Midpoint equation

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As midpoint equation the deviation of the uneven movement of will since the ancient astronomy moon and planets from a uniform movement along a circular path designated. As Johannes Kepler showed in 1609, it depends on the eccentricity e of the respective orbit ellipse . Its maximum amount is called the great inequality .

It results from the Kepler's equation as the difference between medium anomaly M and true anomaly V . The latter is the current angular distance of the celestial body from its periapsis (point closest to the earth or the sun of the orbit ellipse), while the angle M progresses evenly over time and begins with zero in the periapsis. Because the Kepler equation can only be solved iteratively, V - M today is mostly calculated by a series expansion . For degree measures, the result is an approximation of the second order

or in an approximation of the third order

The maximum occurs at the angles 90 ° and 270 ° - i.e. H. at quarter or ¾ of the period of rotation - and is called great inequality . It corresponds to the 1st term 2e of the above series and is ± 6.3 ° for the moon , ± 1.9 ° for the earth's orbit or the apparent solar orbit, 24 ° for Mercury , 0.8 ° for Venus, 10 ° for Mars , 7 °, with Jupiter 5 ° and with Saturn 6 °. These values ​​were already well known to Claudius Ptolemy ; probably already Apollonios of Perge around 200 BC. Derived from long-term observations. Similar research has been carried out in ancient India, Babylonia, and Persia.

The largest term of the midpoint equation , the sine oscillation 2e · sinM of the above series expansion, was taken into account in the Greek planetary theory by means of epicycles . The epicyclic center point was allowed to run on an eccentric in such a way that the movement appears uniform when viewed from a compensation point. However, the Babylonians did not calculate it using epicyclic theory , but using arithmetic series.

Ptolemy wrote in his Almagest that the lunar orbit cannot yet be calculated satisfactorily with this . As a correction he introduces evection , a disturbance of 1.3 °, which depends on the mutual position of the sun and moon. 1500 years later, Tycho de Brahe discovered two further perturbations ( variation and annual equation ) in his observations accurate to 0.02 ° , which were confirmed by Newton's law of gravitation . Today, the theory of the lunar orbit takes into account well over 1000 periodic perturbation terms, to which there are also secular effects (e.g. rotation of the lunar orbit plane).

The equation of the center of the planets also describes the irregular velocity due to the ellipticity of the orbits, but only for Mercury ( e = 0.206) and Mars (0.093) does it exceed that of the moon. The other disturbances are less because the earth and other planets are far away.


  • Karl Stumpff , H.-H. Vogt: The Fischer Lexicon Astronomy . Revised 8th edition, Fischer Taschenbuch Verlag, Frankfurt / Main 1972
  • Wolfgang Schroeder: Practical astronomy for star friends , chapter lunar and planetary orbits. Kosmos-Verlag, Stuttgart 1960

Individual evidence

  1. van der Waerden: Balance point, "Persian method" and Indian planetary calculation