Space Probe Rescue

It is the year 2020 and the space probe Galileo has been on its way to the Oort cloud for exactly a year. For the last six months, scientists at UNSA (the United Nations Space Authority) have been trying, without success, to free the stuck transmission antenna on Galileo. In a brilliant PR move, UNSA has announced that a manned rescue mission will be mounted to repair the antenna. You have been called on to do some preliminary feasibility calculations. UNSA has provided you with the following data.

1. The space probe Galileo has been traveling in a straight line away from the earth at a velocity of 20,000 km/hr ever since it was launched.

2. Due to budgetary constraints and development problems, the only spacecraft that is capable of such a mission is the Riemann. This fusion-powered craft has nearly unlimited range but its engines can only deliver a thrust of 0.1 g. Furthermore, the engine cannot be throttled; it is either on or off. There are, of course, small steering engines that can be used to turn the Riemann around so that its engine can be used for braking.

3. UNSA estimates that in a year the spacecraft Euler will be ready. This craft is essentially similar to the Riemann, but its engines will be able to deliver a thrust of 0.2 g.

The UNSA would like you to estimate the time it will take for Riemann to catch up to and match velocities with Galileo. To simplify your calculations, you may treat the problem as motion in one dimension and assume that Riemann will be launched exactly one year after Galileo. Your results should include a time when Riemann should start braking in order to match velocities with Galileo just as it catches up. To minimize the duration of the mission, UNSA requests that Riemann's engine should be on until you catch up to the probe.

UNSA would also like you repeat your calculation for the Euler, assuming it can be launched exactly two years after Galileo was launched. This would give the scientists more time to try to fix Galileo from earth.

Your report to UNSA should at least address the following points.

1. Why is a braking period required? That is, what mathematical theorem says that Riemann will have to be going faster than Galileo for some portion of the mission?

2. Is it possible to obtain an analytical solution for both the time it will take for Riemann to match velocities with Galileo and the time Riemann should begin braking? UNSA is very interested in this kind of result, since it will be useful if plans have to be changed.

3. Will the increased thrust of Euler make a shorter mission time possible? If not, what thrust would Euler have to be capable of to make a shorter mission possible?