Lagrange points for space telescope · CAP 2003 #B3
Physics · mechanics
Problem
The next-generation large space telescope is scheduled for launch in 2010. It will be put in orbit around the Sun, in a special zone where its distance relative to the Earth and to the Sun can remain constant. The location of such zones was first calculated in the XVIIIth century by the French-Italian mathematician Joseph-Louis Lagrange.
Even though it relies on approximations, Lagrange's full solution is fairly involved, but you should still be able to make a good qualitative guess at a partial solution. Consider two point masses, M1 and M2, referring respectively to the Sun and the Earth. Both orbit around their common centre of mass at angular velocity ω and with a period of one year. These orbits are circular to a good approximation, and the distance R between M1 and M2 is constant. Since M1≫M2, the motion of M1 is not detectable at the scale of the figure below and can be neglected.
[Figure: Diagram showing M1 (Sun) and M2 (Earth) separated by distance R, with x-axis along the line joining them]
We wish to find where, on the line that joins M1 and M2, an object of mass m can sit so that it also orbits the centre of mass with the same period as M1 and M2. Solve for x and where roughly the Lagrange zones are positioned on the x axis. Provide a qualitative analysis.