Gauss's method
 | This article, Gauss's method, has recently been created via the Articles for creation process. Please check to see if the reviewer has accidentally left this template after accepting the draft and take appropriate action as necessary.
Reviewer tools: Inform author |
In orbital mechanics (subfield of celestial mechanics), Gauss' method is used for preliminary orbit determination from at least three observations (more observations increases the accuracy of the determined orbit) of the 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.
Gauss developed important mathematical techniques (summed up in Gauss' methods) which was 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. For more details on Ceres' orbit determination via Gauss' method, refer to Tennenbaum.[1]
Observer Position Vector
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 (e.g., Earth) via either:
R_n= [R_e/√(1-(2f-f^2 ) sin^2〖ϕ_n 〗 )+H_n ] cos〖ϕ_n (cos〖θ_n I ̂+sin〖θ_n J ̂ 〗 〗 )〗+[(R_e (1-f)^2)/√(1-(2f-f^2 ) sin^2〖ϕ_n 〗 )+H_n ] sin〖ϕ_n K ̂ 〗
or
R_n= R_e cos〖〖ϕ'〗_n 〗 cos〖θ_n 〗 I ̂+R_e cos〖〖ϕ'〗_n 〗 sin〖θ_n 〗 J ̂+R_e sin〖〖ϕ'〗_n 〗 K ̂
where, Rn is the respective observer position vector (in Equatorial Coordinate System) Re is the equatorial radius of the body (e.g., Earth's Re is 6,378 km) f is the oblateness (or flattening) of the body (e.g., Earth's f is 0.003353) φn is the respective geodetic latitude φ'n is the respective geocentric latitude Hn is the respective altitude θn is the respective local sidereal time
References
- Tennenbaum, J and Director, B. "How Gauss Determined the Orbit of Ceres."
- Curtis, Howard D.. Orbital mechanics for engineering students. Oxford: Elsevier Butterworth-Heinemann, 2005. Print.
- Gronchi, Giovanni F.. "Classical and modern orbit determination for asteroids." Proceedings of the International Astronomical Union2004.IAUC196 (2004): 1-11. Print.
- Der, Gim J.. "New Angles-only Algorithms for Initial Orbit Determination." Advanced Maui Optical and Space Surveillance Technologies Conference. (2012). Print.