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Orbit determination is the estimation oforbits of objects such as moons, planets, and spacecraft. One major application is to allow tracking newly observedasteroids and verify that they have not been previously discovered. The basic methods were discovered in the 17th century and have been continuously refined.

Observations are the raw data fed into orbit determination algorithms. Observations made by a ground-based observer typically consist of time-taggedazimuth,elevation,range, and/orrange rate values. Telescopes orradar apparatus are used, because naked-eye observations are inadequate for precise orbit determination. With more or better observations, the accuracy of the orbit determination process also improves, and fewer "false alarms" result.

After orbits are determined, mathematical propagation techniques can be used to predict the future positions of orbiting objects. As time goes by, the actual path of an orbiting object tends to diverge from the predicted path (especially if the object is subject to difficult-to-predictperturbations such asatmospheric drag), and a new orbit determination using new observations serves to re-calibrate knowledge of the orbit.

Satellite tracking is another major application. For theUnited States and partner countries, to the extent thatoptical andradar resources allow, theJoint Space Operations Center gathers observations of all objects in Earth orbit. The observations are used in new orbit determination calculations that maintain the overall accuracy of thesatellite catalog.Collision avoidance calculations may use this data to calculate the probability that one orbiting object will collide with another. A satellite's operator may decide to adjust the orbit, if the risk of collision in the present orbit is unacceptable. (It is not possible to adjust the orbit for events of very low probability; it would soon use up thepropellant the satellite carries fororbital station-keeping.) Other countries, includingRussia andChina, have similar tracking assets.

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

The mathematical methods for orbit determination originated with the publication in 1687 of the first edition ofNewton'sPrincipia, which gave a method for finding the orbit of a body following aparabolic path from three observations.^[1] This was used byEdmund Halley to establish the orbits of variouscomets, including that which bears his name.Newton's method of successive approximation was formalised into an analytic method byEuler in 1744, whose work was in turn generalised to elliptical and hyperbolic orbits byLambert in 1761–1777.

Another milestone in orbit determination wasCarl Friedrich Gauss's assistance in the "recovery" of thedwarf planetCeres in 1801.Gauss's method was able to use just three observations (in the form ofcelestial coordinates) to find the sixorbital elements that completely describe an orbit. The theory of orbit determination has subsequently been developed to the point where today it is applied inGPS receivers as well as the tracking and cataloguing of newly observedminor planets.

In order to determine the unknown orbit of a body, someobservations of its motion with time are required. In early modern astronomy, the only available observational data for celestial objects were theright ascension anddeclination, obtained by observing the body as it moved in itsobservation arc, relative to thefixed stars, usingan optical telescope. 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 aunit vector.

Withradar, relativedistance measurements (by timing of the radar echo) and relativevelocity measurements (by measuring theDoppler effect of the radar echo) are possible usingradio telescopes. However, the returned signal strength from radar decreases rapidly, as the inversefourth power of the range to the object. This generally limits radar observations to objects relatively near the Earth, such asartificial satellites andNear-Earth objects. Larger apertures permit tracking of transponders on interplanetary spacecraft throughout theSolar System, andradar astronomy of natural bodies.

Various space agencies and commercial providers operate tracking networks to provide these observations. SeeCategory:Deep space networks for a partial listing. Space-based tracking of satellites is also regularly performed. SeeList of radio telescopes#Space-based andSpace Network.

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 the motion of the Earth around theSun, 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 areconic sections, with the attracting body (such as the Sun or the Earth) in theprime 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 theorbital plane.

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, invented in the 18th century. SeeLambert's problem for details.

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.Gauss's method, made famous in his 1801 "recovery" of the firstlost minor planet,Ceres, has been subsequently polished.

One use is in the determination of asteroid masses via thedynamic method. In this procedure Gauss's 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.^{[citation needed]}

The basic orbit determination task is to determine the classicalorbital elements orKeplerian elements,${\displaystyle a,e,i,\mathrm{\Omega },\omega ,\nu }$, from theorbital state vectors [${\displaystyle \overrightarrow{r},\overrightarrow{v}}$], of an orbiting body with respect to thereference frame of its central body. The central bodies are the sources of the gravitational forces, like the Sun, Earth, Moon and other planets. The orbiting bodies, on the other hand, include planets around the Sun, artificial satellites around the Earth, and spacecraft around planets. Newton'slaws of motion will explain the trajectory of an orbiting body, known asKeplerian orbit.

The steps of orbit determination from one state vector are summarized as follows:

• Compute thespecific angular momentum${\displaystyle \overrightarrow{h}}$ of the orbiting body from its state vector:$${\displaystyle \overrightarrow{h}=\overrightarrow{r}\times \overrightarrow{v}=\left|\overrightarrow{h}\right|\overrightarrow{k}=h\overrightarrow{k},}$$ where${\displaystyle \overrightarrow{k}}$ is the unit vector of the z-axis of the orbital plane. The specific angular momentum is a constant vector for an orbiting body, with its direction perpendicular to the orbital plane of the orbiting body.
• Compute theascending node vector${\displaystyle \overrightarrow{n}}$ from${\displaystyle \overrightarrow{h}}$, with${\displaystyle \overrightarrow{K}}$ representing the unit vector of the Z-axis of the reference plane, which is perpendicular to the reference plane of the central body:$${\displaystyle \overrightarrow{n}=\overrightarrow{K}\times \overrightarrow{h}.}$$ The ascending node vector is a vector pointing from the central body to theascending node of the orbital plane of the orbiting body. Since the line of ascending node is the line of intersection between the orbital plane and the reference plane, it is perpendicular to both the normal vectors of the reference plane (${\displaystyle \overrightarrow{K}}$) and the orbital plane (${\displaystyle \overrightarrow{k}}$ or${\displaystyle \overrightarrow{h}}$). Therefore, the ascending node vector can be defined by thecross product of these two vectors.
• Compute theeccentricity vector${\displaystyle \overrightarrow{e}}$ of the orbit. The eccentricity vector has the magnitude of the eccentricity,${\displaystyle e}$, of the orbit, and points to the direction of theperiapsis of the orbit. This direction is often defined as the x-axis of the orbital plane and has a unit vector${\displaystyle \overrightarrow{i}}$. According to the law of motion, it can be expressed as:$${\displaystyle e=\left|\overrightarrow{e}\right|}$$ where${\displaystyle \mu =GM}$ is thestandard gravitational parameter for the central body of mass${\displaystyle M}$, and${\displaystyle G}$ is theuniversal gravitational constant.
• Compute thesemi-latus rectum${\displaystyle p}$ of the orbit, and its semi-major axis${\displaystyle a}$ (if it is not aparabolic orbit, where${\displaystyle e=1}$ and${\displaystyle a}$ is undefined or defined as infinity):$${\displaystyle p=\frac{h^2}{\mu }=a(1-e^2)}$$$${\displaystyle a=\frac{p}{1-e^2},}$$ (if${\displaystyle e\ne 1}$).
• Compute theinclination${\displaystyle i}$ of the orbital plane with respect to the reference plane: where${\displaystyle h_K}$ is the Z-coordinate of${\displaystyle \overrightarrow{h}}$ when it is projected to the reference frame.
• Compute thelongitude of ascending node${\displaystyle \mathrm{\Omega }}$, which is the angle between the ascending line and the X-axis of the reference frame: where${\displaystyle n_I}$ and${\displaystyle n_J}$ are the X- and Y- coordinates, respectively, of${\displaystyle \overrightarrow{n}}$, in the reference frame.
  Notice that${\displaystyle \cos (A)=\cos (-A)=\cos (360-A)=C}$, but${\displaystyle \arccos (C)}$ is defined only in [0,180] degrees. So${\displaystyle \arccos (C)}$ is ambiguous in that there are two angles,${\displaystyle A}$ and${\displaystyle 360-A}$ in [0,360], who have the same${\displaystyle \cos }$ value. It could actually return the angle${\displaystyle A}$ or${\displaystyle 360-A}$. Therefore, we have to make the judgment based on the sign of the Y-coordinate of the vector in the plane where the angle is measured. In this case,${\displaystyle n_J}$ can be used for such judgment.
• Compute theargument of periapsis${\displaystyle \omega }$, which is the angle between the periapsis and the ascending line: where${\displaystyle e_K}$ is the Z-coordinate of${\displaystyle \overrightarrow{e}}$ in the reference frame.
• Compute thetrue anomaly${\displaystyle \nu }$ at epoch, which is the angle between the position vector and the periapsis at the particular time ('epoch') of observation: The sign of${\displaystyle \overrightarrow{r}\cdot \overrightarrow{v}}$ can be used to check the quadrant of${\displaystyle \nu }$ and correct the${\displaystyle \arccos }$ angle, because it has the same sign as thefly-path angle${\displaystyle \varphi }$. And, the sign of the fly-path angle is always positive when${\displaystyle \nu \in [0,{180}^{\circ }]}$, and negative when${\displaystyle \nu \in [{180}^{\circ },{360}^{\circ }]}$.^[1] Both are related by${\displaystyle h=rv\sin (90-\varphi )}$ and${\displaystyle \overrightarrow{r}\cdot \overrightarrow{v}=rv\cos (90-\varphi )=h\tan (\varphi )}$.
• Optionally, we may compute theargument of latitude${\displaystyle u=\omega +\nu }$ at epoch, which is the angle between the position vector and the ascending line at the particular time: where${\displaystyle r_K}$ is the Z-coordinate of${\displaystyle \overrightarrow{r}}$ in the reference frame.

• Curtis, H.;Orbital Mechanics for Engineering Students, Chapter 5; Elsevier (2005)ISBN 0-7506-6169-0.
• Taff, L.;Celestial Mechanics, Chapters 7, 8; Wiley-Interscience (1985)ISBN 0-471-89316-1.
• Bate, Mueller, White;Fundamentals of Astrodynamics, Chapters 2, 5; Dover (1971)ISBN 0-486-60061-0.
• Madonna, R.;Orbital Mechanics, Chapter 3; Krieger (1997)ISBN 0-89464-010-0.
• Schutz, Tapley, Born;Statistical Orbit Determination, Academic Press.ISBN 978-0126836301
• Satellite Orbit Determination, Coastal Bend College, Texas

• Astrometry
• Spaceflight technology
• Orbits
• Astrodynamics

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