Confidential manuscript submitted to JGR

1 Pitch Angle Dependence of Energetic Electron :

2 Energy Deposition, Backscatter, and the Bounce Loss Cone

1 2 3 R. A. Marshall and J. Bortnik

1 4 Ann and H. J. Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 5 80309, USA. 2 6 Department of Atmospheric and Oceanic Sciences, University of California Los Angeles, Los Angeles, CA 90095, USA.

7 Key Points:

• 8 We characterize energy deposition and atmospheric backscatter of radiation belt 9 electrons as a function of energy and pitch angle • 10 We use these simulations to characterize the bounce loss cone and show that it is 11 energy dependent • 12 The simulated backscatter of precipitation is characterized by field aligned beams 13 of low energies which should be observable

Corresponding author: R. A. Marshall, [email protected]

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14 Abstract

15 Quantifying radiation belt precipitation and its consequent atmospheric effects re- 16 quires an accurate assessment of the pitch angle distribution of precipitating electrons, as 17 well as knowledge of the dependence of the atmospheric deposition on that distribution. 18 Here, Monte Carlo simulations are used to investigate the effects of the incident electron 19 energy and pitch angle on precipitation for bounce-period time scales, and the implica- 20 tions for both the loss from the radiation belts and the deposition in the upper atmosphere. 21 Simulations are conducted at discrete energies and pitch angles to assess the dependence 22 on these parameters of the atmospheric energy deposition profiles and to estimate the 23 backscattered distributions. We observe that the atmospheric response is both 24 energy and pitch angle dependent. These effects together result in an energy-dependent 25 bounce loss cone angle, which can vary by 2–3 degrees with particle energy when consid- 26 ered at low-Earth orbit. This modeling also predicts that a significant fraction of the input 27 electron distribution will be backscattered, and should be observable by low-Earth-orbiting 28 satellites as field aligned beams emerging from the atmosphere at energies lower than the 29 input distribution, and having pitch angles distributed just inside the loss cone.

30 1 Introduction

31 Energetic Particle Precipitation (EPP) from the inner into the Earth’s 32 atmosphere is both a source of energy for the upper atmosphere, and a sink for the radia- 33 tion belts and ring current. Both the upper atmospheric and magnetospheric communities 34 require an accurate assessment of EPP fluxes and spectra to understand the role played by 35 EPP in their respective domains.

36 EPP can significantly change the properties, dynamics, and chemical composition 37 of the upper and middle atmosphere. The chemical changes induced by EPP have long 38 been found to have implications for the production of atmospheric nitric oxides (NOx) 39 and reactive hydrogen oxides (HOx) [e.g., Sinnhuber et al., 2012], both of which can lead 40 to significant ozone losses in the stratosphere and mesosphere [e.g., Randall et al., 2007; 41 Rozanov et al., 2012; Seppälä et al., 2015]. HOx produced in the mesosphere has a short 42 lifetime, but can cause days-long ozone depletion of up to 90% [Andersson et al., 2014]. 43 The energy deposited by EPP in the mesosphere and lower thermosphere (MLT) can also 44 result in massive production of NOx. During the polar winter, these EPP-induced NOx 45 descends in the polar vortex into the stratosphere, leading to catalytic ozone destruction 46 [Callis et al., 1998; Randall et al., 2007].

47 Despite the overwhelming evidence of EPP’s influence on the atmosphere, numeri- 48 cal models are incapable of capturing the effects satisfactorily. For example, in the Arctic 49 spring of 2004, an enormous influx of EPP-induced NOx was observed to descend from 50 the MLT into the polar stratosphere [Natarajan et al., 2004]. NOx mixing ratios in the 51 upper stratosphere increased by as much as a factor of 4, causing localized catalytic re- 52 ductions in ozone of more than 60% [Randall et al., 2005]. Randall et al. [2016] com- 53 pared the observed increase in NO2 in the upper stratosphere with results calculated using 54 the National Center for Atmospheric Research (NCAR) Whole Atmosphere Community 55 Climate Model (WACCM) [e.g., Marsh et al., 2007]. Modeling results were inconsistent 56 with the observed flux of NOx that descends into the stratosphere and the discrepancy was 57 attributed to a combination of inaccurate specification of the EPP flux and distribution, 58 and inadequate simulation of the vertical transport [Randall et al., 2016]. In particular, 59 WACCM simulation underestimated NOx in the mesosphere between about 60 and 80 km, 60 where electrons with energies near 50–300 keV typically deposit their energy [Codrescu 61 et al., 1997].

62 Underestimation of EPP-induced NOx by WACCM is not entirely surprising, since 63 only those electrons with auroral energies (i.e. a few keV) were included in this model,

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64 and higher energies (which are known to play a role) were omitted. This underestima- 65 tion is partly attributable to the lack of a satisfactory data set of higher-energy precip- 66 itation electron to be included in the modeling. Precipitating electrons in the relevant 67 energy range are mainly measured by the Medium Energy Proton and Electron Detector 68 (MEPED) [Evans and Greer, 2004] onboard the NOAA POES spacecraft. More recently, 69 observations have been made by the Balloon Array for Radiation-belt Relativistic Electron 70 Losses (BARREL) balloon mission via X-ray observations. The inversion from BARREL 71 measurements to precipitating electrons is extremely difficult to obtain uniquely as dif- 72 ferent precipitating electron distributions can fit the observed X-ray spectra equally well 73 [Halford et al., 2015; Clilverd et al., 2017]. On the other hand, MEPED measurements suf- 74 fer from proton contamination and poor spectral resolution, and do not adequately sample 75 the loss cone [Nesse Tyssøy et al., 2016; Peck et al., 2015; Rodger et al., 2010]. While the 76 Arctic winter of 2004 was remarkable with regard to the amplitude of the NOx descent 77 [Randall et al., 2005], such processes occur every winter at some level.

78 From a magnetospheric perspective, EPP is known to be one of the key loss mecha- 79 nisms for radiation belt enhancements following geomagnetic storms and substorms [e.g., 80 Tu et al., 2010; Thorne, 2010]. Radiation belt fluxes can be enhanced by orders of magni- 81 tude during these events [e.g., Baker et al., 2012], and high-energy above a few 82 hundred keV can be particularly damaging to high-altitude spacecraft [Baker et al., 2004]. 83 Understanding and predicting the loss rates of radiation belt fluxes during the recovery 84 phase of storms thus has direct implications for the overall understanding of radiation belt 85 dynamics, its impact on technology and hence better decision making in spacecraft opera- 86 tions.

87 Currently, uncertainties in the theoretical precipitation loss rates lead to large dis- 88 crepancy in electron lifetimes used in radiation belt models. The uncertainty in lifetimes is 89 attributed to a number of factors, including a lack of adequate observation of the waves 90 [e.g., Engebretson et al., 2008], the validity of the linear approach for modeling wave- 91 particle coupling [e.g., Bortnik et al., 2008], and uncertainty in the pitch angle distribu- 92 tion of electrons near the loss cone [e.g., Friedel et al., 2002; Millan et al., 2007; Tu et al., 93 2009]. The electron loss rates or lifetimes used in current papers vary by an order of mag- 94 nitude. For example, Shprits et al. [2005] used a constant lifetime of 10 days inside the 95 plasmapause and an empirical function of Kp (i.e., τ = 3/Kp) outside the plasmapause. 96 Conversely, Barker et al. [2005] used an L-dependent lifetime varying from 3 days at 97 L = 6 to 29 days at L = 4. Thorne et al. [2005] determined that the effective lifetime 98 is about one day in the outer radiation belt, based on microburst observations. Clearly, the 99 radiation belt electron lifetimes and their dependencies on magnetospheric conditions are 100 poorly understood.

101 Progress has been made in the area of wave distributions and the resulting diffusion 102 coefficients. Meredith et al. [2012] used multiple satellite observations to build a model of 103 chorus waves based on satellite observations; Glauert et al. [2014] used diffusion coeffi- 104 cients for whistlers, chorus, and hiss to build a global model of radiation belt dynamics. 105 Lifetimes based on wave distributions have been estimated by Agapitov et al. [2014] based 106 on Akebono wave data for the inner belt and slot region, Orlova and Shprits [2014] based 107 on chorus waves, and by Orlova et al. [2016] based on a model of hiss distribution derived 108 by Spasojevic et al. [2015]. These studies determined lifetimes ranging from ∼0.1–1000 109 days, varying with L-shell, energy, and Kp. It should be noted, however, that all of these 110 studies considered only quasi-linear (weak) diffusion and incoherent , neglect- 111 ing the coherent interactions between intense waves that can have dramatic effects on pitch 112 angles within a single bounce period [e.g., Meredith et al., 2012].

113 The precipitation of energetic particles in the upper atmosphere has been extensively 114 studied using parameterization methods [e.g., Roble and Ridley, 1987; Lummerzheim, 115 1992; Fang et al., 2008, 2010] and physics-based Monte Carlo simulations [e.g., Solomon, 116 2001; Xu et al., 2018]. Roble and Ridley [1987] first developed a parameterization method

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117 using empirical ionization profiles [Lazarev, 1967] for the National Center for Atmo- 118 spheric Research (NCAR) thermospheric general circulation model.

119 In this paper, we revisit the basic calculations of energy deposition in the upper at- 120 mosphere by EPP, using a Monte Carlo model of electron propagation in air. The use of 121 Monte Carlo modeling enables investigation of arbitrary and discrete energies and pitch 122 angles as input parameters, in turn producing output distributions as a function of energy 123 and pitch angle. The Electron Precipitation Monte Carlo (EPMC) model used here addi- 124 tionally simulates the backscatter of energetic particles and their energetic secondaries, as 125 discussed by Cotts et al. [2011]. This added capability provides a far more detailed and 126 realistic picture of the bounce loss cone (BLC) structure, its true effectiveness in remov- 127 ing energetic electrons, their potential effects on electron populations of other energies and 128 pitch-angles, and a more realistic quantification of the BLC for different energies. Quan- 129 tifying the backscattered flux as a function of the input energy and pitch angle provides 130 a unique calculation of the BLC and quantifies the BLC for different energies. Note that 131 our calculations are applicable to precipitation on short timescales, unlike the quasi-linear 132 diffusion calculations described above. In the course of the dynamic scattering process, on 133 the timescale of seconds or even minutes, it makes a large difference if a substantial frac- 134 tion of the “precipitated” electron flux is in fact not precipitated as assumed, but returns to 135 the radiation belts.

136 In this paper we define “backscatter” as those electrons propagating upward from the 137 atmosphere, within the bounce loss cone. These may include particles that have undergone 138 collisions in the atmosphere but escaped with a fraction of their input energy, but also in- 139 cludes particles that within the atmosphere without colliding with any atmospheric 140 neutrals.

141 2 Simulation Procedure

142 For a prescribed distribution of electrons in both energy and pitch angle, propagat- 143 ing down magnetic field lines towards the atmosphere, we wish to determine the fraction 144 of the input particle and energy flux that is deposited in the atmosphere and at what alti- 145 tudes, as well as the fraction of the energy flux that is returned back up the field line, ei- 146 ther by mirroring or scattering from the upper atmosphere. The EPMC model used here is 147 adapted from the Monte Carlo model of Lehtinen et al. [1999]. This model has been used 148 for a variety of purposes in recent years, including calculating the atmospheric effects of 149 an artificial beam of relativistic electrons [Marshall et al., 2014] and studying the effects 150 of bremsstrahlung propagation into the stratosphere [Xu et al., 2017].

151 The EPMC model propagates electrons one at a time through a simulation space 152 that includes a background magnetic field, a constant field gradient providing a mirror- 153 ing force, and an atmosphere responsible for electron-neutral collisions. Particle scat- 154 tering is calculated through elastic and inelastic collisions; in inelastic collisions, some 155 fraction of the electron’s energy is deposited at its current location in configuration space. 156 When an electron’s energy is reduced through collisions to a specified low energy thresh- 157 old (2 keV in the simulations herein), particle tracking in the Monte Carlo model stops, 158 and the final 2 keV are deposited at the electron’s location. The model also accounts for 159 secondary electron production through ionizing collisions and X-ray production through 160 bremsstrahlung. Both secondary electrons and bremsstrahlung photons are then propagated 161 through the atmosphere, the latter taking into account Compton scattering and photoelec- 162 tron production.

5 163 For the purposes of this paper, we begin a single simulation with, say, 10 elec- 164 trons at a discrete energy and pitch angle, similar to the procedure followed by Cotts et al. 165 [2011]. Our background simulation space is designed to roughly match that at Poker Flat, 166 Alaska, at nighttime. The background magnetic field is B0 = 50, 000 nT, and the gradient

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4 −1 167 of the vertical magnetic field component is (dBz /dz)/(2B0) = −2.4 × 10 km , similar to 168 the value above Poker Flat. The background atmosphere is specified by the NRLMSISE00 169 model [Picone et al., 2002] for a winter nighttime at Poker Flat.

170 Electrons are injected downwards from some nominal altitude above the , 171 typically set between 300 and 660 km; this upper altitude is used to simulate a distribution 172 of electrons as measured by the DEMETER spacecraft. The EPMC model determines en- 173 ergy deposited versus altitude, as well as a table of electrons that “escape” through spec- 174 ified maximum and minimum altitude boundaries (e.g., a minimum of 35 km to simulate 175 a high-altitude balloon, and a maximum of 300–500 km to simulate a LEO instrument). 176 Depending on the input pitch angle, the escaping electrons are a combination of mirrored 177 and scattered electrons, and so will have a pitch angle and energy distribution that may or 178 may not match the input distribution. From the table of electrons escaping, we can assess 179 the fraction of energy deposited versus the fraction backscattered, as well as the backscat- 180 tered energy and pitch angle distributions (PADs).

181 In the following sections we investigate the precipitation response of different energy 182 and pitch angle distributions. We begin by looking at discrete energies and pitch angles. 183 We then discuss the implications of these results on the definition of the bounce loss cone. 184 Finally, we investigate the response of realistic radiation belt distributions, as measured by 185 the DEMETER spacecraft or specified by simple analytical models. These results can be 186 used as predictions of the atmospheric response to different particle fluxes and distribu- 187 tions during both quiet-time and storm-time conditions.

188 3 Energy Deposition

189 The fraction of incident particle energy that is deposited in the atmosphere provides 190 a quantitative measurement of the effect of the atmosphere on different incident pitch an- 191 gles and energies. We begin with a series of simulations, through which we vary the in- 192 put pitch angle at the initial 500 km altitude from 40 to 90 degrees in 1-degree steps. We 193 have also simulated initial pitch angles between 0 and 40 degrees – deep inside the loss 194 cone – but for these angles the deposition profiles do not vary as significantly. In Fig- 195 ure 1, the deposition profiles for pitch angles 0–39 degrees lie nearly on top of the darkest 196 blue curve shown for 40 degrees.

197 We also vary the input particle energy from 100 keV to 10 MeV in order to focus on 198 radiation belt electrons; the model has recently been updated with the capability of simu- 199 lating as low as 10 eV electrons. Figure 1 shows energy deposition profiles as a function 200 of altitude. The top two panels show profiles for 100 keV incident energy, while the lower 201 row shows results for 1 MeV incident energy. The left panel shows that for pitch angles of 202 69 degrees and above, particles mirror at altitudes above 100 km, and only a small amount 203 of energy is deposited; in addition, most of the energy deposition occurs at the mirror 204 altitude, where the particles persist for some time before mirroring. The jagged lines to 205 the left of the main deposition profiles are caused by the few electrons that are scattered 206 downwards, depositing energy below their nominal mirror altitude.

207 Given an initial altitude and a magnetic field gradient, we define the “nominal” 208 bounce loss cone angle αBLC as the angle given by the “naïve” assumption that parti- 209 cles which mirror below 100 km are precipitated. In this case, with an initial altitude of 210 500 km, the nominal αBLC is 66.3 degrees.

211 The zoomed-in views in Figure 1 demonstrate two interesting effects. First, the en- 212 ergy deposition profiles show variation with input pitch angle. Lower input pitch angles 213 (blue curves), closer to field-aligned, deposit energy at lower altitudes, and show a notably 214 higher peak energy deposition. The energy deposition profile is both lower in altitude and 215 sharper, spanning a narrower altitude range. We return to this effect in the discussion of 216 Figure 2.

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217 Second, there is a noticeable difference between the two energies shown. While 218 the rapid transition from precipitating to trapped occurs between 66–68 degrees for the 219 100 keV incident energy (observe the curves labeled with incident pitch angles of 66 and 220 68 degrees), the same effect occurs between ∼64–66 degrees for the 1 MeV case.

(b) Deposition Profiles for 100 keV electrons (c) 150 95

90

100 85 α = 68º Input Pitch Angle (deg) Altitude Altitude 40 50 60 70 80 90 α = 66º

80 500 (a) α = 40º 450 50 75 10-4 10-3 10-2 10-1 100 101 10-3 10-2 10-1 100 400 α = 80º Energy Deposited (eV/m/electron) Energy Deposited (eV/m/electron)

350 (d) Deposition Profiles for 1 MeV electrons (e) 70 α = 75º 300 140

250 120 65 200

Altitude (km) α = 70º 100 α = 66º 150 60

α = 68º Altitude Altitude α = 64º 100 80

55 α = 40º 50 60

0 10-6 10-1 104 40 50 10-4 10-3 10-2 10-1 100 101 10-2 10-1 100 101 Energy Deposited (eV/m/electron) Energy Deposited (eV/m/electron) Energy Deposited (eV/m/electron)

221 Figure 1. Energy deposition profiles for a range of pitch angles. a) Profiles for pitch angles ranging from 40

222 to 90 degrees for 1 MeV incident energies; The nominal loss cone angle is 66.3 degrees, above which profiles

223 show marked mirror altitudes. b-d) zoomed in views of deposition profiles for 100 keV (b-c) and 1 MeV (d-e)

224 incident energies.

225 After simulating all energies and pitch angles, we measure the following parameters 226 for the energy deposition profile in each simulation, shown in the top row of Figure 2: a) 227 the fraction of total incident energy that is deposited in the atmosphere; b) the altitude 228 where the energy deposition peaks; and c) the full-width at half-maximum (FWHM) of 229 the energy deposition profile in altitude, a measure of the alitude extent of the deposition 230 profile. In each panel, the vertical dashed line denotes the nominal αBLC of 66.3 degrees.

231 Figure 2a) demonstrates that while αBLC is relatively sharp as expected, it is not a 232 perfect step function; a fraction of energy is backscattered for all angles, even as high as 233 10% for 40 degree pitch angles. Figure 2b) shows that the altitude of peak energy depo- 234 sition varies with pitch angle; while the nearly-field-aligned particles at 10 MeV deposit 235 most of their energy at 41 km, the particles closer to αBLC deposit most of their energy 236 above 50 km. This effect demonstrates that the pitch angle distribution is critical to the 237 energy deposition profile in the atmosphere. Figure 2c) furthermore shows that pitch an- 238 gles closer to αBLC also distribute their deposited energy over a wider range of altitudes.

239 The bottom row of Figure 2 summarizes measures of the backscattered distributions 240 for each simulation. These measures include d) the fraction of particles that are backscat- 241 tered; e) the mean backscattered energy as a fraction of the input energy; and f) the mean 242 backscattered pitch angle. An interesting effect is observed in Figure 2d), where for 10 243 MeV input energy, there is a range of pitch angles, near αBLC, where more particles are 244 backscattered that were input. This is due to the creation of a large number of secondary 245 particles, near αBLC, which then backscatter along with the primary particles. Figure 2e) 246 shows that higher energy particles lose more of their energy to the atmosphere before scat- 247 tering compared to lower energy particles; this is primarily due to the fact that they pen- 248 etrate deeper into the atmosphere, and thus experience more collisions. Figure 2f) shows

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249 that the mean backscattered pitch angle is not a strong function of the input electron en- 250 ergy. We investigate the backscattered distributions in more detail in the next section.

a) Fraction of energy deposited b) Altitude of peak energy deposition c) FWHM of altitude deposition 1 80 30 100 keV 0.9 200 keV 75 25 0.8 500 keV 70 1.0 MeV 0.7 2.0 MeV 65 5.0 MeV 20 0.6 10.0 MeV 60 0.5 55 15 0.4 50 Altitude (km) 10

0.3 45 Alitude width (km) Fraction Deposited 0.2 40 5 0.1 35 0 30 0 40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90 Pitch angle (deg) Pitch angle (deg) Pitch angle (deg)

d) Fraction of Particles Backscattered e) Mean energy backscattered f) Mean pitch angle backscattered 1.2 1.4 90

80 1.2 1 70 1 0.8 60

0.8 50 0.6 0.6 40 0.4 30 0.4

Fraction Backscattered 20 0.2 Mean energy / Input 0.2 10 Mean Pitch angle backscatted (deg) 0 0 0 40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90 Input Pitch Angle (deg) Pitch angle (deg) Pitch angle in (deg)

251 Figure 2. Atmospheric response to precipitation at different energies as a function of pitch angle. a) frac-

252 tion of energy deposited in the atmosphere; the remainder is backscattered or mirrored. b) Altitude of the

253 peak in the energy deposition profile. c) FWHM of the energy deposition profile. d) Fraction of incident par-

254 ticles that are backscattered. e) Mean energy of backscattered electrons. f) Mean pitch angle of backscattered

255 electrons.

256 4 Backscattered Distributions

257 Figure 3 shows the pitch angle and energy distributions of backscattered electrons ◦ 258 for a few key input pitch angles. The left panels show an input pitch angle of 50 , well 259 inside the loss cone (again, the nominal loss cone angle or αBLC for this case, based on a 260 100 km mirror altitude, is 66.3 degrees). The PADs here have been normalized to units of 261 particles per steradian. We observe that the backscattered distribution is relatively uniform, 262 with a rapid dropoff above the nominal αBLC, as expected. The energy distribution, how- 263 ever, shows a peak at a fraction of the input energy, with a slow dropoff towards zero and 264 a rapid dropoff just below the input energy of 1 MeV. ◦ ◦ 265 The next columns in Figure 3 show results for input pitch angles of 62 , 65 , and ◦ 266 68 , values on both sides of the nominal αBLC. We observe that below and very near 267 αBLC, both the pitch angle and energy distributions skew towards the discrete input values. 268 Above αBLC, the backscattered distributions resemble the input distributions (i.e. delta 269 functions), as expected.

274 The backscattered energy distributions for the same input pitch angles are shown in ◦ 275 the bottom panels. For an input pitch angle inside the loss cone (50 ), the backscattered 276 energy distribution is broad in energy between 0 and 1 MeV, though not quite uniform. ◦ 277 Closer to αBLC, at 62 , the backscattered distribution is peaked just below 1 MeV, with a 278 broad distribution at lower energies. This implies that most of the backscattered particles

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Backscattered Pitch Angle Distribution 1 Input PA = 50° Input PA = 62° Input PA = 65° Input PA = 68° 0.8

0.6

0.4 Particles/str (norm.) 0.2

0 020406080 020406080 020406080 020406080 Pitch Angle (deg) Pitch Angle (deg) Pitch Angle (deg) Pitch Angle (deg) Backscattered Energy Distribution (Normalized) 1 Input PA = 50° Input PA = 62° Input PA = 65° Input PA = 68°

0.8

0.6

0.4 Particles (norm.)

0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Energy (MeV) Energy (MeV) Energy (MeV) Energy (MeV)

270 Figure 3. Backscattered electron distributions for 1 MeV electron input. Top row: backscattered pitch angle

271 distributions, normalized to particles per steradian, for four different input pitch angles (PA) as shown. Bot-

272 tom row: energy distributions. The nominal loss cone angle is 66.3 degrees, but the loss cone angle for this

273 energy, is more accurately described as somewhere in the range 64–66 degrees.

279 lost some of their energy through collisions, but emerged with most of their initial energy; 280 while a significant population backscatters with random energy. As the input pitch angle 281 trends towards the loss cone angle and into trapped pitch angles, the backscattered distri- 282 bution shifts towards the monoenergetic input distribution as expected.

283 Figure 3 demonstrates that for input pitch angles near or within the loss cone, the 284 backscattered distributions have a significant population at lower energies, due to colli- 285 sional losses in the atmosphere. These electrons are “lost” from the point of view of their 286 original energy, in this case 1 MeV, but add to the distribution at lower energies. However, 287 since realistic distributions have much higher fluxes at lower energies, they do not con- 288 tribute appreciably to the lower energy populations. As such, from the point of view of 289 the radiation belts, these particles can be considered “lost”.

290 5 The Bounce Loss Cone

291 Numerous studies performed over the past many decades utilize a basic definition of 292 the bounce loss cone (BLC) angle. Conventionally, electrons with mirror altitudes above 293 100 km are considered trapped, and those below this boundary are considered to be in the 294 loss cone. The specific wording of the bounce loss cone definition is similar througout 295 the literature. Vampola and Kuck [1978] state that “it is assumed that a particle that has a 296 local mirror point at or below 100 km will be immediately lost into the atmosphere; i.e., 297 it is in the bounce loss cone.” Studying Lightning-induced Electron Precipitation, Chang 298 and Inan [1985] state that “the loss cone angles are derived from the dipole magnetic field 299 model by using a mirror height of 100 km.” Imhof and Gaines [1993] apply the same ar- 300 gument to the drift loss cone, stating that “such electrons have a minimum drift altitude, 301 hmin, of less than 100 km.” Abel and Thorne [1998] state more specifically that “the loss 302 cone boundary corresponds to a mirror altitude of 100 km; electrons with bounce trajecto- 303 ries mirroring below this point are assumed to be lost to the atmosphere within a quarter

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304 bounce period (of the order of a second or less) due to collisions with the neutral atmo- 305 sphere.” Similarly, Blake et al. [2001] state that “An altitude of 100 km is traditionally 306 taken to be the boundary below which energetic particles are lost into the atmosphere.”

307 To our knowledge no quantitative justification has been made for this altitude. How- 308 ever, as we have shown, radiation belt electrons undergo elastic and inelastic collisions 309 upon reaching the upper atmosphere, and these collisions result in changes to their trajec- 310 tories and energies. Electron trajectories undergo small-angle scattering events, but the ac- 311 cumulation of many scattering events can modify their trajectories significantly, with only 312 minor changes to the particle energies. Many particles well inside the loss cone, then, can 313 be backscattered by the atmosphere and returned to space, retaining much of their incident 314 energy. Similarly, particles with mirror altitudes above 100 km may be lost due to colli- 315 sions. By any definition, the boundary between trapped and loss-cone particles must be 316 soft.

317 Upon their return to space, those electrons backscattered from below 100 km are 318 necessarily located in the conventionally-defined bounce loss cone, since they originate 319 from altitudes below 100 km; as such, they are likely to precipitate in one of their succes- 320 sive bounce periods. If energetic particles are continuously scattered into the loss-cone at 321 some given energy, we expect a quasi-field-aligned distribution of electrons to emerge over 322 a range of energies below the incident energy, which might have further implications for 323 wave excitation and wave-particle interactions [e.g., Nishimura et al., 2015]. This backscat- 324 ter also implies that only a fraction of the loss cone electrons are deposited in the upper 325 atmosphere, modifying estimates of the energy deposition that is important to upper atmo- 326 sphere chemistry and dynamics.

327 At this point, it is clear that to define αBLC, we must set a threshold for the frac- 328 tion of energy (or, if preferred, particles; here we choose energy) that is returned to space, 329 through a combination of mirroring and backscattering. A high threshold is justified, since 330 many of the backscattered particles at low pitch angles are likely to be lost in subsequent 331 bounce periods. In Figure 4, we plot a few results versus this defined threshold, ranging 332 from 0.5 (i.e., half of the energy is backscattered) to 0.995 (nearly all of the energy is 333 backscattered).

334 Figure 4 provides loss cone results for energies from 100 keV to 10 MeV. Further, 335 we compute some interesting parameters related to the bounce loss cone. Figure 4a) shows 336 the effective mirror altitude as a function of the backscatter (or loss-cone angle, LCA) 337 threshold. This is the altitude which replaces the naïve mirror altitude of 100 km when 338 considering our quantified definition of the bounce loss cone. For example, if we define 339 the threshold to be where 50% of the input energy is backscattered, then for 10 MeV, the 340 effective mirror altitude is about 5 km; one would need to use this altitude, and the as- 341 sociated magnetic field strength, to calculate αBLC somewhere in space. The effective 342 mirror altitude can even be below 0 km altitude; in this case particles appear as though 343 they will reach the ground, based on mirroring alone, but backscatter causes them to be 344 turned back. We see that for high thresholds (90% and above), the mirror altitude depends 345 strongly on energy, with higher energies having lower mirror altitudes, due to the fact that 346 higher energy electrons have smaller collision cross sections with atmospheric molecules. 347 This observation has implications for determining loss rates of radiation belt particles, es- 348 pecially when the strong diffusion limit is calculated, which considers the entire loss cone 349 to be filled [Kennel, 1969].

350 Figure 4b) shows the resulting equatorial αBLC as a function of the backscatter thresh- 351 old, based on an equatorial magnetic field strength of 300 nT. We observe that the angle 352 does not change significantly compared to the nominal αBLC (shown as a black dashed 353 line). Figure 4c), however, shows the resulting αBLC in LEO, at the original altitude of 354 500 km. Here we find that αBLC varies by 2–3 degrees, as a function of energy and of the 355 defined threshold. If future LEO spacecraft missions are able to resolve pitch angles to

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356 within 1–2 degrees or better, with the intention of resolving the loss cone, then knowledge 357 of this loss cone effect is imperative to estimating the precipitated flux.

358 Finally, Figure 4d) shows αBLC at 500 km altitude as a function of energy, for a 359 number of different backscatter thresholds. This panel highlights a key result of this work, 360 that the bounce loss cone is energy dependent. For a high threshold such as 0.95 or 0.99, 361 αBLC in LEO varies by about two degrees between 100 keV and 10 MeV, though it ap- 362 pears to flatten at the higher energies. The effect is less pronounced for lower thresholds.

a) Mirror altitude vs. Threshold b) Equatorial Loss Cone Angle vs. Threshold 200 4.16

100keV 4.14 100keV 200keV 200keV 150 500keV 4.12 500keV 1MeV 1MeV 2MeV 4.1 2MeV 5MeV 5MeV 100 10MeV 4.08 10MeV

4.06

50 4.04

4.02 Equatorial LCA (deg) Effective Mirror Altitude 0 4

3.98

-50 3.96 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 Backscatter threshold Backscatter threshold

c) Loss Cone Angle at 500 km vs. Threshold d) Loss Cone Angle at 500 km vs. Energy 69 68 Threshold = 0.5 67.5 0.75 68 0.9 67 0.95 0.99 66.5 67 66

66 65.5

65 65 Local LCA (deg) Local LCA (deg) 64.5

64 64 63.5

63 63 0.5 0.6 0.7 0.8 0.9 1 0.10.3 1 3 10 Backscatter threshold Input Energy (MeV)

363 Figure 4. (a) Effective mirror altitude vs. backscatter threshold. A backscatter threshold of 0.9, for ex-

364 ample, implies that 90% of the input energy is backscattered. The effective mirror altitude is defined as the

365 altitude at which particles will mirror in the absence of collisions. The effective mirror altitude then directly

366 maps to an equatorial loss-cone angle (b) and a loss-cone angle at 500 km altitude (c). (d) The loss cone angle

367 as a function of electron energy, for a range of selected thresholds.

368 6 Response to Realistic Distributions

369 In this section we investigate the effects of atmospheric backscatter and the results 370 described above on realistic energy and pitch angle distributions, rather than monoen- 371 ergetic, monodirectional beams. We use exponential energy distributions with e-folding 372 factors E0 between 100–400 keV, as derived from DEMETER observations by Whittaker 373 et al. [2013]. Similar results are observed for power law distributions. We use an input en- 374 ergy distribution corresponding to DEMETER channels 2–127, i.e. energies from 91 keV 375 to 2.23 MeV; energies outside of this range are not considered. The initial PADs are spec- 376 ified as either uniform or sinusoidal PADs, and are defined at 500 km altitude. That is, 377 for a uniform distribution, particles at 500 km altitude are uniformly distributed between 0

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402 cone, assuming αBLC = 66.3 degrees. In the numerical simulation, the energy deposition 403 is calculated in the model. We find that for the isotropic PAD, the naïve assumption over- 404 estimates the deposited energy by 13–15%, and for the sinusoidal PAD by 22–24%. The 405 lower end of each range corresponds to the input distribution with E0 = 400 keV, and the 406 higher end with E0 = 100 keV. This result shows that the naïve assumption overestimates 407 energy deposition because it falsely assumes that the energy of all electrons in the nomi- 408 nal loss cone is fully deposited.

409 We also find that for these input distributions, about 36% (isotropic PAD) or 30% 410 (sine PAD) of the backscattered electrons are inside the nominal loss cone; unless they 411 are scattered in their next bounce period, they should precipitate in a subsequent bounce. 412 These electrons are seen in the blue curves in the right panels of Figure 5; everything to 413 the left of the red dashed line is inside the loss cone.

414 An interesting result of this backscatter distribution is that since these electrons are 415 propagating upwards, but with pitch angles inside αBLC, they appear to originate from be- 416 low 100 km altitude, and in fact some of them do. Hence, a spacecraft with an energetic 417 particle detector in LEO pointing down, towards the Earth parallel to B0, with a field-of- 418 view inside the loss cone, would observe these electrons. These electrons would be direct 419 evidence of particle precipitation, – no mirrored electrons could originate from within the 420 loss cone – although they represent the fraction of the loss-cone distribution that is not 421 precipitated. Compared to an upward-pointing detector such as the MEPED instrument on 422 POES [Evans and Greer, 2004], a downward-pointing instrument could potentially provide 423 a better-constrained estimate of the precipitating energy distribution, since the backscat- 424 tered, inside-the-loss-cone flux is directly related to the precipitating flux. An even more 425 accurate measurement could be made with an instrument pointing both up and down B0, 426 to measure the incident and backscattered fluxes together.

427 7 Summary and Discussion

428 In this paper we have investigated the response of the atmosphere to energetic elec- 429 tron precipitation, with a focus on the backscatter of particles from the atmosphere to- 430 wards the magnetosphere. The backscattered distribution inevitably returns to the magne- 431 tosphere with a different pitch angle and energy distribution compared to the input distri- 432 bution, as shown in Figure 5. In general, the backscattered particles will be deeper in the 433 loss cone (i.e., lower pitch-angles) than the incident particles, as pointed out by Cotts et al. 434 [2011].

435 A critical result that emerges from this work is that the bounce loss cone is energy 436 dependent. Higher-energy particles typically have a lower equatorial and local (500 km) 437 loss-cone angle, although again, this loss-cone angle depends on our definition of the LCA 438 threshold.

439 While the analysis shows that a fraction of particles that is nominally in the loss- 440 cone may be mirrored or backscattered from the atmosphere, those particles necessar- 441 ily remain within the loss cone angle, and may thus precipitate during their subsequent 442 bounces. However, those low pitch-angle particles may also encounter various waves dur- 443 ing their bounce and could be scattered either deeper into the loss cone or back into the 444 trapped population. In fact, early nonlinear modeling results indicated that very low pitch- 445 angle particles may in fact experience an effect dubbed “loss-cone reflection” whereby 446 their pitch angle would be very quickly raised into the trapped region, even by moderate 447 whistler waves [Inan et al., 1978]. Numerical modeling of the wave-particle interactions 448 over a range of wave situations would be necessary to quantify this effect.

449 We further observe from Figure 5 that the backscattered population is primarily lo- 450 cated “just” inside the loss cone, i.e., having pitch angles that are immediately within or 451 near the loss cone angle; but additionally, this population is generally a lower-energy dis-

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452 tribution compared to the input population. Thus, a satellite located at low-Earth-orbit car- 453 rying a downward-looking particle instrument would expect to see a roughly field-aligned 454 population of particles emerging from the atmosphere at energies lower than the precipi- 455 tating distribution.

456 In this work we have only considered the effects of atmospheric collisions in remov- 457 ing precipitating electrons and thereby creating the bounce loss cone; we have not consid- 458 ered the effects of longitudinal drift, and removal of particles through the drift loss cone. 459 In the paradigm of quasilinear theory, where weak diffusion driven by incoherent pitch- 460 angle scattering of electrons by averaged wave distributions (as is considered in most of 461 the work on radiation belt lifetimes as discussed in Section 1), a sizeable fraction of the 462 particles can be removed through the drift loss cone at large pitch-angles. However, when 463 particles scattering is driven by intense waves over short durations, such as chorus-driven 464 microbursts or lightning-induced electron precipitation, the local, immediate pitch-angle 465 distribution is most relevant and the bounce loss cone controls the precipitation events.

466 Acknowledgments 467 R. Marshall was supported by NSF INSPIRE award 1344303 and NSF MAG award 1732359. 468 J. Bortnik is supported by NSF MAG award 1732367. The simulation outputs used in the 469 analysis and results herein are available from www.github.com/ram80unit/newblc/.

470 References

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