We have shown that for simulations in an atmosphere with a varying refractive index the radio symmetry center is systematically displaced from the MC impact point. In this section we show that this displacement is in agreement with refraction of radio waves in a refractive atmosphere as described by Snell’s law (cf. Eq. (4)). For this purpose we develop a model simulating the propagation of a single electromagnetic wave. Furthermore we summarize and validate the treatment of the refractivity in CoREAS and discuss the validity of our model.
Description of refraction using Snell’s law
We study the propagation of a single electromagnetic wave through the Earth’s atmosphere described by a curved trajectory undergoing refraction according to Snell’s law. For this purpose we assume discrete changes of the refractive index along the edges of imaginary layers throughout the atmosphere. The propagation within a layer with an upper edge height \(h_i\) is described by a straight uniform expansion with the phase velocity \(c_n = c_0 / n(h_i)\) given the refractive index as function of the height above sea level n(h). We adopt the refractive index as frequency-independent (i.e., non-dispersive) in the band from 30 to 200 MHz that we consider here. The change of direction between two layers with \(n_1\) and \(n_2\) is described in terms of the incidence angle (\(\vartheta \)) from \(\vartheta _1\) to \(\vartheta _2\) following Snell’s law
$$\begin{aligned} \frac{\sin \vartheta _2}{\sin \vartheta _1} = \frac{n_1}{n_2}. \end{aligned}$$
(4)
The refraction is calculated in a curved atmosphere. The relationship between the geometrical distance from ground \(d_\text {g}\) and height above ground \(h_\text {g}\) is given for a zenith angle \(\theta \), observation level \(h_\text {obs}\) and the Earth’s radius \(r_\text {earth} = 6371 \mathrm{km}\) by
$$\begin{aligned} d_\text {g}^2 + 2 (r_\text {earth} + h_\text {obs}) (d_\text {g} \cos \theta - h_\text {g}) - h_\text {g}^2 = 0. \end{aligned}$$
(5)
By solving this quadratic equation, one can calculate the height above sea level for every given distance to the ground by \(h = h_\text {g}(d_\text {g}, \theta ) + h_\text {obs}\). The refractivity \(N \equiv n - 1\) at a given height is calculated according to the density profile of the given atmospheric model and the given refractivity at sea level (\(N_{0}\)),
$$\begin{aligned} N(h) = N_{0} \frac{\rho (h)}{\rho (0)}. \end{aligned}$$
(6)
We employ an atmospheric model with four exponential layers and one linear layer as used in CORSIKA/CoREAS, implementation from
[26, 27]. The thickness of each layer is set to 1 m assuring a high accuracy of the calculation.Footnote 3
To predict the magnitude of a symmetry center displacement by refraction, we simulate the propagation of an electromagnetic wave along a bent trajectory with an initial direction aligned to the MC axis for a shower with a given zenith angle, towards the ground plane. The intersection between the bent trajectory and the ground plane is compared to the intersection between the ground plane and the MC axis. Given these two points, the symmetry center displacement can be inferred as depicted in Fig. 8. In Fig. 9 the predicted symmetry center displacement along the ground plane is shown as function of the geometrical distance along the MC axis for shower geometries with a zenith angle between \(65^{\circ }\) and \(85^{\circ }\). The orange line symbolises the displacement for a source at a fixed slant depth of \(750\,\,\hbox {g}/\hbox {cm}^2\) (e.g., shower maximum, average depth of maximum of our set of simulated showers). For a given slant depth, this distance translates to a zenith angle (top x-axis). Our model predicts a displacement of the order of 1.5 km for the most inclined showers at \(\theta = 85^{\circ }\). In orange squares the displacement is shown for different slant depths between 620 and \(1000\,\,{\hbox {g}/\hbox {cm}^2}\) (typical range in our set of simulated showers) along the MC axis for 5 different zenith angles (\(\theta \) = \(65^{\circ }\), \(75^{\circ }\), \(80^{\circ }\), \(82.5^{\circ }\), \(85^{\circ }\)). The model predictions are compared to the displacements determined from the CoREAS simulation set (colored circles: cf. Sect. 3, Fig. 4). The displacement is reasonably described by our model in terms of the overall magnitude (orange line) as well as the slope as function of the source’s slant depth (orange squares). In the bottom frame we show the absolute residuals between CoREAS displacement and refractive model. For their calculation we interpolated the model prediction along the orange squares to match the actual slant depth of the shower maximum of the simulated air showers. The residuals show no strong correlation with depth of shower maximum and increase up to \(\sim 250\,\,\mathrm{m}\) for the most inclined showers. Furthermore, our model predicts a displacement always towards the shower incoming direction. This corresponds to a refraction towards the ground, i.e., decreasing angle of incidence, which is given by a radially symmetric atmosphere (cf. Fig. 8). In Fig. 5 this behaviour was also observed for CoREAS simulations as the simulated showers exhibit a radio symmetry center displacement almost entirely in the incoming direction of the shower. As emphasised earlier, an East-West asymmetry as seen in CoREAS simulations, cf. Figs. 4 and 5, cannot be described by refractivity.
We verified that the impact of the atmospheric model, i.e. the density profile, is below 3% between the US Standard Atmosphere after Keilhauer and the Malargüe October atmosphere
[20]. Comparing different observer altitudes we find no difference for the displacement as function of \(d_\text {max}\). As already shown in Fig. 1, the refractivity at sea level has an influence on the predicted displacement. The yearly fluctuations of the air refractivity at the site of the Pierre Auger Observatory amount to 7%
[2]. Varying \(N_0\) over a range of \(\pm 15\%\) we find the displacement to scale linearly with \(N_0\).
Refraction and its treatment in CoREAS
For the numerical calculation of the radio emission of an extensive air shower for an observer at ground, the refractive index has to be taken into account for two processes: first, in the generation of the radio emission for each particle; and second, in the the propagation of each electromagnetic wave from a source to an observer. In CoREAS, the former is realistically included in the calculation of the radio emission from each particle using the endpoint formalism
[17, 28, 29]. However, the treatment of the propagation is approximated. Since electromagnetic waves in the radio regime do not suffer from any significant attenuation effects while propagating through air, this propagation is described entirely by two quantities. First, the geometrical distance (d), that the radio wave passes between source and observer, as the intensity of the emission scales with this distance. And second, the light propagation time (\(t_n\)) between source and observer which is of crucial importance as it governs the coherence of the signal seen by an observer from the full air shower. In CoREAS, \(t_n\) is calculated taking into account a refractive index dependent (phase-) velocity of the emission
$$\begin{aligned} t_n = \frac{1}{c} \int _{\text {source}}^{\text {observer}} n(h(\ell ))\,\mathrm {d}\ell . \end{aligned}$$
(7)
To calculate both quantities, CoREAS assumes a straight path between source and observer (cf. Fig. 8: dashed line). This approximation has implications for the geometrical distance between source and observer as a straight line underestimates the real distance along a curved trajectory. For the calculation of the light propagation time an additional implication arises from the fact that the average refractivity along a straight line varies from the refractivity along a curved trajectory. We find that the average refractivity and consequently the light propagation time is overestimated along straight trajectories.
We stress that the description of the propagation of the radio emission along straight trajectories in CoREAS is not in contradiction with the above-established refraction of the radio emission and the resulting displacement of the radio symmetry center in CoREAS simulations. In fact, the refraction of radio waves is a consequence of the fact that the propagation velocity changes with the refractive index \(c_n\). It is in fact possible to achieve a displacement of the whole coherent signal pattern at ground by an accurate description of the light propagation time along straight trajectories, as we will demonstrate below.
To verify if the calculation along straight trajectories between source and observer is sufficiently accurate to calculate the radio emission seen from a full extensive air shower, we determine the geometrical distance and light propagation time following bent and straight trajectories (cf. Fig. 8) for several geometries. We simulate the propagation of an electromagnetic wave given incoming direction and atmospheric depth along a bent trajectory towards the ground plane. Once the trajectory intersects with this ground plane the process is stopped and \(d_{\text {curved}}\) and \(t_{\text {curved}}\) are calculated via a sum of \(d_i\), \(t_{n_i}\) over all layers. Given the intersection and the initial starting point in the atmosphere, a straight trajectory is defined and \(d_{\text {straight}}\) and \(t_{\text {straight}}\) are calculated for comparison.
In Fig. 10 (Top) the geometrical distance is compared between curved and straight trajectories in absolute terms of \(d_{\text {curved}} - d_{\text {straight}}\), given the ambient conditions used for the above introduced simulation set. The comparison is shown as function of the geometrical distance along the straight trajectory between source and observer. The source positions are set to be at an atmospheric depth of \(X = 750\,\,{\hbox {g}/\hbox {cm}^{2}}\) for incoming directions with zenith angles between \(65^{\circ }\) and \(85^{\circ }\). We obtain a maximal error of around 4 cm for the most inclined geometries with a path distance of \(\sim \) 150 km. With a relative deviation of less than \(1\times 10^{-6}\) this approximation is therefore completely suitable.
For the light propagation time, the relative difference for two source positions and one observer position between curved and straight trajectories \(\sigma _t = \varDelta t_\text {curved} - \varDelta t_\text {straight}\) is of relevance as it governs the coherence of the total signal seen by a given observer. We do not have an analytic description for curved trajectories, however we can employ our model to determine the observer position at ground \(\mathbf {O}(\mathbf {P}, \hat{\theta })\) for every given source position \(\mathbf {P}\) and initial direction \(\hat{\theta }\). Hence, to find two sources connected with curved trajectories to one observer at ground we have to find the initial direction \(\hat{\theta }_2\) for a given second source which defines a trajectory that connects this source to an observer given by the first source and direction \(\mathbf {O}_1(\mathbf {P}_1, \hat{\theta }_1)\). For this purpose we employ a root-finding algorithm that solves the following equation for \(\hat{\theta }_2\): \(\mathbf {O}_2(\mathbf {P}_2, \hat{\theta }_2)- \mathbf {O}_1(\mathbf {P}_1, \hat{\theta }_1) = 0\) (\(\mathbf {P}_1,\mathbf {P}_2,\hat{\theta }_1\) are fixed). When the correct \(\hat{\theta }_2\) is found, i.e., \(\mathbf {O} = \mathbf {O}_1 = \mathbf {O}_2\), the propagation between \(\mathbf {P}_1\) or \(\mathbf {P}_2\) and \(\mathbf {O}\) is evaluated for curved and straight trajectories and \(\sigma _t\) is calculated. Figure 10 (bottom) shows \(\sigma _t\) for different geometries and configurations of \(\mathbf {P}_1\) and \(\mathbf {P}_2\). The timing error \(\sigma _t\) between two sources which are located on one axis with a zenith angle between \(65^{\circ }\) and \(85^{\circ }\) and depths of 1000 and \(400\,\,{\hbox {g}/\hbox {cm}^2}\) is shown by the orange line. This range of atmospheric depths covers the bulk of the radiation energy release from the longitudinal development of an extensive air shower
[2, Fig. 5]. In the most extreme case, the error in the relative arrival times for a source at the beginning of the shower evolution and a source at the end of the shower evolution, estimated using straight tracks, amounts to \(\sigma _t \lesssim 0.1 \,\,\mathrm{ns}\). This is well below the oscillation time of electromagnetic waves in the MHz regime. The blue line in the same figure demonstrates the errors made for sources laterally displaced by a shift of \(\pm 655\,\,\mathrm{m}\) above and below the shower axis along an axis perpendicular to the shower axis at a depth of \(750\,\,{\hbox {g}/\hbox {cm}^2}\). This value was chosen such that it matches the Molière radius expected for a shower with \(\theta = 85^{\circ }\) at \(X_\text {max} = {750}\,\,{\hbox {g}/\hbox {cm}^2}\)
[30]. The errors due to the straight-line approximation are even much smaller.
While it may seem paradoxical on first sight that a calculation approximating propagation of electromagnetic waves along straight tracks can yield refractive ray bending, we have shown that the relevant calculation of relative arrival times is described well within the needed accuracy, i.e., is fully adequate for this purpose. We note that, similar to our findings, it was already found based on analytic calculations in reference
[31, cf. Fig 9] that a straight-line approximation is sufficient for the calculation of relative arrival times of radio waves in extensive air showers.
Additionally the refractive ray bending changes the incoming direction of the radio emission. This has implications for the reconstruction of the radio emission with real radio antennas as their response pattern is direction-dependent. We find a maximum change of direction of \(\sim 0.14^{\circ }\), which is in agreement with
[32]. This is below current experimental accuracy as well as the change in the incoming direction between early and late observers on the ground plane, estimated as \(\mathcal {O}(1^{\circ })\) for a \(85^{\circ }\) shower.