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.
Before the modern analytical theories of the Moon, astronomers relied on large numerical tables — the Lunar and Planetary Theories of Brown, Hill, Hansen, and many others — to predict the Moon’s position and librations. These tables were empirical, derived from centuries of observation and limited mathematical frameworks.
By the early twentieth century, E.W. Brown refined the lunar theory into a mathematically rigorous but still fundamentally tabular system, which remained the standard for decades. It was not until the latter half of the twentieth century that Jean Chapront and Michelle Chapront-Touzé at the Bureau des Longitudes replaced these purely numerical approaches with compact trigonometric series. Their formulation expressed each physical quantity (longitude, latitude, distance, libration) as a sum of periodic terms based on the Delaunay arguments, paving the way for fully analytical and computer-ready lunar ephemerides:
This reformulation allowed lunar ephemerides to be generated analytically, reducing human error and paving the way for digital computation.
By the early 1990s, Jean Chapront and Michelle Chapront-Touzé at the Bureau des Longitudes developed a fully analytical lunar theory—ELP 2000—that unified centuries of empirical terms into a consistent modern framework.
Each lunar quantity—longitude (\(\lambda\)), latitude (\(\beta\)), distance (\(\Delta\)), and the three physical librations (\(l, b, c\))—is represented as a sum of periodic series built from the five Delaunay arguments.
Each term’s amplitude (\(a_j, b_j\)) and argument coefficients (\(\alpha_j … \varepsilon_j\)) were determined from a combination of classical theory, numerical integration, and spacecraft data. The resulting model reproduces the Moon’s motion to better than 1 arcsecond in position and 0.1 arcsecond in libration over 1900–2150—adequate for nearly all astronomical, navigational, and eclipse applications.
Unlike earlier tables, Chapront’s series are deterministic and differentiable: each variable is a continuous analytic function of time, not an interpolated dataset. This structure makes the model compact enough for real-time evaluation in browsers or embedded systems, yet faithful to the full physical dynamics encoded in modern ephemerides.
By the 1990s, NASA’s SPICE system (developed at the Jet Propulsion Laboratory) became the authoritative framework for storing and serving planetary and lunar ephemerides. Instead of analytic series, SPICE expresses each quantity—position, orientation, and libration—as a piecewise Chebyshev polynomial, valid over fixed time segments.
For librations, the general form is:
At runtime, SPICE evaluates the polynomial using Clenshaw’s recurrence, a numerically stable algorithm that constructs recursively:
Each segment of coefficients is stored in a binary PCK kernel, typically spanning 32–64 days. During evaluation, SPICE chooses the proper segment, normalizes the time variable τ, and reconstructs the libration or coordinate value with machine-precision accuracy.
This approach eliminates all trigonometric evaluation, greatly reducing runtime cost. SPICE kernels can represent any physical quantity—rotations, orientations, positions, or their time derivatives—within a unified data structure. For the Moon, SPICE achieves residuals of a few milliarcseconds, far exceeding the accuracy of analytic theories.
While SPICE provides the highest-fidelity representation of lunar motion, its binary kernel format and complex evaluation chain are not ideal for lightweight or browser-based tools. To bridge that gap, we generated a uniformly sampled dataset from SPICE—taking values every 15 minutes over the years 1900–2150 CE — and developed a tool which reconstructs intermediate values using cubic Hermite interpolation as follows:
For any quantity and its derivative known at endpoints :
Hermite interpolation matches both the function value and first derivative at the endpoints, ensuring continuity in value and rate of change across samples— an essential property for libration curves.
In effect:
This dual system retains SPICE-level smoothness and sub-arcsecond precision while allowing fast evaluation in JavaScript, Python, or other client-side environments.
Even within the 1900–2150 range, small secular drifts are expected because the
lunar rotation and orbit models continue to evolve. Each new SPICE kernel
incorporates refined dynamical constants, updated tidal parameters, and
long-baseline lunar laser-ranging (LLR) fits. These incremental adjustments—
typically a few milliarcseconds per decade in orientation or tens of meters
in position—reflect improved knowledge, not errors in prior kernels.
Consequently, a set of Hermite coefficients generated from one kernel
(for example, moon_pa_de440_2000.tpc) will diverge slightly from
those derived from a later release (de441, de442,
etc.).
For high-precision research, users should periodically refresh their coefficient tables using the latest SPICE data. For visualization, eclipse prediction, and educational tools, the current implementation remains more than adequate: the cumulative drift over two centuries remains well below any observable threshold.
Although both Chapront’s analytical model and SPICE’s numerical kernels describe the same physical motion of the Moon, they differ fundamentally in how that motion is represented.
| Aspect | Chapront (Analytic) | SPICE (Numerical) |
|---|---|---|
| Representation: | Finite trigonometric series in Delaunay arguments | Piecewise Chebyshev polynomials |
| Form: | Explicit functions of time | Stored coefficients per time segment |
| Continuity: | Global and analytic | Local to each segment (continuous, differentiable) |
| Accuracy: | ≈1″ in position, 0.1″ in libration (1900–2150) | Milliarcsecond level, spacecraft-calibrated |
| Storage: | Text-based coefficients (~40 MB total) | Binary PCK (~2 MB for same coverage) |
| Evaluation: | Trigonometric sums | Polynomial recursion (Clenshaw) |
| Best suited for: | Education, browsers, long-term trends | Research, navigation, high-precision timing |
Our implementation allows both systems to be evaluated from a single time input (Julian Date TT). When a user supplies the epoch:
Between the 15-minute SPICE samples, the Hermite interpolator provides continuous values. This enables a live comparison of analytic and numerical librations at arbitrary times without requiring the user to load or execute SPICE kernels directly.
To verify that the Hermite interpolation reproduces SPICE-kernel values at arbitrary times, we performed a dense sampling test between all 15-minute grid points across the 1900–2150 span. For each interval, the analytic Hermite reconstruction was compared to direct SPICE evaluation at 1-minute cadence.
The results show that the interpolated values remain indistinguishable from direct SPICE evaluation to within 0.01″ RMS and 0.05″ maximum deviation, well below the measurement noise of the underlying kernel and far beneath any visual significance in eclipse or libration visualizations. In terms of rotation angles, this corresponds to a fractional accuracy better than 1 × 10⁻⁸ radians, confirming that the Hermite representation preserves both positional and angular continuity at the numerical precision of double-precision arithmetic.
This means that for browser or educational use, the interpolation effectively is the SPICE result—delivering the same precision without any kernel dependencies or binary parsing on the client side.
SPICE achieves its accuracy by integrating the full n-body equations of motion and fitting to modern spacecraft tracking data. The Chapront model, though purely analytical, remains invaluable for education, legacy reproduction, web presentation, and verification.
Because its form is explicit, it can be differentiated, expanded, or transformed symbolically—something far more difficult with opaque binary kernels. In practical use, Chapront provides intuition and traceability; SPICE provides precision.
Chapront’s analytical outputs \((\lambda,\,\beta,\,\Delta)\) are expressed in the mean ecliptic of date, while SPICE provides data in the true equator and equinox of date. To compare them directly, we must convert the Chapront coordinates to the same frame by applying nutation in longitude (Δψ) and nutation in obliquity (Δε), both functions of the Moon’s ascending node (Ω).
The mean obliquity in arcseconds is evaluated as:
where \(t\) is in Julian centuries (TT) from J2000.0.
The IAU 2000A or 1980 nutation model provides Δψ and Δε as trigonometric series in the Delaunay arguments. The leading terms, driven by the regression of the Moon’s node Ω, dominate:
Ω, the Moon’s nodal longitude, appears in both Chapront’s series and the nutation model. It decreases by ≈ 19.35° per year, completing a full retrograde cycle in 18.6 years. This nodal regression drives the dominant term in Δψ and Δε, coupling the Moon’s orbital motion with the Earth’s axial precession.
Therefore,
The corrected ecliptic coordinates \((\lambda',\,\beta)\) are then rotated to the true equatorial frame using the obliquity \(\varepsilon'\):
Finally, the right ascension and declination follow as:
A comprehensive comparison between Chapront’s analytical results and SPICE kernel outputs over the 1900–2150 CE interval shows:
These results demonstrate that the Hermite interpolation preserves the full fidelity of SPICE while maintaining a smooth, differentiable representation ideal for web and real-time visualization.
The present implementation spans 1900–2150 TT, aligning with the interval over which both the Chapront ELP 2000/82 analytical theory and the standard JPL SPICE lunar orientation kernels maintain uniform accuracy. This range encompasses all modern observations while avoiding epochs where historical constants or incomplete tidal models dominate. It also coincides with the period for which Earth–Moon parameters have been most rigorously constrained by lunar laser-ranging data.
Even within these limits, users should expect very small cumulative drifts between successive kernel releases. SPICE is not a frozen model —it is a living framework continuously refined as new retroreflector data, spacecraft tracking, and planetary constants become available. These revisions slightly alter the Moon’s computed orientation and libration history, producing minute differences (on the order of milliarcseconds) between kernel generations.
For typical scientific and visualization work, the current dataset provides a consistent and stable foundation. However, the same framework can be extended seamlessly: by sampling any newer SPICE kernel at 15-minute cadence and regenerating the Hermite coefficients, users can carry the model forward indefinitely. The interpolation and visualization code remain unchanged; only the underlying coefficient tables are replaced. In this way, the system is both self-contained and future-proof, mirroring the continuous refinement that defines modern lunar ephemerides.
The history of lunar ephemerides is a continuum from empirical observation to analytical theory to numerical integration. Chapront’s analytic series preserved the elegance and transparency of classical astronomy, while SPICE represents the digital culmination of that lineage — embedding the same physics into binary kernels read by spacecraft, web servers, and scientific pipelines alike.
From Brown’s Tables of the Moon to Chapront’s ELP 2000, the goal has always been the same: To describe the Moon’s motion compactly, reproducibly, and with increasing fidelity. The analytic theories bridged human computation and digital automation, but SPICE completed the transition by encoding the dynamics in machine-readable form.
In this project, both worlds meet again: analytic insight from Chapront and numerical precision from SPICE coexist, allowing any user — from student to researcher — to see how the two systems converge, diverge, and together form the modern understanding of lunar motion.
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.
The following technical works include both direct sources used in this page and additional papers originally cited by Yuk Tung Liu in his Notes on Libration Calculation. They are repeated here for convenience and continuity.