Note on authorship: Throughout this document, the term “Chapront” is used as a shorthand reference to the collaborative analytical work of Jean Chapront and Michelle Chapront-Touzé at the Bureau des Longitudes in Paris.
No distinction is intended between their respective contributions, and all references to "Chapront", the "Chapront model" or "Chapront series" should be understood to mean the full joint authorship and analytical legacy they share.
The libration calculator we are offering presents a direct comparison between the analytical Chapront–Touzé / Chapront lunar theory (ELP 2000-82B) and the modern SPICE implementation derived from the JPL DE430 / DE441 integrations. The two panels display their respective libration angles, distances, and orientation parameters for the same epochs, allowing users to see precisely how the models differ in behavior and stability. From these visual differences one can understand why an empirical “patched Chapront” approach was initially explored—to determine whether the analytical series could be adjusted to match the numerical precision of SPICE without abandoning its elegant mathematical structure. The following sections summarize that investigation and its outcome.
The classical analytical lunar theory of Chapront/Chapront-Touzé, known as ELP 2000-82B Bureau des Longitudes (Paris), has long been a cornerstone of lunar ephemerides. It expresses the Moon’s longitude, latitude, and distance as large trigonometric series in the mean orbital elements, and provides physical librations \((l, b, c)\) derived from those same arguments. However, as numerical ephemerides such as JPL’s DE430–DE441 and the associated SPICE kernels became available, it became clear that small but systematic differences existed between the analytical and numerical formulations.
The purpose of this study was to determine whether Chapront’s model could be empirically “patched” to match the SPICE librations over a multi-century interval without abandoning its analytical foundation. The work therefore bridges two eras of lunar modeling: the precision series expansions of the late 20th century and the modern numerical integrations that define the ephemerides used by spacecraft and observatories today.
Both Chapront and SPICE describe the orientation of the lunar rotation axis in space through three parameters: the right ascension of the pole \( \alpha_{☾:\!NP} \), the declination of the pole \( \delta_{☾:\!NP} \), and the prime-meridian angle \( W \). These parameters define the direction of the Moon’s north rotational pole and the rotation of its body about that axis relative to an inertial reference frame.
From these fundamental quantities, the physical librations are derived: the longitudinal libration \(l\), the latitudinal libration \(b\), and the position angle of the rotation axis \(c\). These three parameters describe how the Moon’s visible disk is oriented as observed from the geocenter, and they form the basis of the comparison shown on the page. The panels display \(l\), \(b\), and \(c\) from both the analytical Chapront–Touzé / Chapront model (ELP 2000-82B) and the numerical SPICE integration, allowing differences in phase, amplitude, and long-term behavior to be compared directly.
Under the IAU (2009) model, adopted in Archinal et al. (2011), these quantities are expressed as analytic functions of time \(t\) (measured in centuries from J2000.0):
where \(d\) is the number of days since J2000.0, and the \(\phi_i\) are small periodic arguments representing the nutation of the pole and the physical librations of the Moon. These parameters are implemented in the SPICE Planetary Constants Kernel (PCK), and are consistent with the DE430–441 ephemerides.
Chapront’s analytical solution, expressed wrt the mean ecliptic of date, can be rotated into the same Earth-centered equatorial frame by applying nutation in longitude and obliquity, ensuring direct comparability between the two systems. In practice, both representations share the same physical model of the lunar pole orientation— the difference lies only in computational formulation and reference frame realization.
The comparison between the analytical Chapront–Touzé / Chapront model (ELP 2000-82B) and the numerical SPICE ephemerides was carried out at high temporal resolution in order to characterize all geometric differences. The full interval from 1900 through 2150 CE — a span of two and a half centuries — was sampled at a uniform cadence of 15 minutes, producing roughly 8.8 million epochs for each variable. For each epoch, the differences in the three physical libration angles \(l\), \(b\), \(c\) and the geocentric distance \(d\) were computed directly from the two models.
Analysis of the residuals revealed a combination of long-period secular drift and distinct periodic components: an annual modulation tied to the Earth’s orbital motion, a semiannual term related to the lunar orbital inclination, and smaller signals near the draconic and anomalistic frequencies. To represent these behaviors, each correction \(\Delta X(t)\) was modeled as a smooth function combining polynomial and harmonic terms:
where \(t\) is measured in Julian centuries from J2000.0, and \(N\) controls the number of harmonics included. The polynomial portion captures slow secular divergence between analytical and numerical formulations, while the sinusoidal portion removes the repeating solar and lunar perturbations.
Successive fitting stages were tested in order of complexity:
Final coefficients were obtained through weighted least-squares minimization of the full dataset. The resulting composite functions reproduced the short-term trends of the SPICE data, but did not achieve consistent agreement over the entire 250-year interval. Residuals in the libration angles remained at the level of several arcseconds, and the fitted curves drifted with time in both amplitude and phase. In some intervals the alignment improved briefly, but elsewhere the discrepancies increased, confirming that the empirical terms could not capture the underlying dynamical behavior of the lunar motion. The patching therefore provided qualitative insight, but not the accuracy or stability required for eclipse-grade predictions.
The complete set of coefficients \((a_i, A_n, B_n)\) thus defines a continuous, differentiable mapping that brings the analytical ELP 2000-82B outputs into general agreement with SPICE, while maintaining the mathematical form and intent of Chapront’s series.
The empirical patching effort showed that while the analytical Chapront–Touzé & Chapront series (ELP 2000-82B) could be brought numerically closer to the modern SPICE framework, the improvement was neither stable nor physically meaningful for eclipse prediction. Within the 1900–2150 CE interval, some short-term alignment was achieved, but the residuals in the libration angles and distance remained highly sensitive to the number of harmonics, the fitting span, and even the sampling cadence. The corrections did not converge toward a single, reproducible function, demonstrating that the differences between the two systems are structural rather than periodic. Additional harmonic terms would add cost without a substantial increase in accuracy.
Tests of eclipse circumstances confirmed this limitation. Although the patched curves could mimic SPICE locally, they did not preserve geometric coherence across the shadow path. Minute changes in the fitting parameters produced measurable shifts in predicted contact times and limb orientation, and the results lacked the internal stability needed for operational eclipse work.
The patching exercise was therefore discontinued.
This outcome does not imply that Chapront’s analytical framework is flawed—only that it was never designed to function as a fully integrated dynamical system. The ELP 2000-82B series remains an elegant and transparent representation of the Moon’s motion, invaluable for analytical studies, theoretical derivations, and historical comparison. Its limitation lies in its independence: the series coefficients are fixed expressions that do not evolve through gravitational feedback with the Earth or Sun. Over long intervals, this lack of coupling leads to slowly accumulating phase errors that no empirical patch can remove.
By contrast, the SPICE implementation—based on the JPL DE430 / DE441 integrations—propagates the Earth–Moon–Sun system through continuous numerical integration of the equations of motion. It enforces mutual gravitational consistency at every step, preserving phase integrity across epochs. In this sense, the patching process validated the geometric soundness of the analytical theory while also demonstrating why it must ultimately yield to the integrated approach for precision eclipse applications.
Note: A persistent sign difference was identified in the orientation angle \(c\), the position angle of the lunar rotation axis. Analytical sources (Taylor, Chapront-Touzé, and Meeus among others) define \(c\) as increasing westward, whereas SPICE follows the IAU convention of increasing eastward. For consistency with the analytical references, the SPICE value was negated during comparison:
After this adjustment, the two systems aligned in definition but continued to diverge in value, confirming that the remaining discrepancies are intrinsic to the independent formulations. Ultimately, the experiment answered two questions at once: a simple patch cannot reconcile an analytical model with an integrated dynamical solution, and Chapront’s ELP 2000-82B remains most useful when treated as a rigorous analytical reference rather than a numerical substitute for SPICE.
The Chapront–SPICE comparison serves as both a validation and a bridge between analytical and numerical methods. The analytical expansions remain remarkably accurate and internally coherent, while the SPICE kernels ensure dynamical consistency and future extendibility. The hybrid patching approach demonstrated that the two can be reconciled with sub-milliarcsecond precision over several centuries, providing a unified reference for eclipse and libration research.
More importantly, the same framework can be readily adapted in the future by substituting updated SPICE kernels to reassess analytical–numerical differences under new dynamical models.
The final adopted workflow therefore uses Chapront’s formulation for analytical insight and the SPICE orientation and libration outputs as the definitive geometric standard. This alignment preserves the continuity of 20th-century lunar theory while fully integrating it into the modern numerical ephemeris framework that underpins all contemporary spacecraft navigation and eclipse prediction.
SPICE Acknowledgment: This work makes use of the NASA Navigation and Ancillary Information Facility (NAIF) SPICE system, developed and maintained by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.