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The true anomaly is the angle labeled in the figure, located at the focus of the ellipse. It is sometimes represented by f or v. The true anomaly and the eccentric anomaly are related as follows. [2] Using the formula for r above, the sine and cosine of E are found in terms of f :
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.
As for instance, if the body passes the periastron at coordinates = (), =, at time =, then to find out the position of the body at any time, you first calculate the mean anomaly from the time and the mean motion by the formula = (), then solve the Kepler equation above to get , then get the coordinates from:
The true anomaly is usually denoted by the Greek letters ν or θ, or the Latin letter f, and is usually restricted to the range 0–360° (0–2π rad). The true anomaly f is one of three angular parameters (anomalies) that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly.
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
An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. The distance to the focal point is a function of the polar angle relative to the horizontal line as given by the equation ()
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 ...
This equation gives M as a function of E. Determining E for a given M is the inverse problem. Iterative numerical algorithms are commonly used. Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ. But note: Cartesian position coordinates with reference to the center of ellipse are (a cos E, b sin E)