Appendix/Ramblings/Radiation/CodeUnits - Revision history

Jet53man: /* Super-Eddington Accretion */

2022-04-11T21:31:45Z

Super-Eddington Accretion

← Older revision		Revision as of 17:31, 11 April 2022
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	</tr>		</tr>
	</table>		</table>
	where, {{Math/C_GravitationalConstant}}, {{Math/C_SpeedOfLight}}, {{Math/C_ProtonMass}}, and {{Math/C_ThompsonCrossSection}} are physical constants defined in [[Appendix/~~Variables_templates~~\|an accompanying appendix]]; <math>M_\mathrm{a}</math> and <math>R_\mathrm{a}</math> are the mass and radius of the accreting star; and <math>\dot{M}</math> is the mass accretion rate. Expressed in code units, these two expressions become,		where, {{Math/C_GravitationalConstant}}, {{Math/C_SpeedOfLight}}, {{Math/C_ProtonMass}}, and {{Math/C_ThompsonCrossSection}} are physical constants defined in [[Appendix/VariablesTemplates\|an accompanying appendix]]; <math>M_\mathrm{a}</math> and <math>R_\mathrm{a}</math> are the mass and radius of the accreting star; and <math>\dot{M}</math> is the mass accretion rate. Expressed in code units, these two expressions become,
	<table align="center" border="0" cellpadding="5">		<table align="center" border="0" cellpadding="5">
	<tr>		<tr>

Jet53man: /* Opacities */

2022-04-11T21:31:21Z

Opacities

← Older revision		Revision as of 17:31, 11 April 2022
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	</math>		</math>
	</div>		</div>
	where, {{Math/C_ThompsonCrossSection}}, {{Math/C_AtomicMassUnit}}, and {{Math/C_ProtonMass}} are all physical constants defined in an [[Appendix/~~Variables_templates~~\|accompanying appendix]]. Therefore, for '''Case A''' (which assumes pure helium, so <math>X = 0)</math>, the value of this free-free (Thompson) opacity in code units is,		where, {{Math/C_ThompsonCrossSection}}, {{Math/C_AtomicMassUnit}}, and {{Math/C_ProtonMass}} are all physical constants defined in an [[Appendix/VariablesTemplates\|accompanying appendix]]. Therefore, for '''Case A''' (which assumes pure helium, so <math>X = 0)</math>, the value of this free-free (Thompson) opacity in code units is,
	<div align="center">		<div align="center">
	<math>\kappa_\mathrm{T}\|_\mathrm{code} = 0.2003101~\mathrm{cm}^2~\mathrm{g}^{-1} \biggl[ \frac{\kappa_\mathrm{cgs}}{\kappa_\mathrm{code}} \biggr]^{-1} = 8.434\times 10^{12}</math> .		<math>\kappa_\mathrm{T}\|_\mathrm{code} = 0.2003101~\mathrm{cm}^2~\mathrm{g}^{-1} \biggl[ \frac{\kappa_\mathrm{cgs}}{\kappa_\mathrm{code}} \biggr]^{-1} = 8.434\times 10^{12}</math> .

Jet53man at 20:57, 11 April 2022

2022-04-11T20:57:05Z

← Older revision		Revision as of 16:57, 11 April 2022
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	===Joel's Check of Dominic's Logic and Numbers===		===Joel's Check of Dominic's Logic and Numbers===
	Let's plug in values of the physical units that we have tabulated in a [[Appendix/~~Variables_templates~~\|Variables Appendix]] to see if we agree with Dominic's conversions.		Let's plug in values of the physical units that we have tabulated in a [[Appendix/VariablesTemplates\|Variables Appendix]] to see if we agree with Dominic's conversions.

	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">
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	===Chandrasekhar Mass and Radius===		===Chandrasekhar Mass and Radius===
	====Review====		====Review====
	The characteristic mass, length, and time scales that are associated with a self-gravitating, degenerate-electron gas are identified [[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, {{Math/C_FermiPressure}}, and characteristic Fermi density, {{Math/C_FermiDensity}}, that are familiar to the condensed-matter community. As recorded in our [[Appendix/~~Variables_templates~~\|accompanying variables appendix]], the definition of these two ''condensed-matter relevant'' quantities is, respectively,		The characteristic mass, length, and time scales that are associated with a self-gravitating, degenerate-electron gas are identified [[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, {{Math/C_FermiPressure}}, and characteristic Fermi density, {{Math/C_FermiDensity}}, that are familiar to the condensed-matter community. As recorded in our [[Appendix/VariablesTemplates\|accompanying variables appendix]], the definition of these two ''condensed-matter relevant'' quantities is, respectively,
	<div align="center">		<div align="center">
	<math>		<math>

Jet53man at 20:55, 11 April 2022

2022-04-11T20:55:18Z

← Older revision		Revision as of 16:55, 11 April 2022
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	===Logic Used by Dominic Marcello===		===Logic Used by Dominic Marcello===
	At our group meeting on 21 July 2010, Dominic explained how he had established the values of various coupling constants in his first, long, <math>q_0 = 0.7</math> simulations. In place of the physical constants,		At our group meeting on 21 July 2010, Dominic explained how he had established the values of various coupling constants in his first, long, <math>q_0 = 0.7</math> simulations. In place of the physical constants,
	{{~~User:Tohline/~~Math/C_GravitationalConstant}}, {{~~User:Tohline/~~Math/C_SpeedOfLight}}, {{~~User:Tohline/~~Math/C_GasConstant}}, and {{~~User:Tohline/~~Math/C_RadiationConstant}}, Dominic used the following code-unit		{{Math/C_GravitationalConstant}}, {{Math/C_SpeedOfLight}}, {{Math/C_GasConstant}}, and {{Math/C_RadiationConstant}}, Dominic used the following code-unit
	values — hereafter referred to as '''Case A''':		values — hereafter referred to as '''Case A''':
	*<math>\tilde{g} = 1</math>		*<math>\tilde{g} = 1</math>
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	* <math>[R_\mathrm{Accretor}]_\mathrm{cgs} = 0.56 R_\odot</math>; and		* <math>[R_\mathrm{Accretor}]_\mathrm{cgs} = 0.56 R_\odot</math>; and
	* <math>[P_\mathrm{orbit}]_\mathrm{cgs} = 28~\mathrm{minutes}</math>.		* <math>[P_\mathrm{orbit}]_\mathrm{cgs} = 28~\mathrm{minutes}</math>.
	Conversely — assuming pure helium, that is, a mean molecular weight {{~~User:Tohline/~~Math/MP_MeanMolecularWeight}} of 2 — since the Thompson cross-section is <math>[\sigma_T]_\mathrm{cgs} = 0.2~\mathrm{cm}^2~\mathrm{g}^{-1}</math>, Dominic determined that, in the code, he needed to set the Thompson cross-section value to <math>[\sigma_T]_\mathrm{code} = 8\times 10^{12}</math>.		Conversely — assuming pure helium, that is, a mean molecular weight {{Math/MP_MeanMolecularWeight}} of 2 — since the Thompson cross-section is <math>[\sigma_T]_\mathrm{cgs} = 0.2~\mathrm{cm}^2~\mathrm{g}^{-1}</math>, Dominic determined that, in the code, he needed to set the Thompson cross-section value to <math>[\sigma_T]_\mathrm{code} = 8\times 10^{12}</math>.
	Finally, Dominic pointed out that the characteristic size of a grid cell in the code is <math>[\Delta z]_\mathrm{code} = 0.025</math>. Hence, if only the Thompson cross-section is relevant, the mean-free-path of a photon will equal the size of one grid cell if,		Finally, Dominic pointed out that the characteristic size of a grid cell in the code is <math>[\Delta z]_\mathrm{code} = 0.025</math>. Hence, if only the Thompson cross-section is relevant, the mean-free-path of a photon will equal the size of one grid cell if,
	<div align="center">		<div align="center">
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	===Joel's Check of Dominic's Logic and Numbers===		===Joel's Check of Dominic's Logic and Numbers===
	Let's plug in values of the physical units that we have tabulated in a [[~~User:Tohline/~~Appendix/Variables_templates\|Variables Appendix]] to see if we agree with Dominic's conversions.		Let's plug in values of the physical units that we have tabulated in a [[Appendix/Variables_templates\|Variables Appendix]] to see if we agree with Dominic's conversions.

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	* <math>[a_\mathrm{separation}]_\mathrm{cgs} = 0.302 R_\odot</math>.		* <math>[a_\mathrm{separation}]_\mathrm{cgs} = 0.302 R_\odot</math>.

	Two other pieces of information are needed in order to reconcile our numbers. First, Dominic has included a value of {{~~User:Tohline/~~Math/MP_MeanMolecularWeight}}<math>= 4/3</math> in his ''cgs'' value of {{~~User:Tohline/~~Math/C_GasConstant}}, that is, he has set <math>\Re/\bar{\mu} = 6.236\times 10^7~\mathrm{cgs}</math>. Second, the length-scale he has adopted in his rad-hydro code is different from the one that Wes provided straight from the SCF code. In particular, Dominic ''thinks'' Wes sets,		Two other pieces of information are needed in order to reconcile our numbers. First, Dominic has included a value of {{Math/MP_MeanMolecularWeight}}<math>= 4/3</math> in his ''cgs'' value of {{Math/C_GasConstant}}, that is, he has set <math>\Re/\bar{\mu} = 6.236\times 10^7~\mathrm{cgs}</math>. Second, the length-scale he has adopted in his rad-hydro code is different from the one that Wes provided straight from the SCF code. In particular, Dominic ''thinks'' Wes sets,
	<div align="center">		<div align="center">
	<math>		<math>
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	<td colspan="1" align="center">		<td colspan="1" align="center">
	<font color="blue"><b>General Relations (taking {{~~User:Tohline/~~Math/MP_MeanMolecularWeight}} into account)</b></font>		<font color="blue"><b>General Relations (taking {{Math/MP_MeanMolecularWeight}} into account)</b></font>
	</td>		</td>
	</tr>		</tr>
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	===Chandrasekhar Mass and Radius===		===Chandrasekhar Mass and Radius===
	====Review====		====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,		The characteristic mass, length, and time scales that are associated with a self-gravitating, degenerate-electron gas are identified [[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, {{Math/C_FermiPressure}}, and characteristic Fermi density, {{Math/C_FermiDensity}}, that are familiar to the condensed-matter community. As recorded in our [[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>
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	</math>		</math>
	</div>		</div>
	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,		and the characteristic, ''astrophysically relevant'' mass <math>(M_\mathrm{Ch})</math> and length <math>(\ell_\mathrm{Ch})</math> scales [[SSC/Structure/WhiteDwarfs#Chandrasekhar_mass\|identified by Chandrasekhar]] are,
	<div align="center">		<div align="center">
	<math>		<math>
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	</math>		</math>
	</div>		</div>
	where, {{~~User:Tohline/~~Math/C_ThompsonCrossSection}}, {{~~User:Tohline/~~Math/C_AtomicMassUnit}}, and {{~~User:Tohline/~~Math/C_ProtonMass}} are all physical constants defined in an [[~~User:Tohline/~~Appendix/Variables_templates\|accompanying appendix]]. Therefore, for '''Case A''' (which assumes pure helium, so <math>X = 0)</math>, the value of this free-free (Thompson) opacity in code units is,		where, {{Math/C_ThompsonCrossSection}}, {{Math/C_AtomicMassUnit}}, and {{Math/C_ProtonMass}} are all physical constants defined in an [[Appendix/Variables_templates\|accompanying appendix]]. Therefore, for '''Case A''' (which assumes pure helium, so <math>X = 0)</math>, the value of this free-free (Thompson) opacity in code units is,
	<div align="center">		<div align="center">
	<math>\kappa_\mathrm{T}\|_\mathrm{code} = 0.2003101~\mathrm{cm}^2~\mathrm{g}^{-1} \biggl[ \frac{\kappa_\mathrm{cgs}}{\kappa_\mathrm{code}} \biggr]^{-1} = 8.434\times 10^{12}</math> .		<math>\kappa_\mathrm{T}\|_\mathrm{code} = 0.2003101~\mathrm{cm}^2~\mathrm{g}^{-1} \biggl[ \frac{\kappa_\mathrm{cgs}}{\kappa_\mathrm{code}} \biggr]^{-1} = 8.434\times 10^{12}</math> .
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	</tr>		</tr>
	</table>		</table>
	where, {{~~User:Tohline/~~Math/C_GravitationalConstant}}, {{~~User:Tohline/~~Math/C_SpeedOfLight}}, {{~~User:Tohline/~~Math/C_ProtonMass}}, and {{~~User:Tohline/~~Math/C_ThompsonCrossSection}} are physical constants defined in [[~~User:Tohline/~~Appendix/Variables_templates\|an accompanying appendix]]; <math>M_\mathrm{a}</math> and <math>R_\mathrm{a}</math> are the mass and radius of the accreting star; and <math>\dot{M}</math> is the mass accretion rate. Expressed in code units, these two expressions become,		where, {{Math/C_GravitationalConstant}}, {{Math/C_SpeedOfLight}}, {{Math/C_ProtonMass}}, and {{Math/C_ThompsonCrossSection}} are physical constants defined in [[Appendix/Variables_templates\|an accompanying appendix]]; <math>M_\mathrm{a}</math> and <math>R_\mathrm{a}</math> are the mass and radius of the accreting star; and <math>\dot{M}</math> is the mass accretion rate. Expressed in code units, these two expressions become,
	<table align="center" border="0" cellpadding="5">		<table align="center" border="0" cellpadding="5">
	<tr>		<tr>

Jet53man: Created page with "FORCETOC =Marcello's Radiation-Hydro Simulations= ==Determining Code Units== ===Logic Used by Dominic Marcello=== At our group mee..."

2021-07-23T16:35:20Z

Created page with "__FORCETOC__  =Marcello's Radiation-Hydro Simulations= ==Determining Code Units== ===Logic Used by Dominic Marcello=== At our group mee..."