Equations: The Earth and rocks

SI; if the density is taken in g/cm³ and the velocity in km/s, $\rho V^2$ comes out directly in gigapascals

Holds when: A homogeneous, isotropic, elastic medium (the same at every point and in every direction, and springing back after a load). The moduli are dynamic: they apply to the small, fast strains that the wave produces.

any consistent units: $r$ and $h$ in the same unit of length, the densities in the same unit

Holds when: The density of the crust is the same everywhere and only its thickness varies (the Airy model). In the Pratt model it is the other way round: the density varies and the depth of the base stays the same. Compensation is local: each column is balanced on its own, and the strength of the lithosphere is ignored. The mantle is the same under the mountains and under the plain.

any consistent units

Holds when: The basin is filled up to the former water level, and sea level has not changed. Compensation is local; the lithosphere beneath the basin does not stretch or cool while the basin is filling. The mean density of the sediments is constant (in reality it increases as they compact).

SI: $q_0$ in W/m², $A$ in W/m³, $\lambda$ in W/(m·K), $z$ in m; temperature in °C

Holds when: Steady one-dimensional transfer of heat by conduction, with no movement of groundwater. $\lambda$ and $A$ are constant in a layer of thickness $D$; below it, where $A \approx 0$, the gradient is constant and equal to $(q_0 - A D)/\lambda$. There has been no recent heating or cooling: rifting, rapid sedimentation and erosion disturb the steady-state profile.

any consistent units

Holds when: The boundary is horizontal and the layers are homogeneous. The source and receivers are at the surface.

$A$ in µW/m³, $\rho$ in kg/m³, $C_U$ and $C_{Th}$ in ppm, $C_K$ in % by mass

Holds when: The coefficients are the heat released by the decay of 1 kg of natural uranium (about 95 µW), thorium (about 26 µW) and potassium (about 0.0035 µW), converted to the units of content used. Different tables of decay constants differ by a few per cent. The uranium and thorium decay series are in radioactive equilibrium.

dimensionless

Holds when: All the tetrahedra are built around silicon. If aluminium replaces part of the silicon, the denominator is the sum of the silicon and aluminium atoms in the tetrahedra.

scale readings in the same unit, densities in the same unit

Holds when: The fragment is a single mineral, without pores and cracks that the liquid enters (otherwise the density includes the voids). The liquid does not dissolve or react with the fragment, and air bubbles have been removed. The buoyancy of air is neglected (the correction is less than 0.1%).

mL and g; find $k$ by calibration under the same conditions

Holds when: There is an excess of hydrochloric acid. Calcite reacts almost completely in the first minute and dolomite over tens of minutes, and the time boundary between them is arbitrary. Siderite and other slowly reacting carbonates are counted as “dolomite”. Part of the gas dissolves in the acid; calibration on pure calcite under the same conditions allows for this.

kilometres and millions of years: 1 km/million years = 1 mm/yr

Holds when: The sea floor forms at a narrow ridge axis, and its magnetisation does not change after cooling. The speed is the mean over the time $t$; if it has changed, different anomalies give different values. The distance is measured along the direction of opening, parallel to the transform faults. An oblique profile across the stripes distorts the speed. If the ridge axis is perpendicular to the opening, the oblique profile makes the speed too high.

SI; with $a$ and $t_c$ in kilometres, $S_i$ comes out in kilometres

Holds when: The stretching is instant and uniform. The crust and the mantle part of the lithosphere thin by the same factor, and temperature in the lithosphere still rises linearly with depth. Isostasy is local; the strength of the lithosphere is ignored. Rock density falls with temperature as $\rho(T) = \rho_0(1 - \alpha_V T)$. The depression is filled with water, and the initial surface is at sea level.

SI ($\tau$ is converted from seconds to millions of years: 1 million years ≈ $3.156\cdot10^{13}$ s)

Holds when: The lithosphere cools by vertical conduction; the basin is wider than about 100 km, and no heat escapes sideways. The series that describes the cooling is cut to its first, slowest term. For $\beta$ from 1.3 to 3, the limiting thermal subsidence differs from the sum of the full series by a few per cent. For $\beta$ near 1.2 it is about 7% too low. In the first 20 million years after the stretching the difference is larger, up to 10–25%, but the thermal subsidence itself is then still small. Radiogenic heat is not included, nor is the fact that sediments of low thermal conductivity “insulate” the lithosphere and slow its cooling.

SI

Holds when: A thin elastic plate of constant thickness on a fluid base (over the time it takes a basin to form, the asthenosphere behaves as a fluid). The whole basin is filled with material of a single density $\rho_i$; the plate is pushed up by the difference $\rho_m - \rho_i$.

any consistent units: $x$ and $\alpha$ in the same unit of length

Holds when: The plate is broken beneath the thrust front, and the load is a line load (it acts along a line) at the end of the plate. This is the usual way to model foreland basins in front of mountain belts. If the plate is taken as continuous, the edge of the basin is farther ($x_0 = 3\pi\alpha/4$), the crest is at $x_b = \pi\alpha$, and the forebulge is lower — 4.3% of $w_0$.

SI

Holds when: Local isostasy: at depth $a$ the pressure beneath the original and the stretched columns is the same.

km and millions of years

Holds when: The plates are rigid; the Earth is a sphere.

$\lambda$ and $t$ in reciprocal units (yr⁻¹ and years); $D^*/P$ is a ratio of numbers of atoms; ln is the natural logarithm

Holds when: The system is closed: after the mineral formed, neither parent nor daughter isotope entered or left it. There was no daughter isotope at the start, or its initial amount is known (otherwise the isochron method is used — see the advanced section). Every decay gives an atom of exactly this daughter isotope; for ⁴⁰K only about 10% of decays turn into ⁴⁰Ar (the rest give ⁴⁰Ca), so for potassium–argon ages $D^*/P$ is divided by this fraction. The decay constant is known exactly; the age is counted from the moment when the mineral cooled enough for the daughter isotope to stop escaping from it.

m/Myr (metres per million years, the same as mm per thousand years)

Holds when: Compaction of previously deposited sediment is included in the subsidence. The bending of the lithosphere under the weight of the sediment and the water is not separated from the tectonic subsidence (isostasy).

ratios of numbers of atoms, dimensionless

Holds when: The samples are contemporaneous and had the same initial strontium isotope ratio. The system remained closed after it formed.