New Work

1. Generated skeleton of paper including most of the "polished" figures and put on sharelatex.
2. "Settled" the "enthalpy vs. thermal energy" issue and also the "gas bound to itself" issue with Eric's help.
3. Wrote up method to calculate velocities in the corotating frame.
4. Made polished plots of edge-on (to the orbital plane) views of gas density at two different zoom levels and inserted into paper.
5. Made zoomed in plots of face-on gas density similar to the edge-on ones.
6. Plots of the potential in the frame corotating with the particle orbit, centered on the companion, with and without the gas potential.
7. Plots of (\bm{\nabla}\Phi_{\mathrm{tot}}+\bm{\nabla}P/\rho)\cdot\bm{r}_2, which relates to the force per unit mass along the radial direction with respect to the companion.
8. Plot of square of tangential velocity v_\phi^2 with respect to the companion, in the corotating frame, but normalized with respect to -(\bm{\nabla}\Phi_{\mathrm{tot}}+\bm{\nabla}P/\rho)\cdot\bm{r}_2.

Skeleton of paper

Here is the ​pdf so far.

"Enthalpy vs. thermal energy" and "gas bound to itself" issues

1. Enthalpy vs thermal energy

• When assessing whether the gas is bound we evaluate whether a given quantity relating to the gas energy density is >0 (unbound) or <0 (bound).

• The question arose whether it should be the internal energy density that enters the equation (more specifically, the internal translational kinetic energy density of the gas particles related to the gas kinetic temperature), or the enthalpy per unit volume (the work per unit volume required to place a fluid element at that position within its environment, equal to the thermal energy density + the pressure).

• Since the pressure can also contribute to the outward motion of the gas (work against gravity), it makes sense to include the enthalpy per unit volume in the calculation, not just the thermal energy per unit volume.

2. Gas bound to itself

• Similarly, the question arose whether to include the negative contribution of potential energy arising from gas self-gravity.

• On one hand, one would think yes because this gravity contributes to the binding of the gas-particle system.

• However, some of the gas may be unbound and escape, and the binding energy associated with this unbound gas should not contribute. For example, for a gas parcel that is bound to itself but not bound to the rest of the system, the potential energy associated with its self-gravity should not be included.

• To resolve this conundrum it helps to be precise about what we really want to call "bound". What we are probably most interested in is whether gas is bound to the particles (the RG core and companion). We are not as interested in whether the gas is bound to itself. Therefore, it is best NOT to include the potential energy due to the gas self-gravity.

The conclusion then is that 1) we should consider the enthalpy per unit volume rather than just internal translational kinetic energy per unit volume, and 2) we should consider the potential energy associated with the gas-particle interaction only, not including the gas-gas interaction. Both of these choices work in the direction of making the gas less "bound".

Method to calculate velocities in corotating frame

Here are some ​notes.

Edge-on plots of density

• Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right. Units are grams/cm³. Color bar is the same for both plots.

• Zoomed-in with different color scheme. Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right. Units are grams/cm³. Color bar is the same for both plots.

New face-on plots of density

• As above but now face-on. Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right. Units are grams/cm³. Color bar is the same for both plots.

New face-on plots of potential

• Total potential in frame rotating with orbital angular velocity and centered on the companion in cgs units. Includes gas potential. Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right.

• Same as above but NOT including gas potential in cgs units. Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right.

• Ratio of gas potential to total potential not including gas potential in cgs units. Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right.

• Comments: the gas potential contributes at about the 20% level at most.

Plots relating to the force

• The quantity (\bm{\nabla}\Phi_{\mathrm{tot}}+\bm{\nabla}P/\rho)\cdot\bm{r}_2 in cgs units. Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right.

• Comments: Here \Phi includes the particle and gas gravitational potentials in the frame corotating with the orbit and centered on the companion. The force \bm{\nabla}\Phi therefore includes the centrifugal force due to the rotating frame. HOWEVER, it does not include the Coriolis force/unit mass -2\bm{\Omega}\times\bm{v}.

• Positive values indicate that the radial force with respect to the companion is directed outward (in the corotating frame), while negative values indicate that it is directed inward (in the corotating frame).

Plot of tangential velocity squared (with respect to the companion) normalized against the quantity plotted above (with a minus sign inserted)

• The quantity -\frac{v_\phi^2}{(\bm{\nabla}\Phi_{\mathrm{tot}}+\bm{\nabla}P/\rho)\cdot\bm{r}_2} in cgs units. Run 143 (no sub-grid accretion) on the left and Run 132 (Krumholz sub-grid accretion) on the right.

• Comments: Values are plotted from 0 to 1. Below 0, the radial force is outward (white). The colored region is for inward radial force. (But note that the colored region does have parts that are almost white, which is a drawback of the color scheme used).

• For circular motion we would expect this quantity to be equal to unity because the centrifugal force in the frame of a moving gas parcel would balance the inward radial force. Therefore < 1 suggests accretion while > 1 suggests outflow.

• However, we have not considered the radial or vertical gas motions, so it is not clear from the above quantity alone whether gas will end up moving toward or away from the companion.

• Moreover, we have neglected the Coriolis force -2\bm{\Omega}\times\bm{v}, which should be included in the analysis. Positive (counter-clockwise) values of v_\phi would produce an outward radial acceleration.

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.