Minimum Uncertainties in Position and Velocity Determination using x-ray Photons From Millisecond Pulsars

Abstract

Position determination of a spacecraft far from earth can be aided by binning the times of arrival of x-ray photons from a millisecond pulsar and matching the pulse shape to a standard shape by adjusting the phase of the observed signal. Usually studies of such measurements assume the frequency of the incoming signal is known, and show that the uncertainty in the component of position vector along the line of sight varies as 1/2 obs T − where obs T is the total observation time. Here we extend such analyses to include determination of velocity. Cramer-Rao lower bounds on the uncertainties of position and velocity are derived based on analysis of the Fisher Information Matrix. In one spatial dimension, the results can be expressed in terms of a minimal ellipse in positionvelocity space whose size and orientation are determined by the observation time and by a single quantity depending on the standard pulse shape. For long observation times, the area of this ellipse decreases as 2 obs T − . If position were known, the uncertainty in velocity would decrease as 3/2 obs T − . If the spacecraft acceleration can be neglected, then by Liouville’s theorem between measurements the area of the ellipse remains constant while its shape elongates in the position dimension. Another measurement of position and velocity may be combined with this minimum uncertainty ellipse to reduce the area further. Sequences of measurements can be used either as inputs to a Kalman filter, or described in terms of a set of recurrence relations for the parameters of the minimum uncertainty ellipse. We shall discuss the derivation of these results and some simulations based on this theory.