Simulation of propagating EAS Cherenkov radiation over the ocean surface
SSimulation of propagating EAS Cherenkov radiation over the ocean surface
Olga P. Shustova a Faculty of Physics, Lomonosov Moscow State University,Leninskie gory 1/2, Moscow 119991, Russia
Nikolai N. Kalmykov b , Boris A. Khrenov Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,Leninskie gory 1/2, Moscow 119991, Russia
Abstract
We present computing results of the Cherenkov light propagation in air and water from ex-tensive air showers developing over the ocean. Limits on zenith angles of the showers, at whichthe registration of flashes of reflected Cherenkov photons by the satellite–based detector TUS ispossible, are analyzed with consideration for waves on the ocean surface.
Introduction
The space experiment TUS is going to be launched in the near future for studying ultrahigh–energycosmic rays (UHECRs) [1]. Its main purpose is to search for UHECRs particles over the whole celestialsphere by recording the fluorescence that is generated in developing of extensive air showers (EASs).This radiation is isotropic, which allows some photons to reach the satellite–based detector.In addition, development of UHE showers is accompanied by the intensive flux of Cherenkovradiation, propagating mainly along the shower axis. Nevertheless, when an EAS passes over a high–albedo surface the Cherenkov radiation can be detected at high altitudes. An example of this isthe balloon experiment [2] for recording Cherenkov photons scattered from snow–covered surfaces.According to [3], the total signal of the Cherenkov radiation gives an exact estimate of the primaryenergy, and its spatial distribution enables to measure the shower maximum depth and therefore toestimate the mass of the primary particle.In the case of the satellite–based detector, recording Cherenkov photons reflected from the oceansurface is of great interest as their flash at the end of the fluorescence track would allow one to deter-mine the shower trajectory more precisely. Besides, Cherenkov photons scattered in the atmosphereproduce an additional background. Calculations show that they will be superimposed on the fluores-cence signal and blur its maximum. Thus dependence of the “noise” on the shower parameters shouldbe found and subtracted from the total signal. Estimates of different components of the Cherenkovlight, which arise in developing of showers over the ocean, are presented in [4].This paper presents a computing technique for propagating EAS–produced Cherenkov photonsin air and water. It was used in [5] for estimating the ability of the TUS detector to record thereflected component of the Cherenkov light. These calculations may also be useful for experimentaldata processing and for constructing next–generation detectors (see, for instance, the JEM-EUSOproject [6]).
The photon propagation task can be conventionally divided into two parts: 1) generation of thephotons by electrons of the shower and 2) their subsequent travel in a medium. We apply the Monte–Carlo method in both cases, with the destiny of a large number of photons (henceforth, just a photon)simultaneously predicted. Each photon is assigned by a certain weight, i.e. the number of photons a [email protected] b [email protected] a r X i v : . [ a s t r o - ph . H E ] O c t survived” at a given calculation stage with respect to the maximum possible number of photons atthe point of their generation.We use the laboratory reference system (LRS) in which the origin O is located at the intersectionpoint of the shower axis with the plane Oxy coinciding with the flat ocean surface, and Oz –axis isdirected to the top of the atmosphere. The LRS and systems to be introduced below are right–handed.First we follow the “scheme of photon generation” (section 2), where the energy and direction of theparent electron, the generation depth, direction and weight of the photon are defined. Thus we obtaincharacteristics of the photon at the generation point, which are further used as input parameters insimulating its travel. The “scheme of photon propagation in a medium” (section 3) presents a sequenceof scattering events in which we define the path length of the photon to the next scattering point,its direction and weight. Calculations continue until the photon reaches a given level (for instance,the height of the satellite) or its weight becomes less than a certain preset value. Reflection andrefraction of the photons at the “air–water” interface are considered with allowance for waves on theocean surface (section 4). When an EAS develops over the ocean surface Cherenkov radiation is generated not only in air.Some part of the cascade, depending on depth of the primary interaction as well as energy and directionof the shower, penetrates into water and also produces Cherenkov photons. Therefore we consider twomedia: air and water.The discussion below will require direction cosines of the shower in the LRS defined in section 1: µ sh x = − sin θ sh cos ϕ sh , µ sh y = − sin θ sh sin ϕ sh , µ sh z = − cos θ sh , where the zenith θ sh and azimuth ϕ sh angles are input parameters. Sought characteristics of a photon,namely its coordinates, direction cosines and weight, will be primed.We assume that the photon is produced along the shower axis, consequently coordinates of itsgeneration point can be calculated from z (cid:48) = (cid:40) − h ln( X/X ) (air) ,µ sh z ( X − X ) /ρ wat (water) , x (cid:48) = z (cid:48) µ sh x µ sh z , y (cid:48) = z (cid:48) µ sh y µ sh z . Here, h = 8 . · cm is the standart atmosphere height, X = 1020 / cos θ sh g/cm is the ocean leveldepth, ρ wat = 1 g/cm is the water density. The depth X of the photon generation point is simulatedfrom the Gaisser–Hillas formula (see, e.g., [7]) N ( X ) = N max (cid:20) XX max (cid:21) X max /λ exp (cid:104) ( X max − X ) /λ (cid:105) with parameters describing the average cascade curve of a shower with primary energy E = 10 eV: N max = 7 · , X max = 800 g/cm , λ = 80 g/cm .For finding the photon direction in the LRS, it is convenient to introduce local reference systemsassociated with the shower axis (the reference system of the shower, RSS) and with the parent electron(the reference system of the electron, RSE). In the RSS the Oz –axis is steered along the shower axisand angles θ e and ϕ e characterize the direction of the electron, which, in turn, coincides with the Oz –axis of the RSE. The direction of the Cherenkov photon in the RSE is described by angles θ ch and ϕ ch . Then its direction cosines µ (cid:48) x , µ (cid:48) y and µ (cid:48) z in the LRS are given by µ (cid:48) x µ (cid:48) y µ (cid:48) z = γ sh µ sh x µ sh z − γ sh µ sh y µ sh x γ sh µ sh y µ sh z γ sh µ sh x µ sh y − /γ sh µ sh z × γ e µ ex µ ez − γ e µ ey µ ex γ e µ ey µ ez γ e µ ex µ ey − /γ e µ ez × µ ch x µ ch y µ ch z , γ α = (cid:2) − ( µ αz ) (cid:3) − / ( α = sh , e ) and direction cosines of the electron and the photon are definedby standart formulae: µ βx = sin θ β cos ϕ β , µ βy = sin θ β sin ϕ β , µ βz = cos θ β ( β = e, ch) . In calculations we distinguish between “forward–current” ( θ e (cid:54) ◦ ) and “back–current” ( θ e > ◦ )electrons, with the latter taken into account only in generating in water. Zenith angles of the back–current electrons are distributed isotropically, while that of the forward–current electrons are simulatedfrom the appropriate distribution suggested in [8]. Azimuth angles of electrons of both types areuniformly distributed.For photons, azimuth angles obey the uniform distribution and zenith angles are given by theTamm–Frank formula as a function of the electron energy E :sin θ ch = ( E/E th ) − E/m e c ) − . Here, E th = m e c (1 − n − ) − / is the threshold generation energy of Cherenkov photons by an electronin a medium with refractive index n . To determine the energy of the forward–current electrons weuse the suitable distribution from [8]. The energy of the back–current electrons is found from thedistribution (cid:101) N ( > E ) N ( > E th ) = 0 . (cid:20) . EE th − m e c (cid:21) − . , which was specially derived by means of the Geant4 code [9] (the calculations were performed witha lower value of E ). The number (cid:101) N of the back–current electrons with energies E > E th is ≈ . N .Finally, the photon is assigned an initial value of its weight. According to the Tamm–Frank theory,the number of photons with wavelength λ , dq/dλ , produced per unit depth, depends on the energy E and depth X of the parent electron and is given by dqdλ ( X, E ) = 2 παλ ρ ( X ) sin θ ch ( E, E th ( X )) . (1)Therefore the number of the photons in a “packet”, the fate of which is simulated, varies from oneiteration to another. We assume here that the unit–weight photon is generated by an electron withinfinite energy at the ocean level. Then the weight at any generation point can be defined as P (cid:48) = ρ max ρ (cid:20) sin θ ch sin θ maxch (cid:21) . We note that a value of P (cid:48) , averaged over a large number of events simulated, allows us to determinethe total number of Cherenkov photons generated by electrons of an EAS quite easily (see app. A).Thus, applying this scheme, we are able to specify the coordinates x (cid:48) , y (cid:48) , z (cid:48) , direction cosines µ (cid:48) x , µ (cid:48) y , µ (cid:48) z and weight P (cid:48) of a photon at its generation point, which are used further as input parametersin simulating its propagation in a medium. The brightest lines of fluorescent radiation in the atmosphere lie in the wavelength range λ =300 −
400 nm. So the TUS detector is assumed to be sensitive to photons with these wavelengths andwe, in turn, restrict our calculations within this range.The process of photon propagation in a medium is usually represented as a sequence of scatteringevents, in which the coordinates of a new scattering point and direction of the photon are defined.We consider scattering in water and air separately because of differences in their physical properties.The input parameters are the coordinates, direction cosines and weight of the photon.3 .1 Water
In simulating the photon travel we take scattering and absorption processes into account. Thecorresponding free path lengths l s and l a are given in [10]. The density and refractive index of waterare assumed to be constant: ρ wat = 1 g/cm and n wat = 1 . x i = x (cid:48) i + µ (cid:48) x i l ( x i = x, y, z ). Here, l is the photon path length between succes-sive scattering events, which, in the case of a medium with constant density, has the exponentialdistribution: p ( l ) dl = exp( − l/l s ) dl/l s . (2)The weight of the photon is attenuated due to absorption on the path l : P = P (cid:48) exp( − l/l a ). Then wecompute new direction cosines of the photon in the LRS: µ x µ y µ z = γ (cid:48) µ (cid:48) x µ (cid:48) z − γ (cid:48) µ (cid:48) y µ (cid:48) x γ (cid:48) µ (cid:48) y µ (cid:48) z γ (cid:48) µ (cid:48) x µ (cid:48) y − /γ (cid:48) µ (cid:48) z × sin θ s cos ϕ s sin θ s sin ϕ s cos θ s for | µ (cid:48) z | < , µ x µ y µ z = sign( µ (cid:48) z ) sin θ s cos ϕ s sin θ s sin ϕ s cos θ s for (cid:0) − | µ (cid:48) z | (cid:1) (cid:54) − , (3)where γ (cid:48) = (cid:2) − µ (cid:48) z (cid:3) − / . The angles θ s and ϕ s characterize deflection of the photon from its initialdirection. ϕ s is uniformly distributed and θ s is simulated from the Heneye–Greenstein distribution(see, e.g., [10]): p ( θ s , g ) d Ω s = 14 π (1 − g ) sin θ s (1 + g − g cos θ s ) / dθ s dϕ s with g = 0 . P (cid:62) P min holds or until the photon reaches the ocean surface. Travelling of a photon in air is considered disregarding its “true” absorption. This assumption isquite acceptable in the wavelength range of photons under study here. Besides, we limit to Rayleighscattering by presuming the absence of clouds, aerosols, etc. The atmosphere is supposed to beisothermal.The coordinates, direction and weight of the photon are determined in the same way as for water.However the atmosphere is nonhomogeneous, which prohibits application of (2) for simulating thepath length to the next scattering point. Nonetheless, within the assumptions indicated, the verticaldepth | ∆ X v | traversed by the photon has the following simple exponential distribution (see app. B): p (∆ X v ) d ∆ X v = κ µ (cid:48) z exp( − κ ∆ X v /µ (cid:48) z ) d ∆ X v . Hence we can find the vertical depth X v and coordinate z of the scattering point: X v = X (cid:48) v − ∆ X v , z = − h ln( X v /X v ) , where X v = 1020 g/cm is the vertical depth at the sea level. The photon escapes the atmosphereboundary when X v < X v > . In their turn, the path lengthand other coordinates of the scattering point are calculated as l = ( z − z (cid:48) ) /µ (cid:48) z , x = x (cid:48) + µ (cid:48) x l, y = y (cid:48) + µ (cid:48) y l. ϕ s distributeduniformly and θ s simulated from p ( θ s ) d Ω s = 316 π (1 + cos θ s ) sin θ s dθ s dϕ s . A new value of the photon weight is calculated with allowance for its attenuation due to scatteringon the traversed path: P = P (cid:48) − exp( − κ ∆ X v )1 − exp( − κ X (cid:48) v ) for µ (cid:48) z > ,P = P (cid:48) − exp( − κ ∆ X v )1 − exp (cid:2) − κ ( X v − X (cid:48) v ) (cid:3) for µ (cid:48) z < ,P = 0 for µ (cid:48) z ≈ . The weight does not change if the photon leaves the atmosphere boundary or reaches the ocean surface.This method of weight calculation makes it possible to “select” photons with low number of scatteringevents.Further we simulate the next scattering event. The procedure continues if the condition P (cid:62) P min is true or until the photon reaches a required level. We deal with two models of wavy ocean surface: with periodic structure (PS) and with chaoticstructure (CS). In the first model the surface is represented in the LRS by a motionless wave withlength λ w and amplitude h w along the Ox –axis. The shape of the wave is described by the parametricequations of a truncated cycloid: x = r w t − h w sin t, z = h w cos t, where r w = λ w / π and r w > h w , without dependence on the y –coordinate. In the second model theocean surface is divided into many small flat cells with height η . The quantities η u = ∂η/∂x and η c = ∂η/∂y (the subscripts u and c indicate the upwind and crosswind directions of the wave) followthe two-dimensional normal distribution [10] p ( η u , η c ) = 12 πσ u σ c exp (cid:20) − (cid:16) η u /σ u + η c /σ c (cid:17)(cid:21) , (4)with variances σ u = 3 . · − u w , σ c = 1 . · − u w depending on the wind velocity u w , in m/s. Bothmodels assume that photons are reflected and refracted at the interface according to the geometricoptics laws. Let us consider the calculation procedure by the example of a reflected photon. Periodic structure of the ocean surface.
Initially we define the projections x (cid:48) and z (cid:48) of thecoordinates x and z = 0 of the point at which the photon crosses the plane Oxy , onto the wavy surfacewith the assumption that one of the wave maxima coincides with the origin of the LRS. Hence the x (cid:48) –coordinate is the residue of x/λ w and z can be derived from the following equations: x (cid:48) = r w arccos( z (cid:48) /h w ) − (cid:112) h w − z (cid:48) for x (cid:48) (cid:54) λ w / ,x (cid:48) = 2 πr w − r w arccos( z (cid:48) /h w ) + (cid:112) h w − z (cid:48) for x (cid:48) > λ w / . Further we proceed to the reference system Ox (cid:48) y (cid:48) z (cid:48) in which the normal to the wavy surface at thepoint of the photon incidence serves as the Oz (cid:48) –axis. Let µ x , µ y and µ z be initial direction cosines of5he photon in the LRS. Then its direction cosines in the Ox (cid:48) y (cid:48) z (cid:48) system are defined as µ (cid:48) x µ (cid:48) y µ (cid:48) z = cos α cos β sin α cos β − sin β − sin α cos α α sin β sin α sin β cos β × µ x µ y µ z , where α = 0 for x (cid:48) (cid:54) λ w / α = π for x (cid:48) > λ w /
2, and β is the angle between axes Oz (cid:48) and Oz : β = arccos r w − z (cid:48) (cid:112) h w + r w − r w z (cid:48) . Finally we determine the direction cosines ν (cid:48) x , ν (cid:48) y , ν (cid:48) z of the photon reflected and appropriatereflection coefficient. Transformation to the LRS is performed from ν x ν y ν z = cos α cos β − sin α cos α sin β sin α cos β cos α sin α sin β − sin β β × ν (cid:48) x ν (cid:48) y ν (cid:48) z . Chaotic structure of the ocean surface.
At the first step we simulate the tangents η u and η c of the angles between a cell and Ox and Oy axes, respectively, from (4) and determine the directioncosines of the normal in the LRS at each point of the photon incidence: ζ x = η u (cid:112) η u + η c + 1 , ζ y = η c (cid:112) η u + η c + 1 , ζ z = 1 (cid:112) η u + η c + 1 . Further the direction cosines of the incident photon in the system of the normal are calculated: µ (cid:48) x µ (cid:48) y µ (cid:48) z = γ ζ x ζ z γ ζ y ζ z − /γ − γ ζ y γ ζ x ζ x ζ y ζ z × µ x µ y µ z , (5)where γ = (1 − ζ z ) − / . The direction cosines of the reflected photon and its reflection coefficient arefound from the laws of geometric optics.Finally transformation to the LRS is implemented by means of the rotation matrix inverse to thematrix from (5). The simulation algorithm presented above was applied in [5] for estimating the signal from reflectedEAS Cherenkov light by the TUS detector with field of view of ≈ . ◦ . Within the model of periodicstructure of a wave, with amplitude h w = 0 . λ w = 40 m, the registration of flashes of thereflected photons was found to be possible for showers with zenith angles (cid:46) ◦ . The wave parametersindicated were chosen on the basis of the following data. Satellite observations [11] show that, onaverage, winds under 8 m/s in speeds predominate over the ocean. According to the Beaufort scale,they are matched to waves with heights up to 2 m. It is known that there exists a wavelength spectrumat a given amplitude h w . Using a two–dimensional distribution over half–lengths and amplitudes,suggested in [12], we chose the mean wavelength, λ w = 0 . h w = 0 . h w = 0 . λ = 40 m in the PS model are matchedto the wind velocity u w = 6 m/s in the CS model. Fig. 1 presents angular distributions of reflectedCherenkov photons near the ocean surface for vertical showers. Photons with zenith θ and azimuth ϕ angles impinge on proper cells with area r ∆ r ∆ ϕ , where ∆ r = 0 .
05 and ∆ ϕ = π/
18. A value of r varies from 0 to 1 and specifies sin θ . The z –coordinate specifies the decimal logarithm of the photonrelative surface density. As illustrated, values of θ in the CS model lies in a much wider range. Figure 1: Angular distribution of reflected Cherenkov photons near the ocean surface for showers with angles θ sh = 0 ◦ and ϕ sh = 0 ◦ , obtained within the models of a) periodic and b) chaotic structures of the ocean surface.The Oz –axis specifies the relative density of the photons on logarithmic scale. The restriction on EAS zenith angles, θ sh (cid:46) ◦ , has been obtained within the PS model withallowance to the condition ϕ sh = 90 ◦ , when the shower axis is in one plane with the wavefront andtherefore more photons are reflected from a practically flat surface. This is also true for the CS modelbecause σ c < σ u . The azimuthally average number of the photons that will hit the mirror of the TUSdetector is shown in Fig. 2 for showers with θ sh = 25 ◦ , ϕ sh = 0 ◦ and ϕ sh = 90 ◦ . Figure 2: Number of reflected Cherenkov photons that hit the detector mirror, as a function of its location R ,in the CS model for showers with angles θ sh = 25 ◦ , ϕ sh = 0 ◦ and ϕ sh = 90 ◦ . The point R = 0 at the height of400 km is projected onto the center of the LRS. Time resolution of the TUS detector ( ≈ µ s) should be taken into account in order to estimate theobserved Cherenkov signal value. For small zenith angles of showers, the photons can be considered7 , u w , m/skm 2 4 6 80 17 . ± . . ± . . ± . . ± .
04 21 . ± . . ± . . ± . . ± .
78 21 . ± . . ± . . ± . . ± .
412 22 . ± . . ± . . ± . . ± .
316 23 . ± . . ± . . ± . . ± .
220 24 . ± . . ± . . ± . . ± .
224 25 . ± . . ± . . ± . . ± .
228 27 . ± . . ± . . ± . . ± . Table 1: Number of reflected Cherenkov photons that impinge upon the mirror of the TUS detector in the CSmodel for showers with angles θ sh = 25 ◦ and ϕ sh = 90 ◦ . to arrive at the detector mirror simultaneously. However, the signal duration grows starting from θ sh = 25 ◦ − ◦ and therefore the signal amplitude reduces even despite of the fact that photons growin number with an increase of the shower zenith angle. Our calculations show that in the CS model,with u w = 6 m/s, flashes of reflected Cherenkov photons at the end of the fluorescence track can berecorded by the TUS detector provided that θ sh (cid:46) ◦ .With a change in wind velocity the restriction on θ sh will also change because of proportionalityof the variances σ u and σ c to the wind velocity. In tab. 1 are listed the results of the calculations,obtained within the CS model at different values of u w for showers propagating at angles θ sh = 25 ◦ and ϕ = 90 ◦ . The entries of the columns indicate the number of reflected Cherenkov photons that fallwithin the field of view of the TUS detector, as a function of R . As evident, the photons are quitesmall in number at u w = 2 m/s so the maximum value of the shower zenith angle should be decreased. Conclusions
In this paper we have presented the simulation algorithm for generating and propagating EASCherenkov photons over the ocean. We have considered two different models for describing the wavyocean surface and imposed restrictions on zenith angles of the showers, at which the TUS detectorwould be able to identify the flashes of reflected Cherenkov photons. Our calculations have shownthat despite the difference in number of the photons impinging on the detector, both models provideapproximately the same results.
Acknowledgments
This work was supported by Russian Foundation for Basic Research (grant 09-02-12162-ofi m).
A Number of Cherenkov photons in a shower
The number of photons with wavelength λ , produced in a layer ( X − X ), is written as dN γ dλ = X (cid:90) X dX (cid:48) N ( X (cid:48) ) ∞ (cid:90) E th ( X ) dE (cid:48) dNdE (cid:48) ( X (cid:48) , E (cid:48) ) dqdλ ( X (cid:48) , E (cid:48) ) . (6)The distributions for Cherenkov photons, dq/dλ , and for electrons, N ( x ) and dN ( X, E ) /dE , areintroduced in section 2. 8e recall that every electron is assigned not a packet but rather a single photon with weight P ( X, E ) = ρ max ρ (cid:20) sin θ ch sin θ maxch (cid:21) . Therefore, (6) can be rewritten as dN γ dλ = 2 παλ sin θ maxch ρ max X (cid:90) X dX (cid:48) N ( X (cid:48) ) ∞ (cid:90) E th ( X (cid:48) ) dE (cid:48) dNdE (cid:48) ( X (cid:48) , E (cid:48) ) P ( X (cid:48) , E (cid:48) ) ≈≈ (cid:101) P παλ sin θ maxch ρ max X (cid:90) X dX (cid:48) N ( X ) . The approximation sign takes in account the fact that (cid:16) (cid:80) Ni =1 P i (cid:17) /N tends to (cid:101) P with an increase inthe number N of events simulated. B Distribution for vertical depth
Let us consider a change in intensity of a light beam on a path length dl : dI = − I dl/l s . (7)For Rayleigh scattering, the mean free path length l s depends on the wavelength λ of radiation anddensity ρ of the medium: l s ∝ λ ρ . Introducing the quantity κ = ( l s ρ ) − and using the following relationship between the verticaldepth X v of the atmosphere and the height h above the sea level: X v ( h ) = ∞ (cid:90) h ρ ( h (cid:48) ) dh (cid:48) , we integrate (7) over a layer with thickness ( h − h ):ln (cid:20) I ( h ) I ( h ) (cid:21) = − κ h (cid:90) h ρ ( h (cid:48) ) dh (cid:48) = − κ (cid:104) X v ( h ) − X v ( h ) (cid:105) = − κ ∆ X v . (8)If the beam is deflected at an angle θ from the vertical direction, (8) transforms toln (cid:20) I ( l ) I ( l ) (cid:21) = − κ l (cid:90) l ρ ( l (cid:48) ) dl (cid:48) = − κ (cid:104) X v ( h ) − X v ( h ) (cid:105) / cos θ = − κ ∆ X v /µ z , where ( l − l ) = ( h − h ) /µ z . We implicitly assume here the case of the plane-parallel atmosphere,which is in reality valid up to θ ≈ ◦ . In our calculations photons with µ z ≈ | ∆ X v | without scattering obeysthe following exponential law: p (∆ X v ) d ∆ X v = κ µ z exp( − κ ∆ X v /µ z ) d ∆ X v . eferences [1] B.A. Khrenov, V.V. Alexandrov, D.I. Bugrov et al. , Phys. Atom. Nucl. , 2058 (2004).[2] S.B. Shaulov, S.P. Besshapov, N.V. Kabanova et al. , Nucl. Phys., B, Proc. Suppl. , 403 (2009).[3]
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