Astronomy & Celestial Mechanics
the sky, computed
Concept

Tidal circularisation — where it appears

The damping of an orbit's eccentricity by dissipation in the tides each body raises on the other. It acts at pericentre and scales very steeply with separation, so it circularises close orbits within a system's lifetime and leaves wide ones untouched.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

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.

exoplanets · Planet migration Relativity switches the cycle off, halving its reach at a ratio of 0.80. The greatest eccentricity a Kozai–Lidov cycle reaches, against the strength of the orbit's own relativistic pericentre precession, measured in units of the cycle's own precession rate at zero eccentricity. The horizontal axis is logarithmic and spans three decades. At the left the relativistic term is negligible and the cycle reaches 0.838, which is the closed-form value for a start at 65 degrees and is what the integration is checked against. At the right it is gone. The mechanism depends on the pericentre staying put while the outer body pulls on the same side of the orbit for a whole cycle, and the relativistic precession is a competing rotation of that same pericentre; when it is faster, the pull averages away. The threshold sits near one by construction and the transition is sharp rather than gradual, with the reach halved at 0.80. What makes it matter is where the relativistic term is largest: it grows as the pericentre falls, so it strengthens exactly as the cycle drives the orbit inward, and it therefore sets a floor on the pericentre distance that this mechanism can deliver a body to.

The precession that switches the cycle off

A distant companion can trade an orbit's inclination for its eccentricity, over and over, and drive a pericentre almost onto the central body. General relativity's own precession competes with the mechanism for the same pericentre, and when it wins the cycle stops — sharply, at a ratio of one, which puts a floor on how close anything can be delivered.

orbits · Kozai–Lidov A period below which every orbit is round. Orbital eccentricity against period for binaries in four clusters of 0.125, 0.625, 6, 4 billion years, with the eccentricities drawn from one seeded distribution and then damped by exp(−age/τ), where τ rises as the 5.333 power of the period. Each cluster shows the same thing: below a boundary period nothing survives eccentric, above it the original distribution is untouched, and there is almost nothing in between because the timescale is so steep. The boundary is a clock. It moves as the three-sixteenths power of the age, which the figure checks against the drawn curves, and the calibration puts it at 6.5 days at 125 million years, 8.8 at 625 million and 12.5 at four billion — against measured cut-offs near 7.2, 8.5 and 12.5 days in the Pleiades, the Hyades and M67. The boundaries are also read back off the plotted points rather than trusted, and required to move outward with age. This is the cleanest measurement of tidal dissipation in ordinary stars that exists, and its cleanliness comes from the ages: a cluster's age is read off its main-sequence turn-off and owes nothing whatever to the tide being measured.

A cut-off period that is an age

Plot eccentricity against orbital period for the binary stars of one cluster and the picture has a wall in it. Below a certain period every orbit is circular; above it, the original spread survives untouched. The wall moves outward as the cluster ages, and where it stands is a measurement of how stars dissipate a tide.

orbits · Circularisation A wall measures a ratio, and a ratio is a line. The tidal quality factor against the system's age, with each line the locus of one measured circularisation boundary. For a planet the eccentricity damping time τₑ goes as Q′P^(13/3), so the cut-off period goes as (age/Q′)^(3/13) and only the ratio of the two appears. A measured wall is therefore a straight line of slope exactly 1 in this plane and never a point on it: a 5-day boundary is consistent with Q′ = 2·10⁵ in a one-billion-year-old system and with Q′ = 2·10⁶ in a ten-billion-year-old one, and nothing in the light curve chooses. The stellar version of this measurement escapes because the cluster supplies the age, from a main-sequence turn-off that owes the tide nothing — which is why a cut-off period read off four clusters is a dissipation measurement and the same wall in the hot-Jupiter plane is not. The shaded band is what a field star's age is actually worth: known to a factor of 3, it leaves Q′ known to a factor of 3 and no better, against a quantity whose published values for giant planets span 10⁴ to 10⁹. What would break the degeneracy is a second measurement with a different power of P in it — an orbital decay rate, which goes as Q′⁻¹ with no age in it at all, and which has now been measured for one planet.

A wall measures a ratio, and a ratio is a line

The same circularisation boundary drawn for planets probes the dissipation inside the planet rather than inside the star. But the boundary depends only on age over Q′, so with no cluster to date the system the measurement is a line in a plane and never a point on it.

orbits · Circularisation Two damping times, one crossing, and the slope that separates them. Circularisation timescale against orbital period for the two tidal mechanisms, both logarithmic, normalised to 1.217 Gyr at 10 days. On logarithmic axes a power law is a straight line and the index is its slope, so the figure's content is that the two lines have different slopes and one crossing. The equilibrium tide gives 5.33, the bulge raised on a convective envelope being dragged ahead by a viscosity that is turbulent convection itself; the dynamical tide gives 7, gravity waves launched at the convective boundary carrying angular momentum to wherever they break. The horizontal lines are the ages of populations a boundary can be read in: where each curve crosses one is the wall that population shows — at 0.125 Gyr, 6.5 days against 7.2; at 0.625 Gyr, 8.8 days against 9.1; at 4 Gyr, 12.5 days against 11.9; at 10 Gyr, 14.8 days against 13.5. One cluster measures one number and both theories have a free normalisation, so one cluster cannot choose. The separation across every age available is a factor of 1.11. The discriminator that does not depend on the normalisation is mass: the equilibrium tide needs a convective envelope, and above about 1.3 solar masses there is not one, so the two predict different behaviour on either side of a boundary the theory names in advance. That is a measurement about where the wall stops behaving, not about where it is — and it is the reason the samples had to grow from tens of binaries per cluster to hundreds.

Two damping times, one crossing, and the slope that separates them

The equilibrium tide gives a damping time going as the sixteen-thirds power of the period and the dynamical tide as the seventh. One cluster measures one number and both theories have a free normalisation, so one cluster cannot choose.

orbits · Circularisation An eccentricity of 0.4 swings the surface by 126 K or by 0.1, depending on the length of the year. The peak-to-trough swing in surface temperature over one orbit, against orbital period, for a planet with eccentricity 0.4 receiving on average the flux the Earth does, for surface layers of 1, 10, 50 metres of water. The dashed line is the 126 K the surface would swing through with no heat capacity. Every curve rises from near zero at short periods, where the orbit is over before the layer can respond and the planet feels only the average flux, towards the full swing at long periods, where every part of the orbit lasts long enough to be felt in full. The crossover is where the orbital period is comparable with the layer's thermal time. The vertical marks are the orbital periods of the Earth-flux orbit round stars of 0.1 M☉ (7 days), 0.5 M☉ (0.2 years), 1 M☉ (0.9 years). At a fixed eccentricity a planet in the habitable zone of a small star is thermally averaging almost regardless of how much water it has, and one round a Sun-like star is not unless it has an ocean — the ordering is set by the star through the period, which is the one quantity the flux-averaged habitable zone discards.

A year too short to feel its own eccentricity

A planet on an eccentric orbit can have a comfortable average and murderous extremes, and the habitable zone is drawn from the average. Whether the surface lives on the average or on the extremes is not decided by the flux at all — it is the ratio of how long the surface takes to change temperature to how long the year lasts, and the star sets the year.

exoplanets · Habitable zone Circularised at twice the closest approach. Eccentricity against semi-major axis, measured in units of the planet's Roche distance 2.16 Rₚ(M⋆/Mₚ)^⅓, for planets delivered on orbits of eccentricity 0.97 with periastra of 1.2, 1.6, 2.2, 3 Roche distances. Tides raised on the planet dissipate orbital energy at periastron and exert almost no torque over the rest of the orbit, so the orbit's angular momentum is nearly conserved while its energy is removed: a(1 − e²) stays fixed and each track runs down and to the left along it. Each ends on a circle of radius q(1 + e₀), which for e₀ near one is twice the periastron — 1.2 → 2.36, 1.6 → 3.15, 2.2 → 4.33, 3 → 5.91. The shaded region above the curve is where the periastron is inside the Roche distance and the planet is stripped on its first close passage. So the channel cannot deliver a circular orbit inside twice the Roche distance: the inner edge of the hot Jupiters is a prediction, stated in units that depend on each planet's own radius and mass rather than on its period.

A planet circularised at twice the distance that would destroy it

A giant planet thrown onto an orbit of eccentricity 0.97 by a distant companion loses its orbital energy to tides at every close passage and keeps almost all of its angular momentum. The orbit therefore shrinks to a circle at exactly twice its closest approach — and since a closest approach inside the Roche distance destroys the planet, the hot Jupiters delivered this way should begin at twice the Roche distance, measured in units of each planet's own size.

exoplanets · Planet migration Half and half, before and after the errors. The distribution of hot Jupiters' semi-major axes in units of each planet's own Roche distance, for a population half delivered on eccentric orbits and half halted in the disc. Shaded: each channel's share when every radius and mass is exact — the eccentric planets begin sharply at twice the Roche distance and pile up between 2 and 5, the disc planets spread over a broader hump peaking near 5.1. Line: the same population measured with 10% errors on radius and mass and with radii that have changed by ±15% since the planets arrived. The edge at 2 is gone: 6% of the eccentric planets now appear inside it, where with exact parameters only the 2% delivered on the least eccentric orbits did. The two channels' shapes still differ, but the step that told them apart has become a slope.

A fraction read in each channel's own units

Hot Jupiters arrive by two routes, one that ends at twice the planet's Roche distance and one that ends where the disc stopped. Fitting the mixture in Roche distances alone, as the edge invites, gives a fraction that drifts by almost half if the planets weigh more than assumed. Each route has to be read in the units in which it does not move — and then the answer is limited not by how well the planets are measured but by how their radii have changed since they arrived.

exoplanets · Planet migration A wall that leans the way the damping body says. The orbital period inside which a hot Jupiter's eccentricity is damped within 3 billion years, against the planet's mass, around a Sun-like star with planet Q′ = 10⁶, star Q′ = 10⁷ and a planetary radius of 1.1 Jupiter radii. If the planet's own tide did all the damping, the wall would fall with mass, from 5.5 days at one Jupiter mass to 3.2 at ten. If the star's tide did, it would rise, from 1.5 to 2.5 days. Together they give 5.5 and 3.4. Points are measured planets: filled where the orbit is eccentric, open where it is circular. The massive planets HAT-P-2 b, XO-3 b and WASP-14 b keep eccentric orbits at periods where lighter planets are all circular, which is the lean the planet's tide predicts and the opposite of the star's.

A wall that leans towards the body that damps

A hot Jupiter's eccentric orbit is rounded off by two tides at once, one dissipated in the planet and one in its star, and they depend on the planet's mass in opposite ways. So the period inside which planets are circular should fall with planet mass if the planet does the work and rise if the star does — and the heavy hot Jupiters that keep eccentric orbits where light ones never do say which.

orbits · Circularisation The ground an eccentric Jupiter sweeps on its way in. The periastron and apoastron of a Jupiter delivered on an eccentric orbit, against its semi-major axis as tides shrink it at fixed angular momentum from 5 au to 0.060 au, both on logarithmic scales. The periastron stays at 0.03 au throughout; the apoastron falls from 10.0 au toward the final circular orbit, a 5.3-day hot Jupiter. Every radius between the two curves is crossed twice an orbit for the whole of the delivery. The shaded strip below the dashed line is where a small companion would have to orbit to be found transiting with the hot Jupiter — periods under 50 days, inside 0.27 au — and the migrator's orbit covers it until the semi-major axis has fallen to 0.15 au, after which the periastron alone keeps passing through its inner edge. A planet there is scattered out, thrown into the star, or swallowed; none survives a delivery that takes thousands of orbits.

A neighbour an eccentric arrival cannot leave

A hot Jupiter that arrived on an orbit of eccentricity near one swept every distance between its closest and farthest point, thousands of times, while tides shrank its orbit — and anything small in that range was scattered, swallowed or thrown into the star. A hot Jupiter carried in by the disc can keep its neighbours. So a hot Jupiter with a close companion came through the disc, and the share of them that have one puts a floor under how many did — a floor whose height is set less by the count than by three chances nobody has measured.

exoplanets · Planet migration

Named alongside it

The objects these essays reach for when they reach for this one.

Hot jupiterTidal dissipationEccentricityEquilibrium tidePlanet migrationTidal quality factorEccentricity dampingHigh-eccentricity migrationOrbital eccentricityRadius inflationRoche limitAngular momentum

All concepts