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!
An elliptical orbit is a type of orbit in which an object, such as a planet or satellite, follows a path that is shaped like an ellipse. This means that the distance between the object and the body it is orbiting around is not constant, but instead varies at different points along the orbit.
An elliptical orbit can be parameterized by time using Kepler's laws of planetary motion. This involves using mathematical equations to describe the shape, size, and orientation of the ellipse, as well as the speed at which the object travels along its orbit at different points in time.
The key parameters used to describe an elliptical orbit include the semi-major axis, eccentricity, inclination, longitude of the ascending node, argument of periapsis, and time of periapsis passage. These parameters help determine the size, shape, and orientation of the ellipse, as well as the position of the object along its orbit at a given time.
The eccentricity of an elliptical orbit determines how elongated or circular the orbit is. A higher eccentricity means a more elongated orbit, while a lower eccentricity means a more circular orbit. This affects the object's motion by causing it to move faster when it is closer to the body it is orbiting, and slower when it is farther away.
No, an object in an elliptical orbit cannot have a constant speed. This is because the object is constantly changing its distance from the body it is orbiting, which affects its speed due to the laws of gravity. However, the object's speed will be constant at a given point along its orbit if it is in a circular orbit.