Exercises: Structure and composition of the geosphere

Name the layers of the geosphere ordered from the outside to the centre of the planet, and state the physical state (solid, fluid) of each one.

Cite two direct methods of studying the geosphere and explain why, even with the deepest boreholes, we directly know only a tiny fraction of the Earth's radius.

Compare continental crust and oceanic crust in terms of their thickness, composition, and density, and relate these differences to their behaviour when two plates collide.

Distinguish between crust, lithosphere and asthenosphere: indicate which criterion defines each boundary (composition versus mechanical behaviour) and which layer contains which.

Explain what the geothermal gradient is, how much the temperature increases when descending through the first kilometres (about $30\ \text{°C/km}$ as an average value) and what consequence this has for deep mines.

1. a) Name the four most abundant chemical elements in the Earth's crust.
2. b) Explain why silicate minerals dominate surface rocks.

Explain the principle of isostasy using the iceberg analogy, and discuss why high mountain ranges have deep crustal "roots" and what happens when erosion thins them.

Classify these methods of studying the Earth's interior as direct or indirect and justify your answer: a deep borehole, the recording of seismic waves, the analysis of xenoliths brought up by a volcano and the study of meteorites.

Explain how P and S seismic waves differ (type of movement and media they pass through) and how their shadow zones allowed the existence and state of Earth's core to be deduced.

What is a seismic discontinuity? State which layers are separated by the Mohorovičić and Gutenberg discontinuities and at roughly what depths they are located.

The diagram represents a cross-section of the concentric layers of the geosphere. a) Name the layers marked with numbers 1 to 4. b) State the missing approximate depth ("?") and which discontinuity is found at that depth. c) Indicate the physical state of layers 3 and 4 and the evidence that proves it.

The outer core is liquid and the inner core, although even hotter, is solid. Explain this apparent paradox by resorting to pressure, and discuss which essential planetary phenomenon (the magnetic field) is generated in the moving outer core.

The Kola borehole (Russia) reached a depth of about $12\ \text{km}$, the drilling record. Calculate what percentage of the Earth's radius ($6\,370\ \text{km}$) this depth represents and conclude why knowledge of the planet's interior depends on indirect methods.

Volcanoes sometimes bring up fragments of mantle rock (peridotite xenoliths). Explain what direct information they provide about the composition of the mantle, why they are so valuable, and what limitation they have (from what maximum depth do they come?).

A P seismic wave travels at $8\ \text{km/s}$ from a focus located $600\ \text{km}$ deep. Calculate how long it takes to reach the station located exactly at the vertical of the epicentre, and explain how, by measuring arrival times at many stations, seismologists locate the hypocentre.

Compare the dominant chemical composition of the crust (aluminium-rich silicates), of the mantle (magnesium-rich silicates, such as olivine) and of the core (iron and nickel), and relate it to the differentiation of a young, molten planet: which materials sank and which floated?

A geophysical exploration probe deploys a network of seismometers on a newly discovered rocky planet. Following a major meteorite impact at its north pole ($0^\circ$), the seismographs record:

• Direct P-waves between $0^\circ$ and $105^\circ$ of angular distance.

• A total shadow zone (without direct P- or S-waves) between $105^\circ$ and $142^\circ$.

• Between $142^\circ$ and the antipodes ($180^\circ$), P-waves are recorded again, but they arrive with a noticeable delay, and no direct S-waves are recorded.

1. a) Interpret these records by applying the laws of seismic wave propagation and refraction to deduce the existence of the core, its physical state, and its relative propagation velocity compared to the mantle.
2. b) Predict which new direct seismic phases you would expect to record in the antipodal zone ($142^\circ$-$180^\circ$) if, with the progressive cooling of the planet, a solid inner core were to begin nucleating and growing.

A student proposes the following hypothesis to simplify calculating the travel time of seismic waves: «Given that Earth's mantle is a continuous solid medium down to a depth of $2\,900\ \text{km}$, seismic rays traveling from the focus to a distant seismological station propagate in a straight line at a constant average velocity; therefore, the arrival time is simply calculated by dividing the straight-line geometric distance by said velocity».

1. a) Reasonably refute their hypothesis by explaining why the actual path of seismic rays through the mantle is curved and concave toward Earth's surface (support your reasoning with the variation of elastic properties with depth and Snell's law of refraction).
2. b) Based on the principle of least time (Fermat's principle), justify why, at epicentral distances greater than a few hundred kilometers, a wave following this curved and deep path arrives at the receiving station sooner than a hypothetical wave traveling in a straight line through shallower levels.

Earth's overall average density is $5{,}5\ \text{g/cm}^3$, whereas that of crustal rocks ranges between $2{,}7$ and $3{,}0\ \text{g/cm}^3$ (granite, basalt) and upper mantle peridotites barely reach $3{,}3\ \text{g/cm}^3$.

1. a) Justify why this contrast conclusively rules out Earth being homogeneous and demonstrates that most of its interior must contain materials with densities far higher than those observed at the surface.
2. b) Two types of meteorites are analyzed: a chondrite (silicate, density $\approx 3{,}5\ \text{g/cm}^3$) and an iron meteorite (Fe-Ni alloy, density $\approx 7{,}8\ \text{g/cm}^3$). Interpret the cosmochemical origin of both based on the process of accretion and differentiation of parent asteroids, and reason why iron meteorites constitute the classic indirect evidence supporting the chemical nature proposed for Earth's core.

Consider two terrestrial planets born simultaneously from the same protosolar nebula with an identical initial homogeneous composition: Planet Alpha (radius of $6\,400\ \text{km}$) and Planet Beta (a smaller body with a radius of $1\,200\ \text{km}$).

1. a) Explain why Planet Alpha underwent global heating that culminated in the iron catastrophe and its differentiation into concentric density layers, while Planet Beta barely managed to partially melt or remained largely undifferentiated. Analyze the role of initial heat sources (accretion energy, gravitational compression, and radioactive decay) versus the rate of heat loss as a function of the surface-to-volume ratio.
2. b) Predict the internal geodynamic state and the presence or absence of present-day volcanic activity in both bodies $4\,600$ million years after their formation, justifying your deduction based on residual heat and the abundance of radioisotopes.

A team of astrobiologists investigates the habitability of a rocky extrasolar planet. Their measurements reveal that it currently lacks a discernible global magnetic field, but analysis of its oldest basalts shows that $3\,500$ million years ago it possessed an intense dipolar field that experienced periodic polarity reversals.

1. a) Justify why the remanent magnetism in the basaltic lavas (acquired below the Curie temperature) proves that the field was generated by a dynamo mechanism in a convective fluid core and rules out the hypothesis of a permanent magnet of magnetized rock in the deep interior.
2. b) Propose a plausible geophysical hypothesis regarding what happened to that planet's interior for the geodynamo to cease, and predict the sequence of impacts that this magnetic collapse caused on its atmosphere, its surface liquid water bodies, and its biological potential, using the geological evolution of Mars as a reference.

Earth has a total mass of $5{,}97 \times 10^{24}\ \text{kg}$ and a volume of $1{,}08 \times 10^{12}\ \text{km}^3$ (average density of $5{,}52\ \text{g/cm}^3$). The core extends from a depth of $2\,900\ \text{km}$ to the center of the planet ($r = 3\,480\ \text{km}$), with an estimated average density of $11{,}0\ \text{g/cm}^3$.

1. a) Calculate the fraction of the planet's total volume occupied by the core and determine what percentage of Earth's total mass is confined within it.
2. b) At the surface, pure metallic iron has a density of $7{,}87\ \text{g/cm}^3$, but high-pressure physics calculations indicate that pure iron compressed to core pressures would reach between $11{,}5$ and $13{,}0\ \text{g/cm}^3$. Deduce what this discrepancy (known as the «core density deficit») reveals about its actual chemical composition regarding the presence of light elements (such as Si, O, S, or H).
3. c) Reason what geodynamic consequences it would have on surface gravity and mantle convection if that same mass of iron were homogeneously dispersed throughout the mantle instead of concentrated in the core.

In a debate on internal geodynamics, a student states: "The Mohorovičić discontinuity is the plane where the lithosphere detaches and slides over the asthenosphere. Furthermore, since oceanic crust is silicate and less dense than mantle peridotites, it is impossible for a lithospheric plate to sink spontaneously into the mantle during subduction."

Identify the geological errors in this statement and refute them by answering the following:

1. a) Precisely differentiate the nature of the crust-mantle boundary versus the lithosphere-asthenosphere boundary, indicating which compositional layers make up a mature oceanic lithospheric plate.
2. b) Resolve the buoyancy paradox: justify why an oceanic lithospheric plate as a whole does become denser than the underlying asthenosphere and ends up subducting spontaneously despite including the oceanic crust.
3. c) Justify why the peridotites of the lithospheric mantle behave as a brittle, rigid solid while the peridotites of the asthenosphere, chemically identical, behave in a ductile manner, analyzing the relationship between the geothermal gradient and the solidus curve.

Global seismological catalogues show that earthquakes in subduction zones are located along the Wadati-Benioff zone down to a limiting depth of about $670$-$700\ \text{km}$; below that level, seismicity ceases completely. However, seismic tomography images confirm that fragments of lithospheric plates continue to sink beyond $700\ \text{km}$, in some cases reaching the core-mantle boundary (D'' layer).

1. a) Justify the existence of earthquakes at depths of several hundred kilometres: why is the subducting plate capable of rupturing in a brittle manner at depths where the surrounding mantle flows plastically?
2. b) Explain the cause of the sharp cessation of earthquakes at $700\ \text{km}$, arguing the dual effect of the progressive heating of the plate by thermal diffusion and the mineralogical phase transformations of olivine (to bridgmanite and periclase).
3. c) If the lower mantle is a solid medium that allows plates to sink without generating seismicity, deduce by what physical mechanism this deformation is accommodated, and debunk the common misconception that the mantle must be molten (magma) to allow such flow.