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Figure 1.-Details of a collision event. k is impact vector, g and g are relative velocities before and after the impact, respectively.
Conservation of momentum requires that
Kinetic energy is not conserved. To simplify the problem, we do not care what happens to this lost energy by simply prescribing the amount lost in a given collision by the relation
Here 3 is a restitution parameter describing varying degrees of inelasticity. It is allowed to take any value in the range (1, 2). The upper limit, 3 = 2, corresponds to elastic collisions. For the lower limit, 3 = 1, a head-on collision will be completely inelastic. The amount of energy lost decreases as the collision takes place more and more off axis, and in the limit of a grazing collision there is no energy loss.
Proceeding in a manner similar to the derivation of the classical Boltzmann operator, the following form of the collision operator can be derived from equations (5) and (6):
Here the explicit r and t dependence of the distribution function was suppressed and
The operator (eq. (7)) reduces to the Boltzmann operator for the case 3 = 2. The apparent singularity of equation (7) as the extreme case 3 = 1 is approached is not real. For the product f(V)f(V') to be nonvanishing, both v and v' must lie inside the support off in velocity space. This requires the quantity (v'-v) k to be of order 3–1. Thus we get one factor 3–1 from the volume element in velocity space and one from the factor (v'-v) k already present in the integrand, making the operator (eq. (7)) tend toward a finite limit as 3 -> 1.
Entropy arguments through Boltzmann's H theorem play an important role in the discussion of the classical Boltzmann equation. With elastic collisions, the entropy source function is always positive except for local thermodynamic equilibrium (LTE) distributions where it vanishes. For the motion in a central gravitational force field with a nonvanishing total angular momentum, however, it is not possible to construct a time-independent, physically relevant state from LTE distributions. For a physically relevant situation, the average flow velocity should increase with decreasing distance from the central body if all the grains are moving in the prograde direction. If, on the other hand, an LTE distribution function is substituted into the time-independent kinetic equation, the only allowed average velocity goes as U = r x w where w is a constant vector; that is, a fixed body rotation with a velocity increasing with increasing distance from the center (Chapman and Cowling, 1960). This result does not imply, however, that the system would not evolve asymptotically toward a time-independent final state. An additional possibility is that the collisions tend to make themselves less important by making the individual orbits more and more parallel, the whole system evolving toward a Saturnian ring configuration.
For the case of inelastic collisions, we no longer have a description of a closed system. We do not have a description of what happens to the lost kinetic energy. No extension of the H theorem has been given for this case. In a way, the kinetic energy function plays the role of Boltzmann's H function, giving the allowed direction of evolution of the system. Again, the only timeindependent state of the system is the singular state consisting of parallel individual orbits only.
THE INITIAL RATE OF CHANGE PROBLEM
The most usual way of extracting information about a system by linearizing the equations of motion around a time-independent equilibrium state is hardly of interest here for two reasons. We have already seen that this state is rather singular. Further, accretion would probably play a crucial role long before this state is approached. A complete analytic study of the dynamics seems out of the question at the present time. A more modest undertaking is to study an initial rate of change problem. How will the system start to change from a prescribed initial state F(r, v)? The crucial point now is to choose initial states that make it possible to extract interesting properties of the system. Because we are interested in the influence of collisions on the dynamics, F(r, v) should be chosen to be a time-independent solution of the collisionless kinetic equation; that is, F should satisfy AF = 0. Any function depending on r and v through only time-independent constants of motion for a central gravitational force field satisfies this requirement. Such constants are (Danby, 1962)
being the semimajor axis, the angular momentum vector, and the perihelion vector, respectively. Only five of these constants are independent because the following two relations exist between them:
P = L = 0 L2 = a(1 - P2) (10) The collisions give rise to an additional mass flux in the stream. If 6f(r, v, t)
designates the deviation of the distribution function from the initial distribution, this flux is given by
NUMERICAL RESULTS AND DISCUSSION
For the special case of an azimuthally symmetric stream, the sixdimensional velocity integration in equation (13) was evaluated numerically. For this case, F does depend only on a, P2, and Lz; and it is enough to study the situation in one cross section of the stream. Instead of a, Po, and L, the distribution function could also be expressed in terms of a, e” = P2, and cos i. where i is inclination and e eccentricity.
The results for a series of distribution functions of the type
for a 1 < a < a.2, e <eo, and i < io are presented below. The values for the parameters a 1, a2, eo, and io were chosen to be
The distribution functions were normalized such that the number of particles in the stream are the same for each case. In figure 2 the density profile corresponding to case I along a radius in the equatorial plane of the stream is plotted. The shape of the profile is the same for the other cases except that it gets narrower with decreasing value of the maximum eccentricity eo. The kinetic pressure tensor takes a diagonal form in a spherical coordinate system. In figure 3, the three pressure components over density are plotted. The pressure in the azimuthal direction Poe is less than the two transverse pressure components. This is a general result, valid for all distribution functions that have been studied. By varying the maximal eccentricity and inclination, the ratio of the two transverse pressure components can be varied. A general property of the collisions is to try to make the pressure tensor more isotropic. This gives rise to a most unusual property of the jetstream configuration that can be seen in figure 4. Here the mass flux vector 6(n, U) is plotted for two points in a cross section of the stream, at r = 0.9 and r = 1.1 and at the same distance above the equatorial plane, represented by the broken line. The flux vector is plotted for three different values of the restitution parameter 3: 3 = 2, which is the elastic case; 3 = 1, which is the opposite extreme; and 3 = 1.5, the intermediate value. The x's indicate the density maximum in the stream. Consider the extreme cases I and IV, the former characterized by an excess pressure in the radial direction, the latter by an excess pressure in the polar
Figure 2.—Particle density in the stream along a radius vector in the symmetry plane of the stream, case I; arbitrary units.
direction. Common physical experience tells that if a system wants to eliminate excess pressure, it has to expand in some way. This is a statement of the Boyle-Marriotte ideal gas law. The jetstream behaves in the opposite manner. It eliminates excess pressure by contracting. This can be understood from a consideration of the individual orbits in the stream. The central force field has the property of twisting these orbits very effectively. A particle that is on the bottom of the stream at one side will be on the top on the other side. A particle in a high-eccentricity orbit that is on the inner side of the stream at one time will be on the outer side half a period later. The pressure component in the polar direction is generated mainly by the orbits having the highest inclination. If the stream expands in this direction, this means that the distribution of inclinations must get wider. This again implies that the polar pressure increases. Thus for case I, to take care of the deficit in the polar pressure, the stream expands in this direction, contracting in the radial direction. Case IV represents the opposite case. Here the stream expands in the radial direction while contracting in the polar direction to eliminate excess pressure in the polar direction. Note that whereas an expansion or contraction in the polar direction is mainly an effect in inclination alone, the similar process in the radial direction is a more complicated phenomenon depending on the distribution of both eccentricity and the semimajor axis. The effect of inelasticity always is to make the stream approach a more narrow configuration relative to the corresponding elastic case. This is in