Hydrogen bound by only grav force (Bohr theory etc)

[kpou]

Homework Statement

If electric charge did not exist, and protons and electrons were only bound together by gravitational forces to form hydrogen, derive the expressions for a_0 and E_n and compute the energy and frequency of the H_alpha line and limit of Balmer series.

Homework Equations

E_n=mc^2 * alpha * 1 / (2n^2)
E_n=-mk^2Z^2e^4/(2hbar*n^2)=-E_0Z^2/n^2
a_0=hbar^2/(mc*alpha)=hbar^2/mke^2
1/lambda=Z^2R(1/nf^2-1/ni^2)

The Attempt at a Solution

If electric charge did not exist, then the balance of electron orbit and distance would change. Bohr's radius would just be zero i would imagine since it depends on electron charge. E_n would suffer the same fate... Would E_n just become mc^2? The radius would have to still be a number as the two do still have mass and gravity would affect them.

Edit:
F=Gm1m2/r^2 is the force on either exerted by the other and
E=-Gm1m2/r is the gravitational potential energy

The energy of the electron E_n must be strictly dependent on this.

How do I "derive" an expression from E_n or a_0 when their original formulas must be completely nixed?

Is there supposed to be a way of expressing it with n levels?

Otherwise I propose the energy of the electron is -Gm1m2/r as there is no kinetic energy.

 

[Redbelly98]

One of the key components of the Bohr model is the electron's angular momentum must be an integer multiple of Planck's constant divided by 2 pi:

L = n h / 2 π . . . with n = 1, 2, 3, ...​