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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.
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    $\begingroup$ There is no general theory about the radiation belts of gas giants. $\endgroup$
    – ProfRob
    Sep 19, 2023 at 15:57
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    $\begingroup$ I'd expect that the strength of the radiation field and belt depends a lot on the gas giant's magnetic field which very likely is tied to its rotation speed. Assuming a slow rotator (for whatever reason) might make an assumption of a small or less harmful radiation belt more likely $\endgroup$ Sep 19, 2023 at 16:19
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    $\begingroup$ Cold gas giants don't get much larger in radius than Jupiter, they just get denser. (Hot gas giants are a bit less dense). See astronomy.stackexchange.com/q/15066/16685 $\endgroup$
    – PM 2Ring
    Sep 19, 2023 at 18:29
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    $\begingroup$ There are a lot of particles in Jupiter's magnetosphere because of Io. (Of course, it also grabs a lot of material from the solar wind). However, it seems reasonably likely that a planet more massive than Jupiter will have a similar tidally stressed moon that spews out particles. $\endgroup$
    – PM 2Ring
    Sep 19, 2023 at 18:36

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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.

"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/

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/

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    $\begingroup$ I am sorry, but this ultimately doesn't answer my question as it is mostly saying "a range does exist", without supplying a said range, or giving me a way for me to find it myself. Can you please try to provide how I would find it? I have the mass and radius of both bodies. Those graphs are also very poorly labled, but that is more of a criticism of the people maing the paper. Edit: Wait, I missed the 10x the host radius. $\endgroup$ Sep 19, 2023 at 17:17
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    $\begingroup$ But also everything I find says even Europa is so close to Jupiter that it would recieve a very strong dose of radiation. To the extent that it literally glows. $\endgroup$ Sep 19, 2023 at 17:23
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    $\begingroup$ @DanceroftheStars I know, I saw that too. I think it's just because Europa is already below the 10-radius limit, and it's not massive enough to host surface life (which is normally what's being considered). I imagine the 10r limit is meant to account for radiation belts as well. $\endgroup$
    – Kazon
    Sep 19, 2023 at 17:40
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    $\begingroup$ I see, thank you for your help. It is much appreciated. $\endgroup$ Sep 19, 2023 at 17:43
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    $\begingroup$ @DanceroftheStars As for your first comment (and I am grateful for the courtesy there!), I agree that it's not easy to understand if you don't know what you're looking at. The graph is logarithmic in both axes: basically, the bars between 1 and 10 are 2, 3, 4, etc., but after 10, it's 20, 30, 40 (or, with Y, 0.001, 0.002, etc., then 0.01...). "Rp" is the distance from the host in parent radii (Radius planet). The Y axis is the flux that the moon will experience, which can be understood as how much heat it will get. $\endgroup$
    – Kazon
    Sep 19, 2023 at 17:44

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