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

2. The probability of observing YSO occultation

2.1. Disk model

The basic accretion disk model used in this investigation was described and used extensively earlier (e.g., Bertout et al. 1988; Basri & Bertout 1989). The main modification introduced for the present study concerns the disk's vertical structure. Local hydrostatic equilibrium is assumed at each disk radius, and reprocessing of stellar and accretion radiation is taken into account when computing the local disk temperature and the disk "flaring". Instead of calculating the disk vertical structure in full detail (e.g., Bell et al. 1997) we follow here the more tractable approach advocated by Hartmann & Kenyon (1987) and recently by Chiang & Goldreich (1997) (hereafter CG97); that is, we assume that the disk is isothermal at each radius, with the disk temperature resulting from balancing the locally dissipated and emitted energy fluxes. While this is a good approximation for optically thin and passive disks, it can be questioned for the optically thick disks envisioned here, since the disk equatorial temperature is expected to be given in the grey approximation by

where is the frequency-integrated optical depth, the energy flux due to accretion and the flux of reprocessed radiation (Malbet & Bertout 1992; Dubus et al. 1999). However, one finds that departures from isothermality are much more important in the inner disk, where most of the accretion energy is deposited, than in the outer parts of the disk. Since we are primarily interested here in the disk flaring produced at a large distance from the star, the assumption of isothermality thus does not appear critical. We nevertheless monitored the ratio of height scales given by and throughout the computation as a check on this hypothesis' consistency.

The treatment of flaring basically follows CG97, but with two differences. First, the computation of the angle of incident stellar light on the disk allows for the fact that physical quantities describing an accretion disk vary as a function of radius ². Second, we explicitly take into account the effect of accretion luminosity on the flaring, assuming that half of the accretion luminosity is radiated away in an optically thick, axially symmetric fraction f of the stellar surface (here 1%). The `equivalent effective temperature' which replaces the stellar effective temperature in the expression giving the flaring angle is then

where , and and are the stellar mass and radius, respectively.

The vertical frequency-integrated optical depth ³ at height x is given by

where is the disk surface density, is the Rosseland opacity and x is the height in units of , h being the local scale height. This expression is used to find the photospheric height which enters the equation giving the angle of incident stellar radiation (Eq. (5) in CG97). In writing Eq. (3), we assumed that is a function of temperature only, which is true as long as dust dominates the opacity, i.e., for K. The temperature, opacity, mean molecular weight, and angle of incident stellar radiation are found iteratively, using a Newton-Raphson scheme. In the horizontal plane, the disk is assumed to extend up to the radius where the temperature goes down to 10K. Fig. 1 shows the resulting disk profile for a mass accretion rate of 3 yr^-1 (valid for a typical CTTS) and for a viscosity parameter . This value of is currently favored in theoretical work on accretion disk MHD turbulence (cf. Balbus & Hawley 1998). Stellar parameters entering the computation represent a typical CTTS: , , and K7 spectral type. The dark area shows the optically thick disk (in the vertical direction) while the thin dotted lines delineate levels of constant vertical optical depths in the disk atmosphere, corresponding to and .

Fig. 1. CTTS disk profile. The dark-shaded (dark red) area shows the vertical disk photosphere () while the thin dotted lines show contours of equal vertical optical depth with values ranging from 10-1 down to 10 The medium-shaded (yellow) area indicated the range of view angles over which the central star is totally obscured by the disk. The light-shaded (light blue) area indicates the range of view angles where partial occultation of the star can be expected. The thick dotted (red) line, corresponding to , shows how deep into the disk an outside observer sees at each view angle s.

Once the disk structure is computed, we set up a grid of sight lines and compute the total optical depth through the disk as a function of view angle s. It is then straightforward to estimate the probability

to observe the central star through the optically thick disk, in which case the YSO will be completely obscured, and the probability

to observe the central star through the disk outer layers.

In these definitions, and are respectively the minimum view angle at which the disk is still optically thick and the maximum view angle for seeing the star free from disk obscuration (see Fig. 1). If the disk atmosphere is inhomogeneous, a likely possibility, we then expect variable occultation of the star by disk matter when the system is viewed between and . The recurrence time-scale of occultations will depend on both the size of obscuring clumps and their location in the disk.

For the sake of illustration, we define somewhat arbitrarily by the range of view angles over which the disk has frequency-integrated optical depths ranging from 0.1 to 10 (light-shaded area in Fig. 1). We should mention here that the exact values of the integration limits are not critical in Eqs. (4) and (5) above for and . For the example shown in Fig. 1, the view angle for is while it is for , and the difference in for these two values of the integration limit is only 0.03. Similarly, would be larger by 0.05 if the lower limit of integration were 0.01 instead of 0.1.

Note that could be determined empirically from the observed number of stars displaying occultations in a statistically significant sample of young stars.

2.2. Results

Fig. 2 displays three different observation probabilities as a function of mass accretion rate for the stellar parameters given above. They are , which represents the fraction of CTTSs that are missed in optical surveys since they are extincted by more than magnitudes, which is the probability of observing occultation events with the assumptions discussed above, and , the probability to view a disk either edge-on or through its atmosphere. In order to bracket the plausible range of observation probabilities, we consider two cases: (a) reprocessing due to stellar luminosity only, and (b) reprocessing due to both stellar and accretion luminosity, in the approximation explained above. These two values bracket the plausible range of probabilities, shown as shaded (colored) areas in Fig. 2. We consider two values of the (uncertain) viscosity parameter , which controls the disk density. The left panel of 2 is for , while the right papel is for . Typical values of and are 0.1-0.15 and 0.3-0.35 for moderately active CTTSs with /yr and , while the fraction of (optically) unseen stars is At these accretion rates, the accretion luminosity is smaller than the stellar luminosity and does not play an important role in the disk flaring properties. At the highest accretion rates envisioned here, the accretion luminosity is larger than the stellar luminosity and the disk mass becomes comparable to the mass of the central object.

Fig. 2. Probabilities , , and as a function of mass accretion rate for computation parameters appropriate for a typical CTTS, shown for two values of . The shaded (colored) areas indicate the range of probabilities obtained when reprocessing only the stellar luminosity (lower values) and when both the stellar and the accretion luminosities are taken into account in the flaring computation (upper values).

Fig. 2 shows that all probabilities increase with the accretion rate, or more precisely with the ratio , which determines the disk surface density. This is because the outer radius of the optically thick disk (and consequently the extent of flaring, which goes approximately as ) increases with disk surface density. Additionally, the flaring increases with increasing reprocessed accretion luminosity. When decreases, the outer disk regions are less dense and flaring increases accordingly.

We now turn to stellar parameters representative of the UX Ori class:  ,  ,  K (Natta et al. 1999). Fig. 3 displays the results; typical values of and are 0.2 and 0.45 for /yr and , while the fraction of optically unseen stars is close to . Note that for a given mass accretion rate, the occultation probability is somewhat lower for UX Ori than for a CTTS. This is because the gravitational pull exerted by the star on the disk is stronger (stellar radius does not increase as fast as mass), resulting in a smaller extent of the flare. In the framework of our simple model, the frequency of Type III variability is thus controlled primarily by the mass accretion rate. The different frequencies of Type III variability observed in HAEBESs and CTTSs (Herbst et al. 1994) then implies that mass accretion rates in UX Ori stars must be larger by one to two orders of magnitude than in CTTSs, in rough agreement with current estimates.

Fig. 3. Same as Fig. 2, but for computation parameters appropriate for UX Ori.

Summarizing, we find that the probability of observing occultation events in YSOs is typically 0.10-0.15 in late-type, low-mass CTTSs and 0.15-0.25 in higher mass UX Ori objects. These results suggest that direct observational study of the disk structure may be possible for a sizable fraction of YSOs and that stellar occultations by flaring circumstellar disks might be the only physical process needed to account for CTTS Type III and UX Ori variability. This last conclusion, however, needs to be confirmed by detailed studies of UX Ori objects in the light of the occultation model, as well as by statistical studies of the occultation occurrences.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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