The shadow that shows the core is liquid

The deepest hole ever drilled, on the Kola Peninsula in the Soviet Union, stopped at a little over twelve kilometres — two-tenths of one per cent of the way to the Earth’s centre. Everything known about the rest of the interior, the rocky mantle and the iron core, the core’s being liquid and the solid ball at its centre, was read from earthquake waves travelling through the planet and recorded at its surface. And the two observations that settled the core’s existence and its state were not of waves arriving, but of waves that failed to.

A shape that travels noted in passing that the transverse wave of an earthquake cannot cross the outer core and that the shadow it casts is the evidence the core is liquid. This essay traces the rays that cast that shadow and the other one, works out where the shadows fall, and shows why a few waves arriving where they should not were the clue to what lies at the very centre.

Two kinds of wave, two speeds #

An earthquake sends out two kinds of body wave. The compressional or P wave is a wave of squeezing and stretching along the direction of travel, like sound; it travels through anything that resists compression, solid or liquid. The shear or S wave is a wave of side-to-side distortion; it needs the material to resist being sheared, and a liquid does not. The medium decides the speed found the rule for both: the speed is the square root of a stiffness over a density. For P waves the stiffness is a combination of the material’s resistance to compression and to shear; for S waves it is the shear stiffness alone, so that

$$v_S = \sqrt{\mu/\rho},$$

and in a liquid, where the shear modulus $\mu$ is zero, there is no S wave at all. Within the rock of the mantle both exist, P always the faster — near the surface about 8 kilometres a second against 4.5, which is why every seismogram shows P arriving first and S some minutes later.

The profile in the figure is a smoothed reading of a reference model built in 1981 from thousands of earthquakes, and every feature of it matters below. Through the mantle both speeds rise steadily with depth, because the rock is compressed by the weight above it and its stiffness grows faster than its density. At 2891 kilometres the rock ends and the iron of the core begins; there the P speed falls by more than a third and the S speed falls to zero. Deeper still, at 5150 kilometres, the iron becomes solid again and S reappears.

Why the speeds rise with depth #

The steady rise of speed through the mantle is not obvious, since rock deeper down is also hotter, and hot rock is softer. Pressure wins. At the base of the mantle the pressure is about 136 gigapascals, over a million atmospheres, set by the weight of rock above in the way the pressure that only knows depth found for any fluid at rest. Squeezing a solid stiffens it far faster than it raises its density, so the square root of stiffness over density grows. The same compression that the sea that stands lower because water gives found lowering the ocean by thirty metres raises the speed of sound in seawater with depth; in the mantle the effect is larger, because the pressures are a thousand times greater.

The rise is what makes the rays curve back to the surface, and so it is what makes the whole method work. In a planet whose speeds were the same at every depth, rays would run in straight chords, and every station would see waves that had sampled only the straight line from the source; the depth reached by each ray would be pure geometry, and the speeds at depth would have to be inferred some other way. Because the speeds rise, each ray turns at a definite depth fixed by its ray parameter, and the arrival time at each distance is dominated by the speed near the deepest point of its ray. A curve of travel time against distance can therefore be inverted, depth by depth, for the speeds — the Herglotz–Wiechert method, worked out in 1907 and 1910, which is how the mantle’s profile was first found.

Rays that curve back up #

A wave front from a point source spreads in all directions, and its progress through the Earth is easiest to follow as rays — lines everywhere perpendicular to the fronts. In a medium whose speed changes with depth, a ray bends, because the part of a front in faster material outruns the part in slower material. The ray that bends without a surface found this in air above a hot road, where light curves up into the denser, slower air; inside the Earth the speed rises with depth, so a ray heading downward at a slant is continuously bent away from the vertical, flattens out at some depth, and curves back up to the surface.

For a spherically layered Earth the bending obeys a single rule, the spherical form of Snell’s law:

$$\frac{r\sin i}{v(r)} = p,$$

where $r$ is the distance from the centre, $i$ the angle the ray makes with the vertical and $v$ the local speed. The quantity $p$, the ray parameter, is the same all along the ray; the bend at the boundary is this rule applied at a single flat interface. A ray turns at the depth where $r/v$ has fallen to $p$, so a ray that leaves the source steeply, with a small $p$, dives deep before it turns, and comes up far away. The figures trace rays by applying the rule shell by shell to the speed profile, each shell a few hundred metres thick.

The mantle rays fan out across the near side of the Earth, each landing further away the more steeply it left. The steepest of them grazes the top of the core and lands about 97° of arc from the source — a little more than a quarter of the way round the planet. Any ray steeper than that strikes the core.

Why a slower core casts a shadow #

A ray crossing into slower material bends towards the perpendicular to the boundary, as light does entering glass. At the core’s surface the P speed drops from 13.7 to 8.1 kilometres a second, so a ray that strikes the core at a glancing angle is bent sharply inward, towards the centre. It crosses the core along a much steeper path than it arrived on, is bent outward again as it leaves into the faster mantle, and comes up on the far side of the Earth. The core acts as a strong lens.

The figure shows where every ray lands. As the leaving angle steepens, the mantle rays land further and further away, up to the grazing ray at 97°. The very next ray, a fraction of a degree steeper, enters the core, is bent inward, and lands not at 98° but on the far side of the Earth, near 180°. As the rays steepen further, their landing points first move back towards the source, to about 144°, and then forward again to 180° for a ray going straight down. There is a gap. No ray that stays in the mantle or turns in the outer core lands between 97° and 144°: the ring of the surface between those distances is in shadow.

The point where the red curve turns back on itself, at 144°, is a caustic: many rays, leaving at slightly different angles, land at nearly the same distance, so the energy piles up there, and the far edge of the shadow is marked by unusually strong arrivals. It is the same focusing that makes the bright line of a rainbow, where rays through a raindrop bunch at one angle, and that makes the bright curves of light on the bottom of a swimming pool.

Beno Gutenberg used exactly this geometry in 1914. From the distances at which direct P waves faded out and core waves began, he computed that the core’s surface lay at a depth of 2900 kilometres — within ten kilometres of the modern value. Richard Dixon Oldham had argued for a core in 1906, from waves arriving late at large distances; Gutenberg’s shadow put its radius on a number.

The cap no shear wave reaches #

The S waves tell a different story about the same core.

The mantle carries S waves just as it carries P waves, and their rays curve back up in the same way, the steepest grazing the core and landing about 103° away. Steeper S rays strike the core and stop. A shear wave arriving at a liquid has nothing to drive: the liquid cannot push back sideways, so the wave’s sideways motion cannot continue into it. Some of the energy is reflected back into the mantle, and some is converted at the boundary into a P wave in the core, which can travel on — but no shear wave crosses. Beyond 103°, over a cap covering 39 per cent of the Earth’s surface, no direct S wave from the earthquake ever arrives.

That is not a ring but a whole hemisphere-sized region, and its existence is the most direct evidence that the outer core is liquid. Harold Jeffreys had argued in 1926 that it must be, from a different observation altogether: the solid Earth yields to the Moon’s tides by an amount that implies an average rigidity lower than the mantle’s, so something deep inside must have none. The S shadow says the same thing in the language of waves. A solid core, however slow or soft, would carry S waves through; this one does not.

Distance from the delay #

The same speeds make a single seismometer a range-finder.

Because S is slower than P, the gap between the two arrivals grows steadily with distance: about eight minutes at 60°, and a station that records the gap can read its distance from the earthquake off the curves. Three stations’ distances fix the earthquake’s location by triangulation. This was the routine use of travel-time tables for most of the twentieth century, and the tables themselves — compiled by Jeffreys and Keith Bullen in 1940, and refined ever since — are the curves in the figure, built up from thousands of earthquakes whose locations were fixed by the same method. The speed profile and the travel-time tables were worked out together, each refining the other.

The figure also shows the shadow in time. Direct P waves stop at 97°, about thirteen and a half minutes after the earthquake. They resume at 144° as core waves. And in the times of the core waves near the antipode there is a second, earlier branch: waves that have crossed the inner core, where the speed jumps up again, arriving a little sooner than they would through liquid alone.

Waves where there should be none #

In the 1930s a Danish seismologist, Inge Lehmann, studying records of large earthquakes in the Pacific at stations in Europe, noticed weak P arrivals inside the shadow zone, at distances where the rays of a single uniform core should have left nothing. She proposed in 1936 that the core has a smaller core within it, in which the speed is higher, and that rays just grazing into it are bent back out early, landing inside the shadow. The dashed rays in the first figure are those rays, traced through the modern speed profile; they land as close as 115° from the source, well inside the ring that a uniform core would leave dark.

Her argument was a short paper with a terse title — a single letter, P′, the notation for core waves — and it was right. The inner core’s surface is at 5150 kilometres depth, about 1220 kilometres from the centre. Whether the inner core is solid was harder to establish. Its higher P speed suggested it, but the decisive evidence came from the Earth’s free oscillations: the tone the whole Earth rings at after a great earthquake includes modes whose frequencies depend on whether the inner core can resist shear, and in 1971 Adam Dziewonski and Freeman Gilbert showed that they require a solid inner core. Direct detection of S waves inside the inner core, converted from P at its surface and back again, has been claimed since 2005 and remains difficult, because the signals are tiny.

Rays are not the whole story #

Every figure here treats the waves as rays, and rays cast perfectly sharp shadows. Waves do not. Where rays stop being enough found light bending round the edge of an obstacle into its geometric shadow, and earthquake waves do the same round the edge of the core: P waves diffracted along the core’s surface arrive, weakened, well into the shadow zone, and their decay with distance beyond 97° is used to measure the structure just above the core. The edges of the shadows in a real seismogram are therefore not lines but gradual fadings, and the textbook figures — a P shadow from about 103° to 143° — reflect where the arrivals become too weak to see as much as where the rays stop.

The model used here also leaves things out. It has no crust, which adds a few seconds to every travel time; the steps in the upper mantle’s speeds near 410 and 670 kilometres are smoothed into one gradient, though in reality they make rays land in overlapping groups between about 15° and 30°; and it is spherically symmetric, when the real mantle has hotter and colder regions whose speeds differ by a few per cent, which bend the rays and spread the arrivals. Those differences are now themselves mapped, by the same kind of reasoning applied to millions of arrival times — seismic tomography — and they show the slow, hot upwellings and the cold, sinking slabs of the mantle’s convection.

The same method on other worlds #

The method has been taken to other planets. Seismometers left on the Moon by the Apollo astronauts recorded moonquakes and meteorite impacts for eight years, and reanalysis of those records in 2011 found evidence of a small, partly liquid core. On Mars, the InSight lander’s seismometer recorded hundreds of marsquakes from 2018 to 2022; in 2021 its team reported S waves reflected from the surface of a Martian core about 1830 kilometres in radius — larger, and therefore less dense, than expected — and liquid. With a single station, the method had to rely on reflections rather than shadows, but the principle was the one Gutenberg used: wave speeds read from arrival times, and boundaries read from where the arrivals change.

The same logic runs at a smaller scale in the oceans, where the channel with no walls found sound trapped by a minimum in its speed and carried across ocean basins, and at the largest scale in the Sun, whose interior is mapped by the frequencies of its own oscillations. In each case a wave that cannot be followed into the interior is read from where and when it comes out.

Still open: what the inner core is doing #

The inner core is the least accessible part of the planet, and it has turned out to be the strangest. P waves cross it faster along the Earth’s rotation axis than across it, by a few per cent — it is anisotropic, probably because its iron crystals are aligned — and the anisotropy differs between its eastern and western halves. Since 1996 several groups have reported that waves crossing the inner core along the same paths decades apart arrive at slightly different times, which has been interpreted as the inner core rotating slightly faster than the mantle, then perhaps slower, though the size and even the existence of the rotation are disputed. And the inner core is growing, as the outer core slowly freezes onto it, releasing the latent heat and light elements that drive the convection behind the Earth’s magnetic field. When it began to form — within the last billion years, or much earlier — is not agreed.

The method that found it is the general one. When a wave’s speed changes with depth, its rays curve, and where they cannot go is as informative as where they arrive: a drop in speed casts a ring of shadow by bending rays away, and a zero in shear speed casts a cap of shadow by leaving nothing to carry the wave. The Earth’s core was found by its shadows, measured by their edges, and shown to be liquid by the waves it would not pass.