Observing a comet and predicting its trajectory I'm planning to observe a comet of apparent magnitude around $18.5$ using an observatory telescope. I plan to figure out the trajectory of the comet $-$ all its orbital elements. Given that I only know the positions of the comet on respective nights, how many and what kind of observations should be sufficient for me to get a rough estimate of the trajectory? Would the light curve of the comet be of any help in determining the trajectory?
 A: An observation gives you two data, usually right ascension and
declination. Since orbital elements are 6, theoretically 3
observations would suffice.
But things are much less favourable in practice, for several reasons. I'll not refer to nongravitational forces. The main issue are observational uncertainties and the only way to reducing their adverse effect is to make more than 3 observations. How to use them, which computations are needed, is excluded in a short answer. I advise you to read a book on celestial mechanics, where this problem is generally
discussed, both in its history ad in present practice.
An historical note. A major contributor to solve the problem was
Gauss, who in 1809 wrote a book about it: Theoria motus corporum
coelestium ... (Theory of the Motion of Celestial Bodies ...). His motivation was the recent discovery of the first asteroids. A reliable determination of their orbits was needed to recognize one of them when it reappeared near Earth after it had disappeared because of distance or because having come in a direction too near to Sun.
To that purpose Gauss in his book founded the theory of errors:
stated the law of distribution of accidental errors (the Gaussian) and fully exploited the method of least squares.
In case of a comet a wide set of observations, made when it is both near and as far as possible from perihelion, is needed because of large eccentricity.
