INTRODUCTION

In the last class we derived formulae for the gravitation effect of several simple geometrical bodies and discussed several ways we could use them. Today we’ll continue on the same path.

INFINITE SLAB (BOUGUER SLAB)

Part of the standard model for the Bouguer anomaly is the gravitational effect of an infinite slab used to approximate that of material between the station and sea-level. We can also use the formula to estimate the anomaly at the center of a wide sedimentary basin or any other extensive sheet or slab.

In this derivation our basic mass element is a horizontal, circular disk with a hole in it. We next evaluate a double integral; first over z from the top of the sheet to the bottom and secondly over the disk radius from zero to infinity. In this manner we expand the disk to include all the mass of the sheet. The result is the familiar Bouguer slab formula. As in the other formulas we cannot separate the density from a size factor, in this case the thickness of the slab. Notice that this formula is valid for any place on the sheet.

By using a finite radius for the upper limit of integration, we get a formula valid at the center of a circular disk. If the radius is ten times the thickness then the anomaly is 98 % that of an infinite slab. The distant parts of the slab have essentially no effect as their attraction is mainly horizontal. Calculations like this demonstrate that the infinite slab formula is generally a good approximation to the actual attraction of rock above sea-level.

HORIZONTAL MASS SHEET, STEP OR FAULT

Now we want to derive a formula for gravity anomalies on a profile extending normal to the straight edge of a semi-infinite horizontal sheet. Here we "condense" the mass of the sheet onto the central plane of the sheet, thus saving ourselves one step of integration. Note that we have two sets of horizontal coordinates; one for the station location and one for a little patch of the mass sheet.

For stations not over the sheet the anomaly goes to zero far from the edge. Over the edge the value is just half that of an infinite sheet just as symmetry would require. Over the sheet itself, the anomalies rise to the infinite slab value far from the edge. It’s a little hard to comprehend but a semi-infinite sheet is in some sense as infinite as an infinite one!

Note that we cannot break down the density times thickness product. Also note that the laterally uniform material over the sheet and below the sheet has no detectable gravitational effect. Detectable gravity anomalies require lateral density contrasts, not vertical ones.

HORIZONTAL MASS STRIP OR RIBBON

By cleverly overlapping a semi-infinite sheet of positive density contrast and another of negative density contrast we can convert the previous formula into one for a strip of finite width and infinite length. Simply for the sake of elegance we shift the origin of the horizontal coordinate to the center of the strip.

Besides approximating valley fills, lava flows and so on we can use this formula as a building block for a computer program for any two-dimensional body. We just pile up a set of thin strips to simulate the cross-section of the body.

HORIZONTAL SHEETS AND SOLID ANGLES

We can show that the gravitational effect of any thin horizontal sheet of any shape (in map view) is proportional to the "solid angle" subtended by the sheet at the gravity station. Solid angles (in "steradians") correspond to areas on a unit sphere in the same way that regular angles (in radians) correspond to arc lengths on a unit sphere. The solid angle of a sphere equals 4 pi steradians and that of a hemisphere is 2 pi radians. The patch of light from the eye of a Jack O’Lantern makes the same solid angle regardless of the distance and orientation of any planar surface on which it falls. I think the illustration will clarify the concept.