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Define ℓ as the mean longitude, the angular distance of the body from the same reference direction, assuming it moves with uniform angular motion as with the mean anomaly. Thus mean anomaly is also [6] = . Mean angular motion can also be expressed, = , where μ is the gravitational parameter, which varies with the masses of the objects, and a ...
Such functions can be expressed as periodic series of any continuously increasing angular variable, [6] and the variable of most interest is the mean anomaly, M. Because it increases uniformly with time, expressing any other variable as a series in mean anomaly is essentially the same as expressing it in terms of time.
where M 1 and M 0 are the mean anomalies at particular points in time, and Δt (≡ t 1-t 0) is the time elapsed between the two. M 0 is referred to as the mean anomaly at epoch t 0, and Δt is the time since epoch.
Instead of the mean anomaly at epoch, the mean anomaly M, mean longitude, true anomaly ν 0, or (rarely) the eccentric anomaly might be used. Using, for example, the "mean anomaly" instead of "mean anomaly at epoch" means that time t must be specified as a seventh orbital element. Sometimes it is assumed that mean anomaly is zero at the epoch ...
An orbiting body's mean longitude is calculated L = Ω + ω + M, where Ω is the longitude of the ascending node, ω is the argument of the pericenter and M is the mean anomaly, the body's angular distance from the pericenter as if it moved with constant speed rather than with the variable speed of an elliptical orbit.
where M is the mean anomaly, E is the eccentric anomaly, and is the eccentricity. With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of from periapsis is broken into two steps: Compute the eccentric anomaly from true anomaly
where the epoch is expressed in terms of Terrestrial Time, with an equivalent Julian date. Four of the elements are independent of any particular coordinate system: M is mean anomaly (deg), n: mean daily motion (deg/d), a: size of semi-major axis (AU), e: eccentricity (dimensionless).
Solving for is more or less equivalent to solving for the true anomaly, or the difference between the true anomaly and the mean anomaly, which is called the "Equation of the center". One can write an infinite series expression for the solution to Kepler's equation using Lagrange inversion , but the series does not converge for all combinations ...