Abstract
The concept of observing pulsars for space autonomous navigation has already caught the attention of space agencies. Driven by the extremely stable nature of pulsar radiation, many research works and in-orbit demonstrations have been performed, demonstrating the suitability of these sources for navigation. The core concepts of the in-orbit demonstrated X-ray pulsar-based navigation systems and the recently proposed space navigation by optical pulsars (SNOP) systems are based on the capability to accurately define the arrival times of a pulsar signal. Therefore, the performance of a pulsar-based navigation system depends on the timing accuracy of the measured signals, which is a function of the characteristics of the navigation payload onboard the satellite. The aim of this paper is to investigate the impact of the optical parameters of a photometer-based instrument on the timing accuracy of a SNOP system; moreover, a first optical design for the payload is proposed.
1 INTRODUCTION
A pulsar is a highly magnetized neutron star that emits beams of broadband electromagnetic radiation out of its magnetic poles (Lyne & Graham-Smith, 2005). Formed from the collapse of a star with a mass of approximately 8–25 solar masses, a pulsar is a rapidly rotating object, with periods ranging from a few milliseconds to seconds, emitting a light beam from the pulsar magnetosphere (Becker, 2009). Pulsars are extremely faint objects, usually embedded in an emitting nebula. The pulsating component is much weaker than the continuous nebula background, making pulsar detection difficult because of a very poor signal-to-noise ratio (SNR). Pulsars are typically discovered at radio wavelengths, but the size of a radio telescope dedicated to pulsar observation would be impracticable for any spacecraft (Emadzadeh & Speyer, 2011). Moreover, the low signal intensity of radio pulsars would require very long observations to achieve a sufficiently high SNR (Sala et al., 2004). Therefore, a large number of feasibility studies and in-orbit demonstrations have been performed considering X-ray pulsars (see, e.g., works by Zhang et al. (2017) and Zheng et al. (2019)), showing the suitability of these sources for autonomous navigation. Nevertheless, such systems require either complex and heavy instrumentation (Winternitz et al., 2016) or very long observations to achieve a significant SNR (Cacciatore et al., 2023). Recently, the authors have investigated the use of optical pulsars for navigation (Zoccarato et al., 2023), proposing a space navigation with optical pulsars (SNOP) system able to reduce the weight and power requirements of the navigation payload, while retaining short pulsar-signal integration times by filtering the nebular background noise. SNOP systems lead to the highest SNR ever achieved in pulsar navigation.
To reconstruct the pulse shape, a pulsar signal must be acquired at a very high rate and folded over a single period, in order to increase the SNR by means of a technique known as epoch folding (Wang & Zhang, 2016). Then, to allow the light curve to be processed, the signal phase is partitioned into a given number of bins. The estimation of the minimum observation time and the most convenient number of phase bins needed to obtain the required SNR are not easy to predict. In fact, the SNR of the pulsating component of the light collected from the pulsar depends on several parameters. Firstly, it is obvious that a longer integration time will correspond to a higher SNR. Secondly, reducing the number of phase bins increases the signal amplitude, but also simultaneously reduces the timing resolution. Finally, the field of view (FoV) (of the instrument if a photometer is used or of the pixels of interest if an imaging system is used) defines the amount of nebular background contributing to the noise. Because very low FoV values are considered in a SNOP system, it is reasonable to assume that the nebular flux is uniformly distributed over the FoV.
This paper investigates how the characteristics of a SNOP system affect the accuracy of the estimated arrival time of a pulsar signal and, thus, the satellite ranging accuracy. In particular, the impact of adapting different telescope apertures, FoVs, and observing times in a pulsar navigation system is quantitatively estimated. In all cases, the chosen parameters are suitable for a potential satellite application. This analysis indicates the limits in the attainable satellite ranging accuracy along the line of sight of a pulsar, considering present optical technologies and reasonably possible satellite resources. More precisely, a sub-μs accuracy (corresponding to a line-of-sight ranging error of just a few hundreds of meters) cab be reached by the proposed optical payload for the Crab pulsar. A three-dimensional (3D) positioning error as low as 5 km (one sigma) can be reached by a SNOP system with such performance after five days of observations of the Crab pulsar (Zoccarato et al., 2023).
This paper is organized as follows. Section 2 discusses the achievable accuracy of the pulse arrival time estimation (PAT) by a SNOP system, determined by varying the optical parameters of the payload. Then, Section 3 presents the proposed optical design, and Section 4 presents the conclusions of this work.
2 PARAMETRIC ANALYSIS
The periodic nature of pulsar signals enables the PAT to be accurately predicted. The PAT is defined as the time at which a given phase value of the pulsed signal (e.g., that of the pulse maximum) reaches the detector; clearly, the PAT depends on the position of the detector along the line of sight of the observed source. Thus, in principle, PAT analysis of a pulsar signal allows one to provide spatial information along a specific direction. PAT estimation of the reconstructed signals is a key process in a pulsar-based navigation system; measurements from one or more pulsars (i.e., measurements along different directions) can be joined together to estimate the satellite position and velocity (Buist et al., 2011).
Emadzadeh and Speyer (2011) proposed a technique to simulate the pulsar signals attainable with a given effective area and FoV. The effective area accounts for the the reflectivity of the optics and the quantum efficiency of the detector, which, in this paper, have been considered to be equal to those of the reference observatory (Zampieri et al., 2015). The photon fluxes for both the pulsar and nebular contributions have been scaled, considering the sole geometric area ratio between the reference observatory and the simulated space instrument. By simulating a large number of pulsar signals, one can show that the phase measurement errors, defined as the difference between the imposed and estimated phase shifts, are well approximated by a Gaussian distribution with zero mean and variance σ2. The PAT estimation is carried out by cross-correlating the simulated signal with an analytical template (as presented by Zoccarato et al. (2023)), and no Doppler or clock errors are considered for the purpose of this paper. Figure 1 displays the logarithm of σ (expressed both in seconds and in meters; the value given in meters is obtained by multiplying σ by the speed of light) as a function of the time when using Crab pulsar observations, obtained via a telescope with a geometric area of 0.07 m2 or 0.2 m2 with different FoV values dividing the pulsar period in 1000 bins. The results are based on Monte Carlo simulations with at least 100 runs per data point and are equally spaced on the observation time axis for a better representation. The results show the advantages of a narrow FoV, which allows the rejection of a large number of photons from the nebula, increasing the SNR. It is evident that a larger telescope area and smaller FoV result in a better attainable timing accuracy.
σ as a function of the observation time for different FoV values, considering a telescope area of 0.07 m2 (left) or 0.2 m2 (right) for the Crab pulsar
It can be clearly seen that σ decreases more rapidly for a larger effective area and a smaller FoV. For larger FoV values, σ decreases slowly, and longer observation times are needed to reduce the PAT estimation error. σ decreases more slowly as the observation time increases; consequently, the improvement in the PAT estimation accuracy is not significant for long observation times, especially if the FoV is large. For this reason, it may be not convenient to choose an overly long observation time in a SNOP system. Conversely, this analysis shows that it could be more convenient to choose shorter observation times and reconstruct a larger number of signals to fit the satellite state parameters along the satellite orbit. Table 1 shows the one-sigma ranging error (obtained by multiplying σ by the speed of light) attainable for the three brightest optical pulsars (Crab, Vela, and B0540-69).
Attainable Line-of-Sight Positioning Error for Different Pulsars with a Telescope Area of 0.07 m2 Over an FoV of 0.79 arcsec2
3 OPTICAL DESIGN
The proposed optical payload consists of three main elements: the optical telescope (OT), the pulsar camera (PC), and the context camera (CC). The OT is responsible for collecting the incoming photons and directing them to the two channels of the PC and CC. The choice of a double-channel optical system comes from the need to both collect the pulsar photons over a narrow FoV (to limit the pulsar background noise) and provide larger FoV images to track the optical source, pointing the OT to the correct direction by means of an appropriate pointing system. The two tasks are performed by the PC and CC, respectively. A telescope diameter of 0.3 m (i.e., with a geometric area of 0.07 m2) has been deemed suitable for the navigation payload. A Ritchey–Chretien configuration (i.e., an optical telescope with two hyperbolic mirrors) with a focal length of 700 mm has been adopted (Figure 2). Because a Ritchey–Chretien telescope is usually characterized by a narrow FoV (a few arcminutes), a split doublet (i.e., two positive lenses) has been added to reach the FoV specified for the CC. The split doublet is added in such a way that the two lenses and their mechanical support are integrated into the primary mirror of the OT. The light beam coming from the OT is split into the two channels by means of a folding mirror with a small central hole with a diameter of 1 mm. The light passing through the hole is directed to the PC channel, whose FoV is of 19.63 arcsec2. The light is then re-focalized after the telescope focus by means of a double-lens system, for a total focal length of 2062.7 mm. A 50-μm-diameter photon-counting detector is assumed in the optical design. On the other side, the light reflected by the folding mirror goes to the CC channel. The FoV of the CC is 1° × 1°, with a 10-mm square detector. To adjust the focal length of the CC, a Cooke triplet with a 190-μm out-of-focus focal plane has been selected.
2D (left) and 3D (right) representations of the proposed optical system
A 3D positioning error as low as 5 km (one sigma) would be reached by a SNOP system with the chosen parameters after five days of observing a signal similar to that of the Crab pulsar (i.e., the brightest optical pulsar) (Zoccarato et al., 2023). Better accuracies can be obtained by increasing the number of observations. The authors believe that this system represents the best trade-off between the achievable accuracy and the total size of a SNOP payload.
4 CONCLUSIONS
This paper has reported, for the first time, the effect of adapting different telescope areas and FoVs on the timing accuracy of a SNOP system. Additionally, a first optical design has been proposed, leading to the highest SNR ever achieved in pulsar navigation. Its compact size (with respect to current X-ray solutions) makes this instrument attractive for future deep space missions. In future work, clock and Doppler errors will be considered to assess their impact on system accuracy.
HOW TO CITE THIS ARTICLE
Larese, S., Naletto, G., Zoccarato, P., Zeni, G., & Zampieri, L. (2025). Performance analysis and possible design of an optical system for pulsar navigation. NAVIGATION, 72(1). https://doi.org/10.33012/navi.682
ACKNOWLEDGMENTS
This work has been co-funded by the European Space Agency, contract n. 4000136103/21/NL/GLC/my.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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