Problem $4\mathrm{.}20$ Consider the earth$$sun system as a gravitational analog to the hydrogen atom. $($a$)$ What is the potential energy function $($replacing Equation $4\mathrm{.}52)?($Let mE be the mass of the earth, and M the mass of the sun.$)($b$)$ What is the $$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$28$ a coincidence?$)$