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Why Halley­Types Resonate but Long­Period Comets Don't:
A Dynamical Distinction between Short and Long­Period Comets
J. E. Chambers
Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road
NW, Washington DC 20015, USA.
Tel 202­686 4370 (x4440)
FAX 202­364 8726
e­mail: chambers@agamemnon.ciw.edu
and
Harvard­Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138.
Abstract
Several recent studies have noted that the orbital evolution of many comets is influenced by mean­motion
resonances with Jupiter. However, the distribution and relative importance of these resonances and the
orbital characteristics of the comets affected have not been addressed to date. Here I show analytically
that cometary orbits with periods greater than a critical value, P c , (which depends upon the orbital
inclination) are prevented from undergoing librations about a mean­motion resonance. Conversely,
numerical integrations indicate that resonances play an important role in the dynamics of comets with
P ! P c . The inclination­averaged value of P c approximately coincides with the traditional and arbitrary
dividing line between Halley­type and long­period comets, which explains why many of the former are
currently observed to be in resonance whereas the latter are not. Thus, we now have a dynamical
justification for separating comets into those of short and long period.
1 Introduction
Many small bodies in the solar system are affected by mean­motion resonances. The Kirkwood gaps in
the main asteroid belt, corresponding to resonances with Jupiter, and the precise integer ratios between
the orbital periods of the jovian moons Io, Europa and Ganymede are prominent examples that have
been known for more than a century. More recently, it has become apparent that resonances also
influence the motion of comets (Marsden 1970, Emel'yanenko 1985, Carusi et al. 1987), although the
role played by resonances in the long­term evolution of comets is not yet clear.
Intriguingly, a significant fraction of the known short­period comets (period P ! 200 years) are
currently librating about resonances with Jupiter, whilst there is no known example of a long­period
comet displaying such behaviour. The current record holder for the resonant comet with the longest
orbital period is a Halley­type (20 ! P ! 200 years), namely P/Swift­Tuttle, which presently resides
in the 1:11 resonance with Jupiter (Chambers 1992, 1995). This striking difference between the Halley­
types and long­period comets, often considered to be very similar physically and dynamically, is of
more than passing interest since resonant librations alter the frequency of close encounters with Jupiter,
change the form of the secular perturbations on a comet's orbit (Chambers 1994) and may alter the
Lyapunov time (Chambers 1995). In addition, we would expect that a comet temporarily trapped in
a resonance will experience different physical evolution via dust­mantling (due to differing evolution of
1

the perihelion distance) than one which is not.
In this paper I address the stability of cometary resonances by examining, from an analytic viewpoint,
small­amplitude librations in the restricted three body problem (Section 2). A fundamental result which
emerges is that only certain cometary orbits can librate about a resonance. Orbits whose semi­major
axes exceed a critical value (which is a function of the orbital inclination) are prevented from remaining
close to a resonance, even for a short time. This conclusion is supported by numerical surveys of highly­
eccentric orbits in the restricted problem and cometary orbits integrated in a more realistic model for
the solar system (Section 3). These results are discussed in Section 4.
2 Small Librations in the Restricted Three­Body Problem
The restricted three­body problem (RTB), in which a comet of negligible mass moves under the gravi­
tational influence of the sun and Jupiter---themselves moving along circular orbits---is a good starting
point from which to examine the phenomenon of comets in resonance. The RTB can be described by
six canonically­conjugate variables: (X; oe), (Y; $) and
(Z;\Omega\Gamma3 where
X = \Gamma(¯a) 1=2 =q
Y = \GammaqX (1 \Gamma e 2 ) 1=2 + pX
Z = (pX \Gamma Y )(1 \Gamma cos i)
oe = p– J \Gamma ql \Gamma p$
$ = !
+\Omega and a, e, i,
!,\Omega and l are the comet's semi­major axis, eccentricity, inclination, argument of perihelion,
longitude of ascending node and mean anomaly respectively, ¯ = GM fi , G is the gravitational constant,
M fi is the solar mass and – J is Jupiter's mean longitude. The parameters p and q are integers (p ! q)
and from now on we will consider the motion of a comet in the vicinity of the p : q external resonance
with Jupiter. (For a discussion of why we are unlikely to observe comets in resonance with the other
planets see Chambers 1995, Section 7.) In addition, the RTB contains an integral of the motion, the
Jacobi constant J , given by
J = ¯ 2 =(qX) 2 + 2n J (Y + Z \Gamma pX) + 2GM J =\Delta
where M J and n J are the mass and mean motion of Jupiter, and \Delta is the distance of the comet from
the planet.
The rate of change of the critical argument, oe, is generally much greater than that of either $
or\Omega (completing one cycle in a few orbits compared to a few hundred orbits) and to a good approximation
we can neglect the motion in these two degrees of freedom over one libration cycle, treating Y and Z
as constants. This allows us to reduce the problem to a one­degree­of­freedom mapping that gives the
values of X and oe after every p revolutions of the comet:
X 1 = F (X 0 ; oe 0 )
oe 1 = G(X 0 ; oe 0 )
where (X 0 ; oe 0 ) are the initial elements, (X 1 ; oe 1 ) are the elements one iteration later, and F , G are
unknown functions.
2

Figure 1: Upper: successive points generated by Eqs. 1 lie on an ellipse, progressing by an angle ~
š at
each iteration of the mapping. Lower: 100 revolutions of a comet, librating about the 1:6 resonance in
the planar RTB (J = 1:5), calculated numerically. Here the progression angle is ¸ 80 ffi .
3

We now define the p : q resonance to be the point in phase space, (XR ; oe R ), at which X and oe
are unchanged after one iteration of the map (i.e. after p revolutions of the comet, corresponding to q
revolutions of Jupiter). Thus F (XR ; oe R ) = XR and G(XR ; oe R ) = oe R . Expanding the mapping about
the resonance gives
X 1 = F (X 0 ; oe 0 ) = F (XR ; oe R ) + (X 0 \Gamma XR )
/
@F
@X
!
R
+ (oe 0 \Gamma oe R )
/
@F
@oe
!
R
+ \Delta \Delta \Delta
oe 1 = G(X 0 ; oe 0 ) = G(XR ; oe R ) + (X 0 \Gamma XR )
/
@G
@X
!
R
+ (oe 0 \Gamma oe R )
/
@G
@oe
!
R
+ \Delta \Delta \Delta
If we consider small librations we can neglect the higher­order terms and the mapping becomes:
/
X 1 \Gamma XR
oe 1 \Gamma oe R
!
=
/
A BXR
C=XR D
!/
X 0 \Gamma XR
oe 0 \Gamma oe R
!
(1)
where A, B, C and D are constant for a particular resonance and particular values of Y and Z, and
A = (@F=@X)R etc. In addition, for the mapping to be canonical (and thus energy conserving) we
require that the matrix determinant is unity and hence
AD \Gamma BC = 1 (2)
The nature of the motion close to a resonance depends upon the values of A, B, C and D. To see
this we rewrite Eqs. 1 as
(X \Gamma XR ) = XR ~a cos(~št + ~
OE)
(oe \Gamma oe R ) = ~ a(1 \Gamma ~
e 2 ) 1=2 cos(~št + ~
OE + ~
!) (3)
where
cos ~
š = (A +D)=2
tan ~
! = [4 \Gamma (A +D) 2 ] 1=2 =(D \Gamma A)
1 \Gamma ~
e 2 = (1 \Gamma AD)=B 2 (4)
and t is an integer representing the iteration index of the mapping. (That these are the same as Eqs. 1
can be verified by substitution.) The libration amplitude, ~a, and phase, ~
OE, are detemined by the initial
values of X and oe, whereas ~
e, ~ š and ~
! are the same for all orbits close to the resonance.
Equations 3 describe an ellipse centred on (XR ; oe R ) with semi­major axis ~
a, eccentricity ~ e and
orientation ~
!. Successive points generated by the mapping advance around the ellipse by an angle ~
š
(see Fig. 1). However, when (A +D) 2 ? 4 the step angle ~ š becomes imaginary and the orbit diverges
hyperbolically from the resonance. Thus cometary orbits can only librate about those resonances for
which (A +D) 2 ! 4.
2.1 Resonances with p = unity
First we consider resonances with p = 1. In this case the mapping period is equal to one revolution of
the comet, and we can derive approximate expressions for some of the constants in Eqs. 1 by making use
of the fact that Halley­type and long­period comets undergo rapid orbital evolution close to perihelion,
4

Figure 2: A versus q 3 X 2
R calculated numerically for small librations about resonances of the form 1 : q,
where 4 Ÿ q Ÿ 9, in the planar RTB with J = 1:5. The predicted value of A for the 1:10 resonance lies
in the unstable region (i.e A 2 ? 1) and librations do not occur for this resonance.
with little change at other times (Carusi et al. 1987). We idealize this by assuming that the comet's
orbit remains fixed except for an impulse at each perihelion.
We begin the comet at aphelion with orbital elements (X 0 ; oe 0 ). The elements just before the following
perihelion will be X = X 0 and
oe = oe 0 + p\Delta– J \Gamma q\Deltal \Gamma p\Delta$
= oe 0 + pn J P 0 =2 \Gamma qú \Gamma 0 (5)
where P 0 = \Gamma2úq 3 X 3
0 =¯ 2 is the comet's initial orbital period, and \Deltax denotes the change in quantity
x.
At perihelion the comet receives an impulse, making X = X 1 . Using an expression similar to Eq. 5
for the further change in oe by the time of the next aphelion gives the total change over one revolution:
(oe 1 \Gamma oe R ) = (oe 0 \Gamma oe R ) \Gamma (pn J úq 3 =¯ 2 )(X 3
0 +X 3
1 )
' (oe 0 \Gamma oe R ) \Gamma (3pn J úq 3 X 2
R =¯ 2 )[(X 0 \Gamma XR ) + (X 1 \Gamma XR )] \Gamma 2pn J úq 3 X 3
R =¯ 2
5

where the last step uses the approximation X 3 ' X 3
R + 3(X \Gamma XR )X 2
R which is valid for values of X
close to XR .
Using the identity n J q 3 X 3
R =¯ 2 = \Gammaq=p, plus the first of Eqs. 1, yields an equation for the new value
of oe:
(oe 1 \Gamma oe R ) = (3qú=XR )(A + 1)(X 0 \Gamma XR ) + (1 + 3qúB)(oe 0 \Gamma oe R )
Thus, using this and Eqs. 1,2, we get an approximate mapping for the motion of a comet close to a
resonance that contains only a single parameter:
/
X 1 \Gamma XR
oe 1 \Gamma oe R
!
=
/
A (A \Gamma 1)XR =(3qú)
3qú(A + 1)=XR A
!/
X 0 \Gamma XR
oe 0 \Gamma oe R
!
(6)
We can now assess how A varies from one resonance to another by considering the corresponding
expression for the change in orbital energy, E = \Gamma¯ 2 =(2q 2 X 2 ) ' ER + [¯ 2 =(q 2 X 3
R )](X \Gamma XR ). Using the
first of Eqs. 6 the change in energy is:
\DeltaE = (E 1 \Gamma E 0 )
= (A \Gamma 1)(E 0 \Gamma ER ) + [¯ 2 (A \Gamma 1)=(3q 3 úX 2
R )](oe 0 \Gamma oe R )
For the case when E 0 = ER the change in energy during one revolution of the comet is simply
proportional to (oe 0 \Gamma oe R ). As we consider resonances with larger q (i.e. increasing orbital period)
the mean orbit tends towards a parabola, and we expect that the ratio \DeltaE=(oe 0 \Gamma oe R ) will tend to a
constant, other things being equal. Thus
` \DeltaE
oe 0 \Gamma oe R
'
E 0 =ER
= ¯ 2 (A \Gamma 1)
3q 3 úX 2
R
' const (7)
and hence
A ' 1 +Kq 3 X 2
R (8)
where K is a constant.
As we increase q, the quantity q 3 X 2
R also increases, and there will come a critical orbital period,
P c , at which A 2 exceeds unity and librations become impossible (c.f. first of Eqs. 4). To see this in
practice Fig. 2 shows a plot of A versus q 3 X 2
R for resonances with p = 1 and 4 Ÿ q Ÿ 9, calculated
numerically in the planar RTB with J = 1:5. (This value of the Jacobi constant is chosen to make
the perihelion distance comparable with that of a typical Halley­type comet.) Extrapolating the data
in Fig. 2 suggests that A ! \Gamma1 for the 1:10 resonance and indeed numerical experiments confirm that
orbits close to this resonance rapidly diverge away from it---that is, librations do not occur beyond the
1:9 resonance in this case.
The trend in the value of A seen in Fig. 2 produces a change in the ratio of the libration period,
P lib , to the comet's orbital period, P , via the first of Eqs. 4:
P lib =P = 2ú= cos \Gamma1 A (9)
At low values of q this ratio is large and librations about a resonance resemble continuous oscillations
of a pendulum---Fig. 3a. Conversely, for large values of q (i.e. long orbital periods) librations appear
more like pendulum oscillations viewed using a stroboscope---Fig. 3b. Librations break down altogether
when P is large enough that the ratio P lib =P falls below the critical value of 2.
6

Figure 3: Librations about the 1:2 (upper) and 1:8 (lower) resonances in the planar RTB, with J = 1:5.
Each data point represents one orbit of the comet. Note that the number of orbits required to complete
one libration cycle decreases with increasing orbital period. When this number falls to 2 librations
become impossible.
7

Figure 4: The ratio of the libration period to the orbital period for small librations about resonances
of the form 2 : q in the planar RTB with J = 1:5. The trend suggests that the 2:15 resonance will lie
below the critical value P lib =P = 4, and indeed librations do not occur for this resonance.
2.2 Resonances with p ? 1
For resonances with p ? 1 the situation is somewhat more complicated as the comet undergoes more
than one revolution per iteration of Eqs. 1, and thus only one in every p revolutions contributes to the
libration ``ellipse'' of Fig. 1. This implies that the transition from stable to unstable librations occurs
when P lib =P = 2p, and numerical experiments confirm this. For example, Fig. 4 shows P lib =P versus P
calculated for resonances with p = 2 in the planar RTB with J = 1:5. As in the p = 1 case the ratio
P lib =P decreases monotonically. The trend displayed in Fig. 4 suggests that the ratio will fall below
the critical value P lib =P = 4 for the 2:15 resonance, preventing librations from occurring about this or
resonances with larger values of q, and in fact this is what happens.
One consequence of this dependence of P c on p is that the resonances with large values of p tend
to disappear at smaller orbital periods than those with small p. For example, in the planar RTB, with
J = 1:5, librations about resonances for which p = 3 cease beyond the 3:10 resonance (orbital period
41 years). For the p = 2 case the last resonance is the 2:13 (P = 77 years), while for the p = 1 case
8

Inclination (degrees) Outermost Resonance Orbital Period (years)
0 1:9 107
30 1:10 119
60 1:12 142
80 1:13 154
100 1:15 178
120 1:17 202
150 1:21 249
180 1:21 249
Table I: Identity and orbital period of the outermost stable resonance in the restricted three­body
problem, as a function of inclination, for orbits with perihelion of 1 au, argument of perihelion 0.
librations can occur for all resonances up to and including the 1:9 (P = 107 years). Hence, any Halley­
type comet with i ' 0 and an orbital period greater than about 80 years will only undergo librations
about resonances of the form 1 : q, and it is worth noting that all of the known Halley­types currently
undergoing librations are located in p = 1 resonances.
2.3 The Effect of Inclination
In a quantative model for the motion of Halley­type comets, Chambers (1994) has shown that indirect
planetary perturbations---the driving force behind the orbital evolution of these objects---diminish in
strength with increasing orbital inclination of the comet. Equations 7 and 9 imply that this will reduce
the rate of change of P lib =P as q increases (recall that A is negative) and thus increase the critical
orbital period, P c .
To test this hypothesis I performed numerical integrations of the RTB for orbits located close to
each of the resonances of the form 1 : q, where 1 Ÿ q Ÿ 25, for various orbital inclinations. In each
case, the semi­major axis of the initial orbit was within one part in 10 7 of the resonance centre. The
initial eccentricity was chosen to make the perihelion distance equal to 1 au---typical for a Halley type
comet---and the initial argument of perihelion was set to zero.
Table 1 shows the results of the survey. Clearly there is a positive correlation between inclination
and the maximum orbital period at which librations can be sustained. This suggests that we are most
likely to observe resonant behaviour in comets with retrograde orbits, and that physical and dynamical
processes resulting from librations will be more prevalent in comets with highly­inclined orbits.
9

Figure 5: The orbital period, as a function of inclination, of the longest­lived librations exhibited by
each of 100 comets integrated for 10 5 years in a solar system containing the sun and planets Jupiter
thru Neptune. The dashed line represents the maximum orbital period at which librations can occur
for orbits in the RTB with perihelia of 1 au and ! = 0.
3 Resonances in Realistic Solar­System Models
So far we have restricted our attention to comets moving in the RTB, whereas we would like to know
whether the effects described above persist for comets in the real solar system. In practice the sample
of known Halley­type comets is rather small, so I have chosen to consider a batch of 100 randomly
generated Halley­type orbits, with a, cos i,
!,\Omega and l distributed uniformly, 12 ! P ! 200 years,
and perihelion distances uniformly chosen between 0 and 2 au. These comets were integrated for 10 5
years along with the sun, Jupiter, Saturn, Uranus and Neptune, with the initial planetary coordinates
and velocities taken from JPL ephemeris DE 200 at Julian Day 2,446,000.5. The integrations were
performed using Everhart's 15 th ­order RADAU integrator (Everhart 1985), with accuracy parameter
LL = 8 and an initial stepsize of 10 days.
An interesting result which emerged is that librations about resonances are very common: the
fictitious Halley types typically spent more than one third of the integration span undergoing mean­
motion librations, and half of the objects logged at least 10 4 years of uninterrupted librations about
10

a single resonance. Similarly, integrations by Levison and Duncan (1994) indicate that objects with
orbits similar to those of the real short­period comets spend a large fraction of their lives in resonances.
Clearly it is not coincidence that many of the known Halley­types are in resonances (the actual number
is uncertain since several of the comets have poorly determined orbits).
Of the time spent in librations roughly 60% involved resonances of the form 1 : q, whilst 2 : q
resonances occupied another 20% of the time, and all other resonances accounted for the remaining
20%. Thus it appears that resonances with p = 1 are most important for the evolution of Halley­type
comets, and these may be considered to be the analogues of the first order (i.e. p + 1 : p) resonances
that dominate the behaviour of many asteroids and planetary satellites.
Looking now at the question of whether libration regions disappear at large orbital periods, Fig. 5
shows the period, P , as a function of inclination i, for the resonance producing the longest­lived set of
librations for each of the 100 comets. The dashed line gives the period of the outermost resonance for
which librations can occur in the RTB survey of Table I. (In cases where i varied significantly whilst
the comet remained in a resonance the minimum value of i was chosen.)
Long­lived librations are fairly uniformly distributed over orbital periods less than the maximum
predicted from the RTB survey, suggesting that no particular values of P are favoured. Thus the
apparent clustering of known Halley­types about the 1:6 resonance (Emel'yanenko 1985) is probably
just a coincidence. However, very few librations are seen at resonances beyond the critical value of
P , and all of these occur in orbits that differ substantially from those used in the RTB survey, most
notably by having a larger perihelion distance, which tends to increase the maximum orbital period at
which librations can occur. These results strongly suggest that the mechanism described in Section 2
also operates to limit the types of orbits that are affected by resonances in the real solar system.
4 Discussion
In the previous sections we have seen that small librations of a cometary orbit about a mean­motion
resonance in the restricted three­body problem can be described by a simple linear mapping, and that
this mapping predicts that librations do not occur at resonances with large orbital periods. This result
is confirmed by numerical integrations using a more realistic model for the solar system. I now offer an
alternative viewpoint on the cause of instability for resonances with long orbital periods.
Consider a cometary orbit with a period exactly equal to a resonance with Jupiter (i.e. X 0 = XR ),
and a critical argument slightly displaced from exact resonance such that oe 0 ? oe R . In the absence of
planetary perturbations (A = 1 in Eqs. 6) the elements of the comet's orbit will remain unchanged,
since the comet and Jupiter each complete an integer number of orbits per mapping iteration.
Now introduce planetary perturbations. We begin by considering a resonance with small q, in which
case the perturbations are small compared to the orbital binding energy of the comet, and produce
only small changes in X. The resulting negative feedback between (X \Gamma XR ) and (oe \Gamma oe R ) produces
low­frequency librations about the resonance, similar to oscillations of a pendulum except that `gravity'
is not continuous but acts as a series of discrete kicks. After one mapping iteration the value of the
displacement joe \Gamma oe R j is less than the initial value---a pendulum dropped from rest never rises above
its initial height.
As we look at resonances with larger q the comet's orbital binding energy gets smaller, whereas the
planetary perturbations remain roughly constant for given values of X and oe. Hence the perturbations
increase in strength relative to the binding energy. This is equivalent to increasing the strength of
`gravity' in our discrete pendulum analogy, and thus increasing the size of the kicks. As long as the
11

relative strength of the planetary perturbations remains small enough that joe \Gamma oe R j does not exceed the
initial value, finite librations can take place, albeit of an increasingly discrete nature.
However, when q is large enough the relative strength of the planetary perturbations is sufficiently
strong (i.e A ! \Gamma1) that joe \Gamma oe R j exceeds its initial value after one mapping iteration, and hence
the `libration' increases in amplitude. This is like dropping a pendulum from a particular height and
having it come to rest at a greater height on the opposite side of the equilibrium point. A cometary
orbit initially close to the resonance now oscillates about it with exponentially increasing amplitude
and rapidly moves away. Thus we will not see a comet in the vicinity of such a resonance for more than
a few orbital periods.
The transition from stable to unstable librations occurs at an orbital period of roughly 200 years,
which coincides with the traditional and rather arbitrary boundary between Halley­type and long­period
comets. The fact that the former spend a substantial fraction of their lives in resonances, whereas the
latter do not, may result in significant differences in the physical and dynamical evolution of objects in
these two classes.
For example, a cometary orbit librating about a resonance undergoes different secular evolution than
one of similar period that is not in resonance (Chambers 1994). This may alter the fraction of these
objects that experience the sungrazing endstate as a result of the secular Kozai mechanism (Bailey
et al. 1992). Secondly, when a comet is in resonance with Jupiter it is prevented from undergoing
close encounters with that planet. This will prolong the dynamical lifetime of Halley­type orbits. In
addition, close encounters with Saturn assume a greater importance in regions of phase space containing
resonances---a fact that should be incorporated in Monte Carlo models for capture of Oort cloud comets
into short­period orbits.
These differences in the dynamics of resonant and non­resonant orbits have implications for the
physical evolution of Halley­type and long­period comets. The smooth secular changes in eccentricity
and the reduction in close­encounter frequency for comets in resonance will promote dust mantling
of their nuclei. Dust mantles are most likely to be shed following a sudden decrease in perihelion
distance (Rickman et al. 1991), which will occur less often for Halley­types than long­period comets.
This, coupled with the longer dynamical lifetimes of Halley­types (measured in terms of the number
of revolutions) implies that a larger fraction of Halley­type comets will be dormant than those of long
period. Thus, the currently observed Halley­types may be just the tip of the iceberg, with most objects
more closely resembling the ostensibly­asteroidal object 1991 DA.
Given that the presence of resonances is likely to be an important factor in determining the phys­
ical and dynamical evolution of a comet, the coincidence of the critical threshhold for librations with
the boundary separating short and long­period orbits now provides a justification for the previously
arbitrary distinction between these two classes.
Acknowledgements
Many thanks to Jane Arthur, Mark Bailey, Andrea Carusi, Fred Franklin, Will Henney, Franz Kahn,
Brian Marsden, Gonzalo Tancredi and Gareth Williams for helpful comments and discussions at various
stages in the preparation of this paper. This work was supported by a predoctoral fellowship at the
Smithsonian Astrophysical Observatory---thankyou.
12

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