ORBITAL ELEMENTS OF THE SYMBIOTIC STAR Z ANDROMEDAE FROM OPTICAL LINEAR POLARIZATION DURING THE QUIESCENT PHASE

We observed Z And between 1998 December and 2000 August, during which this object was quiescent. The observations were carried out using a low-resolution spectropolarimeter, HBS (Kawabata et al. 1999) at the Dodaira Observatory (DO) and the Okayama Astrophysical Observatory (OAO) of the National Astronomical Observatory of Japan. At the DO, HBS was attached to the Cassegrain focus of the 0.91 m telescope (f/18; 124 mm⁻¹ at the focal plane). At the OAO, HBS was attached to the Cassegrain focus of the 1.88 m telescope (f/18; 61 mm⁻¹ at the focal plane). HBS has a superachromatic half-wave plate and a quartz Wollaston prism; the orthogonally polarized spectra are simultaneously recorded on either a front-illuminated TI CCD (1024 × 1024 pixels, 12 μm square pixel⁻¹, used before 1999 August) or a back-illuminated SITe CCD (512 × 512 pixels, 24 μm square pixel⁻¹, after 1999 August). In 1999 August, with the introduction of an SITe CCD, all of the lenses in HBS were replaced with specially designed UV-achromatic ones. The log of the observations is given in Table 1.

We used the 1.4 mm diameter circular hole (D1) as the focal diaphragm. This diaphragm has two holes of the same dimension; we put a target star in one hole and the nearby sky in the other. The spectral resolution depends on the seeing conditions on each night (typically 10 nm) because the stellar images were sufficiently smaller than the aperture of this diaphragm as shown in Table 1. All of observations were carried out in three shared-use runs: from 1998 December 21 to 1999 January 31, from 1999 November 6 to 1999 December 21, and from 2000 August 25 to 2000 September 3. The calibration data such as the instrumental polarization, the instrumental depolarization, and the zero point of the P.A. are produced using all of standard stars observed in each run.

The instrumental polarization was derived from unpolarized standard stars (UPs) observed almost every night in each run. The level of instrumental polarization was well expressed by a smooth function of wavelength and vectorially removed from the observed Stokes Q and U spectra. In any run, the instrumental polarization was less than 0.7% over the 380– 900 nm range, and its stability (1σ) was within 0.05%. The factor of the instrumental depolarization, the ratio of the decrease of polarization of the incident radiation through the instrument, was measured by observing UPs through a Glan–Taylor prism (GTs) that practically produces P = 100% linear polarization. This observation also gave us the wavelength-dependent P.A. of the equivalent optical axis of the superachromatic half-wave plate. The zero point of the P.A. on the sky (north on the celestial plane) was determined by observing strongly polarized standard stars (SPs). The observations of SPs and GTs were carried out roughly once every three nights in each run. The UPs and SPs we used are selected from Serkowski (1974a, 1974b), Schmidt et al. (1992), and Wolff et al. (1996); these UPs and SPs are listed in Tables 2 and 3, respectively. The number of standard stars (UPs, SPs, and GTs) observed in each run is also listed in Table 1.

A unit of the observing sequence consisted of successive integrations at four P.A.s, 0°, 225, 45°, and 675, of the half-wave plate. The obtained images were processed using the reduction package for HBS data, which was outlined in Kawabata et al. (1999). The instrumental polarization, the instrumental depolarization, and the zero point of the P.A. were properly corrected for in the reduction package. Since the uncertainties of the reduced Stokes q and u are the unbiased standard deviation of the average for all observed frames, these uncertainties include all random errors. The systematic error of the calibration data is basically constant in each run. Since these calibration data are obtained by averaging multiple measurements, the systematic error is negligibly small compared to the random error of single measurement. We achieved an overall calibration accuracy of ΔP < 0.1% (see Kawabata et al. 1999, for details of the performance of HBS).

Several solutions have been presented for the orbital ephemeris of Z And; photometrically determined solutions (Formiggini & Leibowitz 1994; Skopal 1998) and spectroscopically determined ones (Mikolajewska & Kenyon 1996; Fekel et al. 2000). Except for the ephemeris given by Skopal (1998), these are consistent with one another to within 0.011 in phase during 1998–2000; the difference is less than the uncertainty in each ephemeris. Owing to the small difference, the choice of the ephemeris among these three ephemerides does not affect the results; we used the most recent ephemeris by Fekel et al. (2000),

$$\begin{equation} T_\mathrm{conj} = \mathrm{JD} \ 2,450,260.2 (\pm 5.4) + 759.0(\pm 1.9)\phi, \end{equation} \tag{ 2 }$$

for conjunction with the cool companion in front, where ϕ denotes the orbital phase.

3.1.1. Variation of Polarization

As seen in Figure 1, the observed continuum polarization seems to show a temporal variation. To investigate this variation in more detail, we calculated wideband averages of Stokes q and u using Equation (9) in Kawabata et al. (1999). The binning uses boxcar functions for five wavelength ranges of 400–500, 500–600, 600–640, 720–800, and 800–850 nm, excluding Hα and Raman emission lines and covering a wide wavelength region; the binning improved photon noise and the random one while retaining wavelength dependence information. The derived Stokes q, u for each band are listed in Table 4 and plotted as a function of the orbital phase ϕ in Figure 2. This figure indicates that the variation of Stokes parameters is more significant with shorter wavelength bands. This tendency is clearly seen in the wavelength dependence of Δq and Δu, the differences between the maximum and the minimum values among all observations, as listed in Table 5; Δq and Δu increase with shorter wavelength bands.

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Table 4. Observed Polarizations at Five Wavelength Bands

Phase	Band (nm)	q (%)	u (%)	P (%)	PA (°)
1.201	400–500	−0.23 ± 0.09	1.41 ± 0.09	1.43 ± 0.09	49.7 ± 1.9
	500–600	−0.17 ± 0.04	1.41 ± 0.04	1.42 ± 0.04	48.5 ± 0.7
	600–640	−0.18 ± 0.04	1.39 ± 0.04	1.40 ± 0.04	48.6 ± 0.8
	720–800	−0.11 ± 0.03	1.22 ± 0.03	1.22 ± 0.03	47.7 ± 0.7
	800–850	−0.10 ± 0.03	1.09 ± 0.03	1.10 ± 0.03	47.5 ± 0.7
1.229	400–500	−0.50 ± 0.18	1.34 ± 0.18	1.44 ± 0.18	55.3 ± 3.6
	500–600	−0.27 ± 0.04	1.36 ± 0.04	1.39 ± 0.04	50.5 ± 0.8
	600–640	−0.30 ± 0.03	1.29 ± 0.03	1.33 ± 0.03	51.5 ± 0.6
	720–800	−0.12 ± 0.02	1.16 ± 0.03	1.17 ± 0.03	48.1 ± 0.5
	800–850	−0.13 ± 0.05	1.14 ± 0.05	1.15 ± 0.05	48.2 ± 1.1
1.657	400–500	−0.15 ± 0.07	1.39 ± 0.07	1.39 ± 0.07	48.0 ± 1.4
	500–600	−0.13 ± 0.06	1.29 ± 0.06	1.29 ± 0.06	47.8 ± 1.2
	600–640	−0.13 ± 0.06	1.29 ± 0.07	1.30 ± 0.07	47.8 ± 1.3
	720–800	−0.09 ± 0.06	1.14 ± 0.07	1.15 ± 0.07	47.2 ± 1.5
	800–850	−0.06 ± 0.07	1.01 ± 0.07	1.02 ± 0.07	46.7 ± 1.9
2.005	400–500	0.10 ± 0.08	1.22 ± 0.07	1.22 ± 0.07	42.7 ± 1.8
	500–600	0.01 ± 0.08	1.27 ± 0.07	1.27 ± 0.07	44.8 ± 1.8
	600–640	0.01 ± 0.09	1.23 ± 0.08	1.23 ± 0.08	44.8 ± 2.2
	720–800	0.03 ± 0.10	1.10 ± 0.09	1.10 ± 0.09	44.3 ± 2.6
	800–850	0.02 ± 0.15	1.02 ± 0.13	1.02 ± 0.13	44.6 ± 4.2

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Using the averaged Stokes q, u in Table 4, we evaluated the variability by χ² test for q and u, respectively. A reduced chi-square χ²_ν(q) was calculated by

$$\begin{equation} \chi _{\nu }^2 (q) = \frac{1}{\nu } \sum _{j = 1}^{4} \frac{(q_j - \overline{q})^2}{\sigma _j^2}, \end{equation} \tag{ 3 }$$

where ν is the degree of freedom, q_j and σ_j are Stokes q and its uncertainty at a certain band in jth observing run, and $\overline{q}$ is a weighted average over all observing runs from j = 1 to 4, calculated by

$$\begin{equation} \overline{q} = \sum _{j = 1}^4 \frac{q_j}{\sigma _j^2} \bigg/ \sum _{j = 1}^{4} \frac{1}{\sigma _j^2}. \end{equation} \tag{ 4 }$$

Similarly, we calculated χ²_ν(u). The results are listed in Table 5. We adopted the criterion for variability as either χ²_ν(q) or χ²_ν(u) being larger than that at a significance level of 0.01 (= 3.78 for ν = 3). From this criterion, we recognized that the continuum polarizations of the bluer three bands (400–500, 500–600, and 600–640 nm) were variable during the quiescent phase. This recognition is consistent with the wavelength dependence of Δq and Δu; the variability is detected in the shorter three wavelength bands that have larger Δq.

Table 5. Results of the Variability Test

Band (nm)	$\overline{q}$ (%)	Δqa (%)	χ2ν(q)b	$\overline{u}$ (%)	Δua (%)	χ2ν(u)b
400–500	−0.11 ± 0.04	0.60	4.63	1.33 ± 0.04	0.20	1.34
500–600	−0.18 ± 0.02	0.28	3.88	1.36 ± 0.02	0.14	1.62
600–640	−0.23 ± 0.02	0.31	5.95	1.32 ± 0.02	0.16	1.81
720–800	−0.11 ± 0.02	0.15	0.82	1.18 ± 0.02	0.12	1.00
800–850	−0.10 ± 0.02	0.14	0.42	1.09 ± 0.02	0.13	0.93
Raman 683	−0.11 ± 0.03	0.20	2.34	1.17 ± 0.04	0.19	0.23

Notes. ^aΔq ≡ q_max − q_min and Δu ≡ u_max − u_min. ^bThe degree of freedom ν = 3 for both q and u.

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3.1.2. Model Fit

For a binary embedded in circumstellar matter, two possible processes can result in a temporal variation of polarization: (1) a secular change in the density distribution of scattering particles in circumstellar space, (2) a periodic modulation due to the orbital motion of binary components without any change in the circumstellar environment. By investigating the periodicity in the variation, we can determine which process causes the detected variation; the investigation needs observations covering at least a few orbital cycles. However, it is reasonable to suppose that during the quiescent phase the circumstellar environment is stable. Thus, in this study, we examined the latter possibility, that is, the temporal variation being due to orbital motion, by fitting this variation with a scattering model.

As for the scattering model, we assumed that the scattering particle was a free electron, a neutral hydrogen atom, or a dust grain with a size much smaller than the optical wavelength. These scattering particles are abundant in symbiotic binary systems, although circumstellar dust grains are probably not abundant in Z And, which is classified as an S-type symbiotic star (Belczyński et al. 2000). The scattering phase functions of these particles have the same dependence on a scattering angle as that of a free electron (Thomson scattering). Thus, this assumption enabled us to use the formulae of Brown et al. (1978, BME) for symbiotic binaries, describing the polarization originating from a binary embedded in a free electron envelope with arbitrary distribution. Additionally, we adopted the geometrical model of Seaquist et al. (1984) for the distribution of the scattering particles; in this model, the density distributions of both ionized and neutral regions have symmetries not only about the orbital plane but also about the axis joining both binary components (binary axis). Although the geometries of both ionized and neutral regions depend on many variables through the dimensionless parameter X (Seaquist et al. 1984), the density distribution symmetry of scattering particles about the binary axis (simultaneously satisfying the symmetry about the orbital plane) holds for any X. Thus, we can use this model for Z And without regard to the value of X.

With this additional constraint for the distribution, the BME formulae can be rewritten as

$$\begin{eqnarray} &&q(\lambda, \phi) = q_{\rm co}(\lambda) - P_{\rm v}(\lambda,\phi,i) \times \cos {2\Theta _{\rm v}(\Omega,\phi,i)}, \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&u(\lambda, \phi) = u_{\rm co}(\lambda) - P_{\rm v}(\lambda,\phi,i) \times \sin {2\Theta _{\rm v}(\Omega,\phi,i)}, \end{eqnarray} \tag{ 6 }$$

where q_co(λ) and u_co(λ) denote the constant part of the polarization independent of orbital motion, composed of interstellar and/or intrinsic components, and P_v(λ, ϕ, i) and Θ_v(Ω, ϕ, i) are defined as

$$\begin{eqnarray} &&P_\mathrm{v}(\lambda,\phi,i) = P_\mathrm{max}(\lambda) \times (1 - \sin ^2{i} \cos ^2{(2 \pi \phi)}), \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&\Theta _\mathrm{v}(\Omega,\phi,i) = \Omega + \arctan { \left(\frac{\tan {(2 \pi \phi)}}{\cos {i}} \right) }, \end{eqnarray} \tag{ 8 }$$

with an orbital phase ϕ and the maximum polarization P_max(λ) at ϕ = 0.25 (and ϕ = 0.75). The polarization of this model has a periodicity of half of the orbital period, and the loci drawn by orbital motion in the Stokes q–u plane are an ellipse.

The modulation due to orbital motion is expected to be independent of the wavelength, although the amplitude and the constant component depend on the wavelength. Hence, we applied the formulae to the bluer three bands simultaneously. This multiband fit had 11 free parameters, q_co(λ), u_co(λ), and P_max(λ) for each band, and i and Ω, which were common among the three bands. The multiband solution is listed in Table 6 and shown in Figure 2. As shown in this figure, the model accurately reproduces the variation in all three bands despite the simplicity of the model. This indicates that observed polarimetric variation in the continuum may originate from orbital motion.

Table 6. Fit Solutions for Continuum and Raman Line Polarizations

Model	Band	i (deg)	Ω (deg)	χ2ν	ν
BME	3 bandsa	72.9 ± 13.7	79.7 ± 4.5	1.01	13
Raman	683	40.5 ± 8.3	81.6 ± 1.6	0.21	2

Note. ^aMultiband fit for the bluer three bands (400–500, 500–600, and 600–640 nm).

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The inclination derived from the continuum i_c differs by about twice the error (2σ_i) from that found from the Raman lines by Schmid & Schild (1997), although the orientation Ω_c is comparable with that of Raman lines. As described in the opening paragraph in Section 3, significant changes across the Raman line λ683 are detected in our observed polarization spectra. Thus, to confirm the possible inconsistency in the inclination, we extracted the intrinsic polarization of Raman line λ683 and estimated the orbital elements from our data in Section 3.2.

To estimate orbital elements from the Raman line, we must extract the intrinsic polarization of the line itself. The intrinsic P.A. of the line P.A._L is

$$\begin{equation} \mbox{P.A.}_\mathrm{L} = \frac{1}{2} \arctan { \left\lbrace \frac{ (1 + R) u_\mathrm{l}^\mathrm{i} - u_\mathrm{C}^\mathrm{i} }{ (1 + R) q_\mathrm{l}^\mathrm{i} - q_\mathrm{C}^\mathrm{i} } \right\rbrace }, \end{equation} \tag{ 9 }$$

where qⁱ_l and uⁱ_l are intrinsic Stokes parameters of the line region, which consist of both line and continuum components and are derived by subtracting interstellar polarization from the observed one; qⁱ_C and uⁱ_C are those of the continuum interpolated by the nearby continuum; R is the flux ratio of the line itself to the continuum. For estimating orbital elements, we used only the P.A. of the line because (1) accurate measurement of R is difficult in our low-resolution spectra and (2) the P.A. is less subject to R than the polarization degree when R < 0.1, as shown below.

To extract P.A._L by Equation (9), we needed to determine the interstellar polarization around the Raman line. The observed continuum polarization around the Raman line does not vary significantly with the orbital phase. This is supported by the negative detection of the variability in χ² test using Equation (3) in Section 3.1.1, as listed in Table 5. Moreover, this continuum polarization, calculated from the weighted average of q and u, $\overline{P} = 1.18\% \pm 0.04$% and $\overline{\mbox{PA}} = 47\mbox{$.\!\!^\circ $}7 \pm 0\mbox{$.\!\!^\circ $}7$, is in good agreement with the estimate of Schmid & Schild (1997), P = 1.20% ± 0.04% and PA = 469 ± 12 derived from the spectral range 673–719 nm. Therefore, we supposed that the continuum polarization at this wavelength is mainly interstellar in origin and accordingly adopted the weighted average of the continuum polarizations as the interstellar polarization.

Using this interstellar polarization, the intrinsic polarizations of the line region and the continuum were extracted (Table 7). We confirmed that the intrinsic polarization of the line region is considerably larger than that of the continuum. In addition, the measured flux ratio R, as listed in Table 7, is small (R < 0.1) for all observing runs. Under both conditions, Equation (9) is approximated as

$$\begin{equation} \mbox{PA}_\mathrm{L} \sim \frac{1}{2} \arctan { \left(\frac{ u_\mathrm{l}^\mathrm{i} - u_\mathrm{C}^\mathrm{i} }{ q_\mathrm{l}^\mathrm{i} - q_\mathrm{C}^\mathrm{i} } \right) }. \end{equation} \tag{ 10 }$$

The extracted P.A. is also listed in Table 7.

The orbital elements are derived by the same formula used by Schmid & Schild (1997),

$$\begin{equation} \Theta _\mathrm{r} = \arctan { \left(\frac{\tan {(2 \pi \phi)}}{\cos {i_\mathrm{r}}} \right) } + \Omega _\mathrm{r}. \end{equation} \tag{ 11 }$$

The best solution, i_r = 405 ± 83 and Ω_r = 816 ± 16, is listed in Table 6 and shown in Figure 3. The curve in this figure is in excellent agreement with the modulation in the extracted P.A.; as shown in the latter panel in the figure, the scatter of the residuals between the model curve and the measurements is small compared to the uncertainty (1σ). Derived elements agree with those of Schmid & Schild (1997); i = 47° ± 12° and Ω = 72° ± 6°.

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In the previous sections, we estimated the orbital elements i and Ω from the continuum and the Raman line polarizations, respectively. The orientation Ω from the continuum is in good agreement with that from the Raman line, but the inclination i from the continuum is inconsistent with that from the Raman line. We consider that the orbital elements estimated from the Raman line are more reliable than those estimated from the continuum for the following three reasons.

• 1.  
  Our understanding of the scattering particle types and geometry for the Raman line is better than our understanding of the same for the continuum. As described in Section 1, the scattering particle for the Raman line is already known and the line formation scattering geometry is relatively simple. In contrast, the scattering particle for the continuum polarization is not clearly identified, although circumstellar dust grains are likely the major candidate, if present (Schulte-Ladbeck et al. 1990). Therefore, the scattering geometry producing the continuum polarization is also unclear. Moreover, if the continuum polarization is produced through Mie scattering by the (spherical) circumstellar dust grains with a circumference comparable to the optical wavelength, the modulation is expected to be more complicated than that of our scattering model (Simmons 1983; Manset & Bastien 2001); this complication would affect the accuracy of estimating i. The functional form of the modulation also depends on the scattering geometry. Although our model assumes a symmetry for the distribution of the scattering particles about the binary axis, this symmetry would not be realized in the actual circumstances (e.g., Gawryszczak et al. 2003; Mitsumoto et al. 2005, and references therein). Whether our scattering model is too simple for the continuum polarization can be verified by examining the functional form of the observed modulation; this examination requires more data points, that is, more frequent observations during the quiescent phase.
• 2.  
  The estimation by Schmid & Schild (1997) is based on more abundant observations (six epochs) than ours (four epochs). In addition, the amplitude of the modulation for the Raman line is much larger than that for the continuum polarization. Therefore, the inclination from the continuum may be influenced more strongly by biasing due to low data quality, as is reported by Wolinski & Dolan (1994).
• 3.  
  The orbital elements from our Raman line data agree with those of Schmid & Schild (1997).

On the basis of the above points, we consider that the inconsistency of i may be due to the simplicity of our adopted model or may be caused by a bias effect as reported by Wolinski & Dolan (1994). To investigate the origin of this inconsistency, more frequent observations during the quiescent phase are urgently required. In addition, to confirm that the variability in the continuum polarization originated from the orbital motion, monitoring observations covering at least more than a few orbital cycles are needed. Moreover, the amplitude of the modulation in the continuum polarization depends strongly on the scattering geometry, which would be changed by the activity of the hot component. Therefore, the monitoring should be carried out during the quiescent phase, uninterrupted by the activity of the hot component.