Reconstruction of Longitudinal Profiles of Ultra-High Energy Cosmic Ray Showers from Fluorescence and Cherenkov Light Measurements
aa r X i v : . [ a s t r o - ph ] J a n Reconstruction of Longitudinal Profiles of Ultra-High Energy Cosmic RayShowers from Fluorescence and Cherenkov Light Measurements
M. Unger a , ∗ , B.R. Dawson b , R. Engel a , F. Sch¨ussler a and R. Ulrich a a Institut f¨ur Kernphysik, Forschungszentrum Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany b Department of Physics, University of Adelaide, Adelaide 5005, Australia
Abstract
We present a new method for the reconstruction of the longitudinal profile of extensive air showers induced by ultra-high energycosmic rays. In contrast to the typically considered shower size profile, this method employs directly the ionization energy depositof the shower particles in the atmosphere. Due to universality of the energy spectra of electrons and positrons, both fluorescenceand Cherenkov light can be used simultaneously as signal to infer the shower profile from the detected light. The method isbased on an analytic least-square solution for the estimation of the shower profile from the observed light signal. Furthermore, theextrapolation of the observed part of the profile with a Gaisser-Hillas function is discussed and the total statistical uncertainty ofshower parameters like total energy and shower maximum is calculated.
Key words:
Cosmic rays, extensive air showers, air shower reconstruction, air fluorescence, Cherenkov light
PACS:
1. Introduction
The particles of an extensive air shower excite nitrogenmolecules in the atmosphere, which subsequently radiateultraviolet fluorescence light isotropically. This fluorescencelight signal can be measured with appropriate optical de-tectors such as the fluorescence telescopes of HiRes [1], thePierre Auger Observatory [2] or the Telescope Array [3].The number of emitted fluorescence photons is expectedto be proportional to the energy deposited by the showerparticles. Recent measurements of the fluorescence yield inthe laboratory confirm this expectation within the experi-mental uncertainties [4,5,6]. Non-radiative processes of ni-trogen molecule de-excitation lead to a temperature, pres-sure and humidity dependence of the fluorescence yield (seee.g. [7]). For atmospheric parameters of relevance to thereconstruction of air showers of ultra-high energy cosmicrays, the pressure dependence of the ionization energy de-posit per meter track length of a charged particle is almostperfectly canceled by the pressure dependence of the fluo-rescence yield (see, for example, [8]). Therefore only a weakpressure and temperature dependence has to be taken intoaccount if the number of emitted photons is converted toa number of charged particle times track length, as has ∗ corresponding author, [email protected] been done in the pioneering Fly’s Eye experiment [9]. Thereconstructed longitudinal shower profile is then given bythe number of charged particles as function of atmosphericdepth.The approximation of assuming a certain number of flu-orescence photons per meter of charged particle track andthe corresponding expression of the longitudinal showerdevelopment in terms of shower size are characterized bya number of conceptual shortcomings. Firstly the energyspectrum of particles in an air shower changes in the courseof its development. A different rate of fluorescence pho-tons per charged particle has to be assumed for early andlate stages of shower development as the ionization en-ergy deposit depends on the particle energy [10]. Secondlythe tracks of low-energy particles are not parallel to theshower axis leading to another correction that has to be ap-plied [11]. Thirdly the quantity “shower size” is not suitedto a precise comparison of measurements with theoreticalpredictions. In air shower simulations, shower size is de-fined as the number of charged particles above a given en-ergy threshold E cut that cross a plane perpendicular to theshower axis. Setting this threshold very low to calculatethe shower size with an accuracy of ∼
1% leads to verylarge simulation times as the number of photons divergesfor E cut →
0. Moreover, the shower size reconstructed fromdata depends on simulations itself since the shower size is
Preprint submitted to Elsevier 28 January 2008 ot directly related to the fluorescence light signal.These conceptual problems can be avoided by directlyusing energy deposit as the primary quantity for showerprofile reconstruction as well as comparing experimentaldata with theoretical predictions. Due to the proportional-ity of the number of fluorescence photons to the energy de-posit, shower simulations are not needed to reconstruct thetotal energy deposit at a given depth in the atmosphere.Another advantage is that the calorimetric energy of theshower is directly given by the integral of the energy de-posit profile [12]. Furthermore the energy deposit profileis a well-defined quantity that can be calculated straight-forwardly in Monte Carlo simulations and does not dependon the simulation threshold [13].Most of the charged shower particles travel fasterthan the speed of light in air, leading to the emission ofCherenkov light. Thus, in general, the optical signal of anair shower consists of both fluorescence and Cherenkovlight contributions. In the traditional method [9] for thereconstruction of the longitudinal shower development, theCherenkov light is iteratively subtracted from the measuredtotal light. The drawbacks of this method are the lack ofconvergence for events with a large amount of Cherenkovlight and the difficulty of propagating the uncertainty ofthe subtracted signal to the reconstructed shower profile.An alternative procedure, used in [14], is to assume afunctional form for the longitudinal development of theshower, calculate the corresponding light emission and varythe parameters of the shower curve until a satisfactoryagreement with the observed light at the detector is ob-tained. Whereas in this scheme the convergence problemsof the aforementioned method are avoided, its major dis-advantage is that it can only be used if the showers indeedfollow the functional form assumed in the minimization.It has been noted in [15] that, due to the universality ofthe energy spectra of the secondary electrons and positronswithin an air shower, there exists a non-iterative solutionfor the reconstruction of a longitudinal shower profile fromlight detected by fluorescence telescopes.Here we will present an analytic least-square solution forthe estimation of the longitudinal energy deposit profile ofair showers from the observed light signal, in which bothfluorescence and Cherenkov light contributions are treatedas signal. We will also discuss the calculation of the statis-tical uncertainty of the shower profile, including bin-to-bincorrelations. Finally we will introduce a constrained fit tothe detected shower profile for extrapolating it to the re-gions outside the field of view of the fluorescence telescope.This constrained fit allows us to always use the full set ofprofile function parameters independent of the quality ofthe detected shower profile.
2. Fluorescence and Cherenkov Light Signals
The non-scattered, i.e. directly observed fluorescencelight emitted at a certain slant depth X i is measured at the detector at a time t i . Given the fluorescence yield Y f i [4,16,17,5] at this point of the atmosphere, the number ofphotons produced at the shower in a slant depth interval∆ X i is N f γ ( X i ) = Y f i w i ∆ X i . (1)Here, w i denotes the energy deposited per unit depth atslant depth X i (cf. Fig. 1) and is defined as w i = 1∆ X i Z π d ϕ Z ∞ r d r Z ∆ z i d z d E dep d V , (2)where d E dep / d V is the energy deposit per unit volume and( ϕ, R, z ) are cylinder coordinates with the shower axis at R = 0. The distance interval ∆ z i along the shower axisis given by the slant depth interval ∆ X i . The fluorescenceyield Y f i is the number of photons expected per unit de-posited energy for the atmospheric pressure and tempera-ture at slant depth X i . The photons from Eq. (1) are dis-tributed over a sphere with surface 4 π r i , where r i denotesthe distance of the detector. Due to atmospheric attenu-ation only a fraction T i of them reach the detector aper-ture with area A . Given a light detection efficiency of ε , themeasured fluorescence light flux y f i can be written as y f i = d i Y f i w i ∆ X i , (3)where the abbreviation d i = ε T i A π r i is used. For the sakeof clarity the wavelength dependence of Y , T and ε will bedisregarded in the following, but discussed later.The number of Cherenkov photons emitted at the showeris proportional to the number of charged particles abovethe Cherenkov threshold energy. Since the electromagneticcomponent dominates the shower development, the emittedCherenkov light, N C γ , can be calculated from N C γ ( X i ) = Y C i N e i ∆ X i , (4)where N e i denotes the number of electrons and positronsabove a certain energy cutoff, which is constant over the fullshower track and not to be confused with the Cherenkovemission energy threshold. Details of the Cherenkovlight production like these thresholds are included in theCherenkov yield factor Y C i [15,18,19,20].Although Cherenkov photons are emitted in a narrowcone along the particle direction, they cover a consider-able angular range with respect to the shower axis, becausethe charged particles are deflected from the primary par-ticle direction due to multiple scattering. Given the frac-tion f C ( β i ) of Cherenkov photons per solid angle emittedat an angle β i with respect to the shower axis [18,20], thelight flux at the detector aperture originating from directCherenkov light is y Cd i = d i f C ( β i ) Y C i ∆ X i N e i . (5)Due to the forward peaked nature of Cherenkov light pro-duction, an intense Cherenkov light beam builds up alongthe shower as it traverses the atmosphere (cf. Fig. 1). Ifa fraction f s ( β i ) of the beam is scattered towards the ob-server it can contribute significantly to the total light re-ceived at the detector. In a simple one-dimensional model2 i r i T ji εβ i X i T i direct Cherenkovscattered Cherenkov T i t i r i β i ε i X i w s ho w e r a x i s direct fluorescence Fig. 1. Illustration of the isotropic fluorescence light emission (solid circles), Cherenkov beam along the shower axis (dashed arcs) and thedirect (dashed lines) and scattered (dotted lines) Cherenkov light contributions. the number of photons in the beam at depth X i is just thesum of Cherenkov light produced at all previous depths X j attenuated on the way from X j to X i by T ji : N beam γ ( X i ) = i X j =0 T ji Y C j ∆ X j N e j . (6)Similar to the direct contributions, the scattered Cherenkovlight received at the detector is then y Cs i = d i f s ( β i ) i X j =0 T ji Y C j ∆ X j N e j . (7)Finally, the total light received at the detector at the time t i is obtained by adding the scattered and direct light con-tributions: y i = y f i + y Cd i + y Cs i . (8)
3. Analytic Shower Profile Reconstruction
The aim of the profile reconstruction is to estimate theenergy deposit and/or electron profile from the light fluxobserved at the detector. At first glance this seems to behopeless, since at each depth there are the two unknownvariables w i and N e i , and only one measured quantity,namely y i . Since the total energy deposit is just the sumof the energy loss of electrons, w i and N e i are related via w i = N e i Z ∞ f e ( E, X i ) w e ( E ) d E, (9)where f e ( E, X i ) denotes the normalized electron energydistribution and w e ( E ) is the energy loss per unit depth of asingle electron with energy E . As is shown in [15,19,20], theelectron energy spectrum f e ( E, X i ) is universal in showerage s i = 3 / (1 + 2 X max /X i ), i.e. it does not depend on theprimary mass or energy, but only on the relative distance tothe shower maximum, X max . Eq. (9) can thus be simplifiedto w i = N e i α i . (10) where α i is the average energy deposit per unit depth perelectron at shower age s i . Parameterizations of α i can befound in [10,20]. With this one-to-one relation (Eq. 10)between the energy deposit and the number of electrons,the shower profile is readily calculable from the equationsgiven in the last section. For the solution of the problem,it is convenient to rewrite the relation between energy de-posit and light at the detector in matrix notation: Let y =( y , y , . . . , y n ) T be the n -component vector (histogram)of the measured photon flux at the aperture and w =( w , w , . . . , w n ) T the energy deposit vector at the showertrack. Using the expression y = Cw (11)the elements of the Cherenkov-fluorescence matrix C can befound by a comparison with the coefficients in equations (3),(5) and (7): C ij = , i < jc d i + c s ii , i = jc s ij , i > j, (12)where c d i = d i (cid:0) Y f i + f C ( β i ) Y C i /α i (cid:1) ∆ X i (13)and c s ij = d i f s ( β i ) T ji Y C j /α j ∆ X j . (14)The solution of Eq. (11) can be obtained by inversion, lead-ing to the energy deposit estimator b w : b w = C − y . (15)Due to the triangular structure of the Cherenkov-fluorescence matrix the inverse can be calculated quicklyeven for matrices with large dimension. As the matrix ele-ments in Eq. (12) are always ≥ C is never singular.The statistical uncertainties of b w are obtained by errorpropagation: V w = C − V y (cid:0) C T (cid:1) − . (16)3 ime slots [100 ns]
270 280 290 300 310 320 330 340 350 / n s ] d e t ec t e d li gh t [ pho t on s / m light at aperturetotal lightMie Cherenkovdirect CherenkovRayleigh Cherenkov (a) Light at aperture. ] slant depth [g/cm )] d E / d X [ P e V / ( g / c m generatedreconstructedGaisser-Hillas fit /Ndf= 81.19/83 χ (b) Energy deposit profileFig. 2. Example of a simulated 10 eVproton shower. It is interesting to note that even if the measurements y i are uncorrelated, i.e. their covariance matrix V y is diag-onal, the calculated energy loss values b w i are not. This isbecause the light observed during time interval i does notsolely originate from w i , but also receives a contributionfrom earlier shower parts w j , j < i , via the ’Cherenkovlight beam’.
4. Wavelength Dependence
Until now it has been assumed that the shower induceslight emission at a single wavelength λ . In reality, thefluorescence yield shows distinct emission peaks and thenumber of Cherenkov photons produced is proportional to λ . In addition the wavelength dependence of the detectorefficiency and the light transmission need to be taken intoaccount. Assuming that a binned wavelength distributionof the yields is available ( Y ik = R λ k +∆ λλ k − ∆ λ Y i ( λ ) d λ ), theabove considerations still hold when replacing c d i and c s ij in Eq. (12) by˜ c d i = ∆ X i X k d ik (cid:0) Y f ik + f C ( β i ) Y C ik /α i (cid:1) (17)and ˜ c s ij = ∆ X j X k d ik f s ( β i ) T jik Y C jk /α j , (18)where d ik = ε k T ik π r i . (19)The detector efficiency ε k and transmission coefficients T ik and T jik are evaluated at the wavelength λ k .
5. Validation with Air Shower Simulations
In order to test the performance of the reconstructionalgorithm we will use in the following simulated fluores-cence detector data. For this purpose we generated protonair showers with an energy of 10 eV with the CONEX [21] event generator. The resulting longitudinal charged parti-cle and energy deposit profiles were subsequently fed intothe atmosphere and detector simulation package [22] of thePierre Auger Observatory. The geometry and profile of theevents in this simulated data sample was then reconstructedwithin the Auger offline software framework [23].Only events satisfying basic quality selection criteria havebeen used in the analysis. In order to assure a good re-construction of the shower geometry, the angular lengthof the shower image on the camera was required to belarger than nine degrees. Moreover, we only selected eventswith at least one coincident surface detector tank (so-calledhybrid geometry reconstruction [24]). Furthermore we re-jected under-determined measured longitudinal profiles bydemanding an observed slant depth length of ≥
300 g cm − and a reconstructed shower maximum within the field ofview of the detector.An example of a simulated event is shown in Fig. 2, illus-trating that the shape of the light curve at the detectorcan differ considerably from the one of the energy depositprofile due to the scattered Cherenkov light detected atlate stages of the shower development. The reconstructedenergy deposit curve, however, shows on average a goodagreement with the generated profile.Since longitudinal air shower profiles exhibit similar shapeswhen transformed from slant depth X to shower age s (seefor instance [25]), a good test of the profile reconstructionperformance is to compare the average generated and re-constructed energy deposit profiles as a function of s nor-malized to the energy deposit at shower maximum. As canbe seen in Fig. (3), the difference between these averagesis ≤ ax ( d E / d X ) / ( d E / d X ) generated reconstructed shower age m ax ( d E / d X ) / ( d E / d X ) ∆ -0.04-0.0200.020.04 Fig. 3. Average generated and reconstructed energy deposit profiles.
6. Shower Age Dependence
Due to the age dependence of the electron spectra f e ( E, s i ), the Cherenkov yield factors Y C i and the averageelectron energy deposits α i depend on the depth of showermaximum, which is not known before the profile has beenreconstructed. Fortunately, these dependencies are small:In the age range of importance for the shower profile recon-struction ( s ∈ [0 . , . α varies by only a few percent [20]and Y C by less than 15% [15]. Therefore, a good estimateof α and Y C can be obtained by setting s = 1 over the fullprofile or by estimating X max from the position maximumof the detected light profile. After the shower profile hasbeen calculated with these estimates, X max can be deter-mined from the energy deposit profile and the profile canbe re-calculated with an updated Cherenkov-fluorescencematrix. The convergence of this procedure is shown inFig. 5. After only one iteration the X max (energy) differsby less than 0.1 g cm − (0.1%) from its asymptotic value.Note that age dependent effects of the lateral spread of theshower on the image seen at the detector [26,27], thoughnot discussed in detail here, have also been included in thesimulation and reconstruction.
7. Gaisser-Hillas Fit
A knowledge of the complete profile is required for thecalculation of the Cherenkov beam and the shower energy.If due to the limited field of view of the detector only a partof the profile is observed, an appropriate function for theextrapolation to unobserved depths is needed. A possiblechoice is the Gaisser-Hillas function [28] f GH ( X ) = w max (cid:18) X − X X max − X (cid:19) ( X max − X ) /λ e ( X max − X ) /λ , (20) number of iterations ] [ g / c m m ax X ∆ -4 -3 -2 -1 E / E ∆ -6 -5 -4 -3 -2 energy max X Fig. 5. X max and energy difference with respect to the tenth showerage iteration. which was found to give a good description of measuredlongitudinal profiles [29]. It has four free parameters: X max ,the depth where the shower reaches its maximum energydeposit w max and two shape parameters X and λ .The best set of Gaisser-Hillas parameters p can be ob-tained by minimizing the error weighted squared differencebetween the vector of function values f GH and b w , which is χ = [ b w − f ( p )] T V w − [ b w − f ( p )] . (21)This minimization works well if a large fraction of theshower has been observed below and above the shower max-imum. If this is not the case, or even worse, if the showermaximum is outside the field of view, the problem is under-determined, i.e. the experimental information is not suffi-cient to reconstruct all four Gaisser-Hillas parameters. Thiscomplication can be overcome by constraining X and λ totheir average values h X i and h λ i . The new minimizationfunction is then the modified χ χ = χ + ( X − h X i ) V X + ( λ − h λ i ) V λ , (22)where the variances of X and λ around their mean valuesare in the denominators.In this way, even if χ is not sensitive to X and λ , theminimization will still converge. On the other hand, if themeasurements have small statistical uncertainties and/orcover a wide range in depth, the minimization function isflexible enough to allow for shape parameters differing fromtheir mean values. These mean values can be determinedfrom air shower simulations or, preferably, from high qual-ity data profiles which can be reconstructed without con-straints.Eq. (22) can be easily extended to incorporate correlationsbetween X and λ and the energy dependence of their meanvalues. Air shower simulations indicate a small logarithmicenergy dependence of the latter ( ≤
25% and 5% per decadefor X and λ respectively [30]). In practice it is sufficient touse energy independent values determined at low energies,because at high energies the number of measured points5 herenkov fraction [%] g e n ) / E g e n - E r ec ( E -0.15-0.1-0.0500.050.10.15 Cherenkov fraction [%] g e n ) / E g e n - E r ec ( E -0.15-0.1-0.0500.050.10.15 (a) Energy. Cherenkov fraction [%] ] ) [ g / c m g e n m ax - X r ec m ax ( X -40-2002040 Cherenkov fraction [%] ] ) [ g / c m g e n m ax - X r ec m ax ( X -40-2002040 meanrms (b) X max .Fig. 4. Energy and X max reconstruction accuracy as function of the amount of detected Cherenkov light. is large and thus the constraints do not contribute signifi-cantly to the overall χ .The accuracy of the reconstructed energy, obtained byintegrating over the Gaisser-Hillas function (see below),and that of the depth of shower maximum, are displayedin Fig. 4 as a function of the relative amount of Cherenkovlight. Note that the good resolutions of ≈
7% and 20 g cm − are of course not a feature of the reconstruction methodalone, but depend strongly on the detector performanceand quality selection. The mean values of difference to thetrue shower parameters, however, which are close to zero forboth fluorescence and Cherenkov light dominated events,indicate that both light sources are equally suited to recon-struct the longitudinal development of air showers.A slight deterioration of the resolutions can be seen forevents with a very small Cherenkov contribution of <
8. Error Propagation
A realistic estimate of the statistical uncertainties of im-portant shower parameters is desired for many purposes,like data quality selection cuts, or the comparison betweenindependent measurements like the surface and fluores-cence detector measurements of the Pierre Auger Obser-vatory or the Telescope Array. The uncertainties of w max , X max , X and λ obtained after the minimization of Eq. (22),reflect only the statistical uncertainty of the light flux,which is why these errors will be referred to as ’flux uncer-tainties’ ( σ flux ) in the following. Additional uncertaintiesarise from the uncertainties on the shower geometry ( σ geo ),atmosphere ( σ atm ) and the correction for invisible energy( σ inv ). 8.1. Flux uncertainty of the calorimetric energy
Even with the flux uncertainties of the Gaisser-Hillasparameters it is not straightforward to calculate the fluxuncertainty of the calorimetric energy, which is given bythe integral over the energy deposit profile: E cal = Z ∞ f GH ( X ) d X . (23)To solve this integral one can substitute t = X − X λ and ξ = X max − X λ (24)in the Gaisser-Hillas function Eq. (20) to get f GH ( t ) = w max (cid:18) eξ (cid:19) ξ e − t t ξ , (25)which can be identified with a Gamma distribution. There-fore, the above integral is given by E cal = λ w max (cid:18) eξ (cid:19) ξ Γ( ξ + 1) , (26)where Γ denotes the Gamma-function. Thus, instead of do-ing a tedious error propagation to determine the statisticaluncertainty of E cal one can simply use it directly as a freeparameter in the fit instead of the conventional factor w max : f GH ( t ) = E cal λ Γ( ξ + 1) e − t t ξ (27)In this way, σ flux ( E cal ) is obtained directly from the χ + 1contour of Eq. (22).8.2. Geometric uncertainties
Due to the uncertainties on the shower geometry, thedistances r i to each shower point are only known withina limited precision and correspondingly the energy de-posit profile points are uncertain due to the transmissionfactors T ( r i ) and geometry factors 1 / (4 π r i ). Further-more, the uncertainty of the shower direction, especially6he zenith angle θ , affects the slant depth calculation via X slant = X vert / cos θ and thus X max . Finally, the amountof direct and scattered Cherenkov light depends on theshower geometry, too, via the angles β i .The algorithms used to reconstruct the shower geome-try from fluorescence detector data usually determine thefollowing five parameters [9], irrespective of whether thedetectors operate in monocular, stereo or hybrid mode: α = { θ SDP , Φ SDP , T , R p , χ } . (28) θ SDP and Φ
SDP are the angles of the normal vector of aplane spanned by the shower axis and the detector (the socalled shower-detector-plane), χ denotes the angle of theshower within this plane and T and R p are the time anddistance of the shower at its point of closest approach tothe detector.For any function q ( α ) standard error propagation yields thegeometric uncertainty σ ( q ) = X i =1 5 X j =1 d q d α i d q d α j V αij , (29)where V α denotes the covariance matrix of the axis param-eters. As the calorimetric energy and X max depend non-trivially on the shower geometry, the above derivatives needto be calculated numerically, i.e. by repeating the profilereconstruction and Gaisser-Hillas fitting for the ten newgeometries given by α i ± p V αii ≡ α i ± σ i to obtain∆ i = d q d α i σ i ≈
12 [ q ( α i + σ i ) − q ( α i − σ i ) ] , (30)with which Eq. (29) reads as σ ( q ) = X i =1 5 X j =1 ∆ i ∆ j ρ ij (31)where ρ ij = V αij p V αii V αjj (32)denote the correlation coefficients of the geometry param-eters α i and α j .8.3. Atmospheric uncertainties
Whereas the Rayleigh attenuation is a theoretically wellunderstood process, the molecular density profiles andaerosol content of the atmosphere vary due to environ-mental influences and need to be well monitored in orderto determine the slant depth and transmission coefficientsneeded for the profile reconstruction. Uncertainties inthese measured atmospheric properties (see for instance[31,32]) can be propagated in the same way as the geo-metric uncertainties by determining the one sigma showerparameter deviations via Eq. (30). ) rec (lg E σ )/ gen - lgE rec (lgE -4 -2 0 2 4 f r ac t i on o f eve n t s Mean -0.16RMS 1.07 (a) Energy. ) recmax (X σ )/ genmax -X recmax (X -4 -2 0 2 4 f r ac t i on o f eve n t s Mean -0.07RMS 0.88 (b) X max .Fig. 6. Pull distributions Invisible energy
Not all of the energy of a primary cosmic ray particle endsup in the electromagnetic part of an air shower. Neutrinosescape undetected and muons need long path lengths tofully release their energy. This is usually accounted for bymultiplying the calorimetric energy, Eq. (23), with a cor-rection factor f inv determined from shower simulations toobtain the total primary energy E tot = f inv E cal . (33)The meson decay probabilities, and thus the amount of neu-trino and muon production, decrease with energy, therefore f inv depends on the primary energy. For instance, in [33] itis parameterized as f inv = ( a + bE c cal ) − , (34)where a , b and c denote constants depending on the pri-mary composition and interaction model assumed . Thisenergy dependence needs to be taken into account whenpropagating the calorimetric energy uncertainty to the to-tal energy uncertainty.Due to the stochastic nature of air showers, the correc-tion factor is subject to shower-to-shower fluctuations. Thestatistical uncertainty of f inv was determined in [34] andcan be parameterized as follows: σ ( f inv ) ≈ . · · lg( E tot / eV) − . . (35)Typical values are 2.5% at 10 eV and 0.9% at 10 eV.8.5. Total statistical uncertainty
Summarizing the above considerations, the statisticalvariance of the total energy is σ ( E tot ) = E σ ( f inv )+ (cid:18) d f inv d E cal E cal + f inv (cid:19) X i σ i ( E cal ) , (36) Note that here only the statistical uncertainties of the invisibleenergy correction are discussed. For an estimate on the related sys-tematic uncertainties see [34] i runs over the geometric, atmospheric and flux un-certainties. Since the invisible energy correction does notaffect the depth of shower maximum, its uncertainty is sim-ply given by σ stat ( X max ) = sX i σ i ( X max ) . (37)Again we use simulated events to verify the validity of theabove considerations. The pull distributions of the recon-structed energy and shower maximum, shown in Fig. 37,both have a width of approximately one, which means thatthe total uncertainties from Eqs. (36) and (37) are goodestimators for the actual event-by-event measurement un-certainties.
9. Conclusions and Outlook
In this paper a new method for the reconstruction oflongitudinal air shower profiles was presented. With thehelp of simulations we have shown that the least squaresolution yields robust and unbiased results and that un-certainties of shower parameters can be reliably calculatedfor each event.Events with a large Cherenkov light contribution are cur-rently usually rejected during the data analysis (see forinstance [14,35]) However, as we have shown, there is nojustification for rejecting such showers, once experimen-tal systematic uncertainties are well understood. Becauseevents with a large Cherenkov contribution have differentsystematic uncertainties to those dominated by fluores-cence light, both event classes can be compared to studytheir compatibility.At energies below 10 . eV, where new projects [36,37]are planned to study the transition from galactic to ex-tragalactic cosmic rays, events with a large fraction ofdirect Cherenkov light will dominate the data samples,because the amount of light, and thus trigger probability,of these events is much larger than that of a fluorescencedominated shower. If at these energies it is still possible tomeasure an accurate shower geometry, the fluorescence de-tectors should in fact be used as Cherenkov-Fluorescencetelescopes.Acknowledgments The authors would like to thank theircolleagues from the Pierre Auger Collaboration, in partic-ular Frank Nerling and Tanguy Pierog, for fruitful discus-sions. References [1] T. Abu-Zayyad et al. [HiRes Collaboration], Nucl. Instrum.Meth. A450 (2000) 253.[2] J. 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