Astronomy & Celestial Mechanics
the sky, computed
Exoplanets

A planet where one cannot form

A Jupiter at four days orbits inside a region that was too hot to hold ice and too small to hold the material. It did not form there — and the distribution of eccentricities says which of two journeys brought it in.

Assumes Tides and Reflex velocity.

19 min read 10 figures Small perturbations, large consequencesTwo bodies, and no more

51 Pegasi b has a mass around half Jupiter’s and an orbital period of 4.23 days. Its orbit is 0.05 AU across — an eighth of Mercury’s — and at that distance the material it is made of could not have been there.

A giant planet needs a solid core of several Earth masses before it can accrete gas, and solids condense in quantity only beyond the snow line, where water freezes: a few astronomical units out. Inside that, there is not enough rock, and the temperature is high enough that the gas would not stay bound to a small core anyway.

So the planet moved. The question the field spent twenty years on is how, and the answer is written in a distribution rather than in any single object.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 4.5 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 1 Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed rather than remembered. Inside 4.5 days nothing has a measurable eccentricity; outside it, eccentricities run to 0.95. The wall is sharp because the circularisation timescale carries the thirteen-halves power of the orbital distance.

Two ways to move a planet inward

Disc migration. While the protoplanetary disc is still present, a planet exchanges angular momentum with the gas around it through spiral density waves. The torques from the inner and outer disc nearly cancel, and the small residual drives the planet inward on a timescale of 10410^4–10610^6 years — fast, and gentle. It is gentle in a specific and testable sense: the gas damps eccentricity and inclination as it migrates, so a planet delivered this way arrives on a circular orbit in the plane of the disc.

High-eccentricity migration. Long after the disc has gone, a planet’s orbit can be driven to a very large eccentricity by a distant companion — the Kozai–Lidov mechanism, in which an inclined outer perturber trades the inner planet’s inclination for eccentricity cyclically — or by scattering off another planet. The planet then passes close to the star at each periastron, tides raised on it dissipate orbital energy, and the orbit shrinks and circularises over hundreds of millions of years. It is violent: the planet arrives with whatever inclination the process left it, which may be anything at all, including retrograde.

Both deliver hot Jupiters. They differ in what they leave behind.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 0.5 billion years. It falls at 3.0 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 2 The same plane half a billion years after the disc has gone, which is the epoch at which the two routes are still distinguishable. The tidal wall has barely formed: the circularisation time at ten days is over a gigayear, so only the very closest orbits have been erased, and everything else still carries whatever eccentricity it arrived with. A survey of systems this young would therefore see the two populations unmixed — disc-migrated planets circular at every period, violently migrated ones eccentric at every period outside a couple of days. That such a survey is not possible, because young stars are active and their radial velocities are noisy, is the practical reason this argument has to be made from old systems where most of the evidence has already been destroyed.

The boundary that decides the argument

Tides raised on a planet by its star dissipate energy and drain eccentricity. The timescale is

τe=221Q′nmpM⋆(aRp)5,\tau_e = \frac{2}{21}\frac{Q'}{n}\frac{m_p}{M_\star}\left(\frac{a}{R_p}\right)^5,

with nn the mean motion. Because n∝a−3/2n \propto a^{-3/2}, the whole expression goes as a13/2a^{13/2} — a factor of two in distance is a factor of ninety in the timescale.

That is why the boundary in the figure is a wall rather than a slope. Computed for a Jupiter with Q′=106Q' = 10^6 and a three-billion-year-old system, it falls at 4.5 days, and the observed distribution changes character there: inside, every measured eccentricity is consistent with zero; outside, they run to 0.95. The population just outside the boundary is the evidence. If hot Jupiters arrive by disc migration they should be circular at every period, since the gas damped them long before they got close. If they arrive by high-eccentricity migration they should be circular inside the boundary — tides finished the job — and eccentric just outside it, caught mid-circularisation.

Both populations exist. Planets like HAT-P-2 b, at 5.6 days and e=0.52e = 0.52, are exactly the objects the second mechanism predicts and the first forbids: too close in to have formed there, too eccentric to have arrived quietly, and just far enough out that tides have not yet finished. And planets like HD 209458 b, circular at 3.5 days with a well-aligned orbit, are what the first predicts.

The current reading is that both routes operate, with disc migration probably dominant and high-eccentricity migration responsible for a substantial minority — and the strongest single piece of evidence for the second is not eccentricity at all but alignment.

The orbits that go the wrong way

Disc migration cannot change a planet’s orbital plane: the disc defines the plane, and the star formed from the same collapsing cloud spinning the same way. So a disc-migrated planet’s orbit should be aligned with the star’s equator.

High-eccentricity migration has no such constraint. The Kozai–Lidov mechanism converts inclination into eccentricity and back, and a planet delivered by it can end up at any angle — including orbiting backwards.

The measurement is the Rossiter–McLaughlin effect. As a planet crosses a rotating star it covers first the approaching limb and then the receding one, distorting the star’s line profiles and producing an apparent velocity anomaly whose shape gives the sky-projected angle between the orbit and the spin. Around a third of hot Jupiters turn out to be misaligned, and a handful are retrograde.

A planet cannot form in a retrograde orbit. That observation, more than any eccentricity, established that violent migration happens.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 5.0 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 3 The boundary around a half-solar-mass star, where the same tide acts on a different orbit. The circularisation time carries the stellar mass through the mean motion and through the ratio mp/M⋆m_p/M_\star, so a lighter star pushes the wall to a longer period — a planet at five days around an M dwarf is inside a boundary that a solar-type star would put at four. The consequence for the alignment argument is that the erasure of the record is faster around low-mass stars, which are also the ones with the thickest convective envelopes and therefore the most efficient tidal re-alignment. The two erasures compound, and the stars around which hot Jupiters are rarest are also the ones whose hot Jupiters would tell us least.

Where the inward journey stops

A migrating planet does not fall all the way in. Several things stop it, and which one dominates is unsettled.

The disc has an inner edge, cleared by the star’s magnetic field, and a planet migrating through the gas stalls there — which happens to be at a few stellar radii, or a period of a few days. Tides raised on the star by the planet transfer angular momentum outward if the star spins slower than the planet orbits, pushing the planet in; the reverse if faster.

And there is a hard floor. Inside the Roche limit, the tidal difference across the planet exceeds its own self-gravity and it is pulled apart. WASP-12 b sits close enough that it is measurably deforming and losing mass, and its orbit is decaying at about 30 milliseconds per year — a decay measured, remarkably, as a transit timing variation, the same technique that finds unseen planets, applied here to an orbit that is genuinely shrinking rather than merely being pulled about. It will be destroyed in a few million years. That such an object exists at all is a statement about how recently the inward journey ended.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 5 billion years. It falls at 6.2 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 4 The same boundary five billion years on and for a slightly inflated planet. Both the age and the radius push the cut-off outward — the radius strongly, because the tidal rate carries it to a high power — so a planet that arrived on an eccentric orbit and has been inflated by its own irradiation circularises faster than a cold one at the same distance. The desert’s inner edge is therefore not a fixed period: it is a period that depends on what the planet became after it got there.

The desert between the two populations

There is a second feature of the period distribution that any migration story has to account for, and it is an absence.

Plot giant planets in period against mass and there is a sub-Jovian desert: at periods under about three days, planets between roughly 0.02 and 0.8 Jupiter masses are almost entirely missing, while both larger and smaller planets are present. The boundary is sharp on both sides, and it is not a selection effect — a hot Neptune produces a deeper transit than a hot super-Earth and a larger velocity amplitude, so if they existed they would be the easiest objects in the sample after the hot Jupiters themselves. The favoured reading is that the desert’s upper edge is drawn by tidal disruption — objects that migrated too far in were destroyed — and its lower edge by photoevaporation, since a low-mass planet that close is stripped of its envelope entirely and reappears as a bare core below the desert. If that is right, the desert is the same story as the radius valley told for a different mass range, and the two features are the two ends of one process.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 5.9 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 5 The wall computed for a planet of a third of Jupiter’s mass at the same radius — which is to say for an object in the desert, if one existed. The timescale carries mp/M⋆m_p/M_\star linearly, so a lighter planet circularises three times faster and the boundary moves outward. That is the awkward part of reading the desert as a tidal-disruption edge: the objects supposedly destroyed by migrating too far in are also the ones whose orbits are circularised soonest, so their absence has two candidate explanations that this figure cannot separate. Distinguishing them needs the mass distribution at periods just outside the wall, where objects survive and their eccentricities are informative.
Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+5 and an age of 1 billion years. It falls at 8.0 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 6 The same plane with a tenfold more dissipative planet, a billion-year-old system and an inflated radius. The boundary moves out to a longer period, because a larger planet raises a bigger tide on itself and a smaller Q′Q' dissipates it faster. That sensitivity is the honest caveat on the whole argument: the wall’s position is a prediction with an order-of-magnitude parameter in it, and the reason it is convincing is that the observed transition is sharp and lands within that range, not that the calculation is precise.

What was actually measured

The eccentricity distribution. Giant planets beyond about 0.1 AU have a broad eccentricity distribution with a median around 0.25 and a tail to 0.95 — quite unlike the solar system’s giants, all below 0.06. Inside 0.1 AU the distribution collapses to zero. Both features are robust against the selection effects that favour eccentric orbits, which if anything work in the opposite direction for the inner population.

HD 80606 b, 2001 and 2009. Eccentricity 0.933 at a period of 111 days, in a wide binary system — precisely the configuration Kozai–Lidov requires. In 2009 its secondary eclipse was caught with Spitzer, and the planet’s day side was watched heating by 700 K in six hours as it swung through periastron. The mechanism was not inferred; the heating was observed.

The Rossiter–McLaughlin surveys. Around 200 systems now have measured projected spin–orbit angles. Hot Jupiters around cool stars (below about 6,250 K) are mostly aligned; those around hot stars are frequently misaligned. The interpretation is that cool stars have thick convective envelopes that dissipate tides efficiently and re-align the orbit over time, while hot stars do not — so the misalignment distribution is a record of the arrival geometry only for the hot ones.

The occurrence rate, which is the constraint everything must satisfy. Hot Jupiters orbit about 1 per cent of Sun-like stars. Any mechanism proposed has to produce them at that rate and no faster, which turns out to be a serious constraint: high-eccentricity migration channels are typically efficient enough to overproduce them unless the required companion configurations are rare.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+7 and an age of 3 billion years. It falls at 2.6 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 7 And the same age with the dissipation ten times weaker, which is the width of the honest uncertainty. The tidal quality factor for a gas giant is known to about an order of magnitude, and the cut-off moves with it — so the boundary this essay draws is a band rather than a line, and a planet found inside it is evidence about Q′Q' rather than about migration. Every inference here runs through a number nobody has measured directly for any planet.

The population between the two

There is a third group of giant planets that neither mechanism was designed to explain, and it has become the sharpest test of both: the warm Jupiters, at periods of ten to a hundred days.

They are too far in to have formed where they are, by the same snow-line argument that applies to the hot ones. They are too far out for tides to have circularised them — the boundary is at four or five days and the timescale goes as the thirteenth-halves power of distance, so at thirty days it exceeds the age of the universe by a wide margin. Whatever eccentricity they arrived with, they still have.

That makes them the un-erased sample. The population inside the tidal wall has had its history dissipated; the warm Jupiters have not, and their eccentricity distribution is therefore a direct record of how giant planets are delivered inward.

What it shows is a broad distribution with a substantial fraction at low eccentricity. If high-eccentricity migration produced the whole giant-planet population, the warm Jupiters would be systems caught partway through — planets whose periastra are already inside the tidal radius and whose apastra are still far out — and they would be uniformly eccentric. They are not: many are nearly circular at thirty days, which no version of the violent route produces, since a planet on such an orbit has no periastron passage close enough to have been tidally delivered.

The reading is that the warm Jupiters are mostly disc-migrated, and that they are the parent population of the hot ones — a planet delivered quietly to thirty days and then moved further in by some slower process. What that slower process is remains open, and the candidates are the same ones that stop the inward journey: interaction with a residual disc, tides raised on the star, or secular perturbations from an outer companion acting over billions of years.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 10 billion years. It falls at 5.9 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 8 Ten billion years, which is older than any of the systems in the sample and is drawn to show what the wall converges to. It hardly moves. The timescale goes as a13/2a^{13/2}, so doubling the available time moves the boundary by two to the power two-thirteenths — about eleven per cent in period. That is the property that makes the wall useful: its position is a strong function of the orbit and a very weak function of the age, so a boundary observed in a sample of mixed and mostly unknown ages is still sharp. The same insensitivity is why the boundary cannot be used the other way round to date anything.

There is a further discriminant in the same population and it is a count rather than a distribution. High-eccentricity migration needs a perturber — an outer companion, stellar or planetary, inclined enough to drive the cycles — so systems delivered that way should have one, and systems delivered by a disc need not. Searching for outer companions around warm Jupiters is therefore a direct test of the channel, and it is a search that radial velocities can make over a decade of monitoring.

The results so far are mixed in an informative way: the eccentric warm Jupiters have outer companions at a high rate and the circular ones do not, which is what a two-channel picture predicts and what a single-channel one cannot produce. That is a correlation between two independently measured quantities in the same systems, which is a stronger form of argument than either distribution alone.

The population that cannot be circularised is the one that carries the information, which inverts the usual situation — the objects that are hardest to explain are the ones whose history survives.

Where the picture stops

Q′Q' is not measured. The tidal quality factor is uncertain by orders of magnitude for gas giants and is probably not a constant. The boundary computed in the first figure moves by a factor of two in period for a factor of ten in Q′Q', so its agreement with the observed transition is a consistency rather than a measurement.

The planet’s radius enters to the fifth power. τe\tau_e carries (a/Rp)5(a/R_p)^5, and hot Jupiters are inflated by an unexplained energy source to radii 20–50 per cent larger than a cold hydrogen sphere. That is a factor of two to eight in the timescale, in the direction that makes circularisation faster — so the boundary and the inflation problem are not independent, and a planet’s own puzzle feeds into the argument about how it arrived.

Circularisation timescales are calculated for equilibrium tides. Real dissipation in a fluid, rotating, possibly resonant body is far more complicated, and dynamical tides can be orders of magnitude more efficient at particular frequencies.

Projected angles are not angles. Rossiter–McLaughlin gives the sky-projected obliquity, not the true one. An apparently aligned system may be inclined along the line of sight.

A snow line is not a fixed distance. It moves inward as the disc cools and evolves, over exactly the timescale in which the giant planets form, so “beyond the snow line” is a statement about a moving boundary and its position at any epoch is model-dependent. What is robust is the ordering — solids condense further out first — rather than the number of astronomical units.

And the solar system is one system. Whether Jupiter migrated is a question about a single object with no distribution to appeal to. The Grand Tack hypothesis has Jupiter moving inward to 1.5 AU and back out again, on evidence from the asteroid belt’s mass distribution and Mars’s small size — which is a completely different kind of argument from anything above, and correspondingly harder to test.

The eccentricity that the boundary erases

There is an awkwardness in using the period–eccentricity plane as evidence, and it deserves stating plainly rather than being left in the caption.

Tides do not merely circularise; they destroy the record. A hot Jupiter that arrived by the violent route and has since been sitting inside the boundary for two billion years is indistinguishable from one that arrived quietly — the eccentricity that would have told them apart has been dissipated as heat inside the planet. So the population inside the wall is uninformative by construction, and everything rests on the handful of objects caught in transit between the two states.

That is a small sample. It is also a biased one, because an eccentric orbit is easier to detect at fixed mass, so the mid-circularisation population is over-represented relative to the truth. Both effects have to be modelled before a ratio of the two channels can be quoted, and the published ratios differ by more than their stated uncertainties — which is usually a sign that the modelling, rather than the data, is the limiting step.

The mass of the planet enters the erasure as well, and it enters steeply.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 3.5 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 9 The same boundary computed for a planet three times Jupiter’s mass at the same radius. The circularisation time carries the planet’s mass in the numerator, so a heavier planet at the same period circularises more slowly and the wall moves inward — a three-solar-Jupiter planet keeps an eccentricity at a period where a one-Jupiter planet would have lost it. The observed points do not move; only the line the argument draws through them does.

That dependence is the reason the boundary cannot be drawn once and read for the whole sample. Every point on the diagram has its own wall, set by its own mass and radius, and the single line is a line for a fiducial planet rather than a threshold any particular object crosses. A survey that finds an eccentric planet inside the fiducial wall has not found a contradiction; it has found a planet whose own wall sits further in. The honest version of the argument compares each object with its own circularisation time, and the diagram is the summary of that comparison rather than the comparison itself.

The same erasure applies to obliquity around cool stars, for the same reason and on a similar timescale: the tide that circularises an orbit also re-aligns it. So the two cleanest fingerprints of violent migration both fade, and both fade fastest exactly where the objects are most numerous.

The generalisation

Reading a history from a distribution rather than from an object is what this essay does, and it is a standard move wherever the events are unobservable and their outcomes survive.

Resonances in the asteroid belt record the migration of the giant planets: the Kirkwood gaps are where the resonances are now, and their detailed structure records where they have been. Binary star eccentricities show the same circularisation cut-off as hot Jupiters, at a period of about ten days for solar-type pairs, and it was measured decades earlier — the same physics, the same wall, a different population.

The general shape is: a process operates on a timescale that depends steeply on a parameter, so a population develops a sharp boundary in that parameter, and the boundary’s position measures the timescale. The steeper the dependence, the sharper the boundary and the better the measurement. A thirteen-halves power is very steep indeed.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 4.2 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 10 The same construction around a star of 1.4 solar masses — an F star, above the 6,250 K divide the obliquity surveys found. The wall moves inward, because a heavier star means a faster mean motion at a given period and a smaller mp/M⋆m_p/M_\star, and the two pull in the same direction. So hot Jupiters around hot stars are erased more slowly than those around cool ones, which is the same conclusion the convective-envelope argument reached from tides raised on the star rather than on the planet. Two independent effects, both making the hot-star sample the informative one, is why the misalignment statistics are quoted split at a temperature rather than pooled.

It is worth stating what would falsify the whole picture, because a boundary drawn from a formula is only evidence if it could have failed to match. The prediction is not merely that eccentric planets are rare inside the wall; it is that the rate at which they thin out matches the fifth power of the radius in the circularisation time. A population that fell off gradually across two decades in period, or that showed the same eccentricity distribution inside and outside, would refute it outright. The observed transition happens over roughly a factor of two in period, which is what a fifth power looks like, and that is the quantitative claim the diagram is making rather than the qualitative one it appears to make. Testing it properly needs the eccentricities measured well enough to distinguish a distribution truncated at the wall from one merely depleted near it, and radial-velocity eccentricities below about 0.1 are notoriously biased upward by noise, so the sharpest part of the test is the part the data support worst.

Where this goes next

Migration leaves one further fingerprint, and it is the most specific of all: a chain of planets in successive resonances, which cannot be assembled in place and requires the orbits to have converged slowly and stayed converged.

Later rungs on this anchor: the two regimes of disc migration, with and without a cleared gap. Kozai–Lidov cycles worked through. Planet–planet scattering. Tidal dissipation and the value of Q′Q'. The Rossiter–McLaughlin effect. Obliquity and the 6,250 K divide. Orbital decay in WASP-12. The desert of sub-Jovian planets at short periods. The Grand Tack. And the question of why the solar system has no hot Jupiter, which is a question about the alternatives that did not happen here.

Where this goes next

The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one. The tide is a difference, which is why there are two of them The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting. The tide across a moon, against the moon's own gravity. The tidal acceleration across a satellite and the satellite's own surface gravity, both in units of that surface gravity, against distance from the primary in planet radii. The tide falls as the inverse cube and the self-gravity does not fall at all, so they cross once — at 2.23 radii for the density ratio drawn. Inside the crossing the tide wins and a body held together only by its own weight comes apart. The distance at which a moon stops holding together The tide across a body falls as the inverse cube; the body's own gravity does not fall at all. There is therefore exactly one crossing, and Saturn's rings end within a few per cent of it. What each method can see. Planet mass against orbital distance, both logarithmic, with the detection threshold of each method drawn as the boundary it actually is. Radial velocity at 1 m/s needs mass rising as √a; astrometry at 20 µas needs it falling as 1/a, which is the only method that gets easier further out; a 100 ppm transit is a threshold on radius and so a horizontal line at about 1.4 Earth masses, cut off at 1.21 AU by the need for three transits in 4 years; direct imaging begins outside the diffraction limit, 0.6 AU at 10 parsecs for a 39 m aperture at 10 µm. The solar system is drawn on top: for two decades every one of its planets except Jupiter lay outside every region, which is the whole of what the early census was measuring. Every survey draws a different sky The first exoplanets found were enormous and impossibly close to their stars. That was not a discovery about planets. It was a measurement of what a 10 m/s spectrograph watching for three years is able to see.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 22 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

EccentricityHot jupiterKozai–LidovMigrationOrbital decayPeriod distributionProtoplanetary discRoche limitThe snow lineTidal circularisation