Although it is clear from Fig. 3 that using frequency bands that are higher than the standard band (30–80 MHz) will help us in enhancing the signal-to-noise ratio, the exact band that should be used for maximizing the chances of observation is still unclear. It is of course, possible to measure in wide band frequencies, and then to digitally filter into the required frequency range. But this will increase the cost of the experiment considerably, since the usage of higher frequencies require a greater sampling rate and hence better communication facility, memory, ADC, etc. Thus, a detailed study is made to estimate the frequency range that will give a maximum signal-to-noise ratio (and thereby maximize the detection probability), and can hence be used for the experiment, which is the focus in the following section.
A close inspection of the shower footprint at higher frequencies reveals that there are three regions of interest: on the Cherenkov ring, inside the Cherenkov ring, and outside the Cherenkov ring. It is desirable to have a high value of SNR in all of these regions for maximizing the probability of detection in the entire antenna array.
A scan of the possible frequency bands that can be used for the measurement of air showers of energy 10 PeV is made. That is, we can construct a heat map of the SNR in different frequency bands. The frequencies for the heat map range from 30 to 150 MHz for the lower edge of the frequency band and from 80 to 350 MHz for the upper edge of the band. Such a scan is made for antenna stations at each region mentioned above. This is shown in Fig. 4 for a typical gamma-ray shower with a zenith angle of \(61^\circ \) and an energy of 10 PeV.
It is obvious that the typical frequency band of 30–80 MHz (lower left bins in Fig. 4) is not ideal for obtaining an optimal level of SNR. In the figure, the brightest zone for each region on the shower footprint shows the ideal frequency band, where a maximum range of SNR is obtained. Taking measurements at frequencies like 100–190 MHz gives a higher SNR. All bands where a value of SNR less than 10 is obtained are set to the color white, since this is the typical threshold for detection in an individual antenna station [33]. The bands with high SNR become especially crucial, when the energy threshold is attempted to be lowered. A map of the SNR that is measurable by the antennas is shown in Fig. 5. The black dots represent the 81 antennas considered in the simulations. The antennas considered for the frequency band scan in Fig. 4 are also marked here. The SNR map shown in the figure is obtained for the frequency band 100–190 MHz.Footnote 1
Showers of other zenith angles and other primaries also show a similar behavior in the frequency band scan. As the zenith angle and the primary type changes, there is a variation in the scaling of SNR. This results from the change in the total electromagnetic content (for different primary type) and the different spread of the signal strength on the ground (for different zenith angle). There is a direct relation between the spread in the diameter of the Cherenkov ring and the inclination of the shower. Thus, the frequency bands with a higher value of SNR are the same for showers of other primaries and other zenith angles as that for a shower of zenith angle \(61^\circ \) (shown in Fig. 4).
The observed signal-to-noise ratio in the antennas will depend on the energy of the shower, the zenith angle, and the azimuth angle (resulting in varying values of the Geomagnetic angle). The study of SNR in these parameter spaces is described in the following sections. The variation of the SNR with respect to the changing position of the shower maximum is not taken into account over here.
Dependence on the zenith angle
The evolution of the SNR with the zenith angle can be looked at for different frequency bands. This evolution is looked at for antenna stations at various perpendicular distances to the shower axis (which is equivalent to the radial distance of the antennas to the shower axis in the shower plane). Such an evolution is shown for zenith angles ranging from \(0^\circ \) to \(70^\circ \) in Fig. 6, for the bands 30–80, 100–190 and 50–350 MHz.
For the standard band of 30–80 MHz, the signal-to-noise ratio is significantly lower than that for the bands 50–350 and 100–190 MHz. Among all the three bands, the highest level of signal-to-noise ratio is obtained for 100–190 MHz for all zenith angles, as expected. In particular, for showers of greater inclination, a higher signal-to-noise ratio is achieved in most of the antennas if we use the higher frequency bands. The areas where the Cherenkov ring falls on the antennas are visible for the higher frequencies. These are the really bright regions seen for each zenith angle and appears only for the more inclined showers.
At lower zenith angles, a major part of the shower is lost because of clipping effects. The high observation level at the South Pole is the reason for the showers getting clipped off. The distance to the shower maximum at these zenith angles is about a few kilometers, while that for showers of \(70^\circ \) inclination is in the order of tens of kilometers. The clipping of the shower at lower zenith angles causes the radio emission to be underdeveloped for detection. This is also the reason for the appearance of the Cherenkov ring only for zenith angles \(\gtrsim 30^\circ \).
In Fig. 6, the distances of the antennas from the shower axis fall within the range of 50 m to approximately 520 m, but only the antennas with a SNR \(>10\) can detect these showers. For vertical showers, these are the antennas with distances of \(\approx \) 100 m and for inclined showers, these are the antennas that are even as far away as 500 m. This range corresponds to the required minimum spacing to detect these showers. That is, for vertical showers the antennas could at most have a spacing of 100 m and for inclined showers with \(\theta \gtrsim 60^\circ \) a spacing of 300 m is sufficient to achieve a threshold of 10 PeV.
It is a known feature that the farther the shower maximum is from the observation level, the greater is the radius of the Cherenkov ring. This is purely due to geometric effects of shower propagation. The propagation of the Cherenkov ring signature in the figure as the zenith angle increases is a manifestation of this. For an observation level of 2838 m above sea level, the average distance at which the Cherenkov ring falls is \(\mathrm {d}_{\mathrm {Ch}} \approx 250\) m for a shower of zenith angle \(70^\circ \) and is \(\mathrm {d}_{\mathrm{Ch}} \approx 150\) m for a shower of zenith angle \(60^\circ \).
The total energy fluence of the radio signal at the ground increases up to the zenith angle where clipping effects are no longer observed. On an average, it was seen that for 10 PeV gamma-ray showers, the total radiated energy does not get clipped-off for zenith angles greater than \(50^\circ \). For zenith angles greater than this, the total energy in the radio footprint remains nearly the same, but the area increases. This results in a lower power per unit area on the ground, causing a decrease in the SNR. The relatively lower signal-to-noise ratio for the \(70^{\circ }\) shower in Fig. 6 as compared to the \(60^{\circ }\) shower is an effect of this.
Dependence on the azimuth angle
Another parameter that the signal-to-noise ratio depends on is the azimuth angle of the shower. Variations in the azimuth angle result in changes in the geomagnetic angle. As a shower of zenith angle \(61^{\circ }\) covers a range of azimuth angles from −180 to 180 degrees, the geomagnetic angle (at the South Pole, where the magnetic field is inclined to the vertical direction by \(18^{\circ }\)) varies from \(43^{\circ }\) to \(79^{\circ }\). This leads to an amplitude variation by a factor of \(\frac{\sin (43^{\circ })}{\sin (79^{\circ })} = 0.7\). We find that for gamma-ray showers with these range of orientations and with an energy of 10 PeV, the maximum value of the SNR varied with a standard deviation of \(\sigma _{\mathrm{SNR}} = 264\) with a mean value 1518. That is, with changing the azimuth angle there is a variation in the maximum value of the SNR by 17.4\(\%\) about the mean. Apart from this, there is also a variation of the amplitude at a fixed azimuth angle due to shower-to-shower fluctuations which comes to 3.7\(\%\) on an average. This will be further discussed in Sect. 4.3.
We can thus infer that for inclined air showers at the South Pole, there is not a strong variation of the signal-to-noise ratio as the azimuth angle varies. This is shown in Fig. 7. Here, gamma-ray showers each with an energy of 10 PeV and an inclination of \(61^{\circ }\) and with varying azimuth angles are shown. Thus it is justified to study the other effects only at one particular azimuth angle.
Dependence on the primary energy
The signal that is observed by the antennas will obviously depend on the energy of the primary particle. The SNR becomes weaker as the energy of the primary particle decreases. The signal-to-noise ratio of showers with gamma-ray and proton primaries with energies ranging from 1 to 9 PeV, are shown in Fig. 8. These are showers with zenith angles of \(61^{\circ }\), \(40^{\circ }\) and \(70^{\circ }\), and are filtered to the band 100–190 MHz.
If we use the optimal frequency band like 100–190 MHz, we will be able to lower the threshold of detection down to 1 PeV for gamma-ray showers which have a zenith angle of \(61^{\circ }\). For detection, it is required that a minimum of three antennas have a SNR above 10. For \(61^{\circ }\) showers, we can achieve this, provided we have at least three antennas within a distance of \(\sim \) 50–180 m from the shower axis. This is mainly the area where the Cherenkov ring falls on the antenna array that gives a higher level of SNR. For proton showers of \(61^{\circ }\) inclination, it is possible to lower the energy threshold to the level of 2 PeV in the band 100–190 MHz.
In a similar manner, the energy threshold can be lowered for showers with zenith angles \(40^{\circ }\) and \(70^{\circ }\) as shown in Fig. 8. For showers with \(\theta = 40^{\circ }\), we need at least three antennas within a distance of \(\sim 80\) m from the shower axis. This means that a much denser array is needed in this case. In the case of the \(70^{\circ }\) showers, the minimum energy that can be detected is 2 PeV and is nearly independent of how dense the array spacing is.
The showers shown in Fig. 8 are sample showers in these energy ranges. They will also have shower-to-shower fluctuations, because of which the amplitude detected in each antenna station will differ. Taking such fluctuations into account, gamma-ray induced air showers with zenith angles of \(61^{\circ }\) and azimuth angles of \(0^{\circ }\) were simulated with 11 simulations at each energy. Figure 9 shows the fluctuations in the maximum SNR and the maximum amplitude for these gamma-ray showers with energies ranging from 1 to 9 PeV. This is shown in the figure for a frequency range of 100–190 MHz. These showers were seen to have an average relative standard deviation of the maximum SNR of 7.6\(\%\) for all energies. Similarly, a 3.7\(\%\) variation in the maximum amplitude is obtained.
It is seen that there is a clear correlation between the maximum SNR (or maximum amplitude) obtained and the energy of the primary particle. The maximum SNR was seen to be proportional to \(E^2\) and the maximum amplitude \(\propto E\). A fit of \(\mathrm {SNR}_{\max } = (17.04 \pm 0.43) \times E^{2.03 \pm 0.02})\) was obtained. Similarly, the maximum amplitude was seen to be related to the energy as \(\mathrm {Amp}_{\max } = (8.04\pm 0.10) \times E^{1.01 \pm 0.01})\)
Detection of air showers using the radio technique in the PeV energy range is something that has not been achieved so far. This study shows that such a detection is possible if the measurement is taken in the optimum frequency range, e.g. 100–190 MHz. This means that by using this frequency range, for radio air shower detectors, it is possible to search for gamma rays of PeV energy arriving at the South Pole, from the Galactic Center.
Such a method can indeed be used at other locations on the Earth, for increasing the probability of detection of lower energy air shower events. However, the Galactic Center may not be visible at all times. The exact threshold for detection may vary depending on the observation level, the magnetic field at these locations and the dimensions of the antenna array. The noise conditions of these areas will also affect the measurement. The use of the optimum frequencies will nevertheless increase the detection rate of inclined air showers and will lower the energy threshold. In addition, by using interferometric methods, the very conservative condition of SNR > 10 in 3 antennas can certainly be achieved.