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Delta T Calculator: Method & Data Documentation

Definitions, data provenance, interpolation, time-scale handling, and update procedure.

1. What the calculator computes

\[\Delta T = TT - UT1\]\[DUT1 = UT1 - UTC\]

For modern dates for which TAI − UTC is known, TT is related to International Atomic Time by \(TT = TAI + 32.184\ \mathrm{s}\). Therefore

\[\Delta T = 32.184 + (TAI-UTC) - DUT1\]

The calculator accepts an instant expressed as UTC, UT1, or TT. A selected solar eclipse loads its greatest-eclipse epoch as TT. The output source label identifies the data series or model used for the result.

1.1 Using query strings and permalinks

The calculator can also be opened with the date, time, and input time scale supplied in the URL. This makes it possible to save or share a direct link to a specific calculation. The supported query-string parameters are date, time, and scale.

The date is written as YYYY-MM-DD, the time as HH:MM:SS with optional fractional seconds, and the scale as UTC, UT1, or TT. For example:

delta-t.html?date=2027-01-01&time=00:00:00.000&scale=UTC

When a valid query string is present, the calculator loads those values and performs the calculation automatically. After a manual calculation, the page URL is updated in the same form so that it can be copied and used as a permalink to that calculation.

2. Definitions and abbreviations

The calculator uses terminology from IERS, USNO, BIPM, and the Astronomical Almanac. The following definitions are used throughout this page.

ΔT: The difference between Terrestrial Time and UT1: \( \Delta T = TT - UT1 \). It measures the accumulated difference between a uniform terrestrial time scale and the irregular rotation of the Earth.
DUT1: The quantity \(UT1-UTC\), in seconds. USNO also refers to this as ΔUT1. It tells how far Earth-rotation time is ahead of or behind UTC.
UT1: Universal Time 1, the Earth-rotation time scale. USNO describes UT1 as the modern form of mean solar time on the Greenwich meridian. It is derived from observations of the Earth's orientation and is affected by irregularities in the Earth's rotation.
UTC: Coordinated Universal Time, the basis of modern civil time. UTC uses atomic seconds and is kept close to UT1 by internationally defined adjustments. Since 1972 those adjustments have taken the form of leap seconds.
TAI: International Atomic Time, a continuous atomic time scale maintained by the BIPM. Modern UTC has the same rate as TAI and differs from it by an integral number of seconds in the post-1972 system.
TT: Terrestrial Time, the uniform terrestrial time coordinate used in modern astronomical ephemerides. For the purposes of this calculator, \(TT = TAI + 32.184\ \mathrm{s}\).
ET: Ephemeris Time, the uniform astronomical time scale introduced in the twentieth century because Earth rotation was too irregular to serve as a uniform time argument. ET was later superseded by dynamical and relativistic time scales, including TT.
LOD: Length of Day, as used in IERS Earth-orientation products. In this context LOD means the excess or deficit of the observed rotational day relative to exactly 86,400 SI seconds. IERS tabulates LOD in seconds or milliseconds depending on the product. A positive LOD means the Earth took slightly longer than 86,400 SI seconds to complete the corresponding rotational day; a negative LOD means slightly less. In the equations below the calculator converts LOD to seconds per day before using it as a derivative.
MJD: Modified Julian Date, defined by \(MJD = JD - 2400000.5\). Because the subtraction places the day boundary at midnight, MJD is convenient for daily Earth-orientation tables.
EOP: Earth Orientation Parameters, the set of quantities published by IERS that describe the orientation and rotation of the Earth, including UT1−UTC, polar motion, and LOD.
IERS: International Earth Rotation and Reference Systems Service, the international service responsible for Earth-orientation and reference-system products used by this calculator.
C04: The IERS combined Earth-orientation series. The calculator uses C04 for the highest-quality retrospective/operational Earth-orientation values available in the local data set. Recent values can be revised as additional observations and improved analyses become available.
Bulletin A: The IERS Rapid Service/Prediction Center product containing recent Earth-orientation estimates together with short-term predictions. Values identified as predictions are not treated as equivalent to finalized retrospective measurements.
Hermite interpolation: Cubic interpolation that uses both the value and the first derivative at each end of an interval. Here, the values come from Earth-orientation data and the endpoint derivatives come from LOD.

3. Time scales and historical dates

The modern names UTC, UT1, TAI, and TT did not all exist throughout the 1900–2200 range of this calculator. The terminology evolved as astronomy moved from mean-solar time, through ephemeris time, into atomic and relativistic time scales. The dates below are therefore important when interpreting historical records.

Date or period	Time-scale development
Before 1925	Astronomical Greenwich Mean Time in almanacs traditionally began at noon rather than midnight. The U.S. Naval Observatory notes that this older convention is now called Greenwich Mean Astronomical Time.
1925	Astronomers changed the astronomical day to begin at midnight, matching the civil day. The U.S. Nautical Almanac history records the introduction of Greenwich Civil Time (GCT) for this convention.
Mid-20th century	Universal Time terminology became the technical replacement for ambiguous uses of GMT. UT1 is the modern Earth-rotation form of mean solar time on the Greenwich meridian. UT0 and UT2 were also used for specific observational and corrected forms of Universal Time.
1952–1956	Ephemeris Time (ET) was introduced for astronomical work because Earth rotation was not uniform enough to define a uniform time argument. The ephemeris second was adopted as the unit of time in 1956 and served as the SI second until the atomic definition took effect in 1967.
1958-01-01	Atomic time and the coordinated broadcast time system were placed in agreement with UT1 at the beginning of the atomic-time era. Modern retrospective TAI is referenced to this epoch.
1961	Coordinated atomic time signals were already in operational use. Pre-1972 UTC did not yet have today's integer-leap-second form; it used small frequency offsets and occasional steps. Historical TAI−UTC therefore cannot be represented by leap seconds alone.
1972-01-01	Modern UTC began with UTC = TAI − 10 s. From this date onward, UTC has the same rate as TAI and differs from it by an integer number of seconds, changed by leap seconds when required.
1976 / 1979	The IAU introduced a new family of dynamical time scales in 1976 to replace Ephemeris Time. In 1979 the geocentric scale was formally designated Terrestrial Dynamical Time (TDT).
1991	IAU Resolution A4 renamed Terrestrial Dynamical Time (TDT) as Terrestrial Time (TT) and placed the modern definition in a relativistic framework.
2000	The IAU refined the definition of TT. The present conventional relation used by this calculator remains \(TT = TAI + 32.184\ \mathrm{s}\).

UT1 is the modern Earth-rotation time scale and is the quantity on the rotational side of ΔT. TT is the modern uniform terrestrial time coordinate on the other side. For dates preceding the historical existence of UTC, the calculator permits “UTC” as a convenient proleptic input coordinate so that one interface can be used across the full date range. That label should not be interpreted as a claim that UTC was actually used on that historical date. Historical work should distinguish the physical instant, the modern coordinate used by the calculator, and the civil or astronomical convention appearing in the original observation.

4. Data-source hierarchy and result colors

Dated Earth-orientation values are not constants embedded in the calculator program. The browser code contains the equations and interpolation rules; the replaceable delta-t-data.json file contains the dated source values. This allows new observations and revisions to be installed without editing the calculation code.

Color	Meaning	Typical source
Gray	Historical/reconstructed	USNO/IERS historical ΔT and LOD series
Green	Observed/calculated Earth orientation	IERS C04 values and, where used, IERS rapid observational estimates. Recent operational values may still be revised.
Blue	Predicted Earth orientation or ΔT	the current weekly IERS Bulletin A daily prediction table (`ser7.dat`) and the IERS/USNO long-term ΔT prediction product.
Red	Long-range model	The named model shown with the result

Boundaries between these regimes are determined from the contents of the current data file. They are not tied to hardcoded calendar dates in the JavaScript. When newer IERS data extend the observed range, the boundary moves automatically after the data file is replaced.

5. Why interpolation is necessary

IERS Earth-orientation values are tabulated at discrete epochs, commonly 0h UTC for daily data. An eclipse or a manually entered time will usually fall between two tabulated epochs. Simply returning the nearest daily value would discard real sub-day variation, while ordinary linear interpolation would use the endpoint values but ignore information that IERS also supplies about the rate of change. Where LOD is available at both endpoints, this calculator therefore uses cubic Hermite interpolation. The daily Bulletin A prediction table does not publish LOD for each future row, so those daily predicted UT1−UTC values are interpolated linearly between adjacent published days rather than assigning an invented derivative.

6. Cubic Hermite interpolation

Hermite interpolation is especially appropriate here because at each endpoint we know both a value and a physically related slope. A cubic polynomial is the lowest-degree polynomial that can satisfy four endpoint conditions simultaneously: the value at the beginning, the value at the end, the slope at the beginning, and the slope at the end.

5.1 Quantity interpolated for modern IERS data

Across a leap second, \(DUT1 = UT1-UTC\) contains a deliberate discontinuity because UTC itself jumps by one second. Interpolating DUT1 directly across such a boundary would therefore be undesirable. The calculator instead forms the continuous quantity

\[Q = UT1-TAI = (UT1-UTC) - (TAI-UTC)\]

Since TAI is continuous, Q remains continuous through a UTC leap second. Once Q has been interpolated to the requested instant,

\[\Delta T = 32.184 - Q\]

and, when TAI − UTC is defined for that instant,

\[DUT1 = Q + (TAI-UTC)\]

6.2 How LOD supplies the slope

LOD means Length of Day. In IERS Earth-orientation work it is not the full 86,400-second day written out as a large number; it is the small excess or deficit of the observed rotational day relative to exactly 86,400 SI seconds. Thus an LOD entry of +0.001 s means a rotational day approximately one millisecond longer than 86,400 SI seconds. Before using LOD in the interpolation, the calculator expresses it in seconds per day.

Let \(Q=UT1-TAI\). If the Earth's rotational day is longer than 86,400 SI seconds, UT1 accumulates slightly more slowly than TAI, so Q decreases. Consequently, when the independent variable is MJD measured in days and Q is measured in seconds,

\[\frac{dQ}{d(\mathrm{MJD})} = -LOD\]

This derivative relation is the standard first-order connection used here over a one-day interpolation interval. The minus sign is physically important: positive LOD means slower Earth rotation relative to atomic time, so \(UT1-TAI\) becomes more negative with time.

Thus the slope supplied to the Hermite interpolator for modern Q data is −LOD, in seconds per day. Equivalently,

\[\frac{d(\Delta T)}{d(\mathrm{MJD})} = LOD\]

This is why LOD is useful for more than describing the Earth's rotation separately: it gives the local slope of ΔT itself. In the historical ΔT series, where ΔT and LOD are supplied directly, the calculator interpolates ΔT using LOD as the endpoint derivative.

5.3 The Hermite polynomial used

Let the two tabulated epochs be x₀ and x₁ in MJD, with values y₀ and y₁ and derivatives m₀ and m₁ in seconds per day. Define

\[h=x_1-x_0 \qquad t=\frac{x-x_0}{h}\]

so that t runs from 0 at the first row to 1 at the second. The interpolated value is

\[ y(t) = (2t^3-3t^2+1)y_0 +(t^3-2t^2+t)h\,m_0 +(-2t^3+3t^2)y_1 +(t^3-t^2)h\,m_1 \]

The factors multiplying y₀, m₀, y₁, and m₁ are the cubic Hermite basis functions. The multiplication of each derivative by h is essential: the derivatives are expressed per day, whereas t is dimensionless. This keeps the units of every term in seconds.

The construction exactly reproduces both tabulated endpoint values. Its derivative also exactly reproduces the supplied endpoint slopes. Therefore the interpolated curve joins consecutive daily values smoothly while respecting the measured change in Earth rotation at both ends of the interval. Linear interpolation reproduces only the two endpoint values and assumes one constant slope between them; Hermite interpolation makes use of the additional LOD information already present in the IERS data.

5.4 Exact tabulated epochs and missing LOD

If the requested epoch exactly matches a tabulated row, the calculator returns that row's value; no interpolation is performed. If LOD is unavailable at either end of an otherwise usable interval, the engine falls back to linear interpolation for that interval rather than inventing a derivative.

5.5 Uncertainty

Formal uncertainty values supplied with the source data are retained in the generated data set. Between tabulated epochs the current engine linearly interpolates the tabulated uncertainty. This is a practical representation of the source uncertainty and is distinct from the Hermite interpolation used for the Earth-rotation quantity itself. These uncertainty values are not currently displayed in the calculator result box.

7. Converting the requested input scale

UTC input can be evaluated directly against modern UTC-tagged Earth-orientation data. For a TT input, the engine iterates the relation \(TT = UTC + (TAI-UTC) + 32.184\ \mathrm{s}\) to identify the corresponding UTC instant. For a UT1 input, it iterates using \(DUT1 = UT1-UTC\). These corrections are small, and the iteration rapidly converges.

The TAI − UTC history is itself data in the generated file. It is not represented by a list of calendar-specific leap-second constants in the calculator code. Historical pre-1972 relations can include non-integer offsets and rates, reflecting the timekeeping system actually used in that era.

8. Predictions and long-range values

When the requested instant lies beyond the latest calculated Earth-orientation data, the calculator first uses the current IERS Bulletin A daily prediction table. Bulletin A is reissued regularly, and its future UT1−UTC values can change as new observations and updated prediction models become available. The updater therefore downloads the current Bulletin A file on every run and replaces the previous prediction table rather than appending to it. After the last daily Bulletin A prediction, the calculator uses the longer-term IERS/USNO ΔT prediction product while it is available. Those results are shown in blue and the source is named with the result. Beyond the official prediction range, a named long-range model is used and the result is shown in red.

Long-range ΔT can be predicted even when future DUT1 cannot be stated meaningfully, because DUT1 depends on future UTC and therefore on future leap-second or UTC-policy decisions. In such cases the calculator may return ΔT while displaying DUT1 as unavailable rather than inventing a future UTC offset.

9. Updating the data

The numerical engine and dated observations are deliberately separated. Updating the calculator does not require changing an embedded ΔT value for a particular date. The updater downloads the authoritative source files, including the current Bulletin A (ser7.dat), rebuilds the rapid-observation and prediction portions of the data, validates the result, and writes a new delta-t-data.json. Because Bulletin A predictions themselves are revised, the current Bulletin A prediction table is replaced in full on every update.

This matters because recent Earth-orientation values can be revised as additional observations become available. An older spreadsheet or a value copied from an earlier rapid solution may therefore differ by tens or hundreds of microseconds from the current IERS series without either calculation having contained an arithmetic error. Rebuilding from the current source incorporates those revisions as well as newly added dates.

During initial operation the updater may be run locally and the resulting JSON file uploaded by FTP. The same updater can later be scheduled on the web host. In either case, ordinary data updates replace the JSON; the HTML and calculation JavaScript do not require daily editing.

10. Precision, revision status, and interpretation

The number of digits displayed by the calculator should not be confused with physical certainty. For modern IERS data, interpolation can be carried out numerically to microsecond-level resolution, but the meaningful accuracy of the result is limited by the uncertainty and revision status of the underlying Earth-orientation data. Recent operational estimates may change after additional observations are incorporated.

Historical ΔT values have substantially larger uncertainties than modern observations, and long-range predictions are intrinsically uncertain because future changes in the Earth's rotation cannot be known exactly. A red model value therefore has a very different evidentiary status from a green modern IERS value even if both are printed with the same number of decimal places.

The source label and color are part of the numerical result. They indicate whether a value is historical reconstruction, observational/operational Earth orientation, an IERS prediction, or a long-range model. Users should retain that provenance when quoting a result.

11. Reproducibility

A result can be reproduced independently by identifying the source named below the displayed result, obtaining the two source epochs bracketing the requested instant, converting DUT1 to \(Q = UT1-TAI\) when modern UTC data are involved, using the corresponding LOD values as Hermite derivatives, evaluating the polynomial above, and finally applying \(\Delta T = 32.184-Q\). At an exact source epoch, the source row itself should be recovered.

This design intentionally keeps provenance visible. The calculator does not treat historical reconstruction, calculated Earth orientation, active prediction, and long-range modeling as equivalent kinds of information even though all can produce a numerical value of ΔT.

12. Data sources

The production updater records the exact source family in the generated JSON metadata. Principal numerical sources are the International Earth Rotation and Reference Systems Service (IERS) Earth Orientation Centre and the U.S. Naval Observatory IERS Rapid Service/Prediction Center. These include the IERS C04 series, Bulletin A, the historical ΔT/LOD series, the TAI−UTC history, and the long-term ΔT prediction. Long-range modeled values are explicitly labeled with the model name used by the calculator.

13. References

The following sources define the time scales, historical conventions, Earth-orientation quantities, and data products used or discussed on this page.

U.S. Naval Observatory, Astronomical Applications Department. Universal Time. Defines UT1 as the modern form of mean solar time on the Greenwich meridian and discusses the historical ambiguity of Greenwich Mean Time, including the pre-1925 astronomical day beginning at noon. https://aa.usno.navy.mil/faq/UT
U.S. Naval Observatory, Astronomical Applications Department. History of The Nautical Almanac. Historical chronology including the 1925 change to a midnight astronomical day and the introduction of Greenwich Civil Time. https://aa.usno.navy.mil/publications/na_history
U.S. Naval Observatory, Astronomical Applications Department. Astronomical Almanac Glossary. Definitions of ΔT, ΔUT1/DUT1, UT, UT1, UT0, UTC, and related astronomical quantities. https://aa.usno.navy.mil/faq/asa_glossary
National Institute of Standards and Technology. Time and Frequency from A to Z: Ephemeris Time. Summary of Ephemeris Time and the ephemeris second, including its use as the SI second from 1956 to 1967. https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-e
Bureau International des Poids et Mesures. SI Brochure, Annex 1: Decisions relating to time. Includes CIPM Resolution 1 (1956) on the definition of the second, CGPM Resolution 1 (1967) establishing the atomic SI second, CCTF Recommendation 2 (1970) defining TAI, CGPM Resolution 1 (1971) assigning responsibility for TAI, and CGPM Resolution 5 (1975) concerning UTC. https://www.bipm.org/en/publications/si-brochure/annex-1/time
Bureau International des Poids et Mesures. Time Metrology. Current BIPM definitions and realization of TAI, UTC, UTCr, and TT(BIPM). https://www.bipm.org/en/time-metrology
Bureau International des Poids et Mesures. Mise en pratique for the definition of the second in the SI, Appendix 2. Describes TAI as a continuous time scale, UTC as having the same rate as TAI while differing by an integral number of seconds, and the relationship of TAI to TT. https://www.bipm.org/documents/20126/41489667/SI-App2-second.pdf
Bureau International des Poids et Mesures. Recent insights into the Earth rotation rate unveil possible consequences for global timekeeping, 29 March 2024. Notes that TAI and UTC were set in agreement with UT1 in 1958 and that the present integer-leap-second method began in 1972. https://www.bipm.org/en/-/2024-03-29-nature-timekeeping
International Astronomical Union. Proceedings of the XVIth General Assembly, 1976. Defines the contemporary terminology for TAI, UT0, UT1, UT2, and UTC and records the introduction of the dynamical-time framework that replaced Ephemeris Time. https://www.iau.org/static/resolutions/IAU1976_French.pdf
International Astronomical Union. IAU Recommendation 5 (1976), IAU Resolution 5 (1979), and IAU Resolution A4 (1991), as summarized in later IAU resolutions. These introduced the dynamical time scales, designated Terrestrial Dynamical Time (TDT), and subsequently renamed it Terrestrial Time (TT). https://www.iau.org/Iau/Iau/Publications/List-of-Resolutions.aspx
U.S. Naval Observatory. Kaplan, G. H. The IAU Resolutions on Astronomical Reference Systems, Time Scales, and Earth Rotation Models: Explanation and Implementation, USNO Circular 179, 2005. Defines and discusses UT1, TAI, TT, Earth-rotation quantities, and IAU 2000/2003 conventions. https://aa.usno.navy.mil/downloads/Circular_179.pdf
International Earth Rotation and Reference Systems Service. Petit, G. and Luzum, B., eds. IERS Conventions (2010), IERS Technical Note No. 36, Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie, 2010. Official conventions for reference systems, Earth rotation, and time-related quantities. https://iers-conventions.obspm.fr/
International Earth Rotation and Reference Systems Service, Earth Orientation Centre. EOP C04. Combined Earth-orientation series used for calculated/final Earth-rotation values. https://datacenter.iers.org/
IERS Rapid Service/Prediction Center, U.S. Naval Observatory. Bulletin A: Rapid Service/Prediction of Earth Orientation. Provides recent estimates and predictions of UT1−UTC, LOD, polar motion, and related Earth-orientation parameters. https://maia.usno.navy.mil/products/bulletin-a
IERS Rapid Service/Prediction Center, U.S. Naval Observatory. Historic Delta T and Length-of-Day Data. Historical ΔT and LOD values used by the calculator's historical regime. https://maia.usno.navy.mil/ser7/historic_deltat.data
IERS Rapid Service/Prediction Center, U.S. Naval Observatory. Long-term Predictions of Delta T. Published long-range IERS/USNO prediction product used beyond the short Bulletin A prediction interval while available. https://maia.usno.navy.mil/ser7/deltat.preds
U.S. Naval Observatory, IERS Rapid Service/Prediction Center. TAI−UTC. Historical relation between International Atomic Time and Coordinated Universal Time, including the pre-1972 non-integer offsets and rates. https://maia.usno.navy.mil/ser7/tai-utc.dat

Accessed 13 September 2026. Because IERS operational and prediction products are revised, the numerical data file used by the calculator should be treated as versioned by its own generation date rather than by the access date of this documentation.

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