On the Radiation Field of large Gas Giants

Question

I am woring on creating a fictional star system, and I need to find an answer to a question to find an accurate way to depict this.

I am aware that Gas Giants create a dangerous field of radiation around themselves that would be harmful to anyone within a certain radius. I now my moon has to be within the Hill Sphere, which sets a solid outer boundary. But my question is, how would I find how far away my moon should be in order to properly maintain an atmosphere and be habitable, not being effected by the radiation to a harmful degree.

I don't need an exact number given, I moreso just need help finding out how I would calcualte the required distance to be able to remain safe. If any sort of formula or way to calculate it exists at all. But it would also be helpful and appreciated if you would do more help to find the numbers as well.

My parametres currently are around 5.9441197891 Jupiter Masses and 1.81 Jupiter Radii -1. (Please tell me if this isn't reasonable). How would I be able to find how far a moon would have to be in order to be habitable, have a atmosphere, and not be basked in deadly radiation.

Your help is much appreciated.

1. I found these numbers by just choosing a radius of 1.81, and finding what mass it would have to be to have the same density as Jupiter. And found it was ~5.9441197891. This is likely wrong as density of larger gas giants likely changes. Although it could also be reasonable as, looking at lists, density seems to have a very large range. It also likely would be different as it is closer to the star than Class I gas giants.

Answer 1

I've been interested in exomoons for a while as well, and having read multiple studies (I admit I only fully read about two), I don't recall seeing any mention of Van-Allen belts being an issue. The most commonly mentioned threat to life was a runaway greenhouse due to extra heating from the host, whether from tidal forces, planetshine (the heat from the star reflecting from its surface), or both. It seems generally agreed that:

1. A moon must orbit about a distance of 10 times the host's radius in order to be safe from additional tidal heating;
2. The moon's orbit must not be too eccentric, or it will slowly spiral into its host due to tidal braking.
3. The habitable zone around the actual star will be slightly larger. This moon-habitable zone (which I like to arbitrarily name "MHZ") varies depending on the study, but one from July 2022 ("A target list for searching for habitable exomoons") refers to "Pierrehumbert 2010, Chapter 4" for the inner limit, and "Kopparapu et al. 2014" for the outer limit. I should add that it doesn't seem like the greenhouse potential of hydrogen (and thus a larger habitable zone) is considered here.

An exomoon is considered habitable, if the global heating flux of the moon is between the runaway greenhouse (𝐹RG) and maximum greenhouse fluxes (𝐹MG). The runaway greenhouse flux was calculated with a method that is dependent on the surface gravity of the exomoon (Pierrehumbert 2010, Chapter 4). The advantage of this method is that the runaway greenhouse flux scales with the radius and the mass of the body. This is in contrast to the standard calculation method described by Kopparapu et al. (2014), in which the calculations only apply to certain masses and radii. It was shown, however, that the outer boundary of the circumstellar habitabile zone has a weak dependence on the mass of the exomoon (Kopparapu et al. 2014), and for this reason, the maximum greenhouse limit described by Kopparapu et al. (2014) was used as a lower limit for habitability.

EDIT: looks like the description I gave for the image above hasn't been included. This is from a DIFFERENT study.

"Contours of summed absorbed stellar irradiation and tidal heating (in logarithmic units of W/m2) as a function of semimajor axis aps and eccentricity eps on an Earth-like (upper row) and a Super-Ganymede (lower row) exomoon. In the left panels, the satellite orbits a Jupiter-like planet, in the right panels a Neptune-mass planet, in both cases at 1 AU from a Sun-like host star. In the white area at the right, tidal heating is negligible and absorbed stellar flux is 239 W/m2. The right-most contours in each panel indicate Io's tidal heat flux of 2 W/m2, a tidal heating of 10 W/m2, and the critical flux for the runaway greenhouse (295 W/m2 for the Earth-like moon and 243 W/m2 for the Super-Ganymede)." - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3549631/