Expert: Courtney Seligman - 11/26/2008

Question
Hello,
In my current Msc course in Astronomical Climate Forcing we briefly touched the subject of Orbit-Orbit resonance and it's bearing on solar system Chaos.
Basically, I wonder about the following hypothetical situation: if two planetary bodies have orbits with periodicities in a ratio of e.g. 1:3. At a certain point, these bodies will be relatively close, and influence each others orbit. However, will this not change their orbit, their respective periodicities, and therefore the ratio of these orbits?
Thank you for your time.

Answer
It is true that each interaction between two planets will alter their orbits, but that doesn't alter the basic ratio you're describing (which is referred to as commensurate orbital periods). What happens is that the period of each orbit will slowly swing back and forth relative to the commensurate orbital period, but on the average, the ratio involved stays the same.

As an example, Neptune laps Pluto about once every 500 years, at intervals which are twice the orbital period of Pluto and three times the orbital period of Neptune, for a ratio of 3:2 (which is the strongest interaction ratio possible). This is a 'good' example, in that Pluto has so little mass that it has no measurable effect on Neptune's orbit, so all we have to do is discuss the effects on Pluto's orbit.

Let's suppose that Neptune were to lap Pluto on that side of its orbit where it is headed outward, away from the Sun. Neptune's gravitational pull would slightly slow Pluto, which would make its orbit slightly smaller (with a slower outward velocity, it wouldn't be able to go as far out as before). According to Kepler's Third Law, this would also make its orbital period shorter, making it harder for Neptune to catch up to it, at later passages. Over a number of such passages, the shorter and shorter orbital period for Pluto would cause it to 'gain' on Neptune, in the sense that Neptune would lap it further and further along their respective orbits, until eventually, Neptune wouldn't lap Pluto until it was on the inward-moving portion of its orbit, on the other side of aphelion.

At this point, the process would reverse. For a while, Pluto would continue to 'gain' on Neptune, being lapped further and further along its path, but every time, being lapped at a point where it is headed inward, toward the Sun, so that Neptune's inward pull would make it go faster. This would slightly increase the size of Pluto's orbit, and after many passages, the larger orbital size would give it a longer orbital period which would allow Neptune to lap Pluto earlier and earlier, so that after a very long time, things would return to the way they were, tens or hundreds of thousands of years earlier.

In other words, over very long periods of time, the resonance is preserved, by slowly swinging the shape, size and period of the orbit, one way, then the other, relative to a stable ratio. (Other orbital elements, such as the angles that describe the orientation of the orbit, also 'wobble', so that the orbits swing around in space, but that doesn't affect the resonance).

Courtney Seligman
Professor of Astronomy
Long Beach City College

P.S. You might also refer to the Wikipedia article on Pluto, or to the discussion of orbital regularities on my website, at http://cseligman.com/text/planets/regularities.htm