An analytical approach to the multiply scattered light in the optical images of the extensive air showers of ultra-high energies MariaGillerandAndrzejS´miałkowski TheUniversityofLodz,DepartmentofHighEnergyAstrophysics,Pomorska149/153,90-236,Lodz,Poland 2 1 0 Abstract 2 n Oneofthemethodsforstudyingthe highestenergycosmicraysistomeasurethefluorescencelightemittedbythe a extensiveairshowersinducedbythem. Toreconstructashowercascadecurvefrommeasurementsofthenumberof J photonsarrivingfromthesubsequentshowertrackelementsitisnecessarytotakeintoaccountthemultiplescatterings 9 thatphotonsundergoontheirwayfromtheshowertothedetector. Incontrastto theearlierMonte-Carlowork,we 1 presenthereananalyticalmethodtotreattheRayleighandMiescatteringsintheatmosphere. Themethodconsists in considering separately the consecutive ’generations’ of the scattered light. Starting with a point light source in ] E a uniform medium, we then examine a source in a real atmosphere and finally - a moving source (shower) in it. H We calculate the angulardistributionsof the scattered lightsuperimposedonthe notscattered lightregisteredfrom . a shower at a given time. The analytical solutions (although approximate) show how the exact numerical results h should be parametrised what we do for the first two generations(the contribution of the higher ones being small). p Notallowingfortheconsideredeffectmayleadtoanoverestimationofshowerprimaryenergyby 15%andtoan - o underestimationoftheprimaryparticlemass. ∼ r t s Keywords: ultrahighenergyextensiveairshowers,cosmicrays,fluorescencelight,showerreconstruction a [ 1 1. Introduction shower image had contained the fluorescence photons v only.Thisisbecausethereexistsexperimentalevidence 2 Oneofthemethodsforstudyingextensiveairshow- that the number of fluorescence photons induced by a 5 ersofhighenergies( 1017 eV)istoregistertheirim- charged electron in the atmosphere is proportional to 0 ≥ agesintheoptical(mainlyfluorescence)light. Thiscan 4 the energy lost by it for ionisation [4]. As practically bedonebyobservingshowersfromthesideinorderto . all primary particle energy is eventually used for ioni- 1 avoidthemoreintenseCherenkovlightemittedrougly sation,thisenergycanbedeterminedbymeasuringthe 0 intheshowerdirection. Theobservationsaremadeby 2 fluorescencelightemittedalongtheshowertrackinthe 1 anumberofopticaltelescopes,eachcontainingalarge atmosphere. : mirror and a camera with a matrix of photomultipliers However,thereareseveralproblemsinderivingtheflux v (PMTs)placedatthefocusoftheopticalsystem(HiRes i ofthefluorescencelightemittedbyashowerfromthat X [1], The Pierre Auger Observatory [2], The Telescope arriving at the detector. Firstly, the arriving light con- r Array[3]),sothatphotonsarrivingfromagivendirec- tainsnotonlythefluorescencebutalsoCherenkovpho- a tionontheskyarefocusedonaparticularPMT(pixel). tons.Iftheviewingangle(theanglebetweenthelineof Photon arrival time can also be measured if the time sightandtheshowerdirection)islarge(say,>30 )then ◦ structureofthePMTsignalsisrecorded. Acosmicray it is mainlythe Cherenkovlightscattered in the atmo- induced shower produces at a given time a light spot sphere region just passed by the shower and observed onthecamerawhichmovesacrossitastheshowerde- bythe detector(thislight, beforebeingscattered, trav- velopsin the atmosphereso that succeedingPMTs are elsroughlyalongthedirectionsoftheshowerparticles). beinghit. Typicallyitsfractionatthedetectorisabout15%ofthe It would have been ideal if the light producing a fluorescence flux. For smaller viewing angles it is the Cherenkov light produced at the observed part of the showerthatmaydominateeventhefluorescencesignal. Emailaddress:[email protected]() PreprintsubmittedtoAstroparticlePhysics January20,2012 The contribution of the Cherenkov light, which has to Asouraimistoapplyourresultstocosmicrayshowers besubtractedfromthetotalsignal,hasbeenextensively weneedtoconsideranon-uniformmediumliketheat- studied[5,6,7,8]. mosphere. Assuminganexponentialdistributionofthe The subject of this paper is another phenomenon, af- gasdensity and similarly for aerosolswe show that an fectingshowerimages,mostcommonlycalledthemul- effectivescatteringlengthbetweenanytwopointsinthe tiple scattering(MS)of light. Photonsproducedatthe atmosphere can be easily calculated analytically. Sig- observed shower element, whatever their origin (fluo- nalsofthe first two generationsarrivingata particular rescenceorCherenkov),mayundergoscatteringinthe detector within a given angle ζ to the direction to the atmosphereontheirwayfromtheshowertothedetec- sourcearefoundasafunctionoftime(Section3). tor, causing an attenuation of the light flux arriving at Using these it is straightforward to derive the corre- the detector and a smearing of the image. This scat- sponding distributions if the source moves across the tering may take place on the air molecules (Rayleigh atmosphere, integrating the point source distributions scattering) or on larger transparent particles, aerosols over changing distance and time of light emission. In (Miescattering). Mostofthescatteredphotonschange Section4weconsideramovinglightsource,modelling theirdirections,sothattheynolongerarriveatthepixel a distantshower. We calculate angulardistributionsof registering the not scattered (direct) light. Moreover, MSlightarrivingatadetectoratthesametimeasthedi- theyarrivelaterhavinglongerpathlengthstopass. On rect(notscattered)photonsemittedbytheshower.This the other hand, photons emitted by the shower at ear- particular approach is quite natural because the data liertimesandscatteredsomewhere,mayfallinthefield fromopticaldetectorsconsistoftherecordedsignalsby ofviewofthepixelsjustregisteringthedirectphotons thecameraPMTswithinshorttimeintervals∆t sothat emittedatalatertime.Theneteffectisthatthescattered oneneedstoknowhowmuchoftheMSlighthastobe light formsits own instantaneousimage superimposed subtractedfromthemain,directsignal. Themethodfor onthatinthedirectlight. calculating images in the MS light simultaneous with Ouraimistocalculatetheshowerimagesinthemulti- images in the direct light is relatively simple (for the ply scattered light, so that this effect could be allowed firstandthesecondgenerations)asitisbasedonthege- for (subtracted) when determining the shower primary ometryofthescatteredphotonsintheparticulargener- energyfromthePMTsignals.Thisproblemwasalready ation. Itdoesnotrequiretime-consumingMonte-Carlo studied by Roberts [9], by us in several, short confer- simulationsthatweredoneforvariousshower-observer encecontributions[10]andmorerecentlybyPe¸kalaet geometries. Making some approximations we derived al [11]. The approach of the other authors was based analyticalformulaforashowerimageproducedbythe on Monte Carlo simulations of photons emitted by a firstgeneration(Section4.2).Ouranalyticalderivations shower. Photons were followed up to 5-6 scatterings allowedustochooseeasilythevariablesonwhichand andtheirarrivaldirectionsandtimewereregisteredby howtoparametrisetheMSsignals. Theyalsomadeus thedetector. Manyshowersimulationswereneededto torealisethatthedependenceofitontheviewingangle obtain the MS images for various distances, heights, was different for Rayleigh and Mie scatterings. Thus, viewing angles of the observed shower parts. Finally, weintroducedanew,simpleparametrisationofthefrac- a phenomenological parametrisation of the number of tion of the scattered photons arriving at the telescope MS photonswas made as a functionof the parameters withinagivenviewingcone,dependingontheviewing foundasrelevant. angle of the shower (what has not been done before), IncontrasttoRobertsandPe¸kalaetalthisapproachis separatelyforthetwoscatterings. Forthesamereason based on an analytical treatment. The main idea is to we also parametrised separately the second generation considerthearrivingMSlightasa sumofthe photons (Section4.3). scatteredonlyonce(thefirstgeneration),ofthosescat- AdiscussionoftheresultsandtheimplicationoftheMS tered two times (the second generation)and so on and effectonthederivationofshowerparametersisgivenin calculate separately the angular and temporal distribu- Section5. ThelastSection(6)containsasummaryand tionsforeachgeneration. conclusions. Westart(Section2)withaconsiderationofthesimplest situation when a point source of isotropic light flashes 2. Point source flashing isotropically in uniform for a very short time in an uniform medium. We de- medium riveanalyticalexpressionsfortheangularandtemporal distributionsofthefirstandnextgenerationsoflightar- Atanyfixedtimeadistantshowercanberegardedas rivingataparticulardistancefromthesource. a point source emitting isotropically fluorescence light 2 (aboutCherenkovlightseelater). Asexplainedabove, and we treat the light scattered in the medium as a sum R τ2 2τcosθ+1 of consecutive generations consisting of photons scat- x= − (3) 2 τ cosθ teredonlyonce,twice,andsoon,ontheirwayfromthe − sourcetotheobservationpoint. whereτ=ct/Randcisthespeedoflight. TheJacobianofthetransformationgives 2Rcosθ sinαdαdx= dτdcosθ (4) τ2 2τcosθ+1 | | − Thus,weobtain c f(α)sinθcosθ dn1(θ,t)=4πλe−cλt τ2 2τcosθ+1dθdt (5) − Finally,thenumberofphotonsarrivingataunitsurface at an angle (θ,θ +dθ) (all azimuths) at time (t,t+dt) equals dn (θ,t)= c e−cλt f(α)sinθ|cosθ|dθdt (6) 1 λR2 τ2 2τcosθ+1 − and 1 d2n j = 1 (7) Figure1: Geometryofthefirstscatteringinauniformmedium.The 1 2πsinθ cosθ dθdt lightsourceisatthecentre OofaspherewithradiusR. Tworays | | shownarescatteredatpointsS1andS2,correspondingly,arrivingat 2.1.1. Rayleighscattering thesurfaceofthespherewithradiusRatanglesθ1andθ2. FortheRayleighscatteringwehave 3 f(α)= fR(α)= (1+cos2α) (8) 2.1. Firstgeneration 16π Letusconsiderthefirstgeneration,consistingofpho- Expressingαasafunctionofθandτ(Eq.2)weobtain tons scattered once only. We shall calculate the flux of these photons, j (θ,t;R), at a distance R from the 3 2sin2θ 2sin4θ 1 fR = 1 + (9) source, such that j1(θ,t;R)dΩdtdS is the number of 8π(cid:20) − y y2 (cid:21) photonsscattered only once, arrivin⊥gat time (t,t+dt) wherey =τ2 2τcosθ+1. Thus,theflux jR(θ,t;R)of aftertheflash,withinasolidangledΩ(θ)atthesurface − 1 thefirstgeneration,definedabove,equals dS (perpendiculartothearrivaldirection)locatedata ⊥ distanceR. Todothisweshallcalculatefirstthenum- berofphotonscrossingthesphereofradiusR(fromin- jR(θ,t;R)= 3ce−cλt 1 2sin2θ + 2sin4θ 1 16π2λ R2y · − y y2  tsiimdee)(att,tan+adntg);lese(θe,Fθi+g.d1θ).wThiteharevsepraegctetnoutmhebneorromfaplh,oa-t R  (10) tons, per one photon emitted, interacting at a distance whereλ isthemeanfreepathlengthfortheRayleigh (x,x+dx) fromthe sourceandscattered atanangleα R scattering. However, λ in the exponentdependson all withindΩ(α),equals the scattering processes active. In general it is deter- dx minedby dn1(x,α)=e−λx λ f(α)dΩ(α)·e−xλ′ (1) n 1 1 where λ is the mean scattering path length, f(α)dΩ is = (11) λ λ the probability that, once the scattering has occured, Xi i thescatteringangleisαwithindΩ(α)=2πsinαdα. To for n processes. Thus, if both molecular and aerosol each pair of variables (x,α) there corresponds another scatteringsareactivebutonewantstocalculatetheflux pair(θ,t)relatedtotheformerby ofphotonsscatteredbytheRayleighprocessonly,λin theexponentequalsλ=( 1 + 1 ) 1butinthedenomi- tgα = τ−cosθ (2) natoronehasλ . λR λM − 2 sinθ R 3 From Eq. 10 one can find the number of photons In addition to the Rayleigh process one has to take dN1R(t;ζ,R) arriving per unit time at a unit surface within into account the Mie scattering occurring on particles dt agivenangleζ,asafunctionoftime. (aerosols)largerthatthelightwavelength. TheMiean- Wehave gulardistributionisconcentratedatrathersmallangles, incontrasttotheRayleighcase.Moreover,inthedeeper dNR(t;ζ,R) ζ 1 = jR(θ,t;R) 2πsinθcosθdθ partsoftheatmospherethemeanfreepathlengthforthe dt Z0 1 · | | MiescatteringmaybecomparabletothatforRayleigh, (12) sothatitisnecessarytocalculatethedistributionofthe lightscatteredbytheMieprocessonly. Theintegralcanbefoundanalytically,givingtheresult As before, we start with a simpler case - a uniform dNR(t;ζ,R) 1 = medium. The angulardistributionoflightscatteredon dt particleswithsizeslargerthanthelightwavelengthde- = 3ce−cλt 1 τ2+1 a−2 1 1 + pkennodwsnofnunthcetiodnis.triRboubtieorntso[f9t]haedsoipzetssaanfdunisctnioont aofwtehlel 8πλ R2 · 32τ6 2 y2 − y2 R (cid:16) (cid:17) yi+(cid:16)12 yi+11(cid:17) form +i=X2,0,1(cid:20)(τ2+1)ai+1−ai(cid:21)· 2i+−11 + f(α)∼e−Bα+CeDα (16) − Here, however,we preferan expressionallowingusto y y3 y3 + (τ2+1)a a ln 2 a 2− 1 (13) performsomeintegrationsanalytically. Mostcrucialis where(cid:20)y = (τ 01−)2,−y1(cid:21) =yτ12− 22τco3sζ +1, a = 1, toonhdagveentehreantiuomnbaesrfoefwnausmpeoriscsaibllien.teWgreatsiohnalslfsoereththeastecto- 1 − 2 − 2 find jR(θ,t;R)fortheRayleighscattering(Section2.3) a = 4, a = 6(τ4 + 1) 4τ2, a = 4(τ2 1)2, 2 a1 =(−τ2 10)4. − −1 − − thereisonlyoneintegration(overx′)tobedonenumer- W−e2 have −also calculated analytically a similar distri- icallysincetheformof fR(α′)enablesonetointegrate analytically overφ and θ (Eq. 25 and 26). Thus, we bution dNis/dt if the scattering was isotropic. i.e. if ′ ′ 1 adoptthe following form for the Mie angulardistribu- f(α)= 1 (AppendixA). 4π tion: One can also find analytically the angular distribution π ddMθ1R ofthe arrivinglight(integratedovertime), butfor f1M(α) = a1cos8α+b for 0≤α≤ 2 smallanglesonly(AppendixB).Theresultis π fM(α) = a cos8α+b for α π (17) dMR(θ;R) 2 2 2 ≤ ≤ 1dθ =2πsinθcosθZ ∞ jR1(θ,t;R)dt= wherea1 =0.857,a2 =0.125,b=0.025. R/c Thisfunctionisnormalisedasfollows = 9kRe−kR 1 4θ + 8kRθ ln(k θ)+C 1 π 64R2  − 3π 3π (cid:20) R Eu− 2(1(cid:21)4) Z0 f(α)·2πsinαdα=1 (18) It describes quite reasonably the distribution used by whereθ 1,k = R,ifthereisnoMiescatteringand Roberts. ≪ R λR C 0.577istheEulerconstant. FromEq.6and7wehave Eu ≃ Theratioofallphotonsarrivingwithinasmallangleθ ce kτ fM α(θ,t) tothosenotscatteredN0,equals j1M(θ,t;R)= 2πλ R−2(τ·2 i 2hτcosθi+1) (19) N10 Z0θ ddMθ′1Rdθ′ ≃ 4eπ−RkR2694kRRe−2kRθ= (15) wherei=1iftgα2 = τ−Msicnoθsθ <1− andi=2iftgα >1 = 9πkRθ(rad) 3.1 10−2kRθ(deg) Since 2 16 ≃ · whereterms θ2 havebeenneglected. cosα= 1−tg2α2 = 2sin2θ 1 (20) ∼ 1+tg2α τ2 2τcosθ+1 − 2 − 2.1.2. Miescattering weobtain Inthenextparagraphweshallconsiderthescattering of light emitted by showers developing in the real at- jM(θ,t;R)= ce−kτ a 2sin2θ 1 8+b (21) mosphere, i.e. with the density depending on height. 4 1 2πλMR2y i(cid:18) y − (cid:19)  wherey=τ2 2τcosθ+1,andiisdeterminedasbefore. sphericalshell of thicknessdR by an angle α (within ′ ′ − Inprinciple,itispossibletofindanalyticallythenumber dΩ(α))equals ′ dNM(t;ζ,R) ofphotons 1 arrivingwithinanangleζaftertime dx tperunittime.dHt owever,eachofthenineintegrals j1A(θ′,t′;R′)dΩ′dt′∆S′|cosθ′|λ fB(α′)dΩ (24) B n ζ 1 sin2θ where dx = dR′/cosθ′. As now both processes are I = sinθcosθdθ (22) | | n Z0 y y  saicotnivefothrej1Ameisanthinegeofffeλctiinvethmeefaacntopraeth−cλtl′eningtthhefoexrpbroetsh- containsmanytermsitself,sothatananalyticaldepen- processes. denceonζ and/orτwouldbepracticallylost. Thus,we The direction of the scattered photonsis at an angle α havefound dN1M byintegratingtheflux(Eq. 21)numer- totheradiusofthesphereanddΩ= sinαdαdφ,where dt ically(similarlytoEq. 12). φistheazimuthofthephotonsscatteredforthesecond Finally,thetotalfluxofthefirstgenerationisthesumof time. For any given direction (θ ,φ) before and (α,0) ′ ′ thetwo fluxesarisingfromthe two activemechanisms afterthesecondscattering,thescatteringangleα fulfils ′ ofthescattering therelationcosα = cosαcosθ sinαsinθ cosφ. The ′ ′ ′ ′ − onlyfunctiondependingontheazimuthangleφ ofthe j1(θ,t;R)= jR1 + j1M (23) incidentphotonsis fB(α′). Denoting ′ 2.2. Thesecondgeneration 2π FB(θ,α)= fB(α)dφ (25) ′ ′ ′ Z 0 andintegrating(25)overθ weobtainforthenumberof ′ photons incident on ∆S and scattered within dR into ′ ′ thesolidangledΩ(α)thefollowingexpression π dt ∆S dRdΩ jA(θ ,t;R)FB(θ,α)sinθ dθ = ′ ′ ′ Z 1 ′ ′ ′ ′ ′ ′ 0 =GAB(R,α) dt ∆S dRdΩ (26) ′ ′ ′ ′ · wherethefunctionGAB(R,α)isdefinedbytheintegral ′ inthel.h.s. of(26). ThepairoffixedvariablesR andαdefinesanotherpair ′ x and θ, where x is the photon path length after the ′ ′ Figure2: Geometryofthesecondandhigherscatteringsinauniform second scattering. The Jacobian of the transformation medium.ThelightsourceisatOandthedetectoristhesurfaceofthe givestherelation spherewithradiusR.Thepictureshowsthelastscattering. R2 These are the photons scattered exactly two times. dR sinαdα= sinθcosθdxdθ (27) ′ ′ R2 | | We shall consider first a general case when there are ′ more than one scattering processes (as Rayleigh and Putting∆S = 4πR2, the contributionofphotonsscat- ′ ′ Mie). Let us call the process of the first scattering as teredforthesecondtimeatadistance(R,R +dR)to ′ ′ ′ AandthisofthesecondoneasB. BothAandBcanbe arriveatanangle(θ,θ+dθ)atthespherewithradiusR eitherRayleighorMie.Wedenotetheirmeanscattering equals pathlengthsbyλ andλ , andtheangulardistribution A B functions of the scattering by fA(α) and fBα), corre- dn2AB(θ,t,x′;R)=e−xλ′ ·GAB(cid:20)R′(x′,θ),α(x′,θ)(cid:21)· spondingly.Thelightsourceflashesisotropicallyatthe 4πR2 2πsinθ cosθ dxdθdt (28) centreofa spherewithradiusRattimet = 0(Fig. 2). · · | | ′ As beforewe want to calculate the numberof photons Thefactore−x′/λ multipliedbye−ct′/λ in theexpression crossingthesurfaceofthespherefrominsideatagiven for jA gives e ct/λ, independentof x . Integrationover 1 − ′ angleθ,attimet,perunittime. x givesthetotalnumberoftheabovephotons ′ Let us consider the photons scattered for the second time at a distance R′ from the source. The number of dn2AB(θ,t;R)= (29) (pwhoitthoinnsdiΩnci)daetnttimonea(ts,mta+lldsutr)faacned∆sSca′ttaetreadnwanitghlienθa′ =e−cλt x′maxGABdx′ 4πR2 2πsinθ cosθdθdt ′ ′ ′ ′ Z0 ∗ · · | | 5 Figure3: Comparisonofthefirst(dN1/dτ)(threeuppercurves)andthesecond(dN2/dτ)(threelowercurves)generationsasfunctionsoftime (ǫ =τ 1=ct/R 1). Numberofphotonsarewithinangleζ. Uniformmedium,R=1. a). Rayleigh(solidlines)andisotropic(dashedlines) − − scattering.b).Twoscatteringprocessesatwork:RayleighandMie,eachwithλ=2R. whereGAB =GAB/e−ct′/λ. 2.3. Thenextgenerations Themax∗imumvalueofx′resultsfromfixingtimet. We Anynextgenerationof the scatteredphotonscan be havethat calculatedinthesamewayasthesecondonehasbeen R τ2 1 foundfromthepreviousone(thefirst). Tocalculatethe x′max =ct−R′ = 2 · τ c−osθ (30) flux ji(θ,t;R)ofthei thgeneration,given ji 1(θ,t;R) − weproceedasbefore−whencalculatingthesec−ondgen- Thus,thefluxofthesecondgenerationABequals eration from the first one (Eq. 24). The number of 1 d2nAB photons, incident on ∆S at an angle θ ,φ within dΩ jAB = 2 = (31) ′ ′ ′ ′ 2 4πR2 2πsinθ cosθ dθdt at time (t′,t′ + dt′) and scattered along dx into dΩ(α) · | | =e−cλt Z0x′maxG∗ABdx′ equaljs: (θ,t ;R)dtdΩ∆S cosθ dxf(α)dΩ (33) and,integratedoverθforθ < ζ,givesdNAB(t;ζ,R)/dt. i−1 ′ ′ ′ ′ ′ ′| ′|λi i ′ 2 With the Rayleigh and Mie processes active we must Iftherearetwoscatteringprocesses,R+M,then takeintoaccountallfourcasesA=RorMandB=Ror f(α) fR(α) fM(α) M. Finally,thefluxofthecombinedsecondgeneration i ′ = ′ + ′ (34) λ λ λ photonsisasumofallspecificfluxes i R M isthescatteringprobabilitybyanangleα byanypro- j = jRR+ jRM+ jMR+ jMM (32) ′ 2 2 2 2 2 cess per unit distance per unit solid angle. The rest of Some of the integrals defined in this Section can be thederivationof j(θ,t;R)isthesameasintheprevious i found as analytical functions (Appendix C). It is of Section. However, for each next generation the num- some importance when calculating higher generations berofnumericalintegrationsincreases,unlessvaluesof (seethenextSection). j (θ,t;R)arestoredasa3-dimensionmatrix. Thus,it i 1 − 6 isconvenienttofindanalyticalsolutionsoftheintegrals F(θ,α)and/orG(R,α),ifpossible. ′ ′ 2.4. Resultsofcalculations Fig. 3ashowsthenumberofphotonsarrivingatthe detector within an angle ζ < 1 ,3 ,10 per unit area ◦ ◦ ◦ perunitτ = ct/Rasafunctionofǫ = τ 1. Theupper − curvesrefertothefirstgeneration,thelower-tothesec- ond one. We also compare here the time distributions obtained for the Rayleigh with those for the isotropic scattering. Thedistance detector-sourceequalsto one scatteringlength(k=R/λ=1). Firstofallwenoticethatforshorttimes(ǫ 0.01)the ≤ firstgenerationdominatesoverthesecondone,and(as wecanguess)overthehigherones. Itcanbeseenfrom the formulaefor the isotropic scattering (AppendixA) Figure4: Verticalcross-sectionthroughtheatmosphere. Linescor- dNis ∆Nis respondtoconstantvaluesofk = R/λPD,shownbynumbers,look- thattheratio 2 / 1 foranygiventimeshouldbepro- ingfromthe detector (at x = 0, h = 0)toapoint ontheline, for dτ dτ portionaltok = R/λ,sothattheimportanceofthesec- λR = 18km and λM = 15km at the ground and the scale heights HR=9kmandHM=1.2km. ond(andthehigher)generationwillbebiggerforlarger k. We can also see that the number of photons arriving of two sorts of matter, molecules and aerosols, each withinanopeningangleζ reachesthedependence ζ2 havingitsdensitydecreasingwithheightexponentially ∼ only at later times. This reflects the fact that the ini- with a different scale heights, H - for molecules and R tial angular distribution of light is steep and becomes H for aerosols, and having the corresponding mean M almostflatattimesτ 1.1orso. Whencomparingthe pathlengthsforscatteringatthegroundλR andλM. It ≥ D D Rayleighcurveswiththeisotropiconesonecanseethat isnotdifficulttoderivethattheeffectivemeanfreepath the latter are slightly flatter for shorter times, as might forascatteringforlighttravellingbetweentwoarbitrary beexpectedbutbecomeparallelto theformerforlater pointsPandS equals times. Next,weconsiderasituationwhentherearetwoscatter- 1 1 1 λ = + − = (36) ingprocesses,RayleighandMiewithquitedifferentan- PS (cid:18)λR λM (cid:19) PS PS gulardistributions f(α)(asdiscussedbefore).Weadopt h h = P− S k= R =R( 1 + 1 )=1 (35) HλRR e−hS/HR −e−hP/HR + HλMM e−hS/HM −ehP/HM λtot λR λM D(cid:16) (cid:17) D (cid:16) (cid:17) andλ =λ forsimplicity. wherehPandhS aretheheightsofpointsPandS above R M the level, where the Rayleigh and Mie scattering path The result for the first and the second generation de- lengths are correspondingly λR and λM. If the source pending on time is shown in Fig. 3b. There is now D D (point P) is at a distance R from the detector (point D morelightatearliertimesthaninthepreviouscase(Fig. on the ground) then the ratio k, of R to the mean free 3a) due to the strong Mie scattering in the forwarddi- pathlengthalongPDequals rections. However, the flux of the first generation de- creasesabout3timesquickerovertheconsideredtime R region.Althoughtheratioofthesecondtothefirstgen- k= = (37) λ erationispracticallythesameatǫ =10 3inbothcases PD − 1 H H (la≤te1r%tim).etshewihmepnotrhteanMceieosfctahtetesreincgonisdporneeseinstr.eachedat = cosθZ(cid:20)λRDR(cid:16)1−e−hP/HR(cid:17)+ λDMM(cid:16)1−e−hP/HM(cid:17)(cid:21) Itcan be seen that increasingthe distanceR to infinity 3. Pointlightsourceintheatmosphere (keeping θ constant) the ratio k reaches its maximum Z finitevalue Now we shall study the situation when a pointlight sourceflashesinanon-uniformmedium,suchastheat- 1 H H k (θ )= R + M (38) mosphere.Weassumethattheatmosphereiscomposed max Z cosθZ(cid:18)λRD λDM (cid:19) 7 This situation is illustrated in Fig. 4. Here a vertical thezenithangleθ ofthesourceandontheazimuthan- Z cross-section of the atmosphere is shown. Detector is gle φ around the direction towards it. Fig. 5 shows a at x = 0,h = 0 and lines represent constant values of trajectoryPSDofafirstgenerationphotonscatteredat k corresponding to the straight path from the detector S. We wantto calculatethe angulardistributionofthe to the point on the line. We have adopted the follow- firstgenerationasa functionoftime, fora fixedRand ing values: λRD = 18km,HR = 9km, λDM = 15km and θZ, ddΩ2nd1t(θ,φ,t;R,θZ),crossingaunitareaperpendicular HM = 1.2km. Thesevaluesdescribeapproximatelythe tothedirectiontowardsthesource. atmosphericconditionsatthePierreAugerObservatory We notice that for a fixed arrival direction of photons [2]. (θ,φ) andtime t, the scatteringpointS is uniquelyde- FromFig. 4onecanalsodeducethatrelevantvaluesof termined.Toarriveatthedetectoratangles(θ,φ)within k,iflightsourcesareatdistances (10 30)km(asex- dΩ = sinθdθdφ after time (t,t+dt), photons have to ∼ − tensiveairshowersseenbyAuger)are1/2 k 3/2. crossthesurfaceda(shadedin thefigure)andbescat- ≤ ≤ Thus,thiswillbetheregionofourinterest. teredalongapathlengthdxbytheangleαdetermined byEq.2.Thenumberofsuchphotonsequals dn1(θ,φ,t)= x′sinθ4dπφxd2x′cosγe−λPxS · (39) ·λdxf(α)∆ΩDe−λSx′D S where x = PS, x = SD, γ is the angle between ′ the normal to the surface da and the direction of the incident photons PS, ∆Ω is the solid angle deter- D mined by the unitarea at D and the scattering pointS (∆Ω = cosθ/x2). Thereisnoneedtocalculateγ be- D ′ causedx = xdθ/cosγ, sothatitcancelsout. Itcanbe ′ shownthat Figure5: Geometryofthefirstgeneration intherealatmosphere. LightsourceisatP,detectoratD.ScatteringtakesplaceatS. dx 2 dτ ′ = (40) x2 τ2 2τcosθ+1 R − Insertingthisinto(40)weobtain d2n j (θ,φ,t) cosθ= 1 = (41) 1 · dΩdt = c f(α)cosθ e−(λPxS+λSx′D) 2πλ R2 τ2 2τcosθ+1 · S − Onecanseethatthisformulaispracticallythesameas (7) for the uniform medium, the only difference being inthescatteringpathlengthsdependingnotonlyondis- tancesbutalsoonthegeometry. TheheightofthescatteringpointS necessarytocalcu- lateλ ,λ andλ equals: S PS SD sinθ h = ct R (42) iFniggusriete6s:oTfhfiersetllgipesnoeirdastio(tnhepihroctroonsss-aserrcitviionngsaatreDshaoftwern)tismheowR(s1ca+tteǫr)-. s (cid:18) − sinα(cid:19)· c (cosθ cosθ+sinθ sinθcosφ) Thecorrespondingnumbersareequaltoǫ. LightsourceisatP,de- · Z Z tectoratD. Wecalculatenumericallythetimedistributionsoflight dNreal(t;ζ)/dt,arrivingatthedetectoratanglessmaller 1 thanζ fordifferentzenithanglesofthesource. There- 3.1. Thefirstgeneration sults,intheformoftheratio: As the medium is non-uniform, we cannot use the idea of a sphere to be crossed by the scattered pho- dNreal/dt F (τ;ζ)= 1 (43) tons, as in Section 2. Now their flux will depend on 1 dNuni/dt 1 8 are presented in Figs. 7, 8 and 9, where dNuni/dt are 1 the distributions obtained in the previous section for a uniformmedium. Figure9:AsinFig. 8butfortwovaluesofk. Eachgroupoflinesis forζ=1◦,2◦,3◦,5◦,and10◦frombottomtotop(atǫ=10−3). distancePD)musthavebeenscatteredonthesurfaceof anellipsoidwiththeeccentricityeequal Figure7:RatioofthefirstgenerationdNreal/dtinrealatmosphereto 1 thatinuniformmediumdNuni/dtasafunctionoftime(ǫ=ct/R 1) 1 − 1 fortheRayleighscatteringonlyforvariousvaluesofzenithangleof e= (44) thesource.Solidlines-ζ=1 ,dashedlines-ζ=10 ,k=R/λR = 1+ǫ ◦ ◦ PD 1/2. The ellipses refer to ǫ = 10 3,10 2,10 1 and 3 10 1 − − − − · keepingtherightproportions.Thedetectorfieldofview cutsonlyapartoftheellipsoidsurfacewherethepho- tonsregisteredaftertime1+ǫmusthavebeenscattered. WenoticethatallratiosF inFig. 7aresmallerthan1 1 and decrease with time (althoughthose for ζ = 1 are ◦ practically constant) and ratios for ζ = 10 are larger ◦ thanthoseforζ = 1 . Allthisbecomesclearwhenin- ◦ specting Fig. 6 and the correspondingscattering sites. Forexample-theconstancyof F (τ;ζ = 1 )fortimes 1 ◦ τ 1 = ǫ = 10 3 10 1 reflectsthe factthatthescat- − − − ÷ tering takes place very close to point P during all this time andstarts to moveaway fromit(higherin theat- mosphere)only for largertimes i.e τ 1.3. Since the ≥ scatteringpathlengthintheuniformmediumischosen Figure8:AsinFig.7butwithMieincluded;k=R/λPD=1/2. equaltotheeffectivepathlengthintheatmosphereλPD (Eq.36),wehavethatλ >λ andthescatteringprob- P PD Tocomparethelightfluxobtainedfortherealatmo- abilityatPissmallerintheexponentialatmosphere. sphere with that for a uniform medium we adopt the Let us consider now a more realistic atmosphere with samevalueofk andthesamedistancefromthesource bothprocesses,RayleighandMieatwork.Fig.8shows to detector for both cases. To understandthe effect of timedependenceoftheratioF (τ;ζ)forζ =1 and10 1 ◦ ◦ a purely exponentialatmosphere we start with consid- andzenithanglesθ = 10 75 . Letustakea closer Z ◦ ◦ ÷ ering only the Rayleigh scattering (Fig. 7), neglecting lookatthecaseζ =1 andθ =75 (uppersolidcurve). ◦ Z ◦ Mie (λ = ). Understanding the behaviour of the It may seem strange that the ratio F increases since M 1 ∞ curves in this figure is easier with the help of Fig. 6. inthecaseoftheRayleighscatteringonlyitdecreases, Each ellipse is a cross-section of a rotational ellipsoid although very slowly. We have checked that a similar withthesymmetryaxisdeterminedbypointD-thede- slow decreasetakesplaceif the scatteringisonlyMie. tectorandpointP-thelightsource.Thesearethefocal Thebehaviourofthecurvesinthisfigurewouldbehard pointsofalltheellipses. Photonsarrivingatthedetec- tounderstandwithoutthepresentationofthescattering toratDaftersomefixedtimeτ = 1+ǫ (inunitsofthe sites in Fig. 6. Itcan be seen thatfor ǫ smaller thana 9 few 10 2 thescatteringsitesareclosetothesourceat λ s as defined before. In our example λ 33km, × − ′ P1D ≃ P. Thus,wemayapproximatetheratioF asfollows λ 113km,λ 15.3km,λ 25.6kmsothat 1 P1 ≃ P1/2D ≃ P1/2 ≃ the abovefactor equals 0.49 whereasthe exactratio F = λfRPR + λfPMM = λRPD 1+ λλPMRP ffMR (45) fromFig. 9≃0.48. ≃ 1 ≃ fR + fM λRP · 1+ λRPD fM 3.2. Thesecondgeneration λR λM λM f PD PD PD R where f aresomeeffectiveangulardistributionsof R(M) photonsscatteredbyRayleigh(Mie)onthecutsurface. Astimeincreases,noneoftheλ schangesmuch.How- ′ ever,thetypicalscatteringanglesincrease,whataffects muchmore f than f ,sothat f /f decreases.Since M R M R λR λR P < PD (46) λM λM P PD thenumeratordecreasesbyasmallerfactorthanthede- nominatorsothattheratio F increases. Itcanbeseen 1 from Fig. 6 that for ǫ 2 10 2 the scattering angles − ≥ · of the registered photonsdo not change much (the de- nominator stays constant) but now λR and λM have to be substituted by λRS and λSM, wherePS is anPeffective Fasigaurfeun1c0t:ionRoatfiotimofeth(ǫe=seccot/nRddN12)/fdrotmtoathfleafishrst(agtetne=ra0t)ioonfdaNp1o/idntt scatteringpointwithgrowingheight. AsHM < HR,the sourceatzenithangleθZ = 75−◦ intherealatmosphere. Fluxesare ratioλRS/λSM decreasesandsodoesF1. Tinhteregeracteudrvewsitfhoirneζach=ζ1r◦ef(esrotloidkli=ne1s/)2a,n1d,3ζ/2=(fr1o0m◦ b(doattsohmedtolintoeps)).. AdifferentbehaviourofF (τ;ζ =10 )canbeexplained 1 ◦ DottedlinereferstoauniformmediumwithRayleighscatteringonly, also with thehelp ofFig. 6. Atfirst F1 decreases(the fork=1,ζ=1◦. smaller θ - the stronger decrease) because the detec- Z tor field of view cuts out a growing part of the deep We proceed like in the case of the first generation atmospherewherethescatteringisstrongintherealat- (Fig. 5), but now point S refers to the second scatter- mosphere. At ǫ 0.02 F starts to increase for the ing.Photonsscatteredonlyoncearriveatthesurfaceda 1 ≥ same reason as just described in the case ζ = 1 . It from all directions according to j (θ ,φ ,t ;x), where ◦ 1 1 1 1 must finally decrease since the scattering takes places t = t x /c. Thus, the number of photons dn inci- 1 ′ 2 − furtherandfurtherbehindthesource,whereλR andλM dent on da and scattered for the second time towards aregrowingintherealatmosphere. thedetector(toarrivetherewithindΩ (θ,φ)aftertime D InFig.9weshowF (τ;ζ)forθ =60 forseveralinter- (t,t+dt)equals 1 Z ◦ mediatevaluesofζ, andfortwovaluesk = 1/2and1. -Ntohteedthisattacnhcaentgoitnhgekso=urRc/eλ.PTDhmisuisstthreesmulatiinnrcehaasnogninwghRy dn2(θ,φ,t;x′)=ZΩ1 j1(θ1,φ1,t;x)dΩ1dtdacosγ· Ithtecacnurbveessefeonrfkro=m1Faigre. 5lotwheartftohranθZth=os6e0f◦o(rthke=sc1a/le2s. · λdSl f(α2)dΩDe−λSx′D (48) onbothaxesarethesamesointhefiguretheanglesare wheretheintegrationhastobedoneoverfullsolidangle correctlyrepresented)thedistanceRismuchshorterfor (0 < φ < 2π,0 θ π),dlisthepathlengthforthe 1 1 ≤ ≤ k = 1/2 than for k = 1, implyingthat the correspond- second scattering to occur (cosγdl = x dθ) and α is ′ 2 ingheightsofthesourcedifferconsiderably.Inspecting theangleofthesecondscattering.Nowthepairofvari- Fig. 6wecanestimatethatforǫ 0.01andζ = 1◦ the ables,θandt,doesnotdetermineuniquelythepositions ≃ curvesfork = 1 shouldbe downwith respectto those ofthesecondscattering,sincethetimest elapsedfrom 1 fork=1/2byafactor photonemission to theirarrivalat S (orstrictly speak- ing,atda)havesomedistribution. However,t cannot λ λ 1 λP1D/ λP1/2D (47) besmallerthanx/c,thusx′max =ct−x. Expressingx′max P1 P1/2 asafunctionofθ,tandRonlyweobtain whereP1/2 and P1 arepositionsofthesourcereferring (ct)2 R2 to k = 1/2 and 1 respectively, with the meaningof all x′max = 2(ct R−cosθ) (49) − 10