Mars is unpleasantly cold for human habitation. It would be easier to terraform Mars if it were at the earth's distance from the sun, either opposite the earth or perhaps in a Trojan orbit relative to that of the earth.
Moving Mars with rockets would take unpleasantly long - hundreds of millions of years according to some preliminary calculations if it were feasible at all. Here's how to do it in a some tens of thousands of years.
The orbit of an asteroid that crosses the orbits of Jupiter, Mars, earth and Venus would be chaotic, i.e. small deviations in initial course have large effects. Chaotic systems are inexpensively controllable if one is in a position to provide external forces.
Our tame asteroid passes by Jupiter, Mars and Venus repeatedly. It transfers energy and angular momentum between Mars and these planets. We need both Venus and Jupiter, because both energy and angular momentum must be conserved. The orbit is supposed to be such that only very, very small delta-v s must be infrequently be applied to the asteroid to keep it in the desired orbit. If you like analogies with fundamental physics, you can regard the asteroid as the carrier of an exchange force among Mars, Jupiter and Venus. I'd be surprised if the analogy turned out to be more than metaphorical.
How small the forces have to be depends on how long in advance you apply them.
Here's how to get a lower bound on the time required. Ceres, the largest asteroid, has a mass about 1/1,000 that of Mars. The delta-v of the asteroid at an encounter with Mars is at most twice the escape velocity from Mars which is about 5 km/sec. Therefore, Mars itself can experience a delta-v of at most 0.01 km/sec. If we want a total 10 km/sec, we need at least 1,000 encounters. The time for an orbit is at least about the period of Jupiter, i.e about 12 years. This give 12,000 years, which must be multiplied by an inefficiency factor, maybe 5, maybe 50, maybe 500.
Here are some questions. (1) Does a suitable orbit for the asteroid exist that can be maintained with small forces? (2) If so, what is the inefficiency factor? How do we tame Ceres in the first place? Preliminary calculations suggest that it would be expensive unless we can make Jupiter do most of the work.
In connection with the idea of moving the earth a billion years from now to get farther from an increasingly hot sun, Korycanski, Laughlin and Adams propose using a Kuiper belt asteroid to carry energy and angular momentum. This is a better idea than using an asteroid belt asteroid, because the Kuiper belt asteroids, being much farther away from the sun, are much easier to get started on a suitable trajectory. See the detailed paper mentioned below for a full reference.
It looks like it would be easier to use a Kuiper belt asteroid rather than a main belt asteroid, because a very small delta-v out there produces a large change in course when the asteroid comes in. Also there are asteroids bigger than Ceres in the Kuiper belt. The price is that instead of the 12 year Jupiter orbit we are dealing with hundreds of years.
I've done some of the mathematics, specifically that having to do with conservation of energy and angular momentum.
There are html, dvi, postscript, and pdf versions of a more detailed paper.
Q.Isn't a project taking tens of thousands of years ridiculous on the face of
A. No. Present humanity, only a few tens of years into space technology, won't initiate such a project. However, conditions on earth can be kept suitable for humanity for a few billion years. When humanity is 100,000 years into the space age, moving Mars won't look ridiculous just from the time it would take.
If speculation on a large scale offends you, too bad for you.
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