4. Details of the equivalent width method
4.1. Selection of spectral lines
Fe i lines in the range from 5260 to 9240 Å were selected according to the following criteria:
Lines free of blends were chosen from a visual inspection of the spectrum of the sharp-lined K3 V star HD 16160 and checked by comparison with the atlas of the solar spectrum by Moore et al. (1966). We thus found about 60 lines usable for HD 16160, but many lines were excluded later in the following reduction process. Some were excluded because of a large deviation (more than ) from the obtained mean empirical curve of growth or because of an unusually large correction to the laboratory value (Sect. 4.2), and others because they were blended (late K7 star LkCa 16, faster rotating star GW Ori) or not measurable (disturbed by emission in the cTTSs). Table 2 lists all the Fe i lines finally used.
Table 2. Compilation of all Fe i lines finally used. All values describing the line transitions are taken or calculated from the Fe i multiplet tables by Nave et al. (1994). We give wavelength, Rowland multiplet number R, effective Landé factor , the number of Zeeman components n, and lower excitation potential E. Lab are the laboratory -values. The next three columns give the corrections we applied to the laboratory values for the three comparison stars, namely the Sun, HD 16160 (K3) and AGK3+630096 (K7). In the few cases where no laboratory values were found, we give here the absolute derived from our iterative procedure, these values are in parenthesis. The last nine columns list the EWs measured for the three comparison stars (solar values are taken from the atlas of the solar spectrum by Moore et al. 1966), and for our target stars HD 42807(G2), T Tau(TT), UX Tau A(UXT), LkCa 15(L15), LkCa 16(L16), and VY Ari(VYA).
4.2. Effective -values
Laboratory -values have been taken from the Fe i multiplet tables by Nave et al. (1994). These values are mostly based on the work of Fuhr, Martin & Wiese (1988). As already discussed by Basri et al. (1992), the use of laboratory -values as input parameters to LTE atmosphere models can lead to discrepancies of several percent between calculated and observed EWs. In a similar way to Basri et al. (1992), we try to compensate for inadequate assumptions in our LTE model atmosphere by adjusting the laboratory -values by correction terms, C, to reproduce the observed EWs for a non-magnetic template star. Starting with laboratory -values as the initial values, the corrections C are iteratively adjusted in order to exactly match the observed EWs with the calculated ones using the appropriate temperatures for the template stars (), as listed in Table 3. We finally end up with a model which reproduces the spectrum of the nonmagnetic template star with the corrected -values. However, during this process some spectral lines are rejected. We use a curve of growth derived according to Milne-Eddington theory and then reject all lines that fall more than off this mean curve of growth (see Fig. 2). To construct this curve, we optimse the mean excitation temperature of all Fe i lines in such way that the scatter of the curve is minimised. The excitation temperture is listed in Table 3. After the -values have been corrected, the scatter around the curve of growth is much smaller than with the uncorrected -values. As will be described in Sect. 4.6, the reduction of the scatter around the curve in the magnetic and non-magnetic case is also used as a criterion for the detection of the fields. Table 2 gives the correction terms C computed for each of the -values.
Table 3. Results from the equivalent width method. Beside the apparent filling factor , the constant of Eq. 7 is given. is the maximum correlation coefficient following from the linear regression, N is the number of suitable lines per star. For each spectral type we list as expected from the statistical relations by de Jager & Nieuwenhuijzen (1987) assuming Luminosity Class V for the T Tauri stars, followed by the effective temperature of the applied model atmosphere . The next two columns list the mean excitation temperature of the Fe i lines obtained from the empirical curve of growth before and after correcting the measured EWs for the expected magnetic intensification. The last column gives the ratio of the scatter of the curve of growth after to before the correction. As this value is larger than one for UX Tau A, we are thus cautious about the result for this star. For LkCa 16 the number of lines with a high sensitivity to the magnetic field is rather small.
In order to measure the EWs in the same way in the template and the T Tauri stars, we broaden the spectra of the template stars to the same as that of the T Tauri stars. Solar Fe i lines were chosen from the solar atlas by Moore et al. (1966) in such a way that no blends are expected for the broader lines of the nonmagnetic G2 template star ( = 10 km/s). For the T Tauri stars we use values derived for the appropriate template star which is for LkCa 16 and for all the other (classical) T Tauri stars.
Our procedure has two advantages. Firstly, it allows us to exclude lines with unusually large residuals compared with the mean scatter around the final curve of growth, especially lines for which the applied radiative transfer code is not fully compatible. Secondly, it is possible to calculate empirical -values for a (not too large) number of lines were no values have be measured in the laboratory. Since the correction of the -values compensates for any uncertainties of the atmosphere model, we compute these corrections separately for all of our three template stars of different spectral type (Table 2). The mean difference between the corrected and the laboratory values is in the same range as the mean difference obtained by Basri et al. (1992) ( dex) for their LTE atmosphere. The corrections for the Fe i lines we get for the K3 star HD 16160 also correspond very well to the corrections applied by these authors on the basis of their measured EWs of Fe i lines in the G8 V star 61 UMa.
Fig. 2 shows the empirical curves of growth obtained with our method for the EWs from the solar atlas by Moore et al. (1966). Since EWs are given for the centre of the solar disk, we did not integrate over the disk in this case but calculated the EWs for the disk centre. For the Fe i lines we find a mean excitation temperature of 5310 K for the Sun using the corrected -values and 5790 K using the laboratory -values. The internal accuracy of our temperature determination is of about K. As expected, the derived excitation temperature is quite different from the effective temperature of the Sun. In contrast to , the excitation temperature describes the mean temperature in the layers of the photosphere were the Fe i lines are generated, rather than the continuum temperature of the photosphere. Since the lines arise from smaller optical depths than the continuum, the excitation temperatures is generally lower than . A commonly used value for the excitation temperature of solar lines is 5140 K, as given by Cowley & Cowley (1964). The authors derived this temperature from a composite curve of growth for lines of seven elements based on the oscillator strengths taken from Corliss & Bozman (1962). The agreement between the excitation temperature which we derive and the excitation temperature given by Cowley & Cowley (1964) is clearly better if we use the corrected -values instead of the uncorrected, laboratory -values. The corrections to the -values thus seem to be realistic. The effective temperatures of our model atmospheres, on the other hand, have been chosen according to the spectral types of the stars and have not been derived from the obtained Fe i excitation temperatures, since the relation between both temperatures is not very well known for all spectral types. In Table 3 we give, for each star, according to its spectral type, the effective temperature adopted in the model atmosphere calculations, and the derived Fe i excitation temperature .
4.3. The line transfer program for polarized light
We used a program by Staude (1982), which is based on a polarized radiative transfer code which calculates the line profiles for all 4 Stokes parameters under the influence of a magnetic field (given field strength B, inclination to the line-of-sight , and azimuth) for given spectral line properties (defined by parameters such as oscillator strength, quantum numbers of the transition, possible non-LTE effects, and blends with other lines). The depth dependence of all quantities is also taken into account. The program, originally developed for the investigation of solar magnetic fields, computes intensities arising from a single point on the stellar surface. Opacities are also calculated by using the program by Staude (1982), LTE model atmospheres are taken from the Kurucz 9 program. Assuming a model atmosphere with fixed , and microturbulence the program computes the line profiles , where describes a radially symmetric geometry, described below. To obtain the EWs we have to integrate over the stellar disk, taking limb darkening into account. Results depend on the particular Zeeman pattern of each line as well as on the assumed surface geometry and strength of the magnetic field.
4.4. Field geometry and integration over the stellar disk
For simplicity we consider a two-component model atmosphere consisting of regions with and without magnetic fields, and we assume that each of the surface elements contains the same amount of regions with magnetic field, and of regions which are field free. The effective temperatures of these two regions can differ. The fraction of the surface covered by the field is given by the filling factor f. (This symbol f represents a different quantity from that for oscillator strength which is used in .) The two-component atmosphere allows a time-saving integration over the stellar disk assuming radial symmetry and, as shown below, it is sufficient to compute the line profiles for f = 1.
The magnetic enhancement of a spectral line depends on the orientation of the magnetic field. For a given field strength, the increase in EW for a transverse field ( = ) is much stronger than for a longitudinal field ( = ) (Fig. 3). In some cases authors treating the two component model atmosphere assumed a mean inclination of the field of (e.g. Marcy 1982, Gray 1984, Bopp et al. 1989). However, as various geometries may lead to different values for , a more thoroughly discussion is needed.
For a magnetic dipole would be between and , depending on the orientation of the dipole axis to the observer. More complex fields often result in a mean inclination in this range. Pillet et al. (1997), for example investigated the magnetic structure of solar plage regions. They found that the magnetic field vector is orientated almost vertical to the solar surface throughout the solar disk. If we assume a limb darkening coefficient of = 1.5 2, the mean inclination of such a radial field observed in integrated light is about . We thus conclude that a mean inclination of the field in integrated light differing from by more than can only be possible if the field is entirely confined to a few large spots. In order to avoid any prejudice on the geometry of the fields we prefer to leave as a free parameter, and we calculate a grid of EWs for all reasonable values of and B.
To take limb darkening into account, we integrate over the stellar surface dividing the disk into rings of equal area and compute the mean intensity profile of each ring. After the summation of all contributions the result is normalised to the sum of all continuum contributions.
4.5. Determining B, , and f from the observed EWs
Let represent the intensity profile of a line arising from a field-free region and the intensity profile arising from a region with a field B of inclination , and and the corresponding continuum intensities. We further define the mean ratio of the continuum intensities arising from the magnetic regions, and from the non-magnetic regions as
The factor represents the differences between the atmospheres of the magnetic and non-magnetic regions of the star (effective temperature etc) for the line i. We designate the ratio of the calculated EWs, , for a field free region to the measured EWs, , for an (assumed field free) template star as
The factor represents our limited ability to model line i in the atmosphere with no magnetic field. The EW of absorption line i from the field free and field regions are respectively (to avoid cumbersome notation we omit the subscript i in the following equations)
We define the line veiling v as the ratio of the veiling intensity to the continuum intensity
(for simplicity we assume here that the veiling v is constant over the spectral range used, so that the veiling for the ith line is ; a more general discussion is given in Sect. 6.3)
We take into account any continuum veiling (which we assume applies equally to the magnetic and non-magnetic regions) by multiplying both sides of Eq. (6) by to give the ratio of the (veiled) EW to
If our line transfer calculations were perfect then would be 1 for all lines i, and in an idealised case there would be no difference between the atmospheres of the magnetic and non-magnetic regions of template star and then would be 1 for all lines i. Then (if there were no veiling) we would have and . In general the filling factor f is related to and by
from which it is clear that f can determined from independent of the veiling, at least for the case that the veiling is not wavelength dependent. In Sect. 6.3 we will discuss the implication if the veiling is wavelength-dependent.
From our simple model it is also clear that our determination of the true filling factor depends somewhat on the differences in the atmospheres in the magnetic and field-free regions (). As we cannot be sure of the precise value of we define an apparent filling factor
The assumptions are very rough, but are a first step to account for differences in the atmospheres of magnetic and non-magnetic regions. In fact the quantities , , and are wavelength dependent and so are the and . We use linear regression according to Eq. (7) to find the average values of and .
The two-component atmosphere model has a computational advantage in that to use Eq. (7) for the determination of the magnetic field parameters it is obviously sufficient to calculate from the line transfer code a grid of , so we do not have to take different filling factors into account in our radiative transfer computations.
4.6. Magnetic effects on the empirical curve of growth
The magnetic intensification of the EWs affects the position of the curve of growth. In the range of of the T Tauri stars the EWs of the Fe I lines increase with decreasing temperature, so the magnetic enhancement will shift the curve of growth to larger EWs and lower temperatures. In other words, the apparent excitation temperature of a magnetic star obtained from the curve of growth according to Sect. 4.2 can be much lower than the true excitation temperature. To verify this, we computed `zero field' EWs of the T Tauri stars trying to remove the calculated magnetic intensification effects from the observed . This we can do by using Eq. (8) for and subtracting the contribution from the magnetic regions () and replacing it with the equivalent contribution from the regions () we find:
So, after determining the optimised parameters and from a linear regression using Eq. (7), we compute the `zero field' EWs from Eq. (11) and obtain the true excitation temperature from a new empirical curve of growth. This method is very useful for estimating the effects of the magnetic field on the spectral type determined from line strengths and for choosing the right atmosphere model and the right template stars for our method.
On the other hand, one could use Eq. (11) directly to minimise the scatter of the curve of growth by optimising the parameters and . Compared to the method of a linear regression according to Eq. (7), there is the computational time consuming fact that we have to optimise all three parameters, whereas the constant follows as a result of the linear regression if we use Eq. (7). However, the results of this method should be virtually identical to the results of the method previously discussed.
4.7. Atmospheric parameters used
In a first step we determine three sets of empirical -values from the two non-magnetic comparison stars HD 16160, and AGK3+63 0096, and from the solar lines, for which we derive Fe i excitation temperatures of 4750 K, 4140 K and 5310 K, respectively. EWs are computed from Kurucz model atmospheres for which we choose effective temperatures corresponding to the known spectral types of the template stars (see Table 3). We use solar metallicity, = -4.51, for all of the stars. Our model atmosphere treats collisional damping using van de Waals damping in the Unsöld approximation, which we modified by applying a correction factor of 2 to the Unsöld damping constant .
We tested the method using the K3 template star HD 16160 and VY Ari, and model atmospheres of different and microturbulence. The results obtained are almost insensitive to . Even for temperatures differing by 400 K from the assumed correct value of 4775 K we get nearly the same values of B, , and - only the significance of the results decreases. The reason for this low sensitivity of the magnetic field determination to the temperature dependence of the Fe i line strength is that our method is based on the magnetic sensitivity of the spectral lines and so it is explicitly sensitive to the magnetic enhancement of the EWs. However, we point out that the robustness of our method depends on a sufficiently large number of lines being used, which is not the case for all of our target stars. The same assertion can be made for a low dependence on microturbulence. Tests with = 0 km/s give nearly the same results as with = 2 km/s, with only slightly less sharp maxima for the optimum values found for B, . Microturbulence was generally chosen to be 2 km/s which is a mean value based on the values observed for several T Tauri stars by Padgett (1996). Since the dependence of the EWs or the curve of growth on gravity is much smaller than the temperature dependence, we decided to use a constant of 3.9 for all of the T Tauri stars. This value represents a mean value taken from literature (Finkenzeller & Basri 1987; Magazzu et al. 1992), the total variation of for dwarf stars within the spectral sequence K 0 to K 7 being about 0.07 (Schmidt-Kaler 1982). Appropriate values were used for the template stars.
4.8. Detection procedure
The magnetic field strength is derived in the following way: For all reasonable combinations of B and , the normalised observed EWs are plotted versus the normalised computed EWs . The best values of B and are found when the scatter of the linear regression according to Eq. (7) is minimised. The constants and are the coefficients of the linear regression, and using Eq. (9) the apparent filling factor is then obtained.
To illustrate the use of Eq. 7 we have plotted in Fig. 4 the ratio Y = = 2kG, f = 0.5, = = 0) of a synthetic test star versus the values computed from our model, X = = 2kG, = = 0). As a test star we used our non-magnetic template star HD 16160. We then calculated the Fe i lines for a given configuration of B, f, and . The resulting equivalent widths produce a straight line with slope 0.5 ( = 0.50, = 0.50, = 0.50) in Fig. 4. Since here Y and X are in fact the same quantity, there is no scatter of the values (crosses in Fig. 4) about this line. In practice, we would have chosen the absolutely correct model atmosphere and parameters. On the other hand, if we plot Y versus = 1kG, = = 0), we get the scattering values marked by dots and a regression line with slope 1 ( = 0.00, = 1.03, = 1.00). Both results give the same but they are distinguishable by the RMS (or correlation coefficient) of the linear regression. Fig. 4 thus illustrates how we can disentangle B and as well as B and by searching for that pair of B, for which the scatter about the regression line is a minimum ( follows then from the obtained constants and ).
4.9. Estimating the uncertainties
One problem is the estimation of the significance level or probable uncertainty of the obtained results. Because of the different uncertainty sources and the very complex nature of the propagation of uncertainties, we do not try to model this propagation. Instead we estimate the significance from the confidence level of the correlation coefficient obtained from the linear regression for each pair of B, . The correlation coefficient is a measure of the coincidence of the regression lines following from the regression Y with X, and from X with Y, as is illustrated in Fig. 7. For all further derivations we use the results of the regression Y with X according to Equ. 7. This is illustrated for the case of maximum correlation coefficient for all stars in the left parts of Figs. 5 and 7 to 12.
The Fisher -transformation (Fisher 1925) normalises the statistical distribution of , where
From the distribution of it is possible to estimate the probability that two determined values of the correlation coefficients are different at a given confidence level (e.g. Sachs 1998). This is the case if
where N is the number of data points and is the value of the Students-t distribution for N-2 degrees of freedom at confidence level .
The right parts of Figs. 5 and 7 to 12 show two-dimensional contour plots for . Inside the areas enclosed by the dotted lines the correlation coefficients obtained from the linear regression are not significantly distinguishable. Outside this area the regression coefficients are different from the maximum value at a statistical 95% level. In the following we regard the enclosed areas as the areas of confidence of our results.
Fig. 6 shows the contour plot for the product for VY Ari. Although the correlation coefficients of the linear regression may occasionally be relatively high even for a slope of larger than one, physically meaningful solutions are restricted to slopes of less than one, because the filling factor has to be less than or equal to one. These acceptable solutions are represented in Fig. 6 by the part to right of the thick, almost vertical line. Inside the area of confidence there is only a very small variation of of about 0.3 kG. That means that the filling factor varies in an opposite sense to the magnetic field strength. From the contour plots we can derive the (well-defined) product as the mean inside the area of confidence, and the limits for B from the section of the area of confidence with 1. Since B and f are strongly correlated, we will not give mean values and error bars for these values, instead we will only quote the derived limits.
© European Southern Observatory (ESO) 1999
Online publication: December 16, 1998