True anomaly


In astronomy, Kepler’s laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

1- The orbit of a planet is an ellipse with the Sun at one of the two foci.
2- A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3- The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Most planetary orbits are nearly circular, and careful observation and calculation are required in order to establish that they are not perfectly circular. Calculations of the orbit of Mars, whose published values are somewhat suspect, indicated an elliptical orbit. From this, Johannes Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits.

Kepler’s work (published between 1609 and 1619) improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets’ speeds varied, and using elliptical orbits rather than circular orbits with epicycles.

Isaac Newton showed in 1687 that relationships like Kepler’s would apply in the Solar System to a good approximation, as a consequence of his own laws of motion and law of universal gravitation.
Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler’s equation.

The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the time t since perihelion. Shown here the equation of true anomaly θ. This is the third step in the connection between time and position in the orbit.

Related formulas


ϵeccentricity (dimensionless)
θMean anomaly (degree)
EEccentric anomaly (degree)