Forward Modeling of Solar Coronal Magnetic Field Measurements Based on a Magnetic-field-induced Transition in Fe x

For a detailed description of the MIT process in Fe x, we refer to Li et al. (2015, 2016). Here we just present a brief introduction. An electric dipole (E1) transition can occur from the 3p⁴3d ⁴D_5/2 level to the ground state 3p^5 2P_3/2. In the absence of an external magnetic field, the 3p⁴3d ⁴D_7/2 level can only decay to the ground state through a forbidden magnetic quadruple (M2) transition. In the presence of an external field, through a quantum mixing of the ⁴D_7/2 level with the close-by ⁴D_5/2 level, an MIT decay channel from the ⁴D_7/2 level to the ground state can be induced, with a transition rate increasing with the magnetic field strength. The transition probability of MIT, $A_{\mathrm{MIT}}$, according to the first-order perturbation theory, depends on both the magnetic field strength (B) and the energy separation ($\Delta E$) between the ⁴D_7/2 and ⁴D_5/2 levels:

It is worth noting that the wavelength difference between the E1 (257.259 Å) and M2/MIT (257.261 Å) lines is too small to be resolved in observations. So in the following the 257 Å line intensity refers to the total intensity of the E1, M2, and MIT lines.

The diagnostic technique is based on a simple concept: the intensity of the MIT emission and thereby the 257 Å (MIT+M2+E1) line intensity is directly and significantly affected by the local magnetic field strength; it therefore can be used to measure the magnetic field strength by measuring the intensity ratio of a line pair, i.e., the 257 Å line and a reference Fe x line that is not sensitive to the magnetic field. The Fe x 174, 175, 177, 184, and 255 Å lines can be observed simultaneously along with the 257 Å line by Hinode/EIS, and therefore could be adopted as candidates of the reference line. The atomic transitions corresponding to these lines are listed in Table 1. We used CHIANTI database (version 10.0; Dere et al., 1997; Del Zanna et al., 2021) to calculate the line emissivities. The original source for these data are Del Zanna et al. (2012) for level energies and collision data, Wang et al. (2020) for radiative transition data, and Li et al. (2021) for $A_{\mathrm{MIT}}$ values.

These line ratios also depend on the electron density and temperature. Taking the 257 and 174 Å line pair as an example, we show the intensity ratio ($I_{257}/I_{174}$) as a function of the magnetic field at different densities and temperatures in Figure  1. Figure  1(a) shows the $I_{257}/I_{174}$ intensity ratio as a function of the magnetic field strength for four different values of the electron density, i.e., 10^8.6, 10^9.0, 10^9.4, and 10^9.8 cm^-3. These densities values are typical for different structures in the solar corona. A fixed temperature of 10^6.0 K, around which the contribution functions of the Fe x lines peak, was adopted. As expected, since the upper ⁴D_7/2 level of the 257 Å line is metastable, the $I_{257}/I_{174}$ ratio is also sensitive to electron density. In Figure  1(b) we took a typical density of active regions (10^9.4 cm^-3) and four different temperatures of 10^5.96, 10^6.00, 10^6.04, and 10^6.08 K. It is clear that the line ratio also changes with temperature. A similar behavior is also seen for the ratios of 257 Å and other reference lines. Figure  1 indicates that we need to determine both the temperature and density before deriving the magnetic field strength using the MIT technique. Fortunately, the Fe x 184/345 Å line pair can be used to diagnose the electron temperature, and the Fe x 174/175 Å line pair is good for coronal density diagnostics (Phillips et al., 2008; Del Zanna & Mason, 2018).

Another important atomic parameter that affects the accuracy of the coronal magnetic field measurement is the energy separation $\Delta E$ between the ⁴D_7/2 and ⁴D_5/2 levels (see Equation (1)). There have been several attempts to estimate the values of $\Delta E$ from solar observations as well as laboratory measurements (Sandlin & Tousey, 1979; Thomas & Neupert, 1994; Brosius et al., 1998; Li et al., 2016; Judge et al., 2016). More recently, based on observations with the Solar Ultraviolet Measurements of Emitted Radiation (SUMER, Wilhelm et al., 1995) onboard the Solar and Heliospheric Observatory (SOHO), Landi et al. (2020b) obtained a value of $\Delta E=2.29\pm 0.50~{}$cm^-1, which was adopted for the computation of $A_{\mathrm{MIT}}$ and thereby the magnetic field diagnostics in the present work. Most analyses in this study were performed using $\Delta E=2.29$ cm^-1. In Section 5, we adopted $1.79$ and $2.79$ cm^-1 as the lower and upper limits of $\Delta E$, respectively, to investigate the impact of the uncertainty in the energy separation on magnetic field measurements.

Finally, we created lookup tables of the contribution functions $G(T,n_{e},B)$ for the Fe x lines listed in Table 1 at a large number of electron temperatures (log${}_{10}T$, from 5.6 to 6.4 with an interval of 0.002), electron densities (log${}_{10}n_{e}$, from 6 to 15 with an interval of 0.008), and magnetic field strengths ($B$, from 0 to 6000 G with an interval of 5 G). Since these Fe x spectral lines are emitted from the same ion, it is reasonable to assume that they form in roughly the same layers/structures of the corona. In other words, these lines sample almost the same plasma in the line of sight (LOS). In such a case, the intensity ratio of two lines may be approximated by the ratio of their contribution functions at a certain density, temperature and magnetic field.

Figure  1 shows that the line ratio $I_{257}/I_{174}$ is more sensitive to the magnetic field when the density is higher, which implies that this technique is more suitable for magnetic field diagnostics in active regions where the density is usually higher. So far, the MIT-based magnetic field measurements were all made for active regions (Si et al., 2020; Landi et al., 2020a, 2021; Brooks & Yardley, 2021). Therefore, we constructed a 3D MHD model of an active region to verify the suitability of this new technique and evaluate the accuracy of the measured magnetic field strength.

We used the MURaM radiation MHD code (Vögler et al., 2005; Rempel, 2017) to simulate an active region. The model contains an area of 98.304$\times$49.152 Mm² in the horizontal direction and extends from 7.488 Mm below to 41.6 Mm above the photosphere in the vertical direction. The horizontal and vertical grid spacings in the model are 192 and 64 km, respectively. The lateral boundary are periodic. The top boundary is half open and only allows for outflows. A torus flux tube imported from the bottom boundary (Rempel & Cheung, 2014) emerges into the photosphere and gives to the active region.

When the active region evolves into a stable state, we took a snapshot from the model. Figure  2(a) shows the vertical component of the photospheric magnetic field in the snapshot. There is a bipolar sunspot pair with magnetic field strengths of several thousand Gauss in the simulation box. The plasma temperatures at coronal heights are of the order of $\sim$10^5.9-6.3 K, which are typical for the solar corona. Under the situation of ionization equilibrium, the Fe x lines are known to form in this temperature range.

In order to perform forward modeling, we synthesized emissions of the Fe x lines listed in Table 1. The emissivity $\epsilon_{i}$ of the Fe x lines can be written as

where $G_{i}(T,n_{e},B)$ is the contribution function and the subscript $i$ ($i=$174, 175, 177, 184, 255, 257, and 345) corresponds to the wavelength of an Fe x line in the unit of angstrom. We first calculated the emissivity of each line at each grid point and then integrated the emissivity along the LOS. Hence, the line intensity can be written as

Figure  2(b) shows the Fe x 174 Å line intensity map after integration of the emissivity along the vertical direction, assuming location of the active region at the solar disk center. Typical coronal loop structures are clearly present in the synthesized coronal image. Similar structures are also seen in the intensity maps of other Fe X lines.

Furthermore, we defined an emission-weighted formation height of the Fe x 174 Å line as

where $\epsilon$ and $z$ are the emissivity and height at each grid point, respectively. It is worth noting that the emission can extend over a wide range of height, and thus the formation height obtained from Equation (4) can only be regarded as an average height of the Fe x 174 Å–emitting plasma. A map of the formation height of the Fe x 174 Å line is shown in Figure  2(c), from which we can see that $h_{em}$ varies at different regions and extends from below 6 Mm near the sunspots (at lower parts of the coronal loops) to $10-20$ Mm in higher parts of the loops. Similar results were also obtained for other Fe x lines.

As discussed in Section 2.1, the intensity ratios of the Fe x line pairs used to derive the magnetic field strength are also temperature- and density-sensitive. Thus, we need to determine the temperature and density before conducting magnetic field measurements. We first adopted a fixed temperature of $T_{0}=10^{6.0}$ K, around which the contribution functions of the Fe x lines reach their peaks; then we used the 174/175 Å line ratio for the density estimation. Since the intensity ratio $I_{174}/I_{175}$ is barely sensitive to the magnetic field, we can determine the electron density using the ratio of the contribution functions of the Fe x 174 and 175 Å lines calculated in Section 2.1, as

Once the density at each pixel (e.g., each [x,y] position in Figure  2) in the coronal image, $n_{e}^{*}$, is determined, the intensity ratio between the Fe x 257 Å line and a reference line ${I_{257}}/{I_{i}}$ ($i=174,175,177,184,255$) only depends on the magnetic field strength,

meaning that we can derive the coronal magnetic field strength from the ${I_{257}}/{I_{i}}$ value at each pixel.

Because all the spectral lines we used in this work are emitted from the Fe x ions, the derived magnetic field strengths should be the field strengths at heights of strong Fe x line emissions. However, the magnetic field strengths along the LOS are often inhomogeneous. Considering the optically thin nature of the coronal EUV emission, the derived magnetic field strengths should be the emission-weighted averaged field strengths. For the purpose of comparison, we defined the coronal magnetic field strength at each pixel in the model ($B_{0}$) as the emissivity-weighted field strength,

Equation (7) can provide a precise estimation of the magnetic field strength when the field strength and emissivity are both constant within the line-formation height range along the LOS, or the emissions of the Fe x lines are from an infinitely thin layer along the LOS. However, the Fe x line emissions in both the real corona and our model are from a wide range along the LOS. Nevertheless, Equation (7) still gives one of the best estimations of the coronal magnetic field strength in the model. We took the vertical direction as the LOS and present the $B_{0}$ map in Figure  3(a), which shows that the coronal magnetic field strengths range from about one hundred to a thousand Gauss. Since the emission of an Fe x line roughly scales with the electron density squared, and the density generally decreases with height, $B_{0}$ should reflect the field strength in the lower corona.

Having determined the electron density and magnetic field strength under a fixed temperature using the technique introduced in Section 3.1, we can compare the measured magnetic field to the model field. Figure  3 (b-f) presents the maps of magnetic field strengths obtained from the 257/174, 257/175, 257/177, 257/184, and 257/255 Å line ratios, respectively. The ratios involving the 174, 175, 177, and 184 Å lines can only provide measurements of magnetic field at lower parts of the coronal loops, or around sunspot regions. While for the 257/255 Å line ratio, the MIT technique can yield a value of field strength at almost every pixel. However, the field strengths in the upper parts of loops are largely overestimated compared to the model values shown in Figure  3 (a).

To explore the reasons why the MIT technique fails in regions far away from the sunspots and why the magnetic field measurements using the 255 Å line and other reference lines behave differently, we present the intensity ratios of the 257/174, 257/175, 257/177, 257/184, and 257/255 Å line pairs as a function of temperature in Figure  4. We can see that all the line ratios are temperature-dependent, i.e., $I_{257}/I_{255}$ increases monotonically with temperature while the others decrease monotonically with temperature, meaning that the choice of temperature in the calculations will significantly affect the results of magnetic field measurements. Furthermore, the temperature in the real coronal varies in space, and the formation temperature of an Fe x emission line at each pixel in the model can be different from the peak temperature of its contribution function (Peter et al., 2006). To verify this, similar to Equations (4) and (7), we defined the emissivity-weighted coronal density and temperature in the model as:

The calculated density and temperature maps are presented in Figure  5 (a) and (b), respectively. At each pixel these values could be regarded as the average density and temperature within the formation height range of the Fe x 174 Å line. It is obvious that the emissivity-weighted temperature changes significantly at different locations. As a result, if we underestimate the temperature at one location, the magnetic field strength obtained using the 257/255 Å line ratio will be overestimated since $I_{257}/I_{255}$ increases monotonically with temperature (see Figure  4). While it will be underestimated using the other line ratios. Note that all these ratios increase monotonically with the magnetic field strength (e.g., Figure  1). Also, if the temperature is underestimated at locations of very weak magnetic field, we will fail to obtain a field strength value using the 257/174, 257/175, 257/177, and 257/184 Å line ratios, as the ratio values will be lower than those corresponding to a field strength of zero Gauss (see Figure  1). This exactly explains the behavior in the upper parts of the coronal loops, where the magnetic field and temperature are expected to be weaker and higher (Figure  5(b)), respectively. Thus, the assumption of a fixed temperature we made in the calculations is not appropriate, and we have to determine the local temperature at each pixel for magnetic field diagnostics.

As discussed in Section 3.3, the uniform temperature distribution assumption has obvious limitations and introduces significant uncertainties in the magnetic field measurements. So, in addition to the density, we also need to determine the local formation temperature of the Fe x emission lines before the magnetic field measurement at each pixel.

The Fe x 174 or 184 Å lines can provide temperature diagnostics when taken as a ratio with respect to any of the 257, 345, and 6374 Å lines (Del Zanna & Mason, 2018). In this work, we adopted the 184/345 Å line ratio for temperature measurements because (1) the 257 Å line involving the MIT transition is used to measure the magnetic field; (2) the 174 Å line is used for density diagnostics; (3) the 6374 Å line is difficult to be detected on the solar disk and affected by photoexcitation. In addition, the large wavelength difference between the 6374 Å line and the 174 or 184 Å line poses a formidable challenge on simultaneous well-calibrated observations.

Figure  6 presents the theoretical relationships between the 174 and 175 Å ratio and electron density, and between the 184 and 345 Å ratio and electron temperature. We can see that the 174/175 (184/345) Å line ratio also changes slightly with the temperature (density). To provide a simultaneous estimation of the density and temperature, we defined a norm value ($L$), based on the least-squares method, as

where $I_{174}^{*}$, $I_{175}^{*}$, $I_{184}^{*}$, and $I_{345}^{*}$ are, respectively, intensities of the Fe x 174, 175, 184, and 345 Å lines at each pixel. The temperature ($T^{*}$) and density ($n_{e}^{*}$) can thus be simultaneously determined when $L$ reaches a minimum at each pixel. To do this, we created a lookup table for $L$ at a large number of electron densities and temperatures and then selected the minimum value from the table. The maps of the determined density and temperature are shown in Figure  5 (c-d). It is obvious that the derived density and thermal structures match very well with the ones calculated from Equations (8) and (9) (Figure  5 (a-b)). Furthermore, we defined the differences between the measured temperature/density and the values obtained from Equations (8) and (9) as:

The histograms of $n_{err}$ and $T_{err}$ obtained using the least-squares method and the assumption of a fixed temperature of 10⁶ K discussed in Section 3.1 are shown in Figure  7. We can see that the least–squares method can significantly reduce the differences. Besides, the differences in density and temperature are less than 10% for more than 95% pixels using the least–squares method, demonstrating that the least–squares method can provide a reliable estimation of the coronal density and temperature. Using $T^{*}$ and $n_{e}^{*}$ at each pixel, we can then determine the magnetic field strength using the MIT technique, i.e., from the line ratios between the 257 Å line and other reference lines as shown in Equation (6).

We first performed forward modeling assuming that the LOS is along the vertical ($z$) direction. The modeling result, shown in Figure  8, could help us evaluate the suitability of the MIT technique for measurements of the coronal magnetic field in disk-center observations. It is obvious that the new method has significant improvement compared to the results in Figure  3. First, the new method provides a more accurate estimation of the magnetic field strength. Second, the new method can provide a measurement of the magnetic field strength not only around the footpoints but also in higher parts of the coronal loops. However, we also noticed that similar to the results shown in Figure  3, the 257/174, 257/175, 257/177, and 257/184 Å line ratios appear to underestimate the field strengths around the loop footpoints. While in the same regions the field strengths derived from the 257/255 Å line pair are closer to the model values. It is worth mentioning that our technique fails in regions without obvious Fe x emissions, and thus the temperature, density, and magnetic field measurements show lanes at $x\approx 25$ and 75 Mm. We examined the magnetic field structures in these regions and found that the field lines are mostly vertical and extend to the top boundary of the simulation box. Continuous plasma outflows along these field lines evacuate materials in these regions, leading to very low densities and intensities in these regions. When the density is too low, the ${I_{257}}/{I_{i}}$ ratio becomes insensitive to the magnetic field (Figure  1(a)). In such a case, a reliable magnetic field measurement is impossible.

In order to understand the potential and limitations of this method better, we present the joint probability density functions (PDF) of the field strengths in the coronal model $B_{0}$ and the MIT-measured values $B_{1}$ in Figure  9. The solid white line represents the $B_{0}=B_{1}$ reference line, i.e., the magnetic field strengths measured through the MIT technique are the same as the values in the model. In general, there is a good agreement between $B_{0}$ and $B_{1}$ for all the line pairs we used to diagnose the magnetic field. We noticed that the magnetic field strengths are slightly underestimated when the field is stronger than $\sim$600 G. This is because the density-sensitive 174/175 Å line ratio is slightly dependent on the magnetic field strength, which was not considered in our calculation. In the presence of an external magnetic field, the density derived using the above method was actually underestimated. From Figure  6(b) we can see that the temperature was also underestimated. For the 257/174, 257/175, 257/177, and 257/184 Å line ratios, an underestimation of the density or temperature will lead to an underestimation of the magnetic field strength (Figure  1). This effect will be magnified at locations of strong magnetic field, i.e., around loop footpoints. While for the 257/255 Å line ratio, an underestimation of the density and temperature will lead to an underestimation and overestimation of the field strength, respectively. In weak-field regions such as the top parts of coronal loops, the temperature effect dominates, leading to an overestimation of the field strength. Around the loop footpoints where the field is strong, the temperature and density effects may cancel out, which explains the fact that the field strengths derived from the 257/255 Å line ratio are close to the model values there.

Nevertheless, Figure  8 and Figure  9 show that this technique can provide reasonably good measurements of the coronal magnetic field strengths in most parts of the active region. The measurements have a high degree of accuracy when the field strengths are in the range of a few hundred Gauss, which are the typical values of field strength in the lower corona of active regions.

In addition, for each pixel we found the height $h_{B_{0}}$ where the model magnetic field is equal to the emissivity-weighted field strength $B_{0}$ calculated from Equation (7). For most pixels the magnetic field strength decreases with height, and we could easily found $h_{B_{0}}$. However, at some pixels there might be multiple heights where the model field is equal to $B_{0}$. In such cases, we took the largest height as $h_{B_{0}}$. The result is presented in Figure  10 (a). We found that $h_{B_{0}}$ increases from below 6 Mm near the footpoints of coronal loops to above 20 Mm in upper parts of the loops, which is roughly consistent with the average formation height of the Fe x 174 Å line ($h_{em}$) shown in Figure  2 (c). For the purpose of comparison, we also found the height ($h_{B_{1}}$) where the model magnetic field is equal to the MIT-measured field $B_{1}$ at each pixel (see Figure  10 (b–f)). For all the line pairs, $h_{B_{1}}$ distributions roughly match the $h_{B_{0}}$ distribution. Around the sunspots regions (footpoints and lower parts of the coronal loops), the heights are 3–9 Mm. The heights of the magnetic field calculated using the 257/174, 257/175, 257/177, and 257/184 Å line pairs are higher than 20 Mm around the loop tops, which are comparable to the values of $h_{B_{0}}$. The 257/255 Å line pair overestimate the magnetic field strengths at loop tops. Since the field strength decreases with height, the heights where the model field strengths equal to the MIT-measured field strengths are lower than $h_{B_{0}}$.

So far, all the analyses have assumed that the active region in the model is located at the disk center, i.e., the LOS is along the vertical (z) direction. In this section, we chose a different LOS, i.e., along the y-direction, which corresponds to an off-limb observation of the active region. By integrating the emissivity along this new LOS, the intensities of the Fe x lines were calculated (Equation (3)). As an example, the intensity map of the 174 Å line is shown in Figure  11. We can see loop structures that are similar to coronal loops in off-limb coronal observations of the real Sun. The coronal loops are rooted in the sunspots and extend to heights of over 20 Mm. In the sunspot regions and above, the majority of the Fe x emission originates from a height lower than 5 Mm, around the loop footpoints. While between the two sunspots, a significant fraction of the Fe x emission comes from the upper parts of the coronal loops (above 10 Mm). These are consistent with the findings shown in Figure  2 (c) as well as in Figure  10. The intensity images of other Fe x lines exhibit a similar pattern.

The same technique described in Section 4.1 was applied to derive the temperature, density, and magnetic field strength. A comparison between the model magnetic field and the MIT-measured field is presented in Figure  12. Figure  12 (a) shows the $B_{0}$ map in the coronal model and Figure  12 (b-f) are the results obtained using the 257/174, 257/175, 257/177, 257/184, and 257/255 Å line pairs, respectively. Overall, the MIT technique provides a reasonably good estimation of the magnetic field in the coronal loop structures. Similar to the results in Section 4.2, the MIT-measured magnetic field strengths in the lower parts of coronal loops agree well with the field strengths in the coronal model; the 257/255 Å line pair appear to overestimate the magnetic field at the loop tops. At larger heights ($z\geqslant$ 20 Mm) and in regions directly above the sunspots ($x\approx 25$ and 75 Mm), the electron densities and Fe x line intensities are very low. From Figure  1(a) we can see that the ${I_{257}}/{I_{i}}$ ratio becomes insensitive to the magnetic field at very low densities. So in these regions our technique could not provide reliable measurements of the magnetic field.

In summary, the new technique developed in Section 4.1 was applied to both synthetic disk-center (see Section 4.2) and off-limb (see Section 4.3) observations and provided an overall good estimation of the coronal magnetic field. However, there are some limitations caused by the choice of the Fe x lines when applying to real observations.

A critical point to note is that different lines used in this work are often observed by different detectors or even by different instruments. For example, among the line pairs 184/345 and 174/175 Å which are used for the temperature and density measurements, the 345 Å line was often not observed with the other three lines using the same instrument. This poses a formidable challenge on the relative radiometric calibration of different instruments.

The line pairs used to derive the magnetic field have been observed simultaneously by Hinode/EIS but using different detectors. The 257 and 255 Å lines are observed by the long-wavelength detector, while the other reference lines, i.e., 174, 175, 177, and 184 Å lines, are taken by the short-wavelength detector. This makes the relative calibration for the two detectors a critical factor that could affect precise measurements of the coronal magnetic field. Radiometric calibration has been proved to be one of the main sources of uncertainties in the measured magnetic field (Landi et al., 2020a). In this respect, 257/255 Å is the best choice for magnetic field measurements. However, the Fe x 255 Å line is often too weak to have enough signal-to-noise ratios for spectral analysis.

The limitations of current EUV spectrometers impede the application of the least-squares diagnostic technique and thereby magnify the uncertainties of the magnetic field measurements based on the MIT method. Thus, next-generation EUV spectrometers that can simultaneously observe the Fe x lines mentioned above and allow an accurate radiometric calibration in a wide range of wavelength, e.g., 170–350 Å, are highly desired.