Parametrization of CoREAS simulations of radio emission from extensive air showers

Transcription

1 Parametrization of CoREAS simulations of radio emission from extensive air showers Bachelorarbeit von Anton Huber Am Institut für Experimentelle Kernphysik Karlsruher Institut für Technologie (KIT) Erstgutachter: Betreuender Mitarbeiter: Prof. Dr. J. Blümer Dr. T. Huege KIT University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association

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3 Eidesstattliche Erklärung Hiermit versichere ich an Eides statt, dass die vorligende Arbeit selbständig und ohne fremde Hilfe - abgesehen von der Beratung durch meine wissenschaftlichen Betreuer - angefertigt wurde. Örtlich übernommene Ausführungen anderer Autoren sowie eng an den Gedankengänge Anderer anlehnende eigene Ausführungen sind entsprechend gekennzeichnet. Karlsruhe, den (Anton Huber) iii

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5 Abstract High energy cosmic rays, interact in the Earth s atmosphere and generate a cascade of secondary particles. The charged particles interact with the geomagnetic field and induce radiation in the MHz regime (radio emission). To get a better understanding of these processes and to help planning and improving radio detection experiments, Monte Carlo based simulations such as CoREAS 1 are necessary. Because of the long computing time that is needed for CoREAS simulations, it is useful to parametrize the simulation results. The Parametrization can be used for fast calculations, which are necessary for the early stages of experiment planning and for getting a first overview of an expected radio signal. The following bachelor thesis finds a parametrization for cosmic ray air showers based on simulation results from CoREAS. Zusammenfassung Hochenergetische, geladene Teilchen, erzeugen in der Erdatmosphäre durch Wechselwirkungsprozesse Teilchenschauer. Elektrisch geladene Schauerteilchen verursachen im Erdmagnetfeld Strahlung im MHz-Bereich (Radio-emission). Um diese physikalischen Prozesse besser zu verstehen und um Planungen bzw. Weiterentwicklungen der Radio-Messungen zu erleichtern sind Monte-Carlo-gestütze Simulationen notwendig. Da diese Simulationen zeit- und rechenaufwändig sind, ist es sinnvoll, die Ergebnisse zu parametrisieren und somit eine schnelle Berechnung zu ermöglichen. In der nachfolgenden Bachelorarbeit werden die quantitativen Abhängigkeiten des mit CoREAS simulierten Radiosignals ausgedehnter Luftschauer von verschiedenen physikalsichen Parametern untersucht und eine Parametrisierung für das Radiosignal ausgedehnter Luftschauer gefunden. 1 COrsika based Radio Emission from Air Showers v

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7 Contents 1. Introduction 1 2. Cosmic Rays and Radio Emission Models Simulation of Cosmic Ray Air Showers and used models Coordinate System Convention Previous Parametrizations Experimental Results Parametrization Primary Particle Energy Lateral Dependence The Gaussian Distribution and the Parameter p The Offset Parameter p Dependence on the Shower s Geometry Asymmetry Effects of the radio signal The sin α Dependence Parameters for arbitrary Geometries Overall Parametrization Quality Check Polarization and the v B - Model Conclusion 25 Bibliography 27 Appendix 29 A. Log-ratio and Comparison-plots for all used Simulations A.1. Vertical simulations A.2. Showers with Θ = A.3. Showers with Θ = A.4. Other Showers Danksagung 35 vii

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9 1. Introduction In 1912 Victor Hess first detected ionizing radiation coming from cosmic sources [Hes12]. In the following decades, a lot of approaches explaining the radiation were made. In the mid 1960s the first observation of radio pulses emitted from cosmic ray air showers was made [Jea65]. Today s cosmic ray research is still dominated by surface detector methods which are detecting the secondary particles of comic rays and fluorescence telescopes, detecting fluorescence light emitted by nitrogen excited by extensive air showers. Prototype experiments like the LOPES 1 experiment [KC04] at the KIT showed that the radio detection technique, i.e. the detection of cosmic rays by their radio radiation, is an alternative, effective and reliable way to detect the high energy particles. AERA 2 at the Pierre Auger Observatory in Argentina [Fea10] is with a planned area of 20km 2 and 160 antennas, the first larger scaled experiment including radio detection. Beside AERA, there are also other experiments (eg. Tunka-Rex 3 ) [Sea12] detecting cosmic rays by their radio emission. A good understanding of the processes emitting radio signals in the Earth s atmosphere is necessary for further developments of the radio detection technique. Therefore, Monte-Carlo-based simulations are useful tools to plan such experiments, and compare theory with experimental data. A parametrization of the simulation results is important for several reasons: it is helpful for planing new experiments it can save time and computational resources (a CoREAS simulation that simulates the radio signal for 64 observers for a primary particle of ev energy, requires full 2 days on a standard CPU at the time of writing) it allows easier comparisons with experimental data REAS 3.1 is a recent Monte Carlo based simulation code, developed by Tim Huege and Marianne Ludwig in 2011 [LH11]. An important modification to the previous REAS version was the inclusion of the end-point formalism, which takes the previously missing radio emission due to the variation of the number of charged particles during the air shower evolution into account [Jea10]. The simulation results are in an acceptable agreement with experimental data from the LOPES experiment [Lud11]. 1 LOfar PrototypE Stadion 2 Auger EngineeringRadio Array 3 Tunka Radio EXtension 1

10 2 1. Introduction As REAS 3.1 is not able to calculate the development of the cosmic ray cascade in the atmosphere by itself, the information is made available by the simulation code CORSIKA 4 [Hec98]. Because of the very high number of secondary particles, not every particle is tracked individually in REAS 3.1 simulations. The shower cascade is recorded in histograms provided by CORSIKA simulations. Due to histograming, information gets lost. CoREAS is a newly developed simulation code which combines CORSIKA simulated showers and the radio signal calculations from REAS 3.1. The contribution of every secondary particle is taken into account for the overall radio signal. The parametrization found in this thesis relies on CoREAS results. 4 COsmic Ray SImulation for KAscade

11 2. Cosmic Rays and Radio Emission Models After the discovery of radioactivity at the beginning of the 20th century it was noticed that the air is being ionized frequently. Victor Hess was the first who proved with balloon experiments in 1912, that the ionization rate increases with growing altitude [Hes12]. Hence the ionizing radiation could not only have a terrestrial source but also had to be of cosmic origin. The very new fields of cosmic ray physics and astroparticle physics were founded. In the following years, several experiments proved his theory (e.g. Milikans detection of muons from extensive air showers). With the development of cloud chambers, Geiger-Müller counters and other new methods for particle measurements the cosmic rays could be tracked and observed. It was possible to measure their amount of ionization and their energy. Whith the fast progress of particle physics in the 1950s and 60s, the so called standard model of cosmic rays was found and studied in several ways. Primary particles coming from cosmic sources interact in the Earth s atmosphere and induce extensive air showers with up to secondary particles such as photons, muons, neutrinos, electrons and positrons. The charged particles (mainly electrons and positrons) are deflected by the Earth s magnetic field, hence a current is induced and due to its time-variation, coherent electromagnetic radiation is emitted [LK66]. Many experiments measured this radio emission from cosmic rays, eg. the LOPES experiment in Karlsruhe. To test the reliability of the new detection method, the radio detected air showers were compared with other detecting methods in coincidence experiments (e.g. surface detectors like KASCADE, which detects secondary particles like muons, positrons and electrons) [KC05]. The radio detection technique has turned out to be a promising alternative to common detection methods. As opposed to other cosmic ray detection methods like fluorescence telescopes (measuring the fluorescence light of atmospheric nitrogen excited by charged secondary particles) the radio detection technique has the big advantage of almost 100% duty cycle. While fluorescence detectors only work during cloudless nights, the meassurement of radio emission is almost always possible (only exception are thunderstorms). These measurements can help to get a better understanding of the cosmic rays, their composition, direction of arrival and interaction in the atmosphere. With particle energies up to ev, cosmic rays reach a much higher energy than any particle made in particle accelerators. Therefore, ultra-high energy cosmic rays (UHECR) give the chance to study interesting topics for particle physics like search for physics beyond the standard model. 3

12 4 2. Cosmic Rays and Radio Emission Models In the following chapter, different models and simulation techniques are discussed. We take a look at former parametrizations and experimental results to get a brief overview of the relevant parameters, before we develop a parametrization for the present simulations Simulation of Cosmic Ray Air Showers and used models In 1966 Kahn and Lerche introduced the effect of transverse currents which has been exposed as the dominating part of the radio emission process [LK66]. Electrons and positrons are accelerated in the Earth s magnetic field and decelerated by interactions with atmospheric molecules. These effects, and the fact that the number of secondary particles in a extensive air shower raises till it reaches a maximum, lead to a time-variation of the transverse current and therefore to a radio emission. The opening angle of the radiation is proportional to 1/γ (where γ is the Lorentz factor) [FG02]. The REAS simulation code, which was developed by T. Huege in 2004 and the following years, is a simulation code based on Monte Carlo techniques to simulate the radio emission from extensive air showers coming from cosmic rays [Hue04]. While in REAS 1 the air showers were parametrized, REAS 2 used the tracks of the primary and secondary particles as simulated by CORSIKA, a simulation code for cosmic ray air shower development in the Earth s atmosphere [Hec98]. But it turned out, that both, REAS 1 and REAS 2, used an incomplete model for the radio emitting process. They did not take into account that the varying number of charged particles (time-varying charge excess) has an non-negligible contribution to the radio emitting process. This emission process was first proposed by G.A. Askaryan in 1962 [Ask66]. Also the effect of the time-varying transverse current was included incorrectly. The radiation contribution caused by the time-variation of the transverse current is linearly polarized with the electric field vector aligned in the direction of the Lorentz force vector (for detailed calculations see chapter 3.6), and is therefore independent of the observer position. The contribution due to the time-varying charge excess causes a linearly polarized radiation, with electric field vectors radially oriented to the shower axis. Hence, the polarization angle varies with the observer position [Ask66]. Figure 2.1 shows the alignment of the electric field vectors as a function of the observer position (for a vertical shower). The electric field of the two different emitting processes interfere with each other and cause asymmetric effects in the radio signal (mainly in the east-west direction). This asymmetric effects can be seen in the lateral distribution in the radio signal and are discussed in the next chapter. Using an end-point formalism [Jea10], REAS 3 take the radiation coming from a variation in the number of charged particles during the air shower into account. Due to that, it allows a contribution of the time-varying charge excess to the radio signal [Lud11] and includes both major effects that cause the radio emission during the shower development, the time-varying transverse currents and charge excess. The transverse current effect leads straight to the v B- model which is described in detail in chapter 3.7. For designated geometries, the two effects are in the same order. In this case, the asymmetric effects become very significant (the contribution of the time-varying charge excess is big for geomagnetic angles close to 0 and 180 ).

13 2.2. Coordinate System Convention 5 Figure 2.1.: Alignment of the electric field vectors caused by a vertical shower as a function of the observer position. Left: radio emission from transverse currents linearly polarized in the direction of the Lorentz force vector, right: charge excess emission linearly polarized and radially orientated, from [Hue13] 2.2. Coordinate System Convention To describe the arrival direction of cosmic ray air showers, two angles, the zenith angle Θ and the azimuth angle ϕ, are necessary. The zenith angle Θ denotes the angle between the z-axis (perpendicular to the ground) and the shower axis. For vertical showers Θ = 0, if Θ > 0, the shower is inclined (see also figure 2.2). The cardinal direction is given by the azimuth angle ϕ. The CoREAS convention has the following angles for showers coming from: south: ϕ = 0 east: ϕ = 90 north: ϕ = 180 west: ϕ = 270 The geomagnetic field is the major cause of the particle acceleration during the shower development. Hence it is useful to introduce the angle between the shower axis and the Earth magnetic field the geomagnetic angle α. According to the location on the Earth, the direction of the Earth s magnetic field changes. Considering the Pierre Auger Observatory we use the direction of the magnetic field in Argentina with Θ B = 53. Due to the fact that the magnetic field lines lead from south to north, the magnetic field s azimuth angle is always ϕ B = Previous Parametrizations The radio signal depends on different parameters, given by the arrival direction and the energy of the primary particle. The aim of a parametrization is to qualify and quantify these dependences. Before a new parametrization is found, it is necessary to take a look at former ones. In this subsection three parametrizations based on simulations and experimental data are introduced and discussed. The first one uses the data of Allan (1971). It was the first formula which was able to estimate the electric field at a specific distance R SA to the shower axis at a frequency of

14 6 2. Cosmic Rays and Radio Emission Models Figure 2.2.: Inclined air shower with the zenith Θ and the azimuth angle ϕ in a shower based reference frame, from [Fuc12] 55MHz [All71]. E(Θ, RSA, E p, α) ( ) ( Ep = sin α cos Θ exp R ) SA ev R 0 (ν, Θ) (2.1) This formula contains a linear dependence of the electric field amplitude on the primary particle energy. The E p is used as a factor to scale the amplitude of the radio signal, which decreases exponentially with the distance of the observer to the shower axis. The amplitude of the radio signal depends also on the geometry of the shower. It scales with the cosine of the zenith angle Θ and the sine of the geomagnetic angle α. R 0 is a scale factor which depends on the shower geometry and the frequency of interest. The first simulation based parametrization was found in 2004 by Tim Huege [Hue04]. E(Θ, R SA, ν, E p, α) ( ) 0.96 ( Ep = E Θ exp 200m(α(X ) max) 1) R SA α(x max )l Θ (2.2) ( ) ν/mhz 10 exp exp( R SA /b Θ ) In this case α denotes a function of the shower maximum (X max ) and not the geomagnetic angle: ( ) Xmax α(x max ) = gcm 2 (2.3) with E Θ, l Θ and b Θ being tabulated parameters depending on the zenith angle Θ. The almost linear dependence of the primary particle energy and the exponential relation between the distance to the axis and electric field strength is, for the most part, the same as in Allan s parametrization. However, the REAS 1 code had major problems because of the missing charge excess emission and other missing processes. Therefore, the complex parametrization shown in this chapter has just a limited validity. In the following chapter, a parametrization based on experimental data is introduced.

15 2.4. Experimental Results Experimental Results To test the above parametrization, the results must be compared to experimental data. In this chapter the LOPES experiment is introduced, a recently finished prototype experiment, and the results are compared with the previous presented parametrizations. Furthermore, a parametrization based on experimental data is discussed [Hor06] and compared with the parametrizations from the previous chapter. The LOPES experiment [KC04] is a radio detection prototype experiment measuring the radio emission of extensive air showers with dipole antennas. It is based at the Campus North of the Karlsruhe Institute of Technology (KIT) within the KASCADE experiment [KC05]. The KASCADE experiment consists of 252 detector stations arranged in a 200m x 200m detector array. The KASCADE experiment uses electron, gamma and muon detectors to detect secondary particles coming from extensive air showers. LOPES measures the north-south and the east-west polarized electric field components of the radio emission in an effective bandwidth of MHz [Hub10]. It is triggered by the KASCADE experiment which means measuring coincidences with a well calibrated experiment. The aim of LOPES was to prove the principle of the radio detection technique, and to test different detection antennas and their arrangement for advanced experiments. In contrast to the parametrizations in chapter 2.2, the LOPES parametrization only parametrizes the east-west component of the electric field. The results show that the east-west polarized part of the radio signal depends mainly on four different quantities. Three of them were expected because of former REAS simulations, the primary particle energy E p, the geomagnetic angle α and the mean distance from the shower axis to the antenna R SA. The experiment showed further more that the influence of the shower direction cannot be expressed only by the geomagnetic angle. Also the zenith angle of the shower must be taken in account to parametrize the results. From all analyses the following parametrization [Hor06] was found: E east = (11 ± 1.)[(1.16 ± 0.025) cos α] cos Θ (2.4) [ exp R SA (236 ± 81)m ] ( ) (0.95±0.04) [ ] Ep µv ev m MHz The given errors are statistical errors from the fits. The result from the LOPES experiment mostly confirm Allan s formula and show similarities with the REAS 1 parametrization. All results agree with the exponential dependence of the signal on the distance to the shower axis. An almost linear dependence to the E p is also found. The difference of the parameter in the exponent of E p (γ = 0.95) to 1 (γ = 1 would be the perfect linear case) is explained in chapter 3. Instead of a simple sin α correlation we have an almost (1-cos α) dependence (which is not a well- motivated approach and no longer used today). Furthermore, a liner sin α dependence is only expected for the absolute shower signal E and not specifically for the east-west polarized component E east. The comparison of the LOPES results and the REAS 3 reconstructed events shows a good agreement. Almost 80 % of the results have the exponential lateral distribution which REAS 3 suggest [Lud13]. As a consequence of former parametrizations we try to find a power law correlation between the electric field and the primary particle energy and a linear dependency on sin α.

16 8 2. Cosmic Rays and Radio Emission Models Concerning a signal flattening for small distances to the shower axis, which is caused by Cherenkov effects (not unity refracting index) we try to find a modified exponential dependence (Gaussian distribution) on the distance to the shower axis with parameters only depending on the shower s zenith angle Θ (more detailed description of the Cherenkov effect can be found in chapter 3.2).

17 3. Parametrization In the previous chapter we described the occurrence of cosmic rays in the atmosphere and how they can be detected by their radio emission, different emission processes were discussed and the simulations were introduced. In this chapter we develop a parametrization for CoREAS simulations. The simulations used with different parameters can be found in figure 3.1. As default parameter we used the Pierre Auger Observatory location at the altitude of 1400m and the geomagnetic field setup with Θ B = 53 and ϕ B = 0 and a strength of 23 µt (0.23 Gauss). The 64 simulated observers are aligned in 8 circles with different radii, starting from 10m to 1200m distance to the shower core. The shower development is calculated by CORSIKA using the QGSET II.03 [Ost06] model for high energy hadronic interactions and the UrQMD [Bea98] for low energy hadronic interactions. The primary particles were protons. The simulated signal is filtered to a bandwidth from MHz and the maximum amplitudes of the filtered pulses are determined for every observer. 9

18 10 3. Parametrization Figure 3.1.: The simulations used with different geometries and primary particle energies Nr. Θ [ ] φ [ ] E p [ev] α [ ] E E E E E E E E E E E E E E E E In the following chapter, we will discuss the dependences for the individual shower parameters and quantify them to find an overall parametrization at the end Primary Particle Energy One major aim of UHECR research is the knowledge of the sources of cosmic rays. Therefore, the reconstruction of the cosmic ray energy is essential. Because of coherent emission, which is caused by the charged secondary particles, the electric field should scale linearly with the number of particles during the shower development. The number of particles grows approximately linearly with the primary particle energy and therefore a liner scaling between the primary particle energy and the electric field of the radio signal is expected. Figure 3.2 shows the absolute electric field strength for an observer at a distance of 200m to the shower core for four vertical showers with different E p (number 1-4 from figure 3.1). As other parametrizations suggest, the scaling of the total field strength with the primary particle energy E p follows a power law given by: E abs E 0.98 p (3.1) The exponent γ = 0.98 confirms the other parametrizations and has only a small variance to the experimental data. The difference of the calculated γ and the originally assumed (eg. by Allan) linear dependence (with γ = 1.0) can be explained by geometric effects. The shower maximum (shower depth) depends on the primary particle energy E p. Because the shower depth undergoes strong shower to shower fluctuations it is set to a typical (average) value in the simulations of 3.1. With a change of the shower depth, the geometry is changing and due to that the lateral distribution of the radio signal. The higher the energy of the primary particle the steeper

19 3.2. Lateral Dependence 11 is the radial emission pattern (and therefore the lateral distribution). Therefore we expect a little variance from the power law with γ = 0.98 and the first predicted linear law [HF04]. Figure 3.2.: electric field strength simulated for the same observer for vertical showers with different E p. The green dashed line is the powerlaw fit E E 0.98 p 3.2. Lateral Dependence Caused by projection effects, the emission patterns from inclined showers become broader and asymmetric with higher zenith angles. Thus it is helpful to change from groundbased coordinates to a shower-based coordinate system, which reduces the influence of the projection effects significantly as discussed in [Hue04]. The position of the observers (normally given in ground distance to shower core and cardinal direction) are converted into perpendicular distances to the shower axis using the following relation: R SA = r 1 cos 2 (ϕ 0 ϕ) sin 2 Θ (3.2) with r the ground distance to the shower core, the azimuth angle of the observer ϕ 0 and the shower azimuth ϕ and zenith angle Θ (from [Hue04]). Figure 3.3 illustrates the change of the reference frame The Gaussian Distribution and the Parameter p 2 As discussed in the previous chapter, former parametrizations used and exponential dependence like: E total E 0 e R SA R 0 (3.3) However, the CoREAS simulation results show a small but non-negligible flattening of the radio signal close to the shower axis. Figure 3.4 shows a near-vertical shower with Θ=25 (shower no.5 in 3.1).

20 12 3. Parametrization Figure 3.3.: relation between ground based an shower based coordinates with R SA the distance from the shower axis to the observer, from [Fuc12] The radio signal has its maximum at the shower axis (lateral distance is 0) and shows a exponential dependence like in equation (3.3). On figure 3.5 a more inclined shower with Θ = 60 is shown (shower no.15 in 3.1). The radio signal shows a flatting close to the shower axis. The maximum of the radio signal (at 180 m) is no longer reached at the closest point to the shower axis. The flattening can be explained by Cherenkov effects. It is caused by the fact that the refractive index of the Earth s atmosphere is not unity (n at sea level [Hue13]) and decreases with height. Therefore, the radiation emitted at different times and locations can reach suitable observers at the same time. This leads to a change of the signal time pulse (compression) and respectively an extension of the frequency spectra to higher frequencies. The diameter of this Cherenkov Ring (in the frequency space) depends on the distance of the observer to the shower axis. However, this effect should not be mixed up with the classical know Cherenkov effect caused by particles with β > 1 n. For a detailed discussion please read [DVea12]. Because of the flattening of the radio signal close to the shower axis a lateral distribution like in equation 3.3 leads to an inaccurate parametrization. Therefore a new distribution has to be found. In our case we use a Gaussian distribution in the form of: E total p 0 e ( p2 R SA p 1 ) 2 (3.4) The parameter p 2 gives the option to have the distribution s maximum not only at the shower core and provides the opportunity to fit the flattening. However, the scale parameter p 0 can not longer be directly interpreted as the shower s maximum radio signal.

21 3.2. Lateral Dependence 13 Figure 3.4.: lateral distribution of an near-vertical shower with Θ=25. The maximum signal is reached at the core of the shower. Figure 3.5.: lateral distribution of an inclined shower with Θ = 60. The maximum signal is reached for an axis distance bigger than 0.

22 14 3. Parametrization The Offset Parameter p 3 Every simulation result shows an almost constant radio signal for large distances to the shower core (depending on the shower geometry). This energy dependent offset is an artifact, caused by thinning effects in the simulation process and should not be handled as a measurable signal. However, it influences the fitting procedure. The distance where the thinning-caused offset dominates the radio signal varies for almost all showers, which makes it inconvenient to limit the range of the fit to valid values of the signal for every analyzed shower. To reduce the influence of this offset to the fitting procedure, another parameter p 3 is introduced which leads to the lateral distribution used in the parametrization of this thesis: E total p 0 e ( p2 R SA p 1 ) 2 + p 3 (3.5) As the offset parameter p 3 only scales with the energy, it has a constant value for different geometries and is not handled as a free fitting parameter. In figure 3.6 the advantage of the offset parameter p 3 can be seen. Since the offset parameter p 3 has only a technical use in the fitting procedure, it is excluded in the final parametrization. Figure 3.6.: inclined shower with Θ=60 coming from east, the blue line is the fit with the offset parameter p 3 using (3.5) as the fit function and green using (3.4) as the fit function with out the parameter p Dependence on the Shower s Geometry Beside the above discussed energy dependence there is a dependence of the radio signal with the shower s geometry Asymmetry Effects of the radio signal Figure 2.1 (in chapter 2.1) shows the electric field alignment for vertical showers caused by the effect of time-varying transverse currents (left) and time-varying charge excess

23 3.3. Dependence on the Shower s Geometry 15 (right). The superposition of the field vectors caused by the two dominating radio emitting processes, leads to an asymmetric electric field (mainly east-west asymmetry). For designated geometries, the two effects are in the same order and the lateral distribution is dominated by asymmetric effects. Figure 3.7 shows a shower with Θ = 50 and ϕ = 180. With sin α 0 the geomagnetic contribution becomes small and in the same order as the time-varying charge excess. Therefore, the signal is strongly asymmetric. Observers with the same distance to the shower axis but different cardinal direction have deviations in the electric field of up to a factor of 10. Such showers were left out of the parametrization process. Figure 3.7.: Strongly asymmetric shower with sin α 0 In general, all showers show an asymmetry in the signal up to %. These small variations of the electric field strength were not taken into account in the parametrization The sin α Dependence The geomagnetic angle α denotes the angle between the shower axis and the magnetic field (chapter 2.2). Most of the former parametrizations suggest a linear dependence between the sin α and the electric field strength induced by radio emission. This dependence has its origin in one of the key aspects of the radio emission processes, the v B-model. It suggest that the charged particles propagate along the shower axis and feel an accelerating force in the geomagnetic field - the Lorentz force. The force is calculated by: FL = q c v B (3.6) with v the velocity vector of the particle (on average corresponding to the shower axis), q its charge and B the vector of the magnetic field. Due to the fact that the secondary particles are accelerated a transverse current is induced (see chapter 2.1).

24 16 3. Parametrization A major part of the emitted radiation coming from extensive cosmic ray air showers is caused by this effect. The magnitude of the v B can be written as v B = v B sin α (3.7) an the electric field as this: E total v B sin α (3.8) Therefore the radio signal should show a linear dependence on the sin α. To study only the sin α dependence, we choose 4 inclined showers ( no. 5,6 and 8,9 in figure 3.1) with the same zenith angle Θ = 25 and the same primary particle energy E p = ev. As a result of changing the shower s azimuth angle the geomagnetic angle differs for every shower. Figure 3.8 shows the total field strength for an observer at a distance R SA = 50 m to the shower axis in dependence of the sin α of the different showers. If there was no observer for 50m, the electric field was interpolated using the next two closest observer. The electric field scales linearly with sin α as expected. Figure 3.8.: The absolute field strength for the same observer for four showers with different geomagnetic angles. The signal scales linearly with sin α. This dependence is used in the next chapter to find constant parameters for showers with different geomagnetic angle Parameters for arbitrary Geometries As discussed in chapter 3.2, the parameter p 2 allows the parametrization to take Cherenkov effects into account, which causes a rise of the radio signal at certain distance to the shower axis. As the Cherenkov effect is caused by the non-unity refractive index, it should only scale with the showers zenith angle (longer distance to the ground and therefore bigger influence of the refractive index) and the primary energy. All simulation results were fitted with the Gauss function (3.5). Figure 3.9 shows the resulting fit parameters for three showers with Θ=50 and various azimuth angles ϕ and

25 3.3. Dependence on the Shower s Geometry 17 therefore different geomagnetic angles α (number in 3.1). Against the expectations, the parameter p 2 changes significantly for different α. A reason can be found in the strong correlation of the three fitting parameters. The fitting results, with the deviation and the correlation matrix (calculated by GNUPLOT) is for shower no.13 (Θ = 50, ϕ = 135, α = 152 ): p 0 = ± ( 2.448% ) p 1 = ± ( 13.38% ) p 2 = ± ( 8.727% ) p 0 p 1 p 2 p p p The parameters have a small error but are strongly correlated. Because of the correlation, the independent parameters p 1 and p 2 are varying for the different showers. Another combination of the two parameters could lead to a fit with the same quality. However, other possible reasons for a dependency of the two parameters to the shower s azimuth angle ϕ, and respectively to its geomagnetic angle α, can not be excluded by this thesis. An investigation would be interesting for the further understanding of the phenomena (dependency on the shower s maximum or the Cherenkov effects), but is beyond the scope of this thesis. As a consequence, the mean values of the p 1 and p 2 are used for the parametrization. The fitting procedure was repeated with both parameters fixed, to find a new p 0. Figure 3.9.: The fit parameter p 0, p 1, p 2 for Θ = 50 for various ϕ and α. With the mean and the standard deviation (STD). The last column shows the p 0,fixed for values of p 1 and p 2 fixed to the mean for all 3 showers ϕ[ ] α[ ] p 0 [ µv m ] p 1 [m] p 2 [m] p 0,fixed [ µv m ] Mean: STD: % STD: The fitting procedure with the two fixed parameters shows almost no disadvantage to the one with the three free parameters as can be seen in figure At the last step we can find the dependence of the amplitude scaling parameter p 0 on the geomagnetic angle α. Figure 3.11 shows the correlation between the parameter p 0 and the sin α of 4 inclined showers with Θ = 25 and various azimuth angles ϕ. The linear dependence of the p 0 parameter on sin α is what we expected from chapter 2.2. Now we can find a constant p 0 for every Θ independent of the showers azimuth angle ϕ by dividing p 0 by sin α.

26 18 3. Parametrization Figure 3.10.: Comparison of both fitting procedures. method with the fixed parameters. There is no disadvantage to the Figure 3.11.: the linear dependence of p 0 and geomagnetic angle Table 3.12 shows the final parameters found for the different shower geometries. There is no obvious dependency of the two parameter p 1,p 2, but there is a clear sin α dependency for parameter p 0.

27 3.4. Overall Parametrization 19 Figure 3.12.: Overview of parameters for different shower geometries Θ [ ] p 0 [ µv m ] p 1 [m] p 2 [m] Overall Parametrization In this chapter we find an overall formula that represents the parametrization of a radio signal of extensive cosmic ray air showers. The parametrization is based on the simulations listed in table 3.1. The parametrized electric field is the magnitude of the three-dimensional vector of the electric field (eastwest, north-south and vertical). The field components are the pulse amplitudes filtered to a bandwidth of MHz. We assume that all dependences are uncorrelated and independent of each other and can be separated. We set together all the different dependences found in the previous chapters and get the overall parametrization: E(E p, R SA, Θ, α) ( ) [ 0.98 ( ) ] Ep p2 R 2 SA = p sin α exp (3.9) ev The unit of the electric field E is [ µv m ]. Where α is the geomagnetic angle, E p the primary particle energy, p 0 the scaling parameter found in chapter 3.4, p 2 and p 1 the Gaussian parameters which govern the steepness of the Gaussian decay of the radio signal and the position of the maximum electric field. A full list of the parameters can be found in table Figure 3.14 and 3.16 show lateral distributions of simulated data and the parametrization for two different sets of parameters. The shower in 3.14 is in good agreement to the parametrization, while the shower in 3.16 has problems with asymmetry effects. A precise quality check follows in the next chapter Quality Check To verify the quality of the parametrization, we calculate the absolute electric field for every observer with the parametrization and compare it with the simulated field by forming the decadic logarithm of the ratio: ( ) Esim r log = log 10 (3.10) E para The r log allows a direct quantification of the percental deviation of the parametrized to the simulated signal, even if deviations are large (the conversion can be seen in table 3.18). p 1

28 20 3. Parametrization Figure 3.13 shows the log-ratio plot for a shower with Θ = 60, E p = ev and a geomagnetic angle of α = 79 and figure 3.14 its lateral distribution of the simulated radio signal red crosses) and the parametrization (green dotted line). With a mean log-ratio of the systematic offset is under 0.2%. The maximum deviation of 17% shows, that the parametrization is in a good agreement with the simulated data. Most of the other shower from table 3.1 were in a good agreement with the found parametrization. Table 3.17 shows the calculated mean values of the log-ratios and their standard deviation, for showers with different energies and geometries. Most of the r log are smaller than 0.1 which means a average deviation of under 26%. However, vertical showers show a bigger deviation for distances bigger than 300m. Therefore, the parametrization is limited for distances to the shower axis smaller than 300m for vertical showers. In general, the validity of the parametrization is limited to the distance where the thinning effects (see chapter ) dominates the radio signal. Figure 3.15 and 3.16 show the log-ratio plot and the lateral distribution (with simulated data and parametrization) of shower no. 7. With α = 162, it is a shower almost parallel to the magnetic field (close to 180 ), where the effect of the time-varying charge excess has its biggest contribution to the radio signal emitted by the shower (same effect for showers with α close to 0 ). Therefore, it shows big asymmetries in its radio signal (chapter 3.3.1). With a mean of the log-ratios of r log = (with a standard deviation of σ rlog = 0.248) it reaches deviations of up to 1000% which means a factor of 10 over-estimated signal. The parametrization is not able to take asymmetries into account, and therefore is not valid for shower with big (or small) geomagnetic angles. Overall, the parametrization is in an satisfying agreement with the simulated data, and therefore can be used as tool to estimate the radio signal, emitted by extensive air showers coming from cosmic rays. For all showers, the log-ratio plots and the lateral distribution of the simulated data compared to the parametrization can be found in appendix A. 26 % 12.2 % 0 % 12.2 % 26 % Figure 3.13.: The log-ratio has a close distribution to the value of 0. With a mean log-ratio of the average percental deviation is under 0.2%. With a maximum deviation of 17%, the parametrization is in a good agreement with the simulated data.

29 3.5. Quality Check 21 Figure 3.14.: Shower with Θ = 60, α = 79 and E p = ev. The parametrization and the simulated data are in good agreement. The signal can be reconstructed even for high distances to the shower axis % 1000 % 316 % 0 % 316 % Figure 3.15.: Strongly asymmetric shower with the setting Θ = 25, E p = ev and α = 162. The parametrization has a tendency to over-estimate the signal and has an average deviation of more than 60%. The log-ratios reach values up to 0.9 which means a over-estimation of a factor of 8.

30 22 3. Parametrization Figure 3.16.: Shower with α = 162. The radio signal shows big asymmetries for observer with the same distances to the shower axis. The parametrization is not able to estimate the signal in a satisfying way and is therefore of limited use for showers with big (or small) geomagnetic angles α. Figure 3.17.: The quality check with the mean log-ratio and the standard deviation and their conversion to percental deviations. The last column is the range of validity in distance to the shower axis. Number Θ [ ] α [ ] E p [ev] rlog σ rlog rlog in [%] σ rlog in [%] validity R SA [m] E < E < E < E < E < E < E < E < E < E < E < E < E < E < E < E < 600

31 3.6. Polarization and the v B - Model 23 Figure 3.18.: Conversion from r log to the percental deviation. Because of the symmetry properties of the decadic logarithm, negative and positive values of the same magnitude belong to the same percental deviation. rlog deviation in % Polarization and the v B - Model For now, we only considered the magnitude of the electric field. But experiments and simulations provide the chance to measure and simulate the three different components of the electric field which are: E = north west vertical (3.11) Equation (3.8) shows the dependence between the electric field and the magnitude of the v B-vector. The polarization vector has the same direction as the Lorentz force vector: sin Θ sin ϕ cos Θ B FL = cos Θ sin Θ B sin Θ cos ϕ cos Θ B (3.12) sin Θ sin ϕ sin Θ B with Θ B, the zenith angle of the magnetic field. The unit vector of F L gives us the expected polarization vector of the electric field for pure geomagnetic emission: sin Θ sin ϕ cos Θ B cos Θ sin Θ B sin Θ cos ϕ cos Θ B sin Θ sin ϕ sin Θ B Ê(Θ, ϕ, Θ B ) = (3.13) sin 2 Θ sin 2 ϕ + (cos Θ sin Θ B sin Θ cos ϕ cos Θ B ) 2 The polarization vector Ê(Θ, ϕ, Θ B ) combined with the electric field parametrization from the previous chapter E(E p, R SA, Θ, α) leads to the formula: E(E p, R SA, Θ, α) = Ê(Θ, ϕ, Θ B ) E(E p, R SA, Θ, α) (3.14) Now we can split up a parametrized electric field in the three different components and compare it with the simulated data. For a proof of principle we chose a number of simulations. The polarization described by the v B is caused by the time-varying transverse currents. To analyze this model, it is useful to choose showers were the effect of transverse currents dominates over the effect

32 24 3. Parametrization of time-varying charge excess and the signal shows only a small asymmetry. Table 3.19 shows some examples with r n-s, r e-w and r vert the ratios of the simulated field component and the component calculated by (3.14): r = E sim E v B (3.15) Figure 3.19.: Ratios of the simulated and calculated field components for showers with different geomagnetic angles α and various distances R SA α[ ] R SA [m] r n-s r e-w r vert The results are contrary to the expectations of the v B-model. Even for a shower with α 90, the splitting of the signal is not reliable. There is no obvious reason, why this model does not work for the parametrized data. This fact suggest, that the simple v B-model is not longer valid for simulations like CoREAS and should be further investigated.

33 4. Conclusion Highly energetic particles from cosmic sources create extensive cosmic ray air showers in the Earth s atmosphere. The charged particles of the shower cascade interact with the geomagnetic field and induce time-varying transverse currents. The currents cause radiation in the MHz regime. The effect of time-varying charge excess in the shower cascade also causes radiation during the shower development. The sum of the radio emission can be measured with radio antennas. For further development of the radio detection technique and for verifying physical theories, which describe extensive air showers, Monte Carlo based simulations like REAS 3 and CoREAS are necessary. Models predict a variety of correlations of the measured radio signal with the properties of the cosmic ray air shower (eg. the primary particle energy and the shower geometry). These dependences should be found in simulation results and experimental data. This thesis studied several dependences of the radio signal simulated by CoREAS and found a parametrization, which allows a fast and easy way to estimate the radio signal for a given shower without time-consuming simulations, with the following parametrization: E(Ep, R SA, Θ, α) ( ) [ 0.98 ( ) ] Ep p2 R 2 SA = p sin α exp ev The parameters can be found in figure However, the parametrization is only valid for shower with an Auger like setup (1400m above sea level and the magnetic field configurations from chapter 3). As the quality check in chapter 3.5 showed, the calculated and simulated results are in a good agreement and the parametrization is only limited by showers near-parallel to the geomagnetic field (α close to 0 or 180 ). Therefore, the parametrization can be used as a good tool for approximative calculations, which was the goal of this thesis. Furthermore, the found dependences confirm most of the former parametrizations and support the underlying physical theories. p 1 25

34

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36 28 Bibliography [KC04] [KC05] [LH11] [LK66] [Lud11] [Lud13] KASCADE-Collaboration. The KASCADE-grande experiment and the LOPES project. Nucl.Phys.Proc.Suppl. 136 (2004) , KASCADE-Collaboration. Detection and imaging of atmospheric radio flashes from cosmic ray air showers. Nature, M. Ludwig and T. Huege. REAS3: Monte carlo simulations of radio emission from cosmic ray air showers using an end-point formalism. Astropart.Phys., 34: , I. Lerche and F.D. Kahn. Radiation from cosmic ray air showers. Proceedings of the royal society of London serieas a-mathematical and physical siences, M. Ludwig. Modelling of radio emission from cosmic ray air showers. PhD thesis, Karlsruher Institut fuer Technologie, M. Ludwig. Comparison of LOPES measurements with CoREAS and REAS 3.11 simulations. Proceeding ARENA Conference 2012, [Ost06] S. Ostapchenko. Quark gluon string jet - II model. Phys. Lett. B636, [Sea12] F. Schroeder et al. Tunka-rex: a radio antenna array for the tunka experiment (ARENA-conference 2012). Nuclear Instruments and Methods in Physics Research A., 2012.