# 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:

### Flattening

**oblatenessellipticityflattened**

### 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:

### 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.

### Cross product

**vector cross productvector productcross-product**

### Dot product

**scalar productdotinner product**

### 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.