Astronomy & Celestial Mechanics
the sky, computed
Galaxies

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.

Assumes Interstellar medium and Synchrotron radiation.

17 min read 10 figures Scales that defeat intuitionMeasuring the unreachable

Every other messenger in astronomy travels in a straight line. Photons do, neutrinos do, gravitational waves do; the whole apparatus of the subject rests on the fact that pointing a telescope at a direction means looking at what is in that direction.

Cosmic rays do not. They are charged, the galaxy is magnetised, and a charged particle in a magnetic field spirals. By the time one arrives its direction of travel is a statement about the last field line it happened to be on, and nothing else. The cosmic-ray sky, measured with enormous care by detectors covering thousands of square kilometres, is almost perfectly uniform — anisotropies of a part in a thousand — and that uniformity is the single most important thing it says.

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.
Fig. 1 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. Below the energy at which the radius exceeds the disc, a particle is stored and stirred for millions of years; only above the halo crossing does it travel in anything like a straight line.
The field a collapse would arrive with, and the one it has. Field strength against hydrogen density, both logarithmic, over the eight decades between diffuse gas and a protostellar core. The steeper line is what perfect flux freezing demands: a sphere collapsing conserves both mass and flux, so B goes as R⁻² while ρ goes as R⁻³, and therefore B goes as the two-thirds power of the density exactly — an exponent with no free parameter in it. The shallower locus is what Zeeman measurements find: a flat branch at about 10 µG up to 300 per cubic centimetre, where the density is rising and the field is not, and a rise as the 0.65 power of the density above it. By the density of a core the two differ by a factor of 1, and a star built at the frozen-flux value would carry a field four orders of magnitude beyond anything measured on one. The flat branch is the important half: it says the gas is moving along field lines without dragging them, which is what gravity does to a cloud that is still magnetically supported, and it locates where the freezing has to break.
Fig. 2 And how the field strength that sets the gyroradius is arrived at. Flux freezing ties the field to the gas, so compression raises it as a power of the density near two thirds — which means a field measured in a diffuse cloud predicts the field in a dense one without any dynamo being invoked. Every field strength in this essay is inferred that way rather than measured in place, and the exponent is the whole of the inference.

One formula, and two crossings

The radius of the helix is the particle’s momentum divided by its charge and by the field strength. In units the subject uses, a proton of one petaelectronvolt in a one-microgauss field has a gyroradius of about one parsec.

That is a startling number and it is worth sitting with. A petaelectronvolt is a macroscopic energy — comparable to a well-struck tennis ball — carried by a single proton, and the galaxy’s field bends its path on a scale of a parsec. Multiply the energy by a thousand and the radius is a kiloparsec, which is beginning to be comparable to the thickness of the disc.

The disc’s half-thickness is about a hundred and fifty parsecs, and at three microgauss that corresponds to a proton energy near half a petaelectronvolt. The halo extends to perhaps fifteen kiloparsecs, corresponding to about fifty. Above the first the particle can no longer be held by the disc alone; above the second, not by the galaxy.

Those two crossings are the whole architecture of the problem, and they are computed from one formula with one field strength in it. Everything the measured spectrum does, it does near them.

The confinement is not simply a matter of the gyroradius fitting, though, and the distinction matters. A particle whose gyroradius is small compared with the system does not merely circle; it also scatters off irregularities in the field, changing which field line it is on, and the resulting motion is a random walk rather than a helix. So the storage time is a diffusion time rather than a crossing time, and it is far longer — millions of years rather than the thousand a straight crossing of the disc would take.

That random walk is what erases the direction so thoroughly. A particle that had merely spiralled would retain some memory of where it entered; one that has scattered ten thousand times retains none, and the residual anisotropy of a part in a thousand is a measurement of how nearly complete the erasure is.

There is one complication that matters enormously later. The gyroradius depends on the charge as well as the energy, so a fully stripped iron nucleus with twenty-six protons’ worth of charge has a gyroradius twenty-six times smaller at the same energy. Confinement is therefore a statement about rigidity — momentum over charge — rather than about energy, and any feature produced by confinement should appear at an energy proportional to the nuclear charge.

The spectrum, and what its bends mean

The all-particle spectrum runs as a power law over eleven decades of energy and thirty-two of flux, which makes it one of the most extraordinary measurements in physics. To see any structure in it at all the flux has to be multiplied by a power of the energy.

Ten decades of one power law, with two places where it bends. The cosmic-ray spectrum, multiplied by energy to the 2.7 so that its features can be seen at all — undivided, it falls by thirty orders of magnitude across this plot and every bend in it is invisible. The knee at 3·10¹⁵ electronvolts is where the spectrum steepens from E^−2.7 to E^−3.1, and it sits within a factor of a few of the energy at which a proton's gyroradius in a microgauss field becomes comparable to the thickness of the galactic disc: above it, confinement begins to leak. The ankle at 3·10¹⁸ is where it flattens again, which is read as a galactic population running out and an extragalactic one taking over. Above 5·10¹⁹ the spectrum is cut off, because a proton that energetic loses energy to the microwave background within about fifty megaparsecs and cannot have come from further. Three features, each of them a statement about a magnetic field: one about the galaxy's, one about where the galaxy's ends, and one about the fact that empty space is not empty.
Fig. 3 The spectrum, multiplied by energy to the 2.7 so that a straight line means the plain power law and any bend is visible. The knee at three petaelectronvolts is a steepening, the ankle at three exaelectronvolts a flattening, and the cut-off above fifty exaelectronvolts is where the microwave background starts absorbing.

The knee is at about three petaelectronvolts and it is a steepening, from an index near 2.7 to one near 3.1. It sits within a factor of a few of the energy at which a proton’s gyroradius reaches the disc’s thickness, which is the natural reading: above the knee, protons begin to leak out, so the spectrum steepens.

The reading has a test built into it, from the rigidity point above. If the knee is a confinement effect, each nuclear species should have its own knee at an energy proportional to its charge, and the composition should therefore become heavier through the knee region as the protons leave first. Measurements of composition around the knee — which are hard, because at those energies the particle is detected only through the air shower it makes — do show a shift toward heavier nuclei, and that is the main evidence that the knee is about confinement rather than about acceleration running out.

The ankle at three exaelectronvolts is a flattening, and the usual reading is a handover: the galactic population is dying away and a harder extragalactic population is showing through. That reading is consistent with the arrival directions, which stay isotropic through the ankle rather than concentrating toward the galactic plane as a galactic population would.

A galactic population above the ankle would be a strange thing to have. Its gyroradius would exceed the size of the galaxy, so it could not be confined and would stream away at nearly the speed of light — which means it would have to be replenished continuously by sources inside the disc, and those sources would be visible as a concentration of arrival directions toward the plane. No such concentration is seen at the required level, which is a null result doing real work.

There is a second, subtler argument from the same data. If the highest-energy particles were galactic, the anisotropy should grow with energy as the confinement weakens. What is observed is a dipole that grows slowly and points nowhere near the galactic centre, which is what an extragalactic population viewed from a moving observer would produce.

Above about fifty exaelectronvolts the spectrum cuts off. A proton that energetic can photoproduce pions off the microwave background — the target photons are cold, but in the proton’s frame they are gamma rays — so it loses energy over a few tens of megaparsecs. The cut-off was predicted in 1966, immediately after the microwave background was discovered, and observed forty years later.

What is doing the accelerating

Confinement explains where the particles are kept and not where they got their energy. The standard answer is supernova remnant shocks, and the argument for it is largely one of energetics.

The galaxy’s cosmic rays carry an energy density of about one electronvolt per cubic centimetre and are replaced every twenty million years or so. Multiply out and the power required is a few times ten to the fortieth ergs per second. Supernovae release ten to the fifty-first ergs each and occur a few times a century, giving ten to the forty-second — so the requirement is about a tenth of the available power, which is demanding but not absurd.

The mechanism is diffusive shock acceleration, and it is a magnetic mechanism through and through. A particle upstream of a shock is scattered back by magnetic irregularities, crosses into the downstream flow, is scattered back again, and each round trip gains it energy because the two fluids are approaching each other. Repeated crossings give a power law, and for a strong shock the predicted index is exactly two. An index of exactly two is a striking prediction and it deserves a note, because it comes from almost nothing. The compression ratio of a strong shock is four, the fractional energy gain per crossing and the escape probability per crossing both depend on that ratio, and the resulting power-law index is a simple function of it that evaluates to two. No property of the medium, the field or the shock speed enters. That is why the mechanism is believed to operate in supernova remnants, in the solar wind’s termination shock, in the Earth’s own bow shock, and in the lobes of radio galaxies — four settings sharing nothing but a compression ratio.

Two is not 2.7, and the difference is the propagation. A population injected with an index of two and then stored for a time that falls with energy — because more energetic particles diffuse out faster — is observed with a steeper index, and the steepening required is about 0.6 in the index. That is a statement about how the diffusion coefficient depends on energy, and it is one of the numbers the whole picture has to get right.

A radial wind from a rotating star draws a spiral. The interplanetary field out to 5 astronomical units, drawn as the Archimedean spiral it is. Nothing here rotates: the plasma moves radially outward at 400 kilometres a second and the field is frozen into it, so each parcel remembers the longitude it left from and the pattern winds up while the material does not. The pitch angle is arctan(Ωr/v), which is 47° at one astronomical unit — the radius where the star's rotation has carried the footpoint through one radian in the time the wind takes to arrive. The practical consequence is a matter of hours: a flare's particles follow the field rather than the line of sight, so the ones that reach a given planet left a longitude about 61° to the west of it. A magnetically well-connected flare on the western limb delivers a particle storm and a larger one at disc centre does not, and the difference is this geometry rather than anything about the flare.
Fig. 4 The shape the field takes once the source is rotating. A wind blowing radially from a turning star drags the frozen-in field into a spiral, so the field direction at any radius makes an angle with the radial that grows outward — 45° at the Earth, nearly transverse by Jupiter. A particle following that field is not travelling in a straight line from the source, which is the whole reason arrival direction and source direction are different questions.

How long they stay, measured with a clock

The residence time is not a free parameter. It is measured, and the measurement is one of the most elegant in the subject.

Cosmic rays passing through interstellar gas occasionally strike a nucleus and fragment. That is spallation, and it produces nuclei that are essentially absent from stellar nucleosynthesis — lithium, beryllium, boron — in proportion to the amount of material traversed. Measuring the boron-to-carbon ratio at Earth therefore measures the grammage: about ten grams per square centimetre.

That gives a path length, not a time. To convert one to the other requires knowing the density along the path, which is not known — unless one of the spallation products happens to be radioactive with a suitable half-life.

Beryllium-10 is, with a half-life of one and a half million years. Comparing the amount of beryllium-10 that survives with the amount of stable beryllium-9 produced alongside it gives the elapsed time directly, independent of density. The answer is about fifteen million years. Combine the two: ten grams per square centimetre traversed in fifteen million years at the speed of light implies a mean density along the path of about a fifth of a particle per cubic centimetre. The interstellar medium in the disc is around one per cubic centimetre. So the particles spend most of their time somewhere thinner than the disc — in a halo, several kiloparsecs thick, which is exactly where the radio synchrotron emission of edge-on galaxies is seen to extend.

That is a remarkable conclusion to have reached from a chemical measurement. The size of the volume in which cosmic rays are stored — a structure whose existence cannot be established from within the disc by any direct means — was determined by counting boron nuclei in a detector and comparing two isotopes of beryllium. The halo is inferred, and it is inferred from spallation products.

The consistency check is that the same halo shows up in the radio. Synchrotron emission from cosmic-ray electrons traces where those electrons are, and edge-on spiral galaxies show radio haloes extending a few kiloparsecs above and below their discs. Two measurements of the same volume, one made in a laboratory and one made with a telescope pointed at another galaxy.

The crossing the essay turns on moves with the field, and it is worth reading it at two more.

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 0.001 µG a proton's gyroradius equals the disc's half-thickness at 1.4·10¹⁴ electronvolts and the halo's radius at 1.4·10¹⁶, 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.
Fig. 5 The gyroradius against energy for fields five orders of magnitude apart — an intergalactic field, a weak interstellar one, and a stellar surface. The energy at which a particle’s gyroradius exceeds the system’s own size moves in proportion, which is why the same particle is confined in a galaxy and free in the space between them.
The field a collapse would arrive with, and the one it has. Field strength against hydrogen density, both logarithmic, over the eight decades between diffuse gas and a protostellar core. The steeper line is what perfect flux freezing demands: a sphere collapsing conserves both mass and flux, so B goes as R⁻² while ρ goes as R⁻³, and therefore B goes as the two-thirds power of the density exactly — an exponent with no free parameter in it. The shallower locus is what Zeeman measurements find: a flat branch at about 3 µG up to 300 per cubic centimetre, where the density is rising and the field is not, and a rise as the 0.5 power of the density above it. By the density of a core the two differ by a factor of 7, and a star built at the frozen-flux value would carry a field four orders of magnitude beyond anything measured on one. The flat branch is the important half: it says the gas is moving along field lines without dragging them, which is what gravity does to a cloud that is still magnetically supported, and it locates where the freezing has to break.
Fig. 6 The field a collapse arrives with, computed for a weaker starting field and a shallower scaling index. The endpoint moves and the shape does not: flux freezing carries whatever field the cloud began with down to the star, and every observed stellar field is orders of magnitude below what that predicts.

Why they matter to everything else

A population that is invisible in every image and carries an energy density comparable to everything else is not a curiosity, and cosmic rays turn up in three places in this collection.

They are a pressure. Their energy density of about one electronvolt per cubic centimetre is comparable to the thermal, turbulent and magnetic terms, so they are a significant part of what holds the gas disc open against its own weight — a term alongside the magnetic pressure and the turbulence and comparable to both. They are the ionising agent in dense clouds. Ultraviolet light does not penetrate a shielded core, and the ionisation fraction there — one part in ten million — is maintained by cosmic rays alone. That fraction sets the rate at which a magnetised cloud can shed its field, so the pace of star formation is set by a particle flux from elsewhere in the galaxy.

And they drive winds. A cosmic-ray population that streams outward faster than the gas exerts a force on it through the waves it generates, and the resulting pressure gradient can lift gas out of a galactic disc entirely. Cosmic-ray-driven winds are currently one of the more promising explanations for why galaxies retain so much less of their gas than a straightforward accounting allows, and therefore for why the gas runs out before the galaxy does.

The one thing they carry that photons cannot

There is a compensation for the loss of directional information, and it is not small.

A photon tells its observer about the state of the matter that emitted it. A cosmic ray is matter from somewhere else — a sample of nuclei from another part of the galaxy, delivered intact, and measurable element by element and isotope by isotope.

The composition, corrected for spallation, is close to solar with systematic differences that are themselves informative: elements with low first ionisation potential are enhanced by about a factor of four, which says something about what part of the source material gets picked up and accelerated. There is a small excess of neon-22, pointing at Wolf–Rayet star winds contributing to the source — the same mass a massive star does not keep, showing up as a signature in particles that arrive here. And there is a measured, and startling, absence of the isotopes that decay by electron capture, which only makes sense if the particles were accelerated more than about a hundred thousand years after the nucleosynthesis that made them — so the accelerator is not the explosion itself but something acting on already-cooled material. That is a level of detail no photon delivers, and it comes from a messenger whose direction is worthless.

How a particle is detected at all

The instruments deserve a section, because what counts as a measurement changes completely across the spectrum and the changeover is where most of the systematic disagreement lives.

Below about a hundred teraelectronvolts the flux is high enough that a detector above the atmosphere can catch the particles themselves — a stack of silicon and calorimetry on a balloon or a spacecraft, measuring charge, energy and sometimes isotope. That is where the composition and the spallation ratios come from, and the measurements are direct in the strict sense: the particle enters the instrument.

Above that the flux falls below one particle per square metre per year and no affordable detector is large enough. What is measured instead is the air shower: the particle strikes a nucleus in the upper atmosphere, the collision makes pions, the pions decay and interact in turn, and a cascade of billions of particles reaches the ground spread over square kilometres. The primary’s energy is reconstructed from the shower’s size and the primary’s identity from the depth at which the shower peaked. That reconstruction depends on a hadronic interaction model extrapolated well beyond any accelerator’s reach, which is the largest systematic in the field. Two experiments measuring the same sky have disagreed on the absolute energy scale by twenty per cent for years, and twenty per cent on a spectrum this steep is a factor of two in flux.

And two more readings of the geometry the particles actually travel through.

A radial wind from a rotating star draws a spiral. The interplanetary field out to 5 astronomical units, drawn as the Archimedean spiral it is. Nothing here rotates: the plasma moves radially outward at 800 kilometres a second and the field is frozen into it, so each parcel remembers the longitude it left from and the pattern winds up while the material does not. The pitch angle is arctan(Ωr/v), which is 28° at one astronomical unit — the radius where the star's rotation has carried the footpoint through one radian in the time the wind takes to arrive. The practical consequence is a matter of hours: a flare's particles follow the field rather than the line of sight, so the ones that reach a given planet left a longitude about 31° to the west of it. A magnetically well-connected flare on the western limb delivers a particle storm and a larger one at disc centre does not, and the difference is this geometry rather than anything about the flare.
Fig. 7 The Parker spiral for a fast solar-wind stream at eight hundred kilometres a second rather than four hundred. The spiral unwinds: a faster wind carries the field further out before the Sun has rotated, so the angle at the Earth falls from about forty-five degrees to nearer thirty.
The field a collapse would arrive with, and the one it has. Field strength against hydrogen density, both logarithmic, over the eight decades between diffuse gas and a protostellar core. The steeper line is what perfect flux freezing demands: a sphere collapsing conserves both mass and flux, so B goes as R⁻² while ρ goes as R⁻³, and therefore B goes as the two-thirds power of the density exactly — an exponent with no free parameter in it. The shallower locus is what Zeeman measurements find: a flat branch at about 30 µG up to 1000 per cubic centimetre, where the density is rising and the field is not, and a rise as the 0.65 power of the density above it. By the density of a core the two differ by a factor of 1, and a star built at the frozen-flux value would carry a field four orders of magnitude beyond anything measured on one. The flat branch is the important half: it says the gas is moving along field lines without dragging them, which is what gravity does to a cloud that is still magnetically supported, and it locates where the freezing has to break.
Fig. 8 And the same collapse from a denser, more strongly magnetised cloud. The discrepancy at the end grows rather than shrinking, which is the point: no choice of starting condition makes flux freezing produce a star with the field a star actually has, so something must destroy magnetic flux during the collapse.

Where the picture is thin

Three difficulties are worth stating, because the account above is a consensus rather than a proof.

No supernova remnant has been shown to accelerate protons to the knee. Gamma-ray observations establish that remnants accelerate protons — the signature is a spectral feature from neutral pion decay, seen in two remnants — but the energies reached are hundreds of teraelectronvolts rather than petaelectronvolts. Whether a remnant can reach the knee depends on the field being amplified far above the interstellar value by the streaming particles themselves, which is expected and is hard to confirm. The remnants that do show the sharpest X-ray filaments imply fields of a hundred microgauss or more, thirty times the interstellar value, and the synchrotron cooling time in such a field is short enough to explain how thin the filaments are.

Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 3·10⁸ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 10 billion years the budget is 33 e-foldings, which is a factor of 3·10¹⁴. That is the finding: the three tracks reach the same ceiling within 5.5 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived.
Fig. 9 Amplification in the setting where it is established. A dynamo multiplies a seed field by an enormous factor over a galaxy’s life; the amplification a shock needs is far smaller and far faster, and the mechanism is different — driven by the accelerated particles rather than by the flow.

The diffusion coefficient is inferred rather than computed. Its energy dependence is fixed by requiring the observed spectrum and the observed boron-to-carbon ratio to come out right, which is two constraints for a function, and the resulting scattering rate is larger than plausible field irregularities easily supply.

And the highest-energy particles have no identified source. Above the ankle the arrival directions should start to point, and the largest datasets show a dipole anisotropy of about seven per cent and a hint of clustering toward nearby galaxies. That is progress and it is not identification, and the flux at those energies — one particle per square kilometre per century — is what makes it slow.

The one number that decides whether a field is a fluid. The magnetic Reynolds number, vL/η, for five plasmas, on a logarithmic axis spanning 22 decades. It is the ratio of the term in the induction equation that carries a field with the flow to the term that lets it slip through, so a large value means a field line is a material line — it moves with the gas, it cannot break, and its flux through any surface carried along with the fluid is conserved. Astronomical values run from 10⁹ to 3·10²⁰; the laboratory value is 100, which is why the behaviour that dominates every plasma in this collection is the one that is hardest to arrange on a bench. The size of the number is doing the work: at 10¹⁶ the diffusion time across a molecular cloud is ten million times the age of the universe, so freezing is not an approximation to be checked but a constraint to be worked around, and the interesting question everywhere below is where and how the coupling is allowed to fail.
Fig. 10 Why the field the particle is turning in exists at all. A magnetic Reynolds number above unity means the field is carried with the flow rather than diffusing out of it, and in every astrophysical plasma it is above unity by many orders of magnitude — so the field is frozen in, and its strength follows the gas it is frozen to. That is what makes a galactic field a property of the galaxy rather than a relic, and it is why the gyroradius argument below can be made with a field strength inferred from the gas.

There is a fourth place they matter that is easy to overlook: they are a background. Every detector flown above the atmosphere sees them, and for most instruments they are the dominant source of noise — a cosmic ray depositing charge in a pixel is indistinguishable from a very bright, very small source, and the standard defence is to require a detection in two exposures. That is a nuisance rather than a physics result, and it is the reason cosmic rays were discovered at all: Victor Hess went up in a balloon in 1912 to find out why an electroscope discharged faster than it should, and found that the rate increased with altitude.

The honest summary is that a particle whose direction is erased has told astronomy the thickness of its halo, the age of its own confinement, the composition of matter it never saw made, and the pressure holding a galaxy’s gas open — much as a cluster’s unseen mass is weighed by an average rather than by an image. Losing the direction cost less than it should have.

What made that possible is worth naming, because it is the general lesson. A messenger that arrives with its direction scrambled still arrives with everything else intact: its energy, its charge, its mass, its isotope. Astronomy is so thoroughly built around direction that a source of information carrying none looks at first like no information at all, and the discipline had to learn to ask different questions of it — not where did this come from, but how long was it stored, what was it made of, and how much of it is there. Each of those is a question about an ensemble rather than about an object, and an ensemble is exactly what a scrambled messenger delivers well. Losing the individual is the price of measuring the population, and in this case the population was the thing worth knowing. A single cosmic ray of known origin would be a curiosity; ten thousand of unknown origin, sorted by isotope, are a measurement of the galaxy. Every one of those questions has been answered better by cosmic rays than by anything that travels in a straight line.

Where this goes next

A layer several times thicker than its heat can explain. The vertical density profile of the neutral gas layer, drawn twice. The narrow curve is what thermal pressure alone supports: an 8-kilometre-a-second sound speed against the vertical gravity of the stellar disc gives a scale height of 47 parsecs. The broad curve adds the three pressures that are not heat — turbulent motion, the magnetic field and the cosmic rays — which together roughly triple the total and, because the scale height goes as the square of the effective dispersion, thicken the layer by a factor of 3.1 to 145 parsecs. The observed half-thickness of the H I layer is about 150. The narrow curve is a real prediction and it is wrong, and what is missing from it is precisely the two terms that emit nothing: a field measured by Zeeman splitting and Faraday rotation, and a particle population measured by what it does to a detector on a mountain. A galaxy's gas disc is inflated by things that cannot be photographed, and the thickness is how the inflation is measured. A disc held open by what cannot be photographed Solve hydrostatic equilibrium for the galaxy's gas layer using its temperature and the disc's gravity, and the answer is a layer several times thinner than the one that is there. What is missing from the calculation is turbulence, a magnetic field and a population of cosmic rays — three pressures of comparable size, two of which emit nothing. Three slopes: 2.5, −0.75 and −1.25, and only the middle one is about the electrons. A radio source's spectrum across five decades of frequency, computed for an electron population with index p = 2.5 in a uniform field. Nothing in this curve is a temperature, because a power law has no scale and therefore nothing a thermometer could read. The straight section between the two bends has slope −0.75, measured here off the drawn curve, and the electron index follows from it and from nothing else: α = (p − 1)/2, so a flux ratio between two frequencies is a measurement of the energy distribution of particles in a place no detector will ever visit. The two bends are the other two measurements. Below 80 MHz the source is opaque to its own radiation and rises as ν^2.5 — a slope fixed at 5/2 by the geometry alone, whatever the electrons are doing — and the frequency at which that happens gives an angular size for a source nothing has resolved, because the turnover is where the brightness temperature meets the electrons' own. Above 12 GHz the spectrum steepens by 0.50: the electrons that radiate at high frequency lose their energy fastest, so the top of the distribution has already emptied, and the frequency of the break is a clock. What no part of this curve gives is the magnetic field. Only the field and the particle density together enter the emission, and every field strength ever quoted for a radio source comes from assuming the two share the energy equally. A spectrum with no temperature in it Every spectrum in this collection so far has been a thermometer. A radio lobe's is a power law, and a power law has no scale — so there is nothing for a thermometer to read, and what the shape carries instead is the energy distribution of the particles that made it. The field a collapse would arrive with, and the one it has. Field strength against hydrogen density, both logarithmic, over the eight decades between diffuse gas and a protostellar core. The steeper line is what perfect flux freezing demands: a sphere collapsing conserves both mass and flux, so B goes as R⁻² while ρ goes as R⁻³, and therefore B goes as the two-thirds power of the density exactly — an exponent with no free parameter in it. The shallower locus is what Zeeman measurements find: a flat branch at about 10 µG up to 300 per cubic centimetre, where the density is rising and the field is not, and a rise as the 0.65 power of the density above it. By the density of a core the two differ by a factor of 1, and a star built at the frozen-flux value would carry a field four orders of magnitude beyond anything measured on one. The flat branch is the important half: it says the gas is moving along field lines without dragging them, which is what gravity does to a cloud that is still magnetically supported, and it locates where the freezing has to break. A field that would have arrived ten thousand times too strong A collapsing cloud carries its magnetic field with it, and the arithmetic of that is not negotiable — the field grows as the two-thirds power of the density. Run it from a diffuse cloud to a protostellar core and the answer is four orders of magnitude above anything measured on a young star.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 14 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Cosmic raysGreisen zatsepin kuzmin cutoffGyroradiusInterstellar mediumKneeMagnetic confinementResidence timeShock accelerationSpallationSupernovaeSynchrotron radiation