It's not that hard to apply Newtonian mechanics to a comet's orbit. "They" can predict the time and place of an eclipse of the sun to the very minute, decades in advance. Predicting the arrival of a comet, to within a thousand years or so is really not difficult. Halley never saw his eponymous comet, but he predicted its return based on a handful of imprecise ancient observations.
Newtonian mechanics is anything but an occult art. There are literally hundreds, probably thousands, of astronomy programs that can process the observations, and make consistent (and verifiable) predictions about its orbit.
