Energy and Flux Measurements of Ultra-High Energy Cosmic Rays Observed During the First ANITA Flight
H. Schoorlemmer, K. Belov, A. Romero-Wolf, D. García-Fernández, V. Bugaev, S. A. Wissel, P. Allison, J. Alvarez-Muñiz, S. W. Barwick, J. J. Beatty, D. Z. Besson, W. R. Binns, W. R. Carvalho Jr., C. Chen, P. Chen, J. M. Clem, A. Connolly, P. F. Dowkontt, M. A. DuVernois, R. C. Field, D. Goldstein, P. W. Gorham, C. Hast, C. L. Heber, T. Huege, S. Hoover, M. H. Israel, A. Javaid, J. Kowalski, J. Lam, J. G. Learned, K. M. Liewer, J. T. Link, E. Lusczek, S. Matsuno, B. C. Mercurio, C. Miki, P. Miočinović, K. Mulrey, J. Nam, C. J. Naudet, J. Ng, R. J. Nichol, K. Palladino, B. F. Rauch, J. Roberts, K. Reil, B. Rotter, M. Rosen, L. Ruckman, D. Saltzberg, D. Seckel, D. Urdaneta, G. S. Varner, A. G. Vieregg, D. Walz, F. Wu, E. Zas
EEnergy and Flux Measurements of Ultra-High Energy Cosmic RaysObserved During the First ANITA Flight
H. Schoorlemmer a,b, ∗ , K. Belov c,d , A. Romero-Wolf d , D. Garc´ıa-Fern´andez e , V. Bugaev f , S. A. Wissel c,g ,P. Allison h , J. Alvarez-Mu˜niz e , S. W. Barwick i , J. J. Beatty h , D. Z. Besson j,k , W. R. Binns f ,W. R. Carvalho Jr. e , C. Chen l , P. Chen l,p , J. M. Clem m , A. Connolly h , P. F. Dowkontt f ,M. A. DuVernois a , R. C. Field p , D. Goldstein i , P. W. Gorham a , C. Hast p , T. Huege n , C. L. Heber a ,S. Hoover c , M. H. Israel f , A. Javaid m , J. Kowalski a , J. Lam c , J. G. Learned a , J. T. Link f , E. Lusczek a ,S. Matsuno a , B. C. Mercurio h , C. Miki a , P. Mioˇcinovi´c a , K. Mulrey m , J. Nam a , C. J. Naudet d , J. Ng p ,R. J. Nichol o , K. Palladino h , B. F. Rauch f , J. Roberts a , K. Reil p , B. Rotter a , M. Rosen a , L. Ruckman a ,D. Saltzberg c , D. Seckel m , D. Urdaneta c , G. S. Varner a , A. G. Vieregg r , D. Walz p , F. Wu i , E. Zas e a University of Hawaii at Manoa, Department of Physics and Astronomy, Honolulu, Hawaii 96822, USA. b Max-Planck-Institut f¨ur Kernphysik, 69117, Heidelberg, Germany c Dept. of Physics and Astronomy, Univ. of California, Los Angeles, Los Angeles CA 90095, USA d Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109. USA e Departamento de F´ısica de Part´ıculas & Instituto Galego de F´ısica de Altas Enerx´ıas, Universidade de Santiago deCompostela,15782 Santiago de Compostela, Spain. f Dept. of Physics and McDonnell Center for the Space Sciences, Washington Univ. in St. Louis, MO 63130,USA g Dept. of Physics, California Polytechnic State Univ., San Luis Obispo CA 93407, USA h Dept. of Physics, Ohio State Univ., Columbus, OH 43210, USA i Dept. of Physics, Univ. of California, Irvine, CA 92697, USA j Dept. of Physics and Astronomy, Univ. of Kansas, Lawrence, KS 66045, USA k National Research Nuclear University (Moscow Engineering Physics Institute), Moscow Russia, 115409 l Dept. of Physics, Grad. Inst. of Astrophys.,& Leung Center for Cosmology and Particle Astrophysics, National TaiwanUniversity, Taipei, Taiwan. m Dept. of Physics, Univ. of Delaware, Newark, DE 19716, USA n Institut f¨ur Kernphysik, Karlsruhe Institute of Technology (KIT), Germany o Dept. of Physics and Astronomy, University College London, London, United Kingdom. p SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA q School of Physics and Astronomy, Univ. of Minnesota, Minneapolis, MN 55455, USA. r Dept. of Physics, Enrico Fermi Institute, Kavli Institute for Cosmological Physics, Univ. of Chicago , Chicago IL 60637,USA
Abstract
The first flight of the Antarctic Impulsive Transient Antenna (ANITA) experiment recorded 16 radio signalsthat were emitted by cosmic-ray induced air showers. The dominant contribution to the radiation comesfrom the deflection of positrons and electrons in the geomagnetic field, which is beamed in the direction ofmotion of the air shower. For 14 of these events, this radiation is reflected from the ice and subsequentlydetected by the ANITA experiment at a flight altitude of ∼
36 km. In this paper, we estimate the energy ofthe 14 individual events and find that the mean energy of the cosmic-ray sample is 2 . × eV, which issignificantly lower than the previous estimate. By simulating the ANITA flight, we calculate its exposurefor ultra-high energy cosmic rays. We estimate for the first time the cosmic-ray flux derived only from radioobservations and find agreement with measurements performed at other observatories. In addition, we findthat the ANITA data set is consistent with Monte-Carlo simulations for the total number of observed eventsand with the properties of those events. Keywords:
Cosmic Rays; Air Shower; Radio Detection;
Preprint submitted to Elsevier February 9, 2016 a r X i v : . [ a s t r o - ph . H E ] F e b . Introduction Ultra-high energy cosmic rays (UHECRs) have been observed for over half a century; however, theirnature and origin remain uncertain. Modern observatories, like the Pierre Auger Observatory in Argentina[1, 2] and the Telescope Array in Utah [3, 4], try to unravel the mysteries of UHECRs not only by increasingthe number of recorded UHECRs, but also by increasing the measurement accuracy of their properties.Simultaneous observations of the fluorescence light emitted by the air shower while it passes through theatmosphere and the particle footprint on the ground provide cross calibration of the primary particle’s prop-erties including its energy. This method significantly reduces the systematic uncertainties in the observationof UHECRs. However, there still remains a discrepancy between the observed cosmic-ray energy spectrumat the Pierre Auger Observatory and the Telescope Array (see for example [5]). In this paper, we developan independent method to estimate the energy of UHECRs and apply this to cosmic rays observed duringthe first Antarctic Impulsive Transient Antenna (ANITA) flight, ANITA-I.When the cosmic-ray sample of the first ANITA flight was reported [6] none of the models for calculatingradio emission from air showers was able to predict significant radiation in the frequency band of ANITA-I. A preliminary energy estimate of this sample was performed using a Bayesian approach with a simplemodel as the prior for the emission pattern. In this paper the cosmic-ray data set is re-analyzed usinga realistic model for the radio emission of air showers in order to estimate the cosmic-ray energy. Theanalysis presented here shows consistency with cosmic-ray flux observations made with other experimentsand results in consistency when comparing distributions of parameters obtained from a full flight simulationto the observed distributions. The resulting energy scale of the cosmic-ray sample is about a factor of fourlower than the previous estimate.We first introduce the ANITA-I experiment and briefly discuss its cosmic-ray data set. In the followingsection, we introduce our energy estimation method and apply it to the cosmic-ray observations. Thereafter,we perform a full Monte Carlo simulation of the ANITA-I flight to calculate the exposure and cosmic-ray fluxand test for consistency between predictions from simulations and cosmic-ray observations. We concludewith a discussion of the obtained results and a brief discussion as to how these results could impact futureexperiments.
2. The cosmic-ray data set obtained with ANITA-I
The objective of the ANITA-I experiment was to observe impulsive Askaryan radiation resulting fromultra-high-energy neutrino interactions in the Antarctic ice. However, during its first flight the ANITAexperiment collected a data set of 16 radio signals that were emitted by cosmic-ray air showers [6]. Theseevents remained after careful rejection of man-made pulsed signals, continuous waves, and thermal noiseevents. They all exhibited the polarization signature typical of the geomagnetic radiation from cosmic-rayair showers [7, 8].The arrival directions of two events were from above the horizon. This indicated that these signals camedirectly from air showers that skimmed through the atmosphere without reaching the Earth’s surface. Thearrival directions of the radiation of the other 14 events pointed back toward locations on the Antarctic icesheet. The polarity of the signal was inverted with respect to the two direct events. These signals came fromdown-going air showers where the radio emission is reflected off the Antarctic ice sheet toward the payload.Understanding the particle shower and the resulting radiation from the two direct events is very interesting.These direct events correspond to particle showers which develop in significantly lower atmospheric densityand never reach the ground. Therefore, the shower development is expected to differ significantly fromstandard down-going particle showers. To understand these particle showers, and the resulting radiationfrom them, dedicated shower simulations are required, which means modification of existing simulationcodes. This is out of the scope of the analysis presented in this paper. ∗ Corresponding author
Email address: [email protected] (H. Schoorlemmer) . Description of the ANITA instrument and its first flight The ANITA-I experiment was suspended from a high altitude balloon that flew in the 2006-2007 Australsummer over Antarctica. The total flight took 35 days in which the payload made almost four revolutionsaround the south pole. However, due to stability issues with the flight computer and periods in highbackground environments (near the base after take off) the effective uptime of the experiment was 17.25days. The payload had a float altitude of ∼
36 km and an antenna array provided a panoramic view on theice sheet below it. The antenna array consisted of 32 dual-polarized quad-ridged horn antennas optimized toobserve radiation in the 200-1200 MHz band. Each antenna had a vertical and horizontal polarization feed,and the full-width-half-maximum beamwidth was about 45 ◦ . The antennas were arranged in an azimuthally-symmetric array with one ring of 16 antennas at lower part of the payload and two rings of antennas (8+8)at the top of the payload. Each antenna is tilted 10 ◦ downward.Signals from each antenna feed were amplified and bandpass-filtered into 64 channels. These channelswere then split into a trigger channel and a channel for digitization. The signal going into the trigger channelwas decomposed into left- and a right-handed circular polarization. The two circularly-polarized signals weresplit into four frequency bands that were connected to tunnel diodes that functioned as power detectors. Atrigger signal was issued when a threshold crossing was observed in several frequency bands (three or moreout of eight). When a combination of neighboring antennas (minimal 4) in the array issued a trigger in timecoincidence, signals from all of the antennas in the array were digitized and written to disk for a durationof 100 ns and with a sampling frequency of 2.6 GSa s − .Orientation, timing, and location information was provided by an array of GPS units. In addition, redun-dant systems using sun sensors and magnetometers were used to provide additional orientation information.Trigger efficiency and direction reconstruction were validated inflight by ground pulser stations. For a moredetailed description of the instrument and its performance see [9].
4. Energy estimation of individual events
A plane wave arriving at the antenna array will induce a pattern of arrival times corresponding to theincoming direction. We use these relative arrival times to calculate the coherent power corresponding to acertain direction. By calculating the coherent power from multiple directions a two dimensional interfer-ometric map is generated. The location of the maximum coherent power within this map is used as thereconstructed incoming direction of the signal. This technique is explained in more detail in [10].After direction reconstruction, the electric field at the antenna feed is estimated by deconvolving theresponse function of both the signal chain and the effective height of the antennas from the measured voltage.Note that the effective height of each antenna depends on both frequency and the incoming direction of theradiation. As an example of the reconstructed time-dependent electric field at the payload we show thecalibrated measurement of one cosmic-ray signal in the left panel of Figure 1. As indicated, the pulsedsignal is fully contained within a window of about 10 ns. The samples in this window are used to obtain theamplitude spectrum as shown in the right panel of Figure 1.The amplitude spectrum below 300 MHz suffers from loss of power due to filtering of emission fromsatellites between 220 and 290 MHz. In addition, the antenna model is less accurate at the lower frequencies.The antenna model is obtained from preflight measurements, and the lab setup provides less accurate resultsfor the low frequency part of the antenna sensitivity due to near field effects. Therefore, frequencies below300 are not considered in the spectral fitting. Around 1000 MHz, the sensitivity of the digitizer drops rapidly,which makes signal recovery above this frequency unreliable. Therefore, we choose to evaluate the amplitudespectrum between 300 and 1000 MHz.The observed amplitude at a given frequency is the sum of the signal and a thermal noise background witha random phase. The probability distribution of the resulting amplitude is described by a Rician distribution.We use this probability distribution function to estimate a central 68 % confidence interval in which the signalis contained given the measured amplitude and the background. The background is estimated from several(4 or 5) 10 ns time windows in the same waveform as the signal while excluding the signal region. We average3 ime [ns]20 40 60 80 A m p lit ud e [ m V / m ] - - measurementsimulation Frequency [Hz]200 400 600 800 1000 · ] - . M H z - m . [ p W f A c – A = 0.94 -9 · – = (-2.0 g meas.sim.fitBkg. Figure 1: Left: The time dependent electric field induced by an air shower at the location of the ANITA payload. The impulsivesignal is contained within a window of about 10 ns as indicated in red. Right: The background corrected amplitude spectrumof the signal window. The black markers represent the estimated signal and their 68 % confidence interval (see text for details).A simple exponential fit (see Eq. (1)), in the range 300-1000 MHz, to the signal spectrum is also shown. In addition we showan example of simulated radio pulse obtained with the ZHAireS Monte Carlo (with an arbitrary time-offset). the amplitude spectrum from these sideband samples to get a frequency-dependent amplitude spectrum ofthe background. We observed that within statistical fluctuation this background amplitude spectrum is flatover the band (as shown in Figure 1 (right)). We then averaged the background over the full amplitudespectrum to get a single estimation of the background level for the complete frequency band. The range ofobserved background levels are between 0.07 and 0.2 pW . m − MHz − . .From the signal spectrum, A f , as a function of frequency, f , we obtain two parameters by fitting a simpleexponential function A f ( f ) = Ae γ ( f −
300 MHz) . (1)Parameter A gives the amplitude of the electric field at 300 MHz (at the location of ANITA) and parameter γ describes the frequency dependence of the amplitude spectrum. An exponential function is motivated bythe shape of the amplitude spectrum obtained from ZHAireS simulations [11]. The fit parameters for the14 cosmic-ray observations are listed in Table 1.In addition to the measured pulse, we show in Figure 1 also an example of a ZHAireS simulation withan energy and off-axis angle close to reconstructed values (see section 4). A simple rectangular bandpassfilter between 300 and 1000 MHz is applied to the time domain of the simulations . The amplitude, polarityand duration of the simulated pulse are comparable to the measurement. However, small deviations of thepulse shape in the time domain are expected due to the simple bandpass filter applied to the simulation. During the ANITA-I flight in 2006-2007 none of the available models were able to predict observablesignals in the 300-1000 MHz frequency band. However, the recent inclusion of the effect of the atmosphericrefractive index increased the frequency at which the simulations are coherent up to several gigahertz [11, 12,13, 14]. This makes it now possible to compare the observation directly to the models and this is the mainmotivation to reanalyze the ANITA-I data set. The effect of the atmospheric refractive index on propagationof the radiation from the air shower results in a cone-like beam around the shower axis. The variation inarrival time of the radiation originating from different regions of the air shower shrinks to a minimum forobservers located at the Cherenkov angle corresponding to the location of the region of maximum emission:Therefore, radiation with short wavelengths adds more coherently in the direction of the Cherenkov angle ψ c , resulting in the flattest frequency spectrum at the Cherenkov cone. As an observer moves away fromthe Cherenkov angle, coherence is lost at the higher frequencies causing the frequency spectra to steepen.This frequency dependence as a function of distance to the Cherenkov angle plays a fundamental role inthe method that we developed to estimate the energy of cosmic-ray particles. In the frequency band of4vent θ A γ numbers [ ◦ ] [pW . m − MHz − . ] [Hz − ]1 84.6 0.25 ± ± . × − ± ± . × − ± ± . × − ± ± . × − ± ± . × − ± ± . × − ± ± . × − ± ± . × − ± ± . × −
10 78.8 0.61 ± ± . × −
11 70.5 0.94 ± ± . × −
12 79.1 0.39 ± ± . × −
13 81.9 0.72 ± ± . × −
14 78.6 0.57 ± ± . × − Table 1: Incident angle of the radiation on the ice, θ , and the fitted amplitudes at 300 MHz ( A ) and spectral slopes ( γ ) (seeEq. (1)) for the 14 cosmic-ray events. the ANITA experiment the coherence due the Cherenkov effect, and therefore the atmospheric refractiveindex, is crucial to obtain measurable radiation. The dependence of the refractive index with altitude has animpact on the observed radiation. The refractive index falls off as a function of altitude. As a consequence,more inclined showers will emit their radiation within a smaller off-axis angle since they develop at higheraltitude. At lower frequencies the conditions for coherent radiation are easier satisfied. Therefore, thetime compression due to the Cherenkov-effect does not play a dominant role anymore. As a result, thecharacteristic ring pattern is less prominent at lower frequencies. For examples of the shape of the radiationpattern over a large frequency range see [15]. The geometry for reflected radiation from a cosmic-ray-inducedair shower is illustrated in Figure 2.All of the standard simulation packages of radio emission from air showers have been developed tocalculate the emission for antenna arrays on the ground. However, recently the ZHAireS code [16] wasupgraded [15] to incorporate a reflection off a dielectric medium, such as the Antarctic ice. The code reflectsthe radiation contribution from each particle track individually on the ice and propagates it to the payloadlocation where they are summed to calculate the total electric field. Therefore, we can compare the simulatedelectric field properties directly to the observed ones at the location of the payload. At the reflection pointthe Fresnel coefficients are applied.This method avoids the problem of simulating the electric field in the near field on the ground, whilethe payload observes it in the far field. When both are in the far field one can simply extrapolate groundcalculations to payload locations but this method does not apply in the range of incident angles as observedby ANITA (see [15]). This is why we had to use the upgraded ZHAireS package to properly calculate electricfield strength at the payload. However, to study the dependency on the choice of simulation package wecompared the ZHAireS simulation to another simulation package in Appendix B for antennas located on theground. In this section, we will explain the basic concept for estimating the cosmic-ray energy and then introducethe refinements that go into the final energy estimation. The method presented here builds upon conceptsthat were presented in [17, 18].For a given simulated air shower we calculate the electric field at different off-axis angles, ψ (see Figure2). The falloff of the amplitude spectrum depends strongly on the off-axis angle and is reasonably describedby a simple exponential in the frequency range of 300-1000 MHz [11]. As with the measurements, we fit the5 NITA X max ψ c ψθ s h o w e r a x i s Figure 2: Illustration of the footprint of the Cherenkov-ring on the ground and the geometry system used in the analysis. Fromthe observation of radiation at the location of ANITA we can calculate the incident angle of the radiation on the ice θ . Aftersimulating an air shower, we can define a line from the location of X max that after reflection with angle θ would end up at thelocation of the payload. The angle this line makes with respect to the shower axis is defined as the off-axis angle ψ . A rangeof off-axis angles is probed for a single air shower simulation by varying the location of ANITA in simulation while keepingthe payload at a fixed altitude above sea-level. The off-axis angle of the Cherenkov ring is indicated by ψ c . The color map isobtained by calculating the radio signal at 300 MHz for an incoming cosmic ray with a zenith angle of 70 ◦ using the ZHAireScode and the interpolation technique as discussed in [19]. Note, due to the shape of the radiation pattern there will aways bean offset between the direction of the observed radiation and the direction of the shower-axis. This will be discussed in moredetail in Section 4.4. (cid:176) [ y ]) - . M H z - m - . ( A / [ p W l og -1-0.8-0.6-0.4-0.20 ] (cid:176) [ y ] - [ H z g -20-15-10-50 -9 · Figure 3: Left: The amplitude at 300 MHz, A , as a function of the off-axis angle ψ of the location of the payload. Right: Thespectral slope γ as a function of off-axis angle ψ of the location of the payload. These results are obtained from a simulatedcosmic ray with a 70.5 ◦ zenith angle and an energy of 10 . eV. Note, the error bars are smaller than the marker size. TheCherenkov-angle ψ c is at the off-axis angle where γ and A reach their maximum value. function in Equation (1) to estimate the amplitude A at 300 MHz and the spectral slope γ at each off-axisangle. In Figure 3, we show an example of the amplitude distribution and spectral slope as a function ofoff-axis angle for payload locations along a line perpendicular to the shower axis. Both distributions peakat the location of the Cherenkov angle ψ c . As we move away from the Cherenkov cone, we loose power atthe higher frequencies and the spectra become steeper. At some point we hit the noisy, mostly incoherentpart of the spectrum, which jeopardizes the fitting of meaningful spectrum slopes.In the left panel of Figure 4, we show the correlation between γ and A from Figure 3 in the regionnear the Cherenkov cone. The color code shows the result of repeating the air shower simulations withdifferent cosmic-ray energy. At different energies the markers can be on slightly different locations becauseof differences in shower maximum (see Section 4.4 for more details) We observe that the relation betweenspectral slope and its amplitude is remarkably simple and can be described by a linear relation:log ( A ) = log ( A c ) + b ( γ − γ c ) , (2)in which A c gives the amplitude at 300 MHz on the Cherenkov cone, γ c gives the spectral slope on theCherenkov cone, and parameter b describes the dependence of A on γ . This means that if we determine b and γ c from the simulations, we can derive for each measured spectral slope, γ , and amplitude, A , thecorresponding amplitude on the Cherenkov cone, A c . The value of γ c is determined as the flattest frequencyspectrum from the set of simulations at different off-axis angles. There is no significant dependence of γ c on energy, therefore we average it over the different energies. Parameter b is used as a free fit parameter inEquation (2). It is clear from Figure 4 that it is quite independent of energy. We thus average its value overthe different energies considered.We use the values of b and γ c to calculate for each simulation point in the left panel of Figure 4 the valueof A c . The right panel of Figure 4 shows the dependence of the obtained mean value of A c as a function ofcosmic-ray energy (markers). The (tiny) error bars represent the root mean square of the A c distribution.These values are fitted with a linear relationshiplog (cid:18) A c [pW . m − MHz − . ] (cid:19) = p + p × log (cid:18) En eV (cid:19) (3)7 -1 [Hz g -5 -4 -3 -2 -1 -9 · ]) - . M H z - m . ( A / [ p W l og -101 (En/eV): log17.718.018.318.618.919.219.519.8 log (En/ eV)17.5 18 18.5 19 19.5 20 ]) - . M H z - m . / [ p W c ( A l og -1-0.500.511.5 /ndf = 4.6/6 c – p0 = -18.3 0.007 – p1 = 0.991 Figure 4: Left: The amplitude A as a function of the slope γ of the amplitude spectra obtained from electric field calculationsnear the Cherenkov cone for a cosmic ray with zenith angle of 70.5 ◦ . The Cherenkov cone is reached where γ and A are at amaximum. Each marker corresponds to a different off-axis angle. We show this dependency for air showers induced by cosmicrays of different energies. Right: The relation between the amplitude on the Cherenkov cone and the energy of the primarycosmic-ray particle are shown (see the text for detailed explanation). with p = − . ± . p = 0 . ± .
007 (statistical uncertainties from the fit). This linear relationshipis used as the cosmic-ray-energy calibration curve for this particular event. The fit results show that theobtained values of A c are directly proportional to the energy of the cosmic-ray particle.For each cosmic-ray event, we use the measured values of γ and A to calculate the value of A c using thecalibration constants b and γ c obtained from simulations. Using the value of A c we estimate the cosmic-rayenergy from the calibration curve generated specifically for the event.It must be noted that the simple relation between γ and A in Equation (2) starts to break down as onemoves further away from the Cherenkov cone. However, the steepest spectral slope obtained from the cosmic-ray observations is -4.0 × − Hz − , which is well in the range where Equation (2) is a good approximation.As an example, there is still a good agreement of the markers with the linear fit at γ ∼ × − as shownin Figure 4 (left) . Although the ZHAireS simulations are tailored to calculate reflected radio signals for a high altitudeballoon, there are still two effects that will alter the electric field observed after the reflection and these aretaken into account in this analysis:
Defocusing due to curvature of the Earth:
Since the Earth’s surface is curved, parallel incident rayswill diverge after reflection. This means that the amount of power-per-area will decrease after thereflection. This decrease depends only on the incident angle of the radiation. It becomes moresignificant near the horizon where the incident angles of the radiation are large with respect to thesurface. To calculate the loss we used the analytical model from [20] and validated its accuracy withthe ray tracing algorithm explained in [15].
Loss of coherence due to surface roughness:
If the roughness of a reflecting surface is of a similar orlarger size than the wavelength of the radiation, a reflection will disturb the wavefront and there will bea loss of coherence. We estimated the loss of coherence by performing a physical optics calculation overa surface using the method described in [21]. The model of surface roughness used in the calculationis based on a parameterization [22] of surface roughness measurements performed over length scalesbetween 0 and 120 m at the Antarctic Taylor Dome station. The parameterization is rescaled to reflectthe roughness at other locations using the digital elevation model from [23] which provides elevationinformation on a 200 m spaced grid. From evaluating our physical optics calculation it turns out that8 (cid:176) [ q
50 60 70 80 90 r e f l ec ti on c o e ff i c i e n t CurvatureRoughnessFresnelCombined
Figure 5: The various contributions to the reflection coefficient as a function of incident angle. The green dashed line showsthe scaling factor for the combination of all the contributions. For illustration purposes, the surface roughness factor is takenat 650 MHz. In the full calculations the frequency dependency of this factor is taken into account as described in Appendix A. we are mostly interested in surface roughness on length scales between 10 and 30 m. Therefore, weassign a significant uncertainty to the roughness correction.A rougher surface with respect to the currently used values induces a smaller amplitude and a steeperspectrum. The smaller amplitude implies that the energy of the event would be underestimated, buton the other hand the steeper spectrum implies that the off-axis angle would be reconstructed closerto the Cherenkov angle resulting in an overestimate of the energy. These two effects almost canceleach other out and the resulting uncertainty on the energy due to the uncertainty in roughness is therange 1%-6%.However, surface roughness becomes more important when simulating the ANITA-I flight as both thesteepening of the amplitude spectrum and the reduced amplitude will reduce the likelihood of a triggerat the ANITA-I instrument.These two effects, in combination with the Fresnel coefficients, give the total scaling factor for simulatedelectric fields. The Fresnel coefficients are already included in the simulation package. In Figure 5 we showhow the scaling factors depend on the incident angle of the radiation.There are several parameters that influence the cosmic-ray energy estimation that cannot be constrainedfrom the observations made by ANITA. As a result, these parameters contribute to the uncertainty in thecosmic-ray energy estimate. Below we describe each of these contributions and how we treated them toassess their contribution to the uncertainty.
Ambiguity in the direction of the air shower:
The radio signal is emitted in a cone-like beam aroundthe shower axis. Therefore, there is an offset between the direction of the radiation and the directionof the shower axis. There are also two effects that cause asymmetries in the radio beam pattern thatdepend on which side of the shower axis the detector is located: Firstly, there is a radially-polarizedcontribution,with respect to the air shower axis, due to charge-excess radiation (Askaryan radiation)that interferes with the dominant geomagnetic radiation [8]. At the zenith angle range of the airshowers and the frequency range considered here, this interference alters the observed emission only at9he percent level. Secondly, a stronger source of asymmetry is due to whether the zenith angle of theshower axis was smaller or larger than the observed incident angle of the radiation ( θ ). The altitudeat which the air shower develops becomes more strongly dependent on the zenith angle of the showeraxis when the zenith angle increases. The zenith angle will influence the distance from the detector tothe emitting region and the amount of radiation emitted by the air shower.To sample these asymmetries associated with the payload location with respect to the air shower axiswe pick four air shower offset directions. If we describe the direction of the radiation by the incidentangle on the surface, θ , and azimuthal angle, φ , then the following offsets were chosen for the directionof the shower axis: ( θ + ψ c , φ ),( θ − ψ c , φ ),( θ, φ − ψ c ) and ( θ, φ + ψ c ). The payload locations are distributedalong lines that extend from the location of the offset through the location of observed radiation toan off-axis angle of 2.4 times ψ c . We estimate the calibration constants γ c and b and calibration lines(as in Figure 4) on each of these four air showers individually to reconstruct the energy. We reportthe average value as the energy of the cosmic ray and use the spread in these values as an uncertaintyon the cosmic-ray energy. The uncertainty on the individual cosmic-ray energy varies from 4% to 15%and are listed in Table 2.The uncertainty on angular reconstruction is small δθ ≈ . ◦ and δφ ≈ . ◦ and compared to theoffsets discussed above the resulting uncertainty on the energy is negligible. Uncertainty on the atmospheric depth of air shower development:
Most of the radiation is ex-pected to come from near the region where the air shower reaches its maximum number of particles,usually referred to as an atmospheric depth X max expressed in g cm − . The location of X max fluctuatesfrom shower to shower and the distribution of X max depends on the energy of the cosmic ray. Themost accurate measurement of the X max distributions in the energy range of interest are performed bythe Pierre Auger Observatory [24]. We force the air shower simulations used for the energy estimationof individual events to be near ( ∼ ) the observed mean (cid:104) X max (cid:105) as measured by the PierreAuger Observatory. In this way the simulations are tuned toward observations and not the choice ofinteraction model.To estimate the uncertainty on the energy reconstruction due to the spread of X max we generated aset of 50 proton air shower simulations to obtain a distribution in X max values (without forcing thesimulations to be near (cid:104) X max (cid:105) observed at Pierre Auger Observatory). By estimating the the cosmic-ray energy for each of these showers individually we obtain a distribution of reconstructed energies.The spread of this distribution is used as the uncertainty on the energy estimation due to variation inshower development and is of the order of 4%. The variation in X max is largest for proton induced airshowers, therefore the uncertainty of 4% can be considered as an upper limit. Note, the uncertaintyon the energy of only 4% due to proton X max fluctuations indicates that there is little dependency(and sensitivity) on the exact depth of shower maximum whatsoever. Variation in atmospheric refractive index:
From atmospheric measurements made throughout the aus-tral summer, we model the atmospheric refractive index. To the average atmospheric refractive indexprofile we fit an exponential function that is used in the simulation of the radio signals. The deviationsfrom this fit are ∼
3% at an altitude relevant for air showers. To estimate the impact of these devia-tions on the energy estimation, we varied the refractive index model in the simulations and evaluatedthe impact on A c . This resulted in uncertainty on the cosmic-ray energy estimates that range from4% to 9%. Variation of the refractive index of the snow surface:
The Fresnel coefficients depend directly onthe dielectric properties of the surface. For the Fresnel coefficients used in our analysis an aver-age density of the firn surface is assumed which leads to a refractive index of 1.35. A layer of freshsnow reduces the average density which can lead to a 10% lower refractive index. The effect of a 10%decrease of the refractive index lowers the reflection coefficient by 25 % at an incident angle of 57 ◦ and 8% at an incident angle of 85 ◦ . Because of the lack of knowledge of the exact refractive index of10he snow surface at the location of each reflection, we assign an uncertainty to the Fresnel reflectioncoefficient corresponding to 10% deviation in the refractive index. Calibration of the ANITA-I instrument:
The observations have been corrected for instrument response.The instrument response is characterized by measurements of the effective height of the antenna andthe impulse response of the system after the antenna. An uncertainty of 1 dB is assigned to each indi-vidual event to take into account the imperfections in the gain corrections per signal chain. In addition,we adopt a 1 dB (12%) systematic uncertainty on the energy scale to account for the uncertainty inthe method to measure the antenna effective height.
Uncertainty due the radio simulation package used:
We compared the electric field obtained withthe ZHAireS and CoREAS packages for antennas on the ground, for details see Appendix B. Deviationsin A c were found from 5% up to 30% depending on the zenith angle of the air shower. We adopt thesedeviations as an uncertainty in the energy reconstruction per individual event. Uncertainty on parameters derived from the measurements:
The statistical uncertainty on the fitparameters A and γ are propagated to the uncertainty on the energy of the cosmic rays. This uncer-tainty contributes between 6 and 38% of the cosmic-rays energy. Uncertainty on calibration constants derived from the simulations:
While generating the simula-tions to obtain the calibration constants we have to a make few pragmatic choices. The simulationshave a finite number of locations where the electric field is calculated and the values of A and γ arelinearly interpolated between these locations. In addition, there is a slight variation in the depth ofshower maximum for each different cosmic-ray energy. To estimate the impact on calibration constants b and γ c , the spread on these constants are calculated for the different cosmic-ray energies. The spreadon these constants are propagated to the uncertainty on the cosmic-ray energy and varies between 2%and 10% of the cosmic-ray energy. We applied the method and corrections discussed in the previous sections to the recorded signals toobtain the energy of the individual cosmic rays. The measured cosmic-ray energy distribution is shown inFigure 6 and the individual energy estimates are listed in Table 2.From Table 2 we calculate the weighted mean energy of the cosmic-ray sample to be 2.9 EeV. Theweights are set by quadratic sum of contributions to the uncertainty (as listed in Table 2) that are expectedto cause random event-to-event fluctuations in the energy estimate. On the mean energy we estimate an95% confidence interval σ i due to these uncertainties of 0.4 EeV. In addition to this uncertainty, we estimatean 95% confidence interval σ s due to sources of uncertainty that could systematically influence the overallenergy scale to be 0 . A c between the largest roughness scalingparameter that resulted in a reconstructable energy and the nominal used roughness scaling factor is usedto set an additional uncertainty on the energy reconstruction of 30%. This event underlines the importanceof having accurate values for the local surface roughness.In addition to the cosmic-ray energy estimations, we also present the off-axis angle of the Cherenkovcone ψ c in Table 2 as obtained from the simulations. From the spectral slope distributions obtained fromsimulations, e.g., the right panel of Figure 3, we obtain the two off-axis angles where the measurement11 nergy [EeV]0 2 4 6 8 10 12 14 E D N / D – – = (2.9 s s – i s – m Figure 6: Distribution of number of events ∆ N per energy bin ∆ E (with ∆ E = 1 EeV) for the 14 cosmic-ray events. Shownare the weighted mean µ , the uncertainties on it due to uncertainties on the individual events σ i and the uncertainty on it dueto the absolute scale σ s . The quoted uncertainties represent 95% confidence intervals. Event Energy Uncertainty Unc. Amb. dir. ψ c ∆ ψ c ◦ ] [ ◦ ]1* 2.1 43 14 0.43 02* 3.1 49 12 0.52 03 2.3 30 4 0.78 0.134 3.1 28 5 0.78 0.155 9.0 27 5 0.80 0.166 2.1 25 6 0.73 0.067 1.0 27 4 0.62 0.078 9.9 55 15 0.93 0.209 1.0 29 5 0.63 0.0510 2.6 24 4 0.55 0.0911 3.9 24 5 0.70 0.0812 1.9 28 6 0.55 0.0313 5.6 34 12 0.49 0.1214 2.7 23 3 0.56 0.07 Table 2: Energy estimate on the individual events and the total uncertainty on it due to the several sources of uncertaintylisted in Section 4.4. The column labelled Unc. Amb. dir. gives the fraction of the uncertainty due to the effect discussedin
Ambiguity in the direction of the air shower in Section 4.4. Also listed are the off-axis angle of the Cherenkov coneand the reconstructed offset with respect to the Cherenkov angle. The two events marked with an ‘*’ had flatter observedspectral slopes than obtained from the simulations and are therefore assumed to be on the Cherenkov cone (see the text formore details). ψ c .
5. Simulating the ANITA flight
In order to estimate the exposure, we need to calculate the trigger efficiency of ANITA to an isotropiccosmic-ray flux. The method to calculate the exposure is partially inspired by the calculations performedin [25] and the detector simulation is based on the flight performance as reported in [9]. Below we describethe strategy of the flight simulation in a (semi-) sequential order:1. The payload is located at a fixed location with an altitude of 36 km above sea level, which correspondsto the average altitude during the flight.2. The Antarctic surface is chosen at altitude of 2 km above sea level, which corresponds to the averageheight of the Antarctic ice sheet overlooked by ANITA during the flight.3. Seeds for cosmic rays are generated in the following way: We choose a surface 200 km above sea levelparallel to the surface of the Earth. The size of the surface (4.5 × km ) extends slightly over thehorizon that ANITA can see from 36 km altitude. On this surface seed locations for cosmic rays aregenerated by drawing from a uniform distribution function. To each seed a direction is assigned bydrawing a direction from an isotropic distribution. In total 6.7 × seeds are generated.4. A cosmic ray is tracked into the atmosphere and the location is calculated where it reaches a slantdepth of 735 g cm − , which corresponds roughly to the average value of X max above 10 eV. Thislocation is used to make a decision: if the off-axis angle with the location of the payload is less than2 ◦ we will start the full simulation of the particle shower and the radio emission. If it is larger than2 ◦ , it is unlikely that it will trigger the instrument and therefore it will be rejected (see Figure 7 forvalidation of this cut). The number of cosmic rays passing this cut is 16968.5. The full simulations are run at a fixed cosmic-ray energy of 10 . eV and protons as a primary cosmic-ray particle. To explore the energy dependency of the exposure, the radio signal is scaled proportionalto the cosmic-ray energy afterward (this linear scaling is a valid assumption see for example Figure 4(right)). The geomagnetic field input parameters of the simulations are chosen such that they mimicthe geomagnetic field configurations along the flight path. The orientation of the shower directionwith respect to the geomagnetic field is the dominant factor that determines the polarization of thesignal. By varying the geomagnetic field configurations and the incoming directions the polarizationdependency of the trigger efficiency is taken into account.6. The roughness corrections are applied to the resulting electric field obtained from the simulations.Several scenarios for the reflection properties of the surface are simulated to estimate the influence ofthe surface properties on the trigger efficiency.7. The electric field at the payload, corrected for reflection effects, is propagated through the antennamodel and the analog chain taking into account the polarization of the signal and the antennas.8. To the signal, several noise contributions are added. The noise seen in the field of view of the antennais the combination of the noise temperature of the ice and the sky which is roughly 190 K over thefull frequency band. Narrowband emission between 220 and 290 MHz is also present in the field ofview, arising from satellites. This emission was observed toward the north side of the payload andtherefore their contributions are simulated into the north facing antennas. The power is set to matchobservations during the flight. In addition to the noise seen by the antennas, the noise generated bythe full trigger chain is simulated based on measurements performed before flight and is about 140 Kover the frequency band.9. The combination of signal and noise is fed into the trigger simulation. The trigger simulation is setupwith typical thresholds and multiplicity of frequency bands (minimal three out of eight) and antennamultiplicity (four or more adjacent antennas) as used during flight. The thresholds are set such thatthey correspond to a fix rate of threshold crossings per single antenna, therefore it depends on the13 (cid:176) [ y yD N / D -1
10 110 All eV
10 eV
10 eV c y Data
Figure 7: The distribution of the number of events ∆ N per off-axis bin ∆ ψ (with ∆ ψ = 0 . ◦ ) obtained from the flightsimulation. We compare the off-axis distribution of all simulated events to the distributions of events that resulted in a trigger.The distributions are shown for cosmic rays simulated with different energies. In addition we show the distribution of valuesof ψ c for the measurements (Table 2) amount of noise seen by an antenna. We ran the simulation once with the nominal noise setup (asdescribed in the previous item), once reflecting a more noisy environment (near a base) and oncereflecting a quiet region of the flight.10. The fraction of accepted cosmic rays (not rejected in step 4 or 9) with respect to the total number ofcosmic rays simulated in step 3 is used to calculate the effective aperture of the ANITA experiment.Using the total life time of the experiment of 17.25 days the exposure is estimated. The trigger efficiency from the flight simulation determines the total exposure of the flight, which isdisplayed in the top panel of Figure 8. The energy scale at which the exposure grows rapidly is approximatelythe same energy scale as the average energy from the individual events. The fast rise in exposure below10 . eV basically reflects the energy regime where some of the air showers start to be above the detectionthreshold. At a higher energy, the growth in exposure slows down and the steady increase is due to theincreasing possibility of detecting events further away from the Cherenkov cone, as seen in Figure 7.In the bottom panel of Figure 8 we compare the cosmic-ray energy flux J ( E ) as observed by the PierreAuger Observatory and the Telescope Array to the flux observed by ANITA-I. Due to the small numberof events measured by ANITA, we decided to calculate the flux using a single bin that contains the wholeenergy range of the observations and use the average reconstructed energy to set the energy scale. Thevertical error bar is the quadratically summed combination of the Poisson uncertainty on the number ofevents and uncertainty on the exposure.To assign an uncertainty to the exposure we performed the full flight simulation in several scenarios byvarying the roughness model (twice as rough or half as rough as at Taylor Dome), the background model(quiet conditions and noisy conditions) and the average refractive index of the ice (n = 1.31 and n = 1.35).Increasing the surface roughness by a factor of two results in a significant decrease in the exposure by afactor 0.3 at 2 . × eV, while reducing it by a factor of 0.5 increases the exposure only by a factor1.05. Changing the refractive index of the ice from 1.35 to 1.31 leads to a 3% increase of the exposure at14 (Energy/eV) log18 18.5 19 19.5 20 s r y r] E xpo s u r e [ k m -2 -1
10 1 (Energy / eV) log18 18.2 18.4 18.6 18.8 19 - y r s r e V ] J ( E ) [ k m -19 -18 -17 ANITAAugerTAANITAAugerTA
Figure 8: Top: ANITA-I’s exposure to UHECRs, derived from Monte Carlo simulations. In additional to the nominal values(black line) for the reflective properties of the ice we ran several scenarios varying the properties of the reflective surface. Allthe scenarios are contained within the green shaded area (see the text for more details). Bottom: Comparison between theUHECR flux as observed by ANITA-I (see the text for details), the Pierre Auger Observatory [26] and the Telescope Array [5]. . × eV. Using a background model that reflects quiet conditions increases the exposure by 6% whilemore noisy conditions reduce the exposure by 11% at 2 . × eV. To be conservative, we used the twomost extreme combinations of these scenarios as an uncertainty on the exposure.Using this approach we find that the observed cosmic-ray flux by ANITA at 2.9 EeV is 1.4 +1 . − . km − yr − sr − eV − .The horizontal error bars are given by the 95% confidence interval σ i due to the uncertainties on the indi-vidual events of 0.4 EeV while the brackets indicate the 95% confidence interval σ s due to the uncertaintyon the overall scale of the radio signal of 0 . Using the exposure, we estimate the number of expected cosmic rays via Monte Carlo simulations.To do so, we integrate the parameterized cosmic-ray-energy spectrum, as measured at the Pierre AugerObservatory [26] (or at the Telescope Array [5]), multiplied with the exposure function over the full energyrange. The resulting number of expected events is 16 (Pierre Auger spectrum) or 20 (Telescope Arrayspectrum), compared to the 14 observed events.In addition to the expected total number of events, we can also compare the distributions of severalvariables obtained from the measurements to those of the flight simulation. We compared the followingparameters: A , the amplitude of the frequency spectrum of the signal at 300 MHz, γ , the spectral slope fittedbetween 300-1000 MHz, and θ , the incident angle of the radiation on the ice. The results are given in Figure9. The distributions obtained from the Monte Carlo simulations are weighted as a function of energy by thenumber of events expected from the cosmic-ray energy spectrum as observed at the Pierre Auger Observatoryor the Telescope Array. To test the compatibility of the measured and simulated distributions we normalizedthem both and perform a Kolmogorov-Smirnov test. In Figure 9, we report the probability of havingthe calculated Kolmogorov-Smirnov test statistic or larger, P KS , using the Pierre Auger energy spectrum.This indicates that there is statistical agreement between the distributions obtained from simulations andobservations. Using the cosmic-ray energy spectrum as observed by the Telescope Array, we find the followingprobabilities P KS = 0 . A , P KS = 0 . γ and P KS = 0 . θ .
6. Discussion
The distribution of observed amplitudes contains one obvious outlier, this event has been identified asnumber 5 in Table 2. We inspected this event carefully however, and except for its exceptionally large ampli-tude, the other parameters of this event are in the bulk of their corresponding distributions. Thunderstormconditions are known to cause enhanced radio signals from air showers [27, 28, 29] and therefore provide afeasible candidate to explain outliers in the amplitude distribution. However, thunderstorm conditions areextremely rare in Antarctica [30] and therefore it is an unlikely explanation for this outlier.Having provided the ANITA-I flight simulation with the most realistic treatment of radio signals, weinvestigate the impact of the corrections on the electric field values from ZHAireS simulations. Not includingthe correction due to the reflection on the curved ice surface results in a significant increase of sensitivityfor events near the horizon; therefore, the total number of expected events increased to 21 using the PierreAuger Observatory spectrum and 25 for the Telescope Array spectrum. Assuming a completely smoothsurface results in tension between the simulated and observed distributions of the spectral slope: P KS =0.03 using the Pierre Auger Observatory energy spectrum and P KS = 0.01 when using the Telescope Arrayenergy spectrum. These results underline the importance of having a correct treatment of the reflection.Some studies of the reflection properties of the Antarctic ice in the ANITA-I frequency band have beenperformed using the direct and reflected emission from the Sun [31]. These studies indicate that the reflectioncoefficient roughly follows the expectation for Fresnel coefficients at low incident angles, but smaller valuesthan predicted by Fresnel coefficients are observed at larger angles of incidence. This is qualitatively what isexpected for both the defocusing due to surface curvature and the loss of coherence due to surface roughness.However, to exactly calculate the reflective properties of the ice the location dependent surface model needsimprovement. Additional refinement of the models could include local curvature, local slope and local16 -0.5 MHz -1 m A [pW0 1 2 3 -4 -3 -2 -1
10 1
ObservedMC-simulation = 0.3 KS P ] -1 [Hz g -10 -5 0 -9 · -5 -4 -3 -2 -1
10 1
ObservedMC-simulation = 0.3 KS P ] (cid:176) [ q -5 -4 -3 -2 -1 ObservedMC-simulation = 0.7 KS P Figure 9: Comparison between distribution of parameters obtained from the flight simulation to observations. The bin sizes arechosen of equal size for simulations and observations and the vertical scale is normalized so that integral over each distributionequals 1. Top: Distribution in amplitude A . Middle: Distribution in spectral slope γ . Bottom: Distribution in incidentangle on the ice. To the entries in the Monte-Carlo distributions weights are applied to reflect the energy dependence of thecosmic-ray spectrum. We used a weight proportional to the number of events as a function of energy as expected from PierreAuger Observatories cosmic-ray-energy spectrum. ∼
10 % [26] and ∼
16 % [34, 35] deviations between simulations and observations. Recently, the CROME experiment ob-served several air showers using gigahertz receivers [36]. The distribution of shower impact locations was inagreement with expectations from simulations. Which indicates that current simulation models are capableof describing radio emission over a large frequency range.The agreement that was found between number of predicted events from simulation and the observednumber of events is encouraging for the validity of the absolute scale of the radio signal from the simula-tions. However, in the near future the radio emission codes will be absolutely calibrated by the antennaarrays incorporated in one of the existing cosmic-ray observatories (LOFAR [33], AERA [8], TunkaRex [37],CODALEMA [7], and by dedicated particle-beam experiments such as T-510 at SLAC [38].The energy scale of the analysis presented here is significantly lower than the first results publishedon this data set [6]. The main reason for this discrepancy is that at the time of that publication therewas no radio emission model known that could produce coherent radiation in the ANITA frequency band.Therefore all observables were fitted together in a Bayesian approach with a simple parameterization of theradio beam. The input parameters were based on observations made at lower frequencies. Therefore, thefit to the data preferred a wider beam with significantly less power at higher frequencies than what themodels predict now resulting in higher energy estimates and larger exposure at high energy. However, therecent incorporation of the Cherenkov-like effects in the emission models significantly altered the lateral andfrequency dependencies of the radio beam. We have shown that we now can independently estimate theenergy of the cosmic rays and get reasonable agreement between the distributions from observations andflight simulations. Therefore, a parameterized fit is no longer necessary and both cosmic-ray energy and thedetector exposure can be obtained directly from simulations.
7. Conclusion and Outlook
In this paper, we present a new method of energy reconstruction for cosmic rays using observations inthe frequency range of 300 and 1000 MHz and applied this to observations made during the ANITA-I flight.The mean energy of cosmic rays observed by the first ANITA flight is 2 . ± . σ i ) ± . σ s ) × eV.The analysis in this paper shows that by using antennas sensitive between 300-1000 MHz there is astraightforward method to estimate the energy of the cosmic-ray particle. Since this method of energyestimation is independent of energy estimation from other techniques it might become useful in the crosscalibrating of the different techniques when implemented in existing cosmic-ray observatories.We simulated the full ANITA flight to estimate the sensitivity to cosmic rays that produce radio signalsthat are reflected from the ice. From this simulation, we find that the expected number of events is inagreement with the observed number of events. We estimated the total exposure of the flight and estimated,for the first time, the cosmic-ray flux based on radio observations only. This cosmic-ray flux estimate is inagreement with observations made at the Pierre Auger Observatory and Telescope Array.The flight simulation is also used to compare the distributions of simulated events that trigger theANITA instrument to the distributions obtained from the measurements. We found reasonable agreementbetween simulations and measurements in the distribution of the incident angle, the spectral slope, and the18mplitude. This provides us with some confidence that the simulations are consistent and allows us to studythe importance and necessity of the assumptions made in the flight simulation. However, the limited sizeof our data set prevents us to go into very deep details and therefore we provide a single flux point ratherthan an energy spectrum. In the 2014-2015 Austral summer the third ANITA flight flew with a significantlyhigher sensitivity for cosmic-ray signals than the previous flights. Therefore, a significant increase of thenumber of cosmic-ray observations is expected. This data set will allow for an energy spectrum and studymore of the details of the reflections.The exposure of the ANITA-I flight is orders of magnitude too small to significantly contribute to thecollection of cosmic rays with energy above the GZK limit ( ∼ × eV). However, it was shown that witha measurement at a single location with respect to the shower axis we were able to measure the cosmic-ray energy with a reasonable accuracy. Therefore, with an array of antennas on the ground, broadbandobservations might be used to constrain the energy and other parameters, such as the longitudinal airshower development [39], with only a few antennas per air shower. It should be noted that the accuracyto determine air shower properties is in the case of ANITA largely set by uncertainties on the reflectionproperties of the surface, these can of course be discarded in the case of a ground array of antennas. Inaddition, a ground array will point up to the sky and therefore the antenna temperature is expected to dropsignificantly with respect to the downward pointing antennas of ANITA. At lower frequencies (50 MHz) theantenna temperature is more than two orders magnitude higher than in the ANITA frequency range due tothe galactic background. The straightforward way of deriving shower parameters with only a few broadbandantennas in combination with the low antenna temperature are beneficial characteristics to be utilized ina cosmic-ray ground array. However, since the footprint on the ground is rather small, it is unlikely thatthis technique is scalable to a large array for detecting UHECRs at a low cost. However, it could be eitherincorporated in an existing cosmic-ray observatory to provide a cross-calibration technique for the energyestimation of air showers or as an array making accurate measurements of air showers around 10 - 10 eV. Acknowledgments
We would like to thank Marianne Ludwig from Karlsruhe Institute of Technology for helpful discussionsand in the early days of this work. We are grateful to NASA, the U.S. National Science Foundation, the U.S.Department of Energy, and the Columbia Scientific Balloon Facility for their generous support of these ef-forts. We would like to extend our thanks to the 2006-2007 on-ice LDB and McMurdo crews for their support.Part of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, un-der a contract with the National Aeronautics and Space Administration. J. A-M, W.R.C., D.G-F and E.Z.thank Ministerio de Econom´ıa (FPA2012-39489), Consolider-Ingenio 2010 CPAN Programme (CSD2007-00042), Xunta de Galicia (GRC2013-024), Feder Fundsand Marie Curie-IRSES/ EPLANET (EuropeanParticle physics Latin American NETwork), 7 th Framework Program (PIRSES- 2009-GA-246806).
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50 60 70 80 90 r e f l ec ti on c o e ff i c i e n t
265 MHz435 MHz650 MHz990 MHz
Figure 10: The amplitude multiplication factors due to surface roughness. They are shown for four frequencies within theANITA frequency band as a function of incident angle on the ice.
Appendix A: Frequency dependency of surface roughness reflection coefficient
To address the effect of surface roughness we followed the method as described in [21]. As an input wetake a parameterization of the surface roughness, as a function of length scales, that is fitted to observationsfrom [23] and measurements taken at the Antarctic Taylor Dome Station.The calculation of the frequency dependent roughness effect is very CPU time consuming. Therefore,it has been calculated at a few specific frequencies and over a range of incident angles. Afterwards, theresults are parametrized with smooth functions to intermediate distance values. The multiplication factorsas a function of incident angle are shown in Figure 10 at four different frequencies. We perform a linearinterpolation between the results for the different frequencies to obtain a frequency dependent multiplicationfactor.
Appendix B: Comparing ZHAireS to CoREAS
The ZHAireS simulation package used in this paper is the only one that can estimate the electric field atlocation of the payload after reflection on the ice cap [15]. However, we made a comparison to the CoREASpackage [13] to estimate the discrepancy between the two. This forced us to perform calculations on locationson the ground only in order to compare them.The implementation of the end-point algorithm for the calculation of the electric field in CoREAS and thecorresponding
ZHS algorithm in ZHAireS result in identical amounts of radiation when applied to identicalcharge distributions. However, since the two algorithms are implemented in different air shower simulationpackages, differences in the calculated radiation may occur. To minimize these differences we modified theCoREAS code in order to use the same model of refractive index as in ZHAireS which was tuned to Antarcticatmospheric data. We also selected air showers from both simulations that have X max that deviates lessthan 5 g cm − from a fixed value.In Figure 11, we compare the distributions of A as a function of ψ (off-axis angle) on the ground asobtained with ZHAireS and CoREAS for three different shower zenith angles. Overall the distributions inamplitude look rather similar for the two packages. The 10% and 5% deviations observed at the two smallest22 (cid:176) [ y c , Z HA i r e S A / A ZHAireS (cid:176) =57 q CoREAS (cid:176) =57 q ZHAireS (cid:176) =75 q CoREAS (cid:176) =75 q ZHAireS (cid:176) =85 q CoREAS (cid:176) =85 q Figure 11: Comparison of A values from CoREAS and ZHAireS on the ground for a 57 ◦ zenith angle shower with cosmic-rayenergy of 10 eV. We normalized the A values to that obtained from ZHAireS on the Cherenkov cone A c, ZHAireS . The linescorrespond to an average value of A obtained from a set of 8 to 25 simulations (depending on the geometry).obtained from a set of 8 to 25 simulations (depending on the geometry).