Astronomy & Celestial Mechanics
the sky, computed
Series

Aberration — the series

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

    part 1 · sky
  2. 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.

    part 2 · sky
  3. Where every star appears to an observer moving fast. The apparent angle of a star from the direction of motion, against its angle in the rest frame of the stars, for observers moving at 0.3c, 0.6c, 0.9c. The diagonal is a stationary observer. Every star except the two on the line of motion is displaced forward, and the displacement grows with speed until, at 0.9c, a star at right angles to the motion appears 26° from the direction of travel and one directly behind still appears behind. A star at a true angle of 90° is seen at the angle whose cosine is β — 72.5°, 53.1°, 25.8°. At the Earth's orbital speed, ten thousandths of c, the curve is indistinguishable from the diagonal, and the whole of annual aberration is its first-order departure from it.

    The sky crowded forward by the observer's speed

    Annual aberration is the first-order term of a law that has no higher terms Bradley could have seen. Written exactly, it maps the whole sky onto itself, crowding every star towards the direction of motion — by half a milliarcsecond at the Earth's speed, which a modern astrometric catalogue must carry, and at half the speed of light by enough to put three-quarters of the stars in the forward half of the sky.

    part 3 · sky
  4. The Sun bends light from every direction in the sky. The gravitational deflection of starlight by the Sun, as seen from the Earth, against the angle between the star and the Sun, as general relativity predicts it (solid) and as a Newtonian particle of light would be bent (dashed), exactly half. At the Sun's limb the deflection is 1.75 arcseconds, the value measured at the eclipse of 1919. It falls only as the cotangent of half the angle, so it is 4.07 milliarcseconds at right angles to the Sun and 9.8 at 45°, the closest a space astrometry mission looks; it reaches zero only directly opposite the Sun. Every star in a modern catalogue has been displaced by an amount far larger than the precision of its position, and the shading marks the angles such a mission actually observes.

    Every star in the catalogue is bent by the Sun

    The bending of starlight by the Sun is remembered as an eclipse measurement of 1.75 arcseconds at the solar limb. The same bending reaches every direction in the sky, falling only slowly away from the Sun, and at right angles to it is still four milliarcseconds — two hundred times the precision of a modern star position. Every catalogue corrects every star for it, and the correction has become the most precise test of how much space a mass curves.

    part 4 · sky
  5. A reflector made worse so the light lands where it is wanted. The intensity returned by a 3.8 cm fused-silica corner cube against angle from the exact retro-direction, on a logarithmic scale relative to a perfect cube's peak, at 532 nm. A perfect cube returns an Airy pattern whose central lobe is 17 microradians in radius. Offsetting each of its three dihedral angles by 1.25 arcseconds splits the return into six beams, each from a sixth of the face, deflected by (4√6/3)·n·ε = 29 µrad. In the band of offsets a LAGEOS station sees, 32–38 µrad (shaded), the spoiled cube returns 10 times as much light as the perfect one, along the direction of each lobe. Each lobe is drawn here as the Airy pattern of a circular aperture of a sixth of the area, which is an approximation to the true sector shape.

    A reflector built wrong so the light comes home

    A corner-cube reflector sends light straight back the way it came — in its own frame. On a satellite crossing the sky the return is aberrated by twice the satellite's sideways speed over the speed of light, fifty microradians for a low orbit, which is further than a small perfect cube spreads its light. So the cubes on geodetic satellites are made imperfect on purpose, their angles offset by about an arcsecond to throw the returning beam out to where the telescope will be.

    part 5 · sky

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