He i Vector Magnetic Field Maps of a Sunspot and Its Superpenumbral Fine-Structure

Abstract

Advanced inversions of high-resolution spectropolarimetric observations of the He i triplet at 1083 nm are used to generate unique maps of the chromospheric magnetic field vector across a sunspot and its superpenumbral canopy. The observations were acquired by the Facility Infrared Spectropolarimeter (FIRS) at the Dunn Solar Telescope (DST) on 29 January 2012. Multiple atmospheric models are employed in the inversions because superpenumbral Stokes profiles are dominated by atomic-level polarization, while sunspot profiles are Zeeman-dominated, but also exhibit signatures that might be induced by symmetry-breaking effects of the radiation field incident on the chromospheric material. We derive the equilibrium magnetic structure of a sunspot in the chromosphere and furthermore show that the superpenumbral magnetic field does not appear to be finely structured, unlike the observed intensity structure. This suggests that fibrils are not concentrations of magnetic flux, but are instead distinguished by individualized thermalization. We also directly compare our inverted values with a current-free extrapolation of the chromospheric field. With improved measurements in the future, the average shear angle between the inferred magnetic field and the potential field may offer a means to quantify the non-potentiality of the chromospheric magnetic field to study the onset of explosive solar phenomena.

Appendix: The Radiation Field Tensor Near a Sunspot

The pumping radiation field responsible for the development of population imbalances and quantum coherences in the orthohelium atomic system can be fully specified by the irreducible components of the statistical spherical tensor given in Equation (1). In the presence of symmetry-breaking structure, the radiation field tensor can be determined by numerically integrating the radiation field given by the observed continuum structure of the region. Our observations give the normalized continuum intensity near 1083 nm. Thus, the absolute photometric intensity across the region can be inferred from the observations using the known limb-darkening law at 1083 nm. Then, by transforming first to the local frame of reference of a given point in the atmosphere at height, h, we find the nonzero radiation field tensor components by applying the following expressions assuming the radiation field is unpolarized,

$$\begin{aligned} J_{0}^{0}(\nu) = & \oint\frac{d\Omega}{4\pi} I(\nu,\vec{ \Omega}), \end{aligned}$$

(3)

$$\begin{aligned} J_{0}^{2}(\nu) = & \frac{1}{2\sqrt{2}} \oint \frac{d\Omega}{4\pi} \bigl(3\cos^{2}\theta- 1\bigr) I(\nu,\vec{\Omega}), \end{aligned}$$

(4)

$$\begin{aligned} J_{\pm1}^{2}(\nu) = & \mp\frac{\sqrt{3}}{2} \oint \frac{d\Omega}{4\pi } \sin\theta\cos\theta e^{\pm i\chi} I(\nu,\vec{\Omega}), \end{aligned}$$

(5)

$$\begin{aligned} J_{\pm2}^{2}(\nu) = & \frac{\sqrt{3}}{4} \oint \frac{d\Omega}{4\pi} \sin^{2}\theta e^{\pm2 i\chi} I(\nu,\vec{ \Omega}), \end{aligned}$$

(6)

which are the expansion of Equation (1) for i=0. We follow the geometry used in Section 12.3 of Landi Degl’Innocenti and Landolfi (2004). Iterated Gaussian quadrature bivariate integration is used to numerically integrate these equations. The results are illustrated in Figure 9 for a height of h=1750 km. Only the real portion of the Q=±1,2 components is shown.

Figure 9

Multipole moments of the local radiation field tensor for a slab of material located at a height of 1750 km above the solar surface. Only the real part of Q=±1,2 components is displayed. NBAR represents the zero moment, where NBAR is defined as $\bar{n} = (c^{2}/2h\nu^{3})J_{0}^{0}$.

The symmetry breaking is most pronounced within the sunspot. Contours of the inner and outer penumbral edges show the sunspot’s photospheric extent (Figure 9). The mean intensity structure given by $\bar{n}$ (see Figure 9) is slightly offset from the photospheric structure due to the observational geometry of the region, and the assumed height of the helium slab. The weak ring structure in the map of ω best illustrates the extent of the radiative symmetry breaking in the superpenumbra. The half-angle of the heliographic extent of the light cone at this height is 4.06^∘, whereas the entire heliographic extent of the observed region is 4.44^∘×5^⋅. The effect of the Q=±1 components is minimal beyond the sunspot edge, whereas the Q=±2 components show pronounced changes within the inner superpenumbral region. However, the magnitude of these components is an order of magnitude below that of the $J^{2}_{0}$ component. Figure 9, of course, only shows the radiation field tensor at the frequency of one transition within the orthohelium atomic system. As a result of the variation of the Planck function, we might expect the magnitude of the Q≠0 components to increase at shorter wavelengths where the sunspot contrast is enhanced; but overall, the effect of this symmetry breaking around the penumbral edge is expected to be an order of magnitude below the $J^{2}_{0}$ component, which itself induces very weak polarization signatures of ≲ 0.1 % of the incoming intensity.