© The Authors 2024

1. Introduction

Among the various spectral lines used in the diagnostic of the solar chromosphere and prominences, we have the Mg II h and k lines, located at 280.3 nm and 279.6 nm, respectively. This ultraviolet (UV) resonance doublet is known for its strong sensitivity to the physical conditions of the plasma, making it a valuable tool for probing.

Recently, there has been increased interest in these UV lines thanks to the spectroscopic observations obtained by the Interface Region Imaging Spectrograph (IRIS; De Pontieu et al. 2014) and the spectropolarimetric data collected by the two Chromospheric LAyer SpectroPolarimeter missions, CLASP2 (Narukage et al. 2016) and CLASP2.1 (McKenzie et al. 2021). These suborbital space experiments have provided unprecedented observations of Mg II h and k lines, confirming previous theoretical predictions and providing new insights into the structure and dynamics of the upper solar atmosphere (see the review by Trujillo Bueno & del Pino Alemán 2022).

Building on the success of these missions, the scientific community is now planning the Chromospheric Magnetism Explorer (CMEx) space telescope (Bryans & The CMEx Team 2023), which will focus on the spectropolarimetry of the spectral region of the Mg II h and k doublet. This new mission is expected to provide detailed information on the magnetic field and dynamics of the plasma in the chromosphere and prominences.

For the last 40 years, many studies have attempted to understand the formation of the Mg II h and k lines in prominences to explain the available prominence observations. The first 2D freestanding slab models of prominences were developed by Vial (1982) using the complete frequency redistribution (CRD) approximation for calculating the Mg II k line intensity. Later, Paletou et al. (1993) found similar results for the core of the h and k lines with both the CRD approximation and accounting for partial frequency redistribution (PRD) effects in prominences. Heinzel et al. (2014) conducted a thorough analysis using 1D prominence slab models and confirmed that the results obtained using a two-level Mg II atomic model without a continuum agree with those found using a multilevel plus continuum model. Their investigation indicated that prominences are generally optically thick in the k line, with line-center thicknesses reaching up to 10³ or 10⁴. Jejčič et al. (2018) compared 1D models with IRIS data of prominences and find that the integrated line intensities require optically thick plasma.

The magnetic field strength in quiescent solar prominences presumably ranges from a few to several tens of gauss, although it can occasionally reach significantly higher values (Hillier 2018). While in the chromosphere the h and k doublet exhibits remarkable linear polarization signals in the profile wings due to J-state interference and magneto-optical effects (Belluzzi & Trujillo Bueno 2012; Alsina Ballester et al. 2016; del Pino Alemán et al. 2016, 2020), only the line core is visible in prominences. Therefore, our study focused on the core of the Mg II k line. This line is sensitive to both the Zeeman and Hanle effects. Its circular polarization degree due to the Zeeman effect scales linearly with ℛ = 1.5 × 10⁻⁴B (see Eq. 1 of Trujillo Bueno & del Pino Alemán 2022), allowing for the determination of the line-of-sight (LOS) component of the magnetic field, B_LOS, while the transversal component in the plane of the sky (which scales with ℛ²) would be very difficult (or impossible) to determine via the Zeeman effect at the expected magnetic field strengths. The linear polarization of the Mg II k line arises dominantly from the scattering of anisotropic radiation and the Hanle effect. The critical Hanle field of the line is B_H ≈ 22 G (see Eq. 2 of Trujillo Bueno & del Pino Alemán 2022) and is therefore suitable for quiescent prominence diagnostics. However, diagnosing magnetic fields becomes challenging when the field is not constant along the LOS and when radiation transfer needs to be considered because the prominence plasma is optically thick (τ > 1) and not in local thermodynamic equilibrium (LTE), that is, when it is in the so-called non-LTE regime.

The main aim of this paper is to theoretically investigate the Stokes inversion problem of the Mg II k line in solar prominences, taking the effects of 3D radiative transfer (RT) into account. To this end, we used an optically thick 3D prominence model with a magnetic field that varies spatially along the model’s loop-like structure. Because the model’s physical properties vary along the three spatial directions, the axial symmetry of the incident radiation field can break at each point within the medium without the need of a magnetic field. Such nonmagnetic causes of symmetry breaking can have an important impact on the linear polarization signals caused by the scattering of anisotropic radiation (e.g., Štěpán & Trujillo Bueno 2013; Jaume Bestard et al. 2021), which at the line center are sensitive to the presence of magnetic fields via the Hanle effect. In addition to anisotropic radiation pumping and the Hanle effect, our study also includes the Zeeman effect, which dominates the line’s circular polarization. Given that in prominences the Mg II k line does not show extended wings, where the effects of PRD and J-state interference are very important (Belluzzi & Trujillo Bueno 2012), in this investigation we solved the 3D non-LTE RT problem assuming CRD without J-state interference. Our approach to the 3D Stokes spectral synthesis and inversion problem without assuming LTE can be found in Štěpán et al. (2022).

Here we focused on investigating the suitability of the Mg II k line for spectropolarimetric diagnostics of optically thick prominences, which requires understanding the magnetic field’s impact on the polarization of the emergent spectral line radiation. In particular, we ask if our 3D Stokes inversion, which consistently accounts for RT without assuming LTE, can uncover the global magnetic field geometry of an optically thick prominence using the Mg II k line. Section 2 describes the prominence model and the spectral synthesis of the emergent Stokes profiles, taking the effects of RT into account in the 3D model. In Sect. 3 we apply the weak field approximation (WFA; see Sect. 9.6 in Landi Degl’Innocenti & Landolfi 2004), a Bayesian approach based on the constant-property slab model, and the full 3D Stokes inversion to the synthetic data, assessing the goodness of the inference. Finally, Sect. 4 outlines our conclusions.

2. Prominence model and synthesis of the Stokes profiles

Figure 1 illustrates the prominence model we used to investigate the performance of the three inference methods we applied to the synthetic Stokes profiles. Although it is an academic prominence model with a relatively simple magnetic field geometry, it features the main ingredients that can affect the performance of the inversion, namely a nontrivial variation of the physical quantities both across the field of view (FOV) and along the LOS, and a relatively large optical thickness to manifest the effects of RT in 3D geometry.

Fig. 1. Geometry of the model prominence. Top panel: LOS of the observation. The symmetry plane of the prominence model is rotated by 45° with respect to the LOS. The red curve indicates the geometry of the magnetic field. Bottom panel: Visualization of the magnetic field lines in the 50 × 50 × 50 Mm3 spatial domain of the 3D prominence model.

For simplicity, we assumed that all Mg atoms are in the Mg II ionization stage. At chromospheric temperatures the Mg II ion is indeed the dominant species (Leenaarts et al. 2013), and at about 15−20 kK the Mg II/Mg III fraction quickly decreases and Mg III becomes the dominant species (Heinzel et al. 2014). We modeled the Mg II k resonance line using a two-level atom model and in the limit of CRD. These two approximations can be justified because we can directly specify the total population in the lower and upper levels of the k line of Mg II (avoiding the need to account for an equation of state and for ionization and recombination processes) and because the line center, dubbed k₃, is not strongly affected by PRD effects (Leenaarts et al. 2013; Trujillo Bueno et al. 2017)¹. Finally, we assumed an abundance of 7.544 for magnesium. Given the academic nature of our 3D prominence model, we assumed that hydrogen, which is the most important donor of electrons under the considered thermodynamic conditions, is fully ionized. We could then estimate the electron volume density directly from the volume density of magnesium and its abundance. The inelastic collisional rates in the Mg II k line transition have been calculated using the data from Sigut & Pradhan (1995).

The spatial domain of our prominence model is 50 × 50 × 50 Mm³; it contains a loop-like structure whose legs start at the top of the chromosphere and extends up to about 40 Mm above it; the spatial domain sits on top of the C model of Fontenla et al. (1993), hereafter the FAL-C model, at about 2.2 Mm above the visible solar surface. The Mg II number density is prescribed and decreases outward from the inner part of the prominence body. The optical thickness is maximum at the central part of the apex of the loop-like structure and exceeds 100 at the Mg II k line center. The prominence is isothermal, with a temperature of 10 kK, with a microturbulent velocity of 5 km/s (Heinzel et al. 2014). For simplicity, we do not include bulk velocities in this model. Finally, the magnetic field is such that the field lines follow the loop-like structure (see Fig. 1), with a strength reaching about 100 G close to the chromosphere and decreasing to about 40 G at the apex. These relatively strong magnetic fields have been chosen to produce a circular polarization signal above the considered noise level.

The prominence is illuminated by the underlying chromosphere, not limited to the computational domain. This illumination is axially symmetric and only depends on the angle between the propagation direction and the local vertical. The spectral profiles and their center-to-limb variation (see Fig. 2) have been calculated with HanleRT-TIC² (del Pino Alemán et al. 2016) using the FAL-C model. We note that in our solution we did not approximate the geometry of the solar chromosphere by an infinite plane, but we took the curvature of the solar surface into account.

Fig. 2. Stokes I and Q/I profiles of the emergent Mg II k line radiation calculated in the FAL-C model atmosphere for various cosines (μ) of the heliocentric angle, from 0.1 to 1. The Stokes I and Q profiles were used to obtain the illumination of the boundaries of the 3D prominence model.

We solved the non-LTE RT problem in 3D by applying the spectral synthesis mode of our code (Štěpán et al. 2022), obtaining the J_Q^K radiation field tensor components everywhere in the domain, and the emergent Stokes profiles for the chosen LOS (see Fig. 1). The resulting FOV is shown in Fig. 3. The linear polarization is due to the scattering of anisotropic radiation and the Hanle effect, while the circular polarization is caused by the Zeeman effect. For the application of the inference methods presented in Sect. 3, we selected two particular positions in the FOV (black and orange dots in Fig. 3). The spectra corresponding to these two locations, after adding Gaussian polarimetric noise with σ = 5 × 10⁻¹¹ erg s⁻¹ cm⁻² Hz⁻¹ sr⁻¹, on the order of 5 × 10⁻⁴ of the line-center intensity, are shown in Fig. 4. The material behind the black dot LOS has an optical thickness at the line center of about 160, showing the typical self-reversal in its intensity profile, while the orange dot LOS has a more modest optical thickness of about 2.5 at the line center, showing no self-reversal. In Fig. 5 we show how B_LOS and the line center opacity change with the optical thickness at the line center, τ, along the LOS for these two selected positions.

Fig. 3. Emergent radiation at each point in the 128 × 128 pixels FOV in the observation as illustrated in the top panel of Fig. 1. The I, Q, and U signals are shown at the line-center wavelength, while the V signal is shown at Δλ = −0.079 Å from the line center. The white vectors in the intensity panel show the orientation of the linear polarization at the line center. The black and orange dots indicate two particular locations in the FOV analyzed in the text.

Fig. 4. Stokes profiles with an added Gaussian noise with σ = 5 × 10−11 erg s−1 cm−2 Hz−1 sr−1 at the points indicated by the black and orange dots in Fig. 3.

Fig. 5. Variation of the LOS component of the magnetic field (blue line) and of the line-center opacity (magenta line) at the black (top panel) and orange (bottom panel) spatial points indicated in Fig. 3. The horizontal axis gives the line-center optical depth along the LOS.

3. Magnetic field inference

In this section we apply three methods to infer the magnetic field in the prominence model described in Sect. 2. In Sects. 3.1 and 3.2 we apply two commonly used inference techniques, both of them based on strong assumptions for the RT. These two methods are applied pixel by pixel, and for the presentation of the results we chose two particular positions in the FOV (see the black and orange dots in Fig. 3), which are representative of a LOS with significant optical thickness at the line center (> 100) and a LOS with a modest optical thickness at the line center (∼2.5). In Sect. 3.3 we apply our 3D Stokes inversion to infer the magnetic field vector in the whole prominence body and its surroundings. In particular, we are interested in assessing the goodness of the fits in the different methods and discussing their advantages and disadvantages.

3.1. Weak field approximation (WFA)

When the Zeeman splitting of the line’s levels produced by the magnetic field is much smaller than the spectral line width, and under other certain assumptions, it is possible to find a closed and simple expression for the circular polarization, the so-called WFA (see, e.g., Sect. 9.6 of Landi Degl’Innocenti & Landolfi 2004),

$$\begin{array}{c}\hfill V(\lambda )=-C{\lambda }_0^2{B}_{\mathrm{LOS}}{g}_{\mathrm{eff}}\frac{\partial I}{\partial \lambda }(\lambda ),\end{array}$$(1)

where C = 4.6686 × 10⁻¹³ G⁻¹ Å⁻¹, λ₀ is the wavelength of the spectral line in Å, g_eff is the effective Landé factor (7/6 for the Mg II k line), and B_LOS is the longitudinal component of the magnetic field in gauss.

The first assumption that must be fulfilled for the applicability of the WFA is that the Zeeman effect must be the only mechanism contributing to the polarization. While in solar prominences the linear polarization of the Mg II k line is produced by the scattering of anisotropic radiation and the Hanle effect, the circular polarization is dominated by the Zeeman effect. Consequently, in this subsection we focus only on the circular polarization profiles.

The second assumption for the applicability of the WFA is that B_LOS must be constant along the LOS. While this condition seems really restrictive and close to impossible to fulfill, in practice it means that B_LOS must be approximately constant along the LOS in those regions actually contributing to the emergent profiles. It is also clear from Eq. (1), which relates Stokes V and the derivative of the intensity I, that when the profiles are formed in extensive regions, the contributions to both Stokes parameters must come from the same regions along the LOS.

If we assume uncorrelated and Gaussian noise, we can use Eq. (1) to calculate B_LOS as follows (Martínez González et al. 2012a),

$$\begin{array}{c}\hfill B_{\mathrm{LOS}}=-\frac{1}{C}\frac{{\sum }_{j}V({\lambda }_j)I\prime ({\lambda }_j)}{{\sum }_j{(I\prime ({\lambda }_j))}^2}\pm \frac{\sigma }{C\sqrt{\sum _j{(I\prime ({\lambda }_j))}^2}},\end{array}$$(2)

where $I\prime ({\lambda }_j)={\lambda }_0^2{g}_{\mathrm{eff}}\frac{\partial I}{\partial \lambda }({\lambda }_j)$, λ_j are the observed wavelengths, σ is the standard deviation of the Gaussian distribution of the noise in Stokes V, and the error is computed from the covariance matrix assuming a wavelength-independent standard deviation and a confidence level of 68.3% (1σ).

In Fig. 6 we show the WFA fit to the Mg II k circular polarization profiles shown in Fig. 4, corresponding to the black and orange dots in Fig. 3, by applying Eq. (2). The fits are quite good and the corresponding B_LOS are −34.9 ± 2.3 and −36.6 ± 3.6 G. From Fig. 5, we see that the retrieved B_LOS correspond approximately to the magnetic field around the maximum of the line-center opacity. We note that the τ scale in Fig. 5 corresponds to the line center wavelength and that the opacity quickly decreases for the line wings. Therefore, even if at the line core we cannot “see” the regions with τ ∼ 80 in Fig. 5, at the near wing wavelengths we can.

Fig. 6. WFA best fits (dashed curves) of the spectra shown in Fig. 4 (solid curves). A noise level of σ = 5 × 10−11 erg s−1 cm−2 Hz−1 sr−1, corresponding to approximately to 6 × 10−4 ILC, has been added to the profiles, where ILC is the line-center intensity. The inferred longitudinal components of the magnetic field are −34.9 ± 2.3 G for the top panel and −36.6 ± 3.6 G for the bottom panel.

The WFA has the advantage of providing an estimation of B_LOS with negligible computing effort given that in our 3D model the required conditions are satisfied. However, the WFA returns a single number and an uncertainty that only accounts for the noise; there is thus no information about the magnetic field geometry and stratification, nor about other sources of uncertainty in the inferred values. For the particular cases we studied in this work, it turns out that B_LOS is relatively constant along the formation region of the profiles, keeping the same polarity along the whole LOS, and thus the inferred B_LOS are rather good estimates. However, for more complex magnetic field geometries, with potential cancellation effects (different polarities along the LOS) or strong source function gradients, the WFA does not guarantee a good estimation of B_LOS.

3.2. Constant-property slab Bayesian inversion

Most of the magnetic field inference in prominences (and filaments) over the last two decades relies on the modeling of spectropolarimetric observations in the He I triplet at 10830 Å and D₃ triplet at 5877 Å assuming a constant-property slab illuminated by the solar radiation of the underlying quiet Sun disk (see, e.g., the review by Trujillo Bueno & del Pino Alemán 2022, and references therein). Several inference methods based on this model can be found in the literature, such as look-up tables based on principal component analysis (e.g., Casini et al. 2003), minimization methods such as that implemented in the Hanle and Zeeman light (HAZEL) code (Asensio Ramos et al. 2008), or through Bayesian statistical approaches (e.g., Díaz Baso et al. 2019).

One of the main assumptions of this modeling approach is that the radiation pumping within the slab is fully dominated by the cylindrical symmetric illumination from the underlying solar disk (i.e., that the excitation of the atoms within the slab is not affected by RT within the slab). In order for this approximation to be reasonable, the optical thickness of the slab plasma should be small enough (Trujillo Bueno & Asensio Ramos 2007; Vicente Arévalo et al. 2023). The model then assumes that all properties of the plasma within the slab are constant along the LOS. Typically, a single slab is assumed in the modeling, but several components – both side by side (e.g., Xu et al. 2010) and one after the other along the LOS (e.g., Martínez González et al. 2012b) – have been considered.

When the modeling assumptions are satisfied, these inference methods can provide estimations of the magnetic field vector while allowing for the study of ambiguities and uncertainties. However, when the optical depth is on the order of, or larger than, the unity along any direction within the prominence (or filament), or if the plasma properties are not constant in the region along the LOS where the line forms, the accuracy of the inference can be severely compromised. Moreover, the inference methods based on Bayesian statistics, while providing a clear picture of the uncertainties and ambiguities, are computationally heavy, especially when considering more than a single slab.

In prominences, the Mg II k line investigated in this work typically shows larger optical thickness than the He I triplet lines (Jejčič et al. 2018). Consequently, an unsuitable performance of the constant-property slab model is, a priori, expected. At the pixel marked with a black dot in Fig. 3 the plasma of our 3D model is very optically thick at the k-line center, with an optical depth of over one hundred. Its intensity profile (black curve in Fig. 4) shows a clear self-reversal, which cannot be reproduced assuming a single constant-property slab. On the contrary, at the orange dot pixel in Fig. 3 the plasma of our 3D model has a total optical depth of about 2.5 along the LOS. This pixel is near the prominence “edge” and it can thus “see” most of the underlying chromosphere. We note, however, that the prominence body blocks some of the chromospheric radiation, so the assumption of cylindrically symmetric illumination is not fully valid. The B_LOS is approximately constant along the LOS for the orange dot pixel (see the bottom panel of Fig. 5), but the inclination of the magnetic field vector changes along the LOS.

We calculated the Bayesian posterior distribution for the case of a single-component constant-property slab model inversion of the emergent Stokes profiles at the location of the orange dot pixel. The parameters of our model are (i) B_LOS, (ii) the cosine of the polar angle of the magnetic field inclination with respect to the solar radius (μ_B), (iii) its azimuth in the plane normal to the radius with respect to the projection of the LOS on such a plane (χ_B), (iv) the thermal width of the line (Δv_D), and (v) and the line-center optical thickness (τ). We used a Jeffreys prior³ for τ and uniform priors for the rest of the parameters, between 0 and 1000 G for B_LOS, between −1 and 1 for μ_B, between 0 and 2π for χ_B, and between 0.1 and 10 km/s for Δv_D.

Figure 7 shows the marginalized posterior distributions for B_LOS, μ_B, and χ_B. Although the physical properties of the 3D model at the selected pixel do not fulfill the applicability conditions, because the optical depth is larger than unity, the illumination is not cylindrically symmetric, and μ_B changes along the LOS, the inferred B_LOS turns out to be as good as with the WFA (see Sect. 3.1) and, moreover, the constant-property slab approach is capable of finding χ_B up to the ambiguities. This inversion method is much slower than the WFA, but in exchange it provides additional physical information.

Fig. 7. Marginal posteriors of the Bayesian model parameters. The orange curves in the bottom panels indicate the variation of the actual parameters along the chosen LOS in the spatial domain of the original model. See the main text for details.

Even though the B_LOS inference is rather good, the magnetic field strength is overestimated by about a factor of 2 (hence the magnetic field energy density by a factor of 4) due to the significant uncertainty in μ_B. Due to the symmetry assumed in the 3D model, there are ambiguous solutions for both μ_B and χ_B.

For an optically thin prominence with a not-too-complex magnetic field geometry, approaches based on this constant-property slab model (Bayesian inference, principal component analysis, etc.) seem to be optimal since they can provide maximum information on uncertainties even when there are no self-consistent RT constraints.

3.3. 3D Stokes inversion

The third inference method we applied to the synthetic Stokes profiles that we calculated by solving the non-LTE RT problem in our 3D prominence model is the 3D Stokes inversion approach described in Štěpán et al. (2022). This method approaches the inverse problem by finding the physical quantities in the whole spatial domain of the 3D model simultaneously. Different regions of the model’s spatial domain are coupled by the transfer of polarized radiation and, in addition, the solution can have additional constraints such as those from the magneto-hydrodynamic equations. As described in detail in the aforementioned paper, this mesh-free method does not rely on calculating a sequence of self-consistent forward models that lead to the minimum of a merit function. Instead, it follows an unconstrained minimization method in which unphysical solutions are allowed but penalized via regularization terms in the merit function. This allows relatively accurate solutions to be obtained within a significantly shorter computing time. This method does not only provide a solenoidal magnetic field vector B everywhere in the model’s spatial domain, but for our particular case also other thermodynamic quantities such as the atomic number density.

We solved the inversion problem using 480 CPU cores of the OASA computer of the Astronomical Institute in Ondřejov. The solution shown here was reached in about 20 hours or 10⁴ CPU hours. As mentioned above, we penalized unphysical (non-self-consistent) solutions and magnetic field vector distributions not fulfilling ∇ ⋅ B = 0. The initial state of the magnetic field vector components has been chosen so that all the amplitudes of the basis functions were randomly sampled from a normal distribution with zero mean and a standard deviation of 20 G. The initial guess of the atomic number density has been a constant function equal to log₁₀N = 1 in the units of [N] = cm⁻³.

To evaluate the goodness of the inversion, we quantified the error in the inferred magnetic field with

$$\begin{array}{c}\hfill e=\frac{\Vert {\mathit{B}}_{\mathrm{inv}}-\mathit{B}\Vert }{\Vert \mathit{B}\Vert },\end{array}$$(3)

where B is the true magnetic field vector (see Fig. 1) and B_inv is magnetic field vector resulting for the inversion. Figure 8 shows the histogram of this quantity evaluated at 10⁵ randomly located points in the whole 3D domain (blue histogram) and with the same number of random points within the prominence body (orange histogram). As the Stokes parameters of the Mg II k line give us information about the “visible” surface of the prominence, whose size is comparable to that of the model’s domain, the two histograms are relatively close.

Fig. 8. Histogram of the errors (see Eq. 3) in the magnetic field inferred via the 3D Stokes inversion.

We find a typical relative error in the inferred magnetic field vector of about 20−30%. However, it is important to emphasize that using this relative error as a measure of the quality of the inference can be misleading. For instance, a small spatial displacement of a magnetic loop, for example at position (X, Z) = (0, −25) Mm, can lead to a very significant relative error. In Fig. 9 we show a cut in the X–Z plane of the original and inferred spatial distribution of the magnetic field vector and of the atomic number density, demonstrating that the inversion does a pretty good job in recovering the overall physical model.

Fig. 9. Comparison of vertical slices through the Y = 0 Mm plane in the original model and in the model resulting from the 3D inversion. The left panel shows the magnetic field lines and the density in the original model, and the right panel shows the same quantities in the inferred model. The background color in both panels represents the log-density of Mg II ions. Our 3D Stokes inversion successfully captures the essential morphology of the original model.

Given the available CPU time, we performed about two dozen inversions with different initializations of the model and different setups of the inversion algorithm. This is not enough to make any quantitative conclusions about the ability of different setups to perform the inversion. But overall, we have found that the model has (i) converged to a solution similar to the one presented in this section, (ii) ended up in a local minimum of the merit function, or (iii) has diverged. This last situation has occurred in cases where the inversion algorithm setup can be considered too aggressive (i.e., too large iteration steps, etc.). Importantly, the solution did never converge to any solution fundamentally different and ambiguous from the one presented here. This is indicative that, at least for this 3D academic prominence model, the global consistency imposes constraints strong enough to remove the ambiguities present in approaches assuming either no coupling between the FOV pixels or unrealistic symmetric conditions. Consequently, the nontrivial spatial coupling through RT seems to impose strong constraints on the solutions that are possible, leading to robust results.

Concerning the linear polarization signals caused by scattering processes, it is important to emphasize that they are sensitive mostly to regions not much deeper than optical depth unity along the LOS, where the anisotropy of the radiation can be substantial so as to produce observationally relevant linear polarization signals. Given that the Mg II k line can be very optically thick in this prominence model, the inversion cannot be expected to perfectly recover the magnetic field in these optically inaccessible regions for which the Stokes profiles do not give us enough information. Nevertheless, the 3D inversion provides a sufficiently good estimate of the global structure and strength of the magnetic field.

4. Discussion and conclusions

We solved the non-LTE problem of the generation and transfer of polarized radiation in the Mg II k line in an academic 3D prominence model, where the magnetic field lines follow an optically thick loop-like structure. We chose this relatively simple geometry to facilitate the comparison between different magnetic field inference approaches. We have found that, despite 3D RT effects, for this relatively simple prominence model both the WFA and the constant-property slab approaches can give good estimations of the longitudinal component of the magnetic field from the observed Stokes V profile. To some degree, the constant-property slab approach is able to estimate the magnetic field transversal component from the linear polarization. To recover the full 3D picture, a full Stokes inversion method that includes the effects of RT in 3D is necessary, such as the one we applied in this study.

We have found that the WFA provides a fast estimation of B_LOS that is suitable at least for a simple structure such as our prominence model. For profiles with small enough optical thickness along the LOS, methods based on the constant-property slab model seem to be the best approach in terms of recovered information and computational time required, if the prominence has a simple enough geometry. When the optical thickness exceeds unity, we can take advantage of the RT coupling by applying our full 3D RT approach and inferring the global structure of the prominence.

Due to the significant optical thickness in our 3D model, also found in actual prominences (e.g., Jejčič et al. 2018), the Stokes Q and U profiles are mostly sensitive to the outermost layers of the prominence (in the direction toward the observer). This is more critical in prominences because, unlike on-disk observations, the wings are not observed. In contrast with the He I lines more popularly used in prominence diagnostics, the Mg II k line is a strong resonance line with a spectral structure. This entails that the chromospheric radiation illuminating the prominence plasma is sensitive to velocities, introducing frequency shifts between the absorption profile and the incoming illumination spectrum, and to variations in the chromospheric surface that produce significant changes in the intensity of this line.

The significant optical thickness can be even more problematic if the real physical scenario is that of many small-scale threads with their own prominence-corona transition regions (e.g., Gunár et al. 2007). In this case, the inversion can become extremely challenging. Therefore, spectral lines with smaller optical thickness in prominences, which provide more spatially averaged information on the magnetic field, may provide very valuable additional information.

The scientific importance of developing a space telescope that enables routine spectropolarimetric observations in the near-UV region of the Mg II h and k lines cannot be overestimated, because the polarization signals that the combined action of scattering processes and the Hanle and Zeeman effects introduce in this spectral region encode a wealth of information on the magnetism and geometry of chromospheric and prominence plasmas. Equally important is the development of advanced plasma diagnostic techniques capable of providing reliable information on the magnetic field vector.

Acknowledgments

J.Š. acknowledges the financial support from project RVO:67985815 of the Astronomical Institute of the Czech Academy of Sciences. T.P.A.’s participation in the publication is part of Project RYC2021-034006-I, funded by MICIN/AEI/10.13039/501100011033, and the European Union “NextGenerationEU”RTRP. T.P.A. and J.T.B. acknowledge support from the Agencia Estatal de Investigación del Ministerio de Ciencia, Innovación y Universidades (MCIU/AEI) under grant “Polarimetric Inference of Magnetic Fields” and the European Regional Development Fund (ERDF) with reference PID2022-136563NB-I00/10.13039/501100011033.

References

All Figures

	Fig. 1. Geometry of the model prominence. Top panel: LOS of the observation. The symmetry plane of the prominence model is rotated by 45° with respect to the LOS. The red curve indicates the geometry of the magnetic field. Bottom panel: Visualization of the magnetic field lines in the 50 × 50 × 50 Mm3 spatial domain of the 3D prominence model.
In the text

	Fig. 2. Stokes I and Q/I profiles of the emergent Mg II k line radiation calculated in the FAL-C model atmosphere for various cosines (μ) of the heliocentric angle, from 0.1 to 1. The Stokes I and Q profiles were used to obtain the illumination of the boundaries of the 3D prominence model.
In the text

Fig. 3. Emergent radiation at each point in the 128 × 128 pixels FOV in the observation as illustrated in the top panel of Fig. 1. The I, Q, and U signals are shown at the line-center wavelength, while the V signal is shown at Δλ = −0.079 Å from the line center. The white vectors in the intensity panel show the orientation of the linear polarization at the line center. The black and orange dots indicate two particular locations in the FOV analyzed in the text.

In the text

	Fig. 4. Stokes profiles with an added Gaussian noise with σ = 5 × 10−11 erg s−1 cm−2 Hz−1 sr−1 at the points indicated by the black and orange dots in Fig. 3.
In the text

	Fig. 5. Variation of the LOS component of the magnetic field (blue line) and of the line-center opacity (magenta line) at the black (top panel) and orange (bottom panel) spatial points indicated in Fig. 3. The horizontal axis gives the line-center optical depth along the LOS.
In the text

	Fig. 6. WFA best fits (dashed curves) of the spectra shown in Fig. 4 (solid curves). A noise level of σ = 5 × 10−11 erg s−1 cm−2 Hz−1 sr−1, corresponding to approximately to 6 × 10−4 ILC, has been added to the profiles, where ILC is the line-center intensity. The inferred longitudinal components of the magnetic field are −34.9 ± 2.3 G for the top panel and −36.6 ± 3.6 G for the bottom panel.
In the text

	Fig. 7. Marginal posteriors of the Bayesian model parameters. The orange curves in the bottom panels indicate the variation of the actual parameters along the chosen LOS in the spatial domain of the original model. See the main text for details.
In the text

	Fig. 8. Histogram of the errors (see Eq. 3) in the magnetic field inferred via the 3D Stokes inversion.
In the text

Fig. 9. Comparison of vertical slices through the Y = 0 Mm plane in the original model and in the model resulting from the 3D inversion. The left panel shows the magnetic field lines and the density in the original model, and the right panel shows the same quantities in the inferred model. The background color in both panels represents the log-density of Mg II ions. Our 3D Stokes inversion successfully captures the essential morphology of the original model.

In the text