User:Tohline/Appendix/Ramblings/Photosphere - Revision history

Tohline: /* Mass-Transfer Rates */ Fix tau_chirp determination

2011-05-24T13:47:55Z

Mass-Transfer Rates: Fix tau_chirp determination

← Older revision		Revision as of 13:47, 24 May 2011
Line 159:		Line 159:
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{\dot{M}}{M_a} = 4.7\times 10^{-13} \approx 1.5\times 10^{-5} ~~~M_\odot/~~\mathrm{yr} \, .		\frac{\dot{M}}{M_a} = 4.7\times 10^{-13} ~\mathrm{s}^{-1} \approx 1.5\times 10^{-5} ~\mathrm{yr}^{-1} \, .
	</math>		</math>
	</div>		</div>
	~~A number of physical relations shown in this section are drawn from~~ [http://iopscience.iop.org/0004-637X/655/2/1025/ Kopparapu & Tohline (2007)] (hereafter, KT07)~~. For~~ stable, mass-transferring double-white dwarfs~~, the mass-transfer rate~~ is,		According to [http://iopscience.iop.org/0004-637X/655/2/1025/ Kopparapu & Tohline (2007)] (hereafter, KT07), the mass-transfer rate for stable, mass-transferring double-white dwarfs is,
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: /* Derivation from Juhan Frank */ Define R_9

2011-05-24T13:39:06Z

Derivation from Juhan Frank: Define R_9

← Older revision		Revision as of 13:39, 24 May 2011
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	\biggl\{		\biggl\{
	\biggl[ \frac{\pi^4}{20} \biggr] \biggl[ \frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}} \biggr]^{2} \biggl[ \frac{\kappa M_\odot}{4\pi R_9^2 } \biggr]		\biggl[ \frac{\pi^4}{20} \biggr] \biggl[ \frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}} \biggr]^{2} \biggl[ \frac{\kappa M_\odot}{4\pi R_9^2 } \biggr]
	\biggr\}^{1/6} \biggl[ \frac{f^{2/3}}{\dot{m}^{1/6}} \biggl( \frac{M_a}{M_\odot} \biggr)^{1/2} \biggl( \frac{R_9 }{R_a} \biggr) ^{1/3}\biggr] \, .		\biggr\}^{1/6} \biggl[ \frac{f^{2/3}}{\dot{m}^{1/6}} \biggl( \frac{M_a}{M_\odot} \biggr)^{1/2} \biggl( \frac{R_9 }{R_a} \biggr) ^{1/3}\biggr] \, ,
	</math>		</math>
	</div>		</div>
	Again, since,		where <math>R_9 \equiv 10^9~\mathrm{cm}</math>. Again, since,
	<div align="center">		<div align="center">
	<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, ,</math>		<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, ,</math>

Tohline: /* Better Temperature Estimate */ add link to Chandrasekhar mass presentation

2011-05-24T13:36:00Z

Better Temperature Estimate: add link to Chandrasekhar mass presentation

← Older revision		Revision as of 13:36, 24 May 2011
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	</math>		</math>
	</div>		</div>
	~~Note that~~,		Now, based on the accompanying discussion of the [[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass\|Chandrasekhar mass]],
	<div align="center">		<div align="center">
	<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, .</math>		<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, .</math>

Tohline: /* Mass-Transfer Rates */ Fine tune final numbers

2011-05-24T01:54:27Z

Mass-Transfer Rates: Fine tune final numbers

← Older revision		Revision as of 01:54, 24 May 2011
Line 159:		Line 159:
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{\dot{M}}{M_a} = 4.7\times 10^{-13} \, .		\frac{\dot{M}}{M_a} = 4.7\times 10^{-13} \approx 1.5\times 10^{-5} ~M_\odot/\mathrm{yr} \, .
	</math>		</math>
	</div>		</div>

Tohline: /* Mass-Transfer Rates */ Numerical mass-transfer values

2011-05-24T01:40:07Z

Mass-Transfer Rates: Numerical mass-transfer values

← Older revision		Revision as of 01:40, 24 May 2011
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	===Mass-Transfer Rates===		===Mass-Transfer Rates===
			Juhan chose the situation where everything inside the final square brackets is unity, in which case,
			<div align="center">
			<math>
			f = 1; ~ \dot{m} = 1; ~ M_a = 1M_\odot; ~ R_a = 10^9\mathrm{cm}; ~ \mathrm{and}~ R_{ph} \approx 2.63 R_\odot \, .
			</math>
			</div>
			If I use the same values, I get,
			<div align="center">
			<math>
			\frac{\dot{M}}{M_a} \cdot \frac{R_a}{c} = \dot{m} \biggl[ \frac{4\pi R_a^2}{\kappa M_a} \biggr] = 1.57\times 10^{-14} \, .
			</math>
			</div>
			This means,
			<div align="center">
			<math>
			\frac{\dot{M}}{M_a} = 4.7\times 10^{-13} \, .
			</math>
			</div>
	A number of physical relations shown in this section are drawn from [http://iopscience.iop.org/0004-637X/655/2/1025/ Kopparapu & Tohline (2007)] (hereafter, KT07). For stable, mass-transferring double-white dwarfs, the mass-transfer rate is,		A number of physical relations shown in this section are drawn from [http://iopscience.iop.org/0004-637X/655/2/1025/ Kopparapu & Tohline (2007)] (hereafter, KT07). For stable, mass-transferring double-white dwarfs, the mass-transfer rate is,
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{\dot{M}}{M_a} \approx \tau_\mathrm{chirp}^{-1} \, .		\frac{\dot{M}}{M_a} \approx \tau_\mathrm{chirp}^{-1} \, .
			</math>
			</div>
			Combining these last two expressions means,
			<div align="center">
			<math>
			\tau_\mathrm{chirp} \approx 2\times 10^{12}~ \mathrm{s} = 6.7\times 10^4~\mathrm{yr} \, .
	</math>		</math>
	</div>		</div>

Tohline: /* Locating the Photosphere of Stably Accreting Double White Dwarf Binaries */ Back to original derivation

2011-05-24T01:11:23Z

Locating the Photosphere of Stably Accreting Double White Dwarf Binaries: Back to original derivation

← Older revision		Revision as of 01:11, 24 May 2011
Line 96:		Line 96:
	</math>		</math>
	</div>		</div>
	~~Notice~~ that ~~every term inside the curly brackets on the right-hand-side of this expression is of order unity except the term involving~~ <math>\~~dot~~{M}</math>. Hence~~, we may also write~~,		Note that,
			<div align="center">
			<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, .</math>
			</div>
			Hence,
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{R_{ph}}{R_a} \approx \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c } \biggr]^{-1/6} \, .		\frac{R_{ph}}{R_a} \approx 0.920 \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c } \biggr]^{-1/6} f^{2/3} \approx \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c } \biggr]^{-1/6} \, .
	</math>		</math>
	</div>		</div>
Line 128:		Line 132:
	</math>		</math>
	</div>		</div>
	~~Note that~~,		Again, since,
	<div align="center">		<div align="center">
	<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, .</math>		<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, ,</math>
	</div>		</div>
	~~Hence~~,		we obtain,
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: /* Derivation from Juhan Frank */ More manipulation of J. Frank solution

2011-05-24T00:52:51Z

Derivation from Juhan Frank: More manipulation of J. Frank solution

← Older revision		Revision as of 00:52, 24 May 2011
Line 124:		Line 124:
	\biggr\}^{1/6} =		\biggr\}^{1/6} =
	\biggl\{		\biggl\{
	\biggl[ \frac{\pi^4}{20} \biggr] \biggl[ \frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}} \biggr]^{2} \biggl[ \frac{\kappa M_\odot}{4\pi } \biggr]		\biggl[ \frac{\pi^4}{20} \biggr] \biggl[ \frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}} \biggr]^{2} \biggl[ \frac{\kappa M_\odot}{4\pi R_9^2 } \biggr]
	\biggr\}^{1/6} \biggl[ \frac{f^{2/3}}{\dot{m}^{1/6}} \biggl( \frac{M_a}{M_\odot} \biggr)^{1/2} R_a^{-1/3} \biggr] \, .		\biggr\}^{1/6} \biggl[ \frac{f^{2/3}}{\dot{m}^{1/6}} \biggl( \frac{M_a}{M_\odot} \biggr)^{1/2} \biggl( \frac{R_9 }{R_a} \biggr) ^{1/3}\biggr] \, .
			</math>
			</div>
			Note that,
			<div align="center">
			<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, .</math>
			</div>
			Hence,
			<div align="center">
			<math>
			\frac{R_{ph}}{R_a} = 183.9 \biggl[ \frac{f^{2/3}}{\dot{m}^{1/6}} \biggl( \frac{M_a}{M_\odot} \biggr)^{1/2} \biggl( \frac{R_9 }{R_a} \biggr) ^{1/3}\biggr] \, .
	</math>		</math>
	</div>		</div>

Tohline: /* Derivation from Juhan Frank */ Simplify J.F. expression

2011-05-24T00:33:29Z

Derivation from Juhan Frank: Simplify J.F. expression

← Older revision		Revision as of 00:33, 24 May 2011
Line 122:		Line 122:
	\frac{R_{ph}}{R_a} \approx \biggl\{		\frac{R_{ph}}{R_a} \approx \biggl\{
	\biggl[ \frac{20}{\pi^4} \biggr]^{-1} \biggl[ \frac{\mu_e^2 M_{Ch}}{ m_3 M_a} \biggr]^{-2} \biggl[ \frac{4\pi R_a^2}{\kappa M_a} \biggr]^{-1} {\dot{m}}^{-1} f^{4}		\biggl[ \frac{20}{\pi^4} \biggr]^{-1} \biggl[ \frac{\mu_e^2 M_{Ch}}{ m_3 M_a} \biggr]^{-2} \biggl[ \frac{4\pi R_a^2}{\kappa M_a} \biggr]^{-1} {\dot{m}}^{-1} f^{4}
	\biggr\}^{1/6} \, .		\biggr\}^{1/6} =
			\biggl\{
			\biggl[ \frac{\pi^4}{20} \biggr] \biggl[ \frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}} \biggr]^{2} \biggl[ \frac{\kappa M_\odot}{4\pi } \biggr]
			\biggr\}^{1/6} \biggl[ \frac{f^{2/3}}{\dot{m}^{1/6}} \biggl( \frac{M_a}{M_\odot} \biggr)^{1/2} R_a^{-1/3} \biggr] \, .
	</math>		</math>
	</div>		</div>

Tohline: /* Better Temperature Estimate */ Insert comparison with Juhan Frank's derivation

2011-05-23T18:58:19Z

Better Temperature Estimate: Insert comparison with Juhan Frank's derivation

← Older revision		Revision as of 18:58, 23 May 2011
Line 102:		Line 102:
	</math>		</math>
	</div>		</div>
	This relation behaves in a very different way from the initially derived relation. For small mass-transfer rates, this relation predicts that the photospheric radius will be quite large relative to the radius of the accretor.		This relation behaves in a very different way from the initially derived relation. For small mass-transfer rates, this relation predicts that the photospheric radius will be quite large relative to the radius of the accretor.

			===Derivation from Juhan Frank===
			In his independent derivation, Juhan substitutes the following dimensionless mass-transfer rate:
			<div align="center">
			<math>
			\dot{m} \equiv \frac{\dot{M}}{\dot{M}_{int}} = \frac{\kappa\dot{M}}{4\pi c R_a} \, ,
			</math>
			</div>
			where, <math>\kappa = \sigma_T/m_p</math>. Hence,
			<div align="center">
			<math>
			\frac{\dot{M}}{M_a} \cdot \frac{R_a}{c} = \dot{m} \biggl[ \frac{4\pi R_a^2}{\kappa M_a} \biggr] \, ,
			</math>
			</div>
			and,
			<div align="center">
			<math>
			\frac{R_{ph}}{R_a} \approx \biggl\{
			\biggl[ \frac{20}{\pi^4} \biggr]^{-1} \biggl[ \frac{\mu_e^2 M_{Ch}}{ m_3 M_a} \biggr]^{-2} \biggl[ \frac{4\pi R_a^2}{\kappa M_a} \biggr]^{-1} {\dot{m}}^{-1} f^{4}
			\biggr\}^{1/6} \, .
			</math>
			</div>

	===Mass-Transfer Rates===		===Mass-Transfer Rates===

Tohline: Discuss mass-transfer rates

2011-05-16T15:54:42Z

Discuss mass-transfer rates

← Older revision		Revision as of 15:54, 16 May 2011
Line 103:		Line 103:
	</div>		</div>
	This relation behaves in a very different way from the initially derived relation. For small mass-transfer rates, this relation predicts that the photospheric radius will be quite large relative to the radius of the accretor.		This relation behaves in a very different way from the initially derived relation. For small mass-transfer rates, this relation predicts that the photospheric radius will be quite large relative to the radius of the accretor.

			===Mass-Transfer Rates===

			A number of physical relations shown in this section are drawn from [http://iopscience.iop.org/0004-637X/655/2/1025/ Kopparapu & Tohline (2007)] (hereafter, KT07). For stable, mass-transferring double-white dwarfs, the mass-transfer rate is,
			<div align="center">
			<math>
			\frac{\dot{M}}{M_a} \approx \tau_\mathrm{chirp}^{-1} \, .
			</math>
			</div>

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