This technique for finding exomoons, called the Orbital Sampling Effect, was developed by René Heller and involves looking for the subtle signature of a moon’s shadow alongside the shadow of its transiting planet host, as depicted in the image below.
The dark cloud shown around the planet represents the exomoon’s shadow, averaged over several orbits. At epoch (1), a satellite transits just before the planet. At epoch (2), the planet’s transit begins, inducing a large dip in the measured stellar brightness. At epoch (3), the satellite modifies the planet’s transit light curve slightly but measurably.
This simple technique has advantages over alternative exomoon searches in that it doesn’t require significant computational resources to implement. It can also use data already available from the Kepler and K2 missions. However, on its own, the technique can’t provide a moon’s mass, only its size, and it requires many transits of the host planet to find the moon’s quite subtle transit signature.
No exomoon has been found yet in spite of tremendous efforts to find them, so the search continues.
Heller and Pudritz modeled the conditions in circumplanetary disks around Jupiter-like planets to find where temperatures are right for icy moons like Jupiter’s to form. Like Goldilocks, moon formation requires conditions that are juuust right: the planet can’t be too close to its star or too small.
But given the right conditions, moons will happily accrete around a gas giant and the most massive circumplanetary disks around super-Jovian planets can form moons the size of Mars.