The shape of the radio wavefront of extensive air showers as measured with LOFAR
Introduction
A high-energy cosmic ray that enters the atmosphere of the Earth will interact with a nucleus of an atmospheric molecule. This interaction produces secondary particles, which in turn interact, thereby creating a cascade of particles: an extensive air shower. The origin of these cosmic rays and their mass composition are not fully known.
Due to the high incident energy of the cosmic ray, the bulk of the secondary particles propagate downward with a high gamma factor. As this air shower passes through the atmosphere and the Earth’s magnetic field, it emits radiation, which can be measured by antennas on the ground in a broad range of radio frequencies (MHz–GHz) [1], [2], [3]. For a review of recent developments in the field see [4]. The measured radiation is the result of several emission processes [5], and is further influenced by the propagation of the radiation in the atmosphere with non-unity index of refraction [6]. Dominant in the frequency range considered in this study is the interaction in the geomagnetic field [7], [8], [3], [9]. An overview of the current understanding of the detailed emission mechanisms can be found in [10].
The radio signal reaches the ground as a coherent broadband pulse, with a duration on the order of 10 to 100 ns (depending on the position in the air shower geometry). As the radio emission originates effectively from a few kilometers in altitude, the incident wavefront as measured on the ground is non-planar. Geometrical considerations suggest that the amount of curvature and the shape of the wavefront depend on the height of the emission region, suggesting a relation to the depth of shower maximum, . The depth of shower maximum is related to the primary particle type.
Assuming a point source would result in a spherical wavefront shape, which is used for analysis of LOPES data [11]. It is argued in [12] that the actual shape of the wavefront is not spherical, but rather conical, as the emission is not point-like but stretched along the shower axis. In a recent further refinement of this study, based on CoREAS simulations, evidence is found for a hyperbolic wavefront shape (spherical near the shower axis, and conical further out) [13]. Hints for this shape are also found in the air shower dataset collected by the LOPES experiment [14]. However, due to high ambient noise levels, the timing precision of these measurements did not allow for a distinction between spherical, hyperbolical and conical shapes on a shower-by-shower basis, and only statistically was a hyperbolic wavefront shape favored.
We use the LOFAR radio telescope [15] to measure radio emission from air showers, in order to measure wavefront shapes for individual showers. LOFAR consists of an array of two types of antennas: the low-band antennas (LBA) sensitive to frequencies in a bandwidth of 10–90 MHz, and the high-band antennas (HBA) operating in the 110–240 MHz range. While air showers have been measured in both frequency ranges [16], [17], this study only uses data gathered with the 10–90 MHz low-band antennas. A combination of analog and digital filters limits the effective bandwidth to 30–80 MHz which has the least amount of radio frequency interference. For detecting cosmic rays we use the (most densely instrumented) inner region of LOFAR, the layout of which is depicted in Fig. 1. LOFAR is equipped with ring buffers (called Transient Buffer Boards) that can store the raw-voltage signals of each antenna for up to 5 s. These are used for cosmic-ray observations as described in [16].
Inside the inner core of LOFAR, which is a circular area of 320 m diameter, an array of 20 scintillator detectors (LORA) has been set up [18]. This air shower array is used to trigger a read-out of the Transient Buffer Boards at the moment an air shower is detected. The buffer boards provide a raw voltage time series for every antenna in a LOFAR station (a group of typically 96 LBA plus 48 HBA antennas that are processed together in interferometric measurements), in which we identify and analyze the radio pulse from an air shower. Analysis of the particle detections delivers basic air shower parameters such as the estimated position of the shower axis, energy, and arrival direction.
The high density of antennas of LOFAR, together with a high timing resolution (200 MHz sampling rate) are especially favorable for measuring the wavefront shape.
Section snippets
Simplified model for the wavefront shape
Inspection of the pulse arrival times in our datasets (as explained in the following sections) shows that while the shape at larger distances from the shower axis might be described by a conical wavefront, in many measured air showers there is significant curvature near the shower axis. A natural choice for a function of two parameters that describes this behavior is a hyperbola. In Fig. 2 a toy model is sketched, where the wavefront is formed by assuming the emission to be generated by a point
Measurements
For this analysis we have used air-shower measurements with LOFAR accumulated between June 2011 and November 2013. These consist of 2 to 5 ms of raw voltage time series for every antenna of the LOFAR core stations; we identify the air shower’s radio pulse in every individual trace, and measure its strength and arrival time.
In order to have a dense, high-quality sampling of the radio wavefront, and a substantial distance range of more than 150 m, we require an air shower to be detected in at
Reconstructing the wavefront shape
From the arrival time of the pulse in different radio antennas, and the information from the particle detector array, we find the shape of the wavefront using the following procedure.
Results
An example shower is shown in Fig. 5. This plot shows the layout of the LOFAR low-band antennas in the inner-core region. The colors show deviations from the best-fitting plane-wave solution, increasing outward from the center of the array.
Conclusions
We have shown that the wavefront of the radio emission in extensive air showers is measured to a high precision (better than 1 ns for each antenna) with the LOFAR radio telescope. The shape of the wavefront is best parametrized as a hyperboloid, curved near the shower axis and approximately conical further out. A hyperbolic shape fits significantly better than the previously proposed spherical and conical shapes.
Reconstruction of the shower geometry using a hyperbolic wavefront yields a more
Acknowledgments
The authors thank Eric Cator for a useful discussion on statistical tests, and thank the anonymous referee for constructive comments. The LOFAR Key Science Project Cosmic Rays very much acknowledges the scientific and technical support from ASTRON.
Furthermore, we acknowledge financial support from the Netherlands Research School for Astronomy (NOVA), the Samenwerkingsverband Noord-Nederland (SNN) and the Foundation for Fundamental Research on Matter (FOM) as well as support from the Netherlands
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