A&A 472, 643-647 (2007)
DOI: 10.1051/0004-6361:20077876
F. Marzari1 - M. Barbieri2
1 - Dipartimento di Fisica, University of Padova, Via Marzolo 8, 35131 Padova, Italy
2 - LAM, Traverse du Siphon, BP 8, Les Trois Luc,
13376 Marseille Cedex 12, France
Received 14 May 2007 / Accepted 31 May 2007
Abstract
Aims. If a significant fraction of binary star systems spent some time as inclined triple systems, either during their formation process or as the outcome of several close dynamical encounters in a crowded stellar environemnt, then the number of planets in binaries would be significantly lower than around single stars. The stellar chaotic phase preceding the instability of the triple system and the wide oscillations in eccentricity and inclination of the companion star due to the high mutual inclination between the companion and the singleton would quickly eject planets orbiting the binary in S-type orbits.
Methods. We perform numerical simulations of the dynamical evolution of hierarchical triple star systems with planets hosted around the primary star of the inner binary. Different values of mutual inclination, binary separation and singleton initial semimajor axis are explored in a statistical way.
Results. We find that a significant mutual inclination
between the singleton and the binary is a key factor for instability of the planetary system. When
is larger than
the fraction of planets in the binary surviving the chaotic phase of the triple declines dramatically. The combination of eccentricity and inclination oscillations of the binary companion induced by the secular perturbations of the singleton and the sequence of close encounters preceding the ejection of one star fully destabilize a planetary system extending beyond 1 AU from the star. For
around
the percentage of surviving planets is lower than 20% for all binaries with a semimajor axis smaller than 200 AU.
Conclusions. The frequency of planets in binaries with low separation may be strongly reduced by the residence of the pair in the past in a temporary inclined hierarchical triple.
Key words: methods: N-body simulations - stars: planetary systems
Over 65 percent of the main sequence stars in the solar neighborhood are members of binary or multiple star systems (Duquennoy & Mayor 1991). As a consequence, answering the question of whether planets can form and persist near one of the stars in a binary has far-reaching implications for the overall frequency of planetary systems. Studies on the long term stability of planets in binaries have shown that a planet cannot be located too far away from the host star or its orbit will be destabilized by the gravitational perturbations of the companion star. Holman & Wiegert (1999) found that the stable/unstable boundary depends on the mass ratio and eccentricity of the binary, but for a wide range of parameters stable orbits may extend well beyond one tenth of the binary semimajor axis. However, in terms of the probability of finding a planet in binary systems, the dynamical stability analysis is not exaustive since it does not take into account the profund influence that stellar dynamic interactions may have had on the early evolution of a planetary system in a binary.
It has been suggested that most binaries originate from the decay of multiple systems (Reipurth 2000; Larson 1972, 1995, 2001; Kroupa 1995). The most common configuration among multiple systems is the hierarchical triple, where a singleton orbits around the baricenter of a binary system. A hierarchical triple can become unstable after some time, depending on its initial orbital parameters, leading to the disintegration of the triple (Eggleton & Kiseleva 1995; Kiseleva et al. 1996). This disintegration occurs via a phase of chaotic evolution whose outcome is the ejection of one of the three stars (typically the least massive body) on an unbound trajectory. The other two stars, members of the original binary, are left in a more tightly bound binary. In a previous paper (Marzari & Barbieri 2007, herein after MB1) we showed that the orbital changes of the binary and the strong gravitational perturbations during the chaotic phase prior to the singleton ejection can influence the final configuration of a planetary system hosted by the primary star of the pair. However, in the context of near-coplanarity between the binary and the singleton, planets can survive the triple decay in most cases and adapt to the new orbital parameters of the binary. The major effect would be a significant change in the orbital configuration of the system after the triple instability with respect to the original configuration, as an outcome of the planet formation process.
In this paper we consider the dynamic effects of the
decay of inclined
hierarchical triples on planetary systems.
In particular, we will
focus on planet survival during the unstable triple
configuration.
At present, determinations of the mutual inclinations
of the two orbits
in hierarchical triple stellar systems are available
only for a very limited number of cases and are often
ambiguous.
Fekel (1981) determined that at least 1/3 of a sample of
20
triple star systems have an inclination exceeding
and are not coplanar.
Sterzik & Tokovinin (2002) analysed a different set of 22 visual triples
finding an average mutual inclination of
.
However, in both the studies the
mutual inclination was derived from incomplete
observational data. To compute unambiguous mutual inclinations
for triples, both radial velocities and visual orbits are
required
for the inner and outer system.
So far, only six nearby systems have been observed with both methods and
have direct and precise measured orbits
(Muterspaugh et al. 2006). The values of mutual inclination for
these systems range from
to
but the sample is too small
to give
hints on the real
distribution of inclinations among triples.
The mutual inclination of triples may either be
primordial and related to the
formation process of the triple by fragmentation of a molecular cloud
or it may form
at later times because of
dynamical interactions, like encounters, between single stars and binaries
in a dense cluster-like environment. In the latter case the inclination
is due to the encounter geometry between the binary and the single
star and should be randomly distributed.
Any deviation from randomly oriented orbits may be an important indication
of the relative importance of the two formation
mechanisms.
Assuming that planets can form
in the binary before the bound hierarchical triple becomes unstable,
the dynamical interactions between the stars during the chaotic
phase can strongly
affect the stability of the planetary system.
We can envision two different scenarios for planet formation and subsequent destabilization within an inclined hierarchical triple:
We will not explore in this paper the full complexity of the hierarchical triple dynamics as performed in Ford et al. (2000). We are interested on the consequences of the large variations of the star orbital elements on the planets and we perform statistical numerical simulations giving the fraction of planets surviving the chaotic phase of unstable triples. We also do not investigate the planetary formation process in detail, but we assume that planets can form by either of the two mechanisms, core-accretion or gravitational instability.
In Sect. 2. we describe the numerical model adopted for the numerical integration of the trajectories of the stars and planets. Section 3 is devoted to the statistical analysis of the survival of planets in S-type orbits around the primary star. In Sect. 4 we present our conclusions.
Our numerical model consists of 3 stars, two locked in a binary system
and the third orbiting the barycenter of the pair. A set of 10 massless
bodies started on circular orbits around the primary star simulate a
planetary system that formed in the early phases of evolution of
the binary. The semimajor axes of the test bodies are regularly spaced
from 1 to 10 AU and the initial inclinations are all set to with respect to the binary orbital plane. The trajectories of the stars and of
the "planets'' are computed with the numerical integrator RADAU (Everhart 1985).
It handles in a very precise manner close encounters between massive bodies
and it does not require a fixed hierarchical structure such as HJS
(Beust 2003) or SYMBA5 (Duncan et al. 1998).
To model the outcome of the triple instability in all possible configurations
is a difficult task since the parameter space to explore is
wide. For this reason we select a limited number of parameters
to be sampled while the others are left unchanged. To better compare
our results with those presented in MB1, we adopt the same masses for
the stars i.e. 1 and 0.4 solar masses for the binary, 0.4 solar mass
for the singleton. An eccentricity of 0.2 is adopted for both the binary
and the singleton, taking into account that the orbit of the
singleton is defined with respect to the barycenter of the binary.
The mutual inclination is sampled between
and
including in this way retrograde orbits of the
singleton. For any value of the semimajor axis of the binary
,
we sample
different values of the semimajor axis of the singleton
and
of the orbital angles other than those giving the mutual
inclination. For any set of
we perform 20 simulations with random initial orbital
angles to increase the statistics on the star and planet
dynamical behaviour.
In this section we discuss in detail the two mechanisms leading to destabilization
of a putative planetary system extending beyond 1 AU around the primary star
of a binary in an inclined temporary triple.
In Fig. 1 we show the evolution of a model with
AU,
AU and initial mutual inclination
.
In this configuration, the critical semimajor axis for
long-term stability of planetary orbits around the primary is,
according to Holman & Wiegert (1999), around 21 AU. Our initial planetary system,
extending out to 10 AU, is well within the stability region.
The perturbation of the singleton induces
Kozai cycles on the binary companion that achieves an
eccentricity of almost one
over a timescale of
yrs.
This behaviour is well described by quadrupole and octupole-level
secular equations presented in Ford et al. (2000); Mazeh & Shahan (1979).
All the planets beyond
2 AU are ejected from the system after the first cycle, while that orbiting
at 2 AU is destabilized after the second cycle. Starting from
yrs the singleton and the companion star have
mutual close encounters that quickly lead to the ejection of the
last inner planet, that lived through the Kozai cycles of the companion.
Finally, after about
yrs, the outer star is ejected on a
hyperbolic orbit and the the binary system is left with a
smaller separation but no planets.
This kind of behaviour, typical of systems with high mutual inclination
,
places in jeopardy not only the stability of planets around the primary
but also the possibility that they can form. According to Boss (1997),
several gaseous protoplanets can rapidly form by disk instability
in a marginally gravitationally unstable protoplanetary disk. Within this
scenario
in a few hundreds years we might witness the formation of the
unstable triple and of a planetary system made of gas giant planets
around the primary before the Kozai cycle increases the eccentricity
for the companion star. On the other hand, core-accretion would
not have enough time to accumulate a core by planetesimal accretion, and even
planetesimals may have failed to form on such a short timescale.
A protoplanetary disk around the primary star would be strongly perturbed
and almost fully
destroyed during the first Kozai cycle in eccentricity and
inclination of the companion star. However,
if the binary system was isolated during its formation and it became part of
an unstable triple later on because of repeated stellar encounters
in a dense star-forming region, then
planets might have the time to grow even by core-accretion, before
the onset of the strong perturbations related to the stellar interactions
in the triple phase.
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Figure 1:
Orbital evolution of the planets around the
primary star of the binary (plot a) under the perturbations of the
companion star, in turn affected by the gravitational
pull of the outer singleton star (plot b). The initial
semimajor axis of the binary is 70 AU, the eccentricity
of the binary 0.2, that of the singleton 0.2, and the mutual
inclination ![]() ![]() |
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Only systems with large values of
are fully destabilized by
the stellar perturbations of the triple.
When the mutual inclination is lower than
,
close encounters between the stars, and the consequent impulsive changes
of the orbital elements, are a source of instability for the
planets but often not strong enough to destabilize the full
planetary system.
In Fig. 2 we illustrate the evolution of a model with
AU,
AU, as in the previous case, but with a lower initial mutual
inclination
.
The triple quickly becomes unstable and
the singleton has frequent close approaches with the binary companion
marked by sudden steps in eccentricity and semimajor axis. The
changes in the orbital elements of the companion leads to unstable planetary
orbits as shown in Fig. 2. However, contrary to the case shown in
Fig. 1 the planetary system is not fully destroyed and planets
within 5 AU of the star survive the chaotic phase.
Further perturbations by the binary companion
after the triple disruption do
not destabilize the planetary survivors since they are well
within the critical semimajor axis for stable orbits (Holman & Wiegert 1999).
![]() |
Figure 2:
Same as in Fig. 1 but for a lower value of
![]() ![]() |
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If the companion and the singleton are on retrograde orbits, instability
builds up in a similar way. For mutual inclinations lower than
,
large amplitude oscillations of the eccentricity
begin to destabilize the planetary system which is finally
destroyed by the stellar encounters in the chaotic phase. Contrary to
the prograde case, the oscillations of eccentricity and
inclination are not in phase, as predicted by the quadrupole theory.
Apparently, the two orbital parameters are no longer bound in an
invariant and they evolve with independent frequencies. For inclinations
in between
and
the oscillations
in eccentricity are moderate to low but some instability of planetary orbits
is driven by the large inclination oscillations of the companion.
In Fig. 3 we show the evolution of the inclinations of the
singleton and companion star when the mutual initial inclination is
.
We plot the inclination of each individual
star, referred to the initial plane of the binary, because this is
the plane where the planets also begin to orbit the primary.
The inclination of the companion becomes very high and
becomes retrograde for a short while. The behaviour is characterized also
by the libration of the angle
(see Fig. 3, lower panel) with the same frequency as the inclination
oscillations. Most of the planetary orbits are destabilized during these
large inclination excursions of the companion star and after 1 Myr only
the two inner planets survive. The onset of the chaotic phase of
the stars ejects finally also these two survivors.
![]() |
Figure 3:
Destabilization of planets around the primary
when the singleton is on a retrograde orbit relative to the
companion. The initial mutual inclination between the
two outer stars is
![]() ![]() ![]() |
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To test the chances of a planetary system in a binary
to survive a period of stellar interactions typical of
an unstable triple, we have run several models with
the binary semimajor axis fixed to
AU. The orbital eccentricities of the stars are both set
to 0.2. In Fig. 4 we plot the percentage
of
dynamical systems that, at the end of the period as a
hierarchical triple, retain at least one of the initial 10 planets
vs.
,
the initial mutual inclination between
the two outer stars. This percentage is very high for low inclinations
confirming the results presented in MB1 for low-inclination
systems,
while it declines very quickly when the inclination approaches
.
This is a consequence of both the Kozai cycle that pushes the
binary companion closer to the planets, and of the more complex
orbital behaviour during close approaches between the stars when
their orbits are inclined.
![]() |
Figure 4:
Fraction ![]() ![]() |
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![]() |
Figure 5:
Fraction ![]() ![]() ![]() ![]() |
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Retrograde orbits of the singleton also lead to
fast instability of the planets when the mutual inclination is
close to
.
Wide oscillations of the eccentricity up
to large values are observed, even if not related to the known
Kozai type mechanism: there is no phasing between eccentricity
and inclination. However, even in this case when
the eccentricity is at its peak value most of the
planets are destabilized. Only when the mutual inclination has values
beyond
the planets around the primary
are partly spared by the oscillations in eccentricity of the binary
companion. However, as observed in the previous section,
for mutual inclinations in the range
140
-180
large inclination oscillations of the companion star destabilize
planets even if to a lesser extent than the
eccentricity oscillation. As a result,
the percentage of planets surviving the chaotic phase grows
for inclinations larger than
but it does not
return to 100%, halting at about 30%.
If we increase the semimajor axis of the binary ,
the fraction of systems with surviving planets
increases in an almost
linear way. In Fig. 5
we show the fraction of systems retaining planets vs.
for the worst case, i.e. with mutual inclination equal to
.
The triple instability is a mechanism that easily destroys
planetary systems of close binaries while it is less effective for
wide binaries. For larger values of
the planetary systems
that survive are also more extended in semimajor axis.
In most cases for
AU all the planets up to
AU survive the stellar chaotic phase.
The fraction of binary systems hosting planets in S-type orbits can be lower than expected. If the binary is part of a crowded stellar environment, encounters with other stellar objects can lead to the formation of a transitional triple with large mutual inclination between the singleton and the binary. The subsequent dynamic evolution of the triple, in particular the large oscillations in eccentricity of the companion star in the binary and the chaotic evolution during the triple destruction, destabilize planetary orbits around the main star. Even if the binary was born as part of an unstable inclined triple, the planetary system is fated to be disrupted.
Observing a binary system without planets in S-type orbits does not necessarily imply that the stars did not posses circumstellar disks in their early phases or that planet formation did not occur. The history of the binary and of its primordial environment must be taken into account since in most cases it may be the cause of the absence of planets. Planet formation might be a very efficient process also in the presence of external perturbations, but the survival of planetary systems may be threatened by the binary dynamical history.
Acknowledgements
We thank P. Eggleton for stimulating us to perform this work.