Posted: Jun 10, 2011 1:05 pm
Interestingly (if you're that way inclined...) this is, as it were, a reverse slingshot effect.
The slingshot effect used - for instance - by the two Voyagers is like this. We have a large object (Jupiter, say) which is in orbit around the Sun, and rotates around its axis as it does so. A much smaller object (Voyager 2) approaches it, in the direction, more or less, of Jupiter's rotation, with a speed which is more than necessary to escape Jupiter's gravitational field, but not enough to escape that of the Sun, even at this distance. As Voyager passes Jupiter, it is attracted by the latter's gravitational field and its trajectory is bent around Jupiter into a partial orbit; as it does so, it picks up speed from Jupiter (Jupiter drags it along and speeds it up). (Relative to Jupiter, it gains kinetic energy and loses gravitational potential energy, at first, but once its trajectory starts to point away from Jupiter the reverse occurs.) Having gone part of the way around Jupiter, it escapes Jupiter's gravitational well with a speed - relative to Jupiter - identical to that with which it entered it: the gravitational well is symmetrical. Relative to the Sun, however, it has picked up a lot of Jupiter's speed, and its speed relative to the Sun now exceeds that necessary to escape the solar system.
The Melancholia thing is slingshot in reverse, because the object on its way away from the Sun is more massive than the one in a (stable) orbit. (Also, the two objects are far more similar in mass than Jupiter and Voyager!) But the important thing is how they are moving relative to each other. In that frame, the Earth is approaching Melancholia (as Melancholia crosses Earth's solar orbit) and it too is deflected, for the briefest of times, into an orbit around Melancholia. Like Voyager, Earth's speed is far greater than that necessary to escape Melancholia's gravitational field (as my back-of-an-envelope calculation indicated), but its solar orbit would be bent away from its current, almost circular one.
The slingshot effect used - for instance - by the two Voyagers is like this. We have a large object (Jupiter, say) which is in orbit around the Sun, and rotates around its axis as it does so. A much smaller object (Voyager 2) approaches it, in the direction, more or less, of Jupiter's rotation, with a speed which is more than necessary to escape Jupiter's gravitational field, but not enough to escape that of the Sun, even at this distance. As Voyager passes Jupiter, it is attracted by the latter's gravitational field and its trajectory is bent around Jupiter into a partial orbit; as it does so, it picks up speed from Jupiter (Jupiter drags it along and speeds it up). (Relative to Jupiter, it gains kinetic energy and loses gravitational potential energy, at first, but once its trajectory starts to point away from Jupiter the reverse occurs.) Having gone part of the way around Jupiter, it escapes Jupiter's gravitational well with a speed - relative to Jupiter - identical to that with which it entered it: the gravitational well is symmetrical. Relative to the Sun, however, it has picked up a lot of Jupiter's speed, and its speed relative to the Sun now exceeds that necessary to escape the solar system.
The Melancholia thing is slingshot in reverse, because the object on its way away from the Sun is more massive than the one in a (stable) orbit. (Also, the two objects are far more similar in mass than Jupiter and Voyager!) But the important thing is how they are moving relative to each other. In that frame, the Earth is approaching Melancholia (as Melancholia crosses Earth's solar orbit) and it too is deflected, for the briefest of times, into an orbit around Melancholia. Like Voyager, Earth's speed is far greater than that necessary to escape Melancholia's gravitational field (as my back-of-an-envelope calculation indicated), but its solar orbit would be bent away from its current, almost circular one.