A mountain's gravity says what holds it up

A rotating planet bulges at its equator by an amount set by its spin and its interior, and what is left after the bulge is subtracted is shape the spin cannot explain: mountains, basins, continents and volcanic provinces that stand above or below where a fluid would put them. That residue is a load the planet is carrying, and something has to carry it. It is also a nuisance to anyone weighing the planet’s core from its moment of inertia, because a load held up by strength adds to the flattening without telling anything about the density at depth. There are two candidates, and they are the two answers to the oldest question in geophysics.

One is strength. A mountain can sit on a rigid shell that holds it up the way a table holds up a book, with no change beneath. The other is buoyancy. A mountain can float, like an iceberg, on a root of light crust pushed down into the denser mantle beneath, so that the column under the mountain weighs the same as the column under the plains — the condition called isostasy. Both answers are right, for different mountains, and the gravity field can tell which is which, because the two answers leave different amounts of mass behind.

A mountain held up by strength adds its own mass to the planet and nothing else, and the gravity above it is stronger by the attraction of that mass. A mountain floating on a root adds its mass above and removes an equal mass below, where light crust has displaced denser mantle, and the gravity above it is nearly unchanged. Between the two lies every real case, and the measure of where a feature sits between them is the ratio of its gravity to its height.

Strength holds up the small, buoyancy the large #

The outer shell of a rocky planet — the lithosphere — behaves on geological timescales like an elastic plate floating on a fluid mantle. Put a load on it and it bends, and how much it bends depends on the load’s size compared with the plate’s stiffness.

The stiffness is the flexural rigidity, which grows as the cube of the plate’s elastic thickness: a plate twice as thick is eight times as hard to bend. The resistance to bending is strongest at short wavelengths, because bending a plate into a tight curve takes far more energy than bending it into a gentle one — the energy grows as the fourth power of the curvature’s wavenumber. The buoyancy that supports a bent plate, on the other hand, depends only on how far it is pushed down into the mantle, not on the wavelength.

Those two scalings compete, and the result is a crossover.

For a load much narrower than the crossover wavelength, the plate’s resistance to bending wins: it hardly deflects, nothing is compensated, and the load is held up by strength. For a load much broader, the plate bends freely, sinks until the load is buoyantly supported, and the topography is fully compensated — Airy isostasy, as if the plate had no strength at all. The crossover, where half the load is compensated, is at a wavelength of about 140 kilometres for a plate 5 kilometres thick, 460 for one 25 kilometres thick and 1,300 for one 100 kilometres thick, growing as the thickness to the three-quarters power.

The Himalaya and the Tibetan plateau sit at the long end: features thousands of kilometres across, standing on crust seventy kilometres thick whose root extends far into the mantle. A single seamount sits at the short end, a cone a few tens of kilometres across held up by the oceanic plate around it, which bends into a moat beneath its flanks. The same physics covers both, and the same parameter — the thickness of the elastic part of the plate — decides where any given feature falls.

The admittance is the crossover, measured #

The gravity anomaly a feature produces is the attraction of its own mass minus the attraction of the mass its root displaces. The root is deeper than the topography, so its attraction at the surface is weaker by a factor that depends on its depth and the wavelength, and it is smaller in proportion to how much of the load is compensated. The free-air gravity anomaly per unit topography — the admittance — is therefore

$$Z(k) = 2\pi G\rho_c\left(1 - C(k)\,e^{-k z_m}\right),$$

where $\rho_c$ is the crust’s density, $C(k)$ the compensated fraction at wavenumber $k$, and $z_m$ the depth of the compensating boundary, the Moho.

At short wavelengths $C$ is nearly zero, and the admittance is the attraction of an uncompensated slab of crust, $2\pi G\rho_c$: about 117 milligals per kilometre of topography for crust of density 2,800 kilograms per cubic metre. At long wavelengths $C$ is nearly one and $e^{-kz_m}$ is nearly one too, so the two terms cancel and the admittance falls to a few milligals per kilometre. Between them the curve falls, and the opening figure shows that where it falls depends on the plate: the thicker the plate, the longer the wavelength at which the gravity starts to disappear.

The gravity field is read off the orbits of the spacecraft that cross it, and the topography off altimeters on the same spacecraft. A measured admittance is computed by transforming both the gravity field and the topography into wavelength components — spherical harmonics on a planet, Fourier components on a regional map — and dividing one by the other at each wavelength, averaging over orientation and over the region. The ratio is then fitted with the flexural model, and the fitted thickness is the elastic thickness of the lithosphere where the loads were applied.

That is a remarkable thing to be able to measure from orbit. The elastic thickness is a property of the planet’s outer shell some tens of kilometres down: how much of the lithosphere is cold enough to behave elastically over geological time. It is set by the temperature profile, and therefore by the heat flowing out of the interior, and it is measured by dividing two maps.

A load from below #

The model so far has the load on top: a volcano piled on the surface, a mountain belt stacked by thrusting. Loads can also be applied from below — a body of dense rock intruded at the base of the crust, a region of hot, buoyant mantle pushing the plate up — and a buried load produces topography of its own, by bending the plate from underneath.

For a buried load the relationship between gravity and topography is very different. At long wavelengths the two kinds of load give the same small admittance, because both are compensated. At shorter wavelengths the plate’s strength means that a large buried load raises only a small hill — the plate resists being bent upward as much as downward — and the gravity of the large buried mass, divided by the height of the small hill, gives an admittance far from that of a surface load. For the plate drawn it is minus 120 milligals per kilometre at two hundred kilometres’ wavelength, where the surface load gives plus 116, and it falls without limit at shorter wavelengths.

That has a practical consequence and a cautionary one. The practical one is that the loading can sometimes be read directly: a region whose admittance is negative at intermediate wavelengths is being pushed up from below. The cautionary one is that most regions have both kinds of load, in unknown proportions, and an admittance fitted with surface loading alone will return the wrong thickness wherever buried loads are present. The modern treatment fits both, using the coherence between gravity and topography — how consistently their phases agree — as a second observable, and the debate over the elastic thickness of the continents, which has run for thirty years, turned partly on whether buried loads had been allowed for.

Every kilometre of distance is a filter #

Gravity is measured from orbit, and orbit is above the surface. That changes what can be measured, and in a way that cuts against this method in particular.

A gravity anomaly with wavelength $\lambda$ decays upward as $e^{-2\pi z/\lambda}$: the field of a short-wavelength mass pattern is averaged out with height, because at a distance its positive and negative parts blend. The same factor appears in the admittance itself, as the root’s weaker attraction, and it appears again between the surface and the spacecraft. A satellite at 250 kilometres keeps half of a signal at 2,300 kilometres’ wavelength and almost none below a thousand; one at 50 kilometres keeps half at about 450.

The admittance’s diagnostic part — the fall from the uncompensated slab to the compensated plateau, which sits at a few hundred kilometres for a typical plate — is therefore exactly the part that a high orbit cannot see. A gravity field mapped from a few hundred kilometres up constrains only thick plates, whose crossovers are long; thin plates are invisible to it. This is the same altitude filter that limits every satellite gravity field, and it is why the missions that changed the subject flew low. Twin spacecraft mapping the Moon flew at fifty kilometres and then lower, and resolved its field to a few kilometres; earlier lunar fields, from higher orbits, had been unable to separate the crust’s structure from its loads.

Mountains on smaller worlds #

The crossover depends on the plate’s thickness and also on the planet. Buoyancy is a force per unit area proportional to gravity times the density contrast between crust and mantle, so a world with weaker gravity supports a bent plate less strongly, and a plate of a given stiffness can carry broader loads before it floats.

The effect is modest — the crossover scales as one over the fourth root of the gravity — but the planets’ elastic thicknesses differ by far more than their gravities do, and the two effects compound. Olympus Mons, twenty-two kilometres high and six hundred across, stands on a Martian lithosphere whose elastic thickness beneath it is estimated at close to a hundred kilometres. On the Earth, a volcano that size would bend the plate until much of it sank: the Hawaiian islands, a fraction of its volume, sit in a moat several kilometres deep that they have pushed into the ocean floor.

Mars’s largest load is Tharsis, a volcanic province thousands of kilometres across that is responsible for most of the part of Mars’s shape the spin cannot explain. Its admittance shows it largely uncompensated at the wavelengths of the individual volcanoes and partly compensated at the scale of the whole rise, and the elastic thicknesses fitted to different parts of it differ, because they were loaded at different times: the older loads sat on a thinner, warmer lithosphere and kept the flexure it allowed them.

The Moon shows the most extreme case. Its large impact basins carry positive gravity anomalies — mass concentrations, discovered in 1968 because they disturbed the orbits of the first lunar orbiters — where a fully compensated basin would show none. They are dense mantle that rebounded into the basin floor after the impact and lava that later filled the depression — lava whose depth the craters it buried still record — held up by a lithosphere that had cooled enough to support them. The admittance across them records the thickness of a plate that no longer exists in that form, as it was billions of years ago.

A plate remembers when it was loaded #

The elastic thickness measured beneath a feature is the thickness when the feature was built, not now, and the ocean floor shows why.

Oceanic lithosphere is made hot at a mid-ocean ridge and cools as it spreads away, conducting heat to the sea floor above. The temperature at a given depth falls with age, and the depth of any isotherm grows as the square root of the age, from nothing at the ridge to tens of kilometres after a hundred million years. Rock behaves elastically on geological timescales below about 400 to 600 degrees Celsius and flows above, so the elastic thickness should follow one of those isotherms.

It does. Elastic thicknesses fitted to the flexure around hundreds of seamounts and ocean islands lie mostly between the 300 and 600 degree isotherms of a cooling half-space, at the age the floor had when each volcano was built. A seamount built on young, thin, warm floor near a ridge sits on a weak plate and is largely compensated; one built on old, thick, cold floor stands on a stiff one and has a large gravity signal. And a seamount built young keeps its weak-plate compensation after the floor beneath it has aged and stiffened, because the bending was done when it was loaded and the plate’s later strength does not undo it — the flexure is frozen in.

That is the sense in which an admittance is a historical measurement. It records the plate’s state at the time of loading. A planet whose loads were applied at different epochs carries a record of how its lithosphere thickened as it cooled, and the elastic thickness, read from gravity and topography alone, is one of the few ways of reading the thermal history of a planet nobody has drilled.

What was actually measured #

The Earth. Admittance and coherence analyses of satellite gravity and of shipboard and land gravity, with topography from altimetry and surveys, give elastic thicknesses from a few kilometres beneath young ocean floor and active rifts to more than a hundred beneath the oldest continental shields. Whether the continental values are really that large, or are an artefact of unmodelled buried loads, was argued for two decades; opinion has moved towards thick shields where the loading is well characterised, though not unanimously.

Venus. Radar topography and a gravity field from an orbiter whose low point was a couple of hundred kilometres up give elastic thicknesses of tens of kilometres beneath the highlands, with large regional variation. Venus’s high surface temperature makes its crust weaker than the Earth’s at the same depth, and its admittance suggests the mantle beneath it is pushing up much of its topography from below.

Mars. Orbiters tracked for two decades, combined with laser altimetry, give elastic thicknesses ranging from tens of kilometres beneath the old southern highlands to well over a hundred beneath the young polar caps — the caps being among the youngest loads on the planet, applied to a lithosphere that has had billions of years to cool.

The Moon. Twin spacecraft flying in formation at fifty kilometres and lower mapped the gravity field to a resolution of a few kilometres, and the admittance of the highlands showed a crust far more porous than expected, its bulk density reduced by the fracturing of billions of years of impacts. It was a measurement of the crust’s density, not just its strength, made from the short-wavelength end of the admittance — the part only a low orbit can reach.

Where the plate model stops #

A thin elastic plate. The lithosphere is modelled as a thin elastic sheet of uniform thickness floating on an inviscid fluid. Real lithosphere has a strength that varies with depth, is weakened by faults, and relaxes viscously over long times, so the elastic thickness is an effective parameter, not a layer that exists.

Flat geometry. The figures use the Cartesian relations. On a small body, or at wavelengths comparable to the planet’s radius, the curvature of the shell adds membrane support — a curved shell resists bending more than a flat one — and the spherical formulation is needed.

Fixed densities and depths. The crust’s density, the mantle’s and the depth of compensation are fixed for each world. In reality all three vary, and they trade against the elastic thickness in a fit; the crust’s density, especially, is often what an admittance analysis actually constrains at short wavelengths.

Isotherms, not strength. The ocean-floor figure draws the depths of isotherms in a simple cooling model. Equating the elastic thickness with an isotherm assumes a rheology that flows sharply above a temperature, which is a simplification of a gradual transition.

A ratio that turns two maps into a temperature #

The general shape is a measurement made by dividing two quantities at each scale, so that what they share cancels and what differs between scales is left. Gravity alone mixes the mass of every feature with the mass of its root. Topography alone says nothing about roots. Their ratio, wavelength by wavelength, separates the strength that holds a feature up from the buoyancy that floats it, and the scale at which one gives way to the other is the plate’s thickness.

A transfer function measured across scales says more than either signal alone, because the physics that connects the two usually depends on scale in a known way — here, bending resistance as the fourth power of the wavenumber against buoyancy that does not depend on it at all. It is the outer shell’s version of the gravity field turned into an instrument pointed inward. The same structure runs through the Love numbers that measure how much a whole planet gives under a tide, which are a response measured at the tidal frequency instead of a wavelength.

The other reading is about time. The thickness an admittance measures is the thickness at the time of loading, and loads of different ages carry different values. A planet’s outer shell is a record of its own cooling, written into the gravity field by every volcano, basin and plateau it has had to carry, and readable from orbit without landing anywhere.

Still open: a shell that is not rock #

The method assumes a lithosphere floating on a fluid mantle. Several icy moons have a different structure — an ice shell floating on a liquid ocean, with loads of ice topography on top — and the admittance, measured from flybys, constrains the shell’s thickness and its strength in the same way. The flybys are fast and high, the gravity fields reach only low degrees, and whether an ocean’s depth can be separated from its shell’s is still an open question. The orbiters now travelling to Jupiter’s moons will fly low enough to measure admittances at the wavelengths where an ice shell’s strength shows, and the answer will say how thick the ice is above the oceans that are the best candidates for life beyond the Earth.