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
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.
(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.