The discussion focuses on parameterizing elliptical orbits in terms of time, contrasting it with circular orbits. While circular orbits can be expressed analytically using the equations $$(x,y) = (cos(\omega t),sin(\omega t))$$ with $$\omega = \sqrt{GM/r^3}$$, elliptical orbits do not yield a straightforward analytic solution for position as a function of time. The conversation highlights the limitations of Newton's laws in providing a time-based parametric equation for elliptical orbits, directing participants to explore "The Kepler Equation" for deeper insights.
PREREQUISITESAstronomers, physicists, and students of orbital mechanics seeking to understand the complexities of elliptical orbits and their time parameterization.
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!