6 Polarization and scattering light

Due to the multicomponent structure of CS shells around young stars several sources of observed polarization exist. These are: - scattering of radiation in nonspherical disk-like dust shells; - Thomson scattering in CS gaseous shells; - Thomson scattering in gas jets; - scattering on gas and dust inhomogeneities rotating around a star etc.

It is important that large IR excesses ( $>1\mbox{$\,.\!\!\!^{\rm m}$ }5$) at 3.5 $\mu$m cannot be explained by free-free and free-bound emission. This is in good agreement with the data for classical Be stars (most of them show near IR excesses $<1^{\rm m}$). Thus the main source of IR excesses in young stars is the absorption and re-emission of radiation by dust grains. Because strong correlations between polarization and IR colour excess E(V-L) exist for the majority of young stars we suppose that the main source of polarization in young stars is the same as for near IR excesses i.e. CS dust. Therefore we should not consider here the mechanisms of polarization which are connected with CS gaseous shells. In the framework of this assumption the observed polarization in young stars can be described as a vector sum of several components:

$${P}_{\rm obs}={P}_{\rm is}+{P}_{\rm CSdisk}+{P}_{\rm CSinhom}$$ (5)

where ${P}_{\rm is}$ - is the interstellar component of polarization, ${P}_{\rm CSdisk}$ - is polarization arising in the nonspherical dust envelope and ${P}_{\rm CSinhom}$ - is polarization due to scattering CS dust inhomogeneities. As a first step we assume that ${P}_{\rm CSinhom}$ does not affect strongly the value of observed polarization. Although this component of polarization is rather small, as will be shown later, the presence of these inhomogeneities in CS shells is an important factor that may result in large polarimetric variations indirectly. We also exclude from our study the interstellar component of polarization.

Grinin et al. ([1995]) in the context of the observational behaviour of UXOrs have discussed two possibilities for the intrinsic polarization in the framework of the dust model, namely optical dichroism of CS nonspherical grains, and the scattering of stellar radiation in CS dust shells. They noted that in the first case a linear dependence must be observed between the amplitude of brightness variations and polarization degree:

$$P_{\rm int}=\alpha\times\Delta{m}$$ (6)

where $\alpha$ - is the factor dependent on the degree of alignment of the dust particles in a SC cloud. In the second case, the dependence is quite different and has the form:

$$P_{\rm int}=P_{\rm int}^{0}\times10^{0.4\Delta{m}}.$$ (7)

As may be inferred from observations of UXOrs as well as from our analysis, the second case is much more acceptable as an explanation for the behaviour discussed in the present study.

Figure 18: $\log p/E(V-L)$ relations for different sources of polarization

In fact, $P_{\rm int}^{0}$ is dependent on the amount of hot dust in CS shells and the geometry of the envelopes. Note that it is impossible to explain the observed polarization in all young stars assuming that the dust envelopes have a spherical shape. Therefore we will consider the disk-like envelopes.

Let us consider two components ( $P_{\rm int}\sim P_{\rm int}^{0}$ and $P_{\rm int}\sim 10^{0.4\Delta{m}}$) in Eq. (7) separately. In the general case polarization for the disk-shaped envelope can be described by the following formula (see Dolginov et al. [1995]):

$$P_{\rm int}^{0}(\%)\approx300/16\mid\bar{b}\mid \times f(d/R) \times \tau\sin^{2}{\theta}$$ (8)

where $\tau$ - is the optical thickness of the envelope, f(d/R) - a function dependent on the ratio of envelope geometrical thickness to the radius, i the inclination angle of the disks to the line of sight and $\bar{ b}=1/q=1-\sigma_{\rm a}/\sigma_{\rm t}$ ( $\sigma_{\rm a}$ and $\sigma_{\rm t}$ are absorption and scattering cross-sections). Let us re-construct this formula in terms of E(V-L). Using the canonical equations  $A_{V}=1.086\tau_{V}$and the relation between E(B-V) and E(V-L) the optical thickness $\tau_{V}\approx0.98\times E(V-L)$. Thus

$$P_{\rm int}^{0}(\%)\approx300/16\times f(d/R) \times E(V-L)\times \sin^{2}{\theta}.$$ (9)

Numerical values for f(d/R) are tabulated in Dolginov et al. ([1995]). For favorable geometry and $\sin^{2}\theta=1$

$$P_{\rm int}^{0}(\%)\approx E(V-L)$$ (10)

or for the intermediate case

$$P_{\rm int}^{0}(\%)\approx0.2\times E(V-L).$$ (11)

The relations (10, 11) are plotted in Fig. 18. Note that these relations cannot explain our dependence adequately. Moreover these relations are valid only for the optically thin case which corresponds to $E(V-L)\leq1^{\rm m}$. Note however that in the case of the presence of optically thick dust condensations in the midplane of the envelope instead of the optically thick narrow dust disk, the average optical thickness of this envelope may reach the value of $\tau\approx1$ at values of $E(V-L)\approx3^{\rm m}$ or even larger. Incidentally, the relation (10) is in agreement with the data for classical Be stars for which the polarization is due to the scattering of free electrons in nonspherical optically thin disk-like CS gaseous shells. The relations (10, 11) are also valid for some TT stars and describe well the position of most Vega-type and post HAEBE stars. This fact is in good agreement with the above suggestion on the presence of optically thin and homogeneous disks around them.

Let us consider now the second term of Eq. (7) assuming $P_{\rm int}^{0}\leq1$ (this corresponds to the observed values for UXOrs in their bright states).

Because of the assumption on nonvariable IR fluxes we can re-write $\Delta{m}\approx\Delta(V-L)=E(V-L)$ and

$$P_{\rm int}(\%)=10^{0.4E(V-L)}$$ (12)

or

$$P_{\rm int}(\%)={\rm e}^{E(V-L)}.$$ (13)

This relation is also shown in Fig. 18. Note that as is evident from the present statistical study the polarization is proportional to near IR excess E(V-L) in the form $P\sim {\rm e}^{k_{1}E(V-L)}$ or $P\sim10^{k_{2}E(V-L)}$. However the coefficient k in our relation (3) differs from those in Eqs. (12, 13). To make the coefficients consistent we must assume that the extinction curve for dust clouds/condensations is different from that of the standard curve. Put another way the extinction in near IR must be higher: $\tau_{V}\approx0.6\times E(V-L)$ which corresponds to the ratio $A_{L}/A_{V}\approx0.2-0.4$ i.e. the extinction curve is flatter than the interstellar one (see also Mitskevich [1995]). This partly nonselective extinction can occur only due to the presence of large dust grains in clouds from which stellar radiation is obscured. Rostopchina et al. ([1997]) noted recently that the modelling of the behaviour in UXOrs requires the minimum size of the dust particles to be about ten times larger than that of the interstellar matter. They have discussed this fact in terms of the growth of particles in these kinds of objects and noted that these particles are still smaller than in the "old" protoplanetary disks of Vega-type stars. Simultaneous optical and IR photometry for some UXOrs which was carried out by Hutchinson et al. ([1994]), clearly indicates the decreasing of IR fluxes during a visual fade with the ratio that is in excellent correspondence with $A_{L}/A_{V}\approx0.3$. According to Chini et al. ([1990]) the minimum size of the dust particles in CS shells around Vega, $\alpha$ PsA and $\epsilon$ Eri is $a_{\rm min}>20~\mu$m. Similar conclusions were recently made for HD 45677 by de Winter & van den Ancker ([1997]) who suggested that photometric variations appeared in the star due to obscuration by CS condensations with large ($>1~\mu$m) dust grains. Some evidence for unusually large grains in front of some Orion stars (most of them are in our list of post HAEBE and Vega-type stars, see Appendices 1-2) has been discussed by Breger ([1977]) and Breger et al. ([1981]). Moreover, there is much evidence that the wavelength dependence of polarization in most young stars diverges considerably from that determined by the Serkowski law, namely in the sense that polarization in the red and near IR is often higher that might be expected for standard interstellar grains (see Tamura & Sato [1989]; Vrba et al. [1979]; Garrison & Anderson [1978]).

Taking into account the above discussion the expression for the degree of intrinsic polarization for the majority of young stars has the form:

$$P_{\rm int}(\%)\approx0.2~E(V-L)\times {\rm e}^{0.6E(V-L)}$$ (14)

for the region of $0^{\rm m}<E(V-L)<1^{\rm m}$ or $\leq3^{\rm m}$ and

$$P_{\rm int}(\%)\approx0.2\times {\rm e}^{0.6E(V-L)}$$ (15)

for the region of $3^{\rm m}\leq E(V-L) <8^{\rm m}$.

Note that the expression derived above corresponds well with the statistical dependence (see Fig. 18). It is also important that even for optically thin disk-like dust envelopes the polarization degree can be greater than $\sim6$% if the stellar radiation is screened (see series of papers by Daniel [1980a], [1980b], [1982] or Voshchinnikov & Karjukin [1994]).

Finally, the change in the average line inclination should be caused by multiple scattering if the optical thickness of the shell $\tau>$1 (this may occur in some young stars). This leads to a decrease in the fraction of scattering radiation and decreasing of observed polarization (Voshchinnikov & Karjukin [1994]).

In addition note that the conclusion of the existence of large dust particles in CS envelopes of young stars, which follows from the analysis of the behaviour of young stars on the diagram discussed here, is beyond the reach of classical $p - \Delta m_{\rm visual}$ diagrams. Thus, the diagram discussed here is more informative.