MNRAS 437, 1420–1433 (2014)
doi:10.1093/mnras/stt1974
Advance Access publication 2013 November 8
Saturn Trojans: a dynamical point of view
X. Y. Hou,1‹ D. J. Scheeres2 and L. Liu1
1 School
of Astronomy and Space Science, Nanjing University, Nanjing 210093, China
of Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309, USA
2 Department
Accepted 2013 October 11. Received 2013 October 7; in original form 2013 September 4
ABSTRACT
Different from the usual approach in the inertial frame, the stability problem of fictitious Saturn
Trojans is studied in the synodic frame in this paper. First, some numerical facts are shown
to allow us to simplify the force model. Then, motion equations centred at the geometrical
triangular libration points for the planar S-JS model are derived. Using these equations, the
resonance mechanism that causes the instability is studied. We confirm the opinion that the secular resonances and the near-commensurability between the libration frequency and the great
inequality are the reasons to cause the instability of motions close to the triangular libration
point. By studying the survivability of the long period family and the short period family in
the S-JSUN model, the planar stable region far away from the triangular libration point is
studied. By frequency analysis of the orbits in the stable region, we are able to find two secular
resonances associated with the boundary of the stable region. Three-dimensional motion is
also discussed, by starting with the survivability of the vertical period family in the S-JSUN
model. The secular resonance that causes the orbit inclination restriction on the Trojans is
qualitatively discussed. Lastly, the effects of planetary migrations are briefly studied. With
the contribution in this paper, a global picture of the dynamics around the triangular libration
points in the Sun–Saturn system perturbed by Jupiter is presented.
Key words: minor planets – asteroids: general.
1 I N T RO D U C T I O N
Trojans are a special kind of small bodies in the Solar system. They
are trapped around the triangular libration points of the restricted
three-body systems formed by the Sun and the major planets, and
some of them may survive a very long time. The first object of this
kind Achilles (588) was found in 1906 in the Sun–Jupiter system by
Wolf. Recently, with the discovery of Trojans in the Sun–Neptune
system (Brasser, Mikkola & Huang 2004), the Sun–Uranus system (Alexandersen et al. 2013) and even in the Sun–Mars system
(Mikkola et al. 1994) and the Sun–Earth system (Connors, Wiegert
& Veillet 2011), a renewed interest in the Trojans arises. Many
works have been performed to explain the origin of these observed
Trojans. To list a few, please see Marzari, School & Farinella (1996),
Marzari et al. (1997), Gomes (1998) and Morbidelli et al. (2005).
Interestingly, till now no Saturn Trojans have been found. This
seems weird because Saturn is the second largest planet in the
Solar system. If we neglect the possible origin from the evolution
history of the early Solar system and just study the problem by
the present state of the planets, perturbations from Jupiter may be
the culprit to this phenomenon. This should be the first study of
this kind (Innanen & Mikkola 1989; Mikkola & Innanen 1992;
E-mail: [email protected]
Holman & Wisdom 1993). By integrating the orbits of the four
outer planets and the fictitious Trojans simultaneously, they found
an unstable ‘hole’ in the distribution of fictitious Saturn Trojans:
Trojans initially put exactly at the triangular libration point or very
close to it escape very quickly, while the ones far away can stay
longer or even be stable. By slightly changing the semimajor axis
of Jupiter or Saturn, this phenomenon does not occur. This leads
the authors to believe that the near 2:5 orbital resonance between
Jupiter and Saturn (also known as GI – great inequality) might be
the reason to this phenomenon. In later works, de la Barre, Kaula &
Varadi (1996) used their own simplified model to show that not only
the GI but also the secular resonance between the procession rates
of Saturn and the Trojans can cause the instability. In Nesvorný
& Dones (2002), using the so-called ‘bi-circular’ model, the chaos
caused by the overlap of the GI and the 1:1 mean motion resonance
between the Trojans and Saturn are to be blamed for the unstable
zone. In Marzari & School (2000), another secular resonance 2g6 −
g5 is also found to cause the instability of the motion. Besides the
planar instability, the orbital inclinations of the long-lived fictitious
Saturn Trojans are generally smaller than a critical value. In Marzari
& Tricarico (2002), the secular resonance is to be blamed for this
phenomenon.
Reading through these works, it is striking to us that nearly all
the works intending to find the reasons of instability are performed
in the Sun-centred inertial frame, by numerically integrating or
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Saturn Trojans: a dynamical point of view
analytically analysing the complete or the simplified system, obtaining their proper elements and studying their possible resonances
(Milani 1993; Beaugé & Roig 2001). This is understandable because there is no doubt that the proper elements are a powerful tool
to study the long-time dynamics of the Solar system (Milani 1993;
Murray & Dermott 2000). But they are not the only tool. For the
specific case of Trojans, a different approach is taken in our work.
We did not use the concept of proper elements (although they can
be directly obtained from our results), and we studied the problem
in the synodic frame instead of the usual inertial frame. The inspiration of this paper is the work of the Barcelona group on the
triangular libration points of the Earth–Moon system perturbed by
the Sun (Gómez et al. 2001), where the problem is also studied in
the synodic frame.
In our study, we first study the unstable region close to the triangular libration point and confirm the opinion that both the GI and the
secular resonances (g6 and 2g6 − g5 ) contribute to the instability.
Then, we study the stable regions far away from the triangular libration point. Two secular resonances associated with the size of the
stable region are found. After that, the vertical motions are studied.
At the end, the effects of planetary migrations are briefly studied.
The aim of this paper is to qualitatively explain the unstable and
stable regions of the Saturn Trojans but not to get their accurate
sizes. With the efforts in this paper, we believe that a better picture of the global dynamics around triangular libration points in the
Sun–Saturn system perturbed by Jupiter is presented.
2 S O M E N U M E R I C A L FAC T S
For studies on small bodies in the outer Solar system, it is a convention to ignore the inner planets’ orbits and simply add their masses
to the Sun. In our studies, we follow this convention. We integrate
the orbit of a fictitious Saturn Trojan along with the orbits of the
four outer planets (Jupiter, Saturn, Uranus and Neptune) in the Suncentred sidereal coordinate. We call this model as S-JSUN. The
initial conditions of the four outer planets are taken from the numerical ephemeris DE 406, corresponding to the epoch of J2000.0.
If we only consider Jupiter and Saturn, we call this force model as
S-JS. For the planar case studied below, we simply set the initial orbital inclinations of Jupiter and Saturn as zero in the ecliptic frame.
The numerical integrators used in our study are the usual RKF-78
one and the multistep one described in Quinlan & Tremaine (1990).
For short time integrations (∼Myr), we usually use the RFK-78
integrator and believe that it is accurate enough. For longer time integrations (∼Gyr), we use the multistep one. Throughout the paper,
unless we specify, the mass unit is the sum of the masses of the Sun
and four inner planets, and the length unit is 1 au.
2.1 Planar region
Initial conditions of fictitious Saturn Trojans are given in the following way: they have the same instantaneous orbital elements as
Saturn except for the semimajor axis and the mean anomaly. The
semimajor axis of the Trojans is between 9.08 and 10.08 au, with a
step size of 0.02 au. The difference of the mean anomaly between
the Trojans and Saturn is between 3◦ and 180◦ , with a step size of
3◦ . The left-hand frame of Fig. 1 shows the results of an integration
time of 0.5 Myr, while the right-hand frame shows the results of
5 Myr. The dots in the frames indicate the Trojans which survive
the integrations. With a longer integration time, the size of the stable
region shrinks a little bit but retains the main features: a big ‘hole’
1421
Figure 1. Planar stable regions in the S-JSUN model. The left-hand frame is
for an integration time of 0.5 Myr. The right-hand frame is for an integration
time of 5 Myr.
Figure 2. Planar stable regions in the planar S-JS model (left) and the
simplified model (right) for 0.5 Myr.
exists in the stable region. The criterion for an orbit to be unstable
in our work is that it crosses the x−z plane.
Since Saturn Trojans are relatively further away from Uranus
and Neptune and these two planets have smaller masses, omitting
them in the integration does not change the results too much . By
‘omitting’, we also mean that the two planets have no effects on the
orbits of Jupiter and Saturn. Since the orbit inclinations of Jupiter
and Saturn in the ecliptic frame are also small, they are simply taken
to be zero in our studies. The left-hand frame of Fig. 2 shows the
planar region corresponding to this model, for an integration time
of 0.5 Myr. The shape agrees with the result of the S-JSUN model
in a perfect way, which means that the planar S-JS model may be a
good choice to study the dynamics of Saturn Trojans.
We further simplify the force model by neglecting the joint perturbations between Jupiter and Saturn (i.e. their orbits are invariant
ellipses) but considering their perturbations on Saturn Trojans. In
the following, we call this model the simplified model. Again, an unstable hole is found in the planar region, as shown in the right-hand
frame of Fig. 2. Since the orbits of Jupiter and Saturn are fixed in this
case, we cannot expect a planar region exactly similar to the one of
the planar S-JS. But the feature of a big hole is retained, which is an
implication that the mutual perturbation between Jupiter and Saturn
may not be the reason or at least not the sole reason that caused the
inner unstable hole. Besides, the difference between the two frames
at the outer part may be caused by the secular resonances which
can only appear when the mutual perturbations between Jupiter and
Saturn are considered. This phenomenon agrees with the following
studies.
2.2 Vertical region
Now we concentrate on the vertical motions. Initial conditions of
fictitious Saturn Trojans are given in this way. They have the same
instantaneous orbital elements as Saturn except for the semimajor
axis, the orbit inclination and the mean anomaly. The mean anomaly
of the Trojans is fixed, with a value of 60◦ ahead of Saturn. The
1422
X. Y. Hou, D. J. Scheeres and L. Liu
3 DY N A M I C A L E Q UAT I O N S
From above results, we know that the planar S-JS model is a very
good approximation to the complete S-JSUN model. Although the
simplified model is simpler and also preserves the key dynamical
properties of Saturn Trojans, we may lose some details due to its
incoherency. As a result, in the following, we focus on the planar
S-JS model. In a Sun-centred sidereal frame, the small body in this
model follows
Figure 3. Vertical stable regions in the S-JSUN model. The left-hand frame
is for an integration time of 0.5 Myr. The right-hand frame is for an integration time of 5 Myr.
R̈ = −μC R/R 3 − μJ (J /3J + RJ /RJ3 )
−μS (S /3S + RS /RS3 ),
(1)
where μC , μJ , μS are gravitational constants of the Sun, Jupiter
and Saturn. R, RJ , RS indicate the position vectors of the fictitious
Trojan, Jupiter and Saturn. J = R − RJ and S = R − RS . Denoting their companions in the Sun-centred synodic frame rotating
with Saturn as r, r J , r S , δ J , δ S , we have
r̈ = −2CT Ċ ṙ − CT C̈ r
− μC r/r 3 − μJ (δ J /δJ3 + r J /rJ3 )
− μS (δ S /δS3 + r S /rS3 ),
Figure 4. Vertical stable regions in the planar S-JS model (left) and the
simplified model (right) for 0.5 Myr.
semimajor axis of the Trojans is between 9.08 and 10.08 au, with a
step size of 0.02 au. The difference of the orbit inclination between
the Trojans and the Saturn is between −30◦ and 30◦ , with a step size
of 2◦ . Fig. 3 shows the results for the complete S-JSUN model. The
left-hand frame corresponds to an integration time of 0.5 Myr, while
the right-hand frame corresponds to 5 Myr. With longer integration
times, the maximum orbit inclination reduces from 20◦ to about
15◦ .
We also performed these numerical integrations in the planar
S-JS model (left-hand frame in Fig. 4) and the simplified system
(right-hand frame in Fig. 4). Again, except for a few points, the
planar S-JS model agrees with the complete S-JSUN model very
well. While the simplified model (the right-hand frame) has more
or less some differences from the left-hand frame, the basic feature
is retained: the maximum orbit inclination is restricted.
Combing the numerical results in the above two subsections, we
know that the planar region close to the triangular libration point
is forbidden for stable motions. The orbit inclinations are also restricted. In the following, we will explain these phenomena. We have
to mention that the above numerical results are far from a complete
numerical survey of the stable regions of fictitious Saturn Trojans.
First, the orbit eccentricity, the longitude of the ascending node
and the argument of the perigee of the Trojans remain unchanged;
Secondly, for Figs 3 and 4, we restrict M = 60◦ ; Thirdly, the
initial configuration of Jupiter and Saturn is fixed; Fourthly, the
integration time of 5 Myr may be too short compared with the age
of the Solar system. Nevertheless, we satisfy ourselves with these
results. First, a complete numerical survey takes a prohibiting long
time; Secondly, the results after a very long integration might not
be trustable; Thirdly, the features of the stable regions such as the
‘hole’ structure and the orbit inclination restriction agree with the
ones in previous works by other authors; Fourthly, we do not aim
to identify the exact stable regions of fictitious Saturn Trojans. This
section is only presented to state some dynamical features of planar
and three-dimensional motions and to introduce the force models
that we are going to use in our studies.
(2)
where C is the matrix transforming a vector in the synodic frame
to the sidereal frame (Gómez et al. 2001). Denote the position of
the triangular libration point in the Sun-centred synodic frame as r 0
and write r as
r = r 0 + ρ.
(3)
We can rewrite equation (2) as (Hou & Liu 2010)
ρ̈ = F 1 + F 2 ,
(4)
where
⎧
T
3
T
3
⎪
⎪ F 1 = −2C Ċ ρ̇ − C C̈ρ − μC (r/r − r 0 /r0 )
⎪
⎪
⎪
⎨
− μJ δ J /δJ3 − δ 0J /(δJ0 )3 − μS δ S /δS3 − δ 0S /(δS0 )3
⎪
F 2 = −μC r 0 /r03 − r̈ 0 − 2CT Ċ ṙ 0 − CT C̈ r 0
⎪
⎪
⎪
⎪
⎩
− μJ δ 0J /(δJ0 )3 + r J /rJ3 − μS δ 0S /(δS0 )3 + r S /rS3 .
(5)
The two new variables in equation (5) are defined as
δ 0J = r 0 − r J ,
δ 0S = r 0 − r S ,
(6)
indicating the vectors from Jupiter and Saturn to the triangular
libration point. It is a plain fact that F 1 can be expanded as a
formal series of ρ, while F 2 is irrelevant to ρ. Due to the term F 2 ,
the triangular libration point is no longer a dynamical equilibrium
point.
Equation (5) can be further simplified. Denoting the companion
of r 0 in the sidereal frame as R0 , we have
R0 = D RS ,
D = Rz (−60◦ ),
(7)
where Rz ( · ) denotes a rotation matrix along the z-axis. Since
R̈0 = D R̈S
= D −(μC + μS )RS /RS3 − μJ SJ /3SJ − μJ RJ /RJ3 ,
(8)
where SJ = RS − RJ , we have
−r̈ 0 − 2CT Ċ ṙ 0 − CT C̈ r 0 = −CT R̈0
= CT D (μC + μS )RS /RS3 + μJ SJ /3SJ + μJ RJ /RJ3 .
(9)
Saturn Trojans: a dynamical point of view
where the matrix E = B−1 (AB − Ḃ) is a constant matrix. After
solving equation (15), we immediately have the solution to the
original system by X = B · Y . Usually, finding the matrix B is a
laborious process (Jorba, Ramı́rez & Villanueva 1997; Hou & Liu
2011), especially when the system has many basic frequencies. In
this work, we do not pursue the exact form of B, but only need to
know that for the problem at hand, it should be of the following
form
For the planar case, we have CT D = DCT . As a result, we can
reduce F 2 to
F 2 = μJ D
δ0
rJ
δ SJ
rJ
+ μJ D 3 − μJ 0J 3 − μJ 3 .
3
δSJ
rJ
(δJ )
rJ
(10)
4 S TA B I L I T Y A N A LY S I S
B = I6×6 + B ,
Equation (4) describes fully the motion of a small particle around
point L4 in the planar S-JS model. Since F 1 is a function of ρ
and equals zero when ρ = 0, point L4 is still an equilibrium point
without the external perturbation term F 2 . We first study the system
without the term F 2 . The term F 1 can be rewritten as
F 1 = −2CT Ċ ρ̇ − CT C̈ρ +
∂
(1 + 2 + 3 ),
∂ρ
J
S
wshort = 0.033 9003,
(12)
wlong = 0.001 4867,
wvertical = 0.033 9635.
S
where the three angles C , J , S are the angles between r 0 , δ 0J , δ 0S
and −ρ. Denoting ρ = (ξ, η, ζ )T and X = (ρ, ρ̇), the linear form
of equation (11) is
Ẋ = A · X,
where the 6 × 6 matrix A can be easily obtained from equations
(11) and (12). Under the assumption that Jupiter and Saturn are
in quasi-periodic motions, the matrix A is also quasi-periodic. For
the planar S-JS model, there are four basic frequencies: the mean
angular velocity of Jupiter θ̇J , the procession rate of Jupiter’s perigee
ω̇J , the mean angular velocity of Saturn θ̇ S and the procession rate of
Saturn’s perigee ω̇S . With the Fast Fourier Transform (FFT) analysis
of Jupiter’s and Saturn’s motions, these four basic frequencies are
found to be
θ̇S = 0.033 9491
ω̇J = 0.000 0031,
ω̇S = 0.000 0217.
Ẋ = A · X + (0, F 2 )T .
(14)
(18)
Equation (15) in this case is changed to
According to the quasi-Floquet theory (Jorba & Simó 1996),
a quasi-periodic matrix B can be found and the transformation
X = B · Y can transform equation (13) to
Ẏ = B−1 (AB − Ḃ)Y = EY ,
(17)
We use ‘short, long and vertical’ to indicate that they come from
the short period term, the long period term and the vertical period
term of the unperturbed case. We notice that wshort is very close
to θ̇S − 2ω̇S + ω̇J = 0.033 9089 (also θ̇S − ω̇S = 0.033 9274) and
wlong is very close to −(2θ̇J − 5θ̇S + 3ω̇S ) = 0.001 0700.
Now, we concentrate on the term F 2 which is obviously the
culprit of the instability zones in Figs 1 and 2. The term F 2 also has
the four basic frequencies in equation (14). FFT analysis is used to
analyse its components. Table 1 shows the three frequencies with
the largest amplitudes and some other specific frequencies. The first
two columns are for the x component and the last two columns are
for the y component. Judging from Table 1, the term F 2 itself is
very small. The question is why such a small term can cause the
instability of the motion.
First, we add the term F 2 to equation (13):
(13)
θ̇J = 0.084 3052,
(16)
where the matrix B is the perturbation matrix caused by Saturn’s
orbit eccentricity and Jupiter’s gravitation. It is a small quantity
compared with the unitary matrix. Obviously, for the circular restricted three-body problem (CRTBP), B ≡ 0.
The system ρ̈ = F 1 is stable, which means that the linearized
system equation (13) should be stable. Integrating an orbit with
equation (13) and frequency analysing the orbit, we can get the
basic frequencies associated with the system equation (13) (also
equation 15). The three basic frequencies are
(11)
where (under the assumption that ρ < r0 , δJ0 , δS0 )
n
⎧
ρ·r 0
μC
ρ
1
⎪
=
=
μ
+
Pn (cos C ),
⎪
1
C
n≥2
3
r
r0
r0
⎪
r0
⎪
⎪
⎨
n
ρ·δ 0
2 = μJ δ1J + (δ0 )J3 = μδ0J n≥2 δρ0 Pn (cos J ),
⎪
J
J
J
⎪
⎪
n
⎪
0 ⎪
⎩ 3 = μS 1 + ρ·δS = μS n≥2 ρ
Pn (cos S ),
δS
(δ 0 )3
δ0
δ0
1423
Ẏ = B−1 (AB − Ḃ)Y = EY + B−1 · (0, F 2 )T ,
(19)
−1
where 0 = (0, 0, 0) . As stated before, the matrix B can be approximately taken as the unitary matrix. Unless F 2 has exactly
T
(15)
Table 1. Several components of the perturbation term F 2 .
Frequency
cos
sin
cos
sin
θ̇J − θ̇S
2(θ̇J − θ̇S )
3(θ̇J − θ̇S )
θ̇S − ω̇S
θ̇S − 2ω̇S + ω̇J
2θ̇J − 5θ̇S + 3ω̇S
θ̇S − ω̇J
θ̇S + ω̇S − 2ω̇J
2θ̇J − 5θ̇S + 4ω̇S − ω̇J
2θ̇J − 5θ̇S + 5ω̇S − 2ω̇J
2θ̇J − 5θ̇S + 2ω̇S + ω̇J
2.2613 × 10−6
7.6932 × 10−6
−7.032 × 10−7
1.80 × 10−8
−2.53 × 10−8
−6.68 × 10−8
−7.150 × 10−7
−3 × 10−10
−6.01 × 10−8
−2.70 × 10−8
−2.8 × 10−9
−3.517 23 × 10−5
7.9582 × 10−6
7.9457 × 10−6
2.662 × 10−7
1.57 × 10−8
8.7 × 10−9
−4.729 × 10−8
−2.11 × 10−8
1.68 × 10−8
2.47 × 10−8
−1.99 × 10−8
4.070 91 × 10−5
1.337 37 × 10−5
8.8552 × 10−6
−1.987 × 10−7
−3.6 × 10−9
−5.15 × 10−8
−6.734 × 10−7
1.90 × 10−8
−4.33 × 10−8
−3.40 × 10−8
2.11 × 10−8
−2.057 78 × 10−5
−4.5865 × 10−6
3.8536 × 10−6
2.342 × 10−7
−1.4 × 10−9
−7.14 × 10−8
−8.406 × 10−7
−1.98 × 10−8
−4.57 × 10−8
−1.57 × 10−8
−1.49 × 10−8
1424
X. Y. Hou, D. J. Scheeres and L. Liu
Figure 5. Local details of the frequency map of a quasi-periodic orbit by integrating equation (18).
the same basic frequencies as equation (17), the perturbed linearized model described by equation (18) is still stable. Nevertheless, if F 2 has frequencies close to the basic frequencies in
equation (17), large excursions due to small denominators can be
expected. These large excursions, when non-linear terms are considered, may cause the instability of the motions. Fig. 5 shows the
local details of the frequency map of a quasi-periodic orbit by integrating equation (18). The initial point of this orbit is put static
at the triangular libration point. The components containing wlong
and wshort are caused by the long and short period components.
Their amplitudes can be arbitrarily chosen and are irrelevant to
F 2 . Other components denoted in Fig. 5 are obviously forced terms
caused by the term F 2 . Considering the very small amplitude of F 2 ,
the amplification caused by small denominators is obvious. Even
adding the second-order terms of F 1 [i.e. non-linear terms O(X2 )] to
equation (18) will cause the instability of the triangular libration
point. This means that the non-linear effects are necessary to onset
the instability.
Next, we consider the complete F 1 by adding specific components of F 2 . For the first three rows in Table 1, although they have
larger amplitudes, adding them to the system does not cause any
instability. (The stability is determined by integrating the motion
equations by putting the Trojan initially static at the triangular libration point. If the Trojan crosses the x−z plane within 1 Myr,
it is unstable. Otherwise, it is stable.) For the 4th–6th rows, no
matter which one of them is added to the system, the instability
arises within 1 Myr. For the rest of the rows in Table 1, although
the frequencies are also very close to wshort (or wlong ) and very
large excursions are also excited, the instability does not occur.
This is interesting because some of these terms even have amplitudes larger than those in the 4th–6th rows. Our explanation for this
phenomenon is that this is a complex non-linear system. Whether
the Trojan will be scattered out of this system depends not only on
the closeness of the perturbing frequency to wshort (or wlong ) but also
on its coefficients and their interactions with the non-linear terms
of F 1 .
One remark should be made here. From equation (16), we know
that the main part of matrix B−1 is the unitary matrix. However,
there is also a small perturbation part with various frequencies.
Multiplying this perturbation part of B−1 with the terms in F 2 of
frequencies different from θ̇S − ω̇S , θ̇S − 2ω̇S + ω̇J or 2θ̇J − 5θ̇S +
3ω̇S may also generate frequencies close to wshort and wlong , but
they are generally smaller and may not be strong enough to scatter
the Trojans to the unstable region or the scatter process requires a
much longer time.
For the big unstable ‘hole’ in the stable region in Figs 1 and
2, we reach our conclusions here: the near-commensurability between some frequencies of the perturbing term F 2 and the two
planar basic frequencies of the non-perturbed system ρ̈ = F 1 cause
large excursions to motions originally near the triangular libration
points and gradually sends them to the unstable region and at last
kicks them out of this system. Not all the components in F 2 with
frequencies close to wshort and wlong cause instability. Our results
confirmed two secular resonances (g6 and 2g6 − g5 , where g5 and
g6 are ω̇J and ω̇S in this paper) found by other authors, and the
near-commensurability between the long period frequency (1:1 resonance libration frequency) and one of the GI frequencies. See the
discussions part for more details.
5 P L A N A R S TA B L E R E G I O N
The planar motions of Trojans are composed of forced components
(the ones that have only the frequencies in equation (14) as the basic
ones) and free components (the ones that contain the long period
component or the short period component or both). Due to the fact
that some forced frequencies are very close to those of the free components, most of the orbits, even stable, are chaotic. This prevents
us from constructing a valid global analytical solution. Nevertheless, one heuristic opinion is that the general phase structure is not
distorted too much by Jupiter compared with CRTBP. This prompts
us to study the survivability of some Kolmogorov-Arnold-Moser
(KAM) curves of CRTBP in the S-JSUN model. Two basic kinds of
planar KAM curves of CRTBP are naturally the long period family
and the short period family. In the following, we will study the
survivability of these two families.
5.1 The long period family
Nearly all the stable orbits we find in the Sun–Saturn system perturbed by Jupiter describe the shape of a banana, resembling the long
period orbits around the triangular libration points in the CRTBP
model. A primitive thought is that some of the long period orbits survive the perturbations and become these stable ones in the S-JSUN
model. To test this thought, we study the survivability of the long
period orbits by directly integrating them in the S-JSUN model.
One argument to justify our approach is Rabe’s work (Rabe 1967).
Saturn Trojans: a dynamical point of view
1425
Figure 6. Left-hand frame: T–C curves of the long period family and some of the families B(k + l, k + l + 1). Right-hand frame: the tested orbits that survive
the perturbations for 1 Myr.
In his work, by expanding the motion equations around a mother
long period orbit, he analytically described the maximum allowable
short period displacements. Here, the short period displacements
are replaced by the perturbing term F 2 . Besides, another difference
is that we do not fix the long period orbit and vary the perturbations.
We fix the forced perturbations and numerically study the allowable
long period orbit that can hold this perturbation.
First, we have to compute the long period family around point
L4 in the CRTBP model, with the Sun and Saturn as primaries. The
long period family terminates on to a short period orbit travelling
k times (for the Sun–Saturn system, k = 23). Starting from this
k-bifurcation short period orbit, a series of periodic families B(k +
l, k + l + 1) (l ≥ 0) consecutively appear. See Hou & Liu (2009) for
more details. The left-hand frame of Fig. 6 shows the T −C curves
of the long period family and some of the families B(k + l, k +
l + 1). The short period family, the long periodic family and the
periodic families B(k + l, k + l + 1) form the backbone of various
planar periodic families around triangular libration points (Hou &
Liu 2009). Taking the distance between the Trojan and the triangular
libration point (when it is at the far side of the line connecting the
Sun and L4 ) as the family parameter λ, the right-hand frame of Fig. 6
shows the T −λ curves of the same families in the left-hand frame.
To show the details of the following results, only part of the curves
is given. We have to mention that the unit ‘adim’ in Fig. 6 has a
different meaning from previous figures. It means the conventional
non-dimensional units for the restricted three-body problem. That
is to say, the length unit is the mean (or instantaneous) distance
between the Sun and Saturn, the mass unit is the sum of the masses
of the Sun and Saturn, and the time unit is chosen such that the
gravitational constant G = 1 (Szebehely 1967).
We notice that for most members in these families with a large λ,
they are exponentially unstable, i.e. chaotic. A heuristic opinion is
that orbits with a large λ (even the stable ones in CRTBP) are sensitive to perturbations and are not likely to survive the perturbations.
We performed a numerical test under the planar S-JS model. We
did not test all the members of the families but just some of them
because otherwise too much computation time would be required.
The tested orbits are of smooth shapes and all of them are stable
in the CRTBP model. These orbits correspond to those computed
by Rabe (1961, 1962). They are expressed by the thick curve (not
the horizontal lines) in the right-hand fame. The thick dots on the
thick curve indicate the orbits that survive the perturbations (for an
integration time of 1 Myr). This is how we do the numerical tests: in
the synodic frame centred at point L4 , we denote the state vector of
the Trojan in CRTBP (when it is at the far side of the line connecting
the Sun and L4 ) as X = (ρ̄, ρ̄˙ ). Denote the instantaneous distance
between the Sun and Saturn as [L]. Instead of the mean distance
between the Sun and Saturn, we use [L] as the length unit (i.e. we
take it as an elliptic restricted three-body problem). As a result, we
have
ρ = ρ̄[L],
ρ̇ = ρ̄˙ [L] + ρ̄[L̇].
(20)
Then, we use equation (20) as the initial condition and integrate the
orbit to see whether it can survive the perturbations.
According to Érdi (1988), the amplitude of the long period orbit
is an index of the semimajor axis of an orbit with orbit eccentricity
close to that of Saturn. In fact, we can take [L](1 ± λ) as an approximation of the maximum and minimum of the semimajor axis of
the Trojan. In this meaning, Fig. 6 agrees with Figs 1 and 2 where
we keep the orbit eccentricity fixed (to the value of Saturn) but vary
the semimajor axis of the Trojans.
Denote the region surviving the perturbations in Fig. 6 as [λmin ,
λmax ]. The exact values of λmin and λmax vary with different initial
relative geometry of Jupiter and Saturn. In order to get different
initial relative geometry between Jupiter and Saturn, we integrate
only the orbits of Jupiter and Saturn for a time T. Obviously, the
integration time T can be taken as a parameter describing the relative
geometry between Jupiter and Saturn. The left-hand frame of Fig. 7
shows the values of λmin and λmax for different initial phases of the
two planes (expressed by the integration time of the two planets).
Similar to Fig. 6, the unit of the ordinate is the instantaneous distance
between the Sun and Saturn. The time to generate the left-hand
frame of Fig. 7 is 1 Myr. The reasons that cause the vibrations in
λmin and λmax are explained as follows: the parameter λ of the tested
orbits is given in the CRTBP problem while we are dealing with
the ‘real’ force model, for which Saturn’s orbit eccentricity and
Jupiter’s gravitation will cause perturbations to these values. This
is somewhat similar to the way of using instantaneous Keplerian
elements to describe a perturbed ellipse. As a result, the parameter
λ is an instantaneous element instead of a mean one. It of course
vibrates quasi-periodically in the ‘real’ force model but can still be
used as an index of the magnitude of the long period component.
Note that although λmin and λmax vibrate for different initial relative
geometry between Jupiter and Saturn, λmax − λmin remains nearly
1426
X. Y. Hou, D. J. Scheeres and L. Liu
Figure 7. Left: values of λmin and λmax for different initial relative geometry between Jupiter and Saturn; right: values of λmin and λmax for different integration
times.
Figure 8. Left: local details of the frequency map around the short period frequency for four orbits; right: local details of the frequency map around the long
period frequency for one of the four orbits (denoted as λ1 in the left-hand frame).
unchanged. This will be addressed in the following section for
vertical motions.
The values of λmin and λmax also vary with different integration
times. Obviously, with longer integration times, λmin will increase
and λmax will reduce. This will reduce the size of the stable region
λmax − λmin . A natural question is whether it is possible that λmax −
λmin → 0 for a very long integration time. Previous studies (Melita
& Brunini 2001) showed that there are fictitious Saturn Trojans with
a lifetime comparable to the age of the Solar system. So, we think
that the stable region does not approach zero. To demonstrate this,
we performed the following computations. For a fixed integration
time, we obtain the values of λmin and λmax . Then, we gradually
increase the integration time to see how the values of λmin and λmax
change with the increasing integration time. The right-hand frame
of Fig. 7 shows the corresponding results. Even with an integration
time of 1 Gyr, the stable region does not approach zero and the
extent of reduction is small. Besides, with longer integration times,
the reduction speed approaches zero (i.e. λmin or λmax approaches a
constant).
Now, we turn our attention to the mechanism that causes the
lower and upper boundaries of the stable region. Even the stable
region does not approach zero with increasing integration times,
nearly all the orbits in the stable region between λmin and λmax are
chaotic. The reason for the chaos lies in the very close values of the
free frequencies and the forced ones. The overlap between various
resonances causes the chaos. Many of these orbits, even chaotic,
can stay around the triangular libration points for a very long time
(at the scale of the age of the Solar system), causing the well-known
phenomenon – stable chaos (Milani & Nobili 1992). The left-hand
frame of Fig. 8 shows the local details of the frequency map (around
wshort ) for four stable orbits, with λ1 = 0.010 0038, λ2 = 0.012 0561,
λ3 = 0.014 0910 and λ4 = 0.016 0115. The unit of the ordinate in
Figs 8–12 is the instantaneous distance between the Sun and Saturn.
The chaos around the short period component for all the four orbits
is not obvious. However, for each one of them, the frequency map
around the long period component shows chaos (Laskar, Froeschlé
& Celletti 1992). Taking λ1 as an example, the right-hand frame
of Fig. 8 shows the details around the long period component.
The reasons for this different behaviour around the short period
component and the long period component lies in the fact that the
side peaks around the long period component is much larger than
the ones around the short period component (Hou, Scheeres & Liu
2013).
What will happen when λ approaches λmin or λmax . From Fig. 8,
we notice that with increasing λ, the short period frequency wshort
moves from θ̇S − 2ω̇S + ω̇J to θ̇S − 3ω̇S + 2ω̇J . A natural opinion is
Saturn Trojans: a dynamical point of view
1427
Figure 9. Local details of the frequency map around the short period component (left) and the long period component (right) for an orbit close to the inner
boundary of the stable region.
Figure 10. Local details of the frequency map around the short period component (left) and the long period component (right) for an orbit close to the outer
boundary of the stable region.
that when λ approaches λmin , wshort is closer to θ̇S − 2ω̇S + ω̇J ; and
when λ approaches λmax , wshort is closer to θ̇S − 3ω̇S + 2ω̇S . Fig. 9
shows the local details of the frequency map for λ = 0.007 5094,
which is very close to λmin in Fig. 7. We see that the resonance
effect is so strong that different resonances wlong ± i(ω̇S − ω̇J ),
i = 1, 2, 3, . . ., which are close to the long period components
overlap with each other. This makes the phase space around the
long period component completely chaotic. The completely chaotic
structure may be the reason that causes the instability for λ < λmin .
Fig. 10 shows the corresponding results for λ = 0.017 7031, which
is very close to λmax in Fig. 7. Similar phenomenon happens. This
time, the overlap of resonances around the long period component
is so severe that we cannot even tell which should be the long period
frequency. In generating these figures, a sample of 222 points, with
an interval of 4 yr was used.
The above three figures show the process of resonance overlap
with varying long period amplitudes. Now, we qualitatively describe the resonance overlap process with increasing short period
amplitude. Introduce the parameter β which is defined as the norm
of the coefficients corresponding to the frequency wshort of the x
component. Fig. 11 shows local details of the frequency map of an
orbit with a very small β. In 222 × 4 yr, the orbit appears regular.
Fig. 12 shows the corresponding results for an orbit of a relatively
larger β. The effects of chaos, which is not obvious when β is small,
appears now.
Two remarks should be made here. The first one: For the inner
boundary of the stable region, it is really hard to say definitely that
it is determined by the resonance between wshort and θ̇S − 2ω̇S +
ω̇S , because the other two resonances (between wshort and θ̇S −
ω̇S , and between wlong and 2θ̇J − 5θ̇S + 3ω̇S ) also contribute to the
instability in the inner regions. Judging from the right-hand frame in
Fig. 2, the resonance between wlong and 2θ̇J − 5θ̇S + 3ω̇S seems to
play the dominant role. But from the studies in Section 4, the other
two resonances also play their roles. Maybe a prudent statement is
that their joint effects determine the inner boundary of the stable
region. The second remark: Although the two secular resonances
2ω̇S − ω̇S and 3ω̇S − 2ω̇J are believed to be associated with the
lower and upper boundaries of the stable region, they may not be
the resonances that directly cause the instability of the Trojans. The
mix of them with the GI frequency (see Robutel & Gabern 2006;
Hou et al. 2013) may be the direct culprit. Unfortunately, we are
unable to identify these mix resonances or make a definite statement
1428
X. Y. Hou, D. J. Scheeres and L. Liu
Figure 11. Local details of the frequency map of a ‘regular’ orbit.
Figure 12. Local details of the frequency map of a chaotic orbit, with a relatively larger β.
about this. But one statement is true: the inner and outer boundaries
of the stable region are closely connected with these four resonances
– three secular resonances and one secondary resonance.
We reach our conclusions here: the stable region of Saturn Trojans
is nearly chaotic everywhere due to the very close values of perturbing frequencies and the free frequencies. With increasing long
period component λ, the frequency wshort changes its value accordingly due to the non-linear effects. When λ approaches the lower
boundary of the stable region λmin , wshort approaches θ̇S − 2ω̇S + ω̇J
(also ω̇S ). The resonance width becomes so wide that the long period frequency and nearby frequencies overlap. This makes the
phase space completely chaotic and provides an efficient way for
Trojans to leave this system, as what happens to the outer boundary
of stable regions in the Sun-Jupiter system (Hou et al. 2013). Similar mechanism applies when λ approaches the upper boundary of
the stable region λmax and wshort approaches θ̇S − 3ω̇S + 2ω̇J . For
λ lies in the middle between λmin and λmax , the amplitude of the
short period libration is restricted. It cannot be too large, or otherwise the resonance width also increases and similar phenomenon
happens as λ approaches λmin or λmax . For all the stable orbits we
computed, a limit value around 0.05 exists. Considering the fact that
the amplitude of the short period component is approximately the
free orbit eccentricity (see the discussion part) and adding Saturn’s
orbit eccentricity to the free orbit eccentricity, the orbit eccentricity
of Saturn’s Trojans seems to be smaller than 0.1, a conclusion in accordance with Nesvorný & Dones (2002) and Marzari & Tricarico
(2002). One additional conclusion is that for small amplitude motions close to the geometrical triangular libration point, the nearcommensurability between wlong and 2θ̇J − 5θ̇S + 3ω̇S , and between
wshort and θ̇S − ω̇S also contributes to the instability of the motions,
which is already stated in Section 4.
5.2 The short period family
Same numerical simulations have been performed with the short
period family. In the CRTBP model, the short period family terminates on to a planar Lyapunov orbit around the collinear libration
point L3 and all the members in this family are stable (Hou & Liu
2009). However, none of them can survive the perturbations. Fig. 13
shows the λ–frequency curve for small to moderate amplitudes of
this family in the CRTBP model. The unit of the abscissa is the mean
distance between the Sun and Saturn, which means that the proper
orbit eccentricity can be as large as 0.5. The basic frequency of the
short period family moves towards the frequencies θ̇S − 2ω̇S + ω̇J
with increasing short period amplitude, which means that the strong
resonance between wshort and θ̇S − 2ω̇S + ω̇J (also θ̇S − ω̇S ) always
Saturn Trojans: a dynamical point of view
1429
Figure 13. λ−T curve of the short period family in the CRTBP model.
exist. We believe that it is the reason which causes the instability of
the whole short period family in the real force model. The difference
between this case and the above case lies in the long period component. In the above case, the amplitude of the long period component
is large enough to drive wshort far away enough from θ̇S − 2ω̇S + ω̇J
to avoid strong resonances, only if it is not too large to draw wshort
to another frequency θ̇S − 3ω̇S + 2ω̇J .
6 V E RT I C A L S TA B L E R E G I O N
Now, we concentrate on the vertical motions of fictitious Saturn
Trojans. We performed the same thing with the vertical period
family as we did with the long period family and the short period
family. Again no vertical period orbit can survive the perturbations.
This is weird because in the CRTBP model, most members of the
vertical period family are stable and the stable ones can reach very
large out-of-plane amplitude (Hou & Liu 2008). In our opinion, the
instability is not caused by the vertical motion itself, but by the
planar motion. From discussions in the above section, we know that
the planar instability mainly depends on how close the frequency
wshort is to θ̇S − 2ω̇S + ω̇J or θ̇S − 3ω̇S + 2ω̇S . Obviously, due to
the non-linear effects, wshort varies with long period amplitudes,
short period amplitudes and vertical period amplitudes. The general
dynamics, exempting the regions where resonances happen, should
differ little from that of the CRTBP model. Denote, respectively, the
amplitudes of long period component, the short period component
and the vertical period component as λ, β and γ . The meanings of
the parameters λ and β are already given in previous sections. The
parameter γ is defined as the norm of the coefficients corresponding
to the frequency wvertical of the z component. The short frequency,
similar to the CRTBP model, can be expanded as a literal series of
these amplitudes. Truncated at the second order, this relation can be
written as (Lei & Xu 2013)
0
+ Aλ2 + Bβ 2 + Cγ 2 ,
wshort = wshort
(21)
= 0.033 9024,
A
=
−0.029 7089
and
where
B = −0.000 6367. These values are obtained as follows: we
randomly choose 300 planar orbits with λ between λmin and λmax ,
analyse the short period frequency and the short period amplitude
β. Then, we numerically fit the relation between wshort and the two
0
, and A and B. This is somewhat
parameters λ and β to get wshort
awkward because β is the mean element (an index of the proper
orbit eccentricity), while λ is an instantaneous parameter. The
reason for this awkward choice lies in the chaotic nature around
the long period component. For most orbits, we cannot obtain the
0
has a
accurate value of wlong and its associated amplitude. wshort
0
wshort
Figure 14. Coupling between the planar and the vertical motions, and the
allowable region for the vertical motion.
value different from that in equation (17) because this is a fitted
value. (The relation is truncated at the second order and more
importantly λ is an instantaneous parameter.) Similarly, judging
from Fig. 13, it is not possible for B to be a negative value, even
in the perturbed Sun–Saturn system. This means that the accuracy
of the coefficients A and B is only of the order of 10−3 . As a
0
= 0.033 9003, A = −0.023 and B = 0. The
result, we set wshort
coefficient C, judging from the results of the CRTBP model, should
be positive and small. Theoretically speaking, we can also obtain
it by numerically fitting the orbits with vertical displacements.
However, under such a bad accuracy of fitting, we cannot trust the
fitted result. We arbitrarily take C = 0.0001, λmin = 0.008 and
λmax = 0.018 as an example. Fig. 14 shows two curves. They are
2
0
min
Aλ + Cγ 2 = wshort
− wshort
,
(22)
0
max
2
2
Aλ + Cγ = wshort − wshort ,
0
0
max
min
= wshort
+ Aλ2min and wshort
= wshort
+ Aλ2max are
where wshort
lower and upper boundaries for the planar stable region. The unit
in Fig. 14 is the instantaneous distance between the Sun and Saturn. The shadowed area is the allowable region for vertical motions.
From the right-hand frame of Fig. 7, we know that λmax reduces with
longer integration times, so does the upper boundary of γ . This
explains the reduction of orbit inclination restriction with longer
integration times in Fig. 3. For the vertical period family whose
long period component λ ≈ 0 < λmin , its members are not in the
shadowed region and cannot survive the perturbations. Fig. 14 also
explains another feature of Fig. 3: the maximum inclination is obtained at the outer boundary of the semimajor axis (corresponding
to larger values of λ). This figure also agrees with the results in
Marzari & Tricarico (2002): when the orbit inclination (the vertical
amplitude) increases, the planar frequency wshort approaches the
resonance curve of θ̇S − 2ω̇S + ω̇J . From Fig. 7, we know that the
value of λmax − λmin is generally fixed for different initial relative
geometry of Jupiter and Saturn, so the orbit inclination restriction
does not heavily depend on the initial relative geometry of Jupiter
and Saturn.
One remark should be made here. The shadowed region in Fig. 14
closely depends on the values of C, λmin and λmax . The two parameters λmin and λmax can be obtained accurately from Fig. 7, but not
the parameter C. As a result, Fig. 14 is somewhat arbitrary. A different set of values may give a different shape, but it will not change
qualitatively as long as C > 0.
1430
X. Y. Hou, D. J. Scheeres and L. Liu
7 P L A N E TA RY M I G R AT I O N
Nowadays, a commonly accepted opinion is that the outer planets in
our Solar system are not formed at their present positions. They have
undergone different extents of migrations in the early Solar system
(Malhotra 1995; Tsiganis et al. 2005). According to the planetary
migration models, present Jupiter Trojans were captured during
the migration process (Morbidelli et al. 2005; Lykawka & Horner
2010). If this is true, Saturn should also be able to capture its Trojans
during this process. However, no Saturn trojans haven been found
till now. This may be due to the longer distance (and corresponding
worse observation conditions) than the Jupiter Trojans, but the late
stage of the planetary migration process may be the more intrinsic
reason.
To study the effects of the late stage of the planetary migration
process, the following numerical simulations were performed. We
still use the S-JSUN model. The initial conditions of the four planets are still given by the numerical ephemeris DE 406 at the epoch
J2000.0. Except the semimajor axis of Saturn, we keep the other
orbital elements of the four planets unchanged. Denote the semimajor axis of Saturn given by DE 406 at the epoch J2000.0 as aS0 . The
actual semimajor axis of Saturn is chosen as aS = k · aS0 , where k is
smaller than but very close to 1. Fig. 15 shows the longperiod orbits surviving the perturbations (for an integration time of 1 Myr).
The upper frames, from left to right, correspond to k = 0.9982
and 0.9987. The lower frames, from left to right, correspond to
k = 0.9991 and 0.9995.
The dashed vertical line in these frames approximately indicate
the place where the secular resonance wshort = θ̇S − 3ω̇S + 2ω̇J happens. Since such a small amount of displacement of Saturn’s semimajor axis causes little variations to ω̇S and ω̇J , the position of this
secular resonance can approximately be taken as being unchanged.
First, we discuss the case of k = 0.9982. Fig. 16 shows the local
details of the frequency map of a typical orbit close to the resonance wshort = θ̇S − 3ω̇S + 2ω̇J . The left-hand frame is around the
short period component, while the right-hand frame is around the
long period component. For orbits at the right of the dashed curve
in Fig. 15, wshort is at the left of frequency θ̇S − 3ω̇S + 2ω̇J . For
orbits at the left of the dashed curve, wshort is at the right of this
frequency. As stated in the above discussions, the long period component overlaps not only with its side peaks, but also overlaps with
nearby frequencies. These orbits are generally unstable. Although
they can survive the integration time of 1 Myr, they cannot survive
a longer integration time. In fact, nearly all the orbits at the right
of the dashed curve diverge for an integration time of 100 Myr. Actually, even if the right part could survive the planetary migration
process, according to our studies above, it cannot survive in the
Figure 15. The members in the long period family that survive the perturbations in 1 Myr for different initial states of Saturn.
Saturn Trojans: a dynamical point of view
1431
Figure 16. Local details of the frequency map of a typical orbit close to the secular resonance wshort = θ̇S − 3ω̇S + 2ω̇J .
Figure 17. Local details of the frequency map of an orbit initially static at the triangular libration point by integrating equation (18), for k = 0.9975.
present state of the Solar system. In the following, we will not focus
on this region.
For the part on the left of the dashed line in Fig. 15, with k
increasing, it shrinks from inwards to outwards, to a complete extinction (k = 0.9987) and then appears again with k increasing even
more (k = 0.9991 and 0.9995). In the following, we give some
plots to show what causes these changes. For k between 0.9978 and
0.9991 (approximate values), integration of the linearized system in
equation (13) shows that the triangular libration point is unstable,
which means that the constant matrix E in the transformed system
equation (15) has eigenvalues that are not pure imaginaries. As a
result, we can only obtain quasi-periodic orbits for k smaller than
0.9978 or larger than 0.9991. Fig. 17 shows the frequency analysis
of an orbit by integrating equation (18) starting from the triangular
libration point for k = 0.9975. Fig. 18 shows the corresponding
results for k = 0.9992.
With k increasing, the frequencies wlong cross the frequency 5θ̇S −
2θ̇J − 3ω̇S . During the crossing phase, the strength of the resonance
increases and causes the instability. For the frequency wshort , it may
not cross the frequency θ̇S − 2ω̇S + ω̇J . It is quite possible that wshort
first approaches θ̇S − 2ω̇S + ω̇J and then leaves away but always
remains at the left of this frequency. During the approach stage, the
resonance strength also increases and causes the instability. Similar
to the right-hand frame in Fig. 2, we performed the same numerical
survey in the simplified force model where the orbits of Jupiter and
Saturn are invariant ellipses. It seems that the resonance between
wlong and 5θ̇S − 2θ̇J − 3ω̇S plays the dominant role. But according
to the studies in Section 4, both resonances may contribute to the
instability, even including the resonance between wshort and θ̇S − ω̇S .
The actual planetary migration process is of course not so simple
as the above simulations, but one point can be sure that the positions
of resonances (especially the secondary resonance between wlong
and 5θ̇S − 2θ̇J − 3ω̇S ) migrate with the migration of planets. The
migration of these resonances may have cleared out any existing
Saturn Trojans, leading to the current non-existence in this area.
The study in this section is only preliminary. Further studies are
required.
8 DISCUSSIONS
In Section 4, we reach the conclusion that the nearcommensurability between θ̇S − ω̇S and wshort , and between 2θ̇J −
5θ̇S + 3ω̇S and wlong can cause the instability close to the
1432
X. Y. Hou, D. J. Scheeres and L. Liu
Figure 18. Local details of the frequency map of an orbit initially static at the triangular libration point by integrating equation (18), for k = 0.9992.
triangular libration points. The first near-commensurability is actually the secular resonance between the procession rates of Saturn’s
perigee and the Trojan’s perigee. For generality, we write the term
of the motion with frequency wshort in the synodic frame as
ξ = ξc cos θ + ξs sin θ,
η = ηc cos θ + ηs sin θ,
(23)
where θ = wshort t + θ 0 . We transform this term to the inertial
frame. To simplify, we take Saturn’s orbit as a circle, rotating with
a constant angular velocity θ̇S . As a result, we have
x = cos θS · ξ − sin θS · η,
η = sin θS · ξ + cos θS · η.
(24)
Two frequencies appear in equation (24). They are w1 = θ̇S − wshort
and w2 = θ̇S + wshort , w2 w1 . Averaging over time, we get the
long period terms of frequency w1 . Obviously, w1 can be taken as
the procession rate of the Trojan’s perigee. As a result, the nearcommensurability between θ̇S − ω̇S and wshort in the synodic frame
is actually the near-commensurability between ω̇S and w1 in the
inertial frame. For the other two frequencies θ̇S − 2ω̇S + ω̇J and
θ̇S − 3ω̇S + 2ω̇J , similar discussions apply.
Now, we turn to the secondary resonance. The frequency wlong indicates the long libration period of the Trojan’s mean motion around
the triangular libration point. Similar to equations (23) and (24),
transforming this term back to the inertial frame, we have two frequencies w3 = θ̇S − wlong and w4 = θ̇S + wlong . Since wlong θ̇S ,
w3 ≈ w4 ≈ θ̇S . As a result, the GI between Jupiter and Saturn should
also apply to the Saturn Trojan (de la Barre et al. 1996). Or, we can
view this as an ‘overlap’ between the GI, and the 1:1 resonance
between Saturn and the Trojan (Nesvorný & Dones 2002).
Even though the time-scale in this study seems too short compared with the age of the Solar system, the results are reasonable.
To demonstrate this, Fig. 19 shows an orbit surviving 4.6 Gyr in the
complete S-JSUN model (integrating backwards, every 0.1 Myr a
point is plotted). The unit in Fig. 19 is the instantaneous distance
between the Sun and Saturn. The integrator is the multistep one in
Quinlan & Tremaine (1990), with a step size of 29 d. To select an
initial condition which can survive such a long time, we choose the
long periodic orbit with λ far away from λmin and λmax in Fig. 7. The
chosen one is λ = 0.014 745 748 704 132. For readers to check the
results, the initial orbital elements of the orbit in the J2000 ecliptic
Figure 19. One orbit surviving 4.6 billion years in the S-JSUN model
(integrating backwards).
frame are
a = 9.735 862 850 358 16,
e = 0.057 178 712 476 877,
i = 2.◦ 484 483 693 605 65
= 113.◦ 511 795 925 292,
ω = 35.◦ 966 979 107 6818,
M = −39.◦ 379 870 343 8223.
The mass unit is the sum of the masses of the Sun and four inner
planets, and the length unit is 1 au. The initial conditions of the
four outer planets are given by the numerical ephemeris DE 406
at the epoch of J2000.0. Many long period orbits with λ between
λmin and λmax in Fig. 7 are believed to also survive such a long time
integration. Even the probability of finding a Saturn Trojan might
be very low due to the possible planetary migration; in our opinion,
if astronomers intend to search for Saturn Trojans, their observation
regions should be close to that in Fig. 19. Of course, the actual
libration width could be smaller or larger depends on the parameter
λ which should be confined between λmin and λmax in Fig. 7. Also,
vertical motions with an orbit inclination restricted to be within 15◦
should also be considered.
9 CONCLUSIONS
The stability problem of fictitious Saturn Trojans is studied in this
paper. Restricted to the planar S-JS model, we deduced the motion
equations of the Trojans in the synodic frame. With the aid of FFT
Saturn Trojans: a dynamical point of view
analysis, we are able to identify the resonances that caused the
instability of motions close to the triangular libration points. For
larger amplitude motions, by studying the survivability of the long
period family and the short period family, we are able to identify
the resonance mechanism that causes the chaos in the stable region
and the boundaries of the stable region. As for the vertical motions,
an explanation is given for the orbit inclination restriction of the
Trojans. The effects of planetary migration are also briefly studied.
AC K N OW L E D G E M E N T S
This work was supported by national Natural Science Foundation of
China (11322330, 11078001), National Basic Research Program of
China (2013CB834100) and National High Technology Research
and Development Program 863 of China (2012AA121602). It was
finished during the first author’s visit to the Colorado Center for
Astrodynamics Research (CCAR).
REFERENCES
Alexandersen M., Gladman B., Greenstreet S., Kavelaars J. J., Petit J.-M.,
Gwyn S., 2013, Sci, 341, 994
Beaugé C., Roig F., 2001, Icarus, 153, 391
Brasser R., Mikkola S., Huang T.-Y., 2004, MNRAS, 347, 833
Connors M., Wiegert P., Veillet C., 2011, Nat, 475, 481
de la Barre C. M., Kaula W. M., Varadi F., 1996, Icarus, 121, 88
Érdi B., 1988, Celest. Mech., 43, 303
Gomes R. S., 1998, AJ, 116, 2590
Gómez G., Llibre J., Martı́nez R., Simó C., 2001, Dynamics and Mission
Design near Libration Point Orbits, Vol. I, Fundamentals: The Case of
Collinear Libration Points. World Scientific Press, Singapore
Holman M. J., Wisdom J., 1993, AJ, 105, 1987
Hou X. Y., Liu L., 2008, Celest. Mech. Dyn. Astron., 101, 309
Hou X. Y., Liu L., 2009, Celest. Mech. Dyn. Astron., 104, 241
1433
Hou X. Y., Liu L., 2010, Celest. Mech. Dyn. Astron., 108, 301
Hou X. Y., Liu L., 2011, Celest. Mech. Dyn. Astron., 110, 71
Hou X. Y., Scheeres D. J., Liu L., 2013, Celest. Mech. Dyn. Astron., submitted
Innanen K. A., Mikkola S., 1989, AJ, 97, 900
Jorba À., Simó C., 1996, SIAM J. Math. Anal., 27, 1704
Jorba À., Ramı́rez R., Villanueva J., 1997, SIAM J. Math. Anal., 28, 178
Laskar J., Froeschlé C., Celletti A., 1992, Phys. D., 56, 253
Lei H.-L., Xu B., 2013, MNRAS, 434, 1376
Lykawka P. S., Horner J., 2010, MNRAS, 405, 1375
Malhotra R., 1995, AJ, 110, 420
Marzari F., School H., 2000, Icarus, 146, 232
Marzari F., Tricarico P., 2002, ApJ, 579, 905
Marzari F., School H., Farinella P., 1996, Icarus, 119, 192
Marzari F., Farinella P., Davis D. R., Scholl H., Bagatin A. C., 1997, Icarus,
125, 39
Melita M. D., Brunini A., 2001, MNRAS, 332, L17
Mikkola S., Innanen K., 1992, AJ, 104, 1641
Mikkola S., Innanen K., Muinonen K., Bowell E., 1994, Celest. Mech. Dyn.
Astron., 58, 53
Milani A., 1993, Celest. Mech. Dyn. Astron., 57, 59
Milani A., Nobili A. M., 1992, Nat, 357, 569
Morbidelli A., Levison H. F., Tsinganis K., Gomes R., 2005, Nat, 435, 462
Murray C. D., Dermott S. F., 2000, Solar System Dynamics. Cambridge
Univ. Press, Cambridge
Nesvorný D., Dones L., 2002, Icarus, 160, 271
Quinlan G. D., Tremaine S., 1990, AJ, 100, 1694
Rabe E., 1961, AJ, 66, 500
Rabe E., 1962, AJ, 67, 382
Rabe E., 1967, AJ, 72, 10
Robutel P., Gabern F., 2006, MNRAS, 372, 1463
Szebehely V., 1967, Theory of Orbits. Academic Press, New York
Tsiganis K., Gomes R., Morbidelli A., Levison H. F., 2005, Nat, 435, 459
This paper has been typeset from a TEX/LATEX file prepared by the author.