Astronomy & Celestial Mechanics
the sky, computed
Concept

Hot jupiter — where it appears

A giant planet orbiting within a few days of its star, at a distance where it cannot have formed and must therefore have arrived from further out. About one solar-type star in two hundred has one, which is rare enough that they are not the ordinary outcome and common enough that the mechanism has to work reliably.

Named by 19 essays across 3 fields — each of them below, with the objects they name alongside it.

A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the uniform stellar disc over the region the planet covers. The depth is 1.05%, which is exactly (Rp/R⋆)² = 0.01055. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii.

A planet measured by the light it removes

A transit gives a depth, and the depth is a ratio of two radii rather than a size. Everything a transit says about a planet is said in units of a star nobody has visited either.

exoplanets · Transits The wobble of a star with a circular companion. The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.

The star moves, and the mass is a lower bound

A planet is found by watching its star fall towards it. The wobble gives a mass multiplied by the sine of an angle nobody has measured — and the shortfall is not a rounding error.

exoplanets · Reflex velocity Mass against radius, and what lies between the curves. Planetary radius against mass on logarithmic axes, in Earth units, with composition curves computed from interior models rather than drawn through the points. The solid-planet curves are R ∝ M^(1/3.7) — flatter than a constant density because a heavier planet compresses itself — and the hydrogen curve turns over near three Jupiter masses, where degeneracy pressure takes over and adding mass makes the planet smaller. A mass alone or a radius alone places a planet on a line; only both together place it between two curves, and that is the whole argument for measuring a planet twice.

Two methods, and one density

A transit gives a radius. A wobble gives a mass. Neither says what a planet is made of, and the two together say it in one number — which is the only reason both are worth doing on the same object.

exoplanets · Planet composition 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.

exoplanets · Detection bias 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 An edge where two torques balance, 21 kilometres from the shepherd. Two torques on the edge of a ring, against distance from a shepherding moon, both axes logarithmic and both scaled by the same combination of surface density, radius and orbital rate so that only their shapes are being compared. The flat line is the viscous torque, which comes from collisions between ring particles and does not care how far away anything is; it is drawn for a kinematic viscosity of 12 square centimetres a second, within the range ring seismology gives. The falling line is the moon's, summed over the first-order resonances that crowd together as the gap narrows, which makes it an inverse cube. A flat curve and an inverse cube cross once, and the crossing is where an edge can sit: closer in the moon wins and pushes the material back, further out viscosity wins and the ring spreads. For Daphnis and the Keeler gap the balance lands 20.8 kilometres out against a measured half-width of 21, which is agreement to well inside the uncertainty on the viscosity — and it is the only handle anybody has on that viscosity, since the quantity being inferred is the collision rate among particles a metre across, a billion kilometres away.

A torque that nearly cancels

A planet embedded in a gas disc pulls on the material inside its orbit and outside it, and the two torques are almost equal and opposite. What survives the subtraction is a per cent of either, and it is still enough to carry a planet from where it formed to its star in less time than the disc lasts.

exoplanets · Planet migration The inflation threshold at 2·10⁵ W m⁻², and the 0.69 R_J above it. Radius against the starlight received, for a population of Jupiter-mass planets. The horizontal line is what a structural model gives for a cold, old, Jupiter-mass ball of hydrogen and helium: 1.06 Jupiter radii, and it hardly depends on mass at all in this range, because degeneracy is beginning to set in and the mass–radius relation is flattening toward its turnover. Planets receiving less than about 2·10⁵ watts per square metre sit on that line, with a median of 1.06, which is the control the rest of the figure depends on: the models are not wrong in general. Above the threshold the radii climb, reaching a median of 1.75 — half again the size a cold planet of the same mass can be — and the onset is sharp enough to be called a threshold rather than a trend. Starlight by itself will not do this. Irradiation is absorbed high in the atmosphere and re-emitted from there; it slows the escape of heat from below, which delays contraction, but it cannot deposit energy beneath the radiative–convective boundary, and it is the interior entropy that sets the radius. So the excess is evidence for a mechanism that carries roughly half a per cent of the incident flux down to pressures of tens of bars — ohmic dissipation of currents driven through a partly ionised atmosphere, breaking gravity waves, and tidally forced turbulence are the candidates, and the threshold is the number each of them has to reproduce. The points are a synthetic population from a seeded generator, not a catalogue; what is real is the threshold, the size of the excess, and the fact that the un-irradiated planets sit exactly where they should.

A radius no cold planet is allowed

A Jupiter-mass ball of hydrogen has a maximum size, and it is about 1.06 Jupiter radii however old or young it is. Hundreds of hot Jupiters are half again that, and the excess switches on sharply above a threshold in the starlight they receive — which means something is putting energy in deep.

exoplanets · Planet composition The same orbit, realigned by one star and not by the other. The time an equilibrium tide takes to bring a planet's orbit into the plane of its star's equator, against orbital separation in stellar radii, for a planet of 1e-3 stellar masses. Both axes are logarithmic. The two curves differ only in how efficiently the star dissipates the tide, by a factor of 10⁴ — the contrast between a star with a convective envelope, where turbulence turns the tidal flow into heat, and one hotter than about 6,250 K, which has almost none. The lower curve is calibrated so that a Jupiter at 8 stellar radii realigns a cool star's orbit in 1 billion years, which is what the aligned systems require; the tidal quality factor of a star is not known from first principles and this is the honest way to say so. Everything else follows from the sixth power of the separation, which is measured off the drawn curve as 6.000. The two curves cross a Hubble time at 12.4 and 2.7 stellar radii, and their ratio is the sixth root of the dissipation contrast. Hot Jupiters sit between those two numbers. So the same arrival distribution of orbital tilts is erased around cool stars and preserved around hot ones, and a survey that finds cool hosts aligned and hot hosts scattered has measured the filter rather than the arrivals.

A misalignment only cool stars forget

A third of hot Jupiters orbit at a large angle to their star's equator, and some go round backwards. Sort the same planets by the temperature of their host and the picture changes — below about 6,250 kelvin almost all are aligned, and above it almost none are. The boundary is not about the planets.

exoplanets · Spin–orbit alignment 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 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 A loss rate that stops following the starlight. The rate at which a hot Jupiter of 1.38 Jupiter radii and 0.69 Jupiter masses loses its atmosphere, in grams per second, against the ionising flux it receives, in erg per square centimetre per second, both on logarithmic axes. The dashed line is the energy-limited rate with a heating efficiency of 0.3: linear in the flux, 1.04·10¹⁰ g s⁻¹ at 10³. The solid curve follows it at low flux and turns over near 10⁴, where the base of the wind becomes nearly fully ionised and each recombination radiates its energy as Lyman-α instead of heating the flow; above that the rate grows only as the square root of the flux. At 10⁶ erg cm⁻² s⁻¹, the flux such a planet receives from a young, active star, the energy-limited formula overestimates the loss by a factor of 11. A single efficiency cannot describe both ends of the curve: fitted at one flux, it is wrong at the other.

An escape that stops following the starlight

The standard estimate of how fast a planet loses its atmosphere assumes the loss grows in proportion to the ultraviolet light it absorbs. It does, up to a point. Beyond about ten thousand erg per square centimetre per second the escaping gas is fully ionised, its heat leaves as Lyman-α light instead of lifting anything, and the loss grows only as the square root of the flux — so the one efficiency every calculation needs is really a function of where the planet sits.

exoplanets · Atmospheric escape How much of a population is driven how close. The fraction of an isotropic population of inner orbits whose Kozai–Lidov cycling brings 1 − e below a given value within 150 secular timescales, against that value, both logarithmic, for octupole strengths ε = 0, 0.01, 0.03; each curve is 128 integrated orbits with random pericentres and nodes. The thin line is the quadrupole closed form, √((3/5)(1 − (1 − q)²)), which the ε = 0 population follows. At 1 − e = 0.01 the fractions are 9.4, 20.3, 40.6 per cent; at 0.001, 3.1, 18.0, 38.3. The octupole matters most where the quadrupole is weakest — at the extreme closeness a planet needs to be captured by tides, or a pair of compact objects needs to merge — because it adds orbits whose quadrupole cycle stopped far short of it.

Eleven in a hundred by geometry, and the rest by waiting

A distant companion drives an inner orbit through cycles of eccentricity, and whether a given system reaches a dangerously close approach depends on its inclination. Counted over a population with random orientations, the answer at quadrupole order is pure geometry — about eleven per cent reach one per cent of their semi-major axis, all at the same inclination. An eccentric outer orbit multiplies that share, spreads the inclinations, and makes the count depend on how long the systems have been waiting.

orbits · Kozai–Lidov A nearly aligned population, seen in projection and in three dimensions. 60,000 transiting systems whose stars are tilted from their planets' orbits by a small random amount: each of the tilt's two components is Gaussian with a width of 5°, and its direction around the orbit normal is random. Orange: the true obliquity ψ, which is never negative and is almost never near zero — its median is 5.9°, because a tilt with two components is small only when both of them are small at once. Blue: the projected angle λ the line profile measures, which is symmetric about zero and peaked there. The projected angles look better aligned than the true ones, and they are not hiding anything: for small tilts λ is one component of a two-component tilt, so the rms true angle, 7.04°, is √2 times the rms projected angle, 5.01°, with no knowledge of any star's inclination.

A spread the projection does not hide

A single projected spin–orbit angle says little about the true one, because the star's tilt is unknown. A population of nearly aligned systems is different: for small tilts the projected angle is one component of a two-component vector, so the rms true obliquity is exactly √2 times the rms projected angle — no star's inclination needed.

exoplanets · Spin–orbit alignment 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 The polarised light HD 189733 b can send. The polarised part of HD 189733 b's reflected light, as a fraction of its star's light in parts per million, against orbital phase from one transit to the next, for an orbit of radius 0.0314 AU, a planet of 1.138 Jupiter radii and an inclination of 85.7°. A planet's reflected light is its geometric albedo times (R/a)² — here 3·10⁻⁴ — times a phase function, and the polarised part is that times the degree of polarisation, which for a scattering atmosphere peaks near quadrature where the planet is half lit. With an albedo of 0.4 and a peak polarisation of 0.5 the signal reaches 24 ppm (solid). The ceiling (dashed) gives the planet the largest albedo a scattering atmosphere can have, 0.75, and perfect polarisation: 91 ppm. An amplitude of 100 ppm (dotted), the order of the signal claimed in 2008, would need the planet to reflect and polarise more light than falls on it.

A polarisation capped by the light the planet catches

Light scattered by a planet's atmosphere is polarised, most strongly when the planet is half lit, and the polarisation angle turns as the planet orbits — so a planet's polarised light would weigh its atmosphere and draw its orbit on the sky without a transit. In 2008 it was reported for a hot Jupiter at a part in ten thousand of its star's light. That amplitude is more than the planet can send: no atmosphere reflects more than three-quarters of the light that falls on it or polarises more than all of it, and the light that falls on it is set by its size and its distance. The photons for a real detection are cheap; what limits it is a polarimeter stable to a part in a million.

sky · Apparent motion 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 One parabola, two causes. Transit times of a planet on a 1.0914-day orbit over 12 years, minus the best straight-line ephemeris, with timing errors of 30 seconds — the shape a period shrinking by 29 milliseconds a year produces, a parabola 116 seconds deep. The curve is drawn once because it is the same curve for both explanations: an orbit decaying at that rate, or a star accelerating towards the observer at 2.9e-6 m/s², which shortens the light-travel time to each successive transit by the same steadily growing amount. Nothing in the transit times can tell them apart.

A clock that slows, or a star that falls

Transit times that curve away from a straight line can mean a planet's orbit is shrinking — the tides draining it into its star — or that the star itself is accelerating along the line of sight under a distant companion's pull. Both draw the same parabola, and the transit times cannot choose. The star's velocities can, but only for the shortest orbits: the acceleration that would fake a given decay grows as one over the period, and beyond a day or two an unseen Saturn can mimic more decay than the tides produce.

exoplanets · Transit timing 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.

Tidal circularisationPlanet migrationHigh-eccentricity migrationTidal dissipationTidal quality factorDisc migrationEccentricityEquilibrium tideKozai–LidovRadius inflationRoche limitStellar obliquity

All concepts