Interferometric Techniques for Binary Stars

**Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** c **Copyright Year** Astronomical Society of the Pacific Interferometric Techniques for Binary Stars Brian Kloppenborg 1 1 Department of Physics and Astronomy, University of Denver, 2112 East Wesley Ave., Denver, CO 80208 Abstract. Binary stars are important astrophysical laboratories that permit the determination of fundamental parameters. The masses and radii of the components can be derived if a good understanding of the orbits are known. In the best case, even a distance may be determined if results from spectroscopic and astrometric orbits are combined. Single line spectroscopic binaries are an exception to this statement, because the secondary is not detected. In this work we review traditional orbital solution routines and propose a method to find the undetermined semi-major axis, and therefore the mass ratio, for single line eclipsing spectroscopic binaries using interferometric techniques. 1. Types of Binary Stars The plethora of subtypes of binary stars is staggering. They range from quiescent eclipsing binary systems like Algol to cataclysmic variables which exhibit multiple explosive outburst throughout their lifetime. In the context of this review we consider all binary stars as members of four groups: Visual binaries: These systems are observable as binaries by eye, binocular, or a telescope. Their orbits may be traced out by traditional astrometric means (i.e. van de Kamp 1981). Spectroscopic binaries: These systems show the Doppler effect in its emitted light. There are two classes. SB1 systems show signs of a single component undergoing Doppler shift. This is often due to an unseen companion. SB2 systems show two sets of Doppler shifted lines. Eclipsing binaries: are systems whose orbital plane is in the line of sight. They feature periodic dimming due to obscuration caused by one or both of the components. Many of the most important physical parameters can be derived for these systems (Russell 1912a,b) using only light curve data. Astrometric binaries: these are systems where the orbital motion of one component can be observed, but no companion is visible. Analyzing and reducing data for each of the above categories is beyond the scope of this publication. Instead we will focus on how to determine orbital parameters once the data is reduced. 1

2 Kloppenborg 2. Deriving Orbital Parameters The orbital parameters that may be obtained depend on the type of data available. For astrometric data one may derive the position angle of the ascending node (Ω), the inclination of the orbital plane relative to the plane of the sky (i), the argument of periastron (ω), the angular orbital semi-major axis (α), the eccentricity (e), the time of periastron passage (τ), and the orbital period (T). If wide-field (i.e. greater than a 2 x2 field of view) astrometric data is available several additional parameters relating to the motion of the system may also be determined. Radial velocity (RV) data from spectroscopy yields ω, e, T, τ, plus the system velocity (γ), and the projected semi-major axis (a sin(i)). By combining the RV and spectroscopic solutions one may also determine a distance for the system via. d = a/ tan(α). 2.1. Astrometric/Interferometric Orbits These data would consist of a series of positions (relative or absolute) as a function of time. If the data are taken relative to other star systems (i.e. long-focus photographic astrometry), then the data likely includes proper motion, parallax, or other spurious accelerations. In this case, these effects must be fit and removed during the reduction process. Normally this procedure is undertaken in planar coordinates (i.e. the gnomonic projection of spherical coordinates onto the plane of the photographic plate or CCD) although in principle it can be done in spherical coordinates too. Following convention if we let (X, Y) be the planar position of a star relative to some background, then X = c x + µ x t + µ x t 2 + πp α + ORBIT x (Ω, ω, α, e, i, T, τ) (1) Y = c x + µ y t + µ y t 2 + πp δ + ORBIT y (Ω, ω, α, e, i, T, τ) (2) where c x and c y are zero point offsets, µ x and µ y are proper motion, µ x and µ y are spurious acceleration terms, π is the parallax, P α and P δ are parallax factors (see van de Kamp 1981, for further discussion), and ORBIT may be found using one of the methods described below. 2.1.1. The Thiele-Innes Method A very eloquent approach for determining the orbital parameters is the Thiele-Innes method (see any spherical astronomy textbook like Smart & Green 1977; Green 1985; Eichhorn 1974). This makes use of projections of the orbital ellipse onto an auxiliary ellipse or Keplerian circle. Let ( X, Y, Z) be the residuals from the application of Equation 2 (i.e. only the orbital motion about some reference point), x = cos(e) e and y = 1 e 2 sin(e) be the elliptical rectangular coordinates in a unit circle orbit, and E the eccentric anomaly, then we may find the value of the Thiele-Innes (A, B, F, and G) constants by: X = Bx + Gy (3) Y = Ax + Fy (4) A, B, F, and G are related to the orbital parameters by successive application of the rotation matrix. Their algebraic expressions are

Binary Stars: Astrometric and Interferometric Techniques 3 B = a(cos(ω) sin(ω) + sin(ω) cos(ω) cos(i)) (5) A = a(cos(ω) cos(ω) sin(ω) sin(ω) cos(i)) (6) G = a( sin(ω) sin(ω) + cos(ω) cos(ω) cos(i)) (7) F = a( sin(ω) cos(ω) cos(ω) sin(ω) cos(i)) (8) where a, ω, Ω, and i have their traditional meaning. These constants are of a very physical nature. (B, A) is the projection of the rectangular equatorial coordinates of periastron onto the auxiliary circle, whereas (G, F) is the position where the eccentric anomaly, E, is 90 (see van de Kamp (1981) or Green (1985) for treatment of these elements in more detail). With a little algebra, the orbital parameters may be obtained from tan(ω + Ω) = (B F)/(A + G) (9) tan(ω Ω) = ( B F)/(A G) (10) a 2 ( 1 + cos 2 (i) ) = A 2 + B 2 + F 2 + G 2 (11) cos(i) = (AG BF)/a 2 (12) where ω and Ω have ±180 ambiguity that may be broken using a spectroscopic orbital solution. 2.1.2. Direct Orbit Fitting At times it is more convenient to work with the full orbital equations rather than an intermediate solution method, particularly when translational velocity information is available. Following the convention for orbital equations found in Roy (2005), the ( X, Y, Z) position of a star in its orbit at time t is: X = a (L 1 cos(e) + βl 2 sin(e) el 1 ) (13) Y = a (M 1 cos(e) + βm 2 sin(e) em 1 ) (14) Z = a (N 1 cos(e) + βn 2 sin(e) en 1 ) (15) where E E(t) is solved via. Kepler s equation and the remaining constants are: n = 2π/T (16) L 1 = cos(ω) cos(ω) sin(ω) sin(ω) cos(i) (17) M 1 = sin(ω) cos(ω) + cos(ω) sin(ω) cos(i) (18) N 1 = sin(ω) sin(i) (19) L 2 = cos(ω) sin(ω) sin(ω) cos(ω) cos(i) (20) M 2 = sin(ω) sin(ω) + cos(ω) cos(ω) cos(i) (21) N 2 = cos(ω) sin(i) (22) and in the case of elliptical orbits: β = 1 e 2. (23)

4 Kloppenborg 2.2. Spectroscopic Orbits Spectroscopic orbits are found by measuring the heliocentric radial velocity of specific spectral lines as a function of time. Frequently the spectral lines are blended, therefore it is necessary to use spectral disentanglement methods (e.g. KOREL, Hadrava 2004) to measure the RV of individual lines. The resulting radial velocities are fit to the radial velocity equation: d dt ( Z 1) = γ + n a 1 sin(i) [ ] 1 e 1 e cos(e) 2 cos(ω) cos(e) sin(ω) [ ] = γ + K 1 1 e 2 cos(ω) cos(e) sin(ω) (24) (25) where γ is the system velocity and the other parameters have traditional meaning. For the second star in system a 1 and K 1 are replaced with a 2 and K 2 to account for differing semi-major axes and the opposite relative motion of the two components. 3. Interferometry and Multiple Star Systems A simple model of a non-interacting, non-eclipsing resolvable binary system may be derived from the superposition of uniform disks of various brightness. Recalling the derivation in Lawson (2000), the normalized visibility of a uniform disk of radius θ centered at (α, β) V(B, λ, θ) = e 2πi(uα+vβ) 2J 1 (πθb /λ) πθb /λ where J 1 is the first-order Bessel function, B is the projected baseline length, and λ is the wavelength of observation. Because of noise considerations, normally the squared visibility, (26) ( ) 2 V 2 2J1 (πθb /λ) (B, λ, θ) = (27) πθb /λ is used. In a multiple-star system the wavefronts coming from the individual components are uncorrelated, therefore we may apply the superposition of Fourier transforms to yield the visibility of a multi-source system: V = V k = k j 2J 1 (πθ k B /λ) P k e 2πi(uα k+vβ k ) πθ k B /λ where P k is the flux from component k. The normalized visibility for such a system is simply V = P j V j j P where P is the total flux. For a binary system with flux ratio r = P 2 /P 2 with relative source coordinates α α 2 α1, β β 2 β 1 the normalized squared visibility is (28) V 2 = V2 1 + r2 V 2 2 + 2r V 1 V 2 cos(2π B s/λ) (1 + r) 2 (29)

Binary Stars: Astrometric and Interferometric Techniques 5 where B is the baseline and s = ( α, β) is the binary separation vector. Clearly one may fit the interferometric data to this simple model to derive positions necessary for the orbital equations listed above. 4. Interferometric Methods for SB1s If both components are visible or resolvable, the methods described above can be directly applied and all of the orbital information can be extracted given enough time and sufficiently robust decorrelation methods. For binaries where only one components is resolved the above methods cannot be so easily applied. Below we detail existing and new methods for improving the orbital elements of SB1s. 4.1. Resolving the Companion The easiest approach is to simply resolve out the companion using either higher resolution techniques or methods that enhance the contrast ratio. For example, NPOI recently resolved the 2.9 mag fainter companion of HD 4180 in V-band (Koubský et al. 2010). Much longer baselines like those found at CHARA have resolved even more SB1 systems (Taylor et al. 2003). If the component separation is sufficiently small, or the flux ratio too great, all hope is not lost. One can use traditional nulling techniques (i.e. visibility or closure phase, Danchi et al. 2006) or phase closure nulling. The latter relies models of the closure phase around visibility nulls in the primary component. Any deviations from normal closure phase behavior in these regions is treated as signal from the companion. This technique was recently applied by (Duvert et al. 2009) using AMBER/VLTI to detect the companion of HD 59717 which was found to be 5 magnitudes fainter in K, at a distance of 4 stellar radii from the primary. Yet another method for resolving the companion comes using high resolution (R 30, 000) spectro-interferometry. Here the missing secondary could be detected in a deep absorption line seen in the primary s spectrum. If such a signal were detected, it would manifest as an increased visibility and a shift in differential phase that implies there is something not centered on the photocenter of the primary. 4.2. Observe an Eclipse If the companion emits little to no radiation detection by any technique may be impossible. If the observer is lucky enough to have a system which undergoes periodic eclipses much of the orbital information may be recovered. For example, consider the epsilon (ɛ) Aurigae system which has recently undergone an eclipse. Kloppenborg et al. (2010) found the eclipsing object moved 0.62 ± 0.14 mas west and 0.34 ± 0.06 mas north in 30 days. If a distance were accurately known, then the angular velocities could be converted to linear and directly used in the velocity equations below (Roy 2005) to yield a solution for the semi-major axis of the binary orbit. d na ( X) = dt η (βl 2 cos(e) L 1 sin(e)) (30) d na ( Y) = dt η (βm 2 cos(e) M 1 sin(e)) (31)

6 Kloppenborg where η = 1 e cos(e). If a 1 sin(i) is known from spectroscopy or a 1 = d tan(α 1 ) the unknown semi-major axis, a 2 is easily algebraically obtained when the orbital elements are found via. a minimization algorithm. 4.3. Extension to planets Exoplanets are intrinsically dim when compared with their host star. Burrows et al. (2008) estimates this ratio to be 10 5 10 6. Until a beam combining technology is developed that can deliver this sort of precision, direct detection of exoplanets using typical interferometric techniques may be difficult. The use of nulling beam combiners can significantly improve the contrast ratio, therefore increasing the likelihood of detection. The use of closure phase can also be a powerful tool in determining the orbits of exoplanets in that it is highly sensitive to small asymmetries in the source. To detect planets however, accurate limb darkening laws must be adopted and the host-star well characterized for surface anomalies such as spots or large convective cells. In both cases with high resolution spectro-interferometry a change in differential visibilities and closure phase could be detected, akin to the well-known Rossiter-McLaughlin effect (Rossiter 1924; McLaughlin 1924). Acknowledgments. The author is grateful for discussions with Gail Schaefer, Paul Hemenway and Robert Stencel; financial support for this work was provided in part under NSF grant 10-16678, and the bequest of William Hershel Womble in support of astronomy at the University of Denver. References Burrows, A., Budaj, J., & Hubeny, I. 2008, The Astrophysical Journal, 678, 1436 Danchi, W. C., Rajagopal, J., Kuchner, M., Richardson, L. J., & Deming, D. 2006, The Astrophysical Journal, 645, 1554 Duvert, G., Chelli, A., Malbet, F., & Kern, P. 2009, Astronomy and Astrophysics, 509, A66 Eichhorn, H. 1974, Astronomy of star positions - A critical investigation of star catalogues, the methods of their construction and their purpose (Frederick Ungar Publishing Co. New York) Green, R. M. 1985, Spherical astronomy (Cambridge and New York) Hadrava, P. 2004, In Spectroscopically and Spatially Resolving the Components of the Close Binary Stars, 318, 86 Kloppenborg, B., Stencel, R., Monnier, J. D., Schaefer, G., Zhao, M., Baron, F., McAlister, H., Ten Brummelaar, T., Che, X., Farrington, C., Pedretti, E., Sallave-Goldfinger, P. J., Sturmann, J., Sturmann, L., Thureau, N., Turner, N., & Carroll, S. M. 2010, Nature, 464, 870 Koubský, P., Hummel, C., Harmanec, P., Yang, S., Bozic, H., Tycner, C., & Zavala, R. 2010, The Interferometric View on Hot Stars (Eds. Th. Rivinius & M. Curé) Revista Mexicana de Astronomía y Astrofísica (Serie de Conferencias) Vol. 38, 38, 87 Lawson, P. R. 2000, Principles of Long Baseline Stellar Interferometry, -1 McLaughlin, D. B. 1924, The Astrophysical Journal, 60, 22 Rossiter, R. A. 1924, The Astrophysical Journal, 60, 15 Roy, A. E. 2005, Orbital motion (Institute of Physics Publishing) Russell, H. N. 1912a, The Astrophysical Journal, 35, 315 1912b, The Astrophysical Journal, 36, 54 Smart, W. M., & Green, R. M. 1977, Textbook on Spherical Astronomy (Cambridge University Press)

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