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Ephemeris time

Ephemeris time (ET) is a uniform astronomical timescale defined by the orbital motion of Earth around the Sun, specifically as a fraction of the tropical year at the epoch of 1900 January 0, providing a stable reference for celestial mechanics independent of Earth's irregular rotation.^[1] The ephemeris second, its fundamental unit, was established as 1/31,556,925.9747 of the tropical year length determined from Simon Newcomb's orbital tables for 1900.0.^[1] Introduced by the International Astronomical Union (IAU) in 1952 at its Rome General Assembly, ET addressed the gradual slowing of Earth's rotational period due to tidal friction and other geophysical effects, which made Universal Time (UT) unsuitable for precise ephemeris calculations.^[2] Historically, ET served as the standard for predicting positions of the Sun, Moon, and planets in dynamical theories and ephemerides, with its adoption into national almanacs planned for 1960 to ensure consistency in astronomical data.^[2] Unlike UT, which is based on Earth's rotation relative to distant stars, or atomic time scales like International Atomic Time (TAI), ET derived its uniformity from observed orbital periods, though it required empirical corrections (such as ΔT) to align with mean solar time.^[3] By the mid-20th century, advancements in atomic clocks revealed ET's limitations, including uncertainties of about 0.05 seconds over nine years, prompting its replacement.^[4] In 1976, the IAU endorsed Barycentric Dynamical Time (TDB) and Terrestrial Dynamical Time (TDT) as successors, with ET fully phased out by 1984 in favor of these more precise scales based on atomic standards.^[3] TDT was later refined into Terrestrial Time (TT) in 1991, defined as TAI plus 32.184 seconds and scaled to 86,400 SI seconds per day on Earth's geoid, while TDB accounts for relativistic effects relative to the solar system's barycenter.^[3] Though obsolete today, ET's legacy endures in historical astronomical computations and as a foundational step toward modern relativistic time systems used in space missions and global positioning.^[1]

Historical Development

Origins in Astronomy

The foundations of celestial mechanics, as established by Isaac Newton's laws of motion and universal gravitation in his 1687 Philosophiæ Naturalis Principia Mathematica, presuppose a uniform, absolute time scale to accurately predict the motions of celestial bodies under gravitational influences.^[5] This Newtonian framework treats time as flowing equably without regard to external factors, enabling the integration of differential equations that describe planetary orbits and other astronomical phenomena.^[6] However, reliance on Earth's rotation for timekeeping introduced challenges, as solar time—derived from the apparent motion of the Sun—proved insufficiently uniform for high-precision calculations due to inherent variabilities in diurnal measurements.^[7] In the 19th century, astronomers increasingly documented irregularities in Earth's rotation that compounded errors in planetary ephemerides, which are tabular predictions of celestial positions essential for navigation and observation. Tidal friction, first quantitatively analyzed in this era as a dissipative force slowing Earth's spin through interactions with ocean tides, was identified as a primary cause of secular deceleration, contributing approximately 2.3 milliseconds per century to the lengthening of the day.^[8] Concurrently, polar motion—the oscillatory wobble of Earth's rotational axis relative to its crust—was observed by Seth Carlo Chandler in 1891, with amplitudes up to 20 milliseconds affecting the length of the day and introducing periodic discrepancies in positional data.^[7] These variations, including seasonal atmospheric influences, led to cumulative errors in ephemerides exceeding several seconds over decades, rendering traditional mean solar time unreliable for verifying theoretical models against observations.^[9] A pivotal advancement came from American astronomer Simon Newcomb, whose 1895 Tables of the Motion of the Earth on Its Axis and Around the Sun—adopted internationally in 1896 by directors of major national ephemerides—provided refined expressions for Earth's orbital elements, implicitly accounting for rotational inconsistencies by basing computations on a more idealized solar motion.^[10] Newcomb's work, drawing on global observations since 1750, highlighted the need for a corrected time scale, as discrepancies between predicted and observed planetary positions, particularly for the Moon, pointed to non-uniformity in Earth's rotation rather than flaws in gravitational theory.^[9] His investigations into latitude variations, confirming a 14-month Chandler wobble period, further underscored these issues, earning recognition from the Astronomical Journal in 1895.^[10] Prior to the 1952 formalization, proposals like that of Dutch astronomer Willem de Sitter in 1918 addressed lunar acceleration, attributing observed secular changes in the Moon's mean motion partly to tidal friction's impact on Earth's rotation and advocating for a standardized uniform time derived from orbital dynamics to mitigate such effects.^[7] De Sitter's analysis linked these irregularities to broader implications for time measurement, influencing subsequent efforts to decouple astronomical predictions from terrestrial variabilities. These historical motivations culminated in the 1952 International Astronomical Union standard, which resolved the longstanding need for a dynamical time scale.^[9]

1952 IAU Standard

In 1952, the International Astronomical Union (IAU) adopted Ephemeris Time (ET) as a new uniform time scale for astronomical ephemerides during its 8th General Assembly in Rome, held from September 4 to 13.^[2] The resolution, proposed by figures such as Gerald Clemence, addressed the need for a time measure independent of Earth's irregular rotation, defining ET as the scale in which Simon Newcomb's 1898 theory of the Sun's apparent geocentric motion precisely matches observed positions of the Sun.^[11] This framework ensured that ephemerides could be computed with theoretical consistency, using Newcomb's tables as the foundational model for solar motion.^[2] The core of the 1952 resolution recommended that the unit of time for ephemerides be the "ephemeris second," defined as 1/86,400 of the mean solar day according to Newcomb's theory at the epoch of 1900 January 0 at 12h ephemeris time.^[2] This ephemeris second was initially realized through analysis of Earth's orbital motion, specifically by fitting observed solar and lunar positions from the period 1900 to 1930 and extrapolating to other dates using empirical corrections derived from historical data.^[12] The choice of the 1900.0 tropical year as the reference stemmed from its well-observed baseline, with the ephemeris second equating to approximately 1/31,556,925.9747 of that year, providing a stable unit for reducing precise astronomical observations of the Sun, Moon, planets, and stellar coordinates.^[11] Early implementation faced significant computational challenges, as realizing ET required reconciling observational data with Newcomb's non-relativistic theory, which showed discrepancies due to unmodeled planetary perturbations and secular accelerations in lunar motion.^[2] Astronomers relied primarily on lunar observations for practical determination, given the Moon's faster apparent motion compared to the Sun, but this introduced inconsistencies between solar and lunar ephemerides, necessitating revised planetary theories to improve accuracy in extrapolating the time scale beyond the 1900–1930 calibration period.^[12] These hurdles delayed full integration into national ephemerides until 1960, as stipulated in the resolution, allowing time for theoretical refinements.^[2]

Theoretical Foundation

Core Definition

Ephemeris time (ET) is a theoretical time scale in astronomy that measures intervals such that the orbital motion of the Earth around the Sun proceeds uniformly in accordance with classical Newtonian mechanics, independent of perturbations from Earth's irregular rotation.^[1] This idealized scale ensures that ephemerides—tabulated positions of celestial bodies—align precisely with dynamical predictions without incorporating rotational variations.^[2] The foundational unit of ET, the ephemeris second, is defined relative to the tropical year, which is the interval between two consecutive vernal equinoxes as observed from Earth.^[13] The tropical year provides a stable reference for seasonal cycles and serves as the basis for calibrating the uniformity of ET.^[14] The ephemeris second is precisely

$1/31{,}556{,}925.9747

of the tropical year for 1900 January 0 at 12 hours ephemeris time, a value derived from Simon Newcomb's 1895 solar tables that model the Sun's mean longitude to achieve dynamical consistency.^[14] These tables compute the tropical year's length by integrating the equations of motion for the Earth-Sun system under classical gravity, yielding the fractional division that standardizes the second for uniform orbital progression.^[1] In contrast to apparent solar time, which tracks the Sun's observed position across the sky and fluctuates due to Earth's axial tilt, orbital eccentricity, and rotational inconsistencies, ET abstracts these effects to emphasize purely dynamical uniformity.^[1] This distinction allows ET to serve as a consistent argument for gravitational theories of the solar system, free from terrestrial influences.^[2] The International Astronomical Union adopted this framework in 1952 to formalize ET as the standard for astronomical computations.^[14]

Relation to Earth's Orbit

Ephemeris time (ET) serves as a practical realization of dynamical time, the uniform time parameter employed in the differential equations of celestial mechanics to model the motions of solar system bodies. In this framework, ET is calibrated such that the angular position of the Sun, as viewed from Earth, adheres to Keplerian orbital elements augmented by planetary perturbations, but excludes irregularities arising from Earth's non-uniform rotation. This ensures that predictions of celestial events remain consistent with a hypothetical clock unaffected by rotational variations, thereby providing a stable basis for computing ephemerides.^[15] The precise relation between ET and Earth's orbit is established through the mean longitude of the Sun in the ecliptic plane, defined relative to the vernal equinox. According to the 1952 International Astronomical Union (IAU) standard, ET is the time scale in which the geometric mean longitude L of the Sun satisfies the expression derived from Simon Newcomb's planetary perturbation theory: at the epoch 1900 January 0, 12h ET, L = 279^\circ 41' 48''.04, increasing uniformly thereafter based on integrated orbital elements that account for gravitational interactions among the planets. This alignment allows ET to integrate the effects of perturbations on Earth's elliptic orbit while maintaining a linear progression of the mean anomaly and longitude, free from short-term fluctuations in observed positions.^[15]^[16] In constructing ephemerides, ET corrects for secular changes in Earth's rotational period induced by tidal friction, which lengthens the day by approximately 2.3 milliseconds per century, ensuring that orbital predictions do not accumulate errors from these cumulative delays. Without such a correction, the irregular slowing of rotation would progressively misalign Universal Time with the dynamical model of Earth's orbit. Regarding coordinate frames, ET functions primarily as a geocentric dynamical time scale for Earth-centered ephemerides, whereas barycentric coordinates in modern solar system models employ Barycentric Dynamical Time (TDB), which adjusts ET for relativistic effects and the solar system's center of mass, introducing small periodic differences of up to 1.7 milliseconds.^[17]^[3]

Realizations and Implementations

Lunar Observation Methods

Lunar observation methods provided a practical realization of ephemeris time (ET) by measuring the Moon's position against fixed stars, exploiting the Moon's rapid orbital motion—approximately 13 degrees per day—to detect small discrepancies between uniform dynamical time and irregular universal time (UT). Since ET is defined as the time scale in which Newton's laws govern the solar system's motions without irregularities from Earth's rotation, lunar observations served as a sensitive probe for deriving ET corrections.^[18] The core technique relied on lunar theory to predict the Moon's position in ET and compared it to timings of the Moon's observed position in UT, solving for the offset ΔT = ET - UT. Ernest William Brown's lunar tables, developed from 1897 to 1908 based on George W. Hill's variational method, formed the foundational theory by providing series expansions for the Moon's coordinates in terms of mean longitude, latitude, and parallax. These tables expressed the Moon's right ascension α and declination δ as Fourier series in the mean anomaly, mean elongation, and other arguments, enabling computation of expected positions for matching against observations.^[19]^[20] In 1954, Wallace J. Eckert, Rebecca R. Jones, and H. K. Clark published the Improved Lunar Ephemeris 1952–1959 (ILE), which refined Brown's theory through numerical integration on early computers like the IBM Selective Sequence Electronic Calculator (SSEC). The ILE incorporated empirical corrections to fit post-1923 observations, including adjustments for the secular tidal acceleration of the Moon's mean longitude due to Earth's tidal friction. This acceleration manifests as a quadratic term in the mean longitude L, with the full empirical correction given by ΔL = 8''.720 + 26''.750 T + 11''.220 T² (in arcseconds), where T is the time in Julian centuries from JD 2433282.423 (1950 January 1.5 ET). This expression accounted for the difference between the observed slowing of the Moon in UT and the uniform motion in ET, allowing ΔT to be inferred from residuals in lunar longitude.^[21]^[22] The United States Naval Observatory (USNO) implemented these methods from 1956 to 1983, primarily through timings of lunar transits and occultations. Starting in 1956, William Markowitz's dual-rate moon camera—mounted on the USNO's 12-inch refractor—captured paired photographs of the Moon at slow (sidereal) and fast (lunar) drive rates, enabling precise measurement of the Moon's angular velocity relative to stars with timings accurate to about 0.01 seconds. Equations for ΔT were derived by minimizing residuals between observed transits and ILE predictions: ΔT ≈ (O - C) / (∂α/∂t), where O - C is the residual in right ascension (in seconds of time) and ∂α/∂t is the Moon's hourly motion (approximately 132 seconds of time per hour). Lunar occultations of stars, timed visually or photoelectrically, supplemented this, yielding ΔT values with similar ~0.01 s precision by comparing immersion/emersion times to ephemeris predictions. Over this period, thousands of such observations accumulated, providing ΔT catalogs essential for almanac computations.^[21] Despite these advances, the method had inherent limitations stemming from its reliance on the accuracy of the underlying lunar ephemeris. Brown's theory and Eckert's ILE, while fitting observations to within a few arcseconds, could not fully model all perturbations, such as higher-order tidal interactions, planetary gravitational influences, or relativistic effects, leading to systematic errors in predicted positions that propagated to ΔT uncertainties of up to 0.1 seconds over decades. Additionally, observational challenges like atmospheric refraction and instrumental stability further constrained long-term precision, necessitating ongoing empirical adjustments.^[23]^[24]

Atomic Clock Adjustments

Following the adoption of the atomic second in 1960, which was defined to match the ephemeris second in duration, atomic clocks provided a practical means to realize Ephemeris Time (ET) more precisely than previous astronomical methods alone. In the 1960s and 1970s, ET was computed by adding offsets to International Atomic Time (TAI), where the offsets were derived from comparisons between atomic time scales and dynamical positions inferred from lunar observations. This integration allowed for continuous realization of ET, bridging the irregular Earth's rotation-based Universal Time with the uniform dynamical scale required for celestial mechanics.^[25] The Bureau International de l'Heure (BIH) played a central role starting in 1962, when it began coordinating international atomic time scales from cesium clocks contributed by various laboratories, achieving accuracies better than 1 second by the late 1960s through systematic comparisons of atomic seconds against ephemeris seconds derived from lunar ephemerides and observations. These computations refined the relationship ET ≈ TAI + ΔT, where ΔT represented the cumulative offset accounting for historical differences between atomic and dynamical time at the epoch of January 1, 1958. By 1977, the offset was established as ET ≈ TAI + 32.184 seconds, based on accumulated lunar data analyses that ensured continuity with prior ET realizations while incorporating the superior stability of atomic standards.^[25]^[7] This atomic-based approach marked a transitional phase, enhancing ET's precision to sub-second levels and paving the way for atomic time's dominance in dynamical scales by the 1980s. Datasets from the BIH, including annual bulletins of atomic-to-ET differences, supported ephemeris computations until 1984, when ET was superseded by Terrestrial Dynamical Time (TDT), later renamed Terrestrial Time (TT), defined as TT = TAI + 32.184 seconds to maintain exact continuity. The shift highlighted atomic clocks' role in eliminating reliance on variable astronomical observations for routine timekeeping in astronomy.^[25]^[26]

Revisions and Modern Equivalents

Shift to Dynamical Time Scales

In the mid-1970s, the International Astronomical Union (IAU) recognized the limitations of Ephemeris Time (ET) for modern astronomical computations, prompting a conceptual shift toward relativistic dynamical time scales. At the XVIth General Assembly in Grenoble in 1976, the IAU adopted resolutions introducing Terrestrial Dynamical Time (TDT) as the independent argument for apparent geocentric ephemerides and Barycentric Dynamical Time (TDB) for barycentric ephemerides, explicitly positioning these as successors to ET.^[27] This transition addressed ET's reliance on Earth's irregular orbital motion, which had been approximated using the tropical year at the epoch 1900.0, by establishing scales better suited to general relativity and precise planetary dynamics.^[27] Subsequent refinements culminated in the 1982 IAU resolutions, which formalized the adoption of TDT (later renamed Terrestrial Time, or TT, in 1991) to further decouple dynamical time from specific Earth-orbit parameters. TDT was defined with a fixed offset from International Atomic Time (TAI), specifically TT = TAI + 32.184 seconds, ensuring continuity with historical ET while incorporating atomic regularity.^[27] This offset was calibrated to match ET at the epoch of 1977 January 1, facilitating seamless integration into ephemeris calculations without abrupt discontinuities.^[27] The primary motivations for this shift included advancements in planetary theories that diminished the need for ET's Earth-centric tropical year basis and the incorporation of relativistic effects. For instance, the VSOP87 theory, published in 1988, provided high-precision analytical expressions for planetary positions over millennia, relying on uniform dynamical time rather than ET's observational approximations from the early 20th century. Relativistic considerations were central, as TDB and TDT accounted for gravitational time dilation and coordinate transformations in the solar system barycenter, with TDB differing from TT by only small periodic terms (maximum amplitude ~1.6 milliseconds over a 1-year period).^[27] By 1984, the IAU formalized these changes through the implementation of the 1976 System of Astronomical Constants in national ephemerides, marking the full operational replacement of ET with TT and TDB starting January 1. ET was thus approximated as continuous with TT, adjusted by small irregular corrections derived from historical lunar and planetary observations to bridge pre-1984 data.^[27] This framework enabled more accurate predictions in dynamical astronomy, free from ET's inherent irregularities tied to Earth's rotation and orbit.

Integration with Atomic Time

The 13th General Conference on Weights and Measures (CGPM), held in 1967 and ratified in 1968, redefined the SI unit of time, the second, as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom at rest and at a temperature of 0 K.^[28] This atomic definition was deliberately calibrated to align closely with the existing ephemeris second, drawing on experimental measurements of the caesium hyperfine frequency relative to ephemeris time conducted between 1955 and 1965, achieving agreement within 1 part in 10^{10}.^[29] The redefinition stabilized the unit of time against the irregularities of Earth's rotation and orbital motion, enabling precise, reproducible measurements via atomic clocks. However, it did not eliminate the need for ephemeris time in celestial mechanics, where uniform dynamical timescales are essential; instead, it necessitated continuous computation of ΔT, the difference between universal time (UT, tied to Earth's rotation) and ephemeris time (ET), to bridge atomic and astronomical references. Calibration data from the mid-20th century established the foundational relation, with the caesium frequency fixed at 9,192,631,770 Hz to match one ephemeris second, though ongoing lunar and planetary observations were required to refine ΔT values over time. In the long term, the atomic redefinition facilitated the development of International Atomic Time (TAI) as a continuous, uniform scale realized by an ensemble of atomic clocks, rendering ET largely historical by the 1980s. To reconcile atomic precision with practical civil timekeeping, Coordinated Universal Time (UTC) was introduced in 1972, incorporating occasional leap seconds to keep UTC within 0.9 seconds of UT1 and thus maintain synchronization with solar day lengths.^[30] As of 2025, ephemeris time has been fully superseded by modern atomic-based scales such as Terrestrial Time (TT), though it persists in legacy ephemerides for historical astronomical computations.

Specific Applications

JPL Ephemeris Argument

The Jet Propulsion Laboratory (JPL) employs a specific time argument known as Teph, which serves as a linear approximation to classical ephemeris time (ET) while incorporating modern relativistic considerations, primarily for constructing high-precision solar system ephemerides in the DE series.^[31] This adaptation ensures a uniform, coordinate-based timescale suitable for numerical integrations of planetary and lunar orbits, aligning closely with barycentric dynamical time (TDB) but optimized for practical computations in space mission contexts. Teph is defined with an adjusted constant to ensure its average rate matches that of TT, avoiding secular accumulation of the small difference between TDB and TT.^[32] For instance, Teph was integral to the DE430 ephemeris released in 2013, which spans from 1550 to 2650 CE and supports detailed modeling of solar system dynamics.^[32] The computation of Teph begins with the Julian Date (JD) and is expressed as T_{\eph} = \frac{\JD - 2451545.0}{36525}, where the reference epoch is J2000.0 (JD 2451545.0), yielding Teph in units of Julian centuries for polynomial evaluations in the ephemeris.^[31] Relativistic adjustments relate Teph to Terrestrial Time (TT), approximating Teph ≈ TDB, with the difference TDB - TT given by

\TDB - \TT = L_B (\TDB - T_0) + 0.001657 \sin D + 0.000589 \sin 2D + \text{smaller terms},

where L_B = 1.550519768 \times 10^{-8}, T_0 = 2443144.5003725 JD(TDB), D is the mean anomaly of Earth's orbit, and additional terms incorporate velocities and positions of Earth, Moon, and Sun for post-Newtonian corrections.^[32] The conversion from TT to Teph accounts for secular and periodic variations, while Teph differs from TCB by a constant offset and rate. These adjustments ensure Teph maintains consistency with observed data while minimizing secular drifts relative to terrestrial time (TT).^[33] In practice, Teph provides a stable temporal framework for mission planning, enabling precise numerical integrations of spacecraft trajectories relative to planetary positions. For example, during the Voyager missions (1977–present) and Cassini mission (1997–2017), JPL's DE-series ephemerides relying on Teph facilitated trajectory optimizations, gravity-assist maneuvers, and real-time navigation by offering uniform time scales for propagating orbital elements across vast heliocentric distances.^[34] This uniformity is critical for handling the relativistic effects inherent in deep-space environments, where small timing discrepancies could impact flyby accuracies or instrument pointing.^[31] As of 2025, JPL's DE-series, including recent iterations like DE442, continues to evolve in coordination with international efforts such as the Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE)'s INPOP series, ensuring interoperability through shared observational constraints and relativistic frameworks.^[35] This alignment maintains Teph's accuracy at approximately 1 ms relative to TT over mission-relevant intervals, supporting ongoing deep-space operations like those involving the Europa Clipper.^[33]

Role in Official Ephemerides

In modern official astronomical almanacs, such as the joint US-UK Astronomical Almanac, Terrestrial Time (TT) serves as the primary time scale for tabulating positions of celestial bodies, Earth's orientation, and dynamical phenomena, reflecting its role as the successor to Ephemeris Time (ET). However, ET concepts are referenced for legacy data interpretation, particularly when reconciling historical observations or ephemerides predating the 1984 transition to relativistic time scales.^[36] The International Earth Rotation and Reference Systems Service (IERS) conventions incorporate ΔT—the difference between TT and Universal Time UT1—into ephemeris computations to correct for Earth's irregular rotation, enabling precise alignment of dynamical models with observational data from techniques like very long baseline interferometry (VLBI).^[37] The Nautical Almanac, a companion publication for navigational purposes, tabulates planetary positions (e.g., Greenwich hour angles and declinations for the Sun, Moon, Venus, Mars, Jupiter, and Saturn) using TT-derived equivalents that maintain continuity with ET-based frameworks, ensuring compatibility for celestial navigation. Recent editions, including the 2025 volume, integrate corrections for ΔT informed by GPS-derived Earth orientation parameters, enhancing accuracy for time-dependent positions to within arcminutes.^[36]^[38] International standards, as defined by the International Astronomical Union (IAU), support these applications through the Standards of Fundamental Astronomy (SOFA) software library, which includes functions for converting between ET and TT to facilitate ephemeris software handling of both historical and contemporary datasets.^[39] JPL's Teph represents one specialized implementation bridging ET legacies in barycentric ephemerides.^[31] The persistence of ET in older datasets stems from its foundational role in pre-1984 solar system tables, such as those derived from Newcomb's 1898 solar theory and Brown's 1919 lunar theory, which formed the basis for The Astronomical Almanac until the adoption of JPL's DE200 ephemeris. These tables required ET as the independent variable for uniform motion predictions, and modern analyses provide conversion formulas to align them with TT; for example, the correction to lunar mean longitude is given by -8.72 - 26.74T - 11.22T^2, where T denotes Julian centuries from JED 2415020.0 (1900 January 0.5 ET).^[31]

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### Summary of SOFA Library Functions for Time Scale Conversions