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planets themselves, and extended series of observations of the Sun and the principal planets, have been available for a long time and have indeed been used for the determination of improved masses on the basis of their mutual attractions. The attainable accuracy has been rather limited, however, because until recently the degree of precision of the orbital theories of some of the major planets was not very high, and the comparative smallness of the perturbations involved magnified the relative seriousness of orbital theory defects in the resulting mass corrections. Also, on the observational side, the attainable accuracy of positions of the Sun and major planets is severely hampered by the difficulties inherent in ascertaining the observed coordinates of the centers of such disk-shaped and more or less diffuse objects. Asteroids, on the other hand, can be observed accurately and easily because of their lightpoint images, at least if they are relatively bright. As to their orbits, it is frequently possible to select minor planets that are strongly perturbed by one or several of the major planets, so that optimum conditions exist for the desired mass determinations, provided that well-distributed observations can be obtained. This proviso is important because corrections to the orbital elements of the asteroid in question must be determined simultaneously with the sought-for mass corrections. If the asteroid's heliocentric orbit remains poorly determined, for instance, because it can be observed only near perihelion, then the uncertainties of the orbital elements will unavoidably affect also those of the intimately related mass corrections. An additional and rather important advantage of the use of asteroids arises from the fact that their own masses are completely negligible, compared to those of the major planets. When dealing with the variations produced in the orbit of an asteroid by the sought-for correction to the mass of a given major planet, one does not have to consider any second-order effects of these orbital changes on the motion of the disturbing or any other major planet, nor does one have to determine or correct the mass of the asteroid. Furthermore, one needs to analyze only the observations of the asteroid, regardless of the number of major planets whose mass corrections are introduced into the observation equations. In contrast to these simplifications, the O - C of any major planet cannot normally be treated separately and independently from those of all the others whose orbits may be affected by the mass corrections involved in the solution. At least when considering substantial mass corrections, one also has to investigate the magnitude of any possibly noticeable second-order effects produced by the first-order orbital variations. It is true that for an asteroid with very large perturbations, a second iteration to the solution may be necessary if the orbital changes produced by the initial correction of the disturbing mass are also substantial, but this additional procedure is still limited to the same set of observations of just one object, even if several planetary masses are being corrected simultaneously. For the reasons mentioned in the preceding two paragraphs, mass determinations using asteroid observations have long been considered to be more accurate and also more convenient than those using observations of major planets and of the Sun. Recently, though, the availability of radar observations of Venus and Mercury has made it possible to increase greatly the accuracy with which the motions and masses of these planets and of Earth can be determined from solutions based on a combination of radar and optical observations of these inner planets and on rigorous numerical integrations of their motions. An even more striking increase in accuracy became apparent when the radio tracking data of the Mariner 2 and 4 space probes were analyzed to determine improved mass values for Venus and Mars, respectively. Obviously the high accuracy with which the very large perturbations produced by the close approaches can be observed by means of Doppler tracking data is superior to the accuracy obtainable from optical observations of even the most favorable asteroid orbits. In passing, it should be noted that the observed orbits of natural satellites can also be used, together with Kepler's third law, to determine the mass of the primary, but that in practice the observational difficulties have tended to limit the attainable accuracy rather severely.


So far only mass variations have been mentioned as affecting the observable motion of an asteroid, aside from any necessary corrections to its orbital elements. Because all observations are made from the moving Earth, any thorough analysis of the O - C of asteroids approaching relatively close to Earth has to consider also the possible need for correcting some of the elements of Earth's orbit. In this connection, it should be noted again that corrections to the masses of disturbing planets will affect not only the motion of the asteroid under consideration but also the motion of the Earth-Moon barycenter. Consequently, the observation equation coefficients providing for such effects on the O - C may have to be augmented by the relevant (normally much smaller) effects due to the adjusted perturbations of the Earth+Moon orbit. The basic elements for which corrections may be necessary are the mean longitude or mean anomaly at some zero epoch, the longitude of perihelion, the orbital eccentricity, and the obliquity (inclination) of the ecliptic relative to the equator. The mean motion or the semimajor axis, on the other hand, is known much more accurately from long series of observations of the Sun, while the longitude of the node on the equator is intimately connected with the basic definition of the fundamental reference system and thus with the effects of precession. The constants related to the reference frame will be considered in another section. Obviously, the four element corrections to be considered for the orbit of the Earth-Moon barycenter are easily introduced into the observation equations by the same principles as those to the elements of the asteroid orbit. It is evident that especially asteroids of the Eros type will be well suited for actual determinations of such corrections because of the magnification of their effects on the computed positions during all close approaches to Earth.

The orbital elements just considered are those of the Earth-Moon barycenter, while Earth itself moves about this barycenter in accordance with the Moon's orbital revolutions around Earth. Consequently, the geocentric position of any asteroid is also a function of the so-called constant of the lunar equation, which is the coefficient of the periodic displacement of an object at a distance of 1 AU in the plane of the Moon's orbit, caused by the motion of Earth's center about the barycenter (with the Earth-Moon distance equal to its mean value). This constant L can therefore also be determined from asteroid observations, in particular from O - C of asteroids like Eros or Amor during close approaches. L in turn is related to the Moon/Earth mass ratio p through an equation involving also the parallaxes of Sun and Moon. Because these parallaxes were supposed to be known more accurately than pu, many determinations of p have been made by deriving L from close-approach residuals of minor planets and then calculating pi from this equation. Today, however, p is one of the primary constants in the newly adopted IAU system of fundamental constants, essentially because of its more accurate and more direct determination from radar observations and space probes, whereas L is now a derived constant. It still enters the computation of geocentric ephemeris positions of planets and asteroids, but it is pointless to try to improve it from observed asteroid residuals in right ascension and declination.


A similarly reversed situation exists today with regard to the solar parallax m, and the astronomical unit, which are related through the definition of tre as the angle subtended by Earth's equatorial radius Re at a distance of 1 AU. (The mean distance of the Earth-Moon barycenter from the Sun does not equal 1 AU, but differs from it by a very small and well-defined amount.) Because asteroids are observed from locations on the surface of Earth, and not from its center, the resulting parallactic displacements on the sky are inversely proportional to the geocentric distance, and thus they increase again with the object's approach to Earth. Consequently, asteroids like Eros could be and have been used to determine tre, in this fashion, by the “trigonometric method.” The astronomical unit, expressed in meters, could then be calculated from it, and the known value of Re. Today, however, radar observations of Venus are used, for instance, to determine its varying distance from Earth in meters (actually in light-seconds, converted into meters by means of the rather accurately known velocity of light). Because the interplanetary distances are well known in astronomical units, the comparison yields a relatively direct determination of the astronomical unit, and to becomes a derived constant.

A very important relation between the mass m, t c of the Earth-Moon barycenter, the solar parallax tre, and the Moon/Earth mass ratio u results from the combination of two equations: the first one governing the acceleration of gravity at the distance Re from Earth's center and the second one representing Kepler's third law for a particle moving around the Sun in circular orbit at a distance of 1 AU. This relation can be used to compute tre and thus the astronomical unit from an improved mass m, c of Earth and Moon, as obtained from the observed motion of an asteroid such as Eros or Amor. This approach is known as the “dynamical method” for determining tre and the astronomical unit from asteroid observations because it is actually a determination of metc. Because p enters the relation between met ( and tre only in the form of a factor 1 + pu, the uncertainty of the adopted value of pu was not very significant in these earlier determinations of tre, and the astronomical unit through met [ . In today's IAU system of astronomical constants, however, the astronomical unit as directly determined from radar observations of major planets is a primary constant. Because not only tre, but also me, c is a function of the astronomical unit, the merc value consistent with the adopted value of the astronomical unit will eventually be used as a derived constant. Presently the IAU system still lists conventional but clearly outdated values for the planetary masses, essentially for practical reasons related to the preparation and publication of ephemerides.


Because the comparison of calculated and observed asteroid positions involves the use of a given system of celestial coordinates, so that any changes in the positions of the equator and equinox relative to the stars, as well as in the precessional rates of change, would affect the resulting O - C, it is clear that the constants defining orientation and motion of the reference frame can also be determined or corrected by means of asteroids. Moreover, because there may be local distortions and systematic errors even in the best available fundamental star catalogs, which in practice define and represent the adopted reference system in the various areas of the sky, it is possible to determine such errors of a local nature also from the observed positions of asteroids (referred to the catalog stars) as they move across sufficiently large parts of the celestial sphere. If the computed positions are based on excellent and dynamically definitive orbits, their comparison with a sufficient number of observed positions of high accuracy will reveal any local distortions in the right ascensions and declinations of the adopted fundamental system of reference.

Most of the asteroid observations, whether photographic or visual, are relative ones, referred to nearby catalog stars. There is considerable merit, therefore, in making and using absolute meridian circle observations of the first four minor planets, Ceres, Pallas, Vesta, and Juno, for which ephemerides of high internal accuracy are published in the American Ephemeris and Nautical Almanac for each year since 1952. Determinations of the equator, equinox, and the annual precession in longitude from such observations have the advantage of being independent of star catalogs. Compared to similar determinations from observations of the Sun, Mercury, and Venus, the starlike appearance of the asteroids again holds the promise of higher accuracy. Finally, such observations can easily be connected to similar fundamental observations of neighboring stars, so that catalogs can be improved at the same time. In any such projects aiming for perfect rigor, corrections to the orbital elements of the asteroids and of the Earth-Moon barycenter will have to be determined with the desired corrections to the constants defining the reference system and its precessional motion.


References to the older determinations of masses and other fundamental constants can be found in a paper by Harkness (1891). Here it may suffice to mention the determination (probably the first) of a planetary mass by means of an asteroid, namely that by Gauss of Jupiter from the motion of Pallas, leading to a result of 1/1042.86 for Jupiter's mass in units of the solar mass. Many astronomical constant determinations made subsequent to Harkness' compilation are listed and discussed in an encyclopedia article by Bauschinger (1920), whereas a number of more recent results, up to the year 1963, are considered by Böhme and Fricke (1965).

Soon after the discovery of 433 Eros in 1898 it became clear that this minor planet was exceptionally well suited for the determination of the solar parallax tre by the trigonometric method, as well as for the derivation of met ( , because of its rather close approaches to Earth and its observability in all parts of its heliocentric orbit. It was also pointed out by Russell (1900) that because of its substantial perturbations by Mars, this asteroid should be able to yield an accurate determination of the mass of Mars. Because for quite some time Eros has actually been used as the principal tool for determinations of the solar parallax and of met c, the history of these results may be outlined here in some detail. As to the direct, trigonometric determinations of tre from Eros, Hinks found its = 8'807+0'003 (probable error) from the photographic right ascensions of the 1900-01 close approach (Hinks, 1909) and 8"806+0"004 from the micrometric ones (Hinks, 1910), whereas Spencer Jones (1941) obtained the total result 8"790+0"001 from the well-prepared 1930-31 approach. The first dynamical determinations of tre, through met ( , were based on relatively short orbital arcs, but from observations from 1893 (prediscovery positions) through 1914. Noteboom (1921) derived 1/met t = 328370+68. This value for the reciprocal of the mass of Earth and Moon was changed only slightly when Witt (1933) found 328 390+69 from the much longer time interval 1893-1931. The related value of the solar parallax is 8'7988+0'0006. The subsequent determination by Rabe (1950) from the more recent time interval 1926-45, with the results 1/m, c = 328 452+43 and To = 8"7984+0'0004, essentially seemed to confirm the preceding dynamical results, and thus to maintain the inexplicably large discrepancy with the formally also very accurate trigonometric determination by Spencer Jones. This disagreement became even more puzzling when the first reliable radar measurements of the Earth-Venus distance were all found to point to a solar parallax of very nearly 8"7940, about halfway between the presumably best

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