New Work

1. Figure of radial force/unit mass (=instantaneous acceleration) pseudocolor with force/unit mass vectors overplotted.
2. Figure of Bernouilli parameter pseudocolor with contours of potential overplotted.

Plots relating to the force

• The quantity being plotted is

-\bm{\nabla}\Phi_1 -\bm{\nabla}\Phi_2 -\bm{\nabla}\Phi_\mathrm{centrif} -\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

• \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}= -\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;

• -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:
  • Vectors have been ommited from inside the softening circle of the secondary for clarity as the magnitudes were large in some cases.
  • It would be much easier to simply plot the instantaneous acceleration instead of computing it, but we do not have this quantity available in the chombo.
  • 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.
  • What we need are streamlines (or actually, pathlines) ​https://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines to really get a sense for what kinds of orbits the fluid is making in the vicinity of the secondary.

• Below are the plots of the various contributions to the force per unit mass, first for run 143 (no sub-grid accretion). In order from left to right, we have
  • Pressure term -\bm{\nabla}P/\rho;
  • Gravity terms -\bm{\nabla}\Phi_1 -\bm{\nabla}\Phi_2 -\bm{\nabla}\Phi_\mathrm{centrif} -\bm{\nabla}\Phi_\mathrm{2,CM} -\bm{\nabla}\Phi_\mathrm{gas};
  • Gravity terms -\bm{\nabla}\Phi_1 -\bm{\nabla}\Phi_2 -\bm{\nabla}\Phi_\mathrm{centrif} -\bm{\nabla}\Phi_\mathrm{2,CM} (excluding gas);
  • Gravity terms -\bm{\nabla}\Phi_1 -\bm{\nabla}\Phi_2 -\bm{\nabla}\Phi_\mathrm{gas} (lab frame);
  • Centrifugal term [-\bm{\Omega}_\mathrm{corot,2}\times(\bm{\Omega}_\mathrm{corot,2}\times\bm{r}_{,2})]\cdot\frac{\bm{r}_{,2}}{r_{,2}}.
  • Coriolis term -2\bm{\Omega}\times\bm{v}_\mathrm{corot,2};

• Now for Run 132:

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.

• We set \Phi_\mathrm{ext}= \Phi_1 +\Phi_2 +\Phi_\mathrm{centrif} +\Phi_\mathrm{2,CM} (so exclude the gas).

• Contours show \Phi_\mathrm{ext}.

• Both pseudocolor and contours are normalized by G(M_1+M_2)/r_{1,2}.

• Comments:
  • Contours show the values -2.5, -2.25, -2, -1.75, -1.5, -1.25. The contour -1 is plotted but does not appear, which means that the values are < -1 everywhere on the plot.
  • The B=0 divide gives a sense for which gas is bound to the particles (red) or unbound (blue).
  • The gas in the vicinity of the particles is mostly bound.
  • To get a sense of whether the gas is bound to the secondary, in particular, we can refer to the Roche potential contours.
  • Note that alternatively we could have omitted \Phi_1 from the calculation, to try to get a sense of whether material is bound to the secondary (rather than to both the secondary and primary together). However, this would not really be correct because it would imply that any "red" material near the primary was bound to the secondary when in reality it would be more tightly bound to the primary than to the secondary.

Discussion

• I think it might be worth including three plots in the paper (for each of the two runs). All are in the frame corotating with the particles' orbit and rotating around the secondary:
  • Tangential velocity (blue/red pseudocolor), normalized with Keplerian speed for a point potential, with velocity vectors overplotted (see ​Feb 15 post).
  • Force/unit mass (ie. instantaneous acceleration) along radial direction (blue/red pseudocolor, normalized) with acceleration vectors overplotted (this post).
  • Bernouilli parameter (blue/red pseudocolor, normalized) with external potential overplotted (this post).

• We should also have the pathlines, ideally, but if this is not possible, then streamlines, to get a better sense of the trajectory of the gas near the secondary.

Next steps

1. Finalize the remaining plots and insert into paper.
2. ADAF papers; read and discuss.
3. Text of paper.
4. Explore energy conservation in the simulations.
5. Is there enough power to drive a jet through the envelope? Estimates.