A positron that cannot have come far

The cosmic rays that reach the Earth are mostly protons and heavier nuclei, and for them the Galaxy is a storage volume with a leak: they are made in the disc, scattered by its magnetic field for tens of millions of years, and lost when they wander out of the halo. Nothing happens to a proton in that time except the occasional collision with a nucleus of gas. It arrives with the energy it was given.

About one cosmic ray in a hundred is an electron, and a smaller number are positrons, and for them the Galaxy is a different place. A light charged particle in a magnetic field radiates, and it scatters the photons it meets up to high energy, and both processes drain its energy at a rate that rises as the square of the energy itself. A high-energy electron does not live long enough to leak out of the halo. It fades where it is. That one difference changes what an electron or a positron at the Earth can be evidence of, and it has turned a measurement of the ratio between them into one of the more surprising results in astrophysics.

A lifetime that falls as the energy rises #

An electron moving through a magnetic field spirals around it and radiates synchrotron light — the same emission that makes supernova remnants and radio galaxies visible. An electron passing through a bath of photons scatters them, and if it is moving close to the speed of light the scattered photons come out with energies multiplied by the square of its Lorentz factor: inverse-Compton scattering, which turns microwave background photons into X-rays and starlight into gamma rays. Both processes take energy from the electron at a rate proportional to the energy density of what it is moving through — the field’s energy density for synchrotron, the radiation’s for inverse Compton — and to the square of its own energy.

In the Galactic disc near the Sun those energy densities are comparable: a few microgauss of field is a few tenths of an electronvolt per cubic centimetre, the microwave background a quarter, and starlight and the infrared glow of dust a few tenths more. Taking one electronvolt per cubic centimetre in all, the loss rate is $dE/dt = -b_0E^2$ with $b_0 \approx 10^{-16}$ per gigaelectronvolt per second, and the time to lose most of an energy E is $1/(b_0E)$.

The crossing at a few gigaelectronvolts divides two regimes. Below it, electrons behave like protons: they leak out of the Galaxy before they fade, and their spectrum is steepened by the same energy-dependent escape. Above it, they fade first, and their spectrum is steepened by one more power of the energy — which is what is measured, an electron spectrum distinctly steeper than the proton spectrum at tens of gigaelectronvolts. Protons do not radiate appreciably at any energy here: the loss rate goes as the inverse fourth power of the particle’s mass, and a proton is two thousand times heavier.

A horizon set by two rates #

A lifetime is not a distance. In a few hundred thousand years a particle diffusing through the Galaxy’s magnetic turbulence random-walks a certain distance, and that distance is the horizon. Over a lifetime $\tau$ with a diffusion coefficient D the typical displacement is $2\sqrt{D\tau}$, and since the energy is changing as the particle goes, the proper expression integrates the diffusion over the energy history.

The diffusion coefficient is not free. It is the one the cosmic-ray nuclei require — normalised to the column of gas the boron-to-carbon ratio measures, with the slope that ratio’s fall implies. At a teraelectronvolt that coefficient is about $10^{29}$ square centimetres per second, and the horizon comes out at a little over a kiloparsec. At ten teraelectronvolts, half a kiloparsec.

The comparison that matters is with the Galaxy. The Sun is eight kiloparsecs from the centre, the disc of young stars is a few hundred parsecs thick, and a kiloparsec around the Sun contains a small fraction of the Galaxy’s supernova remnants and pulsars — a few dozen objects of each kind, of which only a handful are young enough to matter. An electron at a teraelectronvolt arriving at the Earth was made by one of them. The steeper slope, 0.6, stretches the horizon to three kiloparsecs at a teraelectronvolt, but the conclusion is the same: above a few hundred gigaelectronvolts the electrons and positrons are a local population.

The horizon also depends on the energy density of field and radiation, and that is uncertain by a factor of about two near the Sun.

The same inverse square root makes the horizon forgiving: doubling the losses shrinks it by forty per cent, not by half. The starlight part of the losses is also weakened above a few hundred gigaelectronvolts, where the scattering enters the Klein–Nishina regime and becomes less efficient — so the true horizon at the highest energies is somewhat larger than drawn. Neither correction moves it from a kiloparsec to the Galactic scale.

A positron fraction that should fall #

Positrons are made in two ways that can be calculated with confidence. Cosmic-ray protons strike nuclei of interstellar gas, make pions, and the charged pions decay to muons and then to electrons and positrons in nearly equal numbers. And electrons and positrons are made in pairs by high-energy photons in strong fields. The first process is certain to happen everywhere cosmic rays and gas coexist, and its yield can be computed from the measured proton spectrum and the measured gas.

The positrons made by protons have a spectrum that is fixed by the protons’: each inherits a fraction of a proton’s energy, so their source spectrum is about as steep as the protons’, $E^{-2.7}$. Electrons from the sources — supernova remnants accelerating whatever electrons the shock sweeps up — are injected harder, with the same index as the protons at the source, near 2.3. Both then lose energy and diffuse identically. So the ratio of secondary positrons to primary electrons should fall with energy, by about the difference of those indices, and above a few gigaelectronvolts the positron fraction should decline steadily.

It does not. A satellite-borne spectrometer measured it in 2009 rising from about five per cent at ten gigaelectronvolts to well over ten per cent at a hundred, and a larger instrument on the International Space Station has since measured it with far greater precision, rising to about a sixth near three hundred gigaelectronvolts and turning over somewhere beyond. The rise means that positrons above ten gigaelectronvolts are not mostly secondaries. Something makes them directly, with a spectrum harder than the electrons’.

What a single burst delivers #

The horizon says where to look for that something: within a kiloparsec. The question is what a single nearby source would look like in the spectrum.

The shapes are characteristic. A burst’s particles have been losing energy since the moment they were released, and none can have more than $1/(b_0t)$ after a time t, so each spectrum has a sharp cut-off at an energy set by the source’s age. Below the cut-off, the particles that have diffused far enough to reach the Earth are the ones with the largest diffusion coefficient — the most energetic — so the spectrum rises towards the cut-off. A nearby young source therefore delivers a hard spectrum that climbs and then stops. A distant old one delivers a little, at low energy, because its higher-energy particles have long since faded and its lower-energy ones have spread over so large a volume that few are at the Earth.

A rising positron fraction is exactly what a hard nearby source of positrons would add to the falling secondary fraction. The source has to make positrons as well as electrons, and it has to make them hard.

The clock in the cut-off #

The cut-off energy is a clock, and it is worth reading on its own.

The positron fraction’s turnover, then, says something about the age of whatever dominates it. A turnover near a few hundred gigaelectronvolts — or, in the positron spectrum alone, a cut-off near eight hundred, which is how the space-station measurement describes it — points to a source of a few hundred thousand years, which is the age of Geminga. The total electron-plus-positron spectrum, measured from the ground by imaging gamma-ray telescopes that detect the showers electrons make in the atmosphere and from space by calorimeters, shows a sharp break near a teraelectronvolt, beyond which the flux falls steeply. That break too is what a small number of local sources of a few hundred thousand years would produce, and it would be hard to produce with a smooth Galactic population, which would cut off gradually.

The candidates #

Pulsars are the natural candidates, and they were proposed before the measurement. A young pulsar’s magnetic field, spinning in vacuum, generates electric fields strong enough to tear electrons off the surface; the electrons radiate gamma rays in the field, the gamma rays make pairs, and the result is a wind of electrons and positrons in nearly equal numbers. The wind is accelerated further at the termination shock where it meets its surroundings, the pulsar wind nebula, whose synchrotron spectrum shows a hard injection, flatter than $E^{-2}$. The energy available is the pulsar’s spin-down power, which its slowing rotation measures, and a few per cent of the spin-down of Geminga over its life is enough to supply the excess.

Geminga and Monogem have been seen in exactly the way the argument requires. Ground-based water-Cherenkov arrays sensitive to tens of teraelectronvolts found extended haloes of gamma rays around both, several degrees across: inverse-Compton emission from electrons and positrons that have left the pulsar and are diffusing outward. So the pulsars are making the particles, and the particles are escaping. The surprise was in the size of the haloes. They are far smaller than particles diffusing with the Galactic average coefficient would make; the diffusion near the pulsars has to be hundreds of times slower than average to keep the particles that close.

Diffusion that slow everywhere between Geminga and the Earth would stop its positrons from arriving in time, and the first reading of the haloes was that the nearby pulsars could not explain the excess. The later reading is that the slow diffusion is a local condition — the pulsar’s own particles exciting turbulence that traps them, in a zone a few tens of parsecs across — and that beyond it the particles diffuse at the Galactic rate. A two-zone picture of that kind lets Geminga contribute a substantial part of the excess, and a gamma-ray halo detected around Geminga at lower energies, larger and fainter, supports it. How large the slow zone is decides how much of the positron excess Geminga supplies, and that is not settled.

A stranger possibility #

The other proposed source is dark matter. If dark-matter particles annihilate or decay into charged leptons, they make electrons and positrons in equal numbers with a hard spectrum ending at the particle’s mass, and the local dark-matter density would make some of them within the horizon. The positron excess drew a great deal of attention for this reason, and it remains a possibility in principle.

The difficulties are quantitative. The annihilation rate needed to produce the excess is hundreds of times larger than the rate that would leave the right amount of dark matter over from the early universe, so a boost has to be invoked. The same annihilations would deposit energy in the gas at recombination and distort the microwave background, and they would produce gamma rays from dwarf galaxies and the Galactic centre, where any cusp of dark matter is densest; both are measured and both limit the rate below what the excess needs, for the simplest channels. And the sharp cut-off that the horizon argument predicts for an old burst is not what a dark-matter spectrum, ending at a fixed mass, looks like in detail — although with only one cut-off measured, the two cannot yet be told apart by shape alone.

The pulsar explanation has the advantage of needing nothing new. The dark-matter one has the attraction of being something new. The horizon argument is neutral between them: it says only that whatever makes the excess is near, which a pulsar must be and a dark-matter halo, which surrounds the Sun, automatically is.

Ages the pulsars state for themselves #

The cut-off is a clock only if the source’s age is known independently, and for pulsars it is — approximately. A pulsar slows down as its rotation drains into its wind and its radiation, and the ratio of its period to twice the rate at which the period is growing is a time: the characteristic age, which equals the true age if the pulsar was born spinning much faster than it is now and has braked in the standard way since. For Geminga that ratio is about 340 thousand years; for Monogem about 110 thousand. Both are measured from radio and gamma-ray timing to many significant figures, and both are uncertain in the way every characteristic age is — by the unknown birth period and by departures from the assumed braking.

Read against the clock in the figure, those ages give cut-offs near nine hundred gigaelectronvolts and near three teraelectronvolts. The first is where the positron spectrum’s cut-off sits. The second is beyond anything the positron measurements reach, and inside the range where the combined spectrum of electrons and positrons shows its break. That is not a demonstration — the ages are approximate, the loss rate is uncertain by a factor of two, and the cut-off of a source that released its particles over its life is smeared rather than sharp — but it is the kind of agreement an explanation needs and a coincidence would rarely supply: two independently measured clocks, one in a pulsar’s rotation and one in a cosmic-ray spectrum, pointing at the same few hundred thousand years.

Two instruments, one of which cannot tell the charges apart #

The positron measurements come from magnetic spectrometers in orbit: a particle’s track bends in a strong magnet one way for a positron and the other way for an electron, and the curvature gives the momentum. The method is clean and it is limited by the magnet, because a more energetic particle bends less and above a teraelectronvolt the bend is too small to measure its sign reliably. The positron fraction is therefore known only below about a teraelectronvolt, and its behaviour beyond the turnover is uncertain.

The combined spectrum of electrons and positrons reaches much higher, because it can be measured from the ground. An electron entering the atmosphere makes a shower of particles and Cherenkov light that the imaging telescopes built for gamma-ray astronomy record; an electron’s shower is almost indistinguishable from a gamma ray’s, so electrons are a background those telescopes measure in order to remove it, and in doing so they measure the electron spectrum to twenty teraelectronvolts and beyond. They cannot tell an electron from a positron, since the atmosphere has no magnet. A calorimeter in orbit measures the same combined spectrum directly, and both find the break near a teraelectronvolt and a steep decline above it.

So the two halves of the story are measured by different instruments in different energy ranges. The positron fraction rises, peaks and turns over below a teraelectronvolt, measured by charge-separating spectrometers; the combined spectrum breaks near a teraelectronvolt, measured without charge separation. Whether the break in the combined spectrum belongs to the same source as the positron excess is an inference, and the horizon makes it a plausible one: above a teraelectronvolt only a few sources can contribute to either.

What the horizon cannot show #

The figures use a smooth diffusion coefficient, the same everywhere and the same in every direction, and a homogeneous energy density. Each of those is an idealisation. Diffusion in the Galaxy is faster along the field than across it, and the local field near the Sun has a definite direction, so the horizon is an ellipsoid rather than a sphere — longer along the field. Sources in the plane of the disc at the right orientation can reach the Earth from further than the figures suggest; sources off that axis, from less far. The slow-diffusion haloes around pulsars show that the coefficient can vary by two orders of magnitude over tens of parsecs, and the Sun itself sits in a local bubble of hot, thin gas whose magnetic turbulence is not measured.

The single-burst spectra treat each source as releasing all its particles at once. A pulsar releases them over its whole life, mostly early, and the spectrum at the Earth is a sum of bursts of different ages with different cut-offs. That smears the sharp cut-offs into steep declines. And the figures leave out the Sun’s own wind, which modulates all charged cosmic rays below a few gigaelectronvolts and is the reason the positron fraction at the lowest energies changes with the eleven-year cycle.

A messenger that cannot travel #

There is a pleasing inversion in the whole subject. Charged cosmic rays are useless for locating their sources, because the Galaxy’s field scatters their directions until they arrive from everywhere at once. Every argument about their origin is therefore an argument from composition, spectrum and timing rather than from direction. For protons that makes the sources a statistical abstraction — the supernova remnants of the Galaxy, as a population.

Electrons and positrons are scrambled just as thoroughly, but they cannot have come far. The loss that makes them short-lived turns their spectrum into a map of distance: the higher the energy, the nearer the source. Above a teraelectronvolt the population at the Earth is made by a few objects that can be named, at distances that are known, with ages that can be compared with the cut-offs. A messenger that has lost its direction and its range together has, for that reason, become specific again. The positron fraction is not a property of the Galaxy that happens to be measured locally; it is a property of the Sun’s neighbourhood that happens to be measured in the Galaxy’s cosmic rays.

Still open: how slow the diffusion is around a pulsar, and how far that slowness reaches #

The positron excess will be explained by nearby pulsars if their particles escape the slow-diffusion zones that the gamma-ray haloes reveal, and it will need something else if they do not. The haloes measure the diffusion coefficient within a few tens of parsecs of each pulsar; what lies between that and the Galactic average, over the remaining two hundred parsecs to the Earth, is not measured by anything except the positrons themselves. The test is the spectrum at the highest energies. Particles from Geminga above a teraelectronvolt have left its slow zone only recently, if at all, and their arrival at the Earth depends on how far the zone extends. A measurement of the positron spectrum beyond a teraelectronvolt — which needs a magnetic spectrometer larger than any yet flown, or a way of separating positrons from electrons in air showers that does not yet exist — would say whether the nearest pulsar is the answer or merely the nearest candidate.