So I have 2 objects in orbit around the same body, and I have all the orbital elements associated with each body. How do I calculate the angle between them both or figure out when they are a certain angle apart using only the all the orbital elements I have?
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1This seems like a question in celestial mechanics, not mathematics per se. There are standard formulas for computing the rectangular coordinates of each of the orbiting objects. Once you know the distances between each of the objects, I suppose you can use the law of cosines to obtain the angle subtended by the two objects from the central body: $\cos C = \frac{a^2+b^2-c^2}{2ab}$. – Brian Tung Jan 21 '16 at 00:46
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Law of Cosines would be helpful here – Varun Iyer Jan 21 '16 at 01:01
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I presume you're talking about Newtonian mechanics with only one spherical body of non-negligible mass (let's say the Sun), and the others in elliptical orbits. Now unfortunately if these ellipses are not circles the position as a function of time is not available in "closed form": you have to solve an equation
$$ t = \dfrac{a^{3/2}}{\mu^{1/2}} (E - e \sin(E)) $$
to get the eccentric anomaly $E$ as a function of time $t$, and then the coordinates of the body can be obtained from that. However, if the eccentricity $e$ is small, this is well approximated by a series in powers of $e$.
Once you have the positions, you get the angle using the Law of Cosines, as others have mentioned.
Robert Israel
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