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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.

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closed as unclear what you're asking by sammy gerbil, Jon Custer, Emilio Pisanty, glS, Yashas Jun 12 '18 at 16:16

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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)

Please see it here:

https://www.maplesoft.com/content/EngineeringFundamentals/49/MapleDocument_46/Orbita_Mechanics.pdf

https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec17.pdf

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    $\begingroup$ Please use Mathjax as it is the site standard for mathematics. $\endgroup$ – StephenG Jun 5 '18 at 7:47
  • $\begingroup$ This answer is outside the scope of the question asked, due to the fact that what I'm looking for is the relation between the mean angular velocity to the instantaneous angular velocity at given points. I don't need to calculate the velocity given a radius, I need to calculate the difference in velocity from a given mean angular velocity. $\endgroup$ – Raxmo Jun 5 '18 at 15:52

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