List of Footnotes

1 The symbol † indicates the transpose.
2 Many of the codes that have been compared in these papers are publicly available to the community. Minimum energy method: External Link; Non-Potential field calculation: External Link; and AZAM: External Link
3 δ-sunspots are commonly defined as those where the umbra possesses two difference polarities.
4 The unit vectors of the local reference frame are defined as follows: eρ is the unit vector that is perpendicular to the tangential plane on the solar surface at the point of observation, while eα and eβ are inside this plane (see Figure 42View Image).
5 Note that the most general momentum equation would also include, in the right-hand term of Equation (17View Equation), the terms corresponding to the viscous forces: μ1∇2v and [μ2 + (1∕3)μ1]∇ (∇v ), where the coefficients μ1 and μ2 are often referred to as shear viscosity and bulk viscosity, respectively.
6 According to Equation (12View Equation), z and τc have opposite signs. This indicates that τc decreases when z increases and, therefore, τc decreases from the photosphere to the corona.
7 We shall mention here that, in order to provide the values of the derivatives in terms of the geometrical height instead of the optical depth, we have assumed that hydrostatic equilibrium holds (see Section 1.3.3).
8 Thin in this context means that the thin flux-tube approximation (Spruit, 1981) can be applied. This reduces the problem to a 1D problem: it does so by assuming that the flux tube’s radius is much smaller than the pressure scale height.
9 The elevation angle is defined as (1− ζ) (see Equation 10View Equation). Thus, small elevation angles correspond to horizontal magnetic fields (contained in the solar surface) and large elevation angles indicate magnetic fields that are rather vertical (perpendicular to the solar surface).
10 SIR-like inversions refer to inversions of spectropolarimetric data where the physical parameters are allowed to vary with optical depth X (τc) (Equation 2View Equation). This is discussed in some detail in Section 1.3.