Earth Structure and Physical Properties Questions and Answers

Answer: D

The seismic moment, from which moment magnitude is calculated, is slip × area × elastic modulus.

\[ M_0 = \mu A D \]

Here \(\mu\) is the shear modulus (rigidity) of the rock, A the area of the fault rupture and D the average slip. The moment magnitude is then \(M_w = \tfrac{2}{3}\log_{10} M_0 - 6.07\), with \(M_0\) in newton-metres.

Density does not appear in the moment, Young's modulus is the wrong elastic constant and 'rupture' duplicates the area, and the third option multiplies two elastic constants by a velocity, which has the wrong dimensions.

Answer: A

The shadow zone exists because the outer core is liquid.

S waves cannot travel through a liquid, so they do not pass through the outer core, producing an S-wave shadow beyond about 103° from the epicentre. P waves do enter the liquid core, but its sharp drop in velocity refracts them strongly, so no direct P waves arrive between about 103° and 142°. Both effects follow from the outer core being molten.

The waves do obey Snell's law at the core-mantle boundary; their refraction is exactly what that law predicts. The D'' layer is a thin zone at the base of the mantle and does not cause the shadow, and the solid inner core lies inside the shadow-forming boundary, its discovery coming from faint arrivals within the shadow.

Answer: B

The long-wavelength geoid is controlled mainly by density variations in the deep mantle.

The largest geoid highs and lows, thousands of kilometres across (such as the Indian Ocean geoid low and the highs over the Pacific and Africa), correlate with the large low-velocity provinces and subducted slabs imaged by seismic tomography in the lower mantle, and with the dynamic topography of mantle flow.

Crustal density variations produce short-wavelength geoid features, and the liquid outer core cannot sustain lateral density differences of significance. The inner core lies too deep and is too symmetrical to shape the long-wavelength geoid.

Answer: A

The general equation for seismic wave velocity is \(\sqrt{\text{elastic modulus}/\text{density}}\).

The speed of an elastic wave depends on the stiffness of the medium and its inertia:

\[ \begin{aligned} V_P &= \sqrt{(K + \tfrac{4}{3}\mu)/\rho} \\ V_S &= \sqrt{\mu/\rho} \end{aligned} \]

with K the bulk modulus, \(\mu\) the shear modulus and \(\rho\) the density. Both have the form of the first option.

Density over time does not give a velocity, cycles per second is frequency, and the square of a ratio of two moduli is dimensionless.

Answer: B

The key's answer is 55.9°C, which follows from a linear geothermal gradient with a surface temperature of about 25°C.

The question does not state the surface temperature, so one has to be assumed. Taking a mean surface temperature of 25°C:

\[ \begin{aligned} G &= \frac{80 - 25}{3.2} = 17.19\ ^{\circ}\mathrm{C/km} \\ T &= 25 + 17.19 \times 1.8 \\ &= 25 + 30.94 = 55.9\ ^{\circ}\mathrm{C} \end{aligned} \]

85.9°C is impossible, because temperature rises with depth and 1.8 km is shallower than the 3.2 km where 80°C was recorded. 65.9°C and 75.9°C would need surface temperatures of about 48°C and 71°C, which are not realistic.

Answer: D

The equatorial bulge is caused by the centrifugal effect of the Earth's rotation.

Because the Earth rotates, a point on the equator moves in a large circle and experiences the greatest centrifugal effect, which reduces effective gravity there and lets the Earth's material settle into an oblate shape. The equatorial radius (about 6378 km) is roughly 21 km greater than the polar radius (about 6357 km).

The Moon's attraction raises tides but does not create the permanent bulge. Plate tectonics moves the lithosphere but does not shape the planet as a whole, and the sense of rotation does not matter; it is the rotation itself.

Answer: D

Heat flux requires the temperature gradient and thermal conductivity.

By Fourier's law, conductive heat flow through rock is the product of the thermal conductivity of the rock and the vertical temperature gradient:

\[ q = -k\,\frac{dT}{dz} \]

So heat flow studies measure temperatures at several depths in a borehole to get the gradient, and measure the conductivity of core samples in the laboratory.

Temperature alone is not enough without the gradient, 'heat transfer' is the quantity being calculated, and rheology describes deformation, not heat conduction.

Answer: B

The moment of inertia of the Earth is derived from the precession of the equinoxes.

The Sun and Moon pull on the Earth's equatorial bulge and make its rotation axis precess, slowly tracing a cone in about 26,000 years. The rate of precession depends on the dynamical ellipticity (C − A)/C. Combining it with the oblateness term J₂ of the gravity field gives the polar moment of inertia C, about 0.33 MR². A value below 0.4 (that of a uniform sphere) shows that density increases towards the centre.

Density variation is what is inferred from the moment of inertia, not its source. Seismic waves give velocities and, indirectly, densities, and observations from the Moon are not the means of deriving it.

Answer: C

The relation is \(\frac{dp}{dr} = -\rho g\), the hydrostatic equation.

Consider a thin shell of thickness dr and unit area. The weight of the material in it, \(\rho g\, dr\), must be supported by the difference in pressure between its lower and upper faces. Because pressure falls as the radius increases outward, the change is negative:

\[ dp = -\rho\, g\, dr \]

Inside the Earth both \(\rho\) and g vary with r, but the form of the equation is the same at every radius.

The other options are not dimensionally possible as a pressure gradient: adding a density to an acceleration, taking the square root of their product, or squaring g does not give pressure per unit length. Only the product \(\rho g\) has those units.

Answer: C

Granite generates the most radiogenic heat.

The heat-producing elements uranium, thorium and potassium are incompatible: they stay in the melt during partial melting and fractionation, and so are concentrated in evolved, silica-rich rocks. A typical granite produces about 2.5 to 3 microwatts per cubic metre.

Dolerite and basalt produce several times less, and mantle peridotite, which has lost these elements to the crust, produces about a hundredth as much. Chondritic meteorites represent undifferentiated solar-system material with low concentrations of U, Th and K, so they too produce far less than granite.