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Meteoritics & Planetary Science 34, 975--978 (1999) [1999 November issue]
c
fl Meteoritical Society, 1999.
Variation of Leonid maximum times with location of observer
R.H. McNAUGHT 1 and D.J. ASHER 2\Lambda
1 Siding Spring Observatory, Coonabarabran, NSW 2357, Australia
2 Armagh Observatory, College Hill, Armagh, BT61 9DG, UK
\Lambda Correspondence author's e­mail address: dja@star.arm.ac.uk
(Received 1999 June 29; accepted in revised form 1999 August 25)
(Part of a series of papers on the 1998 Leonid shower)
Abstract
A time adjustment from the Earth's centre to the observer's location is quantified and shown to improve
the residuals between the observed time of pronounced Leonid showers and the time calculated from
perturbed dust trails. A correlation with the exact time of closest approach to the nominal dust trail
orbit would allow a further improvement in the residuals. However, there is no evidence for such a
correlation, suggesting that the dust trail cross section is substantially elongated within the orbital plane.
These results indicate that approaches to dust trails within 0.0005 AU are predictable with an uncertainty
of around 5 minutes. Application of this correction to satellites confirms the utility of two strategies to
minimise the impact risk. The time adjustment could be applied to any meteor stream.
1 Introduction
Predictions of the time of Leonid maximum for 1999--2002 have been made by Kondrat'eva and Reznikov
(1985) and McNaught and Asher (1999) based on the perturbed orbits of dust trails. A trail is generated
each time Comet 55P/Tempel­Tuttle returns to perihelion, and the most intense Leonid outbursts (i.e.,
reaching the highest zenithal hourly rates) correspond to encounters of the Earth with young trails (less
than a dozen or so revolutions old).
It has been shown that the time of maximum of Leonid storms lies rather close to the time of passage
through the node of 55P/Tempel­Tuttle (Yeomans, 1981; Kres'ak, 1993). This has been interpreted to
mean that the dust is predominantly distributed within the orbital plane of the comet. However, the
dust trail theory (Asher, 1999) predicts the times much more closely (Kondrat'eva and Reznikov, 1985;
McNaught, 1999a; McNaught and Asher, 1999), and although prominent storms of the past two centuries
have happened to be within a few hours of passage through the comet's orbital plane, the years 2000--2002
in particular are expected to show much greater time discrepancies between meteor activity peaks and
the comet plane (see also Kondrat'eva et al., 1997).
In the present paper we investigate refinements to the predictions of times of maximum, to see whether
the existing accuracy of around 10 minutes can be improved. Previous calculations were based on the
centre of the Earth and the position of the centre of the dust trail at its node. To obtain a specific
prediction for any locality, it is necessary to compensate for the offset of the observer from the Earth's
centre (Sec. 2). An additional adjustment may be required, due to the skewed passage of the Leonid
orbit to the Earth's orbit (Sec. 3); the closest approach to the dust trail is offset from the node, although
the exact time of maximum (when the highest density is encountered) depends on the oblateness of the
dust trail. The technique of considering approaches to orbits was used by Kres'ak (1993), but with the
comet's orbit, rather than dust trails. (We note for reference that in his Fig. 5, the line C, giving the
slant distance to the comet orbit, is in error; also Fig. 3 is incorrectly constructed and the order of the
plotted passages inconsistent with the geometry of the various returns of 55P/Tempel­Tuttle.)
We are investigating the oblateness of the individual dust trails both theoretically and observationally;
the initial indication is that the dust trails are themselves somewhat flattened in the plane of the orbit.
The theoretical approach is model dependent and examines the stream profile resulting from the ejection
process. One observational approach is to compare the dust trail transect from any one year with the
in­plane dust trail profile found from fitting storm data to our (McNaught and Asher, 1999) density
model. The offset of the time of maximum towards the closest approach to a dust trail is a third method
of investigating the oblateness of the dust trail and will be briefly examined here (Sec. 3).
1

2 Topocentric correction
2.1 Derivation
A Leonid dust trail is represented by a nominal orbit, with a density profile that falls off both as one
moves out of the trail's orbital plane, and within the plane away from the orbit. During an encounter
with a trail that produces storm level activity, the Earth's closest approach distance to the nominal orbit
is small, typically !0.0005 AU.
Representing a dust trail to a first approximation by a sheet of dust within the orbital plane (the plane
of the dust sheet having inclination to the ecliptic i ú 162.5 ffi and being encountered at the descending
node), the position of the observer in relation to the centre of the Earth can easily be converted to a
time adjustment for passage through the dust sheet. The observer's location must first be converted to
geocentric ecliptic latitude and longitude (fi, –).
The observer is offset from the Earth's centre by a distance H in the direction of the Earth's velocity
vector, given by
H = RE cos fi cos(– \Gamma – apex )
where RE is the radius of the Earth. The observer is also displaced by a distance Z out of the ecliptic,
and the centre of the Earth must travel a further distance D, to the point where the dust sheet is a
distance Z out of the ecliptic:
D = Z= tan i = RE sin fi= tan i
neglecting an additional factor of 1= sin(– Sun \Gamma – apex ) as it is always very close to 1.000.
The overall correction \Deltat to be added to the time the Earth's centre crosses the dust sheet (nominal
prediction), to give the topocentric time, is
\Deltat = \Gamma(H + D)=vE
where the Earth's velocity vE = 30.13 km/s at this part of its orbit. O--C's from previous showers should
be altered by \Gamma\Deltat (Sec. 2.2).
A time adjustment of this form can be applied to any meteor shower, using the appropriate values for
i and vE (replacing tan i by --tan i for encounters at the ascending node). It is however only particularly
relevant for showers with a very narrow peak or inclination close to the ecliptic.
Contours at 1­minute intervals are calculated for the predicted time of passing the node of the 3 rev­
olution old trail in 1999 (the closest trail that will be encountered this year), and plotted in Fig. 1. The
pattern of contours is broadly determined by D; the effect of the component H, which reaches a maxi­
mum of +3.5 minutes close to the centre of the figure, is to compress the contours upwards. The value
of the overall adjustment for a given location can be calculated directly, or interpolated from Fig. 1. For
other years when the inclination of the Leonids is close to 162.5 ffi , the contours will be similar but the
continents will have rotated (cf. McNaught 1999b).
Fig. 1 shows that the \Deltat range (e.g., northeast Europe vs southern Africa) should be sufficient to be
detectable, as peak times can be determined from observations to within a few minutes. Nevertheless,
it may be that closer trail encounters than in 1999, especially those in 2001 and 2002 (Kondrat'eva and
Reznikov, 1985; McNaught and Asher, 1999) will afford a better test of this topocentric correction, as
the peaks in these years may be more pronounced.
2.2 Application to past data
Since 1833, there have been attempts to accurately determine the time of maximum of the Leonids,
particularly in years of potential storms. With \Deltat differing by up to 23 minutes between two extreme
locations, it would be necessary to apply a time adjustment to each observation to bring them into uni­
formity. Failure to do this in a global analysis would result in the width of the shower being broadened
and the peak intensity lowered. For some of the Leonid showers discussed by Brown (1999), the geo­
graphical distribution of the observers is small, allowing an appropriate adjustment to the observed time
of maximum. The time of maximum has however been published for a number of individual observations
and these will also be examined here.
Table 1 lists the `Old' O--C without the application of the adjustment \Gamma\Deltat, and the `New' O--C with
the adjustment. It is notable that the Old O--C's derived from the global analyses of Brown (1999) are
all positive, and apply to regions of the globe above the ecliptic. They thus have positive \Deltat. The 1869
shower was the only one observed exclusively from below the ecliptic and its Old O--C was negative. In
most cases, the effect of the adjustment is to reduce the magnitude of the O--C. The uncertainty in the
2

New O--C is the same as the Old O--C if the dust sheet assumption holds, and is typically several to
10 minutes, although observations over the next few years should give better data to assess the overall
predictions.
This adjustment to the Earth's centre thus appears to be well confirmed, and is especially well
demonstrated by the Soviet observations in 1966. It is however dependent on the assumption that the
dust is spread predominantly within the plane of the dust trail orbit. This will be examined in Sec. 3.
2.3 Application to satellites
In McNaught and Asher (1999), two strategies are discussed to minimise the threat to satellites capable of
manoeuvre. One is to adjust the orbit to place the satellite deepest in Leonid `shadow' at the predicted
time of maximum, should such a geometry be possible. The other, which we briefly discuss here, is
to place the satellite at the furthest point in its orbit from the centre of the dust trail at the time of
maximum, thus minimising the incident flux. McNaught and Asher (1999, their Table 10) calculated this
position using an approximation accurate to a few degrees. A rigorous derivation of the direction of the
radial density gradient, being perpendicular to the trail orbit within the orbital plane, gives ff 57 ffi ffi +20 ffi
(J2000) as the direction towards which a satellite should be most closely positioned at maximum, for
trails passing inside the Earth's orbit, and ff 237 ffi ffi --20 ffi (J2000) for trails outside. This equates to the
geographical longitude being increased by 3 ffi and the latitude moved poleward by 1 ffi as compared with
their Table 10. Both the strategies for satellites depend on the time of maximum being predictable to
within about 20 minutes, and the adjustment for location in this paper brings the uncertainty well below
this. In the calculation for a satellite, the geocentric distance of the appropriate part of the satellite orbit
must replace the Earth's radius in the formula for \Deltat in Sec. 2.1. Given the corrected time of maximum,
the satellite would be manoeuvred to arrive at the relevant point at that time (see McNaught and Asher,
1999). A combination of the two strategies could also be considered.
3 Close approach to the orbit
In the preceding analysis, we assumed a dust sheet, regarding the dust trail as effectively having an
oblateness of eccentricity 1. Now let us assume instead that the dust trail is cylindrical, i.e., with circular
profile (eccentricity 0), perpendicular to the nominal orbit. The maximum density would then coincide
with the closest approach to this orbit. The time of closest approach can be simply calculated from
the shortest distance between two straight lines defined by the position and velocity vectors of both the
Earth and the dust trail at its node: since the closest approach, for distances within 0.0010 AU (for which
high rates may occur), occurs within half an hour or so of the Earth passing the dust trail node, it is
not necessary to incorporate the curvature of the orbits in the calculation. The angle of the dust trail
orbit outwards from perihelion is the relevant factor here; the marginally skewed passage of the Leonid
orbit to the Earth's orbit displaces the closest approach time by a small amount \DeltaT from the time when
the Earth passes the node. The magnitude of this correction scales approximately linearly with distance
(4.5 min per 0.0001 AU, though slightly different depending exactly how far from perihelion the node
is). For dust trails that pass outside the Earth's orbit (negative r E \Gamma r D , where r D is the heliocentric
radial distance of the dust trail descending node, and r E is the Earth's heliocentric distance as it passes
the node) the time of closest approach is after passing the dust trail node and for positive r E \Gamma r D , it is
before. In 1999, the time of closest approach will be 29 minutes after passing the 3­revolution dust trail
node, i.e., \DeltaT = +29.
The years considered in Table 1 have values of jr E \Gamma r D j Ÿ 0.0005 AU. By tabulating the New O--C
against \DeltaT (Table 2), any correlation between these, related to the dust trail oblateness, could be found.
The New O--C is used, as it will still be relevant, although it is not a rigorous correction in this case.
Only the values from the global analyses by Brown (1999) are used as these should have less scatter,
despite the limitations noted above. The uncertainties in these O--C's are of the order of several minutes
and for the outburst years alone, there is no correlation (Table 2).
It was believed that the signatures of distant dust trails may be present in observations from 1965
and 1998 (McNaught, 1999a); so we give values of \DeltaT in Table 2. The two closest 1 to 4­revolution trails
(McNaught and Asher, 1999) in 1965 and 1998 are considered, there being no other suitable candidates.
The O--C's for these two trails in the respective years are calculated using the same observed faint meteor
peak (Plavcov'a, 1968; Arlt, 1998). There appears little reason, based on these distant dust trails data, to
believe the observed peak is actually associated with these trails, although if it were, it would again tend
3

to discount any correlation with the closest approach to the dust trail orbit. Altogether, the evidence
points towards the dust trails being flattened within the orbital plane.
4 Conclusions
Calculations of the time of Leonid maxima are based on the centre of the Earth and must be adjusted to
give the observed time of maximum from any specific location. For locations on the Earth's surface, this
adjustment can amount to \Sigma12 minutes. The calculation of the adjusted time for orbiting satellites can
be used to place the satellite either within the Leonid `shadow', or furthest from the dust trail centre at
the predicted time of maximum. This could minimise the impact risk.
The retrograde orbit of the Leonids, encountered at their descending node, results, to a close ap­
proximation, in maximum occurring respectively before/after the Earth passes the dust trail node for
observers with negative/positive ecliptic latitude (very roughly, the southern/northern hemisphere). Any
global analysis of a short duration shower must incorporate this effect, or it will widen the activity curve
and lower the peak intensity.
For the Leonid showers examined, there is a close correspondence between the time of maximum and
passage through the plane of the nominal dust trail orbit. It appears that the local time of maximum
fits passage through a dust trail orbit with around 5 minutes uncertainty. No correlation was found
between the time of maximum and the closest approach to the dust trail. This indicates that the dust
trails have an elongated cross section with the major axis within the plane of the orbit. The relatively
shallow intersection of the Leonid and Earth orbits, however, does not make this a strong test, although
the conclusion seems fairly robust. Showers with known parent objects, perihelia well inside the Earth's
orbit, inclinations close to the ecliptic, and capable of producing storms would make a better test, but
unfortunately there are currently no such showers.
References
Arlt R. (1998) Bulletin 13 of the International Leonid Watch: The 1998 Leonid meteor shower. WGN
(Journal of the International Meteor Organization) 26, 239--248.
Asher D.J. (1999) The Leonid meteor storms of 1833 and 1966. Mon. Not. R. Astron. Soc. 307,
919--924.
Bell P.H. (1967) Meteoric storm over Texas. Rev. Popular Astron. 61, 9--11.
BronŸ sten V.A. (1968) Observations of the Leonid meteor shower in November 1966 in the U.S.S.R.
In IAU Symp. No. 33, Physics and Dynamics of Meteors (eds. Ÿ
L. Kres'ak and P.M. Millman), pp.
440--445. Reidel, Dordrecht, Holland.
Brown P. (1999) The Leonid meteor shower: historical visual observations. Icarus 138, 287--308.
Kondrat'eva E.D. and Reznikov E.A. (1985) Comet Tempel­Tuttle and the Leonid meteor swarm.
Sol. Syst. Res. 19, 96--101.
Kondrat'eva E.D., Murav'eva I.N. and Reznikov E.A. (1997) On the forthcoming return of the
Leonid meteoric swarm. Sol. Syst. Res. 31, 489--492.
Kres' ak Ÿ
L. (1993) Cometary dust trails and meteor storms. Astron. Astrophys. 279, 646--660.
McNaught R.H. (1999a) On predicting the time of Leonid storms. The Astronomer 35, 279--283.
McNaught R.H. (1999b) Visibility of Leonid showers in 1999--2006 and 2034. WGN (Journal of the
International Meteor Organization) 27, 164--171.
McNaught R.H. and Asher D.J. (1999) Leonid dust trails and meteor storms. WGN (Journal of the
International Meteor Organization) 27, 85--102.
Millman P.M. (1967) Meteor news. J. R. Astron. Soc. Can. 61, 89--92.
Millman P.M. (1970) Meteor news. J. R. Astron. Soc. Can. 64, 55--59.
Plavcov' a Z. (1968) Radar observations of the Leonids in 1965--66. In IAU Symp. No. 33, Physics and
Dynamics of Meteors (eds. Ÿ
L. Kres'ak and P.M. Millman), pp. 432--439. Reidel, Dordrecht, Holland.
Yeomans D.K. (1981) Comet Tempel­Tuttle and the Leonid meteors. Icarus 47, 492--499.
4

Tables
Table 1. O--C residuals in minutes with and without topocentric correction.
Year Observer Location Reference Old O--C Adjustment New O--C
1866 U.K. Brown (1999) +6 --4.3 +2
1867 Global E. North America Brown (1999) +4 (--0.3) (+4)
1869 Meldrum Mauritius McNaught & Asher (1999) --5 +9.8 +5
1966 Global W. North America Brown (1999) +3 (--1.5) (+2)
1966 Milon Kitt Peak Millman (1967) +2 +0.8 +3
1966 Bell Lubbock, Texas Bell (1967) --5 +0.3 --5
1966 Soviet Arctic BronŸsten (1968) +12 (--9 ) (+3)
1969 Global North America Brown (1999) +7 (--1 ) (+6)
1969 Millman Springhill Millman (1970) +0.5 --1.8 --1.3
The adjustments in brackets are representative of a large geographical area with a spread of a few minutes.
Table 2. Approaches to trail orbits.
Year r E \Gamma r D \DeltaT New O--C
Outburst years
1969 0.0000 0 +6
1966 --0.0001 +7 +2
1867 --0.0002 +10 +4
1866 --0.0004 +16 +2
1869 --0.0005 +28 +5
Distant trails
1965 +0.0015 --65 --410 (3­rev)
1965 +0.0017 --75 +45 (2­rev)
1998 +0.0040 --160 --460 (4­rev)
1998 +0.0055 --230 +60 (2­rev)
r E \Gamma r D is miss distance in AU.
\DeltaT is expected O--C in minutes if trail had
circular cross section; next column is actual
O--C (cf. Table 1).
In 1965 & 1998, these are the two closest
trails 4 or less revolutions old. There is no
evidence that the faint meteor peak in 1965
or 1998 is related to the young dust trails
(see text).
5

Figure 1
Topocentric adjustment \Deltat in minutes to Leonid encounter times, shown in orthographic projection
viewed from the direction of the radiant. North Pole (cross) and lines indicating onset of astronomi­
cal, nautical and civil twilight and daylight are shown, as are the continental outlines for the time the
Earth's centre crosses the orbital plane of the closest dust trail in 1999. The continents are shown more
prominently when the sky is dark so as to provide an idea of observability. The two dashed lines show
where the moon respectively goes 0 ffi and 6 ffi below the horizon. The radiant becomes higher in the sky
the nearer one is to the centre of the plot.
6