Jet53man at 21:59, 4 July 2024

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	Constant in the polytropic equation of state, {{~~User:Tohline/~~Math/EQ_Polytrope01}}		Constant in the polytropic equation of state, {{Math/EQ_Polytrope01}}
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__FORCETOC__


=Polytropic Models of Close Binary Star Systems=

Over the past half-a-dozen years, Patrick Motl, Mario D'Souza, and Wes Even have used the Hachisu SCF technique to construct 3D equilibrium models of synchronously rotating, tidally distorted binary polytropes. To date, four of these models have been used extensively as initial states for our dynamical simulations of binary mass-transfer. Various properties of these four SCF-code-generated models are summarized in the following table; the listed parameters are:
<table border="0" align="center" cellpadding="2">
<tr>
<td align="right">
<math>q \equiv M_d/M_a</math>  :
</td>
<td align="left">
System mass ratio
</td>
</tr>

<tr>
<td align="right">
<math>M</math>  :
</td>
<td align="left">
Mass
</td>
</tr>

<tr>
<td align="right">
<math>a</math>  :
</td>
<td align="left">
Binary separation
</td>
</tr>

<tr>
<td align="right">
<math>\Omega</math>  :
</td>
<td align="left">
Orbital angular velocity
</td>
</tr>

<tr>
<td align="right">
<math>J_\mathrm{tot}</math>  :
</td>
<td align="left">
Total angular momentum
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\mathrm{max}</math>  :
</td>
<td align="left">
Maximum (central) density
</td>
</tr>

<tr>
<td align="right">
<math>K_\mathrm{n}</math>  :
</td>
<td align="left">
Constant in the polytropic equation of state, {{User:Tohline/Math/EQ_Polytrope01}}
</td>
</tr>

<tr>
<td align="right">
<math>V</math>  :
</td>
<td align="left">
Volume occupied by the star or by the Roche Lobe (RL) surrounding the star
</td>
</tr>

<tr>
<td align="right">
<math>R\equiv [3V/(4\pi)]^{1/3}</math>  :
</td>
<td align="left">
Mean stellar radius
</td>
</tr>

<tr>
<td align="right">
<math>f_\mathrm{RL} \equiv V/V_\mathrm{RL}</math>  :
</td>
<td align="left">
Roche-lobe filling factor
</td>
</tr>

</table>

<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">
'''Model'''
</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">
 
</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>\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">
'''Q13'''
</td>
<td align="center" colspan="1">
1.323
</td>
<td align="center" colspan="1">
0.0309
</td>
<td align="center" colspan="1">
0.8882
</td>
<td align="center" colspan="1">
0.2113
</td>
<td align="center" colspan="1">
<math>1.40\times 10^{-3}</math>
</td>

<td align="center" colspan="1">
0.0133
</td>
<td align="center" colspan="1">
1.0000
</td>
<td align="center" colspan="1">
0.0264
</td>
<td align="center" colspan="1">
0.2672
</td>

<td align="center" colspan="1">
0.0176
</td>
<td align="center" colspan="1">
0.6000
</td>
<td align="center" colspan="1">
0.0372
</td>
<td align="center" colspan="1">
0.3509
</td>
<td align="center" colspan="1">
0.968
</td>
</tr>

<tr>
<td colspan="1" align="center">
'''Q07'''
</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">
0.20144
</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">
'''Q05'''
</td>
<td align="center" colspan="1">
0.500
</td>
<td align="center" colspan="1">
<math>9.216\times 10^{-3}</math>
</td>
<td align="center" colspan="1">
0.8764
</td>
<td align="center" colspan="1">
0.1174
</td>
<td align="center" colspan="1">
<math>1.97\times 10^{-4}</math>
</td>

<td align="center" colspan="1">
<math>6.143\times 10^{-3}</math>
</td>
<td align="center" colspan="1">
1.0000
</td>
<td align="center" colspan="1">
0.016
</td>
<td align="center" colspan="1">
0.2067
</td>

<td align="center" colspan="1">
<math>3.073\times 10^{-3}</math>
</td>
<td align="center" colspan="1">
0.235
</td>
<td align="center" colspan="1">
0.016
</td>
<td align="center" colspan="1">
0.2689
</td>
<td align="center" colspan="1">
0.898
</td>
</tr>

<tr>
<td colspan="1" align="center">
'''Q04'''
</td>
<td align="center" colspan="1">
0.4085
</td>
<td align="center" colspan="1">
0.02399
</td>
<td align="center" colspan="1">
0.8169
</td>
<td align="center" colspan="1">
0.2112
</td>
<td align="center" colspan="1">
<math>7.794\times 10^{-4}</math>
</td>

<td align="center" colspan="1">
0.01703
</td>
<td align="center" colspan="1">
1.0000
</td>
<td align="center" colspan="1">
0.03119
</td>
<td align="center" colspan="1">
0.2918
</td>

<td align="center" colspan="1">
0.006957
</td>
<td align="center" colspan="1">
0.71
</td>
<td align="center" colspan="1">
0.01904
</td>
<td align="center" colspan="1">
0.2453
</td>
<td align="center" colspan="1">
0.996
</td>
</tr>

<tr>
<td align="left" colspan="15">
References:
* Model '''Q13''' (<math>q = 1.323</math>): Table 4 in [http://iopscience.iop.org/0004-637X/643/1/381/pdf/63230.web.pdf publication DMTF06]
* Model '''Q07''' (<math>q = 0.700</math>): 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>
* Model '''Q05''' (<math>q = 0.500</math>): Table 5 in [http://iopscience.iop.org/0004-637X/643/1/381/pdf/63230.web.pdf publication DMTF06]
* Model '''Q04''' (<math>q = 0.4085</math>): Table 1 in [http://iopscience.iop.org/0004-637X/670/2/1314/pdf/71427.web.pdf publication MFTD07]
</td>
</tr>
</table>

All of the parameter values listed in these tables are specified in dimensionless ''polytropic units'', defined as follows:

<table border="1" cellpadding="20" align="center" width="75%">
<tr>
<td align="center">
<font color="blue"><b>Polytropic Units</b></font>
</td>
</tr>
<tr>
<td align="left">
Here, ''Polytropic Units'' are defined such that the radial extent of the computational grid for the self-consistent-field (SCF) model, <math>R_\mathrm{edge}</math>,
the maximum density of one binary component, <math>\rho^\mathrm{max}_\mathrm{Accretor}</math>, and the gravitational
constant, <math>G</math>, are all unity, that is,
<div align="center">
<math>G = \rho^\mathrm{max}_\mathrm{Accretor} = R_\mathrm{edge} = 1</math>.
</div>
In an [http://www.phys.lsu.edu/~tohline/clayton/PolytropicUnits.pdf accompanying PDF document], we explain how to convert from this set of dimension code units to real (''e.g.,'' cgs) units.
</td>
</tr>
</table>

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Appendix/PolytropicBinaries - Revision history

Jet53man at 21:59, 4 July 2024

Jet53man: Created page with "__FORCETOC__ =Polytropic Models of Close Binary Star Systems= Over the past half-a-dozen years, Patrick Motl, Mario D'Souza, and Wes E..."

Jet53man: Created page with "FORCETOC =Polytropic Models of Close Binary Star Systems= Over the past half-a-dozen years, Patrick Motl, Mario D'Souza, and Wes E..."