Astronomy & Celestial Mechanics
the sky, computed
Concept

Allan deviation — where it appears

A measure of a clock's stability: the typical fractional difference in frequency between successive averages of a given length. Plotted against the averaging time it shows where a clock is best, since white noise averages down while drifts and slow wanders grow.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Also named here as clock ensemble — the same set of essays touches all of them, so they are one junction rather than several.

How steady each kind of clock is, by how long it is averaged. The Allan deviation — the typical fractional difference in frequency between successive averages of length τ — against τ, for representative noise models of caesium beam, hydrogen maser, caesium fountain, optical lattice clock. Each falls as τ^−½ while random frequency noise dominates, then levels off at a floor set by slower, correlated noise, and a clock with a systematic drift eventually rises again. A caesium beam is still improving at the longest averaging drawn, reaching 10⁻¹⁴; a hydrogen maser reaches its best, 7.3·10⁻¹⁶, after about 27 hours; a caesium fountain is still improving at the longest averaging drawn, reaching 1.2·10⁻¹⁶; an optical lattice clock is still improving at the longest averaging drawn, reaching 10⁻¹⁸. No clock is best at every averaging time — the maser is steadiest over hours and wanders over weeks, the fountain is noisier over minutes and more accurate over months — and a time scale is built by using each where it is best. The curves are typical shapes, not the specification of any particular instrument.

A time scale that is finished after it is used

The world's reference time is not kept by any clock. It is an average of about four hundred and fifty, weighted by how steady each has been, published a month late, and then recomputed a year later against the handful of clocks that define the second — so the best available time for any moment is known only long after the moment has passed. The best clocks are now so good that averaging them requires knowing each one's height to a centimetre.

sky · Timescales The stability of a pulsar clock against atomic clocks, over averaging times from a day to thirty years. Fractional frequency instability, on a logarithmic axis, against averaging time from a day to thirty years. A single millisecond pulsar timed to 300 nanoseconds every 14 days improves as τ^(−3/2) while its white timing noise averages down, and flattens where its slow spin noise, drawn at 10⁻¹⁵ at ten years, takes over. An ensemble of 20 such pulsars is 4.5 times better if their noises are independent. The atomic curves are representative: a hydrogen maser, excellent at hours and drifting at years, and a steered atomic time scale limited near 3·10⁻¹⁶ at long times. In this representative comparison the pulsar ensemble becomes the steadier of the two beyond about 3 years of averaging. At a day a pulsar is a billion times worse than an atomic clock; over decades it is competitive, because nothing inside a neutron star changes on the timescales on which laboratory clocks are rebuilt.

A clock made of neutron stars

The best atomic clocks are rebuilt every decade and averaged every month. A millisecond pulsar has kept its own time for a billion years. Averaged over years, an ensemble of pulsars is as steady as the world's atomic time scale — steady enough to find its errors — but it can never say how fast that time scale runs.

sky · Timescales One pulsar's stability, limited by the background every pulsar shares. The cubic-fit stability of a single millisecond pulsar, on logarithmic axes, against averaging time from 0.3 to 40 years. White timing noise of 100 ns every 14 days averages down as τ^(−3/2) (falling dashed line). The gravitational-wave background at the amplitude the timing arrays reported in 2023, A = 2.4·10⁻¹⁵, raises the curve as τ^(2/3) (rising dashed line). Their sum (solid) is lowest at 2.8 years, at 2.8·10⁻¹⁵; averaging longer than that makes this pulsar a worse clock, not a better one, because the background grows faster in the residuals than the white noise shrinks. At ten years the background alone is 5.5·10⁻¹⁵.

A clock whose best interval is three years

A pulsar time scale was expected to improve for as long as anyone kept timing it. The background the timing arrays found in 2023 says otherwise — a single pulsar is steadiest over about three years and worse over every longer interval, and how far an array of them can push that floor down depends less on how many pulsars it holds than on how evenly they cover the sky.

sky · Timescales

Named alongside it

The objects these essays reach for when they reach for this one.

Clock ensembleGravitational-wave backgroundInternational Atomic TimeMillisecond pulsarPulsar time scaleTerrestrial timeAtomic clockGeoidGravitational redshiftHellings downs curveOptical clockPrimary frequency standard

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