An Explanation of Remarkable Emission-line Profiles in Post-flare Coronal Rain

The purpose of the present paper is to analyze some curiously broad, redshifted profiles of emission lines obtained with the Interface Region Imaging Spectrograph (IRIS; De Pontieu et al. 2014) a few minutes after the impulsive phase of a flare. The X2.1 flare of 2015 March 11 (SOL2015-03-11T16:22) occurred in the active region (AR) NOAA 12297 and was accompanied by a filament eruption. Detectable hard X-ray emission observed with RHESSI (Lin et al. 2002) began near 16:16 UT, reached a peak at 16:21 UT, and returned to its near-quiescent state near 16:35 UT. The broad, redshifted emission of interest here started abruptly near 16:26 UT. Typical profiles of the Mg ii h and k (at air wavelengths 2802.7 and 2795.5 Å, respectively) lines during this phase are shown in Figures 1 and 2. These profiles appear to differ qualitatively from all earlier theoretical and most observational studies on coronal rain, the emission lines being unusually broad.

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By themselves, these data admit several possibilities of interpretation. Fortuitously, some partly resolved "fine structure," corresponding to plasma flows, with similar spectral properties has been observed in another solar flare (Jing et al. 2016). Their work focused on observations of spectral lines with $0\buildrel{\prime\prime}\over{.} 03$ sampling, through narrowband filters. Although not discussed by them, Figure 7(b) in their paper also shows Mg ii spectral data from IRIS during the post-flare (PF) stage that, without doubt, arise from the same phenomenon we analyze here. Thus, the broad, redshifted emission features of Figures 1 and 2 are coincident with the onset of a return of emitting plasma along PF loops (see Figures 7(c) and (d) of Jing et al. 2016). These phenomena shown by Jing et al. (2016) and in the present work are unquestionably a form of coronal rain.

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Usually seen as plasma emission in chromospheric and transition region lines (partially ionized or neutral plasma), or as absorption in EUV images, coronal rain forms as a consequence of instability in coronal loops. In nonflaring AR loops, thermal instability leads to catastrophic cooling (Cally & Robb 1991). Numerous observations and models have studied coronal rain (Moschou et al. 2015; Kohutova & Verwichte 2016, and references within), finding typical redshifts of ∼80 km s⁻¹. These authors have highlighted multistranded and multithermal structure, but without reference to broad line widths. Plasma condensation after eruptive flares can produce similar phenomena (Song et al. 2016), although the physical processes may differ. Kleint et al. (2014) have found bursty flows associated with bright points at the edge of the sunspot umbra that show a striking similarity to the profiles analyzed here, but lasting only ∼20 s.

Here we study the Mg ii and other lines from IRIS to try to constrain the physical nature of the unusually broad lines associated with raining plasma. The various observed parameters, when judiciously combined with the results of Jing et al. (2016), constrain the nature of the raining plasma and its origins more tightly than has been achieved in the past. The primary lines of interest here are listed in Table 1.

Table 1.  Atomic Transitions of Interest

Ion	λ (Å)	Upper Level	Lower Level	gf	${\rm{\Upsilon }}({T}_{{\rm{e}}})$
Mg ii	2802.705	$3p{}^{2}{{\rm{P}}}_{1/2}^{o}$	$3s{}^{2}{{\rm{S}}}_{1/2}$	0.64	5.6
	2795.528	$3p{}^{2}{{\rm{P}}}_{3/2}^{o}$	$3s{}^{2}{{\rm{S}}}_{1/2}$	1.3	12.3
	2790.777	$3d{}^{2}{{\rm{D}}}_{3/2}$	$3p{}^{2}{{\rm{P}}}_{1/2}^{o}$	1.9	9.1
	2797.930	$3d{}^{2}{{\rm{D}}}_{3/2}$	$3p{}^{2}{{\rm{P}}}_{3/2}^{o}$	0.39	4.8
	2797.998	$3d{}^{2}{{\rm{D}}}_{5/2}$	$3p{}^{2}{{\rm{P}}}_{3/2}^{o}$	3.5	18.3
	⋯	$3d{}^{2}{{\rm{D}}}_{5/2}$	$3p{}^{2}{{\rm{P}}}_{1/2}^{o}$	⋯	2.5
	⋯	$3d{}^{2}{{\rm{D}}}_{3/2}$	$3s{}^{2}{{\rm{S}}}_{1/2}$	⋯	1.2
	⋯	$3d{}^{2}{{\rm{D}}}_{5/2}$	$3s{}^{2}{{\rm{S}}}_{1/2}$	⋯	1.8
C ii	1334.5323	$2{s}^{2}2p{}^{2}{{\rm{D}}}_{3/2}$	$2{s}^{2}2p{}^{2}{{\rm{P}}}_{1/2}^{o}$	0.26	1.4
	1335.6625	$2{s}^{2}2p{}^{2}{{\rm{D}}}_{3/2}$	$2{s}^{2}2p{}^{2}{{\rm{P}}}_{3/2}^{o}$	0.051	0.9
	1335.7077	$2{s}^{2}2p{}^{2}{{\rm{D}}}_{5/2}$	$2{s}^{2}2p{}^{2}{{\rm{P}}}_{3/2}^{o}$	0.46	2.9
	⋯	$2{s}^{2}2p{}^{2}{{\rm{D}}}_{5/2}$	$2{s}^{2}2p{}^{2}{{\rm{P}}}_{1/2}^{o}$	⋯	0.5
Si iv	1402.77	$3p{}^{2}{{\rm{P}}}_{1/2}^{o}$	$3s{}^{2}{{\rm{P}}}_{1/2}$	0.54	7.3

Note. Data are from the NIST spectroscopic database (Kramida et al. 2015). Wavelengths above 2000 Å are in air; otherwise, they are in vacuum. Collisional data are from Sigut & Pradhan (1995), Blum & Pradhan (1992), and Zhang et al. (1990). The Maxwellian-averaged collision strengths ${\rm{\Upsilon }}({T}_{{\rm{e}}})$ are given for ${T}_{{\rm{e}}}={10}^{4}\,{\rm{K}}$ for the singly charged ions, and at $T={10}^{5}\,{\rm{K}}$ for Si iv.

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Being a consequence of the unknown mechanisms by which mass and energy are transported into the solar corona, there are many outstanding questions concerning coronal rain. Here we focus on extracting as much information from the IRIS data to study the origins of the cool material arising from the flare and its remarkable thermal structure. PF chromospheric spectra have been obtained for decades in lines such as Hα. The Mg ii data analyzed here have advantages over Hα. The line emissivities and opacities are more simply related to thermal properties of the emitting plasma (discussed below).

We study five Mg ii lines (two of which are blended); collectively they are sensitive to different thermal conditions. Thus, we can perform a simple quantitative analysis of the data with minimal assumptions. Our work complements much recent work (e.g., Antolin et al. 2015; Jing et al. 2016) that studies broad- or narrowband imaging, by analyzing the behavior of line profiles. These profiles differ so dramatically from those previously modeled using existing numerical methods that we adopt analytical methods. Our work should later inspire further numerical simulations, such as that of Fang et al. (2013), which might attempt to better understand the origin of the enormous line widths spanning over 300 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$, pointing to energetic magnetic waves or (less likely) turbulence as the culprit.

On 2015 March 11, IRIS acquired data of AR NOAA 12297 during a flare watch campaign. Data were downlinked from nine spectral windows. Here we analyze those containing the lines of Mg ii, C ii, and Si iv, together with data obtained by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) and Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) instruments on the Solar Dynamics Observatory spacecraft (SDO; Pesnell et al. 2012). The IRIS slit was moved sequentially and repeatedly to four positions offset by 0'', 2'', 4'', and 6'' in the E–W direction on the solar surface. A total of 1230 such pointings were acquired. We focus on data from exposures 110 to 490. At each slit position, spectral data with exposures of 4 s were acquired. To complete one cycle of four slit positions took $\approx 21$ s. During the flare, the exposure times were decreased for the far-UV (FUV) wavelengths by an automatic flare-triggered mechanism, to prevent overexposure, without affecting the total raster duration. The projected slit is 033 wide. The spatial and spectral samplings were 033/pixel and 0.05092 Å/pixel, respectively, for the near-UV (NUV) region. For the FUV regions the spectral sampling was 0.02544 Å/pixel (FUV2) and 0.02596 Å/pixel (FUV1).

We applied iris_orbitvar_corr_l2s.pro, a routine in the IRIS SolarSoft package, to correct for spacecraft orbital variations. Following the procedure in Liu et al. (2015), we then obtained wavelength-integrated emission-line intensities in physical units (see Table 2), which refer to the obvious components in emission above the photospheric absorption line and continua. For comparison we include the values for a quiet-Sun (QS) patch outside the AR alongside those from a representative position within the peculiar spectrum region at different time instances. These values are the sum over both line core and extended redshifted profile, including multiple lines where this is the case. All velocities are determined relative to the average QS spectrum.

Table 2.  Observed Frequency-integrated Emission-line Intensity (erg cm⁻² s⁻¹ sr⁻¹)

Multiplet Phase	Mg ii h + k	Mg ii $3d-3p$	C ii 1334+1335	Si iv 1403
QS	5.36 × 105	⋯	6070	572
Pre-flare	1.83 × 106	⋯	5.16 × 104	2840
Impulse	4.36 × 107	≈1.36 × 107	1.73 × 107	>3.86 × 106
Relax	2.77 × 107	4.81 × 106	4.39 × 106	3.51 × 105
Post-flare	1.82 × 107	≈4.51 × 105	1.45 × 106	2.04 × 105

Note. QS data are from the darkest regions along the IRIS slit. We used calibration factors returned by iris_get_response.pro: dn2phot_sg = 4.0 (FUV) and 18.0 (NUV), and effective areas of 0.474, 1.042, and 0.23 cm² for the 1335, 1400 and 2800 Å wavelengths, respectively. The intensities in boldface will be used throughout this article.

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IRIS slit-jaw images (SJI) at 1330, 1400, and 2832 Å were also acquired during the raster steps 0, 3, and 1, respectively. Each SJI set had a temporal sampling of 20.76 s on average and spatial sampling of 033/pixel, spanning a total field of view (FOV) of $126^{\prime\prime} \times 119^{\prime\prime}$. Four such images are shown in Figure 3.

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The flare appears to have been triggered by the accelerated rise of a filament starting near 16:13 UT. Flare emissions started at 16:16 UT, the flaring reaching the emission maxima at 16:20 UT. This maximum high-energy flare emission, as observed by RHESSI, was located at (X, Y) = (−354, −171)'', a region not covered by the spectral slit positions.

We examine the evolution of the spectra at different locations along the flare ribbon. We found locations of extended very broad and redshifted emission at the same locations swept over by the evolving and propagating flare ribbon, lasting for more than an hour after the flaring event. To highlight typical Doppler shifts and line widths, in Figure 4 we show the first velocity-weighted moment (Doppler shift) and second moment (line width) of the Mg ii k line, normalized to the zeroth moment.³ The figure shows data only for the first slit position. In passing, we note that the smaller-amplitude blueshifts in the umbra probably correspond to umbral flashes, studied, for example, by Bard & Carlsson (2010).

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To highlight the evolution of the line width over the whole region of interest, we represented the normalized intensity of the profile as a function of the line width in Figure 5, along with an inset of the same dependence for a QS patch outside the AR. The color represents the time of the exposure, with early times being masked by later emission. The QS distribution is centered around 400 mÅ and shows little scatter. The distribution for the peculiar emission region is centered around 500 mÅ before the flare, and it shows an increase in both intensity and line width after the flare. The intensity decreases in time to pre-flare values, while the width still has an extended tail up to about 1100 mÅ, suggesting that the raining phenomenon is still ongoing.

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Figure 6 shows the position of the four IRIS slits projected onto an image constructed from data from the SDO/HMI instrument. The HMI data are the standard reductions of the "720s" product, including vector magnetic field data. The field azimuth in the plane perpendicular to the line of sight (LOS) has not been disambiguated. Note that the four slit positions extend across the entire sunspot centered near (X, Y) = (−340, −182)''. The broadened profiles are found above the penumbra; they are marked as the solid components of the dashed lines shown in the figure. The lengths and positions of these lines correspond to the average locations where the anomalously broad profiles occur. These regions are quite separate from the magnetic neutral line. Instead, they lie above regions of modest LOS field, in a large perpendicular field, whose direction follows parallel or antiparallel to the axis of a "dark intrusion" (Figure 6) centered at (X, Y) = (−343, −176)'', a location where systematic upward-directed photospheric motions of ≈1–1.5 km s⁻¹ are present (but not shown here).

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Intensity profiles of the Mg ii h and k lines are shown in Figure 2. Representative data of interest here are from exposure number 100 in our sample obtained at 16:32:31 UT. In the figure these are shown as dashed lines. The time dependence of these profiles is shown in Figure 1. The IRIS instrument's compensation for the Sun's average rotation was used, so that the images show the changing conditions over approximately the same areas of the Sun.

For convenience, in Figure 1 we identify four episodes in the evolution of the Mg ii: a pre-flare phase (before 16:20 UT), an impulsive phase (16:21 UT), a relaxation phase (16:21 to 16:27 UT), and a PF phase (16:25 UT to beyond 17:00 UT). The PF phase is the subject of the present paper. These phases are self-evident in the data. The relaxation phase is simply the transient response of the line profiles to the sudden release of energy in the impulsive phase, judging by the timescale of the decay, which seems appropriate. The chromosphere has a thickness of $\approx 1500\,\mathrm{km}$, and pressure perturbations (shocks) will propagate a little above the sound speed of ≈7–10 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$. A few sound-crossing times corresponds to about 5 minutes. The relaxation phase is most clearly seen in the narrow $3d-3p$ lines of Mg ii, close to 2791 and 2798 Å in this figure. The h and k resonance lines of Mg ii are at 2796.3 and 2803.5 Å. Profiles of C ii resonance lines near 1335 Å show similar spectral line shapes to h and k, as does the 1403 Å line of Si iv. Both lines are slightly broader (in Doppler units) than the h and k lines. However, the durations of the broad PF profiles in both cases are appreciably shorter. The intensities of these lines are also 10–100× smaller than h and k (Table 2).

3.1. Summary of Observed Properties

3.2. A Reference Model

We examine the PF profiles using parameters in a reference model of the emitting plasma. We will assume that the plasma exists in strands with a width $W\approx 100\,\mathrm{km}$ (Jing et al. 2016), that it has number densities of hydrogen nuclei n_H close to 10¹² particles per cm³, and that these particles are all at temperatures of $\approx {10}^{4}\,{\rm{K}}$. If T_e exceeds (2−3) × 10⁴K, Mg becomes doubly ionized. The justification for the adopted density is weak. It exceeds that at the very top of the pre-flare chromosphere by an order of magnitude, but flare models propel mass from deeper layers of the chromosphere higher into the corona, on timescales of a minute or less for M- and X-class events (e.g., Abbett & Hawley 1999; Allred et al. 2005). In an X-class flare (with $\approx {10}^{11}$ erg cm⁻² s⁻¹ of energy flux directed toward the chromosphere), the mass per unit of ejected material is expected to be about $m={10}^{-2}{F}_{11}$ g cm⁻², estimated from the last panels of Figures 3 and 5 of Allred et al. (2005). Here F₁₁ is the downward-directed energy flux in units of 10¹¹ erg cm⁻² s⁻¹. If this mass is spread uniformly along a tube of constant area with length L, then the number density on average is $\approx m/\mu {m}_{{\rm{H}}}L$, where $\mu =1.36$ is the mean atomic weight of the largely neutral pre-flare plasma and m_H is the mass of the hydrogen atom. With $L={10}^{9}{L}_{10}$ cm and with L₁₀ corresponding to 10 Mm, we have

$$\begin{eqnarray}&&{n}_{{\rm{H}}}\approx 5\,\times \,{10}^{12}{L}_{10}^{-1}{F}_{11}\ \ {\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 1 }$$

With ${L}_{10}\approx 5$, corresponding to one coronal pressure scale height, and assuming that the material is ejected along flux tubes of this length or longer, this provides a crude justification for our adopted estimate of ${n}_{{\rm{H}}}={10}^{12}$ cm⁻³ for the raining plasma after this X2 class flare. We use variables normalized to the average values: $W={10}^{7}{W}_{100}$ cm, $n={10}^{12}{n}_{12}$ cm⁻³, and $T={10}^{4}{T}_{{\rm{e}}4}\,{\rm{K}}$. We will also assume that the unresolved "microturbulent" velocities are of order $\xi =10{\xi }_{10}\,\,\mathrm{km}\,{{\rm{s}}}^{-1}$, with ${\xi }_{10}=1$, which is of order the sound speed at ${T}_{4}=1$. In this reference model, ${W}_{100}={n}_{12}={T}_{{\rm{e}}4}={\xi }_{10}$=1. Note that the observed line widths are $w=\xi =100\,\,\mathrm{km}\,{{\rm{s}}}^{-1}$, 10 times broader than the reference width, for reasons to be discussed below.⁴

The model is used below to explore conditions under which these lines might form, understanding that the chosen values are educated guesses.

3.3. Radiative Transfer

We consider the formation of the Mg ii lines in a structure that consists of an array of strands that constitute an episode of coronal rain. Thus, we look first at the spectrum emitted by one elemental strand of width ${W}_{100}=1$.

3.3.1. Strand Optical Depths

Two observations imply that the optical depth of the k line across the emitting stands is ≳1, at line center. First, the emission is self-excited within the raining plasma itself, i.e., there is no identifiable dominant external source of irradiation. Second, the k/h line ratios are far from the optically thin ratio of 2:1 (Figure 2).

With an absorption oscillator strength f, the transition between levels 1 and 2 has an opacity at line center (in units of cm⁻¹) of (e.g., Mihalas 1978)

$$\begin{eqnarray}&&{\kappa }_{0}=\displaystyle \frac{\pi {e}^{2}}{{m}_{e}c}{n}_{1}f/{\rm{\Delta }}\nu =0.0264\ {n}_{1}f/{\rm{\Delta }}\nu ,\end{eqnarray} \tag{ 2 }$$

where m_e is the electron mass, e is the elementary charge, n₁ is the number density of Mg ii ions in the lower level, and ${\rm{\Delta }}\nu$ is the Doppler width of the line in Hz. Then, the optical depth across a strand of width W whose axis has an angle ϑ with the LOS is given by

$$\begin{eqnarray}&&{\tau }_{0}\,=\,0.0264\ \displaystyle \frac{{n}_{1}{fW}}{\mu \,{\rm{\Delta }}\nu },\ \mu =\cos \vartheta .\end{eqnarray} \tag{ 3 }$$

Let us assume that most Mg atoms in the strands are in the ground state of Mg ii; then, with a logarithmic abundance of Mg of 7.42 relative to hydrogen (Allen 1973), we have ${n}_{1}={n}_{{\rm{H}}}\times {10}^{7.42-12}=3\times {10}^{6}{n}_{12}$ cm⁻³. With a Doppler width ξ of 10 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$, ${\rm{\Delta }}\nu \approx 3.6\times {10}^{10}{\xi }_{10}$ Hz, and for k, f = 0.6, then

$$\begin{eqnarray}&&{\kappa }_{0}\,=\,1.1\,\times \,{10}^{-5}{n}_{12}/{\xi }_{10}\ {\mathrm{cm}}^{-1}.\end{eqnarray} \tag{ 4 }$$

With ${W}_{100}=W/100\,\mathrm{km}$ we have

$$\begin{eqnarray}&&{\tau }_{0}\,=\,110\displaystyle \frac{{n}_{12}}{{\xi }_{10}}\displaystyle \frac{{W}_{100}}{\mu }.\end{eqnarray} \tag{ 5 }$$

In this reference model the h and k lines have ${\tau }_{0}\approx {10}^{2}$ across the strands.

The observation that the k:h ratio in the PF profile is 1.15, far from the thin ratio of 2:1, has two possible explanations. The first is that the lines might be effectively thick, a condition that is true for the Mg ii h and k lines formed in the stratified chromosphere (e.g., Linsky & Ayres 1978), causing the Sun's k/h ratio from various nonflaring regions on the Sun to be close to 1.1–1.5 (e.g., Kerr et al. 2015). The second possibility is that the radiation emerging from the strands is anisotropic, and that the h and k lines have different anisotropy under conditions of modest optical depth.

The latter case seems far more likely, as can be seen by comparing the escape probability of line photons from the strand core with the destruction probability. In the Appendix we show that the absorption of Mg ii photons by background continuum is negligible in the reference model. The probability of destruction by collisions from the k line can be evaluated simply as

$$\begin{eqnarray}&&\varepsilon =\displaystyle \frac{{C}_{31}}{{A}_{31}+{C}_{31}},\end{eqnarray} \tag{ 6 }$$

where C₃₁ is the collision rate from level 3 to level 1, the dominant collision rate out of the line's upper level (level 3), and A₃₁ is the Einstein coefficient for spontaneous radiative emission (Allen 1973). Using parameters from Table 1, we find ${A}_{31}=2.7\times {10}^{8}$ s⁻¹ and ${C}_{31}\approx 4\times {10}^{4}{n}_{{\rm{e}}12}{T}_{{\rm{e}}4}^{-1/2}$ s⁻¹. Here we must use the electron density ${n}_{{\rm{e}}12}$ because electrons are responsible for the bulk of the collisions:

$$\begin{eqnarray}&&\varepsilon \approx \displaystyle \frac{{C}_{31}}{{A}_{31}+{C}_{31}}\approx {10}^{-4}{n}_{{\rm{e}}12}/\sqrt{{T}_{{\rm{e}}4}}.\end{eqnarray} \tag{ 7 }$$

The electron density of raining material is determined from the hydrogen nuclei density n_H primarily through the electron temperature T_e when ${T}_{{\rm{e}}}\gtrsim {10}^{4}\,{\rm{K}}$, as we will find below. Then ${n}_{e}\approx {n}_{{\rm{H}}}$. A reasonable approximation for P_esc for resonance lines is

$$\begin{eqnarray}&&{P}_{\mathrm{esc}}\approx \displaystyle \frac{1-\exp (-{\tau }_{0})}{{\tau }_{0}}\end{eqnarray} \tag{ 8 }$$

(e.g., Frisch & Transfer 1984), so that in the reference model for the k line, using Equation (5), then

$$\begin{eqnarray}&&{P}_{\mathrm{esc}}/\varepsilon \approx \displaystyle \frac{{10}^{2}{\xi }_{10}\mu \sqrt{{T}_{{\rm{e}}4}}}{{n}_{{\rm{e}}12}{n}_{12}{W}_{100}}.\end{eqnarray} \tag{ 9 }$$

It seems that ${P}_{\mathrm{esc}}\gg \varepsilon$ for reasonable physical parameters.

Together with the observation that the broad profiles are not self-reversed, we conclude that the lines are effectively thin in the coronal rain.

3.3.2. k:h Ratio and Radiation Anisotropy

How can radiation anisotropy explain the line ratios in the rain phenomenon? The lines of h and k differ in opacity by a factor of two. We can imagine a simple case of cylindrical strands where the optical depth of h is $\approx 1/2$ and that of k is $\approx 1$. The h line photons can escape directly from deeper within the strand than the k-line photons. In essence, the h line will radiate more isotropically from the core of the strand, whereas k photons will scatter preferentially into the path of shortest escape, i.e., perpendicularly to the main axis of the strand. The observed line ratio will change from 2:1 even if the total emission into all directions is in the ratio 2:1. We would expect k:h > 2:1 perpendicular to the strand's main axis of symmetry and k:h < 2:1 in other directions.

If the optical depths are substantially higher (${\tau }_{0}(k)\approx 10$, say), then the differential effect is reduced to the outermost cylinders composing the strand (assuming that it is a cylinder; see Lipartito et al. 2014), the radiation being more isotropic in the opaque deeper core of the strand.

We conclude that we are seeing the strands at a slant angle far from 90° and 0° to the strand's main axis, and that ${\tau }_{0}$ should be $\gt 1$ but less than $\approx 10$ in the k line, in these strands. The value of 110 computed from our reference calculation lies close to the necessary optical depth, when we note that the nearest boundary of the strand for escape is at a distance of at most $W/2$, not W, and, as we will argue below, ${\xi }_{10}$ is an underestimate by a factor of 10; thus, the k-line core photons will see a boundary ≲6 photon mean free paths distant.

Related to this problem is the effect of atomic polarization when scattering dominates the source function ($\varepsilon \ll 1$). The $J=1/2$ upper level of h is not polarizable, and the scattered intensity of the $J=1/2\to J=1/2$ transition (h line) is isotropic. However, the polarizable $J=3/2$ upper level's atomic polarization can change the observed intensity ratios under conditions of anisotropy even by the "diffuse" photons self-generated within the cylinder (Landi Degl'Innocenti & Landolfi 2004). We estimated these effects to be small, on the order of a few percent using calculations similar to those applied to He i lines by Judge et al. (2015).

3.4. Frequency-integrated Intensities

If a line is effectively thin across an emitting strand, the emission is simply the sum of the emission along the LOS. To an accuracy of a factor of two, we can ignore the anisotropy, and we find, for the k line,

$$\begin{eqnarray}&&{I}_{\mathrm{thin}}\approx \displaystyle \frac{h{\nu }_{31}}{4\pi }{n}_{1}{C}_{13}\displaystyle \frac{W}{\mu }\ \mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}{\mathrm{sr}}^{-1},\end{eqnarray} \tag{ 10 }$$

where C₁₃ is the collisional excitation rate (Gabriel & Jordan 1971; Burgess & Tully 1992). Within a factor of 2 or so, we can set $\mu =0.5$, and this expression gives, multiplying by two for the sum of the two lines,

$$\begin{eqnarray}{I}_{\mathrm{thin}} & \approx & \displaystyle \frac{h{\nu }_{31}}{4\pi }{n}_{1}{C}_{13}\displaystyle \frac{W}{\mu }\\ & \approx & 1.4\times {10}^{6}{n}_{12}{n}_{{\rm{e}}12}\exp \left(4.86\left(1-\displaystyle \frac{1}{{T}_{{\rm{e}}4}}\right)\right){W}_{100}\end{eqnarray} \tag{ 11 }$$

for our reference calculation. The observed intensities for both lines are of the order of $1.6\times {10}^{7}$ erg cm⁻² s⁻¹ sr⁻¹. With ${T}_{{\rm{e}}4}=2$ we find ${I}_{\mathrm{thin}}\approx 1.1\times {10}^{7}$ erg cm⁻² s⁻¹ sr⁻¹. We regard this as reasonable agreement with our adopted reference calculation, noting that the calculation depends exponentially on T_e4 and on the product ${n}_{12}{n}_{{\rm{e}}12}$, and allowing for the possibility that there is more than one such strand of emission along each LOS observed by IRIS.

3.5. Doppler Widths and Shifts

3.6. Mg ii h and k and the $3d-3p$ Transitions

Unlike the $3p-3s$ transitions, the $3d-3p$ transitions, when in emission, are certainly not optically thick within the strands. The lower ($3p$) levels have a population of $\lesssim \exp (-4.86/{T}_{{\rm{e}}4})$ times the $3s$ lower level of the h and k lines; thus, the $3d-3p$ optical depths are orders of magnitude smaller than h and k. The observed intensity ratios (the blend of the 2797.930 and 2797.998 Å lines, relative to the 2790.930 Å line) are indeed close to the optically thin ratios of 2:1.

The $3d-3p$ transitions differ from h and k in another essential way. Electron excitation to the $3d$ levels from the $3s$ ground level is optically forbidden. The cross sections are dominated by core penetration of the Mg+ ion by the incoming electrons. There is no long-range interaction. When the $3p$ level populations are far below Boltzmann relative to $3s$, as is the case when the lines are effectively thin (photon escape in h and k reducing the $3p$ populations), collisional excitation of the $3d$ levels occurs predominantly only from the $3s$ level. Then, again ignoring anisotropies in the radiation field,

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{3{\rm{d}}-3{\rm{p}}}}{{I}_{3{\rm{p}}-3{\rm{s}}}}\approx \displaystyle \frac{{C}_{3{\rm{s}}-3{\rm{d}}}}{{C}_{3{\rm{s}}-3{\rm{p}}}},\end{eqnarray} \tag{ 12 }$$

which becomes

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{3{\rm{d}}-3{\rm{p}}}}{{I}_{3{\rm{p}}-3{\rm{s}}}}\approx \displaystyle \frac{{{\rm{\Upsilon }}}_{3{\rm{s}}-3{\rm{d}}}}{{{\rm{\Upsilon }}}_{3{\rm{s}}-3{\rm{p}}}}\exp \left(-\displaystyle \frac{4.86}{{T}_{{\rm{e}}4}}\right)\approx \displaystyle \frac{1}{6}\exp \left(-\displaystyle \frac{4.86}{{T}_{{\rm{e}}4}}\right).\end{eqnarray} \tag{ 13 }$$

Inserting the observed ratio $\approx 1/40$ into Equation (13), we find ${T}_{{\rm{e}}4}\approx 2.5$, or $\approx 2.0$ allowing for the radiation anisotropy in the k line.

We note that the work of Pereira et al. (2015) forces the formation of the $3d-3p$ transitions into the dense, lower chromosphere, through their choice of model atmosphere. In their model, the h and k lines are thermalized where the $3d-3p$ transitions form, and so the $3d$ levels are populated largely through the two allowed transitions $3s-3p$ followed by $3p-3d$.

3.7. Dielectronic Recombination

3.8. Sudden Onset of Post-flare Coronal Rain Emission

Inspection of Figure 1 shows that the PF profiles begin abruptly near 16:26 UT. They show a twofold rise in intensity across all wavelengths that takes between two and three time steps, say, t = 40 s. Fast changes are also seen throughout the PF phase in Figure 1, but with smaller amplitudes.

These data are difficult to reconcile if we assume that the line widths of the emitting plasmas arise from a mixture of LOS velocities on macroscopic scales. At a given slit position, various plasma elements would arrive at different times, as they each follow their own trajectory. Let us suppose that plasma is ejected into the corona from one or both flare footpoints. On their way to crossing the slit, they acquire various LOS velocities as a result of the (unknown) dynamics in the PF tubes of magnetic flux. Depending on the trajectories, if one assumes that the PF line profiles are superpositions of macroscopic flows, then the elements will cross the slit with a broad distribution of arrival times. This is contrary to the observations. The short rise time $t\approx 40$ s across the entire profile implies that, if caused by different arrival times, the plasma elements would have to coordinate themselves such that 50 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ upward-moving plasma would "know" when to cross the slit as well as the 150 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ downward-moving plasma. Simply put, there would have to be a linear relationship between velocity and projected distance d from the slit,

$$\begin{eqnarray}&&v\cos \vartheta =d\cos \vartheta /t.\end{eqnarray} \tag{ 14 }$$

It does not make sense that the Sun would know about the special position of the IRIS slit in such a fashion.

What then is the cause of the twofold rise in intensity across all wavelengths within t = 40 s? We can conceive of two explanations: (1) the rain plasma has small-scale motions (with a magnitude of $\approx 100$ $\,\mathrm{km}\,{{\rm{s}}}^{-1}$) within a coherent large-scale flow, making ${\xi }_{10}\approx 10$, or (2) the plasma is indeed moving at a variety of macroscopic velocities, but there is a special thermodynamic process making the plasma visible on a timescale of 40 s.

The second explanation can be discounted because the history of each plasma element determines whether lines of a certain element will become visible at a certain time. For example, if the raining plasma is unheated and is at densities close to 10¹² cm⁻³, the cooling time is of order 10 s (Anderson & Athay 1989). At 100 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ the plasma can travel $\approx 1\,\mathrm{Mm}$ before it cools significantly. Again, we are faced with the untenable position that the IRIS slit would have to be a "special" place on the Sun for this explanation to hold water, each element evolving along its own trajectory. A change in opacity is discounted because we know of no mechanism to suddenly reveal only the PF broad profiles at 16:26 UT.

Thus, we are left with the first explanation, which is similar to real rain on Earth.

3.9. Origins of the Broad Lines

We have examined all salient features of the peculiar IRIS PF line profiles of Mg ii, C ii, and Si iv, with accompanying slit-jaw and magnetic data. By elimination we are led, remarkably, to a consistent picture in which unresolved Alfvénic motions—waves or turbulence—are generated and gradually decay over a period of more than 1 hr. The estimated time-integrated decay of the magnetic energy is about an order of magnitude larger than the radiation losses from the strong Mg ii lines, suggesting a causal connection between them. The peculiar line ratios and profiles are consistent with strands of plasma of width 100 km, and they require an optical depth across them of between 1 and 10 in the k line.

If we apply the observed intensities and ratios to Equations (13) and (11) to solve for model parameters, we have ${T}_{{\rm{e}}}\approx 2\times {10}^{4}\,{\rm{K}}$ and ${n}_{12}^{2}{W}_{100}\approx 1$.

The total flux radiated during the coronal rain phenomenon seen in h and k is $\approx \pi {{It}}_{r}=5\times {10}^{7}\times 600\approx 3\times {10}^{10}$ erg cm⁻². Here we have used a relaxation lifetime of 10 minutes, appropriate for the first part of the relaxation phase where these line intensities were measured (Figure 1). The total radiation losses from chromospheric plasma (due to losses in calcium, iron, hydrogen, etc.) are between $4\times$ and $10\times$ larger, with larger values occurring at lower plasma temperatures (Anderson & Athay 1989, Figure 4). We adopt a value of $4\times$ the h and k losses, to arrive at a total time-integrated radiative flux of 10¹¹ erg cm⁻².

Above, we found $\rho {\xi }^{2}L\approx 2.3\times {10}^{11}{n}_{12}{L}_{10}$ erg cm⁻². Remarkably, these rough estimates are within a factor of 2 of one another. Perhaps more noteworthy, Figure 1 shows that as the broad-line intensities decrease, so does the power in the fluctuations (line widths). It appears that the Alfvénic fluctuations excited during the impulsive phase might decay ultimately into radiation via a process that slowly (compared with dynamical timescales $\approx L/{V}_{A}$) converts the wave into thermal energy.

Various dynamical mechanisms might achieve this on timescales of an hour, such as (slow) phase mixing or resonant absorption (e.g., Narain & Ulmschneider 1990, 1996), although detailed fully dynamical numerical simulations need to be performed to address the very long damping times suggested by Cargill et al. (2016). Gas-kinetic processes can also efficiently damp oscillations when neutral H or He is abundant in the raining plasma. Ion–neutral collisions will efficiently damp out high-frequency wave energy. The damping time ${\tau }_{D}$ for these kinetic processes can be computed following, for example, Holzer et al. (1983). Consider, simply for reference, an Alfvén wave with a period of 10 s, which has $\omega \approx 0.6$ rad s⁻¹, oscillating in the plasma with ${n}_{12}\approx 1$ and the ambient neutral density in units of 10¹² particles cm⁻³ ${n}_{{\rm{n}}12}\approx 0.1$, consisting entirely of neutral helium. The ion–neutral helium collision time, ${\tau }_{\mathrm{ni}}$, is $\approx {10}^{-3}{n}_{{\rm{n}}12}^{-1}\approx {10}^{-2}$ s⁻¹, which gives $\omega {\tau }_{\mathrm{ni}}\approx 6\times {10}^{-3}$. Under this regime, Equation (17) of Holzer et al. (1983) applies, yielding a wave energy dissipation time of

$$\begin{eqnarray}&&{\tau }_{{\rm{D}}}\approx \displaystyle \frac{{n}_{12}}{{n}_{{\rm{n}}12}}\displaystyle \frac{1}{{\tau }_{\mathrm{ni}}{\omega }^{2}}\approx 2500\,{\rm{s}}.\end{eqnarray} \tag{ 15 }$$

The observed decay time of $\approx {10}^{3}$ s may or may not be naturally explained without needing to invoke dynamical MHD processes, depending on the (unknown) frequency ω of the oscillations.

Changing the initial assumption for the hydrogen number density in the reference model to a lower value, for example, to ${n}_{{\rm{H}}}={10}^{11}$ particles per cm³, does not change the conclusions of our analysis. Although we obtain a lower temperature and lower energy requirements, the consequences of the density change tend to compensate each other. The greatest difference is in the required Alfvén wave period compatible with observations, which, for the lower density, can be of 3 minutes compared to the 10 s estimated for the higher density.

Lastly, the IRIS profiles resemble those seen in flares of very active stars (Linsky et al. 1989). It is likely that our solar analysis can help better understand such enormous flares.

IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre.

For absorption in the background continuum with opacity ${\kappa }_{C}$, effectively thick conditions prevail when

$$\begin{eqnarray}&&{P}_{\mathrm{esc}}\lt {\kappa }_{C}/{\kappa }_{0}.\end{eqnarray} \tag{ 16 }$$

At 2800 Å, the dominant source of continuous opacity within the strands is probably the Balmer continuum of hydrogen. This can be computed approximately assuming that the n = 2 levels of hydrogen are populated not too far from LTE relative to the proton density n_p, because the radiation temperature in the Balmer continuum ($\approx 6000\,{\rm{K}}$) and electron temperature are within a factor of two or so. Then

$$\begin{eqnarray}&&{\kappa }_{C}={n}_{2}\sigma \approx \displaystyle \frac{{n}_{2}^{* }}{{n}_{{\rm{p}}}^{* }}{n}_{{\rm{p}}}\sigma .\end{eqnarray} \tag{ 17 }$$

Let n_e be the electron density in cm⁻³ and T_e be the electron temperature in K. With (Mihalas 1978)

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{2}^{* }}{{n}_{{\rm{p}}}^{* }}=2.07\,\times \,{10}^{-16}{n}_{{\rm{e}}}\displaystyle \frac{{g}_{2}}{{g}_{{\rm{p}}}}{T}_{{\rm{e}}}^{-3/2}\exp \left(\displaystyle \frac{{I}_{2}}{{{kT}}_{{\rm{e}}}}\right),\end{eqnarray} \tag{ 18 }$$

using ${I}_{2}=3.399\,\mathrm{eV}$, and using scaled values ${n}_{{\rm{e}}12}\,={n}_{{\rm{e}}}/{10}^{12}\,{\mathrm{cm}}^{-3}$, ${T}_{{\rm{e}}4}={T}_{{\rm{e}}}/{10}^{4}\,{\rm{K}}$, and with $\sigma \approx {10}^{-17}$ cm² (Allen 1973), then

$$\begin{eqnarray}&&{\kappa }_{C}\approx 1.7\times {10}^{-14}{n}_{{\rm{e}}12}{T}_{{\rm{e}}4}^{-3/2}\exp (3.945/{T}_{{\rm{e}}4}){n}_{{\rm{p}}12}.\end{eqnarray} \tag{ 19 }$$

Then we find, further assuming ${n}_{{\rm{p}}}\approx {n}_{{\rm{e}}}$ and dropping subscripts p, e for convenience,

$$\begin{eqnarray}&&{\kappa }_{C}/{\kappa }_{0}\approx 2.5\times {10}^{-9}{n}_{12}\exp (3.945/{T}_{4}){\xi }_{10}/{T}_{4}^{3/2}.\end{eqnarray} \tag{ 20 }$$

Given Equation  (5), this ratio would need to be $\approx 0.01$ for the line to be effectively thick due to background continuum absorption. If we set extreme conditions (photosphere-like) of ${n}_{12}\approx {10}^{1}$ and ${T}_{4}\approx 0.5$, we can obtain a ratio of 7 × 10⁻⁵.

Therefore, we can safely ignore continuum absorption at 2800 Å in the coronal rain.