A Review of Theories for
Leonid Storm Predictions
R. H. McNaught, last edited 99Nov09



The following are notes were started as preparation for a lecture. However I felt that they might be of general interest, given that many of the works cited below often go unmentioned, or are quoted without analysis. I am aware that I have not mentioned some works (e.g. Kresak's 1993 study), but will include these at a later date. As I'm leaving for overseas in several hours, my attempts to knock this into better shape has come to an end. Despite many helpful comments by David Asher on a much earlier draft, about half of what follows has not been submitted to anyone or comment. It would thus be inappropriate to quote anything from what follows as if it were from a refereed journal. It is my intention to work on this much more after my return from the Leonids trip and submit it for publication.


The original mail was sent by Rob McNaught to the "imo-news" and "meteorobs" mailinglists on november 10, 1999



A Review of Theories for Leonid Storm Prediction
R. H. McNaught, last edited 99Nov09


Introduction

There has been little critical evaluation of the various Leonid storm predictions, either in the professional literature or in the popular astronomy media. This has resulted in speculative methods with no theoretical basis or historical validation, being presented side by side with theoretically rigorous approaches that have been carefully validated against the historical record. I shall discuss some of these methods here, so that a clearer assessment can be made about the various prediction methods.


Studies using the comet node

Yeomans (1981) demonstrated an obvious correlation between the timing of storms and the time and position of the Earth in relation to the comet node. In terms of predictive power this model fails, with years of high and low activity intermixed. It is also only an approximation, as the nodal distance of the comet is only of physical relevance when the comet is actually at the node. Differential perturbations between the comet, and the ejected dust, lead to the dust having a different nodal longitude and distance from the comet. Also, should one choose the osculating orbit at the time of the comet being at the node, or the time of observation? The value of the dust node need only be the same as the comet at the time of ejection. Beyond that, the orbits must be treated independently. Using the node of the comet gives an approximation of the time of storms to a few hours for storms in the last 200 years, using nodal values of either osculating orbit. This is discussed more fully in McNaught (1999).

That the storm years, do indeed cluster in one quadrant of the Yeoman's plot indicates that it does have some predictive value, but there are both false positives and flase negatives. Although both axes of the plot are qualitatively reasonable, only the time axis is also quantitative. The effect of solar radiation pressure is to push particles into longer period orbits, and therefore they return after the comet. The density of the dust with age since ejection but this is not accommodated in the diagram. The radial distance axis is problematic, as noted above. David Asher comments "as for the inside/outside distance being a factor, while the idea is qualitatively correct (i.e. radiation pressure does tend to cause particles to be on marginally bigger orbits in space) it's qualitatively irrelevant, compared to the effect on the nodal distance of gravitational perturbations, for visual meteor size particles."

Cooke (1997) looks at the Yeomans' diagram through a statistician's eye. He tries to derive probabilities of storm conditions in various years. To some extent this must be seen as a failure of this general approach using the comet node. No amount of math can compensate for not undertaking a rigorous dynamical analysis of the ejected dust. To understand Leonid storms, or any physical phenomenon for that matter, one needs both maths and a physical understanding of the phenomenon involved.

Ferrin (1999) uses a similar form of analysis as Yeomans, but gives the Yeomans diagram an additional dimension of ZHR intensity at maximum. Whilst one can argue about the values of ZHR used in the diagram (e.g. the almost certainly spurious storm values for 1900 and 1901), and the way individual values were selected from the available data (e.g. 1866 and 1867) the idea is initially reasonable, given the limitations noted for the Yeomans (1981) paper above. The intensity of the Leonid activity of the last 200 years has isolines of shower intensity empirically fitted. A "ridge" of uniform high intensity (ZHR = 150,000) is identified crossing the diagram in a curve from the comet.

Given the small amount of data for high intensity storms (ZHR > 10,000), it is notable that one of these lies significantly away from the ridge and is too high by a factor of 10 over the fit. Given that the fit is completely empirical, this is a major problem for such sparse data. Probably the most unusual thing about the fit is the assumption that the ridge of high intensity is of uniform intensity. This is clearly false in the close vicinity of the comet. The ZHR immediately beside the comet would be enormous, such dust hardly having time to dissipate. However there is a big difference between the dust density near the comet and that a year or so behind. It is well known that solar radiation pressure causes particles of the size that produce visual meteors, to orbit more slowly. Thus an initially uniform ejection of dust will, one revolution later, result in a mass separation with most "visual meteoroids" being concentrated away from the comet. Thus, even with the probably spurious storm level values for 1900 and 1901, these facts immediately suggests to the eye a series of closed loops off-centered from the comet. The consequence of this would be that the rates during the current epoch would be considerably lower than the values Ferrin suggests, from his unjustified empirical fit. Whilst a number of theoretical considerations are made, there is no attempt to look at the actual spatial distribution of dust through rigorous orbital integrations.

Brown (1999) has analysed the available historical observations of the Leonids, deriving the time and ZHR of maximum and the width of Leonid activity. This represents a major achievement and all Leonid storm prediction method should be demonstrated to be consistent with this historical data. Utilising this data Brown uses the same idea as Ferrin, but allows a contour plotting program to contour the ZHR data. For the limitations presented above, and the reasons given below, the use of the comet node cannot succeed. The fundamental reason is that the dust behaves independently of the comet and detailed dynamical studies of the ejected dust must be used.


Dynamical studies of ejected dust

Wu and Williams (1996) present an analysis of the orbits of dust ejected from comet 55P/Tempel-Tuttle. They apply rigorous corrections for planetary perturbations. Part of their argument is that high ejections velocities of several hundred metres/sec are necessary to produce the orbits of observed Leonid meteors in 1965-66. These orbits remain stable over the past 100 years and do not converge to a common origin. This is in stark contrast to their later modelling where they assume the activity in 1933, 1966, and predictions for 1998-99, can be based solely on dust ejected from the comet on the previous two revolutions.

Using the high velocities of ejection derived from the meteor orbits, they believe particles can be ejected into orbits as short as 17 years or as long as 120 years. This provides pathways for particles to make one, two or three revolutions in 66 years. However, if Leonid activity is dominated by recently ejected particles, then the meteor orbits should converge to the comet orbit at either of the previous two returns. That they do not indicates that either a) the orbits are too uncertain to be useful in this analysis and/or b) the assumption that activity is dominated by the most recent returns of the comet, is false.

If we assume for the moment that these high ejection velocities are possible, it is reasonable to assume that the extreme orbits of both shorter and longer period are likely to be significantly less populated than those closer to the orbital period of the comet. They specifically make this point in section 4. This is most important when they come to assess the number of test particles that pass close to the Earth. They take 20 test particles from one revolution of the comet earlier, with a 33 year period and 60 from 2 revolutions of the comet earlier, 20 each from particle periods of 22, 33 and 66 years. Simple summing of these 80 particles has no validity. It is probable that there will be many more particles with periods of 33 years than 22 or 66 years.

The orbits they integrate have starting orbital periods that make an integral number of revolutions during the time taken for one or two comet orbits. However, despite a claim that they did, there is no evidence in their work that they have iterated these orbital periods to correct for changes due to planetary perturbations resulting in the particles not arriving at the node at the same time as the Earth. The nodal distance is irrelevant if the particle orbit cannot produce a close approach to the Earth. Looking at their Fig 7, the last two bars for each year give the relative number of particles within a nodal distance of 0.002 AU and within a distance from the Earth of 0.005 AU. If the correct orbital period is chosen, then the closest approach will always be (slightly) inside the nodal distance, so the bar giving passage within 0.005 AU of the Earth must always be equal to, or greater than, the bar showing particles within 0.002 AU nodal distance. In two of the four cases they are less, one substantially. Thus most test particles do not in fact have the correct period to have a close encounter with the Earth, and the integrated particles are irrelevant in determining the approach distances and relative numbers. It was found by McNaught and Asher (1999) (see below) that the density of dust trails can vary substantially on scales of the order of an Earth diameter, so the bin sizes used are substantially too coarse to be useful indicators of storm activity. Any conclusions based on Fig 7 are necessarily invalid.

Even assuming the Figure is valid, the comparison of these relative numbers of particles for various years shows 1933 coming in at a little under 10% of 1966, in the important quantities (number of particles with nodal distance near Earth, and number with small distance of closest approach to the Earth). However, the ZHR in 1933 was around 3 orders of magnitude smaller than in 1966 (Brown (1999)), so their statement that these figures "roughly mirror the observations" really has little meaning.

Overall, the assumptions behind this work are reasonable, but in restricting the calculations to only the previous two orbits, and not choosing the precise orbital period to make a close encounter, the work has no validity as a predictive tool. Also they do not attempt to derive the time of storms from the nodal longitudes of the dust orbits. This is a necessary test of any theory, as it would have available some of the best data for comparison.

Kondrat'eva and Reznikov (1985) were the first group to determine meteoroid orbits that had the precise orbital period to arrive at their descending node at the same time as the Earth. Their work has been largely overlooked. The idea is extremely simple. The only meteoroids we can experience as meteors, are ones that have an orbital path from the comet at or near perihelion, to the Earth in some specific November. The application of rigorous planetary perturbations and the consideration of solar radiation pressure, give a nominal orbital solution from which the nodal longitude and distance is derived. Meteoroids with any other orbital period don't pass the node at the same time as the Earth and thus could not become meteors. It is the component of the ejection velocity along the comet's velocity vector that causes the change in orbital period. The spread of the meteoroids about this nominal solution are a result of other components of the ejection velocity that are orthogonal to the comet's velocity vector and of solar radiation pressure.

Their work shows a great consistency with the historical data for the years presented. Their predicted time for 1966 is exact, to the resolution of their prediction, which is 0.01 day. In 1993, Reznikov predicted the time of Giacobinid activity as 1998 Oct. 08.550 UT. This was confirmed within observational error! Clearly the group had the ability to make predictions with high time resolution.

Kondrat'eva, Murav'eva and Reznikov (1997) update this work by extending for dust ejected at earlier passages of 55P/Tempel-Tuttle through perihelion and derive the nodal longitudes and distances for the dust during the period 1760-2002. Curiously, they only give the predictions of the time of maximum activity to one decimal of a day (+/- 1.2 hours). There is an exceptionally strong correlation between the close approaches to dust "swarms" with moderate ejection velocities (<40 m/s), and years with observed storms. All their derived times for the storm years agree with the observed times derived by Brown (1999) to within +/- 1 hour. This was clearly a major advance in Leonid storm prediction.

Asher (1999) was unaware of the Kondrat'eva et al. studies when he basically replicated their early work with his own similar technique. However, he did this with higher precision in nodal longitude than the later Russian study. This led to the realisation that the derived times from the "dust trail" nodal longitude were almost identical to the times of Leonid storm maxima derived by Brown (1999). This was initially discussed by McNaught (1999).

McNaught and Asher (1999a) extended the Asher (1999) results by looking at dust trails up to 6 revolutions old (plus some older trails identified by Kondrat'eva et al. (1997)). This indicated that the times of maxima were consistent to within +/-10 minutes for all storms and short duration outbursts that had well defined times of maximum (1866, 1867, 1869, 1966 and 1969. Additionally, they derived a density model based on the ejection velocity (change in semi-major axis) required to produce passage close to the Earth and the nodal distance of the dust trail. This approach also took into account the mass distribution of the ejected dust encountered in a specific year (which is correlated with ejection velocity) and the dilution of the trail density with age. It was demonstrated from test integrations of dust ejected isotropically from the comet, that the resulting trail width remains essentially constant over several revolutions, dilution of the trail density being by stretching alone. Using this model of trail density, they were able to show a remarkable consistency (+/- 20%) between the calculated relative density and the observed ZHR for the storm data of 1833, 1866, 1867, 1869 and 1966. Earlier storm years were not included due to poor data quality and contamination from additional dust trails. The fit to the data was by a double Gaussian. This will limit the predictive value, as it is believed that the dust trails are not symmetrical in radial distance mostly due to the action of solar radiation pressure. Until a theoretically derived dust trail profile in radial distance is developed, the data is too sparse to suggest what improvement may be achieved.

McNaught and Asher (1999b) derived a topocentric correction for the observer being offset from the center of the Earth which had been used in the earlier calculations. This indicated that the times calculated from the dust trails could be improved from +/-10 minutes to +/-5 minutes against the observed times calculated by Brown (1999).

Lyytinen (1999), unaware of the Kondrat'eva et al. and Asher and McNaught studies, came up with the same results, but a very different starting point. Using van Flandern's satellite model of comets, he derived the times of closest encounter with dust trails through to 2007. Despite that radically different initial assumption, the dynamical analysis was done rigorously and the results of the time of maximum agreeing within minutes with the results of the earlier studies. Lyytinen himself did not do any rigorous historical validation, but his results were clearly very consistent with the historical data.

All three groups (Kondrat'eva et al., Asher and McNaught, and Lyytinen) found a small number of dust trail encounters missed by others. These were mostly of older trails. Calculations by other groups confirmed these.

This "dust trail" approach to predicting Leonid storms is clearly very powerful and has demonstrated a very close correspondence to the time of storms (+/-5 minutes) and to their ZHR (20% error in the fit to 5 storm ZHRs).


Background activity

Although one work of Brown (1999) was mentioned above, he and collegues including Jones, have continued with their studies of the Leonid stream as a whole. The above dynamical studies only addressed the storm peak, whereas Brown et al. are not considering storms in isolation. As I have not seen their latest work, I can only comment on what I believe is their current approach. By ejecting meteoroids (using an ejection model they derived) over a period around perihelion and for many revolutions of the comet into the past, they try to derive the overall activity of the Leonid shower. This requires substantial computing power, but is probably the only way to approach the overall structure. The limitation in this method may be that it lacks adequate temporal and spatial resolution. One could liken this approach to a general geological survey of an area where sampling at coarse intervals can miss narrow dense veins. It may however be the case that the resolution is adequate to identify the dust trails, although the results presented by Brown el al. earlier this year do not confirm many of the dust trail predictions for the coming years. They do however predict the same time of maximum as the dust trail theory in 1999, although the nature of this prediction is unknown to me.


Conclusion

Leonid storms are predictable from dust trail calculations based on the orbit of the parent comet 55P/Tempel-Tuttle. The ZHR predictions are limited by the lack of storm ZHR data, but the dust trail density model of McNaught and Asher (1999) is very consistent with data available.


  1. Asher, D.J. (1999) "The Leonid meteor storms of 1833 and 1966.", MNRAS, 307, 919-924
  2. Brown, P. (1999) "The Leonid meteor shower: historical visual observations.", Icarus, 138, 287-308
  3. Cooke, W. (1997) "Estimation of Meteoroid Flux for the Upcoming Leonid Stroms.", http://see.msfc.nasa.gov/see/mod/leonids.html
  4. Ferrin, I. (1999) "Meteor storm forcasting: Leonids 1999-2001.", Astron. Astrophys., 348, 295-299
  5. Kondrat'eva, E.D. & Reznikov, E.A. (1985), "Comet Tempel-Tuttle and the Leonid meteor swarm.", Solar System Research, 199, 96-100
  6. Kondrat'eva, E.D., Murav'eva, I.N. and Reznikov, E.A. (1997) "On the forthcoming return of the Leonid meteoric swarm.", Solar System Research, 31, 489-492
  7. Lyytinen, E. (1999) "Leonid Predictions for the Years 1999-2007 with the Satellite Model of Comets.", Meta Research Bulletin, 31, 489-492
  8. McNaught, R.H. (1999) "On predicting the time of Leonid storms.", The Astronomer, 35, 279-283
  9. McNaught, R.H. & Asher, D.J. (1999a) "Leonid dust trails and meteor storms.", WGN, 27, 85-102
  10. McNaught, R.H. & Asher, D.J. (1999b) "Variation of Leonid maximum times with location of observer.", Meteorit. Planet. Sci. (in press)
  11. Wu, Z. and Williams, I. P. (1996) "Leonid meteor storms", MNRAS, 280, 1210-1218
  12. Yeomans, D.K. (1981) "Comet Tempel-Tuttle and the Leonid meteors", Icarus, 47, 492-499


The original mail was sent by Rob McNaught to the "imo-news" and "meteorobs" mailinglists on november 10, 1999


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