Astronomy & Celestial Mechanics
the sky, computed
Concept

Rotational splitting — where it appears

The frequency difference between the components of an oscillation multiplet in a rotating star. To first order it is the angular velocity averaged over the mode's own cavity, so modes of different degree report the rotation at different depths and their differences resolve a profile.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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 shear layer that should have spread

The Sun's convection zone turns differentially — its equator laps its poles about once every three months — and the radiative interior below turns as one rigid piece. Between them is a transition four per cent of the radius thick. Nothing in hydrodynamics keeps a velocity discontinuity that thin for four and a half billion years.

stars · Internal rotation 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

Named alongside it

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

AsteroseismologyAngular momentum transportHelioseismologyConvectionDifferential rotationInversionMixed modeTachoclineAcoustic cavityBuoyancy frequencyConditioningConvective envelope

All concepts