Averting Doom Part 2

File: UFO275

Taming Ceres

I confess I haven't yet computed how large a delta-v must be imparted to Ceres to tame it. Making its orbit eccentric enough to cross that of Mars would suffice, because it could be arranged to encounter Mars in an appropriate way to tame it. Besides doing this with a single delta-v, it would probably be cheaper to put Ceres into a suitable resonance with Jupiter, so that Jupiter would do the work of making the orbit of Ceres eccentric enough to tame it. I don't know how to do this kind of computation.

The most straightforward way to impart a delta-v to Ceres is to install a large number of nuclear reactors on it, and use the energy to expel fragments of Ceres in a desired direction and at an appropriate velocity.If maximal energy efficiency is wanted, i.e. to impart the maximum momentum per unit of energy used, then the appropriate velocity is a simple optimization problem, and it turns out to be sqrt(2) vec, where vec is the escape velocity from Ceres. The actual velocity with which the matter leaves the Ceres gravitational field is then vec.

The number of reactors needed is large. A million reactors each of 1000MWE, would give Ceres a delta-v of 1 km/sec in 1,000 years - if I did the arithmetic correctly. If our descendants installed one reactor every thousand years, they would install a million in a billion years, and that is time enough to avert the prophesied doom. A non-trivial fraction of the mass of Ceres would be consumed if energy is optimized and if 1 km/sec is needed. Our descendants might decide to use a higher exhaust velocity. More likely, it will turn out that a considerably smaller delta-v will do with the aid of Jupiter. Our descendants might decide to use a smaller asteroid.

The Broken Kepler Approximation

The full computations required to determine all these trajectories will be quite large (though within the capacity of present computers). Moreover, the required information about the gravitational fields won't be available for a long time. Here is an approximation that can give a qualitative picture and an estimate of the time required to move earth out a given distance or to move Mars in (a shorter time project).

Make the following assumptions:

1. The planets and the asteroid are point masses.

2. The system is planar.

3. The planets have masses small compared to the that of the sun.

4. The asteroid has a mass small compared to that of any planet involved.

5. The encounters between the asteroid and a planet are elastic collisions of point masses. However, we assume that the possible angles of the collisions are limited by the radii of the planets.

6. Any individual collision has a small effect on the trajectory of the planet.

7. The asteroid departs from each collision with a velocity that ensures a subsequent collision with the same or a different planet.

Making these assumptions leads to the following conclusions.

1. The trajectory of the asteroid is a sequence of segments of Keplerian ellipses about the sun. That's why we call the above set of assumptions the broken Kepler approximation.

2. The segments are separated by elastic collisions with the planets conserving energy and momentum.

3. For each collision there is a discrete set of deflections that ensure subsequent collisions. They form a sort of spectrum.

4. Computing the next collision does not require the solution of differential equations. Instead one has transcendental equations to solve analogous to Kepler's equation (the one used to compute the position of a planet as a function of time). However, it looks like the transcendental equation will involve two unknown parameters instead of the one that appears in Kepler's equation.

It would be nice to have a program that would compute broken Kepler trajectories and display them for our contemplation.

I hope I have convinced you that our distant descendants can survive the warming up of the sun until it becomes an actual nova.

By the way, it seems to me that if the above idea is sound, it settles the question of the stability of the solar system - in the negative. Very likely an asteroid could be tamed over a sufficiently long time with as small an expenditure of delta-v as might be desired. Once tamed it could be used with infinitesimal external force to expel a planet from the system. This tells us that the current trajectory of the solar system is arbitrarily close to one in which a planet is expelled. Of course, the probability that a planet actually would be expelled by this mechanism in some particular finite time is extremely low, because maintaining the required sequence of encounters requires an improbable precision in the initial conditions. I suppose a lower bound on the probability could be computed and from it an expected upper bound on the gravitational lifetime of the solar system could be obtained.

Criticism and comments are welcome. For a certain reason, I even welcome comments, however uninformed, to the effect that the whole idea is preposterous. I prefer such comments to be postings rather than email.

Mike Fitzgibbons