A scheme with minimal total

Consider an asteroid that starts very far out. The further the better as concerns energy spent on adjusting the orbit of the asteroid, but the further out the longer the whole process will take. The asteroid makes one encounter with a planet per trip into the inner solar system. On successive trips it encounters Mars, Venus and Jupiter but not necessarily in a fixed order. The encounters are on the correct side to get the sign of the of the planet correct. The distance of the encounter from the planet is large enough so that the asteroid will come back out again after the encounter. The successive encounters arranged so that energy and angular momentum are transferred in the correct proportions in two senses. The first sense is that the asteroid should always have its perihelion inside the orbit of Venus and its aphelion way out. The second sense is that energy and angular momentum are exchanged in the correct ratio among the three planets. Because of conservation of energy and angular momentum, these two conditions will agree.

When the asteroid is very far out very little is required to adjust the phase of its next planetary encounter. In the limit of the asteroid going to infinity, zero is required. Because there is only one encounter per trip, only one condition has to be satisfied. The adjustment can take into account the inclination of the orbit of the planet.

2007-10-06