“”Take three observations of an asteroid not separated from one another by more than 15 days, or three of a comet not separated from one another by more than 6 days, and compute the elements of the orbit by both the method of Laplace and also that of Gauss.
|— Sample problem from Celestial Mechanics course
Julius Bauschinger specialized in the field of determining the orbital paths of heavenly bodies. He fitted in well at Berlin, where many people worked on the measurement and calculation of locations, orbits and distances; the calculation of ephemerides (planetary tables); and the refinement of methods for correcting observational errors.
Bauschinger's "older theories of celestial mechanics" course began with Johannes Kepler, who made the first determination of the laws of planetary motion, and Issac Newton, who had spent much effort on determination of cometary orbits. It then passed through the elaboration of celestial mechanics by Pierre Simon de Laplace, Joseph Louis Lagrange, Wilhelm Olbers and finished with the methods of Karl Friedrich Gauss. In all, it covered the period from 1600 to about 1850. This approach allowed Bauschinger to develop the course as a history of orbital calculation and error reduction. He was in the midst of writing what would become the standard text of orbital determinations, "Die Bahnbestimmung der Himmelskörper" (1906). He was also spending much time documenting the history of the field in Germany and producing translations of hard-to-find earlier works.
The fundamental problem in this branch of celestial mechanics was to find the orbit -- and thus the position relative to Earth at any given time -- of an object -- such as a comet or an asteroid that was moving around the Sun -- and to find it with a minimum number of observations -- usually three. Restricting observations to the absolute minimum also minimized the immense amount of trigonometric calculation required to solve the equations involved in the problem. One took the celestial latitude and longitude (the right ascension and declination) of a celestial object on three successive occasions. Then, using these coordinates and the time of observation, generated a total of nine equations that had to be solved for nine unknowns. There were three principal approaches to this problem, all treated in Bauschinger's course. Each approach had strengths and weaknesses, depending on the character of the orbit being studied, especially its eccentricity and the number and character of perturbations.