Astronomy & Celestial Mechanics
the sky, computed
Concept

Acoustic cavity — where it appears

The region inside a star where a sound wave of a given frequency and degree is trapped, between a lower turning point set by the rising sound speed and the surface. Its size decides which layers a mode samples, and so what its frequency measures.

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

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 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.

AsteroseismologyConvectionTurning pointConditioningHelioseismologyInversionKernelLarge separationRegularisationRotational splittingScaling relationSolar calibration

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