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Astron. Astrophys. 339, 113-122 (1998)

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2. The method

2.1. Description

The main idea behind the method is that if both components in a binary have an axisymmetric circumstellar medium and have a detectable polarization, then this polarization reliably traces the orientation of the symmetry axis of their circumstellar environments, and therefore their rotation axes. In order to get a sufficiently large intrinsic polarization, both components must be surrounded by a sufficient amount of dust and gas, and the inclination with respect to the line-of-sight must be large enough. Our method is thus likely to give good results when applied to binaries where both components are CTTS.

Models of bipolar reflection nebulae by Bastien and Ménard (1990) have shown that the position angle of the integrated linear polarization of the scattered starlight is parallel to the equatorial plane of the disk, provided the inclination is sufficiently large.

Indeed if the star is seen at low inclination, say less than [FORMULA], the integrated polarization level will decrease toward [FORMULA], a value reached if the system is seen face-on. On the other hand, for larger values of the inclination and assuming all CTTS are associated with small, unresolved, bipolar nebulae, one can estimate the projected orientation of the symmetry or rotation axis of the system by measuring the position angle of the linear polarization vector.

Our calculations assume that the star+disk system is surrounded by a large bipolar dust enveloppe scattering a fraction of the photons received by the observer, and making the polarization position angle appear parallel to the equatorial plane. Fig. 1 shows the resulting polarization level P and position angle [FORMULA] computed for a bipolar reflection nebula with a multiple scattering code (Ménard 1989). We use the density structure given by Galli & Shu (1993), for the [FORMULA] case (see their Table 5 and Fig. 1c). The reflection nebula has a radius of 3000 AU ([FORMULA] in Taurus), and an accretion disk was added to that prescription, the disk being defined as an abrupt density increase by a factor of 20, over a radius R=250 AU and a "flaring" angle of [FORMULA]. The results show that if one can reach a polarization uncertainty [FORMULA](P) of [FORMULA]% or better, the position angle of the polarization is well defined and parallel to the disk (PA=[FORMULA]) for all inclinations larger than [FORMULA]. Therefore, this method gives a reliable estimate of the disk orientation for inclination angles larger than approximately [FORMULA].

[FIGURE] Fig. 1. Polarization Model. The polarization level P (histogram) and its position angle [FORMULA] (solid line) are computed for a bipolar nebula surrounding a circumstellar disk. The position angle is not well defined for P[FORMULA], i.e., below [FORMULA] when [FORMULA] (the horizontal dashed line at the bottom of the polarization histogram corresponds to P=0.3%; it intersects the polarization histogram at an inclination [FORMULA]). The position angles below [FORMULA] are not shown.

If no envelope is present, the received photons will be scattered by the disk and this different scattering geometry results in a integrated polarization perpendicular to the disk plane instead. Whitney & Hartmann (1992) and more recently Wood et al. (1998) have modeled such situations, and their results show that in all cases, for a large enough inclination, the polarization is perpendicular to the disk plane rather than parallel to it. However, this [FORMULA] shift on the position angles of the polarization in the models does not impede our ability to estimate the orientation of T Tauri stars. Two T Tauri stars having different orientations within a binary system will have different polarization position angles, whether or not a dust envelope is dominant. In the following, we assume that all the CTTS of our sample are associated with small, unresolved, bipolar nebulae with a dusty envelope surrounding their accretion disk, and that the polarization measurement integrates the whole star+disk+envelope system.

Note that the measured polarization level cannot be used, in conjunction with Fig. 1, to evaluate the inclination angle directly. The integrated polarization levels are sensitive to the exact density distribution: the total extinction on the line-of-sight, the radial dependence of the density profile, the grain size and type, so that Fig. 1 is valid only for one given model. Note also that the exact value of the inclination angle for which [FORMULA] ie. [FORMULA], is model dependent. Nevertheless, two features appear general and useful to derive the orientation of stars in binary systems: 1) the integrated polarization level rises when the inclination increases, 2) within our hypotheses, the integrated polarization position angle remains parallel to the disk plane whenever the polarization is large enough for the angle to be well defined. It is this latter result that we will use to assert the projected orientation in the plane of the sky of the CTTS of our sample for which a significant polarization is detected.

2.2. Linear polarization imaging

The polarization measurements obtained with classical photoelectric polarimeters are limited to binary separations of a few arcsecs because of the diaphragm sizes used, typically 3-4" or more. We have tested the method proposed in Sect.  2.1 on closer binaries by using imaging linear polarimetry instead. The seeing is then the limiting factor to resolve binaries. In practice it allows a study of tighter systems than classical aperture polarimetry does.

In imaging polarimetry, one way to measure the polarization is to obtain 3 images through linear polarizers oriented differently, say at [FORMULA], [FORMULA] and [FORMULA] from the north celestial pole. Each image yields a measure of the polarized intensities [FORMULA], [FORMULA] and [FORMULA] for each component of the binary. This polarimeter is very easy to implement on any imaging camera. Instrumental polarization can be limited to a minimum by installing the polarimeter at the Cassegrain focus and by rotating the whole instrument instead of the polarizers only. The details of the computations to transform [FORMULA], [FORMULA] and [FORMULA] into P and [FORMULA] are given in Appendix A.

This formalism assumes that the only variations between the subsequent frames at [FORMULA], [FORMULA] and [FORMULA] come from polarization effects and not from sky transparency fluctuations (i.e., observations are made in perfect photometric conditions). This puts a strong constraint on the method as we will see; it limits its ability to measure low polarizations.

To avoid this limitation, one can use the fact that one or many field stars can also be detected on the images. By assuming these stars to be unpolarized, because for example they are foreground stars, one can monitor and compensate the photometric variations between each exposure. Unfortunately, T Tauri stars are found in molecular clouds and nearby unpolarized field stars are quite rare, and often too faint for accurate photometry, making this photometric monitoring method difficult most of the time.

Limits on these affirmations can be given by numerical simulations. To estimate the signal-to-noise ratio needed on the field star in each individual image to reach accurate polarimetry, we built artificial images similar to the ones observed (i.e., same separation between stars, same FWHM, same readout noise...). We considered a bright primary, a fainter secondary, and a much fainter reference star. The primary and the secondary are polarized, the field star is not. The results are presented in Table 1. Column 1 gives the signal-to-noise ratio at which the field star used as a photometric reference is detected. In all cases, the primary T Tauri star has SNR=1900, and the secondary has SNR=1250. Columns 3, 4, and 5 give the polarization characteristics of the primary, the secondary and the reference star respectively. The top line gives the polarization level P, the bottom line gives the position angle [FORMULA]. The polarization level on the primary and the secondary are set identical while the angles differ by [FORMULA]. Columns 6 and 7 give the calculated polarization, with the errors in parenthesis, of the primary and secondary assuming the faint reference field star is unpolarized. The formalism of Appendix A is used.


Table 1. Numerical simulation results; see text for details.

The results presented in Table 1 show that in order to get an absolute precision better than [FORMULA] on the linear polarization level, photometric measurements with S/N [FORMULA] are needed on each individual image of the reference star if it is truly unpolarized. This would be valid for example for a CTTS binary located inside a molecular cloud and a reference star located well in front of it and not suffering from interstellar polarization.

Reference stars effectively unpolarized are hard to find in practice, as the stars detected in the same frames as the target will most likely be affected by interstellar polarization. Further simulations were thus performed to estimate the effect of a reference star with a non-zero polarization. This would be expected for example for a reference star located also within the molecular cloud and suffering from the same interstellar polarization as the binary. In that case, if we assume the reference star to be unpolarized while its actual polarization is similar to that of the binary, the polarization of the primary and secondary cannot be recovered. This is true even if the S/N is larger than 580. The use of a reference star with a known and constant polarization is also possible. However, in practice we do not expect to find many of them in the same field than our sources, and this requires more than excellent photometric conditions.

To summarize, a S/N = 580 or more is necessary to measure the polarization of a star with a [FORMULA] accuracy of 0.3% on P, the polarization level. This is valid only if the reference star used to monitor the sky transparency is unpolarized, or if the sky is perfectly photometric if no on-frame reference star is used. The impact of non-photometric conditions will be discussed in Sect.  4.3.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998