An inclination that turns into an eccentricity

A perturbation is supposed to be small. That is what the word means and it is the whole justification for treating it as a correction: the planets nudge each other, the nudges accumulate slowly, and the orbits wander about their unperturbed shapes by amounts that stay modest for a very long time.

There is a case where a perturbation that is small in every reasonable sense — a companion a hundred times further away, contributing a millionth of the central body’s pull — drives an orbit from a circle to an eccentricity of 0.98 and back again, indefinitely, without ever changing its energy or its size. The condition for it is a single angle.

The quantity that does not move #

Three things are conserved in the averaged problem, and it is worth seeing why each is.

The semi-major axis is conserved because the perturbation has been averaged over both orbital periods. Averaging removes every term that depends on where the bodies are along their orbits, and the energy exchange between two orbits is exactly such a term. What is left changes the shape and the orientation of the inner orbit and never its size — which is the first surprise, since a mechanism that takes an orbit to $e = 0.98$ sounds as though it ought to be a large disturbance and is, in energy terms, none at all.

The z-component of the inner orbit’s angular momentum is conserved because the averaged perturbation is axially symmetric about the outer orbit’s plane, and a symmetry gives a conserved quantity — the same trade that produces the third conserved vector of the pure inverse square, arrived at from the other direction. In dimensionless form that component is $\Theta = \sqrt{1-e^2}\,\cos i$, and it is the whole story: $\sqrt{1-e^2}$ is the angular momentum in units of its circular value, so an orbit that wants to become eccentric must shed angular momentum, and the only way to shed it while keeping the $z$ part fixed is to tilt.

And the averaged energy of the perturbation itself is conserved, since the averaged system is autonomous. That third one turns the problem into a single degree of freedom, which is why it can be drawn.

Why that particular angle #

The critical inclination looks arbitrary and is not. It falls out of asking where the averaged system acquires a stable equilibrium.

Written in $j = \sqrt{1-e^2}$ and the argument of pericentre $\omega$, the averaged perturbing function is

$$F(j,\omega) = (5-3j^2)\left(\frac{3\Theta^2}{j^2}-1\right) + 15(1-j^2)\left(1-\frac{\Theta^2}{j^2}\right)\cos 2\omega,$$

and $j$ and $\omega$ are a conjugate pair, so the motion is Hamiltonian in one degree of freedom — which is the exception rather than the rule, since the unaveraged three-body problem admits no such reduction. Setting both derivatives to zero gives a fixed point at $\omega = 90°$ with $j^4 = \tfrac53\Theta^2$ — and that has a solution with $j \le 1$ only when $\Theta^2 < 3/5$. For an orbit starting circular, $\Theta = \cos i_0$, so the condition is $\cos^2 i_0 < 3/5$: the angle is $\arccos\sqrt{3/5}$ and nothing else enters.

The separatrix is worth dwelling on because it explains a fact that would otherwise be a coincidence. In the opening figure, the minimum inclination reached is 39.25° — and it is 39.25° for a start at 50°, at 65° or at 89°. At maximum eccentricity $j = \sqrt{5/3}\,\Theta$, so $\cos i = \Theta/j = \sqrt{3/5}$ whatever $\Theta$ was. Every Kozai cycle bottoms out at the critical angle, arriving at the inclination that would just barely have started it.

The clock, and what sets it #

The amplitude belongs to the geometry alone; the timescale belongs to the perturber entirely. It is

$$t_{\rm K} = \frac{8}{15\pi}\,\frac{m_{123}}{m_3}\,\frac{P_{\rm out}^2}{P_{\rm in}}\,(1-e_{\rm out}^2)^{3/2},$$

and the $P_{\rm out}^2/P_{\rm in}$ is the part worth reading twice. The cycle takes longer the longer the outer period, as expected — but it depends on the outer period squared and on the inner period inversely, so a tight inner orbit with a distant companion has an enormously long Kozai time and the effect is not thereby weakened, only slowed.

That is the second surprise in this mechanism, and it is what makes it astrophysically important rather than a curiosity: there is no distance at which a companion stops mattering, only a distance at which it stops mattering quickly. A perturber ten times further away takes a thousand times longer and gets to the same eccentricity. Given a few billion years, “a few billion times slower” is still inside the budget.

What was actually observed #

There is a measurement of this mechanism that requires no modelling at all, and it is a gap.

The giant planets have two kinds of satellite. The regular ones are close in, nearly circular and nearly equatorial, and formed in place — some in resonant chains that could not have been assembled where they now sit. The irregular ones are far out — a fifth to a half of the Hill radius — on eccentric, strongly inclined orbits, and were captured. There are of order two hundred of them known, and their inclinations are distributed across the whole range except between roughly 55° and 130°. The band is not thinly populated; it is empty.

That is the Kozai window. A captured satellite in it has its eccentricity driven up until its pericentre reaches the planet or its apocentre reaches the Hill boundary, and it is lost — collides, or is stripped. Everything outside the window survives. The distribution of inclinations of the outer satellites of Jupiter, Saturn, Uranus and Neptune is therefore a direct, four-times-repeated measurement of a critical angle predicted from a conserved quantity, and the edges of the observed gap sit within a few degrees of 39° and 141° once the finite mass of the Sun and the planets’ own oblateness are allowed for.

The distinguishing test has been made. Roughly a third of hot Jupiters have orbits misaligned with their star’s rotation, some retrograde, and that angle is measured from the shape of a line profile during transit rather than from any model of migration. A smooth inward migration through a disc cannot produce a retrograde planet; a mechanism whose whole content is the trading of inclination for eccentricity can, and does.

Two discoveries, four years and one iron curtain apart #

The mechanism carries two names because it was found twice, for two entirely different reasons, by people who could not read each other’s journals.

Mikhail Lidov published first, in 1961, and his problem was artificial: the Soviet space programme wanted to know how long a satellite would stay in a high orbit, and he was computing the evolution of orbits perturbed by the Moon and the Sun. What he found was that an artificial satellite of the Earth placed at high inclination would have its perigee driven down into the atmosphere by the lunisolar perturbation, on a timescale of months, and would re-enter. That is a Kozai cycle used as an engineering result: an orbit can be disposed of by tilting it, and the technique is still used to deorbit spacecraft from high orbits without carrying the fuel to lower them.

Yoshihide Kozai published in 1962 and his problem was natural: the asteroids, and specifically the high-inclination ones, whose secular evolution nobody had computed. He derived the same conserved quantity and the same critical angle, and applied it to the asteroid belt.

The two papers describe the same phase portrait. They share no reference, no notation and no application, and they were separated by exactly the political circumstance that made the first of them about a satellite and the second about a rock.

The averaging, which is the assumption underneath all of it #

Every statement above is about the averaged problem, and the averaging is where the mechanism’s respectability and its limits both come from.

The procedure is to replace each body by its orbit — to smear the mass around the ellipse in proportion to how long it spends at each point — and then compute the interaction between two rings rather than between two points. Anything that depends on where the bodies actually are at a given moment vanishes, and what survives depends only on the shapes and orientations of the orbits.

That is legitimate when the two periods are widely separated, because then the fast motion really does complete many cycles while the slow quantities barely move, and its average is what the slow quantities feel. The formal statement is that the neglected terms are smaller by roughly the ratio of the periods, so a hierarchy of a hundred to one leaves errors at the per cent level.

It is not legitimate when the hierarchy is mild, and the failure has a specific shape. The averaged equations describe a smooth exchange between eccentricity and inclination over many outer orbits; the unaveraged system additionally receives a kick at each outer pericentre passage, when the perturber is closest and its influence is briefly large. Those kicks do not average to zero over a Kozai cycle if the cycle is short.

The consequence is that a system near the edge of the hierarchical regime can reach eccentricities the averaged theory says are impossible, in brief spikes lasting one outer orbit. For a mechanism whose astrophysical importance rests entirely on how close to one the eccentricity gets — because tides and gravitational-wave emission are such steep functions of pericentre distance — a rare brief spike can matter more than the whole smooth cycle.

The averaged picture is a theory of the envelope, and some of the physics lives in the excursions the envelope does not contain.

The practical consequence is that a long-term integration of a hierarchical triple cannot be done with the averaged equations if the answer depends on the eccentricity maximum, which it usually does. The averaged form is what makes the mechanism comprehensible; the direct integration is what makes a number, and the two are used for different purposes rather than as an approximation and its refinement.

The other critical inclination #

There is a second critical inclination in orbital dynamics, it is 63.435°, and confusing the two is common enough to be worth a paragraph.

That one comes from the Earth’s own oblateness. The equatorial bulge makes an orbit’s pericentre precess, at a rate proportional to $5\cos^2 i - 1$, and at $\arccos\sqrt{1/5}$ the rate is zero: the orbit’s pericentre stays put. That is why Molniya orbits are flown at 63.4°, so that apogee remains over the northern hemisphere for the satellite’s whole life instead of drifting round to the south.

The two angles come from different terms, are different numbers, and do opposite things — one starts an exchange and the other stops a precession. What they share is that both are a statement about a coefficient vanishing, and both are quoted as bare numbers with no units, which is what invites the confusion.

They also interact, and that is the operationally important part. The oblateness precession competes with the Kozai precession exactly as general relativity does: whichever is faster wins, and the loser’s effect is suppressed. Close to the Earth the bulge dominates and a satellite at high inclination is safe; far out, where Lidov was working, the lunisolar term dominates and it is not.

The dividing line is a few Earth radii, which places most operational orbits on the safe side and puts high-eccentricity science orbits, geostationary transfer orbits and the disposal orbits above the geostationary belt on the other. Spacecraft in those regimes have their long-term evolution computed with the lunisolar terms included as a matter of course, and the eccentricity growth Lidov described is used deliberately — an orbit can be given an inclination that will drive its pericentre into the atmosphere in a few decades, which disposes of it with no propellant at all.

A satellite designer therefore has to know which of the two regimes an orbit sits in before deciding whether its inclination is a free parameter, and the boundary depends on the orbit’s size rather than on anything about the spacecraft. That is an unusual shape for a design constraint: it is not a limit on what the vehicle can do but a statement about which perturbation will dominate its life.

Almost every close binary has a third star #

The strongest population-level evidence for the mechanism is a correlation that has no other good explanation.

Binaries with very short periods — a few days or less — are overwhelmingly found to have a third star in a wide orbit around them. The fraction is far higher than for binaries of longer period, and it is high enough that the tertiary looks like a requirement rather than an accident.

The Kozai account explains it directly. Two stars cannot form at a separation of a few stellar radii, because their progenitor cloud fragments on much larger scales. They must therefore have started wide and been brought in, and the mechanism that brings them in is the same one that brings in a hot Jupiter: a distant companion drives the inner eccentricity up, the pericentre passes close enough for tides to act, and the orbit shrinks and circularises. The third star is left where it was, wide and now dynamically irrelevant, which is exactly the configuration observed.

The account makes a prediction beyond the correlation, and it is a slightly awkward one: the inner and outer orbital planes should be misaligned, since the mechanism works by tilting, and the misalignment should survive because nothing acts to remove it. Measuring the mutual inclination of a triple is difficult and has been done for a modest number of systems, and the distribution is broad rather than clustered near coplanar.

The evidence is a shape in a population rather than an observation of the process, which is the usual situation for a mechanism whose timescale is longer than the history of astronomy.

Where the model stops #

The picture above is the quadrupole approximation with a test particle, and it has two well-understood failures.

The first is that a perturber on an eccentric orbit adds an octupole term, and the octupole does not conserve $\Theta$. With it the eccentricity can be driven far higher, the inclination can flip from prograde to retrograde, and the neat closed forms are gone; what is left is a chaotic secular evolution that has to be integrated, with the sensitivity to initial conditions a three-body system generally has. Since real triple systems mostly have eccentric outer orbits, the octupole case is the common one and the quadrupole picture is the teaching case.

The second is competition. Anything else that makes the inner orbit’s pericentre precess — general relativity, the tidal and rotational bulges of the bodies themselves, a fourth body — competes with the Kozai precession, and if it wins, the resonance is broken and the cycle is suppressed. The relativistic precession that shows up as forty-three arcseconds a century at Mercury scales as $a^{-5/2}$ and the Kozai rate as $a^{3/2}$, so there is a critical inner separation below which the mechanism simply stops working. General relativity protects close binaries from their distant companions, which is a sentence worth pausing over. The threshold and the phase plane are the two ways of seeing the same conserved quantity, and each is worth drawing at a second set of inclinations.

Where this ladder goes next #

This is the base rung, and it is the quadrupole test-particle case: one small body, one distant perturber on a circular orbit, everything averaged. Each of those simplifications is a rung of its own.

The nearest is the octupole — what an eccentric perturber does, why the flips appear, and why a mechanism with an exact conserved quantity becomes chaotic when a term of the next order is added. Beyond it sits the question of what happens when the inner orbit is not a test particle, so that both orbits exchange angular momentum and the outer one tilts too. And running alongside both is the general question this rung is one instance of: a perturbation described as a slow drift of the elements works only while the drift stays small, and here is a case where the same formalism, honestly applied, produces a drift that goes all the way.