Illustrated Physics
physics, drawn
The collection

Every essay — page 24

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Mechanics

Motion, force, and the quantities that refuse to change.

Why the meander has to move west. A line of fluid columns at 45° pushed into a meander 3,000 km long and 300 km in amplitude, seen from above with north up and east to the right. Columns pushed north find the planet spinning faster beneath them and turn clockwise; columns pushed south turn anticlockwise. Each turning column drives the fluid north on one side of it and south on the other, and the arrows are that flow, from integrating the columns' spin along the line: it peaks at 2.32 m/s, a quarter of a wavelength west of each crest, so the columns just west of a crest are carried north and the crest itself is rebuilt further west. The dashed meander is the same pattern 2 days later, moved 638 km west at β/k² = 3.69 m/s. Nothing in the argument depends on which way the meander was first pushed.

The wave that can only travel west

A column of air or water on a spinning planet keeps one quantity as it moves: its total spin divided by its depth. Carry it north, where the planet's own spin about the local vertical is larger, and it must turn the other way to compensate. That single bookkeeping rule makes a wave that exists nowhere else — one whose crests can only ever move west — and the same rule holds the jet stream's meanders still over the mountains, makes the ocean take decades to notice a change in the wind, and pins the Gulf Stream to the western side of the Atlantic.

5 figures · part 5 on Circular motion The noisy orbit a true orbit stays beside. A computation of the doubling map with an error of up to 0.001 added at every one of 400 steps, and two exact orbits compared with it. The exact orbit started from the computation's own first point parts from it at once, by more than 0.1 after 8 steps, because each error is doubled at every step after it. The shadow — the exact orbit built by running the map backwards from the computation's last point, where each error is halved instead — stays within 8.0 × 10⁻⁴ of the computation at every one of the 400 steps, closer than the noise added at any single step. Every point of the noisy computation is therefore a faithful picture of a real orbit of the map — just not of the one that started where the computation did.

The wrong orbit beside a right one

Every computer simulation of a chaotic system is wrong within a few dozen steps: rounding errors are stretched exponentially, and after fifty iterations of the simplest chaotic map two computations of the same equation disagree completely. Yet chaotic simulations are trusted everywhere, and for a reason that is a theorem rather than a hope. For a large class of chaotic systems the computed sequence, though it follows no true orbit from where it started, stays close to some other true orbit for as long as it runs.

5 figures · part 6 on Chaos The area under the spacings is the bond. The gap between each vibrational level and the next, against level number, for a Morse well and a Lennard-Jones well with the same depth and the same curvature at the bottom, Λ = 100, in units of the well depth. The sum of the gaps from the lowest level up is the energy needed to break the bond from there. For the Morse well the gaps fall in a straight line to zero, and a straight line through the first 5 of them, extended to zero, encloses 0.9473 against the true 0.9473. For the Lennard-Jones well the gaps start almost the same and fall slightly faster at first — which is what the straight line sees — then trail off in a long tail of small gaps near the top; the same extrapolation from the first 5 encloses 0.7988 against the true 0.9474 — 15.7 per cent short, because the straight line cannot see the crowded levels its data never reached.

The spectrum that weighs a bond

A vibrational spectrum measured near the bottom of a molecule's well gives its curvature, and the curvature says nothing about how deep the well is. Measured further up, it says everything. The levels of a real bond crowd together towards the top, the gap between neighbours falls towards zero, and the sum of all the gaps is the energy it takes to break the bond. A straight line drawn through the gaps that have been measured, and extended to zero, was for decades how bond energies were found — and for a bond with the tail of two neutral atoms it comes out a sixth short.

5 figures · part 5 on Harmonic approximation

Waves

Oscillation, and everything that turns out to be an oscillation.

A smooth swing that sharpens into pulses. The shape of a string whose far end moves back and forth by 1% of its length at 2 times its fundamental frequency, started swinging in its fundamental, after 0, 10, 20, 30, 40 round trips of a wave. The smooth arch gathers into two ever-narrower pulses running in opposite directions, drawn at the moments they cross in the middle, with their height unchanged and their sides ever steeper; each reaches the moving end just as it moves inwards fastest. The energy has become 1.00, 1.09, 3.15, 10.9, 38.3 times the starting value. Every shape is computed exactly from the moving end, with no approximation to the wall's motion.

The string whose end will not keep still

Move one end of a vibrating string slowly and each mode simply retunes. Move it quickly, at twice the note the string is sounding, and something else happens: the smooth swing gathers into pulses, the energy grows by the Doppler factor of the moving end on every round trip, and it climbs through the modes without stopping — because the modes of a string are evenly spaced, and a wall that couples one pair couples them all.

5 figures · part 8 on Standing waves The path the tip of the field traces. The tip of the total field over one cycle, for four arrangements of coils, with each coil's axis as a faint line and the field at one moment as an arrow. One coil: forward-turning part 0.50, backward-turning part 0.50; Two coils, 90° and a quarter cycle apart: forward-turning part 1.00, backward-turning part 0.00; Three balanced coils: forward-turning part 1.50, backward-turning part 0.00; Three coils, the third at 70 per cent current: forward-turning part 1.35, backward-turning part 0.15. One coil traces a line: equal forward and backward parts, a field that pulses and does not turn. Two coils at right angles a quarter cycle apart trace a circle of radius 1, three balanced coils a circle of 1.5, and a weakened third coil an ellipse, the sum of a large forward circle and a small backward one.

The field that turns with nothing turning

A coil fed with alternating current makes a magnetic field that grows, shrinks and reverses along the coil's axis and never leaves it. Put three such coils at 120° to one another and feed them currents a third of a cycle apart, and their fields add to one of constant strength that turns steadily round the centre — a rotating magnet made of nothing that moves. The trick is superposition read backwards: a field pulsing along a line is already two fields turning opposite ways, and the three coils are arranged so that one pair cancels and the other adds. Everything that goes wrong afterwards is a field turning the wrong way.

7 figures · part 6 on Superposition

Optics

Light, and the small number of rules it obeys.

A strong oscillator lifts a picowatt to the quantum limit. The signal-to-noise ratio for detecting a 1 pW signal at 1.55 µm in a 1 MHz bandwidth, with a detector of 80 per cent quantum efficiency whose amplifier adds 2 pA/√Hz of noise, against the power of the local oscillator. Detected directly, the signal is 66 dB below the amplifier's noise. Heterodyned, it rises with the oscillator's power — −3.3 dB with a microwatt, 7.4 dB with a hundred — until the oscillator's own shot noise swamps the amplifier's, above about 12 µW, and then it stops rising at 8.0 dB: ηPₛ/(hνB), the number of detected signal photons per resolution time, which no receiver can beat. A stronger oscillator raises the signal and its own noise together.

The faint light a bright one makes measurable

A photodetector does not respond to a light wave's field; it responds to the square of it. Shine a strong beam onto the detector together with a weak one at a slightly different frequency, and the square contains a term that is the weak field multiplied by the strong one — oscillating at the difference frequency, carrying the weak light's phase, and as large as the strong beam cares to make it. That product lifts a picowatt out of an amplifier's noise to the limit set by counting photons, turns an optical frequency shift into a radio one, and carries with it a quantum price that decides why radio telescopes and optical telescopes are built so differently.

5 figures · part 7 on Coherence Stopping down removes one colour error and not the other. The two chromatic errors of a single 50 mm N-BK7 lens with its stop 20 mm in front, at an image point 10° off axis, against the f-number. Axial colour — blue and red focusing at different distances, seen as a coloured blur at the yellow focus — is about 775/N µm across: 387 µm at f/2, 97 at f/8, 35 at f/22. Lateral colour — blue and red images of the point centred at different heights — is 55.0 µm at every f-number, because it is carried by the ray through the centre of the stop. The two are equal near f/14, and past that stopping down leaves only the error it cannot touch.

The colour fringe no aperture can close

A lens bends blue light more than red, and that one fact produces two different faults. One puts the blue focus nearer the lens than the red, and closing the aperture shrinks it. The other puts the blue image of an off-axis point nearer the centre of the frame than the red one, and closing the aperture does nothing at all, because it is carried by the one ray every aperture keeps. It is why photographs have coloured fringes in their corners and not in their middles, and why a lens designer who cannot remove it with glass can sometimes remove it with symmetry, or with a distance.

5 figures · part 7 on Imaging Three conditions and the window they leave. For a ray entering a diamond's table straight down, the angle from the normal at which it meets the first pavilion face, the second pavilion face and the top again, against the pavilion angle; diamond's critical angle, 24.4°, is the dashed line. The first two must lie above it, so that the ray is totally reflected; the third must lie below it, so that the ray gets out. The first holds for every pavilion angle drawn. The second fails above 51.9°, where the ray would leak out of the far side of the pavilion. The third fails outside 38.9° to 51.1°, and it is the one that decides. Together they leave a window from 38.9° to 51.1°, and Tolkowsky's 40.75° sits inside it, towards the lower edge.

The window a diamond is cut inside

A well-cut diamond sends back out of its top most of the light that falls on it, and it does so without a single silvered surface. Light entering the top is totally reflected at one sloping face of the pavilion below, again at the opposite face, and then has to leave through the top — where it must not be totally reflected. Each of those three is a condition on one angle, and together they leave a window a dozen degrees wide inside which a stone looks bright and outside which it looks glassy. The window is wide for diamond and nearly closed for glass, which is why a cutter's skill cannot make glass sparkle.

5 figures · part 5 on Total internal reflection

Electromagnetism

Charge, field, and the lines drawn between them.

The same nudge in four kinds of space. A body on a circular orbit given 5 per cent more speed, integrated for up to 6 turns in the field a point mass would have in spaces of 2, 3, 4, 5 dimensions. In 2 dimensions it stays between 1.00 and 1.10 of the original radius; in 3 dimensions it stays between 1.00 and 1.23 of the original radius; in 4 dimensions it leaves after 1.93 turns; in 5 dimensions it leaves after 1.19 turns. In three dimensions the orbit is an ellipse that closes; in two it is a rosette that does not close but never wanders far; from four dimensions up there are no stable orbits at all, and a planet nudged by anything would spiral away or plunge.

The number of dimensions a planet can orbit in

Gauss's law says that a point source's field is its strength spread over the surface around it, and the size of that surface depends on how many dimensions space has. In three it grows as the square of the distance, and the field falls as the inverse square. Count differently and the field falls differently — and above three dimensions it falls so steeply that no planet could hold a stable orbit and no atom could hold its electron. The inverse-square law is not a fact about gravity or electricity. It is a fact about space, and it is one of the conditions for there being anything in space to orbit.

5 figures · part 5 on Gauss's law Three ships, three local times, one clock. The fair-weather field at sea level, in the same model, as recorded by ships at longitudes 120° W, 0°, 120° E, each plotting it against its own local solar time. An ordinary daily effect — sunshine, heating, turbulence — would peak at the same local time everywhere. This one peaks at 11.4, 19.4, 3.4 hours local time in turn, because every ship is reading the same global potential, which follows the world's storms and peaks at one moment in Universal Time. Seeing the maxima line up in Greenwich time rather than local time, over clean ocean air far from any storm, was how the global circuit was demonstrated: the field in fair weather is the far end of a circuit whose generators are thunderstorms thousands of kilometres away.

The field that keeps Greenwich time

Over a calm ocean, a thousand kilometres from any storm, the air carries a downward electric field of about a hundred volts per metre, and it rises and falls by a sixth once a day. It peaks at the same instant on every ship on every ocean — near seven in the evening in Greenwich, whatever the local clock says. Nothing local can do that. The field is the far end of a circuit whose upper plate is a single conductor round the whole planet, and a conductor has one potential, so every patch of fair-weather sky is reading the sum of all the world's thunderstorms at once.

5 figures · part 4 on Potential A field strings a suspension into chains. 48 identical polarisable spheres, two radii apart at least, scattered at random through a periodic cell twenty radii square — 38 per cent of its area — and the same spheres after the field (vertical) has been on long enough for the viscous liquid to let them move. Each moves in proportion to the dipolar force from all the others and is stopped only by contact. At the start no two are touching; at the end 45 are bonded end to end along the field and 48 of the 48 spheres belong to chains, lying along the field lines and kept apart sideways because side-by-side dipoles repel. Chains that reach from one electrode to the other are what make the liquid resist shear, and they form in the milliseconds it takes a sphere to travel its own diameter.

The liquid a field turns solid

Stir fine starch or glass beads into silicone oil and the result pours like cream. Put a few kilovolts across it and within milliseconds it holds its shape and will not flow until it is pushed hard. Nothing in it is charged. Each particle becomes a dipole in the field, dipoles pull on each other end to end and push apart side by side, and so the particles string themselves into chains from one electrode to the other — a solid made of forces that exist only while the voltage is on.

5 figures · part 6 on Dielectrics

Thermodynamics

Heat, disorder, and the one law with a direction in it.

Hydrogen comes apart far below its binding temperature. The fraction of hydrogen that is ionised in equilibrium, against temperature on a logarithmic axis, for densities of hydrogen nuclei of 10²³, 10¹⁸, 10¹², 10⁶ per cubic metre, from Saha's equation. Hydrogen's binding energy, 13.6 eV, corresponds to 157,803 K. It is half ionised at 13,980 K at 10²³ per cubic metre, 7,235 K at 10¹⁸ per cubic metre, 4,519 K at 10¹² per cubic metre, 3,270 K at 10⁶ per cubic metre — at a temperature where kT is between an eleventh and a fiftieth of the energy needed to free the electron. The thinner the gas, the cooler the temperature at which it comes apart, because a free electron in a thin gas has vastly more room to be in than a bound one.

The electron that leaves because there is room

It takes 13.6 electronvolts to pull the electron off a hydrogen atom, the energy of a particle at a temperature of 158,000 kelvin. Hydrogen is half ionised at the surface of a star at 14,000 kelvin, in a thin nebula at 7,000, and in the early universe at under 4,000 — a fortieth of its binding energy. The electron leaves not because it has enough energy but because a free electron has so much more room to be in than a bound one, and at equilibrium room counts as much as energy. Saha's equation is that trade written down, and it turned out to be the key to what stars are made of.

5 figures · part 6 on Chemical potential The melting curve that turns back, then lies flat. The pressure at which liquid and solid helium-3 coexist, against temperature from 0.9 millikelvin to 1 kelvin on a logarithmic axis, from the international provisional low-temperature scale PLTS-2000, which is defined by this curve. Above 315.2 mK the curve behaves like any other: a higher temperature needs a higher pressure to freeze the liquid. Below it the curve turns back — at its minimum, 2.93113 MPa, it takes a higher pressure to freeze colder helium — and it rises to 3.4391 MPa at 1 mK, where it lies almost flat, as the third law requires of every coexistence line. The kinks that mark the liquid becoming superfluid and the solid's spins ordering sit on that flat end, at 2.444, 1.896 and 0.902 mK.

The melting curve that has to arrive flat

The slope of every line on which two phases coexist is their difference in entropy divided by their difference in volume. The third law says that entropy differences vanish at absolute zero, so every such line — melting, boiling, a superconductor's critical field — must reach the bottom of the temperature scale lying flat. Helium-3 reaches it in the strangest way available: below a third of a kelvin its solid is more disordered than its liquid, the melting curve runs backwards, and squeezing the liquid into solid absorbs heat. That backwards curve was the refrigerator in which superfluid helium-3 was found, and it is now the definition of temperature below one kelvin.

5 figures · part 7 on Third law

Relativity

Space and time, drawn on the same axes.

The stiffness that falls below four-thirds. The first adiabatic exponent Γ₁ — how steeply pressure rises when a gas is squeezed without heat escaping — of hot oxygen with its radiation and its equilibrium electron–positron pairs, against temperature, at densities of 10³, 10⁴, 10⁵, 10⁶ g/cm³. At 10³ g/cm³ its least value is 1.222, near 1.2 billion kelvin; at 10⁴ g/cm³ its least value is 1.235, near 1.3 billion kelvin; at 10⁵ g/cm³ its least value is 1.285, near 1.8 billion kelvin; at 10⁶ g/cm³ its least value is 1.339, near 3.1 billion kelvin. A star can hold itself up only if its core's Γ₁ averages above 4/3, the value for a gas of light alone; a core that dips below it has no stable size, and the pair production that pulls it below is strongest where the gas is hot and thin — the cores of the most massive stars.

The heat that turns into matter

Thermodynamics counts particles and relativity lets them be made. Heat a gas to a billion kelvin and its light carries enough energy to create electrons and positrons, which the gas then holds in equilibrium like any other ingredient. The heat spent making them raises neither the temperature nor the pressure, so the gas grows soft — softer, in the cores of the most massive stars, than a star can survive. Those stars collapse, ignite their oxygen all at once and blow themselves entirely apart; and the same pairs, vanishing a few seconds after the Big Bang, left the universe's light hotter than its neutrinos by a factor that is the cube root of eleven over four.

5 figures · part 5 on Relativistic thermodynamics Two straight worldlines that meet twice. A spacetime diagram of a universe whose space is a circle 10 light-years round, drawn unrolled: the shaded strip is the whole of space, and the strips beside it are the same space again, so the dashed lines are the same stay-at-home twin A seen in the copies. Twin B leaves A at 0.6c and never changes speed or direction. After 16.67 years by A's clock B has gone once round and meets A again — the line reaches A's copy at 10 light-years. Neither twin has accelerated at any moment; both worldlines are straight. A has aged 16.67 years and B 13.33, less by the factor 1/γ = 0.800. The usual resolution of the twin paradox, that the travelling twin is the one who turned round, is not available: here nobody turned.

The twin who never turns round

The twin paradox is usually settled by pointing at the turn: the travelling twin is the one who fired the engines and came back, and so the one whose worldline is not straight. Put the twins in a universe whose space closes on itself, like the surface of a cylinder, and the traveller can come home by going straight on. Neither twin ever accelerates, both worldlines are straight, and the traveller still comes back younger. What decides it is not any motion but the shape of space, which picks out one state of motion as special — and hands every observer a way to measure it.

5 figures · part 8 on Time dilation

Quantum

Where the continuous picture runs out, and what replaces it.

Each unit of angular momentum costs a factor of tens of thousands. Weisskopf's estimates of the half-life of a gamma-ray transition in a nucleus of mass number 100, against the photon's energy, for electric multipoles of order 1 to 5 — the photon carrying away 1 to 5 units of angular momentum. At 1 MeV the estimates are 10⁻¹⁵ s for E1, 10⁻¹¹ s for E2, 2.0 µs for E3, 292.5 ms for E4, 17.3 h for E5. Each extra unit multiplies the half-life at 1 MeV by between 60,000 and 210,000: about a thousand of that is (λ/2πR)², the square of the photon's reduced wavelength over the nucleus's radius, which grows as the energy falls, and the rest is a numerical factor that grows with the order. A transition that must carry away five units at 100 keV has an estimated half-life longer than the age of the universe; the same energy carried off with one unit takes less than a nanosecond.

The spin a photon has to carry away

An excited nucleus sheds its energy as a gamma ray in a millionth of a millionth of a nanosecond — or in minutes, or centuries, or never within the age of the universe. The energy available hardly matters. What matters is how much angular momentum the photon must carry off, because a source much smaller than its wavelength is very bad at launching a wave that twists, and each extra unit of twist costs a factor of tens of thousands in time. Selection rules are usually stated as what cannot happen; they are really a price list, and it spans forty decades.

5 figures · part 6 on Decay A temperature the circuit refuses to go below. The escape temperature — the temperature that would give the measured escape rate if escape were purely thermal — against the actual temperature of the junction, for a 10 µA, 5 pF junction at biases of 0.985 and 0.99 of its critical current. Hot, the two are equal: the phase is kicked over the barrier by thermal noise. Cooled below the crossover, 39 mK at 0.985 and 36 mK at 0.99, the escape temperature stops falling and levels off at 42 mK and 44 mK: the rate no longer depends on the temperature at all. This levelling-off, measured by Michel Devoret, John Martinis and John Clarke in 1985 with the junction's electrical environment carefully characterised, was the evidence that a macroscopic electrical variable tunnels.

The circuit that goes through a wall

Tunnelling is usually told about electrons and alpha particles — single, small, quantum things passing through barriers they cannot climb. In 1985 it was seen in an electrical circuit: the phase difference across a Josephson junction, a coordinate shared by all the electrons in two pieces of superconducting metal, trapped in a well and escaping from it through the barrier rather than over it. The evidence was a temperature below which the circuit's escape rate stopped caring how cold it was. The same circuit showed discrete energy levels, and its descendants are the qubits of superconducting quantum computers.

5 figures · part 6 on Tunnelling

Fluids

Matter that will not hold a shape, and the forces that act in it anyway.

The thin parcel that keeps sinking. A parcel pushed one centimetre down a column in which heat stabilises and salt destabilises, with a density ratio of 2, integrated for three hours, for parcels whose pattern of sinking and rising water repeats every 2 m, 20 cm, 6 cm. The 2 m parcel keeps its heat and its salt and swings back and forth with the buoyancy period of 23.6 minutes. The 20 cm parcel crosses its starting level 8 times while its swing dies away, and is left sinking slowly, at the 0.48 e-folds an hour its width allows; the 6 cm parcel never comes back and passes five centimetres below it after 60 minutes. The thinnest loses its heat to its surroundings in 11 minutes and its salt in 18 hours, so it arrives salty and cooled, heavier than the water around it, and runs away at 2.33 e-folds an hour — the rate the linear dispersion relation gives for that width.

The salt that sinks through a stable sea

Warm salty water lying on cold fresh water is heavier at the bottom at every depth, and by every test of static stability it should stay where it is. It does not. Heat leaks sideways out of a thin tongue of sinking water a hundred times faster than salt does, so the tongue arrives cooled and still salty — heavier than its new surroundings — and keeps going. A column can be stable by density and unstable by diffusion, and most of the subtropical ocean is.

6 figures · part 7 on Stratification The flow past a thread takes thousands of widths to recover. The speed of the fluid beside a body, at right angles to the stream, as a fraction of the free stream, against distance from the body's axis in body radii, in slow viscous flow. Past a sphere, from Stokes's solution, the flow is back to 90 per cent of the stream by 7.5 radii whatever the Reynolds number. Past a cylinder, from Lamb's inner solution, the speed grows only as the logarithm of distance, with a slope set by the Reynolds number: it reaches 90 per cent of the stream at 29, 232, 1,843 and 1.5 × 10⁴ radii for Re = 10⁻¹, 10⁻², 10⁻³ and 10⁻⁴ in turn. Each cylinder curve is drawn out to where the inner solution reaches the stream speed, beyond which Oseen's outer flow takes over. Each tenfold drop in the Reynolds number pushes the recovery ten times further out, because what finally stops the disturbance is inertia, at a distance of order ν/U.

The drag a distant wall decides

A sphere moving slowly through a liquid has a drag that belongs to the sphere: six pi times the viscosity, the radius and the speed, and the liquid a few radii away hardly knows it is there. An infinitely long cylinder has no such drag. The equations of slow viscous flow have no solution past it at all — Stokes found this in 1851 — because a force spread over circles instead of spheres never finishes spreading. What rescues the cylinder is whatever distant thing supplies a largest length, and its drag carries the logarithm of that length for ever after.

5 figures · part 8 on Viscosity Shake more gently and the grains choose a side. The steady share of the grains in each of two shaken compartments joined by a slot, against B, the parameter that grows as the shaking gets gentler. Below B = 4 the only steady state is an equal share, and it is stable. At B = 4 it becomes unstable and two clustered states appear, continuously, one with each compartment full: the fuller compartment holds 0.855 of the grains at B = 5, 0.929 at 6 and 0.979 at 8. Which compartment fills is decided by the first fluctuation. Nothing in the rules prefers either side, and the equal state that the rules would suggest is the one the grains avoid.

The sand that chooses a side

Divide a box of grains into two compartments with a slot in the wall between them and shake it from below. Shaken hard, the grains share themselves equally, as molecules of a gas would. Shaken a little more gently, they gather in one compartment and leave the other almost empty, and the crowded side is cold and the empty side hot. Nothing sorted them. Every collision between grains loses energy, and a compartment that happens to hold a few more grains loses more of it and stops throwing grains over the wall.

5 figures · part 8 on Granular matter

Astrophysics

Gravity read as geometry, and the laws carried where no laboratory can follow.

Tadpoles and a horseshoe, drawn in the turning frame. Orbits of small bodies near the circle of a planet's orbit, integrated in the frame turning with it, with the distance from that circle exaggerated 12 times so the shapes can be seen. With Jupiter's mass ratio, 0.000954, a body started 10° and 25° ahead of L4 at rest in the turning frame circles round L4 rather than falling off the hill: it swings between 51° and 70°, and between 42° and 85° from the planet — a tadpole. With a mass ratio of 10⁻⁴, a body started opposite the planet sweeps round through L4, L3 and L5 and back, never coming within 24° of the planet — a horseshoe, the kind of orbit Saturn's moons Janus and Epimetheus follow relative to each other, exchanging places every four years. In every case the Jacobi constant, the one quantity conserved in the turning frame, is held by the integration to better than a part in a million.

The hilltop that holds the Trojans

Sixty degrees ahead of Jupiter and sixty behind it, thousands of asteroids share the planet's orbit, held at the corners of the equilateral triangles that Jupiter and the Sun make. Seen from the frame that turns with Jupiter, those corners are the tops of two hills in the effective potential — the worst places to try to keep anything still. They hold asteroids anyway, because in a turning frame a body sliding off a hill is swung sideways by the Coriolis force into an orbit round the top. The trick works only if the planet is light enough: above a mass ratio of about one in twenty-six, the hilltops let go.

5 figures · part 5 on Orbit stability The heat an eccentricity buys. The power tides dissipate inside Io against the eccentricity of its orbit, from (21/2)(k₂/Q) G M² R⁵ n e²/a⁶ with k₂/Q = 0.015: it grows as the square of the eccentricity. At Io's forced eccentricity, 0.0041, it is 9.3 × 10¹³ W, against about 1.0 × 10¹⁴ W measured from Io's infrared glow — the same to within the uncertainty of both — and against 4.7 × 10¹³ W for the whole of the Earth's internal heat, from a body forty-three times Io's volume. Halve the eccentricity and the heating falls fourfold; take it away and Io would freeze.

The moon kept molten by its neighbours

Io is the most volcanic body known, with hundreds of active vents and lava hotter than any erupting on the Earth, on a moon smaller than the Earth's own and far too small to have kept any heat from its formation. The heat is made continuously, by tides: Jupiter stretches Io, and because Io's orbit is slightly eccentric the stretch changes every day and a half, flexing the rock and warming it. Tides should have rounded the orbit off in a hundred thousand years and switched the heating off. They cannot, because two other moons keep tugging the orbit back into its oval, and the heat Io radiates is paid for by Jupiter's spin.

5 figures · part 6 on Self-gravity Five images of one source, and the curves where they are made. The sky around a lens with a core of 0.10 and an external shear of 0.15, in Einstein radii, with its two critical curves: an outer one near the Einstein ring and a small inner one round the core. A source at (0.06, 0.04) in the source plane — inset, with the caustics the critical curves map to — lies inside the diamond-shaped caustic and has 5 images. Their magnifications are +4.99, −3.76, +3.42, −2.63, +0.01 — a minus sign for an image turned over — and the parities sum to +1. Four sit near the outer curve and carry nearly all of the 14.8-fold total; the fifth, at the centre, is magnified by only 0.013. A source further out, at (0.55, 0.1), has 3.

The fifth image a galaxy hides

A single star bends light into two images of whatever lies behind it. A galaxy, whose mass is spread out, has to make an odd number — one, three or five — because the map from the sky to the source is smooth and has nowhere to put an even count. The famous cross of four quasar images is therefore five images, with the fifth thousands of times too faint to see, sitting at the centre of the galaxy that made it. Images are born and destroyed only in pairs, and only on the curves where the map folds.

5 figures · part 5 on Light deflection The self-bias set by a phase alone. The dc self-bias against the phase between the fundamental and its second harmonic, each of amplitude A, for three chambers: symmetry parameter ε = 0.5, 1, 2, where ε = 1 is a chamber with equal sheaths on both sides. In the symmetric chamber the bias runs from −0.438 A at 0° through zero at 45° to 0.438 at 90°, and back — −43.8 V to 43.8 V at A = 100 V — staying within 0.011 A of a straight line between 0° and 90°. An asymmetric chamber adds a fixed offset of the kind unequal electrodes produce and leaves the phase control in place, so the geometry and the waveform add rather than compete.

The bias a waveform can choose

A plasma between two identical electrodes, driven by a sine wave, has no reason to prefer either side, and it does not: no direct voltage appears. Add the second harmonic and it does, although the chamber is still symmetric, no direct current flows and the power spectrum does not depend on the phase between the two. The phase alone sets a bias of up to 0.44 of each harmonic's amplitude, of either sign, and moves the energy of the ions from one electrode to the other while their number stays put.

5 figures · part 7 on Plasma oscillation

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