I know Jupiter has powerful radiation belts, but I'm wondering if there are places within the magnetosphere that are relatively calm. I'm asking about Jupiter (a gas giant we know) because I'm curious as to whether a gas giant (in general) could have a magnetosphere that would protect a moon from solar radiation. If a moon of a gas giant is in a calmer area of the planet's magnetic field - not inside a radiation belt - would the planet's magnetosphere protect it from the star's radiation? Or are all parts of the planet's magnetosphere heavily irradiated?
I'm sure that some of the other answers will have data about the shapes and dimensions of radiation belts around Jupiter and the other giant planets in the solar system.
Possibly you should study the structure of the Van Allen Belts, regions of dangerous charged particles orbiting the Earth which manned space vehicles avoid staying in for long. If you get a map of the spacing of the dangerous radiation belts around Earth, you can suppose that Jupiter might have similar radiation belts with safer regions between and surrounding the belts.
Of course the structure of Jupiter's magnetosphere would be different in some ways from Earth's magnetosphere. But it would probably be very unusual for a gas giant planet to have radiation belts which extend down to the surface and out to the orbits of the outermost moons, and also completely cover the entire surface of the planet, instead of being concentrated at the equatorial plane of the planet's magnetic field.
Here is a link to the Wikipedia article on Jupiter's magnetosphere, which apparently has a complex structure.
Thus some regions of the magnetosphere should have higher densities of charged particles than other regions, and thus some regions should be safer for the hypothetical atmospheres and lifeforms of moons orbiting within the magnetosphere.
Unlike the other moons of Jupiter - and the moons of other planets in the solar system - Ganymede as a magnetosphere of its own, so the interactions between that of Ganymede and that of Jupiter are complex.
The interaction of the Jovian magnetosphere with Ganymede, which has an intrinsic magnetic moment, differs from its interaction with the non-magnetized moons. Ganymede's internal magnetic field carves a cavity inside Jupiter's magnetosphere with a diameter of approximately two Ganymede diameters, creating a mini-magnetosphere within Jupiter's magnetosphere. Ganymede's magnetic field diverts the co-rotating plasma flow around its magnetosphere. It also protects the moon's equatorial regions, where the field lines are closed, from energetic particles. The latter can still freely strike Ganymede's poles, where the field lines are open. Some of the energetic particles are trapped near the equator of Ganymede, creating mini-radiation belts. Energetic electrons entering its thin atmosphere are responsible for the observed Ganymedian polar aurorae.
If you ask about the possibility of moons of Jupiter being shielded from trapped charged particles in the magnetosphere of Jupiter, you are probably considering the possibility of life on the surfaces of those moons, or hypothetical alien exomoons orbiting giant planets in other solar systems.
There is no need to worry about charged particles in the radiation belts of planets when discussing hypothetical lifeforms in the subsurface oceans of large ice covered moons of those planets. Since the layers of ice covering the internal oceans would be many kilometers or miles thick, the hypothetical lifeforms in those internal oceans should be protected from charged particles striking the surfaces of those moons.
It is only lifeforms on the surfaces of moons and planets and other types of worlds which would need protection from charged particles from stellar winds or trapped in the radiation belts of companion worlds.
All surface life with Earth like biochemistry would need liquid water on the surface, which would require an atmosphere, of whatever gases, dense enough to keep the water liquid instead of all evaporating into water vapor.
The ability of a world to retain a specific gas in its atmosphere for geological eras of time, long enough for life to evolve, depends first of all on the world's escape velocity (not the same as its surface gravity). It also depends on the ratio of the escape velocity divided by the root-mean-square velocity of atoms and molecules of that gas in the exosphere of the atmosphere.
Page 34 of Habitable Planets for Man, Stephen H. Dole, 1964, has a formula for calculating the amount of time it would take for the gas in an atmosphere to drop from its original amount to only 0.368 of the original amount. It says that there are more complicated formulas available for more specific situations but I think most of us will find that one complicated enough.
According to the table 5 on page 35 a comparatively small change in the ratio between the escape velocity and the root-men-square velocity of particles of a substance in the exosphere can change the retention time of that gas from zero to millions or billions of years.
I think that is the primary factor determining how long a world could possibly retain an atmosphere. Other factors, like stellar wind, could make the world lose atmosphere faster, but nothing can make the world lose atmosphere slower.
Of course a world might be able to replace atmospheric gases lost into space as fast or faster than they are lost, but there should obviously be limits to how fast any likely process would produce more gas.
A world's escape velocity depends on its mass and radius. On page 54 Dole decided that under some conditions a planet with a mass of 0.195 Earth mass might be able to retain 0.368 of its original oxygen for 100 million years. Many or most of the probable common atmospheric gases would probably have atmospheric retention periods of roughly the same order of magnitude. So a minimum mass of about 0.195 Earth mass would be fairly close to minimum mass of a world with both atmosphere and surface temperature similar enough to Earth's to have liquid water on the surface, and thus life in general.
According to Dole that would correspond to a world with 0.63 Earth's radius and a surface gravity of 0.49 g.
There have been other calculations of the minimum mass of world capable of retaining a significant atmosphere. Rene Heller and Roy Barnes in "Exomoon habitability constrained by illumination and tidal heating", 2013, pages 3 to 4discuss the mas range of hypothetical habitable exomoons.
A minimum mass of an exomoon is required to drive a magnetic shield on a billion-year timescale (Ms ≳ 0.1M⊕, Tachinami et al. 2011); to sustain a substantial, long-lived atmosphere (Ms ≳ 0.12M⊕, Williams et al. 1997; Kaltenegger 2000); and to drive tectonic activity (Ms ≳ 0.23M⊕, Williams et al. 1997), which is necessary to maintain plate tectonics and to support the carbon-silicate cycle. Weak internal dynamos have been detected in Mercury and Ganymede (Kivelson et al. 1996; Gurnett et al. 1996), suggesting that satellite masses > 0.25M⊕ will be adequate for considerations of exomoon habitability. This lower limit, however, is not a fixed number. Further sources of energy – such as radiogenic and tidal heating, and the effect of a moon’s composition and structure – can alter our limit in either direction. An upper mass limit is given by the fact that increasing mass leads to high pressures in the moon’s interior, which will increase the mantle viscosity and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed around 2M⊕ (Gaidos et al. 2010; Noack & Breuer 2011; Stamenković et al. 2011). Summing up these conditions, we expect approximately Earth-mass moons to be habitable, and these objects could be detectable with the newly started Hunt for Exomoons with Kepler (HEK) project (Kipping et al. 2012).
A lower limit of 0.12 Earth mass to maintain a significant atmosphere is a lot lower than Dole's, but possibly their source considered an atmosphere composed of heavier gases than oxygen to retain water on the surface. Heller and Barnes combine several factors to say that worlds with more than 0.25 Earth mass can potentially be habitable depending on various factors, but say that is not a hard limit.
So a world with at least 0.12 or 0.195 Earth Mass should be able to retain a dense enough atmosphere for liquid water to exists on the surface and thus for life to possible survive there. Such an atmospheric density would also probably at least partially protect the surface of the planet from charged particles in cosmic rays, stellar wind, and the radiation belts of any large companion world.
And of course any magnetosphere which the world itself generates might also protect the world from charged particles in the radiation belts of a companion world. Barnes and Heller cite a source claiming that a mass of only 0.1 Earth mass may be sufficient to generate a magnetic field.
Of course at the present time the forces which create a world's magnetic field and give it is specific strength are not known very well. But there seems to be a general impression that the more massive a world is, the stronger is magnetic field will usually be, and also that the more rapidly the world (or part of it) rotates, the stronger the magnetic field will be.
So the more rapidly a moon of a giant planet rotates, the more likely it will be to generate a strong magnetic field to protect it from charged particles coming from its star - and also charged particles in the radiation belt of its planet, which will be important if the moon orbits in one of those radiation belts.
Most large moons of giant planets would probably orbit close enough to their planets to become tidally locked to the planets, so that the rotation rates of the moons would be equal to their orbital periods around their planets. Since a rapid rotation rate would be desirable for generating a magnetic field, it would be desirable for a moon to orbit as close to its planet as possible to have as rapid a rotation as possible.
I note that Dole, on pages 58 to 61, discusses the possible rotation rates for planets habitable for humans. Worlds with day night cycles too long might get too hot in the day and too cold at night, and plants might die without light during long nights. Dole guessed that 96 hours or 4 Earth days would be about the limit for the rotation period of a world habitable for humans - and possibly for other Earth lifeforms.
Heller and Barnes discuss the length of day of giant exomoons of giant exoplanets on page 3.
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.
It is interesting that it is theoretically possible for exomoons to rotate faster than their orbital periods.
I note that Iapetus is the outermost moon of Saturn which is tidally locked to Saturn. All the moons inside the orbit of Iapetus have stronger tidal forces from Saturn and so should be tidally locked. And in fact Titan and all the moons closer than Titan are tidally locked to Saturn.
But Hyperion, which orbits between the orbits of Titan and Iapetus, is not tidally locked to Saturn.
The Voyager 2 images and subsequent ground-based photometry indicated that Hyperion's rotation is chaotic, that is, its axis of rotation wobbles so much that its orientation in space is unpredictable. Its Lyapunov time is around 30 days. Hyperion, together with Pluto's moons Nix and Hydra, is among only a few moons in the Solar System known to rotate chaotically, although it is expected to be common in binary asteroids. It is also the only regular planetary natural satellite in the Solar System known not to be tidally locked.
Hyperion is unique among the large moons in that it is very irregularly shaped, has a fairly eccentric orbit, and is near a much larger moon, Titan. These factors combine to restrict the set of conditions under which a stable rotation is possible. The 3:4 orbital resonance between Titan and Hyperion may also make a chaotic rotation more likely. The fact that its rotation is not locked probably accounts for the relative uniformity of Hyperion's surface, in contrast to many of Saturn's other moons, which have contrasting trailing and leading hemispheres.
The article by Heller and Barnes mentions tidal heating between an exomoon and the planet it orbits in its title. If an exomoon receives too little heat from its star to make surface water liquid, the extra heat from tidal heating might warm up the exomoon enough for water to be liquid. And on the other hand, if an exomoon gets too much heat from tidal heating its surface and atmosphere may become hot enough for a runaway greenhouse effect, causing the world to lose all its surface water and become barren.
In the article Heller and Barnes coined the term "habitable edge" for the minimum distance between an exoplanet and its exomoon for the exomoon to avoid a runaway greenhouse effect. "Magnetic shielding of exomoons beyond the circumplanetary habitable edge" by Rene Heller and Joge Zuluaga, 2018, discusses the problem of potentially habitable exomoons that don't have their own magnetospheres. If such exomoons orbit within the magnetospheres of their planets they can suffer bad effects from the planetary radiation belts under some circumstances but also be protected by the planetary magnetospheres from cosmic radiation and stellar winds from their star.
Since smaller astronomical bodies are more common than larger ones, most of the exomoons large enough to potentially habitable would be near the smaller side of the mass range, and thus less likely to produce strong magnetospheres of their own to protect their atmospheres from being destroyed by charged particles and life $o teri$ surfaces being killed by galactic cosmic rays.
Thus smaller mass potentially habitable exomoons would have to orbit within the magnetospheres of their planets if they can't generate their own magnetospheres.
Their abstract says:
....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 large enough moons between about 5 and 20 planetary radii from a giant planet could possibly be habitable.
For Jupiter itself, with a polar radius of 66,854 kilometers (41,541 miles)] and an equatorial radius of 71,492 kilometers (44,423 miles), the habitable zone for its moons would extend from about 334,270 to 357,460 kilometers (207,705.7 to 222,115.3 miles) on the inner edge to about 1,337,080 to 1,429,840 kilometers (830,822.9 to 888,461.3 miles) on the outer edge.
The World building Stack Exchange has a number of questions from writers asking about the possibilities of habitable moons in other solar systems.
Here is a link to questions and answers about habitable moons of planets at the Worldbuilding Stack Exchange: