Astron. Astrophys. 363, 984-990 (2000)

3. Discussion and preliminary comparison with observations

3.1. Validity of model assumptions

The observation probabilities derived above depend on the properties of the investigated disk model. The main assumptions made in the computation are the vertical isothermality and the homogeneous mixing of gas and grains. As mentioned above, the assumption of isothermality in the vertical direction is incorrect in the innermost part of the disk for the highest values of the mass accretion rates envisioned here. Beyond a disk radius of a few AUs, however, heating due to dissipation of viscous energy becomes smaller than heating due to reprocessing of stellar and accretion luminosity and the vertical structure becomes isothermal in the outer disk regions where flaring is important. We therefore do not expect the isothermal hypothesis to strongly affect our results, which depend mainly on the outer disk structure. Similarly the assumption made in deriving Eq. (3) above is valid everywhere except in the innermost disk region, and is therefore uncritical here. There is some concern, however, that homogeneous mixing of gas and grains may lead to an overestimate of YSO extinction. Indeed, D'Alessio et al. (1999) find that their detailed vertical structure models assuming well-mixed gas and dust grains lead to an overestimate of extinction due to disk flaring. It is difficult to compare the simple model investigated here with the much more elaborate D'Alessio et al. (1999) model, which considers additional heating mechanisms and treats the radiative transfer in considerable detail. We note, however, that the extent of flaring we derive here is comparable or less than observed in disks imaged by HST and CFHT adaptive optics (see next section), so that the given probabilites to observe occultations are likely to be lower limits as well. Finally, we should mention that the disk colors will depend in good part on the scattering of stellar photons in the disk atmosphere, a process which is not currently included in the model but will be considered in subsequent work. Obviously, scattering should also work as to increase the probabilities of observing variability caused by non-axisymmetric disk structure in the photometric light curves.

3.2. Comparison with observed disks

The current status of YSO disk observations has recently been reviewed by Ménard & Bertout (1999). From their compilation, we reproduce in Table 1 the properties of two disks surrounding apparently single stars, HH30 and HK Tau/c, which we compare with typical properties of two models computed as described above. The three indices and q are defined by and , while other table entries are self-explanatory. Stellar parameters are , and K7 spectral type. The accretion disk model used here appears to approximately account for the observed p values; and the flaring resulting from vertical hydrostatic equilibrium is consistent with the value observed in HK Tau/c, although it is much lower than observed in HH30, where an additional physical mechanism must contribute to the local pressure. The probability of observing HH30 through its disk is therefore a lower limit. Index q is approximately equal to 0.54, which means that the disk IR spectrum is almost flat from 5 to 100 µm.

Table 1. Properties of observed and model disks

The probability of observing HH30 edge-on or through the disk atmosphere is , a very high value indeed. It is 0.5 for HK Tau/c, a difference that can be attributed to the lower mass accretion rate. The opening half-angle of the optically thick disk is 31^o in Model 1, and 23^o in Model 2. From the observations, we know that both disks are seen nearly edge-on, so that their central stars might well display Type III photometric variability. Note that we used a viscosity parameter in the above comparison, as smaller values would result in disk masses larger than a solar mass or so for the high accretion rates and extended disk radii envisioned here.

3.3. Disk flaring and Type III photometric variability

Whether occultation of the star by disk material will occur when the disk is seen at a grazing angle depends on the way matter is distributed within the disk, both in the equatorial plane and in the vertical direction. As discussed above, the model assumes that dust and gas are well-mixed throughout the disk, but it is likely that real disks are inhomogeneous (e.g., there are some indications for inhomogeneities in the HST HH30 images) and that some optically thick clumps will at times be located in the otherwise optically thin atmospheric layers.

While an investigation of the physical properties of clumpy disks requires numerical techniques well beyond the scope of this work, it is obvious that once a clump finds itself in the optically thin disk atmosphere, its frequency of oscillation about the disk plane in the local gravitational field is equal to its period of revolution around the star. If one assumes that there are several such clumps at different radii in the inner disk, apparently aperiodic partial or full occultation may occur depending on both the clump size and its altitude at time of observation. Also, successive occultations will display a cyclic behavior mirroring the location of the main contributing clumps in the disk. These are very much the characteristics of UX Ori variability.

Alternatively, disk warps might easily produce partial or total periodic occultation of the central star when looking toward the disk at a grazing angle. In support of this possibility, Terquem & Papaloizou (2000) recently showed that the torque exerted on the disk by the stellar magnetic field, assumed to be an inclined dipole, leads to formation of a warp in the innermost parts of the disk that can explain the photometric observations of the CTTS AA Tau.

As a third possibility, one might also envision massive disks in which instabilities lead to the formation of fragmented spiral arms, perhaps with similar observational consequences (Pickett et al. 2000). In order to find clues allowing us to distinguish between these various possibilities for producing occultations, we will study in a forthcoming paper the mid-term variability of a reasonably large sample of YSOs.

As a first test of the ideas discussed here, we performed a preliminary study of available V photometric data for a number of YSOs, using Bill Herbst's photometric database (Herbst et al. 1994; Herbst & Schevchenko 1999) and looking for cyclic variability on time-scales of a few weeks to a few years. Because YSOs are known to display shorter-term variability, uncovering mid-term cycles is not an easy task, and we defer the full description of the time-series analysis to Paper II. Here, we merely describe preliminary results based on a careful, CLEANed periodogram analysis (Roberts et al. 1987) of the V data corroborated by an extensive search of preferred cycles using both minimization techniques and robust wave variogram analysis (Eyer & Genton 1999).

We find that cyclic photometric variability occurs in a small but sizable fraction of young stars. Out of 36 (20) investigated CTTSs (HAEBESs), we find 6 (4) stars displaying cyclic V variations that deserve further study. This is apparently in rough agreement with the observation probabilities given above but there are severe biases in the data which need to be investigated further before a definite conclusion is drawn. Also, a more detailed study of photometric colors is needed to confirm that all suspected cyclic variations can be explained by occultations of stellar radiation by circumstellar matter.

Three examples of cyclic occultations are shown in Fig. 4, which displays the phase-folded V data for the K7 CTTS AA Tau (top panel), the HAEBES UX Ori (central panel), and the G2 CTTS SU Aur (bottom panel). The cycle length for AA Tau over the 11 years observation time span is 8.19d, in good agreement with the value found by Bouvier et al. (1999). Note that the irregular luminosity drops, which are resolved in the Bouvier et al. (1999) data, are unresolved in the present observations, and introduce much scatter in the light-curve. This is because these short time-scale occultation events, which are more likely to occur when the star is faint, are not regularly distributed in the phase diagram. Thus, the picture which emerges from the phase-folded data is a low-amplitude sine-like light variation between V=12.5 and 13.3 mag, with individual short-term occulting events superimposed. As mentioned above, a disk warp induced by the inclined stellar magnetic field accounts successfully for the observations (Terquem & Papaloizou 2000).

Fig. 4. Phase-folded V lightcurves for three Type III YSOs. Top Panel: AA Tau data folded with period 8.19d. Central Panel: UX Ori data folded with period 319.1d. Bottom Panel: SU Aur data folded with period 443.5d.

The preferred cycle for UX Ori has period 319.1d. Here, the data span 16 years and the light-curve resembles that of AA Tau but the deep minimum appears better defined in phase, suggesting a larger projected size of the occulting screen, located at 0.8AU from the star. The amplitude of the cyclic variability reaches 2 mag in V, compared to about 1 mag in AA Tau and SU Aur. The nature of the UX Ori occulting screen, which apparently is present on the sight line during about 40% of the phase, is quite puzzling.

Finally SU Aur displays a cycle length of 443.5d on V data spanning 16 years. The very low scatter of the sine-like low-amplitude light variation is truly remarkable (there are 1088 data points in this figure). The matter responsible for occultation is located at 1.1 AU from the star. Besides the sharp drops in the light curve, which are characteristic of individual occultation events, there is again a low amplitude sine-like variability with the same period, and the sharp drops occur most often when the star is fainter, i.e., when the obscuring "wall" is highest on the observer's horizon. This is reminiscent of what one would expect from occultation by a disk warp (cf. AA Tau), with the sharp drops due to clumpy material located in the disk atmosphere flying across the stellar disk. The physical conditions needed to form and maintain a warp at 1AU are unclear at present, although a low-mass companion orbiting the primary star in a trajectory non-coplanar with the disk might be able to produce warps at such distances from the central star (C. Terquem, private communication).

As a conclusion, we can emphasize the following predictions of the model investigated here, which can all be tested by statistical analysis of YSO properties and detailed models of UX Ori-type objects.

• Direct observation of the circumstellar disk structure using photometric and spectroscopic techniques appears possible for 1/10th to 1/5th of all YSOs, and strong constraints on the nature of physical processes at work in protoplanetary disks are likely to follow from such studies.

• YSOs surrounded by disks with high mass-accretion rates are more likely to be observed edge-on than low mass-accretion rates objects.

• The larger frequency of UX Ori variability in early T Tauri stars and HAEBESs (compared to late-type, low-mass CTTSs) is caused by their larger mass-accretion rates.

• We expect to find Type III variability in perhaps 1/10th of late-type CTTSs, unless their mass accretion rates are much smaller than /yr. In other words, Type III vari-ability might occur in a sizable fraction of late-type stars that also display the Type II variability typical of CTTSs. This is apparently more than currently observed (see Sect. 1), but the time-scales of both processes can be so different that a new examination of available data is in order.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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