A comparison study of CORSIKA and COSMOS simulations for extensive air showers
Soonyoung Roh, Jihee Kim, Dongsu Ryu, Hyesung Kang, Katsuaki Kasahara, Eiji Kido, Akimichi Taketa
aa r X i v : . [ a s t r o - ph . H E ] J a n A comparison study of CORSIKA and COSMOS simulations forextensive air showers
Soonyoung Roh a,b , Jihee Kim a , Dongsu Ryu a, ∗ , Hyesung Kang c , Katsuaki Kasahara d , EijiKido d , Akimichi Taketa e a Department of Astronomy and Space Science, Chungnam National University, Daejeon 305-764, South Korea b Department of Physics, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan c Department of Earth Sciences, Pusan National University, Pusan 609-735, South Korea d Institute for Cosmic Ray Research, University of Tokyo, Chiba 277-8582, Japan e Center for High Energy Geophysics Research, Earthquake Research Institute, University of Tokyo, Tokyo 113-0032,Japan
Abstract
Cosmic rays with energy exceeding ∼ eV are referred to as ultra-high energy cosmic rays(UHECRs). Monte Carlo codes for extensive air shower (EAS) simulate the development ofEASs initiated by UHECRs in the Earth’s atmosphere. Experiments to detect UHECRs uti-lize EAS simulations to estimate their energy, arrival direction, and composition. In this paper,we compare EAS simulations with two di ff erent codes, CORSIKA and COSMOS, presentingquantities including the longitudinal distribution of particles, depth of shower maximum, kineticenergy distribution of particle at the ground, and energy deposited to the air. We then discussimplications of our results to UHECR experiments. Keywords: extensive air shower, Monte Carlo simulation, ultra-high energy cosmic rays
1. Introduction
The nature and origin of ultra-high energy cosmic rays (UHECRs) with energy above ∼ eV are outstanding problems of modern physics. Many studies have been performed to unravelthe problems: where do UHECRs come from, what is the composition of UHECRs, and howare UHECRs accelerated to such extreme energies? UHECRs are believed to be the result ofextremely powerful cosmic phenomena [1]; the most powerful astrophysical events, such asactive galactic nuclei (AGNs) [2], gamma ray bursts (GRBs) [3], and shock waves around clustersof galaxies [4], have been suggested as possible sources of UHECRs. Yet, the nature and originof UHECRs remain unsolved (see [5, 6] for review).Cosmic rays (CRs), which are electrically charged particles, do not travel in straight lines inspace. Their trajectories are bent by intergalactic and interstellar magnetic fields that are known ∗ Corresponding author.
Email addresses: [email protected] (Soonyoung Roh), [email protected] (Jihee Kim), [email protected] (Dongsu Ryu), [email protected] (Hyesung Kang), [email protected] (Katsuaki Kasahara), [email protected] (Eiji Kido), [email protected] (Akimichi Taketa)
Preprint submitted to Astroparticle Physics November 8, 2018 o exist between galaxies and between stars [7, 8]. For this reason, even though we may guesstheir arrival directions at the Earth, we do not know exactly where they come from. However,this problem may be resolved if enough events of UHECRs are observed.Directly detecting UHECRs at the top of the Earth’s atmosphere is practically impossibleowing to the rarity of UHECR events. On average, only a few particles hit a square kilometerof the atmosphere per century. Such low flux of UHECRs demands experiments covering hugeareas to increase opportunities for their detection. In the last few decades, UHECRs have beenobserved by Akeno Giant Air Shower Array (AGASA) [9], High Resolution Fly’s Eye Experi-ment (HiRes) [10], Pierre Auger Observatory (AUGER) [11], Telescope Array (TA) [12] and soon. These experiments detect extensive air showers (EASs) created by UHECRs.When UHECRs enter the Earth’s atmosphere, they first collide with air molecules of oxygenor nitrogen; subsequently through complex interactions and cascades, EASs, which are made ofup to hundreds of billions of secondary particles (See Table 1 for the case of 10 . eV primary),are generated [13, 14]. By detecting the photons produced by secondary particles and / or thesecondary particles arriving at the ground, the properties of primary particles such as the energy,arrival direction, and composition are inferred.To observe UHECRs, AGASA used a ground array of scintillation detectors, HiRes usedfluorescence telescopes, AUGER uses a hybrid facility of water Cherenkov tanks and fluores-cence telescopes, and TA also uses a hybrid facility of scintillation detectors and fluorescencetelescopes. Fluorescence telescopes measure the ultra-violet (UV) fluorescence light producedthrough interactions between air molecules and secondary particles in EASs, and arrays of scintil-lation detectors and water Cherenkov tanks recode the secondary particles arriving at the ground.From these, AGASA reported 58 events above 40 EeV [15]. HiRes observed 13 events above 56EeV [16]. AUGER and TA have so far reported 69 events [17] and 15 events [18] above 55 EeVand 57 EeV, respectively.Along with observations of UHECRs, EASs need to be investigated by performing MonteCarlo (MC) simulations. EAS simulations form an essential part of UHECR experiments. TheTA experiment employs two existing MC codes for the simulations, CORSIKA (COsmic RaySImulations for KAscade) [19] and COSMOS [20, 21]. In this paper, we report a comparisonstudy of CORSIKA and COSMOS simulations for TA, presenting the longitudinal distributionof particle, depth of shower maximum, kinetic energy distribution of particle at the ground, andenergy deposited to the air. We then discuss implications of our results to the TA experiment.
2. EAS simulation
CORSIKA and COSMOS follow the development and evolution of EASs in the atmosphere;they describe the spatial, temporal, and energy distributions of secondary particles. To compareCORSIKA and COSMOS, we generated 50 EAS simulations with CORSIKA for each set ofparameters (see below) and another 50 EAS simulations with COSMOS for each set of param-eters. Primary energies of E = . eV, 10 . eV, 10 eV, 10 . eV, 10 . eV, 10 . eV and10 eV were considered. And zenith angles of θ = ◦ , 18.2 ◦ , 25.8 ◦ , 31.75 ◦ , 45 ◦ , 70 ◦ for protonand iron primaries were employed assuming the fat Earth. All together about 8,400 EASs weregenerated. Events with θ ≤ ◦ have been analyzed in the TA experiment [18], so we focus onthe cases with θ ≤ ◦ . Simulations with θ = ◦ are used to estimate the energy deposited intothe air in Section 3.4. Version 6.960 was used for CORSIKA simulations with θ = ◦ , 18.2 ◦ ,25.8 ◦ , 31.75 ◦ , and 45 ◦ , and version 6.980 was used for θ = ◦ . In COSMOS, for θ = ◦ , 18.2 ◦ ,25.8 ◦ , 31.75 ◦ , and 45 ◦ , 30 simulations were generated with version 7.54 and 20 simulations with2ersion 7.581; the di ff erence between versions 7.54 and 7.581 is small, so they were mixed. For θ = ◦ , version 7.581 was used. For high-energy ( E >
80 GeV), the hadronic interaction generator QGSJETII-03 [22] wasused for both CORSIKA and COSMOS. QGSJETII-03 is one of most widely used interactiongenerators for UHECR EAS simulations.For low-energy ( E <
80 GeV), the hadronic interaction generator CORSIKA-FLUKA [23]was used in CORSIKA, while the Bertini and JQMD interaction models included in the PHITScode ( E < < E <
80 GeV) [25] were used in COSMOS.We simply use the term “PHITS” for the two generators and the nelst routine managing the elasticscattering.In CORSIKA, the interactions of electro-magnetic (EM) particles (i.e., photons and elec-trons) were calculated with the EGS4 model [26]. On the other hand, in COSMOS, the interac-tions were calculated with the Tsai’s formula [27] and Nelson’s formula [26] which are based onthe basic cross-sections of particles.
We employed the following simulation parameters, trying to compare CORSIKA and COS-MOS simulations in parallel. First, in both CORSIKA and COSMOS, the Earth’s magnetic fieldat the TA observation site (39.1 ◦ N and 112.9 ◦ W, just west of Delta, Utah) was used. The groundwas fixed at the height of the TA site, 1430 m above the sea level, corresponding to the verti-cal atmospheric depth x v =
875 g / cm . Second, in both CORSIKA and COSMOS, the samethreshold energies, E threshold , were applied to secondary particles. Particles having energy below E threshold were not tracked in simulations. E threshold =
500 keV was used for EM particles, while E threshold =
50 MeV for muons and hadrons. Most particles reach the ground with energy largerthan the E threshold (see Figures 4 and 5 for the case of iron primary with E = . eV and θ = ◦ ). Third, the Landau-Pomeranchuk-Migdal (LPM) e ff ect [28, 29, 30] is included in both COR-SIKA and COSMOS. The LPM e ff ect causes a reduction of bremsstrahlung and pair productioncross sections at high energies.In principle, all secondary particles can be tracked along their trajectories and their physicalproperties can be stored until they reach the ground. Then, the number of particles can becometoo large to be comfortably accommodated with available computational resources. To alleviatethis problem, most EAS simulations introduce the Hillas thinning algorithm [31]. The algorithmpicks up only a small fraction for secondary particles with energy smaller than the product ofthe primary energy ( E ) and a thinning level ( L th ), i.e., for particles with E ≤ E × L th . At eachvertex of interaction, one secondary particle is selected in a way that more energetic particles arepicked up with higher probabilities, and further tracked. A weight, which is defined as the ratioof the energy of the selected particle to that of all secondary particles at the vertex, is assigned tothe selected particle to represent untracked particles. Eventually, the total number of secondaryparticles are recovered by counting tracked particles multiplied by their weights [32].In CORSIKA, it is recommended to take a value between 10 − and 10 − for L th . We chose L th = − . On the other hand, in COSMOS, it is recommended to use L th = A × − , where A is the mass number. Hence, for proton primary, L th = × − was used for both CORSIKAand COSMOS; for iron primary, L th = × − and L th = . × − were used for CORSIKAand COSMOS, respectively. In addition, COSMOS applies a smaller thinning, L ′ th = − × L th ,3here a higher accuracy is required. At the upper atmosphere of x v <
400 g / cm or near theshower core of r <
20 m, L th is used, while L ′ th is applied to the region of x v = −
875 g / cm and r ≥
20 m.CORSIKA and COSMOS both have an upper limit on the weight, the so-called maximumweight value, W max . As an EAS develops, through interactions of particles, weights of trackedparticles are continuously accumulated. When accumulated weights reach W max , the thinningalgorithm is no longer applied, and particles are tracked without further thinning. We used W max which is di ff erently for CORSIKA and COSMOS: W max = L th × ( E [eV] / ) for CORSIKA[14] and W max = E [eV] / for COSMOS [20].With our choices of thinning and weighting, overall more particles are tracked in COSMOSthan in CORSIKA. The computation time is roughly proportional to the number of tracked parti-cles, so for the same primary COSMOS simulations presented here took longer than CORSIKAsimulations.We also note that the data dumping is di ff erent between CORSIKA and COSMOS. In COR-SIKA, the grid points of the vertical atmospheric depth have a spacing of ∆ x v = / cm . Onthe other hand, in COSMOS, the grid points are defined at x v =
0, 100, 200 g / cm , and after200 g / cm they have a spacing of ∆ x v =
25 g / cm . So the data from CORSIKA simulations aredumped in every ∆ x v = / cm , while the data from COSMOS are dumped in every ∆ x v = / cm for x v ≤
200 g / cm and in every ∆ x v =
25 g / cm for x v >
200 g / cm .
3. Comparison of CORSIKA and COSMOS simulation results
When UHECRs strike the atmosphere, most of the particles initially generated are neutraland charged-pions. Neutral-pions quickly decay into two photons. Charged-pions (positivelyor negatively charged) survive longer, and either collide with other particles or decay to muonsand muon neutrinos. Those particles produce the so-called EM and hadronic showers. In EMshowers, photons create electrons and positrons by pair-production, and in turn electrons andpositrons create photons via bremsstrahlung, and so on. EM showers continue until the averageenergy per particle drops to ∼
80 MeV. Below this energy, the dominant energy loss mechanismis ionization rather than bremsstrahlung. Then, EM particles are not e ffi ciently produced any-more, and EASs reach the maximum (see the next subsection). In hadronic showers, muons andhadrons are produced through hadronic interactions and decays. Here, hadrons include nucleons(neutrons and protons), pions, and kaons.The number of secondary particles created by EM and hadronic showers initially increasesand then decreases, as an EAS develops through the atmosphere. The distribution of particlesalong the atmospheric depth is called the longitudinal distribution [33, 34]. Here, we first com-pare the longitudinal distributions from CORSIKA and COSMOS simulations, and analyze thedi ff erences in photon, electron, muon, and hadron distributions.Figures 1 and 2 show the typical longitudinal distributions as a function of slant atmosphericdepth, x s = x v / cos θ . Lines represent the numbers of particles averaged for 50 EAS simulations, h N i , and error bars mark the standard deviations, σ , defined as σ = vt n sim n sim X i = ( N i − h N i ) . (1)4ere, n sim =
50 is the number of EAS simulations for each set of parameters and N i is thenumber of particles at x s in each simulation. The EASs shown are for proton and iron primaries,respectively, with E = . eV and θ = ◦ and 45 ◦ . Numbers for photons, electrons, muons, andhadrons are shown. Table 1 shows the numbers of particles at peaks, which are again averagesof 50 EAS simulations. If the peaks are located beyond the maximum depth, the values at themaximum depth are shown. Note that di ff erence species have peaks at di ff erent x s ’s.In the cases shown, COSMOS predicts slightly more particles in the early stage of EASs(except for photons in the upper-left panel of Figure 1), while CORSIKA predicts slightly moreparticles in the late stage. But the di ff erence is within the fluctuation (that is, less than σ inEquation (1)). Quantitatively, the di ff erence between CORSIKA and COSMOS results in thepeak numbers of particles is at most 7 − The depth of shower maximum, denoted by X max , is defined as the slant atmospheric depthat which the number of secondary electrons reaches the maximum in EASs. X max is a functionof the primary energy, but it has di ff erent values and dispersions for di ff erent primary particles.For a given primary energy, proton primary has larger values and dispersions of X max than ironprimary. The average and standard deviation of X max , h X max i and σ X max , are known as the keyquantities that discriminate the composition of primary particles in UHECR experiments.In calculating X max , the longitudinal distribution of electrons along the shower axis was fittedto the Gaisser-Hillas function (GHF) [35], N electron ( x s ) = N electron , max x s − x s X max − x s ! X max − xs λ exp (cid:18) X max − x s λ (cid:19) , (2)where N electron , max is the maximum number of electrons at X max . We sought X max by treating x s and λ as well as X max and N electron , max as fitting parameters. We note that originally x s was meantto be the depth at which the first interaction occurs and λ to be the proton interaction mean freepath. But in practice, they were regarded as fitting parameters. It was shown that the resulting X max is not sensitive to whether λ is used as a free parameter or set to a fixed value [36, 37].Figure 3 and Table 2 show h X max i and σ X max in our simulations for proton and iron primarieswith di ff erent E ’s. h X max i and σ X max in Table 2 were calculated for 250 simulations includingthose of five di ff erent zenith angles ( θ ≤ ◦ ). Solid and dashed lines in Figure 3 are the leastchi-square fits of h X max i and σ X max in Table 2. The result of Wahlberg et al. with CORSIKA [38]is included with dot-dashed lines for comparison.We note that in some simulations the shower maximum occurred beyond the maximum depth.In such cases, X max ’s from fits to the GHF may have larger errors. And for CORSIKA results,the data dumped in every ∆ x v = / cm were used, while for COSMOS results, the data in every ∆ x v =
25 g / cm were used. So a larger systematic error may exist in COSMOS results.The results for h X max i and σ X max in Figure 3 and Table 2 are summarized as follows. First,the di ff erence between CORSIKA and COSMOS results in h X max i is at most ∼
16 g / cm forboth proton and iron primaries. It is smaller than the fluctuation, σ X max . Second, the di ff erencebetween h X max i ’s for proton and iron primaries is typically ∼ −
80 g / cm , which is beyondthe fluctuations both in CORSIKA and COSMOS simulations as well as the di ff erence betweenCORSIKA and COSMOS results. Third, σ X max is ∼ −
60 g / cm in for proton primary, while it is ∼ −
25 g / cm for iron primary. σ X max is somewhat larger in CORSIKA than in COSMOS, as isclear in Figure 3; the di ff erence is larger for proton primary. Fourth, our CORSIKA results agree5ith those of Wahlberg et al. Yet ours are smaller by up to ∼
10 g / cm . A number of possiblecauses can be conjectured. Our simulations performed with versions, models, and parametersdi ff erent from those of Wahlberg et al. In our work h X max i is defined as the depth of the peakin the number of electrons above 500 keV, while in Wahlberg et al. it was defined as the depthof the peak in overall energy deposit. Also the error in the fitting could be in the level of ∼ / cm . Although not shown here, we found that h X max i for di ff erent zenith angles varies by up to ∼
10 g / cm . In EASs, a fraction of secondary particles reach the ground. Those particles deposit a partof their energy to ground detectors, such as scintillation detectors or water Cherenkov tanks. Inexperiments, by measuring the amount and spatial distribution of the deposited energy, the pri-mary energy and arrival direction of UHECRs are estimated [39]. Here, we present the kineticenergy (i.e., the total energy subtracted by the rest-mass energy) distributions of secondary par-ticles over the entire ground; the amount of energy deposited to detectors is determined by thekinetic energy.Figure 4 shows the typical kinetic energy distributions of photons, electrons, muons, andhadrons, including particles in the shower core; here the EAS is for iron primary with E = . eV and θ = ◦ . Lines are the averages of 50 EAS simulations, and error bars mark the standarddeviations, σ , defined similarly as in Equation (1). Tables 3, 4, 5, and 6 show the total kineticenergies ( E ) and numbers ( N ) of particles reaching the ground for each particle species. Again,they are the averages of 50 EAS simulations. To further analyze the kinetic energy distributionsof di ff erent components, hadrons were separated into nucleons, pions, and kaons, and Figure 5shows their distributions.We first point that although N photon ≫ N electron ≫ N muon ≫ N hadron for all the cases wesimulated as shown in Tables 5 and 6, the energy partitioning depends on EAS parameters andvaries significantly as shown in Tables 3 and 4. For instance, in the EAS of iron primary with E = . eV and θ = ◦ which is shown in Figures 4 and 5, the partitioning of the kineticenergies of particles reaching the ground is E EM : E muon : E hadron ∼ .
18 : 0 .
11. On the otherhand, in the EAS of proton primary with E = . eV and θ = ◦ , E EM : E muon : E hadron ∼ . . ff erence between CORSIKA and COSMOS results in Figures 4 and 5 isup to 30 %, but yet the di ff erence is within the fluctuation at most energy bins. Tables 3, 4, 5,and 6 indicate di ff erences of up to 30 % in the integrated kinetic energies and numbers. Thereare following general tends: 1) For most cases, CORSIKA predicts larger energies for photonsand electrons, while COSMOS predicts larger energies for muons. 2) The di ff erence is largerfor proton primary than for iron primary. 3) The di ff erence is larger for larger E and for larger θ . We note that larger numbers of particles do not necessarily mean larger energies; this point isparticularly clear for muons. Interactions between air molecules and secondary particles yield UV fluorescence light,which is observed with fluorescence telescopes in UHECR experiments [40, 41]. The energyestimated through observation of UV fluorescence light is called the calorimetric energy, and itis used to infer the primary energy of UHECRs [42]. The energy released as the fluorescencelight is determined by the energy deposited to the air, E air . So in order for the primary energy tobe accurately estimated in UHECR experiments, E air needs to be precisely known [43].6e compare the energy deposited to the air due to EM particles, muons, and hadrons inCORSIKA and COSMOS simulations. Both CORSIKA and COSMOS follow E air by the parti-cles with E > E threshold along the atmospheric depth in simulations. But the codes does not trackparticles and their contribution any more, if their energy drops below the threshold energy. Sowe compare E air by particles with E > E threshold . Figure 6 shows E air as a function of the slantatmospheric depth, x s , for proton primary with E = . eV and θ = ◦ , 31 . ◦ , 45 ◦ and 70 ◦ .Lines are the averages of 50 EAS simulations. Table 7 shows the average of the fraction of theenergy, h E air i / E , and the relative standard deviation, σ E air / h E air i , for proton and iron primarieswith di ff erent primary energies and θ = ◦ at the ground; for θ = ◦ the slant atmosphericdepth at the ground is large enough that E air has reached the maximum (see Figure 6). Thevalues in Table 7 were calculated with 50 EAS simulations for each set of parameters.There is a clear trend in Figure 6 that COSMOS predicts larger E air ( x s ) than CORSIKA. Theenergy deposited to the air by the particles with E > E threshold in Table 7 is h E air i / E = . − . h E air i / E = . − .
82 in COSMOS simulations. The di ff erenceis ∼
15 %, which is larger than the fluctuation. The relative standard deviation, σ E air / h E air i , issmall and typically ∼ E < E threshold , as well as that by the particles with E > E threshold , should be counted. Yet, the di ff erenceof ∼
15 % is substantial. It means that the UV fluorescence light assessed with CORSIKA andCOSMOS simulations could di ff er by a similar amount, so does the primary energy of UHECRevents estimated with CORSIKA and COSMOS simulations.
4. Summary
EAS simulations form an essential part of experiments to detect UHECRs; they are used toestimate the energy, arrival direction, and composition of primary particles. The TA experimentemploys two codes, CORSIKA and COSMOS, for the simulations. In this paper, we comparedCORSIKA and COSMOS simulations by quantifying the di ff erences in the longitudinal distribu-tion of particles, depth of shower maximum, kinetic energy distribution of particle at the ground,and energy deposited to the air.Most of all, we should point that the simulation results of CORSIKA and COSMOS agreewell with each other, despite of all the complexities and di ff erences in the models and elementsinvolved in the codes. Such agreement should be quite an achievement. Nevertheless, there arenon-negligible di ff erences in the quantities we presented. Those may be regarded as systematicuncertainties in the EAS simulation part of UHECR experiments.1) The di ff erence between CORSIKA and COSMOS results in the longitudinal distribution ofparticles is less than ∼
10 % for most cases, which is within the fluctuation. The di ff erence in thepeak numbers of particles is at most 7 − h X max i , which are con-sistent with each other. The di ff erence between h X max i ’s from CORSIKA and COSMOS is upto ∼
16 g / cm , which is noticeable but smaller than the fluctuation, σ X max . The di ff erence be-tween h X max i ’s for proton and iron primaries, which is typically ∼ −
80 g / cm , is beyond thefluctuations both in CORSIKA and COSMOS simulations as well as the di ff erence between theCORSIKA and COSMOS results. σ X max is ∼ −
60 g / cm in for proton primary, while it is7 −
25 g / cm for iron primary. σ X max is somewhat larger in CORSIKA than in COSMOS; thedi ff erence is larger for iron primary.3) There are di ff erences of up to 30 % between CORSIKA and COSMOS results in thenumbers and energies of the particles reaching the ground; the di ff erence is larger for protonprimary with larger E ’s and larger θ ’s. It implies that the amount of the energy deposited toground detectors could be di ff erent up to 30 % or so in CORSIKA and COSMOS simulations.The exact response to the particles passing through ground detectors, however, depends on thedetails of detectors, and need simulations, for instance, with the GEANT code. We leave thisissue for a follow-up study.4) The energy deposited to the air, E air , by the particles with E > E threshold is larger by ∼
15 %in COSMOS simulations than in CORSIKA simulations. This implies that the UV fluorescencelight assessed with CORSIKA and COSMOS simulations could di ff er by a similar amount. Acknowledgments
We thank the referee for critical reading of the manuscript and constructive comments. Wethanks the CORSIKA user support team for help in using CORSIKA. KK thanks K. Niita forcourtesy and help in implementing the PHITS code. The work of SR, JK, DR, and HK wassupported by the National Research Foundation of Korea through grant 2007-0093860. Thework of KK, EK, and AT was supported by Grant-in-Aid for Scientific Research on PriorityAreas (Highest Energy Cosmic Rays: 15077205).
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200 400 600 80002•10 < N had r on > CORSIKACOSMOS 200 400 600 800 1000 1200
Atmospheric slant depth [g/cm ] 05.0•10 < N m uon > < N e l e c t r on > < N pho t on > Figure 1: Longitudinal distribution of photons, electrons, muons, and hadrons for EASs of proton primary with E = . eV and θ = ◦ (left panels) and 45 ◦ (right panels). Lines represent the averages of 50 simulations, and error barsmark the standard deviations. For clarity, only the error bars of CORSIKA results are shown.
200 400 600 80002.0•10 < N had r on > CORSIKACOSMOS 200 400 600 800 1000 1200
Atmospheric slant depth [g/cm ] 05.0•10 < N m uon > < N e l e c t r on > < N pho t on > Figure 2: Longitudinal distribution of photons, electrons, muons, and hadrons for EASs of iron primary with E = . eV and θ = ◦ (left panels) and 45 ◦ (right panels). Lines represent the averages of 50 simulations, and error bars markthe standard deviations. For clarity, only the error bars of CORSIKA results are shown. < X m a x > [ g / c m ] ProtonIron H.Wahlberg et alCOSMOSCORSIKA σ X m a x [ g / c m ] ProtonIron COSMOSCORSIKA
Figure 3: Average of shower maximum, h X max i (upper panel), and standard deviation, σ X max (lower panel,) as a functionof primary energy. Lines are the least chi-square fits of the values in Table 2, which were calculated for 250 simulationsfor all zenith angles. The result reported in [38] is included for comparison. -4 -3 -2 -1 0 1 2 3 4 5 E d N / d E [ pe r s ho w e r ] log Energy [GeV] photonelectronmuon hadron CORSIKACOSMOS
Figure 4: Kinetic energy distribution of photons, electrons, muons, and hadrons at the ground for EASs of iron primarywith E = . eV and θ = ◦ . Solid lines are CORSIKA results and dashied lines are COSMOS. Violet, blue, green andred colors indicate photons, electrons, muons and hadrons, respectively. Lines are the averages of 50 simulations, anderror bars mark the standard deviations. For clarity, only the error bars of CORSIKA results are shown. -4 -3 -2 -1 0 1 2 3 4 5 E d N / d E [ pe r s ho w e r ] log Energy [GeV] pionkaonnucleon CORSIKACOSMOS
Figure 5: Kinetic energy distribution of nucleons, pions, and kaons at the ground for EASs of iron primary with E = . eV and θ = ◦ . Solid lines are CORSIKA results and dashied lines are COSMOS. Blue, red and green colorsindicate nucleons, pions and kaons, respectively. Lines are the averages of 50 simulations, and error bars mark thestandard deviations. For clarity, only the error bars of CORSIKA results are shown. tmospheric slant depth [g/cm ]0 200 400 600 8000 4.0•10 E a i r [ G e V ] CORSIKACOSMOS θ = 0 o E a i r [ G e V ] θ = 31.75 o E a i r [ G e V ] θ = 45 o E a i r [ G e V ] θ = 70 o Figure 6: Energy deposited to the air by the particles with energy above the threshold energy as a function of slantatmospheric depth, x s , for EASs of proton primary with E = . eV. Shown are for θ = ◦ , θ = ◦ , θ = ◦ , and θ = ◦ from top to bottom. Lines are the averages of 50 simulations. / θ simulation photons electrons muons hadronsproton CORSIKA 1.19e +
11 1.93e +
10 1.61e +
08 8.07e + ◦ COSMOS 1.19e +
11 2.00e +
10 1.54e +
08 8.36e + +
11 1.92e +
10 1.42e +
08 7.20e + ◦ COSMOS 1.22e +
11 2.04e +
10 1.36e +
08 7.30e + +
11 1.97e +
10 2.10e +
08 1.02e + ◦ COSMOS 1.21e +
11 2.04e +
10 1.96e +
08 1.02e + +
11 1.97e +
10 1.83e +
08 8.97e + ◦ COSMOS 1.23e +
11 2.05e +
10 1.74e +
08 9.71e + Table 1: Number of particles at peak for proton and iron primaries with E = . eV and θ = ◦ and 45 ◦ . Averages of50 simulations are listed. Depth of shower maximum, X max (units: g / cm )primary log E (eV) 18.5 18.75 19 19.25 19.5 19.75 20CORSIKA h X max i σ X max h X max i σ X max h X max i σ X max h X max i σ X max Table 2: Average and standard deviation of X max , which were calculated for 250 simulations for all zenith angles. θ log E (eV) simulation photons electrons muons hadrons all18.5 CORSIKA 6.56e +
08 3.71e +
08 1.26e +
08 8.75e +
07 1.24e + +
08 3.46e +
08 1.45e +
08 9.07e +
07 1.20e + ◦
19 CORSIKA 2.29e +
09 1.32e +
09 3.51e +
08 2.70e +
08 4.24e + +
09 1.26e +
09 3.96e +
08 2.73e +
08 4.11e + +
09 4.68e +
09 1.01e +
09 8.44e +
08 1.44e + +
09 4.35e +
09 1.11e +
09 7.83e +
08 1.36e + +
07 4.20e +
07 1.33e +
08 1.32e +
07 2.85e + +
07 3.89e +
07 1.59e +
08 1.56e +
07 3.01e + ◦
19 CORSIKA 4.04e +
08 1.88e +
08 3.89e +
08 4.74e +
07 1.03e + +
08 1.58e +
08 4.32e +
08 4.46e +
07 9.83e + +
09 7.31e +
08 1.13e +
09 1.47e +
08 3.55e + +
09 5.02e +
08 1.24e +
09 1.21e +
08 2.95e + Table 3: Total kinetic energy of particles reaching the ground, for proton primary. Averages of 50 simulations are listed.
Kinetic energies of particles at the ground for iron primary (units: GeV) θ log E (eV) simulation photons electrons muons hadrons all18.5 CORSIKA 4.52e +
08 2.38e +
08 1.89e +
08 9.54e +
07 9.75e + +
08 2.23e +
08 1.99e +
08 9.24e +
07 9.40e + ◦
19 CORSIKA 1.59e +
09 8.54e +
08 5.25e +
08 2.94e +
08 3.27e + +
09 8.81e +
08 5.43e +
08 2.94e +
08 3.35e + +
09 3.12e +
09 1.45e +
09 9.04e +
08 1.12e + +
09 2.89e +
09 1.54e +
09 9.14e +
08 1.06e + +
07 2.61e +
07 2.01e +
08 1.40e +
07 3.01e + +
07 2.25e +
07 2.14e +
08 1.36e +
07 3.02e + ◦
19 CORSIKA 2.26e +
08 9.96e +
07 5.62e +
08 4.59e +
07 9.34e + +
08 8.87e +
07 5.89e +
08 4.34e +
07 9.24e + +
08 3.70e +
08 1.57e +
09 1.44e +
08 2.92e + +
08 2.90e +
08 1.68e +
09 1.21e +
08 2.76e + Table 4: Total kinetic energy of particles reaching the ground, for iron primary. Averages of 50 simulations are listed. θ log E (eV) simulation photons electrons muons hadrons all18.5 CORSIKA 1.12e +
10 1.72e +
09 1.89e +
07 8.50e +
06 1.30e + +
10 1.74e +
09 1.94e +
07 8.83e +
06 1.27e + ◦
19 CORSIKA 3.67e +
10 5.69e +
09 5.47e +
07 2.56e +
07 4.25e + +
10 5.72e +
09 5.45e +
07 2.20e +
07 4.14e + +
11 1.83e +
10 1.61e +
08 7.70e +
07 1.35e + +
11 1.86e +
10 1.53e +
08 7.92e +
07 1.34e + +
09 3.71e +
08 1.29e +
07 3.34e +
06 3.35e + +
09 3.60e +
08 1.41e +
07 4.19e +
06 3.06e + ◦
19 CORSIKA 1.14e +
10 1.49e +
09 3.92e +
07 1.08e +
07 1.29e + +
10 1.40e +
09 3.99e +
07 1.23e +
07 1.18e + +
10 5.57e +
09 1.19e +
08 3.48e +
07 4.77e + +
10 4.35e +
09 1.11e +
08 3.56e +
07 3.66e + Table 5: Total number of particles reaching the ground, for proton primary. Averages of 50 simulations are listed.
Numbers of particles at the ground for iron primary θ log E (eV) simulation photons electrons muons hadrons all18.5 CORSIKA 9.33e +
09 1.37e +
09 2.54e +
07 9.96e +
06 1.07e + +
09 1.38e +
09 2.39e +
07 1.01e +
07 1.05e + ◦
19 CORSIKA 3.15e +
10 4.68e +
09 7.28e +
07 2.97e +
07 3.63e + +
10 4.95e +
09 6.92e +
07 3.19e +
07 3.70e + +
11 1.60e +
10 2.10e +
08 8.97e +
07 1.23e + +
11 1.59e +
10 1.96e +
08 9.02e +
07 1.19e + +
09 2.35e +
08 1.71e +
07 3.60e +
06 2.12e + +
09 2.17e +
08 1.67e +
07 3.96e +
06 1.88e + ◦
19 CORSIKA 6.95e +
09 8.81e +
08 5.01e +
07 1.13e +
07 7.89e + +
09 8.51e +
08 4.85e +
07 1.21e +
07 7.33e + +
10 3.24e +
09 1.45e +
08 3.46e +
07 2.89e + +
10 2.80e +
09 1.38e +
08 3.60e +
07 2.41e + Table 6: Total number of particles reaching the ground, for iron primary. Averages of 50 simulations are listed. E (eV) 18.5 18.75 19 19.25 19.5 19.75 20CORSIKA h E air i / E σ E air / h E air i h E air i / E σ E air / h E air i h E air i / E σ E air / h E air i h E air i / E σ E air / h E air i Table 7: Average of fraction and relative standard deviation of E air , the energy deposited to the air by the particles withenergy above the threshold energy, in EASs of proton and iron primaries at the ground for θ = ◦ , which correspondsto x s = (875 / cos 70 ◦ ) g / cm . The average and standard deviation were calculated for 50 simulations for each set ofparameters.. The average and standard deviation were calculated for 50 simulations for each set ofparameters.