Gauss's 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. 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. 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 (for example, the Earth) via either: Orbiting body  direction cosine vector. 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: 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 (which states that an orbit lies in a two dimensional plane in three dimensional space), 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 (i.e., insignificant curvature in the arc between observations, refer to Gronchi for more details) 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 (order matters): Step 3. Calculate common scalar quantity (scalar triple product), take the dot product of the first observational unit vector with the cross product of the second and third observational unit vector: Step 4. Calculate nine scalar quantities (similar to step 3): Step 5. Calculate scalar position coefficients: Step 6. 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 (i.e., not negative nor complex) 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 (which is the slant distance multiplied by the slant direction vector): Step 11. Calculate the Lagrange coefficients: