AO Imaging of Binary Stars: From Calibration to Orbital ScienceAren N. Heinze (Swarthmore College; aheinze1@swarthmore.edu) and Phil M. Hinz (University of Arizona)

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Orbit 2 - AO Astrometry over 1/4 Period

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Orbit 2 -- Fits to Position Angle

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Abstract: We have obtained observations of a number of binary stars on the 6.5 meter MMT with adaptive optics (AO) in the L’ band, and also the 1.5 meter Kuiper Telescope, without AO, in the U, V, and I bands. Both sets of observations were carried out to obtain precise astrometric calibration for the Clio camera on the MMT telescope, which has been used for several AO planet-imaging surveys, the results of which have been presented elsewhere. Astrometric calibration was needed to determine if candidate planets were common proper motion companions of the parent star. We find that extremely precise astrometry of binaries is possible using AO instruments. We discuss the potential of AO astrometry to determine the masses of binary stars with great accuracy even when observations span 25% or less of the full orbital period. A number of challenges must be met to achieve the full potential accuracy. We explain how these can be addressed. Finally, we present measurements of a set of binary stars that may serve as a beginning for a catalog of wide, long-period binaries useful for precise calibration of AO instruments.

Figure 1: Above left, UVI color image of the 6.16 asec binary HD 100831 from the Kuiper Telescope. Left, image of the same object in L’ using Clio with AO at the MMT. Below, same comparisons for the 7.54 asec binary HR 2176. The enormously higher resolution of the AO images makes it clear why AO astrometry is far more precise.

AO astrometry is powerful! AO astrometry enables new science with binary stars.

Binary Star Simulations: We find that our uncertainties in seeing-limited astrometry with the Kuiper Telescope are about 0.02 asec in separation and 0.3° in position angle for double stars with separations larger than about 3 arcsec. The internal precision of AO astrometry using Clio at the MMT is far better: uncertainties are about 0.0015 asec in separation and 0.007 ° in position angle. We have created simulated binary stars and simulated observations of them using both telescopes. We developed a minimum χ² orbit-fitting program using a modified version of the downhill simplex method described in Numerical Recipes in C by Press et al. We fit 7-parameter orbits to the simulated data using a binary orbit code from Matt Kenworthy (the 7 parameters are the period, date of periastron, semimajor axis, eccentricity, inclination, longitude of periastron, and PA of ascending node). To determine how well the simulated data sets constrained the period and semimajor axis, we performed 6-parameter downhill simplex fits with period or semimajor axis fixed, and determined the bounds of a confidence interval defined by Δχ² < 6, corresponding approximately to a 99% confidence interval assuming Gaussian errors. For our first simulation, we did a blind test in which an accurate orbit was found, and confidence intervals containing the true values used in the simulations were determined, without prior knowledge what the input parameters to the simulation had been.

Figure 2: (Left) We first simulated a binary for which observations covered about half of the full orbital period. The figure shows the AO data lying essentially on the true orbit, and the non-AO data only slightly scattered. Despite the lack of data spanning a full orbit, the period and semimajor axis were well-constrained based on simulated observations with or without AO. For the non-AO observations, the fractional uncertainties on period and semimajor axis were 2.9% and 1.3%, respectively, giving a fractional uncertainty of 7.0% on the total system mass (this assumes a good distance measurement is available so the semimajor axis can be converted from asec to physical units). For the AO observations, the period and semimajor axis were constrained with an accuracy of 0.09% and 0.05%, for a fractional uncertainty of only 0.24% in the system mass -- this based on measurements through only half the orbit!

Figure 3: (Right) Next, we simulated a binary for which observations were made over only about a quarter of a period. Here, the simulated non-AO observations failed to produce useful constraints. Periods ranging over nearly a factor of 3 were statistically permitted, precluding a meaningful mass determination. The figure shows the orbit corresponding to the minimum statistically permitted period (red) and the maximum permitted period (blue).

Figure 4: (Left) The simulated AO data from the second simulation, even though they covered only about one fourth of the orbital period, nevertheless allowed extremely accurate determination of the period (fractional uncertainty 0.27%), semimajor axis (0.21%), and hence the system mass (0.82%). This impressive accuracy was achieved with measurements spanning only a quarter of an orbit – while astrometry of the accuracy typical of good non-AO observations failed to produce any meaningful constraint at all. The figure shows the orbit corresponding to the minimum period permitted by the AO data (red) and the maximum period (blue), but the permitted range is so small that the orbits overlap.

Figure 5: (Above) This figure illustrates how the fits to the non-AO data in the second simulation failed to constrain the orbit. Simulated observations cover the years 2005-2019, while the system period is 55.708 years. Here, position angle vs. time is shown, relative to the ‘true’ orbit input to the simulations. The red and blue curves are the same minimum-period and maximum-period fits shown in Figures 3 and 4. The simulated AO data are not shown to avoid confusion. The non-AO fits match all the data points but then rapidly diverge. The much more precise AO data has forced the AO fits to yield a tolerable match to the whole orbit.

Challenges to precision astrometry: In many astrometry projects, background stars serve as a reference. In most cases these will not be available for AO observations. The telescope itself must be the reference. Plate scale calibrations are relatively straightforward, but position angle calibrations present more challenges.

Challenge 1: Parallactic Rotation: If the telescope image rotator is not used, the images must be digitally de-rotated based on very accurate parallactic angle values referenced to the time at the exact center of each exposure. An error of even a few seconds can bias the results at the precision we consider here – this effect has actually been seen with Clio on the MMT.

Challenge 2: Rotational Calibration: Even if the telescope image rotator is used, or the telescope is an equatorial, the true direction of celestial North on the images must be identified with high accuracy – and this may change slightly every time the instrument is re-mounted on the telescope.

Challenge 3: Precession: The precession of celestial coordinates must also be considered. Epoch of date coordinates should be used at the telescope, since both altazimuth and equatorial telescopes are basically aligned with the precessing Earth. However, in data reduction, coordinates should ultimately be transformed back to a standard epoch – otherwise precession can introduce a small but relevant spurious rotation.

Challenge 4: Finally, proper motion of a binary star across lines of RA can cause a non-dynamical, spurious rotation of the celestial position angle. This is easily corrected, as it is exactly equal to the amount the star has moved in RA.

Solutions: All of the challenges listed here essentially carry their own solutions, except for Challenge 2. What is needed is a celestial object with position angle known to about 0.005˚. There are cases where two accurately measured Hipparcos stars are found within a few arcminutes of each other. These are excellent for rotational calibration of optical CCDs, but useless for AO instruments with their small fields of view. Well-studied binary stars are numerous, but the precision of the measurements is insufficient. Also, distant binaries are preferred because their orbital periods will be so long that a simple linear fit to PA vs. time may suffice almost indefinitely. AO position angles of interesting binaries could be measured relative to the average PA of a known set of distant, very long period binary stars with known motions. In the hopes of fostering the development of such a set of rotational standards, we offer the table at upper right, which presents measurements of wide binary stars made with the Kuiper Telescope. Not all pairs in this table are known to be physically bound, though statistical arguments make it likely. Many of these stars have the potential to become useful members of a well-measured reference set, but they can also be used more immediately for approximate calibrations.

Star Name Separation(asec) PA(degrees) Date measured

HR 8549 4.353 ± 0.030 0.383 ± 0.200 Nov 21, 2005

HD 217890 4.124 ± 0.030 150.509 ± 0.200 Nov 21, 2005

HD 223718 5.651 ± 0.015 86.257 ± 0.150 Nov 21, 2005

HD 13480 3.932 ± 0.040 69.668 ± 0.300 Nov 21, 2005

HD13480 3.969 ± 0.017 68.725 ± 0.300 Feb 5, 2006

HD18715 8.649 ± 0.025 7.598 ± 0.100 Nov 21, 2005

HD 25411 4.473 ± 0.025 268.933 ± 0.300 Nov 21, 2005

HR 3208 6.095 ± 0.026 72.666 ± 0.200 Feb 5, 2006

HR 2176 7.538 ± 0.014 357.352 ± 0.040 Feb 6, 2006

HD 47888 7.166 ± 0.013 329.402 ± 0.200 Feb 6, 2006

HD 58246 6.740 ± 0.020 45.122 ± 0.030 Feb 6, 2006

HD 71151 5.202 ± 0.020 217.996 ± 0.500 Feb 6, 2006

HD 100831 6.159 ± 0.011 165.810 ± 0.150 Feb 7, 2006

HD115404 7.532 ± 0.050 104.868 ± 0.200 Feb 7, 2006

Background Image: Mount Wrightson, AZ, seen from near the MMT. Photo by Aren Heinze.

Conclusion: AO astrometry of binary stars can yield accurate masses even with observations spanning only a quarter of an orbit. Such accurate mass determinations of wide binaries could be of great benefit to precision stellar astrophysics.