Perhaps you have already anticipated the key to this final mystery: retrograde motion.
An astronomer tracking the movement of planets through the star field watches not so much on the scale of minutes, but on the longer scale of days, weeks and months.
On this scale of time, Jupiter did stop.
On December 25 of 2 BC as it entered retrograde, Jupiter reached full stop in its travel through the fixed stars. Magi viewing from Jerusalem would have seen it stopped in the sky above the little town of Bethlehem.