How do you calculate the average distance (over TIME) of an orbiting body from its primary, given the periapsis and apoapsis (or equivalently, the orbit’s semimajor axis and its eccentricity)?
I know in abstract that the solution should be:
R_avg = ?R(t)dt / T
where R(t) is the body’s distance as a function of time; and T is its period. However, I don’t think there’s a closed-form solution for R(t), is there? Failing that, is there a (short) formula that gives a fair numerical approximation?
Also, my intuition is that “T” should cancel out of the final equation (i.e. R_avg should depend just on the geometry of the orbit), since the geometry uniquely determines the fraction of time that the body spends in any particular segment of the orbit.