  The frequencies and phases of emission from extra-solar sources measured by
Earth-bound observers are modulated by the motions of the observer with respect
to the source, and through relativistic effects. These modulations depend
critically on the source's sky-location. Precise knowledge of the modulations
are required to coherently track the source's phase over long observations, for
example, in pulsar timing, or searches for continuous gravitational waves. The
modulations can be modelled as sky-location and time-dependent time delays that
convert arrival times at the observer to the inertial frame of the source,
which can often be the Solar system barycentre. We study the use of reduced
order modelling for speeding up the calculation of this time delay for any
sky-location. We find that the time delay model can be decomposed into just
four basis vectors, and with these the delay for any sky-location can be
reconstructed to sub-nanosecond accuracy. When compared to standard routines
for time delay calculation in gravitational wave searches, using the reduced
basis can lead to speed-ups of 30 times. We have also studied components of
time delays for sources in binary systems. Assuming eccentricities $< 0.25$ we
can reconstruct the delays to within 100s of nanoseconds, with best case
speed-ups of a factor of 10, or factors of two when interpolating the basis for
different orbital periods or time stamps. In long-duration phase-coherent
searches for sources with sky-position uncertainties, or binary parameter
uncertainties, these speed-ups could allow enhancements in their scopes without
large additional computational burdens.
