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2 Dynamics of the Planetary System INTRODUCTION Two recent developments have added new impetus to the study of the mo- tions of the planets and their satellites. Planetary radar has added the dimen- sion of distance to measurements previously confined to angular position, and the digital computer has permitted numerical solution of the exact equations of motion with far greater precision and speed than ever before possible. Because the computer has become an integral part of every scientific laboratory, it may not be necessary to stress its central importance to the solution of problems in celestial mechanics: classes of problems can now be undertaken which would otherwise remain inaccessible. The significance of ground-based radar in the study of planetary motions rests on its ability to measure, with high resolution, the power-density distribu- tion in time (delay) and frequency (Doppler) of the planetary echo. From these data the two-way light time to the surface of the planet can be inferred with an error which in some situations can be as small as a few microseconds. For the planet Venus, with which the most precise work has been done, the maximum echo delay is nearly 1,700 sec, yielding measurements that have a fractional precision of a few parts in 10". Using measurements of the Doppler- shifted frequency of the echo, the radial motion of the planet with respect to the radar can, under favorable conditions, be estimated to above 2 mm/sec, for a fractional precision approaching 10~7 near elongation. (For comparison
8 PLANETARY ASTRONOMY it may be noted that a planetary angular determination can approach 0'.'2 arc accuracy, which represents a fractional precision of about 4x 10~7.) A series of echo delay and Doppler measurements in combination with angular posi- tions obtained by traditional optical techniques can be reduced to yield an improvement of several orders of magnitude in the accuracy of determination of some of the orbital parameters. Granted that any improvement in the accuracy with which planetary param- eters may be estimated will be eagerly awaited by students of orbital me- chanics, what significance would it have to the larger scientific community? As planetary orbits, masses, and rotations become known more precisely, theories that purport to explain the dynamical history of the planets and their satellites will also require refinement. Certainly any tests that can verify or disprove the several theories of gravitation will have wide interest. The high accuracy of delay measurements obtainable with radar makes possible a number of such tests, as described below. Improved orbital parameters are basic to the design of accurate space-probe trajectories to the planets and comets. The recent successes of the Mariner series of probes would have been compromised without access to the results of ground-based planetary radar, and the planning of missions to Jupiter and its satellites should similarly benefit from improved determinations of their orbits. In the process of reducing radar measurements of a planet's motion, a number of other characteristics of the planet, such as its mass, radius, and shape, must also be taken into account. In some cases, these by-products may have interest equal with the improvement in the basic orbital parameters. Because the systematic errors associated with radar measurements affect the results in a manner qualitatively different from those accompanying optical observations, the presence of systematic errors in both types of data can be detected more accurately through cross-checking. A few of the ramifications of the new techniques in celestial mechanics which appear to be of exceptional interest have been selected for more detailed discussion. These topics include the testing of gravitational theories and some studies involving the orbits of planets and comets. TESTING OF GRAVITATIONAL THEORIES Since the time of Newton, the solar system has served as a testbed for gravita- tional theory. Three of the classic tests of the general theory of relativity in- volve the solar system, namely, the gravitational deflection of starlight, the
DYNAMICS OF THE PLANETARY SYSTEM 9 additional advance in the perihelion of Mercury, and the red shift of solar spectral lines. A way to more definitive testing of the general theory and its alternatives is now open with the precise measurement of propagation time by planetary radar. In discussing the ways by which gravitational theory may be tested, it is useful to distinguish between (1) gravitational effects on the propagation of electromagnetic radiation and (2) perturbations (as compared with New- tonian theory) introduced into orbital motions in the solar system. The first category comprises not only the classical attempts to measure the apparent bending of starlight that passes near the solar limb but also the more recent proposal to measure the apparent retardation in propagation velocity along a similar path, made possible by the use of radar time-of-flight data. In the second category are included the additional secular advances of planetary perihelia and periodic orbital perturbations. A number of attempts have been made during eclipses to observe the apparent displacement of a star when viewed near the solar limb. Because observing conditions for astrometry near the Sun are far from ideal even during an eclipse, and suitable target stars are often not favorably located, a positional accuracy of not much better than a few tenths of a second of arc can be obtained. Since the effect is typically of the order of about 1 sec of arc, the level of verification is very poor. It has been suggested that interferometric observations at centimeter radio wavelengths of quasi-stellar sources (such as 3C279) might yield a positional accuracy near the Sun considerably better than that achieved optically. Because of the perturbing effects of the coronal plasma at radio frequencies, it will be necessary to obtain measurements at several wavelengths to separate dis- persive plasma effects from nondispersive gravitational effects. Nevertheless, measurements can be made more regularly since eclipse conditions are not required and since optical limitations such as weather and seeing are not present to the same degree. The level of accuracy to which an angular displace- ment test based on radio observation could be pressed is not clear, but it very probably would surpass the present level of optical verification. A second and potentially more precise verification of the predicted effects of gravity on the propagation of electromagnetic waves involves their measur- able retardation near the Sun. A two-way radar ray path connecting Earth and Venus at superior conjunction and grazing the solar limb should show a retardation of about 200 /u,sec in addition to that calculated on the basis of uniform propagation at the speed of light. As the angular distance of the target from the Sun increases, the relativistic contribution to delay diminishes approximately as the logarithm of the inverse angle of elongation. At a dis-
10 PLANETARY ASTRONOMY tance of four solar radii (1°) the effect still amounts to about 150 /*sec and represents a feasible measurement. In the second category, involving dynamical effects on planetary motions, nearly two centuries of optical observation of Mercury has verified the addi- tional secular advance of perihelion to an accuracy of about one percent of the effect (43 sec of arc per century) predicted by the general theory of rela- tivity. Radar observations to an accuracy of 10 /usec, when continued over three years, should reduce the measurement error to less than 0"2 and should provide a useful determination of the magnitude of the effect in the orbits of Earth and Mars. In the general theory, the effect, expressed as perihelion advance per unit time, varies with the orbital semimajor axis, a, as <r5/2. The advance resulting from a quadrupole moment in the solar gravitational field, on the other hand, varies as <r7/2. Given sufficient accuracy and duration of measurement it should be possible to distinguish these two predicted contribu- tions by comparing the values obtained for Mercury, Earth, and Mars and perhaps Venus as well. Also in the second category of dynamical effects are included periodic perturbations of the orbits of the terrestrial planets, which if confirmed may establish the presence of a quadrupole term in the solar gravity. A precise calculation demonstrating a distinction between the effect of these perturba- tions and that of, say, general relativity has not been completed. Estimates, however, seem to place the periodic terms at a magnitude amenable to radar investigation. Improved determinations of planetary orbits may also be applied to a closer examination of the possible time-dependence of the gravitational constant. It is known from observations that are presently available that the time- dependence cannot be greater than a part in 10~9 per year, whereas theory suggests a variation of as much as 1Q-10 per year. With the accuracy in delay measurement improved to the level of a microsecond, it should be possible after several years of radar observation to achieve a sensitivity of the order of 10"11 per year in this variation. Many of the tests discussed in this section conceivably could be performed by an artificial solar satellite, suitably instrumented with radio and perhaps optical transponders. One advantage of the satellite would be our ability to determine its position at any moment with very high precision. Observations over a period of time are required, however, to establish with certainty many of the effects of interest here, and one must be concerned during this interval with vehicle accelerations arising from the unpredictable effects of solar radia- tion pressure, residual gas leakage, and the like. If the probe were in orbit around a planet, however, or better yet, landed on the surface, highly precise measurements of the planetary orbit would be facilitated.
DYNAMICS OF THE PLANETARY SYSTEM 11 OUTER PLANETS, ASTEROIDS, COMETS, AND SELECTED PLANETARY SATELLITES At great distances and for the study of small objects, optical observations have major importance. More optical observations, extended to fainter mag- nitudes, are needed to determine, for example, the positions of the more re- mote major planets, of the smaller and fainter planetary satellites, and of asteroids and comets. Four satellites have been discovered within the last two decades, but fewer than a dozen positions of one of them, Jupiter XII, have been measured since the first determination of its orbit in 1952. All the comets and some of the asteroids discovered each year have orbits of sufficient interest to warrant more than superficial attention. Even so, not a single accurate position was reported for more than a month following the discovery of a recent naked-eye comet. Preliminary computation of its orbit was complicated by the need to adjust a collection of positional observations which by modern standards were of very low precision. Positional observations of planetary satellites are important, first, to deter- mine two-body orbital elements, then, as increased precision is gained through continued observations, to evaluate (1) secular effects arising from deviations from spherical symmetry in the internal density distribution in the planet and (2) gravitational interactions among satellites. The asteroids, which are generally confined to the belt between the orbits of Mars and Jupiter, show striking avoidance, in the detailed distribution of their orbits, of certain simple fractional values of the period of Jupiter and a clustering around other values. The present distribution of asteroid orbits is believed to reflect conditions that prevailed during the formation of the solar system as modified by subsequent interactions of asteroids with each other and with other bodies in the solar system. Extension of precise observations of position to fainter magnitudes and to the longest possible orbital arcs is fundamental to a study of the dynamical evolution of asteroids and comets and to a determination of the place where comets originate. Interested astronomers are concerned over the paucity of current observations, which is sufficiently serious for some objects to risk losing track of them. It seems well established that collisions resulting in progressive fragmenta- tion occur between asteroids. Chemical and physical characteristics of most meteorites suggest that they have undergone processing within bodies of asteroidal size, and the few reliable atmospheric trajectories available are not inconsistent with asteroidal origin of meteorites. The marked variation among meteorites raises questions of whether asteroids have comparable differences and variations in composition and whether
12 PLANETARY ASTRONOMY their physical properties are related to orbital characteristics. The relative importance of collision processes can be evaluated from an analysis of the asteroidal size-frequency distribution. As a basis for such analysis, statistically interpretable physical and orbital data extending to fainter magnitude limits are needed. In addition to the broader questions on physical and dynamical character- istics, certain asteroids are of special interest because of their unusually close approaches to the Sun, to the Earth, to Mars, or to another asteroid. Certain kinds of orbits are of special theoretical interest in celestial mechanics. Among them are those that exemplify the stable special solution of the restricted problem of three bodies and those whose perturbations reinforce through commensurabilities of the mean motion with Jupiter. It is useful to investigate the secular stability of some of these special classes of orbits, since the findings may be related to the capture or loss of planetary satellites. Relatively short- lived asteroid classes, such as those that cross the Earth's orbit, are of interest as the most immediate source of meteorites. The presence of large amounts of volatiles in comets suggests that they are recent arrivals in the inner planetary system that have never spent an appreci- able time in the vicinity of the Sun. Otherwise, most or all of the easily vapor- ized constituents would long since have been lost. The location of the reservoir from which they come, and such evolutionary effects on periodic comets as rates of mass loss, can be investigated by a careful study of comet motions. It must be noted, however, that the observed arc, even when observations are extended to the brightness limit of available telescopes, represents only a very small portion of the complete orbit of a nearly parabolic comet. Observations of even short-period comets have extended over as much as half of the orbit in only two cases. The complex gravitational interactions of major planets with each other and with comets will require careful evaluation as improved determinations of distances, planetary masses, and orbital elements are obtained through refinement of classical techniques and from radar observations. To separate inadequacies in the application of gravitational theories from the effects as- sociated with physical evolution of asteroids or comets is an important and challenging task. Execution of the complex calculations to the required order of precision has become practical only through use of modern high-speed digital computers. Observational-selection effects strongly influence discovery of small asteroids and comets. Asteroid orbits, and orbits of short-period comets, are concen- trated near the plane of the Earth's orbit, while orbits of nearly parabolic comets have, as far as is known, random orientations. The paucity of observ- atories in the Southern Hemisphere militates against discovery of comets
DYNAMICS OF THE PLANETARY SYSTEM 13 having only a small part of their orbits north of the ecliptic plane and against detecting asteroids in orbital longitudes that correspond to opposition during the short nights of the northern summer season. A search program designed to sample volumes of space to a specific magnitude would add important weight to statistical data on sizes and orbit distributions of asteroids and comets. Special efforts would also be usefully directed toward the discovery and observation of asteroids crossing the Earth's orbit.
