Conservation of Energy and Angular Momentum

Since the asteroid is small compared to planets, and our goal is to directly apply very little total for the asteroid, the main effect is exchange of energy and angular momentum among the three planets.

Before trying to design orbits for the tame asteroid, we compute the changes in the orbits of Jupiter and Venus required to move Mars to one AU from the sun. We assume that energy and angular momentum are conserved, i.e. that the asteroid itself overall contributes nothing, because of its small size.

Here are some equations in which we assume that the planets have circular orbits. We derive relations between the energy , the angular momentum and the distance of a planet from the sun. We use the speed of the planet in its orbit as an intermediate variable.

We start with equations for a single planet in a circular orbit.

Circularity also gives

Having expressed in terms of , we are ready to write the conservation laws for the total energy and the total angular momentum . We use subscripts for the quantities associated with the three planets.

Conservation of energy for the three planets, e.g. Mars, Venus and Jupiter gives for the new configuration

Conservation of angular momentum gives for the new configuration

We assume that planet 1, e.g. Mars, is to be moved to a desired circular orbit, and therefore we know and . We need to solve for , , , and . We use (8) to express the energies in terms of the angular momenta, so (9) then becomes

(10) and (11) must be solved for and , which will then allow determining and by substituting in (7).

These equations have the form

Substituting gives

Solving these equations for moving Mars to 1.0 AU with the aid of Venus and Jupiter yields

new Venus distance = 1.99886e+09 compared to 1.07700e+11

new Jupiter distance = 7.79062e+11 compared to >7.78000e+11

Venus comes out distressingly close to the sun, whose radius is 6.96000e+08. Oh well, nobody we know lives on Venus.

All distances are in meters.

2007-10-06