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A great many questions have been written about a habitable moon around a gas giant. This is my take on it.

Somehow (extra points to whoever figures out exactly how it is possible) Jupiter gets knocked out of orbit and plunges towards the inner Solar System. At first we think it will eat the Earth and as you can imagine, people go haywire and mass-panic is unavoidable. As it turns out, though, Jupiter (miraculously) added Earth and Mars to its collection of moons.

I realize how unlikely this is but bear with me. Society is in ruins because of our panicking. Jupiter (another miracle---I'm starting to think we have someone looking out for us;) settles into an orbit slightly closer to the sun than our current orbit, going in an odd elliptical orbit around the sun.

I have a few questions:

  • How might the radiation from Jupiter effect the population of Earth? Will it render Earth uninhabitable or will we only see an increase in cancer rates?

  • What location around Jupiter would be suitable for Earth to come in without colliding with another moon?

  • Would space flight to our Martian bases be significantly easier or would the amount of gravity cancel out the benefit of closer proximity?

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    $\begingroup$ Does Jupiter really emit radiations, or is it some thing specific to your setting? $\endgroup$
    – Maxime Lucas
    Mar 18, 2015 at 15:20
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    $\begingroup$ @skysurf3000 Jupiter actually does emit radiation. Similar to our very own Van Allen belts generated by Jupiter's magnetosphere. $\endgroup$
    – Jax
    Mar 18, 2015 at 15:25
  • $\begingroup$ If you are unfamiliar with Worlds in Collision, I recommend you become familiar with it, as many of these ideas are reminiscent of it. (Note that Velikovsky puts forth his crazy theory as historical fact rather than science fiction.) $\endgroup$
    – Eric Lippert
    Mar 18, 2015 at 22:52

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Radiation

The bulk of the radiation we have to deal with around Jupiter would be held within its Van Allen belts. If Earth settled into an area between (or beyond) the radiation belts, it would be fairly safe from those radioactive waves. Also, we have our own magnetic field that would provide us with shielding from Jupiter's radiation. With the amount of luck you are calling for here, landing in the right spot wouldn't be too hard. We are currently working on maps of Jupiter's radiation belts in an effort to figure out where to send probes...but as we know from Earth's Van Allen belts, there are gaps between them.

Location

Space is big™, there is Plenty of space around Jupiter for Earth and Mars to find an orbital path where they won't actually impact one of Jupiter's moons. However, getting both planets to orbit Jupiter in a stable fashion would be harder. Simply because Earth and Mars would be gravitationally pulling at each other on a regular basis... So Earth and Mars would have to orbit at very different distances from Jupiter, or they would destabilize each other's orbits. See this other question for math and numbers on how to more specifically position a 'habitable satellite' around a gas giant. But, all said, we're going to need to be a pretty good distance away from Jupiter in order to have a safe, stable orbit.

Travel

Getting out of Earth's gravity well wouldn't change much as far as clearing the atmosphere goes. The biggest difference would be the location of the Earth-Jupiter Lagrange points, particularly the L1 point (the point at which, if you stopped, you'd fall towards Jupiter instead of Earth). Once you hit that point, you could use orbital dynamics to throw yourself at Mars by slingshotting off of Jupiter (make sure you have good radiation shielding, since you'll probably hit a Van Allen belt. Jupiter's can kill an unshielded human in minutes). Since the Earth/Jupiter L1 point would be closer to Earth than to Jupiter, it would require less energy to reach than it would require to otherwise travel to Mars. Somewhat. Since force of gravity is related to the distance between two objects squared...most of your energy is expended at close range to whatever gravity well you are trying to leave, rather than once you are further away.

Also, travel would be much easier one way, and harder the other. Simply because Earth and Mars would have to orbit at different distances from Jupiter in order to be stable. So, to get to the one closer to Jupiter, you'd simply have to 'fall' towards Jupiter, and then stop at the right place. To get to the outer planet, you'd need to burn against Jupiter's gravity well as well. Again, this wouldn't be a huge difference, because of the distances involved...but still noteworthy.

Other Concerns

Some theorize that Jupiter is one of the reasons that our inner solar system is habitable. It acts like a big gravitational linebacker and intercepts long period comets, often flinging them clear out of the solar system rather than letting them come into the inner system where Impact is a risk. With us sitting in orbit around our gravitational linebacker, we are likely to see more things hitting our atmosphere as Jupiter accretes space junk. Not to mention the fun to be had with any asteroids it picked up on its way through the asteroid belt.

Tides on Earth are also going to be interesting. We have another major player in the tidal-force game added in, in addition to our moon and the sun. Additionally, our moon (if we get to keep it) would pick up a more eccentric orbit around us as it is tugged on by Jupiter, and would result in varying strengths of tides as the moon drifted closer and further to us in our orbit. And depending on our distance from Jupiter, and the tidal forces exerted on us, we may also see a rise in volcanic activity.

Weather would also get interesting, as (with Jupiter) our orbit around the sun would become more eccentric, leading to more extreme seasons. I'm not sure how Earth's energy budget would be impacted, with the addition of the energy and forces produced by Jupiter (some of its moons are internally heated to above 0C simply from Tidal Forces) but it would likely change.

It would take a really long time for this to happen, but eventually, days would get weird. The question I linked above includes equations to determine Tidal Locking...which would happen eventually. We would be tidally locked to Jupiter (so on side of the planet would always face it) and so our day would end up the same length as our orbital period around Jupiter. This would take much less time than is projected for Earth to become tidally locked to the sun.

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    $\begingroup$ I'm thinking we'd lose the moon. Jupiter could easily pass between the earth and moon (Jupiter diameter: 139,822 km, distance of moon to earth: 384,400 km), so the moon would either be captured in its own orbit around Jupiter, get flung away into space, or crash into Jupiter. Not nitpicking, it's a great answer. Also, this is a fun picture: i.imgur.com/Ae9hbU1.jpg $\endgroup$
    – AndyD273
    Mar 18, 2015 at 18:39
  • $\begingroup$ We might not, it all depends on where we end up orbiting and how we get there. The inner moons or the resonance moons would steal a moon, but if we were out by callisto or a little farther there's space to have a moon of your own. Not forever, though, the jovian system is really complicated and we would probably get hit by something eventually. $\endgroup$
    – Resonating
    Mar 18, 2015 at 19:38
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    $\begingroup$ @Resonating Very good point. Though at 17 days that would be a rough orbit to be in when tidal locking happened. If we're in Jupiter's shadow for 6-8 old earth days, solar insolation would be way down, so things could get cold... Or maybe that far out the umbra isn't that big. Depending on the Van Allen belts, it seems like a Europa orbit would be better. Actually, I wonder how close would Jupiter have to come in order to get Earth to follow it? $\endgroup$
    – AndyD273
    Mar 18, 2015 at 20:07
  • $\begingroup$ Jupiter is only about 71 thousand km wide and even Io orbits at 400 thousand km. You don't need a lot of inclination to dodge that shadow, and even Io is only shaded for 100 minutes or so at worst The hill sphere for Jupiter is 57 million km, so there's plenty of space between Callisto and freedom for an Earth to sit. $\endgroup$
    – Resonating
    Mar 19, 2015 at 0:56
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I will divide this answer into 3 parts

Radiation

Jupiter's radiation isn't actually a risk to us. Jupiter's radiation is simply a result of Io's charged volcanic plumes forming a radiation belt around it. Our Earth does possess these radiation belts (though they are generated by solar wind and not volcanoes). We call them as Van Allen Radiation Belts.

Jupiter's radiation belts are only dangerous to astronauts on a moon with no atmosphere, i.e. Europa or Ganymede. The lack of an atmosphere makes it susceptible to Chernobyl-level radiation.

For the Earth, this is not so, because we have an atmosphere, which is pretty much opaque to radiation. However the radiation is a minor problem, compared to the most difficult problem we will ever face.

The Jupiter-Mars-Earth-Moon system will be dangerously unstable

Earth has a Moon, and Jupiter has more than 60 moons in its system. Our system would be unique because, Earth would be the only moon, which also has its own moon, i.e. a submoon system.

However, this means that the system would also be really unstable, as our Moon is still more massive than Europa. This would either mean that Moon would drag it into an orbit around the Earth (highly unlikely), or else it would redirect it to crash on Mars. Remember that although most of Jupiter's moons are massive than our Moon, our Moon is still comparatively massive *(for e.g. Ganymede is only 2x as massive as our Moon).

Furthermore, the combined gravitational pull of Jupiter, Earth and Mars, proves to be too much for our Moon, and can trigger volcanoes to erupt on our Moon. This would imply that our Earth's radiation belt would become stronger and be a major problem for future space explorers.

Furthermore, Earth's magnetic field and Jupiter's magnetic field interact producing complex phenomena. Although Jupiter's magnetic field is already big, we must not forget that Earth too also has a large magnetic field, while its magnetosphere might stretch out to only 36,000-60,000 (6-10 Earth Radii), its magnetotail stretches out to 6 million miles, about 6 times the Sun's size. Pictures of Earth's and Jupiter's magnetic field-enter image description here enter image description here

As seen above, both the magnetic fields are already comparatively large. When you get multiple magnetic fields to interact with each other, you can get really complex phenomena, like magnetic eddies, and as Io's volcanic plumes enter Earth's domain they would generate strong auroras which would be intensely radioactive to flight passengers.

Also, as Jupiter, Moon and Earth exert gravitational pull on Mars, this can also result in Mars's core remelting and generating a magnetic field. Also, the Moon itself is now being intense flexed by Jupiter, Earth and Mars, generating a lunar magnetic field. The Earth is not intact, as Jupiter, Mars and Moon also pull on it, perhaps causing a stronger magnetic field.

When Jupiter's magnetic field, Earth's magnetic field, Mars's magnetic field and Moon's magnetic field interact, they no longer behave like a protective shield and instead behave like a giant, oversized particle accelerator, occasionally bursting in energy due to the combined effect of both the solar wind, Io's and Moon's volcanic plumes erupting trillions of tons of matter out into space. This forms an region of intense radiation which is potentially lethal for future space explorers. However I think I may be digressing a bit so let's move to the next point.

Location

Jupiter's hill sphere is about 57 million km across as pointed out by @Resonating in the comments, but that doesn't mean that you would have a comfy space to place a orbit in. Gravity decreases gradually as you come closer to the Hills Sphere, and at the Langrange point, which is always at the edge of the Hills Sphere, the orbit is extremely unstable, like a pencil on its back-end, it can fall given the slightest disturbance. So we need to be much closer to Jupiter to get a (still unstable) proper orbit, same for Mars.

Mars Bases

This question is significantly more difficult to answer, and it can vary. Jupiter's gravity at that distance cannot affect small objects like spacecraft, so I guess that my answer would be yes, though that would be backhanded by the fact that the intense radiation produced would make space travel extremely risky.

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