2.1 As seen at constant τ-level

In Figures 2View Image and 3View Image in Section 1.3, and Figures 4View Image8View Image in Section 1.3.2, we have presented the 3 components of the magnetic field both in the observer’s reference frame and in the local reference frame. Those maps were obtained from the inversion of spectropolarimetric observations employing a Milne–Eddington (ME) atmospheric model (see Section 1.3.1). This means that the results from a ME inversion should be interpreted as an average of the magnetic field vector over the region where the lines are formed ¯τ: Bβ(x β,xα,¯τ), Bα (x β,xα,¯τ), and B ρ(xβ,xα,τ¯). This makes the results from the ME inversion ideal to study the magnetic field at a constant τ-level. The coordinates xα and x β refer to the local reference frame: {eβ,eα,eρ} as described in Section 1.3.2. Note that the optical depth τ is employed instead of x ρ, which is the coordinate representing the geometrical height. For convenience let us now consider polar coordinates in the α β-plane: (r,𝜃) with r being the radial distance between any point in the sunspot to the center of the umbra. 𝜃 is defined as the angle between the radial vector that connects this point with the umbra center and the e β axis (see, for example, Figure 4View Image). With this transformation we now have: --2------------2------- Bh(r,𝜃,¯τ ) = B β(r,𝜃,¯τ) + B α(r,𝜃,¯τ) (horizontal component of the magnetic field) and B ρ(r,𝜃,¯τ) (vertical component of the magnetic field).

We will now focus on the radial variations of the Ψ-azimuthally averaged (see Equation (11View Equation)) components of the magnetic field vector. Since sunspots are not usually axisymmetric we will employ ellipses, as illustrated in Figure 10View Image, to calculate those averages. The ellipses are determined by first obtaining the coordinates of the center of the umbra: {xβ,u;xα,u}, and then fitting ellipses with different major and minor semi-axes, such that the outermost blue ellipses in Figure 10View Image provides a good match to the boundary between the penumbra and the quiet Sun. The upper panel in Figure 10View Image shows the ellipses for AR 10923 observed on November 14, 2006 at Θ = 8.7°, whereas the lower panel shows AR 10933 observed on January 9, 2007 at Θ = 49.0°.

The radial variation of the azimuthal averages is presented in Figure 11View Image. The vertical bars in this figure represent the standard deviation for all considered points along each ellipse’s perimeter. Note that, although the scatter is significant, the radial variation of the different components of the magnetic field vector are very well defined. Furthermore, both sunspots (AR 10923 in the upper panels; AR 10933 in the lower panels) show very similar behaviors of the magnetic field vector with r∕Rs (Rs refers to the total sunspot radius). This happens for all relevant physical quantities: the total magnetic field strength B (r,¯τ) tot (green curve), the vertical component of the magnetic field vector B ρ(r,¯τ) (red curve), as well as for the horizontal component of the magnetic field vector ------------------- Bh (r) = B2β(r,¯τ ) + B2α(r,¯τ) (blue curve). Consequently, the radial variation of the inclination of the magnetic field vector with respect to the vertical on the solar surface ζ, which can be obtained from Bρ and Bh (Equation (10View Equation)) is also very similar for both sunspots (right panels in Figure 11View Image).

The vertical component of the magnetic field vector B ρ monotonously decreases with the radial distance from the center of the umbra, while the transverse component of the magnetic field, Bh, first increases until r∕Rs ∼ 0.5, and decreases afterward. In both sunspots the vertical and the transverse component become equally strong close to r∕Rs ∼ 0.5, which results in an inclination for the magnetic field vector of ζ ≃ 45∘ exactly in the middle of the sunspot radius (right panels in Figure 11View Image). This location is very close to the umbra-penumbra boundary, which occurs at approximately r∕R ≃ 0.4 s (vertical dashed lines in Figure 11View Image). The inclination of the magnetic field ζ monotonously increases from the center of the sunspot, where it is considerably vertical (ζ ≃ 10– 20∘), to the outer penumbra, where it becomes almost horizontal (ζ ≃ 80 ∘). Furthermore, the inclination at individual regions at large radial distances from the sunspot’s center can be truly horizontal (ζ = 90∘) or, as indicated by the vertical bars in Figure 11View Image, the magnetic field vector can even point downwards in the solar surface, with Bρ < 0 at certain locations. This is also clearly noticeable in the black contours in Figures 4View Image and 7View Image. Before Hinode/SP data became available, detecting these patches where the magnetic field returns into the solar surface (Bellot Rubio et al., 2007bJump To The Next Citation Point) was not possible unless more complex inversions (not ME-like) were carried out (see Section 2.2). Nowadays with Hinode’s 0.32” resolution, these patches which sometimes can be as long as 3 – 4 Mm, are detected routinely (see Figure 4View Image; also Figure 4 in Bellot Rubio et al., 2007bJump To The Next Citation Point). Note that theoretical models for the sunspot magnetic field allow for the possibility of returning-flux at the edge of the sunspot (Osherovich, 1982; Osherovich and Lawrence, 1983; Osherovich, 1984Jump To The Next Citation Point).

All these results are consistent with previous results obtained from Milne–Eddington inversions such as: Lites et al. (1993Jump To The Next Citation Point); Stanchfield II et al. (1997Jump To The Next Citation Point); Bellot Rubio et al. (2002Jump To The Next Citation Point, 2007b). Although most of these inversions were also obtained from the analysis of spectropolarimetric data in the Fe i line pair at 630 nm, a few of them also present maps of the magnetic field vector obtained from other spectral lines such as C i 538.0 nm and Fe i 537.9 nm (Stanchfield II et al., 1997Jump To The Next Citation Point), or Fe i 1548 nm in Bellot Rubio et al. (2002Jump To The Next Citation Point). Analysis of spectropolarimetric data employing other techniques such as the magnetogram equation, which yields the vertical component of the magnetic field at a constant τ-level, have also been carried out by other authors (Bello González et al., 2005). The consistency between all the aforementioned results is remarkable, specially if we consider that each work studied different sunspots and employed different spectral lines.

The picture of a sunspot that one draws from these radial variations is that of a vertical flux tube, with a diameter of 30 – 40 Mm (judging from Figure 10View Image), where the magnetic field is very strong and vertical at the flux tube’s axis (umbral center), while it becomes weaker and more horizontal as we move towards the edges of the flux tube. Even though these results were obtained only for a fixed τ-level on the solar photosphere, they clearly indicate that the flux tube is expanding with height as the magnetic field encounters a lower density plasma. The overall radial variations of the components of the magnetic field seem to be independent of the sunspot size, although the maximum field strength (which occurs at the sunspot’s center) clearly does, as illustrated by Figure 11View Image, where the magnetic field strength for AR 10923 peaks at about 3300 Gauss (large sunspot), whereas for AR 10933 (small sunspot) it peaks at around 2900 Gauss. This has been further demonstrated by several works that employed data from many different sunspots (Ringnes and Jensen, 1960; Brants and Zwaan, 1982; Kopp and Rabin, 1992; Collados et al., 1994; Livingston, 2002; Jin et al., 2006).

View Image

Figure 10: Map of the continuum intensity for two sunspots. The top panel shows AR 10923 observed at Θ = 8.7°, whereas the bottom panel shows AR 10933, observed at Θ = 49.0°. These are the same sunspots as discussed in Sections 1.3 and 1.3.2. The blue ellipses are employed to determine the azimuthal averages (Ψ-averages) of the magnetic field vector. Note that the outermost ellipse tries to match the boundary between the penumbra and the quiet Sun. The orange arrow points towards the center of the solar disk.
View Image

Figure 11: Left panels: Azimuthally averaged components of the magnetic field vector as a function of the normalized radial distance r∕Rs from the sunspot’s center. The magnetic field corresponds to a constant τ-level. In green the total magnetic field strength Btot(r,¯τ) is presented while red and blue refer to the vertical B ρ and horizontal B h components of the magnetic field. Top panel shows the radial variations for AR 10923 and the bottom panel refers to AR 10933 (see Figure 10View Image for details). Right panels: inclination at a constant τ-level of the magnetic field vector with respect to the vertical direction on the solar surface, as a function of the normalized radial distance from the sunspot’s center: ζ (r,τ¯) (see Equation (10View Equation)). The horizontal dashed line is placed at ζ = 90°, indicating when the magnetic field points downwards on the solar surface. The vertical dashed line at r∕Rs ≃ 0.4 is placed at the boundary between the umbra and the penumbra.

As explained in Section 1.3.1, Figures 4View Image, 5View Image, 6View Image, and 11View Image refer to the average magnetic field vector in the photosphere: τ¯∈ [1,10− 3]. This is because they were obtained from the Milne–Eddington inversion of spectropolarimetric data for the line Fe i pair at 630 nm. The investigations of the magnetic field vector in the chromosphere is far more complicated, since Non-Local Thermodynamic Equilibrium (NLTE) conditions make the interpretation of the Stokes parameters more difficult. However, in the last years a number of works have addressed some of these issues. For example, Orozco Suarez et al. (2005Jump To The Next Citation Point) analyzes data from the Si i and He i spectral lines at 1083 nm, which are formed in the mid-photosphere and upper-chromosphere, respectively. They find very similar radial variations of the magnetic field vector in the chromosphere and the photosphere, with the main difference being a reduction in the total magnetic field strength. Furthermore, Socas-Navarro (2005aJump To The Next Citation Point) has presented an actual NLTE inversion of the Ca ii lines at 849.8 and 854.2 nm. These two spectral lines are formed in the photosphere and chromosphere: ¯τ ∈ [1,10− 6].


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