Coulomb orbit

A classical two-body trajectory in an inverse-distance Coulomb potential. With an effectively fixed centre and coupling $C$, a circular orbit has $v^2=C/(mr)$ and mechanical energy $-C/(2r)$.

An accelerating classical charge radiates according to the Larmor formula. For a slowly evolving nonrelativistic circular electron orbit, energy balance gives $d(r^3)/dt=-4c{r}_0^2$, with $r_0$ the classical electron radius. The finite formal collapse time exposes classical atomic instability; it is not a model of a stable quantum atom.