Bohr atomic model: does the electron fall?

Question

I know this is a classical system, and thus not compliant with the quantum nature of real atoms. But please bare with me.

I have heard this before: the orbiting electron should radiate in the classical Bohr atomic model (point-charge electron and point charge proton orbiting common center of mass).

When searching around, I found it in wikipedia and also in a very popular question and answer here.

Yet I can't see how. I have been checking the literature:

Using Lorentz Force where we account for both the electrical and magnetic fields $\textbf{F}_L = e\textbf{E} + e\textbf{v} \times \textbf{B}$ you find it radiates only if you fix the position of the proton.

If you are committed to be accurate, and work on the center of mass, then magnetic field terms cancel and only electric fields remain. This then leads to classical movement in central field where momentum conservation prevents the electron from falling into the nucleus.

Now for a more serious consideration,

Using Pointing Theorem and then trying to find the radiated energy coming out of the system, with a bit more work, I find that there is nothing coming out. Which is consistent with above, but inconsistent with calculations of radiation coming from an equivalent system: rotating dipole

Where is the mistake?

Answer 1

I will turn my comment into an answer:

One is dealing with classical physics in this question, at the level of the Bohr model for the atom special relativity is unknown.

An inertial frame can be defined as a frame where the laws of physics have the same mathematical form and measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation).

This means that predictions for physical observables made in one inertial frame hold for all inertial frames, so one uses the simplest frame. In the frame where the proton is at rest and the electron is turning around it like a planet , classical electrodynamics predicts that there will be electromagnetic radiation. Transforming this prediction to the rest mass system of the initial conditions of (proton+electron) will still show in the time progress electromagnetic radiation. The original center of mass can only be defined if the momentum of the departing radiation is included for the Galilean transformation.

In the quote with no link that you give :

This then leads to classical movement in central field where momentum conservation prevents the electron from falling into the nucleus.

The discrepancy is in this statement: Momentum conservation does not prevent the electron to radiate in an electric field. The radiation takes away momentum. Again ignoring that the center of mass system will change.

I cannot check the Poynting vector comment because the maths are not shown.

But is is enough that the laws of physics are the same in all inertial frames to show that the assumed center of mass frame is not stable either, so some mistake will be there also.