We will now focus on the radial variations of the -azimuthally averaged (see Equation (11)) components of the magnetic field vector. Since sunspots are not usually axisymmetric we will employ ellipses, as illustrated in Figure 10, to calculate those averages. The ellipses are determined by first obtaining the coordinates of the center of the umbra: , and then fitting ellipses with different major and minor semi-axes, such that the outermost blue ellipses in Figure 10 provides a good match to the boundary between the penumbra and the quiet Sun. The upper panel in Figure 10 shows the ellipses for AR 10923 observed on November 14, 2006 at = 8.7°, whereas the lower panel shows AR 10933 observed on January 9, 2007 at = 49.0°.
The radial variation of the azimuthal averages is presented in Figure 11. 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 ( refers to the total sunspot radius). This happens for all relevant physical quantities: the total magnetic field strength (green curve), the vertical component of the magnetic field vector (red curve), as well as for the horizontal component of the magnetic field vector (blue curve). Consequently, the radial variation of the inclination of the magnetic field vector with respect to the vertical on the solar surface , which can be obtained from and (Equation (10)) is also very similar for both sunspots (right panels in Figure 11).
The vertical component of the magnetic field vector monotonously decreases with the radial distance from the center of the umbra, while the transverse component of the magnetic field, , first increases until , and decreases afterward. In both sunspots the vertical and the transverse component become equally strong close to , which results in an inclination for the magnetic field vector of exactly in the middle of the sunspot radius (right panels in Figure 11). This location is very close to the umbra-penumbra boundary, which occurs at approximately (vertical dashed lines in Figure 11). The inclination of the magnetic field monotonously increases from the center of the sunspot, where it is considerably vertical (), to the outer penumbra, where it becomes almost horizontal (). Furthermore, the inclination at individual regions at large radial distances from the sunspot’s center can be truly horizontal () or, as indicated by the vertical bars in Figure 11, the magnetic field vector can even point downwards in the solar surface, with at certain locations. This is also clearly noticeable in the black contours in Figures 4 and 7. Before Hinode/SP data became available, detecting these patches where the magnetic field returns into the solar surface (Bellot Rubio et al., 2007b) 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 4; also Figure 4 in Bellot Rubio et al., 2007b). 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, 1984).
All these results are consistent with previous results obtained from Milne–Eddington inversions such as: Lites et al. (1993); Stanchfield II et al. (1997); Bellot Rubio et al. (2002, 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., 1997), or Fe i 1548 nm in Bellot Rubio et al. (2002). Analysis of spectropolarimetric data employing other techniques such as the magnetogram equation, which yields the vertical component of the magnetic field at a constant -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 10), 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 -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 11, 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).
As explained in Section 1.3.1, Figures 4, 5, 6, and 11 refer to the average magnetic field vector in the photosphere: . 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. (2005) 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 (2005a) 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: .
Living Rev. Solar Phys. 8, (2011), 4
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