I am interested to see a derivation of Kepler's laws from Newtonian mechanics, which would be a suitable exercise or demonstration for a first-semester course in calculus. Therefore I'd like to avoid any use of the cross-product since this is usually only introduced much later in a calculus curriculum.
I've looked at Newton's original derivations. They don't use any objects or theories which a first-semester calculus course couldn't use. But they do involve enough geometry that it seems "a bit much" for a first-semester calculus student.
So I recognize that any such derivation will probably be hairy in one way or another. But in the off chance that there is some known elementary and simpler derivation, I wanted to ask. Thanks!