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Mon. Not. R. Astron. Soc. 307, 919--924 (1999) [1999 August 21 issue]
The Leonid meteor storms of 1833 and 1966
D.J. Asher ?
Armagh Observatory, College Hill, Armagh, BT61 9DG
Accepted 1999 March 23. Received 1999 March 15; in original form 1998 November 18
ABSTRACT
The greatest Leonid meteor storms since the late eighteenth century are generally
regarded as being those of 1833 and 1966. They were evidently due to dense meteoroid
concentrations within the Leonid stream. At those times, Comet 55P/Tempel­Tuttle's
orbit was significantly nearer the Earth's orbit than at most perihelion returns, but
still some tens of Earth radii away. Significantly reducing this miss distance can be
critical for producing a storm. Evaluation of differential gravitational perturbations,
comparing meteoroids with comet, shows that in 1833 and 1966 respectively, the Earth
passed through meteoroid trails generated at the 1800 and 1899 returns.
Key words: comets: individual: 55P/Tempel­Tuttle -- meteors, meteoroids.
1 PREDICTING LEONID STORMS
The most spectacular meteor storms of the past two hun­
dred years were probably the Leonids of 1833 and 1966
(Kres'ak 1993a,b, Rao 1998). The Leonids' parent comet,
55P/Tempel­Tuttle (Table 1), presently has orbital period
ú 33.2 yr, and storms occur in years around the comet's
perihelion passages.
However, the occurrence of storms is not the same at
every return to perihelion. Firstly, the comet's period not be­
ing a near­integer number of years immediately shows that
the Earth--comet configuration is not repeated at successive
returns. Moreover, the distance between the orbits of comet
and Earth changes slightly over an orbital period. This dis­
tance, and also the time lag between comet and Earth pass­
ing their near­intersection point (meteoroids affected by ra­
diation pressure tending to fall behind the comet), have to­
gether been used as quite good predictors of enhanced me­
teor activity (Yeomans 1981, Rao 1998).
The storms, the greatest of which can produce 10 4 times
the meteor rate of Leonids in normal years, and even 10 2
times that of some years that are themselves classified as
storm years, show that there are narrow, dense concentra­
tions within the Leonid stream. They presumably comprise
material on orbits close to that of the comet, relative to the
scale of the stream as a whole. If many meteoroids had been
ejected on to orbits further away, intervening regions of or­
bital element parameter space would also be filled, rather
than narrow, high density regions existing in space. Thus
this material is recently released, not having had time to
disperse throughout the stream. These compact trails of me­
teoroids and dust were discovered in the orbits of other pe­
riodic comets by IRAS (Sykes & Walker 1992) and their
potential to produce meteor storms when the Earth inter­
? E­mail: dja@star.arm.ac.uk
Table 1. Orbit of 55P/Tempel­Tuttle calculated by Nakano
(1998). Epoch 1998 Mar. 8.0 TT. The transverse nongravitational
parameter A 2 is more reliably determined than the radial com­
ponent A 1 (cf. Table I of Yeomans et al. 1996).
Perihelion passage time T = 1998 Feb. 28.0982 TT
Perihelion distance q=0.976577
Semi­major axis a=10.337486
Eccentricity e=0.905531
Inclination (J2000.0) i=162.4860
Longitude of ascending node
\Omega\Gamma42240/­2 Argument of perihelion !=172.4988
Nongravitational parameters A 1 = --0.80\Theta10 \Gamma8
(au day \Gamma2 ) A 2 = +0.0090\Theta10 \Gamma8
section geometry is right has been emphasized by Kres'ak
(1993a,b).
One trail is created during each return of the comet,
surviving until it is dispersed into the stream. As these are
distinct entities from the comet itself, albeit they are re­
lated, considering their separate orbital evolution could give
greater accuracy for predicting the most intense storms than
considering the comet's orbit. For a few revolutions, enough
trail material may stay coherent enough that systematic dif­
ferences from the comet's orbit can be precisely calculated,
these differences dominating the randomness that arises be­
cause the exact initial orbits of individual particles at ejec­
tion are unknown.
The Leonid stream as a whole is not studied here; see
Brown & Jones (1996) for a fuller model of the stream. That
is, while a model meteor flux is often generated statistically
in dynamical meteoroid stream studies because the flux re­
sults from a wider range of stream orbits than can be explic­
itly considered individually, or because the timescale is long
enough for the chaotic dynamics which affects all planet ap­
proaching orbits to occur, the idea below is to consider just a
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fl 1999 RAS

2 D.J. Asher
few individual trails and to follow particles on short enough
timescales that orbits are predictable.
2 DIFFERENTIAL PERTURBATIONS
The heliocentric distance rD of the descending node is
rD = a(1 \Gamma e 2 )
1 \Gamma e cos !
(1)
while the Earth's distance at the longitude where it passes
through the Leonid stream is rE ú 0.9886 au (200 yr ago,
rE ú 0.9889 au, precession having marginally altered the
value of the relevant longitude). As the solar--antisolar com­
ponents of the velocity vectors of Leonid and Earth are both
small near this point, the difference between rD and rE ap­
proximately gives the orbit--orbit approach distance. For ref­
erence, one Earth diameter is about 0.0001.
From (1),
@rD
@! = \Gamma a(1 \Gamma e 2 ) e sin !
(1 \Gamma e cos !) 2
ú \Gamma0:001 au deg \Gamma1
where appropriate a, e, ! have been used to give the numer­
ical value. Similarly, writing (1) as
rD = q(2 \Gamma q=a)
1 \Gamma (1 \Gamma q=a) cos !
and using aú10, qú1 and cos !ú--0.988,
rD ú q ) @rD
@q ú 1
and
@rD
@a ú 3 \Theta 10 \Gamma5
can soon be derived. Hence
\Deltar D ú \Deltaq + 3 \Theta 10 \Gamma5 \Deltaa \Gamma 0:001\Delta! (2)
(cf. Pecina & Ÿ Simek 1997).
Although meteoroids released on to similar orbits to the
comet will undergo similar orbital evolution for a while, the
question arises as to whether even rather small differences
in elements could crucially affect how closely a meteoroid
can approach Earth. This can be investigated by trial inte­
grations that only need cover quite short timescales.
Only meteoroids with the right mean anomaly M to
impact Earth at whatever date of interest need be consid­
ered (cf. Wu & Williams 1996). Specifying which perihelion
passage a meteoroid is ejected at, and a given year not too
many revolutions later in which it produces a meteor, tightly
constrains the value a0 of the semi­major axis at ejection,
since the time elapsed between ejection and meteor impact
fixes the orbital period. The period is not taken to be con­
stant, gravitational perturbations being known to change
the comet's semi­major axis by amounts of order 0.1 au
between successive returns, but because the perturbations
over a short enough timescale are a rather smooth function
of a0 , just a few iterations (integrations) allow a0 to be de­
termined. The key point is that although the \Deltaa term is
negligible in (2) for relevant \Deltaq, \Deltaa and \Delta! (Sec. 4), varia­
tions in a are the primary cause of the differential planetary
perturbations that lead to differences in all the elements.
Furthermore, those perturbations, and so the heliocentric
Figure 1. Orbital element evolution, for just over two revolu­
tions, beginning at perihelion in 1899, of 55P/Tempel­Tuttle and
4 particles with initial q, i, \Omega\Gamma ! matching the comet but different
a (comet 10.39, particles 10.5564 to 10.5567).
distance (through \Deltaq and \Delta!) and longitude of the descend­
ing node after a few revolutions, are precisely calculable (by
means of integrations) functions of the initial elements (pri­
marily a0 ), if there are no close approaches to planets.
Everhart's (1985) 15th order Radau integrator has been
used, in a program by Chambers & Migliorini (1997), with
initial planetary elements in each integration taken from
JPL's DE403. The conclusions below can be found using an
accuracy parameter of 10 \Gamma8 in Radau and including Mer­
cury in the Sun, allowing the computer programs to be run
quickly, but as computer time was available, the results in
Table 2 were generated with 10 \Gamma9 and explicitly integrating
Mercury (altogether taking ¸6 times longer).
Fig. 1 shows an example of differential perturbations
between comet and meteoroids. The fact that the individual
meteoroids cannot be resolved until the Earth approach in
1966 causes their semi­major axis to separate (initial orbits
were specifically chosen because they undergo this approach)
confirms that over just a couple of revolutions, the effect of
planetary perturbations, in particular on rD , is quite deter­
ministic (chaotic dynamics affects evolution after the close
approach). As the range in mean anomaly, M , in 1966 brack­
ets the M that would give Earth impact, checking rD tells
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1833 and 1966 Leonid storms 3
whether impact actually occurs. This example shows the sig­
nificant differences that can result from the perturbations,
rD moving away from the comet's value by some tens of
Earth diameters.
At aphelion, 55P/Tempel­Tuttle reaches approximately
Uranus's orbit, but Jupiter and Saturn are its main per­
turbers (Yeomans, Yau & Weissman 1996). The differential
perturbations in Fig. 1 are, in the main, not induced dur­
ing the outer part of the orbit. Saturn's orbit is crossed at
around 35 ffi and 325 ffi in M ; significantly beyond this, the
evolutions of comet and meteoroids are almost indistinguish­
able in the first revolution, and separated but parallel in the
second. (The slow oscillations while the objects are away
from the inner solar system, about two cycles being appar­
ent in each revolution, are because elements are heliocentric,
and Jupiter causes the Sun's barycentric position to oscillate
with period ¸12 yr.) As the comet and meteoroids return
towards perihelion the first time, they are already several
degrees apart in M , which can correspond to well over an
au, and the second time, the difference is greater still. This
causes perturbations to differ. Conversely, if two particles
remain close together in M , then even if (Sec. 3) they are
subject to moderately different radiation pressures, their or­
bital precession will be similar.
3 EARTH ENCOUNTER CIRCUMSTANCES
Further examples show rD moving away from the comet's
value by greater or lesser amounts than in Fig. 1. For var­
ious years near the comet's returns, particles were found
with the right M to produce a meteor (a necessary but not
sufficient condition to do so), to see if their nodal crossing
points were also near meteor producing values (Table 2).
At first, the comet was integrated, including the nongrav­
itational parameters, to give elements at each return. The
main integrations then involved starting at one perihelion
time (in turn, 1, 2 or 3 revolutions earlier) and integrating
4 particles, covering a small range in a0 , displaced from the
comet's a with other elements fixed, forward an approxi­
mately whole number of revolutions until the relevant date
in November. The relative behaviour of 4 particles confirms
that over these fairly short timescales, the elements are gen­
erally well behaved functions of the initial elements, thus
that the a0 found really is the value that gives the right M
rather than being random, and therefore that significance
can be attached to the values of rD
and\Omega determined. A
few iterations usually converged on the desired a0 , which is
listed in Table 2 as \Deltaa 0 , i.e., relative to the comet's value
at ejection.
Naturally all particles found in each final iteration ap­
proach Earth, so that their evolution afterwards would be­
come less predictable, but as it was predictable until that
point, the determination of their nodal points was reliable.
Only a few cases were badly behaved strictly before the en­
counter; having multiple particles was then useful to show
up something being wrong. The partial absence of data for
1865, 1967, 1968 and 1998 is because meteoroids that would
have had an appropriate a0 came quite close to Earth in
respectively 1832, 1933, 1934 and 1965 (the 1934 case was
the least disruptive, perhaps because of greater orbital dis­
tance from Earth). For example, the ``missing'' value for 1865
is \Deltaa 0 ú--0.02, which appears (as the 2 revolutions entry)
against 1832. Thus these encounters perturb the orbital ele­
ments erratically, whereas usually, for some revolutions, the
elements evolve as smooth functions of the elements at ejec­
tion.
The calculations were repeated with a nonzero value
of fi, the ratio of the forces of radiation pressure and solar
gravity, fi=0.001 being reasonable for meteoroids that pro­
duce visual Leonids (Williams 1997). This had virtually no
effect on rD
and\Omega at the accuracy given in Table 2; a0 was
changed as shown, the two values being consistent with what
is expected (Williams, loc. cit.), and giving a reasonable idea
of what \Deltaa 0 would be for other fi.
The Table 2 calculations had ejection at perihelion,
meaning that q and thus rD are not affected by nonzero fi.
In fact, a particle with fi=0.001 on an unperturbed elliptical
orbit will have q (defining this, as with a0 , as being calcu­
lated from instantaneous position and velocity with GM fi
taking its usual value, not modified by fi) 0.001 au greater
at perihelion than away from perihelion (cf. equation (2) of
Yeomans 1981). More than half the difference occurs within
r!2 and so fi=0.001 meteoroids ejected at r¸2 could have
rD increased by ? ¸ 0.0005 au. Repeating some integrations
(a column from Table 2), this time beginning 100 days be­
fore perihelion, and checking the rD values, confirms this.
Depending on the meteoroid production rate as a function
of r, the average increase in rD relative to Table 2 will
be of order one to a few times 0.0001. This is much less
than the rD range due to differential gravitational pertur­
bations (Table 2) which are therefore primarily what bring
orbits to Earth intersection. In so far as radiation pressure
is important in affecting rD for visual meteor size particles,
it is more indirectly, through changing the orbital period
with the consequent effect on differential gravitational per­
turbations, than directly in pushing particles out towards
Earth intersecting orbits. Quite often radiation pressure, in
increasing the period, happens to lead to perturbations that
decrease rD .
Whether values in Table 2 signify major storms depends
on rE \Gamma rD being as small as possible, or certainly within
the trail width. Also the values of fi must apply to the real
meteoroids, and \Deltaa 0 , for appropriate fi, must be within an
acceptable range. \Deltaa 0 near zero will always be a good fit
since most ejection scenarios will tend to produce a range of
a0 centred on the parent. The permitted (positive or nega­
tive) \Deltaa 0 depends on ejection processes (Sec. 4). Here, ra­
diation pressure can move the value of \Deltaa 0 nearer to zero
and will tend to make better meteor displays lag the comet,
as is well known.
The small values of rE \Gamma rD for the 1800 and 1899 trails
respectively producing meteors in 1833 and 1966 are quite
striking. In both cases, 1966 especially, the miss distance
is smaller than that based on the comet, the values
of\Omega (cf. Kres'ak 1993a, Mason 1995) being correct either way.
Although in a typical year it is not the case that most me­
teors have originated at a single perihelion passage, there is
evidence that such was the case in those two great storm
years.
The only other similarly small rE \Gamma rD is for 1867, a
storm year but lesser than 1833 and 1966. The 1833 trail
could be largely responsible for the 1867 storm, meteoroids
either having somewhat larger fi, or being somewhat away
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4 D.J. Asher
Table 2. Nodal crossing points of recently (Ÿ3 revolutions previously) ejected particles that go through the ecliptic at the same time
as the Earth passes through the Leonid stream in the given year, and of 55P/Tempel­Tuttle when it passes through the ecliptic (which
happens between the 2nd & 3rd of each set of 6 times). It is the values relating to the particles that are physically meaningful with regard
to assessing whether meteor storms can occur. Listed are the difference \Deltaa 0 in semi­major axis from the comet at the time of ejection
1, 2 or 3 revolutions before; the difference between the heliocentric distances of Earth and particles at the longitude of the particles'
descending node; and the longitude of the ascending node (which equals the longitude of the Sun, since the crossing point in question
is the descending node). The 2 values of \Deltaa 0 are for fi=0 (purely gravitational motion) and fi=0.001 (ratio of radiation pressure force
to Sun's gravity). Changing fi had little effect on r E \Gamma r D
and\Omega\Gamma The value
of\Omega is relative to the standard equinox J2000.0 for ease of
comparison with data generally in the literature (e.g., Table 1 of Kres'ak 1993a), but the nodal crossing is through the ecliptic of date as
the deviation from the J2000 ecliptic can lead to values differing by, e.g., ¸0.035 ffi in 1900 and ¸0.07 ffi in 1800. The omitted data were
unreliable because the relevant particles were significantly perturbed by Earth during an intervening perihelion passage.
Especially low values of jr E \Gamma r D j, the miss distance from Earth, are expected to be associated with storms. Smaller j\Deltaa 0 j may also
be advantageous, in the sense of needing smaller ejection speeds from the comet.
1 revolution before 2 revolutions before 3 revolutions before Comet
Year \Deltaa 0 r E \Gamma r D
\Omega \Deltaa 0 r E \Gamma r D
\Omega \Deltaa 0 r E \Gamma r
D\Omega r E \Gamma r
D\Omega (au) (au) (deg) (au) (au) (deg) (au) (au) (deg) (au) (deg)
1798 --0.28 --0.48 0.0043 233.04 --0.15 --0.36 0.0057 233.02 --0.09 --0.31 0.0017 232.15
1799 --0.07 --0.28 0.0032 233.04 --0.04 --0.25 0.0036 233.03 --0.02 --0.24 0.0018 232.84
1800 0.14 --0.08 0.0028 233.03 0.07 --0.15 0.0020 233.06 0.02 --0.19 0.0061 233.33 0.0030 233.04
1801 0.35 0.12 0.0029 233.02 0.19 --0.04 0.0007 233.17 0.06 --0.16 0.0105 233.58
1802 0.56 0.32 0.0030 233.04 0.31 0.08 --0.0011 233.50 0.09 --0.13 0.0134 233.95
1803 0.76 0.52 0.0022 233.18 0.42 0.19 --0.0009 234.44 0.12 --0.10 0.0209 234.84
1831 --0.25 --0.44 0.0034 233.16 --0.13 --0.34 0.0051 233.17 --0.10 --0.31 0.0068 233.16
1832 --0.04 --0.24 0.0014 233.17 --0.02 --0.23 0.0017 233.18 --0.02 --0.23 0.0019 233.18
1833 0.17 --0.04 --0.0003 233.18 0.09 --0.12 --0.0016 233.18 0.07 --0.15 --0.0028 233.21 0.0012 233.18
1834 0.38 0.16 --0.0017 233.18 0.20 --0.01 --0.0043 233.17 0.15 --0.07 --0.0070 233.27
1835 0.59 0.36 --0.0026 233.18 0.31 0.09 --0.0065 233.17 0.24 0.01 --0.0107 233.42
1836 0.79 0.55 --0.0033 233.18 0.42 0.20 --0.0083 233.18 0.33 0.10 --0.0142 233.81
1864 --0.25 --0.44 0.0124 233.97 --0.13 --0.33 0.0138 233.94 --0.10 --0.31 0.0156 233.95
1865 --0.04 --0.24 0.0072 233.32 --0.02 --0.23 0.0074 233.32 -- -- -- --
1866 0.17 --0.04 0.0036 233.30 0.09 --0.12 0.0026 233.31 0.07 --0.15 0.0012 233.31 0.0065 233.30
1867 0.37 0.15 --0.0002 233.42 0.20 --0.01 --0.0026 233.43 0.15 --0.07 --0.0057 233.42
1868 0.58 0.35 0.0011 234.06 0.31 0.10 --0.0045 234.03 0.24 0.02 --0.0096 234.01
1869 0.78 0.54 0.0102 233.43 0.43 0.21 0.0055 233.49 0.32 0.10 --0.0006 233.54
1897 --0.35 --0.54 --0.0020 234.24 --0.18 --0.37 0.0008 234.93 --0.12 --0.32 0.0013 235.27
1898 --0.14 --0.34 0.0155 234.84 --0.07 --0.28 0.0167 234.97 --0.05 --0.26 0.0176 235.05
1899 0.07 --0.14 0.0138 235.03 0.04 --0.17 0.0132 234.98 0.03 --0.18 0.0126 234.98 0.0117 234.63
1900 0.28 0.06 0.0199 234.07 0.15 --0.06 0.0182 234.02 0.11 --0.10 0.0167 234.05
1901 0.48 0.25 0.0146 233.85 0.25 0.04 0.0124 233.82 0.19 --0.02 0.0096 233.85
1902 0.68 0.44 0.0114 233.85 0.36 0.14 0.0086 233.85 0.28 0.06 0.0035 234.03
1930 --0.36 --0.55 0.0075 235.10 --0.17 --0.38 0.0071 235.25 --0.12 --0.32 0.0019 235.39
1931 --0.14 --0.35 0.0065 235.09 --0.08 --0.29 0.0105 234.90 --0.06 --0.26 0.0125 235.09
1932 0.07 --0.15 0.0060 235.09 0.03 --0.18 0.0060 235.36 0.02 --0.18 0.0060 235.44 0.0061 235.08
1933 0.28 0.06 0.0054 235.15 0.11 --0.11 0.0119 236.01 0.07 --0.14 0.0135 235.99
1934 0.49 0.25 0.0040 235.50 0.16 --0.06 0.0173 236.25 0.10 --0.11 0.0182 235.95
1935 0.69 0.45 0.0183 235.97 0.23 0.01 0.0342 236.21 0.16 --0.05 0.0327 235.66
1963 --0.31 --0.51 0.0059 235.09 --0.17 --0.37 0.0077 235.11 --0.13 --0.33 0.0136 235.10
1964 --0.10 --0.30 0.0038 235.12 --0.05 --0.26 0.0043 235.12 --0.04 --0.25 0.0063 234.95
1965 0.11 --0.10 0.0023 235.13 0.06 --0.16 0.0017 235.14 0.04 --0.17 0.0015 235.45 0.0031 235.13
1966 0.32 0.10 0.0015 235.13 0.17 --0.05 --0.0002 235.16 0.09 --0.13 0.0034 235.94
1967 0.53 0.30 0.0012 235.13 -- -- -- -- 0.13 --0.09 0.0063 236.21
1968 0.73 0.49 0.0010 235.15 0.39 0.17 --0.0037 235.44 -- -- -- --
1996 --0.28 --0.47 0.0099 235.30 --0.15 --0.35 0.0126 235.27 --0.11 --0.32 0.0149 235.27
1997 --0.06 --0.27 0.0085 235.26 --0.04 --0.24 0.0091 235.26 --0.03 --0.24 0.0095 235.26
1998 0.14 --0.07 0.0068 235.26 0.08 --0.13 0.0055 235.27 -- -- -- -- 0.0081 235.26
1999 0.35 0.13 0.0047 235.28 0.19 --0.02 0.0019 235.27 0.14 --0.08 --0.0007 235.29
2000 0.55 0.33 0.0031 235.29 0.30 0.08 --0.0012 235.27 0.22 0.00 --0.0050 235.32
2001 0.76 0.52 0.0021 235.29 0.41 0.19 --0.0034 235.25 0.30 0.08 --0.0087 235.39
from the central values of a0 produced on ejection (the latter
would impose a lower bound on ejection speeds needed). The
values
of\Omega in the reasonable storm years 1866--68 match ob­
servations quite well, the 1868 value especially differing from
the comet. Depending on what spread in elements within a
trail is reasonable, Table 2 might be able to explain why
despite the comet's similar rD , Leonid activity was greater
in 1866--68 than around the 1933 return.
As trails gradually lengthen in M , the spatial density
will tend to become progressively diluted with each revolu­
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1833 and 1966 Leonid storms 5
tion, reducing the potential meteor flux. Moreover, increas­
ingly many sections of the trail will be disrupted. Neverthe­
less, nodal crossings were checked, for a few years of possible
interest, for larger numbers of revolutions (up to 6) between
ejection and appearance of meteors. It was found that the
1600 trail gave an excellent nodal distance match ( ! ¸ 0.0001)
in 1799 (i.e., 6 revolutions), the values for 4 and 5 revolu­
tions being similar to those for 3 shown in Table 2. It is
hard to be sure of the trails' relative importance in produc­
ing the impressive 1799 storm, but a clearly better fit
of\Omega (well constrained by eyewitness reports; Rao 1998) than the
cometary value was notable. For reference, the
calculated\Omega is 232.80. Regarding the similarly good 1832 storm, 4, 5 or
6 revolutions all give rE \Gamma rD ú 0.0008.
No especially close nodal distances are found for any
trail and 1998 meteors, but a distance of 0.0010 is found for
the 1866 trail and 2000 meteors (4 revolutions). For 1999
meteors, the 1899 trail listed in Table 2 is much closer than
for the 3 earlier trails. Despite the similar preferred values of
fi to the 1833 and 1966 cases, both the greater miss distance
and greater number of revolutions make a storm at the level
of those two unlikely. However, there may be some kind of
storm, the estimated distance from the centre of the trail
being less than 10 Earth diameters. It may also be desirable
to calculate as accurately as possible the 3­dimensional posi­
tion of the trail as a function of time over a day or so around
predicted passage near (through) it so as to assess whether
there is a risk to satellites (cf. Beech, Brown & Jones 1995).
One year of possible interest not in Table 2 is 1969,
where a sharp outburst, notable for being rich in small
particles (McIntosh 1973), was observed. It turns out that
the 1932 trail gives both a perfect nodal distance match ú
0.0000,
and\Omega = 235.27 precisely in accord with the timing
of the peak (Millman 1970). The required \Deltaa 0 is 0.9 au if
fi=0, with fi=0.004 giving \Deltaa 0=0. Such a value of fi corre­
sponds to small particles. This fit impressively demonstrates
the applicability of the trail model.
4 DISPERSION
In addition to radiation pressure causing position to vary
with particle size, the range in orbital elements induced
on ejection affects the extent of each trail. The Poynting­
Robertson effect is negligible at this early evolutionary stage
in a stream (Kres'ak 1993b).
Some repeat integrations (a column from Table 2 at a
time) were done but in each case altering one of the initial
elements q, i,
\Omega\Gamma ! (respectively by 0.002, 0.02, 0.2, 0.2).
To a very good approximation, differences in these elements
were conserved over the timescales in question. The range in
elements at ejection time is therefore of interest as it carries
through to a range in rD
(and\Omega\Gamma at meteor observation time
via equation (2).
A detailed investigation, e.g., using analytical formulae
for changes in elements (Pecina & Ÿ
Simek 1997) in addition
to the Monte Carlo type procedure used for Fig. 2, or consid­
ering a range of physical models (cf. Brown & Jones 1998),
is not attempted at this stage, but Fig. 2 gives an approx­
imate idea of spreads in elements that may be expected.
For simplicity, ejection is isotropic (rather than having sun­
ward hemisphere only, or jets opposite to the known sense
of the transverse nongravitational force). Consideration of
cometary ejection processes (Whipple 1951, Jones 1995) sug­
gests the velocities in Fig. 2 are reasonable, given a nuclear
radius ú 1.8 km (Hainaut et al. 1998). There are correla­
tions among elements that will not be explored here, but
the overall dispersions should tell us something useful.
A trail's range
in\Omega gives the length of the Earth's path
through it, which is a factor of ¸3 longer than the stream's
vertical cross section, since the Earth passes through the
Leonid stream at an angle of 17 ffi , this angle being largely
in a vertical plane (cf. first paragraph of Sec. 2).
The\Omega range in Fig. 2 corresponds to a few hours, consistent per­
haps with some past storms. The width across the Earth's
path depends via equation (2) on q and !. In Fig. 2, the
former converts to ? ¸ 0.001 au in cross section and the latter
rather less. In the absence of further modelling, ¸0.001 au
may be tentatively adopted as the cross section. Increases
in density towards the centre would be unsurprising. The
distance is reasonably in accord with the attempts to inter­
pret the rE \Gamma rD distances from Table 2 in Sec. 3. For visual
meteor size particles, the random dispersion due to ejection
seems clearly less than the range in rD due to perturbations,
emphasizing the value of accurately calculating the latter. It
can be noted that the range in a and both cross sections will
all tend to increase and decrease together as the maximum
ejection velocity is varied in the model.
Studying past meteor activity profiles (Jenniskens 1995)
would be useful. Observations from 1966 may yield informa­
tion about the 1899 trail, perhaps of interest before 1999
November.
5 DUST TRAILS AND METEOR STORMS
The purpose of this paper has been to do the calculations
that identify as easily as possible the fejection time, me­
teor yearg pairs likely to produce the greatest storms. Kon­
drat'eva et al. (1997) were the first, so far as I am aware,
to make a significant number of such identifications relat­
ing to the Leonids, but as that paper is quite short, it has
been worth presenting here an independent derivation and
discussion not given there.
More detailed modelling of the lengthening and then
disruption of each trail (cf. Kres'ak 1993a,b) could be done.
Before a trail is disrupted, perturbations at a distance from
Jupiter and Saturn are probably responsible for its gradual
evolution away from the comet's orbit. Though small, this
difference from 55P/Tempel­Tuttle's orbit has a critical ef­
fect on the occurrence of meteor storms. The actual disrup­
tion of cometary trails is generally due to the giant planets
(Kres'ak 1993a,b), but the Earth also affects the Leonids,
since its orbit passes through the stream. During each peri­
helion passage, a total of as much as 10% (comprising sepa­
rate trail sections a year apart) of each trail can be seriously
affected by the Earth. Williams (1997) has shown how in the
present few centuries, a near commensurability makes trails
near the comet unusually safe from disruption by Uranus.
Whatever the cause of disruption, the various trails eventu­
ally merge into the background Leonid stream.
Given records of meteor storms over the past millen­
nium (Hasegawa 1993, Mason 1995), this study could be
extended to see if they can be related to particular trails.
The orbit of 55P/Tempel­Tuttle may be known accurately
c
fl 1999 RAS, MNRAS 307, 919--924

6 D.J. Asher
Figure 2. Orbital elements of 250 test particles randomly (isotropically) ejected from 55P/Tempel­Tuttle at its 1998 return, 50 at each
of M=--4,--2,0,2,4 ffi (plotted as 2,+,\Lambda,\Theta,ffi). Ejection velocities 25, 15 and 10 m/s at r = 1, 1.5, 2.2 au were adopted.
enough for this purpose. This could help to confirm a general
lifetime for coherent, dense trails of order a few revolutions.
Numerical integrations, then, can be used to find the
nodal crossing points of those parts of trails that cross the
ecliptic when the Earth is nearby. These locations can be de­
termined at the accuracy given in Table 2, and need not nec­
essarily match the point where the comet crosses the ecliptic.
The occurrence of a storm is dependent on the distance from
the Earth being small enough. Based both on relating cal­
culated miss distances to past storms, which successfully ex­
plained the observed storms with the highest zenithal hourly
rate (Sec. 3), and on estimates of orbital element dispersion
(Sec. 4), the allowed miss distance for a storm appears to
be a few times 0.0001 au. A procedure therefore exists to
determine reliably when Leonid storms occur.
ACKNOWLEDGMENTS
I am grateful to Rob McNaught for discussions about the
Leonids and the idea leading to this work. Comments from
Mark Bailey, Brian Marsden, and the referee Iwan Williams
are also appreciated. John Chambers kindly made available
his Mercury integrator package. The research was supported
by PPARC and Starlink.
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