In article ,
Bill Bogen wrote:
It's the "lose some velocity" part that's hard...

But an object _could_ (very small chance, I admit) be in heliocentric
orbit and yet pass over the Earth at just the right speed to enter a
circular polar orbit at 20310.8 km radius, could it not?

Unfortunately, no, because Earth's gravity will accelerate it as it
approaches. If it arrives from infinity, its speed must be at least
escape velocity (for that distance), which is about 1.4x circular-orbit
velocity.

The only exception to this, which is what got J002E3 captured temporarily,
is if the whole thing is happening at the outer fringes of Earth's sphere
of influence, where the Sun's gravity is quite significant and three-body
complications invalidate simple concepts like "escape velocity". But
that's out around 900000km radius.

(Oh, there's one other exception, if it chances to make a lunar flyby
that robs it of some energy. But that will necessarily leave it in an
orbit that goes out, at least, nearly to the Moon's orbit.)

In general, three-body systems which are not "hierarchical" -- one close
pair plus a distant third -- over time have a strong tendency to lose one
body by ejection.

And yet Jupiter has a number of moons in pretty stable orbits, over
millenia.

Those cases are not really three-body systems, because Jupiter dominates
the situation so overwhelmingly. Interactions between the moons are minor
by comparison. (Sometimes they are non-trivial -- e.g. the resonance with
Europa and Ganymede that maintains the slight eccentricity of Io's orbit
and hence its internal tidal heating -- but not to the extent of actually
altering another moon's orbit substantially.)
--
MOST launched 30 June; first light, 29 July; 5arcsec | Henry Spencer
pointing, 10 Sept; first science, early Oct; all well. |