Astronomy & Celestial Mechanics
the sky, computed
Concept

Parallax — where it appears

The angular shift of a nearby star against distant ones as the Earth moves, half the full annual swing, and the only geometric distance beyond the solar system. One arcsecond corresponds to one parsec, and the whole geometric distance scale rests on the largest such angle being under a single arcsecond.

Named by 13 essays across 5 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 One over a noisy parallax, at three precisions. The distribution of the distance obtained by inverting a parallax, for a star truly at 100 parsecs measured with fractional errors of 5, 10, 20 per cent. At five per cent the distribution is nearly symmetric and inverting is harmless. At twenty per cent it is strongly skewed: the mean sits at 105 parsecs rather than 100, and the tail runs to distances several times the truth, because a parallax scattered a little towards zero is a distance scattered a long way outward. The asymmetry is a Jacobian and nothing else — the parallax measurement is unbiased and symmetric throughout. Above about twenty per cent the mean of the distribution stops existing at all, because the density falls only as the inverse square of the distance and the integral of d times that diverges.

The distance is not one over the parallax

A parallax is measured with symmetric errors and a distance is one over it. Inverting a noisy positive quantity is not a change of units — it is a change of distribution, and the one that comes out is skewed, biased outward, and above about twenty per cent error has no mean at all.

starlight · Parallax A gap 1 per cent deep that became 63. The distribution of planet radii, drawn three ways: the underlying distribution with a gap in it, the same distribution convolved with a 25 per cent stellar radius error, and convolved with a 5 per cent one. Every planet radius is the transit depth's square root multiplied by a stellar radius, so an error in the star is an error in the planet, and a population of planets inherits the population of stellar errors as a smearing. The gap is 1 per cent deep at the old precision and 63 at the new one, and its centre does not move — a symmetric smearing hides a feature without displacing it. That is what happened when parallaxes for a hundred thousand planet hosts arrived: no new planets were observed, and a feature that had been marginal became unambiguous.

A planet radius is a stellar radius

A transit measures a ratio and nothing else. Every planet radius ever published is that ratio multiplied by a stellar radius that came from somewhere entirely different, so a population of planets inherits the errors of a population of stars — and when the stars were measured better, a feature nobody could see became unmistakable.

exoplanets · Planet composition A light curve pulled out of shape by the Earth's own orbit. Magnification against time for a 90-day event at impact parameter 0.3, computed with the observer's orbital motion included at three values of the microlens parallax. The dashed curve is π_E = 0 and is exactly symmetric in time, because a straight-line track past a point lens has to be. The others are not. The Earth's displacement over the months the event lasts adds a term π_E times its projected orbital motion to the lens–source trajectory, so one wing is pushed closer to the lens and the other further away, and the curve acquires an asymmetry of 16 per cent at π_E = 0.15 and 34 per cent at π_E = 0.35. That asymmetry is the whole measurement. An ordinary event gives one dimensioned number, t_E, which mixes the lens mass with two distances and a proper motion and therefore weighs nothing; the parallax gives a second, and two constraints on the same lens are what a mass requires. It is only available on long events — the Earth has to move appreciably while the magnification is changing — which is why parallaxes are measured for the timescales above about fifty days and not for the short ones that a low-mass lens produces.

An asymmetry that the Earth's own orbit puts in

A microlensing event delivers one number with dimensions, and one is not enough to weigh anything. The Earth's motion over a long event distorts the light curve, and the distortion is the second constraint — after which the mass follows with no distance in it at all.

exoplanets · Microlensing The position closure discards, and what it costs to buy back. The astrometric error left after phase referencing, against how far away the calibrator is, for three switching cadences, on a 8000-kilometre baseline at 1.3 mm. Referencing gives up the immunity closure quantities have: instead of forming a combination the atmosphere cannot enter, it measures the atmosphere on a nearby source and subtracts it, which recovers the absolute position and leaves behind whatever differs between the two lines of sight. That difference is governed by the Kolmogorov structure function, which rises as the five-thirds power of a separation, so the phase residual and therefore the position error rise as the five-sixths power of the separation, which the drawn spatial term reproduces exactly. Below about 1.7 degrees at the fastest cadence the curves flatten, because there the atmosphere's change between one visit to the calibrator and the next is the larger of the two differences and the separation has stopped mattering. The consequence is a premium on finding a close calibrator: at one degree the error is 2 microarcseconds and at eight it is 5, and the sky is not dense in compact bright sources. The three curves are the other half of the trade — a faster cycle freezes the atmosphere better and spends more of the observation looking at the calibrator, so the optimum is where the two losses meet rather than as fast as the hardware allows.

The position closure throws away

A closure phase is immune to the atmosphere because it cannot see anything that looks like a per-antenna error — and a displacement of the source on the sky looks exactly like one. Recovering where a source is means giving that immunity up on purpose, measuring the atmosphere on a neighbour instead of cancelling it, and paying for the difference between two directions.

starlight · Interferometry 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 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 Sun's motion, written across a group of stars. Proper motions of 400 stars all at 300 pc, against their angle from the solar apex, each with its own random velocity of 25 km/s in every direction. Filled dots are the component along the great circle away from the apex, υ; open dots are the component at right angles to it, τ. One star's υ is dominated by its own motion — the scatter is 17.6 mas/yr against a reflex of at most 13.7 — so no single star gives a distance. Together the υ components follow the Sun's reflex, 19.5 km/s times sin λ, and the fitted amplitude, 13.70 ± 1.06 mas/yr, gives a distance of 300 pc ± 8 per cent against the true 300 pc. The τ components scatter about zero at every angle, because the Sun's motion has no part across the great circle.

A baseline that grows four astronomical units a year

The Earth's orbit gives a parallax a baseline of two astronomical units and never more. The Sun's own motion through the nearby stars gives one that lengthens by four a year without limit — but the stars move too, by as much, so the long baseline yields a distance only for a group whose motions average away, and for a century it was the only way to reach past a few hundred parsecs.

starlight · Parallax 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 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 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 How far a wrong metallicity term moves the Hubble constant. The fractional change in the Hubble constant against the error in the adopted metallicity term, in magnitudes per dex, for supernova hosts whose Cepheids are 0.1, 0.2, 0.4 dex richer in metals than the geometric calibrators'. A term too weak (too close to zero) by 0.2 mag/dex with hosts 0.2 dex richer moves H₀ by 1.9%. The shaded band is the ±8% disagreement between the local measurement and the microwave background's prediction: reaching it with a 0.2-dex difference needs the term wrong by 0.9 mag/dex.

A metallicity term too small to close a gap

Cepheids of the same period are slightly brighter when they are richer in metals, by about a fifth of a magnitude per factor of ten in metal content. The galaxies whose supernovae calibrate the Hubble constant hold Cepheids a little richer than the galaxies whose distances are known geometrically, so the term enters the answer. How much it can move it is simple to compute: a few per cent at most for any plausible error — and the eight per cent separating the local Hubble constant from the early universe's would need the term to be wrong by four times its own size.

stars · Variable stars

Named alongside it

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

AstrometryProper motionReference frameAberrationBinary starsBlack holeDegeneracyLutz kelker biasMass functionParallax zero pointPriorRadial-velocity

All concepts