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 $-C/(2r)$.

A projectile of mass $m_1$, electric charge $q_1$, and nonrelativistic laboratory kinetic energy $K_{\mathrm{lab}}$ approaches an initially stationary free target of mass $m_2$ and like-sign electric charge $q_2$. The relative-motion kinetic energy is $K_{\ast }=m_2{K}_{\mathrm{lab}}/(m_1+m_2)$. At a head-on turning point it is converted to the repulsive Coulomb potential $k_e{q}_1{q}_2/r_{\min }$, proving the displayed separation. The remaining laboratory kinetic energy is the translation of the centre of mass; it cannot be converted into the pair's separation energy. An externally fixed target is the infinite-target-mass limit.

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.