Mon. Not. R. Astron. Soc. 324, 1109–1116 (2001)
Some properties of a two-body system under the influence of the Galactic
tidal field
R. BrasserP
Tuorla Observatory, 21500, Piikkiö, Finland
Accepted 2001 February 5. Received 2000 November 13
A B S T R AC T
In this paper the effect of the Galactic tidal field on a Sun– comet pair will be considered
when the comet is situated in the Oort cloud and planetary perturbations can be neglected.
First, two averaged models were created, one of which can be solved analytically in terms of
Jacobi elliptic functions. In the latter system, switching between libration and circulation of
the argument of perihelion is prohibited. The non-averaged equations of motion are integrated
numerically in order to determine the regions of the (e, i) phase space in which chaotic orbits
can be found, and an effort is made to explain why the chaotic orbits manifest in these regions
only. It is evident that for moderate values of semimajor axis, a , 50 000 au, chaotic orbits
are found for ðe , 0:15, 408 # i # 1408Þ as determined by integrating the evolution of the
comet over a period of 104 orbits. These regions of chaos increase in size with increasing
semimajor axis. The typical e-folding times for these orbits range from around 600 Myr to
1 Gyr and thus are of little practical interest, as the time-scales for chaos arising from passing
stars are much shorter. As a result of Galactic rotation, the chaotic regions in (e, i) phase space
are not symmetric for prograde and retrograde orbits.
Key words: celestial mechanics – comets: general – Solar system: general.
1
INTRODUCTION
In 1983, Byl investigated the role of Galactic perturbations on the
flux of comets that enter the inner Solar system, both analytically
and numerically. He found relations between the initial perihelion
position and the positions of successive perihelia, and that comets
with semimajor axis around 25 000 au make only a few passages
through the inner Solar system on a time-scale of 1 Gyr. Heisler &
Tremaine (1986) continued these studies by examining the effects
of the Galactic tide and of stellar encounters on the flux of comets
passing through the inner Solar system and coming from the Oort
cloud (Oort 1950). They based their work on Monte Carlo
simulations of comets in the cloud being perturbed by the Galaxy
and estimated the rate of comets entering the loss cylinder. Morris
& Muller (1986) concluded at the same time that Galactic
perturbations of the angular momentum vector of the comets would
create a steady stream of comets with an isotropic distribution.
Matese & Whitman (1989) studied the problem of the non-random
distribution of observed Oort cloud comets and concluded that the
work by Morris & Muller contained an error. However, all of these
papers concentrated on near-parabolic orbits and included only the
z-component of the Galactic tidal field, under the assumption that
the other components can be ignored because the z perturbation is
at least one order of magnitude stronger. In this paper we will
P
E-mail: [email protected]
q 2001 RAS
summarize some of the known results and will focus particularly
on chaotic orbits in the Oort cloud as induced by the Galaxy.
The aim of this paper is not to predict the flux of comets coming
from the Oort cloud that enter the inner Solar system. Rather, we
attempt to describe the motion of a comet or planet orbiting the Sun
with a semimajor axis of typically 50 000 au under the
perturbations of the Galactic tide. We also try to find any regions
in which this motion may be chaotic. We have neglected the actions
of passing stars and molecular clouds, as their actions are not the
purpose of this study. However, for typical Oort cloud comets with
semimajor axes of 30 000 au, the effects of passing stars and
molecular clouds are comparable to that of the Galaxy (Fernández
& Ip 1991). In that same paper, Fernández & Ip conclude that a star
passes through the Oort cloud with a frequency comparable to the
orbital frequency of the comet about the Sun. The action of such
stars is that the velocities of the comets are randomized with each
encounter, which will make the orbits chaotic on time-scales of the
order of a few orbits. These encounters also cause so-called ‘comet
showers’ (Fernández & Ip 1991). The action of molecular clouds is
to disrupt the Oort cloud of comets, which will randomize the
orbits even further. In this paper we shall concentrate on the action
of the Galaxy only.
In Section 2 the equations of motion are solved in two extreme
cases. In Section 3 we present two averaged models from which
some properties of the motion can easily be derived. Section 4
deals with the properties of the motion, and the differential
1110
R. Brasser
equation for the eccentricity as a function of time is derived and
solved in terms of Jacobi elliptic functions (Matese & Whitman
1989, 1992) using an averaged model. The equations of motion for
inclination and argument of perihelion can be derived from it. In
Section 5, the averaged models are abandoned and orbits are
integrated numerically using a non-averaged model to show which
regions are chaotic in eccentricity and inclination phase space. A
discussion concerning these regions follows in the same section.
We draw our conclusions in Section 6.
Consider a comet close enough to the Sun that the Galactic
influence can be neglected, but at the same time far enough away to
ignore the effects of the planets, in which case the motion is purely
two-body motion. However, when the comet is at Galactic
distances from the Sun, the influence of the Sun itself can be
ignored with respect to that of the Galaxy and the equations of
motion become
x€ 2 2Gr y_ 2 G2r x ¼ 2G11 x;
y€ 1 2Gr x_ 2 G2r y ¼ G22 y;
2
z€ ¼ G33 z:
B A S I C E Q UAT I O N S
We consider a system consisting of the Sun and a massless body in
the Galactic tidal field. The lighter body will be referred to as a
comet hereafter.
We derive three models to describe the perturbations of the
Galactic tidal field. Two models are simplified models in which
the perturbing potential is averaged over the mean anomaly M in
one case and over both M and the longitude of the ascending node,
V, in the second case. The Hamiltonian for a system consisting of
the Sun and a comet in inertial coordinates, (X, Y, Z ), centred on the
Sun and experiencing an external perturbing force is
M(
H ¼ 12 ðp2X 1 p2Y 1 p2Z Þ 2
1 UðX; Y; Z; tÞ;
ð1Þ
r
where U(X, Y, Z, t) is the perturbing potential, in our case the
Galactic tidal potential, which is a function of time arising from
Galactic rotation; ðpX ; pY ; pZ Þ are the momenta of the comet; M(
is the mass of the Sun, and r is the distance between the Sun and the
comet. The unit of distance is 1 au, one year is 2p and M( ¼ 1.
The influence of the Galactic tidal potential is best described in a
coordinate system that corotates with the local standard of rest and
with fixed axes centred on the Sun. The positive x axis is directed
towards the Galactic Centre, the positive z axis is directed towards
the North Galactic Pole and the y axis is perpendicular to these two,
such that z^ x^ ¼ y^ . The Galactic tidal potential in this coordinate
system is of the form (Heisler & Tremaine 1986; Matese, Whitman
& Whitmire 1999)
U ¼ 12 G11 x 2 1 12 G22 y 2 1 12 G33 z 2 :
ð2Þ
This system can be solved in terms of standard functions and is
just a harmonic oscillator with two frequencies, one of which is
hyperbolic. However, this is only true for a certain region in space
as the x component of the Galactic tidal force is repulsive and
beyond a critical distance in x the comet will escape from the Sun.
The distance at which this occurs is around 143 000 au, which can
be calculated by equating the potential arising from both the Sun
and the Galaxy in the x-direction and solving for x. Then we have
x ¼ ð2/ G11 Þ1=3 .
3
T H E AV E R AG E D M O D E L S
A typical Oort-cloud comet originating in the cloud has a <
30 000 au and hence the orbital period of the comet is several
millions of years. This is at least one order of magnitude shorter
than the oscillation period of the Sun about the Galactic midplane,
85 Myr, or its orbital period about the Galactic Centre, 240 Myr, so
that one can average the potential in equation (2) over the mean
anomaly M. The last term in equation (3) is vLz, where Lz is the
magnitude of the z component of the angular momentum vector
which can easily be expressed in terms of the usual orbital
elements, which are actually Galactic elements but will be
treated as normal orbital elements. Hence the quantity
U 0 ¼ U 1 vðx_y 2 y_x) will be averaged over the mean anomaly
M. With the values of ðG11 ; G22 ; G33 Þ as given in equations (4), this
results in
kU 0 l ¼
1 2
8 a G33
0
The new Hamiltonian, H , in this rotating coordinate system
becomes
1
ð3Þ
H0 ¼ 12 ðp2x 1 p2y 1 p2z Þ 2 1 Uðx; y; zÞ 1 vðxpy 2 ypx Þ;
r
where v is the angular frequency of rotation of the coordinate
system. The last term just states that the z component of the angular
momentum is a constant of motion. The equations of motion
derived from the Hamiltonian in equation (3) are those of the nonsimplified model, which we will call Model I from now on. For the
Galactic tidal potential (see e.g. Matese et al. 1999),
4p2
<
Myr22 ;
2402
G22 ¼ 2G11 ;
ð4Þ
4p2
G33 ¼ 24pGr0 < 2 Myr22 ;
85
where G is the gravitational constant and r0 is the local density of
matter. Its value is r0 ¼ 0:098 ^ 0:011 M( pc23 (Holmberg &
2
2
22
Flynn 2000) so that |G33 | < 4p
pffiffiffiffi/ffiffi85 Myr . The angular frequency
of rotation is of the order G11 and
pffiffiffiffiffiffiwill be called Gr hereafter.
Numerically we have taken Gr ¼ G11 .
sin2 i½2 1 e 2 ð3 2 5 cos 2vފ
1 58 a 2 G11 e 2 cos 2v cos 2Vð1 1 cos iÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 Gr að1 2 e 2 Þ cos i;
ð6Þ
1
0.9
0.8
0.7
e
G11
ð5Þ
0.6
0.5
0.4
0.3
0.2
0
5e+09
1e+10 1.5e+10 2e+10 2.5e+10 3e+10 3.5e+10 4e+10 4.5e+10
Time [yr]
Figure 1. The behaviour of the eccentricity of a sample orbit with a ¼
30 000 au and inclination starting at 808. Model I was used.
q 2001 RAS, MNRAS 324, 1109–1116
Some properties of a two-body system
Figure 2. The behaviour of the inclination of a sample orbit with a ¼
30 000 au and eccentricity starting at 0.3. Model I was used.
1
element V can, in principle, be eliminated from the problem.
However, it is still present in equation (6). The only way to
eliminate it is if one can average the quantity kU 0 l over V too. We
will show with the help of some figures that this procedure is
reasonable. Fig. 1 shows the behaviour of the eccentricity as a
function of time and Fig. 2 shows the same for the inclination. Both
figures were calculated using Model I. Figs 3 and 4 show the
comparison between Model I and Model II.
From these figures it can be seen that the typical time-scale for
the evolution of the orbit is the age of the Solar system, which is at
least three orders of magnitude longer than that of V, as the latter
processes with the same frequency as the rotation of our coordinate
system. The typical time-scale of evolution of the other orbital
elements changes with semimajor axis as Pe / a 23=2 . Hence we
can simplify the problem even further by averaging equation (6)
over V too so that our system has only one degree of freedom and
can be integrated analytically. The resulting quantity becomes
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kU 00 l ¼ 18 a 2 G33 sin2 i½2 1 e 2 ð3 2 5 cos 2vފ 1 Gr að1 2 e 2 Þ cos i:
ð7Þ
0.9
This we refer to as Model III and it will be used to analytically
derive some properties of the motion. It is convenient to express the
quantity kU00 l in terms of the usual Delaunay elements as these are
canonical. This results in
1
H2
G2
kU 00 l ¼ L 4 G33 1 2 2 2 1 1 2 2 ð3 2 5 cos 2vÞ
G
L
8
0.8
0.7
e
1111
0.6
0.5
0.4
0.3
0.2
0
5e+09
1e+10 1.5e+10 2e+10 2.5e+10 3e+10 3.5e+10 4e+10 4.5e+10
Time [yr]
Figure 3. Comparison between models I and II for the eccentricity. Starting
conditions are e ¼ 0:3, i ¼ 808, v ¼ 908, a ¼ 30 000 au. The solid line is
for Model I while the dashed line is for Model II.
2 Gr H;
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffi
where L ¼ a, G ¼ að1 2 e 2 Þ and H ¼ G cos i. The latter
quantity is a constant of motion. This model assumes that a_ ¼ 0 as
the potential is averaged over mean anomaly M. In reality, of
course, a_ – 0.
The perturbations induced by the x 2 y components of the
Galactic tidal field are effectively zero. The z component of
the Galactic tidal field induces Kozai resonance (Kozai 1962) in the
system, which is generally well known and its equations can be
solved analytically (Kozai 1962; Kinoshita & Nakai 1999). The
analytic solutions of the current problem are the same as those
given by Kozai (1962) and Kinoshita & Nakai (1999) with only
some modified constants.
4
PROPERT IES OF T HE MOTION
The quantity kU00 l is a constant of motion, so the motion of the
comet satisfies
C1 ¼ H
ð9Þ
2
1
H
C2 ¼ L 4 G33 1 2 2
G
8
Figure 4. Comparison between models I and II for the inclination. Starting
conditions are e ¼ 0:3, i ¼ 808, v ¼ 908, a ¼ 30 000 au. Model I is the
solid line while Model II is the dashed line.
where a is the semimajor axis of the comet orbit, e is its
eccentricity, i is its inclination, v is its argument of perihelion and
V is its longitude of the ascending node. All the elements are
heliocentric. This averaged model will be called Model II. The
rotation of our coordinate system causes the z component of the
angular momentum to be a constant of motion, so that the orbital
q 2001 RAS, MNRAS 324, 1109–1116
G2
2 1 1 2 2 ð3 2 5 cos 2vÞ ;
L
ð10Þ
where C1, C2 are arbitrary constants. There exists an upper limit of
H for librations of v to occur, which is derived from the equation
›kU 00 l
¼ 0;
ð11Þ
›G cos 2v¼21;e¼0
pffiffi
with the solution H 0 ¼ ð2 5Þ=5. This means that libration can only
occur in the region in (i, v) phase space for which H , H 0 . The
critical inclination, ic, for the orbit to liberate, is a function of v and
satisfies the relation
ÿ pffiffi
ð12Þ
ic ¼ arcsin 15 5 csc v :
1112
R. Brasser
For v ¼ ^908, ic ¼ 26:578. Note that in the case of Kozai
resonance, this critical inclination becomes ic ¼ 39:238 (Kozai
1962). This critical inclination can be observed from Figs 5 and 6.
These show the values of e versus v for various values of i. The
crosses show the evolution of v for i ¼ 108 and i ¼ 208 while the
solid lines are for i ¼ 308 – i ¼ 808. For i , ic, v circulates. In the
other cases it librates around its mean value of 908. Motion is
clockwise in the figures. From equation (12) one finds that no
libration is possible for v , 26:578, v . 153:448 and its
complementary values, a result noted by Matese & Whitman
(1989, 1992).
The libration amplitude of v can be obtained from the relation
v_ ¼ 0. This results in
cos 2v ¼
3G 4 2 5H 2
:
5ðG 4 2 H 2 Þ
ð13Þ
cos 2v oscillates between the values 21 and the value given in
equation (13). As can be seen from Fig. 6, there is a point in the
e 2 v plane where the libration amplitude is zero, a result found by
Heisler & Tremaine (1986). This only occurs for v ¼ ^908. Here
the Kozai mechanism is suppressed, meaning the perturbations
average out to zero. When i , if , where if is the critical inclination
at which no libration occurs, if is a minimum and e decreases with
time. Consequently, when i . if , if is a maximum and e will
increase with time. The value for if is a function of e and satisfies
the relation
sin2 if ¼ 15 ð1 1 4e 2 Þ:
Some more properties of the motion can be found from the
differential equation for G. We follow Matese & Whitman (1989)
to derive the equations. The energy integral, equation (10), can be
written as
1 4
4 L G33
sin2 i½x 1 5ð1 2 xÞ sin2 vŠ ¼ C;
2
ð15Þ
2
where x ¼ G / L and C is a constant. Define x0 the value of
equation (15) when v ¼ 0 and we have that
x0 ¼ x 1 5ð1 2 xÞ sin2 v:
ð16Þ
Note that 0 , x0 # 5 in these units. The equation for Ġ can be
transformed into one for ẋ, which is
x_ ¼ ^
4pP pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx0 2 xÞðx 2 x1 Þðx2 2 xÞ;
Tz
ð17Þ
where
x1 1 x2 ¼ 14 ð5 1 5u 2 x0 Þ;
x1 x2 ¼ 54 u;
1
ð14Þ
ð18Þ
and u ¼ H/ L, Tz is the oscillation period of the Sun about the
Galactic midplane and P is the orbital period of the comet.
Equation (17) can be rewritten as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x_ ¼ ^ ðx 2 x0 Þy;
ð19Þ
0.9
0.8
0.7
0.6
e
where y ¼ 4x 2 2 xð5 1 5u 2 x0 Þ 1 5u. The solution of equation
(17) is
!
2 xl 2 x1 2pP pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xg 2 x1 t ;
ð20Þ
;
xðtÞ ¼ x1 1 ðxl 2 x1 Þ sn
xg 2 x1 T 2z
0.5
0.4
0.3
0.2
0.1
0
20
40
60
80
100
120
140
160
180
ω
Figure 5. Eccentricity versus argument of perihelion with the initial values
for ðe0 ; v0 Þ ¼ ð0:2; 908Þ for values of i0 ¼ 108–808 in steps of 108. The
critical value of i0 for libration to occur is 26.68.
Pe ¼
1
0.9
T 2z K½ðxl 2 x1 / xg 2 xl ފ
;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pP xg 2 xl
ð21Þ
which scales as a 23/2.
There are four possible types of motion, according to the value
of x0, which we summarize below.
0.8
0.7
e
where (xg, xl) is the greater/lesser of (x1, x2). One can derive the
equations of motion for the other orbital elements from equation
(20) with the use of equations (9) and (10). Since the eccentricity in
this simplified model evolves sinusoidally, no switching between
libration and circulation of v is allowed. The period of one
oscillation in x(t) is
0.6
0.5
0.4
0.3
0.2
0
20
40
60
80
100
120
140
160
180
ω
Figure 6. Same as Fig. 4 but now e ¼ 0:8. Note that there exists a critical
point where the libration amplitude is zero.
(1) When x0 ¼ u, y has the roots x ¼ u and x ¼ 5=4. The latter is
physically impossible and hence the eccentricity is fixed and v
circulates. The inclination is zero in this case.
(2) When x0 ¼ 1, y has the roots x ¼ 1 and x ¼ ð5=4Þu. So either
x ¼ 1 always or it varies between 1 and ð5=4Þu for u # ð4=5Þ. The
equation u ¼ 4=5 is that of the limiting value of H. There is a
stationary solution for i ¼ 26:578 and v ¼ ^908. If u # ð4=5Þ, v
has to librate as it cannot make a complete revolution without
equation (17) staying real. Since the motion is real, no roots with a
non-zero imaginary part are allowed.
(3) When x0 [ ðu; 1Þ, the quadratic y has two solutions, one of
which is greater than 1. The argument of perihelion circulates and x
q 2001 RAS, MNRAS 324, 1109–1116
Some properties of a two-body system
5
250
200
ln |dR|
oscillates, with a minimum when v ¼ p and a maximum when
v ¼ 0.
(4) If x0 . 1 then cos 2v oscillates between 21 and the value
given in equation (13). The equation y ¼ 0 has two roots, both
between 1 and ð5=4Þu. Hence a real solution can only exist for
u # 4=5 and v librates.
1113
150
100
C H AO S
In order to investigate whether any orbits in the Oort cloud under
the influence of the Galactic tidal field show significant chaotic
behaviour, the variational equations of motion were integrated
together with the orbit.
The integrations were carried out using Model I over a period of
10 000 orbits. The equations of motion were integrated
numerically, using the logarithmic Hamiltonian method of
Mikkola & Tanikawa (1999). Their formulation gives correct
results for the two-body problem if the leapfrog symplectic
integrator is used. We have used the fourth-order leapfrog
symplectic integrator (see, e.g., Yoshida 1990) to integrate the
equations of motion derived from equation (3). The value of the
semimajor axis was set equal to 50 000 au while the initial value of
v was set at 08, 26.568, 608 and 908. The range of inclination was
divided into two sections: one set of integrations were for 08 #
i # 908 and another for 908 # i # 1808. In all cases, the set of
values for the eccentricity was taken to be 0 # e # 0:99.
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e
Figure 7. The value of ln |d r| versus starting eccentricity for a ¼ 50 000 au.
v ¼ 908. Prograde orbits.
5.1 Prograde orbits
Figs 7 to 18 show the value of ln |dr|, where |r| is the distance to the
Sun and |dr| is the variation of the distance, at the end of the
integration versus the initial value of the eccentricity or inclination
of the orbit (see the captions for details). Fig. 19 shows the value of
ln |dr| versus time of a sample orbit. From this, one can make a
rough estimate of the Lyapunov exponent and hence the
characteristic e-folding time of the chaos. As ln |d r| , lt for
chaotic orbits, hence
l ¼ lim
t !1
ln |d r|
:
t
Figure 8. The value of ln |d r| versus starting inclination for a ¼ 50 000 au.
v ¼ 908. Note the two sets of chaotic orbits for inclinations of 26.568 and
for 153.448, which are discussed in Section 5.3.
250
ð22Þ
Typical values are 1/ l ¼ ð600–1000Þ Myr in the system studied, as
obtained from the figures.
As can be seen from Figs 7 to 18, the regions for chaotic retrograde
orbits are, as expected, almost a mirror image of those for prograde
orbits. The slight asymmetry is caused by Galactic rotation, which
is particularly evident in the case for v ¼ 608 for large eccentricity.
For v ¼ 08, the ‘islands’ for large e are smaller and lower for
retrograde orbits than for prograde orbits. Retrograde orbits have a
higher relative velocity with respect to the rotation of the Galaxy
and hence the magnitude of the perturbations is smaller than for
prograde orbits.
5.3 Discussion of the chaotic orbits
The chaotic orbits are clustered in a parameter space roughly
defined by e , 0:1, 408 # i # 1408 for v ¼ ^608; ^908 and
e , 0:2, 408 # i # 1408 for v ¼ 08. The case for v ¼ 26:568 is
clearly different as the chaos does not seem to depend on the
q 2001 RAS, MNRAS 324, 1109–1116
ln |dR|
5.2 Retrograde orbits
200
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e
Figure 9. The value of ln |dr| versus starting eccentricity for a ¼ 50 000 au.
v ¼ 908. Retrograde orbits.
eccentricity. This fact is to be expected from equation (12), as any
orbit near the barriers v ¼ ^26:568 with high inclination must be
chaotic per se and it must be independent of eccentricity.
Contrary to the solution derived from the averaged theorem,
equation (7), switching between libration and circulation does
R. Brasser
250
250
200
200
150
150
ln |dR|
ln |dR|
1114
100
100
50
50
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0.1
0.2
0.3
0.4
e
0.5
e
0.6
0.7
0.8
0.9
1
Figure 13. The value of ln |d r| versus starting eccentricity for
a ¼ 50 000 au. v ¼ 26:568. Prograde orbits.
Figure 11. The value of ln |d r| versus starting inclination for a ¼ 50 000 au.
v ¼ 08.
Figure 14. The value of ln |d r| versus starting inclination for a ¼ 50 000 au.
v ¼ 26:568.
250
250
200
200
150
150
ln |dR|
ln |dR|
Figure 10. The value of ln |d r| versus starting eccentricity for
a ¼ 50 000 au. v ¼ 08. Prograde orbits.
100
100
50
50
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e
0
0.1
0.2
0.3
0.4
0.5
e
0.6
0.7
0.8
0.9
1
Figure 12. The value of ln |d r| versus starting eccentricity for
a ¼ 50 000 au. v ¼ 08. Retrograde orbits.
Figure 15. The value of ln |d r| versus starting eccentricity for
a ¼ 50 000 au. v ¼ 26:568. Retrograde orbits.
occur – see Figs 20 and 21. This behaviour is intrinsically chaotic
(Laskar 1990) because the switching occurs at random when the
evolution of the orbit is fastest, which occurs when i is a minimum
and hence when e is a maximum, i.e. this coincides when the
aphelion distance, Q ¼ að1 1 eÞ, is a maximum and the comet is
more susceptible to Galactic influence. From equation (20) and
Fig. 2 it is known that i is almost static for high values of i, which
we are interested in here, and then shows rapid evolution towards a
sharp minimum. The variations of the eccentricity are smoother as
it appears as e 2 in equation (9).
q 2001 RAS, MNRAS 324, 1109–1116
Some properties of a two-body system
1115
200
180
160
ln |dR|
140
120
100
80
60
40
20
0
0
Figure 16. The value of ln |d r| versus starting eccentricity for
a ¼ 50 000 au. v ¼ 608.
2e+10
4e+10
6e+10 8e+10
Time [yr]
1e+11
1.2e+11
Figure 19. The value of ln |d r| versus time. a ¼ 50 000 au. v ¼ 908, e ¼ 0
and i ¼ 788.
350
300
ω [˚]
250
200
150
100
50
0
0
Figure 17. The value of ln |d r| versus starting inclination for a ¼ 50 000 au.
v ¼ 608. The two sets of chaotic orbits for 31.108 and 148.908 are discussed
in Section 5.3.
2e+10
4e+10
6e+10 8e+10
Time [yr]
1e+11
1.2e+11
Figure 20. An example of switching between libration and circulation. For
this orbit, the starting values are ðe; i; vÞ ¼ ð0; 788; 908Þ and a ¼ 50 000 au.
350
250
300
250
ω [˚]
ln |dR|
200
150
200
150
100
100
50
50
0
0
0
0
0.1
0.2
0.3
0.4
0.5
e
0.6
0.7
0.8
0.9
1
Figure 18. The value of ln |d r| versus starting eccentricity for
a ¼ 50 000 au. v ¼ 608. Retrograde orbits.
As a !0, the chaotic orbits are found for e !0, i !908 because
with these conditions e !1 and i !08 at some time because
equation (9) holds. Furthermore, the libration regions of v are
larger for i !908 according to equation (12). Since e and i are
q 2001 RAS, MNRAS 324, 1109–1116
2e+10
4e+10
6e+10 8e+10
Time [yr]
1e+11
1.2e+11
Figure 21. Second case of switching between libration and circulation. For
this orbit, the starting values are ðe; i; vÞ ¼ ð0; 468; 908Þ and a ¼ 50 000 au.
coupled according to equation (9), and V processes almost with
constant speed, the only orbital element that can show chaotic
behaviour is v. An example of this fact is observed in Fig. 8, where
there are some chaotic orbits observed for i ¼ 268. It is again
confirmed in Fig. 17 for i ¼ 318, which follows from equation (12)
when v ¼ 608. This is an obvious phenomenon because from
1116
R. Brasser
equation (12) it is known that libration will occur for i . 26:568
when v ¼ ^908, but for any small changes in inclination it will
randomly switch between libration and circulation. In the limit
e ¼ 0, v is undefined, so that for small e large, random changes in
v may occur for only small perturbations, so that it may randomly
switch between circulation and libration. When i is also small, H is
maximal and hence the orbit will become near-parabolic.
Similar calculations have been done for a ¼ 30 000 au – not
shown here – but there was no significant chaotic motion for a
period of 104 orbits. As a is increased, the regions where chaos acts
will also increase, as a result of the stronger action of the Galactic
field.
6
CONCLUSIONS
We have derived three models in order to study the effects of the
Galactic tidal field on a comet in the Oort cloud, two of which are
simplified. We have also investigated chaotic orbits in the Oort
cloud.
By averaging the disturbing potential over the mean anomaly M
and then again over the angle V, the system is reduced to one
degree of freedom and can be solved analytically. Averaging over
M is plausible because the orbital period of the comet is short
compared to the orbital evolution of the other elements. A similar
argument applies for the longitude of the ascending node, V.
The typical evolution period of an orbit in the Oort cloud
(a < 30 000 auÞ is comparable to the age of the Solar system and
scales with a 23/2, if passing stars and molecular clouds are
ignored.
When v ¼ ^908, there is a certain value of inclination, if, for
given eccentricity, where the perturbations average out in the
averaged models and the orbit is static.
Libration of the argument of perihelion is only possible in the
region defined by i $ 26:578 and 26:578 # v # 153:448. The
libration region in the (i,v ) phase space is given by equation (12).
Chaotic orbits seem to be clustered around high inclination and
low eccentricity for a ¼ 50 000 au over a period of 104 orbits. This
region is roughly defined by ðe , 0:1, 408 # i # 1408Þ for v ¼
^608; ^908 and ðe , 0:2; 408 # i # 1408Þ for v ¼ 08. For the
critical value v ¼ 26:568, the chaotic orbits are found roughly in a
region 408 # i # 1408 and are independent of the eccentricity.
The regions in the (e, i) phase space where chaos is observed are
not the same for prograde and retrograde orbits. This asymmetry is
caused by Galactic rotation.
Chaoticity of the orbits seems to be a result of the random
switching between libration and circulation of the argument of
perihelion. This angle is less well defined for small e and undefined
for e ¼ 0.
The averaged models forbid switching between circulation and
libration of the argument of perihelion. In the real model, Model I,
this phenomenon is observed.
The regions where chaos is evident increase in the (e, i) space
towards larger e and smaller i for increasing a. This is a result of
Galactic perturbations being stronger at larger distances.
Characteristic e-folding times for the chaos in the Oort cloud at
a ¼ 50 000 au are 600 Myr to 1 Gyr, which is much longer than the
time-scale of randomization arising from passing stars (Fernández
& Ip 1991) and is therefore of low practical interest.
AC K N O W L E D G M E N T S
I am indebted to Seppo Mikkola for offering helpful comments and
suggestions and for providing a large part of the computer code. I
am also grateful for comments and suggestions provided by John
Matese, Mauri Valtonen, Jia-Qing Zheng, Chris Flynn and Pasi
Nurmi.
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This paper has been typeset from a TEX/LATEX file prepared by the author.
q 2001 RAS, MNRAS 324, 1109–1116