110 lines
4.2 KiB
Markdown
110 lines
4.2 KiB
Markdown
---
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title: Leap seconds
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summary: Why UTC has step discontinuities and how the embedded table handles them.
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tags: time, utc, leap-seconds, iers
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---
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# Leap seconds
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UTC is kept close to the rotation of the Earth, which is not uniform, by
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inserting or removing whole seconds. Each insertion is a step
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discontinuity in the UTC clock. This document explains the mechanism and the
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convention the kernels use to avoid the discontinuity in arithmetic.
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## Definition
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A leap second is a one second adjustment to UTC announced by the
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International Earth Rotation and Reference Systems Service in a Bulletin C
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notice. The adjustment keeps
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$$
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\lvert \text{UT1} - \text{UTC} \rvert < 0.9\,\text{s}
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$$
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UT1 is the scale defined by the rotation of the Earth. The Earth's rotation
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rate is not constant, so the number of seconds that have elapsed since an
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epoch and the angle the Earth has turned do not stay in step. Leap seconds
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close the gap.
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## Mathematical meaning
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The offset between TAI and UTC is a step function of the date:
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$$
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\Delta\text{TAI}(t) = 10\,\text{s} + N(t)
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$$
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where `N(date)` is the number of leap seconds inserted between 1972-01-01
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and the date. The kernel stores an instant as TAI, so the offset is applied
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only when an instant is converted to or from UTC:
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$$
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\begin{aligned}
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\text{tai} &= \text{utc} + \frac{\Delta\text{TAI}(\text{utc})}{86400} \\
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\text{utc} &= \text{tai} - \frac{\Delta\text{TAI}(\text{utc})}{86400}
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\end{aligned}
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$$
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The second relation needs the offset at the UTC instant, which is what is
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being computed, so the table is searched for the interval that contains the
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approximate instant and the result is checked for consistency. The
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implementation does this by locating the table entry that applies.
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An inserted second makes a UTC calendar day contain 86,401 seconds. A minute
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that contains an insertion runs from second 0 to second 60.999, so the
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seconds field of a calendar value may exceed 59.
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## Unit convention
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The table stores, for every insertion, the Modified Julian Date at which the
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offset changes and the offset in seconds after the change. Both are exact in
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the sense that the Modified Julian Date is the one the notice publishes,
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converted to a day number.
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## Applicability
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The table starts at 1972-01-01, the beginning of the modern UTC system.
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Before that date the offset was not an integer number of seconds and the
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concept of a leap second did not apply; a conversion of such an instant
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returns no value.
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The table ends at the most recent insertion it contains. A later instant is
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converted with the last known offset, which is correct until the next
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insertion. A caller that needs certainty about the far future must treat
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that as a forecast.
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A leap second may in principle be negative, which would remove a second and
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make a minute 59 seconds long, but no negative insertion has ever been
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announced. The implementation handles the offset as a signed number, so a
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negative insertion would be represented correctly, though it would produce a
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UTC minute with no second 59.
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## Why an instant is not stored on UTC
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Storing an instant on UTC makes subtraction wrong across an insertion. Two
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UTC seconds that are one tick apart in the table region can be 1 second or 0
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seconds of elapsed time depending on which side of the step they fall. The
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kernels avoid the question by storing TAI and converting at the edges. See
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[[time-scales]].
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## Comparison with the alternatives
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| Approach | Elapsed time exact | Matches civil time | Handles insertions | Used here |
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|----------|--------------------|--------------------|--------------------|-----------|
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| Store TAI, convert at the edges | Yes | On display | Yes, from the table | Yes |
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| Store UTC, subtract directly | No | Yes | Incorrectly at a step | No |
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| Store TAI seconds since a fixed epoch | Yes | Needs conversion | Yes, from the table | For durations |
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| Ignore leap seconds, use a fixed offset | Yes | No, off by up to a minute | No | No |
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| Use UT1 throughout | Yes for rotation | No | Not applicable | No |
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## Source
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- Leap second announcements: IERS Bulletin C.
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- Offset history: the table published with the bulletins, embedded by
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`apsis-leap`.
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## See also
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- [[time-scales]] for the scales and the storage convention
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- [[calendar-and-mjd]] for the date conversions
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