Orbit determination
Encyclopedia
Orbit determination is a branch of astronomy
specialised in calculating, and hence predicting, the orbit
s 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 asteroid
s 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.
s 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
, 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
.
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 planet
s.
s 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 object
s.
, 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 section
s, 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
.
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.
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
specialised in calculating, and hence predicting, the orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s of objects such as moons, planets, and spacecraft . These orbits could be orbiting the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
, or other bodies. The determination of the orbit of newly observed asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s 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
Global Positioning System
The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...
signals to determine position.
History
Orbit determination has a long history, beginning with the prehistoric discovery of the planetPlanet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s and subsequent attempts to predict their motions. Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
used Tycho Brahe
Tycho Brahe
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...
's careful observations of Mars
Mars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
to deduce the elliptical shape of its orbit and its orientation in space, deriving his three laws of planetary motion
Kepler's laws of planetary motion
In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....
in the process.
The beginning of modern understanding of orbit determination is considered to be Anders Johan Lexell
Anders Johan Lexell
Anders Johan Lexell was a Swedish-born Russian astronomer, mathematician, and physicist who spent most of his life in Russia where he is known as Andrei Ivanovich Leksel .Lexell made important discoveries in polygonometry and celestial mechanics; the latter led to a comet named in...
's work on computing the orbit of the comet discovered in 1770 that later was named Lexell's Comet
Lexell's Comet
D/1770 L1, popularly known as Lexell's Comet after its orbit computer Anders Johan Lexell, was a comet discovered by astronomer Charles Messier in June 1770. It is notable for having passed closer to the Earth than any other comet in recorded history, approaching to a distance of only...
, in which Lexell
Anders Johan Lexell
Anders Johan Lexell was a Swedish-born Russian astronomer, mathematician, and physicist who spent most of his life in Russia where he is known as Andrei Ivanovich Leksel .Lexell made important discoveries in polygonometry and celestial mechanics; the latter led to a comet named in...
computed the interaction of comet with Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
that first made the comet fly close to Earth and then would have expelled it from the Solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
.
Another milestone in orbit determination was Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
' assistance in the "recovery" of the dwarf planet
Dwarf planet
A dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be spherical as a result of its own gravity but has not cleared its neighboring region of planetesimals and is not a satellite...
Ceres in 1801. He introduced a method which, when given three observations (in the form of pairs of right ascension
Right ascension
Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.-Explanation:...
and declination
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...
), would result in the six orbital elements
Orbital elements
Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used...
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 planet
Minor planet
An asteroid group or minor-planet group is a population of minor planets that have a share broadly similar orbits. Members are generally unrelated to each other, unlike in an asteroid family, which often results from the break-up of a single asteroid...
s.
Observational data
In order to determine the unknown orbit of a body, some observationObservation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...
s of its motion with time are required. In early modern astronomy, the only available observational data for celestial objects were the right ascension
Right ascension
Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.-Explanation:...
and declination
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...
, 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
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
, relative distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
measurements (by timing of the radar echo) and relative velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
measurements (by measuring the doppler effect
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...
of the radar echo) are possible. However, the returned signal strength from radar decreases rapidly, as the inverse fourth power
Fourth power
In arithmetic and algebra, the fourth power of a number n is the result of multiplying n by itself four times. So:Fourth powers are also formed by multiplying a number by its cube...
of the range to the object. This limits radar observations to objects relatively near the Earth, such as artificial satellites and Near-Earth object
Near-Earth object
A near-Earth object is a Solar System object whose orbit brings it into close proximity with the Earth. All NEOs have a perihelion distance less than 1.3 AU. They include a few thousand near-Earth asteroids , near-Earth comets, a number of solar-orbiting spacecraft, and meteoroids large enough to...
s.
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 SunSun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
, 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 section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s, 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
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...
.
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 planetDwarf planet
A dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be spherical as a result of its own gravity but has not cleared its neighboring region of planetesimals and is not a satellite...
Ceres, has been subsequently polished.
One use of this method is in the determination of asteroid masses via the dynamic method
Dynamic method
The dynamic method is a procedure for the determination of the masses of asteroids. The procedure gets its name from its use of the Newtonian laws of the dynamics, or motion, of asteroids as they move around the solar system. The procedure works by taking multiple position measurements to...
. 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.
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