Radio interferometry applied to the observation of cosmic-ray induced extensive airshowers.

Harm Schoorlemmer1, ∗ and Washington R. Carvalho Jr.2

1Max-Planck-Institut fur Kernphysik Saupfercheckweg 1, 69117, Heidelberg, Germany2Physics Institute, University of So Paulo, Rua do Mato 1371, So Paulo, Brazil

(Dated: June 19, 2020)

We developed a radio interferometric technique for the observation of extensive air showers initi-ated by cosmic particles. In this proof-of-principle study we show that properties of extensive airshowers can be derived with high accuracy in a straightforward manner. Direction reconstructionresolution of < 0.2◦ and resolution on the depth of shower maximum of < 10 g/cm2 are obtainedover the full parameter range studied, with even higher accuracy for inclined incoming directions. Inaddition, by applying the developed method to dense arrays of radio antennas, the energy thresholdfor the radio detection of extensive air showers can be significantly lowered. The method can beapplied to several operational experiments and offers good prospects for planned cosmic particleobservatories.

Interferometry is a method to expose coherence proper-ties in wave phenomena and is applied in many branchesof physics. In this study we show how interferometry canbe used to derive properties of a relativistic cascade ofparticles initiated by an (ultra-) high-energy cosmic par-ticle in the atmosphere, a so-called extensive air shower(EAS).By observing properties of EASs, like the primary axis ofpropagation and the particle density along that axis or atground level, the energy, incoming direction and compo-sition of the cosmic particles can be derived. Classically,the characteristics of an EAS can be observed by record-ing the flash of faint fluorescence and/or Cherenkov lightemitted during its propagation through the atmosphereand/or by sampling the footprint of the EAS using par-ticle detectors at ground level. However, when an EASpropagates through the atmosphere it also emits radio-waves which can be observed by an array of antennas.For recent reviews on the radio detection of cosmic rayinduced air shower see [1, 2]. From a macroscopic point ofview, the emission can be understood as the sum of twomajor contributions, one arising from the deflection ofthe secondary electrons and positrons in the geomagneticfield and the other arising from negative charge build-upin the shower front [3, 4]. Radiation emitted from differ-ent regions in the EAS development will add up coher-ently when the differences in arrival time at the receiverare smaller than a quarter of the wavelength. The powerof radiation that is received coherently depends on thelocation of the observation. As the relativistic movingemitter propagates through the refractive atmosphere itleads to a Cherenkov-cone feature in the radiation pat-tern where the received radiation is maximally coherent[5, 6].Although radio interferometry is widely used in radio as-tronomy, the properties of signals from radio emissionfrom EAS mandate for a different approach. A key dif-ference is that the emitting region is typically not in thefar-field where the wavefront can be approximated by a

plane. As will be shown, this difference will actually en-able three dimensional reconstruction of the EAS prop-erties. Another difference is that the signal is impulsive(with a typical duration in the order of nano-seconds),which has the benefit that aliasing features occurring attime differences that are multiples of the wavelength aresuppressed in comparison to continuous emitting sources.The final difference is that the polarisation of the nega-tive charge build-up emission depends on the location ofthe receiver. We choose not to address this issue for nowas it is typically the sub-dominant contribution to theoverall emission, but we recognise that a proper treat-ment could be implemented as an improvement to theproof-of-principle study presented here.In our method, waveforms Si(t) measured with receiversat different locations i = (1, .., n) are added in the fol-lowing way

Sj(t) =

n∑i

Si(t− ∆i,j), (1)

with ∆i,j the light-propagation time from location j toi. Sj(t) present the coherent waveform at location j andis the key quantity in the interferometric method. Sincewe are interested in impulsive signals, we need the actualtravel time of the light ∆i,j , rather than the geometricalphase difference which is sufficient for continuous emit-ting source in the far-field and commonly used in radioastronomy. By evaluating the power in Sj at differentlocations j a three-dimensional measure of the coherenceis obtained. The light travel time ∆i,j from each antennalocation i to location j is approximated by

∆i,j ≈di,j ni,jc

, (2)

with c the speed of light in vacuum, di,j the distance be-tween location i and j, and n the average atmosphericrefractive index along the path from i to j, assumingthat the light travels in straight paths between i and j.

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This is a valid approximation for the calculations of ra-dio emission from air showers (see appendix A of [7]).The atmospheric refractive index can be parametrised asan exponential function of altitude z, n = 1 + ae−bz,and for simplicity in the remainder of the paper all thecalculations are performed using a = 325 × 10−6 andb = 0.1218 km−1.

To study the spatial distribution of Sj , we used

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a) y=0

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b) z=7 km

c) z=5.25 km

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x (m)

y (m

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FIG. 1. The Radio interferometric technique applied to asimulation of a cosmic-ray induced air shower initiated by a1017 eV proton with an incoming direction one degree fromzenith. Panel a) shows the normalized power of Sj(t) in Eq.1 mapped onto the vertical plane that contains the showeraxis (chosen to be at y=0), while b), c) and d) show horizontalplanes at different heights. As receivers (measuring each Si(t)in Eq. 1) we used an array of 25 antennas on a grid with equalspacing of 100 m. In b), c) and d) the orange dot marks thelocation of the true shower axis, while the blue dot marks themaximum in the map.

the ZHAireS simulation code [8] to calculate the ra-dio emission from air showers simulated by the Airespackage. The ZHAireS calculations provides us witha time-dependent electric field vector at a specifiedantenna position. To obtain signal traces Si(t), a30–80 MHz bandpass filter has been applied to mimicroughly the frequency response of currently operationalarrays like LOFAR[9], AERA[3], Tunka-REX[10], andOVRO-LWA[11]. For simplicity, we decided to evaluatethe component of the electric-field in the direction per-pendicular to the geomagnetic field orientation and tothe direction of propagation of the particle shower. Thiscorresponds to the axis of polarisation of the geomagneticemission, which is the dominant contribution in EASs.In Figure 1, we show the mapping of the normalisedpower of Sj in several planes for a simulated 1017 eV pro-

ton entering the atmosphere one degree from zenith. Themapping in the xz-plane (panel a) shows clearly the co-herent power being mapped near the shower axis andit reaches a maximum along the shower-axis. The hor-izontal slices (panels b, c, d) show that the location ofmaximum power in these maps only deviate a few me-tres from the true position of the shower axis.

We are going to use the observation that the powermapping strongly correlates with the air shower axis toreconstruct air shower properties. We applied the fol-lowing reconstruction approach to identify an axis in thepower mapped space:

1. A horizontal plane is defined at an altitude corre-sponding to the average depth of shower maximumfor a reference energy of the primary particle. Inthis plane an iterative search for the position of themaximum power is conducted. In each iteration theresolution is adjusted to zoom in on the ”hottest”pixel.

2. Based on the location of the maximum in step 1,multiple horizontal planes (like in figure 2) are de-fined and in each plane we search again for the lo-cation of the maximum. We fit a track to the loca-tions of the set of maxima in the horizontal planes,what gives the first estimate of the parameters ofthe shower axis: the impact point at ground levelx0, y0, z = 0 and an azimuth φ and zenith θ angle.

3. In a final step, the estimated parameters from step2 are used as starting point to maximise the inte-grated power along a track of predefined length byvarying the four axis parameters.

We calculated with ZHAireS the radio emission of sets ofproton and iron induced air showers at different zenithangles. A rectangular array of antennas was used inwhich we increased the long-edge of the rectangular gridunit as a function of the zenith angle of the shower. Inthis way, the number of antennas in the radio-footprinton the ground was kept roughly constant in the differentzenith angle sets. We randomised the impact point ofthe shower-axis uniformly within the central rectangulargrid unit. The reconstruction approach has been appliedto this simulated set of air showers and the main resultsare summarised in Figure 2. The result of evaluatingthe power along the reconstructed axis for four simulatedair showers that reach shower maximum at depths thatincrease in steps of roughly 100 g/cm2 is shown in theleft panel in Figure 2. The distribution of the observedpower along the reconstructed axis depends clearly onthe longitudinal development of the air shower. In themiddle panel of Figure 2 the comparison is shown be-tween the depth XRIT of the maximum power along thereconstructed axis (now expressed as slant-depth in unitsof g/cm2 ) and the true depth Xmax where the air show-ers reach their maximum values. The relation between

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+protoniron

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"68% = 0.182!

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"68% = 0.037!

FIG. 2. Air shower development traced by the radio-interferometric-technique (RIT). Left panel, examples of the powerevaluated at distance d measured from the ground along the shower axis. Four simulated proton induced air showers with azenith angle of θ = 30◦ are shown. Middle panel, correlation between Xmax and the slant-depth XRIT of the on-axis maximumof the power obtained by the reconstruction for four simulation sets at different zenith angles. Each set consists of 30 iron and100 proton induced air showers. Right panel, angular deviation δ between reconstructed and true direction for the same foursets of simulations as in the middle panel.

XRIT and Xmax at a specific angle can be described bya simple first order polynomial. We used this fit to es-timate an Xmax value for each shower. This estimatewas then compared to the true shower maximum to ob-tain a distribution of the deviations. We then took theRMS of this distribution as the scatter σ, which repre-sents a measure of the resolution that is achievable bythe method. The obtained overall resolution is encour-aging, with σ < 10 g cm−2 and decreasing significantlywith zenith angle. For comparison, the resolution ofthe Fluorescence detection technique by the Pierre AugerObservatory [12] ranges from 25 g cm−2 at 1017.8 eV to15 g cm−2 towards the highest energies (> 1019 eV ), whilethe Low Frequency Array (LOFAR), using radio mea-surements and a reconstruction method based on fullMonte-Carlo simulations, reported a mean uncertaintyof 16 g cm−2 in the energy range from 1017 to 1017.5 eV[13].In the right panel of Figure 2 the direction of the re-constructed axis is compared to the location of the trueshower axis. The obtainable angular resolution is indi-cated by the 68% containment level δ68%. For both thedirection and Xmax reconstruction we observed that theresolution decreases with decreasing zenith angle (andincreasing depth of shower maximum), i.e. when theshower is physically closer to the receivers the methodworks poorer. This is not yet fully understood and in-vestigating this trade-off in more detail will be part of afollow-up study. One plausible reason might be that co-herence in the wavefront is not achieved over large scalesof the emitting air shower when it is too nearby. Werecognise that the resolution on direction and depth ofshower maximum presented here does not include ad-ditional measurement uncertainties and the impact ofspecific observatory designs. However, a constrain on

observatory design to use this method comes from thecoherence condition. It implies that the timing uncer-tainty ∆t and position uncertainty ∆x of each receiverneeds to be known to an accuracy of smaller than quar-ter of the wavelength, for a signal at 50 MHz this yields√

∆2t + (∆x/c)2 < 5 ns. This limit has been verified

by randomising the timing at each antenna in the re-construction of Xmax. No significant degradation hasbeen observed on the resolution when randomising upto the order of 3 ns, but the power mapping is signifi-cantly disturbed when crossing the 5 ns coherence limit.Dedicated studies of simulated showers including full in-strumental response and array configurations are neededto estimate the performance of this method in the dif-ferent operational and planned experiments. However,the results from the relatively simple reconstruction ap-proach given here yield a promising starting point formore elaborate approaches including more realistic in-strument modelling.This radio interferometric technique provides a straight-

forward way of reconstructing EAS parameters, with theonly assumption being the atmospheric refractive indexmodel. As shown, the mapping of the power in Sj givesaccess to shower parameters like the primary axis andlongitudinal development parameters. However, this in-terferometric method also has its limitations and arte-facts. For example, one artefact shows up in the leftpanel of Figure 2, where there is a long tail in the powerof Sj that remains even in the region of large distancesalong the shower axis, where the number of particles fromair shower development vanishes. This is caused by thefact that the gradient of arrival times reduces to zerofor antennas in the array for large distances d along theshower axis. Therefore there is some remainder power,which corresponds to the power of Sj under a plane wave

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Summed Waveformat location of maximum

FIG. 3. Properties of the summed waveform Sj,max with the maximum power along the shower axis for a 1017 eV protoninduced air shower 60◦ from zenith. At the location of each antenna a waveform with white-noise is simulated with a root meansquare of 14µV/m. Left panel, dependency of the amplitude of Sj at XRIT on the antenna density of the array. Middle panel,the waveform of the location on the ground that received the maximum signal from the air shower. Right panel, the summedwaveform Sj,max for an array density of 22 antennas per km2.

approximation.The radio interferometric technique has the appearanceof being a tomographic method as it can be applied inthree dimensions. However, one should use this termwith some caution, since the mapping of the power inSj(t) is not a direct representation of the emitted radia-tion, the spatial data points are correlated over scales setby coherence condition for a given wavelength. Applyingthe method at higher frequencies than used in this studywill shorten the coherence length and therefore will give amore direct mapping of the emitting regions of the EAS.In addition, it might be possible to expand the methodwith deconvolution techniques in order to map the emis-sion directly.One important aspect of interferometric methods is thatsignal-to-noise ratio scales as the square-root of the num-ber of receiving elements. This means that the detectionthreshold of the radio-detection technique can be low-ered by increasing the antenna density in the array ac-cordingly. This behaviour is illustrated in Figure 3 inwhich we vary the antenna array density for a 60◦ simu-lated air shower. At each receiver, we simulate a randomwhite noise realisation resulting in an average root meansquare of 14µV/m. In the left panel of Figure 3 thesummed waveform Sj,max at XRIT is evaluated as a func-tion of receiver density in the array. We show two cases,once simulating noise and once only simulating signal.As expected for the coherent signal, the peak amplitudeof Sj,max scales linearly with the density of the receiverswhile the incoherent noise level increases as the square-root of the receiver density. As an example of how thiscould lower the energy threshold for estimating air showerparameters, we show in the middle panel the waveformof the antenna that receives the strongest signal in thearray. As is visible, when the noise is added to the signal(black line) it is hard to identify the presence of any signal

at all. However, a clear signal can be seen in the coherentsum, when all the antennas in the array are combined toobtain Sj,max, as shown in the right panel. There are ex-isting high-density arrays that are capable of measuringEAS, like the LOFAR detector where this method canbe directly tested. New facilities, like the square kilo-metre array, might have a density of 60000 antennasper km2[14]. For such arrays, this analysis technique willenable the study of air shower parameters over a signifi-cantly larger range of energies.Several upcoming experiments are focusing on radio ob-servation of inclined air showers, where the interferomet-ric method is showing its best performance. The PierreAuger Observatory is currently being upgraded which in-cludes a radio-antenna on each water-Cherenkov tank ofthe surface detector in order to observe the radio emis-sion of inclined events [15]. The determination of showermaximum with the method developed here will enableaccurate hybrid measurements of inclined air showers tobe used in composition studies. Another project is theGiant Radio Array for Neutrino Detections [16], wherethe radio interferometric method might both help withair shower reconstruction and with the identification ofair showers among other impulsive events. Since boththese arrays are deployed over large areas, the challengewill be to obtain synchronisation between the individualstations accurately enough to satisfy the coherence con-ditions. This might for example be achieved with the useof external continuous narrow-band transmitters [17, 18].We would like to remark that the presented method hereis generic and is applicable to a wider range of observa-tions. A similar method has already been applied to theobservation of lightning [19], revealing substructures withunprecedented resolution. Another obvious candidate toapply the radio interferometric technique are the in-iceradio antenna arrays deployed for neutrino observations

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[2]. In this case, the presented method here might needto be complemented by ray-tracing algorithms in orderto determine the light travel time to sufficient accuracy.To conclude, the method presented here provides a wholenew perspective on observations of extensive air showersover a wide energy range, which could lead to significantimprovement on the accuracy of the measurements of cos-mic particle properties. Therefore this method can play acrucial role in understanding the origin and propagationof these cosmic particles in the near future.

ACKNOWLEGEMENTS: We would like to thankJaime Alvarez-Muiz, Stephanie Ann Wissel and EnriqueZas for their insightful comments. W.R.C. thanks grant2015/15735-1, So Paulo Research Foundation (FAPESP).

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