SOLAR FLARE ELEMENT ABUNDANCES FROM THE SOLAR ASSEMBLY FOR X-RAYS (SAX) ON MESSENGER

SAX data were normally collected continuously over 5-minute intervals, although on some occasions the instrument was commanded into a high-cadence mode with 20 s or 40 s data-gathering intervals close to periapsis of the orbit about Mercury and during some of the more intense flares. Only those time intervals in which the count rate was $\gt 5000$ s⁻¹ in the 6.3–7.0 keV energy range were selected for analysis to ensure that the Fe line complex at 6.7 keV was well observed. This resulted in the selection of 656 flares or significant increases in X-ray flux for analysis. Of these, events lasting for only a single 5-minute data-gathering interval were eliminated from this data set, leaving a total of 526 flares with at least two contiguous data intervals that were used to determine element abundances. The results from these flares were used in computing the average abundances listed in Table 2. A full catalog of flares analyzed with best-fit spectral parameters and more extensive plots are available at the following location: http://hesperia.gsfc.nasa.gov/messenger/results/plots_and_tables/.

The analyzed flares covered the period from 2007 May 28 to 2013 August 19. They are not uniformly distributed in time for three main reasons: the varying solar activity through the deep minimum in 2009, intermittent detector operations necessitated by the various spacecraft maneuvers and periods of high detector temperatures, and particle contamination when the spacecraft was presumably well connected magnetically to the flaring active region. As a result, there are 27 flares analyzed up until 2007 December 31 and the remainder after 2010 February 6. An analysis of 11 flares in 2011 has previously been given by Nittler et al. (2011). Four of these satisfy our selection criteria and so are included in our lists of analyzed events. The others either were too small to be included or occurred during a period of high particle contamination.

A detailed analysis of an M2.8 flare observed by SAX on 2007 June 1 is first given as an illustration of the methods we used. Figure 1 shows X-ray light curves as recorded by SAX and GOES. Since the event was from NOAA active region 10960 at S08E78 and Mercury was at a Stonyhurst longitude of $49{}^\circ$ East, both instruments would have seen the event equally well. The GOES light curves show an initial peak at 14:22 UT followed by the main event peaking at 14:59 UT, then a gradual decay to the pre-flare level at about 16:15 UT. As would be expected from their sensitivity to similar energy ranges, the SAX light curves (corrected for the light travel time from MESSENGER to 1 AU) closely match the GOES light curves, although with a time resolution of 300 s compared with 3 s for GOES (2 s after 2009 December 2 with GOES 14 and GOES 15).

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We describe the fitting procedure for the spectrum taken during the 5-minute interval beginning at 14:56:29.9 UT, at flare maximum. The observed spectrum, shown in the two panels of Figure 2, is dominated by continuum (the sum of free–free and free–bound emission), with line complexes due to Ca ions (∼3.9 keV) and Fe ions (∼6.7 and ∼8 keV) also evident. The temperature governs the slope of the spectrum when plotted against energy, with higher temperatures giving rise to spectra with flatter slopes, while the emission measure determines the absolute value of the flux at any energy.

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The measured count rate spectrum was fitted with theoretical photon spectra calculated from the chianti package (v. 7.0) (Landi et al. 2013). These spectra are defined by values of the electron temperature, T, and the volume emission measure, ${\rm EM}=N_{e}^{2}\;V$ (where ${{N}_{e}}\;=\;$ electron density, $V\;=$ emitting volume), for a set of abundances which in our analysis were chosen to be relative to the coronal abundances of Feldman et al. (1992) with ion fractions from Mazzotta et al. (1998). In order to speed up the calculation of the theoretical thermal spectra, base isothermal spectra were obtained from an abstraction of the chianti routines. Tables were generated for each chianti spectral line for a logarithmically spaced temperature grid with the grid spacing dependent on the significance of that line. When a particular spectrum was generated from that database, the lines were found in the energy range of interest and the emissivity computed by interpolating from the table. The total line spectrum was then computed by summing over all line contributions with each line weighted by its assumed element abundance. The continuum was similarly computed and made available either as a separate spectrum or combined with the total line spectrum. In this way, EM, T, and the abundances of Fe, Ca, S, Si, Ar, and other elements could all be used as free parameters and adjusted as necessary in the fitting procedure.

As the temperature structure of the flare emission is unknown, functional forms must be assumed. The simplest case is an isothermal function defined only by T and EM. This appears to apply most nearly to X-ray spectra over limited energy ranges taken near maximum and in the decay of a flare, as has been shown in analysis of GOES flares by Garcia (1994) and with temperature-dependent spectral line ratios from the Yohkoh Bragg Crystal Spectrometer by Phillips et al. (2005). In the analysis of Nittler et al. (2011), SAX spectra were fitted assuming an isothermal emitting region for the 1–8 keV spectral range on the grounds that two-temperature models gave worse fits. However, isothermal fits to the entire spectral range of SAX are not likely to be valid in principle for most spectra. This is because previous flare observations with high-resolution crystal spectrometers have shown that temperatures determined from the Ca lines at 3.9 keV (due to Ca xix lines with relatively weaker satellites) are found to be lower by typically ∼5 MK than those determined from the Fe-line complex at 6.7 keV (due to Fe xxv lines with temperature-dependent dielectronic satellites) (e.g., Feldman et al. 1980). An isothermal assumption is more likely to be valid over limited spectral regions, particularly on either side of the 6.7 keV Fe-line complex, where the neighboring continuum has a slope determined by a temperature that is likely to approximate that derived from satellite-to-resonance line ratios in the Fe xxv lines and nearby satellite lines.

For the SAX spectrum in Figure 2, four forms of the temperature dependence of the differential emission measure, defined as DEM $=\;N_{e}^{2}dV/dT$, were assumed: (a) an isothermal model ("1-T") described by single values of T and EM; (b) a two-temperature model ("2-T"), with two values of T and EM; (c) a multithermal model with power-law dependence of the differential emission measure on T ("multitherm_pow"), constant $\times {{T}^{-\alpha }}$ (where α is positive); (d) a multithermal model with the differential emission measure having an exponentially decreasing dependence on T ("multitherm_exp"), ${\rm constant}\times {\rm exp} (-T/\tau )$ (where τ is positive). In our fitting procedure, theoretical spectra calculated from chianti were folded through each of these functional forms with free parameters (e.g., T, EM, α, and τ) and the abundances of elements significantly affecting the spectrum in this energy range, viz., Fe, Ca, S, Si, and Ar. For the 2-T model the abundances of the higher temperature component were allowed to vary, but the abundances of the lower temperature component were held fixed at the chianti "coronal" values (Feldman et al. 1992) listed in Table 1.

The fitting routines are much the same as those used by Phillips & Dennis (2012) to obtain the Fe abundance from RHESSI spectra. Data files containing the count-rate spectrum and detector response matrix were read by a program called Object Spectral Analysis Executive (OSPEX) (http://hesperia.gsfc.nasa.gov/rhessi3/software/spectroscopy/spectral-analysis-software/). Theoretical spectra calculated from each of the four model spectral functions ((a)–(d)) were then folded through the detector response function with first-guess values of the free parameters (e.g., for model (a) T, EM, and abundances of Fe, Ca, S, Si, and Ar) to calculate predicted count rates. Fits to the background-subtracted SAX count-rate spectrum (with the background determined before or after the flare) were accomplished with OSPEX by continual adjustment of the fitting parameters to improve the goodness of fit, determined by the reduced chi-squared statistic:

$$\begin{eqnarray}&&\chi _{{\rm red}}^{2}=\frac{1}{\nu }\mathop{\sum }\limits_{i}\left[ {{({{C}_{i,{\rm obs}}}-{{C}_{i,{\rm exp} }})}^{2}}/\sigma _{i,{\rm exp} }^{2} \right],\end{eqnarray} \tag{ 1 }$$

where ν is the number of degrees of freedom (number of data points minus the number of free parameters), ${{C}_{i,{\rm obs}}}$ and ${{C}_{i,{\rm exp} }}$ are the measured and predicted count rates, respectively, in the ith energy bin, and ${{\sigma }_{i,{\rm exp} }}$ is the standard deviation of the expected count rate assuming Poisson statistics.

Figure 2 shows the 1-T and 2-T fits to the measured SAX count-rate flux spectrum with normalized residuals below. (Equivalent plots for the multitherm_pow and multitherm_exp fits are not shown but have similar appearance, and all have acceptable values of $\chi _{{\rm red}}^{2}$.) Count rate flux is the measured count rate converted to units of counts s⁻¹ cm⁻² keV⁻¹ by dividing by the effective area projected to 1 AU and by the energy bin width. This enables the varying distance of the MESSENGER spacecraft from the Sun to be accounted for as it passed from Earth to Mercury and during its elliptical orbit about Mercury. For the 1-T fit, the energy range (indicated in the figure) was confined to 2.3–8.5 keV because a good fit could not be achieved at lower energies. For the 2-T function, a satisfactory fit was achieved over the range 1.5–8.5 keV, i.e., including lower energies. The values of $\chi _{{\rm red}}^{2}$ for each fit are both less than 1.5, indicating a satisfactory fit to the observed spectrum in each case. For the 1-T fit, $\chi _{{\rm red}}^{2}=1.21$ with a probability $P=3.6\%$ of exceeding this value (165 energy bins each 0.0375 keV wide and five free parameters giving $\nu =160$). For the 2-T fit, $\chi _{{\rm red}}^{2}=1.18$, giving $P=5.2\%$ (187 energy bins and 12 free parameters giving $\nu =175$).

The abundances of the elements were set as free parameters and determined for each of the functional forms given above. The Fe and Ca abundance estimates are expected to have higher precision (of order ±20%) than for S, Si, and Ar since Fe and Ca spectral lines form recognizable line features in the measured spectrum (Figure 2). Separate line features from S, Si, and Ar are not evident with the SAX energy resolution, but their contributions to the spectrum are considerable, as can be seen from Figure 3. Here, the contributions to the SAX count flux spectrum of the thermal continuum emission and the line fluxes from the different elements are shown with the SAX energy resolution and sensitivity. The photon spectrum used to calculate these different components using chianti was the 1-T model with a temperature of 22 MK (1.9 keV) that gave the best fit to the measured count flux spectrum in the 5-minute interval used for Figure 2. Note that the contributions from the different elements peak at the following energies: Fe—1.5, 6.7, and 8 keV; Ca—3.9 keV; S—2.5 keV; Si—2.0 keV; and Ar—3.2 keV. Other elements (notably Mg) give rise to emission peaking at ∼1.6 keV. The SAX sensitivity is poorly known at such low energies because of the uncertain absorption of overlying material in the instrument and the presence of a steep and time-varying background spectrum, so spectral fits were limited to energies above 1.5 keV and no attempt was made to determine the Mg abundance.

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As mentioned above, the continuum is made up of free–free and free–bound radiation, and so varying element abundances in the spectral fits in principle affects free–bound emission. Phillips & Dennis (2012) found that varying the Fe abundance from the (e.g.) photospheric (Asplund et al. 2009) to coronal (Feldman et al. 1992) values leads to only a few percent difference in the total continuum at 10 keV for a temperature of 25 MK; for the lower temperatures and energies mostly appropriate here, the effect will be less and can be ignored.

Estimates of abundances and FIP biases of Fe, Ca, S, Si, and Ar from the two model fits to the spectrum of Figure 2 are given in Table 1. Also given are the corresponding FIP biases from the abundance sets of Feldman et al. (1992) and Fludra & Schmelz (1999).

Figure 4 shows the temperatures, emission measures, and estimated abundances of Fe, Ca, S, Si, and Ar for all time intervals using the 2-T fit procedure over the duration of the 2007 June 1 flare. The temperatures and emission measures derived from the flux ratio of the two GOES channels are also shown. From 14:30 to 15:30 UT, there is a hotter component with a temperature generally above 11.6 MK (1 keV) and a cooler component with a temperature below 6 MK (0.5 keV). Apart from the first few time intervals of the flare during its impulsive phase, the GOES temperature is only slightly less than the hotter component (labeled "SAX T1" in Figure 4).

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The abundance values for all elements in Figure 4 show variations apparently larger than the statistical uncertainties given by the OSPEX routine. These uncertainties are obtained from the error matrix and correspond to $\chi _{{\rm red}}^{2}+1$ values (Bevington & Robinson 2003). However, they do not take into account the interdependencies of the different parameters, which have been recently discussed by Ireland et al. (2013), and hence should be considered as lower limits. The horizontal line in the plot for each element shows the weighted average abundance for the time interval represented by the line length when the count flux spectrum could be measured over the full energy range.

The detailed analysis of the 2007 June 1 flare above was generalized to spectra taken in flares detected by SAX in the 2007–2013 period and selected according to criteria given in Section 2.1. Care was taken to avoid selection biases and effects of varying instrument performance and the changing environment in which MESSENGER was located, both in its cruise phase and in orbit about Mercury. A semi-automatic routine was written in the Interactive Data Language (IDL) to analyze all the larger flares detected with SAX up to 2013 September 13 with the same basic techniques described using the 1-T and 2-T temperature functions. (The multitherm_pow and multitherm_exp functions were not used at this stage since acceptable fits were generally achieved with the 1-T or 2-T models.) The procedure handled the variations in the performance and background spectrum over this extensive period, analyzing spectra in each time period when the count rate was above a given threshold.

The data were analyzed in three stages. First, a variable background spectrum was determined for each 24 hr period of the observations. This was achieved by selecting the 10% of all the observational time periods with the lowest count rates in the 1.3–5.3 keV energy range and using those values to determine a smoothed time variation by applying a Savitzky & Golay (1964) filter with a width of 128 intervals. This same time profile, normalized by the average rate in the background intervals, was used to estimate the background flux during a flare interval for each energy bin. (See http://hesperia.gsfc.nasa.gov/ssw/packages/spex/doc/ospex_explanation.htm#Background for details.) The background spectrum estimated in this way was then subtracted from the measured spectrum in each flare time interval for analysis in OSPEX. Only those background-subtracted spectra having count rates in the 6.3–7.0 keV range of $\gt 5000\;{{{\rm s}}^{-1}}$ were selected for analysis, this being the limit at which the Fe-line complex at 6.7 keV could be reliably measured.

In the second stage, OSPEX was used to analyze the background-subtracted count flux spectrum for each selected time interval using both the 1-T and 2-T temperature models. A FITS file was generated for each model covering an entire 1 yr period of observations, containing the count flux spectra and best-fit model spectra with all parameters for each analyzed time interval.

For the 2-T model, the spectral fit was performed over the energy range 1.5–8.5 keV as with the analysis of the 2007 June 1 flare discussed above. This range avoids both low energies ($\lt 1.5$ keV) with a variable instrumental component (see Figure 2) and high energies with variable background count rates from charged particles. The free parameters include two emission measures (EM₁ and EM₂) and temperatures (T₁ and T₂ where ${{T}_{1}}\;\lt \;{{T}_{2}}$), as well as abundances of Fe, Ca, S, Si, and Ar (in order of decreasing measurement certainty). For 35% of the time intervals, the emission measure of the cooler (T₁) component fell to values of $\lt {{10}^{43}}$ cm⁻³, whereupon the fitting routine automatically set its value to zero and continued fitting with a single temperature and emission measure. A pseudo-function (drm_mod) was also included to permit the automatic adjustment of the SAX nominal energy resolution and energy calibration. The instrument energy calibration can be found from the recognizable peaks at the known energies of the Fe and Ca line complexes, allowing both the intercept and slope of the energy calibration function to be estimated. The observed width of the Fe-line complex gives the detector resolution (FWHM, as a fraction of the measured pre-launch value). The number of free parameters in this case is therefore 12 (10 when EM₁ was set to zero). Some 186 energy bins (each 0.0375 keV wide) are included in the 1.5–8.5 keV fit energy range, giving $\nu =174$.

For the 1-T case, we performed spectral fits over the energy range 5.5–8.5 keV (80 energy bins), which is similar to the range used for the fits to RHESSI spectra done in the analysis by Phillips & Dennis (2012). In this range, only the Fe-line complexes at 6.7 and 8 keV are included, so only the Fe abundance can be estimated (ionized Ni lines make a small contribution to the 8 keV complex). Also, since the Ca line complex is not included, only the resolution and intercept of the instrument's energy calibration function can be determined. These, with T, EM, and the Fe abundance, are the free parameters, so $\nu =75$.

Once the FITS files were made for all spectra and flares, selection and statistical analysis of the spectra were made in a third and final stage. Several criteria were used for including those spectra that were free from unwanted environmental, instrumental, or other effects. Thus, spectra were only included if (a) they were in flare decay periods (those in the initial stages of flares had evidence of time-varying element abundances as suggested by the plots in Figure 4); (b) for the 1-T case, temperatures were in the range 12.8–23.2 MK (1.1–2.0 keV) (at temperatures $\lesssim 12.8$ MK the emission function of the Fe 6.7 keV line complex falls sharply so the measured intensity of the line above the continuum is uncertain, while spectra having measured temperatures $\gtrsim 23$ MK were nearly always the result of charged particle contamination of the X-ray spectrum); (c) emission measure EM $\gt {{10}^{48}}$ cm⁻³ (lower emission measures gave rise to count rates too low to obtain reliable abundance estimates); (d) there were at least two contiguous 5-minute time intervals; (e) the values of $\chi _{{\rm red}}^{2}$ of the fitted spectra were in the range 0.5–1.5 (values outside this range indicated poor fits or low count rates); (f) there was no evidence of charged particle contamination (identified by the count rate ratio in the 9–9.5 keV to 8–8.5 keV ranges being larger than 20).

The Fe FIP bias estimates for all spectra analyzed with the 1-T function over the energy range 5.5–8.5 keV are plotted against time in Figure 5 (top panel). The period analyzed covered the declining portion of Cycle 23 (2007–2008) and the rising portion of Cycle 24 (2010–2013). With fewer free parameters, the 1-T fits are likely to be more reliable than those from the 2-T function (Section 3.2). Plotted in Figure 6 (top panel) are FIP bias values plotted against flare sequence number, with the FIP bias averaged over the duration of each flare. This plot shows the existence of flare-to-flare variations since the scatter of points is significantly larger than the uncertainties on individual points. The uncertainties are a combination of the uncertainties given by the OSPEX routine and variations between different times during the same flare. There is at least one group of flares, flare numbers 238–261 occurring between 2011 October 28 and 30, that have mostly smaller-than-average Fe abundances. These flares may all come from a single active region, although MESSENGER was then viewing the solar hemisphere partly directed away from the Earth, so we cannot be sure of this. STEREO B was viewing the Sun from a similar angle to MESSENGER at that time and showed an active region at a Stonyhurst heliographic longitude of $\sim -100{}^\circ$ that could have been the origin of at least some of these flares. None of them are evident in the GOES light curves, except for six on 2011 October 28, all at the C1 to C2 level.

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Figure 7 (top panel) shows the frequency distribution of FIP bias estimates obtained for all spectra. From this distribution, the mean FIP bias from the 1-T function is 1.62 ± 0.28 (s.d.), corresponding to an Fe abundance (on a logarithmic scale with H = 12) of $A({\rm Fe})=7.71\;\pm \;0.07$. The standard deviations include any real-time variations over the period of observations.

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To check for any trends with flare parameters, the Fe FIP bias is plotted against both emission measure and temperature in Figure 8. There appears to be a slight tendency for lower FIP biases to result from larger emission measures ($\gtrsim {{10}^{49}}$ cm⁻³) and lower temperatures, but the significance of the trend is low.

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The occurrence distribution of $\chi _{{\rm red}}^{2}$ values is of interest as it checks whether there is a match with that expected for a chi-squared distribution with the appropriate number of degrees of freedom ($\nu =75$; Section 2.3). Figure 9 (top panel) shows the observed distribution for the 1-T fitting function and the statistically expected distribution for $\nu =75$. There is a very close agreement of the observed and expected distributions, showing that $1\sigma$-normalized residuals for each energy bin are close to expected and that there are no significant instrumental effects in the 5.5–8.5 keV energy range affecting the distributions.

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Abundance estimates of Fe, Ca, S, Si, and Ar were made using the 2-T fitting function over the broader energy range of 1.5–8.5 keV. The three free parameters describing the SAX detector energy resolution and calibration were found to vary by less than 20% over the entire 7 yr period of the observations, indicating the stability of the SAX instrument.

Estimates of the Fe FIP bias from the 2-T analysis are summarized in Figure 5 (bottom panel) and Figure 6 (middle panel). Figure 7 (bottom panel) shows the frequency histograms for Fe FIP bias values from this analysis. While there is little difference in the mean value of FIP bias, 1.66 ± 0.50 (s.d.), or $A({\rm Fe})=7.72\;\pm \;0.11$, the amount of scatter is larger. As with the 1-T case, there is little apparent dependence of Fe FIP bias on emission measure or temperature of the higher-temperature component. The frequency distribution of the reduced chi-squared values obtained for all analyzed time intervals in the 2-T case deviates significantly from the expected distribution for $\nu =174$ (Figure 9). This difference did not show up for the 1-T fits probably because of the narrower energy range. It most likely reflects some small instrumental effect not included in the instrument response matrix. The values of the $1\sigma$-normalized residuals averaged over all analyzed spectra for each energy bin show that certain bins consistently have larger or smaller count fluxes at the few percent level, but the origin of these small discrepancies is unknown. Since the difference is only ∼0.1 in the reduced chi-squared value, it is does not result in any significant deviation in the mean values of the free parameters obtained from the fits.

Abundance and frequency distribution plots were also obtained from the 2-T function for Ca, S, Si, and Ar. Figure 6 (bottom panel) shows the mean Ca FIP bias for each flare plotted against flare sequence number. There is a slight but statistically significant decrease for events 238–261, as with the Fe FIP bias values shown in this plot (top two panels). Frequency histograms for Ca and also for S, Si, and Ar are given in Figure 10. The mean Ca FIP bias is 3.89 ± 0.76, corresponding to $A({\rm Ca})=6.9\;\pm \;0.1$ with the uncertainty including any possible real variations from flare to flare.

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Figure 10 (top right and bottom panels) shows frequency histograms for S, Si, and Ar. As these elements do not have recognizable line complexes in SAX spectra, the FIP bias determinations are less reliable. We obtained mean values for all flares to be 1.23 ± 0.45 (S), 1.64 ± 0.66 (Si), and 2.48 ± 0.90 (Ar). These correspond to abundances (on a logarithmic scale with H = 12) of $A({\rm S})=7.2\;\pm \;0.2$, $A({\rm Si})=7.7\;\pm \;0.2$, and $A({\rm Ar})=6.8\;\pm \;0.2$.

The mean FIP bias values and corresponding element abundances determined from this analysis are given in Table 2. The FIP bias is plotted against the FIP for each element in Figure 11.

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