I’ve been looking into this lately, and most sources I have used solve this problem numerically. I was thus wondering if there is a proper equation to solve for position without numerical, especially since the motion of planets and satellites seems to follow harmonic motion.
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2$\begingroup$ The problem is Kepler's equation en.wikipedia.org/wiki/Kepler%27s_equation $\endgroup$– PM 2RingApr 14 at 6:32
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$\begingroup$ @PM2Ring That's an actual answer as short as it may be... $\endgroup$– planetmakerApr 14 at 9:11
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2$\begingroup$ See this answer to What is the analytical closed-form solution of the two-body problem to verify its numerical integration results? in Space Exploration SE. For Keplerian orbits you can solve for $t(\theta)$ analytically, but the inverse $\theta(t)$ needs numerical or a power series approximation. $\endgroup$– uhohApr 14 at 20:18
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1$\begingroup$ In space.stackexchange.com/questions/55356/… I have a short Python demo that compares different initial approximations when using Newton's method to solve Kepler's equation. $\endgroup$– PM 2RingApr 14 at 22:39
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