Problem 9. Satellites can exist in orbits as shown in the diagram. The satellite of mass m is at what is referred to as the L1 point or the first Lagrange point. At this extremely specific location,the satellite will orbit the Sun with the exact same period as the Earth. Given that the satellite is closer to the Sun than the Earth is, you would normally expect the satellite to have a shorter period. However, the satellite is influenced by gravitational forces from the Sun and the Earth.The combination of those forces leads to the same orbital period as the Earth when the satellite is at just the right distance from the Sun. (a) Write out Newton's 2nd Law for the satellite. Your expression must be written in terms of G, M sun, M Earth, R, d and T where T is the Earth's orbital period. This expression could be solved for the distance d. Do not attempt to do that since you would obtain a fifth order polynomial for d. (b) Is this satellite in a stable or unstable orbit? In other words, if it is slightly perturbed (for example, imagine that it is pushed slightly closer to the Sun), would it continue to orbit with the same period as the Earth? Explain.
Note that the Solar & Heliospheric Observatory (SOHO) is at the L1 point.
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