### 2.2 Vertical- variations

The determination of the vertical variations of the magnetic field in sunspots has been a recurrent topic in Solar Physics for decades. Traditionally, this determination had been done through a combination of spectropolarimetric observations, where the magnetic field is measured at different heights in the solar atmosphere (Kneer, 1972; Wittmann, 1974b), and theoretical considerations such as employing a given sunspot model, applying the condition, etcetera (Hagyard et al., 1983; Osherovich, 1984, and references therein). Those first attempts were usually limited to the vertical component of the magnetic field :
where is the coordinate along the direction that is perpendicular to the solar surface (see Figure 42) and has been referred to as in Section 1.3.3. In those early works, inferences of the vertical gradient of the vertical component of the magnetic field could differ by as much as an order of magnitude: 1 – 10 G km–1 (Kotov, 1970), 0.5 – 2 G km–1 (Makita and Nemoto, 1976). Here we will refer, however, to the gradients of the magnetic field in terms of the optical depth scale (Equation (12)):

If and (or alternatively and ) are sufficiently far apart ( 1000 km), the gradient refers to the average gradient between the chromosphere and the photosphere. This can be done, for example, employing pairs of lines where one of them is photospheric and another one is chromospheric: Fe i 525.0 nm and C iv 154.8 nm (Hagyard et al., 1983), Fe i 1082.8 nm and He i 1083.0 nm (Kozlova and Somov, 2009), Fe i 630.2 nm and Na i 589.6 nm (Leka and Metcalf, 2003). Through a Milne–Eddington-like inversion (or applying a magnetrogram calibration) the vertical component of the magnetic field can be inferred separately for each line and, thus, separately for and . Another way is to employ a single spectral line whose formation range is very wide. Examples of such lines are: Ca ii 393.3 nm or Ca ii 854.2 nm. These lines are sensitive to , with being the photosphere and the chromosphere (Socas-Navarro, 2005a,b).

Since the theory of spectral line formation in the chromosphere is not currently fully understood (see Section 1.3), in this review we will focus mostly in the photospheric gradient of the magnetic field. To that end, we will employ the Fe i line pair at 630 nm observed with Hinode/SP. These two spectral lines are both formed within a range of optical depths of . We perform an inversion of the Stokes vector in these two spectral lines, assuming that each of the physical parameters in (Equation (2)) change linearly with the logarithm of the optical depth:

where refers to the -component of . Note that the inversion cannot be carried out with a Milne–Eddington-like inversion code, since those assume that the physical parameters do not change with optical depth: (see Sections 1.3.1 and 1.3.2). Instead, we employ an inversion code that allows for the inclusion of gradients in the physical parameters. In this case we have used the SIR inversion code (Ruiz Cobo and del Toro Iniesta, 1992), but we could have also employed SPINOR (Frutiger et al., 1999) or LILIA (Socas-Navarro, 2002). Applying this inversion code allows us to determine two-dimensional maps of the three components of the magnetic field vector , , and at different optical depths (cf. Figures 2 and 3).

Once those maps are obtained, the 180°-ambiguity in the azimuth of the magnetic field can be resolved at each optical depth following the prescriptions given in Section 1.3.2. This allows to obtain (vertical component of the magnetic field on the solar surface) and (horizontal component of the magnetic field). By the same method as in Section 2.1 we then employ ellipses to determine the angular averages of these physical parameters as a function of the normalized radial distance in the sunspot: . However, as opposed to the previous section, it is now possible to determine this radial variations at different optical depths. The results are presented in Figure 12, in red color for the deep photosphere (), blue for the mid-photosphere (), and green for the high-photosphere ().

Figure 12 shows two distinct regions. The first one corresponds to the inner part of the sunspot: , where the total magnetic field strength (upper-left panel) decreases from the deep photosphere (red color) upwards. This is caused by an upwards decrease of the vertical (upper-right), and horizontal (lower-left) components of the magnetic field. Also, in this region the inclination of the magnetic field vector (lower-right) remains constant with height. From the middle-half of the sunspot and outwards, , the situation, however, reverses. The total magnetic field strength, as well as the vertical and horizontal components of the magnetic field, increase from the deep photosphere () to the higher photosphere (). In this region, the inclination of the magnetic field vector no longer remains constant with but it decreases towards the higher photospheric layers. The actual values of the gradients are given in Figure 13. These values are close to the lower limits () obtained in early works (Kotov, 1970; Makita and Nemoto, 1976; Osherovich, 1984, and references therein). However, Figures 12 and 13 extend those results for the three components of the magnetic field vector and not only for its vertical component . In addition, these figures show a clear distinction between the inner and the outer sunspot. Although Figures 12 and 13 show only the results for AR 10923, the other analyzed sunspot (AR 10933) presents very similar features.

Similar studies have been carried out in a number of recent works. For instance, our results are in very good agreement with those from Westendorp Plaza et al. (2001b) (see their Figure 9) in the value and sign of the gradients in the different components of the magnetic field. In our case, as well as theirs, the total magnetic field strength decreases towards the deep photosphere for . At the same time the inclination (with respect to the vertical) increases towards deeper photospheric layers. This can be interpreted in terms of the existence of a canopy (see also Leka and Metcalf, 2003), and is perfectly consistent with a picture in which sunspots are vertical flux tubes where the magnetic field lines fan out with increasing height as they meet a plasma with lower densities. Another interesting result concerns the fact that, once the physical parameters are allowed to vary with optical depth (Equation (20)), the evidence for return-flux () in the deep photosphere becomes more clear: compare lower-right panels in Figures 11 and 12. It is important to note that Westendorp Plaza et al. (2001b) also employed in their inversions spectropolarimetric data from the Fe i line pair at 630 nm.

Other spectral lines, such as the Fe i line pair at 1564.8 nm were employed by Mathew et al. (2003), who instead found that the magnetic field strength increases towards deeper layers in the photosphere at all radial distances in the sunspot: (see their Figure 15). In addition, they found that the inclination of the magnetic field decreases towards deep layers: at all radial distances. These results are, therefore, consistent with ours as far as the inner part of the sunspot is concerned, but they are indeed opposite to ours (and to Westendorp Plaza et al., 2001b) for the sunspot’s outer half. Furthermore, Sánchez Cuberes et al. (2005), as well as Balthasar and Gömöry (2008), analyzed two Fe i lines and one Si i line at 1078.3 nm to study the magnetic structure of a sunspot. From their spectropolarimetric analysis (see their Figure 11) they inferred a total magnetic field strength that was stronger in the deep photospheric layers: at all radial distances from the sunspot’s center (in agreement with Mathew et al., 2003). As far as the inclination of the magnetic field is concerned, Sánchez Cuberes et al. (2005) obtained different behaviors depending on the scheme employed to treat the stray light in the instrument. However, they lend more credibility to the results obtained with a constant amount of stray light. In this case, they concluded that for and for , which supports our results and those from Westendorp Plaza et al. (2001b), but not Mathew et al. (2003). Results from all the aforementioned investigations are summarized in Table 1. It is important to mention that, although Balthasar and Gömöry (2008) did not find in the outer half of the sunspot (considered as evidence for a canopy), they did indeed find this trend outside the visible boundary of the sunspot.

Table 1: Sign of the gradients of the different components of the magnetic field vectors: total magnetic field strength , vertical component of the magnetic field vector , horizontal component of the magnetic field vector , and inclination of the magnetic field vector with respect to the vertical direction on the solar surface . The sign of the gradients are split in two distinct regions: inner sunspot , and outer sunspot . Figure 13 gives the actual values.
 This work and Westendorp Plaza et al. (2001b)
 all

In the light of these opposing results it is critical to ask ourselves where do these differences come from. One possible source is the spatial resolution of the observations. Westendorp Plaza et al. (2001b), Mathew et al. (2003), and Sánchez Cuberes et al. (2005) employed spectropolarimetric observations at low spatial resolution (about 1”). The data employed here (Hinode/SP) possess much better resolution: 0.32”. However, this should not be very influential to the study of the global properties of the sunspot, since we are discussing azimuthally or -averaged quantities. Another possible explanation lies in the different formation heights of the employed spectral lines. Since each set of spectral lines samples a slightly different region in the solar photosphere, they might be sensing slightly different magnetic fields, thereby yielding gradients (see Table 1). This is a plausible explanation because the Fe i line pair at 1564.8 nm sample a deep and narrow photospheric layer: , as compared to for the Fe i lines at 630 nm (see, for example, Figures 3 and 4 in Mathew et al., 2003, and Figure 3 in Bellot Rubio et al., 2000). Indeed, the different formation heights have been exploited by numerous authors (Bellot Rubio et al., 2002; Mathew et al., 2003; Borrero et al., 2004; Borrero and Solanki, 2008) in order to explain the opposite gradients obtained from different sets of spectral lines in terms of penumbral flux tubes and the fine structure of the sunspot (see, also, Sections 3.2.1 and 3.2.5). The role of the sunspot’s fine structure is emphasized by the fact that the scatter bars (produced by the inversion of individual pixels; see Figures 12 and 13) are of the order of, or even larger than, the differences between the magnetic field at the different atmospheric layers chosen for plotting. To solve this problem one would like to analyze, ideally, many different spectral lines formed at different heights (Section 1.3.1). This approach has been already followed by the recent works of Cabrera Solana et al. (2008) and Beck (2011), where simultaneous and co-spatial spectrolarimetric observations in Fe i 630 nm and Fe i 1564.8 nm where analyzed. Their results further emphasize the role of the fine structure of the sunspot in the determination of the vertical gradients of the magnetic field vector.

A final possibility to explain the difference in the gradients obtained by different authors could be the different treatments employed to model the scattered light in the instrument. Arguments in favor of this possibility are given by Sánchez Cuberes et al. (2005) and Solanki (2003). Arguments against the results being affected by the treatment of the scattered light have been presented in Borrero and Solanki (2008). Moreover, in Cabrera Solana et al. (2006), Cabrera Solana (2007), and Cabrera Solana et al. (2008) a careful correction for scattered light was performed, and still the fine structure of the sunspot had to be invoked to explain the observed gradients in the magnetic field vector.