Meteor storms or sharp meteor outbursts occur when the Earth passes through dense, narrow dust trails of the type discovered by the Infra-Red Astronomical Satellite IRAS [1,2], Such structures exist since the dispersion in orbital period among meteoroids that have been ejected from a cometary nucleus leads to particles getting progressively further ahead of or behind the comet, thus stretching into a trail . A particle's orbital period differs from the comet's both because of the velocity relative to the nucleus induced during the ejection process, and because of the radiation pressure acting on the particle during its subsequent motion. The latter is parameterised by /?, the ratio of radiation pressure to solar gravity.
The ejection process induces a spread in all the orbital elements, not only the period. This causes a trail to have a nonzero width. For meteor producing streams, it is most relevant to measure this width (two dimensional cross section) in the ecliptic near r = 1 AU. Ejection velocities expected for the size of particles that produce visual meteors [4-6] are generally small enough that a trail's width, although nonzero, is rather narrow compared to the cross section of the stream as a whole. For particles having /? ^ 0, ejection away from r = 1 AU can also affect the position in the orbit at r = 1 AU and thus increase the trail width slightly.
Gravitational perturbations are a very important influence on the evolution of meteor streams, and trails within streams. Even over a single orbital revolution, they can alter the nodal position of particles by an amount that is significant compared to the stream's width, and even by several times the width of a single trail. It follows that perturbations can move a trail away from or towards Earth intersection, and that calculating them is essential for determining the occurrence of meteor storms.
As the node of the comet itself is similarly perturbed, particles released at different returns of the comet to perihelion have significantly different initial orbits. Additionally the configuration of the planets differs between different returns, and so the various sets of particles have different perturbation histories. Therefore the trails embedded in a stream, one corresponding to each previous return of the comet, tend to be separated in space, and the Earth may or may not encounter any of them during its annual passage through the stream. The trails only tend to converge in the vicinity of the comet itself.
Although different particles are perturbed differently, the perturbations on particles at a single point along a single trail are very similar. This is because particles in a single trail have been perturbed over the same interval of time (i.e., since ejection), and moreover in order to be at the given point along that trail, they have been comoving (whereas a different point along the trail may be months or years ahead or behind). Comoving particles are always in nearly the same position relative to each perturbing planet. The orbital period is continuously perturbed, but it is perturbed in the same way for comoving particles, which consequently remain comoving.
If, however, the particles at some point along a trail undergo a close approach to a planet, then the position relative to the planet is not the same for all those particles, even though they had been comoving around their orbits. This can cause that part of the trail to be scattered into the stream as a whole. This effect limits the lifetime of trails. But before a (part of a) trail is dispersed in such a way, it almost exactly retains its original width (this width being due to ejection velocities and radiation pressure), and the dilution of the density of particles in space is due only to the gradual lengthening of the trail in the along orbit dimension. This provides the potential for a sharp, intense meteor outburst if the Earth encounters a trail. While the trail still exists, the perturbations on any part of it can be precisely evaluated, so that the nodal position can be calculated and the occurrence or non-occurrence of a storm predicted.
The main purpose of the remainder of this paper is to exemplify the above principles in the case of the Leonids by means of suitably chosen numerical integrations.
2. EARLY EVOLUTION: PREDICTABILITY
As a first example we consider the Leonid trail generated when 55P/Tempel-Tuttle returned to perihelion in 1800. To produce Figure 1, particles were ejected at perihelion and tangential to the direction of motion, as this is sufficient to generate particles of any orbital period, and therefore any subsequent perturbation history (before the onset of chaotic behaviour). The quantity Aao, the difference in semi-major axis from the comet at the time of ejection, can be used to parameterise the period.
In the absence of differential perturbations and radiation pressure and if Aao is small, the difference in mean anomaly M after n revolutions is
Figure 1 illustrates this, approximately showing a slope of M against Aao that increases proportionally to the number of revolutions. However, there are two further features of
Figure 1. Early evolution of 1800 Leonid trail. Mean anomaly M shown as function of Aao (difference in semi-major axis from comet at ejection), after 1, 2, 3 and 4 revolutions of particles at Aao = —0.3 AU (after 4 revolutions, particles at Aao = +1-0 are over half a revolution behind). Comet shown by circle.
Figure 1 to be noted. Firstly, after a few revolutions there are gentle changes in the M vs Aa0 slope along the trail's length. This is because points at different distances along the trail have become substantially separated in space and therefore the gravitational effect (even moderately distant, i.e. away from close approaches) of Jupiter and Saturn is quite different. In particular, the orbital period is perturbed differently and so the rate at which different parts of the trail stretch (i.e., at which AM increases) varies. The second feature is small gaps at various points along the trail, to be discussed in Section 3.
The aim of Figure 1 is to illustrate dynamical evolution, not spatial density variations along the trail. The latter depend on the ejection process from the nucleus, since a distribution in ejection velocities corresponds to a distribution in Aao. For reasonable ejection models, the distribution will be centred around Aao = 0. However, a meteoroid susceptible to radiation pressure and with a given Aao (defining Aao as being calculated from instantaneous position and velocity without adjusting the attraction towards the Sun by ¡3) comoves around its orbit with a (3 = 0 particle of higher Aa0. An typical value of (3 for meteoroids that produce visual Leonids is ~0.001, corresponding to Aao~+0.2 . Therefore the peak in the particle density as a function of Aao will effectively be at ~0.2 (depending on ¡3) with significant levels at up to a few x 0.1 AU on either side.
Figure 1 shows that M is largely a well behaved function of Aao for a young trail, suggesting that the trail's evolution is predictable. Section 3 will confirm that the same is true of the nodal position.
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