Atmospheric effects on extensive air showers observed with the Surface Detector of the Pierre Auger Observatory
Abstract
Atmospheric parameters, such as pressure (), temperature () and density (), affect the development of extensive air showers initiated by energetic cosmic rays. We have studied the impact of atmospheric variations on extensive air showers by means of the surface detector of the Pierre Auger Observatory. The rate of events shows a seasonal modulation and diurnal one. We find that the observed behaviour is explained by a model including the effects associated with the variations of and . The former affects the longitudinal development of air showers while the latter influences the Molière radius and hence the lateral distribution of the shower particles. The model is validated with full simulations of extensive air showers using atmospheric profiles measured at the site of the Pierre Auger Observatory.
keywords:
extensive air showers, UHECR, atmosphere, weatherPacs:
96.50.sd, 96.50.sb, 96.50.sfThe Pierre Auger Collaboration
J. Abraham,
P. Abreu,
M. Aglietta,
C. Aguirre,
E.J. Ahn,
D. Allard,
I. Allekotte,
J. Allen,
P. Allison,
J. AlvarezMuñiz,
M. Ambrosio,
L. Anchordoqui,
S. Andringa,
A. Anzalone,
C. Aramo,
E. Arganda,
S. Argirò,
K. Arisaka,
F. Arneodo,
F. Arqueros,
T. Asch,
H. Asorey,
P. Assis,
J. Aublin,
M. Ave,
G. Avila,
T. Bäcker,
D. Badagnani,
K.B. Barber,
A.F. Barbosa,
S.L.C. Barroso,
B. Baughman,
P. Bauleo,
J.J. Beatty,
T. Beau,
B.R. Becker,
K.H. Becker,
A. Bellétoile,
J.A. Bellido,
S. BenZvi,
C. Berat,
P. Bernardini,
X. Bertou,
P.L. Biermann,
P. Billoir,
O. BlanchBigas,
F. Blanco,
C. Bleve,
H. Blümer,
M. Boháčová,
C. Bonifazi,
R. Bonino,
N. Borodai,
J. Brack,
P. Brogueira,
W.C. Brown,
R. Bruijn,
P. Buchholz,
A. Bueno,
R.E. Burton,
N.G. Busca,
K.S. CaballeroMora,
L. Caramete,
R. Caruso,
W. Carvalho,
A. Castellina,
O. Catalano,
L. Cazon,
R. Cester,
J. Chauvin,
A. Chiavassa,
J.A. Chinellato,
A. Chou,
J. Chudoba,
J. Chye,
R.W. Clay,
E. Colombo,
R. Conceição,
B. Connolly,
F. Contreras,
J. Coppens,
A. Cordier,
U. Cotti,
S. Coutu,
C.E. Covault,
A. Creusot,
A. Criss,
J. Cronin,
A. Curutiu,
S. DagoretCampagne,
R. Dallier,
K. Daumiller,
B.R. Dawson,
R.M. de Almeida,
M. De Domenico,
C. De Donato,
S.J. de Jong,
G. De La Vega,
W.J.M. de Mello Junior,
J.R.T. de Mello Neto,
I. De Mitri,
V. de Souza,
K.D. de Vries,
G. Decerprit,
L. del Peral,
O. Deligny,
A. Della Selva,
C. Delle Fratte,
H. Dembinski,
C. Di Giulio,
J.C. Diaz,
P.N. Diep,
C. Dobrigkeit ,
J.C. D’Olivo,
P.N. Dong,
D. Dornic,
A. Dorofeev,
J.C. dos Anjos,
M.T. Dova,
D. D’Urso,
I. Dutan,
M.A. DuVernois,
R. Engel,
M. Erdmann,
C.O. Escobar,
A. Etchegoyen,
P. Facal San Luis,
H. Falcke,
G. Farrar,
A.C. Fauth,
N. Fazzini,
F. Ferrer,
A. Ferrero,
B. Fick,
A. Filevich,
A. Filipčič,
I. Fleck,
S. Fliescher,
C.E. Fracchiolla,
E.D. Fraenkel,
W. Fulgione,
R.F. Gamarra,
S. Gambetta,
B. García,
D. García Gámez,
D. GarciaPinto,
X. Garrido,
G. Gelmini,
H. Gemmeke,
P.L. Ghia,
U. Giaccari,
M. Giller,
H. Glass,
L.M. Goggin,
M.S. Gold,
G. Golup,
F. Gomez Albarracin,
M. Gómez Berisso,
P. Gonçalves,
M. Gonçalves do Amaral,
D. Gonzalez,
J.G. Gonzalez,
D. Góra,
A. Gorgi,
P. Gouffon,
E. Grashorn,
S. Grebe,
M. Grigat,
A.F. Grillo,
Y. Guardincerri,
F. Guarino,
G.P. Guedes,
J. Gutiérrez,
J.D. Hague,
V. Halenka,
P. Hansen,
D. Harari,
S. Harmsma,
J.L. Harton,
A. Haungs,
M.D. Healy,
T. Hebbeker,
G. Hebrero,
D. Heck,
C. Hojvat,
V.C. Holmes,
P. Homola,
J.R. Hörandel,
A. Horneffer,
M. Hrabovský,
T. Huege,
M. Hussain,
M. Iarlori,
A. Insolia,
F. Ionita,
A. Italiano,
S. Jiraskova,
M. Kaducak,
K.H. Kampert,
T. Karova,
P. Kasper,
B. Kégl,
B. Keilhauer,
E. Kemp,
R.M. Kieckhafer,
H.O. Klages,
M. Kleifges,
J. Kleinfeller,
R. Knapik,
J. Knapp,
D.H. Koang,
A. Krieger,
O. Krömer,
D. KruppkeHansen,
D. Kuempel,
N. Kunka,
A. Kusenko,
G. La Rosa,
C. Lachaud,
B.L. Lago,
P. Lautridou,
M.S.A.B. Leão,
D. Lebrun,
P. Lebrun,
J. Lee,
M.A. Leigui de Oliveira,
A. Lemiere,
A. LetessierSelvon,
M. Leuthold,
I. LhenryYvon,
R. López,
A. Lopez Agüera,
K. Louedec,
J. Lozano Bahilo,
A. Lucero,
R. Luna García,
H. Lyberis,
M.C. Maccarone,
C. Macolino,
S. Maldera,
D. Mandat,
P. Mantsch,
A.G. Mariazzi,
I.C. Maris,
H.R. Marquez Falcon,
D. Martello,
J. Martínez,
O. Martínez Bravo,
H.J. Mathes,
J. Matthews,
J.A.J. Matthews,
G. Matthiae,
D. Maurizio,
P.O. Mazur,
M. McEwen,
R.R. McNeil,
G. MedinaTanco,
M. Melissas,
D. Melo,
E. Menichetti,
A. Menshikov,
R. Meyhandan,
M.I. Micheletti,
G. Miele,
W. Miller,
L. Miramonti,
S. Mollerach,
M. Monasor,
D. Monnier Ragaigne,
F. Montanet,
B. Morales,
C. Morello,
J.C. Moreno,
C. Morris,
M. Mostafá,
C.A. Moura,
S. Mueller,
M.A. Muller,
R. Mussa,
G. Navarra,
J.L. Navarro,
S. Navas,
P. Necesal,
L. Nellen,
C. NewmanHolmes,
D. Newton,
P.T. Nhung,
N. Nierstenhoefer,
D. Nitz,
D. Nosek,
L. Nožka,
M. Nyklicek,
J. Oehlschläger,
A. Olinto,
P. Oliva,
V.M. OlmosGilbaja,
M. Ortiz,
F. Ortolani,
N. Pacheco,
D. Pakk SelmiDei,
M. Palatka,
J. Pallotta,
G. Parente,
E. Parizot,
S. Parlati,
S. Pastor,
M. Patel,
T. Paul,
V. Pavlidou,
K. Payet,
M. Pech,
J. Pȩkala,
R. Pelayo,
I.M. Pepe,
L. Perrone,
R. Pesce,
E. Petermann,
S. Petrera,
P. Petrinca,
A. Petrolini,
Y. Petrov,
J. Petrovic,
C. Pfendner,
R. Piegaia,
T. Pierog,
M. Pimenta,
T. Pinto,
V. Pirronello,
O. Pisanti,
M. Platino,
J. Pochon,
V.H. Ponce,
M. Pontz,
P. Privitera,
M. Prouza,
E.J. Quel,
J. Rautenberg,
O. Ravel,
D. Ravignani,
A. Redondo,
S. Reucroft,
B. Revenu,
F.A.S. Rezende,
J. Ridky,
S. Riggi,
M. Risse,
C. Rivière,
V. Rizi,
C. Robledo,
G. Rodriguez,
J. Rodriguez Martino,
J. Rodriguez Rojo,
I. RodriguezCabo,
M.D. RodríguezFrías,
G. Ros,
J. Rosado,
T. Rossler,
M. Roth,
B. Rouilléd’Orfeuil,
E. Roulet,
A.C. Rovero,
F. Salamida,
H. Salazar,
G. Salina,
F. Sánchez,
M. Santander,
C.E. Santo,
E.M. Santos,
F. Sarazin,
S. Sarkar,
R. Sato,
N. Scharf,
V. Scherini,
H. Schieler,
P. Schiffer,
A. Schmidt,
F. Schmidt,
T. Schmidt,
O. Scholten,
H. Schoorlemmer,
J. Schovancova,
P. Schovánek,
F. Schroeder,
S. Schulte,
F. Schüssler,
D. Schuster,
S.J. Sciutto,
M. Scuderi,
A. Segreto,
D. Semikoz,
M. Settimo,
R.C. Shellard,
I. Sidelnik,
B.B. Siffert,
A. Smiałkowski,
R. Šmída,
B.E. Smith,
G.R. Snow,
P. Sommers,
J. Sorokin,
H. Spinka,
R. Squartini,
E. Strazzeri,
A. Stutz,
F. Suarez,
T. Suomijärvi,
A.D. Supanitsky,
M.S. Sutherland,
J. Swain,
Z. Szadkowski,
A. Tamashiro,
A. Tamburro,
T. Tarutina,
O. Taşcău,
R. Tcaciuc,
D. Tcherniakhovski,
N.T. Thao,
D. Thomas,
R. Ticona,
J. Tiffenberg,
C. Timmermans,
W. Tkaczyk,
C.J. Todero Peixoto,
B. Tomé,
A. Tonachini,
I. Torres,
P. Travnicek,
D.B. Tridapalli,
G. Tristram,
E. Trovato,
V. Tuci,
M. Tueros,
R. Ulrich,
M. Unger,
M. Urban,
J.F. Valdés Galicia,
I. Valiño,
L. Valore,
A.M. van den Berg,
J.R. Vázquez,
R.A. Vázquez,
D. Veberič,
A. Velarde,
T. Venters,
V. Verzi,
M. Videla,
L. Villaseñor,
S. Vorobiov,
L. Voyvodic,
H. Wahlberg,
P. Wahrlich,
O. Wainberg,
D. Warner,
A.A. Watson,
S. Westerhoff,
B.J. Whelan,
G. Wieczorek,
L. Wiencke,
B. Wilczyńska,
H. Wilczyński,
C. Wileman,
M.G. Winnick,
H. Wu,
B. Wundheiler,
T. Yamamoto,
P. Younk,
G. Yuan,
E. Zas,
D. Zavrtanik,
M. Zavrtanik,
I. Zaw,
A. Zepeda,
M. Ziolkowski
Centro Atómico Bariloche and Instituto Balseiro (CNEA
UNCuyoCONICET), San Carlos de Bariloche, Argentina
Centro Atómico Constituyentes (Comisión Nacional de
Energía Atómica/CONICET/UTNFRBA), Buenos Aires, Argentina
Centro de Investigaciones en Láseres y Aplicaciones,
CITEFA and CONICET, Argentina
Departamento de Física, FCEyN, Universidad de Buenos
Aires y CONICET, Argentina
IFLP, Universidad Nacional de La Plata and CONICET, La
Plata, Argentina
Instituto de Astronomía y Física del Espacio (CONICET),
Buenos Aires, Argentina
Observatorio Meteorologico Parque Gral. San Martin (UTN
FRM/CONICET/CNEA), Mendoza, Argentina
Pierre Auger Southern Observatory, Malargüe, Argentina
Pierre Auger Southern Observatory and Comisión Nacional
de Energía Atómica, Malargüe, Argentina
University of Adelaide, Adelaide, S.A., Australia
Universidad Catolica de Bolivia, La Paz, Bolivia
Universidad Mayor de San Andrés, Bolivia
Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro,
RJ, Brazil
Pontifícia Universidade Católica, Rio de Janeiro, RJ,
Brazil
Universidade de São Paulo, Instituto de Física, São
Carlos, SP, Brazil
Universidade de São Paulo, Instituto de Física, São
Paulo, SP, Brazil
Universidade Estadual de Campinas, IFGW, Campinas, SP,
Brazil
Universidade Estadual de Feira de Santana, Brazil
Universidade Estadual do Sudoeste da Bahia, Vitoria da
Conquista, BA, Brazil
Universidade Federal da Bahia, Salvador, BA, Brazil
Universidade Federal do ABC, Santo André, SP, Brazil
Universidade Federal do Rio de Janeiro, Instituto de
Física, Rio de Janeiro, RJ, Brazil
Universidade Federal Fluminense, Instituto de Fisica,
Niterói, RJ, Brazil
Charles University, Faculty of Mathematics and Physics,
Institute of Particle and Nuclear Physics, Prague, Czech
Republic
Institute of Physics of the Academy of Sciences of the
Czech Republic, Prague, Czech Republic
Palacký University, Olomouc, Czech Republic
Institut de Physique Nucléaire d’Orsay (IPNO),
Université Paris 11, CNRSIN2P3, Orsay, France
Laboratoire AstroParticule et Cosmologie (APC),
Université Paris 7, CNRSIN2P3, Paris, France
Laboratoire de l’Accélérateur Linéaire (LAL),
Université Paris 11, CNRSIN2P3, Orsay, France
Laboratoire de Physique Nucléaire et de Hautes Energies
(LPNHE), Universités Paris 6 et Paris 7, CNRSIN2P3, Paris Cedex 05,
France
Laboratoire de Physique Subatomique et de Cosmologie
(LPSC), Université Joseph Fourier, INPG, CNRSIN2P3, Grenoble,
France
SUBATECH, CNRSIN2P3, Nantes, France
Bergische Universität Wuppertal, Wuppertal, Germany
Forschungszentrum Karlsruhe, Institut für Kernphysik,
Karlsruhe, Germany
Forschungszentrum Karlsruhe, Institut für
Prozessdatenverarbeitung und Elektronik, Germany
MaxPlanckInstitut für Radioastronomie, Bonn, Germany
RWTH Aachen University, III. Physikalisches Institut A,
Aachen, Germany
Universität Karlsruhe (TH), Institut für Experimentelle
Kernphysik (IEKP), Karlsruhe, Germany
Universität Siegen, Siegen, Germany
Dipartimento di Fisica dell’Università and INFN,
Genova, Italy
Università dell’Aquila and INFN, L’Aquila, Italy
Università di Milano and Sezione INFN, Milan, Italy
Dipartimento di Fisica dell’Università del Salento and
Sezione INFN, Lecce, Italy
Università di Napoli “Federico II” and Sezione INFN,
Napoli, Italy
Università di Roma II “Tor Vergata” and Sezione INFN,
Roma, Italy
Università di Catania and Sezione INFN, Catania, Italy
Università di Torino and Sezione INFN, Torino, Italy
Istituto di Astrofisica Spaziale e Fisica Cosmica di
Palermo (INAF), Palermo, Italy
Istituto di Fisica dello Spazio Interplanetario (INAF),
Università di Torino and Sezione INFN, Torino, Italy
INFN, Laboratori Nazionali del Gran Sasso, Assergi
(L’Aquila), Italy
Benemérita Universidad Autónoma de Puebla, Puebla,
Mexico
Centro de Investigación y de Estudios Avanzados del IPN
(CINVESTAV), México, D.F., Mexico
Instituto Nacional de Astrofisica, Optica y
Electronica, Tonantzintla, Puebla, Mexico
Instituto Politécnico Nacional, México, D.F., Mexico
Universidad Michoacana de San Nicolas de Hidalgo,
Morelia, Michoacan, Mexico
Universidad Nacional Autonoma de Mexico, Mexico, D.F.,
Mexico
IMAPP, Radboud University, Nijmegen, Netherlands
Kernfysisch Versneller Instituut, University of
Groningen, Groningen, Netherlands
NIKHEF, Amsterdam, Netherlands
ASTRON, Dwingeloo, Netherlands
Institute of Nuclear Physics PAN, Krakow, Poland
University of Łódź, Łódz, Poland
LIP and Instituto Superior Técnico, Lisboa, Portugal
J. Stefan Institute, Ljubljana, Slovenia
Laboratory for Astroparticle Physics, University of
Nova Gorica, Slovenia
Instituto de Física Corpuscular, CSICUniversitat de
València, Valencia, Spain
Universidad Complutense de Madrid, Madrid, Spain
Universidad de Alcalá, Alcalá de Henares (Madrid),
Spain
Universidad de Granada & C.A.F.P.E., Granada, Spain
Universidad de Santiago de Compostela, Spain
Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, Oxford, United Kingdom
School of Physics and Astronomy, University of Leeds,
United Kingdom
Argonne National Laboratory, Argonne, IL, USA
Case Western Reserve University, Cleveland, OH, USA
Colorado School of Mines, Golden, CO, USA
Colorado State University, Fort Collins, CO, USA
Colorado State University, Pueblo, CO, USA
Fermilab, Batavia, IL, USA
Louisiana State University, Baton Rouge, LA, USA
Michigan Technological University, Houghton, MI, USA
New York University, New York, NY, USA
Northeastern University, Boston, MA, USA
Ohio State University, Columbus, OH, USA
Pennsylvania State University, University Park, PA, USA
Southern University, Baton Rouge, LA, USA
University of California, Los Angeles, CA, USA
University of Chicago, Enrico Fermi Institute, Chicago,
IL, USA
University of Hawaii, Honolulu, HI, USA
University of Nebraska, Lincoln, NE, USA
University of New Mexico, Albuquerque, NM, USA
University of Pennsylvania, Philadelphia, PA, USA
University of Wisconsin, Madison, WI, USA
University of Wisconsin, Milwaukee, WI, USA
Institute for Nuclear Science and Technology (INST),
Hanoi, Vietnam
(‡) Deceased
(a) at Konan University, Kobe, Japan
(b) On leave of absence at the Instituto Nacional de Astrofisica, Optica y Electronica
(c) at Caltech, Pasadena, USA
1 Introduction
Highenergy cosmic rays (CRs) are measured by recording the extensive air showers (EAS) of secondary particles they produce in the atmosphere. As the atmosphere is the medium in which the shower evolves, its state affects the lateral and longitudinal development of the shower. Pressure () and air density () are the properties of the atmosphere that mostly affect the EAS. An increase (or decrease) of the ground corresponds to an increased (or decreased) amount of matter traversed by the shower particles; this affects the stage of the longitudinal development of the shower when it reaches the ground. A decrease (or increase) of increases (or decreases) the Molière radius and thus broadens (or narrows) the lateral extent of the EAS.
The properties of the primary CR, e.g., energy, mass and arrival direction, have to be inferred from EAS, which can be sampled by an array of detectors at ground level. Therefore the study and understanding of the effects of atmospheric variations on EAS in general, and on a specific detector in particular, is very important for the comprehension of the detector performances and for the correct interpretation of EAS measurements.
We have studied the atmospheric effects on EAS by means of the surface detector (SD) of the Pierre Auger Observatory, located in Malargüe, Argentina (35.2 S, 69.5W) at 1400 m a.s.l. [1]. The Pierre Auger Observatory is designed to study CRs from eV up to the highest energies. The SD consists of 1600 waterCherenkov detectors to detect the photons and the charged particles of the showers. It is laid out over 3000 km on a triangular grid of 1.5 km spacing [2] and is overlooked by four fluorescence detectors (FD) [3]. The SD trigger condition, based on a 3station coincidence [4], makes the array fully efficient above about eV. For each event, the signals in the stations are fitted to find the signal at 1000 m from the shower core, , which is used to estimate the primary energy [5]. The atmosphere is continuously monitored by different meteorological stations located at the central part of the array and at each FD site. In addition, balloonborne sensors are launched at regular intervals to measure the atmospheric temperature , pressure and humidity as a function of the altitude above the detector [6].
In section 2, we develop a model of the expected atmospheric effects on . The modulation is described by means of three coefficients that depend on the EAS zenith angle (). They are related to variations of and , measured at ground level, on slower (dailyaveraged) and faster (within a day) time scales. The dependence of on and implies a modulation of the counting rate of events. In section 3, we study the behaviour of the recorded rate of events as a function of and . On the base of the model defined previously, we derive the and coefficients. In section 4, we perform full simulations of EAS developing in various realistic atmospheres (based on measurements from balloon soundings above the site of the Pierre Auger Observatory) in order to compare, in section 5, the results from data and simulations with the predictions of the model. We conclude in section 6.
2 Model of atmospheric effects for the surface detector of the Auger Observatory
2.1 Atmospheric effects on the measured signal
The waterCherenkov detectors are sensitive to both the electromagnetic component and the muonic component of the EAS, which are influenced to a different extent by atmospheric effects, namely by variations of and . These in turn influence the signal measured in the detectors: for the Auger Observatory, we are in particular interested in the effects on the signal at 1000 m from the core, .
The continuous measurement of atmospheric and is available only at ground level. We will show that the variation of can be fully described in terms of variation of air pressure and air density measured at the altitude of the Observatory site. If not otherwise stated, and refer to the values at ground level.
In the following, we first describe separately the effects on due to , section 2.1.1, and , section 2.1.2, and then in section 2.1.3 we provide the full parameterisation of its variations as a function of changes in and .
2.1.1 Effect of air pressure variations on the SD signal
From the point of view of (which measures the vertical air column density above ground), an increase (decrease) corresponds to an increased (decreased) matter overburden. This implies that the shower is older (younger), i.e. in a more (less) advanced stage when it reaches the ground level.
The longitudinal profile of the electromagnetic component of the EAS is exponentially attenuated beyond the shower maximum and can be described by a GaisserHillas profile [7] (see Fig. 1). We are interested in the value of the electromagnetic signal measured at 1000 m from the core, referred hereafter as . The longitudinal development of the shower far from the core is delayed with respect to the one at the core, and can be parameterised as
where is the primary energy, the slant depth, the average maximum of the shower at 1000 m from the core with being the shower maximum^{1}^{1}1 g cm for eV showers according to the elongation rate measurement with the FD at the Pierre Auger Observatory [10], 150 g cm is the typical increase of the shower maximum at 1000 m from the core [8] and g cm is the effective attenuation length after the maximum [9]. Therefore, a change in affects :
(1) 
where d is used, with the acceleration of gravity, and the shower zenith angle. Due to the flat longitudinal development of the muons (see Fig. 1), no significant pressure dependence is expected for the muonic component.
2.1.2 Effect of air density variations on the SD signal
Regarding , this affects the Molière radius
where MeV is the energy constant characterising the energy loss due to multiple Coulomb scattering, MeV is the critical energy in air and g cm is the radiation length in air. A variation in affects the lateral distribution of the electromagnetic component of the EAS, which can be approximately described with a NishimuraKamataGreisen (NKG) profile [11, 12]. At a large distance from the core, it behaves as , where and is the age of the shower. Hence, a change in affects :
(2) 
In fact, the relevant value of is the one corresponding to the air density two radiation lengths above ground [12] in the direction of the incoming shower. This corresponds to above the site of the Pierre Auger Observatory. On time scales of one day or more, the temperature gradient (d/d) in the lowest layers of the atmosphere (the planetary boundary layer, which extends up to about 1 km above ground level) can be described by an average value of C km at the site of the Auger Observatory. Therefore the variation of on temporal scales of one day essentially follows that of . An additional effect is related to the diurnal variations of d/d, because during the day the surface of the Earth is heated by solar radiation, producing a steeper d/d in the boundary layer. On the other hand, during the night the surface is cooled by the emission of long wavelength radiation: d/d becomes smaller and even inversions can be observed before sunrise. As a result, the amplitude of the diurnal variation in (and ) is smaller at two radiation lengths above ground than at ground level. It is then useful to separate the daily modulation from the longer term one introducing the average daily density and the instantaneous departure from it, . Therefore, the dependence of on can be modeled by
where = 1.06 kg m is chosen as the reference value of and is the average value measured at the site of the Pierre Auger Observatory over more than three years (1 Jan 2005  31 Aug 2008).
Concerning the muonic component of the signal at 1000 m from the core, , its dependence on can be parameterised as
The dependence is written in terms of only because, as the muons are produced high in the atmosphere, their contribution to signal is not expected to depend on the daily modulations taking place in the boundary layer.
2.1.3 Model of atmospheric effects on S(1000)
The dependence of the total signal at 1000 m from the core, , upon and can hence be written as
(3) 
where hPa is the reference at the site of the Pierre Auger Observatory, is the value of the total signal at reference pressure and density ( and ), and
(4) 
where is the electromagnetic fraction of the signal at 1000 m from the core. The values of are obtained by means of protoninitiated showers simulated with CORSIKAQGSJETII (see section 4): they decrease approximately linearly with sec for all the simulated primary energies (see Fig. 2).
We will adopt hereafter
(5) 
where varies between at eV and at eV. We note that since the inferred electromagnetic fraction depends on the hadronic model adopted and on the CR composition assumed, the actual value of may be different. As shown in [9], for ironinduced showers the simulated is 40% higher than in the case of protons, while the SIBYLL model [13] predicts a muonic signal 13% lower than QGSJETII for both proton and iron primaries. The corresponding variation at a primary energy of would be for iron with respect to proton, and for SIBYLL simulations with respect to QGSJETII.
Finally, with respect to the coefficients in eq. 4:
(i) for the pressure coefficient, we have from eq. 1
where and g cm is the atmospheric depth at the site of the Pierre Auger Observatory.
(ii) From eq. 2
where , with for eV primaries. Pressure effects associated to the change in the slope of the lateral distribution function due to the dependence of are negligible.
(iii) The coefficient should be smaller than (in absolute value) reflecting the reduction in the amplitude of the variations two radiation lengths above ground level. The difference should also depend on . For instance, assuming an exponential decrease of the density amplitude with the height
would lead to
(6) 
where parameterises the amplitude of the daily density variation in the lower atmosphere and is completely independent of the shower development. It characterises the scale height for the decrease of the daily thermal amplitude, which becomes of its ground value at a height m. The value of is expected to be of order unity.
(iv) The coefficient is expected to be small, and will be assumed to be independent of , because of the relatively flat longitudinal development of the muons as shown in Fig. 1. Its value will be taken to be zero since the air shower simulations are consistent with a vanishing coefficient (see section 4).
2.2 Atmospheric effects on the event rate
The dependence of the measured signal on variations of and produces also a modulation of the rate of recorded events. The trigger probability, , is a well defined function of the signal [4]. As atmospheric variations correspond to signal variations, this implies that the same primary particle (in particular, with the same primary energy) will induce different signals depending on and . This in turn affects the probability for the shower to trigger the SD array.
The effect can be quantified starting from the relation between and the energy of the primary cosmic ray. In the case of the Pierre Auger Observatory, the primary energy is reconstructed as
where is derived from the calibration of the SD energy using the FD energy measurement [14]. Following eq. 3, the primary energy that would have been obtained for the same shower at the reference pressure and density , is related to as follows
(7) 
In a zenith angle bin d, the rate of events per unit time and unit solid angle above a given signal can be written as
where is the geometrical aperture and is the flux of cosmic rays.
Assuming that the cosmic ray spectrum is a pure power law, i.e. d, using eq. 7, and neglecting the small energy dependence of the weather coefficients, we find that
From the dependence on the atmosphere of the measured CR flux above a given signal, we derive the corresponding dependence of the rate of events. If is the minimum required signal at 1000 m from the core to trigger the array
(8) 
with the integral on the right hand side being independent of the weather variations. The coefficients , and are then related to the coefficients describing the modulation of the signal by and .
3 Atmospheric effects on the experimental rate of events
To study the modulation of the rate of events, we use data taken by the SD from 1 January 2005 to 31 August 2008. All events with are used, for a total of about 960000 showers with a median energy eV. These are selected on the basis of the topology and time compatibility of the triggered detectors [4]. The station with the highest signal must be enclosed within an active hexagon, in which all six surrounding detectors were operational at the time of the event.
At the site of the Pierre Auger Observatory, the ground temperature and pressure are measured every five minutes. The air density is given by: where is the molecular mass of air, the gas constant. The daily average density is obtained with a smoothing procedure consisting in taking, for each time, the average value of over a 24 h interval centered at the time of interest. The daily and diurnal variations of the ground and are shown in Fig. 3 (upper and lower panels respectively). The pressure exhibits less than variation during the period considered, while changes up to a maximum of with an additional diurnal variation of density which is of on average with maximum values of .
In the period under study, the number of surface detectors steadily increased from about 700 to about 1590. To take this into account, rather than using the raw number of triggering events, we compute the rate every hour normalized to the sensitive area, which is calculated every second from the total area of the active hexagons. The daily and the diurnal rate of events are presented in Fig. 4 (black points), where it is evident that they both follow qualitatively the corresponding modulations of pressure and density from Fig. 3.
We use the expression given by eq. 8 to fit the measured rate of
events. Assuming that the number of events observed in each hour bin
follows a Poisson distribution of average , a maximum likelihood fit is
performed to estimate the coefficients , and .
The
likelihood function is . The
expected number of events in bin is given by
where is the average rate we would have observed if the atmospheric parameters were always the reference ones, i.e. , with the sensitive area in the bin and, according to eq. 8, is
The fitted parameters are:
(9)  
corresponding to a reduced of , where . The result of the fit is shown in Fig. 4, compared to the daily averaged and the shorter term modulations of the measured event rate.
To check the stability of the coefficients with respect to the energy, the same study has been done for the subset of events with a reconstructed energy above eV, corresponding to of the total statistics. The fitted coefficients are consistent within the fit uncertainties. A more detailed study of the energy dependence of the coefficients will become feasible in future with increased statistics.
4 Atmospheric effects on simulated air showers
To complete the study of atmospheric effects, we performed full EAS simulations in different atmospheric conditions. We simulated protoninitiated showers using the CORSIKA code [15] with hadronic interaction models QGSJETII [16] and Fluka [17].
We considered four fixed energies of the primary particle ( = eV, eV, eV and eV) and seven fixed zenith angles between and . For the air density profiles, we used five parameterisations (shown in Fig. 5) of the seasonal average of radio sounding campaigns carried out at the site of the Pierre Auger Observatory [6] over a wide range of variation in temperature^{2}^{2}2The atmospheric profiles are implemented in the CORSIKA code through the dependence of on . , and profiles can be derived from: and . The ground values in Fig. 5 are computed at an observation level m ( 880 g cm), corresponding to the altitude of the Pierre Auger Observatory.. The set of simulations consists of 60 showers for each combination of atmospheric profile, energy and angle with an optimal statistical thinning level of [18, 19].
To compare with model predictions and data, we need to determine for each
combination (, ) the dependence of on the variations of
and . The signal can be estimated through simplified assumptions about the
energy deposited by particles on the basis of their kinetic energy :
(i) ee deposit , where keV is the energy threshold for Cherenkov emission in water.
(ii) photons deposit .
(iii) muons deposit 240 MeV corresponding to the average energy
released by a vertical muon crossing a 1.2 m high waterCherenkov tank.
The contribution of each particle is multiplied by the weight assigned by the
thinning algorithm. We obtain the Cherenkov signal per unit area perpendicular to
the shower plane .
For the muons, the Cherenkov signal is proportional to the track length in the
station so that: , whereas for the electromagnetic component:
.
The left panel of Fig. 6 shows the lateral distribution , which is proportional to , for four atmospheres (relative to the Spring one) in the case of = eV and . The effect related to the Molière radius can be clearly seen as a broadening of the lateral distribution with increasing temperature.
To derive the atmospheric coefficients, we correlate the simulated (taken as the average signal between 950 m and 1050 m) with and (see eq. 3). Since we are using seasonal atmospheric profiles, we do not have access to the diurnal variation of and thus we cannot determine the coefficient related to the diurnal variation of . The two coefficients and can be determined for each fixed energy and angle with a two dimensional fit of the , obtained for the five atmospheric profiles, as function of and . As an example, we show in Fig. 6 (right) the results of the fit for the case of = eV and , projected on the (, ) plane for the sake of clarity. Moreover, in the case of simulations we are able to separate the electromagnetic and the muonic contribution to the signal and thus to determine the atmospheric coefficients for each component (see Fig. 7).
5 Comparison among model, data and simulations
In this section, we compare the atmospheric coefficients derived from data with those expected from the model and simulations. We recall that with the simulations we cannot access the coefficient , as we use average seasonal profiles for the atmosphere, while we can investigate the behaviour of separate coefficients for the electromagnetic and muonic components of EAS. On the other hand, with experimental data we cannot separate the electromagnetic and muonic components, while we can fully investigate the diurnal effects of atmospheric changes and compare measurements and expectations for all of the three coefficients.
The comparison between atmospheric coefficients for the electromagnetic and muonic components of EAS from simulations and model is shown in Fig. 7, as a function of sec . With respect to the electromagnetic part, the model predictions for both the and coefficients, and their dependence on the shower zenith angle, are reasonable at all energies. Concerning the muonic component of the signal and its dependence on , is compatible with zero at all energies, as expected from the flat longitudinal development of the number of muons. For the dependence on , the model is not predictive but from the simulations we get a value of compatible with zero. This justifies the adoption in the model of vanishing coefficients for the muonic component.
The comparison of the global coefficients as a function of sec is done for , and in Figs. 8 and 9. In the case of the data, the dependence on has been studied by dividing the data set in subsets corresponding to five bins of equal width in sec . For each subset the same fitting procedure as illustrated in section 3 is used. The signal coefficients are then derived by dividing the rate coefficients by (see the end of section 2.2). Since the bulk of the triggering events have an energy eV, we used , as measured with the Auger Observatory below eV [20].
The comparison among data, simulations and model is shown for the pressure coefficient and the daily component of the density coefficient in Fig. 8 (top and bottom respectively). In the model, we use the value of measured by the Auger Observatory at the median energy of the triggering events [10], and a , corresponding at the same energy, obtained under the assumption that scales linearly with the logarithm of the primary energy. The reduced for the datamodel comparison is 3.3 for and 11.0 for . For the instantaneous density coefficient , the comparison between data and model is shown in Fig. 9. The datamodel comparison gives in this case a reduced of 0.6.
6 Conclusions
We have studied the effect of atmospheric variations (in , and ) on extensive air showers using about 960000 events collected by the surface detector of the Pierre Auger Observatory from 1 January 2005 to 31 August 2008. We observe a significant modulation of the rate of events with the atmospheric variables, both on a seasonal scale ( 10%) and on a shorter time scale ( 2% on average during a day). This modulation can be explained as due to the impact of the density and pressure changes on the shower development, which affects the energy estimator , the size of the shower signal 1000 m from the shower axis. This affects the trigger probability and the rate of events above a fixed energy.
The dominant effect is due to the change with the air density of the Molière radius near ground. It induces a variation of the rate of events with associated correlation coefficients of kgm and kgm on long and short time scales, respectively.
The second effect is due to the pressure changes, which affect, through the variation of the amount of matter traversed, the stage of development of the showers when they reach ground. The impact of the pressure variation on the rate amounts to hPa.
Comparing the coefficients obtained from data, shower simulations in different atmospheric profiles and expectations from the model built, a good agreement is obtained, not only for the overall size of the effect but also for the zenith angle dependence.
Taking into account the atmospheric effects will allow to reduce the systematics in the energy reconstruction. Furthermore, it will be possible to correct for the seasonal modulation, which can affect the search for large scale anisotropies.
7 Acknowledgements
The successful installation and commissioning of the Pierre Auger Observatory would not have been possible without the strong commitment and effort from the technical and administrative staff in Malargüe.
We are very grateful to the following agencies and organizations for financial support: Comisión Nacional de Energía Atómica, Fundación Antorchas, Gobierno De La Provincia de Mendoza, Municipalidad de Malargüe, NDM Holdings and Valle Las Leñas, in gratitude for their continuing cooperation over land access, Argentina; the Australian Research Council; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de Estudos e Projetos (FINEP), Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (FAPERJ), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Ministério de Ciência e Tecnologia (MCT), Brazil; AVCR AV0Z10100502 and AV0Z10100522, GAAV KJB300100801 and KJB100100904, MSMTCR LA08016, LC527, 1M06002, and MSM0021620859, Czech Republic; Centre de Calcul IN2P3/CNRS, Centre National de la Recherche Scientifique (CNRS), Conseil Régional IledeFrance, Département Physique Nucléaire et Corpusculaire (PNCIN2P3/CNRS), Département Sciences de l’Univers (SDUINSU/CNRS), France; Bundesministerium für Bildung und Forschung (BMBF), Deutsche Forschungsgemeinschaft (DFG), HelmholtzGemeinschaft Deutscher Forschungszentren (HGF), Finanzministerium BadenWürttemberg, Ministerium für Wissenschaft und Forschung, NordrheinWestfalen, Ministerium für Wissenschaft, Forschung und Kunst, BadenWürttemberg, Germany; Istituto Nazionale di Fisica Nucleare (INFN), Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR), Italy; Consejo Nacional de Ciencia y Tecnología (CONACYT), Mexico; Ministerie 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. 1 P03 D 014 30, N202 090 31/0623, and PAP/218/2006, Poland; Fundação para a Ciência e a Tecnologia, Portugal; Ministry for Higher Education, Science, and Technology, Slovenian Research Agency, Slovenia; Comunidad de Madrid, Consejería de Educación de la Comunidad de Castilla La Mancha, FEDER funds, Ministerio de Ciencia e Innovación, Xunta de Galicia, Spain; Science and Technology Facilities Council, United Kingdom; Department of Energy, Contract No. DEAC0207CH11359, National Science Foundation, Grant No. 0450696, The Grainger Foundation USA; ALFAEC / HELEN, European Union 6th Framework Program, Grant No. MEIFCT2005025057, European Union 7th Framework Program, Grant No. PIEFGA2008220240, and UNESCO.
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