Astronomy & Celestial Mechanics
the sky, computed
Concept

Aberration — where it appears

The displacement of every star towards the direction of the observer's motion, of size v/c, independent of how far away the star is. It is a property of the observer rather than of the star, and its annual ellipse is the same size for every object on the sky, which is what distinguishes it from parallax.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

One path, five numbers. Left: the apparent path of a star over 4 years, with a proper motion of 193 mas a year and a parallax of 50 mas, at ecliptic latitude 42°. It is one curve and there is nothing in the sky it can be compared against — the reference stars have paths of their own. Right: the same path with a straight line taken out of it. What is left is an ellipse of semi-major axis 50.0 mas and semi-minor axis 33.5 mas, closed and repeating once a year. The two are separated by their time signatures and by nothing else: proper motion is secular and parallax is annual, in a phase the Earth's position fixes in advance. That is why the five parameters can be told apart at all, and why an astrometric catalogue quotes five rather than two — a position without them is a position at one instant, which is not a direction to anything.

Five numbers from one wiggle

A star's path across a plate is a straight line with a one-year ellipse laid on it. Nothing measures either alone — one fit yields five parameters at once, and they are separable only because their time signatures differ.

sky · Parallax 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 Three passes, and the fourth is below the noise. How far the position is still wrong after each pass of the light-time iteration, for five targets, on a logarithmic scale of kilometres. The first pass uses the body's position now and is wrong by exactly the distance the body travels while its light is in transit — 6254 kilometres for Mars near opposition, which is about twenty-five arcseconds and larger than any residual in an ephemeris fit. Each further pass multiplies the error by the body's speed divided by the speed of light, so the lines are straight and their slopes are that ratio and nothing else. Three passes take every one of these below a metre, which is why the loop in an ephemeris code has a fixed trip count rather than a convergence test. The same argument run backwards is why the correction cannot be applied to the observation instead: the direction light arrived from is what the observation is, and the position it corresponds to is not known until the body's orbit is.

Where a planet is and where it is seen

An ephemeris is fitted to observations, and no observation is of a position. It is of a direction light arrived from, at a time that is not the time the light left, bent by a Sun that is nowhere near the line of sight. Three corrections stand between the two, and all three are larger than the residuals.

orbits · Ephemerides A dipole 187 times the signal, and its own harmonics under it. The amplitude of each harmonic of the observer's own motion imprinted on the microwave sky, against the anisotropies of the sky itself. Moving at 369.8 kilometres a second through a blackbody field makes it hotter ahead and cooler behind by a fraction β = v/c, giving a dipole of 3.36 millikelvin — 187 times the 18 microkelvin anisotropies. Each further harmonic is smaller by another factor of β, so the kinematic quadrupole is 4.15 microkelvin, which is comparable to the real quadrupole and has to be subtracted separately. The dipole is not a nuisance in one respect: it is the measurement of the solar system's motion with respect to the radiation, and it is the most precisely known velocity in astronomy.

A dipole a hundred times the signal

The largest structure in the microwave sky is the observer. Moving through a blackbody radiation field makes it hotter ahead and cooler behind by three and a third millikelvin — nearly two hundred times the anisotropies that all of cosmology is read from — and removing it is the first operation on any map.

cosmology · Microwave background The dipole a 370 km/s motion has to put into counts of distant sources. The amplitude of the dipole in the number of sources per unit solid angle expected from the Sun's motion at 369.82 km/s relative to the microwave background, β = 1.234e-3, split into its two parts: aberration, 2β, which crowds sources towards the direction of motion, and the Doppler boost, x(1 + α)β, which brightens sources there and lifts fainter ones above the flux limit. For radio sources, with counts steepening as S^(−1) and spectra falling as ν^(−0.75), the expected dipole is 0.0046; for mid-infrared quasars, with counts steepening as S^(−1.7) and spectra falling as ν^(−1.26), the expected dipole is 0.0072, and the measured one is 0.0155 — 2.16 times larger, which would need a speed of 797 km/s. A dipole of a few parts in a thousand needs a catalogue of more than a million sources to see at all. The disagreement is not with the direction, which lies close to the microwave background's, but with the size, and it is not yet explained: either the samples carry a systematic nobody has found, or the matter and the radiation do not share one rest frame on these scales — in which case the assumption that the universe looks the same from everywhere is wrong in a way it has never been caught being wrong before.

The Sun's speed counted in quasars comes out twice too large

The microwave background is warmer in one direction by a part in a thousand, and that dipole is read as the Sun's motion through it at 370 kilometres per second. The same motion must crowd the counts of distant galaxies and quasars towards the same direction by an amount that can be calculated exactly. When a million quasars were counted, the direction agreed and the size came out more than twice too large — as though the Sun were moving at 800 kilometres per second relative to the matter.

cosmology · Expansion

Named alongside it

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

AstrometryProper motionReference frameCMB dipoleParallaxPeculiar velocityQuasarSolar apexAnisotropyApparent placeAstrometric placeAstrometric solution

All concepts