And now that we have a compact query for Mr Google, now comes the inevitable application of your formula to other bodies. Why are all these ratios not 1.0? All from Google search results for (G*m_*planet*)^2*m_e/(8*hbar*c^3) / r_*planet*:

((((G * mass of Mercury)^2) * m_e) / (8 * hbar * (c^3))) / radius of Mercury =

0.00797276022

((((G * mass of Venus)^2) * m_e) / (8 * hbar * (c^3))) / radius of Venus =

0.698777299

((((G * mass of Earth)^2) * m_e) / (8 * hbar * (c^3))) / radius of Earth =

0.998204362

((((G * mass of Mars)^2) * m_e) / (8 * hbar * (c^3))) / radius of Mars =

0.0216372774

((((G * mass of Jupiter)^2) * m_e) / (8 * hbar * (c^3))) / radius of Jupiter =

8 995.76321

((((G * mass of Saturn)^2) * m_e) / (8 * hbar * (c^3))) / radius of Saturn =

956.62368

((((G * mass of Neptune)^2) * m_e) / (8 * hbar * (c^3))) / radius of Neptune =

75.5975102

((((G * mass of Uranus)^2) * m_e) / (8 * hbar * (c^3))) / radius of Uranus =

52.6310715

And, of course, why are the ratios all over the place, from .007973 to 8996?

*0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...*

*Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan*