IRIS Mg ii Observations and Non-LTE Modeling of Off-limb Spicules in a Solar Polar Coronal Hole

We used the data of a coronal hole in the southern polar region observed by IRIS (see Figure 1). The observation was conducted from 12:29 UT to 13:29 UT on 2016 February 21, in the medium sit-and-stare mode (ObsID: 3600257402). The slit was in the north–south direction. The width of the IRIS slit is 033. In the observation that we use in the present study, the length of the slit was 60'', the cadence of the spectral data was 5.4 s, the spatial pixel size along the slit was 017, and the spectral pixel size was 25.6 mÅ. In this work, we focused on the Mg ii h and k spectra. In Figure 1(b), we show the IRIS slit-jaw image taken in the 2796 Å passband with a field of view (FOV) of 60'' × 65'' and a pixel size of 017. Note that the view in Figure 1(b) is rotated by 180°.

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To process the IRIS Mg ii h and k data, we used the spatial and wavelength information in the header of the IRIS level-2 data and derived the rest wavelengths of the Mg ii k and h as 2796.35 and 2803.52 Å from the reversal positions of the averaged spectra at the disk. For our analysis, we rotated the spectral data by 180° from the original north–south direction. Therefore, in all spectra shown hereafter, south is up and north is down.

In Figure 1, we show the overview of the observed region. From panel (a), it is clear that the southern polar region was a coronal hole at the time of the observations. This panel shows a 193 Å image obtained by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al. 2012). The limb location from the AIA header information is indicated by a dashed curve in both panels of Figure 1. The FOV of the IRIS slit-jaw image in the 2796 Å passband (Figure 1(b)) is indicated by a small square in Figure 1(a).

In this section, we present the results of our analysis of the entire data set of the Mg ii h and k observations described in the previous subsection. This data set contains 660 exposures, and, in an average exposure, spicules are covered by ≈70 pixels along the slit. An example of the Mg ii k spectrum is shown in Figure 2. The studied Mg ii line profiles show a complex behavior at different heights above the limb. The line profiles at lower altitudes are broad and generally double peaked (Figure 2(b)), profiles at middle altitudes are rather flat topped (Figure 2(c)), and those at upper parts of spicules tend to be narrower and single peaked (Figures 2(d), (e)). This complex behavior can be seen also in the online movie showing the entire observed time series. In our analysis, we first converted the data count rate (DN s ${}^{-1}$ pix ${}^{-1}$) to radiance (erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹) using the effective area of the near-UV (NUV) passband obtained from the IDL function "iris_get_response.pro." Note that we adopted the spatial information in the header of the IRIS level-2 data and the SDO AIA 193 Å data for the spatial alignment between IRIS and AIA data. We adopted the solar radius in the AIA header information to all data and defined the position of the solar radius on the IRIS slit as the altitude Y = 0''.

From the observed spectra in each exposure and at all pixels covering spicules, we extracted quantitative information about the integrated intensity, the shift of the line profile with respect to the rest wavelength, and the line width in the following way. We measured the integrated intensity in both h and k lines (${E}_{{\rm{h}}}$ and ${E}_{{\rm{k}}}$). In addition, we calculated the ratio of the integrated intensities in the h and k lines (${R}_{\mathrm{kh}}$ = ${E}_{{\rm{k}}}$/${E}_{{\rm{h}}}$). The measured integrated intensities in the Mg ii k line and the line ratios are shown in Figures 3(a) and (b). The black color means that the integrated intensity is too small to show. The intensity ${E}_{{\rm{k}}}$ is larger at the lower altitudes and smaller at the higher altitudes. The upper parts of tall spicules especially have small values of integrated intensity (e.g., the tall feature around time 1750 s). The line ratio shows a large value only for the highest part of spicules. Note that data around Y = 85–90 are inadequate (probably due to some dust on the slit). The distribution of these values as a function of altitude is shown in Figures 4(a) and (b). In this figure, we show the detection counts at each altitude bin normalized at each altitude bin. Along the altitude we use a bin size of 033, which is twice the size of the original binning size of observational data. The red line shows the median and the blue line the mean of the plotted values at each altitude with an averaging bin size of 1''. The median and mean are not shown for Y = 8'' and 9'' in Figures 4(a) and (b) and for Y = 11'' in Figure 4(b) owing to the inadequate data. The median and mean of the integrated intensities clearly decrease with altitude, from about 10⁵ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹ at Y ∼ 2'' to about 1.5 × 10⁴ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹ at Y ∼ 15''. The line ratio is nearly constant (around 1.3) below Y = 7'' but increases to values of up to 2 or more above Y = 7''.

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To measure the shift of the line profiles, we used the 50% bisector method. In the bisector method, we defined λ_blue and λ_red as the two wavelengths at the 50% maximum intensity level and then defined the line shift as ${\lambda }_{\mathrm{bis}}=({\lambda }_{\mathrm{blue}}+{\lambda }_{\mathrm{red}})/2$. The corresponding LOS velocity was defined as ${V}_{\mathrm{LOS}}=c\ ({\lambda }_{\mathrm{bis}}-{\lambda }_{0})/{\lambda }_{0}$, where c is the speed of light and λ₀ is the rest wavelength. The derived line shifts in the Mg ii k line ${V}_{\mathrm{LOS},{\rm{k}}}$ are shown in Figure 3(c). The most important result of this analysis is that the derived LOS velocities are significantly smaller at lower altitudes of the observed spicules than at higher ones. In addition, the derived LOS velocities at the upper part of the observed spicules do not have distinct blue–red asymmetry. To better compare the amplitude of the derived LOS velocities, we show in Figure 4(c) the distribution of the unsigned values of the derived LOS velocities ∣${V}_{\mathrm{LOS},{\rm{k}}}$∣ as a function of altitude Y. The displayed plots correspond to the LOS velocities shown in the time–distance plots in Figure 3(c). The mean and median of the measured unsigned LOS velocities can be more than 10 km s⁻¹ at the higher altitudes, while they are typically around 2 km s⁻¹ at the lower altitudes. Figure 4(c) also shows that the absolute values of LOS velocities can reach more than 20 km s⁻¹ at the higher altitudes, which is consistent with the velocity amplitude of transverse motions in spicules reported in De Pontieu et al. (2007). This suggests that we observe a few or even individual spicules within the pixel in the higher altitudes.

In addition to the analysis of line shifts, we studied here the line widths using the 50% bisector method. We defined the line width as ${\rm{\Delta }}{\lambda }_{\mathrm{bis}}=({\lambda }_{\mathrm{red}}-{\lambda }_{\mathrm{blue}})/2$ in the wavelength units and $W=c\times {\rm{\Delta }}{\lambda }_{\mathrm{bis}}/{\lambda }_{0}$ in the velocity units. The results of the analysis of the measured line widths in the Mg ii k line ${W}_{{\rm{k}}}$ are shown in the time–distance plot in Figure 3(d). Figure 3(d) shows clear stratification of the measured line widths with the altitude. These results are clearly visible also in the distribution plots of the measured line widths as a function of the altitude, which we show in Figure 4(d). The line widths at lower altitudes are significantly greater (e.g., ∼100 km s⁻¹; ∼0.9 Å at Y ∼ 2'') than those measured at the upper part of spicules (e.g., ∼30 km s⁻¹; ∼0.3 Å at Y ∼ 15''). It is also important to note that ${W}_{{\rm{k}}}$ increased in the range Y < 3'' and then decreased in the range Y > 3'' toward higher altitudes, respectively.

Note that in Figure 3 we can clearly identify a very tall individual spicule in the upper part of the observed spectra at $\sim 1750\,{\rm{s}}$ after the beginning of the observations (12:29:07 UT), and this feature is the main contributor to the dense distribution in the upper altitudes in Figure 4. Such a spicule would be easy to identify in high-resolution imaging observations.

The 1D non-LTE radiative transfer code MALI (Heinzel et al. 2014) was used here to synthesize the Mg ii h and k line profiles from a single-slab model. The adopted 1D isothermal and isobaric slab stands vertically on the solar surface and has a finite geometrical thickness along a horizontal LOS (see Figure 5(a)). The slab is illuminated at both sides from the solar disk. Under the spicule conditions, this incident radiation plays a crucial role in determining the line-source functions through the scattering. For the Mg ii incident radiation coming from the solar disk, we adopt the quiet-Sun observation from the balloon experiment RASOLBA (Staath & Lemaire 1995), which is similar to that obtained by IRIS (see Figure 4 in Liu et al. 2015). We adopted results of this quiet-Sun observation as a model of the illumination from the disk, even though the spicule observations used here are from the polar coronal hole. We acknowledge that this assumption could cause some differences in the optical thicknesses, in the source functions, and consequently in the specific intensities. However, such differences could be expected to be in the few tens of percent or less. This would not affect the overall results or conclusions of this paper. We also adopt center-to-limb variations of the disk radiation. The slab illumination varies with height above the surface owing to the dilution factor. In the present study, we used a fixed height above the surface, H = 5000 km, which corresponds to half of the typical maximum height of spicules. However, the resulting synthetic profiles produced by the model do not depend significantly on the adopted height within the range of typical spicule altitudes. The MALI code first solves the non-LTE problem for a five-level plus continuum hydrogen atom and then for a five-level plus continuum atom for Mg ii. Partial frequency redistribution (PRD) is considered for all hydrogen and magnesium resonance lines. Input parameters in the model are the temperature T, gas pressure P, geometrical thickness D, microturbulent velocity ${V}_{\mathrm{turb}}$, and radial (vertical) velocity ${V}_{\mathrm{radial}}$. ${V}_{\mathrm{radial}}$ is important for calculations of the incident radiation owing to the Doppler dimming/brightening effects (see Heinzel et al. 2014). By assuming specific values of these five input parameters, we obtain the ionization equilibrium of hydrogen and thus the electron densities, which are then used to solve the non-LTE problem for magnesium. Finally, we get the synthetic intensities of Mg ii lines together with their optical thicknesses. We note that a 1D vertical slab approximates rather well the line-source function of a cylindrical spicule. The difference in the line-source function between the 1D slab and 1D axially symmetric cylinder roughly amounts to 20%, as demonstrated by Heasley (1977). In the present work, we thus use the 1D slab approximation, which is computationally less demanding.

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The optical thickness of a single slab as a function of wavelength τ(Δλ) is calculated by considering thermal and nonthermal broadening and damping parameters of the natural, Stark, and Van der Waals broadenings. The mean thermal velocity V_th that corresponds to the slab temperature T and the (nonthermal) microturbulent velocity ${V}_{\mathrm{turb}}$ of the slab were adopted to calculate the total Doppler width in frequency units ${\rm{\Delta }}{\nu }_{D}=({\nu }_{0}/c)\sqrt{(}{V}_{\mathrm{th}}^{2}+{V}_{\mathrm{turb}}^{2})$, where ν₀ is the rest frequency and c is the light speed. The employed natural broadening parameter (Einstein A-coefficient) Γ_n was 2.55 × 10⁸ s⁻¹ and 2.56 × 10⁸ s⁻¹ for the h and k lines, respectively. The Stark broadening was calculated as ${{\rm{\Gamma }}}_{s}=4.8\times {10}^{-7}\times \,({n}_{e}$[cm⁻³]) s⁻¹ for both the h and k lines. The Van der Waals damping is ${{\rm{\Gamma }}}_{{vw}}=6.6\,\times \,{10}^{-11}\times {(T[{\rm{K}}])}^{0.3}\times {n}_{{\rm{H}}{\rm{I}}}$, where n_{H i} is the density of neutral hydrogen atoms (Milkey & Mihalas 1974). Note that in these off-limb structures having rather low densities the main broadening is the Doppler one with a dominant microturbulent component in the Mg ii lines. Then, the total damping parameter is defined as ${\rm{\Gamma }}={{\rm{\Gamma }}}_{n}+{{\rm{\Gamma }}}_{s}+{{\rm{\Gamma }}}_{{vw}}$, and the Voigt function takes the form

$$\begin{eqnarray}&&H(a,x)=\displaystyle \frac{a}{\pi }{\int }_{-\infty }^{\infty }\displaystyle \frac{{e}^{-{y}^{2}}}{{\left(x-y\right)}^{2}+{a}^{2}}{dy}.\end{eqnarray} \tag{ 1 }$$

Here, a is the damping parameter given as $a={\rm{\Gamma }}/(4\pi {\rm{\Delta }}{\nu }_{D})$, x is the dimensionless frequency offset given as $x=(\nu -{\nu }_{0})/{\rm{\Delta }}{\nu }_{D}$, and y is defined as $y={\rm{\Delta }}\nu /{\rm{\Delta }}{\nu }_{D}$. The optical thickness as a function of wavelength can be expressed as

$$\begin{eqnarray}&&\tilde{\tau }(x)={\tau }_{0}\displaystyle \frac{H(a,x)}{H(a,0)},\end{eqnarray} \tag{ 2 }$$

or

$$\begin{eqnarray}&&{\left.\tau ({\rm{\Delta }}\nu )=\tilde{\tau }(x)\right|}_{x=x({\rm{\Delta }}\nu )}.\end{eqnarray} \tag{ 3 }$$

The synthesized specific intensity (radiance) for the single-slab model (I_s) was computed as

$$\begin{eqnarray}&&{I}_{{\rm{s}}}={\int }_{{t}_{\nu }=0}^{{t}_{\nu }={\tau }_{\nu }}{S}_{\nu }({t}_{\nu }){e}^{-{t}_{\nu }}{{dt}}_{\nu },\end{eqnarray} \tag{ 4 }$$

where S_ν is the line-source function and τ_ν is the optical thickness of the slab.

Figure 6(a) shows an example of the synthetic Mg ii k line profile from a single-slab model, with P = 0.1 dyn cm⁻², T = 10⁴ K, D = 250 km, ${V}_{\mathrm{turb}}$ = 10 km s⁻¹, ${V}_{\mathrm{radial}}$ = 0 km s⁻¹, which is a set of typical parameters of spicules. With these parameters, the electron density at the center of the slab is about 2.7 × 10 ${}^{10}$ cm${}^{-3}$, the optical thickness at the line center is about 8.8, and the Mg ii k line profile shows a little reversal, the peak intensity level of ∼1.0 × 10⁵ erg s⁻¹ cm⁻² sr⁻¹ Å${}^{-1}$, and the line width measured by the 50% bisector is 34 km s⁻¹ (0.32 Å). This line width is partially due to the opacity broadening and enough to explain the width of the observed profile at the upper part of the spicules—for more details see discussion in Section 5.1. Note that in Figure 6(a) we also show the illumination intensity profile at H = 5000 km used here.

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Our choice of the typical geometrical width of spicules D = 250 km is based on the work by Pereira et al. (2012). These authors obtained the mean diameter of quiet-Sun and coronal hole spicules in Ca ii H images as 250 and 340 km, respectively, using space-borne observations by Hinode/SOT. Although in the models used in the present work we compute the ionization equilibrium of hydrogen to obtain the electron density used for non-LTE modeling of magnesium, we may assume that the observable geometrical width of spicules in Ca ii H is similar to that in the Hα line. This is because the formation temperature of Hα is similar to the formation temperature of Ca ii H and the optical thickness of the Hα line is also similar to the optical thickness of the Ca ii H line. In fact, Pereira et al. (2013) show in their Figure 3 that both the structure of spicules and their time evolution are very similar in Hα and Ca ii H observed by Hinode/SOT. However, we note that Pasachoff et al. (2009) measured quiet-Sun spicule widths of around 660 km, using ground-based Hα observations. These observations were performed in a different region and at a different time from those of Pereira et al. (2012). Also, the Hα observations by Pasachoff et al. (2009) were obtained using a narrowband filter, while the Ca ii H observations by Pereira et al. (2012) were obtained using a broadband filter. The difference between the filters may result in different apparent widths of the observed spicules. Moreover, the seeing-affected ground-based observations might result in a smearing of the observed fine structures and thus larger perceived widths of these structures. Therefore, we adopt the results of Pereira et al. (2012) in the present work. Additionally, the conclusions of the present paper would not be significantly affected, even if we assumed a large width (e.g., D = 500 km) as the typical width of spicules.

In this model, we experimented with placing additional static slabs along the LOS to study how the synthetic Mg ii profiles change when more than one slab is assumed. The multi-slab model here consists of a set of N identical 1D slabs along the LOS (Figure 5(b)). In this subsection, we assume static slabs to first investigate the superposition effect, although we acknowledge that this situation is not realistic (for a more realistic case, see Section 3.3). Using  the synthetic intensity of the single-slab model I_s (Δλ) described in the previous section, the intensity of the multi-slab model without LOS velocities I_m (Δλ) is obtained as

$$\begin{eqnarray}&&{I}_{{\rm{m}}}({\rm{\Delta }}\lambda )={I}_{{\rm{s}}}({\rm{\Delta }}\lambda )\displaystyle \sum _{i=1}^{N}\exp \{-(i-1)\tau ({\rm{\Delta }}\lambda )\},\end{eqnarray} \tag{ 5 }$$

where the intensity from the ith slab is attenuated by the slabs that stand in front of it.

Figure 6(b) shows an example of the synthetic Mg ii k line profile obtained by multi-slab modeling using a set of N identical slabs. Each slab assumes the same set of input parameters as the single-slab model described in Section 3.1. The number of slabs varies from 1 to 10, and the result with N = 1 is the same as shown in Figure 6(a). As we add additional slabs, the resulting intensity profiles get enhanced at the wavelengths in which the total optical thickness is less than unity, i.e., we can see more spicules along the LOS in the line wings. The width of the synthetic profile produced by the static multi-slab model does not increase after a certain number of slabs are added. In the example shown in Figure 6(b), the width of the resulting profile becomes saturated after adding fewer than 10 slabs. The line width measured by the 50% bisector method for the case of N = 10 is 46 km s⁻¹ (0.43 Å). Such a width is not sufficient to explain the observed line widths at lower and middle altitudes, where the line width measured by the same method is up to 110 km s⁻¹ (∼1.0 Å). For more details see Section 5.2.

In this subsection, we simulate a more realistic situation with multiple slabs to which we randomly assign LOS velocities (Figure 5(c)). The origin of the LOS velocities used in this model is as follows. Because spicules are typically inclined with respect to the solar surface (an inclination of 20°–37° is given by Tsiropoula et al. 2012), we need to take into account motions both parallel and perpendicular to the spicule axis. The typical amplitude of transverse velocities (i.e., perpendicular to the spicule axis) is ∼25 km s⁻¹ (De Pontieu et al. 2007). Typical apparent upward velocities (i.e., parallel to the spicule axis) are of the order of 25 km s⁻¹ on the quiet Sun (Pasachoff et al. 2009). We assume here that the same is true in a coronal hole. Moreover, from our analysis of the IRIS observations presented in Section 2.2 we obtained in the upper parts of spicules typical unsigned LOS velocities up to 25 km s⁻¹. Therefore, in the present study, we adopted an LOS velocity that is randomly selected from a uniform distribution ranging from −25 to 25 km s⁻¹.

In the case of a multi-slab model with randomly assigned LOS velocities, the specific intensity I_r(Δλ) is obtained as

$$\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{r}}}({\rm{\Delta }}\lambda ) & = & {I}_{{\rm{s}}}({\rm{\Delta }}\lambda -{\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},1}})\\ & & +\displaystyle \sum _{i=2}^{N}\left[{I}_{{\rm{s}}}({\rm{\Delta }}\lambda -{\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},i}})\right.\\ & & \left.\times \ \exp \left\{-\displaystyle \sum _{l=1}^{i-1}\tau ({\rm{\Delta }}\lambda -{\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},l}})\right\}\right],\end{array}\end{eqnarray} \tag{ 6 }$$

where N ≧ 2 and ${\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},i}}$ is the Doppler shift in the wavelength due to the LOS velocity of the ith slab (see Figure 5(c)). A similar scheme was applied in Gunár et al. (2008) for 2D multi-thread modeling of prominence fine structures. Due to the LOS velocity, the entire intensity profile emerging from the ith slab and the absorption profile of the ith slab are shifted by ${\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},i}}$ with respect to the rest wavelength. Such a Doppler-shifted intensity profile from the ith slab ${I}_{{\rm{s}}}({\rm{\Delta }}\lambda -{\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},i}})$ is then attenuated by all foreground slabs with the optical thickness ${{\rm{\Sigma }}}_{l=1}^{i-1}\tau ({\rm{\Delta }}\lambda -{\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},i}})$ as ${I}_{{\rm{s}}}({\rm{\Delta }}\lambda -{\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},i}})\exp \{-{{\rm{\Sigma }}}_{l\,=\,1}^{i-1}\tau ({\rm{\Delta }}\lambda -{\rm{\Delta }}{\lambda }_{{V}_{\mathrm{LOS},i}})\}$.

Figure 6(c) shows, as an example, the result of the multi-slab modeling with LOS velocities using a set of $N(=\,10)$ identical slabs modeled with the MALI code. The details are shown in Figure 7 and its caption. For individual slabs we again use the same set of input parameters as in Section 3.1. The number of slabs N varies from 1 to 10. The LOS velocities of the 1st slab to 10th slab used are 16, −24, 22, −1, −13, −14, −9, −22, 22, and −19 km s⁻¹. As we add additional slabs, the resulting intensity profiles get enhanced at the wavelengths in which the total optical thickness ${{\rm{\Sigma }}}_{i=1}^{N}\exp \{-(i-1)\tau ({\rm{\Delta }}\lambda )\}$ is less than unity, which is the same mechanism as demonstrated in the multi-slab modeling without LOS velocities in the previous subsection. However, the line width in the case of N = 10 is significantly larger (83 km s⁻¹ or 0.78 Å) compared to the case of N = 10 without LOS velocities (46 km s⁻¹ or 0.43 Å) shown in Figure 6(b). This is due to the relative shifts of the intensity profiles and optical thickness profiles for each slab. Because the optical thickness in the wing of each profile is small, the intensity from the added slab can appear there if the added slab has the Doppler velocity that corresponds to the wing wavelength. The line width in this case is dependent on the minimum and maximum LOS velocities, which are −24 and 22 km s⁻¹ in the 2nd and 9th slab, respectively. This can help to explain the broad and flat profiles observed at middle heights, as we discuss in Section 5.2.

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The IRIS NUV instrumental profile has an FWHM of Δλ_inst = 50.54 mÅ. This instrumental broadening affects the observed line profile and thus needs to be considered also for the synthetic profiles. We take the instrumental broadening into account in the following way:

$$\begin{eqnarray}&&\begin{array}{l}I({\rm{\Delta }}\lambda )=\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{\mathrm{inst}}}}\\ \times \ \displaystyle \int {I}_{\mathrm{synt}}({\rm{\Delta }}{\lambda }^{{\prime} })\exp \left[-\displaystyle \frac{{\left({\rm{\Delta }}\lambda -{\rm{\Delta }}{\lambda }^{{\prime} }\right)}^{2}}{2{\sigma }_{\mathrm{inst}}^{2}}\right]d({\rm{\Delta }}{\lambda }^{{\prime} }),\end{array}\end{eqnarray} \tag{ 7 }$$

where ${\sigma }_{\mathrm{inst}}={\rm{\Delta }}{\lambda }_{\mathrm{inst}}/(2\sqrt{2\mathrm{ln}2})$. This convolution is applied to all synthetic profiles obtained by the single-slab model, the multi-slab model without LOS velocities, and the multi-slab model with random LOS velocities, which are presented in this work.

Synthetic Mg ii k profiles produced by the single-slab model are shown in Figure 8. In Figure 8(a), we show the dependence on the different values of the pressure (the used pressure values are indicated in the figure). From the plotted synthetic profiles, we can clearly see that the synthetic profiles become very intense with increasing pressure. The optical thickness ${\tau }_{M}$ increases and is almost proportional to the pressure. The integrated intensities ${E}_{{\rm{k}}}$ also increase with the pressure. The ratio of the integrated intensities of the h and k lines is large and more than 2 for the lowest pressure value (0.02 dyn cm⁻²). Line widths become larger with increasing pressure, but they do not exceed 50 km s⁻¹ (∼0.5 Å) even for a very high pressure of 0.5 dyn cm⁻².

In Figure 8(b), we demonstrate the dependence of the synthetic Mg ii k profiles on the temperature. The intensity steeply decreases when the temperature reaches above 18 kK. Optical thickness τ_M also decreases with temperature. This is because of the ionization of the Mg ii ion to Mg iii. With the temperature of 22 kK, the most Mg ii ions are ionized into Mg iii (Heinzel et al. 2014), and the modeled spicule becomes optically thin (τ_M ∼ 0.1) in the Mg ii k line. In the temperature range T > 18 kK, the number of Mg ii ions decreases, the integrated intensity ${E}_{{\rm{k}}}$ becomes significantly smaller, and the line intensity ratio ${R}_{\mathrm{kh}}$ becomes somewhat larger than two. The line widths ${W}_{{\rm{k}}}$ are always less than 40 km s⁻¹ (∼0.4 Å) and are not too sensitive to the temperature, and they decrease slightly (from 38 to 19 km s⁻¹) between temperatures of 6 and 22 kK.

Figure 8(c) shows how the Mg ii k profiles change when we modify the geometrical thickness of the modeled slab, from 250 to 2500 km. The optical thickness is nearly proportional to the geometrical thickness. The Mg ii k line is more and more reversed for larger geometrical thicknesses. This is because with the increasing geometrical (and thus also the optical) thickness, the layer of τ ∼ 1 in the line core is located closer and closer to the surface where the source function decreases. On the other hand, the line widths ${W}_{{\rm{k}}}$ do not change significantly and stay below 50 km s⁻¹ (∼0.5 Å). Although the integrated intensity ${E}_{{\rm{k}}}$ increases with the geometrical thickness, the ratio of integrated intensities of h and k lines ${R}_{\mathrm{kh}}$ is nearly constant.

In Figure 8(d), we change the microturbulent velocity. The integrated intensity ${E}_{{\rm{k}}}$ increases with the microturbulent velocity but practically saturates above ${V}_{\mathrm{turb}}$ of 20 km s⁻¹. The line width ${W}_{{\rm{k}}}$ increases dramatically with the microturbulent velocity, and the resulting Mg ii line profiles have significantly broadened wings. As the microturbulent velocity increases, the line ratio ${R}_{\mathrm{kh}}$ increases because the line profile broadens and the optical thickness at the line center τ_M decreases.

In Figure 8(e), we show the dependence of synthetic Mg ii profiles on the upward (radial) velocity, which causes the Doppler dimming/brightening effect. In this case, the plasma properties such as temperature, pressure, and density do not change, and thus the optical thickness remains the same. However, the peak intensity changes with the change of the upward velocity. This is because the intensity at each wavelength changes owing to the Doppler shifts of the illumination profile (indicated by the dotted curve in the left panel of this row). We show here for the first time how the Mg ii h and k line intensity of spicules is sensitive to the radial velocity via the Doppler brightening/dimming effects. Zhang et al. (2012) reported that typical upward velocities of spicule tops are about 13 km s⁻¹ in a quiet region and about 38 km s⁻¹ in a coronal hole. We demonstrate here that upward velocities above 30 km s⁻¹ cause Doppler dimming.

In Figure 9, we show synthetic Mg ii k profiles obtained by a multi-slab model without LOS velocities (for more details see Section 3.2). Because we assume 10 identical slabs, the optical thickness for each set of model input parameters is 10 times larger than that of the corresponding single-slab model. Therefore, in almost all cases, the optical thickness at the line center is very large, and the Mg ii k line has reversed profiles. Due to the large total optical thickness of multiple slabs, line widths ${W}_{{\rm{k}}}$ for each set of input parameters are only slightly larger than in the case of a single-slab model and do not exceed 60 km s⁻¹ (0.56 Å). The exceptions are the very broad profiles with large values of microturbulent velocities. The integrated line intensities ${E}_{{\rm{k}}}$ are mostly less than twice as intense as those from corresponding single-slab models even though we use 10 identical slabs. This is again caused by the large optical thickness of individual slabs. Exceptions are the cases of the largest temperatures of T = 18 and 20 kK, in which, however, the optical thickness is small. The ratio of the integrated intensities in the k and h lines ${R}_{\mathrm{kh}}$ is ~1.3–1.5 except in the very high temperature case with T = 2.2 kK, in which the Mg ii k line has optical thickness of ~1.

Results of the multi-slab model with random LOS velocities (see Section 3.3) are shown in Figure 10. The maximum optical thickness τ_M for each set of input parameters is significantly lower (about half) than that of the corresponding multi-slab model without LOS velocities. This is because now each slab has its own LOS velocity and both the intensity and optical thickness profiles of individual slabs are mutually Doppler shifted. Even so, the Mg ii k line is optically thick for all sets of input parameters except the set with the highest temperature (T = 22 kK). The line width ${W}_{{\rm{k}}}$ is now about 1.5 times larger compared to the corresponding multi-slab model without LOS velocities and typically reaches values between 70 km s⁻¹ (0.65 Å) and 90 km s⁻¹ (0.84 Å). Such large widths are more than twice the widths produced by the single-slab model and cannot be achieved by either the single-slab model or the multi-slab model without LOS velocities unless we use very large microturbulent velocities. We note that line widths between 80 and 120 km s⁻¹ are typically obtained by the bisector method from the observations at middle or lower altitudes around Y = 0''–10'' analyzed here (see Figures 3(d), 4(d)). The integrated intensity ${E}_{{\rm{k}}}$ is also about 1.5 times larger than that in multi-slab models without LOS velocities, but the ratio of integrated intensities ${R}_{\mathrm{kh}}$ is almost the same. The exceptions are the input parameter sets with the highest temperatures (18 and 22 kK). Note that in Figure 10 we show results with identical radial velocities applied to all slabs in the multi-slab model with LOS velocities. We acknowledge that such an assumption may not be entirely realistic.

The ratio of the integrated intensities of Mg h and k lines is sensitive to the optical thickness and can be larger than 2 only when the lines have optical thickness around unity or below. From the set of the single-slab models that we show here, this is the case only for models with the smallest pressure (0.02 dyn cm⁻²) or the highest temperatures (18 or 22 kK). For the multi-slab models, it is only the model with the highest temperature of 22 kK. We note that higher values of ${R}_{\mathrm{kh}}$ are identified only at higher altitudes in the observed data set studied here (see Figure 3(b)). At these altitudes (Y > 1''), the majority of the obtained ${R}_{\mathrm{kh}}$ values is below 1.8. However, a fraction of the observed data exhibits ${R}_{\mathrm{kh}}$ values around 2 or higher (see Figure 4(b)).

Our analysis of IRIS Mg ii observations presented in Section 2 shows that at higher altitudes (Y ∼ 15'') Mg ii h and k profiles are relatively narrow and generally exhibit lower integrated intensities than profiles obtained at lower altitudes. Typical widths of the Mg ii k profiles ${W}_{{\rm{k}}}$ at higher altitudes are a few $\times 10\,{\rm{k}}{\rm{m}}\,{{\rm{s}}}^{-1}$ (see Figures 3(d) and 4(d)), and the integrated line intensities ${E}_{{\rm{k}}}$ reach up to a few ×10⁴ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹ (see Figures 3(a) and 4(a)). The ratio of integrated intensities of h and k lines (${R}_{\mathrm{kh}}$) reaches up to 2.0 (Figures 3(b) and 4(b)). The observed profiles at higher altitudes are often strongly Doppler shifted and exhibit unsigned LOS velocities up to 20 km s⁻¹ (Figures 3(c) and 4(c)). In the present work, we argue that such Mg ii profiles represent an emission of individual spicules that reach above the spicular forest. To support this argument, we show here that these profiles can be reproduced by narrow single-slab models representing a single spicule. In Figure 11(a), we show an example of a typical Mg ii k line profile obtained at the higher altitude—in this particular case at altitude Y = 15'' at time step 324. The width of this profile ${W}_{{\rm{k}}}$ is 21 km s⁻¹, the integrated line intensity ${E}_{{\rm{k}}}$ = 1.5 × 10⁴ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹, the ratio of integrated intensities ${R}_{\mathrm{kh}}$ = 1.9, and the line shift ${V}_{\mathrm{LOS},{\rm{k}}}$ = −19 km s⁻¹. In the present work, we do not aim to find a model with the best fit to this observed profile. Rather, from the set of profiles shown in Figures 8 and 10, we have selected one profile that closely resembles the observed profile. In Figure 11(a), we plot the synthetic profile (the red dashed line) redshifted by 0.15 Å (∼16 km s⁻¹). This synthetic profile corresponds to a single-slab model with P = 0.1 dyn cm⁻², T = 18 kK, D = 250 km, ${V}_{\mathrm{turb}}$ = 10 km s⁻¹, and ${V}_{\mathrm{radial}}$ = 0 km s⁻¹. The used geometrical thickness (250 km) is consistent with the observations (e.g., Pereira et al. 2012). The profile has the width ${W}_{{\rm{k}}}$ = 22 km s⁻¹, ${E}_{{\rm{k}}}$ = 1.2 × 10⁴ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹, and ${R}_{\mathrm{kh}}$ = 2.2. These profile parameters are quantitatively comparable to the parameters of the selected observed profile.

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When we assume the geometrical thickness of a spicule to be 250 km, only an optically thin slab with high temperature or low pressure plasma can achieve the observed single-peaked profile with small integrated intensity (see Figure 8). In our observations, a fraction of the upper part of spicules has large values of the ratio of the k and h lines ${R}_{\mathrm{kh}}$ (around 2 or higher). Such large line ratios can be achieved only when the line is optically thin owing to its high temperature (∼20 kK) or low pressure (≲0.02 dyn cm⁻²) as we argue in Section 4.4. A pressure of 0.02 dyn cm⁻² represents a rather low, coronal value that might be too low for spicules in the static case. However, in a more realistic dynamic situation, cooling due to an expansion might occur in the upper parts of spicules. Such a scenario could lead to rather low pressure values. On the other hand, the combination of pressure of 0.1 dyn cm⁻² (typical in spicules) and temperature of 20 kK (typical chromospheric values) might be more realistic. Moreover, recent imaging observations suggest that upper parts of spicules are visible in the higher-temperature, transition region lines (Pereira et al. 2014). If optically thin plasma is caused by high temperature (∼20 kK), one possible heating mechanism could be a shock heating. In such a case, the possible causes of the shocks in relation to spicule dynamics need to be investigated. To gain a better understanding of why and how such high-temperature or low-pressure plasma is produced, we will need to follow individual spicules throughout their lifetimes with multi-line spectroscopy and high-resolution imaging.

The observations at the middle altitudes (Y ∼ 5'') show Mg ii profiles that are significantly broader than profiles obtained at higher altitudes. Figures 3(d) and 4(d) show line widths ${W}_{{\rm{k}}}$ between 80 and 110 km s⁻¹. The integrated intensities ${E}_{{\rm{k}}}$ are around 8 × 10⁴ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹, and the ratio of integrated intensities ${R}_{\mathrm{kh}}$ is 1.2–1.4 (see Figures 3(a), (b) and 4(a), (b)). The LOS velocities derived from the observed Doppler shifts ${V}_{\mathrm{LOS},{\rm{k}}}$ are between −5 and +5 km s⁻¹ (Figures 3(c) and 4(c)), which are significantly lower than the LOS velocities derived at higher altitudes. In Figure 11(b), we show an example of a typical Mg ii k line profile obtained at altitude Y = 5'' at time step 350. The parameters of this profile are ${W}_{{\rm{k}}}$ = 100 km s⁻¹, ${E}_{{\rm{k}}}$ = 7.6 × 10⁴ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹, and ${R}_{\mathrm{kh}}$ = 1.3. The selected profile is flat topped and shows multiple small intensity variations at the top.

To reproduce the observed Mg ii k profile shown in Figure 11(b), we selected a model from the grid of models shown in Figure 10. We again note that we do not aim here to find a model with the best fit to the observation. The selected model is shown with the blue dashed line in Figure 11(b), which represents the multi-slab model with randomly assigned LOS velocities in an interval of −25 to +25 km s⁻¹, P = 0.1 dyn cm⁻², T = 10 kK, turbulent velocity ${V}_{\mathrm{turb}}$ = 15 km s⁻¹, ${V}_{\mathrm{radial}}$ = 0 km s⁻¹, and 10 identical threads with a width D = 250 km. This model produces a synthetic Mg ii k profile with parameters ${W}_{{\rm{k}}}$ = 98 km s⁻¹, ${E}_{{\rm{k}}}$ = 9.3 × 10⁴ erg s⁻¹ cm⁻² sr⁻¹ Å⁻¹, and ${R}_{\mathrm{kh}}$ = 1.4 that are quantitatively comparable to the observed profile. The used LOS velocities with the maximum unsigned value of 25 km s⁻¹ are consistent with the Hinode Ca ii H observations of the apparent velocity amplitude reported by De Pontieu et al. (2007). In addition, the maximum line shifts at upper altitudes in our spectroscopic observations are also about 25 km s⁻¹ (see Figure 4(c)). Therefore, we suggest that the apparent velocities in the Hinode Ca ii H imaging data correspond to the chromospheric gas motions in spicules and that velocities in the LOS direction are typically between ±25 km s⁻¹. This multi-slab modeling with LOS velocities succeeds in producing broad Mg ii k profiles without using the large ${V}_{\mathrm{turb}}$ values like 25 km s⁻¹. We note that such a large ${V}_{\mathrm{turb}}$ is above the local sound speed and thus may not be entirely realistic. The multi-slab models with random LOS velocities achieve broad Mg ii h and k profiles by Doppler-shifting the intensity and optical thickness profiles of individual slabs with respect to each other. As we demonstrate in Figure 7, such mutual shifts lead to significantly broad profiles even when the profiles produced by individual slabs are narrow. Therefore, each individual slab can have turbulent velocity below the local sound speed. Moreover, the synthetic profiles produced by multi-slab models with LOS velocities have an added advantage that they exhibit flat tops with a certain level of fine structuring, albeit not as pronounced as in the observed profiles. The intensity level at the top of the modeled profile is slightly higher than the observed profile. However, the intensity level of the modeled profile would naturally decrease if we considered the Doppler dimming/brightening effects for each spicule slab depending on the relative values between the velocities of illuminating materials and the velocities of the slabs. Another possibility is that if the spicule would be modeled as a vertical cylinder, it would emit radiation in all directions and the source function would thus decrease. Then, the line core emission, which is roughly equal to the source function, will be lowered (Heasley 1977).

We note that the broad observed Mg ii h and k profiles, such as the one shown in Figure 11(b), cannot be reproduced just by increasing the temperature in the models. This is because the Mg ii ion is heavy and the Mg ii h and k lines are thus not sensitive to the thermal broadening. In fact, the thermal width of the Mg ii k line is only 2 km s⁻¹ (0.02 Å) for the temperature of 6 and 4 km s⁻¹ (0.04 Å) for 22 kK. When we compare this to the turbulent broadening of 10 km s⁻¹ (0.09 Å) due to ${V}_{\mathrm{turb}}$, we can see that the Mg ii h and k line widths observed at middle altitudes must be caused by some form of dynamics. One can either consider a large microturbulent velocity such as 25 km s⁻¹ or assume macroscopic LOS velocities of multiple components along an LOS.

Recently, Alissandrakis et al. (2018) have investigated the Mg ii h and k line profiles observed by IRIS in a quiet-Sun polar region. To understand the averaged line profiles at all altitudes above the limb, these authors conducted 1D single-slab modeling with the fixed thickness of 500 km to synthesize the spectra and derived physical quantities for different heights by comparing temporally averaged observed spectra and synthetic spectra. To achieve a good fit, Alissandrakis et al. (2018) considered a large turbulent velocity exceeding 20 km s⁻¹. However, it can be expected that the Mg ii h and k line profiles at the lower part of the chromosphere are a product of a superposition of several spicules along the LOS. As we have shown in the present work, in the case of such a superposition of spicules, the large turbulent velocity is not necessary to achieve the broad observed profiles if individual spicules have different LOS velocities.

At altitudes near the limb (Y ∼2''), the observed Mg ii profiles are broad and usually distinctly double peaked with a deep central reversal (see the example in Figure 2(b)). The intensity of the peaks varies significantly over time as can be seen in the online movie. The 1D models in any of the configurations used in the present work are not able to reproduce such a type of profiles. To synthesize these profiles, we need to include more complex geometry in our models of spicules. We will investigate these profiles in a future study.