Astronomy & Celestial Mechanics
the sky, computed
Concept

Asteroseismology — where it appears

The reading of a star's mass, radius and age out of the frequencies of the sound modes standing in its own interior. The two large-scale quantities it uses are the spacing between consecutive overtones, which is a mean density, and the frequency of maximum power, which is a surface gravity.

Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.

14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here.

The interior read from a comb of frequencies

A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

stars · Asteroseismology Two stars at the same point of every other diagram, 4.1 times apart in one. Above: the gravity-mode period spacing against the large frequency separation, for 90 shell-burning giants and 55 core-burning ones. These are the same stars in every other measurement. They have the same luminosity, the same temperature, the same colour, the same surface gravity and — inside the band drawn — the same Δν, which means the same mean density; the global scaling relations return the same mass and the same radius for both. What separates them is a quantity that comes from nowhere near the surface: ΔΠ₁ is set by the buoyancy frequency integrated across the core, and a core that has ignited helium is expanded and convective, so its integral is smaller and its period spacing larger. The two sequences do not touch — 61 seconds against 248 — and the gap sorts a catalogue of tens of thousands of giants into stars burning hydrogen in a shell and stars burning helium in a core, by a Fourier transform of a light curve. Below: what that looks like in the spectrum itself, drawn over two radial orders. The number of mixed ℓ = 1 modes between consecutive radial modes is Δν/(ΔΠ₁ν²), so the shell burner has about 12 of them and the core burner about 3: the star with the larger period spacing has the sparser spectrum. The picture cannot show what the same modes are also used for and cannot settle — the splitting of each mixed mode gives the rotation rate of the core separately from the envelope, and the cores come out spinning some ten times faster than the surface and a hundred times slower than any model of angular-momentum transport predicts.

Two stars only a Fourier transform can tell apart

A giant burning hydrogen in a shell and one burning helium in its core sit at the same luminosity, the same temperature and the same mean density. Every scaling relation returns the same mass and radius for both. The gravity-mode period spacing is fifty seconds for one and three hundred for the other.

stars · Asteroseismology A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the limb-darkened stellar disc over the region the planet covers. The depth is 1.26%, deeper than (Rp/R⋆)² = 0.01055 because the planet crosses a limb-darkened disc whose centre is brighter than its average. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii.

Four contact points, and what they fix

The depth of a transit gives a radius ratio. The shape gives the impact parameter, and then — through nothing but Kepler's third law — the mean density of the star being crossed.

exoplanets · Transits The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives.

A fluid that turns as one piece

The Sun's surface turns faster at its equator than at its poles, and everyone expected the inside to do something similar. Below seven-tenths of the way down it does not — the radiative interior rotates as rigidly as a bell, and nothing in hydrodynamics makes a fluid do that.

stars · Internal rotation Where each mode turns back: ℓ = 0 through the centre, ℓ = 300 in the outer 2 per cent. Why a set of frequencies is a depth profile and a single frequency is not. An acoustic wave travelling into a star meets a rising sound speed and is refracted back; it turns where its horizontal phase speed matches the local sound speed, which happens at c(r)/r = 2πν/√(ℓ(ℓ+1)). The horizontal axis is the angular degree on a logarithmic scale and the vertical axis is the fractional radius of that turning point, drawn at 2000, 3090, 4000 microhertz. The ordering is the content. A radial mode, ℓ = 0, has no horizontal phase speed at all and passes straight through the centre. Degrees one and two turn deep in the core. By ℓ = 300 the mode is trapped in the outer 2 per cent and knows nothing about anything below. So a frequency measured to a part in ten thousand constrains an average of the interior weighted in a way the mode itself decides, and measuring thousands of modes of different degree gives thousands of differently weighted averages — which is a solvable inverse problem, and is how the base of the convection zone was located at 0.713 of the radius rather than assumed. The sound speed here is a polytrope's rather than a tabulated solar model's, so the curve is the right shape and the wrong star in its outer tenth, where the real Sun is convective and this one is not. What the picture cannot show is the frequency dependence at fixed degree, which is weaker but not negligible: a higher-frequency mode of the same degree turns slightly deeper, and the three curves separating toward the right is that effect.

Every note turns back at its own depth

A sound wave heading into a star is refracted by the rising sound speed and turns around before it reaches the middle. Where it turns depends on which mode it is — so a list of frequencies is not one average of the interior but thousands of differently weighted ones, and that is a solvable problem.

stars · Asteroseismology A period, a colour, and an age. Rotation period against colour for stars of 125, 625, 1000, 2500, 4570 million years, under the empirical relation P = t^0.5189 × 0.7725(B−V − 0.4)^0.601. The isochrones do not cross and are separated at every colour by exactly the age ratio raised to 0.5189, which is what allows a single measured period to be inverted for an age once the colour is known. The Sun, at B−V = 0.653 and 4570 million years, is placed by the relation at 26.8 days against the 25.4 days it is observed to have. Three clusters are marked at a common colour to show the spacing directly. The dashed boundary at the left is where the Rossby number — the period divided by the convective turnover time — passes 2 on the oldest isochrone, at B−V = 1.35: past that point the braking weakens and the relation is known to over-predict the age, which is the one place a rotation period stops being a clock. What the figure cannot show is the scatter, which is a few days at fixed colour and age and is the real error bar on any single star.

The clock that starts by forgetting

An ordinary star tells nothing about its age. It sits on the main sequence for billions of years at almost fixed brightness and colour, and the one property that changes monotonically is how fast it turns — but only because the braking law destroys the initial condition first, and only until it stops.

stars · Magnetic braking Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.

A neutron star born turning too slowly

Collapse an iron core a few thousand kilometres across down to twelve, and conservation of angular momentum multiplies its rotation rate by about eighteen thousand. A model with no transport in it delivers a newborn pulsar at the break-up limit; the ones that are observed turn twenty times slower, which is a measurement of the core before it fell.

stars · Supernovae A hundredfold expansion is a four-order slowdown. Equatorial rotation speed against radius for three stars leaving the main sequence at 2, 10, 100 km/s, on the single assumption that nothing exerts a torque. Both axes are logarithmic. The specific angular momentum is held fixed along each track and checked at four radii rather than assumed, so the speed falls as the reciprocal of the radius and then faster, as the moment of inertia coefficient slides from 0.073 on the main sequence to 0.02 in a centrally condensed giant envelope. A star at 100 km/s crosses the 8 km/s line — the usual boundary for calling a giant a rapid rotator — at 45.6 solar radii, and everything larger is slower. That is why the observed giants are almost all under two kilometres a second, and why the one or two per cent that are not cannot be explained by anything the star did on its own: the angular momentum has to have been delivered, by a swallowed companion or a merger. The figure assumes no mass loss, which for the largest radii drawn is the weakest of its assumptions.

A surface that slowed because the star grew

A star leaving the main sequence expands by a factor of a hundred. Conserve its angular momentum and its equator slows by four orders of magnitude, which is why every red giant is a slow rotator. One or two per cent are not, and a giant turning at fifty kilometres a second has to have been given the angular momentum by something else.

stars · Stellar evolution A transit that lasts 4.0 times longer at one end of the orbit than the other. The duration of a transit, relative to what a circular orbit of the same period around the same star would give, against the orientation of the orbit. A planet transiting near perihelion is moving fastest and its transit is shortest; one transiting near aphelion is slowest and its transit is longest. The two extremes are exact reciprocals — the circular duration is their geometric mean, whatever the eccentricity — and at e = 0.6 they differ by a factor of (1+e)/(1−e), which is 4.0. That is an enormous, easily measured effect, and it means a transit duration is not a stellar density unless the orbit is circular. Turned round, it is a measurement: given a stellar density from asteroseismology or from a parallax and a spectrum, the duration anomaly gives the eccentricity — from photometry alone, with no radial velocities at all.

A duration that measures an eccentricity

A transit's length is a measurement of how fast the planet was moving when it crossed, and that speed depends on where it was on its orbit. For a circular orbit the duration gives the star's density; for an eccentric one it gives the density times a factor of up to four — and if the density is known independently, the factor is the eccentricity.

exoplanets · Transits The true obliquity a projected angle allows, against the tilt of the star. The true angle ψ between a transiting planet's orbit and its star's spin axis, against the inclination of that spin axis to the line of sight, for projected angles λ of 0°, 30°, 60°, 90°, 130° — from cos ψ = sin i★ cos λ for an orbit seen edge-on. At i★ = 90°, a star seen equator-on, the projection loses nothing and ψ = λ. As the star tips towards pole-on the true angle moves towards 90° from whichever side it started, so every curve ends at 90° when i★ = 0. Two things follow exactly. The projected angle is a lower bound for an aligned-looking system and an upper bound for a retrograde-looking one — ψ always lies between λ and 90°. And λ = 90° needs no inclination at all: a planet whose projected orbit crosses the stellar equator at right angles is on a polar orbit whatever the tilt, which is why the pile-up of polar orbits could be seen in projected angles alone.

An angle hidden by the tilt of the star

The line profile of a transited star gives the angle between orbit and spin as projected on the sky. The true angle needs the tilt of the star's spin axis, and every way of measuring that tilt is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits.

exoplanets · Spin–orbit alignment Four transiting planets, one star, and a density that must agree. The stellar density implied by each of four transiting planets of one simulated star, each fitted as though its orbit were circular, relative to the star's true density, against orbital period, both logarithmic, with one-sigma errors of 8, 10, 14, 20 per cent from the transit fits. Three planets have eccentricities of a few hundredths and agree with one another and with the truth (dashed line: their weighted mean, 0.950). The planet at 15.9 days has e = 0.2 with its periastron 70° from the line of sight, which makes it cross the star faster than a circular orbit would, and its implied density is 2.10 times the star's — 5.2 standard deviations from the others. A single planet's density can be reconciled with any star by choosing its eccentricity; several planets must share one star, and the odd one out is caught.

Every planet of a star must agree on its density

A single transiting planet gives the density of its star only if its orbit is known to be circular, and an eccentric orbit can disguise almost any star as almost any other. A star with four transiting planets has four such measurements of one number. They must agree — and the way they disagree measures how eccentric the orbits are, catches the candidate that orbits some other star, and pins down the star better than its own spectrum does.

exoplanets · Transits How many stars it takes to tell a shared axis from chance. The number of stars whose measured sin i, with an error of 0.1, is needed on average before an aligned cluster is preferred over random axes by a likelihood ratio of 370 to one — three standard deviations — against the inclination of the common axis, on a logarithmic scale, for perfectly aligned, κ = 20, κ = 5. Perfect alignment needs a handful of stars at any angle — 4 at 30°, 10 at 85°. A loose alignment, κ = 5, needs 29 about an axis at 30° and 38 about one at 85°, and 985 about one at 55°: a loose cone about an axis near 54.7°, where 3cos² i₀ = 1, has the same mean cos² i as random axes and is the hardest of all to tell from them.

A shared axis hides at the angle where it does no harm

If the stars of a cluster were born spinning about a common axis, their inclinations would show it. A tight alignment shows in a handful of stars at any orientation. A loose one about an axis tilted 54.7° from the line of sight has the same average inclination as random axes and takes a thousand stars to find. At that same angle, a shared axis also stops biasing the cluster's brightnesses.

starlight · Gravity darkening Two eccentricity distributions for the planets that transit alone. Two hypotheses for the orbital eccentricities of planets that transit their stars alone, as probability densities on a logarithmic axis. Solid: a mixture of two populations, 70 per cent at σₑ = 0.04 and 30 per cent at σₑ = 0.25 — a cold majority like the planets in multi-planet systems and a hot minority. Dashed: a continuum, each planet's Rayleigh scale drawn evenly in its logarithm between 0.015 and 0.34, chosen as the continuum whose densities most resemble the mixture's. The two differ most between eccentricities of 0.1 and 0.15, where the mixture dips between its two populations and the continuum does not: 5.6 and 12.0 per cent of planets respectively. A trough there is what a separate hot population would leave, and it is the feature a measurement has to find.

A gap in eccentricity a density can only just see

Planets that transit alone have more eccentric orbits than planets with transiting siblings, and the excess can be fitted as a separate hot population or as a smooth spread with no gap. The two eccentricity distributions look plainly different. The stellar densities through which they are measured look almost the same, and how many planets it takes to tell them apart is set less by the star than by the transit.

exoplanets · Transits Four clocks for a 1 M☉ dwarf, and the best of them. The fractional age uncertainty each method reaches for an isolated 1 solar-mass main-sequence star, against its age, logarithmic, for isochrones, rotation, oscillations and chemical clock, with representative published precisions. Rotation is best while the star is young, once its spin has forgotten its start and before its braking weakens; isochrones and oscillations improve with age as the star moves further on the diagram. The combination (heavy line) is 14 per cent at 1 Gyr, 9 at 3 and 4 at 8. It is better than ten per cent (dashed) from 2.6 to 9.4 Gyr.

Four clocks and the stars none of them reads

An isolated dwarf star can be dated four ways — by its position on the Hertzsprung–Russell diagram, by how fast it spins, by its oscillations, and by the ratio of two elements its galaxy made at different rates. Each clock has an age range where it works and masses it cannot read. Laid side by side for a Sun-like star they cover most of its life to ten per cent or so. For a star of eight-tenths of the Sun's mass only one clock is left, it works only while the star is young, and the commonest stars in the Galaxy's disc are dated one at a time no better than fifteen per cent at any age.

stars · The HR diagram What a hundred kelvin moves, for a star like the Sun. The change in eight quantities derived for a Sun-like star and its planet when every temperature is raised by 100 K — 1.7% — as the uncertainty in the absolute temperature scale allows, against the precision each is quoted to (grey bars). The star's radius from its flux and parallax falls by 3.4%, because radius goes as the inverse square of temperature; the planet's radius follows, and its density rises by +12.8%. The oscillation-based mass rises by +2.6%, the main-sequence mass by +3.5% and the age falls by 9.8%. The iron abundance read from neutral iron lines rises by 0.08 dex, +20.7%. In every case the shift is comparable to or larger than the precision quoted, and in every case it moves all stars of a sample together.

A hundred kelvin under every radius

The temperatures of cool stars are quoted to tens of kelvin, and the scale they are all measured on is uncertain by about a hundred. That hundred is under almost everything else: a star's radius moves by twice its fractional error, a transiting planet's density by six times, a main-sequence age by ten per cent and an iron abundance by a fifth. And because a scale error moves every star of a sample the same way, no amount of averaging removes it.

starlight · Stellar colour Ridges that bend where the model's surface is wrong. An échelle diagram of a Sun-like star's oscillation frequencies — each mode's frequency against its remainder after dividing by the large separation of 135.1 µHz — for radial, dipole and quadrupole modes (red, green, blue), as a one-dimensional model predicts them (open) and as the star produces them (filled), if the model's outer layers are wrong in the way the Sun's models are. Low in the diagram the two coincide. With increasing frequency the star's modes fall behind the model's, by an amount that grows steeply and is the same for every degree, so all three ridges bend to the left together: by 19.3 µHz at 4104 µHz. The bend is the surface effect, and it is larger than the uncertainty of any measured frequency by a factor of a hundred.

A correction that grows as the fifth power of the frequency

Every model of a star gets its outermost layers wrong, because one-dimensional models cannot follow the turbulent convection that fills them. The error shows up in the star's oscillations as an offset that is nothing at low frequency and twenty microhertz at high, two hundred times a frequency's measurement error. It is corrected with a power law nobody can derive, and the correction recovers the star only when its shape happens to be right — which is why the ages of stars are fitted not to their frequencies but to ratios of differences the surface cannot reach.

stars · Asteroseismology

Named alongside it

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

Stellar densityConvectionEccentricityRotational splittingAngular momentum transportGyrochronologyHelioseismologyPhotoeccentric effectTransitAcoustic cavityAngular momentumArgument of periastron

All concepts