Compton scattering in terrestrial gamma-ray flashes detected with the Fermi gamma-ray burst monitor

Terrestrial gamma-ray flashes (TGFs) are short intense flashes of gamma rays associated with lightning activity in thunderstorms. Using Monte Carlo simulations of the relativistic runaway electron avalanche (RREA) process, theoretical predictions for the temporal and spectral evolution of TGFs are compared to observations made with the Gamma-ray Burst Monitor (GBM) on board the Fermi Gamma-ray Space Telescope. Assuming a single source altitude of 15 km, a comparison of simulations to data is performed for a range of empirically chosen source electron variation time scales. The data exhibit a clear softening with increased source distance, in qualitative agreement with theoretical predictions. The simulated spectra follow this trend in the data, but tend to underestimate the observed hardness. Such a discrepancy may imply that the basic RREA model is not sufficient. Alternatively, a TGF beam that is tilted with respect to the zenith could produce an evolution with source distance that is compatible with the data. Based on these results, we propose that the source electron distributions of TGFs observed by GBM vary on time scales of at least tens of microseconds, with an upper limit of $\sim 100\mu \mathrm{s}$.

Figure 1

Evolution of the duration (crosses) and hardness ratio (circles) of the RREA simulations at an altitude of 565 km, as a function of the TGF source to subsatellite distance for a 15 km altitude wide beam source model. The duration is measured using the $T_{50}$, the time interval in which 50% of the flux occurs, starting and ending at 25% and 75% levels. The hardness ratio is given by the number of events with energy greater than 300 keV divided by the number of events with energy less than 300 keV. As the source off-axis distance is increased, a clear elongation in time and spectral softening is visible. This is due to the increased Compton scattering experienced by the photons as they propagate through a greater integrated density of atmosphere.

Figure 2

Photon fluence in BGO detectors as a function of the source to satellite distance. These values have not been corrected for dead time losses. The horizontal grey line represents the cut that was applied to ensure robust statistics for the analysis (see Sec. 4).

Figure 3

Results of the Bayesian block algorithm for TGF081123. The top panel shows the time profile, with the integrated counts in $20\mu \mathrm{s}$ bins shown in grey. The solid line is the optimum representation of the data. The bottom panel shows the probability distribution for each change point.

Figure 4

Comparison between duration calculated using Bayesian blocks ($T_{BB}$) and $T_{90}$ values for the intersection of TGFs used in this work and Connaughton et al. [17]. The shaded region indicates the 68% containment region.

Figure 5

Distribution of the observed delay between soft and hard counts.

Figure 6

Delay between soft and hard counts as a function of the distance between the source and the Fermi subsatellite position. The individual TGFs are plotted in grey, and the average in 50 km bins is plotted as cyan squares with the standard deviation plotted as an error bar. The values obtained in the simulations are plotted in orange. For the sake of clarity, the simulated values are offset from the data in 5 km steps. To indicate the spread of the simulated values, the standard deviation of the fit is plotted as an error bar. The different smearings for the simulations are indicated by the marker, circles for no smearing and squares for $100\mu \mathrm{s}$.

Figure 7

Hardness ratio as a function of the distance between the source and the Fermi subsatellite position. The data are plotted in the same fashion as Fig. 6. The expected softening with increased offset is evident. The values derived from the simulations with no smearing are significantly softer than the data. Simulations with longer smearing time scales ($>50\mu \mathrm{s}$) are more representative of the data, but still tend to underestimate the hardness. In the interest of clarity, only the results from the $100\mu \mathrm{s}$ simulations are overplotted.

Figure 8

Calculated duration as a function of the distance between the source and the Fermi subsatellite position. The data are plotted in the same fashion as Fig. 6. The durations from the simulations exhibit little variation with increased offset, and encompass the range of observed durations within their spread. The results from the $50\mu \mathrm{s}$ smearing agree with the mean of the data up to 300 km.