Thoughts on binary sims

So Eric and I were discussing the specific angular momenta in the gas yesterday and you can calculate the velocity of the wind at any given point (assuming the secondary does not influence it's motion) by solving for the characteristic that leaves from the surface of the primary with a velocity vector pointed at the position. It is difficult to solve exactly and some approximations must be made… If we assume that the wind velocity is much greater than the orbital velocity, and that the distance to a given position is much greater than the primary's orbital radius, then we can easily esimate the time delay (t-t0) between the wind leaving the primary at t=t0 and arriving at a position x at time t as t-t0=|x|/vwind. We can then determine the location of the primary at this time, and then assuming that the primary's radius is << |x|, we can estimate the direction of the wind as being the same as the vector going from the primary towards |x| or v(x,t) x (x-xp(t0)) = 0. Then to calculate the magnitude of the wind we know that it will be constrained between vw +- vorbit depending on the degree to which vp(t_0) is aligned with (x-xp(t0))

See the following illustration

The upshot, is that the secondary tends to be in a region where the specific angular momenta from the primary is positive (which is retrograde)…

Here is the 'corrected' approximate solution plotted (here the orbit is clockwise and the z-axis if pointing out of the plane)

Click here​ for the original version. Note where |x| = 1 - there is no difference

And here is another more exact solution where the 'time delay' is not calculated based on the distance from the origin, but by the distance to the current location of the primary. The exact solution would require solving a transcendental equation to find the distance to the position of the primary at the retarted time…

and here is the calculations that went into the above approximation

And here is the exact solution calculated from the simulation

Here the yellow corresponds to positive specific angular momenta - and the system as a whole has negative angular momenta - so yellow ⇒ retrograde disk

Here is a few frames later after the secondary has begun to torque the wind…

The secondary should likely give a net torque that is prograde (so more blue then yellow) but positive specific angular momenta does not automatically form a disk since the disk has to be orbiting the system's center of mass as well as orbiting the secondary. So does the infalling material have to have a net specific angular momenta that is greater than the secondary to form a prograde disk?

And here is a zoomed out version of the approximate solution… Note the whole thing rotates with the same pattern speed - so just spin your monitor to see a movie :)

Also was thinking about what the flow looked like from the point of view of the secondary - I don't see any shear in the flow, but there is of course divergence. But in the co-rotating frame, the secondary sees a nearly uniform flow subject to a Coriolis force as well as a gravitational force… something like

So setting up the initial solution can be accomplished by the following:

First start with the assumption that the retarted time is the current time

DO

Calculate the displacement vector from the primary at the retarded time

Calculate the wind normal so that

It is possible to solve for the unit vector

Calculate the wind velocity from the primary

Update the retarded time using the new distance and wind speed

END DO

The only problem occurs when there are multiple solutions for the retarded time…

This will occur once we reach distances of order

If we switch to a rotating frame that rotates counter to the orbit so the angular speed is , then

and

so that

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