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

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A light curve that is a decay chain. Bolometric luminosity against days since explosion, for three Type Ia models synthesising 0.3 M☉, 0.6 M☉, 0.9 M☉ of ⁵⁶Ni, from Arnett's one-zone diffusion model. Nothing in the shape is fitted: the two timescales are the laboratory half-lives of ⁵⁶Ni and ⁵⁶Co, 8.8 and 111.3 days, and the rise is those decays seen through an envelope that takes 11.3, 15.5, 18.6 days to leak. Each peaks at 14, 18, 21 days, and at maximum the luminosity equals the instantaneous deposited power to within 0.6% — Arnett's rule, and the only reason a peak brightness can be read as a mass of nickel. After maximum the curves settle to nearly straight lines on this logarithmic axis, declining at 0.0360 magnitudes a day against ⁵⁶Co's own 0.0098. The difference is gamma-ray escape: the ejecta become transparent to the very photons that are supposed to be heating them, and the tail is therefore steeper than the isotope. The one thing the tail is not is a property of the star — it is a half-life, plus a column density that is falling as t⁻².

    The candle that has to be standardised

    A Type Ia's light is the decay of half a solar mass of nickel-56 seen through an expanding envelope. Their peak brightnesses span six-tenths of a magnitude — and it is the width of the light curve, not anything else, that says which.

    part 1 · stars
  2. Four kinds of explosion, told apart by which line is missing. Spectra of 4 supernova types — Ia, II, Ib, Ic — near maximum light, stacked, over the same wavelength range and drawn with the same photospheric expansion speed of 10,000 kilometres per second. Every feature is a P Cygni profile: a trough blueshifted by the material expanding towards the observer and a peak at rest wavelength from the material moving across the line of sight, and the width of both is the expansion speed. Reading the silicon trough of the type Ia off the drawing puts it at 614.0 nanometres, which recovers 10,131 km/s from a rest wavelength of 635.5. The classification is a decision tree on absences. Hydrogen present makes it a type II; no hydrogen but silicon makes it a Ia; no hydrogen and no silicon but helium makes it a Ib; none of the three makes it a Ic. That tree was written down before anybody knew what these objects were, and it separates a detonating white dwarf from a collapsing massive core almost perfectly — because what the letters are actually reading is how much of the star's outer envelope was still there when it exploded. The type II kept its hydrogen, the Ib had lost it and shows the helium beneath, the Ic had lost that too.

    Two explosions told apart by a missing line

    The classification of supernovae is a decision tree on absences — is hydrogen there, is silicon there, is helium there — and it was drawn up decades before anybody knew what any of these events were. It nevertheless separates a detonating white dwarf from a collapsing stellar core almost perfectly, and the reason it does is worth the essay.

    part 2 · stars
  3. Two bodies at a mass ratio of 2.6 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 2.6 to 1, so the heavier body's path is 2.6 times smaller.

    The companion survives, and it is moving

    When one star of a close pair explodes, the other is left holding an orbital velocity and no orbit. Whether the pair stays bound turns on a single question — did more than half the total mass leave? — and the stars that answer no are running across the Galaxy at a hundred kilometres a second with nothing visible behind them.

    part 3 · stars
  4. Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.

    A neutron star born turning too slowly

    Collapse an iron core a few thousand kilometres across down to twelve, and conservation of angular momentum multiplies its rotation rate by about eighteen thousand. A model with no transport in it delivers a newborn pulsar at the break-up limit; the ones that are observed turn twenty times slower, which is a measurement of the core before it fell.

    part 4 · stars

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