# Orbit determination

Under orbit determination refers to the calculation of the orbit of a celestial body (star, planet, moon, comet, or satellite small body ) from the measurement results of earthly or being in space observatories.

For this standard task of celestial mechanics it is not sufficient to determine only the six Keplerian orbital elements and perform the path computation by solving the Kepler equation. The Keplerian orbital elements are only valid for the case of a single central body ( sun or planet ), which still would have to be exactly spherical. Precise orbit determination must also include the effect of the sun ( ideal Keplerian orbit ) take into account the perturbations by the attraction of other larger masses and for satellites the Earth flattening. Added to this comes in the collection of observational data the problem is that all measurements are based on an apparently moving background.

- 2.1 Perturbation theory of Kepler orbits
- 2.2 Refinement by compensation calculation

## History

Since at least 5000 years, astronomers and mathematicians deal with it, to calculate the paths of the stars in advance. The enigmatic annual planetary grinding provided the astronomers in Mesopotamia and elsewhere remains a mystery, they could solve on the basis of the then state of knowledge only through intervention of deities. Other explanations are not known.

### Early guesses and explanations

In ancient Greece, was then found geometric- mathematical models that could explain the seemingly complicated planetary orbits. It solved the problem with the roundest in the sense of Aristotle geometries, there is - with circles on them and running additional circles, the epicycles.

Then the known planets Mercury, Venus, Mars, Jupiter and Saturn, but also the sun and moon should move on ideal orbits around the Earth, namely on circles, each of which an epicycle is attached. If, as we now know, elliptical planetary orbits were not good enough representable with a epicycles, you just sat since Ptolemy's epicycles another at first. This happened at Mercury and Mars several times ( from today's perspective, almost a Fourier analysis). All of this was two-dimensionally on the background of a spherical shell, the celestial sphere.

### Brahe, Kepler, Newton

The very precise observations of Tycho Brahe (especially on Mars), which took place even without optical aids, made it possible to find Johannes Kepler, his three laws of Kepler. So now you could describe the orbits of the major planets well in a spatial planetary system. However, the orbits of new celestial bodies could therefore not be accurately calculated.

1687, almost one hundred years later, succeeded Isaac Newton - based on the findings of Kepler - set up his law of gravitation. Thus the cause of the movement of the heavenly bodies was indeed recognized, but on mathematical methods for the specific calculation of orbital elements still lacking.

## Laplace, Gauss: The analytical orbit determination

Complete the two-body problem (motion of two bodies around each other ) was dissolved in 1800 by Laplace and Gauss. For example, to a new comet to be determined from three measured positions of its orbital elements, they found almost simultaneously the solution in a very different ways:

- On Pierre- Simon Laplace, the direct method returns, which the Keplerian elements on the left side of - represents equations - but extremely complicated.
- Carl Friedrich Gauss invented the indirect method, which operates with small changes to approximations (especially the spatial distances). It is slightly easier to solve by their iterative procedure.

This method succeeded Gauss, the path of the lost asteroid ( 1) Ceres to calculate what led to its sensational rediscovery. Even today, in the age of computers, this method is applied. It amounts to a numerical integration of the equations of motion, allowing it to incorporate all known forces with little extra effort into the physical-mathematical model.

Important theoretical contributions to the orbit determination were made by Leonhard Euler and Joseph -Louis Lagrange. The first reliable determination of a highly elliptical comet train succeeded in 1780 to the later asteroid discoverer Wilhelm Olbers.

### Perturbation of the Kepler orbits

In order to calculate the de facto always existing railway third-body perturbations, it fell in 1800 on the model of the osculating ( snug ) tracks. If the - ideal after Kepler - conic shaped path of a celestial body was too variable, the currently valid data set of six orbital elements was taken as the reference system for the changes (.. days, weeks ) from this system condition emerged after a few hours.

The deviations from the osculating ellipse can be calculated as a function of the disturbing force. Thus, the method of variation of elements was born. She allowed with former computing aids an arbitrary precision orbit determination, if only the effort has been correspondingly high. Their consistent application led to the discovery of Neptune in 1846 and presented - the Age of Enlightenment - a true " triumph of celestial mechanics " dar. Neptune probable position was calculated from small perturbations of Uranus, and he found himself less than 1 ° from it.

### Refinement by minimization process

If the path of a new celestial body was determined by three good observations for the first time, it can be refined by balancing invoice or collocation in the presence of other observations. Thus, the inevitable in certain systems small contradictions are repaid by by small variation of the orbital elements of the square sum of the remaining deviations minimized ( method of least squares).

The same principle can also include the perturbation theory: based on the first web, the web disturbances ( especially in comets by Jupiter) calculated this attached to the measurements and used to determine a next best path.

## Methods and Applications

The most important application of certain new tracks is the ephemeris, the prediction of the positions for several future points in time.

In the determination of the orbit itself, a distinction

- The first calculation of a Keplerian orbit on the basis of the two-body problem
- The refined web of more than 3 observations by least squares curve fitting
- Advanced models and weighting for different observation types and accuracies - eg Speed and travel time measurements, relativistic effects
- With perturbation theory by other celestial bodies

In the treatment of the three- body problem:

- Perturbations by Jupiter Lagrange points and the Trojans
- Asteroid tracks and the Kirkwood gaps

- Satellite Tracking
- Geostationary instability and orbital maneuvers, debriefing

In the multi -body problem:

- Advance and backward calculations in the solar system for centuries to millions of years
- Modeling of star clusters, galaxies

Theory Chaotic orbits: Many cars, particularly of minor planets, extending over centuries " regular", and then suddenly drifting in one direction. In principle, all orbits are unstable in the long term, but changes are corrected out by train resonances, which is why the solar system is non- chaotic, with its eight major planets over billions of years. Systems in which not adjust those self-regulating mechanisms, (after a cosmic scale ) not old.

- Celestial mechanics