A feeding zone, and the spacing it forces

The rung below used the Hill radius as a boundary on what a planet may keep: a moon inside it stays, a moon outside it is taken away by the Sun, and the numerically determined limits are about half that radius for a prograde orbit and rather more for a retrograde one.

The same radius, asked the opposite question, becomes a statement about what a planet could reach. A body growing in a disc of planetesimals can capture material whose orbits come within a few Hill radii of its own and cannot capture anything further out, because further out the Sun’s tidal field wins the tug of war and the encounter is a flyby rather than an accretion.

That gives the growing body a feeding zone with a width set by its own mass, which sets a mass at which it runs out of food, which sets how far away the next one has to be.

The radius, from the other side #

The Hill radius of a body of mass $m$ orbiting a primary of mass $M$ at distance $a$ is

$$R_H = a\left(\frac{m}{3M}\right)^{1/3},$$

and it is the distance at which the body’s own gravity and the primary’s tidal field are comparable. Rung one derived it as a stability boundary, and the same expression is what limits a growing embryo’s reach.

Isolation mass #

Suppose an embryo sweeps a zone $b\,R_H$ wide on each side, with $b$ of order 5 to 10 — the value comes from integrations rather than from an argument. It accretes everything in an annulus of area $2\pi a \cdot 2bR_H$ carrying surface density $\Sigma$, and it stops when it has eaten it all. Setting the mass equal to the mass in the annulus and solving,

$$M_{\text{iso}} = \frac{\left(4\pi b\,\Sigma\,a^2\right)^{3/2}}{(3M_\star)^{1/2}}.$$

The $a^3$ inside the three-halves power is what makes this interesting. Isolation mass grows steeply with distance, and it jumps where the surface density does.

For a minimum-mass solar nebula with $b = 10$, the number at one astronomical unit is about 0.07 Earth masses — a Moon-sized to Mars-sized body, and nothing bigger. That is the well-known embarrassment of terrestrial planet formation: isolation alone cannot make the Earth, and the last stage has to be a chaotic phase in which a few dozen isolated embryos have their orbits stirred until they collide with each other.

Past the snow line, where water ice condenses and the solid surface density jumps by a factor of three or four, the same expression gives around ten Earth masses at five astronomical units. Ten Earth masses is roughly the threshold at which a core can begin to accrete gas from the disc directly, and the coincidence between the two numbers is the standard explanation for why the giant planets are outside the snow line and the terrestrial ones are inside it.

Oligarchic growth #

The feeding zone has one more consequence, and it explains why the picture is a set of comparable bodies rather than one winner.

While the embryos are small, growth is runaway: a body slightly larger than its neighbours has a larger gravitational cross-section, sweeps material faster, and pulls further ahead. The ratio of masses in a population diverges.

That cannot continue, because a growing embryo stirs the planetesimals around it. Once its Hill velocity exceeds their random velocities, further growth heats the local swarm faster than it accretes from it, and the gravitational focusing that made growth run away is lost. Growth then becomes oligarchic: each embryo dominates its own zone, they all grow at comparable rates, and the ratio of masses stops diverging and starts converging.

The end state is a set of embryos of roughly the isolation mass, separated by roughly the feeding-zone width, each one a local oligarch. That is where the spacing in Hill radii is set, before anything about long-term stability has come into it, and it is the reason the observed floor and the formation prediction land in the same part of the axis rather than by accident.

The solar system in its own unit #

Applying the definition to the eight planets gives a list worth reading, because it is not uniform.

Mercury to Venus is 63 mutual Hill radii; Venus to Earth, 26; Earth to Mars, 40. The terrestrial planets are extravagantly spaced — an order of magnitude clear of the survival threshold — and Mars to Jupiter is 16, with the belt in the gap.

Jupiter to Saturn is 7.9, below the empirical floor of about ten. Saturn to Uranus and Uranus to Neptune are both 14.

The Jupiter–Saturn pair is the interesting entry. It is the tightest adjacent pair in the solar system in this unit, it is close to a 5:2 near-commensurability, and the long-term integrations that show the solar system to be marginally stable find that the giant planets are where the marginality lives. It is also the pair that most reconstructions of the early solar system have moving: models in which Jupiter and Saturn crossed a resonance during migration use exactly this proximity as the mechanism for stirring everything else.

TRAPPIST-1, by contrast, runs 10.0, 12.8, 11.8, 10.2, 6.6, 10.5 — a system sitting almost exactly on the empirical floor for its whole length, with one pair below it. The pair below it is held by a resonance.

The theorem, and the fit #

Two planets that end up too close together do not stay. There are two statements about how close is too close, and they are of quite different kinds.

The theorem applies to two planets on nearly circular, nearly coplanar orbits. It comes out of the same conserved quantity that governs the restricted three-body problem: for a sufficiently large separation, the zero-velocity surfaces close around each planet and orbit crossing is topologically forbidden, whatever else happens for however long. In terms of the mutual Hill radius

$$R_{H,\text{mut}} = \left(\frac{m_1+m_2}{3M_\star}\right)^{1/3}\frac{a_1+a_2}{2},$$

the condition is $\Delta = (a_2-a_1)/R_{H,\text{mut}} > 2\sqrt{3} \approx 3.46$. Above that separation the pair is Hill stable — proven, not fitted. Hill stability is weaker than it sounds. It forbids the two planets from swapping places or encountering each other; it does not forbid one of them from being ejected, and it says nothing at all about systems of three or more.

The fit covers what actually happens. Integrate systems of three or more planets for as long as patience allows, record how long they survive before the first orbit crossing, and the survival time rises roughly exponentially with $\Delta$. Getting to billions of years takes about $\Delta = 10$, and the exact number depends on the number of planets, their eccentricities and their mutual inclinations.

Why the observed distribution has a floor #

Put the two together and the shape of the figure at the top follows.

Nothing forbids a system from forming with $\Delta = 4$. Such systems presumably did form, in quantity. They are not observed because they did not last: the packing is above the theorem’s boundary, so the planets cannot immediately swap, but well below what a chaotic three-planet system needs, so within a few tens of millions of years the eccentricities have grown, an orbit has crossed another, and the survivors are further apart than they started.

The observed floor near ten is therefore a survival filter and not a formation preference. What is seen is not the distribution of systems that formed; it is that distribution with everything below a threshold removed by time.

The upper end has a different explanation, and it is close to a tautology: a pair of planets very far apart in Hill radii has an empty gap between them, and an empty gap is a place where nothing formed or where something was removed. The solar system’s own widest gap in this unit is between Mars and Jupiter, and it contains the asteroid belt.

What the unit is doing #

The change of variable is worth pausing on, because it is the whole content of the rung.

A separation in astronomical units compares a gap against a standard nobody in the system cares about. A separation in mutual Hill radii compares it against the distance over which the two planets’ own gravity dominates the star’s, which is the quantity that decides whether they interact. The first is a description; the second is a dynamical statement.

The clearest demonstration is TRAPPIST-1. Its seven planets fit inside the orbit of Mercury — the outermost is at 0.062 astronomical units — and in kilometres they are extraordinarily crowded. In mutual Hill radii they are spaced much like the solar system’s, because their star is a tenth of a solar mass and their own masses are of order the Earth’s, and the Hill radius scales with both. The system is not densely packed; it is normally packed, in a small box.

Planets in a system resemble each other #

There is an empirical regularity in the same data that the spacing argument does not predict, and it is worth setting beside the floor because the two together say more than either alone.

Take the multiplanet systems with several transiting planets and ask how similar the planets within one system are, compared with how similar two planets drawn at random from the whole catalogue are. The answer is that a system’s planets are markedly more alike in radius than random draws would be, and their spacings are more uniform than random draws would be. Neighbouring planets in a system tend to have radii within about twenty per cent of one another, and the gaps between successive pairs tend to be within about twenty per cent of one another too.

That is a stronger statement than the floor. A floor is a statement about a threshold; this is a statement about correlation within a system, and it survives the obvious objections. It is not a detection bias, because a system with one large planet and one small one is easier to notice than a system with two of the same size, not harder. It is not an artefact of the mass–radius relation, since it shows up in radius directly. And it holds across the range of stellar types the survey covers.

The isolation-mass argument gives a natural account of it. Oligarchic growth produces embryos of comparable mass at comparable spacing, because each one eats a zone whose width is set by its own mass and each one stops when its zone is empty — so the outcome of a smooth disc is a regular comb rather than a random assortment. What the argument does not naturally give is the strength of the regularity: simulations of the last chaotic stage of terrestrial planet formation, in which embryos collide with one another, tend to erase some of it, and reproducing what is observed requires that stage to be gentler than the solar system’s own appears to have been.

So the two facts pull in slightly different directions. The floor is a survival filter and says that what is seen has been through a destructive phase. The similarity within systems says that phase cannot have been very destructive, or the regularity imposed at formation would not have survived it. The solar system, with its factor-of-twenty spread in terrestrial planet masses and its irregular spacing, is on the disorderly end of the distribution its own formation theory was built to explain.

What the argument does not settle #

Three things are worth flagging.

$b$ is not derived. The width of a feeding zone in Hill radii comes from numerical experiments on planetesimal accretion, and it depends on the velocity dispersion of the planetesimals, which depends on how strongly the embryos have already stirred them. Isolation mass inherits that uncertainty to the three-halves power.

Resonances break the rule in both directions. A pair locked into a mean-motion resonance can survive at separations well inside the empirical floor, because the resonance keeps the conjunctions away from the dangerous longitudes. Chains of such planets exist and are evidence of migration rather than of in-place formation. Conversely, a pair near a resonance without being in it is less stable than the spacing alone suggests.

The masses are usually not measured. Most of the Kepler multiplanet systems have radii and no masses, so the mutual Hill radii quoted for them come from a mass–radius relation. That is a defensible estimate and it is not an observation, and any statement about the distribution of $\Delta$ carries whatever bias the relation has. The dependence is at least gentle: $R_H$ goes as the cube root of the mass, so a factor of two in an assumed mass is 26 per cent in the spacing — which is small against a distribution that spans a factor of ten, and not small against a floor quoted to one significant figure.

And the sample is a survey’s sample. A system is counted as a multiplanet system only if several of its planets transit, which requires them to be nearly coplanar as seen from here — so the census is drawn from a biased sky in exactly the way that matters, since mutual inclination is one of the quantities the stability criterion depends on. The observed distribution of $\Delta$ is the distribution for flat systems, and whether it is the distribution for all systems is not something transits can answer.

The same spacing statistic and the same gap structure are worth reading at other settings, because both are claims about a distribution rather than about any one system.

A floor read as a prediction #

If systems really are packed close to the stability limit, then a gap much wider than the floor is not a fact about the system but a gap in the observations — somewhere a planet should be, and is not seen.

That reading has been made into a method. Take a known multiplanet system, compute the separations in mutual Hill radii, find any pair far above the floor, and ask what mass of planet could sit between them without destabilising anything. Integrate a grid of trial planets, keep the ones that survive, and the surviving region is a prediction: a range of periods and masses in which an undetected body is dynamically permitted.

The predictions are testable and several have been tested. Radial-velocity follow-up of systems with conspicuous gaps has found planets in the permitted regions in a number of cases and has found nothing in others, and the ratio between those two outcomes is the actual measurement — it says how nearly the packing hypothesis holds rather than whether it is true.

The result is that systems are often but not always packed. Some gaps are occupied by a planet the original survey missed, usually because it was too small or its orbit too inclined to transit. Some gaps are genuinely empty, and the empty ones tend to sit near strong resonances with a neighbour, which is the same clearing mechanism the asteroid belt demonstrates at a smaller scale.

A criterion derived to explain what survives has therefore been turned into a search strategy for what has not been found, which is a better use of a stability threshold than the descriptive one this essay began with — and it is falsifiable in a way a statement about a distribution’s floor is not.

And the same construction applied to a different planet’s resonances entirely.

One more reading widens each resonance to see whether the gaps stay attached to them.

What the gaps and the spacings have in common is that both are set by a mutual Hill radius rather than by any absolute distance, which is why the same arithmetic describes a belt around a star and a ring around a planet.

Where this ladder goes next #

This rung has taken the Hill radius from a boundary on ownership to a unit of dynamical distance, and used it to explain why planetary spacings look structureless in one unit and sharply bounded in another.

The rung above is capture: how a body on a solar orbit gets inside a Hill sphere at all, given that a purely gravitational two-body encounter cannot leave it. The answers involve a third body, a drag, or a collision, and which one operated is written in the inclinations of the irregular satellites.

Beside it lies the inclination dependence of every limit quoted here, which is the largest thing this rung leaves out: a mutually inclined pair is more stable than a coplanar one at the same separation, and the reason is that the encounters are rarer rather than gentler.

And below it, the habit: a distance means nothing until it is divided by something the system itself supplies. Measured in kilometres, planetary spacings are a list. Measured in the radius the planets define, they are a distribution with a theorem underneath it.