2.3 Is the sunspot magnetic field potential?

The potentiality of the magnetic field vector in sunspots is often studied by means of the current density vector j = 1-∇ × B μ0 (in SI units). Theoretical models for sunspots usually come in two distinct flavors attending to the vector j: those where the currents are localized at the boundaries of the sunspot (current sheets) and the magnetic field vector is potential elsewhere (Simon and Weiss, 1970; Meyer et al., 1977Jump To The Next Citation Point; Pizzo, 1990), and those where there are volumetric currents distributed everywhere inside the sunspot (Pizzo, 1986). From an observational point of view, in order to evaluate j it is necessary to calculate the vertical derivatives of the three components of the magnetic field vector: dBα ∕dxρ, dB β∕dxρ, and dB ρ∕dxρ. This is not possible through a Milne–Eddington inversion, because it assumes that the magnetic field vector is constant with height: τc or x ρ (Equation (12View Equation)). In this case it is only possible to determine the vertical component of the current density vector, jρ (aka jz), because it involves only the horizontal derivatives:
1 dB α dB β jρ = (∇ × B )z = --- ---- − ---- . (21 ) μ0 dx β dx α

An example of the vertical component of the current density vector, j ρ, obtained from a ME inversion is presented in Figure 14View Image. This corresponds to the sunspot observed in November 14, 2006 with Hinode/SP at Θ = 8.7°. The derivatives in Equation 21View Equation have been obtained from Figures 5View Image and 6View Image. Because the magnetic field vector was obtained from a ME inversion, these derivatives of the magnetic field vector refer to a constant optical depth ¯τ in the atmosphere. As long as the ¯τ (x ρ)-surface (Wilson depression) is not very corrugated (small pixel-to-pixel variations) and that the vertical-τ variations of the magnetic field vector are not very strong (Equation (6View Equation)), it is justified to assume that the maps in Figures 5View Image and 6View Image also correspond to a constant geometrical height x ρ. If these assumptions are in fact not valid, artificial currents in jρ might appear as a consequence of measuring the magnetic field at different heights from one pixel to the next one.

Note that prior to the calculation of currents, the 180°-ambiguity in the azimuth of the magnetic field vector must be solved (see Section 1.3.2). The final results for j ρ are displayed in Figure 14View Image, where it can be seen that the vertical component of the current density vector is highly structured in radial patterns resembling penumbral filaments. The values of the current density are of the order of −2 |jρ| < 75 mA m. This value is consistent with previous results obtained with different instruments and, therefore, different spectral lines and spatial resolutions: |jρ| < 50 mA m− 2 (Figure 10 in Li et al., 2009; 2” and Fe i 630 nm), − 2 |jρ| < 40 mA m (Figure 8 in Balthasar and Gömöry, 2008; ≈ 0.9” and Si/Fe i 1078 nm), −2 |jρ| < 100 mA m (Figure 3 in Shimizu et al., 2009; 0.32” and Fe i 630 nm). These various results show a weak tendency for the current to increase with increasing spatial resolution. However, this result is to be taken cautiously, since at low spatial resolutions two competing effects can play a role. On the one hand, a better spatial resolution can detect larger pixel-to-pixel variations in the magnetic field and, thus, yields larger values for jz. On the other hand, a worse spatial resolution can leave certain magnetic structures unresolved and, in this case, the finite-differences involved in the Equation (21View Equation) can produce artificial currents where originally there were none.

A curious effect worth noticing is the large and negative values of jρ in Figure 14View Image around the sunspot center that describe an oval shape (next to the light bridges). This is an artificial result produced by an incorrect solution to the 180°-ambiguity in the azimuth of the magnetic field close to the umbral center (see Section 1.3.2). An incorrect choice between φ and φ + π (see for instance Equation (27View Equation)) can lead to very large and unrealistic pixel-to-pixel variations in dB α∕dxβ or dB β∕dx α. Thus, regions where large values of jρ are consistently obtained can sometimes be used to identify places where the solution to the 180°-ambiguity was not correct. Indeed, many methods to solve the 180°-ambiguity minimize jρ in order to choose between the two possible solutions in the azimuth of the magnetic field vector (Metcalf et al., 2006, see also Section 1.3.2).

In order to compute the horizontal component of the current density vector j = j2-+-j2 h α β, it is necessary to analyze the spectropolarimetric data employing an inversion code that allows to retrieve the stratification with optical depth in the solar atmosphere (see Section 2.2). Even in this case, the derivatives must be evaluated in terms of the geometrical height xρ instead of the optical depth τc. Because the conversion from these two variables assuming hydrostatic equilibrium is not reliable in sunspots (see Section 1.3.3), j h is not something commonly found in the literature. As a matter of fact, most inferences of jh were performed through indirect means (Ji et al., 2003; Georgoulis and LaBonte, 2004). Very recently, however, Puschmann et al. (2010cJump To The Next Citation Point) have been able to determine the full current density vector j from purely observational means (inversion of Stokes profiles including τc-dependence) plus a proper conversion between τc and xρ (Puschmann et al., 2010bJump To The Next Citation Point, see also Section 1.3.3). In the latter two works, the authors found that the horizontal component of the current density vector is about 3 – 4 times larger than the vertical one: jh ≈ 4jρ. Figure 15View Image reproduces Figure 1 from Puschmann et al. (2010cJump To The Next Citation Point), which shows the j vector in a region of the penumbra. jρ (they refer to it as jz) also shows radial patterns as in our Figure 14View Image. More importantly, jh is strongest in the vicinity of the regions where jρ is large.

View Image

Figure 14: Same as Figures 4View Image8View Image but for the vertical component of the current density vector jz (or jρ) in sunspot AR 10923.
View Image

Figure 15: Same as Figure 9View Image but for the vertical component of the current density vector: jz (or j ρ). The arrows indicate the horizontal component of the current density vector: j β and j α. The white contours enclose the area where the vertical component of the magnetic field, B ρ, is equal to 650 G (solid) and 450 G (dashed) (from Puschmann et al., 2010cJump To The Next Citation Point, reproduced by permission of the AAS).

Currents in the chromosphere have also been studied, although to a smaller extent, by Solanki et al. (2003) and Socas-Navarro (2005b). The latter author finds values for the vertical component of the current density vector in the chromosphere which are compatible with those in the photosphere: |jz| < 250 mA m −2. In addition, the detected currents are distributed in structures that resemble vertical current sheets, spanning up to 1.5 Mm in height. The mere presence of large currents within sunspots clearly implies that the magnetic field vector is not potential: ∇ × B ⁄= 0.


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