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where a, b, c, and d are polynomials in eo. Because the total number of particles and the total angular momentum are conserved, we have
It can be shown that c > 0. Using equation (6), we find that
Equation (20) looks something like a diffusion equation with a negative diffusion coefficient. Thus h(L) grows at maxima of L-2h?(L) and decreases where L *h”(L) is at a minimum as shown in figure 3(a). Note that this
Figure 3.-(a) Initial smooth distribution function with local maxima separated by approximately eoL. (b) A distribution function in which most of the particles are in groups, separated in angular momentum by about eoL.
equation will never allow h(L) to become negative. The fastest growth is experienced by the narrowest peaks. These fine scale peaks eventually dominate the distribution function, and the particles concentrate at the angular momenta where the narrowest peaks were originally, as in figure 3(b). We initially used a distribution function that was smooth on a scale length eoL (i.e., initial peaks in F(L) and, consequently, in h(L) and h”(L) were separated in angular momentum by distances eoL). Because eo is small compared to unity, L is slowly varying by comparison and peaks in L’ho(L) are separated by eoL. Thus grains concentrate in orbits separated in angular momentum by eol; these jetstreams must be circular because of the initial axisymmetry of our distribution. As the grains lose energy because of inelastic collisions, the orbits themselves become more circular. A finer grain distribution function would have finer scale peaks, but our result would not necessarily apply in that case because the calculation depended critically on the Taylor expansion of the original distribution. The fine scale peaks might evolve into distinct subjetstreams, or they might merge into a single jetstream. Now consider what effect this has on the radial density distribution. Because the radius of a circular orbit is related to its angular momentum by
L2 r = — (21) mk then the radial separation is given by 6 ôL or - 2 - - 260 (22) r L
(In our solar system ör/r is roughly 0.4 to 0.6 corresponding to
eo = 0.25 +0.05 (Jeans, 1944).) There are many other properties that may influence the collisional evolution
of an orbiting cloud of grains. Although we have neglected size, shape, and mass differences among grains, effects of rotational degrees of freedom, selfgravitation, actual accretion, or even shattering of particles, our calculation indicates that the inelasticity tends to cause jetstreams.
We wish to thank H. Alfvén for his suggestions and encouragement. This work was supported in part by the United States Atomic Energy Commission, contract no. AT(04-3)-34 PA 85-13.
Alfvén, H. 1970, Jet Streams in Space. Astrophys. Space Sci. 6, 161.
COLLISIONAL FOCUSING OF PARTICLES IN SPACE
Jetstreams probably played an important role at an intermediate stage of the formation of the solar system (Alfvén and Arrhenius, 1970). A jetstream is defined here as a collection of grains moving in neighboring elliptical orbits around a central gravitating body and with the dynamics modified by the action of complicated collision processes among the grains themselves.
Three main types of collisions will take place in such a stream. Hypervelocity impacts will lead to fragmentation of the grains involved. At lower impact velocities, the particles will retain their identities after the collision even if they might be deformed to some degree depending on impact velocity and internal structure of the grains. With still lower impact velocity, accretion can take place, the grains sticking together after collision to form larger grains. A common feature of these collision processes are that they will be partially inelastic. A certain fraction of the kinetic energy of the colliding particles will be spent on changing their internal structure. The internal kinetic energy in an isolated jetstream thus will tend to decrease with time. The mass spectrum of the grains also will vary during the lifetime of a stream, the probability for accretive processes increasing with time.
Qualitative arguments for the collisional focusing effect leading to the formation of streams have been given by Alfvén. The most important points in his argument are
(1) Two orbits after a collision will be more similar than before because of the loss of kinetic energy.
(2) Because of the central force field, a particle having collided with a stream is not easily lost from the stream. It will always return to the place where it last collided, thus making it subject to new collisions with particles in the stream.
As a first step toward a quantitative theory, the idealized situation of a jetstream of identical spherical grains will be studied. Collisions leading to fragmentation and accretion will be neglected together with the selfgravitational effect of the stream. The distribution of grains in the stream will be described by a distribution function f(r, v, t) in the six-dimensional phase space. The first problem is to construct a kinetic equation describing the evolution of this distribution function due to the motion in the central gravitational field and to the mutual collisions.
The equations of motion of a single particle in a central gravitational force field are
The orbits for bound particles are ellipses with the central body at one of the focal points. Time and length units have been chosen such that the orbital period in an ellipse with a semimajor axis equal to unity is 2m time units.
The rate of change of the distribution function due to the motion in the gravitational field is
If the effect of collisions is described by a nonlinear operator C.
Already Boltzmann gave the form of the collision operator for the case of number-conserving elastic collisions (Chapman and Cowling, 1960). Because we are interested in partially inelastic collisions, a somewhat modified expression is needed.
DERIVATION OF COLLISION OPERATOR
The details of a collision process are described in figure 1. A collision takes place as soon as the distance between the centers of two particles is equal to their diameter D. The direction between the centers at impact is given by the impact vector k which is a unit vector. Particles 1 and 2 have velocities v1b and v2b before the collision. Afterward, the corresponding velocities are v1a and