Astronomy & Celestial Mechanics
the sky, computed
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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.

Assumes Pulsars, Timescales and Parallax.

16 min read 9 figures The sky as a clockSmall perturbations, large consequencesLight is the only messenger

The rung below took a pulsar’s period and its derivative as given and showed how much of what is usually quoted about a pulsar — an age, a magnetic field, a luminosity — is derived from those two numbers through a model.

Where those two numbers come from is a subject of its own, and it is one of the most elaborate inverse problems in observational astronomy. Nobody measures a pulsar’s period. What is recorded is a sequence of times of arrival: the moment a particular pulse, or more usually an average of a few thousand pulses, reached a particular telescope. Everything else is inferred by requiring a model to predict those times, and what is examined is the difference.

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.
Fig. 1 The observable. Timing residuals over six years, one curve per kind of error in the model: a position error leaves a sinusoid of period exactly one year, an unmodelled proper motion leaves the same sinusoid with an envelope growing linearly, and a wrong period derivative leaves a parabola. The three shapes do not resemble each other, and that is why a pulsar is an instrument rather than a clock.

What has to be subtracted first

Before a residual can mean anything, the arrival time has to be moved from the telescope to a place that is not accelerating.

The telescope is on a rotating Earth, which is orbiting the solar system’s barycentre at thirty kilometres a second. The light-travel time across the Earth’s orbit is 499 seconds each way, so an arrival time referred to an observatory contains an annual term of up to about 500 seconds that has nothing to do with the pulsar. That is the Roemer delay, and removing it requires the observatory’s position on the Earth, the Earth’s position in the solar system, and the pulsar’s direction — all three to considerable precision.

Two smaller relativistic terms follow. The Einstein delay accounts for the varying gravitational potential and velocity at the observatory over a year, some 1.7 milliseconds peak to peak. The Shapiro delay accounts for the extra light travel time through the Sun’s gravitational field, up to about 120 microseconds for a line of sight passing near the Sun. Then there is the interstellar medium. A radio pulse travels more slowly at lower frequencies, delayed by

Δt∝DMν2,\Delta t \propto \frac{\text{DM}}{\nu^2},

where DM is the integrated electron column density along the line of sight. The delay is enormous — seconds, at metre wavelengths — and it is removed by observing at two or more frequencies and solving for DM, which is why a timing programme never uses a single band. DM is not constant either: the line of sight sweeps through the interstellar medium as the pulsar and the Sun move, so it drifts by parts in 10410^4 per year and has to be re-fitted continuously. That drift is itself a measurement — a map of electron density along a line that nothing else can probe — and it is one of several places where a nuisance parameter in one analysis is the data of another. The same reversal happens with dust, which is a correction to a magnitude and a measurement of the medium.

The residual as a diagnostic

With all of that removed, the model predicts each arrival to within a few microseconds, and the residuals are examined.

Their power is that each way of being wrong has a distinct signature.

A position error displaces the assumed direction to the pulsar, which changes the projection of the Earth’s orbital position onto that direction. The residual is a sinusoid of period exactly one year and amplitude (a/c) δθcos⁡β(a/c)\,\delta\theta\cos\beta, which for a milliarcsecond and an ecliptic latitude of 20° is about two microseconds. Fitting it out delivers a position — and pulsar positions obtained this way are among the most precise in astronomy, at the sub-milliarcsecond level, from a source that is a point at every wavelength.

A proper motion is a position error growing linearly with time, so it leaves an annual sinusoid whose amplitude grows linearly. Two decades of timing routinely deliver proper motions to microarcseconds per year.

A parallax leaves a signal at twice per year, because the curvature of the arriving wavefront over the Earth’s orbit is a second-order effect. It is small — of order a2/(2cd)a^2/(2cd), a few microseconds at a kiloparsec — and it is measured.

Parallax for a star at 1.3 parsecs. The same star observed from two ends of a baseline. The two sight lines are 1.54″ apart, so the parallax — half of that, the shift seen from a one-astronomical-unit baseline — is 0.769″ for a star 1.3 parsecs away. The definition of the parsec is the distance at which it would be exactly one.
Fig. 2 The same geometry as the triangle that reaches the stars, measured with a clock rather than a camera. The baseline is the Earth’s orbit either way; what differs is that a timing parallax is read off the curvature of the wavefront rather than the angular displacement, so it needs no imaging at all and is unaffected by everything that limits astrometry.

A wrong period leaves a straight line, since the phase error accumulates at a constant rate. A wrong period derivative leaves a parabola, and the sensitivity is startling: a fractional error of one part in 10910^9 in P˙\dot P already gives a residual of several microseconds over six years, which is why the quantity is measured to as many figures as it is. And a pulsar in a binary leaves the orbit’s own Roemer delay — the light-travel time across the pulsar’s orbit — which is a periodic signal of the orbital period whose amplitude gives apsin⁡ia_p\sin i directly, in kilometres.

Folding, and why a single pulse is not used

One step has been passed over and it is the reason any of this is possible at all.

An individual pulse from a pulsar is not a clean copy of the average. Its shape and its arrival phase vary substantially from rotation to rotation — some pulsars null entirely for hundreds of rotations, some switch between two distinct modes, and almost all show sub-pulses drifting across the profile. A single pulse is a poor timing standard.

What is stable is the average. Fold a few thousand or a few hundred thousand rotations at the current best period, and the resulting integrated profile is reproducible to a remarkable degree — the same shape, at the same phase, from the same telescope, year after year. That average profile is cross-correlated with a template to give one arrival time per observation, and the timing analysis works entirely on those.

The stability of the folded profile is an empirical fact rather than a theoretical one, and it is the assumption the whole subject rests on. Where it fails — in pulsars that switch modes, or whose profile changes on a timescale of years — the timing degrades in a way that no amount of integration repairs, and the object is quietly dropped from the precision programmes.

A measurement standard that is stable only in the average is still a standard, and recognising that is what turned an erratic radio source into a clock.

The fit, and what it delivers

All of these are fitted simultaneously. A modern timing model for a binary millisecond pulsar carries the rotation frequency and its derivative, the position, the proper motion, the parallax, the dispersion measure and its own derivatives, five Keplerian orbital elements, and a handful of post-Keplerian parameters.

The precision is what makes the exercise worthwhile. For the best millisecond pulsars the residuals are a few tens of nanoseconds over a decade, which corresponds to knowing the number of rotations since the start of the campaign — some 101110^{11} of them — without ever losing count.

That is the property the whole enterprise depends on and it is worth naming: a timing solution is coherent. The model does not fit a period; it assigns an integer pulse number to every observation, and getting one wrong is not a small error but a broken solution. Establishing coherence over a gap in the observations is the hardest practical problem in pulsar timing, and it is why campaigns are never allowed to lapse.

Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.
Fig. 3 What the two fitted rotation parameters are then used for. The period–period-derivative plane with its contours of characteristic age and inferred field: only the two axes are measurements, and every contour is a model. The millisecond pulsars at the lower left — the ones with the nanosecond residuals — have period derivatives of order 10−2010^{-20}, which is a measurement of a change of one part in 101710^{17} per second.

A number, to fix the scale

The precisions involved are worth writing down, because they are unlike anything else in the collection.

A millisecond pulsar’s period might be 5.757451924362137 milliseconds. That is sixteen significant figures, and it is not a display artefact: a decade of timing at 100-nanosecond residuals determines the period to about 10−1610^{-16} of itself, because the uncertainty in the period is the residual divided by the total elapsed time.

The period derivative for the same object is of order 10−2010^{-20} — a change of one part in 101710^{17} per second — and it is measured to two or three figures.

The consequence is that quantities which would be hopeless anywhere else become routine. The pulsar’s transverse velocity produces a kinematic contribution to the apparent P˙\dot{P}, because a source moving across the sky is slowly changing its distance and therefore its Doppler factor. That term is μ2d/c\mu^2 d/c times the period, and for a nearby fast-moving millisecond pulsar it can exceed the intrinsic spin-down entirely — so an apparent period derivative can be dominated by proper motion, and correcting for it requires an independent distance.

Even the Galaxy’s own gravity shows up. A pulsar accelerating in the Galactic potential relative to the solar system contributes a further term of order 10−1910^{-19} per second, which for the quietest millisecond pulsars is a measurable fraction of the total. Two of the terms in a pulsar’s spin-down are properties of where it is rather than of what it is, and separating them is not optional at these precisions.

Parameters that hide behind each other

Fitting all of those at once has a consequence the list conceals: the parameters are not independent, and some pairs are very nearly indistinguishable over a short baseline.

Position and proper motion are the clearest case. Both produce annual sinusoids, and over a single year they are the same signal — only the growth of the amplitude separates them, so a proper motion needs several years before it detaches from the position it is correlated with. The parallax is in the same situation, its semi-annual term partly absorbed by an error in the assumed ecliptic latitude.

Geometry makes one case much worse than the rest. The amplitude of the annual term goes as the cosine of the ecliptic latitude in one coordinate and as the sine in the other, so for a pulsar close to the ecliptic plane one of its two position coordinates is barely constrained at all, and everything correlated with it inherits the weakness. Timing programmes report ecliptic rather than equatorial coordinates for exactly this reason: the covariance is then nearly diagonal in the coordinates the data actually determine, and a badly measured quantity is visible as a large error bar rather than hidden inside two moderate ones.

A fitted uncertainty is a statement about a whole model rather than about one number. A dispersion-measure drift, an orbital period derivative and a spin-down term can each partly absorb the others, so a residual that appears after twenty years of timing is as often a parameter finally separating from its neighbour as it is a new effect arriving.

Each of the three residual shapes is produced by a different error and each of them scales differently with the size of that error, so it is worth drawing two of them at other magnitudes.

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 3 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 6.8 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.
Fig. 4 The same six years of residuals with a position error nearly three times larger. The annual sinusoid grows in proportion and the other two shapes are untouched, which is the property that lets a fit separate them — three shapes, three independent amplitudes, and no combination that mimics another.
Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.
Fig. 5 The plane with field contours a hundred times lower than the usual set. The measured pulsars do not move; only the contours drawn over them do, and the whole interpretation of the diagram is those contours — which are computed from a model of a rotating magnetic dipole in a vacuum, an object no pulsar is.

What is left when the shapes are gone

The residuals that survive the fit are the science, and they come in three kinds.

Glitches are sudden increases in the rotation rate, by a fractional amount between 10−1110^{-11} and 10−510^{-5}, followed by a partial recovery over days to months. They occur in young pulsars, they are unpredictable, and they are the best available probe of the interior: the standard interpretation is a sudden transfer of angular momentum from a superfluid component that had been rotating faster than the crust.

Timing noise is a slow, wandering residual with far more power at low frequencies than white noise has — red noise, in exactly the sense that makes a false-alarm probability untrustworthy. It is present in almost all young pulsars and in some millisecond ones, and it limits the precision of every parameter with a long-timescale signature: a quadratic term is hard to measure when the noise itself wanders on the timescale of the observation.

A characteristic age of 1257 years for a pulsar 972 years old. The Crab pulsar's period against time, integrated backwards from today's measured P = 33.39 ms and Ṗ = 4.21·10⁻¹³ under ν̇ = −kν^n, at n = 3 and n = 2.51. Differencing each drawn curve returns 3.000 and 2.510, so the curves really are solutions of the law they are labelled with. The characteristic age P/2Ṗ is 1257 years and the true age is 972, because the supernova was seen and recorded; the discrepancy is not an error in the timing but the assumption buried in τ_c, which is that the pulsar was born spinning infinitely fast under a pure dipole. Feeding the measured index of 2.51 and the known age into the spin-down integral instead gives a birth period of 18.7 ms, against 15.9 ms for a vacuum dipole. Three measurements — a period, its derivative, and a date in a chronicle — produce a fourth that nothing observed directly.
Fig. 6 And the quantity that only a long coherent solution can give. Measuring the second derivative of the rotation frequency requires a decade of timing and yields the braking index n=νν¨/ν˙2n = \nu\ddot{\nu}/\dot{\nu}^2, which is exactly 3 if the star is a rotating magnetic dipole in a vacuum. It is 2.51 for the Crab, so the star is not one — and the same measurement, combined with a historical date, returns a birth period that nothing observed directly.

And, in binaries, relativity. The post-Keplerian parameters — periastron advance, gravitational redshift, Shapiro delay and orbital decay — are each measurable, and each is a different function of the two masses. Measuring three of them over-determines the system and turns the binary into a test.

The Shapiro delay is worth drawing out, because whether it can be measured at all is a question about geometry rather than about precision. The pulses passing closest to the companion are delayed most, so the signal is a sharp spike in the residuals once per orbit, centred on superior conjunction. Its two parameters are conventionally the range and the shape: the range is proportional to the companion’s mass, the shape is the sine of the inclination.

The two are strongly correlated except when the orbit is nearly edge-on. In a system inclined at forty-five degrees the delay is a smooth, low-amplitude curve that small adjustments to the Keplerian elements almost entirely absorb, and the mass emerges with an uncertainty of order itself. Within a degree or two of edge-on the spike is narrow and tall, nothing else in the model has that shape, and both parameters separate cleanly.

That is why the sharpest neutron-star masses come from a handful of systems selected for their inclination rather than for their brightness. J1614−2230, at 89.2 degrees, gave a pulsar mass near two solar masses from a delay of about thirty microseconds — a result that immediately excluded several proposed equations of state for matter at nuclear density, obtained by recording when pulses arrived.

A parameter’s measurability can depend on an angle nobody chose. Nothing about the timing of that system is better than the timing of the others; it is simply oriented so that one effect cannot be mimicked by any other. The three kinds are not independent nuisances. A glitch leaves a permanent change in the frequency and often in its derivative, so a timing solution must be broken at the glitch and re-established afterwards; and the recovery from a glitch looks, over a few years, very like timing noise. Distinguishing the two on a young pulsar is largely a matter of how densely it has been observed.

The array

The last use is the one the technique was not built for.

A gravitational wave passing between the Earth and a pulsar changes the light-travel time between them, and therefore appears in the residuals. A single pulsar cannot show it, because a residual of the right size is indistinguishable from timing noise.

But the effect is correlated between pulsars in a specific way. Two pulsars close together on the sky are affected almost identically; two on opposite sides are anti-correlated at a particular level; and the expected correlation as a function of the angle between them is a specific curve with no free parameters in its shape. Measuring that curve — by timing several dozen millisecond pulsars for decades and cross-correlating their residuals — is a detection of a background of gravitational radiation, and the amplitude is a statement about the population of supermassive black-hole binaries in the universe.

The instrument is a few dozen rotating neutron stars, thousands of light years apart, and a set of radio telescopes writing down when the pulses arrived.

What makes the correlation curve so valuable is that it cannot be faked by anything local. A clock error at the observatory is common to all pulsars and appears as a monopole; an error in the solar system ephemeris — a wrong mass for Jupiter, say — appears as a dipole, because it displaces the barycentre in a particular direction. An ephemeris is a fit with correlated parameters, and pulsar timing is now precise enough to test one: the residuals of an array have been used to bound the errors in planetary masses, and a Jupiter mass wrong by one part in 10810^8 would be visible. The quadrupole pattern is the signature that belongs to nothing in the solar system, which is why it, and not the amplitude, is what a detection claim rests on.

The same age, wrong by 21× one way and 2.5× the other. The 7 pulsars whose age is known from something other than their own timing — a supernova seen from Earth in the Crab's case, a remnant's expansion for three, and a transverse velocity carrying the pulsar away from its birthplace for the last two — with the age their timing gives on the vertical axis and the independent one on the horizontal. The diagonal is where the model would be right. Nothing is on it. τ_c = P/2Ṗ assumes a birth period of zero and a braking index of exactly 3, and both push the estimate upwards — the Crab, whose true age is a date in a chronicle rather than a model, sits 1.29 times high. But the scatter runs the other way too: J0538+2817's timing age is 21 times its kinematic one, and B1757−24's is 0.40 of it. So a characteristic age is not an upper bound with a known sign; it is an order-of-magnitude estimate whose error is not even one-sided, and it is the only age available for the ninety-nine per cent of pulsars with no remnant left to date them by.
Fig. 7 And a reminder of what is measured and what is not. Independently known ages against the age each pulsar’s own timing gives: the ratios run from 0.4 to 21, in both directions. Every derived quantity in pulsar astronomy inherits the model it was derived through, and the discipline the timing analysis enforces — separate the fitted parameters from the assumed ones, and quote which is which — is the same discipline the rest of this collection keeps asking for.

And the braking history at the measured index rather than the dipole one, which is the single sharpest discrepancy the timing model has produced.

A characteristic age of 1257 years for a pulsar 972 years old. The Crab pulsar's period against time, integrated backwards from today's measured P = 33.39 ms and Ṗ = 4.21·10⁻¹³ under ν̇ = −kν^n, at n = 3 and n = 2. Differencing each drawn curve returns 3.000 and 2.000, so the curves really are solutions of the law they are labelled with. The characteristic age P/2Ṗ is 1257 years and the true age is 972, because the supernova was seen and recorded; the discrepancy is not an error in the timing but the assumption buried in τ_c, which is that the pulsar was born spinning infinitely fast under a pure dipole. Feeding the measured index of 2 and the known age into the spin-down integral instead gives a birth period of 20.5 ms, against 15.9 ms for a vacuum dipole. Three measurements — a period, its derivative, and a date in a chronicle — produce a fourth that nothing observed directly.
Fig. 8 The Crab’s spin history under a braking index of three and of two. The two tracks diverge substantially over the object’s known age, and the measured index is 2.51 — so neither curve is right and the characteristic age computed from either is wrong by a knowable amount.

One more reading covers the exponent every one of these fits assumes and none of them measures.

A formula everyone uses, and the number no pulsar has. Above: the braking index measured for the 4 pulsars whose spin-down has been followed long enough to give a second derivative, against the value a magnetic dipole rotating in vacuum requires. That value is exactly 3, and it is what every catalogued field strength and every characteristic age assumes. Not one measurement reaches it: they run from 1.4 to 2.839, and all of them fall short in the same direction, which is the signature of a systematic rather than of noise. Below: what that costs. The age a spin-down history gives is the period divided by (n − 1) times its derivative, so the ratio to the quoted characteristic age is 2/(n − 1) — 5.00 for Vela. The numbers are not thereby useless: an exponent recovered from the data is exactly the kind of correction a measurement can absorb. What has gone is the claim that the field strength printed beside a pulsar is a measurement of a field. It is a measurement of a spin-down rate, read through a model the same pulsar refutes.
Fig. 9 The measured braking indices over the range they occupy. Not one reaches the dipole value of three, and the timing model that produces every other number in this essay is fitted with that value held fixed.

Where this ladder goes next

This rung has established that the observable is a residual, that each error has its own shape, and that what survives the fit is where the physics is.

The rung above is the noise modelling: treating timing noise as a stochastic process with a fitted spectrum rather than as a nuisance, which is what makes a gravitational-wave detection statistically defensible and which changes the uncertainties on every other parameter.

Beside it lies the pulsar as a probe of the interstellar medium: the dispersion measure and its variations map the electron density along the line of sight, and the scattering that broadens a pulse measures the turbulence in it.

And below it, the habit: the observable is rarely the quantity. A pulsar’s period is not measured, a residual is; and the discipline of asking which shape a discrepancy has, before asking what caused it, is what turns a clock into an instrument for measuring positions, orbits, masses and spacetime.

Where this goes next

Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway. A clock that runs down and says what it is A pulsar hands over two measured numbers, a period and its derivative. Everything else usually quoted about it — an age, a magnetic field, a luminosity — is derived from those two through a model, and the one measurement that tests the model refutes it. A clock that has lost 5.7 hours in 2720 years. ΔT = TT − UT1, the accumulated difference between a uniform time scale and the Earth's own rotation, from 700 BC to 2020, on a logarithmic scale. The points are the published record; the smooth curve is the parabola 31.94·u² fitted to the entries at or before 1000, with u in centuries from 1820. It is a parabola and not a line because the Earth is not merely slow, it is slowing: a day lengthening at a constant rate makes a clock fall behind by the integral of the lag. That coefficient is a length-of-day rate and nothing else — 31.94 = ½ × r × 36525 days per century gives r = 1.75 ms per century, against the 1.78 ms computed from tidal angular momentum with no eclipse anywhere in the derivation. At 700 BC the offset is 5.7 hours, which is 85° of the Earth's rotation, and that is why the ancient measurements are records of the place a total eclipse was seen from rather than of the hour: the hour was never written down accurately enough to matter, and the shadow's track on the ground was. Six kinds of second The Earth is a clock that loses, and it has lost five and a half hours since 700 BC. That number was measured from the places ancient eclipses were seen from, not the times they were seen at — because the record carries a longitude and no clock. PSR B1913+16: 45 years of periastron arriving 88 seconds early. The cumulative shift of periastron passage for PSR B1913+16, Δt = ½(Ṗ_b/P_b)T², over 45 years from its discovery in 1975. The orbital period is 0.322997449 days and is measured to be shortening at 2.423e-12 seconds per second — a change in the twelfth decimal place, which over a career accumulates to 87.5 seconds. That is the whole reason the effect is measurable. Ṗ_b is taken here as a measured quantity and no radiated power is derived from it; the parabola is the general-relativistic prediction as published, and the points are that prediction scaled by the published ratio of observed to predicted decay, 0.997 ± 0.002 (Hulse & Taylor 1975; Weisberg & Huckins 2016), which is how the agreement is quoted. The error bars are drawn: at the last point the bar is 1.25 px tall against a dot 9 pixels across, so they are invisible, and their invisibility is the result. The campaign has run 45 years. An orbit measured to be shrinking A 7.75-hour orbital period that shortens by 68 nanoseconds each turn is beyond any single measurement, and unmissable after fifty thousand of them, because the shift accumulates as the square of the elapsed time. Forty-five years of pulse arrival times have made it 87.5 seconds, which is why the rate is a measured quantity rather than an inferred one.

What links here

The 8 of 11 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Barycentric correctionDispersion measureGlitchProper motionA pulsar timing arrayRed noiseThe Roemer delayThe Shapiro delayTime of arrivalTiming modelTiming parallaxTiming residual