Barnhill, D

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29th International Cosmic Ray Conference Pune (2005) 7, 291294Measurement of the Lateral Distribution Function of UHECR AirShowers with the Pierre Auger ObservatoryD. Barnhill, P. Bauleo, M.T. Dova, J. Harton, R. Knapik, J. Knapp, J. Lee,M. Mancen˜ido, A.G. Mariazzi, I.C. Maris¸, D. Newton, M. Roth, T. Schmidt andA.A. Watson for the Pierre Auger CollaborationPierre Auger Observatory, Av San Mart·n Norte 304,(5613) Malarg¤ue, ArgentinaPresenter: P. Bauleo (bauleo@lamar.colostate.edu), usa-bauleo-PM-abs2-he14-poster1. IntroductionThe Pierre Auger Observatory [1] is being used to study cosmic rays with energies larger than 	withunprecedented precision and statistics. An essential quantity that must be deduced from data is the lateral dis-tribution function (LDF) that describes the decreasing of the signals in the water-tanks as a function of distance.Knowledge of the LDF is important for the reconstruction of the shower core and the shower direction. It canalso be compared with model calculations to give useful information relating to primary mass. Here we de-scribe how the LDF is measured using the large sample of events recorded with the surface detector (SD) arrayand with a small sample observed with the fluorescence detectors (FD). For hybrid events, in which SD and FDmeasurements of the same shower are available, the core position is much better constrained than for SD-onlyevents, thus providing an important cross-check on the LDF determined from SD measurements alone.2. The Fitting MethodThe water-Cherenkov detectors provide information about Cherenkov photons, which are produced whencharged particles cross the tanks. The number of Cherenkov photons collected is to a good approximationproportional to the energy deposit in the tank. The signal is calibrated in units of vertical equivalent muons(VEM) [9]. The energy deposit, however, depends strongly on the particle type and the conversion from theCherenkov signal back to the number of particles in the tank is not obvious. For large tank signals (  	)this is not crucial since the uncertainty  of a signal  (in VEM) was determined from data of two detectorspositioned 11 meters apart [5] to be  ﬁﬀ ﬁﬂﬃ . But for small tank signals the number of effectiveparticles,  , is needed since their Poissonian fluctuation dominates the uncertainty of the signal and is re-quired for the maximum likelihood fit. We have introduced a function that gives  for a measured signal  "!# :$&%'(!*),+-)/.')102 34 "!# where the conversion factor P is called the Poisson factor and is presently assumedto be independent of the primary energy, . , and mass, 0 , for any distance, ! , and zenith angle, + . The factorreflects the different energy deposits of different secondaries and is determined by simulations. Finally we setup a maximum likelihood fit to determine the parameters of a trial LDF functional form and, at the same time,the position of the shower core, by comparing each tank signal, with its fluctuations, to the value expectedfrom the trial function, 65(7 . 89;:4<ﬁ= P ( < ),> < 3:@?= G ( ? ),> ? 3A:CB*D sat " B ),> B E3:4F$D zero ( F )1> F ﬀTheindividual factors of the likelihood function are determined using the information of tanks at their respectivedistance ! . The Poissonian probability density, = P " < ),> <  , is calculated for small signals (  <G H	2E). Forlarger signals the Gaussian approximation is used, = G " ? )1> ?  . The effective particle number  of a saturatedtank represents a lower limit of the actual signal and we have to integrate = G over all possible values larger than , to estimate the detecting probability of a signal larger than  . In case of tanks without a signal we have tosum over all Poissonian probabilities with a predicted particle number > < and actual effective particle number<IKJ .292 D. Barnhill et al.lg(r/[m])2.5 2.6 2.7 2.8 2.9 3 3.1lg(S/S1000)-0.500.511.52lg(E/[eV]) < 18.018.0 < lg(E/[eV]) < 18.518.5 < lg(E/[eV]) < 19.019.0 < lg(E/[eV]) < 19.519.5 < lg(E/[eV])Golden Hybrid (fixed hybrid core)Figure 1. Averaged LDF for LNM,OQPSR4T UWV XHYUWV Z\[ (NKG fit andfloating slope).)-1θsec(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1β11.21.41.61.822.22.42.62.833.2 19.5 < lg(E/eV)19   < lg(E/eV) < 19.5 18.5 < lg(E/eV) < 19 18   < lg(E/eV) 18   < lg(E/eV)  (Hybrid)Figure 2. Slope ] vs zenith angle as function of energy forSD-only or hybrid geometry. Fits to the slope parameterfor different energy bins are shown3. LDF MeasurementsIn contrast to S(1000) the shape of the lateral distribution does not change much with energy [8]. Therefore, thenormalisation constant is decoupled from the shape parameter and showers of different energies are combined.The LDF was deduced from experimental data using SD-only and Hybrid events. Data from January 2004 toApril 2005 have been used for the following analysis. High-quality events have been selected, which had asuccessful directional reconstruction with + Gﬂﬁ_^, at least 6 stations with signal above detection threshold,and a core position well inside the SD array. Using SD-only events the following LDFs have been investigated:(i) a power law:  (!*`a  ﬁbc-"!bd Hbﬁ eWfg , with a + dependent index h@jiSkml*(no+qp  , (ii) aNKG-like function [6]:  (!#r;0mcQst(!bd!uc_ kv!bd!Huxwfzy with 0{&| ﬁb}cJﬀ ~_y , i`kl*(no+Sp and ! u *ﬁe(since  and ! u are strongly correlated, we have fixed ! u *beand left  to vary),and (iii) a function used by the Haverah Park experiment [3]: |"!#AE!-fŁ\11, , if ! Gbe, else11r!QfŁ\11"Ł with fixed  , the shape parameter  varying with zenith angle, and ! u ~Qbbe. Theseforms were fitted to individual events using a maximum likelihood fit of core location and LDF at the sametime (see section 2). Two SD-only analyses were performed. First, in a four-parameter fit, besides the corelocation x and y, the slope parameters h ,  and  , respectively, have been varied together with the scale factor| ﬁb . Then a parameterisation of h ,  and  as function of + was determined, which was then used ina second analysis fitting only x, y, and   ﬁb . Figure 1 shows the averaged LDF for no+;s bﬀ) bﬀ ~wwhen the NKG assumption is used. For comparison a hybrid derived average LDF is shown too (see belowfor details). An energy dependent threshold effect is apparent at large radii and reflects upward fluctuationsof signals close to the trigger threshold of single tanks. In case of this NKG-like function with a free slopeparameter,  , the fit results for . 	are given in Table 2. The energy dependence of  is shown inTable 1. Moments (mean,  ) of residual distribution of exp. data with various LDFs. Only events with $ stations were used in thepresent analysis. For the NKG-like LDF both SD and Hybrid moments are shown, denoted by “sd” and “hy” respectively. "¡£¢b¤ number of NKG power law Haverah Parkrange eventsyfree (hy)yfree (sd)yfixed (sd)gfreegfixedfreefixedhy sd m ¥ m ¥ m ¥ m ¥ m ¥ m ¥ m ¥¦\§¨W§ ©ª 5 367 0.27 2.12 0.04 0.48 -0.07 1.45 -0.03 0.55 -0.07 1.45 -0.17 1.27 -0.21 1.0¦\§ ©¨W§ «,ª 14 549 -0.18 1.71 0.06 0.53 -0.04 1.30 -0.27 0.81 -0.04 1.30 -0.03 0.95 0.14 1.2¦\§ «¨W§ ¬ª 17 624 -0.07 2.00 0.07 0.55 0.04 1.02 -0.12 0.65 0.05 1.02 -0.09 1.04 0.03 1.6¦\§ ¬¨W§ª 8 576 -0.04 1.40 0.09 0.59 -0.07 0.80 -0.07 0.81 -0.14 0.92 0.01 1.27 0.23 1.4¦\§­¨©­§ª 6 493 -0.26 1.34 0.11 0.61 -0.11 0.98 -0.11 0.98 -0.21 1.15 -0.21 1.28 -0.24 1.9Measurement of the LDF of UHECR Air Showers with the Pierre Auger Observatory 293Figure 2 and described by EN.S®i(.Sk¯l*NnHo+Sp  , with i(.qﬀ#ﬂkzﬀt H°ﬁ±²b³N.qdE­	 and b =-0.98. To quantify the quality of the fits the residuals, Ap´µ(¶b1d#µ(¶ , and their distributions are computed. Fora good LDF the residuals should scatter symmetrically around 0 with v . Means and standard deviationsof the residual distribution are used to compare different LDFs and are given in Table 1. For simplicity onlyresiduals up to 1500 m are taken into account, resulting in a variance smaller than the expectation value of 1 toavoid systematic biases of upward fluctuating signals. The NKG-like function fits the data best, which can beseen from the smallest mean residuals and the smallest residual variances.Complementarily to the SD analysis, a hybrid LDF analysis was performed. The hybrid reconstruction exploitsthe independent knowledge of the core position to determine the shower axis geometry and distance from eachdetector to the shower core. A maximum likelihood fit (section 2) is used to determine only S(1000) and theLDF slope parameter (  ). A NKG-like function was studied in the hybrid analysis.Hybrid triggers usually include a relatively large number of accidental stations, which in the case of low multi-plicity events could even outnumber the number of stations that are part of the event making the identificationof accidentals and candidate stations a difficult task. Therefore, strict quality cuts were imposed and only eventswith at least 6 triggered stations were included in the analysis. That reduces the sample size, but at the sametime, selecting events with a large number of active stations, prevents biasing the LDF slope due to signal fluctu-ations. Moreover, as the quality cuts imposedTable 2. Comparison between SD andHybrid analysis on the parameterisationof the LDF slope (NKG-like function;·m¸v¹Wº#»(¼¾½\¿ ).SD HybridIntercept À 2.24 Á 0.01 2.26 Á 0.17Slope Â -0.98 Á 0.02 -1.1 Á 0.3on the hybrid analysis are similar to those used on the SD-only analysisthe comparison is straightforward. Figure 2 shows also how  varieswith + for hybrid events. Despite the limited statistics of the hybridsample the agreement between SD and Hybrid is encouraging. TheHybrid data sub-sample is not large enough as to accurately quantifythe depencence of  with energy at the time of writing.4. Uncertainty in S(1000)High statistics are required to accurately describe the LDF, both to reduce the statistical and systematic uncer-tainties. Hybrid measurements (though with much reduced statistics) are a useful tool to help identify sourcesof systematic uncertainty, and it is certain that, as the Auger exposure increases, the functional form of the LDFwill evolve and become increasingly accurate. Increasing accuracy will lead to much smaller uncertainties inthe reconstructed core position, but the ground parameter,   ﬁb which is used to determine the energy ofthe primary CR, is very robust to innacuracies in the LDF. It can be shown that by measuring  "!# at 1000m, fluctuations in the ground parameter due to a lack of knowledge of the LDF are minimised. The result ofanalysing one SD event many times, whilst allowing the slope parameter to vary by Ã _Ä is shown in figure3. This was chosen as a reasonable value for the magnitude of the shower-to-shower fluctuations, based onmeasurements made at Haverah Park where the precision was sufficient to measure intrinsic shower-to-showerfluctuations [4]. Analysing the shower with different assumed values for the slope parameter, results in a shiftin the reconstructed core position, but by choosing to measure the ground parameter  (!# at the point wherethe LDFs intersect, (at Å ﬁbm), the effect of the changing slope parameter is minimised. At this pointthe ground parameter is independent of the LDF. Analysing many showers in this way shows that !ﬁÆ£Ç 5 , theoptimum groundparameter has very little dependence on zenith angle, energy, or the form of the LDF used to reconstruct theshowers. For example, an analysis of ÅbbAuger SD events with energies H-§ ÈeV G . G _,eV andzenith anglesQ^G+Gﬂﬁﬁ^gives a distribution of ! Æ£Ç 5 with mean 940 m and a rms of 110 m. An analysis ofsimulated events at ﬁ©eV gives a mean of 930 m and a rms deviation of 40 m. The prescence of a saturatedstation (predominantly in vertical, high energy events) tends to push !bÆ£Ç 5 out by several hundred metres, andafter an analysis of many showers, at different zenith angle and energy, S(1000) was chosen as a robust groundparameter to measure all showers at. At 1000 m from the core the mean uncertainty in  "!# (across all events)294 D. Barnhill et al.r/m0 1000 2000Signal / VEM10210Array Spacing / m1000 1200 1400 1600 1800 2000 / moptr700800900100011001200Figure 3. LDFs of an Auger SD event, analysed with different slope parameters. Red points indicate stations with zerosignal. The inset shows how the optimum ground parameter É­ÊxË\Ì varies with the array spacing.is minimised, and furthermore, for the few showers where !#Æ£Ç 5 lies far from 1000 m, the uncertainty in S(1000)can easily be estimated.5. Summary and OutlookThe lateral distribution function of EAS observed using the Auger Observatory has been derived. Differentfunctions have been tested and it is concluded that an NKG-like LDF describes the data well. The dependenceof the function on atmospheric depth has been described. It should be emphasized that the global showerobservables, like the lateral distribution of particles, are not affected by the geomagnetic field for zenith angles+G*Q^. However, for the case of very inclined showers which are dominated by muons, the density at groundis rendered quite asymmetric by the geomagnetic field and the LDF approach is not longer valid [2, 7].References[1] Abraham J. et al., (Auger Collaboration), Nucl. Instr. Meth. A 523 (2004) 50-95[2] Ave M. et al., Astropart. Phys. 14 (2000) 91[3] Coy R.N. et al., Astropart. Phys. 6 (1997) 263[4] Coy R.N. et al., Proc. 17 5(7 ICRC, Paris (1981) vol.6.[5] Ghia P. et al. for the Auger Collaboration, these proceedings (ita-ghia-P-abs1-he14-oral)[6] Greisen K, Progress in Cosmic Ray Physics 3 (1956) North Holland Publ.Kamata K and Nishimura J. Prog. Theoret. Phys. Suppl. 6 (1958) 93[7] Hillas A.M. et al., Proc. 11 5(7 ICRC, Budapest (1969), Acta Physica Academiae Scietiarum Hungaricae29, Suppl. 3, (1970) 533[8] Roth M. for the Auger Collaboration, Proc. 28 5(7 ICRC, Tsukuba (2003) 333.[9] Salazar H. for the Auger Collaboration, Proc. 27 5(7 ICRC, Hamburg (2001) 752.

Measurement of the Lateral Distribution Function of UHECR Air shower with the Pierre Auger Observatory

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