# Satellite Motion

1. Dec 6, 2012

### msteve

Does anybody know if there is an analytical solution for satellite motion with drag
force F=-k V ?

thanks
Steve

2. Dec 6, 2012

### K^2

I'm almost certain there is no closed-form general solution. The equations will not separate nicely in either coordinate system, and it's going to be a mess.

There are solutions for special cases. There is certainly one for zero angular momentum. Also known as satellite falling straight down. Probably not of interest to you, though.

There is probably a monotone descent solution as well. Basically, what would become a circular orbit solution as k->0. That might be possible to work out if you figure out exactly what this means. And this might be the type of solution you are looking for.

Finally, it's certainly possible, and should be relatively easy, to find an approximate solution of the above. Under assumption that V is very high compared to descent rate, you can assume that only angular velocity factors into drag. In that case, you can compute the rate at which energy is lost to drag and get descent rate from that. Simply use angular velocity for circular orbit at given energy/radius to get velocity. This will give you a sufficiently straight forward differential equation that should have a simple enough solution.