Reconstruction of Longitudinal Profiles of Ultra-High Energy Cosmic Ray Showers from Fluorescence and Cherenkov Light Measurements

0.32 MB·English

by  M. Unger

Reconstructionof LongitudinalProfilesofUltra-HighEnergy CosmicRay
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0 Showers fromFluorescence andCherenkov Light Measurements
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2
M. Ungera,∗,B.R.Dawsonb,R.Engela,F.Schu¨sslera andR.Ulricha

n
aInstitut fu¨rKernphysik, Forschungszentrum Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany
a
J bDepartment of Physics, Universityof Adelaide, Adelaide 5005, Australia

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2

]Abstract
h
pWe present a new method for thereconstruction of the longitudinal profile of extensiveair showers induced by ultra-high energy
-cosmicrays.Incontrasttothetypicallyconsideredshowersizeprofile,thismethodemploysdirectlytheionization energydeposit
o
rof the shower particles in the atmosphere. Dueto universality of theenergy spectra of electrons and positrons, both fluorescence
tand Cherenkov light can be used simultaneously as signal to infer the shower profile from the detected light. The method is
s
abasedonananalyticleast-squaresolutionfortheestimationoftheshowerprofilefromtheobservedlightsignal.Furthermore,the
[extrapolation oftheobservedpartof theprofilewith aGaisser-Hillas function isdiscussed andthetotal statistical uncertaintyof

shower parameters liketotal energyand shower maximumis calculated.
1
v
9
Keywords: Cosmicrays, extensive air showers,air shower reconstruction, airfluorescence, Cherenkov light
0
PACS: 96.50.sd
3
4
.
1
01. Introduction beendone inthe pioneeringFly’s Eyeexperiment[9].The
8 reconstructed longitudinal shower profile is then given by
0 The particles of an extensive air shower excite nitrogen thenumberofchargedparticlesasfunctionofatmospheric
:
vmolecules in the atmosphere, which subsequently radiate depth.
iultravioletfluorescencelightisotropically.Thisfluorescence Theapproximationofassumingacertainnumberofflu-
X
light signal can be measured with appropriate optical de- orescencephotons per meter of chargedparticle trackand
r
atectorssuchasthefluorescencetelescopesofHiRes[1],the the corresponding expression of the longitudinal shower
PierreAuger Observatory[2]orthe TelescopeArray[3]. development in terms of shower size are characterized by
Thenumberofemittedfluorescencephotonsisexpected a number of conceptual shortcomings. Firstly the energy
to be proportional to the energy deposited by the shower spectrumofparticlesinanairshowerchangesinthecourse
particles.Recentmeasurementsofthefluorescenceyieldin of its development. A different rate of fluorescence pho-
the laboratoryconfirmthis expectationwithinthe experi- tons per charged particle has to be assumed for early and
mentaluncertainties[4,5,6].Non-radiativeprocessesofni- late stages of shower development as the ionization en-
trogenmoleculede-excitationleadtoatemperature,pres- ergydepositdepends onthe particleenergy[10].Secondly
sureandhumiditydependenceofthefluorescenceyield(see the tracks of low-energy particles are not parallel to the
e.g. [7]). For atmospheric parameters of relevance to the showeraxisleadingtoanothercorrectionthathastobeap-
reconstruction of air showers of ultra-high energy cosmic plied[11].Thirdlythe quantity“showersize”isnotsuited
rays,the pressuredependence ofthe ionizationenergy de- to a precise comparison of measurements with theoretical
positpermetertracklengthofachargedparticleisalmost predictions. In air shower simulations, shower size is de-
perfectly canceledby the pressure dependence of the fluo- finedas the number ofchargedparticlesabovea givenen-
rescenceyield(see,forexample,[8]).Thereforeonlyaweak ergythresholdEcut thatcrossaplaneperpendiculartothe
pressureandtemperaturedependencehastobetakeninto shower axis. Setting this threshold very low to calculate
account if the number of emitted photons is converted to the shower size with an accuracy of ∼ 1% leads to very
a number of charged particle times track length, as has large simulation times as the number of photons diverges
forE →0.Moreover,theshowersizereconstructedfrom
cut
∗ correspondingauthor, [email protected] data depends on simulations itself since the shower size is
Preprintsubmitted toElsevier 28 January 2008
notdirectly relatedtothe fluorescencelightsignal. the detector at a time t . Given the fluorescence yield Yf
i i
These conceptual problems can be avoided by directly [4,16,17,5] at this point of the atmosphere, the number of
using energy deposit as the primary quantity for shower photons produced at the shower in a slant depth interval
profile reconstruction as well as comparing experimental ∆X is
i
datawiththeoreticalpredictions.Duetotheproportional- Nf(X )=Yfw ∆X . (1)
γ i i i i
ityofthenumberoffluorescencephotonstotheenergyde- Here, w denotes the energy deposited per unit depth at
i
posit,showersimulationsarenotneededtoreconstructthe slantdepth X (cf.Fig.1)andis definedas
i
total energy deposit at a given depth in the atmosphere.
1 2π ∞ dE
Another advantage is that the calorimetric energy of the w = dϕ rdr dz dep, (2)
i
shower is directly given by the integral of the energy de- ∆Xi Z0 Z0 Z∆zi dV
posit profile [12]. Furthermore the energy deposit profile wheredE /dV istheenergydepositperunitvolumeand
dep
is a well-defined quantity that can be calculated straight- (ϕ,R,z) are cylinder coordinates with the shower axis at
forwardlyinMonteCarlosimulationsanddoesnotdepend R = 0. The distance interval ∆z along the shower axis
i
onthe simulationthreshold[13]. is givenby the slantdepth interval∆X . The fluorescence
i
Most of the charged shower particles travel faster yield Yf is the number of photons expected per unit de-
i
than the speed of light in air, leading to the emission of posited energy for the atmospheric pressureand tempera-
Cherenkov light. Thus, in general, the optical signal of an ture at slant depth X . The photons from Eq. (1) are dis-
i
air shower consists of both fluorescence and Cherenkov tributedoveraspherewithsurface4πr2,wherer denotes
i i
light contributions. In the traditional method [9] for the the distance of the detector. Due to atmospheric attenu-
reconstructionofthelongitudinalshowerdevelopment,the ation only a fraction T of them reach the detector aper-
i
Cherenkovlightisiterativelysubtractedfromthemeasured turewithareaA.Givenalightdetectionefficiencyofε,the
total light. The drawbacks of this method are the lack of measuredfluorescencelightflux yf canbe writtenas
i
convergence for events with a large amount of Cherenkov
yf =d Yfw ∆X , (3)
light and the difficulty of propagating the uncertainty of i i i i i
the subtractedsignaltothe reconstructedshowerprofile. wheretheabbreviationd =εT A isused.Forthesake
An alternative procedure, used in [14], is to assume a ofclaritythewavelengthidependie4nπcrei2ofY,T andεwillbe
functional form for the longitudinal development of the disregardedinthe following,butdiscussedlater.
shower,calculatethecorrespondinglightemissionandvary The number of Cherenkov photons emitted at the shower
the parameters of the shower curve until a satisfactory is proportional to the number of charged particles above
agreement with the observed light at the detector is ob- theCherenkovthresholdenergy.Sincetheelectromagnetic
tained. Whereas in this scheme the convergence problems componentdominatestheshowerdevelopment,theemitted
of the aforementioned method are avoided, its major dis- Cherenkovlight,NC,canbe calculatedfrom
γ
advantageisthatitcanonlybe usediftheshowersindeed
followthe functionalformassumedinthe minimization. NγC(Xi)=YiCNie∆Xi, (4)
It hasbeennotedin [15]that,due tothe universalityof where Ne denotes the number of electrons and positrons
i
theenergyspectraofthesecondaryelectronsandpositrons aboveacertainenergycutoff,whichisconstantoverthefull
within an air shower, there exists a non-iterative solution shower track and not to be confused with the Cherenkov
forthereconstructionofalongitudinalshowerprofilefrom emission energy threshold. Details of the Cherenkov
lightdetectedby fluorescencetelescopes. light production like these thresholds are included in the
Herewewillpresentananalyticleast-squaresolutionfor CherenkovyieldfactorYC [15,18,19,20].
i
theestimationofthe longitudinalenergydepositprofileof Although Cherenkov photons are emitted in a narrow
air showers from the observed light signal, in which both cone along the particle direction, they cover a consider-
fluorescenceandCherenkovlightcontributionsaretreated ableangularrangewithrespecttotheshoweraxis,because
assignal.Wewillalsodiscussthecalculationofthestatis- the charged particles are deflected from the primary par-
ticaluncertaintyoftheshowerprofile,includingbin-to-bin ticle direction due to multiple scattering. Given the frac-
correlations.Finally we will introduce a constrained fit to tion f (β ) of Cherenkov photons per solid angle emitted
C i
the detected shower profile for extrapolating it to the re- at an angle β with respect to the shower axis [18,20], the
i
gionsoutsidethefieldofviewofthefluorescencetelescope. light flux at the detector aperture originating from direct
This constrained fit allows us to always use the full set of Cherenkovlightis
profile function parameters independent of the quality of
yCd =d f (β )YC∆X Ne. (5)
the detected showerprofile. i i C i i i i
Due to the forwardpeakednature of Cherenkovlightpro-
duction, an intense Cherenkovlight beam builds up along
2. Fluorescence andCherenkov LightSignals
the shower as it traverses the atmosphere (cf. Fig. 1). If
a fraction f (β ) of the beam is scattered towards the ob-
s i
The non-scattered, i.e. directly observed fluorescence server it can contribute significantly to the total light re-
light emitted at a certain slant depth X is measured at ceived at the detector. In a simple one-dimensional model
i
2
direct fluorescence
w
X i direct Cherenkov
i
scattered Cherenkov
T
i
β
i
r T
i ε ji T ε
t i
t
i i
s X
h
ow i β
er a i ri
xi
s
Fig. 1. Illustration of the isotropic fluorescence light emission (solid circles), Cherenkov beam along the shower axis (dashed arcs) and the
direct (dashed lines)and scattered (dotted lines)Cherenkov lightcontributions.
thenumberofphotonsinthebeamatdepthX isjustthe where α is the averageenergy deposit per unit depth per
i i
sumofCherenkovlightproducedatallpreviousdepthsX electron at shower age s . Parameterizations of α can be
j i i
attenuatedonthe wayfromX to X by T : found in [10,20]. With this one-to-one relation (Eq. 10)
j i ji
between the energy deposit and the number of electrons,
i
Nbeam(X )= T YC∆X Ne. (6) the shower profile is readily calculable from the equations
γ i ji j j j
given in the last section. For the solution of the problem,
j=0
X
it is convenient to rewrite the relationbetween energy de-
Similartothedirectcontributions,thescatteredCherenkov
positandlightatthedetectorinmatrixnotation:Lety=
lightreceivedatthe detector isthen
(y ,y ,...,y )T be the n-component vector (histogram)
1 2 n
i
of the measured photon flux at the aperture and w =
yCs =d f (β ) T YC∆X Ne. (7)
i i s i ji j j j (w ,w ,...,w )T the energy deposit vector at the shower
1 2 n
j=0
X track.Using the expression
Finally,the totallightreceivedatthe detectoratthe time
t isobtainedbyaddingthescatteredanddirectlightcon- y=Cw (11)
i
tributions:
theelementsoftheCherenkov-fluorescencematrixCcanbe
y =yf +yCd+yCs. (8)
i i i i foundbyacomparisonwiththecoefficientsinequations(3),
(5)and(7):
3. Analytic ShowerProfile Reconstruction 0, i<j
C = cd+cs, i=j (12)
ij  i ii
The aim of the profile reconstruction is to estimate the cs , i>j,
ij
energy deposit and/or electron profile from the light flux
observed at the detector. At first glance this seems to be where 
hopeless, since at each depth there are the two unknown cdi =di Yif +fC(βi)YiC/αi ∆Xi (13)
variables wi and Nie, and only one measured quantity, and (cid:0) (cid:1)
namely yi. Since the total energy deposit is just the sum cs =d f (β )T YC/α ∆X . (14)
ofthe energylossofelectrons,w andNe arerelatedvia ij i s i ji j j j
i i
∞ ThesolutionofEq.(11)canbeobtainedbyinversion,lead-
w =Ne f (E,X )w (E)dE, (9) ingto the energydepositestimatorw:
i i e i e
Z0 w=C−1y. (15)
where f (E,X ) denotes the normalized electron energy
e i b
distributionandw (E)istheenergylossperunitdepthofa
e Due to the triangular structure of the Cherenkov-
singleelectronwithenergyE.Asisshownin[15,19,20],the fluorescence matrix thebinverse can be calculated quickly
electron energy spectrum f (E,X ) is universal in shower
e i evenformatriceswith largedimension.As the matrixele-
agesi =3/(1+2Xmax/Xi),i.e.it doesnotdepend onthe ments inEq.(12)arealways≥0,C is neversingular.
primarymassorenergy,butonlyontherelativedistanceto The statistical uncertainties of w are obtained by error
theshowermaximum,X .Eq.(9)canthusbesimplified
max propagation:
to
w =Ne α . (10) V =C−1V CbT −1 . (16)
i i i w y
(cid:0) (cid:1)
3
ns] 900 light at aperture 2)]m 20 χ2/Ndf= 81.19/83
2/100m 780000 tMdoiriteea clC tli hgCehhrteerneknokvov V/(g/c 15 greecnoenrsattreudcted
s/ 600 Rayleigh Cherenkov Pe Gaisser-Hillas fit
oton 500 dX [ 10
h 400 E/
ht [p 300 d 5
ed lig 120000
ct 0
e 0
et
d 270 280 290 300 310 320 330 340 350 0 200 400 600 800 1000 1200 1400 1600 1800
time slots [100 ns] slant depth [g/cm2]
(a) Lightat aperture. (b) Energydeposit profile
Fig.2. Example ofasimulated 1019 eVproton shower.
It is interesting to note that even if the measurements y event generator.The resulting longitudinal charged parti-
i
are uncorrelated, i.e. their covariance matrix V is diag- cle and energy deposit profiles were subsequently fed into
y
onal, the calculated energy loss values w are not. This is theatmosphereanddetectorsimulationpackage[22]ofthe
i
because the light observed during time interval i does not PierreAugerObservatory.Thegeometryandprofileofthe
solely originate from w , but also receives a contribution eventsinthissimulateddatasamplewasthenreconstructed
i b
from earlier shower parts w , j < i, via the ’Cherenkov withinthe Auger offlinesoftwareframework[23].
j
lightbeam’. Only events satisfying basic quality selection criteria have
been used in the analysis. In order to assure a good re-
construction of the shower geometry, the angular length
4. WavelengthDependence of the shower image on the camera was required to be
largerthanninedegrees.Moreover,weonlyselectedevents
withatleastonecoincidentsurfacedetectortank(so-called
Until now it has been assumed that the shower induces
hybrid geometry reconstruction [24]). Furthermore we re-
light emission at a single wavelength λ. In reality, the
jectedunder-determinedmeasuredlongitudinalprofilesby
fluorescence yield shows distinct emission peaks and the
demandinganobservedslantdepthlengthof≥300gcm−2
numberofCherenkovphotonsproducedisproportionalto
and a reconstructed shower maximum within the field of
1 .Inadditionthewavelengthdependenceofthedetector
λ2 viewofthe detector.
efficiency andthe light transmissionneed to be takeninto
An example of a simulated event is shown in Fig. 2, illus-
account. Assuming that a binned wavelength distribution
of the yields is available (Y = λk+∆λY (λ)dλ), the trating that the shape of the light curve at the detector
ik λk−∆λ i can differ considerably from the one of the energy deposit
above considerations still hold when replacing cd and cs
R i ij profile due to the scattered Cherenkov light detected at
inEq.(12)by
late stages of the shower development. The reconstructed
c˜d =∆X d Yf +f (β )YC/α (17) energy deposit curve, however, shows on average a good
i i ik ik C i ik i
agreementwiththe generatedprofile.
k
X (cid:0) (cid:1) Sincelongitudinalairshowerprofilesexhibitsimilarshapes
and
whentransformedfromslantdepthX toshowerages(see
c˜s =∆X d f (β )T YC/α , (18)
ij j ik s i jik jk j for instance [25]), a goodtest of the profile reconstruction
k
X performance is to compare the average generated and re-
where
ε T constructed energy deposit profiles as a function of s nor-
k ik
d = . (19)
ik 4πr2 malizedtotheenergydepositatshowermaximum.Ascan
i
be seen in Fig. (3), the difference between these averages
Thedetectorefficiencyε andtransmissioncoefficientsT
k ik is≤1.5%anditcanbeconcludedthatthematrixmethod
andT areevaluatedatthe wavelengthλ .
jik k introducedhereperformswellinreconstructingairshower
profileswithout a priorassumption abouttheir functional
5. ValidationwithAirShower Simulations
shape.
In order to test the performance of the reconstruction
algorithm we will use in the following simulated fluores-
cencedetectordata.Forthispurposewegeneratedproton
air showers with an energy of 1019 eV with the CONEX [21]
4
X)/(dE/dX)max 0.81 greecnoenrsattreudcted 2] [g/cmmax 10-11 Xenmeaxrgy 1100--32∆E/E
d X
E/ ∆
(d 0.6 10-2 10-4
0.4 10-3 10-5
0.2 10-4 10-6
max 0 1 2 3
dX) 0.04 number of iterations
E/ 0.02
d
X)/( 0
E/d -0.02 Fig.5.Xmax andenergydifferencewithrespecttothetenthshower
∆ (d-0.04 age iteration.
0.4 0.6 0.8 1 1.2 1.4 1.6
shower age
which was found to give a good description of measured
longitudinalprofiles[29].Ithasfourfreeparameters:X ,
Fig.3.Averagegeneratedandreconstructedenergydepositprofiles. max
the depth where the shower reaches its maximum energy
6. Shower AgeDependence depositw andtwoshape parametersX andλ.
max 0
The best set of Gaisser-Hillas parameters p can be ob-
Due to the age dependence of the electron spectra tainedbyminimizingtheerrorweightedsquareddifference
fe(E,si), the Cherenkovyield factors YiC and the average betweenthe vectoroffunctionvalues fGH andw,whichis
electronenergydepositsαi dependonthedepthofshower χ2 =[w−f(p)]T V −1 [w−f(p)]. (21)
maximum, which is not knownbefore the profile has been GH w b
reconstructed. Fortunately, these dependencies are small: This minimization works well if a large fraction of the
b b
Intheagerangeofimportancefortheshowerprofilerecon- showerhasbeenobservedbelowandabovetheshowermax-
struction(s∈[0.8,1.2])αvariesbyonlyafewpercent[20] imum. If this is not the case, or even worse, if the shower
and YC by less than 15%[15]. Therefore,a goodestimate maximumisoutsidethefieldofview,theproblemisunder-
ofαandYC canbeobtainedbysettings=1overthe full determined, i.e. the experimental information is not suffi-
profileorbyestimating Xmax fromthe positionmaximum cienttoreconstructallfourGaisser-Hillasparameters.This
of the detected light profile. After the shower profile has complicationcanbeovercomebyconstrainingX andλto
0
been calculated with these estimates, X can be deter- their average values hX i and hλi. The new minimization
max 0
mined from the energy deposit profile and the profile can functionis thenthe modified χ2
be re-calculated with an updated Cherenkov-fluorescence
(X −hX i)2 (λ−hλi)2
matrix. The convergence of this procedure is shown in χ2 =χ2 + 0 0 + , (22)
GH V V
Fig. 5. After only one iteration the Xmax (energy) differs X0 λ
by less than 0.1 gcm−2 (0.1%) from its asymptotic value. wherethe variancesofX andλaroundtheirmeanvalues
0
Notethatagedependenteffectsofthelateralspreadofthe areinthe denominators.
shower on the image seen at the detector [26,27], though In this way, even if χ2 is not sensitive to X and λ, the
GH 0
notdiscussedindetailhere,havealsobeenincludedinthe minimization will still converge.On the other hand, if the
simulationandreconstruction. measurements have small statistical uncertainties and/or
cover a wide range in depth, the minimization function is
7. Gaisser-HillasFit flexibleenoughtoallowforshapeparametersdifferingfrom
their mean values. These mean values can be determined
fromairshowersimulationsor,preferably,fromhighqual-
A knowledge of the complete profile is required for the
ity data profiles which can be reconstructed without con-
calculationoftheCherenkovbeamandtheshowerenergy.
straints.
Ifduetothelimitedfieldofviewofthedetectoronlyapart
Eq.(22)canbeeasilyextendedtoincorporatecorrelations
of the profile is observed, an appropriate function for the
betweenX andλandtheenergydependenceoftheirmean
extrapolation to unobserved depths is needed. A possible 0
values.Airshowersimulationsindicateasmalllogarithmic
choiceis the Gaisser-Hillasfunction[28]
energydependenceofthelatter(≤25%and5%perdecade
f (X)=w X−X0 (Xmax−X0)/λe(Xmax−X)/λ, forX0andλrespectively[30]).Inpracticeitissufficientto
GH max X −X useenergyindependentvaluesdeterminedatlowenergies,
(cid:18) max 0(cid:19)
(20) because at high energies the number of measured points
5
gengen 22]m]m mean
)/E)/Enn 00..1155 g/cg/c 4400 rms
-EE-EEgegerecrec 00..0000..5511 gengen) [X) [Xmaxmax 2200
(( 00 recrec-X-Xmaxmax 00
((
--00..0055
--2200
--00..11
--4400
--00..1155
00 1100 2200 3300 4400 5500 6600 7700 8800 9900 00 1100 2200 3300 4400 5500 6600 7700 8800 9900
CChheerreennkkoovv ffrraaccttiioonn [[%%]] CChheerreennkkoovv ffrraaccttiioonn [[%%]]
(a) Energy. (b) Xmax.
Fig.4. Energy andXmax reconstruction accuracy as function ofthe amount of detected Cherenkov light.
is large and thus the constraints do not contribute signifi- 8.1. Flux uncertaintyof thecalorimetric energy
cantlyto the overallχ2.
The accuracy of the reconstructed energy, obtained by Even with the flux uncertainties of the Gaisser-Hillas
integrating over the Gaisser-Hillas function (see below), parameters it is not straightforward to calculate the flux
and that of the depth of shower maximum, are displayed uncertainty of the calorimetric energy, which is given by
inFig.4asafunctionoftherelativeamountofCherenkov the integraloverthe energydepositprofile:
light.Notethatthegoodresolutionsof≈7%and20gcm−2
∞
are of course not a feature of the reconstruction method E = f (X)dX . (23)
cal GH
alone, but depend strongly on the detector performance Z0
andqualityselection.Themeanvaluesofdifferencetothe Tosolvethis integralonecansubstitute
trueshowerparameters,however,whichareclosetozerofor
X−X X −X
0 max 0
both fluorescence and Cherenkov light dominated events, t= and ξ = (24)
λ λ
indicatethatbothlightsourcesareequallysuitedtorecon-
inthe Gaisser-HillasfunctionEq.(20)to get
structthe longitudinaldevelopmentofairshowers.
A slight deterioration of the resolutions can be seen for ξ
e
eventswithaverysmallCherenkovcontributionof<10%. fGH(t)=wmax e−ttξ, (25)
ξ
Such showers are either inclined events which developed (cid:18) (cid:19)
highintheatmospherewherethelightscatteringprobabil- whichcanbeidentifiedwithaGammadistribution.There-
itiesarelowordeepverticalshowers,forwhichmostofthe fore,the aboveintegralis givenby
latepartoftheshowerisbelowgroundlevel.Bothtopolo- ξ
e
giesresultinasomewhatworseresolution:Theformercor- E =λw Γ(ξ+1), (26)
cal max
ξ
respond to larger than average distances to the detector (cid:18) (cid:19)
andthe latterto shorterobservedprofiles. whereΓdenotestheGamma-function.Thus,insteadofdo-
ingatediouserrorpropagationtodeterminethestatistical
uncertainty ofE one cansimply use it directly as a free
cal
8. ErrorPropagation parameterinthefitinsteadoftheconventionalfactorwmax:
E
f (t)= cal e−ttξ (27)
Arealisticestimateofthestatisticaluncertaintiesofim- GH λΓ(ξ+1)
portant shower parameters is desired for many purposes,
Inthisway,σ (E )isobtaineddirectlyfromtheχ2+1
likedataqualityselectioncuts,orthecomparisonbetween flux cal
contourofEq.(22).
independent measurements like the surface and fluores-
cence detector measurements of the Pierre Auger Obser-
vatory or the Telescope Array.The uncertainties of w , 8.2. Geometric uncertainties
max
X ,X andλobtainedaftertheminimizationofEq.(22),
max 0
reflect only the statistical uncertainty of the light flux, Due to the uncertainties on the shower geometry, the
whichiswhytheseerrorswillbereferredtoas’fluxuncer- distances r to each shower point are only known within
i
tainties’ (σ ) in the following. Additional uncertainties a limited precision and correspondingly the energy de-
flux
arisefromtheuncertaintiesontheshowergeometry(σ ), posit profile points are uncertain due to the transmission
geo
atmosphere (σ ) and the correction for invisible energy factors T(r ) and geometry factors 1/(4πr2). Further-
atm i i
(σ ). more, the uncertainty of the shower direction, especially
inv
6
tXhselaznten=ithXavenrgt/lecoθ,sθaffaencdtsththuessXlamnatx.deFpinthallcya,lctuhleataimonouvniat events00..0067 MRMeaSn - 01..1067 events00..0089 MRMeaSn - 00..0878
of direct and scattered Cherenkov light depends on the n of 0.05 n of 0.07
showergeometry,too,via the anglesβi. ctio0.04 ctio0.06
a a0.05
fr0.03 fr0.04
The algorithms used to reconstruct the shower geome- 0.02 0.03
try from fluorescence detector data usually determine the 0.02
0.01
following five parameters [9], irrespective of whether the 0.01
0 -4 -2 0 2 4 0 -4 -2 0 2 4
detectorsoperateinmonocular,stereoorhybridmode: (lgErec- lgEgen)/σ(lg Erec) (Xrmeacx-Xgmeanx)/σ(Xrmeacx)
α={θ ,Φ ,T ,R ,χ }. (28)
SDP SDP 0 p 0 (a) Energy. (b) Xmax.
θ and Φ are the angles of the normal vector of a
SDP SDP
Fig.6. Pull distributions
plane spannedbythe showeraxisandthe detector(the so
called shower-detector-plane),χ0 denotes the angle of the 8.4. Invisible energy
shower within this plane and T and R are the time and
0 p
distance of the shower at its point of closest approach to
Notalloftheenergyofaprimarycosmicrayparticleends
the detector.
upinthe electromagneticpartofanairshower.Neutrinos
Foranyfunctionq(α)standarderrorpropagationyieldsthe
escape undetected and muons need long path lengths to
geometricuncertainty
fully release their energy. This is usually accounted for by
5 5 dq dq multiplying the calorimetric energy, Eq. (23), with a cor-
σ2 (q)= Vα, (29)
geom dα dα ij rection factor finv determined from shower simulations to
i j
Xi=1Xj=1 obtainthe totalprimaryenergy
whereVαdenotesthecovariancematrixoftheaxisparam-
E =f E . (33)
eters. As the calorimetric energy and X depend non- tot inv cal
max
triviallyontheshowergeometry,theabovederivativesneed Themesondecayprobabilities,andthustheamountofneu-
to be calculated numerically, i.e. by repeating the profile trinoandmuonproduction,decreasewithenergy,therefore
reconstruction and Gaisser-Hillas fitting for the ten new finv dependsontheprimaryenergy.Forinstance,in[33]it
geometriesgivenbyα ± Vα ≡α ±σ to obtain isparameterizedas
i ii i i
dq p finv =(a+bEccal)−1, (34)
∆ = σ
i i
dαi where a, b and c denote constants depending on the pri-
1 mary composition and interaction model assumed1. This
≈ [q(α +σ )−q(α −σ )], (30)
2 i i i i energy dependence needs to be taken into account when
propagatingthe calorimetricenergyuncertaintytothe to-
withwhichEq.(29)readsas
talenergyuncertainty.
5 5 Due to the stochastic nature of air showers, the correc-
σ2 (q)= ∆ ∆ ρ (31)
geom i j ij tionfactorissubjecttoshower-to-showerfluctuations.The
Xi=1Xj=1 statistical uncertainty of finv was determined in [34] and
where canbe parameterizedasfollows:
Vα
ρij = ViαjVα (32) σ(finv)≈1.663·106·lg(Etot/ eV)−6.36. (35)
ii jj
denote the correlationcoefficients of the geometry param- Typicalvaluesare2.5%at1017 eV and0.9%at1020 eV.
p
etersα andα .
i j
8.3. Atmospheric uncertainties 8.5. Total statisticaluncertainty
WhereastheRayleighattenuationisatheoreticallywell Summarizing the above considerations, the statistical
understood process, the molecular density profiles and varianceofthe totalenergyis
aerosol content of the atmosphere vary due to environ-
σ2 (E )=E2 σ2(f )
mental influences and need to be well monitored in order stat tot tot inv
to determine the slant depth and transmissioncoefficients df 2
+ invE +f σ2(E ), (36)
needed for the profile reconstruction. Uncertainties in dE cal inv i cal
these measured atmospheric properties (see for instance (cid:18) cal (cid:19) Xi
[31,32]) can be propagated in the same way as the geo- 1 Note that here only the statistical uncertainties of the invisible
metric uncertainties by determining the one sigma shower
energycorrection arediscussed. For anestimate onthe relatedsys-
parameterdeviations viaEq.(30). tematicuncertainties see[34]
7
where iruns overthe geometric,atmosphericandflux un-
certainties. Since the invisible energy correction does not
affectthedepthofshowermaximum,itsuncertaintyissim- References
ply givenby
[1] T. Abu-Zayyad et al. [HiRes Collaboration], Nucl. Instrum.
Meth. A450(2000) 253.
σ (X )= σ2(X ). (37)
stat max i max [2] J. Abraham etal. [PierreAuger Collaboration], Nucl. Instrum.
s i Meth. A523(2004) 50.
X
[3] H. Kawai etal. [TACollaboration], Proc. 29th ICRC (2005).
Againweusesimulatedeventstoverifythevalidityofthe
[4] F. Kakimoto etal.,Nucl. Instrum. Meth. A372 (1996) 527.
above considerations. The pull distributions of the recon- [5] T. Waldenmaier, J. Blu¨mer,H.Klages, astro-ph/0709.1494.
structed energy and shower maximum, shown in Fig. 37, [6] M.Aveetal.[AIRFLYCollaboration],Proc.30thICRC(2007)
bothhaveawidthofapproximatelyone,whichmeansthat and arXiv:0711.4519 [astro-ph].
[7] P. Privitera et al. [AIRFLY Collaboration], Proc. 30th ICRC
the total uncertainties from Eqs. (36) and (37) are good
(2007).
estimators for the actual event-by-eventmeasurementun-
[8] B. Keilhauer, J. Blu¨mer, R. Engel, H. O. Klages, Astropart.
certainties. Phys. 25 (2006) 259.
[9] R.M.Baltrusaitisetal.[Fly’sEyeCollaboration],Nucl.Instrum.
Meth. A240(1985) 410.
9. Conclusionsand Outlook
[10]C. Song etal.,Astropart. Phys. 14 (2000) 7.
[11]J. Alvarez-Mun˜iz, E. Marques, R. A. Vazquez, E. Zas, Phys.
In this paper a new method for the reconstruction of Rev. D67 (2003) 101303.
longitudinal air shower profiles was presented. With the [12]J. Lindsley, Proc.27th ICRC (2001).
[13]M. Risse,D.Heck, Astropart. Phys. 20 (2004) 661.
help of simulations we have shown that the least square
[14]R. U. Abbasi et al. [HiRes Collaboration], Astrophys. J. 622
solution yields robust and unbiased results and that un-
(2005) 910.
certaintiesofshowerparameterscanbereliablycalculated [15]M. Gilleret al.,J. Phys. G30(2004) 97.
foreachevent. [16]M. Nagano etal.,Astropart. Phys. 22 (2004) 235.
Events with a large Cherenkov light contribution are cur- [17]M. Ave et al. [AIRFLY Collaboration], Astropart. Phys. 28
(2007) 41.
rently usually rejected during the data analysis (see for
[18]A. M.Hillas,J. Phys. G8(1982) 1461.
instance [14,35]) However, as we have shown, there is no
[19]A. M.Hillas,J. Phys. G8(1982) 1475.
justification for rejecting such showers, once experimen- [20]F. Nerlinget al.,Astropart. Phys. 24(2006) 421.
tal systematic uncertainties are well understood. Because [21]T. Bergmannet al.,Astropart. Phys. 26(2007) 420.
events with a large Cherenkov contribution have different [22]L. Pradoet al.,Nucl.Instrum. Meth. A545 (2005) 632.
[23]S. Argiroet al.,Nucl.Instrum. Meth. A580 (2007) 1485.
systematic uncertainties to those dominated by fluores-
[24]P. Sommers,Astropart. Phys. 3(1995) 349.
cence light, both event classes can be compared to study
[25]M. Gilleret al.,J. Phys. G31 (2005) 947.
their compatibility. [26]D. Goraet al.,Astropart. Phys. 24(2006) 484.
At energies below 1017.5 eV, where new projects [36,37] [27]M. Gilleret al, Astropart.Phys. 18 (2003) 513.
are planned to study the transition from galactic to ex- [28]T. K.Gaisser,A. M.Hillas,Proc. 15th ICRC (1977).
[29]Z. Caoet al.,Proc. 28th ICRC (2003).
tragalactic cosmic rays, events with a large fraction of
[30]L. Perrone, INFN Lecce, privatecommunication.
direct Cherenkov light will dominate the data samples,
[31]S. Y. BenZvi et al. [Pierre Auger Collaboration], Proc. 30th
because the amount of light, and thus trigger probability, ICRC (2007), astro-ph/0706.3236.
of these events is much larger than that of a fluorescence [32]B. Keilhauer et al. [Pierre Auger Collaboration], Proc. 29th
dominatedshower.Ifattheseenergiesitisstillpossibleto ICRC (2005), astro-ph/0507275.
[33]H. M.J. Barbosaet al.,Astropart.Phys. 22 (2004) 159.
measureanaccurateshowergeometry,thefluorescencede-
[34]T. Pieroget al.,Proc.29th ICRC (2005).
tectors should in fact be used as Cherenkov-Fluorescence
[35]T.Abu-Zayyadetal.[HiRes/MiaCollaboration],Astrophys.J.
telescopes. 557 (2001) 686.
[36]H.Klagesetal.[PierreAugerCollaboration],Proc.30th ICRC
(2007).
[37]J.Belzetal.[TA/TALECollaboration],Proc.30thICRC(2007).
AcknowledgmentsTheauthorswouldliketothanktheir
colleaguesfrom the Pierre Auger Collaboration,in partic-
ular Frank Nerling and Tanguy Pierog,for fruitful discus-
sions.
8