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Astron. Astrophys. 333, 721-731 (1998)

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2. Hydromagnetic model and line transfer

2.1. Magnetohydrostatic model

We describe the magnetic structure of the network or an active region plage by a periodic array of rotationally symmetric, vertical flux tubes in magnetohydrostatic (MHS) equilibrium. They are embedded in a static field-free gas. The magnetic structure is obtained as a numerical solution of the MHS equations for prescribed internal and external stratification of the gas pressure. The procedure is described in detail by Steiner et al. (1986). The radius [FORMULA] of the flux tube at [FORMULA] (i.e. unit continuum optical depth [FORMULA] at 5000 Å in the embedding atmosphere), the filling factor [FORMULA] (i.e. the area occupied by the flux-tube magnetic field at [FORMULA] divided by the total area) and the pressure stratification may be prescribed freely. The vertical component of the magnetic field is kept constant across the flux tube radius at the bottom of the computational box, which then is still true to within a few percent at [FORMULA]. We have determined the value of the internal gas pressure at [FORMULA], [FORMULA] by prescribing [FORMULA], the field strength on the flux tube axis at that level. Here we keep [FORMULA] fixed at the typically observed value of 1600 G (e.g., Rüedi et al. 1992). The internal gas pressure stratification is given by the temperature [FORMULA] of the plage flux-tube model PLA of Solanki (1986) and the condition of hydrostatic equilibrium, while the external atmosphere is described by HSCOOL, which is similar to the HSRA (Gingerich et al. 1971), except that it is cooler by approximately 250 K at all heights. This choice of [FORMULA] demonstrates the influence of the non-magnetic atmosphere between the flux tubes on the Stokes profiles particularly clearly. The flux-tube models calculated and analyzed during the present investigation are listed in Table 1. For comparison we often use a 1-D plane-parallel atmosphere with data from PLA and with a homogeneous, vertical magnetic field [FORMULA] equal to the values along the flux-tube axis of the other models. It turns out that model C gives results very similar to model B for almost all parameters, except the magnetic line ratio (Sect. 3.5) and is not discussed separately except in that section.


Table 1. Summary of the analyzed magnetohydrostatic models

2.2. Radiative transfer along multiple rays

The procedure employed to diagnose the hydromagnetic models described in Sect. 2.1 is basically the same as that described by Bünte et al. (1993). To summarize, we first determine the atmospheric quantities required for the radiative transfer of Zeeman split lines along a set of parallel rays inclined to the vertical by a prescribed angle [FORMULA]. The rays pass through a periodic two-dimensional array of identical model flux tubes and are distributed such that the whole set samples a full period of the flux-tube array at a given height. This is appropriate for simulating low spatial resolution observations. Fig. 1 shows outlines of two arrays of flux tubes of different [FORMULA] and the bundle of rays passing through them in the central plane at two different µ values each. Note how the number of flux tubes (or flux-tube walls) that a ray crosses depends on both µ and the flux-tube radius.

[FIGURE] Fig. 1. Cut along the central plane (i.e. the plane spanned by the flux-tube axis and the line of sight) through arrays of two flux-tube models, showing the flux-tubes shaded), and a sample of rays in the central plane, along which line profiles were calculated. For the line profile calculations more rays were used. The thick line delineates the continuum optical depth unity ([FORMULA]) level at the appropriate viewing angle. The thick vertical bar on the right border frame indicates roughly the height range over which Fe I 5250.2 Å is on average formed. a Model D with [FORMULA] km, b model E with [FORMULA] km. Profiles are also calculated along rays lying within other planes parallel to the central plane, as described in the text and illustrated by Bünte et al. (1993). The angle [FORMULA] between the rays and the flux-tube axis is the same in both figures, and corresponds to [FORMULA]. c and d show the same models as Figs. 1a and b, respectively, but now with rays having an inclination corresponding to [FORMULA].

We use [FORMULA] rays in all of the following calculations. We also consider Stokes Q and U in addition to Stokes V and therefore distribute rays over a full period also in the direction perpendicular to the plane of Fig. 1.

We use an adaptive step size grid along each ray. It ensures that the number of grid points along any ray section that pierces a flux tube is larger than some minimum value (recall that the Stokes parameters Q, U and V obtain constructive contributions only within the magnetized part of the atmosphere) and that a maximum optical step size of [FORMULA] is not exceeded, which is important to maintain the sharpness of the flux tube boundary on the optical depth scale.

Once the atmosphere has been computed along each ray, the Stokes parameters are calculated using the radiative transfer code described by Solanki (1987), which, for the solution of the Unno-Rachkovsky equations, now incorporates the Diagonal Element Lambda Operator (DELO) routines of Rees et al. (1989).

We have calculated three spectral lines for each model, Fe I 5250.2 Å, Fe I 5247.1 Å and Fe I 15648 Å, all of which have in the past been extensively used to study solar magnetism. For the two visible lines we use the same atomic data as Solanki et al. (1987), while for the infrared line we follow Solanki et al. (1992).

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Online publication: April 20, 1998