Orbital equations, and power required to run them

Question

Due to a discussion on the SO IRC today, I'm curious about orbital mechanics, and

• The equations needed to solve orbital problems
• The computing power required to solve complex problems

The question in particular is calculating when the Earth will plow into the Sun (or vice versa, depending on the frame of reference).

I suspect that all the gravitational pulls within our solar system may need to be calculated, which makes me wonder what type of computer cluster is required, or can this be done on a single box?

I don't have the experience to do a back of the napkin test here, but perhaps you do?

Also, much thx to Gortok for the original inspiration (see comments).

Answer 1

See Three Body Problem on Wikipedia. When you have more than two bodies in a gravitational field and you cannot simplify the problem, it is very difficult :)

Answer 2

In one of the Feynman lectures, he talks about doing orbital calculations with 1960s-era computers, and how good that was. No computer from the early 1960s has anywhere near the power of my phone or DS, and the stuff I actually buy for use as computers is much more powerful.

You've got the computrons, friend. The forces are easy to calculate, too, since it's all gravitational and the planets can be treated as point masses. It might be easier to calculate planetary orbits analytically, and treat gravitational perturbations as discrete pushes. Go for it. If you want help, find something on orbital mechanics or talk to a physicist or astronomer.

This isn't going to help you find when the Earth hits the Sun, since our orbit is extremely stable. However, in a few billion years, the Sun is going to expand a lot, and might reach our orbit. Still, it might be a fun project.

Answer 3

With more than three bodies, there is no closed form solution. There are various methods of approximation (look at the N-Body simulation articles here or here). Depending on how much accuracy you'll require, you'll need anywhere from seven to hundreds of bodies. Because of the relative scale (compared to, say, galaxy simulations), you won't be able to get much simplification from clustering.

As far as the specific question, though, you'd also have to work on estimates for changes in the Sun's diameter. I think the red giant phase would happen sooner than orbital decay, and that will make the Sun's diameter larger than the current Earth orbit.

Answer 4

Although the problem you posed undoubtedly requires significant computation with great accuracy (ignoring red giant phase issues, would/will the orbit of the Earth decay into the Sun?), there are software tools readily available to perform other celestial path/orbit calculations. Here are just a few:

• OrbFit
• ORSA - Orbit Reconstruction, Simulation and Analysis

Lastly, although the software that NASA uses to calculate future orbits and trajectories for spacecraft, etc, falls under ITAR restrictions, it apparently does make some past trajectory information publicly available for various bodies in our solar system as well as major past and current missions. This is handled through the Navigation and Ancillary Information Facility (NAIF).

NAIF provides software and data for the above purpose:

• SPICE toolkit (not to be confused with the other SPICE tool familiar to every EE major)
• SPICE data kernels

If you try out all of the above, you might learn something about orbital equations and the computing power involved. ;-)