Astronomy & Celestial Mechanics
the sky, computed
Concept

Albedo — where it appears

The fraction of incident light a surface reflects rather than absorbs. It enters an equilibrium temperature to the quarter power, so a factor of two in it is only a fifth in the answer, which is why a habitable zone is quoted as a range rather than a boundary.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

7 clocks, and a body 233 kilometres across. A stellar occultation reduced. Each horizontal segment is one observer's chord: the star vanished, the star came back, and the interval multiplied by the shadow's 21.4 km/s across the ground is the length drawn. The longest, at an offset of -28 km, is 252.3 kilometres. The dashed ellipse is the limb fitted to the chords by least squares in the half-length squared, and its equivalent-area diameter is 233.9 km against the silhouette's true 232.9 — the residual is the shape the fitted ellipse cannot hold, not an error in any timing. The two open marks are observers inside the predicted path who saw nothing, and they are measurements: they bound the limb inside their own offsets, which is what fixes the extent when the positive chords all fall on one side. At 0.12 s per contact the length precision is 2.6 km, 1.10 per cent of the body — a size measured with a clock rather than with an angle, on an object no telescope resolves.

A shape measured by the edge of a shadow

When a small body passes in front of a star the observable is two times. Multiply the interval by the body's speed across the sky and it becomes a chord across a silhouette no telescope can resolve — and enough chords give a profile to a kilometre or two, for an object a few hundred across.

sky · Occultations 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.

exoplanets · Direct imaging 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.

exoplanets · Habitable zone 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.

exoplanets · Exoplanet atmospheres Drift against thermal inertia: a peak at Γ = 93, in the same place for every size. How fast an asteroid's orbit drifts under its own re-radiated heat, against the thermal inertia of its surface, at a rotation period of 4.3 hours and 1.13 astronomical units. Both axes are logarithmic, and the curves are four diameters. The non-monotonic shape is the content. A surface that conducts nothing re-radiates its heat the instant it receives it: the emission is then symmetric about the sub-solar point and the transverse push cancels exactly. A surface that conducts perfectly is isothermal, has no temperature contrast at all, and again pushes nowhere. The force lives between those two nothings, and peaks where the surface's thermal time constant is comparable to the rotation period — here at Γ = 93 in SI units, and at the same place on every curve, because the size scales the drift without moving the optimum. That separation is what makes the effect a measurement. A drift rate on its own is a single number with several unknowns in it; a drift rate together with a size from radar, a spin from a light curve and a density from a flyby leaves the thermal inertia as the only thing not measured, and solving for it says what the surface is made of. Fine dust sits near 50, bare rock in the thousands, and the values measured for the bodies spacecraft have visited — Bennu at 310, Ryugu at 225, Itokawa at 700 — straddle the peak, with the two rubble piles a factor of two or three above it and the Moon's dust well below. Being past the optimum is not a small effect but it is a gentle one: the curve falls as one over the thermal inertia on that side, so a surface three times more conductive than optimal still drifts at a third of the best rate, while one three times more insulating drifts at a third as well. The shape is symmetric in the logarithm, which is why the measurement is a good one for telling dust from pebbles and a poor one for telling pebbles from boulders. The curve is one-dimensional linear theory for a rotating half-space: it has the right limits and the right peak, and it omits the body's shape, which for an irregular asteroid changes the answer by tens of per cent.

A drift rate that says what the surface is made of

The thermal recoil that moves an asteroid's orbit depends on how long its surface holds heat, and the dependence is not monotonic — a perfect insulator and a perfect conductor both push nothing. The peak in between means a measured drift is a measurement of thermal inertia, which is a measurement of grain size.

orbits · Non-gravitational forces 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.

exoplanets · Direct imaging What colour removes, and what it can only estimate. The share of a synthetic family's linked membership that is not part of the family, against the clustering cutoff, with the parts a colour test can and cannot see. The true interloper fraction rises with the cutoff, from 0 per cent at 20 m/s to 84 per cent at 110. A colour cut removes the interlopers of the other type — 12 per cent of the list at 60 m/s — and leaves those of the family's own type, 6 per cent, which can only be estimated by scaling the detected ones by the background's mix. That estimate (dashed) tracks the truth on average and scatters about it, because a few dozen interlopers are a few dozen random draws.

A family checked by a colour it was not chosen by

Asteroid families are found by clustering orbits, and the clustering absorbs background asteroids that happen to orbit nearby. Fragments of one body share its composition, and the survey that found them measured their colours for free — so colour is a second membership test the clustering never used. It removes the interlopers that look different, and it can only estimate the ones that look the same.

orbits · Asteroid families

Named alongside it

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

CoronagraphHabitable zoneTidal lockingYarkovsky effectAdaptive opticsAngular separationAsteroid familyAstrometric predictionBlackbodyBrightness temperatureThe central flashClimate model

All concepts