Astronomy & Celestial Mechanics
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Magnetic braking — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A lever 12 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 12 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 3·10⁻¹¹ of its angular momentum in the same year — a ratio of 1,315, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 3.3·10¹⁰ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.

    A wind that takes no mass and all the spin

    The Sun loses about a ten-thousandth of itself to its own wind over its whole life. It loses most of its rotation to the same wind, and the whole of that asymmetry is one geometric factor — the gas is forced to keep turning with the star until it is a dozen or more radii out from the surface it left.

    part 1 · stars
  2. 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.

    part 2 · stars
  3. In a mixed field the torque follows the dipole and ignores the rest. The torque of a field that is part dipole and part quadrupole, with the total surface strength fixed, as a fraction of a pure dipole's, against the fraction of the field strength in the dipole, for wind magnetisations of 300, 3,000, 30,000, using the broken power law fitted to simulated mixed-field winds. Above a dipole fraction of 0.39, 0.24, 0.15 respectively, the torque is exactly what the dipole component alone would give: the quadrupole has fallen below the dipole long before the wind reaches its Alfvén radius, and contributes nothing to the lever. Below it, the quadrupole sets a floor. A star whose surface is mostly quadrupole by strength can therefore be braked almost entirely by its small dipole — and a measurement of the total field says little about the torque unless it says how much of the field is dipolar.

    A lever set by the dipole and not the field

    A star's wind carries away its spin because the magnetic field holds the gas in step with the surface out to the Alfvén radius, and the torque goes as the square of that radius. How far out the field can hold depends less on how strong it is than on how it is arranged: a quadrupole of the same surface strength holds the Sun's wind to half the distance and brakes it at a fifth of the rate. In a mixed field only the dipole reaches far enough to matter — so the field that brakes a star is the part of it that is hardest to measure.

    part 3 · stars
  4. A binary that goes silent for part of its shrinking. The orbital period of a white dwarf of 0.8 M☉ and a donor from 0.5 M☉, in a one-zone evolution, against time. Mass transfer drives the period down fast while magnetic braking acts. At 3.17 hours the donor reaches 0.2 M☉ and loses its radiative core and with it the brake. Its radius, swollen by the fast transfer, relaxes and it falls inside its Roche lobe; transfer stops (dashed, no accretion, nothing to see). Gravitational waves alone close the orbit, until contact returns at 2.30 hours, 0.60 billion years later. The observed gap runs from 2.15 to 3.18 hours. The donor's thermal time at 0.2 M☉, set here to 3×10⁸ years, is chosen to place the upper edge, so that edge is fitted and the lower one is not. The period reaches a minimum of 68 minutes against an observed 80, a shortfall every model of this simplicity shares.

    A binary that goes silent for an hour of period

    Binaries of a white dwarf feeding off a small star can be found at almost any orbital period down to about eighty minutes, except between about two and three hours, where they are strikingly rare. They have not stopped existing there. They have stopped shining. The moment the donor loses its radiative core, the wind's magnetic brake appears to switch off. The star, swollen by the transfer it was being driven through, shrinks back inside its own Roche lobe, and the feeding stops. Only gravitational waves remain to close the orbit, and it takes them about half a billion years.

    part 4 · stars

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