Astronomy & Celestial Mechanics
the sky, computed
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Internal rotation — the series

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

    part 1 · stars
  2. 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.

    part 2 · stars
  3. X-ray activity against Rossby number, with and without a tachocline. The X-ray luminosity of a star as a fraction of its total, against its Rossby number — the rotation period divided by the convective turnover time — for a synthetic sample of 600 stars of 0.4 to 1.2 solar masses, which have a radiative core and a tachocline below their convection zones, and 220 of 0.1 to 0.35 solar masses, which are convective to the centre and have neither. Both are drawn from one relation: saturated at a ratio of 10⁻³·¹³ below a Rossby number of 0.13 and falling as its −2.7th power above, with 0.25 dex of scatter. The saturated stars of the two kinds average −3.09 and −3.16 in the logarithm; 56 of the fully convective stars rotate slowly enough to lie on the unsaturated branch. This is the shape the measured relation has: the stars with no tachocline lie on the same line as those with one, saturated at the same level and falling along the same slope.

    A dynamo that did not need the shear layer

    The Sun's magnetic cycle has long been placed in the thin layer of shear beneath its convection zone, where a rigidly rotating interior meets a differentially rotating envelope. Stars below about a third of a solar mass have no such layer — they are convective to the centre — and they are as magnetically active as the Sun, saturate at the same level, and fade with slowing rotation along the same line. The saturation is the cleaner half of that evidence, because the slope depends on a turnover time that was itself calibrated from activity.

    part 3 · stars

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