Astronomy & Celestial Mechanics
the sky, computed
Concept

Solar apex — where it appears

The direction on the sky towards which the Sun is moving relative to a chosen reference. Aberration displaces every source towards it, and because the Sun's velocity is not constant that displacement drifts rather than repeating annually.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The same ellipse at 955 times the size, and a quarter of a year out of step. The aberration ellipse (solid) and the parallax ellipse (dashed) for γ Draconis at four ecliptic latitudes, drawn to one scale. Aberration is v/c towards the Earth's own direction of travel, so its ellipse has semi-major axis 20.49551″ for every star in the sky; parallax is 1/d towards the Sun, so its semi-major axis is that star's own 0.02147″ — 955 times smaller, and drawn 955 times smaller here. At the ecliptic pole both are circles; on the ecliptic both collapse to lines; in between both are ellipses of semi-minor axis sin β times the major, and the ratio is identical at every latitude because both effects project the same way. The marks are the same four dates on each: they are a quarter of a year apart between the two, because the aberration displacement follows the Earth's velocity and the parallax displacement follows its position, and velocity leads position by 90° on a circular orbit. That quarter-year is the only thing distinguishing the two phenomena on the sky, and it is why Bradley, looking for the dashed ellipse in 1728, spent months unable to interpret the solid one he had found instead.

The other twenty arcseconds

Every star in the sky traces a small ellipse over a year, of the same shape as its parallax ellipse and 90° out of step with it — and the same size for all of them, near or far. Bradley found it in 1728 while hunting for parallax, and it proved the Earth moves a century before anything's distance was known.

sky · Aberration Every quasar in the sky streaming at 5.23 µas a year towards one point. Above: the apparent proper motion of distant quasars, drawn in Galactic coordinates with the centre of the Galaxy at the origin. Quasars do not move — at their distances a real transverse velocity of a thousand kilometres a second would be a hundredth of a microarcsecond a year — so a pattern in their apparent motions is a statement about the observer. Annual aberration displaces every source by v/c and returns it a year later; the Sun's velocity is not constant, and a changing displacement does not return. The Sun is being accelerated towards the centre of the Galaxy at 2.4·10⁻¹⁰ m s⁻², so the aberration vector rotates at a/c and the whole sky streams towards the same point, at (a/c) sin θ for a source θ from it. Below: that amplitude against angle from the apex, with the fitted dipole and the measurement. A circular speed of 248 km s⁻¹ at 8.28 kiloparsecs predicts 5.23 microarcseconds a year; the measured dipole in the proper motions of 1.6 million quasars is 5.05 ± 0.35, pointing to within a few degrees of the Galactic centre. The picture cannot show what took so long: the effect is a twenty-thousandth of annual aberration, it accumulates over the whole mission rather than over a year, and it is degenerate with any real rotation of the quasar frame — so a measurement of the acceleration of the solar system is also, unavoidably, an assumption that the distant universe does not turn.

The whole sky drifting towards one point

Annual aberration is the Earth's velocity, and it closes every year. The Sun's velocity is not constant, so the same effect leaves a residue that never closes — every quasar in the sky creeping towards the Galactic centre at five microarcseconds a year, which is a direct measurement of the Sun's acceleration.

sky · Aberration The Sun's motion, written across a group of stars. Proper motions of 400 stars all at 300 pc, against their angle from the solar apex, each with its own random velocity of 25 km/s in every direction. Filled dots are the component along the great circle away from the apex, υ; open dots are the component at right angles to it, τ. One star's υ is dominated by its own motion — the scatter is 17.6 mas/yr against a reflex of at most 13.7 — so no single star gives a distance. Together the υ components follow the Sun's reflex, 19.5 km/s times sin λ, and the fitted amplitude, 13.70 ± 1.06 mas/yr, gives a distance of 300 pc ± 8 per cent against the true 300 pc. The τ components scatter about zero at every angle, because the Sun's motion has no part across the great circle.

A baseline that grows four astronomical units a year

The Earth's orbit gives a parallax a baseline of two astronomical units and never more. The Sun's own motion through the nearby stars gives one that lengthens by four a year without limit — but the stars move too, by as much, so the long baseline yields a distance only for a group whose motions average away, and for a century it was the only way to reach past a few hundred parsecs.

starlight · Parallax

Named alongside it

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

Proper motionAberrationAstrometryParallaxReference frameCelestial coordinatesCepheidDipoleDoppler effectEclipticGalactic accelerationGalactic rotation

All concepts