Once the technology is ready to reach and set up a base in Mars (perhaps 10 years if SpaceX expectations come true) could it be possible to build a route among the asteroids to reach Jupiter's moons?

The problem of the time of the travel would be more or less the same if you would have bases in the following places, and if you start the travel when the planets/asteroids are closer,

• Earth - Mars 0,52 AU


• Mars - Vesta 0,63 AU


• Vesta - Sylvia 0,64 AU


• Sylvia - 588 Achilles 0,67 AU


• 588 Achiles - Jupiter 0,57 AU

But how about the problems of setting permanent bases in those asteroids, even when some are large ones of almost of the size of a dwarf planet, would it be feasible? Setting up a base in Mars it has to be easier. I suppose they plan to obtain water from Mars, may be they would find a way to obtain Oxigen from the CO2 in the atmosphere, also I suppose Mars is warmer than those asteroids and have more protection (even though a tiny not enough protection) from radiation than atmosphereless asteroids, and how about energy supply, if the way they plan to obtain energy in Mars is somehow related with the sun (solar energy), that has to be harder when you are far away.

Could a route among the asteroids be built to reach Jupiter's moons once the technology to reach and set up a base in Mars is ready?

Think Delta-V, not Distance

It makes absolutely no sense to stop at a base on the way. What matters is the fuel/reaction mass to get into a transfer orbit, and to leave it again. The time spent coasting in between is relatively cheap by comparison. Sure, you might be able to tank a couple tons of oxygen for your life support, but what if you need to expend hundreds of tons of fuel to do that? Nothing gained!

According to SpaceX, the answer is yes. The SpaceX program is banking on the idea that people can derive rocket fuel from the elements on Mars almost as easily as we can engineer it here on Earth, and water/food/oxygen could be readily processed in bulk, even if it's not as easy when you don't have a livable atmosphere.

If this proves to be true, then Mars would become a much better base of operation for solar exploration and habitation than Earth because the lower gravity would allow us to launch large payloads into space VS Earth which is so big that it is near the limit of how large a planet can be an still reach space with chemical fuel.

You are clearly aware of the fact that a route would not be constant - everything orbits with different periods, and although the asteroids have orbits that are 'close' to one another, they still cover a huge range. Perturbations make life even more complicated. Jupiter and Mars have wildly different orbital periods from one another, of course.

The presents a difficulty with setting up staging posts in the asteroids. Even if you always head from Mars to Jupiter around conjunction (the optimal launch window does not precisely coincide with conjunction), the asteroids that will be "on the way" will vary. So you would need a lot of such posts scattered around the belt, and you'd have to be careful in selecting larger bodies such that they wouldn't sometimes cluster all together.

It would also be expensive to build and equip these points, and keep them supplied. Even if they were unmanned, the whole point of them would be to hold supplies for such journeys. What is there to be gained from such expense? Is there any advantage to shipping supplies out there to be picked up by people en route? If you shipped them in unmanned shipments, they could be shipped more cheaply, but then it would be a challenge to get them into a depot at the staging post.

However, even if supply caches are desired, they don't have to be in the asteroids themselves. We don't need to build a route through the asteroid belt - it's nowhere near as hard to navigate as sci-fi shows like to depict. The distance between them averages as 2.5 times the distance between Earth and the Moon. If we need supply caches put part way along the journey by unmanned flights, we may as well just have containers that fly themselves out there and then do a burn to put themselves in a stable solar orbit, or grapple on to an asteroid if you prefer, to be picked up later. Permanent or semi-permanent installations would simply be unnecessary.

You could, but not a permanent set of bases. Instead, you would need to have some form of mobile, anchorable bases that hopped from asteroid to asteroid as the asteroid they were anchored to fell farther and farther behind the optimal location for a Mars-Asteroid X orbit.

Each of the intermediate stages would need to move on independent schedules, since the outermost asteroids take longer than the innermost ones. The orbital period of asteroids in the asteroid belt runs from three to six years, give or take a bit. Mars has an orbital period of about 1.88 years, so Mars would orbit the Sun 3+ times before the outermost asteroids orbited the Sun once.

Unless the bases moved on the basis of an ion drive (spewing out asteroid atoms as propellant), solar sails, or some equally low energy, low thrust option, it is unlikely that anything even semi-permanent could be set up to make the flight in hops. It could be possible for limited, exploratory missions to the moons of Jupiter, but except for being the first man to walk on 10 or 20 moons, why?

Well, I can actually think of one possible reason why. The closer you are to the planet, the faster you can control a Jupiter probe. With delays of a second or two (assuming you can safely get that close), you could actually hope to control and direct a flying probe (with the assistance of a lot of AI flight tech) and direct it to interesting features. Except for this possibility (and the possibility of moon probes directed similarly), sending humans out there without a specific target that actually needs human adaptability / flexibility seems a bit foolhardy.