Astronomy & Celestial Mechanics
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Gravitation

Every pair arrives circular

Gravitational radiation drains a binary's energy and its angular momentum at rates that do not keep step, and the mismatch destroys eccentricity far faster than it shrinks the orbit. A pair that starts at 0.99 and spirals in from a fifth of an astronomical unit is round to better than a part in a hundred million by the time a detector can hear it.

Assumes Gravitational waves, Relativistic orbits and Orbital averages.

17 min read 8 figures Two bodies, and no moreMeasuring the unreachable

A merger’s distance is read off its own waveform with no ladder under it, and everything else about it is read off the same signal. A binary radiating gravitational waves loses two things at once, and the ratio of the two rates is the whole of this essay.

It loses energy, which shrinks the orbit. And it loses angular momentum, which — for a fixed energy — would round the orbit out. The two losses are not independent: both are computed from the same quadrupole formula, both are dominated by what happens near periastron where the acceleration is largest, and the ratio between them depends on the eccentricity. It works out that the angular momentum goes disproportionately fast, and the consequence is that eccentricity is destroyed on the way in.

Every pair arrives circular. Eccentricity against gravitational-wave frequency for four binaries of 30 and 30 solar masses, each starting at 0.2 astronomical units with an eccentricity of 0.3, 0.7, 0.9, 0.99. Both axes are logarithmic and the tracks run left to right as the orbit shrinks. The curves are Peters' closed solution a(e), and they are checked against Peters' differential equation at three eccentricities on each track rather than against the integral they came from. The ordering is preserved — a pair that starts rounder stays rounder — but by the time the orbit is radiating at 10 hertz, where a ground-based detector begins to hear it, the four eccentricities are 8.8·10⁻⁸, 5.6·10⁻⁷, 3.6·10⁻⁶, 1.5·10⁻⁴. All four are far below anything a detector could measure. That is the figure's whole content and it is a strong statement: radiation reaction removes angular momentum faster, relative to energy, than a circular orbit would need, so eccentricity is destroyed on the way in. A binary observed to be eccentric in band therefore cannot have spent long shrinking quietly, and must have been put on that orbit recently — by a third body, or in the crowded centre of a cluster. The figure assumes the two bodies are points and nothing else acts on them, which is exactly the assumption an eccentric detection would refute.
Fig. 1 Eccentricity against gravitational-wave frequency for four equal-mass pairs of thirty solar masses each, starting at a fifth of an astronomical unit with eccentricities from 0.3 to 0.99. Both axes are logarithmic and the tracks run left to right as the orbit shrinks. By the time the orbit radiates at ten hertz, where a ground-based detector begins to hear it, the four eccentricities span from ten to the minus eight down to ten to the minus twelve — all of them far below anything measurable.

The two rates, and why they do not match

Peters wrote the orbit-averaged equations in 1964. For a pair of masses m1m_1 and m2m_2 with total MM:

⟨dadt⟩=−645G3m1m2Mc5a3  1+7324e2+3796e4(1−e2)7/2,\left\langle\frac{da}{dt}\right\rangle = -\frac{64}{5}\frac{G^3 m_1 m_2 M}{c^5 a^3}\;\frac{1 + \tfrac{73}{24}e^2 + \tfrac{37}{96}e^4}{(1-e^2)^{7/2}},

⟨dedt⟩=−30415 e G3m1m2Mc5a4  1+121304e2(1−e2)5/2.\left\langle\frac{de}{dt}\right\rangle = -\frac{304}{15}\,e\,\frac{G^3 m_1 m_2 M}{c^5 a^4}\;\frac{1 + \tfrac{121}{304}e^2}{(1-e^2)^{5/2}}.

Both are averaged over one orbit, which is legitimate as long as the orbit changes little in one period — true everywhere except the last few cycles.

Dividing one by the other removes the constants, the masses and the speed of light, and leaves a relation between the two elements alone:

dade=1219 ae  1+7324e2+3796e4(1−e2)(1+121304e2).\frac{da}{de} = \frac{12}{19}\,\frac{a}{e}\;\frac{1 + \tfrac{73}{24}e^2 + \tfrac{37}{96}e^4}{(1-e^2)\left(1 + \tfrac{121}{304}e^2\right)}.

At small eccentricity that reduces to da/a=(12/19) de/eda/a = (12/19)\,de/e, which says the semi-major axis changes by twelve nineteenths of a per cent for every per cent the eccentricity changes — so the eccentricity falls about 1.6 times faster, fractionally, than the orbit shrinks. That factor compounds over many decades of shrinkage.

At large eccentricity the (1−e2)(1-e^2) in the denominator makes it far more dramatic. A pair at e=0.99e = 0.99 passes very close at periastron, radiates furiously there, and rounds out quickly.

The integral of that relation is Peters’ closed solution,

a(e)∝e12/191−e2(1+121304e2)870/2299,a(e) \propto \frac{e^{12/19}}{1-e^2}\left(1 + \frac{121}{304}e^2\right)^{870/2299},

and the exponents are not decoration: they come from partial fractions on the expression above and they are what the drawn tracks are checked against.

Why the angular momentum goes faster

The asymmetry between the two rates has a physical reading that the algebra hides, and it is worth extracting.

Gravitational radiation from an eccentric orbit is not emitted smoothly around the orbit. The power goes as the sixth power of the orbital speed and the square of the acceleration, both of which peak sharply at periastron — so almost all of the emission from a very eccentric orbit happens in a small fraction of the period, in a short burst near closest approach.

Now ask what a burst of emission at periastron does. It removes energy, which lowers the semi-major axis. It also removes angular momentum, which lowers the periastron distance and the apastron distance together. But a kick applied at periastron mostly changes the apastron: an impulsive change in speed at one point of an orbit leaves that point on the new orbit and moves the opposite side. So the apastron comes down, the periastron barely moves, and the orbit becomes rounder — while the mean of the two, the semi-major axis, has come down by only half as much.

That is the whole mechanism. Emission concentrated at one point of the orbit preferentially shrinks the far side, and shrinking the far side of an eccentric orbit is circularisation. The same reasoning explains why the effect vanishes for a circular orbit: the emission is then uniform around the orbit, there is no preferred point, and the shrinkage is symmetric. A circular orbit stays circular, which is a fixed point rather than a coincidence.

What the numbers come to

Take a pair of thirty-solar-mass black holes — a mass combination that would be read straight off the one number the early waveform delivers — at a fifth of an astronomical unit — a plausible outcome of two massive stars evolving through a common envelope — with an eccentricity of 0.9. The orbit shrinks by a factor of about ten million before it enters the band of a ground-based detector. Over that shrinkage the eccentricity falls by a factor of ten to the eleventh.

The smallest eccentricity a current detector could measure in such a signal is around 0.05, and optimistically 0.01 for a loud event. The predicted value is twelve orders of magnitude below that.

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. 2 The path through frequency and amplitude that a detector actually follows. As the orbit shrinks, the wave frequency rises and the amplitude rises with it, tracing a curve across the detector’s sensitivity band — and most of the signal-to-noise accumulates over hundreds of cycles at the low-frequency end rather than at the merger. That is why the early inspiral, which is exactly where circularisation has had the longest to act, is where the parameters are measured.

Why that is useful rather than dull

A prediction of “zero” would be uninteresting if there were no way for it to fail. There is.

The circularisation argument assumes the pair has been isolated — that nothing has disturbed the orbit since it was formed, and that the only thing acting is radiation reaction. Two black holes that were once two stars in a binary satisfy that. Two black holes that met in the crowded core of a globular cluster do not. If the last encounter leaves a pair sufficiently tight and sufficiently eccentric, radiation reaction takes over from a starting point so deep that there is no room left to circularise. Such a binary enters the band with measurable eccentricity — and there is no other way to produce one.

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. 3 The population this argument is about in the band where it is not resolved. A pulsar timing array does not see individual binaries; it sees the summed background of many supermassive ones, as a correlation between pulsars that depends on the angle between them. Eccentricity survives differently there: the sources are wide, the circularisation time is longer than the age of the universe, and the background’s spectrum at the lowest frequencies is bent by whatever eccentricity the population retains. The same physics, measured on a population instead of an event, and reading the residual eccentricity rather than its absence.

There is a second dynamical route, and it produces eccentricity by a completely different mechanism. That is the mechanism this collection takes up under an inclination turning into an eccentricity, and here it becomes an observational discriminant: a triple-driven merger and a cluster-assembled merger both produce eccentricity, and an isolated binary produces none.

What a detection would have to overcome

Measuring a small eccentricity is not a matter of looking harder. It is a matter of having a template.

Detection works by matched filtering: the data are correlated against a bank of predicted waveforms, and a signal is found when the correlation is large. The bank is built from circular templates, because the circular waveform is a two-parameter family in the masses and an eccentric one is not — adding eccentricity and the orientation of the periastron multiplies the size of the bank by a large factor.

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. 4 The parameter the circular templates are indexed by. Hundreds of inspiral cycles depend on the two masses only through one combination of them, so a whole contour in the mass plane radiates an identical sweep and one number rather than two is measured from the early signal. That degeneracy is what makes a circular template bank small enough to search with. An eccentric bank has no equivalent simplification, and that — rather than any physical argument — is why eccentric searches lag.

The consequence is a mild circularity in the observational programme: the searches are most sensitive to exactly the signals the isolated-binary channel produces, and least sensitive to the ones that would distinguish the dynamical channel. Dedicated eccentric searches exist and are less sensitive than the main pipeline, so a modest population of eccentric mergers could be present and under-counted.

There is a worse version of the same problem, and it has been the subject of some argument. A circular template correlated against a mildly eccentric signal does not simply return a lower score; it returns a good score at slightly wrong parameters. Eccentricity affects the rate at which the frequency sweeps, and so does the mass ratio, and so do the spins — so a bank without eccentricity in it can absorb a small eccentricity into a biased estimate of something else. Distinguishing a genuinely eccentric event from an unusual mass ratio therefore requires a targeted reanalysis rather than a number that falls out of the search, and the handful of candidate eccentric events so far have all been contested on exactly that ground.

The system that measured the effect first

None of this would be believed on theory alone, and it does not have to be.

How long PSR B1913+16 has left, against how far apart it is. The coalescence time of PSR B1913+16 against its separation, log–log, from the published closed form t_c = (5/256)c⁵a⁴/(G³m₁m₂M) (Peters 1964) — quoted, not derived: this figure computes no radiated power. Because t_c goes as the fourth power of the separation, both lines are exactly slope 4, which is the cheapest possible test of the drawing. The upper line is the circular-orbit form; the lower is the same form multiplied by Peters' eccentricity factor (1−e²)^(7/2), which at e = 0.6171334 is 0.1868 — a factor of 5.4, and by far the largest thing about this system's fate. At the present separation of 1.95·10⁶ km the eccentric form gives 306 Myr against a published 301 Myr from integrating with the eccentricity allowed to evolve, and the circular form gives 1637 Myr, which is wrong by more than a factor of five. A year from coalescence the two stars are 14,741 km apart — about two Earth radii — and the closed form stops describing anything in the final seconds.
Fig. 5 The measurement that established radiation reaction as a fact rather than a prediction. The orbital period of a binary pulsar shortens because the pair is radiating, and the cumulative shift of periastron passage — 88 seconds over 45 years — matches the prediction with no free parameters to within a fraction of a per cent. The same equations that produce that curve are the ones integrated to make the opening figure, so the circularisation claim rests on machinery that has been tested to four significant figures.

The pulsar timing also delivers the eccentricity to seven decimal places, which is worth noting because it is the only binary in either category whose eccentricity is measured rather than inferred. That precision comes from the arrival times of pulses rather than from the waveform, and it is the reason the system serves as a calibration for the theory rather than merely as an illustration of it.

It is also worth being clear about which half of the prediction that system tests. The timing measures the rate at which the orbit shrinks, and the shrinking is driven by the energy loss; the circularisation is driven by the angular momentum loss, and the two are different integrals of the same quadrupole formula. A pulsar binary confirms the first to a fraction of a per cent and constrains the second only weakly, because the eccentricity is changing far too slowly to be detected over forty-five years — the damping timescale is of order the merger timescale, which is hundreds of millions of years. So the circularisation claim inherits its credibility from a shared derivation rather than from a direct measurement, and no observed system has yet been watched becoming rounder.

That is a smaller gap than it sounds, because the two integrals differ only in a weighting and the weighting is not adjustable. But it is the reason the argument is stated here as a consequence of Peters’ equations rather than as an established fact about any particular pair, and it is why a confirmed eccentric merger would be informative about the assembly channel rather than about the radiation.

That system’s eccentricity is 0.617 and its merger is three hundred million years away. It is circularising as it goes, on exactly the track the opening figure draws, and by the time it merges it will be round. Its present eccentricity is not a counterexample: it is a point near the left-hand edge of one of those curves.

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.
Fig. 6 The signal at the end of it. The inspiral is a sweep whose rate delivers the mass combination; the merger is the part no closed form describes; the ringdown is the settling of the object left behind. Eccentricity, if present, would appear as a modulation of the inspiral’s amplitude and phase at the orbital period — a periodic imprint on an otherwise monotonic chirp, which is exactly the kind of structure a template bank built without it would smear out rather than reject.

None of the three is a hypothesis about what the dissipation is; each is a statement about when it happens, and that is what the averaging over an orbit turns into a rate.

Where the two rates came from

The equations at the top of this essay are quoted, and it is worth saying where they came from, because their derivation was contested for longer than the result has been used.

The quadrupole formula — the statement that a system’s gravitational-wave luminosity is proportional to the square of the third time derivative of its mass quadrupole moment — was written down by Einstein in 1918, correcting a factor in his own paper of 1916. For the next four decades it was not clear that it meant anything physical. The difficulty was that gravitational waves in general relativity are a change in the coordinates as much as in anything else, and a coordinate change carries no energy; several respected arguments held that the waves were an artefact of the choice of gauge and that no detector could respond to one.

The argument that settled it was a thought experiment about a bead on a stick. If a passing wave moves two rings on a rod relative to each other, and the rod is not frictionless, the sliding does work and heats the rod. Heat is not a coordinate artefact. So a wave carries energy, and a system emitting one must lose it.

The eccentric case was computed by Peters and Mathews in 1963 and completed by Peters the following year, and it is a lengthy but entirely mechanical calculation: write the Keplerian orbit, differentiate the quadrupole moment three times, square, average over one period. The averaging over an eccentric orbit is where the polynomials in ee come from, and the awkward exponents in the closed solution come from partial fractions on the ratio of the two rates.

Nothing in that chain has a free parameter in it. The rates are fixed by the masses and the orbit, so the tracks in the opening figure are predictions rather than fits, and the binary pulsar’s agreement to a fraction of a per cent is a test of the whole apparatus rather than of a coefficient.

The same outcome, by an entirely different mechanism

Circularisation is not peculiar to gravitational radiation, and the parallel case is worth stating because it produces the same signature by physics that has nothing to do with relativity.

An ordinary binary of two stars close enough to raise tides on each other also circularises. The mechanism is dissipation: a tidal bulge raised on a star is dragged out of alignment, friction inside the star converts the deformation into heat, and the energy comes out of the orbit. Because the tide is enormously stronger at periastron than at apastron — it falls as the inverse cube of the separation — the dissipation is concentrated there, and a loss concentrated at periastron rounds the orbit out for exactly the reason described above.

The observable consequence is a sharp feature in a population. Survey the binaries in a star cluster and plot eccentricity against orbital period, and there is a period below which every system is circular and above which the eccentricities are spread. That transition period is the circularisation period, and it is a clock: the longer the cluster has existed, the further out the boundary has moved.

Measured across clusters of different ages, the boundary does move in the right direction. The rate at which it moves is a measurement of tidal dissipation inside stars — a quantity nothing else constrains — and it comes out larger than the standard theory of turbulent damping predicts, which has been an open discrepancy for decades.

So two utterly different processes, one relativistic and one hydrodynamic, produce the same rule: a loss concentrated at closest approach destroys eccentricity faster than it shrinks the orbit. The signature belongs to the geometry of where the dissipation happens rather than to what is doing the dissipating, and that is why the argument in this essay could be made without ever computing a radiated power.

That generality is the reason the prediction is worth stating as a sharp zero rather than as an expectation, and the next section says what would have to be true for it to fail.

What the argument cannot rule out

Two caveats belong here rather than in a footnote.

The orbit averaging in Peters’ equations fails in the last few cycles, and for a pair that arrives at merger with any eccentricity at all, the final orbits are not adiabatic. The tracks drawn are therefore reliable through the long inspiral and not through the plunge.

And the calculation is for two point masses in vacuum. A pair embedded in the gas disc of an active galactic nucleus — a proposed assembly site — is subject to torques from the gas, and those can pump eccentricity as well as damp it. A merger detected with eccentricity would be evidence for a dynamical origin; the converse, that a circular merger proves isolation, is weaker, because most dynamical channels also produce circular mergers most of the time.

There is a third case in the same family, and it is the one that makes the rule general rather than a coincidence of two examples. A body losing mass, or gaining it, from a stream concentrated near one point of its orbit is subject to the same asymmetry, and the orbit responds the same way. What all three share is not a force law but a schedule: whatever acts, acts hardest where the body moves fastest, and an impulse delivered at periastron rearranges the far side of the orbit.

The circularisation argument is a comparison of two power laws, and it is worth drawing at a second mass and a second set of starting eccentricities.

Every pair arrives circular. Eccentricity against gravitational-wave frequency for four binaries of 10 and 10 solar masses, each starting at 0.2 astronomical units with an eccentricity of 0.5, 0.9, 0.99. Both axes are logarithmic and the tracks run left to right as the orbit shrinks. The curves are Peters' closed solution a(e), and they are checked against Peters' differential equation at three eccentricities on each track rather than against the integral they came from. The ordering is preserved — a pair that starts rounder stays rounder — but by the time the orbit is radiating at 10 hertz, where a ground-based detector begins to hear it, the four eccentricities are 1.2·10⁻⁷, 2·10⁻⁶, 8.2·10⁻⁵. All four are far below anything a detector could measure. That is the figure's whole content and it is a strong statement: radiation reaction removes angular momentum faster, relative to energy, than a circular orbit would need, so eccentricity is destroyed on the way in. A binary observed to be eccentric in band therefore cannot have spent long shrinking quietly, and must have been put on that orbit recently — by a third body, or in the crowded centre of a cluster. The figure assumes the two bodies are points and nothing else acts on them, which is exactly the assumption an eccentric detection would refute.
Fig. 7 A pair of ten-solar-mass objects starting from eccentricities up to 0.99. Every track arrives at the detector band with an eccentricity below a thousandth, because the eccentricity falls as a high power of the separation while the separation falls slowly — the conclusion is insensitive to the masses.
GW150914: 27 Hz to 250 Hz in 0.35 seconds. The strain of GW150914 — two black holes — through the last 0.35 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 27 Hz to 250 Hz, and the envelope — the outer curve — grows by a factor of 4.4, 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.
Fig. 8 And the waveform that arrives as a result. It is a clean chirp with no modulation on it at all, which is what a circular inspiral looks like — an eccentric one would show a periodic amplitude and frequency modulation that no detection has ever shown.

Where the ladder goes

The measurement this points to is a population one rather than an individual one. Isolated binaries, cluster assembly and triple-driven mergers make different predictions for the joint distribution of eccentricity, spin alignment and mass ratio — and eccentricity is the cleanest of the three, because the isolated channel predicts identically zero rather than merely a preference.

The same equations run at much lower frequency lead somewhere else. A space-based detector — or an array of pulsars acting as one instrument the size of the galaxy — would hear binaries years or millennia before they reach a ground-based band, when their eccentricities are correspondingly higher and the circularisation has had less time to work. The tracks in the opening figure are steep enough that shifting the observing frequency down by four decades raises the predicted eccentricities by many orders of magnitude — so the same argument that makes eccentricity undetectable at ten hertz makes it a routine measurement at a millihertz.

Where this goes next

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. 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. One number where two masses were The hundreds of orbits an inspiralling binary completes inside a detector's band depend on its two masses only through one combination of them. Every pair on that contour radiates an identical sweep, so the early signal — which carries nearly all the signal-to-noise — cannot say which pair it was. An eccentricity and an inclination trading, at 65° of mutual tilt. The secular equations integrated from a nearly circular orbit (e = 0.02) inclined at 65° to a distant perturber's plane, over three oscillations. Above, the eccentricity; below, the inclination, with the constant √(1−e²)cos i drawn as the flat line it is. The eccentricity climbs to 0.8380 and the inclination falls to 39.25° at the same instant, and neither is a coincidence: the product is fixed, so one can only rise as the other falls. That floor is the same for every starting tilt — at maximum eccentricity j = √(5/3)Θ, so cos i = √(3/5) and the inclination arrives at 39.23° whether the orbit began at 50° or at 89°. The closed form for a circular start is e_max = √(1 − (5/3)cos²i₀) = 0.8380, which contains nothing about the perturber — not its mass, not its distance. Those set the clock and not the amplitude, and the period here is 4.83 Kozai times. What the figure cannot show is what happens at the top of the cycle in a real system: at e = 0.838 the pericentre is 0.1620 of the semi-major axis, where tides, general relativity or a stellar surface all intervene, and the quadrupole picture ends. An inclination that turns into an eccentricity A distant companion cannot change an orbit's size or its energy. It can take a circular orbit tilted past 39.23 degrees and drive it to an eccentricity near one, and back, over and over — and the companion's mass and distance set only the clock.

About the same objects

Not linked from either essay — found by the objects both name.

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.

Angular momentumBinary black holeChirp massCircularisationDynamical formationGlobular clusterGravitational radiationInspiralKozai lidov mechanismMatched filteringOrbital eccentricityPeters equationsRadiation reactionWaveform template