Eleven in a hundred by geometry, and the rest by waiting

Every argument about the Kozai–Lidov mechanism so far has been about one system: an inner orbit, a distant companion, a starting inclination, and what the cycle does to them. A tilted orbit trades its inclination for eccentricity and back; a relativistic precession fast enough switches the trade off; an eccentric outer orbit lets the inner one turn over, and at the moment of turning over its pericentre can shrink by a factor of a thousand. Each of those is a sharp result about a single system.

The questions that observations can answer are not about single systems. They are about populations: what fraction of planets with a stellar companion should end up as hot Jupiters, what fraction of black-hole triples should merge with measurable eccentricity, how often stars near a black-hole binary should be torn apart. Nobody observes a system’s starting inclination, its argument of pericentre or its node; they are random variables drawn from some distribution, and the prediction is an integral over them. That integral turns out to have a simple answer in the quadrupole limit and a much richer one when the octupole is included, and the difference between the two is where the population statistics carry information.

A closed form for the quadrupole population #

At quadrupole order, for an inner orbit that starts nearly circular, the maximum eccentricity reached depends only on the starting inclination,

$$e_{\max} = \sqrt{1 - \tfrac{5}{3}\cos^2 i_0},$$

and is independent of the arguments of pericentre and node. That makes the population integral trivial once the distribution of inclinations is chosen. The natural choice, for companions that formed independently of the inner orbit, is isotropic: orientations spread uniformly over the sphere, which is uniform in $\cos i_0$, not in $i_0$ itself.

Plotted against $\cos i_0$, the population is spread evenly along the horizontal axis, and the fraction reaching any closeness is the width of the band where the curve falls below it. Inclinations within 39.2 degrees of the outer orbit’s plane, either way, do not cycle at all: they are the quarter of the population near the poles of the axis. The rest cycle, but mostly mildly. The curve touches zero only at exactly perpendicular, and near there it is a square-root cusp, so the width of the band below $1 - e = q$ is

$$f(q) = \sqrt{\tfrac{3}{5}\left[1 - (1-q)^2\right]} \approx \sqrt{\tfrac{6q}{5}}.$$

A third of an isotropic population reaches within a tenth of its semi-major axis, about eleven in a hundred within a hundredth, and three and a half in a hundred within a thousandth. Those are small numbers only in the last case. A planet at five astronomical units needs a pericentre of about 0.05 AU for tides to capture it as a hot Jupiter — a closeness of one per cent — so, at quadrupole order and with nothing to stop the cycle, about one planet in nine with a distant companion is sent close enough.

The integrations reproduce it, and then depart #

The closed form is exact for the quadrupole and approximate for everything else. The next figure integrates the same secular equations as the flipping orbits of the octupole — the test-particle Hamiltonian, doubly averaged — for 128 orbits with isotropic inclinations and random pericentres and nodes, at three strengths of the octupole.

The quadrupole population follows the closed form to within the sampling noise of 128 orbits — 9.4 per cent against 10.9 at a closeness of one per cent, 3.1 against 3.5 at a tenth of that. The octupole curves do something the quadrupole curve cannot. They flatten. At ε = 0.03 the fraction reaching a thousandth of the semi-major axis is almost the same as the fraction reaching a hundredth, and the fraction reaching a ten-thousandth is barely lower. An orbit driven by the octupole to a closeness of one per cent does not stop there; it continues, because the octupole’s extremes happen at flips, where the component of angular momentum the quadrupole conserves passes through zero and the eccentricity goes with it.

That flattening is the most important feature of the population result. In the quadrupole limit the population’s closest approaches are spread smoothly over many decades, a few per cent per decade. With the octupole a large block of the population is delivered all the way to collision. For planets that means disruption or tidal capture; for stars, collision; for compact objects, a merger in a single close passage. The integrations stop at a closeness of $10^{-5}$ because the averaged equations mean nothing beyond it, and a real system would have met a short-range force long before.

The shape of the outer orbit decides the rate #

The octupole’s strength is one parameter, $\varepsilon = (a/a_{\rm out})\,e_{\rm out}/(1-e_{\rm out}^2)$, which combines the ratio of the orbits’ sizes with the outer orbit’s eccentricity. Its effect on the population’s rate is large and nearly linear.

At ε = 0.01 — a planet at five AU with a companion at three hundred on an orbit of eccentricity 0.4 — the rate of close approaches doubles. At ε = 0.04 it is five times the quadrupole rate at one per cent and seventeen times at a tenth of a per cent, and more than half of the whole population goes through an extreme approach. The added orbits come from outside the narrow band near perpendicular: the octupole slowly changes the component of angular momentum that the quadrupole holds fixed, so an orbit starting at sixty or seventy degrees can wander into the band after enough cycles, and once in it is driven through a flip.

This curve is what turns a census of companions into a predicted rate. The ratio of the orbits’ sizes comes from the two periods through Kepler’s third law, and the outer eccentricity from the companion’s orbit, so ε can be estimated for each system in a survey. What cannot be estimated for a single system is the mutual inclination, which is why the rate has to be a population statement; and the population statement depends, through this curve, on the distribution of ε the survey finds. Surveys of planets in binaries find companions with outer eccentricities spread broadly between zero and 0.8, which puts most systems with separation ratios of a hundred or less in the regime where the octupole doubles the quadrupole rate or more.

Every quadrupole extreme at one angle #

Where in the cycle the extreme occurs has a consequence of its own, because an extreme approach can freeze the orbit. A planet captured by tides at pericentre stops cycling: it circularises close to its star, and the companion can no longer drive it. Its final orbit keeps the orientation it had at capture. A pair of compact objects driven close enough to radiate gravitational waves merges before the cycle can turn round. In both cases the population is left with the inclinations it had at its closest approaches.

In the quadrupole limit the answer is a single number. The conserved component of angular momentum, $\sqrt{1-e^2}\cos i$, is fixed, and the energy of the averaged motion is fixed; at the maximum eccentricity the two together force $\cos^2 i = 3/5$. Every orbit in the quadrupole population that reaches its extreme does so at 39.2 degrees or at 140.8, and a population frozen there has a bimodal distribution of mutual inclinations with nothing between. That prediction was made for close binary stars with distant tertiaries, where tides between the inner pair play the capturing role, and the observed distribution of mutual inclinations in such triples does show an excess near those two angles, though the samples are small and the inclinations are hard to measure.

With the octupole the spikes survive, fed by the orbits that reach their extreme on an ordinary quadrupole-like cycle, but most of the population is spread across the range between them, and a large share sits near perpendicular — the orbits that reached their extreme in the middle of a flip. A population of frozen systems therefore carries a record of which mechanism froze it. Spikes mean a quadrupole population; a broad distribution with spikes on it means the octupole did most of the work. For hot Jupiters the relevant angle is not quite this one — it is the angle between the planet’s orbit and its star’s spin, which also depends on how the star’s spin axis precessed during the cycles — but the same logic applies, and the measured obliquities of hot Jupiters, with their broad spread and their retrograde tail, are closer to the octupole’s picture than to the quadrupole’s.

A rate that depends on the age of the system #

The last difference is in time.

A quadrupole cycle reaches its maximum in its first period, and every orbit in the population that is going to reach a given closeness does so at once. The curve rises in the first few secular timescales and never moves again. With the octupole the delivery continues. Orbits that start outside the perpendicular band drift into it only after many cycles, as the octupole’s slow change in the conserved angular momentum accumulates, and they arrive one at a time over hundreds of timescales. At ε = 0.03 half of the eventual close approaches within 150 timescales happen after the tenth.

A secular timescale for a planet at five astronomical units with a solar-mass companion at five hundred is of order a few million years, and the octupole’s timescale is longer by a factor of roughly $1/\sqrt{\varepsilon}$. So a population of such systems a billion years old has been through a few hundred secular timescales, and a younger one fewer. The rate of hot-Jupiter production from this channel should therefore depend on the host stars’ ages, rising with time for systems where the octupole matters. The same is true for the rate of stars driven into a black hole by a second black hole orbiting further out: a burst at first from the stars already in the perpendicular band, then a continuing trickle that the octupole feeds.

Why the band near perpendicular is so thin #

The square root in the population fraction is the most consequential number in the problem, and it has a plain origin. The quadrupole cycle conserves $\sqrt{1-e^2}\cos i$, so an orbit that starts nearly circular at inclination $i_0$ can never have $\sqrt{1-e^2}$ smaller than about $|\cos i_0|$ times a constant of order one. To reach $1 - e = q$ an orbit needs $\sqrt{1-e^2} \approx \sqrt{2q}$, and therefore needs $|\cos i_0|$ below a number proportional to $\sqrt{q}$. An isotropic population is uniform in $\cos i_0$, so the fraction that qualifies is proportional to $\sqrt{q}$ too. Asking for an approach ten times closer cuts the eligible population by a factor of three, not ten, and asking for a hundred times closer cuts it by ten.

That is a generous law. A mechanism that needed a fine-tuned orientation, with a fraction proportional to $q$ itself, would deliver one system in ten thousand to a closeness of $10^{-4}$; the Kozai cycle delivers about one in a hundred. It is why the mechanism is taken seriously as a channel for events as extreme as a stellar collision or a merger within a single passage, and why it is invoked so often. It is also why the quadrupole’s prediction is robust against the details of the eccentricity cap. Moving the cap by a factor of ten moves the fraction by a factor of three, so the population fraction is known to within a factor of a few even when the short-range physics is uncertain by an order of magnitude.

The phases do not enter at all in the quadrupole, and that is equally worth noticing. The maximum eccentricity depends on the starting inclination and nothing else, so a population that is isotropic in inclination gives the same answer whatever the distribution of pericentres and nodes. The octupole ends that. Two orbits at the same starting inclination but different nodes can have completely different fates — one flipping within a few cycles, one never — and the population fraction becomes an average over the phases as well, which is why the octupole curves in the figures are noisier than the quadrupole one at the same number of orbits.

A single system read as a draw #

The best-known candidate for the mechanism shows what population thinking does to a single case. HD 80606 b is a planet of four Jupiter masses on an orbit with eccentricity 0.93, passing within 0.03 AU of its star every 111 days and swinging out to 0.9 AU. Its star has a stellar companion about a thousand AU away. The planet is almost certainly in the middle of a Kozai-driven migration: driven to its present extreme by the companion, and being circularised by tides at each passage, on its way to becoming a hot Jupiter in a few billion years. Its orbit is inclined to its star’s equator by about forty degrees, measured as it transits.

Taken alone, the system is a demonstration that the mechanism works. Taken as a draw from a population, it says something else as well. The companion’s orbit and the mutual inclination are not known, but whatever they are, the system had to be in the perpendicular band — or to have been carried into it by the octupole — to be where it is now, and the chance of that for a random orientation is the fraction drawn in the figures: about a tenth at quadrupole order, several tenths if the companion’s orbit is eccentric. A system like it is therefore not a rarity requiring special pleading. What the population figures add is the expectation that for every HD 80606 b there should be several systems with the same architecture and a companion whose orientation did not qualify, whose planet is still on a mild cycle at a few AU — and surveys of wide companions to stars with cold giant planets are where those should be found.

What a population survey can and cannot test #

The three predictions — the geometric floor, the octupole’s multiplication, and the continued delivery — are each testable only statistically, and each is entangled with things the figures leave out.

The first is short-range forces. The integrations let every orbit go as close as the averaged equations allow. Real systems have a precession from general relativity, from tides and from the inner body’s rotational bulge, and each of these caps the eccentricity at a value that depends on the system. For planets around Sun-like stars the cap usually lies below the tidal-capture closeness, so it does not remove the channel; for the most compact triples it does. A population prediction has to fold in the distribution of that cap, which depends on the planets’ masses and radii and on the stars’ spins.

The second is that the inner orbit is not a test particle. For stellar triples the inner pair’s angular momentum can be comparable to the outer orbit’s, the outer orbit then responds, and the octupole takes a different form. The test-particle result is an approximation whose accuracy depends on the mass ratios, and the population’s mass-ratio distribution enters the rate through it.

The third is the isotropic assumption. Companions that formed in the same collapsing cloud as the inner system probably share some of its angular momentum and are more often nearly aligned than isotropy allows, which puts fewer systems in the perpendicular band and lowers the quadrupole floor; companions acquired later, by capture in a cluster, are closer to isotropic. The measured distribution of mutual inclinations in stellar triples shows an excess of aligned systems at small separations and something near isotropy at large ones. The floor of eleven per cent is therefore an upper bound for close companions and a fair estimate for wide ones.

Three samples of the same fraction #

The observations that bear on the population fraction come from three quite different kinds of system, and they have not yet been made to agree.

Hot Jupiters are the most numerous. About one Sun-like star in a hundred has one, and a substantial fraction of them have distant stellar or planetary companions. Whether the Kozai channel produced a tenth of them or most of them depends on the population fraction, the cap from short-range forces, and how many cold Jupiters there were to start with, and estimates range across that span. The obliquity distribution is the discriminating evidence, and it favours a large octupole contribution for the misaligned ones — though the misalignments are erased round cool stars, which complicates the reading.

Merging black holes detected by gravitational waves are the second sample. Pairs formed in isolation arrive circular, having radiated away their eccentricity over billions of years; pairs driven to merge by a third body in a single extreme approach can arrive still eccentric in the detectors’ band. A handful of events show hints of eccentricity. The fraction that do constrains the fraction of mergers that came through triples, and through this curve the distribution of ε among black-hole triples in the dense clusters where they are thought to form.

Tidal disruptions of stars by black holes are the third. Their rate in galaxies known to host binary black holes should be elevated by the Kozai channel, with a burst after the binary forms and a slow decline as the perpendicular band empties — the time-dependence of the last figure, played out over tens of millions of years in a galactic nucleus. The observed rate of disruptions is a few per galaxy per hundred thousand years, too few in any one galaxy to see a burst, but post-starburst galaxies — often the aftermath of a merger, and so plausibly the recent owners of a second black hole — are strongly over-represented among the hosts of disruptions. That is the direction the argument predicts, though a dense central star cluster left by the starburst predicts it too, and the two have not been separated.

Still open: whether the three samples imply one distribution of companions #

Each of the three samples measures the product of a population fraction and a population of triples — the distribution of ε, the mass ratios, the ages, the short-range forces — and each has its own selection effects. The octupole’s rate curve is the common factor, and it is well understood; the distributions it multiplies are not. Whether the fraction of misaligned hot Jupiters, the fraction of eccentric black-hole mergers and the excess of disruptions in post-merger galaxies can be explained by one physically sensible set of triple populations, or require three different ones, has not been worked out with the precision the rate curve would allow. Astrometric surveys are now measuring the orbits of wide companions in three dimensions for tens of thousands of systems, which will fix the distribution of ε directly for the planet hosts, and turn the first of the three samples from a consistency argument into a test.