3. Analysis technique, calculations and results
Our determination of the element abundances is based on the assumption of LTE and makes use of empirical models of the solar network (which we shall call NET) and plage flux tubes (PLA) constructed using Fe I, Fe II and C I lines (Solanki & Brigljevi 1992), as well as the standard quiet sun atmosphere (HSRA) of Gingerich et al. (1971). The HSRA has been preferred over newer models since the flux-tube models were constructed relative to it. To ensure that quiet-sun profiles of stronger spectral lines are well reproduced the chromospheric part of the HSRA was changed so as to correspond to a steadily decreasing temperature with height (Solanki 1986). The Stokes profile calculations are carried out with a modified version of the Stokes radiative transfer code described by Sheminova (1990), which is based on a code written by Landi Degl'Innocenti (1976). The thin-tube approximation is used to describe the field strength stratification and shape of the flux tubes. The profiles are calculated along a set of 30 vertical rays piercing the cylindrical model flux tube at different radial distances from its axis using the scheme of Solanki & Roberts (1991). The line profiles formed along each of the rays are weighted according to the area on the solar disk which that ray represents and then added together to give a combined line profile which is compared with the spatially unresolved observations (we call this procedure 1.5-D radiative transfer).
A height-independent micro- and macroturbulence of 1 km s-1 and 2 km s-1, respectively, were introduced in the flux tubes, and of 0.8 km s-1 and 1.7 km s-1 in the quiet sun. The empirical factor to the Van der Waals damping constant, calculated using the formula given by Unsöld (1955), is chosen to be . These are the same values as those used to construct the empirical flux tube models we use for the line calculations (Solanki 1986; Solanki & Brigljevi 1992). However, we have also tested the influence of varying these and other parameters (see Sect. 4). The Landé factors and splitting patterns of the selected lines of Al, C, O, Na, Si, Ca, Y, Zn have been determined assuming LS coupling. For Sc, Ti, Cr, Fe, Ni the empirical values of and , taken from the tables of Sugar & Corliss (1985), were employed instead. Here and are the Landé factors of the lower and upper state of the transition. Wherever available with sufficient accuracy, i.e. of the Fe I, Ti I, and Cr I lines, the statistically weighted oscillator strengths, gf, obtained by Blackwell's group (e.g., Blackwell et al. 1982) have been used, otherwise we employed those of Gurtovenko & Kostik (1989). We have determined the amplitudes of the blue and red wings, , and , of the observed Stokes V profiles and found the average amplitudes for each line in the network (FTS2, FTS3) and in the plage (FTS4, FTS5). The central line depths observed in the quiet sun were taken from Gurtovenko & Kostik (1989).
In a first step we determined the element abundances for the quiet photosphere by fitting the observed central depths of all selected lines using the HSRA model. The obtained abundances, , as well as their standard deviation are listed in the 4th column of Table 1. The remaining columns list the following: The element and ion, the corresponding first ionization potential, element abundances derived using 1.5-D radiative transfer in the network and in plages with the NET and PLA models of Solanki & Brigljevi (1992) and with the plage flux-tube model (PLAOLD) of Solanki (1986) (respectively labeled , and ) and abundances obtained using 1-D radiative transfer with the PLA model of Solanki & Brigljevi (1992) (). Finally, abundances taken from Grevesse & Sauval (1998) are tabulated under . N is the number of analysed lines of each element.
The absence of a standard deviation value for a particular element signifies that only a single spectral line could be used. A comparison with the last column of Table 1 shows that the HSRA gives consistently too low A values compared to those published by Grevesse & Sauval (1998). The differences are on average a factor of 4-5 larger than the standard deviation and are most probably a result of the difference in temperature stratification between the HSRA and the Holweger & Müller (1974) model employed by most investigators determining photospheric abundances.
When deducing the elemental abundance inside flux tubes we are faced by a further problem. In addition to their dependence on abundance and thermal stratification, the Stokes V profiles scale almost linearly with , where is the magnetic filling factor and is the angle between the line-of-sight and the magnetic vector. They also depend on the intrinsic magnetic field strength. To counter the latter problem the intrinsic field strength in the observed regions was determined using the Fe I 5250.2 Å (Landé ) and 5247.1 Å line pair (in the spectra FTS2 and FTS4) and the Fe I 6301.5 Å and Fe I 6302.5 Å (Landé ) line pair (FTS3 and FTS5). Such combinations of large and small Landé factor lines provide good diagnostics of the field strength (e.g., Stenflo 1973). The dependence on the filling factor poses a more fundamental problem, however. It is not possible to simultaneously determine the filling factor and the abundances of all the elements. However, it is sufficient to assume an abundance for one of the elements and to determine the abundances of all the other elements relative to it. Since the flux tube thermal stratifications were derived mainly on the basis of Fe I and II lines assuming an iron abundance of 7.46 (corresponding roughly to the quiet sun value) we have fixed the iron abundance to this value for the current analysis as well.
We have then constructed ratios between the V amplitudes of each line of each element and of each Fe I and II line. By fixing the iron abundance to and comparing calculated with observed V ratios we then obtained estimates of the abundance of a given element separately from each single line ratio involving lines of this element in the numerator. These abundances, after averaging over all ratios involving a particular element, are presented in Table 1 (columns 5-8). Note that .
The iron abundances listed in columns 5-8 of Table 1 require explanation. These values were determined separately from Fe I lines and Fe II lines in the same way as the ones of the other elements. Hence the abundance of, e.g., Fe I is determined by forming the ratio of the V profile of each Fe I line with each Fe I and II line. The abundance of the latter (i.e. the lines in the denominator) is fixed at 7.46, while that of the former is varied until the computed line ratios correspond to the observed values. In this way one obtains an abundance value from each line ratio. The abundances obtained from ratios involving Fe I in the numerator are then averaged together to give the Fe I abundance, similarly those involving Fe II. The Fe I and II abundances deduced in this manner differ from the assumed iron abundance in our flux tube models (7.46) due to differences between Fe I and II lines [note that Fe IFe II) for all models, except NET, for which Fe IFe II)].
We also considered the possibility of using the areas under the V profile blue and red lobes instead of their amplitude. We decided against their use, however, since in the network the noise in the wings of the observed profiles affects approximately half of the lines from our list with sufficient severity to render them of limited use.
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999