User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings

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:
<math> \frac{m_\mathrm{cgs}}{m_\mathrm{code}} </math> <math> = </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} </math> <math>= ~~2.810\times 10^{33}~\mathrm{g} </math> <math> \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} </math> <math> = </math> <math> 4.438\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} </math> <math>=~~ 8.179\times 10^{9}~\mathrm{cm}</math> <math> \frac{t_\mathrm{cgs}}{t_\mathrm{code}} </math> <math> = </math> <math> 2.9261\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} </math> <math>= ~~54.1~\mathrm{s}</math>
where:	<math> \mu_e^2 M_\mathrm{Ch} = 1.14169\times 10^{34}~\mathrm{g} </math>;     <math> \mu_e \ell_\mathrm{Ch} = 7.71311\times 10^{8}~\mathrm{cm} </math>;     <math> \mu_e^{1/2} t_\mathrm{Ch} = 3.90812~\mathrm{s} </math>
Case A   <math>\Rightarrow ~~~\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>; <math>\rho_\mathrm{max} = 1</math>; <math>(\Delta R) = \frac{\pi}{128}</math>

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 (<math>n=3/2</math>) Polytropic Binary Systems

Q071

Binary System

Accretor

Donor

<math>q</math>

<math>M_\mathrm{tot}</math>

<math>a</math>

<math>P = \frac{2\pi}{\Omega}</math>

<math>J_\mathrm{tot}</math>

<math>M_a</math>

<math>\rho^\mathrm{max}_a</math>

<math>K^a_{3/2}</math>

<math>R_a</math>

<math>M_d</math>

<math>\rho^\mathrm{max}_d</math>

<math>K^d_{3/2}</math>

<math>R_d</math>

<math>f_\mathrm{RL}</math>

SCF units

0.70000

0.02371

0.83938

31.19

<math>8.938\times 10^{-4}</math>

0.013945

1.0000

0.02732

0.2728

0.009761

0.6077

0.02512

0.2888

0.998

conversion2

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3 </math>

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr) </math>

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^5 </math>

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3 </math>

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^2 </math>

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr) </math>

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3 </math>

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^2 </math>

<math> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr) </math>

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

<math>1.924\times 10^{33}</math>

<math>2.106\times 10^{10}</math>

<math>1.687\times 10^{3}</math>

<math>1.924\times 10^{33}</math>

<math>1.132\times 10^{33}</math>

<math>5.136\times 10^{3}</math>

<math>6.845\times 10^{9}</math>

<math>7.921\times 10^{32}</math>

<math>3.121\times 10^{3}</math>

<math>7.247\times 10^{9}</math>

0.996

Other units

<math>0.967 M_\odot</math>

<math>0.303 R_\odot</math>

<math>28.1~\mathrm{min}</math>

<math>0.569 M_\odot</math>

<math>0.0984 R_\odot</math>

<math>0.398 M_\odot</math>

<math>0.1042 R_\odot</math>

1Model Q07 (<math>q = 0.700</math>): 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, <math>(\ell_\mathrm{code}/\ell_\mathrm{SCF}) = \pi(128 - 3)/128 = 3.068</math>; see more detailed, accompanying discussion.

Here are some additional useful relations:

General Relation			Case A:
<math> f_\mathrm{Edd} \equiv \frac{L_\mathrm{acc}}{L_\mathrm{Edd}} </math> <math> = </math> <math> 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} </math> <math>= ~~6.74\times 10^9 \biggl[ \frac{\dot{M}}{R_a} \biggr]_\mathrm{code}</math> <math> \frac{\rho_\mathrm{threshold}}{\rho_\mathrm{max}} \equiv \frac{1}{\rho_\mathrm{max}\kappa_\mathrm{T} (\Delta R)} </math> <math> = </math> <math> 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} </math> <math>=~~ 4.83\times 10^{-12}</math> <math> \Gamma \equiv \frac{P_\mathrm{gas}}{P_\mathrm{rad}} </math> <math> = </math> <math> \biggl( \frac{3\tilde{r}}{\tilde{a}} \biggr) \biggl[ \frac{ \rho }{T^3} \biggr]_\mathrm{code} </math> <math>= ~~30 \biggl[ \frac{ \rho }{T^3} \biggr]_\mathrm{code}</math> <math> \frac{v_\mathrm{circ}}{c} \equiv \frac{2\pi a_\mathrm{separation}}{c P_\mathrm{orbit}} </math> <math> = </math> <math> \frac{2\pi}{\tilde{c}} \biggl[\frac{a_\mathrm{sep}}{P_\mathrm{orb}}\biggr]_\mathrm{code} </math> <math>= ~~0.032 \biggl[\frac{a_\mathrm{sep}}{P_\mathrm{orb}}\biggr]_\mathrm{code}</math>
Case A   <math>\Rightarrow ~~~\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>; <math>\rho_\mathrm{max} = 1</math>; <math>(\Delta R) = \frac{\pi}{128}</math>

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., <math>f_\mathrm{Edd} > 1</math>) when

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

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

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

(3)  The system is weakly relativistic because,

<math> \frac{v_\mathrm{circ}}{c} = 0.0026 . </math>

 

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation