Short Answer: HykranianBlade should consider where they want the story to be within the Mohs scale of Science Fiction Hardness trope.
A writer who wanted their story to be level one on the scale wouldn't worry at all about scientific plausibility.
But HykranianBlade seems to want their story to be a least a little, and possibly a lot, more scientifically plausible than a level one story. In fact, there seem to be scientific calculations showing that the maximum possible orbital period of a habitable moon of a giant planet would be only a little more than about 17 earth days, perhaps no more than about 20 Earth days.
So HykranianBlade should probably read my long answer.
First I point out the fictional habitable exomoon of a gas giant exoplanet in another star system is likely to orbit in the gas giant's equatorial plane and also to rotate in the same plane as the giant planet rotates. Tidal interactions between the exomoon and the exoplanet are likely to realign its orbit and rotation that way just a few million years after they form, and it should take thousands of times that long for the exomoon to become as habitable as I think it will be in the story.
So the 14 days of light followed by 14 days of darkness would only happen during the equinoxes of your exomoon. During some seasons at latitudes the light periods could be many times as long as dark, and in others the dark periods many times as long as the light periods, like on Earth.
On Earth there are seasons because the axis of Earth's rotation is titled 23 degrees away from perpendicular to the Earth's orbital plane around the Sun. Thus there are reversed seasons in the northern and southern hemispheres of Earth.
And seasons change the relative lengths of days and nights, especially at higher northern and southern latitudes.
This table shows the axial tilts of the eight planets in our solar system, varying from 3.13 degrees to 82.23 degrees.
A day night cycle that lasts for 28 Earth days may have other implications beside how plants will adapt to long periods of alternating constant light and increasing temperatures and constant darkness and lowering temperatures.
HyrkanianBlade, like every writer of stories set on other planets, moons, and other types of worlds, should research current speculation and calculations about various possibilities.
And if HyrkanianBlade wants to depict lifeforms on those worlds, he should study research about what is necessary for a world to have life.
And if HyrkanianBlade wants to depict humans from Earth walking around on the planet without environmental protective suits, or native intelligent beings who have requirements similar to those of Earth humans, then he should study the specific requirements for Earth humans.
If a demon offered to teleport someone to a randomly selected location and bring them back after a month there, the person might be clever and restrict the possible locations to those within Earth's biosphere, so that he wouldn't be teleported into outer space and die.
But Earth's biosphere includes all locations where some lifeforms can live, including several kilometers or miles high in the atmosphere, or beneath the ocean, or deep within solid rock. If the person restricts the random locations to the surface of the Earth, most locations on the surface of the Earth are in the ocean many kilometers and miles from the nearest land. If the persons restricts the random locations to the land surface of Earth, they might wind up in a desert or arid location and die of thirst, or a hot or cold enough place to die of heat or cold.
Some Earthly lifeforms flourish where humans would die within weeks, days, hours, minutes, or seconds.
So when astrobiologists discuss the conditions necessary for life, they often do not restrict themselves to conditions necessary for human survival. They often discuss conditions were life could exist but where humans and similar alien beings would die almost instantly if unprotected.
Fortunately for science fiction writers who tend to concentrate on alien worlds where humans or aliens with similar needs could flourish, I know of at least one scientific study dedicated to that specific sub category of astrobiology: Habitable Planets for Man, Stephen H. Dole, 1964, 2007.
The 1964 edition is online here:
Although the 2007 edition may be updated and more accurate.
On page 53 Dole begins the discussion of the mass range for a planet habitable for humans.
On page 53 Dole said that a surface gravity of about 1.5 g seemed like the maximum that humans would tolerate, and that corresponded to a planet with a mass of 2.35 earth masses, a radius of 1.25 Earth radii, and an escape velocity of 15.3 kilometers per second.
The minimum mass for a habitable planet would be the minimum mass necessary to have an escape velocity high enough relative to the average velocity of air particles to retain an atmosphere for billions of years.
On page 54 Dole calculated the minimum size of a planet that could retain a breathable atmosphere for billions of years as 0.195 Earth's mass, with of 0.63 of Earth's radius and a surface gravity of 0.49 g. But Dole believed such a planet would be unable to produce an atmosphere dense enough to be breathable.
...To prevent atomic oxygen from escaping from the upper layers of its atmosphere, the planet's escape velocity must be of the order of five times the root-mean-square velocity of the oxygen atoms in the atmosphere. This is shown in figure 12 (see page 37)...then the escape velocity of the smallest planet capable of retaining atomic oxygen may be as low as 6.25 kilometers per second (5 X 1.25). Going back to figure 9, this may be seen to correspond to a planet having a mass of 0.195 Earth mass, a radius of 0.63 Earth radius, and a surface gravity of 0.49 g. Under the above assumptions, such a planet could theoretically hold an oxygen-rich atmosphere, but it would probably be much too small to produce one, as will be seen below.
Dole calculated via various lines of reasoning two figures for the minimum mass necessary to produce a breathable atmosphere, 0.253 Earth mass, which he believed too low, and 0.57 Earth Mass, which he believed too high:
With 0.25 being too low, and 0.57 being too high, the appropriate value of mass for the smallest habitable planet must lie between those figures, somewhere in the vicinity of 0.4 Earth mass.
...This corresponds to a planet having a radius of 0.78 Earth radius and a surface gravity of 0.68 g.
So if you want your alien exomoon to have an oxygen rich atmosphere that humans or similar beings could breath and survive in, it should be at least as massive as Dole's 0.4 Earth mass. Or if one disagrees with Dole's reasoning, one might think that the minimum possible mass for a habitable exomoon might be somewhere between 0.253 and 0.57 Earth mass. Possibly someone might believe the minimum possible mass would be the minimum possible mass to retain oxygen n the atmosphere, which Dole calculated at 0.195 Earth mass.
The minimum mass for a world with a dense and oxygen rich atmosphere is especially important in the case of an exomoon orbiting an exoplanet in another star system, because there is a question whether the maximum possible mass of an exomoon would be enough for it to retain an oxygen rich atmosphere for geological lengths of time.
The most massive moon in our Solar System, Ganymede, has a mass of only 0.0248 that of Earth, which is barely more than 12 percent of the minimum mass necessary for a world to retain an oxygen rich atmosphere.
But the moon of a giant planet with the most mass relative to its primary is Triton, the moon of Neptune, with a mass 0.003599 of Earth, orbiting Neptune, with a mass 17.147 Earths. Thus the ratio is as high as 0.0002098, so if Jupiter, with a mass of 317.8 Earths, had a moon with that relative mass that moon would have mass 0.0666744 of Earth.
Giant planets can be much more massive than Jupiter. The theoretical division between highly massive planets and brown dwarfs is about 13 times the mass of Jupiter, while the theoretical division between brown dwarfs and low mass stars is about 75 to 80 times the mass of Jupiter. Thus a giant planet about 13 times the mass of Jupiter, or 4,131.4 times the mass of Earth, could have a moon with a 0.0002098 mass ratio and thus a mass of 0.8667677 that of Earth.
And there are other possibilities for giant exoplanets to have much more massive exomoons than Ganymede.
You want your exomoon to be more covered with water than Earth. It is believed that as a general trend the larger an Earth like world is, the more water it will have, which may require that your exomoon be more massive than Earth. However, I note that on Earth the proportion of the surface covered by water has varied significantly over time as sea levels rise and fall and cover more or less of the surfaces of continents, and as the sizes of continents change over eons due to geological forces.
Many of the moons of the outer planets are tiny irregular objects thought to be captured asteroids. In our Solar System the longest orbital period of any moon of a giant planet that probably formed with the planet instead of being captured later is the orbital period of Iapetus, 79.3215 Earth days. Thus your period of 28 Earth days for your exomoon is within the limits of possibility.
But there may be some problems with such an orbital period. the closer a moon orbits its planet, the smaller its orbit will be, and the faster it has to orbit to avoid falling into the planet. Those two factors will make its orbital period shorter. The farther a moon orbits from its planet, the larger its orbit will be, and the slower it will have to move to avoid escaping from the planet. Those two factors will make its orbital period longer. Moons that orbited planets of different mass at the same distance would have different orbital speeds and periods.
The formula for calculating the distance that a body would have to orbit another body of a specified mass to have a specified orbital period is here:
A moon of a planet, including an exomoon of an exoplanet, will have to orbit within the Hill Sphere of the planet in order to remain in orbit.
The formula for calculating the Hill Sphere of a planet relative to its star is found here:
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for retrograde orbits at a large distance from the primary is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated.5
So therefore a fictional exomoon should orbit its fictional exoplanet within 0.5000 or even 0.3333 of the maximum calculated Hill sphere of the fictional exoplanet, in order to have an orbit stable for the billions of years of time necessary to become habitable.
The size of the Hill sphere of a planet depends on its mass, the mass of its star, and the distance between them. Adjusting those parameters will change the size of a fictional planet's Hill sphere, and thus of its smaller zone where an exomoon can have a necessary stable orbit.
You need to increase the size of the possible orbit of the exomoon around its exoplanet, so that the exomoon's orbital period will be as long as your desired 28 days. But there are a few "catch 22" problems to watch for.
Making your fiction exoplanet more massive relative to its star will increase the size of its Hill sphere and its inner zone of true stability. But the more massive a planet is, the farther away its moon will have to be in order to have an orbital period of 28 days.
Increasing the distance that your fictional exoplanet orbits its star will increase the size of the exoplanet's zone of stability. But your fictional exoplanet will have to orbit within the star's circumstellar habitable zone.
To find the size of a star's circumstellar habitable zone, find the inner and outer limits of the Sun's circumstellar habitable zone and then multiply by the square root of the star's luminosity relative to the Sun.
Unfortunately there is considerable uncertainty about the inner and outer edges of the Sun's circumstellar habitable zone. This table of estimates of the Sun's habitable zone illustrates the uncertainty:
Unless a writer's research convinces them that a specific estimate for the size of the Sun's Habitable zone is very probably correct, they should make their habitable worlds receive exactly as much radiation from their star as Earth gets from the Sun, in order to be certain that will be the right amount of luminosity. Then all they have to do is multiply one Astronomical Unit (AU), the distance between Earth and the Sun, by the square root of the star's luminosity relative to the Sun's luminosity, to get the distance between their exoplanet and its star to calculate their exoplanet's Hill sphere.
What determines how luminous a main sequence star (the only type of star suitable for a writer who wants a habitable planet to consider using) is relative to the Sun? The mass of the star, slightly modified by its age, will determine how luminous the star is relative to the Sun. And a slight change in the mass of the star will make a significantly larger change in its luminosity.
A writer wanting an exoplanet's moon to have an orbital period as long as 28 days will want the exoplanet to orbit as far from the star as possible for the planet to have a Hill Sphere as large as possible, and thus will want the star to be as luminous as possible. But increasing the luminosity of a star means increasing its mass, which tends to decrease the size of it's planet's Hill sphere. Since small increases in mass cause large increases in luminosity, the mass of a star necessary for a planet to have as large a Hill sphere as possible will have to be calculated.
There is an inner limit to how closely an object held together by its gravity, such as a moon, can orbit a planet.
In celestial mechanics, the Roche limit, also called Roche radius, is the distance within which a celestial body, held together only by its own force of gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction.3 Inside the Roche limit, orbiting material disperses and forms rings, whereas outside the limit material tends to coalesce. The term is named after Édouard Roche (pronounced [ʁɔʃ] (French), /rɔːʃ/ rawsh (English)), who was the French astronomer who first calculated this theoretical limit in 1848.4
The formula for calculating the Roche limit is here:
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The Roche limit will probably not be a problem for someone who wants their exomoon to have an orbital period as long as 28 days.
There are other factors which narrow down the orbital distances for a habitable exomoon, that create a sort of "circumplanetary habitable zone" around an exoplanet where an exomoon can be habitable.
The possibility of habitable exomoons has been discussed in scientific papers. such as:
Heller, René; Rory Barnes (2012). "Exomoon habitability constrained by illumination and tidal heating". Astrobiology. 13 (1): 18–46.
Heller, René (September 2013). "Magnetic shielding of exomoons beyond the circumplanetary habitable edge". The Astrophysical Journal Letters. 776 (2): L33.
As Heller and Barnes say in section 2 of their 2012 paper:
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 (1.7–16.7 d) and as Titan's orbital period around Saturn (≈16 d) (NASA/JPL planetary satellite ephemerides)4.
So the desired orbital period of 28 Earth days would be about 16.47 to 1.6788 times as long as the observed orbital periods of large satellites around giant planets in our solar systems. And Heller and Barnes are clearly concerned about the possibility that day-night cycles that are too long would have negative impact on the habitability of giant exomoons.
In that section Heller and Barnes also say that:
The longest possible length of a satellite's day compatible with Hill stability has been shown to be about Pp/9, Pp being the planet's orbital period about the star (Kipping, 2009a).
Therefore a natural satellite can not have an orbital period around its planet longer than one ninth of the planet's orbital period around it star. Since an exomoon orbital period of 28 Earth days around its exoplanet is desired, that exoplanet would have to have an orbital period around it's star that was at least about nine times that long, or at least about 252 Earth days.
Of the few known exoplanets orbiting in their stars' habitable zones, Kepler-1638 b has the orbital period closest to 252 days, being 259.337 Earth days long, and orbiting 0.745 AU from Kepler-1638,. Kepler-62 f has a similar period of 267.291 Earth days, orbiting Kepler-62, a spectral type K2V star with a mass of about 0.69 that of the Sun, at a distance of 0.718 AU.
Thus the minimum possible mass of a star with a planet orbiting within the star's habitable zone with a period of 252 Earth days would probably be about 0.65 of the mass of the Sun. If a habitable exomoon has an orbital period of 28 Earth days, then the exoplanet it orbits should have an orbital period of at least about 252 days, and thus the star should have a mass of at least about 0.65 of the Sun's mass.
On the other hand, if your fictional exomoon had an orbital period only 1.0222 Earth days long, it could orbit an exoplanet with an orbital period around its star of only 9.2 earth days. Exoplanet TRAPPIST-1 f orbits the star TRAPPIST-1 within its habitable zone with a period of 9.2 Earth days, and TRAPPIST-1 is a spectral class M8V star with a mass of about 0.089 times that of the Sun. So if your fictional exomoon had an orbital period only 1.0222 Earth days long the star that its planet orbited could have a mass as low as about 0.089 of the mass of the Sun.
In their section 2.1 Heller and Barnes mention that it has been shown that moons formed in the circumplanetary disc around a planet will have no more than 0.0001 of the mass of the planet. Jupiter has a mass 317.8 times Earth's. the largest planets would have about 13 times the mass of Jupiter or about 4,121.4 times the mass of Earth. So an exomoon formed in the circumplanetary disc around the most massive possible exoplanet could have no more than about 0.43134 of the mass of Earth, just about what Dole calculated was the minimum possible mass for a world to form a dense oxygen rich atmosphere and be habitable for humans.
Fortunately Heller and Barnes discuss several suggested methods for exoplanets to acquire Earth mass exomoons.
Heller and Barnes also introduce the "habitable edge", an inner limit to how closely an otherwise habitable exomoon can orbit an exoplanet without light reflected from the planet onto the moon, and tidal heading of the moon, providing too much energy and leading to a runaway greenhouse effect as on the planet Venus. They work out formulas for calculating whether an exomoon will suffer a runaway greenhouse effect.
So the "habitable edge" concept for the orbits of habitable exomoons leads to the concept of a circumplanetary habitable zone for moons.
Planetary-mass natural satellites have the potential to be habitable as well. However, these bodies need to fulfill additional parameters, in particular being located within the circumplanetary habitable zones of their host planets. More specifically, moons need to be far enough from their host giant planets that they are not transformed by tidal heating into volcanic worlds like Io, but must remain within the Hill radius of the planet so that they are not pulled out of the orbit of their host planet. Red dwarfs that have masses less than 20% of that of the Sun cannot have habitable moons around giant planets, as the small size of the circumstellar habitable zone would put a habitable moon so close to the star that it would be stripped from its host planet. In such a system, a moon close enough to its host planet to maintain its orbit would have tidal heating so intense as to eliminate any prospects of habitability.
In Heller, René (September 2013). "Magnetic shielding of exomoons beyond the circumplanetary habitable edge". The Astrophysical Journal Letters. 776 (2): L33.
Heller discusses whether a giant planet's magnetic field would extend far enough to protect its moon's from negative effects due to particle radiation from outer space and from the star. For smaller giant planets, the protection of the planetary magnetic field will take a long time to extend as far as the orbits of exomoons that are far enough from the planet to avoid a runaway greenhouse effect, and thus those exomoons will lose their atmospheres and water and become uninhabitable. Larger giant planets can extend their magnetic fields out to exomoons orbiting beyond the habitable edge in time to protect those exomoons from loss of water and atmosphere.
Moons between 5 and 20 Rp can be habitable, depending on orbital eccentricity, and be affected by the planetary magnetosphere at the same time.
So Heller calculates that a exomoon could be habitable if orbiting between 5 and 20 Rp, where Rp is the exoplanet's radius. The 20 planetary radii outer limit should usually be much closer than the Hill sphere limit, and thus be the significant factor in the outer edge of a circumplanetary habitable zone.
Uranus has a mass of 8.6810 times ten to the 25th power kilograms, or 14.536 Earths and an equatorial radius of 25,559 kilometers miles. Five to twenty times the equatorial radius would be 127,795 to 511,180 kilometers. 127,795 kilometers would be inside the orbit of Miranda, which has an orbital period of 1.413 Earth days, and 511,180 kilometers would be between the orbits of Titania and Oberon, which have orbital periods of 8.705 and 13.463 Earth days.
Neptune has a mass of 1.024 times ten to the 26th power kilograms, or 17.147 Earths and an equatorial radius of 24,764 kilometers. 5 to 20 times the equatorial radius is a distance of 123,820 kilometers, and 20 times the equatorial radius is a distance of 495,280 kilometers. A distance of 123,820 kilometers is farther than the orbit of Proteus, which has an orbital period of 1.122 Earth days, and a distance of 495,280 kilometers is inside the orbit of Triton, which has an orbital period of 5.877 Earth days.
Saturn has a mass of 5.6834 times ten to the 26th power kilograms, or 95.2 Earths, and an equatorial radius of 60,268 kilometers, or 37,449 miles. So a distance of 5 to 20 times the radius of Saturn would be a distance of 301,340 to 1,205,360 kilometers. A distance of 301,340 kilometers would be between the orbits of Calypso and Dione, which have orbital periods of 1.887 and 2.736 Earth days. A distance of 1,205,360 kilometers would be inside the orbit of Titan, which has an orbital period of 15.945 Earth days.
Jupiter has a mass of 1.8982 times ten to the 27th power kilograms, or 317.8 Earths, and an equatorial radius of 71,492 kilometers or 44,423 miles. A distance of 50 to 20 times the equatorial radius would be 357,460 to 1,429,840 kilometers. A distance of 357,460 kilometers would be between the orbits of Thebe and Io, which have orbital periods of 16 hours and 1.7691 Earth days. A distance of 1,429,840 kilometers would be between the orbits of Ganymede and Callisto, which have orbital periods of 7.1546 and 16.689 Earth days.
These examples indicate that the best exoplanet for a habitable exomoon to orbit with an orbital period as long as 28 Earth days would be one both more massive and with a larger radius than Jupiter.
Unfortunately, Jupiter has almost the largest possible radius for a planet. When planets get a little more massive than Jupiter, they become compressed to greater and greater densities by their increasing gravity.
But there’s also a more literal take on the question: Is there a limit on how physically large a planet can be? Here there is a definite and rather surprising answer. Jupiter is 11 times the diameter of Earth, and it turns out that is about as large as any planet can be! If you kept dumping more matter on Jupiter, it would not get any larger. Instead, gravity would crush its mass together more tightly and efficiently.
Through the whole range from a Jupiter-mass planet to the brown dwarf boundary, all the way up to the lowest-mass dwarf stars (about 70 times the mass of Jupiter, the point at which sustained lithium and hydrogen fusion occurs), the size barely budges. All of these objects are within about 15 percent of the same diameter. That constancy has some odd consequences.
Take, for example, the star Trappist-1A, which was in the news recently because it has seven Earth-size planets orbiting it. Trappist-1A is a red dwarf, just 1/2000th as bright as the sun, but it’s genuine star, no question. It is powered by steady, sustained nuclear reactions that will burn for a trillion years or more. It is 80 times as massive as Jupiter.
On the other hand, Trappist-1A is less than 10 percent larger in diameter than Jupiter. Put those two details together, and you quickly realize that this little star must be extremely dense–as indeed are all extremely dim, cool red dwarf stars...
...Even more extreme is the red dwarf star EBLM J0555-57Ab, recently measured to be 15 percent smaller than Jupiter, about the size of Saturn. It is the tiniest known mature star (as opposed stellar cinders like white dwarfs or neutron stars), and it is 17 times the density of lead–188 times the density of water!
This means that even the most massive exoplanet will have a radius - and thus a circumplanetary habitable zone - not much larger than that of Jupiter, while having many times the mass of Jupiter and thus forcing the moons in the circumplanetary habitable zone around the exoplanet to orbit much faster and have much shorter orbital periods than the moons of Jupiter within Jupiter's circumplanetary habitable zone.
So the current calculations indicate that unless an exomoon is large enough to have its own magnetic field to protect it from particle radiation, it will have to orbit within 20 planetary radii of the exoplanet to be protected by the planet's magnetic field, and thus it will not be able to have an orbital period much more than 17 Earth days long, at a guess not more than about 20 Earth days long.