# Derive an expression of Bohr radius in gravitational case

1. Sep 7, 2014

### Frioz

1. The problem statement, all variables and given/known data
Both Newton's gravitational law and Coulomb's law are inverse-square laws: The force of attraction
between the sun (S) and Earth(E) has (G*m_S*m_E)/r^2, whereas the force of attraction between an electron and a proton in a hydrogen atom is (e^2)/(4*pi*epsilon_0*r^2). Derive an expression for the equivalent of the Bohr radius for the gravitational case. What is the value of the quantum number of Earth's orbit? Would distance differences between individual quantum states in the solar system be observable?

2. Relevant equations
F = (G*m_S*m_E)/r^2
F= (e^2)/(4*pi*epsilon_0*r^2)
KE = 1/2 * m*v^2
PE = -e^2 / (4*pi*epsilon_0*r)
U=(G*m_S*m_E)/r
E_total = nhv

3. The attempt at a solution

My attempt was to use the virial theorem, that total energy is equal to one of half of the potential energy. Therefore, since U=-(G*m_S*m_E)/r, then E_total = -(G*m_S*m_E)/2r.

Then I equated E_total = -(G*m_S*m_E)/2r = nhv, and solved for v to get v = -(G*m_S*m_E)/(2rnh).

I then used F = (G*m_S*m_E)/r^2 = (m*v^2) / r and substituted what I got for v, and solved for r. This yielded:

r = (G*m_S*(m_E^2))/(4n^2*h^2).

Is this the correct way to get the Bohr radius?

I also solved for n to get n = sqrt((G*m_S*m_E^2)/(4*h^2*r)) which gave me n=3.533x10^63, which seems to be off by a factor of 10^11 from the actual answer. Any direction would be valuable, and I thank you in advance!

2. Sep 7, 2014

### Staff: Mentor

That looks like a very complicated way, but I guess it is possible, if you fix an error:
The last formula cannot be right, it gives inverse meters as unit: WolframAlpha
This is easy to fix - do you expect a larger r to give a smaller n, as your formula suggests? What could have went wrong?

3. Sep 7, 2014

### Frioz

I cannot seem to find this error.. I saw that I neglected a negative sign when solving for r, but that doesn't change the magnitude. Am I just missing something algebraic or is there a huge concept that's flying over my head?

4. Sep 8, 2014

### Staff: Mentor

Well, the idea to find the error is easy: check the units in every equation. At some point it starts to be wrong.