2 Coordinate systems

Figure 1: A vector, $\vec{B}$, with polar coordinates $(r,\psi )$w.r.t. the center of the spot can be described within a local Cartesian coordinate system (x',y',z') of which the (x',y')-plane lies in the sunspot surface. The x'-axis is parallel to the line which connects disk and spot center pointing toward the solar limb. The figure sketches the inclination, $\gamma '$ w.r.t. the surface normal $\vec{\hat n}$ (which is parallel to the z'-axis), and the azimuth, $\phi '$w.r.t. the x'-axis, of $\vec{B}$. Note that the LOS is within the (x',z')-plane.

A position in the surface plane of a sunspot can be given in polar coordinates $(r,\psi )$, with r being the distance from spot center and with $\psi=0^\circ$ and $180^\circ$ corresponding to the line which connects disk and spot center. This line is also referred to as the line-of-symmetry, since, e.g., the map of the line-of-sight velocity component of a radial outflow is symmetric w.r.t. this line. As depicted in Fig. 1, we introduce a local Cartesian coordinate system (x',y',z') at $(r,\psi )$. The z'-axis is parallel to the surface normal $\vec{\hat n}$ and the x'-axis is parallel to the line-of-symmetry. A vector $\vec{B}$ is described by the inclination $\gamma '$ w.r.t. $\vec{\hat n}$, and the azimuth, $\phi '$. The coordinates of $\vec{B}$ in the local Cartesian system are $B_{x'}=B \sin\gamma' \cos\phi'$, $B_{y'}=B \sin\gamma' \sin\phi'$, and $B_{z'}=B \cos\gamma'$, with $B\equiv\vert\vec{B}\vert$.

For the calculation of the emanating Stokes vector, the relevant angles of the magnetic and flow field are the angles for the inclination, $\gamma$, and the azimuth, $\phi$, w.r.t. the LOS. Hence, the local coordinate system has to be rotated around the y'-axis by the heliocentric angle, $\theta$, which is the angle between the LOS and $\vec{\hat n}$ (see also Title et al. 1993). In the LOS coordinate system, the inclination, $\gamma$, and the azimuth, $\phi$, of $\vec{B}$are given by

$$% \gamma= \arccos \left( \cos\gamma' \cos\theta - \sin\gamma' \cos\phi' \sin\theta \right),$$ (1)

$$% \phi = \arctan \left( \frac{\sin\gamma' \sin\phi'} {\co... ...' \sin\theta + \sin\gamma' \cos\phi' \cos\theta} \right)\cdot$$ (2)

Note that $\phi'=0$ implies $\phi=0$. The LOS component of the flow velocity, v, with v₀ being the absolute velocity, is given by

$$v= v_0 \cos\gamma = v_0 \left( \cos\gamma' \cos\theta - \sin\gamma' \cos\phi' \sin\theta \right) .$$ (3)