Usually, positions of spacecraft on interplanetary or deep space missions are determined by radar tracking from ground stations, a method by which uncertainty increases with distance from Earth. As an alternative, a spacecraft equipped with e.g. an X-ray telescope could determine its position autonomoulsy via onboard analysis of X-ray pulsar signals. In order to find out which pulsars are best suited for this approach and what accuracy can be achieved, we build up a database containing the temporal emission characteristics of the ~ 60 X-ray pulsars for which a pulsed radiation has been detected by mid 2010.
Deep Dive into Timing X-ray Pulsars with Application to Spacecraft Navigation.
Usually, positions of spacecraft on interplanetary or deep space missions are determined by radar tracking from ground stations, a method by which uncertainty increases with distance from Earth. As an alternative, a spacecraft equipped with e.g. an X-ray telescope could determine its position autonomoulsy via onboard analysis of X-ray pulsar signals. In order to find out which pulsars are best suited for this approach and what accuracy can be achieved, we build up a database containing the temporal emission characteristics of the ~ 60 X-ray pulsars for which a pulsed radiation has been detected by mid 2010.
The conventional way to obtain data for orbit determination is tracking the spacecraft by ground stations on Earth, e.g. with NASA's Deep Space Network (DSN). This technique yields very accurate range and range-rate data along the Earth-spacecraft line, but due to limited angular resolution large errors can occur in perpendicular directions, resulting in positional errors that grow with distance. Interferometric measurements can augment the angular resolution, thereby achieving positional accuracies in the order of 4 km per AU of distance between Earth and spacecraft [5]. Thus, especially -but not exclusively -for interplanetary and deep space missions, it is desirable to have an autonomous navigation solution that works independently from ground stations.
Pulsars can be used as natural beacons for navigation by comparing pulse arrival times measured onboard the spacecraft with predicted arrival times at an inertial reference location -e.g. the barycenter of the solar system. The phase difference between the expected and the measured pulse arrival corresponds to a run-time difference along the line of sight towards the pulsar and hence to a range difference in this direction. Full three-dimensional position information can be deduced from the range information along the pulsar lines-of-sight of at least three different pulsars. In general, this procedure results in multiple solutions, which can be reduced by either constraining the possible spacecraft positions to a finite volume around an initial position assumption, or by observing additional pulsars [8,9].
In order to decide which pulsars are best suited for spacecraft navigation, we re-analyzed all pulsar timing data from the X-ray satellites XMM-Newton, Chandra and the ROSSI X-ray Timing Explorer. The individual photon arrival times of each data set were corrected for the orbital motion of the detector around Earth/Sun via transformation to the solar system barycenter and for the orbital motion of the pulsar in case of a binary system. To account for the gradual increase of pulse period the spin frequency was modelled as a Taylor expansion,
with f 0 , ḟ0 , f0 being the rotation frequency and its first and second time derivative at some reference epoch t 0 , and a pulse number Φ was assigned to each individual arrival time. Since the frequency equals the rate of change of pulse number, f = dΦ/dt, integration of (2.1) yields
where Φ 0 denotes the pulse number at t 0 . The pulse numbers are used to generate a mean pulse profile reflecting the temporal emission characteristics of the pulsar: (1) For each photon its phase φ := (Φ mod 1) is computed, i.e. the fraction of full pulse period at which the photon was detected;
(2) the domain [0, 1) of φ is divided into n finite intervals ∆φ i := i-1 n , i n for i = 1, 2, . . . , n; (3) finally, the mean pulse profile is given by the histogram of photon counts per phase interval. The optimal value of n depends on the number of recorded photons and the harmonic content of the pulse profile. It can be computed as follows: For each set of arrival times t i with i = 1, 2, . . . , N and phases φ i := φ (t i ) we calculated the statistical variable
which is a measure for the periodicity of the signal [3], and evaluated the expression
to get the optimal number M of harmonics [4]. Then, the formula
with Z 2 0 := 0 yields an estimate for the optimal number of phase intervals [2]. Since equation (2.1) may not allow to compute f (t) for times t outside the validity range of the ephemerides, the values of f 0 , ḟ0 , f0 have to be known at an epoch sufficiently close to the measured photon arrival times. We used pulsar ephemerides from the ATNF database [6] for the period folding and produced pulse profiles with high signal-to-noise ratios by superposing phase values from several obvservations of the same pulsar.
The navigation algorithm is based on the comparison of pulse arrivals measured onboard the spacecraft with those measured at the solar system barycenter. 1 For this purpose, representation of the reference profiles by analytical functions is of advantage. Using a least-squares-fit procedure, we found approximations of the profiles by either a sum of Gaussians, 2 , or of sine functions, ∑ i A i sin(B i φ +C i )+D, depending on the individual shape of the profile. Concerning the use of Gauss or sine functions, the resulting M value of the H-test is a good indicator: For M ≤ 2 the profile is well approximated by a sum of two or three sine functions, whereas for M ≥ 3 it is often better to use a sum of two or more Gaussians. Examples of pulse profiles and their analytical representations are shown in Figure 1. These pulse profile templates allow us to measure pulse arrivals with high accuracy even for sparse photon statistics -again by using a least-squares fit of an adequately adjusted template to the profile in question. The typical error of measuring a pulse phase lies in the order of 10 -3 , which corresponds
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