Download Overlapping of secular resonances in a Venus horseshoe orbit

ASTRONOMY
AND
ASTROPHYSICS
Letter to the Editor
Overlapping of secular resonances in a Venus horseshoe orbit
Patrick Michel
Observatoire de la Côte d’Azur, Dept. Cassini, UMR 6529, B.P. 4229, F-06304 Nice Cédex 4, France
Received 21 August 1997 / Accepted 16 September 1997
Abstract. A numerical N–body integration of the asteroid
(4660) Nereus over 3.5 Myr shows the presence of the Kozai
resonance and two secular resonances inside a horseshoe orbit with Venus. This 1/1 mean motion resonance with Venus
remains stable during the whole time span though a secular increase of the orbital inclination of the small body occurs due to
the overlapping of two secular resonances, namely ν13 and ν14 .
This result shows that the dynamics of horseshoe orbits can be
very complex since an interplay with some secular mechanisms
is possible. Whether such interplay can occur frequently with
Venus or the Earth and does not either affect or increase the
long–term stability of this kind of orbits is still not clear and
needs further investigations.
Key words: Minor planets, asteroids – Celestial mechanics
1. Introduction
Recently, it has been discovered that the Near–Earth Asteroid
(3357) 1986 T0 could currently evolve on a horseshoe orbit with
the Earth (Wiegert et al. 1997). This discovery has increased the
motivation to understand the dynamics of such orbits. In this
paper, we report a peculiar result which shows the presence of
the Kozai resonance and two secular resonances inside a Venus
horseshoe orbit, which remains stable at least for several millions of years. This horseshoe orbit has already been shown in
a previous paper in which the possible orbital evolution of the
asteroid (4660) Nereus was studied (Michel et al. 1996) by integrating numerically over 2 Myr its current orbital elements as
well as “dynamical clones” defined by slightly changing one of
the initial orbital elements. After 1.4 Myr, one clone entered into
the Kozai resonance, then left this resonance and evolved on a
Venus horseshoe orbit until the end of the 2 Myr integration time
span. This Letter presents the result obtained by continuing this
integration over 3.5 Myr and shows a new behaviour. Roughly,
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we have found that the orbit remains in a stable state, switching
between the Kozai resonant orbit and the 1/1 resonant one with
Venus, and also that an overlapping of the Kozai resonance and
two secular resonances inside the 1/1 mean motion resonance
occurs. This overlapping causes a secular increase of the inclination of the asteroid’s orbit. Therefore, though the evolution
of this object was strongly affected by close approaches with
the Earth during the first 1.4 Myr, this new regime has provided
a protection mechanism which increased the body’s dynamical
lifetime. Up to now, no study has shown that some other dynamical mechanisms could take place inside a horseshoe orbit.
Here, we shall show that the Kozai resonance and some secular
resonances can be present without affecting the orbital stability
and lead the horseshoe orbit to high inclinations.
Although tadpole and horseshoe orbits are well known features of the three-body problem (e.g. Everhart 1973; Hollabaugh
& Everhart 1973; Dermott & Murray 1981; Mikkola & Innanen 1990, 1992), many puzzles remain concerning their origins
and evolutions. The studies of the stability of the triangular Lagrangian points in the circular restricted three–body problem
(see Szebehely 1967) and then in the elliptic one (Danby 1963)
have shown that the triangle is stable in the circular problem
and that it oscillates but the stability remains in the elliptic one.
In the Solar System, several bodies evolving on tadpole orbits have already been observed. The more famous ones are
the Trojan asteroids which are companions of Jupiter and librate around one of the Lagrangian equilibrium points, either
L4 or L5 . In 1990, the asteroid (5261) Eureka was also discovered orbiting in a 1/1 resonance near the trailing point of the
Sun–Mars system (Mikkola et al. 1993). These real cases have
suggested the conservation, though external perturbations are
present, of tadpole orbits. However, it has been shown (Innanen
& Mikkola 1989) that in the case of Saturn the triangular point
itself is unstable, while stable orbits exist. This has suggested
that the stability must be studied separately for every planet.
An object librating on paths which encompass the Lagrangian points L4 , L5 and L3 evolves on a horseshoe orbit.
The only real examples of objects moving on such orbits are the
saturnian satellites Janus and Epimetheus (Dermott & Murray
LETTER
Astron. Astrophys. 328, L5–L8 (1997)
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P. Michel: Secular resonances in horseshoe orbits
1981) and the recent discovery of the Earth horseshoe orbit of
the asteroid (3357) 1986 T0. According to Dermott & Murray
(1981), who studied the properties of horseshoe orbits in the circular and elliptic restricted three–body problem with small mass
ratio, the relative width of the horseshoe and tadpole regions
is a function of the planetary mass, with the horseshoe region
increasing relative to the tadpole one as the mass decreases.
This suggests that horseshoe orbits are more likely when the
mass is small, which is the case of the inner planets. In addition,
horseshoe orbits provide a protection mechanism from close approaches with the considered planet. Mikkola & Innanen (1990,
1992) have studied the stability of horseshoe orbits by means
of numerical integrations on a 2.25 Myr time span. They have
shown that the ones with Venus or the Earth can remain stable at
least over this time span, indicating the possibility of long–term
stability. Therefore studies dedicated to these kinds of orbits
are important to have a better knowledge of the dynamics and
lifetimes of Near–Earth Asteroids (NEAs).
2. Numerical method and initial conditions
We refer to Michel et al. (1996) for a complete discussion about
the choice of the numerical method as well as of a detailed definition of the initial conditions. Here, we just recall that the integrations have been performed with a Bulirsch–Stoer variable
stepsize integrator (Stoer & Bulirsch 1980) optimized for dealing accurately with close approaches to planets. The adopted
model of the Solar System was purely gravitational and included
all the planets from Venus to Neptune, the mass of Mercury being added to that of the Sun and with the Earth–Moon pair
considered as a single body.
Since the nature of Nereus’ orbit is strongly chaotic (see
Michel et al. 1996), we have integrated over 3.5 Myr not only
the initial conditions listed in the ephemerides, but also a sample of clones, i.e. “indistinguishable” orbits obtained either by
small variations of the nominal initial conditions or by using
two different computers (implying different round–off errors).
This permitted us to follow different possible orbital evolutions
of this asteroid. The initial conditions are presented in Table 1
of Michel et al. (1996). In the following we study the evolution
of the clone called Nereus Ia, meaning that the initial semimajor
axis has slightly been changed.
3. Result
Initially on a deeply Earth–crossing orbit, the asteroid suffers
many close approaches to the Earth which drastically changes
its semimajor axis a down to a value close to that of Venus’orbit
(a ≈ 0.723 AU). The small body is then injected into the Kozai
resonance at time t ≈ 1.4 Myr. In this region and at small inclination (i < 20◦ ), this resonance is characterized by a libration
of the argument of perihelion around 0◦ (Michel & Thomas
1996; Michel & Froeschlé 1997) and coupled oscillations of
the eccentricity e and the inclination i, e being minimum when
i is maximum, and vice versa (Kozai 1962). This resonance provides a protection mechanism from close approaches to planets
(Michel & Thomas 1996) during 2 × 105 years, and just afterwards the body is trapped in the 1/1 mean motion resonance
with Venus. Then, until the end of the 3.5 Myr integration time
span, the semimajor oscillates around that of Venus. As shown in
Fig. 6 of Michel et al. (1996), the critical argument of the resonance σ1/1 (σ1/1 = λ−λV , λ and λV being the mean longitudes
of Nereus and Venus respectively) librates around the unstable
Lagrangian point L3 at 180◦ with a large amplitude, allowing
the body to come quite close to Venus but never enough to suffer
destabilizing close encounters for at least the total integration
time span.
This typical property of horseshoe orbits is already well
known. However, we have found an interesting new behaviour
shown in Fig. 1: while being in the 1/1 resonance with Venus,
at t ≈ 1.6 Myr, the asteroid is injected into the secular resonances ν13 and ν14 (which correspond to commensurabilites
between the mean precession frequencies of the nodal longitudes of the orbits of the asteroid and of the Earth and Mars,
respectively), as indicated by the wide libration of their corresponding critical angles (Fig. 1). Michel (1997) has shown
that the overlapping of these two resonances can cause drastic
variations of the orbital inclination of the small body. Thus, the
inclination increases from ≈ 17◦ to ≈ 35◦ (Fig. 1). Mikkola
and Innanen (1990) have studied the effect of the inclination on
the stability of horseshoe orbits with the Earth and have found
that the orbit becomes unstable at i > 6◦ . However here, at
t ≈ 2 Myr, the inclination reaches a value (i ≈ 24◦ ) where
the Kozai resonance is present. Here, this resonance appears in
the same regime as that occurring in the asteroid belt (Kozai
1962), with the argument of perihelion librating at 270◦ , due to
the growing importance of jovian perturbations at high inclinations (see Michel & Thomas 1996 for a full explanation). Thus,
from t ≈ 2 Myr to t ≈ 2.45 Myr, the integration shows for the
first time the occurrence of an overlapping between the Kozai
resonance and two secular resonances inside a Venus horseshoe
orbit (Fig. 1). This dynamical mechanism causes a secular increase of the inclination and does not affect the stability of the
orbit since the semimajor axis remains librating around that of
Venus and shows its typical feature in horseshoe orbits. If this
interplay of resonances is also possible inside horseshoe orbits
with the Earth, it could be the mechanism at the origin of the
high inclined orbit of the asteroid (3357) 1986 T0. Moreover,
the Kozai resonance protects the asteroid from close encounters
with Venus (Michel and Thomas 1996).
Dermott & Murray (1981) have derived an analytical expression of the width of the horseshoe region, in the frame of
the circular restricted three–body problem with small mass ratio. They have found that the width is ∝ m1/3 , m being the
planetary mass in solar mass units. In our case, Venus’ mass
being approximately 2.4478 × 10−6 solar mass, the width is
of the order of 0.0135 AU. We can compare this value to the
libration amplitude, in our integration, of the semimajor axis of
Nereus inside the horseshoe orbit, which is equal to 3.5 × 10−3
AU. The difference is certainly due to the assumptions made
by the authors to derive their theoretical expression and since
the width found in our integration is smaller than the theoretical
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P. Michel: Secular resonances in horseshoe orbits
Fig. 1. Evolution of the orbital elements of Nereus Ia from time t = 1.4 Myr to t = 2.6 Myr. Besides semimajor axis, eccentricity and inclination,
the figure shows also the two critical arguments Ω − ΩM and Ω − ΩE of the ν14 and ν13 secular resonances (ΩM and ΩE being the longitudes
of the node of Mars and the Earth, respectively), and the argument of perihelion ω.
one, this suggests that a small increase of the libration amplitude
of Nereus’ orbital semimajor axis should not affect the stability.
These authors have also derived an analytical expression which
gives the lifetime of a horseshoe orbit, i.e. the expected time
scale after which a particle is lost due to a random walk of the
semimajor axis caused by successive encounters with the planet.
The lifetime Γ of the horseshoe orbit is given by: Γ ≤ T /m5/3 ,
T being the orbital period of the planet (in years) and m being the planetary mass in solar mass units. The orbital period
of Venus being ≈ 0.615 years, the lifetime of the horseshoe
orbit can be as great as 1.38 × 109 years. This suggests that
the orbit of Nereus can in principle remain stable on a much
longer timescale than our integration time span. However, this
expression is also based on several assumptions which do not
take into account the possible interactions with other dynamical
mechanisms such as the Kozai resonance or secular ones. Although these mechanisms do not seem to destabilize the orbit
of the small body on a short time scale, they can cause secular
changes of its inclination and/or eccentricity which can eventually eject the body from the horseshoe orbit.
4. Conclusion
A numerical integration over 3.5 Myr of the orbit of the near–
Earth asteroid (4660) Nereus has allowed us to show that the
presence of an overlapping of secular resonances inside horseshoe orbits is possible and provides a transport mechanism to
high inclinations. When the orbital semimajor axis of the small
body gets close to that of Venus, the orbit is first injected into
the Kozai resonance which protects it from close approaches to
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P. Michel: Secular resonances in horseshoe orbits
Venus. Then it leaves this resonance and is trapped in the 1/1
mean motion resonance with Venus, showing the typical feature
of a horseshoe orbit. The overlapping of two secular resonances,
ν13 with the Earth and ν14 with Mars, causes a secular increase of
the inclination which allows the orbit to get into another regime
of the Kozai resonance. Then the two secular resonances and the
Kozai one overlap until this last only remains inside the horseshoe state. These complex dynamics do not affect the stability
of the horseshoe orbit at least during the 3.5 Myr time span,
since the semimajor axis remains librating from one border of
the 1/1 resonance to the other. If this mechanism is also possible with the Earth, it may explain the high inclination of the
asteroid (3357) 1986 T0. However, the dynamical lifetime of
this kind of state is still not clear and needs longer term integrations currently in progress. Further studies shall also be done to
determine whether this dynamical state is possible in horseshoe
orbits with the Earth and is statistically significant. In this case,
since it can lead to high inclinations inside horseshoe orbits,
it could suggest that many real objects can be found evolving
currently or in the future on such orbits.
Acknowledgements. We are grateful to Ch. Froeschlé, P. Farinella, M.
Lo, C. Murray and P. Wiegert for fruitful discussions.
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