  The Hipparcos catalog and its Double and Multiple System Annex (DMSA) lists
4099 components with individual proper motions and coordinates on the epoch
1991.25. Many of these long-period binary stars are also present in the Gaia
Data Release 2 (DR2). Using the available relative positions and proper motions
separated by 25.25 years, the equations of relative orbital motion can be
solved for the two epoch eccentric anomalies, orbital period, and eccentricity.
The method employs elimination of the linear Thiele-Innes unknowns and
nonlinear optimization of the remaining condition equations. The quality of
these solutions is compromised by the insufficient condition and modest
precision of the Hipparcos astrometric data, as revealed by Monte-Carlo
simulations with artificially perturbed data points. The presence of multiple
systems and optical pairs can also perturb the results. Limited experiments
with artificial data indicate that useful estimates can be obtained with a
25-year epoch difference for wide binaries with orbital periods up to $\sim
500$ years. The prospects of this method dramatically improve with the proposed
next-generation space astrometry missions such as Gaia-NIR and Theia,
especially when additional conditions are included from astrometric or
spectroscopic measurements. A ancillary catalog of cross-identification and
astrometric information for 1295 double star pairs cross-matched in Gaia DR2
and Hipparcos is published.
