Influence of atmospheric electric fields on the radio emission from extensive air showers

The atmospheric electric fields in thunderclouds have been shown to significantly modify the intensity and polarization patterns of the radio footprint of cosmic-ray-induced extensive air showers. Simulations indicated a very nonlinear dependence of the signal strength in the frequency window of 30–80 MHz on the magnitude of the atmospheric electric field. In this work we present an explanation of this dependence based on Monte Carlo simulations, supported by arguments based on electron dynamics in air showers and expressed in terms of a simplified model. We show that by extending the frequency window to lower frequencies, additional sensitivity to the atmospheric electric field is obtained.

Figure 1

A schematic structure of a thundercloud is given where charge is accumulated at the bottom and the top layers. An air shower (in red) is passing through the thundercloud. The LOFAR core is seen as a circular structure on the ground, where a few LOFAR antenna stations can be distinguished. The structure of the induced electric field is given schematically on the right-hand side.

Figure 2

Intensity footprints of ${10}^{15}\mathrm{eV}$ vertical showers for the 30–80 MHz band for the cases of no electric field (top), $E_{\parallel }=50\mathrm{kV}/\mathrm{m}$ (middle), and $E_{\parallel }=100\mathrm{kV}/\mathrm{m}$ (bottom).

Figure 3

Intensity footprints of ${10}^{15}\mathrm{eV}$ vertical showers for the 30–80 MHz band for the cases of $F_{\perp }=50\mathrm{keV}/\mathrm{m}$ (top) and $F_{\perp }=100\mathrm{keV}/\mathrm{m}$ (bottom).

Figure 4

The number of electrons and positrons at the shower maximum (blue left axis), the number of electrons at the shower maximum (yellow left axis), and the maximum $\sqrt{I}$ (black right axis) for vertical ${10}^{15}\mathrm{eV}$ showers (dashed lines) and for vertical ${10}^{16}\mathrm{eV}$ showers (right solid lines) as a function of the parallel electric fields. For the ${10}^{16}\mathrm{eV}$ showers, the number of particles and the pulse amplitude are scaled down by a factor of 10.

Figure 5

The number of electrons (solid blue line, left axis) and their drift velocity at $X_{\max }$ (dashed black line, right axis) of vertical ${10}^{15}\mathrm{eV}$ showers as a function of the net-transverse forces.

Figure 6

The square root of the power $\sqrt{I}$ at the ring of maximal intensity for vertical ${10}^{15}\mathrm{eV}$ showers (dashed line) as well as for vertical and inclined ${10}^{16}\mathrm{eV}$ showers (solid lines) as a function of the net transverse force. For the ${10}^{16}\mathrm{eV}$ showers, the $\sqrt{I}$ is scaled down by a factor of 10.

Figure 7

The median distance from the shower front of electrons for vertical ${10}^{15}\mathrm{eV}$, ${10}^{16}\mathrm{eV}$ and ${10}^{17}\mathrm{eV}$ showers in the absence of electric fields. The simple prediction based on Eqs. (5) and (13) is also shown.

Figure 8

The number of electrons is shown as a function of their energy for electric fields $E=0$, ${\mathbf{E}}_{\parallel }=50\mathrm{kV}/\mathrm{m}$, and ${\mathbf{E}}_{\parallel }=100\mathrm{kV}/\mathrm{m}$ as obtained from the analytical calculations (solid and dashed curves) and from full corsika simulations (markers) at $X=500\mathrm{g}/{\mathrm{cm}}^2$.

Figure 9

The distance by which the electrons travel in the absence and in the presence of the parallel electric fields from analytical calculations (solid and dashed curves) and from corsika simulations (markers).

Figure 10

The energy-loss time of electrons at $X=500\mathrm{g}/{\mathrm{cm}}^2$ (at $X_{\max }$) in the absence and in the presence of parallel electric fields.

Figure 11

The energy spectrum of electrons within a distance of 3 m (thin curves) and from 3 m to 10 m (thick curves) behind the shower front for $E=0$, $E_{\parallel }=50\mathrm{kV}/\mathrm{m}$ and $E_{\parallel }=100\mathrm{kV}/\mathrm{m}$ at the shower maximum from analytical calculations (a) and from full corsika simulations (b).

Figure 12

The signal strength, as obtained from CoREAS simulations, as a function of frequency for a vertical shower of ${10}^{15}\mathrm{eV}$ at 50 m from the core in the absence and in the presence of parallel electric fields. The black vertical lines represent the LOFAR LBA frequency window.

Figure 13

The median distance by which the electrons trail behind the shower front as a function of their energy in the absence and in the presence of a perpendicular electric field as obtained from corsika simulations (markers) in comparison to the model expectation (solid and dashed curves).

Figure 14

The number of electrons as a function of their energy that are within 3 m behind the shower front with and without perpendicular electric fields at $X=500\mathrm{g}/{\mathrm{cm}}^2$ as calculated from Eq. (29) (solid and dashed curves) and from the full corsika simulations (markers).

Figure 15

The drift velocity in the unit of the speed of light at $X_{\max }$ of the electrons lagging within a distance $\mathrm{\Delta }=1\mathrm{m}$ (a, b), $\mathrm{\Delta }=3\mathrm{m}$ (c, d) and $\mathrm{\Delta }=10\mathrm{m}$ (e, f) behind the shower front for small ($F_{\perp }=5\mathrm{k}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}$, (a, c, e)) and large ($F_{\perp }=100\mathrm{k}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{m}$, (b, d, f)) transverse net forces from analytical estimates (blue curves) and from corsika simulations results (markers). The colors of the dots represent the number of electrons.

Figure 16

The number of electrons within 3 m behind the shower front (left panel) and their mean drift velocity (right panel) as a function of transverse net forces from analytical calculations and from corsika simulations.

Figure 17

The same as displayed in Fig. 12 for different strengths of the net-transverse force.

Figure 18

Intensity footprints of ${10}^{15}\mathrm{eV}$ vertical showers for the 2–9 MHz band for the cases of no electric field (top), $E_{\parallel }=50\mathrm{kV}/\mathrm{m}$ (middle), and $E_{\parallel }=100\mathrm{kV}/\mathrm{m}$ (bottom).

Figure 19

Intensity footprints of ${10}^{15}\mathrm{eV}$ vertical showers for the 2–9 MHz band for the cases of $F_{\perp }=50\mathrm{keV}/\mathrm{m}$ (top) and $F_{\perp }=100\mathrm{keV}/\mathrm{m}$ (bottom).

Figure 20

The number of electrons with kinetic energy larger than 401 keV as a function of atmospheric depth for ${10}^{15}\mathrm{eV}$ vertical showers, for electric fields as indicated at several altitudes. The depth of 0 means very high up in the atmosphere, and the ground level is at the depth of $1036\mathrm{g}/{\mathrm{cm}}^2$.