page_title: Ter-Antonyan function
hierarchical_section_title: Ter-Antonyan function
caption_attribution_description: English: Examples of the “knee” shaping functions illustrating the change of the spectral slope at different sharpness parameters of the “knee”.
context_page_description: The Ter-Antonyan function parameterizes the energy spectra of primary cosmic rays in the "knee" region by the continuously differentiable function of energy taking into account the rate of change of spectral slope. The function is expressed as:
  where is a scale factor, and are the asymptotic slopes of the function in a logarithmic scale at and respectively for a given energy. The rate of change of spectral slopes is set in function by the "sharpness of knee" parameter. Function was proposed in ANI'98 Workshop
  by Samvel Ter-Antonyan
  for both the interpolation of primary energy spectra in the energy range 1—100 PeV and the search of parametrized solutions of inverse problem to reconstruct primary cosmic ray energy spectra.
  Function is also used for the interpolation of observed Extensive Air Shower spectra in the knee region.
  Function can be re-written as:
  where and
  is the “knee” shaping function describing the change of the spectral slope. Examples of for are presented above.
  The rate of change of spectral slope from to with respect to energy is derived from as:
  ,
  where
  ,
  ,
  and
  is the sharpness-independent spectral slope at the knee energy.
  Function coincides with B. Peters
context_section_description: The Ter-Antonyan function parameterizes the energy spectra of primary cosmic rays in the "knee" region ( eV) by the continuously differentiable function of energy  taking into account the rate of change of spectral slope. The function is expressed as:
  where  is a scale factor,  and  are the asymptotic slopes of the function (or
  spectral slopes) in a logarithmic scale at  and  respectively for a given  energy (the so-called "knee" energy). The rate of change of spectral slopes is set in function (1) by the "sharpness of knee" parameter, . Function (1) was proposed in ANI'98 Workshop (1998) 
  by Samvel Ter-Antonyan 
  for both the interpolation of primary energy spectra in the energy range 1—100 PeV and the search of parametrized solutions of inverse problem to reconstruct primary cosmic ray energy spectra. 
  Function (1) is also used for the interpolation of observed Extensive Air Shower spectra in the knee region. 
  Function (1) can be re-written as:
  where  and
  is the “knee” shaping function describing the change of the spectral slope. Examples of  for  are presented above.
  The rate of change of spectral slope from  to  with respect to energy () is derived from (1) as:
  ,
  where 
  , 
  ,  
  and 
   
  is the sharpness-independent spectral slope at the knee energy.
  Function (1) coincides with B. Peters
  spectra for  and
  asymptotically approaches the broken power law of cosmic ray energy spectra for :
  ,
  where
caption: Example of the “knee” shaping function
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