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(meteorobs) [long] Theories for Leonid Storm Prediction (draft)



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.


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.


Asher, D.J. (1999) "The Leonid meteor storms of 1833 and 1966.",
  MNRAS, 307, 919-924

Brown, P. (1999) "The Leonid meteor shower: historical visual observations.",
  Icarus, 138, 287-308

Cooke, W. (1997) "Estimation of Meteoroid Flux for the Upcoming Leonid
  Stroms.",  http://see.msfc.nasadot gov/see/mod/leonids.html

Ferrin, I. (1999) "Meteor storm forcasting: Leonids 1999-2001.",
  Astron. Astrophys., 348, 295-299

Kondrat'eva, E.D. & Reznikov, E.A. (1985), "Comet Tempel-Tuttle and the
  Leonid meteor swarm.", Solar System Research, 199, 96-100

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

Lyytinen, E. (1999) "Leonid Predictions for the Years 1999-2007 with the
  Satellite Model of Comets.", Meta Research Bulletin, 31, 489-492

McNaught, R.H. (1999) "On predicting the time of Leonid storms.",
  The Astronomer, 35, 279-283

McNaught, R.H. & Asher, D.J. (1999a) "Leonid dust trails and meteor storms.",
  WGN, 27, 85-102

McNaught, R.H. & Asher, D.J. (1999b) "Variation of Leonid maximum times
  with location of observer.", Meteorit. Planet. Sci. (in press)

Wu, Z. and Williams, I. P. (1996) "Leonid meteor storms",
  MNRAS, 280, 1210-1218

Yeomans, D.K. (1981) "Comet Tempel-Tuttle and the Leonid meteors",
  Icarus, 47, 492-499
~

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