# Angular velocity by mean velocity and ecentricity of orbit [closed]

Okay, so I'm working on a project that will include a track of a star's position in the sky of an arbitrary planet. I've thus far managed to work out all the factors (save refraction) using Quaternions but this model is only valid for a circular orbit. I've spent a few hours trying to think of how to introduce eccentricity, and looked online for some answers, but I've not really been able to find much of anything. I know of Kepler's law, but unsure of how to use it to derive what I need. So far, I've come to the conclusion that the only thing that actually matters with tracing a host star's path in the sky is the eccentricity and orbital period relative to the number of days/year relative to the target planet. The answer I am looking for is how to find the angular velocity of a planet at a given point in it's orbit given the orbit's eccentricity.

EDIT: I've come to the realization that I've been approaching this problem incorrectly, I'll review my notes and research and abandon this question for the time being.

EDIT: after some thought and research I've come to a possible solution:

$$v_{inst} = \frac{v_{mean}}{1-e cos(\theta)}$$

given $v_{mean}$ is the mean angular velocity of the orbit, $v_{inst}$ is the instantaneous angular velocity of the orbit at position $\theta$ in the orbit where $\theta = 0$ is at periapsis, is this assumption correct? because if it is, then the variance will simply be $v_{mean} - v_{inst}$ and I can easily apply it. I am unsure just how I would be able to test it properly however, which is why I'm posting this finding here before implementing it.

• Possible duplicate of Satellite in Elliptical orbit. In particular see answer by David Hammen. Jun 5, 2018 at 0:10
• Welcome to Physics! Please do not make your post look like a revision table, instead just seamlessly integrate the new material into the post. There is an edit history button at the bottom of the post for those interested in seeing what changed. Jun 6, 2018 at 10:02

There are four cases:

1. e>1 hyperbolic, v>sqrt(2*G*M/r0)

2. e=1 parabolic, v=sqrt(2*G*M/r0)

3. 0> e >1 ellipse, v < sqrt(2*G*M/r0)

4. e=0 circle, v=sqrt(G*M/r0)