1.Introduction

In this paper we report astrometric results for double stars obtained by speckle interferometric observations carried out with the 2.1 m telescope of Sierra San Pedro Mártir National Astronomical Observatory (OAN-SPM) in June of 2016. This is the seventh in a series of publications that started with speckle interferometric measurements performed with the OAN telescopes in 2008 (^{Orlov et al. 2009}). As in our previous publications, we focus on double stars from the Washington Double Star (WDS) catalog (^{Worley & Douglass 1997}).

The Speckle Interferometry (SI) (^{Labeyrie 1970}) is one of the most used high resolution techniques. This method allows the observer to obtain information about relative positions in close binary stars systems with diffraction-limited accuracy. This technique was most widely used in the study of binary and multiple stars (^{Tokovinin et al. 2020}; ^{Guerrero et al. 2020}; ^{Mitrofanova et al. 2020}). The observation methodology and data processing of SI is very well studied and described by ^{Tokovinin et al. (2010)}.

The 2.1 m telescope of OAN-SPM has a thin primary mirror; its shape is corrected by air bags. The process of correction takes about one hour and is performed only once before observations. During the night, the temperature of the primary mirror and of the telescope mount change, which leads to thermal deformations. Also, corrections introduced by the airbags depend on the hour angle and the zenith distance of the target. As a result, we have different aberrations for each object (Figure 1). Because of this, it is unfeasible to construct a universal synthetic speckle interferometric transfer function or even to use a reference star. This fact limits the possibility of finding both astrometric and photometric parameters of double stars. In addition, the telescope’s vibration distorts the specklegrams. All these factors have a greater impact on the ability to recover photometric parameters than on the recovery of astrometric parameters. Therefore, in this study we focus on improving the recovery of astrometric parameters.

Fig. 1 Long exposure image of WDS 20480+3917 (a). Long exposure image of WDS 18003+2154 (b). Both images show very strong coma aberration with different angles. 

In order to estimate astrometric parameters, we designed an algorithm which allows one to recover each measurement from the distorted power spectrum. In section 3.2 we describe the calculation of the high resolution autocorrelation function in polar coordinates. This algorithm allows for blind searching of the astrometric parameters ρ and θ of double stars, since it finds the coordinates of the absolute maximum of a two-dimensional discrete function.

2.OBSERVATIONS

Speckle interferograms were taken during four nights in the summer of 2016, from June 28 to July 1 at the 2.1 m telescope of the Observatorio Astronómico Nacional (OAN), which is located at the astronomical site Sierra San Pedro Mártir, México.

The observations were performed using the EMCCD iXon Ultra 888 from Andor Technology. This is a low-noise, high-sensitivity EMCCD camera that can be cooled thermoelectrically down to -95^o which provides excellent elimination of dark noise, even for the short time exposures. The detector has quantum efficiency higher than 80% in the range of 450 - 750nm, with a maximum of 95% at 550nm (V-band). This camera allows a fast frame rate so it can be used for speckle interferometry. The detector has 1024 x 1024 square pixels of 13μ per side.

The observations were carried out using broadband filters V(538/98 nm), R(630/118 nm) and I(894/330 nm) from the Johnson-Cousins set. The size of the diffraction-limited speckle (λ/D) for the 2.1m telescope is approximately 70 mas at this filter wavelength. Given these parameters, we need an angular pixel scale of about 35 mas to obtain a Nyquist sampling of specklegrams. To provide a suitable sampling, we used the f/7.5 secondary mirror combined with a microscope objective lens x 4.

We recorded 500 speckle frames of 400 x 400 pixel per object, taken with exposure times of 29.5ms. We use EM gain of 1/300 photons/e^- for all observations.

The seeing was better than 1over all the observing nights. However, aberrations introduced by the telescope have a larger effect (Figure 1). As a result, long exposure images have a resolution of about 1.5.

3. Data Processing

The first step of the data processing is the dark field correction of detected images In'(x):

where x is a 2D spatial coordinate, In(x) is the corrected image, Dark_(x) is the average dark image captured with a closed shutter (Figure 2 left). In order to remove the reading noise, we also set to zero all values less than 4σ of dark (Figure 2 right).

Fig. 2 The average Dark image (left) and σ of Dark image (right). 

3.1.Unshifted Power Spectrum

The next step is to calculate the averaged power spectrum (PS) for each star:

PS(f)=⟨|FT{In(x)}|2⟩, (2)

where f is a spatial frequency, FT{…} is the Fourier transform and ... denotes averaging over all images.

In the case of low light images, the averaged power spectrum can be expressed as (^{Kerp et al. 1992}):

PS(f)=P(f)⋅|G(f)|2+q|G(f)|2, (3)

where P(f) is the unshifted estimation of the power spectrum, q is some constant, |G(f)|2 is the power spectrum of the photon event shape function, also known as photon bias. The photon bias |G(f)|2 can be determined as the normalized power spectrum of the night sky. |G(f)|2 is constant in the 𝑌 direction for this camera. Thus, it can be determined directly from PS(f) (Figure 3, left) by analysis of its part beyond the cut-off frequency of telescope. The unshifted power spectrum of specklegrams P(f) is shown in Figure 3 (right). Therefore, it can be presented as:

P(f)=|O(f)|2⟨|Sn(f)|2⟩, (4)

where |O(f)|2 is the power spectrum of the object, and ⟨|Sn(f)|2⟩ is the speckle interferometric transfer function. The speckle interferometric transfer function can be obtained by observing a reference star, or one can construct a universal synthetic speckle interferometric transfer function (^{Tokovinin et al. 2010}). If one needs only astrometric parameters, they can be obtained without the speckle interferometric transfer function, directly from P(f).

Fig. 3 Power spectrum of WDS 20312+1116 before photon bias correction (left) and after correction (right). The separation is 0:´´3. 

3.2.Autocorrelation Function in Polar Coordinates

In order to find astrometric parameters from the unshifted power spectrum we calculated the high resolution autocorrelation function in polar coordinates ACF_p:

ACFp(ρ,θ)=const∫0∞∫02πcos(2πrρcos(θ-ϕ))××P(r,ϕ)W(r,ϕ)rdrdϕ, (5)

where W(r,ϕ) is the window which excludes part of P(r,ϕ) beyond the cut-off frequency of the telescope f_T and for frequencies lower than the atmospheric cutoff f_A. Also, taking in to account central symmetry of P(r,ϕ) equation 5 can be rewritten as:

ACFp(ρ,θ)=const∫fAfT∫0πcos(2πrρcos(θ-ϕ))××P(r,ϕ)rdrdϕ. (6)

One example of ACF_p is shown in Figure 4 The position of the maximum gives us ρ and θ which determine the position of the component in the coordinates of the detector.

Fig.4. The ACF in polar coordinates for WDS20312+1116. The separation is 0.´´3. 

Now let us see if ACF_p allows us to find astrometric parameters when the power spectrum is distorted by vibrations and strong aberrations of the telescope. As shown in Figure 5 (left) the power spectrum loses the high frequencies in the vertical direction. However, the high resolution ACF_p has a strong maximum (Figure 5, right). The precision of determining astrometric parameters of the binary system depends on the accuracy with which we can determine the coordinates of the maximum of the discrete function ACF_p. Thus, we can recover the astrometric parameters from the distorted power spectrum. Although measurements can be carried out without the speckle interferometric transfer function correction, its use improves their accuracy.

Fig. 5 Power spectrum P(f ) of WDS 14394-0733 distorted by vibrations and aberrations of the telescope (left). ACFp of WDS 14394-0733 obtained from its P(f ) (right). The separation is 0:´´55. 

3.3.180 Degree Ambiguity

The power spectrum has a 180^o ambiguity. To deal with this issue, we used the self-calibrating shift-and-add technique (^{Christou et al. 1986}). The technique allows us to get diffraction-limited images without using any reference star. When the components have similar magnitudes, the result of this technique is similar to the diffraction-limited autocorrelation, as in Figure 6 (left), contrary to the case in which there is a clear difference between the components, as shown in Figure 6 (right). This double star has a difference of one magnitude between components. This technique allows us to overcome the common 180 degree ambiguity, thus obtaining a reconstruction of the close double star system. Then we can obtain the real 𝜃 (position angle) and ρ (separation) by calibration.

Fig. 6 Example of the reconstruction of WDS 20312+1116 (left) and WDS 19326+0435, Δm = 1(right) 

3.4.Calibration

To perform the calibration, we need to find the pixel scale and the position angle offset. There are two common ways to do this. The first one is by observing some binary stars which have known orbits of grade 1 and calculating ephemerides from the orbital elements. The second way is by observing double stars with very slow relative motion of the components. In this case, ephemerides are calculated by linear approximation of the component motion, or by using the last known value of ρ and θ if there is no evidence of motion over more than 20 years. Most suitable for this method are optical doubles with slow proper motion. This method is preferable to the first one, because the accuracy of speckle interferometric measurements with 2-meter telescopes exceeds the accuracy of even the best orbits (^{Tokovinin et al. 2015}).

For the astrometric calibration, we selected 21 systems with a separation ranging from 4´´ to 6´´ which had more than one reliable observation from the Fourth Catalog of Interferometric Measurements of Binary Stars and from the WDS catalog. These 21 systems also have very slow movements and a long time base of observations. A comparison with our data (Figure 7) gives us the following offset for the position angle θ₀ =-0.42^o ± 0.14^o and a pixel scale s=0´´.0326 ± 0´´.00009 per pixel.

Fig. 7 Calibration 

4.Astrometric Measurements

The astrometric measurements we obtained for double stars are displayed in four tables (Tables 1-4). Table 1 presents astrometric measurements of 21 double stars used for calibration. All these systems show slow motions of components. The first column contains the epoch-2000 coordinates in the format used in the WDS Catalog (^{Worley & Douglass 1997}). The second column gives the official binary star discoverer designation. The third column gives the epoch of the observation in fractional Julian years. The fourth column indicates the filter used. The two following columns contain the measured position angles given in degrees, with the errors of their determination, and the angular separation in arcseconds, with the errors of its determination.

The astrometric measurements of close double stars without known orbits are displayed in Table 2. The symbol (*) indicates that this system was previously discovered but never confirmed. We confirm these systems as double stars. However, for many of them, the current position of the component is far away from the one reported previously. Therefore, it is uncertain to determine whether it is a confirmation or a new pair. The second column gives the official binary star discoverer designation. The last four columns give the position angle θ (Column 5) with its error σθ (Column 6) in degrees, and the angular separation ρ (Column 7) with its error σρ (Column 8) in arcseconds.

Furthermore, we have observed 65 close binary stars with known orbits from the Sixth Catalog of Orbits of Visual Binary Stars (OC6) (^{Hartkopf et al. 2001}). The astrometric measurements are displayed in Table 3. The first 8 columns are the same as in Table 2. The last three columns give the difference between our measurements and the ephemeris calculated for the date of observation, as well as references in the format of OC6. The orbital elements and the complete list of references may be found in the current electronic version of OC6: http://ad.usno.navy.mil/wds/orb6.html.

The last Table 4 displays the astrometric parameters of three new close double stars with separation less than one arcsecond.

The astrometric results include errors arising in the process of recovering the component positions from the power spectrum. In addition, the position angle measure (θ) can have a systematic error of 0.14^o and the separation measure (p) has an additional error pertaining to the pixel scale.

5.Conclusions

We present results of double star speckle interferometric observations focused on close binaries from the WDS catalog. We present the astrometric results for 468 resolved stars. We confirm 59 stars as doubles.

For astrometric measurements, we calculate the high resolution autocorrelation function in polar coordinates. It allows one to perform astrometric measurements even for a distorted power spectrum. The coordinates of the global maximum of ACF_p corresponds to the ρ and θ of the component. The measurements can be carried out without a speckle interferometric transfer function correction, because we exclude atmospheric distortion by using the window W(r,φ). Finally, the self-calibrating shift-and-add technique solves the 180 degree ambiguity.