A New Approach to Dynamical Evolution of Interplanetary Dust

We introduce the continuity equation written in the coordinate space of the orbital elements (e.g., semimajor axis a versus eccentricity e, etc.). This equation can serve as an effective tool to analyze the transport of interplanetary dust particles as well as their dynamical evolution, and offers a very useful complement to the approach using purely numerical integration of orbits. Using the continuity equation and suitable analytical and numerical approximations, statistically useful results can be achieved very quickly, and new integrals of the motion can be sought to simplify the description of large-scale phenomena. This paper describes the method, illustrates it with a simple example of multiple gravitational scatterings of particles on planets in circular orbits in two dimensions, and outlines the program for further development with more accurate approximations. We describe the particle dynamical evolution due to gravitational scattering by means of the "scattering matrix" W(a, e, a', e') in the continuity equation. This matrix determines both the probability of transition and the value of the particle's shift from the point a, e to the point a', e'. For purposes of illustration, two cases of the zodiacal particle diffusion due to gravitational scattering are computed, which are characterized by the initial conditions (1) (asteroid case), a0 = 2.5 AU, e0 = 0.4 (resonance 1:3 with Jupiter), and (2) (comet case), a0 = 2.22 AU, e0 = 0.846 (comet Encke). We discuss the approximations of the example and how they might be improved in future work. These include treatments of the Poynting-Robertson drag, interparticle collisions, secular perturbations, three-dimensional orbits, and resonance capture by the planets. For instance, analytical expressions are available for the rates of gradual change of orbital elements due to the Poynting-Robertson and solar wind drags, which can be incorporated easily in the "div" terms of the continuity equation.