INTRODUCTION

In introductory physics class we learned that the acceleration of gravity, "g", equals 9.8 meters per second squared. This is true as a first approximation but it turns out that the acceleration of gravity varies slightly depending on the local density underground. We can use these differences in g to infer local densities provided we account for variations in gravity caused by the shape of the earth, elevation with respect to sea-level, topography, tidal effects and instrument drift.

We define a gravity anomaly as the difference between measured g and that of some standard earth model. Great confusion occurs because several standard earth models are used.

Unlike the active seismic and resistivity methods we have studied, the gravity and magnetic methods are passive. Rather than actively provoking a response from the earth we simply measure an existing field.

NEWTON’S LAW OF GRAVITATIONAL ATTRACTION

In 1665 at the age of 23 Isaac Newton showed that there is an attractive force between any two masses. In particular his inverse square law was used to explain the orbits of the planets around the sun. The force between any two masses is equal to a constant times the product of the two masses divided by the square of the distance between them. In the SI system of units the force is expressed in Newtons, the masses in kilograms and the distances in meters.

There is some fine print to read with respect to Newton’s law. As stated above it refers to point masses, spheres of uniform density, spherical masses made of shells of uniform density and bodies small compared to the distance between them. For extensive bodies we’ll have to break them into little parts, apply Newton’s law, and integrate to get the total attraction (classes 28 and 29).

GRAVITATIONAL ACCELERATION OF THE EARTH

Let’s take one mass equal to the mass of the earth and the other as one kilogram. We define the gravitational acceleration , "g" (m/s^2), as the gravitational force on our unit mass at the surface of the earth. Therefore g equals the gravitational constant times the mass of the earth divided by the square of the radius of the earth. How can we determine these quantities?

Over two thousand years ago, Erastothenes estimated the radius of the earth based on the angular elevation of the sun at two points on a meridian in Egypt. The distance between Aswan and Alexandria was estimated by the number of days it took a camel caravan to make the journey. His result was in the right ballpark. Modern measurements indicate that the polar radius is about 6356 km and the equatorial radius about 6378 km.

In 1798 Cavendish used a torsion balance and a huge lead sphere to determine the gravitational constant. His result was very close to modern estimates (6.67x10^-11 Nm^2/kg^2). The errors of the best modern measurements are about plus or minus one percent. Thus this is one of the most poorly known of all the fundamental physical constants.

Historically the acceleration of gravity was determined by timing the oscillations of a pendulum. Nowadays the fall of an object in a vacuum can be timed using lasers to get better values. The result is the famous 9.8 m/s^2.

With these results in hand we can now calculate the mass of the earth. The result, about 6x10^24 kg, corresponds to a mean density of about 5500 kg/m^3. Given that this value is almost twice that of rocks at the surface, we conclude that the interior of our globe is denser than the outer parts.

EFFECT OF LATITUDE

On a non-rotating earth gravity would be the same at any latitude. However the real earth spins, influencing gravity directly and indirectly. We can estimate the direct effect using the concept of "centrifugal force". At the equator the centripetal acceleration directly opposes g whereas at the poles this effect is zero. This direct effect of spin is responsible for about two-thirds of the roughly 5000 milligal increase in g from the equator to the pole ( one milliGalileo=milligal=one mgal=10^-5 m/s^2).

The indirect effect of spin is that it causes the earth to "bulge out" at the equator and to be "flattened out" at the poles. Thus the polar radius is smaller than the equatorial by about 21 km or one part in three hundred. Therefore g is a little larger at the pole than at the equator as indicated by Newton’s law. This indirect effect is about one-third of the total 5000 mg difference.

The effect of latitude (at sea level) is described by a conventional formula. The most common one in use today is the "International Formula of 1967".

From this formula we can readily calculate the rate of change of g with latitude. This is zero at both the equator and the poles and reaches a maximum of about .81 mgal/km and 45 degrees latitude.

EFFECT OF ELEVATION

Let’s use Newton’s law to compare g at sea-level with g at a distance "h" above sea-level. We’ll assume that the second measurement is in "free air" with no material between the two sites. The result indicates that g decreases with height at a rate of about .31 mgal/m. For detailed work we should use a more exact value for the latitude and elevation of our survey.

GRAVITY ANOMALIES

In the next class we’ll examine how these and other effects are combined to define a variety of gravity anomalies, each with it’s own purpose and utility.