Astronomy & Celestial Mechanics
the sky, computed
Concept

Timing residual — where it appears

What is left of a pulsar's pulse arrival times once the best-fitting timing model has been subtracted. It is the observable a timing array works in, and anything the model can absorb — a position error, a spin-frequency error — is invisible in it by construction.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Nothing visible in any pulsar, and a quadrupole in the angle between them. Above: 4 millisecond pulsars' timing residuals over 15 years, at the few hundred nanoseconds a good one reaches. Each wanders, and none of them shows anything a reader could call a signal; a gravitational-wave background of amplitude 2.4·10⁻¹⁵ at one cycle per year contributes a common part to all of them that is smaller than each pulsar's own red noise. Below: the correlation between pairs, against the angle on the sky between them. 2211 pairs out of 67 pulsars, binned into 15 angles, against three curves with no free parameters between them. A quadrupolar background gives the Hellings–Downs shape — positive for nearby pulsars, negative near 83°, and back up to exactly half its zero-separation value at 180° because a background looks the same in opposite directions. An error in the observatory clock would give a flat line, because it shifts every pulsar identically. An error in the solar-system ephemeris would give a cosine, because it moves the barycentre in one direction. The drawn points prefer the quadrupole over the flat line by Δχ² = 358. That is the detection: not a waveform, not an event, not a moment — a shape in an angle, accumulated over fifteen years, on data taken for another purpose entirely.

A detector the size of the galaxy

At a nanohertz no instrument can be built, so the clocks already in the sky are used instead. The signal is in no single pulsar's data — it is in the correlation between pairs as a function of the angle between them, and that curve has no free parameters at all.

gravitation · Gravitational waves Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 6 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 6 years as at 3. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 4e-12 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find.

A clock read against a model of everything in between

A pulsar delivers arrival times and nothing else. Everything else is a model, and what is measured is the difference between the model and the arrivals — a residual whose shape says which term is wrong, and whose annual sinusoid is a position measured from pulses rather than from an image.

stars · Pulsars 14 glitches, and the 1.5 per cent of Vela that is not slowing down. Accumulated fractional spin-up against time for a Vela-like pulsar over 40 years, in parts per million. The underlying spin-down has been removed, so a perfectly braking pulsar would be a flat line at zero. What is drawn instead is a staircase: 14 sudden jumps of a few parts per million, each rising in less than a minute and then relaxing partway back over a couple of months, leaving a permanent step behind. Nothing outside the star can deliver angular momentum on a timescale of seconds, so the source is internal, and the only internal component that could have any to give is one that has not been slowing down with the rest. That is a neutron superfluid: it carries its rotation in quantised vortices, the vortices pin to the crustal lattice and cannot migrate outward, and so the superfluid keeps the spin it had while the crust brakes past it. The reservoir grows until the pinning fails somewhere, and a glitch is the unpinning. The straight line through the staircase is the glitch activity, 0.68 parts per million per year, and it converts directly into an interior measurement: the crust cannot on average take more than the superfluid stores, so the decoupled component must hold at least 2τ_c times the activity of the star's moment of inertia, which here is 1.5 per cent. It is a lower bound rather than a value, and it is one of the very few quantitative statements about the inside of a neutron star that needs no equation of state at all. The individual glitch times and sizes here are drawn from a seeded generator rather than a catalogue; what is real is the staircase's shape, the partial healing, and the arithmetic that turns a slope into a fraction.

The part of a star that never slowed down

A pulsar's spin decays smoothly for years and then jumps upward inside a minute. Nothing outside can deliver angular momentum that fast, so something inside has been storing it — and the rate at which the jumps accumulate is a lower bound on how much of the star is not braking with the rest.

stars · Pulsars The delay a companion's gravity puts in a pulsar's pulses, over one orbit. The Shapiro delay of a pulsar's pulses against orbital phase, for a companion of 0.5 solar masses and orbital inclinations of 89.17°, 85°, 70°, from Δ = −2r ln(1 − sin i · sin Φ), where r is the companion's mass times G/c³ = 2.46 µs. Superior conjunction, when the pulsar is behind its companion, is at 90°. For the orbit nearly edge-on (89.17°, the inclination measured for PSR J1614−2230) the delay rises in a narrow spike 48.5 µs high as the pulses graze the companion; at 85° the spike is 30.8 µs and broad, at 70° 17.1 µs and nearly a sinusoid. The height is set by the companion's mass and the sharpness by the inclination, which is how one curve returns two numbers the orbit alone cannot give.

The part of a delay no orbit can imitate

A binary pulsar's pulses are delayed by up to fifty microseconds as they pass its companion, and most of that delay is invisible — a timing fit folds it into the orbit. What survives is the third orbital harmonic and above, whose size goes as the cube of the tangent of half the inclination, and that cube, not the delay, decides which neutron stars can be weighed.

gravitation · Relativistic orbits One parabola, two causes. Transit times of a planet on a 1.0914-day orbit over 12 years, minus the best straight-line ephemeris, with timing errors of 30 seconds — the shape a period shrinking by 29 milliseconds a year produces, a parabola 116 seconds deep. The curve is drawn once because it is the same curve for both explanations: an orbit decaying at that rate, or a star accelerating towards the observer at 2.9e-6 m/s², which shortens the light-travel time to each successive transit by the same steadily growing amount. Nothing in the transit times can tell them apart.

A clock that slows, or a star that falls

Transit times that curve away from a straight line can mean a planet's orbit is shrinking — the tides draining it into its star — or that the star itself is accelerating along the line of sight under a distant companion's pull. Both draw the same parabola, and the transit times cannot choose. The star's velocities can, but only for the shortest orbits: the acceleration that would fake a given decay grows as one over the period, and beyond a day or two an unseen Saturn can mimic more decay than the tides produce.

exoplanets · Transit timing

Named alongside it

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

GlitchNeutron starPulsar timingA pulsar timing arrayRed noiseThe Roemer delayThe Shapiro delayApsidal precessionBarycentric correctionBinary pulsarCharacteristic strainChirp mass

All concepts