Ter-Antonyan function

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The Ter-Antonyan function parameterizes the energy spectra of primary cosmic rays in the "knee" region (${\displaystyle 10^{15}-10^{17}}$ eV) by the continuously differentiable function of energy ${\displaystyle E}$ taking into account the rate of change of spectral slope. The function is expressed as:

where ${\displaystyle \Phi }$ is a scale factor, ${\displaystyle \gamma _{1}}$ and ${\displaystyle \gamma _{2}}$ are the asymptotic slopes of the function (or spectral slopes) in a logarithmic scale at ${\displaystyle E\ll E_{k}}$ and ${\displaystyle E\gg E_{k}}$ respectively for a given ${\displaystyle E_{k}}$ energy (the so-called "knee" energy). The rate of change of spectral slopes is set in function (1) by the "sharpness of knee" parameter, ${\displaystyle \epsilon >0}$. Function (1) 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 (1) is also used for the interpolation of observed Extensive Air Shower spectra in the knee region.^[2]

Function (1) can be re-written as:

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

where ${\displaystyle \Delta \gamma =\gamma _{2}-\gamma _{1}}$ and

${\displaystyle Y(E,\epsilon ,\Delta \gamma )\equiv \left(1+\left({\frac {E}{E_{k}}}\right)^{\epsilon }\right)^{-{\frac {\Delta \gamma }{\epsilon }}}}$

is the “knee” shaping function describing the change of the spectral slope. Examples of ${\displaystyle Y(E,\epsilon ,\Delta \gamma =0.5)}$ for ${\displaystyle \epsilon \equiv 0.5,1,2,\cdots 500}$ are presented above.

The rate of change of spectral slope from ${\displaystyle -\gamma _{1}}$ to ${\displaystyle -\gamma _{2}}$ with respect to energy (${\displaystyle E}$) is derived from (1) as:

${\displaystyle {\frac {df(E)}{dx}}=-\gamma _{1}-{\frac {\Delta \gamma }{1+(E_{k}/E)^{\epsilon }}}}$,

where

${\displaystyle f=\ln \left({\frac {dF}{dE}}\right)}$,

${\displaystyle x=\ln({\frac {E}{E_{k}}})}$,

and

${\displaystyle \left({\frac {df}{dx}}\right)_{E=E_{k}}=-{\frac {\gamma _{1}+\gamma _{2}}{2}}}$

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

Function (1) coincides with B. Peters^[3] spectra for ${\displaystyle \epsilon =1}$ and asymptotically approaches the broken power law of cosmic ray energy spectra for ${\displaystyle \epsilon \gg 1}$:

${\displaystyle \left({\frac {dF}{dE}}\right)_{\epsilon =\infty }\propto \left({\frac {E}{E_{k}}}\right)^{-\gamma }}$,

where

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