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Figure 2.—Collection efficiency versus inertia parameter V as function of parameter @. Dotted curve: viscous flow.
Let us then assume that chondrules have a radius of 0.05 cm and decide that particles of 1/3 this radius (diameter 1/30 cm) should not impact our asteroid of radius S moving at velocity vs through the primeval nebula of viscosity n. Then s = 1/60 cm, VW = 0.2, and
if we take p, as 3 g/cm3; employ cps units; and adopt a “solar mix” of primeval gas with mass distribution XH = 74 percent, YHe = 24 percent, and Zother = 2 percent, hydrogen being in the form of neutral molecules at a temperature of 550 K. The viscosity becomes approximately m = 1.6 X 10° dyne-s-cm-2
Numerical values for equation (6) are given in table I.
TABLE I.—Numerical Values for Equation (6)
Diameter Limiting asteroid, 2S, velocity vi, Re km km/s 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.0034 0.04 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .034 .4 10.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 4.0 100.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4? 40.0
DISCUSSION OF RESULTS
Before drawing conclusions from the first two columns of table I, we must check to see that the Reynolds numbers involved are not too high for the Stokes extrapolation to be valid; i.e., Re ‘ 10°. This check involves an assumption as to the density near the plane of the solar nebula in the asteroid belt, say at 2.5 AU. Few theorists place the gas pressure here much greater than 1 kN/m2 (10-2 atm). (See, for example, Cameron, 1962.) For a central mass equal to the Sun and allowing for the gravitational attraction of the gas itself, we find the corresponding density, p - 5 X 1077 g/cm3, and the surface density integrated perpendicular to the plane throughout the nebula some 3X 103 g/cm2, or about 1/30 solar mass per square astronomical unit. The Reynolds number (eq. (2)) is then given at s = 1/60 cm by R. = 11 X 10–3 v cm/s, values of which are tabulated in the third column of table I, safely within our limits for asteroids up to 100 km in diameter. The limiting velocity, however, becomes supersonic at vio 1.8 km/s. Hence the condition of subsonic velocity limits our present conclusions to asteroids less than about 50 km in diameter. The condition of molecular mean free path not exceeding the limiting dimensions of the chondrules restricts the theory to p > 10−8 g/cm3, a somewhat higher density than is sometimes assumed for the solar nebula in the asteroid belt. It is evident that a more complete theory is needed to cover the case of low densities in the solar nebula and that the transonic case should be developed before the present suggestion for the selective accumulation of chondrules by asteroids can be wholeheartedly accepted. The latter situation probably requires numerical analysis. The case of lower density can be roughly approximated by means of Epstein's law of drag (see, e.g., Kennard, 1938), which, for mean free paths that are large compared with the dimension of the body, gives a drag force roughly ps3pe that of Stokes' law, where pc is the critical gas density at which the mean free path of the molecules equals the dimension of the body. Because V varies inversely as the drag force, the critical value of V for accumulation in equation (4) will also vary approximately as ps3pe for relatively low gas densities. Hence the limiting velocity in equation (6) and in table I can be corrected as to order of magnitude by a factor of ps3pe, or about p X 107 g cm-3, for p < 10-8 g cm-3. The simple solution involving Stokes' law covers a considerable range of possible physical conditions and fairly reasonable ranges for planetoid velocities. In equation (6) the square of the radius of the limiting particle size varies inversely as the velocity and directly as the radius of the asteroid or planetoid. This indicates that the process of selective accumulation is fairly sharply defined in particle size when measured by velocity or size of the planetoid. If the process of chondrule formation (e.g., by lightning) is inherently limited for large dimensions, but not at small dimensions, the aerodynamic selection factor could frequently produce a fairly narrow range in chondrule dimensions. Furthermore, extremely small chondrules could easily lose their identity in some meteorites by chemical differentiation during subsequent heating of the asteroid. Together, upper limits to dimensions in chondrule formation, a selective accumulation process, and perhaps some partial differentiation seem capable of relieving the theorist from the undesirable postulate that chondrules once constituted a sizable fraction of the mineral content in any part of the solar nebula.
Note that aerodynamic forces will prevent the accumulation of finely divided minerals on planetoids in motion with respect to the gaseous medium, thus greatly reducing the accumulation rates calculated on the basis of simple cross-sectional areas and velocities.
Cameron, A. G. W. 1962, The Formation of the Sun and Planets. Icarus 1, 13-74.
THE ALINEMENT OF ASTEROID ROTATION
JOSEPH A. BURNS
Data have been accumulating since the beginning of this century that indicate that most, if not all, large asteroids have periodic lightcurves. The variations that are seen have periods of the order of several hours and can be understood as being caused by bodily rotation, accompanied by changes in shape and/or surface properties. Because corresponding color changes are usually absent, the former explanation of a variation in cross section is probably the correct one.
The lightcurves of the asteroids do not exhibit photometric beat phenomena and, as Kopal (1970) has argued, this indicates that the rotation is about only one axis. In point of fact, the pole of the rotation axis can be determined from observations." The principle behind interpreting these observations is easily understood: If, for simplicity, we assume that an asteroid orbits in the ecliptic and that its brightness is proportional to the surface area seen, then any variation in brightness (after corrections for distance and phase effects have been made) must correspond to a variation in the projected surface area. There will be no change in the brightness variation over one orbital period if the rotation pole is normal to the orbit plane, for then the differences in surface area seen over one rotation period are the same everywhere on the orbit. On the other hand, the maximum changes in surface area and, hence, the maximum brightness variation will occur when the rotation pole lies in the orbit plane; intermediate variations will correspond to intermediate positions. So, by observing the variation in the magnitude of the brightness over one orbital period, one can evaluate the longitude and latitude of the asteroid's rotation pole. An approximate technique, based on this idea but using only a few observations, has been applied to many asteroids in a series of papers primarily by Kuiper and Gehrels with others. (See Dunlap and Gehrels, 1969; Gehrels, 1967; Gehrels and Owings, 1962; Gehrels et al., 1970; Vesely;” and Wood and Kuiper, 1962.) The results, which could be further refined, indicate that the rotation axes may be clustered in ecliptic longitude and that almost all asteroids have large obliquities; the only one of the 15 or so whose rotation
axis lies more than 20° from the ecliptic is the large and nearly spherical Vesta (Gehrels et al., 1970) whose pole appears to be at about 65° ecliptic latitude. The result that the rotation is about only one axis is truly surprising. According to rigid body dynamics, only when a principal axis lies along the direction defined by the body's angular momentum vector H will there be no precession. Otherwise, the principal axis system xyz fixed to the body should freely precess about H. We define CX B > A to represent the moments of inertia about the z, y, and x axes, respectively, and consider as an example the case where the angular velocity a lies near the maximum axis z of inertia. Then the precession has an angular velocity (a6)”o, where
(Symon, 1960). For typical asteroid shapes, a and 6 range from 10° to 107? and thus the free precession will occur on a time scale that is within an order of magnitude or two of the rotation time scale; in other words, it would be observable if it existed and to were not closely alined with H. Dynamics also tells that the rotation will be stable only if it is about z, the axis of maximum moment of inertia, or x, the minimum axis. Observations are in agreement with this: The asteroids appear to be spinning about the maximum axis. This latter fact indicates that energy dissipation may be taking place because convergence of the maximum axis with H generally will occur whenever energy is dissipated internally in a quasi-rigid body (Pringle, 1966). Kopal (1970) has argued that the absence of any precession indicates that the asteroids could not have arisen from collisions because then one should expect a random distribution of their angular momenta with respect to their body axes; thus he believes asteroids must have been formed in their present alined state. We wish to present a different interpretation of the peculiar alinement phenomenon. We will present directly below calculations showing that at least a few major impacts should have taken place on the large asteroids after their formation. Such collisions will change each asteroid's H and will, in general, misaline H from w. Thus, even if the rotation axes were perfectly alined originally, precession of some asteroids should be observed today. Because it is not seen, an alining mechanism must be (or must have been) at work if the collision calculations are correct. This idea receives some further support from the unusual ordering of the orientations of the rotation axes that itself speaks of an alinement process; it is quite difficult to explain Gehrels' large obliquities and the clustering in ecliptic longitude without some such process. Before discussing possible alining mechanisms, let us first consider the collision process. Collisions between at least small asteroids are generally