Classically, how would an electron orbiting a proton in a perfectly circular orbit emit photons?

Question

Obviously, with quantum mechanics, we know particles don't really orbit in electric fields. From what I know, clasically, Bremsstrahlung radiation would occur if an electron was deflected or accelerated around a proton, and the energy of that photon would be equivalent to the energy lost. But if an electron did classically orbit a proton in a perfectly circular orbit, it would never change energy and could never emit a photon, yet it would be accelerated to form a circular path. How would the electron emit any radiation if it was not changing energy in a circular path?

Answer 1

Classically, the accelerating electron would emit electromagnetic radiation, lose energy, and rapidly spiral into the nucleus. For example, the classical orbit corresponding to the quantum ground state of hydrogen would have a lifetime of only 16 trillionths of a second. (The fact that this does not happen is a fine falsification of classical physics!) The classical orbit would not be circular.

For the calculation, see Classical Lifetime of a Bohr Atom.

Answer 2

The correct answer to your question is that "classically", nobody emits a photon... Either you consider non-quantum physics, and yes, an accelerated charge emits electromagnetic waves, and here the electron would spiral towards the proton, or you use quantum mechanics, and apparently you already know what the answer is in this case, the electron being unable to go below the fondamental level of energy and so won't emit a real photon.