Photon production at Kepler escape threshold

We obtain an exact closed-form total photon count for a prescribed Newtonian Kepler parabola, with a fully relativistic treatment of its radiation. We also obtain closed expressions for the angularly resolved photon spectrum and elementary expressions for the total radiated energy and momenta. The count is finite, scale-independent, and analytic in subluminal speed at closest approach. At large maximum rapidity $\eta$, $N\sim (7\alpha \mathrm{/}\sqrt{6})\eta$, while $E\propto e^{3\eta }$ at fixed passage timescale: the radiation chiefly hardens. For slow motion at fixed coupling and closest distance, a nearby ellipse emits fewer photons but more energy and angular momentum per orbit. The hyperbolic photon count remains IR divergent arbitrarily close to threshold, while the count at the exact parabolic threshold is finite.