1.4 Young Stars
2.1.4 Orbits
The third major use of astrometry, apart from parallaxes and proper motions, is the deter- mination of orbits. This takes two very different forms: the astrometric orbit, and the visual orbit.
Visual orbits are derived from resolved observations of the two stars, either from visual observations (including Speckle Interferometry, Lucky Imaging, Adaptive Optics), or short- and long-baseline interferometry. In these cases the observers record the actual separations and positions of both stars in two dimensions. Separations and position angles can either be recorded relative to other stars in the field of view (as with parallax) or simply relative to the primary component. The product of solving for the orbit (via the Thiele-Innes method) are the actual orbital elements of a system.
An astrometric perturbation describes the orbit of the photocenter about the barycenter, not the motion of any individual component. The amplitude α of an astrometric binary orbit is dependent on a number of factors:
α∝M2, M1+M2, a,
L1
L2
(2.3) Astrometric perturbations are therefore greater for larger luminosity differences between the components (L1
L2), larger companion masses (M2), larger system masses (M1 +M2), and larger semi-major axes of the orbits (a). Note that the photocenter of an equal-mass equal-
luminosity binary will not move at all. The signal of an astrometric binary is a multivariable problem, and it is thus difficult to describe the sensitivity limits of a given program.
The CTIOPI parallax reduction pipeline fits the astrometric positions of a star to a linear proper motion and a parallax ellipse of known shape and unknown size. Any further motion caused by the orbit of a companion remains, appearing as a perturbation in the astrometric residuals (Figures 5.1,5.2, and 5.3). The product of solving for this kind of orbit is an orbit with a semimajor axis scaled down compared to the visual orbit, and flipped 180 degrees to the real one. In a visual orbit, the moving body is the secondary component relative to the fixed primary; in an astrometric orbit, the photocenter moves, and is on the primary star’s side of the barycenter.
CTIOPI’s companion detection capabilities are limited by several factors. Systems are typically observed 1–4 times a year and thus the data are insensitive to periods less than a year. The program has only been running since 1999, and cannot wrap orbits with periods longer than the period of observations. CTIOPI also has a 3–6 mas nightly precision error (depending on the specifics of the reference field) that limits our sensitivity to low-amplitude binaries. Fortunately, astrometric perturbations are typically self-confirming; genuine orbital motion will show up in both the RA and DEC axes unless the orbit is nearly north-south or east-west on the sky. Nevertheless, for the three astrometric binaries in Riedel et al. (2010) (see §5.2.2.2) we reduced the three brightest reference stars in each of our astromet- ric perturbation fields as if they were the parallax targets, to make sure the perturbation signal was not some unnoticed systematic of the field. Infrared AO observations (currently
unpublished) later resolved two of the astrometric binaries.
To solve astrometric orbits, RECONS uses an iterative Thiel-Innes least-squares solver (Hartkopf et al. 1989) to fit the astrometric orbit, with points from nights with only a single CCD image (generally obtained for the purpose of photometry) removed. As orbital motion can bias the parallax and proper motion fits (up to, hypothetically, a one-year orbit can- celing out the parallax ellipse entirely), we calculate and subtract the fitted orbit from the initial positional data and re-fit the parallax (Riedel et al. 2010). This second reduction also demonstrates the potential accuracy of the fitted orbit: the residuals with the orbit removed should be flat, to within the signal-to-noise ratio of our astrometric data, 3-6 mas per night. As a general rule, the orbit-removed proper motion and parallax are not substantially differ- ent, but the errors are far smaller. Unfortunately, it appears that we do not yet understand our errors in terms of perturbations, and our fitted astrometric orbits do not quite reproduce published orbits. Investigations are ongoing.
Obtaining a full orbit from an astrometric one can be accomplished with a single obser- vation: The instantaneous resolved separation p between the two components at a time t can be compared to the expected astrometric separation ρbetween the photocenter and the barycenter at the same time t. The ratio between p and ρ is the same as between the real semimajor axis a and the photocentric semimajor axis α.
The derivation of Thiel-Innes elements (the coefficients A, B, F and G) is a method of relating conventional spherical coordinates for an orbit to measured (projected) Cartesian orbital data. The Thiel-Innes elements describe the positions of a binary star as a function
of time t relative to some epoch of periastron passage T0, and dx and dy are the differences
between the predicted X and Y positions and the actual measured positions.
dx=XdA+Y dF +Pxde−QxdT +Qx(t−T0)dT (2.4)
dy=XdB+Y dG+Pyde−QydT +Qy(t−T0)dT (2.5)
Minimizing dx and dy is an iterative process solved by recalculating several differential equations: Px =A dX de +F dY de (2.6) Py =B dX de +G dY de (2.7) Qx =A dX dM +F dY dM (2.8) Qy =B dX dM +G dY dM (2.9)
which can themselves be calculated from the “normal equations”:
dX de =−1− sin2E 1−ecosE (2.10) dY de = XsinE √ 1−e2(1−ecosE) (2.11) dX dM = −sinE 57.296(1−ecosE) (2.12) dY dM = √ 1−e2cosE 57.296(1−ecosE) (2.13)
where e is the orbital eccentricity, E is the eccentric anomaly calculable (iteratively) from 360
P ×(t−T0) = E−esinE (2.14)
where P is the period, T0 is the time of periastron passage, and t is the variety of times at
which data are available.
At each step, dx and dy are calculated as the difference between the real orbit and a calculated orbit (X,Y). The X (dA, dF and de) and Y (dB, dG, and de) unknowns are solved for in matrix form. Those values are added to A,B, F, G, andeto produce the next step’s values, which are then used to come up with new differential equations. Eventually, the answers converge on values of A, B, F, G and e, which are then turned into the seven orbital elements: a, i, ω, Ω, e, period P (original) and T0 (original). This basic method
has a constant P and T0 throughout all the operations, which is why more advanced orbital
solvers sample a range of T0 and P.