  We study the stability of the circular orbits of the electromagnetic two-body
problem of classical electrodynamics. We introduce the concept of resonant
dissipation, i.e. a motion that radiates the center-of-mass energy while the
interparticle distance performs bounded oscillations about a metastable orbit.
The stability mechanism is established by the existence of a quartic resonant
constant generated by the stiff eigenvalues of the linear stability problem.
This constant bounds the particles together during the radiative recoil. The
condition of resonant dissipation predicts angular momenta for the metastable
orbits in reasonable agreement with the Bohr atom. The principal result is that
the emission lines agree with the predictions of quantum electrodynamics (QED)
with 1 percent average error even up to the $40^{th}$ line. Our angular momenta
depend logarithmically on the mass of the heavy body, such that the deuterium
and the muonium atoms have essentially the same angular momenta, in agreement
with QED. Analogously to QED, our stability analysis naturally uses the
eigenvalues of an infinite-dimensional linear operator; the
infinite-dimensionality is brought in by the delay of the electromagnetic
interaction.
