Astronomy & Celestial Mechanics
the sky, computed
Generator

The aberration-ellipse generator

The same ellipse at 955 times the size, and a quarter of a year out of step
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 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.

8 essays call aberration-ellipse. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. The observed sky

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.

 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 observed sky

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.

 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 observed sky

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.

 Parallax for a star at 1.3 parsecs. The same star observed from two ends of a baseline. The two sight lines are 1.54″ apart, so the parallax — half of that, the shift seen from a one-astronomical-unit baseline — is 0.769″ for a star 1.3 parsecs away. The definition of the parsec is the distance at which it would be exactly one. Starlight

The triangle that reaches the stars, and stops

Parallax is the only distance measurement in astronomy that assumes nothing. It is also the only one with a hard ceiling, and everything beyond that ceiling rests on it.

 A frame whose precision stops improving because its sources move. The uncertainty in the orientation of a celestial reference frame, in microarcseconds, against the number of extragalactic sources it is built from. The falling line is what averaging alone would give: each source's position is measured to about 200 microarcseconds and combining N of them improves the frame as one over the square root of N. The upper curve adds the part that does not average down in the same way — the wander of a quasar's radio centroid as new components are ejected along its jet, which is a real motion of the thing being used as a fixed point. Three catalogue generations are marked. The frame gained an order of magnitude in a quarter of a century, and it gained it by observing more sources rather than by observing any of them better, which is the signature of a limit that is in the objects rather than in the instrument. The observed sky

A frame made of things that are not points

Every position in astronomy is measured against a set of objects declared to be fixed. The objects chosen are quasars, because they are the most distant things there are — and each of them is a jet whose radio brightness centre wanders by tens of microarcseconds as new material is ejected from the core.

 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 observed sky

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.

 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. The observed sky

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.

 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. The observed sky

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.

The whole library · All essays