Astronomy & Celestial Mechanics
the sky, computed
Concept

Transit depth — where it appears

The fraction of a star's light a transiting body removes, equal to the square of the ratio of radii. It is a ratio and therefore immune to most calibration errors, which is why transit photometry reaches parts per million while absolute photometry does not.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

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

A planet measured by the light it removes

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

exoplanets · Transits The 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.

exoplanets · Atmospheric escape A transit depth of 1.200 per cent for a planet of area 1.000 per cent. Three transits of the same planet across the same star, differing only in how the star's brightness falls toward its edge. A planet of radius ratio 0.1 covers 1.000 per cent of the stellar disc's area, and if the disc were uniformly bright that would be the depth. It is not uniformly bright: a sight line near the limb leaves the photosphere at a shallow angle and therefore from a cooler layer, so the edge is dimmer than the centre, and a planet crossing near the middle blocks light that is brighter than average. The transit drawn with realistic coefficients is 1.200 per cent deep — 20 per cent deeper than the area — and it is also rounder, because the covered brightness changes through the crossing instead of staying flat. The consequence is stated in the numbers beside the curves. Each is a least-squares fit of the radius ratio to the realistic curve, performed with a different assumed limb-darkening law, and the recovered radius moves by up to 3.6 per cent depending on which law is assumed. Fitting with the law the curve was made from returns the input to five figures, which is the control: the bias is the mis-specification and not the fitter. Since the coefficients come from a model atmosphere rather than from the light curve, every published planetary radius carries a systematic from stellar physics that no amount of photometric precision removes — and it is the dominant one for the best-measured planets. The picture holds the impact parameter fixed; a grazing transit is worse, because it samples only the limb, where the disagreement between laws is largest.

The depth is not the area

A planet covering one per cent of its star's disc does not make a transit one per cent deep. The star is brighter in the middle, so a planet crossing the middle blocks more than its share — and the correction depends on coefficients that come from a stellar atmosphere model rather than from the light curve.

starlight · Limb darkening 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.

exoplanets · Detection bias 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.

exoplanets · Detection bias Where the photons stop mattering and the star starts. Noise in a 13-hour window — the length of an Earth-like planet's transit across a Sun-like star — against the star's brightness, for a space telescope that reaches 17 parts per million in 6.5 hours on a twelfth-magnitude star. Photon noise falls by a factor of ten for every five magnitudes of brightness; the star's granulation, averaged over the window, stays at 5.1 parts per million whatever the brightness. The two are equal at magnitude 10.1; brighter than that, the star is the floor. The dashed line is an Earth's transit depth across the Sun, 84 parts per million: a single transit on a twelfth-magnitude star is detected at 6.4 standard deviations, and on the brightest stars at no better than 17, set by the star alone — and supergranulation, spots and faculae add more on these timescales.

A floor the star sets for itself

Above the atmosphere, with comparison stars and a big enough telescope, every noise in a measurement of a star's brightness can be beaten down except one: the star's own surface, a mosaic of convective cells each brighter or fainter than the rest. That flicker does not average away against anything, and it is larger for stars that are bigger and puffier than the Sun. For the brightest stars, an Earth's transit is limited by the star, not the photons — and the stars Kepler watched turned out twice as noisy as the Sun.

starlight · Photon noise 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.

exoplanets · Detection bias What a hundred kelvin moves, for a star like the Sun. The change in eight quantities derived for a Sun-like star and its planet when every temperature is raised by 100 K — 1.7% — as the uncertainty in the absolute temperature scale allows, against the precision each is quoted to (grey bars). The star's radius from its flux and parallax falls by 3.4%, because radius goes as the inverse square of temperature; the planet's radius follows, and its density rises by +12.8%. The oscillation-based mass rises by +2.6%, the main-sequence mass by +3.5% and the age falls by 9.8%. The iron abundance read from neutral iron lines rises by 0.08 dex, +20.7%. In every case the shift is comparable to or larger than the precision quoted, and in every case it moves all stars of a sample together.

A hundred kelvin under every radius

The temperatures of cool stars are quoted to tens of kelvin, and the scale they are all measured on is uncertain by about a hundred. That hundred is under almost everything else: a star's radius moves by twice its fractional error, a transiting planet's density by six times, a main-sequence age by ten per cent and an iron abundance by a fifth. And because a scale error moves every star of a sample the same way, no amount of averaging removes it.

starlight · Stellar colour 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.

exoplanets · Detection bias

Named alongside it

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

Occurrence rateDilutionSelection effectBinary starsDetection thresholdImpact parameterLimb darkeningPhotometryPhoton noiseRadius ratioRadius valleySignal-to-noise

All concepts