User:Tohline/Appendix/Ramblings/Radiation/SummaryScalings
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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 radhydrocode. The following tables summarize some of the mathematical relationships that have been derived in that accompanying discussion.
General Relation 
Case A: 

 
where: 
; ; 

Case A ; ; ; ; ; ρ_{max} = 1; 
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 

Q07^{1} 
Binary System 
Accretor 
Donor 


q 
M_{tot} 
a 

J_{tot} 
M_{a} 


R_{a} 
M_{d} 


R_{d} 
f_{RL} 
SCF units 
0.70000 
0.02371 
0.83938 
31.19 

0.013945 
1.0000 
0.02732 
0.2728 
0.009761 
0.6077 
0.02512 
0.2888 
0.998 
conversion^{2} 














RadHydro 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 












0.996 
Other units 














^{1}Model Q07 (q = 0.700): Drawn from the first page of the accompanying PDF document. NOTE: In this PDF document, Rochelobe volumes appear to be too large by factor of 2. 
Here are some additional useful relations:
General Relation 
Case A: 

 
Case A ; ; ; ; ; ρ_{max} = 1; 
Combining the above Case A relations with the RadHydrocode 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 "superEddington" accretion (i.e., f_{Edd} > 1) when
(2) The meanfreepath, , of a photon will be less than one grid cell (ΔR)_{code} when
(3) The system is weakly relativistic because,
© 2014  2020 by Joel E. Tohline 