1.3 Current tools to infer sunspot’s magnetic field

Not much has changed since Hale’s discovery of magnetic fields in sunspots (Hale, 1908). The broadening of the intensity profiles of spectral lines he saw on his photographic plates was produced by the Zeeman splitting of the atomic energy levels in the presence of the sunspot’s magnetic field. Hale estimated a magnetic field strength of about 2600 – 2900 Gauss. This basic technique is still widely used nowadays. The addition of the polarization profiles: Stokes Q, U, and V , besides the intensity or Stokes I, allows us to determine not only the strength of the magnetic field but the full magnetic field vector B. This is done thanks to the radiative transfer equation (RTE):
dI-λ(X-[τc]) dτ = 𝒦ˆλ(X [τc])[Iλ(X [τc]) − S λ(X [τc])], (1 ) c
where Iλ (X [τc]) = (I,Q, U, V)†1 is the Stokes vector at a given wavelength λ. The variation of the Stokes vector with optical depth τc appears on the right-hand side of Equation (1View Equation). The dependence of Iλ with τc arises from the fact that X is a function of the optical depth itself: X = X [τc]. Here X represents the physical parameters that describe the solar atmosphere:
X (τc) = [B (τc),T (τc),Pg (τc),Pe(τc),ρ(τc),Vlos(τc),Vmic(τc),Vmac(τc)], (2 )
where B (τ ) c is the magnetic field vector, T(τ ) c is the temperature stratification, P (τ) g c and P (τ) e c are the gas and electron pressure stratification, ρ(τc) is the density stratification, and Vlos(τc) is the stratification with optical depth of the line-of-sight velocity. In addition, macro-turbulent Vmac (τc) and micro-turbulent Vmic(τc) velocities are often employed to model velocity fields occurring at spatial scales much smaller than the resolution element. Finally, on the right-hand side of Equation (1View Equation) we have the propagation matrix ˆ 𝒦 λ(X [τc]) and the source function S λ(X [τc]) at a wavelength λ. The latter is always non-polarized and, therefore, only contributes to Stokes I:
Sλ(X [τc]) = (Sλ(X [τc]),0,0,0)†. (3 )

The radiative transfer equation has a formal solution in the form:

Z I (X [τ ]) = ∞ 𝒪ˆ (X [τ ])ˆð’¦ (X [τ ])S (X[τ ])d τ, (4 ) λ c 0 λ c cλ c λ c c
where ˆ 𝒪 λ(0, τc) is the evolution operator, which needs to be evaluated at every layer in order to perform the integration. During the 1960s and early 1970s, the first numerical solutions to the radiative transfer equation for polarized light became available (Beckers, 1969a,b; Stenflo, 1971; Landi Degl’Innocenti and Landi Degl’Innocenti, 1972; Wittmann, 1974aJump To The Next Citation Point; Auer et al., 1977). Techniques to solve Equation (1View Equation) have continued to be developed even during the past two decades (Rees et al., 1989; Bellot Rubio et al., 1998; López Ariste and Semel, 1999a,bJump To The Next Citation Point).

Figure 1Watch/download Movie shows an example of how the Stokes vector varies when the magnetic field vector changes. In that movie we use spherical coordinates to represent the three components of the magnetic field vector: B = (B, γ,φ), where B is the strength of the magnetic field, γ is the inclination of the magnetic field with respect to the observer, and φ is the azimuth of the magnetic field in the plane perpendicular to the observer’s line-of-sight. In Figure 1Watch/download Movie we assume that the observer looks down along the z-axis, but this does not need to be always the case.

A major milestone was reached when these methods to solve the RTEs (1View Equation) and (4View Equation) were implemented into efficient minimization algorithms that allow for the retrieval of magnetic field vector in an automatic way (Ruiz Cobo and del Toro Iniesta, 1992Jump To The Next Citation Point; Ruiz Cobo, 2007; Socas-Navarro, 2002Jump To The Next Citation Point; del Toro Iniesta, 2003a; Bellot Rubio, 2006). This retrieval is usually done by means of non-linear minimization algorithms that iterate the free parameters of the model X (τc) (Equation (2View Equation)) while minimizing the difference between the observed and theoretical Stokes profiles (measured by the merit function 2 χ). The X (τc) that minimizes this difference is assumed to correspond to the physical parameters present in the solar atmosphere:

X4 MX 2 obs syn 32 χ2 = ---1---- 4Ii--(λk)-−-Ii--(λk,-X-[τc])5 . (5 ) 4M − L i=1 k=1 σik

Here Ioibs(λk ) and Isiyn(λk, X [τc]) represent the observed and theoretical (i.e., synthetic) Stokes vector, respectively. The latter is obtained from the solution of the RTE (4View Equation) given a particular set of free parameters X (Equation (2View Equation)). The letter L represents the total number of free parameters in X and, thus, the term 4M − L represents the total number of degrees of freedom of the problem (number of data points minus the number of free parameters). In Equation (5View Equation), indexes i and k run for the four components of the Stokes vector (I,Q,U,V ) and for all wavelengths, respectively. Finally, σik represents the error (e.g., noise) in the observations Iobs(λ ) i k.

Traditionally, the 2 χ-minimization has been carried out by minimization algorithms such as the Levenberg–Marquardt method (Press et al., 1986). However, more elaborated methods have also been employed in recent years: genetic algorithms (Charbonneau, 1995; Lagg et al., 2004), Principal-Component Analysis (Rees et al., 2000; Socas-Navarro et al., 2001), and Artifical Neural Networks (Carroll and Staude, 2001Jump To The Next Citation Point; Socas-Navarro, 2003).

Get Flash to see this player.


Figure 1: mpg-Movie (908 KB) The change of the synthetic emergent Stokes profiles (I,Q,U,V ) when the magnetic field present in the solar plasma varies. The magnetic field vector is expressed in spherical coordinates: B moduli of the magnetic field vector, γ inclination of the magnetic field vector with respect to the observer’s line-of-sight (z-axis in this case), and φ azimuth of the magnetic field vector in the plane perpendicular to the observer’s line-of-sight. Results have been obtained under the Milne–Eddington approximation.

Due to our limited knowledge of the line-formation theory, these investigations have usually been limited to the study of the photospheric magnetic field, where Local Thermodynamic Equilibrium and Zeeman effect apply for the most part. However, recent advancements in the line-formation-theory under NLTE conditions (Mihalas, 1978Jump To The Next Citation Point), scattering polarization (Trujillo Bueno et al., 2002; Manso Sainz and Trujillo Bueno, 2003; Landi Degl’Innocenti and Landolfi, 2004), etc., allow us to extend these techniques to the study of the chromosphere (Socas-Navarro et al., 2000a; Asensio Ramos et al., 2008; Trujillo Bueno, 2010; Casini et al., 2009). Indeed, some recent works have appeared where the magnetic structure of sunspots in the chromosphere is being investigated (Socas-Navarro et al., 2000b; Socas-Navarro, 2005aJump To The Next Citation Point,bJump To The Next Citation Point; Orozco Suarez et al., 2005Jump To The Next Citation Point).

Techniques to study the coronal magnetic field from polarimetric measurements of spectral lines are also becoming available nowadays (Tomczyk et al., 2007, 2008). These observations, carried out mostly with near-infrared spectral lines, are recorded using coronographs (to block the large photospheric contribution coming from the solar disk) and, therefore, limited to the solar limb. Other possibilities to observe polarization on the solar disk involve EUV (Extreme Ultra Violet) lines, which are only accessible from space, and radio observations of Gyroresonance and Gyrosynchrotron emissions, which can show large polarization signals: White (2001Jump To The Next Citation Point, 2005), Brosius et al. (2002), and Brosius and White (2006). Unfortunately, so far radio measurements have allowed only to infer the magnetic field strength in the solar corona. Interestingly, opacity effects in the gyroresonance emission (see Equations (1) and (2) in White, 2001) might also permit to infer the inclination of the magnetic field vector with respect to the observer’s line-of-sight, i.e., γ. However, this possibility has not been yet successfully exploited.

1.3.1 Formation heights

According to Equations (1View Equation) and (2View Equation) the solution to the radiative transfer equation depends on the stratification with optical depth τc of the physical parameters. The range of optical depths in which the solution X (τc) will be valid depends on the region of the photosphere in which the analyzed spectral lines are formed. In the future we will refer to this range as ¯τ = [τc,min,τc,max ]. ¯τ can be determined by means of the so-called contribution functions (Grossmann-Doerth et al., 1988; Solanki and Bruls, 1994Jump To The Next Citation Point) and the response functions (Landi Degl’Innocenti and Landi Degl’Innocenti, 1977; Ruiz Cobo and del Toro Iniesta, 1994). In the literature, it is usually considered that the range of optical depths, that a given spectral lines is sensitive to, is so narrow that the physical parameters do not change significantly over [τc,min,τc,max]. This can be mathematically expressed as:

Xf-(τc,max)-−-Xf-(τc,min) << 1, (6 ) Xf (τc,max) + Xf (τc,min)
where X f refers to the f-component of X (Equation (2View Equation)). When the conditions in Equation (6View Equation) are met for all f’s, a Milne–Eddington-like (ME) inversion can be applied. The advantage of ME-codes is that an analytical solution for the RTE (1View Equation) exists in this case. ME-codes assume that the physical parameters are constant in the range τ¯. One way to determine the magnetic field at different heights in the solar atmosphere is to perform ME-inversions of spectropolarimetric data in several spectral lines that are formed at different average optical depths ¯τ’s, with each spectral line yielding information in a plane at a different height above the solar surface.

As an example of the results retrieved by a Milne–Eddington-like inversion code we show, in Figures 2View Image and 3View Image, the three components of the magnetic field vector, for two different sunspots, in the observer’s reference frame. B or magnetic field strength is shown in the upper-right panels, γ or the inclination of the magnetic field vector with respect to the observer’s line-of-sight in the lower-left panels, and finally, φ or the azimuthal angle of the magnetic field vector in the plane perpendicular to the observer’s light-of-sight in the lower-right panels. The first sunspot, AR 10923 (Figure 2View Image), was observed very close to disk center (∘ Θ â‰ƒ 9) on November 14, 2006. The second sunspot, AR 10933 (Figure 3View Image), was observed on January 9, 2007 very close to the solar limb (Θ â‰ƒ 50 ∘). In both cases, the magnetic field vector was obtained from the VFISV Milne–Eddington-type inversion (Borrero et al., 2010) of the Stokes vector recorded with the spectropolarimeter on-board the Japanese spacecraft Hinode (Suematsu et al., 2008; Tsuneta et al., 2008; Ichimoto et al., 2008a). The observed Stokes vector corresponds to the Fe i line pair at 630 nm, which are formed in the photosphere. As explained above, Milne–Eddington inversion codes assume that, among others, the magnetic field vector does not change with optical depth: B ⁄= f(τ ) c (see Equation (2View Equation)). Therefore, Figures 2View Image and 3View Image should be interpreted as the averaged magnetic field vector over the region in which the employed spectral lines are formed: −3 ¯τ ≃ [1,10 ].

When the conditions in Equation (6View Equation) are not met, it is not possible to perform a ME-line inversion. If we do, the results should be interpreted accordingly, that is, the inferred values for X correspond to an average over the region τ¯∈ [τc,min,τc,max] where the spectral line is formed. A different approach consists in the application of inversion codes for the radiative transfer equation that consider the full τc dependence of the physical parameters X. In this case, the solution of the radiative transfer equation can only be found numerically (cf. López Ariste and Semel, 1999b). Examples of these codes are: SIR (Ruiz Cobo and del Toro Iniesta, 1992Jump To The Next Citation Point), SPINOR (Frutiger et al., 1999Jump To The Next Citation Point), and LILIA (Socas-Navarro, 2002Jump To The Next Citation Point). This allows to obtain the optical depth dependence (τc-dependence) of the physical parameters with one single spectral line. Ideally, in order to increase the range of validity of the inferred models, one still wants to employ different spectral lines.

View Image

Figure 2: These plots show the magnetic field vector in the sunspot AR 10923, observed on November 14, 2006 close to disk center (Θ = 8.7° at the umbral center). The upper-left panel displays the normalized (to the quiet Sun value) continuum intensity at 630 nm. The upper-right panel displays the total magnetic field strength, whereas the lower-left and lower-right panels show the inclination of the magnetic field vector γ with respect to the observer’s line-of-sight, and the azimuth of the magnetic field vector in the plane perpendicular to the line-of-sight φ, respectively. The white contours on the colored panels indicate the umbral boundary, defined as the region in the top-left panel where I ∕Iqs < 0.3. These maps should be interpreted as the average over the optical depth range in which the employed spectral lines are formed: ¯τ ≃ [1,10−3].
View Image

Figure 3: Same as Figure 2View Image but for the sunspot AR 10933, observed on January 9, 2007 close to the limb (Θ = 49.0° at the umbral center).

1.3.2 Azimuth ambiguity

The elements of the propagation matrix ˆ 𝒦 λ(X[τc]) (Equation (1View Equation)) for the linear polarization (see, e.g., del Toro Iniesta, 2003b, Chapter 7.5) can be written as:

η ∝ cos 2φ (7 ) Q ηU ∝ sin 2φ, (8 )
where φ corresponds to the azimuthal angle of the magnetic field vector in the plane perpendicular to the observer’s line-of-sight. Equations (7View Equation) and (8View Equation) also hold for the dispersion profiles (magneto-optical effects) ρQ and ρU present in the propagation matrix 𝒦ˆλ. Note that these matrix elements remain unchanged if we take φ + π instead of φ. Because of this the radiative transfer equation cannot distinguish between these two possible solutions for the azimuth: [φ,φ + π]. This is the so-called 180°-ambiguity problem in the azimuth of the magnetic field. Because of this ambiguity, the azimuthal angle of the magnetic field φ (as retrieved from the inversion of spectrolarmetric data) in Figures 2View Image and 3View Image (lower-right panels) is displayed only between 0°and 180°. A number of techniques have been developed to solve this problem. These techniques can be classified in terms of the auxiliary physical quantity that is employed:

In recent reviews by Metcalf et al. (2006Jump To The Next Citation Point) and Leka et al. (2009) several of these techniques are compared against each other, employing previously known magnetic field configurations and measuring their degree of success employing different metrics when recovering the original one. It is important to mention that in these reviews, some other very successful methods (which do not necessarily fall into the aforementioned categories) are also employed2: the non-potential magnetic field calculation method by Georgoulis (2005) and the manual utility AZAM by Lites et al. (private communication), which is part of the ASP routines (Elmore et al., 1992). In those reviews it is found that acute-angle methods perform well only if the configuration of the magnetic field is simple, whereas interactive methods (AZAM) tend to fail in the presence of unresolved structures below the resolution element of the observations. Current free and null divergence methods tend to work better when both conditions (Canfield et al., 1993; Metcalf, 1994) are applied instead of only one (Gary and Demoulin, 1995; Crouch and Barnes, 2008), with local minimization being more prone to propagate errors than global minimization techniques.

Several of these techniques are very suitable to study complex regions, in particular outside sunspots. However, in regular sunspots (excluding those with prominent light bridges or δ-sunspots3) the magnetic field is highly organized, with filaments that are radially aligned in the penumbra. We can use this fact to resolve the 180°-ambiguity in the determination of the azimuthal angle φ. This is done by finding the coordinates of the magnetic field vector B in the local reference frame: {e α,eβ,eρ}4 and taking whichever solution, B (φ) or B (φ + π), minimizes the following quantity:

B ⋅ r min |B--⋅ r| ± 1 , (9 )
where the vector r corresponds to the radial direction in the sunspot or, in other words, r is the vector that connects the center of the umbra with the point of observation. Because the condition of radial magnetic fields (Equation (9View Equation)) can only be safely applied in the local reference frame, it is important to describe how B and r are obtained. A detailed account is provided in Appendix 5 of this paper.

Because we aim at minimizing the above value (Equation (9View Equation)) this method can be considered as an acute-angle method where the reference magnetic field is not obtained from a potential extrapolation but rather assumed to be radial. Note that if the sunspot has positive polarity, the magnetic field vector and the radial vector tend to be parallel: Br > 0 and, therefore, the − (minus) sign should be used in Equation (9View Equation). If the sunspot has negative polarity, then the magnetic field vector and the radial vector are anti-parallel and, therefore, the sign + (plus) should be employed. However, this is only a convention: we can choose to represent the magnetic field vector as if a sunspot had a different polarity as the one indicated by Stokes V .

As an example of the method depicted here we show, in Figures 4View Image, 5View Image, and 6View Image, the vertical B ρ and horizontal Bβ and B α components of the magnetic field vector (Equation (46View Equation)), once the 180°-ambiguity has been resolved for two sunspots: AR 10923 and AR 10933 (same as in Figures 2View Image and 3View Image). B ρ, B β, and B α are the components of the magnetic field vector in the local reference frame. Note that strictly speaking, the unit vectors eβ and eα shown in these figures correspond to the unit vectors at the umbral center. Although differences are small, at other points in the image the unit vectors have different directions since those points have different (Xc,Yc) and (α, β) coordinates (Equations (29View Equation) – (33View Equation)). Once the 180°-ambiguity has been solved we can obtain, in the local reference frame, the inclination and the azimuth of the magnetic field, ζ (Figure 7View Image) and Ψ (Figure 8View Image) as:

2 3 −14 ----Bρ----5 ζ = cos B2 + B2 , (10 ) α β −1 B-β Ψ = tan B α . (11 )

It is important to notice that because the ambiguity has now been solved, the angle Ψ varies between 0° and 360° (see Figure 8View Image), whereas before, lower-right panels in Figures 2View Image and 3View Image, φ ranged only between 0° and 180°.

As already mentioned, the method we have described here works very well for regular (e.g., round) sunspots. There is, however, one important caveat: when the retrieved inclination γ (in the observer’s reference frame) is close to 0, the azimuth φ is not well defined. In this case, applying Equation (9View Equation) does not make much sense. Here we must resort to other techniques (Metcalf et al., 2006Jump To The Next Citation Point) to solve the ambiguity. The region where γ = 0° occurs usually at the center of the umbra for sunspots close to disk center, and it shifts towards the center-side penumbra as the sunspot is closer to the limb. A similar coordinate transformation as the one depicted here have been described in Hagyard (1987) and Venkatakrishnan et al. (1988), with the difference that no attempt to solve the 180°-ambiguity was made. Bellot Rubio et al. (2004Jump To The Next Citation Point) and Sánchez Almeida (2005a) employ a smoothness condition to solve the 180°-ambiguity, however their coordinate transform is done in two dimensions, whereas here we consider the Sun’s spherical shape. In addition, only one heliocentric angle Θ was considered in their transformation, whereas here Θ changes for each point on the solar surface (Equation (39View Equation)). One might think that the variation of the angle Θ across the field-of-view (FOV) are negligible. However, for a FOV with 100 × 100 arcsec2 this variation can be as large as 4 – 5°. These differences can be important, for instance, when searching for regions in the sunspot penumbra where the magnetic field points down into the solar surface: B ρ < 0.

View Image

Figure 4: Vertical component of the magnetic field Bρ in the local reference frame in two different sunspots: AR 10923 (top; Θ = 8.7°) and AR 10933 (bottom: Θ = 49.0°). The black contours highlight the regions where the magnetic field points downwards towards the solar center: B ρ < 0. The white contours surround the umbral region, defined as the region where the continuum intensity (normalized to the quiet Sun intensity) I∕I < 0.3 qs. The horizontal and vertical directions in these plots correspond to the eβ and eα directions, respectively.
View Image

Figure 5: Same as Figure 4View Image but for the B β component of the magnetic field vector in the local reference frame. The arrow field indicates the direction of the magnetic field vector in the plane tangential to the solar surface.
View Image

Figure 6: Same as Figure 5View Image but for the B α component of the magnetic field vector.
View Image

Figure 7: Same as Figure 4View Image but for the inclination of the magnetic field with respect to the normal vector to the solar surface eρ: ζ (see Equation (10View Equation)). The black contours indicate the regions where ∘ ζ > 90 and coincide with the regions, in Figure 4View Image, where B ρ < 0.
View Image

Figure 8: Same as Figure 7View Image but for the azimuthal angle of the magnetic field in the plane of the solar surface: Ψ (see Equation (11View Equation)).

1.3.3 Geometrical height and optical depth scales

Traditionally, inversion codes for the RTE (1View Equation) such as: SIR (Ruiz Cobo and del Toro Iniesta, 1992Jump To The Next Citation Point) and SPINOR (Frutiger et al., 1999Jump To The Next Citation Point), provide the physical parameters as a function of the optical depth, X (τc) (Equation (2View Equation)). The optical depth is evaluated at some wavelength where there are no spectral lines (continuum), hence the sub-index c. When this is done for each pixel in an observed two-dimensional map, the inversion code yields X (xβ,xα, τc). However, it is oftentimes convenient to express them as a function of the geometrical height x ρ. To that end, the following relationship is employed:

dτc = − ρ (x ρ)χc [T (xρ),Pg(xρ),Pe(x ρ)]dxρ, (12 )
where χc is the opacity evaluated at a continuum wavelength and depends on the temperature, gas pressure, and electron pressure. Now, these thermodynamic parameters barely affect the emergent Stokes profiles Iλ and, therefore, are usually not obtained from the inversion of the polarization profiles themselves. Instead, other kind of constraints are usually employed to determine them, being the most common one, the application of the vertical hydrostatic equilibrium equation:
dPg-(xρ) = − gρ(xρ), (13 ) dx ρ
which after applying Equation (12View Equation) becomes:
dPg(τc) g --------= ------. (14 ) dτc χc(τc)

Note that, since Equations (13View Equation) and (14View Equation) do not depend on (xβ,x α), they can be applied independently for each pixel in the map. Hence, the geometrical height scale (at each pixel) can be obtained by following the next steps:

  1. Given a boundary condition for the gas pressure in the uppermost layer of the atmosphere, P (τ ) g min, we can employ the fixed-point iteration described in Wittmann (1974a) and Mihalas (1978) to obtain the electron pressure in this layer: Pe(τmin).
  2. From the inversion, the full temperature stratification T(τc) and, thus, T (τmin) are known. Since the continuum opacity χc depends on the electron pressure, gas pressure, and temperature, it is therefore possible to obtain χc(τmin).
  3. A predictor-corrector method is employed to integrate downwards Equation (14View Equation) and obtain Pg(τmin−1). This is done by first assuming that χc is constant between τmin and τmin −1:
    Pg,1(τmin− 1) = Pg (τmin) −----g---[τmin − τmin− 1] (15) χc (τmin)
    and with Pg,1(τmin−1), we apply step #1 to calculate Pe,1(τmin−1).
  4. Since we also know T (τ ) min−1, we repeat step #2 to recalculate χ (τ ) c min− 1, which is then employed to re-integrate Equation (14View Equation) as:
    2g Pg,2(τmin −1) = Pg(τmin) −--------------------- [τmin − τmin−1]. (16) [χc(τmin + χc(τmin−1)]
    Step #4 is repeated k-times until convergence: |Pg,k(τmin−1) − Pg,k−1(τmin −1)| < 𝜖.
  5. We now have Pg (τmin −1). In addition, T (τc) and, thus, T(τmin) are known. Consequently, we can repeat steps #1 to #3 in order to infer Pg (τmin −2).
  6. Thus, repeating steps #1 through #5 yields: Pg(τc), Pe (τc), and χc(τc).
  7. The equation of ideal gases can be now employed to determine ρ(τc). And, finally, the integration of Equation (12View Equation) yields the geometrical depth scale as: τc(xρ). To integrate this equation, a boundary condition is needed. This is usually taken as x ρ(τc = 1) = 0, which sets an offset to the geometrical height such that the continuum level τc = 1 coincides with x = 0 ρ.

Applying the condition of hydrostatic equilibrium to obtain the density, gas pressure, and the geometrical height scale z is strictly valid only when the Lorentz force are small and the velocities are much smaller than the speed of sound. In the chromosphere and corona this is certainly not the case. In the solar photosphere the assumption of hydrostatic equilibrium is, in general, well justified. One exception are sunspots, where the large velocities and magnetic fields might break down this assumption. In these case, a more general momentum (force balance) equation must be employed5:

ρ(v∇ )v = − ∇P + 1j × B + ρg. (17 ) g c

Trying to solve this equation to obtain the gas pressure, density, and geometrical height scale is not an easy task. In the hydrostatic case, the horizontal derivatives did not play any role, thus simplifying Equation (17View Equation) into:

8 >< dPg ∕dxρ = − ρg hydrostatic : dPg ∕dxβ = 0 >: dPg ∕dxα = 0.

However, if the Lorentz force j × B and the advection term (v∇ )v cannot be neglected, the horizontal components of the momentum equation must be considered. In addition, the horizontal derivatives of the gas pressure mix the results of the magnetic field and velocity from nearby pixels. Thus, the determination of the gas pressure, density, and geometrical height scale cannot be achieved individually for each pixel of the map. Instead, a global technique must be employed. This can be done by shifting the z-scale at each pixel in the map (effectively changing the boundary condition mentioned in step #7 above) in order to globally minimize the imbalances in the three components of the momentum equation and the term ∇ ⋅ B. The shift at each pixel, Zw (xβ,x α), represents the Wilson depression. This kind of approach has been followed by Maltby (1977), Solanki et al. (1993), Martínez Pillet and Vazquez (1993), and Mathew et al. (2004Jump To The Next Citation Point). However, changing the boundary condition in step #7 does not change the fact that the vertical stratification of the gas pressure still complies with hydrostatic equilibrium (Equation (13View Equation)). A way out of this problem has not been figured out until very recently with the work of Puschmann et al. (2010a,bJump To The Next Citation Point), who have devised a technique that takes into account the general momentum equation (17View Equation) when determining the gas pressure and establishing a common z-scale. Figure 9View Image shows a map for the Wilson depression in a small region of the inner penumbra of a sunspot (adapted from Puschmann et al., 2010bJump To The Next Citation Point). Another interesting technique has been proposed recently by Carroll and Kopf (2008), where the vertical height scale can be obtained, instead of a posteriori as in Puschmann et al. (2010bJump To The Next Citation Point), directly during the inversion of the Stokes profiles. This is achieved by performing the inversion employing Artificial Neural Networks (ANNs; Carroll and Staude, 2001, see Section 1.3) that have been previously trained with snapshots of MHD simulations, which are given in the z-scale.

View Image

Figure 9: Map of the Wilson depression Zw (x, y) in a small region of the inner penumbra in AR 10953 observed on May 1, 2007 with Hinode/SP. The white contours enclose regions where upflows are present: Vlos > 0.3 km s–1. Negative values of Zw correspond to elevated structures. In this figure x and y correspond to our coordinates xβ and xα, respectively (from Puschmann et al., 2010bJump To The Next Citation Point, reproduced by permission of the AAS).

  Go to previous page Go up Go to next page