Beyond the Atmosphere:
Early Years of Space Science
CHAPTER 11
SATELLITE GEODESY
[186] 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 [187] 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 mid-summer 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....
[188]
Figure 36. Eratosthenes' method for measuring the circumference
of the earth.
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 6 378 388
meters, while the flattening-that is, the ratio of the difference
between the equatorial and polar radii to [189] 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-6,378,388 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,
[190] 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,....
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.
[191]
Figure 39. The geoid in the case of a homogeneous, plastic,
rotating earth. In this case, also an idealized one, the geoid is
a perfect oblate ellipsoid of revolution.
....thus forming a bulge. In a similar
vein, near a mass deficiency, the level surface would show a
depression.
For a rotating, nonhomogeneous earth, the
same reasoning applies. Mass concentrations in the crust produce
bulges in the geoid while mass deficiencies create depressions
(fig.
41). Thus, the actual geoid, which
for the real earth is defined as the level surface that over the
oceans coincides with mean sea level, furnishes a good means of
visualizing variations in the structure of the earth. Since mass
and gravity go together, these structural...
Figure 40. The geoid in the case of a nonhomogeneous,
nonrotating earth. Variations in the earth's density generate
bulges and depressions in the geoid.
[192]
Figure 41. The actual geoid. The real earth's geoid has
numerous bulges and depressions, revealing by their existence
appreciable variations in the material of the planet.
....features are revealed in their
influence on the earth's gravitational field, intensifying the
field near mass excesses and weakening it near mass deficiencies.
These perturbations in the earth's field produce corresponding
perturbations in the orbits of satellites which revolve under the
influence of gravity. A precise analysis of these orbital
perturbations can yield the features of the field. From these one
finally gets back to the geoid and the earth's structure.
Mathematically the earth's gravitational
field can be derived by calculus from what is called the
geopotential function psi [Greek letter]. Physically the surfaces
over which psi [Greek letter] is constant are the level surfaces
discussed earlier in defining the geoid. Thus the makeup of the
geoid and that of the earth's surface geopotential are
identical.
The geopotential function can be expressed
as the sum of an infinite number of terms (in general). Because
these terms can be expressed in sines and cosines of latitude and
longitude, they are referred to as spherical harmonics, by analogy
with the harmonic analysis of a vibrating string where sines and
cosines of the various multiples of the fundamental frequency of
the string are used.^{*} The amount of a specific harmonic in the expansion
of the geopotential is given by a coefficient J_{nm}. The most
general harmonics correspond to distortions of the geoid in both
latitude and longitude. Some, called sectorial harmonics,
reveal major distortions in longitude for example, an ellipticity
in the earth's equator. Of special importance are the zonal
harmonics, which correspond to coefficients J_{nm} for which
m =
0 and which depend only on latitude. The second zonal harmonic
corresponds to the earth's equatorial bulge caused by the earth's
rotation. The third zonal harmonic, if present in the expansion of
the geopotential, [193] would add a
pear-shaped component to the earth's figure, elevating the geoid
at one pole and depressing it at the other.
It was in regard to the reference
ellipsoid and the coefficients in the spherical harmonic expansion
of the earth's potential that satellite measurements could aid the
geodesist. The most straightforward contribution was to provide a
sighting point in the sky that could be used to make direct
connections between remotely separated points of the earth,
supplementing the method of triangulation along the earth's
surface. For this purpose simultaneous sightings of a satellite
from two widely separated points were most useful. But once the
orbit of a satellite had been accurately determined, simultaneous
sightings were not required. One could relate separate sightings
by computing the time and distance along the orbit from one
sighting to the other, and again proceed to compute the distance
between the two observing stations on the earth. By this latter
method continental and transoceanic distances could be spanned,
clearly a powerful aid in tying together different geodetic nets
of the world.
The second major contribution that
satellites could make was to help determine the different harmonic
components of the earth's gravitational field, or alternately of
the earth's gravitational potential. The orbit of a satellite is
determined completely by its initial position and velocity and the
forces operating on it. These forces include the gravitational
influences of the earth, sun, and moon; atmospheric drag; solar
radiation pressures; and self-generated disturbances such as those
caused by gases escaping from the interior of the satellite. For a
satellite near the earth yet well out of the appreciable
atmosphere, the earth's gravity controls the orbit, the other
effects amounting to corrections that have to be taken into
account. As for the earth's field, Newton's inverse square law
term constrains the satellite to an essentially elliptical orbit.
But higher order terms also have their effects. The second order
zonal harmonic or equatorial bulge causes the plane of the
satellite's orbit and the line joining apogee and perigee to
rotate in space. Still higher harmonics produce slight undulations
in the satellite's orbit, which can be measured and analyzed to
determine which harmonics, and how much of each, are producing the
observed effects.
The application was simple in principle,
but mathematically very complicated. Satellite orbits and their
perturbations were directly related to the geoid, while the
positions of the tracking stations and geodetic nets were tied to
the reference ellipsoid, and a major objective was to improve the
quantitative definitions of both geoid and ellipsoid. Because of
the complexities, the modern computer was required to take
advantage of the satellite opportunities. But with the computer
the complexities and important results were quickly sorted
out.
Some of the earliest came from the first
Sputnik and Vanguard satellites. Using Sputnik 2, E. Buchar
of Czechoslovakia was able to make an estimate of the earth's
flattening. From the measured rate of precession of
[194] the satellite's orbit, which could be related
mathematically to the flattening, Buchar obtained the value
f = 1/(297.90±0.18) (12)
which is to be compared to the previously
accepted value of 1/297. From a more extensive analysis of the
Vanguard 1 satellite, workers at the U.S. Army Map Service
obtained
f = 1/(298.38±0.07) (13)
while a U.S. Naval Research Laboratory
group got virtually the same answer.^{34}
While the measured flattening of the earth
was smaller than that which had been in use, it was significantly
greater than that which would exist in a plastic earth rotating at
the present angular velocity of the earth, namely
1/300.^{35} The implication was that the earth's mantle was not
perfectly plastic. For a perfectly plastic earth, the flattening
indicated by the satellite measurements would correspond to an
earlier, faster angular velocity of earth. Instead, changes in the
earth's equatorial bulge lag by a substantial period-tens of
millions of years-behind the changes in the centrifugal forces
producing the bulge.^{36}
Vanguard 1 data also showed that the eccentricity of the
satellite's orbit varied by 0.00042 ± 0.00003 with a period
of 80 days. John O'Keefe and his colleagues concluded that this
variation had to be caused mostly by the third harmonic in the
earth's gravitational field. The distortion corresponding to this
harmonic was very slight, amounting to only about 20 meters
elevation of the geoid at the north pole and an equivalent
depression at the south pole-widely described in the newspapers as
the earth's "pear-shaped" component-but was significant in that it
might imply a considerable strength in the earth's interior.
O'Keefe and his colleagues estimated that a crustal load of 2 x
10^{7} dynes/cm^{2} (2 x
10^{6} n/m^{2}) was implied by
their findings, producing stresses which they said "must be
supported by a mechanical strength larger than that usually
assumed for the interior of the earth or by large-scale convection
currents in the mantle."^{37}
It was possible by a detailed analysis of
the orbital perturbations to derive a chart of the departures of
the geoid above and below the reference ellipsoid, a chart which
could suggest a great deal about the distribution of mass within
the earth's crust. Using observations from five different
satellites, William Kaula of Goddard Space Flight Center produced
the chart shown in figure 42.^{38} The positive numbers give elevations of the geoid
in meters above the reference ellipsoid, which was taken to have
the flattening of 1/298.24 indicated by satellite measurements.
The negative numbers give depressions of the geoid below the
reference ellipsoid. As an example of what one can read from such
a chart, the elevations and depressions of the geoid shown in the
equatorial belt strongly suggest that the earth's equator is not a
circle, but an ellipse. This is consistent with an analysis by....
[195]
Figure 42. The geoid as
revealed by satellite measurements.
[196] ....C. A. Wagner
of data from the communications satellite Syncom 2 in
synchronous orbit over the earth's equator.^{39} Wagner found a difference of 130 ± 4 meters
between the major and minor equatorial diameters, with one end of
the major diameter at 19° ± 6° west of
Greenwich.
While many workers were concerning
themselves with physical, or dynamical, geodesy, others were
working on the problems of mensuration and mapping. Using the
large Echo 1 satellite, French geodesists undertook to check the
tie between the French geodetic net and that of North
Africa.^{40} With radio tracking techniques, the Applied Physics
Laboratory of the Johns Hopkins University determined distances
between stations at various points from the east coast to the west
coast of the United States. From comparison of distances measured
by ground surveys and those determined by satellite techniques,
the APL workers concluded that, with a modest number of satellite
passes and four or more observing stations, relative positions of
ground stations separated by continental distances could be
obtained with a confidence level of about 10
meters.^{41} Using simultaneous photographic observations of the
Anna 1B geodetic satellite taken from different stations,
Air Force workers measured distances between stations separated by
about 1000 km with good accuracy-better than 10 m. They concluded
that their geodetic stellar camera system was "operationally
capable of extending geodetic control to proportional accuracy of
better than 1/100,000 when cameras in a network simultaneously
observe a flashing satellite beacon."^{42}
^{ }
^{
*} More
completely, like the sines and cosines, the basic functions out of
which spherical harmonics are constructed form what mathematicians
call an orthogonal set.