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The Chaotic Dynamics of Comets and the
Problems of the Oort Cloud
ROALD Z. SAGDEEV AND G.M. ZASLAVSK1Y
Institute of Space Research
ABSTRACT
This paper discusses the dynamic properties of comets entering the
planetary zone from the Oort cloud. Even a very slight influence of the
large planets (Jupiter and Saturn) can trigger stochastic cometary dynamics.
Multiple interactions of comets with the large planets produce diffusion of
the parameters of cometary orbits and a mean increase in the semi-major
axis of comets. Comets are lifted towards the Oort cloud, where collisions
with stars begin to play a substantial role. The transport of comets differs
greatly from the customary law of diffusion and noticeably decelerates the
average comet flow. The vertical tidal effect of the galaxy in this region of
motion is adiabatic and cannot noticeably alter cometary distribution. A
study of the sum of forces operating in the region to a ~ 104 AU does not
permit us to explain at this time the existence of a sharp maximum, where
a ~ 104 AU in the distribution of long-period comets. This is an argument
in favor of the suggestion that it was caused by the close passage of a star
several million years ago.
INTRODUCTION
The solar system's new object, the Oort cloud, arose as a source of
long-period comets (Oort 1950) in the planetary system's visible portion (r
< 2 AU). Experimental material generated by processing the trajectory of
a large number of long-period comets (Marsden et al. 1978; Marsden and
Roemer 1982) determined for these comets the region in which they exist,
which reaches a size of up to ~ 2105 AU. Oort proposed that collisions
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PI~4NETARY SCIENCES
with stars passing fairly close to the Sun may be one of the primary causes
for which comets attain the visible region. Research during the ensuing
years greatly complicated the Oort cloud model, inputting Hills cloud (Hills
1981) into the analysis (with an upper boundary of r ~ 2~104 AU) and the
action of various forces such as the galactic tidal effect (Hills 1981; Heisler
and Tremaine 1986; Morris and Muller 1986; Bailey 1986), collision with
molecular clouds (Biermann and Lust 1978; Hut and Romaine 1985), and
interaction with the planets (Oort 1950; Khiper 1951~.
Numerical analysis within the framework of the simplest, initial Oort
model demonstrated the possibility of qualitatively explaining the reason
for which comets enter the visible zone due to the effect of near stellar
passages (Wiesmann 1982~. Subsequent analysis showed that the action of
the galaxy's vertical tidal effect may somewhat modify Oort cloud and Hills
cloud parameters and the number of comets in them (Fernandez and Ip
1987; Duncan et al. 1988~. Oort cloud mass and momentum may fluctuate
more significantly if we assume certain, typical estimates for them, made
after processing the results of the Halley's comet mission (Marochnik et al.
1988~. The large mass of the Oort cloud (Mo ~ 100 Me, r > 2 104 AU)
must affect in the most serious way models of the formation of the solar
system.
It should be added here that the increase in the mass of the Hills cloud
must also bring about an increase in its angular momentum (Marochnik
et al. 1988~. This must be reconciled with the approximate equality of the
number of prograde and retrograde new comets. If it was not a question
of new comets, this equality would be a sufficiently obvious consequence of
the impact of random collisions of stars with comets with highly eccentric
orbits. However, numerical simulation, where the initial angular momen-
tum value of the cometary protocloud is taken into account, also reveals
the considerable impact it exerts on the size of the final Oort cloud and on
the number of comets in it (Lopatnikov et al. 1989~. The obvious reason
for this is tied to the different impacts of stellar collisions on circular and
eccentric orbits.
In accounting for the final angular momentum of cometary distribution
in the Hills and Oort clouds, an anisotropy is created In cometary dynamics
at virtually all of its stages. Anisotropy in the distribution of cometary
aphelia has been experimentally discovered (Delsemme 1987~. It indicates
the correlation between cometary distribution and the effect of galactic
tidal forces. We can consider that these forces exert an influence on both
cometary dynamics in the aphelion region (Heisler et al. 1987) and on
their dynamics in the planetary zone. Hence, all characteristic regions of
cometary orbits must participate in an interrelated manner in the formation
of cometary zones. This makes it necessary to analyze more carefully all of
the processes by which comets interact with the planets and the stars.
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AMERICAN AND SOVIET RESEARCH
261
The interaction of comets with the stars is statistical. Therefore, the
impact of individual collisions is averaged out, while the mechanism itself
by which they have an impact on the zone of cometary aphelia is weakly
dependent on individual details.
The passage of comets through the planetary zone is quite different.
The influence of the large planets, Jupiter and Saturn, on cometary dy-
namics has proven more subtle. A significant portion of comets (about
50%) which pass close to Jupiter are thrown into hyperbolic orbits as early
as the first passage. However, the phase space occupied by comets with
a perhelion ~ 2 AU is not very large. The phase space occupied by
comets with a perihelion similar to the radius of Jupiter's orbit is sub-
stantially greater. The phenomena of chaos may arise for such comets
(Petrosky 1986; Sagdeev and Zaslavskiy 1987; Petrosky and Broucke 1988;
Sagdeev et al. 19883. They consist of the following: in the strictly dynamic,
three-body problem (Sun-Jupiter-comet), the movement of the comets in
varying conditions becomes unstable. This instability is seen, in particu-
lar, in the fact that Jupiter's phase at the moment when a comet passes
through its perihelion is a sequence of random numbers. As a result, a
mechanism accelerating comets begins to operate. This mechanism is anal-
ogous to Fermi's method of stochastic acceleration (Sagdeev et al. 1988~.
It produces diffusive alteration of all of the comet's parameters, a mean
increase of the semi-major axis of cometary orbit, and the expulsion of the
comet from the solar system. The process of stochastization of cometary
movement is considered in detail in Natenson et al. (1989~. Numerical
analysis demonstrated that the region of cometary eccentricity values for
which chaos arises is very broad. Orbits with ~ ~ 0.5 may already become
stochastic. This circumstance should noticeably modify views of cometary
interaction with the planetary zone.
We will discuss below the conditions in which the dynamics of comets
with long-period orbits become stochastic, the role of such comets in the
overall model of the Oort cloud, and the influence of the galactic tidal
forces on cometary dynamics.
THE DYNAMIC CHAOS OF COMETS
We can generate a straightforward idea of this chaos by looking at
how a ball falls on a heavy plate in a gravity field (Figure 1, Zaslavskiy
1985) and if we consider their collision to be absolutely elastic. If the plate
oscillates with an oscillation amplitude of a and a velocity amplitude of v,
then on the condition that:
2v2 > ag,
(g denotes acceleration in the gravity field), the oscillation phase of the
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Representative terms from entire chapter:
cometary dynamics
262
/////////
PLANETARY SCIENCES
g
Q I
iV
FIGURE 1 Illustration of the action of the "gravitational machine" (Zaslavskiy 1985),
triggering the stochastic increase of the energy of a ball bouncing on a periodically oscillating
plate.
plate at the moment of collision is random. The ball bounces irregularly
over the plate and, on the average, rises increasingly higher. Its mean
energy at the moment of impact behaves asymptotically, as in ~
t2/3, and the mean height of lift correspondingly increases, as in ~
t2/3. The time for the ball to return back to the plate, naturally, increases.
However, the acceleration process does not stop.
This example somewhat clarifies what occurs with long-period comets
whose perihelion is in the sphere of influence of, for example, Jupiter.
Let M denote the orbital momentum of a comet, and ~,6 be the phase of
a comet's location relative to Jupiter in the comet's orbital plane. If the
orbits of Jupiter and the comet lie in the same plane (the so-called flat,
limited three-body problem), the Hamiltonian for the comet is equal to
H = 2 (
AMERICAN AND SOVIET RESEARCIl
263
unit, we (Jupiter's rotational frequency). In this two-dimensional motion,
= ~ swat, and a Jacoby motion integral exists:
Go = H—swim.
(2)
Using ratio (2), we can make problem (1) a two-variable, nonstationary
problem. We can select a canonically coupled pair (M, ˘) as the variables.
Let, for example, In denote the moment in time when the comet passes
through its aphelion, Mn be the orbital momentum during passage through
aphelion, and In denote the value of the Jovian phase during cometary
passage through perihelion (preceding the time to). The relationship be-
tween the values (Mn+~, An+ and Mn, Any then defines the expression
(Petrosky 1986; Sagdeev and Zaslavskiy 1987)
Mn+~ = Mn + /\M Sin~bn
An+ = An + Bra w`Ei i,
(3)
where AM denotes fluctuation in the orbital momentum over one cometary
rotation, while the comet's energy, En is determined using the motion
integral (2~:
En=Go+~w'Mn
(4)
The variables (Mn, An) are canonically coupled. Expression (3) is only
defined in the region of negative values of a comet's energy, H = E < 0,
that is, according to (2)
Go ~ swim < 0.
(5)
Inequality (5) is violated and expression (3) becomes meaningless when a
comet is thrown into hyperbolic orbit. lithe value AM in (3) is defined by
the expression
/\M = man /
/~ln+1 AH
(6)
where tn. and tn+l denote the moments of time of two sequential passages
of the apocenter by a comet. Estimates of the value of AM in varying cases
are provided in Petrosky 1986; Sagdeev and Zaslavskiy 1987; Petrosky and
Broucke 1988; Natenson e! al. 1989.
Expression (3) has a frequently encountered form, described in detail
by Sagdeev et al. (1988) and Zaslavskiy (1985~. If a comet does not pass
too far from Jupiter, the duration of its interaction with Jupiter is on the
order of a Jovian period of 2~rh~ I. This time scale determines the duration
264
PLANETARY SCIENCES
of a "collision." It is a great deal less than the time period between two
"collisions" for long-period comets of 2,r/w(E). In view of this circumstance,
we can write a simple form of expression (3), in which the second formula
simply describes alteration of the ~ phase during the time between two
sequential collisions. A similar expression also occurs for the model with
the ball in Figure 1. The velocity, v, plays the role of a generalized pulse,
M, while the collision frequency is proportional to lo. In this case, w(E)
= ~2E~3/2-
We can produce a straightforward assessment of the stochastic dynam-
ics of the comet for problem (3) from the condition that (Sagdeev et al.
1988; Zaslavskiy 1985~:
K = ~ o˘~+i—1~ > 1. (7)
This gives us
I; = 27r 2`E' ~ f'( )~ · ~AMCos˘~ > 1. (8)
Since the perihelion changes little as a result of the collision (AM
<< M), while the comet's rotation frequency w(E) = ~ 2E~3/2 , O where
HER ~ O. condition (8) can be fulfilled for comets with sufficiently low
binding energy BIER ~ O) with a fixed value M. The phase portrait of
cometary movement, corresponding to the Hamiltonian (1), with a fixed
Jacoby integral value is in Figure 2 (Natenson et al. 1989~. It was produced
for true trajectories and demonstrates the complex structure of phase space
with a large number of regions of stability. The region of global chaos is
defined by the estimate in (7) and (8~. A boundary with ~ ~ 0.55 and a
semi-major axis of a ~ 16.5 AU corresponds to this. The comet's perihelion
originally had a value of q ~ 7.5 AU.
Points in the region of global chaos in Figure 2, that is where O <
E < - 0.03 AU, belong to one trajectory. If any initial condition in this
region is selected, with the same Jacoby integral value, the corresponding
movement of a comet will also be stochastic. This is where the significance
of this region of stochasticity is manifested.
Nvo important consequences stem from the results of study (Natenson
et al. 1989~. The region of chaos is very significant, and even Halley's comet
enters the zone where conditions of chaos are applicable. An analogous
comment regarding Halley's comet, based on the use of representation
(3), was made by Chirikov and Vecheslavov (1985~. The region of chaos
in Figure 2 also applies to medium-period comets. Therefore, the phase
magnitude of comets with stochastic dynamics is an order more than the
phase magnitude of comets appearing in the visible portion with r < 2 AU.
AMENCAN AND SOVIET RESEARCH
O~r
— .U.. ,; -,, `, ... ~..........
;_·r~ · f ~ · ~ · - ! ~ il ~ ^.·~ ~ ~ ~-.~ · .~ ; ~ ~ ~ .-
-~. · ~ · ~ : . .. ·t . . ~ ~ ~ r. ~
. . ,~, .~. . .~ ~ ~. - . . . ·_ , ; - ~ .;. .~ ..i - · . ~ '~~:I. ! : J
I. ~ ~ ' ~. ~r' ~ ;. .;; · . ~, r~: ~ ~; ~; ; ·- .. ;;
·~ ~ ·~ 4. .~'! .' ·'~ '~ J ' ~ ~ . :1; :'. ~ .n .~ . . .
~~~ , ~ ;x . ~ ~ , _ , - ~_ ~ ~ ~ ~.~. . ; . . ~ ~ i~ 1t . ~ -r~ - ~ ~ I
:~ ' ~; f .;~; ':~ ~ ,- . r~;; ·.! .- : ~ ~ ~ ·'4r' ~ .; .~; '; ~. ; ' ;~, · -:~ -
' ~ ~ ; ~ ~~ ; ~ r - ~ .~: ~ t. ' r . . . ·..1 ~ ~ . '. 1 .. ~ .,
~ , -~ ·l -;~~ ·.'~ J2I ~~ ; I.! .~ .~.~ · ~ <,.-.; ' ~:J:.;.: ~ -.' ,4~ ':,: I , ~ :,';. ~;,
_~.!~. ~_ ~ ~ ·-..ri"~ ;!y! - -~ ~`.' · r~ ~ ~ ~
,~ ~_; ~ 8. · I · ._.~~ . ;. . ,. ~,~ ; ·^,.;~ ~ ~ ·i ~ :,,) -.f;
; 4'~-' ~ ' ~'--I- ~ ~ 18~-~;-- ^~ . c: ~ ICi~' .*2 ~ ~;;~ ;. ,~ v. .,.·! _~.
. -. · ~~ Cy~ ,r ~ ~ · ~ ·r ~ : · ~ ~ K~; ~ ~'.s ~:4 . ; ~lr ~ . r · ~ ~c · ~ ~ ~ i ._
·~j ;~,,r,~~3~ .~,1~ ~ ~.~r~·ierr~.~;~4 ~L~r.~.~; .- ~~r~,r~--~-+ - :~ ,T.—, ,^.~.'
. ·. ~ l.;i ·- - ·. ~ .~. —~ . . ~ ~ s ~ —~t'~ ~; '. ~,~--4 —-
—0.025 - ;`i~ ' -- I ~` .;.~` ~,,,, . ~ ~ . ~ ,.,;
FIGURE 2 Phase portrait on the trajecto~y plane (˘, E = - 1/2a) of a comet. The orbit
points are plotted at the moment in time when a comet passes through the aphelion point.
For the sake of convenience, the portrait has been broken into two parts.
266
PLANETARY SCIENCES
The second consequence is related to the nature of cometary diffusion.
This process is extremely important, since it is the mechanism by which
comets attain the region of semi-major axes. The usual diffusion formula
(Yabushita 1980)
At
~f(M,t) 1 ~ ng, ~ ( )
is only justified in a region sufficiently remote from the chaos boundary.
The impact of the chaos boundary and regions of stability (see Figure 2)
significantly decelerates diffusion at the initial stage in comparison with the
diffusion defined by formula (9) (Natenson et al. 1989~.
GENERAL COMETARY DYNAMICS AND
THE INFLUENCE OF THE GALACTIC TIDE
The existence of a mechanism of dynamic chaos makes it necessary
to reconsider the general dynamics of comets as they move from the Oort
cloud to the visible zone. The customary route is that collisions with stars
operate in the zone of aphelion of a comet's orbit. Those comets in the
loss cone region enter the visible zone, originally having a ~ 104 AU.
Jupiter's influence throws about one half of these comets into hyperbolic
orbit. Only a small portion of the comets may subsequently return again
directly to the loss cone, in order to set out on the new path from the Oort
cloud to the visible zone.
However, another avenue also exists.
A rather large portion of comets, those that first entered the invisible
zone and have a perihelion comparable to the radius of Jupiter's orbit,
enter the region of stochastic dynamics. The comet begins a long, diffusive
path to the loss cone region. Therefore, an independent way of filling the
loss cone is defined.
Other large planets of the solar system may also play the same role
as Jupiter. Therefore, the portion of comets moving stochastically should
be insignificant. The planetary "barrier," expelling part of the comets,
concurrently makes the dynamics of others stochastic, thereby providing
their route to the loss- cone. At the same tune, the cometary perihelion
changes very slightly, and the comets' orbits are as if "attached" in the zone
of their perihelion.
Vertical tidal galactic forces must play an important role in the process
of cometary stochastic transport described above (Bahcall 1984; Heisler
and Romaine 1986)
Fz = l~rmGp`,z,
(10)
AMERICAN AND SOVIET RESEARCH
2~7
where m denotes comet mass, G is the gravitational constant, ps = 0.186
m<~/pc3 is stellar density, and z is the vertical coordinate in the galactic sys-
tem of coordinates. The tidal force (10) considerably alters the perihelion.
Therefore, it creates the drift of comets from the planetary zone during
one phase of cometary orbit and, inversely, causes cometary perihelia to
flow into the planetary barrier in another orbital phase. These two fluxes
are approximately equal.
We will find the region, on the nonadiabatic influence of the tidal force
Fz, from the condition that perhelion variation under its influence must be
fairly strong. We will assume 10 AU as an example of the characteristic
size of the planetary zone. Then the nonadiabatic condition means that
/`q>lOAU,
(11)
where Aq denotes perihelion alteration under the influence of Fz in a
period of cometary orbit. For eccentric orbits
Aq~ M2/2m2meG.
Thus
Assuming
/\q~ MAM/m2m:,G.
/\M ~ FzaT~ 47rmGp5a2T~ rlmc>Ga2T/pc3,
after substitution of all of these expressions in (11), we yield:
a > 104AU.
This estimate (Duncan et al. 1988) can also be clarified using a more
careful input of numbers. However, it is clear that the effect of tidal
forces in the region of a significant portion of cometary orbits is adiabatic.
Therefore, the tidal forces cannot substantially alter cometary distribution
in the region < 104 AU. However, they exert considerable influence on
the near-boundary processes, where the planetary barrier operates, and
along the border of the Oort cloud, where effective collisions with stars fill
the loss cone (Fernandez and Ip 1987; Duncan et al. 1988~.
CONCLUSION
The dynamic chaos of comets with fairly eccentric orbits, moving in
the Sun's field and perturbed by the fields of Jupiter and Saturn (or by the
fields of other remote planets), forces us to reconsider individual elements
268
PLANETARY SCIENCES
of the Oort cloud theory. Cometary stochastization creates a mechanism by
which comets move away from the planetary belt towards the Oort cloud,
and may be an additional source by which the loss cone is filled. The
process of diffusive cometary transport differs from the usual process of
diffusion and retards the characteristic time scale for the flux of comets
toward the semi-major axes a. The action of the tidal forces does not
alter this time scale significantly and is adiabatic (with the exception of the
regions near the belt of the large planets and near the inner boundary of
the Oort cloud). Therefore, the internal mechanisms of cometary dynamics
cannot explain the existence of a sharp maximum in the distribution of
the observed comets from the Oort cloud (Marsden et al. 1978; Weismann
1982) with a period on the order of several million years. This gives us
reason to suggest that the reason for the appearance of such comets may
have been the last near-Earth passage of a star. This conclusion correlates
with the conclusions of studies Biermann et al. 1983; Lust 1984) on the
possible passage of a star or another large object in the region of cometary
orbit with a ~ 104 AU, triggering the appearance of a coherently moving
cometary cluster.
The global modeling of the dynamics of long-period comets must
include the multiple interactions of comets with the large planets, if the
perihelion of the comets does not greatly exceed the radii of planetary
orbits. These issues and the existence of asymmetry in cometary cloud
distribution are discussed in greater detail in Lopatnikov et al. (1989~;
Natenson et al. (1989~.
ACKNOWLEDGEMENT
The authors wish to thank L.S. Marochnik, NI. Neishtadt, and P.
Veismann for their useful comments.
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