Effects of proper motion of neutron stars on continuous gravitational-wave searches

All-sky and directed continuous gravitational-wave searches look for signals from unknown asymmetric rotating neutron stars. These searches do not take into account the proper motion of the neutron star, assuming that the loss of signal-to-noise ratio caused by this is negligible and that no biases in parameter estimation are introduced. In this paper we study the effect that proper motion has on continuous wave searches, and we show for what regions of parameter space (frequency, proper motion, sky position) and observation times this assumption may not be valid. We also calculate the relative uncertainty on the proper motion parameter estimation that these searches can achieve.