Astronomy & Celestial Mechanics
the sky, computed
Concept

Astrometry — where it appears

The measurement of positions and motions on the sky, in which everything interesting is a residual left after much larger effects have been removed. Its precision is set by how well a source's centroid can be located rather than by the resolution, so a sharper image is a better position on the same photons.

Named by 19 essays across 7 fields — each of them below, with the objects they name alongside it.

Ceres from five directions and no distance. Ceres seen five times over 41 days, from an Earth on a circular orbit, reduced in the plane. Each sighting gives a direction and no range, so the object is somewhere on its sight line; the five lines here span 1.37° of geocentric arc altogether, and Earth's own motion supplies the only baseline there is — 0.403 AU of its 0.691 AU of travel lies across the sight lines. A planar orbit is four numbers, so five angles over-determine it and one orbit comes out: a = 2.7658 AU, e = 0.0785. That is the answer and not an input — the sightings were generated at a = 2.7658 AU and e = 0.0785, and the solve, which sees only the directions and the dates, returns them to 3e-12. The two shaded sectors are what closes the determination: between the first and middle sightings the radius vector sweeps 0.2993 AU² in 21.0 days and between the middle and last 0.2853 AU² in 20.0 days, a ratio of 1.04918 against a time ratio of 1.04918. Slide all three crossings out along their sight lines together and that equality fails at once, so it fixes the distance by itself, with no propagation anywhere in the argument — and it gives a = 2.7658 AU over again. The two dashed curves are candidates that thread the same three sight lines at 85% and 108% of the recovered distance: a = 1.86 AU at e = 0.35, sweeping its areas in a ratio 4.5% wrong; and a = 5.61 AU at e = 0.49, sweeping its areas in a ratio 2.9% wrong. A few per cent in the distance is an orbit of another kind, which is the same fact the conditioning panel measures: one arcsecond of angle error moves a by 0.60% on this arc.

Five directions and no distance among them

An image of a moving point records an angle and throws the range away, so an orbit has to be assembled out of angles alone. How many angles are needed is not a detail of the method — it is the whole of what a determination is.

orbits · Orbit determination One admissible root, 0.01% from the truth. Gauss's reduction of three directions to a distance, drawn as the two relations whose intersection it is. Three observations of Ceres on days 0, 20, 40 of an arc, generated from its elements and used only as sight directions — no range, no radial velocity. The rising curve is geometry: the heliocentric distance a candidate at geocentric distance ρ₂ would have, r₂² = ρ₂² + 2ρ₂(R₂·L̂₂) + R₂², which contains no dynamics at all. The falling curve is dynamics: ρ₂ = A + µB/r₂³, with A and B built from the three sight vectors, the three observer positions and the three times, and containing no orbit. Eliminating ρ₂ between them gives r₂⁸ + a r₂⁶ + b r₂³ + c = 0 — an eighth-degree equation, from a problem with exactly as many equations as unknowns. Here they cross once at a positive ρ₂, at r₂ = 2.5893 AU against the true 2.5890. The other 2 real roots are rejected not by fitting but by sign: the ρ₂ each implies is negative, and an object behind the observer was not the thing observed.

Three observations and no orbit at all

Three directions in space give six numbers for the six elements of an orbit, which sounds like a solved problem. The algebra that solves it is of the eighth degree, and for a near-Earth asteroid three perfect observations can be consistent with three different orbits.

orbits · Orbit determination Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller.

Neither body is still, and the wobble is how planets are found

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.

gravitation · The two-body problem One star, two coordinate systems, at latitude 52°. The equatorial grid and the horizon grid drawn on the same sphere for an observer at latitude 52°. The star marked has declination 20° and hour angle -40° in the first, and altitude 45.5° and azimuth 239.4° in the second. The two frames differ by a single rotation through the co-latitude 38°, which is why the celestial pole stands 52° above the northern horizon.

Where a star is depends on who is asking

The sky needs two coordinate systems because two different things stay still in it — the observer's horizon and the stars themselves. One rotation converts between them, and the angle of that rotation is the time.

sky · Celestial sphere 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 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.

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.

starlight · Parallax 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 A square field imaged as a squashed one. A square grid of stars 1° across, centred 70° from the zenith, and how refraction moves each one relative to the field's centre after the displacement common to the whole field has been removed. The arrows are magnified 200 times. Refraction lifts every star towards the zenith, but it lifts the lower ones more because refraction grows towards the horizon, so the field is compressed vertically by 2377 parts per million; and every star moves along its own vertical circle, which converge at the zenith, so the field is compressed horizontally by 286 parts per million. The corners move by up to 4.4 arcseconds against the centre — a hundred times the precision of a modern relative astrometric measurement, and a distortion that changes as the field crosses the sky.

A square field imaged as a squashed one

Relative astrometry measures positions against neighbouring stars so that the atmosphere's refraction, common to them all, cancels. It does not cancel. Refraction changes across the field, so a square patch of sky seventy degrees from the zenith is imaged squashed by a few parts in a thousand — and stars of different colour are moved by different amounts, by more than the parallaxes being measured.

sky · Refraction 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.

sky · Aberration Curvature turns a curve into an island. The companions — separation across, mass up, on edge-on circular orbits — that fit simulated velocities of a star with a 12 Jupiter-mass companion at 22 AU (period 103 years), within a Δχ² of 6.18 of the best fit, for observing spans of 6, 12, 24 years with 60 velocities of 2 m/s noise each. Over 6 years the data show a slope and the allowed region is a long band along M ∝ a², spanning 3.3 decades in mass. As the span grows the curvature of the orbit appears, and the band contracts — to 1.0 decades at 24 years — towards the true companion. Inclined orbits, which the fit leaves out, would extend every region to higher masses.

A slope that is a whole curve of companions

A companion on an orbit longer than the observations shows in a star's velocity only as a steady drift. A drift is one number, an acceleration, and an acceleration is a mass divided by the square of a distance — so a planet at ten astronomical units and a brown dwarf at fifty produce the same straight line. Turning the line into a companion needs the orbit to bend, a picture, or the other two components of the acceleration.

exoplanets · Reflex velocity 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.

sky · Aberration The path of a star with a dark companion: a 2.66 mas orbit riding on its parallax and proper motion. The track on the sky of a star like the visible member of Gaia BH1 over 3 years, in milliarcseconds, east to the left: proper motion of about 27 mas a year, a yearly parallax loop of 2.09 mas, and on top of both the star's own orbit about the barycentre it shares with an unseen companion, a 185.6-day ellipse of semi-major axis 2.66 mas, eccentricity 0.45, inclined at 126.6°. The dashed line is the path the star would follow alone. The difference between the two, a loop more than 1.3 times the parallax's size, is the orbit — a semi-major axis of 1.27 AU at the parallax's distance, which with the visible star's mass gives the companion's.

A black hole weighed by a wobble on the sky

A velocity curve weighs an unseen companion only up to the tilt of its orbit. The same orbit followed on the sky, as a loop a few thousandths of an arcsecond across riding on a star's parallax, shows the tilt as the shape of the ellipse — and with the parallax turning the angle into astronomical units, Kepler's third law gives the companion's mass outright. That is how the nearest black holes were found: silent, unfed, and on orbits no X-ray survey could have reached.

gravitation · The two-body problem Haloes that agree on the line of sight, and disagree across the sky. The dispersion of the tangential component of motion on the sky — across the radius vector, as a proper motion measures it — predicted by each of the 13 dark haloes that fit the same flat line-of-sight dispersion of 10 km/s, against projected radius in half-light radii, for stars in a Plummer profile. Coloured by the halo's inner slope; the dashed line is the line-of-sight value all of them reproduce. At the half-light radius the haloes' line-of-sight dispersions agree to 2.0 per cent, and their tangential dispersions range from 8.3 to 10.8 km/s, a factor of 1.29. The anisotropy each halo needed to match the line of sight is written into the motion across it, which no line-of-sight measurement sees.

Motions across the sky measure the orbits first

A dwarf galaxy's line-of-sight velocities leave the shape of its stars' orbits free, and that freedom hides whether its dark halo has a core or a cusp. Proper motions add the two components across the sky. They read the orbits' shape almost at once, with a few hundred stars — and they reach the slope of the halo only with ten thousand.

galaxies · Velocity dispersion Dark companions' masses with and without a gap, as an astrometric survey would see them. The distribution of the masses of dark companions in a sample found by their stars' astrometric orbits, per unit of the logarithm of mass, for two hypotheses: neutron stars near 1.35 solar masses and black holes near 8 in equal numbers with nothing between 2 and 5 (solid), or the same with the gap filled at the density the black holes have at 5 (dashed; 16 per cent of the sample in it). Each is drawn as measured with mass errors of 10 per cent (blue) and 30 per cent (orange). With 10 per cent errors the empty gap stays empty: 0.2 per cent of the sample lands between 2.5 and 4.5 solar masses, against 10 per cent if it is filled. With 30 per cent errors the edges spill in — 4.3 per cent against 12 per cent — and the gap becomes a shallow dip that a modest sample cannot distinguish from a filled one.

A gap that counting can test and impostors can fill

The black holes in X-ray binaries leave a gap between the heaviest neutron stars and about five solar masses, and nobody knows whether it is real or an artefact of how those masses are measured. Astrometric orbits give masses that need no light curve, and with errors of ten per cent a few dozen of them would settle it. The difficulty is not the counting. It is that pairs of white dwarfs and neutron stars, dark and close, can weigh exactly what a gap object would.

gravitation · The two-body problem One orbit seen from three angles. An orbit of eccentricity 0.5 with its periastron 40° from the line of nodes, seen face on and tilted by 60° and 75°, each panel centred on its apparent ellipse. Every projection is again an ellipse — of apparent eccentricity 0.88 and 0.97 — but only the face-on view has the star (red) at one of its foci (open circles). Tilted, the star sits 0.48 and 0.57 apparent semi-major axes from the nearer focus. The displacement is what measures the tilt.

An orbit seen at an angle keeps everything but its focus

No orbit outside the solar system is seen face on. It is seen as a shadow on the sky, and the shadow of an ellipse is an ellipse, swept at equal areas in equal times, indistinguishable in shape and timing from a real orbit. Only one thing gives the tilt away. The star the companion circles projects to where the true focus projects, and that is not where the apparent ellipse has its focus.

orbits · Conic sections An association that has been flying apart for ten million years. The velocity of each of 220 stars along one direction against its position along the same direction, for a model association whose stars were born 10 million years ago inside a region 1 pc across (one standard deviation), with random velocities of 1 km/s, and that began coasting apart at once. The fast stars have gone furthest, so position and velocity are correlated, and the slope of the line through them is the inverse of the time they have been moving: the fit gives 10.1 million years. A star at 10 pc moving at 1 km/s has been travelling for ten million years — which is what an expansion age is. Measuring it needs the stars' distances and their motions across the sky, both of which space astrometry now supplies for associations within a few hundred parsecs.

An expansion clock read from the stars' motions

The masses of a young cluster's brown dwarfs are read from their brightness, and a brown dwarf's brightness is a mass multiplied by an age. The age can now be measured without any model of a star: the cluster is flying apart, and the time since it began is its size divided by its speed. But the expansion clock counts only the coasting, not the stars' lives, and it runs fast when the stars began spread out — two biases of opposite sign, and nothing in the motions themselves to say which one wins.

stars · Initial mass function A peak that follows the snow line inwards. Giant planets per star per unit of ln a against orbital distance, for stars of 0.3, 0.6, 1, 1.5 solar masses, if the peak's position scales as the 1 power of the star's mass — snow line, young star's luminosity. The shape is the broken power law measured round Sun-like stars, rising as a⁰·⁸ and falling as a⁻¹, with its height scaled in proportion to the star's mass as the giants' overall rate is. The peaks sit at 0.9 AU, 1.8 AU, 3.0 AU, 4.5 AU: a 0.3-solar-mass star would have its giants piled up 3.3 times closer in than the Sun's. The dots mark each peak.

A peak that should move with the star

Round Sun-like stars, giant planets are commonest at about three astronomical units, close to where water first froze in the disc. If the snow line put them there, the peak should sit closer in round smaller stars — three times closer round a star of a third of the Sun's mass. Every hypothesis agrees about the Sun, which is why the Sun-like stars cannot test them; and one of them makes a Jupiter at the peak wobble its star by the same angle whatever the star's mass.

exoplanets · Occurrence rates

Named alongside it

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

Proper motionParallaxAberrationReference frameRadial-velocityBinary starsBlack holeDegeneracyEpochBarycentreBrown dwarfCelestial sphere

All concepts