Simulation of radio signals from cosmic-ray cascades in air and ice as observed by in-ice Askaryan radio detectors (2024)

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  Figure 1

  A sketch of the three different types of rays: a direct ray (solid line), an indirect refracted ray (dotted line), and an indirect reflected ray (dashed line). In reality, only one of the two indirect rays will be a solution of the ray tracing.

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  Figure 2

  A visualization of an interpolation table used for the in-ice ray tracing, for a given receiver position. Interpolation tables for the in-air ray tracing follow a similar structure, but instead cover a much larger area in air. A single in-air table is used for multiple receivers at the same depth in the ice.

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  Figure 3

  Left: visualisation of several rays traced from the receiver to a line of emitters, including an air-ice boundary at $z=0$. Direct rays are shown in blue. Reflected rays are only considered when total internal reflection occurs and are shown in red. Transmitted rays are shown in green. Right: the boost factor estimations along with the numerically computed value of $\frac{dt}{d{t}^{\prime }}$ as a function of depth. The diamonds correspond to the calculations using $\widehat{r}$ pointing directly from emission point to receiver point. The stars correspond to the values of the boost factor when evaluating $n$ at the emission point and using the launch direction for $\widehat{r}$. The dots correspond to the numerically calculated values of the boost factor. The different regions are associated with direct (blue), reflected (red), and transmitted rays (green).

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  Figure 4

  A visualization of the variables used in the end-point formula including ray tracing [Eq.(3)], in the specific case where the start point, end point, and receiver coincide in the same plane. The circles represent the start and end point of the emitter step; the square represents the receiver. All vectors shown in the illustration lie in the same plane.

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  Figure 5

  A visualization of the relation between a global Cartesian coordinate system and a local spherical coordinate system, defined by the incoming direction of the ray.

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  Figure 6

  The fluence footprint of the simulated cosmic-ray shower at a depth of 100m for the in-air emission (left), in-ice emission (middle), and the combined emission (right), using a primary energy of $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$. The simulation was performed using 121 antennas in a star-shaped grid with eight arms and an antenna spacing of 10m, indicated by the white dots. For the interpolation of the star grid we used the code described in [57].

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  Figure 7

  The electric field components as a function of time for four different antennas positioned along the west axis at a depth of 100m, using a primary energy of $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$. The coordinate of each antenna is indicated at the top of the plots. The blue line indicates the in-air emission, and the red line indicates the in-ice emission. Note that the range on the y axis is different for every plot.

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  Figure 8

  The electric field components as a function of time for four different antennas positioned along the north axis at a depth of 100m, using a primary energy of $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$. The coordinate of each antenna is indicated at the top of the plots. The blue line indicates the in-air emission, and the red line indicates the in-ice emission. Note that the range on the y axis is different for every plot.

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  Figure 9

  The fluence on the west axis of the simulated cosmic-ray shower at a depth of 100m for the in-ice emission only using a constant ice density of $359\mathrm{kg}/{\mathrm{m}}^3$ and a constant index of refraction $n=1.35$, compared to the result using ray tracing with an exponential density and index of refraction profile. In both cases an eighth order digital Butterworth bandpass filter for a frequency band of 300–1000MHz was applied. The simulation was performed using 201 antennas using an antenna spacing of 1.5m, with a primary energy of $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$.

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  Figure 10

  The fluence on the north axis (left) and west axis (right) of the simulated cosmic-ray shower at a depth of 100m for the combined in-air and in-ice emission when applying an eighth order digital Butterworth bandpass filter, for the frequency bands 30–100MHz (upper), 100–300MHz (middle), and 300–1000MHz (lower). The simulation was performed using 201 antennas each on the north and west axis using an antenna spacing of 1.5m, with a primary energy of $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$.

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  Figure 11

  The fluence footprint of the simulated cosmic-ray shower at a depth of 100m for the in-air emission (left), in-ice emission (middle), and the combined emission (right), using a primary energy of $E_p={10}^{18}\mathrm{eV}$ and zenith angle $\theta =0$. The simulation was performed using 121 antennas in a star-shaped grid with eight arms and an antenna spacing of 10m, indicated by the white dots. For the interpolation of the star grid we used the code described in [57].

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  Figure 12

  The electric field components as a function of time for two different antennas positioned along the west axis at a depth of 100m, using a primary energy of $E_p={10}^{18}\mathrm{eV}$ and zenith angle $\theta =0$. The coordinate of each antenna is indicated at the top of the plots. The blue line indicates the in-air emission, and the red line indicates the in-ice emission. Note that the range on the y axis is different for every plot.

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  Figure 13

  The difference in arrival time of air and ice radio pulses as a function of distance to the shower axis for varying depths as indicated by the legend, using a primary energy $E_p={10}^{17}\mathrm{eV}$ and a zenith angle $\theta =0$. The arrival time was defined as the time where the Hilbert envelope reaches 33% of its maximum value, after passing an eighth order digital Butterworth bandpass filter for the frequency band 30–1000MHz, averaged over the three components.

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  Figure 14

  The number of particles (left) and the distribution of the energy (right) in the air shower as a function of depth with primary energy $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$. The particle distributions are obtained over the full radial extent of the in-air particle cascade. Similar results were shown in [22].

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  Figure 15

  The number of particles (left) and the distribution of the energy (right) in the air shower as a function of depth with primary energy $E_p={10}^{18}\mathrm{eV}$ and zenith angle $\theta =0$. The particle distributions are obtained over the full radial extent of the in-air particle cascade. Similar results were shown in [22].

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  Figure 16

  The total energy contained within a given radius from the shower core at the air-ice boundary up to 100m (left) and in more detail up to 5m (right), for the simulated shower with a primary energy $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$ and the simulated shower with a primary energy $E_p={10}^{18}\mathrm{eV}$ and zenith angle $\theta =0$. The energy is expressed as a fraction of the primary energy $E_p$. The air-ice boundary is located at an altitude of 2.835km, which corresponds to a vertical atmospheric depth of $729\mathrm{g}/{\mathrm{cm}}^2$.

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  Figure 17

  The energy deposited in ice by the in-ice particle cascade for the simulated shower with a primary energy $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$ (left) and the simulated shower with a primary energy $E_p={10}^{18}\mathrm{eV}$ and zenith angle $\theta =0$ (right). Shown here is the deposited energy density within a vertical 1-cm wide slice going through the center of the particle shower. The air-ice boundary is located at an altitude of 2.835km, which corresponds to a vertical atmospheric depth of $729\mathrm{g}/{\mathrm{cm}}^2$. Similar results were shown in [22].

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  Figure 18

  A distribution of the CPU time of 591 cores used for the simulation of the in-ice particle cascade and the corresponding radio emission for 121 antennas in the ice, using a primary energy $E_p={10}^{17}\mathrm{eV}$ and zenith angle $\theta =0$.

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