A stability limit with a notch cut in it

The account of the region a planet may keep a moon in ends with a list of the things it left out, and the first entry on it is inclination: the limit depends strongly on inclination, and the two numbers drawn average over a range no planar figure can represent. This essay is that entry.

The two numbers are 0.5 of a Hill radius for a satellite going round the same way as its planet and 0.7 for one going the other way, and both were obtained by integrating orbits in the planet’s orbital plane. The natural expectation is that a tilted orbit sits somewhere between them, since a tilt of ninety degrees is neither prograde nor retrograde.

It does not.

Why a tilt turns into an eccentricity #

The mechanism is the one an inclination that turns into an eccentricity is about, applied to a satellite rather than to a planet. Average the Sun’s gravity over its own orbit and over the satellite’s, and what is left is a quadrupole torque on the satellite’s orbit. That torque conserves the component of the satellite’s angular momentum along the Sun’s direction — which is to say it conserves

$$\sqrt{1-e^2}\,\cos i,$$

while leaving both factors free to change. An orbit that starts circular and tilted can therefore trade its tilt for an eccentricity, and it does, cyclically, provided the tilt is large enough that the trade is available at all.

The condition is $\cos^2 i < 3/5$, which is an inclination between 39.23 and 140.77 degrees, and inside it the greatest eccentricity reached is

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

At the band’s edges that is zero and the orbit does nothing. At ninety degrees it is one, and the orbit is driven to a radial line.

The boundary is an apocentre #

Now put the two together, and the reason the limit drops is almost trivial once stated.

A satellite is lost when it leaves the lobe. Whether it leaves is decided by how far out it actually goes, which is $a(1+e)$ and not $a$. For a coplanar orbit starting circular, the two are the same and the distinction never arises; for a tilted one they differ by a factor that reaches two.

So the stability limit at inclination $i$ is the coplanar limit divided by $1 + e_{\max}(i)$, and the notch in the hero figure is that division. There is nothing else in it — no fit, no integration, no numerically determined coefficient beyond the two coplanar values it started from.

That is a stronger statement than it looks, because it means the inclination dependence of the limit is derived while the limit itself is integrated. The shape of the notch is a theorem; where the notch hangs from is a numerical result.

The discrepancy that figure is about was flagged in that earlier account and left open there. The outermost retrograde satellites sit 1.8 times further out than the outermost prograde one, where the integrated limits stand at a ratio of only 1.4. Raising the retrograde limit to 0.93 makes the ratio 1.9, which agrees — and the reason the two integrations disagree is that a retrograde orbit near the boundary takes a very long time to leave, so an integration that stops early records it as stable and one that runs longer does not. A stability limit is a statement about a duration, and neither of the two published pairs says which duration out loud.

What the sky shows #

The test is a census, and the census exists.

Three things in that figure are worth separating, because they are three different arguments.

The two flat limits hold. No satellite is outside its own planet’s coplanar boundary, and the retrograde population reaches considerably further out than the prograde one. That is the earlier result, measured on forty objects rather than four.

The notch bites at the edges. Exactly one satellite in the census sits above the derived curve — Jupiter’s Carpo, at 0.32 Hill radii against a limit of 0.31 — and Carpo is the object whose eccentricity is known to be oscillating with a large amplitude, which is what a satellite sitting on this boundary is supposed to do. A derived curve that puts one object marginally outside itself, and that object being the one already known to be marginal, is about as good as this kind of agreement gets.

And the middle of the gap is not explained by the curve. At ninety degrees the derived limit is 0.25 Hill radii, and a satellite at 0.15 would be inside it and ought to survive. None is there. The reason is that the escape criterion is not the only way to lose a satellite: at high eccentricity the pericentre falls, and a pericentre driven to a hundredth of the semi-major axis puts the satellite through the planet’s regular satellite system and eventually into the planet. The curve models leaving the lobe; the middle of the gap is about arriving at the planet.

So the observed gap has two causes and the figure draws one of them. That is worth stating plainly rather than presenting a curve that appears to explain a feature it explains only half of.

The clock, and the number that cancels out of it #

A mechanism that takes longer than the age of the solar system is not a stability criterion, so the cycle’s period has to be checked before any of this is allowed to matter. The check has a pleasant answer.

The secular timescale for a hierarchical triple is of order the outer period squared divided by the inner one, times the ratio of the inner pair’s mass to the perturber’s. Here the outer period is the planet’s year, the inner one is the satellite’s, and the perturber is the Sun, so the mass ratio is the planet’s own mass divided by the Sun’s — which is also the quantity in the Hill radius. Writing the satellite’s distance as a fraction $x$ of the Hill radius and substituting, every mass disappears and what is left is

$$\frac{t_{\rm Kozai}}{P_{\rm sat}} \;\approx\; \frac{2}{\pi x^{3}}.$$

The cycle’s length in units of the satellite’s own orbit depends on nothing but how far out it is, measured in the unit the system supplies. Not on the planet’s mass, not on its distance from the Sun, not on which planet it is. A satellite at a third of a Hill radius takes about twenty of its own orbits per cycle wherever it is; one at a tenth takes six hundred.

Put the real numbers in and the mechanism is fast. Jupiter’s Pasiphae, at 0.44 Hill radii, has an orbital period of about two years and a cycle of some twenty-five years. Saturn’s Phoebe, at 0.20, has a period of a year and a half and a cycle of several centuries. Neptune’s Neso, at 0.42, takes twenty-seven years to go round and a few hundred to complete a cycle.

Against four and a half billion years, every one of those is instantaneous. The satellites in the census have each been through millions of cycles, so the census is not a snapshot of a transient — it is what is left after the mechanism has had every opportunity to act, which is exactly the condition under which a gap in it means something.

The same arithmetic says why the regular satellites are untouched. Callisto sits at 0.035 Hill radii, so its cycle would run to fifteen thousand of its own orbits — and long before that matters, the planet’s own bulge has taken over the precession entirely.

Where the Sun stops mattering #

All of this assumes the Sun is the dominant perturber, and inside a certain distance it is not.

Inside that radius a satellite’s orbit precesses about the planet’s equator and the Sun’s secular torque averages away; outside it the orbit precesses about the planet’s orbital plane and everything in this essay applies. The crossing is called the Laplace radius, and the plane a satellite’s orbit actually precesses about — intermediate between the two, weighted by the two torques — is the Laplace plane.

The sorting it produces across the solar system is nearly perfect. Every regular satellite of every giant planet is inside its planet’s Laplace radius and every irregular one is outside it, with a single exception: Saturn’s Iapetus, at 0.054 Hill radii against a crossing at 0.038. Iapetus is consequently the one satellite in the solar system whose orbital plane is neither its planet’s equator nor its planet’s orbit but a compromise between them — inclined by about fifteen degrees to the equator, and precessing about an axis that is neither.

That is also the reason the notch in this essay has no counterpart among the regular satellites. A regular satellite deep inside the Laplace radius cannot undergo the exchange at all, because its pericentre is being precessed by the planet’s bulge far faster than the solar torque can accumulate — which is exactly the quench that stops the same mechanism in a hierarchical triple, arriving here from a planet’s oblateness rather than from general relativity — and the oblateness in question is the one a satellite’s own regressing node measures.

The same notch, one level up #

Nothing in the argument is about satellites. It needs a hierarchical triple, an inner orbit whose tilt can be traded, and a boundary that the apocentre rather than the semi-major axis has to respect — and that description fits two planets and their star as readily as a moon, a planet and the Sun.

The essay that measured planetary spacings in mutual Hill radii found a floor near ten, and named the inclination dependence as the largest thing it left out. This is that dependence, and it runs the opposite way from the satellite case in a manner worth noticing.

A pair of planets at fixed separation is more stable when mutually inclined, not less. The reason is that the encounters are rarer rather than gentler: two orbits that cross in projection but are tilted with respect to one another only approach closely when both bodies are near the line where the planes intersect, so the conjunctions that matter happen on a small fraction of the orbits — the same counting argument that decides where a resonance clears a gap and where it locks a moon. Numerically, the stability threshold in mutual Hill radii falls by roughly a quarter as the mutual inclination goes from zero to thirty degrees.

The satellite case runs the other way because the perturber is external rather than internal. There the tilt is measured against the perturbing orbit, and it is the tilt that makes the secular exchange available; here it is measured against the neighbour, and it is the tilt that keeps the two apart. The same word describes two geometries with opposite consequences, and which one applies is decided by whether the object being tilted away from is the thing perturbing the orbit or the thing the orbit might hit.

Both results are why a stability criterion expressed as a single number in Hill radii should be read with its geometry attached. The floor of ten mutual Hill radii is a floor for a flat system, and the transit surveys that measure it can only see flat systems — so the observed distribution is the distribution for the case the criterion was derived for, and whether it is the distribution for all systems is not something a transit can answer.

What was actually measured #

The satellite census: forty named objects with orbits fitted from astrometric arcs of years, and semi-major axes converted into Hill radii using each planet’s own mass and orbit.

Three things about that conversion are worth stating, because the figure’s horizontal and vertical axes are both derived rather than observed.

The inclination is to the planet’s orbit and the catalogues do not always agree. An irregular satellite’s orbital elements are quoted variously with respect to the ecliptic, the planet’s equator or the local Laplace plane, and for objects well outside the Laplace radius the first and third nearly coincide while the second does not. Uranus is the trap: its equator is tilted 98 degrees to its orbit, so an inclination quoted to its equator is meaningless for this figure and differs from the right number by up to ninety-eight degrees.

The elements oscillate. These orbits are perturbed strongly enough by the Sun that an osculating element changes measurably within one orbit, and the census plots mean elements averaged over the perturbation. For a satellite in the middle of a large cycle that average is a good number and a poor description.

And the population is a survey’s population. Forty objects is what the deepest searches have found, and the searches reach about a kilometre at Jupiter and several kilometres at Neptune. A gap in a census is only a dynamical statement if the census is complete across it, and the case here is that nothing about a search favours one inclination — an argument about the method rather than about the data, and the right one to make, since the data cannot establish their own completeness.

Where the model stops #

The quadrupole is the leading term and not the only one. The expression for the greatest eccentricity comes from averaging at quadrupole order with the perturber on a circular orbit and the satellite massless. The planets’ orbits are not circular; at octupole order the conserved combination is not conserved, the inclination can pass through ninety degrees, and the excursions are larger and chaotic. The notch drawn here is therefore a floor on the damage rather than the whole of it.

The satellites perturb each other. A group of satellites sharing an orbit precess one another, and mutual precession competes with the solar torque in the same way a planet’s bulge does. For the tight families that competition is not negligible, and it is one reason the groups survive at inclinations where a single body might not.

The averaging assumes a separation of timescales that these orbits do not have. At the stability limit a satellite makes three to five orbits in one of its planet’s years, so averaging the Sun’s perturbation over the satellite’s orbit is averaging over something that is not fast. The earlier account says the same thing about its own figure; it applies with more force here, because the secular theory the notch is derived from is exactly an averaging.

And nothing here is a satellite’s history. Every object in the census arrived by a capture that no purely gravitational encounter could have performed, and the orbit it arrived on is not the orbit it has. What the census shows is what survived billions of years, so the limit it respects is the limit for that duration and not for any other. A satellite captured onto a marginal orbit and lost a hundred million years later leaves no trace at all.

The generalisation #

The shape to carry away is that a stability boundary quoted as one number has a hidden argument, and the argument is usually the one the calculation held fixed.

The coplanar limits in Hill radii are correct and they are answers to the question how far out may a satellite orbit in the plane. Read as answers to how far out may a satellite orbit, they are wrong by up to a factor of two, and wrong in a region of parameter space that happens to contain most of the objects the limit is quoted about. Nothing in the number announces which question it answered.

The second reading is about what the excursion does. The Sun cannot change a satellite’s semi-major axis — the averaged problem conserves it exactly — and it cannot therefore add or remove energy. What it does instead is move angular momentum between two orbits, and an orbit that has lost angular momentum at fixed energy is an eccentric one. A mechanism that cannot change how far a satellite is on average can still decide whether it stays, because staying is a question about the extremes of an orbit and not about its average.

That is the same structure as the impossibility that makes capture require dissipation, read forwards instead of backwards. There, a conserved quantity made capture impossible without dissipation. Here, a conserved quantity makes loss possible without any. Both times the useful question was not what the perturbation adds but what it is forbidden to change.

Still open: the orbit that flips #

What comes next is what happens when the averaging that produced the notch is taken one order further. At octupole order the conserved combination is not conserved, the satellite’s orbit can pass through ninety degrees and reverse its sense, and the clean band edges at cos²i = 3/5 soften into a chaotic region. Whether any observed satellite has been through such a flip is not a question the present orbits can answer, and it bears directly on whether the retrograde majority is a capture preference or a survival one.

Beside it lies the Laplace plane treated properly: not as a crossing radius but as a surface, warped between the equator and the orbit, along which a satellite’s orbit actually precesses — and whose shape carries the planet’s $J_2$ and the satellite’s distance together, which makes a satellite’s own plane a measurement of the planet’s interior.