Coherent Suppression of Molecular Bremsstrahlung Radiation at GHz Frequencies in the Ionization Trail of Extensive Air Showers
CCoherent Suppression of Molecular Bremsstrahlung Radiation atGHz Frequencies in the Ionization Trail of Extensive Air Showers
Olivier Deligny
Laboratoire de Physique des 2 Infinis Irène Joliot-Curie, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
Abstract
Several attempts to detect extensive air showers (EAS) induced by ultrahigh-energy cosmicrays have been conducted in the last decade based on the molecular Bremsstrahlung radia-tion (MBR) at GHz frequencies from quasi-elastic collisions of ionisation electrons left in theatmosphere after the passage of the cascade of particles. These attempts have led to the detec-tion of a handful of signals only, all of them forward-directed along the shower axis and hencesuggestive of originating from geomagnetic and Askaryan emissions extending into GHz fre-quencies close to the Cherenkov angle. In this paper, the lack of detection of events is explainedby the coherent suppression of the MBR in frequency ranges below the collision rate due to thedestructive interferences impacting the emission amplitude of photons between the succes-sive collisions of the electrons. The spectral intensity at the ground level is shown to be severalorders of magnitude below the sensitivity of experimental setups. In particular, the spectral in-tensity at 10 km from the shower core for a vertical shower induced by a proton of 10 eV is7-to-8 orders of magnitude below the reference value anticipated from a scaling law convertinga laboratory measurement to EAS expectations. Consequently, the MBR cannot be seen as thebasis of a new detection technique of EAS for the next decades.
1. Introduction
The investigation and understanding of the intensity of cosmic rays with energies in ex-cess of 10 eV, particles discovered nearly 60 years ago [1, 2], has been demanding for moreand more precise data, both from the statistical and from the systematical point of view. Cur-rently, the Pierre Auger Observatory, covering an area of 3000 km in Argentina [3], and the Tele-scope Array, covering an area of 700 km – planned to be extended to 3000 km – in the UnitedStates [4], are the two largest-ever built detectors of EAS induced by cosmic rays. A harvest ofdata is now allowing numerous constraints to be inferred on the acceleration mechanisms op-erating in the extragalactic astrophysical sites producing the particles, and on the energeticsand the location of these sources [5–7]. While the noose is tightening around some nearby ex-tragalactic objects, no discrete source of ultrahigh-energy cosmic rays has been identified so farthrough an intense clustering of arrival directions. This does not preclude that sources may becaptured on a collective basis in a near future, but another jump in statistics appears necessary.The pending challenge for the next generation of ground-based observatories is thus to providethis jump in statistics while preserving equal, or reaching better, performances in accuracy tomeasure the EAS characteristics. Email address: [email protected] (Olivier Deligny)
Preprint submitted to Elsevier February 25, 2021 a r X i v : . [ a s t r o - ph . I M ] F e b istorically, a breakthrough in the detection technique of EAS has been the use of fluores-cence telescope stations that was pioneered first in tests at the Volcano Ranch experiment andthen with the original Fly’s Eye experiment [8] made up of arrays of several hundred of photo-multiplier tubes which, thanks to a set of telescope mirrors, each monitor a small portion of thesky. These sensors detect the fluorescence caused by the de-excitation of nitrogen molecules asa result of their excitation by the many ionising electrons created as the cascade passes throughthe atmosphere. This de-excitation gives rise to weak ultraviolet radiation, but which can bedetected up to 30 or 40 km away on moonless nights and which offers the possibility to observeEAS side-on thanks to the isotropic emission. These telescopes thus allow a measurement ofthe longitudinal profile of the showers, which in turn is used to infer both the energy of theshowers in a calorimetric way, without recourse to external information to calibrate the energyestimator, and the slant depth of maximum of shower development, a proxy, the best up to date,of the primary mass of the particles. Large detection areas can be covered by means of a fewfluorescence stations only, spaced every 20 km or so. However, the flip side of the technique isits low duty cycle, about 10%, due to the need for operating during moonless nights only.Through the passage of charged particles in the atmosphere, the energy of an EAS is de-posited mainly by ionisation. The resulting numerous ionisation electrons can, in turn, pro-duce their own emission such as continuum Bremsstrahlung emission through quasi-elasticscattering with molecular nitrogen and oxygen. Due to the expected isotropic and unpolarisedemission, molecular Bremsstrahlung radiation (MBR) in the GHz band, which propagates in theatmosphere in a quasi-unattenuated way (less than 0.05 dB km − ), is thus providing a mecha-nism, with a 100% duty cycle, for performing shower calorimetry in the same spirit as the flu-orescence technique does, by mapping the ionisation content along the showers through theintensity of the microwave signals detected at ground level.Triggered by microwave emission measurements in laboratory [9], new telescope techniquesbased on the detection of the microwave emission in the GHz frequency range have been subse-quently tested [10–12]. Only a few handful signals forward-directed along the shower axes wererecorded, with in particular no side-on observation of EAS [10, 11]. Hence, the hopes raised adecade ago for a new breakthrough in the detection technique of EAS have been dashed, andthe MBR technique does not even remain on the drawing board nowadays. The goal of this pa-per is to explain the reasons of the faintness of the emission that was not anticipated in previousestimates, which were based on the scaling of the radiation of a single quasi-elastic collision of afree electron with a neutral molecule to the total rate of quasi-elastic collisions in the short-livedelectron/air plasma [13, 14].Coherence effects for Bremsstrahlung processes in dense matter are well-known in the caseof ultra-relativistic electrons undergoing multiple scattering on Coulomb centers: if an electronundergoes multiple scattering while traversing the “formation zone”, the Bremsstrahlung am-plitudes from before and after the scattering can interfere, reducing the probability of photonemission for photon energies below a certain value. This is the Landau-Pomeranchuk-Migdaleffect [15, 16], which leads to a suppression for the Bremsstrahlung cross section compared tothe Bethe-Heitler one of the single-scattering picture. In air, this suppression factor becomesimportant at GHz frequencies for electron kinetic energies greater than 1 MeV. In section 2, ef-fects of the same nature are shown to induce a suppression factor in the case of non-relativisticelectrons and to be responsible, hence, for the faintness of the MBR emission of EAS in the GHzband. Once this suppression mechanism is established, it is straightforward to apply it to theshort-lived electron/air plasma left after the passage of an EAS in section 3 and infer the ex-2ected spectral intensity from EAS at ground level as shown in section 4. A discussion of theresults is given in section 5.
2. Molecular Bremsstrahlung Radiation of Low-Energy Electrons in a Dense Plasma
Let us consider an electron/neutral plasma with low-energy electrons colliding elasticallywith neutral molecules during a finite time and describe each electron as a classical chargedparticle coupled to a Maxwell field. In this framework, the energy radiated by the electron isassociated with the deviations caused by the collisions with the neutral molecules: when anelectron approaches a neutral molecule, the electric field of the electron polarises the neutralmolecule, and this polarisation gives rise to a dipole moment that induces an attractive inter-action potential at a short distance range. For non-relativistic particles, the spectral radiatedenergy per unit solid angle flowing into an elementary cone d Ω and received at distance R as-sumed to be far away from the accelerated charge reads as E ( ω , Ω ) = e π (cid:178) c (cid:191)(cid:175)(cid:175)(cid:175)(cid:175)(cid:90) d t (cid:48) (cid:161) q × (cid:161) q × ˙ v ( t (cid:48) ) (cid:162)(cid:162) exp ( − i ω t (cid:48) ) (cid:175)(cid:175)(cid:175)(cid:175) (cid:192) , (1)where e is the electric charge, (cid:178) is the vacuum permittivity, c is the speed of light, q is a unitvector in the observer direction that changes negligibly during a small acceleration interval and v ( t (cid:48) ) is the electron velocity at retarded time t (cid:48) . The use of the 〈·〉 symbol stands for the aver-age over realisations of the stochastic process that governs the dynamics of v ( t (cid:48) ). An electronappearing free at t (cid:48) = t (cid:48) = τ experiences accelerationsduring each collision. To derive the expected radiation by accounting for the effect of succes-sive collisions, the collisions are modeled as a random series of impulsive velocity changes ∆ v k occurring during the finite time duration τ . This implies that the acceleration of an electroncan be written as ˙ v ( t (cid:48) ) = N coll (cid:88) k = ∆ v k δ ( t (cid:48) , t (cid:48) k ), (2)with N coll a Poisson variable governed by the collision rate Γ coll . On inserting this expressioninto the angular frequency spectrum of the energy radiated by a non-relativistic acceleratedparticle, one gets, after integration over all directions: E ( ω ) = e π (cid:178) c (cid:42)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) v + N coll (cid:88) k = ∆ v k exp ( − i ω t (cid:48) k ) − v N coll exp ( − i ωτ ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) (cid:43) , (3)where, to include the transition radiations associated with the appearance and disappearanceof the electron, an effective acceleration has been introduced at the initial and final times,˙ v (0) = v δ ( t (cid:48) , 0) and ˙ v ( τ ) = − v N coll δ ( t (cid:48) , τ ). In contrast to the traditional recipe to derive E ( ω )for the Bremsstrahlung process consisting in multiplying the radiated energy of one single col-lision, E ( ω ), by the number of collisions N coll , this expression accounts for coherence effects.For ω (cid:192) Γ coll , the random arguments in the exponential ω t k are random numbers since therandom times t k are of the order of 1/ Γ coll : the regime is then incoherent, and the scaling E ( ω ) = N coll E ( ω ) holds. In contrast, for ω (cid:191) Γ coll , the random arguments are close to 0 sothat all random phases are close to 1, and the radiation is then largely suppressed from the in-terference between photons emitted by different elements of electron pathlength.3 requency [Hz] R a d i a t e d e n e r gy s p ec t r u m [ J / H z ] -45 -43 -41 -39 -37 -1 s =10 coll G , -1 s =10 att G -1 s =10 coll G , -1 s =10 att G -1 s =10 coll G , -1 s =10 att G -1 s =10 coll G , -1 s =10 att G Figure 1:
Left: Two-point correlation function of the electron velocity, obtained by Monte-Carlo, for elec-tron kinetic energies of 1 eV, and Γ att = Γ coll = s − . Right: Spectrum of the radiated energy for severalvalues of Γ att and Γ coll . The r.h.s. of eq. (3) provides the relevant framework to simulate by Monte-Carlo the inter-ferences that lead to the Bremsstrahlung suppression at angular frequencies smaller than thecollision rate. However, a more explicit expression can be obtained starting from, similarly tothe Landau-Pomeranchuk way, integrating by parts eq. (1). The antiderivative term is zero since v (0 − ) = v ( τ + ) =
0. After integration over all directions, one thus gets E ( ω ) = e ω π (cid:178) c (cid:207) d t (cid:48) d t (cid:48)(cid:48) (cid:173) v ( t (cid:48) ) · v ( t (cid:48)(cid:48) ) (cid:174) exp ( − i ω ( t (cid:48) − t (cid:48)(cid:48) )), (4)so that the information is actually encompassed in the two-point correlation function of theelectron velocity. The connection between eq. (3) and eq. (4) is illustrated below with two sce-narios for which both the Monte-Carlo and the analytical methods are used.First, let’s consider the case of free electrons during a time duration 0 ≤ t ≤ τ undergoingelastic collisions at a rate Γ coll . The time duration τ is a random variable governed by a processof electron attachment at a rate Γ att , attachment process that does not need to be made explicitat this stage. Averaged over a large number of pathlengths of free electrons, the correspondingtwo-point correlation function of the electron velocity, obtained by Monte-Carlo, is shown inthe left panel of fig. 1 for electron kinetic energies of 1 eV, and Γ att = Γ coll = s − so that theeffect of both processes is made visible. The attachment process leads to a decorrelation ofthe velocities evolving as exp (cid:161) − Γ att max (cid:161) t (cid:48) , t (cid:48)(cid:48) (cid:162)(cid:162) . Besides, the spatial diffusion induced by theelastic collisions leads to a decorrelation that depends only on the time difference between t (cid:48) and t (cid:48)(cid:48) , decorrelation hence evolving as exp (cid:161) − Γ coll (cid:175)(cid:175) t (cid:48) − t (cid:48)(cid:48) (cid:175)(cid:175)(cid:162) . The two-point correlation functionof the electron velocity thus reads as (cid:173) v ( t (cid:48) ) · v ( t (cid:48)(cid:48) ) (cid:174) = v × (cid:189) exp (cid:161) − Γ att t (cid:48) − Γ coll (cid:161) t (cid:48) − t (cid:48)(cid:48) (cid:162)(cid:162) if t (cid:48) ≥ t (cid:48)(cid:48) ,exp (cid:161) − Γ att t (cid:48)(cid:48) − Γ coll (cid:161) t (cid:48)(cid:48) − t (cid:48) (cid:162)(cid:162) otherwise. (5)On inserting this expression into eq. (4), the angular frequency spectrum of the radiated energyreads E ( ω ) = e v π (cid:178) c ( Γ att + Γ coll ) ω Γ att (cid:161) ( Γ att + Γ coll ) + ω (cid:162) . (6)4 requency [Hz] R a d i a t e d e n e r gy s p ec t r u m [ J / H z ] -45 -43 -41 -39 -37 =0 a , -1 s =10 in G , -1 s =10 att G =0.1 a , -1 s =10 in G , -1 s =10 att G =0.5 a , -1 s =10 in G , -1 s =10 att G =1 a , -1 s =10 in G , -1 s =10 att G Figure 2:
Left: Two-point correlation function of the electron velocity, obtained by Monte-Carlo, for elec-tron kinetic energies of 1 eV, Γ att = Γ in = s − , Γ el = α = Γ att = Γ in = s − and several values of α . This expression exhibits the suppression of the emission for ω (cid:191) Γ coll + Γ att . On the other hand,for ω (cid:192) Γ coll + Γ att , the radiation scales as ( Γ coll / Γ att )( e v /3 π (cid:178) c ) ≡ N coll E if Γ coll (cid:192) Γ att , andas E if Γ coll (cid:191) Γ att , as expected. These features are illustrated in the right panel of fig. 1, wherethe spectrum of the radiated energy is shown as a function of the frequency ν = ω /2 π for severalvalues of Γ att and Γ coll . Overlaid on the points obtained from the Monte-Carlo computation bymeans of eq. (3), the continuous line obtained from eq. (6) is in perfect agreement. How thespectral suppression and the overall scale of the radiation are governed by these two parametersis clearly visible.More complex is the case considering inelastic collisions in addition to elastic ones. Forclarity, the corresponding rates are denoted as Γ el (elastic) and Γ in (inelastic). The electron issupposed to loose a constant fraction of velocity, α v , at every inelastic collision. This scenario isnot necessarily of relevant interest in practice, but it provides us with an analytical solution thatallows a direct understanding of the impact of the inelastic processes on the radiated energy.The result of the Monte-Carlo for the (cid:173) v ( t (cid:48) ) · v ( t (cid:48)(cid:48) ) (cid:174) function is shown in the left panel of fig. 2for, as previously, electron kinetic energies of 1 eV, Γ el = Γ att = Γ in = s − , and for α = | t (cid:48) − t (cid:48)(cid:48) | , an additionalevolving term exp (cid:161) − α (2 − α ) Γ in min (cid:161) t (cid:48) , t (cid:48)(cid:48) (cid:162)(cid:162) is observed so that the expression sought for thetwo-point correlation function of the electron velocity reads as (cid:173) v ( t (cid:48) ) · v ( t (cid:48)(cid:48) ) (cid:174) = v × (cid:189) exp (cid:161) − Γ att t (cid:48) − ( Γ el + Γ in ) (cid:161) t (cid:48) − t (cid:48)(cid:48) (cid:162) − α (2 − α ) Γ in t (cid:48)(cid:48) (cid:162) if t (cid:48) ≥ t (cid:48)(cid:48) ,exp (cid:161) − Γ att t (cid:48)(cid:48) − ( Γ el + Γ in ) (cid:161) t (cid:48)(cid:48) − t (cid:48) (cid:162) − α (2 − α ) Γ in t (cid:48) (cid:162) otherwise. (7)The angular frequency spectrum of the radiated energy reads in this case E ( ω ) = e v π (cid:178) c ( Γ att + Γ el + Γ in ) ω ( Γ att + α (2 − α ) Γ in ) (cid:161) ( Γ att + Γ el + Γ in ) + ω (cid:162) , (8)which highlights, through the term Γ att + α (2 − α ) Γ in in the denominator, the additional sup-pression of the emission induced by the inelastic process. This is illustrated in the right panel of5g. 2 for Γ att = Γ in = s − , Γ el = α . For α =
0, the impact of the inelasticcollisions is identical to that of elastic ones, while for α =
1, the impact is that of an attachmentat a rate Γ in .The classical electrodynamics approach used here to derive the Bremsstrahlung emissionis justified by the weakness of the photon energies in the considered frequency range. From aquantum perspective, the production of photons with energies h ν , with h the Planck constant,corresponds to transitions between unquantised energy states of the free electrons (“free-free”transitions). In the framework of non-equilibrium quantum field theory, it can be shown thatthe quasi-free scattering approximation indeed breaks down due to the successive collisionsthrough a careful classification of diagrams and an appropriate re-summation of subsets ofgraphs [17]. The suppression factors obtained in this framework are however depending onthe relaxation rate of the source, which is less straightforward to infer than the collision ratesused in the approach adopted here.
3. Emission from the Ionisation Trail Left After the Passage of an Extensive Air Shower
The energy of an EAS is, as already stressed, deposited mainly through the ionisation pro-cess through the development of the cascade in the atmosphere. Let n EAS be the number ofhigh-energy charged particles per surface unit in the cascade and ρ ( x ) the density of molecularnitrogen or oxygen in the atmosphere at the position x . These high-energy electrons/positronsfrom the cascade are refereed to as “primary electrons” hereafter, in contrast to the ionisationelectrons, the production of which per unit volume, per velocity band and per time unit followsfrom n ( x , v , t ) = ρ ( x ) f ( v , t ) I + 〈 T 〉 (cid:191) d E d X (cid:192) n EAS ( x ). (9)Here, I is the ionisation potential to create an electron-ion pair in air, the bracketed expression 〈 d E /d X 〉 stands for the mean energy loss of the EAS charged particles per grammage unit, and f ( v , t ) is the distribution in velocity and time of the resulting ionisation electrons, which isrelated to that expressed in terms of kinetic energy, f T ( T ), through the Jacobian transformation f ( v , t ) = mv π (cid:161) − ( v / c ) (cid:162) f T ( T ( v ) , t ) , (10)with m the mass of the electron. For primary charged particles in the cascade with ≥ MeVenergies, an expression for the distribution f T ( T , t =
0) that accounts for relativistic effects aswell as indistinguishability between primary and secondary electrons, which modify the low-energy behaviour [18], follows from that provided in [19]: f T ( T ) = π Z R y m (cid:161) β ( T p ) c (cid:162) + C exp ( − T / T k ) T + T , (11)where R y is the Rydberg constant, β ( T p ) is the relativistic factor for the primary electron withenergy T p , T ranges from 0 to T max = ( T p − I )/2 due to the indistinguishability between pri-mary and secondary electrons, the constant C is determined in the same way as in [20] so that (cid:82) d T f T ( T ) reproduces the total ionisation cross section, T k =
77 eV is a parameter acting asthe boundary between close and distant collisions, and T is a measured parameter such that6 [eV] T -1
10 1 10 C o lli s i on r a t e [ GH z ] -2 -1 momentum transferexcitationionisation3-body attachment2-body attachment Figure 3:
Collision rates of free electrons in air as a function of their kinetic energy. T = 〈 T 〉 (cid:39)
40 eV, in agreement with the well-known stopping power. The remaining time depen-dence in t , reflecting the subsequent cascade of ionisation electrons produced by secondaryelectrons themselves as long as their kinetic energy is above I , is derived by Monte-Carlo below.As long as they remain free, ionisation electrons with density n (cid:48) ≡ n ( x (cid:48) , v (cid:48) , t (cid:48) ) can thus pro-duce photons through the process of quasi-elastic collisions with neutral molecules in the at-mosphere with an angular frequency spectrum E ( ω ) = e ω π (cid:178) c (cid:207) d x (cid:48) d x (cid:48)(cid:48) (cid:207) d v (cid:48) d v (cid:48)(cid:48) (cid:207) d t (cid:48) d t (cid:48)(cid:48) (cid:90) ∞ t (cid:48) (cid:90) ∞ t (cid:48)(cid:48) d t (cid:48) d t (cid:48)(cid:48) (cid:173)(cid:161) n (cid:48) v ( t (cid:48) ) (cid:162) · (cid:161) n (cid:48)(cid:48) v ( t (cid:48)(cid:48) ) (cid:162)(cid:174) e − i ω ( t (cid:48) − t (cid:48)(cid:48) ) ,(12)where, compared to the single-particle case presented in section 2, the two-point correlationfunction of the electron velocities must now account for the density of particles. For an in-coherent process between independent particles such as the MBR, this two-point correlationfunction is diagonal in every variable governing the densities: (cid:173)(cid:161) n (cid:48) v ( t (cid:48) ) (cid:162) · (cid:161) n (cid:48)(cid:48) v ( t (cid:48)(cid:48) ) (cid:162)(cid:174) = n (cid:48) δ ( x (cid:48) , x (cid:48)(cid:48) ) δ ( v (cid:48) , v (cid:48)(cid:48) ) δ ( t (cid:48) , t (cid:48)(cid:48) ) (cid:173) v ( t (cid:48) ) · v ( t (cid:48)(cid:48) ) (cid:174) . (13)In this way, the radiation scales with the number of particles once integrating eq. (12) over po-sitions, initial velocities and initial creation time.The radiation is thus determined, as in the simple examples in section 2, by the two-pointcorrelation function of the velocities of a single electron, obtained by Monte-Carlo by simulat-ing a large number of test particles with initial velocities drawn at random from eq. (10) andundergoing collisions, the rate of which being taken from experimental tabulated data in [21].The main features of the different rates, shown in fig. 3, depend on the energy. The total mo-mentum transfer collision rate goes from (cid:39)
100 GHz up to (cid:39)
10 THz in the explored kineticenergy range, with different inelastic contributions depicted by the different curves. Ionisa-tion on N and O molecules dominates the collisions for T ≥
40 eV, causing energy losses ona time scale below the picosecond. Excitation on electronic levels of N and O molecules en-ters into play in a dominant way below 40 eV down to 4 eV, with energy losses that occur on7 requency [Hz] R a d i a t e d e n e r gy s p ec t r u m [ J / H z ] -46 -45 -44 -43 -42 -41 -40 Figure 4:
Frequency spectrum of the radiated energy by the ionisation trail of an EAS. time scales going from picoseconds to a few nanoseconds when going down in energy. Below4 eV down to 1.7 eV, resonances for excitation on N and O molecules through ro-vibrationalprocesses cause energy losses on a time scale of the picosecond. Then, below 1.7 eV down to0.2 eV, resonances for excitation on N and O molecules through ro-vibrational processes andfor two-body attachment process on O molecules enter into play. These processes are quan-tised in energies. The energy losses of the excitation resonances occur on a time scale of a fewtens of picoseconds, while the time scale of disappearance of the electrons through the two-body attachment process is of the order of the nanosecond. Despite their low abundance, CO and H O molecules induce energy losses on a time scale of a nanosecond that degrade electronenergies down to 0.1 eV, where the two-body attachment process make them disappearing ona time scale of a few nanoseconds. Excitations of H O molecules are also considered, the con-centration of which is subject to large variations in the atmosphere; a typical value of 3,000 ppmis used in this study. Below T = f ( v , t ) = f ( v ) δ ( t , 0), which is the relevant quantity after carrying out the changes of vari-ables t (cid:48) → t (cid:48) − t (cid:48) and t (cid:48)(cid:48) → t (cid:48)(cid:48) − t (cid:48)(cid:48) in eq. (12). The resulting spectrum of radiation from theionisation trail left after the passage of an EAS, normalised to the contribution of one singleparticle, is shown in fig. 4. The radiation is observed to be suppressed below THz frequencies,with a quadratic dependence in frequency in the GHz range.
4. Spectral Intensity at GHz Frequencies Expected from Extensive Air Showers
The function that describes the power of the radiation received per unit frequency and pass-ing through any unit area at an observation point x obs is the spectral intensity, Φ ( ν , x obs ). It8 hower core observation pointground level Rxyz raltitude a shower axis (cid:160) ,a) (cid:160) emission point (r, Figure 5:
Geometry of a vertical EAS used throughout the paper. results from the summation of the radiation emitted by all the ionisation electrons producedalong the shower track. Expressed in W m − Hz − units, it is the quantity directly accessible tothe experiment in a frequency band. To provide relevant orders of magnitude for the spectralintensities that can be expected from MBR in the GHz band at the ground level, a crude modelof EAS, limited to a vertical incidence, is used to infer an expression of n EAS ( x ) to be pluggedinto eq. (9), the geometry of which is depicted in fig. 5. The spectral intensity then results from Φ ( ν , x obs ) = (cid:209) r d r d ϕ d a π R ( r , ϕ , a ) P ( ν , a ), (14)where P ( ν , a ) is the frequency spectrum of emitted power, obtained by dividing the radiatedenergy by the mean duration of the emission identified as the “mean lifetime of the plasma”,which amounts, from the simulations described in the previous section, to (cid:39)
30 ns at the groundlevel.Following [13], the EAS is considered as a thin disk of high-energy charged particles prop-agating in the atmosphere at the speed c . The density of particles in the disk depends on thedistance r to the axis. Restricting ourselves to the electromagnetic component, which is thedominant component producing ionisation electrons, the lateral extension of the cascade canbe expressed in terms of the Molière radius R M , which is such that 90% of the energy is con-tained within this distance from the axis. In this way, the number of electrons/positrons persurface unit n EAS at any position x = ( r , ϕ , a ) is known to be well reproduced by the NKG pro-file [22, 23]: n EAS ( x ) = N ( a ) × C ( s ( a )) R − M (cid:181) rR M (cid:182) s ( a ) − (cid:181) + rR M (cid:182) s ( a ) − . (15)Here, s ( a ) stands for the age parameter at altitude a defined as s ( a ) = X ( a )/( X ( a ) + X max ),and C ( s ) is a normalisation factor such that 2 π (cid:82) r d r n EAS = N ( a ), where N ( a ) is the number ofelectrons/positrons at any altitude a . For a given primary type and a given energy E , this latterquantity follows from the Gaisser-Hillas parameterisation of the longitudinal development of9 istance to the shower core [m] ] - H z - S p ec t r a l i n t e n s it y [ W m -33 -31 -29 -27 -25 -23 -21 =1 GHz, sea level altitude n eV, =10 E =4 GHz, sea level altitude n eV, =10 E =1 GHz, 1400 m altitude n eV, =10 E =4 GHz, 1400 m altitude n eV, =10 E Figure 6:
Spectral intensity as a function of the distance to the shower core. the electromagnetic cascade, which depends only on the cumulated slant depth X expressed asthe ratio between the vertical thickness of the atmosphere X vert ( ∼ − at sea level) andthe cosine of the zenith angle of the EAS [24]: N ( a ) = N max (cid:179) X ( a ) − X X max − X (cid:180) X max − X λ exp (cid:179) X max − X ( a ) λ (cid:180) , (16)with X ( a ) the depth corresponding to the altitude a , X the depth of the first interaction, X max the depth of shower maximum, N max the number of particles observed at X max , and λ a param-eter describing the attenuation of the shower.The spectral intensity expected at different distances from the shower core is shown in fig. 6,for two primary energies and two frequencies of interest. The quadratic dependence in fre-quency is seen. The rapid decrease in amplitude for increasing distances is striking. Publishedlimits on the MBR emission are currently at the level of 10 − W m − Hz − [25]. The valuesderived in this study are by orders of magnitude below these current limits. A relevant estimateof the minimal spectral intensity Φ min detectable by an antenna operating in a bandwidth ∆ ν with a noise temperature T sys and an effective area A eff is known to obey Φ min = kT sys A eff (cid:112) τ ∆ ν , (17)where k is the Boltzmann constant and τ the receiver sampling time. For values ∆ ν = τ =
10 ns, T sys =
50 K and A eff approaching 10 cm , values typical of the setups used at thePierre Auger Observatory, for instance, one gets Φ min on the order of a few 10 − W m − Hz − .Based on previous estimates of MBR spectral intensities, such a sensitivity was anticipated to al-low the detection of high-energy showers within a kilometer from the core [13, 14]. By contrast,the results obtained in this study show that the expected signals are out of reach of the exper-imental setups, even for a 10 eV shower sampled at 1400 m altitude level, that of the PierreAuger Observatory. The spectral intensities are 7-to-8 orders of magnitude below the referencevalues anticipated from a scaling law converting the laboratory measurement to EAS expecta-tions put forward in [9] and 5-to-6 orders of magnitude below the values estimated in [13, 14].10 . Discussion The coherent suppression of the MBR in the GHz frequency range as described in section 2is thus prohibitive to allow experimental setups using antennas to detect EAS crossing the fieldof view of the receivers. This coherent suppression stems from the destructive interferences im-pacting the emission amplitude of photons between the successive collisions of the same elec-tron. The spectral intensity at the ground level is several orders of magnitude below the sensitiv-ity of experimental setups. The few detected events over the past years in this frequency rangecannot be due to MBR from the ionisation electrons left along the shower track. Other radio-emission mechanisms, such as the geomagnetic effect, the Askaryan effect or the MBR fromthe primary electrons of the showers, are likely responsible for the observed forward-directedsignals. No side-on observation of EAS is, however, expected from these emission mechanisms,which consequently cannot be seen as the basis of a new breakthrough in the detection tech-nique of EAS for the next decades.For frequencies above the collision rate, the contribution of the MBR to the air-fluorescenceyield, Y , estimated in [26] is also affected by the treatment of the successive collisions presentedin section 2 due to the impact of the inelastic collisions that quench the emission comparedto the simple scaling E = N coll E . Considering n l = ρ f ( v ) 〈 d E /d X 〉 /( I + 〈 T 〉 ) as the numberof ionisation electrons per length and velocity units, the number of emitted photons is thenestimated by plugging n l into eq. (13) and eq. (12), normalised by h ν . The MBR contribution to Y is then obtained by normalising the number of emitted photons to the deposited energy perunit length and by integrating over the frequency range corresponding to the UV [330-400] nmwavelength range: Y = π I + 〈 T 〉 (cid:90) d ν h ν (cid:90) d v f ( v ) (cid:207) d t (cid:48) d t (cid:48)(cid:48) (cid:173) v ( t (cid:48) ) · v ( t (cid:48)(cid:48) ) (cid:174) e − i πν ( t (cid:48) − t (cid:48)(cid:48) ) . (18)This yields to Y (cid:39) − MeV − , which is three orders of magnitude lower than the estimateprovided in [26]. The contribution of the MBR to the fluorescence yield, the world average valuefrom various experiments is (7.04 ± − [18], is thus negligible.Finally, the effect of the successive collisions on the radiation of the ionisation electrons, aspresented in this work, is important to study the possible radar echoes of cascades of particles,which have been recently measured at SLAC in ice to a level that may lead to a viable neutrinodetection technology for energies above 10 eV [27]. The adaptation of the formalism to ac-count for the incoming wave in the two-point correlation function of the electron velocitiesand for the coherence of the re-radiation in eq. (13) can be used to quantify the contribution ofthe re-radiation of the incoming wave by the ionisation electrons [28]. Acknowledgements
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