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Astron. Astrophys. 351, 619-626 (1999)

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3. Orbit adjustment

3.1. The adjustment code

The 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 [FORMULA] 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" [FORMULA] 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 [FORMULA] 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 [FORMULA], 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 [FORMULA]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.

[FIGURE] Fig. 2. Visual orbit of the Gl 570BC system. Separations are in arc seconds, North is up and East is left.


Table 3. Orbital elements and derived parameters.
Epochs are listed as offsets relative to Julian Day 2450000. W1 and W2 are the full widths to half power of the Gaussian function for the two components, while EW1 and and EW2 are their equivalent widths. The other orbital elements have their usual meaning.

The longitude of the periastron is given with the spectroscopic convention, and thus refers to the primary. 180o must be added to [FORMULA] to obtain the visual convention. Table 1 lists the individual speckle and adaptive optics measurements and Table 2 (only available electronically) gives the same information for the radial velocities. Note, though, that the final orbit was directly adjusted to the individual cross-correlation profiles, while the velocities given in Table 2 were determined separately by adjusting two Gaussian curves to each profile. These velocities are given here mostly for illustrative purposes, as plotted in Fig. 1. Their use would give a slightly different (and noisier) solution, but may nonetheless prove convenient. There is no compact way to publish the full information needed to reproduce our analysis, since it does not proceed through determining radial velocities as an intermediate step. We will make the digital correlation profiles available upon request to the first author. Fig. 2 shows the visual data and orbit.

3.2. Spectroscopic orbit

As 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 [FORMULA] 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 [FORMULA] 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 orbit

Inclusion 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 (a ), the inclination (i ), and the orientation of the projected orbit on the sky ([FORMULA]). Individual stellar masses as well as an orbital parallax for the system can then be derived from the full complement of orbital elements, and are also listed in Table 3. The standard errors computed from the covariance matrix were checked through Monte-Carlo simulations, which resulted in confidence intervals that were fully consistent with gaussian errors of the stated dispersion. This verifies that non-linearities in the least square adjustment are negligible for this very well constrained system. The two masses are determined here with 1.6% accuracy, and the orbital parallax to within 2.2 milliarsecond (1.3%).

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, [FORMULA]=0.1693[FORMULA]0.0018". The parallax for Gl 570BC itself has very large error bars ([FORMULA]=0.033") in the Hipparcos catalog (ESA 1997), because the unaccounted orbital motion with P[FORMULA]0.8 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: [FORMULA]=0.1697[FORMULA]0.0010". The two astrometric determinations are mutually consistent and agree with the orbital parallax of 0.1710[FORMULA]0.0022", to within better than 1 [FORMULA].

3.4. Spectroscopic+Visual+Parallax orbit

Instead 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 i , e , P , a and K1+K2. It is then included in the combined least square adjustment, together with the radial velocities and angular separations. This very significantly improves the determinacy of the least square system, and in particular reduces the standard errors of the semi-major axis by a factor of 2. We note here that it would be preferable, at least conceptually, to adjust an orbit directly to the Hipparcos transit data rather than to the Hipparcos parallax. ORBIT does not yet support this data type however, which we plan to add in the near future.

The contributions to the overall [FORMULA] 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 180o ambiguity in the identification of the ascending node (i.e. [FORMULA]) in purely visual orbits. We adopt it as our preferred solution for the rest of the discussion. The inclusion of the trigonometric parallax information improves the relative accuracy of the two masses to 1.2%: M1 = 0.586[FORMULA]0.007[FORMULA] and M2 = 0.390[FORMULA]0.005[FORMULA]. They are consistent with the values obtained by Mariotti et al. (1990) to within a fraction of their quoted standard deviation, as well as with the mass sum derived by Söderhjelm (1999) from an analysis of the Hipparcos transit data, but improve on their accuracy by over a factor of 5. These accuracies are among the best obtained to date for non-eclipsing binaries, but still do not match the [FORMULA] obtained for the spectroscopic [FORMULA]. There is thus still room for improvement in the astrometric measurements. We are conducting some long baseline interferometric observations of Gl 570BC (Segransan et al. in progress) with IOTA (Millan-Gabet et al. 1999), which have the potential to ultimately match the sub-% spectroscopic accuracy for the masses. The "resulting" parallax in the last column of Table 3), 0.1698[FORMULA]0.0009", is computed as an orbital parallax from the elements of this combined orbit, which themselves take the trigonometric parallax into account. It is an optimal combination of the distance information available in the trigonometric parallax and in the orbit. In the present case is mostly determined by the weighted average of the two HIPPARCOS parallaxes, with little contribution from the orbit.

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999