Semi-empirical Models of Spicule from Inversion of Ca ii 8542 Å Line

Spicules are small-scale, jet-like plasma features observed ubiquitously at the solar limb and they have been reviewed by Beckers (1968, 1972), Sterling (2000), and Tsiropoula et al. (2012). Originating in the active regions and network boundaries, they protrude into the corona and can act as conduits channeling energy and mass from the solar photosphere into the upper layers. Despite tremendous observational and theoretical efforts since their discovery in the 19th century (Secchi 1875), many aspects of spicule physics, such as their formation mechanism, magnetism, and thermodynamic properties, remain the central subject of ongoing solar research.

There is a wide variety of spicular jets in the solar atmosphere observed in strong chromospheric lines. Their classification is based on their morphological properties. However, there is no final consensus regarding their categorization. A traditional spicule is defined as a chromospheric jet with height between 6500 and 9500 km, width between 700 and 2500 km, and lifetime of up to 10–15 minutes (Beckers 1972; Pasachoff et al. 2009). They are characterized by rising and falling motions with speeds of ∼20–40 km s⁻¹. Disk observations have revealed similar types of chromospheric fine structures such as quiet Sun mottles (Suematsu et al. 1995) and active region dynamic fibrils (De Pontieu et al. 2004; Hansteen et al. 2006). At the limb de Pontieu et al. (2007) discovered energetic and short-lived spicular features called type II spicules. Soon after, structures with similar properties were identified in high-resolution ground-based observations on the solar disk. These absorption features detected in the blue and red wings of the Ca ii and Hα lines are called rapid redshifted and blueshifted excursions (Langangen et al. 2008; Rouppe van der Voort et al. 2009; Kuridze et al. 2015a).

Spicular structures are often observed with inverted Y-shaped footpoints both at the limb and on the disk (Yokoyama & Shibata 1995; Shibata et al. 2007; He et al. 2009; Morita et al. 2010; Nishizuka et al. 2011; Nelson et al. 2019). Due to their similarity with coronal X-ray jets, Yokoyama & Shibata (1995) and Shibata et al. (2007) refer to them as anemone jets. The topology of these jets suggests that they are formed as a result of magnetic reconnection in the lower solar atmosphere.

A reliable quantitative measurement of the physical parameters in spicules is extremely challenging. The difficulties arise due to the fact that they are very dynamic features, with spatial and temporal scales that are often close to the resolution limit of modern solar telescopes, and the large number of spicules in the solar atmosphere makes overlapping emissions coming from them difficult to separate. Chromospheric spectral lines are sensitive to plasma properties in spicules. Therefore, chromospheric spectroscopy offers the most powerful diagnostic opportunities. However, modeling and interpretation of chromospheric spectral lines are always challenging as they require solving the complex non-LTE (i.e., departures from local thermodynamic equilibrium) radiative transfer problem.

It is widely accepted that the external radiation field irradiating spicules is entirely photospheric (Beckers 1968) and their emission in most chromospheric lines forms under strong non-LTE conditions. The line profiles of spicules in the strong chromospheric spectral lines of the hydrogen Balmer series, single ionized calcium lines, and helium lines very often appear in emission with central reversal and broad, enhanced, asymmetric wings (Beck et al. 2016). As discussed by Beckers (1972) and confirmed by subsequent studies, the central reversals in spicule spectra are not produced by the reduction of the source function at the line core (self-absorption), but rather related to their non-thermal dynamics.

One of the first spectroscopic diagnostics of spicule parameters was carried out by Beckers (1968). He assumed that the spicule is a cylinder with the diameter of 815 km situated vertically on the Sun and irradiated by solar radiation. Theoretical relations between electron temperature T_e and electron densities n_e were calculated for the intensities of Hα, Ca ii, and He spectral lines through solving the radiative transfer equation and the statistical equilibrium equation in non-LTE. Then, for the observed intensities at different heights along spicules, T_e−n_e curves were obtained and interceptions between these curves for different lines were used for diagnostics of temperature and electron density. The temperature of the spicules reported by Beckers (1968) between the heights of 2–8 Mm is ∼9000–16,000 K, and the electron densities between 4 and 8 Mm are 1.5 × 10¹¹ − 4.3 × 10¹⁰ cm⁻³. Alissandrakis (1973) and Krall et al. (1976) implemented the same method and obtained similar results for electron density and temperature as Beckers (1968). Socas-Navarro & Elmore (2005) used multiline full Stokes observations of spicules in the Ca ii 8498, 8542 Å, and He i 10830 Å lines to derive spicule properties. They found that the Ca ii and He i 10830 Å lines have almost identical widths suggesting that most of the broadening in spicules are non-thermal, which in turn indicates that the electron temperature of the spicule should be lower than 13,000 K. Measurements of magnetic fields in spicules using spectropolarimetric single-line observations have been done in the He i 10830 Å line (Trujillo Bueno et al. 2005; Centeno et al. 2010; Orozco Suárez et al. 2015), in the Ca ii 8542 Å line (Kriginsky et al. 2020), and in the He i 5876 Å (D3) line (López Ariste & Casini 2005; Ramelli et al. 2005, 2006). For more details on spicule spectropolarimetry see Trujillo Bueno (2010).

To interpret the spectra of spicular structures observed against the solar disk and appearing as absorbing features in chromospheric lines, Beckers (1964) developed a different method known as the "cloud model." The cloud model is a simple inversion technique that fits the observed intensity contrast profile with four free parameters—the source function, the optical depth, the Doppler width, and the Doppler shift. With these parameters one can determine several other physical parameters of the structure. This method has been successfully applied over the years to chromospheric structures observed in Hα (Alissandrakis et al. 1990; Tsiropoula & Schmieder 1997; Tziotziou et al. 2003).

Since the launch of the Interface Region Imaging Spectrograph (IRIS; De Pontieu et al. 2014), spicules have been intensively observed and investigated in several UV lines by Pereira et al. (2014) and Skogsrud et al. (2015). Alissandrakis et al. (2018) compared the Mg ii k and h line profiles observed with IRIS with computations from a 1D non-LTE model and estimated the temperature and electron density from the lower to the upper part of spicules to be between ∼8000 and 20,000 K and 1.1 × 10¹¹−4 × 10¹⁰ cm⁻³, respectively. More recently, Tei et al. (2020) used 1D non-LTE vertical slab models in single- and multiple slab configurations and concluded that the width of the Mg ii k and h line profiles can be significantly influenced by a superposition of multiple spicules along the line of sight (LOS).

An alternative method of plasma diagnostics in spicules is the so-called magnetoseismology, which relies on observations of magnetohydrodynamic waves to infer the properties of a magnetic flux tube (Zaqarashvili et al. 2007; Zaqarashvili & Erdélyi 2009; Verth et al. 2011; Kuridze et al. 2013; Morton 2014). These methods also depend on the assumed nature of the wave modes (local tube modes versus genuine Alfvén waves in more homogeneous media).

Spicule temperature and density were also determined using radio data obtained with the Atacama Large Millimeter/submillimeter Array (ALMA; Wootten & Thompson 2009) radio telescope by Shimojo et al. (2020). They derived kinetic temperature and the number density of ionized hydrogen in the spicule plasma seen in the ALMA 100 GHz images as 6,800 K and 2.2 × 10¹⁰ cm⁻³.

The vertical stratification of the physical parameters along chromospheric structures can be obtained using semi-empirical models that attempt to reproduce the observed profiles in non-LTE radiative transfer. A powerful way to construct atmospheric models with a semi-empirical approach is to fit the observed Stokes profiles using inversion algorithms (de la Cruz Rodríguez & van Noort 2017). Through such inversions, the ionization equilibrium, statistical equilibrium, and radiative transfer equations are solved numerically to synthesize the Stokes profiles under a set of predefined initial atmospheric conditions. The Ca ii infrared (IR) triplet line at 8542 Å is well suited for the development of such models due to its sensitivity to physical parameters in the solar photosphere and chromosphere (Pietarila et al. 2007; Cauzzi et al. 2008). Furthermore, calcium is singly ionized under typical chromospheric conditions, with negligible partial redistribution effects for the Ca ii IR lines (Uitenbroek 1989), making their modeling and interpretation of the observations more straightforward. The non-LTE radiative transfer code NICOLE (Socas-Navarro et al. 2000, 2015) is a suitable tool for our purposes here since it allows for the synthesis and inversion of Ca ii Stokes profiles and the construction of semi-empirical models. Inversions with NICOLE have been successfully applied to the study of spectropolarimetric Ca ii 8542 Å observations of umbral flashes in sunspots (de la Cruz Rodríguez et al. 2013), granular-sized magnetic elements (magnetic bubbles) in an active region (de la Cruz Rodríguez et al. 2015), and chromospheric flares (Kuridze et al. 2017, 2018). However, NICOLE was not designed to work on off-limb spectroscopic observations as it assumes a geometry in which the emergent radiation comes from the bottom of the atmosphere and hydrostatic equilibrium is imposed at each inversion iteration to compute the plasma density and pressure.

In this paper we use a new version of NICOLE to invert high-resolution imaging spectroscopy in the Ca ii 8542 Å line to construct models of the solar off-limb spicule (Figure 1). This new version first computes the atomic level populations in a vertically stratified atmosphere, including the spicule up to 10 Mm. Then, once all the source functions and opacities are known, a final formal solution computes the spectra in the direction of the LOS, which in this case is horizontal, at a discrete set of heights specified by the user (in our case, the 14 points where observed spectra are available). These 14 profiles are fitted simultaneously with one single model atmosphere. The model has two components, as explained below. The chosen parameterization allows us to retrieve the height stratification of temperature, mass and electron density, LOS velocity, and optical thickness along the spicule.

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2.1. Observational Setup

We observed the west limb of the Sun between 08:20 and 08:43 UT on 2018 June 22 near the active region (AR) NOAA 12714. The heliocentric coordinates of the center of the field of view at the beginning of the observations were [937'', 127'']. Observations were made with the CRisp Imaging SpectroPolarimeter (CRISP; Scharmer 2006; Scharmer et al. 2008) and the CHROMospheric Imaging Spectrometer (CHROMIS) instruments, both based on dual Fabry–Pérot interferometers (FPI) mounted on SST. The imaging setup includes a dichroic beamsplitter with the transmission/reflection edge at 5000 Å. CRISP is mounted in the reflected red beam and CHROMIS in the transmitted blue beam (Löfdahl et al. 2018, Figure 2).

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The CRISP data comprises narrow-band imaging spectropolarimetry in the Ca ii 8542 Å line profile sampled from −1.75 Å to +1.75 Å in 21 line positions ±1.75, ±0.945, ±0.735, ±0.595, ±0.455, ±0.35, ±0.28, ±0.21, ±0.14, ±0.07, 0.0 Å from the line center (hereafter, unless specified otherwise, when referring to the Ca ii line we mean the Ca ii 8542 Å line). Each spectral scan of the Ca ii line had an acquisition time of 16 s but the cadence of the CRISP time series was 33 s due to the inclusion of spectropolarimetric scans in the Fe i 6302 Å photospheric line. However, we note that the present paper includes only the analysis of the Ca ii data, as the spicule analyzed in this study and its footpoints were not detected in the Fe i line. The CRISP data are processed by the CRISPRED reduction pipeline (de la Cruz Rodríguez et al. 2015) and reconstructed with Multi-Object Multi-Frame Blind Deconvolution (MOMFBD; Löfdahl 2002; van Noort et al. 2005).

Simultaneous observations were taken with the CHROMIS imaging spectrometer—a dual FPI observing in the range of 3900–4900 Å. The CHROMIS observations comprise narrow-band and wide-band spectral imaging in the Hβ and Ca ii H 3968.5 Å lines, plus one position in the continuum at 4000 Å. CHROMIS data are processed using the CHROMISRED reduction pipeline, which includes MOMFBD image restoration (Löfdahl et al. 2018).

In this work we focus only on the inversion of the CRISP Ca ii Stokes I profiles.

2.2. Data Radiometric Calibration

The inversion of SST/CRISP spectroscopic observations of the spicule requires careful intensity calibration of the data. Inversion codes, including NICOLE, usually require the input data to be calibrated with respect to the quiet-Sun continuum intensity at the disk center to convert the digital counts in the observations to actual physical units. However, this was not an option in our case due to the absence of a disk-center continuum reference in the data.

To calibrate a single CRISP Ca ii scan, the Stokes I wing images at Δλ = ±1.75 Å are averaged yielding a composite image (Figure 2: left panel). The limb is identified in the composite image by the IDL function sobel.pro and approximated by a limb line. Then a dense mesh of 536 lines parallel to the limb is created covering the whole visible disk. Using the solar disk radius of 944338 and the CRISP spatial sampling of 0057 pixel⁻¹ the position cosines $\mu =\cos \theta$ corresponding to the mesh lines are calculated. A sample of 38 equidistant mesh lines and the position cosines are shown in Figure 2 (left and middle panel). Averaging Stokes I pixel intensities over particular mesh lines at a given μ yields radial intensity gradients I_CRISP(λ_CRISP, μ) at particular CRISP wavelengths, λ_CRISP. Quasi-linear intensity segments I_CRISP(λ_CRISP, M) (not shown here) in the position cosines ranging from 0.17 to 0.20 (hereafter M ≡ μ ∈ (0.17, 0.20)) are adopted for calibration, avoiding the remnant of the AR NOAA 12714 seen at μ ∼ 0.13 (Figure 1: bottom right panel and Figure 2: left panel). The calibrated Ca ii profiles from Linsky et al. (1970) are adopted as a reference extrapolating them for M and convolving them with the transmission profile of the CRISP Fabry–Pérot etalons provided by J. de la Cruz Rodríguez (2017, private communication). The resulting profiles are referred to hereafter as I_LIN(λ_LIN, M). To compensate for differences between the CRISP wavelength scale λ_CRISP and the scale λ_LIN used in Linsky et al. (1970), the precise line minimum positions λ_c of I_CRISP(λ_CRISP, μ) are measured in a broad range of μ values. Then the profile minimum positions of I_LIN(λ_LIN, M) are compared to λ_c at the corresponding μ, allowing the interpolation of I_LIN(λ_LIN, M) on the CRISP wavelength scale. This yields the reference intensity gradients I_LIN(λ_CRISP, M). The ratio of the reference intensity gradient 〈I_LIN(λ_CRISP, M)〉_M and the observed gradient 〈I_CRISP(λ_CRISP, M)〉_M, averaged over the M range, renders a calibration profile (Figure 2: right panel) allowing the conversion of the Ca ii Stokes I intensities from digital numbers to the physical units expressed relative to the quiet-Sun continuum intensity at 8542 Å in the disk center I_c (Figures 3–6). To compensate for possible variations in sky transparency, the calibration procedure is repeated separately for each CRISP scan. The right panel of Figure 2 shows a cluster of profiles used for calibrating particular scans.

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2.3. Corrections for Canopy Emission and Stray Light

Figure 1 shows images of the spicule in the Ca ii and Hβ line core and blue wing images taken at 08:34 UT. The spicule was already developed at the start of the observations at ∼08:20 UT and disappeared after around 20 minutes. The line core images show that the structure extends up to the height ∼9 Mm above the limb, well above the chromospheric canopy populated with shorter type II spicules (Figure 1). The blue wing images nicely illustrate the spicule footpoint shaped as the inverted Y. It suggests an association of the spicule with small-scale magnetic reconnection, which likely launched the spicule. Thus the event discussed here is of the same type and origin as those in Shibata et al. (2007), He et al. (2009), Nishizuka et al. (2011), and Nelson et al. (2019). The spicule has an inclination angle θ ∼ 70° with respect to the limb. The average width of the spicule is around 1 Mm.

We select four line profile scans taken at ∼ 08:34, 08:35, 08:38, and 08:41 UT when the spicule was relatively bright and started becoming fainter. The red circles in Figures 3–6 represent the observed Ca ii line intensities of the spicule for the four selected snapshots at fourteen space-separated positions along its axis. They are normalized with respect to the quiet-Sun continuum intensity at 8542 Å in the disk center I_cont(8542 Å, μ = 1) ≡ I_c. The red lines in the lower right panels of Figures 3–6 show the equidistant heights located at different positions along the spicule. Along each height the pixel with maximum wavelength-integrated intensity are found and line profiles for those pixels are selected for inversions. The lowest (1) and highest (14) positions (the lower right panels in Figures 3–6) are located at the heights ∼2050 and 8600 km above the limb, respectively. It is known that the limb is uplifted by about ∼350 km above the base of the photosphere at τ₅₀₀₀ = 1 measured at μ = 1 (Lites 1983, Table 1). This off-set is due to an increase of optical depth at 5000 Å when looking from disk center (μ = 1) to limb (μ = 0). The edge of the solar disk at the Ca ii±1.75 Å line positions is also expected to be formed at ∼350 km above the τ₅₀₀₀ = 1 at μ = 1. The blue lines in the lower right panel of Figures 3–6 show the location of the true photosphere and the indicated heights of the selected pixels along the spicule are considered as a projection on the normal to the photospheric limb.

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Figures 3–6 show that the line profiles appear in emission with a central reversal and broad, enhanced, asymmetric wings. However, the observed central reversals are not due to self-absorption by spicule plasma itself. The main characteristic of the self-absorption is that blueshifted line core (the location of the central dip) produces a red asymmetry (higher red wing emission) and the redshifted line core produces a blue asymmetry (higher blue wing emission). This is a well-known phenomenon related to the shift in the wavelength of maximum opacity in the line core to shorter and longer wavelengths, which invokes the red and blue asymmetries, respectively (Kuridze et al. 2015b). The observed line profiles do not show this behavior suggesting that the central reversal could be due to strong opposite LOS velocity components within the spicule. The top part of the spicule at 08:38 and 08:41 UT has single peak line profiles without central reversal (Figures 5 and 6). Intensities at all wavelength positions across the Ca ii line decrease as a function of height (Figures 3–6). The maximum line emission in the blue wing at around −0.7 Å from the line center drops by about a factor of ∼3–10 over the heights between ∼2 and 9 Mm above the limb (Figures 3–6).

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4.1. Modified Version of NICOLE

4.2. Double-component Spicule Model

The new version of NICOLE fits simultaneously the observed Ca ii profiles from fourteen different locations along the spicule axis with the double-component model that has a constant temperature and a smooth height-dependent density. The inversion is performed for four selected snapshots of the spicule lifetime. Figures 3–6 show the best-fit synthetic profiles resulting from the inversion. All models in the four selected snapshots characterize the constant temperature of T = 9560 K in both components.

The height stratification of the mass density and the electron density, reproducing the fourteen observed Ca ii profiles at four selected snapshots, are shown in Figure 7 (top left panel). In order to compute a density scale height, we fitted an exponential function (not shown) to the density stratifications retrieved by the inversion. The density scale heights Λ obtained in this manner are shown in the bottom left corner. The height scale drops from about 2100 km to about 1000 km over the interval of 7 minutes. The inversion yields the mass density of components typically in the range from about 4 × 10⁻¹¹ g cm⁻³ at the spicule bottom to about 3 × 10⁻¹⁴ g cm⁻³ at its top. The corresponding electron densities computed from the mass densities by the equation of state are shown on the right y-axis. They decrease from about 1.4 × 10¹³ cm⁻³ at the spicule bottom to about 9 × 10⁹ cm⁻³ at its top. The density drop in time is apparent especially in the upper parts of the spicule.

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The variations of the LOS velocity in the blue (V_LOS,blue) and red (V_LOS,red) components of the spicule model are shown in the bottom left panel of Figure 7. Disregarding the height-time variations, we can take −22 km s⁻¹ as typical for the former and +3 km s⁻¹ for the latter. We interpret the asymmetry in their absolute values as a consequence of vector superposition of counter-oriented Doppler velocity components of spicule tilted toward the observers. The whole-spicule decrease of ∣V_LOS,blue∣ from about 27 km s⁻¹ to 16 km s⁻¹ is followed by an increase of ∣V_LOS,red∣ from about 1 km s⁻¹ to 6 km s⁻¹ apparent below ∼6 Mm. This is compatible with the ceasing of upflow motion and also with the drop of density scale height Λ over time.

The top right panel of Figure 7 shows height-time variations of the filling factor in the blue (F_blue) and red (F_red) component of the spicule model. It suggests roughly balanced ∼50/50 coverage of the resolution element by the spicule model components at all heights. An exception is the last moment (08:41:03 UT) shown in blue, when the red model component (F_red) dominates over the resolution element at the heights above ∼7 Mm.

The bottom right panel of Figure 7 shows the average optical thickness of spicule τ in the Ca ii line center. Due to the same mass densities of both model components, the component optical thicknesses τ_blue and τ_red are also almost the same. Therefore the panel shows their simple average separately for the four selected times. The optical thickness decreases exponentially from about 300 at the spicule bottom to about one at its top. It proves that the spicule plasma is mostly optically thick in Ca ii and validates our assumption about infinitely extended plane-parallel spicule slab in the radiative transfer calculations (Section 4.1). The drop of τ over time in its upper part is also apparent.

We note that, the inversions cannot achieve a good fit at the top part of the spicule (Figures 3–6, points 13–14). Due to the significant change in relative amplitudes of the two components at the top, the code cannot fit the line profiles with the same success using the same parameterization. The ratio of amplitudes along the spicule is nearly constant but at the top one component vanishes and the ratio raises significantly. Possible reason could be the end of the helical structure or spinning/unwinding motion, when the field does not wrap around any more at the top part of the spicule. We plan to explore this in more detail in the follow up paper where we use the 2D models.

5.1. Response Functions (RFs)

To investigate the sensitivity and response of the Stokes I Ca ii profiles to variations of physical parameters, we investigate the numerical RFs for the spicule model implemented in the new version of NICOLE. The RF RF_x(λ) to the given atmospheric parameter x is a measure of how the line intensity at a given wavelength λ reacts to a small perturbation on x. Formally, RF_x(λ) to the parameter x is defined as:

$$\begin{eqnarray}&&{\mathrm{RF}}_{x}(\lambda )=\left[{I}_{\mathrm{pert}}(\lambda )-{I}_{\mathrm{ref}}(\lambda )\right]/({x}_{\mathrm{pert}}-{x}_{\mathrm{ref}}),\end{eqnarray} \tag{ 1 }$$

where x_ref is a reference value of a spicule model parameter producing reference intensity I_ref(λ) at λ and x_pert is the value of perturbed parameter producing the perturbed intensity I_pert(λ). In the limit when x_pert tends to x, the expression becomes the mathematical representation of a derivative. Figure 8 shows the Stokes I RFs of the Ca ii line to perturbations of the temperature, spicule thickness, and mass density for the fourteen segments of the blue component of the spicule model. The curves are normalized to maximum values of corresponding RF_x. All displayed RFs show maxima or minima at Δλ ∼ −0.6 Å. While the RFs suggest different response of Ca ii blue wing intensity to parameter perturbations in the bottom spicule segments 1−6 there is a strong degeneracy of response in most of the upper segments 6−14—the blue wing intensity reacts the same to parameter perturbations of the blue component model of spicule. This is the reason why it would not be possible to invert each spectrum separately. One needs the constraints given by a consistent height-dependent model to fit all the observations at various heights.

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We study an off-limb solar spicule using high-resolution imaging spectroscopy obtained in the Ca ii 8542 Å line by the SST/CRISP instrument. We use a new version of the non-LTE code NICOLE assuming a double-component spicule model to infer the spicule temperature and height stratifications of density, and LOS velocity along the spicule axis defined by pixel positions at the maxima of line integrated intensities.

Our analyses shows that the spicule plasma is isothermal having the uniform temperature of 9560 K along its length in the time interval of 7 minutes. Early and also recent measurements reported temperature variation along the spicule axis (Beckers 1968, 1972; Alissandrakis 1973; Krall et al. 1976; Matsuno & Hirayama 1988; Alissandrakis et al. 2018). However, despite reporting some temperature variations, based on analyses of the T_e−n_e curves at different heights, the pioneering work of Beckers (1968) concluded that the spicule intensities in chromospheric lines do not yield information about temperature variation and hence observed spectral characteristics in principle can be reproduced by a spicule model with uniform temperature (see page 408 therein). In this context Krall et al. (1976) also mentioned that Hα emission is a rather sensitive indicator of electron density and relatively independent of temperature over a large temperature range (see page 100 therein). Our results confirms this as we fitted multiple Ca ii profiles with constant temperature in four selected snapshots from the spicule lifetime. The inverted Y-shaped topology of the spicule footpoint implies a magnetic reconnection in the junction point that may have populated the spicule body with plasma of uniform temperature.

As shown in Figure 7 the mass and electron densities of the spicule plasma decrease exponentially as a function of height. We note that the analyzed structure bears many similarities and dissimilarities with conventional spicules making its unequivocal classification substantially difficult within current taxonomy of diverse small-scale jets. The structure has a similar width, length, and line profiles to the spicules analyzed in Beckers (1968, 1972), providing the main motivation to interpret it as spicule. However, density obtained through the inversion makes this structure more similar to the larger scale chromospheric jets (macrospicule/surges) rather than typical spicules. The values of the electron density n_e at four different times near the base of the spicule at a height of z ∼ 2 Mm is around 10¹³ cm⁻³. Beckers (1972) derived n_e ∼ 1.6 × 10¹¹ cm⁻³ at a height of z ∼ 2 Mm, around 60 times less than our estimates. Furthermore, Beckers (1972) estimates that n_e decreases by 20 times at the top of the spicule at a height of ∼10 Mm, whereas our models show that the mass and electron densities decrease by ∼100 times at a height of z ∼ 9 Mm for the first three models (Figure 7: top left panel). The model at a late phase of the spicule (at 08:41 UT, around 2 minutes before the disappearance) shows a sharper drop in density and pressure. Alissandrakis (1973) performed multiline spectroscopy of 36 spicules and found that the average values of the electron density at a height of 5400 km is around 6 ×10¹⁰ cm⁻³. Our models show that the n_e at the same height varies between 1 and 3 × 10¹² cm⁻³ for the four different models (Figure 7: top left panel).

The density and pressure scale heights of spicules measured from our diagnostics are in the range of Λ ∼ 1000–2100 km over the ∼2−9 Mm length projected to the surface normal (Figure 7: top left panel). This is higher than some previous estimates of the scale heights in limb spicules using eclipse flash spectra (Makita 2003) and magnetoseismology (Verth et al. 2011). Density scale heights derived by Bjølseth (2008) from Ca ii H intensities in spicules vary between ∼2 and 8 Mm (see Figure 4.10 of Bjølseth 2008). However, Bjølseth (2008) assumes that the source function is constant along the spicule, and hence the scale height of the intensity is defined only by the density stratification. In our case, the wavelength-integrated Ca ii intensities (not shown) decrease by a factor of ∼10−16 in the range 2–8 Mm along the spicule, much slower than density and pressure (Figure 7: top left panel). Estimated density/pressure scale heights are also much higher than the scale height of ∼290 km of gas with comparable temperature in hydrostatic equilibrium. This is not surprising as the observed spicule displays a complex dynamic behavior during its lifetime. Inversions distinguish two LOS velocity components along the spicule.

The double-peak Ca ii 8542 Å profiles detected along the jet axis are likely due to the unwinding of highly twisted field lines rooted in magnetic topology with opposite polarities. Unwinding twists can appear as a spinning motion of the plasma strands as found in rotating chromospheric jets (see e.g., Shibata & Uchida 1986; Canfield et al. 1996; Liu et al. 2009, 2011; Pariat et al. 2015). The line-of-sight superposition of such strands may result in the double-peak profiles observed along the jet axis. Liu et al. (2009) has shown the observations of reconnection-driven jet, which involves untwisting helical threads rotating about the axis of a single large cylinder and shedding magnetic helicity into the upper atmosphere. They proposed the model in which flux emergence in an open-field region leads to magnetic reconnection, forming a jet and fan-spine topology. The twists in the reconnected open field tend to unwind themselves and drive these field lines to rotate (Liu et al. 2011). The spicule analyzed in this paper closely resembles the fan-spine magnetic reconnection topology resulting from flux emergence studied in Liu et al. (2011) and displays a spinning motion during its lifetime.

We note that the bidirectional flows along the spicular jet with parallel field lines in unipolar topology can also produce centrally reversed profiles of spicules. An observation of bidirectional flows at tops and bases of spicules is reported in Pasachoff et al. (2009). Signature of counterstreams along a simulated macrospicule can be found in Murawski et al. (2011) for vertical or oblique magnetic field. However, Murawski et al. (2011) consider a localized velocity pulse as a driver of macrospicule, whereas the Y-shaped base of the spicule analyzed in this study suggests the magnetic reconnection for the formation mechanism (Figure 1). Furthermore, there is no clear and obvious evidence of the bidirectional flows in the observed spicule. Therefore, double-peak profiles of the spicule presented in this study are less likely to be a manifestation of counterstreaming or bidirectional flow.

The fact that the two spicule components have different amplitudes of the maximum intensity could be due to the different filling factors within a resolution element. Indeed, inversions with height-dependent filling factors for red and blue components provided the best fit with the observations. The exact relation of the spicule spectra with the dynamic motion and its transverse structure will be studied in a future paper where we plan to perform full 2D analysis and inversions of our observations.

The optical thickness τ in the Ca ii line center, i.e., where opacity reaches maximum, is a few hundreds at the spicule bottom and decreases exponentially toward its top. Over the main part of the spicule the τ is well above one, thus classifying the spicule plasma as optically thick in the Ca ii line center. Krall et al. (1976) performed theoretical calculations of the spicule optical thickness τ₈₅₄₂ in the Ca ii line center for the temperatures and electron densities ranging from 10,000 K to 20,000 K and from 2 × 10¹⁰ cm⁻³ to 1 × 10¹¹ cm⁻³, respectively, but inferring τ₈₅₄₂ mostly of the order of 1 × 10⁻²−1 × 10⁻³. They assumed the spicule thickness of 1425 km (see page 100 therein), thus larger than the thickness of 1000 km employed here (Table 1). The high optical thickness could be related to the measured high density of spicule along its length.

We note that there are only a limited number of radiative transfer codes that can deal with the off-limb emissions of the structures such as spicules. One of the most advanced is non-LTE inversion code HAZEL (Asensio Ramos et al. 2008), which may be of interest for other similar investigations. We have decided to employ NICOLE for this work because of some unique features that are very important in this context, such as the ability to calculate Ca ii populations in an optically thick medium with radiative transfer.

Generally, the main problem of inversion techniques is the possible non-uniqueness of the obtained solutions due to the degeneracy between the different parameters. We have investigated the degeneracy between temperature, density, and thickness of the spicule in the models obtained through inversion by analyzing the RFs of the intensity to the perturbations of these parameters. The results presented in Figure 8 indicate that profiles 8–14 have a strong degeneracy among the three parameters. This means that we could obtain exactly the same solution with different combinations of those three parameters. Therefore, if we were fitting those profiles individually, we would have no way to obtain a unique solution. However, the degeneracy is broken as the RFs are not the same for the profiles of the bottom and middle parts of the spicule (see profiles 1–7 in Figure 8). As long as we fit all of the profiles simultaneously the models obtained are free of this degeneracy and the results are reasonably reliable.

We note that the spectral diagnostics presented here are applicable to the local plasma parameters of a particular spicule and they may not necessarily be considered as typical of all spicular structures. However, it has been demonstrated that inversions can be successfully applied to spicule spectra at multiple height positions. With a wealth of statistics of semi-empirical spicule models, we might gain a better understanding of chromospheric spicules, their properties, and dynamics.

To our knowledge, this is the first spectral diagnostics of the physical parameters of a spicule through line profile fitting with a non-LTE inversion code. The results are very encouraging for future semi-empirical modeling of spicules using inversion techniques and high-resolution spectropolarimetric observations with present and future ground-based observing facilities.

The research leading to these results has received funding from the Sêr Cymru II scheme, part-funded by the European Regional Development Fund through the Welsh Government, and STFC grant ST/S000518/1 to Aberystwyth University. The work of D.K. was supported by Georgian Shota Rustaveli National Science Foundation project FR17_323. H.S.N. acknowledges support from the Spanish Ministry of Economy and Competitivity through project AYA2014-60476-P and PGC2018-102108-B-I00. 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). J.K. acknowledges the project VEGA 2/0048/20. R.O. acknowledges support from the Spanish Ministry of Economy and Competitiveness (MINECO) and FEDER funds through project AYA2017-85465-P. Ellie Nicholson is gratefully acknowledged for language corrections. We acknowledge the support of the Supercomputing Wales project, which is part-funded by the European Regional Development Fund (ERDF) via the Welsh Government.

Facilities: SST(CRISP) - Swedish 1 meter Solar Telescope, NICOLE. -