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:
$\frac{m_{\mathrm{c}\mathrm{g}\mathrm{s}}}{m_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}$ $=$ $0.40375{\mu }_e^2{M}_{\mathrm{C}\mathrm{h}}(\frac{{\tilde{g}}^3\tilde{a}}{{\tilde{r}}^4{\overline{\mu }}^4}{)}^{1/2}$ $=2.8094\times 10^{33}\mathrm{g}$ $\frac{{\ell }_{\mathrm{c}\mathrm{g}\mathrm{s}}}{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}$ $=$ $4.4379\times 10^{-4}{\mu }_e{\ell }_{\mathrm{C}\mathrm{h}}(\frac{{\tilde{c}}^4\tilde{g}\tilde{a}}{{\overline{\mu }}^4{\tilde{r}}^4}{)}^{1/2}$ $=8.179\times 10^9\mathrm{c}\mathrm{m}$ $\frac{t_{\mathrm{c}\mathrm{g}\mathrm{s}}}{t_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}$ $=$ $2.9216\times 10^{-6}{\mu }_e^{1/2}{t}_{\mathrm{C}\mathrm{h}}(\frac{{\tilde{c}}^6\tilde{g}\tilde{a}}{{\overline{\mu }}^4{\tilde{r}}^4}{)}^{1/2}$ $=54.02\mathrm{s}$ $\frac{T_{\mathrm{c}\mathrm{g}\mathrm{s}}}{T_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}$ $=$ $1.08095\times 10^{13}\left(\frac{\tilde{r}\overline{\mu }}{{\tilde{c}}^2}\right)$ $=1.618\times 10^8\mathrm{K}$
where:	${\mu }_e^2{M}_{\mathrm{C}\mathrm{h}}=1.14169\times 10^{34}\mathrm{g}$;     ${\mu }_e{\ell }_{\mathrm{C}\mathrm{h}}=7.71311\times 10^8\mathrm{c}\mathrm{m}$;     ${\mu }_e^{1/2}{t}_{\mathrm{C}\mathrm{h}}=3.90812\mathrm{s}$
Case A   $\Rightarrow \tilde{g}=1$; $\tilde{c}=198$; $\tilde{r}=0.44$; $\tilde{a}=0.044$; $\overline{\mu }=4/3$; ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}=1$; $(\mathrm{\Delta }R)=\frac{\pi }{128}$

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

$q$

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$a$

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

$J_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$M_a$

${\rho }_a^{\mathrm{m}\mathrm{a}\mathrm{x}}$

$K_{3/2}^a$

$R_a$

$M_d$

${\rho }_d^{\mathrm{m}\mathrm{a}\mathrm{x}}$

$K_{3/2}^d$

$R_d$

$f_{\mathrm{R}\mathrm{L}}$

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

$(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}{)}^3$

$\left(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}\right)$

$(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}{)}^5$

$(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}{)}^3$

$(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}{)}^2$

$\left(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}\right)$

$(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}{)}^3$

$(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}{)}^2$

$\left(\frac{{\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}}{{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}}}\right)$

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

$0.967M_{\odot }$

$0.303R_{\odot }$

$28.1\mathrm{m}\mathrm{i}\mathrm{n}$

$0.569M_{\odot }$

$0.0984R_{\odot }$

$0.398M_{\odot }$

$0.1042R_{\odot }$

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, $({\ell }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}/{\ell }_{\mathrm{S}\mathrm{C}\mathrm{F}})=\pi (128-3)/128=3.068$; see more detailed, accompanying discussion.

Here are some additional useful relations:

General Relation			Case A:
$f_{\mathrm{E}\mathrm{d}\mathrm{d}}\equiv \frac{L_{\mathrm{a}\mathrm{c}\mathrm{c}}}{L_{\mathrm{E}\mathrm{d}\mathrm{d}}}$ $=$ $1.25\times 10^{21}\left(\frac{{\tilde{g}}^{1/2}{\tilde{r}}^2{\overline{\mu }}^2}{{\tilde{c}}^5{\tilde{a}}^{1/2}}\right)[\frac{\dot{M}}{R_a}{]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ $=6.74\times 10^9[\frac{\dot{M}}{R_a}{]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ $\frac{{\rho }_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}}}{{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}}\equiv \frac{1}{{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}{\kappa }_{\mathrm{T}}(\mathrm{\Delta }R)}$ $=$ $5.164\times 10^{-21}\left(\frac{{\tilde{c}}^4{\tilde{a}}^{1/2}}{{\overline{\mu }}^2{\tilde{r}}^2{\tilde{g}}^{1/2}}\right)[\frac{1}{{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}(\mathrm{\Delta }R)}{]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ $=4.83\times 10^{-12}$ $\mathrm{\Gamma }\equiv \frac{P_{\mathrm{g}\mathrm{a}\mathrm{s}}}{P_{\mathrm{r}\mathrm{a}\mathrm{d}}}$ $=$ $\left(\frac{3\tilde{r}}{\tilde{a}}\right)[\frac{\rho }{T^3}{]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ $=30[\frac{\rho }{T^3}{]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ $\frac{v_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}}{c}\equiv \frac{2\pi a_{\mathrm{s}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}}{c{P}_{\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{i}\mathrm{t}}}$ $=$ $\frac{2\pi }{\tilde{c}}[\frac{a_{\mathrm{s}\mathrm{e}\mathrm{p}}}{P_{\mathrm{o}\mathrm{r}\mathrm{b}}}{]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ $=0.032[\frac{a_{\mathrm{s}\mathrm{e}\mathrm{p}}}{P_{\mathrm{o}\mathrm{r}\mathrm{b}}}{]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$
Case A   $\Rightarrow \tilde{g}=1$; $\tilde{c}=198$; $\tilde{r}=0.44$; $\tilde{a}=0.044$; $\overline{\mu }=4/3$; ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}=1$; $(\mathrm{\Delta }R)=\frac{\pi }{128}$

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., $f_{\mathrm{E}\mathrm{d}\mathrm{d}}>1$) when

$[\dot{M}{]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}>1.3\times 10^{-10}.$

(2)  The mean-free-path, ${\ell }_{\mathrm{m}\mathrm{f}\mathrm{p}}$, of a photon will be less than one grid cell $(\mathrm{\Delta }R{)}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ when

$[\rho {]}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}>{\rho }_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}}=5\times 10^{-12}.$

(3)  The system is weakly relativistic because,

$\frac{v_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}}{c}=0.0026.$