An analytical approach to the multiply scattered light in the optical images of the extensive air showers of ultra-high energies

0.53 MB·English

by  Maria Giller

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