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 Terrestrial gamma-ray flashes (TGFs) are associated with emission of detectable beams of electrons into space. In this paper we use simulations of TGF and electron beam emission and escape from the atmosphere to determine how the geometry and fluence of such events depend on the angular distribution of the source photons. Given a photon source, the geometry of the electron beam depends on the geomagnetic latitude of the source but can be well-predicted by tracing a disk at 57 km altitude along the geomagnetic field to satellite orbit. The fluence and geometry are then used to infer the relative detection probabilities of TGF and electron beam in the context of a variety of photon sources and intensities. Analysis of detection probabilities and the relative frequency of TGF and electron beam detections suggests the existence of a population of electron beams emitted by TGFs too faint to be detected as photons.
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 In spite of this theoretical and observational effort, the global frequency of TGF production is unknown. The observed frequency of satellite TGF observations sets a hard lower limit on the global frequency of 50/day assuming TGFs can be detected whenever they occur within 1000 km of the subsatellite point [Smith et al., 2005]. Assuming TGFs are only detectable within 300 km as suggested by lightning geolocations cited above requires a global frequency above 500/day [Carlson et al., 2009a]. Note that comparing 500 TGFs per day with the typical global lightning frequency of 45/s [Christian, 2003] indicates that on average at least 1 of every 104 lightning discharges produces a TGF. Such estimates of the true global frequency depend strongly on the width of the initial directional distribution (beaming) of the TGF photons. Narrow beams are less likely to be detected by satellites and require a higher global frequency to account for existing observations. The beaming of TGF photons also carries information about the geometry of the TGF source, with broader beams associated with divergent small-scale fields and narrower beams with relatively uniform large-scale fields. Unfortunately, the beam properties are not resolved in existing satellite measurements, though existing efforts point to broader beams [Carlson et al., 2007; Carlson, 2009; Hazelton et al., 2009; Gjesteland et al., 2011]. As such, new sources of information are needed to address this topic.
 One such source of information is the electrons that escape to space along with the photons during a TGF. These electrons either directly come from the strong electric field regions that produce the TGF or are produced indirectly as a consequence of the physics of gamma-ray propagation in the atmosphere. TGF gamma-rays undergo Compton scattering, pair production, and photoelectric absorption to release energetic electrons. Regardless of the source, such electrons are only likely to escape the atmosphere if they are produced at high altitudes, at least above ∼35 km where the collision frequency is less than the electron gyrofrequency [Lehtinen et al., 1999]. As TGF photon emissions seem to originate at substantially lower altitudes, the electrons that escape to satellite altitude are most likely secondary electrons produced by TGF photons [Dwyer et al., 2008].
 Given that TGF photons both produce and are attenuated less than escaping electrons, relatively few electrons escape to satellite altitude. In spite of the smaller relative population size, the electron fluence remains comparable to the photon fluence because the geomagnetic field confines the escaping electrons to a much smaller region. While photons spread widely as they propagate to satellite altitude, electrons merely gyrate and travel along the geomagnetic field in a narrow beam. The result is a broad region of satellite orbit illuminated by gamma rays and a small region illuminated by electrons. A satellite within the region illuminated by gamma-rays will see a TGF, while a satellite struck by the electron beam will see a pulse of electrons and positrons with a broad energy distribution lasting ∼5–25 ms depending on the distance along the geomagnetic field from source to satellite [Dwyer et al., 2008; Carlson et al., 2009b].
 Such events have been observed, though much less frequently than TGFs due to the small size of the electron beam. For example, observations made by the Burst and Transient Source Experiment (BATSE) on board the Compton Gamma-Ray Observatory of a TGF-like pulse of energetic particles over the Sahara Desert can be well-explained by modeling a beam of electrons produced by a TGF in the conjugate hemisphere, as can an event off the coast of Japan [Dwyer et al., 2008]. The Gamma-ray Burst Monitor (GBM) on board the Fermi Gamma-ray Space Telescope has also seen such events and additionally measures a particle spectrum that shows a clear signature of positron annihilation in the spacecraft as would be produced if ∼15% of the incoming particles were positrons [Briggs et al., 2011]. Anomalous pulses of electrons observed by the SAMPEX spacecraft may also be TGF-associated electron beams [Carlson et al., 2009b].
 In the context of TGF beaming and global frequency, the focus of this paper is the geometry, fluence, and frequency of observation of electron beams as a possible source of constraints on TGF properties. We approach this topic by assuming various initial photon populations and simulating the resulting photon and electron escape to satellite altitude, estimating the properties of satellite observations of these electrons, and calculating average observation probabilities over the observed TGF distributions. The resulting fluence and detection frequency estimates help us interpret the frequency of TGF and electron beam observations, allow us to crudely constrain the properties of the photon source, and will raise questions about satellite detection efficiency and the global frequency of TGFs.
2. Electron Beam Simulations
 The electron emissions described above are simulated with a combination of the GEANT4 Monte Carlo [Agostinelli et al., 2003] and a deterministic guiding center motion simulation. Processes below 150 km altitude are handled with GEANT4, while electron and positron propagation above 150 km are handled with the guiding center motion simulation. The threshold of 150 km is chosen to limit the guiding center motion simulation to regions where electron motion is not significantly affected by collisions. The IGRF11 vector magnetic field model is used throughout [Finlay et al., 2010], while the GEANT4 simulations include all relevant physics of photon and electron propagation in a MSIS atmosphere [Hedin, 1991] assumed constant around a spherical Earth. The simulations are started with a point source of photons directed upward with solid angle distribution dN/dΩ ∝ exp(−θ2/2σ2) where θ is the zenith angle and σ is a measure of the width of the beam. Such point sources provide a lower limit on the size of the emission region, while extended sources can be treated by expanding the size estimates given by point sources. The initial photon energy distribution is taken from relativistic runaway electron avalanche simulations consistent with TGF observations [Carlson et al., 2007]. For simplicity, this spectrum is assumed not to vary significantly with direction as justified by very good spectral fits with nearly direction-independent spectra as in Hazelton et al.'s Figure 2. σ ranges from 5° to 60°, ranging from unrealistically narrow (narrower than allowed by electron scattering and bremsstrahlung beaming) to unrealistically broad (such broad emissions cannot explain the observed average spectra) and including the range that best fits observed TGF spectra [Carlson et al., 2007; Hazelton et al., 2009; Gjesteland et al., 2011]. The source is always placed at 20 km altitude, the highest altitude consistent with TGF spectra [Dwyer and Smith, 2005; Carlson et al., 2007; Gjesteland et al., 2010], though simulations at lower altitudes do not significantly alter our results. Photons that reach 550 km altitude are taken to have reached satellite orbit and are recorded. Electrons are recorded when they cross satellite altitude in both upward and downward directions, and are propagated until they either are absorbed by the atmosphere or are terminated by the simulation when they exceed a propagation time limit of 0.2 s. The net result of these simulations is a single record of photons reaching satellite altitude and a complete record of initial and subsequent electron crossings of satellite orbit up to the time limit.
 As we are interested in the frequency of detection of TGFs and electron beams, two results of the simulations are particularly relevant: the fluence (particles per area) of the emissions at satellite orbit and the area of satellite orbit illuminated by electrons and photons. The fluence of the emissions can easily be assessed by histogram or kernel density estimation. Example photon and electron fluence distributions at satellite altitude are shown in Figure 1. Electron fluence estimates are made in a plane perpendicular to the geomagnetic field and are propagated along the field to determine the corresponding fluence at satellite orbit. Photon fluence estimates are simply collected at satellite orbit. Note that the absolute fluence scale is arbitrary as the number of simulation particles is not the same as the number of particles in a TGF. However, the same fluence scale is used throughout Figure 1 and a single simulation gives both photons and electrons, so the relative fluence measurements in Figure 1 give the proper ratio of peak electron fluence to peak photon fluence. Provided the electron beam reaches satellite altitude, this ratio depends only weakly on the latitude and varies from 1 to 3, consistent with Dwyer et al. , and varies slowly with the initial beam width (σ).
Figure 1 shows electron fluence distributions for several different TGF source latitudes. The variety of areas at satellite orbit illuminated by electrons indicates that electron beams emitted at different latitudes are not all equally likely to be detected. As such, estimates of electron beam detection probability depend strongly on the location of the source. The relevant area at satellite orbit can be determined directly by Monte Carlo particle simulations, but can also be estimated by tracing along the geomagnetic field from a ring placed at 57 km altitude above the source as shown by the lines in Figure 1. This is a straightforward method of estimating the position and size of the region at satellite orbit in which an electron beam is observable.
 The simulations in Figure 1 all used photons beamed upward with σ = 40°. Narrower or broader initial beams of photons will produce different size electron beams and will illuminate different size areas at satellite orbit. The effective size of a particular electron beam is determined by the instrument detection threshold and therefore depends on the absolute luminosity of the source and on the minimum fluence (particles per area) detectable by the instrument in a given time window. The quantity of interest then is the area illuminated by electrons and photons with a fluence higher than the minimum detectable fluence as a function of both σ and the peak fluence. As such, we measure peak fluence in units of the minimum detectable fluence and determine the illuminated area as follows. For a given σ we simulate a TGF and the associated electron beam. The fluence of photons and electrons at satellite altitude is calculated as in Figure 1. Contours of the fluence relative to the peak fluence are then calculated for electrons and photons and the area enclosed by the contours is measured. We then measure the effective radius of the photon beam at satellite orbit by simply calculating the radius of the circle with equivalent area. We measure the electron beam size as the radius of the circle that, when traced from some altitude above the source to satellite orbit, intersects an area that matches the area enclosed by the given relative fluence contour. These circle traces match the electron beam locations found in simulation if the circles are started at 57 km altitude centered above the source as shown in Figure 1. This area measurement procedure is executed for each σ and each relative fluence contour to give a measure of the size of the electron and photon beams as a function of relative fluence and beaming angle, as desired. The relative fluence levels are then inverted to give the peak fluences (measured relative to the minimum detectable fluence) such that the given fluence contour falls at the minimum detectable fluence. These effective sizes are shown for four different σ in Figure 2. Note that for low peak electron beam intensities (e.g. less than 2 × the minimum detectable fluence for σ = 30°), the peak photon fluence is below the detection threshold and no TGF is detectable so the photon beam can be considered to have an effective radius of 0.
3. Detection Probability Estimates
 The effective size of the electron and photon beam does not itself determine the probability of TGF or electron beam detection. This probability depends on the area illuminated by a beam of that size and the likelihood of a satellite passing through that area during a TGF. This likelihood depends on the geographic and geomagnetic distribution of TGFs and the satellite orbit inclination and can be determined as follows. For a given TGF location, the effective size of photon and electron beams determines the area of satellite orbit illuminated by electrons and photons as described above. The probability of observation can then be determined by the fraction of randomly drawn satellite positions that fall within the illuminated areas. Here, satellite positions are drawn by selecting a random orbital phase, calculating the resulting satellite position, and placing Earth beneath the satellite with a random longitude. This scheme faithfully reproduces the density of satellite measurement positions. We model our simulation after the Fermi Gamma-ray Space Telescope with orbit inclination ∼26° and altitude 550 km, and correspondingly remove any randomly chosen positions that happen to occur in the South Atlantic Anomaly [Briggs et al., 2010, Figure 1].
 These probability measurements can be carried out for any TGF and for any effective electron and photon radii. Averaging over the distribution of TGF positions gives the average probability of detection of TGFs and electron beams as a function of the effective radii. We calculate the average over TGF positions simply as a mean over the RHESSI spacecraft locations for the 820 RHESSI TGFs given by Grefenstette et al. . While this is not an unbiased sampling of the true population of TGFs due to variations in sampling time, background, and trigger sensitivity with latitude and longitude, these biases are limited over the ±26° latitude range region outside the South Atlantic Anomaly covered by our simulated satellite. The probability of electron beam detection varies substantially among TGFs. Electron beams that graze satellite altitude illuminate a comparatively large area of satellite orbit and thus are more detectable, while roughly 50% of RHESSI TGFs occur at lower geomagnetic latitudes and thus produce electron beams that do not reach satellite altitude at all. Electron beams that pass through satellite orbit near the maximum latitude visited by the satellite are more likely to be detected as the satellite spends more time at high latitudes. These geometric effects are shown for 40 km effective electron beam size in Figure 3. Figure 4 shows the mean detection probabilities as lines while the box plots indicate the distribution of probabilities as measured for the set of RHESSI TGFs. As seen in Figures 3 and 4, though the electron beam detection probability varies widely due to variations in the size of the region illuminated, the majority of the electron beams that reach satellite altitude occur in regions where the probability of detection is reasonably constant.
 The average detection probability of TGFs and electron beams as a function of the effective radii can be used to predict the ratio of the number of observed TGFs to the number of observed electron beams, a = P(TGF)/P(electron beam).This ratio depends on the effective radii of electron and photon beams as shown by the thin roughly radial contours in Figure 5. The grey region in Figure 5 shows the effective radii resulting from various intensities of initial photon sources with various σ assuming the minimum detectable fluence of electrons and photons are equal. For instance, a TGF with an initial photon beaming angle σ = 40° and an luminosity such that the peak electron beam fluence is 20 × larger than the detection threshold will have an effective electron beam size of ∼55 km and an effective photon beam radius at satellite orbit of ∼600 km, while at the same peak fluence, σ = 10° gives a 22 km electron beam size and a 250 km TGF radius at satellite orbit. These points both fall near the a = 100 contour, indicating that such TGFs are likely to be detected ∼ 100 × more frequently as TGFs than electron beams.
 The main result of this paper is shown in Figure 5, which gives the ratio of the probabilities of detection of TGF and electron beam as a function of their effective sizes in the context of a variety of TGF intensities and initial beaming angles.
 Unfortunately, electron beam observations are still quite rare, so rigorous analysis must wait until more data are collected. Only crude comparisons can be made with available data. For electron beams, the most comprehensive data set thus far comes from the Fermi satellite as reported in Briggs et al. , which includes 77 TGFs and 6 electron beams, a ∼ 13 TGFs per electron beam (a similar fraction of electron beam events is seen in BATSE, see Dwyer et al. ). This is quite a large fraction of electron beams, and falls on Figure 5 near the a = 10 contour, which only intersects the grey region representing the results of our simulations for very small photon and electron beams. Since the GBM triggering algorithm assesses statistical significance over a 16 ms window, long enough to span most electron beams, the total fluence comparison made here should approximate the behavior of GBM: if the total fluence is significantly above the background count rate in the window, the event should trigger, whether TGF or electron beam. If the electron beam deposits its counts across multiple consecutive trigger windows, it is less likely to trigger. The excess of electron beams is therefore not simply an effect of the trigger algorithm unless there are additional nontrivial bias effects. One possible physical explanation, simply reading Figure 5, is that the effective photon beam radius at satellite orbit is ≲ 100 km as it would be if on average TGFs were just barely intense enough to be detected, though this is unlikely given the observed intensity distribution of TGFs [e.g., Grefenstette et al., 2009, Figure 8] and the large distances from which TGFs are sometimes observed [e.g., Cohen et al., 2010, Figures 1b and 1c]. A more likely explanation is that in addition to the distribution of observed TGF intensities, there is a population of low-intensity TGFs that are too faint to be detected as photons but still produce observable electron beams. Such faint TGFs could account for some of the detected electron beams and therefore reduce a to the observed value from the values expected solely from TGF observations. Other, more speculative explanations include a distinct population of exceptionally long-duration photon events or pulses of energetic electrons from some other unknown source.
 These conclusions are admittedly crude and will undoubtedly change as more data are collected and as more detailed analyses of detector response to electron beams become available. One key assumption we have made is that the satellite detection efficiencies for electrons and photons are equal. This implies a balance between lower electron detection efficiency at low energies due to attenuation outside the detector volume and higher electron detection efficiency at high energies for electrons penetrating the detector volume. An exact balance is unlikely, however. If the effective area for electrons is relatively lower than the effective area for photons, it will reduce the effective electron fluence below what we have assumed. Lower effective electron fluence will decrease the relative size of the electron beam, shifting the curves in Figure 2 (left side) to the left (increased effective TGF radius at satellite orbit for a given effective peak electron fluence). This will likewise shift the grey band in Figure 5 downward and decrease the peak fluence associated with the marked points. These effects are reversed if the effective area for electrons is instead relatively larger. For example, assuming a typical TGF radius of ∼300 km as suggested by lightning geolocation results, a ∼ 13 TGFs observations per electron beam observation implies an effective electron beam size of ∼60 km, requiring a very large upward shift of the grey region indicative of approximately 10 × higher detection efficiency for electrons than for photons. The frequency of detection of electron beams is thus not very sensitive to small vertical shifts of the grey region in Figure 5, so it is unlikely that electron detection efficiency concerns alone account for the unexpectedly high frequency of electron beam detection.
 Our analyses also assume upward-directed point sources of photons at 20 km altitude with direction-independent spectra. These assumptions could be relaxed. For instance, removing the assumption of upward-directed photon beams will complicate the analysis of electron and photon peak fluence: while both electron and photon peak fluence is affected by the zenith angle of the photon beam, electron peak fluence will additionally depend on the azimuth angle with respect to geomagnetic field. The relative frequency of observation of TGFs and electron beams should not be significantly affected, however, as a narrow photon beam will make narrow TGFs and narrow electron beams even if not directed upward.
 Regardless of these complications, the methods described above promise to shed new light on such basic properties of TGF production as beaming and overall frequency. For instance, the idea that there is a population of TGFs detectable as electron beams but too faint to be detected as TGFs implies that the effective size of such electron beams is <20 km. Figure 4 gives the mean probability of detection of such electron beams as <6 × 10−6, so the 6 electron beams observed by Fermi in 2 years of data imply at least 106 TGFs occurred globally during that interval. This amounts to ∼1400/day, a factor of 3 higher than the constraints suggested by the TGF observation frequency. We hope this analysis will motivate further searches for electron beam events and detailed comparisons of detection efficiencies for TGFs and electron beams. Upcoming missions with the ability to distinguish energetic electrons and photons, such as TARANIS and ASIM, may even detect both photons and electrons from the same TGF, allowing for even clearer constraint of source properties from such consideration of electron beams as described above.
 The authors gratefully acknowledge the referees for their insightful comments. This work was supported by Norwegian Research Council grant 197638/V30. Robert Lysak thanks the reviewers for their assistance in evaluating this paper.