Fe xii and Fe xiii Line Widths in the Polar Off-limb Solar Corona up to 1.5 R_⊙

We investigated the observations of a southern coronal hole made by the EIS on board the Hinode satellite (Kosugi et al. 2007) during CR 2107 on March 5, 6, and 11, 2011. The main strength of this data set is the extremely long exposure time—over 30,000 s per day. The center of the 2'' wide 512'' long slit was pointed at (0'', − 1242'') during the off-limb observations, covering a region from ∼ 1.00 R_⊙ to 1.54 R_⊙. A few on-disk images are taken when the slit center was pointed at (0'', − 842'') to estimate the stray-light levels in off-limb exposures (see Figure 1). Finally, we obtained 143 frames of the off-limb spectrum and 9 frames of the on-disk spectrum, each of which has an exposure time of 600 s. The position of the EIS slit during on-disk exposures is shown in Figure 1.

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Part of the data set was not converted into fits files by the EIS pipeline for technical reasons. A parallel suite of IDL programs that converted individual data packets telemetered down from the satellite into science-ready IDL save data files was developed by the authors. We compared the results of this suite of codes with eis_prep on the data sets for which both the fits files and the data packets were available. They agree very well. After data reduction, the offset along the y-axis was corrected for by the IDL routine eis_ccd_offset (Young 2011a). The 1σ error in data numbers (DNs) was determined by Poisson statistics and dark-current (readout) noise (Young 2019). The slit tilt in each image was then corrected for using the quadratic function described in the EIS software notes (Young 2010). The radiometric calibration is performed in two steps: first we perform the original laboratory calibration to convert the units from $\mathrm{DN}\,\ {{\rm{s}}}^{-1}$ into erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹, and then the decay of the instrument is corrected for following the calibration work by Warren et al. (2014). It is important to note that Del Zanna (2013) developed an alternative revised intensity calibration correction, which shows some disagreement with the Warren et al. (2014) method we adopted. However, because we are interested in the line widths and the only use we make of the line intensities is to determine the intensity ratios of lines close in wavelength, the choice of intensity calibration correction plays a minor role in the present work. We experimented with many different spatial binnings (see Section 4.3 and Figure 10) and found that by increasing the pixels in each bin, the result did not change, but the noise decreased up to a binning over 16 pixels. Beyond the 16-pixel spatial binning, there was no particular gain in noise, but only loss in resolution. Therefore we binned the data along the slit direction by every 16 pixels to further increase the S/N.

Solar EUV emission from the quiet Sun and active regions can be scattered into the EIS FOV, contaminating the observed line profiles when EIS points above the limb. Ugarte-Urra (2010) measured a 2% stray-light level using observations during an eclipse, and this value is widely used in studies of coronal line broadening (e.g., Hahn et al. 2012; Hahn & Savin 2013). However, a recent study of an orbital eclipse suggested that the amount of stray light in EIS FOV could be higher than 2% above the distance of 1.3 R_⊙ (Hara 2019). In addition, Wendeln & Landi (2018) estimated that about a fraction of 10% radiation from surrounding active and quiet-Sun regions contaminated an equatorial coronal hole observed on the disk. They suggested that the 2% stray-light value might either be underestimated or dependent on the specific configuration of the observation being analyzed, so that the 2% value proposed by Ugarte-Urra (2010) may not apply to all observations.

Unfortunately, it is difficult to determine the contribution of stray light directly from EIS observations. To estimate the stray-light level, the on-disk portion of the slit in the cross-limb observations was averaged along the slit direction and multiplied by a fraction of 2%, 4%, and 10%. We took these three fractions as possible estimates of the stray-light level to estimate the uncertainties introduced by removing the stray light.

The 2% stray-light intensities inferred from on-disk exposures on March 5, 6, and 11 are shown in Figure 2 together with the averaged off-limb intensity at∼1.4 R_⊙ on the same date to show the relative strength of the stray light and the measured emission at the slit location where the latter is weakest. The estimated stray light on March 11 contributes much less to the total line intensity than that on the other two days. This is because the slit was pointing to the brighter quiet-Sun plasma during the on-disk exposures on March 5 and 6 (see Figure 1), rather than the coronal hole on March 11. The stray light was evaluated by fitting a single-Gaussian profile to the on-disk intensity; the fitted profile was rescaled by the stray-light fractions and used to fit the off-limb spectrum, as described in Section 2.3. We note that there is a small wavelength shift (∼0.01–0.02 Å) between the line centroids of the on-disk and off-limb spectrum in Figure 2. We compared the line profiles from the overlapped region (∼1.01–1.12 R_⊙) that was covered by on-disk and off-limb exposures and found that the line profiles match perfectly after slit-tilt correction. Therefore we suggest that this is not a systematic shift in the wavelength calibration or slit-tilt correction and use the same line centroid wavelength of the stray-light profiles in the fitting discussed in Section 2.3. We further discuss this wavelength shift in Section 4.

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To avoid the negative DNs when subtracting the estimated stray-light profiles from the off-limb profiles at high altitudes, we discarded the March 5 and 6 data in the rest of the study, leaving 56 off-limb and 3 on-disk exposures. The total exposure time of the off-limb observations is 33,600 s.

We fitted observed spectral lines to the summation of two Gaussian profiles with the emcee Markov chain Monte Carlo (MCMC) algorithm (Foreman-Mackey et al. 2013). One component is the "real" off-limb spectrum, and the other is the fixed stray-light profile. The off-limb profile can be determined with four parameters: the integrated intensity I₀, the centroid wavelength λ₀, the full width at half maximum (FWHM) Δλ, and the background intensity I_bg,

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{ln}P({I}_{\mathrm{obs}}| \lambda ,{\sigma }_{I},{I}_{0},{\lambda }_{0},{\rm{\Delta }}\lambda ,{I}_{\mathrm{bg}})\\ =\,-\displaystyle \frac{1}{2}\displaystyle \sum _{i}\left\{{\left[\displaystyle \frac{{I}_{\mathrm{obs},i}-g({\lambda }_{i})}{{\sigma }_{i}^{2}}\right]}^{2}+\mathrm{ln}(2\pi {\sigma }_{i}^{2})\right\},\end{array}\end{eqnarray} \tag{ 1 }$$

where g(λ_i ) is the double-Gaussian profile determined by the parameters describing the true emission: I₀, λ₀, Δλ, I_bg, and the stray-light profile. The ${\sigma }_{i}^{2}$ is determined by the Poisson statistical noise and the CCD readout noise. We used a constant prior in the fitting. The uncertainty of each parameter is defined by 90% credible levels.

The FWHM of a Gaussian profile observed by EIS can be written as

$$\begin{eqnarray}&&{\rm{\Delta }}\lambda ={\left[{\lambda }_{{\rm{I}}}^{2}+4\mathrm{ln}2{\left(\displaystyle \frac{{\lambda }_{0}}{c}\right)}^{2}\left(\displaystyle \frac{2{k}_{B}{T}_{i}}{{m}_{i}}+{\xi }^{2}\right)\right]}^{1/2}\end{eqnarray} \tag{ 2 }$$

where λ_I is the instrumental FWHM, T_i is the ion temperature, m_i is the ion mass, ξ denotes the nonthermal widths, c is the speed of light, and k_B is the Boltzmann constant. We calculated the instrumental broadening along the 2'' slit by using the IDL routine eis_slit_width.pro (Young 2011b) in the bottom half of the CCD and removed it from the measured values.

The AWSoM is the representation of the solar corona (SC) and inner heliosphere (IH) components of the Space Weather Modeling Framework (SWMF, Tóth et al. 2012). The simulations in this paper were performed with the three-temperature AWSoM model (AWSoM, van der Holst et al. 2014), where the model solves for isotropic electron and anisotropic proton temperatures. The coronal heating and solar wind acceleration are addressed via low-frequency Alfvén waves that are partially reflected by gradients of the solar wind plasma. The energy partitioning of the model has been improved, so that the electron-to-proton heating ratio increases, resulting in decreased, more realistic polar wind speeds (see B. van der Holst et al. 2021, in preparation). This paper uses the SC component, which uses a spherical grid between 1 and 24 R_⊙ for which the radial coordinate is stretched to numerically resolve the steep gradients near the Sun. We also artificially broadened the transition region similar to Lionello et al. (2009) and Sokolov et al. (2013). The domain is decomposed into adaptive mesh refinement (AMR) blocks, and we applied one additional level of AMR around the heliospheric current sheet. To obtain plasma parameters for the times of the observation, we used a Global Oscillation Network Group magnetogram (GONG, Harvey et al. 1996) processed with Air Force Data Assimilative Photospheric flux Transport (Henney et al. 2012) into magnetograms as the radial magnetic field at the inner boundary, while the solar wind initial condition was the Parker solution. Then we obtained a steady-state solution after 100,000 steps, from which we extracted the observed plasma parameters into Cartesian boxes covering the region of emission observed by Hinode/EIS.

The solar wind model is capable of predicting both in situ and remote-sensing observations. SPECTRUM is a post-processing tool within the SWMF that uses the AWSoM coronal simulation results to calculate synthetic spectra on a voxel-by-voxel basis. To calculate the emergent line profiles, the AWSoM simulation results are first interpolated to a Cartesian grid after the line-of-sight (LOS) direction has been specified by the user. Then the SPECTRUM module performs the calculation voxel by voxel. In each voxel, we calculate the emissivity profiles, which include Doppler shifts due to the plasma motion along the LOS direction (Equation (3)) and the broadening of the emissivity profiles (Equation (4)). Finally, SPECTRUM integrates the emissivity profiles along the LOS direction to obtain the emergent spectrum.

SPECTRUM takes care of two different Doppler-broadening mechanisms:

(1) The first is the macroscopic Doppler broadening due to the LOS integration of the line profile with different line centroids. The centroid of the emissivity profile at each voxel is shifted by the local bulk plasma motion u_LOS

$$\begin{eqnarray}&&{\lambda }_{\mathrm{shifted}}=\left(1-\displaystyle \frac{{u}_{\mathrm{LOS}}}{c}\right){\lambda }_{0},\end{eqnarray} \tag{ 3 }$$

where c is the speed of light.

(2) The second mechanism is the microscopic Doppler broadening in individual voxels caused by the thermal motion and Alfvénic nonthermal motions,

$$\begin{eqnarray}&&{\rm{\Delta }}\lambda ={\left[4\mathrm{ln}2{\left(\displaystyle \frac{{\lambda }_{0}}{c}\right)}^{2}\left(\displaystyle \frac{2{k}_{B}{T}_{\mathrm{LOS}}}{{m}_{p}{A}_{i}}+{\xi }^{2}\right)\right]}^{1/2},\end{eqnarray} \tag{ 4 }$$

where m_p is the proton mass, and A_i is the mass number of the ion. Because AWSoM does not yet predict ion temperatures, we assumed the LOS ion temperature to be given by the LOS proton temperature ${T}_{\mathrm{LOS}}={T}_{\mathrm{perp}}{\sin }^{2}\alpha +{T}_{\mathrm{par}}{\cos }^{2}\alpha$, where T_perp and T_par are proton temperatures perpendicular and parallel to the magnetic field, and α is the angle between the direction of the local magnetic field and the LOS; this assumption is further discussed in Section 4.2.

The nonthermal component is

$$\begin{eqnarray}\begin{array}{rcl}{\xi }^{2} & = & \displaystyle \frac{1}{2}\langle \delta {u}^{2}\rangle {\sin }^{2}\alpha =\displaystyle \frac{1}{2}\displaystyle \frac{{\omega }^{+}+{\omega }^{-}}{\rho }{\sin }^{2}\alpha \\ & = & \displaystyle \frac{1}{8}\left({z}_{+}^{2}+{z}_{-}^{2}\right){\sin }^{2}\alpha .\end{array}\end{eqnarray} \tag{ 5 }$$

Here z_± are the Elsässer variables for forward- and backward-propagating waves, and the respective energy densities are ω_±.

We note that the nonthermal velocity in this paper is defined in a slightly different but equivalent way in comparison to the definitions in Szente et al. (2019). Szente et al. (2019) adopted the standard deviation to describe the line broadening, and the nonthermal velocity is defined as ${v}_{\mathrm{nth}}^{2}=\tfrac{1}{4}\langle \delta {u}^{2}\rangle {\sin }^{2}\alpha =\tfrac{1}{2}{\xi }^{2}$. For the detailed implementation of synthetic spectral calculations, see Szente et al. (2019).

The presence of a streamer at the far side of the Sun is demonstrated in the meridional cut of the AWSoM simulations in Figure 6. To understand how the streamer affects the formation of the monotonically increasing line widths in simulations, we plot the contours of the Fe xii 195.1 Å contribution function normalized to its maximum ${C}_{{\rm{I}},195}/{C}_{{\rm{I}},195,\max }$ over the local FWHMs Δλ along the LOS in the upper left panel of Figure 7. The relative contribution of the Fe xii 195.1 emission from the polar coronal hole or the streamer is shown in the upper right panel. First, the local FWHMs are dominated by nonthermal velocity and increase from ∼0.06 Å to ∼0.1 Å in the coronal hole. Second, photons from the streamer dominate the synthetic profiles with narrower line widths, even up to 1.4 R_⊙.

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The middle and lower panel of Figure 7 show the LOS distribution of the relative contribution of the Fe xii 195.1 emission per voxel along the LOS (i.e., the local contribution function normalized to the total Fe xii 195.1 Å emergent intensity) and the local FWHM at a heliocentric distance of 1.2 R_⊙ and 1.5 R_⊙. The fitted line widths from the synthetic profiles are also provided as references. The line profiles in the AWSoM simulation are much narrower in the lower corona, e.g., at 1.2 R_⊙ , because the SPECTRUM LOS integration causes the streamer emission to provide the bulk of the observed photons. The AWSoM line widths begin to increase with height because the streamer contributes fewer photons in higher altitudes, e.g., 1.5 R_⊙, and the coronal hole FWHMs start to increase with height. So the fitted line width at 1.5 R_⊙ is ∼0.07 Å, which is much larger than that of ∼0.03 Å at 1.2 R_⊙.

The AWSoM simulation shows how the streamer photons are likely to contaminate the spectrum and result in much narrower line profiles. The simulation also shows how complex the variation of local nonthermal velocity along the LOS is even within coronal holes, which means that a simple Gaussian fitting may not explain the observed profiles. In addition, AWSoM has a limited spatial resolution and uses a synoptic magnetogram to calculate the inner boundary, which cannot resolve the small-scale effects or structures of the sizes of the FWHM fluctuations. Therefore we cannot use AWSoM to investigate the nature of the FWHM periodic spatial fluctuations in the line widths shown in Figure 5.

Based on the AWSoM simulations, we suggest that the Fe xii and Fe xiii emission in the observation is also significantly contaminated by the streamer emission. In addition, the streamer was also observed by SECCHI EUVI (Howard et al. 2008) 195 Å imaging on board the STEREO-B spacecraft (Kaiser et al. 2008), as both the STEREO spacecraft were in quadrature with Earth and Sun (see Figure 8). The fluctuations in the line widths may come from the streamer (also see Singh et al. 2003, Figure 4).

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Synthetic line widths are much narrower than the observed ones either because AWSoM underestimates the nonthermal broadening in the streamer or because the observed Fe xii line profiles are less contaminated by the streamer emission than predicted. The possible differences between the variation of FHWM versus height in this study and previous studies (e.g., Bemporad & Abbo 2012; Hahn et al. 2012) may be due to (1) contamination from the streamer, (2) different wave-damping levels in other coronal holes, and (3) wave-damping levels affected by different phases of the solar cycle.

The SPECTRUM module uses the LOS component of the anisotropic proton temperature to evaluate the thermal broadening of the spectral lines in place of the temperature of each ion because the current version of AWSoM does not calculate the temperature of each ion. However, a previous study (e.g., Moran 2003) suggested that there is no uniform ion temperature in the off-limb corona. The ion temperature could also deviate greatly from the local proton and electron temperatures due to some other heating mechanisms such as ion-cyclotron resonance (e.g., Tu et al. 1998). Landi & Cranmer (2009) measured the Fe xii temperature in the coronal hole between 1.03 and 1.17 R_⊙ and obtained a result of $\mathrm{log}{T}_{i}\sim 6.7\mbox{--}6.95$, which corresponds to a thermal FHWM of ∼0.05 Å. The Fe xii ion temperature in the quiet solar corona is about $\mathrm{log}T=6.2\mbox{--}6.6$ (Landi 2007), which is also higher than the proton temperature used in AWSoM/SPECTRUM simulations.

To investigate the influence of a higher ion temperature on the line broadening, we manually changed the temperatures in the meridional cut, as shown in Figure 9(a). We first determined the streamer region using the contribution function of the Fe xii 195 Å line and then arbitrarily assigned a temperature of $\mathrm{log}T=6.4$ to this region. The ion temperature in the remaining grid is set to be $\mathrm{log}T=6.8$. Then we resynthesize the Fe xii 195 Å line profiles using the new thermal broadening. We compare the line widths measured from the modified line profiles with those from the current AWSoM simulation and EIS observations in Figure 9(b). Because most of the emission in the lower corona comes from the streamer, the line widths only increase by ∼0.01 Å below 1.35 R_⊙, which is still insufficient to explain the line broadening at lower altitudes. Above 1.4 R_⊙, the increased ion temperatures ($\mathrm{log}T\sim 6.8$) broaden the line profiles by ∼0.02 Å, which causes the synthetic widths to become much closer to the EIS observations. We have to stress that these results come from a very crude and arbitrary approximation of the real ion temperatures. Nevertheless, they point toward an important parameter that could be responsible for the disagreement between the measured and observed FHWM values. Still, while the much higher ion temperature may account for the widths at heights where the streamer is not present, it cannot reproduce the widths where the streamer dominates the emission.

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Previous studies have used large spatial binning, usually more than 30 pixels, to increase the S/N and obtain Gaussian profiles to fit. Because our observations have an extremely long exposure time of 33,600 s, we fitted the line profiles with different spatial binnings every 2, 4, 8, 16, and 32 pixels. The results are shown in Figure 10. Smaller spatial binnings like 2 or 4 pixels provide a larger uncertainty as well as more fluctuating FHWMs. An 8- or 16-pixel binning removes the smallest-scale fluctuations due to a strong noise reduction, but it maintains variations at the 0.05–0.1 R_⊙ level, making them significant. However, the 32-pixel binning is so large that it smooths these fluctuating fine structures without improving the uncertainty. Therefore we suggest that the fluctuations may not be found in previous studies due to the large spatial binning, even if the emission is contaminated by other structures.

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The measured line centroid wavelengths from EIS observations and AWSoM simulations are shown in Figure 11. By choosing different stray-light levels, we show that a larger stray-light component may result in a larger FWHM. Most of the previous studies used the 2% stray-light level obtained from Ugarte-Urra (2010). However, the wavelength offset between the stray light might be due to the real Doppler shifts in the solar wind flows. In this case, it is possible that the stray light does not dominate the off-limb spectrum even at 1.5 R_⊙, because otherwise, the off-limb profiles should have the same line centroid wavelength as the stray light. The line centroids in AWSoM simulations are first blueshifted because of the bulk motion in the streamer toward the observer. The AWSoM line centroid wavelength increases by ∼0.01 Å at higher altitudes because the photons are increasingly emitted by the redshifted and blueshifted flows from the solar wind, and less from the blueshifted streamer. Because shifts of the AWSoM line centroid show a similar trend as EIS observations, we suggest that the stray light does not significantly affect the line profiles, even at 1.5 R_⊙. Moreover, the 10% stray light would make the redshift even larger and the line widths artificially narrower at large heights, so that it is likely an overestimation. After all, the off-disk configuration of the EIS slit in our observation is much more similar to that of the eclipse configuration from Ugarte-Urra (2010) than the full-disk configuration of Wendeln & Landi (2018). Therefore estimating the stray-light fraction of 2%–4% may be sufficient in this study.

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The Fe xii 192.4, 193.5, and 195.1 Å line ratios may be sensitive to photoexcitation at large heights, where collisional excitation is less efficient and self-absorption of Fe xii emission coming from a lower altitude, brighter area could contribute to populating the parent ^4P levels. In this case, their ratios should be dependent on height. We show the intensity ratios of each two of the three lines in Figure 12 as a function of height using two different radiometric calibration methods: those of Del Zanna (2013) and of Warren et al. (2014). The reference values given by CHIANTI are also plotted. The line ratios do not vary significantly below 1.2 R_⊙. The ratio of the Fe xii 192.4 and 195.1 Å lines and the ratio of the Fe xii 193.5 and 195.1 Å lines increase with height, while the Fe xii 192.4 and 193.5 Å line ratios do not show significant variations. Hahn et al. (2012) calculated the line ratios, but found no notable changes at different heights. Because these ratios are independent of temperature and density, different structures along the LOS should not alter the ratios. Moreover, if photoexcitation were active, the brightest line (i.e., Fe xii 195 Å) should become brighter with height relative to the others, so that the ratios should decrease rather than increase. The systematic increase in the Fe xii 192/195 and 193/195 line ratios may not result from photoexcitation, but may be caused by other instrumental and physical effects.

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Ideally, the Fe xii triplets should have identical line widths at any given height. However, in EIS observations, the Fe xii 195.12 Å line is always broader than the other two. The differences in the Fe xii triplet line widths along the slit measured in this study are shown in Figure 13. We note that below 1.2 R_⊙ (corresponding to CCD pixel ∼300–500), the Fe xii 195.1 Å line is broader than the other two lines.

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Hara (2019) suggested that another weaker Fe xii line at 195.18 Å may blend with the Fe xii 195.12 Å line and broaden the profile. The Fe xii 195.18 Å line was found in laboratory spectra (Arthanayaka et al. 2020), although some previous experiments did not resolve the line (e.g., Träbert et al. 2014). We used CHIANTI to calculate the intensity ratio of these two lines and found that the intensity of the suggested blended line is about 2% of the Fe xii 195.12 Å intensity in the streamer condition ($\mathrm{log}{n}_{e}\sim 8.5$, measured from the Fe xii 195.12 to 186.86 ratio). Young et al. (2009) measured the ratio of the two blended lines by double-Gaussian fitting and showed that the Fe xii 195.18 to 195.12 ratio is about 5% when $\mathrm{log}{n}_{e}\lt 9$. We also attempted to fit the Fe xii 195.12 Å line double-Gaussian profiles, but we found the fitted Fe xii 195.12 Å width only narrows by 0.007 Å, which is insufficient to explain the maximum discrepancy by ∼0.02 Å between the widths of Fe xii 195.12 and 192.39. Furthermore, the fitted Fe xii 195.18 intensity is more than 10% of the Fe xii 195.12 intensity, which disagrees with the density measurements. The inconsistency may be caused by the LOS integration because Young et al. (2009) used on-disk observations of an active region, or some other factors.

In a recent study of EIS line widths in the quiet solar corona, Del Zanna et al. (2019) suggested that the anomalous widths of the strongest Fe xii 193.5 Å and 195.1 Å line are due to instrumental reasons. They called into question whether firm conclusions could be obtained because of the uncertainties on the instrumental broadening described in Young (2011b). Our results show similar patterns in the difference of the FWHMs of Fe xii measured by Del Zanna et al. (2019, see Figure 2) in 2006, but with larger standard deviations. Therefore we cannot exclude the existence of the instrumental broadening that depends not only on the position along the slit, but also on wavelength.