A ceiling the outer orbit imposes

Every statement about this mechanism has rested on one expression:

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

which says how far a distant companion can drive a nearly circular orbit’s eccentricity, given only the angle between the two planes. It appears in the account of how an inclination turns into an eccentricity, it sets the floor the relativistic quench is measured against, and it is the number that makes the merger rate of compact triples large.

It is a test-particle result, and the assumption is not a technicality. A test particle carries no angular momentum, so the cycle can take all of the inner orbit’s away without anything else having to change. A real inner binary of comparable mass carries angular momentum comparable with its companion’s, and what it gives up has to go somewhere.

Where the ceiling comes from #

Two things are conserved by the averaged quadrupole problem, and the test-particle formula uses only a degenerate version of the first.

The total angular momentum — the third thing a two-body problem conserves, here summed over two orbits rather than one. The inner orbit’s angular momentum is $G_1 = L_1\sqrt{1-e_1^2}$ and the outer’s is $G_2$, and with the two making a mutual angle $i$ the total satisfies

$$G_{\rm tot}^2 = G_1^2 + G_2^2 + 2G_1G_2\cos i.$$

For a test particle $G_1 \ll G_2$, so $G_{\rm tot} \approx G_2$ and the equation reduces to $G_1\cos i =$ constant — which is $\sqrt{1-e^2}\cos i$, the quantity every account of the mechanism quotes. The famous conserved combination is the general law in a limit, and outside that limit it is not conserved.

And the averaged energy, the quadrupole Hamiltonian, which for an outer orbit that stays circular is

$$F \propto (2 + 3e_1^2)(3\cos^2 i - 1) + 15\,e_1^2\sin^2 i\,\cos 2\omega_1 .$$

Setting both equal to their initial values and asking for the largest $e_1$ reachable gives one algebraic equation in one unknown. At zero mass ratio it factorises and the closed form falls out; at finite mass ratio it does not, and the answer has to be found numerically — which is what the figure draws.

The physical reading is short. The cycle works by transferring angular momentum from the inner orbit to the outer one. A test particle can be emptied because the outer orbit is infinitely large by comparison and does not notice. An inner orbit carrying a comparable share cannot, because taking its angular momentum tilts the outer orbit, which changes the mutual inclination, which is the quantity the cycle’s strength depends on — so the mechanism throttles itself.

What it changes about the band #

The critical inclination band — the range within which the cycle operates at all — is a test-particle result too, and it moves.

At zero mass ratio the condition is $\cos^2 i_0 < 3/5$, giving 39.2 to 140.8 degrees. At finite mass ratio the boundaries shift, and the shift is asymmetric: the prograde edge moves outward and the retrograde one inward, because the total angular momentum’s magnitude depends on whether the two orbits add or subtract.

That asymmetry is genuinely new physics rather than a correction. For a test particle a prograde orbit at 45 degrees and a retrograde one at 135 behave identically, because only $\cos^2 i$ appears. For a massive inner orbit they do not, and one of the two is more easily driven than the other.

What a ceiling of 0.866 costs #

The number at ninety degrees for equal angular momenta is $\sqrt{3}/2$, and it looks like a mild reduction. It is not, because nothing in this subject depends on the eccentricity linearly.

The quantity that matters is the pericentre, $a(1-e)$. At $e = 1$ the pericentre is zero; at $e = 0.866$ it is 13.4 per cent of the semi-major axis. So the deepest approach a massive inner binary can be driven to is an order of magnitude further out than the test-particle formula allows at the same inclination.

Follow that into the radiation rate, which goes as $(1-e^2)^{-7/2}$. At $e = 0.99$ that factor is 200,000; at $e = 0.866$ it is 74. A ceiling that lowers the eccentricity by thirteen per cent lowers the radiation rate at the peak by a factor of nearly three thousand.

And follow it into the tidal dissipation that makes hot Jupiters, which goes as a still higher power of the pericentre — the tidal heating rate scales as roughly the inverse sixth power of the separation, so the same change is a factor of order $10^{5}$ in what happens at closest approach.

A modest correction to an eccentricity is an enormous correction to everything the eccentricity is used for, and the reason is that the mechanism’s purpose is to deliver a small pericentre while the eccentricity is the quantity being bounded. One minus a number near one is where all the sensitivity is, and a ceiling on the number is a floor on the difference.

That is the practical content of the correction, and it is the reason it is worth making rather than noting. The test-particle formula does not overstate the maximum eccentricity by much; it overstates what the mechanism can deliver by orders of magnitude, in exactly the systems the mechanism is invoked for.

Which systems sit on the wrong side #

The ratio that matters is the inner orbit’s angular momentum divided by the outer’s, which for masses $m_1, m_2$ inside and $m_3$ outside is roughly

$$\eta \simeq \frac{m_1m_2}{m_3(m_1+m_2)}\sqrt{\frac{(m_1+m_2)\,a_1}{(m_1+m_2+m_3)\,a_2}}\,(m_1+m_2+m_3).$$

The square root makes it small for a hierarchical system and the mass factors can make it large, and the two do not always point the same way.

A planet with a distant stellar companion has $\eta$ of order $10^{-3}$: a Jupiter at five astronomical units against a solar-mass star at five hundred. This is the case the test-particle formula was written for and it is essentially exact.

A hot Jupiter’s progenitor with a binary companion is similar and still safe.

A stellar binary with a third star is not. Two solar masses inside at ten astronomical units, one outside at a thousand, gives $\eta$ of order 0.1 — enough to lower the ceiling measurably, and enough to matter for a mechanism whose output depends on $1-e^2$ to the seventh power.

And a compact-object binary with a stellar companion — the configuration the compact-triple merger channel is built on — can have $\eta$ near or above one. Sixty solar masses of black holes inside against one solar mass of star outside is a mass ratio that swamps the square root, and the inner orbit is then the dominant reservoir of angular momentum in the system.

That last case is the awkward one, because it is exactly where the test-particle formula has been used to justify a large merger rate. The channel’s own configuration is the one the approximation fails for, and the correction goes in the direction that reduces the rate.

The other direction it can go #

Being more careful does not always make an estimate smaller, and here there are two effects with opposite signs.

The ceiling this essay is about lowers the maximum eccentricity. Against it, two things raise it.

The octupole term. When the outer orbit is itself eccentric, the conserved combination that produces the ceiling stops being conserved at the next order, and the inner orbit can be driven past the quadrupole limit and flipped through ninety degrees. That extends the mechanism’s reach considerably.

And the failure of averaging. The secular equations assume the inner orbit completes many revolutions during one step of the outer one’s influence. At the top of an excursion the pericentre passage is fast, the change per orbit is not small, and direct integrations find peak eccentricities systematically higher than the averaged equations give — by enough to matter in exactly the regime where the pericentre is deepest.

So the honest statement is that the quadrupole test-particle formula is an upper bound with respect to one correction and a lower bound with respect to two others, and which dominates depends on the system. Corrections to an approximation do not come with a sign, and an estimate improved in one respect is not an improved estimate.

The outer orbit’s own excursion #

There is a second half to the same conservation and the figures above suppress it: while the inner orbit’s eccentricity rises, the outer orbit is doing something.

At quadrupole order with an outer orbit that starts circular, its eccentricity stays zero — that is a genuine result and not an assumption, and it is why the figures can hold the outer orbit fixed in shape. What does not stay fixed is its orientation. The angular momentum the inner orbit gives up is absorbed as a tilt, so the outer orbit’s plane precesses and nutates through the cycle by an angle of order $\eta$ times the inner orbit’s own change.

For a test particle that is nothing. For $\eta$ near one it is tens of degrees, and it means the whole picture of “a fixed outer orbit driving an inner one” stops describing what happens. Both orbits move, about the invariable plane that the total angular momentum defines, and the mutual inclination oscillates as a consequence of both.

There is a tidy consequence for what the invariable plane means. For a triple the total angular momentum vector is fixed in space, and both orbits precess about it. A test-particle treatment identifies that direction with the outer orbit’s normal, which is right to order $\eta$; a careful treatment does not, and the difference is the angle the outer orbit itself swings through.

The reference direction the inclination is measured from is itself a function of the mass ratio, which is an unusually direct way for an approximation to contaminate a measurement: two analyses of the same system that differ only in whether they treat the inner orbit as massless will disagree about what the inclination is, before they disagree about what it does.

What was actually measured #

Nothing about this mechanism is measured directly, and the ceiling is measured least of all — it is a correction to a mechanism whose operation is itself inferred statistically.

What can be constrained is the population the mechanism is supposed to produce, and two quantities bear on it.

The multiplicity of close binaries. A high fraction of short-period binaries have a distant third component, which is what the mechanism predicts since the tight pairs are supposed to have been made by it. Establishing that a pair has a third member is itself hard, and the census is incomplete in a way that depends on the companion’s mass and separation — so a claimed fraction carries a completeness correction larger than the effect.

And the mutual inclinations. The mechanism operates only inside the band, so the distribution of mutual inclinations in triples is a direct test — and one that needs two stars a Fourier transform can barely tell apart to be resolved into two orbits. Measuring a mutual inclination requires both orbits, which requires both to be resolved or both to be spectroscopic, and the sample for which it has been done is a few dozen systems.

The ceiling’s own effect on either is below what those samples can see. What it changes is a theoretical rate, and rates from this mechanism are quoted with uncertainties of an order of magnitude from other sources.

Where the model stops #

The outer orbit is circular here. Allowing it an eccentricity brings in the octupole term, which breaks the conservation the ceiling rests on. Everything in this essay is the quadrupole statement, and the quadrupole statement is exactly the one the octupole term invalidates.

The two orbits are treated as rigid rings. The averaging that produces the Hamiltonian smears each orbit into a ring of mass, which is right on timescales long compared with both periods and wrong on the timescale an excursion’s peak occupies.

Spins are absent. A close binary of stars or compact objects has spin angular momentum as well as orbital, and it couples to the orbits. The coupling is weak for stars and not negligible for rapidly rotating compact objects, where it introduces another reservoir the cycle can draw on or has to fill — which is how a black hole’s spin ends up correlated with its orbit rather than independent of it.

And the figure solves for a maximum rather than integrating to it. What is drawn is the largest eccentricity consistent with the two conservation laws, which a real trajectory reaches only if it starts in the right place — a portrait says which trajectories exist and not which one a system is on. Starting from a nearly circular orbit at the right pericentre argument it is, which is the standard assumption and is a statement about initial conditions that nothing measures.

Where the test-particle limit is safe #

The correction is not always worth making, and knowing when it is not is as useful as knowing when it is.

The ratio that governs it carries a square root of the ratio of semi-major axes, so a strongly hierarchical triple — one whose outer orbit is a hundred times the inner one — suppresses the effect by a factor of ten before the masses are considered at all. That is the ordinary case for the systems the mechanism was invented for.

A planet with a stellar companion has the mass ratio working the same way, since the planet is the light one. Both factors point in the same direction and the ratio is of order a thousandth, so the test-particle formula is exact for every practical purpose.

A moon with a distant perturber is the same again, with a still smaller ratio, so the satellite stability limits the same mechanism sets are safe.

A stellar triple is the first case where the correction matters, and it matters at the ten per cent level in the eccentricity — which, run through the powers priced above, is a factor of a few in what the mechanism delivers.

And a compact-object binary with a light companion is the case where the approximation inverts entirely, because the mass factor overwhelms the separation factor.

So the boundary is not a value of a single parameter but a competition between two, and the practical rule is that a system whose inner pair holds most of the system’s mass is a system to compute rather than to look up.

The generalisation #

The shape to carry is that a conserved quantity quoted for a problem is often the conserved quantity of a limit, and that the limit is usually the one in which something is small enough to ignore.

$\sqrt{1-e^2}\cos i$ is conserved for a test particle and is not conserved in general. The general law is conservation of total angular momentum, of which it is the leading term when one reservoir is negligible. Every consequence drawn from the special version — the closed form, the critical inclination, the symmetry between prograde and retrograde — inherits that limit, and each of them fails differently outside it.

A conservation law is a statement about a system, and a system is a choice about what to include. The useful discipline is to ask, of any conserved combination, which body’s contribution was dropped to obtain it, and then to ask whether that body is small in the case at hand. Here the dropped contribution is the inner orbit’s own angular momentum, and the case at hand — two black holes and a star — is precisely the case where it is not small.

The second reading is about where an approximation’s error lives. The test-particle formula is excellent at low inclination, where the excursion is small, and worst at ninety degrees, where the excursion is complete. That is not a coincidence: the approximation is that the inner orbit gives up a negligible amount of angular momentum, and the excursion’s size is how much it gives up. An approximation that assumes a quantity is small fails hardest where the answer depends on that quantity being large, and the place to check it is never the middle of the range.

Still open: the term that breaks the conservation #

What comes next is the octupole, which is what the outer orbit’s own eccentricity introduces. At quadrupole order the component of the inner orbit’s angular momentum along the outer orbit’s axis has a sign that cannot change, so the inner inclination cannot pass through ninety degrees; at octupole order it can, and the orbit flips from prograde to retrograde and back. The excursions are larger, the reach of the mechanism is wider, and these clean bounds become the outer envelope of something chaotic.

Beside it lies the case the averaging cannot describe at all: a pericentre passage fast enough that the outer body’s influence changes appreciably during it, so the orbit-averaged equations break exactly at the moment they are being used to compute the deepest approach.