Appendix/Ramblings/Radiation/SummaryScalings - Revision history

https://selfgravitatingfluids.education/JETohline/index.php?action=history&feed=atom&title=Appendix%2FRamblings%2FRadiation%2FSummaryScalings Appendix/Ramblings/Radiation/SummaryScalings - Revision history 2026-04-20T15:43:36Z Revision history for this page on the wiki MediaWiki 1.43.1 https://selfgravitatingfluids.education/JETohline/index.php?title=Appendix/Ramblings/Radiation/SummaryScalings&diff=1888&oldid=prev Joel2: Created page with "__FORCETOC__  =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. <!-- ***************************************** * * * FIRST TABLE * * ***********..." 2024-07-04T21:35:37Z

<p>Created page with "__FORCETOC__  =Summary of Scalings= On <a href="/JETohline/index.php/Appendix/Ramblings/Radiation/CodeUnits" title="Appendix/Ramblings/Radiation/CodeUnits">an accompanying Wiki page</a> 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. <br /> <br /> =Summary of Scalings=<br /> <br /> 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.<br /> <br /> <br /> <br /> <table border="4" align="center" cellpadding="8" width="95%"><br /> <tr><br /> <td colspan="3" align="center" width="80%"><br /> <b>General Relation</b><br /> </td><br /> <td colspan="1" align="center"><br /> <b>Case A</b>:<br /> </td><br /> </tr><br /> <br /> <tr><td colspan="4" align="center"><table border="0" cellpadding="3" cellspacing="10"><br /> <br /> <tr><br /> <td align="right"><br /> <math><br /> \frac{m_\mathrm{cgs}}{m_\mathrm{code}}<br /> </math><br /> </td><br /> <td align="center"><br /> <math><br /> =<br /> </math><br /> </td><br /> <td align="left"><br /> <math><br /> 0.40375~\mu_e^2 M_\mathrm{Ch} \biggl( \frac{\tilde{g}^3 \tilde{a}}{\tilde{r}^4 \bar{\mu}^4 } \biggr)^{1/2}<br /> </math><br /> </td><br /> <td align="left"><br /> <math>= ~~2.8094\times 10^{33}~\mathrm{g} </math><br /> </td><br /> </tr><br /> <br /> <tr><br /> <td align="right"><br /> <math><br /> \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}}<br /> </math><br /> </td><br /> <td align="center"><br /> <math><br /> =<br /> </math><br /> </td><br /> <td align="left"><br /> <math><br /> 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}<br /> </math><br /> </td><br /> <td align="left" width="30%"><br /> <math>=~~ 8.179\times 10^{9}~\mathrm{cm}</math><br /> </td><br /> </tr><br /> <br /> <tr><br /> <td align="right"><br /> <math><br /> \frac{t_\mathrm{cgs}}{t_\mathrm{code}}<br /> </math><br /> </td><br /> <td align="center"><br /> <math><br /> =<br /> </math><br /> </td><br /> <td align="left"><br /> <math><br /> 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}<br /> </math><br /> </td><br /> <td align="left"><br /> <math>= ~~54.02~\mathrm{s}</math><br /> </td><br /> </tr><br /> <br /> <br /> <tr><br /> <td align="right"><br /> <math><br /> \frac{T_\mathrm{cgs}}{T_\mathrm{code}}<br /> </math><br /> </td><br /> <td align="center"><br /> <math><br /> =<br /> </math><br /> </td><br /> <td align="left"><br /> <math><br /> 1.08095\times 10^{13} ~\biggl( \frac{\tilde{r} \bar\mu}{\tilde{c}^2} \biggr)<br /> </math><br /> </td><br /> <td align="left"><br /> <math>= ~~1.618 \times 10^8~\mathrm{K}</math><br /> </td><br /> </tr><br /> <br /> </table></td></tr><br /> <br /> <tr><br /> <td colspan="1" align="right" width="10%"><br /> where:<br /> </td><br /> <td colspan="3" align="left"><br /> <math><br /> \mu_e^2 M_\mathrm{Ch} = 1.14169\times 10^{34}~\mathrm{g}<br /> </math>; &nbsp; &nbsp;<br /> <math><br /> \mu_e \ell_\mathrm{Ch} = 7.71311\times 10^{8}~\mathrm{cm}<br /> </math>; &nbsp; &nbsp;<br /> <math><br /> \mu_e^{1/2} t_\mathrm{Ch} = 3.90812~\mathrm{s}<br /> </math><br /> </td><br /> </tr><br /> <br /> <tr><br /> <td colspan="4" align="center"><br /> <b>Case A</b>&nbsp;&nbsp;<br /> <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><br /> </td><br /> </tr><br /> <br /> </table><br /> <br /> <br /> 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.<br /> <br /> <span id="TableProperties"><table align="center" border="1" cellpadding="8" width="95%"><br /> <tr><br /> <td align="center" colspan="15"><br /> '''<font color="darkblue"><br /> Properties of (<math>n=3/2</math>) Polytropic Binary Systems<br /> </font>'''<br /> </td><br /> </tr><br /> <tr><br /> <td colspan="1" align="center"><br /> '''Q07'''<sup>1</sup><br /> </td><br /> <td align="center" colspan="5"><br /> '''Binary System'''<br /> </td><br /> <td align="center" colspan="4"><br /> '''Accretor'''<br /> </td><br /> <td align="center" colspan="5"><br /> '''Donor'''<br /> </td><br /> </tr><br /> <br /> <tr><br /> <td colspan="1" align="center"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math>q</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>M_\mathrm{tot}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>a</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>P = \frac{2\pi}{\Omega}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>J_\mathrm{tot}</math><br /> </td><br /> <br /> <td align="center" colspan="1"><br /> <math>M_a</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>\rho^\mathrm{max}_a</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>K^a_{3/2}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>R_a</math><br /> </td><br /> <br /> <td align="center" colspan="1"><br /> <math>M_d</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>\rho^\mathrm{max}_d</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>K^d_{3/2}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>R_d</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>f_\mathrm{RL}</math><br /> </td><br /> </tr><br /> <br /> <tr><br /> <td colspan="1" align="center"><br /> '''SCF''' units<br /> </td><br /> <td align="center" colspan="1"><br /> 0.70000<br /> </td><br /> <td align="center" colspan="1"><br /> 0.02371<br /> </td><br /> <td align="center" colspan="1"><br /> 0.83938<br /> </td><br /> <td align="center" colspan="1"><br /> 31.19<br /> </td><br /> <td align="center" colspan="1"><br /> <math>8.938\times 10^{-4}</math><br /> </td><br /> <br /> <td align="center" colspan="1"><br /> 0.013945<br /> </td><br /> <td align="center" colspan="1"><br /> 1.0000<br /> </td><br /> <td align="center" colspan="1"><br /> 0.02732<br /> </td><br /> <td align="center" colspan="1"><br /> 0.2728<br /> </td><br /> <br /> <td align="center" colspan="1"><br /> 0.009761<br /> </td><br /> <td align="center" colspan="1"><br /> 0.6077<br /> </td><br /> <td align="center" colspan="1"><br /> 0.02512<br /> </td><br /> <td align="center" colspan="1"><br /> 0.2888<br /> </td><br /> <td align="center" colspan="1"><br /> 0.998<br /> </td><br /> </tr><br /> <br /> <tr><br /> <td colspan="1" align="center"><br /> conversion<sup>2</sup><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3<br /> </math><br /> </td><br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)<br /> </math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^5<br /> </math><br /> </td><br /> <br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3<br /> </math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^2<br /> </math><br /> </td><br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)<br /> </math><br /> </td><br /> <br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^3<br /> </math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)^2<br /> </math><br /> </td><br /> <td align="center" colspan="1"><br /> <math><br /> \biggl( \frac{\ell_\mathrm{code}}{\ell_\mathrm{SCF}} \biggr)<br /> </math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> </tr><br /> <br /> <tr><br /> <td colspan="1" align="center"><br /> '''Rad-Hydro''' units<br /> </td><br /> <td align="center" colspan="1"><br /> 0.70000<br /> </td><br /> <td align="center" colspan="1"><br /> 0.6847<br /> </td><br /> <td align="center" colspan="1"><br /> 2.5752<br /> </td><br /> <td align="center" colspan="1"><br /> 31.19<br /> </td><br /> <td align="center" colspan="1"><br /> 0.24293<br /> </td><br /> <br /> <td align="center" colspan="1"><br /> 0.4027<br /> </td><br /> <td align="center" colspan="1"><br /> 1.0000<br /> </td><br /> <td align="center" colspan="1"><br /> 0.2571<br /> </td><br /> <td align="center" colspan="1"><br /> 0.8369<br /> </td><br /> <br /> <td align="center" colspan="1"><br /> 0.28187<br /> </td><br /> <td align="center" colspan="1"><br /> 0.6077<br /> </td><br /> <td align="center" colspan="1"><br /> 0.2364<br /> </td><br /> <td align="center" colspan="1"><br /> 0.88603<br /> </td><br /> <td align="center" colspan="1"><br /> 0.998<br /> </td><br /> </tr><br /> <br /> <tr><br /> <td colspan="1" align="center"><br /> '''cgs''' units<br /> </td><br /> <td align="center" colspan="1"><br /> 0.70000<br /> </td><br /> <td align="center" colspan="1"><br /> <math>1.924\times 10^{33}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>2.106\times 10^{10}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>1.687\times 10^{3}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>1.924\times 10^{33}</math><br /> </td><br /> <br /> <td align="center" colspan="1"><br /> <math>1.132\times 10^{33}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>5.136\times 10^{3}</math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math>6.845\times 10^{9}</math><br /> </td><br /> <br /> <td align="center" colspan="1"><br /> <math>7.921\times 10^{32}</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>3.121\times 10^{3}</math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math>7.247\times 10^{9}</math><br /> </td><br /> <td align="center" colspan="1"><br /> 0.996<br /> </td><br /> </tr><br /> <br /> <tr><br /> <td colspan="1" align="center"><br /> '''Other''' units<br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math>0.967 M_\odot</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>0.303 R_\odot</math><br /> </td><br /> <td align="center" colspan="1"><br /> <math>28.1~\mathrm{min}</math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <br /> <td align="center" colspan="1"><br /> <math>0.569 M_\odot</math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math>0.0984 R_\odot</math><br /> </td><br /> <br /> <td align="center" colspan="1"><br /> <math>0.398 M_\odot</math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> <td align="center" colspan="1"><br /> <math>0.1042 R_\odot</math><br /> </td><br /> <td align="center" colspan="1"><br /> &nbsp;<br /> </td><br /> </tr><br /> <br /> <tr><br /> <td align="left" colspan="15"><br /> <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 /><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]].<br /> </td><br /> </tr><br /> </table><br /> </span><br /> <br /> <br /> <br /> Here are some additional useful relations:<br /> <table border="4" align="center" cellpadding="8"><br /> <tr><br /> <td colspan="3" align="center"><br /> <b>General Relation</b><br /> </td><br /> <td colspan="1" align="center" width="32%"><br /> <b>Case A</b>:<br /> </td><br /> </tr><br /> <br /> <tr><td colspan="4" align="center"><table border="0" cellpadding="3" cellspacing="10"><br /> <br /> <tr><br /> <td align="right"><br /> <math><br /> f_\mathrm{Edd} \equiv \frac{L_\mathrm{acc}}{L_\mathrm{Edd}}<br /> </math><br /> </td><br /> <td align="center"><br /> <math><br /> =<br /> </math><br /> </td><br /> <td align="left"><br /> <math><br /> 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}<br /> </math><br /> </td><br /> <td align="left"><br /> <math>= ~~6.74\times 10^9 \biggl[ \frac{\dot{M}}{R_a} \biggr]_\mathrm{code}</math><br /> </td><br /> </tr><br /> <br /> <tr><br /> <td align="right"><br /> <math><br /> \frac{\rho_\mathrm{threshold}}{\rho_\mathrm{max}} \equiv \frac{1}{\rho_\mathrm{max}\kappa_\mathrm{T} (\Delta R)}<br /> </math><br /> </td><br /> <td align="center"><br /> <math><br /> =<br /> </math><br /> </td><br /> <td align="left"><br /> <math><br /> 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}<br /> </math><br /> </td><br /> <td align="left" width="30%"><br /> <math>=~~ 4.83\times 10^{-12}</math><br /> </td><br /> </tr><br /> <br /> <tr><br /> <td align="right"><br /> <math><br /> \Gamma \equiv \frac{P_\mathrm{gas}}{P_\mathrm{rad}}<br /> </math><br /> </td><br /> <td align="center"><br /> <math><br /> =<br /> </math><br /> </td><br /> <td align="left"><br /> <math><br /> \biggl( \frac{3\tilde{r}}{\tilde{a}} \biggr) \biggl[ \frac{ \rho }{T^3} \biggr]_\mathrm{code}<br /> </math><br /> </td><br /> <td align="left"><br /> <math>= ~~30 \biggl[ \frac{ \rho }{T^3} \biggr]_\mathrm{code}</math><br /> </td><br /> </tr><br /> <br /> <tr><br /> <td align="right"><br /> <math><br /> \frac{v_\mathrm{circ}}{c} \equiv \frac{2\pi a_\mathrm{separation}}{c P_\mathrm{orbit}}<br /> </math><br /> </td><br /> <td align="center"><br /> <math><br /> =<br /> </math><br /> </td><br /> <td align="left"><br /> <math><br /> \frac{2\pi}{\tilde{c}} \biggl[\frac{a_\mathrm{sep}}{P_\mathrm{orb}}\biggr]_\mathrm{code}<br /> </math><br /> </td><br /> <td align="left"><br /> <math>= ~~0.032 \biggl[\frac{a_\mathrm{sep}}{P_\mathrm{orb}}\biggr]_\mathrm{code}</math><br /> </td><br /> </tr><br /> <br /> </table></td></tr><br /> <br /> <br /> <tr><br /> <td colspan="4" align="center"><br /> <b>Case A</b>&nbsp;&nbsp;<br /> <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><br /> </td><br /> </tr><br /> <br /> </table><br /> <br /> <br /> 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:<br /> <br /> (1)&nbsp;&nbsp;The system will experience "super-Eddington" accretion (''i.e.,'' <math>f_\mathrm{Edd} > 1</math>) when <br /> <div align="center"><br /> <math><br /> [\dot{M}]_\mathrm{code} > 1.3\times 10^{-10} .<br /> </math><br /> </div><br /> <br /> (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<br /> <div align="center"><br /> <math><br /> [\rho]_\mathrm{code} > \rho_\mathrm{threshold} = 5\times 10^{-12} .<br /> </math><br /> </div><br /> <br /> (3)&nbsp;&nbsp;The system is weakly relativistic because,<br /> <div align="center"><br /> <math><br /> \frac{v_\mathrm{circ}}{c} = 0.0026 .<br /> </math><br /> </div><br /> <br /> {{ SGFfooter }}</div>

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