You might want to consider some suggestions and lower the mass of your planet to 1 Jupiter mass or 317.8 Earth masses. And/or circularize the orbit of your moon by avoiding any other large moons or nearby large planets which might keep the orbit elliptical. Those would reduced the strength of the tidal heating on your moon.
And if you want to keep the mass of your planet 7.8 Jupiter mases or 2,478.84 Earth masses, read the long answer where I discuss ways to keep your moon from having a runaway greenhouse effect from too strong tidal heating.
Part One: Sidereal and Synodic Days.
A satellite of a giant planet would probably be tidally locked in 1:1 resonance so that its rotation period would equal its orbital period. Thus one side would always face the planet and one side would always face away from the planet.
As the moon orbited the planet the angle between, the planet, the moon, and the star would constantly change, and thus as seen from the moon the star would seem to rise and set in a day/night cycle approximately as long as the moon's orbital period around the planet.
But not exactly as long as the moon's orbital period around the planet. As the moon orbits around the planet, the planet is orbiting around the star. Thus the time which it takes for the moon, the planet, and the star to return to a previous alignment is longer than it takes for the moon to orbit around the planet.
For example, the Moon's orbital period around the Earth is 27.321661 Earth days, and since it is tidally locked to the Earth, its sidereal day, the time it takes the Moon to rotate 360 degrees relative to the stars is also 27.321661 Earth days. But since the Earth travels a number of degrees in its orbit around the Sun during that period, it takes longer, 29.530589 Earth days, for the Moon to rotate 360 degrees relative to the Sun and for the Sun to return to the same position in the sky of a place on the Moon - a synodic day.
The larger the ratio of a planet's orbit around the star divided by the moon's orbit around the planet is, the smaller will be the difference between the sidereal and synodic days of the moon.
The smaller the ratio of a planet's orbit around the star divided by the moon's orbit around the planet is, the larger will be the difference between the sidereal and synodic days of the moon.
A similar effect makes the sidereal day of planet Earth 0.997 Earth days or 23 hours 56 minutes and 4.1 seconds and the synodic day of Earth 1.00 Earth days or 24 hours, 0 minutes and 0 seconds.
Part Two: The Day Length Range for Habitable Planets.
And for some purposes the sidereal day of a planet or of a moon will be important and for other purposes the synodic day will be important.
For example, in Habitable Planets for Man (1964), Stephen H. Dole considered the length range for a day of a habitable planet on pages 41 to 48 and on pages 58 to 61.
On page 60 Dole guessed that an Earth sized planet would probably be unstable if it rotated in less than 2 or 3 Earth hours. And he decided that days longer than 96 Earth hours or 4 Earth days would be too long.
And obviously Dole wrote about the sidereal days of a planet in the first case and the synodic days of a planet in the second case, but he didn't specify that, which might confuse some readers.
And I think that possibly a planet or moon could be habitable for liquid water using life in general, and even habitable for humans beings and beings with the same requirements as humans in particular, with a da/night cycle, a synodic day, much longer than four Earth days.
What the longest imaginable length for a day/night cycle, a synodic day? Obviously an infinitely long synodic day would be the longest imaginable.
And if a planet is tidally locked in 1:1 resonance to it's star, it will have an infinitely long synodic day since it will take forever for the apparent position of the star in the sky of the planet to change.
One side of the planet will face the star in an eternal day and one side of the planet will face away from the star in eternal night.
And many people believe that a tidally locked planet would be lifeless because all of the water will become frozen on the dark side of the planet and stay frozen forever.
The only ways in which potential life could avoid either an inferno or a deep freeze would be if the planet had an atmosphere thick enough to transfer the star's heat from the day side to the night side, or if there was a gas giant in the habitable zone, with a habitable moon, which would be locked to the planet instead of the star, allowing a more even distribution of radiation over the planet. It was long assumed that such a thick atmosphere would prevent sunlight from reaching the surface in the first place, preventing photosynthesis.
This pessimism has been tempered by research. Studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibars (0.10 atm), for the star's heat to be effectively carried to the night side. This is well within the levels required for photosynthesis, though water would still remain frozen on the dark side in some of their models. Martin Heath of Greenwich Community College, has shown that seawater, too, could be effectively circulated without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Further research—including a consideration of the amount of photosynthetically active radiation—suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.
So if it might be possible for a world to be habitable despite having an infinitely long synodic day, it might be possible for a world to be habitable with a synodic day that is Earth weeks, months, or years long, despite being longer than Dole's rather arbitrary guess of a limit at 4 Earth days.
That means that you can make your moon orbit your planet at a distance far enough that the tidal heating is not too large.
Part Three: The Day Length of a Moon With Various Orbital Distances.
The Gas Giant has a mass of around 7.8 Jupiter Masses, and of course a radius of ~1 Jupiter Radii. The mass of the moon does not appear to be relevant for the question as it does not seem to factor into the tidal heating formula.
Jupiter has an equatorial radius of about 71,492 kilometers. And I guess I will arbitrarily set the radius of your giant planet at 70,000 kilometers.
Five times the radius will be 350,000 kilometers, ten times the radius will be 700,000 kilometers, fifteen times the radius will be 1,050,000 kilometers, twenty times the radius will be 1,400,00 kilometers, fifty times the radius will be 3,500,000 kilometers, and one hundred times the radius will be 7,000,000 kilometers.
The formula for calculating the orbital period of an astronomical object can be found at:
And naturally I use an online orbital period calculator.
The mass of Jupiter is 317.8 times the mass of Earth, so your planet should have 2,478.84 the mass of Earth, while your moon can have 1 Earth mass and the semi-major axis can be in kilometers.
At 5 rad11 (350,000 kilometers) the sidereal period is 0.479 days, at 10 radii 1.3547 Earth days, at 15 radii 2.489 Earth days, at 20 radii 3.832 Earth days, at 50 radii 15.146 Earth days, at 100 radii (7,000,00 kilometers) 42.84 Earth days.
So it should be possible to put your moon in a far enough orbit around the planet that the tidal heating is not excessive while the sidereal day of the moon is not incredibly long - the orbital period of the planet around the star will be important in calculating the length of the synodic day.
Part Four: An Upper Limit to the Orbital Period of a Moon.
In "Exomoon Habitability Constrained by Illumination and Tidal Heating" (2013) Rene Heller and Roy Barnes discussed the orbital periods of potentially habitable tidally locked moons - which would of course be the same as their sidereal days - on page 3:
. However, considering an Earth-mass exomoon around a Jupiter-like host planet,
within a few million years at most the satellite should be tidally locked to the planet – rather than to the star (Porter &
Grundy 2011). This configuration would not only prevent a primordial atmosphere from evaporating on the illuminated side
or freezing out on the dark side (i.) but might also sustain its internal dynamo (iii.). 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.
The upper limit they put on the orbital period and thus sidereal day of a potentially habitable exomoon is:
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).
That means that the moon's orbital period around the planet should be less than about one ninth (1/9 or 0.11111) of the orbital period of the planet around the star.
So if the moon has an orbital period of 15.146 Earth days, the orbital period of the planet should be at least about 136.314 Earth days; if the moon has an orbital period of 40.5833 Earth days, the orbital period of the planet around the star should be at least about 365.25 Earth days or one Earth year; if the moon has an orbital period of 42.84 Earth days, the orbital period of the planet around the star should be at least about 385.56 Earth days; and so on.
Heller and Barnes don't speculate about whether sidereal days 1.7 to 16.6 Earth days long - with somewhat longer synodic days - would be too long for life on such hypothetical Earth size moons of giant planets. They probable believed that most Earth size moons of giant planets that had atmospheres dense enough for surface water to be liquid would also have atmospheres dense enough to transfer heat from the day side to the night side no matter how long the synodic days were.
Part Five: Another Upper Limit on the Orbital Distance and Period of a Moon.
But there is one other limit on how far a potentially habitable exomoon could orbit from its exoplanet. Such a moon would have to orbit within the Hill Radius of the planet to have an orbit stable long enough for life to evolve and to make a breathable oxygen rich atmosphere.
The formula for calculating the Hill radius can be found at:
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. As stated, the satellite (third mass) should be small enough that its gravity contributes negligibly.6: p.26ff
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.
In our solar system Earth has a Hill sphere radius of 1,471,400 kilometers, about 21.02 arbitrary 70,000 kilometer radii of your planet. The true region of stability would be about 490,466 to 735,700 kilometers.
Jupiter has a Hill sphere radius of 50,573,600 kilometers, about 722.5 arbitrary 70,000 kilometer radii of your planet. The true region of stability would be about 16,857,866.7 to 25,286,800 kilometers.
Saturn has a Hill sphere radius of 61,634,000 kilometers, about 880.5 arbitrary 70,000 kilometer radii of your planet. The true region of stability would be about 20,544,666.7 to 30,817,000 kilometers.
Part Six: If Your Planet Orbits 1 AU From a Star with the Mass and Luminosity of the Sun.
Your planet has a mass of about 7.8 Jupiter's mass or about 2,478.84 Earth's mass
According to this Hill sphere calculator
A planet with a mass of 2,478.84 Earth's mass orbiting a star with the mass of the Sun at a distance of 1 AU and with an orbital eccentricity of 0.01 would have a Hill radius of 20,050,969.995648 kilometers. It would have a true region of stability about 6,683,656.6 to 10,025,485 kilometers.
If the planet orbited a star with the mass of the Sun and a semi-major axis of 1 AU, it would have an orbital period of about 365.25 Earth days. A moon orbiting at the limit of orbital stability at about 1/9 of that period would have an orbital period of about 40.58333 Earth days. According to the orbital period calculator, a moon orbiting the planet at 6,752,000 kilometers would have an orbital period of 40.58 days. That gives a rough agreement with 1/3 of the Hill radius being 6,683,656.6 kilometers.
If a 7.8 Jupiter mass planet would have a radius of about 60,000 to 80,000 kilometers, a moon orbiting at exactly the edge of the stable orbit radius would orbit at about 83.5 to 112.5 times the radius of the planet.
Most of the energy which warms Earth comes from the Sun and only a tiny fraction comes from inside the Earth.
The geothermal heat flow from the Earth's interior is estimated to be 47 terawatts (TW) and split approximately equally between radiogenic heat and heat left over from the Earth's formation. This corresponds to an average flux of 0.087 W/m2 and represents only 0.027% of Earth's total energy budget at the surface, being dwarfed by the 173,000 TW of incoming solar radiation.
You say that:
I did find the point in which the tidal heating per square metre is equal to the solar flux of Earth, if that is relevant. ~14R.
So if the moon and planet orbit a star like the Sun at a distance of 1 AU they will receive about 1.631 kilowatts per square meter from the star, about as much as Earth gets from the Sun. And if that moon orbits its planet with 7.8 times the mass of Jupiter at 14 planetary radii, the tidal heating of the moon should be equal to the solar energy received, according to your calculations.
Thus the moon's surface should get twice the energy that Earth gets from the Sun. That would not make the moon twice as hot as the Earth, but its average temperature would be higher, perhaps too high for to be comfortable for humans.
If the moon orbited at twice 14 planetary radii, or 28 planetary radii, the effect of tidal would be smaller, and so the overall temperature of the moon lower. If the moon orbited at 4 times 14 radii, or 56 radii, the heating effect would be smaller still. if the moon orbited at 8 times 14 radii or 112 radii the tidal heating would be even smaller. And that would be about as far as the moon could get from the planet in a stable orbit.
So depending on how much the tidal heating diminished with distance it might be possible for the moon to have an Earthlike average temperature if the planet and moon orbited at 1 AU from a star similar to the Sun.
Part Seven: If Your Planet Orbited a Sun-like Star at a Distance of 5.2038 AU.
What if the planet and moon orbited beyond the habitable zone? Radiation from the star would be inadequate to keep the moon warm enough for liquid surface water. But the light from the star might be enough to energize photosynthesis if some other factor kept the moon warm enough. If tidal heating can be enough to make a world too hot for life a lesser amount of tidal heating can be enough to make a world warm enough for life.
A planet with a mass of 2,478.84 Earth's mass orbiting a star with the mass of the Sun at a distance of 5.2038 AU (the semi-major axis of Jupiter) and with an orbital eccentricity of 0.01 would have a Hill radius of 104,341,338 kilometers. It would have a true region of stability extending out to about 34,780,446 to 52,170,669 kilometers.
If your planet has Jupiter's orbital period of 4,332.59 Earth days or 11.862 Earth years, its moon should have an orbital period no more than 1/9, or 0.111111, of that, or 481.4 Earth days. If the moon orbits the planet at a distance of 35,100,000 kilometers it should have a period of 481 days.
So the moon should have a stable orbit out to about 440 to 580 planetary radii.
A planet and its moon orbiting a star with the luminosity of the Sun at a distance of 5.2038 Au would receive 1/(5.2038 X 532038), or 1/27.079, or 0.0369 as much radiation as Earth gets from the Sun. According to your calculation if the moon orbited the 7.8 Jupiter mass planet at about 14 planetary radii, it would have tidal heating equal to the solar flux of Earth. Thus it would have total of 1.0369 times much energy received as earth receives from the Sun.
If that might be too much, the moon could orbit the planet farther out, and thus receive slightly less tidal heating, so that the total energy is the same as Earth gets. Certainly there be room to move it a long way farther from the planet.
The big question would be whether sunlight only 0.0369 as intense as on Earth would be strong enough for photosynthesis at that distance.
Part Eight: If Your Planet Orbited a Sun-like Star at a Distance of 9.5826 AU.
A planet with a mass of 2,478.84 Earth's mass orbiting a star with the mass of the Sun at a distance of 9.5826 AU (the semi-major axis of Saturn) and with an orbital eccentricity of 0.01 would have a Hill radius of 192,140,425 kilometers. It would have a true region of stability about 64,046,808 to 96,070,212.5 kilometers.
Saturn has an orbital period of 10,755.70 Earth days or 29.4475 Earth years. So a moon with an orbital period no more than one ninth that long would have an orbital period no more than 1,195.07 Earth days long. A moon orbiting a planet with a mass of 7.8 Jupiters with a period of about 1,195.07 days would have a semi-major axis of about 64,388,501.4 kilometers.
A planet orbiting a star with the luminosity of the Sun at a distance of 9.5826 AU would receive only 1/91.826 or 0.01089 of the radiation from its star that Earth gets from the Sun. If the moon orbits a planet with 7.8 times the mass of Jupiter at a distance of about 14 radii it will have tidal heating about equal to the radiation that Earth gets from the Sun, according to your calculation. Thus the moon would receive a total of about 1.01089 as much energy as Earth gets from the Sun.
And if that might be too much energy and might overheat the moon, the moon could always be moved to a farther orbit around the planet to reduce the amount of tidal heating and make the total energy input 1.000 that of Earth. Certainly the moon could be moved much farther than 14 planetary radii from the planet before having an unstable orbit.
If the starlight at the orbit of Jupiter or Saturn might be too weak for photosynthesis, the planet and its moon might be closer to the star to have stronger starlight.
Part Nine: If Your Planet Orbited a Sun-like Star at a Distance of 3.16337766 AU
If the planet orbits a star with the the mass and luminosity of the Sun at a distance of 3.16337766 AU, the starlight will have 0.1 of the intensity of Sunlight at 1 AU. If the moon orbits the planet with 7.8 of the mass of Jupiter at a distance of about 14 radii it should receive as much energy from tidal heating as Earth gets from the Sun, according to your calculations. Thus it would receive a total of 1.1 times as much energy as Earth gets from the Sun.
And if that might be too much energy and overheat the moon, the moon could be moved farther away from the planet to reduce its tidal heating to about 0.9 of the energy that Earth gets from the Sun.
How far from the planet could the moon get and still have a long term stable orbit?
According to the Hill sphere calculator, the planet would have a Hill radius of 57,662,536.859604 kilometers. The true region of stability would extend to about 19,220,845.6 to 28,831,268.3 kilometers.
If the planet & moon orbit a star with a mass of 1 Sun at a distance of 3.16337766 AU, the planet's orbital period around the star will be 2,047.2 days. One ninth of that would be 227.46666 days. If the moon orbits the planet with 7.8 times the mass of Jupiter at a distance of 21,305,000 kilometers it should have an orbital period of 227.47 days.
If the planet has a radius of 60,000 to 80,000 kilometers, the region of stability should extend out to a distance of about 240 to 355 planetary radii.
Part Ten: Magnetic Shielding of Moons.
In "Magnetic Shielding of Exomoons Beyond the Planetary Habitable Edge" Rene Heller and Jorge I. Zuluaga discuss the magnetic shielding of exomoons.
Cosmic rays from interstellar space and the stellar wind of a nearby star contain charged particles which can kill lifeforms on the surface of world. If that world has an atmosphere the atmosphere can stop charged particles and protect the surface and any lifeforms on it. But the interactions of charged particles with the upper atmosphere can knock away into interplanetary space atmospheric particles and gradually erode the world's atmosphere, leading to atmospheric loss sooner than would otherwise happen.
If the planet or other world has a magnetic field the magnetic field deflects the motions of charged particles, often trapping many of them in radiation belts above the atmosphere of the world and protecting both the atmosphere and the surface from charged particles.
In the article Heller and Zuluaga discuss hypothetical Mars sized exomoons of giant exoplanets which might exist and which might be habitable if all conditions for habitability are present.
Considering such exomoons to probably be too small to have their own magnetic fields, they believe that such moons would need to be inside the magnetic fields of their giant planets to be protected from stellar wind and cosmic rays.
If the moons orbit inside the wrong parts of the magnetic fields of their planets, they could be trapped in the radiation belts of their planets and bombarded by even more charged particles than outside the magnetic field. But moons that orbit outside the radiation belts but inside the planetary magnetic fields will be shielded from charged particles.
Heller and Zuluaga says that those exomoons would have to orbit out side of the habitable edges of their planets. If exomoons orbit inside the habitable edges of their planets they will have excessive tidal heating and suffer from Runaway Greenhouse or RG and all their surface water will end up in the atmosphere leaving the surface dry and lifeless.
But habitable moons would have to orbit inside the magnetic fields of their planets if they didn't have their own magnetic fields. Heller and Zuluaga investigate whether it is possible to orbit outside the Habitable edge of a planet while also orbiting inside the planet's magnetic field, modeling the development of the magnetic fields of giant planets.
Their conclusion is:
Here we synthesize models for the evolution of the magnetic environment of giant planets with thresholds from the runaway greenhouse (RG) effect to assess the habitability of exomoons. For modest eccentricities, we find that satellites around Neptune-sized planets in the center of the HZ around K dwarf stars will either be in an RG state and not be habitable, or they will be in wide orbits where they will not be affected by the planetary magnetosphere. Saturn-like planets have stronger fields, and Jupiter-like planets could coat close-in habitable moons soon after formation. Moons at distances between about 5 and 20 planetary radii from a giant planet can be habitable from an illumination and tidal heating point of view, but still the planetary magnetosphere would critically influence their habitability.
So that seems to be the origin of the idea the best distance for a habitable moon to orbit being between 5 and 20 planetary radii.
However, if Saturn mass planets have larger magnetic fields than Neptune mass planets, and Jupiter mass planets have larger magnetic fields that Saturn mass planets, then possibly your planet, with 7.8 times the mass of Jupiter, might have a much bigger magnetic field than Jupiter which might extend many times as far as Jupiter's magnetic field.
And possibly your moon might have its own strong magnetic field adequate to protect it from cosmic rays and solar wind, and so can safely orbit far beyond the magnetic field of the planet, far enough to have a low enough tidal heating.