Ter-Antonyan function

{{Orphan|date=January 2021}}

File:Knee shaping function.png

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:

{{NumBlk|:|

$$

\frac{dF}{dE} = \Phi E^{-\gamma_1}\left(1+\left(\frac{E}{E_{k}}\right)^{\epsilon}\right)^{\frac{\gamma_1-\gamma_2}{\epsilon}}

,

|{{EquationRef|1}}}}

where $\backslash Phi$ is a scale factor, $\backslash gamma\_1$ and $\backslash gamma\_2$ are the asymptotic slopes of the function (or

spectral slopes) in a logarithmic scale at $E\backslash ll\; E\_k$ and $E\backslash gg\; E\_k$ respectively for a given $E\_k$ energy (the so-called "knee" energy). The rate of change of spectral slopes is set in function ({{EquationNote|1}}) by the "sharpness of knee" parameter, $\backslash epsilon>0$. Function ({{EquationNote|1}}) was proposed in ANI'98 Workshop (1998)

by Samvel Ter-Antonyan

{{cite journal

|author=S.V. Ter-Antonyan, L.S. Haroyan

|date=2000

|title=About EAS size spectra and primary energy spectra in the knee region

|arxiv=hep-ex/0003006

}}

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.

{{cite journal

|author=Samvel Ter-Antonyan

|date=2014

|title=Sharp knee phenomenon of primary cosmic ray energy spectrum

|journal=Physical Review D

|volume=89 |issue=12

|pages=123003

|arxiv=1405.5472

|doi=10.1103/PhysRevD.89.123003

|bibcode = 2014PhRvD..89l3003T |s2cid=118459803

}}

Function ({{EquationNote|1}}) is also used for the interpolation of observed Extensive Air Shower spectra in the knee region.

Function ({{EquationNote|1}}) can be re-written as:

$$

\frac{dF}{dE} = \Phi E^{-\gamma_1}Y(E,\epsilon,\Delta\gamma),

where $\backslash Delta\backslash gamma=\backslash gamma\_2-\backslash gamma\_1$ and

$$

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 $Y(E,\backslash epsilon,\backslash Delta\backslash gamma=0.5)$ for $\backslash epsilon\backslash equiv0.5,\; 1,\; 2,\; \backslash cdots\; 500$ are presented above.

The rate of change of spectral slope from $-\backslash gamma\_1$ to $-\backslash gamma\_2$ with respect to energy ($E$) is derived from ({{EquationNote|1}}) as:

$$

\frac{df(E)}{dx}=-\gamma_1-\frac{\Delta\gamma}{1+(E_k/E)^\epsilon},

where

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

$x=\backslash ln(\backslash frac\{E\}\{E\_k\})$,

and

$\backslash left(\backslash frac\{df\}\{dx\}\backslash right)\_\{E=E\_k\}=-\backslash frac\{\backslash gamma\_1+\backslash gamma\_2\}\{2\}$

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

Function ({{EquationNote|1}}) coincides with B. Peters

{{cite journal

|author=B. Peters

|date=1961

|title=Primary cosmic radiation and extensive air showers

|journal=Nuovo Cimento

|volume=22 |issue=4

|pages=800–819

|doi=10.1007/BF02783106

|s2cid=120682656

}}

spectra for $\backslash epsilon=1$ and

asymptotically approaches the broken power law of cosmic ray energy spectra for $\backslash epsilon\backslash gg1$:

$\backslash left(\backslash frac\{dF\}\{dE\}\backslash right)\_\{\backslash epsilon=\backslash infin\}\backslash propto\backslash left(\backslash frac\{E\}\{E\_k\}\backslash right)^\{-\backslash gamma\}$,

where

$\backslash gamma=$

\begin{cases}

\gamma_1, & \text{if } E

\gamma_2, & \text{if } E>E_k.

\end{cases}