Astronomy & Celestial Mechanics
the sky, computed
Gravitation

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.

Assumes Gravitational waves, Pulsars and Ephemerides.

15 min read 8 figures Everything is a conicTwo bodies, and no moreThe sky as a clock

The gravitational waves that have been seen are a fraction of a second long and a few hundred hertz across, and the instrument that catches them is four kilometres of evacuated pipe. The frequency and the size are not independent: an interferometer responds best to waves whose period is about its own light-crossing time, and a detector for a wave of period one year would have to be a light-year across.

There is a band a billion times lower in frequency where the loudest sources in the universe live, and no instrument for it can be built. What exists instead is a set of clocks already in place.

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.
Fig. 1 The measurement, in two panels. Above, the timing residuals of four millisecond pulsars over fifteen years, at the few hundred nanoseconds a good one reaches: each wanders, and none of them shows anything a reader would call a signal. Below, the correlation between pairs of pulsars against the angle between them on the sky — and there the signal is, as a shape with no free parameters in it. A quadrupolar background gives the curve that dips negative near eighty degrees and comes back to exactly half its zero-separation value at a hundred and eighty. A clock error gives a flat line. An error in the solar-system ephemeris gives a cosine.

Why the low band is the interesting one

The rung below this one took a distance off the amplitude of a signal from two black holes of a few tens of solar masses. The frequency at which such a pair merges is set by the size of its own last orbit, and for thirty solar masses that is a few hundred hertz.

Scale the masses up by eight orders of magnitude and the frequencies come down by the same factor. A pair of black holes of a billion solar masses each — the kind that sit at the centres of large galaxies — orbits at periods of years to decades while it is radiating hardest, which is nanohertz. And there is a reason to expect a great many of them: galaxies merge, their nuclei sink together by dynamical friction, and every merger in the history of the universe should have produced one. A superposition of many unresolved sources is not a waveform. It is a background — a stochastic process with a spectrum — and its expected shape is one of the few things about it that is predictable: a population of circular binaries inspiralling under gravitational radiation alone gives a characteristic strain falling as f−2/3f^{-2/3}, because the number of sources per unit frequency and the energy each radiates both follow from the same inspiral rate.

The clocks

A millisecond pulsar is a neutron star spinning several hundred times a second, and the arrival times of its pulses can be predicted for years ahead to a few hundred nanoseconds. That is a stability comparable with a laboratory atomic standard, achieved by an object with no laboratory and no maintenance.

One measured number, and every pair of masses that produces it. The plane of the two component masses of an inspiralling binary, with three curves of constant chirp mass across it. The middle one is GW150914's value of 28.7 solar masses, and the chirp mass recovered from the coordinates of the drawn curve varies along its whole length by 7.4e-14 per cent — which is the point: every binary on that line radiates the same frequency sweep at leading order, so the early inspiral cannot tell them apart. Two of them are marked. An equal pair of 33.0 and 33.0 solar masses and a lopsided pair of 63.6 and 18.4 sit on the same contour, and their total masses differ by a factor of 1.24. What separates them is the mass ratio, which enters the phasing only at the first post-Newtonian order, suppressed by the square of the orbital speed in units of the speed of light — small through the hundreds of cycles that carry most of the signal, and appreciable only in the last few, where that speed approaches a third of c. So the chirp mass is a measurement and the individual masses are an inference from the end of the signal, which is exactly the part a detector's high-frequency noise eats first.
Fig. 2 The quantity a background is a sum over. Each supermassive pair contributes according to its chirp mass, and the background’s amplitude is an integral over the whole population’s mass function and merger rate — so a measured amplitude is a statement about how many such pairs exist and how heavy they are, not about any one of them. That is the same degeneracy the resolved band has between two masses, taken one level up: one number, standing for a population instead of for a pair.
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. 3 What a timing model is. Every pulse arrival is predicted from a model with a spin frequency, its derivative, a position, a proper motion, a parallax, and — for the ones in binaries — five orbital elements; the residual is what the model does not account for. Fitting the model removes certain shapes from the residuals by construction, which is the reason a timing array cannot see a monochromatic signal at exactly one cycle per year: that shape has already been absorbed into the position.

A gravitational wave passing between the Earth and a pulsar changes the light-travel time between them, and the change appears as a slow wander in the residuals. Slow is the point: the wave’s period is years, so the induced residual is a low-frequency, or red, process, and the amplitude expected from the astrophysical background is of order a hundred nanoseconds over a decade.

Two parallel lines of slope 11/3, and the gap between them is a mass ratio. The rate at which the frequency rises, against the frequency, for GW150914 and GW170817. In these coordinates the quadrupole sweep is a straight line of slope exactly 11/3, and the only property of the binary that moves it is the chirp mass, which shifts it vertically by 5/3 of a decade for every decade of mass. So the two lines here are parallel and 2.31 decades apart, and that gap is the 24.2-fold difference in chirp mass between 28.72 and 1.18 solar masses — nothing else about either system enters. Two black holes of 36 and 31 solar masses and a pair of equal masses adding to the same chirp mass would draw the same line, which is why the individual masses are always quoted with error bars several times wider than the chirp mass's. The dots mark where each signal entered the detector band, and the horizontal run to the right of each is the whole observation: GW150914, 183 milliseconds; GW170817, 102 seconds.
Fig. 4 What the two bands have in common. A binary’s frequency and its rate of change trace a line whose offset is the chirp mass, and the relation is the same at a hundred hertz and at a nanohertz — the supermassive pairs a timing array listens for are the same objects, eight orders of magnitude larger and eight slower. What differs is that at nanohertz frequencies the sources overlap in every resolution element, so the array measures a background rather than an event.

The signature that cannot be faked

What makes the measurement possible is that a gravitational-wave background is not merely red — it is correlated between pulsars in a very particular way.

A metric perturbation is quadrupolar. Two pulsars close together on the sky sit in nearly the same part of the wave, so their residuals are positively correlated. Two at right angles sit where the wave’s two polarisation axes have opposite sign, so they are anti-correlated. Two in opposite directions are correlated again, because a plane wave looks the same coming and going, and at exactly half the strength of the zero-separation case.

The full expression is one line:

Γ(ζ)  =  32xln⁡x  −  x4  +  12,x=1−cos⁡ζ2,\Gamma(\zeta) \;=\; \tfrac{3}{2}x\ln x \;-\; \tfrac{x}{4} \;+\; \tfrac{1}{2},\qquad x=\tfrac{1-\cos\zeta}{2},

and it contains no parameter. Not the amplitude of the background, not the spectrum, not the masses, not the distances to the pulsars. It follows from the number of polarisation states of the wave and from the geometry of two lines of sight, and nothing an astrophysicist could adjust appears in it.

That is what makes it a detection rather than an excess. The alternatives have their own angular signatures and none of them is this one:

A clock error is a monopole. If the terrestrial time standard the arrival times are referred to drifts, every pulsar’s residual acquires the same extra term, so the correlation is a constant, independent of angle.

An ephemeris error is a dipole. Arrival times are corrected to the solar system barycentre, and computing that barycentre requires knowing the masses and orbits of the planets. An error in Jupiter’s mass moves the barycentre in one direction, which advances the arrival times from one half of the sky and delays them from the other — a cos⁡ζ\cos\zeta correlation.

Two ends of every arm

There is a feature of the response that the picture of a “detector with arms” hides, and it is what makes the correlation curve possible in the first place.

A wave changes the light-travel time along a line of sight, and the change depends on the metric perturbation at both ends: at the Earth when the pulse is received, and at the pulsar when it was emitted. The residual therefore carries two terms, conventionally called the Earth term and the pulsar term, and they are separated in time by the light-travel distance to the pulsar — thousands of years.

The Earth term is common to every pulsar in the array. Whatever the wave is doing here, it is doing it to all of them at once, and the different lines of sight sample the same perturbation with different projections onto the wave’s polarisation axes. That is precisely the quadrupolar pattern the correlation curve measures.

The pulsar term is not common to anything. Each pulsar samples the wave field at its own position, thousands of years ago, at a place uncorrelated with every other pulsar’s. So the pulsar terms behave as independent noise: they raise each pulsar’s own red variance and contribute nothing to the correlation between pairs.

Two consequences follow, and they run in opposite directions. The pulsar terms are the reason the zero-separation value of the correlation curve is not the same as a pulsar’s own variance — half the power in each residual is uncorrelated with everything, so the curve’s value as the separation goes to zero is defined by a limit taken from the correlated part alone, and the autocorrelation point sits off the curve entirely. Every published figure marks it separately or omits it, and it is the commonest thing to misread on the plot.

The other consequence is about resolved sources rather than backgrounds. Searching for a single binary coherently means fitting the Earth term and the pulsar term together, and the pulsar term’s phase depends on the distance to the pulsar. Getting that phase right requires the distance to a fraction of a gravitational wavelength — which at nanohertz is a few parsecs, against distances known to tens of per cent. So a coherent search either fits an unknown phase for every pulsar, at a cost in sensitivity, or gives up the pulsar term and uses the Earth term alone. Both are done, and neither is satisfactory.

There is a compensation, and it is the reason the pulsar term is pursued despite the difficulty. The two terms sample the source at epochs separated by the light-travel time to the pulsar, which for a binary evolving on a ten-million-year timescale means the pulsar term carries the source at a measurably earlier point in its inspiral. A pair of terms detected together in one pulsar is therefore two measurements of the same binary thousands of years apart, and the frequency difference between them constrains the chirp mass without any amplitude calibration at all. Nothing else in the nanohertz band offers that, and it is one of the arguments for improving pulsar distances by very long baseline interferometry rather than by timing parallax.

The arithmetic of a fifteen-year baseline

Two numbers bound what a timing array can see and both are set by the observing campaign rather than by the sky.

The lowest frequency reachable is one cycle per data span, because anything slower is indistinguishable from a change in the pulsar’s own spin frequency and is absorbed by the timing model. Fifteen years is 2.1 nanohertz. The highest useful frequency is set by the observing cadence, typically a few weeks, but the spectrum falls so steeply that it hardly matters: with a characteristic strain going as f−2/3f^{-2/3}, the induced timing residual goes as f−13/6f^{-13/6}, so almost all the power a decade of data contains is in its first two or three frequency bins.

That is a harsher constraint than it sounds. The sensitivity of the array to the background improves as roughly the thirteenth power of the observing span in the regime where the pulsars’ own white noise dominates — an extra two years is worth far more than an extra telescope — and it improves only as the square root of the number of pairs once the background itself is the limiting noise, which is the regime the arrays have now entered. The instrument’s exposure time is measured in decades and cannot be shortened by building anything.

It also explains the shape of the collaboration. A fifteen-year data set is a fifteen-year commitment to observing the same few dozen objects on the same instruments with the same calibration, and a change of receiver in year eight is a systematic that has to be modelled rather than an improvement. The three arrays combine their data precisely because a pulsar observed by two of them for different halves of the period is worth more than either half.

What was actually measured

Three collaborations reported the same thing within a fortnight in June 2023, having spent between fifteen and twenty-five years accumulating the data.

The number of pulsars is a few dozen: sixty-eight in the North American array’s fifteen-year data set, a similar number in the European and Indian combination, thirty in the Australian. That gives some two thousand pairs, which is what the angular curve is binned from — and it is the pair count rather than the pulsar count that matters, so adding one pulsar to an array of sixty adds sixty pairs.

The amplitude is a characteristic strain of about 2.4×10−152.4\times10^{-15} at a frequency of one cycle per year, with a spectral index consistent with the −2/3-2/3 of an inspiralling population and also consistent with rather more than that. In timing residuals it is of order a hundred nanoseconds accumulated over the span.

The significance is the part worth stating carefully. What is quoted is between three and four sigma for the angular correlation specifically — the evidence that the common red process present in all the pulsars has the Hellings–Downs shape rather than a monopolar or dipolar one. The common red process itself had been visible for several years before that and was not claimed as a detection, precisely because a common red signal is what a clock error looks like too.

A background that has been accumulating for the whole history of the universe was announced at three sigma. Not because the effect is small — it is far larger in strain than anything an interferometer has detected — but because the discriminating statistic is a shape in an angle, and a shape in an angle has to be built out of many independent pairs of a rare kind of object, each measured for a decade.

It is worth being explicit about what the error bars on the binned points mean, because they are not what a plot of measurements against a curve usually implies. Each pair of pulsars gives one estimate of the correlation, and that estimate is dominated by each pulsar’s own noise rather than by the signal — so a single pair carries almost no information and the binned points are averages over dozens of nearly useless numbers. The pairs are also not independent of each other: two pairs sharing a pulsar share its noise, so the covariance between bins is large and the naive chi-squared of the points against the curve is not the significance. Every published number comes from a full likelihood over the pair covariance, and the reason the value is quoted as a range rather than a figure is that the answer depends on how the pulsars’ individual red noise is modelled — a spectral shape assumed for one pulsar’s own irregularity moves the inferred background amplitude by tens of per cent.

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.
Fig. 5 The signature that separates a real background from every noise source a pulsar has. Gravitational waves passing over the array correlate the timing residuals of two pulsars by an amount that depends only on the angle between them — the Hellings–Downs curve — and nothing else does: a clock error correlates all pairs equally, an ephemeris error correlates them dipolar. So the detection is not a signal in any one pulsar; it is a shape in the correlation between sixty-seven of them.

What the picture cannot show

Which sources. The background is unresolved by construction. Whether it is supermassive black-hole binaries at all is inferred from its amplitude and spectrum, and both are compatible with other things — cosmic strings, a first-order phase transition in the early universe, primordial black holes. The astrophysical interpretation is the most economical rather than the only one, and distinguishing among them needs the spectral index measured well enough to see the departure from −2/3-2/3 that a real population’s eccentricity and environmental coupling would produce, which is a decade away.

The distances. Nothing in the measurement locates a source. A background is a statistical quantity and carries no direction and no redshift, so the amplitude constrains the integral of the merger rate over all of cosmic history rather than the rate at any epoch — the same difficulty that makes an integrated brightness a weak constraint on a luminosity function.

The individual binaries. A single sufficiently loud pair would appear as a continuous wave at one frequency in one direction, and none has been found. That is itself informative: a background made of many equal contributors would have shown a resolvable brightest source by now, so either there are more sources than the simplest model has or the loudest ones are quieter than expected.

And whether the pairs merge at all. Two black holes at a few parsecs sink further only by scattering stars or by interacting with gas, and the supply of stars on the right orbits runs out. If nothing else intervenes the pair stalls, radiates nothing, and waits — the final parsec problem. A detection of the background is evidence that they do get through, which is one of the more consequential facts the measurement establishes and one of the least discussed. The same array read over a longer baseline, and the same waveform read over a wider band, bracket what the two detectors can do.

Nothing visible in any pulsar, and a quadrupole in the angle between them. Above: 4 millisecond pulsars' timing residuals over 30 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. 4950 pairs out of 100 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 = 779. 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.
Fig. 6 Thirty years of timing on a hundred pulsars rather than fifteen on sixty-seven. The correlation curve the array is looking for is unchanged and the error bars shrink, because the sensitivity of a pulsar-timing array grows with the baseline much faster than with the number of clocks.
Two parallel lines of slope 11/3, and the gap between them is a mass ratio. The rate at which the frequency rises, against the frequency, for GW150914 and GW170817. In these coordinates the quadrupole sweep is a straight line of slope exactly 11/3, and the only property of the binary that moves it is the chirp mass, which shifts it vertically by 5/3 of a decade for every decade of mass. So the two lines here are parallel and 2.31 decades apart, and that gap is the 24.2-fold difference in chirp mass between 28.72 and 1.18 solar masses — nothing else about either system enters. Two black holes of 36 and 31 solar masses and a pair of equal masses adding to the same chirp mass would draw the same line, which is why the individual masses are always quoted with error bars several times wider than the chirp mass's. The dots mark where each signal entered the detector band, and the horizontal run to the right of each is the whole observation: GW150914, 183 milliseconds; GW170817, 102 seconds.
Fig. 7 The two archetypal ground-based signals over a wider frequency band. Both are straight lines of the same slope on these axes, offset by their chirp masses — which is the statement that a chirp rate is a mass and that nothing else about the system enters it.

The generalisation

The method is older than the application. Any signal buried under noise that is independent between channels can be recovered from the cross-correlation between channels rather than from any channel’s own spectrum, provided the signal’s correlation structure is known in advance and the noise’s is not the same shape.

The same argument is what lets two interferometers on opposite sides of a continent claim a detection that neither could claim alone; it is what a radio interferometer does when it discards the total power and keeps only the correlated part; and it is the reason an atmospheric phase error cancels in a closure relation while surviving in any single baseline. The pattern is always the same: the thing being measured is common to the channels, the thing being avoided is not, and the discriminating statistic is a correlation rather than an amplitude.

What is unusual here is only the geometry of the arms — a few dozen of them, thousands of light-years long, each with one end on the Earth and the other on a neutron star nobody chose the position of.

And the mass degeneracy the chirp leaves behind, over a wider range than a single event occupies.

One measured number, and every pair of masses that produces it. The plane of the two component masses of an inspiralling binary, with three curves of constant chirp mass across it. The middle one is GW150914's value of 28.7 solar masses, and the chirp mass recovered from the coordinates of the drawn curve varies along its whole length by 7.4e-14 per cent — which is the point: every binary on that line radiates the same frequency sweep at leading order, so the early inspiral cannot tell them apart. Two of them are marked. An equal pair of 33.0 and 33.0 solar masses and a lopsided pair of 63.6 and 18.4 sit on the same contour, and their total masses differ by a factor of 1.24. What separates them is the mass ratio, which enters the phasing only at the first post-Newtonian order, suppressed by the square of the orbital speed in units of the speed of light — small through the hundreds of cycles that carry most of the signal, and appreciable only in the last few, where that speed approaches a third of c. So the chirp mass is a measurement and the individual masses are an inference from the end of the signal, which is exactly the part a detector's high-frequency noise eats first.
Fig. 8 Every pair of masses consistent with the chirp mass of a thirty-and-eight-solar-mass pair. The curve is a hyperbola and the measurement fixes a point on it and nothing more, so separating the two masses needs the later part of the waveform where the approximation the chirp rests on breaks down.

Where this ladder goes next

Later rungs on this anchor: continuous waves from a resolved individual binary, and what a detection of one would fix that the background cannot; the anisotropy of the background, which should be dominated by the nearest few sources and is a map rather than a number; memory, the permanent offset a passing burst leaves in the residuals; the polarisation content, since alternative theories of gravity permit scalar and vector modes with their own angular correlation curves that are not Hellings–Downs; and the space-based interferometer band between the two, where the sources are the massive binaries in their final years and the galactic white-dwarf population is a foreground rather than a signal.

Where this goes next

GW150914: 33 Hz to 250 Hz in 0.22 seconds. The strain of GW150914 — two black holes — through the last 0.22 seconds before merger, computed from the quadrupole sweep at the chirp mass its fit returned, 28.716 solar masses, and drawn at the luminosity distance it returned, 440 megaparsecs. Two things rise together and neither is free to rise on its own: the frequency goes from 33 Hz to 250 Hz, and the envelope — the outer curve — grows by a factor of 3.9, because the amplitude goes as f^2/3 and nothing else in it changes over so short a span. The vertical axis is in units of 10⁻²¹, so the peak here is a fractional length change of about 2.9·10⁻²¹: over the four kilometres of an interferometer arm that is 1.2·10⁻¹⁷ metres, a thousandth of the width of a proton. The chirp mass is not fitted to the amplitude at all — it comes from the spacing of these zero crossings, which is why it is the best-determined number in the whole event and why the distance, which does come from the amplitude, is the worst. A distance with no ladder under it The frequency sweep of an inspiral fixes the chirp mass with no distance in it, and the amplitude then gives the luminosity distance directly, because one expression fixes both. That is a distance measured with nothing calibrated beneath it — and its error budget is one angle. 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. S2's orbit, and the mass it implies. The orbit of S2 about the centre of the Milky Way, drawn from its measured elements: a semi-major axis of 0.1251 arcseconds, which at 8.28 kpc is 1036 AU, an eccentricity of 0.8843, and a period of 16.05 years watched round twice. Kepler's third law in solar units — a³/P² — gives 4.31 million solar masses, by exactly the calculation that weighs a planetary system. At periapsis the star is 120 AU from the focus, 1407 Schwarzschild radii, and the mean density inside that radius is 5.3e+15 M☉ per cubic parsec — a million times the densest star cluster known. That density, not the mass, is the argument: there is no configuration of stars that would fit. The mass at the centre that is not stars One star at the centre of the Milky Way has been watched round a complete orbit. Its semi-major axis and its period give four million solar masses by Kepler's third law — and the volume that mass occupies is small enough to rule out every alternative to a black hole.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Characteristic strainChirp massCorrelationGravitational-wave backgroundHellings downs curveMillisecond pulsarA pulsar timing arrayQuadrupoleRed noiseSolar system ephemerisSupermassive black hole binaryTiming residual