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In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the true anomaly and the mean anomaly .
where M 0 is the mean anomaly at the epoch t 0, which may or may not coincide with τ, the time of pericenter passage. The classical method of finding the position of an object in an elliptical orbit from a set of orbital elements is to calculate the mean anomaly by this equation, and then to solve Kepler's equation for the eccentric anomaly.
On the computation of the eccentric anomaly, equation of the centre and radius vector of a planet, in terms of the mean anomaly and eccentricity. Monthly Notices of the Royal Astronomical Society, Vol. 43, p. 345. Gives the equation of the center to order e 12. Morrison, J. (1883). Errata. Monthly Notices of the Royal Astronomical Society, Vol ...
Unlike with mean anomaly, mean longitude is defined relative to the vernal point, which means it is defined for circular orbits. Eccentric anomaly at epoch (E 0) — the eccentric anomaly at the epoch time. Eccentric anomaly is defined at the angular displacement along the auxiliary circle of the ellipse (circle tangent to the ellipse both at ...
Diagram of an orbit. The plane of the orbit is in yellow, the reference plain is in gray, and the reference direction (vernal point) is the arrow in red.Also labeled are the mean anomaly (M) in pink, the true anomaly in red, the argument of periapsis (ω) and periapsis in purple, the longitude of ascending node (Ω) in green, and the inclination (i) in dark green.
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.
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
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 ...