What was the critical scientific methodology Galileo employed immediately after his first sighting of objects near Jupiter?

Answer

He tracked their motion across multiple evenings to prove association.

The true genius lay in applying scientific methodology; Galileo recognized that he had to track the objects' motion across several nights to prove they were celestial bodies associated with Jupiter, not just coincidentally aligned background stars.

What was the critical scientific methodology Galileo employed immediately after his first sighting of objects near Jupiter?
discoveryGalileotelescopeJupiterMoon