Numerical Simulations

Forced elements are the component of a dust particle's osculating (or instantaneous) orbital elements imposed by the presence of the planets. These forced elements vary with heliocentric distance, time and particle diameter, and are responsible for the large-scale asymmetries observed in the distribution of dust particles in the zodiacal cloud. To calculate the current distributions of these forced elements, we employed the new dissipative code discussed above to evolve representative samples of asteroidal dust particles forward in time to the present epoch, along with the planets Jupiter, Saturn, Uranus, and Neptune, from a number of different epochs in the past. As the time scale for a dust particle orbit to decay under the effect of Poynting-Robertson and solar-wind drag is dependent on the particle size, each set of past epochs chosen was dependent on the size of the dust particles considered, and a separate set of integrations had to be carried out for each different particle size. Up to 80 past epochs were selected for each particle size, in order to provide a comprehensive picture of the forced element distribution of asteroidal dust particles across a wide range of semimajor axis values in the inner solar system at the present time. In this paper, we consider asteroidal dust particles (originating in this case from the Eos family, although this is not critical) composed of astronomical silicate of density 2,500 kg m-3 with diameters 10, 100, and 200 pm, for which we calculated (3 values (the ratio of radiation pressure to solar gravity) of 0.04871, 0.00446, and 0.00221 respectively, using Mie theory. The longest integrations performed for the 10-, 100-, and 200-pm diameter dust particles were for time scales of 0.06, 0.6, and 1.2 Myr, respectively.

To obtain initial orbital element distributions for our forward integrations we first employed a standard MVS integration code (incorporating point-mass gravitational forces only) to evolve Eos family asteroids, along with the gas giant planets, backwards in time from the present. Initial osculating orbital elements for 444 Eos family asteroids were obtained from The Asteroid Orbital Element Database [10] for the epoch of Julian Date 2450700.5, using the family classification of Zappala et al. [11]. Osculating orbital elements for the planets were obtained for the same epoch using the data from Sta.ndish et al. [12]. Using a low-order secular perturbation theory 'particle on a circle' approximation [13], we then generated initial osculating orbital elements for 124 dust particles, representative of the whole Eos asteroid family, at each of the past epochs required.

The results of the integrations presented in the next section represent a total of over 4 months CPU time running on a variety of Pentium processors.

Figure 1. Variation of the forced inclination (left) and the forced longitude of ascending node (right) with heliocentric distance at the present epoch (Julian Date 2450700.5) for Eos family dust particles with diameters 10, 100, and 200/im. The dashed lines show the present osculating inclination (left) and osculating longitude of ascending node (right) for Jupiter. Reprinted with permission from Dermott et al. [14]. Copyright 2001, SpringerVerlag.

Figure 1. Variation of the forced inclination (left) and the forced longitude of ascending node (right) with heliocentric distance at the present epoch (Julian Date 2450700.5) for Eos family dust particles with diameters 10, 100, and 200/im. The dashed lines show the present osculating inclination (left) and osculating longitude of ascending node (right) for Jupiter. Reprinted with permission from Dermott et al. [14]. Copyright 2001, SpringerVerlag.

3. RESULTS and CONCLUSIONS

The forced inclinations and longitudes of ascending node of some small (10 /jm diameter) and large (100 and 200 /xm diameter) Eos family dust particles, obtained using the new dissipative code, are shown in Fig. 1. All orbital elements are heliocentric and given with respect to the mean ecliptic and equinox of the standard J2000 reference frame. In the region of the main asteroid belt (between 2.5 and 3 AU), the forced elements of the large particles display similar behaviour to that of the small particles. That is, their forced inclinations and nodes are locked onto Jupiter's inclination and node, respectively. The low dispersion of the inclinations and nodes in this region of the main belt, regardless of particle size, is the fundamental reason why dust bands are observed at those heliocentric distances. However, as the large dust particles encounter the i/ie secular resonance at the inner edge of the asteroid belt (at about 2 AU), the effect of the resonance disperses their forced inclinations and nodes, diffusing the dust band particles into the broad-scale zodiacal background. The secular resonance (also at about 2 AU) produces analogous behaviour in the forced eccentricities and longitudes of pericentre of the dust particles. The effects of these secular resonances are more pronounced for the large dust particles because they are acted on by the resonances for longer periods of time. The orbital element distributions of large asteroidal dust particles produced by intra-family collisional attrition therefore lose their characteristic family signatures in the inner region of the main belt and become indistinguishable from the general background cloud of zodiacal dust. Consequently, the IRAS (Infrared Astronomical Satellite) dust bands have a natural inner edge at about 2.5 AU. The action of these secular resonances also means that large asteroidal dust particles in the inner solar system have orbits with significant eccentricities and inclinations and may be comparable to some cometary orbits.

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