Pulsars as Natural Oscillators for Long-Term Deep-Space Missions

  • NAVIGATION: Journal of the Institute of Navigation
  • January 2026,
  • 73
  • navi.733;
  • DOI: https://doi.org/10.33012/navi.733

Abstract

Pulsars are spinning neutron stars that emit highly stable, periodic signals with periods on the order of milliseconds to seconds. This natural stability makes pulsars promising for forming an independent timing system for spacecraft navigation, especially in deep space. The North American Nanohertz Observatory for Gravitational Waves team has released 15 years of data for 68 millisecond pulsars. We analyze the stability of these pulsars and estimate their range positioning accuracy for autonomous navigation. Several pulsars enable sub-kilometer accuracy over averaging periods of 100 days to more than 15 years. We also demonstrate the feasibility of an onboard pulsar ensemble using a classical weighted algorithm, achieving stability levels of 10−16 over 10 years or more. Currently, no space-based atomic clock can provide these levels of stability over a 10-year period. Therefore, the exceptional natural stability of pulsars could be leveraged for deep-space missions. In addition, pulsars have extremely long lifespans, offering unprecedented reliability for space missions.

Keywords

1 INTRODUCTION

Navigation for deep-space missions is currently achieved via the European Space Agency’s European Space Tracking (ESTRACK) network and the National Aeronautics and Space Administration (NASA) Deep Space Network (DSN) (Bokor, 2000; Curkendall & Border, 2013; Doat et al., 2018). Both ESTRACK and DSN are formed by a global network of ground-based radio antennas. This ground-based radiometric tracking is the state-of-the-art approach for deep-space navigation, generating a position accuracy of 1 m at a distance equivalent to Jupiter’s distance from Earth, i.e., 588 million km (Thornton & Border, 2003). The DSN was developed in 1958 and has since been consistently upgraded to meet the requirements of the ever-increasing number of space missions. The volume of these missions, combined with limited antenna capacity, has resulted in a high demand for DSN tracking in recent years. As a result, the DSN is oversubscribed, with limited antenna capacity (Lucena et al., 2021). Furthermore, this approach necessitates extensive back-and-forth communication, as well as the utilization of specialized ground infrastructure and the participation of flight dynamics teams, resulting in elevated expenses for space missions (Malgarini et al., 2023). The radial positions of spacecraft can be correctly established by using the DSN; however, their perpendicular positions with respect to Earth have significant inaccuracies. Position errors perpendicular to the spacecraft–Earth line are typically approximately 4 km per astronomical unit of distance. For Pluto’s orbit and for the distance of Voyager 1, uncertainty can reach ±200 km and ~ 500 km, respectively (Vivekanand, 2020).

For deep-space missions, autonomous location and velocity determination, along with precision timing by onboard data with in situ determination, is preferred. Autonomous navigation operates without any human intervention and thus requires no resources from Earth. Complete autonomous navigation has been previously proposed based on different optical methods (Franzese & Topputo, 2022). Some of these methods include observing nearby planets and their moons, as their ephemerides are accurately known. However, this approach requires favorable geometric conditions and may not be robust if none of these objects are within view.

The idea of using pulsars for spacecraft navigation dates back more than five decades, beginning with pioneering proposals by Reichley et al. (1971) and Downs (1974). These studies highlighted the potential of pulsars—rapidly rotating neutron stars with remarkably stable periodic signals—as natural celestial clocks for time and position determination. As early as the 1980s, Chester and Butman (1981) explored the feasibility of using X-ray pulsars for spacecraft navigation, emphasizing their practical advantages owing to the smaller detector sizes required compared with radio pulsars. Later, S. I. Sheikh et al. (2006) and S. I. Sheikh et al. (2011) laid the foundation for pulsar-based position, velocity, and time estimation techniques, including both delta-correction and absolute navigation algorithms. These methods demonstrated the theoretical feasibility of autonomous, Global Positioning System (GPS)-independent deep-space navigation using pulsar signals. Graven et al. (2008) also contributed key simulations and system-level analyses in this domain. Building on this foundation, recent efforts (e.g., work by Chen et al. (2020), Fang et al. (2021), Lohan and Putnam (2022), and Zoccarato et al. (2023)) have further refined X-ray pulsar navigation strategies, supported by advances in onboard instrumentation and space-tested X-ray detectors such as those utilized by the Neutron-star Interior Composition Explorer (NICER) and Station Explorer for X-ray Timing and Navigation Technology (SEXTANT). These developments collectively support the emergence of pulsar-based navigation as a viable tool for future deep-space missions.

This paper does not analyze X-ray data but instead focuses on investigating the foundation of using pulsars not only for navigation but also as a precision timing system. To achieve this, we utilize the most recent and high-precision timing data available for radio pulsars, as provided by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) (Agazie et al., 2023). The rationale for using radio pulsar data lies in the fact that long-term, high-cadence timing observations—spanning over a decade—are currently only available at radio wavelengths. This data set enables a rigorous assessment of long-term pulsar stability, which is central to our objective of evaluating their suitability as timing references for deep-space missions. We acknowledge, however, that radio observations of pulsars from spacecraft are not practical because of the large antenna baselines and infrastructure required. For onboard implementation, navigation systems would instead rely on pulsar signals observed in the X-ray spectrum. We anticipate that the stability analysis and ensemble methods developed in this work can be effectively extended to X-ray pulsars, provided that suitable detection technologies are employed onboard.

This paper is organized as follows. Section 2 briefly presents the need for a pulsar-based space clock, including a related literature review describing the advantages and disadvantages of such a system compared with currently used space clocks. Section 3 describes the NANOGrav data, including a classification and visualization of the data. In Section 4, we describe the methods used to analyze the stability of pulsars, along with an estimation of range positioning accuracy. Results and a discussion are presented in Section 5, with a conclusion provided in Section 6.

2 NEED FOR A PULSAR-BASED SPACE CLOCK

An accurate and stable clock is fundamental to spacecraft operations, underpinning navigation, signal tracking, data time-stamping, telemetry, and scientific measurements. For deep-space missions using one-way ranging, clock precision must reach the nanosecond level to maintain sub-meter accuracy—requiring a frequency stability on the order of 10−13 over several hours. This level of performance is achievable with space-qualified atomic clocks, such as NASA’s Deep Space Atomic Clock (DSAC), which was launched in 2019 to close the performance gap between terrestrial and spaceborne time standards (Burt et al., 2021). The DSAC demonstrated excellent short-term stability, achieving an Allan deviation of ~ 3 × 10−15 over one day and a linear frequency drift below 3 × 10−16 (Ely et al., 2022), thereby exceeding all mission expectations. These performance metrics were obtained despite several environmental challenges that affect space clocks, including magnetic field variations, thermal and Doppler effects, ultra-stable oscillator sensitivities, radiation exposure—especially in regions like the South Atlantic Anomaly—and evolving background pressures. However, it is important to note that these stability figures were based on averaging times of up to 50 days, which remains a relatively short duration compared with the timeline of deep-space missions such as Cassini and Europa Clipper or the decades-long Voyager missions. Over such extended timescales, traditional onboard clocks accumulate significant drift and rely on periodic synchronization with Earth-based time standards via telemetry links (Ashby, 2003). While this limitation has historically applied to quartz- and rubidium-based systems, the long-term performance of newer technologies such as the DSAC over mission-relevant durations (years to decades) is yet to be studied. Nonetheless, factors such as frequency drift, aging, and potential cosmic ray-induced degradation, known to affect crystal oscillators (Bloch et al., 2009), may still impact the performance of these clocks over time. The need for periodic synchronization presents a growing challenge for missions expected to operate in communication-limited or communication-denied scenarios because of deep-space distances, limited antenna availability on the oversubscribed DSN, or autonomous operational requirements in planetary and interstellar environments (Han et al., 2023). Historical missions such as Voyager have already encountered long communication gaps, underscoring the need for alternative, self-sustaining timekeeping systems capable of supporting navigation and timing over multi-year to multi-decade durations without ground contact.

Pulsars are natural celestial bodies; therefore, pulsar timing (PT) offers a fundamentally different approach, based on the periodic signals emitted by rapidly rotating neutron stars. Unlike atomic clocks, which rely on quantum transitions, pulsar signals are astrophysical in origin and are not influenced by spacecraft dynamics or the choice of a reference frame. Because pulsar signals are uniformly available throughout the solar system, the range and time accuracies achieved using such a system are not limited by how far the spacecraft is from Earth. For decades, pulsars have served as tools for independently testing the long-term stability of terrestrial time standards (G. Hobbs et al., 2012; Reichley et al., 1971). In addition, pulsar observations can also be used to correct the frequency drifts of a spaceborne atomic clock (Han et al., 2023). All of the above-mentioned points render pulsar timing an excellent choice for long-duration space missions. However, there are a few challenges to pulsar timing, such as the installation of onboard X-ray detectors, which may increase the size, weight, power, and cost of the mission. Moreover, a high signal-to-noise ratio (SNR) is needed to accurately measure the time of arrival (TOA) of the pulses. It is important to note that significant work has recently been conducted to analyze the X-ray detector characteristics required for such a task (Chen et al., 2020; Shemar et al., 2016; Wang et al., 2023). One such study was conducted by NASA’s SEXTANT mission, where the NICER telescope was used to observe pulsars (Winternitz et al., 2016). Table 1 shows the position accuracies obtained using X-ray detectors in various missions (Wang et al., 2023).

View this table:
TABLE 1

Position Accuracy Achieved by Present Missions, as Described by Wang et al. (2023) All of the missions observed pulsars using X-ray detectors. This table also shows the time taken to achieve the resulting accuracy and the X-ray detector area.

It is important to note that all of the missions listed in Table 1 operated in an Earth orbit, where ground-based tracking and gravitational constraints reduce the number of degrees of freedom in position determination. Thus, these missions have a geometric advantage not available in deep-space scenarios, where full four-dimensional (4D) estimation must rely solely on onboard measurements.

3 NANOGRAV: PULSAR DATA SET

NANOGrav is the source of the data used in this work (Agazie et al., 2023). The NANOGrav 15-year data set, which covers data collected from 2004 to mid-2020 using Arecibo, the Green Bank Telescope, and the Very Large Array with ASP/GASP and PUPPI/GUPPI/YUPPI backend instrumentation, contains “narrowband” and “wideband” TOA and timing solutions. Figure 1(a) shows the distribution of data based on observation duration. NANOGrav has a diverse data set, with the longest observation duration for PSR 1855+09 (15.88 years) and the shortest for J0613-0200 (2.42 years). A total of 14 pulsars have observation durations of more than 15 years. Figure 1(b) shows a visualization of all 68 observed pulsars as an Aitoff plot (Snyder, 1997), with the sun at the origin. This figure shows the sky distribution of pulsars in the data set in the galactic frame. In this frame, the “x–y plane” is the plane of the Milky Way. The positive x-direction (i.e., the l=0, b=0 direction) points to the center of the Milky Way, and the z-axis points toward the North Galactic Pole. One can see that there are pulsars in the data set located above or below the ±15° galactic latitude band.

(a) Histogram of total observation duration: the X-axis shows the total observation duration of the pulsars, and the Y-axis shows the number of pulsars corresponding to different bins of observation duration. This histogram shows that more than 30 pulsars have observations longer than 8 years. (b) Distribution of observed pulsars in an Aitoff plot with the sun at the origin: despite its anisotropic nature, the distribution is not strictly limited to the galactic plane.
FIGURE 1:

(a) Histogram of total observation duration: the X-axis shows the total observation duration of the pulsars, and the Y-axis shows the number of pulsars corresponding to different bins of observation duration. This histogram shows that more than 30 pulsars have observations longer than 8 years. (b) Distribution of observed pulsars in an Aitoff plot with the sun at the origin: despite its anisotropic nature, the distribution is not strictly limited to the galactic plane.

One can quantify the quality of the satellite (in this case, pulsar) distribution by calculating a dimensionless metric known as the geometric dilution of precision (GDOP) (Yarlagadda et al., 2000). The GDOP value can be calculated from the trace of the design matrix (Misra & Enge, 2006; Salminen, 2014):

GDOP=trace((GTG)−1)1

where G is an N × 4 design matrix defined as follows:

G=[n1xn1yn1z1n2xn2yn2z1⋮⋮⋮⋮nNxnNynNz1]2

Here, N is the number of pulsars used, (nix, niy, niz,) are the Cartesian components of the unit vector pointing from the spacecraft to the i-th pulsar, and the final column of ones accounts for the unknown clock bias in the timing solution. The data set consists of 68 pulsars, out of which we select 11 pulsars to form an ensemble (see Table 5 in Section 5.3). Below, we summarize the GDOP values for all combinations of four pulsars from our selected 11-pulsar set. The GDOP analysis of all 4-pulsar subsets from the 11 selected pulsars shows a wide variation in geometric strength. The GDOP values for quartets range from a minimum of approximately 2.41 to a maximum exceeding 2500, indicating that some configurations are poorly conditioned and highly sensitive to noise. The median GDOP across all quartets is approximately 8.47, suggesting that typical subsets offer moderate geometric strength. In contrast, when all 11 pulsars are used together, the GDOP significantly improves to 1.77, reflecting a well-conditioned and robust configuration for 4D navigation and timing. While geometry is a key factor in determining navigation precision, our calculation implicitly assumes that measurements from all selected pulsars (typically four) are available simultaneously and continuously. However, in practical scenarios, several mission-specific factors such as detector sensitivity, pulsar brightness, SNR, duty cycles, and observing time allocations can limit the ability to concurrently observe and extract TOAs from all pulsars. These constraints may reduce the effective availability of pulsars at a given time, thereby diminishing the benefits of an optimal geometric configuration. A more complete GDOP analysis, as noted by Salminen (2014), would incorporate such detector-level and signal-level effects to yield a more realistic estimate of overall navigation error.

4 METHODOLOGY

4.1 Pulse Timing

The key measurable parameter in a pulsar-based navigation system is the TOA of the detected pulse. Pulsar rotation results in a precise interval between pulses that varies from milliseconds to seconds. Accordingly, a mathematical timing model may be formulated as a Taylor expansion up to the third order, representing the signal phase progression over time, written as follows:

ϕ(t)=ϕ(t0)+(t−t0)f+(t−t0)2f.2+(t−t0)3f..63

where the frequency of rotation and its derivatives are given as f,f,.f,.., with a known initial phase ϕ(t0) at a reference time t0. With these parameters known, the arrival time of the wavefront can be accurately predicted. Furthermore, each pulsar has its own unique signature because no two neutron stars are formed in precisely the same manner or have the same geometric orientation. These differences result in unique pulse frequencies and signal shapes. Therefore, on a galactic scale, pulsars can function as “celestial lighthouses” or “natural beacons.” The TOA is determined by folding the observed pulse profile data and measuring the temporal shift relative to a predefined standard template, which typically has a high SNR. Earth-based radio observations are generally the source of the most accurate templates. The recorded phase and frequency of a pulse signal at a specific epoch can then be described and simulated at a known position, such as the solar system barycenter (SSB) (Lorimer & Kramer, 2005). This approach has been used in pulsar timing analyses for decades (e.g., see work by Taylor (1992)).

The NANOGrav data set contains parameter (.par) and TOA (.tim) files for each pulsar. A typical .tim file consists of a list of pulse arrival times, and a .par file is parsed in two steps: first, the structure of the timing model is determined (based on the components that comprise the timing model and the number of parameters for each component), and then, the values and settings from the .par file are extracted and input into the model. The NANOGrav team is also developing Pint (Luo et al., 2021)—a Python library that implements a robust pulsar timing solution. Currently in an active development stage, Pint has been extensively used by the NANOGrav collaboration teams and has proven to produce residuals from most “normal” timing models that agree with Tempo and Tempo2 (pulsar timing packages) (G. B. Hobbs et al., 2006) to within ~ 10 ns. Pulsar timing refers to the process of unambiguously, and to high precision, accounting for pulse TOAs at a telescope using a relatively simple timing model. To calculate the pulsar timing, Pint uses the following approach: (i) TOAs are obtained, (ii) the pulse emission and propagation time are modeled, (iii) the model is compared against observed data, and (iv) the model is then improved (Luo et al., 2021).

Figure 2 shows the timing residuals for three pulsars after the model from the .par file has been fitted with the .tim file using Pint software. The residual occurs because of a discrepancy between the predicted model and the observed signal. The residuals can be caused by (but are not limited to) measurement errors, spin-down, proper motion, etc. (Gao et al., 2016). If the residual is caused by measurement errors, it generally has a normal distribution with zero mean.

Timing residuals of three pulsars These residuals are caused by the discrepancy between the model and the observations. The residuals are generally on the order of microseconds, following a zero-mean Gaussian distribution.
FIGURE 2:

Timing residuals of three pulsars

These residuals are caused by the discrepancy between the model and the observations. The residuals are generally on the order of microseconds, following a zero-mean Gaussian distribution.

4.2 Deep-Space Navigation

Efforts have been initiated towards navigation using X-ray pulsars (Graven et al., 2008; S. I. Sheikh et al., 2006; S. Sheikh et al., 2007; S. I. Sheikh et al., 2011). In this section, we briefly reiterate these methods. There are two primary methods for utilizing pulsars in navigation: delta correction and absolute navigation. In the delta-correction approach, a spacecraft begins with a coarse position estimate—typically derived from ground-based tracking (e.g., DSN) or onboard inertial navigation. As the spacecraft receives pulsar signals, it measures a TOA deviation Δt relative to the expected arrival time. This deviation corresponds to a range error along the line of sight (LoS) to the pulsar, calculated as Δr = cΔt, where c is the speed of light. By applying corrections derived from multiple pulsars observed sequentially, the spacecraft can iteratively refine its position and adjust its clock. This method is robust, computationally efficient, and especially effective when at least three pulsars are observed over short intervals (Graven et al., 2008).

In contrast, the absolute navigation method solves for the full 4D space-time position (three spatial coordinates and a clock offset) using simultaneous measurements from at least four pulsars. This technique uses the phase differences between pulse profiles to triangulate the spacecraft’s position in inertial space. This approach does not require any initial guess of position or time, offering complete autonomy. However, the absolute navigation method faces challenges in resolving phase ambiguities and often requires multiple simultaneous X-ray detectors or sophisticated sequencing strategies (Becker et al., 2013; Winternitz et al., 2016). Both techniques depend on high-precision pulsar timing models, which are generated and updated through ground-based radio observatories (e.g., NANOGrav, European Pulsar Timing Array, Parkes Pulsar Timing Array). These models must be uploaded to the spacecraft and regularly synchronized; however, with high-stability onboard clocks and predictable pulsar ephemerides, limited gaps in ground contact are tolerable.

Pulsar timing provides distance-independent positional accuracy and enables efficient pointed communication (through attitude adjustment procedures), offering a cost-effective solution. This approach could potentially conserve power and further extend the mission’s operational lifespan. In this work, we do not provide a solution for autonomous navigation using pulsars. Instead, we focus on estimating the errors in range position determination when employing the delta-correction approach, based on the NANOGrav 15-year data set, as detailed in Section 5.2. Thus, this paper provides an error estimation framework rather than a direct method for onboard navigation.

4.3 Stability Analysis Methods

Evaluating the long-term stability of timing signals is crucial in both clock metrology and pulsar timing. While the Allan deviation is a well-established tool for assessing the performance of atomic clocks (Allan, 1966; Rutman & Walls, 1991), it is not ideally suited for pulsar data. Pulsars experience slow but continuous frequency drifts due to intrinsic spin-down, magnetic braking, interstellar medium propagation effects, and other astrophysical processes (Lorimer & Kramer, 2005). Additionally, pulsar observations are typically irregular and sparsely sampled. These characteristics violate the assumptions underlying the Allan deviation, which relies on regularly spaced measurements and is sensitive to long-term trends such as parabolic phase variation (Petit & Tavella, 1996; Taylor, 1991).

To overcome these limitations, we adopt a statistic more appropriate for irregular data: σz(τ), introduced by Matsakis et al. (1997). This measure is derived from a cubic polynomial fit to segments of timing data of duration τ. By focusing on the cubic term of this fit and appropriately normalizing it, σz(τ) effectively characterizes the stability while being robust to linear and quadratic trends, including slow spin-down. This measure is also less sensitive to data gaps compared with the Allan deviation. This statistic is conceptually related to the Hadamard variance and third-difference approaches, which remove frequency offsets and drifts by computing higher-order differences:

D3(t,τ)=125[x(t+τ2)−3x(t+τ6)+3x(t−τ6)−x(t−τ2)]4

Here, x(t) represents the timing residuals, and τ is the interval length. This expression provides an estimate of phase fluctuations that are insensitive to a fixed frequency and linear drift. When squared and averaged, the resulting value becomes proportional to the Hadamard deviation, rather than representing a new metric. σz(τ) incorporates this third-difference concept via polynomial fitting, yielding the root mean square of the cubic coefficients weighted over multiple segments. To calculate σz(τ), the timing residuals are first fit to a cubic polynomial:

X(t)=c0+c1(t−t0)+c2(t−t0)2+c3(t−t0)35

Then, the root-mean-square average of the third-order coefficient is normalized to obtain the following:

σz(τ)=τ225〈c32〉1/26

More details on the in-depth protocol for computing σz have been reported by Matsakis et al. (1997).

4.4 An Onboard Timescale

Although pulsar observations do not provide an absolute time reference, they enable high-precision phase tracking. Compared with a local oscillator, the phase difference between observed pulses and predicted TOAs can be used to correct the onboard clock, for example, via a phase-locked loop (Zhang et al., 2024). We reason that a pulsar ensemble timescale (PET) has advantages over conventional single-pulsar timing. Some variations in the timing residuals—such as red noise, spin irregularities, or measurement noise—are uncorrelated between different pulsars. By combining multiple pulsar signals, these independent errors tend to average out, yielding a more stable and reliable time reference. This approach is conceptually similar to the free atomic timescale EAL (Arias et al., 2011), where multiple atomic clocks are averaged to create a stable ensemble timescale.

For pulsar-based navigation to achieve accurate functioning, a spacecraft must receive up-to-date pulsar parameters, including timing models, binary parameters, positions, and dispersion measures, from the ground station, typically every few months depending on pulsar stability (S. I. Sheikh et al., 2006). However, if communication with Earth is lost, the spacecraft must autonomously manage its timing solution. To enable model updates without ground contact, we propose a combined solution that jointly refines both the onboard timescale and the pulsar timing models. Our method assumes that the spacecraft is equipped with a chip-scale atomic clock (CSAC) that provides short-term accuracy in the range of 100 ns to 1 μs over an hour, corresponding to an Allan deviation of 10−10 to 10−11. During the early mission phase, when communication with ground stations is still active, pulsar parameters are periodically updated. Over this period (typically a few days to weeks depending on pulsar observability), the PET becomes more stable than the CSAC in terms of long-term performance. After this point, the PET can be used to apply corrections to the onboard clock, effectively yielding a hybrid timescale combining the short-term stability of the CSAC and the long-term stability (months to decades) of the PET. After loss of contact with Earth, the spacecraft autonomously updates pulsar models for the newly observed TOAs by precisely timing the pulses using the onboard clock, which remains sufficiently stable over the short durations (minutes to hours) of individual observations. For each pulsar, the frequency of model fitting is based on its known stability and glitch history, ensuring that timing solutions remain accurate while minimizing parameter uncertainty. Crucially, the long-term stability of the onboard clock—on the order of 10−14 or better (see Figure 5) when corrected using the PET—makes it possible to log pulse TOAs over months and years, thereby enabling correction for long-term pulsar drifts. These drifts include those due to secular spin-down, binary orbital motion, red timing noise, and even low-frequency gravitational wave influences. By maintaining an accurate phase history, the system can detect deviations from predicted pulse behavior and iteratively refine pulsar models, even in the absence of ground updates. Thus, the CSAC enables short-term pulse timing, while the PET provides long-term drift correction and timescale continuity. Furthermore, incorporating a spacecraft dynamics model would enable the implementation of a Kalman filter to jointly estimate position and time from pulsar measurements. This approach would allow for autonomous navigation while accounting for measurement noise, dilution of precision, and system dynamics. While Kalman filter-based solutions can be initiated using three pulsars and initial position/velocity estimates, in practice, additional pulsars would be incorporated over the course of the mission as they come into view, thereby preserving observability and limiting long-term uncertainty growth.

This approach constitutes a tight integration of pulsar timing and timekeeping. Newly observed TOAs refine pulsar models, and in turn, those refined models stabilize the PET. This integrated method allows the spacecraft to autonomously maintain timing accuracy, refine navigation estimates, and predict Earth-relative position, thereby enabling more reliable communication re-linking and efficient attitude control. Ultimately, this method reduces the dependency on strong uplink/ downlink signals and supports sustainable long-term operations at interplanetary or interstellar distances. This study establishes a pulsar-based ensemble timescale from archival data, laying the foundation for future work that will explore real-time integration with CSACs.

4.4.1 Weighting Algorithm

To construct the PET, we use the classical weighted average algorithm (Petit & Tavella, 1996). Pulsar data are typically recorded using an observatory clock, often a hydrogen maser, which can drift relative to International Atomic Time (TAI). To monitor and correct for these deviations, GPS is commonly used to track the clock’s discrepancies. In the case of NANOGrav, the data adhere to Terrestrial Time (TT) standards as defined by the Bureau International des Poids et Mesures (BIPM) (Luo et al., 2021). The PET is defined as follows (Petit & Tavella, 1996):

TT−PET=∑i=0Nwi(TT−PT)i7

where N is the total number of constituent pulsars considered in the ensemble and wi is the weight assigned to individual pulsars. In this algorithm, prior to ensemble averaging, the individual pulsar residuals are derived after fitting and removal of a parabolic trend, which effectively captures the long-term prediction of pulse arrivals. This approach is conceptually similar to the approach used in forming the free-running atomic timescale EAL by BIPM (Arias et al., 2011), which considers both predicted and measured clock readings:

xj(t)=EAL(t)−hj(t)=∑i=1Nwi[hi'(t)−xi,j(t)]8

In our case, we set the weights by taking the inverse of σz(τ) values (refer to Section 4.3) and normalizing the total weight to unity:

wi=1/σzi(τ)∑j=0N1/σzj(τ)9

One must use an arbitrary τ value, denoted as τref from here onward, with respect to which σz values are used to calculate the weights. One can choose this value based on the data available and the computational rigor that works best for a given mission scenario.

One of the main concerns in realizing PET is the synchronization of individual pulsar data. Because the observation times for each pulsar are irregular and unevenly spaced (i.e., the observations are conducted at different timestamps for different pulsars), one cannot directly add the weighted data at different timestamps to obtain the ensemble time. To synchronize the data, we first use an averaging filter with a span of 30 days. As explained by Zhang et al. (2024), a 30-day average span is much larger than the upper bandwidth limit of the clock signal (TT(BIPM15) – TT(TAI)); as a result, the spectrum of the clock signal will remain unchanged. If there is a gap of more than 30 days in the data set, cubic interpolation is used to fill in the gaps, resulting in an equi-spaced data set.

We note that the PET presented in this study is constructed using post-processed data, where the timing model parameters for each pulsar (e.g., spin frequency, derivatives, binary motion) are estimated using the entire 15-year observing span. In this study, we assume that future pulsar observations will maintain the stability trends seen to date and select only those pulsars with a history of minimal glitches. However, this assumption limits the robustness of the proposed system, particularly for long-duration autonomous missions. A more realistic implementation would include an error detection and mitigation mechanism, such as a filtering loop (e.g., adaptive Kalman, Vondrak, Wiener, etc.(Yin et al., 2015)) that monitors residuals or adjusts weights in response to anomalous behavior (e.g., rate jumps or timing irregularities). In future work, we intend to study these different filters.

5 RESULTS AND DISCUSSION

5.1 Stability Analysis

In Figure 3, we show σz(τ) values for six pulsars, all exhibiting stability below 1 × 10−14 across averaging times of one year and beyond. It is important to note that the compared space atomic clocks do not exhibit performance beyond 50 days, and extended operation of these clocks may lead to drifts caused by various perturbing effects, including potential aging of the clock components themselves. Table 2 lists key parameters of these six pulsars, including pulse periodicity, duration of observation, number of data points collected, and one-year stability. Each pulsar is a millisecond pulsar with a long observation history. Although these pulsars were observed over extended durations, the observations are sparse; consequently, the number of data points is lower than would be expected under continuous observation. Despite this, the pulsars demonstrate stabilities that rival those of atomic clocks, albeit over much longer averaging times. In particular, PSR J2043+1711 exhibits an astonishingly low σz value of 3.6−1.9+8.4×10−16 at an integration time of approximately 9 years. For the σz values, we show 1-sigma error bars, following the approximations provided by Matsakis et al. (1997). However, we note that these approximations are biased at longer averaging times, and the reported confidence levels may be understated.

σz(τ) for a few pulsars showing the best stability over all time periods These pulsars have stabilities rivaling that of atomic clocks, albeit on a different averaging interval. The solid blue and red curves show the Allan deviation (σz(τ)) for the DSAC (Burt et al., 2021) and an RAFS (Technologies, 2024), which are state-of-the-art space clocks in the industry.
FIGURE 3:

σz(τ) for a few pulsars showing the best stability over all time periods

These pulsars have stabilities rivaling that of atomic clocks, albeit on a different averaging interval. The solid blue and red curves show the Allan deviation (σz(τ)) for the DSAC (Burt et al., 2021) and an RAFS (Technologies, 2024), which are state-of-the-art space clocks in the industry.

View this table:
TABLE 2

Parameters of the Six Pulsars Showcasing High Stability

Parameters include the derived period P (in ms), total observation duration (in years), data size, and σz evaluated at one year.

All of the pulsars exhibit σz (τ)∝τ−3/2 behavior at shorter averaging times, indicative of a source primarily influenced by Gaussian white noise (Matsakis et al., 1997.) This pattern is consistent with measurements involving the determination of event epochs in the presence of white noise, which is largely attributable to receiver noise due to limited SNRs. Therefore, improvements in detection sensitivity can yield better signals from the same pulsars. Between 1 and 4 years, the behavior of σz(τ) resembles the flicker noise observed in atomic clocks, where σz (τ)∝τ0 , i.e., the value remains constant. To elucidate this phenomenon, a detailed investigation of pulsar noise characteristics is warranted. At longer averaging times (beyond 5 years), the σz(τ) curves for some pulsars begin to drift upward because of red noise, which may arise from several factors, including interstellar medium effects, intrinsic spin noise, pulsar mode changes, and gravitational wave influences. These noise sources can be modeled and mitigated using various Bayesian approaches (Goncharov et al., 2021). Interestingly, not all pulsars exhibit such behavior, highlighting the need for further analysis to better understand these discrepancies.

The solid blue and red curves (see Figure 3) represent the Allan deviations of the DSAC (Ely et al., 2022) and a space-qualified rubidium atomic frequency standard (RAFS)(Technologies, 2024), both state-of-the-art space clocks based on lamp-pumped standards using mercury ions and rubidium atoms, respectively. The DSAC has been tested in low Earth orbit and has demonstrated phenomenal stability on the order of 10−15, with low drift rates approaching 10−16 (Ely et al., 2022). Despite their excellent short-term stability, there are no data on the long-term performance of these space clocks, over time periods of decades or even years, as would be relevant for missions like Voyager. While state-of-the-art clocks exhibit white frequency noise on short timescales, long-term drift (random walk) remains a concern in the absence of ground-based synchronization. This reinforces the potential value of pulsars—some of which exhibit superior long-term stability, as discussed in this and the following section.

5.2 Range Positioning Accuracy

In this section, we estimate the range positioning accuracy (error) associated with the delta-correction approach (Graven et al., 2008). This approach provides the spacecraft’s position relative to a reference point—specifically, the SSB—along the LoS to a pulsar. Accordingly, we aim to assess the accuracy with which this method can determine the spacecraft’s position along the LoS. A key point to note here is that all estimations in this Section are independent of the spacecraft’s location, as pulsar signals are uniformly available throughout the solar system. The accuracy of range measurements can be evaluated using basic statistical principles applied to a frequency distribution (S. I. Sheikh et al., 2006; Sullivan et al., 1990), expressed as follows:

σR=cσTOA10

where σR is the spacecraft’s range error, c is the speed of light, and σTOA is the pulse TOA error. From Riley and Howe (2008), we note that the time deviation σx is related to the modified Allan deviation (MDEV) as follows:

σx=τ3Mod.σy11

The time deviation σx is mathematically equivalent to σTOA, neglecting detector-specific parameters. As reported by Riley and Howe (2008), for a power spectral density of the form Sx(f)∞fα−2, MDEV scales as Mod.σy∞τμ'/2, where μ'=−(α+1). This trend is consistent with the scaling observed for σz in pulsar timing data (Matsakis et al., 1997.) Therefore, we assume that the σz statistic for pulsars is analogous to the MDEV in clocks. We compute the final result as follows:

σR=cτ3σz12

We note that in this study, σR is determined based on pulsar stability, derived from σz values, which serve as lower bounds on σR under ideal conditions—namely, the conditions under which the NANOGrav data set was obtained. In practical applications, detector-specific parameters must be incorporated to compute the actual range error.

Table 3 shows the calculated range accuracies for the seven millisecond pulsars discussed earlier. We observe some temporal variation in the σR values, explained as follows. As reported by Matsakis et al. (1997), σz∞τμ'/2, where μ'=−(α+1) for α < 3, which leads to σR∞τμ'/2+1. In the pulsar timing community, the parameter α characterizes the dominant noise type in the data, including random walk, flicker walk, flicker noise, and white noise. Based on these noise types, Table 4 summarizes how σR(τ) trends vary with α. We find that, except for PSR J1713+0747, all pulsars exhibit significant improvements in range accuracy from 100 days to 1 year. Table 4 shows that all pulsars exhibit white noise behavior (α > 1) during this interval. Over longer durations—from 1.5 years to more than 15 years—we observe diverse trends in σR. For PSRs J1918-0642 and J1713+0747, σR increases, suggesting that α < 1 during this period, corresponding to various forms of red noise, as classified in Table 4. In pulsar timing, such noise types are collectively referred to as red noise and may arise from interstellar medium effects, spin noise, or intrinsic pulsar variability. A dedicated noise analysis is needed to classify these noise sources in detail. For the remaining long-term pulsars, the range accuracy remains relatively constant, indicating the presence of a flicker floor. PSRs J2302+4442 and J2043+1711 show white noise behavior across the entire averaging interval from 100 days to 8 years. Overall, all pulsars maintain sub-kilometer-level σR values.

View this table:
TABLE 3

Range Accuracy for Six Pulsars

This table presents four different sets of σR values, showing the range accuracy obtained for different values of the averaging time τ. The first two pulsars have an observation span of less than 16 years (refer to Table 2). We note that because each pulsar has a different observation duration, their corresponding τ values are different. In this table, the range accuracy is shown to the nearest octave.

View this table:
TABLE 4

Behavior of σR(τ) with Respect to α

As reported by Matsakis et al. (1997), σz is proportional to τμ'/2, where μ'=−(α+1) for α < 3. Here, α represents the type of noise present, namely random walk, flicker walk, flicker, and white noise. FM: frequency modulation; PM: phase modulation

We emphasize that although our estimation of σR is proportional to σz, it also depends on the averaging time τ. Defining the range positioning accuracy in this manner reflects how timing instabilities accumulate over time and provides a meaningful metric for estimating long-term range uncertainty in deep-space missions. Therefore, in addition to acting as an error metric, σR also serves as a measure of the long-term reliability of pulsars for navigation. We apply this interpretation when computing range positioning accuracy for the PET (see Section 5.3). We compute the range uncertainty at such large values of τ for two key reasons. First, determining σR at smaller τ values would require σz values at those intervals, which are unavailable owing to the sparse cadence of pulsar observations. One might consider extrapolating the σz curve to lower τ values, but logarithmic extrapolation yields unreliable results and can lead to unrealistic accuracy estimates. To our knowledge, no robust method has yet been established in the literature for such extrapolation. We find it more reliable to compute σR directly from available σz values. Second, the primary motivation for using pulsars in navigation is to support long-duration space missions, where tracking spacecraft and maintaining communication become increasingly challenging because of the immense distances involved. In such scenarios, it is essential to quantify the long-term stability of pulsars.

5.3 Pulsar Ensemble Time

To implement the PET, we first selected a subset of pulsars from the data set, ensuring a diverse range of observation durations and noise characteristics for assessing the effectiveness of the ensemble timescale. While any number of pulsars could be chosen based on these criteria, we selected 11 pulsars that remained consistent (based on σz values) with the raw data after the processing described in Section 4.4.1. Table 5 presents key parameters for these pulsars, including the derived period in milliseconds, observation span in Modified Julian Dates (MJD), total duration in years, and number of data points. The last row presents the corresponding parameters for the PET. The total MJD range—from 53291 to 59081— spans almost 16 years, providing a suitable baseline for our estimates. Notably, despite the extensive observation span, the final data set is reduced to 194 points owing to the application of a 30-day averaging filter.

View this table:
TABLE 5

Parameters of the Pulsars Considered in the Ensemble

Parameters include the derived period in milliseconds, observation span in MJD, total observation duration in years, and number of data points. The last row shows the result for the PET, after the procedure described in Section 4.4.1 has been applied.

Figure 4 shows the resulting ensemble clock residuals, where weights were computed using a reference averaging time (τref) of 3 years. This value yielded the best results, determined via a trial-and-error procedure discussed later. During implementation, we ensured that the residuals from each pulsar aligned with the same MJD grid. Among the 11 pulsars selected, six have data spans of 15 years (2004/05– 2020), two span 12 years (2007/08–2020), and the remaining three cover 8 years (2011/12–2020). Whenever a pulsar was added to or removed from the ensemble, the weights were re-normalized to unity. This dynamic weighting approach facilitated a smooth evolution of the ensemble, preventing any abrupt discontinuities in the timescale, as shown by the red markers in Figure 4. We found that the average difference between consecutive residual points at transition times was consistent with the overall timescale. As different pulsars begin contributing to the PET at different timestamps, we observe a noticeable reduction in residual variation over the 16-year period. The standard deviation of the residuals decreases by an order of magnitude as more pulsars are added. However, this improvement does not scale indefinitely, as the ensemble’s performance is also affected by the types of noise in individual pulsars and their short- and long-term stability.

Ensemble pulsar-clock residual obtained using the algorithm described in Section 4.4.1 The weights were calculated based on individual pulsar stability for a τref of 3 years. The red markers indicate points where new pulsars were added to the ensemble.
FIGURE 4:

Ensemble pulsar-clock residual obtained using the algorithm described in Section 4.4.1

The weights were calculated based on individual pulsar stability for a τref of 3 years. The red markers indicate points where new pulsars were added to the ensemble.

To quantify the improvement offered by the ensemble approach, we computed σz for the PET, as shown in Figure 5. Because the cubic interpolation used for gap filling introduces high-frequency artifacts, we report PET stability only for averaging times longer than 2 years. As previously mentioned, the PET weights are not static but change dynamically based on pulsar availability. Therefore, for each averaging window, the PET reflects a distinct combination of pulsars and weights. For instance, when σz is evaluated for τ = 3 years, different windows encompass different subsets of pulsars, each contributing according to their dynamic weights. Over the range of averaging times from 2 to 15 years, the PET consistently demonstrates superior stability compared with individual pulsars. The σz curve for PET tracks closely with the best-performing pulsars across all timescales without exceeding their stability floors. This trend confirms that the PET effectively retains the strengths of individual pulsars while mitigating their uncorrelated noise—a primary motivation behind ensemble pulsar timescales.

σz(τ) plot for PET Dashed lines represent constituent pulsars. The weights were calculated based on individual pulsar stability for a τref of 3 years.
FIGURE 5:

σz(τ) plot for PET

Dashed lines represent constituent pulsars. The weights were calculated based on individual pulsar stability for a τref of 3 years.

The rationale for selecting τref = 3 years is also evident from Figure 5. Our weighting scheme relies on the σz values of individual pulsars, with the goal of assigning greater weights to those that are more stable. The chosen σz value used in weight computation must lie within a timescale range in which the pulsar stability is well characterized and consistent. For our data set, this range corresponds to averaging times between 2 and 8 years—the upper bound being set by the shortest data span among the pulsars. Within the 2- to 4-year window, we observe that the σz values of most pulsars (with one exception) are relatively stable and comparable, offering a robust basis for weight assignment. Based on this observation, we selected a τref of 3 years. This choice was further validated through trial-and-error tests across various τ values, with τref = 3 years yielding the most favorable results.

We subsequently computed range accuracies (σR) for each constituent pulsar and the PET, as shown in Table 6. As discussed in Section 5.2, σR serves not only as an error estimate but also as an indicator of the system’s reliability. Because the PET is a paper clock formed by combining pulsars in different sky directions, the scalar σR reported here does not correspond to a specific geometric direction. Rather, this term provides a quantitative measure of how ensemble processing improves the robustness of pulsar-based positioning, navigation, and timing (PNT).

Table 6 shows that the PET consistently maintains a range accuracy below 100 m across averaging times exceeding 15 years. Additionally, the PET accuracy matches or surpasses that of the best-performing individual pulsars at all timescales. Importantly, no single pulsar dominates in stability across the entire range of τ. The results affirm a core principle of the ensemble approach—its capacity to enhance both short-term and long-term stability through a judicious combination of individual signals. These findings underscore the potential of ensemble-based pulsar PNT systems to support autonomous deep-space missions over multi-decadal timescales.

View this table:
TABLE 6

Range Accuracy for the Constituent Pulsars and PET

This table shows four different sets of σR values, indicating the range accuracy obtained for different octaves of τ. For the first five pulsars, the observation duration is less than 15.6 years (refer to Table 5). The tabulated values correspond to the nearest octave.

6 CONCLUSION

In this study, we analyzed the stability of pulsars from the NANOGrav 15-year data set and demonstrated their potential for deep-space navigation. This data set provides long-baseline, high-quality timing data for 68 millisecond pulsars, enabling unprecedented evaluation of pulsar stability over decadal timescales. While the idea of using pulsars for navigation dates back to their discovery in 1967, recent advancements in timing precision and detector technology have brought this concept closer to practical realization (Wang et al., 2023). The primary goal of this work was to validate the feasibility of pulsar-based navigation using the most up-to-date radio timing data available. We began by comparing the stability of space-qualified atomic clocks with that of pulsars to highlight the relevance of pulsar-based PNT systems. Using the σz metric, we estimated range positioning accuracy under ideal conditions for the delta-correction method, across averaging intervals ranging from 100 days to over 15 years. For most pulsars, we observed sub-kilometer position accuracy across all averaging times.

In practical applications, navigation would rely on simultaneous observations of multiple pulsars, enabling not only three-dimensional positioning but also the creation of an ensemble timescale. We implemented such an ensemble—the PET— which exhibited long-term instability below 1 × 10−14, averaging 6 × 10−16 beyond 10 years. While our analysis shows that the long-term stability of the best-performing millisecond pulsars rivals, or even exceeds, that of current atomic clocks, we acknowledge that next‐generation optical and nuclear frequency standards have demonstrated fractional uncertainties below 10−18 on short timescales. However, these emerging technologies have not yet been employed over space mission durations of years to decades. We further computed the range accuracy for PET to quantify the improvement in reliability, demonstrating that ensemble pulsar timing effectively suppresses independent noise sources and instabilities in individual pulsars. These results emphasize the advantages of pulsar-based navigation, especially for long-duration missions where reliance on Earth-based tracking systems becomes increasingly impractical. We interpret PET not merely as a stable clock reference, but as a mission-enabling technology—critical for autonomous onboard navigation, clock correction, and sustained mission operations in deep-space environments.

This work supports the broader application of pulsars for precision timing, in addition to navigation. Pulsar-based timing could complement atomic clocks to enable hybrid PNT systems capable of autonomous operation over decades, without sacrificing stability or reliability. We believe that this study lays the foundation for future investigations into broader applications such as coordinated timekeeping for lunar or planetary systems. In subsequent work, we aim to quantitatively evaluate the contribution of a pulsar-derived timescale to onboard clock correction, position solution accuracy, and time-to-relock in communication-constrained environments. Future directions also include refining ensemble construction algorithms, integrating detector-specific parameters to support onboard X-ray detection, and advancing noise modeling techniques to optimize the use of pulsars in space-based navigation systems.

CONFLICT OF INTEREST

The authors declare that there are no conflicts of interest regarding the publication of this paper.

AUTHOR CONTRIBUTIONS

V. I.: formal analysis, data curation, investigation, methodology, validation, writing—original draft preparation. T.N.B: conceptualization, methodology, writing— review and editing, supervision, project administration.

HOW TO CITE THIS ARTICLE:

Iyer, V., & Bandi, T.N. (2026). Pulsars as Natural Oscillators for Long-Term Deep-Space Missions. NAVIGATION, 73. https://doi.org/10.33012/navi.733

ACKNOWLEDGMENTS

We express our sincere gratitude to the NANOGrav Collaboration for providing their invaluable 15-year pulsar data set, which has been instrumental to this research. We also acknowledge their Pint-Pulsar Python module, a crucial tool for pulsar timing analysis and navigation studies. Additionally, we thank Dr. Demetrios Matsakis, Jahnvi Verma, and Luna Kronzer for their insightful suggestions and constructive feedback. The authors appreciate the University of Alabama College of Arts and Sciences for supporting the Quantime Lab. More information is available at https://Quantime.ua.eduhttps://Quantime.ua.edu.

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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In this Issue

NAVIGATION: Journal of the Institute of Navigation

NAVIGATION: Journal of the Institute of Navigation

Vol. 73

January 2, 2026

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Keywords

deep-space navigation, NANOGrav, positioning, pulsar, stability, timing