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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 :
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 ...
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.
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 ()
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
Although the true anomaly is an analytic function of M, it is not an entire function so a power series in M will have a limited range of convergence. But as a periodic function, a Fourier series will converge everywhere. The coefficients of the series are built from Bessel functions depending on the eccentricity e.
In orbital mechanics, the universal variable formulation is a method used to solve the two-body Kepler problem.It is a generalized form of Kepler's Equation, extending it to apply not only to elliptic orbits, but also parabolic and hyperbolic orbits common for spacecraft departing from a planetary orbit.