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Reconstruction of the energy and depth of maximum of cosmic-ray air-showers from

LOPES radio measurements

W.D. Apela, J.C. Arteaga-Velazquezb , L. Bahrenc, K. Bekka, M. Bertainad, P.L. Biermanne, J. Blumera,f,H. Bozdoga, I.M. Brancusg, E. Cantonid,h,q, A. Chiavassad, K. Daumillera, V. de Souzai, F. Di Pierrod,P. Dolla, R. Engela, H. Falckec,j,e, B. Fuchsf, D. Fuhrmannk,r, H. Gemmekel, C. Grupenm, A. Haungsa,

D. Hecka, J.R. Horandelc, A. Horneffere, D. Huberf, T. Huegea,∗, P.G. Isarn, K.-H. Kampertk, D. Kangf,O. Kromerl, J. Kuijpersc, K. Linkf, P. Luczako, M. Ludwigf, H.J. Mathesa, M. Melissasf, C. Morelloh,

J. Oehlschlagera, N. Palmierif,∗, T. Pieroga, J. Rautenbergk, H. Rebela, M. Rotha, C. Ruhlel, A. Saftoiug,H. Schielera, A. Schmidtl, F.G. Schrodera, O. Simap, G. Tomag, G.C. Trincheroh, A. Weindla, J. Wochelea,

J. Zabierowskio, J.A. Zensuse

aInstitut fur Kernphysik, Karlsruhe Institute of Technology (KIT), GermanybUniversidad Michoacana, Morelia, Mexico

cRadboud University Nijmegen, Department of Astrophysics, The NetherlandsdDipartimento di Fisica Generale dell’ Universita Torino, Italy

eMax-Planck-Institut fur Radioastronomie Bonn, GermanyfInstitut fur Experimentelle Kernphysik, Karlsruhe Institute of Technology (KIT), Germany

gNational Institute of Physics and Nuclear Engineering, Bucharest, RomaniahINAF Torino, Instituto di Fisica dello Spazio Interplanetario, Italy

iUniversidad Sao Paulo, Inst. de Fısica de Sao Carlos, BrasiljASTRON, Dwingeloo, The Netherlands

kUniversitat Wuppertal, Fachbereich Physik, GermanylInstitut fur Prozessdatenverarbeitung und Elektronik, Karlsruhe Institute of Technology (KIT), Germany

mUniversitat Siegen, Fachbereich Physik, GermanynInstitute for Space Sciences, Bucharest, Romania

oNational Centre for Nuclear Research, Department of Cosmic Ray Physics, Lodz, PolandpUniversity of Bucharest, Department of Physics, Romania

qnow at: Istituto Nazionale di Ricerca Metrologica, Torino, Italyrnow at: Universitat Duisburg-Essen, Duisburg, Germany

Abstract

LOPES is a digital radio interferometer located at Karlsruhe Institute of Technology (KIT), Germany, whichmeasures radio emission from extensive air showers at MHz frequencies in coincidence with KASCADE-Grande.In this article, we explore a method (slope method) which leverages the slope of the measured radio lateraldistribution to reconstruct crucial attributes of primary cosmic rays. First, we present an investigation of themethod on the basis of pure simulations. Second, we directly apply the slope method to LOPES measurements.Applying the slope method to simulations, we obtain uncertainties on the reconstruction of energy and depthof shower maximum (Xmax) of 13 % and 50 g/cm2, respectively. Applying it to LOPES measurements, we areable to reconstruct energy and Xmax of individual events with upper limits on the precision of 20-25 % for theprimary energy and 95 g/cm2 for Xmax, despite strong human-made noise at the LOPES site.

Keywords: cosmic rays, extensive air showers, radio emission, LOPES, Xmax, energy

∗Corresponding authors:Email addresses: tim.huege@kit.edu (T. Huege),

nunzia.palmieri@kit.edu (N. Palmieri)

1. Introduction

Hundred years have passed since the discovery ofcosmic radiation, and an accurate reconstruction ofboth the energy and mass of primary cosmic rays athigh energies still remains a compelling need in con-

Preprint submitted to Physical Review D August 12, 2014

temporary astro-particle physics.

Only recently the methods for detection of co-herent MHz radiation from extensive air showers aswell as the understanding of the underlying emis-sion physics have been strongly improved, reachingimportant milestones [1]. A precise reconstructionof both the primary energy and the depth of theshower maximum (Xmax) of air showers is a funda-mental goal for modern radio detection, which aimsto become competitive and complementary to thealready well-established fluorescence and Cherenkovdetection techniques, limited, in contrast to the radiotechnique, to a low duty cycle [2].

In the following, we present a method (slopemethod), which obtains information on the energyand the depth of maximum of air showers from fea-tures of the lateral distribution of the radio signals:On the one hand, a defined distance from the showeraxis exists where the reconstruction of the primaryenergy is affected least by shower-to-shower fluctua-tions. The presence of such a characteristic distancewas previously predicted with REAS2 simulations [3].Here, we demonstrate on the basis of state-of-the-artCoREAS simulations [9] that such a characteristicdistance is still present when refractive index effectsand radiation due to the variation of the number ofcharged particles are taken into account, and thatmeasurements at this distance can thus be exploitedfor an energy determination. On the other hand, theslope of the radio lateral distribution is related tothe geometrical distance between the observer andthe radio source [3, 4, 5]. Therefore, information onthe depth of shower maximum (Xmax) and consecu-tively on the type of primary particle initiating theshower, can be directly extracted from the radio lat-eral distribution measurement: For iron-initiated airshowers, which start earlier and develop faster in theatmosphere compared to proton-initiated air showers,the radio source is typically further away from an ob-server at ground, and thus the slope of the lateraldistribution is flatter in comparison. Recently, thissensitivity of the slope of the measured radio lateraldistribution to the development of air showers hasbeen proven experimentally with LOPES measure-ments [6].

We strive to take a practical approach tailored tothe data quality provided by the LOPES experiment.In particular, we aim to develop an analysis strategywhich can exploit the potential of LOPES in lightof its experimental limitations (relatively low num-

ber of antennas, limited lateral distance range cov-ered in a single measurement, relatively high back-ground noise) without becoming overly involved andcomplex. For experiments with higher data quality,the methods presented here should thus be refinedand further improved.

Another independent method has been separatelydeveloped and applied on LOPES measurements toextract information on Xmax. This method considersthe shape of the radio wave front [7, 8], but lower pre-cision in the reconstruction of Xmax compared to theslope method was achieved. The slope and wavefrontmethods could in the future also be combined witheach other and with further methods such as the de-termination of the pulse width, or spectral slope [30],respectively, or the determination of the position ofthe Cherenkov ring [31].

CoREAS simulations [9], which include a realistictreatment of the refractive index of the atmosphere,are more sophisticated and more complete than sim-ulations with the REAS predecessor codes. The re-markable agreement between CoREAS simulationsand the measured slope of the radio lateral distri-bution [1, 10] makes studies based on CoREAS simu-lations even more promising than previous analyses.

An amplitude scale mismatch between the mea-sured (LOPES) and simulated (CoREAS) radiopulses still exists [10]. However, it does not affect thepart of this analysis in which the slope of the radiolateral distribution, i.e., the ratio of amplitudes at dif-ferent distances, is considered. It only has influencewhen absolute amplitude values are used, i.e., in ourcase on the investigation of the energy reconstruction(sec. 6), in which it will influence the calibration con-stants determined for the linear correlations betweenprimary particle energy and radio amplitude.

Taking the analysis described in [12, 13] as aguideline, we enhance and further improve the slopemethod. As a first step, we apply the slope methoddirectly on pure CoREAS simulations, deriving im-portant calibration parameters. In a second step,we apply the slope method on events measured withLOPES to investigate the reconstruction of the totalprimary energy and the depth the shower maximum.

2. LOPES event selection

LOPES [14, 15] is one of the pioneering experi-ments in the digital detection of radio emission fromair showers. Placed at Karlsruhe Institute of Technol-

2

west-east [m]-150 -100 -50 0 50 100

sout

h-no

rth

[m]

-150

-100

-50

0

50

100KASCADE-Array

shower cores

LOPES antennas

until end of 2006

Grande stations

Figure 1: Shower core positions of the 205 selected events,measured simultaneously by the particle detector KASCADE-Grande and by the LOPES east-west-aligned antennas.

ogy, Germany, LOPES benefits from the co-locationwith the particle detector experiment KASCADE-Grande [16], which provides us with the reconstruc-tion of fundamental air-shower parameters: arrivaldirection, shower core, and primary energy.

Characteristic for the LOPES analysis pipeline isthe interferometric combination of the recorded elec-tric field strength traces: according to the arrival di-rection of the air shower, the traces detected in eachindividual antenna are first shifted in time. After-wards, a cross-correlation beam (CC-beam) and apower-beam, which gives the total power received inall the antennas, are computed [17, 18]. The digitalcross-correlation beam forming plays an importantrole also for the reconstruction of the lateral distri-bution of amplitudes. Indeed, both the CC-beam andthe power-beam are crucial for the selection of eventswith a clear radio signal, and for the determinationof the exact time of the radio pulse (CC-beam time).A Hilbert envelope is applied to the up-sampled traceof each individual antenna, and the maximum instan-taneous amplitude of the radio signal is defined bythe local maximum of the Hilbert envelope closestto the CC-beam time. More details can be found in[7, 10, 17, 18].

The events selected for the slope method ana-lysis presented here have been measured with theLOPES30 and LOPESpol setups [19]. The former

consisted of 30 calibrated dipole antennas, all ori-ented in the east-west direction, while the latter used15 east-west and 15 north-south-aligned antennas.For the purpose of this analysis, only data of theeast-west oriented channels are taken into account,primarily due to the higher available statistics [18].The effective frequency band used for the analysis is43-74 MHz.

The selected events have a primary energy between1017 and 1018 eV, a zenith angle less then 40◦ and acore position at a maximum distance of 90 m fromthe center of the KASCADE array - (0 m west-east, 0 m south-north) in fig. 1. These cuts aremade to achieve high quality in the KASCADE airshower reconstruction. A high signal-to-noise ratioand high coherence for the radio signal in the an-tennas is required as well by forcing the signal-to-noise ratio of the CC-beam, i.e. (amplitude of thecross-correlation beam / RMS of the cross-correlationbeam ·

√

Nant/30)1 to be larger than 9, and thefraction of correlated power in the antennas (CC-beam/power-beam) larger than 80 %. Further qual-ity cuts demand a good fit for the lateral distribu-tion function (sec. 4). All cuts are chosen to achievehigh standards in the KASCADE-Grande data recon-struction and high quality radio measurements, yetsimultaneously acquire good statistics. 205 events ofalmost 900 KASCADE-Grande pre-selected showersremain after applying all cuts.

3. CoREAS simulations of the selected events

The CoREAS code is one of the latest develop-ments of simulation codes reflecting an increasinglyprecise knowledge and an improved modelling of theradio emission from air showers [9]. Through the in-clusion of a realistic treatment of the refractive indexof the atmosphere, CoREAS simulations take into ac-count Cherenkov-like time-compression of the radioemission occurring for observers near the Cherenkovangle. This effect has been recognized to be the mainreason for the previously measured flattening of theradio lateral distribution close to the shower axis [20].The flattening effects depend on the geometrical dis-tance of the shower maximum to the observer, andthus on both the zenith angle of the air shower and on

1The CC-beam is normalized with a factor√

Nant/30, sincenot always all the 30 antennas participate in each event

3

the type of the primary particle. A detailed compar-ison between CoREAS simulations and LOPES mea-surements is presented in [10].

For the following analysis, CORSIKA [21] andCoREAS are used to simulate the particle cascadeand the radio emission from the electromagnetic com-ponent of air showers, respectively. QGSJet II.03 [22]and FLUKA [23] are used as high energy and lowenergy interaction models, respectively.

Specific criteria are applied to generate theCoREAS simulations to facilitate the final applica-tion of the slope method directly on LOPES data. Onthe one hand, the specific individual events measuredby LOPES are simulated: The energy, the incomingdirection and the core position of each LOPES event,reconstructed from the measurements with the par-ticle detector KASCADE(-Grande) [16], are used asinput parameters for the simulations. The simulatedelectric field is filtered with an ideal rectangular band-pass filter (43-74 MHz). The geomagnetic field for thesimulations is set to values valid for the LOPES site(47 µT magnetic field strength, 65◦ inclination angle,negligible declination). With this procedure, the geo-metrical acceptance and important characteristics ofthe LOPES experiment are taken into account. EachLOPES event is simulated as both proton and ironprimary. On the other hand, the observer positions inthe simulations appropriately represent the antennapositions in the LOPES array: For each individualevent, the correct positions of the individual LOPESantennas with respect to the shower core are takeninto account. For the selected events, the cores of theshower typically lie in between the antennas, thus thefit of the lateral distribution function (LDF) does notonly involve observer positions in a specific azimuthaldirection relative to the core. As a consequence, thecharge excess effect which typically contributes to thetotal radio emission at a level of ≈ 10−15 % [4, 11] inAERA, and can be expected to be about half as im-portant in LOPES due to the stronger local magneticfield, is included in the following analysis. We canthus judge the robustness of our method in light ofthe complex asymmetries present in the actual mea-sured radio lateral distributions. No pre-selection oftypical showers (showers with a typical Xmax value)is applied to the CoREAS simulations. Therefore,shower-to-shower fluctuations are included in the in-vestigation for the data set as a whole.

0

1

2

3

4

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6

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Gaussexponential

simulations

axis distance d [m]0 20 40 60 80 100 120 140 160 180 200 220

V/m/MHz]

[∈

fieldstrength

µ

axis distance d [m]0 20 40 60 80 100 120 140 160 180 200 220

V/m/MHz]

µ[∈

fieldstrength

0

2

4

6

8

10

12

14

16

Gaussexponential

data

Figure 2: Lateral distribution functions for an event in the se-lection used for the analysis. A Gaussian and an exponentialfunction are used to fit the points. Top: CoREAS simula-tions. Bottom: LOPES data. The absolute amplitudes exhibita known scale mismatch between LOPES data and CoREASsimulations (see text and ref. [10]).

4. The lateral distribution function

The lateral distribution function (LDF) describesthe measured (or simulated) electric field as a func-tion of the observer position with respect to theshower axis. Due to the superposition of geomag-netic and charge-excess radiation in radio emissionfrom extensive air showers, the LDF can exhibit sig-nificant asymmetries [1]. Ideally, therefore a two-dimensional LDF taking into account the observerazimuth angle in addition to the axis distance shouldbe fitted to the measured radio data. In the caseof LOPES, however, the limited number of antennassampling the true LDF led us to apply a simpler ap-proach. Instead of an asymmetric two-dimensionalLDF, we apply a one-dimensional LDF to our data,

4

thereby effectively averaging out the asymmetries inthe true LDF. Depending on the coverage in the sam-pled antenna locations, a per-event amplitude shiftcan be introduced by this procedure. This effect pos-sibly leads to increased scatter in the fitted ampli-tudes, which is, however, implicitly accounted for inthe following analysis. Another effect of the one-dimensional averaging could be an overall bias inthe calibration of radio-amplitudes to primary par-ticle energies determined by KASCADE. To quantifythese two effects, we compared simulations includingthe charge-excess asymmetry with special simulationsin which the charge-excess contribution has been re-moved. The result of this dedicated study are: Theuse of a one-dimensional LDF increases the scatterobserved in the amplitude-energy correlations by anamount that is negligible with respect to other un-certainties, in particular LOPES experimental uncer-tainties. For the data set as a whole a systematic biasof 4.4% in the estimated average amplitudes arises,which we account for in the following energy corre-lation analysis by adopting an extra systematic un-certainty of 5%, well-below our global experimentalcalibration scale uncertainty of 35%.

The uncertainty of the measured electric field am-plitudes we fit with our one-dimensional LDF is takenas the root sum of squared uncertainties due to theindividual antenna calibration (5 %) and due to thenoise [7]. The distance of the antennas from theshower axis is calculated in the shower-plane coordi-nate system with a correlated uncertainty which de-pends on both the KASCADE precision (∼ 4 m [24])and the geometry reconstruction accuracy (< 0.3◦).In contrast, the CoREAS electric field at each ob-server position is assumed to be exact, thus no un-certainty is associated with the pulse amplitude.

The CoREAS simulations are able to reproducethe complexity of the measured radio LDF, includ-ing asymmetries and flattening due to Cherenkov-like time compression effects. Even at the distancesprobed by the LOPES experiment, this complexityis clearly visible [10, 20]. Since only events with acore position relatively close to the antenna arrayare selected (sec. 2 and fig. 1), the flattening of theradio lateral distribution close to the reconstructedcore becomes an important aspect for our investiga-tion. In previous LOPES analyses, we have used anexponential function to fit data and simulations ofLOPES events [10, 20]. Here, we prefer to use a one-

dimensional Gaussian function

ǫ(d) ≃ ǫG exp

(

(d− b)2

2c2

)

(1)

with ǫG [µV/m/MHz], b [m], and c [m] as free pa-rameters to fit the LOPES LDF. While the aver-age LOPES event can be fitted by either a one-dimensional exponential or a Gaussian with similarquality, there are two advantages in using the Gaus-sian. For one, there are some events for which theGaussian can fit the data well, while the exponentialgives a considerably worse fit. Second, and more im-portantly, we develop our analysis method on the ba-sis of CoREAS simulations, and for simulated eventsthe Gaussian fits considerably better than an expo-nential. We illustrate this with a comparison of aGaussian and an exponential fit to data and simu-lations in fig. 2. We stress that the analysis pre-sented here can be (and has also been) performedon the basis of an exponential LDF, and the re-sults agree within the systematic uncertainties quotedhere. However, for the above reasons, we prefer to usea Gaussian LDF.

Besides the depth of the shower maximum (Xmax),the air-shower inclination influences the slope of theradio LDF. This is related as well to a geometricaleffect: near-vertical events constitute a radio sourcecloser to the observer at ground level than inclinedshowers, and therefore are characterized by a steeperslope of the radio LDF. Five zenith angle bins are con-sidered separately to reduce the dependence on theinclination and to focus on the effects connected tothe depth of shower maximum of the showers. Thesezenith angle bins are chosen such that they cover thesame solid angle.

5. The slope method

We first investigate the slope method using theCoREAS simulations of the LOPES events. The pro-cedure aims to compare the LDF of several eventswith different primary energies and arrival directions,thus normalizations for the radio amplitudes are re-quired. With the well-founded assumption that theleading contribution to the radio emission from airshowers is of geomagnetic origin (> 85 %) [11, 18], theEast-West channel amplitudes for each given eventare corrected for

−→P ew, the east-west component of

the Lorentz vector (−→v ×−→B ), with −→v the arrival di-

rection and−→B the geomagnetic field vector. (Due to

5

axis distance [m]50 100 150 200 250 300 350

V/m

/MH

z/G

eV

]µ

/E

[EW

PEW

ε

1

10

zenith < 19.4 deg

CoREAS p

CoREAS Fe

/

d0

Figure 3: Normalized CoREAS lateral distribution functionfits using a one-dimensional Gaussian for the events with zenithangles smaller than 19.4◦, simulated once as proton (blue), onceas iron (light-red) primaries. The dotted line represents thedistance d0 at which the RMS-spread of the fitted normalizedamplitudes is minimal.

50 100 150 200 250 300 350

10

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40

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rela

tive

sp

rea

d L

DF

fits [%

]

axis distance [m]

0 < zenith < 19.4

19.4 < zenith < 26.8

26.8 < zenith < 32.0

32.0 < zenith < 36.2

36.2 < zenith < 40.0_

__

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0

0

0

0

0

0

0

0

0

0

Figure 4: Relative RMS-spread profile for the CoREAS LDFfits as a function of distance from the shower axis, for the dif-ferent zenith angle bins.

detection threshold effects, there are no events in theLOPES data set with a shower axis parallel to themagnetic field; the smallest geomagnetic angle is 15◦,corresponding to a sin(α) of 0.26, whereas the meansin(α) is 0.7.) In addition, we normalize the elec-tric field amplitudes by the Monte Carlo true energyof each event, which has been adopted as the energyreconstructed by KASCADE(-Grande) for the under-lying LOPES event.

The normalized lateral distribution function fits forthe events in the first zenith angle bin (zenith<19.4◦)are presented here as an example (fig. 3). This figurealready illustrates that the spread of the obtained ra-

V/m/MHz]µ [Pew , 0d

ε10

eV

]17

[1

0tr

ue

(C

oR

EA

S in

pu

t)

E

1

Fit result:

slope par. k 0.299 ± 0.003

CoREAS p

CoREAS Fe

RMS-spread 8.7%

zenith < 19.4 deg

Figure 5: Linear correlation between the Monte Carlo true pri-mary energy and the CoREAS radio pulse amplitude at thedistance d0, with k the free parameter of the fit. The radioamplitude is normalized by the east-west component of theLorentz force vector. The RMS-spread is ∼ 8.7 %.

dio LDF fits drastically varies as a function of the dis-tance from the shower axis. The spread is quantifiedin fig. 4, where the profile of the relative RMS-spreadfor the LDF fits is shown for each of the zenith anglebins. The spread reaches from ≈ 10 % at distances of70-100 m from the shower axis up to more than 50 %at distances larger than 300 m. Simulations [3, 12]previously predicted the existence of a peculiar dis-tance (d0) where the radio amplitudes do not carryinformation on Xmax and the influence of shower-to-shower fluctuations is minimal. Also here, we observethe minimum spread of the LDF fits at a peculiar dis-tance d0. The value of d0 as well as the value of theminimum RMS-spread vary with the zenith angle ofthe air shower and are summarized in table 1. ForLOPES events, d0 lies in the range 70-100 m fromthe shower axis.

6. Primary energy reconstruction

The radio amplitude at the peculiar distance d0,barely affected by shower-to-shower fluctuations, pro-vides direct information on the shower energy [3, 12].

6.1. CoREAS simulations

To illustrate this first on the basis of simula-tions, we take the east-west component of the elec-tric field vector predicted by CoREAS simulations ofthe LOPES events, apply the digital bandpass-filter,

6

CoREAS sim.

∆θ entries d0 [m] RMS-spread [%] total uncertainty [%]

0◦ − 19.4◦ 53 70 8.7 13.319.4◦ − 26.8◦ 48 80 8.1 12.926.8◦ − 32◦ 45 90 7.6 12.532◦ − 36.2◦ 36 100 9.4 13.736.2◦ − 40◦ 23 100 10.0 14.1

Table 1: Distances d0 for the five zenith angle bins as determined from CoREAS simulations. For each bin, the RMS spreadfor the LDF fits at d0 of the different events is given, as well as a conservative estimate of the total uncertainty (computed asquadratic sum of the given RMS spread and the spread of individual points around the fits between 70m and 100m).

CoREAS sim. LOPES data

predicted slope par. k slope par. k

∆θ events d0,CoREAS

√

RMSspread2 + 20%2 ± stat. uncert. RMSspread ± stat. uncert.

[m] [%] [GeV m MHz/µV ] [%] [GeV m MHz/µV ]

0◦ − 19.4◦ 53 70 21.8 0.299 ± 0.003 18.8 0.134 ± 0.00519.4◦ − 26.8◦ 48 80 21.6 0.328 ± 0.003 23.1 0.151 ± 0.00626.8◦ − 32◦ 45 90 21.4 0.358 ± 0.004 24.6 0.141 ± 0.00632◦ − 36.2◦ 36 100 22.1 0.382 ± 0.005 20.3 0.142 ± 0.00736.2◦ − 40◦ 23 100 22.4 0.398 ± 0.006 25.6 0.137 ± 0.008

Table 2: Comparison between the CoREAS prediction and the LOPES measurements for the RMS-spread at the distance d0, andfor the slope paramters kCoREAS and kLOPES determined with linear fits (see figs. 5 and 7 for the first zenith angle bin). In additionto the reported statistical uncertainties of the k-parameters, an additional 5% systematic uncertainty has to be taken into accountbecause of the adoption of a symmetrical one-dimensional LDF. Since the CoREAS results are based on the KASCADE energy asMonte Carlo true input without uncertainty, but the LOPES results implicitly contain the KASCADE energy uncertainty (approx.20%), we quadratically added 20% to the CoREAS RMS spread.

V/m/MHz]µ [Pew,

0d

ε10

eV

]1

7

[10

K

AS

CA

DE

-Gra

nde

E

1

c

k 0.002977– 0.2996

0 < zenith < 19.419.4 < zenith < 26.826.8 < zenith < 32.032.0 < zenith < 36.236.2 < zenith < 40.0_

__

___

0

0

0

0

0

0

0

0

0

0

Figure 6: Linear fits for the energy – radio amplitude in d0

correlation (double-log scale), for all the zenith angle bins. Theslope of each fit clearly depends on the inclination of the airshower.

fit the Gaussian LDF, read off the radio pulse ampli-tude at the distance d0 and normalize this amplitude

with the east-west component of the Lorentz forcevector adequate for the arrival direction (i.e.

−→P ew).

We then plot this normalized amplitude against theMonte Carlo true energy of the air shower, as shownin fig. 5 for the events in the first zenith angle bin.

A simple linear correlation with the total pri-mary energy becomes apparent: eq. 2 with k[GeV · m · MHz /µV] as free parameter is used tofit the filtered radio amplitudes at the distance d0 forproton and iron simulated events.

energy = k · (ǫd0/|−→P ew|) (2)

A measurement of the radio pulse amplitude at thepeculiar distance d0 thus allows us to directly recon-struct the energy of the cosmic ray primary. The in-trinsic uncertainty of this method is composed of twocontributions: the RMS-spread of the amplitude at d0

around the linear correlation with the energy (approx.9 %), and the scatter of the individually measured ra-dio amplitudes at d0 (approx. 10 %). Of course, bothquantities are correlated, but as a conservative esti-mate we added both quadratically and estimate the

7

total energy uncertainty to approximately 13 % (seetable 1). This might be improved by explicitly takingthe asymmetries introduced by the Askaryan effectinto account in the fitting procedure, since they arethe main source of the scatter. However, such an im-provement would be below our testing power which islimited by the energy precision of KASCADE-Grande(∼ 20 %).

On closer look, a systematic shift between the nor-malized radio amplitudes at the distance d0 is visi-ble between the two different primaries (fig. 5). Theradio-amplitudes for proton primaries are slightlyhigher than those for iron primaries. This is relatedto the fact that the radio emission is connected onlyto the electromagnetic component of the air shower.Depending on the primary type, a slightly differentfraction of the total primary energy is transferred tothe electromagnetic and non-electromagnetic compo-nents of the air shower, thus changing the amplitudesof the radio signal [3, 28].

We stress here that this is not a “problem” of theradio detection technique. It simply signifies that ra-dio emission purely probes the energy in the electro-magnetic component of an air shower, with a highintrinsic precision. Thus, combining a radio detectorwith a technique sensitive to the muonic componentof air showers would be a particularly compelling ap-proach potentially giving very detailed informationon the cosmic ray composition. This feature alsomakes the interpretation of radio measurements lesssensitive to hadronic interaction model uncertainties,as the electromagnetic component of air showers isgenerally well-understood.

The linear correlations for all the zenith angle binsare shown in fig. 6. The values for the slope parame-ter k are reported in table 2. Table 2 also denotes theRMS-spread around the linear fit — an estimator ofthe achievable energy precision — expected when tak-ing into account the KASCADE energy uncertainty:The simulations use the KASCADE energy as MonteCarlo true input and thus are not affected by any un-certainty of this energy. In contrast, for LOPES mea-surements, the KASCADE energy uncertainty has asignificant effect on the results (thus, there are y-errorbars in fig. 7 (top), but not in fig. 5). Consequently,for a comparison between simulations and data theKASCADE energy uncertainty has to be added tothe values derived from simulations. For the data setand energy range used here, this uncertainty is notknown with very high precision, and is estimated to

be approximately 20 %. We thus quadratically add aconstant 20 % to the RMS spread of the simulations.As described in section 4, the k-parameters reportedhere have an additional 5% systematic uncertaintydue to the use of a symmetric one-dimensional LDFfunction to estimate the per-event amplitude at thecharacteristic distance.

A deeper look at the absolute values of the k-parameters reveals that they increase with increas-ing zenith-angle. This is due to the evaluation of thefit at the specific characteristic distance d0, whichincreases with zenith angle. As d0 increases, theaverage radio amplitude decreases, and thus the k-parameter increases. If the correlations are evaluatedat a fixed lateral distance for all zenith angle bins, thek-parameters become consistent within uncertainties.

6.2. LOPES measurements

We now apply the energy reconstruction to LOPESdata. The LOPES-measured radio lateral distribu-tions for each event are fit with the Gaussian func-tion. The fit value at the distance d0 is used as theenergy estimator. We adopt the values for the dis-tances d0 at which we evaluate the radio amplitudein each zenith angle bin as those derived previouslyfrom the CoREAS simulations (table 1).

This yields a simple linear correlation between themeasured radio pulse (at distance d0) and the energyreconstructed by KASCADE-Grande, as shown in fig.7 -top panel (zenith angles smaller than 19.4◦ takenas example). The slope parameter k for the linearfit in each zenith bin (reported in table 2, right side)is not compatible within the uncertainties with theCoREAS predicted slope. This illustrates the stillexisting amplitude scale mismatch between CoREASsimulations and LOPES measured radio pulses [10],which on average amounts to a factor of ≈ 2.5.Further comparison between the k-parameters deter-mined for simulations and LOPES data reveals thatthe deviation increases with zenith angle. We haveperformed an independent analysis on a different se-lection of LOPES events and compared the measuredamplitudes at a distance of 100 m with the ampli-tudes predicted by CoREAS simulations. This cross-check confirmed that the amplitude discrepancy be-tween LOPES measurements and CoREAS simula-tions increases with zenith angle. The reason for thisdiscrepancy is currently unknown. It could be causedby a discrepancy between the simulated LOPES an-tenna gain pattern and the actual antenna gain pat-

8

10

1

10

EK

ASC

AD

E-G

rande[1

01

7eV

]

slope par. k 0.134 ± 0.005

LOPES data

zenith<19.4 deg

εd0,PEW

[µV/m/MHz]

axis distance [m]0 50 100 150 200 250 300 350

rela

tive s

pre

ad L

DF fi

ts [

%]

10

15

20

25

30

35

40

45

50

55

60

zenith<19.4 deg

CoREAS

LOPES

Figure 7: Top: Linear correlation of the KASCADE(-Grande)reconstructed primary energy and the normalized LOPES mea-sured radio pulse in d0, for zenith angles less than 19.4◦. TheRMS-spread is bf 18.8 %. Bottom: RMS-spread from theenergy-fit for zenith angle less than 19.4◦, computed at severaldistances from the shower axis. With the LOPES measure-ments (black triangles) the minimum is found at 80 m. Theblue stars represent the CoREAS predictions (fig. 4 for the sameevents, adding 20% for the KASCADE energy uncertainty).

tern, in spite of efforts to cross-check our antennapattern simulations with dedicated calibration mea-surements [29]. Another explanation could be a dete-rioration of the KASCADE-based (not KASCADE-Grande-based) energy reconstruction at energies wellbeyond 1017 eV [10], where punch-through of the elec-tromagnetic air shower component into the shieldedmuon detectors may start to matter, an effect whichis obviously zenith-angle dependent. Finally, therecould also be a problem in the CoREAS simulations,which should thus be compared with data from otherexperiments.

As a next step, we verify the existence of the pecu-liar distance d0 in the LOPES measurements. At thedistance d0 the correlation between the energy and

the radio amplitude exhibits the lowest uncertainty.Thus, we use again the RMS-spread around the linearfit (fig. 7-top panel) of the measured data, but nowevaluate the RMS-spread over the full range of rele-vant lateral distances di from the shower axis. TheRMS-spread around each linear fit is again referredto as the precision for the primary energy reconstruc-tion achievable at each distance di. The RMS-spreadobtained at these several distances, for zenith anglessmaller than 19.4◦, is shown in fig. 7, bottom panel(black triangles).

The measurements show a minimum at 80 m fromthe shower axis, with an RMS value of 19 % (andslightly higher values for the other zenith angle bins,cf. table 2). Although there is a slight deviation ofthe measured d0 value of 80 m with respect to theCoREAS-predicted value of 70 m, one may noticethat the RMS-spread for the measurements differs byonly 1 percentage point between 60 m and 100 m, asthe minimum is rather broad. The shift between thepredicted and the measured d0 is, therefore, not prac-tically relevant, and the radio amplitude measured atthe simulation-predicted distance d0 can still be usedto determine the upper limit on the primary energyreconstruction precision with LOPES radio data.

The RMS-spreads of each of the linear fits of theLOPES measurements at the simulation-predicteddistance d0 are summarized in table 2. These val-ues contain the combination of the LOPES- andKASCADE-Grande energy uncertainties. For com-parison with the simulations, an additional uncer-tainty of 20 %, representative for the KASCADE-Grande energy uncertainty [24], is added to thespread determined from the CoREAS simulations.The resulting spread is shown as well in fig. 7-bottompanel (blue stars) and in table 2. In case of thefirst zenith angle bin, the spread determined from thesimulations with an added 20 % even seems to over-estimate the spread in the LOPES data. At least inthis zenith angle range, the KASCADE energy res-olution might therefore be actually better than 20%. The upper-limit on the energy reconstructionachieved with the LOPES measurements, in all thezenith angle bins, is only slightly higher than theuncertainty of the KASCADE-Grande energy reso-lution.

The LOPES energy resolution is, therefore, verypromising, and the application of similar procedureson experiments with a better reference energy resolu-tion and radio data affected by a lower ambient noise

9

20

0m

/0

dε

]2 [g/cm

maxX

A ± 0.00160.0087 B ± 0.0022C ± 0.0550.842

ε

CoREAS p

CoREAS FeSpread ∆X

max = 54 g/cm

zenith < 19.4 deg

0.0577

500 600 700 800 900 1000

1

10

Fit parameters:

Figure 8: Correlation between the true Xmax from Monte Carlosimulations and the LDF slope of the CoREAS simulated radioamplitudes. The RMS-spread around the fit is ∼54 g/cm2.

200

m/

0dε

]2 [g/cm

maxX

ε

500 600 700 800 900 1000

1

10

0 < zenith < 19.419.4 < zenith < 26.826.8 < zenith < 32.032.0 < zenith < 36.236.2 < zenith < 40.0_

__

___

Figure 9: Fits as in fig. 8 but for each zenith angle bin. Theexpected dependence of the Xmax–radio LDF slope correlationon the zenith angle is clearly visible.

background is highly suggested in order to properlyverify the simulation-based expectations of a high in-trinsic energy resolution of radio measurements.

7. Xmax reconstruction

The feature of the radio lateral distribution func-tion which correlates best with the depth of showermaximum is the slope of the LDF fit. We first “cal-ibrate” the relation of this slope with the depth ofshower maximum on the basis of CoREAS simula-tions. Afterwards, we apply the found relation toLOPES measurements.

7.1. CoREAS simulations

We define the LDF slope ǫratio as the ratio of theradio amplitudes taken at the distance d0 and at thefixed distance of 200 m (ǫratio=ǫd0/ǫ200m). The lat-ter value is chosen for the specific investigation onthe LOPES events, since 200 m is a relatively largedistance but is still in the range of the measurements.

The Xmax values for each event are taken directlyfrom the Monte Carlo simulations, and are consid-ered as the “true” values. The ratio of the radio am-plitudes and the depth of the shower maximum areplotted against each other in fig. 8, for zenith anglesless than 19.4◦. There is a clear correlation which wefit with eq. 3.

ǫratio = B exp[

(A ·Xmax)C]

(3)

This (more precisely the inverse) three-parameterfunction was already introduced in [3] and success-fully applied to previous analyses [12, 13]. The func-tion (eq. 3) appropriately describes the correlationfor the events simulated with CoREAS, and it is ad-equate for all of the zenith bins. The A, B and Cparameters, including the statistical uncertainties ofthe fit, are extracted separately for each zenith anglebin (table 3), and the corresponding fit results areshown in fig. 9. This function has not been optimizedfor minimal correlation between fit parameters, whichcan lead to comparably good fits with more than oneset of parameters, visible as fluctuations in the de-rived parameter values in table 3. This is fine aslong as the resulting fits describe the distribution ofdata points correctly, which they do. However, in thefuture it would be worth investigating better fit func-tions with minimally correlated fit parameters, ide-ally parameters that can be associated directly withphysical parameters of the underlying air shower. Toimprove the determination of the three fitting param-eters and to better constrain the fit over a wide rangeof Xmax values, increased (doubled) statistics for theproton- and iron-simulated showers are used.

We performed the fit procedure for all zenith an-gle bins both without and with associating an uncer-tainty propagated from the LDF fits to the individualdata points, and the results vary only insignificantly.Propagating the resulting fit uncertainties into an un-certainty for the reconstructed Xmax values cannot beachieved by Gaussian error propagation due to thenon-linear nature of the problem. One option to pro-ceed would be to employ an involved procedure for

10

QGSJET II.03 & CoREAS

∆θ entries A [cm2/g] B C ∆Xmax

0.◦-19.4◦ 53 *2 0.0087 ± 0.0016 0.0577 ± 0.0022 0.841 ± 0.055 54.019.4◦-26.8◦ 48 *2 0.0060 ± 0.0034 0.0835 ± 0.0084 0.945 ± 0.016 47.926.8◦-32◦ 45 *2 0.0014 ± 0.0010 1.100 ± 0.092 2.542 ± 0.011 53.232◦-36.2◦ 36 *2 0.0053 ± 0.0018 0.225 ± 0.018 0.691 ± 0.027 65.736.2◦-40◦ 23 *2 0.0013 ± 0.0002 0.98 ± 0.52 2.8 ± 2.0 47.9

Table 3: Xmax-reconstruction parameters derived with the CoREAS simulations and their statistical uncertainties. In the rightmostcolumn, the uncertainty on the Xmax reconstruction (CoREAS) for each zenith angle bin is listed (see text). The values in thistable have been derived by using and increased (double) statistics for the simulations.

the estimation of the uncertainty of the reconstructedXmax values, e.g. by bootstrapping methods. As wewant to keep our analysis practical and are anywaylimited by the lack of an independent Xmax deter-mination to compare the reconstructed values with,however, we choose instead to use the resulting spreadin the distribution around the fit as an estimate forthe uncertainty of the reconstructed Xmax values.

In other words, we calculate the typical Xmax pre-cision of the slope method as the spread of the differ-ence between the true Monte Carlo Xmax values andthe Xmax values reconstructed using the slope methodon CoREAS simulations. Fig. 10 shows the cor-responding distributions for proton showers (RMS-spread 58 g/cm2) and iron showers (RMS-spread 51g/cm2) for the complete selection of events. In table3, the combined spread for proton and iron showersis denoted as ∆Xmax for the individual zenith anglebins. For the CoREAS simulated events with doubledstatistics, and considering the complete zenith anglerange, the RMS-spread is in the range of 45-65 g/cm2.

It is important to note that the results presentedhere apply to the specific situation of LOPES, inparticular the altitude of the LOPES site, the spe-cific observing frequency band used [3], the energyrange probed, and the dimension of the antenna ar-ray. These parameters all affect the amplitude ra-tio and consequently play a role in the determina-tion of the Xmax calibration curves. Indeed, for in-clined showers, larger amplitude ratios may be ob-tained when the far-away distance used to calculatethe ratio is taken well beyond the 200 m distance usedfor this analysis and imposed, here, by the dimensionof the LOPES array.

7.2. LOPES measurements

The three-parameters function (eq. 3) is now ap-plied to reconstruct the Xmax values from LOPES

]2[g/cmmax,CoREAS- Xmax,trueX

-200 -150 -100 -50 0 50 100 150 200

ev

en

ts

0

5

10

15

20

25

30

35proton:

9.0 ± 58.3

iron:

1.1 ± 50.9

Mean ± RMS for

Figure 10: Difference of the true Monte Carlo Xmax and thevalue derived by applying the slope method on CoREAS sim-ulations for proton showers (blue, dash-dotted line) and ironshowers (red, solid line). The RMS spread of this differenceis 58 g/cm2 for proton showers and 51 g/cm2 for iron showers.This spread constitutes an estimate for the intrinsic uncertaintyof the slope method for the reconstruction of Xmax.

measurements, using the CoREAS-derived A, B andC parameters listed in table 3 for the appropriatezenith angle bin of each individual event.

In fig. 11, we show the resulting distribu-tion of Xmax,LOPES values in the black histogram,with 633.2 ± 94.6 g/cm2 as the mean and thestandard deviation of the distribution. A com-parison with the CoREAS proton (blue dottedline) and iron (red line) prediction is shown aswell: Xmax,CoREAS,P = 678.5 ± 76.9 g/cm2 andXmax,CoREAS,Fe = 604.2 ± 56.4 g/cm2. These val-ues are derived by applying eq. 3 to the (CoREAS)simulated events, or, in other words these are the val-ues on the fit in fig. 8 of the corresponding simulatedLDF slopes.

11

]2

[g/cmmaxreconstructed X

events

LOPESMean 633.2

Std.Dev. 94.6

CoREAS pMean 678.5

Std.Dev. 76.9

CoREAS FeMean 604.2Std.Dev. 56.4

uncertainty

on Xmax

proton

iron

40

35

30

25

20

15

10

5

0300 400 500 600 700 800 900 10000

Figure 11: Xmax distribution reconstructed with the slopemethod for the LOPES measurements (black) and for theCoREAS proton (dashed-blue) and iron (light-red) simulationsof the same events, for the complete zenith angle range of 0-40◦.The indicated uncertainty on Xmax is derived from fig. 10.

KASCADE-Grande log (E / GeV)

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9

2 [

g/c

m ]

ma

xre

co

nstr

ucte

d X

450

500

550

600

650

700

750

800

10

Slope Method (CoREAS p ) True simulated p

Slope Method (CoREAS Fe) True simulated Fe

Slope Method (LOPES)

Figure 12: The energy dependence of the reconstructed LOPESXmax distribution (black), compared with the CoREAS valuesfor proton (blue-dashed line) and iron (red-dotted line) pri-maries, and with the Monte Carlo values for proton (light-bluemarker) and iron (light-red marker) primaries.

The standard deviation value for the LOPES dis-tribution of ∼95 g/cm2 can be considered itself asan upper-limit for the LOPES precision on Xmax, al-though it does not explicitly account for systematicuncertainties, e.g., scale uncertainties due to mod-elling the atmosphere or due to hadronic interaction

models used for the shower simulations.

The reconstructed values for Xmax,LOPES are com-patible with the expectations. Compared to previousanalyses [12, 13], in which the simulations used didnot include the refractive index of the atmosphere,a much improved result is now obtained. We candefinitively affirm that the systematic shift observedpreviously in [12] was caused by the discrepancy be-tween the LDF slopes predicted by older simulationcodes and the measured LDF slopes.

In addition, we examine the energy dependence ofthe Xmax distribution (fig. 12), integrated over thewhole zenith angle range: Each cross represents themean value and the standard deviation of the Xmax

distributions, for the slope method applied to LOPESmeasurements and to CoREAS simulations. Additio-nally, the mean and the error of the mean (standarddeviation/

√events) for the true Monte Carlo Xmax

are shown in each energy bin. Good consistency withthe expectations is achieved in all energy bins.

Next, we directly compare the mean LOPES Xmax

values reconstructed for the four energy bins withthe results of other experiments (fig. 13). In thefigure, the solid lines represent the Monte Carlo ex-pectations for QGSJetII-03 for pure iron-simulated(red) and pure proton-simulated showers. The uncer-tainty indicated in this figure on the LOPES averageXmax values is the statistical uncertainty, i.e. stan-dard deviation/

√events. The intrinsic uncertainty of

the slope method, as determined by the spread be-tween the reconstructed and Monte Carlo true values,is indicated as a shaded band. No additional analysison the systematics is carried out.

In summary, the results of the LOPES Xmax de-termination are self-consistent and plausible. Never-theless, we point out that the LOPES uncertaintiesare too large for any conclusion concerning the masscomposition.

8. Discussion

In the following discussion, willful and imposedlimitations which had influence on the slope methodanalysis of the LOPES simulated and measuredevents are pointed out.

The upper limit on the observed energy preci-sion (∼ 20-25 %) includes the convolution of boththe LOPES and KASCADE-Grande (particle de-tector) uncertainties. The good compatibility with

12

(E/TeV)10

log3.5 4 4.5 5 5.5 6 6.5

]2

[g/cm

max

X

450

500

550

600

650

700

750

800LOPES (SlopeMethod)

Auger (ICRC 2011)

Hires-MIA (Phys.Rev.Lett.2000)

Tunka (NIMA56079 2013)

Yakutsk (small array)

Yakutsk (big array)

QGSJ etII.03 - Fe

QGSJ etII.03 - P

SlopeMethod uncertainty

10066

2811

Figure 13: Mean Xmax as a function of the primary cosmic ray energy. The blue points correspond to the LOPES-reconstructedvalues, using the slope method. The errors are calculated as (standard deviation/

√events), and the numbers indicate our statistics

in each energy bin. The lines refer to the Monte Carlo expectations for pure iron (red) and pure proton (blue) composition, usingthe QGSJetII-03 hadronic interaction model. Results from other experiments are indicated by markers (please see legend).

the KASCADE-Grande statistical uncertainty (ap-prox. 20 %) indicates that the combined uncertaintyof the energy is strongly influenced by KASCADE-Grande and that the energy resolution of LOPES islikely significantly better than 20-25 %. The totalenergy precision of the slope method suggested bythe pure CoREAS simulations is approximately 13 %and could potentially be achieved by applying themethod on low-noise radio data.

It is worth stressing once again that radio emissionis purely sensitive to the electromagnetic componentof air showers. If combined with other detection tech-niques which are able to provide information on othercomponents, in particular the muonic one, radio de-tection can help to disentangle the individual com-ponents, which in itself opens access to composition-sensitive information. Moreover, the radio signals arenot critically affected by uncertainties in the hadronicinteraction models used to understand the data.

One restriction of the analysis presented hereconcerns the deliberate exclusion of the detector-simulation in the investigation. A more realistic com-parison with the measurements may include bothsimulations of detector properties as well as simulatedcontinuous and transient noise typical of the specificexperimental site.

Also, we did not try to apply a more sophisti-cated normalization of the radio field strengths in-cluding a charge-excess component. Since we fit aone-dimensional LDF with observers at various az-

imuth angles, the charge-excess asymmetry mostlyaverages out in the LDF fit. We confirmed with a ded-icated simulation study that the additional spread in-troduced by this simplified treatment is negligible andthat there is a bias in the reconstructed k-parametersof less than 5%, which is much smaller than the over-all calibration scale uncertainty of 35% and for whichwe account with an additional systematic uncertaintyof 5%. For experiments with better data quality thanLOPES, however, methods should be investigated totry and correct individual measured amplitudes bythe presumed charge-excess contribution or fit two-dimenaional LDF functions to improve the fit andthus possibly the energy determination. This aspectwas not investigated since the precision we achieve inthe combined analysis anyway seems to be limited bythe reference energy provided by KASCADE-Grandeand not by our method.

The QGSJet II.03 interaction model was employedto generate the CoREAS simulations of the LOPESevents. Other interaction models are available andthey differ both in their implementation and theirpredictions. We used the slope method to show howair shower parameters such as energy and Xmax canbe directly derived from the features of the radioLDF. Regarding the energy reconstruction, differentinteraction models predict different fractions of theenergy going into the electromagnetic component andother components, respectively. This might lead tomore significant shifts between the radio amplitudes

13

observed for different primaries of the same energythan illustrated in this analysis. Such shifts, how-ever, are not a disadvantage of radio detection, sincein fact radio emission with its sensitivity to the pureelectromagnetic component of the air shower couldbe used to study such effects and test hadronic in-teraction models. Regarding the shower evolution,different interaction models will make different pre-dictions for the Xmax values of air showers initiated byspecific primaries with given energies. However, theobtained “calibration” relating a given value of Xmax

to a corresponding radio LDF slope should not bestrongly affected by the choice of interaction model.This is because the slope method exploits an intrinsicproperty of the radio emission: the relation betweenthe geometrical distance of the radio source and theslope of the radio lateral distribution arising from theforward-beaming of the emission. A slight influenceon the calibration curve can be expected by varia-tions of the atmosphere, which relate the geometricaldistance to the depth of shower maximum in g/cm2.Within the precision achievable by LOPES, however,these effects can be neglected.

The Xmax resolution obtained in this analy-sis (∼95 g/cm2) is an upper-limit attained with(LOPES) radio data affected by strong environmentalnoise. Much better results may be achieved by usinghigher quality radio measurements and a denser an-tenna array. Such is the case for LOFAR [25], whichhas recently applied a method based on global fitsof the two-dimensional radio LDF and obtained asensitivity in Xmax comparable with the Fluorescencetechnique (∼20 g/cm2). The underlying principle isthe same, namely the exploitation of the geometricalconnection between the source distance and the radioLDF described above.

In the analysis presented here, we strove to exploitthe potential of the LOPES data while at the sametime keeping the analysis practical. Existing and up-coming more modern experiments such as AERA [26],Tunka-Rex [27] and LOFAR [25] will have higher-quality data with lower background noise, a largernumber of antennas per event and a larger range oflateral distances probed with each event. The meth-ods presented here will have to be refined to achievethe maximum precision these instruments will be ableto deliver in energy and Xmax resolution. In particu-lar, a lateral distribution function accounting for theasymmetries induced by the charge-excess should beused. The function used to fit the Xmax values as a

function of amplitude ratios should be re-investigatedand correlations between fit parameters should beminimized. Finally, a more sophisticated determina-tion of the reconstruction uncertainties, taking intoaccount the non-Gaussian error propagation shouldbe applied. In spite of these limitations, we havedemonstrated that a determination of the energy andXmax from the radio lateral distribution is feasiblewith promising upper limits on the achievable preci-sion.

9. Conclusion

The purpose of this analysis was the reconstruc-tion of the energy and depth of shower maximum ofindividual cosmic rays from radio measurements per-formed with the LOPES experiment. We achievedthis goal by exploiting characteristic features of thelateral distribution function of the radio emission, inparticular its slope.

We have developed and applied a CoREAS-calibrated method to the measurements of theLOPES experiment. Significant results emerged:

• A characteristic distance from the shower axis,i.e. d0 (70–100 m), is predicted by CoREAS sim-ulations of the LOPES events to be the appro-priate place for the primary energy investigationwith radio data. Its existence is also indicated inLOPES measurements.

• A simple linear correlation between the radioamplitude measured at the distance d0 and theKASCADE-Grande reconstructed energy wasfound. A zenith-angle dependent deviationin the correlation parameters remains betweenCoREAS simulations and LOPES data; this dis-crepancy needs to be investigated in a dedicatedanalysis.

• An upper-limit for the LOPES precision onthe total energy reconstruction was determinedfrom the combined LOPES–KASCADE-Grandeenergy uncertainty (∼20-25% depending onzenith angle). The intrinsic energy resolution ofthe radio measurements suggested by the simula-tions is approximately 13 %, and might even beimproved by taking the Askaryan effect explic-itly into accout in the LDF fits. However, anyfurther improvment is beyond the testing powerof LOPES.

14

• The slope of the LOPES lateral distributionfunction was exploited to reconstruct Xmax fromLOPES data. The resulting distributions areplausible and comparable with expectations fromcosmic ray nuclei.

• By itself, and for the specific situation of LOPES,the slope method predicted a precision of Xmax

around 50 g/cm2, depending on the zenith angleof the event. From the analysis of the LOPESmeasurements, an upper limit of ∼95 g/cm2 onthe precision was found.

Xmax is the principle indicator for cosmic ray com-position, and we have demonstrated the possibilityto reconstruct it with the slope method applied toLOPES radio measurements. The importance of thisresult is underlined by the fact that radio detectiondoes not suffer from low duty cycles as is the casefor optical detection techniques. Comparisons of thereconstructed Xmax values between, at least, two dif-ferent detection methods are essential but not possi-ble in the framework of the LOPES and KASCADE-Grande experiments. This opportunity is offered bythe new generation of antenna arrays, such as AERA[26] (at the Pierre Auger Observatory) and Tunka-Rex [27], where the radio reconstruction can be crosschecked with the fluorescence and Cherenkov experi-mental results, respectively.

Acknowledgments

LOPES and KASCADE-Grande have been sup-ported by the German Federal Ministry of Educa-tion and Research. KASCADE-Grande is partly sup-ported by the MIUR and INAF of Italy, the Pol-ish Ministry of Science and Higher Education andby the Romanian Authority for Scientific ResearchUEFISCDI, IDEI, grant 271/2011. This research hasbeen supported by grant number VH-NG-413 of theHelmholtz Association.

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