Scientific Frontiers in Research on Extrasolar Planets
ASP Conference Series, Vol. 294, 2003
D. Deming and S. Seager, eds.
Orbit Evolution of Planetary Systems in Stellar Clusters
Astronomisches Rechen-Inst., M¨ onchhofstr. 12?14, 69120 Heidelberg,
Douglas N. C. Lin
UCO/Lick Observatory, University of California, Santa Cruz, CA
formed in star clusters. Dynamical interaction between stars in clusters
affects the stability and dynamical evolution of their companion planetary
systems. We study systematically the statistical evolution of planetary
orbit parameters self-consistently with the evolution of their parent star
cluster using direct high-accuracy N-body simulations.
Stars and with them their planetary system are generally
Recent detection of jupiter-mass extrasolar planets indicate that they are at-
tached to at least ∼ 8% of nearby solar-type stars and that they have diverse
dynamical properties (http://www.exoplanets.org). An important challenge
to the theory of planetary formation and evolution is to account for the dynam-
ical diversity of extrasolar planets. The standard theoretical models of runaway
and oligarchic growth of planets from kilometer size protoplanets (Kokubo & Ida
1998, 2000) predict that most planets have small eccentricities. Observations,
however, reveal that there are even many massive planets with highly eccentric
orbits. One of a number of explanations would be the influence of encounters
between host stars of planets and field stars (de la Fuente-Marcos & de la Fuente-
Marcus 1997, 1998; Laughlin & Adams 1998). This hypothesis arises naturally
because most young stars are formed in clusters (cf. e.g., Kroupa 1995).
Strong dynamical perturbations resulting from close encounters between
stars may also have led to the loss of their planetary companions. Using the
equation of restricted three body motion to approximate encounters between
planetary systems and single stars, Smith and Bonnell (2001) inferred that only
a small fraction of planets would become detached from their host stars during
the characteristic life spans of open and young stellar clusters.
Dynamical instability resulting from the interaction between planets may
lead to both eccentricity excitation and orbital migration (Lin & Ida 1997).
Again, in this context of freely floating planets, there is a need to investigate the
effect of stellar encounters on planetary systems with more general initial orbits
than previously considered.
Spurzem & Lin
While Davies & Sigurdsson (2001) focus on the effect of close encounters
over relatively long time scales, and Heggie & Rasio (1996) only discuss in an
approximate way very wide encounters, we study here systematically the influ-
ence of stellar encounters with all impact parameters occurring in a real cluster
over nearly one crossing time, and use the results to elaborate probabilities for
the variations of planetary orbit parameters. Future work will be extended to
larger particle numbers, longer integration times, and include stellar binaries.
2. Computational Methods and Initial Model Parameters
We use the direct integration code NBODY6++ (Spurzem 1999), which is based
on full computations of individual pairwise particle forces, and is an offspring
of the family of NBODY codes developed by Aarseth (1999). The regularisa-
tion methods of Mikkola & Aarseth (1996, 1998) are applied to integrate the
planetary systems, which are initialized as so-called KS pairs in the simulation
(Kustaanheimo & Stiefel 1965). We attach only one planet to each host star
and in most simulations all planets have the same eccentricity. In regular short
time intervals of one N-body time unit (in these units the cluster crossing time
is 2√2) all changes of planetary eccentricities and semi-major axes are checked.
If the relative change of semi-major axis is larger than 10−5or the change of
eccentricity larger than 10−10an encounter has taken place and the initial and
final parameters of the planetary orbit are stored. From each run we thus get
a data bank of orbital changes; then we are binning all events according to the
initial semi-major axis and the eccentricity changes.
For all models, we adopt an isotropic Plummer model for the phase space
distribution function. This model provides a reasonable approximation for open
cluster potentials. In most runs all stars had equal masses, whereas in one
case for comparison we adopt an initial mass function following the prescription
of Kroupa, Tout & Gilmore (1993). All models are in dynamical equilibrium
initially, all stars are assumed to be point masses neglecting stellar evolution
effects. As model units we take as G = 1, M = 1, E = −0.25, for the gravi-
tational constant, initial total mass and energy, respectively (standard Nbody
units). We use 5000 stars.
Table 1. Model Parameters
15 · 103
25 · 103
35 · 103
45 · 103
55 · 103
65 · 103(ktg)
Models 1-4 are the most idealized ones, having equal stellar masses and
equal initial planetary eccentricities. Model 6 is yet the only one utilizing a
realistic stellar mass distribution. For planetary orbits, we adopt a range of
Orbit Evolution of Planetary Systems in Stellar Clusters
05 10 15 202530 35 40 4550
5000 stars, 1000 planets
tion of semi-major axes of planetary systems, crosses: negative changes,
quadrangles: positive changes.
Relative changes of eccentricities per time unit as a func-
semi-major axes between 3 and 50 AU and constant eccentricities of 0.1, 0.3, 0.6,
0.9. For comparison a model with a so-called thermal eccentricity distribution
f(e) = 2e is also used.
3.Results and Conclusions
For the preliminary results shown here we have followed the evolution of all
planetary systems for about two dimensionless time units, which would be some
4 million years in a low density environment such as the Orion cluster.
Figure 1 all events changing the eccentricities are plotted, with different symbols
for positive and negative changes. In Figure 2 we show the relative probabilities
for eccentricity changes of a certain size to occur separately for models 1-6 (here
for clarity only the positive eccentricity changes have been used). Here only
about 1% of all planets experiences significant changes and 0.1% of the planets
have been liberated. We have more data for higher densities, corresponding to
a globular cluster environment, where these numbers increase dramatically. As
the main conclusion it is stressed that the variation of eccentricities and semi-
major axes in planetary systems embedded in a star cluster behaves more like a
random walk or a diffusion process, where changes in both directions are nearly
equally probable. More details on these and results of further models will be
published in the future (Spurzem & Lin 2003, in prep.).
Spurzem & Lin
5000 stars, 1000 planets, positive changes of eccentricity
models 1-6, for the labels see Table 1.
Probabilities of relative eccentricity changes (positive) for
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