Estimate radiation from a weak encounter

Let a projectile of charge q and mass m pass a heavy target charge Q at speed v and impact parameter b. The target is fixed and recoil is neglected. On the unperturbed straight path the separation squared is b squared plus v squared times time squared, with closest approach at time zero. Define alpha as the magnitude of the Coulomb product divided by four pi epsilon zero. The acceleration magnitude on that trial path is alpha divided by mass and separation squared.

It may sound inconsistent to assign acceleration to a straight path. This is a first-order orbit approximation: evaluate the force on the leading undeflected trajectory and use it to calculate small corrections. The actual projectile bends slightly. Integrating its transverse acceleration gives a velocity change proportional to the inverse of b times v. Require that change to be much smaller than the incoming speed. This condition, rather than the visual straightness of a sketch, controls whether the approximation is useful.

Larmor power on the trial path is proportional to acceleration squared. Its time integral can be performed by substituting v times time divided by impact parameter. The dimensionless integral of one over one plus the squared variable, all squared, equals pi over two. This produces an exact coefficient for this approximate trajectory. A rough estimate using peak power times encounter duration captures the same impact-parameter scaling but misses that numerical factor.

The result decreases as impact parameter cubed and incoming speed to the first power. The strong dependence on impact parameter reflects both stronger forces and shorter encounter duration at small distance. These scalings hold with target charge, projectile properties, and the approximation fixed. They are not a complete material stopping-power formula: a material calculation must integrate over a physically justified distribution of encounters, account for screening and recoil, and determine suitable short-distance limits.

As b shrinks, the trial-path energy diverges while the projectile initially carries finite kinetic energy. This is a failure of extrapolation, not a source of infinite physical energy. The small-deflection condition fails first or another neglected scale becomes important. Independently require radiated energy to be small compared with kinetic energy if an energy-loss-free orbit is used. A better trajectory alone does not justify neglecting radiation reaction when the energy loss becomes substantial.

Classical localization also needs an impact parameter large compared with the projectile's reduced de Broglie wavelength. Characteristic encounter frequencies are of order v divided by b; their photon energies must be small compared with kinetic energy for a classical soft-emission picture. Screening changes distant encounters, and target recoil matters when the target is not sufficiently heavy. The quadratic lab isolates the acceleration-square rule; it cannot compute a Coulomb trajectory, its impulse, or a bremsstrahlung spectrum.