   Next: Conservation of Energy and Up: CHAOS AND MOVING MARS Previous: Chaos and controllability

# A tame asteroid

The idea is to use an asteroid to transfer energy and angular momentum among Mars, Jupiter and Venus in a way that makes Mars move closer to the sun. We would expect Venus and Jupiter to move further away, but not much. We'll see how the mathematics turns out. We hope to do this with only incidental applied to the asteroid by rockets.

The largest asteroid is Ceres, which has a mass about th that of Mars. The most favorable encounter of Ceres with Mars is when Ceres comes as close to Mars as possible and has its velocity reversed in the Mars-centered co-ordinate system. The of Ceres is then approximately the escape velocity from Mars. [We'll give the details later]. The given to Mars is then of this escape velocity. Actually, the encounters will not usually be so favorable, so there will be some efficiency factor less than 1.0. The escape velocity from Mars is about 5,100 meters per second, so a for Mars of 24 km/sec would require at least 4,800 encounters. If we imagine that the orbital period of an asteroid within the orbit of Jupiter is about 10 years, then we are talking about some multiple of 48,000 years. Very likely, our descendants will become interested in starting such a project only after they have had a technological civilization for some tens of thousands of years.

Ceres is the largest asteroid, but it looks like would be very expensive to get Ceres out of its very stable orbit. Therefore, we propose using an asteroid from the Kuiper belt which would be much more easy to manipulate at first. (This idea belongs to [DGK01].) The biggest of these isn't known yet, but we can imagine it to be th the mass of Ceres.

We want a tame asteroid.

Asteroids large enough to have significant gravitational effects are expensive to move much with feasible rockets. Therefore, we want an asteroid orbit that passes by many planets. Passing close to a planet should amplify any perturbation of the orbit. The earlier the perturbation can be made the greater the amplification. Note that you get no amplification out of a full Keplerian ellipse about the sun, because the ellipical orbit amounts to a focussing effect. A small perturbation now will cause a slightly changed ellipse that will repeat itself unless there is a further perturbation.

The one effect that does grow with time in a perturbed elliptical orbit is the time change of arrival at a given point. If the period is changed, then the perturbation will grow linearly with time. This effect can be used to adjust the phase of the next encounter of the asteroid with the orbit of the planet. It looks like by waiting long enough we can achieve the next encounter with an arbitrarily small of the asteroid.

The first mathematical question is whether there can be a tame asteroid. Ideally its orbit would, if unperturbed, graze planets for the indefinite future. Let's call this a tame orbit. The second question is whether an early enough small perturbation could make the asteroid switch to a different tame orbit. If so, the third question is what is the set of tame orbits reachable from an initial orbit. As a first approximation to these questions we regard the asteroid as having infinitesimal mass, so the planets themselves are not perturbed.

The next main question arises when the finite mass of the asteroid is taken into account. What class of planetary configurations can be reached with an arbitrarily small tame asteroid? Can these configurations be reached with an arbitrarily small given enough time?

The specific question is whether an asteroid from the Kuiper belt can be tamed and used to move Mars to an orbit on the opposite side of the sun at the earth's distance from the sun.   Next: Conservation of Energy and Up: CHAOS AND MOVING MARS Previous: Chaos and controllability
John McCarthy
2007-10-06