# Orbit determination

Under orbit determination is meant the calculation of orbit a celestial body ( star , planet , moon , comet , satellite or small bodies) of the measurement results terrestrial or space befindlicher observatories .

For this standard task of celestial mechanics it is not sufficient to determine the six Kepler orbital elements and to calculate the orbit by solving the Kepler equation ; the Kepler orbital elements are only valid for the case of a single central body ( sun or planet), which would also have to be exactly spherical. In addition to the effects of the sun (ideal Kepler orbit), an exact orbit must also take into account the orbital disturbances caused by the attraction of other larger masses and, in the case of satellites, the flattening of the earth . In addition, when recording the observation data, there is the problem that all measurements relate to an apparently moving background.

## history

For at least 5000 years, astronomers and mathematicians have been busy calculating the orbits of the stars as they can be observed from the earth in advance. The roughly annual planetary loops in particular formed a riddle that astronomers in Mesopotamia and elsewhere could only explain on the basis of the state of knowledge at that time through the intervention of deities . No other explanations have come down to us.

### Early guesses and attempts at explanation

In ancient Greece , geometrical-mathematical models were found that could describe the complicated planetary orbits. The problem of the planetary loops and other apparent irregularities was solved with the most rounded geometries there are in the sense of Aristotle - with circles and additional circles running on them, the epicycles , which were all traversed at constant speed.

According to this, the then known planets Mercury , Venus , Mars , Jupiter and Saturn , but also the sun and moon, should move on ideal orbits around the earth, namely in circles, each with an epicyclic. Although, as Copernicus already knew, an elliptical orbit can be represented exactly with one epicyclic if one chooses its radius and rotation speed appropriately (see heliocentric view of the world ), since Ptolemy one simply added another epicyclic to the first to improve accuracy. This happened several times with Mercury and Mars (from today's perspective almost a Fourier analysis ). In addition, since Ptolemy, the requirement that the circular movement should take place uniformly was related to a compensation point outside the center of the circle.

### Brahe, Kepler, Newton

The very exact observations of Tycho Brahe (especially on Mars), which were made without optical aids, made it possible for Johannes Kepler to find his three Kepler laws . With this one could now describe the orbits of the large planets in a spatial planetary system. The orbits of new celestial bodies could not yet be calculated with it.

In 1687, almost a hundred years later, Isaac Newton succeeded - building on the knowledge of Kepler - establishing the law of general mass attraction . This recognized the law for the movement of celestial bodies, but there was still a lack of mathematical methods for the concrete calculation of orbit elements.

## Laplace, Gauss: The analytical orbit determination

The two-body problem (movement of two bodies around each other) was completely solved by Laplace and Gauss around 1800 . To from three measured positions z. For example, to determine the orbit elements of a new comet, they almost simultaneously found the solution in very different ways:

• The direct method goes back to Pierre-Simon Laplace , which represents the Kepler elements on the left-hand side of - albeit extremely complicated - equations .
• Carl Friedrich Gauß invented the indirect method , which operates with small changes to approximate values (especially the spatial distances ). It is somewhat easier to solve due to its iterative approach.

With this method Gauss succeeded in calculating the orbit of the lost asteroid (1) Ceres , which led to its sensational rediscovery. This method is still used today, in the age of computers. It boils down to a numerical integration of the equations of motion and allows all known forces to be incorporated into the physical-mathematical model without much additional effort.

Leonhard Euler and Joseph-Louis Lagrange also made important theoretical contributions to determining orbit . The first reliable determination of a strongly elliptical comet's orbit was made around 1780 by Wilhelm Olbers, who later discovered asteroid .

### Disturbance calculation of the Kepler railways

In order to be able to calculate the de facto always existing path disturbances by third bodies , the model of the osculating (clinging) paths was developed around 1800 . If the - according to Kepler ideal - conic-shaped orbit of a celestial body was too variable, the currently valid data set of the six orbit elements was used as the reference system for the changes that arose from this system state after a few hours (days, weeks ...).

The deviations from the oscillating ellipse can be calculated as a function of the disturbing force . Thus the method of variation of the elements was born. With the arithmetic aids of the time, it allowed an arbitrarily precise orbit determination , if only the effort was increased accordingly. Their consistent application led to the discovery of Neptune in 1846 and represented - in the Age of Enlightenment  - a true "triumph of celestial mechanics". Neptune's presumed position had been calculated from small orbital disturbances of Uranus , and he was barely 1 ° away from it.

### Refinement through equalization calculation

If the orbit of a new celestial body has been determined for the first time by three good observations, it can be refined by means of compensation calculations or collocation if further observations are available. This eliminates the inevitable small contradictions in overdetermined systems by minimizing the sum of squares of the remaining deviations through small variations of the path elements ( method of the least squares ).

The perturbation calculation can also be included according to the same principle : on the basis of the first orbit, the orbital perturbations are calculated (in the case of comets mainly by Jupiter), these are attached to the measurements and a next better orbit is determined from them.

## Methods and Applications

The most important application of newly determined orbits is the ephemeris calculation, the advance calculation of positions for several future points in time.

A distinction is made when determining the orbit itself

• the first calculation of a Kepler orbit based on the two-body problem
• the refined trajectory from more than three observations
• by least squares adjustment calculation
• advanced models and weighting for different observation types and accuracies - e.g. B. speed and transit time measurements , relativistic effects
• with perturbation calculation by other celestial bodies

When treating the three-body problem :

With the multi-body problem :

Theory of chaotic orbits : Many orbits, especially those of minor planets , run “regularly” for centuries, only to then suddenly drift in one direction. In principle, all orbits are unstable in the long term, but changes are corrected by orbital resonances , which is why the solar system with its eight large planets remains non-chaotic for billions of years. Systems in which such self-regulating mechanisms do not occur do not get old (by cosmic standards).

## Determining the orbit of meteors

The trajectory of meteors through the earth's atmosphere is determined by a geometric intersection method . When the light trail on the starry sky has been recorded by the cameras of several meteor stations , the spatial path can be calculated using a kind of forward cut (analogous to surveying). By calculating back, the origin of the meteoroids can be determined, most of which come from the asteroid belt .

In the case of larger bodies that fall as meteorites on the earth's surface, the approximate place of fall can also be determined.

## literature

• Manfred Schneider : Himmelsmechanik (4 volumes) Spektrum-Verlag, Heidelberg 1992ff, in particular
• Kurt Arnold: Methods of Satellite Geodesy (230 p.), Chapter 7 "Determination of the orbital elements"; Akademie-Verlag, Berlin 1970
• Julius Bauschinger : The determination of the orbit of the heavenly bodies , 2nd edition (672 p.), Verlag Wilhelm Engelmann, Leipzig 1928.

## Individual evidence

1. ^ Ernst Zinner: Origin and expansion of the Copernican doctrine . 2nd Edition. CH Beck, Munich 1988, ISBN 978-3-406-32049-1 , p. 199 .