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Concept
Gauss's method
Summary
In orbital mechanics (a 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. 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.
Carl Friedrich Gauss developed important mathematical techniques (summed up in Gauss's methods) which were specifically used to determine the orbit of Ceres. 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.
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:
or
where,
is the respective observer position vector (in Equatorial Coordinate System)
is the equatorial radius of the central body (e.g., 6,378 km for Earth)
is the geocentric distance
is the oblateness (or flattening) of the central body (e.g., 0.003353 for Earth)
is the eccentricity of the central body (e.g., 0.081819 for Earth)
is the geodetic latitude (the angle between the normal line of horizontal plane and the equatorial plane)
is the geocentric latitude (the angle between the radius and the equatorial plane)
is the geodetic altitude
is the local sidereal time of observation site
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:
where,
is the respective unit vector in the direction of the position vector (from observation point to orbiting body in Topocentric Equatorial Coordinate System)
is the respective declination
is the respective right ascension
The initial derivation begins with vector addition to determine the orbiting body's position vector.
Official source
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Related lectures (1)
Matrix Operations: Scaling and Invertibility
Covers scaled matrices, ladder matrices, and the Gauss method for matrix operations.
Related concepts (3)
Orbit determination
Orbit determination is the estimation of orbits of objects such as moons, planets, and spacecraft. One major application is to allow tracking newly observed asteroids 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-tagged azimuth, elevation, range, and/or range rate values.
Kepler orbit
In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on.
Ceres (dwarf planet)
Ceres (pronounced ˈsɪəriːz, ), minor-planet designation 1 Ceres, is a dwarf planet in the asteroid belt between the orbits of Mars and Jupiter. It was the first asteroid discovered, on 1 January 1801, by Giuseppe Piazzi at Palermo Astronomical Observatory in Sicily and announced as a new planet. Ceres was later classified as an asteroid and then a dwarf planet - the only one always inside Neptune's orbit. Ceres's small size means that even at its brightest, it is too dim to be seen by the naked eye, except under extremely dark skies.