Follow radiation from a relativistic bend

For motion with speed v along a path of curvature radius rho, acceleration perpendicular to velocity has magnitude v squared divided by rho. Insert this into the perpendicular Liénard power. In the ultrarelativistic limit v approaches c, giving total power proportional to gamma to the fourth power divided by curvature radius squared. At fixed radius, doubling energy approximately doubles gamma and increases power sixteenfold. At fixed magnetic field the radius also changes, so that comparison has a different scaling.

An external magnetic field supplies a transverse Lorentz force but no direct power because velocity dotted into that force is zero. This does not violate radiation loss. A prescribed constant-energy orbit implicitly includes another source of energy, such as an accelerating electric field. Without replacement, particle energy and curvature evolve. Using the fixed-orbit formula instantaneously can still be useful if the change over one formation interval is small, but an indefinitely constant orbit requires an explicit energy budget.

Relativistic angular factors concentrate much of the emitted power within an angle of order inverse gamma around the velocity. This is a characteristic width, not a hard cone boundary containing all power. As the particle turns, its velocity sweeps through that width over a source-time interval of order radius divided by gamma times c. A stationary observer receives an even shorter interval because the source travels toward the observer between emission events.

The reception-time factor is kappa, which near the forward direction is of order inverse gamma squared. Combining source sweep and arrival compression gives a pulse scale of order radius divided by gamma cubed times c. Its inverse gives the characteristic frequency scaling. A full Fourier calculation fixes the conventional critical angular frequency to three halves times gamma cubed times c divided by radius. That is not ordinary frequency in hertz, and the spectrum extends across a range around it.

Large gamma does not automatically invalidate classical radiation; the relevant recoil compares a characteristic photon energy with particle energy. Require hbar times critical angular frequency to be much smaller than gamma times rest energy. In a transverse magnetic field the related quantum parameter is approximately gamma times field strength divided by the critical field. If it is not small, discrete photon recoil and quantum changes to emission matter, even though the classical orbit geometry remains intelligible.

Small recoil also does not guarantee negligible total energy loss. A very large number of individually soft photons can drain substantial energy over an orbit. Check the fractional energy radiated over the trajectory interval for which constant gamma was assumed. The vector lab only compares angular projections for two small angles; it can illustrate the geometric narrowing of a window. It has no angular power distribution, spectrum, photon recoil, magnetic orbit, or energy replenishment calculation.