PLOTTING ORBITS OF BINARY STARS FROM THE INTERFEROMETRIC DATA

Transcription

1 PLOTTING ORBITS OF BINARY STARS FROM THE INTERFEROMETRIC DATA by Driss Takir University of North Dakota Visiting Research Intern at the Indian Institute of Astrophysics A Report Submitted to the Indian Institute of Astrophysics August 2008 Prof. S.K. Saha Indian Institute of Astrophysics Bangalore India

2 TABLE OF CONTENTS LIST OF FIGURES iii LIST OF TABLES.iv AKNOWLEDGEMENT.v ABSTRACT...vi CHAPTER I. INTRODUCTION..1 Speckle Phenomena 2 Speckle Formation. 4 Speckle Imaging.6 II. SPECKLE INTERFEROMETRY APPLICATIONS: BINARY STARS..8 Binary Stars...8 Plotting Orbits of Binary Stars Algorithms.11 III. RESULTS OF THE COMPARATIVE STUDY Plotting Orbits of Binary Systems..14 Conclusion and Future Work..24 REFERENCES.25 ii

3 LIST OF FIGURES Figure Page 1. Speckles obtained by A. Labeyrie by observing Vega using the 5m Mont Palomar A short-exposure image of a binary star, Zeta Bootis, as seen through atmospheric seeing 3 3. The airy diffraction pattern A specklegram of HD Layout of speckle interferometer Part of the celestial sphere where A is the primary and B is the companion.9 7. Apparent orbit of a binary star Apparent orbit of HD761 generated by the IIA s program Radial velocity curves of HD781 generated by IIA s program Mean anomaly in radian vs. the epoch The Apparent orbit of HD26690 generated by IIA s program Radial velocity curves of HD generated by IIA s program Apparent orbit of HD generated by IIA s program Radial velocity curves of HD generated by IIA s algorithm The Apparent orbit of HD generated by IIA s program Radial velocity curves of HD generated by IIA s algorithm...23 iii

4 LIST OF TABLE Table Page 1. Orbital parameters of HD Orbital parameters of HD Orbital parameters of HD Orbital parameters of HD iv

5 ACKNOWLEDGEMENTS My sincere appreciation and thanks go to my mentor, Prof. S.K. Saha, Prof. A.V. Raveendran, and to all faculty members, students, and staff of the Indian Institute of Astrophysics for their support and their genuine hospitality. I also would like to thank Ms. Libby Pertrick and Dr. Kiran Jain at the National Solar Observatory for making all the necessary arrangements for our trip to India. This research is supported by NSF Grant Number OISE [NSO]. v

6 ABSTRACT The effect of atmospheric turbulence on the diffraction-limited imaging of celestial bodies is one of the major problems in observational astronomy. The speckle interferometric technique was introduced in the 1970s to solve this problem. The technique is used to decode the diffraction-limited spatial Fourier spectrum and image features of the celestial objects, using a series of short-exposure (< 20 ms) images. Since most common binary orbit periods vary from 10 to 30 years, a large number of these binary systems, studied using the speckle data, completed one or two revolutions. In this study, an algorithm developed by the Indian Institute of Astrophysics (IIA) and an algorithm developed by Hartkopf s group at Georgia State University were used to plot the orbits of four binary systems, =Bu 1026AB, WDS = A1938, WDS = Mir 4, and =Ho 276. The orbital parameters of these binary systems were calculated using speckle data and other interferometric data. The former algorithm is based on standard least square technique with iterative improvement of the orbital parameters. Unlike the latter algorithm, the former algorithm does not require any previous knowledge of the period and the eccentricity of the binary systems. The results of this comparative study have shown that both algorithms generate almost the same orbital parameters. However, the algorithm developed by the IIA requires fewer steps to calculate the orbital parameters of these binary systems. vi

7 CHAPTER I INTRODUCTION One of the major problems in observational astronomy is atmospheric seeing, blurring and twinkling of celestial objects. The light from these objects is significantly affected by the micro-thermal fluctuations in the atmosphere, which is highly turbulent and optically inhomogeneous. The resolution of the images of these objects is reduced by a factor of 20 due to the effects of atmospheric seeing (Saha 2000). To reduce the effects of atmospheric seeing, Labeyrie (1970) introduced a new observational technique, speckle interferometry, which is based on taking short exposure (less than 20 ms) images, specklegrams. Speckle interferometry freezes the atmosphere, but produces an instantaneously distorted image. The specklegrams produced are then processed using Fourier-domain methods, which allow regaining the true diffractionlimited image of the object. In this study, orbits of four binary systems, =Bu 1026AB, WDS = A1938, WDS = Mir 4, and =Ho 276, were plotted using a computer program, developed by the Indian Institute of Astrophysics (IIA). The orbital parameters of these binary stars were compared to the orbital parameters generated by another computer program developed by Hartkopf et al. (1989, 1996) at Georgia State University. Chapter 1 is an overview of speckle phenomena, speckle formation, and speckle imaging. Chapter 2 describes the algorithm developed by Hartkopf s group and the algorithm developed by the IIA to plot orbits of binary systems and calculate their orbital 1

8 parameters. Chapter 3 presents the results of the comparison of the orbital parameters of these binary systems generated by both programs. Speckle Phenomena Speckle refers to the grainy structure observed when a laser beam is reflected from a diffusing surface. Speckle is a result of the interference effects in a coherent beam with random spatial phase fluctuations. The Speckle phenomenon is observed when an optically uneven surface of an object is illuminated by a fairly coherent source (Figure 1). The speckle grains can be determined with the coherence domains of the Bose-Einstein statistics (Labeyrie 1970). Figure 1. Speckles obtained by A. Labeyrie by observing Vega using the 5m Mont Palomar. This speckle has the same size as the Airy disk given by the telescope. Source: Speckle pattern can also occur in the image of point stars, observed by large telescopes, due to seeing induced phase fluctuations on the wavefront. In the absence of seeing and aberrations, the minimum grain size of the speckle observed is equal to the 2

9 size of the Airy disk given by the telescope. The speckle-affected images contain more information on smaller features than long exposure images with a blurred speckle. The image of a double star, with spacing smaller than the turbulence angle, consists of a superposition of two identical speckle patterns that are shifted by an amount smaller than the image size. This image is difficult to analyze visually (Figure 2). However, its infinity diffraction pattern, generated by a laser beam, shows a set of parallel equispaced fringes whose spatial frequency is proportional to the double star separation. Figure 2. A short-exposure image of a binary star, Zeta Bootis, as seen through atmospheric seeing. The atmospheric seeing causes the images of the two stars to break up into two patterns of speckles. Source: 3

10 Speckle Formation When observing a point source and a continuum of wave components pass through a telescope s aperture, the superposition of these components leads to a pattern of constructive and destructive interference. In telescopes, the incoming light is approximately a plane wave since the source of the light is so far away. The intensity pattern of these constructive and destructive interference rings is known as the Airy diffraction pattern (Figure 3). Figure 3. The airy diffraction pattern. Source: All telescopes have an inherent limitation to their angular resolution due to the diffraction of light at the telescope s aperture.the resolution of a telescope is characterized by the width of the Point Spread Function (PSF), which is the order of, 1.22 / r o 4

11 is the wavelength of light and r o is the average size of the turbulence cell, which is the order of 10 cm. A wave plane propagating through the atmosphere of Earth is distorted by the microstructures of refractive index inhomogeneity, called eddies. This plane wave reaches the pupil of the telescope with random patches of uniform phases. Each patch phase of the plane wave with a diameter r o, known as Fried s parameter or atmospheric coherence diameter, is independent of the rest of the patch phases of the plane wave. Thus, the aperture of the telescope is subdivided into a set of subapertures. The resultant interference patterns produced by all patch phases of the plane wave consists the speckle image of the object. The measurement of the atmospheric coherence diameter is crucial to estimate the seeing at any astronomical site. This diameter can be calculated using speckle interferometric technique and the following equation, = (r o / ) 2 The wave plane coherence area can be calculated by ratioing the area of telescope aperture to the estimated number of speckles. Generally, the effective resolution of a telescope is affected by two factors, the PSF of the atmosphere and the telescope s aperture. If the telescope s diameter, D, is smaller than the atmospheric coherence diameter, r o, the resolution will be the true diffraction limited resolution, 1.22 /D. On the other hand, if D is larger than r o, the resolution of the telescope will be affected by the atmospheric turbulence, which suppresses the telescope diameters. The seeing disc in a large telescope is equal to 1.22 /r o. 5

12 Speckles form when the atmospheric turbulence causes random phase fluctuations of the incoming optical wave plane. When the incoming light gets diffracted, after it reaches the telescope s aperture, linear interference fringes with a fringe width of /d are produced, where d is the distance between two adjacent fringes. The interference of these fringes result in enhanced bright speckles of width /d. Figure 4 illustrates a specklgram of HD91172 obtained from the Vainu Bappu Telescope (VBT) in Kavalur, India. Figure 4. A specklegram of HD Courtesy of the Indian Institute of Astrophysics. Speckle Imaging Speckle Interferometry is a technique that involves the modulus of the Fourier transform of the object intensity distribution from a set of short exposure specklegrams. When the light from a celestial object reaches an optical system (i.e., telescope) traveling through the atmosphere, the instantaneous 2-D distribution of the image intensity I (x,y) is produced by the convolution of the object intensity O (x,y) and the instantaneous point 6

13 spread function S (x,y), I(x,y) = O(x,y) * S(x,y) Speckle Interferometry involves using high quality imaging instruments and detectors. The speckle interferometer utilized by the Indian Institute of Astrophysics is located at the Vainu Bappu Telescope (VBT) in Kavalur near Bangalore. This interferometer records the specklegrams using a narrow band filter centered at H α at the Cassegrain focus of the VBT. This interferometer also uses a diffraction-limited camera that can record magnified (f/100) short exposure images (Figure 5). Figure 5. Layout of speckle interferometer. The wavefront falls on the focal plane of an optical flat, which is made of low expansion glass with a high precision hole of aperture (of the order of 350-µm), at an angle of 15 o. Courtesy of the Indian Institute of Astrophysics. 7

14 CHAPTER II SPECKLE INTERFEROMETRY APPLICATIONS: BINARY STARS Binary Stars Binary stars are systems of two close stars gravitationally bound together and moving around each other. Generally, the two stars of the system have unequal brightness. The brighter star is more massive and called the primary, while the fainter is less massive and called the secondary or the companion. Binary systems can be classified into four types based on the techniques used for their discoveries- visual, spectroscopic, eclipsing, and astrometric binaries. The relative positions of the visual binary stars can be plotted from long-term observations to determine their orbits. Due to their gravitational boundness, the relative positions of these binaries change over the years. The speckle interferometry technique is used to study this type of the binary stars Since its introduction by Labeyrie (1970), almost four decades ago, the speckle interferometry has been widely used by binary star observers (Hartkopf et al. 1989, 1996; Baize 1992; Saha et al. 2002). This technique allows accurate astrometric study of close visual binary stars. It has revolutionized the field of binary star astronomy. The Center for High Angular Resolution Astronomy (CHARA) of the Georgia State University has been a major contributor of the interferometric measurements of thousands of binary stars since it begun by McAlister (1976). Some binary stars data used for this study are from CHARA. The interferometric measurements of binary stars allow deducing the apparent orbit of these binaries, using Kepler s third law, 8

15 4 2 a 3 = G (m 1 + m 2 ) P 2 Where a is the semi-axis of the orbit, G is the gravitational constant, P is the period, and m 1 and m 2 are the masses of the stars. These stars s masses, as well as their orbital parallax, can also be identified using other spectroscopic elements. The position of the companion of a binary system vis-à-vis the primary is determined by two coordinates, the angular separation,, and the position angle, ρ (Figure 6). Figure 6. Part of the celestial sphere where A is the primary and B is the companion. AN defines the direction of the north celestial pole. Due to the mutual gravitational boundness, both stars move around the mass of the system, barycenter. The motion of the secondary star with respect to the primary describes the true elliptic orbit. Using Kepler s laws, the orbital elements of a binary system can be identified. These elements are crucial in determining the masses and the parallax of the individual stars. 9

16 The projection of the true orbit on the plane of the sky, the tangent plane to celestial sphere, is referred to as the apparent orbit. The apparent orbit can be determined using the semi-axis, eccentricity, position angle of the major axis, and the two coordinates of the center of the ellipse with respect to the primary star. Figure 7 illustrates the apparent orbit of a binary star. Figure 7. Apparent orbit of a binary star. S represents the primary star. The general equation of the ellipse of the orbit illustrated in Figure 5 can be expressed by, Ax H xy + B y G x + 2F y + 1 = 0 The above equation has five independent constants, A, H, B, G, and F. If the companion is at C, the angular separation, ρ, and the position angle,, can be determined using the observation measurements. Finding these parameters allows identifying the rectangular coordinates x and y of C according to, x = ρ cos and y = ρ sin 10

17 Theoretically, five observations spread over the orbit are sufficient to determine the five constants, A, H, B, G, and F. However, a large number of observations spread over many years are required to determine accurate orbit. Plotting Orbits of Binary Stars Algorithms Various algorithms have been used to determine the elements of the orbit of a binary system. Hartkopf et al. (1989, 1996) utilized a method based on 3-D grid search technique and visual measurements along with the interferometric data to calculate and plot the orbits of binary system. If the period, P, eccentricity, e, and the time of the periastron 1 passing,, are given, the four Thiele-Innes elements, A, F, B, and G, semimajor axis, a, orbital inclination, i, the longtitude of ascending node, Ω, the argument of periastron passage,, can be determined by the least square method. Given (P, e, ) and a set of observations (t, x i, y i ), the eccentric anomaly E is found using the following equation, M = E e sin E Where M = 2 /P (t - ) is the mean anomaly of the companion at a time t. The normalized rectangular coordinates X i, Y i, are determined by the following equations, X i = cos (E) e Y i = 1 e2 sin E The four Thiele-Innes elements A, F, B, and G are found by a least squares solution of the equations, X = AX i + FY i Y = BX i + GY i 1 Periastron is the point in the orbital motion of a binary star system when the two stars are closest together. The other extremity of the major axis is called apastron. 11

18 The orbital elements are then deduced from these Thiele-Innes elements. However, Hartkopf s method requires a previous knowledge of the period of the binary system. Saha et al. (2007) used another algorithm based on least square method to obtain the plots and orbital calculations. The normal equations are solved using cracovian matrix elimination technique. This algorithm produces results similar to the results produced by the Kowalsky s algorithm, the inversion method, but it involves a fewer numbers of steps. This algorithm is based on minimizing the sum of squares of residual with respect to each constant and obtaining five equations, which can be written using these matrices, The first matrix can be represented by U, the second matrix by V, and the third matrix by W. The three matrices can then be expressed by, UV = W Therefore, the constants, a, h, b, g, f, can be calculated by inverting the Matrix U and using, V = U -1 W The algorithm used by Saha et al. (2007) is the first algorithm to use cracovian matrix elimination technique in an orbital program. This program was written by Dr. A. 12

19 V. Raveendran from the Indian Institute of Astrophysics. The method has a system of giving different weightage to data obtained from different sources. This algorithm eliminates high residues data. 13

20 CHAPTER III RESULTS OF THE COMPARATIVE STUDY Plotting Orbits of Binary Systems In this study, four binary systems orbits were plotted, =Bu 1026AB, WDS = A1938, WDS = Mir 4, and =Ho 276. The data were obtained from the Fourth Catalog of Interferometric Measurements of Binary Stars, which began in 1982 as an internal database at the Georgia State University Center for High Angular Resolution Astronomy (CHARA). The derivation of orbits of binary stars was carried out using an algorithm developed at the Indian Institute of Astrophysics, and discussed in chapter 2. These orbits were compared to the ones obtained by Hartkopf s team (Hartkopf et al. 1989, 1996) =Bu 1026AB (HD761). Speckle data for this system date back to 1975, so they cover about ½ revolution and define the period well. For this system, 36 interferometric measurements, from 1975 to 2000, were used for the orbit calculations. Most of these observations are speckle measurements. The data shows almost uniform variation of ρ and. This system was found to have a period of about 60 years. The system s approaching periastron 2 is around Figure 8 illustrates the apparent orbit of HD761 and Figure 9 represents the radial velocity curves of the same system. 2 Periastron is the point in the orbital motion of a binary star system when the two stars are closest together, while other extremity of the major axis is called apastron. 14

21 Figure 8. The apparent orbit of HD761 generated by IIA s program. 15

22 Figure 9. Radial velocity curves of HD781 generated by IIA s program. In Figure 8, the x and y scales are in arsecs. The dot-dash line denotes the line of nodes. The shaded circle centered in the orbit represents the Rayleigh limit of the telescope. Figure 10 illustrates the plot of the mean anomaly (radian) vs. the epoch 16

23 Figure 10. The mean anomaly in radian vs. the epoch. The orbital parameters of HD761 generated by IIA s program and Hartkopf s program are given in table 1. Table 1. Orbital parameters of HD761. P (year) a (arcsec) i ( o ) Ω( o ) T e ( o ) Hartkopf s Algorithm IIA s Algorithm Both programs generate slightly different orbital parameters for HD761. The slight difference between these results can be explained by the fact that Hartkopf s team used more visual and speckle measurements in their orbits calculations. 17

24 WDS = A1938 (HD26690). The speckle data for this system date back to 1975, so they cover about 3/2 revolution and define the period very well. For this system, 36 interferometric measurements, from 1975 to 1995, were used for the orbit calculations. Most of these observations are speckle measurements. The data shows almost uniform variation of ρ and. This system was found to have a period of about 7.18 years. The system s approaching periastron 3 is around Figure 11 illustrates the apparent orbit of HD26690 and Figure 12 represents the radial velocity curves of the same system. Figure 11. The Apparent orbit of HD26690 generated by IIA s program 3 Periastron is the point in the orbital motion of a binary star system when the two stars are closest together, while other extremity of the major axis is called apastron. 18

25 Figure 12. Radial velocity curves of HD generated by IIA s program. The orbital parameters of HD generated by IIA s program and Hartkopf s program are given in table 2. Table 2. Orbital parameters of HD Both programs generated almost similar results. P (year) a (arcsec) i ( o ) Ω( o ) T E ( o ) Hartkopf s Algorithm IIA s Algorithm WDS = Mir 4 (HD ). This system was discovered by Muller in It remains under 0.2 separation through its 20-year period. For HD222516, 29 interferometric measurements, from 1980 to 2000, were used for the orbit calculations. Most of these observations are speckle measurements. The data shows almost uniform variation of ρ and 19

26 . This system was found to have a period of about years. The system s approaching periastron 4 is around Figure 13 illustrates the apparent orbit of HD and Figure 14 represents the radial velocity curves of the same system. Figure 13. The Apparent orbit of HD generated by IIA s program. 4 Periastron is the point in the orbital motion of a binary star system when the two stars are closest together, while other extremity of the major axis is called apastron. 20

27 Figure 14. Radial velocity curves of HD generated by IIA s algorithm. The orbital parameters of HD generated by IIA s program and Hartkopf s program are given in table 3. results. Table 3. Orbital parameters of HD Both programs generated almost similar P (year) a (arcsec) i ( o ) Ω( o ) T E ( o ) Hartkopf s Algorithm IIA s Algorithm = Ho 276 (HD ). Baize (1992) found the eccentricity of this system is equal to zero and Hartkopf et al. (1996) found it to be very small but nonzero. However, our results show that the eccentricity of this system is For this system, 32 interferometric measurements, from 1961 to 2000, were used for its orbit calculations. Most of these 21

28 observations are speckle measurements. The data shows almost uniform variation of ρ and. This system was found to have a period of about years. The system s approaching periastron 5 is around Figure 15 illustrates the apparent orbit of HD and Figure 16 represents the radial velocity curves of the same system Figure 15. The Apparent orbit of HD generated by IIA s program. 5 Periastron is the point in the orbital motion of a binary star system when the two stars are closest together, while other extremity of the major axis is called apastron. 22

29 Figure 16. Radial velocity curves of HD generated by IIA s algorithm. The orbital parameters of HD generated by IIA s program and Hartkopf s program are given in table 4. results. Table 4. Orbital parameters of HD Both programs generated almost similar P (year) a (arcsec) i ( o ) Ω( o ) T e ( o ) Hartkopf s Algorithm IIA s Algorithm

30 Conclusion and Future Work In this study, an algorithm developed by the IIA and another algorithm developed by Hartkopf s group were used to plot the orbits of four binary systems, =Bu 1026AB, WDS = A1938, WDS = Mir 4, and =Ho 276. The orbital parameters of these binary systems were calculated using speckle data and other interferometric data. Both programs generate slightly different orbital parameters for the four binary systems. This slight difference can be explained by the fact that Hartkopf s team used more visual and speckle measurements in their orbits calculations. More speckle data are crucial to calculate the orbital parameters of binary systems with more accuracy. 24

31 REFERENCES Baize, P.1992, Orbital elements of 17 binary stars, A&AS, 87, 49. Labeyrie, A., Attainment of diffraction limited resolution in large telescopes by Fouriers analysis speckle patterns in star images, Astron & Astrophysics. 6, McAlister, H.A., Spectroscopic binaries as a source for astrometric and speckle interferometric studies, Astron. Soc. Pac, Vol 88, p Hartkopf, W.L., and McAlister, H.A., Binary star orbits from speckle interferometry. II. Combined visual/speckle orbits of 28 close systems. The Astronomical Journal, Vo. 98, Number 3 Hartkopf, W.L., Mason, B.D., and McAlister, H.A., Binary star orbits from speckle interferometry. VIII. Orbits of 37 close visual systems, Astronomical Journal, Volume 111, Number 1. Saha, S.K., Speckle Interferometry. S K Saha, Ind. J. Phys, 73B, Saha, S.K., Speckle Interferometric Observation of Close Binary Stars, Bull. Astron. Soc. India 30. Saha, S.K., 2003, Speckle Imaging: a Boon for Astronomical Observation in `Recent Trends in Astro and Plasma Physics in India', eds. S K Chakrabarty, S B Das, B Basu. 25