Astronomy & Celestial Mechanics
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Exoplanets

Planets nobody has seen, weighed and measured from a dip, a wobble and a delay.
A fold 0.06 Einstein times wide, and two configurations that make the same one. A binary lens of mass ratio 0.003 at a projected separation of 1.5 Einstein radii. Top left: the source plane, with the caustic — the set of source positions at which the magnification is formally infinite — and the track of a background star across it. A single lens has no such curve; it magnifies smoothly and diverges only at one point. A second mass makes the lens mapping fold, and the fold has edges: crossing one, the number of images changes from 3 to 5, because a pair is created out of nothing on the critical curve. The small closed curve near the origin is the central caustic, always there; the larger one at 0.83 Einstein radii is the planetary caustic, and its distance from the origin is s − 1/s, which is where the planet's own image lies. Top right: the central caustic drawn twice, once for s = 1.5 and once for s = 0.667. They are 0.0184 and 0.0169 Einstein radii across and they lie on top of each other. That is not a coincidence of these numbers: to the order that a central-caustic anomaly is measured, a close binary and a wide one with the reciprocal separation produce the same perturbation, so an event with only a central anomaly returns two separations and no way to choose. Below: the light curve along the track. The smooth part is what a single lens of the same total mass would do; the spikes are the two crossings, 0.06 Einstein times apart, so a few hours inside an event lasting a month. The two curves differ in one thing only — the size of the source. A point source diverges at each fold and reaches 27; a source of angular radius 0.006 Einstein radii averages over its own disc and reaches 9 — an eighth of a source radius inside the fold the two are 8 and 4, with the divergence replaced by a rounded shoulder whose width is the source's own diameter. Everywhere else in this collection the finite size of a star is a nuisance that degrades a measurement. Here it is the ruler: the fold is a straight edge of known sharpness sweeping across a disc, so the shape of that shoulder gives the source's angular radius, and dividing it by the crossing time gives the angular Einstein radius — which is the one quantity a light curve otherwise cannot supply.

A light curve with a fold in it

A single lens magnifies smoothly. A second mass makes the lens mapping fold, and a fold has an edge — a curve across which two images appear out of nothing and the magnification formally diverges. Crossing it turns the finite size of the source star from a nuisance into a ruler.

 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.

 A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the limb-darkened stellar disc over the region the planet covers. The depth is 1.26%, deeper than (Rp/R⋆)² = 0.01055 because the planet crosses a limb-darkened disc whose centre is brighter than its average. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii.

Four contact points, and what they fix

The depth of a transit gives a radius ratio. The shape gives the impact parameter, and then — through nothing but Kepler's third law — the mean density of the star being crossed.

 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.

 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.

 A 6.2 m s⁻¹ signal at the rotation period, made by no planet at all. Above: the apparent radial velocity of a star carrying one dark spot over 0.4 per cent of its disc, rotating with an equatorial velocity of 3.2 km s⁻¹, over 3 rotations — and beside it a circular-orbit planet of the same period fitted to the same amplitude, 6.2 metres a second. That amplitude is several times the precision of a modern spectrograph and squarely inside the range in which warm sub-Neptunes are claimed, so the two are competing on equal terms. The spot signal is not a sinusoid: the spot is in view for half the rotation and hidden for the other half, so the curve is truncated, and its power at half the period is 1.16 of its power at the period. A Keplerian orbit at the same period has none there at all. Below: the diagnostic that actually settles it. A planet moves the whole spectrum bodily, so every line keeps its shape and the bisector — the locus of midpoints up a line profile — does not move; a spot removes light from one side of the profile, so the bisector tilts in step with the velocity and against it. The two clouds correlate at -0.92 and 0.14. What the picture cannot show is why this took so long to become routine: measuring a bisector to a few metres a second needs a signal-to-noise ratio of several hundred per spectrum, so for two decades the diagnostic existed and could not be applied to the faint stars the interesting claims were about.

A planet that was the star's own rotation

A dark spot rotating across a star removes light from the approaching limb and then the receding one, and the line centroid moves. That is several metres a second at the rotation period, from no planet at all — and two of the most celebrated nearby planets were withdrawn on exactly this evidence.

 A magnification, and a spike inside it. The brightness of a background star as a foreground one passes in front of it. The smooth curve is exact: a point mass magnifies a point source by (u² + 2)/(u√(u² + 4)), which peaks at 6.7 for this track. The spike is the planet, of mass ratio 0.001, lensing one of the two images. Its duration is the Einstein time scaled by √q — about 23 hours against 30 days — so the whole planetary signal is a few hours in an event lasting a month, and it never repeats. The deviation is drawn in the approximation that the planet lenses the image in isolation; a real caustic crossing has structure this smooths over, and its true height is set by the source's size rather than by the geometry.

A star magnified by a planet nobody will see again

Microlensing weighs a planet by the way its gravity bends light around it. The measurement lasts a few hours, cannot be repeated, and is the only one that does not require the planet's star to be visible at all.

 Contrast against separation, which is where direct imaging lives. Planet-to-star brightness ratio against apparent separation, both logarithmic, for a system 10 parsecs away. The reflected-light curves are A_g(R_p/a)² and fall as the inverse square of the orbit; the thermal curve is the ratio of two Planck functions at 10 µm and does not, which is why every imaged planet so far is young and hot rather than merely large. The vertical lines are diffraction limits λ/D — nothing inside a telescope's own line is reachable by it at any contrast at all. An Earth at ten parsecs sits at 6.3e-10, which is 4 orders of magnitude below the faintest planet yet imaged. The four imaged planets are plotted at their measured near-infrared contrasts rather than at 10 µm, because the near infrared is the band they were found in — which is itself part of the argument, since a young planet is hot enough to be bright where its star is not.

Nine orders of magnitude, half an arcsecond apart

Photographing a planet is not a resolution problem. It is a contrast problem, and the contrast is set by an inverse square that punishes exactly the planets a telescope can most easily separate.

 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.

 A gap where planets should be. The number of planets per star per interval of log radius, for orbital periods under a hundred days, corrected for detection efficiency. There are two peaks — super-Earths near 1.3 R⊕ and sub-Neptunes near 2.4 R⊕ — and a deficit between them at 1.89 R⊕, where the occurrence falls to 33% of the peak. The gap is not a gap in what can be detected: detection efficiency rises smoothly through it, so a smooth underlying distribution could not produce a dip there.

The planets that were not seen

An occurrence rate is a count divided by a probability, and the probability can be a five-hundredth. Everything difficult about saying how common planets are lives in that denominator.

 A gap where planets should be. The number of planets per star per interval of log radius, for orbital periods under a hundred days, corrected for detection efficiency. There are two peaks — super-Earths near 1.3 R⊕ and sub-Neptunes near 2.4 R⊕ — and a deficit between them at 1.89 R⊕, where the occurrence falls to 33% of the peak. The gap is not a gap in what can be detected: detection efficiency rises smoothly through it, so a smooth underlying distribution could not produce a dip there.

A gap in a histogram that says how planets are built

Between the super-Earths and the sub-Neptunes there is a radius at which planets are markedly rarer. The gap is not a gap in what can be detected, and its slope with orbital period names the process that made it.

 A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±1.7 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.

Planets found by a transit running late

A planet on a fixed orbit transits like a clock. A second planet pulling on it makes the clock run fast and slow by minutes — and fitting that wander weighs a planet that may never cross the star at all.

 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.

 Libration and circulation of a resonant angle. The critical angle of a resonance over time, integrated from the pendulum equation it obeys. Below the separatrix the angle oscillates about a fixed value — the body is locked; above it, the angle runs without bound and the body is not.

A chain that could not have been assembled in place

Seven planets whose successive period ratios are all close to small whole numbers. Capture into a resonance requires the orbits to converge slowly, and converging slowly is something planets can only do in a disc.

 A radius that depends on the colour it is measured in. Transit depth against wavelength for a planet of 1.38 Jupiter radii at 1400 K, whose atmosphere has a scale height of 538 km — computed from H = kT/µg, not assumed. One scale height of extra opacity adds 153 parts per million to a transit of 1.40 per cent, so the whole spectral signal is 862 ppm at its strongest: one part in 16 of the transit that carries it. The features are at real band centres — sodium at 0.589 µm, water at 1.4 µm, carbon dioxide at 4.3 µm — with the rise at the blue end the Rayleigh slope of scattering off the smallest particles.

A radius that depends on the colour it is measured in

Measure a transit in one colour and then another, and the planet is a different size. The difference is a few atmospheric scale heights, which is a few hundred parts per million of an already tiny signal.

 The habitable zone, and the radius inside which a day never ends. The conservative habitable zone — the runaway-greenhouse and maximum-greenhouse limits of Kopparapu's parameterisation — as a band in stellar mass against orbital distance, both logarithmic. For the Sun it runs from 0.99 to 1.71 AU, which is the published result and the check on the arithmetic here. The band moves inward far faster than the mass falls, because luminosity goes as roughly the fourth power of mass: a 0.2 M☉ star's zone is at 0.082–0.158 AU. The dashed line is the distance inside which a planet is tidally locked within 4.5 billion years, computed from τ = ω₀αmQa⁶/(3GM⋆²k₂R³) with an initial ten-hour spin, Q = 100 and k₂ = 0.3. It crosses the inner edge of the zone at 0.67 M☉ — so around every star below that, which is the great majority of stars, a planet in the habitable zone has one hemisphere in permanent daylight. The Earth is outside its own locking radius of 0.53 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 8.

One number sets the zone

The habitable zone is a band of stellar flux, so it scales as the square root of luminosity and moves inward far faster than mass falls. For most stars it lies inside the radius at which a planet is tidally locked.

 One whole orbit. The system's total brightness through one orbit: the transit at phase 0, the slow rise and fall of the planet's illuminated hemisphere between, and the secondary eclipse at phase 0.5 where the planet's own light is removed. The transit is 1.05%; the secondary eclipse is 1800 ppm, about 6 times shallower.

The planet is seen when it disappears

Half an orbit after the transit the planet passes behind its star, and the light that vanishes is the planet's own. Subtracting two brightnesses taken hours apart isolates a body nothing has ever resolved.

 A 27.5 m/s velocity the star does not have. Left: a rotating stellar disc, approaching on one side and receding on the other, with the chord a planet of 0.1 stellar radii takes across it at impact parameter 0.5 and a sky-projected obliquity of 0°. Right: the apparent radial velocity that results, computed by covering the disc cell by cell — the flux hidden at each phase and its mean line-of-sight velocity — rather than from a fitted formula. The star's centre of mass does not move at any point in this: the anomaly is entirely a statement about which parts of the line profile are missing. Its amplitude is 27.5 m/s at v sin i = 4.5 km/s, and the two numbers are related by the depth of the transit, since blocking a fraction f of light of mean velocity v shifts a flux-weighted centroid by f·v. The curve is antisymmetric about mid-transit to 0.00% of its own amplitude, which is what an aligned transit gives: equal time on the blue half and the red. Limb darkening is included at u = 0.6, and it matters: it weights the hidden light towards the centre of the disc, where the rotation velocity is smallest.

A velocity measured from a shape

A transiting planet hides part of a rotating disc, so the star's line profile loses a slice at one velocity and its fitted centroid moves. The star has not moved at all — and the lopsidedness of that motion is the whole measurement of whether the orbit lies in the star's own equatorial plane.

 Two objects, the same 1.055 per cent, and only one of them a planet. A planet of 0.1027 stellar radii transiting at impact parameter 0.3, and a background eclipsing binary of radius ratio 0.62 whose 38 per cent eclipse is diluted by the target's light to 1.055 per cent — the same depth to nine decimal places, because the dilution was chosen to make it so. A blend contributing 2.7 per cent of the light in the aperture can manufacture any planetary depth at all, so the depth is not evidence about what produced it. Two things in the same photometry are. The ingress occupies 20.4 per cent of the planet's transit and 80.3 per cent of the blend's, a factor of 3.9: the shape of the shoulders is set by the radius ratio of whatever is actually eclipsing, and dilution scales a curve without changing its shape. And the duration with the period gives the mean density of the star being crossed — 1.41 g/cm³ here against 0.15, a factor of 9 — so a blend usually implies a host of a completely different kind from the one the spectrum shows. Neither test needs an observation the survey did not already make, and neither of them proves a planet: they reject specific alternatives, and what is left is a probability.

A planet that is never confirmed, only validated

A background eclipsing binary diluted by the target's light reproduces a planetary transit depth exactly, and no amount of better photometry separates the two. Most known planets are therefore the output of a probability calculation rather than a detection, and the honest statement about them is a statement about a false-positive rate.

 Pluto and Triton sit on the nitrogen line, and the Earth sits between helium and nitrogen. Escape speed against exospheric temperature, with a criterion line for each molecular species at v_esc = 6 v_th — the speed at which the Jeans loss time is comparable to the age of the solar system. Every line has slope one half, because the thermal speed goes as √T; a body above a line keeps that gas and a body below it does not. The horizontal axis is the temperature at the exobase, which for the Earth is near 1000 K rather than the 255 K of its equilibrium — the difference is extreme-ultraviolet heating, and using the wrong temperature puts the Earth above the hydrogen line, keeping an atmosphere it observably lost. Three readings are worth making. The Earth falls between helium and nitrogen and does exactly that: it loses helium as fast as radioactive decay supplies it, and keeps nitrogen for ever. Titan sits just above nitrogen and just above methane, which is why it has a thick nitrogen atmosphere and is slowly losing its methane. And Pluto and Triton sit on the nitrogen line, within 1 per cent — which is why both have atmospheres that are marginally bound and measurably escaping. Where it fails it fails in one direction only. Mercury, the Moon and the Galilean satellites all plot above lines for gases they do not have, because retention is necessary and not sufficient: a body also needs a source, and needs to survive the non-thermal losses this criterion says nothing about. Venus is the sharpest case — it sits above the hydrogen line and has still lost an ocean, because that hydrogen left by charge exchange with the solar wind rather than by moving fast enough.

The gas a planet cannot keep

Escape is a statement about the tail of a distribution rather than about its mean, so the threshold is not a speed but a dimensionless number near thirty — and past that number the loss rate falls by twelve orders of magnitude. Then, for a hot Jupiter, the whole picture fails and the atmosphere leaves as a wind.

 Four defensible boxes, and a factor of 2.9 between the answers. Above: the occurrence surface in starlight received against planet radius, with four published definitions of "an Earth-size planet in the habitable zone" drawn on it as rectangles. The surface is a stated parameterisation — two lognormal populations at 1.3 and 2.4 Earth radii, with the radius valley at 1.9 between them, normalised so that the whole of it comes to 0.5 planets per star between one and four Earth radii inside a hundred days. Integrating it over the four boxes gives 17%, 8%, 6%, 10% — a factor of 2.9 between the conservative zone and a broad definition, before any error bar is attached to any of them. The dashed curve is what a transit survey can actually see: at one year around a Sun-like star the smallest detectable planet is 1.6 Earth radii, so the lower-left corner of every box contains no detections at all and the rate there is an extrapolation of a fitted surface rather than a count of anything. Below: the same integral with only one corner moved. Holding the flux range fixed and sliding the radius bound from 1.5 to 1.75 Earth radii — a quarter of an Earth radius, well inside the uncertainty of a measured planetary radius — changes the answer by 43 per cent, which is larger than every error bar quoted with any of these numbers. The published values of η⊕ span two per cent to sixty; roughly a factor of 2.9 of that is definition, and the rest is how far each author was willing to extrapolate past the dashed line.

The part of a rate that is a definition

Published values for the frequency of Earth-size planets in habitable zones span two per cent to sixty. The spread is not measurement error — it is where the box was drawn, on a surface that is steepest exactly at the corner every author has to choose, and outside the last detection.

 The ice line at 2.7 AU, and the 3.4-fold jump in solid material across it. Two temperature thresholds turned into radii, against stellar mass, both axes logarithmic. The shaded band is the habitable zone, where water can be liquid on a planet's surface. The heavy line is the snow line of the disc the planets formed in — the distance at which a passively heated disc, whose temperature falls as the inverse square root of radius, reaches 170 K and water freezes. For a solar-luminosity star it sits at 2.71 AU, just outside the asteroid belt, and it is 1.6 times further out than the outer edge of the habitable zone; around a 0.15 solar-mass star both have moved inwards and the ratio is 5.5. What makes the line matter is what happens as it is crossed. Water is by far the most abundant condensable material after hydrogen and helium, so freezing it raises the surface density of solids by roughly 3.4 times in one step. Everything about the architecture of a planetary system follows from that step: a core massive enough to capture gas can be assembled outside the line and not inside it, which is why the solar system has small rocky planets in and giant ones out, and why a giant planet found at 0.05 AU is a statement about migration rather than about formation. The line is drawn where a mature disc puts it; a young, accreting disc is hotter and its line is several times further out, sweeping inwards as the disc drains.

The line beyond which ice counts as rock

A disc of gas around a young star gets colder outwards, and at about a hundred and seventy kelvin water stops being vapour and becomes a building material. Crossing that one line multiplies the solid mass available by roughly three and a half, in a single step, and the architecture of every planetary system is downstream of it.

 The same wind strips a mini-Neptune inside 0.06 au and leaves a hot Jupiter intact. The fraction of a planet's hydrogen envelope removed in 5 billion years by an energy-limited wind, against orbital distance, at a heating efficiency of 0.15 and an extreme-ultraviolet fluence integrated over the star's own history — saturated at L_XUV/L_bol = 3.2·10⁻⁴ for the first 100 million years and declining as t to the power −1.23 after, which comes to 4.9·10¹⁵ J m⁻² at one astronomical unit and is some 7 times what today's flux would give over the same span. This is a different mechanism from the tail of a Maxwellian, not a correction to it. Close to a star the upper atmosphere absorbs more extreme ultraviolet than it can radiate away, expands, and flows off as a wind whose rate is set by the energy arriving — Ṁ = ηπR³F/GM — so the exponential in the Jeans parameter vanishes entirely and what remains is a ratio of radius cubed to mass, which is one over the density. That is why the three curves are ordered as they are. The hot Jupiter is dense enough to lose 0.0087 of itself even at 0.047 au, where HD 209458 b sits and where its escaping hydrogen makes a transit fifteen per cent deep in Lyman α against one and a half per cent in the optical — an exosphere filling and overflowing the Roche lobe, and still costing the planet almost nothing. The mini-Neptune loses its whole envelope anywhere inside 0.06 au, and what is left when it does is a bare core about 1.5 Earth radii across. That is one of the two standard accounts of the gap in the radius histogram, and this figure is what it looks like before any of the observations are brought in.

A planet ten times larger in one colour

A hot Neptune that blocks one and a half per cent of its star's light in the optical blocks fifteen per cent of it in the ultraviolet line of hydrogen. No bound atmosphere can be that large — the material is well outside the planet's Roche lobe — so the observation is not a measurement of an atmosphere but of one leaving.

 A resonance keeps what convergence brings it and releases what divergence takes away. The resonant angle of a body inside a resonance whose strength is changing, for the two signs of that change. The angle obeys a pendulum, and the strength of the pendulum is set by how close the two orbits are; migration changes it slowly compared with the swing, which is the condition under which the area a trajectory encloses is conserved. Convergent migration strengthens the resonance, so the separatrix grows around a trajectory of fixed area and the swing narrows — from 1.05 radians to 0.77 across the figure, the body ending more deeply locked than it began. Divergent migration weakens it, the separatrix shrinks, and it eventually passes inside the trajectory: the angle stops oscillating and begins to run, 37.5π in the last quarter of the drawing alone, and the lock is gone for good. Nothing here is dissipative and nothing is random. The whole asymmetry is the sign of one derivative, which is why a chain of planets in resonance is direct evidence that they migrated toward one another, and why a chain cannot survive a phase in which they moved apart.

Capture is a direction, not a strength

A resonance holds a body that drifts into it from one side and lets go of one that drifts out the other way, and the asymmetry is not about how strong the resonance is. It is the sign of a derivative — whether the trapped region is growing or shrinking — which is why a chain of planets in resonance is direct evidence that they migrated toward each other.

 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.

 One mass and one radius, and every composition that gives them. A planet of 5 Earth masses and 1.6 Earth radii, and the compositions consistent with it. The horizontal axis is the fraction of the planet's mass in an iron core and the vertical axis the fraction in a water layer outside the rock; the heavy curve is every pair that reproduces the measurement exactly, and the band around it is what the 0.05 Earth-radius uncertainty allows. The answer is a curve, not a point, and that is not a failure of precision. Two numbers cannot determine three components: a planet can be made denser by adding iron or lighter by adding water, and along this locus the two changes cancel exactly. The ends of it are not small variations on one planet. At the left is a body with no iron at all and 0 per cent of its mass in water; at the right, one with an iron core like Mercury's and 23 per cent water. Those have different formation histories, different interiors, different everything, and the same mass and radius to the precision anybody can measure them. Breaking the degeneracy needs an observation that is neither a mass nor a radius. The usual one is a transmission spectrum, which measures the atmosphere's scale height and so its mean molecular weight — a hydrogen envelope and a steam envelope differ by a factor of nine in that, and the corresponding factor in the size of the spectral features. What the picture assumes is that the planet is differentiated into clean layers, which is the standard assumption and is false in detail: water dissolves into silicate melt at these pressures, and a mixed interior sits at neither end of this curve.

One density, and every planet that has it

A mass and a radius are two numbers, and a differentiated planet has at least three components. The set of compositions matching a measurement is therefore a curve rather than a point — and its two ends are a body with no iron and half its mass in water, and a body with a Mercury-like core.

 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.

 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.

 The shield that does not shield. Above: measured ion escape rates for three planets against their surface magnetic field. Venus and Mars have no dynamo at all and Earth has one, and the three rates lie within a factor of 9 — with the magnetised planet losing the most. The intuition that a magnetosphere protects an atmosphere is not a small correction away from being right; the measurement does not support it. Below: why. A dipole's field lines are not all closed. Those emerging within a polar cap reconnect with the wind's own field and lead straight to space, and the cap's area is set by how far the magnetosphere reaches — a boundary at ten planetary radii still leaves 5.1 per cent of the surface open. So a magnetosphere is both a shield and a funnel: it deflects the wind from most of the planet and collects ions from the whole ionosphere into the polar wind, which is exactly what an instrument above the poles measures leaving. Whether the net is protection depends on quantities nobody can compute from the field strength alone, and the three points above are the state of the evidence.

The shield that is also a funnel

A magnetic field is supposed to protect an atmosphere from the stellar wind. Venus and Mars have no dynamo and Earth has one, and their measured ion escape rates lie within a factor of a few — with the magnetised planet losing the most, because a dipole's polar field lines are open and lead straight to space.

 A transit that lasts 4.0 times longer at one end of the orbit than the other. The duration of a transit, relative to what a circular orbit of the same period around the same star would give, against the orientation of the orbit. A planet transiting near perihelion is moving fastest and its transit is shortest; one transiting near aphelion is slowest and its transit is longest. The two extremes are exact reciprocals — the circular duration is their geometric mean, whatever the eccentricity — and at e = 0.6 they differ by a factor of (1+e)/(1−e), which is 4.0. That is an enormous, easily measured effect, and it means a transit duration is not a stellar density unless the orbit is circular. Turned round, it is a measurement: given a stellar density from asteroseismology or from a parallax and a spectrum, the duration anomaly gives the eccentricity — from photometry alone, with no radial velocities at all.

A duration that measures an eccentricity

A transit's length is a measurement of how fast the planet was moving when it crossed, and that speed depends on where it was on its orbit. For a circular orbit the duration gives the star's density; for an eccentric one it gives the density times a factor of up to four — and if the density is known independently, the factor is the eccentricity.

 A gap 1 per cent deep that became 63. The distribution of planet radii, drawn three ways: the underlying distribution with a gap in it, the same distribution convolved with a 25 per cent stellar radius error, and convolved with a 5 per cent one. Every planet radius is the transit depth's square root multiplied by a stellar radius, so an error in the star is an error in the planet, and a population of planets inherits the population of stellar errors as a smearing. The gap is 1 per cent deep at the old precision and 63 at the new one, and its centre does not move — a symmetric smearing hides a feature without displacing it. That is what happened when parallaxes for a hundred thousand planet hosts arrived: no new planets were observed, and a feature that had been marginal became unambiguous.

A planet radius is a stellar radius

A transit measures a ratio and nothing else. Every planet radius ever published is that ratio multiplied by a stellar radius that came from somewhere entirely different, so a population of planets inherits the errors of a population of stars — and when the stars were measured better, a feature nobody could see became unmistakable.

 A planet that subtracts 73 per cent of itself at the inner working angle. The fraction of a planet's flux that survives an angular differential imaging subtraction, against its separation from the star in resolution elements, for a sequence covering 25 degrees of field rotation. The reference image is built from the target's own frames, so a planet that has not moved far between them is present in the reference and is removed along with the speckles. How far it moves is the arc length, which is proportional to the separation — so the self-subtraction is severe close in and negligible far out, and the half-throughput point is at 2.1 resolution elements here. The consequence for any contrast curve is that it is a statement about an algorithm as well as about an instrument: the depth reached has to be measured by injecting fake planets into the data and recovering them, because no calculation predicts what fraction of a real one survives.

A star subtracted using the star

Imaging a planet means removing a halo of scattered starlight a hundred million times brighter than the planet, and no model of that halo is good enough to subtract. So it is built from the star's own exposures — and since the planet is in those exposures too, it subtracts part of itself.

 Three planets that are the same spectrum. A model transmission spectrum, in scale heights of apparent radius, drawn three times: once as it is, once with the reference radius raised by 0.45 scale heights and the abundance reduced to compensate, and once with a cloud deck truncating the features. The three differ by 0.21 scale heights root-mean-square against features of 2.1, which is well inside the error bars of any real observation. The reason is structural rather than observational: a transmission spectrum measures a difference in apparent radius with wavelength and never an absolute radius, so the level is a free parameter, and shifting the level trades against the abundance almost exactly. Adding a cloud deck adds a third parameter that flattens features and trades against both. Three unknowns and one curve is why the quoted abundance uncertainties from transmission spectroscopy are so much larger than the photometric precision suggests.

A spectrum flattened by cloud, or by nothing

A transmission spectrum measures how a planet's apparent radius changes with wavelength, and never the radius itself. That missing level is a free parameter, it trades almost exactly against the abundance of whatever is absorbing, and a cloud deck adds a third unknown to a curve that constrains two.

 A bump that crosses the line from -41 to 25 km/s. The residual of a rotationally broadened line profile during a transit, drawn at five epochs and offset vertically. The planet covers a strip of the stellar disc whose radial velocity is the projected rotation at that position, so it removes light from one velocity and leaves a bump in the residual there. As the planet crosses, the bump travels across the profile — and where it starts and ends is set by the geometry of the chord. An orbit aligned with the star's equator gives a track symmetric about the line centre; this one, tilted by 30 degrees, runs from -41 to 25 kilometres a second and is not. The measurement is of a path rather than of a centroid, which is why it works on rapidly rotating stars where the velocity anomaly is swamped by the line's own width.

A shadow crossing a rotating line

A transiting planet hides a strip of a rotating star, and that strip has a definite velocity. So the planet removes light from one place in the line profile and leaves a bump there — a bump that travels across the line as the transit proceeds, tracing the path the planet took across the disc.

 The habitable zone of a 1 M☉ star, sweeping outwards. The inner and outer edges of the liquid-water zone against time, for a 1 solar-mass star whose main sequence lasts 10.0 Gyr. The star brightens as it burns hydrogen — a heavier core needs a hotter centre to hold the star up — so both edges move outward by a factor of 1.63 across the whole main sequence, and the band drawn here sweeps past any fixed orbit rather than containing it. Two quite different zones can be read off. The instantaneous zone at the age of the present-day Sun is 0.99 to 1.71 AU, which is the band a survey means by "in the habitable zone". The continuously habitable zone over the 10.0 Gyr drawn is the overlap of every instant in it — outside the inner edge at the end and inside the outer edge at the beginning — which is 1.37 to 1.44 AU, 10 per cent of the instantaneous width. The horizontal line is an orbit at 1 AU. It leaves the zone at 4.72 Gyr, when the inner edge overtakes it. None of these edges is a measurement: both come from one-dimensional climate models, and the inner one in particular is where a runaway greenhouse begins in a model whose clouds are prescribed.

The band moves and the orbit does not

A star brightens as it burns, so the distance at which water can be liquid sweeps outwards by a factor of one and a half across a main sequence. The band a survey quotes is an instant; the band a planet needs is the overlap of every instant, and for the Sun it is a tenth as wide and does not contain the Earth.

 Three perturbing masses drawing one curve. Transit-timing residuals for three systems whose perturbing planets are 12, 8, 5 Earth masses — a factor of 2.4 apart — each given the inner-planet eccentricity that the near-resonant theory says will compensate: 0.0000, 0.0166, 0.0363. The three curves have amplitudes of 1.7, 1.7, 1.7 minutes, within 0 per cent of each other, and they are drawn by three separate integrations that were told nothing about the theory used to pick the eccentricities. The eccentricity enters the near-resonant term divided by Δ, the fractional distance from exact resonance — here 0.0267 — so a hundredth of an eccentricity does the work of a factor of two in mass. This is why a transit-timing mass is not a mass until something else fixes the eccentricity, and why the masses that came out of the first years of such fits were systematically lower than the radial-velocity masses of the same planets.

A mass that is only a mass once the eccentricity is known

The near-resonant part of a transit-timing signal carries the perturber's mass and the pair's free eccentricity in the same bracket, divided by the distance from resonance. A hundredth of an eccentricity therefore does the work of a factor of two in mass, and three quite different systems draw one curve.

 A light curve pulled out of shape by the Earth's own orbit. Magnification against time for a 90-day event at impact parameter 0.3, computed with the observer's orbital motion included at three values of the microlens parallax. The dashed curve is π_E = 0 and is exactly symmetric in time, because a straight-line track past a point lens has to be. The others are not. The Earth's displacement over the months the event lasts adds a term π_E times its projected orbital motion to the lens–source trajectory, so one wing is pushed closer to the lens and the other further away, and the curve acquires an asymmetry of 16 per cent at π_E = 0.15 and 34 per cent at π_E = 0.35. That asymmetry is the whole measurement. An ordinary event gives one dimensioned number, t_E, which mixes the lens mass with two distances and a proper motion and therefore weighs nothing; the parallax gives a second, and two constraints on the same lens are what a mass requires. It is only available on long events — the Earth has to move appreciably while the magnification is changing — which is why parallaxes are measured for the timescales above about fifty days and not for the short ones that a low-mass lens produces.

An asymmetry that the Earth's own orbit puts in

A microlensing event delivers one number with dimensions, and one is not enough to weigh anything. The Earth's motion over a long event distorts the light curve, and the distortion is the second constraint — after which the mass follows with no distance in it at all.

 Two signals that peak at different separations. The astrometric centroid shift and the photometric magnification against the source–lens separation in Einstein radii, on a common horizontal axis and their own vertical ones, for θ_E = 1 milliarcseconds. The shift is u/(u²+2) times θ_E: it vanishes at u = 0, because the two images are then symmetric about the lens and their centroid is the lens itself; it vanishes as u grows, because the minor image fades and the major one approaches the source; and it is largest in between, at u = √2 exactly, where it is θ_E/2√2 = 0.3536 mas. The magnification at that separation is only 1.15, which is a nineteen per cent brightening — a signal a survey would barely flag. The two observables are therefore complementary rather than redundant. The photometric event is short, bright and centred on the closest approach; the astrometric one is broad, largest on the wings, and lasts several times longer, falling only as 1/u once the source is well away. That slow decline is why astrometric microlensing needs years of monitoring and why it was a prediction for sixty years before it was a measurement.

A centroid that moves when the brightness does not

The two images a lens makes are never resolved, but their centre of light is displaced from where the source would be — by an amount that is largest at a separation where the magnification is only 1.34, long after the photometric event is over.

 How long an event lasts, against the mass that causes it. Einstein crossing time against lens mass, on logarithmic axes, for a lens at 6.5 kpc in front of a source at 8 moving at 6 milliarcseconds a year — a typical bulge geometry. The slope is a half, exactly, because θ_E goes as the square root of the mass and t_E is θ_E divided by a proper motion: an M dwarf gives 21 days, a Jupiter gives 22 hours, a Neptune gives 4 hours, an Earth gives 1 hours. Nothing else about the event changes with the mass. The peak magnification is set by the impact parameter alone, so a free-floating Earth passing close enough produces exactly the light curve a star would, at exactly the same height, and lasts 1.2 hours instead of a month. That is the entire difficulty of detecting them: the events are ordinary and brief, so what decides whether they are found is the survey's cadence rather than its sensitivity. The dashed line is where the source becomes larger than the Einstein ring — 1.5e-4 solar masses for a 6-microarcsecond source — and below it every event is finite-source-dominated, which flattens the peak and, in the same stroke, measures θ_E.

An event of a few hours and no host

The Einstein crossing time is the square root of the lens mass and nothing else changes, so a free-floating Earth produces exactly the light curve a star does, at exactly the same height, and is over in ninety minutes. What decides whether it is found is cadence rather than sensitivity.

 Two separations that draw the same caustic. The width of the central caustic against the separation, computed from the lens equation for a mass ratio of 0.003, for each separation s and for its reciprocal 1/s. The two curves lie nearly on top of each other. That is the close–wide degeneracy, and it is a theorem rather than a coincidence: expanding the binary lens equation near the primary shows that the central caustic depends on the separation only through s + 1/s to first order in the mass ratio, and that combination is invariant under s → 1/s. It is first order and not exact, which the figure shows rather than hides — the two agree to 0.7 per cent at s = 2.8 and only to 13.1 at s = 1.4, because a smaller separation is closer to the resonant regime where the central and planetary caustics have not yet separated. Reducing the mass ratio to 1.0e-3 brings the worst case to 4.8 per cent, which is the first-order statement being checked rather than quoted. What that means for a measurement is uncomfortable. An event whose planetary signal comes from the source passing near the central caustic — which is most of them, because the central caustic sits where the magnification is already high and the event is already being watched — cannot distinguish a companion at 2.8 Einstein radii from one at 0.357. For a typical lens that is the difference between a planet at four astronomical units and one at less than one. And the disagreement the figure measures is not the way out: at 2.8 Einstein radii the caustics differ in width by 0.7 per cent, which is far below what a light curve sampled through a night's seeing can separate, so the ambiguity is real in the data even where it is not exact in the mathematics.

Two systems that draw the same curve

A binary lens with separation s and one with separation 1/s have central caustics that agree to first order in the mass ratio. The same event is therefore a planet at four astronomical units or one at less than one, and no amount of photometric precision decides between them.

 How long a planet's hydrogen lasts, against how much of it there is. The time a young Sun-like star's saturated X-ray and ultraviolet output would take to remove a planet's whole hydrogen envelope, at 100 times the Earth's insolation, against the envelope's share of the planet's mass, for cores of 3, 5, 8 Earth masses. It is computed with energy-limited escape and an interior fit for the envelope's thickness, and it is not monotonic. A heavy envelope takes long to remove because there is a lot of it. A very light one takes long because the planet is small and intercepts little light. In between the time peaks, and it peaks where the envelope has swollen the planet most for its mass: 3 Earth masses at an envelope of 1.9 per cent, 129 Myr, where the envelope is 1.31 times as thick as the core's radius; 5 Earth masses at an envelope of 2.8 per cent, 364 Myr, where the envelope is 1.30 times as thick as the core's radius; 8 Earth masses at an envelope of 4.0 per cent, 946 Myr, where the envelope is 1.30 times as thick as the core's radius. The peak is what makes a valley. A planet above it losing gas moves up the curve, its remaining envelope lasting longer and longer, and settles; a planet below it moves down, lasting less and less, and loses everything. The horizontal lines are the saturated phase, 100 Myr, and the 300 Myr of saturated-equivalent exposure the whole history delivers: a core whose peak lies below the second line cannot keep any envelope at all.

The envelope that doubles a planet lasts longest

The gap in the radii of small planets is not merely a place where planets are rare — it is nearly empty, and the reason is a peak. The time a young star needs to strip a planet's hydrogen is longest for the envelope that swells the planet to a little over twice its core's size. Anything thinner runs away to nothing, and anything thicker settles back towards the peak.

 When the stripping happens, in a model population. For the same 3,000 model planets, the share that have lost their whole envelope by each age, scaled to the share bare at 5.0 Gyr (35 per cent of the population), beside the share of the star's lifetime XUV energy delivered by then. Half of all the stripping in this model is finished by 72 Myr and nine tenths by 457 Myr, while the star has delivered 27 and 76 per cent of its XUV energy. That is the clock photoevaporation keeps: the valley is essentially finished within the first few hundred million years, because the planets near the boundary are the ones that run away, and they run away early. A mechanism powered instead by the slow cooling of the planets' own cores would keep moving planets across the valley for billions of years. The difference is in when, not where, and it is why the ages of the stars hosting planets on either side of the valley are the measurement that can separate the two.

The stripping runs ahead of the starlight that drives it

If young stars carve the radius valley with their X-ray light, the valley should be finished early — half of it before the star has delivered a third of that light, and nine tenths of it within a few hundred million years. If the planets' own cooling cores carve it instead, planets should still be crossing it billions of years later. The two accounts put the valley in the same place, and they are separated by the one thing a histogram cannot show — when.

 Where photoevaporation puts the valley, round stars of different mass. The radius of the largest core stripped bare, against orbital period, round stars of 0.5 M☉, 0.75 M☉, 1 M☉, 1.25 M☉, in the same energy-limited model, with each star's luminosity taken as its mass to the fourth power and its saturated phase lengthened for smaller stars as the mass to the −1.5, 100 Myr for the Sun. At ten days the valley is at 1.22 Earth radii round the 0.5 M☉ star, 1.38 Earth radii round the 0.75 M☉ star, 1.50 Earth radii round the 1 M☉ star, 1.60 Earth radii round the 1.25 M☉ star, and every one of them tilts as the −0.176 power of the orbital period, because the tilt comes from the exponents of the escape law and the interior fit, which do not depend on the star. A lower-mass star is far fainter, so at a given period its planets receive much less light, and even its longer active phase does not make up the difference: this model puts the valley at smaller radii round smaller stars while keeping the same sign of tilt. The stellar scalings are rough, and they are the least certain part of the calculation; what is robust is that photoevaporation ties the valley to the XUV energy a planet received, which falls with the star's mass at fixed period, and a mechanism tied to something else would tie it differently.

A smaller star puts the valley lower

Round stars of half the Sun's mass, the gap between bare rocky cores and sub-Neptunes should sit at a smaller radius than round the Sun, because at the same orbital period their planets receive a tenth of the light. Their stars also stay young and active far longer, which pushes the other way. Photoevaporation weighs those two against each other in a definite proportion, and the answer is a valley that scales as the star's mass to about the power three tenths.

 The XUV energy that reached one astronomical unit, for three young Suns. The cumulative X-ray and extreme-ultraviolet energy delivered per square metre at the Earth's distance from the Sun, against age, for three histories that differ only in how long the young Sun stayed magnetically saturated: 20 Myr for a slow rotator, 100 Myr for a medium rotator, 300 Myr for a fast rotator. After saturation each declines to a common track by 1 Gyr, as rotation histories are observed to converge. By 4.5 Gyr the totals are 2.06·10¹⁵ J/m² for the slow rotator, 3.66·10¹⁵ J/m² for the medium rotator, 6.47·10¹⁵ J/m² for the fast rotator: the fast rotator delivered 3.1 times as much as the slow rotator, nearly all of it in the first few hundred million years. The Sun's own rotation at that age is not measured; it is inferred from the spread of rotation periods in young clusters, and all three histories are consistent with a Sun that ends up rotating as it does now.

The young Sun's spin decides what the Earth kept

The calculation that strips sub-Neptunes applies just as well to a planet with a trace of hydrogen. An Earth that captured a few hundredths of a per cent of its mass from the gas it formed in — several times the hydrogen now in its oceans — would have lost all of it, or kept most of it, depending on something nobody has measured — how fast the Sun was spinning in its first few hundred million years.

 An ephemeris fitted to 120 days, 43 minutes wrong within a year against a band of ±5.1. Transit times of a 6 Earth-mass planet on a 10-day orbit, perturbed by a 14 Earth-mass planet at 15.24 days, integrated for 1460 days and compared with a straight-line ephemeris fitted only to the transits in the first 120 days — the shaded window. Inside the window the line fits to 2.1 minutes. Outside it the pair's 317-day super-period carries the transits away from the line, and within a year of the window closing the prediction is 42.8 minutes early of the observed transit, 347 days after the last fitted one. The narrow band is the formal three-sigma uncertainty of the same line for a timing precision of 0.5 minutes per transit, which at that date is ±5.1 minutes: the error is 8.4 times the band. A statistical uncertainty assumes the residuals are noise, and these are a signal, so the band describes a planet that does not exist.

A forecast that fails on a schedule

A transiting planet perturbed near a resonance keeps a clock that wanders, and a straight-line ephemeris fitted to part of the wander predicts the next transit with a confidence the wander does not deserve. The error is not noise and does not average down; it grows on the pair's super-period, it is many times the formal uncertainty within a year, and how soon it appears depends on which stretch of the wander happened to be observed. When a model that includes the known perturber still fails, the failure has a period, and the period is a planet.

 Io's eclipses, early at opposition and late at conjunction. The delay in the timing of Io's eclipses by Jupiter's shadow caused by the changing distance between the Earth and Jupiter, against days from an opposition, for circular orbits at 1 and 5.2026 AU. At opposition the two planets are closest and the eclipses arrive 8.3 minutes early against the average; 199 days later, near conjunction, they are furthest apart and arrive 8.3 minutes late. The whole swing, 16.6 minutes, is the time light takes to cross the diameter of the Earth's orbit, and the pattern repeats every 399 days, the synodic period. This is transit timing done on a moon in 1676, with a clock that ran on Io's 42.5-hour orbit and a residual that had nothing to do with Io.

A transit late by the width of an orbit

A transiting planet's clock can run fast and slow for a reason that has nothing to do with gravity acting on the planet. If its star is itself in orbit about a distant companion, each transit's light has further or less far to travel, and the timing wanders by the light-travel time across the star's orbit. It is the measurement that first showed light has a speed, made again on a different kind of clock.

 5 perturbers that draw one 58-day timing signal. Every perturbing planet that gives a 3-day transiting planet the same timing signal — a sinusoid with a 57.8-day super-period and an amplitude of 1.27 minutes — placed wide of the nearest first-order commensurabilities inside and outside its orbit, with its mass found by integrating until the amplitude matched. outside, near 4:3 at 4.070 days needs 7.6 Earth masses and would move the star by K = 3.1 m/s; outside, near 3:2 at 4.620 days needs 12.0 Earth masses and would move the star by K = 4.8 m/s; outside, near 2:1 at 6.329 days needs 25.6 Earth masses and would move the star by K = 9.2 m/s; inside, near 3:2 at 1.966 days needs 6.9 Earth masses and would move the star by K = 3.7 m/s; inside, near 2:1 at 1.462 days needs 45.8 Earth masses and would move the star by K = 26.7 m/s. The period ratio is along the bottom on a logarithmic axis and the required mass up the side. A super-period fixes the distance from some resonance and not which resonance it is, and the amplitude then fixes a mass for each guess — so the timing alone returns a list rather than a planet. The velocity semi-amplitudes differ by a factor of 8.5 across the list, which is one of the two ways the list is shortened.

One timing curve and five planets that could draw it

A transiting planet whose times wander at a 58-day period, by just over a minute, is being pulled by something — but the period says only how far from some resonance the pull comes, not from which. Perturbers inside and outside the orbit, near four different commensurabilities, each with its own mass, reproduce the same curve to a fraction of a per cent. Timing alone returns a list, and even the detail that shortens it hides a coincidence of its own.

 A 10 Earth-mass Trojan started 10° from its point, seen in the planet's transits. A Jupiter-mass planet on a 4-day orbit about a 1 solar-mass star, with a 10 Earth-mass companion started 10 degrees beyond the leading Lagrange point of the same orbit, integrated for 160 days. Above, the angle between the companion and the planet as seen from the star: it swings about 60 degrees, between 51.1 and 70.3, with a period of 48.5 days measured off the curve, against 49.8 from the small-amplitude formula P/√(27μ/4). Below, the planet's own transit times minus a straight line: they swing by ±300.0 seconds at the same period, because the planet and the companion orbit their common centre of mass and the companion's libration moves that centre along the orbit. The companion shares the planet's period exactly, so it produces no timing signal at any orbital period of its own and would not transit on a schedule distinct from the planet's; the libration period is the only clock it has.

A companion on the same orbit, seen in the planet's clock

A body sharing a planet's orbit at one of its Lagrange points has the planet's period exactly, so no search for periodic dips or wobbles can find it at a period of its own. It shows instead in two ways the planet's own signals carry — a slow swing of the transit times at the companion's libration period, and a fixed offset between the planet's transit and its star's velocity curve.

 The solid mass available, as a staircase of 5 fronts. The share of the condensable material that is solid, against distance from a 1 solar-mass star, on a logarithmic radius axis. Each riser is one species freezing out, at the radius where the disc's temperature — falling as the inverse square root of the distance — reaches that species' condensation point. silicates and iron at 1400 K and 0.04 AU; water ice at 170 K and 2.71 AU; carbon dioxide at 70 K and 16.00 AU; methane and ammonia at 30 K and 87.11 AU; carbon monoxide at 20 K and 196.00 AU. Inside every front the solid surface density is 22 per cent of what is available, which is the refractories alone; water alone contributes 53 per cent, more than twice everything else combined, which is why one of these steps is called the snow line and the others are not. A body's composition is decided by which pair of risers it formed between, and the steps are narrow because a vapour pressure is exponential in the inverse temperature.

There is not one line, there is a staircase

Water freezing is the biggest step in a protoplanetary disc and it is one of five. Each condensable species has its own temperature and therefore its own radius, and what a body is made of is decided not by which side of a line it formed on but by which pair of risers it formed between.

 The ice line sweeps from 6.4 to 2.7 AU while the disc drains. The radius at which a disc around a 1 solar-mass star reaches 170 K, against the disc's age, both axes logarithmic. The temperature is the fourth root of the sum of two fluxes: starlight, which does not change, and the disc's own accretion, which releases gravitational energy at a rate set by how fast material is flowing inward. The accretion rate decays as the disc drains — taken here as 5e-5 solar masses a year falling off as t^(−3/2) beyond 0.1 million years — so the viscous term fades and the line sweeps in. It starts at 6.38 AU and ends at 2.73, the passive value the closed form gives. A body at three astronomical units formed dry if it formed early and icy if it formed late, so a composition dates a formation rather than locating one, and the dating is only as good as the accretion history assumed.

A composition that dates a formation rather than placing it

The ice line in a young disc starts six astronomical units out and sweeps inward to under three as the disc drains. A body at four astronomical units therefore formed dry or wet depending only on when — so what it is made of is a clock, and reading it as a map is the mistake the moving line makes easy.

 A carbon-to-oxygen ratio that steps at every front. The carbon-to-oxygen ratio of the gas and of the solids against distance from a 1 solar-mass star, each in units of the star's own ratio. Crossing a condensation front moves one element or both out of the gas and into the solids, so the two curves step in opposite directions at water at 2.7 AU, carbon dioxide at 16.0 AU, carbon monoxide at 196.0 AU. Water takes oxygen and no carbon, so beyond it the gas is carbon-rich — 3.20 times the stellar ratio — and the solids are oxygen-rich. Carbon monoxide takes both, in a ratio of one to one, so beyond that front the gas ratio rises again. A giant planet's atmosphere is made mostly of gas it accreted, so measuring its ratio and inverting this staircase gives a formation radius — and the inversion is not unique, because more than one interval returns the same value once the solids a planet also swallowed are allowed for.

Two elements in a ratio, and a birthplace read off it

Carbon and oxygen freeze out at different places, so the gas between the fronts is carbon-rich and the solids are oxygen-rich. A giant planet is made mostly of gas it accreted, so measuring the ratio in its atmosphere and inverting the staircase should give the radius it formed at — and the inversion turns out not to be unique.

 What comes back is a ramp, not a threshold. Detection efficiency against signal-to-noise: the fraction of synthetic transits injected into real photometry that the pipeline afterwards finds. The measured curve is a gamma cumulative distribution of shape 4.65 and scale 0.98 beginning at 4.1, which is the form a survey's own injection tests are fitted with; the dashed line is the step at 7.1 that a threshold calculation assumes instead. Half the injections are recovered at 8.33, 1.2 units above the nominal threshold — the ramp is a property of the search and the cut is a separate decision, so the two need not meet anywhere in particular. The rest of the disagreement is the area between the curves. The pipeline does not reach 99 per cent efficiency until 14.9, four units above the threshold, and it recovers 45 per cent one unit above it. Over a population whose signal-to-noise falls as s^-2 — which is what a planet population looks like, because there are far more small planets than large ones — the step function counts 1.21 times as many detections as the ramp does. That factor is not an error bar. It multiplies every occurrence rate computed without it, and it is larger for the small planets than for the large ones, because the small ones live where the ramp is.

The threshold that is not a threshold

A survey's detection limit is quoted as a number — seven point one — and a pipeline does not behave that way. Half the injected signals come back at the threshold, and full efficiency arrives four units above it.

 Three biases against eccentricity, and they do not agree. Four quantities against orbital eccentricity, each relative to a circular orbit of the same semi-major axis, averaged over the argument of periastron. The transit probability rises as (1 − e²)⁻¹, because an eccentric planet spends part of its orbit inside its own semi-major axis: at e = 0.5 a transit is 1.33 times as likely. The transit duration falls as √(1 − e²), so the event carries less signal-to-noise, and the two together — probability times the square root of the time in transit — come to 1.24 at the same eccentricity. They very nearly cancel, and that is the surprise: a transit survey has almost no eccentricity bias at all. The radial-velocity curve is the one that does. A Keplerian of eccentricity e puts less of its variance in the fundamental and more into harmonics no sinusoidal search is looking at — 68 per cent remains at e = 0.6 and 47 per cent at e = 0.8 — so a velocity survey loses amplitude exactly where a transit survey does not. What no figure here can show is which of these the measured eccentricity distribution is made of, because the correction depends on a detection pipeline rather than on geometry, and the two surveys have to be corrected separately before their answers can be compared.

Every method prefers a circle, and not for the same reason

A transit is more likely on an eccentric orbit and shorter when it happens, and the two very nearly cancel. A velocity curve loses amplitude to harmonics no sinusoidal search is looking at, and that one does not cancel at all.

 How many planets a star has is the hardest thing a catalogue measures. The multiplicity distribution a transit catalogue would contain, for systems that all truly hold 5 planets, at four mutual inclination dispersions. 40,000 systems are drawn per dispersion with an isotropic viewing direction and Rayleigh-distributed inclinations about a common plane, at semi-major axes of 12, 16, 21, 27, 34 stellar radii; the bars are conditioned on at least one planet transiting, which is what makes a system appear in a catalogue at all. At 0.5° of dispersion 33 per cent of the detected systems show all 5 planets and the mean apparent multiplicity is 3.13; at 10° it is 1.39, with 68 per cent of them showing exactly one. Every one of those systems has 5 planets. The entire difference between a catalogue of singles and a catalogue of compact multiples is one number that nothing in the light curve measures. And the two effects run in opposite directions: the fraction of stars showing any planet RISES with the dispersion — 8%, 9%, 12%, 19% across the four — because scattering the orbits gives more of them a chance to cross the line of sight, while the number seen per detected star falls by a factor of 2.2. A survey that scatters its systems finds more stars with planets and fewer planets per star, and neither number on its own says which has happened. What no figure here can show is the true dispersion, because the observable is the ratio of those two and a system with fewer planets and a tighter plane reproduces it exactly.

How many planets a star has is not a measurement

Draw five thousand identical five-planet systems, scatter their orbital planes by half a degree, and a third of the detections show all five. Scatter them by ten degrees and two thirds show exactly one. Every system has five.

 A thermostat that passes 40 per cent of the change it is meant to cancel. Surface temperature against the flux a planet absorbs, in units of the Earth's, for a planet round a 1 M☉ star, with and without the carbonate–silicate cycle. With carbon dioxide held at 280 µbar the temperature follows the flux directly. With weathering allowed to adjust — rock dissolves faster when it is warm and when there is more CO₂, and in the steady state it must remove exactly what volcanoes supply at 1 times today's rate — a colder planet accumulates CO₂ until the balance is restored. The feedback is real and it is not a set point. Near S = 1 it passes 40 per cent of a flux change through to the surface: the loop gain is k s / β = 1.49, with weathering rising one e-fold for every 9.7 K, a greenhouse of 4.33 K per e-folding of CO₂ and a CO₂ exponent of 0.3. The required CO₂ would reach 8.8 bar — the point at which more of it scatters sunlight faster than it traps heat, and the controller has nothing left to add — at S = 0.249. The steady state reaches 273 K at S = 0.531, before the CO₂ has run out — the outer edge of this planet's habitable zone is where the thermostat saturates or freezes, whichever comes first. The climate law is logarithmic in CO₂, which is right near today's values and only a calibration at several bar; ice-albedo feedback, which makes a cooling planet able to jump to a frozen state, is left out.

A thermostat that only halves the error

The carbonate–silicate cycle is credited with keeping a planet's water liquid across the whole width of its habitable zone. Written down with its own measured exponents, it is a proportional controller that cancels about three-fifths of a change in sunlight, takes half a million years to do it — and reaches the published outer edge only on a planet with an order of magnitude more volcanism than the Earth.

 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.

 A planet in the middle of a 0.08 M☉ star's zone spends 94 Myr too hot to keep an ocean. How long a planet spends receiving more flux than the runaway-greenhouse limit while its star contracts onto the main sequence, against stellar mass, for planets at the inner edge, in the middle, at the outer edge of the zone the star will have once it settles. The luminosity is the contraction law of a fully convective star, falling as t^(−2/3) from an age of 1 Myr until arrival; the limit is the runaway flux for the star's temperature. A planet at the inner edge is too hot for 313 Myr round a 0.08 M☉ star and 17.8 Myr round a 0.6 M☉ one; a planet in the middle is too hot for 94 Myr round a 0.08 M☉ star and 5.5 Myr round a 0.6 M☉ one; a planet at the outer edge is too hot for 39 Myr round a 0.08 M☉ star and 2.0 Myr round a 0.6 M☉ one. A runaway greenhouse is not a hot climate; it is a state in which the ocean is entirely in the atmosphere as steam, where ultraviolet light splits it and hydrogen escapes. A planet in the eventual zone of the commonest stars in the galaxy begins its life in that state for tens to hundreds of millions of years — a span comparable with the whole assembly of the Earth. The durations are measured from 1 Myr; a rocky planet may take tens of millions of years to finish forming, and one that formed later misses the start of its exposure, while the smallest stars' arrival times are somewhat short in this model, which lengthens the end of it.

Steam before the zone existed

The smallest stars take hundreds of millions of years to contract onto the main sequence, shining at many times the luminosity they will settle at. A planet in the habitable zone such a star will eventually have spends that time with its ocean in the air as steam, while starlight splits the water and the hydrogen leaves — so the zone of the commonest star in the galaxy is a place that had to survive being too hot first.

 Two references, failing in opposite places. The faintest companion each technique can detect, against separation in resolution elements, in units of the speckle noise a perfectly matched reference would leave. Angular differential imaging builds its reference from the target's own exposures, so the reference matches the instrument's state exactly and the residual speckle is small — and the companion is in every frame of it, so close in the companion subtracts most of itself and the achievable contrast collapses. Reference-star differential imaging builds the reference from archival exposures of other stars. The companion is in none of them, so the throughput is one at every separation and the curve is flat; what is lost is the match, because two stars observed at different times through different air with a differently flexed telescope have different speckles. The two cross, and where they cross depends on how large the library is: 1.1 resolution elements at 20 frames, 1.4 resolution elements at 200 frames, 1.8 resolution elements at 2000 frames. A library of thousands of archival exposures is therefore worth building, and the thing it buys is the inner working angle rather than the depth.

A reference built from other stars

Subtracting a star's halo using the star's own exposures removes part of the planet with it. Building the reference from a library of archival exposures of other stars removes none of the planet and matches the halo worse — so the two techniques fail in opposite places, and which one wins at a given separation depends on how many exposures somebody has kept.

 Where a disc pushes a planet outward. The sign of the total torque on a low-mass planet, as a function of the power-law indices of the disc's surface density (Σ ∝ r^−α, horizontal) and temperature (T ∝ r^−β, vertical), from the fitting formulae of Paardekooper and collaborators for an adiabatic disc with γ = 1.4 and the horseshoe torque unsaturated. Above the line the total is positive and the planet migrates outward; below it the planet migrates inward. The line is straight because every part of the torque is linear in the two indices, and it is steep in α because a shallow density profile strengthens the vortensity part and weakens nothing. The minimum-mass nebula (α = 1.50, β = 0.50) sits at a total of −3.76 in units of Γ₀/γ; The viscously heated inner disc (α = 0.60, β = 0.90) sits at a total of +0.74 in units of Γ₀/γ; The irradiated outer disc (α = 1.07, β = 0.43) sits at a total of −2.65 in units of Γ₀/γ. The minimum-mass nebula every textbook starts from is on the inward side by a wide margin, and a disc heated by its own accretion is on the outward side: the same planet can migrate either way depending on how the gas is heated, which the Lindblad torque alone could never say.

The gas that turns round beside the planet

The two spiral wakes a planet raises nearly cancel, and what decides the direction it drifts is a third, smaller torque from the gas sharing its orbit. That gas makes a U-turn in front of the planet and behind it, and whether the turn pushes the planet outward depends on how the disc was heated and on a viscosity nobody has measured.

 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.

 One eccentric planet, drawn as two circular ones. Top: the velocity of a star with one companion on an orbit of eccentricity 0.25 (argument of periastron 0.9 radians), in units of its semi-amplitude K, over two orbits — and on top of it the sum of two circular orbits, one at the same period and one at half of it, fitted by taking the first two terms of the exact curve's Fourier series. The second orbit's amplitude comes out at 0.246 of the first, which is the eccentricity to first order. Bottom: the difference between the two, on a scale 12 times magnified; its largest excursion is 0.0889 K and its rms 0.0473 K, second order in the eccentricity. For a planet with K = 30 m/s that is 1.42 m/s rms — below the noise of most instruments — so an eccentric single planet and a pair of planets in a 2:1 period ratio are the same measurement unless the data are good enough to see a term of order e².

One eccentric planet, or two circular ones

The velocity curve of a star with one planet on an eccentric orbit is, to first order in the eccentricity, exactly the curve of two planets on circular orbits with periods in the ratio two to one. The difference between the two readings is a term in the eccentricity squared, and for the modest eccentricities most planets have, it is smaller than the noise.

 An exposure time chosen by the star. The fraction of a star's p-mode velocity signal that survives averaging over an exposure of the length on the horizontal axis, for an envelope of modes centred on νₘₐₓ with the stated width. The dashed curve is a single oscillation at νₘₐₓ = 3.1 mHz, a period of 5.4 minutes: it averages to exactly zero after one period and at every multiple. The solid curve is the real case, a band of modes 1.2 mHz wide, which cannot all be nulled at once: its first minimum is at 5.8 minutes, where 15% of the rms survives, and beyond it the residual falls only as one over the exposure time. Short exposures on bright stars sample the oscillations rather than averaging them, which is why a bright star observed quickly can be noisier than a faint one observed slowly.

An exposure time chosen by the star

A Sun-like star's surface rings at five minutes and boils on timescales from ten minutes to a day, and each of those motions shifts its spectral lines by more than the nine centimetres a second an Earth would. The ringing can be averaged away by an exposure tuned to the star; the slowest boiling cannot be averaged away within a night at all, and it sets the number of nights an Earth twin costs.

 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.

 A fringe that is dark on the star and bright on the planet. The fraction of a point source's light an interferometric nuller transmits, against its angle from the star along the baseline, in milliarcseconds, for two telescopes 10.3 m apart observing at 10 μm — the baseline that puts the first bright fringe exactly on a planet 1 AU from a star 10 pc away, 100 milliarcseconds out. The solid curve is the two-telescope null, sin²: it is zero on the star and rises as the square of the angle. The dashed curve is a quartic null, sin⁴, made by combining four apertures: it is darker near the star and narrower at the planet. The shaded strip is the star itself, 0.93 milliarcseconds across. It looks negligible at this scale, but the null is not perfect across it, and what leaks from that strip is the quantity the whole technique has to beat.

A null filled in by the star itself

Two telescopes combined out of phase cancel a star on the axis and pass a planet beside it, and the cancellation rises only as the square of the angle away from the axis. A star is not a point, so its own edges sit off the axis — and for the Sun seen with the baseline that finds the Earth, they leak through at a hundred and fifty times the Earth's brightness, at every distance, because the leak depends only on the size of the star compared with the planet's orbit.

 How planet occurrence rises with the host star's iron, by kind of planet. Planet occurrence relative to its value at solar metallicity, on a logarithmic axis, against the host star's iron abundance [Fe/H] (zero is the Sun; one unit is a factor of ten). Each class is a power law, occurrence ∝ 10^(β[Fe/H]). Giant planets within a few astronomical units follow β = 2, the relation found in the first large radial-velocity survey to measure it: 12 per cent of stars at [Fe/H] = +0.3 have one, against 3 per cent at solar and under 0.3 per cent at −0.5 — a factor of 158 across the range drawn. The small planets found by transits rise far more gently; the slopes drawn, β = 0.5 for sub-Neptunes and 0.2 for super-Earths, are representative of published fits, which range from nearly flat to about a doubling across this interval depending on orbital period.

A threshold that makes iron matter only to giants

A star with twice the Sun's iron is four times as likely to have a giant planet, and barely more likely to have a small one. The steep law and the flat one look like two mechanisms. A core racing its disc, run over a spread of discs, produces both — steep where winning is rare, flat where it is common.

 The true obliquity a projected angle allows, against the tilt of the star. The true angle ψ between a transiting planet's orbit and its star's spin axis, against the inclination of that spin axis to the line of sight, for projected angles λ of 0°, 30°, 60°, 90°, 130° — from cos ψ = sin i★ cos λ for an orbit seen edge-on. At i★ = 90°, a star seen equator-on, the projection loses nothing and ψ = λ. As the star tips towards pole-on the true angle moves towards 90° from whichever side it started, so every curve ends at 90° when i★ = 0. Two things follow exactly. The projected angle is a lower bound for an aligned-looking system and an upper bound for a retrograde-looking one — ψ always lies between λ and 90°. And λ = 90° needs no inclination at all: a planet whose projected orbit crosses the stellar equator at right angles is on a polar orbit whatever the tilt, which is why the pile-up of polar orbits could be seen in projected angles alone.

An angle hidden by the tilt of the star

The line profile of a transited star gives the angle between orbit and spin as projected on the sky. The true angle needs the tilt of the star's spin axis, and every way of measuring that tilt is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits.

 Brightness temperature across the infrared, for three profiles. The brightness temperature of the planet's emission against wavelength from 1 to 12 microns, for the three temperature profiles, with water and carbon monoxide bands in the opacity. In a band the opacity is high, so the light comes from higher, lower-pressure layers; in the windows between bands it comes from deeper. Over a profile falling outward the bands are cooler than the windows: at 6.3 microns 1457 K against 1570 K at 3.6. Over the inverted profile they are hotter: 1851 against 1538. Over the isothermal profile every wavelength has the same brightness temperature and the spectrum is a blackbody, whatever molecules are present.

Molecules seen only through a change of temperature

The heat a planet gives off, measured colour by colour as it disappears behind its star, carries the fingerprints of the molecules in its atmosphere — but only because the atmosphere's temperature changes with height. Water absorbs where the air cools upward and emits where it warms, and in an atmosphere at one temperature every molecule vanishes from the spectrum, however much of it there is. An emission spectrum is a sounding of a temperature gradient, and the molecules are what it is taken with.

 What shares a resolution element with an Earth twin. Light collected in one resolution element around an Earth-like planet in the habitable zone of a Sun-like star, against the system's distance, both on logarithmic scales, for a 6 metre telescope at visible wavelengths, in units of the planet's own light at 10 parsecs. The planet and the leaked starlight both dim as the square of distance. The exozodiacal dust, at 3 times the Solar System's level, and the Solar System's own zodiacal light do not dim at all: each is a surface brightness, and a resolution element at a greater distance simply contains more dust. The exozodi equals the planet at 9.1 parsecs and is 11 times it at 30.

Dust that does not dim with distance

An Earth-like planet around a nearby star is seen against the glow of the dust in its own system, and the two behave differently with distance. The planet is a point, and it fades as the square of how far away it is. The dust is a surface, and a surface does not fade at all — a more distant system packs more dust into each resolution element in exactly the proportion its light is diluted. So the dust around a star, not the telescope's contrast, sets how far away an Earth can be found, and how many stars a search can reach.

 Small planets and giant planets per star, against the mass of the star. Planets per star against the host star's mass, both on logarithmic axes, for small planets (one to four Earth radii, orbital periods under 200 days) and giant planets (above a third of Jupiter's mass, within 2.5 astronomical units). The small-planet rate is drawn as 1.1 per star at one solar mass rising as M^−0.6; the giant rate as 7 per cent at one solar mass falling towards small stars as M^1. Both are representative of published fits, not fits of their own. At 0.3 solar masses a star has 2.3 small planets and a 2.1 per cent chance of a giant; at one solar mass 1.1 and 7 per cent. The ratio of small planets to giants changes by a factor of 7 between the two, so a statement about how common planets are depends on which star it is about as much as on which planet.

More small planets round smaller stars

A red dwarf has about twice as many small planets as the Sun's kind of star and a third as many giants. The two populations move in opposite directions along the same axis, and the small planets were built from discs several times poorer.

 The chance that a candidate is a planet, against how strong its signal is. The reliability of a transit candidate — the fraction of candidates at a given detection statistic that are planets rather than false alarms — against the multiple-event statistic, the combined signal-to-noise of all its transits, for orbital periods of 10, 100, 350 days. Below the 7.1 threshold nothing is a candidate. At a statistic of 9 the reliability is 100 per cent, 80 per cent, 23 per cent; at 12 it is 100 per cent, 95 per cent, 44 per cent. The true planets and the false alarms are modelled, not fitted: the planets as a falling power law above the survey's completeness curve, the false alarms as an exponential tail above the threshold that grows longer and taller with period, because a long-period signal is two or three transits and two or three transits average away little of the non-Gaussian noise that makes false alarms. At short periods a candidate near threshold is nearly always real; at a year, a candidate near threshold is usually not.

At a year, most candidates are noise

A transit survey's threshold is set so that Gaussian noise yields less than one false planet in the whole catalogue. The noise is not Gaussian, and at year-long periods the candidates just above the threshold are more often artefacts than planets — so the rate of Earth analogues is partly a measurement of the noise.

 Giant planets per logarithmic interval of orbital distance, and who can see where. Giant planets (a third of Jupiter's mass to thirteen Jupiters) per star per unit of the natural logarithm of orbital distance, against distance, round Sun-like stars. The curve is a broken power law representative of the long radial-velocity surveys: rising as a^0.8 to a peak at 3 AU and falling beyond it as a^−1, which integrates to 2.7 per cent of stars inside 1 AU and 12 per cent in all. The shaded bands are where each method has the sensitivity to measure it: radial velocities out to about 7 AU, where two decades of monitoring cover one orbit, and direct imaging beyond about 10 AU, where a young giant can be separated from its star. The peak sits near the outer edge of the radial-velocity window, where a planet has completed one or two orbits in the data, and the fall beyond it lies entirely outside that window — so the peak's position is where the two methods have to be joined rather than a place either can measure alone.

A peak that neither survey can see alone

Giant planets become commoner with distance from their star out to a few astronomical units, and then rarer. The rise is measured by radial velocities and the fall by direct imaging, and the peak between them sits where neither method is comfortable.

 Four transiting planets, one star, and a density that must agree. The stellar density implied by each of four transiting planets of one simulated star, each fitted as though its orbit were circular, relative to the star's true density, against orbital period, both logarithmic, with one-sigma errors of 8, 10, 14, 20 per cent from the transit fits. Three planets have eccentricities of a few hundredths and agree with one another and with the truth (dashed line: their weighted mean, 0.950). The planet at 15.9 days has e = 0.2 with its periastron 70° from the line of sight, which makes it cross the star faster than a circular orbit would, and its implied density is 2.10 times the star's — 5.2 standard deviations from the others. A single planet's density can be reconciled with any star by choosing its eccentricity; several planets must share one star, and the odd one out is caught.

Every planet of a star must agree on its density

A single transiting planet gives the density of its star only if its orbit is known to be circular, and an eccentric orbit can disguise almost any star as almost any other. A star with four transiting planets has four such measurements of one number. They must agree — and the way they disagree measures how eccentric the orbits are, catches the candidate that orbits some other star, and pins down the star better than its own spectrum does.

 Two models through the same points, one with a planet. 60 simulated velocities over 200 days of a star rotating every 25 days, with spot activity drawn from a quasi-periodic process of 3 m/s amplitude and a 40-day decay, white noise of 1 m/s, and a planet of 2.5 m/s semi-amplitude on a 27.0-day orbit — 1.08 times the rotation period. Two models are fitted, each with its activity amplitude and decay time chosen by maximum likelihood: activity plus the planet (solid), whose planet comes out at 3.3 m/s, and activity alone (dashed), which has no planet in it. Both pass through the points; the likelihoods differ by 3.6 in the logarithm, a ratio of 34.9, which is not decisive. Near the rotation period a flexible model of the star can carry the planet's signal by itself.

A model of the star that can carry the planet

Starspots are modelled as correlated noise with the star's rotation period built in, and the model is good enough to remove signals that would otherwise be announced as planets. It is also good enough to remove planets. Near the rotation period no amount of data in one season separates the two, and away from it a flexible noise model quietly takes most of the evidence a planet would have had.

 The red velocity against the blue one. Simulated velocities of two stars, each measured simultaneously in a blue channel at 0.45 micrometres (across) and a red one at 0.8 (up), with 0.6 m/s of noise in each. One star (filled) has a 4 m/s planet on a 9.3-day orbit: its points lie along the diagonal, slope 0.97, because a Doppler shift is the same velocity in every colour. The other (open) has no planet and a spot 900 K cooler than its 5100 K photosphere, rotating every 12 days and evolving: its points lie along a shallower line, slope 0.64, which is the ratio of the spot's contrasts in the two channels. Neither star's blue velocities alone say which it is. The flux-deficit spot model is the simplest case; a spot whose signal is mostly suppressed blueshift lies closer to the diagonal.

A planet has one velocity in every colour

A Doppler shift moves every line in a spectrum by the same velocity, because it is the whole star that moves. A starspot does not move anything; it removes light, and how much light it removes depends on the colour it is seen in. Measuring the velocity in the blue and the red at once separates the two in a single night — until the star's magnetic field, which grows more visible towards the red, takes part of the separation back.

 Curvature turns a curve into an island. The companions — separation across, mass up, on edge-on circular orbits — that fit simulated velocities of a star with a 12 Jupiter-mass companion at 22 AU (period 103 years), within a Δχ² of 6.18 of the best fit, for observing spans of 6, 12, 24 years with 60 velocities of 2 m/s noise each. Over 6 years the data show a slope and the allowed region is a long band along M ∝ a², spanning 3.3 decades in mass. As the span grows the curvature of the orbit appears, and the band contracts — to 1.0 decades at 24 years — towards the true companion. Inclined orbits, which the fit leaves out, would extend every region to higher masses.

A slope that is a whole curve of companions

A companion on an orbit longer than the observations shows in a star's velocity only as a steady drift. A drift is one number, an acceleration, and an acceleration is a mass divided by the square of a distance — so a planet at ten astronomical units and a brown dwarf at fifty produce the same straight line. Turning the line into a companion needs the orbit to bend, a picture, or the other two components of the acceleration.

 The deuterium in the solar system's water, body by body. The ratio of deuterium to ordinary hydrogen in water, on a logarithmic axis, for the Earth's oceans and for the bodies it has been measured in, with the protosolar hydrogen and the water around young protostars as the two ends of the range. The oceans, 1.6·10⁻⁴, are 7.8 times the protosolar value; the carbonaceous and enstatite chondrites sit within ten per cent of them, as do two comets from the Jupiter family. The comets from the Oort cloud, and 67P from the Jupiter family, are two to three and a half times richer. Water made in a cold molecular cloud is richer still, near 0.001. The ratio is fixed when the water forms or last exchanges with hydrogen gas, and is not changed by anything that happens afterwards except escape, so it is a label that each body's water carries from wherever it was made.

Water that remembers how cold it was

Every water molecule in the solar system carries a label: the share of its hydrogen that is deuterium, fixed when the water last exchanged with hydrogen gas. Water made in the cold cloud the Sun formed from is rich in it; water that passed through the hot inner disc is poor. The Earth's oceans sit within ten per cent of the carbonaceous meteorites and twice below most comets — a record of where the water was assembled, and of how much of it had ever been hot.

 The mass at which a growing core cuts off the pebbles drifting past it. The pebble-isolation mass — the mass at which a core perturbs the gas around it enough to raise a pressure maximum outside its orbit, where drifting solids stop — against distance from the Sun, for a passively heated disc whose thickness grows as the distance to the 2/7, with α = 10⁻³. The mass goes as the cube of the disc's aspect ratio, because a planet has to push the gas by a fraction of its own pressure, and so it grows outward: 7 Earth masses at 1 AU, 30 at 5 AU where Jupiter's core formed, 138 at 30. A core reaching this mass stops growing from pebbles, starts accreting gas, and becomes a dam: everything drifting inward from outside its orbit is held there.

A core that shut the door on the pebbles

Solids in a gas disc drift inward, and a growing planet eats them as they pass until it is heavy enough to bend the gas around it. Then a pressure maximum forms outside its orbit, the drift stops there, and everything beyond is cut off from everything within. Jupiter's core reached that mass early, and the solar system's meteorites still fall into two families that never mixed.

 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.

 A star brighter at its poles, and three paths a planet might take across it. A star rotating at 0.6 of its critical angular velocity, seen with its spin axis tipped 30° towards the observer from the plane of the sky, drawn with that axis vertical. Its outline is the Roche surface, 1.05 times as wide as it is tall in projection. The contours are the surface brightness, from von Zeipel's law with exponent 0.2 and a linear limb darkening: gravity at the equator is weaker, so the equator is cooler and its intensity there is 81 per cent of the pole's before limb darkening. Three transit chords at the same impact parameter, 0.3 polar radii, cross at projected spin–orbit angles of 0°, 45°, 90°. Only the first is symmetric about the star's axis; the others run from bright regions to dark ones or the reverse, and the light curve records which.

A transit that runs differently backwards

A star spinning near break-up is brighter at its poles than its equator, so a planet crossing it at an angle covers bright surface on the way in and dim surface on the way out. The light curve stops being symmetric in time, and the part that differs forwards and backwards measures the true spin–orbit angle from photometry alone — on exactly the stars where the line-profile method struggles.

 The radius a rocky planet would have, by core fraction: at 5 M⊕ an Earth-like and a Mercury-like planet differ by 8%. Radius against mass for rocky planets of one to eight Earth masses, from Zeng and collaborators' interior models, for core mass fractions of 0, 0.33, 0.5, 0.7, 1 — from a planet with no iron core to one that is all iron. The shaded band is the range of core fractions planets would have if they kept their host stars' iron, magnesium and silicon in the stars' own proportions, for stars whose iron-to-magnesium ratio is within 0.15 dex of the Sun's: 0.26 to 0.41. The Earth, at 0.325, sits in it. At five Earth masses an Earth-like planet and one with a core fraction of 0.7, like Mercury, differ in radius by 8 per cent — a difference that transit and radial-velocity measurements can now resolve.

More iron than the star could give

A rocky planet is made of what its star was made of, minus the gases: iron, magnesium and silicon in the star's proportions, which for the Sun predict an iron core of a third of the planet's mass — and the Earth's core is 32.5 per cent. A handful of small exoplanets are so dense that their cores must hold seven-tenths of their mass, more iron than any star known to host planets could have supplied. Either something removed most of their rock after they formed, or the rock and the metal were separated before they did.

 Four rocky planets against the bare-rock limit: TRAPPIST-1b at 0.99, TRAPPIST-1c at 0.87, LHS 3844b at 1.01, GJ 367b at 0.99. Published day-side brightness temperatures of four small planets orbiting M dwarfs, each divided by the temperature its day side would have as dark bare rock re-radiating where it absorbs (dot, with its uncertainty), with the value for heat spread evenly over the whole planet marked for comparison (grey tick, at 0.71). TRAPPIST-1b: 503 ± 26 K against a bare-rock 508 K; TRAPPIST-1c: 380 ± 31 K against a bare-rock 434 K; LHS 3844b: 1040 ± 40 K against a bare-rock 1031 K; GJ 367b: 1728 ± 90 K against a bare-rock 1741 K. Three sit at the bare-rock limit within their errors, which leaves room for no thick atmosphere on dark rock. TRAPPIST-1c sits about one in eight below it, which a thin atmosphere could explain, and so could a moderately reflective surface. None comes near the even-redistribution value that a thick atmosphere would give.

A day side as hot as bare rock

A planet with no air can be no hotter on its day side than bare rock that re-radiates every photon where it lands. An atmosphere carries some of that heat to the night side, so the day side comes out cooler. For a rocky planet too small for its spectrum to be read, that difference is the test: three of the first four small planets around red dwarfs measured this way sit at the bare-rock limit, and the fourth sits below it by an amount that a thin atmosphere, or a brighter surface, could explain equally well.

 How much too small a planet looks when a second star shares its light. The factor by which a planet's true radius exceeds the radius inferred from its transit, against the mass of an unresolved companion star in units of the Sun's, for a planet transiting a Sun-like primary and for one transiting the companion, with the radius computed in both cases from the diluted depth and the primary's radius. Logarithmic vertical axis. A planet on the primary is diluted by the companion's light alone, and the factor is √(1 + F₂/F₁), never more than √2 = 1.41, reached for an equal twin; for a companion of half a solar mass, whose 3943 K surface gives 4.5 per cent of the primary's light in a Kepler-like band, it is 1.022. A planet on the companion is diluted by the primary's much larger light and measured against the wrong star's radius, and the factor is (R₂/R₁)√(1 + F₁/F₂): 2.6 for the half-solar-mass companion and 4.3 for one of 0.3 solar masses. The two curves meet at √2 for twins. The stars are a main-sequence sketch — power-law radii, a smoothed luminosity law and blackbody spectra — good to tens of per cent in flux ratio.

The Earth-sized bin is where diluted planets land

A star with an unresolved companion shares its aperture with that companion's light, and every planet found around either star is measured too small. The error runs one way only, so the smallest radius bin receives planets from every larger one and returns none. With a companion beside three stars in ten, more than a quarter of the planets called Earth-sized are something larger.

 The cadence that finds a planet and the cadence that weighs it. The fraction of caustic crossings — seven tracks across the planetary caustic at offsets spanning it, each sampled at 160 phases — in which a survey of a given cadence detects the anomaly (three points deviating from the single-lens curve by more than five times a 1 per cent photometric error, solid) and in which it also measures the source size (at least two of those points where the source's finite size changes the magnification by more than five per cent, which happens only during a fold crossing; dashed), for a Jupiter and an Earth around a third-of-a-solar-mass star. For the Jupiter, detection stays at 100 per cent at a four-hour cadence and 48 per cent at one day, while the source size is measured in 43 per cent of crossings at fifteen minutes, 15 at four hours and 0 at a day. For the Earth what limits the measurement is the track rather than the cadence: the source size is measured in 43 per cent of crossings at fifteen minutes and still 43 hourly — the tracks that cross a fold — while detection falls from 100 to 54 per cent, and at four hours both are 0. A cadence that finds giant planets reliably measures few of them; for small planets the cadence that finds them is already the one that measures them.

A cadence that finds a planet is not one that weighs it

A planetary anomaly on a microlensing event is found by a handful of points that deviate, and it is turned into a mass only by points taken while the source crosses the caustic — a few hours for a Jupiter whose anomaly lasts days. Sampled every four hours, every Jupiter crossing in a simulated survey is found and fewer than one in six is weighed; the field layout that finds the most planets is not the one that weighs the most.

 The heaviest atom a hydrogen wind can carry, against the strength of the wind. The crossover mass — the heaviest atom an escaping hydrogen flow drags out with it — against the hydrogen flux at the base of the flow, on a logarithmic axis, for the Earth's gravity and gas at 1000 K. Atoms lighter than the crossover leave with the hydrogen; heavier ones stay. The crossover rises in proportion to the flux: lifting neon takes 9.7·10¹² hydrogen atoms per square centimetre per second, argon 1.8·10¹³, krypton 4.3·10¹³ and xenon 6.6·10¹³. The Earth loses about 3·10⁸ today, which lifts nothing heavier than hydrogen. Driven by a young Sun at its saturated X-ray and ultraviolet output, an energy-limited flow would reach 2.2·10¹³ on Mars, a crossover of 114; 2.3·10¹³ on Venus, a crossover of 50; 10¹³ on the Earth, a crossover of 21 — the smaller planet's weaker gravity both raises the flux per unit area and lowers the force holding the heavy atoms back.

The heaviest atom a hydrogen wind can lift

A planet losing hydrogen fast enough to blow off as a wind drags other gases with it, but only those lighter than a crossover mass set by how strong the wind is. Below it every gas leaves in proportion to its distance from the limit and its isotopes are sorted by the same amount per mass unit; above it nothing goes. The Earth's xenon is sorted and its krypton is not, which no wind of neutral atoms can arrange.

 Two eccentricity distributions for the planets that transit alone. Two hypotheses for the orbital eccentricities of planets that transit their stars alone, as probability densities on a logarithmic axis. Solid: a mixture of two populations, 70 per cent at σₑ = 0.04 and 30 per cent at σₑ = 0.25 — a cold majority like the planets in multi-planet systems and a hot minority. Dashed: a continuum, each planet's Rayleigh scale drawn evenly in its logarithm between 0.015 and 0.34, chosen as the continuum whose densities most resemble the mixture's. The two differ most between eccentricities of 0.1 and 0.15, where the mixture dips between its two populations and the continuum does not: 5.6 and 12.0 per cent of planets respectively. A trough there is what a separate hot population would leave, and it is the feature a measurement has to find.

A gap in eccentricity a density can only just see

Planets that transit alone have more eccentric orbits than planets with transiting siblings, and the excess can be fitted as a separate hot population or as a smooth spread with no gap. The two eccentricity distributions look plainly different. The stellar densities through which they are measured look almost the same, and how many planets it takes to tell them apart is set less by the star than by the transit.

 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.

 One planet's helium line, around four kinds of star. The excess absorption at the centre of the 1083 nm helium line during transit, against the mass-loss rate, for one sub-Neptune — 26 Earth masses, 4.4 Earth radii, 0.0525 AU — losing an isothermal wind at 8,000 K, around four representative stellar spectra. Both scales logarithmic. The same outflow absorbs very differently: at 1 × 10¹⁰ g s⁻¹ the depth is 0.06%, 0.14%, 0.35%, 0.40% around an F star, a Sun-like star, an active K dwarf, an active M dwarf. Read the other way, a measured depth of 1% means 1.1 × 10¹¹ g s⁻¹ around an F star, 6.3 × 10¹⁰ g s⁻¹ around a Sun-like star, 3.0 × 10¹⁰ g s⁻¹ around an active K dwarf, 2.9 × 10¹⁰ g s⁻¹ around an active M dwarf — a factor of 4 between the extremes, from the star alone.

A helium line that weighs the star first

The metastable helium line at 1083 nanometres made the escape of a planet's atmosphere measurable from the ground. But the atoms it sees exist only in a balance between the star's extreme-ultraviolet light, which fills the level, and its mid-ultraviolet, which empties it — so the same outflow is six times deeper around an active K dwarf than around an F star, one depth allows a fifty-fold range of mass-loss rates across the wind's possible temperatures, and a rate inferred from it inherits a share of every error in a spectrum that cannot be observed.

 The same worlds, placed by the light that strips. The bodies of the shoreline plane, each moved from its present instellation to the cumulative X-ray and extreme-ultraviolet energy it has received over its star's life, both relative to the Earth's (arrows). The Sun's planets do not move, by construction. A planet of a small red dwarf moves up by a factor of 7 to 16: its star stayed at its saturated activity level — a thousandth of its light in X-rays and extreme ultraviolet — for 2.3 billion years for TRAPPIST-1 against 0.1 for the Sun, and its planets have absorbed that for longer. TRAPPIST-1 e, which receives 0.65 of the Earth's sunlight, has received 10 times the Earth's lifetime XUV and moves from well inside the line to above it. Every planet measured as bare, TRAPPIST-1 b, TRAPPIST-1 c, LHS 3844 b, now lies above the line. The saturation times are a scaling with stellar mass rather than a fit to any one star.

A shoreline measured in light already spent

In the solar system one line of slope four, in the plane of escape velocity against sunlight, separates every world with air from every world without. Drawn against the light a planet receives today it puts the temperate planets of a small red dwarf safely on the side with air. Drawn against the stripping light their star has already spent, it moves them an order of magnitude toward the airless side — and leaves most of them inside a band that the line's unknown height cannot resolve.

 A second peak, seen through three precisions. The distribution of measured core fractions for a population in which 90 per cent of planets sit at their stars' prediction (0.32, with an intrinsic scatter of 0.03) and 10 per cent at 0.65 (scatter 0.08), each planet compared with its own star, observed with measurement errors of 0.04, 0.08, 0.15 in core fraction. At 0.04 the dense group is a separate hump; at 0.08 it is a shoulder; at 0.15 it has become a long upper tail on a single peak, indistinguishable by eye from a population that is simply broad. The current measurements sit between the second and third curves.

A second peak the errors have to let through

Whether the densest small planets are a population of their own or the far edge of the ordinary one is not a question about any single planet. It is a question about the shape of a histogram seen through measurement errors as wide as the gap it is looking for, and the number of planets it needs rises fifteen-fold for every factor of three in lost precision — which puts the answer a few dozen planets beyond the samples that first suggested it.

 Where a ten-micron zodi comes from. How much each logarithmic interval of radius contributes to the ten-micron glow of a Sun-like star's exozodiacal dust, for blackbody grains in the Solar System's radial profile, seen face-on from 10 pc: with no instrument (dashed) and through a 14.4 m nulling interferometer whose fringes suppress the light near the star (solid, with the averaged transmission dotted). Without the null the glow is concentrated far inside the habitable zone at 1 AU (marked), because the grains' emission at ten microns rises steeply with their temperature: half of it comes from inside 0.34 AU. The null removes the innermost part and pushes the sensitivity outward, but half the nulled signal still comes from inside 0.69 AU, peaking at 0.74. A zodi level measured this way is a measurement of the dust between a few tenths of an AU and the habitable zone, extrapolated to the habitable zone with the Sun's profile.

A zodi measured in heat and needed in light

The dust around nearby stars that will limit the search for Earth-like planets in reflected light has been measured — but in its own glow at ten microns, by an interferometer whose fringes hide the dust nearest the star. Converting that measurement into the visible brightness at the habitable zone needs the dust's radial profile and its grains' albedo and temperature, none of which was measured. The same ten-micron zodi can mean a quarter or ten times the visible dust a coronagraph will see.

 An envelope worn down by a young star's ultraviolet. The mass of a primordial hydrogen envelope, as a fraction of the planet's mass, against age, for an Earth-mass planet in the middle of the eventual habitable zone of a 0.1 M☉ star, starting with 0.1%, 0.3%, 1%, 3% of its mass in hydrogen. The star contracts toward the main sequence and stays saturated in X-rays for 1.5 Gyr; the envelope inflates the planet, so it intercepts more of that light and loses mass faster than a bare rock would. The shaded span is the 75 Myr during which the planet receives more light than the runaway-greenhouse limit, when any exposed ocean would be steam. 0.1% is gone at 463 Myr; 0.3% is gone at 1.5 Gyr; 1% outlasts 5 Gyr; 3% outlasts 5 Gyr. An envelope that outlasts the shaded span has shielded the ocean through the whole of the runaway; one that also outlasts the planet's present age has left it a small gas-rich planet rather than a rocky one.

An envelope spent so an ocean could stay

A planet in the eventual habitable zone of a small red dwarf spends its first tens of millions of years too close to a star that is still contracting and still bright, with its ocean boiled into steam and the steam being split and lost. A wisp of hydrogen left over from the nebula — a few hundredths of a per cent of the planet's mass — would change that, because the star's ultraviolet would spend itself removing the hydrogen first. The envelope that shields the ocean and the envelope that never leaves bracket a range a factor of twenty wide, and the amounts a rocky planet is expected to capture fall inside it.

 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.

 More pairs on the target list than in the sky. The fraction of a magnitude-limited survey's targets that are unresolved binaries, against the fraction of stars in the volume that are, for companions with mass ratios spread evenly between a tenth and one. For solar-type stars, of which about 45 per cent have a companion, the target list holds 53 per cent binaries if the stars fill space uniformly, 50 if the survey reaches beyond the disc. The excess is modest on average because most companions are faint; it is concentrated in the near-equal pairs, which a magnitude limit over-represents by up to a factor of 2.8.

A target list that counts pairs twice

A planet survey chooses its stars by how bright they look, and an unresolved pair of stars looks brighter than either alone. A twin pair is seen from 2√2 times the volume a single star is, so a magnitude-limited target list holds more binaries than the Galaxy does, weighted towards exactly the near-equal pairs that dilute a transit most. The survey then counts its planets per target, and a target is not a star.

 An envelope that boils away in a million years, or keeps for longer than the universe. The time a 5-Earth-mass planet takes to lose an envelope of 5.0% of its mass through an isothermal wind driven by its own heat and the star's light, against the escape parameter Λ at the envelope's outer edge — the ratio of the gravitational binding of a molecule there to its thermal energy — for equilibrium temperatures of 500, 800, 1200 K, on a logarithmic time axis. The rate is exponential in Λ, so the time grows by a factor of e for every unit: each step of ten in Λ is more than four decades of time. The envelope goes in under a million years below Λ ≈ 19, 18, 18 and lasts longer than a billion years above Λ ≈ 26, 26, 26. The window between a million and a billion years is a band only about seven units of Λ wide, so a planet is either boiling or effectively sealed.

An envelope that boils down before the starlight strips it

The usual story of how small planets lose their hydrogen starts with the star's ultraviolet light and runs for a hundred million years. It skips the first million. When the gas disc that fed a young planet disappears, an envelope that was swollen and held in by the disc's pressure is suddenly exposed, and it pours off as a wind powered by ordinary heat — until it has shrunk to the size at which the same number near thirty that governs every escaping atmosphere shuts the flow off.

 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.

 A peak that follows the snow line inwards. Giant planets per star per unit of ln a against orbital distance, for stars of 0.3, 0.6, 1, 1.5 solar masses, if the peak's position scales as the 1 power of the star's mass — snow line, young star's luminosity. The shape is the broken power law measured round Sun-like stars, rising as a⁰·⁸ and falling as a⁻¹, with its height scaled in proportion to the star's mass as the giants' overall rate is. The peaks sit at 0.9 AU, 1.8 AU, 3.0 AU, 4.5 AU: a 0.3-solar-mass star would have its giants piled up 3.3 times closer in than the Sun's. The dots mark each peak.

A peak that should move with the star

Round Sun-like stars, giant planets are commonest at about three astronomical units, close to where water first froze in the disc. If the snow line put them there, the peak should sit closer in round smaller stars — three times closer round a star of a third of the Sun's mass. Every hypothesis agrees about the Sun, which is why the Sun-like stars cannot test them; and one of them makes a Jupiter at the peak wobble its star by the same angle whatever the star's mass.

 Where each test can name the host. The plane of a pair's separation and brightness difference, divided by which measurement can say which star a 1000-ppm transit crossed. Above Δm ≈ 4 a sub-Neptune on the companion is diluted below detection and there is nothing to assign. Imaging resolves the pair to the right of an illustrative contrast curve, Δm = 5 log(s/0.04″). The centroid's shift, read against the pair's separation and flux ratio, names the host to the right of s = σ(1 + f)/δ for a precision σ of 30 μas, about 0.05″ for near twins. The density from the transit's shape names it between Δm ≈ 1.1, below which the two stars are too alike, and the detection limit — at any separation, but only for strong transits on nearly circular orbits. The corner of close, nearly equal pairs, under about 0.04″ and 1.1 magnitudes, is reached by none.

A transit that says which star it crossed

When a planet transits one star of an unresolved pair, the catalogue usually assumes the brighter star. The light curve itself can say otherwise, in two ways: the transit's shape measures the density of the star crossed, and a red dwarf is three times denser than the Sun; and the centre of the pair's light moves during the transit, away from whichever star was dimmed. Each test works where the other fails, and between them they leave a corner of close, nearly equal pairs that no measurement of the transit reaches.

 Rise, steady state and decline. The mass of magnesium (solid) and iron (dashed) in a white dwarf's mixing layer, each as a fraction of its steady-state value for magnesium, during an accretion episode that delivers both at equal rates for 5 magnesium sinking times and then stops; time in units of the magnesium sinking time. Iron sinks 1.6 times faster. While they rise, both grow together and their ratio is the accreted ratio; once accretion has run for a few sinking times each element saturates at its rate times its own sinking time, so the layer holds 0.63 times the accreted iron-to-magnesium ratio; after accretion stops iron vanishes first, and 7 sinking times later the layer holds 0.009 of the accreted ratio. The same body seen at three moments reads as three compositions.

A planet read in the ashes of its star

A white dwarf's gravity drags every heavy element out of its visible surface, so a white dwarf showing magnesium, iron and silicon is swallowing rock now. Its spectrum is the only place where the bulk composition of a rocky body round another star can be read element by element. But each element sinks at its own rate, so the ratio seen depends on when in the meal the spectrum was taken — and only the hottest hydrogen white dwarfs say.

 The bands of sky from which each planet would be seen to transit. Ecliptic latitude against ecliptic longitude, with the vertical scale stretched 11 times: the band from which each of Mercury, Venus, Earth, Mars, Jupiter would be seen to cross the Sun's disc. The Earth's band lies along the ecliptic, ±0.266° wide. Mercury's is the widest, ±0.69°, and tilted by its 7° inclination; the giants' are narrow, ±0.051° for Jupiter. Each crosses the Earth's at the two nodes of its orbit, and only near those crossings can an observer see two planets transit. From nowhere on the sky can more than 3 be seen.

The stars that could catch the Earth in transit

The geometry that lets one Earth twin in two hundred be seen crossing its star works in reverse: the Earth crosses the Sun for observers in a band of sky half a degree wide, under half a per cent of the sky. About two thousand stars within a hundred parsecs sit in it. The band is fixed to the Earth's orbit and the stars are not, so the list changes on the timescale of recorded history — and from no direction can more than three of the eight planets be seen transiting at once.

 A shift in the time and a change in the length, a quarter-cycle apart. The transit timing shift and duration change of a planet of 0.5 Jupiter masses in a 287-day orbit, carrying a moon of 10 Earth masses at 20 planetary radii, against the moon's orbital phase at the moment of transit, for two cycles. The timing swings by ±42 minutes and the duration by ±10.1 minutes. The moon's period is 12.9 days. Timing is largest when the moon is farthest along the track, and duration when the moon is moving fastest along it, so they are in quadrature; a perturbing planet would produce nothing of that kind in the duration.

A moon read from how late and how long

A moon pulls its planet round their common centre, and at each transit the planet is somewhere slightly different along that small orbit, moving slightly differently. So the transit arrives early or late, and it lasts longer or shorter. The two effects depend on the moon in different ways and they run a quarter of a cycle apart. Neither alone says anything about the moon. Their ratio gives its period, which a series of transits sampled once an orbit cannot otherwise reach.

 Water that is on the star and not on the planet. Excess transit depth, in parts per million, against wavelength, for a planet of depth 7000 ppm around a star of 2600 K. The upper curve is a bare rock with no atmosphere on a star whose disc is 10% covered by 2300 K spots that hold more water than the rest of the photosphere: it shows a broad slope and a band near 1.4 μm, 36 ppm from its shoulder. The lower curve is the signal of an Earth-like air column with water on an Earth-sized planet, about 93 ppm at the band. The band in the spot spectrum is an imposed illustration rather than a model atmosphere, but its sign is measured: it adds where a planet's own water would add.

Water that is on the star and not the planet

A transit depth is the planet's shadow divided by the brightness of the whole star, and the shadow falls on only one strip of it. If the strip is cleaner than the rest of the disc, the depth comes out too large, by an amount that changes with colour. The star's own spots hold water, so the error has water's fingerprint. On the smallest red dwarfs, whose rocky planets are the best targets there are, that false signal is larger than the real atmosphere being looked for.

The ladders in this field

16 anchors · one idea each

MicrolensingTransitsReflex velocityPlanet compositionDirect imagingDetection biasOccurrence ratesRadius valleyTransit timingPlanet migrationResonanceExoplanet atmospheresHabitable zoneSpin–orbit alignmentAtmospheric escapeThe snow line

All fields · All essays