Spectral Diagnostics of Cool Flare Loops Observed by the SST. I. Inversion of the Ca ii 8542 Å and Hβ Lines

Solar flares are rapid energy releases within active regions (ARs) caused by the reconnection of the coronal magnetic field, resulting in plasma heating to temperatures beyond tens of millions of kelvins. A significant amount of the released energy is transported along the magnetic loops to the lower solar atmosphere via accelerated particles, magnetohydrodynamic waves, and thermal conduction (Hirayama 1974; Aschwanden 2005, see chapter 16). The bulk of flare energy dissipates upon reaching the dense loop footpoints. As most of the flare energy is deposited in the chromosphere, a strong evaporation of the heated plasma occurs (e.g., Neupert 1968; Fisher et al. 1985a, 1985b, 1985c; Graham & Cauzzi 2015). A detailed EUV spectroscopic analysis of explosively evaporating plasma following an M1.6-class flare by Gömöry et al. (2016) found electron densities up to 3.16 × 10¹⁰ cm⁻³ at a temperature of 1.56 MK. The evaporation provides an injection of plasma into flare coronal loops (hereafter flare loops) and leads to the formation of bright high-density structures at the apex of the flare loop arcade. Later, the evaporated plasma cools and subsequently drains down along the loops toward the chromosphere in the form of discrete plasma blobs called "coronal rain" (Antolin et al. 2010; Vashalomidze et al. 2015; Jing et al. 2016). In this context we refer to Švestka (2007) who criticized the widespread misnomer "post-flare loops." Observations show that such loop systems are not truly post-flare phenomena but are an integral part of the flare event and therefore the terms "eruptive flare loop system" or "flare loops" are more relevant. However, quantitative measurements of the fundamental plasma parameters of flare loops and their bright apices such as temperature, density, gas pressure, or magnetic field strength are rare.

Since its launch on 2013 June 23, the Interface Region Imaging Spectrograph (IRIS; De Pontieu et al. 2014) has brought new, important insights into the structure and dynamics of flare loops during the evaporation, cooling, and subsequent draining phases. The study of Lacatus et al. (2017) showed remarkably broad and redshifted emission profiles of Mg ii, C ii, and Si iv lines observed by IRIS in on-disk flare loops of the X2.1-class flare. The authors interpreted the very broad line profiles as due to unresolved Alfvénic motions, waves, or turbulence whose energy exceeds the radiation losses in the IRIS Mg ii lines by an order of magnitude. Extremely broad Mg ii emission profiles, observed in on-disk flare loops, have been also reported in Mikuła et al. (2017) and explained as due to large unresolved non-thermal motions. Analyzing IRIS observations of on-disk flare loops of an M-class flare, Brannon (2016) also found significant non-thermal broadening of the Si iv 1402.772 Å line, ranging from ≈12 km s⁻¹ near the loop apices up to ≈60 km s⁻¹ near the footpoints, and an approximately uniform and constant electron density of 10¹¹ cm⁻³ in the loops.

In September 2017, the AR NOAA 12673 produced a series of powerful X-class flares as it rotated from the disk center to the limb (Verma 2018; Romano et al. 2019). On 2017 September 10, the AR was just behind the western limb when it produced the SOL2017-09-10T16:06 X8.2-class flare, ranked as the second-largest flare in the solar cycle 24. The flare triggered a very fast coronal mass ejection (Veronig et al. 2018) followed by significant space weather and heliospheric effects including a solar energetic particles event (Kurt et al. 2018). Analysis of elemental abundances of the flare loops was performed in Doschek et al. (2018) using spectra obtained by the Hinode's Extreme-ultraviolet Imaging Spectrometer (EIS; Culhane et al. 2007). In some loops, they found that the abundances are coronal at the loop apices or cusps, gradually changing to photospheric toward the loop footpoints (see Figure 4 therein). Remarkably, using the intensity ratio of the Ca xiv and Ar xiv lines near 194 Å they found electron densities of (2–5.8) × 10¹⁰ cm⁻³ in the bright cusp of flare loops. On the other hand, Jejčič et al. (2018) found relatively high electron densities ranging from 10¹² to 10¹³ cm⁻³ in their analysis of the same flare loops observed in white-light (WL) continuum by the Solar Dynamics Observatory's (SDO; Pesnell et al. 2012) Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012; Schou et al. 2012) under the assumption of optically thin flare loop plasma in the continuum at 6173 Å. They concluded that the hydrogen Paschen and Brackett recombination continua are dominant in cool flare loops (T ≲ 2 × 10⁴ K), while the hydrogen free–free continuum emission is dominant for warmer and hot loops (T > 2 × 10⁴ K) (Figure 4 therein). Finally, Kuridze et al. (2019, hereafter KMM) presented a unique measurement of the magnetic field of this flare loop arcade, yielding a field strength as high as 350 G at heights up to 25 Mm.

An extensive compilation of earlier high electron density measurements in flares is given in Švestka (1972). More recently, high electron densities of the order of 10¹² cm⁻³ and gas pressure higher than 3 dyn cm⁻² in on-disk flare loops were considered by Heinzel & Karlicky (1987) using non-local thermal equilibrium (non-LTE) modeling of the Hα spectral line. Similar modeling was also performed by Švestka et al. (1987). High electron densities of 10^12.8 cm⁻³ for the off-limb flare loops were reported in Hiei et al. (1983). They also interpreted wide Fe xiv 5303 Å flare profiles as arising from turbulent velocities of 30–40 km s⁻¹. Hiei et al. (1992) studied a 1989 August 16 X20-class WL flare and the corresponding off-limb flare loops, concluding that the WL emission of the apex was due to hydrogen free–bound/free–free emission. The electron density was estimated to be about 10^12–13 cm⁻³. Analyzing WL observations of flare loops of a X2.8-class off-limb flare obtained by SDO/HMI, Saint-Hilaire et al. (2014) estimated the free electron density of the WL loop system to be as high as 1.8 × 10¹² cm⁻³.

In contrast, using Hα observations of various on-disk and off-limb flare loops acquired by the Multichannel Subtractive Double-Pass spectrograph at Pic du Midi, Heinzel et al. (1992a, 1992b) and Schmieder et al. (1996a, 1996b) inferred relatively low electron densities of (2–10) × 10¹⁰ cm⁻³ and gas pressures of the order of 0.1–0.5 dyn cm⁻². A 1D radiation–hydrodynamic model of flare loops by Tsiklauri et al. (2004), based on the Bastille Day flare of 2000 July 14, showed that flare loop densities up to 5 × 10¹¹ cm⁻³ can be achieved (Aschwanden 2005, Chapter 16). Recently, Firstova & Polyakov (2017) analyzed Hα spectral measurements of off-limb M7.7-class flare loops, finding electron densities of 10¹¹ cm⁻³.

This paper presents an analysis of high-resolution imaging spectroscopy data of off-limb cool loops of the X8.2-class flare acquired by the Swedish 1 m Solar Telescope (SST; Scharmer et al. 2003a, 2003b). We use the 1D non-LTE radiative-transfer code based on the Multilevel Accelerated Lambda Iterations technique (MALI; see, e.g., Heinzel 1995) and invert the SST spectra using an extensive grid of models. These inversions are used to obtain diagnostic information on key plasma parameters of selected bright patches at the flare loop apex at three times during the gradual phase, focusing mainly on electron densities and gas pressures.

2.1. Target, Event, and Observational Setup

2.2. Spatial Alignment of Data

Figure 1 shows an overview of the MOMFBD-processed SST spectral imagery of the flare loops acquired on 2017 September 10. The CRISP (top frames) and CHROMIS (bottom) images were acquired at 16:28:27 UT and 16:26:43 UT, respectively. Although made close in time, their FoV centers are not spatially co-aligned. Due to highly variable and less-than-optimum seeing, only three quasi-simultaneous pairs of line profile scans, taken at the moment of the best viewing conditions at ≈(16:25, 16:27, 16:28) UT, are used for analysis. To co-align the loop multi-spectral imagery, images in wavelength-integrated intensities over Hβ and Ca ii profiles are created for the three times of observation, allowing clearly recognizable common features to be identified, mainly at the bright loop apex (Figure 2). Through cross-correlation, a satisfactory spatial alignment of Ca ii and Hβ data is achieved as shown in Figure 2.

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2.3. Data Radiometric Calibration

Diagnostics of plasma parameters through the non-LTE radiative-transfer computations requires careful calibration of spectral data in absolute intensity units. Observed data, yielded by the CRISPRED pipeline, are represented by a Ca ii profile $\langle {I}_{\mathrm{CRED}}\rangle$ (Figure 3: top left frame) taken as the spatial average over the 165 × 145 px² = 94 × 83 ≈ 6.8 × 6.0 Mm² rectangle centered at a quiet-Sun area (Figure 1: top frames) at (x, y) = (34, 4) Mm at the direction cosine μ = 0.2032. The average profile $\langle {I}_{\mathrm{CRED}}\rangle$ is compared with the reference profile I_REF taken from Linsky et al. (1970). The latter profile is: (i) extrapolated for the given μ, (ii) intensity calibrated by the disk-center absolute continuum intensity taken from Cox (2000), (iii) corrected for limb darkening by the formula given therein, and finally (iv) convolved with the transmission profile of the CRISP Fabry–Pérot etalons provided by J. de la Cruz Rodríguez (2017, private communication). The ratio of the reference profile and the observed average renders a calibration profile ${I}_{\mathrm{REF}}/\langle {I}_{\mathrm{CRED}}\rangle$ (Figure 3: bottom left frame) allowing the conversion of the Ca ii flare loop profiles from digital to absolute units (Figure 4: top frames).

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The CHROMISRED reduction pipeline implicitly performs calibration in disk-center absolute intensity units using the Hamburg disk center spectral atlas (Neckel 1999), disregarding the position angle of CHROMIS science data (Löfdahl et al. 2018, Section 4.3). We verify the calibration by the Hβ profile taken as a spatial average over the 400 × 200 px² = 150 ×75 ≈ 10.9 × 5.4 Mm² rectangle centered at a quiet-Sun area (Figure 1: bottom frames) at (x, y) = (37, 3.5) Mm at the direction cosine μ = 0.1789. The Hβ disk intensity is corrected for limb darkening by the formula of Cox (2000), and further corrected for air mass increase through a disk-center flat field measurement at 14:45 UT and science measurements at ≈16:27 UT. The CHROMISRED pipeline used the flat fields in radiometric calibration. The resulting Hβ profile $\langle {I}_{\mathrm{CHRED}}(\lambda ,\mu )\rangle$ is compared with the Hβ profile I_DAV(λ, μ) taken from David (1961) (Figure 3: top right frame). The latter profile is interpolated for the given μ and further processed with identical steps to CRISP (ii)–(iv), but with a transmission profile provided by M. Löfdahl (2019, private communication). The coefficients of limb darkening of 0.399, the air mass change of 1.083, and the ratio I_DAV(λ, μ)/$\langle {I}_{\mathrm{CHRED}}(\lambda ,\mu )\rangle$ (Figure 3: bottom right frame) are applied in re-calibrating the Hβ flare loop profiles yielded by CHROMISRED (Figure 4: bottom frames).

2.4. Flare Loop Appearance

The line core images show very prominent bright apices of axially quasi-symmetric flare loops (Figure 1: left column), whose axes lie almost along the line of sight (LoS), with the right leg and footpoint displaced toward the observer (Figure 5; KMM, Figure 12). The loop legs appear faint due to significant Doppler shifts (KMM, Figure 11(a)) and presumably due to lower density compared to the apex. The right leg is partially obscured behind a prominence that is situated at the limb.

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The Hβ near-wing and wing images show bright narrow rims at the loop footpoints (Figure 1: middle and right frames at the bottom), which closely follow the limb. We interpret them as Hβ flare ribbons seen edge-on. They are absent in the Hβ and Ca ii core images due to the opaque chromospheric canopy with the maximum opacity at the line cores (Figure 1: left column). Remarkably, the Hβ ribbon at the right leg is substantially brighter than that at the left leg (Figure 1: bottom right frame). It suggests that the right leg and ribbon are closer to the observer than the left one (Figure 5; KMM, Figure 12), which is likely partially obscured behind the limb. The blueshifts and redshifts detected in the right and left legs, respectively, support this scenario (KMM, Figure 11(a)). However, the ribbons are absent in the Ca ii near-wing and wing images (Figure 1: middle and right frames at the top). This behavior of footpoints will be analyzed in another paper.

2.5. Basic Characteristics of Selected Line Profiles

2.6. Double-Gaussian Fitting and Symmetrization

Because our observations were not tailored for very broad profiles occurring in solar flares (Section 2.1) and our non-LTE modeling assumes a static plasma without any bulk motions, thus yielding symmetric profiles, the observed profile intensities (the red circles in Figure 4) were further processed to be comparable with the results of modeling. The observed intensities of both lines I(λ) were fitted by the double-Gaussian model function. The double-Gaussian fits are displayed in Figure 4 by the black solid lines. For the fitting the SolarSoft function mpfitfun.pro is used. The function calls the core procedure mpfit.pro, which performs Levenberg–Marquardt least-squares minimization of a χ² merit function (Markwardt 2009; Moré 1978; Moré & Wright 1993). Further, the fits are symmetrized with respect to the zero wavelength by taking their average 0.5[I(Δλ) + I(−Δλ)]. This compensates for the peak intensity differences and possible small profile asymmetries and shifts with respect to the rest line-center wavelength. The symmetric fits within the scanning ranges of ±1.75 Å and ±1.2 Å of CRISP and CHROMIS, respectively, are shown in Figures 6 and 7 by the black solid lines.

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As explained, the symmetric fits are used as a proxy of the real profiles in the inversion. It is important to consider the effect of symmetrization on their basic characteristics and their spatio-temporal variability (Table 1). Comparing the integrated intensities E, the reversal depths r of the observed Ca ii profiles, and their symmetric fits shows differences typically less than ≈6%. This confirms the validity of using symmetric fits as proxies of the observed profiles in the inversion. Despite using just three E and r values of the Ca ii profiles, their spatio-temporal variability over bright patches at loop apex is less than ≈9%. On the other hand, the Hβ profiles and their symmetric fits manifest larger variability. Obviously, due to missing wings in the observed Hβ profiles, their integrated intensities are smaller than the values inferred by the symmetric fits. But as proven earlier by the Ca ii profiles, the symmetric Hβ fits can be considered as a good proxy in the inversion. This is also confirmed by the close similarity of the observed Hβ reversal depths r and the depths of the symmetric fits. The symmetric fits indicate a decrease of Hβ and Ca ii integrated intensities over the loop apex within a time interval of ≈3 min, but this is too small a statistical sample and too short a time interval for any firm conclusions.

The inversion is performed by exploring a multidimensional space of plasma parameters using several grids of models. Each model is characterized by a unique set of four free parameters (see Section 3.1). A grid of models comprises an extensive set of individual models, usually more than several tens of thousands, whose parameters change by regular increments. The grids differ by widths of parameter ranges, sizes of sampling steps, and number of models. For each model grid a library of synthetic line profiles is computed. These are convolved with the instrumental characteristics in order to simulate the observed profiles. Comparing the library of synthetic profiles I_syn(λ_i) with the observed profiles I_obs(λ_i) identifies a model that yields the best match. A similarity of the profiles, sampled in the N wavelengths λ_i (i = 1 ... N) with the measurement uncertainties of σ(λ_i), is quantified by the χ² merit function weighted by σ(λ_i). The function is computed within the wavelength range sampled by the CHROMIS Hβ observations but only up to 2.25 Å for Ca ii. Its global minima define the optimal model parameters that represent a plausible solution of the problem. In the following we assume, for both the Ca ii and Hβ lines, constant and wavelength-independent value of σ = 2 × 10⁻⁶ erg s⁻¹ cm⁻² sr⁻¹ Hz⁻¹, which converts into 0.8 × 10⁵ and 2.5 × 10⁵ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹ for Ca ii and Hβ, respectively. Double- and single-line inversions are performed separately. For the former, the merit function is evaluated simultaneously for both lines concatenating their I_obs(λ_i) into one vector as well as I_syn(λ_i). For the latter the merit function is evaluated separately for each line. The corresponding results are distinguished by the symbols ‡ and † in Tables 1, 3, and 4. Both I_obs(λ_i) and I_syn(λ_i) are calibrated in units of erg s⁻¹ cm⁻² sr⁻¹ Hz⁻¹ and resampled on a denser wavelength grid with the step of 0.01 Å yielding N(Ca ii) = 226 and N(Hβ) = 121 intensity samples for the χ² computation.

Table 1 compares the integrated intensities E and the central reversal depths r of the observed profiles, their double-Gaussian and symmetric fits, and the synthetic profiles. For the last, the line center optical thicknesses τ₀ are also listed. The reversal depth r is defined as: r = mean(I_b, I_r)/I_min, where I_b and I_r are the intensities of the blue and red peaks, respectively, around the reversal minimum intensity I_min.

3.1. MALI Radiative-transfer Code

3.2. Model Grids and Inversion Process

Figure 6 displays symmetric double-Gaussian fits, as proxies of the observed Ca ii and Hβ line profiles, and their fits corresponding to the model 1 and 2 in Table 3 that minimize the merit function within the fine grids in Table 2 for the photospheric Ca abundance. Both models roughly reproduce the basic features of the observed profiles, such as the central reversal of the Hβ line, and central and wing intensities of the Ca ii line. However, the models fail to capture central reversals of Ca ii, prominent in the observed profiles (Figure 4: top frames). The integrated intensities E of the best-fit synthetic Ca ii profiles and symmetric fits agree well both for the double- and single-line inversions (Table 1). However, the integrated intensities E of the best-fit synthetic Hβ profiles are always larger than the symmetric double-Gaussian fits, while the synthetic Hβ reversals are mostly smaller than the observed reversals. These differences are due to the Gaussian wings of the symmetric Hβ fits (Figure 4: bottom frames), with intensities lower than the wing intensities of the Voigt profiles that represent the synthetic Hβ profiles at high electron densities (Figure 6), due to a significant pressure broadening computed by the MALI code.

Resulting model parameters of the double-line inversion in Table 3 show constancy of the plasma kinetic temperature at 8.7 kK and decrease of gas pressure from 9.5 to 8.7 dyn cm⁻², surface electron density from 0.72 × 10¹² to 0.67 × 10¹² cm⁻³, and core electron density from 2.74 × 10¹² to 2.53 × 10¹² cm⁻³ over the time interval of 3 min. High electron densities are consistent with the results of Jejčič et al. (2018) based on SDO/HMI continuum observations. The geometrical slab thickness and the microturbulent velocity do not show any trend. Reproducing the observed line profiles requires thicknesses of 4.5–5.1 Mm and a microturbulence of 24 km s⁻¹. The line center optical thicknesses τ₀ of the Ca ii and Hβ lines are given in the last column of Table 1 separately for the model 1 and 2 (in parentheses), for double- and single-line inversions marked by ‡ and †, respectively. Because the synthetic Ca ii profiles lack the central reversals (r = 1.00) the optical thicknesses τ₀(Ca ii) are lower limits. Similarly, the optical thicknesses τ₀(Hβ) are lower limits because the synthetic Hβ reversal depths are always smaller than the observed reversals.

However, there is a systematic difference between the resulting model parameters yielded by the single-line inversion of Ca ii and Hβ profiles (compare rows in Table 3 starting with †). For both models, fitting the Ca ii profiles requires higher temperatures of approximately 1 kK, smaller gas pressures, higher microturbulence, about two times higher surface electron densities, and about 0.7 × 10¹² cm⁻³ higher core electron densities than those needed for fitting single Hβ profiles. This suggests that the Ca ii and Hβ lines may originate from different parts of the inhomogeneous loops although similar geometrical thicknesses are required in the single-line inversions. Thus it is likely that the model parameters yielded by the double-line inversion are weighted averages of the parameter distributions along the LoS. Most of the double-line model parameters fall between the values inferred by single-line inversions.

There is some ambiguity in that models 1 and 2 differ in the values of gas pressure and geometrical thickness, yet produce very similar profiles as shown in Figure 6 by the red lines and blue circles and also by similar χ² values in the last column of Table 3. Models 1 and 2 thus represent a range of valid solutions, which only improved future diagnostics can improve.

To examine the effect of an increase in Ca abundance, the inversion is repeated for the fine grids of Table 2, but using synthetic Ca ii profiles computed for the coronal Ca abundance. The resulting best-fit parameters are given in Table 4 for comparison with Table 3. Although the inversions often hit the limits of parameter ranges in Table 2 yielding a temperature of 10 kK, gas pressure of 7 and 10 dyn cm⁻², and slab geometrical thicknesses of 4.5 and 2 Mm, the coronal Ca abundance is still compatible with high electron number densities of about (1–2) × 10¹² cm⁻³.

The uniqueness and precision of the inversion within a given model grid can be checked by exploring the distribution of χ² values over the parameter space. Figure 8 displays three cuts through the χ²(T, p_g, D, v_tur) hypercube out of six possible combinations (T − p_g, T − v_tur, T − D, p_g − v_tur, p_g − D, v_tur − D) for the parameters in the first row of Table 3. The contour maps show the χ² isosurfaces for pairs of complementary parameters (D, v_tur; T, D; etc.) fixed to the values in Table 3. They demonstrate that within fine grids 1 and 2 there is only one optimal solution. The well-localized and isolated minima prove that temperature, microturbulent velocity, gas pressure, and also electron densities are well defined by the inversion process. A comparison of the top and bottom frames of Figure 8 provides an interesting insight into relations between individual parameters. Narrowing the slab from 4.5 Mm (top frames) to 2.2 Mm (bottom frames) invokes an increase of gas pressure from 9.5 to 12.4 dyn cm⁻² and also an increase of the central electron density n_e(c) from about 2.7 × 10¹² to 3.3 × 10¹² cm⁻³. This also holds for other double-line inversion results (Table 3). Considering the fine-thread nature of the flare loops (which apparently consist of threads with cross-sectional widths about 100 km (Jing et al. 2016), possibly small volume filling factor, and natural anti-correlations p_g − D and n_e(c) − D, then the given values of the gas pressure and electron densities are only lower limits.

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Gouttebroze & Heinzel (2002) investigated a diagnostic potential of the ratio of the Ca ii and Hβ integrated intensities E(8542)/E(Hβ) employing an extensive grid of low-pressure models up to 1 dyn cm⁻², characteristic of quiescent prominences. They demonstrated that the ratio is less than one for most of the models (Figure 9 therein) and there exists a statistical correlation between the ratio E(8542)/E(Hβ), the gas pressure p_g, and the temperature T for p_g ranging from ≈0.1 to 1 dyn cm⁻². Inspired by these conclusions, we investigate the possible existence of such a correlation in high-pressure regimes beyond 1 dyn cm⁻² using the fine grids in Table 2.

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The left frame of Figure 9 illustrates the dependence "ratio–p_g–T" for synthetic Ca ii and Hβ profiles computed by the fine grid 1 taking D = 4.5 Mm and v_tur = 24 km s⁻¹, i.e., for results of the double-line inversion at ≈16:25 UT. The right frame shows the ratio E(8542)/E(Hβ) for the fine grid 2 taking D = 2.2 Mm and v_tur = 26 km s⁻¹ (Table 3: first row). The plots show that: (i) the ratio E(8542)/E(Hβ) is constrained within the limits 0.1–0.7, (ii) higher ratios correspond to lower temperatures, and (iii) there is a slight increase of the ratio with the gas pressure at higher temperatures. The dotted lines show the ratios for the parameters of model 1 and 2 (Table 3: first row). The ratio of corresponding symmetric Ca ii and Hβ fits 4.90/10.66 ≈ 0.46 (Table 1) is indicated in Figure 9 by the short horizontal bar. The ratio discrepancy is due to the Gaussian wings of the symmetric Hβ fits (Figure 4: bottom frames) with intensities which are lower than the wing intensities of Voigt profiles representing synthetic Hβ profiles at high electron densities (Figure 6). Obviously, observations of full Hβ profiles, including near and far wings, are needed for better comparison of synthetic and observed E(8542)/E(Hβ) ratios in the high-pressure regime.

We present Ca ii 8542 Å and Hβ imaging spectroscopy of well-developed off-limb flare loops pertinent to the X8.2-class flare acquired about 20 min after the flare peak over an interval of about 3 min. We analyze quasi-simultaneous emission line profiles averaged over small bright patches at the top of the flare loop apex. The Ca ii and Hβ lines show central reversals surrounded with slightly asymmetric peaks. Following the absolute calibration and spatial alignment of the data, extended grids of cool loop models are constructed using the 1D non-LTE code MALI. The grids are then used for the spectral inversions. Double- and single-line inversions are made, treating the Ca ii and Hβ lines together or separately.

Estimates of the kinetic temperature 8.7 kK, gas pressure 9.1 dyn cm⁻², microturbulent velocity 24 km s⁻¹, and electron number density 2.6 × 10¹² cm⁻³ are obtained for a geometrical thickness of 4.5 Mm of the bright region at the top of the flare loop arcade. These median values gained from double-line inversions (Table 3) are representative of the cool component of multithermal flare loops composed of cool and hot strands, as suggest by Hinode's EIS and SDO/Atmospheric Imaging Assembly observations (Doschek et al. 2018; Jejčič et al. 2018; Kuridze et al. 2019). The profile characteristics and resulting model parameters imply relatively high electron density of the order of 10¹² cm⁻³, which is very likely a lower limit estimate due to: (i) significant fine-scale structuring of the flare loops, revealed e.g., by Jing et al. (2016), with presumably small volume filling factor, and (ii) underestimation of reversal depths r by our 1D isothermal–isobaric models. Nevertheless, our inversion-based estimate confirms high electron densities in cool loops as reported in Jejčič et al. (2018) and also in a number of earlier studies (Švestka 1972; Hiei et al. 1983, 1992; Heinzel & Karlicky 1987; Švestka et al. 1987).

Another noteworthy feature of the flare loop plasma is a strong turbulence manifested through a large non-thermal broadening of lines. Here we find that a microturbulent velocity about 24 km s⁻¹ is needed to explain the profiles of Ca ii and Hβ lines. Our values of the microturbulent velocity agree quite well with those obtained by Brannon (2016, see Figure 3(d)) and Mikuła et al. (2017, see Figures 8, 9, and Section 5) at the flare loop apices.

There is a two-orders-of-magnitude difference between electron densities found in this study and in Jejčič et al. (2018) with those inferred in Doschek et al. (2018) using the Hinode's EIS spectra of hot loops. The latter study reports a conservative density of the order of 10¹⁰ cm⁻³ in the hot loops within the same multithermal flare loop system. Remarkably, it also shows a coronal Ca abundance at a cusp height ≈42 Mm, which steadily changes from coronal to photospheric toward the solar limb. Figure 4 therein suggests an intensity ratio Ca xiv/Ar xiv of about 2–3, indicating transformative coronal-to-photospheric Ca abundance for the bright patches at the cool loop apex at a height of ≈17 Mm. Our inversions show that high electron number densities of about (1–2) × 10¹² cm⁻³ are consistent with the upper limit of coronal Ca abundance.

Interpreting the Ca ii and Hβ line profiles (Figures 4 and 6) requires a median optical thickness of 4.5 for the former and ≈70 or ≈44 for the latter, depending on the adopted geometrical thickness. Due to the unsatisfactory fits of central reversals of both lines (i.e., absent or not deep enough) these values are very likely lower limits.

To explain the absence of Ca ii reversals, we constrained the fine grids 1 and 2 with a lower microturbulent velocity of 20 km s⁻¹, and photospheric Ca abundance. The inversion yields the best fits shown in Figure 7 with the Ca ii profiles having central reversals that are very similar to the observed. However, the overall Ca ii fit of the profile wings is less satisfactory than if the whole range of microturbulent velocities is considered (Figure 6). Fitting the Ca ii reversal requires the following best-fit parameters for the fine grid 1 (grid 2): T = 8.6 (8.5) kK, p_g = 9 (11.9) dyn cm⁻², D = 6 (3) Mm, and n_e(c) = 2.5 (3.04) cm⁻³ corresponding to χ² = 4.48 (6.05) and τ₀(Ca ii) = 6.9 (6.4). We interpret these results as a signature of insufficient pressure broadening in the wings of the synthetic Ca ii profiles that is compensated by an increased microturbulence. The latter and the instrumental broadening remove the Ca ii reversal but yield a better overall fit. High electron densities produce enhanced and extended pressure-broadened Hβ profiles (Figures 6 and 7). For Ca ii lines we used the Stark and van der Waals broadening parameters suggested by Shine & Linsky (1974) but this ideally should be based on more modern damping parameters.

Our future study will repeat the density diagnostics of cool flare loops, employing more sophisticated modeling of the whole loops using the 2D non-LTE radiative transfer code MALI2D (Heinzel & Anzer 2001; Heinzel et al. 2005; Gunár et al. 2007). The current study will provide initial estimates of the plasma parameters. This should give a more satisfactory coincidence of the observed and synthetic profiles, a better fit to the central reversals, and a more realistic reproduction of the peak asymmetries of the Ca ii and Hβ line profiles through including bulk motions along the loops. Such results will also improve our understanding of the importance of flare loops on other flaring stars, particularly those that produce superflares (Heinzel & Shibata 2018).

This research has received funding by the Sêr Cymru II, part-funded by the European Regional Development Fund through the Welsh Government. H.M. is partly funded by an STFC consolidated grant to Aberystwyth University. The Swedish 1 m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. The Institute for Solar Physics is supported by a grant for research infrastructures of national importance from the Swedish Research Council (registration number 2017-00625). The work of D.K. was supported by Georgian Shota Rustaveli National Science Foundation project FR17_323. J.K. acknowledges the project VEGA 2/0004/16. This work was also supported by the project ITMS No. 26220120029, based on the operational Research and Development Program financed from the European Regional Development Fund. The authors acknowledge support from the grant 19-17102S of the Czech Funding Agency. P.H. was supported also by the grant 19-09489S. S.J. acknowledges the financial support from the Slovenian Research Agency No. P1-0188. The data were acquired within Spanish SST time allocation. This research has made use of NASA's Astrophysics Data System. The authors wish to thank the anonymous referee who provided constructive remarks that greatly improved the quality of this paper.

Facilities: SST (CRISP - , CHROMIS). -