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COMMON ENVELOPE SIMULATIONS

New Work

Corrected the main figures from last post

Plots relating to the force

The quantity being plotted is

-\bm{\nabla}\Phi_1 -\bm{\nabla}\Phi_2 -\bm{\nabla}\Phi_\mathrm{centrif,2} -\bm{\nabla}\Phi_\mathrm{2,CM} -\bm{\nabla}\Phi_\mathrm{gas} -\bm{\nabla}P/\rho -2\bm{\Omega}\times\bm{v}_\mathrm{corot,2} -[\bm{\Omega}_\mathrm{corot,2}\times(\bm{\Omega}_\mathrm{corot,2}\times\bm{r}_{,2})]\cdot\frac{\bm{r}_{,2}}{r_{,2}}.

where (repeated from last post):

\Phi_1=-GM_1/r_1 and \Phi_2=-GM_2/r_2 are the potentials due to the primary (RG core) and secondary (companion), respectively. Actually, inside the softening radius, we have also corrected for the spline potential by including extra terms, not written out here;

\Phi_\mathrm{gas} is the potential of the gas;

\Phi_\mathrm{centrif,2}= -\frac{1}{2}\Omega^2r_{,2}^2 is the centrifugal potential in the frame corotating with the particles' orbit. Here (0,0,\Omega) is the angular velocity of this frame with respect to the lab frame and r_{,2} is the distance from the secondary in the x-y plane;

\Phi_\mathrm{2,CM}= \Omega^2[(x_\mathrm{CM}-x_2)(x-x_2) +(y_\mathrm{CM}-y_2)(y-y_2)] is the potential due to the shift of the rotation axis of the rotating frame from the CM to the secondary;

Last post, there was an error in \Phi_\mathrm{2,CM} which had the incorrect sign.

-2\bm{\Omega}\times\bm{v}_\mathrm{corot,2} is the Coriolis term;

\bm{\Omega}_\mathrm{corot,2}= [0, 0, (\bm{r_{,2}}\times\bm{v}_\mathrm{corot,2})_z]/r_{,2}^2 is the angular velocity of gas about the secondary in the frame corotating about the secondary with the instantaneous orbital angular velocity of the particles, and -\bm{\Omega}_\mathrm{corot,2}\times(\bm{\Omega}_\mathrm{corot,2}\times\bm{r}_{,2})\cdot\frac{\bm{r}_{,2}}{r_{,2}} is the centrifugal force due to the motion of the gas IN THE COROTATING FRAME (i.e. we have already accounted for the rotation of the reference frame, but here we account for the rotation of the gas within the rotating reference frame);

We normalize with respect to GM_2/r_{1,2}^2, that is, the magnitude of g due to the secondary alone at the location of the primary.

Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right.

Comments (itmes repeated from last post):
- Vectors have been ommited from inside the softening circle of the secondary for clarity as the magnitudes were large in some cases.
- In both cases, some of the gas around the secondary is accelerating inward while some is accelerating outward.
- So it is probably incorrect to conclude that gas is bound to the secondary.

Plots relating to the energy

The quantity being plotted in pseudocolor is the Bernouilli parameter B = \frac{1}{2}v^2 +h +\Phi_\mathrm{ext}, where the first term is the specific kinetic energy, the second term is the specific enthalpy (=(E_\mathrm{thermal}+P)/\rho), and the last term is the external potential.

Since E_\mathrm{thermal}/\rho= \frac{1}{\gamma-1}\frac{k_\mathrm{B}T}{\mu} and P/\rho= \frac{k_\mathrm{B}T}{\mu}, we have h= \frac{\gamma}{\gamma-1}\frac{k_\mathrm{B}T}{\mu}= \gamma E_\mathrm{thermal}. Here E_\mathrm{thermal} is just the total energy density minus the bulk KE density of the gas.

For plots in the frame rotating around the secondary, we set \Phi_\mathrm{ext}= \Phi_1 +\Phi_2 +\Phi_\mathrm{centrif} +\Phi_\mathrm{2,CM} (so exclude the gas).