The line-center photons of the hydrogen Lyα line originate just at the boundary of the model’s TR. In a 1D model atmosphere (e.g., Fontenla et al. 1993), the vector normal to the TR lies along the vertical, which coincides with the symmetry axis of the incident radiation field. In a 3D model atmosphere (e.g., Carlsson et al. 2016), the model’s TR delineates a corrugated surface, so that the vector normal to the model’s TR changes its direction from point to point (see Figure 7 of Štěpán et al. 2015). On the other hand, at each point on such corrugated surface the stratification of the physical quantities along the local normal vector 





 is much more important than along the perpendicular direction. In order to estimate how the line-center fractional polarization of the hydrogen Lyα line is at each point of the field of view, it is reasonable to assume that the incident radiation field has axial symmetry around the direction of the normal vector 





 corresponding to the spatial point under consideration (see also Štěpán et al. 2018). Taking a reference system with the Z-axis directed along the normal vector 





, and recalling Equation (1), at each point of the corrugated TR surface we can estimate the line-center fractional polarization signals through the following formula
2





where the positive Stokes Q reference direction is now the perpendicular to the plane formed by 





 and the LOS, 





 is the cosine of the angle between 





 and the LOS, and 





 is the anisotropy factor calculated in the new reference system. Clearly, because the direction of 





 changes as we move through the corrugated surface that delineates the TR, the Q/I signals estimated with Equation (2) do not share the same reference direction for the quantification of the linear polarization. In order to arrive at equations for Q/I and U/I having the parallel to the nearest limb as the positive Stokes Q reference direction, we have applied suitable rotations of the reference system. Specifying the direction of the local TR normal vector through its inclination 





 with respect to the vertical (with 





 between 0° and 90°) and azimuth 





 (with 





 between 0° and 360°), our analytical calculations show that the line-center Q/I and U/I signals of the hydrogen Lyα line can be estimated by the following formulas:
3







4





Figure 2 shows examples of CLV of the Q/I and U/I Lyα line-center signals for several topologies of the model’s TR, with random azimuth 





. The first three rows of the figure correspond to the indicated fixed inclinations 





, while the bottom row panels show the case in which also 





 has random values (i.e., the probability distribution 





 for 





 and 





 for 





). Note that the CLV of the fractional linear polarization line-center signals is very sensitive to the geometry of the model’s TR, and that there is no CLV at all when the normal vector 





 has random orientations. Moreover, it is also very important to point out that the Q/I and U/I amplitudes are sensitive to the magnetic field strength of the chromosphere-corona TR, through the Hanle factor 





 in Equations (3) and (4).
Citation: Štěpán et al. 2018
Citation function: Uses