arX

iv:1

201.

4052

v1 [

astr

o-ph

.HE

] 19

Jan

201

2

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

Maria Giller and AndrzejSmiałkowski

The University of Lodz, Department of High Energy Astrophysics, Pomorska 149/153, 90-236, Lodz, Poland

Abstract

One of the methods for studying the highest energy cosmic rays is to measure the fluorescence light emitted by theextensive air showers induced by them. To reconstruct a shower cascade curve from measurements of the number ofphotons arriving from the subsequent shower track elementsit is necessary to take into account the multiple scatteringsthat photons undergo on their way from the shower to the detector. In contrast to the earlier Monte-Carlo work, wepresent here an analytical method to treat the Rayleigh and Mie scatterings in the atmosphere. The method consistsin considering separately the consecutive ’generations’ of the scattered light. Starting with a point light source ina uniform medium, we then examine a source in a real atmosphere and finally - a moving source (shower) in it.We calculate the angular distributions of the scattered light superimposed on the not scattered light registered froma shower at a given time. The analytical solutions (althoughapproximate) show how the exact numerical resultsshould be parametrised what we do for the first two generations (the contribution of the higher ones being small).Not allowing for the considered effect may lead to an overestimation of shower primary energy by∼ 15% and to anunderestimation of the primary particle mass.

Keywords: ultra high energy extensive air showers, cosmic rays, fluorescence light, shower reconstruction

1. Introduction

One of the methods for studying extensive air show-ers of high energies (≥ 1017 eV) is to register their im-ages in the optical (mainly fluorescence) light. This canbe done by observing showers from the side in order toavoid the more intense Cherenkov light emitted rouglyin the shower direction. The observations are made bya number of optical telescopes, each containing a largemirror and a camera with a matrix of photomultipliers(PMTs) placed at the focus of the optical system (HiRes[1], The Pierre Auger Observatory [2], The TelescopeArray [3]), so that photons arriving from a given direc-tion on the sky are focused on a particular PMT (pixel).Photon arrival time can also be measured if the timestructure of the PMT signals is recorded. A cosmic rayinduced shower produces at a given time a light spoton the camera which moves across it as the shower de-velops in the atmosphere so that succeeding PMTs arebeing hit.It would have been ideal if the light producing a

Email address:maria.giller@kfd2.phys.uni.lodz.pl ()

shower image had contained the fluorescence photonsonly. This is because there exists experimental evidencethat the number of fluorescence photons induced by acharged electron in the atmosphere is proportional tothe energy lost by it for ionisation [4]. As practicallyall primary particle energy is eventually used for ioni-sation, this energy can be determined by measuring thefluorescence light emitted along the shower track in theatmosphere.However, there are several problems in deriving the fluxof the fluorescence light emitted by a shower from thatarriving at the detector. Firstly, the arriving light con-tains not only the fluorescence but also Cherenkov pho-tons. If the viewing angle (the angle between the line ofsight and the shower direction) is large (say,> 30) thenit is mainly the Cherenkov lightscatteredin the atmo-sphere region just passed by the shower and observedby the detector (this light, before being scattered, trav-els roughly along the directions of the shower particles).Typically its fraction at the detector is about 15% of thefluorescence flux. For smaller viewing angles it is theCherenkov lightproducedat the observed part of theshower that may dominate even the fluorescence signal.

Preprint submitted to Astroparticle Physics August 28, 2018

The contribution of the Cherenkov light, which has tobe subtracted from the total signal, has been extensivelystudied [5, 6, 7, 8].The subject of this paper is another phenomenon, af-fecting shower images, most commonly called the mul-tiple scattering (MS) of light. Photons produced at theobserved shower element, whatever their origin (fluo-rescence or Cherenkov), may undergo scattering in theatmosphere on their way from the shower to the detec-tor, causing an attenuation of the light flux arriving atthe detector and a smearing of the image. This scat-tering may take place on the air molecules (Rayleighscattering) or on larger transparent particles, aerosols(Mie scattering). Most of the scattered photons changetheir directions, so that they no longer arrive at the pixelregistering the not scattered (direct) light. Moreover,they arrive later having longer path lengths to pass. Onthe other hand, photons emitted by the shower at ear-lier times and scattered somewhere, may fall in the fieldof view of the pixels just registering the direct photonsemitted at a later time. The net effect is that the scatteredlight forms its own instantaneous image superimposedon that in the direct light.Our aim is to calculate the shower images in the multi-ply scattered light, so that this effect could be allowedfor (subtracted) when determining the shower primaryenergy from the PMT signals. This problem was alreadystudied by Roberts [9], by us in several, short confer-ence contributions [10] and more recently by Pekala etal [11]. The approach of the other authors was basedon Monte Carlo simulations of photons emitted by ashower. Photons were followed up to 5-6 scatteringsand their arrival directions and time were registered bythe detector. Many shower simulations were needed toobtain the MS images for various distances, heights,viewing angles of the observed shower parts. Finally,a phenomenological parametrisation of the number ofMS photons was made as a function of the parametersfound as relevant.In contrast to Roberts and Pekala et al this approach isbased on an analytical treatment. The main idea is toconsider the arriving MS light as a sum of the photonsscattered only once (the first generation), of those scat-tered two times (the second generation) and so on andcalculate separately the angular and temporal distribu-tions for each generation.We start (Section 2) with a consideration of the simplestsituation when a point source of isotropic light flashesfor a very short time in an uniform medium. We de-rive analytical expressions for the angular and temporaldistributions of the first and next generations of light ar-riving at a particular distance from the source.

As our aim is to apply our results to cosmic ray showerswe need to consider a non-uniform medium like the at-mosphere. Assuming an exponential distribution of thegas density and similarly for aerosols we show that aneffective scattering length between any two points in theatmosphere can be easily calculated analytically. Sig-nals of the first two generations arriving at a particulardetector within a given angleζ to the direction to thesource are found as a function of time (Section 3).Using these it is straightforward to derive the corre-sponding distributions if the source moves across theatmosphere, integrating the point source distributionsover changing distance and time of light emission. InSection 4 we consider a moving light source, modellinga distant shower. We calculate angular distributions ofMS light arriving at a detector at the same time as the di-rect (not scattered) photons emitted by the shower. Thisparticular approach is quite natural because the datafrom optical detectors consist of the recorded signals bythe camera PMTs within short time intervals∆t so thatone needs to know how much of the MS light has to besubtracted from the main, direct signal. The method forcalculating images in the MS lightsimultaneouswithimages in the direct light is relatively simple (for thefirst and the second generations) as it is based on the ge-ometry of the scattered photons in the particular gener-ation. It does not require time-consuming Monte-Carlosimulations that were done for various shower-observergeometries. Making some approximations we derivedanalytical formula for a shower image produced by thefirst generation (Section 4.2). Our analytical derivationsallowed us to choose easily the variables on which andhow to parametrise the MS signals. They also made usto realise that the dependence of it on the viewing anglewas different for Rayleigh and Mie scatterings. Thus,we introduced a new, simple parametrisation of the frac-tion of the scattered photons arriving at the telescopewithin a given viewing cone, depending on the viewingangle of the shower (what has not been done before),separately for the two scatterings. For the same reasonwe also parametrised separately the second generation(Section 4.3).A discussion of the results and the implication of the MSeffect on the derivation of shower parameters is given inSection 5. The last Section (6) contains a summary andconclusions.

2. Point source flashing isotropically in uniformmedium

At any fixed time a distant shower can be regarded asa point source emitting isotropically fluorescence light

2

(about Cherenkov light see later). As explained above,we treat the light scattered in the medium as a sumof consecutive generations consisting of photons scat-tered only once, twice, and so on, on their way from thesource to the observation point.

Figure 1: Geometry of the first scattering in a uniform medium. Thelight source is at the centreO of a sphere with radiusR. Two raysshown are scattered at pointsS1 andS2, correspondingly, arriving atthe surface of the sphere with radiusR at anglesθ1 andθ2.

2.1. First generation

Let us consider the first generation, consisting of pho-tons scattered once only. We shall calculate the fluxof these photons,j1(θ, t; R), at a distanceR from thesource, such thatj1(θ, t; R)dΩdtdS⊥ is the number ofphotons scattered only once, arriving at time (t, t + dt)after the flash, within a solid angledΩ(θ) at the surfacedS⊥ (perpendicular to the arrival direction) located at adistanceR. To do this we shall calculate first the num-ber of photons crossing the sphere of radiusR (from in-side) at an angle (θ, θ+dθ) with respect to the normal, attime (t, t + dt); see Fig. 1. The average number of pho-tons, per one photon emitted, interacting at a distance(x, x + dx) from the source and scattered at an angleαwithin dΩ(α), equals

dn1(x, α) = e−xλdxλ

f (α)dΩ(α) · e−x′λ (1)

whereλ is the mean scattering path length,f (α)dΩ isthe probability that, once the scattering has occured,the scattering angle isα within dΩ(α) = 2π sinαdα. Toeach pair of variables (x, α) there corresponds anotherpair (θ, t) related to the former by

tgα

2=τ − cosθ

sinθ(2)

and

x =R2τ2 − 2τ cosθ + 1τ − cosθ

(3)

whereτ = ct/Randc is the speed of light.The Jacobian of the transformation gives

sinαdαdx=2Rcosθ

τ2 − 2τ cosθ + 1dτ|d cosθ| (4)

Thus, we obtain

dn1(θ, t) = 4πcλ

e−ctλ

f (α)sinθ cosθτ2 − 2τ cosθ + 1

dθ dt (5)

Finally, the number of photons arriving at a unit surfaceat an angle (θ, θ + dθ) (all azimuths) at time (t, t + dt)equals

dn1(θ, t) =cλR2

e−ctλ f (α)sinθ |cosθ|τ2 − 2τ cosθ + 1

dθ dt (6)

and

j1 =1

2πsinθ |cosθ|d2n1

dθdt(7)

2.1.1. Rayleigh scatteringFor the Rayleigh scattering we have

f (α) = f R(α) =3

16π(1+ cos2α) (8)

Expressingα as a function ofθ andτ (Eq. 2) we obtain

f R =38π

[

1−2sin2θ

y+

2sin4θ

y2

]

(9)

wherey = τ2 − 2τ cosθ + 1. Thus, the fluxjR1 (θ, t; R) ofthe first generation, defined above, equals

jR1(θ, t; R) =3c e−

ctλ

16π2λRR2 y·

1− 2sin2θ

y+

2sin4θ

y2

(10)

whereλR is the mean free path length for the Rayleighscattering. However,λ in the exponent depends on allthe scattering processes active. In general it is deter-mined by

1λ=

n∑

i

1λi

(11)

for n processes. Thus, if both molecular and aerosolscatterings are active but one wants to calculate the fluxof photons scattered by the Rayleigh process only,λ inthe exponent equalsλ = ( 1

λR+ 1λM

)−1 but in the denomi-nator one hasλR.

3

From Eq. 10 one can find the number of photonsdNR

1 (t;ζ,R)dt arriving per unit time at a unit surface within

a given angleζ, as a function of time.We have

dNR1 (t; ζ,R)

dt=

∫ ζ

0jR1(θ, t; R) · 2π sinθ|cosθ|dθ

(12)

The integral can be found analytically, giving the result

dNR1 (t; ζ,R)

dt=

=3ce−

ctλ

8πλRR2· 1

32τ6

(

τ2 + 1)a−2

2

( 1

y22

− 1

y21

)

+

+∑

i=−2,0,1

[

(τ2 + 1)ai+1 − ai

]

·yi+1

2 − yi+11

i + 1+

+

[

(τ2 + 1)a0 − a−1

]

lny2

y1− a2

y32 − y3

1

3

(13)

wherey1 = (τ − 1)2, y2 = τ2 − 2τ cosζ + 1, a2 = 1,

a1 = −4, a0 = 6(τ4 + 1) − 4τ2, a−1 = −4(τ2 − 1)2,a−2 = (τ2 − 1)4.We have also calculated analytically a similar distri-bution dNis

1 /dt if the scattering was isotropic. i.e. iff (α) = 1

4π (Appendix A).One can also find analytically the angular distributiondMR

1dθ of the arriving light (integrated over time), but for

small angles only (Appendix B). The result is

dMR1 (θ; R)

dθ= 2πsinθ cosθ

∫ ∞

R/cjR1(θ, t; R)dt =

=9kRe−kR

64R2

1− 4θ3π+

8kRθ

3π

[

ln(kRθ) +CEu −12

]

(14)

whereθ ≪ 1, kR =RλR

, if there is no Mie scattering andCEu ≃ 0.577 is the Euler constant.The ratio of all photons arriving within a small angleθto those not scatteredN0, equals

1N0

∫ θ

0

dMR1

dθ′dθ′ ≃ 4πR2

e−kR

964

kRe−kR

R2θ = (15)

=9π16

kR θ(rad) ≃ 3.1 · 10−2kR θ(deg)

where terms∼ θ2 have been neglected.

2.1.2. Mie scatteringIn the next paragraph we shall consider the scattering

of light emitted by showers developing in the real at-mosphere, i.e. with the density depending on height.

In addition to the Rayleigh process one has to takeinto account the Mie scattering occurring on particles(aerosols) larger that the light wavelength. The Mie an-gular distribution is concentrated at rather small angles,in contrast to the Rayleigh case. Moreover, in the deeperparts of the atmosphere the mean free path length for theMie scattering may be comparable to that for Rayleigh,so that it is necessary to calculate the distribution of thelight scattered by the Mie process only.As before, we start with a simpler case - a uniformmedium. The angular distribution of light scattered onparticles with sizes larger than the light wavelength de-pends on the distribution of the sizes and is not a wellknown function. Roberts [9] adopts a function of theform

f (α) ∼ e−Bα +CeDα (16)

Here, however, we prefer an expression allowing us toperform some integrations analytically. Most crucial isto have the number of numerical integrations for the sec-ond generation as few as possible. We shall see that tofind jR2 (θ, t; R) for the Rayleigh scattering (Section 2.3)there is only one integration (overx′) to be done numer-ically since the form off R(α′) enables one to integrateanalytically overφ′ andθ′ (Eq. 25 and 26). Thus, weadopt the following form for the Mie angular distribu-tion:

f M1 (α) = a1cos8α + b for 0 ≤ α ≤

π

2

f M2 (α) = a2cos8α + b for

π

2≤ α ≤ π (17)

wherea1 = 0.857,a2 = 0.125,b = 0.025.This function is normalised as follows

∫ π

0f (α) · 2π sinαdα = 1 (18)

It describes quite reasonably the distribution used byRoberts.From Eq. 6 and 7 we have

jM1 (θ, t; R) =

ce−kτ · f Mi

[

α(θ, t)]

2πλMR2(τ2 − 2τ cosθ + 1)(19)

wherei = 1 if tgα2 =τ−cosθ

sinθ < 1andi = 2 if tgα2 > 1Since

cosα =1− tg2 α

2

1+ tg2 α2

=2 sin2θ

τ2 − 2τ cosθ + 1− 1 (20)

we obtain

jM1 (θ, t; R) =

c e−kτ

2πλMR2 y

ai

(2 sin2θ

y− 1

)8+ b

(21)

4

wherey = τ2−2τ cosθ+1, andi is determined as before.In principle, it is possible to find analytically the number

of photonsdNM

1 (t;ζ,R)dt arriving within an angleζ after time

t per unit time. However, each of the nine integrals

In =

∫ ζ

0

1y

sin2θ

y

n

sinθ cosθ dθ (22)

contains many terms itself, so that an analytical depen-dence onζ and/or τ would be practically lost. Thus, we

have founddNM

1dt by integrating the flux (Eq. 21) numer-

ically (similarly to Eq. 12).Finally, the total flux of the first generation is the sum ofthe two fluxes arising from the two active mechanismsof the scattering

j1(θ, t; R) = jR1 + jM1 (23)

2.2. The second generation

Figure 2: Geometry of the second and higher scatterings in a uniformmedium. The light source is atO and the detector is the surface of thesphere with radiusR. The picture shows the last scattering.

These are the photons scattered exactly two times.We shall consider first a general case when there aremore than one scattering processes (as Rayleigh andMie). Let us call the process of the first scattering asA and this of the second one asB. Both A andB can beeither Rayleigh or Mie. We denote their mean scatteringpath lengths byλA andλB, and the angular distributionfunctions of the scattering byf A(α) and f Bα), corre-spondingly. The light source flashes isotropically at thecentre of a sphere with radiusR at timet = 0 (Fig. 2).As before we want to calculate the number of photonscrossing the surface of the sphere from inside at a givenangleθ, at timet, per unit time.Let us consider the photons scattered for the secondtime at a distanceR′ from the source. The number ofphotons incident on a small surface∆S′ at an angleθ′

(within dΩ′) at time (t′, t′ + dt′) and scattered within a

spherical shell of thicknessdR′ by an angleα′ (withindΩ(α′)) equals

jA1 (θ′, t′; R′)dΩ′dt′∆S′|cosθ′|dxλB

f B(α′)dΩ (24)

wheredx = dR′/|cosθ′|. As now both processes areactive the meaning ofλ in the factore−

ct′λ in the expres-

sion for jA1 is the effective mean path length for bothprocesses.The direction of the scattered photons is at an angleαto the radius of the sphere anddΩ = sinαdαdφ, whereφ is the azimuth of the photons scattered for the secondtime. For any given direction (θ′, φ′) before and (α, 0)after the second scattering, the scattering angleα′ fulfilsthe relationcosα′ = cosα cosθ′ − sinα sinθ′ cosφ′. Theonly function depending on the azimuth angleφ′ of theincident photons isf B(α′). Denoting

FB(θ′, α) =∫ 2π

0f B(α′)dφ′ (25)

and integrating (25) overθ′ we obtain for the number ofphotons incident on∆S′ and scattered withindR′ intothe solid angledΩ(α) the following expression

dt′∆S′dR′dΩ∫ π

0jA1 (θ′, t′; R′)FB(θ′, α)sinθ′dθ′ =

= GAB(R′, α) · dt′∆S′dR′dΩ (26)

where the functionGAB(R′, α) is defined by the integralin the l.h.s. of (26).The pair of fixed variablesR′ andα defines another pairx′ and θ, wherex′ is the photon path length after thesecond scattering. The Jacobian of the transformationgives the relation

dR′ sinαdα =R2

R′2sinθ|cosθ|dx′dθ (27)

Putting∆S′ = 4πR′2, the contribution of photons scat-tered for the second time at a distance (R′,R′ + dR′) toarrive at an angle (θ, θ + dθ) at the sphere with radiusRequals

dnAB2 (θ, t, x′; R) = e−

x′λ ·GAB

[

R′(x′, θ), α(x′, θ)]

·

·4πR2 · 2πsinθ |cosθ| dx′dθdt (28)

The factore−x′/λ multiplied bye−ct′/λ in the expressionfor jA1 givese−ct/λ, independent ofx′. Integration overx′ gives the total number of the above photons

dnAB2 (θ, t; R) = (29)

= e−ctλ

∫ x′max

0GAB∗ dx′ · 4πR2 · 2π sinθ |cosθ|dθ dt

5

Figure 3: Comparison of the first (dN1/dτ) (three upper curves) and the second (dN2/dτ) (three lower curves) generations as functions of time(ǫ = τ − 1 = ct/R− 1). Number of photons are within angleζ. Uniform medium,R = 1. a). Rayleigh (solid lines) and isotropic (dashed lines)scattering.b). Two scattering processes at work: Rayleigh and Mie, each with λ = 2R.

whereGAB∗ = GAB/e−ct′/λ.

The maximum value ofx′ results from fixing timet. Wehave that

x′max= ct− R′ =R2·τ2 − 1τ − cosθ

(30)

Thus, the flux of the second generationABequals

jAB2 =

14πR2 · 2πsinθ |cosθ|

d2nAB2

dθ dt= (31)

= e−ctλ

∫ x′max

0GAB∗ dx′

and, integrated overθ for θ < ζ, givesdNAB2 (t; ζ,R)/dt.

With the Rayleigh and Mie processes active we musttake into account all four casesA = Ror M andB = RorM. Finally, the flux of the combined second generationphotons is a sum of all specific fluxes

j2 = jRR2 + jRM

2 + jMR2 + jMM

2 (32)

Some of the integrals defined in this Section can befound as analytical functions (Appendix C). It is ofsome importance when calculating higher generations(see the next Section).

2.3. The next generations

Any next generation of the scattered photons can becalculated in the same way as the second one has beenfound from the previous one (the first). To calculate theflux j i(θ, t; R) of the i − th generation, givenj i−1(θ, t; R)we proceed as before when calculating the second gen-eration from the first one (Eq. 24). The number ofphotons, incident on∆S′ at an angleθ′, φ′ within dΩ′

at time (t′, t′ + dt′) and scattered alongdx into dΩ(α)equals:

j i−1(θ′, t′; R′)dt′dΩ′∆S′|cosθ′|dxλi

fi(α′)dΩ (33)

If there are two scattering processes,R+ M, then

fi(α′)λi=

f R(α′)λR

+f M(α′)λM

(34)

is the scattering probability by an angleα′ by any pro-cess per unit distance per unit solid angle. The rest ofthe derivation ofj i(θ, t; R) is the same as in the previousSection. However, for each next generation the num-ber of numerical integrations increases, unless values ofj i−1(θ, t; R) are stored as a 3-dimension matrix. Thus, it

6

is convenient to find analytical solutions of the integralsF(θ′, α) and/or G(R′, α), if possible.

2.4. Results of calculationsFig. 3a shows the number of photons arriving at the

detector within an angleζ < 1, 3, 10 per unit areaper unitτ = ct/R as a function ofǫ = τ − 1. The uppercurves refer to the first generation, the lower - to the sec-ond one. We also compare here the time distributionsobtained for the Rayleigh with those for the isotropicscattering. The distance detector- source equals to onescattering length (k = R/λ = 1).First of all we notice that for short times (ǫ ≤ 0.01) thefirst generation dominates over the second one, and (aswe can guess) over the higher ones. It can be seen fromthe formulae for the isotropic scattering (Appendix A)

that the ratiodNis

2dτ /

∆Nis1

dτ for any given time should be pro-portional tok = R/λ, so that the importance of the sec-ond (and the higher) generation will be bigger for largerk.We can also see that the number of photons arrivingwithin an opening angleζ reaches the dependence∼ ζ2only at later times. This reflects the fact that the ini-tial angular distribution of light is steep and becomesalmost flat at timesτ ≥ 1.1 or so. When comparing theRayleigh curves with the isotropic ones one can see thatthe latter are slightly flatter for shorter times, as mightbe expected but become parallel to the former for latertimes.Next, we consider a situation when there are two scatter-ing processes, Rayleigh and Mie with quite different an-gular distributionsf (α) (as discussed before). We adopt

k =Rλtot= R(

1λR+

1λM

) = 1 (35)

andλR = λM for simplicity.The result for the first and the second generation de-pending on time is shown in Fig. 3b. There is nowmore light at earlier times than in the previous case (Fig.3a) due to the strong Mie scattering in the forward di-rections. However, the flux of the first generation de-creases about 3 times quicker over the considered timeregion. Although the ratio of the second to the first gen-eration is practically the same atǫ = 10−3 in both cases(≤ 1%). the importance of the second one is reached atlater times when the Mie scattering is present.

3. Point light source in the atmosphere

Now we shall study the situation when a point lightsource flashes in a non-uniform medium, such as the at-mosphere. We assume that the atmosphere is composed

Figure 4: Vertical cross-section through the atmosphere. Lines cor-respond to constant values ofk = R/λPD, shown by numbers, look-ing from the detector (atx = 0, h = 0) to a point on the line, forλR = 18km and λM = 15km at the ground and the scale heightsHR = 9kmandHM = 1.2km.

of two sorts of matter, molecules and aerosols, eachhaving its density decreasing with height exponentiallywith a different scale heights,HR - for molecules andHM for aerosols, and having the corresponding meanpath lengths for scattering at the groundλR

D andλMD . It

is not difficult to derive that the effective mean free pathfor a scattering for light travelling between two arbitrarypointsP andS equals

λPS =

( 1

λRPS

+1

λMPS

)−1= (36)

=hP − hS

HR

λRD

(

e−hS/HR − e−hP/HR

)

+HM

λMD

(

e−hS/HM − ehP/HM

)

wherehP andhS are the heights of pointsP andS abovethe level, where the Rayleigh and Mie scattering pathlengths are correspondinglyλR

D andλMD . If the source

(point P) is at a distanceR from the detector (pointDon the ground) then the ratiok, of R to the mean freepath length alongPD equals

k =RλPD= (37)

=1

cosθZ

[HR

λRD

(

1− e−hP/HR)

+HM

λMD

(

1− e−hP/HM)

]

It can be seen that increasing the distanceR to infinity(keepingθZ constant) the ratiok reaches its maximumfinite value

kmax(θZ) =1

cosθZ

(HR

λRD

+HM

λMD

)

(38)

7

This situation is illustrated in Fig. 4. Here a verticalcross-section of the atmosphere is shown. Detector isat x = 0, h = 0 and lines represent constant values ofk corresponding to the straight path from the detectorto the point on the line. We have adopted the follow-ing values:λR

D = 18km,HR = 9km, λMD = 15km and

HM = 1.2km. These values describe approximately theatmospheric conditions at the Pierre Auger Observatory[2].From Fig. 4 one can also deduce that relevant values ofk, if light sources are at distances∼ (10− 30)km(as ex-tensive air showers seen by Auger) are 1/2 ≤ k ≤ 3/2.Thus, this will be the region of our interest.

Figure 5: Geometry of the first generation in the real atmosphere.Light source is atP, detector atD. Scattering takes place atS.

Figure 6: The ellipsoids (their cross-sections are shown) show scatter-ing sites of first generation photons arriving atD after time R

c (1+ ǫ).The corresponding numbers are equal toǫ. Light source is atP, de-tector atD.

3.1. The first generation

As the medium is non-uniform, we cannot use theidea of a sphere to be crossed by the scattered pho-tons, as in Section 2. Now their flux will depend on

the zenith angleθZ of the source and on the azimuth an-gle φ around the direction towards it. Fig. 5 shows atrajectoryPS Dof a first generation photon scattered atS. We want to calculate the angular distribution of thefirst generation as a function of time, for a fixedR andθZ, d2n1

dΩdt(θ, φ, t; R, θZ), crossing a unit area perpendicularto the direction towards the source.We notice that for a fixed arrival direction of photons(θ, φ) and timet, the scattering pointS is uniquely de-termined. To arrive at the detector at angles (θ, φ) withindΩ = sinθ dθ dφ after time (t, t + dt), photons have tocross the surfaceda (shaded in the figure) and be scat-tered along a path lengthdx by the angleα determinedby Eq.2. The number of such photons equals

dn1(θ, φ, t) =x′sinθ dφ dx′cosγ

4πx2e−

xλPS · (39)

·dxλS

f (α)∆ΩDe−x′λS D

where x = PS, x′ = S D, γ is the angle betweenthe normal to the surfaceda and the direction of theincident photonsPS, ∆ΩD is the solid angle deter-mined by the unit area atD and the scattering pointS(∆ΩD = cosθ/x′2). There is no need to calculateγ be-causedx = x′dθ/cosγ, so that it cancels out. It can beshown that

dx′

x2=

2τ2 − 2τ cosθ + 1

dτR

(40)

Inserting this into (40) we obtain

j1(θ, φ, t) · cosθ =d2n1

dΩdt= (41)

=c

2πλSR2

f (α)cosθτ2 − 2τ cosθ + 1

· e−( xλPS+ x′λS D

)

One can see that this formula is practically the same as(7) for the uniform medium, the only difference beingin the scattering path lengths depending not only on dis-tances but also on the geometry.The height of the scattering pointS necessary to calcu-lateλS, λPS andλS D equals:

hs =

(

ct− Rsinθsinα

)

· (42)

·(cosθZ cosθ + sinθZ sinθ cosφ)

We calculate numerically the time distributions of lightdNreal

1 (t; ζ)/dt, arriving at the detector at angles smallerthanζ for different zenith angles of the source. The re-sults, in the form of the ratio:

F1(τ; ζ) =dNreal

1 /dt

dNuni1 /dt

(43)

8

are presented in Figs. 7, 8 and 9, wheredNuni1 /dt are

the distributions obtained in the previous section for auniform medium.

Figure 7: Ratio of the first generationdNreal1 /dt in real atmosphere to

that in uniform mediumdNuni1 /dt as a function of time (ǫ = ct/R− 1)

for the Rayleigh scattering only for various values of zenith angle ofthe source. Solid lines -ζ = 1, dashed lines -ζ = 10, k = R/λR

PD =

1/2.

Figure 8: As in Fig. 7 but with Mie included;k = R/λPD = 1/2.

To compare the light flux obtained for the real atmo-sphere with that for a uniform medium we adopt thesame value ofk and the same distance from the sourceto detector for both cases. To understand the effect ofa purely exponential atmosphere we start with consid-ering only the Rayleigh scattering (Fig. 7), neglectingMie (λM = ∞). Understanding the behaviour of thecurves in this figure is easier with the help of Fig. 6.Each ellipse is a cross-section of a rotational ellipsoidwith the symmetry axis determined by pointD - the de-tector and pointP - the light source. These are the focalpoints of all the ellipses. Photons arriving at the detec-tor atD after some fixed timeτ = 1+ ǫ (in units of the

Figure 9: As in Fig. 8 but for two values ofk. Each group of lines isfor ζ = 1, 2, 3, 5, and 10 from bottom to top (atǫ = 10−3).

distance PD) must have been scattered on the surface ofan ellipsoid with the eccentricitye equal

e=1

1+ ǫ(44)

The ellipses refer toǫ = 10−3, 10−2, 10−1 and 3· 10−1

keeping the right proportions. The detector field of viewcuts only a part of the ellipsoid surface where the pho-tons registered after time 1+ǫ must have been scattered.We notice that all ratiosF1 in Fig. 7 are smaller than 1and decrease with time (although those forζ = 1 arepractically constant) and ratios forζ = 10 are largerthan those forζ = 1. All this becomes clear when in-specting Fig. 6 and the corresponding scattering sites.For example - the constancy ofF1(τ; ζ = 1) for timesτ − 1 = ǫ = 10−3 ÷ 10−1 reflects the fact that the scat-tering takes place very close to pointP during all thistime and starts to move away from it (higher in the at-mosphere) only for larger times i.eτ ≥ 1.3. Since thescattering path length in the uniform medium is chosenequal to the effective path length in the atmosphereλPD

(Eq. 36), we have thatλP > λPD and the scattering prob-ability at P is smaller in the exponential atmosphere.Let us consider now a more realistic atmosphere withboth processes, Rayleigh and Mie at work. Fig. 8 showstime dependence of the ratioF1(τ; ζ) for ζ = 1 and 10

and zenith anglesθZ = 10 ÷ 75. Let us take a closerlook at the caseζ = 1 andθZ = 75 (upper solid curve).It may seem strange that the ratioF1 increases sincein the case of the Rayleigh scattering only it decreases,although very slowly. We have checked that a similarslow decrease takes place if the scattering is only Mie.The behaviour of the curves in this figure would be hardto understand without the presentation of the scatteringsites in Fig. 6. It can be seen that forǫ smaller than a

9

few ×10−2 the scattering sites are close to the source atP. Thus, we may approximate the ratioF1 as follows

F1 =≃f R

λRP+

f M

λMP

f R

λRPD+

f M

λMPD

=λR

PD

λRP

·1+

λRP

λMP

f M

f R

1+λR

PD

λMPD

f M

f R

(45)

where f R(M) are some effective angular distributions ofphotons scattered by Rayleigh (Mie) on the cut surface .As time increases, none of theλ′schanges much. How-ever, the typical scattering angles increase, what affectsmuch moref M than f R, so thatf M/ f R decreases. Since

λRP

λMP

<λR

PD

λMPD

(46)

the numerator decreases by a smaller factor than the de-nominator so that the ratioF1 increases. It can be seenfrom Fig. 6 that forǫ ≥ 2 · 10−2 the scattering anglesof the registered photons do not change much (the de-nominator stays constant) but nowλR

P andλMP have to

be substituted byλRS andλM

S , whereS is an effectivescattering point with growing height. AsHM < HR, theratioλR

S/λMS decreases and so doesF1.

A different behaviour ofF1(τ; ζ = 10) can be explainedalso with the help of Fig. 6. At firstF1 decreases (thesmallerθZ - the stronger decrease) because the detec-tor field of view cuts out a growing part of the deepatmosphere where the scattering is strong in the real at-mosphere. Atǫ ≥ 0.02 F1 starts to increase for thesame reason as just described in the caseζ = 1. Itmust finally decrease since the scattering takes placesfurther and further behind the source, whereλR andλM

are growing in the real atmosphere.In Fig. 9 we showF1(τ; ζ) for θZ = 60 for several inter-mediate values ofζ, and for two valuesk = 1/2 and 1.Note that changingk = R/λPD must result in changingR- the distance to the source. This is the main reason whythe curves fork = 1 are lower than those fork = 1/2.It can be seen from Fig. 5 that forθZ = 60 (the scaleson both axes are the same so in the figure the angles arecorrectly represented) the distanceR is much shorter fork = 1/2 than fork = 1, implying that the correspond-ing heights of the source differ considerably. InspectingFig. 6 we can estimate that forǫ ≃ 0.01 andζ = 1 thecurves fork = 1 should be down with respect to thosefor k = 1/2 by a factor

λP1D

λP1

/λP1/2D

λP1/2

(47)

whereP1/2 andP1 are positions of the source referringto k = 1/2 and 1 respectively, with the meaning of all

λ′s as defined before. In our exampleλP1D ≃ 33km,λP1 ≃ 113km, λP1/2D ≃ 15.3km, λP1/2 ≃ 25.6kmso thatthe above factor equals≃ 0.49 whereas the exact ratiofrom Fig. 9≃ 0.48.

3.2. The second generation

Figure 10: Ratio of the seconddN2/dt to the first generationdN1/dtas a function of time (ǫ = ct/R− 1) from a flash (att = 0) of a pointsource at zenith angleθZ = 75 in the real atmosphere. Fluxes areintegrated withinζ = 1 (solid lines) andζ = 10 (dashed lines).Three curves for eachζ refer tok = 1/2, 1, 3/2 (from bottom to top).Dotted line refers to a uniform medium with Rayleigh scattering only,for k = 1, ζ = 1.

We proceed like in the case of the first generation(Fig. 5), but now pointS refers to the second scatter-ing. Photons scattered only once arrive at the surfacedafrom all directions according toj1(θ1, φ1, t1; x), wheret1 = t − x′/c. Thus, the number of photonsdn2 inci-dent onda and scattered for the second time towardsthe detector (to arrive there withindΩD(θ, φ) after time(t, t + dt) equals

dn2(θ, φ, t; x′) =∫

Ω1

j1(θ1, φ1, t; x)dΩ1dt da cosγ·

· dlλS

f (α2)dΩDe−x′λS D (48)

where the integration has to be done over full solid angle(0 < φ1 < 2π, 0 ≤ θ1 ≤ π), dl is the path length for thesecond scattering to occur (cosγ dl = x′ dθ) andα2 isthe angle of the second scattering. Now the pair of vari-ables,θ andt, does not determine uniquely the positionsof the second scattering, since the timest1 elapsed fromphoton emission to their arrival atS (or strictly speak-ing, atda) have some distribution. However,t1 can notbe smaller thanx/c, thusx′max= ct− x. Expressingx′maxas a function ofθ, t andRonly we obtain

x′max=(ct)2 − R2

2(ct− R cosθ)(49)

10

and the distribution of the second generation equals

d2n2(θ, φ, t)dΩ dt

=

∫ x′max

0dx′

∫ 2π

0dφ· (50)

·∫ π

0dθ1 j1 sinθ1 ·

f (α2)λS

e−x′λS D cosθ

Finally, the number of photons within an angleζequals

dN2(t; ζ)dt

=

∫ 2π

0

∫ π

0

d2n2

dΩ dtsinθ dθ dφ (51)

The above integrals have been calculated numerically.The ratiodN2

dt /dN1dt as a function ofτ−1 in the real atmo-

sphere, where the source is atθZ = 60 at the distanceR = 34km corresponding tok = 1, within two open-ing angles of the detectorζ = 1 and 10, is presentedin Fig. 10. For comparison we have also drawn therethe same ratio for the uniform medium, fork = 1 withRayleigh scattering active only (following from Fig. 3afor ζ = 1). First we see that the contribution of thesecond generation increases with time as it should beexpected. Next, it is considerably larger for the uniformmedium than for the real atmosphere, which may notbe so obvious at first sight. In fact, atǫ = 10−2 ÷ 10−1

the curve for the uniform medium is∼ 2.3 times higherthan that for the real atmosphere. This factor equals (for simplicity we neglect the operatorsddt)

η =Nuni

2 (R)

Nuni1 (R)

/Nreal

2 (R+ M)

Nreal1 (R+ M)

= (52)

=Nreal

1 (R+ M)

Nuni1 (R)

/Nreal

2 (R+ M)

Nuni2 (R)

The numeratorNreal

1

Nuni1∼ λP1D

λP1= 0.29 as explained in the

previous paragraph. The denominator refers to photonsscattered exactly two times. For smallǫ the scatteringsmust take place close to the source or along the field ofview (ζ = 1 in our example). Thus we should have that

Nreal2 (R+ M)

Nuni2 (R)

≥(

λP1D

λP1

)2= 0.084 (53)

Our calculations show thatNreal

2

Nuni2≃ 0.13 for ǫ = few

×10−2 so thatη ∼ 0.290.13 ≃ 2.2, in agreement with the

exact calculations.

4. A moving point source - a shower

So far we have been interested in the light signalsfrom a flash of a stationary point source in a detector

at a given distance, within a given field of view, as afunction of time.Now, we shall consider a moving point emitting lighton its way through the atmosphere. This is a reason-able model of a distant cosmic ray shower exciting at-mosphere what results in emitting isotropic fluorescencelight. Moreover, the Cherenkov light produced by∼ 1/3of shower electrons [5], propagating alongside the par-ticles will be scattered to the sides by the Rayleigh andMie processes in an anisotropic way. Thus, in principle,a shower is a moving source, emitting light anisotrop-ically. Our aim is to find an instantaneous optical im-age of the shower, produced by the scattered light atthesame timeas the shower image in the direct (not scat-tered) light.The image obtained for a small integration time∆t is aradially symmetric light spot, corresponding (roughly)to the emission distribution at the shower lateral cross-section [12]. It is produced mainly by the direct light.But at the same time as the direct light, there arrive alsophotons produced by shower particles at an earlier stageof shower evolution and scattered in the atmosphere.As shower parameters follow straightforwardly from theamount of the direct light, it is important to calculatethis effect and to take it into account when deriving theflux of the direct light.Let us assume that we know the geometry and the tim-ing of the shower. If the source is at pointQ at time tand emitsC photons per unit length, the telescope willrecord the following number of the direct photons attime t + R/c, in the time bin∆t:

∆n0 = C ·c∆t

1− cosδfP(δ)

AR2

e− RλQD (54)

whereδ is the angle between the shower direction andthe direction from pointQ to the detectorD, λQD is theeffective mean free path for attenuation along the dis-tanceR, A is the diaphragm area determining the collec-tion solid angle andfP(δ) is the angular distribution ofthe emitted light.In the case of isotropic fluorescence lightfP(δ) = 1/4πandC equals to the number of photons produced perunit length (∼ 4m−1, per one electron). In the case ofthe scattered Cherenkov photons, or those from a laser,fP(δ) = 3

16π (1 + cos2δ) for the Rayleigh scattering (ora corresponding distribution for Mie) andC = N/λQ,whereλQ is the mean free path (in length units) forthe process in consideration at pointQ and N is thenumber of Cherenkov photons at the observed showerlevel propagating (approximately) in the direction of theshower. The number of the scattered Cherenkov pho-tons to the fluorescence ones is typically 10− 15%, and

11

a lot of the former are scattered by Rayleigh, what isnot far from isotropic. Thus, with a good approxima-tion one can treat a shower as a moving isotropic lightsource. This is what has been adopted in this paper:fP(δ) = 1

4π .Simultaneously with the direct light some photons pro-duced earlier (above pointQ) will also arrive at the tele-scope. However, as we already mentioned, they musthave been scattered on their way to the telescope. Ourearlier calculations [10] showed that if the distance tothe shower is not very much longer than the mean scat-tering path, then the main contribution to the scatteredlight is due to the photons scattered only once (the firstgeneration). Here, we will calculate the fraction of thetotal light consisting of the first and the second genera-tions.

4.1. The first generationLet us consider photons produced at pointP, along a

shower path elementdl (Fig. 11). The first generationphotons arriving at the detectorD at the same time asthe direct photons produced atQ must have been scat-tered on the surface of the rotational ellipsoid with thefocal points atD andP, with theDP line being the ro-tational symmetry axis of the ellipsoid.As we have already mentioned, the main interest of theshower experiments is to measure the direct light. Thus,we are interested in the light arriving from the directionof point Q and the region around it. In practice (as it isin The Pierre Augere Experiment) the light signal arriv-ing simultaneously with the direct light from pointQ ismeasured within a certain angleζ around the lineDQ.The cone with the opening angleζ cuts out on the sur-face of the ellipsoid a region of our interest (the shadedsurface). It is at this part of the ellipsoid that the pho-tons produced atP have to be scattered to arrive simul-taneously with the direct light produced atQ, within anangle smaller thanζ with respect to the latter. Thus, tofind the contribution of the first generation to the directlight one has to integrate the number of the scatteredphotons over the cutout surface, and then integrate theresult over the distanceQP = l (the upper part of theshower).

The proportions in Fig. 11 have not been preserved.For example the actual distance from pointP to thedetector atD is much longer thanl = QP (althoughin principle the integration overl should go far up theshower, the distances contributing to the total scatteredlight are rather close to pointQ). Also the angleζ doesnot need to be larger than a few degrees, but on the fig-ure it is much larger for the sake of clarity.

Figure 11: First generation from a shower. The shaded surface showssites of scattering of photons produced atP, scattered once and arriv-ing at detectorD within its field of view simultaneously with directphotons produced atQ. PQ+ QD = PP1 + P1D.

Let us first find the contribution to the first generationfrom photons produced by the shower at pointP along apath length elementdl, assuming thatP lies outside thedetector field of view.

a) θP > ζ (Fig. 11).

d(dn1

dt

)

=cCdl2πR′2

·∫ θP+ζ

θP−ζdθ

f (α) sinθ cosθτ′2 − 2τ′cosθ + 1

·

·∫ φmax(θ)

0

e−(

PP1λPP1+

P1DλP1D

)

λP1

dφ

(55)

where τ′ = l+RR′ , tg(α/2) = τ′−cosθ

sinθ , cosφmax =

Figure 12: Fluorescence light produced at pointP and scattered onthe shaded surface (inside field of view of detectorD) of the ellipsoidarrives atD simultaneously with the direct light produced atQ.

12

cosζ−cosθP cosθsinθP sinθ , θ is the arrival angle (with respect to

the direction to the source atP), and R′ = PD =√R2 + l2 − 2Rlcosδ.

The azimuth angleφ is measured in the plane perpen-dicular to the axisPD and φ = 0 refer to points onthe shower-detector plane (Fig. 11 plane). The lengthPP1 can be found from the trianglePP1D: PP1 =

R′sinθ/sinα andP1D = l + R− PP1.

b) θP < ζ. (Fig. 12).It is clear from the figure that now the integration limitsof θ andφ are different so that one has to add to theintegral in case a) the following term

cCdl2πR′2

∫ ζ−θP

0dθ

f (α) sinθ cosθτ′2 − 2τ′cosθ + 1

· (56)

·2∫ π

0

e−(

PP1λPP1+

P1DλP1D

)

λP1

dφ

and the lower limit ofθ in the term a) changes sign, sothatθP − ζ → ζ − θP. To find the total flux of the firstgeneration arriving at the same time as the direct lightfrom point Q one has to integrate overl the contribu-tion from case b) from 0 tolmax = Rsinζ/sin(δ − ζ),(whereδ ≥ ζ), and add to it the contribution from a)integrated fromlmax to∞. (It turns out that contributionfrom points withl ≃ 0.1Rare negligible).If there are two scattering processes the first generationmeans the sum of the number of photons scattered byRayleigh and those by Mie and thenf (α)

λP1=

f R(α)λR

P1

+f M (α)λM

P1

,

and in the attenuation exponentλPP1 andλP1D are the ef-fective mean free paths for both processes.

4.2. Analytical calculation ofdnR

1 (ζ)dt

By making some approximations we have found ananalytical solution of the shower image in the first gen-eration light scattered by the Rayleigh process.We shall treat separately light produced within the de-

tector field of viewdnR

in

dt and that outside it,dnRout

dt , startingwith the former (Fig. 12). As the scattering points lieclose to pointQ we assume thatλP1 = λQ. For the samereason we assume that the exponential factor describ-

ing light attenuation equalse− RλQD . Our main assump-

tion, however, consists in integrating over the symmet-ric (with respect to axisPD) part of the ellipsoid so thatwe could solve the integrals overφ, θ and finallyl. Ex-pressing scattering angleα as a function ofθ andτ′ (Eq.

2) we obtain that the contribution from a shower pathelementdl equals

d(dnR

in

dt

)

≃ BdlR′2

2π38π· (57)

·∫ θmax

0

(

1− 2sin2θ

y′+

2sin4θ

y′2) sinθ cosθ dθ

y′

wherey′ = τ′2 − 2τ′cosθ + 1, B = cC2πλQ

e− RλQD .

We adoptθmax= ζ being the mean of the limiting valuesof θ in the exact integration.In the air shower experiments it is the direct light whichshould be measured so that the viewing angleζ is small(a few degrees) and so are anglesθ. This leads toτ′−1 ≡ǫ ≪ 1. Taking all this into account the result of theintegration overθ is

d(dnR

in

dt

)

≃cCe

− RλQD

λQ

38π· (58)

·12

lnx2 + 1

x2− 1

(x2 + 1)2

dlR′2

wherex = ǫ/ζ. From geometrical considerations it canbe derived that

dlR′2=

1R

11− cosδ

− ǫ cosδ(1− cosδ)2

dτ′ ≃ (59)

≃ dτ′

R(1− cosδ)=

ζdxR(1− cosδ)

where we have dropped the second term∼ ǫ.To obtain

dnRin

dt it remains to integrate (59) overx (includ-ing (60) from 0 toxmax, corresponding to pointP lyingon the edge of the viewing cone (θP = ζ). Again, as-suming thatζ ≪ 1, it can be derived thatxmax ≃ tgδ−ζ2(for δ > ζ). The result of the integration is the following

dnRin

dt≃

cCe− RλQD ζ

4πRλQ(1− cosδ)· (60)

· 316

(

3δ′ − sinδ′ − 8tgδ′

2· ln sin

δ′

2

)

whereδ′ = δ − ζ.To calculate the contribution from the outer part of the

shower i.e. outside the field of view we keep the previ-ous approximate assumptions. However, since now thescattering surface looks differently (Fig. 11) we makeanother simplifications. We assume that all scatteringangles equalδ so that f (α) = f (δ) = const. Next, asζis small we assume that allθ = θP. Integration overΩD,the solid angle of the field of view, is then reduced to

13

Figure 13: Second generation from a shower. Photon, emittedat Pmust be scattered for the second time (pointP2) within detectorDfield of view. First scattering (P1) must occur on the surface of theellipsoid with focal points atP andP2. Direct light fromQ arrives atD at the same time:PQ+ QD = PP1 + P1P2 + P2D.

multiplying the integrand forθ = θP byΩD = πζ2. Thus

we have

d(dnR

out

dt

)

=cCe

− RλQD dl

2πλQR′2·∫

ΩD

f (α)dΩ(θ, φ) cosθτ′2 − 2τ′cosθ + 1

≃

≃ cCe− RλQD

2πf R(δ)λQ

πζ2cosθPτ′2 − 2τ′cosθ + 1

dlR′2

(61)

It can be derived that the denominator in the above ex-pression equals

R′2(τ′2 − 2τ′cosθ + 1) ≃ (1+ sin2δ − cosδ)l2 (62)

Assuming1 thatcosθP = 1, we obtain

d(dnR

out

dt

)

≃ cCe− RλQD

2πf R(δ)λQ·

· πζ2

1+ sin2δ − cosδ

∫ ∞

lmin

dll2

(63)

Sincelmin ≃ Rζsin(δ−ζ) we have

dnRout

dt≃ cCe

− RλQD ζ

2πRλQ

316

(1+ cos2δ)sin(δ − ζ)1+ sin2δ − cosδ

(64)

1TakingcosθP ≃ 1− 12 sinδ( l

R)2 needs to allow for light attenuationalong l in the integration overl. The correction todnout/dt is a fewpercent, being negligible for the totaldnR

1/dt.

anddnR

1

dt=

dnRin

dt+

dnRout

dt(65)

Expressing the number of photons∆nR1 =

dnR1

dt ∆t as theratio to that of the direct light (Eq. 54 forf (δ) = 1/4π)we finally obtain

∆nR1

∆n0≃

316

RλQζ

3δ′ − sinδ′ − 8tgδ′

2· ln(sin

δ′

2)

+ 2sinδ′(1+ cos2δ)(1− cosδ)

1+ sin2δ − cosδ

(66)

whereδ′ = δ − ζ (in radians). Terms∝ ζ2 and of higherorder have been neglected. We can see that, with the ap-proximations adopted, the ratio∆nR

1/∆n0 is proportionalto Rζ/λQ reflecting the fact (as we shall see below) thatmost scatterings take place close to pointQ sinceRζis proportional to the shower path segment seen by thedetector. Forζ ≪ δ, δ′ ≃ δ and the dependence onδ separates from that on other parameters. It becomesalso obvious that it must be different for different an-gular distributionsf (α), of photons at scattering. Asf R(α) ∝ 1+ cos2α is quite different fromf M(α) whichis peaked in forward directions, we expect another de-pendence onδ for the latter.While deriving the analytical formula (66) we have alsoassumed that the length of the shower segment cut outby the detector field of view is small when compared tothe distanceR. Forζ ≪ 1 it is fulfilled for almost allδ,apart from the case whenδ is close to 0 orπ. We havecalculated analytically the ratio∆nR

1/∆n0 for δ = π − γfor γ → 0. However, as the derivation is lengthy and(as it will be seen later) agrees with the exact, numeri-cal results only atδ = π, we present here only the finalresult:

∆nR1

∆n0−−−−−→δ→ π

916

kπ ζ(

1− 73πζ)

(67)

These values are marked as stars (forδ = π) in Fig. 15.We have not tried to calculate∆nM

1 /dt because the func-tion f M(α) is not well known, depending on the sizesof the aerosol particles and, anyway, our analytical ap-proach is approximate. Thus, it seems better to calculateit exactly numerically and, being led by our analyticalsolutions for Rayleigh, find an appropriate parametrisa-tion of the numerical results.

4.3. Numerical (exact) calculations of∆n1/∆n0 and∆n2/∆n0

The ratio of the first generation to the direct light ar-riving simultaneously is obtained by numerical integra-tion of point contributions overl (shower track above

14

Figure 14: Ratio of the first generation to the direct light asa function of distanceR to point Q on the shower in units of the scattering lengthλQat that point.a). Rayleigh onlyb). Mie only. Curves correspond toζ = 1, 3, 5 (from bottom to top) andδ = 90. Various point shapes refer todifferent distancesR=12, 24, 32km. Lines are power law fits.

point Q ) and dividing the result by∆n0. Fig. 14 showsthe ratio for the Rayleigh a) and Mie b) scattering as afunction of kQ = R/λQ for ζ = 1, 3 and 5 and forseveral different distancesR. It can be seen that for theRayleigh case the ratio is proportional tokQ andζ, as ithas been derived analytically. For Mie the dependencefollows a power law, with the indices depending slightlyon ζ. However, in each case there is practically no de-pendence on the distanceR itself. These results are forδ = 90.The dependence onδ is shown in Fig. 15a,b forkQ =

1 andζ = 1, 3 and 5. One can see the difference ofthe behaviour of this dependence between Rayleigh andMie. In the figure a comparison of our analytical calcu-lations with the exact ones is also shown. The biggestdifference reaches some 10% at largeδ. There is nonormalisation there. We find this agreement quite satis-factory.Nevertheless, as the agreement is not perfect we haveparametrised the exact numerical results. For theRayleigh scattering the ratio∆nR

1/∆n0 can be expressed

as

∆nR1

∆n0= 0.024· kQ · ζ gR(x) (68)

wherex = δ/100, ζ is in degrees, and

gR(x) = 0.112+ 1.86x− 1.33x2 + 0.383x3 (69)

For Mie the factorisation is not as complete as forRayleigh, and our fit to the numerical results is the fol-lowing:

∆nM1

∆n0= (70)

= 0.096·( kQ

0.7

)0.93−0.04ζ·( ζ

3)0.93+0.05 δ

90 · gM(x)

x ≤ 0.6 gM(x) = x(7.76x2 − 11.25x+ 5.625)

x > 0.6 gM(x) = 1 (71)

These fits are also shown in Fig. 14 and 15. It is seenthat they do not deviate from the exact values by morethan a few percent.The second generation has been calculated numerically.

15

Figure 15: Ratio of the first generation to the direct light asfunction of angleδ between shower and line of sight.a). Rayleigh onlyb). Mie only.Solid lines - exact numerical calculations, dotted lines and stars in a) - analytical (approximate) calculations, dashed lines - our parametrisation ofthe numerical curves. All curves are forkQ = R/λQ = 1.

Fig. 13 should be helpful for understanding the calcula-tion. The thin solid line shows a path of a photon pro-duced atP, scattered at pointsP1 andP2 and arriving atD. The pointP2 of the second scattering must lie some-where inside the viewing cone. Fixing it at a distancer3

from the detector corresponds to fixing the value of theremaining photon path lengthPP1P2. We have that

PP1 + P1P2 = R+ l − r3 (72)

wherel = PQ andR= DQ.In order to assure that the photons scattered twice ar-rive at D at the same time as the direct light emittedat Q, point P1 must lie anywhere on the surface of theellipsoid with focal points atP andP2. Thus, to calcu-late the number of photons produced atP and scatteredtwice one has to integrate the contributions like in (49)over the position of the first scattering (the whole sur-face of the ellipsoid) and then integrate the result overthe volume of the viewing cone. Finally, the integrationhas to be performed over the distancel = PQ.The ratio∆n2/∆n0, including Rayleigh and Mie scatter-ings as a function ofkQ = R/λQ for δ = 90 and as a

function ofδ for kQ = 1 is presented in Fig. 16. Variouspoint signs refer to different distancesR. As in the caseof the first generation the ratios depend onR/λQ ratherthanR itself. However, now they are proportional to thesecond power of it. We have parametrise the obtainednumerical results as follows

∆n2

∆n0= 1.2 · 10−3 k2

Q ζ1.77(deg) ·

δ(deg)90

(73)

5. Discussion of shower results

5.1. Dependence of∆n1+∆n2∆n0

on height

As we see the numbers of the scattered photons,∆n1

and∆n2, depend on the angleζ within which they ar-rive at the detector. This angle has to be chosen in sucha way as to encompass the total direct signal from theshower. It is obvious that for closer showersζ has to belarger and for those more distant - smaller, so thatRζ- meaning the lateral spread of the direct light - shouldremain constant. Our approximations show that in thiscase (Rζ = const) nR

1/n0 depends only on height (byλQ)andδ. The power indices in the dependence of∆n1/∆n0

16

Figure 16: a). Ratio of the second generation to the direct light as function of kQ. b). Ratio of the second generation to the direct light as functionof angleδ between shower and line of sight. Points - results of numerical calculations, lines - our parametrisation. Different point signs refer todifferent distances (R= 12, 24, 32km); kQ = R/λQ = 1.

on R andζ are close to 1, whereas∆n2/∆n0 ∼ (R · ζ)2

(roughly), so that their dependence on the distanceR it-self should be weak. Fig. 17 presents the contributionsof the individual components of the scattered light as afunction of height, forRζ = 30km·deg, δ = 90 and two(quite different) values ofR: 12 and 36km. The depen-dence of the sum∆n1+∆n2

∆n0onR is very weak, indeed, the

more so as the ratios∆nM1 /∆n0 and∆n2/∆n0 change in

opposite directions. The chosen valueRζ = 30km degcorresponds to a shower lateral radius of∼ 500m, equalto ∼ 5 rM (Moliere radii). This may seem too large tocontain the total fluorescence signal. However, deep inthe atmosphere it is the shower Cherenkov light whatdetermines its lateral dimensions and it goes up to 5rM

[8].As the typical viewing anglesδ are not far from 90 onecan draw a conclusion that the maximum contributionof the MS light is∼ 14% forRζ = 30km degand theadopted atmospheric parameters.

5.2. Influence on a reconstruction of shower parame-ters, E0 and Xmax

Let us estimate how not allowing for the MS effectwould affect a reconstructed value of a shower primaryenergyE0. The total signal would then be treated asthe direct light, therefore the primary energy would beoverestimated. To estimate how much it would be let usconsider a typical (for Auger) shower withE0 = 1019

eV, falling to the ground 25kmaway from the light de-tector, atθZ = 40 and azimuthal angle 90 (angle be-tween the direction from the detector to the shower coreand the projection of the shower axis on the ground). Ifthe primary is a proton the mean depth of its maximumequalsXmax ≃ 790g cm−2, corresponding to the heightabove Auger level (∼ 860g cm−2) hmax ≃ 3.2km. FromFig. 17 we see that the contribution of the scatteredlight is ∼ 4.5% if Rζ = 30km deg, what correspondsto ζ = 1.2. However, in the Auger telescopes the indi-vidual pixel has a diameter of 1.5, so that it would benecessary to choose also the neighbouring pixels aroundthe one with the stronger signal to collect the total directlight. Thus, we chooseζ = 3

2 · 1.5 = 2.25. Within this

angle the MS light grows to∼ 9%, overestimating the

17

Figure 17: Ratio of the first two generations to the direct light as afunction of height of the observed shower. Solid lines -R = 12km,dashed linesR = 36km. Curves denoted by ”Rayleigh” and ”Mie”refer to the first generation, by ”n2” - to the second one. If a constantlateral dimension is chosen (Rζ = const, δ = 90) the ratios do not(practically) depend on shower distanceR.

number of particles atXmax by this value. For showerparts near the ground, where shower age equals 1.25,this number grows to∼ 27%. To find the shower pri-mary energyE0 (by integrating the energy deposit overdepthX0) it is necessary to extrapolate the shower forlarger ages, so that the overestimation from these partswould be even larger. We estimate that not allowing forthe MS effect would increase the reconstructedE0 by∼ 15%.Not allowing for the MS light would also lead to anoverestimation of the depth of the shower maximumXmax. Fitting a new cascade curve to that from the pre-vious example with the increased values according tothe MS contribution, we obtain that the overestimationequals∆Xmax≃ 35g cm−2. With the difference betweenthe depths of an average proton and iron showers be-ing ∼ 100g cm−2, this is not a negligible number fordrawing conclusions about the mass composition of thehighest energy cosmic rays.

5.3. Contribution of higher generations

At first sight it may seem odd that it is only the firsttwo generations that contribute to the scattered light.One might expect that the main contribution shouldcome from the generation with the number closest tothe valueR/λQD, according to the Poisson distribu-tion (although we have shown that the first two gener-ations scale withR/λQ rather than withR/λQD). ForλR = 18km and λM = 15km and a shower close tothe ground we haveλQD = 8.2km . Thus, if the dis-tanceR = 24km thenR/λQD ≃ 3, so should one expectthe third and the fourth generations to contribute even

more than the first one? The answer is no, since thescattered photons should fall into the detector field ofview, so that it is not the path length of a scatteringbyany angle(whatλQD means ) that counts but by someparticular valuesof it. This decreases the probabilityof such a scattering and can be viewed as an increaseof the (actually) effective mean free path lengthλe f f forthe processwe are interested in(i.e. photons arrivingat the detector). One could estimate its value from theration1/n0 or n2/n0 adopting for this aim a Poisson dis-tribution of the number of contributing generations. Thetwo obtained values differ typically by 30-60%, so thatan estimation of the share of the remaining generationson the basis of this uncertainλe f f is uncertain as well.However, let us take an example:R = 25km, eleva-tion angle 15, ζ = 2, what are typical conditions forregistered showers in the Auger Observatory. The twovalues of the mean number of (effective) scatterings are0.034 (fromn1/n0) and 0.061 (fromn2/n0). Assumingthe larger one, one gets 2.7% for a fraction of the thirdand higher generations.However, as our earlier work for a uniform medium[10]b) shows, where the third generation was calculated,its share was two times smaller than that deduced fromthe Poisson model and the ratiosn1/n0 and n2/n0, sothat this estimation may be too high. Thus, we estimatethat the contribution of the higher number generationsshould be small, at least for the conditions of Auger.

5.4. Comparison with other work

Figure 18: Ratio of the scattered to direct light∆n/∆n0 integratedwithin angleζ, for shower 24kmaway from the detector, at elevationangle 10. Our results (first and second generation) are representedby thick solid lines forδ = 90 and thin solid lines - forδ = 30 and150 (second generation). Parametrisation from Monte-Carlo works:Roberts - dashed line and Pekala et al - dotted line.

The problem of the influence of the scattered light onthe shower image in the direct light was first undertaken

18

by Roberts [9]. He treated it by Monte-Carlo simula-tions of tracks of individual photons emitted by verti-cal showers. Such shower (although, strictly speaking,non-existing) enabled the author to introduce a detectorin a form of a ring around the shower as its symmetryaxis, increasing dramatically the number of registeredphotons. Although the author appreciate that the singlyscattered photons dominated the total MS signal, hissimulations went to more than five scatters, includingboth Rayleigh and Mie. He obtained a parametrisationof the ratio of the MS signal to the total signal (MS+direct) in the form

K = 77.4(OD× α · R1/2 · ζ1.1)0.68 (74)

where OD is the total optical depth (molecular andaerosol) between the detector and the light source,α -total scattering coefficient (inm−1) at the source. In ourtermsOD = R/λQD, α = 1

λQ= 1λR

Q+ 1λM

Q, whereλQD

is determined by formula (36) andλQ depends only onthe height in the stable atmosphere, so that (74) wouldmean the following

K ∼ R · ζ0.75( 1λQDλQ

)0.68(75)

This parametrisation differs from those obtained in thepresent work although not very much. It has a weakerdependence on the angleζ, which in our case is as(about)∼ ζ1. The smaller power index atλ−0.68

Q (∼ λ−1Q

in our case) is to some extent compensated by the termλ−0.68

QD .Another work, dealing with the present problem wasthat of Pekala et al [11]. The calculation method wasMonte-Carlo as well. This paper did consider inclinedshowers so that the cylindrical symmetry, as in the pre-vious case did not apply. Instead large ”packages” ofphotons were assumed to behave in the same way, ac-cording to simulated points and directions of their scat-tering. After extensive simulations the authors arrivedat a parametrisation of the ratioM of the MS signal tothe direct one as follows

M ∝ RλQDζ e−

hG (76)

with G = 5.43km. For a one component exponentialatmosphere with the scale heightG we would have

e−hG ∝ 1λQ

(77)

Indeed, it is not difficult to check that an effective scaleheight of the two-component atmosphere used by the

authors for 2≤ h ≤ 6km is about 5km what approxi-mately equalsG. Thus, their parametrisation would beas

M =∝ Rζ1

λQDλQ= kQζ

1λQD

(78)

It is not far from our derivation for the first generation,where the term 1

λQDdoes not appear. A comparison of

the ratio of the total MS signal to the direct one as afunction ofζ, obtained in this study with that of Robertsand Pekala et al is presented in Fig. 18. The agreementis quite good forζ ≤ 4 but for largerζ the other authorsseem to have underestimated a little the effect.

6. Summary and conclusions

Previous studies of the influence of the scattering ofphotons on their way from the shower to the detector onthe shower image used Monte-Carlo simulations. Theeffect has been called ”the multiple scattering” sinceit was not clear how many times the photons arrivingsimultaneously with the direct light actually scatter.Although Roberts [9] appreciated the importance ofphotons scattered one time only he went on withsimulating till several scatterings.The main idea of this study was to divide the scatteredlight into separate generations: the first generationbeing photons scattered exactly one time, the secondone two times, and so on. This enabled us to findanalytically the angular distributions of photons as afunction of time for the first generation which turnedout to be the most important one. We also showed thatany next generation can be found by some numericalintegrations of the previous one. These results wereobtained for a point source in a uniform mediumas well as in the actual atmosphere (although morenumerical integrations were needed in the latter case).By considering separately the first generation it waseasy to understand the differences between the twocases. The ratio of the second to the first generationis very small for small times after the light emissionwhich will turn out to be relevant when consideringthe scattered light from extensive air showers what is afinal aim of this study .In particular, this aim was to calculate the angulardistributions of scattered photons arriving simultane-ously with the direct (not scattered) light emitted byan extensive air shower and find relevant parametersdetermining these distributions. The solutions for apoint source were useful here since a shower could betreated as a series of consecutive (in space and time)point sources. For the first and the second generations

19

(the most important) we have explained where actuallythe scattering points took place, so that we were able tointegrate over all of them.We have derived an analytical expression (althoughapproximate) for the angular distribution of the firstgeneration photons, scattered by the Rayleigh process,fitting quite well the exact numerical distributions. Theanalytical solutions have led us to a proper choice of theindependent variables for a final parametrisation of thenumerical results. For example the number of photonsin the n-th generation turns out to be proportional to(R/λQ)n, at least forn ≤ 3 (although for Mie with somecorrection). Knowing the positions of the scatteringswe were able to show that there should be a dependenceon the viewing angleδ, not taken into account in theconsiderations of the previous authors. Moreover,this dependence should be different for the Rayleighand the Mie scatterings. Thus, we parametrised thefirst generation separately for these two processes.Concerning the second generation we combined allfour possibilities of the two scatterings (RR, RM, MRand MM) to give just one parametrisation since itscontribution was rather small.From our parametrisation it follows that forRζ = const(corresponding to a constant observed lateral dimensionof a shower) the contribution of the MS component tothe direct light depends practically only on the height ofthe observed shower element and on the viewing angle(the latter dependence being rather mild). A typicaloverestimation of the primary shower energy (if theMS effect was ignored) would be∼ 15% and of thedepth of shower maximum∆Xmax ≃ 35g cm−2. Ouranalytical and numerical approach gives similar resultsas the previous Monte-Carlo work of other authors,although the parametrisation obtained in this paper isdifferent. Only for larger opening angles (ζ > 4) theprediction of the MS contribution obtained here is abit larger, what would have some implications for areconstruction of very close showers.Summarising we conclude that the MS effect is notto be neglected while reconstructing parameters ofthe extensive air showers. To be properly allowed fora good knowledge of the atmospheric conditions isnecessary [13] for determining values ofλQ, as theshower traverses the atmosphere.

Acknowledgements.This work has been financiallysupported by Polish Ministry of Science and Higher Ed-ucation, grant No N N202 200239. It was stimulated byour participation in the Pierre Auger Collaboration. Wethank the Collaboration for fruitful discussions.

References

[1] T. Abu-Zayyad et. al, Astroparticle Physics 12 (1999) 121-134[2] The Pierre Auger Collaboration: J. Abraham et al, Nucl. In-

strum. Meth. A 620 (2010) 227-251[3] T. Nonaka et al, Nuclear Physics B Proc. Suppl. 190 (2009)26-

31[4] The MACFLY Collaboration: P. Colin et al, Astroparticle

Physics 30 (2009) 312-317[5] M. Giller et al, J. Phys. G: Nucl. Part. Phys. 30 (2004) 97-105[6] M. Giller et al, J. Phys. G: Nucl. Part. Phys. 31 (2005) 947-958[7] F. Nerling at al, Astroparticle Physics 24 (2006) 421[8] M. Giller and G. Wieczorek, Astroparticle Physics 31 (2009)

212-219[9] M. D. Roberts, J. Phys. G: Nucl. Part. Phys. 31 (2005) 1291-

1301[10] a) M. Giller and A.Smiałkowski, (2005) Proceedings of the 29th

Int. Cosmic Ray Conference, Pune, Indiab) M. Giller and A.Smiałkowski, (2007) Proceedings of the 30thInt. Cosmic Ray Conference, Merida, Mexicoc) M. Giller and A. Smiałkowski, (2011) Proceedings of the32nd Int. Cosmic Ray Conference, Beijing, China

[11] J. Pekala et al, Nucl. Instrum. Meth. A605 (2009) 388-398[12] P. Sommers, Astroparticle Physics 3 (1995) 349-360[13] The Pierre Auger Collaboration: J. Abraham et al, Astroparticle

Physics 33 (2010) 108-129

Appendix A. Isotropic scattering of light in a uni-form medium

It may be of some value to know how much the fluxesjR(θ, t; R) for the Rayleigh scattering differ from thosewhen the scattering functionf (α) = const, so when thescattering is isotropic. To calculate it we proceed as inSection 2 puttingf (α) = 1

4π in Eq. 6. We obtain

j is1 (θ, t; R) =c

8π2λR2· e−kτ

τ2 − 2τcosθ + 1(A.1)

and

dNis1 (ζ; t)

dt=

c e−kτ

16πλR2τ2· (A.2)

·[

(τ2 + 1)ln(

τ2 − 2τcosζ + 1(τ − 1)2

)

− 2τ(1− cosζ)]

The second generation can be obtained by proceedingas in Section 2.2, giving the result

j is2 (θ, t; R) =c e−kτ

16π2λ2

∫ x′max

0

1τ′

lnτ′ + 1τ′ − 1

dx′

R′2(A.3)

where τ′ = ct−x′

R′ ; R′ =√

x′2 − 2Rcosθx′ + R2 and

x′max = ct − R′ = R2τ2−1τ−cosθ . To find

dNis2 (ζ;t)dt one must

integrate the flux overθ numerically. How does the ra-tio of the two generations depend on the distance ? Wecan represent the first generation (A.2) in the form

dNis1 (ζ, t)

dt∝ k

R3f1(t; ζ) (A.4)

20

where f1 depends ont andζ only.The flux of the second generation equals

j is2 ∝k2

R3e−kτ· (A.5)

·∫ u′max(τ,θ)

0

1τ′

lnτ′ + 1τ′ − 1

·du′

u′2 − 2u′cosθ + 1

whereu′ = x′

R , u′max=12τ2−1τ−cosθ , τ

′ = τ−u′√u′2−2u′cosθ+1

.We see that the integral depends onτ andθ only so that

dNis2

dt∝ k2

R3f2(τ; ζ) (A.6)

Thus

dNis2

dt/

dNis1

dt∝ k f(τ; ζ) (A.7)

So that for a given time the ratio is proportional to thedistanceR measured in units ofλ. It can be verifiedthat this relation holds for any scattering process in auniform medium, not only for isotropy.

Appendix B. Derivation of an analytical expressionof the angular distribution of the firstgenerationdMR

1/dθ for small anglesθ

The formula for the angular distribution of the firstgeneration integrated over time, can be rewritten in thefollowing form

dM1

dθ=

k2|cosθ|

R2· (B.1)

·∫ π−θ

0exp

[

− ksinβ + sinθsin(β + θ)

]

f (β + θ)dβ

where f (β + θ) = f (α) is the angular distribution of thescattering angleα, normalised to 1 when integrated overthe full solid angle.For the Rayleigh scattering we obtain

dMR1

dθ=

3k32π|cosθ|

R2· (B.2)

·∫ π−θ

0exp

[

− ksinβ + sinθsin(β + θ)

][

1+ cos2(β + θ)]

dβ

To find the above integral forθ ≪ 1 we divide the inte-gration region into two parts:

dMR1

dθ= C

[

∫ π−θ−ǫ

0...dβ +

∫ π−θ

π−θ−ǫ...dβ

]

=

= C(I1 + I2) (B.3)

whereC = 3k32π|cosθ|

R2 .Choosingθ ≪ ǫ ≪ 1 it will be possible to find analyt-ically the integralsI1 andI2. Each of them is a sum ofthe two integrals:

I i =

∫ βi2

βi1

exp[

− ksinβ + sinθsin(β + θ)

]

dβ+

+

∫ βi2

βi1

exp[

− ksinβ + sinθsin(β + θ)

]

cos2(β + θ)dβ =

= I i1 + I i2

(B.4)

for i = 1, 2 andβ11 = 0,β12 = β21 = π−θ−ǫ, β22 = π−θ.Usingθ ≪ 1 we have

I11 ≃∫ π−θ−ǫ

0exp

[

− ksinβ(1+ θ

sinβ )

sinβ(1+ θ ctgβ)

]

dβ ≃

≃∫ π−θ−ǫ

0exp

[

− k(1+θ

sinβ− θ ctgβ)

]

dβ =

= e−k∫ π−θ−ǫ

0exp

(

− kθ tgβ

2

)

dβ (B.5)

where we have neglected all terms∝ θ2. Note that in theintegration regionθ ctgβ < θ

ǫ≪ 1 so that our approxi-

mation

(1+ θ ctgβ)−1 ≃ 1− θ ctgβ (B.6)

is justified. For the same reason the exponential indexkθ tgβ2 < kθ 2

ǫ≪ 1 so that

I11 ≃ e−k∫ π−θ−ǫ

0

(

1− kθ tgβ

2

)

dβ = (B.7)

= e−k[

π − θ − ǫ + 2kθ ln(

sinǫ + θ

2

)]

≃

≃ e−k(π − θ − ǫ + 2kθ lnǫ

2)

The integralI12 can be found easily:

I12 ≃ e−k∫ π−θ−ǫ

0

(

1− kθ1− cosβ

sinβ

)

· (B.8)

· (cos2β − 2θsinβcosβ)dβ ≃

≃ e−k[

π

2− θ − ǫ + 2kθ (ln

ǫ

2+ 1)

]

Thus

I1 = I11 + I12 = (B.9)

= e−k[32π − 2θ − 2ǫ + 2kθ(2ln

ǫ

2+ 1)

]

When calculatingI2 we introduce an angleχ = π − θ −β to take advantage ofχ being small in the integration

21

region ofI2.We have

I21 =

∫ ǫ

0exp

− kθ + (θ + χ)[1 − (θ+χ)2

6 ]

χ(1− χ2

6 )

dχ ≃

≃ e−k∫ ǫ

0exp

− 2kθx

dx= e−k2kθ∫ ∞

2kθǫ

e−ydyy2

≃ e−k[

2kθ(

CEu − lnǫ

2+ ln kθ − 1

)

+ ǫ

]

(B.10)

whereCEu ≃ 0.577 is the Euler constant.The last integral,I22, equals

I22 =

∫ ǫ

0exp

− k2θ + χχ

(1− χ2)dχ ≃ (B.11)

≃ e−k[

2kθ∫ ∞

2kθǫ

e−ydyy2− (2kθ)3

∫ ∞

2kθǫ

e−ydyy4

]

The second integral is of the order ofǫ2, so thatI22 = I21

and

I2 = e−k

4kθ[CEu − lnǫ

2+ ln(kθ) − 1] + 2ǫ

(B.12)

Thus

I1 + I2 = e−k3

2π − 2θ + 4kθ[ln(kθ) +CEu −

12

]

(B.13)

and finally

dMR1

dθ=

9ke−k

64R2

[

1− 4θ3π+

8kθ3π

(

ln(kθ) +CEu −12

)

]

(B.14)

As should be expected, the terms depending onǫ havecancelled in the final result.The ratio of the scattered light arriving within a cer-tain angleζ to the point source direction to that non-scattered (direct) equals

MR1

N0=

9π16

k

ζ +4kζ2

3π

[

ln(kζ) +CEu − 1− 12k

]

(B.15)

Appendix C. Towards an analytical solution ofthe second generation in a uniformmedium

Here we present results of some analytical integra-tions of functionsF(θ′, α) andG(R′, α) defined in Sec-tion 2.2. For the Rayleigh scattering we have

f R(α′) =3

16π(1+ cos2α′) (C.1)

Sincecosα′ = cosα cosθ′ − sinα sinθ′ cosφ′, FR(θ′, α)(Eq. 25) can be easily found analytically

FR(θ′, α) =∫ 2π

0f R(α′)dφ′ = (C.2)

=38

[

1+12

sin2α + (cos2α − 12

sin2α)cos2θ′]

The calculation of a corresponding functionFM(θ′, α)for the Mie scattering is a little more complicated. Theadopted here form off M(α′) has been chosen in sucha way as to make the integration overφ′ analyticallypossible. We have (Eq. 16)

FM(θ′, α′) =∫ 2π

0f M(α′)dφ′ = (C.3)

= 2∫ π

0(ai cos8α′ + b)dφ′

The angle of the second scatteringα′ changes withinlimits α′min ≤ α

′ ≤ α′max where

α′min = |θ′ − α|

α′max = θ′ + α if θ′ + α < π

α′max = 2π − (θ′ + α) if θ′ + α > π (C.4)

Sinceα′ = π2 separates the two regions ofα′ where

f M(α′) has different shapes, the solution of (C.3)depends on the positions ofα′min on theα′ axis withrespect to the pointα′ = π2. There are three cases to beconsidered

a). α′max<π2 . Then, of course,α′min <

π2 as well and the

function f M(α′) = f M1 (α′) applies to the whole region

of φ′ (0 ≤ φ′ ≤ 2π). Thus we have

FM = 2[

∫ π

0a1

(

A+ B cosφ′)8

dφ′ + πb]

= 2[

a1

8∑

k=0

Ck

∫ π

0coskφ′ dφ′ + πb

]

(C.5)

where Ck =(

8k

)

A8−kBk, A = cosα cosθ′ and B =sinα sinθ′.For oddk the integrals in (C.5) vanish. For evenk = 2mwe have

J2m =

∫ π

0cos2mφ′dφ′ =

(2m− 1)!!2m!!

π (C.6)

Hence

FM = 2[

a1

4∑

m=0

C2mJ2m + πb]

(C.7)

22

b). α′min <π2 < α

′max. This case is a bit more com-

plicated since there are both partsf M1 (α′) and f M

2 (α′)involved. The valueφ′∗ corresponding toα′ = π2 equalscosφ′∗ = ctgα · ctgθ′.We have

FM = 2[

∫ φ′∗

0f M1 (α′)dφ′ +

∫ π

φ′∗

f M2 (α′)dφ′

]

=

= 28

∑

k=0

Ck

[

a1

∫ φ′∗

0coskφ′dφ′ +

+ a2

∫ π

φ′∗

coskφ′dφ′]

+ 2πb (C.8)

Solving the integrals we finally obtain

FM = 2

4∑

n=0

C2n

(a1 − a2)[ 122n

(

2nn

)

φ′∗ +Gn(φ′∗)]

+

+a2

22n

(

2nn

)

π

3∑

n=0

C2n+1(a1 − a2)G′n(φ′∗) + bπ

(C.9)

where

Gn(φ′∗) =1

22n−1

n−1∑

i=0

(

2ni

)

sin[(2n− 2i)φ′∗]2n− 2i

(C.10)

G′n(φ′∗) =1

22n

n∑

i=0

(

2n+ 1i

)

sin[(2n− 2i + 1)φ′∗]2n− 2i + 1

c). α′min >π2 . Then, of courseα′min >

π2 as well and for

all values ofφ′ we have thatα′ > π2 , so thatf M

1 (α′) =f M2 (α′). The calculation ofFM is the same as in case a)

buta1→ a2. Thus

FM = 2[

a2

4∑

m=0

C2mJ2m+ πb]

(C.11)

The next integration overθ′ (Eq. 26) definingGAB(R′, α) can be found analytically only2 for GRR. Thecalculation is lengthily so that we will present here onlythe final result for the flux of the second generation

jRR2 (θ, t) =

12

( 38π

)2 c

λ2R

e−ctλ

∫ x′max

0I (τ′, α)

dx′

R′2

(C.12)

where

2actually forGMR, GRM andGMM the result could also be foundanalytically but due to the high power ofcosα (index= 8) there wouldbe very many terms making such an effort worthless.

I (τ′, α) ≡∫ π

0

1y′

(

1− 2sin2θ′

y′+

2sin4θ′

y′2

)

·

·(

a+ b cos2θ′)

sinθ′ dθ′ = (C.13)

=1

16τ′2×

[

a+b(τ′2 + 1)2

4τ′2

]

·( 2∑

i=−2i,0

aixi

2 − xi1

i+

+ a0 lnx2

x1+

b4τ′2

4∑

i=1

ai−2 ·xi

2 − xi1

i+ a−2ln

x2

x1+

− 2(τ′2 + 1) ·( 3∑

i=−1i,0

ai−1xi

2 − xi1

i+ a−1ln

x2

x1

)

The coefficientsai are the same as in (13);x1 = (τ′−1)2,x2 = (τ′ +1)2, a = 1+ 1

2 sin2α, b = cos2α− 12 sin2α. The

integral overx′ has to be found numerically.

23