I've dug into the idea of a habitable moon getting daily eclipses due to its gas giant getting in the way of the sun; but most examples I've found specify a tidally locked moon. I'm no astrophysicist so I'm curious what this would mean more for a moon that still rotates on its axis.
Would the giant stay in the sky due to its proximity, or would it also appear/disappear like the sun?
I will use the term exomoon to described a moon in another star system orbiting an exoplanet in another star system, and refer to Earth's moon as The Moon.
If a large, habitable exomoon of a giant expplanet is tidally locked to the planet, that means that the the rotation period of the exomoon around its axis in relation to the distant stars has the same length as the exomoon's orbit around its planet. The exomoon will turn 360 degrees once while it orbits the planet once.
So one side of the exomoon will always face the exoplanet and one side of the exomoon wiil always face away from the planet, and the planet will be on the horizon as seen from the dividing line between those two hemispheres of the exomoon.
That is a situation like the situation on the Moon, which is tidally locked to the Earth. One side always faces the Earth, and one side always faces away from the Earth. I think that was first mentioned in Kepler's Somnium in 1634.
If an exoplanet doesn't orbit its star but stays in the same spot relative to the star (which is impossible), the exomon will have a cycle of light and dark which has the same length as the period of its orbit around the planet. Thus the sun will appear to rise and set on any spot on the exomoon, and any spot on the exomoon will have a succession of days and nights.
But since real planets orbit around their stars, during one orbit of the exomoon around its planet, the planet will have moved partially around the star.
That means that the sidereal day of the exomoon, the time it takes to turn 360 degrees with respect to the distant stars, which be the same as its orbital period, but the stellar or synodic day of the exomoon, the time it takes to turn 360 degrees with respect to its own star & thus the length of cycles of light and dark on it's surface, will be different.
In the case of Earth's Moon, it's sidereal day is 27.321661 Earth days, and its synodic day is 29.530589 Earth days.
If the orbital period of the exomoon is a small fraction of the exoplanet's orbital period around its star, the sideral day and the synodic day of the exomoon will be very similar. If the orbital period of the exomoon is a large fraction of the exoplanet's orbital period around its star, the sideral day and the synodic day of the exomoon will be very different in length.
So what sort of day length will a hyothetical habitable exomoon of a giant exoplanet have, considering that it will be orbiting in the habitable zone of its star?
The synchronized rotation periods of putative Earth-mass exomoons around giant planets could be in the same range as the orbital periods of the Galilean moons around Jupiter (1.7d−16.7d) and as Titan’s orbital period around Saturn (≈16d) (NASA/JPL planetary satellite ephemerides)4. The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P∗p/9, P∗p being the planet’s orbital period about the star (Kipping 2009a). Since the satellite’s rotation period also depends on its orbital eccentricity around the planet and since the gravitational drag of further moons or a close host star could pump the satellite’s eccentricity (Cassidy et al. 2009; Porter & Grundy 2011), exomoons might rotate even faster than their orbital period.
https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf
So that give the impression that the sidereal day of a habitable exomoon might be about 1.6 to 16 Earth days long.
You should also note their statement that:
The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P∗p/9, P∗p being the planet’s orbital period about the star (Kipping 2009a)
It means that if the exomoon has a stable orbit for billions of years to become habitable for oxygen breathers - like humans, for example - the orbital period of the exoplanet around the star will have to be at least 9 times the orbital period of the exomoon around the exoplanet.
Since a potentally habitable exomoon might not have a sufficient magnetic field to shield it from stellar wind and cosmic radiation, it might have to orbit within the magnetic field of its exoplanet.
This article suggests that a potentially habitable exomoon would have to orbit its exoplanet at a distance of about 5 to 20 planetary radii.
https://arxiv.org/abs/1309.0811
According to my calculations, the planet would thus appear to have an angular diameter of about 2.9 to 11 degrees of arc as seen from a habitable exomoon, which is about 6 to 22 times the angular diameter of the Moon as seens from Earth.
So if the orbit of the exomoon around its exoplanet is in the same plane, or a similar plane, to the orbit of the exoplanet around its star, the exoplanet should eclipse the star once in every orbit of the exomoon. Of course if the exomoon is tidally locked to the exoplanet the eclipses should only be visible from the side of the exomoon which faces the exoplanet.
I hope this helps you a bit.
