Here is a frame challenge.
Is it possible for a gas giant exoplanet to appear to have 7 times the apparent diameter of the Moon, as seen from Earth, in the night sky of a giant, habitable exomoon of that gas giant?
Would the apparent diameter of a gas giant have to appear less than 7 times that of the Moon from Earth?
Or would the apparent diameter of the gas giant planet have to appear more than 7 times that of the Moon as seen from Earth?
Short Answer:
Maybe it can be posible for a habitable exomoon orbiting a Saturn-like exoplanet at a distance at which the exomplanet has an angular diameter 7 times that of the Moon. But it might be necessary to make a habitable exomoon much closer to the exoplanet, and thus where the exoplanet will have a much larger apparent magnitude.
Long Answer in Six Parts:
Part One: The distance at which a habitable exomoon would have to orbit Saturn in order for Saturn to have an angular diameter seven times that of the Moon as seen from Earth.
Because the orbit of the Moon is elliptical, and it thus becomes nearer to and father from the Earth, its apparent or angular diameter varius from 29.3 to 34.1 arcminutes. Since there are 60 arc minutes in a degree of arc, the angular diameter of the Moon varies between slightly less than half a degree to slightly more than half a degree.
So an astronomical object which has 7 times the angular diameter of the moon would have an angular diameter of about 3.5 degrees. Since a full circle has a circumference of 360 degrees, an object with an angular diameter of 3.5 degrees would go about 102.857 times withinn the circumerence of a circle.
Since the circumference of a circle is 2 Pi times the radius, or about 6.28318 times the radius, the circumference divided by 6.28318 gives the radius at which a moon would have to orbit for the planet to appear 3.5 digrees wide. That radius is 16.3702 times the diameter of the planet, or 32.7404 times the radius of the planet.
Part Two: Would an exomoon orbiting an exoplanet at such a distance be within the habitable zone around that exoplanet, where an exomoon could potentially be habitable?
As I pointed out in my answer to the question:
https://worldbuilding.stackexchange.com/questions/198117/what-if-a-moon-orbits-far-away-and-extremely-fast/198135#198135[1]
In "Magnetic Shielding of Exomoons Beyond the Circumplanetary Habitable Edge", Heller & Zuluaga calculate that an exomoon around a Neptune sized planet would either be too close to it and suffer a runaway greenhouse effect or be outside the protection of the planetary magnetic field. They calculate that, other things being favorable, a large exomoon orbiting a jupiter sized planet at a distance of 5 to 20 times the radius of the planet could avoid a runaway greenhouse effect while also being shielded by the planet's magnetic field.
If a planetary mass exomoon orbits its exoplanet at a distance of 5 to 20 planetary radi, or 2.5 to 10 planetary diameters, the circumerence of the exomoon's orbit around the exoplanet should be 15.707 to 62.8318 times the diameter of the planet. Since there are 360 degrees in a full circle, the exoplanet should have an angular diameter of between about 5.729 and 22.919 degrees of arc as seen from the exomoon, which is about 1.63 to 6.5 times the angular diameter specified in your question.
Part Three: Could an exomoon orbting far enough away that the planet had an angular diameter seven times that of the Moon as seen from Earth generate a magnetic field strong enough to keep the exomoon habitable?
Of course a habitable exomoon could orbit outside of the protective magnetic field of the exoplanet if it had a strong magnetic field of its own to deflect the charged particles of the stellar wind of its star. Such an exomoon would have to be large enough and rotate fast enough to generation a strong magnetic field.
But would an exomoon rotate fast enough to generate a strong magnetic field if it was tidally locked to its exoplanet?
An exomoon with an orbital radius of 16.3702 planetary diameters or 32.7404 planetary radii around a Saturn-like eoxplanet, with a Saturn-like equatorial radius of 60,268 kilometers, would have to orbit at a distance of 1,973,198.427 kilometers.
Iapetus, which orbits Saturn at a distance of 3,560,820 kilometers, is tidally locked to Saturn and has a rotation period of 79.3215 Earth days. Titan, which orbits Saturn at a distance of 1,221,930 kilometers, is tidally locked to Saturn and has a rotation period of 15.9454 Earth days.
Most solar system bodies with magnetic fields rotate fairly rapidly. Earth rotates at 1.0 Earth day, Uranus backwards at 0.72 Earth days, Neptune at 0.67 Earth days, Saturn at 0.44 Earth days, and Jupiter at 0.41 earth days.
Ganymede is the only moon with a magnetic field, and has a rotation period of 7.154 Earth days.
Mercury also has a magnetic field, and a rotation period of 58.65 Earth days. But the magnetic field of Mercury is only about 1.1 percent as strong as Earh's magnetic field, and so might not be stronge enough to protect a world against the stellar wind.
Part Four A rapid rotation is also needed for a short enough day and night sycle.
I also note that if the rotation period of an exomoon is too long it will tend to have extremes of temperature during the day and light. It is possible that photosynthasizing planets could die during the lightless nights, which would make the ones that grew during the day very small because the day wouldn't be long enough for them to grow much before they died.
Stephen H. Dole, in Habitable Planets for Man, 1964, page 60, estimated that the maximum posible day length for a planet habitable for humans would be four Earth days or 96 hours.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf[2]
Just what extremes of Rotation rate are compatible with habitability is difficult to say. These extremes, however, might be estimated at, say, 96 hours (4 Earth days) per revolution at the lower end of the scale and 2 to 3 hours at the upper end, or at angular velocities where the shape becomes unstable due to the high rotation rate.
Dole's words make it hard to believe that a world could be habitable with a day/night cycle much longer than 5 or 6 Earth days (120 or 144 hours).
The moon Rhea orbits Saturn at a distance of 527,108 kilometers and has a rotation period of 4.51821 Earth days, while the next outer moon, Titan, orbits Saturn at a distance of 1,221,930 kilometers and has a rotation period of 15.9454 Earth days. And a moon would have to orbit saturn (or an exoplanet identical to Saturn) at a distance of about 1,973,198.427 kilometers in order for the planet to have an angular diameter of about 3.5 degrees as seen from that moon.
Thus an exomoon orbiting a Saturn-like exoplanet at a distance necessary for the exomplanet to have an angular diameter of 3.5 degrees should, if tidally locked to the planet, have a day too long to be habitable forhumans and similar lifeforms.
Part Five: A More Massive Exoplanet could have atidally locked exomoon at the first distance which roatted fast enough.
But a much more massive exoplanet could have a tidally locked moon orbiting at a distance sufficient for the exoplanet to have an angular diameter of 3.5 degrees while the tidally locked moon orbits fast enough to have a day short enough to be habitable.
Compare Jupiter and Saturn.
Saturn has an equatorial radius of 60,268 kilometers, and thus a moon would have to orbit at a distance of 1,973,198.427 kilometers for Saturn to have an angular diameter of 3.5 degrees of arc.
Jupiter has an equatorial radius of 71,492 kilometers, and thus a moon would have to orbit at a distance of 2,340,676.677 kilometers for Jupiter to have an angular diameter of 3.5 degrees of arc.
Since the radius and diameter of Jupiter is 1.186 times that of Saturn, the volume of Jupiter is 1.668 times that of Saturn. So if Jupiter had the same overall density as Saturn, it would have 1.668 times the mass of Saturn. But the mass of Saturn is 95.159 times that of Earth, while the mass of Jupiter is 317.8 Earths. Jupiter has over 3.33 times the mass of Saturn in 1.688 times the volume of Saturn, and so has a much greater overall density.
A moon of Jupiter would have to orbit at a 2,340,676.677 kilometers for Jupiter to have an angular diameter of 3.5 degrees of arc.
Callisto orbits Jupiter at a distance of 1,882,709 kilometers and has a rotation period of 16.689 Earth days. The next moon out from Jupiter, Themisto, orbits at a distance of 7,405,000 kilometers, and if tidally locked has a rotation period of 130.18 Earth days.
Note that the orbit of Callisto has a radius, and thus a circumference, which is 1.540 times that of Titan, but has an orbital period, and thus a rotation period, which is only 1.046 times that of Titan, because it has to move faster to stay in orbit around the more massive planet Jupiter.
What happens if more and more mass is added to a planet with the mass of Jupiter? the radius, diameter, and volume of the planet will increase, but not as much as the mass increases. The increased mass and gravity will compress the planet's material more and more. Eventually the diameter and volume of the planet will cease to expand as the mass increases, and will eventually stop increasing as more massis added, and then the planet might decrease in volume with added mass.
I have read that, except for hot jupiter type gas giants whch are very close to their stars and very hot and whch expand in the heat, no planet can get much larger than Jupiter no matter how massive they are.
I note that objects with up to about 13 times the mass of jupiter are giant planets, and object with up to 75 or 80 times the mass of Jupiter are brown dwarfs, and objects with over 75 or 80 times the mass of Jupiter are low mass stars, tiny red dwarfs.
Giant planets like Jupiter and Saturn, and brown dwarfs, have their volumes supported by the pressure of the degenerate electrons in their cores.
The material inside a degenerate object like Saturn is softer than in the smaller planets; unlike the solid rock of Earth, the material at the center of Saturn gives when it is squeezed. This means that as the mass of a degenerate object increases, which increases the pressure required to counter the object's self-gravity, the density also increases. The consequence is that the radius can decrease as the mass increases. For cold bodies of the same composition, the radius goes as the inverse of the cube root of the mass. For bodies with some internal heat?and generally there is some internal heat left over from the creation of the body?the radius decreases more slowly than for the cold bodies as the mass rises. This residual heat causes Jupiter to be slightly larger than Saturn, and it causes most of the known brown dwarfs to be about the size of, rather than much smaller than, Jupiter. The trend of smaller radius with larger mass extends up to the degenerate dwarfs, which are about the size of Earth. The ratio of Sirius B's (the white dwarf in the Sirius binary system) radius to Saturn's radius is the ratio of their masses to the ?0.28 power, which is reasonably close to the ?1/3 power expected from simple arguments that ignore differences in composition and the structure of the outer, non-degenerate regions of each body. The giant gaseous planets are therefore fundamentally linked to the degenerate dwarfs through electron degeneracy.
https://astrophysicsspectator.org/topics/overview/SizeStarsPlanets.html[3]
So if your exomoon orbits a planet or a brown dwarf with many times the mass of Jupiter, the distance at which the exomoon has to orbit in order for the primary to have an angular diameter of 3.5 degrees, will not incease or decrease much, while the orbital speed needed to orbit the object at that distance will increase greatly as the mass of the primary increases, thus greadly reducing the time for the exomoon to orbit the primary.
So it might be possible for a tidally locked exomoon to have an oribital period, and thus a rotation period, short enough for life while orbiting its primary at a distance where the primary has an angular diameter of 3.5 dgrees, if the primary is a massive enough planet or brown dwarf.
Part Six: Can a habitable exomoon orbit an exoplanet at the right distance and not be tidally locked and thus have a day short enough?
Can a habitable exomoon orbit a Saturn mass planet at a distance such that the planet has an angular diameter of about 3.5 degrees, and not be tidally locked, but have a much faster rotation rate than its orbital period, a rotation rate fast enough for the exomoon to be habitable?
As I remember, Rene Heller and Roy Barnes say something on the subject in their article "Exomoon Habitabiity Constrained by Illumination and Tidal Heating".
https://faculty.washington.edu/rkb9/publications/hb13.pdf[4]
In section 2, Habitabiity of exomoons, page 20, they write:
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
and Grundy, 2011), exomoons might rotate even faster than
their orbital period.
They cite:
Cassidy, T.A., Mendez, R., Arras, P., Johnson, R.E., and Skrutskie, M.F. (2009) Massive satellites of close-in gas giant exoplanets. Astrophys J 704:1341–1348.
https://iopscience.iop.org/article/10.1088/0004-637X/704/2/1341/meta[5]
And:
Porter, S.B. and Grundy, W.M. (2011) Post-capture evolution of
potentially habitable exomoons. Astrophys J 736:L14.
https://iopscience.iop.org/article/10.1088/2041-8205/736/1/L14/meta[6]
I am not quite certain how Heller and Barnes deduce that exomoons could rotate faster than their orbital periods from those articles.
Anyway, Titan orbits Saturn at a distance of 1,221,930 kilometers and an orbital period of 15.9454 Earth days, and Iapetus orbits Saturn at a distance of 3,560,820 kilometers and an orbital period of 79.3215 Earth days. And both those moons are tidally locked and have rotation periods equal to their orbital periods.
But Hyperion, which orbits Saturn at a distance of 1,481,010 kilometers, a little farther than Titan and much closer than Hyperion, and with a rotation period of 21.2766 days, is not tidally locked. It has a rotation period of about 13 Earth days, only about 0.61 as long as its obital period.
So when and if the reasons for Hyperion not being tidally locked ae fully understood, it may be possible to calculate what masses and orbits of other moons in the system might be necessary for a large exomoon orbiting a Saturn-like exoplanet to have a roation period short enough to be habitable.
I personally would simply move the habitable exomoon closer to the giant planet and have the giant planet appear much larger in its sky that merely seven times the angular diameter of the Moon as seen from Earth.
