Variation of the elements

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The variation of the elements is a method developed in the 19th century for the precise orbit determination of celestial bodies. It is still used today to model railway disruptions .

In the idealization of the two-body problem , the orbit of a planet around the sun or a moon around a planet runs exactly on a Kepler ellipse . The prerequisite for this is that both bodies are spherical, move in a vacuum and no other celestial bodies or forces act. For the calculation of such Kepler ellipses, six orbital elements and three Kepler's laws are sufficient . The first 6 numerical values ​​remain constant - that is, the orbit ellipse and its plane do not change with respect to the central body and the fixed star sky .

De facto, however, orbit disturbances are always effective: third bodies, interplanetary gases and dust , radiation pressure from the sun, atmospheres and flattening of planets etc. These disturbing forces slowly change the 6 orbit elements and cause additional deviations from the Kepler orbit .

The procedure

Lagrange and other astronomers therefore developed the model of the oscillating orbits . If the orbit of a celestial body was too variable, an elliptical orbit is adapted to the currently valid one, which fits all observations as closely as possible. In the course of time, a series of oscillating orbital elements is created that continuously merge into one another. Each of these sets of elements represents a path on which the celestial body would continue to fly if the disturbing forces ceased.

In the calculations, the elements are approximately determined at certain times using differential equations. The envelope curve is then determined from the calculated paths, which then results in the “theoretical” path of the object. The temporal change of the path elements can be secular , periodic and to a certain extent also irregular - depending on the force and path element causing it . For each of the 6 numerical values, time-dependent terms can therefore be determined, which characterize the change in the elements.

The deviations of the actual orbit from the oscillating ellipse valid just before can be calculated as a function of the disturbing forces. In this way, disturbances of the Uranus orbit were modeled for the first time in 1846 , which led to the discovery of the planet Neptune . Conversely, for space probes , for example , the gravitation of all known bodies can be taken into account or the effect of short corrective maneuvers can be calculated.

Today's computers allow an arbitrarily precise orbit determination, if only the effort is increased accordingly. In this case, it is preferable to iterative methods and used for so-called track improve the compensation calculation or the co-location . After a first (approximate) orbit determination from a few measurements (at least 3), the locations of the celestial body at the times of all of its observations are calculated . The deviations from this ephemeris are the input variables of the orbit improvement . By suitably varying the elements, it brings the calculated positions into line with the measurements . As a result, and by including distance measurements with radar , the computational accuracy in the inner solar system has increased to 1:10 million and better (earth's orbit to km).

Instead of changing the orbital elements in a targeted manner, the result is equivalent methods such as varying the geocentric distance . The resulting differential equations are solved , for example, according to Runge-Kutta . The numerical integration of the Jet Propulsion Laboratory ( JPL - program DE200 / LE200) works according to a similar method . It is used to calculate the positions of all major planets and some asteroids for the Astronomical Almanac every year .

literature

  • Satellite geodesy: Basics, methods and applications by Günter Seeber, 1989; Verlag de Gruyter ( ISBN 978-3110100822 )

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