SPAM: Solar Spectrum Prediction for Applications and Modeling

Abstract

Solar Spectrum Prediction for Applications and Modeling (SPAM) is a new empirical model of solar X-ray, extreme ultraviolet and far ultraviolet radiation flux at the top of the Earth’s atmosphere. The model is based on 14 years of daily averaged TIMED spacecraft measurements from 2002 to 2016, when its sensors were regularly calibrated. We used a second-order parametrization of the irradiance spectrum by a single parameter—the F${}_{10.7}$ index—which is a reliable and consistently observed measure of solar activity. The SPAM model consists of two submodels for general and specific use. The first is the Solar-SPAM model of the photon energy flux in the first 190 spectral bands of 1 nm each, which can be used for a wide range of applications in different fields of research. The second model, Aero-SPAM, is designed specifically for aeronomic research and provides a photon flux for 37 specific wavelength intervals (20 wave bands and 16 separate spectral lines within the range of 5–105 nm, and an additional 121.5 nm Ly-alpha line), which play a major role in the photoionization of atmospheric gas particles. We provide the full set of parameterization coefficients that allows for the immediate implementation of the model for research and applications. In addition, we used the Aero-SPAM model to build a ready-to-use numerical application for calculating the photoionization rates of the main atmospheric components N${}_2$, O${}_2$, O, N and NO with known absorption and ionization cross sections.

1. Introduction

Operating at low Earth orbit, TIMED (Thermosphere Ionosphere Mesosphere Energetics and Dynamics) spacecraft provides measurements of the solar radiation spectrum in the shortwave range of 0–190 nm [1]. This exact radiation range is the main source of ion production at 80–200 km altitudes in the sunlit atmosphere (e.g., [2]). The small part of the solar radiation spectrum from 0–190 nm, measured by the TIMED spacecraft, makes a decisive contribution to the chemistry and dynamics of the Earth’s thermosphere and ionosphere [3,4], becoming one of the governing parameters for space weather [5] and global climate dynamics, including ozone variability [6,7,8].

In the D-region and the lower part of the E-region of the ionosphere, the extreme ultraviolet (EUV) band, including the Ly-alpha continuum (centered on 121.5 nm), usually plays a major role in the photoionization process, with minor contributions from the X-ray (0–10 nm) and far ultraviolet (FUV, 122–190 nm) irradiance. However, during solar flares, the X-ray flux can increase by several orders of magnitude, which leads to a significant increase in the photoionization of the D-region [9,10]. The ultraviolet (UV) radiation from 100–190 nm induces molecular oxygen dissociation in mesosphere, contributing to ozone layer formation [11]. The EUV range of the radiation spectrum is absorbed by the upper atmosphere, creating ionosphere regular E and F1 layers [12,13,14], which play a crucial role in the propagation of high-frequency radio waves [15,16].

Shortwave irradiance permanently changes together with the changing Sun (solar cycle, solar rotation, solar flares and other sources of solar variability), which, in turn, leads to congruent variation in the chemical composition of the upper atmosphere [17,18,19,20,21,22]. The variability of shortwave radiation flux in specific spectral ranges can exceed 50% during 27-day solar rotation, 200% in the course of the solar cycle and several orders of magnitude during solar flares [23].

Solar X-rays and UV irradiance are completely absorbed by the upper atmosphere, so spacecraft measurements remain the only reliable source of information on the shortwave radiation flux [24,25,26]. Spacecraft lifetime limits, their possible planned and unplanned outages, sensor degradation and other reasons lead to the need for empirical modeling and spectrum reconstruction used for many applications. This task is far from new, but given the strong variability of the shortwave spectrum, the limited possibilities of its measurement and its strong influence on the chemistry of the upper atmosphere, the problem of solar spectrum modeling remains relevant [27,28,29,30].

There are a number of EUV models specially designed for atmospheric studies. The publicly available solar EUV flux model for the aeronomic calculations (EUVAC) empirical model [31] is based on the Atmosphere Explorer E spacecraft measurements from 1977 to 1981, and is widely used for aeronomy research as a source of EUV radiation, covering a wavelength range from 5 to 105 nm. The model is parameterized by function P${}_{10.7}=$ (F${}_{10.7}$A + F${}_{10.7}$)/2, where F${}_{10.7}$ is the daily solar activity index and F${}_{10.7}$A is the 81-day average around the day of calculation. This formulation significantly narrows the applicability of the model, making it impossible to use for real-time observations and forecasts. Moreover, Ref. [32] revealed a significant systematic deviation in the EUVAC results. The discrepancy between the simulated and measured total EUV radiation is 20–40% depending on the solar activity level [33].

With the launch of the TIMED spacecraft, the EUVAC model was further upgraded to the HEUVAC (high-resolution version of the solar EUV irradiance model for aeronomic calculations) version using measurements from 2003 to 2010, but its parametrization remained the same as in EUVAC [34]. However, [35] found that, during the solar minimum, the HEUVAC soft X-ray irradiance is ∼65% larger and the Ly-alpha continuum flux is ∼30% smaller than spacecraft measurements. They suggested their own model, but parameterized by the same function P${}_{10.7}$, which again led to the inability of real-time calculations and forecasts.

The flare irradiance spectral model (FISM) and the improved version, FISM2, consider both the quiet Sun variation and the solar flare component of irradiance spectrum [36,37,38]. FISM is based on a modification of the solar minimum reference spectrum by summarizing the irradiance variability from the solar cycle, solar rotation and two solar flare components due to gradual and impulsive phase variations. All contributions are modeled independently, taking into account various proxies: F${}_{10.7}$, the MgII core-to-wing ratio and Ly-alpha emission line and the 17.1 and 30.4 nm emission lines from the Solar Dynamics Observatory (SDO), as well as the Geostationary Operational Environmental Satellite (GOES) X-ray radiation flux and its time derivative. FISM2 is currently available through the Interactive Solar Irradiance Data Center (https://lasp.colorado.edu/lisird/data/fism_p_ssi_earth/, accessed on 20 November 2022), but the online version has a two-year lag.

Ref. [39] made a simple EUV and FUV spectrum model using TIMED satellite data, which was parameterized by radiation flux in the Ly-alpha line. They propose using their model for the reconstruction of the shortwave radiation spectrum from the measurements of spaceborne spectral-selective photometers with a reduced spectral resolution.

The complexity of parameterization, model inaccessibility and the unavailability of control parameters in real time make it difficult to use solar irradiance models in aeronomic research and especially in real-time monitoring and forecasting. For the needs of aeronomy and other applications, we present a simple but accurate empirical model of the solar shortwave radiation spectrum SPAM (Solar Spectrum Prediction for Applications and Modeling), parameterized by a single ground-based F${}_{10.7}$ solar activity index. The model is based on 14 years of TIMED measurements and divided into two separate submodels: the Solar-SPAM model of the photon energy flux spectrum for the wavelength range of 0–190 nm with an initial TIMED resolution of 1 nm; and the Aero-SPAM model of photon flux in 37 wavelength intervals, including 20 wave bands and 16 separate spectral lines within the range of 5–105 nm, with an additional Ly-alpha line 121.5 nm, intended for aeronomic calculations.

2. Data

2.1. F${}_{10.7}$ Index

The solar activity index F${}_{10.7}$ is the radio emission at 10.7 cm (2800 MHz) measured by the ground-based receiver. The F${}_{10.7}$ index has a high correlation with sunspot numbers and ultraviolet and visible solar irradiance, which makes it an excellent measure of solar activity. The F${}_{10.7}$ observational series has a fairly long and continuous record. Everyday measurements have been publicly available since 1947 [40]. In addition, the reliable forecast of the F${}_{10.7}$ index [41,42,43,44,45] gives an opportunity to predict the upper atmosphere state up to 55 days.

In this study, the daily solar F${}_{10.7}$ index was taken from the OMNI database (https://omniweb.gsfc.nasa.gov/ow.html, accessed on 20 November 2022). The F${}_{10.7}$ index traditionally measures in solar flux units (s.f.u.), 1 s.f.u. = ${10}^{-22}·$W·m${}^{-2}·$ Hz${}^{-1}$.

2.2. TIMED Spacecraft Data

The TIMED spacecraft was commissioned on 7 December 2001 and is still in operation. One of the scientific objectives of the TIMED mission is to study the mesosphere and lower thermosphere dynamics under the influence of solar shortwave irradiance.

The SEE device (Solar EUV Experiment) (Woodraska et al., 2004) was developed for the TIMED mission at the University of Colorado. The SEE data represent the solar spectrum from 0 to 190 nm covering X-ray (0–10 nm), EUV (10–122 nm) and FUV (122–190 nm) spectrum ranges with 1 nm spectral and 97 min temporal resolution. The SEE device observes the Sun for approximately 3 min on each orbit, which gives 14–15 measurements per day. The SEE data are then processed to Level 3A by applying a correction for atmospheric absorption and sensor degradation and averaging over each 3-min observational interval. Here, we used the Level 3A SEE data (http://lasp.colorado.edu/home/see/data, accessed on 20 November 2022) to develop the empirical model of solar shortwave irradiance SPAM.

The SEE data represent the total energy flux in each 1 nm wide spectral interval. In the following, we will use the center point to define a specific interval, e.g., speaking of the spectral channel of 11.5 nm, we mean the same as the spectral band of 11–12 nm.

Several sounding rocket launches were undertaken during the mission to detect degradation trends and calibrate the SEE detector. The last suborbital calibration flight was carried out on 1 June 2016. Analyzing the spacecraft data, we found suspicious behavior of the radiation flux in some of the SEE spectral channels beginning from December 2016, soon after the last sensor calibration (more details in the Section 5). Although the TIMED spacecraft is still in operation, we limited the SEE data used in the development of the SPAM model to the interval from 22 January 2002 to 24 November 2016.

3. Solar-SPAM: Shortwave Energy Spectrum Model

The model is based on the long time series of X-ray, EUV and FUV solar irradiance received by the TIMED SEE instrument from 22 January 2002 to 24 November 2016. The histogram in Figure 1 shows the distribution of 5368 daily averaged spectra samples against the SPAM control parameter F${}_{10.7}$. Since the number of samples clearly decreases with an increasing solar activity, we set the limits on the data used in the SPAM model development to $65<F_{10.7}<200$ s.f.u., which includes 95% of all measured data. The excluded 5% of solar radiation measurements related to the tail of the F${}_{10.7}$ distribution (over 200 s.f.u.) are very sparse, have a significant scatter and often have random skewness. These data limits can also be considered as the limits of the SPAM model applicability.

Figure 2a shows the average energy spectrum based on TIMED measurements from 2002 to 2016. The vertical bars denote the standard deviation of measurement data in each spectral channel from 0 to 190 nm. The average spectrum has a complex structure with sharp changes in the energy flux by an order of magnitude even in neighboring channels.

It is worth noting how strongly the level of variability changes with an increasing wavelength. If the values of the X-ray energy flux vary more than an order of magnitude, then, for the EUV range, the energy flux changes only several times, and changes even less at the edge of the measured FUV range—within a few tens of percent (see examples in Figure 3 and Supplementary Material for the full set of scatterplots). To illustrate the level of irradiance variability more specifically, we show the relative standard deviation of the measured energy flux as a function of wavelength (Figure 2b). The standard deviation value drops exponentially from 100% to 2% with the wavelength increasing from 0 to 190 nm.

The naturally low variability of the FUV energy flux (Figure 3) in the course of the solar cycle also leads to low correlation coefficients between the model and measurement data in the spectral interval from 170 to 190 nm. The distribution of correlation coefficients over wavelengths is shown in Figure 2c. Most of the spectral bands demonstrate a high accuracy of the fitting procedure, and the value of the correlation coefficient between the model and measurement data exceeds 0.7 for most spectral bands and 0.9 for many spectral bands.

A continuous TIMED observation time series makes it possible to obtain a reliable functional dependence between the differential energy flux and solar activity index F${}_{10.7}$. The second-order polynomial is suggested as a best fit function by [46] for the parameterization of the flux intensity in separate spectral lines of HeII (30.4 nm), HeI (58.4 nm), CIII (97.8 nm) and FeXVIII (9.4 nm) in the 24-th solar cycle by the F${}_{10.7}$ index. As a result of data analysis, we found that the second-order parameterization is also valid for the entire shortwave radiation spectrum measured by TIMED spacecraft, including X-ray, EUV and FUV radiation. Thus, when developing the Solar-SPAM model, we used the parametrization of the energy flux F in each individual SEE spectral channel in the following form:

$$F=P_1·F_{10.7}^2+P_2·F_{10.7}+P_3,$$

(1)

where $P_1$, $P_2$ and $P_3$ are regression coefficients and $F_{10.7}$ is a daily averaged value of the solar activity index.

Figure 3 shows the scatterplots of energy flux measurements as a function of the F${}_{10.7}$ index and their quadratic fitting functions for several X-ray, EUV and FUV spectral channels: 0.5, 2.5, 4.5, 58.5 (contains HeI 58.4 nm line), 97.5 (contains CIII 97.8 nm line), 102.5 (contains Ly-beta 102.6 nm line), 139.5, 150.5 and 189.5 nm.

The entire set of regression coefficients of the Solar-SPAM model (Equation (1)) is collected in

Table A1. The Solar-SPAM model describing the solar irradiance spectrum (F, W·m${}^{-2}·$ nm${}^{-1}$) depending on the daily F${}_{10.7}$ index, $F=P_1·F_{10.72}+P_2·F_{10.7}+P_3$. $P_1$, $P_2$ and $P_3$ are the regression coefficients for the X-ray, EUV and FUV spectral intervals ($\lambda$) with 1 nm resolution. R is a correlation coefficient between F${}_{10.7}$ index and measured photon energy flux, RMSE: root-mean-square error calculated for each wavelength $\lambda$.

$\mathit{\lambda }$	${\mathit{P}}_1$	${\mathit{P}}_2$	${\mathit{P}}_3$	R	RMSE
1.5	−4.94024181$\times {10}^{-09}$	3.22175699$\times {10}^{-06}$	−1.80553335$\times {10}^{-04}$	0.93	2.67305197$\times {10}^{-05}$
2.5	−1.51234762$\times {10}^{-09}$	8.72873177$\times {10}^{-07}$	−3.59300652$\times {10}^{-05}$	0.96	4.64649582$\times {10}^{-06}$
3.5	−8.19051896$\times {10}^{-10}$	4.76131943$\times {10}^{-07}$	−1.75069528$\times {10}^{-05}$	0.96	2.57208191$\times {10}^{-06}$
4.5	−1.25299002$\times {10}^{-09}$	7.18966612$\times {10}^{-07}$	−2.30913459$\times {10}^{-05}$	0.96	3.78321360$\times {10}^{-06}$
5.5	−1.05306909$\times {10}^{-09}$	6.00386365$\times {10}^{-07}$	−1.72945105$\times {10}^{-05}$	0.96	3.13381799$\times {10}^{-06}$
6.5	−1.24449248$\times {10}^{-09}$	7.07626722$\times {10}^{-07}$	−2.22302072$\times {10}^{-05}$	0.96	3.69094684$\times {10}^{-06}$
7.5	−2.59694610$\times {10}^{-09}$	1.45374662$\times {10}^{-06}$	−6.31698909$\times {10}^{-05}$	0.96	7.43469515$\times {10}^{-06}$
8.5	−2.16866639$\times {10}^{-09}$	1.21455389$\times {10}^{-06}$	−4.6505577$\times {10}^{-05}$	0.96	6.21501074$\times {10}^{-06}$
9.5	−1.52361462$\times {10}^{-09}$	8.68533191$\times {10}^{-07}$	−3.0246892$\times {10}^{-05}$	0.96	4.53196359$\times {10}^{-06}$
10.5	−1.01521500$\times {10}^{-09}$	5.78430109$\times {10}^{-07}$	−1.79161041$\times {10}^{-05}$	0.96	3.02874105$\times {10}^{-06}$
11.5	−5.92986152$\times {10}^{-10}$	3.39846459$\times {10}^{-07}$	−7.46323908$\times {10}^{-06}$	0.96	1.86757596$\times {10}^{-06}$
12.5	−3.91620799$\times {10}^{-10}$	2.22633150$\times {10}^{-07}$	−6.85281881$\times {10}^{-06}$	0.96	1.16787269$\times {10}^{-06}$
13.5	−3.68662392$\times {10}^{-10}$	2.14384736$\times {10}^{-07}$	−5.53046688$\times {10}^{-06}$	0.96	1.20433914$\times {10}^{-06}$
14.5	−8.15473279$\times {10}^{-10}$	4.56904530$\times {10}^{-07}$	−1.28641555$\times {10}^{-05}$	0.96	2.34443139$\times {10}^{-06}$
15.5	−1.05314223$\times {10}^{-09}$	5.90337830$\times {10}^{-07}$	−2.51992115$\times {10}^{-05}$	0.96	3.02143819$\times {10}^{-06}$
16.5	−1.71833912$\times {10}^{-09}$	9.58528730$\times {10}^{-07}$	−2.37694140$\times {10}^{-05}$	0.96	4.91384954$\times {10}^{-06}$
17.5	−7.84248508$\times {10}^{-09}$	4.35156360$\times {10}^{-06}$	−6.77832612$\times {10}^{-05}$	0.96	2.24071577$\times {10}^{-05}$
18.5	−5.35989584$\times {10}^{-09}$	2.98171616$\times {10}^{-06}$	−7.78638812$\times {10}^{-05}$	0.96	1.53019138$\times {10}^{-05}$
19.5	−5.64009182$\times {10}^{-09}$	3.15119971$\times {10}^{-06}$	−1.24435988$\times {10}^{-04}$	0.96	1.61351843$\times {10}^{-05}$
20.5	−4.90601610$\times {10}^{-09}$	2.74377445$\times {10}^{-06}$	−1.31174042$\times {10}^{-04}$	0.96	1.40380060$\times {10}^{-05}$
21.5	−4.26086127$\times {10}^{-09}$	2.38514084$\times {10}^{-06}$	−1.14409417$\times {10}^{-04}$	0.96	1.22091037$\times {10}^{-05}$
22.5	−3.19159017$\times {10}^{-09}$	1.77982999$\times {10}^{-06}$	−5.49342607$\times {10}^{-05}$	0.96	9.12119600$\times {10}^{-06}$
23.5	−1.75266637$\times {10}^{-09}$	9.77475268$\times {10}^{-07}$	−2.67796958$\times {10}^{-05}$	0.96	5.01372608$\times {10}^{-06}$
24.5	−2.43810818$\times {10}^{-09}$	1.36192347$\times {10}^{-06}$	−4.57043670$\times {10}^{-05}$	0.96	6.97290978$\times {10}^{-06}$
25.5	−3.98387625$\times {10}^{-09}$	2.22551933$\times {10}^{-06}$	−7.70072803$\times {10}^{-05}$	0.96	1.13958300$\times {10}^{-05}$
26.5	−2.00773403$\times {10}^{-09}$	1.12849376$\times {10}^{-06}$	−5.67602506$\times {10}^{-05}$	0.96	5.77108499$\times {10}^{-06}$
27.5	−1.19730867$\times {10}^{-09}$	5.65394676$\times {10}^{-07}$	−5.85964757$\times {10}^{-06}$	0.94	3.19981683$\times {10}^{-06}$
28.5	−1.46295195$\times {10}^{-09}$	1.01404261$\times {10}^{-06}$	−3.58365291$\times {10}^{-05}$	0.93	8.68678105$\times {10}^{-06}$
29.5	−8.15084563$\times {10}^{-10}$	4.34467210$\times {10}^{-07}$	3.65433670$\times {10}^{-06}$	0.93	3.03848274$\times {10}^{-06}$
30.5	−7.05077152$\times {10}^{-09}$	4.04741769$\times {10}^{-06}$	1.43586031$\times {10}^{-04}$	0.94	2.75155631$\times {10}^{-05}$
31.5	−7.77266741$\times {10}^{-10}$	4.68802536$\times {10}^{-07}$	1.08074811$\times {10}^{-05}$	0.93	3.64885891$\times {10}^{-06}$
32.5	−5.61602010$\times {10}^{-10}$	2.83431306$\times {10}^{-07}$	−3.27491491$\times {10}^{-06}$	0.92	2.03502960$\times {10}^{-06}$
33.5	6.11880133$\times {10}^{-10}$	6.52060422$\times {10}^{-07}$	−2.90103367$\times {10}^{-05}$	0.91	1.14428500$\times {10}^{-05}$
34.5	−2.13887412$\times {10}^{-09}$	9.00810935$\times {10}^{-07}$	−1.41575668$\times {10}^{-05}$	0.93	5.08999204$\times {10}^{-06}$
35.5	−2.29655541$\times {10}^{-09}$	1.06445240$\times {10}^{-06}$	−3.23960927$\times {10}^{-05}$	0.94	6.31089521$\times {10}^{-06}$
36.5	−1.69356060$\times {10}^{-09}$	1.14642019$\times {10}^{-06}$	−1.49427472$\times {10}^{-05}$	0.93	9.15296502$\times {10}^{-06}$
37.5	−4.44754965$\times {10}^{-10}$	2.24144087$\times {10}^{-07}$	2.29852502$\times {10}^{-06}$	0.94	1.39955751$\times {10}^{-06}$
38.5	−2.57264554$\times {10}^{-10}$	1.20451838$\times {10}^{-07}$	1.08315857$\times {10}^{-07}$	0.95	6.47189724$\times {10}^{-07}$
39.5	−1.90799471$\times {10}^{-10}$	8.54129718$\times {10}^{-08}$	−3.66570920$\times {10}^{-07}$	0.95	4.29404481$\times {10}^{-07}$
40.5	−1.63110170$\times {10}^{-10}$	7.67917532$\times {10}^{-08}$	2.48479107$\times {10}^{-06}$	0.94	4.43317217$\times {10}^{-07}$
41.5	−2.64676272$\times {10}^{-10}$	1.61050402$\times {10}^{-07}$	−5.64014188$\times {10}^{-06}$	0.94	1.09306430$\times {10}^{-06}$
42.5	−8.80727378$\times {10}^{-11}$	5.09406819$\times {10}^{-08}$	1.64104876$\times {10}^{-06}$	0.94	3.40898878$\times {10}^{-07}$
43.5	−1.29810569$\times {10}^{-11}$	3.70763160$\times {10}^{-08}$	8.20072441$\times {10}^{-06}$	0.82	7.51972352$\times {10}^{-07}$
44.5	−1.92736419$\times {10}^{-10}$	9.35286333$\times {10}^{-08}$	2.78944317$\times {10}^{-08}$	0.95	5.03560346$\times {10}^{-07}$
45.5	−1.07354187$\times {10}^{-10}$	5.33853880$\times {10}^{-08}$	3.23226206$\times {10}^{-06}$	0.94	3.25167635$\times {10}^{-07}$
46.5	7.14925370$\times {10}^{-11}$	2.28505089$\times {10}^{-08}$	1.60620921$\times {10}^{-05}$	0.85	7.85818717$\times {10}^{-07}$
47.5	−1.65354143$\times {10}^{-10}$	7.61012429$\times {10}^{-08}$	3.10560608$\times {10}^{-06}$	0.95	3.96736808$\times {10}^{-07}$
48.5	−3.12973348$\times {10}^{-10}$	1.39150026$\times {10}^{-07}$	3.17305408$\times {10}^{-06}$	0.95	6.77782659$\times {10}^{-07}$
49.5	−8.48676578$\times {10}^{-10}$	3.82780831$\times {10}^{-07}$	−8.99955570$\times {10}^{-06}$	0.95	1.84750385$\times {10}^{-06}$
50.5	−6.92148698$\times {10}^{-10}$	3.05273387$\times {10}^{-07}$	−2.21022224$\times {10}^{-06}$	0.95	1.42588987$\times {10}^{-06}$
51.5	−3.52451773$\times {10}^{-10}$	1.49567940$\times {10}^{-07}$	−2.77810793$\times {10}^{-06}$	0.95	6.67205223$\times {10}^{-07}$
52.5	−4.88285423$\times {10}^{-10}$	2.06597500$\times {10}^{-07}$	−5.71616451$\times {10}^{-06}$	0.95	9.23676788$\times {10}^{-07}$
53.5	−1.51950475$\times {10}^{-10}$	6.22135728$\times {10}^{-08}$	4.00946070$\times {10}^{-06}$	0.92	3.69972956$\times {10}^{-07}$
54.5	−1.47270077$\times {10}^{-10}$	5.88009296$\times {10}^{-08}$	1.18884773$\times {10}^{-06}$	0.95	2.61720547$\times {10}^{-07}$
55.5	−2.67639507$\times {10}^{-10}$	9.72464276$\times {10}^{-08}$	2.22162082$\times {10}^{-05}$	0.63	1.37799232$\times {10}^{-06}$
56.5	−1.01960607$\times {10}^{-10}$	4.67610258$\times {10}^{-08}$	3.41494514$\times {10}^{-06}$	0.93	2.80937628$\times {10}^{-07}$
57.5	−1.61974440$\times {10}^{-10}$	6.72143036$\times {10}^{-08}$	1.17127082$\times {10}^{-06}$	0.95	3.07827728$\times {10}^{-07}$
58.5	−1.01763203$\times {10}^{-09}$	4.54759489$\times {10}^{-07}$	1.65241671$\times {10}^{-05}$	0.91	3.09599754$\times {10}^{-06}$
59.5	−1.56307209$\times {10}^{-10}$	6.36007172$\times {10}^{-08}$	3.86630485$\times {10}^{-06}$	0.92	3.58530411$\times {10}^{-07}$
60.5	−6.00806861$\times {10}^{-10}$	2.63372194$\times {10}^{-07}$	5.64617215$\times {10}^{-07}$	0.94	1.45080898$\times {10}^{-06}$
61.5	−6.04959878$\times {10}^{-10}$	2.45041951$\times {10}^{-07}$	−1.62883930$\times {10}^{-06}$	0.94	1.21428067$\times {10}^{-06}$
62.5	−6.49554222$\times {10}^{-10}$	2.79301792$\times {10}^{-07}$	1.91126658$\times {10}^{-05}$	0.89	2.08911118$\times {10}^{-06}$
63.5	−4.82850036$\times {10}^{-10}$	1.71382019$\times {10}^{-07}$	2.02470800$\times {10}^{-05}$	0.75	1.67842699$\times {10}^{-06}$
64.5	−1.06653492$\times {10}^{-10}$	4.22914685$\times {10}^{-08}$	1.50553273$\times {10}^{-06}$	0.94	2.03369385$\times {10}^{-07}$
65.5	−8.18714565$\times {10}^{-11}$	3.40516816$\times {10}^{-08}$	1.66550540$\times {10}^{-06}$	0.94	1.76539694$\times {10}^{-07}$
66.5	−9.90389288$\times {10}^{-11}$	4.04518744$\times {10}^{-08}$	1.56394010$\times {10}^{-06}$	0.94	1.96591466$\times {10}^{-07}$
67.5	−9.41472141$\times {10}^{-11}$	3.78395774$\times {10}^{-08}$	8.82049866$\times {10}^{-07}$	0.94	1.80481835$\times {10}^{-07}$
68.5	−8.29145674$\times {10}^{-11}$	3.45290364$\times {10}^{-08}$	5.19960242$\times {10}^{-06}$	0.82	3.44043640$\times {10}^{-07}$
69.5	−1.37956080$\times {10}^{-10}$	5.60239688$\times {10}^{-08}$	1.65024766$\times {10}^{-06}$	0.94	2.65869617$\times {10}^{-07}$
70.5	−1.01440668$\times {10}^{-10}$	4.37751602$\times {10}^{-08}$	9.44754103$\times {10}^{-06}$	0.79	4.90083335$\times {10}^{-07}$
71.5	−8.82187053$\times {10}^{-11}$	3.44490090$\times {10}^{-08}$	2.44945919$\times {10}^{-06}$	0.92	1.89763276$\times {10}^{-07}$
72.5	−1.38052514$\times {10}^{-10}$	5.94292750$\times {10}^{-08}$	4.19292916$\times {10}^{-07}$	0.95	2.87817244$\times {10}^{-07}$
73.5	−4.91176916$\times {10}^{-11}$	2.32864269$\times {10}^{-08}$	1.44437460$\times {10}^{-06}$	0.94	1.39806401$\times {10}^{-07}$
74.5	−6.46953103$\times {10}^{-11}$	2.83325477$\times {10}^{-08}$	2.59377737$\times {10}^{-06}$	0.93	1.66218244$\times {10}^{-07}$
75.5	−7.34309741$\times {10}^{-11}$	3.29461087$\times {10}^{-08}$	4.00460892$\times {10}^{-06}$	0.90	2.45820894$\times {10}^{-07}$
76.5	−3.05657856$\times {10}^{-11}$	2.93068006$\times {10}^{-08}$	1.40723930$\times {10}^{-05}$	0.74	6.32637967$\times {10}^{-07}$
77.5	5.43557600$\times {10}^{-11}$	2.19484499$\times {10}^{-08}$	1.18134495$\times {10}^{-05}$	0.89	5.64382540$\times {10}^{-07}$
78.5	−8.51779066$\times {10}^{-11}$	5.79401798$\times {10}^{-08}$	1.63692915$\times {10}^{-05}$	0.82	8.43951888$\times {10}^{-07}$
79.5	−2.21722606$\times {10}^{-10}$	8.88851417$\times {10}^{-08}$	8.27783879$\times {10}^{-06}$	0.89	6.07167204$\times {10}^{-07}$
80.5	−1.65576278$\times {10}^{-10}$	7.79627572$\times {10}^{-08}$	5.66367756$\times {10}^{-06}$	0.94	4.44114376$\times {10}^{-07}$
81.5	−1.92487027$\times {10}^{-10}$	8.70304770$\times {10}^{-08}$	5.81788594$\times {10}^{-06}$	0.93	5.12896384$\times {10}^{-07}$
82.5	−2.51845067$\times {10}^{-10}$	1.14456368$\times {10}^{-07}$	6.30086163$\times {10}^{-06}$	0.94	6.49515749$\times {10}^{-07}$
83.5	−4.38140196$\times {10}^{-10}$	1.97591643$\times {10}^{-07}$	1.99037756$\times {10}^{-05}$	0.90	1.48395770$\times {10}^{-06}$
84.5	−3.73488547$\times {10}^{-10}$	1.70436208$\times {10}^{-07}$	8.14608828$\times {10}^{-06}$	0.94	9.68131273$\times {10}^{-07}$
85.5	−4.33366783$\times {10}^{-10}$	2.09399580$\times {10}^{-07}$	9.28690005$\times {10}^{-06}$	0.94	1.21724813$\times {10}^{-06}$
86.5	−4.84412766$\times {10}^{-10}$	2.36665683$\times {10}^{-07}$	1.09788284$\times {10}^{-05}$	0.94	1.44126953$\times {10}^{-06}$
87.5	−4.83254202$\times {10}^{-10}$	2.68350608$\times {10}^{-07}$	1.39973835$\times {10}^{-05}$	0.94	1.85920378$\times {10}^{-06}$
88.5	−6.21282703$\times {10}^{-10}$	3.37561898$\times {10}^{-07}$	1.45738166$\times {10}^{-05}$	0.94	2.27765383$\times {10}^{-06}$
89.5	−6.68966301$\times {10}^{-10}$	3.86462205$\times {10}^{-07}$	1.63709915$\times {10}^{-05}$	0.93	2.80715372$\times {10}^{-06}$
90.5	−9.40447671$\times {10}^{-10}$	4.91524881$\times {10}^{-07}$	1.81668436$\times {10}^{-05}$	0.93	3.38519060$\times {10}^{-06}$
91.5	−7.89804624$\times {10}^{-10}$	4.23827993$\times {10}^{-07}$	1.56806295$\times {10}^{-05}$	0.94	2.81369582$\times {10}^{-06}$
92.5	−2.51717440$\times {10}^{-10}$	1.25207391$\times {10}^{-07}$	7.46452895$\times {10}^{-06}$	0.95	7.21226067$\times {10}^{-07}$
93.5	−2.52576412$\times {10}^{-10}$	1.27365730$\times {10}^{-07}$	7.86103295$\times {10}^{-06}$	0.95	7.08295267$\times {10}^{-07}$
94.5	−1.68630601$\times {10}^{-10}$	8.97996451$\times {10}^{-08}$	6.01452925$\times {10}^{-06}$	0.95	5.30470738$\times {10}^{-07}$
95.5	−1.71120327$\times {10}^{-10}$	8.22690427$\times {10}^{-08}$	4.20933726$\times {10}^{-06}$	0.95	4.36420483$\times {10}^{-07}$
96.5	−1.16976002$\times {10}^{-10}$	5.79214134$\times {10}^{-08}$	3.16068207$\times {10}^{-06}$	0.95	3.17857028$\times {10}^{-07}$
97.5	−9.42103907$\times {10}^{-10}$	7.84156346$\times {10}^{-07}$	6.35681772$\times {10}^{-05}$	0.89	9.05302834$\times {10}^{-06}$
98.5	−1.74979936$\times {10}^{-10}$	9.39505851$\times {10}^{-08}$	8.37957920$\times {10}^{-06}$	0.94	6.18586716$\times {10}^{-07}$
99.5	−3.06874683$\times {10}^{-10}$	1.63261897$\times {10}^{-07}$	1.00132945$\times {10}^{-05}$	0.95	9.92576035$\times {10}^{-07}$
100.5	−2.08354284$\times {10}^{-10}$	1.12300476$\times {10}^{-07}$	1.82943275$\times {10}^{-06}$	0.95	6.70876385$\times {10}^{-07}$
101.5	−2.22515562$\times {10}^{-10}$	1.04514427$\times {10}^{-07}$	5.86589870$\times {10}^{-06}$	0.95	5.33557018$\times {10}^{-07}$
102.5	−1.34390246$\times {10}^{-09}$	8.98461854$\times {10}^{-07}$	3.81054263$\times {10}^{-05}$	0.92	7.66619612$\times {10}^{-06}$
103.5	−1.01901721$\times {10}^{-09}$	6.79827099$\times {10}^{-07}$	4.51530159$\times {10}^{-05}$	0.94	5.22679531$\times {10}^{-06}$
104.5	−4.18859821$\times {10}^{-10}$	1.89817197$\times {10}^{-07}$	3.18890523$\times {10}^{-06}$	0.95	9.38122529$\times {10}^{-07}$
105.5	−2.54275033$\times {10}^{-10}$	1.20088332$\times {10}^{-07}$	6.60009868$\times {10}^{-06}$	0.96	5.89125254$\times {10}^{-07}$
106.5	−2.43809593$\times {10}^{-10}$	1.18160635$\times {10}^{-07}$	7.74794070$\times {10}^{-06}$	0.96	5.90449142$\times {10}^{-07}$
107.5	−2.33163874$\times {10}^{-10}$	1.20516330$\times {10}^{-07}$	9.87441885$\times {10}^{-06}$	0.95	6.87291920$\times {10}^{-07}$
108.5	−3.69477413$\times {10}^{-10}$	2.17542411$\times {10}^{-07}$	1.10904940$\times {10}^{-05}$	0.96	1.26819431$\times {10}^{-06}$
109.5	−2.81707873$\times {10}^{-10}$	1.45887138$\times {10}^{-07}$	9.82531461$\times {10}^{-06}$	0.95	7.99552250$\times {10}^{-07}$
110.5	−9.10934796$\times {10}^{-11}$	1.04423228$\times {10}^{-07}$	1.44644224$\times {10}^{-05}$	0.92	1.08652364$\times {10}^{-06}$
111.5	−1.66854605$\times {10}^{-10}$	1.18197160$\times {10}^{-07}$	1.27341149$\times {10}^{-05}$	0.95	8.46551633$\times {10}^{-07}$
112.5	−1.97256040$\times {10}^{-10}$	1.22023811$\times {10}^{-07}$	1.39870362$\times {10}^{-05}$	0.95	8.24895534$\times {10}^{-07}$
113.5	−1.84532308$\times {10}^{-10}$	1.04923564$\times {10}^{-07}$	7.42101884$\times {10}^{-06}$	0.95	6.63927760$\times {10}^{-07}$
114.5	−1.82255068$\times {10}^{-10}$	1.00964734$\times {10}^{-07}$	1.09676338$\times {10}^{-05}$	0.95	5.83301149$\times {10}^{-07}$
115.5	−1.79977830$\times {10}^{-10}$	9.70059057$\times {10}^{-08}$	1.45142487$\times {10}^{-05}$	0.95	5.79507535$\times {10}^{-07}$
116.5	−1.14063699$\times {10}^{-10}$	6.14789756$\times {10}^{-08}$	2.11095772$\times {10}^{-05}$	0.95	3.67271760$\times {10}^{-07}$
117.5	−1.19813939$\times {10}^{-09}$	5.28441463$\times {10}^{-07}$	5.22879409$\times {10}^{-05}$	0.95	2.46827715$\times {10}^{-06}$
118.5	−3.17606335$\times {10}^{-10}$	1.43250820$\times {10}^{-07}$	2.42879319$\times {10}^{-05}$	0.95	6.91404625$\times {10}^{-07}$
119.5	−5.07968379$\times {10}^{-10}$	2.73788903$\times {10}^{-07}$	3.79524397$\times {10}^{-05}$	0.95	1.63559867$\times {10}^{-06}$
120.5	−3.91694650$\times {10}^{-09}$	1.76667070$\times {10}^{-06}$	4.67276673$\times {10}^{-05}$	0.95	8.52689227$\times {10}^{-06}$
121.5	−2.10473407$\times {10}^{-08}$	3.12348452$\times {10}^{-05}$	4.59160841$\times {10}^{-03}$	0.91	3.68717499$\times {10}^{-04}$
122.5	−9.24069369$\times {10}^{-10}$	4.16785440$\times {10}^{-07}$	4.09787537$\times {10}^{-05}$	0.95	2.01162816$\times {10}^{-06}$
123.5	−6.22067015$\times {10}^{-10}$	2.80572523$\times {10}^{-07}$	2.53756139$\times {10}^{-05}$	0.95	1.35419220$\times {10}^{-06}$
124.5	−4.72854326$\times {10}^{-10}$	2.13272731$\times {10}^{-07}$	1.79398692$\times {10}^{-05}$	0.95	1.02936761$\times {10}^{-06}$
125.5	−2.37088928$\times {10}^{-10}$	1.27788104$\times {10}^{-07}$	2.05651276$\times {10}^{-05}$	0.95	7.63398559$\times {10}^{-07}$
126.5	−7.26808556$\times {10}^{-10}$	3.27814376$\times {10}^{-07}$	1.76920985$\times {10}^{-05}$	0.95	1.58220650$\times {10}^{-06}$
127.5	−1.57815999$\times {10}^{-10}$	8.50609426$\times {10}^{-08}$	1.52935833$\times {10}^{-05}$	0.95	5.08149030$\times {10}^{-07}$
128.5	−1.17377545$\times {10}^{-10}$	6.32650980$\times {10}^{-08}$	1.23296767$\times {10}^{-05}$	0.95	3.77941945$\times {10}^{-07}$
129.5	−5.73834119$\times {10}^{-11}$	7.57079693$\times {10}^{-08}$	1.75886126$\times {10}^{-05}$	0.93	7.63296694$\times {10}^{-07}$
130.5	−1.24053248$\times {10}^{-09}$	5.70638506$\times {10}^{-07}$	1.20965081$\times {10}^{-04}$	0.91	4.16222369$\times {10}^{-06}$
131.5	−2.12721218$\times {10}^{-10}$	9.61118248$\times {10}^{-08}$	1.95150657$\times {10}^{-05}$	0.91	6.62435421$\times {10}^{-07}$
132.5	−1.07001857$\times {10}^{-10}$	6.29334120$\times {10}^{-08}$	1.62571838$\times {10}^{-05}$	0.90	5.67873985$\times {10}^{-07}$
133.5	−8.94382834$\times {10}^{-10}$	8.71939234$\times {10}^{-07}$	1.31975911$\times {10}^{-04}$	0.94	7.54203153$\times {10}^{-06}$
134.5	−8.35494200$\times {10}^{-11}$	6.28665121$\times {10}^{-08}$	1.35754292$\times {10}^{-05}$	0.88	7.29144198$\times {10}^{-07}$
135.5	−1.25088026$\times {10}^{-10}$	8.54845405$\times {10}^{-08}$	3.77824430$\times {10}^{-05}$	0.90	8.81264590$\times {10}^{-07}$
136.5	−9.79967900$\times {10}^{-11}$	7.22954986$\times {10}^{-08}$	2.10731394$\times {10}^{-05}$	0.90	7.68205303$\times {10}^{-07}$
137.5	−1.24013546$\times {10}^{-10}$	7.97949440$\times {10}^{-08}$	2.37165611$\times {10}^{-05}$	0.91	7.11357554$\times {10}^{-07}$
138.5	−7.39876006$\times {10}^{-11}$	5.55570063$\times {10}^{-08}$	2.54832371$\times {10}^{-05}$	0.87	6.99090195$\times {10}^{-07}$
139.5	−9.49205508$\times {10}^{-10}$	4.89958571$\times {10}^{-07}$	4.33705249$\times {10}^{-05}$	0.96	2.54613277$\times {10}^{-06}$
140.5	−$5.67937224\times {10}^{-10}$	3.05432744$\times {10}^{-07}$	$4.62582464\times {10}^{-05}$	0.95	1.88073609$\times {10}^{-06}$
141.5	−$2.43621952\times {10}^{-10}$	1.13736869$\times {10}^{-07}$	$3.27628394\times {10}^{-05}$	0.90	8.95784614$\times {10}^{-07}$
142.5	−$1.95955912\times {10}^{-10}$	1.01825495$\times {10}^{-07}$	$3.75119030\times {10}^{-05}$	0.83	1.18926525$\times {10}^{-06}$
143.5	−$2.46624207\times {10}^{-10}$	1.15175318$\times {10}^{-07}$	$4.23845795\times {10}^{-05}$	0.88	1.00877166$\times {10}^{-06}$
144.5	−$1.61278923\times {10}^{-10}$	8.93582633$\times {10}^{-08}$	$4.38438448\times {10}^{-05}$	0.86	9.81989792$\times {10}^{-07}$
145.5	−$2.75726953\times {10}^{-10}$	1.38512452$\times {10}^{-07}$	$4.32208222\times {10}^{-05}$	0.89	1.19372564$\times {10}^{-06}$
146.5	−$2.43757781\times {10}^{-10}$	1.36559581$\times {10}^{-07}$	$5.45324615\times {10}^{-05}$	0.89	1.27366067$\times {10}^{-06}$
147.5	−$2.46147842\times {10}^{-10}$	1.39066818$\times {10}^{-07}$	$7.11427892\times {10}^{-05}$	0.89	1.35250470$\times {10}^{-06}$
148.5	−$3.12927956\times {10}^{-10}$	1.56053330$\times {10}^{-07}$	$7.20397030\times {10}^{-05}$	0.87	1.51332205$\times {10}^{-06}$
149.5	−$2.31230729\times {10}^{-10}$	1.24944106$\times {10}^{-07}$	$6.58375866\times {10}^{-05}$	0.83	1.51364906$\times {10}^{-06}$
150.5	−$2.43791216\times {10}^{-10}$	1.21205065$\times {10}^{-07}$	$7.57760769\times {10}^{-05}$	0.80	1.52268475$\times {10}^{-06}$
151.5	−$1.94987017\times {10}^{-10}$	1.16236962$\times {10}^{-07}$	$8.38650446\times {10}^{-05}$	0.80	1.65675848$\times {10}^{-06}$
152.5	−$4.33361801\times {10}^{-10}$	2.37558655$\times {10}^{-07}$	$9.63535070\times {10}^{-05}$	0.91	2.01495486$\times {10}^{-06}$
153.5	−$1.73871412\times {10}^{-10}$	1.71988954$\times {10}^{-07}$	$1.12191003\times {10}^{-04}$	0.88	2.21903105$\times {10}^{-06}$
154.5	−$1.16988909\times {10}^{-09}$	6.16487236$\times {10}^{-07}$	$1.70987119\times {10}^{-04}$	0.93	4.36900801$\times {10}^{-06}$
155.5	−$2.55020760\times {10}^{-10}$	2.92214578$\times {10}^{-07}$	$1.64242591\times {10}^{-04}$	0.92	3.20494890$\times {10}^{-06}$
156.5	−$4.53027932\times {10}^{-10}$	2.51428725$\times {10}^{-07}$	$1.68458783\times {10}^{-04}$	0.85	2.85808410$\times {10}^{-06}$
157.5	−$1.72784194\times {10}^{-10}$	1.48582781$\times {10}^{-07}$	$1.57486859\times {10}^{-04}$	0.77	2.79008511$\times {10}^{-06}$
158.5	−$2.29772147\times {10}^{-10}$	1.46877406$\times {10}^{-07}$	$1.54423418\times {10}^{-04}$	0.74	2.69140338$\times {10}^{-06}$
159.5	−$9.39400590\times {10}^{-11}$	1.10698536$\times {10}^{-07}$	$1.54968996\times {10}^{-04}$	0.73	2.59598688$\times {10}^{-06}$
160.5	−$8.39077851\times {10}^{-11}$	1.26638098$\times {10}^{-07}$	$1.71073746\times {10}^{-04}$	0.77	2.80522066$\times {10}^{-06}$
161.5	−$1.98721268\times {10}^{-10}$	1.37610016$\times {10}^{-07}$	$2.05647262\times {10}^{-04}$	0.65	3.39322701$\times {10}^{-06}$
162.5	−$4.73652596\times {10}^{-10}$	2.82618134$\times {10}^{-07}$	$2.23170447\times {10}^{-04}$	0.82	3.76275218$\times {10}^{-06}$
163.5	−$1.55191039\times {10}^{-10}$	2.30331717$\times {10}^{-07}$	$2.39080251\times {10}^{-04}$	0.83	4.07902394$\times {10}^{-06}$
164.5	−$4.04165128\times {10}^{-10}$	3.37772150$\times {10}^{-07}$	$2.81318936\times {10}^{-04}$	0.86	4.56771102$\times {10}^{-06}$
165.5	−$7.73823055\times {10}^{-12}$	2.97227900$\times {10}^{-07}$	$4.63250839\times {10}^{-04}$	0.76	7.93202201$\times {10}^{-06}$
166.5	$4.07530120\times {10}^{-11}$	8.57496540$\times {10}^{-08}$	$3.29404588\times {10}^{-04}$	0.51	5.05815432$\times {10}^{-06}$
167.5	−$2.04674881\times {10}^{-10}$	3.39319234$\times {10}^{-07}$	$3.67674446\times {10}^{-04}$	0.81	6.73876290$\times {10}^{-06}$
168.5	−$4.43360209\times {10}^{-10}$	2.79935936$\times {10}^{-07}$	$4.04891772\times {10}^{-04}$	0.66	6.38829474$\times {10}^{-06}$
169.5	$1.21153174\times {10}^{-10}$	2.01744392$\times {10}^{-07}$	$5.46199225\times {10}^{-04}$	0.64	8.70736220$\times {10}^{-06}$
170.5	−$5.85190609\times {10}^{-10}$	4.32779772$\times {10}^{-07}$	$6.13206686\times {10}^{-04}$	0.68	1.01094842$\times {10}^{-05}$
171.5	−$5.06522331\times {10}^{-10}$	4.47020613$\times {10}^{-07}$	$6.17992427\times {10}^{-04}$	0.70	1.06043009$\times {10}^{-05}$
172.5	−$3.24768827\times {10}^{-10}$	3.72615949$\times {10}^{-07}$	$6.98235220\times {10}^{-04}$	0.60	1.23238287$\times {10}^{-05}$
173.5	$5.76888817\times {10}^{-11}$	2.79814116$\times {10}^{-07}$	$6.97522794\times {10}^{-04}$	0.61	1.18627791$\times {10}^{-05}$
174.5	−$1.17218795\times {10}^{-10}$	3.92089797$\times {10}^{-07}$	$8.58418577\times {10}^{-04}$	0.61	1.50705494$\times {10}^{-05}$
175.5	−$1.37607193\times {10}^{-10}$	5.06896902$\times {10}^{-07}$	$1.04781557\times {10}^{-03}$	0.60	1.97468460$\times {10}^{-05}$
176.5	−$8.96657609\times {10}^{-10}$	6.78269058$\times {10}^{-07}$	$1.12910891\times {10}^{-03}$	0.59	2.01888857$\times {10}^{-05}$
177.5	$1.32627992\times {10}^{-09}$	1.79727927$\times {10}^{-07}$	$1.42010797\times {10}^{-03}$	0.47	$2.90878237\times {10}^{-05}$
178.5	$1.98167580\times {10}^{-09}$	$8.00937083\times {10}^{-08}$	$1.59925973\times {10}^{-03}$	0.48	$3.18302836\times {10}^{-05}$
179.5	$1.18969584\times {10}^{-09}$	$3.52847586\times {10}^{-07}$	$1.62399680\times {10}^{-03}$	0.54	$3.13644235\times {10}^{-05}$
180.5	−$1.61188575\times {10}^{-09}$	$1.46962414\times {10}^{-06}$	$2.01940644\times {10}^{-03}$	0.62	$4.36405768\times {10}^{-05}$
181.5	−$3.61654690\times {10}^{-09}$	$2.50708549\times {10}^{-06}$	$2.31968020\times {10}^{-03}$	0.66	$5.98817371\times {10}^{-05}$
182.5	$1.44002194\times {10}^{-10}$	$8.58952566\times {10}^{-07}$	$2.34991118\times {10}^{-03}$	0.46	$5.34544458\times {10}^{-05}$
183.5	−$5.43430484\times {10}^{-10}$	$9.53841592\times {10}^{-07}$	$2.52448745\times {10}^{-03}$	0.45	$5.15037308\times {10}^{-05}$
184.5	−$1.30322879\times {10}^{-09}$	$9.76484389\times {10}^{-07}$	$2.20300301\times {10}^{-03}$	0.47	$3.95649715\times {10}^{-05}$
185.5	−$1.62506964\times {10}^{-09}$	$1.13882183\times {10}^{-06}$	$2.50603519\times {10}^{-03}$	0.44	$4.83075423\times {10}^{-05}$
186.5	−$1.89245396\times {10}^{-09}$	$1.37388136\times {10}^{-06}$	$2.88406355\times {10}^{-03}$	0.44	$5.91202561\times {10}^{-05}$
187.5	$6.84720021\times {10}^{-10}$	$6.10050906\times {10}^{-07}$	$3.29718092\times {10}^{-03}$	0.35	$6.56653209\times {10}^{-05}$
188.5	$1.53237630\times {10}^{-09}$	$8.38490308\times {10}^{-07}$	$3.18211645\times {10}^{-03}$	0.46	$7.35219971\times {10}^{-05}$
189.5	$7.23049173\times {10}^{-09}$	−$4.24163156\times {10}^{-07}$	$2.42571457\times {10}^{-03}$	0.51	$6.85476734\times {10}^{-05}$

in Appendix A and can be instantly applied to reconstruct the model of the solar radiation spectrum. Table A1 also includes the correlation coefficients (R) and root mean square errors (RMSEs) to demonstrate the level of confidence of the Solar-SPAM model.

The high variability of the shortwave solar radiation, especially in the X-ray and EUV range, naturally provides a wide range of possible ionospheric and thermospheric conditions and should be taken into account in aeronomic modeling and research. The next section is devoted to the development of a specific model intended for aeronomic studies.

4. Aero-SPAM: EUV Photon Flux Model for Aeronomic Applications

Here, we present the development of an empirical model of solar irradiance suitable for aeronomic studies, as well as its ready-to-use application for calculating photoionization rates in the vertical column of the atmosphere. There is a set of well-defined spectral bands and individual spectral lines responsible for the photoionization of the main atmospheric neutrals with known absorption and ionization cross sections [47,48,49]. According to this set, we reduced the TIMED data to 20 spectral intervals from 5 to 105 nm with a width of 5 nm each and 16 individual spectral lines, in correspondence to the EUVAC model [31]. Additionally, we included the strong spectral line Ly-alpha ($\lambda$ = 121.5 nm) in our model, which is responsible for the nitric oxide ionization in the lower ionosphere. The radiation flux in each 5 nm interval is calculated by direct summation of the flux in the SEE channels that fall into this interval, whereas the flux in an individual spectral line is assumed to be equal to the flux in the corresponding 1 nm wide SEE channel.

For aeronomic applications, such as the calculation of the photoionization rate in the upper atmosphere, solar irradiance is usually expressed in units of photon flux. Therefore, we converted the differential energy flux (F, W·m${}^{-2}·$nm${}^{-1}$) measured by the TIMED SEE device into a differential photon flux (I, m${}^{-2}·$s${}^{-1}·$nm${}^{-1}$) by dividing the original flux by the photon energy, corresponding to each spectral channel: $I=F/(hc/\lambda )$, where h is Planck’s constant, c is the speed of light and $\lambda$ is the central wavelength of each 1 nm SEE channel.

The further development of the model is fully consistent with the procedure used in the previous chapter. We used daily averaged data and a second-order polynomial fit to formulate the Aero-SPAM model:

$$I=P_1·F_{10.7}^2+P_2·F_{10.7}+P_3,$$

(2)

where $P_1$, $P_2$ and $P_3$ are regression coefficients and F${}_{10.7}$ is a daily averaged value of the solar activity index. The photon flux I was calculated in 20 spectral intervals and 17 individual spectral lines listed in

Table A2. The Aero-SPAM model describing the photon flux (I, m${}^{-2}·$s${}^{-1}·$nm${}^{-1}$) in 37 specified spectral channels depending on the daily $F_{10.7}$ index, $I=P_1·F_{10.7}^2+P_2·F_{10.7}+P_3$. $P_1$, $P_2$ and $P_3$ are the regression coefficients for the 37 EUV spectral intervals ($\lambda$), including 17 lines and 20 bands. R is a correlation coefficient between $F_{10.7}$ and photon flux. RMSE is the root-mean-square error for the measured and simulated I values.

No	${\mathit{\lambda }}_{\mathbf{min}}$, nm	${\mathit{\lambda }}_{\mathbf{max}}$, nm	${\mathit{P}}_1$	${\mathit{P}}_2$	${\mathit{P}}_3$	R	RMSE
1	5	10	−$7.22814128\times {10}^{+06}$	$4.34844365\times {10}^{+09}$	−$1.63154083\times {10}^{+11}$	0.96	$2.49833157\times {10}^{+10}$
2	10	15	−$1.72793713\times {10}^{+08}$	$1.06538527\times {10}^{+11}$	−$2.83695953\times {10}^{+12}$	0.96	$6.38391466\times {10}^{+11}$
3	15	20	−$1.79873111\times {10}^{+09}$	$1.05716281\times {10}^{+12}$	−$2.74337230\times {10}^{+13}$	0.96	$6.09484906\times {10}^{+12}$
4	20	25	−$1.67014302\times {10}^{+09}$	$9.88384185\times {10}^{+11}$	−$3.90160466\times {10}^{+13}$	0.96	$5.65031654\times {10}^{+12}$
5	25.6		−$2.42136993\times {10}^{+08}$	$1.44676220\times {10}^{+11}$	−$7.27167455\times {10}^{+12}$	0.96	$8.22340432\times {10}^{+11}$
6	28.4		−$2.15749026\times {10}^{+08}$	$1.47186233\times {10}^{+11}$	−$5.25550336\times {10}^{+12}$	0.92	$1.36257538\times {10}^{+12}$
7	25	30	−$8.49047253\times {10}^{+08}$	$4.39836923\times {10}^{+11}$	−$1.07767192\times {10}^{+13}$	0.96	$2.37564549\times {10}^{+12}$
8	30.3		−$1.05374887\times {10}^{+09}$	$6.15749059\times {10}^{+11}$	$2.21870265\times {10}^{+13}$	0.94	$4.36129879\times {10}^{+12}$
9	30	35	−$5.78821182\times {10}^{+08}$	$4.09300016\times {10}^{+11}$	−$7.39277758\times {10}^{+12}$	0.93	$3.67860958\times {10}^{+12}$
10	36.8		−$3.67641064\times {10}^{+08}$	$2.23500665\times {10}^{+11}$	−$3.44107714\times {10}^{+12}$	0.93	$1.76712865\times {10}^{+12}$
11	35	40	−$5.27393084\times {10}^{+08}$	$2.60815376\times {10}^{+11}$	−$4.81963679\times {10}^{+12}$	0.94	$1.61468490\times {10}^{+12}$
12	40	45	−$1.76485806\times {10}^{+08}$	$9.43602417\times {10}^{+10}$	$1.20746026\times {10}^{+12}$	0.95	$5.78446944\times {10}^{+11}$
13	46.5		−$9.16428947\times {10}^{+06}$	$1.10576870\times {10}^{+10}$	$3.46127070\times {10}^{+12}$	0.84	$1.99324850\times {10}^{+11}$
14	45	50	−$3.28417068\times {10}^{+08}$	$1.54464379\times {10}^{+11}$	$2.70338559\times {10}^{+11}$	0.96	$7.65976697\times {10}^{+11}$
15	50	55	−$4.35029980\times {10}^{+08}$	$1.94267789\times {10}^{+11}$	−$9.12633707\times {10}^{+11}$	0.96	$9.17807984\times {10}^{+11}$
16	55.4		−$7.45540942\times {10}^{+07}$	$2.70143268\times {10}^{+10}$	$6.20498828\times {10}^{+12}$	0.60	$4.21493552\times {10}^{+11}$
17	58.4		−$2.67242090\times {10}^{+08}$	$1.26513904\times {10}^{+11}$	$5.22846617\times {10}^{+12}$	0.91	$9.60988214\times {10}^{+11}$
18	55	60	−$1.11331394\times {10}^{+08}$	$4.91943896\times {10}^{+10}$	$2.59480808\times {10}^{+12}$	0.94	$2.86378660\times {10}^{+11}$
19	60.9		−$1.75317009\times {10}^{+08}$	$7.84311521\times {10}^{+10}$	$2.56019089\times {10}^{+11}$	0.94	$4.50958343\times {10}^{+11}$
20	62.9		−$1.95380036\times {10}^{+08}$	$8.57116193\times {10}^{+10}$	$6.10932407\times {10}^{+12}$	0.89	$6.88675932\times {10}^{+11}$
21	60	65	−$3.18739604\times {10}^{+08}$	$1.31380748\times {10}^{+11}$	$7.10288562\times {10}^{+12}$	0.90	$8.93006924\times {10}^{+11}$
22	65	70	−$1.54464890\times {10}^{+08}$	$6.57942854\times {10}^{+10}$	$3.90335230\times {10}^{+12}$	0.93	$3.82370215\times {10}^{+11}$
23	70.3		−$3.92892316\times {10}^{+07}$	$1.62255869\times {10}^{+10}$	$3.31133072\times {10}^{+12}$	0.78	$1.84543928\times {10}^{+11}$
24	70	75	−$1.17284653\times {10}^{+08}$	$5.16415922\times {10}^{+10}$	$2.62008424\times {10}^{+12}$	0.94	$2.82746960\times {10}^{+11}$
25	76.5		−$2.80655392\times {10}^{+07}$	$1.48549365\times {10}^{+10}$	$5.22726113\times {10}^{+12}$	0.72	$2.63413854\times {10}^{+11}$
26	77		−$1.42682444\times {10}^{+07}$	$1.64116032\times {10}^{+10}$	$4.19804672\times {10}^{+12}$	0.88	$2.42046487\times {10}^{+11}$
27	78.9		−$5.13752841\times {10}^{+07}$	$2.67845754\times {10}^{+10}$	$6.25669926\times {10}^{+12}$	0.81	$3.56475663\times {10}^{+11}$
28	75	80	−$1.09163517\times {10}^{+08}$	$4.63578518\times {10}^{+10}$	$4.91238247\times {10}^{+12}$	0.89	$3.39676578\times {10}^{+11}$
29	80	85	−$5.71385988\times {10}^{+08}$	$2.65288858\times {10}^{+11}$	$1.93295978\times {10}^{+13}$	0.94	$1.62478581\times {10}^{+12}$
30	85	90	−$1.26716263\times {10}^{+09}$	$6.53242857\times {10}^{+11}$	$2.77927431\times {10}^{+13}$	0.94	$4.18839876\times {10}^{+12}$
31	90	95	−$1.14503862\times {10}^{+09}$	$5.87977430\times {10}^{+11}$	$2.50221903\times {10}^{+13}$	0.95	$3.62200988\times {10}^{+12}$
32	97.8		−$4.60750790\times {10}^{+08}$	$3.84479195\times {10}^{+11}$	$3.11115764\times {10}^{+13}$	0.90	$4.64901797\times {10}^{+12}$
33	95	100	−$3.84402107\times {10}^{+08}$	$1.97008185\times {10}^{+11}$	$1.26370475\times {10}^{+13}$	0.95	$1.14344491\times {10}^{+12}$
34	102.6		−$7.45028477\times {10}^{+08}$	$4.75205812\times {10}^{+11}$	$1.89621526\times {10}^{+13}$	0.93	$4.08952635\times {10}^{+12}$
35	103.2		−$6.18608147\times {10}^{+08}$	$3.73585739\times {10}^{+11}$	$2.24459796\times {10}^{+13}$	0.94	$2.81357100\times {10}^{+12}$
36	100	105	−$4.16550795\times {10}^{+08}$	$2.04940624\times {10}^{+11}$	$5.82827246\times {10}^{+12}$	0.96	$1.07697392\times {10}^{+12}$
37	121.6		−$2.81408845\times {10}^{+10}$	$2.25475006\times {10}^{+13}$	$2.62203706\times {10}^{+15}$	0.92	$2.35540620\times {10}^{+14}$

in Appendix A, together with model coefficients and R and RMSE values.

The photoionization of the atmosphere by solar EUV is a main source of the formation of regular ionospheric layers E and F${}_1$. The Aero-SPAM model is specifically developed for the calculation of photoionization rates in the Earth’s upper atmosphere and can be used in many aeronomic applications and research. As an example, it is already integrated into the AIM-E numerical model of the ionosphere [33,50] as a part of the photoionization module. Below, we provide a description, as well as a ready-to-use numerical module, for calculating the photoionization rates of atmospheric gases using the Aero-SPAM model.

The photoionization rate $q_j\left(z\right)$ of the neutral gas j-th component at the altitude z is the number of photoionization acts per unit volume per unit time. Here, we determine the photoionization rates for N, O, NO, N${}_2$ and O${}_2$ using the expression (Shunk and Nagy, 1980):

$$q_j\left(z\right)=n_j\sum _{\lambda }{\sigma }_{j\lambda }^i{I}_{n\lambda }^{\infty }exp(-\sum _n{\sigma }_{n\lambda }^a{\int }_z^{\infty }Ch\left(\chi \right)n_n\mathrm{d}z),$$

(3)

where $n_j$ is the concentration of the j-th component of atmospheric gas, ${\sigma }_{j\lambda }^i$ is its photoionization cross section at wavelength $\lambda$, ${\sigma }_{j\lambda }^a$ is its photoabsorption cross section at wavelength $\lambda$, $Ch\left(\chi \right)$ is the extended Chapman function [12,51] and $I_{\lambda }^{\infty }$ is photon flux at wavelength $\lambda$ at the top of the atmosphere, provided by the Aero-SPAM model.

To illustrate the aeronomic application of the Aero-SPAM model, we reconstructed the solar irradiance spectrum and calculated the vertical profiles of the photoionization rates (Equation (3)) for days with low and high solar activity (20 June 2009 and 18 June 2015 correspondingly) (Figure 4). The neutral atmosphere composition and temperature, required for the calculations, were taken from the NRLMSISE-00 model [52] hosted at the Community Coordinated Modeling Center (https://ccmc.gsfc.nasa.gov/, accessed on 20 November 2022), whereas the nitric oxide density, which is not represented in the MSISE model, was obtained from the E Region Auroral Ionosphere Model (AIM-E) [14].

The values of the F${}_{10.7}$ solar activity index for the selected days 20 June 2009 and 18 June 2015 are 65 s.f.u. and 155 s.f.u., respectively. The differential photon flux changes significantly between low and high solar activity levels (Figure 4a). It is worth noting that the ratio of variation is not the same for different bands and spectral lines. For example, as shown in Figure 4a, the change in the photon flux in the 25.6 nm line $\Delta$I${}_{25.6}$∼ 700%, whereas, in the Ly-alpha line, $\Delta$I${}_{121.6}$∼ 50%. Such a notable change in the spectrum of solar radiation leads to significant changes in atmospheric photochemistry and in the overall dynamics of the ionosphere.

Figure 4b,c shows the vertical distribution of photoionization rates q between 90 and 250 km above the subauroral station Gorkovskaya (60.27${}^{\circ }$ N, 29.38${}^{\circ }$ E) at 12 h MLT for the molecular oxygen and nitrogen (O${}_2$ and N${}_2$), atomic oxygen (O) and nitric oxide (NO). The altitude distribution of the total photoionization rate has two peaks at ∼100 km and ∼160 km. The first peak (∼100 km) of the photoionization rate leads to the formation of the ionospheric regular layer E in approximately the same altitudes. The photoionization here is totally dominated by $q_{O_2}$ and changes more than 50% between low (F${}_{10.7}$ = 65 s.f.u) and high (F${}_{10.7}$ = 155 s.f.u) solar activity. The second peak (∼160 km) of the photoionization rate is one of the main sources of the formation of the regular layer F1, together with the vertical plasma drift. It is formed mainly due to the $q_{N_2}$ component, which differs by approximately 70% for low and high solar activity. While the $q_{NO}$ value is several times lower than the photoionization rates of other components, the difference between the $q_{NO}$ profiles shown in Figure 4b,c is huge: ∼250% for the E-layer heights and ∼400% for the F-layer heights. Despite the low photoionization rate, $q_{NO}$ has a significant effect on the electron concentration in the ionosphere due to the relatively high density of neutral NO content in the upper atmosphere.

For the chemical models of the ionosphere (e.g., [14,53,54], it is extremely important to take into account the changes in the shape of the solar radiation spectrum and the corresponding change in the vertical distribution of photoionization rates. Our Solar-SPAM and Aero-SPAM models accurately track these changes over the course of the changing Sun. Both models, as well as the photoionization rate module, are available on GitHub as ready-to-use Matlab scripts (https://github.com/magnetophys/SPAM, accessed on 20 November 2022).

5. Discussion

One of the main requirements for building a good empirical model is the reliability of the measurements, which becomes especially important in the case of solar shortwave irradiance. The point is that the Earth’s atmosphere completely absorbs solar irradiance in the range of 0–190 nm, so measurements can only be carried out on board the spacecraft, which makes it difficult to continuously control the quality of received data. The sensitivity of the photosensors may suffer during a prolonged operation of instruments in space, leading to systematic errors and false trends in observations [55]. To build an accurate model, it is necessary to exclude data distorted due to sensor degradation.

One of the few ways to calibrate instruments onboard a spacecraft (e.g., [56,57,58]) is to launch a sounding rocket (e.g., [59]) with a copy of the instrument on board. This method was used for the calibration of the TIMED SEE device.

There were nine SEE suborbital calibration flights during 2002–2013 years, and, after them, the last one was launched on 1 June 2016 (http://lasp.colorado.edu/data/timed_see/level3a/README_SEE_L3A_012.TXT, accessed on 20 November 2022). Thus, the TIMED SEE data from 2002 to 2016 can be considered well-calibrated and reliable for model development.

We noticed non-typical trends in spacecraft measurements made after the last calibration rocket launch in 2016, apparently related to the SEE detector degradation, that are not indicated in the “Known Issues/Problems” section of the actual instrument’s release notes (http://lasp.colorado.edu/data/timed_see/SEE_v12_releasenotes.txt, accessed on 20 November 2022).

Figure 5 shows several examples of the solar irradiance time series in different spectral lines. Panels a–d demonstrate the TIMED SEE measurements (red and green) for the entire period of the spacecraft operation from 2002 to 2022, divided by the vertical blue line that denotes the date of the last calibration launch of a suborbital rocket on 1 June 2016. Half a year after the last calibration, artificial trends appeared in different spectral lines, which are easy to recognize by the abrupt shift in the measured values. Data obtained before sensor degradation are shown in red, and after, in green. The black curve denotes the Solar-SPAM calculations based on the daily averaged F${}_{10.7}$ index.

Examples in Figure 5 demonstrate that different spectral lines of the detector were affected to varying degrees. Thus, for the 3.5 nm line (Figure 5a), there is no visible jump in the measured radiation flux and there is a good agreement between the data and model calculations. This is generally true for the first 27 SEE channels covering the 0–28 nm spectral range. The situation changes dramatically in the case of all other SEE channels. For example, the sharp drop in the radiation flux can be seen in 53.5 and 159.5 nm lines shortly after the last calibration in 2016 (Figure 5b,d). At the same time, there is an unusual growth in the 97.5 nm line (more than 200% in comparison with the previous solar cycle) throughout the following years. We conclude the presence of an anomaly in the TIMED SEE data after the last calibration in 2016, which is not associated with the solar activity variations.

All measurement time series for each individual SEE channel can be found in Supplementary Materials in the same format as in Figure 5.

6. Conclusions

SPAM is an empirical model for the solar irradiance spectrum that has been developed using 14 years of TIMED spacecraft observations from 2002 to 2016. The model covers the X-ray (0–10 nm), EUV (10–122 nm) and FUV (122–190 nm) spectral intervals with a 1 nm resolution. The model is parametrized by a single F${}_{10.7}$ index—the solar radio flux at 10.7 cm—which gives a number of advantages:

1.

The F${}_{10.7}$ index is an excellent indicator of solar activity (Tapping, 2013) and may serve as a reliable proxy for the solar spectrum variations within the limits of the SPAM model applicability 65 < F${}_{10.7}$ < 200 s.f.u.

2.

Since the atmosphere is transparent to radiation at a wavelength of 10.7 cm, the F${}_{10.7}$ radio flux can be reliably measured from the Earth’s surface in any weather. The time series of the daily F${}_{10.7}$ has been available continuously since 1947 for seven solar cycles, so the SPAM model can be applied to large-scale retrospective studies that require solar radiation data.

3.

The F${}_{10.7}$ index is predictable, which allows SPAM to forecast the solar spectrum for various operational tasks. There are a number of services providing the forecast of the daily averaged F${}_{10.7}$ index, e.g., up to 55 days ahead (http://spaceweather.izmiran.ru/eng/forecasts.html, accessed on 20 November 2022), up to 27 days ahead with open-access forecast (https://www.swpc.noaa.gov/products/27-day-outlook-107-cm-radio-flux-and-geomagnetic-indices, accessed on 20 November 2022) and up to 45 days ahead with 5-day resolution (https://www.swpc.noaa.gov/products/usaf-45-day-ap-and-f107cm-flux-forecast, accessed on 20 November 2022). In addition, there is a long-term monthly average F${}_{10.7}$ forecast up to 20 years, including the next solar activity cycle, that can be used for future aeronomy estimations. (https://www.swpc.noaa.gov/products/predicted-sunspot-number-and-radio-flux, accessed on 20 November 2022).

4.

SPAM’s single-variable parameterization is easy to implement. Despite the model simplicity, our results are in good agreement with the measurements.

A special part of the study is given to the aeronomy-oriented model Aero-SPAM. It provides photon flux values in 17 spectral lines and 20 bands, covering the 5–105 nm EUV range and additionally including the Ly-alpha 121.5 nm spectral line, which is the main source of NO+ formation. The Aero-SPAM model is used to calculate the ionization rates of the neutral atmosphere gasses (N${}_2$, O${}_2$, O, N and NO) with well-known absorption and ionization cross sections. The Aero-SPAM model for aeronomy calculations can be applied for the long-term forecast of the ionospheric regular E and F1 regions.