Expert: Courtney Seligman - 1/24/2010

Question
Hi Courtney,
I can visualize how the planets corkscrew through space relative to the sun's moving path. My question is how could we set up the solution to finding when and where the major planetary paths might intersect at the same relative "points" in space?
Thanks so much!

Answer
The Sun's motion through space is much faster than any of the planetary motions, and is at an angle to the plane of their motion around it, so there cannot be any actual intersections of the sort I believe you mean. You could probably find some plane in which the projected paths would intersect, but the intersections wouldn't have any physical significance, and such planes might be few and far between.

As an example, if you project the orbits onto the plane of the Earth's orbit, the Sun's motion through space becomes irrelevant, and the projected orbits become the ones you typically see in textbooks, in which the orbits look like more or less concentric circles, save for those of Neptune and Pluto, which appear to overlap (such diagrams are said to show the "curtate" orbits). But if you projected them onto a plane halfway between the planes of Neptune and Pluto's orbits, even those orbits wouldn't intersect.

Presuming you still wanted to try to find a solution, you would need to decide how accurately you want to represent the motions (the more accurate, the more complicated the equations of motion become), and the coordinate system you want to use ("simple" motions are usually done with polar coordinates, but more exacting ones are usually done with Cartesian coordinates, as shown below). The calculations would be done in pairs, comparing the equations for one planet to those for another, by setting up a limiting equation in which a particular point in space-time has the same value for both planets (the "event" at point x, y, z, and t for one planet being identical to that for the other planet). As noted above, there would be no solutions, but that is the idea involved.