In orbital mechanics, Gauss's method is used for preliminary orbit determination from at least three observations 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, the direction cosine vector of the orbiting body from the observation points and general physical data. Carl Friedrich Gauss developed important mathematical techniques 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 of the observation points can be determined from the latitude and local sidereal time at the surface of the focal body of the orbiting body via either:
The orbiting body direction cosine vector can be determined from the right ascension and declination of the orbiting body from the observation points via:
Gauss's method of preliminary orbit determinations algorithm
The initial derivation begins with vector addition to determine the orbiting body's position vector. Then based on the conservation of angular momentum and Keplerian orbit principles, a linear combination of said position vectors is established. Also, the relation between a body's position and velocity vector by Lagrange coefficients is used which results in the use of said coefficients. Then with vector manipulation and algebra, the following equations were derived. For detailed derivation, refer to Curtis. NOTE: Gauss's method is a preliminary orbit determination, with emphasis on preliminary. The approximation of the Lagrange coefficients and the limitations of the required observation conditions causes inaccuracies. Gauss's method can be improved, however, by increasing the accuracy of sub-components, such as solving Kepler's equation. Another way to increase the accuracy is through more observations.
Step 1
Calculate time intervals, subtract the times between observations:
Step 2
Calculate cross products, take the cross products of the observational unit direction :
Step 3
Calculate common scalar quantity, take the dot product of the first observational unit vector with the cross product of the second and third observational unit vector:
Calculate the squared scalar distance of the second observation, by taking the dot product of the position vector of the second observation:
Step 7
Calculate the coefficients of the scalar distance polynomial for the second observation of the orbiting body:
Step 8
Find the root of the scalar distance polynomial for the second observation of the orbiting body: Various methods can be used to find the root, a suggested method is the Newton-Raphson method. The root must be physically possible and if multiple roots are suitable, each must be evaluated and compared to any available data to confirm their validity.
Step 9
Calculate the slant range, the distance from the observer point to the orbiting body at their respective time:
Step 10
Calculate the orbiting body position vectors, by adding the observer position vector to the slant direction vector :
Step 11
Calculate the Lagrange coefficients:
Step 12
Calculate the velocity vector for the second observation of the orbiting body:
Step 13
The orbital state vectors have now been found, the position and velocity vector for the second observation of the orbiting body. With these two vectors, the orbital elements can be found and the orbit determined.
