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### Crustal Structure:Gravity and Topography

Oded Aharonson

Reference Ellipsoid

- Planets are flattened by rotation.
- They can be represented roughly by ellipsoids.
- More irregular bodies can be represented by tri-axial ellipsoids, with two different equatorial radii.

Coordinate System

- Geocentric latitude is measured by the angle that a line to the COM makes with the equatorial plane.
- Geographic latitude is measured by the angle that the surface normal makes with the equatorial plane.
- Geographic (aka geodetic) latitude is what is used in most map projections.
- Most space based measurements are in the geocentric form.

- For a non-spherical body the idea of latitude becomes ambiguous. Many auxiliary latitudes exist such as conformal, authalic, rectifying, geocentric, parametric (aka reduced) latitude. Most of these are just mathematical constructs.

Map Projections

e.g. Albers Equal Area e.g. Mercator e.g. Stereographic

- Map projections convert spherical and Cartesian coordinates
- There are three common categories of map projections
- Find x/y coordinates by taking a line from the center of projection through the spheres surface and intersecting the map surface.

Gravity

Gravitational force:

Or in vector form:

Gravitational attraction at the surface of the Earth was determined by Galileo, hence the unit we use today is a “gal”.

1 gal = 1 cm/s2 = 0.01 m/s2 ~ 10-3 g

Gravity

Since gravity is a conservative field, its far more convenient to work with a scalar potential instead of the vector field:

The potential is 0 at r=.

Things “fall” down potential gradients.

The force per unit mass, g, is:

Gravity

- Write potential:

Spherical Harmonics

- Zonal Harmonics: m=0
- Sectorial Harmonics: l=m
- Tesseral Harmonics: l≠m

History

- Gravity: a probe of internal structure
- 1778: Cavendish torsion balance (111 years after Newton’s law of gravity) → determine G → ‘weighing of the Earth’Earth~5500 kg/m3 > rocks→ deduce Earth must increase in density with depth
- 1800’s: studies of regional gravity perturbations

Geoid Height

- The geoid undulates slowly over long distances.
- Mass excesses in the lithosphere cause “Lumps” in the geoid.
- Mass deficits in the lithosphere cause “depressions” in the geoid.

Topography

Topography is always measured relative to the geoid: “orthometric height” or height above sea-level

This is just as well since we'd like our definition of downhill to be the direction that gravity acts.

Topography and geoid height are usually correlated.

The ratio between the two is sometimes is related to the admittance.

Gravity

In order to interpret observed gravity measurements, a series of corrections are often applied.

1. Free-Air correction: accounts for the elevation of the measurement point:

Free-Air Anomaly:

Free-air assumes there’s massless “air” between the observer and the reference surface

On the surface of the Earth, we obtaindgfa~3.1×10-6 m/s2 per meter, or ~310 mgal/km

Gravity

In order to interpret observed gravity measurements, a series of corrections are often applied.

2. Bouguer correction:accounts for the extra attraction of mass due to observed topographic features:

Bouguer Anomaly:

Bouguer assumes there’s a homogeneous thin layer between the observer and the reference surface.

For G=6.67×10-11 m3/kg3s2, =2,700 kg/m3, we obtain dgB~1.1×10-6 m/s2 per meter, or ~110 mgal/km

Gravity

In order to interpret observed gravity measurements, a series of corrections are often applied.

3. Terrain correction: accounts for the proximity of mass anomalies to the absovation points, for example near mountain tops. Typically applied by a Fourier method.

4. Isostatic correction: effect of masses that support loads

5. Tidal correction: effect of time-dependent shapes in Earth’s shape

6. Eötvös correction: effects of the motion of the observation point (such as a moving ship)

7. Other corrections: effects of other assumed crust or mantle density anomalies (“geology” or “geodynamic”)

Gravity

You might expect that near a mountain gravity would increase due to all that extra mass, however the increase is not as much as you would expect. This is due to what is called isostatic compensation.

The Airy explanation of isostasy is the familiar iceberg effect. A rigid lithosphere floats upon a viscous asthenosphere. (Note: This is not the same as crust vs. mantle, which is a compositional distinction)

The Pratt explanation of isostasy relies upon columns of lithosphere having different densities.

Compensated?

A strong positive free-air anomaly with a weak Bouguer anomaly indicates a structure is supported by the strength of the lithosphere i.e. no compensation.

A weak free-air anomaly with a strong negativeBouguer anomaly indicates that the structure is compensated.

Geoid

Solid planets are not perfectly smooth ellipsoids, they are lumpy irregular objects.

A gravitational equipotential surface is used to define the shape of the planet. Its known as the Geoid.

This surface coincides with mean sea level in a best fit sense.

Shown here is a representation of the geoid in meters relative to a best fit ellipsoid.

This is a horizontal surface even though it has “lumps” and “holes” in it.

Gravity

The geoid is an integrated quantity, so it is optimal for studying long wavelength effects.

The Gravity anomaly is a differential quantities, so it exhibits small scale structure.

Interpretation

Bouguer anomalies can be interpreted as:

Density fluctuations below the surface

OR

A constant density layer that varies in thickness

A) Mars topography B) Mars free-air gravity anomaly C) Bouguer anomaly converted to crustal thickness. Figure from Zuber et al., Science, 287, 1788-1793,2000.

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