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Barycentric Dynamical Time

Barycentric Dynamical Time (TDB) is a relativistic coordinate time scale defined by the International Astronomical Union (IAU) for use as the independent argument of ephemerides and dynamical theories referenced to the solar system barycenter. It accounts for general relativistic effects, such as gravitational time dilation, by serving as a linear transformation of Barycentric Coordinate Time (TCB), ensuring that TDB remains closely aligned with Terrestrial Time (TT) through only small periodic variations with amplitudes less than 2 milliseconds over millennia.^[1]^[2] The development of TDB traces back to the IAU's 1976 System of Astronomical Constants, which introduced it alongside Terrestrial Dynamical Time (TDT, later renamed TT) as a replacement for the irregular Ephemeris Time to provide a more uniform scale for solar system dynamics.^[3] In 1979, the IAU formally adopted the names TDB and TDT for these barycentric and geocentric scales, respectively.^[3] Subsequent refinements integrated TDB into a general relativistic framework via IAU Resolution A4 in 1991, established its role within the Barycentric Celestial Reference System (BCRS) through 2000 resolutions B1.3 and B1.5, and provided a precise redefinition in 2006 Resolution B3 to eliminate ambiguities in its scaling relative to TCB.^[3]^[2] This evolution addressed the need for consistency in relativistic modeling while maintaining practical compatibility with earlier ephemerides like the Jet Propulsion Laboratory's DE405.^[3] In practice, TDB functions as the time coordinate for barycentric positions and velocities in modern solar system ephemerides, such as those produced by JPL and used in The Astronomical Almanac since 2003, enabling high-precision predictions of planetary and satellite motions with relativistic corrections.^[3] Its relation to TT is given approximately by periodic terms, such as TDB ≈ TT + 0.001657 sin(628.3076T + 6.2401) seconds (where T is the Julian century from J2000.0), plus higher-order effects, ensuring the scale's rate matches TT on average without secular drift.^[3] TDB's adoption facilitates user convenience in astronomical computations, particularly for astrometry and celestial mechanics, while the underlying TCB provides the strict coordinate time in the BCRS for fundamental theoretical work.^[1]^[2]

Fundamentals

Definition

Barycentric Dynamical Time (TDB) is a relativistic coordinate time scale defined within the solar system barycentric reference frame (BCRS), a coordinate system centered at the solar system barycenter—the center of mass of the Sun and all orbiting bodies—and oriented in alignment with the International Celestial Reference System (ICRS).^[1] This frame provides a uniform spatial and temporal reference for describing the positions and motions of celestial objects in the solar system, excluding perturbations from Earth's rotation or orbit to focus on dynamical purposes. TDB serves as the independent argument for ephemerides and dynamical theories referenced to the solar system barycenter, enabling consistent calculations of orbital elements and planetary positions while accounting for general relativistic effects such as gravitational time dilation and velocity-induced time variations.^[4] The primary purpose of TDB in dynamical astronomy is to provide a stable time standard that integrates relativistic corrections into astronomical models, ensuring that predictions of solar system dynamics remain accurate over long periods without secular drifts that could arise from non-relativistic approximations.^[5] As a successor to the earlier non-relativistic Ephemeris Time (ET), TDB relates to other scales like Terrestrial Time (TT) and Barycentric Coordinate Time (TCB) by incorporating periodic adjustments for Earth's motion around the barycenter.^[4] A key property of TDB is that it is formulated as a linearly scaled version of TCB, with the scaling chosen to match the rate of TT while applying an offset that eliminates long-term drift between TDB and terrestrial scales.^[5] This design ensures that the difference between TDB and TT remains bounded by less than 2 milliseconds over millennia centered on the present epoch, based on ephemeris models like JPL DE405.^[5]

Relation to Other Time Scales

Barycentric Dynamical Time (TDB) is defined as a scaled variant of Barycentric Coordinate Time (TCB), designed to align closely with Terrestrial Time (TT) over extended periods. While TCB serves as the uniform coordinate time at the solar system barycenter, advancing at a constant rate defined by proper seconds in the Barycentric Celestial Reference System, TDB incorporates a linear scaling factor to ensure its average rate matches that of TT, facilitating continuity in astronomical computations. This scaling, established by the International Astronomical Union (IAU), prevents secular divergence between TDB and TT, with the relation given without periodic terms to maintain long-term stability.^[6] In contrast to TT, which is the realized uniform time scale for clocks on Earth's geoid surface and serves as the standard for geocentric ephemerides and precise timekeeping, TDB accounts for relativistic effects in a barycentric frame. The primary differences arise from periodic variations due to Earth's orbital motion around the solar system barycenter, resulting in fluctuations of approximately 1.6 milliseconds annually. However, the secular drift between TDB and TT is minimal, limited to less than 0.002 seconds over 1,000 years around the J2000 epoch, ensuring practical interchangeability for most applications.^[6]^[7] TDB functions as the modern successor to Ephemeris Time (ET), the pre-1984 dynamical time scale used for solar system ephemerides based on Earth's orbital motion. Adopted following IAU resolutions in the late 1970s and refined in subsequent decades, TDB replaced ET for calculations in post-1984 ephemerides while preserving continuity through aligned epochs and rates, allowing seamless transition in planetary theories without significant offsets.^[6]^[7]

Historical Development

Origins in Ephemeris Time

In the early 20th century, astronomical timekeeping relied primarily on Universal Time (UT), which was derived from the Earth's rotation relative to the Sun, but this scale proved inadequate due to the irregular slowing of Earth's rotation caused by tidal friction and other geophysical effects.^[8] Observations of lunar and planetary motions revealed discrepancies accumulating to seconds over decades, necessitating a more uniform time standard for precise ephemerides.^[9] By the 1940s, astronomers recognized that a dynamical time scale based on the predictable orbital motion of celestial bodies would better serve astronomical computations. To address these limitations, Ephemeris Time (ET) was introduced in 1952 by the International Astronomical Union (IAU) as a uniform scale defined by the Earth's orbital motion around the Sun, specifically the tropical year at the epoch 1900.0.^[8] The ephemeris second was established as 1/31,556,925.9747 of that tropical year, drawing on Simon Newcomb's solar tables from 1895 to ensure consistency with existing planetary theories.^[9] This definition allowed ET to serve as the independent argument for ephemerides in almanacs starting from 1960, decoupling time from Earth's variable rotation and improving the accuracy of predicted celestial positions.^[10] By the 1970s, however, the advent of atomic clocks and increasingly precise radar ranging observations highlighted the shortcomings of ET, which neglected general relativistic effects such as gravitational time dilation and the varying potentials within the solar system. Calculations using Einstein-Infeld-Hoffmann (EIH) equations for post-Newtonian motion revealed that relativistic perturbations in planetary orbits required a revised time framework to maintain sub-arcsecond accuracy in ephemerides.^[11] ET's non-relativistic foundation thus became insufficient for high-precision solar system dynamics, prompting calls for a barycentric standard that incorporated these effects.^[9] In response, the IAU proposed Barycentric Dynamical Time (TDB) at its 1976 General Assembly in Grenoble as a relativistic successor to ET, defined to provide a uniform time argument for barycentric ephemerides based on the solar system's center of mass.^[10] This scale integrated dynamical theories with general relativity, adjusting for periodic variations due to the Sun's gravitational potential to enhance accuracy in computing planetary positions and orbital parameters. The 1976 system of astronomical constants formalized TDB's role, marking a pivotal shift toward relativistic timekeeping in astronomy.^[9] The International Astronomical Union (IAU) first formally defined Barycentric Dynamical Time (TDB) in 1976 during its XVI General Assembly in Grenoble, establishing it as a dynamical time scale for equations of motion referred to the solar system barycenter, replacing the earlier Ephemeris Time (Teph) while aligning with the Jet Propulsion Laboratory's (JPL) ephemeris time implementation. This definition specified that TDB advances at the same average rate as Terrestrial Time (TT), differing only by periodic variations with a maximum amplitude of 0.0016 seconds over one year, to ensure continuity in astronomical computations. The resolution was part of the broader IAU (1976) System of Astronomical Constants, providing a standardized framework for barycentric ephemerides.^[3]^[12] In 1991, at the XXI General Assembly in Buenos Aires, the IAU refined the relativistic foundations of time scales through Resolution A4, introducing Barycentric Coordinate Time (TCB) as the proper coordinate time for the Barycentric Celestial Reference System (BCRS) and distinguishing it from TDB to better incorporate general relativistic effects such as gravitational potentials and metric tensors. Recommendation V of the resolution recognized TDB's role for maintaining continuity with pre-relativistic dynamical practices, defining it to differ from TCB solely by periodic terms rather than a secular drift, while mandating SI units (e.g., the SI second) for all scales including TDB and TCB to enhance precision in solar system dynamics. These adjustments addressed limitations in the 1976 definition, where TDB lacked a robust relativistic specification, enabling more accurate transformations between geocentric and barycentric frames.^[13]^[5] The 2006 IAU Resolution B3, adopted at the XXVI General Assembly in Prague, redefined TDB as a linear scaling of TCB to eliminate long-term drift relative to TT and ensure seamless integration with existing ephemerides, using specific constants L_B = 1.550519768 × 10^{-8}, T_0 = 2443144.5003725 (Julian Date at 1977 January 1, 00:00:00 TAI), and an offset of -6.55 × 10^{-5} seconds. This formulation preserved the periodic nature of TDB-TT differences (under 2 milliseconds over millennia) while aligning with the practical Teph scale from JPL's DE405 ephemeris (covering 1600–2200).^[5] Post-2006, the redefined TDB has maintained continuity in major ephemerides, with JPL's DE405 serving as a benchmark for the scaling constants, followed by DE430 (2013, spanning 1550–2650) and DE441 (2021, extending to -13200 to +17191 for specialized lunar fits), all integrated using TDB as the time argument to support high-precision planetary and lunar orbit predictions without introducing secular offsets from TT. These updates incorporate additional data and relativistic models while adhering to the 2006 constants for ongoing relevance in dynamical astronomy.^[5]^[14]^[15]

Computation

Scaling from Barycentric Coordinate Time

Barycentric Dynamical Time (TDB) is defined through a linear scaling of Barycentric Coordinate Time (TCB) to align its average rate with Terrestrial Time (TT) while avoiding long-term divergence between the two scales. This transformation ensures that TDB remains suitable for dynamical calculations in the solar system, preserving the uniformity of barycentric coordinates but adjusting for the relativistic differences in clock rates. The scaling factor is derived from the specific gravitational potential and velocity effects in the solar system barycenter reference frame, as adopted by the International Astronomical Union (IAU).^[5] The precise relation is given by the formula:

\text{TDB} = \text{TCB} - L_\text{B} (\text{JD}_\text{TCB} - T_0) \times 86400 + \text{TDB}_0

where \text{JD}_\text{TCB} is the Julian Date in TCB seconds past the J2000.0 epoch (January 1, 2000, 12:00 TT), L_\text{B} = 1.550519768 \times 10^{-8} is the dimensionless scaling factor that adjusts the rate of TDB to match that of TT on average, T_0 = 2443144.5003725 is the Julian Date corresponding to January 1, 1977, 00:00:00 TAI at the Earth's geocenter, and \text{TDB}_0 = -6.55 \times 10^{-5} s is a fixed offset to maintain continuity with prior definitions of TDB.^[5] This formulation results in TDB advancing at a rate of $1 - L_\text{B} relative to TCB, preventing secular accumulation of differences with terrestrial-based time scales over millennia (with drift limited to under 2 ms).^[5] The constants L_\text{B} and \text{TDB}_0 were specifically chosen based on the JPL Development Ephemeris DE405 to eliminate linear drift between TDB and TT at the geocenter; when using other ephemerides, a small residual drift (not exceeding about 1 ns per year) may occur but is considered negligible for most applications.^[5] This linear approximation captures the dominant relativistic effects for long-term uniformity without introducing periodic terms, which are addressed separately. In practice, this scaling is implemented in astronomical software toolkits for accurate time conversions in ephemeris computations. For example, NASA's SPICE toolkit, used for mission planning and solar system geometry, incorporates the TCB-to-TDB transformation to handle ephemeris times in barycentric frames, ensuring consistency with IAU standards.^[16]

Periodic Relativistic Effects

The periodic relativistic effects in Barycentric Dynamical Time (TDB) arise primarily from two sources: gravitational redshift due to the varying gravitational potential generated by the masses in the solar system, particularly the Sun, and special relativistic time dilation effects stemming from the velocity of the Earth-Moon barycenter in its orbital motion around the solar system barycenter.^[17]^[18] These effects cause the difference between TDB and Terrestrial Time (TT) to exhibit periodic variations rather than a secular drift, as the underlying scaling from Barycentric Coordinate Time (TCB) is adjusted to match the average rate of TT.^[17] The dominant periodic term is annual, with an amplitude of approximately 1.6 milliseconds resulting from Earth's orbit around the barycenter, while smaller contributions from the Moon's orbit (monthly variations) and other planets add perturbations on the order of tens of microseconds.^[17]^[18] Overall, these variations cause the total difference TT - TDB to cycle within about 2 milliseconds over the year.^[17] For practical computations, an empirical approximation models the primary periodic component as:

\text{TT} - \text{TDB} \approx 0.001658 \sin\left(2\pi \frac{(D - 0.5)}{365.25}\right) + \text{smaller terms (s)},

where D is the number of days elapsed from J2000.0 (Julian Date 2451545.0).^[19] This low-order fit captures the leading annual term, with higher-order series (e.g., involving lunar and planetary perturbations) providing sub-microsecond accuracy for extended periods.^[19]^[17] Over long timescales, the linear scaling factor between TCB and TDB ensures no net relativistic drift relative to TT, but the periodic terms must be explicitly corrected in high-precision applications such as pulsar timing or spacecraft navigation to avoid cumulative errors.^[18]^[17]

Applications

In Solar System Ephemerides

Barycentric Dynamical Time (TDB) serves as the fundamental time argument in both analytical and numerical ephemerides for determining the positions of major solar system bodies, including the planets, the Moon, and the solar system barycenter. In the Jet Propulsion Laboratory's (JPL) Development Ephemeris (DE) series, such as DE430 and DE440, ephemeris data are integrated using days of TDB as the time unit, ensuring consistency in relativistic coordinate computations for orbital predictions.^[15] Since the release of DE405 in 1998, TDB has been adopted as the standard time scale across the JPL DE series, facilitating high-precision orbit determinations essential for spacecraft navigation and astronomical observations. This integration allows for accurate modeling of planetary and lunar motions by aligning ephemeris arguments with barycentric relativistic dynamics, supporting missions like Cassini and Mars Surveyor.^[20]^[21] The use of TDB provides significant advantages in solar system simulations by inherently accounting for relativistic effects, such as post-Newtonian corrections and frame transformations, which yield superior accuracy over non-barycentric or geocentric time scales like Terrestrial Time (TT). For instance, in DE440, TDB-based integrations incorporate parameterized post-Newtonian parameters fixed to general relativity values, improving perihelion advance predictions for inner planets by incorporating effects like the Sun's Lense-Thirring precession. Ephemeris updates have maintained TDB's continuity while enhancing coverage and fidelity: DE405, spanning Julian Ephemeris Days (JED) 2305424.5 to 2525008.5 (approximately 1600 to 2201), was the active standard until the 2013 release of DE430, which extended the high-accuracy interval from JED 2287184.5 to 2688976.5 (1550 to 2650). Subsequent advancements in DE440 (2020) further refined TDB-integrated models with additional observational data and dynamical perturbations, extending reliable predictions through the 27th century without altering the core time scale.^[15]

In Exoplanet and Stellar Timing

Barycentric Dynamical Time (TDB) plays a crucial role in exoplanet and stellar timing through the Barycentric Julian Date in TDB (BJD_TDB), which serves as the standard timestamp for precise observations of distant astronomical events. BJD_TDB corrects for the light-travel time delays caused by Earth's orbital motion around the solar system barycenter, ensuring that timings from different observatories or missions are aligned on a common reference frame independent of the observer's location. This correction is essential for achieving sub-minute accuracy in measuring event epochs, preventing systematic errors that could mimic orbital variations or skew period determinations. The adoption of BJD_TDB was formalized in seminal work emphasizing its necessity for high-precision astrophysics, where uncorrected timestamps can introduce errors up to several minutes.^[22] In exoplanet detection and characterization, BJD_TDB is indispensable for analyzing transit and eclipse light curves from space-based missions. For the Kepler mission, all photometric data are provided in BJD_TDB, enabling accurate computation of orbital periods and transit timing variations (TTVs) without light-travel time artifacts that could otherwise compromise mass and eccentricity estimates for thousands of confirmed planets. Similarly, the Transiting Exoplanet Survey Satellite (TESS) employs BJD_TDB for its timestamps, as seen in the discovery of thick-disk planet LHS 1815b, where precise barycentric corrections facilitated radial velocity follow-up and orbital parameter refinement. The James Webb Space Telescope (JWST) also utilizes BJD_TDB for exoplanet observations, such as the transmission spectroscopy of WASP-39b, ensuring homogeneous timing across its near-infrared instruments to model atmospheric features during transits. These applications highlight TDB's role in mitigating geocentric biases, allowing global collaboration on exoplanet catalogs with consistent ephemerides.^[23]^[24]^[25] For stellar timing, TDB via BJD_TDB corrects pulsation and eclipse epochs in variable stars, enhancing period analysis and evolutionary studies. In pulsating variables like Cepheids, which serve as standard candles for distance measurements, BJD_TDB timestamps are used to construct light curves free from Earth's motion effects, as demonstrated in surveys of young open clusters where mid-exposure times were converted to BJD_TDB for identifying new Cepheids with periods of 3–80 days. For eclipsing binaries, TDB-based corrections are applied to eclipse minima timings to derive dynamical masses and orbits accurately, such as in studies of systems like V1388 Ori, where all observations were transformed to BJD_TDB to model light-travel time variations. This barycentric standardization is vital for long-term monitoring of stellar variability, reducing phase errors in period-luminosity relations.^[26]^[27] The modern relevance of TDB in these fields is underscored by its endorsement in international standards, promoting homogeneity across large-scale surveys. The International Astronomical Union (IAU) defined TDB in its 2006 resolutions to support relativistic ephemerides, and subsequent guidelines from astronomical journals recommend BJD_TDB as the preferred format for publishing timings in exoplanet and variability studies. This is evident in the Gaia Data Release 3 (DR3, 2022), where BJD_TDB is advocated for consistent handling of exoplanet host star parameters and variable star epochs, bridging post-2010 missions like TESS and JWST with legacy data for improved astrophysical modeling.^[2]^[28]

References

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[PDF] Glossary - IERS Conventions Centre
that is, for practical purposes, the same as Barycentric Dynamical Time (TDB). TDB is related to Barycentric Coordinate Time (TCB) by TDB = TCB − LB × (JDTCB −.
[2]
[PDF] IAU 2006 Resolution B3
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[3]
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Barycentric Dynamical Time (TDB), is the independent argument of ephemerides and dynamical theories that are referred to the solar system barycenter. TDB ...
[5]
[PDF] IAU 2006 Resolution 3
IAU 2006 Resolution 3 redefines Barycentric Dynamical Time (TDB) as a linear transformation of Barycentric Coordinate Time (TCB), using specific constants.
[6]
Glossary - Astronomical Applications Department
TDB is a linear function of Barycentric Coordinate Time (TCB) that on average tracks TT for an extended time period around the current standard epoch, JD 245 ...
[7]
[PDF] Fundamental Concepts
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[8]
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[9]
[PDF] Historical Reflections on the Work of IAU Commission 4 ... - arXiv
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[10]
[PDF] NFA WG EXPLANATORY DOCUMENT B) EXPLANATION OF THE ...
Dec 23, 2005 · Barycentric Dynamical Time (TDB): a time scale originally intended to serve as an independent time argument of barycentric ephemerides and ...
[11]
Relativistic scaling of astronomical quantities and the system of ...
Context. For relativistic modeling of high-accuracy astronomical data, several time scales are used: barycentric and geocentric coordinate times (TCB and TCG) ...<|control11|><|separator|>
[12]
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The EIH equations have been used to construct accurate solar system ephemerides starting from the middle of the 1970s. Equation (6) describes the relativistic.
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[PDF] The JPL Planetary and Lunar Ephemerides DE440 and DE441
Feb 8, 2021 · Compared to the previous general-purpose ephemerides DE430, seven years of new data have been added to compute DE440 and DE441, with improved.
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The integration time units are days of barycentric dynamical time (TDB). ... DE405 : Created May 1997; includes both nutations and librations. Referred ...
[16]
[PDF] SPICE Tutorials
Mar 1, 2010 · – A SPICE kernel that, in conjunction with Toolkit DBK software, provides a self-contained SQL-like database capability. Page 44. Navigation ...
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Abstract. Over the past decades, the IAU has repeatedly at- tempted to correct its definition of the basic fundamental argu- ment used in the ephemerides.
[18]
[PDF] 10 General relativistic models for space-time coordinates and ...
Barycentric Dynamical Time (TDB) and Terrestrial Dynamical Time (TDT; Ka ... TE405 is available in a Chebyshev form at <2> and at the. IERS Conventions Center ...
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[21]
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Jun 28, 2001 · Finally, the TDBtoTT computation is performed based on the Fairhead & Bretagnon et al algorithm, with one value per day.<|control11|><|separator|>
[22]
Achieving better than 1 minute accuracy in the Heliocentric and ...
May 24, 2010 · The BJD_TDB is the most practical absolute time stamp for extra-terrestrial phenomena, and is ultimately limited by the properties of the target ...Missing: IAU | Show results with:IAU
[23]
BJD_TDB vs HJD? | aavso
Jan 25, 2018 · The Kepler mission data are in BJD TDB as the most accurate time stamp. BJD in TDB (BJDTDB) is usually the best time stamp to use in practice.
[24]
[PDF] LHS 1815b: The First Thick-disk Planet Detected by TESS
Time stamps are given in barycentric Julian Date in the barycentric dynamical time. (BJD TDB). We searched the HARPS-TERRA RVs for the Doppler reflex motion ...
[25]
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... Barycentric Julian Date Dynamical Time (BJDTDB.). These values are different from the FIREFLy-reduced white light parameters, and these differences will be ...
[26]
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The mid-times of observation of each frame were then converted to BJD(TDB).2 Using these, light curves could be constructed for all targets believed to be ...
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