Astronomy & Celestial Mechanics
the sky, computed
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Cosmic rays — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The energy at which the sky begins to point. Gyroradius against energy for a singly charged particle, in three field strengths, with the thickness of the galactic disc and the size of its halo marked. A cosmic ray is not an image of anything: its path is a helix about a field line, and by the time it arrives the direction it came from has been erased. The erasure is quantitative. At 3 µG a proton's gyroradius equals the disc's half-thickness at 4.2·10¹⁷ electronvolts and the halo's radius at 4.2·10¹⁹, so below the first the particle is stored and stirred for tens of millions of years, and only above the second does it travel in something like a straight line. The measured spectrum has a break — the knee — at about 3×10¹⁵ eV, which is within a factor of a few of the first of those; the sky only starts showing structure above 10¹⁹, which is the second. Two features of a spectrum measured on the ground, both located by one straight line on this plot.

    The energy at which a sky begins to point

    A cosmic ray arrives from a direction that has nothing to do with where it came from. Its path is a helix about a galactic field line, and only above the energy at which that helix is bigger than the galaxy does the arrival direction start to mean anything — which is one particle per square kilometre per century.

    part 1 · galaxies
  2. The distance a proton keeps its energy, against the energy. The energy-loss length of a cosmic-ray proton — the distance over which it would lose all its energy at its current rate — against its energy, computed from the photon density of the microwave background at 2.725 K and the cross-section for photoproducing a pion, which peaks at the Δ resonance. The dashed line is the loss to cosmic expansion alone, about 4,283 Mpc. Only above about 6.1·10¹⁹ eV does the combined loss length fall below a gigaparsec, because below that few background photons are energetic enough in the proton's frame to cross the 145 MeV threshold. Above it the loss length collapses: 144,327 Mpc at 3.16·10¹⁹ eV, 155 Mpc at 10²⁰ and 19 Mpc at 10²¹, where every proton is above threshold and the loss is set by the photon density alone. A proton seen above 10²⁰ eV has come from within a hundred megaparsecs or so — a horizon drawn by light at three kelvin. Pair production on the same photons, which costs about a gigaparsec near 10¹⁹ eV, is not drawn.

    A horizon drawn by light at three kelvin

    A proton above about fifty exaelectronvolts cannot cross the universe, because the microwave background, seen from the proton, is a bath of gamma rays that it loses energy to by making pions. The cut-off in the cosmic-ray spectrum sits where that horizon predicts. But a source that simply cannot accelerate particles any harder would put a cut-off in the same place, and what the particles turn out to be made of says the second explanation may be the right one.

    part 2 · galaxies
  3. The anisotropy each slope of escape predicts. The dipole anisotropy of cosmic rays diffusing out of a halo 4 kpc high, 3D/(cH), against energy, for diffusion slopes of ⅓, 0.5, 0.6, each normalised to the boron-to-carbon grammage at 10 GV — so all three agree at low energy and part above it. The shaded box is the range of dipole amplitudes measured between 1 and 100 TeV, a few parts in ten thousand to about a part in a thousand. At 10 TeV the three predict 0.0043, 0.014, 0.027: the slope that would let shocks inject an index of two overshoots the measurement by 27 times, and even Kolmogorov's ⅓ by 4. The normalisation of the prediction is uncertain by a factor of a few, through the arrangement of the sources and the shape of the halo; its rise with energy is not, because it is the same slope that the spectrum and the secondary ratio require.

    One slope read twice, and the reading that disagrees

    How fast cosmic rays leak out of the Galaxy depends on their energy through a single power, and that power is measured twice: in the falling ratio of boron to carbon, and in how unevenly the particles arrive across the sky. The first reading says one-third. The second, taken at face value, says the leak must be slower still — and neither leaves room for the spectrum shock acceleration predicts.

    part 3 · galaxies
  4. How far an electron can have come, by its energy. The distance an electron or positron can diffuse in its radiative lifetime, 2√(∫D dt), against its energy, for the diffusion coefficient fitted to the nuclei with slopes ⅓ and 0.6, and 1 eV/cm³ of field and radiation. At 100 GeV the horizon is 2.7 or 4.7 kpc; at 1 TeV, 1.24 or 2.95; at 10 TeV, 0.57 or 1.86. The marks are the distances to the nearest middle-aged pulsars and to the Galactic centre. Above a teraelectronvolt the electrons and positrons arriving at the Earth were made within about a kiloparsec — inside the Sun's own neighbourhood of the disc — and a feature in their spectrum is a feature of a handful of nearby sources, not of the Galaxy.

    A positron that cannot have come far

    A proton at a teraelectronvolt can wander the Galaxy for millions of years. An electron or a positron at the same energy radiates its energy away in a few hundred thousand, and in that time it can diffuse about a kiloparsec. So the rising fraction of positrons among high-energy cosmic rays is not a fact about the Galaxy. It is a fact about a few objects within a thousand parsecs of the Sun.

    part 4 · galaxies

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