## 3. Orbit adjustment## 3.1. The adjustment codeThe program used for the orbit adjustment, ORBIT, derives from the code of Tokovinin (1992). Like its progenitor, it performs a least square adjustment to all available spectroscopic and "visual" observations, with weights inversely proportional to the square of their standard errors. Besides a number of cosmetic changes, the modifications and additions to Tokovinin (1992) include allowance for multiple radial velocity zero-points, the use of the more robust Levenberg-Marquardt minimisation algorithm, as well as direct support for triple systems. Standard errors for derived parameters (masses, parallax, etc) are computed from the full covariance matrix of the linearized least square adjustment, rather than from just the standard errors of the orbital elements (i.e. the diagonal terms of the covariance matrix). In addition ORBIT can estimate confidence intervals for both derived parameters and orbital elements through Monte-Carlo experiments. For well constrained binaries (like Gl 570BC) these intervals are consistent with the analytic standard errors and gaussian statistics, incidentally providing a check against gross errors in the analytic gradients. For noisier systems though, the vicinity of the minimum of the surface becomes significantly non-quadratic. The error statistics for the parameters are then substantially non-gaussian and asymetric, and the Monte-Carlo confidence intervals become an essential feature. ORBIT also accepts some additional data types, beyond spectroscopic velocities and "visual" or (X,Y) pairs. These additions, all of which are used in the present paper, include: -
1D projected separation measurements, produced by early IR scanning speckle observations and lunar occultations; -
parallaxes, which can be seen as an additional link between the velocity amplitudes, the eccentricity, the period and the semi-major axis; they enter in the least square adjustment with their standard error, providing an optimal unified description between orbital parallaxes of SB2+visual pairs and mass ratios from SB1+visual pairs with a known parallax; -
cross-correlation "dips", produced by correlation velocimeters and cross-correlation analysis of echelle spectra; to support these profiles ORBIT complements the usual list of 10 orbital elements by the width and the equivalent width of one gaussian profile for each spectroscopic component, and adjusts the 10 orbital elements and the profile parameters directly to the ensemble of all cross-correlations, rho-theta pairs, parallaxes, etc. In a similar manner, modern analyses of long baseline interferometric data sometimes adjust orbital elements (together with magnitude differences) directly to the *uv*data, avoiding the intermediate step of extraction (Hummel & Amstrong 1992), as do adjustments of visual orbits to Hipparcos Transit Data (Quist & Lindegren 1999; Söderhjelm 1999). A practical inconvenient of this bypassing of intermediate steps is that no individual radial velocities (or , or separations on great circles) are available for compact publication, and that the usual figures of radial velocity curve (or visual orbit) loose sense and become illustrative only. On the positive side however, direct orbital adjustment to the profiles greatly decreases the effective number of degrees of freedom, compared with orbital adjustment to velocities measured from individual profiles. We found that this often improves the error bars on the spectroscopic orbital elements by a factor of two, or better. Global adjustment also considerably reduces the susceptibility of the orbital solution to "pulling" of the velocity of the weaker component towards and away from that of its brighter companion for profiles which are only imperfectly gaussian. In addition, and to further diminish this sensitivity, we produce an average recentered profile for each component, determine residuals to the best fitting gaussian, and subtract those residuals from the individual measurements to then determine iteratively improved parameters.
ORBIT runs on Unix systems and can be obtained from the authors upon request. To illustrate our methods and the power of combining different data types for the same binary, as probably first advocated by Morbey (1975), we present 3 distinct orbits for Gl 570BC, incorporating successively more of the information available to us. The speckle and adaptive optics data alone turn out to be insufficient to properly define a visual orbit, largely because they never resolve this eccentric system close to its periastron (Fig. 2), where the minimal separation is only 12 milliarcsecond. As a consequence these data alone leave indeterminate a combination of inclination and semi-major axis with eccentricity. The most restrictive orbit we can present in Table 3 is therefore a spectroscopic one. We then present a combined spectroscopic and visual orbit, and finally an orbit which in addition uses the independent trigonometric parallax of the Gl 570 system.
The longitude of the periastron is given with the spectroscopic
convention, and thus refers to the primary. 180 ## 3.2. Spectroscopic orbitAs mentioned above, the spectroscopic orbit presented in Table 3 was adjusted directly to the correlation profiles rather than to the extracted radial velocities. As can be seen, the spectroscopic quantities are determined with very high accuracies of 0.2%. The relatively few ELODIE measurements contribute considerably to the overall precision of the orbital solution, and ignoring them would for instance degrade the standard errors of the by an order of magnitude. The much more numerous CORAVEL measurements of the primary star by contrast only contribute to an improved orbital period, thanks to their much longer timespan. They otherwise carry very little weight in the solution. ## 3.3. Spectroscopic+visual orbitInclusion of the speckle and adaptive optics data in the adjustment
leaves all spectroscopic elements essentially unchanged
(Table 3), but determines the three otherwise unknown orbital
elements: the semi-major axis ( The availability of Hipparcos parallaxes represents an opportunity to independently verify the orbital parallax, and thus to globally check the orbital solution for systematic errors. Though somewhat noisier than typical for a V=5.7 star, the Hipparcos catalog parallax of Gl 570A is well determined, =0.16930.0018". The parallax for Gl 570BC itself has very large error bars (=0.033") in the Hipparcos catalog (ESA 1997), because the unaccounted orbital motion with P0.8 year strongly couples into the parallax solution over the limited lifetime of the Hipparcos satellite. Fortunately, Söderhjelm (1999) recently reanalysed the Hipparcos intermediate transit data, accounting for the orbital motion within Gl 570BC, and obtained sharply reduced error bars for the trigonometric parallax: =0.16970.0010". The two astrometric determinations are mutually consistent and agree with the orbital parallax of 0.17100.0022", to within better than 1 . ## 3.4. Spectroscopic+Visual+Parallax orbitInstead of using the independent trigonometric parallaxes as a
sanity check for the orbital solution, ORBIT offers the option to
consider it as an additional observation, linking The contributions to the overall
of the different data types included in the solution are approximately
consistent with their respective number of measurements. This
indicates that there are no large systematic errors in any one data
type, and that the adopted standard errors are at least approximately
correct. This orbit (last column of Table 3) has the smallest
errorbars (and smallest covariances) for all orbital elements. It is
consistent with all previously published orbits to within their stated
error bars, after allowing for the 180 © European Southern Observatory (ESO) 1999 Online publication: November 3, 1999 |