Astronomy & Celestial Mechanics
the sky, computed
Concept

Centroid — where it appears

The brightness-weighted centre of light of an image or a blend of sources. During a transit or an eclipse it moves towards the star that was not dimmed, so its shift names which of two unresolved stars was crossed.

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

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.

exoplanets · Microlensing 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.

Angular einstein radiusAstrometric microlensingAstrometryBinary starsBlack holeDegeneracyDilutionEccentricityImage multiplicityM dwarfMicrolens parallaxMicrolensing

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