# Gravity and Geodesy

# GRAVITY AND GEODESY

## CONCEPT

Thanks to the force known as gravity, Earth maintains its position in orbit around the Sun, and the Moon in orbit around Earth. Likewise, everything on and around Earth holds its place—the waters of the ocean, the gases of the atmosphere, and so on—owing to gravity, which is also the force that imparts to Earth its nearly spherical shape. Though no one can really say what gravity *is,* it can be quantified in terms of mass and the inverse of the distance between objects. Earth scientists working in the realm of geophysics known as geodesy measure gravitational fields, as well as anomalies within them, for a number of purposes, ranging from the prediction of tectonic processes to the location of oil reserves.

## HOW IT WORKS

### Gravitation

Not only does gravity keep Earth and all other planets in orbit around the Sun, it also makes it possible for our solar system to maintain its position in the Milky Way, rather than floating off through space. Likewise, the position of our galaxy within the larger universe is maintained because of gravity. As for the universe itself, though many questions remain about its size, mass, and boundaries, it seems clear that the cosmos is held together by gravity.

Thanks to gravity, all objects on Earth as well as those within its gravitational field remain fixed in place. These objects include man-made satellites, which have grown to number in the thousands since the first was launched in 1957, as well as the greatest satellite of them all: the Moon. Even though people are accustomed to thinking of gravity in these large terms, with regard to vast bodies such as Earth or the Moon, every object in the universe, in fact, exerts some gravitational pull on another.

This attraction is proportional to the product of the mass of the two bodies, and inversely related to the distance between them. Bodies have to be fairly large (i.e., larger than an asteroid) for this attraction to be appreciable, but it is there, and thus gravity acts as a sort of "glue" holding together the universe. As to what gravity really is or exactly why it works, both of which are legitimate questions, the answers have so far largely eluded scientists.

Present-day scientists are able to understand how gravity works, however, inasmuch as it can be described as a function of mass and the gravitational constant (discussed later) and an inverse function of distance. They also are able to measure gravitational fields and anomalies within them. That, in fact, is the focus of geodesy, an area of geophysics devoted to the measurement of Earth's shape and gravitational field.

### Discovering Gravity

As discussed in several places within this book (see the entries Earth, Science, and Nonscience and Studying Earth), the physical sciences made little progress until the early sixteenth century. For centuries, the writings of the Greek philosopher Aristotle (384-322 b.c.) and the Alexandrian astronomer Ptolemy (*ca.* a.d. 100-170) had remained dominant, reinforcing an almost entirely erroneous view of the universe. This Aristotelian/Ptolemaic universe had Earth at its
center, with the Sun, Moon, and other planets orbiting it in perfect circles.

The discovery by the Polish astronomer Nicolaus Copernicus (1473-1543) that Earth rotates on its axis and revolves around the Sun ultimately led to the overturning of the Ptolemaic model. This breakthrough, which inaugurated the Scientific Revolution (*ca.* 1550-1700), opened the way for the birth of physics, chemistry, and geology as genuine sciences. Copernicus himself was a precursor to this revolution rather than its leader; by contrast, the Italian astronomer Galileo Galilei (1564-1642) introduced the principles of study, known as the scientific method, that govern the work of scientists to this day.

#### GALILEO AND GRAVITATIONAL ACCELERATION.

Galileo applied his scientific method in his studies of falling objects and was able to show that objects fall as they do, not because of their weight (as Aristotle had claimed) but as a consequence of gravitational force. This meant that the acceleration of all falling bodies would have to be the same, regardless of weight.

Of course, everyone knows that a stone falls faster than a feather, but Galileo reasoned that this was a result of factors other than weight, and later investigations confirmed that air resistance, rather than weight, is responsible for this difference. In other words, a stone falls faster than a feather not because it is heavier but because the feather encounters greater air resistance. In a vacuum, or an area devoid of all matter, including air, they would fall at the same rate.

On the other hand, if one drops two objects that meet similar air resistance but differ in weight—say, a large stone and a smaller one—they fall at almost exactly the same rate. To test this hypothesis directly, however, would have been difficult for Galileo: stones fall so fast that even if dropped from a great height, they would hit the ground too soon for their rate of fall to be tested with the instruments then available.

Instead, Galileo used the motion of a pendulum and the behavior of objects rolling or sliding down inclined planes as his models. On the basis of his observations, he concluded that all bodies are subject to a uniform rate of gravitational acceleration, later calibrated at 32 ft. (9.8 m) per second per second. What this means is that for every 32 ft. an object falls, it is accelerating at a rate of 32 ft. per second as well; hence, after two seconds it falls at the rate of 64 ft. per second, after three seconds it falls at 96 ft. per second, and so on.

#### NEWTON'S BREAKTHROUGH.

Building on the work of his distinguished fore-bear, Sir Isaac Newton (1642-1727), who was born the same year Galileo died, developed a paradigm for gravitation that even today explains the behavior of objects in virtually all situations throughout the universe. Indeed, the Newtonian model reigned supreme until the early twentieth century, when Albert Einstein (1879-1955) challenged it on certain specifics.

Even so, Einstein's relativity did not *disprove* the Newtonian system as Copernicus and Galileo had disproved Aristotle's and Ptolemy's theories; rather, it showed the limitations of Newtonian mechanics for describing the behavior of certain objects and phenomena. In the ordinary world of day-to-day experience, however, the Newtonian system still offers the key to how and why things work as they do. This is particularly the case with regard to gravity.

### Understanding the Law of Universal Gravitation

Like Galileo, Newton began in part with the aim of testing hypotheses put forward by an
astronomer—in this case, Johannes Kepler (1571-1630). In the early years of the seventeenth century, Kepler published his three laws of planetary motion, which together identified the elliptical (oval-shaped) path of the planets around the Sun. Kepler had discovered a mathematical relationship that connected the distances of the planets from the Sun to the period of their revolution around it. Like Galileo with Copernicus, Newton sought to generalize these principles to explain not only *how* the planets moved but also *why* they did so.

The result was Newton's *Philosophiae naturalis principia mathematica* (Mathematical principles of natural philosophy, 1687). Usually referred to simply as the *Principia,* the book proved to be one of the most influential works ever written. In it, Newton presented his law of universal gravitation, along with his three laws of motion. These principles offered a new model for understanding the mechanics of the universe.

### The Three Laws of Motion

Newton's three laws of motion may be summarized in this way:

- An object at rest will remain at rest, and an object in motion will remain in motion at a constant velocity unless and until outside forces act upon it.
- The net force acting upon an object is a product of its mass multiplied by its acceleration.
- When one object exerts a force on another, the second object exerts on the first a force equal in magnitude but opposite in direction.

The first law of motion identifies inertia, a concept introduced by Galileo to explain what kept the planets moving around the Sun. Inertia is the tendency of an object either to keep moving or to keep standing still, depending on what it is already doing. Note that the first law refers to an object moving at a constant velocity: velocity is speed in a certain direction, so a constant velocity would be the same speed in the same direction.

Inertia is measured by mass, which—as the second law states—is a component of force and is inversely related to acceleration. The latter, as defined by physics, has a much broader meaning than it usually is given in ordinary life. Acceleration does not mean simply an increase of speed for an object moving in a straight line; rather, it is a change in velocity—that is, a change of speed or direction or both.

By definition, then, rotational motion (such as that of Earth around the Sun) involves acceleration, because any movement other than motion in a straight line at a constant speed requires a change in velocity. This further means that an object experiencing rotational motion *must* be under the influence of some force. That force is gravity, and as the third law shows, every force exerted in one direction is matched by an equal force in the opposing direction. (This law is sometimes rendered "For every action there is an equal and opposite reaction.")

#### NEWTON'S GRAVITATIONAL FORMULA.

The law of universal gravitation can be stated as a formula for calculating the gravitational attraction between two objects of a certain mass, *m* _{1} and *m* _{2}: *F* _{grav} = *G* × (*m* _{1}*m* _{2}) */r* ^{2}. In this equation, F_{grav} is gravitational force, and r^{2} is the square of the distance between the two objects.

As for *G,* in Newton's time the value of this number was unknown. Newton was aware simply that it represented a very small quantity: without it, (*m* _{1}*m* _{2})/*r* ^{2} could be quite sizable for objects of relatively great mass separated by a relatively small distance. When multiplied by this very small number, however, the gravitational attraction would be revealed to be very small as well. Only in 1798, more than a century after Newton's writing, did the English physicist Henry Cavendish (1731-1810) calculate the value of *G* using a precision instrument called a torsion balance.

The value of *G* is expressed in units of force multiplied by distance squared, and then divided by, mass squared; in other words, *G* is a certain value of (N × m^{2})/kg^{2}, where *N* stands for newtons, *m* for meters, and *kg* for kilograms. Nor is the numerical value of *G* a whole number such as 1. A figure as large as 1, in fact, is astronomically huge compared with *G,* whose value is 6.67 × 10^{−11}—in other words, 0.0000000000667.

### Physical Geodesy

Within the realm of geodesy is that of physical geodesy, which is concerned specifically with the measurement of Earth's gravitational field as well as the geoid. The latter may be defined as a surface of uniform gravitational potential covering the entire Earth at a height equal to sea level. ("Potential" here is analogous to height or, more specifically, position in a field. For a discussion of potential in a gravitational field, see Energy and Earth.)

Thus, in areas that are above sea level, the geoid would be below ground—indeed, *far* below it in mountainous regions. Yet in some places (most notably the Dead Sea and its shores, the lowest point on Earth), it would be above the solid earth and waters. The geoid is also subject to deviations or anomalies, owing to the fact that the planet's mass is not distributed uniformly; in addition, small temporary disturbances in the geoid may occur on the seas as a result of wind, tides, and currents.

Generally speaking, however, the geoid is a stable reference platform from which to measure gravitational anomalies. It is a sort of imaginary gravitational "skin" covering the planet, and in the past, countries conducting geodetic surveys tended to choose a spot within their boundaries as the reference point for all measurements. With the development of satellites and their use for geodetic research, however, it has become more common for national geodetic societies to use global points of reference such as the planet's center of mass.

#### MEASUREMENTS FROM SPACE, LAND, AND SEA.

The geoid can be determined by using such a satellite, equipped with a radar altimeter, but there is also the much older technique of terrestrial gravity measurement. The terrestrial method is much more difficult and prone to error, however, and calculations require detailed checking and correction to remove potential anomalies due to the presence of matter in areas above the points at which gravitational measurements were obtained.

Also highly subject to error are measurements made from a vessel at sea. This has to do not only with the effect of the ship's pitch and roll but also with something called the Eötvös effect. Named for the Hungarian physicist Baron Roland Eötvös (1848-1919), who conducted extensive studies on gravity, the effect is related to the Coriolis force, which causes the deflection of atmospheric and oceanic currents in response to Earth's rotation. Measurements of gravity from the air are also subject to the Eötvös effect, though the use of GPS (global positioning system) information, obtained from satellites, can improve greatly the accuracy of seaborne measurements.

#### HOW GRAVITY IS MEASURED.

Scientists can obtain absolute terrestrial gravity measurements by measuring the amount of time it takes for a pellet to fall a certain distance within a vacuum—that is, a chamber from which all matter, including air, has been removed. This, of course, is the same technology Galileo used in making his observations more than 400 years ago. It is also possible to obtain relative gravity measurements with the use of mechanical balance instruments.

As noted earlier, the acceleration due to gravity is 9.8 m/s^{2}, or 9.8 m s^{−2}. (Scientists sometimes use the latter notation, in which the minus sign is not meant to indicate a negative but rather is used in place of "per".) This number is the measure of Earth's gravitational field. In measuring gravitational anomalies, scientists may use the Gal, named after Galileo, which is equal to 0.01 m/s^{2}. Typically, however, the milligal, equal to one-thousandth of a Gal, is used. Note that "Gal" sometimes is rendered in lowercase, but this can be confusing, because it looks like the abbreviation for "gallon. ")

#### WHY MEASURE GRAVITY?

Why is it important to measure gravity and gravitational anomalies? One answer is that weight values can vary considerably, depending on one's position relative to Earth's gravitational field. A fairly heavy person might weigh as much as a pound less at the equator than at the poles and less still at the top of a high mountain. The value of the gravity field at sea level has a range from 9.78 to 9.83 m/s^{2}, a difference of about 50,000 g.u., and it is likely to be much lower than 9.78 m/s^{2} at higher altitudes.

Indeed, the higher one goes, the weaker Earth's gravitational field becomes. At the same time, the gases of the atmosphere dissipate, which is the reason why it is hard to breathe on high mountains without an artificial air supply and impossible to do so in the stratosphere or above it. At the upper edge of the mesosphere, Earth's gravitational field is no longer strong enough to hold large quantities of hydrogen, lightest of all elements, which constitutes the atmosphere at that point. Beyond the mesosphere, the atmosphere simply fades away, because there is not sufficient gravitational force to hold its particles in place.

Back down on Earth, gravity measurements are of great importance to the petroleum industry, which uses them to locate oil-containing salt domes. Furthermore, geologists, in general, remain acutely interested in measurements of gravity, the force behind tectonics, or the deformation of Earth's crust. Thus, gravity, responsible for fashioning Earth's exterior into the nearly spherical shape it has, is key to the shaping of its interior as well.

## REAL-LIFE APPLICATIONS

### Gravity on Earth

Using Newton's gravitational formula, it is relatively easy to calculate the pull of gravity between two objects. It is also easy to see why the attraction is insignificant unless at least one of the objects has enormous mass. In addition, application of the formula makes it clear why *G* is such a tiny number.

Suppose two people each have a mass of 45.5 kg—equal to 100 lb. on Earth, though not on the Moon, a matter that will be explained later in this essay—and they stand 1 m (3.28 ft.) apart. Thus, *m* _{1}*m* _{2} is equal to 2,070 kg (4,555 lb.), and *r* ^{2} is equal to 1 m squared. Applied to the gravitational formula, this figure is rendered as 2,070 kg^{2}/1 m^{2}. This number then is multiplied by the gravitational constant, and the result is a net gravitational force of 0.000000138 N (0.00000003 lb.)—about the weight of a single-cell organism!

#### WEIGHT.

What about Earth's gravitational force on one of those people? To calculate this force, we could apply the formula for universal gravitation, substituting Earth for m_{2}, especially because the mass of Earth is known: 5.98 × 10^{24} kg, or 5.98 septillion (1 followed by 24 zeroes) kg. We know the value of that mass, in fact, through the application of Newton's laws and the formulas derived from them. But for measuring the gravitational force between something as massive as Earth and something as small as a human body, it makes more sense to apply instead the formula embodied in Newton's second law of motion: *F = ma.* (Force equals mass multiplied by acceleration.)

For a body of any mass on Earth, acceleration is figured in terms of *g* —the acceleration due to gravity, which, as noted earlier, is equal to 32 ft. (9.8 m) per second squared. (Note, also, that this is a lowercase *g,* as opposed to the uppercase *G* that represents the gravitational constant.) Using the metric system, by multiplying the appropriate mass figure in kilograms by 9.8 m/s^{2}, one would obtain a value in newtons (N). To perform the same calculation with the English system, used in America, it would be necessary first to calculate the value of mass in slugs (which, needless to say, is a little-known unit) and multiply it by 32 ft./s^{2} to yield a value in pounds.

In both cases, the value obtained, whether in newtons or pounds, is a measure of weight rather than of mass, which is measured in kilograms or slugs. For this reason, it is not entirely accurate to say that 1 kg is equal to 2.2 lb. This is true on Earth, but it would not be true on the Moon. The kilogram is a unit of mass, and as such it would not change anywhere in the universe, whereas the pound is a unit of force (in this case, gravitational force) and therefore varies according to the rate of acceleration for the gravitational field in which it is measured. For this reason, scientists prefer to use figures for mass, which is one of the fundamental properties (along with length, time, and electric charge) of the universe.

### Why Earth Is Round—and Not Round

Everyone knows that Earth, the Sun, and all other large bodies in space are "round" (i.e., spherical), but why is that true? The reason is that gravity will not allow them to be otherwise: for any large object, the gravitational pull of its interior forces the surface to assume a relatively uniform shape. The most uniform of three-dimensional shape is that of a sphere, and the larger the mass of an object, the greater its tendency toward sphericity.

Earth has a relatively small vertical differential between its highest and lowest surface points, Mount Everest (29,028 ft., or 8,848 m) on the Nepal-Tibet border and the Mariana Trench (−36,198 ft., −10,911 m) in the Pacific Ocean, respectively. The difference is just 12.28 mi. (19.6 km)—not a great distance, considering that Earth's radius is about 4,000 mi. (6,400 km).

On the other hand, an object of less mass is more likely to retain a shape that is far less than spherical. This can be shown by reference to the Martian moons Phobos and Deimos, both of which are oblong—and both of which are tiny, in terms of size and mass, compared with Earth's Moon. Mars itself has a radius half that of Earth, yet its mass is only about 10% of Earth's. In light of what has been said about mass, shape, and gravity, it should not be surprising to learn that Mars is also home to the tallest mountain in the solar system, the volcano Olympus Mons, which stands 16 mi. (27 km) high.

#### EARTH'S 'FLAT TOP' (AND BOTTOM).

With regard to gravitation, a spherical object behaves as though its mass were concentrated near its center, and indeed, 33% of Earth's mass is at its core, even though the core accounts for only about 20% of the planet's volume. Geologists believe that the composition of Earth's core must be molten iron, which creates the planet's vast electromagnetic field.

It should be noted, however, that Earth is not really a perfect sphere, and the idea that its mass is concentrated at its center, while it works well in general, poses some problems in making exact gravitational measurements. If Earth were standing still, it would be much nearer to the shape of a sphere; however, it is not standing still but instead rotates on its axis, as does every other object of any significance in the solar system.

Incidentally, if Earth were suddenly to stop spinning, the gases in the atmosphere would keep moving at their current rate of about 1,000 MPH (1,600 km/h). They would sweep over the planet with the force of the greatest hurricane ever known, ripping up everything but the mountains. As to why Earth spins at all, scientists are not entirely sure. It may well be angular momentum (the momentum associated with rotational motion) imparted to it at some point in the very distant past, perhaps because it and the rest of the solar system were once part of a vast spinning cloud.

At any rate, the fact that Earth is spinning on its axis creates a certain centripetal, or inward-pulling, force, and this force produces a corresponding centrifugal (outward) component. To understand this concept, consider what happens to a sample of blood when it is rotated in a centrifuge. When the centrifuge spins, centripetal force pulls the material in the vial toward the center of the spin, but the material with greater mass has more inertia and therefore responds less to centripetal force. As a result, the heavier red blood cells tend to stay at the bottom of the vial (or, as it is spinning, on the outside), while the lighter plasma is pulled inward. The result is the separation between plasma and red blood cells.

Where Earth is concerned, this centrifugal component of centripetal force manifests as an equatorial bulging. Simply put, Earth's diameter around the equator is greater than at the poles, which are slightly flattened. The difference is small—the equatorial diameter of Earth is about 26.72 mi. (43 km) greater than the polar diameter—but it is not insignificant. In fact, as noted later, a person of fairly significant weight actually would notice a difference if he or she got on the scales at the equator (say, in Singapore) and then later weighed in near one of the poles (for instance, in the Norwegian possession of Svalbard, the northernmost human settlement on Earth).

Owing to this departure from a perfectly spherical shape, the Sun and Moon exert additional torques on Earth, and these torques cause shifts in the position of the planet's rotational axis in space. An imaginary line projected from
the North Pole and into space therefore, over a period of time, would appear to move. In the course of about 25,800 years, this point (known as the celestial north pole) describes the shape of a cone, a movement known as Earth's *precession.*

### Satellites

Why, then, does Earth move around the Sun, or the Moon around Earth? As should be clear from Newton's gravitational formula and the third law of motion, the force of gravity works both ways: not only does a stone fall toward Earth, but Earth also actually falls *toward it.* The mass of Earth is so great compared with that of the stone that the movement of Earth is imperceptible—but it does happen.

Furthermore, because Earth is round, when one hurls a projectile at a great distance, Earth curves away from the projectile. Eventually, gravity itself forces the projectile to the ground. If one were to fire a rocket at 17,700 mi. per hour (28,500 km per hour), however, something unusual would happen. At every instant of time, the projectile would be falling toward Earth with the force of gravity—*but* the curved Earth would be falling away from it at the same rate. Hence, the projectile would remain in constant motion around the planet—that is, it would be in orbit.

The same is true of an artificial satellite's orbit around Earth: even as the satellite falls toward Earth, Earth falls away from it. Change the names of the players, and this same relationship exists between Earth and its great natural satellite, the Moon. Furthermore, it is the same with the Sun and *its* many satellites, including Earth: Earth plunges toward the Sun with every instant of its movement, but at every instant, the Sun falls away.

### WHERE TO LEARN MORE

Ardley, Neil. *The Science Book of Gravity.* San Diego, CA: Harcourt Brace Jovanovich, 1992.

*Ask the Space Scientist* (Web site). <http://image.gsfc.nasa.gov/poetry/ask/askmag.html>.

Beiser, Arthur. *Physics,* 5th ed. Reading, MA: Addison-Wesley, 1991.

*Exploring Gravity* (Web site). <http://www.curtin.edu.au/curtin/dept/phys-sci/gravity/>.

*Geodesy for the Layman* (Web site). <http://www.nima.mil/GandG/geolay/TR80003A.HTM>.

*The Gravity Society* (Web site). <http://www.gravity.org>.

Hancock, Paul L., and Brian J. Skinner. *The Oxford Companion to the Earth.* New York: Oxford University Press, 2000.

Riley, Peter D. *Earth.* Des Plaines, IL: Heinemann Interactive Library, 1998.

Smith, David G. *The Cambridge Encyclopedia of Earth Sciences.* New York: Cambridge University Press, 1981.

Wilford, John Noble. *The Mapmakers.* New York: Knopf, 2000.

## KEY TERMS

### ACCELERATION:

A change in velocity over time. The acceleration due to gravity, for instance, is 32 ft. (9.8 m) per second per second, meaning that for every second an object falls, its velocity is increasing as well.

### ATMOSPHERE:

In general, an atmosphere is a blanket of gases surrounding a planet. Unless otherwise identified, however, the term refers to the atmosphere of Earth, which consists of nitrogen (78%), oxygen (21%), argon (0.93%), and other substances that include water vapor, carbon dioxide, ozone, and noble gases such as neon, which together comprise 0.07%.

### CENTRIFUGAL:

A term describing the tendency of objects in uniform circular motion to move outward, away from the center of the circle. Though the term centrifugal force often is used, it is inertia, rather than force, that causes the object to move outward.

### CENTRIPETAL FORCE:

The force that causes an object in uniform circular motion to move inward, toward the center of the circle.

### FORCE:

The product of mass multiplied by acceleration.

### GEODESY:

An area of geophysics devoted to the measurement of Earth's shape and gravitational field.

### GEOID:

A surface of uniform gravitational potential covering the entire Earth at a height equal to sea level.

### GEOPHYSICS:

A branch of the earth sciences that combines aspects of geology and physics. Geophysics addresses the planet's physical processes as well as its gravitational, magnetic, and electric properties and the means by which energy is transmitted through its interior.

### INERTIA:

The tendency of an object at rest to remain at rest or an object in motion to remain in motion, at a uniform velocity, at a uniform velocity, unless acted upon by some outside force.

### MASS:

A measure of inertia, indicating the resistance of an object to a change in itsmotion.

### POTENTIAL:

Position in a field, such as a gravitational force field.

### SCIENTIFIC METHOD:

A set of principles and procedures for systematic study that includes observation; the formation of hypotheses, theories, and laws; and continual testing and reexamination.

### SCIENTIFIC REVOLUTION:

A period of accelerated scientific discovery that completely reshaped the world. Usuallydated from about 1550 to 1700, the Scientific Revolution saw the origination of the scientific method and the introduction of ideas such as the heliocentric (Sun-centered) universe and gravity.

### TORQUE:

A force that produces, or tends to produce, rotational motion.

### UNIFORM CIRCULAR MOTION:

The motion of an object around the center of a circle in such a manner that speed is constant or unchanging.

### VACUUM:

An area devoid of matter, even air.

### VELOCITY:

Speed in a certain direction.

### WEIGHT:

A measure of the gravitational force on an object. Weight thus would change from planet to planet, whereas mass remains constant throughout the universe. A pound is a unit of weight, whereas a kilogram is a unit of mass.

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