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A&A
Volume 703, November 2025
Article Number A195
Number of page(s) 14
Section Celestial mechanics and astrometry
DOI https://doi.org/10.1051/0004-6361/202554869
Published online 18 November 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1 Introduction

Neptune, as the outermost planet in the Solar System and one of only two ice giants (the other being Uranus), plays an important role in our understanding of planetary evolution and the diversity of exoplanetary systems (Beddingfield et al. 2021; Fortney et al. 2021). Despite its scientific significance, deep-space explorations of ice giants remain limited, with only the Voyager 2 flyby in the 1980s providing high-resolution direct observations (Smith et al. 1986, 1989). The absence of detailed in situ observations has positioned ice giants as high-priority targets for international deep-space exploration (Guillot et al. 2021), including the Neptune Odyssey mission concept (Rymer et al. 2021) and China’s prospective Neptune orbiter missions (Yu et al. 2021). These probes are planned to reach Neptune around 2040–2050.

Numerically integrated planetary ephemerides, foundational to interplanetary missions since the 1960s (e.g., Mariner 4 to Mars) (Newhall et al. 1983; Moiseev & Emelyanov 2024), have evolved significantly through advances in celestial reference frames (e.g., ICRF, HCRF, and Gaia-CRF) (Fienga et al. 2011; Gaia Collaboration 2022; Deram et al. 2022; Liu et al. 2023), Earth orientation models, and observational techniques (e.g., spacecraft radiometry and CCD astrometry) (Standish 1990; Park et al. 2021; Fienga et al. 2021; Pitjeva et al. 2022). The accuracy of planetary ephemerides is an important factor in planning and navigations of interplanetary space missions and often requires a specific assessment for each mission. This assessment requires a characterization of the observation accuracy and the possible systematic errors. Cross-validation methods, which compare orbits derived from non-overlapping or partial datasets, are essential to quantifying these errors (Folkner 2011).

For Neptune, achieving high-accuracy ephemerides is not only necessary to optimize its space missions and expansion potentials, but also to advance scientific objectives such as gravitational wave detection by pulsar timing (Arzoumanian et al. 2018; Lazio et al. 2018; Vallisneri et al. 2020) and planetary dynamics. To enhance the ephemeris accuracy and alignment with Gaia-CRF, we introduce the following new data:

  • six-year Gaia Focused Product Release (FPR) astrometry for Triton (2014–2019) in sub-mas accuracy (Gaia Collaboration 2023);

  • forty-two-year stellar occultation astrometry (1981–2022) reduced with Gaia DR3 (Gaia Collaboration 2021), enhancing the long-term accuracy;

  • a Triton ephemeris (Yuan et al. 2021a,b) independently updated with new ground-based CCD observations based on Gaia catalogs until 2024, reducing the systematics from Triton orbit.

In addition, the reanalysis of historical data (e.g., relative angular measurements against nearby stars) based on the Gaia catalogs helps correct systematics introduced by legacy star catalogs. The present study focuses on the systematics analysis and refinement of the orbit of the Neptune system barycenter (NSB). The methodologies and analytical insights derived from this work will serve as a reference for our future planetary ephemeris development. We describe the construction method and results of the new orbital ephemeris for NSB in Sect. 2, followed by details of treatments and residuals of observational data in Sect. 3. Section 4 presents analysis and discussion of the post-fit results, including the consistency between new observations, the Triton ephemeris impact, the uncertainties of the new NSB ephemeris, its comparison with existing planetary ephemerides, and the contributions from different data. We conclude in Sect. 5 with a summary of improvements and future perspectives, including strategies to further reduce systematic errors.

thumbnail Fig. 1

Comparison of NEP097 and FORCES-8-MAIN-2024 (F8M24) ephemerides for Triton’s orbit relative to the NSB. Positional differences are presented in radial (R), transvers (T), and out-of-plane (N) directions. The norm of the 3D differences (Δd) are also shown.

Ephemeris construction

2.1 Background and data strategy

The latest widely used planetary ephemerides have been developed by three institutions: DE440 (Park et al. 2021) by NASA’s Jet Propulsion Laboratory (JPL); EPM2021 (Pitjeva et al. 2022) by the Institute of Applied Astronomy of the Russian Academy of Sciences (IAA/RAS); and ΓNPOP2la (Fienga et al. 2021) by the Observatoire de la Côte d’Azur of the Institut de Mécanique Céleste et de Calcul des Éphémérides (OCA/IMCCE) in France. All the three ephemerides were released in 2021, succeeding their previous versions (DE438 (Folkner & Park 2018), EPM2017 (Pitjeva & Pitjev 2Ol8a,b), and INPOP19a (Fienga et al. 2019)) published between 2017 and 2019.

In our ephemeris development, DE440 provided the positions and masses of the bodies in the perturbation model. For NSB orbit, a correction was implemented by adjusting initial conditions in the numerical integration to achieve a weighted least-squares fit to observational data. We prioritized the observational data adopted in the widely used DE440 planetary ephemeris and applied necessary corrections. Currently, aside from the three-dimensional (3D) normal point observation from Voyager 2, which is already reduced to the NSB, all other observations require transformation from targets (Neptune or its moon Triton) to the NSB using main satellite ephemerides of Neptune. Moreover, to further enhance the ephemeris accuracy, we incorporated Gaia FPR astrometry for Triton (sub-mas accuracy) and stellar occultation astrometry of Neptune and Triton reduced with Gaia DR3 (accuracy ranging from sub-mas to tens of milliarcseconds). Thus, the accuracy of main satellite ephemeris of Neptune becomes a critical factor in maximizing the effective utility of these new datasets.

2.2 Triton ephemeris as an independent update: FORCES-8-MAIN-2024

The latest Neptune satellite ephemeris from JPL is NEP0971, providing Neptune and Triton positions relative to the NSB. Additionally, our team developed the independent FORCES-8-MAIN-2021 (F8M21) ephemeris (Yuan et al. 2021a,b), which has been updated to FORCES-8-MAIN-2024 (F8M24) with the following changes: 1) the inclusion of recent ground-based observations (Zhang et al. 2021, 2022; Yan et al. 2020, 2022) and new Yaoan Telescope data (2021-2024); and 2) planetary and satellite masses and Neptune J2, J4 from DE440/NEP0972 (Park et al. 2021; Brozović & Jacobson 2022). The update is available via the PMO website3 and will be detailed in another paper (in prep.). Notably, both F8M24 and NEP097 use identical positions and masses of Sun and planets from DE440 for perturber modeling. The differences between NEP097 and F8M24 are shown in Fig. 1. Then, in the subsequent NSB ephemeris construction (Sect. 2.3), the updated F8M24 was adopted as the main satellite ephemeris.

2.3 Neptunian system barycenter ephemeris: FORCES-8-BARY-2024

2.3.1 Dynamical modeling

The dynamical model for NSB orbit includes: (1) Newtonian point-mass attractions from Sun, Mercury, Venus, Earth, Moon, and the system barycenters of Mars, Jupiter, Saturn, Uranus, and Pluto; and (2) figure perturbations from the Sun’s second-degree zonal harmonic, where Sun J2 are adopted from DE440. Notably, the adopted masses are identical to those used in F8M24 and DE440/NEP097. The numerical integration method is detailed in Sect. 2.3.2. Other perturbations modeled in DE, EPM, and INPOP ephemerides are excluded from this baseline model, but they are evaluated in Sect. 2.3.3.

thumbnail Fig. 2

Positional discrepancies in stepwise forward-backward NSB orbit integration tests across forward spans ranging from −200 to +200 years. Sub-kilometer discrepancies validate the numerical integrator’s accuracy.

2.3.2 Numerical integrator settings

The numerical integration was performed using MRA15 (Yuan et al. 2021a), a modified version of the IAS 15 integrator (Rein & Spiegel 2015), which has been successfully employed in developing the F8M21 and F8M24 satellite ephemerides. The predictor-corrector and step-size control parameters are set as 10−16 and 10−14, respectively, to adapt to the highest observation accuracy. Its numerical accuracy was validated through stepwise forward-backward integration tests, with the maximum positional discrepancy across all tests remaining below 1 km, as shown in Fig. 2. In particular, for each forward integration span of n (assigned to every integer from −200 to +200) years, we integrate NSB orbit for n years and back to the initial epoch (J2000.0 TDB) using the initial condition obtained from DE440, finding the maximum positional difference between the two integrating orbits.

2.3.3 Impact of dynamical model systematics on post-fit orbits

A key practical consideration is to assess the impact of dynamical model systematics on the reliability of orbit extrapolation. These systematics include variations in perturbers, perturbation factors (e.g., main-belt asteroids, Kuiper-belt objects (KBO), and post-Newtonian effects), Solar System barycenter (SSB) positions, dynamical parameters, and other potential unknowns. To address these aspects, we fit our dynamical models to synthetic observations generated from multiple reference planetary ephemerides (e.g., DE440, DE438, EPM2021, EPM2017, INPOP21a, INPOP 19a), which represent diverse dynamical modeling approaches.

To preserve the observational constraints, the synthetic datasets are constructed to replicate the types, epochs, and weights of the actual observations (see Sect. 3). We applied identical weighting schemes to fit our dynamical model to these synthetic datasets and compared the resulting post-fit orbits with the reference ephemerides. The discrepancies illustrated in Fig. 3 reflect biases introduced by differences in the dynamical models, thereby serving as a clear indicator of their impact on the fitted orbits. The observation weights are also shown in Fig. 3, as proxies for observation accuracy.

These discrepancies at the epochs of actual observations are negligible compared to the corresponding observational accuracy. Moreover, for the period between 2040 and 2050, when planned Neptune missions are expected to arrive, positional discrepancies for DE and EPM models remain consistent at ~50 km, while the INPOP21a model exhibits larger deviations up to ~300 km. This highlights the significant influence of KBO handling differences on the NSB orbit. Notably, DE440, EPM2021, and INPOP 19a use ring approximations, whereas INPOP21a adopts individual elliptic orbits for KBOs. Given that all observed discrepancies are smaller than the NSB positional uncertainty of ~500 km in our final orbit-fit solution (Sect. 4.3), we retained the baseline dynamical model described in Sect. 2.3.1. However, when enough high-accuracy data become available in the future, the impact of KBO might not be negligible and would merit further investigation.

2.3.4 Final orbit-fit solution

Based on the data treatments detailed in Sect. 3, we determined NSB orbit using DE440 data, complemented by new Gaia FPR astrometry and stellar occultation measurements. Additionally, the updated F8M24 Triton ephemeris significantly enhances the utilization of high-accuracy Gaia FPR and stellar occultation astrometry for Triton. The post-fit initial conditions (position and velocity relative to the DE440 SSB in the ICRF) and the correlation matrix, defined at the J2000.0 TDB epoch, are provided in Table 1 and Fig. 4, respectively. In addition, the residuals for all datasets are presented in Sects. 3 and Appendix A. The distributions of these residuals conform to expectations, confirming the robustness of our fitting procedure. The generated NSB ephemeris, F8B24, is available via the PMO website4.

3 Observational data treatments and residuals

3.1 DE440 observational data with necessary corrections (1904–2016)

The DE440 ephemeris incorporates observational data spanning 1904 to 2016, targeting Neptune and its largest moon Triton. Except for the relative angular measurements from Yerkes Observatory and Triton CCD astrometry from Yunnan Kunming 1-meter telescope (Wang et al. 2017), all data are publicly available via the JPL Solar System Dynamics (SSD) website5.

thumbnail Fig. 3

Post-fit NSB orbit sensitivity to dynamical model variations (DE, EPM, INPOP) in simulated orbit-fit tests, as well as the adopted actual observation weights serving as proxies for observation accuracy. Geocentric positional differences in the ICRF are presented in right ascension (α), declination (δ), and range (D) directions. The norm of the 3D differences (Δd) are also shown. The yellow region marks the approach window of currently planned Neptune missions (2040–2050).
Table 1

Initial conditions at the J2000.0 TDB epoch for the final orbit-fit solution of NSB, with (px, py, pz) and (υx, υy, υz) representing the NSB position and velocity relative to the DE440 SSB in the ICRF, respectively.

thumbnail Fig. 4

Diagonal correlation matrix of the initial conditions of the new NSB ephemeris FORCES-8-BARY-2024.

3.1.1 Relative angular measurements of Neptune against nearby stars corrected by Gaia catalogs (1904–1922)

DE440 incorporates relative angular measurements of Neptune against nearby stars obtained from the Yerkes Observatory between 1904 and 1922 by Barnard (1906, 1907, 1909, 1912, 1916, 1919, 1927), expressed as position angles (PA) and angular separations (Sep.) of apparent positions of stars relative to Neptune. While modern star catalogs provide stellar positions in right ascension (RA) and declination (Dec), their accuracy is limited by uncertainties in stellar proper motions. These observations are not available via the JPL SSD website and must be extracted from original publications, requiring a manual cross-matching of the observed stars. In contrast to the approach taken in DE440, we utilized the latest Gaia catalog positions for star matching, establishing them as reference points for Neptune’s relative angular measurements. The matching criteria include Neptune positions from DE440 and angular constraints from the observational data, deriving the following matching results. From January 18–22, 1922: stellar positions were sourced from Gaia DR2 (Gaia Collaboration 2018), with catalog-derived positional uncertainties of ~ 100 mas; from other epochs: stellar positions were sourced from Gaia DR3, with all catalog-derived positional uncertainties <6 mas. After excluding stars unmatched in Gaia catalogs and two outliers with post-fit residuals larger than 3000 mas, 48 data points were retained across the two angular measurement directions, as presented in Table A.1. The residuals are shown in Fig. 5, with their standard deviations in PA and Sep. are 330 mas and 390 mas, respectively.

In our orbit determination process, we conservatively set the initial weights to 500 mas to account for unmodeled systematic errors in stellar proper motions. Subsequent fitting results revealed no need for further weight adjustments, as the root-mean-square (RMS) of residuals remained consistent with the initial weighting scheme.

thumbnail Fig. 5

Relative angular measurements in position angles (θ) and angular separations (ρ) for Neptune against nearby stars. The pre-flt residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared.

3.1.2 Meridian transit astrometry of Neptune (1926–1998)

DE440 incorporates the following meridian transit observations of Neptune: (1) Washington Observatory 6-inch telescope: data spanning 1926–1993; (2) La Palma Observatory: data spanning 1984–1998; and (3) Bordeaux Observatory: data spanning 1985–1993. These observations, shown in Fig. A.1, provide astrometric positions (RA and Dec) in the true equator and true equinox of date (TETE) reference frame. To align these positions with the ICRF, the IAU 2006/2000A precession-nutation model (Wallace & Capitaine 2006) was applied.

For Washington Observatory data prior to 1976, which were originally tied to the FK4 system, we corrected the systematic offsets between the FK4 and FK5 systems using the equinox correction formula proposed by Fricke (1982), as recommended by JPL (Standish 1990; Standish & Williams 2012).

In our orbit determination process, meridian transit observations are assigned weights of 500 mas following the VFCC17 (Vereš et al. 2017) weighting scheme. This conservative value slightly exceeds the standard deviation of residuals relative to DE440, ensuring robustness against unmodeled systematic errors.

3.1.3 Photographic and CCD astrometry of Neptune and Triton (1961–2016)

DE440 incorporates the following photographic and CCD astrometric datasets: (1) Mykolaiv Observatory: Photographic astrometry of Neptune spanning 1961–1999 reduced with the Extra-Galactic Reference Frame (EGRF) catalog (Johnston et al. 1995); (2) Flagstaff Observatory: CCD astrometry of Neptune and Triton spanning 1995-2015 reduced with the EGRF, ACT (Urban et al. 1998), and Tycho-2 (Høg et al. 2000) catalogs; (3) Table Mountain Observatory: CCD astrometry of Neptune and Triton spanning 1998-2013 reduced with the Tycho-2 catalog; and (4) Yunnan Kunming 1-meter Telescope: CCD astrometry of Triton spanning 2014–2017 reduced with the Gaia DR1 catalog (Lindegren et al. 2016). While the first three datasets are accessible via the JPL SSD website, the Kunming data are sourced from the Natural Satellites Database (NSDB) (Arlot & Emelyanov 2009; Arlot et al. 2024). These observations are shown in Figs. A.2 and A.3.

The weighting scheme in our orbit determination process follows VFCC17: (1) Photographic data (post-1950): 2500 mas; (2) Flagstaff CCD data: 500 mas; and (3) Table Mountain CCD data: 300 mas. Furthermore, following Eggl et al. (2020), no catalog biases are applied, as the ECRF, ACT, Tycho-2, and Gaia DR1 catalogs show no large scale structures in local position and proper motion systematics.

Notably, the Kunming data exhibit a significant positive systematic offset (3–4σ) in right ascension during NSB orbit determination, regardless of whether they are included in the fit. These data are retained in the construction of the F8M24 Triton ephemeris, where such systematics can be effectively mitigated by introducing adjustable bias parameters that account for ephemeris-, catalog-, and observer-specific effects, following common practices in satellite orbit fitting (e.g., Yuan et al. 2021a; Jacobson & Park 2025). However, since the exact source of the RA bias remains unclear, we chose not to apply such bias adjustments in the final F8B24 solution for NSB at this stage, though such corrections may be considered in future iterative refinements, pending further investigation.

3.1.4 Voyager 2 3D normal point observation of the Neptunian system barycenter (1989)

During its 1989 Neptune ffyby, Voyager 2 provided the high-precision 3D normal point observation for NSB, derived from a synthesis of spacecraft tracking measurements through its trajectory determination. This observation includes the geometric RA, Dec, and two-way range of NSB relative to Earth at a specific Barycentric Dynamical Time (TDB) epoch, along with their uncertainties. These values were adopted by the DE440 ephemeris without any additional stellar aberration or light-time corrections. The residuals relative to DE440 are shown in Fig. A.4. In our orbit determination process, the weights assigned to the observation are set to their reported uncertainties.

3.2 Gaia FPR astrometry of Triton (2014–2019)

The Gaia mission released its Focused Product Release (FPR) in 2024, marking its second publication of natural satellite astrometry. Triton observations span 2014–2019 and include: (1) stellar-aberration-corrected positions of Triton relative to the Gaia spacecraft, incorporating light-time and gravitational deflection effects; and (2) Gaia spacecraft positions and velocities relative to Earth. Pre-fit residuals of the Gaia FPR data relative to DE440/NEP097 ephemerides (shown in Fig. 6) reveal systematic offsets of ~7 mas (1σ level) and distinctly bimodal distributions in the along-scan (AL) direction. Such asymmetric profiles and bimodal structures indicate significant astrometric discrepancies.

Given the sub-mas accuracy of Gaia FPR observations, systematic errors in the transformation from Triton to NSB become significant. The primary sources are Triton’s orbit systematics, as well as its photocenter-barycenter offset arising from phase effects and nonuniform albedo distribution. For phase effects, theoretical models constrain these shifts to ~ 1 mas (Lindegren 1977). For nonuniform albedo distribution, V-band observations show 0.1 mag peak-to-peak variability (Hicks et al. 2022) during Triton’s 5.9-day synchronous rotation. While this effect has been modeled for Pluto (Buie et al. 2012; Brozović & Jacobson 2024), we treated it as high-frequency noise relative to Neptune’s low-frequency orbital systematics over the Gaia FPR baseline, approximating it as random error. Accordingly, we conservatively assigned Triton a 5-mas contribution to the weights. As shown in Fig. 6, post-fit residuals have near-zero means and standard deviations slightly smaller than this weight increment, validating our treatment of these systematics.

thumbnail Fig. 6

Gaia FPR astrometry of Triton in along-scan (AL) and across-scan (AC) directions. The pre-fit residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared.

3.3 Stellar occultation astrometry of Neptune and Triton (1981–2022)

Stellar occultations are accurate timings of stars being occulted by Solar System bodies, which provide accurate astrometric positions of the occulting bodies relative to the stars. Combined with accurate stellar positions, these measurements yield accurate astrometry for outer Solar System objects, overcoming limitations imposed by distance and photocenter-barycenter shift (Sicardy et al. 2024; Yuan et al. 2024).

Historically, the DE405 ephemeris incorporated stellar occultations by Neptune in 1968 and 1981–1985 (Standish & Williams 2012), reduced using pre-1992 star catalogs with assumed uncertainties of ~200 mas (available via the Astro-metric Planetary Database, APDB6). Subsequent ephemerides (e.g., DE421, DE430, DE440) excluded these data (Folkner et al. 2009, 2014; Park et al. 2021); however, an extended dataset through 1989 (Kovalevsky & Link 1969; French et al. 1983, 1985; Hubbard et al. 1985; Covault et al. 1986; Nicholson et al. 1990; Sicardy et al. 1991) was reused in the 2008 Voyager 2 trajectory reconstruction to refine DE42l ephemeris for NSB (Jacobson 2008). Astrometry from the 1968 and 1983 occultations was reduced using the Hipparcos catalog (van Leeuwen 2007), while others relied on UCAC2 (Zacharias et al. 2004).

Since 1989, more stellar occultations by Neptune and Triton were observed and published (Roques et al. 1994; Elliot et al. 2000; Uckert et al. 2014; Marques Oliveira et al. 2022; Yuan et al. 2024). Notably, the ground-based stellar occultations by Triton have not been used for the construction of a NSB ephemeris before. This work will introduce this astrometry for the first time.

Modern Gaia stellar positions, with an improvement in accuracy of up to two orders of magnitude over UCAC2, allow us to use the occultation data as one of the most accurate astrometric sources for outer Solar System objects (Desmars et al. 2019). We analyzed occultation data from 1981 to 2022, deriving the astrometry based on Gaia DR3 listed in Table A.2. Process for the astrometry reconstruction is described below.

For occultations providing reduced half-light immersion and emersion times, we fit a five-parameter elliptical model to derive geocentric ICRF-referenced RA–Dec offsets and uncertainties using the SORA library (Gomes-Júnior et al. 2022). Model systematics were conservatively propagated through Monte Carlo simulations with wide-enough uniform parameter sampling. The model parameters were set as follows: (1) both the ƒ and 𝑔 coordinates of the ellipse center are set as 0 ± 5000 km; (2) the equatorial radius and oblateness of the ellipse are set as 25 159 ± 100 km7 and 0.018 ± 0.01, respectively (French et al. 1998, 1985); (3) the pole position angle is adopted from the IAU model with ±10° interval; and (4) the model systematic uncertainty is set at 100 km. In addition, time uncertainties are assigned as 5 s (if reported significant figure are given to 0.1 s) or 10 s (otherwise). Post-fit residuals exhibit a small per-degree-of-freedom χ2 < 0.1, prompting a halving of the derived astrometric uncertainties to systematically correct their overestimation.

For occultations providing solved values and uncertainties of closest-approach angular separations and times, we simply used the coordinate transformation and corresponding error propagation formulae to convert them into RA–Dec offsets and uncertainties.

In addition, to make proper use of occultations by Triton, the theoretical positional uncertainties of Triton were also included and set to 5 mas, as in the Gaia FPR weighting scheme. The 1968 data were excluded due to potential binary motion of the occulted star HIP 766058 (flagged as a Gaia non-single-star, NSS, acceleration solution). The 1981 occulted star, classified as a single-lined spectroscopic binary (period ~23.5 days) in Gaia DR3, was retained as its astrometric NSS signal is not significant enough to generate an astrometric NSS solution.

The renormalized unit weight error (RUWE)9 given by Gaia catalogs is expected to be ~ 1.0 for sources well-fit by a single-star model, while a value greater than 1.4 may indicates NSS behavior or problematic astrometric solutions. Although RUWE serves as an NSS indicator, tests presented in Figs. 7 and 8 show negligible differences in NSB orbital solutions when applying RUWE cuts, justifying their relaxation in the final analysis.

thumbnail Fig. 7

Stellar occultation astrometry of Neptune and Triton based on Gaia DR3 in geocentric ICRF-reſerenced right ascension (α) and declination (δ). Residuals relative to DE440 ephemeris, the occultation-only orbit-fit tests using various RUWE cuts, and our orbit-fit solutions, using NEP097 and FORCES-8-MAIN-2024 (F8M24), are presented.
thumbnail Fig. 8

Post-fit residual distributions of Gaia FPR Triton astrometry in the along-scan (AL) direction. Occultation-only orbit-fit solutions using three RUWE selection criteria (no limit, <1.4, <1.0) and the all-data solution are presented. Each panel compares solutions using NEP097 (left) and FORCES-8-MAIN-2024 (F8M24; right) Triton ephemerides. The all-data F8M24-based solution corresponds to our final FORCES-8-BARY-2024 (F8B24) NSB ephemeris.

4 Analysis and discussion

4.1 Impact of Triton ephemeris choices

We evaluated the impact of Triton ephemeris selection by comparing Gaia FPR residuals in the high-accuracy AL direction from solutions based on NEP097 and F8M24, as shown in Fig. 8. For all the occultation-only solutions, NEP097 yields mean AL residuals around 2 mas, whereas F8M24 achieves improved performance with mean residuals at sub-mas levels. For the all-data solutions, while both Triton ephemerides yield sub-mas AL residual means, only the F8M24-based solution (F8B24) exhibits a unimodal, near-Gaussian residual distribution. One possible explanation is that F8M24 employs a slightly more consistent dynamical model for Triton, incorporates new high-precision datasets, and handles observations more effectively (Yuan et al. 2021a,b), despite both NEP097 and F8M24 being constructed on the basis of DE440.

thumbnail Fig. 9

Final orbit-fit solution for NSB. Least-squares propagated formal uncertainties are derived using a nonlinear mapping of the covariance matrix (CM) based on a MCCM/RP method. Uncertainties based on the Jackknife resampling method are also presented. Geocentric positional differences in the ICRF are presented in right ascension (α), declination (δ), and range (D) directions. The norm of the 3D differences (Δd) are also shown. Exc. means excluding, V2 3D means Voyager 2 3D, Occ. means occultation, Rel. means relative, Mer. means meridian, Pho. means photographic, and Obs. means observations.

4.2 Consistency between occultation astrometry and Gaia FPR

Building on the above comparison, we then examined the consistency between Gaia DR3-based occultation astrometry and Gaia FPR. Theoretically, they are both aligned to the Gaia-CRF and should therefore exhibit mutual consistency as long as they are properly reduced and applied.

To validate this alignment, we focused on the Gaia FPR AL residuals resulting from occultation-only orbit-fit solutions for NSB using the F8M24 Triton ephemeris. As shown in Figs. 6 and 8, the post-fit AL residual means are significantly reduced (from ~7 mas to sub-mas levels) across all F8M24-based solutions. These results are consistent with the final F8B24 solution and confirm a high-accuracy alignment between occultation astrometry and Gaia FPR.

4.3 Orbital uncertainty analysis

Least-squares propagated formal uncertainties are useful for assessing orbital precisions, but they often underestimate the realistic uncertainties, as they rely on the assumption of well-characterized random errors and do not account for potential systematic biases. Accordingly, JPL recommends adopting realistic uncertainties up to three times the formal values for the Uranus system barycenter (USB) ephemeris (Jacobson & Park 2025). Figure 9 presents the 3σ formal uncertainties of NSB’s geocentric position (RA, Dec, and range) in the ICRF, derived using a nonlinear mapping of the covariance matrix (CM) shown in Table 1 and Fig. 4. This was achieved using the Monte Carlo random parameter (MCCM/RP) method (e.g., Desmars et al. 2009; Emelyanov 2010) and extracting the 0.15th and 99.85th percentiles of the distribution of 1000 simulated solutions.

Another method that JPL adopts to estimate realistic uncertainties is comparing orbits derived from non-overlapping or partial datasets, which helps to reveal and quantify systematic errors. To implement a similar strategy, we applied a Jackknife resampling method: observational data were grouped by type, date, and target, and a total of 1000 orbital fitting solutions were computed, each excluding a randomly selected 5% subset of the data. The 3σ uncertainties were then derived from the 0.15th and 99.85th percentiles of the distribution of these Jackknife solutions and displayed in Fig. 4.

The difference in uncertainty between the Jackknife and MCCM/RP methods occurred mainly in the period leading up to the Voyager 2 ffyby of Neptune. However, both methods suggest that NSB position uncertainty is at the level of hundreds of kilometers over the 1950–2050 interval. In particular, during the critical 2040–2050 interval that the planned Neptune mission is expected to reach, the uncertainty is ~500 km, which is larger than the dynamical model systematics evaluated in Sect. 2.3.3.

4.4 Contributions from the various observational datasets

To determine the contribution of each dataset to the overall orbital accuracy, we implemented a cross-validation approach: datasets were partitioned by type and the orbits were repeatedly fitted, while excluding one subset at a time. The orbital differences are displayed in light solid lines in Fig. 9, where the largest ones indicate that the long-term uncertainty of NSB orbit is primarily constrained by 42 years of Gaia DR3-based occultation astrometry. The two next most influential factors are the 1989 Voyager 2 3D normal point observation and 6 years of Gaia FPR astrometry. Overall, the former contributes more significantly to the accuracy of the past orbit, while the latter enhances that of the present orbit and provides comparable constraints to Voyager 2 for future orbital extrapolation.

To verify the dominance of these three factors in constraining the overall orbital accuracy, we derived a simplified orbital solution using only these three datasets. The resulting solution deviates from the final orbit at the level of 1.5σ uncertainty, consistent with expectations. However, removing Voyager 2 data, retaining only the two new datasets (Gaia FPR and occupations), leads to rapidly growing deviations, exceeding 1000 km in post-2040 and pre-1980 orbits.

For comparison, a solution excluding the two new datasets but incorporating all other data (including Voyager 2) remains compatible with the DE440 ephemeris at the several-thousand-kilometer accuracy level, with trends similar to DE440. While the deviation growth rate is slower than DE440 (likely due to differences in datasets and weighting), discrepancies exceeding 1000 km from the final orbital solution emerge even in the present epoch. Notably, as expected, the inclusion of Voyager 2 data ensures superior accuracy for past orbits compared to the solution based on the two new datasets only.

thumbnail Fig. 10

Comparison of the latest two versions of the DE, EPM, and INPOP ephemerides and our final FORCES-8-BARY-2024 (F8B24) solution for NSB orbit. Geocentric positional differences in the ICRF are presented in right ascension (α), declination (δ), and range (D) directions. The norm of the 3D differences (Δd) are also shown.

4.5 Comparison with latest planetary ephemerides

While the latest planetary ephemerides differ slightly in terms of observational data, dynamical models, and fitting techniques, their orbital differences provide a complementary insight into solution accuracy and reliability. Figure 10 shows the inter-ephemeris differences for NSB. Most positional differences between 1950 and 2050 remain at the level of several thousand kilometers, consistent with the accuracy reported for DE430 (Folkner et al. 2014). However, EPM2021 shows significantly larger discrepancies, exceeding 15 000 km before 2040, and deviates markedly from all the other ephemerides. These large discrepancies in EPM2021 have also been reported by Moiseev & Emelyanov (2024), who advise caution when using it for high-precision applications. With corrections applied to legacy data and the inclusion of new high-accuracy observations and updated Triton orbit (F8M24), the final F8B24 solution achieves a positional correction of ~5000 km over 2040–2050, aligning with DE430’s claimed accuracy and demonstrating the necessity of our refinement process for future mission planning.

5 Conclusions

By analyzing and using DE440 data along with new Gaia FPR and Gaia DR3-based occultation astrometry, as well as our independently developed and updated F8M24 Triton ephemeris, we have successfully refined the NSB orbit and robustly evaluated its accuracy. The post-fit residuals for each dataset are summarized as follows:

  • Gaia FPR astrometry: the residuals follow a distribution closely approximating a normal distribution centered at zero. Specifically, the mean and median of residuals in the along-scan (AL) direction are both within 0.5 mas, with a standard deviation of ~2 mas;

  • Stellar occultation astrometry based on Gaia DR3: the residuals fall well within the expected range explained by the corresponding weights;

  • Other data: the distributions of residuals are consistent with our initial expectations.

And key findings include:

  • Sub-mas alignment consistency between stellar occultations and Gaia FPR asatrometry: the alignment discrepancies between the two datasets, both theoretically aligned with Gaia-CRF, are at the sub-mas level. This demonstrates their high accuracy and mutual consistency, ensuring that the NSB orbit can be accurately constrained;

  • Near-Gaussian AL residuals in Gaia FPR astrometry using F8M24: by replacing the NEP097 ephemeris with F8M24, the AL residuals of Gaia FPR data approximate a unimodal, Gaussian-like distribution, with mean and median residuals consistently below 0.5 mas;

  • Uncertainty quantification for mission planning: the orbital uncertainty of NSB during the anticipated mission arrival window (2040–2050) is constrained to ~500 km, as a refinement for future spacecraft navigation.

To further advance the accuracy and reliability of NSB ephemeris, we will:

  • Incorporate additional high-accuracy astrometric data from a wider range of sources over longer time spans, including more data on future stellar occultations and the fully reduced Gaia DR5 release;

  • Continue to analyze sources of systematic errors, particularly those arising from dynamical modeling assumptions or observational biases;

  • Improve orbit consistency through joint integrations and data fitting with satellites and other planets, considering mutual gravitational effects.

These efforts are aimed at establishing a next-generation NSB ephemeris capable of accommodating longer-term, higher-accuracy requirements, supporting both fundamental Solar System dynamics research and high-stakes interplanetary exploration.

Acknowledgements

We would like to thank the anonymous referee for thoughtful reviews. This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant Nos. XDA0350300 and XDA0350303), the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 12203105), the Major Program of the National Natural Science Foundation of China (Grant Nos. 62394350 and 62394351). This work has made use of data from the 80-cm Yaoan High Precision Telescope (YAHPT). This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

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7

Here, 25 159 = 25 263 – 104 km, where the subtrahend 104 km subtracts the combined effects of atmospheric refractive bending at the half-light level (50 km) and gravitational deflection of starlight by Neptune (54 km).

Appendix A Observational data summary

Relative angular measurements of Neptune against nearby stars from 1904 to 1922 corrected by Gaia catalogs are summarized in Table A.1. Meridian transit, photographic, and CCD astrometry, and Voyager 2 3D normal point observations adopted by DE440 are summarized in Figs. A.1-A.4. Stellar occultation astrometry of Neptune and Triton from 1981 to 2022 based on Gaia DR3 catalog is summarized in Table A.2.

Table A.1

Summary of relative angular measurements of Neptune against nearby stars corrected by Gaia catalogs.

thumbnail Fig. A.1

Meridian transit astrometry of Neptune in right ascension (α) and declination (δ) from Washington Observatory (786) 6-inch telescope. La Palma Observatory (950), and Bordeaux Observatory (999). The pre-fit residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared.
thumbnail Fig. A.2

Photographic astrometry of Neptune in right ascension (α) and declination (δ) from Mykolaiv Observatory (089). The pre-fit residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared.
thumbnail Fig. A.3

CCD astrometry of Neptune and Triton in right ascension (α) and declination (δ) from Flagstaff Observatory (689), Table Mountain Observatory (673), and Yunnan Kunming 1-meter Telescope (286). The pre-fit residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared. Kunming data are excluded from F8B24 orbital fitting.
thumbnail Fig. A.4

Voyager 2 3D normal point observation of the Neptune system barycenter in right ascension (α), declination (δ), and range (D). The 1σ uncertainties of observations are shown in errorbars. The pre-fit residuals relative to the DE440 ephemeris and the post-fit ones relative to our final FORCES-8-BARY-2024 (F8B24) ephemeris are compared.
Table A.2

Summary of stellar occultation astrometry of Neptune and Triton based on Gaia DR3 catalog.

All Tables

Table 1

Initial conditions at the J2000.0 TDB epoch for the final orbit-fit solution of NSB, with (px, py, pz) and (υx, υy, υz) representing the NSB position and velocity relative to the DE440 SSB in the ICRF, respectively.

In the text
Table A.1

Summary of relative angular measurements of Neptune against nearby stars corrected by Gaia catalogs.

In the text
Table A.2

Summary of stellar occultation astrometry of Neptune and Triton based on Gaia DR3 catalog.

In the text

All Figures

thumbnail Fig. 1

Comparison of NEP097 and FORCES-8-MAIN-2024 (F8M24) ephemerides for Triton’s orbit relative to the NSB. Positional differences are presented in radial (R), transvers (T), and out-of-plane (N) directions. The norm of the 3D differences (Δd) are also shown.
In the text
thumbnail Fig. 2

Positional discrepancies in stepwise forward-backward NSB orbit integration tests across forward spans ranging from −200 to +200 years. Sub-kilometer discrepancies validate the numerical integrator’s accuracy.
In the text
thumbnail Fig. 3

Post-fit NSB orbit sensitivity to dynamical model variations (DE, EPM, INPOP) in simulated orbit-fit tests, as well as the adopted actual observation weights serving as proxies for observation accuracy. Geocentric positional differences in the ICRF are presented in right ascension (α), declination (δ), and range (D) directions. The norm of the 3D differences (Δd) are also shown. The yellow region marks the approach window of currently planned Neptune missions (2040–2050).
In the text
thumbnail Fig. 4

Diagonal correlation matrix of the initial conditions of the new NSB ephemeris FORCES-8-BARY-2024.
In the text
thumbnail Fig. 5

Relative angular measurements in position angles (θ) and angular separations (ρ) for Neptune against nearby stars. The pre-flt residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared.
In the text
thumbnail Fig. 6

Gaia FPR astrometry of Triton in along-scan (AL) and across-scan (AC) directions. The pre-fit residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared.
In the text
thumbnail Fig. 7

Stellar occultation astrometry of Neptune and Triton based on Gaia DR3 in geocentric ICRF-reſerenced right ascension (α) and declination (δ). Residuals relative to DE440 ephemeris, the occultation-only orbit-fit tests using various RUWE cuts, and our orbit-fit solutions, using NEP097 and FORCES-8-MAIN-2024 (F8M24), are presented.
In the text
thumbnail Fig. 8

Post-fit residual distributions of Gaia FPR Triton astrometry in the along-scan (AL) direction. Occultation-only orbit-fit solutions using three RUWE selection criteria (no limit, <1.4, <1.0) and the all-data solution are presented. Each panel compares solutions using NEP097 (left) and FORCES-8-MAIN-2024 (F8M24; right) Triton ephemerides. The all-data F8M24-based solution corresponds to our final FORCES-8-BARY-2024 (F8B24) NSB ephemeris.
In the text
thumbnail Fig. 9

Final orbit-fit solution for NSB. Least-squares propagated formal uncertainties are derived using a nonlinear mapping of the covariance matrix (CM) based on a MCCM/RP method. Uncertainties based on the Jackknife resampling method are also presented. Geocentric positional differences in the ICRF are presented in right ascension (α), declination (δ), and range (D) directions. The norm of the 3D differences (Δd) are also shown. Exc. means excluding, V2 3D means Voyager 2 3D, Occ. means occultation, Rel. means relative, Mer. means meridian, Pho. means photographic, and Obs. means observations.
In the text
thumbnail Fig. 10

Comparison of the latest two versions of the DE, EPM, and INPOP ephemerides and our final FORCES-8-BARY-2024 (F8B24) solution for NSB orbit. Geocentric positional differences in the ICRF are presented in right ascension (α), declination (δ), and range (D) directions. The norm of the 3D differences (Δd) are also shown.
In the text
thumbnail Fig. A.1

Meridian transit astrometry of Neptune in right ascension (α) and declination (δ) from Washington Observatory (786) 6-inch telescope. La Palma Observatory (950), and Bordeaux Observatory (999). The pre-fit residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared.
In the text
thumbnail Fig. A.2

Photographic astrometry of Neptune in right ascension (α) and declination (δ) from Mykolaiv Observatory (089). The pre-fit residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared.
In the text
thumbnail Fig. A.3

CCD astrometry of Neptune and Triton in right ascension (α) and declination (δ) from Flagstaff Observatory (689), Table Mountain Observatory (673), and Yunnan Kunming 1-meter Telescope (286). The pre-fit residuals relative to the DE440/NEP097 ephemerides and the post-fit ones relative to our final FORCES-8-BARY-2024/FORCES-8-MAIN-2024 (F8B24/F8M24) ephemerides are compared. Kunming data are excluded from F8B24 orbital fitting.
In the text
thumbnail Fig. A.4

Voyager 2 3D normal point observation of the Neptune system barycenter in right ascension (α), declination (δ), and range (D). The 1σ uncertainties of observations are shown in errorbars. The pre-fit residuals relative to the DE440 ephemeris and the post-fit ones relative to our final FORCES-8-BARY-2024 (F8B24) ephemeris are compared.
In the text

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