Ionization in the Earth’s Atmosphere Due to Isotropic Energetic Electron Precipitation: Ion Production and Primary Electron Spectra

: Energetic electron precipitation (EEP) via atmospheric ion production rates is a natural force acting on the atmosphere and climate systems. The correct estimation of EEP ion production and spectra for the computation of ionization rates is an important issue for estimating climate forces. In the present paper, we propose a favorable method for the computation of ionization rates forced by EEP using the new parameterization of ion production and a new spectrum shape, which allow one to take into account the range of precipitating particles from tens of keV to several MeV. A new function of spectral ﬁt will also be helpful in obtaining information about EEP from satellite and balloon observations. Presented here, the parameterization of atmospheric ionization in the Earth’s atmosphere includes a new yield function of isotropically precipitating monoenergetic electrons and ionization via Bremsstrahlung radiation. Look-up tables with ion production/yield function for isotropically precipitating monoenergetic electrons (30 keV–5 MeV) can be easily used for the computation of ionization rates and can further be used by atmospheric and chemistry-climate models for accurate quantiﬁcation of atmospheric parameters during energetic electron precipitation.


Introduction
Energetic electron precipitation (EEP) is a wide population of electrons with various energy ranges that are well registered by satellite-borne detectors, balloon-borne platforms, and ground-based instrumentations. Satellites usually observe electron fluxes for short periods (seconds/minutes) in each orbit, providing an almost instant picture for EEP energies and flux from space, e.g., [1,2]. In contrast, balloon measurements register the Bremsstrahlung, which is received in the atmosphere and gives a basis to derive the EEP energies and flux in the atmosphere, e.g., [3][4][5][6]. However, balloon measurements are limited in the spatial extent to the region of the balloon flight. Ground-based instrumentation such as incoherent scatter radars can observe only electron content mostly in the upper altitudinal range with a small viewing region close to the zenith, e.g., [7]. Therefore, a combination of various types of observations can lead to new information about the EEP spectra (energies and flux).
Energetic electron precipitation plays an important role in ion production into the Earth's atmosphere and affects the neutral chemistry of the middle atmosphere. In turn, the number of ion productions per second (or in other words, ionization rates) leads to the formation of odd nitrogen and hydrogen, which are important catalysts that participate in ozone loss reactions in the upper stratosphere and mesosphere [8,9]. Then, the ozone affects the radiative balance, temperature, and dynamics of the atmosphere, and an EEP-ozone-climate link can be estimated via chemistry-climate models [10,11]. Various chemistryclimate models [12,13] covered an altitudinal range from the ground to about 100 km, where an important source of the ionization atmosphere is EEP with energy ranges from about 30 keV to about 5 MeV. Therefore, it is important to know the energy range and intensity of fluxes of precipitating electrons as well as the EEP ionization rates in the atmosphere. However, despite the current progress in the understanding of the impact of EEP on the chemistry of the middle atmosphere and climate, the effect is still an outstanding question [5,12,13]. To obtain an answer to this question, a correct description of ionization rates induced by energetic particle precipitation is required. In its turn, ionization rates require knowledge of the energy spectral distribution and precalculated ion production or ionization yield function.
Estimating and calculating EEP ionization in the ionosphere and atmosphere system has a long history that started from the pioneer works of Rees M.H. (1963) [14], Lazarev V. I. (1967) [15], Berger M.J. and Seltzer S.M. (1972) [16], etc. Ress (1963) [14] analytically calculated ion production in the Earth's atmosphere produced by energetic primary auroral electrons in the energy range 0.4 keV to 300 keV, taking into account auroral excitation between 60 km and 300 km. Lazarev (1967) [15] presented an analysis the absorption of the energy of an electron beam in the upper atmosphere up to 32 keV. Roble R.G. and Ridley E.C. (1987) [17] adopted an analytical formula of Lazarev to Maxwellian energy spectral distributions that could be used for the NCAR thermospheric general circulation model. Later, Sergienko T.E. and Ivanov V.E. (1993) [18] proposed a new approach to calculate the excitation of atmospheric gases via an auroral electron impact that is based on the Monte Carlo method. All previous works were concentrated on auroral electrons, which is the input of interest to the ionospheric community. Fang et al. (2008) [19] were interested in extending the ionization parametrization models to evaluate the effect of the medium energy (30 keV-1000 keV) electrons on the ionosphere and atmosphere. Fang et al. (2008) [19] proposed parameterization methods for only a Maxwellian energy spectra distribution. EEP energy deposition in terms of ion-pair production rates by isotropic electron precipitation was proposed by  [16] were some of the first to describe the spectral distribution and the emission of Bremsstrahlung by electrons in the upper atmosphere generated for electron beams incident to the atmosphere with energies between 20 keV and 2 MeV. It was clearly shown that EEP with energies more then 30 keV produce Bremsstrahlung at altitudes below 100 km. As it is seen up until now, there is a gap with the homogenious EEP parametrization of ion production/yield function that covers Bremsstrahlung radiation for medium-and high-energy electrons with the energy ranging from 30 keV to several MeV, well described by satellite and balloon observations. The purpose of this paper is to show a stable method of computing the ionization rates of the atmosphere (below about 100 km) using various spectral functions distribution covering the satellite and balloon observations of the EEP and the ion production/yield function (presented as a new look-up table located in the Supplementary Materials) for isotropically precipitated monoenergetic electrons in the energy range 30 keV to 5 MeV, taking into account Bremsstrahlung radiation.

Computation of EEP Ionization Rates
The computation of EEP ionization rates requires knowledge of the parametrization of ion production via ionization yield functions (see Supplementary Material and Section 3) and the energy spectra (see Section 4).
The ionization yield function Y(x, E) (ion pairs cm 2 g −1 ) at the atmospheric depth x (g cm −2 ) is a number of ion pairs created by one precipitating electron with the initial energy E at the upper boundary of the atmosphere. In this study, we used modified ionization yield functions for mono-energetic electrons with initial energy from tens of keV to several MeV. Both direct ionization by primary electrons as well as the secondary Bremsstrahlung electromagnetic emissions are considered in this model.
The ionization rates I(x) (ion pairs g −1 s −1 ) can be computed as follows: where F(E) is a spectral distribution (cm −2 s −1 keV −1 ) of precipitating electrons at the top of atmosphere, and E x and E n are the minimum and maximum energies of electrons in a flux. In this article, we consider the isotropic flux of energetic electrons on the upper boundary of the atmosphere (x = 0 g cm −2 ). The calculations of the yield function Y(x, E) are discussed in the Section 3 and presented by a look-up table in the Supplementary Materials of this paper. The spectral distributions of F(E) are considered in Section 4 and can be used for the calculation ionization rates induced by EEP.

Atmospheric Ionization Yield Function for EEP with Isotropic Incidence
Here, we perform a calculation of the yield function Y(x, E) of monoenergetic electron beams with the energy of primary particles E in the atmospheric depth x (in g cm −2 ). Direct ionization by primary electrons is calculated analytically. Within this approach, we started our computations using the formalism presented in detail by [25]. Here, we assumed that electrons with initial kinetic energy E penetrate from the upper layer of the atmosphere (x = 0) from the zenith angle ϑ. The elastic scattering is neglected, and only the ionization process is considered.
The electron energy E after passing the thickness l (g cm −2 ) along the electron trajectory can be calculated as follows: where R(E) is the path length of an electron with energy E due to ionization losses. The differential ionization rate at the atmospheric depth x is defined as follows: where E is defined from Equation (2). Note that dx = cos ϑ · dl and dE/dl(E )-the electron stopping power due to ionization losses. We assume that, on average, one ion-electron pair is produced per each 35 eV of deposited energy [26]. For the isotropic unit flux of precipitating electrons impinging on the top of the atmosphere from the solid angle Ω at dF/dΩ = cosϑ, the ionization yield function Y(x, E) is defined as Figure 1 presents the calculated yield function (red lines with circles) for E: 100 keV and 1000 keV. For the comparison in Figure 1, the black lines with squares depict direct ionization yield functions for isotropic precipitating electrons according to Reference [20].
A sharp break in the yield function with an increase in thickness is due to the stopping power energy range of primary electron deposition. At lower altitudes (greater thicknesses), ionization is performed by the secondary Bremsstrahlung radiation. Ionization due to the Bremsstrahlung (blue triangles in Figure 1) was computed using the GEANT4 simulation tool PLANETOCOSMICS (http://cosray.unibe.ch/~laurent/planetocosmics/ accessed on 10 September 2021) with the NRLMSISE 00 atmospheric model [27]. The results of these calculations are approximated by the standard deviation function on a logarithmic scale (red curves with circles in Figure 1).   Figure 2 shows the ionization yield function Y(x, E) (ion pairs cm 2 g −1 ) vs. atmospheric depth x (g cm −2 ) due to the isotropic monoenergetic one-electron precipitation with an energy electron range of 30 keV-5000 keV. These ion productions or ionization yield function Y(x, E) (ion pairs cm 2 g −1 ) are presented as a look-up table in the Supplementary Materials of this paper.
The bottom panel, (b) of Figure 2 shows the dependence of the product Y(x, E) · ρ (ion pairs cm −1 ) on height (not thickness) in the atmosphere. ρ(h) is the atmospheric density at a height of h. Multiplying the yield function Y(x, E) (presented in the Supplementary Materials) by the real density of the atmosphere ρ(h), one can calculate the ionization rates of the atmosphere at a given height of h.

Spectra Function for Energetic Electron Precipitation
Retrieving ionization rates requires knowledge of the energy spectra and parameterization of ion production via ionization yield functions. The parametrization of ion production into atmosphere or yield function Y(x, E) was discussed previously in Section 3 and presented in Supplementary Material. The distribution of the energy spectra requires different fitting functions covering the same energy range, which makes the calculation of ionization rates ambiguous [28]. In this section, we consider various types of spectra distributions that allow us to fit the EEP fluxes and energy range as well as to propose a new combined spectral distribution F c (E) covering satellite and balloon observations in the EEP energy range.

Exponential Spectral Distribution
Generally, the exponential-law function of energy spectra is used for fitting balloonborne [4] and rocket [29] observations. In this case, the form of the electron energy spectrum is fitted using a function of exponential law: where F e (E) is the differential flux of precipitating electrons with kinetic energy E, E o is the characteristic energy of precipitating electrons, and F o is a parameter of the flux of incident electrons (cm −2 s −1 keV −1 ). The characteristic energy of the electron spectra is from several keV to tens of MeV [4]. Such a type of electron spectra has also been used in many other studies fitting balloon-borne measurements, e.g., [3,30].

Maxwellian Spectral Distribution
The calculation of the ionization rates from the Maxwellian type of electron energy spectra F M (E) (cm −2 s −1 keV −1 ) can be expressed based on satellite and ground-based measurements [20,[31][32][33]. The Maxwellian energy distribution is specified by where E o is characteristic energy (in keV), Q o is the total precipitating energy flux (keV cm −2 s −1 ). Auroral electrons of low energies (< 30 keV) are in a quasi-equilibrium state, which is described by the Maxwellian energy distribution [34][35][36].

The Generalized Lorentzian (or κ-) Spectral Distribution
The source of precipitating electrons is the plasma sheet region of the magnetotail. The suprathermal populations of electrons are well described by the so-called Kappa κ or generalized Lorentzian distributions [37]. Due to these facts, one can suggest that the shape of electron energy distribution in the plasma sheet should be reflected in the energy distribution of precipitating electrons [33]. The κ distribution [20], which is Maxwellian at low energies and power-law at high energies, can be presented as follows: where E o is characteristic energy and κ is the spectral gradient. F L (E) is the κ distribution of the differential energy spectrum. Q o is the total precipitating energy flux (keV cm −2 s −1 ). Direct satellite-based spectral measurements can be approximated using one of these types of the spectral distribution. Reference [38] suggested that, for some cases, a fit by kappa distribution is more appropriate. We note that the κ distribution of differential energy spectrum (Equation (7)) becomes the Maxwellian distribution (Equation (6)) as κ → ∞ [39]. The generalized Lorentzian distribution can be used with Maxwellian as additional high energy component of an energy spectrum of precipitating electrons [32].
An example of a spectra, in power-law format is the one presented by [44]: where F p (E) is the power-law integral energy spectrum, F o is a parameter of the flux of incident electrons (cm −2 s −1 keV −1 ), and k is the spectral gradient that can be calculated using a scheme described by [45]. It should be noted that the lower energy limit for this particular approximation was 30 keV and that the upper limit was 1 MeV [44].

Combined Spectral Distribution
Here, we propose a new form of combined spectral distribution that allows us to cover an energy range from tens of keV up to several MeV based on observation by satellite and balloon data.
Here, δ = E b E o , E b is a characteristic energy of electron's spectrum and E o is the characteristic energy on a part of the spectrum of electrons where the spectrum decreases sharply. For an energy E = E b , parts of the function F c (E) and their first derivatives are stitched together. F o is a parameter of the flux of incident electrons (cm −2 s −1 keV −1 ). For E < E b , the power spectrum uses a spectral index γ. The function F c (E) is continuous at the respective energy along with its first derivative.
An example of such a type of combined spectral distribution F c (E) can be seen in Figure 3. In Figure 3, on the top and bottom panels, the energies band from 10 keV to 1000 keV that can be covered by satellites, as for example, by MEPED instrument of NOAA POES satellite, that usually are fitted by the power-law spectral distribution and balloon observations that usually are fitted by the exponential spectral distribution. E 0 = 1000 keV is characteristic energy on a part of the spectrum of electrons where spectrum decreases sharply; see Figure 3. Using proposed combined spectral distribution allows for describing balloon and satellite observations simultaneously.

Conclusions
In this paper, we presented a new parameterization of ion production and new spectra function for computation of ionization rates induced by energetic electron precipitation. A new parameterization of atmospheric ionization in the Earth's atmosphere includes a yield function for isotropically precipitating monoenergetic (30 keV to 5 MeV) electrons and ionization via Bremsstrahlung radiation. We propose a favorable method for computation of the ionization rates forced by EEP using the new parameterization of ion production and a new spectrum shape, which allows one to take into account the range of precipitating particles from tens of keV to several MeV. A new function of spectra fit will also be helpful for obtaining information about EEP from satellite and balloon observations. A new lookup table with ion production/yield function for isotropically precipitating monoenergetic electrons can be easily used for computation of ionization rates and can further be used by atmospheric and chemistry-climate models for accurate quantification of atmospheric parameters during energetic electron precipitation.

Conflicts of Interest:
The authors declare no conflict of interest.