I assume that it isn't actually orbiting the comet, but instead is just adjusting to follow the same orbit as the comet, right?
"Rosetta will, follow, at least for now, a three-legged triangular orbit that requires a small thruster burn at each apex. The legs are about 100 km long and it will take Rosetta between three and four days to complete each one."
http://blogs.esa.int/rosetta/2014/08/04/whats-happening-in-r...
We're not talking about propellant residue, that was very well planned ahead.
The spacecraft carried about 1.5 tons of propellant at launch. That gives the spacecraft a delta-v budget that is several kilometers per second.
The next thruster burns (on the triangular "hyperbolic orbits") will be tiny, in the order of centimeters per second.
When the mass and mass distribution has been established, the spacecraft will eventually settle in an orbit about 20 kilometers from the comet and finally down to 10 kilometers.
Yes, exactly. But since the comet is highly irregular in shape and mass distribution (both are still unknown at this point!), the orbit won't be a perfect ellipse, not even close.
For example, the perturbations from Earth's equatorial bulge causes the perigee of an orbit to shift eastwards (see: Molniya orbit). The Moon's gravity field is very irregular and orbiting the Moon at low altitude is just black magic.
That approximation can describe an orbit for a while, but after time the position will differ from the predicted orbit, until at some point you have very little idea of where it will be.
The time it takes for this to happen is called the Lyapunov time. For example, the Solar System is chaotic that way in the ~100 MY time scale. See http://en.wikipedia.org/wiki/Stability_of_the_Solar_System .
A Rosetta goal now is to figure out the comet's mass distribution, so that the orbital predictions can be better. The orbit won't be "stable" but it will be more "predictable."
Even then, we expect Rosetta to release gas, dust, and water vapor which affects the orbit. I expect there will be orbital corrections over the next few years.
One definition for the "stability" of an orbit is that if you give the orbiter a small nudge, it will not radically change its orbit. Lagrangian points L4 and L5 are not stable (with this definition) while L1, L2 and L3 are (only in theory, though, in practice perturbations matter a lot).
Since the mass distribution is unknown, we can't quite predict if the orbit will be stable or not. The mass distribution may be very uneven and there may be spikes in the distribution (due to shape and porosity of the comet) that could cause the orbit to be unstable. The comet also rotates and makes things more difficult.
Now the goal of the crew is to find an orbit that is reasonably stable and predictable given the knowledge of mass distribution they have. Nevertheless, the space craft will probably have to do thrust maneuvers to stay in orbit around the comet.
And once the comet comes close enough to the sun and starts to lose mass to the vapor trail, all bets are off. That almost certainly implies some kind of unstability due to mass changes as well as the drag from the vapor trail. The comet may even break down to several pieces when it comes close to the sun.
When orbiting a uniformly-dense sphere in an otherwise empty universe, this holds. When orbiting a body that approximates a uniform sphere in a universe where all other large bodies are relatively far away, this mostly holds. But significant deviations from a spherical shape can easily make most orbits unstable.
Little known fact: the Earth's Moon deviates enough from a uniform sphere to make long-term orbits tricky. Wikipedia article:
http://en.wikipedia.org/wiki/Lunar_orbit
Nice bit about a small satellite released by Apollo 16:
"Instead, something bizarre happened. The orbit of PFS-2 rapidly changed shape and distance from the Moon. In 2-1/2 weeks the satellite was swooping to within a hair-raising 6 miles (9.7 km) of the lunar surface at closest approach. As the orbit kept changing, PFS-2 backed off again, until it seemed to be a safe 30 miles away. But not for long: inexorably, the subsatellite's orbit carried it back toward the Moon. And on May 29, 1972—only 35 days and 425 orbits after its release"—PFS-2 crashed into the Lunar surface.
And this is all without any of the additional trickery coming from outgassing.
My understanding is that it's unlikely that they would ever recombine into a solid object since the gravity between them is so weak.
http://www.newscientist.com/article/dn25995-cometmapping-ros...
No, not quite. Remember, we're talking about a three body problem here so any intuition from two body dynamics doesn't really hold.
The spacecraft travels 100km closer to the sun than the comet, but that's a very small fraction of the total distance. When you factor in the small acceleration from the comet body, we can no longer think of the orbit as being a slightly smaller orbit with a shorter period (and thus overtaking the comet).
My best understanding is that the triangular "orbit" is to measure the mass and gravitational properties of the comet in order to orbit as well as scientific goals.