Second Catalog of Rectilinear Elements

(last update: Nov 15 2018)

William I. Hartkopf* & Brian D. Mason
U.S. Naval Observatory, Washington, DC
(* retired)

Introduction & Purpose

Many systems in the Washington Double Star Catalog (henceforth WDS) have shown significant relative motion since their discovery. This catalog provides linear fits for those systems whose motions do not appear to be Keplerian. While a few of these may in fact be very long-period physical pairs whose orbital motion is not yet apparent, most are probably optical pairs (i.e., chance alignments of unrelated stars). These linear fits, then, just describe the relative proper motions between these pairs of stars. The purposes of this catalog are threefold. First, the very well-defined relative motion of some of these systems means they may prove useful for scale calibration for imaging systems. Also, these differential proper motions may allow us to improve upon proper motions of individual components. Finally, these fits, especially in comparison with Hipparcos and Gaia proper motions, may be useful in searches for submotions due to closer components.

Definition of Terms

A very straightforward method was used to define the motion of the secondary star relative to the primary:

First, define:

            x = sin(theta) * rho                                                          
            y = cos(theta) * rho  ,                                                       

where x is the East-West ("horizontal") axis and y the North-South ("Vertical") axis.

(Note two changes made to our definitions for the Second Catalog (henceforth LIN2). First, the x term above was defined as [-sin(theta) * rho] in the First Catalog (LIN1). The minus sign was unnecessary and confusing, so has been removed. Also, in the initial version of LIN1, we defined (x0, y0, t0) or (rho0, theta0, t0) as the relative location and time of closest approach. This definition of T0 was of course arbitrary, and intended to make it roughly equivalent to the definition of T0 as time of periastron in an orbital solution. However, in a linear fit the time of closest approach may occur thousands of years earlier or later than the range of observations. Andrei Tokovinin pointed out that this may result in a very large "lever arm" which may degrade the accuracy of ephemerides. We therefore redefined T0 as the weighted mean date of all measures. We have continued to use that definition for T0 in LIN2, with (x0, y0) or (rho0,theta0) the corresponding location at that date.

Next, solve for:

            x = xa * (t-t0) + x0                                                          
            y = ya * (t-t0) + y0                                                          

A standard weighted least squares fit is made, with weights of individual measures determined using the technique outlined in the Sixth Catalog of Orbits of Visual Binary Stars.

Catalog Files

Available files are as follows:
Figures are also provided for each system, and are linked from the main catalog by the letter "P" in one of the righthand columns. The figures include all measures in the WDS, as well as the linear fits whose elements are given in the main catalog (shown as a solid black line). Individual measures in these figures are color-coded as in the Orbit Catalog; green, blue, and purple indicate micrometric, interferometric, and photographic/CCD measures, respectively, while red symbols indicate measures from space-based instruments (Hipparcos, Tycho, Gaia, etc.). Measures are connected to their predicted locations by "O-C" lines; measures given zero weight are indicates by dotted O-C's. A dashed line runs from the origin to the predicted relative separation and angle at time T0.

For those pairs with two sets of proper motions in the WDS, a red line indicates the differential proper motion predicted by these values. Since most of these proper motions are currently from the Hipparcos or Tycho-2 catalogs, the red lines intersect the least-squares fit to the data at the Hipparcos epoch of 1991.25. This will be modified as more Gaia proper motions are ingested into the WDS, however. The lengths of both the black and red lines correspond to a similar range of dates, usually roughly corresponding to the range of observation dates.


LIN2 currently includes fits for 1507 systems. Qualities of these fits vary considerably; some undoubtedly need more data, while others have not shown enough motion for a meaningful fit to be made in our lifetimes. These systems will be weeded out through further examination. Other systems in the WDS do show enough motion for a fit to be attempted; additional pairs will be added as they are discovered. Additional notes, astrometric data and proper motion information, as well as updated figures, will be added regularly, as well.

(Note: The initial version of LIN1 did not include a reference code for each solution, as all were done "in house". However, publication of linear elements by others prompted the editors to make a slight format change in July 2011 to include those references. Due to severe manpower constraints, however, we have now reverted back to including only in-house solutions for LIN2, as this allows for simpler, more automated updates.

Your comments regarding format, as well as notification of errors, are welcome.