  The observed orbits of emission-line stars may be affected by systematics
owing to their broad emission lines being formed in complex and extended
environments. This is problematic when orbital parameter probability
distributions are estimated assuming radial-velocity data are solely comprised
of Keplerian motion plus Gaussian white noise, leading to overconfident and
inaccurate orbital solutions, with implications for the inferred dynamical
masses and hence evolutionary models. We present a framework in which these
systems can be meaningfully analysed. We synthesise benchmark datasets, each
with a different and challenging noise formulation, for testing the performance
of different algorithms. We make these datasets freely available with the aim
of making model validation an easy and standardised practice in this field.
Next, we develop an application of Gaussian processes to model the
radial-velocity systematics of emission-line binaries, named marginalised
$\mathcal{GP}$. We benchmark this algorithm, along with current standardised
algorithms, on the synthetic datasets and find our marginalised $\mathcal{GP}$
algorithm performs significantly better than the standard algorithms for data
contaminated by systematics. Finally, we apply the marginalised $\mathcal{GP}$
algorithm to four prototypical emission-line binaries: Eta Carinae, GG Carinae,
WR 140, and WR 133. We find systematics to be present in several of these case
studies; and consequently, the predicted orbital parameter distributions, and
dynamical masses, are modified from those previously determined.
