Elliptical orbit parameterized by time

[DuckAmuck]

TL;DR

How to express an eliptical orbit in terms of time

It is fairly trivial to do this with a circular orbit: $$(x,y) = (cos(\omega t),sin(\omega t))$$
where t is time, and $$\omega = \sqrt{GM/r^3}$$
How this parametric equation look for an elliptical orbit?

 

[PeroK]

Summary:: How to express an eliptical orbit in terms of time

It is fairly trivial to do this with a circular orbit: $$(x,y) = (cos(\omega t),sin(\omega t))$$
where t is time, and $$\omega = \sqrt{GM/r^3}$$
How this parametric equation look for an elliptical orbit?

I don't think it works out very well. Have you tried?

 

[DuckAmuck]

Yes. I get something non-analytic.

 

[PeroK]

Yes. I get something non-analytic.

I think that's what you get. The analysis from Newton's laws bails out at a certain point and describes the shape of an orbit in terms of $r$ and $\theta$, but eliminates $t$. You get the shape (ellipse) and the period, but not an analytic formula for position as a function of time.

 

[gneill]

Look up "The Kepler Equation". You're in for an interesting ride!

 

[DuckAmuck]

Look up "The Kepler Equation". You're in for an interesting ride!

Thank you!