The Problem of Electron Orbits

With gravitational orbits, it seems, an object can be in any orbit. That is to say, if a mass, m is at a distance (radius) R, with tangential velocity v, feeling an inward (centripetal) force,

$$F_c = m \displaystyle \frac{{v^2}^{\phantom{1^1\!\!\!}}}{{R}_{\phantom{1_1\!\!\!}}}$$ (1)

then it can be in a stable orbit. For any radius, you calculate the gravitational force

$$F_G = G \displaystyle \frac{{Mm}^{\phantom{1^1\!\!\!}}}{{R^2}_{\phantom{1_1\!\!\!}}}$$ (2)

and then give the object a velocity equal to

$$v = \sqrt{\displaystyle \frac{{R \, F_c}^{\phantom{1^1\!\!\!}... ...frac{{G M}^{\phantom{1^1\!\!\!}}}{{R}_{\phantom{1_1\!\!\!}}}}$$ (3)

(found from plugging Eq. 2 into Eq. 1 and solving for v.) The orbit will be stable, for any R you choose. The problem for electron orbits is that when electrons accelerate, they emit light. Since an orbiting electron is accelerating,³ physicists expect it to always be emitting light. Well, that would mean that everything would glow in the dark! It would also mean that the electrons constantly give up energy. That implies that the electrons would drop to a lower radius, eventually spiraling into the nucleus. Atoms just couldn't live like this!