Derive an expression of Bohr radius in gravitational case

[Frioz]

Homework Statement

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?

Homework 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

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!

 

[mfb]

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?

 

[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?

 

[mfb]

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

 

[paranormal]

The approach you took to derive the Bohr radius in the gravitational case is correct. However, there are a few things to consider when comparing it to the actual value of the Bohr radius.

Firstly, the value of the quantum number n in the gravitational case may not necessarily be the same as in the atomic case. In the Bohr model of the atom, n represents the energy level of the electron, which is related to its distance from the nucleus. In the case of Earth's orbit around the sun, n may not have the same physical interpretation. Therefore, it is not accurate to directly compare the value of n in the two cases.

Secondly, the value of the gravitational constant G is significantly smaller than the value of the Coulomb constant k (which is related to the permittivity of free space). This means that the force of attraction between the sun and Earth is much weaker compared to the force of attraction between an electron and a proton in a hydrogen atom. This leads to a much larger value for the Bohr radius in the gravitational case.

Lastly, the Bohr model of the atom is a simplified model that does not take into account the effects of relativity and quantum mechanics. In reality, the electron does not orbit the nucleus in a circular path, and the concept of a well-defined orbit may not apply in the gravitational case. Therefore, the concept of a Bohr radius may not have the same significance in the gravitational case.

In conclusion, while your derivation is correct, it may not accurately represent the actual value of the Bohr radius in the gravitational case. The differences in scale, physical interpretation of n, and the limitations of the Bohr model should be taken into consideration.