Gauss's method

efficient method
In orbital mechanics (subfield of celestial mechanics), Gauss's method is used for preliminary orbit determination from at least three observations (more observations increases the accuracy of the determined orbit) of the orbiting body of interest at three different times.wikipedia
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Orbital mechanics

astrodynamicsastrodynamicistorbital dynamics
In orbital mechanics (subfield of celestial mechanics), Gauss's method is used for preliminary orbit determination from at least three observations (more observations increases the accuracy of the determined orbit) of the orbiting body of interest at three different times.
Gauss's method was able to use just three observations (in the form of pairs of right ascension and declination), to find the six orbital elements that completely describe an orbit.

Ceres (dwarf planet)

Ceres1 CeresAtmosphere of Ceres
Carl Friedrich Gauss developed important mathematical techniques (summed up in Gauss's methods) which were specifically used to determine the orbit of Ceres.
To recover Ceres, Carl Friedrich Gauss, then 24 years old, developed an efficient method of orbit determination.

Orbit determination

assess its orbitdetermined orbitGaussian method
In orbital mechanics (subfield of celestial mechanics), Gauss's method is used for preliminary orbit determination from at least three observations (more observations increases the accuracy of the determined orbit) of the orbiting body of interest at three different times.
Gauss's method was able to use just three observations (in the form of celestial coordinates) to find the six orbital elements that completely describe an orbit.

Carl Friedrich Gauss

GaussCarl GaussCarl Friedrich Gauß
Carl Friedrich Gauss developed important mathematical techniques (summed up in Gauss's methods) which were specifically used to determine the orbit of Ceres.
Gauss's method involved determining a conic section in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law).

Least squares

least-squaresmethod of least squaresleast squares method
There are techniques/methods available that can be used but why not use Gauss's own method, least squares method (still popularly used today).
An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid Ceres.

Celestial mechanics

celestialcelestial dynamicscelestial mechanician
In orbital mechanics (subfield of celestial mechanics), Gauss's method is used for preliminary orbit determination from at least three observations (more observations increases the accuracy of the determined orbit) of the orbiting body of interest at three different times.

Equatorial coordinate system

equatorial coordinatesequatorialCoordinates
The required information are the times of observations, the position vectors of the observation points (in Equatorial Coordinate System), the direction cosine vector of the orbiting body from the observation points (from Topocentric Equatorial Coordinate System) and general physical data. The observer position vector (in Equatorial coordinate system) of the observation points can be determined from the latitude and local sidereal time (from Topocentric coordinate system) at the surface of the focal body of the orbiting body (for example, the Earth) via either:

Earth

Earth's surfaceterrestrialworld
The method shown following is the orbit determination of an orbiting body about the focal body where the observations were taken from, whereas the method for determining Ceres' orbit requires a bit more effort because the observations were taken from Earth while Ceres orbits the Sun.

Sun

solarSolThe Sun
The method shown following is the orbit determination of an orbiting body about the focal body where the observations were taken from, whereas the method for determining Ceres' orbit requires a bit more effort because the observations were taken from Earth while Ceres orbits the Sun.

Sidereal time

sidereal daysidereallocal sidereal time
The observer position vector (in Equatorial coordinate system) of the observation points can be determined from the latitude and local sidereal time (from Topocentric coordinate system) at the surface of the focal body of the orbiting body (for example, the Earth) via either:

Latitude

latitudesSouthlatitudinal
The observer position vector (in Equatorial coordinate system) of the observation points can be determined from the latitude and local sidereal time (from Topocentric coordinate system) at the surface of the focal body of the orbiting body (for example, the Earth) via either:

Altitude

Right ascension

RAR.A.α
The orbiting body direction cosine vector can be determined from the right ascension and declination (from Topocentric Equatorial Coordinate System) of the orbiting body from the observation points via:

Declination

DecDec.declinations
The orbiting body direction cosine vector can be determined from the right ascension and declination (from Topocentric Equatorial Coordinate System) of the orbiting body from the observation points via:

Angular momentum

conservation of angular momentumangular momentamomentum
Then based on the conservation of angular momentum and Keplerian orbit principles (which states that an orbit lies in a two dimensional plane in three dimensional space), a linear combination of said position vectors is established.

Kepler orbit

Keplerian orbitKeplerianKeplerian ellipse
Then based on the conservation of angular momentum and Keplerian orbit principles (which states that an orbit lies in a two dimensional plane in three dimensional space), a linear combination of said position vectors is established.

Kepler's equation

formula for this inverseinverse radial Kepler equationradial Kepler equation
This can be done by solving the universal Kepler's equation.

Triple product

scalar triple productvector triple productLagrange's formula

Standard gravitational parameter

gravitational parametergeocentric gravitational constantheliocentric gravitational constant

Newton's method

Newton–Raphson methodNewton-RaphsonNewton–Raphson
Various methods can be used to find the root, a suggested method is the Newton-Raphson method.

Slant range

rangeSlant directionslant distance
Calculate the slant range, the distance from the observer point to the orbiting body at their respective time:

Lagrange polynomial

Lagrange interpolationLagrange formLagrange polynomials
Also, the relation between a body's position and velocity vector by Lagrange coefficients is used which results in the use of said coefficients.