Answer Happy

Parts c and d
Problem 4 (20 points) As a prelude to our more general study of orbital motion, let's have a look at the Lagrangian for an object in a gravitational field. A satellite of mass m is in the vicinity of a planet of mass M. We'll study the motion in the plane containing the planet and the path of the satellite, so we can treat the problem as two-dimensional. Part A Using polar coordinates (r, o), write the Lagrangian for the motion of the satellite under the in- fluence of the planet's gravity. (Make sure you use the more general 1/r form of the gravitational potential, don't assume it's constant!) Part B Write the Euler-Lagrange equations for r and . Your Lagrangian from part A should satisfy ac/a0 = 0; identify the corresponding conserved quantity, and use the fact that this quantity is a constant of the motion to eliminate 8 and obtain a single equation of motion for i. Show that there is one equilibrium value of r, corresponding to a circular orbit. Part C A geostationary orbit is a particular circular orbit in which a satellite remains fixed in position above a spot above the planet's surface, i.e. the motion of the satellite satisfies the conditions j = 0 and 0 = 2, (1) where is the angular speed of the planet's rotation. Find the fixed radius R at which a geosta- tionary orbit is possible; it should depend only on the gravitation of the planet, i.e. on G, M, and 12. Now suppose the satellite is perturbed from its geostationary orbit by a small amount, i.e. take T= R +€. (2) Using your result from part B, Taylor expand the acceleration to first order in e, and show that the geostationary orbit is a point of stable equilibrium. What is the frequency w of small oscillations in (Hint: here is a helpful calculus identity: (1 + 2)" m = 1- nx + (...) for small, where (...) indicates terms of order 3and higher.) Part D The presence of a small oscillation in r will induce fluctuations in the previously constant angular speed as well. Again working to first order in E, solve for the modified 0(t). The orbital period T is the time required for to change by 2n; show that even with the added e term, the orbital period is unchanged to this order.