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Whitworth's (1981) Isothermal Free-Energy Surface
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Summary of Scalings

On an accompanying Wiki page we have explained how to interpret the set of dimensionless units that Dominic Marcello is using in his rad-hydrocode. The following tables summarize some of the mathematical relationships that have been derived in that accompanying discussion.


General Relation

Case A:


\frac{m_\mathrm{cgs}}{m_\mathrm{code}}

=


0.40375~\mu_e^2 M_\mathrm{Ch} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 \bar{\mu}^4 } \biggr)^{1/2}

= ~~2.8094\times 10^{33}~\mathrm{g}


\frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}}

=


4.4379\times 10^{-4}~ \mu_e \ell_\mathrm{Ch}~\biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}  {\bar{\mu}^4 \tilde{r}^4} \biggr)^{1/2}

=~~ 8.179\times 10^{9}~\mathrm{cm}


\frac{t_\mathrm{cgs}}{t_\mathrm{code}}

=


2.9216\times 10^{-6}~\mu_e^{1/2} t_\mathrm{Ch} ~\biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}  {\bar{\mu}^4 \tilde{r}^4} \biggr)^{1/2}

= ~~54.02~\mathrm{s}


\frac{T_\mathrm{cgs}}{T_\mathrm{code}}

=


1.08095\times 10^{13} ~\biggl( \frac{\tilde{r} \bar\mu}{\tilde{c}^2} \biggr)

= ~~1.618 \times 10^8~\mathrm{K}

where:


\mu_e^2 M_\mathrm{Ch} = 1.14169\times 10^{34}~\mathrm{g}
;     
\mu_e \ell_\mathrm{Ch} = 7.71311\times 10^{8}~\mathrm{cm}
;     
\mu_e^{1/2} t_\mathrm{Ch} = 3.90812~\mathrm{s}

Case A   \Rightarrow ~~~\tilde{g} = 1; \tilde{c} = 198; \tilde{r} = 0.44; \tilde{a} = 0.044; \bar\mu = 4/3; ρmax = 1; (\Delta R) = \frac{\pi}{128}

Now let's convert all of the system parameters listed on the accompanying page that details the properties of various polytropic binary systems.

Properties of (n = 3 / 2) Polytropic Binary Systems

Q071

Binary System

Accretor

Donor

 

q

Mtot

a

P = \frac{2\pi}{\Omega}

Jtot

Ma

\rho^\mathrm{max}_a

K^a_{3/2}

Ra

Md

\rho^\mathrm{max}_d

K^d_{3/2}

Rd

fRL

SCF units

0.70000

0.02371

0.83938

31.19

8.938\times 10^{-4}

0.013945

1.0000

0.02732

0.2728

0.009761

0.6077

0.02512

0.2888

0.998

conversion2

 


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)

 


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^5


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3

 


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^2


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3

 


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^2


\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)

 

Rad-Hydro units

0.70000

0.6847

2.5752

31.19

0.24293

0.4027

1.0000

0.2571

0.8369

0.28187

0.6077

0.2364

0.88603

0.998

cgs units

0.70000

1.924\times 10^{33}

2.106\times 10^{10}

1.687\times 10^{3}

1.924\times 10^{33}

1.132\times 10^{33}

5.136\times 10^{3}

 

6.845\times 10^{9}

7.921\times 10^{32}

3.121\times 10^{3}

 

7.247\times 10^{9}

0.996

Other units

 

0.967 M_\odot

0.303 R_\odot

28.1~\mathrm{min}

 

0.569 M_\odot

 

 

0.0984 R_\odot

0.398 M_\odot

 

 

0.1042 R_\odot

 

1Model Q07 (q = 0.700): Drawn from the first page of the accompanying PDF document. NOTE: In this PDF document, Roche-lobe volumes appear to be too large by factor of 2.
2For this model, (\ell_\mathrm{code}/\ell_\mathrm{SCF}) = \pi(128 - 3)/128 = 3.068; see more detailed, accompanying discussion.


Here are some additional useful relations:


General Relation

Case A:


f_\mathrm{Edd} \equiv \frac{L_\mathrm{acc}}{L_\mathrm{Edd}}

=


1.25\times 10^{21} \biggl( \frac{\tilde{g}^{1/2} \tilde{r}^2 \bar{\mu}^2 }{\tilde{c}^5 \tilde{a}^{1/2}} \biggr) \biggl[ \frac{\dot{M}}{R_a} \biggr]_\mathrm{code}

= ~~6.74\times 10^9 \biggl[ \frac{\dot{M}}{R_a} \biggr]_\mathrm{code}


\frac{\rho_\mathrm{threshold}}{\rho_\mathrm{max}} \equiv \frac{1}{\rho_\mathrm{max}\kappa_\mathrm{T} (\Delta R)}

=


5.164\times 10^{-21}~\biggl( \frac{\tilde{c}^4 \tilde{a}^{1/2}}{\bar{\mu}^2 \tilde{r}^2 \tilde{g}^{1/2}} \biggr) \biggl[ \frac{1}{\rho_\mathrm{max}(\Delta R)} \biggr]_\mathrm{code}

=~~ 4.83\times 10^{-12}


\Gamma \equiv \frac{P_\mathrm{gas}}{P_\mathrm{rad}}

=


\biggl( \frac{3\tilde{r}}{\tilde{a}} \biggr) \biggl[ \frac{ \rho }{T^3} \biggr]_\mathrm{code}

= ~~30 \biggl[ \frac{ \rho }{T^3} \biggr]_\mathrm{code}


\frac{v_\mathrm{circ}}{c} \equiv \frac{2\pi a_\mathrm{separation}}{c P_\mathrm{orbit}}

=


\frac{2\pi}{\tilde{c}} \biggl[\frac{a_\mathrm{sep}}{P_\mathrm{orb}}\biggr]_\mathrm{code}

= ~~0.032 \biggl[\frac{a_\mathrm{sep}}{P_\mathrm{orb}}\biggr]_\mathrm{code}

Case A   \Rightarrow ~~~\tilde{g} = 1; \tilde{c} = 198; \tilde{r} = 0.44; \tilde{a} = 0.044; \bar\mu = 4/3; ρmax = 1; (\Delta R) = \frac{\pi}{128}


Combining the above Case A relations with the RadHydro-code properties of the Q0.7 polytropic binary that serves as an initial condition for Dominic's simulations, we conclude the following:

(1)  The system will experience "super-Eddington" accretion (i.e., fEdd > 1) when


[\dot{M}]_\mathrm{code} > 1.3\times 10^{-10} .

(2)  The mean-free-path, \ell_\mathrm{mfp}, of a photon will be less than one grid cell R)code when


[\rho]_\mathrm{code} > \rho_\mathrm{threshold} = 5\times 10^{-12} .

(3)  The system is weakly relativistic because,


\frac{v_\mathrm{circ}}{c} = 0.0026 .

 

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2019 by Joel E. Tohline
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