The orbit that has to shrink

An orbit is usually presented as the standing example of something that persists. Two masses go round each other with a conserved energy and a conserved angular momentum, and nothing about the arrangement suggests a direction for time. That is exactly right in Newtonian gravity and exactly wrong in general relativity, where an accelerating mass distribution radiates and radiation carries energy away.

Every bound orbit in the universe is therefore temporary, and the arrow that gives it a direction is the same kind of arrow a diffusion equation carries: a conservative system coupled to something that carries energy away and does not bring it back. The question is only whether the timescale is shorter than anything else that will happen first, and for almost everything it is not.

Why the third term is the first that radiates #

An accelerating electric charge radiates, and the leading term in its radiation is dipolar — proportional to the second derivative of the dipole moment. Gravity has no such term, and the reason is a conservation law rather than an accident.

The consequence is quantitative and severe. A dipole radiator’s power goes as the square of a second derivative of a first moment; a quadrupole’s goes as the square of a third derivative of a second moment, which carries two extra factors of velocity over the speed of light. Combined with the smallness of Newton’s constant — and gravity’s weakness relative to every other interaction is the fact that decides the scale of everything large — it gives the famous result that the whole Earth–Sun system radiates about 200 watts — comparable with a few light bulbs, from a system of 2 × 10³⁰ kilograms.

The power for a circular binary is

$$P = \frac{32}{5}\frac{G^4}{c^5}\frac{m_1^2m_2^2(m_1+m_2)}{a^5},$$

and the extraordinary steepness in $a$ is what makes the subject. Every quantity in the essay follows from that line and from the energy of a circular orbit.

The runaway, and why it is one #

A circular orbit’s total energy is $-Gm_1m_2/2a$, which becomes less negative as $a$ grows. So losing energy means shrinking, and shrinking means a larger orbital frequency and a larger radiated power, which means losing energy faster.

Setting the radiated power equal to the rate of change of the orbital energy gives

$$\frac{da}{dt} = -\frac{64}{5}\frac{G^3}{c^5}\frac{m_1m_2(m_1+m_2)}{a^3},$$

which integrates to a time to coalescence proportional to $a^4$. That fourth power is what the hero figure measures off its own curves, and it is the reason a binary spends almost all of its life at nearly its initial separation and then finishes suddenly.

The numbers are worth stating because they span so much. The Earth and the Sun will merge in about $10^{23}$ years, which is thirteen orders of magnitude longer than the universe has existed and is therefore a statement about an expression rather than about the Earth. The Hulse–Taylor binary pulsar, two neutron stars in an eight-hour orbit, has about 300 million years left. A pair of stellar-mass black holes at a thousandth of an astronomical unit has minutes.

What has to happen first, for anything to get close enough #

The fourth power cuts both ways, and it produces a difficulty that is not obvious from the formula. Two stars that form together in a wide binary have a coalescence time far longer than the age of the universe; two that form close enough to merge would have been inside one another as main-sequence stars. Neither route works, and the systems exist.

An orbit’s shape is a sensitive statement about what is acting on it, which is why anything that changes it — a third body, a tidal bulge, a departure from the inverse square — shows up as a precession or a drift long before it shows up as anything else. That sensitivity is what makes the decay measurable at all: the energy lost per orbit is minute, and the accumulated change in the orbital period is not.

The candidate mechanisms are all about losing orbital energy some way other than by radiation: a common-envelope phase in which the two stars spiral in through each other’s outer layers, dynamical exchanges in a dense cluster, or a distant third body driving the eccentricity up so that the closest approach becomes small enough for radiation to take over. Which of them dominates is a question about populations and is settled by counting detections rather than by deriving anything, and it belongs to the collection that owns the sky.

What this essay can say is the arithmetic those mechanisms are up against. To merge within ten billion years, two thirty-solar-mass black holes must start within about 0.2 astronomical units of one another — a fifth of the Earth’s orbit, for objects that were once stars a hundred times that size.

What a detector actually measures #

Because the frequency rises as the orbit tightens, the signal from an inspiral is a chirp: a sweep upward in frequency and amplitude, ending abruptly.

Working out the sweep rate gives an expression in which the two masses appear only through

$$\mathcal{M} = \frac{(m_1m_2)^{3/5}}{(m_1+m_2)^{1/5}},$$

the chirp mass. That is a genuinely remarkable fact about the measurement. A detector watching a sweep for a fraction of a second learns one number about a pair of objects it cannot see, and the number is neither mass and neither the sum nor the product.

It also explains the shape of the published uncertainties. The chirp mass of a detected event is typically known to a per cent or better; the individual masses to tens of per cent. The mass ratio enters only through the post-Newtonian corrections that become significant near the end of the inspiral, so it is measured from the least of the signal rather than the most.

The pulsar that proved it before anything was detected #

The first evidence that gravitational radiation carries energy was not a wave. It was an orbit.

A binary pulsar is a clock in an orbit, and the arrival times of its pulses measure that orbit to a precision nothing else in astronomy approaches. That is what proved the radiation before any detector saw a wave: Hulse and Taylor watched the orbital period of PSR B1913+16 shorten by 76 microseconds a year, and general relativity predicts 76 microseconds a year. The agreement now stands at better than a part in a thousand over four decades.

Hulse and Taylor found a pulsar in a binary in 1974 and timed it. The orbital period was decreasing, by 76 microseconds a year, and general relativity’s prediction from the quadrupole formula — with no free parameters, every quantity in it determined from the same timing data — matched to within a fraction of a per cent. That agreement, published in 1979 and improved continuously since, was the reason nobody doubted the waves existed for the thirty-six years before one was directly detected. It is a good illustration of how a prediction becomes knowledge: not by being confirmed in the form it was made, but by being confirmed in a form nobody was looking for — the way a gas’s density-independent viscosity was.

The measurement is a good example of a general point about how a weak effect becomes measurable. The energy loss per orbit is a part in 10¹⁴; what is measured is not that but its accumulated effect on the phase of the orbit, which builds up over decades. A shift in period integrates to a shift in time that grows as the square of the elapsed time, and forty years of it is a shift of tens of seconds — enormous, on a clock good to microseconds.

The distance that needs no ladder #

The chirp measures a mass and it measures something else at the same time, and the something else is the quantity astronomy has spent a century building an elaborate chain of methods to obtain.

The amplitude of the wave arriving at a detector depends on the chirp mass, on the frequency, and on how far away the source is — and the first two are read directly from the sweep. So dividing the observed amplitude by what those two predict gives the distance, with no calibration, no assumed standard candle and no reference to anything else in the sky.

That is unlike every other distance measurement in astronomy. The usual route is a ladder: parallax for the nearest stars, calibrating a class of variable star, calibrating a class of supernova against those, and using the supernovae far away — with every rung’s uncertainty inherited by the ones above it. A binary inspiral is not a rung on that ladder; it is a self-calibrating measurement, because general relativity supplies the absolute amplitude and there is nothing to standardise.

Sources measured that way are called standard sirens, by analogy with standard candles, and the term is better than the analogy: a candle’s brightness has to be established from somewhere, and a siren’s does not.

The catch is the same kind of degeneracy the mass ratio suffers. The amplitude also depends on the orientation of the orbit — a binary seen face-on is louder than one seen edge-on — and a single detector cannot separate a distant face-on source from a nearer inclined one. Breaking that needs the two polarisations, which requires several detectors, or an independent handle on the inclination, which is what an electromagnetic counterpart supplies.

When one arrives, the pairing is decisive. A merger of two neutron stars in 2017 was seen both as a gravitational-wave chirp and as a burst of light, so its distance came from the siren and its recession speed from the host galaxy’s spectrum — and the ratio of the two is the expansion rate of the universe, measured by a route with no ladder in it at all.

The mass that is not the mass #

There is a second degeneracy, more fundamental than the others, and it is one no amount of detector improvement will remove.

A source at cosmological distance is receding, so every frequency in its signal arrives lower than it left, by a factor of one plus the redshift. And the chirp mass is inferred from the frequencies. So what a detector measures is not the chirp mass but the chirp mass multiplied by that factor — the redshifted chirp mass — and no observation of the wave alone can separate the two.

The consequence is exact and slightly startling: a merger of two thirty-solar-mass black holes at redshift one produces a signal identical to a merger of two sixty-solar-mass black holes at redshift zero. Not similar; the waveforms are the same waveform, because the whole of general relativity has no length scale in it that a mass could be compared against. Scaling every mass in the problem by a factor and every time by the same factor gives a solution of the same equations.

So a gravitational-wave catalogue is a catalogue of redshifted masses, and converting to intrinsic masses requires a distance — which the amplitude gives, subject to the inclination degeneracy above. The two degeneracies are therefore entangled: getting a mass needs a distance, getting a distance needs an orientation, and getting an orientation needs more than one detector.

That is why the field’s mass estimates are quoted with the redshift assumption stated, and why the statistical properties of the population — how many events at what masses — have to be extracted with a model of the distance distribution folded in rather than read off.

Why the band matters #

One more consequence of the same scaling explains why several quite different instruments are being built for the same phenomenon.

Since the equations have no scale, everything about a merger scales with the total mass: the size of the final object, the duration of the last cycles, and the frequency at which it all happens. Heavier means lower and slower. A pair of stellar-mass black holes finishes at a few hundred hertz; a pair of a million solar masses finishes at a few millihertz — five orders of magnitude down, from a mass five orders up.

Ground-based detectors cannot reach millihertz, and the obstacle is not engineering. Seismic motion of the ground swamps everything below a few hertz, and no isolation system removes it, so the low-frequency band is closed on Earth in principle rather than in practice. Reaching it requires an instrument in space, with arms of millions of kilometres rather than kilometres — because the arm length has to be comparable with the wavelength of interest.

And the very lowest frequencies, nanohertz, are reached by an instrument nobody built: an array of millisecond pulsars, whose pulse arrival times are perturbed by waves passing through the Galaxy. The “detector” is thousands of light years across, its “mirrors” are neutron stars, and its band corresponds to binaries with orbital periods of years — supermassive black holes in merging galaxies.

Three bands, three technologies, three populations of source, and one scaling law connecting them. What a detector can see is decided by its size, and what wants to be seen is decided by its mass, and the two are related by the absence of any scale in the theory.

The energy, and where it goes #

It is worth asking what the radiated energy is, because the numbers for a merger are outside ordinary experience — and because what is radiated is mass turned into energy, which is the only currency the sum can be stated in.

The first detected event, two black holes of about 36 and 29 solar masses, produced a remnant of about 62. The missing three solar masses were radiated as gravitational waves in about two tenths of a second. Converting that to power gives roughly 3.6 × 10⁴⁹ watts — greater, briefly, than the combined light output of every star in the observable universe.

Chemical burning converts parts in $10^9$ of a mass, fission and fusion a fraction of a per cent, and a black-hole merger several per cent — radiated as gravitational waves in a fraction of a second. That is where the energy goes and how much of it there is: the merger that LIGO first detected radiated three solar masses, briefly outshining the entire visible universe, and none of it arrived as light.

None of that power was visible. Gravitational waves pass through matter almost without interacting — the same weak coupling that makes them hard to detect makes the universe transparent to them — so an event brighter than every star produced no photons at all and heated nothing on its way out.

Where the Newtonian description stops #

Everything above uses the quadrupole formula, which is a first-order result derived on a flat background, together with a Newtonian orbit. Both fail at the end.

The Schwarzschild radius of thirty solar masses is 88 km and the innermost stable orbit is three times that, so the last few orbits before a merger happen at separations of a few hundred kilometres at a substantial fraction of light speed. That is where the Newtonian description stops entirely — not by becoming inaccurate, but by having no term for the radiation that is by then carrying away most of the energy.

Below the innermost stable circular orbit there are no circular orbits at all — a threshold with no Newtonian counterpart, and one of the few places where the strong-field theory says something qualitatively new rather than numerically different, since the general-relativistic term abolishes them — so the slow inward drift ends and the two bodies plunge together in less than an orbital period. What follows is a merger and then a ringdown, in which the single remaining object settles to a stationary state by radiating away its deformations, and neither can be computed by any expansion. Numerical relativity, which solves Einstein’s equations on a grid, was developed for precisely that stretch, and the first stable simulation of a binary merger was achieved in 2005 — ten years before the first detection, and only just in time to interpret it.

The Newtonian and relativistic predictions for light deflection differ by exactly a factor of two, and the factor between the Newtonian and relativistic accounts of orbital decay is the same kind of thing: a term that has no Newtonian counterpart at all rather than a correction to one that does. Where the Newtonian description stops is not where it becomes inaccurate but where it becomes silent — it predicts no radiation whatever.

Where the model stops #

The orbits here are circular and most are not. An eccentric binary radiates far more strongly, because the emission is concentrated at closest approach and the power goes as the inverse fifth power of separation. The enhancement factor is enormous for high eccentricity — a factor of a thousand at $e = 0.9$ — and it also circularises the orbit, so any pair that has been inspiralling for a long time is very nearly circular by the time it is loud.

Nothing here treats the bodies as extended. Neutron stars are deformed by their companion’s tide near the end, and the deformation absorbs energy and changes the sweep. That tidal signature is a measurement of the star’s internal structure and is one of the few routes to the equation of state of matter above nuclear density.

Spin has been left out entirely. A rotating black hole drags spacetime around with it, which shifts the innermost stable orbit inward or outward by a large factor depending on whether the spins are aligned with the orbit, and adds a precession of the orbital plane that modulates the observed amplitude. The spin is measurable from the signal, badly, and is the second-largest source of uncertainty after the mass ratio.

And the sources are named and not observed here. Which pairs exist, how many, how they formed and what their populations imply are questions about the sky, and they belong to the collection that owns celestial mechanics and observation. What is derived here is what any two masses must do.

What the pictures cannot show #

The chirp figure draws a frequency against time and cannot draw the amplitude, which rises together with it — so the signal is not merely getting higher but louder, and the two rises are not independent. A drawing that showed both would need a second axis and would obscure the sweep, which is the thing being measured.

Nor can the ring figure convey the size of the strain. A stretch of 10⁻²¹ drawn to scale on any figure is invisible, so the deformation is exaggerated by twenty orders of magnitude, and the drawing is a diagram of a shape rather than a picture of a motion.

Where this ladder goes next #

The rung below established what a gravitational wave is: a transverse quadrupolar strain with two polarisations, stretching one way while squeezing the other. This rung is what happens to the system that emitted it — the recoil on the source, which is the part any radiation argument eventually has to face, and which turns an orbit from a permanent arrangement into a countdown.

The rungs above it are the ones this essay named and left. The merger and ringdown, where perturbation theory fails and the remnant’s own resonances — quasinormal modes, determined by its mass and spin and nothing else — are what the last few cycles carry. The stochastic background, the superposition of every unresolved binary in the universe, which is a noise floor and a measurement at once. And the memory effect, the permanent displacement a passing wave leaves behind, which is the strangest prediction in the subject and has not been detected.

The habit worth carrying away is about steep powers. When a rate goes as a high power of a shrinking quantity, the process is not gradual and its history is not informative about its future. A fourth-power lifetime means a binary looks unchanged for 99 per cent of its existence and then finishes within a fraction of a per cent of it. The same shape governs a bearing’s wear, a crack’s growth and a resonance’s decay, and in every case the mistake to avoid is extrapolating from the long quiet part.