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on fall back disk
Circumbinary Disk
Typical circumbinary disks are mainly made of dusts and accompanied with some gas (Jura & Kahane 1999). The disk is at low temperature, usually 100 K to several hundreds K (Bujarrabal et al. 2005). Our module is purely hydro, and we can not simulate dust formation. However, if the disk is purely gas, it will disperse or be reaccreted onto the stars.
The inner radii of a disk is 2-20 AU, the outer radii of a disk is 100-2000 AU. (I am looking for specific datas)
People has not found a stable circumbinary disk in simulation yet. It is possible that certain physics is not included in all the simulations (M. Akashi & N. Soker 2007).
Tidal Force
Strong tidal force will make the boundary of the primary star invalid. If we set the upper limit ratio of tidal force to gravitational force to be , the radius of the primary is , separation is , primary mass is , secondary mass is , then:
\[\frac{2 r G m_2}{d3}∧frac{m_2 G}{d2}<\gamma\]
That means:
\[\frac{2r}{d}<\gamma\]
We set our , if let , then . This is an example of lower limit of binary separation.
Separation, Mass Ratio and Accretion Mode
Different binary separation and mass ratio have impact on shape of the Roche Lobe. If the wind emitted from the primary star is subsonic at L1 point, it will become wind-RLOF, if the wind is supersonic at L1 point, it is more like wind accretion picture, if the envelop its self is filling the Roche Lobe of the primary, it is RLOF.
In simulations, we want to fix the initial mass of the primary and the secondary and to vary the effective mass of the primary to adjust the Roche Lobe while keeping the orbital period the same. In the Temperature section we can see how the sonic radii behaves when we change the effective gravity coefficient - the greater the effective gravity, the farther the sonic radii. Therefore, to get a RLOF, we need a high , to get a wind accretion mode, we need a low .
Radiation Pressure
The effective gravity of the primary exerted on the gas is:
\[f_g=\frac{\alpha G m_1}{r2}\]
Where
Prof. Morris pointed out that we can make alpha to be a function of radius, I believe this is true because in reality the dust formation and condensation will change the opacity and momentum coupling.
Another fact is that will decrease to zero during the helium flash since the luminosity of AGB star is exceeding the Eddington Luminosity but then increase slowly during the decaying phase and reach a certain value at quiet phase, thus making time dependent, too. If we have enough computational resource to simulation the whole period of a pulsation, I think we can see the envelop pushed out and then fall back.
Feed Back from the Secondary
As more material accrete onto the secondary, more gravitational energy will be released from the star. It is a good reason for us to let the high accretion rate to create a strong photon flux and offset some secondary's gravity (similar to primary's ). This will make set an upper limit of the accretion and slow down the reaccretion onto the secondary.
Temperature
Wind-RLOF require the wind speed to be subsonic at L1 point, this set a lower limit on temperature. According to Parker's solution:
\[r_s=\frac{\alpha G m_1}{c_{s}2}=\frac{\alpha G m_1}{k T/m}\]
Where is the mean particle mass. At temperature between to atoms start to condensate into molecules and dusts. So we can take .
The temperature in the photonsphere of an AGB star is about .
On the other hand, we do not want the dispersion to be too large, we can incorporate some cooling, especially low temperature cooling.
Mass Loss Rate
Typical AGB mass loss rate is between . We can make the quiet phase mass loss rate to be and He-flash mass loss to be . The duration of He-flash is several years.
Wind Speed
Quiet phase wind speed is greater than the combined escape velocity of the binary stars. Pulsation wind speed is less than the primary's escape velocity.
Pulsation
Some reference say the typical duration of pulsation is 200 days to 2000 days (Kyung-Won Suh 2014), which is more friendly for simulation. So, I assume the pulsation duration to be 1000 days at this moment and decays one E fold in 10000 days. The density during pulsation is 500 times higher than the quiet phase, so after 50000 days, the density will drop to quiet phase density.
From my experience, we can choose following parameters:
| parameters | wind-RLOF | wind accretion | ||
| separation | 20 | 50 | 20 | 50 |
| primary mass | 1.5 | 1.5 | 1.5 | 1.5 |
| secondary mass | 0.5 | 0.5 | 0.5 | 0.5 |
| effective gravity | 0.12 | 0.3 | 0.06 | 0.16 |
| sonic radii (if isothermal) | 12.6 | 31.4 | 6.3 | 16.8 |
| temperature | 2000 | 2000 | 2000 | 2000 |
| quiet phase wind speed | 20.0 | 30.0 | 13.0 | 17.0 |
| pulsation wind speed | 17.0 | 24.0 | 11.0 | 15.0 |
| pulsation duration (days) | 1000 | 1000 | 1000 | 1000 |
| pulsation decay time (days) | 10000 | 10000 | 10000 | 10000 |
| period (days) | 23100 | 91300 | 23100 | 91300 |
A typical simulation box for separation binary will be which will be divided by base cells. Therefore a base cell has physical dimension of . Using 2 levels of AMR will make the finest grid be , thus the primary star will have a radii of 8 cells. This is actually 8 times larger than my previous simulations.
If the simulation time range for this simulation is 1000 yrs. I guess it could take 200*120 CPU hrs or more.
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