Pulsar timing array observations of gravitational wave source timing parallax

Pulsar timing arrays (PTAs) act to detect gravitational waves by observing the small, correlated effect the waves have on pulse arrival times at the Earth. This effect has conventionally been evaluated assuming the gravitational wave phase fronts are planar across the array, an assumption that is valid only for sources at distances R >> 2 pi L-2/lambda, where L is physical extent of the array and lambda is the radiation wavelength. In the case of PTAs, the array size is of the order of the pulsar-Earth distance (kpc) and lambda is of the order of parsec. Correspondingly, for point gravitational wave sources closer than similar to 100 Mpc, the PTA response is sensitive to the source parallax across the pulsar-Earth baseline. Here, we evaluate the PTA response to gravitational wave point sources including the important wavefront curvature effects. Taking the wavefront curvature into account, the relative amplitude and phase of the timing residuals associated with a collection of pulsars allow us to measure the distance to, and the sky position of, the source.