Chromospheric and photospheric properties of sunspots as inferred from Stokes inversions under magneto-hydrostatic and non-local-thermodynamic equilibrium

Abstract

Sunspots are crucial for exploring how magnetic fields and plasma flows interact in the solar atmosphere, spanning from the stable photosphere to the shock-dominated chromosphere. To determine the thermal, magnetic, and kinematic properties of a sunspot across these layers and to investigate transient phenomena like umbral flashes, we analyzed high-resolution spectropolarimetric data from the CRISP instrument at the Swedish Solar Telescope. By applying the FIRTEZ inversion code, which incorporates non-local thermodynamic equilibrium (non-LTE) and 3D magneto-hydrostatic (MHS) equilibrium, to full Stokes measurements of multiple spectral lines (Mg I, Na I, Fe I, and Ca II), we successfully mapped the atmospheric parameters in a 3D domain. Our analysis reveals that the photospheric Evershed flow actually reverses into an inflow in the upper photosphere. In contrast, the surrounding moat flow persists as an outflow at similar heights, indicating that it is not a direct continuation of the Evershed flow. Furthermore, observations of an umbral flash event uncovered supersonic upflows (Mach numbers $\|M\|\geq 1.5$) and thermodynamic conditions characteristic of shock fronts. Ultimately, combining 3D MHS equilibrium and non-LTE effects across multiple spectral lines proves highly effective for simultaneously constraining parameters in both the photosphere and chromosphere. These findings provide clear evidence of shock dynamics in umbral flashes, supporting the theory that converging supersonic flows act as the primary driving mechanism while shifting optical depth iso-surfaces.

Figures (8)

• Figure 1: Representative Stokes profiles at three different locations in the diffraction limited observations. The three locations are: penumbra (green), umbra (blue), umbral flash (red). The points display the observed data whereas solid lines correspond to the fitted profiles. The profiles are normalized to the quiet sun continiuum and in order to better visualize the data, vertical shifts have been applied. These are: $[+0.2, 0, -0.2]$ in Stokes $I$ and [+0.02, 0, -0.02] in Stokes $Q$, $U$ and $V$. In addition, Stokes $I$ signals in the umbra (blue) and umbral flash (red) have been multiplied by a factor 2 before the shift applied to better display the variations.
• Figure 2: Observations and inversion results. Top panels display maps of the observed intensity at three different wavelengths (from left to right): continuum of the Fe i line at 630 nm, core of the Mg i line at 517 nm, and core of the Ca ii line at 854 nm. Middle panels: temperature inferred from the inversion at three different optical depth levels (from left to right): $\tau_{\rm c}=1$ (photospheric continuum), $\tau_{\rm c}=10^{-2.5}$ (high photosphere), and $\tau_{\rm c}=10^{-5}$ (chromosphere). Bottom panels: same as middle panels but showing the inferred line-of-sight velocity. Inner and outer blue circles are located at radii $r/R_{\rm s} = 0.4, 1.0, 1.4$, respectively, where $R_{\rm s}$ is defined as the sunspot's radius as measured from the umbral center (red cross). The direction of the solar disk center is indicated by the bold arrow. Black dashed lines depict a cone of $\pm 15^{\circ}$ around the line of symmetry of the spot.
• Figure 3: Inversion results. Similar to middle and bottom panels of Fig. \ref{['fig:results_tau_1']} but displaying the $B_x$ (top), $B_y$ (middle) and $B_z$ (bottom) components of the magnetic field in the local reference frame at three different optical depths.
• Figure 4: Average sunspot properties. Azimuthal averages of the physical parameters, as inferred from the Stokes inversion, as a function of the normalized sunspot radius $r/R_{\rm s}$ at different optical-depth levels: $\log\tau_{\rm c} = 0$ (red), $-1$ (orange), $-2$ (yellow), $-3$ (green), $-4$ (blue), $-5$ (cyan), $-6$ (purple). Typical deviations around the mean are represented by the vertical color bars. All panels, expect (h) and (i) were produced by averaging over $2\pi$ radians. Panels h ($v_{\rm los}$ in the center side penumbra) and i ($v_{\rm los}$ in the limb side penumbra) were obtained by averaging only over $\pm \pi/12$ radians around the sunspot's line of symmetry (see cone denoted by dashed lines in Figs. \ref{['fig:results_tau_1']} and \ref{['fig:results_tau_2']}).
• Figure 5: View of the umbral flash inferred from the diffraction limited data. Upper panel: divided map of temperature $T(x,y)$ (bottom left) and line-sight-velocity $v_{\rm los}$ (top right) at $z \sim 1$ Mm. The dotted horizontal line represents the slice selected for the three lower panels, and the cross indicates where the umbral flash happens (i.e. where the strongest supersonic velocities are found). Bottom panels: temperature $T(x,z)$, Mach number $M(x,z)$, and magnetic field ${\bf B}(x,z)$ on the plane of the slice indicated on the upper panel ($y=88$ Mm). As a reference, the curves of constant optical depth $\log\tau_{\rm c} = 0, -2, -3, -4, -5$ are indicated by the black or white dashed lines. Green contour indicates the regions where supersonic flows are found: $\|M > 1\|$.