Astronomy & Celestial Mechanics
the sky, computed
Series

Transits — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · exoplanets
  2. 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.

    part 2 · exoplanets
  3. 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.

    part 3 · exoplanets
  4. 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.

    part 4 · exoplanets
  5. 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.

    part 5 · exoplanets
  6. 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.

    part 6 · exoplanets

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