Editing Appendix/Ramblings/Radiation/CodeUnits (section)

===Joel's Check of Dominic's Logic and Numbers===
Let's plug in values of the physical units that we have tabulated in a [[Appendix/VariablesTemplates|Variables Appendix]] to see if we agree with Dominic's conversions.

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>
\frac{c^2}{\Re}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{(3\times 10^{10})^2}{8.314\times 10^7}~\mathrm{cgs}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
1.083\times 10^{13}~\mathrm{K}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\biggl(\frac{\Re^4}{c^4 G a_\mathrm{rad}}\biggr)^{1/2}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{(8.314\times 10^7)^2}{(3\times 10^{10})^2 (6.674\times 10^{-8})^{1/2}(7.566\times 10^{-15})^{1/2}}~\mathrm{cgs}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
3.418\times 10^{5}~\mathrm{cm}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\biggl(\frac{\Re^4}{c^6 G a_\mathrm{rad}}\biggr)^{1/2}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{(8.314\times 10^7)^2}{(3\times 10^{10})^3 (6.674\times 10^{-8})^{1/2}(7.566\times 10^{-15})^{1/2}}~\mathrm{cgs}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
1.139\times 10^{-5}~\mathrm{s}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\biggl(\frac{\Re^4}{G^3 a_\mathrm{rad}}\biggr)^{1/2}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{(8.314\times 10^7)^2}{(6.674\times 10^{-8})^{3/2}(7.566\times 10^{-15})^{1/2}}~\mathrm{cgs}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
4.609\times 10^{33}~\mathrm{g}
</math>
  </td>
</tr>
</table>

Hence,

<table border="1" align="center" cellpadding="8">
<tr>
  <td colspan="3" align="center">
<font color="blue"><b>General Relations</b></font>
  </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>
1.083\times 10^{13}~\mathrm{K} \biggl( \frac{\tilde{r}}{\tilde{c}^2} \biggr)
</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>
3.418\times 10^{5}~\mathrm{cm} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
</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>
1.139\times 10^{-5}~\mathrm{s} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
</math>
  </td>
</tr>

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

For the '''Case A''' parameter values adopted by Dominic, above, and for the particular SCF-code-generated model provided by Wes, I derive,


<table border="1" align="center" cellpadding="8">
<tr>
  <td colspan="5" align="center">
<b>Case A</b>
  </td>
</tr>

<tr>
  <td align="right">
<math>
R_\mathrm{Accretor}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
3.418\times 10^{5}~\mathrm{cm} \biggl[ \frac{198^4 \times 0.044 }{(0.44)^4 } \biggr]^{1/2} \times 0.40
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
5.8\times 10^{9}~\mathrm{cm} = 0.083~\mathrm{R}_\odot 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
P_\mathrm{orbit}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
1.139\times 10^{-5}~\mathrm{s} \biggl[ \frac{198^6 \times 0.044 }{(0.44)^4} \biggr]^{1/2} \times 31
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
2.97\times 10^{3}~\mathrm{s} = 49.5 ~\mathrm{minutes}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
M_\mathrm{total}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
4.609\times 10^{33}~\mathrm{g} \biggl[ \frac{0.044 }{(0.44)^4 } \biggr]^{1/2} \times 0.85
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
4.245\times 10^{33}~\mathrm{g} = 2.1~\mathrm{M}_\odot
</math>
  </td>
</tr>
</table>

These values do not agree with the ones derived by Dominic.

<table border="1" width="75%" cellpadding="8" align="center">
<tr>
  <td align="center">
<font color="red">Possible Point of Confusion/Disagreement</font>
  </td>
</tr>
<tr><td align="left">
NOTE:  Either Dominic wrote the wrong values on my whiteboard or I copied them down incorrectly, but based on the [http://www.phys.lsu.edu/~tohline/clayton/q07.pdf SCF-code parameters] that were given to me by Wes Even, in dimensionless code units the model parameters should be: <math>[M_\mathrm{total}]_\mathrm{code} = 0.0237</math> and <math>[R_\mathrm{Accretor}]_\mathrm{code} = 0.273</math> and <math>[P_\mathrm{orbit}]_\mathrm{code} = 31.19</math>; the orbital ''separation'' is <math>[a_\mathrm{separation}]_\mathrm{code} = 0.83938</math>.  Combining ''these'' values with Dominic's '''Case A''' parameter values gives:
* <math>[M_\mathrm{total}]_\mathrm{cgs} = 0.059 M_\odot</math>;
* <math>[R_\mathrm{Accretor}]_\mathrm{cgs} = 0.057 R_\odot</math>; 
* <math>[P_\mathrm{orbit}]_\mathrm{cgs} = 50~\mathrm{minutes}</math>; and
* <math>[a_\mathrm{separation}]_\mathrm{cgs} = 0.174 R_\odot</math>.
</td></tr>
<tr>
  <td align="left">
On 7/24/2010, Joel checked this boxed-in group of numbers against a "polytropic unit conversion spreadsheet" that he developed while at the Lorentz Institute in the Fall of 2010.  They are all consistent with Wes Even's SCF-generated Q07 model.
  </td>
</tr>
</table>