the condensed-matter community. As recorded in our [[User:Tohline/Appendix/Variables_templates\|accompanying variables appendix]], the definition of these two ''condensed-matter relevant'' quantities is, respectively,
	<div align="center">		<div align="center">
	<math>		<math>
Line 862:		Line 863:
	</math>		</math>
	</div>		</div>
	and the characteristic, ''astrophysically relevant'' mass <math>(M_\mathrm{Ch})</math> and length <math>(\~~ell_1~~)</math> scales [[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass\|identified by Chandrasekhar]] are,		and the characteristic, ''astrophysically relevant'' mass <math>(M_\mathrm{Ch})</math> and length <math>(\ell_\mathrm{Ch})</math> scales [[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass\|identified by Chandrasekhar]] are,
	<div align="center">		<div align="center">
	<math>		<math>
Line 879:		Line 880:
	</div>		</div>

			====Application to Unit Conversion Expressions====

	Rewriting <math>M_\mathrm{Ch}</math> only in terms of the fundamental physical constants, we obtain,		Rewriting <math>M_\mathrm{Ch}</math> only in terms of the fundamental physical constants, we obtain,
Line 980:		Line 981:

	</table>		</table>


	And,		And,

Tohline: /* Chandrasekhar Mass and Radius */ Correct mistake on t_Ch and evaluate various physical constants

2010-08-03T03:40:13Z

Chandrasekhar Mass and Radius: Correct mistake on t_Ch and evaluate various physical constants

← Older revision		Revision as of 03:40, 3 August 2010
Line 853:		Line 853:
	<div align="center">		<div align="center">
	<math>		<math>
	A_\mathrm{F} \equiv \frac{\pi m_e^4 c^5}{3 h^3} ,		A_\mathrm{F} \equiv \frac{\pi m_e^4 c^5}{3 h^3} = 6.00233\times 10^{22}~\mathrm{erg}~\mathrm{cm}^{-3},
	</math>		</math>
	</div>		</div>
Line 859:		Line 859:
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{B_\mathrm{F}}{\mu_e} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 ;		\frac{B_\mathrm{F}}{\mu_e} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 = 9.81019\times 10^{5}~\mathrm{g}~\mathrm{cm}^{-3};
	</math>		</math>
	</div>		</div>
Line 865:		Line 865:
	<div align="center">		<div align="center">
	<math>		<math>
	\mu_e^2 M_\mathrm{Ch} = 4\pi m_3 \biggl(\frac{2 A_\mathrm{F}}{\pi G}\biggr)^{3/2} \frac{\mu_e^2}{B_\mathrm{F}^2} ,		\mu_e^2 M_\mathrm{Ch} = 4\pi m_3 \biggl(\frac{2 A_\mathrm{F}}{\pi G}\biggr)^{3/2} \frac{\mu_e^2}{B_\mathrm{F}^2} = 1.14169\times 10^{34}~\mathrm{g},
	</math>		</math>
	</div>		</div>
Line 871:		Line 871:
	<div align="center">		<div align="center">
	<math>		<math>
	\mu_e \ell_\mathrm{Ch} \equiv \biggl( \frac{2 A_\mathrm{F}}{\pi G} \biggr)^{1/2} \frac{\mu_e}{B_\mathrm{F}} ,		\mu_e \ell_\mathrm{Ch} \equiv \biggl( \frac{2 A_\mathrm{F}}{\pi G} \biggr)^{1/2} \frac{\mu_e}{B_\mathrm{F}} = 7.71311\times 10^8~\mathrm{cm} ,
	</math>		</math>
	</div>		</div>
	where the dimensionless coefficient <math>m_3 = 2.01824</math>. We could just as well define a characteristic dynamical timescale associated with white dwarfs as,		where the dimensionless coefficient <math>m_3 = 2.01824</math>. We could just as well define a characteristic dynamical timescale associated with white dwarfs as,
	<div align="center">		<div align="center">
	<math>\mu_e^{1/2} t_\mathrm{Ch} \equiv \biggl[\frac{\mu_e}{GB_\mathrm{F}} \biggr]^{1/2} = \biggl[ \frac{3h^3}{8\pi G m_p m_e^3 c^3} \biggr]^{1/2} = 3.~~9081~~~\mathrm{s}</math> .		<math>\mu_e^{1/2} t_\mathrm{Ch} \equiv \biggl[\frac{\mu_e}{GB_\mathrm{F}} \biggr]^{1/2} = \biggl[ \frac{3h^3}{8\pi G m_p m_e^3 c^3} \biggr]^{1/2} = 3.90812~\mathrm{s}</math> .
	</div>		</div>

Line 1,001:		Line 1,001:
	<td align="left">		<td align="left">
	<math>		<math>
	\mu_e^{1/2} t_\mathrm{Ch}\cdot \frac{5^{1/2}}{\pi^4 } \biggl( \frac{m_e^3 m_p}{m_u^4}\biggr)^{1/2} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}		\mu_e^{1/2} t_\mathrm{Ch}\cdot \frac{5^{1/2}}{\pi^2 } \biggl( \frac{m_e^3 m_p}{m_u^4}\biggr)^{1/2} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}
	</math>		</math>
	</td>		</td>
Line 1,015:		Line 1,015:
	<td align="left">		<td align="left">
	<math>		<math>
	2.~~960~~\times 10^{-7} ~\mu_e^{1/2} t_\mathrm{Ch} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .		2.9261\times 10^{-6} ~\mu_e^{1/2} t_\mathrm{Ch} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .
	</math>		</math>
	</td>		</td>
Line 1,021:		Line 1,021:

	</table>		</table>

	[NOTE: (2 August 2010) This leading coefficient and the expression for <math>t_\mathrm{Ch}</math> need to be checked because their product gives a slightly different (by less than one percent) answer from earlier scaling values.]

	==Opacities==		==Opacities==

Tohline: /* Chandrasekhar Mass and Radius */ Add discussion of Chandrasekhar timescale

2010-08-03T00:53:57Z

Chandrasekhar Mass and Radius: Add discussion of Chandrasekhar timescale

← Older revision		Revision as of 00:53, 3 August 2010
Line 871:		Line 871:
	<div align="center">		<div align="center">
	<math>		<math>
	\~~ell_1~~ \~~mu_e~~ \equiv \biggl( \frac{2 A_\mathrm{F}}{\pi G} \biggr)^{1/2} \frac{\mu_e}{B_\mathrm{F}} ,		\mu_e \ell_\mathrm{Ch} \equiv \biggl( \frac{2 A_\mathrm{F}}{\pi G} \biggr)^{1/2} \frac{\mu_e}{B_\mathrm{F}} ,
	</math>		</math>
	</div>		</div>
	where the dimensionless coefficient <math>m_3 = 2.01824</math>.		where the dimensionless coefficient <math>m_3 = 2.01824</math>. We could just as well define a characteristic dynamical timescale associated with white dwarfs as,
			<div align="center">
			<math>\mu_e^{1/2} t_\mathrm{Ch} \equiv \biggl[\frac{\mu_e}{GB_\mathrm{F}} \biggr]^{1/2} = \biggl[ \frac{3h^3}{8\pi G m_p m_e^3 c^3} \biggr]^{1/2} = 3.9081~\mathrm{s}</math> .
			</div>



	Rewriting <math>M_\mathrm{Ch}</math> only in terms of the fundamental physical constants, we obtain,		Rewriting <math>M_\mathrm{Ch}</math> only in terms of the fundamental physical constants, we obtain,
Line 955:		Line 960:
	<td align="left">		<td align="left">
	<math>		<math>
	\~~ell_1~~ \~~mu_e~~ \biggl[ \frac{2^2 \cdot 5}{\pi^3 } \biggr]^{1/2} \frac{m_e m_p}{m_u^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}		\mu_e \ell_\mathrm{Ch} \biggl[ \frac{2^2 \cdot 5}{\pi^3 } \biggr]^{1/2} \frac{m_e m_p}{m_u^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}
	</math>		</math>
	</td>		</td>
Line 969:		Line 974:
	<td align="left">		<td align="left">
	<math>		<math>
	4.438\times 10^{-4} ~\~~ell_1~~ \~~mu_e~~ \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .		4.438\times 10^{-4} ~\mu_e \ell_\mathrm{Ch} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .
	</math>		</math>
	</td>		</td>
Line 975:		Line 980:

	</table>		</table>


			And,
			<div align="center">
			<math>
			\frac{\Lambda}{c^3} = \biggl[ \frac{3\cdot 5}{2^3 \pi^5} \biggl( \frac{m_p}{m_u} \biggr)^4 \cdot \frac{ h^3}{m_p^4 c^3 G} \biggr]^{1/2} ,
			</math>
			</div>
			so,
			<table align="center" border="0" cellpadding="8">
			<tr>
			<td align="center">
			<math>
			\frac{t_\mathrm{cgs}}{t_\mathrm{code}}
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\mu_e^{1/2} t_\mathrm{Ch}\cdot \frac{5^{1/2}}{\pi^4 } \biggl( \frac{m_e^3 m_p}{m_u^4}\biggr)^{1/2} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="center">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			2.960\times 10^{-7} ~\mu_e^{1/2} t_\mathrm{Ch} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .
			</math>
			</td>
			</tr>

			</table>

			[NOTE: (2 August 2010) This leading coefficient and the expression for <math>t_\mathrm{Ch}</math> need to be checked because their product gives a slightly different (by less than one percent) answer from earlier scaling values.]

	==Opacities==		==Opacities==

Tohline: /* Chandrasekhar Mass and Radius */ Chandrasekhar radius

2010-08-02T00:39:40Z

Chandrasekhar Mass and Radius: Chandrasekhar radius

← Older revision		Revision as of 00:39, 2 August 2010
Line 859:		Line 859:
	<div align="center">		<div align="center">
	<math>		<math>
	B_\mathrm{F} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 ;		\frac{B_\mathrm{F}}{\mu_e} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 ;
	</math>		</math>
	</div>		</div>
Line 930:		Line 930:
	<math>		<math>
	0.4038 ~\mu_e^2 M_\mathrm{Ch} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .		0.4038 ~\mu_e^2 M_\mathrm{Ch} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .
			</math>
			</td>
			</tr>

			</table>

			Similarly,
			<div align="center">
			<math>
			\frac{\Lambda}{c^2} = \biggl[ \frac{3\cdot 5}{2^3 \pi^5} \biggl( \frac{m_p}{m_u} \biggr)^4 \cdot \frac{ h^3}{m_p^4 c G} \biggr]^{1/2} ,
			</math>
			</div>
			so,
			<table align="center" border="0" cellpadding="8">
			<tr>
			<td align="center">
			<math>
			\frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}}
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\ell_1 \mu_e \biggl[ \frac{2^2 \cdot 5}{\pi^3 } \biggr]^{1/2} \frac{m_e m_p}{m_u^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="center">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			4.438\times 10^{-4} ~\ell_1 \mu_e \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .
	</math>		</math>
	</td>		</td>

Tohline: /* Chandrasekhar Mass and Radius */ Relating M_Ch to Dominic's dimensionless parameters

2010-08-01T23:48:13Z

Chandrasekhar Mass and Radius: Relating M_Ch to Dominic's dimensionless parameters

← Older revision		Revision as of 23:48, 1 August 2010
Line 865:		Line 865:
	<div align="center">		<div align="center">
	<math>		<math>
	\mu_e^2 M_\mathrm{Ch} = 4\pi m_3 \biggl(\frac{2 A_\mathrm{F}}{\pi G}\biggr)^{2/3} \frac{\mu_e^2}{B_\mathrm{F}^2} ,		\mu_e^2 M_\mathrm{Ch} = 4\pi m_3 \biggl(\frac{2 A_\mathrm{F}}{\pi G}\biggr)^{3/2} \frac{\mu_e^2}{B_\mathrm{F}^2} ,
	</math>		</math>
	</div>		</div>
Line 875:		Line 875:
	</div>		</div>
	where the dimensionless coefficient <math>m_3 = 2.01824</math>.		where the dimensionless coefficient <math>m_3 = 2.01824</math>.

			Rewriting <math>M_\mathrm{Ch}</math> only in terms of the fundamental physical constants, we obtain,
			<div align="center">
			<math>
			\mu_e^2 M_\mathrm{Ch} = \biggl[ \frac{3 m_3^2}{2^5 \pi^2} \cdot \frac{c^3 h^3}{G^3 m_p^4} \biggr]^{1/2} .
			</math>
			</div>
			But also note that,
			<div align="center">
			<math>
			\frac{\Lambda}{G} = \biggl[ \frac{3\cdot 5}{2^3 \pi^5} \biggl( \frac{m_p}{m_u} \biggr)^4 \cdot \frac{c^3 h^3}{G^3 m_p^4} \biggr]^{1/2} .
			</math>
			</div>
			Hence, we can also write,
			<table align="center" border="0" cellpadding="8">
			<tr>
			<td align="center">
			<math>
			\frac{m_\mathrm{cgs}}{m_\mathrm{code}}
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			M_\mathrm{Ch} \biggl[ \frac{3\cdot 5}{2^3 \pi^5} \biggl( \frac{m_p}{m_u} \biggr)^4 \cdot \frac{2^5 \pi^2 }{3 m_3^2 } \biggr]^{1/2} \frac{\mu_e^2}{\bar{\mu}^2} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 } \biggr)^{1/2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="center">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\mu_e^2 M_\mathrm{Ch} \biggl[ \frac{2^2 \cdot 5}{\pi^3 m_3^2 } \biggr]^{1/2} \frac{m_p^2}{m_u^2} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="center">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			0.4038 ~\mu_e^2 M_\mathrm{Ch} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .
			</math>
			</td>
			</tr>

			</table>

	==Opacities==		==Opacities==

Tohline: /* Determining Code Units */ Relate scalings to the Chandrasekhar mass and radius

2010-08-01T17:43:49Z

Determining Code Units: Relate scalings to the Chandrasekhar mass and radius

← Older revision		Revision as of 17:43, 1 August 2010
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	</tr>		</tr>
	</table>		</table>

			===Chandrasekhar Mass and Radius===
			The characteristic mass and length scales that are associated with a self-gravitating, degenerate-electron gas are identified [[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass\|in an accompanying Wiki page]] in the context of our discussion of the structure of spherically symmetric white dwarfs and the Chandrasekhar mass. Both scales depend on the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, and characteristic Fermi density, {{User:Tohline/Math/C_FermiDensity}}, that are familiar to the condensed-matter community. As recorded in our [[User:Tohline/Appendix/Variables_templates\|accompanying variables appendix]], the definition of these two ''condensed-matter relevant'' quantities is, respectively,
			<div align="center">
			<math>
			A_\mathrm{F} \equiv \frac{\pi m_e^4 c^5}{3 h^3} ,
			</math>
			</div>
			and,
			<div align="center">
			<math>
			B_\mathrm{F} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 ;
			</math>
			</div>
			and the characteristic, ''astrophysically relevant'' mass <math>(M_\mathrm{Ch})</math> and length <math>(\ell_1)</math> scales [[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass\|identified by Chandrasekhar]] are,
			<div align="center">
			<math>
			\mu_e^2 M_\mathrm{Ch} = 4\pi m_3 \biggl(\frac{2 A_\mathrm{F}}{\pi G}\biggr)^{2/3} \frac{\mu_e^2}{B_\mathrm{F}^2} ,
			</math>
			</div>
			and,
			<div align="center">
			<math>
			\ell_1 \mu_e \equiv \biggl( \frac{2 A_\mathrm{F}}{\pi G} \biggr)^{1/2} \frac{\mu_e}{B_\mathrm{F}} ,
			</math>
			</div>
			where the dimensionless coefficient <math>m_3 = 2.01824</math>.

	==Opacities==		==Opacities==

Tohline: /* Corrected Logic */ Add "span" anchor to properties table

2010-07-30T21:10:51Z

Corrected Logic: Add "span" anchor to properties table

← Older revision		Revision as of 21:10, 30 July 2010
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	Hence, beginning with the values of various binary system parameters [http://www.phys.lsu.edu/~tohline/clayton/q07.pdf as generated by the SCF code], we conclude that the initial model used by Dominic in his '''Case A''' rad-hydro simulations has the following properties:		<span id="Q0.7properties">Hence, beginning with the values of various binary system parameters [http://www.phys.lsu.edu/~tohline/clayton/q07.pdf as generated by the SCF code], we conclude that the initial model used by Dominic in his '''Case A''' rad-hydro simulations has the following properties:</span>

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


	==Opacities==		==Opacities==

Tohline: /* Opacities */ Define density threshold

2010-07-30T19:35:34Z

Opacities: Define density threshold

← Older revision		Revision as of 19:35, 30 July 2010
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	<div align="center">		<div align="center">
	<math>		<math>
	\rho_\mathrm{code} \ge \frac{128}{\pi (0.200 ~\mathrm{cm}^2~\mathrm{g}^{-1}) } \biggl[ \biggl(\frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^2 \biggl(\frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr)^{-1} \biggr] = 5.164\times 10^{-21} \biggl[ \frac{\tilde{c}^4 \tilde{a}^{1/2}}{\tilde{r}^2 \bar{\mu}^2 \tilde{g}^{1/2}} \biggr] .		\rho_\mathrm{code} \ge \rho_\mathrm{threshold} \equiv \frac{128}{\pi (0.200 ~\mathrm{cm}^2~\mathrm{g}^{-1}) } \biggl[ \biggl(\frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^2 \biggl(\frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr)^{-1} \biggr] = 5.164\times 10^{-21} \biggl[ \frac{\tilde{c}^4 \tilde{a}^{1/2}}{\tilde{r}^2 \bar{\mu}^2 \tilde{g}^{1/2}} \biggr] .
	</math>		</math>
	</div>		</div>
	To check this relation, note that when '''Case A''' parameter values are used, the combination of factors inside the last set of square brackets gives <math>9.367\times 10^{8}</math>, which produces the same value for ~~the~~ threshold ~~density~~ (in code units) as before.		To check this relation, note that when '''Case A''' parameter values are used, the combination of factors inside the last set of square brackets gives <math>9.367\times 10^{8}</math>, which produces the same value for <math>\rho_\mathrm{threshold}</math> (in code units) as before.

	==Ratio of Gas Pressure to Radiation Pressure==		==Ratio of Gas Pressure to Radiation Pressure==