User:Tohline/Appendix/PolytropicBinaries
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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:
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<math>q \equiv M_d/M_a</math> : |
System mass ratio |
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<math>M</math> : |
Mass |
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<math>a</math> : |
Binary separation |
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<math>\Omega</math> : |
Orbital angular velocity |
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<math>J_\mathrm{tot}</math> : |
Total angular momentum |
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<math>\rho^\mathrm{max}</math> : |
Maximum (central) density |
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<math>K_\mathrm{n}</math> : |
Constant in the polytropic equation of state, <math>~P = K_\mathrm{n} \rho^{1+1/n}</math> |
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<math>V</math> : |
Volume occupied by the star or by the Roche Lobe (RL) surrounding the star |
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<math>R</math> : |
Mean stellar radius, where <math>R \equiv [3V/(4\pi)]^{1/3}</math> |
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<math>f_\mathrm{RL} \equiv V/V_\mathrm{RL}</math> : |
Roche-lobe filling factor |
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Properties of (<math>n=3/2</math>) Polytropic Binary Systems |
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Model |
Binary System |
Accretor |
Donor |
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<math>q</math> |
<math>M_\mathrm{tot}</math> |
<math>a</math> |
<math>\Omega</math> |
<math>J_\mathrm{tot}</math> |
<math>M_a</math> |
<math>\rho^\mathrm{max}_a</math> |
<math>K^a_{3/2}</math> |
<math>R_a</math> |
<math>M_d</math> |
<math>\rho^\mathrm{max}_d</math> |
<math>K^d_{3/2}</math> |
<math>R_d</math> |
<math>f_\mathrm{RL}</math> |
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Q13 |
1.323 |
0.0309 |
0.8882 |
0.2113 |
<math>1.40\times 10^{-3}</math> |
0.0133 |
1.0000 |
0.0264 |
0.2672 |
0.0176 |
0.6000 |
0.0372 |
0.3509 |
0.968 |
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Q07 |
0.70000 |
0.02371 |
0.83938 |
0.20144 |
<math>8.938\times 10^{-4}</math> |
0.013945 |
1.0000 |
0.02732 |
0.2728 |
0.009761 |
0.6077 |
0.02512 |
0.2888 |
0.998 |
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Q05 |
0.500 |
<math>9.216\times 10^{-3}</math> |
0.8764 |
0.1174 |
<math>1.97\times 10^{-4}</math> |
<math>6.143\times 10^{-3}</math> |
1.0000 |
0.016 |
0.2067 |
<math>3.073\times 10^{-3}</math> |
0.235 |
0.016 |
0.2689 |
0.898 |
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Q04 |
0.4085 |
0.02399 |
0.8169 |
0.2112 |
<math>7.794\times 10^{-4}</math> |
0.01703 |
1.0000 |
0.03119 |
0.2918 |
0.006957 |
0.71 |
0.01904 |
0.2453 |
0.996 |
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References:
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All of the parameter values listed in these tables are specified in dimensionless polytropic units, defined as follows:
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Polytropic Units |
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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, <math>G = \rho^\mathrm{max}_\mathrm{Accretor} = R_\mathrm{edge} = 1</math>. In an accompanying PDF document, we explain how to convert from this set of dimension code units to real (e.g., cgs) units. |
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© 2014 - 2021 by Joel E. Tohline |