page_title: Synchrotron radiation
section_title: Frequency distribution of radiated energy
hierarchical_section_title: Synchrotron radiation / Formulation / Example 1: bending magnet / Frequency distribution of radiated energy
caption_attribution_description: English: Frequency_distributions_of_radiated_energy.png
context_page_description: Synchrotron radiation is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.
  In astrophysics, synchrotron emission occur, for instance, due to ultra-relativistic motion of a source around a black hole. When the source performs a circular geodesic around the black hole, the synchrotron radiation occur for orbits close to the photosphere where the motion is in ultra-relativistic regime.
context_section_description: From Eq. (5), we observe that the radiation intensity is negligible for . Critical frequency is defined as the frequency when ξ =  and θ = 0. So,
  and critical angle is defined as the angle for which  and is approximately
  For frequencies much larger than the critical frequency and angles much larger than the critical angle, the synchrotron radiation emission is negligible.
  Integrating on all angles, we get the frequency distribution of the energy radiated.
  If we define
  where y = . Then
  Note that , if , and , if 
  The formula for spectral distribution of synchrotron radiation, given above, can be expressed in terms of a rapidly converging integral with no special functions involved (see also modified Bessel functions) by means of the relation:
caption: Frequency distribution of radiated energy
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