A cycle that builds what the wave needs

The precession that switches the cycle off is about a competition that the cycle loses: relativity precesses the same pericentre the cycle needs held still, and when it wins the cycle stops. That essay ends by naming the other competition, the one the cycle wins.

Gravitational radiation removes energy from an orbit at a rate that depends violently on the eccentricity. Peters’ expressions give

$$\frac{\mathrm{d}a}{\mathrm{d}t} \propto -\frac{1}{a^3}\frac{1 + \tfrac{73}{24}e^2 + \tfrac{37}{96}e^4}{(1-e^2)^{7/2}},$$

and the factor at the end is the whole story: at $e = 0.9$ it is 260, at $e = 0.99$ it is 200,000. A mechanism that drives eccentricity is a mechanism that turns radiation on.

Why the two feed each other #

It would be possible to treat this as two timescales and take the shorter, and that would miss what happens.

The cycle raises the eccentricity. The radiation, at the peak of each cycle, removes energy — which shrinks the semi-major axis without changing the eccentricity much, because the removal is concentrated at pericentre where the orbit’s shape is least affected. The orbit is therefore left smaller and still eccentric, and the next cycle starts from a tighter orbit.

Three things change when it does.

The cycle slows. Its timescale goes as the outer period squared over the inner one, so a tighter inner orbit cycles more slowly. The mechanism that is driving the shrinkage is weakened by the shrinkage.

The relativistic precession strengthens, as the inverse five-halves power of the semi-major axis and as the inverse of one minus the square of the eccentricity. So the quench described there is approaching from below at the same time.

And the radiation strengthens, as the inverse fourth power of the semi-major axis and far more steeply in eccentricity.

The outcome is a race between the third and the first two, and which wins decides the whole character of the system. If the radiation gets there first the binary merges during an excursion, eccentric. If the quench gets there first the cycle stops, the eccentricity relaxes, and the binary is left as an ordinary circular pair shrinking by radiation alone on a timescale that may exceed the age of the universe.

The staircase, and what each step is #

Watching the hero figure’s solid curve rather than reading its endpoints shows the mechanism working, and the shape is worth naming.

The eccentricity runs up, peaks, and falls, over and over — the ordinary cycle. Superimposed on it, the semi-major axis falls in steps: nearly flat while the orbit is round, dropping sharply at each peak. Every step is one pericentre-focused burst of radiation, lasting a small fraction of a cycle and doing almost all of that cycle’s work.

The steps get smaller. Not because the radiation weakens — it strengthens as the orbit tightens — but because the peaks themselves fall: as the orbit shrinks, the relativistic precession rises as the inverse five-halves power of the semi-major axis, the relativistic quench approaches, and each excursion reaches a little less far than the last.

So the system is running two clocks against each other and the figure shows which wins. If the steps shrink faster than the orbit does, the cycle is quenched before the merger and the binary is left to a radiation-only inspiral of essentially infinite length. If the orbit shrinks faster, the steps accelerate and the merger happens.

The boundary between the two outcomes is not a boundary in any one parameter — it is where two functions of the orbit cross, and the crossing depends on the masses, the separations, the mutual inclination and the outer eccentricity all at once. That is why population estimates for this channel are simulations rather than formulae, and why they disagree with one another by more than any of them quotes.

The eccentricity the binary arrives with #

The observable consequence is not the merger rate, which is a number every channel can be tuned to produce. It is a shape.

Radiation alone is a circularising process: the eccentricity falls faster than the semi-major axis, so a binary that has shrunk by a factor of ten has lost almost all of its eccentricity. That is why a binary formed at a wide separation and left alone arrives at a detector’s frequency band with an eccentricity below $10^{-4}$, which is indistinguishable from zero.

A binary driven by a cycle does not. It is being fed eccentricity from outside throughout, and the residual it carries into the band is orders of magnitude larger.

That difference is the channel’s signature, and it is a hard one to observe. The eccentricity at ten hertz for a system that merged through this route is of order $10^{-3}$ to $10^{-2}$, against $10^{-6}$ for an isolated pair — a large ratio between two small numbers, and current detectors constrain eccentricity at ten hertz to about $10^{-1}$. The measurement that would identify the channel is one detectors cannot yet make, and the one that would make it is the one sensitive below ten hertz, where the binary spends far longer and the eccentricity is far larger.

The rate, and why it is a large number #

The reason anybody integrated this system in the first place is that the factor in the first figure is a factor in a merger rate.

Take a population of black-hole binaries at separations of a tenth of an astronomical unit, each with a distant stellar companion on an inclined orbit. Radiation alone would merge essentially none of them in the age of the universe: at that separation the inspiral time for a pair of thirty-solar-mass holes is of order $10^{14}$ years.

The cycle changes that by whatever the factor in the hero figure is for each system, and the factor depends steeply on the mutual inclination through the eccentricity it reaches. Across a population with isotropic inclinations, a few per cent of triples have inclinations inside the band and near enough to perpendicular for the factor to be enormous — and those few per cent merge. The rest are quenched, and the same population supplies both outcomes — which is why the portrait the cycle lives in has to be read for a distribution rather than for a system.

Where the mechanism operates, and what else does the same job #

The compact-triple channel is one of several ways to merge two black holes, and it is worth setting beside the others because they are distinguished by different observables and none by this one.

Isolated binary evolution. Two massive stars in a close pair, which exchange mass, go through a common envelope, and emerge as two compact objects close enough to merge by radiation alone. This is the best-developed channel and its weak point is the common-envelope phase, which nobody can compute.

Dynamical assembly in a cluster. Two black holes that meet by chance in the core of a globular cluster and are hardened by successive three-body encounters until radiation takes over. Its signature is isotropic spins, and it needs the cluster’s core to be dense enough for long enough.

The triple channel, which this essay is about. A hierarchical triple in the field, or a binary in a cluster with a third body bound to it, driven by the cycle. Its distinctive prediction is a residual eccentricity nothing else produces.

And the nuclear-cluster variant, in which the third body is the supermassive black hole at a galaxy’s centre and the inner pair is anything orbiting it. The same mechanism with a mass ratio of a million, a much faster cycle, and a population that is small and concentrated.

The four have overlapping predictions for the merger rate and the mass distribution, and the rate is the quantity most easily tuned in each. A channel is identified by something it predicts that the others cannot produce, and for the triple channel that is the eccentricity — which, as this essay has spent a section explaining, is the quantity that arrives smallest. The nuclear variant has a better signature, because its mergers should be spatially concentrated in galactic nuclei and should carry a measurable Doppler shift from orbiting the central mass, and neither of those has been observed either.

The honest position is that the triple channel is a mechanism that certainly operates, whose contribution to the observed population is unmeasured, and whose distinguishing observable is a factor of ten below the current limit.

What was actually measured #

Nothing here is an observation. Both mechanisms are calculated, and each has been tested separately in a way the combination has not.

The radiation is measured, to a fraction of a per cent. A binary pulsar’s orbit has been observed to shrink at exactly the rate Peters’ expressions give, for fifty years, and that measurement is among the most precise in astronomy.

The cycle is not measured at all. Its period in a stellar triple is ten thousand to a million years, so nothing has been watched through one. The evidence for it is statistical: the fraction of close binaries with distant companions, the obliquity distribution of hot Jupiters, and the pile-up in orbital period near three days.

And the combination is evidence for a channel rather than for itself. What a population of mergers can constrain is the mixture of formation routes, and the quantity that separates them is the spin distribution rather than the eccentricity — because the spins carry the assembly history and the eccentricity has been erased. That the discriminating observable is the one this mechanism does not affect is an awkward fact about the subject.

The integration itself is checked against the one case with a closed form. With the cycle switched off and the orbit circular, the semi-major axis to the fourth power falls linearly with time, so the merger occurs at a computable moment — and the integration reproduces it to a few per cent, which bounds how much of everything else is numerical.

A number for a real triple #

The dimensionless parameters in the figures are easier to trust with a system behind them, so it is worth pricing one.

Take a pair of thirty-solar-mass black holes with a semi-major axis of a tenth of an astronomical unit, and a solar-mass star at fifty astronomical units on an orbit inclined by seventy-two degrees. That is an ordinary hierarchical triple and nothing about it is contrived.

The radiation-only inspiral time for the inner pair is about $2\times10^{13}$ years — a thousand times the age of the universe, so without the companion nothing happens ever.

The cycle’s period is of order the outer period squared over the inner one, times the mass ratio: with an inner period of two days and an outer of three hundred years, that comes to a few hundred thousand years — and the harmonic law is doing all of the work in that sentence.

At seventy-two degrees the excursion reaches an eccentricity of about 0.91, which multiplies the radiation rate by roughly seventy. The figure’s integration then gives a merger after eighteen cycle times, or a few million years — from $2\times10^{13}$ to $10^{7}$, for a companion a thirtieth as massive as the pair, five hundred times further away.

Two things about that arithmetic are worth holding. The companion contributes no energy at all: it changes the shape of an orbit and the radiation does the rest. And the factor is not a property of the companion’s mass, which appears only in the cycle’s clock; it is a property of the inclination, which sets how far the excursion goes. A weaker perturber acting for longer reaches the same eccentricity, so the mechanism’s reach is set by geometry and its speed by mass — which is why the population that merges this way is selected on inclination and barely at all on the companion’s mass.

Where the model stops #

The secular equations are quadrupole and test-particle. The inner binary is treated as massless with respect to the outer companion, which for a pair of black holes and a stellar companion is wrong in the direction that matters — and it is what the ceiling the outer orbit imposes is about. At octupole order, with an eccentric outer orbit, the excursions are larger and the mechanism reaches further.

Averaging fails at the peak. The secular equations average over both orbits, which requires the inner orbit to complete many revolutions during one step of the outer one’s influence. At the top of an excursion the pericentre passage is fast and the change per orbit is not small, so the averaged equations underestimate the eccentricity reached — by enough that direct integrations of the full three-body problem give systematically higher peaks and shorter merger times.

The radiation is a leading-order expansion. Peters’ expressions are the quadrupole formula averaged over an orbit, valid while the orbit is slow and wide. At the pericentre of a very eccentric compact binary neither holds well, and the last few orbits before a merger are outside the theory entirely.

And nothing here has a tidal interaction in it. For a triple whose inner pair is two stars rather than two black holes, tides raised at the pericentre passages dissipate energy far faster than radiation does, and the same mechanism with a different sink produces hot Jupiters and close binaries rather than mergers.

The generalisation #

The structure worth extracting is about what happens when a mechanism’s output is another mechanism’s steepest input.

Two processes acting on one system are usually analysed by comparing timescales, and that is right when each proceeds independently. It is wrong here, because the cycle’s product — a high eccentricity — is the argument of a function the radiation depends on to the seventh power. The combined rate is not the larger of the two; it is the radiation rate evaluated at a state the radiation would never have reached on its own.

The arithmetic of that is worth stating once, because it is the whole reason the factor is large. Averaging a steeply convex function over a driver’s cycle is not the function of the average: the peak contributes almost everything, and the peak is where the driver put it. A merger rate computed from the mean eccentricity a cycle produces is therefore not a poor estimate of the right quantity — it is a good estimate of a different one, and it is smaller by whatever the convexity is worth.

When one process supplies the variable another is most sensitive to, the pair is not a sum. The same shape appears wherever a slow driver feeds a steep response: a tidal heating rate that goes as a high power of an eccentricity a resonance maintains, a nuclear burning rate that goes as a high power of a temperature convection delivers, a reaction rate fed by a fluctuation. In each case the average of the steep function over the driver’s cycle is enormously larger than the function of the average, and computing it as though it were the latter understates the answer by orders of magnitude.

The second reading is about what a fast process erases. Radiation circularises, so any information about how a binary was assembled that was carried in its eccentricity is destroyed by the same process that makes it detectable. A channel identified by a quantity the channel’s own physics removes is a channel that cannot be identified that way, and the useful observables are the ones the intervening physics does not touch — which for a merger means the spins and the masses rather than the shape of the orbit.

Still open: what the companion has to pay #

What comes next removes the assumption that has been in every equation so far. The inner orbit has been treated as a test particle, so the angular momentum the cycle takes out of it goes nowhere in particular. A real inner binary of comparable mass carries comparable angular momentum, the outer orbit has to absorb what the inner one gives up, and the mutual inclination the cycle depends on therefore changes as the cycle runs.

Beside it lies the octupole term, which is what happens when the outer orbit is itself eccentric: the conserved combination that has bounded every excursion here stops being conserved, the inner orbit can flip from prograde to retrograde, and the reach of the whole mechanism extends to configurations the quadrupole theory forbids.