the solar wind, although later measurements by Explorer 35 would show that the lunar wake extends only a few lunar radii downstream, instead of to the vicinity of the earth as originally supposed.30 The moon presented the case of a planetary body with very little magnetic field and no atmosphere. Solar wind particles might be expected, then, to strike the lunar surface directly. In the case of Venus, which also has little magnetic field but which has an atmosphere perhaps 100 times that of Earth, the solar wind would impinge on the top of the atmosphere but would not be able to reach the planet's surface. Mars would present the case of a planet with little magnetic field and an atmosphere about one percent that of Earth. Jupiter, on the other hand, with its very strong magnetic field would have a huge magnetosphere. If radio bursts that were observed to come from Jupiter were from trapped particles, the Jupiter radiation belt would prove much more intense than Earth's. At the end of 1964 these were principally ideas for future research. Knowledge of Earth's magnetosphere invested that future research with considerable promise.

SATELLITE GEODESY

Satellite geodesy also made a substantial contribution to the deepening perspective in which men could view their own planet. But the new perspective differed in an interesting way from that provided in magnetospheric physics. For the latter, rockets and satellites revealed a wide range of hitherto unknown phenomena. In contrast the subject matter and problems of geodesy were well known; it was increased precision, the ability to measure higher order effects, and the means for constructing a single global reference system that space methods helped to provide.

Geodesy may be divided into two areas: geometrical geodesy and dynamical, or physical geodesy. The former seeks by geometrical and astronomical measurements to determine the precise size and shape of the earth and to locate positions accurately on the earth's surface. The latter is the study of the gravitational field of the earth and its relationship to the solid structure of the planet. As will be seen, geometrical and physical geodesy are intimately related.

Geodesy offers many practical values. Accurate maps of the earth's surface depend on a knowledge of both size and shape. Into the 20th century the requirements for precision were rather modest. Individual countries could choose their own reference systems and control points and, using geodetic measurements made within their own territories, produce maps of sufficient accuracy for national purposes. The appreciable differences among the various geodetic systems did not appear to matter. As late as 1947, disagreements among Danish, Swedish, German, Norwegian, French, and English systems ranged from 95 meters to 250 meters, while in the absence of adequate surveys, errors between the various continents and

ocean islands could be a kilometer or more.31 For demands in the mid-20th century, the most obvious being those of air and marine navigation and missilery, such errors could at times appear enormous, and there was a growing agitation among geodesists to generate a world geodetic system that would use a common reference frame and tie all nets around the world into a single system. At this point the artificial satellite appeared on the scene and was able to provide some help. To understand how the satellite could contribute, a few basic concepts are needed.

The science of geodesy began when the Greek Eratosthenes (c. 276-c. 192 B.C.), believing the earth to be spherical, combined astronomical observation with land measurement to estimate the size of the globe (fig. 36).32 He had learned (actually incorrectly) that at noon in midsummer the sun shown vertically down into a well in Syene (now Aswan). Observing that at the same time the sun as seen from Alexandria was 7.2° south of the zenith, he concluded that the arc along the earth connecting Syene and Alexandria had to subtend an angle of 7.2° at the earth's center. The arc accordingly had to be 7.2/360 or 1/50 of a total meridian circle. He was told that a camel caravan took 50 days to travel from Alexandria to Syene. Using a reasonable camel speed he deduced a length for the arc, which multiplied by 50 gave him a rough estimate (16% too large) of the length of a whole meridian circle. Such estimates of the earth's dimensions improved over the centuries as different persons used better measurements, and eventually better techniques.

Concerning techniques, the next major step in geodesy came when Tycho Brahe conceived of the method of triangulation, which was developed into a science by Willebrord Snell. In this technique (fig. 37) the points A and P, between which the distance is to be determined, are connected by a series of interlinking triangles. The length of one side of one of the triangles that is convenient to measure-say the side AB of the first triangle is then measured to a high degree of accuracy. One then measures the angles of the first triangle, which can be done with precision much more easily than measuring length. Using the law of sines, the initial side of the next triangle down the chain can then be calculated. The process can be repeated to get the length of the initial side of the third triangle of the chain. Moving step by step from triangle to triangle, one finally gets to the last triangle, of which P is a vertex. With the lengths of all the sides of the triangles known, it is then possible to compute the distance between A and P along the terrestrial sphere. For great distances one must, of course, introduce appropriate corrections to take into account that the sum of the angles of a triangle on a sphere is greater than 180°. With care a high degree of accuracy can be achieved. By using nets of triangles one can proceed outward along one chain to the selected point P, and back along a different chain to calculate the measured baseline AB. If the calculated value of AB is sufficiently close to the originally measured

Figure 37. Triangulation. This technique provided a step-by-step method of accurately determining the distance between widely separated points on the earth's surface.

value, the confidence in the calculated value of AP can be high. Jean Picard (1620-1682) employed this technique in obtaining the value of the earth's radius that Isaac Newton used in deriving his law of gravitation.

The period from Eratosthenes to Picard has been referred to as the spherical era of geodesy. During that time it was assumed that the earth was a sphere. This made the geodetic problem quite simple, for one had only to determine the radius of the terrestrial sphere, and the rest came out of simple geometry (spherical trigonometry). But in the 17th century it became clear that the earth was not spherical. From this period on the earth was visualized as essentially an ellipsoid of revolution, with its major axis in the equatorial plane and minor axis along the earth's axis of rotation. The bulge in the equatorial plane could be explained as due to centrifugal forces from the earth's rotation. Thus, the 18th and 19th centuries could be thought of as the ellipsoidal period of geodesy, and a prime task was to find the ellipsoid of proper size and flattening to best represent the earth. By the mid-20th century the equatorial radius of the reference ellipsoid had been determined as 6378 388 meters, while the flattening— that is, the ratio of the difference between the equatorial and polar radii to

the equatorial radius-was put as 1/297.33 The tasks of modern geodesy grew out of this historical background.

Those seeking a single geodetic net for the world had to agree on a suitable reference frame. It is natural to take this as a rectangular coordinate system with origin at the earth's center and three mutually perpendicular axes, one along the earth's rotational axis and the other two in the equatorial plane. Alternatively one could use spherical polar coordinates locating a point by its distance from the origin of coordinates, and its latitude and longitude. In principle all measurements and calculations could be made in terms of these coordinates without any intermediate reference. To visualize the geometry, however, a reference surface approximating the actual surface of the earth is helpful. The most useful reference surface should satisfy two important criteria. First, it should be of such a size and shape that all locations on the earth are close to the reference. Secondly, the surface should be one on which calculations of positions, angles, and distances are mathematically simple. A sphere would satisfy the second criterion very nicely, since one could use the ordinary spherical trigonometry of air and marine navigation. But for any chosen sphere, many locations on earth would be unacceptably far from the reference. By flattening the sphere at the poles, however, to produce an oblate ellipsoid of revolution, both criteria can be met. For calculations the methods of analytic geometry can be used, and an ellipsoid of the equatorial radius and flattening given in the preceding paragraph-6378 388 meters and 1/297 respectively-provides a good first order approximation to the actual size and shape of the earth. This ellipsoid of revolution, with center at the origin of coordinates, was often used as reference ellipsoid before the advent of satellites. As will be seen later, satellite geodesy provided an improved estimate of the size and flattening of this reference ellipsoid.

By furnishing the means of accurately positioning different sites and features on the earth, geometrical geodesy provides essential data for map makers, the fixing of political boundaries, civil engineering, and military targeting. But, the data also raise numerous scientific questions, such as why various features are where they are and what forces cause observed irregularities in the shape of the earth. Dynamical geodesy addresses itself to such questions.

Among the factors affecting the shape of the earth are the distribution of matter in the crust and mantle, centrifugal forces due to the earth's rotation, and gravity. The dominant factor is gravity, and an investigation of the earth's gravitational field lies at the heart of dynamical geodesy. To understand why, the concepts of geoid and spherical harmonics will be helpful.

First the geoid. To start, consider a simplified case. Suppose the earth to be perfectly homogeneous, plastic, and nonrotating. Then the earth would assume the shape of a perfect sphere (fig. 38). More significantly,

level surfaces around the earth would also be perfect spheres. By level surface is meant a surface to which the force of gravity is perpendicular everywhere. At any point on the surface the bubble of a spirit level held tangent to the surface would be centered. A pool of water on a level surface would experience no sidewise, or "downhill," gravity forces urging the water to flow (and were it not for internal pressures in the water and adhesion to the material of the surface, the pool would stay where it was).

The level surface that coincides with the actual surface of the earth is called the geoid. In the idealized case treated here, the geoid is a perfect sphere.

Now suppose a homogeneous, plastic earth rotates around a fixed axis. In this case the centrifugal forces of the rotation combine with gravity lessening the gravity and causing a bulge at the equator and producing a flattening at the poles (fig. 39). The earth's figure becomes that of an oblate ellipsoid of revolution, the surface of which is level. If the surface were not level, sideways forces on the plastic material would keep the material flowing until those forces were reduced to zero. Thus, for this case, the geoid is the ellipsoid of revolution comprising the earth's surface.

Next, return to the nonrotating earth, but this time suppose that near the surface is a large mass of material much denser than the rest of the earth (fig. 40). In this case, near the dense mass the level surfaces are no longer spherical. For, if one imagines holding a spirit level near the intruding mass, its extra gravitational pull draws the fluid of the level toward the mass thus forcing the bubble away. To counter this effect the end of the level nearer the mass must be tipped up to recenter the bubble. In other words, the level surface tips upward as one approaches the mass,

Level Surfaces

Figure 38. The geoid in the case of a homogeneous, plastic, nonrotating earth. For the idealized case depicted here, the geoid is a perfect sphere.