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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

Rainer Spurzem

Astronomisches Rechen-Inst., M¨ onchhofstr. 12?14, 69120 Heidelberg,

Germany

Douglas N. C. Lin

UCO/Lick Observatory, University of California, Santa Cruz, CA

95064, USA

Abstract.

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

1.Introduction

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.

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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

Model

15 · 103

25 · 103

35 · 103

45 · 103

55 · 103

65 · 103(ktg)

N∗

M∗(M⊙)

1

1

1

1

1

1

Np

103

103

103

103

103

103

ap(AU)

3-50

3-50

3-50

3-50

3-50

3-50

ep

0.1

0.3

0.6

0.9

thermal

thermal

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

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Orbit Evolution of Planetary Systems in Stellar Clusters

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1e-05

0.0001

0.001

0.01

0.1

1

05 10 15 202530 35 40 4550

|de/e|

a [AU]

5000 stars, 1000 planets

Figure 1.

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.).

In

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0.0001

0.001

0.01

0.1

1

1e-061e-050.00010.001

de/e

0.010.11

N/Ntot

5000 stars, 1000 planets, positive changes of eccentricity

2e-ktg

2e

e=0.3

e=0.6

e=0.9

e=0.1

Figure 2.

models 1-6, for the labels see Table 1.

Probabilities of relative eccentricity changes (positive) for

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