Expert: Steve Nelson - 3/18/2009

Question
Hi,

I'm interested in creating a mathematical model of the solar system via a computer program. But I'd like to make sure I'm doing it as best I can before I start. I'd like to make it as accurate as possible. I will assume that the Sun is static and so I'm only interested in the planets orbiting the Sun.

Would you be able to tell me...

Are Newton's Laws all I need? If I use them to calculate the forces between objects then how do I use this force to plot/calculate their course around the Sun? I know that planets trace out an ellipse so what type of co-ordinate system should I use? And would Newton's laws account for the acceleration felt by the planets as their orbit gets closer to the Sun in the ellipse? Also, how much of a problem is Mercury? Can the General Relativity solution be applied without too much trouble?

This may seem like a college project or assignment but it is actually just something I'm doing as an interest in my spare time. I'm not asking for all the equations to be given to me but if you have any good websites or even just an explanation of what I need that would be great. And if you have any other advice I'd appreciate it greatly.

Thanks in advance.

Answer
I don't mean to rain on your parade here, but I'm not the bearer of good news for this hobby of yours.  Technically yes, Newton's Laws and General Relativity are pretty much all you need, but applying those to calculate long-term orbits including the interaction of all the planets with one another and their effect on the motion of the Sun is extremely challenging.  Newton's Laws: simple.  Their application to this problem: dizzyingly difficult.  If you're looking for high accuracy in particular then you're in for a real challenge, experienced astrophysics researchers develop such models for their full-time jobs.  The numerical integration necessary such models presents many unique difficulties for the accurate prediction of the motion of objects in the solar system due to the scale of what you're trying to simulate.  Not working directly on orbit problems, I'm also not sure where you'd even go to get accurate data for inputs.  For that you should ask an astronomer.

If you're going to assume the Sun is static and the planets don't interact much via gravity, then you don't need numerical integration or a computer.  Just plug in your orbital data to orbital equations via Kepler's laws and you're done.  Don't worry about the precession of Mercury, you're already making a gross enough approximation that you don't need to bother.  If you're going the computer model route, then I'd start by carefully studying Runge-Kutte methods for numerical integration and see if you can find precise orbit data from an astronomer or from some public NASA database.  That should give you a good idea how to proceed, once you've created differential equations describing the acceleration of the planets which you can turn into (via integration) velocities and positions over time.