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Pro ceedings IMC Frasso Sabino 1999

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Leonid Dust Trail Theories
D.J. Asher
Armagh Observatory, College Hill, Armagh BT61 9DG, UK dja@star.arm.ac.uk

Abstract

At the end of the 19th century, Irish astronomers Johnstone Stoney and A.M.W. Downing showed in principle how to predict meteor storms. Their work and later developments applying similar ideas are reviewed here. The theories essentially involve determining the dynamical b ehavior of dust trails, which are narrow, dense structures within a broader stream. The formation of dust trails and the main features of their evolution are explained. Some Leonid dust trails are p erturb ed by the planets so as to result in meteor storms at the start of the 21st century.

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A hundred years: from Stoney and Downing to the 1999 Leonid storm
IRAS trails and meteor storms

1.1

In 1983 the Infra-Red Astronomical Satellite (IRAS) discovered narrow, dense trails of meteoroids and dust near the orbits of some known short-perio d comets. It was clear that there are similar trails asso ciated with many perio dic comets, those detected by IRAS generally being in cases where the comet was near perihelion during the IRAS mission. It was recognized that the Earth's passages through such dense trails correspond to the o ccurrence of meteor storms (see Sykes & Walker 1992, Kresґk 1993). As the trails are a narrow, it follows that at any given time most trails will not intersect the Earth's orbit in 3-dimensional space. However, the gravity of the planets continuously perturbs the orbits of dust and meteoroid particles in the trails, and from time to time, a trail (or a subsection of a trail) is brought to Earth intersection. A meteor outburst results, provided that the Earth and the particles in that subsection of the trail reach their intersection point at the same time. Therefore by calculating the gravitational perturbations on sections of trails that might come near the Earth, meteor storms can be predicted. This idea has been considered at various times, albeit not always using the term `dust trail' to describe the theory involved. Indeed, some of the studies in question date back to before IRAS detected the cometary dust trails. The orbit of the Leonid parent comet, 55P/Tempel-Tuttle, comes closer than most comet orbits to the Earth's orbit. The total perturbation required to bring Leonid dust trails to Earth intersection is therefore smaller than for trails asso ciated with most other comets, and the Leonid stream provides a primary candidate for meteor storms.

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1.2

Stoney and Downing's ortho-Leonids

From the impressive Leonid displays of the 1860s, and others at earlier epo chs, it was known that during every cycle of 33 yr, there were at least two or three successive years carrying some possibility (but not certainty) of a Leonid storm. Stoney & Downing (1899a) wrote The ortho-Leonids at present form a compact stream of such a length that it takes nearly three years to pass each point of its orbit, and so narrow that when the earth passes obliquely through it the transit o ccupies only some five or six hours; whereas the clino-Leonids form a less dense and wider stream, which has spread itself the whole way round the ring, and which pro duces in every November, when the earth passes through it, a feeble meteoric shower that lasts for several days. The authors use the term `ortho-Leonids' to refer to what, in the light of the phenomenon observed by IRAS, I am calling a dust trail, and `clino-Leonids' to refer to the background Leonid stream. Current knowledge of cometary physics makes it clear that new meteoroidal material, capable of forming into a trail, is being released at 33 yr intervals, every time 55P/Tempel-Tuttle returns to perihelion. In addition, the availability of digital computers, to enable fast calculation of the dynamical evolution of the meteoroids, allows us to know now that not just a single compact stream of ortho-Leonids, but rather several separate Leonid trails (each formed at a different perihelion passage), exist simultaneously, and that they can lengthen to take even more than three years to pass a given point (see Section 2.1). Of course, Stoney and Downing recognized that some kind of dense, compact structure, which gives rise to storms, exists within the background stream. But more than this, they identified all the essential points about the gravitational perturbations on this `ortho-stream': . . . the meteors which o ccupy successive positions in the pro cession are differently affected by the surrounding planets . . . The dense part of the stream, with which we are chiefly concerned, and which we may call the ortho-stream, is now so long that it takes between two and three years to pass each point in its orbit, so that the configurations in which the several parts are presented to the disturbing planets are markedly different. Accordingly, perturbations must have pro duced in this long stream both sinuosities and an unequal distribution of density . . . The most important idea is differential perturbations. If two points are two or three years apart, their separation in space can increase to several astronomical units, and there may at some time be substantial perturbations from, for example, Saturn, at one point, while the other point is at that time to o distant from Saturn for the same effect to o ccur. As successive sections of a trail are perturbed differently, only a particular section might be perturbed to intersect the Earth's orbit. As regards meteor storms, the relevant question is whether the various sections of each Leonid trail that reach the ecliptic in each mid-November (i.e., short sections of each trail spaced one year apart) intersect the Earth's orbit or whether they miss it (Section 2.2). Stoney and Downing showed how, in principle, exact predictions of meteor storms can be made. In the above quote they refer to `sinuosities'. This is a corollary of the differential perturbations: particles at different points along a trail are each perturbed on to slightly different elliptical orbits, and the perturbations can shift trail sections either inside or outside of an

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Figure 1: Schematic diagram of part of a trail, illustrating Stoney and Downing's ideas. The comet was lost in 1899, but is now known to have crossed the ecliptic in 1899 July. The segments of the trail indicated B and A respectively cross the ecliptic in 1899 November and 1900 January. Different segments of a trail undergo different planetary perturbations and so intersect the ecliptic at different points, although if one segment is not to o far along the trail from another then the ecliptic crossing points may be close to each other. Since the Earth go es through the Leonid stream in November, it is B that is relevant for determining whether a meteor storm o ccurs. In fact, all trail segments, that crossed the ecliptic in 1899 November, did so at a large distance from the Earth's orbit (Figure 3), and so no meteor storm o ccurred.

idealized `mean orbit'. Knowing that several trails co exist, the overall dust trail structure can be pictured as multiple, sinuous (or wavy) strands flowing through the Leonid stream as a whole. Stoney and Downing applied the theory to the eagerly awaited 1899 Leonids, taking as their starting data point the section of the trail (ortho-stream) observed in the great 1866 display, which they termed `segment A'. They adopted the orbit that Adams (1867) determined, from the 1866 radiant and an orbital perio d based on the long term cyclicity in Leonid storms. Evaluating perturbations through to the 1899 epo ch, they calculated that segment A was shifted more than 100 Earth diameters inside the Earth's orbit, and noted that the ortho-stream would therefore have to be at least that wide for the Earth to encounter it in 1899. Stoney & Downing's (1899b) suggestion, that the ortho-stream probably was that wide, was based on a theory (discussed by Stoney 1899) that the Leonids were originally drawn into the solar system by Uranus. We know to day that Leonid meteoroids originate during cometary activity around perihelion and that trails are not that wide. Stoney & Downing (1899a) also calculated the longitude at which segment A would pass through the ecliptic. If there were to be an encounter (i.e., if the ortho-stream were wide enough), the date at which the Earth reached this longitude would give the peak time of the meteor outburst. This calculation of the ecliptic crossing point (cf. Figure 3 below), both its longitude, and its miss distance inside or outside the Earth's orbit, is the same pro cedure that enabled the prediction of the 1999 Leonid storm (Figure 8 in Section 2.2). The perturbation calculations showed that segment A would pass through the ecliptic on 1900 January 27, and that strictly, more relevant for 1899 November would be the ecliptic crossing point of a section of the ortho-stream (`segment B') a couple of months ahead of A (Figure 1). The authors noted that their conclusion therefore

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Figure 2: Cross sections of dust trails that passed through the ecliptic in 1866 November. Each trail is labeled by the year (of perihelion return of 55P/Tempel-Tuttle) when it was generated. Trail cross sections are discussed by McNaught & Asher (1999a, b), but here they are idealized as ellipses. The cross indicates where the comet passed through the ecliptic (at its descending no de) in 1866 January, and do es not represent a dust trail; although an incipient trail was present in 1866 January (due to material being released at that perihelion passage), it would not have extended far from the comet and so was not there in 1866 November. The Earth's orbit is shown, and the Earth's position, exaggerating its physical size 10 times, at dates (Universal Time) in 1866. If pro jected on to the ecliptic, Leonid trails would in this region run nearly parallel to the Earth's orbit, with Leonid meteoroids traveling in the opposite direction to the Earth; Leonid orbits are inclined 18 out of the ecliptic. . . . rests on two assumptions: (1) That segments A and B were, in 1866, moving in orbits that did not much differ; (2) That the perturbations which segments A and B have since suffered have not much differed. To day we adjust our starting data point (analogously to considering segment B instead of segment A) and quickly run another iteration of a computer program that evaluates perturbations. The expense of carrying on the work has been met partly Grant administered by the Royal So ciety, and partly out Donation Fund. The computations have been made by J. H. Bell, and W. H. Walmsley, members of the staff of office . . . out of the Government of the Royal So ciety's Messrs. F. B. Co oper, the Nautical Almanac

A hundred years ago, one would probably have had to request an extra research grant! The authors therefore drew their conclusions from their segment A calculations, although later (Stoney & Downing 1900) they performed calculations for ortho-Leonid segments a year ahead and a year behind, i.e., relating to the 1898 and 1900 Leonids. But in any case, an extra iteration, i.e., computing segment B to check their assumption (2), could not have solved the problem completely, because the accuracy of the result is limited by the accuracy

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Figure 3: Cross sections of dust trails that passed through the ecliptic in 1899 November. All these trails except 1866 appear in Figure 2 but the cross sections are at different points along the same trails. The cross indicates where the comet passed through the ecliptic in 1899 July. of the starting data -- cf. their assumption (1). Stoney and Downing realized the inherent uncertainty in an orbit determined from the 1866 radiant, and a direct determination of an orbit for the unobserved segment B was not, of course, possible. To achieve the highest precision, the essential starting data are the orbital elements of the comet at each of its most recent perihelion times (i.e., not only in 1866, but in 1833, 1800 and a few more), since it releases new meteoroids on each return. However, the known observations of 55P/TempelTuttle at the end of the 19th century covered just seven weeks during the discovery apparition in 186566. Although Oppolzer's (1867) orbit was the best fit to those available observations, and indeed allowed the connection between Tempel-Tuttle and the Leonids to be recognized, more recent determinations using observational arcs of centuries rather than weeks show that Oppolzer's orbital perio d differed by 0.3 yr from the genuine value in 186566. This would therefore have limited the accuracy of the findings even if Stoney and Downing had used the comet orbit. Nevertheless, Stoney & Downing's (1899a) paper can fairly be regarded as a landmark in understanding the cause of meteor storms, demonstrating the effect of planetary perturbations on no dal crossing points. For the purpose of visualizing trail crossing points, it is perhaps of interest to plot trail cross sections in the ecliptic plane. Figures 2 and 3 depict the situations in November 1866 and 1899. Figure 2 shows that the largest contribution to the 1866 storm came from meteoroids in the 1733 trail, which was then 4 revolutions old. The plotted cross section of that trail can be taken as analogous to Stoney and Downing's segment A (since A was based on the 1866 storm), in which case the cross section of the 1733 trail in 1899 November (Figure 3) corresponds to their segment B. The correspondence should not be regarded as exact, since data for these Figures have been calculated by using accurate orbital elements for the comet at 1733 and other epochs, which are available now (e.g., Yeomans et al. 1996, Nakano 1997, 1998) but were not in 1899. However, the effect of perturbations calculated by Stoney and Downing for segment A, the shifts in position both inside the Earth's orbit and to greater longitude, are within 10% of the displacement

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between the 1733 trail cross sections in Figures 2 and 3. This quantitatively validates their assumptions (1) and (2) mentioned above, albeit discrepancies of this size must be removed for high precision storm prediction.

1.3

Recovery of Temp el-Tuttle, and further trail studies

Based on Schubart's (1965) prediction, observers were just in time to recover 55P/TempelTuttle at its 1965 apparition before it disappeared back to the outer solar system for another thirty years. With the resulting greatly increased accuracy in its orbit, significant advances in Leonid storm calculations were possible. Knowledge of the comet orbit (which is perturbed just as particles in dust trails are) at each perihelion return constrains the initial orbits of meteoroids in the trail generated at that return, and so is necessary for predicting meteor storms. Of course, the comet orbit alone is not sufficient, because the perturbations on the comet and on (the relevant sections of ) trails are different, i.e., one must evaluate the latter. Orbits (i.e., sections of trails) have to be identified with an appropriate perio d that brings them to cross the ecliptic in mid-November. Although identifying these orbits is not completely trivial because perturbations cause the perio d to change continuously (Section 2.1), they can generally be identified reasonably quickly using an iterative pro cedure. Evdokimov & Kondrat'eva (1972) found that a suitable orbital difference from the comet in 1800 could lead to an ecliptic crossing very near the Earth in 1833 November (the comet itself crossed the ecliptic in 1833 January), and similarly traced an orbital difference in 1899 through to 1966, although the accuracy achievable at the time meant that these solutions were not proved uniquely. Among the papers noted in Rao's (1999) review of Leonid dust trail work is that of Upton (1977). Upton drew attention to the differing perturbations acting on material released at different perihelion returns (i.e., in separate trails), finding which trails caused the great 1833 and 1966 displays, and correctly predicting the occurrence and cause of the 1999 storm. The accuracy of Evdokimov & Kondrat'eva's study was improved by Kondrat'eva & Reznikov (1985), and the calculations of Asher (1999) and Brown (1999a), the latter based on a full mo del of the Leonid stream including detailed consideration of the ejection mo del, also identified the 1800 and 1899 trails as the primary components respectively contributing towards the 1833 and 1966 storms. Kondrat'eva & Reznikov (1985) also predicted the Earth's encounters with the 1899 trail in 1999, and with the 1866 trail in 2000, 2001 and 2002. Those last three encounters are thus with sections, spaced a year apart, along the same trail. By the time of the 1999 Leonids, Kondrat'eva et al. (1997), McNaught & Asher (1999a), and Lyytinen (1999) had each generated a list of those and other encounters. They all evaluated the differential gravitational perturbations on orbits of different perio d, it being the different perio d, whether due to ejection velo city or radiation pressure, that causes particles to separate along a trail (Section 2.1). The calculations yield a theoretical ecliptic crossing point, whose longitude and miss distance from the Earth's orbit (cf. Sections 1.2, 2.2) are of interest. Kondrat'eva & Reznikov (1985), McNaught & Asher (1999a) and Lyytinen (1999)'s predictions of the 1999 Leonid peak time all agreed, ranging from Nov 18, 02h 08m to 02h 10m UT, with Upton's (1977) calculated time being `close to 2 am.' Other research applied the same principles to other streams capable of pro ducing outbursts. Davies & Turski (1962) considered whether ejections from 19th century perihelion passages of 21P/Giacobini-Zinner could have led to the 20th century storms and outbursts in the

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Draconid (Giacobinid) shower. Although later work showed that 20th century ejections (i.e., dust trails generated during 20th century returns of the comet) were relevant for such meteor displays as 1933 and 1946, so that Davies and Turski's paper was not the definitive answer to the Draconid problem, they illustrated how orbits of differing perio d undergo differential gravitational perturbations. Their work was thus similar to so-called dust trail theories. Further investigations (references in Reznikov 1993) were done by Evdokimov and also by Reznikov into the cause of the 1933 and 1946 Draconid storms, and subsequently Reznikov's (1993) paper was quite an extensive study of the possibilities for Draconid outbursts from 1985 to 2025. Reznikov's perfect calculation in advance of the 1998 Draconid outburst (relating to the dust trail generated at the 1926 return of 21P/Giacobini-Zinner) demonstrated the accuracy of the storm prediction technique. Similar investigations can be done into other streams. The technique has been applied by Reznikov (references in Kondrat'eva et al. 1997) to the Bielids and June Bo otids, and by Lyytinen (2000) to the Perseids. Altogether, meteor storm prediction has now become achievable at a high level of precision.

2
2.1

Dust trails
Why dust trails exist; density distribution along trail

When meteoroids escape from a comet, the velo city they have relative to the comet nucleus causes their orbits to differ slightly from the comet's orbit. Although there are differences in all the orbital elements, the important parameter, as regards the formation of a trail, is the orbital perio d. If a meteoroid's perio d differs by P0 from the comet's perio d P0 , then they gradually separate along the orbit, the amount of time by which the meteoroid lags the comet after n revolutions being T = n P0 (1 ) More often than not, the semi-ma jor axis a is used rather than P , in work on orbit evolution. They are equivalent by Kepler's 3rd law of planetary motion P = a3/2 Using (2), (1) becomes T n 3 1/2 a a0 20 = constant в n в a0 (2 )

(3 )

a0 can be negative or positive: obviously, meteoroid particles with smaller and larger perio d than the comet respectively get ahead of and fall behind the comet. Since the particles released during any one perihelion passage will cover a range of values of a0 , the result is that the particles must gradually stretch into a trail (Figure 4), with the extent of the range in a0 determining the rate at which the trail lengthens. The quantities T , a0 and n are related by equation (3). If one considers a fixed displacement (in time) behind the comet, e.g. T = +1 yr, then the particles at that point are continuously changing, i.e., different

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Figure 4: Evolution of a Leonid dust trail, shown after 1, 2, 3 and 4 revolutions, neglecting radiation pressure and gravitational perturbations. The x-axis is the time by which particles lead or follow the comet. In this representation, particle density in the trail is inversely proportional to spacing between lines. Two points to note are: (i) At any one time, the density profile along the trail shows a concentration close to the comet. The extent of this concentration could be adjusted (compressed or expanded) depending on the assumed mo del of ejection from the comet nucleus (in general, smaller ejection velo cities give a smaller range of meteoroids' orbital perio ds, and compress the concentration towards the comet more), but here is in reasonable accord with expected ejection velo cities for particles in a Leonid trail that would pro duce visual meteors. (ii) The density becomes diluted with time, since the separation between particles increases (any particle with a longer orbital perio d gets progressively further behind a particle whose perio d is shorter). In this approximation of no gravitational perturbations, the trail necessarily lengthens at a constant rate [see equation (3)]. particles are present at n=1, 2, etc. If one considers a fixed group of particles, e.g., those with a0 = +0.1, then their time lag relative to the comet is greater when n=2 than n=1, and so on. Figure 4 schematically shows the variation in particle density along the trail. There is also a density variation across the trail, discussed later in Section 2.3. The density along the trail is the pro duct of two terms: fa (a0 ) fM (4 ) The first depends on the initial distribution of particles as a function of a0 [cf. point (i) in Figure 4 caption]. The second term depends on the evolution of the trail [cf. point (ii)], McNaught & Asher (1999a) using the notation fM for `mean anomaly factor', since particles gradually stretch along the trail in mean anomaly, diluting the density, as the trail evolves. To a first approximation, the trail stretches linearly with time as shown by equation (3), and so fM = 1/n.

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Figure 5: Evolution of a Leonid dust trail, shown after 1, 2, 3 and 4 revolutions, neglecting gravitational perturbations, but allowing for radiation pressure at a suitable average value for particles that pro duce visual meteors. Cf. Figure 4.

Considering a particle distribution as a function of a0 rather than of T has the advantage that fa (a0 ) can be related directly to ejection mo dels, whereas T immediately after ejection is zero for all particles. Meteoroid ejection pro cesses determine the distribution fa and so the adopted fa is mo del dependent. However, realistic ejection pro cesses are likely to lead to a distribution that concentrates towards the cometary value a0 . In Figure 4 an idealized distribution (Gaussian) in a0 was assumed (cf. Figure 2 of McNaught & Asher 1999a), with a width that would result from ejection speeds of up to a few tens of m s-1 . These speeds are reasonable for the size of Leonid meteoroid capable of pro ducing a typical visual meteor (Arlt & Brown 1999). If solar radiation pressure is negligible, then fa is expected to be centered at zero, a0 = 0 corresponding to `Position of comet' in Figure 4. However, sunlight falling on meteoroids exerts a force on them. As radiation pressure obeys an inverse square law like gravity, the effect for any given meteoroid is equivalent to reducing the Sun's gravity by a constant proportion (by a higher proportion for smaller particles). That is, the Sun is effectively pulling the meteoroid around its orbit less strongly, and so the meteoroid takes longer to complete each revolution. The effect is small but not negligible for visual meteor sized particles. Therefore imposed on the range of orbital perio ds above and below the cometary value (Figure 4) is an overall tendency for perio ds to be longer, and trails to shift backwards relative to the comet (Figure 5). If a section of a given trail intersects the ecliptic in mid-November of a given year, the value of a0 of this trail section is essentially specified by equation (3), since T is the time lag from the comet reaching the ecliptic to mid-November. In a rigorous calculation, gravitational perturbations are included, rather than simply using (3). It seems easiest (cf. McNaught & Asher 1999a) to quote a0 for purely gravitational solutions and to be aware that a

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gravitational solution at, e.g., a0 = +0.2, corresponds to a solution including gravitational perturbations and radiation pressure at a0 = 0. Since the peak density in reality is at a0 = 0, and since in reality radiation pressure exists, the peak density (i.e., the maximum of the function fa ) when the mo del has been expressed in terms of purely gravitational solutions will be at some positive value of a0 . Thus the offset of the maximum of fa from a0 = 0 depends on radiation pressure, while we saw above that the width (in a0 ) of the fa distribution depends on ejection pro cesses. Both these depend on the size and density of particles, but as a first approximation, suitable values for visual meteor sized particles (i.e., a single value for the width and a single value for the offset) can be assumed (in fact can be fitted from past storms; Section 2.3). For the first couple of revolutions in the evolution of a trail, (3) usually holds to a very go o d approximation, i.e., T is proportional to a0 at any one time. However, as n increases, (3) becomes more affected by planetary perturbations, which continuously change a of the comet and of meteoroids. Moreover, the effect on a varies along the trail, and therefore different sections of a trail stretch by different amounts (as the amount of stretching is related to T ). Equation (3) is helpful for understanding how trails form, allowing a convenient expression for T since a0 is a constant for a particular section of a trail (a0 relates to a single time, namely the ejection time, whereas a would be a continuously changing function), but in practice, perturbations must be calculated. Thus perturbations cause fM to be a function of position along a trail, and the density to be uneven. This effect on fM can, however, be evaluated precisely, i.e., whereas fa is mo del dependent, fM is not. Although (3) becomes increasingly inaccurate, T do es at least remain a smo othly increasing function of a0 for quite a few revolutions, except in a few short, isolated ranges of a0 . A trail tends to be disrupted in these short ranges, as was suggested by Stoney & Downing (1 8 9 9 a ): Through one of these [segments of the ortho-Leonids] the earth passed in November 1866, and on that o ccasion there was withdrawn from it that small portion which consisted of meteors which either encountered or passed close to the earth. Those that actually plunged into the earth's atmosphere were destroyed: those that passed near were deflected, and were also either accelerated or retarded, and they thus became clino-Leonids. Thus on each successive perihelion passage, small sections (spaced a year apart) of each trail are scattered into the Leonid background. Although the pro cess requires more detailed mo delling, it appears that trails are significantly scattered, to the extent that it is no longer appropriate to describe them as single, coherent entities, after a dozen revolutions or so. Therefore perturbations firstly cause density variations along trails by affecting fM , and secondly limit the lifetimes of trails. The limited lifetime means that about a dozen Leonid trails can exist simultaneously, one for each of the last dozen perihelion passages of the comet.

2.2

Trail intersections with ecliptic

Intersection points of dust trails with the ecliptic were shown in Figures 2 and 3 for 1866 and 1899, and can be calculated for mid-November of other years. For this purpose, an

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idealized mo del is used, in which transverse ejection at perihelion is assumed, because this is sufficient to generate particles with values of a0 differing from the cometary value. An effective difference in a0 (i.e., an actual difference in orbital perio d) can equivalently be pro duced by radiation pressure (cf. Kondrat'eva & Reznikov 1985, Williams 1997, Asher 1999, Lyytinen 1999). The calculated intersection points are very similar whether a0 is due to ejection velo city, radiation pressure, or a combination of the two. Results are shown in Figures 611. In Figures 67 (and also in Figures 23 earlier), trails 1 to 6 revolutions old are plotted, and in Figures 811, trails 1 to 9 revolutions old are shown. Within these limits, trail cross sections are absent only if no part of the trail crosses the ecliptic in mid-November of that year. For example, in Figure 7, the section of the 1899 trail that supposedly had the correct orbital perio d to bring it to the ecliptic in 1998 November had its evolution disrupted because of a fairly close approach to the Earth in 1965 (cf. last paragraph of Section 2.1). For ease of comparison in seeing no dal displacements caused by perturbations, all the Figures depict the same area in absolute space, the drift in dates over a hundred years being due to precession of the equinoxes. Progressing through Figures 711, a pattern is evident whereby after the comet has passed, the individual dust trails separate, i.e., for larger T , the ecliptic intersection points of the various trails are distributed over a wider region. In Figure 3, with T being only four months, the trails were quite close to the comet no de. In 3-dimensional space, the trails are wavy lines converging on the comet. Encounters with the same trail, the one generated at the 1899 return, when it was respectively 2 and 3 revolutions old pro duced the 1966 and 1999 storms (Figures 6, 8). This theory of young dust trails predicts no storm in 1998 (Figure 7), and indeed the peak Leonid ZHR was well below storm level (Arlt & Brown 1999). The impressive fireball outburst was due to larger meteoroids sub ject to different dynamical behavior asso ciated with a resonance, over longer time-scales than the lifetimes of a few centuries for the dust trails being described here (Asher et al. 1999, Arlt & Brown 1999). This material ejected a longer time ago can also be regarded as forming into trails, but these are older, `resonant trails', as contrasted with the younger trails that pro duce storm level activity and that are the sub ject of the present article. The `resonant zone' containing 55P/Tempel-Tuttle and the highest concentration of large, resonant meteoroids has moved on and will not return until the next apparition of the comet. More relevant for the next few years are the young dust trails whose cross sections are shown in the Figures. It is possible that more detailed modelling can determine whether the miss distances involved in the trail encounters in 2000 (Figure 9) will be small enough to pro duce storm level activity; however, although outbursts will o ccur, it may be better at present to be cautious about their intensity (see Section 2.3). In 2001 and 2002 (Figures 10, 11), there will certainly be encounters at small enough miss distances to pro duce storms.

2.3

Density cross section

Although planetary perturbations can be accurately evaluated, there is more inherent uncertainty in defining a meteoroid ejection mo del. A priori it is unclear whether this uncertainty would invalidate the dust trail mo del, for practical purposes. The pro of, of whether the idealized mo del can genuinely predict meteor storms, lies in comparisons with past storms (McNaught 1999a): a successful theory must predict storms when they o ccur, must predict lack of storms when they do not o ccur, and must be able to predict the timings of storms that o ccur. The mo del determines the helio centric distance and longitude of ecliptic crossing points of trails (Sections 1.2, 2.2). The distance component relates to the o ccurrence or

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Figure 6: Cross sections of dust trails that passed through the ecliptic in 1966 November. The comet reached its descending no de, shown as a cross, in 1965 May. Cf. Figure 2 for more details.

Figure 7: Cross sections of dust trails that passed through the ecliptic in 1998 November. The comet reached the point shown in 1998 March.

Figure 8: Cross sections of dust trails that passed through the ecliptic in 1999 November.

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Figure 9: Cross sections of dust trails passing through the ecliptic in 2000 November.

Figure 10: Cross sections of dust trails passing through the ecliptic in 2001 November.

Figure 11: Cross sections of dust trails passing through the ecliptic in 2002 November.

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non-o ccurrence of storms (they o ccur if the distance from the Earth's orbit is small) while the longitude component relates to the timing. Accurate timings of the peaks of past storms are available (see Jenniskens 1995, Brown 1999b). Comparisons of mo del predictions to observed timings were done by McNaught (1999a) and McNaught & Asher (1999a, b). Including a topo centric correction, past outbursts with sharply defined peak times are fitted to an accuracy of 5 minutes. A similar accuracy applies to other streams to o, e.g., the time of the observed peak of the 1998 Draconid outburst (Arlt 1998) compared to the prediction by Reznikov (1993). These accuracies of a few minutes can be compared to the hours or days that the Earth takes to traverse streams, during which outbursts can potentially o ccur at any time. It was therefore clear that the peak of the 1999 Leonid display would o ccur within a few minutes of the predicted time. In 1999 this was also within 20 minutes of the time when the Earth crossed the orbital plane of 55P/Tempel-Tuttle. The years 20002002 will show discrepancies of order a day compared to crossing the comet plane. It remains to comment on how close a trail encounter has to be in order to pro duce a sharp outburst, the `sufficiently close' encounters being those used in the timing comparisons with past storms. The quantity of interest is rE - rD , where rE and rD are the helio centric distances of respectively the Earth at the encounter, and the descending no de of trail particles involved in the encounter. In Figures 2, 3, 611, rE - rD is the distance from the center of an idealized trail cross section to the Earth's orbit. The observational evidence is that meteor storms o ccur when the miss distance |rE - rD | is less than a few Earth diameters or so. This is in accord with the size of trail cross section expected, for ejection velo cities that correspond to visual meteor sized particles (cf. Section 2.1). The cross section is increased just a little by radiation pressure. McNaught & Asher (1999a) considered whether the strength of storms, as measured by the ZHR at maximum, relates quantitatively to rE - rD , given that a density profile across a trail would be expected that increases sharply towards the center of the trail. The intensity of a meteor outburst would have a strong dependence on how closely the Earth encounters the center of the trail. Although numerical simulations of the ejection pro cess could pro duce a rather detailed mo del, a more straightforward mo del is to assume that the peak ZHR depends only on rE - rD (as well as on the along-trail parameters discussed in Section 2.1). Moreover, it is assumed that cross sections at all points along all trails are the same, sub ject to an overall normalizing constant accounting for the along-trail density given by (4); there are in fact theoretical reasons to expect this assumption not to be exactly true, but again the alternative would be to generate a detailed ejection mo del (probably a range of such mo dels to cover uncertainties about the ejection pro cess). The maximum density (or peak ZHR) reached during a trail encounter (or meteor outburst) is therefore [cf. (4)]: (5 ) fr (rE - rD ) fa (a0 ) fM with fr corresponding to a density profile that increases towards the center of the trail. McNaught & Asher (1999a) to ok fr as a Gaussian centered at zero. They also considered a Gaussian with the center slightly outside (i.e., further from the Sun than) the idealized rD of the trail, as radiation pressure could move the true center very slightly outside the idealized center. Calculation of the idealized center of course includes gravitational perturbations, which for visual meteor sized particles have a much greater effect than radiation pressure on the lo cation of the ecliptic crossing point. McNaught & Asher (1999a) fitted the offset (from

Pro ceedings IMC Frasso Sabino 1999

19

Table 1: Leonid trail encounters in forthcoming years. Visibility maps were plotted by McNaught (1999b). Time of maximum (UT) 2000 Nov 18, 03h 44 2000 Nov 18, 07h 51 2 2 2 2 0 0 0 0 0 0 0 0 1 1 1 2 Nov Nov Nov Nov 1 1 1 1 8 8 8 9 , , , , 1 1 1 0 0h 7h 8h 4h 0 3 1 0 1 1 9 0
m m

Trail 8-rev 4-rev 7 9 4 7 r r r r

Peak ZHR 100? 100? 5 0 0 0 0 0 0 0 0? 0 0 0

Mo on Visible from age 22 22 3 3 3 15 15 28 W. Africa, W. Europe, NE S. America N. America, Central America, NW S. America N. & Central America Australia, E. Asia W. Australia, E., SE & Central Asia W. Africa, W. Europe, N. Canada, NE S. America N. America W. Europe, W. Africa

m m m m

ev 2 ev 9 ev 15 ev 15

2002 Nov 19, 10h 36 2006 Nov 19, 04h 45

m m

4-rev 30 000 2-rev 100

a0 = 0) and width of fa , and the width of fr , from the estimated observed peak ZHRs of the 1833, 1866, 1867, 1869 and 1966 storms and outbursts. The fitted parameters can then be applied to forthcoming trail encounters to predict the peak ZHR. The 1969 outburst is not used in the fit because it was at the comparatively large a0 of +0.9 and it is unlikely that a single Gaussian function fa is valid over such a wide range of a0 . For reference, data relating to the next few years are listed in Table 1. The estimated ZHRs may in due course be adjusted if the work is refined. The times are unlikely to be revised, and certainly not by more than a few minutes. In 2006 there will be a similar a0 to 1969, and so the peak ZHR for the 2006 encounter alone is estimated simply from comparison with 1969 alone. The ZHRs in 2000 are especially uncertain using a fit based on equation (5) because the rE - rD values in the years used in the fit (1833, 1860s, 1966) all happened to be negative, whereas rE - rD will be around +0.0008 AU in both 2000 encounters, so that the calculation is very sensitive to asymmetries in fr . A storm or lack of storm in 1801 may be relevant to 2000 (McNaught & Asher 1999a). Meteor storms will o ccur in both 2001 and 2002, with 2001 being favored observationally owing to the ideal lunar phase. In (5) there is no evolutionary term relating to the across-trail profile (corresponding to fM for the along-trail profile): dynamical simulations show that the width of a Leonid trail at the descending no de do es not change with each successive revolution, for the first few centuries. This point is actually quite important with regard to why dense, narrow trails exist at all. Particles the same distance along a trail are basically co-moving around the Sun and so undergo similar perturbations. The differential perturbations do not apply to particles in a single, short section of a trail. A short trail section can be perturbed towards or away from the Earth's orbit but the particles are perturbed together rather than dispersing into a wider area. If trails became wider, then the Earth would run into them more often, but the ZHR would tend to be lower. Narrow trails increase the chances of storms with extremely high ZHR.

20

Pro ceedings IMC Frasso Sabino 1999

Acknowledgments
I am grateful to Vacheslav Emel'yanenko for bringing to my attention many of the papers that I have discussed here, to Ilia Roussev for translations from Russian, and to Rob McNaught for helpful comments on the material in this article.

References
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