Comparative Study of the Three-Dimensional Thermodynamical Structure of the Inner Corona of Solar Minimum Carrington Rotations 1915 and 2081

Abstract

Using differential emission measure tomography (DEMT) based on time series of EUV images, we carry out a quantitative comparative analysis of the three-dimensional (3D) structure of the electron density and temperature of the inner corona ($r<1.25\,\mathrm{R}_{\odot}$) between two specific rotations selected from the last two solar minima, namely Carrington Rotations (CR)1915 and CR-2081. The analysis places error bars on the results because of the systematic uncertainty of the sources. While the results for CR-2081 are characterized by a remarkable north–south symmetry, the southern hemisphere for CR-1915 exhibits higher densities and temperatures than the northern hemisphere. The core region of the streamer belt in both rotations is found to be populated by structures whose temperature decreases with height (called “down loops” in our previous articles). They are characterized by plasma $\beta\gtrsim1$, and may be the result of the efficient dissipation of Alfvén waves at low coronal heights. The comparative analysis reveals that the low latitudes of the equatorial streamer belt of CR-1915 exhibit higher densities than for CR-2081. This cannot be explained by the systematic uncertainties. In addition, the southern hemisphere of the streamer belt of CR-1915 is characterized by higher temperatures and density scale heights than for CR-2081. On the other hand, the coronal hole region of CR-1915 shows lower temperatures than for CR-2081. The reported differences are in the range ${\approx}\,10\,\mbox{--}\,25\%$, depending on the specific physical quantity and region that is compared, as fully detailed in the analysis. For other regions and/or physical quantities, the uncertainties do not allow assessing the thermodynamical differences between the two rotations. Future investigation will involve a DEMT analysis of other Carrington rotations selected from both epochs, and also a comparison of their tomographic reconstructions with magnetohydrodynamical simulations of the inner corona.

Appendix: Uncertainty Analysis

Following the study developed by Nuevo et al. (2015), the systematic uncertainty of the results derived from the DEMT + PFSS study that are due to the tomography regularization level and the EUV image radiometric calibration is quantitatively investigated.

The EUVI and EIT data were prepared using the latest processing tools and calibration corrections provided by their teams through the SolarSoft package. The EUVI channels have an estimated relative radiometric calibration uncertainty between the different filters on the order of ${\lesssim}\,15\%$ (Article I), and the EIT channels are thought to have a similar relative calibration uncertainty (Frédéric Auchère, private communication). Since the LDEM is sensitive to the ratios of the channel intensities, the relative calibration uncertainty propagates into the estimated electron density and temperature.

The two telescopes also have an absolute radiometric calibration uncertainty that corresponds to a common global factor that affects all channels simultaneously. Comparisons of EIT and EUVI observations indicate that they agree to within about ${\approx}\,30 \%$ (Frédéric Auchère, private communication). For a practical comparison between the results based on data from both telescopes, we assume that the absolute radiometric calibration of each telescope is half of that value, i.e. ${\approx}\,15\%$. The effect of the absolute radiometric uncertainty is a global correction factor for the intensity of all EUV channels, and thus on the determined LDEM area. As the derived electron density scales with the square root of the LDEM, the absolute radiometric calibration uncertainty carries an uncertainty on the order of ${\approx}\,8\%$ for the electron density, and has no effect on the determination of either the electron density scale height or the electron temperature (as they depend on the shape of the LDEM only, not on its area).

The regularization level of the tomographic inversion is controlled by a single dimensionless amplitude parameter $p$, called the regularization parameter. For $p=0$, no regularization is applied. Its optimal value $p_{\mathrm{opt}}$ and its uncertainty $\Delta p$ are determined through tomography cross-validation studies (Frazin and Janzen, 2002; Frazin, Vásquez, and Kamalabadi, 2009) for each band independently. The resulting values depend on the particular instrument, band, and period under analysis. Based on the previous studies by Frazin, Vásquez, and Kamalabadi (2009), Vásquez, Frazin, and Manchester (2010), Vásquez et al. (2011) for EUVI data, by Nuevo et al. (2015) for SDO/AIA data, and our own studies for EIT data, we here used characteristic average values for all EUV bands, specifically, $p_{\mathrm{opt}}=0.75$ and $\Delta p=0.40$.

The effect of the sources of uncertainty on the results was then investigated through an “error box” analysis, which consists of varying the relevant parameters within their range of uncertainty in a controlled fashion. The DEMT analysis was applied to over-regularized ($p = p_{\mathrm{opt}} + \Delta p$) and under-regularized ($p = p_{\mathrm{opt}} - \Delta p$) FBE sets, with the regularization level varied in unison for all bands. In each case, the relative radiometric uncertainty of the EUVI (or EIT) bands was simultaneously considered by applying a ${\pm}\,15\%$ relative correction to the FBE values of each band. As three EUV bands were used, the possible combinations for this relative correction were $C=2^{3}$. In summary, for each rotation, $C$ under-regularized and $C$ over-regularized DEMT studies were performed, with each of the $N=2C$ cases corresponding to a “corner” case of the error box analysis.

For each of the resulting $N$ DEMT data sets we performed the analysis. For each coronal region, the median value of the $N$ statistical distributions of each analyzed physical parameter ($N_{\mathrm{CM}}$, $\lambda_{N}$, $\langle T_{\mathrm{m}}\rangle$) can be computed. Then, the mean value and standard deviation (over all $N$ studies) of these medians was obtained. The results are fully detailed in the first three columns of Tables 7 and 8 for rotations CR-2081 and CR-1915, respectively. For each coronal region, the first row summarizes the “base” DEMT results reported in the article (i.e. without applying corrections to the FBEs). The second row shows the average $\mu$ of the results of the $N$ DEMT + PFFS studies, while the third row shows their fractional standard deviation $\sigma /\mu$. This fractional standard deviation is a measure of the uncertainty of the corresponding physical parameter due to the systematic uncertainties. The last three columns show the results for the fractional standard deviation of the $N$ statistical distributions of each analyzed physical parameter. The results of the tables are discussed in Section 3.6.