Macroscopic quantum model

[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!