Although useful analyses of the line fluxes and equivalent widths require the use of detailed models, some constraints on the sizes and locations of the line-emitting regions can be obtained using very simple means. Our purpose is not to present plausible detailed models for the emission regions but to place limits on the types of possible models consistent with all of the data.
6.1. Kinematic evidence
Unfortunately, the few high resolution spectra were not taken simultaneously with the low resolution spectra, so it is not surprising that they provide few constraints on where the line-emitting regions might be. In DI Cep, we observed that the redshifted wing of the emission line profile is variable, indicating infalling matter, but no quasi-periodic variations of the star are observed. In DR Tau, we observed blue-shifted component () in H and H and quasi-periodic variations but Johns & Basri (1995) report that the central part of the profile rather than the blue-shifted component is quasi-periodic in DR Tau. Thus, it appears that we cannot extract any further information from so few high-resolution spectra.
6.2. The area of the emission line regions
Valenti, Barsi & Johns (1993) show that the Balmer jumps, the higher Balmer emission lines, and the veiling continuua in a large number of TTS can be reproduced by relatively small regions with temperatures around 9000 K. However, they were unable to fit the large line strengths of the lower Balmer lines with their simple slab model. While it is undoubtably possible that a complicated model could be constructed which fits all of the Balmer lines, the suggestion is, that the higher Balmer lines are produced in the region responsible for the veiling continuum and the lower Balmer lines - particularly H - are produced in a separate region. In order to address this question, we can derive crude but interesting lower limits to the sizes of the H emission regions. This line is particularly well-suited to this task, since it is the strongest of all the optical emission lines and should be the one with the largest optical depths.
First, it is not unreasonable to assume that H is optically thick and that the gas is in LTE. For instance, the models for the source of the veiling continuum (e.g. Guenther & Hessman 1993; Kenyon et al. 1994; Valenti, Basri & Johns 1993) predict small emitting regions and continuum optical depths of order unity (i.e. high densities). More importantly, by assuming large optical depths, we minimize the emitting area: an optically thin region would have to be much larger.
Second, noting that the maximum specific line intensity for a gas in LTE is given by the Planck function, the total line flux emitted by an optically thick slab of gas with negligible continuum optical depth is roughly equal to , where is the projected area of the emission region and is the intrinsic line width. If the line is very optically thick, this line width will, of course, be larger that the thermal or "micro-turbulent" width. The observed line width, , is larger than due to the smearing of the line by, e.g., rotation or other large-scale motions. By using the former, we will thus over-estimate the total line flux for a given area or under-estimate the area for a given flux. The same effect will occur if the continuum has a significant optical depth, since this will lower the intensity.
Finally, since the ratios of the observed emission line and stellar continuum fluxes are proportional to the ratios of the corresponding projected areas, we can use the definition of the equivalent width and the assumption that the star also radiates like a black-body to compute the ratio of the minimum line-emitting area to the projected area of the star:
where and the factor corrects the observed EW for the presence of the non-stellar continuum.
The most serious objection to this simple model is that the line flux can be significantly affected by the division of the emitting region into many line components with kinematic velocity separations greater than the intrinsic line widths - for instance, in a stellar wind. Many authors have attempted to fit the high-resolution line profiles of cTTS with stellar wind models (see Johns-Krull & Basri 1996 and references therein) with only modest success. However, if the variations are quasi-periodic (and hence presumably due to rotational modulation), the kinematic structures responsible for an increased line strength at one rotation phase will "flatten out" roughly 1/4 of a period later. In this case, our simple formula for the line-emitting area using the mean line strengths will again be quite adequate.
Applying Eqn. 2 to our data, we see that, if we take and the values from Table 2, we can derive the following effective minimum areas:
All of the realistic complications make the size of the region where H is formed and the star even larger. In the context of the rotational modulation model, we have to conclude that at least H is formed at a considerable distance from the star, and that the region emitting the veiling continuum is physically detached from the regions that are emitting the lines. The amount of observed variability in H (Table 2) suggests either a variation in the projected surface area of 30-50%, a variation in the temperature of 7-11%, or - of course - simultaneous variations in both of unknown but well-constrained magnitude. The minimum areas for the H emission regions correspond to the sizes of the accretion disk holes invoked in magnetic disk accretion scenarios (e.g. Königl 1991). Ultimately, however, we need proof that the line variations are due to rotational effects rather than random variations in a stellar wind in order to use the derived emitting areas to constrain the origin of the lines.
6.3. Evidence from the time-variability constraints
Our time-resolved moderate-resolution data place much more severe constraints on wind models than isolated high-resolution spectra, since the wind must be capable of producing the observed 10-20% variations in the line fluxes on timescales of days and - depending upon the line - occasionally up to 100% variations on timescales of minutes to hours. The wind-crossing time
is potentially short enough to explain much of the longer timescale variations but not those on the shortest time-scales (see Fig. 2). Since most of the lines are seen in emission, the more characteristic timescale for wind emission line variations is even longer. Thus, the observed variability is unlikely to be due to a variable, quasi-spherical stellar wind, and the line-emitting areas estimated in the previous section are likely to be realistic. This result corroborates Valenti, Basri & Johns' (1993) suggestion, that the hot, dense regions responsible for the veiling continuua and the higher Balmer line emission are separate from those producing most of the H emission.
If the observed modulations of the emission lines are truly quasi-periodic and not due to random "flares", we have to assume that some fraction of the line-emitting regions are almost rigidly rotating with the central star in order to explain the similar quasi-periods and phases of the lines. Rigid rotation of a large region like that responsible for the H line is only possible if large-scale magnetic field lines keep the rotation rate almost constant. Any variations in the line emission from such a region due to geometric effects would lead to changes on timescales of the stellar rotation period and should show up in the strength of the veiling continuum - just like that seen, e.g., in Fig. 4. However, the veiling continua show no quasi-periodicities: while the equivalent widths are weakly correlated with the veiling continuum strengths, the correlation is either too weak to show up as a clear rotationally modulated veiling continuum or the veiling source is not simple enough (e.g. one or two distinct star-spots) to produce a measurable modulation.
Given that the observed modulation fraction in H (Fig. 4) is both small - of order 20% in DR Tau - and significantly smaller than that of He I , H and the Ca II H lines (Table 3), it would seem quite possible that a minor emission component produced at the site of the veiling continuum is highly modulated (e.g. active hot star spots coming in and out of view) and that the extended component is unmodulated. However, the short time-scale (minutes to hours) variations in the lines are not seen in the veiling continuua.
Thus, we are confronted with an unexpected result: the time-scale for variations in the extended region appears to be short while that for the accretion region at the bottom of the local potential well is long!. There are several possible explanations for this strange behavior:
The last possibility can be tested by simultaneous high time-resolution B- and I-band photometry and high-resolution spectroscopy. However, the high-speed UBVRI monitoring of BP Tau by Gullbring et al. (1996) suggests that the relative amplitude of the B- and I-band variations on timescales of minutes to hours is the same, so this is unlikely to be the explanation in different but quite similar systems. To test the second hypothesis, one would require much more time coverage to isolate the timescales of the various spectral components and to determine which variations are due to "flaring" and which to quasi-period rotational modulation - a daunting task, indeed. Fortunately, observations of the type presented here but with much higher spectral resolution and complete (relative) photometric coverage may provide enough additional information to disentangle the spectral components in these very complex systems.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998