Thus we see that the traditional (two-body) concept of an embryo growing by sweeping out a tube of matter of ever-increasing cross section is actually a valid concept, although the mechanics are rather involved. Because the scattering jetstream is itself stable (particles are either stored or captured), the presence of embryos in primordial jetstreams does not destroy their stability. REFERENCES Alfvén, H., and Arrhenius, G. 1970, Structure and Evolutionary History of the Solar System, I. Astrophys. Space Sci. 8,338-421. Giuli, R. T. 1968a, On the Rotation of the Earth Produced by Gravitational Accretion of Particles. Icarus 8,301-323. Giuli, R. T. 1968b, Gravitational Accretion of Small Masses Attracted From Large Distances as a Mechanism for Planetary Rotation. Icarus 9, 186-190. DISCUSSION DOHNANYI: Have you had a chance to consider the influence on your accretion rates of competing processes; e.g., planetary perturbations, the Poynting-Robertson effect, and lifetime due to collisional breakup or erosion? GIULI: No. We do not consider accretion rates in this model. To do that would require an estimate of the particle density during planet formation. That consideration will be an elaboration to the model. SINGER: Is there a simple dimensional argument that can be put forth to explain the qualitative nature of the results for the gravitational accretion theory? GIULI: Because the numerical integrations do display the asymptotic development of rotation with mass so dramatically, I feel strongly that there should be a simple explanation. As yet I have not found it. ALFVEN: One can show that if accretion occurs for any size body for which the total angular momentum contributed by the accreted material is some constant fraction of the angular momentum contributed by a particle that grazes the body tangentially with the body's escape speed, then the rotation speed acquired by the body is proportional to the square root of the body's density. Giuli's calculations show that the asymmetry of the impacts give about 1 percent of the tangential angular momentum for all masses. WHIPPLE: It seems to me we are putting too much emphasis on the assumption that the process of formation of bodies produces a particular rotation rate. The limitation on rotation rates is density, not mass, and the solid bodies of the solar system have a small range of densities. The reason the observed rotation rates appear to cluster around certain values is that accumulation processes tend to give rapid rotation rates. Those bodies that tried to form with much higher rotation rates were disrupted and are not observed. Some bodies that were formed with lower rotation rates are observed. Forces that change rotation rates thus may destroy the bodies or reduce rotation rates, leading to the observed distribution. GIULI: Getting back to Singer's question, and elaborating somewhat on Alfvén's comment: The rotational angular momentum per unit mass (specific angular momentum) contributed by an impacting particle that grazes a body with the body's escape speed (or with some factor of the escape speed) is easily shown to vary as the two-thirds power of the mass of the body, for a given body density. If any accretion process adds matter to an embryo in such a way that the sum of the contributions of specific angular momentum of added matter is some constant C times the two-thirds power of the embryo mass, then it is easy to show analytically that the asymptotic development of rotation rate with mass is an inevitable result, along with the period-density relation stated above by Alfvén. These points are developed by Giuli (1968a,b). The gravitational accretion calculations provide this relation between contributed angular momentum and embryo mass, for any particular body that grows with constant mean density, at any distance from the Sun. This fact is true over at least the seven orders of magnitude of mass for which I did the calculations. This is a result of the fact that the geometry of the impacting particle trajectories scales linearly with the radius of the body. I have no simple explanation of why this should be the case, but probably it is connected with the fact that all embryo masses for which I did the calculations were small compared to the solar mass. (The largest embryo mass considered was Jupiter's mass.) I should mention that one failure of the current model is that it gives a different value of C for the different bodies of the solar system. Fish (1967) and Hartmann and Larson (1967) have shown that most of the bodies of the solar system have the same value of C over a mass range of 11 orders of magnitude. UREY: MacDonald was the first to consider the relation between specific angular momentum and mass, and he obtained a power of 0.83 rather than two-thirds. This came about because he included Mars. The value two-thirds applies only if all terrestrial planets are excluded. Is this justifiable? GIULI: Mercury, Venus, and Earth are excluded because of the apparent tidal effects upon their rotation rates subsequent to their formation. Mars is a very serious problem. If no subsequent process has affected Mars’ rotation rate, and if Mars and the other bodies have formed by the same process, then the validity of the present gravitational accretion model as representing the process of formation may be in doubt. On the other hand, the present model can explain a retrograde rotation for, e.g., Venus if some peculiar condition restricted the eccentricity of heliocentric particle orbits in Venus' vicinity to low values during its growth. HAPKE: The final rotation state of the body after accretion is strongly influenced by the initial density distribution in the nebula. This follows from consideration of the conservation of angular momentum and is independent of the details of the accretion process. Consider small particles in orbit about the proto-Sun that later condense into a larger body. The material initially inside the final orbit will have been moved outward during accretion and the material outside will have moved inward. The orbital angular momenta of both sets of particles will have changed in opposite senses. In general, the net change will not be zero, and thus the orbital angular momentum difference will show up as the spin angular momentum of the body. The direction and amplitude of the rotation depends on the original density distribution and final orbit. GIULl: Perhaps. I am currently investigating the question of whether an embryo captures particles from their primordial heliocentric orbits or redistributes them before capture. The current investigation suggests that the latter situation applies to most of the captured particles. Also, the work of Trulsen" suggests that an intermediate particle state may occur before embryo formation; namely, a viscous jetstream that modifies the primordial particle distribution over the distances of interest. DISCUSSION REFERENCES Fish, F. F., Jr. 1967, Angular Momenta of the Planets. Icarus 7,251-256. Giuli, R. T. 1968a, On the Rotation of the Earth Produced by Gravitational Accretion of Particles. Icarus 8, 301-323. Giuli, R. T. 1968b, Gravitational Accretion of Small Masses Attracted From Large Distances as a Mechanism for Planetary Rotation. Icarus 9, 186-190. Hartmann, W. K., and Larson, S.M. 1967, Angular Momenta of Planetary Bodies. Icarus 7, 257-260. MacDonald, G. J. F. 1963, The Internal Constitutions of the Inner Planets and the Moon. Space Sci. Rev. 2,473-557. ACCUMULATION OF CHONDRULES ON ASTEROIDS FRED L. WH/PPLE It is suggested that aerodynamic forces played a significant role in the selective accumulation of chondrules on asteroids moving with respect to the gas in a primeval solar nebula. Particles smaller than millimeter chondrules would sweep by an asteroid moving in a critical velocity range, whereas larger particles could be accumulated by impact. Theory and calculation cover the case of subsonic velocity and asteroidal diameter up to 50 km for a nebula density up to 10 °g/cm3, or higher for smaller asteroids. Chondrules, roughly millimeter spherules found abundantly in many meteorites, have long been aptly described in Eucken's (1944) terms as products of a “fiery rain” in a primeval solar system nebula. Chondrules are clearly mineral droplets that have cooled rapidly, some showing evidence of supercooling. On the basis of the quantitative loss of volatile elements, Larimer and Anders (1967) deduced that chondrules were formed in an ambient temperature of some 550 K. Because melting temperatures are roughly 1300 K greater, some violent heating mechanism must have been involved. Noteworthy is a suggestion by Wood (1963) that the quick heating was produced by shock waves in a primitive solar nebula. Volcanic and impact processes have been suggested, as has the pinch effect in lightning (Whipple, 1966). Whatever the source of droplet formation, a major evolutionary problem concerns the high abundance of chondrules among several classes of meteorites; in some the percentage of chondrules exceeds 70 percent by mass. Accepting the concept that meteorites are broken fragments of asteroids that were originally accumulated from solids in a gaseous solar nebula, one's credulity is taxed by the added assumption that a substantial fraction of the solid material should have been in the form of spherules. Thus, the purpose of this paper is to explore the possibility that chondrules may have been selectively accumulated on some asteroidal bodies, thereby eliminating the undesirable supposition that chondrules constituted a major fraction of the dispersed solids in any part of the nebula. Almost axiomatic is the assumption that the accumulation process for smaller asteroids essentially ceased when the solar nebula was removed, presumably by the effect of the solar wind from the newly formed Sun in its brilliant Hayashi phase (Hayashi et al., 1960). Possibly the largest asteroids can still continue to grow in vacuum conditions, but the relative velocities of particle impact on asteroids less than perhaps a hundred kilometers in dimension would be generally dissipative rather than accumulative because of the low velocities of escape against gravity. where L is the mean free path of the atoms or molecules, assumed to be neutral. Because the flow about the forward surface of the moving body is relatively streamlined at rather high values of the Reynolds number, the reference Reynolds number Re is calculated for the small particles and given by the expression R. = * (2) Figure 1.-Flow pattern in which larger particles may strike the moving body. S = radius of sphere. While the solar nebula was present, however, small bodies moving through the gas would have exhibited aerodynamic characteristics. At a given body velocity and gas density, solid particles having a mass-to-area ratio below a certain value would be carried around the body by the inertia and viscosity of the gas currents so as not to impinge or accumulate on the moving body (fig. 1). The physical conditions for certain such accumulation processes will be established in the following sections of this paper. IMPACT OF SMALL PARTICLES ON A SPHERE MOVING Taylor (1940) dealt with this basic problem for a cylinder. Langmuir and Blodgett (1945) derived numerical results by theory and calculation for cylinders, wedges, and spheres moving through air containing water droplets or icy spheres. Fuchs (1964) and Soo (1967) summarized the subject for subsonic flow and included both theoretical and experimental results by various investigators. Probstein and Fassio (1969) investigated “dusty hypersonic flows.” The transonic case has apparently not been attacked seriously. The following discussion is based on the presentations by Langmuir and Blodgett augmented by the summaries by Fuchs and Soo. A sphere of radius S is assumed to move at velocity v through a gas of density p and viscosity n containing in suspension small spheres of radius s and density ps. The gas viscosity is given by the classical approximation The applicable Reynolds number is reduced greatly from this value because the small particles will not be thrown violently into the full velocity of the gasflow v, except perhaps near the stagnation point. As will be seen, the relatively small value of the applicable Reynolds number permits the application of the simple Stokes' law of particle drag at values of v far above those for which the law might intuitively appear to be valid. Because the Stokes force F on a sphere of radius s moving at a velocity v, through a gas is independent of the gas density when L 3 s, the impact equation for particles impinging on the larger sphere of radius S is widely applicable in a solar nebula where the density cannot be accurately specified. The Stokes approximation begins to fail significantly for Re P 10, but the drag force is overestimated only by about a factor of 3 at Re= 10°. (See Probstein and Fassio, 1969.) An inertia parameter W is defined as which is the ratio of the inertia force to the viscous force for small particles in the stream. Theory and experiment show (fig. 2) that particles of radius <s do not impact the sphere for VW 3.0.2 for potential flow and 0.8 for viscous flow. At higher values of Re, when Stokes' law deteriorates, the behavior of the impact changes as a function of another parameter p defined by Figure 2 illustrates the changes in impact efficiency for values of p up to q = 10°. Note that the limiting value of VW, initiating the impacts, is nearly independent of p, whereas the efficiency of impact is not greatly dependent on 4). Hence limiting conditions for impaction of particles of radius s, on a sphere of radius S, up to Re somewhat less than 10°, can be confidently given by equation (4) when Vlimit - 0.04. |