Orbit determination

Orbit determination is a branch of astronomy specialised in calculating, and hence predicting, the orbits of objects such as moons, planets, and spacecraft . These orbits could be orbiting the Earth, or other bodies. The determination of the orbit of newly observed asteroids is a common usage of these techniques, both so the asteroid can be followed up with future observations, and also to check that it has not been previously discovered. Among other uses, it is critical for making use of GPS signals to determine position.

History

Orbit determination has a long history, beginning with the prehistoric discovery of the planets and subsequent attempts to predict their motions. Johannes Kepler used Tycho Brahe's careful observations of Mars to deduce the elliptical shape of its orbit and its orientation in space, deriving his three laws of planetary motion in the process.

The beginning of modern understanding of orbit determination is considered to be Anders Johan Lexell's work on computing the orbit of the comet discovered in 1770 that later was named Lexell's Comet,^[1] in which Lexell computed the interaction of comet with Jupiter that first made the comet fly close to Earth and then would have expelled it from the Solar system.^[2]

Another milestone in orbit determination was Carl Friedrich Gauss' assistance in the "recovery" of the dwarf planet Ceres in 1801. He introduced a method which, when given three observations (in the form of pairs of right ascension and declination), would result in the six orbital elements that completely describe an orbit. The theory of orbit determination has subsequently been developed to the point where today it is applied in GPS receivers as well as the tracking and cataloguing of newly observed minor planets.

Observational data

In order to determine the unknown orbit of a body, some observations of its motion with time are required. In early modern astronomy, the only available observational data for celestial objects were the right ascension and declination, obtained by observing the body as it moved relative to the fixed stars. This corresponds to knowing the object's relative direction in space, measured from the observer, but without knowledge of the distance of the object, i.e. the resultant measurement contains only direction information, like a unit vector.

With radar, relative distance measurements (by timing of the radar echo) and relative velocity measurements (by measuring the doppler effect of the radar echo) are possible. However, the returned signal strength from radar decreases rapidly, as the inverse fourth power of the range to the object. This limits radar observations to objects relatively near the Earth, such as artificial satellites and Near-Earth objects.

Methods

Orbit determination must take into account that the apparent celestial motion of the body is influenced by the observer's own motion. For instance, an observer on Earth tracking an asteroid must take into account both the motion of the Earth around the Sun, the rotation of the Earth, and the observer's local latitude and longitude, as these affect the apparent position of the body.

A key observation is that (to a close approximation) all objects move in orbits that are conic sections, with the attracting body (such as the Sun or the Earth) in the prime focus, and that the orbit lies in a fixed plane. Vectors drawn from the attracting body to the body at different points in time will all lie in the orbital plane.

Lambert's method

If the position and velocity relative to the observer are available (as is the case with radar observations), these observational data can be adjusted by the known position and velocity of the observer relative to the attracting body at the times of observation. This yields the position and velocity with respect to the attracting body. If two such observations are available, along with the time difference between them, the orbit can be determined using Lambert's method. See Lambert's problem for details.

Gauss' method

Even if no distance information is available, an orbit can still be determined if three or more observations of the body's right ascension and declination have been made. A method, made famous by Gauss in his "recovery" of the dwarf planet Ceres, has been subsequently polished.

One use of this method is in the determination of asteroid masses via the dynamic method. In this procedure Gauss' method is used twice, both before and after a close interaction between two asteroids. After both orbits have been determined the mass of one or both of the asteroids can be worked out.

References

1. ^ Valsecchi, G. '236 Years Ago...' in Near Earth Objects, Our Celestial Neighbors: Opportunity and Risk : Proceedings of the 236th Symposium of the International Astronomical Union, Cambridge University Press, 2006, xvii-xviii
2. ^ J. A. Lexell (1779). "Disquisitio De Tempore Periodico Cometae Anno 1770 Observati". Philosophical Transactions of the Royal Society of London 69: 68–85. Bibcode 1779RSPT...69...68L. 

Further reading

• Curtis, H.; Orbital Mechanics for Engineering Students, Chapter 5; Elsevier (2005) ISBN 0750661690.
• Taff, L.; Celestial Mechanics, Chapters 7, 8; Wiley-Interscience (1985) ISBN 0471893161.
• Bate, Mueller, White; Fundamentals of Astrodynamics, Chapters 2, 5; Dover (1971) ISBN 0486600610.
• Madonna, R.; Orbital Mechanics, Chapter 3; Krieger (1997) ISBN 0894640100.
• Orbit Determination and Satellite Navigation
• Satellite Orbit Determination