Consider the earth–sun system as a gravitational analog to the hydrogen atom.
(a) What is the potential energy function (replacing Equation 4.52)? (Let mE be the mass of the earth, and M the mass of the sun.)
(b) What is the "Bohr radius," ag, for this system? Work out the actual number.
(c) Write down the gravitational "Bohr formula," and, by equating En to the classical energy of a planet in a circular orbit of radius ro, show that n=ro/ag. From this, estimate the quantum number n of the earth.
(d) Suppose the earth made a transition to the next lower level (n−1). How much energy (in Joules) would be released? What would the wavelength of the emitted photon (or, more likely, graviton) be? (Express your answer in light years—is the remarkable answer a coincidence?)