The collection

Every essay — page 27

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.

Every path a rolling ball can take on a turntable, seen from the room. The paths, seen from the room, of four solid balls rolling without slipping on a turntable of radius 15 cm turning at 33.3 revolutions a minute (grey disc), over 6.3 seconds. A ball set down at rest relative to the room stays exactly where it is, spinning on the spot while the table turns under it. Every other ball moves on a circle, turning the same way as the table at 2/7 of its rate — once every 3.5 turns of the table — and comes back to where it started, whatever the push. A ball set down moving with the table does not ride with it: it circles on a path three and a half times the radius it started at, and the circle's centre is on the other side of the axis.

The ball a turntable will not throw off

Set a marble down on a spinning turntable and the expectation is that it will be flung off the edge. It is not. Placed at rest relative to the room, it stays exactly where it is, spinning on the spot while the table turns beneath it. Given a push, it moves in a circle, turning the same way as the table at two-sevenths of the table's rate, and comes back to where it started. Tilt the turntable and the marble does not roll downhill: it wanders across the slope in a chain of loops. A rolling ball on a rotating surface obeys exactly the equation of an electric charge in a magnetic field, and the two-sevenths is how much of every push goes into turning the ball rather than moving it.

5 figures · part 6 on Circular motion

The frequency of a mass on a spring that weighs something. The angular frequency of a mass M on a spring of stiffness k and mass mₛ, in units of √(k/M), against the ratio mₛ/M on a logarithmic axis: exact, from β tan β = mₛ/M (solid); Rayleigh's estimate √(k/(M + mₛ/3)) (dashed); and the textbook answer that ignores the spring's mass, 1 (dotted). For a spring a tenth of the mass the exact value is 0.9836, Rayleigh's 0.9837 and the textbook's 1 — 1.7 per cent too fast. With spring and mass equal the exact frequency is 0.8603, Rayleigh's 0.8660: the third is good to 0.66 per cent where the textbook answer is 16 per cent out.

The third of itself a spring carries

Every textbook gives the frequency of a mass on a spring as the square root of stiffness over mass, and every real spring has mass of its own, which the formula leaves out. The repair is famous and oddly specific: add a third of the spring's mass to the load. The third is not a fudge. It is what the spring's kinetic energy comes to if each coil moves in proportion to its distance from the fixed end, and it is good to two-thirds of a per cent even when the spring weighs as much as the mass. Where it fails, it fails because the spring has stopped being a spring and become a column of waves, with notes of its own above the bounce — the notes that make the valve springs of an engine lose control.

5 figures · part 7 on Harmonic approximation

The drift is the area under the height. Height against time for the two balls: the throw (dashed curve) rising to 100 m in 4.52 s and falling back in as long again, and the drop (solid) falling from 100 m. For vertical motion the east–west velocity is always −2Ω cos λ times the height above the starting point, so the east–west drift is −2Ω cos λ times the area between the path and the starting height. The throw's area (lower shading) is (4/3)·h·t = 602 m s, above its start, so it drifts west; the drop's (upper shading) is h·t/3 = 151 m s, below its start, so it drifts east. The ratio of the two areas is four.

The ball thrown up that lands to the west

Drop a ball from a tower a hundred metres high and, because the Earth turns, it lands a centimetre and a half east of the point directly below. Throw a ball straight up so that it just reaches the same height, and it lands six centimetres west of where it was thrown — four times as far, the other way. The asymmetry has a one-line explanation that turns a problem in rotating frames into a problem about areas: for anything moving straight up or down, the sideways drift is proportional to the area under its graph of height against time. A dropped ball spends its flight below where it started, a thrown one above, and the thrown one's area is four times as large.

5 figures · part 7 on Projectile

Waves

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

$The same absorber, spread evenly and gathered into patches. The fraction of light transmitted by a layer holding a fixed amount of absorber, on a logarithmic axis, against the optical depth it would have if spread evenly, for the absorber spread evenly (solid) and gathered into patches covering 50 per cent and 10 per cent of the area, the rest left clear. At an average optical depth of 2 the even layer transmits 13.5 per cent; with the absorber in patches over half the area, 50.9 per cent; over a tenth, 90.0 per cent — almost all of it through the gaps. The average of e^(−τ) over a layer is never less than e^(−average τ): Jensen's inequality, which holds for any convex function. Beer's law describes a uniform absorber and overstates the attenuation of every other kind.$

The pigment that absorbs less for being packed

Dissolve a dye evenly in water and it absorbs a certain fraction of the light passing through. Gather the same dye into grains — the same molecules, the same amount, in the same volume — and it absorbs less. The light has not changed and neither have the molecules. The grains shadow their own insides, and the light that would have been caught by the shaded molecules slips through the gaps between grains instead. The effect is called packaging, and it is why the chlorophyll inside large algae is worth less per molecule than the chlorophyll in small ones, and why an ocean's colour does not say directly how much chlorophyll is in it.

5 figures · part 7 on Attenuation $A lever that makes air and water meet. The fraction of sound power crossing from air into water through an ideal lossless transformer that multiplies pressure by a factor n, against n on a logarithmic axis. With no transformer, n = 1, it is 0.11 per cent. It rises as the transformer makes the water look less stiff and air look stiffer, reaching everything at n = 60, and falls again beyond, where the water is made to look softer than air. The middle ear's 22 gives 43 per cent in this picture — a gain of 26 dB over no transformer at all. The curve is flat near its top: a lever within a factor of two of the ideal still passes about two-thirds of the power, which is why a mechanism need not be precise to be effective.$

The lever that lets air speak to water

Sound in air striking a surface of water is almost entirely reflected: 99.9 per cent bounces off, which is why a swimmer underwater hears the world above as a muffled murmur. The inner ear is filled with fluid, and the eardrum faces air, so every land animal has the problem of getting sound across that boundary. The answer is three tiny bones and a difference in area, which together multiply the pressure about twenty-fold and turn a thousandth of the power into nearly half. The middle ear is an impedance transformer, the mechanical twin of the one in every radio, and it works best over just the band where speech carries its meaning.

5 figures · part 7 on Impedance

Optics

Light, and the small number of rules it obeys.

The beams that add only if they differ

Take two identical lasers and try to merge their beams into one beam twice as bright. No arrangement of lenses, mirrors and prisms will do it: the brightness of light cannot be increased by passive optics, and two copies of the same beam can only ever be put side by side. The escape is to make the beams not copies. Give them crossed polarisations and two will merge. Give each a different colour and a grating will stack dozens into a single beam. Lock their phases so that they stop being separate beams at all, and they add — but only as well as the gaps between them allow and only while their phases are held to a twentieth of a wavelength.

5 figures · part 7 on Etendue $A scan into a layered sample, with a broad source and a narrower one. The envelope of the interference signal against depth in a sample with five reflecting boundaries (marked), at 60, 72, 150, 158 and 260 µm, scanned with a 50 nm source (solid) and a 10 nm source (dashed), both centred at 840 nm. The broad source, with a 6.2 µm resolution, separates every boundary, including the two that are only 8 µm apart. The narrow one, with 31 µm, merges the pairs into single blurred echoes. Every boundary in a real eye or skin sample is a small change of refractive index that reflects a tiny fraction of the light; the scan is a map of where those changes are.$

The depth a broad spectrum can see

Interference is usually spoiled by a broad spectrum: many colours drift out of step, and fringes wash out a few wavelengths from where two paths are equal. Turned round, that failure is a ruler. Light whose colours stay in step only when two paths match to within a few micrometres can report the depth of every reflecting layer inside something translucent — a retina, a skin, a coat of paint — to that precision, without cutting it open. Optical coherence tomography reads depth off the width of a spectrum, and it has become one of the most used imaging methods in medicine; its resolution is set by how many colours the light contains, and the finest instruments use sources whose spectra span hundreds of nanometres.

5 figures · part 8 on Coherence

Electromagnetism

Charge, field, and the lines drawn between them.

Rotation is a magnetic field, and a superconductor cancels it. Magnetic fields along the spin axis inside a spinning metal, in units of 2mω/e, as seen by its electrons in the frame turning with it. In that frame the Coriolis force on an electron is exactly the force a magnetic field of 2mω/e would exert, pointing against the spin for a negative charge (Larmor's theorem). A normal metal spun in no field has no real field inside, and its electrons feel the rotation's equivalent field in full. A superconductor will not let its carriers feel any field in their interior — that is the Meissner effect — so it generates a real field of exactly the same size pointing the other way, and the sum vanishes. The real field is the London moment.

The field a spinning superconductor makes of itself

Spin a lump of superconductor with no magnet anywhere near it, and a magnetic field appears inside it, pointing along the axis of spin: about eleven trillionths of a tesla for every radian per second, the same in lead, niobium or a copper-oxide ceramic. The field is not left over from anything. In a turning frame the Coriolis force on a moving charge is indistinguishable from a magnetic force, a superconductor will not let its carriers feel any field inside it, and so it makes a real field to cancel the one rotation imitates. The size depends on nothing but the carriers' mass and charge, which makes the field a balance for weighing them — and the one time it was read to a part in a hundred thousand, it gave an answer theory has not matched.

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

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.

5 figures · part 6 on Gauss's law

Thermodynamics

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

What one trajectory says, and what the crowd says. For the walk with α = 0.5: the time-averaged mean-square displacement of each of 12 single trajectories, each watched for 10000 time units — the average of the squared displacement over every window of a given lag along that one path — against the lag, on logarithmic axes (thin lines), and the mean-square displacement of an ensemble of 800 walkers at the same times (thick). The ensemble spreads as t^0.5. Every single trajectory's time average grows nearly in proportion to the lag — mean slope 0.82, where the ensemble's is 0.5 — as if it were ordinary diffusion, and the trajectories disagree with one another about the diffusion coefficient by more than a factor of ten. A time average along one path and an average over many paths, which for ordinary diffusion are the same thing, here give different laws. That is weak ergodicity breaking, and it is what a single-molecule tracking experiment has to be read against.

The diffusion that slows the longer it is watched

A random walker whose steps come at irregular times still diffuses normally, provided the waits have a finite average. Let the waits have no average — mostly short, occasionally enormous, as they are for a molecule that keeps getting stuck — and three things change at once. The spread grows more slowly than time. A diffusion coefficient measured by following one particle depends on how long it was followed. And following one particle for a long time no longer tells what many particles do: averages along a path and averages over a crowd give different laws. Tracking single molecules inside living cells runs straight into all three.

5 figures · part 9 on Diffusion

The melting line of water, through five ices. The pressure at which water freezes, in megapascals on a logarithmic axis, against temperature, along each of the five solid phases that meet the liquid, from the international standard equations. Along ordinary ice, Ih, the melting pressure rises as the temperature falls, from 611 Pa at 273.16 K to 209 MPa at 251.165 K — the backwards slope of the familiar melting line. There the liquid meets ice III, and from then on every ice melts at a higher pressure as the temperature rises: ice V, ice VI, and ice VII, which reaches 20.6 GPa at 715 K. The dots are the four triple points where two ices and the liquid coexist. The lowest temperature at which liquid water can exist in equilibrium, 251.165 K, is at the first of them.

The melting line that turns round

Squeeze water and it freezes at a lower temperature — the backwards slope of its melting line, and the reason ice floats. Keep squeezing and the slope reverses. Past two thousand atmospheres ordinary ice gives way to a denser ice, then another, and from there on every ice melts at a higher temperature under more pressure, exactly as a normal substance's does. At twenty thousand atmospheres water freezes into an ice that is still solid at a hundred degrees Celsius; at two hundred thousand, one still solid at four hundred. The famous anomaly belongs to one ice out of five, and the same relation that explained it explains its end.

5 figures · part 10 on Phase change

Relativity

Space and time, drawn on the same axes.

The heat hidden in a Doppler shift

An observer who accelerates without stopping sees empty space as warm. The usual derivation takes quantum field theory and several pages. There is a shorter one, and it needs no quanta at all. Watch a single wave of a single frequency from a steadily accelerating rocket: it is redshifted, and because the rocket keeps speeding away, the redshift grows exponentially. Take the spectrum of that fading signal, and it is exactly a Planck spectrum, at a temperature set by the acceleration and nothing else — the same whatever frequency the wave started with. The thermal distribution was hidden in the geometry of the horizon all along.

5 figures · part 5 on Accelerated frames

A beam of light pulls hardest on what comes towards it. The sideways pull of a long, thin beam of light on a body moving parallel to it at velocity βc, as a multiple of the pull the same body would feel from a static mass with the same energy per unit length, against β: 2(1 − β)²/(1 + β²). A body at rest beside the beam is pulled twice as hard as by a static mass — the beam's pressure along its length counts. A body moving with the beam is pulled less, and a ray of light running alongside it is not pulled at all (β = 1). A ray running against it is pulled 4 times as hard as a static mass of the same energy would pull it. Tolman, Ehrenfest and Podolsky found this in 1931. Two parallel laser beams do not attract each other; two opposed ones do.

The light that pulls twice as hard as its mass

Energy has mass, E/c², and a box of light weighs that much more than an empty one. But the gravity something produces is not set by its energy alone. Pressure counts too, three times over, and light's pressure is a third of its energy — so a beam of light pulls on things beside it twice as hard as its mass would suggest. The box weighs only E/c² because its walls, straining to hold the light in, pull with a negative amount that exactly cancels the difference. Two parallel beams of light do not attract each other at all; two opposed ones attract four times as hard. And a substance whose pressure is negative enough does not pull at all — it pushes.

5 figures · part 7 on Mass-energy $The orbit an observer would reconstruct. The star's velocity along the line of sight, measured from the Doppler shift of its spectrum, as a fraction of its true orbital speed, plotted against the time the light arrives, in orbital periods, if light left the star at c plus the star's velocity. With light at c (dotted) the curve is the sine wave of a circular orbit. At 0.6 times the critical distance (solid) the same circular orbit arrives as a lopsided curve — a slow rise and a steep fall — that an astronomer would fit with a highly eccentric orbit that is not there. At twice the critical distance (dashed) the curve doubles back on itself: at some moments the spectrum would show two or three velocities at once, as if the star were in several places. Willem de Sitter pointed out in 1913 that spectroscopic binaries show nothing of the kind.$

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.

5 figures · part 7 on Velocity-addition $The stiffness a two-solar-mass star demands. The heaviest neutron star the model matter can support, in solar masses, against the constant square of the sound speed assumed above 300 MeV/fm³, as a fraction of c². At the conformal value, one-third (dashed), it is 1.92 solar masses; the heaviest star the model can make rises to 2.97 at the causal limit. To hold up the 2.08 solar masses of J0740+6620 (shaded) the square of the sound speed must exceed 0.394 of c² somewhere above that density — above the conformal one-third, by this model's reckoning. The conclusion depends on the model's low-density part, which the next figure varies.$

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.

5 figures · part 7 on Relativistic thermodynamics

The spin that runs ahead of its own motion

A charged particle circling in a magnetic field has two directions that turn: the direction it is moving, and the direction its spin points. If the particle's magnetism were exactly what Dirac's equation first gave — a g-factor of exactly two — the two would turn together at every speed, the spin locked to the motion as a compass needle is locked to a car. The muon's g is not quite two, and its spin creeps ahead by a fixed rate that is the same at a crawl and at 99.94 per cent of the speed of light. That constancy is a delicate cancellation between a magnetic field made stronger by motion and a rotation that exists only because the particle is being turned, and it is what made the most precise measurements of the muon possible — at one magic speed, where the electric fields that steer the particles stop affecting the spin at all.

5 figures · part 8 on Field transformation

Quantum

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

$What correlation is worth, to a helium atom and to H⁻. The energy needed to remove the second electron, by three descriptions, as a fraction of the measured value, for the negative hydrogen ion and for helium. Helium: one shared orbital 23.07 eV, one electron in and one out 23.83 eV, measured 24.59 eV — the simple picture is 94 per cent right. H⁻: −0.74, 0.362 and 0.755 eV — the simple picture is not merely inaccurate but has the wrong sign. The energy the simple picture misses — the electrons avoiding each other — is about 1.5 eV in both; in helium it is a small correction to 24.6 eV, and in H⁻ it is everything.$

The ion that exists because its electrons keep apart

Add a second electron to a hydrogen atom and it holds on, by three-quarters of an electronvolt. The simplest respectable calculation says it should not — that the two electrons, sharing one orbital, repel each other more than the proton can hold them, and the ion falls apart. The same calculation gets helium's energy right to two per cent. The difference is not a failure of the method but a measure of what it leaves out: the two electrons avoiding each other. In helium that is a small correction. In the negative hydrogen ion it is the whole of the binding, and that ion is what makes the Sun opaque.

5 figures · part 5 on Atomic structure

Three identical quarks in one state, and the label that rescues them. The Δ⁺⁺ particle, mass 1,232 MeV and charge +2, drawn as the quark model describes it: three up quarks, all in the lowest orbital state, all with their spins pointing the same way, giving spin 3/2. Left: with nothing else to tell them apart, the state is unchanged by swapping any two quarks — symmetric in where they are, in their spins and in their flavour — which the exclusion principle forbids for identical fermions. Right: give each quark a label that takes three values, and put the three in the one combination of the three values that changes sign under every swap. The whole state is then antisymmetric, as it must be, and the particle is allowed. The label is called colour, and three is the smallest number of values that can do this for three quarks.

The label three identical quarks forced

In 1964 the quark model had one embarrassment it could not explain away. A particle called the Δ⁺⁺, well measured and entirely ordinary, had to be three identical up quarks in the same state with their spins pointing the same way — which the exclusion principle forbids outright. The way out was to give quarks a hidden property that takes three values, so that the three could be different after all. Nobody could see the property, and for years it looked like a bookkeeping trick. Then three independent measurements counted it, and every one came out at three. The exclusion principle, which keeps electrons in shells, had predicted the colour of the strong force.

5 figures · part 5 on Exclusion $Only a slow grating sends everything one way. The largest fraction of particles, arriving at Bragg's angle, that a standing light wave can put into the first diffracted order, choosing the wave's depth freely, against how long it acts, in units of ħ divided by the two-kick recoil energy, on a logarithmic axis. A brief pulse acts as a thin phase grating and can put at most 33.9 per cent into that order (dashed), the Raman–Nath limit, spreading the rest over the others. A pulse lasting longer than a few units has time to notice that only one exchange conserves energy, and can put nearly all of it there: 99.9 per cent at the longest time drawn. The same distinction between thin and thick gratings decides whether sound in a crystal diffracts light into many orders or one.$

The grating made of light

A diffraction grating is ordinarily made of matter and diffracts light. In 1933 Pyotr Kapitza and Paul Dirac proposed the opposite: a standing wave of light, whose bright and dark bands repeat every half wavelength, should diffract a beam of electrons. It took fifty-three years to see the effect with atoms and sixty-eight to see it with electrons, because a free electron barely notices light. The grating works by momentum: it can only change a particle's motion in whole pairs of photon kicks. A brief flash scatters particles into a spread of orders in proportions fixed by Bessel functions; a slow, gentle one sends every particle one way, which is how laser light became the mirrors and beam splitters of atom interferometers.

5 figures · part 5 on Matter waves

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.

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.

5 figures · part 6 on Blackbody

Fluids

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

Why the mixture of two equal waters is heavier than either. A temperature–salinity diagram for seawater near the surface, with lines of equal density (curves) 0.3 kg/m³ apart. Two waters lie on the same line: one at 4 °C and 33.93 g/kg, the other at 16 °C and 36.53 g/kg, both 1026.00 kg/m³. Mixing them in any proportion moves along the straight line between them, because temperature and salt add linearly; the lines of equal density curve, because cold water's density responds less to temperature than warm water's. The straight line lies on the dense side of the curve. An even mixture, at 10 °C and 35.23 g/kg, is 0.182 kg/m³ denser than either parent, and sinks between them.

The mixture heavier than either water

Two bodies of seawater can have exactly the same density — one cold and fresh, one warm and salty — and sit side by side without either wanting to sink. Mix them, and the mixture is denser than both, and sinks. Nothing has been added and nothing removed; the extra weight comes from the fact that water's density is a curved function of temperature while mixing averages temperature in a straight line. The same curvature, at its most extreme, is why a lake overturns twice a year and freezes from the top, and why the open ocean, too salty to have a density maximum above freezing, does neither.

5 figures · part 8 on Stratification

Five square logs of different densities, as they float. Cross-sections of five long square logs in their stable floating positions, with relative densities 0.1, 0.25, 0.5, 0.75 and 0.9, each drawn with its water line and its submerged part shaded, and a dot at its centre. The lightest floats flat and high; at 0.25 it has tilted to about 27°; at 0.5 it floats on its corner with a diagonal vertical and the water line across the middle; at 0.75 it tilts to the same angle as at 0.25, and at 0.9 it floats flat again, low in the water. The heavy logs are the light ones turned upside down, with air and water exchanged.

The log that floats on its corner

A long square beam of very light wood floats flat, one face up, as anyone would expect. A beam of wood half as dense as water does not: it rolls onto a corner and floats with a diagonal vertical, like a diamond. Between the two, at a quarter of water's density, it floats tilted at twenty-seven degrees, neither flat nor on edge. Heavier beams repeat the sequence in reverse, and the whole pattern is symmetric: a beam of density s floats at exactly the angle of one of density 1 − s. None of it depends on anything but a square, a density and Archimedes, and the same arithmetic decides which way an iceberg rolls when it breaks.

5 figures · part 7 on Buoyancy

Astrophysics

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

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.

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.

5 figures · part 7 on Horizons $The band in which radiation feeds a mode instead of draining it. The speed of a mode's pattern as seen from far away, in units of the star's own spin rate, against the spin rate as a fraction of the fastest a star can turn. The shaded band, between zero and one, holds patterns that run forward across the sky and backward through the star; gravitational radiation drives any mode inside it and damps any mode outside. The r-modes with ℓ = m = 2, 3 and 4 sit at 0.667, 0.833 and 0.900 at every spin, inside the band however slowly the star turns. The dashed curves are schematic modes whose crests run backward on the star at a fixed 0.2, 0.5 and 0.9 of the Kepler rate: each is dragged into the band only once the star turns faster than its crest runs, which is why an ordinary oscillation needs a star near break-up and an r-mode does not.$

The wobble that radiating makes larger

Radiation normally drains a vibration: a struck bell goes quiet because it sends its energy away as sound, and a ringing black hole settles in milliseconds because it sends its energy away as gravitational waves. A spinning star can have the opposite. A pattern that runs backward through the star's material, while the star carries it forward across the sky, has less angular momentum than the star would have without it, and every wave it radiates takes away angular momentum it did not have to give. The pattern grows. One family of such patterns, the r-modes, runs backward on every rotating star, so every spinning neutron star is unstable unless friction wins — and that may be why none has been found spinning faster than 716 turns a second.

5 figures · part 7 on Gravitational waves

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