• Recall that we now think that many extrasolar giant planets have migrated inward toward their parent stars.
• What has not been considered before is what happens to their moons. Do they get lost? Do the change orbits or find their orbital eccentricity pumped up?

To get a rough idea of what might happen, let's examine Ganymede around Jupiter in current and instantaneous Hill radii.

The Hill¹ radius is given by

Clearly, the only part of this that changes is a.

First, m_J/m_S = 1/1048. So for Jupiter, the Hill radius is 0.35 AU. In other words, huge.

Ganymede orbits at 0.02 r_H. At this distance, it is well within Jupiter's Hill sphere, and so 3-body effects are neglegible. (For those playing the home game, the Moon is at 0.25 Hill radii, and 3-body effects are important.)

What if Jupiter moved in to around 1 AU? Then Ganymede's orbit would now be at 0.1 r_H. And if Jupiter were to become, say, 51 Pegasi, then Ganymede's orbit would be at 2.6 r_H. At this distance, we're well outside the Hill sphere and the moon is gone.

If giant planets migrate in, it is quite possible that they might lose their moons as 3-body interactions become more significant.

In the late 19^th century, George Hill first derived a set of equations that approximately govern the motion of a small mass in a hierarchical 3-body system.

In the 2-D case, Hill's equations become (in non-dimensional terms)

Where the frame is fixed on m₂ and x points away from m₁. d is the distance to m₂ (sqrt(x²+y²)).

• Orbital Times
  • Planet orbits are from 1 year-ish (inner solar system) to 10 years (outer solar system)
  • Moon orbits are of order a few days.
  • So right up until the planet is really close to the star, the moon orbits much faster than the planet orbits the star.
• Migration time
  • Migration probably takes from 1,000 to 10,000 [Earth] years.
  • This means that the moon's orbit can adjust more or less adiabtically.
  • Tidal damping timescales are of order one million years, so tidal damping is not an important effect.

• Hill's equations were integrated with a fourth-order Runge-Kutta integrater.
• Time steps were set to be 1/1000 of a moon-orbit.
• Output were dumped every planet orbit.

A more insightful plot

A more insightful plot

• Prograde moons migrate outwards and the planet moves in.
• Retrograde moons move inwards at the same time.
• Both types of moons get lost, because the Hill sphere overtakes them.

To understand an analyze these odd results, we see an invariant of the motion. That is, a quanitity that won't change as the planet migrates.

Starting with the Hill Hamiltonian and jumping over a lot of gorey details, under the assumption of an elliptical orbit we get

• As a planet migrates in:
  • Prograde moons migrate outwards
  • Retrograde moons migrate inwards
• This motion can be predicted for moons with low eccentricity using a simple invariant of motion.
• It matters naught in the long run, because the Hill sphere collapses faster than the retrograde moons can migrate inward. All moons at reasonable distances are lost.
• Points at which moons are lost agree with work by Doug Hamilton.
• Things are worse than you think: when the eccentricities of outer large moons get pumped up near their end, they'll start to mess with the inner moons, with disastrous (and hilarious?) results.

• Glen, my advisor, for giving me advice and being patient.
• Doug Hamilton, for his very helpful insight.
• Lynette Cook, who made the wonderful background image. (And gave me permission to use it.
• Murray and Dermott. If you don't know why, don't ask.
• The good people at Starburst^TM, makers of Skittles^TM candies.