Editing Appendix/Ramblings/Radiation/CodeUnits

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=Marcello's Radiation-Hydro Simulations=

==Determining Code Units==
===Logic Used by Dominic Marcello===
At our group meeting on 21 July 2010, Dominic explained how he had established the values of various coupling constants in his first, long, <math>q_0 = 0.7</math> simulations.  In place of the physical constants, 
{{Math/C_GravitationalConstant}}, {{Math/C_SpeedOfLight}}, {{Math/C_GasConstant}}, and {{Math/C_RadiationConstant}}, Dominic used the following code-unit
values &#8212; hereafter referred to as '''Case A''':
*<math>\tilde{g} = 1</math>
*<math>\tilde{c} = 198</math>
*<math>\tilde{r} = 0.44</math>
*<math>\tilde{a} = 0.044</math>
This means that any temperature in the simulation that has a value <math>T_\mathrm{code}</math> in code units must represent an actual physical temperature <math>T_\mathrm{cgs}</math> in cgs units (''i.e.,'' measured in Kelvins) of,
<div align="center">
<math>
T_\mathrm{cgs} = \biggl[ \biggl(\frac{c^2}{\Re}\biggr)\biggl(\frac{\tilde{c}^2}{\tilde{r}}\biggr)^{-1} \biggr] T_\mathrm{code} ;
</math>
</div>
any length-scale in the simulation that has a value <math>\ell_\mathrm{code}</math> must represent an actual physical length <math>\ell_\mathrm{cgs}</math> in cgs units of,
<div align="center">
<math>
\ell_\mathrm{cgs} = \biggl[\biggl(\frac{\Re^4}{c^4 G a_\mathrm{rad}}\biggr)\biggl(\frac{\tilde{r}^4}{\tilde{c}^4 \tilde{g}\tilde{a}}\biggr)^{-1}\biggr]^{1/2} \ell_\mathrm{code} ;
</math>
</div>
any time in the simulation that has a value <math>t_\mathrm{code}</math> must represent an actual physical time <math>t_\mathrm{cgs}</math> in cgs units of,
<div align="center">
<math>
t_\mathrm{cgs} = \biggl[\biggl(\frac{\Re^4 }{c^6 G a_\mathrm{rad}}\biggr)\biggl(\frac{\tilde{r}^4}{\tilde{c}^6 \tilde{g}\tilde{a}}\biggr)^{-1}\biggr]^{1/2} t_\mathrm{code} ;
</math>
</div>
and, finally, 
any mass in the simulation that has a value <math>m_\mathrm{code}</math> must represent an actual physical mass <math>m_\mathrm{cgs}</math> in cgs units of,
<div align="center">
<math>
m_\mathrm{cgs} = \biggl[\biggl(\frac{\Re^4}{G^3 a_\mathrm{rad}}\biggr)\biggl(\frac{\tilde{r}^4}{\tilde{g}^3 \tilde{a}}\biggr)^{-1}\biggr]^{1/2} m_\mathrm{code} .
</math>
</div>


Now, the SCF-code-generated polytropic binary that Wes Even gave to Dominic had the following properties, in dimensionless code units:
* <math>[M_\mathrm{total}]_\mathrm{code} = 0.85</math>;
* <math>[R_\mathrm{Accretor}]_\mathrm{code} = 0.4</math>; and
* <math>[P_\mathrm{orbit}]_\mathrm{code} = 31</math>.
According to Dominic's calculations this means that his simulation represents a real binary system with the following properties:
* <math>[M_\mathrm{total}]_\mathrm{cgs} = 0.1 M_\odot</math>;
* <math>[R_\mathrm{Accretor}]_\mathrm{cgs} = 0.56 R_\odot</math>; and
* <math>[P_\mathrm{orbit}]_\mathrm{cgs} = 28~\mathrm{minutes}</math>.
Conversely &#8212; assuming pure helium, that is, a mean molecular weight {{Math/MP_MeanMolecularWeight}} of 2 &#8212; since the Thompson cross-section is <math>[\sigma_T]_\mathrm{cgs} = 0.2~\mathrm{cm}^2~\mathrm{g}^{-1}</math>,  Dominic determined that, in the code, he needed to set the Thompson cross-section value to <math>[\sigma_T]_\mathrm{code} = 8\times 10^{12}</math>.
Finally, Dominic pointed out that the characteristic size of a grid cell in the code is <math>[\Delta z]_\mathrm{code} = 0.025</math>.  Hence, if only the Thompson cross-section is relevant, the mean-free-path of a photon will equal the size of one grid cell if,
<div align="center">
<math>
\biggl[\frac{1}{\sigma_T\rho}\biggr]_\mathrm{code} = [\Delta z]_\mathrm{code}
</math><br/><br />

<math>
\Rightarrow ~~~~~ [\rho]_\mathrm{code} = \biggl[\frac{1}{\sigma_T(\Delta z)}\biggr]_\mathrm{code} = \frac{1}{2\times 10^{11}} .
</math>
</div>

===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>


===Response from Dominic===
What he wrote on my whiteboard contained some mistakes.  For example, the correct code units for various quantities are:
* <math>[\rho_\mathrm{max}]_\mathrm{code} = 1.000</math>;
* <math>[M_\mathrm{Accretor}]_\mathrm{code} = 0.403</math>;
* <math>[R_\mathrm{Accretor}]_\mathrm{code} = 0.850</math>;
* <math>[P_\mathrm{orbit}]_\mathrm{code} = 31.2</math>;
* <math>[M_\mathrm{total}]_\mathrm{code} = 0.685</math>; and
* <math>[a_\mathrm{separation}]_\mathrm{code} = 2.58</math>.
And when he applies the unit conversions, he gets:
* <math>[\rho_\mathrm{max}]_\mathrm{cgs} = 5.12\times 10^3~\mathrm{g}~\mathrm{cm}^{-3}</math>;
* <math>[M_\mathrm{Accretor}]_\mathrm{cgs} = 0.569 M_\odot</math>;
* <math>[R_\mathrm{Accretor}]_\mathrm{cgs} = 0.100 R_\odot</math>;
* <math>[P_\mathrm{orbit}]_\mathrm{cgs} = 28.12~\mathrm{minutes}</math>; 
* <math>[M_\mathrm{total}]_\mathrm{cgs} = 0.968 M_\odot</math>; and
* <math>[a_\mathrm{separation}]_\mathrm{cgs} = 0.302 R_\odot</math>.

Two other pieces of information are needed in order to reconcile our numbers.  First, Dominic has included a value of {{Math/MP_MeanMolecularWeight}}<math>= 4/3</math> in his ''cgs'' value of {{Math/C_GasConstant}}, that is, he has set <math>\Re/\bar{\mu} = 6.236\times 10^7~\mathrm{cgs}</math>.  Second, the length-scale he has adopted in his rad-hydro code is different from the one that Wes provided straight from the SCF code.  In particular, Dominic ''thinks'' Wes sets,
<div align="center">
<math>
[\Delta R]_\mathrm{Wes\_code} = \frac{1}{128-3} = 8\times 10^{-3} ,
</math>
</div>
whereas, in order to conform to the constraints imposed by HAD, Dominic sets,
<div align="center">
<math>
[\Delta R]_\mathrm{Nic\_code} = \frac{\pi}{128} = 2.454\times 10^{-2} .
</math>
</div>
Hence, in order to transform from the ''code units'' used by Wes (and the SCF code) to ''code units'' used by Dominic, every quantity that includes a unit of length must be multiplied by,
<div align="center">
<math>\biggl[ \frac{\ell_\mathrm{Nic}}{\ell_\mathrm{Wes}}\biggr]_\mathrm{code} = \frac{\pi (128-3)}{128} = 3.067962.
</math></div>


===Other Thoughts===

Notice that Dominic's method for converting from code units to cgs units frequently involves the following ratio of physical constants:
<div align="center">
<math>
\Lambda \equiv \biggl( \frac{\Re^4}{G a_\mathrm{rad}} \biggr)^{1/2} = 3.076 \times 10^{26}~\mathrm{cm}^3~\mathrm{s}^{-2}.
</math>
</div>
In terms of this ''new'' physical constant,

<table border="0" align="center" cellpadding="12">
<tr>
  <td align="center">
<math>
\ell \sim \frac{\Lambda}{ c^2 \bar{\mu}^2} ;
</math>
  </td>
  <td align="center">
<math>
t \sim \frac{\Lambda}{ c^3 \bar{\mu}^2} ;
</math>
  </td>
  <td align="center">
<math>
\mathrm{and} ~~~~~ m \sim \frac{\Lambda}{ G \bar{\mu}^2} .
</math>
  </td>
</tr>
</table>


===Corrected Logic===
Taking all of the above into consideration, the expressions that should be used to convert from Dominic's code units to real units are the following:


<table border="1" align="center" cellpadding="8">
<tr>
  <td colspan="1" align="center">
<font color="blue"><b>General Relations (taking {{Math/MP_MeanMolecularWeight}} into account)</b></font>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>
\frac{T_\mathrm{cgs}}{T_\mathrm{code}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{c^2}{(\Re/\bar{\mu})} \biggl( \frac{\tilde{r}}{\tilde{c}^2} \biggr)
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
1.083\times 10^{13}~\mathrm{K} \biggl( \frac{\tilde{r} \bar{\mu}}{\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>
\frac{\Lambda}{ c^2 \bar{\mu}^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
</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 \bar{\mu}^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>
\frac{\Lambda}{ c^3 \bar{\mu}^2} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
</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 \bar{\mu}^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>
\frac{\Lambda}{ G \bar{\mu}^2} \biggl( \frac{\tilde{g}^3 \tilde{a} }{\tilde{r}^4} \biggr)^{1/2}
</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 \bar{\mu}^4} \biggr)^{1/2}
</math>
  </td>
</tr>

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


Hence, for Dominic's first simulation ('''Case A'''), the following conversions apply.

<table border="1" align="center" cellpadding="8">
<tr>
  <td colspan="1" align="center">
<b>Case A</b>:<br />
<math>\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>
  </td>
</tr>

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

<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{0.44 (4/3)}{198^2} \biggr]
</math>
  </td>
  <td align="left">
<math>= 1.621\times 10^8~\mathrm{K}</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{198^4 (0.044) }{(0.44)^4 (4/3)^4} \biggr]^{1/2}
</math>
  </td>
  <td align="left">
<math>= 8.167\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>
1.139\times 10^{-5}~\mathrm{s} \biggl[ \frac{198^6 (0.044) }{(0.44)^4 (4/3)^4} \biggr]^{1/2}
</math>
  </td>
  <td align="left">
<math>= 5.334\times 10^1~\mathrm{s}</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{ 0.044 }{(0.44)^4 (4/3)^4} \biggr]^{1/2}
</math>
  </td>
  <td align="left">
<math>= 2.809\times 10^{33}~\mathrm{g}</math>
  </td>
</tr>

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


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

<tr>
  <td align="right">
<math>
\frac{\rho_\mathrm{cgs}}{\rho_\mathrm{code}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr)
\biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{-3}
</math>
  </td>
  <td align="left">
<math>= 5.157\times 10^{3}~\mathrm{g}~\mathrm{cm}^{-3}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{\kappa_\mathrm{cgs}}{\kappa_\mathrm{code}}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{2}
\biggl( \frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr)^{-1}
</math>
  </td>
  <td align="left">
<math>= 2.375\times 10^{-14}~\mathrm{cm}^2~\mathrm{g}^{-1}</math>
  </td>
</tr>

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

</table>

When using the above tabulated '''Case A''' conversion units, it must be understood that the "code unit" values refer to units used in Dominic's rad-hydro code.  But it should also be appreciated, as discussed above, that the initial model provided to Dominic by Wes &#8212; which had been generated by the SCF code &#8212; used a different unit of length from Dominic.  The conversion factor from SCF-code lengths to the length's used in Dominic's code is:

<div align="center">
<math>
\frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} =  \biggl[ \frac{\pi (128-3)}{128} \biggr] = 3.068 .
</math>
</div>

<span id="Q0.7properties">Hence, beginning with the values of various binary system parameters [http://www.phys.lsu.edu/~tohline/clayton/q07.pdf as generated by the SCF code], we conclude that the initial model used by Dominic in his '''Case A''' rad-hydro simulations has the following properties:</span>

<table align="center" border="1" cellpadding="5">
<tr>
  <td align="center" colspan="5">
'''Properties of Initial Q0.7 Polytropic Binary'''
  </td>
</tr>

<tr>
  <td align="center">
Quantity
  </td>
  <td align="center">
SCF-code<br/>
Value
  </td>
  <td align="center">
Conversion<br/>
Factor
  </td>
  <td align="center">
RadHydro-code<br/>
Value
  </td>
  <td align="center">
'''Case A'''<br/>
physical units
  </td>
</tr>

<tr>
  <td align="center">
<math>M_\mathrm{Accretor}</math>
  </td>
  <td align="center">
<math>0.01394</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^{3}</math>
  </td>
  <td align="center">
<math>0.4025</math>
  </td>
  <td align="center">
<math>0.565~M_\odot</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>M_\mathrm{Donor}</math>
  </td>
  <td align="center">
<math>0.009761</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^{3}</math>
  </td>
  <td align="center">
<math>0.2819</math>
  </td>
  <td align="center">
<math>0.396~M_\odot</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\rho_\mathrm{Accretor}</math>
  </td>
  <td align="center">
<math>1.000</math>
  </td>
  <td align="center">
<math>1</math>
  </td>
  <td align="center">
<math>1.000</math>
  </td>
  <td align="center">
<math>5.16\times 10^{3}~\mathrm{g}~\mathrm{cm}^{-3}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>a_\mathrm{separation}</math>
  </td>
  <td align="center">
<math>0.8394</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)</math>
  </td>
  <td align="center">
<math>2.575</math>
  </td>
  <td align="center">
<math>0.300~R_\odot</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>P_\mathrm{orbit}</math>
  </td>
  <td align="center">
<math>31.19</math>
  </td>
  <td align="center">
<math>1</math>
  </td>
  <td align="center">
<math>31.19</math>
  </td>
  <td align="center">
<math>27.7~\mathrm{min}</math>
  </td>
</tr>
</table>

===Chandrasekhar Mass and Radius===
====Review====
The characteristic mass, length, and time scales that are associated with a self-gravitating, degenerate-electron gas are identified [[SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|in an accompanying Wiki page]] in the context of our discussion of the structure of spherically symmetric white dwarfs and the Chandrasekhar mass.  All three of these scales depend on the characteristic Fermi pressure, {{Math/C_FermiPressure}}, and characteristic Fermi density, {{Math/C_FermiDensity}}, that are familiar to the condensed-matter community.  As recorded in our [[Appendix/VariablesTemplates|accompanying variables appendix]], the definition of these two ''condensed-matter relevant'' quantities is, respectively,
<div align="center">
<math>
A_\mathrm{F} \equiv \frac{\pi m_e^4 c^5}{3 h^3} = 6.00233\times 10^{22}~\mathrm{erg}~\mathrm{cm}^{-3},
</math>
</div>
and,
<div align="center">
<math>
\frac{B_\mathrm{F}}{\mu_e} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 = 9.81019\times 10^{5}~\mathrm{g}~\mathrm{cm}^{-3};
</math>
</div>
and the characteristic, ''astrophysically relevant'' mass <math>(M_\mathrm{Ch})</math> and length <math>(\ell_\mathrm{Ch})</math> scales [[SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|identified by Chandrasekhar]] are,
<div align="center">
<math>
\mu_e^2 M_\mathrm{Ch} = 4\pi m_3 \biggl(\frac{2 A_\mathrm{F}}{\pi G}\biggr)^{3/2} \frac{\mu_e^2}{B_\mathrm{F}^2} = 1.14169\times 10^{34}~\mathrm{g},
</math>
</div>
and,
<div align="center">
<math>
\mu_e \ell_\mathrm{Ch} \equiv \biggl( \frac{2 A_\mathrm{F}}{\pi G} \biggr)^{1/2} \frac{\mu_e}{B_\mathrm{F}} = 7.71311\times 10^8~\mathrm{cm} ,
</math>
</div>
where the dimensionless coefficient <math>m_3 = 2.01824</math>.  We could just as well define a characteristic dynamical timescale associated with white dwarfs as,
<div align="center">
<math>\mu_e^{1/2} t_\mathrm{Ch} \equiv \biggl[\frac{\mu_e}{GB_\mathrm{F}}  \biggr]^{1/2} = \biggl[ \frac{3h^3}{8\pi G m_p m_e^3 c^3} \biggr]^{1/2} = 3.90812~\mathrm{s}</math> .
</div>

====Application to Unit Conversion Expressions====

Rewriting <math>M_\mathrm{Ch}</math> only in terms of the fundamental physical constants, we obtain,
<div align="center">
<math>
\mu_e^2 M_\mathrm{Ch} = \biggl[ \frac{3 m_3^2}{2^5 \pi^2} \cdot \frac{c^3 h^3}{G^3 m_p^4} \biggr]^{1/2} .
</math>
</div>
But also note that,
<div align="center">
<math>
\frac{\Lambda}{G} = \biggl[ \frac{3\cdot 5}{2^3 \pi^5} \biggl( \frac{m_p}{m_u} \biggr)^4 \cdot \frac{c^3 h^3}{G^3 m_p^4} \biggr]^{1/2} .
</math>
</div>
Hence, we can also write,
<table align="center" border="0" cellpadding="8">
<tr>
  <td align="center">
<math>
\frac{m_\mathrm{cgs}}{m_\mathrm{code}} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{Ch} \biggl[ \frac{3\cdot 5}{2^3 \pi^5} \biggl( \frac{m_p}{m_u} \biggr)^4 \cdot \frac{2^5 \pi^2 }{3 m_3^2 } \biggr]^{1/2} \frac{\mu_e^2}{\bar{\mu}^2} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 } \biggr)^{1/2} 
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mu_e^2 M_\mathrm{Ch} \biggl[ \frac{2^2 \cdot 5}{\pi^3 m_3^2 } \biggr]^{1/2} \frac{m_p^2}{m_u^2} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

</table>

Similarly,
<div align="center">
<math>
\frac{\Lambda}{c^2} = \biggl[ \frac{3\cdot 5}{2^3 \pi^5} \biggl( \frac{m_p}{m_u} \biggr)^4 \cdot \frac{ h^3}{m_p^4 c G} \biggr]^{1/2} ,
</math>
</div>
so,
<table align="center" border="0" cellpadding="8">
<tr>
  <td align="center">
<math>
\frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mu_e \ell_\mathrm{Ch} \biggl[ \frac{2^2 \cdot 5}{\pi^3 } \biggr]^{1/2} \frac{m_e m_p}{m_u^2} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4.438\times 10^{-4} ~\mu_e \ell_\mathrm{Ch} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .
</math>
  </td>
</tr>

</table>

And,
<div align="center">
<math>
\frac{\Lambda}{c^3} = \biggl[ \frac{3\cdot 5}{2^3 \pi^5} \biggl( \frac{m_p}{m_u} \biggr)^4 \cdot \frac{ h^3}{m_p^4 c^3 G} \biggr]^{1/2} ,
</math>
</div>
so,
<table align="center" border="0" cellpadding="8">
<tr>
  <td align="center">
<math>
\frac{t_\mathrm{cgs}}{t_\mathrm{code}} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mu_e^{1/2} t_\mathrm{Ch}\cdot \frac{5^{1/2}}{\pi^2 } \biggl( \frac{m_e^3 m_p}{m_u^4}\biggr)^{1/2} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2}
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2.9261\times 10^{-6} ~\mu_e^{1/2} t_\mathrm{Ch} \biggl( \frac{\tilde{c}^6 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr)^{1/2} .
</math>
  </td>
</tr>

</table>

==Opacities==
How should an opacity coefficient be introduced into Dominic's rad-hydrocode?  Let's examine the simplest case of free-free absorption, <math>\kappa_\mathrm{T}</math> (''i.e.,'' [http://en.wikipedia.org/wiki/Thomson_scattering Thompson scattering]):
<div align="center">
<math>
\kappa_\mathrm{T}|_\mathrm{cgs} = \frac{\sigma_\mathrm{T}}{m_p} \biggl[\frac{1}{2}(1+X) \biggl(\frac{m_p}{m_u}\biggr)\biggr] = 0.2003101 (1+X)~\mathrm{cm}^2~\mathrm{g}^{-1},
</math>
</div>
where, {{Math/C_ThompsonCrossSection}}, {{Math/C_AtomicMassUnit}}, and {{Math/C_ProtonMass}} are all physical constants defined in an [[Appendix/VariablesTemplates|accompanying appendix]].  Therefore, for '''Case A''' (which assumes pure helium, so <math>X = 0)</math>, the value of this free-free (Thompson) opacity in code units is,
<div align="center">
<math>\kappa_\mathrm{T}|_\mathrm{code} = 0.2003101~\mathrm{cm}^2~\mathrm{g}^{-1} \biggl[ \frac{\kappa_\mathrm{cgs}}{\kappa_\mathrm{code}} \biggr]^{-1} = 8.434\times 10^{12}</math> .
</div>

When Thompson scattering dominates the opacity, the mean-free-path of a photon is,
<div align="center">
<math>
\ell_\mathrm{mfp} = \frac{1}{\kappa_\mathrm{T}\rho}.
</math>
</div>
This means that, in Dominic's rad-hydrocode, <math>\ell_\mathrm{mfp}</math> will be less than or equal to the size of one radial grid zone, <math>(\Delta R)_\mathrm{Nic\_code}</math>, whenever,
<div align="center">
<math>
[\kappa_\mathrm{T}]_\mathrm{code} ~\rho_\mathrm{code} \ge \frac{1}{(\Delta R)_\mathrm{Nic\_code}} = \frac{128}{\pi}
</math>
<br />&nbsp; <br />
<math>
\Rightarrow ~~~~~\rho_\mathrm{code} \ge 4.83\times 10^{-12} .
</math>
</div>

It is perhaps more instructive to write this last expression in a form that will permit us to determine how this threshold value of <math>\rho_\mathrm{code}</math> depends on the chosen set of scaling parameters.  Specifically, we can write,
<div align="center">
<math>
\rho_\mathrm{code} \ge \rho_\mathrm{threshold} \equiv \frac{128}{\pi (0.200 ~\mathrm{cm}^2~\mathrm{g}^{-1}) } \biggl[ \biggl(\frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^2 \biggl(\frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr)^{-1} \biggr] = 5.164\times 10^{-21} \biggl[ \frac{\tilde{c}^4 \tilde{a}^{1/2}}{\tilde{r}^2 \bar{\mu}^2 \tilde{g}^{1/2}} \biggr] .
</math>
</div>
To check this relation, note that when '''Case A''' parameter values are used, the combination of factors inside the last set of square brackets gives <math>9.367\times 10^{8}</math>, which produces the same value for <math>\rho_\mathrm{threshold}</math> (in code units) as before.

==Ratio of Gas Pressure to Radiation Pressure==

Let's define the following pressure ratios:

<div align="center">
<math>
\Gamma \equiv \frac{P_\mathrm{gas}}{P_\mathrm{rad}} = \frac{3 (\Re/\bar\mu)}{a_\mathrm{rad}} \biggl[ \frac{\rho}{T^3} \biggr]_\mathrm{cgs} ,
</math>
</div>
and,
<div align="center">
<math>
\beta \equiv \frac{P_\mathrm{gas}}{(P_\mathrm{gas}+P_\mathrm{rad})} = \frac{\Gamma}{1 + \Gamma} .
</math>
</div>

Following Dominic's definition of code units, above:  <math>T^3</math> should be normalized by <math>[ c^6/(\Re/\bar\mu )^3]</math>; the mass density should be normalized by the quantity <math>[a_\mathrm{rad} c^6/(\Re/\bar\mu)^4]</math>; and <math>(\Re/\bar\mu)</math> should be replaced by <math>\tilde{r}</math>.  Hence, the ratio of gas pressure to radiation pressure can be written as,
<div align="center">
<math>
\Gamma = \frac{3(\Re/\bar\mu)}{a_\mathrm{rad}} \biggl[\frac{\rho}{T^3}\biggr]_\mathrm{code} \biggl[\frac{c^6 a_\mathrm{rad}}{(\Re/\bar\mu)^4}
\biggl( \frac{\tilde{r}^4}{\tilde{c}^6 \tilde{a}} \biggr)\biggr] \biggl[\frac{(\Re/\bar\mu)^3}{c^6}
\biggl( \frac{\tilde{c}^6 }{\tilde{r}^3} \biggr)\biggr]
= 3\biggl[ \frac{\rho_\mathrm{code}}{T_\mathrm{code}^3} \biggr] \frac{\tilde{r}}{\tilde{a}} .
</math>
</div>

We might, in addition, ask what the central temperature is in an <math>n=3/2</math> polytrope.  Well, if the gas pressure dominates (''i.e.,'' if <math>\Gamma \gg 1</math>),
<div align="center">
<math>
P_\mathrm{cgs} \approx \frac{\Re}{\bar\mu} \rho_\mathrm{cgs} T_\mathrm{cgs};
</math>
</div>
and in code units,
<div align="center">
<math>
P_\mathrm{code} = \kappa_\mathrm{code} \rho_\mathrm{code}^{5/3} .
</math>
</div>

Hence,

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
T_\mathrm{code}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} T_\mathrm{cgs} \approx \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \frac{1}{(\Re/\bar\mu)} \biggl(\frac{P_\mathrm{cgs}}{\rho_\mathrm{cgs}}\biggr)

</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\Re/\bar\mu)} \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \biggl(\frac{P_\mathrm{cgs}/P_\mathrm{code}}{\rho_\mathrm{cgs}/\rho_\mathrm{code}}\biggr) \biggl(\frac{P_\mathrm{code}}{\rho_\mathrm{code}}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\Re/\bar\mu)} \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{2}\biggl( \frac{t_\mathrm{cgs}}{t_\mathrm{code}} \biggr)^{-2} \kappa_\mathrm{code}\rho_\mathrm{code}^{2/3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{\tilde{r}} ~\kappa_\mathrm{code}\rho_\mathrm{code}^{2/3} .
</math>
  </td>
</tr>

</table>

This, in turn, tells us that at the center of a polytropic star,
<div align="center">
<math>
\Gamma \approx \frac{3\tilde{r}^4}{\tilde{a}}~\kappa_\mathrm{code}^{-3} \rho_\mathrm{code}^{-1} .
</math>
</div>

This derivation will need to be modified to handle the more general case when <math>\Gamma</math> is not necessarily large.

==Super-Eddington Accretion==

In the simplest case of spherically symmetric accretion, the [http://en.wikipedia.org/wiki/Eddington_luminosity Eddington luminosity] and accretion luminosity are defined, respectively, as
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
L_\mathrm{Edd}
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
\frac{4\pi G M_\mathrm{a} m_p c}{\sigma_\mathrm{T}} ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
L_\mathrm{acc}
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
\frac{G M_\mathrm{a} \dot{M}}{R_\mathrm{a}} ,
</math>
  </td>
</tr>
</table>
where, {{Math/C_GravitationalConstant}}, {{Math/C_SpeedOfLight}}, {{Math/C_ProtonMass}}, and {{Math/C_ThompsonCrossSection}} are physical constants defined in [[Appendix/VariablesTemplates|an accompanying appendix]]; <math>M_\mathrm{a}</math> and <math>R_\mathrm{a}</math> are the mass and radius of the accreting star; and <math>\dot{M}</math> is the mass accretion rate.  Expressed in code units, these two expressions become,
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
L_\mathrm{Edd}\biggr|_\mathrm{code}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
4\pi \tilde{g} \tilde{c} \biggl[ \biggl(\frac{m_p}{\sigma_\mathrm{T}}\biggr) \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^2 \biggl( \frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr)^{-1} \biggr] [M_a]_\mathrm{code}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
4\pi \tilde{g} \tilde{c} \biggl[ 6.373\times 10^{-23} \biggl( \frac{\tilde{c}^4 \tilde{g} \tilde{a}}{\tilde{r}^4 \bar{\mu}^4} \biggr) \biggl( \frac{\tilde{r}^2 \bar{\mu}^2}{\tilde{g}^{3/2} \tilde{a}^{1/2}} \biggr) \biggr] [M_a]_\mathrm{code}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
8.009\times 10^{-22} \biggl( \frac{\tilde{c}^5 \tilde{g}^{1/2} \tilde{a}^{1/2}}{\tilde{r}^2 \bar{\mu}^2} \biggr) [M_a]_\mathrm{code} ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
L_\mathrm{acc}\biggr|_\mathrm{code}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\tilde{g} \biggl[\frac{M_\mathrm{a} \dot{M}}{R_\mathrm{a}} \biggr]_\mathrm{code} .
</math>
  </td>
</tr>
</table>
Hence,
<div align="center">
<math>
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_\mathrm{a}} \biggr]_\mathrm{code} ,
</math>
</div>
that is, the accretion will be super-Eddington <math>(f_\mathrm{acc} > 1)</math> if,
<div align="center">
<math>
[ \dot{M}]_\mathrm{code} > 8\times 10^{-22}\biggl( \frac{\tilde{c}^5 \tilde{a}^{1/2}}{\tilde{g}^{1/2} \tilde{r}^2 \bar{\mu}^2} \biggr) [R_\mathrm{a}]_\mathrm{code} .
</math>
</div>
For '''Case A''' parameters and for the Q0.7 polytropic binary model in which <math>[R_\mathrm{a}]_\mathrm{code} = 0.85</math>, the condition for super-Eddington accretion is,
<div align="center">
<math>
[ \dot{M}]_\mathrm{code} > 8\times 10^{-22} \biggl[ \frac{198^5 (0.044)^{1/2}}{ (0.44)^2 (4/3)^2} \biggr] [0.85] = 1.26 \times 10^{-10} .
</math>
</div>

For purposes of comparison with results from some previously published mass-transfer simulations, we should normalize <math>[ \dot{M}]_\mathrm{code}</math> to <math>[M_\mathrm{Donor}/ P_\mathrm{orbit}]^\mathrm{code}_0</math>, where the subscript "0" means initial values.  Doing this gives the following condition for super-Eddington accretion:
<div align="center">
<math>
[ \dot{m}]^\mathrm{code}_\mathrm{norm} > 1.26 \times 10^{-10} \biggl[ \frac{P_\mathrm{orbit}}{M_\mathrm{Donor}} \biggr]^\mathrm{code}_0 = 1.26 \times 10^{-10} \biggl[ \frac{31.19}{0.2819} \biggr] = 1.4\times 10^{-8}.
</math>
</div>

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