The discussion revolves around how to express the parametric equations for an elliptical orbit in terms of time, contrasting it with the simpler case of circular orbits. Participants explore the challenges of deriving an analytic formula for position as a function of time for elliptical orbits.
Participants generally agree that deriving a time-dependent parametric equation for elliptical orbits is challenging and does not yield a simple analytic form, but there is no consensus on a specific method or solution.
The discussion highlights limitations in deriving a time-dependent position formula for elliptical orbits, particularly the reliance on parameters like ##r## and ##\theta## without a clear connection to time.
DuckAmuck said: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 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.DuckAmuck said:Yes. I get something non-analytic.
Thank you!gneill said:Look up "The Kepler Equation". You're in for an interesting ride!