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title	summary	tags
Time scales	UTC, TAI, TT, and GPS, and why an instant is stored on one scale.	time, utc, tai, tt, gps

Time scales

A time scale is a rule for assigning a number to an instant. Several are in use at once, they do not agree, and the differences between them are large enough to matter for pointing and ranging. This document defines the scales the kernels support and the storage convention that removes the ambiguity from the arithmetic.

Definition

Scale	Full name	Rate	Reference
UTC	Coordinated Universal Time	Not uniform	Civil time, steered to UT1 by leap seconds
TAI	International Atomic Time	Uniform	Weighted mean of atomic clocks
TT	Terrestrial Time	Uniform	Geocentric coordinate time
GPS	GPS time	Uniform	Maintained by the GPS constellation

UTC is the only scale here that is not uniform. It is kept within 0.9 seconds of UT1, the scale defined by the rotation of the Earth, by inserting or removing whole seconds. A UTC minute that contains an inserted second has 61 seconds.

Mathematical meaning

The storage model is one number: a Modified Julian Date on the TAI scale. Every instant is converted to TAI when it is created and converted back to whatever scale is asked for when it is displayed.

The reason is exactness. Two instants stored on TAI differ by


\Delta t = \left( \text{tai}_{\text{later}} - \text{tai}_{\text{earlier}} \right) \times 86400

and that difference is the elapsed SI time between them, whatever scales they were created in. If an instant were stored on UTC, the same subtraction would be wrong across a leap second insertion, because the UTC clock does not have a single well defined numeric value at that point.

The conversions between scales are offsets except for UTC:


\begin{aligned}
\text{TT} - \text{TAI} &= 32.184\,\text{s} && \text{(constant, IAU 2000)} \\
\text{TAI} - \text{GPS} &= 19\,\text{s} && \text{(constant since 1980)} \\
\text{TAI} - \text{UTC} &= 10\,\text{s} + N && \text{(N leap seconds since 1972)}
\end{aligned}

The first two are constants. The third is a step function of the date, and its value comes from the leap second table in apsis-leap, which is the table published in IERS Bulletin C.

Unit convention

Instants are Modified Julian Dates, in days, counted from midnight on 1858-11-17.
A duration is in seconds, and is SI seconds of uniform elapsed time.
A calendar date carries a year, a month, a day, an hour, a minute, and a second with a fraction. The second may be up to 60.999 in a minute that contains an inserted leap second.

Applicability

The leap second table starts at the beginning of 1972, which is where the modern UTC system begins. Before that date a UTC instant has no exact representation here, and the conversion returns no value rather than a guess. The table ends at the most recent entry it contains; a date after that is converted using the last known offset, which is correct until the next insertion and is therefore a forecast rather than a fact.

A date far in the future is a forecast for the same reason. The kernels make this visible rather than hiding it: a conversion that is not covered returns an absent value, and the caller decides what to do.

Duration against calendar arithmetic

Adding seconds to an instant moves it by exactly that much elapsed time. Adding a day in the calendar sense is a different operation: across a leap second insertion, one UTC calendar day is 86,401 SI seconds. The kernels expose only the first operation on the instant, and require a conversion to a calendar for the second. This is deliberate, because the two are easy to confuse and the error is small enough to go unnoticed until it matters.

Comparison of representations

Representation	Uniform rate	Leap seconds	Range	Used here for
TAI MJD	Yes	Not represented	Wide	Storage and arithmetic
UTC MJD	No	Represented as a step	1972 onward	Input and output
TAI seconds since an epoch	Yes	Not represented	Depends on epoch	Duration
Calendar fields	No	Represented	Any	Human input and display
GPS week and seconds	Yes	Not represented	1980 onward	GNSS interfaces

Source

Julian Date and Modified Julian Date conventions: IAU, as described in the Explanatory Supplement to the Astronomical Almanac.
Terrestrial Time offset from TAI: IAU 2000 Resolution B1.9.
Leap seconds: IERS Bulletin C.
Calendar algorithms: Howard Hinnant, chrono-Compatible Low-Level Date Algorithms, 2015.

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