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Figure 2. Integrations of particles ejected tangentially at perihelion in 1800. Time particles reach descending node 2 revolutions later, heliocentric distance of descending node rD, and longitude of ascending node 0 shown as functions of Aa0- Particles with Aa0 near +0.174 reach the ecliptic in 1833 November when the Earth is nearby and are grav-itationally scattered.

Figure 2 illustrates more details of the orbital evolution near one of these breaks in the 1800 trail, namely around the value of Aa0 for which the 1833 close approach occurred. The time of nodal passage is plotted (essentially equivalent to M in Figure 1) along with the two coordinates of the node, one revolution after the approach. The Aa0 for which the closest approach (0.0002 AU) occurred is clear in Figure 2. Particles having Aa0 within about ±0.002 of this value approached to within less than 0.01 AU. The further Aa0 is from that value (ahead or behind in the trail), the greater the miss distance from the Earth and the smaller the effect of the gravitational deflection on the subsequent orbit.

As only one parameter (tangential ejection velocity at perihelion, equivalently Aao with perihelion distance held fixed) is varied in this idealised model, subsequent orbital elements are a function of this parameter. Figure 3 shows that the time of nodal passage and the coordinates of the node are, within narrow limits, unchanged under a more realistic model that includes ejection away from perihelion and in arbitrary directions from the nucleus. No claim is made that the particular model used for Figure 3 is true, but it is representative of such realistic models, and specifically it is useful for demonstrating the how the width of trails is affected by dispersion in orbital elements.

Apart from at the centre of the plots (where the trail has been disrupted by the 1833 Earth approach), the elements in Figure 3 are dispersed by only small amounts about the single valued functions depicted in Figure 2. Thus away from disrupted (widely scattered) sections of trails, orbital elements can be calculated by considering the idealised trail (with tangential ejection at perihelion) only. The idealised model is sufficient for storm prediction, since storms are due to the highest density regions, which have not been widely scattered. This idealised trail has been defined [8] as the 'centre' of the real trail, which has a nonzero width. Although the idealised trail is a convenient representation, avoiding all need for physical models of ejection, it should be noted that the value of rp at the trail centre may be slightly displaced from the real mean ro of particles at that point along the trail [8]. The trail width (dependent on Aa0 and the ejection model used) is given by the ranges in rp and fi in Figure 3.

Moreover, comparing Figure 3 with Figure 4 shows that the width does not increase with time during the first few revolutions of trail evolution, but is instead due only to the range in ejection velocities around the arc of the comet orbit during which ejection occurs. Radiation pressure when coupled with ejection over that arc also makes a small contribution to the trail width, but this increased dispersion in nodal position due to radiation pressure (for Leonids with ¡3 = 0.001) is very small compared to typical shifts in nodal position that subsequently occur due to gravitational perturbations. Radiation pressure (constant (3 on any given particle) and planetary perturbations do not cause trails to widen with time for several revolutions, although it is possible that the Yarkovsky effect on spinning meteoroids can have a widening effect [9].

The fact that all particles in Figure 4 have very similar nodal passage time merely means that a trail has negligible length during the perihelion passage when it is formed. Within a revolution it has elongated greatly, and thereafter is a dense, narrow structure broken only at points where close approaches have occurred. To investigate the effect of approaches, the central part of Figure 3 had extra particles integrated. No significance should be attached to the increased density of points, which are to only provide better number statistics for illustrating the scatter. In reality particles would be expected to

Particles ejected in 1800

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