One slope read twice, and the reading that disagrees

A cosmic ray arriving at the Earth has been scattered so often by the Galaxy’s tangled magnetic field that its direction says nothing about where it came from. It has been stored for tens of millions of years in a volume much larger than the disc — a halo of hot, thin gas held up partly by the cosmic rays’ own pressure — and in that time it has passed through a few grams of interstellar gas per square centimetre — enough to break a small fraction of the carbon and oxygen nuclei into boron, beryllium and lithium, which stars do not make. Those are measurements, not models, and they fix the averages: how much gas, how long.

The averages are not the whole of it. Everything about cosmic-ray storage depends on energy, because the magnetic field scatters a particle according to its rigidity — its momentum divided by its charge — and a more rigid particle is scattered less and leaks out faster. The dependence is usually written as a power law, the diffusion coefficient rising as the rigidity to some power δ, and δ is the single most consequential number in the subject. It decides how the spectrum the sources make is steepened before it is observed, and therefore what the sources must make. It decides the energy at which the Galaxy stops holding particles at all. And it decides how unevenly they should arrive across the sky.

That last consequence is where the trouble is. The same δ that fits the measured composition predicts an anisotropy in arrival directions that grows with energy, and at the energies where the anisotropy has been measured it is much smaller than predicted.

The first reading: what the secondaries say #

The boron in cosmic rays is a clock that reads in grams per square centimetre. Each carbon or oxygen nucleus that strikes a hydrogen nucleus in the interstellar gas has a known chance of fragmenting into boron, so the ratio of boron to carbon at a given rigidity is proportional to the column of gas a typical particle of that rigidity has crossed before escaping. At ten gigavolts that column is about eight grams per square centimetre — the equivalent of passing through the Galaxy’s disc a few thousand times.

If the diffusion coefficient rises as the rigidity to the power δ, particles escape faster in proportion, cross less gas, and the boron-to-carbon ratio falls as the rigidity to the minus δ. The ratio’s slope is the diffusion coefficient’s slope read directly.

The measurement has been made. Magnetic spectrometers flown in space, which measure charge and rigidity particle by particle without the atmosphere in the way, have followed the boron-to-carbon ratio to beyond a teravolt. Above about sixty-five gigavolts it falls as the rigidity to the minus one-third, with an uncertainty of a few hundredths in the exponent. One-third is a famous number: it is the slope Kolmogorov’s theory of turbulence predicts for scattering by a cascade of magnetic irregularities, the same spectrum that describes turbulence in air and water, and that blurs every star seen through the atmosphere. On its face this is a clean result. The Galaxy’s magnetic turbulence is Kolmogorov, and δ = ⅓.

What the sources then have to make #

A slope in the escape steepens the observed spectrum, and the steepening is exactly δ. If sources inject cosmic rays with a power-law spectrum and the Galaxy holds each energy for a time that falls as the energy to the minus δ, the population in storage — the one that is observed — is steeper than the injected one by δ. The observed spectrum of protons, after the corrections for the Sun’s own wind at low energy, has an index near 2.7.

The leading theory of where cosmic rays are accelerated makes a definite prediction for the injected index, and it is two. A particle bouncing back and forth across a strong shock gains a fixed fraction of its energy on each round trip and has a fixed chance of being carried away downstream on each one, and those two fractions, both set by the shock’s compression ratio of four, combine into a power law of index exactly two. The prediction needs nothing about the shock except that it is strong, which is why it is believed to operate in supernova remnants, in the solar wind’s termination shock and in the lobes of radio galaxies alike — the last two seen through the synchrotron light of electrons accelerated the same way.

With δ = ⅓ the sources must inject 2.37, not 2.0. That gap has been absorbed, over the past two decades, by refinements of the shock theory: the accelerated particles push back on the shock and soften it, the highest-energy particles escape the remnant before it has finished accelerating them, and the spectrum released into the Galaxy over a remnant’s whole life is steeper than the spectrum at any moment inside it. Those refinements are physically reasonable and they reproduce an index near 2.3 with plausible parameters. They are also, unavoidably, adjustments made after the measurement to meet it.

The second reading: what the sky says #

The same diffusion that sets the residence time sets a flow. Cosmic rays are made in the disc and leak out through the halo, so there is a net outward current of particles, and any net current is visible as an excess arriving from one direction. A population diffusing out of a slab of half-height H with diffusion coefficient D carries a dipole anisotropy of about $3D/(cH)$. The ratio D/H is not a free parameter: the grammage fixes it, since a particle crosses more gas the longer it lingers, and lingering scales as H/D. So once the boron-to-carbon ratio is matched at one rigidity, the anisotropy at that rigidity is fixed — a few parts in ten thousand at ten gigavolts — and its rise with energy is fixed by δ.

That is the opening figure. At ten teraelectronvolts, a thousand times higher in energy, an anisotropy proportional to D has grown by a thousand to the power δ: ten times for δ = ⅓, sixty-three times for δ = 0.6. Ground-based arrays recording millions of air showers a day — the same technique that, at far higher energies, finds the horizon the microwave background draws — have measured the dipole between one and a hundred teraelectronvolts with great precision, and it is between a few parts in ten thousand and about a part in a thousand — roughly constant across that range, rather than rising, and changing its direction on the sky near a hundred teraelectronvolts.

The diffusion slope that would let shocks inject an index of two predicts twenty-seven times too much anisotropy at ten teraelectronvolts. Even the Kolmogorov slope the secondary ratio prefers predicts four times too much. The normalisation of the estimate is uncertain by a factor of a few, through the arrangement of the sources in the disc and the shape of the halo, so a factor of four is uncomfortable rather than fatal. A factor of twenty-seven is outside anything a geometry can supply. What the anisotropy cannot accommodate is a steep slope: the larger δ, the faster the predicted anisotropy rises, and the measured one does not rise at all.

Two readings that pull the same way #

Put the readings side by side and they agree with each other more than either agrees with the shock theory. The secondary ratio says δ is about a third. The anisotropy says δ is a third or less. The spectrum then says the sources inject 2.37 or steeper. Shock theory, in its simplest form, wants 2.0 and a δ of 0.7.

This is a measurement of a disagreement, and it is worth being exact about which disagreement. It is not a conflict between two observations. It is a conflict between the observations and the theory of the sources, which they constrain from two directions. The composition says the escape is slow to change with energy; the sky says so too; and slow escape leaves the sources making a softer spectrum than the textbook mechanism produces. The anisotropy is the stronger of the two constraints at high energy, because it grows with D while the grammage shrinks with it, and a small number is easier to measure precisely than a large one is to measure a slope of.

Where the leak becomes a flood #

The slope also decides where diffusion stops being the right description at all.

A residence time cannot fall below the time light takes to cross the halo. Where the diffusion estimate reaches that floor, the particles are no longer random-walking out but streaming, and their anisotropy approaches one. For the steepest slope that happens at about eight petaelectronvolts — within a factor of a few of the knee at three, the energy where the cosmic-ray spectrum steepens. That coincidence was one of the attractions of a steep slope: it would explain the knee as the energy at which the Galaxy stops holding its own cosmic rays. With δ = ⅓ the floor is reached only far above the knee, and the knee has to be something else — most likely the maximum energy that supernova remnants can give protons, with heavier nuclei, which carry more charge, reaching proportionally higher energies before they too run out.

The composition measured through the knee favours that second account: the spectrum steepens first for light nuclei and later for heavy ones, in proportion to their charge, which is what a rigidity-limited accelerator does and what a leak from the Galaxy also does. The two explanations differ in what happens above the knee — whether the anisotropy jumps, as a leak requires, or stays small, as an accelerator limit allows — and the measured anisotropy at a few petaelectronvolts stays at the level of a part in a hundred or below.

One sky is one draw #

The anisotropy estimate assumes the sources are smeared smoothly over the disc. They are not. Supernovae happen a few times a century somewhere in the Galaxy — a rate measured from the remnants and neutron stars they leave — and the particles from each one diffuse outward from a point. At the energies where the dipole is measured, the cosmic rays at the Earth come from explosions within a few kiloparsecs over the last few million years, and the gradient at the Sun is set by where the nearest and youngest of those happened to be.

The distribution is wide and lopsided. An average sky has the anisotropy of the smooth estimate; a particular sky can have ten times less or ten times more, depending on whether the nearest recent explosions happened to sit on opposite sides of the Sun or on one. About one history in nine gives an anisotropy as small as the one measured, with the Kolmogorov slope. With a slope of 0.6 the whole distribution moves up by a factor of six, and the measured value is in its far tail.

So the anisotropy does not quite rule out δ = ⅓, and it makes δ = 0.6 very hard to sustain. It also says something about what an anisotropy measurement is. It is not a measurement of the Galaxy’s diffusion coefficient alone; it is a measurement of the coefficient and of the Sun’s particular history. The direction of the dipole carries the second part: a dipole set by a few nearby sources should point roughly towards them, and the measured dipole’s direction, which swings across the sky near a hundred teraelectronvolts, has been read as the signature of exactly that — a nearby source taking over from the bulk as energy rises.

A field that projects the dipole #

There is one further effect that has nothing to do with the sources, and it may be the most important. The Sun sits in a local magnetic field with a definite direction, and cosmic rays diffuse along field lines much faster than across them. A gradient in the cosmic-ray density does not produce a dipole pointing straight down the gradient; it produces one projected onto the local field direction, and if the gradient points mostly across the field, the projected dipole is small.

The local interstellar field’s direction has been measured, by the ribbon of energetic neutral atoms that spacecraft at the edge of the heliosphere see and by the polarisation of nearby starlight, which grains aligned with the field impose, and it lies in a direction that would suppress a dipole pointing towards the inner Galaxy. The measured dipole’s direction between one and a hundred teraelectronvolts sits close to the local field line, as a projected dipole should. That projection can reduce the amplitude by a factor of several, and with it much of the remaining discrepancy for δ = ⅓.

None of these corrections rescues a steep slope. The projection and the source history can each reduce the predicted anisotropy by a factor of a few; together they can make a factor of four disappear. They cannot make a factor of twenty-seven disappear without both acting at their extremes, and even then the predicted rise with energy remains. The anisotropy’s flatness with energy is the feature a steep slope cannot reproduce by any rearrangement of the sources.

What the spectrometers added #

A complication that the simple story leaves out is that δ itself appears not to be one number. Above about two hundred gigavolts the spectra of protons and helium harden — become flatter — by about 0.1 in their index, and the secondary-to-primary ratios flatten at a similar rigidity. Both changes are what would follow if the diffusion coefficient’s slope fell at high rigidity, so that the most energetic particles escape less quickly than an extrapolation from lower energies would suggest.

One explanation is that the turbulence that scatters low-rigidity cosmic rays is generated by the cosmic rays themselves, streaming away from their sources and exciting the waves that scatter them, while at high rigidity the scattering is done by turbulence cascading down from large scales. The first process gives a steep δ at low rigidity, the second a shallow one at high. If that is right, the δ measured in the secondary ratio is a blend of two regimes, and the δ relevant to the anisotropy at teraelectronvolts is the shallower one. That moves the prediction in the right direction for the anisotropy, and away from what shock theory wanted.

Turbulence that has to exist on a scale nobody sees #

The slope of one-third is attractive because it names a mechanism, and the mechanism makes a demand that is worth spelling out. A charged particle is scattered efficiently only by irregularities in the field on the scale of its own gyroradius. A ten-gigavolt proton in a few microgauss gyrates on a scale of about ten trillion kilometres — a few thousandths of a parsec. A ten-teravolt proton gyrates on a scale a thousand times larger. For the diffusion coefficient to follow a single power law across that range, the turbulence has to have a single power-law spectrum across the same range of scales, from a fraction of a light-month to a few light-years, and with the right amplitude.

That range is far below anything that can be imaged. The magnetic field of the Galaxy is mapped on scales of parsecs and more, by Faraday rotation and by polarised dust emission, and its turbulent part is seen to have a roughly Kolmogorov spectrum where it is seen. The scattering of cosmic rays is the only probe of the same turbulence a thousand times smaller. The slope of the boron-to-carbon ratio is, in that sense, a measurement of the shape of the Galaxy’s magnetic turbulence on scales no telescope resolves, made by counting nuclei.

There is a known difficulty with it. Turbulence in a magnetised plasma does not cascade the way turbulence in a fluid does: the eddies stretch along the field and the energy at small scales goes mostly into fluctuations that vary across the field rather than along it. Those are inefficient at scattering particles that gyrate around the field, and the scattering rate they supply falls short of the one the grammage requires by a large factor. Something else must be scattering — the waves cosmic rays excite themselves, or fast magnetosonic modes that cascade differently — and each of those has its own slope. The match to Kolmogorov’s third may be a coincidence of two processes rather than the signature of one.

The general shape of the problem #

The structure here appears whenever a single unobserved function has to account for several observations. The diffusion coefficient is not measured; it is inferred, and each observable is an integral of it against a different kernel. The grammage is weighted towards the gas in the disc. The residence time, from radioactive secondaries, is weighted towards the halo. The anisotropy is weighted towards the gradient at the Sun. Each pins the function at a different place, and agreement among them is a test of whether a single power law is the right form at all.

The answer so far is that it nearly is, with the exponent near a third, and that the sources have to be adjusted to fit rather than the propagation. That is the less comfortable of the two options, because it moves the uncertainty from a part of the problem that can be measured — the Galaxy’s turbulence — to a part that can only be modelled: what a supernova remnant releases over its whole life.

Still open: whether the knee belongs to the sources or the Galaxy #

The measurements that would settle it are of the composition and the anisotropy above the knee, at energies where the flux is a particle per square metre per year and the detectors are arrays of tens of square kilometres. If the knee is an accelerator limit, the anisotropy above it should stay small and the composition should grow heavier steadily, nucleus by nucleus. If it is a leak, the anisotropy should rise sharply once the residence time meets the light-crossing floor. The arrays now running measure both, and the answer so far favours an accelerator limit — which means a shallow slope, sources that inject something steeper than the simplest theory, and an anisotropy at lower energies that owes its smallness partly to the local field and partly to the Sun’s particular neighbourhood. Whether those last two effects are enough to bring the Kolmogorov slope into agreement without strain is the question the direction of the dipole, energy by energy, is beginning to answer.