---
title: "The Layered Earth"
book: "University Earth and Space Sciences — Year 1"
subject: earth
language: en
chapter: 9
exercises: 12
source: https://one-course.com/books/earth/3/en/chapter/9-the-layered-earth
license: CC-BY-NC-SA-4.0
credit: "One Earth and Space Sciences Book, One Course (one-course.com)"
---

# Chapter 9 — The Layered Earth

On 8 October 1909 an earthquake shook the Kupa valley, south-east of Zagreb.

Andrija Mohorovičić, who ran the observatory there, gathered the seismograms recorded across Europe and noticed something odd: up to a few hundred kilometres from the epicentre the first tremor came in one way, farther out a second kind of first arrival took over, as if part of the energy had found a faster road at depth. He explained it by a boundary some fifty kilometres down, below which the waves travel faster: the base of the [crust](#def-b1-layered-earth-layers). A century of such records, millions of them, has turned that first boundary into a reference model of the whole Earth, layer by layer, with its velocities, its density, its gravity and its pressure down to the centre. This chapter shows how a boundary is found from travel times, what the reference model says, how a density profile is drawn from seismic velocities alone and checked against the Earth’s mass and moment of inertia, which [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) make the [mantle](#def-b1-layered-earth-layers) at each depth, what the core is made of, and where the rigid outer shell of the planet ends.

**You already know.**

A P wave is a compression wave, an S wave a shear wave; S waves do not cross liquids, and the P and S shadow zones of the core give the radius of the liquid [outer core](#def-b1-layered-earth-layers) (Book 2, grade 11). Waves are refracted at a boundary according to Snell’s law, $\sin i_1/v_1 = \sin i_2/v_2$ (physics). In a fluid at rest the pressure grows with depth as $\mathrm dP = \rho g\,\mathrm dz$ (physics, University Year 1). The [crust](#def-b1-layered-earth-layers), the [mantle](#def-b1-layered-earth-layers) and the core were introduced in Book 1 (grade 6).

## 9.1 Seismic waves in a layered Earth

The speeds of seismic waves depend on the elastic moduli of the rock and on its density. The physics books derive them from the equations of continuum elasticity; here they are taken as known.

**Proposition 9.1 (Seismic speeds).**

In an isotropic elastic solid of density $\rho$, bulk modulus $K$ and shear modulus $\mu$, P and S waves travel at

$$
v_P = \sqrt{\frac{K + \tfrac43\mu}{\rho}}, \qquad v_S = \sqrt{\frac{\mu}{\rho}}.
$$

In a liquid $\mu = 0$: $v_S = 0$ and no S wave propagates.

**Proof.** *Admitted at this level.* ∎

Since $K$ and $\mu$ are positive, $v_P > v_S$ always: in PREM’s [crust](#def-b1-layered-earth-layers) and [mantle](#def-b1-layered-earth-layers) $v_P$ is about $1.8$ to $1.9$ times $v_S$, and P waves arrive first. Both speeds generally grow with depth, because pressure stiffens the rock more than it compresses it, so that rays bend back towards the surface; and at a few depths they jump.

**Definition 9.2 (Seismic discontinuity, Moho, core–mantle and inner-core boundaries).**

A *seismic discontinuity* is a surface inside the Earth across which the seismic speeds, the density or both change abruptly, over a thickness small compared with a seismic wavelength. The *Mohorovičić discontinuity* (or *Moho*) is the base of the [crust](#def-b1-layered-earth-layers), where $v_P$ jumps to about $8\,\mathrm{km}\,\mathrm{s}^{-1}$. The *core–mantle boundary* is the surface between the solid silicate [mantle](#def-b1-layered-earth-layers) and the liquid [outer core](#def-b1-layered-earth-layers); the *inner-core boundary* is the surface between the liquid [outer core](#def-b1-layered-earth-layers) and the solid [inner core](#def-b1-layered-earth-layers).

### 9.1.1 Refraction at a layer boundary

Take a layer of thickness $h$ and speed $v_1$, the [crust](#def-b1-layered-earth-layers), over a half-space of speed $v_2 > v_1$, the [mantle](#def-b1-layered-earth-layers), and a source at the surface. Three waves reach a receiver at the distance $x$. The direct wave travels in the layer, at $v_1$. The reflected wave bounces off the boundary. And a ray that strikes the boundary at the critical angle $i_c$, with $\sin i_c = v_1/v_2$, is refracted along it, travels in the faster medium just below it at the speed $v_2$, and sends energy back up into the layer at the same angle $i_c$ all along its way.

**Definition 9.3 (Head wave, crossover distance).**

A *head wave* is the wave that travels along a boundary at the speed of the faster medium below it and leaves it upward at the critical angle. The *crossover distance* is the distance from the source beyond which the head wave arrives before the direct wave.

**Theorem 9.4 (Travel times over a two-layer Earth).**

Over a layer of thickness $h$ and speed $v_1$ resting on a half-space of speed $v_2 > v_1$, a surface source and a surface receiver at the distance $x$ give

$$
t_{\text{direct}} = \frac{x}{v_1}, \qquad
  t_{\text{head}} = \frac{x}{v_2} + \frac{2h\cos i_c}{v_1}
  \quad (x \ge 2h\tan i_c), \qquad \sin i_c = \frac{v_1}{v_2},
$$

and the [head wave](#def-b1-layered-earth-head-wave) arrives first beyond the [crossover distance](#def-b1-layered-earth-head-wave)

$$
x_c = 2h\sqrt{\frac{v_2 + v_1}{v_2 - v_1}}.
$$

**Proof.** The [head wave](#def-b1-layered-earth-head-wave) goes down at the angle $i_c$ through the layer, a path of length $h/\cos i_c$, runs along the boundary over the distance $x - 2h\tan i_c$, and comes up along a path of length $h/\cos i_c$. Its travel time is

$$
t_{\text{head}} = \frac{2h}{v_1\cos i_c} + \frac{x - 2h\tan i_c}{v_2}
  = \frac{x}{v_2} + \frac{2h}{v_1\cos i_c}\Bigl(1 - \frac{v_1\sin i_c}{v_2}\Bigr)
  = \frac{x}{v_2} + \frac{2h}{v_1\cos i_c}\,(1 - \sin^2 i_c),
$$

that is $x/v_2 + 2h\cos i_c/v_1$. The path exists only if the horizontal run is positive, $x \ge 2h\tan i_c$. The two times are equal when $x(1/v_1 - 1/v_2) = 2h\cos i_c/v_1$; with $\cos i_c = \sqrt{v_2^2 - v_1^2}/v_2$,

$$
x\,\frac{v_2 - v_1}{v_1 v_2} = \frac{2h\sqrt{(v_2 - v_1)(v_2 + v_1)}}{v_1 v_2},
$$

which gives $x_c$. Beyond it the head-wave line, of smaller slope $1/v_2$, lies below the direct-wave line. ∎

On a plot of travel time against distance, the direct and [head waves](#def-b1-layered-earth-head-wave) are two straight lines. Their slopes are the inverse speeds $1/v_1$ and $1/v_2$; the head-wave line, extended back to $x = 0$, cuts the time axis at the *intercept time* $t_i = 2h\cos i_c/v_1$. The reflected wave, $t = \sqrt{x^2 + 4h^2}/v_1$, is a hyperbola that touches the head-wave line at the critical distance $2h\tan i_c$, where the [head wave](#def-b1-layered-earth-head-wave) is born ([Exercise 9.10](#exo-b1-layered-earth-10)).

![Refraction over a crust 35\, km thick, at 6.0\, km\, s-1, on a mantle at 8.0\, km\, s-1 ((a) at true scale). (a) The direct wave runs along the surface; the reflected wave drawn is the one that strikes the Moho at the critical angle, 48.6; the head wave runs along the Moho and returns at that angle. (b) Arrival times: the head wave appears at the critical distance, 79\, km, tangent to the reflection, and overtakes the direct wave at the crossover distance, 185\, km. The dotted extension gives the intercept time t_i = 7.7\, s.](https://one-course.com/images/onecourse/chapters/earth-3/b1-layered-earth/fig-cb2f55e77045.svg)

*Refraction over a [crust](#def-b1-layered-earth-layers) $35\,\mathrm{km}$ thick, at $6.0\,\mathrm{km}\,\mathrm{s}^{-1}$, on a [mantle](#def-b1-layered-earth-layers) at $8.0\,\mathrm{km}\,\mathrm{s}^{-1}$ ((a) at true scale). (a) The direct wave runs along the surface; the reflected wave drawn is the one that strikes the Moho at the critical angle, $48.6{}^{\circ}$; the [head wave](#def-b1-layered-earth-head-wave) runs along the [Moho](#def-b1-layered-earth-discontinuity) and returns at that angle. (b) Arrival times: the [head wave](#def-b1-layered-earth-head-wave) appears at the critical distance, $79\,\mathrm{km}$, tangent to the reflection, and overtakes the direct wave at the [crossover distance](#def-b1-layered-earth-head-wave), $185\,\mathrm{km}$. The dotted extension gives the intercept time $t_i = 7.7\,\mathrm{s}$.*

**Method 9.5 (The depth of a boundary from a refraction profile).**

1. Plot the first-arrival times against distance, and fit one straight line through the arrivals close to the source and another through the distant ones.
2. The inverse slopes give $v_1$ and $v_2$ ; then $\sin i_c = v_1/v_2$ .
3. Read the intercept time $t_i$ of the distant line at $x = 0$ , and compute $h = t_i v_1/(2\cos i_c)$ ; check it with the [crossover distance](#def-b1-layered-earth-head-wave) , $h = \tfrac12 x_c\sqrt{(v_2 - v_1)/(v_2 + v_1)}$ .
4. A boundary is found only if $v_2 > v_1$ : a slower layer below a faster one produces no [head wave](#def-b1-layered-earth-head-wave) and stays hidden.

Mohorovičić’s two kinds of first arrival were exactly these two lines: the direct wave in the [crust](#def-b1-layered-earth-layers) near the source, the [head wave](#def-b1-layered-earth-head-wave) along the top of the [mantle](#def-b1-layered-earth-layers) farther out. His figure for the thickness of the [crust](#def-b1-layered-earth-layers) in that region, $54\,\mathrm{km}$, has been confirmed in its order of size, but the depth of the [Moho](#def-b1-layered-earth-discontinuity) varies from place to place. Under the oceans the [crust](#def-b1-layered-earth-layers) is only about $5\,\mathrm{km}$ thick; under the continents it averages about $30\,\mathrm{km}$, and it is much thicker under large mountain ranges. For its reference model, PREM took the [Moho](#def-b1-layered-earth-discontinuity) at $11\,\mathrm{km}$ below sea level under the oceans and $35\,\mathrm{km}$ under the continents.

**History — Andrija Mohorovičić.**

![](https://one-course.com/images/onecourse/chapters/earth-3/b1-layered-earth/img-32c3dbb82a2e.jpg)

*Andrija Mohorovičić. Public domain.*

Andrija Mohorovičić (1857–1936) came to seismology from meteorology: a teacher, then from 1891 to 1921 the director of the meteorological observatory of Zagreb, he published his last meteorological study in 1901. On the seismograms of the Kupa valley earthquake of 8 October 1909 he found two distinct pairs of P and S phases: near the source one kind of first tremor arrived first, farther away another, which had travelled faster at depth. His reading of these two arrivals as a direct wave and a wave refracted below a boundary gave the boundary that now bears his name, and the first measure of the thickness of the [crust](#def-b1-layered-earth-layers).

## 9.2 A reference Earth model

The same reasoning, applied to millions of travel times at all distances and to the periods of the Earth’s free oscillations (the whole planet rings like a bell after a large earthquake), gives the speeds at every depth.

**Definition 9.6 (Reference Earth model).**

A *reference Earth model* is a spherically symmetric Earth, given as the seismic speeds $v_P(r)$, $v_S(r)$, the density $\rho(r)$ and, in the fullest models, the attenuation as functions of the radius $r$ alone, that fits the observed travel times, free-oscillation periods, the Earth’s mass and its moment of inertia. The most widely used is PREM, the Preliminary Reference Earth Model (1981).

PREM was fitted to the radius $6371\,\mathrm{km}$, the mass $5.974 \times 10^{24}\,\mathrm{kg}$ and the [moment-of-inertia factor](https://one-course.com/books/earth/3/en/chapter/3-comparative-planetology#def-b1-comparative-planetology-moi) $0.3308$ of the Earth. It divides the planet into a few regions joined by discontinuities, and its profiles (figure below) show at once the main layers.

**Definition 9.7 (Crust, mantle, outer core, inner core).**

The *crust* is the outer silicate layer above the [Moho](#def-b1-layered-earth-discontinuity). The *mantle* is the solid silicate shell between the [Moho](#def-b1-layered-earth-discontinuity) and the [core–mantle boundary](#def-b1-layered-earth-discontinuity), at a depth of about $2891\,\mathrm{km}$. The *outer core* is the liquid metallic shell between the [core–mantle boundary](#def-b1-layered-earth-discontinuity) and the [inner-core boundary](#def-b1-layered-earth-discontinuity), at about $5150\,\mathrm{km}$ depth. The *inner core* is the solid metallic sphere below it, of radius about $1220\,\mathrm{km}$.

![PREM: the speeds of P and S waves and the density against depth. The S speed falls to zero in the liquid outer core and reappears, small, in the solid inner core; at the core–mantle boundary the density almost doubles while v_P drops from 13.7 to 8.1\, km\, s-1; the jumps near 400\, km and 670\, km bound the transition zone.](https://one-course.com/images/onecourse/chapters/earth-3/b1-layered-earth/fig-7078bd2fdabe.svg)

*PREM: the speeds of P and S waves and the density against depth. The S speed falls to zero in the liquid [outer core](#def-b1-layered-earth-layers) and reappears, small, in the solid [inner core](#def-b1-layered-earth-layers); at the [core–mantle boundary](#def-b1-layered-earth-discontinuity) the density almost doubles while $v_P$ drops from 13.7 to $8.1\,\mathrm{km}\,\mathrm{s}^{-1}$; the jumps near $400\,\mathrm{km}$ and $670\,\mathrm{km}$ bound the transition zone.*

| boundary in PREM | depth (km) | radius (km) | what changes |
| --- | --- | --- | --- |
| [Moho](#def-b1-layered-earth-discontinuity) | 24.4 | 6346.6 | [crust](#def-b1-layered-earth-layers) above, peridotite below; $v_P$ 6.8 to 8.1 |
| 220-km discontinuity | 220 | 6151 | base of the [low-velocity zone](#def-b1-layered-earth-lithosphere) in PREM |
| 400-km discontinuity | 400 | 5971 | [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) to wadsleyite ($410\,\mathrm{km}$ in most studies) |
| 670-km discontinuity | 670 | 5701 | ringwoodite to bridgmanite and ferropericlase ($660\,\mathrm{km}$ in most studies) |
| [core–mantle boundary](#def-b1-layered-earth-discontinuity) | 2891 | 3480 | silicate to liquid iron; $v_S$ 7.3 to 0; $\rho$ 5.57 to 9.90 |
| [inner-core boundary](#def-b1-layered-earth-discontinuity) | 5149.5 | 1221.5 | liquid to solid iron; $v_S$ 0 to 3.5; $\rho$ 12.17 to 12.76 |

***Table 9.1.** The main discontinuities of PREM (speeds in $\mathrm{km}\,\mathrm{s}^{-1}$, densities in $\mathrm{g}\,\mathrm{cm}^{-3}$). PREM averages the [crust](#def-b1-layered-earth-layers) to $24.4\,\mathrm{km}$; real [crusts](#def-b1-layered-earth-layers) range from about $5\,\mathrm{km}$ under the oceans to many tens of kilometres under mountain ranges.*

Two facts in these profiles are directly observed. The vanishing of $v_S$ in the [outer core](#def-b1-layered-earth-layers) is the S-wave shadow recalled above: S waves enter the core only as P waves, the core cannot transmit shear, so it is liquid. And the [inner core](#def-b1-layered-earth-layers) transmits S waves, slowly: it is solid. That second fact came from weak P arrivals where no P wave should be.

**History — Inge Lehmann and the inner core.**

![](https://one-course.com/images/onecourse/chapters/earth-3/b1-layered-earth/img-aaf65ddebd5b.jpg)

*Inge Lehmann in 1932, by Even Neuhaus. Public domain.*

Inge Lehmann (1888–1993), head of the seismological department of the Danish geodetic institute, studied the records of the 1929 Buller earthquake in New Zealand and found P waves arriving inside the shadow zone, where the core should have bent them away. In a paper of 1936 with the shortest of titles, *P$'$*, she showed that they could be explained if the core held a smaller [inner core](#def-b1-layered-earth-layers) with a higher P speed, which returns part of the energy towards the surface. Her [inner core](#def-b1-layered-earth-layers) is now known to be solid, and to grow slowly as the Earth cools. She also discovered the discontinuity near $220\,\mathrm{km}$ that bears her name.

## 9.3 Density, gravity and pressure from seismic data

Seismic speeds give two combinations of the moduli and the density, not the density itself. A third relation comes from the assumption that, inside a layer of uniform composition, the rock is simply compressed by the weight of the rock above.

**Definition 9.8 (Seismic parameter).**

The *seismic parameter* at a depth is

$$
\Phi = v_P^2 - \tfrac43 v_S^2 = \frac{K}{\rho},
$$

the bulk modulus divided by the density, computed from the two seismic speeds alone ([Proposition 9.1](#prop-b1-layered-earth-speeds)).

**Theorem 9.9 (Adams–Williamson equation).**

In a region of uniform composition, in hydrostatic equilibrium, compressed adiabatically, the density varies with the radius as

$$
\frac{\mathrm d\rho}{\mathrm dr} = -\frac{\rho(r)\,g(r)}{\Phi(r)},
$$

where $g(r)$ is the gravity at the radius $r$.

**Proof.** Hydrostatic equilibrium gives $\mathrm dP/\mathrm dr = -\rho g$ (the recall, with $\mathrm dz = -\mathrm dr$). The bulk modulus is defined by $K = \rho\,\mathrm dP/\mathrm d\rho$ for the compression at hand; for a rock of fixed composition compressed without exchange of heat, it is the adiabatic modulus that the seismic waves measure, so $\mathrm d\rho/\mathrm dP = \rho/K$. By the chain rule,

$$
\frac{\mathrm d\rho}{\mathrm dr} = \frac{\mathrm d\rho}{\mathrm dP}\,
  \frac{\mathrm dP}{\mathrm dr} = \frac{\rho}{K}\,(-\rho g) = -\frac{\rho g}{K/\rho}
  = -\frac{\rho g}{\Phi}. \qedhere
$$

∎

The equation fails at a discontinuity, where the composition or the crystal structure changes, and wherever the temperature departs from the adiabat; but between the discontinuities it accounts for the observed increase of density with depth very well (panel (c) of the figure below). Starting from a density at the top of each region, integrating it downward, and adjusting the starting values until the mass and the moment of inertia come out right, is how the first density models were built. The two global numbers are integrals of the density.

**Proposition 9.10 (Mass, gravity, pressure and moment of inertia).**

For a spherically symmetric Earth of radius $R$ and density $\rho(r)$, the mass inside the radius $r$, the gravity, the pressure and the moment of inertia about an axis through the centre are

$$
m(r) = \int_0^r 4\pi s^2\rho(s)\,\mathrm ds, \quad
  g(r) = \frac{G\,m(r)}{r^2}, \quad
  P(r) = \int_r^R \rho(s)\,g(s)\,\mathrm ds, \quad
  I = \frac{8\pi}{3}\int_0^R \rho(r)\,r^4\,\mathrm dr.
$$

**Proof.** A thin shell of radius $s$ and thickness $\mathrm ds$ has the volume $4\pi s^2\,\mathrm ds$, hence the first integral. The gravity of a spherically symmetric body outside a radius is that of its mass inside, concentrated at the centre, and the shells outside $r$ pull with no net force (the shell theorem, admitted from the physics books); hence $g(r)$. Integrating the hydrostatic equation from the surface, where $P = 0$, gives $P(r)$. For the moment of inertia, a thin spherical shell of mass $\mathrm dm$ and radius $r$ has, about the three axes $x$, $y$, $z$, moments whose sum is $\int (y^2 + z^2) + (z^2 + x^2) + (x^2 + y^2)\,\mathrm dm = 2r^2\,\mathrm dm$; by symmetry the three are equal, so each is $\tfrac23 r^2\,\mathrm dm$. With $\mathrm dm = 4\pi r^2\rho\,\mathrm dr$, summing the shells gives $I$. ∎

For a sphere of uniform density, $I = \tfrac{8\pi}{15}\rho R^5 = 0.4\,MR^2$. The Earth’s factor $I/MR^2 = 0.3308$ is smaller: its mass is concentrated towards the centre, which is the first argument for a dense core. Computed from PREM’s density alone, the integrals give a mass of $5.976 \times 10^{24}\,\mathrm{kg}$ ($0.06\,\%$ above the measured $5.972 \times 10^{24}\,\mathrm{kg}$), a factor $0.3309$, and a central pressure of $364\,\mathrm{GPa}$, the value tabulated with PREM to $0.1\,\%$.

![Inside the Earth, from PREM. (a) and (b) Gravity and pressure computed from PREM’s density with : gravity stays close to 10\, m\, s-2 through the mantle, peaks at 10.7\, m\, s-2 at the core–mantle boundary (CMB) and falls to zero at the centre; the pressure reaches 136\, GPa at the CMB, 329\, GPa at the inner-core boundary (ICB) and 364\, GPa at the centre. (c) The Adams–Williamson equation, integrated downward from PREM’s density at 771\, km and at the top of the outer core with PREM’s speeds and gravity (circles), follows PREM to within 0.2\,\%: there the density increase is compression of a uniform material; the jumps are not.](https://one-course.com/images/onecourse/chapters/earth-3/b1-layered-earth/fig-cccbf5811b05.svg)

*Inside the Earth, from PREM. (a) and (b) Gravity and pressure computed from PREM’s density with [Proposition 9.10](#prop-b1-layered-earth-integrals): gravity stays close to $10\,\mathrm{m}\,\mathrm{s}^{-2}$ through the [mantle](#def-b1-layered-earth-layers), peaks at $10.7\,\mathrm{m}\,\mathrm{s}^{-2}$ at the [core–mantle boundary](#def-b1-layered-earth-discontinuity) (CMB) and falls to zero at the centre; the pressure reaches $136\,\mathrm{GPa}$ at the CMB, $329\,\mathrm{GPa}$ at the [inner-core boundary](#def-b1-layered-earth-discontinuity) (ICB) and $364\,\mathrm{GPa}$ at the centre. (c) The Adams–Williamson equation, integrated downward from PREM’s density at $771\,\mathrm{km}$ and at the top of the [outer core](#def-b1-layered-earth-layers) with PREM’s speeds and gravity (circles), follows PREM to within $0.2\,\%$: there the density increase is compression of a uniform material; the jumps are not.*

That gravity grows on the way down through the [mantle](#def-b1-layered-earth-layers) is a direct consequence of the dense core: the mass left inside the radius falls more slowly than $r^2$. Through the [mantle](#def-b1-layered-earth-layers) $g$ hardly changes, and the pressure grows almost linearly with depth, by about $4.7\,\mathrm{GPa}$ per hundred kilometres.

## 9.4 The mantle: one rock, several mineral assemblages

The upper [mantle](#def-b1-layered-earth-layers) is peridotite, known from the fragments that volcanoes bring up and from slices of [mantle](#def-b1-layered-earth-layers) thrust onto continents: about half to three fifths [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) by volume ($49\text{ to }62\,\%$ in the proposed compositions), the rest [pyroxenes](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) and an aluminous [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) ([plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), spinel or [garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), depending on depth). The pressure that rises with depth does not change this composition, but it forces its atoms into denser arrangements: [polymorphs](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-polymorph) ([Chapter 7](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#ch-b1-rock-forming-minerals)) appear one after the other.

**Definition 9.11 (Mantle transition zone).**

The *mantle transition zone* is the part of the [mantle](#def-b1-layered-earth-layers) between the seismic discontinuities near $410\,\mathrm{km}$ and $660\,\mathrm{km}$ depth, in which [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) of the upper [mantle](#def-b1-layered-earth-layers) is replaced by its denser [polymorphs](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-polymorph), wadsleyite and then ringwoodite. Above it is the upper [mantle](#def-b1-layered-earth-layers); below it the lower [mantle](#def-b1-layered-earth-layers), down to the [core–mantle boundary](#def-b1-layered-earth-discontinuity).

At about $410\,\mathrm{km}$ ($14\,\mathrm{GPa}$) [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) $\ce{(Mg,Fe)2SiO4}$ becomes wadsleyite, of the same formula; near $520\,\mathrm{km}$ ($17.5\,\mathrm{GPa}$) wadsleyite becomes ringwoodite; the [pyroxenes](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) dissolve into a [garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) rich in silicon, majorite, until by the middle of the transition zone [garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) holds all of them, so that the zone is a little more than half [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) [polymorphs](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-polymorph) and the rest [garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups). At about $660\,\mathrm{km}$, at a pressure near $24\,\mathrm{GPa}$, ringwoodite breaks down into two [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), written for the magnesian [end-member](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-solid-solution)

$$
\ce{Mg2SiO4 -> MgSiO3 + MgO}:
$$

bridgmanite, $\ce{(Mg,Fe)SiO3}$ with the structure of perovskite, and ferropericlase, $\ce{(Mg,Fe)O}$. A lower [mantle](#def-b1-layered-earth-layers) of the same composition as the upper [mantle](#def-b1-layered-earth-layers) is about $75\,\%$ bridgmanite, $17\,\%$ ferropericlase and $8\,\%$ of a calcium silicate perovskite by volume, and fits the densities and speeds of the reference model. Bridgmanite is therefore the most abundant [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) of the Earth; it received its name only in 2014, after natural grains were described in a heavily shocked meteorite.

Each change produces denser [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), so it raises the density and the seismic speeds: the transitions are the discontinuities of the profiles (figure below). Their depths are not fixed. A phase change happens at a pressure that depends on the temperature, and the slope of that dependence (the Clapeyron slope, treated in the next university volume) moves the $410\,\mathrm{km}$ discontinuity up and the $660\,\mathrm{km}$ down in the cold rock of a sinking slab. Mapping the topography of the discontinuities is one way of finding cold and hot regions in the [mantle](#def-b1-layered-earth-layers).

![The minerals of a peridotite mantle with depth, as volume fractions (schematic: the boundaries are drawn sharp, and the fractions are round values within the ranges quoted in the text). The composition does not change; the minerals do, at about 410, 520 and 660\, km.](https://one-course.com/images/onecourse/chapters/earth-3/b1-layered-earth/fig-e9093a7ad7ba.svg)

*The [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) of a peridotite [mantle](#def-b1-layered-earth-layers) with depth, as volume fractions (schematic: the boundaries are drawn sharp, and the fractions are round values within the ranges quoted in the text). The composition does not change; the [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) do, at about 410, 520 and $660\,\mathrm{km}$.*

At the bottom of the [mantle](#def-b1-layered-earth-layers), the layer called D$''$ (D double prime, a name kept from an early model of the Earth) has a velocity gradient different from the rest of the lower [mantle](#def-b1-layered-earth-layers); PREM gives it the lowest $150\,\mathrm{km}$ of the [mantle](#def-b1-layered-earth-layers). It is seismically heterogeneous: patches of very low speed, steep gradients, reflectors. It is thought to be a boundary layer between the hot core and the [mantle](#def-b1-layered-earth-layers), and perhaps a graveyard of old slabs.

## 9.5 The core

The core is mostly iron. Its density, its speeds and its ability to generate a magnetic field all point to a metal, and iron is by far the most abundant element dense enough among those that the meteorites and the Sun show to be common. Nickel, abundant in iron meteorites, accompanies it. But the core is not pure iron–nickel.

- The core is $5\text{ to }10\,\%$ less dense than pure iron would be at the same pressure and temperature: it must contain light elements, among which silicon, sulfur, carbon, oxygen and hydrogen are the candidates.
- The density jumps by about $4.5\,\%$ at the [inner-core boundary](#def-b1-layered-earth-discontinuity) , too much for the freezing of one and the same metal: the [inner core](#def-b1-layered-earth-layers) holds light elements too, but fewer, a few per cent against $5\text{ to }10\,\%$ in the liquid.

The difference has a consequence. As the Earth cools — its [mantle](#def-b1-layered-earth-layers) by about $100{}^{\circ}\mathrm{C}$ per billion years at present — the liquid of the core freezes onto the [inner core](#def-b1-layered-earth-layers), which grows, and

freezing leaves most of the light elements in the liquid. The light liquid released at the bottom of the [outer core](#def-b1-layered-earth-layers) rises, and this buoyancy is thought to help stir the [outer core](#def-b1-layered-earth-layers) and drive the dynamo that makes the Earth’s magnetic field.

## 9.6 Mechanical layers: lithosphere and asthenosphere

The layers above are defined by composition and phase: [crust](#def-b1-layered-earth-layers) and [mantle](#def-b1-layered-earth-layers) are different rocks, outer and [inner core](#def-b1-layered-earth-layers) different states. Plate tectonics needs a different division, by strength, which cuts across the first one.

**Definition 9.12 (Lithosphere, asthenosphere, low-velocity zone).**

The *lithosphere* is the rigid outer shell of the Earth, the [crust](#def-b1-layered-earth-layers) and the uppermost [mantle](#def-b1-layered-earth-layers), which behaves as a strong layer over geological times and is divided into plates. The *asthenosphere* is the weaker layer of the upper [mantle](#def-b1-layered-earth-layers) below it, which deforms by slow flow and lets the plates move. The *low-velocity zone* is the depth range below the lithosphere where the seismic speeds, S speeds especially, are lower than just above and below.

The [lithosphere](#def-b1-layered-earth-lithosphere) is not a rock but a temperature range: [mantle](#def-b1-layered-earth-layers) peridotite is strong where it is cold and creeps where it is hot, and its base is roughly an isotherm. It is thin under ridges, where hot [mantle](#def-b1-layered-earth-layers) rises, about $140\,\mathrm{km}$ thick under the oldest ocean floor, and $40\text{ to }280\,\mathrm{km}$ thick under continents, the thickest under old cratons; the top of the [asthenosphere](#def-b1-layered-earth-lithosphere) lies at about $80\text{ to }200\,\mathrm{km}$. Seismically, the [asthenosphere](#def-b1-layered-earth-lithosphere) shows as the [low-velocity zone](#def-b1-layered-earth-lithosphere), close to its melting conditions: in PREM the S speed decreases slightly from the [Moho](#def-b1-layered-earth-discontinuity) down to $220\,\mathrm{km}$, where it jumps up again. How strength depends on temperature, and why it does so, is the subject of [Chapter 16](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#ch-b1-stress-strain-rheology).

**In the field — A seismic station.**

![](https://one-course.com/images/onecourse/chapters/earth-3/b1-layered-earth/img-c787a9689c94.jpg)

*A broadband seismometer on its pier in an underground vault, with the data logger on the wall (AI-generated illustration).*

The travel times on which reference models rest come from permanent stations. A broadband seismometer, which records ground motions from hundredths of a second to hours in period, sits on a concrete pier set on bare rock and isolated from the floor of the vault, under a cover that keeps its temperature steady: a change of a fraction of a degree, a draught or a passing lorry would show on the record. The data logger stamps every sample with time from satellites, and the records stream to data centres that share them openly. Networks of such stations locate and time every large earthquake; arrival times picked on their records were inverted for the speeds at each depth.

## 9.7 Exercises

**Exercise 9.1 ★.**

With the PREM radii $R = 6371\,\mathrm{km}$, $r_{\text{CMB}} = 3480\,\mathrm{km}$ and $r_{\text{ICB}} = 1221.5\,\mathrm{km}$, compute the fractions of the Earth’s volume occupied by the core and by the [inner core](#def-b1-layered-earth-layers).

**Solution of Exercise 9.1.**

The volume fractions are the cubes of the radius ratios: core $(3480/6371)^3 = 0.1630$, that is $16.3\,\%$; [inner core](#def-b1-layered-earth-layers) $(1221.5/6371)^3 = 0.00705$, $0.70\,\%$.

**Exercise 9.2 ★.**

Just above the [core–mantle boundary](#def-b1-layered-earth-discontinuity) PREM gives $v_P = 13.7166\,\mathrm{km}\,\mathrm{s}^{-1}$, $v_S = 7.264\,66\,\mathrm{km}\,\mathrm{s}^{-1}$, $\rho = 5.566\,45\,\mathrm{g}\,\mathrm{cm}^{-3}$; just below, $v_P = 8.064\,82\,\mathrm{km}\,\mathrm{s}^{-1}$, $v_S = 0$, $\rho = 9.903\,49\,\mathrm{g}\,\mathrm{cm}^{-3}$. Compute the [seismic parameter](#def-b1-layered-earth-seismic-parameter) and the bulk modulus on each side (in GPa). What changes most across the boundary?

**Solution of Exercise 9.2.**

Above: $\Phi = 13.7166^2 - \tfrac43\times7.26466^2 = 188.145 - 70.367 =
117.78\,\mathrm{km}^{2}\,\mathrm{s}^{-2}$, $K = \rho\Phi = 5566.45\,\mathrm{kg}\,\mathrm{m}^{-3}\times
117.78 \times 10^{6}\,\mathrm{m}^{2}\,\mathrm{s}^{-2} = 655.6\,\mathrm{GPa}$. Below: $\Phi = 8.06482^2 =
65.04\,\mathrm{km}^{2}\,\mathrm{s}^{-2}$, $K = 9903.49\times65.04 \times 10^{6} = 644.1\,\mathrm{GPa}$. The bulk modulus hardly changes ($-1.7\,\%$). What changes is the shear modulus, $\mu = \rho v_S^2 = 293.8\,\mathrm{GPa}$ above and $0$ below (a liquid), and the density, which rises by $78\,\%$: hence the drop of $v_P$ despite an almost equal $K$.

**Exercise 9.3 ★.**

A [crust](#def-b1-layered-earth-layers) $30\,\mathrm{km}$ thick, at $6.2\,\mathrm{km}\,\mathrm{s}^{-1}$, rests on a [mantle](#def-b1-layered-earth-layers) at $8.1\,\mathrm{km}\,\mathrm{s}^{-1}$. Compute the critical angle, the critical distance and the [crossover distance](#def-b1-layered-earth-head-wave).

**Solution of Exercise 9.3.**

$\sin i_c = 6.2/8.1 = 0.7654$, $i_c = 49.95{}^{\circ}$. Critical distance $2\times30\times\tan49.95{}^{\circ} = 60\times1.1906 = 71.4\,\mathrm{km}$. [Crossover distance](#def-b1-layered-earth-head-wave) $60\sqrt{14.3/1.9} = 60\times2.743 = 164.6\,\mathrm{km}$.

**Exercise 9.4 ★.**

The core’s mass is $1.94 \times 10^{24}\,\mathrm{kg}$ and its radius $3480\,\mathrm{km}$. Compute the gravity at the [core–mantle boundary](#def-b1-layered-earth-discontinuity) and compare it with the surface gravity, $9.82\,\mathrm{m}\,\mathrm{s}^{-2}$. Why is it larger?

**Solution of Exercise 9.4.**

$g = Gm/r^2 = 6.674 \times 10^{-11}\times1.94 \times 10^{24}/(3.48 \times 10^{6})^2 =
10.69\,\mathrm{m}\,\mathrm{s}^{-2}$, $9\,\%$ more than at the surface. Going down to the core, the radius squared falls to $(3480/6371)^2 = 0.298$ of its surface value, while the attracting mass falls only to $1.94/5.972 = 0.325$ of the total, because the core holds a third of the mass: $0.325/0.298 = 1.09$.

**Exercise 9.5 ★★.**

On a refraction profile the first arrivals are at $10.0\,\mathrm{s}$ at $60\,\mathrm{km}$, and at $33.3\,\mathrm{s}$ at $200\,\mathrm{km}$ and $45.8\,\mathrm{s}$ at $300\,\mathrm{km}$; the line through the first point passes through the origin. Find $v_1$, $v_2$, the intercept time and the depth of the boundary, and check the result with the [crossover distance](#def-b1-layered-earth-head-wave).

**Solution of Exercise 9.5.**

Near the source $v_1 = 60/10.0 = 6.0\,\mathrm{km}\,\mathrm{s}^{-1}$. Far away $v_2 = (300 - 200)/(45.8 - 33.3) = 100/12.5 = 8.0\,\mathrm{km}\,\mathrm{s}^{-1}$; the intercept time is $t_i = 33.3 - 200/8.0 = 8.3\,\mathrm{s}$. Then $\sin i_c =
0.75$, $\cos i_c = 0.6614$, and $h = t_i v_1/(2\cos i_c) = 8.3\times6.0/1.3229
= 37.6\,\mathrm{km}$. Check: $x_c = 2\times37.6\times\sqrt{14/2} = 199\,\mathrm{km}$, where both waves arrive at $199/6.0 = 199/8.0 + 8.3 = 33.2\,\mathrm{s}$.

**Exercise 9.6 ★★.**

Show that a uniform sphere has $I = 0.4\,MR^2$. Then compute the mass and the factor $I/MR^2$ of a two-layer Earth: a core of radius $3480\,\mathrm{km}$ and density $11.0\,\mathrm{g}\,\mathrm{cm}^{-3}$ in a [mantle](#def-b1-layered-earth-layers) of density $4.45\,\mathrm{g}\,\mathrm{cm}^{-3}$, with $R = 6371\,\mathrm{km}$. Compare with $5.972 \times 10^{24}$ kg and $0.3308$.

**Solution of Exercise 9.6.**

For uniform $\rho$, $I = \tfrac{8\pi}{3}\rho\int_0^R r^4\,\mathrm dr =
\tfrac{8\pi}{15}\rho R^5$ and $M = \tfrac43\pi\rho R^3$, so $I/MR^2 = (8/15)/(4/3) = 0.4$. Two layers ($r_c = 3.48 \times 10^{6}\,\mathrm{m}$, $R = 6.371 \times 10^{6}\,\mathrm{m}$): $M = \tfrac43\pi\bigl(11000\,r_c^3 + 4450\,(R^3 -
r_c^3)\bigr) = 5.977 \times 10^{24}\,\mathrm{kg}$, $0.08\,\%$ above the measured mass; $I = \tfrac{8\pi}{15}\bigl(11000\,r_c^5 + 4450\,(R^5 - r_c^5)\bigr)$ gives $I/MR^2 = 0.3457$. The mass is right, but the factor is $4.5\,\%$ too high: the real [mantle](#def-b1-layered-earth-layers) is not uniform, its density grows with depth, which moves mass inward.

**Exercise 9.7 ★★.**

At $1471\,\mathrm{km}$ depth PREM gives $v_P = 12.16\,\mathrm{km}\,\mathrm{s}^{-1}$, $v_S = 6.673\,\mathrm{km}\,\mathrm{s}^{-1}$, $\rho = 4.86\,\mathrm{g}\,\mathrm{cm}^{-3}$, and the gravity is $9.9\,\mathrm{m}\,\mathrm{s}^{-2}$. Compute the density gradient given by the Adams–Williamson equation, in $\mathrm{g}\,\mathrm{cm}^{-3}$ per $1000\,\mathrm{km}$, and compare with PREM’s densities, $4.844$ at $1471\,\mathrm{km}$ and $5.357$ at $2471\,\mathrm{km}$.

**Solution of Exercise 9.7.**

$\Phi = 12.16^2 - \tfrac43\times6.673^2 = 147.87 - 59.37 =
88.49\,\mathrm{km}^{2}\,\mathrm{s}^{-2}$. With depth $z$, $\mathrm d\rho/\mathrm dz = \rho g/\Phi = 4860\times9.9/88.49 \times 10^{6} =
5.44 \times 10^{-4}\,\mathrm{kg}\,\mathrm{m}^{-4}$, that is $0.544\,\mathrm{kg}\,\mathrm{m}^{-3}$ per kilometre, or $0.54\,\mathrm{g}\,\mathrm{cm}^{-3}$ per $1000\,\mathrm{km}$. PREM rises by $5.357 - 4.844 =
0.513$ over these $1000\,\mathrm{km}$, $6\,\%$ less: the gradient computed at the top is applied to the whole interval, while $\Phi$ grows with depth and the gradient decreases.

**Exercise 9.8 ★★.**

Estimate the pressure at $670\,\mathrm{km}$ depth from a mean density of $3.57\,\mathrm{g}\,\mathrm{cm}^{-3}$ and a gravity of $9.9\,\mathrm{m}\,\mathrm{s}^{-2}$. Which [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) change is expected there?

**Solution of Exercise 9.8.**

$P \approx \bar\rho g z = 3570\times9.9\times670 \times 10^{3} = 23.7\,\mathrm{GPa}$ (PREM: $23.9\,\mathrm{GPa}$). It is the pressure, about $24\,\mathrm{GPa}$, at which ringwoodite breaks down into bridgmanite and ferropericlase: the $660\,\mathrm{km}$ discontinuity.

**Exercise 9.9 ★★.**

PREM’s mean density of the [outer core](#def-b1-layered-earth-layers) is $10.90\,\mathrm{g}\,\mathrm{cm}^{-3}$; at the [inner-core boundary](#def-b1-layered-earth-discontinuity) its density is $12.166\,\mathrm{g}\,\mathrm{cm}^{-3}$ above and $12.764\,\mathrm{g}\,\mathrm{cm}^{-3}$ below. (a) If iron at the same conditions is $5\text{ to }10\,\%$ denser than the [outer core](#def-b1-layered-earth-layers), what range of densities does that give for liquid iron there? (b) Compute the density jump at the [inner-core boundary](#def-b1-layered-earth-discontinuity) in per cent, and compare it with the $4.5\,\%$ of recent studies. (c) What does a jump larger than freezing alone would give say about the light elements?

**Solution of Exercise 9.9.**

(a) Liquid iron: $10.90\times1.05 = 11.4$ to $10.90\times1.10 =
12.0\,\mathrm{g}\,\mathrm{cm}^{-3}$. (b) $12.764/12.166 = 1.049$: a jump of $4.9\,\%$ in PREM, a little more than the $4.5\,\%$ of recent studies. (c) Freezing alone would make the solid only slightly denser than the liquid; the larger jump means that the solid holds fewer light elements than the liquid: they prefer the liquid, and freezing at the [inner-core boundary](#def-b1-layered-earth-discontinuity) leaves most of them behind.

**Exercise 9.10 ★★★.**

Show that the reflected-wave time $t_r(x) = \sqrt{x^2 + 4h^2}/v_1$ and the head-wave time of [Theorem 9.4](#thm-b1-layered-earth-two-layer) are equal, and have equal slopes, at $x = 2h\tan i_c$, and that $t_r > t_{\text{head}}$ beyond it. Interpret.

**Solution of Exercise 9.10.**

Let $x_* = 2h\tan i_c$. Then $\sqrt{x_*^2 + 4h^2} = 2h/\cos i_c$ and $t_r(x_*) = 2h/(v_1\cos i_c)$. The [head wave](#def-b1-layered-earth-head-wave) gives $t(x_*) = 2h\tan i_c/v_2 + 2h\cos i_c/v_1 = \frac{2h}{v_1}\bigl(\sin^2 i_c/\cos
i_c + \cos i_c\bigr) = 2h/(v_1\cos i_c)$, using $\sin i_c = v_1/v_2$: the same. The slope of the hyperbola is $t_r'(x) = x/(v_1\sqrt{x^2 + 4h^2})$; at $x_*$ it is $\sin i_c/v_1 = 1/v_2$, the slope of the head-wave line. Since $t_r'' > 0$, the hyperbola is convex and lies above each of its tangents: for $x > x_*$, $t_r > t_{\text{head}}$. At the critical distance the reflection strikes the boundary at exactly $i_c$ and is the [head wave](#def-b1-layered-earth-head-wave)’s birth; beyond, the [head wave](#def-b1-layered-earth-head-wave), which gains by running in the fast medium, comes first.

**Exercise 9.11 ★★★.**

For an Earth of uniform density $\rho$, show that $g(r) = \tfrac43\pi G\rho r$ and $P(r) = \tfrac23\pi G\rho^2(R^2 - r^2)$. Compute the central pressure with $\rho = 5513\,\mathrm{kg}\,\mathrm{m}^{-3}$ and $R = 6371\,\mathrm{km}$, and compare with PREM’s $363.85\,\mathrm{GPa}$.

**Solution of Exercise 9.11.**

$m(r) = \tfrac43\pi\rho r^3$, so $g = Gm/r^2 = \tfrac43\pi G\rho r$. Then $P(r) = \int_r^R \rho\,\tfrac43\pi G\rho s\,\mathrm ds = \tfrac23\pi G\rho^2(R^2
- r^2)$. At the centre $P = \tfrac23\pi\times6.674 \times 10^{-11}\times5513^2\times
(6.371 \times 10^{6})^2 = 172\,\mathrm{GPa}$, $47\,\%$ of PREM’s $363.85\,\mathrm{GPa}$: concentrating the mass towards the centre roughly doubles the central pressure.

**Exercise 9.12 ★★★.**

Show that if $g/\Phi$ is constant through a layer, the Adams–Williamson equation gives $\rho = \rho_0\exp\bigl(g(z - z_0)/\Phi\bigr)$ with depth $z$. Apply it to the [outer core](#def-b1-layered-earth-layers), from $2891\,\mathrm{km}$ ($\rho_0 =
9.90\,\mathrm{g}\,\mathrm{cm}^{-3}$) to $5149.5\,\mathrm{km}$, with mean values $g = 7.9\,\mathrm{m}\,\mathrm{s}^{-2}$ and $\Phi = 87\,\mathrm{km}^{2}\,\mathrm{s}^{-2}$; compare with PREM’s $12.17\,\mathrm{g}\,\mathrm{cm}^{-3}$ at the [inner-core boundary](#def-b1-layered-earth-discontinuity).

**Solution of Exercise 9.12.**

With $z$ the depth, $\mathrm d\rho/\mathrm dz = \rho g/\Phi$; if $g/\Phi$ is a constant $a$, $\mathrm d(\ln\rho)/\mathrm dz = a$ and $\rho = \rho_0
e^{a(z - z_0)}$. Here $a(z - z_0) = 7.9\times2258.5 \times 10^{3}/87 \times 10^{6} =
0.2051$ and $\rho = 9.90\times e^{0.2051} = 9.90\times1.2276 =
12.15\,\mathrm{g}\,\mathrm{cm}^{-3}$, $0.2\,\%$ below PREM’s $12.17$: the whole increase of density through the [outer core](#def-b1-layered-earth-layers) is self-compression.

## 9.8 Problem: Weighing the Earth’s shells

**Problem 9.1.**

Weekend problem — volumes, masses and moment of inertia of a four-shell Earth built from PREM, compared with the measured mass and moment of inertia; gravity and pressure deep inside

The Earth is cut into four shells at the radii of PREM, and each is given its mean density in PREM, rounded:

| shell | from (km) | to (km) | mean density ($\mathrm{g}\,\mathrm{cm}^{-3}$) |
| --- | --- | --- | --- |
| [crust](#def-b1-layered-earth-layers), upper [mantle](#def-b1-layered-earth-layers) and transition zone | 5701 | 6371 | 3.57 |
| lower [mantle](#def-b1-layered-earth-layers) | 3480 | 5701 | 4.90 |
| [outer core](#def-b1-layered-earth-layers) | 1221.5 | 3480 | 10.90 |
| [inner core](#def-b1-layered-earth-layers) | 0 | 1221.5 | 12.89 |

Use $G = 6.674 \times 10^{-11}\,\mathrm{m}^{3}\,\mathrm{kg}^{-1}\,\mathrm{s}^{-2}$. The measured mass of the Earth is $5.972 \times 10^{24}\,\mathrm{kg}$ and its [moment-of-inertia factor](https://one-course.com/books/earth/3/en/chapter/3-comparative-planetology#def-b1-comparative-planetology-moi) $I/MR^2 = 0.3308$.

**Part I — Volumes.**

1. Compute the volume of the Earth.
2. Compute the volume of each shell.
3. What fraction of the volume does each shell occupy? Which shell is the largest?
4. What fraction of the volume is core?

**Part II — Masses.**

5. Compute the mass of each shell.
6. Add them, and compare the total with the measured mass.
7. Deduce the mean density of the model Earth.
8. What fraction of the mass is in the [inner core](#def-b1-layered-earth-layers) ?
9. What fraction of the mass is in the [mantle](#def-b1-layered-earth-layers) and [crust](#def-b1-layered-earth-layers) together?
10. Compare the core’s share of the volume and of the mass, and explain.

**Part III — Moment of inertia.**

11. Show from [Proposition 9.10](#prop-b1-layered-earth-integrals) that a uniform shell between $r_1$ and $r_2$ has $I = \tfrac{8\pi}{15}\rho\,(r_2^5 - r_1^5)$ .
12. Compute the moment of inertia of each shell.
13. Which shell contributes most, and why does the [inner core](#def-b1-layered-earth-layers) contribute almost nothing?
14. Add them and compute $I/MR^2$ for the model.
15. Compare with $0.3308$ and with the $0.4$ of a uniform sphere.
16. Inside each real shell the density increases with depth. Does that explain why the model’s factor is too high? Explain.

**Part IV — Gravity and pressure.**

17. Compute the gravity at the surface of the model.
18. Compute the gravity at the [core–mantle boundary](#def-b1-layered-earth-discontinuity) .
19. Compute the mean density of the [mantle](#def-b1-layered-earth-layers) and [crust](#def-b1-layered-earth-layers) together.
20. Gravity in the [mantle](#def-b1-layered-earth-layers) stays close to $10\,\mathrm{m}\,\mathrm{s}^{-2}$ . Estimate the pressure at the [core–mantle boundary](#def-b1-layered-earth-discontinuity) , and compare with PREM’s $135.75\,\mathrm{GPa}$ .
21. Compute the mean density of the core, and the pressure increase from the [core–mantle boundary](#def-b1-layered-earth-discontinuity) to the centre in a uniform core, $\tfrac23\pi G\rho_c^2 r_c^2$ ( [Exercise 9.11](#exo-b1-layered-earth-11) applied to the core alone).
22. Add the two to estimate the central pressure, and compare with PREM’s $363.85\,\mathrm{GPa}$ . Why is the estimate too low?
23. Which of the model’s numbers are checked against a measurement, and which only against PREM itself?
24. What fraction of the Earth’s mass lies in the core?

**Solution of Problem 9.1.**

**1.** $V = \tfrac43\pi(6.371 \times 10^{6})^3 = 1.0832 \times 10^{21}\,\mathrm{m}^{3}$.

**2.** $\tfrac43\pi(r_2^3 - r_1^3)$: [crust](#def-b1-layered-earth-layers), upper [mantle](#def-b1-layered-earth-layers) and transition zone $3.0706 \times 10^{20}\,\mathrm{m}^{3}$; lower [mantle](#def-b1-layered-earth-layers) $5.9961 \times 10^{20}\,\mathrm{m}^{3}$; [outer core](#def-b1-layered-earth-layers) $1.6890 \times 10^{20}\,\mathrm{m}^{3}$; [inner core](#def-b1-layered-earth-layers) $7.634 \times 10^{18}\,\mathrm{m}^{3}$.

**3.** $28.35\,\%$, $55.36\,\%$, $15.59\,\%$ and $0.70\,\%$: the lower [mantle](#def-b1-layered-earth-layers) is the largest shell, more than half of the Earth’s volume.

**4.** $15.59 + 0.70 = 16.3\,\%$.

**5.** [Crust](#def-b1-layered-earth-layers), upper [mantle](#def-b1-layered-earth-layers) and transition zone $3.0706 \times 10^{20}\times3570 =
1.0962 \times 10^{24}\,\mathrm{kg}$; lower [mantle](#def-b1-layered-earth-layers) $2.9381 \times 10^{24}\,\mathrm{kg}$; [outer core](#def-b1-layered-earth-layers) $1.8410 \times 10^{24}\,\mathrm{kg}$; [inner core](#def-b1-layered-earth-layers) $9.841 \times 10^{22}\,\mathrm{kg}$.

**6.** $5.9737 \times 10^{24}\,\mathrm{kg}$, $0.03\,\%$ above the measured $5.972 \times 10^{24}\,\mathrm{kg}$.

**7.** $5.9737 \times 10^{24}/1.0832 \times 10^{21} = 5515\,\mathrm{kg}\,\mathrm{m}^{-3}$.

**8.** $9.841 \times 10^{22}/5.9737 \times 10^{24} = 1.65\,\%$.

**9.** $(1.0962 + 2.9381)/5.9737 = 67.5\,\%$.

**10.** The core is $16.3\,\%$ of the volume and $32.5\,\%$ of the mass, twice as much: its mean density, $10\,986\,\mathrm{kg}\,\mathrm{m}^{-3}$, is about $2.5$ times that of the [mantle](#def-b1-layered-earth-layers).

**11.** A shell of uniform density gives $I = \tfrac{8\pi}{3}\rho\int_{r_1}^{r_2} r^4\,\mathrm dr = \tfrac{8\pi}{15}\rho
(r_2^5 - r_1^5)$.

**12.** [Crust](#def-b1-layered-earth-layers), upper [mantle](#def-b1-layered-earth-layers) and transition zone $2.676 \times 10^{37}\,\mathrm{kg}\,\mathrm{m}^{2}$; lower [mantle](#def-b1-layered-earth-layers) $4.525 \times 10^{37}\,\mathrm{kg}\,\mathrm{m}^{2}$; [outer core](#def-b1-layered-earth-layers) $9.27 \times 10^{36}\,\mathrm{kg}\,\mathrm{m}^{2}$; [inner core](#def-b1-layered-earth-layers) $5.87 \times 10^{34}\,\mathrm{kg}\,\mathrm{m}^{2}$.

**13.** The lower [mantle](#def-b1-layered-earth-layers): it has half the mass, at large radii. The [inner core](#def-b1-layered-earth-layers) holds $1.6\,\%$ of the mass, and all of it close to the axis: its share is $0.07\,\%$.

**14.** $I = 8.1345 \times 10^{37}\,\mathrm{kg}\,\mathrm{m}^{2}$ and $I/MR^2 = 8.1345 \times 10^{37}/
(5.9737 \times 10^{24}\times(6.371 \times 10^{6})^2) = 0.3355$.

**15.** $1.4\,\%$ above $0.3308$, and well below $0.4$: the model has the right concentration of mass towards the centre, nearly.

**16.** Yes. Inside each shell the real density is higher at the bottom than at the top; a uniform shell of the same mass puts too much of that mass near its outer radius, where it counts with $r^2$, and so overestimates $I$.

**17.** $g = GM/R^2 = 6.674 \times 10^{-11}\times5.9737 \times 10^{24}/
(6.371 \times 10^{6})^2 = 9.822\,\mathrm{m}\,\mathrm{s}^{-2}$.

**18.** The core’s mass is $1.8410 \times 10^{24} + 9.841 \times 10^{22} =
1.9394 \times 10^{24}\,\mathrm{kg}$, and $g = G m/r_c^2 = 10.69\,\mathrm{m}\,\mathrm{s}^{-2}$.

**19.** $(1.0962 \times 10^{24} + 2.9381 \times 10^{24})/(3.0706 \times 10^{20} +
5.9961 \times 10^{20}) = 4.0343 \times 10^{24}/9.0667 \times 10^{20} = 4450\,\mathrm{kg}\,\mathrm{m}^{-3}$.

**20.** $P \approx 4450\times10\times2.891 \times 10^{6} = 128.6\,\mathrm{GPa}$, $5\,\%$ below PREM’s $135.75\,\mathrm{GPa}$.

**21.** $\rho_c = 1.9394 \times 10^{24}/1.7653 \times 10^{20} = 10\,986\,\mathrm{kg}\,\mathrm{m}^{-3}$, and $\Delta P = \tfrac23\pi\times6.674 \times 10^{-11}\times10986^2\times
(3.48 \times 10^{6})^2 = 204.3\,\mathrm{GPa}$.

**22.** $128.6 + 204.3 = 332.9\,\mathrm{GPa}$, $8.5\,\%$ below PREM’s $363.85\,\mathrm{GPa}$. Both estimates ignore the increase of density with depth inside each region, which places more mass deeper, keeps the gravity higher on the way down and raises $\rho g$, the pressure gradient.

**23.** The total mass (question 6), the [moment-of-inertia factor](https://one-course.com/books/earth/3/en/chapter/3-comparative-planetology#def-b1-comparative-planetology-moi) (question 15) and the surface gravity (question 17, against the measured mean $9.82\,\mathrm{m}\,\mathrm{s}^{-2}$) are checked against measurements of the Earth; the gravity at the core and the pressures can only be compared with PREM, itself a model.

**24.** The core holds **$32.5\,\%$** of the Earth’s mass.