The effect of the geomagnetic field on cosmic ray energy estimates and large scale anisotropy searches on data from the Pierre Auger Observatory

Abstract

We present a comprehensive study of the influence of the geomagnetic field on
the energy estimation of extensive air showers with a zenith angle smaller than
$60^\circ$, detected at the Pierre Auger Observatory. The geomagnetic field
induces an azimuthal modulation of the estimated energy of cosmic rays up to
the ~2% level at large zenith angles. We present a method to account for this
modulation of the reconstructed energy. We analyse the effect of the modulation
on large scale anisotropy searches in the arrival direction distributions of
cosmic rays. At a given energy, the geomagnetic effect is shown to induce a
pseudo-dipolar pattern at the percent level in the declination distribution
that needs to be accounted for.

Full text

arXiv:1111.7122v1 [astro-ph.IM] 30 Nov 2011
The effect of the geomagnetic field on cosmic ray energy
estimates and large scale anisotropy searches on data from
the Pierre Auger Observatory
The Pierre Auger Collaboration
P. Abreu74, M. Aglietta57 , E.J. Ahn93, I.F.M. Albuquerque19, D. Allard33, I. Allekotte1, J. Allen96 ,
P. Allison98, J. Alvarez Castillo67, J. Alvarez-Mu˜niz84, M. Ambrosio50 , A. Aminaei68 , L. Anchordoqui109,
S. Andringa74, T. Antiˇci´c27 , A. Anzalone56, C. Aramo50, E. Arganda81, F. Arqueros81, H. Asorey1,
P. Assis74, J. Aublin35, M. Ave41 , M. Avenier36, G. Avila12, T. B¨acker45, M. Balzer40, K.B. Barber13,
A.F. Barbosa16, R. Bardenet34, S.L.C. Barroso22, B. Baughman98 f, J. B¨auml39 , J.J. Beatty98,
B.R. Becker106, K.H. Becker38, A. Bell´etoile37 , J.A. Bellido13 , S. BenZvi108 , C. Berat36, X. Bertou1,
P.L. Biermann42, P. Billoir35, F. Blanco81, M. Blanco82 , C. Bleve38, H. Bl¨umer41,39, M. Boh´aˇcov´a29,
D. Boncioli51, C. Bonifazi25,35, R. Bonino57 , N. Borodai72, J. Brack91, P. Brogueira74, W.C. Brown92,
R. Bruijn87, P. Buchholz45, A. Bueno83, R.E. Burton89, K.S. Caballero-Mora99, L. Caramete42,
R. Caruso52, A. Castellina57 , O. Catalano56, G. Cataldi49, L. Cazon74, R. Cester53, J. Chauvin36,
S.H. Cheng99, A. Chiavassa57, J.A. Chinellato20 , A. Chou93 , J. Chudoba29 , R.W. Clay13, M.R. Coluccia49,
R. Concei¸c˜ao74, F. Contreras11, H. Cook87, M.J. Cooper13 , J. Coppens68,70, A. Cordier34, S. Coutu99 ,
C.E. Covault89, A. Creusot33,79, A. Criss99, J. Cronin101, A. Curutiu42, S. Dagoret-Campagne34,
R. Dallier37, S. Dasso8,4, K. Daumiller39, B.R. Dawson13, R.M. de Almeida26, M. De Domenico52,
C. De Donato67,48, S.J. de Jong68,70 , G. De La Vega10, W.J.M. de Mello Junior20, J.R.T. de
Mello Neto25, I. De Mitri49, V. de Souza18, K.D. de Vries69, G. Decerprit33 , L. del Peral82,
M. del R´ıo51,11, O. Deligny32, H. Dembinski41, N. Dhital95 , C. Di Giulio47,51 , J.C. Diaz95,
M.L. D´ıaz Castro17, P.N. Diep110, C. Dobrigkeit 20 , W. Docters69, J.C. D’Olivo67, P.N. Dong110,32,
A. Dorofeev91, J.C. dos Anjos16 , M.T. Dova7, D. D’Urso50 , I. Dutan42 , J. Ebr29 , R. Engel39 ,
M. Erdmann43, C.O. Escobar20, J. Espadanal74, A. Etchegoyen2, P. Facal San Luis101, I. Fajardo
Tapia67, H. Falcke68,71, G. Farrar96, A.C. Fauth20, N. Fazzini93, A.P. Ferguson89, A. Ferrero2,
B. Fick95, A. Filevich2, A. Filipˇciˇc78,79 , S. Fliescher43 , C.E. Fracchiolla91, E.D. Fraenkel69, U. Fr¨ohlich45,
B. Fuchs16, R. Gaior35, R.F. Gamarra2, S. Gambetta46 , B. Garc´ıa10, D. Garc´ıa G´amez34,83,
D. Garcia-Pinto81, A. Gascon83, H. Gemmeke40, K. Gesterling106, P.L. Ghia35,57, U. Giaccari49,
M. Giller73, H. Glass93, M.S. Gold106 , G. Golup1, F. Gomez Albarracin7, M. G´omez Berisso1,
P. Gon¸calves74 , D. Gonzalez41, J.G. Gonzalez41 , B. Gookin91, D. G´ora41,72, A. Gorgi57, P. Gouffon19,
S.R. Gozzini87, E. Grashorn98, S. Grebe68,70, N. Griffith98 , M. Grigat43 , A.F. Grillo58 , Y. Guardincerri4,
F. Guarino50, G.P. Guedes21, A. Guzman67 , J.D. Hague106, P. Hansen7, D. Harari1, S. Harmsma69,70,
T.A. Harrison13, J.L. Harton91, A. Haungs39, T. Hebbeker43, D. Heck39, A.E. Herve13, C. Ho jvat93,
N. Hollon101, V.C. Holmes13 , P. Homola72, J.R. H¨orandel68 , A. Horneffer68 , P. Horvath30, M. Hrabovsk´y30,29,
T. Huege39, A. Insolia52 , F. Ionita101, A. Italiano52 , C. Jarne7, S. Jiraskova68, M. Josebachuili2,
K. Kadija27, K.H. Kampert38, P. Karhan28, P. Kasper93, B. K´egl34, B. Keilhauer39, A. Keivani94,
J.L. Kelley68, E. Kemp20 , R.M. Kieckhafer95, H.O. Klages39, M. Kleifges40 , J. Kleinfeller39,
J. Knapp87, D.-H. Koang36, K. Kotera101, N. Krohm38, O. Kr¨omer40, D. Kruppke-Hansen38,
F. Kuehn93, D. Kuempel38, J.K. Kulbartz44, N. Kunka40, G. La Rosa56, C. Lachaud33, P. Lautridou37,
M.S.A.B. Le˜ao24, D. Lebrun36 , P. Lebrun93, M.A. Leigui de Oliveira24, A. Lemiere32, A. Letessier-
Selvon35, I. Lhenry-Yvon32, K. Link41, R. L´opez63, A. Lopez Ag¨uera84, K. Louedec34, J. Lozano
1

Bahilo83, L. Lu87 , A. Lucero2,57 , M. Ludwig41 , H. Lyberis32, M.C. Maccarone56, C. Macolino35 ,
S. Maldera57, D. Mandat29 , P. Mantsch93, A.G. Mariazzi7, J. Marin11,57 , V. Marin37, I.C. Maris35,
H.R. Marquez Falcon66, G. Marsella54, D. Martello49, L. Martin37, H. Martinez64 , O. Mart´ınez
Bravo63, H.J. Mathes39, J. Matthews94,100 , J.A.J. Matthews106 , G. Matthiae51 , D. Maurizio53,
P.O. Mazur93, G. Medina-Tanco67, M. Melissas41 , D. Melo2,53 , E. Menichetti53, A. Menshikov40,
P. Mertsch85, C. Meurer43 , S. Mi´canovi´c27, M.I. Micheletti9, W. Miller106 , L. Miramonti48, L. Molina-
Bueno83, S. Mollerach1, M. Monasor101 , D. Monnier Ragaigne34, F. Montanet36, B. Morales67,
C. Morello57, E. Moreno63, J.C. Moreno7, C. Morris98, M. Mostaf´a91, C.A. Moura24,50 , S. Mueller39 ,
M.A. Muller20, G. M¨uller43, M. M¨unchmeyer35, R. Mussa53 , G. Navarra57 †, J.L. Navarro83,
S. Navas83, P. Necesal29, L. Nellen67 , A. Nelles68,70 , J. Neuser38 , P.T. Nhung110, L. Niemietz38,
N. Nierstenhoefer38, D. Nitz95, D. Nosek28, L. Noˇzka29, M. Nyklicek29, J. Oehlschl¨ager39 , A. Olinto101,
P. Oliva38, V.M. Olmos-Gilbaja84, M. Ortiz81 , N. Pacheco82, D. Pakk Selmi-Dei20 , M. Palatka29,
J. Pallotta3, N. Palmieri41, G. Parente84, E. Parizot33, A. Parra84 , R.D. Parsons87, S. Pastor80,
T. Paul97, M. Pech29, J. P¸ekala72, R. Pelayo84, I.M. Pepe23 , L. Perrone54, R. Pesce46, E. Petermann105 ,
S. Petrera47, P. Petrinca51, A. Petrolini46, Y. Petrov91, J. Petrovic70, C. Pfendner108 , N. Phan106,
R. Piegaia4, T. Pierog39, P. Pieroni4, M. Pimenta74, V. Pirronello52, M. Platino2, V.H. Ponce1,
M. Pontz45, P. Privitera101, M. Prouza29, E.J. Quel3, S. Querchfeld38, J. Rautenberg38, O. Ravel37,
D. Ravignani2, B. Revenu37, J. Ridky29, S. Riggi84,52 , M. Risse45 , P. Ristori3, H. Rivera48 ,
V. Rizi47, J. Roberts96, C. Robledo63 , W. Rodrigues de Carvalho84,19, G. Rodriguez84, J. Ro-
driguez Martino11, J. Rodriguez Rojo11, I. Rodriguez-Cabo84, M.D. Rodr´ıguez-Fr´ıas82, G. Ros82 ,
J. Rosado81, T. Rossler30, M. Roth39 , B. Rouill´e-d’Orfeuil101, E. Roulet1, A.C. Rovero8, C. R¨uhle40 ,
F. Salamida47,39, H. Salazar63, F. Salesa Greus91, G. Salina51 , F. S´anchez2, C.E. Santo74, E. Santos74,
E.M. Santos25, F. Sarazin90, B. Sarkar38, S. Sarkar85, R. Sato11, N. Scharf43, V. Scherini48,
H. Schieler39, P. Schiffer43, A. Schmidt40, F. Schmidt101 , O. Scholten69 , H. Schoorlemmer68,70 ,
J. Schovancova29, P. Schov´anek29, F. Schr¨oder39, S. Schulte43, D. Schuster90, S.J. Sciutto7, M. Scuderi52,
A. Segreto56, M. Settimo45 , A. Shadkam94, R.C. Shellard16,17 , I. Sidelnik2, G. Sigl44, H.H. Silva
Lopez67, A. ´
Smia lkowski73, R. ˇ
Sm´ıda39,29, G.R. Snow105, P. Sommers99, J. Sorokin13 , H. Spinka88,93,
R. Squartini11, S. Stanic79 , J. Stapleton98 , J. Stasielak72, M. Stephan43, E. Strazzeri56, A. Stutz36 ,
F. Suarez2, T. Suomij¨arvi32 , A.D. Supanitsky8,67, T. ˇ
Suˇsa27, M.S. Sutherland94,98, J. Swain97,
Z. Szadkowski73, M. Szuba39, A. Tamashiro8, A. Tapia2, M. Tartare36 , O. Ta¸sc˘au38, C.G. Tavera
Ruiz67, R. Tcaciuc45, D. Tegolo52,61, N.T. Thao110 , D. Thomas91 , J. Tiffenberg4, C. Timmermans70,68 ,
D.K. Tiwari66, W. Tkaczyk73, C.J. Todero Peixoto18,24, B. Tom´e74, A. Tonachini53, P. Travnicek29,
D.B. Tridapalli19, G. Tristram33, E. Trovato52, M. Tueros84,4, R. Ulrich99,39, M. Unger39 , M. Urban34 ,
J.F. Vald´es Galicia67, I. Vali˜no84,39, L. Valore50, A.M. van den Berg69, E. Varela63, B. Vargas
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L. Villase˜nor66, H. Wahlberg7, P. Wahrlich13, O. Wainberg2, D. Walz43, D. Warner91, A.A. Watson87,
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L. Wiencke90, B. Wilczy´nska72, H. Wilczy´nski72 , M. Will39 , C. Williams101 , T. Winchen43 , M.G. Winnick13,
M. Wommer39, B. Wundheiler2, T. Yamamoto101 a, T. Yapici95, P. Younk45, G. Yuan94, A. Yushkov84,50,
B. Zamorano83, E. Zas84 , D. Zavrtanik79,78, M. Zavrtanik78,79, I. Zaw96, A. Zepeda64, M. Zimbres
Silva38,20, M. Ziolkowski45
1Centro At´omico Bariloche and Instituto Balseiro (CNEA- UNCuyo-CONICET), San Carlos de
Bariloche, Argentina
2Centro At´omico Constituyentes (Comisi´on Nacional de Energ´ıa At´omica/CONICET/UTN-FRBA),
Buenos Aires, Argentina
3Centro de Investigaciones en L´aseres y Aplicaciones, CITEFA and CONICET, Argentina
4Departamento de F´ısica, FCEyN, Universidad de Buenos Aires y CONICET, Argentina
7IFLP, Universidad Nacional de La Plata and CONICET, La Plata, Argentina
8Instituto de Astronom´ıa y F´ısica del Espacio (CONICET- UBA), Buenos Aires, Argentina
9Instituto de F´ısica de Rosario (IFIR) - CONICET/U.N.R. and Facultad de Ciencias Bioqu´ımicas
y Farmac´euticas U.N.R., Rosario, Argentina
2

10 National Technological University, Faculty Mendoza (CONICET/CNEA), Mendoza, Argentina
11 Observatorio Pierre Auger, Malarg¨ue, Argentina
12 Observatorio Pierre Auger and Comisi´on Nacional de Energ´ıa At´omica, Malarg¨ue, Argentina
13 University of Adelaide, Adelaide, S.A., Australia
16 Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, RJ, Brazil
17 Pontif´ıcia Universidade Cat´olica, Rio de Janeiro, RJ, Brazil
18 Universidade de S˜ao Paulo, Instituto de F´ısica, S˜ao Carlos, SP, Brazil
19 Universidade de S˜ao Paulo, Instituto de F´ısica, S˜ao Paulo, SP, Brazil
20 Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil
21 Universidade Estadual de Feira de Santana, Brazil
22 Universidade Estadual do Sudoeste da Bahia, Vitoria da Conquista, BA, Brazil
23 Universidade Federal da Bahia, Salvador, BA, Brazil
24 Universidade Federal do ABC, Santo Andr´e, SP, Brazil
25 Universidade Federal do Rio de Janeiro, Instituto de F´ısica, Rio de Janeiro, RJ, Brazil
26 Universidade Federal Fluminense, EEIMVR, Volta Redonda, RJ, Brazil
27 Rudjer Boˇskovi´c Institute, 10000 Zagreb, Croatia
28 Charles University, Faculty of Mathematics and Physics, Institute of Particle and Nuclear
Physics, Prague, Czech Republic
29 Institute of Physics of the Academy of Sciences of the Czech Republic, Prague, Czech Republic
30 Palacky University, RCATM, Olomouc, Czech Republic
32 Institut de Physique Nucl´eaire d’Orsay (IPNO), Universit´e Paris 11, CNRS-IN2P3, Orsay,
France
33 Laboratoire AstroParticule et Cosmologie (APC), Universit´e Paris 7, CNRS-IN2P3, Paris,
France
34 Laboratoire de l’Acc´el´erateur Lin´eaire (LAL), Universit´e Paris 11, CNRS-IN2P3, Orsay, France
35 Laboratoire de Physique Nucl´eaire et de Hautes Energies (LPNHE), Universit´es Paris 6 et Paris
7, CNRS-IN2P3, Paris, France
36 Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Universit´e Joseph Fourier,
INPG, CNRS-IN2P3, Grenoble, France
37 SUBATECH, ´
Ecole des Mines de Nantes, CNRS-IN2P3, Universit´e de Nantes, Nantes, France
38 Bergische Universit¨at Wuppertal, Wuppertal, Germany
39 Karlsruhe Institute of Technology - Campus North - Institut f¨ur Kernphysik, Karlsruhe, Ger-
many
40 Karlsruhe Institute of Technology - Campus North - Institut f¨ur Prozessdatenverarbeitung und
Elektronik, Karlsruhe, Germany
41 Karlsruhe Institute of Technology - Campus South - Institut f¨ur Experimentelle Kernphysik
(IEKP), Karlsruhe, Germany
42 Max-Planck-Institut f¨ur Radioastronomie, Bonn, Germany
43 RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany
44 Universit¨at Hamburg, Hamburg, Germany
45 Universit¨at Siegen, Siegen, Germany
46 Dipartimento di Fisica dell’Universit`a and INFN, Genova, Italy
47 Universit`a dell’Aquila and INFN, L’Aquila, Italy
48 Universit`a di Milano and Sezione INFN, Milan, Italy
49 Dipartimento di Fisica dell’Universit`a del Salento and Sezione INFN, Lecce, Italy
50 Universit`a di Napoli ”Federico II” and Sezione INFN, Napoli, Italy
51 Universit`a di Roma II ”Tor Vergata” and Sezione INFN, Roma, Italy
52 Universit`a di Catania and Sezione INFN, Catania, Italy
53 Universit`a di Torino and Sezione INFN, Torino, Italy
54 Dipartimento di Ingegneria dell’Innovazione dell’Universit`a del Salento and Sezione INFN, Lecce,
Italy
3

56 Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo (INAF), Palermo, Italy
57 Istituto di Fisica dello Spazio Interplanetario (INAF), Universit`a di Torino and Sezione INFN,
Torino, Italy
58 INFN, Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila), Italy
61 Universit`a di Palermo and Sezione INFN, Catania, Italy
63 Benem´erita Universidad Aut´onoma de Puebla, Puebla, Mexico
64 Centro de Investigaci´on y de Estudios Avanzados del IPN (CINVESTAV), M´exico, D.F., Mexico
66 Universidad Michoacana de San Nicolas de Hidalgo, Morelia, Michoacan, Mexico
67 Universidad Nacional Autonoma de Mexico, Mexico, D.F., Mexico
68 IMAPP, Radboud University Nijmegen, Netherlands
69 Kernfysisch Versneller Instituut, University of Groningen, Groningen, Netherlands
70 Nikhef, Science Park, Amsterdam, Netherlands
71 ASTRON, Dwingeloo, Netherlands
72 Institute of Nuclear Physics PAN, Krakow, Poland
73 University of L´od´z, L´od´z, Poland
74 LIP and Instituto Superior T´ecnico, Technical University of Lisbon, Portugal
78 J. Stefan Institute, Ljubljana, Slovenia
79 Laboratory for Astroparticle Physics, University of Nova Gorica, Slovenia
80 Instituto de F´ısica Corpuscular, CSIC-Universitat de Val`encia, Valencia, Spain
81 Universidad Complutense de Madrid, Madrid, Spain
82 Universidad de Alcal´a, Alcal´a de Henares (Madrid), Spain
83 Universidad de Granada & C.A.F.P.E., Granada, Spain
84 Universidad de Santiago de Compostela, Spain
85 Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, United Kingdom
87 School of Physics and Astronomy, University of Leeds, United Kingdom
88 Argonne National Laboratory, Argonne, IL, USA
89 Case Western Reserve University, Cleveland, OH, USA
90 Colorado School of Mines, Golden, CO, USA
91 Colorado State University, Fort Collins, CO, USA
92 Colorado State University, Pueblo, CO, USA
93 Fermilab, Batavia, IL, USA
94 Louisiana State University, Baton Rouge, LA, USA
95 Michigan Technological University, Houghton, MI, USA
96 New York University, New York, NY, USA
97 Northeastern University, Boston, MA, USA
98 Ohio State University, Columbus, OH, USA
99 Pennsylvania State University, University Park, PA, USA
100 Southern University, Baton Rouge, LA, USA
101 University of Chicago, Enrico Fermi Institute, Chicago, IL, USA
105 University of Nebraska, Lincoln, NE, USA
106 University of New Mexico, Albuquerque, NM, USA
108 University of Wisconsin, Madison, WI, USA
109 University of Wisconsin, Milwaukee, WI, USA
110 Institute for Nuclear Science and Technology (INST), Hanoi, Vietnam
(†) Deceased
(a) at Konan University, Kobe, Japan
(f) now at University of Maryland
Abstract
4

We present a comprehensive study of the influence of the geomagnetic field on the energy
estimation of extensive air showers with a zenith angle smaller than 60◦, detected at the
Pierre Auger Observatory. The geomagnetic field induces an azimuthal modulation of the
estimated energy of cosmic rays up to the ∼2% level at large zenith angles. We present a
method to account for this modulation of the reconstructed energy. We analyse the effect
of the modulation on large scale anisotropy searches in the arrival direction distributions of
cosmic rays. At a given energy, the geomagnetic effect is shown to induce a pseudo-dipolar
pattern at the percent level in the declination distribution that needs to be accounted for.
5

1 Introduction
High energy cosmic rays generate extensive air showers in the atmosphere. The tra jectories of the
charged particles of the showers are curved in the Earth’s magnetic field, resulting in a broadening
of the spatial distribution of particles in the direction of the Lorentz force. While such effects are
known to distort the particle densities in a dramatic way at zenith angles larger than ∼60◦[1, 2,
3, 4], they are commonly ignored at smaller zenith angles where the lateral distribution function
is well described by empirical models of the NKG-type [5, 6] based on a radial symmetry of the
distribution of particles in the plane perpendicular to the shower axis.
In this article, we aim to quantify the small changes of the particle densities at ground induced
by the geomagnetic field for showers with zenith angle smaller than ∼60◦, focusing on the impacts
on the energy estimator used at the Pierre Auger Observatory. As long as the magnitude of these
effects lies well below the statistical uncertainty of the energy reconstruction, it is reasonable to
neglect them in the framework of the energy spectrum reconstruction. As the strength of the
geomagnetic field component perpendicular to the arrival direction of the cosmic ray, BT, depends
on both the zenith and the azimuthal angles (θ, ϕ) of any incoming shower, these effects are
expected to break the symmetry of the energy estimator in terms of the azimuthal angle ϕ. Such
an azimuthal dependence translates into azimuthal modulations of the estimated cosmic ray event
rate at a given energy. For any observatory located far from the Earth’s poles, any genuine large
scale pattern which depends on the declination translates also into azimuthal modulations of the
cosmic ray event rate. Thus to perform a large scale anisotropy measurement it is critical to account
for azimuthal modulations of experimental origin and for those induced by the geomagnetic field,
as already pointed out in the analysis of the Yakutsk data [7] and the ARGO-YBJ data [8]. Hence,
this work constitutes an accompanying paper of a search for large scale anisotropies, both in right
ascension and declination of cosmic rays detected at the Pierre Auger Observatory, the results of
which will be reported in a forthcoming publication.
To study the influence of the geomagnetic field on the cosmic ray energy estimator, we make
use of shower simulations and of the measurements performed with the surface detector array
of the Pierre Auger Observatory, located in Malarg¨ue, Argentina (35.2◦S, 69.5◦W) at 1400 m
a.s.l. [9]. The Pierre Auger Observatory is designed to study cosmic rays (CRs) with energies
above ∼1018 eV. The surface detector array consists of 1660 water Cherenkov detectors sensitive
to the photons and the charged particles of the showers. It is laid out over an area of 3000 km2
on a triangular grid and is overlooked by four fluorescence detectors. The energy at which the
detection efficiency of the surface detector array saturates is ∼3 EeV [10]. For each event, the
signals recorded in the stations are fitted to find the signal at 1000 m from the shower core, S(1000),
used as a measure of the shower size. The shower size S(1000) is converted to the value S38 that
would have been expected had the shower arrived at a zenith angle of 38◦.S38 is then converted
into energy using a calibration curve based on the fluorescence telescope measurements [11].
The influence of the geomagnetic field on the spatial distribution of particles for showers with
zenith angle less than 60◦is presented in Section 2, through a toy model aimed at explaining
the directional dependence of the shower size S(1000) induced by the geomagnetic field. The
observation of this effect in the data of the Pierre Auger Observatory is reported in Section 3.
In Section 4, we quantify the size of the S(1000) distortions with zenith and azimuthal angles by
means of end-to-end shower simulations, and then present the procedure to convert the shower
size corrected for the geomagnetic effects into energy using the Constant Intensity Cut method. In
Section 5, the consequences on large scale anisotropies are discussed, while systematic uncertainties
associated with the primary mass, the primary energy and the number of muons in showers are
presented in Section 6.
6

u
Figure 1: The shower front plane coordinate system [2, 4]: ezis anti-parallel to the shower direction u,
while eyis parallel to BT, the projection of the magnetic field Bonto the shower plane x-y. (ψ, r ) are the
polar coordinates in the shower plane.
2 Influence of the geomagnetic field on extensive air showers
The interaction of a primary cosmic ray in the atmosphere produces mostly charged and neutral
pions, initiating a hadronic cascade. The decay of neutral pions generates the electromagnetic
component of the shower, while the decay of the charged pions generates the muonic one. Elec-
trons undergo stronger scattering, so that the electron distribution is only weakly affected by the
geomagnetic deflections. Muons are produced with a typical energy Eµof a few GeV (increasing
with the altitude of production). The decay angle between pions and muons is causing only a
small additional random deflection, as they almost inherit the transverse momentum pTof their
parents (a few hundred MeV/c) so that the distance of the muons from the shower core scales as
the inverse of their energy. While the radial offset of the pions from the shower axis is of the order
of a few 10 m, it does not contribute significantly to the lateral distribution of the muons observed
on the ground at distances r≥100 m. Hence, at ground level, the angular spread of the muons
around the shower axis can be considered as mainly caused by the transverse momentum inherited
from the parental pions.
After their production, muons are affected by ionisation and radiative energy losses, decay,
multiple scattering and geomagnetic deflections. Below 100 GeV, the muon energy loss is mainly
due to ionisation and is relatively small (amounting to about 2 MeV g−1cm2), allowing a large
fraction of muons to reach the ground before decaying. Multiple scattering in the electric field of
air nuclei randomises the directions of muons to some degree, but the contribution to the total
angular divergence of the muons from the shower axis remains small up to zenith angles of the
shower-axis of about 80◦.
Based on these general considerations, we now introduce a simple toy model aimed at un-
derstanding the main features of the muon density distortions induced by the geomagnetic field.
We adopt the shower front plane coordinate system depicted in Fig. 1 [2]. In the absence of
the magnetic field, and neglecting multiple scattering, a relativistic muon of energy Eµ≃cpµand
transverse momentum pTwill reach the shower front plane after traveling a distance dat a position
rfrom the shower axis given by
r≃pT
pµ
d≃cpT
Eµ
d. (1)
On the other hand, in the presence of the magnetic field, muons suffer additional geomagnetic
7

Distance to Shower Axis [m]
0 500 1000 1500 2000
x [m]δMagnetic Deviation
0
50
100
150
200 True deviations
Estimated deviations
Figure 2: Magnetic deviations as a function of the distance to the shower axis observed on a simulated
vertical shower (points). Superimposed are the deviations expected from Eq. (3) (line). The shaded region
and the error bars give the corresponding dispersion.
deflections. We treat the geomagnetic field Bin Malarg¨ue as a constant field1,
B= 24.6µT, DB= 2.6◦, IB=−35.2◦,(2)
DBand IBbeing the geomagnetic declination and inclination. The deflection of a relativistic muon
in the presence of a magnetic field with transverse component BTcan be approximated with
δx±≃ ±ecBTd2
2Eµ
,(3)
where eis the elementary electric charge and the sign corresponds to positive/negative charged
muons. The dependence of the geomagnetic deflections δx ≡δx+=−δx−on the distance to the
shower axis r=px2+y2is illustrated in Fig. 2 obtained by comparing the position of the same
muons in the presence or in the absence of the geomagnetic field in a simulated vertical shower of a
proton at 5 EeV. The deviations expected from the expression for δx±are also shown in the same
graph (solid line). It was obtained by inserting muon energy and distance at the production point
of the simulated muons into Eq. (3). It turns out that Eq. (3) estimates rather well the actual
deviations, though the distance between the actual and the predicted deviations increases at large
r. This is mainly because on the one hand dunderestimates the actual travel length to a larger
extent at larger r, while on the other hand the magnetic deviation actually increases while muons
gradually lose energy during travel. Hence, from the muon density ρµ(x, y) in the transverse plane
in the absence of the geomagnetic field, the corresponding density ρµ(x, y) in the presence of such
a field can be obtained by making the following Jacobian transformation, in the same way as in
the framework of very inclined showers [2],
ρµ(x, y) = 
∂(x, y)
∂(x, y)ρµ(x(x, y), y(x, y )).(4)
Here, the term “muon density” refers to the time-integrated muon flux through the transverse
shower front plane associated to the air shower, and the barred coordinates represent the positions
of the muons in the transverse plane in the presence of the geomagnetic field:
x=x+δx±(x, y),
y=y. (5)
1In Malarg¨ue the geomagnetic field has varied by about 1◦in direction and 2% in magnitude over 10 years [12].
8

x [km]
-1500 -1000 -500 0 500 1000 1500
y [km]
-1000
-500
0
500
1000
-4
-2
0
2
4
6
8
[in %]
µ
ρ/
µ
ρ∆
Figure 3: Relative changes of ∆ρµ/ρµin the transverse shower front plane due to the presence of the
geomagnetic field, obtained at zenith angle θ= 60◦and azimuthal angle aligned along DB+ 180◦.
Since Eq. (4) induces changes of the shower size S(1000), it is of particular interest to get an
approximate relationship between ρand ρaround 1000 m. From Fig. 2, it is apparent that around
1000 m the mean magnetic deviation is approximately constant over a distance range larger than
the size of the deviation. When focusing on the changes of density at 1000 m from the shower
core, it is thus reasonable to neglect the xand ydependence of the deviation δx±, which allows
an approximation of the density ρµ(x, y ) around 1000 m as
ρµ(x, y)≃ρµ+(x−δx+, y) + ρµ−(x−δx−, y)
≃ρµ(x, y) + (δx)2
2
∂2ρµ
∂x2(x, y),(6)
where we assumed ρµ−=ρµ+=ρµ/2. The two opposite muon charges cancel out the linear
term in δx and we see that magnetic effects change the muon density around 1000 m by a factor
proportional to (δx)2∝B2
T∝sin2(d
u,b), where uand b=B/|B|denote the unit vectors in the
shower direction and the magnetic field direction, respectively. This is particularly important with
regard to the azimuthal behaviour of the effect, as the azimuthal dependence is contained only in
the B2
T(θ, ϕ) term. This dependency is therefore a generic expectation outlined by this toy model.
The model will be verified in Section 4 by making use of complete simulation of showers. On
the other hand, the zenith angle dependence relies on other ingredients that we will probe in an
accurate way in Section 4, such as the altitude distribution of the muon production and the muon
energy distribution.
3 Observation of geomagnetic effects in the Pierre Auger
Observatory data
To illustrate the differences between ρµand ρµdescribed in Eq. (4), the relative changes ∆ρµ/ρµ
are shown in Fig. 3 in the transverse shower front plane by producing muon maps from simulations
at zenith angle θ= 60◦and azimuthal angle aligned along DB+ 180◦in the presence and in the
absence of the geomagnetic field. A predominant quadrupolar asymmetry at the few percent level
is visible, corresponding to the separation of positive and negative charges in the direction of the
Lorentz force.
This quadrupolar asymmetry is expected to induce to some extent a quadrupolar modulation
of the surface detector signals as a function of the polar angle on the ground, defined here as the
9

ground plane
r
u
B
Figure 4: Definition of angle Φ with respect to the magnetic East Emag and the shower core for a given
shower direction uand a surface detector at r. The azimuthal angle of the magnetic field vector Bdefines
the magnetic North Nmag .
]° Polar Angle on the Ground [
-150 -100 -50 0 50 100 150
exp
SSD
S
∑
1
1.02
1.04
1.06 Simulation without field
Simulation with field
]° Polar Angle on the Ground [
-150 -100 -50 0 50 100 150
exp
SSD
S
∑
1
1.02
1.04
1.06 Real data
Figure 5: Average ratio of the true signal in each surface detector with respect to the expected one as
a function of the polar angle on the ground. Left panel: using simulated showers in the presence (thick
points) and in the absence (thin points) of the geomagnetic field. Right panel: using real data above
4 EeV. The solid lines give the fit of a quadrupolar modulation to the corresponding points.
angle between the axis given by the shower core and the surface detector, and the magnetic East
ϕE
B=−DB=−2.6◦(Fig. 4). The use of this particular angle, instead of the polar angle ψwhich
is defined in the shower front plane (see Fig. 1), allows us to remove dipolar asymmetries in the
surface detector signals, the origin of which is related to the radial divergence of particles from
the shower axis. Such asymmetries cancel out in this analysis, due to the isotropic distribution
of the cosmic rays. To demonstrate the geomagnetic effect, we produced a realistic Monte-Carlo
simulation using 30 000 isotropically distributed showers (with zenith angles less than 60◦) with
random core positions within the array. The injected primary energies were chosen to be greater
than 4 EeV (safely excluding angle dependent trigger probability) and distributed according to a
power law energy spectrum dN/dE ∝E−γwith power index γ= 2.7, so that this shower library is
as close as possible to the real data set. To each shower we apply the reconstruction procedure of
the surface detector, leading to a fit of the lateral distribution function [11]. The lateral distribution
function parametrizes the signal strength in the shower plane, assuming circular shower symmetry.
By evaluating the lateral distribution function at the position of the surface detector, we obtain the
expected signal Sexp. This signal can be compared to the true signal in the surface detector SSD.
The ratio between the observed and expected signals as a function of the polar angle on the ground
in simulated showers is shown in the left panel of Fig. 5, with (thick points) and without (thin
10

points) the geomagnetic field. While a significant quadrupolar modulation with a fixed phase along
DBand amplitude ≃(1.1±0.2)% is observed when the field is on, no such modulation is observed
when the field is off (≃(0.1±0.2)%), as expected. In the right panel, the same analysis is performed
on the real data above 4 EeV, including again about 30 000 showers. A significant modulation of
≃(1.2±0.2)% is observed, agreeing both in amplitude and phase within the uncertainties with
the simulations performed in the presence of the geomagnetic field. This provides clear hints of
the influence of the geomagnetic field in the Auger data.
Note that this analysis is restricted to surface detectors that are more than 1000 m away from
the shower core. This cut is motivated by Fig. 3, showing that the quadrupolar amplitude is
larger at large distances from the shower core. We further require the surface detectors to have
signals larger than 4 VEM2. This cut is a compromise between keeping good statistics and keeping
trigger effects small. Above 4 VEM the measured amplitude does not depend systematically on
the signal strength cut. However a cut in the surface detector signals induces a statistical trigger
bias because showers with upward signal fluctuations will trigger more readily. This explains the
small discrepancy between real and Monte-Carlo data in terms of the global normalisation in Fig. 5
which differs from 1 by ∼3%. Cutting at larger signals reduces this discrepancy.
Most importantly, depending on the incoming direction, the quadrupolar asymmetry is also
expected to affect the shower size S(1000) and thus the energy estimator as qualitatively described
in Eq. (6). Consequently, these effects are expected to modulate the estimated cosmic ray event
rate at a given energy as a function of the incoming direction, and in particular to generate a
North/South asymmetry in the azimuthal distribution3. Such an asymmetry is also expected in
the case of a genuine large scale modulation of the flux of cosmic rays. However related analyses
of the azimuthal distribution are out of the scope of this paper, and we restrict ourselves in the
rest of this article to present a comprehensive study of the geomagnetic distortions of the energy
estimator. This will allow us to apply the corresponding corrections in a forthcoming publication
aimed at searching for large scale anisotropies.
4 Geomagnetic distortions of the energy estimator
4.1 Geomagnetic distortions of the shower size S(1000)
The toy model presented in Section 2 allows us to understand the main features of the influence
of the geomagnetic field on the muonic component of extensive air showers. To get an accurate
estimation of the distortions induced by the field on the shower size S(1000) as a function of both
the zenith and the azimuthal angles, we present here the results obtained by means of end-to-
end simulations of proton-initiated showers generated with the AIRES program [14] and with the
hadronic interaction model QGSJET [15]. We have checked that the results obtained with the
CORSIKA program [16] are compatible. We consider a fixed energy E= 5 EeV and seven fixed
zenith angles between θ= 0◦and θ= 60◦. The dependency of the effect in terms of the primary
mass and of the number of muons in showers as well as its evolution with energy are sources
of systematic uncertainties. The influence of such systematics will be quantified in Section 6.
Within our convention for the azimuthal angle, the azimuthal direction of the magnetic North is
ϕN
B= 90◦−DB= 87.4◦. The zenith direction of the field is θB= 90◦− |IB|= 54.8◦.
To verify the predicted behaviour of the shower size shift in terms of B2
T, we first show the
results of the simulations of 1000 showers at a zenith angle θ=θBand for two distinct azimuthal
angles ϕ=ϕN
Band ϕ=ϕN
B+90◦. Each shower is then thrown 10 times at the surface detector array
with random core positions and reconstructed using exactly the same reconstruction procedure as
2VEM - Vertical Equivalent Muon - is the average charge corresponding to the Cherenkov light produced by a
vertical and central through-going muon in the surface detector. It is the unit used in the evaluation of the signal
recorded by the detectors [13].
3The convention we use for the azimuthal angle ϕis to define it relative to the East direction, counterclockwise.
11

S(1000) [VEM]
0 5 10 15 20 25 30
0
200
400
600
800 0.04 VEM± No field, <S(1000)>=13.63 0.04 VEM± Real field, <S(1000)>=13.62 0.04 VEM±2x real field, <S(1000)>=13.65
S(1000) [VEM]
0 5 10 15 20 25 30
0
200
400
600
800 0.04 VEM± No field, <S(1000)>=13.60 0.04 VEM± Real field, <S(1000)>=13.82 0.04 VEM±2x real field, <S(1000)>=14.40
Figure 6: Distributions of shower size S(1000) obtained by simulating showers at zenith angle θ=θBand
azimuthal angle ϕN
B(left) and ϕN
B+ 90◦(right). Thick histogram: no magnetic field. Dotted histogram:
real magnetic field in Malarg¨ue. Dashed histogram: twice the real magnetic field in Malarg¨ue.
]° [ϕ Azimuth Angle
0 50 100 150 200 250 300 350
[%] S(1000)/S(1000) ∆
0
2
4
6
8
10 Real field
2x Real field
Figure 7: ∆S(1000)/S(1000) (in %) as a function of the azimuthal angle ϕ, at zenith angle θ=θBfor
two different field strengths. Points are obtained by Monte Carlo shower simulation, lines are the expected
behavior (see Section 2).
the one applied to real data. For this specific zenith angle θB, no shift is expected in the North
direction ϕN
Bas the transverse component of the magnetic field is zero. This is indeed the case as
illustrated in the left panel of Fig. 6, showing the distribution of reconstructed S(1000) for three
different configurations of the magnetic field: no field, real field in Malarg¨ue, and twice the real field
in Malarg¨ue. It can be seen that on average all histograms are – within the statistical uncertainties
on the average – centered on the same value. In the right panel of Fig. 6 we repeat the same
analysis with the showers generated in the direction ϕN
B+ 90◦. Since the transverse component of
the field is now different from zero, a clear relative shift in terms of ∆S(1000)/S(1000) is observed
between the three distributions: the shift is ≃1.6% between the configurations with and without
the field, leading to a discrimination with a significance of ≃5.5σ, while the shift is ≃6% between
the configurations with twice the real field and without the field leading to a discrimination with a
significance of ≃20 σ. It can be noticed that the strength of the shift is thus in overall agreement
with the expected scaling B2
T.
For the zenith angle θ=θB, in Fig. 7 we show the shift of the mean S(1000) obtained by
simulating 1000 showers in the same way as previously for eight different values of the azimuth
12

]° [θ Zenith Angle
0 10 20 30 40 50 60 70
) [%]θ G(
0
2
4
6
Figure 8: G(θ) = ∆S(1000)/S(1000)/sin2(
d
u,b) as a function of the zenith angle θ.
angle. Again, the results are displayed for configurations with the real field (bottom) and with twice
the real field (top). The expected behaviours in terms of ∆S(1000)/S(1000) = G(θB) sin2(d
u,b)
are shown by the continuous curves, where the normalisation factor Gis tuned by hand. Clearly,
the shape of the curves agrees remarkably well with the Monte Carlo data within the uncertainties.
Hence, this study supports the claim that the azimuthal dependence of the shift in S(1000) induced
by the magnetic field is proportional to B2
T(θ, ϕ), in agreement with the expectations provided by
general considerations expressed in the previous section on the muonic component of the showers.
The B2
Tterm encompassing the overall azimuthal dependence at each zenith angle, the remain-
ing shift G(θ) = ∆S(1000)/S(1000)/sin2(d
u,b) depends on the zenith angle through the altitude
distribution of the muon production, the muon energy distribution, and the weight of the muonic
contribution to the shower size S(1000). Repeating the simulations at different zenith angles, we
plot Gas a function of the zenith angle in Fig. 8. Due to the increased travel lengths of the muons
and due to their larger relative contribution to S(1000) at high zenith angles, the value of Grises
rapidly for angles above ≃40◦. The superimposed curve is an empirical fit, allowing us to get the
following parametrisation of the shower size distortions induced by the geomagnetic field,
∆S(1000)
S(1000) (θ, ϕ) = 4.2·10−3cos−2.8θsin2(d
u,b).(7)
4.2 From shower size to energy
At the Pierre Auger Observatory, the shower size S(1000) is converted into energy Eusing a two-
step procedure [11]. First, the evolution of S(1000) with zenith angle arising from the attenuation
of the shower with increasing atmospheric thickness is quantified by applying the Constant Inten-
sity Cut (CIC) method that is based on the (at least approximate) isotropy of incoming cosmic
rays. The CIC relates relates S(1000) in vertical and inclined showers through a line of equal
intensity in spectra at different zenith angles. This allows us to correct the value of S(1000) for
attenuation by computing its value had the shower arrived from a fixed zenith angle, here 38 de-
grees (corresponding to the median of the angular distribution of events for energies greater than
3 EeV). This zenith angle independent estimator S38 is defined as S38 =S(1000)/CIC(θ). The
calibration of S38 with energy Eis then achieved using a relation of the form E=ASB
38, where
A= 1.49 ±0.06(stat)±0.12(syst) and B= 1.08 ±0.01(stat)±0.04(syst) were estimated from the
correlation between S38 and Ein a subset of high quality ”hybrid events” measured simultaneously
by the surface detector (SD) and the fluorescence detector (FD) [11]. In such a sample, S38 and E
13

]° [δ Declination
-80 -60 -40 -20 0 20
[%]N/N∆
-4
-2
0
2
°=60
max
θ°=50
max
θ
Figure 9: Relative differences ∆N/N as a function of the declination, for 2 different values of θmax.
are independently measured, with S38 from the SD and Efrom the FD.
This two-step procedure has an important consequence on the implementation of the energy
corrections for the geomagnetic effects. The CIC curve is constructed assuming that the shower
size estimator S(1000) does not depend on the azimuthal angle. The induced azimuthal variation
of S(1000) due to the geomagnetic effect is thus averaged while the zenith angle dependence of the
geomagnetic effects is absorbed when the CIC is implemented. To illustrate this in a simplified
way, consider the case in which the magnetic field were directed along the zenith direction (i.e. in
the case of a virtual Observatory located at the Southern magnetic pole) so that the transverse
component of the magnetic field would not depend on the azimuthal direction of any incoming
shower. Then the shift in S(1000) would depend only on the zenith angle in such a way that the
Constant Intensity Cut method would by construction absorb the shift induced by G(θ) into the
empirical CIC(θ) curve, while the empirical relationship E=ASB
38 would calibrate S38 into energy
with no need for any additional corrections.
This leads us to implement the energy corrections for geomagnetic effects, relating the energy
E0reconstructed ignoring the geomagnetic effects to the corrected energy Eby
E=E0
(1 + ∆(θ, ϕ))B,(8)
with
∆(θ, ϕ) = G(θ)sin2(d
u,b)−Dsin2(d
u,b)Eϕ(9)
where h·iϕdenotes the average over ϕand where Bis one of the parameters used in the S38 to
Econversion described above. This expression implies that energies are under-estimated preferen-
tially for showers coming from the northern directions of the array, while they are over-estimated
for showers coming from the southern directions, the size of the effect increasing with the zenith
angle.
5 Consequences for large scale anisotropy searches
5.1 Impact on the estimated event rate
To provide an illustration of the impact of the energy corrections for geomagnetic effects, we
calculate here, as a function of declination δ, the deviation of the event rate N0(δ), measured if we
14

were not to implement the corrections of the energy estimator by Eq. (8), to the event rate N(δ)
expected from an isotropic background distribution.
The “canonical exposure” [17] holds for a full-time operation of the surface detector array above
the energy at which the detection efficiency is saturated over the considered zenith range. In such
a case, the directional detection efficiency is simply proportional to cos θ,
ω(θ)∝cos(θ)H(θ−θmax) (10)
where His the Heaviside function and θmax is the maximal zenith angle considered. The zenith
angle is related to the declination δand the right ascension αthrough
cos θ= sin ℓsite sin δ+ cos ℓsite cos δcos α(11)
where ℓsite is the Earth’s latitude of the Observatory. The event rate at a given declination δand
above an energy threshold Eth is obtained by integrating in energy and right ascension α,
N(δ)∝Z∞
Eth
dEZ2π
0
dα ω(θ)dN(θ, ϕ, E )
dE(12)
Note that at lower energies this integral acquires an additional energy and angle dependent detec-
tion efficiency term ǫ(E, θ, φ). Hereafter we assume that the cosmic ray spectrum is a power law,
i.e. dN/dE ∝E−γ. From Eq. (8) it follows that if the effect of the geomagnetic field were not
accounted for, the measured energy spectrum would have a directional modulation given by
dN
dE0∝[1 + ∆(θ, ϕ)]B(γ−1) E−γ
0.(13)
This leads to the following measured event rate above a given uncorrected energy Eth ,
N0(δ)∝Z∞
Eth
dE0Z2π
0
dα H(cos θ−cos θmax) cos θ[1 + ∆(θ, ϕ)]B(γ−1) E−γ
0,(14)
where ϕis related to αand δthrough
tan ϕ=sin δcos ℓsite −cos δcos αsin ℓsite
cos δsin α.(15)
The event rate N0(δ) as a function of declination is then calculated using Eq. (13) in Eq. (12).
The relative difference ∆N/N is shown in Fig. 9 as a function of the declination, with spectral
index γ= 2.7. The energy over-estimation (under-estimation) of events coming preferentially from
the Southern (Northern) azimuthal directions, as described in Eq. (8), leads to an effective excess
(deficit) of the event rate for δ.−20◦(δ&−20◦), with an amplitude of ≃2% when considering
θmax = 60◦. It is worth noting that this amplitude is reduced to within 1% when considering
θmax = 50◦, as shown by the dotted line.
5.2 Impact on dipolar modulation searches
The pattern displayed in Fig. 9 roughly imitates a dipole with an amplitude at the percent level.
To evaluate precisely the impact of this pattern on the assessment of a dipole moment in the
reconstructed arrival directions and to probe the statistics needed for the sensitivity to such a
spurious pattern, we apply the multipolar reconstruction adapted to the case of a partial sky
coverage [18] to mock data sets by limiting the maximum bound of the expansion Lmax to 1 (pure
dipolar reconstruction). Since the distortions are axisymmetric around the axis defined by the
North and South celestial poles, only the multipolar coefficient related to this particular axis is
15

Amplitude
0 0.01 0.02 0.03 0.04 0.05 0.06
0
50
100
150
200 N=300,000
N=32,000
]°Declination [
-80 -60 -40 -20 0 20 40 60 80
0
20
40
60
80
100 N=32,000
Figure 10: Dipolar reconstruction of arrival directions of mock data sets with event rates distorted by the
geomagnetic effects. Left: distributions of amplitudes. Right: distributions of declinations. The smooth
lines give the expected distribution in the case of isotropy.
expected to be affected (here: a10 ). Consequently, this particular coefficient has impacts on both
the amplitude of the reconstructed dipole and its direction with respect to the axis defined by the
North and South celestial poles (the technical details of relating the estimation of the multipolar
coefficients to the spherical coordinates of a dipole are given in the Appendix).
To simulate the directional distortions induced by Eq. (8), each mock data set is drawn from
the event rate N0(δ) corresponding to the uncorrected energies, and is reconstructed using the
canonical exposure in Eq. (10). The results of this procedure applied to 1000 samples are shown
in Fig. 10. In the left panel, the distribution of the reconstructed amplitudes rusing N= 300 000
events is shown by the dotted histogram. It clearly deviates from the expected isotropic distribution
displayed as the dotted curve which corresponds to (see Appendix)
pR(r) = r
σpσ2
z−σ2erfipσ2
z−σ2
σσz
r
√2exp −r2
2σ2,(16)
where erfi(z) = erf(iz)/i, and where the width parameters σand σzcan be calculated from
the exposure function [18]. With the particular exposure function used here, it turns out that
σ≃1.02p3/N and σz≃1.59p3/N . This allows us to estimate the spurious dipolar amplitude4
to be of the order of the mean of the dotted histogram, about ≃1.9%. Consequently, we can
estimate that the spurious effect becomes predominant as soon as the mean noise amplitude hri
deduced from Eq. (16) is of the order of 1.9%,
hri=r2
πσz+σ2arctanh(p1−σ2/σ2
z)
pσ2
z−σ2≃1.9%.(17)
This translates into the condition N≃32 000 (solid histogram). Using such a number of events,
the bias induced on the amplitude reconstruction is illustrated in the same graph by the longer
tail of the full histogram with respect to the expected one, and is even more evident in the right
panel of Fig. 10, showing the distribution of the reconstructed declination direction of the dipole
which already deviates to a large extent from the expected distribution.
4Due to the partial sky exposure considered here, the estimate of the dipolar amplitude is biased by the higher
multipolar orders needed to fully describe ∆N/N shown in Fig. 10 [18]. The aim of this calculation is only to
provide a quantitative illustration of the spurious measurement which would be performed due to the geomagnetic
effects when reconstructing a pure dipolar pattern.
16

]° [δ Declination
-80 -60 -40 -20 0 20
N/N [%]∆
-4
-2
0
2
protons, QGSJET01, 5 EeV
iron, QGSJET01, 5 EeV
protons, QGSJET01, 50 EeV
protons, QGSJETII, 5EeV
µ
protons, QGSJET01, 5 EeV, 2xN
Figure 11: Relative differences ∆N/N as a function of the declination, for different primary masses,
different primary energies, different hadronic models and for increased number of muons in showers.
6 Systematic uncertainties
The parametrisation of G(θ) in Eq. (7) was obtained by means of simulations of proton showers
at a fixed energy. The height of the first interaction influences the production altitude of muons
detected at 1000 m from the shower core at the ground level. Moreover, as muons are produced at
the end of the hadronic cascade, when the energy of the charged mesons is diminished so much that
their decay length becomes smaller than their interaction length (which is inversely proportional
to the air density), the energy distribution of muons is also affected by the height of the first
interaction. Because the air density is lower in the upper atmosphere, this mechanism results in
an increase of the energy of muons. The muonic contribution to S(1000) depends also on both the
primary mass and primary energy. For all these reasons, the parametrisation of G(θ) is expected
to depend on both the primary mass and primary energy.
To probe these influences, we repeat the same chain of end-to-end simulations using proton
showers at energies of 50 EeV and iron showers at 5 EeV. Results in terms of the distortions of
the observed event rate N(δ) are shown in Fig. 11. We also display in the same graph the results
obtained using the hadronic interaction model QGSJETII [19]. The differences with respect to the
reference model are small, so that the consequences on large scale anisotropy searches presented
in Section 5 remain unchanged within the statistics available at the Pierre Auger Observatory.
In addition, there are discrepancies in the hadronic interaction model predictions regarding
the number of muons in shower simulations and what is found in our data [20]. Higher number of
muons influences the weight of the muonic contribution to S(1000). The consequences of increasing
the number of muons by a factor of 2 on the distortions of the observed event rate are also shown
in Fig. 11. As the muonic contribution to S(1000) is already large at high zenith angles in the
reference model, this increase of the number of muons does not lead to large differences.
7 Conclusion
In this work, we have identified and quantified a systematic uncertainty affecting the energy deter-
mination of cosmic rays detected by the surface detector array of the Pierre Auger Observatory.
This systematic uncertainty, induced by the influence of the geomagnetic field on the shower devel-
opment, has a strength which depends on both the zenith and the azimuthal angles. Consequently,
we have shown that it induces distortions of the estimated cosmic ray event rate at a given energy
17

at the percent level in both the azimuthal and the declination distributions, the latter of which
mimics an almost dipolar pattern.
We have also shown that the induced distortions are already at the level of the statistical
uncertainties for a number of events N≃32 000 (we note that the full Auger surface detector
array collects about 6500 events per year with energies above 3 EeV). Accounting for these effects
is thus essential with regard to the correct interpretation of large scale anisotropy measurements
taking explicitly profit from the declination distribution.
Acknowledgements
The successful installation, commissioning, and operation of the Pierre Auger Observatory would
not have been possible without the strong commitment and effort from the technical and admin-
istrative staff in Malarg¨ue.
We are very grateful to the following agencies and organizations for financial support: Comisi´on
Nacional de Energ´ıa At´omica, Fundaci´on Antorchas, Gobierno De La Provincia de Mendoza, Mu-
nicipalidad de Malarg¨ue, NDM Holdings and Valle Las Le˜nas, in gratitude for their continuing
cooperation over land access, Argentina; the Australian Research Council; Conselho Nacional de
Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), Financiadora de Estudos e Projetos (FINEP),
Funda¸c˜ao de Amparo `a Pesquisa do Estado de Rio de Janeiro (FAPERJ), Funda¸c˜ao de Am-
paro `a Pesquisa do Estado de S˜ao Paulo (FAPESP), Minist´erio de Ciˆencia e Tecnologia (MCT),
Brazil; AVCR AV0Z10100502 and AV0Z10100522, GAAV KJB100100904, MSMT-CR LA08016,
LC527, 1M06002, and MSM0021620859, Czech Republic; Centre de Calcul IN2P3/CNRS, Cen-
tre National de la Recherche Scientifique (CNRS), Conseil R´egional Ile-de-France, D´epartement
Physique Nucl´eaire et Corpusculaire (PNC-IN2P3/CNRS), D´epartement Sciences de l’Univers
(SDU-INSU/CNRS), France; Bundesministerium f¨ur Bildung und Forschung (BMBF), Deutsche
Forschungsgemeinschaft (DFG), Finanzministerium Baden-W¨urttemberg, Helmholtz-Gemeinschaft
Deutscher Forschungszentren (HGF), Ministerium f¨ur Wissenschaft und Forschung, Nordrhein-
Westfalen, Ministerium f¨ur Wissenschaft, Forschung und Kunst, Baden-W¨urttemberg, Germany;
Istituto Nazionale di Fisica Nucleare (INFN), Ministero dell’Istruzione, dell’Universit`a e della
Ricerca (MIUR), Italy; Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT), Mexico; Min-
isterie van Onderwijs, Cultuur en Wetenschap, Nederlandse Organisatie voor Wetenschappelijk
Onderzoek (NWO), Stichting voor Fundamenteel Onderzoek der Materie (FOM), Netherlands;
Ministry of Science and Higher Education, Grant Nos. N N202 200239 and N N202 207238,
Poland; Funda¸c˜ao para a Ciˆencia e a Tecnologia, Portugal; Ministry for Higher Education, Sci-
ence, and Technology, Slovenian Research Agency, Slovenia; Comunidad de Madrid, Consejer´ıa de
Educaci´on de la Comunidad de Castilla La Mancha, FEDER funds, Ministerio de Ciencia e In-
novaci´on and Consolider-Ingenio 2010 (CPAN), Xunta de Galicia, Spain; Science and Technology
Facilities Council, United Kingdom; Department of Energy, Contract Nos. DE-AC02-07CH11359,
DE-FR02-04ER41300, National Science Foundation, Grant No. 0450696, The Grainger Founda-
tion USA; ALFA-EC / HELEN, European Union 6th Framework Program, Grant No. MEIF-CT-
2005-025057, European Union 7th Framework Program, Grant No. PIEF-GA-2008-220240, and
UNESCO.
Appendix
The p.d.f. of the first harmonic amplitude for a data set of Npoints drawn at random over a
circle is known to be the Rayleigh distribution. In this appendix, we generalise this distribution
to the case of Npoints being drawn at random on the sphere over the exposure ω(δ) of the
Pierre Auger Observatory. Assuming the underlying arrival direction distribution to be of the
18

form Φ(α, δ) = Φ0(1 + D·u), the components of the dipolar vector Dare related to the multipolar
coefficients through
Dx=√3a11
a00
, Dy=√3a1−1
a00
, Dz=√3a10
a00
.(18)
Denoting by x, y, z the estimates of Dx, Dy, Dz, the joint p.d.f. pX,Y, Z (x, y, z) can be factorised in
the limit of large number of events in terms of three centered Gaussian distributions N(0, σ),
pX,Y,Z (x, y , z) = pX(x)pY(y)pZ(z) = N(0, σx)N(0, σy)N(0, σz),(19)
where the standard deviation parameters can be calculated from the exposure function [18]. With
the particular exposure function used here, it turns out that numerical integrations lead to σ≃
1.02p3/N and σz≃1.59p3/N . The joint p.d.f. pR,∆,A(r, δ, α) expressing the dipole components
in spherical coordinates is obtained from Eq. (19) by performing the Jacobian transformation
pR,∆,A(r, δ, α) = 
∂(x, y, z)
∂(r, δ, α)pX,Y ,Z (x(r, δ, α), y(r, δ, α), z(r, δ, α))
=r2cos δ
(2π)3/2σ2σz
exp −r2cos2δ
2σ2−r2sin2δ
2σ2
z.(20)
From this joint p.d.f., the p.d.f. of the dipole amplitude (declination) is finally obtained by
marginalising over the other variables, yielding
pR(r) = r
σpσ2
z−σ2erfipσ2
z−σ2
σσz
r
√2exp −r2
2σ2,
p∆(δ) = σσ2
z
2
cos δ
(σ2
zcos2δ+σ2sin2δ)3/2.(21)
Finally, one can derive from pRquantities of interest, such as the expected mean noise hri, the
RMS σrand the probability of obtaining an amplitude greater than r:
hri=r2
πσz+σ2arctanh(p1−σ2/σ2
z)
pσ2
z−σ2,(22)
σr=q2σ2+σ2
z− hri2,(23)
Prob(> r) = erfcr
√2σz+σ
pσ2
z−σ2erfipσ2
z−σ2
√2σσz
rexp−r2
2σ2,(24)
which are the equivalent to the well known Rayleigh formulas hri=pπ/N , σr=p(4 −π)/N and
Prob(> r) = exp(−N r2/4) when dealing with Npoints drawn at random over a circle [21].
Acknowledgments
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