If Earth was going to be completely destroyed by some natural event and we could not send a sample of humanity to Mars (perhaps due to the same event), I think we should establish a colony on Titan.

• Titan has most of the raw materials that make life possible: nitrogen, hydrogen, carbon, oxygen, etc.
• Titan’s atmosphere is about 60% denser than Earth’s so if you can provide oxygen and heat, you can walk around on the surface without complex pressure suits.
• Parachutes, balloons, and airships can all be used effectively in Titan’s dense atmosphere and low gravity. Air transport would be easier than on Earth
• Titan has large lakes of liquid hydrocarbons allowing use of ships and boats
• Saturn has a magnetic field, but it is not lethally strong like Jupiter’s so Titan should have decent radiation protection
• Gravity is low at about 1/7 the of Earth’s, but it is still higher than most possible bodies
• The main thing lacking on Titan is heat. Typical atmospheric temperature is between -170 C and -180 C, but heat should be easy to provide with nuclear reactors
• Saturn’s atmosphere is rich in Helium-3, so if mankind developed good fusion reactors there is a endless supply of fuel nearby
• Other small Saturnian moons and the rings are nearby for mining whatever materials are not easily available on Titan. Enceladus has loads of water and smaller moons could be mined for the heavy elements at their cores
• The view should be amazing

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I once had a wilderness survival class in high school. The teacher gave us this problem, before breaking us off into groups:

You are in a plane that crashes in the desert. What do you do?

Every group assumed they would need to gather their belongings, and begin marching. Only one group decided to stay put and wait for help.

I thought of this as I was reading some of the other answers.

They all seemed to assume we’d be better off settling on another planet or moon.

But in reality, we’d probably be better off in orbit. Or at least, somewhere the gravity is so low that transporting materials would be easy.

With today’s technology and space infrastructure, we’d first have to lay out intensive mining, manufacturing, and transport on our own moon:

The nice thing about the moon is the low gravity and lack of atmosphere would allow us to build mass drivers that could conveniently shuttle materials deeper into the solar system.

The best target for more mineral exploration would be the asteroid belt:

Because once again, we could access raw materials in a near zero-g environment.

Further infrastructure and staging points would then be built on large asteroids like Vesta and Ceres:

Because fighting gravity is the big obstacle to getting anything off Earth:

The best solution remains getting as much as you can while you’re up in space.

To put the problem in perspective, it took a Saturn V rocket 190 gigawatts of power to lift a 70 ton payload into space:

That is an insane amount of power. That is roughly the power of 200 nuclear power plants.

It was so much power, that for 2.5 minutes it was actually more power than the entire US electrical grid produced.

That’s how hard it was to get the lander off the Earth.

But it was orders of magnitude easier to get the Apollo module off the Moon:

So now that we have access to the raw materials in a zero-g environment, we can build things in zero-g. Big things.

But does it make much sense to drop the things you make in space, onto a planet’s surface?

Why not just build your habitats in space?

I give you the O’Neill Cylinder:

Named for a famous thought experiment in the 1970s, the ONeill Cylinder was a huge metal cylinder that simulated gravity just by rotating slowly:

The inside of the cylinder.

It was supposed to be roughly 10km across and 32km long, and would have the surface area of a large city, and could house as many as 10,000 people.

Because the O Neill cylinder is constructed in a zero g environment, the energy requirements are that much lower than anything built in a gravity well.

So ultimately, we may not ever have much need to settle on any other planets in the solar system. Artificial colonies that remain in orbit might be the better option.