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Where the model stops — part 3 of 3

Every picture in physics has a domain of validity, and the interesting physics usually lives at its edge.
London's attraction, and the distance where it gives out. The dispersion energy between two identical atoms times the sixth power of their separation, relative to London's constant, against the separation in units of c/ω₀ — the reduced wavelength of each atom's strongest transition — on logarithmic axes, from the Casimir–Polder integral with a one-resonance polarisability. Close in, the product is one: London's inverse sixth power. It has fallen to a half by 3.82 c/ω₀, and far out it falls as one over the distance, so the energy goes as the inverse seventh power, reaching 23.2 per cent of London's value at ten reduced wavelengths and 2.44 per cent at a hundred. For two helium atoms, whose strongest line is at 58 nm, c/ω₀ is about 9 nm; for two sodium atoms, 94 nm. Electromagnetism

The attraction that weakens when light is too slow

Two neutral atoms attract because each one's fluctuating electrons polarise the other, and the two fluctuations fall into step. London's calculation of the force, as the inverse sixth power of the distance, assumes the field from one atom reaches the other instantly. It does not. Beyond a distance set by the wavelength of the atoms' own strongest transition — nine nanometres for helium, about a hundred for sodium — the field returns too late to find the fluctuation that made it. The attraction then falls one power faster. Casimir and Polder worked this out in 1948, and it was the starting point for the Casimir force between plates.

 Heat that arrives everywhere at once, and heat that waits. The temperature spread from a point of heat released at time zero, in one dimension, at times 0.5, 2, 8 relaxation times, from the diffusion equation (dashed) and from Cattaneo's telegraph equation (solid) with the same diffusivity, distances in units of the signal speed times the relaxation time. At 0.5, the diffusion equation has put 62 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 78 per cent is still in the two fronts; at 2, the diffusion equation has put 32 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 37 per cent is still in the two fronts; at 8, the diffusion equation has put 5 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 1.8 per cent is still in the two fronts. The fronts, drawn as the vertical marks, carry heat outward at the signal speed and decay; behind them the telegraph solution fills in and, after many relaxation times, becomes the diffusion equation's Gaussian. Relativity

The heat that arrives before it could

Release a pulse of heat at one point, and the diffusion equation — the law that has described heat flow since Fourier — says the temperature rises everywhere at once, however far away. A thermometer a kilometre off registers a change the instant the heat is released. That is an infinite signal speed, and relativity forbids it. The flaw is harmless for a kettle and fatal for a relativistic fluid. The repair is a relaxation time: heat flow takes a moment to respond to a temperature gradient. With it the equation carries heat at a finite speed, sends a sharp front ahead of the spreading warmth, and turns short-wavelength heat into a damped wave.

 A delay the Planck scale would write into a burst. The arrival delay, relative to a low-energy photon emitted at the same moment, of photons of each energy from a gamma-ray burst at redshift 0.903, if light slowed in proportion to its energy over an energy scale, on logarithmic axes, for that scale at the Planck energy and at ten times and a tenth of it. At the Planck energy a 31 GeV photon would arrive 1.16 s late. The 31 GeV photon from GRB 090510 arrived 0.83 s after the burst began (marked), so a delay of more than that is excluded, and with it any scale below 1.4 times the Planck energy on this simple reading. Fermi's own analysis, allowing for when in the burst the photon could have been emitted, set the limit at 1.2 times the Planck energy: a speed that depends on energy at first order has been excluded at the scale where it was expected. Astrophysics

The photons that raced for seven billion years

Nothing on a laboratory bench can reach the Planck scale, but a gamma-ray burst halfway across the universe can amplify a Planck-scale effect until it is a second long. If spacetime has structure at the Planck length, the simplest guess is that light's speed depends slightly on its energy. Then a burst's high-energy photons would arrive measurably later than its low-energy ones, after a race lasting billions of years. In 2009 a 31 GeV photon from a burst 7.5 thousand million light-years away arrived within 0.83 s of the burst's start. That excludes the simplest version of the idea at the Planck energy itself.

 The heaviest star free neutrons can hold up. Mass against radius for cold stars of free neutrons, each point a different central density, held up by the pressure of the neutrons' exclusion alone: in general relativity and, with the same gas, in Newton's gravity. Relativity's curve reaches a highest mass of 0.71 solar masses at a radius of 9.5 km and turns back; Newton's keeps rising towards 5.8 solar masses as the stars shrink. The measured masses of neutron stars reach about 2.08 (dashed), three times what free neutrons allow, so the matter inside has to be stiffer than a gas of free neutrons: the repulsion between nucleons at short range is holding them up. Astrophysics

The pressure that weighs down what it holds up

In Newton's gravity a star is held up by pressure, and stiffer matter can hold up more: a star of free neutrons could reach almost six solar masses. In general relativity the pressure itself has weight. Every layer squeezed harder to hold up the layers above adds to the pull it is resisting, and past a point pressing harder makes things worse. The same gas of neutrons then tops out at 0.71 solar masses. Neutron stars twice as heavy exist, which says what their insides must be like — and no matter of any kind can hold a static body smaller than nine-eighths of its Schwarzschild radius.

 Blood's viscosity, measured in tubes of different sizes. The viscosity of blood at a haematocrit of 45 per cent, as a multiple of the plasma's, measured in glass tubes of each diameter (the fit of Pries and colleagues to many experiments), against the diameter on a logarithmic axis, with the two-layer model of a core inside a cell-free sleeve 2.5 μm thick, the core 4 times as viscous as plasma (dashed). In wide tubes blood is 3.20 times as viscous as plasma. From about half a millimetre down it gets steadily thinner, to 1.25 times at 6.8 μm — about the size of one red cell — and then climbs steeply, to 5.5 at 3 μm, where cells must fold to get through at all. Which viscosity blood has depends on the tube it is in. Fluids

The viscosity that belongs to the tube

Measure the viscosity of blood in a wide glass tube and it is about three times that of the plasma it is made from. Measure it in a tube a tenth of a millimetre across and it is lower; in one a hundredth of a millimetre across, lower still, barely more than plasma's. Then, below about six micrometres, it shoots up. The blood has not changed. The red cells push themselves away from the wall and leave a sleeve of plain plasma a few micrometres thick for the rest to slide in, and in a narrow tube that sleeve is a large part of the flow. A viscosity that depends on the pipe is no longer a property of the fluid alone.

 A target that grows as the traveller slows. The capture cross-section of a non-rotating black hole for particles arriving with speed v, in units of the horizon's own cross-sectional area π(2GM/c²)², against v/c, both on logarithmic axes (solid). For light it is 27/4 = 6.75 times the horizon's area. It grows as the particle slows, following 4c²/v² (dashed) — 404 at a tenth of light speed and 7.43·10⁶ at the 220 km/s of the Galaxy's dark matter. The dotted curve is a Newtonian ball of radius 8GM/c² — four horizon radii — capturing by ordinary gravitational focusing, and for slow particles the hole and the ball cannot be told apart. Astrophysics

The target that grows as the traveller slows

Light aimed at a black hole is swallowed if it passes within about five of the hole's gravitational radii, a disc six and three-quarter times the horizon's own. A particle arriving slowly is swallowed from much further out, and the disc grows as the inverse square of its speed: dark matter drifting at 220 kilometres a second sees a target seven million times the horizon. The reason is that slow matter carries almost no angular momentum for a given aim, and angular momentum is the only thing that keeps anything out. For slow traffic the hole behaves exactly like an ordinary ball four times its own size.

 Two ways to be wrong, and the apparatus that sees each. The fractional sag of the potential at the centre of a charged spherical shell below the shell's own, on a logarithmic axis, against the shell's radius in metres, also logarithmic, for two different departures from Coulomb's law. If the force fell as 1/r^(2+ε) with ε = 10⁻¹⁶, the sag is 3.07·10⁻¹⁷ whatever the shell's size (dashed): the law has no length in it, so neither does its failure. If the photon had a mass of 10⁻¹⁴ eV, the sag is (μR)²/6 (solid): 4.3·10⁻²⁰ for a centimetre shell and 4.3·10⁻¹⁶ for a metre. A mass introduces a length, and the signal grows as the square of the apparatus compared with it — so the strongest bounds on a photon mass come from the largest fields, not the most sensitive laboratories. Electromagnetism

The mass that would make Gauss's law leak

Gauss's law says the flux out of any closed surface counts the charge inside, whatever the surface's size. If the photon had even a tiny mass, that would stop being true: the field of a charge would die off exponentially beyond a length ħ/mc, the flux through a large sphere would fall short of the charge, and the inside of a charged shell would no longer be field-free. There are two different ways Coulomb's law could fail, and they are caught by opposite strategies. A wrong exponent has no length in it and shows equally in any apparatus; a mass has a length, and shows as the square of the apparatus's size. That is why the tightest limit on the photon's mass comes not from a laboratory but from the magnetic field the solar wind carries past Pluto.

 The heat two surfaces exchange when they almost touch. The radiative heat-transfer coefficient between two parallel half-spaces at 300 K, in W/m²K on a logarithmic axis, against the width of the vacuum gap between them, also logarithmic, from fluctuational electrodynamics: silicon carbide (solid) and gold (dashed), with two perfect black bodies, 4σT³ = 6.12 W/m²K, for comparison (dotted). Far apart, silicon carbide exchanges 3.3 W/m²K — less than black — and gold, a good mirror, only 0.024. Closer than the thermal wavelength, about 17 µm, waves that cannot leave either surface begin to tunnel across the gap. At 100 nm silicon carbide exchanges 137 W/m²K, at 10 nm 9338, at 1 nm 9.3·10⁵ — rising as the inverse square of the gap, 1.5·10⁵ times the black-body value at 1 nm. Gold exceeds black only below a few hundred nanometres and then grows much more slowly: 1284 at 10 nm, 2561 at 1 nm. Quantum

The heat that crosses a gap too narrow for light

Stefan and Boltzmann's law is usually read as a ceiling: no surface can radiate more heat than a perfect black body at the same temperature, 6.1 watts per square metre for every degree of difference at room temperature. It is a ceiling only for surfaces that are far apart. Bring two of them closer than the wavelength of their thermal glow and they begin to exchange waves that never leave either surface — waves that cling to it and die away within a wavelength — and those waves tunnel across the gap. Two plates of silicon carbide ten nanometres apart exchange fifteen hundred times what two black bodies can, almost all of it in a single colour, and the colour is the one in which silicon carbide is too shiny to glow at all.

 The orbit an observer would reconstruct. The star's velocity along the line of sight, measured from the Doppler shift of its spectrum, as a fraction of its true orbital speed, plotted against the time the light arrives, in orbital periods, if light left the star at c plus the star's velocity. With light at c (dotted) the curve is the sine wave of a circular orbit. At 0.6 times the critical distance (solid) the same circular orbit arrives as a lopsided curve — a slow rise and a steep fall — that an astronomer would fit with a highly eccentric orbit that is not there. At twice the critical distance (dashed) the curve doubles back on itself: at some moments the spectrum would show two or three velocities at once, as if the star were in several places. Willem de Sitter pointed out in 1913 that spectroscopic binaries show nothing of the kind. Relativity

The binary star that would be seen twice

If light thrown forward by a moving source travelled faster, the way a ball thrown from a moving train does, then light from a star swinging towards the Earth in a binary orbit would catch up with light it had sent out earlier. From far enough away the two would arrive together, and the star would be seen in two places at once. Willem de Sitter pointed out in 1913 that nothing of the kind is seen, and that this settles whether the speed of light depends on its source. The argument had a loophole that took half a century to notice — interstellar gas re-emits light at its own speed — and closing it needed X-ray stars.

 The stiffness a two-solar-mass star demands. The heaviest neutron star the model matter can support, in solar masses, against the constant square of the sound speed assumed above 300 MeV/fm³, as a fraction of c². At the conformal value, one-third (dashed), it is 1.92 solar masses; the heaviest star the model can make rises to 2.97 at the causal limit. To hold up the 2.08 solar masses of J0740+6620 (shaded) the square of the sound speed must exceed 0.394 of c² somewhere above that density — above the conformal one-third, by this model's reckoning. The conclusion depends on the model's low-density part, which the next figure varies. Relativity

The sound speed a neutron star needs

Heat any gas without limit and the speed of sound in it climbs towards a ceiling: not the speed of light, but light's speed divided by the square root of three, the sound speed of a gas of massless particles. For decades it was a natural guess that no matter could beat that ceiling, since the densest matter ought to behave like a gas of free quarks. Neutron stars say otherwise. A star of twice the Sun's mass, packed into a ball twenty-five kilometres across, can only be held up if its core is stiffer than the ceiling allows — unless nuclear physics fails at densities where it is believed to hold. Somewhere inside the heaviest neutron stars, sound almost certainly travels faster than it can in any free gas.

 The Casimir pressure between warm plates, from vacuum to heat. The attractive pressure between two parallel, perfectly reflecting plates at 300 K against their separation, on logarithmic axes, from the Lifshitz sum over Matsubara frequencies. Dashed: the zero-temperature law π²ħc/240a⁴. Dotted: the classical law ζ(3)kT/4πa³, which contains no Planck constant. At 1 μm the full pressure is 0.0013 Pa, within 0.2 per cent of the vacuum value; at 10 μm it is 3.96·10⁻⁷ Pa, 3.05 times the vacuum value and within 0.002 per cent of the classical law. The two laws cross at 3.28 μm; the thermal length ħc/kT is 7.63 μm. The second solid curve removes one term — the static magnetic fluctuation, which a metal of finite conductivity does not reflect — and sits 15 per cent lower at 1 μm and a factor of two lower far out. The plates are otherwise ideal mirrors: a model, chosen to isolate that one term. Electromagnetism

The force that forgets Planck's constant

Two mirrors in a vacuum attract, with a pressure that contains Planck's constant and nothing else about the world but the speed of light. Warm the mirrors and pull them apart, and past a few micrometres at room temperature the Planck constant drops out of the answer. The force is still there and is now larger than the vacuum alone would give, but it has become classical: a count of thermal fluctuations of the static field, with kT of energy each. Half of that classical force comes from one fluctuation — a static magnetic field — and whether a real metal mirror reflects it is a question that has divided the people who measure the force for twenty years.

 How well a diamond-like crystal conducts heat, from 1 K to 1000 K. The thermal conductivity of a model crystal with diamond's sound speed and Debye temperature, from kinetic theory applied to its phonons, against temperature on logarithmic axes: natural carbon in a sample 1 mm across, isotopically enriched carbon, and natural carbon in a sample 0.1 mm across; copper's room-temperature 400 W/m K is marked for comparison. The model is calibrated to natural diamond's 2,200 W/m K at room temperature. At low temperature the conductivity rises as the cube of the temperature, limited by the sample's walls; at high temperature it falls as umklapp collisions shorten the phonons' paths. Between, it peaks: 3.7·10⁴ W/m K at 100 K for the natural crystal; 7.8·10⁴ W/m K at 94 K for the enriched crystal; 1.2·10⁴ W/m K at 126 K for the thin natural crystal. The model gets the shape — the rise as the cube of the temperature, a peak near 100 K, the fall — but overestimates the heights, because it leaves out the phonon collisions that conserve momentum and every defect but isotopes; measured enriched diamond peaks near 4 × 10⁴ W/m K. An insulator with no free electrons still conducts heat five times better than copper at room temperature. Thermodynamics

The gas of sound that carries a diamond's heat

Diamond has no free electrons and conducts heat five times better than copper. What carries the heat is a gas — not of molecules but of quantised sound waves, phonons — and the kinetic theory that gives the conductivity of air gives the conductivity of a crystal with the same formula: a third of heat capacity times speed times mean free path. The gas is strange in two ways. At low temperature its particles cross the whole crystal without meeting anything, so the conductivity depends on how big the sample is. And at room temperature one atom in a hundred being a neutron heavier is enough to scatter its fastest-oscillating members.

 The redshift of a white dwarf weighs it. The gravitational redshift at the surface of a cold white dwarf, expressed as the velocity that would give the same Doppler shift, against the star's mass; computed from the radius each mass is forced to have. The Sun's surface redshift is 636 m/s. A white dwarf of 0.6 solar masses shifts its light by 30.1 km/s, one of 1.0 by 77.6, of 1.2 by 128 and of 1.35 by 223. Because the radius is not free, the curve is single-valued and steepening: a measured shift returns one mass. The cold model puts Sirius B, at 1.018 solar masses, at 80.8 km/s; its measured mass and radius together give 80.0. Astrophysics

The redshift that weighs a dead star

Light leaving the surface of an ordinary star is reddened by gravity, but the shift alone says little, because a star's radius is free and the shift depends on mass over radius. A white dwarf has no free radius. Degenerate electrons fix its size from its mass, so one measured line shift returns one mass — and the gauge grows sharper the closer the star is to collapse. The first measurement of it, in 1925, was hailed as a confirmation of general relativity and was wrong by a factor of four, in agreement with a forecast that was wrong by the same factor.

 Two stiffnesses, positive together only in a band. The stiffness of the trap for a field-following magnet above a ring base, against height in units of the ring's radius: vertically, proportional to the curvature of the field strength along the axis; sideways, proportional to the curvature of |B| across it, B′²/4B − B″/2. Below half a radius the field's fall is still steepening, the vertical stiffness is negative, and a magnet nudged downward keeps going. Above √(2/5) = 0.632 radii the field strength no longer rises away from the axis, the sideways stiffness is negative, and a magnet drifts off to the side. Only between 0.500 and 0.632 radii are both positive — 6.6 mm of height for a 10 cm ring. For the nominal top, floating at 0.565, the trap's natural frequencies are 1.36 Hz vertically and 0.82 Hz sideways. Electromagnetism

The window a spinning magnet floats in

No arrangement of fixed magnets can hold another magnet still in mid-air, and a spinning top made of a magnet floats above a magnetised base for minutes. The spin does not cancel the theorem; it changes what the top's energy depends on, from one component of the field to the field's strength, which can have a minimum. But the minimum exists only in a band of height a few millimetres thick, for a weight right to a couple of per cent, at a spin neither too slow nor too fast, and the well holding the top is about as deep as the energy of a twenty-gram weight dropped fifty micrometres.

 The speed that stops while the energy keeps coming. The speed of an electron, as a fraction of the speed of light, against the kinetic energy given to it, from special relativity (solid) and from Newton's ½mv² (dashed); dots at the five energies Bertozzi used in 1964. Newton's speed passes the speed of light at a quarter of a megaelectronvolt and is 7.7 times it at 15 MeV. Relativity's speed is 0.863 c at 0.5 MeV, 0.967 c at 1.5 MeV, 0.9948 c at 4.5 MeV and 0.99946 c at 15 MeV. Between 1.5 and 15 MeV the energy grows tenfold and the speed by three and a half per cent. The energy does not stop arriving; the speed stops answering it. Relativity

The speed that stops while the energy keeps coming

In 1964 William Bertozzi gave electrons energies from half a megaelectronvolt to fifteen, timed them over eight and a half metres, and caught them in an aluminium disc to measure the heat they brought. By Newton the fastest should have crossed in under four nanoseconds, seven times faster than light. They took twenty-eight, and so did the slower ones, more or less. The heat, meanwhile, rose in proportion to the energy put in. The experiment is famous for showing that nothing outruns light, but its sharper result is the second half: the energy did arrive, all of it, carried at very nearly the same speed.

 The distance at which the air takes over. Evaporation of water at 20 °C into dry air, in kilograms per square metre per second, against the thickness of the layer the vapour must cross before it is carried away, both on logarithmic scales, with the kinetic and diffusive resistances added in series: 1/u + δ/D, where u = √(RT/2πM) is the speed at which the kinetic flux carries vapour off and D the diffusion coefficient. Dashed, the kinetic ceiling. At one atmosphere the two resistances are equal at δ = D/u = 0.17 μm, about two and a half mean free paths of the air; a layer much thinner than that leaves the surface at its ceiling, and a layer much thicker throttles it in proportion to its thickness. At a tenth of an atmosphere diffusion is ten times faster and the crossover moves out to 1.7 μm. Only in vacuum, or across gaps shorter than a mean free path, does the liquid itself set the rate. Thermodynamics

The puddle that could dry in a second

A liquid in equilibrium with its vapour loses molecules exactly as fast as the vapour delivers them, and kinetic theory says how fast that is. For water at room temperature it is two and a half kilograms per square metre every second — a puddle two and a half millimetres deep gone in a second. Puddles take hours because almost every molecule that leaves is knocked back by the air within a fraction of a micrometre. The ceiling binds only where the vapour can escape in a single free flight: in vacuum, where water freezes itself, in the smallest droplets, and on the filament of a lamp.

 The ring where an axisymmetric field vanishes. Field lines of the slowest-decaying dipole field of a conducting sphere, drawn in a plane through its axis (the axis is the left edge, the sphere's surface the arc). Each line is a contour of the flux function, which is zero on the axis and at infinity and positive between, so it must have a maximum. Here the maximum is on the equator at 0.873 of the radius: there the lines shrink to a point, which in three dimensions is a ring round the axis on which the meridional field is exactly zero. 7 of the 11 drawn lines leave the sphere and close outside; the other 4 close inside, round the ring. The toroidal current that makes the field is not zero on the ring: it is 89 per cent of its largest value. Astrophysics

The field no symmetric flow can keep

The Earth's core would forget its magnetic field in about forty thousand years, and the field is at least three and a half thousand million years old, so something must be making it again all the time. The obvious candidate is the flow of liquid iron. In 1934 Thomas Cowling proved that the simplest version cannot work: no motion of a conductor, however fast or however arranged, can keep up a field that is symmetric about an axis. The reason is a single ring inside the field where the current it needs cannot be driven.

 How well a distance can be known, by how long it is. The smallest uncertainty in a distance l, on logarithmic axes, by three rules. Solid: the clock argument, 1.89 (ℓₚ² l)^(1/3), which grows as the cube root of the distance. Dotted: the Planck length, the floor usually quoted, which does not grow at all. Dashed: the guess that Planck-scale jitters add like a random walk, √(ℓₚ l). a metre: 1.2·10⁻²³ m by the clock argument, 4·10⁻¹⁸ m as a random walk; the Earth to the Sun: 6.4·10⁻²⁰ m by the clock argument, 1.6·10⁻¹² m as a random walk; the observable universe: 9.2·10⁻¹⁵ m by the clock argument, 8.4·10⁻⁵ m as a random walk. By the clock argument the size of the observable universe is uncertain by about the size of an atomic nucleus; as a random walk, by about a tenth of a millimetre. All three rules meet at the Planck length. Astrophysics

The clock that cannot be both light and good

The Planck length is usually quoted as the smallest distance that means anything, as though every longer distance could be known to that precision. Time a distance with light against a clock and the floor turns out to rise with the distance. A light clock wanders while the light is away; a heavy clock wanders less but is closer to being a black hole; and the best compromise leaves a metre uncertain by 750 thousand million Planck lengths and the size of the observable universe by the size of an atomic nucleus. The same compromise, counted up, says a region can hold only as many distinguishable places as its surface has Planck areas.

 Water's surface tension, all the way to the critical point. The surface tension of water against its liquid vapour, from the international standard correlation, against temperature from 0 °C to the critical point at 373.946 °C. 20 °C: 72.7 mN/m; 100 °C: 58.9 mN/m; 200 °C: 37.7 mN/m; 300 °C: 14.4 mN/m; 370 °C: 0.4 mN/m. It falls almost in a straight line for most of the range and then bends down to reach zero exactly at the critical point, where liquid and vapour stop being two things. The bend is a power law, the gap to the critical temperature raised to 1.256. Fluids

The tension that vanishes where the phases meet

Surface tension is usually treated as a property of a liquid, like its density. It is a property of a boundary between two things, and it can only exist while there are two things. Heat water in a sealed tube and its surface tension falls steadily, then faster, and reaches exactly zero at 374 °C, where liquid and vapour stop being different. On the way the surface stops being a surface: the step in density spreads from a molecule's width to thousands, and the rate at which the tension dies turns out to be a number the whole family of critical points shares and the simplest theory gets wrong.

 Every steady flow of gas onto a point mass. The Mach number of steady, spherical, isothermal flow round a point mass against distance from it, both on logarithmic axes, distance in units of the sonic radius, where the inflow reaches the speed of sound. Every curve is a contour of Bondi's conserved integral, which ties the Mach number to the distance. The faint curves are flows that never pass the speed of sound, or that exist only close in or only far out. Two curves cross at the saddle (1, 1): the solid one is accretion, starting from rest far away, reaching the speed of sound exactly at the sonic radius and falling supersonically within it; the dashed one is its time-reverse, a wind. Only the crossing flow joins a gas at rest to a hole that swallows it, and it fixes the rate: λ = exp(3/2)/4 = 1.120 in Ṁ = 4πλ(GM)²ρ∞/c³. Astrophysics

The gas a hole catches from far away

A particle drifting past a black hole is captured only if its aim is almost perfect; miss by a little and it swings round and leaves. Gas is different. Streams passing on opposite sides collide, lose their sideways motion as heat, and fall in, so a hole in warm gas draws from the whole sphere inside which its gravity beats the gas's thermal motion. Bondi worked out the flow in 1952: one steady solution passes smoothly from rest to free fall, it crosses the speed of sound at a single radius, and the rate it fixes grows as the square of the hole's mass — fast enough, left alone, to run away in a finite time.

 A heat capacity that goes up and comes down. The heat capacity per system, in units of Boltzmann's constant, against temperature in units of the level spacing over k: for a system with two levels (red), Schottky's hump x²eˣ/(1 + eˣ)², x = Δ/kT, and for a harmonic oscillator with the same spacing (blue), which climbs to one k and stays there. Both are frozen out when kT is well below the gap. The oscillator then fills its unending stack of levels and reaches equipartition's value. The two-level system peaks at 0.439k when kT = 0.417Δ, and then, with both levels nearly equally occupied and nothing higher to go to, it stops absorbing heat: its capacity falls back towards zero as 1/T². Thermodynamics

The heat capacity that goes up and comes down

Equipartition promises every way a system can move a fixed share of heat, and for a harmonic oscillator the promise is kept: once the temperature is high enough, it holds one k per oscillator and goes on holding it. A system with only two levels cannot keep it. It absorbs heat while its upper level is filling, then stops — the two levels equally full and nowhere higher to go — so its heat capacity rises to a peak and falls back to zero. The hump, named after Walter Schottky, is the signature of a degree of freedom with a ceiling, and the area under it counts how many states were there.

 A governor that hunts. The speed of an engine with a centrifugal governor after its load is suddenly increased, against time in units of √(ℓ/g), for three amounts of friction in the governor's linkage: damping 0, 0.25, 0.6, against Vyshnegradsky's minimum for this governor, 0.397. With no friction the speed oscillates about its new, slightly lower value with a growing swing — the governor hunts, opening and closing the throttle out of step with the engine — until the arms hit their stops and the curve ends. At damping 0.25, below the minimum, it still hunts, more slowly. At damping 0.6, above the minimum, the oscillation dies away and the engine settles at its new speed. The friction that seems to spoil the mechanism is what makes it work. Mechanics

The governor that needed friction to work

James Watt's steam engines were kept at a steady speed by two heavy balls on hinged arms, spun by the engine itself: run faster and the balls fly out and close the steam valve; run slower and they drop and open it. The balls stand at a height set by the speed alone. As engines grew and governors were made more sensitive, many began to hunt — speeding and slowing in a growing rhythm, the governor always too late. In 1868 James Clerk Maxwell wrote the equations down and showed why: a governor with no friction cannot settle at all, and the more precisely it is meant to hold the speed, the more friction it needs.

 A sound that survives only in a hot-electron plasma. The damping of an ion-acoustic wave at kλD = 0.3, as the fraction of its amplitude lost per radian of oscillation, −γ/ω, against the ratio of electron to ion temperature, for hydrogen and argon plasmas, on logarithmic axes: from the kinetic dispersion relation (solid) and from the weak-damping formula (dashed), which is valid only where the damping is small. With equal temperatures the wave loses 0.45 of its amplitude per radian in hydrogen — it is gone within a cycle. At Tₑ/Tᵢ = 10 the loss is 0.031; at 100 it is 0.0126, set by the electrons alone. Argon's floor is lower, 0.0020, because its ions are heavier and its electrons a smaller fraction of a sound wave's speed. Astrophysics

The sound a plasma carries on its electrons' heat

Sound in a gas needs two things: something heavy to be pushed, and a pressure to push it. In an ordinary gas the same molecules supply both. In a plasma they can be split — the ions carry the inertia and the electrons, a thousand times lighter and often a hundred times hotter, carry the pressure — and the result is a sound that travels at a speed set by one particle's temperature and the other's mass. It travels only while the two temperatures stay far apart. Warm the ions to match the electrons and the sound dies within a cycle, with nothing to rub against.

 A current limited by the wire's circumference. The largest supercurrent a wire can carry before the field of its own current destroys superconductivity at its surface, Ic = 2πaHc, against the wire's diameter, for lead, mercury and tin at 1.5 K, on logarithmic axes; dashed, a niobium–titanium wire carrying 3,000 A/mm² in a field of 5 tesla, which grows with the cross-section rather than the circumference. A lead wire 1 mm across carries 192 A — 244 A/mm² — and one 1 cm across only ten times as much, at a tenth of the density. Below 81 micrometres lead carries more than niobium–titanium; above it niobium–titanium wins by a factor that grows with the wire, and it does so in a field of 5 tesla, where lead cannot superconduct at all. Electromagnetism

The current a superconductor stops with its own field

A superconducting wire has no resistance, so it seems it should carry any current at all. It carries a definite amount and no more, and the amount is set not by the metal's ability to conduct but by the field the current itself makes: when that field at the wire's surface reaches the critical field, superconductivity fails there. The rule makes the largest current proportional to the wire's radius rather than its area, makes a thick wire worse per square millimetre than a thin one, and explains why the first superconducting magnet, wound in 1913, did not work.

 The suction that holds two wet plates together. The force needed to pull apart two discs 10 cm across, starting 10 micrometres apart with water between them, at constant opening speeds of 1, 10 and 100 micrometres a second, against time; the dashed line is the largest suction the water can transmit, ½πR²pₐₜₘ = 398 N, reached when the pressure at the centre falls to zero and the water cavitates. At 1 µm/s the force starts at 29.5 N and falls as the cube of the widening gap, to 10.7 N after four seconds, when the gap is 14 micrometres. At 100 µm/s the viscous law would demand 2.9 kN; the water cannot pull that hard, and the force sits on the cavitation limit until the gap has opened enough for the viscous suction to fall below it. Fluids

The wet plates that hold on when pulled

Lift a wet glass plate straight up off another and it resists with a force far beyond its weight; slide it sideways and it comes away at a touch. The grip is not surface tension. It is the water that must rush into the widening gap, and the suction it takes to draw it in grows as the inverse cube of the gap — until the water tears open and the atmosphere sets the limit. The same cube makes a flat base float on a wet table where a ball would settle, and gives a micro-machined accelerometer exactly the damping it needs from a few micrometres of air.

 Sturm's conoid: a line, a circle, a line. The cross-section of a parallel beam after a lens whose focal length is 100 mm in its vertical meridian and 125 mm in its horizontal one, traced ray by ray through a round aperture, at seven distances behind the lens: 60.0, 100.0, 105.6, 111.1, 118.1, 125.0, 135.0 mm. There is no point focus anywhere. At 100 mm the vertical meridian has focused and the beam is a horizontal line; at 125 mm the horizontal meridian has, and it is a vertical line. Between them the beam is an ellipse that turns from wide to tall, and at 111.1 mm — the harmonic mean of the two focal lengths — it is a circle, the circle of least confusion, of diameter 22.2 per cent of the aperture's. Optics

The focus that is two lines

A lens curved more steeply in one direction than the other has no focus. It brings light to a short line at one distance, to a line at right angles at another, and in between to an ellipse that turns from wide to tall and is round at exactly one place. Most eyes are built this way to some degree, so do mirrors used at an angle, and the best any ordinary lens can do for them is to put that round patch on the retina. A cylinder, curved in one direction only, can do what no sphere can: move one line onto the other.

 What an amplifier does to a signal's uncertainty. A signal drawn in the plane of its two quadratures — the in-phase and out-of-phase parts of a wave — as a disc one standard deviation across: a coherent input (black), with the vacuum's own uncertainty; and three outputs at a gain of 4: what a noiseless amplifier would give, the input disc scaled up by √G (dashed), which quantum mechanics forbids; what the least noisy phase-insensitive amplifier gives (blue), the same centre with a disc 1.323 times wider again; and what a phase-sensitive amplifier gives (red), the in-phase quadrature stretched by √G and the other squeezed by 1/√G, with no noise added. Referred back to the input, the phase-insensitive amplifier has added 0.375 of a quantum of noise; the signal's distance from the origin, measured in its own spread, has fallen by 0.756. Quantum

The noise every amplifier must add

An amplifier that magnifies a signal without caring about its phase must also add noise, and not because of hot transistors or bad design. However perfect it is, it adds at least half a quantum of noise referred to its input — exactly as much as the vacuum fluctuations the signal already carried — so a faint coherent signal always leaves with half its signal-to-noise ratio. The reason is the same rule that makes position and momentum uncertain, applied to the two halves of a wave, and an amplifier that broke it would be a machine for copying quantum states.

 The fold beyond which no heap is steady. The centre's steady temperature rise, in units of RTₐ²/E, against the Frank-Kamenetskii parameter δ, for a slab, a long cylinder and a sphere, found by integrating the steady heat equation outward from the centre: each shape has two steady states for every δ below a critical value — a cool, stable one and a hot, unstable one — and none above it. The fold is at δ = 0.878, 2.000, 3.322 for the slab, cylinder and sphere: a sphere tolerates nearly four times a slab's δ at the same half-width, because its surface is larger in proportion to its volume and heat from its centre has more ways out. A real heap, which is neither, has a critical δ between, and fixing it for a given shape is the first step in calculating the size at which a store becomes unsafe. Thermodynamics

The heap that sets itself alight

A haystack, a coal stockpile, a bin of oily rags or a silo of damp grain can catch fire with no spark at all. A slow reaction inside makes a little heat; the heap holds it in; the warmer middle reacts faster and makes more. Whether that ends in a warm heap or a burning one is decided by a single number built from the heap's size, the air's temperature and the reaction's speed — and past one value of it there is no steady temperature at all. The critical size shrinks steeply as the weather warms, and the heap gives almost no warning before it goes.

 A whisper that grows into a note. The amplitude of the tube's lowest mode against time, in units of the time sound takes to cross the tube (a period is two), started from a small disturbance, with the gauze at a quarter of the length, at 0.12 of it, and at three-quarters; heat-transfer strength K = 1. At L/4 the disturbance grows some fortyfold over about thirty-five periods and then stops growing, at an amplitude of 0.83 in these units, where the air's to-and-fro speed at the gauze is comparable to the steady upward draught and the heat transfer, which goes as the square root of the air's speed, can no longer follow it in proportion. At 0.12 the gauze is near the edge of the singing range and the sound grows only slowly. At 3L/4 the same disturbance dies away within a few periods. Waves

The tube that sings when heated low down

Hold a wide metal tube upright, push a red-hot wire gauze a quarter of the way up inside it, and take the flame away. The tube begins to howl — a loud, steady note at its lowest resonance, lasting until the gauze cools. Put the gauze a quarter of the way down from the top instead and the tube is silent; lay the tube on its side and it stops. Heat is driving a sound, and whether it drives or damps depends on where the heat is released and when, in a rule Lord Rayleigh stated in 1878 that now governs rocket engines and power-station turbines.

 A peak that keeps growing while the light stays on. The momentum distribution of 20,000 atoms in a one-dimensional dark-state scheme after 100, 1,000 and 10,000 scattering times, as a density per unit of ħk, in bins a hundredth of ħk wide. There is no steady state. The peak at zero grows taller and narrower for as long as the light is on — its height reaches 29 per ħk by the last time — and the share of atoms within a tenth of a photon's momentum climbs from 55 to 87 to 96 per cent. The atoms are not pushed into the peak; they fall into it at random and the light stops touching them. Astrophysics

The atoms the light stops touching

Every photon an atom scatters kicks it by the same small amount in a random direction, so a gas that keeps scattering light cannot be colder than one kick. The way below is not a stronger friction but a weaker touch: arrange for the light to stop scattering off atoms that are nearly at rest. Then the coldest atoms are not cooled, only left alone, for times that have no average — and the gas never settles into a temperature at all, but piles into a peak that narrows for as long as the light stays on.

 The tilt that starts the reversal, in a well of soft crystal. A planar defect three wall parameters wide (three times √(A/K)) in which the crystal's anisotropy falls to zero, drawn as the anisotropy K(x) across it; the curve is the shape of the first tilt of the magnetisation to become unstable as a reverse field is raised, the lowest mode of −A d²/dx² + K(x), and the dashed line is its eigenvalue. The equation is Schrödinger's, with the exchange stiffness A in place of ħ²/2m and the anisotropy as the potential, and the reversal begins in its ground state. Here the ground state lies at 0.373 K, so the magnet reverses at 37 per cent of the anisotropy field. The mode reaches out of the well a distance of about √(A/K) on each side, because exchange will not let the tilt change abruptly. Electromagnetism

The magnet that reverses where its crystal is weakest

A perfect crystal of neodymium-iron-boron should hold its magnetisation against a reverse field of 7.7 tesla. The best magnets made of it give way at one to three. The shortfall, known for eighty years as Brown's paradox, is not a failure of the crystal but of its edges: reversal begins wherever the crystal is weakest, in a layer a few nanometres thick. Working out how weak that layer can afford to be turns out to be a problem already solved in quantum mechanics — the nucleation field is the ground-state energy of a particle in a well shaped like the defect.

 An electron that runs into light and loses its energy to it. The energy of an electron of 511 MeV (γ = 1000) as it passes head-on through a 10-cycle pulse of 0.8 µm light, as a fraction of its starting energy, against the number of laser cycles it has crossed, for normalised field strengths a₀ = 30 and 100: with radiation reaction (solid) and without (dashed). Without it, the electron is shaken sideways while in the light and leaves with exactly the energy it brought, as a plane wave requires. With it, each half-cycle near the pulse's peak radiates away a slice: the electron leaves with 76 per cent of its energy at a₀ = 30 and 22 per cent at a₀ = 100, the rest gone as gamma rays. Relativity

The light that takes back an electron's energy

An electron at rest in the most intense laser light ever made feels its own radiation as a force a millionth of the light's push — the textbook reason radiation reaction never matters. Send the electron into the light at nearly the speed of light and the arithmetic turns over. In its own frame the light is stronger by twice its Lorentz factor, the power it radiates grows as the square of that, and a pulse a few tens of femtoseconds long can strip it of most of its energy as gamma rays. That is where the classical theory of a radiating charge, argued over for a century, finally has to answer to experiment — and where it starts to fail for quantum reasons.

 A floor that has to hit a rod nobody dropped. The normal force on the lower end of a uniform rod at 63° with μ = 1.6, sliding outward at √(gl) and not yet turning, against time on a logarithmic axis, for a floor whose contact stiffness is 10⁴, 10⁵ and 10⁶ times mg/l. The rigid equations have no solution here, and the compliant floor shows why: the end digs in, the deeper it digs the harder friction twists it further down, and the force climbs until the sliding stops. The stiffer the floor, the higher and briefer the spike — 20, 56, 163 times the rod's weight, ending after 41, 18, 7 thousandths of √(l/g) — a collision with no approach. Mechanics

The rod that friction cannot decide about

Newton's laws, a rigid rod and Coulomb's law of friction are the first things a mechanics course teaches, and in 1895 Paul Painlevé showed that together they can fail to give any answer at all. A rod sliding on a rough enough floor, at the right angle, has no motion consistent with the three — or has two. The way out is not a new law but a forgotten fact: real contacts give a little. With that, the rod jams, and the floor delivers a sudden blow to a rod that nothing struck — an impact without a collision, which is how a stick of chalk comes to hop.

 A race between the flame and the end gas. The Livengood–Wu integral of the end gas — the fraction of its ignition delay used up — against time from the spark, for 95-octane fuel at compression ratios 9, 11 and 13; the dotted line marks when nine-tenths of the charge has burned. If the integral reaches one before then, the end gas ignites by itself while a sizeable part of the charge is still unburned, and knocks. At a ratio of 9 the flame wins easily: the integral ends at 0.90. At 11 it reaches one when 92 per cent of the charge has burned, just short of knock; at 13, when only 78 per cent has, and the rest detonates. The integral climbs fastest at the end, where the end gas is hottest and most compressed, which is why knock is heard late in the burn. Thermodynamics

The squeeze a petrol engine is not allowed

Every textbook gives a petrol engine's ideal efficiency as one minus the compression ratio raised to a small negative power, and every reader can see that the efficiency keeps climbing as the ratio rises. Real petrol engines stop near ten or eleven. Nothing in the thermodynamics stops them. The fuel does: the charge being compressed is already fuel and air, and the burning gas behind the flame squeezes the rest until it ignites on its own, in a hammer-blow that wrecks pistons. An octane number buys nothing but time against that, and a diesel escapes it by compressing air and adding the fuel only when it wants ignition.

 The innermost level dives towards the sea. The energy of an atom's innermost (1s) electron, in units of the electron's rest energy mc², against the nuclear charge Z: dashed, Dirac's formula for a point nucleus, √(1 − (Zα)²); solid, the same equation solved for a nucleus of realistic size. Above +mc² lie the free electrons; below −mc², the filled sea of negative-energy states that Dirac's theory requires. For a point nucleus the level stops at Z = 137, where the formula fails. A real nucleus keeps it: it crosses zero near Z = 148 and reaches −mc² at Z ≈ 170, binding the electron by twice its own rest energy. Beyond that the level has nowhere left to go but into the sea. Quantum

The nucleus that would make positrons from nothing

Dirac's equation gives the innermost electron of an atom an energy that falls as the nuclear charge rises, and at 137 protons the formula fails outright. Real nuclei have a size, and with that correction the level does not stop: it falls through zero and on towards minus the electron's rest energy, the edge of the sea of states that Dirac's theory says are already full. It gets there at about 170 protons. Past that point an empty innermost level could be filled from the sea, and the vacuum round the nucleus would emit positrons of its own accord. No nucleus that large exists — but two uranium nuclei, for an instant, come close enough to be one.

 Eight blocks leaning past the edge of a table. 8 identical blocks of length 1 stacked on a table edge (the vertical line), each set as far out as balance allows: the dot on each interface marks the centre of mass of all the blocks above it, which sits exactly on the edge of the block beneath. The top block overhangs the second by 1/2, the second the third by 1/4, the third the fourth by 1/6, and the k-th by 1/(2k); together the stack reaches 1.359 block lengths beyond the table — the top block lies wholly beyond it, since 1.359 is more than one. Every interface is balanced on its edge, so the stack stands, but with no margin at all. Mechanics

The stack that leans past its own table

Stack identical books on the edge of a table, each pushed out as far as it will go without falling, and four of them are enough for the top book to lie entirely beyond the table's edge. Keep going and the overhang grows without limit — the stack can reach as far out as anyone likes — but only as half the harmonic series, so that two book-lengths need 31 books, three need 227 and five need more than twelve thousand. The arithmetic is the same that settles whether a ladder leaning on a wall slips or a wardrobe topples, and the slow growth is the price of having nothing but the stack's own weight to hold it back.

 A cloud that grows wider instead of denser. The density of rubidium atoms in a magneto-optical trap against distance from its centre, for 10⁵, 10⁶, 10⁷, 10⁸ atoms, computed from the trap's spring (200 Hz), the atoms' temperature (100 μK) and the repulsion of rescattered light, with σᵣ/σₗ = 1.30. The smallest cloud is the trap's Gaussian, 78 μm in radius, with a peak of 0.91 × 10¹⁰ per cm³. Larger clouds stop getting denser: each fills a sphere at the same 2.5 × 10¹⁰ atoms per cm³ (dashed), out to 0.21, 0.46, 0.99 mm, with a soft edge about a Debye length, 45 μm, wide. More atoms make a bigger ball, not a denser one. Astrophysics

The light that makes a cold cloud repel itself

Load more atoms into a magneto-optical trap and the cloud does not get denser. It gets bigger, at a density that stays fixed whatever the number. The cause is the trap's own light. A photon one atom scatters can be absorbed by another, which is pushed away, and the push falls off as the inverse square of the distance, like the force between two charges. A cloud of laser-cooled atoms is therefore a plasma of neutral particles. Each behaves as if it carried a ten-thousandth of an electron's charge, and the trap fills with them as a harmonic well fills with charge: at one density, out to whatever radius the number requires.

 A hopper that runs in bursts. Grains discharged against time, on a logarithmic scale, from a silo of 1 mm glass beads through outlets 3.5, 4.0, 4.4 grain diameters wide; each time the outlet jams it is cleared after 2 s (a tap on the silo) and the flow restarts. Avalanche sizes are drawn at random from an exponential distribution with each outlet's mean. At 3.5 diameters the mean avalanche is 53 grains and the outlet jammed 57 times in two minutes, passing 2,537 grains; at 4.0 diameters the mean avalanche is 1,000 grains and the outlet jammed 37 times in two minutes, passing 33,568 grains; at 4.4 diameters the mean avalanche is 45,821 grains and the outlet jammed 3 times in two minutes, passing 116,983 grains. While it flows, each outlet runs at its full Beverloo rate; a narrower outlet does not flow more slowly, it stops more often, and the flat stretches are the waits. Fluids

The outlet that jams by chance

A hopper whose outlet is a few grains wide does not flow slowly. It flows at the full rate and then stops dead, when a handful of grains wedge into an arch across the opening, and it flows again only when something knocks the arch down. The number of grains that pass between one jam and the next is random and exponentially distributed, as if every grain leaving had the same small chance of being the last. Its average rises so steeply with the opening's size that it looks like a threshold, and whether there is one is still argued: two laws agree wherever an outlet can be watched, and they part only after millions of grains.

 A bright core in a set of rings. The intensity across the beam 29 cm beyond an axicon — a glass cone 10 mm across that tilts every ray by 0.5° towards the axis — lit by a 633 nm laser, computed from the Fresnel integral and scaled to its peak (solid). It is the pattern of a cone of plane waves meeting on the axis: a bright core 56 μm across at its first dark ring, at 2.405/kα from the axis, ringed by rings of slowly falling brightness. Dashed: a Gaussian laser beam focused to a spot of the same size, with no rings. The rings are not a flaw of the axicon; they are where the beam keeps the power that the core is short of. Optics

The beam that does not spread and the price it pays

Pass a laser beam through a shallow cone of glass and it emerges as a thin bright line of light, a core a few tens of micrometres wide that keeps its width for half a metre, where a beam focused to the same width by a lens would have doubled in a few millimetres. It seems to defeat diffraction. It does not. The core is not made of the light that was in it a moment before; every point along it is fed by a fresh ring of the aperture, and the core holds about one per cent of the beam's power. The rest is in the rings, and the rings are what pays for the line.

 What one scattering does to a photon's energy. The average fractional change in a photon's energy when it scatters off an electron in a gas at temperature T, against the photon's energy: (4kT − ε)/mc², for gases at 10⁶, 10⁷ and 10⁸ K. A photon below 4kT gains energy, taking it from the electrons' motion; one above loses it, as recoil, to the electrons. The crossing sits at ε = 4kT: 0.34 keV at 10⁶ K, 3.45 keV at 10⁷ K, 34.47 keV at 10⁸ K. The changes per scattering are small, a few per cent at most here, but a gas whose electrons scatter each photon many times, or that is bathed in light for long enough, is heated or cooled until the gains and losses balance over the whole spectrum. Relativity

The temperature a gas takes from the light around it

A thin, ionised gas bathed in light settles at a temperature that has nothing to do with how much light there is and everything to do with its colour. Every time a photon scatters off an electron, the two exchange a little energy: a photon softer than four times the electrons' thermal energy gains, a harder one loses by recoil. Balanced over a spectrum, the rule sets the gas at a quarter of the light's mean photon energy. That puts gas near a quasar at tens of millions of kelvin, and it caps the heating once photons approach the electron's own rest energy. For blackbody light the rule falls four per cent short, and the shortfall is the photon being a boson.

 A light panel that blocks one band as a heavy one would. Sound transmission loss at normal incidence against frequency, on a logarithmic frequency axis. Dashed: a limp panel of 2 kg/m², following the mass law, 13.4 dB at 300 Hz. Dotted: the same panel made 3 kg/m² by adding 1 kg/m² of plain mass, 16.8 dB. Solid: the 2 kg/m² panel with the same added mass hung on springs tuned to 300 Hz (damping 0.02). It blocks 35 dB at 300 Hz — the loss the mass law would give a wall 26 kg/m² heavy — and just above, at 367 Hz, falls to 5.2 dB, far below even the bare panel's 15.1: there the panel lets sound through more easily than if the resonators were not there. Waves

The wall made heavy only at one note

A wall stops sound by being hard to shake, and how hard a plain wall is to shake is its weight: that is the mass law, and it makes quiet walls heavy. Hang small masses on springs inside a light panel, tuned to one frequency, and near that frequency the panel becomes many times harder to shake than it weighs, because the hidden masses swing against it. A two-kilogram panel then blocks a tone as a wall thirteen times heavier would. The price comes just above the tuning, where the panel's dynamic mass passes through zero and it lets sound through more easily than if the masses were not there.

 A quiet resonator growing louder before it sings. The pressure in a resonator's 200 Hz mode over two and a half seconds, kicked continually by the same random noise, at three settings approaching the onset of self-sustained oscillation: the mode's natural decay rate falling from 20 to 6 to 2 per second. Each panel is on the same scale. Far from onset the noise is a faint, ragged hiss, each burst dying almost as soon as it starts. Closer, the bursts last longer and grow larger, because the mode forgets each kick more slowly and the kicks pile up; at the last setting the resonator hums in long swells of a nearly steady tone. Nothing has started oscillating on its own — every setting is below onset — but the noise has already begun to sound like the oscillation it is approaching. Waves

The noise that warns a tube is about to sing

A resonator that is about to break into self-sustained oscillation — a combustor about to hum, a heated tube about to howl — is silent on average right up to the moment it starts. Its noise is not. As the margin of stability shrinks, the random hiss in the mode grows louder, remembers each kick for longer and narrows towards the coming tone, in proportions fixed by the margin itself. The instability can be heard before it exists, except in the case engines most often present, where noise starts the song before the warning has properly begun.

 The shadow a thunderclap's sound cannot enter. Sound rays from a point 3 km up a lightning channel, on an afternoon with 25 °C at the ground and the air cooling 6.5 K per kilometre upwards, so that sound travels slower the higher it goes; the vertical scale is stretched 3.5 times. Every ray curves upwards. Rays aimed below the horizontal reach the ground, reflect and climb away; the last ray to touch the ground grazes it 23.1 km from the channel. Beyond that, shaded, no ray arrives at all: a listener's ears 2 m up are inside the shadow from 23.7 km. The flash is visible there and the thunder is not, though the air between is perfectly clear and absorbs the low notes of thunder hardly at all. Waves

The thunder that passes over the listener's head

Lightning twenty-five kilometres away is plainly visible and usually silent, though the air between is clear and absorbs the low notes of thunder hardly at all. The thunder has not faded; it has gone over the listener's head. Daytime air cools upwards, sound slows as it climbs, and Huygens's wavelets, smaller at the top of every front than at the bottom, tilt each front towards the sky. Beyond the ray that just grazes the ground there is a shadow the sound cannot enter — and a hundred and fifty kilometres further out, the stratosphere can bend it back down.

 The power a cavity holds depends on where its mirror is. The power circulating in a cavity of finesse 10,000, a metre long and lit at 1064 nm, as a fraction of its value on resonance, against the detuning of the laser from the cavity's resonance in units of the cavity's half-linewidth (7.5 kHz). Lengthening the cavity lowers its resonance and moves every point of the curve to the right: 38 half-linewidths for each nanometre of mirror travel. On the blue side (laser above resonance, right) that motion loses power, so the radiation force on the mirror falls as the mirror moves out — a restoring spring, steepest at δ/γ = 0.58 (dashed tangent). On the red side the same motion gains power and pushes harder: a spring of the opposite sign. Astrophysics

The mirror held on a spring of light

Light bouncing between two mirrors pushes them apart, and how hard it pushes depends on how long the gap is, because the gap decides how much light the cavity holds. A mirror free to move therefore sits on a spring made of light: with a kilowatt stored in a metre-long cavity it is four-fifths as stiff as a steel wire a millimetre thick. The spring has a flaw no steel spring has. Stored light takes time to rise and fall, so the push always arrives a little late, and a late spring either winds the mirror up or calms it, depending only on which side of the cavity's resonance the laser sits.

 A lens that cannot focus a wave longer than itself. How much a point mass brightens a wave passing behind it — the factor on the wave's power, on a logarithmic scale — against the dimensionless frequency w = 8πGMf/c³, for a source exactly behind the lens and 0.1, 0.3, 1.0 Einstein radii off it, from the exact diffraction integral. Geometric optics puts the magnification at 10.0, 3.4 and 1.34 for the three offsets and at infinity on the axis, whatever the frequency. The wave disagrees: at w = 0.01 every curve is within 1.6 per cent of 1 — a wave a hundred times longer than the lens passes as if it were not there — and on the axis the power factor is πw/(1 − e^(−πw)): 3.28 at w = 1, 31.4 at w = 10, finite at every frequency. The off-axis curves rise to their geometric values only above w ≈ 1 and then oscillate about them, as the two images begin to interfere. Astrophysics

The lens a long wave walks round

Geometric optics says that a star passing exactly in front of a source magnifies it infinitely. No real wave agrees, because a wave cannot be focused to a point, and for the gravitational waves the detectors hear the disagreement is the whole story. A star, or a black hole of a few tens of suns, is smaller than one of those waves in the only measure that matters — the delay between paths round it — and the wave flows past it almost untouched. What decides whether a mass lenses a wave is not the mass's size, which is millions of kilometres, but its Schwarzschild radius set against the wavelength.

 Two descriptions of gold that agree everywhere but at zero. How strongly gold answers a slowly varying field, (ε(iξ) − 1)ξ²/ωₚ² in the two forms of its permittivity at imaginary frequency ξ, against ξ on a logarithmic axis: the plasma model (dashed), which ignores the electrons' collisions and gives exactly 1, and the Drude model (solid), which includes them at the measured rate ħγ = 35 meV and falls to zero below about γ = 5.3·10¹³ per second. The Lifshitz force samples this curve only at the Matsubara frequencies 2πlkT/ħ (dots, 300 K): the first, 2.47·10¹⁴ per second, lies above γ, and there the two descriptions differ by 18 per cent, at the next by 10, and less beyond. At l = 0, ξ = 0, they part completely: 1 against 0. Everything the two models disagree about is in that one point. Electromagnetism

The static field a metal may or may not reflect

The Casimir force between two gold plates at room temperature has been computed two ways for twenty-five years, and the two answers differ by forty per cent at three micrometres. Both use the same measured optical properties of gold. They part company over a single number: whether gold reflects a magnetic field that does not change in time. Common sense and every electrical engineer say it does not, since a static field passes straight through a copper sheet. The most precise force measurements say it does. The disagreement is not about the Casimir effect at all; it is about what a metal's electrons do with a fluctuation slower than their collisions.

 The same fringes from cooler and hotter molecules. The rate at which molecules arrive across a few periods of the interference pattern behind the last grating, relative to its mean, for molecules at 1000, 2500, 3000 K, each with the visibility the emission model gives: 46 per cent, 14 per cent, 4 per cent. The cooler molecules draw fringes nearly as strong as cold ones; at 3000 K the pattern is a faint ripple on a flat background. The average number arriving is the same in every case: the heated molecules are not lost, and none of them has been deflected out of the beam. They arrive in the same places on average and no longer interfere. Quantum

The molecule that interferes until it glows

A molecule of seventy carbon atoms, sent through a matter-wave interferometer, travels both arms at once and lands in fringes, as an electron or a neutron does. Heat it, and the fringes fade. Nothing has touched it from outside: a hot molecule glows, and each photon it gives off is a messenger that may carry away the information of which arm it was in. Whether it does depends on one comparison — the photon's wavelength against the separation of the arms. Long photons leave the fringes alone; short ones erase them. The loss of interference that is usually described as the onset of the classical world is here a measurement made by the molecule's own light, and its threshold can be computed.

 How long a lens must hold back its central light. The time by which a lens must delay light passing through its centre relative to light passing its edge, so that every part of a short pulse reaches the focus together, against the lens's radius, for focal lengths of 0.1 mm, 1.0 mm, 10.0 mm (solid), in femtoseconds on logarithmic axes: the extra path from the edge, √(R² + f²) − f, over c. The dashed line is what a flat structure 1 μm thick can supply if its building blocks span a group index of 1: 3.3 fs. A 1 mm focal-length lens reaches it at a radius of 45 μm; a 10 mm one at 141 μm. A glass lens of 10 mm focal length and 5 mm radius holds its centre back by 3937 fs, which it does by being 2.27 mm thicker in the middle. A flat lens has no room for that. Optics

The delay a flat lens has no room for

A glass lens is thick in the middle for a reason that is easy to miss. To focus light of every colour at one point, a lens must do more than bend each colour by the right amount: it must make every part of a light wave arrive at the focus at the same moment, and the light passing the edge has further to go. The glass in the middle is what holds the central light back to wait for it. A lens made flat — a kinoform, a metasurface a micrometre thick — has nowhere to keep that delay. The geometry of the focus sets how much delay is needed, the thickness sets how much is available, and between them they cap how large an achromatic flat lens can be: tens of micrometres, not centimetres.

 How much of a wave of gravity a bar can catch. A resonant bar's cross-section for gravitational waves, integrated over its resonance, (8/π)(GM/c)(v/c)², against its mass, on logarithmic axes, for sound speeds of 3.0 km/s, 5.4 km/s, 10.0 km/s (aluminium is about 5.4). It is proportional to the mass, which is why the bars weighed tonnes, and to the square of the speed of sound over the speed of light — the bar answers a strain only by being stretched against its own stiffness, and (v/c)² is about 3 × 10⁻¹⁰. Weber's first bar caught 2.3·10⁻²⁵ m² Hz: to absorb a single joule, a burst must carry 4.4·10²⁴ J per square metre per hertz at the bar's frequency. Dots: Weber's bar, and the two cryogenic aluminium bars that ran longest, ALLEGRO in Louisiana and NAUTILUS near Rome. Astrophysics

The bar a wave of gravity barely touches

The first gravitational-wave detectors were tonnes of aluminium, hung in a vacuum and listened to for ringing. A passing wave stretches and squeezes the bar, and the bar can answer only by vibrating against its own stiffness, so the fraction of the wave it catches is set by the speed of sound over the speed of light, squared: about three parts in ten billion. That one ratio explains why the bars had to be enormous and cold, why Joseph Weber's reported detections of 1969 would have required the Galaxy to be converting dozens of suns a day into gravitational waves, and why the waves were finally found by light instead.

 The height at which hot air stops being light enough. Lift per cubic metre of hot air against height in the standard atmosphere, with the inside at 100 °C and 120 °C, and (dashed) what a 2800 m³ balloon carrying 550 kg plus its envelope needs per cubic metre, 0.241 kg. As the balloon climbs, the outside air thins, and although it also cools — which improves the temperature ratio — the density wins: lift per cubic metre falls by about a tenth for every kilometre. The balloon can climb only to where the curve crosses its need: 6.0 km with the envelope at 120 °C. Above the tropopause the outside temperature stops falling and the lift falls faster still; a hot-air balloon that reaches the stratosphere needs an envelope enormous compared with its load. Fluids

The balloon a cold morning lifts

A hot-air balloon is open at the bottom, so the air inside is at the same pressure as the air outside and lighter only because it is hotter. Its lift is the outside air's density times one minus the ratio of the two absolute temperatures — a small number, a third of a kilogram per cubic metre at best — and since the fabric cannot be heated much above a hundred degrees, the lift depends as much on the cold outside as on the heat inside. That is why balloons are the size of houses, why they fly at dawn, why each has a ceiling, and why the pilot must burn for where the balloon will be half a minute from now.

 How narrow a laser's line can be, against its power. The Schawlow–Townes limit on linewidth, πhνΓ²n/P with Γ the cavity linewidth and n the inversion factor, against output power on logarithmic axes, for a helium–neon laser with a 30 cm cavity of finesse about 300 (Γ = 1.6 MHz) and for a semiconductor laser a third of a millimetre long (Γ about 30 GHz, n = 2), the latter also with Henry's linewidth enhancement 1 + α² for α = 4 (dashed). At a milliwatt the helium–neon laser's limit is 0.0025 Hz — millihertz, a coherence length of tens of millions of kilometres — and the semiconductor laser's 7.2·10⁵ Hz, 1.2·10⁷ Hz with the enhancement. Both fall in proportion to one over the power: more photons in the mode make each spontaneous kick a smaller fraction of the field. The square of the cavity linewidth is why a short, lossy diode cavity has a line ten billion times broader. Optics

The width a perfect laser still has

A laser's light is the nearest thing to a single pure frequency that anything makes, and it is still not one. Every photon spontaneously emitted into the lasing mode adds a kick of random phase, and because nothing in a laser cares what its phase is, nothing pulls the phase back: it wanders, and a wandering phase is a line of finite width. Schawlow and Townes worked out how wide in 1958, before any laser had been built. For a milliwatt helium–neon laser the answer is a few thousandths of a hertz, a coherence length a quarter of the way to the Sun — and no real laser of that kind has ever come close, because mirrors move. For a semiconductor laser the answer is megahertz, and the measured lines were wider still, by a factor nobody expected.

 Why dissolving beats evaporating near absolute zero. Cooling power against temperature on logarithmic axes, for pumping on liquid helium-4, pumping on liquid helium-3 — both with a pump of 100 litres a second at room temperature — and a dilution refrigerator circulating 100 micromoles of helium-3 a second (the T² law, dashed above 0.1 K where it no longer holds). Evaporative cooling is proportional to the vapour pressure, which collapses as e^(−L/T): helium-4 gives 7·10⁻¹¹ W at 0.3 K, helium-3 10⁻⁴ W at 0.3 K and 2·10⁻²⁴ W at 50 mK. The dilution refrigerator's "vapour" is the dilute phase, whose density of helium-3 stays at 6.6% however cold it gets, so its cooling falls only as T²: 2·10⁻⁵ W at 50 mK. Below about 0.3 K nothing that evaporates can compete, and most experiments that must stay continuously below a tenth of a kelvin — superconducting quantum computers, bolometers for the cosmic microwave background, searches for dark matter — run on dilution. Thermodynamics

The mixture that stays mixed at absolute zero

The third law says a mixture's entropy must vanish at absolute zero, so every mixture must either separate completely or order into a compound as it is cooled. Liquid helium-3 dissolved in helium-4 does neither. Below 0.87 kelvin it splits into two liquids, but the one that is mostly helium-4 keeps 6.6 per cent helium-3 however cold it gets — and the third law permits it, because the dissolved atoms are a degenerate Fermi gas with a single ground state and no entropy of mixing to lose. Pushing helium-3 across that boundary absorbs heat the way evaporation does, from a 'vapour' that never thins out. That is the dilution refrigerator, the machine that holds every superconducting quantum computer at ten millikelvin.

 The deeper the line, the closer to gravity. The shift of a solar line in sunlight seen from the Earth, against the line's depth, in the two-stream model with the convective velocities dying away above the photosphere over 150 km and a deeper line taken to form higher (600 km × depth²). Weak lines, formed low among the fastest flows, show 411 m/s — about two-thirds of the gravitational value; the deepest lines, whose cores form in the upper photosphere where the granules' motion has nearly stopped, show 627 m/s and are still climbing towards the 633 m/s that gravity alone gives, drawn as the dashed line. That is the measurement made with integrated sunlight reflected from the Moon and calibrated with a laser frequency comb: plot the shift against line depth, and read the gravitational redshift off where the strongest lines are heading. Its published value, 638 ± 6 m/s, agrees with general relativity. Astrophysics

The redshift the Sun's boiling hides

Light leaving the Sun climbs out of a gravitational well and should arrive redder by 633 metres a second in velocity terms — a prediction Einstein made in 1907, before he had general relativity. It took half a century to confirm, and the reason was not the size of the effect but the Sun's surface, which boils. Bright hot gas rising in granules and fainter cool gas sinking between them blueshift every line by a few hundred metres a second, by an amount that depends on the line and on where on the disc it is seen. The gravitational shift was found by treating that nuisance as a second measurement: deeper lines and lines near the limb both converge on the same number.

 The cloud that stops shrinking. The rms size of a cloud of 10⁵ ideal atoms in a harmonic trap, in units of the Fermi radius, against temperature in units of the Fermi temperature, for fermions, for bosons and for a classical gas. The classical cloud shrinks as the square root of half the temperature ratio, all the way to nothing. The bosons follow it down to their condensation temperature, 0.52 of the Fermi temperature, and then collapse into the trap's ground state, whose rms size is 0.054 Fermi radii. The fermions part from the classical line near the Fermi temperature and level off at one over the square root of eight, 0.354 of the Fermi radius: at a quarter of the Fermi temperature they are 1.17 times the classical size, and cooling further barely changes them. Mixed together in one trap, as lithium-7 and lithium-6 were in 2001, the two clouds start the same size and end 6.5 times apart in width for ideal atoms of these numbers — further still in volume. The fermions are held up by nothing but the rule that no two of them may share a state. Quantum

The cloud that stops shrinking

Two kinds of lithium atom, cooled together in one magnetic trap to a few hundred billionths of a degree, start as clouds of the same size and end very differently. The lithium-7 atoms, which are bosons, shrink with the temperature and then collapse into a condensate. The lithium-6 atoms, which are fermions, stop shrinking at about a quarter of their Fermi temperature and refuse to get any smaller, however much colder they become. Nothing pushes them apart but the rule that no two identical fermions may share a state. The photographs taken in 2001 were the first time Fermi pressure — the pressure that holds up white dwarfs — was seen as a picture.

 Newcomen's dilemma. Two consequences of how cold a Newcomen engine's injection spray leaves its cylinder, against that temperature: the pressure difference the atmosphere then pushes the piston down with, one atmosphere less the vapour pressure of the water left in the cylinder; and the steam the next stroke wastes reheating the cylinder's iron skin, as a fraction of the most it wastes at 30 °C (8.2 cylinder-fulls). A cylinder cooled to 40 °C gives a push of 0.93 atmospheres and wastes 7.0 cylinder-fulls of steam a stroke; one left at 70 °C wastes half as much but pushes with only 0.69. Every choice was bad, because the one vessel had to be both hot, to receive the steam, and cold, to make the vacuum. Newcomen's engineers ran it in between. Thermodynamics

The cylinder that must not be cooled

The first steam engines that did useful work, Thomas Newcomen's pumps of 1712, made a vacuum by spraying cold water into a cylinder full of steam. It worked, and on every stroke it cooled the cylinder, so that the next charge of steam condensed on the cold iron until the walls were hot again — several cylinders' worth of steam spent heating metal that the next spray would cool. James Watt saw in 1765 that the vessel was being asked to be hot and cold at once, and moved the cold to a separate condenser. The improvement was a factor of three or four in coal. A century later the reason had a name: heat passing from hot to cold across a large difference, doing nothing on the way, is work thrown away.

 The product a fusion plasma has to beat. The density, temperature and confinement time multiplied together at which a deuterium–tritium plasma heats itself, against temperature. Multiplying by T turns the previous curve into a pressure times a time, which is what a magnet can actually hold, since the field limits the pressure nT and not the density alone. The curve has a clear minimum of 3.0 × 10²¹ keV s/m³ near 14 keV, and that is why magnetic machines are designed to run hot but not hotter. ITER's design point, a density of 10²⁰ per cubic metre at 8.9 keV and 3.7 s, gives 3.3 × 10²¹: a little under this line, because ITER is meant to be driven by external heating with a tenfold gain and not to ignite outright. Astrophysics

The product a fusion plasma has to beat

A star makes less power per litre than a compost heap, and it has burned for four and a half thousand million years. A fusion plasma makes thousands of times more and has to be held for seconds, and it has taken seventy years to get close. Both are obeying one inequality, written down in a classified report in 1955, in which density and time appear only as their product.

 A drift that twist turns into a circle. An ion's guiding centre at 10 keV in a torus the size of ITER, seen in the plane across the bore, with a vertical drift of 609 m/s from the field's curvature and gradient. With no twist it drifts straight up and reaches the wall in about 2.8 ms, ions one way and electrons the other. With the field lines twisted so that they circle the bore once for every three trips round the torus, the same drift is carried round by the motion along the line, up on the outside and back down on the inside, and the integrated path closes into a circle offset by only 1.3 cm. The offset is the drift speed over the rate of circulation. The twist does not stop the drift; it makes the drift average to nothing, which is all a bottle needs. Astrophysics

The twist a bottle of plasma has to have

A straight magnetic bottle leaks out of its ends, so the obvious repair is to bend it round into a ring. A ring of field with no twist in it loses its plasma in about three millionths of a second. Adding a twist repairs that by making a drift average to nothing, and adding too much makes the whole column kink like an overwound rubber band. A tokamak is built in the narrow band between those two failures.

 A lifetime that oscillates with a distance. The rate at which an excited atom emits a photon, relative to its rate in empty space, against its distance from a perfect mirror in wavelengths, for its dipole pointing across the mirror and along it. On the mirror itself the rate doubles for the first and vanishes for the second. As the distance grows both swing about 1 with a period of half a wavelength and a shrinking size: at half a wavelength the along-mirror rate is 0.96, and at a wavelength 0.99. Nothing in the atom has changed; its excited electron and its transition are the same in every position. An emission rate is a property of an atom and of the room it is in. Quantum

The lifetime that depends on where the mirror is

A half-life is treated as the most private fact about an atom or a nucleus, and for a nucleus that is nearly true. An excited atom beside a mirror decays faster or slower depending on how many half-wavelengths away the mirror stands, and between two mirrors closer than half a wavelength it does not emit at all. The atom has not changed. A decay is a transition to a state with a photon in it, and the room decides whether that state exists.

 The spectrum two rates would make. Balance absorption against spontaneous emission alone, with level populations in the Boltzmann ratio, and the radiation comes out as x³ times e to the minus x, Wien's law; add stimulated emission, and the same balance gives x³ over e to the x minus one, which is Planck's. The two agree on the right, where a photon is much more energetic than the thermal energy and stimulated emission is negligible, and disagree on the left, where only Planck's approaches the classical count of waves. At x = 1 Planck's is 1.58 times Wien's. Measurements around 1900 at long wavelengths had already found that Wien's law failed, so experiment required the third rate seventeen years before anyone saw it in a laser or a maser. Quantum

The third process equilibrium demands

In 1916, with general relativity just finished, Einstein asked how atoms and light could reach equilibrium at all. Absorption and emission, balanced against each other, gave the wrong spectrum. A third process was needed, in which light makes an excited atom emit a copy of itself. Nobody had seen it, and it took forty years to be put to use. It also fixes a ratio that goes as the cube of the frequency, which is why X-ray lasers are so hard.

 The price of a pair depends on who pays it. The lowest energy of a photon that can make an electron–positron pair when struck on a stationary partner, against the partner's mass: 2mc²(1 + m/M), from requiring that the total four-momentum have the invariant mass of what is left at rest in the centre of momentum. For a lead nucleus it is 2mc² to better than one part in a hundred thousand, the figure every textbook quotes. For a proton it is 0.054 per cent above. For a free electron it is exactly twice, 4mc², or 2.04 MeV, even though the partner is the very same mass as the pair's members. The rest energy created is unchanged; what rises is what has to be carried off by the partner and the pair together. Relativity

The partner that sets the price of a pair

Every textbook says it takes 1.02 MeV to make an electron and a positron from light, twice the electron's rest energy. Done in empty space, it cannot be done at any energy at all. Next to a lead nucleus it costs 1.02 MeV. Next to a lone electron, a particle exactly as heavy as the ones being made, it costs 2.04. Nothing about the pair has changed. What changes the price is who has to carry the momentum away.

 The force that does not count the water. The force of a water bridge between two touching spheres of radius R, in units of 2πRγ, against the bridge's volume on a logarithmic scale running over six decades, for three contact angles. Each curve is nearly flat: from a millionth of R³ to a thousandth, the force on a wetting grain falls only from 0.99 to 0.92 of 2πRγ, and it stays close to cos θ for the others. A droplet between two grains of sand pulls almost as hard as one a thousand times larger. This is the reason sand wet with one per cent of water and sand wet with ten per cent hold together with about the same strength. The curves fall where a bridge is large enough for its neck to stop being small compared with the grain, and bridges of that size merge with their neighbours. Fluids

The sandcastle a spoonful of water holds

A dry heap of sand stands at thirty degrees and no steeper. Add a spoonful of water to a bucketful, and the same grains will stand in a vertical wall and carry a person's weight. Add ten times as much water and the wall is no stronger, because a bridge of water between two grains pulls with almost the same force whatever its size. Its strength belongs to the grain and the surface tension, and the amount of water hardly enters.

 A lake a little under half full of gas. Carbon dioxide dissolved in the water of Lake Nyos in Cameroon against depth, in a representative profile of the kind measured in the years after 1986, beside what water at that depth could hold under Henry's law, which rises with the pressure. Everywhere the water holds less than it could: 0.35 mol/L at 200 m, where 0.75 mol/L would be saturated, so the deep water is not on the point of doing anything. It can sit there for decades, fed by a vent in the bed below. But it is only stable where it is. Move a parcel of it up, keeping its gas, and the pressure falls while the gas stays. It meets the saturation line at 88 m, a little over a hundred metres above where it started. Fluids

The lake that holds its gas until it does not

On the night of 21 August 1986, Lake Nyos in Cameroon released about a cubic kilometre of carbon dioxide in a few hours, and some 1,700 people died in their sleep in the valleys below. The lake was not boiling and it was not erupting. Its deep water had been holding the gas safely for decades, in a layering that was stable as long as nothing moved. Lift that water far enough and it cannot stop rising. The same instability, in a pipe, now drains the danger away.

 The band of speeds in which a bicycle rides itself. The growth rates of the small motions of a riderless bicycle against forward speed, from the four roots of the linearised equations for a benchmark machine, with the published eigenvalues reproduced before anything else was drawn. A positive rate means the motion grows. Below 4.29 m/s a pair of roots has positive real part: the weave, a swaying of the steering and lean together, grows. Above 6.02 m/s one real root turns barely positive: the bicycle slowly falls over into a spiral, the capsize. Between these speeds, in the shaded band, all four roots have negative real parts, and a shove dies away on its own with no hand on the bar. Nothing is stabilising the machine except the coupling of lean to steer. Mechanics

The bicycle that steers itself upright

Give a riderless bicycle a shove and, at the right speed, it wobbles and straightens up, and rides on with nobody on it. People have long credited the gyroscopic wheels or the trail of the fork. A machine built in 2011 had neither, and it stayed up. Four numbers, the roots of a quartic, tell the whole story of when a bicycle rides itself, and they do not contain a simple reason.

 A fountain in proportion to the drop. The height of the arch above the jar against the drop below it, for three values of α, the share of each link's momentum that the pile supplies. A steady chain has one tension at the top of the arch, equal to its momentum flux λv². Going down either leg the tension falls by the weight of chain below. At the jar the tension must be (1 − α)λv²; at the floor, with the floor taking all of the arriving momentum, it is zero. That gives v²α = gh₁ and h₁ = h₂ α/(1 − α), a straight line through the origin, and no arch at all if α = 0. A measured α near 0.12 gives 27 cm of fountain at a 2 m drop, and this proportion, about a seventh of the drop, is what the experiments found. Mechanics

The chain that climbs out of the jar

Pull the end of a long bead chain over the rim of a jar and let it fall to the floor, and the rest follows, as a siphon does. That much is unsurprising. What a video showed in 2013 was stranger: the chain does not run over the rim. It leaps up from the jar in a standing arch, tens of centimetres above it, with nothing supporting it. The explanation took a year and turned on something a string does not have: a link is stiff, so the pile has to push back.

 The rings the full solution finds. The light scattered back by water drops of about 10 µm radius, at 550 nm, from the exact Mie series with around 134 terms for each drop, averaged over a ten per cent spread of sizes so that the fine ripple each single drop has is washed out. Instead of the smooth, weak background of the ray picture, the intensity rises sharply toward 180° through a series of rings about one and a half degrees apart, each brighter than the last, to a first ring 1.5° from straight back. These are what an observer sees round the shadow of an aircraft on a cloud, or round a shadow on a mist at a mountain top: the glory. It is present in the same mathematics that gives the rainbow, and it takes the whole series, with its surface-wave terms, to find. Optics

The rings no ray can reach

Look down from an aircraft at a cloud, and round the aircraft's shadow there may be coloured rings, centred exactly on the point opposite the Sun. Mountain climbers have seen them round their own shadows for centuries, and every drop in that cloud is a sphere of water. Yet ray optics says that a drop of water cannot send light straight back. Its best ray falls fourteen degrees short. The glory lives entirely in those fourteen degrees.

 A facet too wide and a facet too narrow. The blur of an insect's compound eye of radius 1 mm against the width of one facet, in green light. A wide facet looks at a wide patch of sky compared with its neighbours, so the eye's sampling is coarse, and that blur rises with the width. A narrow facet is a small aperture, so diffraction spreads what it collects, and that blur falls with the width. Added in quadrature they make a minimum at D = √(λR), here 22 µm, and that is close to the facets of real insects with an eye of that size. No choice of facet width can get below about 1.8°: that figure is forced by the size of the head. Horace Barlow worked this out in 1952 and compared it with bee eyes. Optics

The eye a fly cannot afford

A compound eye is built from thousands of tiny straws, each looking in one direction. Wide straws look coarsely at the world, and narrow ones blur what they collect by diffraction, so there is a best width, and it depends only on the size of the head. To see as sharply as a person, an eye of this kind would need to be about twelve metres in radius. A single-lens eye gets the same result with a pupil of under two millimetres.

 A cooling planet, read from the top. Temperature against depth in a body cooling by conduction from a uniform start at 3870 °C, with a surface held at nought, at four ages: the exact error-function solution for a half-space. Cooling reaches downward as the square root of time, so after a hundred million years the upper few hundred kilometres have cooled and the interior knows nothing about it. The gradient at the surface is what a mine measures, and it falls as the inverse root of time. A thermal gradient therefore gives an age, and this is what Kelvin saw in 1862. He assumed the Earth was a solid conductor from the start, on a good basis, as everyone believed then, and these curves are the mathematical consequence of that belief. Thermodynamics

The age a cooling Earth gives

In 1862 William Thomson, later Lord Kelvin, took a thermometer reading from the bottom of a mine and a piece of mathematics from Fourier and computed the age of the Earth: about a hundred million years. Geologists, needing far longer, had no physics with which to reply. The arithmetic was right. The usual story blames radioactivity for overturning it, but half the heat is not radioactive, and the error lay in assuming the inside of the planet stays still.

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