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Figure 1.-Geometry for the restricted three-body problem showing schematically the positions of the libration (or equilibrium) points. The arrow indicates the direction of rotation of the system. In the Gyldén-Moulton counterglow hypothesis, mi is the Sun, m2 is Earth, and the dust cloud is at the libration point L3. For the Earth-Moon libration clouds, mi is Earth, m2 is the Moon, and the dust clouds are near L4 and Ls. See van de Kamp (1964) or Szebehely (1967) for further discussion.

counterglow was due to a collection of dust around the L3 libration point in the Sun-Earth system (fig. 1). This suggestion was first made by Searle (1882), but it is generally attributed to Gyldén (1884) and Moulton (1900). It also was thought that the counterglow might be due to an Earth's dust tail populated by lunar ejecta (Brandt and Hodge, 1961). All of these suggestions were quite controversial, and in the last 10 yr a prodigious amount of work has been done to test their validity. It now seems safe to say that they are all wrong. Numerous theoretical investigations were carried out to find a justification for the existence of a GDC. The most complete was a series of papers by Lautman, Shapiro, and Colombo (1966) who considered a number of physical processes including gravitational focusing, Jacobi capture, meteor-Moon collisions, and sunlight-pressure air-drag capture. They found that, under any set of reasonable assumptions, none of these mechanisms lead to a significant concentration of material. Peale (1967, 1968) has made an excellent analysis of many dynamical and observational investigations and has set an upper limit of 1 percent on any geocentric contribution to the interplanetary light. Evidence for concentrations of material associated with the Earth-Moon libration points has been sought photographically and photoelectrically by Morris, Ring, and Stephens (1964); Wolff, Dunkelman, and Haughney (1967); Roosen (1966, 1968); Bruman (1969); and Weinberg, Beeson, and Hutchison (1969). None of these workers found any evidence for lunar libration clouds. The last mentioned study concluded that any brightness enhancement due to lunar libration clouds must be less than 0.5 percent of the background brightness. This is 200 times fainter than the brightness reported by Kordylewski (1961). Roosen (1969, 1970) has investigated the Earth-associated theories for the counterglow using the fact that they require such a concentration of material near Earth that Earth's shadow would be visible in the center of the counterglow. Because the shadow was not visible to within an accuracy of 1 percent, dust accumulated at the L3 libration point in the Sun-Earth system can account for no more than 1.2 percent of the counterglow's light. Because the hypothetical dust and gas tails are assumed to have a 3° westward displacement from the antisolar point, the base of the tail in either case would be quite close to Earth (inside the umbra). The lack of a shadow indicates that less than 1 percent of the counterglow light is produced by a dust or gas tail. We can conclude, therefore, that to within an observational limit of 1 percent, there is no evidence for accumulations of material in the near-Earth environment. Thus, for the purposes of this discussion, we can assume that essentially all of the interplanetary dust is in heliocentric orbits.

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RADIAL DISTRIBUTION

A large number of models of interplanetary dust distribution have been built based on observed interplanetary light isophotes and the assumption that the radial distribution of material could be described by a simple power law RTP where R is heliocentric distance. Examples of these can be found in Sandig (1941), Allen (1946), van de Hulst (1947), Fesenkov (1958), Beard (1959), Giese (1962), Ingham (1962-63), Gindilis (1963), Gillett (1966), Aller et al. (1967), Singer and Bandermann (1967), Divari (1967, 1968), Giese and Dziembowski (1967), Powell et al. (1967), Southworth (1967), and Bandermann (1968). Values of p ranging from 0.1 to 3.5 were derived or assumed for the various models.

Southworth (1964) and Bandermann (1968) have shown that if the interplanetary dust is due to cometary debris, then Poynting-Robertson drag causes the dust concentration to vary as R-' for R < q and as R-2 ° for R q, where q is the comet's perihelion distance. Essentially all of the comets that have been suggested as sources of interplanetary dust are short-period comets with perihelia less than 1 AU. In particular, Whipple (1967) has stated that “over the past several thousand years” comet Encke with q = 0.338 has been “quite probably the major support for maintaining the quasi-equilibrium of the zodiacal cloud.” Thus, dust from these comets would be expected to follow an R-23 law outside Earth's orbit. Dust from a cloud of particles injected with perihelia greater than 1 AU would follow an RT' law as long as the injection is a steady-state mechanism (i.e., a large cloud was not injected fairly recently).

Thus the assumption that the radial density follows an inverse power law is based on very reasonable physical arguments. However, Roosen (1969, 1970) has shown that these assumed distributions require such a concentration of material near Earth that Earth's shadow should be visible in the center of the counterglow. Such a shadow is not observed (fig. 2), and hence the spatial density of reflecting material must increase at some distance outside Earth's orbit. The source suggested by Roosen is the asteroid belt, and figure 3 shows the relative density of reflecting material that results. The curves for RTP contributions are upper limits based on the lack of an observed shadow to an accuracy of 1 percent. Note that this result does not say anything about the source or distribution of interplanetary dust inside Earth's orbit. However, models based on an RTP distribution of material outside Earth's orbit are incorrect.

There exists yet another source of information on the radial distribution of interplanetary dust; that is, impact measurements made by two Mariner and two Pioneer spacecraft. Alexander et al. (1965) found that over the heliocentric distance range 0.72 to 1.56 AU the interplanetary dust density was roughly constant. This result is based on two impacts measured by Mariner 2 (Alexander, 1962) and 215 impacts measured by Mariner 4. Berg (1971, personal communication) reports that Pioneers 8 and 9 have ranged in heliocentric distance from 0.75 to 1.1 AU and have measured a total of over 150 impacts. His preliminary analysis also indicates that the interplanetary dust particle density is constant in that range of distances. It is immediately apparent that the number of impacts measured is too small for an R-1 distribution to be detected. However, an R-2-3 distribution should be detectable. Hence the R-2-3 distribution can be questioned on yet another ground.

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DISTANCE IN DECLINATION FROM ANTiSOLAR POINT,
OBSERVATIONS MARCH 12, 1969

Figure 2.—Brightness curves predicted for two possible radial distributions of interplanetary dust. The points are from observations made on a single night. The points that lie well above the mean curve are due to faint stars passing through the field of view. Data are from Roosen (1969, 1970).

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Figure 3.-Relative spatial densities for four possible radial distributions of material. The RTP curves are upper limits with respect to the asteroidal distribution set by Roosen (1969, 1970).

There is an additional simple test to distinguish between the cometary and asteroidal hypotheses (Roosen, 1969, 1970). It requires that a photometer on a space probe traveling toward the outer solar system monitor the counterglow brightness. If the counterglow is due to asteroidal debris, its brightness will remain almost constant until the probe goes further than 2 AU from the Sun. If, on the other hand, cometary debris produces the counterglow, the observed brightness will steadily decrease, and the counterglow will only appear to be a tenth as bright at 2 AU as it is when seen from Earth's distance.

DISTRIBUTION OF INCLINATIONS

Bandermann (1968) and Singer and Bandermann (1967) fit a series of models to the interplanetary light observations reported by Smith, Roach, and Owen (1965) and found that the number of interplanetary dust particles with a given inclination i was best described by a function of the form

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This result was generally confirmed by Zook and Kessler (1968). Results of this type, however, are based on a faulty assumption. The radial distribution assumed by Bandermann and Singer was proportional to R*-*. This means that most of the brightness contribution at elongations greater than 90° is assumed to come from material relatively close to Earth. Let us examine the situation at an elongation of 180°. From figure 4 we see that the closer to Earth the material is, the larger the geocentric latitude

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Figure 4.—The relation between the geocentric latitude 3 and the heliocentric latitude a at elongation 180°.

£ at which one must look to see particles with a given heliocentric latitude o. In fact,

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where s is the projection into the ecliptic plane of the distance of the material from Earth. As an example, let us look at two cases: (1) s = 0.3, the distance within which 50 percent of the counterglow brightness would arise for material distributed according to an R-'o power law, and (2) s = 1.5, the mean distance for an asteroidal contribution. In case 1, in order to see a particle at a heliocentric latitude o of 5°, the observer must look at a geocentric latitude 3 of 21°. For case 2, 3 is 8° (fig. 5). In effect what this means is that if the R-15

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Figure 5.-Spatial density of material at 180° elongation as a function of geocentric ecliptic latitude for various values of s, the projected mean distance of the material from Earth, and Singer and Bandermann's distribution of inclinations.

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