On-sky demonstration of optical polaroastrometry Free

Abstract

1 INTRODUCTION

Quite frequently, some fraction of the radiation of an astronomical object becomes polarized as a result of physical conditions and/or its intrinsic asymmetry and can be measured by means of conventional polarimetry (Tinbergen 1996). For the same reasons, the centroid of an object can depend on the orientation of the transmitted polarization.

This effect is akin to the dependence of the centroid on the wavelength of observation, the measurement of which constitutes the subject of spectroastrometry, as proposed by Beckers (1982). The spectroastrometric signal can be measured on scales smaller than the diffraction limit of the telescope and its analysis allows one to impose additional constraints on the models of binary stars (Bailey 1998), circumstellar discs (Pontoppidan, Blake & Smette 2011), and active galactic nuclei (Gnerucci et al. 2013).

Similarly, measurement of the deviation of the centroid of the polarized flux from the centroid of the total flux – polarimetric astrometry, which we call for brevity polaroastrometry – may provide essentially new information about the object (Safonov 2013; Johnson et al. 2014).

The appearance of an astronomical object can be specified as a dependence of its intensity on the angular coordinate on the sky |$O_I(\boldsymbol {\alpha })$|⁠; by analogy it is possible to introduce the dependences of other Stokes parameters on |$\boldsymbol {\alpha }$|⁠: |$O_Q(\boldsymbol {\alpha })$|⁠, |$O_U(\boldsymbol {\alpha })$|⁠, and |$O_V(\boldsymbol {\alpha })$|⁠.

The total Stokes parameters Q, U, and V are integrals of the functions |$O_Q(\boldsymbol {\alpha })$|⁠, |$O_U(\boldsymbol {\alpha })$|⁠, and |$O_V(\boldsymbol {\alpha })$| over |$\boldsymbol {\alpha }$| and are measured by means of polarimetry. Usually dimensionless Stokes parameters are considered: q = Q/I, u = U/I, and v = V/I.

Let us consider examples of astrophysical objects that differ from the observational point of view for the polaroastrometry.

• If the distribution of the polarized flux of the object is centrally symmetrical (e.g. a main-sequence star), or there is no polarized light, then the polaricentres coincide with the photocentre and the polaroastrometric signal equals zero (first column in Fig. 1).

• If the object is polarized by interstellar dust, then the polaricentres coincide with the photocentre¹ (first column in Fig. 1).

• If the source of polarized light is confined within a small region (relative to the precision of the method), then the polaricentres will coincide with each other, but deviate from the photocentre (second column in Fig. 1).

• In other cases, the coincidence of polaricentres is possible only for special and therefore unlikely object geometry (third column in Fig. 1). For complex objects, the polaricentres can deviate from the photocentre even when the total polarization is close to zero.

Figure 1.

Different types of objects from the point of view of polaroastrometry. Shown in rows are Stokes parameters I, Q, U, photocentres (squares), and polaricentres (crosses). Columns display the cases of a single star, a single star with a scattering object nearby, and a complex object.

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We can conclude that even a qualitative analysis of polaroastrometric data – the detection of a deviation of polaricentres from the photocentre or coincidence of polaricentres – presents a valuable tool for astronomical object diagnosis. A quantitative interpretation of the polaroastrometric signal is also possible by using a priori knowledge about an object.

Here, we propose a method of optical polaroastrometry and study its performance on the basis of observations made with the prototype of the Multimode Fast Camera installed on the 70-cm telescope in Moscow (see description in Section 2). The method used to estimate the dimensionless Stokes parameters q and u from the observational data is presented in Section 3. The polaroastrometric signal is extracted from the same data by using differential speckle polarimetry (DSP; Safonov 2013). Details of the method in an application to polaroastrometry and a noise analysis are given in Section 4. The results in the form of dimensionless Stokes parameters and the polaroastrometric signal are given in Section 5 for unpolarized stars, stars polarized by interstellar dust and stars presumably possessing a detectable signal. Section 6 presents conclusions. Some auxilliary results are presented in Appendices A and B.

2 INSTRUMENT

For the polaroastrometric experiment, we used a prototype of the Multimode Fast Camera under design for the 2.5-m telescope of the Sternberg Astronomical Institute (SAI). A schematic of the instrument is depicted in Fig. 2. The instrument combines features of a two-beam polarimeter with a rotating half-wave plate (HWP) and a speckle interferometer.

Figure 2.

Schematic of the instrument used in the experiment. Numbers are for the following components: (1) pinhole, (2) auxiliary lens, forming an image of the pinhole at the first focal plane (with components 1 and 2 being assembled as a single unit which that be inserted into the beam manually), (3) linear polarizer, which is inserted into the beam manually, (4) field diaphragm, (5) half-wave plate, (6) collimator lens, (7) filter, (8) Wollaston prism, (9) camera lens, and (10) detector.

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We used an Andor iXon 897 electron multiplying CCD (EMCCD) camera as the detector. Owing to electron multiplication technology, readout noise in this type of detector is greatly reduced, at the cost of a twofold increase in photon noise (so-called multiplication noise). The frame rate was chosen to be 35 frames per second, which is fast enough to reduce blurring of the image by atmospheric turbulence.

The first optical component of the system is a HWP, which effectively rotates the polarization plane of incoming radiation. This has two goals: measurement of both parameters of linear polarization and calibration of the instrumental polarization of the optics located after the HWP. The HWP (Edmund Optics, EO-49232) consists of several polymer layers selected so that the total phase retardance between ordinary and extraordinary beams deviates from π rad by ≤0.02 rad inside the V band. A motorized rotation stage Standa 8MRU was used to control the position angle of the HWP.

Two achromatic lenses with a collimated beam between them were used as relay optics. These lenses give a magnification of ≈4.6. A V filter and a Wollaston prism (RIVoptics) were installed in the collimated beam. The prism splits the beam in two, polarizing it horizontally and vertically.

A slit diaphragm is installed in the first focal plane; this precludes any overlap of the beams split by the Wollaston prism and limits the field of view to 13 arcmin × 33 arcmin. An example of an image obtained with the instrument is shown in Fig. 3. The instrument is controlled from a custom command line c++ program running under osgnu/linux.

Figure 3.

A single frame from Vega, obtained with the prototype of the Multimode Fast Camera and an exposure 30 ms. The separation of the images is 210 pixels. The brightness scale at the bottom is in counts.

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The instrument was installed at the Cassegrain focus of the 70-cm telescope AZT-2 located near the SAI building in Moscow. The equivalent telescope focal length is 10.5 m; with relay optics of the instrument this is converted to ≈48.3 m. The angular sampling frequency is f_d = 3.22 × 10⁶ rad⁻¹, whereas the cutoff frequency of the optical system is f_c = D/λ = 1.27 × 10⁶ rad⁻¹. Therefore, the Nyquist criterion f_d > 2f_c is fulfilled for this setup, which is necessary for correct computation of the Fourier spectrum of images.

We adopted the following observation sequence. The detector continuously obtained a series of frames, without gaps between them. Meanwhile, the HWP rotated at a constant angular speed. The frame rate and an HWP speed were matched so that the HWP position angle θ changed by 9° over one exposure. Because it was not possible to measure and record θ for each frame with this setup, we determined it a posteriori. For each series, we introduced a linear polarizer at a known orientation into the beam for 10–15 per cent of the total series duration at its beginning and end. During these periods the polarization of incoming radiation can be considered as known and it is possible to determine corresponding θ. Then θ can be interpolated in the middle of the series, when the linear polarizer was removed from the beam. Data obtained using the described technique allow the extraction of both polaroastrometry and polarimetry information.

Observations have been conducted in 2014 August–October in five sessions, with the instrument being removed from the telescope between sessions. Throughout a session, the angular scale w and the position angle ϕ of the camera were fixed by the equatorial mount of the telescope. We measured these parameters for each session for the transformation of measurements from the instrumental reference system to the equatorial J2000. Because our experiment is not very demanding in terms of w and ϕ accuracy, for their estimation we used binary and double stars with well-known ephemerides: HIP94336, HIP103669, and HIP108917 (Mason et al. 2007; Farrington et al. 2014). We acquired long-exposure images of these stars and then measured a separation vector of the stellar images by approximating one image with the other. By comparing the parameters of these vectors with the ephemerides we obtained w and ϕ. The results of our measurements are given in Table 1. For each session we also observed at least one polarization standard and an unpolarized star.

3 POLARIMETRY

The instrument we used is a two-beam polarimeter, whose main advantage is that it is insensitive to variations in atmospheric transparency and to stellar scintillation. However, it is susceptible to the difference in transmission between the two beams of the polarimeter. To mitigate this effect, switching of the images corresponding to orthogonal polarizations is employed. The switching usually is performed with the help of an HWP installed before the polarimeter (Bagnulo et al. 2009). In our case, images are switched every 140 ms.

One potential drawback of this scheme is its high sensitivity to any imperfection of the HWP. Nevertheless, it is possible to estimate this effect by using a generalization of the double difference method as described in the following. This generalization is closely related to the adopted methodology of polaroastrometry (Section 4).

Preparatory processing of data is performed as follows. First, we average frames with θ differing by ≤3°, which is small compared to the angle through which the HWP passes during one exposure. Then we subtract a bias and a constant background from the averaged frames. Fluxes corresponding to two images, J_Lk and J_Rk, are extracted by using aperture photometry, where k is the number of the averaged image. The relative positions of the apertures for photometry are the same for all frames.

Coefficients ζ and ζ^′ are present in equations (8) and (9) because θ as well as its cosine and sine changes during the course of obtaining the frame, and therefore some averaging occurs. ζ gives the sense of the polarimeter effectiveness (Tinbergen 1996) and equals |$\zeta \approx 1-(2/3)\theta _f^2$|⁠, where θ_f is the angle through which the HWP passes during the exposure. At θ_f = 9° the coefficient ζ equals 0.984.

From equation (11), one can see the difference between a signal carrying information about the linear polarization of the source and instrumental effects such as variations of the system transmission and the HWP imperfection. The signal occupies only one frequency in the spectrum |$\widetilde{P}(f_{\theta })$|⁠; meanwhile, the noise is distributed over all frequencies, including the signal frequency. Therefore, the linear polarization can be estimated as the amplitude of fourth harmonics of |$\widetilde{P}(f_{\theta })$|⁠, but it will be inevitably biased by unknown values ω_{u, q}(4). All that remains is to estimate their amplitudes by interpolation and take these as a measure of the uncertainty of the q_d and u_d estimation.

In Fig. 4, the spectra |$|\widetilde{P}(f_{\theta })|^{2}$| for three presumably unpolarized stars are given. As unpolarized stars we have taken main-sequence stars at distances <66 pc and Galactic latitude >10°, as was done by Berdyugina et al. (2011). One can see that the spectra are quite uniform, and that there are no marked frequencies, which demonstrates the possibility of estimation of the fourth harmonic uncertainty. It is notable that |$|\widetilde{P}(f_{\theta })|^{2}$| slowly rises at low frequencies for the bright star. This effect can be explained by the fact that the photon component of the noise for this star is sufficiently low to make the instrumental effects visible.

Figure 4.

Spectrum modulus squared, |$|\widetilde{P}(f_{\theta })|^{2}$| normalized by the expected level of the photon noise for four objects. Black lines are for presumably unpolarized stars; grey one is for weakly polarized o Cet. The legend gives the total numbers of accumulated photoelectrons.

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The spectrum |$|\widetilde{P}(f_{\theta })|^{2}$| for o Cet (2014 October 2) is also presented in Fig. 4. In this case one sees that the fourth harmonic signal is significantly higher than the noise, indicating the presence of detectable linear polarization. For polarization standard stars the fourth harmonic amplitude exceeds the noise level by a factor of 200–1000.

The advantage of method of spectral decomposition of P(θ) over the double difference method (Bagnulo et al. 2009) is that the former enables an adequate estimation of the uncertainty of the resulting value caused by instrumental effects, especially the HWP imperfection (see Appendix A). For polarimetry such an estimation is important only for the brightest stars, but for polaroastrometry the sensitivity to the HWP imperfection is significantly stronger and a more accurate estimation of its effect becomes critical. A similar approach to analysing data from polarimeters with rotating HWPs was applied in millimeter waves by Johnson et al. (2007).

From q_d and u_d one can estimate the degree of polarization p_d and the angle of polarization χ_d; the latter can then be transformed to the equatorial coordinate system.

To verify the method and test its stability, we observed polarization standards and presumably unpolarized stars in every session. The results are given in Table 2 in the form of p and χ. For comparison, values obtained by Schmidt, Elston & Lupie (1992) are given. For the standards, the degree of polarization coincides with a precision 3 per cent, and the angle of polarization with a precision 1°–2°, both of which are acceptable. The stability of p is 1 per cent, and that of χ is 1°; the difference between sessions exceeds the errors insignificantly.

The small value of the measured polarization for the unpolarized stars provides evidence that the instrumental polarization of the telescope is small and can be neglected, which is expected because the optical design is centrally symmetrical.

4 POLAROASTROMETRY

4.1 Basics

According to a convention accepted in optics, a beam with a Stokes vector (I, I, 0, 0) is polarized horizontally, i.e. the plane of polarization is oriented along the OX axis. In astronomical polarimetry for a such beam, the polarization plane will be oriented towards the north celestial pole (NCP). Therefore, for derivations we assume that the OX axis is directed towards the NCP. Then, for an object with s_q > 0, t_q = 0 the q-polaricentre is displaced towards north; for an object with s_q = 0, t_q > 0 – towards east, which may seem counterintuitive.

4.2 Extraction of the polaroastrometric signal

The polaroastrometric signal |$\boldsymbol {\Delta }$| can be extracted from the observational data in various ways, e.g. by performing a simple centroid computation or by approximating one image with another. We solve this task by using a DSP, a method similar to the double difference approach (Bagnulo et al. 2009), but operating in the Fourier domain (Safonov 2013).

The computation of |$\mathcal {R}(\boldsymbol {f})$| starts with removing bias, constant background and equalizing to zero pixels having counts <5 rms of readout noise. Then two rectangular windows of 100 × 100 pixels each are selected. The left window is centred on the brightest pixel of the left image; the right window is displaced relative to the left by a constant vector. The Fourier spectra are evaluated separately for the left and right images for many frames and are combined using equation (16). Only images having θ differing by ≤3° take part in the combination. In such a manner a set of |$\mathcal {R}(\theta ,\boldsymbol {f})$| for different θ values is formed.

To adequately describe our experiment, the following instrumental effects additionally need to be taken into account: (1) deviations of the beams by the Wollaston prism, (2) dispersion from the Wollaston prism and the atmosphere, (3) differential aberrations of the HWP, and (4) transformation of the polarization state by the HWP.

In Appendix A, it is shown that the HWP effect |$\Delta \mathcal {R}_{\mathrm{HWP}}(\theta ,\boldsymbol {f})$| can be represented as |$1+\Delta \mathcal {R}_{\mathrm{HWP}}(\theta ,\boldsymbol {f}) = \exp \lbrace {{\rm i} \phi _{\mathrm{HWP}}(\theta ,\boldsymbol {f})}\rbrace$|⁠, while ϕ_HWP ≪ 1.

As long as the optical system of the telescope is very slow (F/64) and possesses central symmetry, for our level of precision |$\Delta \mathcal {R}_T$| can be considered equal to unity (Safonov 2013).

The dependence of phase |$\mathcal {R}_n(\theta ,\boldsymbol {f})$| on the frequency |$\boldsymbol {f}$| was approximated by least squares with the plane 4π(c(θ)f_x + d(θ)f_y). To test validity of the model we computed a reduced chi-squared statistic, which is the sum of squared deviations of measurements from the model weighted by their variances and normalized by the number of degrees of freedom, which is the difference between the number of independent measurements of |$\mathcal {R}_n(\theta ,\boldsymbol {f})$| phase and number of model parameters. The reduced chi-squared turned out to be 1–2, which indicates the correspondence of the data to the model.

In Fig. 5, the Fourier spectra |$\widetilde{d}(\theta )$| for o Cet and α Lyr are presented; total number of accumulated photoelectrons is 1.5 × 10⁹ and 3.8 × 10¹¹, respectively. In the spectrum for α Lyr, it is easy to distinguish two regions – of low and high frequency, divided by approximately the fifth harmonics. The high-frequency part results from photon noise and other uncorrelated noises, whereas the low-frequency part can be explained by the HWP imperfections (see Section 4.3 and Appendix A). For the low-frequency noise dominance, to estimate the fourth harmonic amplitude uncertainty one should interpolate the noise from adjacent frequencies (dash–dotted line in Fig. 5). For faint objects, such as o Cet, it is possible to use a mean high-frequency noise amplitude for this purpose. Taking this into account makes clear that α Lyr does not show a distinct polaroastrometric signal whereas the o Cet does. In the next subsection, we analyse briefly the properties of the noise components of the spectra |$\widetilde{c}(\theta )$| and |$\widetilde{d}(\theta )$|⁠.

Figure 5.

Fourier spectra |$\widetilde{d}(\theta )$| for o Cet and α Lyr. Black lines are for the real part and grey are for the imaginary part. The dashed and dash–dotted lines depict level of harmonics rms estimated from the high- and low-frequency parts, respectively.

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The obtained values |$s_q^{\star }, s_u^{\star }, t_q^{\star }$|⁠, and |$t_u^{\star }$| and their errors are converted to the equatorial coordinate system using relations (B1) from Appendix B. The results and their discussion are given in Section 5.

4.3 Polaroastrometric signal noise

As a first step, we identify the dependence of noise estimations from the high- and low-frequency parts of the spectra |$\widetilde{c}(\theta )$| and |$\widetilde{d}(\theta )$| on the object brightness. The low- and high-frequency errors are denoted as σ_HF and σ_LF, respectively. For analysis we use the data for the relatively bright star δ Cyg (see Section 5.1). Additional Poisson noise is introduced into the raw data, effectively dimming the object by 2, 4, and 6 mag. After this, the processing described in the previous section is applied and σ_HF and σ_LF are estimated. The results are given in Fig. 6 for X and Y coordinates separately. In this subsection, it is assumed that the X coordinate is measured along the CCD rows.

Figure 6.

Dependence of the polaroastrometric signal Δ uncertainty on the total number of accumulated photoelectrons. The black symbols are for the X component, the grey ones are for the Y component, empty symbols are for the high-frequency error, and filled ones for the low-frequency error (see text). The dash–dotted line is for the photon noise level of astrometry |$25 \mathrm{pix}/\sqrt{N_{\rm e}}$|⁠.

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As one can see, σ_HF depends on the photoelectrons number N_e as |$25/\sqrt{N_{\rm e}}$| pix (dash–dotted line in Fig. 6), which illustrates the fact that the high-frequency part of the spectra |$\widetilde{c}(\theta )$| and |$\widetilde{d}(\theta )$| is produced by photon noise. It is instructive to compare this with the photon limit of astrometry noise, which is |$\beta /\sqrt{N_{\rm e}}$| (Lindegren 1978), where β is the full width at half-maximum (FWHM) of the image. As long as light is split into two halves in our method, and then the half-difference between image centroids is taken, for the considered signal the photon limit of astrometry noise is also |$\beta /\sqrt{N_{\rm e}}$|⁠. Hence the high-frequency noise of the polaroastrometric signal determination corresponds to the photon limit of astrometry noise at β = 25 pix or β = 1.7 arcsec, which is comparable to the FWHM of a long-exposure image.

Photon noise dominates at N_e ≲ 10⁹; for our telescope and instrument this corresponds to a total integration of 500 s from an object V ≈ 6 mag. For larger N_e the low-frequency part σ_LF gives a larger input. The latter varies from 0.6 to 1.5 mpix and does not show a clear dependence on N_e.

Let us consider the correlation properties of the errors, for which the series for δ Cyg is split into five equal parts, with σ_HF and σ_LF being estimated for each separately (see Fig. 6). From the figure, it is evident that the high-frequency noise is uncorrelated on time-scales comparable to the series duration and can be reduced by increasing the accumulation time. Meanwhile, for the low-frequency noise this is not correct. Therefore, we can conclude that a sample of the low-frequency error within a single series is approximately constant.

As described in Appendix A, the low-frequency error can be explained by the imperfection of the HWP given that the centre of the beam passing through it deviates from its centre by 1 mm. Such a decentring value is to be expected because the mechanical structure of the instrument is susceptible to flexures. This also explains why the low-frequency error is highly correlated for a single series of some object but changes after pointing to another object.

In Fig. 6 σ_HF and σ_LF for some objects considered in Section 5 are given. As one can see polarized stars and Mira-type variables exhibit error properties similar to those of unpolarized stars, except for sporadic cases.

5 RESULTS

To test the method efficiency, we observed objects of several types: presumably unpolarized stars (type N), stars polarized by interstellar dust (type P), and Mira-type pulsating variables (type M). The observation method and data processing were identical for all objects. Observation circumstances and processing results (polaroastrometric signal and Stokes parameters) are given in Table 3.

5.1 Unpolarized stars

Main-sequence stars presumably have a very small polaroastrometric signal owing to their symmetry and low polarization even close to their limb. We observed three single stars (HD 184960, HD 221477, and HD 9562), satisfying the following criteria: spectral class from A to M, luminosity class V, distance to the Sun <66 pc and Galactic latitude >10°. HD 9562 exhibits a small total polarization which could be caused by interstellar dust an effect sometimes detectable even for nearby stars (Bailey, Lucas & Hough 2010).

Three additional observed stars satisfy only some of the listed criteria and have peculiarities but nevertheless they can be classified as unpolarized stars. HD 180161 is a BY Dra type variable of spectral class G8V and has an angular diameter of 330 μas. Brightness variations for BY Dra variables are caused by starspots occupying a large part of the stellar surface and by rotation of the star. Polarization at the limb of its disc can be crudely assumed to be the same as solar, i.e. ≈10⁻³ (Fluri & Stenflo 1999). Therefore, for the most favourable geometry the expected total polaroastrometric signal is 0.3 μas, which is much smaller than the error of our measurements.

HD 210027 = ι Peg is a spectroscopic binary system. The angular size of the semimajor axis of its orbit is 10.3 mas, its period is 10.2 d, and the spectral classes of its components are F5V and G8V (Konacki et al. 2010). One can expect a small total polarization resulting from its ellipticity but we did not detect it. Its polaroastrometric signal is likely to be negligible given the object's symmetry.

The main component of the spectroscopic binary HD 186882 = δ Cyg is brighter than its secondary by 3.4 mag; the former has a spectral class of B9.5IV and falls into the class of rapidly rotating stars. The expected angular diameter is 550 μas, and the expected degree of polarization at the limb is ≲5 per cent (Bochkarev, Karitskaia & Sakhibullin 1985). Therefore, the total expected polaroastrometric signal for this star is ≲ 25 μas, which is undetectable with our instrument. According to Bailey et al. (2010), the total polarization for HD 186882 is (1.08 ± 0.02) × 10⁻⁴.

For the six mentioned stars, the expected polaroastrometric signal is much less than the errors of our measurements. This was proven in the experiment, see Table 3. This allows us to argue that our measurements do not have any bias at least down to 100 μas. A negligible value of bias of this kind is expected if one takes into account the central symmetry of the optical scheme.

5.2 Stars polarized by interstellar dust

An undetectable polaroastrometric signal is expected for stars analogous to type N but polarized by interstellar dust (type P), because the latter changes the polarization of each point of a star image identically. However, a small signal can be generated for observations made through the Earth's atmosphere owing to the joint effect of dependence of the degree of polarization on the wavelength and atmospheric dispersion.

Indeed, for some polarization standards variation of the degree of polarization across the V band reaches 0.2–0.3 per cent in absolute measure (Schmidt et al. 1992). Hence the polarized flux has an effective wavelength differing from the effective wavelength of the total flux by Δλ. For computing the effective wavelength, we used the stellar spectra of Gunn & Stryker (1983), the sensitivity curve of the detector, the transmission curve of the filter, and the dependence of degree of the polarization on the wavelength for the considered stars from Schmidt et al. (1992). Differences in the effective wavelengths are Δλ = −1.027 nm and Δλ = −0.480 nm for HD 204827 and HD 7927, respectively. These stars also display a dependence of the angle of polarization on the wavelength (Schmidt et al. 1992) but this can be neglected since its maximum amplitude over the V band is 0| $_{.}^{\circ}$|8.

Atmospheric dispersion displaces the polaricentres and this displacement is the same for polaricentres q and u, as long as Δλ is the same. The total displacement is |$\sqrt{s_q^2+t_q^2} = \sqrt{s_u^2+t_u^2} = \epsilon \cdot \Delta \lambda \cdot \mathrm{tan}z$|⁠, where ϵ is the angular atmospheric dispersion (ϵ = 2.12 × 10⁻⁸ rad nm⁻¹ for place and time of observations) and z is zenith distance. The amplitude of polaroastrometric signal can be estimated according to its definition. At z = 45° for HD 204827 and HD 7927 the amplitudes are 254 and 69 μas, respectively. In our experiment such a small signal cannot be reliably measured.

5.3 Mira-type variables

Mira-type variables are pulsating stars at late stages of evolution. These objects are expected to demonstrate detectable deviations of their polaricentres from the photocentre for two reasons. First, a significant fraction of the visible light radiation is being scattered by the dust envelope in the close vicinity of a star (Norris et al. 2012), which leads to polarization of radiation. Secondly, interferometric observations show that some Mira variables display significant asymmetry (Thompson, Creech-Eakman & Akeson 2002; Ragland et al. 2006).

We considered three Mira variables, suitable for observations in the Northern hemisphere from August to October of 2014: χ Cyg, R Tri, and o Cet. The measured total polarization and polaroastrometric signal are given in Table 3. For R Tri, we did not detect a significant signal; however, the total polarization for this star is largest in comparison with the other observed Mirae. This can be explained by the axial symmetry of this object detected previously by Thompson et al. (2002) using the Palomar Testbed Interferometer in near-infrared and confirmed by Ragland et al. (2006) using the Infrared Optical Telescope Array in the same region of the spectrum.

χ Cyg and o Cet show significant deviation of values |$s_q^{\star }, s_u^{\star }, t_q^{\star }$|⁠, and |$t_u^{\star }$| from zero, indicating the asymmetry of image, the same conclusion had been reached by Ragland et al. (2006) for χ Cyg and by Karovska et al. (1997) and Chandler et al. (2007) for o Cet (in the visible and mid-infrared). Moreover, for these objects the polaricentres q and u clearly do not coincide: |$s_q^{\star }/q \ne s_u^{\star }/u$| and |$t_q^{\star }/q \ne t_u^{\star }/u$|⁠. Therefore, the polaroastrometric signal cannot be explained by a single polarized source.

For o Cet, the situation is complicated by the presence of the nearby component Mira B, an accreting white dwarf (Sokoloski & Bildsten 2010), which in principle can be polarized. On 2014 October 3, 21:55 ut the components of the u-polaricentre were measured to be s_u = −570 ± 80 mas and t_u = 500 ± 80 mas; in the same reference frame the position of Mira B was s_c = −70 mas and t_c = 480 mas (Prieur et al. 2002). Therefore the u-polaricentre was shifted in roughly the same direction as Mira B, which can be interpreted as the effect of the polarized emission of Mira B seen against the background of the ‘random’ polaroastrometric signal generated by the Mira A atmosphere. A more definitive conclusion on polarimetric properties of Mira B in the visible requires more observations conducted in different phases of the pulsations.

To test the methodology, we repeated observations of χ Cyg and o Cet in nearby epochs; corresponding measurements agree well (see Table 3). In case of o Cet, between the second and third measurements the instrument was rotated counterclockwise with respect to the telescope by 57| $_{.}^{\circ}$|5. As one can see |$s_q^{\star }, s_u^{\star }, t_q^{\star }$|⁠, and |$t_u^{\star }$|⁠, transformed to the equatorial coordinate system, remained the same after rotation. For a larger difference in epochs – 55.0 d (between second and third measurements of χ Cyg), the polaroastrometric signal changes significantly. It is worth noting that the brightness of χ Cyg for the same period decreased from V = 7.5 mag to V = 8.5 mag (AAVSO data).

6 CONCLUSIONS

The goals of this work are to develop a methodology and to study the precision limits of polaroastrometry. Analysis has been performed on the basis of observational data obtained with the help of the Multimode Fast Camera prototype. By analysing the dependence of the half-difference of the photocentres of the images split by a Wollaston prism on the position angle θ of the HWP we obtained estimations of the polaroastrometric signal, i.e. displacements of the centroids of the polarized flux Q or U from the photocentre of the total flux, multiplied by the corresponding dimensionless Stokes parameter, q or u.

We performed an analysis of the polaroastrometric signal error dependence on circumstances of an experiment. It was found that for a total number of accumulated photoelectrons N_e < 10⁹ the photon noise dominates and can be approximated crudely as |$1.7\,{\rm arcsec}/\sqrt{N_{\rm e}}$|⁠. At larger N_e the input of imperfections of the given HWP starts to prevail, having an approximately constant level of 50–100 μas. This effect can be reduced by an order of magnitude if the HWP rotation and beam centre are aligned with an accuracy of 0.1 mm (see Appendix A).

Observations of unpolarized main-sequence stars, which presumably do not have any detectable polaroastrometric signal, showed that the measurements are unbiased at least at the level ≈ 100 μas. Main-sequence stars polarized by interstellar dust also do not display a detectable signal demonstrating the absence of cross-talk between a strong polarization and the polaroastrometric signal.

For method verification purposes, we selected three Mira-type variables χ Cyg, o Cet, and R Tri because they are expected to have detectable signal. For χ Cyg the measured polaroastrometric signal is |$s_q^{\star }=+220\pm 70$| μas, |$t_q^{\star }=+220\pm 50$| μas, |$s_u^{\star }=0\pm 70$| μas, and |$t_u^{\star }=-320\pm 50$| μas in the J2000 coordinates for 2014 August 4, 19:16 ut. For o Cet these values are |$s_q^{\star }=+10\pm 110$| μas, |$t_q^{\star }=-430\pm 120$| μas, |$s_u^{\star }=+850\pm 110$| μas, and |$t_u^{\star }=-750\pm 120$| μas for 2014 October 3, 21:55 ut. A significant polaroastrometric signal gives evidence of asymmetry of the polarized visible flux. Asymmetry was detected for χ Cyg and o Cet earlier in the infrared by Ragland et al. (2006) and Chandler et al. (2007), and for the latter in the visible also by Karovska et al. (1997). For R Tri the signal was not found.

As a test of the repeatability of measurements we observed these objects at different, but close epochs and at different orientations of the instrument relative to the telescope. We did not detect a significant difference in measurements. Also as a by-product we obtained the polarimetry of the mentioned objects with a precision of (1–5) × 10⁻⁴.

Mira-type variables are suitable as test objects because apparently they have the greatest polaroastrometric signal among all celestial objects. Though the obtained data demonstrate the effectiveness of the method, their astrophysical interpretation is obstructed by the complexity of these objects.

Possible applications of polaroastrometry using a large telescope to polarized objects described by a simple geometrical model, e.g. some types of circumstellar environments, active galactic nuclei, including Sgr A*, and gravitational microlensing are very promising.

This work was supported by M. V. Lomonosov Moscow State University Program of Development and Russian Foundation for Basic Research grant no. 14-04-31185. The author wishes to thank N. Shatsky, V. Kornilov, and A. Tokovinin for discussions on methodology, and A. Magnitskiy and D. Cheryasov for help with technical issues during observations. Comments by an anonymous referee helped to improve the presentation and extend the analysis. Use was made of the SIMBAD data base, operated at CDS, Strasbourg, France and AAVSO International data base for preparation of observations and data interpretation.

REFERENCES

APPENDIX A: INFLUENCE OF HWP IMPERFECTION

Let us suppose that the HWP imperfection can be modelled as fluctuations of a phase retardance δ and fast axis orientation θ over its area. To derive these fluctuations for the given HWP we illuminated it through a linear polarizer, then passed the resulting light through another linear polarizer, which we could rotate. With the help of a relay lens the HWP surface was imaged on the detector. For each point on the HWP, values δ and θ were extracted from the dependence of the flux on the second linear polarizer orientation. The results are given in Fig. A1.

Figure A1.

Fluctuations of the basic parameters of the HWP over its area. The left-hand panel is for the fast axis orientation; the right-hand panel is for the contrast at crossed linear polarizers δ^′2/4, where δ^′ is the deviation of the retardance from π.

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As one can see from figure, fluctuations of θ reach 2°, and become more pronounced closer to the border of the plate. For δ, the amplitude of the fluctuation is of the order of 3°–5° in a central part of the HWP. On the basis of these measurements and with the help of the model developed by Safonov (2013) we estimated |$\mathcal {R}_{\mathrm{HWP}}(\theta ,\boldsymbol {f})$| (for the definition see Section 4.2) for different beam parameters. Differential aberrations produced by the HWP are purely phase aberrations, and the same is true for |$\mathcal {R}_{\mathrm{HWP}}(\theta ,\boldsymbol {f})$|⁠. The rms of the |$\mathcal {R}_{\mathrm{HWP}}(\theta ,\boldsymbol {f})$| phase in area |f| < f_c is 0.02 rad, in |f| < 0.2f_c it is 0.002 rad.

By approximating the phase of |$\mathcal {R}_{\mathrm{HWP}}(\theta ,\boldsymbol {f})$| with a plane we evaluated functions G(θ) and H(θ) (see Section 4.2) and then computed their Fourier spectra. These are given in Fig. A2 for a beam diameter 2 mm and different decentring parameter in comparison with the analogous spectrum measured in the polaroastrometric experiment for α Lyr. It is evident that the observed behaviour of the spectra can be explained by the described HWP imperfection at decentring ≈1 mm.

Figure A2.

Spectrum of H_n(θ), black and grey lines show the real and imaginary parts, respectively. Bold lines are for measurements of α Lyr, and thin lines are for what is expected when the HWP is used. Decentring values 0, 1, and 2 mm are shown by solid, dashed, and dash–dotted lines, respectively.

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The useful signal corresponds to a complex amplitude of fourth harmonics of the Fourier spectra |$\widetilde{c}(\theta )$| and |$\widetilde{d}(\theta )$|⁠. The amplitude of the noise of this harmonic is given in Fig. A3 as a function of beam diameter and decentring. As one can see, reducing the noise to the level of 0.1 mpix requires aligning the beam and the centre of the HWP rotation with an accuracy of 0.1 mm, it is also worthwhile to use beams of smaller diameter (<2 mm).

Figure A3.

Dependence of the fourth harmonic amplitude of spectrum H_n(θ) on the decentring parameter for three values of beam diameter on the HWP: 2 (black solid line), 3 (dark grey dashed line), and 4 mm (light grey dash–dotted line).

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APPENDIX B: POLAROASTROMETRIC SIGNAL TRANSFORMATION TO ROTATED REFERENCE SYSTEM

As one can see, sometimes it is impossible to transform the polaricentres because the denominators in the last four expressions may be close to zero. In such cases one should operate with the polaroastrometric signal itself.