Acceleration around Earth, the Moon, and other planets
- Also called:
- gravitation
- Key People:
- Kip Thorne
- Isaac Newton
- Galileo
- John Archibald Wheeler
- Simon Stevin
The value of the attraction of gravity or of the potential is determined by the distribution of matter within Earth or some other celestial body. In turn, as seen above, the distribution of matter determines the shape of the surface on which the potential is constant. Measurements of gravity and the potential are thus essential both to geodesy, which is the study of the shape of Earth, and to geophysics, the study of its internal structure. For geodesy and global geophysics, it is best to measure the potential from the orbits of artificial satellites. Surface measurements of gravity are best for local geophysics, which deals with the structure of mountains and oceans and the search for minerals.
Variations in g
Changes due to location
The acceleration g varies by about 1/2 of 1 percent with position on Earth’s surface, from about 9.78 meters per second per second at the Equator to approximately 9.83 meters per second per second at the poles. In addition to this broad-scale variation, local variations of a few parts in 106 or smaller are caused by variations in the density of Earth’s crust as well as height above sea level.
Changes with time
The gravitational potential at the surface of Earth is due mainly to the mass and rotation of Earth, but there are also small contributions from the distant Sun and Moon. As Earth rotates, those small contributions at any one place vary with time, and so the local value of g varies slightly. Those are the diurnal and semidiurnal tidal variations. For most purposes it is necessary to know only the variation of gravity with time at a fixed place or the changes of gravity from place to place; then the tidal variation can be removed. Accordingly, almost all gravity measurements are relative measurements of the differences from place to place or from time to time.
Measurements of g
Unit of gravity
Because gravity changes are far less than 1 meter per second per second, it is convenient to have a smaller unit for relative measurements. The gal (named after Galileo) has been adopted for this purpose; a gal is one-hundredth of a meter per second per second. The unit most commonly used is the milligal, which equals 10−5 meter per second per second—i.e., about one-millionth of the average value of g.
Absolute measurements
Two basic ways of making absolute measurements of gravity have been devised: timing the free fall of an object and timing the motion under gravity of a body constrained in some way, almost always as a pendulum. In 1817 the English physicist Henry Kater, building on the work of the German astronomer Friedrich Wilhelm Bessel, was the first to use a reversible pendulum to make absolute measurements of g. If the periods of swing of a rigid pendulum about two alternative points of support are the same, then the separation of those two points is equal to the length of the equivalent simple pendulum of the same period. By careful construction, Kater was able to measure the separation very accurately. The so-called reversible pendulum was used for absolute measurements of gravity from Kater’s day until the 1950s. Since that time, electronic instruments have enabled investigators to measure with high precision the half-second time of free fall of a body (from rest) through one meter. It is also possible to make extremely accurate measurements of position by using interference of light. Consequently, direct measurements of free fall have replaced the pendulum for absolute measurements of gravity.
Nowadays, lasers are the sources of light for interferometers, while the falling object is a retroreflector that returns a beam of light back upon itself. The falling object can be timed in simple downward motion, or it can be projected upward and timed over the upward and downward path. Transportable versions of such apparatuses have been used in different locations to establish a basis for measuring differences of gravity over the entire Earth. The accuracy attainable is about one part in 108.
More recently, interferometers using beams of atoms instead of light have given absolute determinations of gravity. Interference takes place between atoms that have been subject to different gravitational potentials and so have different energies and wavelengths. The results are comparable to those from bodies in free fall.
Relative measurements
From the time of Newton, measurements of differences of gravity (strictly, the ratios of values of gravity) were made by timing the same pendulum at different places. During the 1930s, however, static gravimeters replaced pendulums for local measurements over small ranges of gravity. Today, free-fall measurements have rendered the pendulum obsolete for all purposes.
Spring gravimeters balance the force of gravity on a mass in the gravity field to be measured against the elastic force of the spring. Either the extension of the spring is measured, or a servo system restores it to a constant amount. High sensitivity is achieved through electronic or mechanical means. If a thin wire is stretched by a mass hung from it, the tension in the wire, and therefore the frequency of transverse oscillations, will vary with the force of gravity upon the mass. Such vibrating string gravimeters were originally developed for use in submarines and were later employed by the Apollo 17 astronauts on the Moon to conduct a gravity survey of their landing site. Another relatively recent development is the superconducting gravimeter, an instrument in which the position of a magnetically levitated superconducting sphere is sensed to provide a measure of g. Modern gravimeters may have sensitivities better than 0.005 milligal, the standard deviation of observations in exploration surveys being of the order of 0.01–0.02 milligal.
Differences in gravity measured with gravimeters are obtained in quite arbitrary units—divisions on a graduated dial, for example. The relation between these units and milligals can be determined only by reading the instrument at a number of points where g is known as a result of absolute or relative pendulum measurements. Further, because an instrument will not have a completely linear response, known points must cover the entire range of gravity over which the gravimeter is to be used.
Since g is an acceleration, the problem of its measurement from a vehicle that is moving, and therefore accelerating relative to Earth, raises a number of fundamental problems. Pendulum, vibrating-string, and spring-gravimeter observations have been made from submarines; using gyrostabilized platforms, relative gravity measurements with accuracies approaching a few milligals have been and are being made from surface ships. Experimental measurements with various gravity sensors on fixed-wing aircraft as well as on helicopters have been carried out.
Gravimetric surveys and geophysics
As a result of combining all available absolute and relative measurements, it is now possible to obtain the most probable gravity values at a large number of sites to high accuracy. The culmination of gravimetric work begun in the 1960s has been a worldwide gravity reference system having an accuracy of at least one part in 107 (0.1 milligal or better).
The value of gravity measured at the terrestrial surface is the result of a combination of factors:
- The gravitational attraction of Earth as a whole
- Centrifugal force caused by Earth’s rotation
- Elevation
- Unbalanced attractions caused by surface topography
- Tidal variations
- Unbalanced attractions caused by irregularities in underground density distributions
Most geophysical surveys are aimed at separating out the last of these in order to interpret the geologic structure. It is therefore necessary to make proper allowance for the other factors. The first two factors imply a variation of gravity with latitude that can be calculated for an assumed shape for Earth. The third factor, which is the decrease in gravity with elevation, due to increased distance from the center of Earth, amounts to −0.3086 milligal per meter. This value, however, assumes that material of zero density occupies the whole space between the point of observation and sea level, and it is therefore termed the free-air correction factor. In practice the mass of rock material that occupies part or all of this space must be considered. In an area where the topography is reasonably flat, this is usually calculated by assuming the presence of an infinite slab of thickness equal to the height of the station h and having an appropriate density σ; its value is +0.04185 σh milligal per meter. This is commonly called the Bouguer correction factor.
Terrain or topographical corrections also can be applied to allow for the attractions due to surface relief if the densities of surface rocks are known. Tidal effects (the amplitudes are less than 0.3 milligal) can be calculated and allowed for.
James E. Faller Alan H. CookThe Moon and the planets
Although the Apollo astronauts used a gravimeter at their lunar landing site, most scientific knowledge about the gravitational attractions of the Moon and the planets has been derived from observations of their effects upon the accelerations of spacecraft in orbit around or passing close to them. Radio tracking makes it possible to determine the accelerations of spacecraft very accurately, and the results can be expressed either as terms in a series of spherical harmonics or as the variation of gravity over the surface. As in the case of Earth, spherical harmonics are more effective for studying gross structure, while the variation of gravity is more useful for local features. Spacecraft must descend close to the surface or remain in orbit for extended periods in order to detect local gravity variations; such data had been obtained for the Moon, Venus, Mars, and Jupiter by the end of the 20th century.
The Moon’s polar flattening is much less than that of Earth, while its equator is far more elliptical. There are also large, more-local irregularities from visible and concealed structures. Mars also exhibits some large local variations, while the equatorial bulges of Mercury and Venus are very slight.
By contrast, the major planets, all of which rotate quite fast, have large equatorial bulges, and their gravity is dominated by a large increase from equator to pole. The polar flattening of Jupiter is about 10 percent and was first estimated from telescopic observation by Gian Domenico Cassini about 1664. As mentioned above, Edmond Halley subsequently realized that the corresponding effect on gravity would perturb the orbits of the satellites of Jupiter (those discovered by Galileo). The results of gravity measurements are crucial to understanding the internal properties of the planets.
Gravitational theory and other aspects of physical theory
The Newtonian theory of gravity is based on an assumed force acting between all pairs of bodies—i.e., an action at a distance. When a mass moves, the force acting on other masses had been considered to adjust instantaneously to the new location of the displaced mass. That, however, is inconsistent with special relativity, which is based on the axiom that all knowledge of distant events comes from electromagnetic signals. Physical quantities have to be defined in such a way that certain combinations of them—in particular, distance, time, mass, and momentum—are independent of choice of space-time coordinates. This theory, with the field theory of electrical and magnetic phenomena, has met such empirical success that most modern gravitational theories are constructed as field theories consistent with the principles of special relativity. In a field theory the gravitational force between bodies is formed by a two-step process: (1) One body produces a gravitational field that permeates all surrounding space but has weaker strength farther from its source. A second body in that space is then acted upon by this field and experiences a force. (2) The Newtonian force of reaction is then viewed as the response of the first body to the gravitational field produced by the second body, there being at all points in space a superposition of gravitational fields due to all the bodies in it.
Field theories of gravitation
In the 1970s the physicists Abdus Salam of Pakistan and Steven Weinberg and Sheldon L. Glashow of the United States were able to show that the electromagnetic forces and the weak force responsible for beta decay were different manifestations of the same basic interaction. That was the first successful unified field theory. Physicists are actively seeking other possible unified combinations. The possibility that gravitation might be linked with the other forces of nature in a unified theory of forces greatly increased interest in gravitational field theories during the 1970s and ’80s. Because the gravitational force is exceedingly weak compared with all others and because it seems to be independent of all physical properties except mass, the unification of gravitation with the other forces remains the most difficult to achieve. That challenge has provided a tremendous impetus to experimental investigations to determine whether there may be some failure of the apparent independence.
The prime example of a field theory is Einstein’s general relativity, according to which the acceleration due to gravity is a purely geometric consequence of the properties of space-time in the neighborhood of attracting masses. (As will be seen below, general relativity makes certain specific predictions that are borne out well by observation.) In a whole class of more-general theories, these and other effects not predicted by simple Newtonian theory are characterized by free parameters; such formulations are called parameterized post-Newtonian (PPN) theories. There is now considerable experimental and observational evidence for limits to the parameters. So far, no deviation from general relativity has been demonstrated convincingly.
Field theories of gravity predict specific corrections to the Newtonian force law, the corrections taking two basic forms: (1) When matter is in motion, additional gravitational fields (analogous to the magnetic fields produced by moving electric charges) are produced; also, moving bodies interact with gravitational fields in a motion-dependent way. (2) Unlike electromagnetic field theory, in which two or more electric or magnetic fields superimpose by simple addition to give the total fields, in gravitational field theory nonlinear fields proportional to the second and higher powers of the source masses are generated, and gravitational fields proportional to the products of different masses are created. Gravitational fields themselves become sources for additional gravitational fields. Examples of some of these effects are shown below. The acceleration A of a moving particle of negligible mass that interacts with a mass M, which is at rest, is given in the following formula, derived from Einstein’s gravitational theory.
The expression for A now has, as well as the Newtonian expression from equation (1), further terms in higher powers of GM/R2—that is, in G2M2/R4. As elsewhere, V is the particle’s velocity vector, A is its acceleration vector, R is the vector from the mass M, and c is the speed of light. When written out, the sum is 
This expression gives only the first post-Newtonian corrections; terms of higher power in 1/c are neglected. For planetary motion in the solar system, the 1/c2 terms are smaller than Newton’s acceleration term by at least a factor of 10−8, but some of the consequences of these correction terms are measurable and important tests of Einstein’s theory. It should be pointed out that prediction of new observable gravitational effects requires particular care; Einstein’s pioneer work in gravity has shown that gravitational fields affect the basic measuring instruments of experimental physics—clocks, rulers, light rays—with which any experimental result in physics is established. Some of these effects are listed below:
- In the presence of gravitational fields, the spatial structure of physical objects is no longer describable precisely by Euclidean geometry; for example, in the arrangement of three rigid rulers to form a triangle, the sum of the subtended angles will not equal 180°. A more-general type of geometry, Riemannian geometry, seems required to describe the spatial structure of matter in the presence of gravitational fields.
- Light rays do not travel in straight lines, the rays being deflected by gravitational fields. To distant observers the light-propagation speed is observed to be reduced near massive bodies.
