grubaugh "synchronous retrograde orbit" (preliminary status)

I believe the "sphere of influence" in this case is the distance at which the sun's tidal forces are stronger than jupiter's direct gravitational forces. And with jupiter at 1au from the sun, this occurs at about 0.1au from jupiter. (if I've done my arithmetic right). This is exactly why it's misleading to say that "jupiter and saturn are orbiting each other" in this scenario. In fact, saturn's acceleration due to jupiter is 100 times less than its acceleration due to the sun.

Nevertheless, the orbit turns out to be quite stable. When N-body simulated for 1000 simulated-years, jupiter and saturn keep much the same relationship WRT the sun; the line connecting keeps pointed at roughly the same direction against the inertial (fixed stars) background, with very little year-to-year drift (though with a yearly "wobble" of a bit more than 30 degrees).

Basically, jupiter and saturn's mutual perturbations are cyclic, and thus the orbits don't "break up", even over the long term. Briefly, I think Grubaugh's calculations for "synchronous retrograde"ness have this effect, because of the fact that epicycles resemble elipses, but this will no dout be discussed further by Bass/Grubaugh if-and-when.

A graphic of the first 100 megaseconds, with the jupiter-saturn line depiced at 4-megasecond intervals, is available as a gif image. The jupiter/saturn separation distance varied from 54 to 111 gigameters, and the angle from -28 to +11 degrees. The average jup/sat orientation drifts a small amount each orbit, but the general pattern remains constant: they are both on the "same side" of the sun, indefinitely.

initial conditions

    MASS ONE  (Sun)
	Mass = 1.90E +08
	Density = 350E +04
	x coordinate = 0
	y coordinate = 0
	x velocity = 0
	y velocity = 0
	orbits: n   stationary: n
    MASS TWO
	Mass = 1.90E +05
	Density = 3.00E +03
	x coordinate = 1.50E +02
	y coordinate = 0
	x velocity = 0
	y velocity = 9.93E -03
	orbits: y  stationary: n
    MASS THREE
	Mass = 5.68E +04
	Density = 1.00E +03
	x coordinate = 2.02E +02
	y coordinate = 0
	x velocity = 0
	y velocity = 7.32E -03
	orbits: y  stationary: n 

    What are the units?  What is the use of "density"?

    My guesses: mass is in units of 10^22 kg, position in 10^6 km,
                and velocity in 10^6 m/sec, 
                and density has something to do with display parameters
                of the object on the screen
    
    This interpretation makes the most sense, presuming the data is for
    a jupiter/saturn scenario of near 1au radius.

late breaking news

It turns out, I've fudged the masses (and hence the orbit times). I presumed the initial conditions were MKS or CGS, and so just tinkered with the exponents until I got nearest-circular orbits. But... the only way to do that and keep all the mantissas in the cgs/kgs systems, is to make the masses all 1/10 their "real" solar-system values. So, if you try to reproduce what I've done, be aware that I used this initial data:

                 kg       x meters    y meters  vx m/s   vy m/s
       sun     1.9e+29      0           0         0      0
       jup     1.9e+26      1.50e11     0         0      9.93e3
       sat     5.68e+25     2.02e11     0         0      7.32e3