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Recent shower calculations
D.J. Asher
Armagh Observatory, College Hill, Armagh, BT61 9DG, UK Email: dja@arm.ac.uk Meteor shower predictions are now achieving considerable levels of success. Recent examples included the activity peaks in the 2009 Perseids and Leonids providing good matches to forecasts made by various meteor astronomers. It can also be shown, from orbit geometry calculations combined with knowledge of meteoroid ejection processes from cometary nuclei, that young dust trails in different streams have differently shaped cross sections, and therefore different outburst activity profiles when the Earth encounters them. Whereas Leonid trails have somewhat elliptical cross sections in the ecliptic plane, Perseid trail cross sections resemble ­ to those with sufficient imagination ­ the shape of legendary forest creatures.

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Trail encounters

Meteor streams generally p ossess a considerable amount of fine structure. This structure often takes the form of long, dense, narrow trails of dust and meteoroids (see e.g. Asher 2000 for a review). Because they are narrow, the Earth misses most of them when it passes through the stream. But when the Earth passes through a trail, a meteor outburst or storm occurs. Dust trail calculations (e.g. Kondrat'eva & Reznikov 1985; Lyytinen et al. 2001; Maslov 2007; McNaught & Asher 1999; Sato 2003) are essentially a simplification or idealisation of a full dynamical simulation of a stream. This idealised technique is well suited to identifying individual trails that produce particular outbursts, esp ecially when the trails are reasonably young. Over time, the meteoroids in a trail eventually disp erse too much for their distribution to b e calculated by such a technique. However, in situations where dust trail theories can b e used, they are in good agreement (Vaubaillon et al. 2003, 2004) with more detailed dynamical models. Recent showers where the accuracy of dust trail calculations can b e assessed include the 2009 Perseids and 2009 Leonids. (At the time when the IMC talk on which this pap er is based was given, the Perseids had already happ ened and the Leonids were forthcoming.)

1.1

2009 Perseids

Calculations for the three youngest trails give the encounter parameters for the 2009 Perseids shown in Table 1. Taking the 1992 return of the Perseid parent comet 109P/Swift-Tuttle as the current p erihelion passage, the 1, 2 and 3-revolution trails consist of meteoroids released around resp ectively the comet's 1862, 1737 and 1610 returns. The parameters a0 , rE - rD and fM are describ ed in more detail in McNaught & Asher (1999) and Asher (2000). Briefly, a0 parametrises the amount by which the meteoroids' and parent comet's orbital p eriods must differ, at the time when the meteoroids are ejected, in order for the Earth to b e encountered n revolutions later (a denotes semi-ma jor axis, related to p eriod by Kepler's Third Law). The quantity rE - rD , given in astronomical units (au), is p ositive or negative when the Earth passes resp ectively outside or inside the centre (generally the highest density region) of the trail, relative to the Sun. The meaning of `trail centre' in this context is discussed further in Section 2. Closer encounters have smaller values of |rE - rD | or `miss distance'. The fact that the 3-rev trail was encountered (meteors were observed) at a miss distance of 0.0009 au, roughly ten Earth diameters, means that the trail is at least this wide. The parameter fM is prop ortional to the spatial density of meteoroids in the along-trail direction, i.e. smaller fM means the trail is more stretched (the density is more diluted) along the orbit.
IMO bibcode IMC-2009-asher


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Table 1 ­ Encounter parameters of Perseid trails with Earth in 2009, calculated using a similar dust trail computer program to that used by McNaught & Asher (1999). Trail 1 - r ev 2 - r ev 3 - r ev a0 +2.18 +1.49 +0.77 rE +0 -0 -0 - rD .0034 .0047 .0009 fM 1.12 0.18 0.05 Time Aug 12 Aug 12 Aug 12 (UT) , 04:55 , 06:45 , 07:50

The data in Table 1 agree with Mikhail Maslov. Although the an ejection velocity comp onent results are essentially the same. 2008).

trail encounter parameters already calculated over the past few years by data listed in Maslov (2009) are parametrised differently, in particular b eing quoted instead of a0 (the two quantities b eing equivalent), the The results were also indep endently found by Esko Lyytinen (McBeath

At the time of the predicted encounter with the 3-rev Perseid trail, observers were active in Canada, the United States and Venezuela (Rafael Barrios Belisario, William Godley, Jesus Guerrero, Cathy Hall, Peter Kozich, Rob ert Lunsford, Pierre Martin, Gilb ert Sanchez, Wesley Stone, Jhonny Torres, William Watson) and a sharp outburst (Full Width Half Maximum 0.5 hr) occurred on Aug. 12 at 08:06 UT ± 9 min with maximum ZHR 120 ± 30 (Barentsen 2009a; Jenniskens et al. 2009a). This was many hours b efore the usual maximum (McBeath 2008) and impressively confirms the accuracy of outburst predictions. Moreover further individual activity p eaks were observed in the subsequent 24 hours (Barentsen 2009a) and had b een calculated in advance by J´r´mie Vaubaillon to b e associated ee with meteoroids released by Comet Swift-Tuttle at its 1348 and 441 returns resp ectively (Jenniskens et al. 2009a). Although dust in the outlying regions of the 2-rev trail produced some meteors (Jenniskens et al. 2009a), the close match in the timing indicates that the ma jority of meteors comprising the outburst centred around 08:06 UT came from the 3-rev trail. That is, the 3-rev trail's smaller value of the miss distance (and mayb e also the b etter value of a0 ) more than comp ensate for the less favourable fM .

1.2

2009 Leonids

Table 2 lists calculated data for the 2009 Leonids. Similar results had b een obtained by Maslov (2007), who additionally provides results for Leonid trail encounters in all years till the end of the 21st century. Whereas other encounters not listed in Table 2 have some or all parameters much less favourable for producing significant ZHRs, the listed encounters have quite small miss distances. Two of these encounters are with the same trail, formed around the Leonid parent comet 55P/Temp el-Tuttle's return 16 revolutions ago in 1466; such multiple encounters b ecome increasingly p ossible as trails get older and planetary p erturbations disrupt their original, simple structure. Analysis of the visual observations indicates a p eak ZHR of 101 ± 8 centred at Nov. 17, 20:19 UT ± 8 min with FWHM 4 ± 1 hr (Barentsen 2009b; Jenniskens et al. 2009b). The accuracy of the timings in Table 2 and timings calculated by others (see Jenniskens 2009) is of the order of an hour. This apparently confirms that younger trails, such as the 3-rev Perseid trail discussed ab ove, lead to sharp er outbursts at more precisely defined times. Compared to the 3-rev Perseid trail, the 14- and 16-rev Leonid trails are a little older in years, and much older in numb er of revolutions (Temp el-Tuttle's p eriod b eing shorter than Swift-Tuttle's). Meteoroids from these Leonid trails have disp ersed to some extent, as can b e seen by looking at the spatial distribution from a detailed dynamical simulation (Vaubaillon 2009). Despite minor discrepancies compared to the observed activity, the agreement of the various predictions (Jenniskens 2009), with each other and with the observations, is in fact very impressive by the standards of the not too distant past.
Table 2 ­ Encounter parameters of Leonid trails with Earth in 2009. Trail 1 4 - r ev 1 6 - r ev 1 6 - r ev a0 +0.136 +0.105 +0.106 rE -0 +0 +0 - rD .0005 .0011 .0009 fM -0.09 0.10 -0.04 Time Nov 17 Nov 17 Nov 17 (UT) , 21:15 , 21:20 , 22:00


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2

Widths and shap es of trail cross sections

For meteoroids to escap e a comet, the momentum transfer from the gases expanding away from the nucleus must b e enough to overcome the gravity of the nucleus, as shown in Whipple's classic (1951) pap er. Both these forces, gas drag and gravity, increase with the size of the nucleus, assuming other parameters to b e equal (a larger nucleus implies a greater surface area over which gas can b e released). Whipple's derivation shows that the gas drag is more imp ortant except for large meteoroids or extremely huge comets, i.e. the laws of cometary physics mean that (Whipple 1951, p. 472) . . . larger comets should eject particles with greater velocities than smaller comets . . . (when other factors are equal). Whipple continues: Observational evidence to test these conclusions must come from detailed studies of meteor showers. One chance to apply Whipple's great insight, and use shower observations to test models of cometary physics, would b e by comparing the Perseids and Leonids. The effective radius of comet 109P is 13 km and of 55P is 1.8 km (Lamy et al. 2004), while the analysis of meteor lightcurves combined with the theory of quasi-continuous fragmentation suggests that Perseid meteoroids are typically denser than Leonids, 1.2 and 0.4 g/cm3 resp ectively (Babadzhanov & Kokhirova 2009). The two comets have similar p erihelion distances, just under 1 au. Although a full comparison b etween the Perseids and Leonids, and b etween their parent comets, is b eyond the scop e of this pap er, all these numerical values are incorp orated in the trail modelling describ ed b elow, as is Whipple's (1951, p. 472) additional (quantitative) conclusion that . . . the ejection of the meteoritic particles from comets should b e more violent as well as more frequent near p erihelion. In dust trail calculations (Section 1), meteoroids are ejected exactly when the comet passes p erihelion and exactly in the comet's direction of motion at that time. Such meteoroids define a `trail centre' and the calculations tell us by how much the Earth misses this trail centre. In reality meteoroids are ejected over an arc of the parent comet's orbit, covering many months or even ab ove a year, and over a range of directions from the comet nucleus; these effects result in the trail having a cross section (in the ecliptic plane) with a particular shap e and characteristic width. If the trail were really to exist only as the trail centre, it would b e harder for the Earth to encounter any meteors. The cross section defines the spatial distribution of meteoroids, and the Earth follows one path through this cross section. Essentially, comparing the width with the miss distance shows whether the trail is encountered or not. Trail and stream cross sections eventually evolve with time (e.g. Ryab ova 2003) owing to disp ersive forces such as planetary p erturbations or radiative forces. The trails in Table 2 are too old for their density distribution to dep end only on the p oints on the comet orbit where the meteoroids were released and the directions in which they were ejected. Conversely, young enough trails are less disp ersed and their structure more closely reflects specifical ly the ejection processes. The 1999 Leonid storm occurred when the Earth encountered a young trail, namely the one generated 3 revolutions earlier, at Temp el-Tuttle's 1899 return (Arlt et al. 1999). Figures 1 and 2 illustrate the Earth's 1999 path through this trail for two values of the radiation pressure parameter (the relevance of in trail modelling is discussed e.g. by Asher 2008). The two values are chosen to roughly corresp ond to typical visual meteors (ab out magnitude +2) and bright meteors around magnitude 0. For pre-atmospheric sp eeds 60­70 km/s of Leonids or Perseids, magnitude +2 meteors result from meteoroids a couple of millimetres in diameter and magnitude 0 from twice that. The `more violent' ejection near p erihelion is modelled as an inverse dep endence of ejection sp eed on the heliocentric distance r in au, in reasonable accord with Whipple and later authors such as Jones (1995). Apart from the values of and mean ejection sp eed at r = 1, the meteoroid ejection model is the same as used by Asher (2008). In particular, the density distribution in the cross section is assumed to result entirely from meteoroid ejection and not from subsequent dynamical evolution.


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Figure 1 ­ Earth's 1999 encounter with 3-rev Leonid trail; = 0.00070, mean ejection speed = 31/r m/s. Relative to trail centre, x-axis is displacement in nodal longitude (expressed as a distance in au rather than an angle) and y -axis is displacement in heliocentric distance of descending node. Earth is shown actual size at 1 hr intervals, as it moved from right to left through trail.

Figure 2 ­ Earth's 1999 encounter with 3-rev Leonid trail; = 0.00140, mean ejection speed = 44/r m/s.

For the larger of the two meteoroid sizes considered (Figure 1) the Earth passes quite near the edge of the density distribution, according to the model used for this plot. As realised by Whipple, the observation of an outburst such as this constrains whether the model is an accurate representation of the meteoroid ejection processes. The implied outburst duration in Figure 2, and even in Figure 1 to some extent, is too long compared to that observed (Arlt et al. 1999), suggesting that a slightly lower ejection sp eed than that used for these plots is appropriate, as it would narrow the cross section. The Whipple comet model shows that lower ejection sp eeds could result, for example, from a lower sublimation efficiency. This illustrates how cometary physics can b e constrained by meteor observations. The sensitivity of the model activity profile on the meteoroid size (Figures 1 and 2) is interesting considering the observed lack of very faint and the lack of very bright meteors (Arlt et al. 1999). Figures 3 and 4 are for the young (3-rev) Perseid trail which the Earth encountered reasonably closely in 2009. The meteoroid sizes are similar (Figure 3 to Figure 1, and Figure 4 to Figure 2) but different


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Figure 3 ­ Earth's 2009 encounter with 3-rev Perseid trail; = 0.00025, mean ejection speed = 47/r m/s.

Figure 4 ­ Earth's 2009 encounter with 3-rev Perseid trail; = 0.00050, mean ejection speed = 68/r m/s.

values are chosen owing to the presumed difference in meteoroid density (Babadzhanov & Kokhirova 2009). Each individual Figure has its own self-consistent density scale but the density scales for all four plots are calibrated separately, i.e. no attempt is made here to measure the relative meteoroid production rates of the two comets, or the meteoroid size distribution from each comet. As with the Leonids, a slightly lower ejection sp eed than used for these plots seems appropriate to match the observed duration of the outburst describ ed in Section 1.1. A comparison of Figures 3 and 4 with Figures 1 and 2 shows that different streams have cross sections of very contrasting shap es. This is to b e exp ected for parent comets whose orbital geometry is not the same (e.g. inclined at differing angles to the ecliptic). The cross section of a Leonid trail in the ecliptic plane near the Earth's orbit vaguely resembles an ellipse, with the densest region very close to the `trail centre'. The density distribution is not symmetrical ab out this p oint, however. Among the reasons for this is solar radiation pressure, which causes meteoroids to b e slightly less strongly attracted towards the Sun and consequently tends to displace the meteoroids' p ositions in the antisolar direction.


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Figure 5 ­ The part of the 3-revolution Perseid trail that was encountered by Earth in 2009, parameters as in Figure 3.

With Perseid trail cross sections, different readers will no doubt imagine different shap es in Figures 3 and 4, a raindrop or a p ear p erhaps. I see the title character of the delightful 1988 animated motion picture `Tonari no Totoro' (My Neighb our Totoro) by the Japanese director Hayao Miyazaki (Figure 5). The good agreement b etween meteoroid ejection sp eeds determined firstly from cometary physics based on the original Whipple model and its later developments, and secondly from the completely indep endent method of observing the occurrence versus non-occurrence, and duration, of meteor outbursts, is one of the great triumphs of meteor science. Such research has essentially b een successful and is now at the stage of minor improvements, such as the slight required adjustments to ejection sp eeds referred to in the ab ove discussion. This work will b e develop ed further, so that we can discover Totoro's contribution to comet and meteor astronomy. Acknowledgments Research at the Armagh Observatory is supp orted by Northern Ireland's Department of Culture, Arts and Leisure. I am grateful to Dr Miruna Pop escu for help with the artwork.

References
Arlt R., Bellot-Rubio L., Brown P., and Gyssens M. (1999). "Bulletin 15 of the International Leonid Watch: first global analysis of the 1999 Leonid storm". WGN, 27, 286­295. Asher D. (2000). "Leonid dust trail theories". In Arlt R., editor, Proc. IMC 1999, Frasso Sabino, pages 5­21. IMO. Asher D. (2008). "Meteor outburst profiles and cometary ejection models". Earth Moon Plan., 102, 27­33.


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Babadzhanov P. and Kokhirova G. (2009). "Densities and p orosities of meteoroids". Astron. Astrophys., 495, 353­358. Barentsen G. (2009a). "Perseids 2009: http://www.imo.net/live/perseids2009. Barentsen G. (2009b). "Leonids 2009: http://www.imo.net/live/leonids2009. visual visual data data quicklook". quicklook". IMO IMO webpage. webpage.

Jenniskens P. (2009). "Leonid meteors 2009". IAU Central Bureau for Astronomical Telegrams. CBET 2019. Jenniskens P., Genovese L., Barentsen G., and Vaubaillon J. (2009a). "Perseid meteors 2009". IAU Central Bureau for Astronomical Telegrams. CBET 1921. Jenniskens P., Vaubaillon J., Atreya P., Vachier F., and Barentsen G. (2009b). "Leonid meteors 2009". IAU Central Bureau for Astronomical Telegrams. CBET 2064. Jones J. (1995). "The ejection of meteoroids from comets". MNRAS, 275, 773­780. Kondrat'eva E. and Reznikov E. (1985). "Comet Temp el-Tuttle and the Leonid meteor swarm". Solar System Res., 19, 96­101. Lamy P., Toth I., Fern´ndez Y., and Weaver H. (2004). "The sizes, shap es, alb edos, and colors of a cometary nuclei". In Festou M., Keller H., and Weaver H., editors, Comets II, pages 223­264. Univ. Arizona Press, Tucson. Lyytinen E., Nissinen M., and van Flandern T. (2001). "Improved 2001 Leonid storm predictions from a refined model". WGN, 29, 110­118. Maslov M. (2007). "Leonid predictions for the p eriod 2001­2100". WGN, 35, 5­12. Maslov M. (2009). "Perseids 2009­2010: prediction of activity". Mikhail Maslov webpage. http://feraj.narod.ru/Radiants/Predictions/Perseids2009eng.html. McBeath A. (2008). 2009 Meteor Shower Calendar. IMO. McNaught R. and Asher D. (1999). "Leonid dust trails and meteor storms". WGN, 27, 85­102. Ryab ova G. (2003). "Mathematical modeling of meteoroid stream formation". In Olech A., Zloczewski K., and Mularczyk K., editors, Proc. IMC 2002, Frombork, pages 125­134. IMO. Sato M. (2003). "An investigation into the 1998 and 1999 Giacobinids by meteoroid tra jectory modelling". WGN, 31, 59­63. Vaubaillon J. (2009). "2009 Leonids". IMCCE webpage. http://www.imcce.fr/langues/en/ephemerides/phenomenes/meteor/DATABASE/Leonids/2009. Vaubaillon J., Lyytinen E., Nissinen M., and Asher D. (2003). "The 2003 Leonid shower from different approaches". WGN, 31, 131­134. Vaubaillon J., Lyytinen E., Nissinen M., and Asher D. (2004). "The unexp ected 2004 Leonid meteor shower". WGN, 32, 125­128. Whipple F. (1951). "A comet model. I I. Physical relations for comets and meteors". Astrophys. J., 113, 464­474.