I've been asked to write about terraforming Mars. While this was not my area of study in grad school (I was in gravitational physics), I thought I'd write a little about it anyway.
First, it's not necessary to terraform Mars in order to live there. I assume we'll build habitats on the red planet long before we attempt terraforming. One can build domes and underground cities, etc. But let's forget about that and be ambitious.
The major obstacle to living on Mars is the lack of an atmosphere. At one time in its past, Mars did have one, but the planet's magnetic field shut down. Without this protective shield, the solar wind stripped the atmosphere away. If we want to terraform Mars, we have to restore its magnetic field. (BTW, Earth's atmosphere would be in trouble, too, if its magnetic field shut down.)
Building a planetary size magnetic field is probably not possible. It may, however, be possible to put a magnetic shield between Mars and the sun. A comparatively smaller field held closer in could shield the planet from the sun. The idea is roughly the same as when you hold your hand close to a candle and cast a giant shadow on the wall. By moving the magnetic shield closer to the sun, it casts a bigger protective shadow.
https://i.stack.imgur.com/DhxA7.jpg
The next problem is that Mars lacks the gravitational mass and size to hold much of an atmosphere. You would only weigh 38% as much on Mars due to the planet's gravitational pull, and that translates to less ability to prevent oxygen and CO2 from escaping. This means that Mars can't hold the kind of atmosphere Earth has, no matter what you do. It is, however, possible for it to have enough of an atmosphere that liquid water can exist on its surface, and maybe enough to pressure walk around without a space suit for a while. Maybe that's good enough?
In the long run, I think we'll need to adapt people to other planets rather than adapting planets to people (homoforming instead of terraforming). It's probably easier to tinker with our genome than to remake entire worlds.
Maybe we'll have to change both people and planets and meet in the middle?
If we can restore the Martian atmosphere and oceans, maybe we can adapt humans the lower gravity and atmospheric pressure of that strange new environment. We should probably also give them Vulcan-like ears while we're at it. They won't help the Martians survive any better, but they'll look cool.
Ever wonder why Star Trek-like universes have aliens that look like people? Maybe they are people, just adapted to the worlds we find and colonize.
While we're at it, we could also adapt people to interstellar travel. Imagine a human adapted to live in zero-g who could be frozen like an Alaskan wood frog for long journeys. Ideally, we would want such people to lack the ability to feel bored or angry--very inconvenient emotions on long journeys.
Update: @WillFlyForFood suggested that we don't need to put the magnetic shield in space because by which the solar wind strips the atmosphere is slow enough that it doesn't matter much. This seems to be true. The loss of 66% of the once full atmosphere of Mars apparently took about 4 billion years (according to Popular Science https://www.popsci.com/we-finally-know-what-happened-to-most-mars-missing-atmosphere#page-4)
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