Table Of ContentMon.Not.R.Astron.Soc.000,1–4() Printed1February2008 (MNLaTEXstylefilev1.4)
Free-floating planets in stellar clusters?
Kester W. Smith1,2 & Ian A.Bonnell3
1 Institute of Astronomy, ETHZ, Zu¨richCH-8092, Switzerland.
2 Paul Scherrer Institut, CH-5232 PSI-Villigen, Switzerland.
3 School of Physics and Astronomy, Universityof St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland.
Received1December2000,Accepted
1
0
0 ABSTRACT
2
We have simulated encounters between planetary systems and single stars in var-
n
ious clustered environments. This allows us to estimate the fraction of systems liber-
a
J ated, the velocity distribution of the liberated planets, and the separation and eccen-
tricity distributions of the surviving bound systems. Our results indicate that, for an
5
initial distribution of orbits that is flat in log space and extends out to 50AU, 50% of
1 the available planets can be liberated in a globular cluster, 25% in an open cluster,
v and less than 10% in a young cluster. These fractions are reduced to 25%, 12% and
5 2% if the initial population extends only to 20AU. Furthermore, these free-floating
8 planets can be retained for longer than a crossing time only in a massive globular
0 cluster. It is therefore difficult to see how planets, which by definition form in a disc
1 arounda youngstar,could be subsequently liberatedto forma significantpopulation
0
of free floating substellar objects in a cluster.
1
0 Key words: stars: formation – stars: dynamics – planets
/
h
p
-
o
r 1 INTRODUCTION tigate the formation of a population of free-floating planets
t in various cluster environments. We pay particular atten-
s
a The discovery of numerous extrasolar planetary systems in tion to the velocity distribution of this population, and the
v: the solar neighbourhood (Mayor & Queloz 1995, Marcy & question of whether the bulk of the liberated objects could
i Butler1996;Marcy1999)hasrevolutionisedourideasofthe beretainedintheirnatalenvironmentoncetheyareejected
X planet formation process and how it can vary from system from their parent system.
r to system. Specifically, the fact that most of the systems Inthenextsection wediscuss thepropertiesof theini-
a
found contain relatively massive planets at small separa- tial planet population and of the various clusters. We then
tions, in contrast to our solar system, has engendered sig- brieflysummarise theissue of interaction cross sections, in-
nificant research into possible orbital migration (e.g. Lin, cluding discussion of the different possibilities following an
BodenheimerandRichardson1996). Morerecently,thedis- interaction.Wethendescribethesimulationsofthevarious
covery that there appears to be no such close systems in encountersandderivevelocitydispersionsandotherproper-
theglobularcluster47Tucimpliesasignificantdifferencein tiesforboththefreefloatingandboundplanetpopulations.
planetary formation which could be dueto the stellar envi-
ronment (Brown et al 2000, Gilliland et al 2000). Indeed,it
ispossiblethatstellarinteractionsintheearlystagesofthe
2 INITIAL CONDITIONS
globular cluster were able todisrupt the circumstellar discs
before any planets were able to form (Bonnell et al. 2000) Observations indicate that YSO disks are typically 100AU
or that the increased radiation from the expected number in radius (McCaughrean & O’Dell, 1996). Although it isn’t
of O stars was sufficient toremove thesecircumstellar discs clear to what radius in the disk planets generally form, we
beforeanyplanetscouldform (Armitage2000). Encounters canestimatebasedonourownsolarsystemthatplanetand
with passing stars in a dense stellar environment can lead planetessimal formation has occured at radii out to 40-50
to disruption of the planetary system and thus theejection AU. In contrast, the extrasolar planets found so far have
oftheplanets(seee.g.Sigurdsson,1992).Thiscouldleadto beeninorbitsastightas4days.Theseobservationsprovide
apopulationoffree-floatingplanetsinthecluster.Recently uswiththeplausiblerangeofplanetaryorbitstoinvestigate.
there has been a reported detection of a population of sub- Theinnerendofthisrangeisunlikelytobestronglyaffected
stellarobjectsinσOrionis(Zapatero-Osorioetal,2000)that by encounters (Bonnell et al 2000), although it is possible
could be due to stellar encounters. In this letter, we inves- thatinsufficientlydensesystems,stellarencountersareable
(cid:13)c RAS
2 Smith & Bonnell
Cluster Density Vdisp Lifetime b(AU) Cluster Ionised Survived Exchanged
pc−3 kms−1 years Min Max Total Kept Lost
Globular 103 10 109 3.43 24.26 Globular 47.3% 30.1% 17.2% 51.5% 1.3%
Open 102 1 109 33.32 221.22 Open 26.6% 0.5% 26.1% 61.1% 12.3%
Young 5×103 2 5×106 47.27 328.09 Young 7.8% 0.5% 7.3% 90.1% 2.1%
Table 1. The properties of the types of clusters studied. The
minimum and maximum impact parameters, b, are also shown. Table 2. The fate of planets in different cluster environments.
In the case of ionisation, three fractions are shown. The total
Thesecorrespondtoroughly10%and99%encounterprobabilities
percentageofsystemsionised,thepercentagewhichareretained
foreachcluster.
inthecluster,andthepercentagewhichescapewithinacrossing
time.
todisrupttheplanetarydiscsbeforetheplanetshaveformed
orbeforetheyareabletomigratetothesesmallseparations.
approachtypicallyoccurredatatimebetween10%and40%
Furthermore, in the case of a young cluster the close orbits
through the total simulation. A total of 7860 systems were
may not yet be populated as the migration timescale is of
used for each cluster.
order 107 years or more (Lin, Bodenheimer & Richardson,
1996).Wethereforeconsiderplanetaryorbitsbetween1and
50 AU in radius. The initial orbits are all circular, and the
distributionofseparationsisflatinlogspace.Torestrictthe 4 RESULTS
parameterspacestudied,theparentandperturbingstarsare
Table 2 shows the number of planets which became un-
assumedbeofequalmass,either0.7or1.5M⊙.Thesemasses bound, remained bound to the parent star, or were ex-
were chosen as representative of the bulk of the expected
changed during the simulation. As expected, a substantial
stellar population.
number of planets were unbound in the dense, long-lived
We consider three different cluster environments. The
globularclusterenvironment,fewerintheopenclustercase,
properties of these are summarized in Table 1. Our cluster
andlessthan10%intheyoungclustercase.Moredisruption
environmentsareintendedtocorrespondtoaglobularclus-
occurred for thehigh-mass stars than the lower mass case.
ter (dense and long-lived, with a high velocity dispersion),
an open cluster (more diffuse with a much lower velocity
dispersion), and ayoungcluster(intendedto correspond to 4.1 Velocity distributions of free floating planets
conditionsindensestarformingregionssuchastheTrapez-
ium.-seee.g.Clarke,Bonnell&Hillenbrand,2000).Theim- Fortheplanetsthat becameunbound,thevelocity atinfin-
pact parameters were drawn from the expected probability itywasestimatedfromthetotalenergy(kineticpluspoten-
distribution for the cluster environment. This is calculated tial) of the planet and, assuming this energy is conserved
using the mean time between encounters given by Binney while the planet escapes from the gravitational potential,
and Tremaine (1987), then 1/2mv2 = Etot 0. The simulations with high-mass
≥
starsproducedmorehigh-velocityliberatedobjects,butthe
te1nc =16√πnvdispRe2nc(cid:18)1+ 2vd2GisMpR∗enc(cid:19). (1) differGenracpehisnotfhethfienvaellvoecliotycitdyisdtrisibtruitbiuotnisonins wvaarsionuostcllaursgtee.rs
are shown (Figure 1). The distributions are normalised to
Here, tenc is the mean time between encounters within a the total number of planets. It is interesting to compare
distanceRenc,nisthenumberdensityofstarsinthecluster, the distributions to the estimated escape velocity for the
and vdisp is thevelocity dispersion. cluster(verticalline).Itisapparentthatwhilsttheglobular
cluster will retain the bulk of its free floating planets, most
liberated planets in the young cluster or open cluster will
3 SIMULATIONS tendtoescapewithinacrossingtime.InTable2,thefraction
of planets liberated in each cluster has been broken down
Simulationsofrestrictedthreebodymotionwerecarriedout
according to whether the planet subsequently escapes the
using a 4th order Runge-Kuttacode with adaptive stepsize
⋆ cluster or not.
on theETH’s Asgard cluster .The Runge-Kuttacodewas
Wenoteherethatachangeintheassumedouteredgeof
foundtoconserveenergyovertheinteractionstoafewparts
in 105 or better. Initial planetary orbits were selected at theplanetaryorbitdistribution,forexampletruncatingthe
outerorbitalradiuscloserin,wouldofcourseleadtoamod-
random from the log-flat distribution. The planetary orbits
ification of thefinal velocity distribution. The more distant
were isotropically distributed with respect tothe stellar or-
objects are more prone to disruption, but this is offset by
bit.Thestellarorbitwasstartedatapointwherethepoten-
theirbeinglessnumerousduetotheflat-loginitialdistribu-
tialenergyofthestellarsystemwas1%orlessofthekinetic
tion of orbits. Truncating the initial orbits at 20AU rather
energy.Theplanetswereinjectedwhentheforceduetothe
than 50AU would reduce the fraction of liberated objects
perturber reached 1% of the force due to the parent. The
to around 50% for the globular cluster, 25% for the open
simulations ran for a fixed time (usually 24 hours). Closest
cluster or around 2% for the young cluster. This reduction
would also tend to affect the low-velocity population more
⋆
Asgard is an Intel Pentium III Beowulf cluster located at the than the high-velocity tail, since the high velocity objects
ETHinZu¨rich.Itcomprises502CPUson251Dual-CPUnodes. come predominantly from theinnerorbits.
(cid:13)c RAS,MNRAS000,1–4
Free floating planets 3
AlsoshowninFigure1arefitstothedistributions.The
globular cluster case is reasonably well fitted with a Gaus-
sian.TheothertwodistributionsdonotresembleGaussians,
and can only be poorly represented by a Maxwellian. The
distributions shown in thesecases were constructed bytak-
ing the product of the initial planetary orbital velocity dis-
tribution(includingthestellarvelocity dispersion),andthe
observed cross section as a function of initial velocity, and
thenconvolvingwithaGaussian. Theamplitudeofthedis-
tribution and the sigma for the Gaussian were then left as
free parameters in the fit. These distributions do not rep-
resenttheobserved distributionentirelysatisfactorily (they
don’t reproduce the high velocity tail), but they serve to
illustrate the essential difference between the high velocity
globular cluster case and the low velocity clusters. In the
highvelocitydispersionenvironmentoftheglobularcluster,
the emerging planetary velocity dispersion is dominated by
the stellar scattering, whereas in the open cluster or young
clusterenvironmenttheionisedplanetarypopulationretains
a memory of the initial Keplerian orbital velocity distribu-
tion.Itisthiseffectwhichleadstotheliberatedpopulation
escaping from thelow velocity dispersion clusters.
4.2 The effect of varying planetary masses
We tested the effects of the restricted three-body assump-
tionforsomespecificcasesusingathree-bodyRunge-Kutta
codeandvariousplanetarymasses.Itwasfoundthatforsys-
temswheretheplanetswereretainedbytheparentstar,the
final binding energies differed by at most a few percent be-
tweenthemasslessplanetcaseandthethree-bodycodewith
a mass of 0.001M⊙ (i.e. 1 Jupiter mass). Wealso examined
cases where the planetary system was ionised, and investi-
gated to what extent changing the planetary mass affected
thefinalescapevelocity.Theeffect wasfound tobeusually
modest for the range of masses applicable to planets (1 to
10 Jupiters), but could be critical in certain circumstances.
The escape velocity usually decreased as the planet mass
wasincreased,althoughtherewerecaseswheretheopposite
occured. Several cases were found where modest changes of
planetmassproducedcriticalchangesintheescapevelocity
or changed the encounter outcome from ionised to retained
or exchanged. These were all distant interactions, in which
the closest approach of the perturbing star to the parent
star was greater than the initial planetary orbit. In these
cases, ionisation is of course sensitive to theencounter con-
Figure 1. The velocity distributions for the populations of free
ditions, and only a minority of systems in these encounters
floating planets in each of the three cluster environments. The
were ionised. We therefore conclude that the velocity dis-
histogramisthedistributionoftotalvelocities.Theverticallinein
tributionspresentedwould not bechangeddramatically for
eachcaseshowstheestimatedclusterescapevelocity.Ineachcase,
any realistic population of planets (up to 10MJup). a fit to the velocity distribution has been made. The Globular
clustercaseisfittedwithaGaussianandtheothertwocasesare
fittedwithafunctionderivedfromtheinitialvelocitydistribution
4.3 The bound population: Separation and oftheionisedsystems.Seetextfordetails.
eccentricity distributions
In Figure 2 we show the distributions of separation and ec-
centricityfortheplanetswhichsurviveencounters.Separate seen in eccentricity.Theretained planetshavenearly circu-
distributions are shown for the cases where objects are re- larorbits,theexchangedoneshaveaflateccentricitydistri-
tainedbytheparentstarandwheretheyareexchanged.As bution. The highly eccentric systems and captured systems
might beexpected, theplanets retained bythe parent tend with large separations will of course be much more vulner-
tolieincloseorbits.Theplanetscapturedbytheinterloper able to disruption in subsequent encounters. The effects of
occupy a flatter separation distribution. A similar trend is scatteringonthepopulationofboundplanetarypopulations
(cid:13)c RAS,MNRAS000,1–4
4 Smith & Bonnell
Figure 2.Distributionsofsemimajoraxis(left)andeccentricities (right) forsurvivingplanetarysystems. Toppanel: systemsretained
in an encounter, bottom panel: systems exchanged. The solid line is the globular cluster case, the dotted line is the open cluster case,
andthedashedlineistheyoungcluster.Thefrequencyhasbeennormalisedtothetotal numberofsystems.
inopenclusterswasinvestigatedinsomedepthbyLaughlin We note finally that the objects escaping from stellar
& Adams (1998). clusterswillformapopulationoffastmovingunboundplan-
etsintheGalacticdisc.However,thiswouldnotbeexpected
to form a significant contribution to the total mass of the
Galaxy.
5 CONCLUSIONS
Wehaveinvestigatedhowapopulationoffree-floatingplan-
etscanbegeneratedbystellarencountersindifferentcluster 6 ACKNOWLEDGEMENTS
environments.Wehavefoundthatinglobularclustersarel-
We thank the Asgard staff at ETH for providing the com-
atively high fraction of any planetary population is likely
puterresourcesusedinthiswork.WealsothankPeterMess-
to be liberated by encounters over the cluster lifetime, and
mer and Simon Hall for useful discussions, and the referee,
furthermore that the majority of these systems should be
Derek Richardson, for his rapid and helpful report.
retained in the cluster at least until they are lost through
two body relaxation after several thousand crossing times.
In the less dense environments of an open cluster or
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