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Astron. Astrophys. 327, 145-154 (1997)

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4. Discussion

A comparison of RR Tau with other stars of this subclass (see Grinin et al. 1991 and references therein) shows that its photopolarimetric behaviour is similar to that of the other UXORs and agrees well with the variable circumstellar extinction model suggested by Grinin (1986), in which the scattered radiation of the CS dust changes the colour indices and Stokes parameters of the total radiation of the central star and its CS disk, at the times of occurrence of the Algol-type minima.

In the framework of this model the brightest state of the star corresponds to its intrinsic luminosity. On the other hand we have observed in RR Tau two unusual events resembling flares. Such events seems to be incompatible with the considered model, which is based on variable CS extinction. Let us consider briefly what their origin possibly can be.

4.1. "Flares" of RR Tau

An overview of the published photometric observations of HAeBe stars shows that "flares" similar to those observed in RR Tau near J.D. 9224 and 9245 (Fig. 2) are not common. Other flares have been observed in the HAeBe stars V351 Ori by Koval'chuk (1984), in HR 5999 by Pérez et al. (1992) and in BF Ori by Koval'chuk & Pugach (1991). Moreover, a short "outburst" of the HAeBe star V1686 Cyg is shown in Fig. 2.13 of Schevchenko's (1989) book.

According to Fig. 3 during both "flares" of RR Tau the colour indices [FORMULA], [FORMULA] and [FORMULA] changed along lines which continue the upper parts of the colour-magnitude diagrams. The same is true also for the flare event of BF Ori observed by Koval'chuk & Pugach (1991). This means that the flare-like events can be interpreted as the result of the decrease of dust extinction in the line-of-sight due to formation of short-living (few days) "holes" free (or almost free) of dust in the neighbourhood of the star.

Formation of such holes was probably accompanied by the local heating of CS matter and the formation of additional gas emission. This suggestion follows from the behaviour of the colour index [FORMULA]. According to Fig. 3 both flares of RR Tau are very blue: [FORMULA] - 0.m40 which is typical for the case when the hydrogen bound-free emission beyond the Balmer jump begins to dominate. If so, the radiation of RR Tau during the "flares" is not purely photospheric one but includes also some additional hydrogen emission.

A more detailed discussion of the "flares" in RR Tau and other UXORs is beyond the scope of the present paper and will be made somewhere else.

4.2. The energy distribution of RR Tau

Using the colour index [FORMULA] = 0.m30 at the brightest state of RR Tau and the "normal" IS extinction law (R = 3.1) we estimated: [FORMULA] = 1.m2 if the spectral type of the star is B8, and [FORMULA] = 0.m60 for the spectral type A3. Two energy distributions based on the above estimated values of [FORMULA] are shown in Fig. 8. In both cases we have used the brightest UBVRI points from Table 1, the IR data for 12, 25 and 60 [FORMULA] m from the IRAS catalog (Gezari et al. 1987), and the JHKLMN magnitudes from Hillenbrand et al.'s (1992) paper. The energy distributions in the UV part of the spectrum were taken from Kurucz's (1992) models for [FORMULA] = 13500 K (B8), and [FORMULA] = 9400 K (A3) and [FORMULA] = 3. On the basis of these data we have estimated the ratios of the IR excess to the bolometric luminosity of RR Tau [FORMULA] to be 0.17 and 0.40, correspondingly. The last value seems to be more realistic since when estimating the spectral type of RR Tau (A3-A5) Strom et al. (1972) used the profiles of the photosperic Balmer lines [FORMULA] and [FORMULA].

[FIGURE] Fig. 8. The energy distributions of RR Tau. The solid line corresponds to the spectral type A3, dashed line to B8. The filled triangles up are the UBVRI points, the filled circles are the JHKLMN magnitudes, triangles down are the IR data from IRAS catalog. See text for more details.

4.3. The scattered radiation and the screening effect

According to the considered model (see Subsection 4.1), the scattered radiation of the protoplanetary disk, which is responsible for the change in the observed colours and linear polarization of a star during Algol-type minima, restricts the variation amplitude as well: they cannot exceed the value

[EQUATION]

where [FORMULA] and [FORMULA] are the intensities of the direct and scattered radiation at the given wavelength, respectively.

In the case of the photometrically most active UXORs the amplitudes of the deepest minima do not usually exceed 2 - 3 stellar magnitudes in the V band (Herbst 1986), which corresponds to the intensity of the scattered light in units of the stellar bolometric luminosity: [FORMULA] (Grinin 1986). In the case of RR Tau the observed amplitudes of the deepest minima are about [FORMULA] in the V band (see Fig. 1), which corresponds to the ratio [FORMULA] 0.03. This simple estimate shows that the scattered radiation of the CS dust envelope of RR Tau is fainter compared with that of other UXORs.

In order to separate the intrinsic linear polarization of RR Tau, caused by scattered radiation, from that due to the IS matter, we assume here (following our previous papers) that the Stokes parameters of the scattered radiation had not changed significantly during the considered time interval of about 7 years. The observational evidence supporting this assumption is discussed in the previous section.

In this case the linear polarization at any given wavelength must change with an amplitude of the brightness variation at the same wavelength ([FORMULA]) as:

[EQUATION]

where

[EQUATION]

Here [FORMULA] is the interstellar polarization of RR Tau, and [FORMULA] represents the intrinsic polarization at the maximum brightness state of the star, whereas [FORMULA] is the amplitude of the brightness variation from this brightest state.

Using (2) and (3) one can answer the questions:

  1. is the intrinsic polarization indeed caused by scattering in the non-spherical CS dust envelope; and
  2. are the brightness variations of the star caused by the screening effect?

Rewriting Eqs. (2) and (3) for Stokes parameters [FORMULA] and putting in the left side of Eq. (2) the Stokes parameters of RR Tau observed at the different brightness states we can transform these equations into a system of linear algebraic equations. Each equation of this system corresponds to one observation in one pass-band. The unknown values are the Stokes parameters of the intrinsic linear polarization at the brightest state and the IS component. The system of equations so obtained, were then solved for each pass-band applying the least squares method taking into account the weight of each observation. The calculated parameters of the IS and the intrinsic linear polarization of RR Tau are given in Table 3 and depicted in Figs. 5 and 9.


[TABLE]

Table 3. The intrinsic and interstellar components of the RR Tau's linear polarization.


We see from Table 3 that: i) the accuracy of both calculated components of the linear polarization is quite high (except perhaps at the U pass-band in which the flare-like events were only partially caused by variations of the CS extinction), ii) the position angles of [FORMULA] coincide within a 3 [FORMULA] interval in all five pass-bands; the same is also true for the IS component. Since the above described method has been made independently for each pass-band such a coincidence of [FORMULA] indicates that the separation of the IS and the intrinsic components from the observed polarization of RR Tau was correct. The wavelength dependence of the IS polarization confirms this suggestion (Fig. 9): it practically coincides with Serkowski's law, with [FORMULA] and [FORMULA] taken from Table 3.

[FIGURE] Fig. 9. The wavelength dependence of RR Tau's intrinsic linear polarization at the bright state [FORMULA] (0) (lower curve) and the IS polarization [FORMULA] (upper curve), calculated using Eq. (1); the open circles and the connecting solid line give the wavelength dependence of the IS polarization according to Serkowski's law.

According to Fig. 5 the calculated degrees of polarizations using Eqs. (2), (3) and the theoretical dependence of the total polarization of the star on the stellar magnitude agrees well in all five pass-bands with the observed ones, within the observed variability ranges of RR Tau. This confirms our assumption that the screening effect is the main source of the large-scale irregular variability of RR Tau. At the same time the dispersion of the observational points from the theoretical lines in Fig. 5 can be quite large and cannot be explained by observational errors. A possible mechanism for these deviations (which are also observed in other UXORs) was discussed in a review paper by Grinin (1994). It can be caused by dust formation due to the disintegration of star-grazing planetesimal bodies in the vicinity of a star (see also Grinin et al. 1996).

4.4. Circumstellar dust modeling

Simultaneous multi-band photometric and polarimetric observations of UXORs at different brightness states of the stars including deep minima, are a good basis for modeling the dust envelopes surrounding these stars. Their preferential orientation (edge-on) relative to the line-of-sight simplified the numerical simulations substantially, since it permits to avoid the uncertainty in the linear polarization degree proportional to [FORMULA] connected with the orientation of the envelope relative to the plane of the sky, which is usually unknown.

The model considered below consists of two elements:

a) The dust cloud intersecting the line-of-sight. Its optical properties determine the initial reddening of the star and depend on the parameters of the dust mixture. By increasing the optical thickness of the cloud we can simulate the Algol-type minima.

b) The optical parameters of the dust envelope which scatter the radiation of the central star are assumed to be constant. They determined the "turnaround" observed in the colour-magnitude diagrams.

We have used for the calculation the Mie theory and the Monte Carlo numerical code described by Voshchinnikov and Karjukin (1994). We have changed in this code the envelope geometry. Instead of the spheroidal form we have used the simplified model of the CS envelope presented schematically in Fig. 10. Such a model agrees better with the classical model of accretion disks (Shakura and Sunyaev 1973). We have assumed for simplicity that the CS dust envelope is homogeneous and that the dust mixtures are the same both in the envelope and in the dust cloud which intersects the line-of-sight.

[FIGURE] Fig. 10. Schematic picture of the Algol-type minimum of an UXOR according to Grinin's (1986) model. The CS dust cloud which intersects the line-of-sight acts as a natural coronograph. In the deep minimum, when the direct stellar radiation is blocked, the faint scattered radiation of the proto-planetary disk dominates.

The model parameters are:

a) The dust parameters. As in the paper by Voshchinnikov et al. (1988) we used here a dust mixture of graphite and silicate particles having the same the size distributions:

[EQUATION]

where [FORMULA] and the ratio of silicate to graphite particles Si/C are the model parameters.

b) The envelope parameters. These are the opening angle [FORMULA] (see Fig. 10) and the optical thickness in the equatorial plane in the U band: [FORMULA].

A few tens of models of the CS dust envelopes for different combinations of the model parameters were calculated and were compared with the observational data. The best fit to the observed colour-magnitude diagrams and the wavelength dependence of the intrinsic linear polarization of RR Tau, [FORMULA], is found for two models with slightly different parameters:

  • Model 1: [FORMULA] m, [FORMULA]
  • Model 2: [FORMULA] m, [FORMULA]

The other parameters in both models are the same: [FORMULA] m, [FORMULA], and Si/C = 1.07 (such as in the case of the IS medium (see Mathis et al. 1977)).

The results of the numerical simulation of the Algol-type minima based on these models are depicted in Figs. 3 and 11. The first model describes the wavelength dependence of the intrinsic linear polarization of RR Tau (Fig. 11) better. The second one explains better the initial reddening in the observed colour-magnitude diagrams of the star (Fig. 3) excluding the "flares" in the U band (see above).

[FIGURE] Fig. 11. The best theoretical fit of RR Tau's intrinsic linear polarization in the brightest state. Model 1 (solid) and model 2 (dashed) are the same as in the colour-magnitude diagrams of RR Tau in Fig. 3.

4.5. Comparison with other objects

In both cases the CS dust mixture is similar to that of the IS matter (Mathis et al. 1977) but differs from it in the minimum size of the particles: in the mixture of the IS dust applied by Mathis et al. (1977) the value of [FORMULA] m, whereas in the above presented models these parameters are about ten times larger. Similar conclusions was reached earlier in the numerical simulations of the Algol-type minima observed in two other UXORs: UX Ori itself and WW Vul (Voshchinnikov et al. 1988, 1995). It is of interest to mention here, that the same larger minimum size of dust particles in the circumstellar disk of pre-main sequence stars was also found in the study of the extinction law in the direction of such stars by Steenman and Thé (1989, 1991).

The above result can be considered as evidence that the growth of particles has already begun in the young protoplanetary disk of RR Tau, but that the particles are still smaller then in the "old" protoplanetary disks of Vega-like stars (Chini et al. 1990) including [FORMULA] Pictoris (Paresce and Burrows, 1987; Telesko and Knacke, 1991).

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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