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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]

Relativistic radiation is concentrated into an angle of order $1/\gamma$ around the instantaneous velocity. The emission direction sweeps past a tangent observer over an orbital interval of order $\rho /(\gamma c)$. Since the charge moves toward the observer, arrival times are compressed by an additional factor of order $1/{\gamma }^2$, giving the spectral scale $\omega \sim c{\gamma }^3/\rho$.

The conventional numerical factor follows from the radiation Fourier phase. Near the tangent direction, circular motion givesWith $u=c\gamma t/\rho$, this becomes $[\omega \rho /(2c{\gamma }^3)](u+u^3/3)$. The resulting Airy-type integrals are expressible through modified Bessel functions with tangent-plane argument $\xi =\omega \rho /(3c{\gamma }^3)$. The standard angle-integrated synchrotron radiation spectrum has shapeIt therefore identifiesThe beaming argument supplies the physical scaling; the spectral integral fixes the conventional factor $3/2$. Critical energy is a broad spectral scale, not a hard photon-energy cutoff.^[ref]

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]

A shorter radiation flash has a broader Fourier transform. From iii, its characteristic angular-frequency bandwidth isAt fixed bending radius this scales as ${\gamma }^3$ as $v\to c$. The standard synchrotron critical angular frequency makes the same scaling explicit:A periodically circulating electron produces harmonics of its revolution frequency. The very large harmonic range, of order ${\gamma }^3$, forms the familiar effectively continuous broad synchrotron spectrum at ordinary detector resolution. The result concerns the intrinsic spectrum of the radiation, not merely the slow envelope imposed by a long incoherent bunch. Ordinary frequency is $\nu =\omega /(2\pi )$; mixing cycles per second with radians per second would introduce a spurious factor of $2\pi$ into photon-energy estimates.^[ref]

Using the conventional critical scale rather than guessing a precise Fourier-width coefficient,This is extreme ultraviolet and the very soft X-ray boundary, with a broad spectrum extending on either side; it is not a hard, multi-keV X-ray source at the stated large bending radius. The characteristic energy is not a sharp upper cutoff. For orientation, the standard angle-integrated spectral power per unit frequency peaks near $0.29E_c\simeq 31$ eV, also in the extreme ultraviolet; a particular angular acceptance weights that spectrum differently.

The crucial scale follows from both the $3$ GeV energy and the $561$ m radius. Using only the fact that synchrotron sources can make X-rays, or replacing the bending radius by a much smaller unstated value, would misidentify the characteristic photon energies here. The critical-energy convention and the broadband character are described by the primary radiation model.^[ref]

For an ultrarelativistic circular orbit of curvature radius $\rho$, the conventional critical angular frequency is ${\omega }_c=3{\gamma }^{3}c/(2\rho )$. It sets the broad spectral scale rather than a strict maximum frequency.^[ref]