Exploring Kepler's Third Law for Multi-Object Systems in Visual Python

[Suiram]

Hi, I am doing a physics project at school, which consists of recreating our solar system (with the sun and only 2 planets, Earth and Jupiter) with Visual Python. Then we want to measure the speed variation of the sun caused by both planets, to determine its period in a graph, which would allow us to find Earth's distance from the sun and Earth's mass. (In resume, I want to find an exoplanet, from a distance, in a binary system) But I just realized that Kepler's third law generalized (P^2 = (4 Pi^2 a^3)/(G(m1+m2)) )is for a binary system...

I was wondering if any of you know about the equation for a system who has more than one object? I'm looking forward to doing it with the whole solar system if possible.

 

[dauto]

What you're asking for is a solution for a gravitational three-body problem - a problem notorious for its complexity. Fortunately, you can consider the planets masses negligible when applying Kepler's law.

 

[tfr000]

Well there are two ways to approach it.
1) consider each planet separately and calculate its effects on the Sun as if it were the only planet.
2) calculate all of the mutual forces of each object on the others and do a numerical integration.

#1 can be done using Kepler's laws and solving Kepler's equation. You'll have to sum the results somehow.
#2 isn't terribly difficult once you understand how to proceed with a numerical integration, but getting to that point can be tough going.

 

[Suiram]

What we're going to do is make a graph using python, with the speed of the sun due to the interference of both planets, and try to find the individual effects of each since its going to show me 2 superimposed sinusoidal functions. Not really sure about how it's going to work, but I'm pretty sure that the three-body problem is pushing the problem too far. And we can't take both planets separatly, because we're trying to simulate it as if we were the observers and were trying to determine the characteristics with only what we can see from a distance. For the moment, we're going to try it with the Bayes theorem, using 6 factors instead of 3 (observed speed, period and phasing, times 2 because there are two planets). What do you think? Any other ideas?

 

[D H]

Any other ideas?

Numerical integration of the underlying ordinary differential equations. This is the standard approach.

 

[dauto]

Do it that way

Step 1: assume the sun is fixed and find the trajectories of the planets using Kepler's laws and completely neglecting any interaction between the planets.

Step 2: allow the sun to move now, use the planetary motion found in step 1 and find the motion of the sun by requiring the center of mass of the whole system to remain fixed.