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EUV signals associated with O^{+} ions observed from ISSIMAP/EUVI in the nightside ionosphere
Earth, Planets and Space volume 73, Article number: 151 (2021)
Abstract
The extreme ultraviolet (EUV) imager, EUVIB, on board the International Space Station (ISS) under the International Space Station–ionospheremesosphereatmosphere plasmasphere cameras (ISSIMAP) mission was originally intended to observe EUV emissions at 83.4 nm scattered by \({\mathrm O}^+\) ions. During the mission, EUVIB occasionally detected evident EUV signals in the umbra of the Earth. However, the source of the signals has not been verified. To evaluate the effect of the 83.4 nm EUV, we conduct a Monte Carlo simulation which considers multiple scattering of the 83.4 nm EUV by \({\mathrm O}^+\) ions. In addition, we modeled the contribution of the 91.1 nm emission, which is due to recombination of \({\mathrm O}^{+}\) ions and electrons, because the 91.1 nm EUV might affect the measurement from EUVIB due to the wavelength range covered. The results suggest that the effect of the 83.4 nm EUV is likely to be negligible while the 91.1 nm EUV explains the observations from EUVIB morphologically and quantitatively. We therefore conclude that the EUV signals observed by EUVIB in the umbra of the Earth can largely be attributed to 91.1 nm emission due to recombination. This conclusion would facilitate the use of the EUVIB data for reconstructing the \({\mathrm O}^+\) density.
Introduction
Ultraviolet imaging observation is a useful tool for diagnosing the state of the ionosphere. In particular, measurements of emissions associated with \({\mathrm O}^+\) ions, which are the dominant species in the lower ionosphere, provide valuable information on spatiotemporal variations of the ionosphere. The most prominent emission from \({\mathrm O}^+\) is the 83.4 extreme ultraviolet (EUV) emission (e.g., Meier 1991) and several observations of this emission have been conducted (e.g., McCoy et al. 1985; Stephan et al. 2019). The observations of airglow at 91.1 nm and 135.6 nm are used for investigating \({\mathrm O}^+\) distributions, especially on the nightside (e.g., Dymond et al. 2004; Sagawa et al. 2005; Immel et al. 2006; Dymond et al. 2017c; Eastes et al. 2019).
The International Space Station–ionospheremesosphereatmosphere Plasmasphere cameras (ISSIMAP) mission operated two kinds of imagers on board the International Space Station (ISS). One was the Visible and nearInfrared Spectral Imager (VISI) (Sakanoi et al. 2011) and the other was the Extreme Ultraviolet Imagers (EUVI) (Yoshikawa et al. 2011). EUVI consisted of two telescopes, EUVIA and EUVIB. EUVIA was designed to observe the 30.4 nm radiation from \({\mathrm He}^+\) and EUVIB was intended to observe the 83.4 nm radiation from \({\mathrm O}^+\). As illustrated in Uji et al. (2014) and Hozumi et al. (2017), these two telescopes are directed toward the aft direction of the ISS orbit to conduct limb observations of the Earth’s ionosphere. The EUVI telescopes thus acquired information on vertical and horizontal structures of the ionosphere and plasmasphere.
The signals taken from EUVIA were likely to be 30.4 nm EUV due to resonant scattering by \({\mathrm He}^+\) ions (Hozumi et al. 2017). However, the source of the signals from EUVIB has not been verified. Hozumi (2013) reported that EUVIB sometimes detected evident EUV signals in the umbra of the Earth, though these nighttime signals were not satisfactorily explained. Indeed, modeling of the EUVIB measurements is not easy. Since 83.4 nm EUV undergoes multiple scattering by \({\mathrm O}^+\) ions in the ionosphere (e.g., Stephan , 2016; Geddes et al. , 2016), the relation with the \({\mathrm O}^+\) density profile along the line of sight would be complicated. In addition, it should be noted that the wavelength range covered by EUVIB was not narrow enough. Since EUVIB covered the wavelength range from about 70 nm to 110 nm (Yoshikawa et al. 2011; Uji et al. 2014), it did not exclude the contribution of other nightglow emissions, such as the 91.1 nm, 98.9 nm, and 102.6 nm emissions. In particular, since the 91.1 nm emission due to recombination between \({\mathrm O}^{+}\) ions and electrons is one of the prominent emissions on the nightside (Chakrabarti et al. 1984; Meier 1991; Feldman et al. , 1992), it is necessary to take into account its contribution.
The purpose of this paper is to clarify the source of the signals observed by EUVIB to facilitate the use of the EUVIB data. Since the EUVIA and EUVIB telescopes were operated when the ISS orbited in the umbra of the Earth (Uji et al. 2014), the EUV flux observed on the nightside is discussed. We conduct a Monte Carlo simulation to examine the effect of multiple scattering of the 83.4 nm EUV. We also examine the effect of the 91.1 nm EUV. The contribution of each effect is then quantitatively evaluated.
Observation
The operation of ISSIMAP started in October 2012. The altitude of the ISS is about 400 km and the orbital inclination is \(51^{\circ }\). The ISSIMAP telescopes thus conducted imaging observations of the ionosphere at mid and low latitudes. The field of view of the imager EUVIB was \(13.2^{\circ }\times 13.2^{\circ }\). Figure 1 shows the EUV images taken from EUVIB during 21:08:26–21:09:25 UT and 21:18:38–21:19:37 UT on December 26, 2012. In each panel, the orange contours indicate the altitude at the tangential point of the line of sight for each pixel. The altitudes from about 100–400 km were covered by the field of view of this imager. Each pixel shows the count of photons. The expected count C for \(4\pi I\) Rayleighs of EUV flux was
where S is the total sensitivity and T is the exposure time. S was equal to 0.0019 [counts/s/Rayleigh/bin] for EUVIB at 83.4 nm (Uji et al. 2014). The exposure time T was 60 s here. The expected count for 100 Rayleighs is thus about 11.4 for the 83.4 nm EUV. Figure 2 shows the position of the ISS from 21:02 UT to 21:25 UT on December 26, 2012. Since each telescope was directed toward the backward direction of the ISS orbit as described above, Fig. 1 captured the northward (equatorward) view from the ISS location. The ISS was located near the equator around 21:08 UT, and it was located at mid latitude in the southern hemisphere around 21:18 UT.
In Fig. 3, the blue line indicates the time profile of the EUV intensity from 21:02 UT to 21:25 UT on December 26, 2012, which corresponds to the period indicated in Fig. 2. The EUV intensity is shown by the averaged count over the \(20\times 20\) pixels surrounded by the red square in each panel of Fig. 1. The error bars in this figure shows the \(\pm 2\sigma \) intervals where \(\sigma \) is the standard errors of the averaged count. Note that the observation was not necessarily taken every 60 s while the exposure time was 60 s. For reference, we plot the total electron content (TEC) value from IONEX (IOnosphere Map Exchange) at the ISS location with the orange line, and that at the tangential point of the line of sight with the green line. The averaged EUV count showed a bimodal profile, likely corresponding to the electron density structure indicated by the TEC observation. Figure 2 shows the position of the ISS for the corresponding period. The trough in the bimodal structure of the TEC was observed when the ISS crossed the dip equator, and thus it was regarded as the equatorial anomaly (e.g., Appleton 1946). The EUV observation thus corresponds well to the electron density structure. In the following, we work to reproduce this observational result.
Modeling of scattering
The EUV at 83.4 nm is attributed to resonant scattering by \({\mathrm O}^{+}\) ions. The EUV flux due to the scattering, \(F_s {\mathrm [photons/(m^{2}\cdot str\cdot s)]}\), is obtained by the following lineofsight integral:
where \({\varvec{r}}\) denotes the location, \(\ell \) is the distance of \({\varvec{r}}\) from the observation site, \(p(\nu ')\) is the probability density function for the frequency of the scattered EUV \(\nu '\), \(\tau \) is the optical depth, \({\bar{\sigma }}_s\) is the scattering cross section, N is the \({\mathrm O}^{+}\) number density, and \(\Phi \) is the photon flux to be scattered. The photon flux is written as a function of the frequency \(\nu \), the location \({\varvec{r}}\), and the direction \(\omega \). We hereinafter assume that the scattering is isotropic. We take the \({\mathrm O}^{+}\) number density N from the International Reference Ionosphere (IRI) model (Bilitza et al. 2014). The optical depth \(\tau \) is associated with absorption due to neutrals and it is given by combining the lineofsight integral for each species:
where i denotes absorbing species and \(\alpha _i\) and \(n_i\) are the absorbing cross section and number density for each of the species, respectively. In this study, the absorption due to \({\mathrm N}_2\), \({\mathrm O}\), and \({\mathrm O}_2\) is taken into account. The number densities for these absorbing species are given by the Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Radar Exosphere (NRLMSISE00) model (Picone et al. 2002), and the absorbing cross sections are taken to be \(10.1\times 10^{22}\,{\mathrm m}^2\), \(3.9\times 10^{22}\,{\mathrm m}^2\), and \(10.4\times 10^{22}\,{\mathrm m}^2\) for \({\mathrm N}_2\), \({\mathrm O}\), and \({\mathrm O}_2\), respectively, according to Link et al. (1994). The probability density function of \(\nu '\) is given as a Gaussian distribution:
according to Eq. (14) in Appendix 1.
The scattering cross section \({\bar{\sigma }}_s\) should be dependent on the EUV frequency \(\nu \) because of the thermal motion of the scattering \({\mathrm O}^{+}\). We obtain the scattering cross section \({\bar{\sigma }}_s\) by the following equation
as shown in Appendix1, where e is the elementary charge, \(\varepsilon _0\) is the vacuum permittivity, m is the mass of a scattering ion (\({\mathrm O}^+\)), c is the light speed, \(f_{12}\) is the oscillator strength, \(\nu _0\) is the wave frequency at the center of the \(83.4\,{\mathrm nm}\) resonance line, and \(\Delta _\nu \) is the scale of Doppler broadening given by Eq. (15). The oscillator strength is taken at \(f_{12}=0.122\) according to Meier (1991). In obtaining \(\Delta _\nu \), the ion temperature is taken from the IRI model. The photon flux \(\Phi \) in Eq. (2) depends on the direction \(\omega \). It is demanding to obtain \(\Phi \) as a function of \(\omega \) over the line of sight for computing Eq. (2). However, since we assume isotropic scattering, it is not necessary to resolve the dependence of the photon flux on the direction \(\omega \), and instead the number density of photons at \({\varvec{r}}\) is sufficient to obtain \(F_s\). Denoting the number density of photons (per volume and frequency) as \(\rho \), Eq. (2) can be rewritten as
where we denote
While the integral with \({\mathrm d}\ell \) and \({\mathrm d}\nu '\) was computed using the trapezoidal rule, we estimated the integral A by the Monte Carlo simulation in which the effect of multiplescattering was considered. In this Monte Carlo simulation, we computed the propagation of a large number of photons from the EUV seeds on the sunlit ionosphere, and estimated the spatial distribution of the photon density resulting from the scattering by \({\mathrm O}^+\) ions. The details of the Monte Carlo simulation are described in Appendix 2. As explained in the appendix, our Monte Carlo model would tend to overestimate the 83.4 nm flux for the target event on 26 December 2012 because of the choice of the parameter of the incident solar flux. The result of the Monte Carlo simulation would thus provide an upperlimit estimate of the 83.4 nm EUV flux.
Modeling of \({\mathrm O}^{+}\) recombination
The EUV at 91.1 nm is emitted as a result of recombination of \({\mathrm O}^{+}\) ions and electrons. The EUV flux due to the \({\mathrm O}^{+}\) recombination is obtained by the following equation:
where \(n_{\mathrm e}\) is the electron density and \(n_{{\mathrm O}^+}\) is the \({\mathrm O}^{+}\) ion density. The electron density \(n_{\mathrm e}\) and the \({\mathrm O}^{+}\) density \(n_{{\mathrm O}^+}\) are taken from the IRI model; that is, we assume the same \({\mathrm O}^{+}\) density distribution as for the 83.4 nm scattering model. The factor \(\lambda \) is the radiative recombination coefficient and \(\lambda =3.5\times 10^{23}\times (1160/T)\,{\mathrm m^3/s}\) (MeléndezAlvira et al. 1999) where T is the electron temperature in Kelvin. The optical depth \(\tau \) can be obtained according to Eq. (3). The absorbing species taken into account are \({\mathrm N}_2\), \({\mathrm O}\), and \({\mathrm O}_2\). The absorption coefficients for 91.1 nm EUV are set to \(14.5\times 10^{22}\,{\mathrm m}^2\), \(3.93\times 10^{22}\,{\mathrm m}^2\), and \(15.34\times 10^{22}\,{\mathrm m}^2\) for \({\mathrm N}_2\), \({\mathrm O}\), and \({\mathrm O}_2\), respectively, according to Dymond et al. (2017c), which uses the values provided by Kirby et al. (1979) and Conway (1988). The number densities for the absorbing species are given according to the NRLMSISE00 model.
Result
Figure 4 shows the model EUV time profile to be observed by EUVIB for the 83.4 nm EUV as a result of resonant scattering (red line) and for the 91.1 nm EUV as a result of recombination (green line). In this figure, the expected count was obtained by the lineofsight integral taken in the direction of the center of the square region over which the averaged EUV count was calculated in Fig. 3 (i.e., the region surrounded by the red square in each panel of Fig. 1). According to Fig. 3, we simulated the EUV measurement during the period 21:00–21:30 UT on December 26, 2012 with the expected count value for one minute which can be obtained with Eq. (1). The total sensitivity S was 0.0019 at 83.4 nm and 0.0018 at 91.1 nm. The simulation result indicated that the contribution of the 91.1 nm EUV dominated that of the 83.4 nm EUV during the period of interest when the ISS orbited on the nightside. In this figure, the mean count observed by EUVIB, which was shown in Fig. 3, is overplotted (blue line). The expected count value for the simulated 91.1 nm EUV was quantitatively comparable to the observed averaged count although the simulated EUV intensity was a little weaker than the observed intensity for the second peak around 21:18 UT. On the other hand, the EUV intensity at 83.4 nm calculated with the Monte Carlo model was much less than the observed value. As explained in Appendix 2, our Monte Carlo simulation would gives an upperlimit estimate of the 83.4 nm EUV for this event. This result therefore suggests that the 83.4 nm EUV could not explain the observed EUV count in the umbra of the Earth. The simulation result for the 91.1 nm EUV also roughly reproduced the bimodal structure seen in the averaged count in Fig. 3. In contrast, Fig. 4 shows that the 83.4 nm EUV should only be discernable when the ISS orbited near the dusk. When the ISS crossed the equator, the 83.4 nm EUV was not detectable in this event because the ISS was located in the deep umbra. We therefore conclude that the observed signal from EUVIB in Fig. 3 can be attributed to the 91.1 nm EUV. On the righthand side of Fig. 4, we show the scale in Rayleighs which is obtained by assuming that the observed signal is entirely due to the 91.1 nm EUV. The estimated photon flux values might contain systematic errors due to the uncertainty of the sensor sensitivity. However, since the sensor uncertainty cannot be quantified, the error bars in Fig. 4 just represents the \(\pm 2\sigma \) intervals for the standard errors of the estimated flux.
Figure 5 shows the simulated images of the 91.1 nm EUV for EUVIB at 21:08:56 UT and 21:19:08 UT on December 26, 2012. The color scale for each panel was adjusted in accord with Fig. 1. There are some discrepancies between Figs. 1 and 5. For example, while Fig. 1a suggests the source of the EUV emission shifted toward higher altitudes around 21:09 UT, the simulated emissions from the lower ionosphere were not as weak as the observed emissions. Figure 6 display the differences of the observed images in Fig. 1 from the simulated images in Fig. 5. The blue colors indicate that the observed EUV intensity exceeded the simulated intensity. Figure 6a suggests that the simulation underestimated the emissions from the lower altitudes. However, the EUV flux from the higher altitudes were well reproduced. Figure 6b suggests that the model underestimated the observed EUV signals around 21:19 UT as also suggested in Fig. 4. These discrepancies could be caused by variation of local conditions compared to the climatology represented in the models our simulation relied on. We determined the electron and \({\mathrm O}^+\) densities by the IRI model. The densities of the absorbing neutrals were given by the NRLMSISE00 model. Although these empirical models provide the average state for a given condition, they are not necessarily highly accurate for each particular event. In general, spatiotemporal variations tend to be smoothed in empirical models. That would be a reason why the dip of the EUV flux observed around 21:14 UT was less clear in the simulation result. Figure 7 compares the IONEX TEC values shown in Fig. 3 with the reconstructed TEC values derived from the IRI model. The IONEX TEC values at the ISS location and the tangential point are plotted with the thick orange and green lines, respectively. The IRI TEC values at the ISS location and the tangential point are plotted with the pale orange and green lines, respectively. There are visible differences between the IONEX TEC and the IRI TEC, which would introduce errors in our model. The trough in the bimodal structure is less clear in the IRI TEC. It should also be noted that the discrepancies might partly be caused by other nightglow emissions. Considering the wavelength range covered by EUVIB (70–110 nm), EUVIB can also detect 98.9 nm and 102.6 nm emissions. The 98.9 nm and 102.6 nm emissions are typically weaker than the 83.4 nm emission by one order (Abreu et al. 1984; Chakrabarti et al. 1984; Meier 1991; Dymond et al. 2017b). Moreover, the sensitivity of EUVIB was 0.0011 at 98.9 nm and 0.00047 at 102.6 nm, while it was 0.0018 at 91.1 nm. The contributions of the 98.9 nm and 102.6 nm EUV would therefore be too small to explain the entire difference. However, those emissions might, in part, contribute to the difference of the observation from the model prediction which assumes that the EUVIB signals are totally attributed to the 91.1 nm EUV. At any rate, rough patterns in the observations were successfully reproduced by the simulation for the 91.1 nm EUV based on Eq. (8).
Discussion
As shown in the previous section, the EUV signals observed by EUVIB in the umbra of the Earth can most likely be attributed to the 91.1 nm EUV. Although some discrepancies were seen in detailed structures between the simulated signals at 91.1 nm and the observed signals from EUVIB, the observations were mostly reproduced by the model for the 91.1 nm EUV described in “Modeling of O^{+} recombination” section. Figure 8 is a histogram of the observed photon flux to show how often EUV signals were observed by EUVIB for each local time interval in nighttime. This histogram displays the frequency distribution of the photon flux, which was estimated by assuming that EUVIB observed the 91.1 nm EUV, for the period from 21 December 2012 to 3 March 2013. In order to focus on lowlatitude observations, the data taken when the ISS latitude was higher than \(20^{\circ }\) (\(20^{\circ }\)N and \(20^{\circ }\)S) were excluded from this histogram. This figure shows that EUVIB frequently observed EUV signals especially from 20:00 to 0:00 local time. This result is encouraging for the use of IMAP/EUVIB data for studies of electron and \({\mathrm O}^+\) density distributions in the ionosphere. Various efforts have been made to obtain the \({\mathrm O}^+\) density distribution from remote optical measurements (e.g., Dymond et al. 1997; Qin et al. 2015). However, it is complicated to make an estimation from the 83.4 nm EUV data because of multiple scattering (e.g., Geddes et al. 2016). In contrast, estimating the \({\mathrm O}^+\) density from the 91.1 nm EUV would be much easier because the relationship between the observed signals and the \({\mathrm O}^+\) density is relatively simple. In the Fregion, since the electron density is approximately equal to the \({\mathrm O}^+\) density, Eq. (8) can be approximated as
If the densities of the absorbing neutrals are known and thus the optical depth \(\tau \) is known, only the \({\mathrm O}^+\) density \(n_{{\mathrm O}^+}\) is unknown. The problem for obtaining \(n_{{\mathrm O}^+}\) thus becomes simple. Indeed, Dymond et al. (2017a, 2017c) demonstrated the reconstruction of the \({\mathrm O}^+\) density distribution based on EUV data at 91.1 nm. Multiple images from different viewpoints according to the orbital motion can be used to estimate the threedimensional ion distribution (e.g., Nakano et al. 2014). The EUVIB data could thus be used to reconstruct the threedimensional \({\mathrm O}^+\) distribution in the vicinity of the ISS orbit.
When the EUVIB signals are treated as the 91.1 nm EUV, the contribution of the 83.4 nm EUV should be regarded as noise. Although the simulation result in Fig. 4 suggested that the contribution of the 83.4 nm EUV was negligible for most of the period of interest, it would be meaningful to evaluate how deeply the 83.4 nm EUV penetrates the umbra of the Earth. We computed the omnidirectional flux of the 83.4 nm EUV at an altitude of 400 km using the Monte Carlo model described in “Modeling of scattering” section, where the omnidirectional flux is the flux averaged over \(4\pi \) steradian. Figure 9 shows the estimated omnidirectional flux in Rayleighs for the nightside. The sunlit region at 400 km altitude was masked in this figure to highlight the penetration into the umbra. Since the EUV flux on the nightside would not be isotropic, the omnidirectional flux would not match the EUV flux observed by an imager. Figure 9, however, would provide a rough estimate of the contribution of the 83.4 nm EUV. The estimated omnidirectional flux could be discerned only near the terminator, and even there it was less than 1 Rayleigh. Therefore, it would be reasonable to assume the contribution of the 83.4 nm EUV to the EUVIB measurements is negligible.
When using the EUVIB data for estimating the \({\mathrm O}^+\) density, we should take care of other factors which can affect the estimation. One issue is the uncertainty of the sensitivity value S. Although the uncertainty of S is unknown and it has not been quantified, it might introduce some bias to the estimate. We should also take into consideration the possible contributions of the 98.9 nm and 102.6 nm emissions. Although those emissions are inferred to be small as described above, their effects might cause some errors when estimating the \({\mathrm O}^+\) density. However, these problems can be calibrated by comparing other independent ionospheric data such as the TEC data.
Summary
In order to explain the EUV signals observed from IMAP/EUVIB on board the ISS on the nightside, we conducted two numerical modelings. One is a Monte Carlo simulation to examine the effect of the 83.4 nm emission due to the resonant scattering by \({\mathrm O}^+\) ions. The other is a forward model to examine the effect of the 91.1 nm emission due to the recombination between electrons and \({\mathrm O}^+\) ions. The simulation result suggests that the EUV signals observed from EUVIB during nighttime were mainly the 91.1 nm EUV. The contribution of the 83.4 nm emission was typically negligible during nighttime. Since reconstructing the \({\mathrm O}^+\) density distribution from the 91.1 nm EUV would be relatively easy, it is expected that the EUV signals observed from IMAP/EUVIB will be useful for studies of electron and \({\mathrm O}^+\) density distributions in the ionosphere.
Availability of data and materials
ISSIMAP/EUVI data are available via email inquiry to Akinori Saito at Kyoto University (saitoua@kugi.kyotou.ac.jp). The IONEX TEC data were obtained via “https://cddis.nasa.gov/archive/gnss/products/ionex/”. The F10.7 solar radio flux data were obtained from “https://www.spaceweather.gc.ca/forecastprevision/solarsolaire/solarflux/sxen.php”.
Abbreviations
 EUV:

Extreme ultraviolet
 ISS:

International Space Station
 ISSIMAP:

International Space Station–ionospheremesosphereatmosphere Plasmasphere cameras
 EUVI:

Extreme Ultraviolet Imagers
 TEC:

total electron content
 IONEX:

IOnosphere Map Exchange
 IRI:

International Reference Ionosphere
 NRLMSISE:

Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Radar Exosphere
 DMSP:

Defense Meteorological Satellite Program
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Acknowledgements
The Monte Carlo computation in this study was carried out on the Supercomputer System for Statistical Science at ISM under the ISM Cooperative Research Program (2020ISMCRP2003).
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SN conducted the numerical modelings and simulation. YH and AS conceived the study and processed the ISSIMAP/EUVIB data. IY, AY, KY, and GM developed the ISSIMAP/EUVI and calibrated the data. All authors read and approved the final manuscript.
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Appendices
Appendix 1. Scattering cross section
The scattering cross section \({\bar{\sigma }}_s\) is derived by considering the thermal motion of scattering ions. The scattering cross section of one \({\mathrm O}^+\) particle for the EUV at frequency \(\nu \) is
(Meier 1991), where \(\nu _{\mathrm p}\) is the center of resonance frequency in the observer’s frame, \(\varepsilon _0\) is the vacuum permittivity, e is the elementary charge, m is the mass of a scattering ion (\({\mathrm O}^+\)), c is the light speed, and \(f_{12}\) is the oscillator strength. The function L is given by
which is the probability density function of the Lorentz distribution, where the parameter \(\gamma \) determines the scale of the width of the distribution. The center of resonance frequency \(\nu _p\) is perturbed by thermal motion of the scattering \({\mathrm O}^+\) ion, and can be written as
where \(\nu _0\) is the frequency at the center of the resonance line, \({\varvec{n}}_I\) is the unit vector in the incident direction, \({\varvec{v}}_{{\mathrm th}}\) is the thermal velocity of the \({\mathrm O}^+\) ion, and \(v_{{\mathrm th},I}={\varvec{n}}_I\cdot {\varvec{v}}_{{\mathrm th}}\). Assuming that the \({\mathrm O}^+\) ions follows an isotropic Maxwell distribution, the probability density function of \(v_{{\mathrm th},i}\) can be written as follows:
where \(\kappa \) denotes the Boltzmann constant and T is the \({\mathrm O}^+\) ion temperature. The probability density function of \(\nu _p\) thus becomes
where \(\Delta _\nu \) is the scale of Doppler broadening given by
Integrating Eq. (10) over the distribution in Eq. (14), the cross section becomes
Denoting \(x_{\mathrm p}=(\nu _{\mathrm p}\nu _0 )/\Delta _\nu \) and \(x=(\nu \nu _0 )/\Delta _\nu \), Eq. (16) can be rewritten as
Since \(\gamma \ll \Delta _\nu \) for the 83.4 nm EUV in the ionosphere (e.g., Meier 1991), we can adopt the following approximation:
to compute the integral in Eq. (17), where \(\delta \) denotes the Dirac delta function, and Eq. (17) becomes
Therefore, the cross section averaged over the thermal motion of \({\mathrm O}^+\) ions is obtained as follows:
Appendix 2. Monte Carlo model for evaluating EUV scattering
We fix the step size \(\Delta \ell \) for computing the photon propagation at \(0.001R_E\) in this study where \(R_E\) is the Earth radius. Given the position of the photon j at the kth step as \({\varvec{r}}_{j,k}\), the transport of the photon in one step can be written as
if the photon is not scattered during this step, where \({\varvec{d}}_{j,k}\) is the unit vector in the direction of the photon propagation. The probability that one photon is scattered during a transit over a small segment \(d\ell \) is
and hence the length of the path over which the photon does not encounter a scattering event, s, obeys the following exponential distribution:
where \(\lambda ={\bar{\sigma }}_s(\nu ) N\). At each step, we draw a random number \(s_t\) from the distribution in Eq. (23). If \(s_t\ge \Delta \ell \), we regard that the photon does not scatter during this step and the direction of the photon propagation is not changed:
On the other hand, if \(s_t<\Delta \ell \), the direction of the photon propagation is changed at
and \({\varvec{r}}_{j,k+1}\) becomes
where \({\varvec{d}}_{j,k+1}\) is a unit vector in a random direction, which is drawn from the uniform distribution on the unit sphere. As a result of the scattering, the wave frequency \(\nu \) for the photon j changes according to the thermal motion of the scattering \({\mathrm O}^{+}\). In this Monte Carlo model, if the direction of the photon propagation is changed, the wave frequency is also changed and the new frequency \(\nu _{k+1}\) is assumed to obey the following probability distribution independent of the previous frequency as follows:
according to Eq. (14). The computation of the photon propagation is continued until the photon escapes the simulation domain. The lower and upper boundaries of the simulation domain were taken to be 100 km and 3600 km in altitude. If the photon went out of the outer boundary, we assumed that it escaped toward the magnetosphere and we stopped the computation of its trajectory. If the photon dropped out of the inner boundary, we assumed that it was absorbed in the atmosphere and also stopped the computation.
If the sequence of photon positions is recorded at discrete steps \(\{{\varvec{r}}_{j,0}, {\varvec{r}}_{j,1}, {\varvec{r}}_{j,2}, \ldots \}\) for each test photon j, (\(j=1,\ldots , J\)), we can obtain a sample set of the photons flowing through the ionosphere by collecting the sequences of photon positions for all of the J test photons. \(\rho \) can be estimated from this sample set. We divide the simulation domain into bins of \(1^{\circ }\) (in longitude) \(\times \ 0.5^{\circ }\) (in latitude) \(\times \ 10\,{\mathrm km}\) (in altitude), and the photon density for each bin is obtained as follows:
where \(B_m\) denotes the mth bin, \(V_{B_m}\) is the volume of the bin \(B_m\), and \(w_{j,k}\) is the weight of an individual photon sample at \({\varvec{r}}_{j,k}\). \(w_{j,k}\) represents the number of real photons represented by the jth test photon. To consider the absorption due to neutrals, it is updated at each step according to the following equation:
where the optical depth \(\tau _{j,k}\) for one step is approximated as
The initial source of the EUV at 83.4 nm is mainly the photoionization of atomic oxygen due to solar EUV less than 43.6 nm (e.g., Stephan 2016). We draw the initial position of each test photon, \({\varvec{r}}_{j,0}\), according to a uniform distribution over the sunlit simulation domain. However, the EUV emission due to the photoionization would not be uniform over the ionosphere because the atomic oxygen density is not uniform. We set the initial weight \(w_{j,0}\) to represent the amount of initial EUV photons generated by the photoionization. We assume that the number of initial photons generated due to the photoionization per volume per time can be written in the following form:
where \(n_{{\mathrm O}}\) is the atomic oxygen density, \(\tau _s\) is the attenuation coefficient of the solar EUV radiation at \({\varvec{r}}\), and \(\zeta \) is a constant. \(\tau _s\) is determined after considering the absorption of the incident solar EUV due to photoionization due to \({\mathrm O}\). We assume \(\tau _s=0\) at the upper boundary of the simulation domain. We calculate \(\tau _s\) below the upper boundary by integrating the absorption from the upper boundary to the point \({\varvec{r}}_{j,0}\) where the absorption cross section is assumed to be \(1.0\times 10^{21}{\mathrm m}^2\) in order to be comparable to the literature values, such as that of Kirby et al. (1979) and Meier et al. (2007). The initial weight \(w_{j,0}\) is then set as follows:
where \(V_{{\mathrm sunlit}}\) is the volume of the sunlit region of the simulation domain, \(V_{B({\varvec{r}}_{j,0})}\) is the volume of the bin which \({\varvec{r}}_{j,0}\) belongs to, and M is the total number of test photons used in this Monte Carlo simulation. Since \((V_{B({\varvec{r}}_{j,0})}/V_{{\mathrm sunlit}})M\) is the expected number of initial photons in the bin \(B({\varvec{r}}_{j,0})\), \(w_{j,0}\) corresponds to the number of real photons represented by the jth test photon. In our Monte Carlo simulation, the number of test photons M was set to be \(14.4\times 10^{10}\). The simulation took about 4 h by using a computer with 80 CPUS (Intel Xeon Gold 6154) each of which has 18 cores (1440 cores in total).
The parameter \(\zeta \) should be determined according to the solar EUV irradiance and the photoionization cross section over the wavelength band which can contribute to the 83.4 nm emission. However, it is complicated to integrate the effect of EUV irradiance with various wavelengths. To avoid having to model the photoionization over a wide wavelength band, we tune the parameter \(\zeta \) by comparing the simulation result with the dayside observations reported by Stephan (2016). The 83.4 nm EUV observation would include the initial source EUV due to the photoionization as well as the EUV after scattering by \({\mathrm O}^+\) ions, especially on the dayside. Therefore, converting the photon flux into the unit of Rayleighs, the observed 83.4 nm EUV becomes
Figure 10 shows the simulated altitudinal profile of the EUV photon flux for comparing the observation from the Defense Meteorological Satellite Program (DMSP) F19 satellite shown by Stephan (2016). We simulated the observation from 850 km altitude after calibrating the parameter \(\zeta \). The tangent point of the line of sight was set at \(40^{\circ }\) in latitude and 0722 LT on 6 September in accord with Fig. 1 in the paper by Stephan (2016). Since the DMSP observation showed that the EUV intensity peaked around 250 km in altitude and the peak intensity was about 650 Rayleighs, we adjusted \(\zeta \) so that the peak intensity is similar to this DMSP observation. The incident solar flux on 26 December 2012 examined in this study must be different from that on 6 September 2014 examined by Stephan (2016). While the F10.7 flux was 110 on 26 December 2012, it 157 on 6 September 2014. This Monte Carlo model would thus tend to overestimate the 83.4 nm flux for the target event. However, we do not fix the difference in the solar flux between the two events because an upperlimit estimate of the 83.4 nm EUV flux is enough for the purpose of this study.
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Nakano, S., Hozumi, Y., Saito, A. et al. EUV signals associated with O^{+} ions observed from ISSIMAP/EUVI in the nightside ionosphere. Earth Planets Space 73, 151 (2021). https://doi.org/10.1186/s40623021014790
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Keywords
 \({\mathrm O}^+\) ion
 Ionosphere
 Extreme ultraviolet measurement