There was a question back in Yahoo!Answer, which catch my interest. The question was, what will happen if the Moon have a moon. Actually there can be more than one answer to this question, or may be there are unlimited number of answer to this question, since there are so many possibilities on the mass, the initial position vector and the initial velocity vector of the moon of the Moon.
For the sake of simplicity lets assume, that the name of the moon of Moon is Usagi( うさぎ , means rabbit, refering to old belief that there are rabbits in the moon ), which mass is 2.3221E20 kg, or about 3.88688E-5 Earth Mass. Then we can predict what is going to happen by using NCGS or PPNCGS Gravity Simulator
Stable moon of Moon
For Usagi to orbit Moon in stable orbit, it may not be too far from the Moon, or Earth gravity field is going to disrupt the orbit.
For example, if the distance of Usagi and Moon is about 24544 km, and the orbital velocity of Usagi is 447.559 m/s, the orbit of Usagi around Moon is going to be stable.
Geocentric orbit path
Lunacentric orbit path
Unstable moon of Moon (Newtonian Simulation)
1. Geocentric Trajectory
But if Usagi is too close to Earth, then Moon is not going to have its own moon for a long time, since the orbit of Usagi is not going to be stable.
For example, if the distance of Usagi and Moon is about 64544 km, and the orbital velocity of Usagi is 275.991595 m/s, the orbit of Usagi around Moon is going to be unstable.
Usagi is going to escape it's Lunacentric orbit right away, since earth have higher mass and thus, stronger gravity field. Between time 0 until month 8 day 22, Usagi will orbit Earth twice in not so eccentric orbit, and twice in highly eccentric orbit.
Then between month 8 day 22 to year 1 month 7 day 5, Usagi will orbit Earth several time, inside the Earth-Lunar space, then move to higher orbit, and orbiting earth 5 times.
During the period between year 1 month 7 day 5 to year 2 month 9 day 22, Usagi will continue to orbit earth chaotically, moving between orbits in Earth-Lunar space.
During the period between year 2 month 9 day 22 to year 3 month 5 day 20, Usagi will orbit earth, as high as the Moon itself.
2. Lunacentric Trajectory
In order to gain more understanding about what is happening, we should see it from different point of view, that is why we need to see the orbit of Usagi, from Moon's point of view.
As we could see, Usagi is going to be pulled right away by earth gravity, form its Lunacentric orbit. Notice that the path of Usagi is chaotic since the first time, up to month 2, until it escape to higher orbit, with higher eccentricity.
Since between month 2 and month 8, Usagi didn't orbit Moon anymore, but orbit Earth in highly eccentric orbit, while the Moon continue to orbit Earth as if nothing happened, the path of Usagi relative to Moon, show retrograde motions, just like the path of any bodies relative to other bodies in a celestial sisterhood.
The path of Usagi relative to Moon, during month 8 to month 12, take my interests, since the plot result of Usagi's motion, somehow draw a face in the space between moon and earth. Looks like the face of a cute dog to me.
Between month 12 and year 1 month 11, Usagi orbit earth in relatively stable orbit. In that period, the path of Usagi relative to Moon, show common retrograde motion.
Instability begin to occur in Usagi's path after year 1 month 11. During the period between year 1 month 11 and year 3 month 5, Usagi continue to exhibit chaotic movement, sometimes dangerously close to Moon.
Unstable moon of Moon (Post-Parameterized Newtonian Simulation)
After trying to simulate what is going to happen using NCGS, I am curious abuot what is going to happen if I use my other Gravity Simulator, the PPNCGS which take relativistic effect into account.
Just link the simulation result of NCGS, the PPNCGS simulation result also shows, that Usagi is going to "take off" to higher Geocentric orbit, in month 2.
But the simulation result after that predict different movement of Usagi. PPNCGS simulation result, predicts that Usagi is going to orbit Earth in highly eccentric orbit, for about 5 times, during the period of time between month 2 to year 1 month 3.
Then PPNCGS simulation result, predicts that Usagi is going to orbit Earth in close orbit, during the period of time between year 1 month 3 to year 1 month 7, Moon gravity field, throw Usagi away from its orbit.
Between year 1 month 7 and year 2 month 9, Usagi is going to occupy different orbit, around Earth, changing its path as it encounter Moon in close distance, and being thrown by Moon's gravity field.
After year 2 month 9, Usagi is going to orbit earth in close distance, until the 2.3221E20 kg object finally collide with Earth in year 3 month 3.