Astronomy & Celestial Mechanics
the sky, computed
Concept

Trigonometric parallax — where it appears

A distance obtained from the angular shift of a nearby star as the Earth moves around its orbit, using no assumption about the star at all. It is the only geometric distance beyond the solar system, and every other method is calibrated against it.

Named by 7 essays across 4 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 Mars through 8.3 months of sky. The geocentric ecliptic longitude and latitude of Mars over 252 days — 8.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 1.5237 AU inclined 1.850°. The motion reverses for 72.7 days and backs up 15.94° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 149 days the planet visits the same point of the sky twice, and the loop it encloses is 15.9° long and 2.94° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 4.6 times a degree of longitude, because the sweep of 51° in longitude and 4.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Mars is 0.525 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops.

The loop a planet does not make

Mars stops in the sky, backs up for ten weeks, and goes on. Neither orbit reverses anywhere — the loop belongs to the difference of two position vectors seen from one of them, and its width and duration are fixed by the ratio of the two radii and by nothing else.

sky · Apparent motion 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 The fractional distance error, rung by rung. Cumulative fractional uncertainty in a measured distance against the distance itself, both on logarithmic axes. Each rung of the ladder is calibrated against the one below it, so its scatter adds in quadrature to everything already inherited, and the total can only rise: radar to the planets 10⁻⁵%, parallax 1.0%, main-sequence fitting 5.1%, Cepheids 6.5%, Type Ia supernovae 8.2%.

Every distance is measured with the last one

No single method reaches from a planet to a distant galaxy. The ladder is built rung by rung, each calibrated on the one below, and the errors multiply all the way up.

starlight · Distance ladder Fringe visibility for a 47 mas disc at 575 nm. Fringe visibility against the separation of the two apertures, for a disc 47 milliarcseconds across seen at 575 nm. The solid curve is a uniform disc, |2J₁(x)/x| with x = πθB/λ; it is exactly one at zero baseline, where both apertures see the same wavefront, and falls to zero at 3.08 m — read off the drawn samples, and equal to 1.21967 λ/θ to better than one part in a million. That is the measurement: not a brightness, a baseline. The dashed curve is the same disc with linear limb darkening u = 0.4, whose null is 4.9% further out at 3.23 m — so the same observed null implies 47 mas as a uniform disc and 49.3 mas limb-darkened, and a diameter quoted without its model is a number without a unit. At the 2.54 m aperture of the telescope this was done on, the visibility is still 0.17: one mirror cannot reach the null, which is the same statement as saying it cannot resolve the star.

An angle of five hundredths of an arcsecond

No telescope has ever resolved a star other than the Sun, and stellar diameters are measured anyway — by finding the separation of two apertures at which the star's interference fringes vanish. What that returns is an angle; the radius arrives only when a distance is brought in, and the distance is the worse-known half.

starlight · Angular diameter A calibration sample, and the stars a precision cut keeps. Measured parallax against apparent magnitude for one star in four of a simulated sample of 8000 standard candles of absolute magnitude 0.5 and intrinsic scatter 0.15 mag, spread through a disc 15 kpc in radius and 300 pc thick and catalogued to magnitude 16. The parallax axis is stretched near zero so the negative measurements show. The solid curve is the parallax a star of exactly the calibrated luminosity would have at each magnitude, and the dashed curve is the cut ϖ/σ = 5: stars above it are the "good parallaxes" a calibration usually keeps. It keeps 3242 of the 8000, 41 per cent, and at faint magnitudes the curve it cuts through runs through the middle of the stars rather than below them, so the ones kept there are the ones whose errors pushed their parallaxes up. 45 stars have negative parallaxes; no inversion can use them, and a fit in parallax space uses them like any other.

A calibration chosen by the measurement it calibrates

A standard candle is calibrated on stars with good parallaxes, and "good" is decided by the parallax itself — so the calibrators are the stars whose errors happened to push them nearer. Correcting each star's distance afterwards does not repair it, because the correction needs a picture of where the sample's stars really lie, and that picture depends on the luminosity being calibrated. What works is to choose nothing, keep every star with the negative parallaxes included, and fit the candle by predicting the parallaxes rather than inverting them.

starlight · Parallax Arms drawn into a disc that has none. Left: 1400 gas clouds spread uniformly over a disc between 3 and 13 kpc from the Galactic centre (cross), seen from above with the Sun (ring) 8.2 kpc out; the two thin curves mark where a two-armed spiral density wave of pitch 12° has its arms. The clouds have no arms in their positions, only in their motions: each streams by 10 km/s in a pattern locked to the arms. Right: the same clouds placed at their kinematic distances, read from their velocities with a circular rotation curve and on the correct side of every tangent point. The streaming has moved them by up to a few kiloparsecs, along lines of sight, and into bands. Lines of sight within 12° of the Galactic centre and 18° of the anticentre, where the velocities carry no distance, are left out. Counted in square kiloparsecs between 4 and 12 kpc, the densest fifth of the map holds 30 per cent of the clouds in reality and 45 per cent in the kinematic map.

A spiral drawn by velocities alone

A map of the Milky Way's gas is built by turning each cloud's velocity into a distance with a circular rotation curve. Gas in a spiral arm does not move on circles: it streams by ten or fifteen kilometres a second in a pattern locked to the arms, so the distance errors are locked to the arms too. Read that way, a disc with no arms in it at all comes out of the calculation with arms.

galaxies · Galactic structure

Named alongside it

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

Distance ladderAstrometryAstronomical unit (AU)EccentricityParsecAbsolute magnitudeAngular diameterCelestial sphereCluster turnoffDensity waveDiffraction limitDistance modulus

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