Nerling, F

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29th International Cosmic Ray Conference Pune (2005) 7, 127130Parameterisation of Cherenkov light production in high-energy showersF. Nerling , J. Blu¨mer  , R. Engel and M. Risse (a) Forschungszentrum Karlsruhe, Institut f¤ur Kernphysik, 76021 Karlsruhe, Germany(b) Universit¤at Karlsruhe, Institut f¤ur Experimentelle Kernphysik, 76021 Karlsruhe, GermanyPresenter: F. Nerling (frank.nerling@ik.fzk.de), ger-nerling-F-abs1-he14-oralA new analytical description of Cherenkov light production in high energy showers is presented. It describesboth the total amount and the angular distribution of the emitted Cherenkov photons as a function of the showerage. A universal parameterisation of electron energy distribution is used for calculating the total number ofCherenkov photons. Based on universality features of the angular distribution of electrons, a parameterisationof produced Cherenkov photons for a given angle to the shower axis is introduced. Thus, a complete analyticaldescription is derived which allows the calculation of direct and scattered Cherenkov contributions to lightprofiles measured by experiments using the fluorescence technique. The results are compared to CORSIKAsimulations and to other parameterisations.1. IntroductionFor correct shower profile reconstruction from light signals measured using the air fluorescence technique, aprecise knowledge of the Cherenkov light contribution to the fluorescence detector signal is mandatory. In con-trast to the isotropically radiated fluorescence light, the Cherenkov photons are emitted mostly in the forwarddirection. Therefore, the amount of Cherenkov light in the measured signal depends on the shower geometrywith respect to the detector. It leads to systematic uncertainties in the determination of primary propertieslike energy and position of shower maximum unless accounted for properly. The analytical calculation of theamount of Cherenkov light received by fluorescence detectors needs the total number of Cherenkov photonsproduced and their angular distribution with respect to the shower axis. The so-called scattered Cherenkovlight contribution, i.e. those photons which originally would not have reached the detector but are scatteredinto the field of view, depends mainly on the former and is nearly independent of the shower geometry. Theamount of Cherenkov photons directly hitting the detector, the so-called direct Cherenkov light, depends onboth the total amount of produced Cherenkov photons and their angular distribution with respect to the showeraxis.The total number of Cherenkov photons  produced per interval of slant depth d 	 and angle d  with respectto the shower axis can be calculated in good approximation by [1, 2]			ﬁﬀﬂ	ﬃ  !#"$&%('*)+,-/.10	2,-354&6,87 (1)Here, 	 is the charged particle number as function of depth 	 , ,:9<;ﬂ= the local Cherenkov energy thresh-old, which depends on the refractive index >?@>A of air, and . 0	2,- is the differential electron energyspectrum at depth 	 normalised to unity above the energy threshold ,CBEDﬂ9 adopted in the simulation (1 MeV inthe examples shown here), cf. [1, 2].  F	2 is the angular distribution of Cherenkov photons per angularbin with respect to the shower axis, normalised to one photon and averaged over azimuth:F	GAHﬀ	I	F	2J3KMLONPﬃRQSTUWVC7 (2)For a given shower geometry, X	 follows from the atmospheric model (US-StdA used in the following).128 F. Nerling et al.E [GeV]10log-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1/dlnEe dNe1/N0.050.10.150.20.25ener y  [ e ]10 -3 10 -2 10 -1 1 10HillasGiller et al.CORSIKAThis Works = 1.0Figure 1. Mean electron energy spectrum CORSIKA, dif-ferent parameterisations (see text) for YZ?[]\ ^ .]2slant depth [g/cm400 500 600 700 800 900 1000 11002/dX per g/cmγdN1000150020002500300035004000450050008x10Hillas (s fixed)HillasGiller et al.CORSIKANerling et al.Figure 2. Longitudinal Cherenkov photon prole obtainedby CORSIKA, different parameterisations (see text).2. Total number of Cherenkov photons producedApplying a parameterisation of the electron energy spectrum, the total number of Cherenkov photons producedper slant depth d 	 can be calculated analytically based on ansatz (1) when the shower size profile is providedby EAS simulations or by fluorescence observations. A parameterisation of the electron energy spectrumdepending only on the shower age _`aI	cb	edgf	ihHjFk] was first provided by Hillas [3]. Two new approacheshave recently proposed [4, 1] based on CORSIKA [5] simulations applying QGSJET01 [6] as interactionmodel. For high-energy showers l@Vnmo(p eV, the electron spectrum has shown to be universal, i.e. it does notdepend significantly on the primary energy or particle type [4, 1, 2] and is also largely independent of theshower zenith angle [2]. In Fig. 1 different parameterisations of the electron energy spectrum are comparedto the Monte Carlo result for _XqV . The comparison of the full Monte Carlo Cherenkov profile calculationwith the model calculation applying the different parameterisations for ansatz (1) is shown in Fig. 2. Theparameterisation given by Nerling et al. also accounts for different ,CBEDr9 , which is important to be consistentwith 	,qls,MBEDﬂ9 . The calculation labelled “Hillas (s fixed)” employs the parameterisation given in [3]for _CsV only, as often used (see e.g. [7, 8]).3. Angular distribution of electrons and Cherenkov photonsElectrons in a shower undergo multiple Coulomb scattering, which broadens their angular distribution withrespect to the shower axis: The higher the mean electron energy, the smaller is the mean electron angle to theshower axis. In Fig. 3 the energy dependent angular distribution of electrons is shown exemplarily for an indi-vidual proton shower of VnmoFt eV for three different shower ages. It can be seen that the distribution is stronglydepending on the particle energy, and is to a large extent independent of the shower age within the statisticalfluctuations. As the electron energy spectra are universal in high-energy showers and the electron scatteringangle is mostly determined by the particle energy, the electron angular distribution is also approximately inde-pendent of shower energy and primary particle type [2]. This is illustrated in Fig. 4 where the electron angulardistributions of many individual proton and iron showers of different energies ( VnmoFuﬂV]m oFt eV) are shown atParameterisation of Cherenkov light production in high-energy showers 129angle to shower axis [deg]0 10 20 30 40 50 60Ω/de dNe1/N10-610-510-410-310-210-1110250 keV < E <   1 MeV  1 MeV < E <   5 MeV  5 MeV < E <  10 MeV 10 MeV < E <  20 MeV 20 MeV < E < 100 MeV100 MeV < E < 500 MeV500 MeV < E <   1 GeV  1 GeV < E <  10 GeV 10 GeV < E < 100 GeVs = 1.0 (markers)s = 0.8 (dashed)s = 1.1 (solid)Figure 3. Normalised electron angular distribution with re-spect to the shower axis, shown for an individual protonshower of [G^Iv<w eV for three different shower ages and vari-ous different ranges of electron energies.angle to shower axis [deg]0 10 20 30 40 50 60Ω/de dNe1/N10-610-510-410-310-210-1110250 keV < E <   1 MeV  1 MeV < E <   5 MeV  5 MeV < E <  10 MeV 10 MeV < E <  20 MeV 20 MeV < E < 100 MeV100 MeV < E < 500 MeV500 MeV < E <   1 GeV  1 GeV < E <  10 GeV 10 GeV < E < 100 GeVeV18,19Fe, p, 10 s = 1.0(a)(a)(b)(b)(c)(c)(d)(d)(e)(e)(f)(f)(g)(g)(h)(h)(i)(i)Figure 4. Universality of electron angular distributions,shown for numerous individual proton and iron showers of[x^ v<y and [x^ v<w eV and various different ranges of electronenergies.shower maximum, cf. also [9]. The distributions of individual showers do not differ much, larger (statistical)fluctuations occur merely at large angles. The distribution of electrons in the GeV-region show a larger spread,which is in agreement with the energy spectra of GeV-electrons showing larger fluctuations as well [2].The Cherenkov photons are emitted under the Cherenkov emission angle, which slightly changes with altitudeand amounts about V1z in air. This angle is negligible compared to the broad distribution of electrons. Con-sequently, the Cherenkov photon angular distribution is determined mainly by the electron energy spectrumand two dependencies occur. The photon angular distribution depends on height  due to the dependenceof , 9<;n=A , and on the shower age _ because of the dependence of .I0,{_1 . It is common to describe theheight dependence by an exponential function, where the scaling angle  S is a function of ,M9<;n= , see e.g. [7].Traditionally this approximation is applied for calculating the Cherenkov contamination of fluorescence lightsignals from high-energy showers, see e.g. [7, 8]. Generalising this ansatz in order to take into account boththe dependence on  as well as on _ we write  +_1J_1+ﬀr , where  _] is a polynomial ofsecond order in shower age and the exponential term depends on altitude only. To enlarge the range of validityup to |Im z and improve the data description around a}m z this factorised ansatz is extended to [2]:+_1~  _]V1BA`}#GG(F]dŁG_1V]BEBCIGF]7 (3)In this expression, the age dependence is included by the polynomials  _1 and Ł _] , and the height depen-dence is taken into account by  B:sﬀ1,9<;ﬂ=and  BEBﬀ1BA with 8/d++ﬀ_ , where , 9<;ﬂ= isgiven in MeV. As shown in Fig. 5 and Fig. 6 respectively, the CORSIKA results are described properly usingthe parameters given in [2].130 F. Nerling et al.angle to shower axis [deg]0 10 20 30 40 50 60 [1/deg]θ/d γ dNγ1/N10-410-310-210-11Baltrusaitis et al.CORSIKATHIS WORK 0.8≈s  1.0≈s  1.2≈s  1/4× 4×oeV, 3019p, 10Figure 5. Angular distribution of Cherenkov photons withrespect to the shower axis. Shown are the simulated distri-butions for an individual proton shower for different showerages, [7] and the new parameterisation Eq. (3) [2].o45angle to shower axis [deg]6 8 10 12 14 16 18 20 22 24 26 28-40-2002040Baltrusaitis et al.This Work s = 0.8relative difference [%]6 8 10 12 14 16 18 20 22 24 26 28-40-2002040 eV, thin -818Fe, 10s = 1.0o0 ,o15 ,o30, o45, o606 8 10 12 14 16 18 20 22 24 26 28-40-2002040s = 1.2Figure 6. Quality of description: Relative differences of dif-ferent approaches ([7] and Eq. (3) [2]) to CORSIKA results,shown for 5 individual iron showers and different zenith an-gles.4. ConclusionsBased on universality features of high-energy showers, an analytical description of the Cherenkov light pro-duction in EAS has been introduced providing both, the total number of produced Cherenkov photons as wellas their angular distribution with respect to the shower axis. It offers the calculation of the direct and scatteredCherenkov contributions to measured fluorescence light profiles, see [10]. The achieved accuracy in repro-ducing the CORSIKA predictions in terms of the number of Cherenkov photons produced per slant depth andangle with respect to the shower axis is (within shower-to-shower fluctuations) better than a few percent ( 10 % for s=0.8,  5 % for s=1.0,  3 % for s=1.2). The differences compared to the traditionally used approachresult in significant and systematic differences in reconstructed energy and position of shower maximum, if thenew model is applied for shower profile reconstruction as has been shown for Auger hybrid data [10].References[1] F. Nerling et al., Proc. 28 9<; Int. Cos. Ray Conf., Tsukuba (Japan), 2 (2003) 611.[2] F. Nerling et al., astro-ph/0506729 (2005).[3] A. M. Hillas, J. Phys. G: Nucl. Part. Phys. 8 (1982) 1461.[4] M. Giller et al., J. Phys. G: Nucl. Part. Phys. 30 (2004) 97; Proc. 28 9<; Int. Cos. Ray Conf., Tsukuba(Japan), 2 (2003) 619. Note: Parameters used here are different from that published, a new set of param-eters [Giller et al. 2005, private communication] is used.[5] D. Heck et al., Report FZKA 6019, Forschungszentrum Karlsruhe (1998).[6] N. N. Kalmykov, S. S. Ostapchenko, A. I. Pavlov, Nucl. Phys. B (Proc. Suppl.), 52B (1997) 17.[7] R. M. Baltrusaitis et al., Nucl. Instr. Meth. A 240 (1985) 410.[8] T. Abu-Zayyad et al., HiRes Collaboration, Astropart. Phys., 16 (2001) 1.[9] M. Giller et al., Int. J. Mod. Phys. A: Proc. 19 9<; European Cos. Ray Symp., Florence (Italy) (2004).[10] F. Nerling et al., Pierre Auger Collaboration, these proceedings, ger-nerling-F-abs2-he14-poster (2005).

Parameterisation of Cherenkov light production in high-energy showers

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