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Astron. Astrophys. 341, 768-783 (1999)

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6. Discussion

6.1. The detection of strong magnetic fields

The significance of the magnetic field strengths obtained can be estimated in three different ways:

  1. The absolute value of the correlation coefficients and the number of underlying data points can be used to estimate the significance of the results. In this way we test the presence of a strong magnetic field.

  2. We can estimate the confidence level of the obtained correlation coefficients using the Fisher [FORMULA]-transformation (e.g. Fig. 5). This method gives an estimate of the uncertainties in [FORMULA], and of also constrains all possible combinations of the separate values for B and f.

  3. Compared to the EWs calculated without the magnetic intensification, the scatter around the empirical curve of growth has to decrease if we take the magnetic intensification of the EWs into account.

From the first item, we find that the probability is that the correlation obtained is significant, being higher than 99%, for VY Ari, LkCa 15, LkCa 16, T Tau, and UX Tau A. Because of the large [FORMULA] GW Ori is the star with the smallest number of suitable lines and thus the star with the lowest accuracy obtained. The correlation coefficient thus is small, and we conclude that the 16 lines are not sufficient to detect a magnetic field of [FORMULA] kG, as a level of [FORMULA] has to be reached to give a 95% probability for a detection.

The areas given by the dotted lines in the contour plots refer to a 95% significance level. The fact that many lines are needed is further highlighted by the data from VY Ari and LkCa 15. These stars are the ones with the largest number of suitable lines, and correspondingly both upper and lower limits of B could be derived. For T Tau we had to extend our model calculations up to 5 kG to derive the upper limits in B and for LkCa 16 we did not find the upper limit up to a field strength of 6 kG. We thus conclude that the results give some hint that LkCa 16 could have a field with [FORMULA] kG (this lower limit follows from the condition that [FORMULA]) but the detection is only marginal. A similar assertion can be made for UX Tau A. In this case, we find a correlation coefficient which is statistically significant but a zero field is within the range of possible values. The variation of [FORMULA] is small: the variation is only [FORMULA] kG up to a field strength of B = 4 kG.

The results of the third test are given in the last column of Table 3. Taking the magnetic intensification into account, the scatter of the empirical curves of growth is reduced by a factor of more than 2 for VY Ari, LkCa 15 and T Tau, and by 30% for LkCa 16. However for UX Tau A, for which the first two tests yield a marginal detection of a magnetic field, the scatter of the curve of growth increases by a factor of three, rather than decreases. We are thus cautious about the detection of a field on UX Tau A.

6.2. Geometry of the magnetic field

In principle, our method allows us to disentangle B, f and [FORMULA]. As can bee seen the contour plots (right parts of Figs. 5 and 7 to 12) the magnetic field strength is much better constrained than the mean inclination of the field lines, and we derived only lower limits of order of [FORMULA] for all of the stars. The apparent filling factor, on the other hand, is closely related to the derived magnetic field strength. Since the lower limits are already quite high the filling factor really has to be very close to one in most cases. This is emphasised by two additional results:

  1. The large values obtained for the constant [FORMULA] (Table 3) for the case of the optimal regression (recall that if [FORMULA], [FORMULA] then [FORMULA]).

  2. The consistency of the derived B and f was checked by removing the mean inclination of the field as a free parameter in the reduction process by assuming a radial field geometry (see below).

We thus have to conclude that in the cases where we have unambiguous evidence of a strong magnetic field, the filling factor has to be larger than 0.5.

To test if the derived magnetic field strengths and filling factors (calculated above using a homogeneous geometry with constant mean inclination) are sensitive to the assumed field geometry we computed EWs for different field strengths, B, from a radially symmetric model of the magnetic field, setting [FORMULA], which corresponds to a field structure as derived by Pillet et al. (1997) for solar plage regions. We then applied the technique described in Sect. 5 to obtain B and [FORMULA] for the target stars. Table 4 lists the results. All values agree very well with the results of Table 3 obtained from the homogeneous magnetic model with fixed inclination of the magnetic field. This test again confirms the values of [FORMULA], and the large lower limits derived on f. On the other hand, since the results are quite insensitive to [FORMULA], it is quite sufficient to assume a radial symmetry to determine the magnetic field strength.


Table 4. Results from the equivalent width method assuming radial symmetry of the magnetic surface field.

6.3. Possible influence of veiling on the results

The possible presence of veiling in the cTTS may complicate the analysis of the spectra. This is especially true if the amount of veiling depends on wavelength. By defining the veiling v as [FORMULA], where [FORMULA] and [FORMULA] are the equivalent widths of spectral lines that are relatively insensitive to the magnetic field ([FORMULA]), we derive values of [FORMULA] for T Tau, [FORMULA] for UX Tau A [FORMULA] for LkCa 15, and [FORMULA] for LkCa 16. Although none of our T Tauri stars showed any significant veiling, we will discuss any possible influence of the veiling on the magnetic field analysis in this section.

The wavelength dependence of the veiling has been studied by a number of authors (Hartigan et al. (1989), Hartigan et al. (1991), Bertout et al. (1988), Basri & Bertout (1988), Hartmann & Kenyon (1990, Basri & Batalha (1990)). The general result is that the amount of veiling drops rapidly between 3000 and 5000 Å , and is often almost constant between 5000 and 9000 Å. As shown by Hartigan et al. (1991) the spectrum of the veiling can be modelled by the emission from a region with [FORMULA] and temperatures of 6000 to 10000 K. For sources with very high veiling, the emission approaches a blackbody. An optical thin veiling continuum results in a slight increase of the veiling with wavelength, and thus causes no problem for the magnetic field determination if lines in the 5000 to 9000 Å region are used. An optically thick veiling continuum will decrease with wavelength. In fact, since the veiling is typically strongest in the blue, and weakest in the red, the EW of lines from a highly veiled cTTS would appear to grow relative to a standard star as the wavelength increases. Since the Zeeman effect grows as [FORMULA], the question is whether the presence of veiling might mimic the results of our method.

To answer this question, we artificially veiled the spectrum of our non-magnetic K3 template star using the wavelength dependence of the veiling calculated by Basri & Batalha (1990) to model the observed veiling of GG Tau. The veiling declines from v = 1 at 5000 Å to v = 0.25 at 9000 Å. Fig. 13 shows the corresponding wavelength dependence of the veiled to the unveiled EWs (Y) as well as the dependence of this ratio on the `magnetic sensitivity' S, which is [FORMULA] given in units of the magnetic splitting in mÅ /kG. Despite the more or less random distribution of [FORMULA] with wavelength, there is a trend that Y increases with S. This trend vanishes if Y is plotted against the ratio X of of EWs calculated from our model for different magnetic field strengths and of the intrinsic EWs. This ratio X gives the effective magnetic sensitivity of our lines for various field strengths. Fig. 14 shows the result of the application of our method to the artificially veiled template star. The left part gives the (non)correlation of Y with X for a field strength of 0.5 kG, the right part the distribution of the correlation coefficient that does not exceed 0.22, which means that no significant field has been detected. The highest local maximum for the mean [FORMULA] is of the order of 0.2 kG. We thus conclude that the results obtained from our method are hardly influenced by such a veiling. There are two obvious explanations for this behaviour: 1) the spectral energy distribution of the veiling is close to that of a blackbody which leads to an increase in Y with wavelength which is less than linear, whereas the magnetic splitting grows as [FORMULA] and 2) the correlation seen in [FORMULA] vanishes for [FORMULA] if the structure of each Zeeman pattern is taken into account like in the line transfer program which leads to a different magnetic sensitivity than [FORMULA] for the lines.

[FIGURE] Fig. 13. Left: Y versus wavelength. Where Y is the ratio of the EWs of artificially veiled spectral lines to the unveiled EWs (Y = [FORMULA]) Right:  Y versus the magnetic sensitivity [FORMULA].

[FIGURE] Fig. 14. Same as Fig. 5 but for the spectrum of the nonmagnetic K3 template star which has been artificially veiled. We find a [FORMULA] = 0.22 (at 0.5 kG) only, indicating that no field has been detected, and that the results of method are not altered by veiling.

6.4. The influence of the magnetic field on measurements of veiling and spectral type

As was shown in Fig. 3, the EW of photospheric spectral lines can increase by a noticeable amount under the influence of a magnetic field. As explained in Sect. 4, this effect may change the spectral type of a star if it is derived from the curve of growth of Fe i lines. Table 3 lists the temperature as computed from the curve of growth of these lines both taking the enhancement due to the magnetic field into account, and not doing so. For the T Tauri stars studied, accounting for the magnetic enhancement increases the temperature of the star by 100 to 300 K compared to what would otherwise be determined. For T Tau, the star with the largest [FORMULA] of 2.35 kG, this effect is most pronounced and would make the spectral type of T Tau appear to be later than it really is by about one to two subclasses.

The other effect of the enhancement of the EW of lines sensitive to the magnetic field is that any measurement of the veiling using such lines might be influenced. For example, using magnetic field strength of one, two and three kG, we derive that the average EW of 48 Fe i lines used in this work would increase by about 10%, 28%, and 43% for a K3V star. If a template star of the correct spectral type is chosen, the enhancement of the EW would thus result in an underestimation of the veiling. For example, failing to account for a 2 kG field would change the deduced veiling from 2.00 to 1.34. Fig. 15 shows the measured against the true veiling for various magnetic field strengths.

[FIGURE] Fig. 15. Measured versus the true veiling using Fe i lines for various magnetic field strengths.

The presence of magnetic fields on T Tauri stars thus may alter both derived effective temperature and veiling, depending on the particular lines used. Obviously, lines with larger magnetic intensification [FORMULA] are more affected than lines with smaller magnetic intensification. For determining the veiling lines with [FORMULA] should thus be preferred . Alternatively, if the magnetic field strength of the cTTS and the wTTS are the same, wTTSs could be used as templates instead of main sequence stars.

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© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998