Mon. Not. R. Astron. Soc. 000, 000{000 (0000)

Printed 7 October 1996

(MN LaTEX style le v1.3)

Orbital evolution of Comet 1995 O1 Hale-Bopp M.E. Bailey1, V.V. Emel'yanenko1 2, G. Hahn3, N.W. Harris1, K.A. Hughes2, K. Muinonen4 and J.V. Scotti5 ;

1 2 3 4 5

Armagh Observatory, College Hill, Armagh, BT61 9DG, U.K. School of Electrical Engineering Electronics and Physics, Liverpool John Moores University, Byrom Street, Liverpool, L3 3AF, U.K. DLR Institut fur Planetenerkundung, Rudower Chaussee 5, D-12489 Berlin, Germany. Observatorio, PL 14, Tahtitorninmaki, FIN-00014, Helsingin Yliopisto, Suomi-Finland. Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona 85721, USA.

7 October 1996

ABSTRACT Results from a series of long-term numerical integrations of orbits centred on that of C/1995O1 (Hale-Bopp) are presented. Initially, 33 orbits taken from various sources were integrated in realistic models of the solar system for various time-scales about the present in the range (?3 +2)Myr and analyzed to assess the probability of dierent dynamical outcomes, such as a sungrazing state or the number of orbits since it was captured from a long-period orbit in the Oort cloud. Further integrations were performed 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 nd that the ensemble half-life for the comet to be captured or ejected is on the order of 0.5Myr in the backward integrations and 1.2Myr in the forward integrations, although a few members of our ensemble were captured during strong encounters with Jupiter within 10 revolutions. Comet Hale-Bopp has a probability ' 0 15 of evolving to a sungrazing end-state, 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 Hale-Bopp, meteors. ;

p

1 INTRODUCTION The discovery of a 10th magnitude comet by A. Hale and T. Bopp on 1995 July 23 UT at the unprecedented heliocentric 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 spectacular 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 several times larger than the 10 km diameter of that comet. Early estimates of the magnitude of C/1995 O1 (Hale-Bopp) suggest a nuclear diameter in the range 50{150 km, making this object potentially the rst `giant' comet (diameter greater than 100 km) to enter the inner solar system in modern times. Previous giants amongst long-period comets may include the progenitor of the Kreutz sungrazer family, which passed perihelion in 372 BC (Marsden 1989), and Comet Sarabat (C/1729 P1), which became a naked-eye object for almost 6 months despite its perihelion distance of 4.05 AU. Comet Hale-Bopp 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 inner solar system, coming at least in the rst instance from the Centaur population (Hahn & Bailey 1990, Bailey et al. 1994). These objects have low-inclination, intermediateperiod orbits with perihelia close to or just beyond the orbit of Saturn. Given their likely provenance in the outer solar 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 signi cance of the arrival of Comet Hale-Bopp is that it provides an opportunity to observe the development of nuclear activity in a large comet, and its presence con rms the existence of exceptionally large comets in the long-period

ux. Comet Hale-Bopp is unusual in having an orbital inclination, i, close to 90 and a much shorter orbital period than most long-period 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 recently captured on account of strong Jovian perturbations, but it may also have put on a good display for our remote ancestors. We note here that there is a possibility of heightened meteoric activity when the Earth approaches the descending node of Hale-Bopp's orbit in early January 1996 and in previous and subsequent years, although the apparent association 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 dierence between the orbital periods of Comet Hale-Bopp and the Quadrantids. The closeness of the orbital inclination to 90 indicates that secular perturbations by Jupiter may cause HaleBopp to become sungrazing in the future (Bailey, Chambers & Hahn 1992a). In this respect, the similarity of the evolution of Hale-Bopp 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 (Bailey, Chambers & Hahn 1992b), suggests that a study of the dynamical evolution of Comet Hale-Bopp may also help in understanding the source of short-period comets and the origin 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 address these questions and to elucidate the likely long-term history of Comet Hale-Bopp.

2 METHOD

We initially integrated an ensemble of orbits of Comet HaleBopp 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 appeared to be the likely uncertainty in the preliminary orbit determination. Since the orbit is chaotic, it is not possible to determine the long-term orbital evolution of the comet exactly; minute errors in the assumed orbit or in the dynamical model of the solar system eventually become significant. Moreover, with an uncertainty in the orbital period of at least several Jovian years we cannot reliably predict the eects 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 suggests that rapid dynamical evolution may occur, leading to capture or ejection on relatively short time-scales. 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; ; !; M0 )T , corresponding to semi-major axis, eccentricity, inclination, longitude of ascending node, argument of perihelion, and mean anomaly at epoch. The three angular elements i, , and ! are referred to the ecliptic at equinox J2000.0.

The rst 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, Saturn, 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 integrator (Everhart 1985) with tolerance parameter set between 10?8 and 10?12 . In addition to these long-term integrations, we also performed more accurate runs of approximately 1000 days into the future in order to determine the eects of an imminent Jovian encounter during the rst half of 1996 with greater precision. Table 1 presents recent orbit determinations for Comet Hale-Bopp; sources (1) and (2) were used as central orbits in our initial study. As explained above, we subsequently applied the method of orbit determination described by Muinonen & Bowell (1993) to create a further ensemble of 26 orbits.

2.1 Orbit determination

For Comet Hale-Bopp, we chose 645 observations covering the time-period 1993 April 27 to 1995 September 3. Observations 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 eect of that observation on the orbital elements and on the resulting orbital uncertainties. JPL Ephemeris DE200 positions and velocities were incorporated for all the planets (Standish et al. 1992), and the largest asteroids Ceres, Pallas, and Vesta are automatically included in the computation of perturbations. Post-Voyager masses were used for the planets, recognizing that more accurate masses are already available, as are more accurate ephemerides. However, the older values suce for the current orbit determination. The numerical integrator derives from Aarseth (1985), and its accuracy was established by integrating over the time-span of observations using dierent numerical steps. We have determined least-squares orbits for Comet Hale-Bopp using residual cutos of 3.0, 2.5, and 2.0 arcsec for both R.A. and Dec., and including the single 1993 observation. The number of observations included in each solution 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 leastsquares orbits were practically identical, and in what follows we describe the 2.5-arcsec least-squares orbit for which the rms-values in R.A. and Dec. were 0.60 and 0.85 arcsec respectively. This suggests that the t was exceptionally good, however the residual statistics showed strong correlations in both coordinates, and strongly non-Gaussian skewness and kurtosis in R.A. These peculiarities probably derive from

Orbital evolution of C/1995 O1 (Hale-Bopp)

3

Central orbits used in this study. Source codes: (1-pre) and (1-post) Marsden (1995a); (2-post) Marsden (1995b); (3-pre) and (3-post) 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 signi cantly less than the present pre-encounter osculating orbital period (6000yr); 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), identi ed as source (4-post).

Table 1.

a

e

251 164 184 331 196 176

0.996348 0.994441 0.995022 0.997230 0.995331 0.994822

AU

q

T

AU

(JD)

0.916702 0.913023 0.913902 0.918037 0.914401 0.913946

2450539.8921 2450540.1416 2450539.6580 2450539.1861 2450539.4350 2450539.7430

! i

(degrees) (degrees) (degrees)

130.4405 130.6678 130.6007 130.3401 130.5656 130.6126

The least-squaresorbital elements for Comet Hale-Bopp at epoch 1995 October 10.0 TT (JD 2450000.50)togetherwith the N -body and two-body uncertainties  and  0 computed respectively including and excluding the single 1993 observation. Here  denotes the time of perihelion (in this case 1997 March 31.703), and the coordinate system corresponds to equinox J2000.0. Table 2.

Element a (AU) e i ( )

( ) ! ( ) M0 ( )  (JD)

P



0

330 0.99722 88.8997 282.47086 130.341 359.9114 2450539.202

13 0.00011 0.0048 0.00077 0.013 0.0053 0.045

130 0.00110 0.1600 0.02300 0.044 0.0520 1.400

3. The least-squares orbital elements at the epoch 1997 April 1.0 TT (JD 2450539.50), close to perihelion passage, together with the uncertainties  computed including the single 1993 observation.  again denotes the time of perihelion passage (in this case 1997 March 31.951).

Table

Element a (AU) e i ( )

( ) ! ( ) M0 ( )  (JD)

P



195.4 0.99532 89.4313 282.47057 130.566 0.000018 2450539.451

4.8 0.00011 0.0047 0.00077 0.013 0.000020 0.056

cometary activity and uncertain positioning of the cometary nucleus, as well as systematic errors in the astrometric reference catalogue (often the HST Guide Star Catalog). The slippage factor required to enforce the signi cance 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 nal slippage-

88.8797 89.4142 89.4250 88.8999 89.4310 89.4173

282.4733 282.4729 282.4715 282.4708 282.4706 282.4729

Epoch (JD) Equinox 2000.0 2450000.5 2450520.5 2450520.5 2450000.5 2450539.2 2450520.5

Source code (1-pre) (1-post) (2-post) (3-pre) (3-post) (4-post)

Orbital parameters and quality metrics for Comet HaleBopp at the epochs 1995 October 10.0 TT (JD 2450000.5) and 1997 April 1.0 TT (JD2450539.5). The uncertainties  have been computed including the single isolated 1993 observation. Table 4.

Parameter n ( =d) Period (yr) q (AU)  (arcmin)  (arcmin)

1995 October 10.0 Value  0.0001644 0.0000099 6000 360 0.91805 0.00023 0.39 23

1997 April 1.0 Value  0.000361 0.000013 2730 100 0.91441 0.00023 0.46 14

corrected rms-deviations utilized in computing the uncertainties in the orbital elements were 1.80 and 2.55 arcsec in R.A. and Dec. respectively. Table 2 gives least-squares orbital elements computed by this procedure together with their respective 1 slippage-corrected uncertainties at the epoch 1995 October 10.0 TT, for a residual cuto 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 considerable dierence 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 semi-major axis decreases towards perihelion, though it is evident that the semi-major axis remains poorly determined. The 1993 observation is crucial for the orbit determination. We did not search for the global maximum likelihood orbit using only 1995 observations, but we were able to conclude that the 1993{1995 least-squares orbit practically coincides with a local maximum likelihood orbit using only the 1995 observations. Table 2 presents two-body orbital uncertainties (0 ) for the least-squares orbit excluding the 1993 observation. The uncertainties in a, e, and M0 increase by a factor of 10, in i and 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 simpli ed leak metrics  and  (see Muinonen & Bowell 1993) on 1995 October 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 Hale-Bopp, the metric takes values that do not compare realistically with those found for main-belt asteroids (Muinonen et al. 1994). However, the metric  remains realistic. Both quality metrics are strongly altered by the close encounter with Jupiter.

1996

1996.5

500

1997 1

0.8 400

0.6 300 0.4

200 0.2

100 1996

1996.5

0 1997

Time (years AD)

2.2 Sample orbital elements

In studying the past and future evolution of Comet HaleBopp, we should in principle assess the entire probability distribution of orbital elements found in the preceding section. 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 ed orbit sampling method developed by Muinonen (1995). This uses 13 orbits to describe the 68.3% uncertainty ellipsoid about the least-squares orbit in six-dimensional orbital element phase space. We rst diagonalize the covariance matrix, 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 least-squares 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 Hale-Bopp at two epochs: 1995 October 10 TT and 1997 April 1 TT. The elements are given in Table 5. It is evident that variations
P1 along the rst principal axis are signi cantly larger, often by several orders of magnitude, than the others. The probability density is thus tightly concentrated about the rst principal axis, and in studies of orbital evolution it may in practise suce to vary the elements only along this principal axis. It is nevertheless straightforward to make use of a wider range of sample orbits to verify this assumption.

3 DYNAMICAL EVOLUTION

Comet Hale-Bopp currently has orbital nodes close to the orbits 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 predictions of the long-term orbital evolution of Hale-Bopp, whether in the past or future, particularly uncertain. An example of the evolution of semi-major axis during the close approach to Jupiter (within 0.77 AU) in March/April 1996 is shown in Figure 1 together with the corresponding Jovian close approach distance. The pre-encounter and postencounter orbits in this Figure correspond to source orbits 3 in Table 1. The distance of closest approach to Jupiter,

Orbital perturbation of C/1995O1 (Hale-Bopp) during the 1996 Jovian encounter. The top curve indicates the close approach distance to Jupiter (right-hand y axis) and the bottom curve shows the resulting change in orbital semi-major axis (lefthand y axis). Figure 1.

near JD 2450178 (1996.26), is 0.77185 AU or 1600 planetary radii. For the original ensemble of 33 orbits the cumulative number of bodies captured and/or ejected in the past over time-scales less than 10, 50, 150 and 250 kyr respectively were 2, 4, 8 and 12. These gures were identical for the future orbital integrations, indicating symmetry between the past and future orbital evolution over such time-scales. However, for the second ensemble of 26 orbits the corresponding gures were 0, 0, 6 and 8 in the past and 2, 3, 5 and 5 in the future. In the long term, Comet Hale-Bopp, like many planetcrossing bodies, exhibits chaotic motion. The end-states of particular integrations thus vary widely with dierent initial conditions, so we can only comment on the long-term evolution of the real body in a probabilistic sense. For the original 33 orbits, the dynamical half-life for capture in the past, or ejection in the future, was 0.6 Myr and 0.8 Myr respectively (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 hyperbolic 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 gures indicate that after a sucient time for memory of the initial conditions to be lost (around 0.3 Myr), the typical dynamical lifetime of comets in orbits like that of Hale-Bopp is approximately 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 perturbations (e.g. Fernandez 1985, Emel'yanenko 1992); for nearly parabolic orbits of perihelion distance q  1 AU and i ' 90 the r.m.s. dispersion in 1=a per revolution is on the order of 6  10?4 AU?1 , indicating about 40 revolutions to evolve from a parabolic orbit to a semi-major axis 250 AU. For the second set of 26 orbits, the dynamical half-life 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 orbits 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 (Hale-Bopp)

5

Sample orbital elements for Comet Hale-Bopp 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 dierences
Pj (j = 1; : : : ; 6) from the least-squares orbital elements Pls . The 1{ orbital uncertainties are also given for comparison.

Table 5.

Pls



a (AU) e i ( )

329.994426 13.254185 0.997218 0.000111 88.899654 0.004769

( ) 282.470855 0.000773 ! ( ) 130.340872 0.012905 M0 ( ) 359.911429 0.005343 a (AU) 195.362077 4.783881 e 0.995319 0.000114 i ( ) 89.431322 0.004654

( ) 282.470573 0.000770 ! ( ) 130.566359 0.013346 M0 ( ) 0.000018 0.000020


P1 34.549363 0.000290 0.011600 ?0:001911 ?0:033496 0.013926 12.531627 0.000297 0.010995 ?0:001874 ?0:034638 0.000047


P2
P3
P4
P5
P6 ?6:437353 1.398466 ?0:036473 0.002161 0.000052 ?0:000054 0.000012 0.000000 0.000000 0.000000 0.005034 0.000153 0.000481 0.000000 0.000000 ?0:000744 0.000018 0.000083 0.000000 0.000000 0.005670 0.004299 ?0:000193 0.000000 0.000000 ?0:002600 0.000569 ?0:000011 ?0:000001 0.000000 ?1:918284 ?0:512549 0.409385 0.056587 0.000276 ?0:000046 ?0:000011 0.000010 0.000001 0.000000 0.005535 0.000918 ?0:000366 0.000104 0.000000 ?0:000809 ?0:000035 ?0:000054 0.000022 0.000000 0.003900 0.006052 0.001633 0.000119 0.000000 ?0:000020 0.000009 ?0:000001 0.000000 0.000000

past and 8 in the future; the median number of orbital revolutions 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 Halley-type orbits

Table 6 summarizes the evolution to intermediate longperiod and Halley-type orbits for both the initial set of 33 orbits and the second set of 26 orbits. The table is divided into two orbital classes: Halley-type orbits with 7:37 < a < 34:2 AU (20 < P < 200 yr) (denoted HT) and intermediate long-period orbits with 34:2 < a < 100 AU (200 < P < 1000 yr) (denoted ILP). Roughly half the orbits in our rst sample (30/66) had their semi-major axes decreased 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 semi-major 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 Halley-types in the forwards integrations (cf. Table 6), but again the total number of such orbits was small (4/52). In no case did we nd an example of orbital evolution to short-period (P < 20 yr).

3.2 Evolution to sungrazing state

The sungrazing state is widespread amongst the evolution of high-inclination, long-period comets with small perihelion distance (Bailey et al. 1992a). Long-term secular perturbations 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-

Cumulative number of objects which at least once evolved to either Halley-type (20 < P < 200yr) HT or intermediate long-period(200 < P < 1000yr) 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.

Table 6.

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. Considering only the 48 orbits in our initial ensemble that were integrated for at least 1 Myr, 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 HaleBopp 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,
J . In the backwards integration this test particle was captured by a close approach to Jupiter (within 0.15 AU) after 6 revolutions ( ?20 kyr). During the forwards integration the particle is temporarily ejected on to a long-period orbit after around 25 kyr. This phase lasts for 10 orbital revolutions (around 170 kyr) after which the particle enters an intermediate long-period phase (a < 100 AU) followed by a Halley-type 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.

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 < 0:1AU and q < 0:02AU (4 solar radii). The notation (P) and (F) denotes an integration towards the past or future respectively. Table 7.

q < 0:1AU 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 diagram was produced by considering the perturbations from Jupiter, Saturn, Uranus and Neptune. The cross marks the position of the particle in phase-space at this particular epoch. The body describes a path in an anti-clockwise direction around the centre of the libration zone and the minimum value of perihelion distance (0.025 AU) corresponds to ! ' 90 . Table 7 summarizes the sungrazing results for the two orbital data sets. It is clear that those Hale-Bopp variants that evolve to suciently short orbital periods for secular perturbations to become dominant (i.e. P  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 rst orbit in Table 1 (6/7 sungrazing events). Overall, the probability of Comet Hale-Bopp evolving to become 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 long-period comet with q < 1 AU, in the absence of secular eects, but not exceptionally large.

3.3 Associated meteoroid stream

The proximity of the descending node of the present orbit of Hale-Bopp to the orbit of the Earth raises the possibility 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

Orbital evolution of a test particle with orbital parameters similar to those of Comet 1995O1 (Hale-Bopp). The plots show the evolution of the semi-major axis, a, the close approach distances to Jupiter,
J , the perihelion distance, q, the inclination, i, and the argument of perihelion, !. This object was captured 6 revolutions ago, at t '?20 kyr, as a result of perturbations by Jupiter.

Figure 2.

Orbital evolution of C/1995 O1 (Hale-Bopp)

Figure 3. Kozai diagram showing the correlated oscillations in perihelion distance, q, and argument of perihelion, !, as the particle featured in Figure 2 circulates about the centre of libration.

incident meteoroids as the Earth approaches the descending node of Hale-Bopp'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 descending node) is unlikely due to the dierence between the orbital periods of the two streams. The observed Quadrantid stream (from photographic meteor data e.g. the IAU Meteor Data Catalogue, Lindblad and Steel 1994) has orbital semi-major axes in the range 2  a  6 AU whereas any visual or photographic `Hale-Boppids' produced during the last perihelion passage some 4600 yr ago are likely to be con ned to 70  a  320 AU; meteoroids with semi-major axes greater than that of the comet at the last perihelion passage (320 AU) will not yet have had time to complete an orbit. In making these estimates, we have assumed meteoroid ejection velocities V in the range 0  V  0:1 km s?1 . The lower semi-major axis Hale-Boppids could have orbited the Sun up to seven times since the last cometary perihelion passage so it is clear that a complete `loop' of cometary debris may have already been formed and hence (if the meteoroid stream intersects the Earth's orbit) be the source of an annual meteor shower. Of course, the Hale-Bopp 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 experienced considerable gravitational perturbations over this period. We have modelled the formation of the stream and its subsequent evolution in two ways. First we ejected 50 particles 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 near-parabolic orbit by a close approach to Jupiter 14 perihelion 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 investigate the possibility of Earth-orbit intersection and also in order to study the eects of the 1996 Jovian close approach on the meteoroid stream particles in the vicinity of the parent comet. The results are summarized in Figure 4. Figure 4 clearly shows that meteoroids ejected from the last cometary perihelion passage ( lled circles) can have or-

7

Figure 4. The ecliptic plane intersection points for particles ejected from Hale-Bopp during the last perihelion passage ( lled circles) and over the past 14 cometary perihelion passages (open triangles). rdnode 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 particles are at the lower end of the Hale-Boppid semi-major axis range (i.e. those particles ejected with the highest velocities) 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 suer major gravitational perturbations 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 values 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 . 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 HaleBopp, the principal conclusions from this study are: (i) The dynamical half-life for the comet to be captured in the past is on the order of 0.5{0.6 Myr. The corresponding gure 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 intermediate long-period 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 Halley-type orbit (P < 200 yr, a < 34:2 AU) during the future evolution, again suggesting that Comet Hale-Bopp 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 long-period or Halley-type orbit. (iv) The bulk of the Hale-Bopp meteoroid stream intersects the ecliptic plane outside the Earth's orbit at heliocentric distances 1.1{1.2 AU. Planetary perturbations cause some particles (predominantly those of shorter orbital period) to become Earth-crossing, so that a minor meteor shower may occur annually around January 4. Any association with the Quadrantid meteoroid stream appears to be unlikely.

ACKNOWLEDGMENTS

We thank P. Magnusson for helpful comments. NWH and VVE thank the PPARC for nancial support; KAH acknowledges support from a Nueld Foundation Undergraduate Bursary. This work has been supported within an HCM Research Network by the E.U., under contract number CHRXCT9.

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. 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., January, 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, Dynamics of Comets: Their Origin and Evolution (IAU Colloq. 83). Reidel, Dordrecht, p.185 Fernandez, J.A., 1985, in Carusi, A., Valsecchi, G.B., eds, Dynamics 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, Meteors 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 ElectronicCircular 1995-P02 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 Almanac, Seidelmann, P.K., ed., University Science Books, Mill Valley, California, p.279