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One explanation is an upper electron energy imposed by the accelerator: finite age, finite confinement size, escape, or a balance of acceleration and losses limits $\gamma$ to ${\gamma }_{\max }$. Since the synchrotron critical frequency scales as ${\nu }_c\propto B{\gamma }^2\sin \alpha$, the absence of higher-energy electrons produces a corresponding high-frequency cutoff, broadened by the single-particle spectrum.

The other is synchrotron cooling, with inverse Compton scattering often contributing: $\dot{\gamma }=-b{\gamma }^2$ gives $t_{\mathrm{cool}}=1/(b\gamma )$. High-energy electrons disappear fastest, so a population of age $t$ develops a cooling scale ${\gamma }_b\sim 1/(bt)$ and ${\nu }_b\propto B/(b^2{t}^2)$. A burst population can develop a strong cutoff. With continuous power-law injection, the simplest steady cooling model instead steepens the electron energy index by one and the synchrotron radiation spectral index by $1/2$, giving a break rather than necessarily an exponential cutoff. The detailed spectral shape therefore distinguishes finite acceleration from aging and injection history.

Synchrotron radiation is radiation from relativistic charges undergoing transverse acceleration, for example electrons bent by a magnetic field. Relativistic aberration concentrates the radiation within an angle of order $1/\gamma$ about the instantaneous velocity. An observer sees short pulses as this cone sweeps past; synchrotron pulse compression produces a broad spectrum extending far above the orbital frequency.

For an ultrarelativistic circular trajectory of curvature radius $\rho$, the characteristic scales are^[ref] The synchrotron critical frequency sets the spectral turnover; it is not a sharp upper cutoff. The transverse-acceleration Lienard formula givesThe strong energy dependence is a major loss mechanism in electron storage rings. Radiation observed in the orbital plane is linearly polarized in that plane; away from it the two components generally produce elliptical polarization, with opposite handedness on opposite sides. Accelerator light sources exploit its brightness, broad tunable spectrum and pulsed timing. At low speeds the relativistic beaming and high-frequency extension disappear, approaching ordinary cyclotron radiation.^[ref]

For a $5$ GeV electron, ${\gamma }_e=E/(m_e{c}^2)\simeq 9785$. With $\rho =10$ m, the stated synchrotron critical frequency formula givesAt the same total beam energy and bend radius, a proton has ${\gamma }_p=5/0.9383\simeq 5.329$, and the same supplied formula givesThe ratio is $({\gamma }_p/{\gamma }_e{)}^3\simeq (m_e/m_p{)}^3$, explaining the strong suppression for protons. The critical-energy formula is an ultrarelativistic approximation, excellent for the electron and only approximate for this moderately relativistic proton. If “5 GeV proton beam” instead denotes kinetic energy, then ${\gamma }_p=1+5/0.9383$ and the estimate is about $7.50\times 10^{-6}$ eV; the two beam-energy conventions should not be mixed.

For an ultrarelativistic electron of Lorentz factor $\gamma =E/(m_e{c}^2)=3000/0.511=5870.84$, the synchrotron critical frequency gives ${\epsilon }_c=\hslash {\omega }_c=3\hslash c{\gamma }^3/(2\rho )$. Thus, using the supplied constants,The angle-integrated spectral power per unit photon energy is proportional to $x{\int }_x^{\infty }{K}_{5/3}(u)du$, with $x=\epsilon /{\epsilon }_c$. It peaks at $x\simeq 0.286$, soThis is a power spectrum per linear energy interval; power per logarithmic interval and photon-count spectra have different peaks. The magnetic rigidity condition is $p=|e|B\rho$, with $pc\simeq E$, givingThe critical energy sets a broad spectral scale, not a maximum allowed photon energy.^[ref]