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Special Astrophysical Observatory, Nizhnij Arkhyz, 369167 Russia pea@sao.ru

Magnitude Magnitude Differences measurements for speckle Differences measurements for speckle Inter ferometric binaries binaries Inter ferometric E. A. Pluzhnik

A method of magnitude differences measurements for speckle interferometric binary stars is presented. The method is based on standard power spectrum analysis of speckle series without correction speckle interferometric transfer function. Both accuracy and m m sources of systematical errors are analyzed. Photometrical accuracy range within between 0. 02 and 0. 1, depending on the seeing, separation and brightness of the components.
Introduction: Binary stars study is the most useful direct way to connect stellar theoretical models with the actual observational results. At present speckle interferometry become the main method for accurate astrometry of binary and multiple stars (Hartkopf et al. 2001). Unfortunately, high-accuracy photometry of the individual components using this method still remains unsolvable problem (Worley et al. 2001). Among more than 70000 measurements of magnitude differences for binaries components only 676 ones were made with different interferometric techniques (Mason & Wycoff m m 2003). Accuracy of such estimations ranges within 0. 1 and 0. 5, m where it is worse than 0. 2 for most of them. Furthermore, other techniques were and still not able to overcome the problem, espeІ cially for separations smaller than 0. 3. In this poster we describe a new method for determining the magnitude differences, bases on standard power spectrum analysis of speckle series. Measurements of magnitude differences: Determination of magnitude differences Dm by interferometric methods leads to measurements of either peak amplitude ratio of the object autocorrelation function, or fringes contrast of the mean object power spectrum (visibility function). The mean power spectrum of speckle interferometric frames can be expressed as: <|I(n)|2> = |O(n)|2 <|S(n)|2> + N(n) ............................(1) where n is the spatial frequency vector, O(n) are Fourier trans2 forms of the object intensity distribution, <|S(n)| > is the speckle interferometric transfer function (STF), and N(n) represents the mean power spectrum of the noise events. Photon noise Np(n) = Cpnp(n) (Goodman & Belsher 1976) and detector noise Nd(n) predominantly contribute to the function N(n) = Np(n) + Nd(n). For modern photon counting devices the effect of the detector noise is negligible in comparison with the photon bias term and Dm measurements are limited mainly by the effect of the photon bias on the power spectrum estimations. The normalized photon bias np(n) depends on the shape of photon events. It can be easily determined as a normalized power spectrum of the "flat field" frames. Photon bias amplitude Cp can be obtained from the power spectrum beyond the telescope cut-off frequency, where the signal is equal to zero. The main problems of deriving Dm are: 1- The required accuracy of the approximation for the photon bias amplitude is a fraction of a percent, whereas the usual accuracy is about several percents. 2- The function np(n), which derived from the "flat field" frames do not vary appreciably in the power spectrum of speckle interferometric frames, due to some registration nonlinearity for example. 3- The deconvolution is known to be a non-trivial procedure. Binary stars, Circular symmetric STF: The photon bias changes the contrast of the power spectrum fringes and affect Dm estimation. Let us assume, that the STF is circular symmetric. In this case, we may select the annular area near spatial frequency n, which is such narrow, that the STF <|S(n)|2> may be considered to be constant. If the value of the amplitude Cp is fixed, both astrometric and photometric solution can be obtained in the annular area by a least squares fitting with the model function ............................(2) F2(n) = a + b Cos(2pnr) where a and b represent unknown constants, and r is also an unknown vector of the system separation. Weighted mean values of the positional parameters r and q, derived from different annular areas, can be used in the successive Dm determination. Let us determine a contrast function as ............................(3) C(a,n) = 2a/b if the resolution of the detector exceeds the telescope resolution limit, then the photon bias term decreases very slowly in compari2 son with |O(n)| . So, dependence of fringe contrast (magnitude difference) on the annular area radius arises, when the amplitude Cp occurs to differ from its true value (Figure 1). To eliminate such dependence, we should select amplitude Cp under the condition dC(a,b)/dn = 0. This condition must be true in the appropriate range of spatial frequencies, excluding both atmospheric seeing and noisy data influence. The derivative of dC(a,b)/dn forms a slope of the first order weighted least squares fitting for C(n) = Co + C1n dependence. The weights of the measurements are selected according to the relative rms of the coefficient b. Intensity ratio of the components A/B can be obtained, using 2 2 .........................(4) (A +B )/AB = A/B + B/A = Co, when C1 is equal to 0, and .........................(5) Dm = mA - mB = -2.5 log(A/B), respectively. The error of the magnitude difference sDm can be obtained from sCo in a conventional way. Binary stars, Noncircular STF: It is worthy to note that we did not use circular symmetry of the STF, but only demanded it to be constant within areas selected for fitting. That is why, t he above f ormalism i s applicable for any STF, Replacing annular areas with areas, where the STF is constant. Elliptical transfer functions, which are constant within annular areas bounded by ellipces, appear to be rather a good approximation for most of the cases. Note that ellipticity causes some oscillations in the contrast function, obtained from the circular annular areas (Figure 2). Paramete rs of areas where the STF is const ant (CTF A, const ant transfer function Let SW(n) and SB(n) be the power spectra of the speckle series {I(r)WW(r)} and {I(r)WB(r)} respectively. It is easy to understand that the weighte of sum S(n) = w1SW(n) + w2[SW(n) - SB(n)] keeps fringes contrast (autocorrelation peaks ratio) unbiased when w1=NA 2 2 2 2 A +NB B , w2=N'A A +N'B B , A and B are middle intensities of the primary and secondary speckles, NA and NB are numbers of the primary and secondary speckles in the BA contributing to the secondary autocorrelation peaks, and N'A and N'B are numbers of the speckles in the BA not contributing to the secondary peaks. Ratio of weights w1 and w2 may be roughly estimated as: w1/w2=РI dr / РI< I(r)> dr I Fig. 4: sDm distribution during where is the average 1998, 1999 obser vations. image, I and II are two areas near the f rame' s boundary separated by vector r (Figure 6). Determined power spectra S(n) such a way can be used to obtain unbiased Dm values. In Figure 5,b we present our Dm545, which is c o rrected with described algorithm together with the literature Fig. 5: Dm1998 versus Dm1999 (right), and data. Least square c fitting of the relation Dm 545 versus literature data (Dmlit). between corrected Dm545 and HIPPARCOS Dm yields D m H =-0.03[ ± 0.03] + 1.2[±0.03] Dmc545 taking into account both our data and HIPPARCOS data errors as well as the difference between spectral bands, and supposing that consistency c orrected and HIPPARCOS data are excellent. Summary : A new method for magnitude differences measurements, based on a c ommon o rder power spectrum estimations, was developed. The method provide accurate Fig. 6: Definitions of frame areas. photon bias correction procedure, which is necessary to obtain precise parameters of speckle interferometric binaries and multiple stars brighter than m 12 . m m Measurements errors lie between 0. 02 and 0. 15, depending on atmospherical seeing, brightness, separation and magnitude difference of the system. Mean value of magnitude difference errors, based on measurements by Balega et al. (2002) and Balega et al. m (2003), was about 0. 06. There is no need to correct the speckle interferometric transfer function by a deconvolution procedure with the method. Examination of our data obtained during different observational sets and their comparison with Dm from the literature demonstrates the high self-consistancy and reliability of the method. Acknowledgment: I a m g rateful t o V. Vasyuk f rom t he Special Astrophysical Observatory - Russia, who participated in software development at the first stages of the project. I would like to thank Dr. V. Tsvetkova from Institute o f Astronomy - K harkov National University (Ukraine), Prof. Yu . B alega from the Special Astrophysical Observatory - Russia for their usefull notes and Dr. M. Al-Wardat (Jordan) for his help. I'd like also to thank Prof. A. Ghez from UCLA for her support. This work has been supported by the Russian Foundation for Basic Research through grant No. 01-0216563a. References: !Bagnuolo, W.G., Jr., Mason, B.D., Barry, D.J., Hartkopf, W.I., & McAlister, H.A. 1992, AJ, 103, 1399. !Balega, I.I., Balega, Y. Y. , Hofmann, K.-H., Maksimov, A.F. , Pluzhnik, E.A., Schertl, D., Shkhagosheva, Z.U., & Weigelt, G. 2002, A&A, 385, 87. !Balega, I.I., Balega, Y. Y. , Hofmann, K.-H., Maksimov, A.F. , Pluzhnik, E.A.,Schertl, D., Shkhagosheva, Z.U., & Weigelt, G., 2003 in preparetion. !ESA1997, Hipparcos Catalog, 1997, ESA SP-1200, V. 10 !Fabricius, C., & Makarov, V. V. 2000, A&A, 356, 141. !Goodman, G.W., Belsher, J.F. 1976, Proc. SPIE, 75, 141. !Hartkopf, W.I., McAlister, H.A., & Mason, B.D. 2001, AJ, 122, 3480. !Mason, B.D., & Wycoff G.L. The Second Photometric Magnitude Difference Catalog, http://ad.usno.navy.mil/wds/dm2.html

Abstract:

Institute of Astronomy of Kharkov National University, Sumskaya 35, Kharkov, 61022 Ukraine

Fig. 1: Constant functions versus spatial frequency n for m m binaries HIP114922 (r=0."107, V=11. 3, Dm=0. 16 m 1 0 2 ±0. 09), photon bias amplitudes Cp area), and vector of the system separation can be estimated simultaneously by an iteration process. A condition to determine such areas is that the correlation coefficient between the power spectrum in this area and Cos(2pnr0) k = [|O(n)|2 <|S(n)|2> Cos(2pnr0) ]/spsm .........................(6) where r0 , which is a previous estimation of r, must reach the maximum. The quantities sp and sm are rms errors of power spectrum and Cos (2pnr 0 ) in the a rea, respectively . Such approach provides quite reliable results until r and Dm values are uncorrelated

Fig. 2: Noncircular OTF influence for contrast function measurements. A power spectrum (left) and the contrast m function of a binar y star (Dm=0. 753) model (right) are presented. Contrast function was determined assuming both cilcular (dashed) and elliptical (line) STF. The cut-off limit nc is shown (dashed circle).
(r > 2l/D, D is telescope diameter, l is wave length). Otherwise, correlation between areas parameters and r arises, and the algorithm becomes inapplicable. A reference star is needed to determine CTFA in this case. Either a single star or rather a wide binary can be used as a reference one. The contrast function for the binary HR 233 is presented in Fig.1 as an example of this case. Statistics and accuracy of Dm measurements: We used the above method to measure multiple stars parameters during our speckle interferometric observations in 1998 and 1999 (Balega et al. 2002, 2003). The program of the stars lie within magІ І m m nitudes between 2 and13 , separations between 0. 016 and 2 m m and magnitude differences between 0 and 3. 7(Figure 3, up). m As a result, 251 measurements for Dm have been made with 0. 02 m to 0. 15 uncertainties, depending on system's brightness, Dm, separation (Figure 3, down) and atmospheric seeing. Accuracy distribution for the all stars are presented in Figure 4. Median value of the m accuracy is about 0. 06. Initial consistency of the measurements have been tested by comparing Dm1998 and Dm1999, obtained during observations on 1998 and 1999 (Figure 5, left). The statistic analysis confirms a high self-consistency of our measurements and validity of the measurements precision with 47% and 60% importance level respectively. Comparison with others measurements: Reliability of the data was examined also by comparing of our results with the literature data. The results (Fig. 5) clearly show that a bias about 0.08 exists between the speckle interferometric and HIPPARCOS measurements. This is mainly due to the speckle interferometric limited field of view. Let us split a frame into some areas, which are defined by Figure 6, and let us define window functions Wi as: Wi(r)=1 inside the i-th area and Wi(r)=0 outside the i-th area.

Fig. 3: Dm and sDm distributions duringobser vations on 1998, 1999 (Balega et al. 2002, 2003).