Earth's Figure, Gravity and Isostasy

Earth ellipsoid and latitude gravity correction

The Earth is not a sphere but an oblate spheroid, bulged at the equator and flattened at the poles. The flattening is f = (a-b)/a, where a is the equatorial radius and b is the polar radius, and the adopted reference ellipsoid has f \approx 1/298.25. The radius of the ellipsoid at geodetic latitude \varphi is approximately r \approx a(1 - f\sin^2\varphi), so the radius is largest at the equator and smallest at the poles.

Normal gravity also varies with latitude, so observations must be corrected before anomalies are interpreted. The latitude correction is proportional to \sin 2\varphi to first order, which makes its magnitude largest at \varphi = 45^\circ and zero at the equator and the poles.

In the spherical-harmonic expansion of the spheroidal potential, the n=2 zonal term -J_2(R/r)^2 P_2(\cos\theta) represents the equatorial bulge and polar flattening, which is the largest departure of the Earth from a sphere. Odd-degree terms such as J_3 instead describe the pear-shaped asymmetry between hemispheres.

Rotation adds a centrifugal acceleration a = \omega^2 R\cos\varphi, directed away from the rotation axis, where R\cos\varphi is the perpendicular distance from the axis. For example, with \omega = 7.27\times10^{-5} rad/s and R = 6371 km, the value at 30^\circ is about 29\times10^{-3} m/s^2, and it is smaller at higher latitudes.

The geodetic latitude is normal to the ellipsoid, whereas the geocentric latitude \varphi' is measured from the centre, with \tan\varphi' = (1-f)^2\tan\varphi. To first order \varphi - \varphi' \approx f\sin 2\varphi, so the difference peaks near 45^\circ and vanishes at the poles and the equator.

The equivalent sphere has the same volume as the reference ellipsoid. Since the ellipsoid volume is \frac{4}{3}\pi r_1^2 r_2, the equivalent radius is R = (r_1^2 r_2)^{1/3}. For example, if r_1 = 2 and r_2 = 1, then R = 4^{1/3} \approx 1.59.

Bouguer correction

The Bouguer correction removes the attraction of the material between the station and the datum, idealized as an infinite horizontal slab of thickness h and density \rho. Its magnitude is \Delta g_B = 2\pi G\rho h, which equals about 0.0419\rho h mGal when \rho is in g/cm^3 and h is in metres. For a station 100 m above the datum with \rho = 2.67 g/cm^3, the correction is about 11.2 mGal.

The Bouguer correction is applied with the opposite sign to the free-air correction. Because the free-air correction adds about 0.3086h mGal, the combined elevation reduction is governed by the density used for the slab.

After the latitude, free-air, Bouguer and terrain corrections have been applied, the residual Bouguer anomaly along a profile reflects lateral changes in subsurface density, such as differences in rock type, intrusions or density contrasts between adjacent crustal blocks. A uniform regional depth trend or a broad contrast across the Moho produces a smooth regional effect that does not vary along the profile.

At the same latitude, the free-air effect dominates observed gravity, which decreases with elevation at about 0.3086 mGal/m. A station in a valley below sea level therefore records a higher value than a sea-level plane, and a mountain top records the lowest value.

Inside a homogeneous layer, the material above the station pulls downward while the material below pulls upward, so the slab reduction per unit depth is twice the Bouguer term, 4\pi G\rho. The gravity change per metre of depth is therefore (0.3086 - 4\pi G\rho) mGal/m, which is about (0.3086 - 0.0839\rho) mGal/m with \rho in g/cm^3.

Laplacian of scalar and vector fields

The Laplacian is \nabla^2 = \partial^2/\partial x^2 + \partial^2/\partial y^2 + \partial^2/\partial z^2 in Cartesian coordinates. For a vector field it acts component by component, so \nabla^2\mathbf{F} = (\nabla^2F_x, \nabla^2F_y, \nabla^2F_z). The vector Laplacian is zero only when every component Laplacian is zero, and the vanishing of one component alone is not sufficient.

A scalar field \phi is harmonic when it satisfies Laplace's equation \nabla^2\phi = 0. A finite non-zero Laplacian means the field fails Laplace's equation, and such a field is not harmonic.

A potential with spherical symmetry depends only on the radial distance r, so \partial U/\partial\theta = \partial U/\partial\varphi = 0, while \partial U/\partial r is generally non-zero. Spherical symmetry therefore does not make the radial derivative vanish.

If the scalar potential \psi is prescribed on a closed surface S, which is a Dirichlet boundary condition, then Laplace's equation has a unique solution inside S. Conditions on \nabla\cdot\mathbf{F} or \nabla\times\mathbf{F} describe the sources or rotation of the field and do not fix \psi on the boundary.

For a radial field \mathbf{F} = a r^n \hat{r}, the divergence is \nabla\cdot\mathbf{F} = \frac{1}{r^2}\frac{d}{dr}(r^2 a r^n) = a(n+2)r^{n-1}, which vanishes everywhere only when n = -2. A radial field that depends only on r has zero curl for any n, so the field is both solenoidal and irrotational only for n=-2, as in the inverse-square field.

Gravity anomaly of simple bodies

The half-width x_{1/2} is the horizontal distance at which the anomaly falls to half its maximum. Standard depth rules relate the depth d to x_{1/2}: for a sphere d \approx 1.3\,x_{1/2}, for a horizontal cylinder d \approx x_{1/2}, and for a steeply dipping thick sheet d \approx 0.7\,x_{1/2}. For an irregular body, Smith's rule gives an upper bound d \le 0.86\,(g_{max})/(\partial g/\partial x)_{max}.

For an infinite horizontal cylinder of radius r, density contrast \Delta\rho and axis depth z, the maximum anomaly is \Delta g_{max} = 2\pi G\Delta\rho\, r^2/z. Two cylinders give the same maximum when their \Delta\rho\, r^2/z values are equal, so a deep, large cylinder can match a shallow, small one if its density contrast is correspondingly smaller.

For a sphere of radius R and density contrast \Delta\rho, the anomaly is g(x) = K d/(x^2+d^2)^{3/2} with K = \frac{4}{3}\pi G R^3 \Delta\rho, so the maximum is g_{max} = K/d^2. Setting g/g_{max} = 1/5 gives x_{1/5}/d = \sqrt{5^{2/3}-1} \approx 1.39.

Because \Delta g_{max} \propto 1/z^2 for a sphere of fixed radius and density contrast, the product \Delta g_{max} z^2 is constant. For example, if the maximum anomaly is 10 mGal at a depth of 3 km, the anomaly at 6 km is 10 \times (3/6)^2 = 2.5 mGal.

Planetary size comparison

Planetary size is compared mainly through the equatorial radius. The Earth's equatorial radius is about 6378 km, and Venus, with an equatorial radius of about 6052 km, is the closest in size among the inner planets. Mercury (about 2440 km) and Mars (about 3396 km) are much smaller, and Neptune (about 24,800 km) is roughly four times the Earth's radius.

A scaled model keeps all lengths in a fixed ratio to the real object. The scale factor is model length divided by real length, so one centimetre on the model represents real length divided by model length. For example, a 20 cm model of the Earth with a diameter of about 12,742 km gives 12742/20 \approx 637.1 km per centimetre on the model.

Airy isostasy root depth

In Airy's hypothesis, topographic loads are compensated by a crustal root of lower density that extends into the denser mantle, and the columns below a compensation depth have equal mass. If the mountain elevation is h, the crustal density is \rho_c and the mantle density is \rho_m, the root thickness is r = h\rho_c/(\rho_m - \rho_c).

For example, a mountain of 3 km elevation with \rho_c = 2.8 and \rho_m = 3.3 g/cm^3 has a root of r = 3 \times 2.8/0.5 = 16.8 km. Using the same relation with a 4 km elevation and densities of 2700 and 3300 kg/m^3 gives a root of 18 km.

If the observed deflection of the plumb line near a mountain is smaller than the value computed from the visible mass, the excess mass must be offset by a mass deficiency at depth. This points to a low-density root, which reduces the net attraction, rather than a high-density root or the absence of any root.

Planetary mean density

The mean density of a planet is its mass divided by its volume, and it depends on bulk composition and compression. Earth has the highest mean density among Mercury, Venus, Earth and Mars, at about 5.5 g/cm^3, slightly above Mercury's value, while Mars, with a mean density near 3.9 g/cm^3, is among the lowest.

For a planet with a core of density \rho_c and radius r_c surrounded by a mantle of density \rho_m, the mean density is \bar{\rho} = \rho_c (r_c/R)^3 + \rho_m[1-(r_c/R)^3], since the mean is a volume-weighted average. If the mantle thickness is 2R/3, then r_c = R/3 and the core occupies 1/27 of the volume, so \rho_m = (27\bar{\rho} - \rho_c)/26.

As a check, a core of density 8000 kg/m^3 with radius R/2 occupies one-eighth of the volume. If the mean density is 5000 kg/m^3, the mantle density is (5000 - 1000)/0.875 \approx 4571 kg/m^3.

Airy isostasy gravity anomalies

For a mountain range that is fully compensated in the Airy sense, the excess topographic mass is exactly balanced by the root deficiency, so the isostatic anomaly is zero. The observed anomalies are different: the Bouguer correction removes the full slab of topography, while the root deficiency at depth remains, so the Bouguer anomaly is strongly negative.

The free-air anomaly over such a mountain is small and positive, because the topographic attraction is only partly offset by the deeper root. The combination of a small positive free-air anomaly and a negative Bouguer anomaly is the signature of Airy compensation.

Pratt isostasy density compensation

In Pratt's model, the crust has lateral density variations, with higher topography having lower density, and all columns have equal mass above a common compensation depth D measured below sea level. For a column of height h above sea level and density \rho, the compensation condition is \rho(D+h) = \rho_0 D, which gives \rho = \rho_0 D/(D+h), where \rho_0 is the reference density. For example, with \rho_0 = 2800 kg/m^3, D = 100 km and h = 2 km, the density is about 2745 kg/m^3.