How many observations are needed to determine a comet's orbit? Based on the following facts:


*

*We have Kepler's laws of planetary motion.

*We have a good knowledge of the positions and orbits  of the gravitationally significant objects in the Solar System.

*We can numerically calculate orbital variables quickly and accurately using computers.


What is the minimum number of accurate visual observations we need to make in order to calculate the orbital elements of a newly discovered comet or asteroid?
I do appreciate the more observations we get, the more accurate the orbit we can calculate, and we will look for more, especially for example, if a comet produces a tail(s), but theoretically would just two/three/four observations be enough to be, say 90% sure we know it's orbital elements.
This may be a question for AstronomySE,  no problem placing it there if necessary.  
 A: An infinite number of observations are needed because a comet does not have a well defined orbit. It is strongly deflected from ideal Newtonian orbits by outgassing, solar wind etc., so if you want to know where it is, and especially where it will be, at a given time in the future, you have to keep observing.
A: Martin Hoecker-Martinez's Answer is correct for perfectly noiseless observations of a two body Kepler system, i.e. the force between the bodies is directed along the vector linking them and the force magnitude follows and inverse square law with distance. An alternative to Martin's answer is that perfectly known position and velocity will determine all future motions for the two body problem.
However, these assumptions do not strictly hold: a comet not only interacts with the Sun but also the planets (especially Jupiter) and other gravitational sources and moreover there are other nonideal "noises" that one must account for (solar wind and so forth) as in CuriousOne's answer. 
Practically, the way one handles this is to assume a Kepler model (or more elaborate model if you know where all the planets are and can thus account for their effects) and treat other perturbations as additive Gaussian noise. One then finds the current maximum likelihood estimate of the orbit model parameters (position, velocity) using a Kalman Filter Algorithm, which I describe here and here. So you will begin six observations and then make new ones regularly, using the Kalman filter to update your estimates. The Kalman filter will give you current variances on your estimates so you can always put fairly rigorous error bounds on theoretically calculated future positions. So there will be some finite number of observations needed for whatever accuracy you need in your calculations: the Kalman filter will let you know when you have enough.
Historically, exactly your problem motivated the invention of the Kalman filter. For, although we credit Rudolf Kalman as its inventor, it was in fact first published in 1809 by Carl Friedrich Gauss, where he documented his use of it for simplifying hand calculations made in estimating the orbital parameters of celestial bodies.  See
"Recursive Estimation and the Kalman Filter" in D.G.S. Pollock's "Kalman Filters"
A: An ideal Kepplerian orbit is defined by six (6) parameters:


*

*angular momentum (3)

*total energy (1)

*Laplace–Runge–Lenz vector which is perpendicular to angular momentum (2)


Therefore you need at least six (6) independent observations.  Astronomical observations are direction (but not range usually) given by a pair (2) of angles therefore three (3) independent observations are the absolute minimum you need.
("ideal Kepplerian" meaning you ignore general relativity and all gravitational interactions except for the Sun) 
