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University of California, San Diego

An inelastic collision integral is used in a Boltzmann-type equation for a distribution of particles in Kepler orbits. A Fokker-Planck equation is found that leads to radial density clustering.

It has been suggested that in a cloud of grains moving in Kepler orbits in a gravitational field, inelastic collisions will cause the grains to form groups having similar orbits, called jetstreams (Alfvén, 1970). One would expect that a jetstream already formed would contract into a tighter jetstream (Trulsen") because the jetstream would lose total energy because of the inelasticity of collisions, whereas the total angular momentum would be conserved. The grains would move toward a circular orbit because circular orbits have the lowest energy Eo for a given angular momentum L (fig. 1). We consider the question of whether jetstreams will form from an initially smooth distribution function.

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Figure 1.-A jetstream can be thought of as a group of particles filling an effective potential well V*(r), which arises from gravitational and centrifugal forces. As the particles lose energy E in inelastic collisions, they concentrate at the bottom of the well. m = mass of a single grain.

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The essential feature of this suggestion is the inelasticity of the collisions. Accordingly, we look at a particularly simple model with particularly simple inelastic collisions; i.e., perfectly inelastic collisions in which colliding particles stick together. We avoid the consideration of accretion of particles by considering the final velocities of colliding particles to be arbitrarily close while the particles maintain their distinct identities. We consider one species of particles moving in coplanar Kepler orbits.

Particles in an arbitrary distribution of exactly circular orbits would never collide, so such a distribution would be stationary (fig. 2). A thermal equilibrium distribution

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which diverges exponentially at r = 0 and r = co, where r and 6 are polar coordinates, p and L are the corresponding canonical momenta (radial and angular momentum, respectively), m is the mass of a single grain, and qb(r) = -k/r is the gravitational potential energy. A distribution with minimum

Circulor Orbits Ellipticol Orbits Ellipticol Orbits (nominteracting) (interacting)

Figure 2.-Particles in circular orbits (or in nonintersecting orbits with e # 0) experience no collisions. Collisional evolution only occurs when these elliptical orbits intersect. Kepler orbit: p(r) = -k/r.

energy for a given total angular momentum would be one in which a single grain, having all the angular momentum and almost no energy, moves very slowly in an orbit with very large radius, whereas all the other particles collapse into the central body. We consider an initial distribution that depends arbitrarily on angular momentum, has orbits of generally small eccentricity, and is axisymmetric:

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mk2/2L2 is the spatial eccentricity of an orbit that passes through the point (r. 6, p, L) in phase space, and -mk?/2L2 is the energy of a circular orbit with angular

momentum L. We assume that essentially all of the particles are orbiting in the same direction. We use a smooth function F(L) so that

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which has a superficial resemblance to the equation for thermal equilibrium. Note that the initial axisymmetry demands that the final state will be axisymmetric (i.e., only circular jetstreams are possible). We will also have to consider the functions

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where h(L) is the density in angular momentum space, A(60) is the normalization constant, and N is the total number of grains.


We wish to find a differential equation that describes the evolution of h(L), the distribution function in angular momentum space. The equation describing the time evolution of the phase space distribution function is

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where x is the position vector, v = dx/dt, a = dv/dt, and I(f, f) is the collision integral. Our initial distribution fo, being a function of constants of the motion only, is stationary in the absence of collisions. We assume that the mean free path is long compared with the orbital path so that collisions are treated as a perturbation, whence

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where 6f is the perturbation distribution caused by collisions. Linearizing equation (8) we get

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is the relative speed between the two grains, and osg) is the collisional cross section. (Note that a (p, L), (p + p', L + L') collision scatters a particle out of the phase space volume element at (r. 6, p. L) and a (p-p's2, L - L'/2), (p + p"|2, L + L'/2) collision scatters two particles into a phase space volume element at (r. 9, p, L.).)

Inserting equation (3) into equation (11), Taylor-expanding F(L), assuming that og) is a constant, and integrating over p' and L', we get

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where 6L = L - Lo; Lo = Wmkr is the angular momentum of the circular orbit at radius r, E and B are constants; and C, D, G, N, J, K, M, N, and P are polynomials in eleo with coefficients of order unity. The expressions eo?mk and Wooksm, can be thought of as effective available relative momentum space and mean relative velocity, respectively. To find

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we transform (r, p) to (e”, x) where x is the orientation of the major axis of the ellipse through the point (r. 6, p, L) in phase space:

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