Astronomy & Celestial Mechanics
the sky, computed
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Asteroseismology — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · stars
  2. 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 scaling relations of the rung below 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.

    part 2 · stars
  3. 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.

    part 3 · stars
  4. 4 per cent in one observable is 48 per cent in an age. How an error in the calibration of the large frequency separation propagates into the quantities derived from it. The two scaling relations are exact in their exponents, so a fractional error in the separation appears as twice that in the radius, four times in the mass, and — because a main-sequence lifetime falls as roughly the two-and-a-half power of the mass — ten times in an age. At the 4 per cent level, which is about what the theoretical corrections to the relation amount to for a red giant, that is 15 per cent in mass and 48 per cent in age. Nothing about the seismology is uncertain at that level; the frequencies are measured to parts in a thousand. What is uncertain is the constant of proportionality, and it is uncertain because it was calibrated on one star.

    Two scaling relations calibrated on one star

    Asteroseismology gives a star's mass and radius from two numbers read off its oscillation spectrum. The two relations are exact in their exponents and approximate in their constants, and the constants were fixed by requiring that the Sun come out right — so an error of a few per cent in one observable is tens of per cent in a mass and nearly a factor in an age.

    part 4 · stars

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