Can a Satellite's Orbit Around Earth Be Quantized Like a Hydrogen Atom?

[December]

[SOLVED] Macroscopic quantum model

Hi!
I'm supposed to calculate the quantum number of a macroscopic system (The Earth and a satellite).

I should assume that the satellite is moving in a circular motion around earth, and that it fulfills the same quantization conditions as the Bohr model of the Hydrogen atom.

So far, I started by calculating the total energy of the system, but using a gravitational potential instead of the Coulomb potential, which gives me a total energy of:

E=mr^{2}\omega^{2}-G\frac{mM}{r}

Then, by using the centripetal force and the gravitational force (to find an equilibrium between the two), I got:

G\frac{mM}{r^{2}}=mr\omega^{2}

Substituting this into the energy expression gives:

E=\frac{1}{2}G\frac{mM}{r}-G\frac{mM}{r}

The total energy then becomes:

E=-G\frac{mM}{2r}

... After this, I'm stuck. As far as I can see (using the book that I have in this course), Bohr postulated that the emitted radiation from the hydrogen atom has a frequency which is given by:

E_{n}-E_{n'}=hf

... Where E_{n} is given by:

E_{n}=-\frac{Rhc}{n^{2}}

My thought was that the total energy expression which I calculated must be for a specific value of n, so what I tried was to set my expression equal to E_{n} and then calculate the quantum number n from this relationship. This yielded:

n=\sqrt{\frac{2rRhc}{GmM}}

...However, I'm not even sure that this is a reasonable approach. What especially bothers me is the Rydberg Constant. Can I use a standardized value on this, or do I have to recalculate it so that it too depends on a gravitational force?

I'm really stuck on this one (I think)... Any help is truly appreciated!

 

[Gokul43201]

Bohr used the quantization condition that the angular momentum of the "orbitting" electron be an integer multiple of \hbar. I imagine you are expected to apply the same condition to the orbitting satellite.

 

[December]

Well, that sure reduced the calculations a lot :P
Thanks for the help!