Editing Appendix/Ramblings/Radiation/SummaryScalings

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=Summary of Scalings=

On [[Appendix/Ramblings/Radiation/CodeUnits|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.
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<table border="4" align="center" cellpadding="8" width="95%">
<tr>
  <td colspan="3" align="center" width="80%">
<b>General Relation</b>
  </td>
  <td colspan="1" align="center">
<b>Case A</b>:
  </td>
</tr>

<tr><td colspan="4" align="center"><table border="0" cellpadding="3" cellspacing="10">

<tr>
  <td align="right">
<math>
\frac{m_\mathrm{cgs}}{m_\mathrm{code}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
0.40375~\mu_e^2 M_\mathrm{Ch} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 \bar{\mu}^4 } \biggr)^{1/2}
</math>
  </td>
  <td align="left">
<math>= ~~2.8094\times 10^{33}~\mathrm{g} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
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}
</math>
  </td>
  <td align="left" width="30%">
<math>=~~ 8.179\times 10^{9}~\mathrm{cm}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{t_\mathrm{cgs}}{t_\mathrm{code}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
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}
</math>
  </td>
  <td align="left">
<math>= ~~54.02~\mathrm{s}</math>
  </td>
</tr>


<tr>
  <td align="right">
<math>
\frac{T_\mathrm{cgs}}{T_\mathrm{code}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
1.08095\times 10^{13} ~\biggl( \frac{\tilde{r} \bar\mu}{\tilde{c}^2} \biggr)
</math>
  </td>
  <td align="left">
<math>= ~~1.618 \times 10^8~\mathrm{K}</math>
  </td>
</tr>

</table></td></tr>

<tr>
  <td colspan="1" align="right" width="10%">
where:
  </td>
  <td colspan="3" align="left">
<math>
\mu_e^2 M_\mathrm{Ch} = 1.14169\times 10^{34}~\mathrm{g}
</math>; &nbsp; &nbsp;
<math>
\mu_e \ell_\mathrm{Ch} = 7.71311\times 10^{8}~\mathrm{cm}
</math>; &nbsp; &nbsp;
<math>
\mu_e^{1/2} t_\mathrm{Ch} = 3.90812~\mathrm{s}
</math>
  </td>
</tr>

<tr>
  <td colspan="4" align="center">
<b>Case A</b>&nbsp;&nbsp;
<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>
  </td>
</tr>

</table>

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Now let's convert all of the system parameters listed on the [[Appendix/PolytropicBinaries|accompanying page]] that details the properties of various polytropic binary systems.

<span id="TableProperties"><table align="center" border="1" cellpadding="8" width="95%">
<tr>
  <td align="center" colspan="15">
'''<font color="darkblue">
Properties of (<math>n=3/2</math>) Polytropic Binary Systems
</font>'''
  </td>
</tr>
<tr>
  <td colspan="1" align="center">
'''Q07'''<sup>1</sup>
  </td>
  <td align="center" colspan="5">
'''Binary System'''
  </td>
  <td align="center" colspan="4">
'''Accretor'''
  </td>
  <td align="center" colspan="5">
'''Donor'''
  </td>
</tr>

<tr>
  <td colspan="1" align="center">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>q</math>
  </td>
  <td align="center" colspan="1">
<math>M_\mathrm{tot}</math>
  </td>
  <td align="center" colspan="1">
<math>a</math>
  </td>
  <td align="center" colspan="1">
<math>P = \frac{2\pi}{\Omega}</math>
  </td>
  <td align="center" colspan="1">
<math>J_\mathrm{tot}</math>
  </td>

  <td align="center" colspan="1">
<math>M_a</math>
  </td>
  <td align="center" colspan="1">
<math>\rho^\mathrm{max}_a</math>
  </td>
  <td align="center" colspan="1">
<math>K^a_{3/2}</math>
  </td>
  <td align="center" colspan="1">
<math>R_a</math>
  </td>

  <td align="center" colspan="1">
<math>M_d</math>
  </td>
  <td align="center" colspan="1">
<math>\rho^\mathrm{max}_d</math>
  </td>
  <td align="center" colspan="1">
<math>K^d_{3/2}</math>
  </td>
  <td align="center" colspan="1">
<math>R_d</math>
  </td>
  <td align="center" colspan="1">
<math>f_\mathrm{RL}</math>
  </td>
</tr>

<tr>
  <td colspan="1" align="center">
'''SCF''' units
  </td>
  <td align="center" colspan="1">
0.70000
  </td>
  <td align="center" colspan="1">
0.02371
  </td>
  <td align="center" colspan="1">
0.83938
  </td>
  <td align="center" colspan="1">
31.19
  </td>
  <td align="center" colspan="1">
<math>8.938\times 10^{-4}</math>
  </td>

  <td align="center" colspan="1">
0.013945
  </td>
  <td align="center" colspan="1">
1.0000
  </td>
  <td align="center" colspan="1">
0.02732
  </td>
  <td align="center" colspan="1">
0.2728
  </td>

  <td align="center" colspan="1">
0.009761
  </td>
  <td align="center" colspan="1">
0.6077
  </td>
  <td align="center" colspan="1">
0.02512
  </td>
  <td align="center" colspan="1">
0.2888
  </td>
  <td align="center" colspan="1">
0.998
  </td>
</tr>

<tr>
  <td colspan="1" align="center">
conversion<sup>2</sup>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3
</math>
  </td>
  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)
</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^5
</math>
  </td>

  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3
</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^2
</math>
  </td>
  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)
</math>
  </td>

  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3
</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^2
</math>
  </td>
  <td align="center" colspan="1">
<math>
\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)
</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
</tr>

<tr>
  <td colspan="1" align="center">
'''Rad-Hydro''' units
  </td>
  <td align="center" colspan="1">
0.70000
  </td>
  <td align="center" colspan="1">
0.6847
  </td>
  <td align="center" colspan="1">
2.5752
  </td>
  <td align="center" colspan="1">
31.19
  </td>
  <td align="center" colspan="1">
0.24293
  </td>

  <td align="center" colspan="1">
0.4027
  </td>
  <td align="center" colspan="1">
1.0000
  </td>
  <td align="center" colspan="1">
0.2571
  </td>
  <td align="center" colspan="1">
0.8369
  </td>

  <td align="center" colspan="1">
0.28187
  </td>
  <td align="center" colspan="1">
0.6077
  </td>
  <td align="center" colspan="1">
0.2364
  </td>
  <td align="center" colspan="1">
0.88603
  </td>
  <td align="center" colspan="1">
0.998
  </td>
</tr>

<tr>
  <td colspan="1" align="center">
'''cgs''' units
  </td>
  <td align="center" colspan="1">
0.70000
  </td>
  <td align="center" colspan="1">
<math>1.924\times 10^{33}</math>
  </td>
  <td align="center" colspan="1">
<math>2.106\times 10^{10}</math>
  </td>
  <td align="center" colspan="1">
<math>1.687\times 10^{3}</math>
  </td>
  <td align="center" colspan="1">
<math>1.924\times 10^{33}</math>
  </td>

  <td align="center" colspan="1">
<math>1.132\times 10^{33}</math>
  </td>
  <td align="center" colspan="1">
<math>5.136\times 10^{3}</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>6.845\times 10^{9}</math>
  </td>

  <td align="center" colspan="1">
<math>7.921\times 10^{32}</math>
  </td>
  <td align="center" colspan="1">
<math>3.121\times 10^{3}</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>7.247\times 10^{9}</math>
  </td>
  <td align="center" colspan="1">
0.996
  </td>
</tr>

<tr>
  <td colspan="1" align="center">
'''Other''' units
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>0.967 M_\odot</math>
  </td>
  <td align="center" colspan="1">
<math>0.303 R_\odot</math>
  </td>
  <td align="center" colspan="1">
<math>28.1~\mathrm{min}</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>

  <td align="center" colspan="1">
<math>0.569 M_\odot</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>0.0984 R_\odot</math>
  </td>

  <td align="center" colspan="1">
<math>0.398 M_\odot</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
  <td align="center" colspan="1">
<math>0.1042 R_\odot</math>
  </td>
  <td align="center" colspan="1">
&nbsp;
  </td>
</tr>

<tr>
  <td align="left" colspan="15">
<sup>1</sup>Model '''Q07''' (<math>q = 0.700</math>): Drawn from the first page of the [http://www.phys.lsu.edu/~tohline/clayton/q07.pdf accompanying PDF document]. <font color="red">NOTE: In this PDF document, Roche-lobe volumes appear to be too large by factor of 2.</font><br />
<sup>2</sup>For this model, <math>(\ell_\mathrm{code}/\ell_\mathrm{SCF}) = \pi(128 - 3)/128 = 3.068</math>; see [[Appendix/Ramblings/Radiation/CodeUnits#Corrected_Logic|more detailed, accompanying discussion]].
  </td>
</tr>
</table>
</span>


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Here are some additional useful relations:
<table border="4" align="center" cellpadding="8">
<tr>
  <td colspan="3" align="center">
<b>General Relation</b>
  </td>
  <td colspan="1" align="center" width="32%">
<b>Case A</b>:
  </td>
</tr>

<tr><td colspan="4" align="center"><table border="0" cellpadding="3" cellspacing="10">

<tr>
  <td align="right">
<math>
f_\mathrm{Edd} \equiv \frac{L_\mathrm{acc}}{L_\mathrm{Edd}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<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>
  </td>
  <td align="left">
<math>= ~~6.74\times 10^9 \biggl[ \frac{\dot{M}}{R_a} \biggr]_\mathrm{code}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{\rho_\mathrm{threshold}}{\rho_\mathrm{max}} \equiv \frac{1}{\rho_\mathrm{max}\kappa_\mathrm{T} (\Delta R)}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<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>
  </td>
  <td align="left" width="30%">
<math>=~~ 4.83\times 10^{-12}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Gamma \equiv \frac{P_\mathrm{gas}}{P_\mathrm{rad}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3\tilde{r}}{\tilde{a}} \biggr) \biggl[ \frac{ \rho }{T^3} \biggr]_\mathrm{code}
</math>
  </td>
  <td align="left">
<math>= ~~30 \biggl[ \frac{ \rho }{T^3} \biggr]_\mathrm{code}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{v_\mathrm{circ}}{c} \equiv \frac{2\pi a_\mathrm{separation}}{c P_\mathrm{orbit}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2\pi}{\tilde{c}} \biggl[\frac{a_\mathrm{sep}}{P_\mathrm{orb}}\biggr]_\mathrm{code}
</math>
  </td>
  <td align="left">
<math>= ~~0.032 \biggl[\frac{a_\mathrm{sep}}{P_\mathrm{orb}}\biggr]_\mathrm{code}</math>
  </td>
</tr>

</table></td></tr>


<tr>
  <td colspan="4" align="center">
<b>Case A</b>&nbsp;&nbsp;
<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>
  </td>
</tr>

</table>


Combining the above '''Case A''' relations with the ''RadHydro-code'' [[Appendix/Ramblings/Radiation/CodeUnits#Q0.7properties|properties of the Q0.7 polytropic binary]] that serves as an initial condition for Dominic's simulations, we conclude the following:

(1)&nbsp;&nbsp;The system will experience "super-Eddington" accretion (''i.e.,'' <math>f_\mathrm{Edd} > 1</math>) when 
<div align="center">
<math>
[\dot{M}]_\mathrm{code} > 1.3\times 10^{-10} .
</math>
</div>

(2)&nbsp;&nbsp;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
<div align="center">
<math>
[\rho]_\mathrm{code} > \rho_\mathrm{threshold} = 5\times 10^{-12} .
</math>
</div>

(3)&nbsp;&nbsp;The system is weakly relativistic because,
<div align="center">
<math>
\frac{v_\mathrm{circ}}{c} = 0.0026 .
</math>
</div>

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