Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/~ambn/Hale-Bopp.ps Äàòà èçìåíåíèÿ: Thu Oct 10 18:30:07 1996 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 22:21:16 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï

Mon. Not. R. Astron. Soc. 000, 000--000 (0000) Printed 7 October 1996 (MN L a T E X style file v1.3)
Orbital evolution of Comet 1995 O1 Hale­Bopp
M.E. Bailey 1 , V.V. Emel'yanenko 1;2 , G. Hahn 3 , N.W. Harris 1 , K.A. Hughes 2 ,
K. Muinonen 4 and J.V. Scotti 5
1 Armagh Observatory, College Hill, Armagh, BT61 9DG, U.K.
2 School of Electrical Engineering Electronics and Physics, Liverpool John Moores University, Byrom Street, Liverpool, L3 3AF, U.K.
3 DLR Institut f¨ur Planetenerkundung, Rudower Chaussee 5, D­12489 Berlin, Germany.
4 Observatorio, PL 14, T¨ahtitorninm¨aki, FIN­00014, Helsingin Yliopisto, Suomi­Finland.
5 Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona 85721, USA.
7 October 1996
ABSTRACT
Results from a series of long­term numerical integrations of orbits centred on that of
C/1995O1 (Hale­Bopp) are presented. Initially, 33 orbits taken from various sources
were integrated in realistic models of the solar system for various time­scales about the
present in the range (\Gamma3; +2)Myr and analyzed to assess the probability of different
dynamical outcomes, such as a sungrazing state or the number of orbits since it was
captured from a long­period orbit in the Oort cloud. Further integrations were per­
formed using an ensemble of 26 orbits more closely clustered about the present orbit
of the comet, based on a more accurate method of orbit determination. We find that
the ensemble half­life for the comet to be captured or ejected is on the order of 0.5 Myr
in the backward integrations and 1.2 Myr in the forward integrations, although a few
members of our ensemble were captured during strong encounters with Jupiter within
10 revolutions. Comet Hale­Bopp has a probability p'0:15 of evolving to a sungrazing
end­state, and although not a `new' comet (in the sense of being recently captured from
the Oort cloud) may be considered dynamically young.
Key words: celestial mechanics, comets: individual ­ C/1995O1 Hale­Bopp, meteors.
1 INTRODUCTION
The discovery of a 10th magnitude comet by A. Hale and
T. Bopp on 1995 July 23 UT at the unprecedented helio­
centric distance of 7.15 AU (Green 1995) suggests that a
comet bright enough to rival any seen this century may
soon be visible with the naked eye and become a spectacu­
lar object viewed from northern latitudes during March and
April 1997. It is possible that the comet may not brighten
as expected, but the apparent magnitude (¸10 magnitudes
brighter than that of 1P/Halley at the same heliocentric
distance) suggests that the nucleus must be at least sev­
eral times larger than the ¸10km diameter of that comet.
Early estimates of the magnitude of C/1995 O1 (Hale­Bopp)
suggest a nuclear diameter in the range 50--150 km, mak­
ing this object potentially the first `giant' comet (diame­
ter greater than 100 km) to enter the inner solar system in
modern times. Previous giants amongst long­period comets
may include the progenitor of the Kreutz sungrazer fam­
ily, which passed perihelion in 372 BC (Marsden 1989), and
Comet Sarabat (C/1729 P1), which became a naked­eye ob­
ject for almost 6 months despite its perihelion distance of
4.05 AU. Comet Hale­Bopp approaches the Sun much closer,
with perihelion distance q = 0:91 AU.
It is now recognized that large objects such as
(2060) Chiron and (5145) Pholus, with diameters in the
range 100--300 km, must be occasional visitors to the in­
ner solar system, coming at least in the first instance from
the Centaur population (Hahn & Bailey 1990, Bailey et
al. 1994). These objects have low­inclination, intermediate­
period orbits with perihelia close to or just beyond the orbit
of Saturn. Given their likely provenance in the outer so­
lar system, with a correspondingly high content of volatile
material, these bodies should exhibit extensive outgassing
and appear cometary when travelling closer to the Sun.
Cometary activity in such an object has been observed at
a heliocentric distance of about 10 AU during the current
perihelion passage of (2060) Chiron = 95P/Chiron (see, for
example, Campins et al. 1994, Meech & Belton 1990, Luu
& Jewitt 1990).
The significance of the arrival of Comet Hale­Bopp is
that it provides an opportunity to observe the development
of nuclear activity in a large comet, and its presence confirms
the existence of exceptionally large comets in the long­period
flux. Comet Hale­Bopp is unusual in having an orbital in­
clination, i, close to 90 ffi and a much shorter orbital period
than most long­period comets; it last came to perihelion

2 M.E. Bailey et al.
¸4600 yr ago. Moreover, the orbit has ascending and de­
scending nodes close to the orbits of Jupiter and the Earth
respectively. Thus, not only might the comet have been re­
cently captured on account of strong Jovian perturbations,
but it may also have put on a good display for our remote an­
cestors. We note here that there is a possibility of heightened
meteoric activity when the Earth approaches the descend­
ing node of Hale­Bopp's orbit in early January 1996 and in
previous and subsequent years, although the apparent asso­
ciation with the known Quadrantid meteoroid stream (since
the longitudes of the corresponding ascending nodes are less
than a degree apart) is unlikely owing to the large difference
between the orbital periods of Comet Hale­Bopp and the
Quadrantids.
The closeness of the orbital inclination to 90 ffi indi­
cates that secular perturbations by Jupiter may cause Hale­
Bopp to become sungrazing in the future (Bailey, Cham­
bers & Hahn 1992a). In this respect, the similarity of the
evolution of Hale­Bopp to that of the Kreutz sungrazer
group (whose orbits are currently sungrazing) or to that of
96P/Machholz 1 (associated with the Quadrantid meteoroid
stream; McIntosh 1990, Babadzhanov & Obruvov 1993),
which is likely to become a sungrazer in the future (Bai­
ley, Chambers & Hahn 1992b), suggests that a study of the
dynamical evolution of Comet Hale­Bopp may also help in
understanding the source of short­period comets and the ori­
gin of cometary dust in the inner solar system. This paper
presents the results of numerical integrations of variational
orbits centred on the orbit of the real comet in order to ad­
dress these questions and to elucidate the likely long­term
history of Comet Hale­Bopp.
2 METHOD
We initially integrated an ensemble of orbits of Comet Hale­
Bopp following a procedure similar to that adopted by
Hahn & Bailey (1990), varying the orbital elements by small
amounts in a systematic way so as to encompass what ap­
peared to be the likely uncertainty in the preliminary orbit
determination. Since the orbit is chaotic, it is not possible
to determine the long­term orbital evolution of the comet
exactly; minute errors in the assumed orbit or in the dy­
namical model of the solar system eventually become signif­
icant. Moreover, with an uncertainty in the orbital period
of at least several Jovian years we cannot reliably predict
the effects of Jupiter's perturbations even one revolution
ago. Nevertheless, the fact that the orbit has a node close
to Jupiter allows close encounters to this planet and sug­
gests that rapid dynamical evolution may occur, leading to
capture or ejection on relatively short time­scales.
This preliminary study was extended to incorporate
a more accurate method of orbit determination described
by Muinonen & Bowell (1993). We investigated a further
26 orbits with osculating orbital elements chosen so as to
uniformly sample the uncertainties allowed by observations.
The vector of osculating orbital elements at the chosen epoch
t0 is P = (a; e;
i;\Omega ; !; M0 ) T , corresponding to semi­major
axis, eccentricity, inclination, longitude of ascending node,
argument of perihelion, and mean anomaly at epoch. The
three angular elements i, \Omega\Gamma and ! are referred to the eclip­
tic at equinox J2000.0.
The first study considered an ensemble of 33 massless
bodies centred on orbits given by Marsden (1995a,b), using
the improved orbit determination given by Marsden (1995c)
in order to assess the accuracy of these preliminary orbits.
The orbits were integrated in a model solar system including
the Sun and seven planets Venus, Earth, Mars, Jupiter, Sat­
urn, Uranus and Neptune, and the objects were integrated in
several batches for times up to 2 Myr into the future and up
to 3 Myr into the past. The total integration time was about
63 Myr. One batch of orbits (9 in total) was integrated in a
model solar system including only the major planets Jupiter,
Saturn, Uranus and Neptune.
These integrations were carried out on workstations at
Armagh, Liverpool and Tucson using Everhart's RADAU in­
tegrator (Everhart 1985) with tolerance parameter set be­
tween 10 \Gamma8 and 10 \Gamma12 . In addition to these long­term inte­
grations, we also performed more accurate runs of approxi­
mately 1000 days into the future in order to determine the
effects of an imminent Jovian encounter during the first half
of 1996 with greater precision. Table 1 presents recent orbit
determinations for Comet Hale­Bopp; sources (1) and (2)
were used as central orbits in our initial study. As explained
above, we subsequently applied the method of orbit deter­
mination described by Muinonen & Bowell (1993) to create
a further ensemble of 26 orbits.
2.1 Orbit determination
For Comet Hale­Bopp, we chose 645 observations covering
the time­period 1993 April 27 to 1995 September 3. Obser­
vations used were those current through the 1995 September
9 Minor Planet Circulars. Two apparitions are involved, and
all observations except one belong to the 1995 apparition;
there is thus an isolated observation in 1993 which plays
a crucial role in orbit determination and indetermination.
Here we analyze the effect of that observation on the orbital
elements and on the resulting orbital uncertainties.
JPL Ephemeris DE200 positions and velocities were in­
corporated for all the planets (Standish et al. 1992), and the
largest asteroids Ceres, Pallas, and Vesta are automatically
included in the computation of perturbations. Post­Voyager
masses were used for the planets, recognizing that more ac­
curate masses are already available, as are more accurate
ephemerides. However, the older values suffice for the cur­
rent orbit determination. The numerical integrator derives
from Aarseth (1985), and its accuracy was established by in­
tegrating over the time­span of observations using different
numerical steps.
We have determined least­squares orbits for Comet
Hale­Bopp using residual cutoffs of 3.0, 2.5, and 2.0 arc­
sec for both R.A. and Dec., and including the single 1993
observation. The number of observations included in each so­
lution were 628, 621, and 604, respectively. In all cases, the
orbit determination procedure converged in a satisfactory
way. In terms of the orbital uncertainties the resulting least­
squares orbits were practically identical, and in what follows
we describe the 2.5­arcsec least­squares orbit for which the
rms­values in R.A. and Dec. were 0.60 and 0.85 arcsec re­
spectively. This suggests that the fit was exceptionally good,
however the residual statistics showed strong correlations in
both coordinates, and strongly non­Gaussian skewness and
kurtosis in R.A. These peculiarities probably derive from

Orbital evolution of C/1995 O1 (Hale­Bopp) 3
Table 1. Central orbits used in this study. Source codes: (1­pre) and (1­post) Marsden (1995a); (2­post) Marsden (1995b); (3­pre) and
(3­post) orbits corresponding to the close approach to Jupiter shown in Figure 1 and derived from an orbital solution by KM. The
notation `pre' and `post' distinguishes orbits derived from the same set of observations but calculated at an osculating epoch prior to the
Jovian encounter in March/April 1996 and after this encounter respectively. The Julian date JD 2450000.5 corresponds to 1995 October
10.0 and JD 2450520.5 is 1997 March 13; perihelion passage occurs on 1997 April 1. It may be noted that the osculating orbital period
at perihelion (¸2700yr) is significantly less than the present pre­encounter osculating orbital period (¸6000 yr); the osculating orbital
period at the last aphelion was about 4600yr and the comet last passed perihelion about 2600BC. For comparison, we also include an
orbit from a more recent Minor Planet Circular (Marsden 1995d), identified as source (4­post).
a e q T ! i
\Omega Epoch Source
AU AU (JD) (degrees) (degrees) (degrees) (JD) code
Equinox 2000.0
251 0.996348 0.916702 2450539.8921 130.4405 88.8797 282.4733 2450000.5 (1­pre)
164 0.994441 0.913023 2450540.1416 130.6678 89.4142 282.4729 2450520.5 (1­post)
184 0.995022 0.913902 2450539.6580 130.6007 89.4250 282.4715 2450520.5 (2­post)
331 0.997230 0.918037 2450539.1861 130.3401 88.8999 282.4708 2450000.5 (3­pre)
196 0.995331 0.914401 2450539.4350 130.5656 89.4310 282.4706 2450539.2 (3­post)
176 0.994822 0.913946 2450539.7430 130.6126 89.4173 282.4729 2450520.5 (4­post)
Table 2. The least­squares orbital elements for Comet Hale­Bopp
at epoch 1995 October 10.0 TT (JD 2450000.50) together with the
N­body and two­body uncertainties oe and oe 0 computed respec­
tively including and excluding the single 1993 observation. Here
Ü denotes the time of perihelion (in this case 1997 March 31.703),
and the coordinate system corresponds to equinox J2000.0.
Element P oe oe 0
a (AU) 330 13 130
e 0.99722 0.00011 0.00110
i ( ffi ) 88.8997 0.0048 0.1600
\Omega ( ffi ) 282.47086 0.00077 0.02300
! ( ffi ) 130.341 0.013 0.044
M 0 ( ffi ) 359.9114 0.0053 0.0520
Ü (JD) 2450539.202 0.045 1.400
Table 3. The least­squares orbital elements at the epoch
1997 April 1.0 TT (JD 2450539.50), close to perihelion passage,
together with the uncertainties oe computed including the single
1993 observation. Ü again denotes the time of perihelion passage
(in this case 1997 March 31.951).
Element P oe
a (AU) 195.4 4.8
e 0.99532 0.00011
i ( ffi ) 89.4313 0.0047
\Omega ( ffi ) 282.47057 0.00077
! ( ffi ) 130.566 0.013
M 0 ( ffi ) 0.000018 0.000020
Ü (JD) 2450539.451 0.056
cometary activity and uncertain positioning of the cometary
nucleus, as well as systematic errors in the astrometric ref­
erence catalogue (often the HST Guide Star Catalog). The
slippage factor required to enforce the significance of the
correlations to an acceptable level (see Muinonen & Bowell
1993) was 3.0, which is considerably higher than that found
for the case of typical asteroid orbits. The final slippage­
Table 4. Orbital parameters and quality metrics for Comet Hale­
Bopp at the epochs 1995 October 10.0 TT (JD 2450000.5) and
1997 April 1.0 TT (JD 2450539.5). The uncertainties oe have been
computed including the single isolated 1993 observation.
1995 October 10.0 1997 April 1.0
Parameter Value oe Value oe
n ( ffi =d) 0.0001644 0.0000099 0.000361 0.000013
Period (yr) 6000 360 2730 100
q (AU) 0.91805 0.00023 0.91441 0.00023
` (arcmin) 0.39 0.46
' (arcmin) 23 14
corrected rms­deviations utilized in computing the uncer­
tainties in the orbital elements were 1.80 and 2.55 arcsec in
R.A. and Dec. respectively.
Table 2 gives least­squares orbital elements com­
puted by this procedure together with their respective 1­
oe slippage­corrected uncertainties at the epoch 1995 Octo­
ber 10.0 TT, for a residual cutoff of 2.5 arcsec. Table 3 gives
the corresponding elements and their uncertainties close to
perihelion passage at epoch 1997 April 1.0 TT. The con­
siderable difference between the two orbits is the result of
the close encounter with Jupiter at an approach distance of
0.772 AU on 1996 April 5 TT. The close encounter focuses
the orbit: the uncertainty in semi­major axis decreases to­
wards perihelion, though it is evident that the semi­major
axis remains poorly determined.
The 1993 observation is crucial for the orbit determi­
nation. We did not search for the global maximum likeli­
hood orbit using only 1995 observations, but we were able
to conclude that the 1993--1995 least­squares orbit practi­
cally coincides with a local maximum likelihood orbit using
only the 1995 observations. Table 2 presents two­body or­
bital uncertainties (oe 0 ) for the least­squares orbit excluding
the 1993 observation. The uncertainties in a, e, and M0 in­
crease by a factor of 10, in i
and\Omega by a factor of 30, while
the uncertainty in ! is increased by a factor of 3.
Table 4 shows the mean motion n, the orbital period

4 M.E. Bailey et al.
and perihelion distance, and the leak and simplified leak
metrics ` and ' (see Muinonen & Bowell 1993) on 1995 Oc­
tober 10.0 TT and 1997 April 1.0 TT. Note how the single
close encounter with Jupiter decreases the orbital period
from 6000 to 2730 years (see also Figure 1). As discussed in
Muinonen & Bowell (1993), the leak metric ` fails to describe
the orbital quality for highly eccentric orbits; for Hale­Bopp,
the metric takes values that do not compare realistically
with those found for main­belt asteroids (Muinonen et al.
1994). However, the metric ' remains realistic. Both qual­
ity metrics are strongly altered by the close encounter with
Jupiter.
2.2 Sample orbital elements
In studying the past and future evolution of Comet Hale­
Bopp, we should in principle assess the entire probability
distribution of orbital elements found in the preceding sec­
tion. For example, we could sample the probability density
using a Monte Carlo method for the generation of orbital
elements. However, such an approach would require vast
amounts of computing time, and we instead follow a simpli­
fied orbit sampling method developed by Muinonen (1995).
This uses 13 orbits to describe the 68.3% uncertainty ellip­
soid about the least­squares orbit in six­dimensional orbital
element phase space. We first diagonalize the covariance ma­
trix, and then vary the orbital elements in the positive and
negative directions along the principal axes up to the 68.3%­
boundary. Including the central least­squares orbit, 13 orbits
thus sample the entire probability density. This provides an
improved sampling of the orbital uncertainties than that
used previously, although the results from this procedure
should still be more thoroughly investigated using Monte
Carlo methods.
We computed sample orbits for Comet Hale­Bopp at
two epochs: 1995 October 10 TT and 1997 April 1 TT. The
elements are given in Table 5. It is evident that variations
\DeltaP 1 along the first principal axis are significantly larger,
often by several orders of magnitude, than the others. The
probability density is thus tightly concentrated about the
first principal axis, and in studies of orbital evolution it may
in practise suffice to vary the elements only along this prin­
cipal axis. It is nevertheless straightforward to make use of
a wider range of sample orbits to verify this assumption.
3 DYNAMICAL EVOLUTION
Comet Hale­Bopp currently has orbital nodes close to the or­
bits of Earth and Jupiter: the descending node lies ¸0.1 AU
outside the Earth's orbit, whereas the ascending node lies
¸0.03 AU inside the orbit of Jupiter. The orbital evolution
is thus dominated by nodal perturbations, which make pre­
dictions of the long­term orbital evolution of Hale­Bopp,
whether in the past or future, particularly uncertain. An
example of the evolution of semi­major axis during the close
approach to Jupiter (within ¸0.77 AU) in March/April 1996
is shown in Figure 1 together with the corresponding Jo­
vian close approach distance. The pre­encounter and post­
encounter orbits in this Figure correspond to source orbits
3 in Table 1. The distance of closest approach to Jupiter,
1996 1996.5 1997
100
200
300
400
500
Time (years AD)
1996 1996.5 1997
1996 1996.5 1997
0
0.2
0.4
0.6
0.8
1
Figure 1. Orbital perturbation of C/1995O1 (Hale­Bopp) dur­
ing the 1996 Jovian encounter. The top curve indicates the close
approach distance to Jupiter (right­hand y axis) and the bottom
curve shows the resulting change in orbital semi­major axis (left­
hand y axis).
near JD 2450178 (1996.26), is 0.77185 AU or ¸1600 plane­
tary radii.
For the original ensemble of 33 orbits the cumulative
number of bodies captured and/or ejected in the past over
time­scales less than 10, 50, 150 and 250 kyr respectively
were 2, 4, 8 and 12. These figures were identical for the fu­
ture orbital integrations, indicating symmetry between the
past and future orbital evolution over such time­scales. How­
ever, for the second ensemble of 26 orbits the corresponding
figures were 0, 0, 6 and 8 in the past and 2, 3, 5 and 5 in
the future.
In the long term, Comet Hale­Bopp, like many planet­
crossing bodies, exhibits chaotic motion. The end­states of
particular integrations thus vary widely with different ini­
tial conditions, so we can only comment on the long­term
evolution of the real body in a probabilistic sense. For the
original 33 orbits, the dynamical half­life for capture in the
past, or ejection in the future, was 0.6 Myr and 0.8 Myr re­
spectively (0.7 Myr in total); for comparison, the median
lifetime for capture or ejection, considering both past and
future integrations, was 0.75 Myr. ?From the initial 66 orbits
investigated (33 in the past and 33 in the future, integrated
for a total integration time of 63 Myr), 23 became hyper­
bolic in the past and 17 in the future; the median number
of orbital revolutions before comets became unbound was
34 in the past and 49 in the future. These figures indicate
that after a sufficient time for memory of the initial con­
ditions to be lost (around 0.3 Myr), the typical dynamical
lifetime of comets in orbits like that of Hale­Bopp is ap­
proximately 40 revolutions, or roughly 0.5--1.0 Myr. This is
in good agreement with results calculated on the basis of
a random walk of the orbital energy due to planetary per­
turbations (e.g. Fern'andez 1985, Emel'yanenko 1992); for
nearly parabolic orbits of perihelion distance q ¸ 1 AU and
i ' 90 ffi the r.m.s. dispersion in 1=a per revolution is on the
order of 6 \Theta 10 \Gamma4 AU \Gamma1 , indicating about 40 revolutions to
evolve from a parabolic orbit to a semi­major axis ¸250AU.
For the second set of 26 orbits, the dynamical half­life
for capture in the past or ejection in the future was 0.5 Myr
and 1.2 Myr respectively (0.8 Myr in total). From the 52 or­
bits integrated (26 in the past and 26 in the future, a total
integration time of 34 Myr), 16 became hyperbolic in the

Orbital evolution of C/1995 O1 (Hale­Bopp) 5
Table 5. Sample orbital elements for Comet Hale­Bopp at the epochs 1995 October 10.0 TT = JD 2450000.5 (top 6 rows) and 1997
April 1.0 TT = JD 2450539.5 (bottom 6 rows). Twelve sets of orbital elements are obtained by adding and subtracting the differences
\DeltaP j (j = 1; : : : ; 6) from the least­squares orbital elements P ls . The 1--oe orbital uncertainties are also given for comparison.
P ls oe \DeltaP 1 \DeltaP 2 \DeltaP 3 \DeltaP 4 \DeltaP 5 \DeltaP 6
a (AU) 329.994426 13.254185 34.549363 \Gamma6:437353 1.398466 \Gamma0:036473 0.002161 0.000052
e 0.997218 0.000111 0.000290 \Gamma0:000054 0.000012 0.000000 0.000000 0.000000
i ( ffi ) 88.899654 0.004769 0.011600 0.005034 0.000153 0.000481 0.000000 0.000000
\Omega ( ffi ) 282.470855 0.000773 \Gamma0:001911 \Gamma0:000744 0.000018 0.000083 0.000000 0.000000
! ( ffi ) 130.340872 0.012905 \Gamma0:033496 0.005670 0.004299 \Gamma0:000193 0.000000 0.000000
M 0 ( ffi ) 359.911429 0.005343 0.013926 \Gamma0:002600 0.000569 \Gamma0:000011 \Gamma0:000001 0.000000
a (AU) 195.362077 4.783881 12.531627 \Gamma1:918284 \Gamma0:512549 0.409385 0.056587 0.000276
e 0.995319 0.000114 0.000297 \Gamma0:000046 \Gamma0:000011 0.000010 0.000001 0.000000
i ( ffi ) 89.431322 0.004654 0.010995 0.005535 0.000918 \Gamma0:000366 0.000104 0.000000
\Omega ( ffi ) 282.470573 0.000770 \Gamma0:001874 \Gamma0:000809 \Gamma0:000035 \Gamma0:000054 0.000022 0.000000
! ( ffi ) 130.566359 0.013346 \Gamma0:034638 0.003900 0.006052 0.001633 0.000119 0.000000
M 0 ( ffi ) 0.000018 0.000020 0.000047 \Gamma0:000020 0.000009 \Gamma0:000001 0.000000 0.000000
past and 8 in the future; the median number of orbital rev­
olutions before comets became unbound was 24 in the past
and 11 in the future (19 considering past and future orbits
together).
3.1 Evolution to Halley­type orbits
Table 6 summarizes the evolution to intermediate long­
period and Halley­type orbits for both the initial set of
33 orbits and the second set of 26 orbits. The table is
divided into two orbital classes: Halley­type orbits with
7:37 ! a ! 34:2 AU (20 ! P ! 200 yr) (denoted HT) and
intermediate long­period orbits with 34:2 ! a ! 100 AU
(200 !P ! 1000 yr) (denoted ILP). Roughly half the orbits
in our first sample (30/66) had their semi­major axes de­
creased to less than 100 AU (ILP) within 150 kyr from the
present although there is a slightly greater probability for
this to occur during the future orbital evolution (18/33 as
opposed to 12/33 in the past). Conversely a greater number
of orbits appeared to evolve from HTs (a ! 34:2 AU) in the
backwards integrations, but the total number of such orbits
was small (5/66).
In the second case, only about a third of the orbits
(18/52) had their semi­major axes decreased to less than
100 AU (ILP) within 150 kyr from the present and again
this was seen to be more likely during the future orbital
evolution (12/26 compared with 6/26 in the past). For the
second set a greater number of orbits became Halley­types
in the forwards integrations (cf. Table 6), but again the total
number of such orbits was small (4/52). In no case did we
find an example of orbital evolution to short­period (P !
20 yr).
3.2 Evolution to sungrazing state
The sungrazing state is widespread amongst the evolution
of high­inclination, long­period comets with small perihelion
distance (Bailey et al. 1992a). Long­term secular perturba­
tions cause correlated changes in the perihelion distance,
eccentricity and inclination, which can lead to a temporary
sungrazing state with perihelion distance ! ¸ 0:02 AU (¸4 so­
Table 6. Cumulative number of objects which at least once
evolved to either Halley­type (20!P ! 200 yr) HT or intermedi­
ate long­period (200!P !1000 yr) ILP orbits for the past (P) and
future (F) orbital integrations. The top 4 rows give results from
the initial set of 33 orbits, the second 4 rows the corresponding
results from the second set of 26 orbits.
Time from present HT HT ILP ILP
(kyr) P F P F
10 0 0 2 2
50 1 0 7 16
150 2 0 12 18
250 4 1 14 18
10 0 0 0 1
50 0 0 0 9
150 0 1 6 12
250 1 3 6 13
lar radii). During such a time the comet is prone to tidal
disruption or destruction by intense solar heating. Consid­
ering only the 48 orbits in our initial ensemble that were
integrated for at least ¸1Myr, 7 became sungrazing, 4 in
the future and 3 in the past. For the second set of 52 orbits,
5 orbits became sungrazing, 3 in the future and 2 in the
past, although the mean integration time for this ensemble
was only 650 kyr per comet. The probability of Comet Hale­
Bopp becoming sungrazing at some time within 1 Myr of the
present is thus on the order of 15%.
An example of one such orbit is shown in Figure 2,
which shows the evolution of the orbital parameters a, q, i
and !, together with a plot of the close approach distances
to Jupiter, \Delta J . In the backwards integration this test par­
ticle was captured by a close approach to Jupiter (within
0.15 AU) after 6 revolutions (¸ \Gamma20 kyr). During the for­
wards integration the particle is temporarily ejected on to a
long­period orbit after around 25 kyr. This phase lasts for 10
orbital revolutions (around 170 kyr) after which the particle
enters an intermediate long­period phase (a ! 100 AU) fol­
lowed by a Halley­type phase (a ! 34:2 AU) after ¸500 kyr
into the future. At this point the body enters a Kozai li­

6 M.E. Bailey et al.
Table 7. Orbital evolution to a sungrazing state for the initial set
of 33 orbits (top 7 rows) and the second set of 26 orbits (remaining
5 rows). N is the number of revolutions to achieve a sungrazing
state. Both N and the time measured from the present are listed
for q ! 0:1 AU and q ! 0:02 AU (¸4 solar radii). The notation
(P) and (F) denotes an integration towards the past or future
respectively.
q ! 0:1 AU q ! 0:02AU
Time (kyr) N Time (kyr) N
125(P) 1994 130(P) 2118
210(F) 664 430(F) 1696
230(F) 600 440(P) 798
430(P) 737 530(F) 2924
500(P) 824 550(P) 878
600(F) 649 610(F) 709
690(F) 628 710(F) 693
231(F) 595 251(F) 670
261(P) 499 288(P) 553
283(F) 594 417(P) 550
405(P) 494 421(F) 668
547(F) 600 565(F) 712
bration zone during which close approaches to Jupiter are
avoided and the perihelion distance, q, begins to decrease
rapidly. This is accompanied by corresponding oscillations in
inclination, i, and argument of perihelion, !. Figure 3 shows
the relevant Kozai diagram at the epoch where q = 0:1 AU
(i.e. after around 547 kyr or 600 orbital revolutions). This di­
agram was produced by considering the perturbations from
Jupiter, Saturn, Uranus and Neptune. The cross marks the
position of the particle in phase­space at this particular
epoch. The body describes a path in an anti­clockwise di­
rection around the centre of the libration zone and the min­
imum value of perihelion distance (¸0.025 AU) corresponds
to ! ' 90 ffi .
Table 7 summarizes the sungrazing results for the two
orbital data sets. It is clear that those Hale­Bopp variants
that evolve to sufficiently short orbital periods for secular
perturbations to become dominant (i.e. P Ÿ 200 yr; Bailey
& Emel'yanenko 1995), i.e. potential sungrazers, achieve a
sungrazing state (q ! 0:1 AU) after around 600 revolutions
(cf. Bailey et al. 1992a). It should be noted, however, that
in our initial ensemble, the orbits which became sungrazers
were dominated by a particular set of integrations centred
around the first orbit in Table 1 (6/7 sungrazing events).
Overall, the probability of Comet Hale­Bopp evolving to be­
come a sungrazer at some stage of its dynamical evolution
appears to be on the order of 15%, much larger than would
be expected for a randomly captured long­period comet with
q ! 1 AU, in the absence of secular effects, but not excep­
tionally large.
3.3 Associated meteoroid stream
The proximity of the descending node of the present orbit
of Hale­Bopp to the orbit of the Earth raises the possibil­
ity of an associated meteor shower visible annually around
January 4 (contemporaneous with the Quadrantid meteor
shower). A further possibility is a surge in the number of
Figure 2. Orbital evolution of a test particle with orbital pa­
rameters similar to those of Comet 1995O1 (Hale­Bopp). The
plots show the evolution of the semi­major axis, a, the close ap­
proach distances to Jupiter, \Delta J , the perihelion distance, q, the
inclination, i, and the argument of perihelion, !. This object was
captured 6 revolutions ago, at t ' \Gamma20 kyr, as a result of pertur­
bations by Jupiter.

Orbital evolution of C/1995 O1 (Hale­Bopp) 7
Figure 3. Kozai diagram showing the correlated oscillations in
perihelion distance, q, and argument of perihelion, !, as the par­
ticle featured in Figure 2 circulates about the centre of libration.
incident meteoroids as the Earth approaches the descend­
ing node of Hale­Bopp's orbit in early January 1996 and in
subsequent years.
Any association with the Quadrantid meteoroid stream
(other than the coincidence of the longitude of the descend­
ing node) is unlikely due to the difference between the or­
bital periods of the two streams. The observed Quadrantid
stream (from photographic meteor data e.g. the IAU Me­
teor Data Catalogue, Lindblad and Steel 1994) has orbital
semi­major axes in the range 2 Ÿ a Ÿ 6 AU whereas any
visual or photographic `Hale­Boppids' produced during the
last perihelion passage some 4600 yr ago are likely to be con­
fined to 70 Ÿ a Ÿ 320 AU; meteoroids with semi­major axes
greater than that of the comet at the last perihelion pas­
sage (¸320 AU) will not yet have had time to complete an
orbit. In making these estimates, we have assumed mete­
oroid ejection velocities V in the range 0 ŸV Ÿ 0:1 km s \Gamma1 .
The lower semi­major axis Hale­Boppids could have orbited
the Sun up to seven times since the last cometary perihe­
lion passage so it is clear that a complete `loop' of cometary
debris may have already been formed and hence (if the me­
teoroid stream intersects the Earth's orbit) be the source of
an annual meteor shower.
Of course, the Hale­Bopp dust complex is likely to have
been formed over many previous perihelion passages, the
meteoroids being ejected from a parent body on a constantly
evolving orbit. The stream itself is likely to have experi­
enced considerable gravitational perturbations over this pe­
riod. We have modelled the formation of the stream and its
subsequent evolution in two ways. First we ejected 50 par­
ticles at each perihelion passage over the past 14 cometary
apparitions and considered the evolution of these particles
over this period (in this case the comet was captured from a
near­parabolic orbit by a close approach to Jupiter 14 peri­
helion passages before present). Secondly we have considered
the ejection and orbital evolution of 1000 particles ejected
during the last perihelion passage only, in order to investi­
gate the possibility of Earth­orbit intersection and also in
order to study the effects of the 1996 Jovian close approach
on the meteoroid stream particles in the vicinity of the par­
ent comet. The results are summarized in Figure 4.
Figure 4 clearly shows that meteoroids ejected from the
last cometary perihelion passage (filled circles) can have or­
Figure 4. The ecliptic plane intersection points for particles
ejected from Hale­Bopp during the last perihelion passage (filled
circles) and over the past 14 cometary perihelion passages (open
triangles). r dnode is the heliocentric distance of the descending
node. The open triangles illustrate the increased nodal dispersion
of the meteoroid stream as a result of planetary perturbations.
bits that intersect the ecliptic plane close to 1 AU and hence
could be visible as meteors around January 4. These par­
ticles are at the lower end of the Hale­Boppid semi­major
axis range (i.e. those particles ejected with the highest veloc­
ities) and have completed several orbits since their ejection.
Particles that are ejected with low velocities (i.e. those that
remain in the vicinity of the parent comet) intersect the
ecliptic plane outside the Earth's orbit. The Jovian close
encounter of 1996 increases the heliocentric distances of the
descending node of those particles that are close enough to
the cometary orbit to suffer major gravitational perturba­
tions and hence these particles are unobservable as meteors.
Figure 4 also shows how the meteoroid stream becomes
more dispersed over 14 cometary perihelion passages (open
triangles). Particles are perturbed to a wide range of val­
ues of descending node heliocentric distance and longitude
of descending node over this period of time, although the
major concentration of ecliptic plane intersections remains
close to 102 ffi . The absence of particles that intersect the
ecliptic plane close to 1 AU is explained by the low number
of particles considered in the analysis (50 particles ejected
per perihelion passage).
4 CONCLUSIONS
Considering the results from both ensembles of orbits, each
closely associated with the present orbit of Comet Hale­
Bopp, the principal conclusions from this study are:
(i) The dynamical half­life for the comet to be captured
in the past is on the order of 0.5--0.6 Myr. The corresponding
figure for ejection in the future evolution is in the range 0.8--
1.2 Myr. This suggests that the comet may be considered as
dynamically young, with an ejection probability in the past
roughly twice that for the future. Nevertheless, the comet is
not dynamically new in the sense of coming `fresh' from the
Oort cloud.
(ii) Considering the results from the second ensemble of
orbits, the probability that the comet will evolve into an in­
termediate long­period orbit (P !1000 yr, a!100 AU) in the
future is roughly twice that of the comet having evolved from

8 M.E. Bailey et al.
such an orbit in the past. Similarly, the comet is marginally
more likely to evolve into a Halley­type orbit (P ! 200 yr,
a ! 34:2 AU) during the future evolution, again suggesting
that Comet Hale­Bopp is dynamically young.
(iii) The comet has a ¸15% probability of evolving to
a sungrazing state at some point in its evolution. For the
future evolution this state is generally attained within about
600 revolutions from the present and occurs only in those
bodies that have evolved to an intermediate long­period or
Halley­type orbit.
(iv) The bulk of the Hale­Bopp meteoroid stream inter­
sects the ecliptic plane outside the Earth's orbit at heliocen­
tric distances ¸1.1--1.2 AU. Planetary perturbations cause
some particles (predominantly those of shorter orbital pe­
riod) to become Earth­crossing, so that a minor meteor
shower may occur annually around January 4. Any asso­
ciation with the Quadrantid meteoroid stream appears to
be unlikely.
ACKNOWLEDGMENTS
We thank P. Magnusson for helpful comments. NWH and
VVE thank the PPARC for financial support; KAH acknowl­
edges support from a Nuffield Foundation Undergraduate
Bursary. This work has been supported within an HCM Re­
search Network by the E.U., under contract number CHRX­
CT9.
REFERENCES
Aarseth, S.J., 1985, in Brackbill, J.U., Cohen, B.I., eds., Multiple
Time Scales. Academic Press, Orlando, p.377
Babadzhanov, P.B., Obrubov, Yu.V., 1993, in Meteoroids and
their Parent Bodies, eds. Ÿ
Stohl, J., Williams, I.P., Astron.
Inst. Slovak. Acad. Sci., Bratislava, p. 49
Bailey, M.E., Chambers, J.E., Hahn, G., 1992a, A&A, 257, 315
Bailey, M.E., Chambers, J.E., Hahn, G., 1992b, Sky & Tel., Jan­
uary, p.5
Bailey, M.E., Clube, S.V.M., Hahn, G., Napier, W.M., Valsecchi,
G.B., 1994, in Gehrels, T., ed., Hazards due to Comets and
Asteroids, University of Arizona Press, Tucson, p.479
Bailey, M.E., Emel'yanenko, V.V., 1995, MNRAS, in press
Campins, H., Telesco, C.M., Osip, D.J., Rieke, G.H., Rieke, M.J.,
Schulz, B., 1994, AJ, 108, 2318
Emel'yanenko, V.V., 1992, Celest. Mech., 54, 91
Everhart, E., 1985, in Carusi, A., Valsecchi, G.B., eds, Dynam­
ics of Comets: Their Origin and Evolution (IAU Colloq. 83).
Reidel, Dordrecht, p.185
Fern'andez, J.A., 1985, in Carusi, A., Valsecchi, G.B., eds, Dy­
namics of Comets: Their Origin and Evolution (IAU Colloq.
83). Reidel, Dordrecht, p.45
Green, D.W.E., 1995, IAU Circ. No. 6187
Hahn, G., Bailey, M.E., 1990, Nat, 348, 132
Lindblad, B.A. and Steel, D.I., 1994, in Asteroids, Comets, Me­
teors 1993, IAU Symp. No. 160, Milani, A., Di Martino, M.,
& Cellino, A., Kluwer, Dordrecht, p.497
Luu, J.X., Jewitt, D.C., 1990, AJ, 100, 913
Marsden, B.G., 1989, AJ, 98, 2306
Marsden, B.G., 1995a, Minor Planet Electronic Circular 1995­P02
Marsden, B.G., 1995b, Minor Planet Circular 25513
Marsden, B.G., 1995c, Minor Planet Circular 25623
Marsden, B.G., 1995d, Minor Planet Circular 25932
McIntosh, B.A., 1990, Icarus, 86, 299
Meech, K.J., Belton, M.J.S., 1990, AJ, 100, 1323
Muinonen, K., 1995, submitted to MNRAS
Muinonen, K., Bowell, E., 1993, Icarus, 104, 255
Muinonen, K., Bowell, E., Wasserman, L.H., 1994, Planet. Space
Sci., 42, 307
Standish, E.M., Newhall, X.X., Williams, J.G., Yeomans, D.K.,
1992, in Explanatory Supplement to the Astronomical Al­
manac, Seidelmann, P.K., ed., University Science Books, Mill
Valley, California, p.279