Ter-Antonyan function explained

The Ter-Antonyan function parameterizes the energy spectra of primary cosmic rays in the "knee" region (

10 ^{15} -10 ^{17} eV) by the continuously differentiable function of energy E taking into account the rate of change of spectral slope. The function is expressed as:where \Phi is a scale factor, \gamma _{1} and \gamma _{2} are the asymptotic slopes of the function (orspectral slopes) in a logarithmic scale at E\ll E _{k} and E\gg E _{k} respectively for a given E _{k} energy (the so-called "knee" energy). The rate of change of spectral slopes is set in function by the "sharpness of knee" parameter, \epsilon> 0 . Function was proposed in ANI'98 Workshop (1998) by Samvel Ter-Antonyan^[1] 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.^{[1] [2]} Function is also used for the interpolation of observed Extensive Air Shower spectra in the knee region.^[2]

Function can be re-written as:

(dF)/(dE) = \Phi E ^{- \gamma _{1}} Y(E,\epsilon,\Delta\gamma),

where

\Delta\gamma = \gamma _{2-\gamma} _{1} andY(E,\epsilon,\Delta\gamma)\equiv\left( 1+ \left( (E)/(E _{k}) \right) ^{\epsilon} \right) ^{- (\Delta\gamma)/(\epsilon)}

is the “knee” shaping function describing the change of the spectral slope. Examples of

Y(E,\epsilon,\Delta\gamma =0 . 5 ) for \epsilon\equiv 0 . 5 , 1 , 2 , … 500 are presented above. The rate of change of spectral slope from - \gamma _{1} to - \gamma _{2} with respect to energy (E ) is derived from as:(df(E))/(dx) =- \gamma _{1- (\Delta\gamma)/(1+ (E ^{\epsilon} _{k/E)})} ,

where

f =ln \left( (dF)/(dE) \right) ,x =ln ( (E)/(E _{k}) ) ,

and

\left( (df)/(dx) \right) _{E = E _{k}} =- (\gamma _{1+\gamma} _{2})/(2)

is the sharpness-independent spectral slope at the knee energy.

Function coincides with B. Peters^[3] spectra for

\epsilon =1 andasymptotically approaches the broken power law of cosmic ray energy spectra for \epsilon\gg 1 : \left( (dF)/(dE) \right) _{\epsilon=infin} \propto\left( (E)/(E _{k}) \right) ^{-\gamma} ,

where

\gamma = \begin{cases} \gamma _{1,} & if E<E _{k\\ \gamma} _{2,} & if E>E _{k. \end{cases}}

Notes and References

1. S.V. Ter-Antonyan, L.S. Haroyan . 2000 . About EAS size spectra and primary energy spectra in the knee region . hep-ex/0003006 .
2. Samvel Ter-Antonyan . 2014 . Sharp knee phenomenon of primary cosmic ray energy spectrum . . 89 . 12 . 123003 . 1405.5472 . 10.1103/PhysRevD.89.123003 . 2014PhRvD..89l3003T . 118459803 .
3. B. Peters . 1961 . Primary cosmic radiation and extensive air showers . . 22 . 4 . 800–819 . 10.1007/BF02783106. 120682656 .