Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/~ambn/comneos.ps Äàòà èçìåíåíèÿ: Thu Oct 10 18:28:46 1996 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 21:29:34 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: comet tail

Irish Astr. J., 23(2), 151­156, (1996) N. W. HARRIS & M. E. BAILEY
THE COMETARY COMPONENT OF THE NEAR­EARTH OBJECT
POPULATION
N. W. HARRIS & M. E. BAILEY
Armagh Observatory, College Hill, Armagh BT61 9DG
email: nwh@star.arm.ac.uk, meb@star.arm.ac.uk
ABSTRACT. The problem of the origin of Near­Earth Objects (NEOs), whether predominantly cometary or asteroidal, has
attracted much recent interest. In this work, we consider NEOs to have perihelion distances q ! 1:4 AU and separate them
according to their aphelion distances into two broad classes: Class 1 NEOs have aphelia Q ? 4:2 AU, allowing these objects
possibly to have close approaches to Jupiter; whereas Class 2 objects have Q ! 4:2 AU. The latter comprise the majority
(about 90%) of the known NEO population and are dynamically `decoupled' from close Jovian encounters. This paper presents
preliminary results from long­term numerical integrations of hypothetical Jupiter­family comets, which according to these
definitions would, if inactive, be considered as Class 1 `cometary' NEOs, or `cometary asteroids'. In particular, we evaluate
the transfer probabilities to Class 2 orbits, by both gravitational and non­gravitational mechanisms, and estimate the overall
cometary contribution to the number of objects in Class 1 and Class 2. Our results are compared with estimates of the rate of
injection of NEOs from the main asteroid belt.
1. NEAR­EARTH OBJECTS
Near­Earth Objects (NEOs) are small bodies of asteroidal ap­
pearance whose perihelia, q, lie in the region of the terrestrial
planets (q ! 1:4 AU), and whose aphelia, Q, typically lie within
a sphere of radius equal to Jupiter's mean distance from the
Sun. The term `NEO' implies that the body may in principle
draw near to the Earth, either approaching the Earth's aphe­
lion (when it might be described as an `Earth­approacher') or
crossing the Earth's orbit, in which case it might be termed
an `Earth­crosser'. Of course, not every Earth­crosser neces­
sarily collides with the Earth, since the probability of a close
encounter depends on the orientation of the orbit in space,
in particular whether one or the other orbital node is close
to 1 AU from the Sun; and even if this condition is satisfied,
it is still necessary that the relative phases of the object and
the Earth should lie within a very small range. Occasionally,
an Earth­crossing object may lie in a mean­motion resonance
with respect to the Earth, in which case the probability of a
collision may be temporarily enhanced, or (as is more often
the case) a collision is made impossible. In the latter case, the
object --- and the Earth --- are said to be `protected' by the
resonance.
Conventionally, NEOs have been divided into 3 or more
orbital types (Shoemaker et al. 1979, Milani et al. 1989), the
usual separation being into Apollo, Amor and Aten asteroids
depending simply on the semi­major axis and perihelion dis­
tance of the orbit. Apollo asteroids may be regarded as `typ­
ical' Earth­crossers, and have semi­major axes a – 1:0 AU
(hence orbital periods greater than 1 yr) and q Ÿ 1:017 AU
(the Earth's aphelion distance). Amors circulate beyond the
Earth's orbit, making close approaches to the Earth only when
their perihelia are close to the Earth's aphelion, and are char­
acterised by a ? 1:0 AU together with 1:017 ! q Ÿ 1:3 AU.
Atens revolve largely inside the Earth's orbit, with a ! 1:0 AU
(orbital periods less than 1 yr) and aphelia Q ? 0:983 AU. A
review of these orbital classes and of their respective collision
rates with the Earth has been given by Shoemaker et al. (1979),
who slightly broadened the original definition of an `Earth­
crosser' to encompass objects which, during their long­term
evolution, may evolve to become Earth­crossing at different
stages of their secular evolution.
We note that according to these definitions Halley's comet
would also be an Apollo object, and although comets are not
normally included in the Apollo category (the term usually be­
ing applied to objects of asteroidal appearance), it is neverthe­
less an important question where one draws the line between
the different types of Earth­crossing and Earth­approaching
object. Hahn & Bailey (1992) and Emel'yanenko & Bailey
(1996) have predicted a large number of hitherto undiscov­
ered objects of asteroidal appearance, moving in orbits usually
associated with those of Halley­type comets; and in general a
variety of perturbations, other than the secular terms consid­
ered by Shoemaker et al. (1979), could in principle act to bring
outer solar system bodies on to Earth­crossing orbits within a
short time­scale (e.g. Hahn & Bailey 1990, Bailey et al. 1994).
The chaotic long­term dynamical evolution of comets and as­
teroids in the inner solar system also suggests that a wide range
of different source orbits may in principle contribute to the ob­
served NEO flux, and in particular highlights the importance
of assessing the overall cometary component to the observed
population. The question whether NEOs primarily originate in
the asteroid belt or through the dynamical and physical evo­
lution of comets has important implications for understanding
the origin of these inner solar system bodies and for a correct
assessment of the impact hazard presented to the Earth by
small bodies throughout the solar system.
Recognizing these difficulties, we here focus on the ques­
tion of the short­period comet contribution to the observed
short­period NEO population, which we define to include all
bodies of asteroidal appearance which have perihelion distances
q ! 1:4 AU and orbital periods P ! 20 yr, the limit on the or­
bital period being introduced for consistency with the conven­
tional definition of Jupiter­family short­period comets and to
exclude Halley­type asteroids such as (5335) Damocles (Asher
et al. 1994). The observed sample of NEOs is, in fact, domi­
c
fl IAJ 1996 1

Irish Astr. J., 23(2), 151­156, (1996) COMETARY NEAR­EARTH OBJECTS N. W. HARRIS & M. E. BAILEY
Fig. 1. The distribution of minor bodies in the solar system in the (q,
Q)­plane. The open circles, filled circles and dots correspond respec­
tively to active short­period comets, Near­Earth Objects (NEOs)
and main­belt asteroids. The Kirkwood gaps are clearly visible in
the main­belt population. Notice the presence of NEOs in the re­
gion mostly occupied by short­period comets, and a single comet
(2P/Encke) in the Class 2 NEO region.
nated by objects with rather short orbital periods and aphelion
distances, Q, that are sufficiently small that close approaches
to Jupiter are impossible; in most cases Q ! 4:2 AU. A minor
fraction have slightly longer orbital periods and paths with
Q ? 4:2 AU that occasionally cross the orbit of Jupiter.
The observed NEOs may thus be conveniently separated
into two classes. The minority Class 1 NEOs, with Q ? 4:2 AU,
may be regarded in the main as dynamically similar to ob­
served Jupiter­family comets (i.e. `extinct' or `dormant' comets
in the parlance of Shoemaker &Wolfe 1982 and Weissman et al.
1989); whereas the majority Class 2 NEOs, with Q Ÿ 4:2 AU,
may be regarded as protected, or dynamically `decoupled',
from close encounters with Jupiter, and to have generally
longer dynamical lifetimes. In this work, we focus in partic­
ular on Class 2 NEOs and on short­period Class 1 NEOs with
orbital periods, P ! 20 yr. Approximately 400 such objects are
currently known, the number now growing at an ever increas­
ing rate due to the introduction of new and effective survey
techniques (e.g. Scotti 1994) and to growing interest amongst
astronomers generally in the number and origin of the objects
which in principle may crater the Earth. Only about 10% of
the observed sample are Class 1 NEOs, the remaining 90% be­
ing typical Class 2 objects. The distribution of these and other
inner solar system bodies in the (q; Q)­plane is shown in Fig­
ure 1, based on a list of NEOs compiled and kindly provided
by Steel (1996).
1.1. Population Estimates
Class 2 NEOs, which in the main have broadly similar orbital
properties to those objects defined previously as Apollo­Amor­
Aten asteroids, have orbits which undergo significant dynam­
ical evolution as a result of secular perturbations on time­
scales in the range 10 3 --10 4 yr, which periodically allow the ob­
jects to make close approaches to the terrestrial planets. Their
long­term evolution is subject to complex interactions between
mean­motion and secular resonances, and these together with
the effects of close approaches eventually cause the bodies to
be ejected from the Class 2 system, sometimes on to a hyper­
bolic orbit or scattered on to a trajectory that allows a phys­
ical collision with the Sun, a terrestrial planet or one of the
asteroids in the main belt. Depending on the particular orbit
under consideration, the time­scale for these processes is in the
approximate range 10 5 --10 8 yr, short compared to the age of
the solar system, and demonstrating the need for a source to
maintain the observed population.
The total population of Apollo­Amor­Aten asteroids has
been estimated by Shoemaker et al. (1979) as approximately
700\Sigma300 Apollos, 1000--2000 Amors and 100 Atens, these fig­
ures corresponding to objects with absolute visual magnitudes
V (1; 0) ! 18 or diameters, d, greater than about 1 km. A
more recent determination has been given by Rabinowitz et al.
(1994), who estimate that there are approximately 20 Earth­
crossing asteroids with d ? 5 km, 1500 with d ? 1 km, ¸135000
with d ? 0:1 km and 10 8 --10 9 with d ? 10 m. Since it is possible
that high­inclination Halley­type asteroids also make a signifi­
cant contribution to the overall NEO population, and since the
total observed sample is only about 400 objects, the observed
population is clearly very incomplete.
For comparison, a theoretical estimate of the NEO popu­
lation, based on the assumption that NEOs are the products
of collisions between asteroids in the main belt, subsequently
having evolved through chaotic routes associated with various
secular perturbations in the inner solar system, suggests that
there may be about 2000 km­sized bodies, and more than 10 6
bodies larger than 0.1 km in diameter (Menichella et al. 1996).
If such figures are approximately correct, the mean interval be­
tween impacts on the Earth with energies comparable to that
of the famous 1908 Tunguska event could be as short as a few
decades.
1.2. Possible Sources
Two main sources have been considered for observed NEOs:
comets and the asteroid belt. So far as the second is concerned,
although the orbital velocities of main­belt asteroids are on the
order of 20 km s \Gamma1 , their relative velocities are much smaller
and the typical ejection velocities of fragments produced as a
result of inter­asteroidal collisions are expected to be less than
about 1 km s \Gamma1 (Binzel & Xu 1993). The change in velocity
as a result of a single collision is thus at most about 5% of
the orbital velocity, too small to produce significant changes
in the heliocentric elements, and such collisional debris will
typically continue to circulate largely within the main belt and
not achieve significantly non­circular orbits with aphelia close
to Jupiter or perihelia in the region of the terrestrial planets.
For this historical reason, most early investigations into
the problem of the origin of NEOs, particularly when the total
number of known bodies was quite small, appeared to rule out
the main belt as a significant source, and instead focused (al­
beit reluctantly) on the possible contribution made by short­
period comets (e.g. ¨
Opik 1963; cf. Bailey et al. 1990). The
notion of a cometary source for NEOs was also supported by
observations that at least one active comet, namely 2P/Encke,
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Irish Astr. J., 23(2), 151­156, (1996) COMETARY NEAR­EARTH OBJECTS N. W. HARRIS & M. E. BAILEY
was known to move in a Jovian­decoupled Apollo­asteroid or­
bit, and, because the physical lifetime of comets was (and still
is) generally assumed to be much shorter than the dynamical
lifetime (e.g. Rickman 1992), a steady supply of comets from
the Jupiter family into NEO orbits could in principle sustain
the entire population.
Recent work has impinged on these conclusions in sev­
eral respects. First, the total number of NEOs has increased
enormously since the 1970s, when a review by Whipple (1973)
concluded that the total population was on the order of 10 2 .
Secondly, several new dynamical mechanisms for transferring
asteroid fragments from the main belt on to high­eccentricity
planet­crossing orbits have now been identified; and thirdly,
the distinction between comets and asteroids has itself be­
come blurred. There are now many examples of objects, con­
ventionally classified as `asteroids', which move on dynami­
cally unstable `cometary' orbits in the outer solar system, at
least one of which shows evidence for cometary outgassing,
namely Chiron. Furthermore, some bodies in the inner solar
system, which are now classified as asteroids, have former histo­
ries as active comets: for example, the asteroid (4015) Wilson­
Harrington, which was observed to show a fanned tail on pre­
discovery photographic plates taken in 1949, and was pre­
viously known as comet Wilson­Harrington 1949g = 1949 III
(Bowell 1992). Moreover, the existence of an active comet
(2P/Encke) moving on a Class 2 NEO orbit, and of the parent
of the dynamically young Geminid meteoroid stream, namely
(3200) Phaethon, which is currently on a Class 2 NEO orbit
with a semi­major axis of 1.27 AU, further complicates the pic­
ture. How a comet, or even an asteroid from the main belt,
could evolve into such an orbit on a short time­scale is not
yet known. Further examples of apparently cometary Class 2
NEOs are described by Weissman et al. (1989), and include
objects considered as cometary on the basis of associated mete­
oroid streams such as (1685) Toro, (2101) Adonis, (2201) Oljato
and (2212) Hephaistos.
These observations suggest that whereas the discovery of
new dynamical mechanisms for transferring material from the
main belt on to high­eccentricity Earth­crossing orbits pro­
vides an important motivation for considering the main aster­
oid belt as possibly the dominant source of observable NEOs, a
cometary source may also be necessary in order to explain part
of the observed NEO population. The mechanisms underlying
these possible orbital transitions are briefly outlined below.
1.3. Transfer Mechanisms
So far as dynamical evolution from the main belt is concerned,
the principal new idea is that the fragments from collisions in
the main belt may be ejected with velocities which, although
insufficient directly to produce Jupiter­approaching or Earth­
crossing orbits, may nevertheless allow the objects to stray
into chaotic regions of phase space associated with one or an­
other mean­motion or secular resonance with the major plan­
ets Jupiter or Saturn. Indeed, several main­belt asteroid fam­
ilies lie in regions associated with chaotic zones (Morbidelli
et al. 1996), from which mean­motion and secular perturba­
tions are believed to transport the fragments into the region of
the terrestrial planets (e.g. Greenberg & Nolan 1989), an idea
developed in detail by Wisdom (1983), who was one of the
first to identify the 3:1 Jovian mean­motion resonance (near
a = 2:5 AU) as an important source of Apollo asteroids and
meteorites on time­scales as short as 1 Myr.
The importance of this general process as a means to gen­
erate Earth­crossing objects has now been confirmed by de­
tailed studies of the strongly chaotic effect of overlapping sec­
ular resonances (e.g. the š5 and š6) on the motion of parti­
cles in the 4:1, 3:1, 5:2 and 7:3 Jovian mean­motion resonances
(Moons & Morbidelli 1995, Ipatov 1992). It has been estimated
that taken together these processes could inject around 370
km­sized fragments and 2:5 \Theta 10 5 0.1 km­sized fragments per
million years (Menichella et al. 1996). On this basis, the main
asteroid belt is a plausible, fluctuating source of NEOs with
the potential to explain something close to the observed pop­
ulation of km­sized NEOs.
On the other hand, long­term orbital integrations of NEOs
often show evolution on to orbits which superficially resemble
those of Jupiter­family comets (e.g. 2P/Encke; Bailey 1995,
1996), demonstrating the feasibility, in principle, of the reverse
dynamical evolution from a cometary to a NEO orbit. More­
over, whereas cometary nuclei comprise an intimate mixture of
amorphous and crystalline ices, and less volatile organic ma­
terial and dust, sublimation of their surface layers during suc­
cessive perihelion passages could leave behind a non­volatile
organic and dusty residue superficially resembling an aster­
oidal regolith. These arguments suggest that some NEOs may
in fact be devolatised or `dormant' cometary nuclei.
The devolatilization of the surface of a cometary nucleus
is a gradual process, occurring on a time­scale on the order of
1000 orbital revolutions, or ¸10 4 yr for a typical Jupiter­family
short­period comet (e.g. Rickman 1992). This time­scale thus
provides a rough upper limit on the physically active lifetime of
a short­period cometary nucleus circulating in an orbit of small
perihelion distance, and the gradual removal of its volatiles
may also explain the tendency for short­period comets to be
less active than apparently similar objects of longer period.
However, although depletion of cometary surface ices might
lead to deactivation of the central nucleus, it is not known
how long such a `dormant' nucleus might continue to exist
without reactivation. In particular, meteoroidal bombardment
may cause disruption or fracture of the insulating mantle and
allow fragments of the surface to break off, while a sufficiently
large impact might even cause splitting or fragmentation of the
whole nucleus. The physics of this process, while conceptually
similar to that of the collisions expected to occur in the main
belt, leading to the formation of asteroid families, differs in
that the target probably has much lower structural strength
than a typical asteroid (and different chemical composition),
and owing to the different cometary orbits the relative velocity
of bombardment is likely to be higher.
According to this scenario, limited amounts of the sub­
surface cometary ices may occasionally be exposed to so­
lar radiation, thus leading to reactivation of the dormant
cometary nucleus or `cometary asteroid', and producing non­
gravitational forces which might in principle affect the dynam­
ics and lead to transfer of the object on to an orbit similar to
those of either Class 1 or Class 2 NEOs.
2. NUMERICAL EXPERIMENT
2.1. Dynamical Evolution
In order to assess the likelihood of Jupiter­family comets evolv­
ing on to orbits similar to those of NEOs, we have performed
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Irish Astr. J., 23(2), 151­156, (1996) COMETARY NEAR­EARTH OBJECTS N. W. HARRIS & M. E. BAILEY
Fig. 2. The evolution of 360 hypothetical Jupiter­family short­period comets in the (q; Q)­plane at various times times during the numerical
integration. Note that no object evolved into the region occupied by Class 2 NEOs (q ! 1:4AU, Q Ÿ 4:2 AU).
Fig. 3. The evolution of 360 hypothetical Jupiter­family short­period comets in the (q; Q)­plane at various times times during the numerical
integration including the effects of non­gravitational forces. The approximate number of revolutions for a body with an orbital period of
5.5 yr is also given. In this case a small population of Class 2 NEOs is produced within ¸10 4 yr.
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Irish Astr. J., 23(2), 151­156, (1996) COMETARY NEAR­EARTH OBJECTS N. W. HARRIS & M. E. BAILEY
long­term numerical integrations of hypothetical short­period
comets using the RADAU integrator RA15 (Everhart 1985),
in a model solar system including the 7 planets Venus to Nep­
tune and with tolerance parameter set to 10 \Gamma8 . The initial
orbits were chosen to have orbital periods P ! 20 yr (i.e.
semi­major axes a ! 7:37 AU), perihelion distances close to
1 AU and aphelia close to 5.2 AU, allowing initial close encoun­
ters with both Jupiter and the Earth. The motivation for this
choice of initial conditions was to generate an ensemble of typ­
ical Jupiter­family comets from which to calculate the transfer
probability on to Class 2 NEO orbits (i.e. to q ! 1:4 AU and
Q ! 4:2 AU), and to determine the proportion of cometary
bodies which might survive for longer than the physically ac­
tive cometary lifetime (¸10 4 yr) as Class 1 NEOs, presumably
in the form of inactive `cometary asteroids'.
We first integrated an ensemble of 360 bodies with iden­
tical initial orbits (q = 1 AU, Q = 5:2 AU, and inclinations
i = 0 ffi ) except for the arguments of perihelion which were dis­
tributed uniformly in the range 0 ffi !!!360 ffi . The initial Tis­
serand parameter with respect to Jupiter was TJ = 2:81. This
ensemble was integrated for approximately 0.25 Myr includ­
ing only gravitational forces. The resulting evolution is sum­
marised in Figure 2, which shows the location of the surviving
objects in the (q; Q)­plane at several times during the evolu­
tion of the ensemble. These diagrams may be compared with
the observed populations shown in Figure 1.
The originally similar orbits are rapidly dispersed (within
less than 100 revolutions), and the values of TJ spread out to
lie within the approximate range 2.5--3.0, typical of most ob­
served Jupiter­family comets. Notice that objects with small
perihelion distances (q ! 2 AU) tend to avoid the region with
Q ? 8 AU, (i.e. Jupiter­family comets tend to have short pe­
riods when visible). This `forbidden' region of phase space
is instead populated with high­inclination Halley­type orbits
(see Figure 1). In work in progress, Emel'yanenko & Harris
have shown that this result is consistent with Halley­type ob­
jects originating from a source region that contains an initially
isotropic distribution of inclinations, such as the outer Oort
cloud.
Among the 360 objects, we found only two which experi­
enced evolution on to temporary Class 2 NEO orbits, and then
only for short periods lasting in total for 708 yr and 3220 yr re­
spectively. In both cases the aphelion distances were greater
than 4 AU, indicating that even for these objects their classifi­
cation as Class 2 NEOs rather than Class 1 is marginal. There
were no long­term transitions to Class 2 NEO orbits over the
time­scale investigated, so the transition probability for a typi­
cal Jupiter­family short­period comet to evolve on to a Class 2
NEO orbit by purely gravitational means is probably less than
1/360, i.e. ! 0:0028.
Approximately 14% of the sample evolved on to a Class 1
NEO orbit (q ! 1:4 AU, Q ? 4:2 AU) for a period which in
total would have exceeded the assumed active lifetime (10 4 yr)
of a comet in such an orbit. Since the initial ensemble started
with perihelia in the NEO zone this figure might be thought to
be somewhat high, and in order to investigate this possibility
a second ensemble of hypothetical Jupiter­family short­period
comets was integrated, containing 51 objects which had been
captured into short­period orbits (P ! 20 yr) from the inner
core of the Oort Cloud. (Their original orbits had been specified
as having i ! 30 ffi , 2500 ! a ! 3500 AU and 12:5 ! q ! 31 AU).
At the start of this second integration the bodies had P ! 20 yr
and q ? 3 AU, and their Tisserand parameters covered approxi­
mately the same range as the evolved first group. The evolution
of this second ensemble of 51 objects was followed for 1 Myr
using the same model solar system and RADAU parameters as
before.
The general properties of the evolution of the second en­
semble were similar to those of the first group, with particles
tending to avoid the high­inclination Halley­type region, and
with no objects transferred on to Class 2 NEO orbits. About
10% of the bodies evolved on to Class 1 NEO orbits for time­
scales in excess of the assumed cometary lifetime (¸ 10 4 yr),
compared with 14% in the previous sample, and roughly 37%
of the ensemble evolved to perihelion distances q ! 1:4 AU.
The average dynamical lifetime of the objects in both these
ensembles in orbits with q ! 3 AU was approximately 6000 yr
or 33000 yr, depending respectively on whether the assumed
limit on the active lifetime was taken to be 10 4 yr or infin­
ity. This gives an indication of the ratio of active to inactive
cometary objects in the Jupiter­family region. Since there are
approximately 140 active Jupiter­family short­period comets
at the present time we would require an injection rate of about
140 particles every 6000 years in order to maintain the active
short­period comet population in a steady state, corresponding
to about 2.35 `new' comets per century (cf. Fern'andez 1985).
Since roughly 10% of Jupiter­family short­period comets ap­
pear to evolve on to Class 1 NEO orbits for times in excess
of 10 4 yr (taking the figure for the second ensemble as more
representative), we estimate that the cometary contribution to
the Class 1 NEO population is on the order of 2300 objects per
million years. The corresponding figure for Class 2 NEOs is less
than 70 objects per million years. Comparing these figures with
the main­belt asteroid injection rate estimated by Menichella
et al. (1996), namely ¸ 370 km­sized Class 2 NEOs per mil­
lion years, and assuming that the objects being compared are of
roughly similar dimensions, these results suggest that consider­
ing gravitational mechanisms alone the majority (i.e. ? 80%)
of Class 2 NEOs may originate in the main asteroid belt.
The typical dynamical lifetimes of Class 2 NEOs are ¸
10 6 yr for `fast­track' objects (i.e. those that evolve through
secular perturbations or secular resonances to hit the Sun or
are ejected on to hyperbolic orbits after becoming planet­
crossing), and ¸ 10 7 yr for `slow­track' non­resonant NEOs
evolving mainly under the influence of close encounters with
the terrestrial planets (Milani et al. 1989). Since about 80% of
the current NEO population are slow­track objects, we there­
fore expect the steady­state Jupiter­family short­period comet
contribution to Class 2 NEOs to be less than about 600 bodies.
Finally, it is encouraging that the majority of our modelled
Class 1 orbits have aphelia in the range 4:5!Q!7 AU, typical
of observed Class 1 NEOs. However, the objects in our ensem­
ble have relatively short dynamical lifetimes against ejection by
Jupiter, typically in the range 10 4 --10 6 yr (on average approxi­
mately 5:4 \Theta 10 4 yr), and spend only a small fraction of their to­
tal dynamical lifetime on Class 1 short­period orbits (on aver­
age only about 1:5\Theta10 4 yr, including the active phase). We thus
expect Jupiter­family short­period comets to make only a very
small steady­state contribution to the total number of Class 1
NEOs, probably less than 100 objects. On the other hand, the
observed population of Class 1 NEOs is likely to be dominated
by bodies having long dynamical lifetimes on Class 1 orbits
(? 10 5 yr), a possibility which appears to be confirmed when
we consider the results of long­term numerical integrations of
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Irish Astr. J., 23(2), 151­156, (1996) COMETARY NEAR­EARTH OBJECTS N. W. HARRIS & M. E. BAILEY
individual test particles with orbits close to those of various
`real' objects among the apparently cometary Class 1 NEOs,
such as (3552) Don Quixote, (5324) Lyapunov, (5370) Taranis,
1982YA, 1994AB1, 1993TR2, 1983LC, 1984QY1 and 1994LW.
If dormant cometary nuclei can be scattered into these Class 1
orbits of longer dynamical lifetime, then the total cometary
contribution to Class 1 NEOs could possibly be increased by
up to an order of magnitude, implying a steady­state cometary
contribution to the Class 1 NEO population on the order of 10 3
objects.
2.2. Non­Gravitational Forces
These results have been obtained using models of the dynam­
ical evolution that neglect non­gravitational forces. However,
as we have described, the principal distinguishing character­
istic of comets is outgassing, and this inevitably produces a
non­gravitational jet­reaction force on the nucleus. The effect
of such forces may be included in the dynamical model by
adopting the conventional (A1 ; A2 ; A3 ) non­gravitational for­
malism (Marsden et al. 1973). Here we restrict attention in
the first instance to the transverse component of the force,
A2 , which is the dominant factor which systematically af­
fects the semi­major axis. Values of non­gravitational param­
eters for observed Jupiter­family comets are given in the Cat­
alogue of Cometary Orbits (Marsden & Williams 1993), and
in this preliminary investigation we have adopted values of A2
that are randomly distributed in the interval (\Gamma0:40; 0:40) \Theta
10 \Gamma8 AU day \Gamma2 according to a triangle law. The standard de­
viation of this distribution, oe A 2
, is 0:16 \Theta 10 \Gamma8 AU day \Gamma2 .
The initial ensemble of 360 orbits was integrated, includ­
ing non­gravitational forces, in the same model solar system
as before for a timespan of 10 4 yr, consistent with the assumed
active lifetime of a Jupiter­family comet. Figure 3 shows that
after this period of time ¸2% of the ensemble had evolved
to apparently `stable' Class 2 orbits (i.e. to Q ! 4 AU), cor­
responding to a net injection rate of about 470 Class 2 NEOs
per million years. This is comparable to estimates of the rate of
fragment injection from the main belt, and suggests a possible
mechanism by which comets might be inserted on to Class 2
NEO orbits within their physically active lifetime.
3. CONCLUSIONS
Long­term numerical integrations of an ensemble of represen­
tative Jupiter­family comets show that only a small fraction
of such orbits evolve on to stable Class 2 NEO orbits (q !
1:4 AU, Q ! 4:2 AU), suggesting that most such NEOs prob­
ably originate in the main asteroid belt. However, when non­
gravitational forces are included in the calculation a greater
proportion of bodies evolve down to the Class 2 NEO region.
Of course, it is difficult to know whether our adopted non­
gravitational parameters are realistic, or in fact likely to re­
main approximately constant throughout the physically active
`cometary' phase of evolution (¸1000 revolutions). Our pre­
liminary results therefore only show that, in principle, non­
gravitational forces may substantially affect the rate of pro­
duction of NEOs from Jupiter­family cometary orbits. Further
work aimed at clarifying the effects of non­gravitational forces
on the rate of production of NEOs of all orbital types is in
progress.
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