Solving Orbital Plane Alignment of Two Planets

[chronnox]

Im trying to solve (approximately) the following problem: Suppose that i have 2 planets with mass m1 and m2 orbiting around the sun and i take into account the following interactions:
a) Interaction between planet 1 and the sun
b) Interaction between planet 2 and the sun
c) Interaction between planet 1 and planet 2

Where the sun has mas M >> m1,m2 . My physical intuition tells me that this system is similar to a damped harmonical oscillator so at later times the orbital planes of the planets will be aligned, i.e., the angle between the orbital planes will be zero at later times. The problem is that i can't find a book or an article which can help me with this. Does anyone know a good book to start with? or a book with a method to solve this problem?. Finally do you agree with my proposal that the angle between the orbital planes will be zero at later times?.

 

[ChaosKnight]

i'm not so sure the alignment will happen. Where does the damping come from?

 

[pixel01]

It's nice idea. It is not the damping force, i think just potential energy between the 2 planets will get the lowest value when they are aligned.

 

[tony873004]

Assuming that their orbits are far enough apart that their orbits are stable, their nodes will precess, but they will never become co-planar unless some other force is at work.

One way to think about it is that point mass n-body problems are time reversable. So if they did become co-planar, it would imply that a perfectly co-planar system could also break free of its co-planar state.

We observe a lot of systems that are roughly co-planar in our solar system. The planets around the Sun all orbit within < 20 degrees. The inner moons of Jupiter, Saturn, Uranus, and Neptune are roughly co-planar, but they probably formed that way.

Mars has two moons that are roughly co-planar even though they may be captured asteroids. This does imply that there are forces that will make objects co-planar. In this case, I believe it is the tides they raise on Mars, and Mars's equatorial buldge that makes their orbits settle into equatorial orbits. So the moons didn't do it to each other. Mars' non-spherical shape caused them to both become equatorial orbiters.

But the Sun takes 30 days to spin once. It has practically no equatorial buldge. And even if it did, the planets are sufficiently far from the Sun that it could be accurately approximated as a point mass, and the planets would never feel a significant pull from this buldge. Additionally, the planets are too far to pull a significant tide on the surface of the Sun.