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COMMON ENVELOPE SIMULATIONS

Summary of last week's work

Worked on alpha prescription stuff for energy paper and discussed with Eric.
Continued running simulation with secondary mass halved (run 149) separation vs time plot
Started running new simulations to explore effects of softening length and resolution.
- This consisted of 3 new simulations restarted from frame 72 (t=16.7 days) of run 143 (Model A of Paper I), where the softening length and smallest resolution element had been halved.
  1. do not halve softening length nor double resolution (Run 152)
  2. halve softening length but but do not double resolution (Run 153)
  3. do not halve softening length but double resolution (Run 154)
Discussed jets with Jonathan.
Ran low resolution simulations to explore the possibility of simulating Roche lobe overflow (leading up to common envelope evolution).

Energy paper

Ivanova+2013 equation 3:
\frac{\mathrm{G} M_1 M_\mathrm{1,e}}{\lambda R_1} = \alpha_\mathrm{CE} \frac{\mathrm{G} M_\mathrm{1,c} M_2}{2}\left(\frac{1}{a_\mathrm{f}} -\frac{M_1/M_\mathrm{1,c}}{a_\mathrm{i}}\right)

New equation suggested by us:
\mathrm{G} M_\mathrm{1,e}\left(\frac{M_1}{\lambda R_1} +\frac{M_2}{2a_\mathrm{i}}\right) = \alpha_\mathrm{CE}\frac{\mathrm{G} M_\mathrm{1,c} M_2}{2} \left[\frac{1}{a_\mathrm{f}} -\frac{1}{a_\mathrm{i}}\left(2-\frac{M_1^2/M_\mathrm{1,c}+M_2}{M_1+M_2}\right)\right]

Run 143 (Model A of Paper I) has initial separation a_\mathrm{i}=49R_\odot. The Roche limit radius for q=2 as in our case is 109 R_\odot.

With a_\mathrm{final}=7 R_\odot, which is the final separation at t=40 days, we get:

	LHS	RHS
Ivanova+2013 with a_\mathrm{i}=49 R_\odot	1.9	0.23\alpha_\mathrm{CE}
Ivanova+2013 with a_\mathrm{i}=109 R_\odot	1.9	0.64\alpha_\mathrm{CE}
New equation with a_\mathrm{i}=49 R_\odot	3.1	1.23\alpha_\mathrm{CE}
New equation with a_\mathrm{i}=109 R_\odot	2.4	1.09\alpha_\mathrm{CE}

So one would never expect the envelope to be completely unbound at this point in the simulation because this would require \alpha_\mathrm{CE} > 1. Therefore, that the envelope is not completely unbound at the end of the simulation is consistent with the alpha prescription.

Then at what value of a_\mathrm{f} should the envelope be completely unbound, for a given value of \alpha_\mathrm{CE}?

	\alpha_\mathrm{CE}=0.1	\alpha_\mathrm{CE}=0.25	\alpha_\mathrm{CE}=0.5	\alpha_\mathrm{CE}=1
Ivanova+2013 with a_\mathrm{i}=49 R_\odot	a_\mathrm{f}=0.35R_\odot	a_\mathrm{f}=0.83R_\odot	a_\mathrm{f}=1.52R_\odot	a_\mathrm{f}=2.61R_\odot
Ivanova+2013 with a_\mathrm{i}=109 R_\odot	a_\mathrm{f}=0.36R_\odot	a_\mathrm{f}=0.87R_\odot	a_\mathrm{f}=1.67R_\odot	a_\mathrm{f}=3.09R_\odot
New equation with a_\mathrm{i}=49 R_\odot	a_\mathrm{f}=0.22R_\odot	a_\mathrm{f}=0.57R_\odot	a_\mathrm{f}=1.16R_\odot	a_\mathrm{f}=2.42R_\odot
New equation with a_\mathrm{i}=109 R_\odot	a_\mathrm{f}=0.28R_\odot	a_\mathrm{f}=0.71R_\odot	a_\mathrm{f}=1.45R_\odot	a_\mathrm{f}=2.97R_\odot

So if we say that there are no extra energy sources or sinks and take the most optimistic (but not very realistic) case \alpha_\mathrm{CE}=1, then the final separation is predicted to be about 3R_\odot. Compare this to the softening radius at the end of the simulation of 1.2R_\odot.

In the probably more realistic case of \alpha_\mathrm{CE}=0.1, we would have a final separation of about 0.3R_\odot. This would imply a merger if the secondary is a main sequence star, but not necessarily if the secondary is a white dwarf.

An AGB star is less tightly bound so would be more promising for ejecting the envelope.

Roche lobe overflow tests

Face-on density (zoomed in) (Run 156 a=109Rsun), equal to theoretical Roche limit separation for this system.
Face-on density (zoomed in) (Run 157 a=98Rsun)
Face-on density (zoomed in) (Run 159 a=73.5Rsun)
Face-on density (zoomed in) (Run 158 a=49Rsun)

The sound-crossing time for the RG is about 8 days or ~35 frames. This is the approximate time the star would take to deform to fill the Roche lobe.

While this deformation is happening, the secondary accretes from the ambient medium. The freefall time onto the point mass m_2 is t_\mathrm{ff}\sim\frac{\pi}{2}\frac{R^{3/2}}{\sqrt{2\mathrm{G}m_2}}. So in 5 days, or about 20 frames, this corresponds to gas being accumulated out to a sphere of radius \sim40\mathrm{R}_\odot from the secondary, corresponding to about 3\times10^{-4} \mathrm{M}_\odot of gas falling onto the secondary from the ambient medium (\rho_\mathrm{amb}=6.7\times10^{-9} grams/cc).

There are numerical problems in the low resolution runs that cause the RG to bulge out opposite to the direction of motion along the orbit.

It is probably worth starting a full resolution run to see what happens. But at which separation?

astrobear

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COMMON ENVELOPE SIMULATIONS

Summary of last week's work

Energy paper

Roche lobe overflow tests

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