Mass transfer between debris discs during close stellar encounters

4.5 MB·

by  Lucie Jilkova

MNRAS000,1–20(2002) Preprint1February2016 CompiledusingMNRASLATEXstylefilev3.0
Mass transfer between debris discs during close stellar
encounters
Lucie J´ılkova´,1(cid:63) Adrian S. Hamers,1 Michael Hammer,2,3
and Simon Portegies Zwart1
1Leiden Observatory, Niels Bohrweg 2, Leiden, 2333CA, The Netherlands
2Cornell University, 614 Space Sciences Building, Ithaca, NY14853, USA
3Department of Astronomy and Steward Observatory, University of Arizona, Tuscon, AZ85721, USA
6
1
0
Accepted2016January29.Received2016January21;inoriginalform2015November17
2

n
a ABSTRACT
J
We study mass transfers between debris discs during stellar encounters. We carried
9 out numerical simulations of close flybys of two stars, one of which has a disc of
2
planetesimals represented by test particles. We explored the parameter space of the

encounters, varying the mass ratio of the two stars, their pericentre and eccentricity
]
P of the encounter, and its geometry. We find that particles are transferred to the other
E star from a restricted radial range in the disc and the limiting radii of this transfer
. region depend on the parameters of the encounter. We derive an approximate ana-
h
lytic description of the inner radius of the region. The efficiency of the mass transfer
p
generally decreases with increasing encounter pericentre and increasing mass of the
-
o star initially possessing the disc. Depending on the parameters of the encounter, the
r transfer particles have a specific distributions in the space of orbital elements (semi-
t
s major axis, eccentricity, inclination, and argument of pericentre) around their new
a host star. The population of the transferred particles can be used to constrain the
[
encounter through which it was delivered. We expect that many stars experienced

1 transfer among their debris discs and planetary systems in their birth environment.
v This mechanism presents a formation channel for objects on wide orbits of arbitrary
1 inclinations, typically having high eccentricity but possibly also close-to-circular (ec-
7 centricities of about 0.1). Depending on the geometry, such orbital elements can be
1 distinct from those of the objects formed around the star.
8
0 Key words: circumstellar matter – open clusters and associations – planetary sys-
. tems
1
0
6
1
: 1 INTRODUCTION the planetary systems formed from these (Clarke & Pringle
v
i 1993;Ostriker1994).SeveralauthorspresentedN-bodysim-
X Stars form in giant molecular clouds and in most cases, a ulations of planetary systems (for example Hurley & Shara
group containing 10 to 104 stars form at a similar time
r 2002; Spurzem et al. 2009; Parker & Quanz 2012; Zheng
a from the same cloud (Lada & Lada 2003). Depending on
etal.2015)andhydro-dynamicalsimulationsofprotoplane-
the number of stars and their spatial and velocity distri-
tarydiscsinstarclusters(Rosottietal.2014).Theseworks
butions, these groups, with stellar densities relatively high
confirmedtheimportanceofthestarclustersandclosestel-
compared to those of the field stars, are classified as stel-
lar encounters for the dynamics of planetary systems and
lar associations or star clusters (e.g. Bressert et al. 2010;
circumstellardiscs.Therateandparametersoftheencoun-
Gieles & Portegies Zwart 2011). The gravitational interac-
ters depend on the characteristics of the star cluster, such
tions in these crowded environments result in close stellar
asitsmass,density,initialspatialandvelocitydistributions
encounters (for example Binney & Tremaine 1987; Bonnell
(for example Binney & Tremaine 1987; Bonnell et al. 2001;
et al. 2001; Pfalzner et al. 2006; Olczak et al. 2006, 2008,
Adamsetal.2006;Malmbergetal.2007;Olczaketal.2010;
2012) that can strongly influence the properties of proto-
Craig & Krumholz 2013). Using N-body simulations, Craig
planetary discs around the still young stars, and eventually
&Krumholz(2013)tabulatedthenumberandpropertiesof
encounters as a function of the cluster characteristics. For
example, they measured the encounter rate (counting fly-
(cid:63) E-mail:[email protected]
c 2002TheAuthors
(cid:13)
2 J´ılkov´a et al.
bys closer than 1000AU) for a solar-like star (0.8–1.2M(cid:12)) less massive than the two stars and their gravitational in-
of ∼ 1.9×10−6yr−1 experienced in a cluster with mass of fluence—mutualaswellasonthestars—canbeneglected.
1000M(cid:12),typicalradiusofabout2.5pc,virialratioof0.75, Planetesimals are then represented as (zero mass) test par-
and a moderate degree of substructure (fractal dimension ticlesthatliveinthetime-dependentgravitationalpotential
D=2.2). ofthetwostarsthatmoveonaconicsectionorbit.Asimilar
After stars form, gas and dust are present in their cir- approach was used already in the seminal work of Toomre
cumstellar discs, where planets and debris form later. The & Toomre (1972), who pioneered the simulations of merg-
current knowledge of the debris discs has been summarized ers of disc galaxies. The particles of the disc are perturbed
byMatthewsetal.(2014),Krivov(2010)andWyatt(2008). during the encounter and can, in general, stay bound to
Debris discs are result of planet formation around main se- their parent star, become bound to the perturbing star, or
quencestarsandtheyconsistofdustandlargesbodies(such become unbound from the system. Previous work showed
as comets, asteroids, or planetesimals) which determine the that dependingonthe parameters of the encounter andthe
dynamics of the discs. The dust grains are heated by the initial size of the disc, the fate of the particles depends on
central star and they re-radiate in the infrared (IR), pro- their initial location in the disc around the parent star. For
ducing so called IR excess in the spectral energy distribu- example, Clarke & Pringle (1993) noticed that in the pro-
tion of their host star, or (sub)millimeter wavelengths. It grade coplanar parabolic encounter of equal mass stars, all
is the radiation of the dust that is observed. The dust has the particles located closer than ∼ 0.3 of the pericentre of
relativelyshortlifetimeandisconstantlyreplenishedbycol- the encounter to the parent star stay bound. Kobayashi &
lisionsbetweenthelargerbodies(forexampleWyatt&Dent Ida (2001) confirmed this result and gave a more detailed
2002). Debris discs have been observed around hundreds of description of the perturbation in eccentricity and semima-
stars.Thedetectionratesvarydependingonthewavelength, jor axis of the disc particle orbits. A natural result of the
stellar type and age, and the relative sensitivity of the sur- encounter is a truncation of the disc and the dependence
veys (see Matthews et al. 2014, for the summary and ref- of the resulting disc size on the mass ratio between the en-
erences). Cold dust (at ≥60µm) has been detected around counteringstarswasrecentlydescribedindetailbyBreslau
24±5and32±5percentofAstars(Thureauetal.2014at et al. (2014). The encounter can also induce various struc-
100µmandSuetal.2006at70µm,respectively)andaround tures in the disc, such as rings or spiral patterns, or cause
20±2percent of solar type FGK stars (Eiroa et al. 2013). thedisctohaveanellipticalshape(Larwood&Kalas2001;
Dust in the mid-IR wavelengths (≤ 60µm) was detected Pfalzner2003).Simulationsoftheinfluenceofastellarflyby
around ∼ 11percent of solar type stars (Dodson-Robinson on a debris disc were also motivated to explain individual
et al. 2011). observed systems, for example in case of βPictoris by Lar-
Thedebrisdiscsareobservedtodecaywithtime,asthe wood&Kalas(2001)andHD141569byRecheetal.(2009),
planetesimals are depleted by collisions which grind them ortounderstandthescatteredKuiperbeltintheSolarsys-
intodust(Dominik&Decin2003).Theplanetesimalsinde- tem (e.g. Melita et al. 2005; Kobayashi et al. 2005; Punzo
brisdiscsmustbestirredsothattheirrelativevelocitiesare et al. 2014). Results on the change of the disc mass or size
sufficient for grinding. The origin of the stirring is still un- due to the flybys were often applied in the simulations of
der discussionandseveral mechanisms have beensuggested star clusters, where the encounters occur (for example Ol-
(Matthews et al. 2014): pre-stirring as a result of the pro- czak et al. 2006; Pfalzner et al. 2006; Lestrade et al. 2011;
toplanetaryphase(seeWyatt2008,andreferencestherein); Vincke et al. 2015; Portegies Zwart 2016).
stirring by planets in the same system (Kenyon & Bromley
2004a, Mustill & Wyatt 2009); self-stirring by sufficiently
1.2 Previous studies of captured planetesimals
massiveplanetesimals(forexampleKenyon&Bromley2008,
Kennedy&Wyatt2010,andreferencestherein);orstirring Most of the studies that model a debris disc during stellar
byexternalprocess,suchasstellarflybys(Kenyon&Brom- encounters focused on the evolution of the disc itself (its
ley 2002). mass loss, change of its size, morphology, energy or angular
momentum).Dependingontheparametersoftheencounter,
a portion of the disc can also be transferred from the par-
1.1 Simulations of discs during stellar encounters
entstartotheperturber.Howeverasystematicstudyofthe
In the early gas-rich stages, the viscosity and pressure of mass transfer among debris discs during a stellar flyby is
thegasinacircumstellardiscareimportantforthegeneral stillmissing.Aswedescribebelow,capturedbodiespresent
disc dynamics when considering the effects of external per- animportantoutputofstellarencountersandmaterialorig-
turbers and close stellar encounters. For the more distant inating from other stars might be present in many debris
ones,wheretheperiastronislargerthanthesizeofthedisc, discs—including in the Solar system.
gas and dust free simulations are commonly used to study Clarke&Pringle(1993)investigatedtheresponseofan
the perturbations due the encounter and the debris disc is accretiondisctoastellarflyby.Theyrepresentedthediscby
oftenmodeledbytestparticles(Clarke&Pringle1993were test particles with pseudo-viscosity and carried out simula-
among the first to use this approach, while they considered tionsofparabolicencountersofequalmassstarsconsidering
dissipation in the disc through pseudo-viscosity; discussion different initial geometries of the encounter orbit and the
onthe role of self-gravity, pressure andviscosity of thedisk disc—coplanar prograde and retrograde, and orthogonal.
was carried for example by Pfalzner et al. 2005b). Theyfoundthatoutofthesethreeconfigurations,themass
The dynamics of a planetesimal during a stellar en- transferoccursonlyinthecoplanarprogradeencounterand
counter can be approximately described as a general re- thatonlytheparticleslocatedinitially ∼>0.35ofthepericen-
stricted three-body problem. The planetesimals are much tredistancefromtheparentstarcanbecapturedbythestar
MNRAS000,1–20(2002)
Transfer of debris during stellar encounters 3
initiallywithoutdisc.Theseresultswerefurtherextendedby transferred depending on the parameters of the encounter
Hall et al. (1996) who focused on energy and angular mo- and what are the orbital characteristics of the particles be-
mentum exchanged during the encounter. They considered fore and after the transfer (Sect. 3). We derive an analytic
a more extended disc, up to four times the pericentre dis- approximation for the minimal radial distance of the parti-
tance,andshowedthatparticlesaretransferredfromlarger clestobecapturedbytheencounteringstarandcompareit
initialradii(beyondthepericentredistance)alsoinconfigu- with the results from the simulations (Sect. 4). We summa-
rations when the disc is inclined with respect to the orbital rize and conclude in Sect. 5.
plane(inclinationsof45,90,135and180◦,wherethelatter
corresponds to the coplanar retrograde encounter).
Larwood&Kalas(2001)studiedprogradecoplanaren-
2 SIMULATIONS
counters and geometries with inclinations of the encounter
orbit with respect to the disc of 30 and 60◦. They varied We carried out simulations of stellar encounters where one
the eccentricity of the encounter and found that planetesi- of the stars has a planetesimal disc and we followed the
mals can be captured for eccentricities ≤ 2 while none are planetesimalstransferredtothestarinitiallywithoutadisc.
capturedforeccentricities≥5.Thetransferredmaterialap- Becauseweapproximatethediscbyzero-massparticles(see
pearedtoformanasymmetricaldiscaroundtheirnewhost below),theresultswillnotchangeincasebothstarshavea
and showed clustering in eccentricities and semimajor axes. disc.
Fromtheirparameterspacestudyofstellarencounters,
Pfalzneretal.(2005a)concludedthatthemasstransferoc-
curs nearly exclusively in prograde encounters. They inves- 2.1 Numerical method
tigatedthedependenceoftherelativemasscapturedduring
Weassumedthatthemassesofplanetesimalsaresmallcom-
progradeparabolicencountersonthemassratioofthestars
pared to the stars (debris discs are typically less massive
and the pericentre of the encounter. They found that the
than 1M , for example Wyatt 2008). Under this assump-
⊕
captured mass increases with the stellar mass ratio, up to
tion we represented the planetesimals by zero-mass points.
value of about 0.8 when the captured mass becomes almost
Suchparticlesmoveinthegravitationalpotentialofthetwo
independent of the stellar masses (they measured the mass
stars, they do not interact with each other and neither do
ratioasthemassofthestarinitiallywithoutdiscrelativeto
they influence the motion of the two stars.
theoneinitiallywithdisc).Theyconcludedthatthismight
Weintegratetheequationsofmotionsusingacombina-
indicate the existence of an upper limit to the mass trans-
tionofN-bodyandhybridmethods(thesameasinJ´ılkova´
fer for a given pericentre distance. They also found that
et al. 2015 and similarly as in J´ılkova´ & Portegies Zwart
thecapturedmassdecreasesalmostlinearlywithincreasing
2015). The orbit of the two stars is integrated using the
pericentre of the encounter. symplectic N-body code huayno (Pelupessy et al. 2012).
Pfalzner et al. (2005b) presented the most detailed As longas the stars are well separated(atleast three times
studyofthemasstransfersamongcircumstellardiscs.They the disc size), the orbits of the planetesimals around their
compared captures in parabolic and hyperbolic stellar or- parent star at the beginning of the encounter and around
bits and confirmed that the transfer is smaller in the latter the encountering star in the later times, are calculated by
case (negligible for periastron larger than three times the solvingKepler’sequationsusinguniversalvariables(herewe
disc size). They also found that the majority of the cap- adoptedthesolverfromthesakuracode,Gon¸calvesFerrari
tured mass moves on highly eccentric orbits—in the equal etal.2014).Inthishybridapproachthegravitationalinflu-
mass parabolic encounter about 80% of the particles have ence of the other star is considered as a perturbation and
eccentricities above 0.8 and almost none below 0.4. is coupled to the planetesimals using bridge (Fujii et al.
Planetesimals transferred during a stellar encounter 2007). All calculations and the coupling of codes are real-
were also suggested to explain the origin of some Solar sys- ized using the Astronomical Multi-purpose Software Envi-
tem planetesimals in peculiar orbits (Morbidelli & Levison ronmentorAMUSE(PortegiesZwartetal.2013;Pelupessy
2004;Kenyon&Bromley2004b;Levisonetal.2010;J´ılkova´ et al. 2013).1 During the initial stages of the flybys (when
et al. 2015). The formation of the population of a so called the distance between the stars is large and decreasing) the
inner Oort cloud—which includes objects with pericentre influence of the star initially without the disc is considered
∼>50auandsemimajoraxisinrangeof150–1500au(Trujillo as a perturbation, while during the later stages (when the
& Sheppard 2014)—is still not well understood, because distancebetweenthestarsislargeandincreasing)theinflu-
these orbits are too far away from the Sun to be influenced enceofthediscparentstarisconsideredasaperturbationof
byplanetaryperturbationsandtooclosetobesubstantially theparticlescapturedbytheotherstar.Bycomparingwith
perturbedbytheGalacticpotentialandencounterswiththe self-consistentN-bodysimulationsinwhichalltheparticles
field stars (Portegies Zwart & J´ılkova´ 2015). J´ılkova´ et al. are integrated directly, we tested that the hybrid approach
(2015) constrained the parameter space of encounters that treats the captured particles correctly. However during the
could result in a population of planetesimals transferred to later stages of the flyby, larger inaccuracies can be intro-
theSolarsystemthatisconsistentwiththe observedorbits ducedintheorbitsoftheparticlesthatarenotcapturedby
of Sedna-like objects (Brown et al. 2004; Trujillo & Shep- the other star, that is those particles that are still bound
pard 2014). Constraining the encounter is possible due to to the parent star or completely unbound from the system.
specific characteristics of the captured orbits. Inthesecasestheinfluenceoftheparentstariscomparable
Herewepresentasystematicstudyofthemasscaptured
fromadebrisdiscduringastellarencounter.Wecarriedout
simulations (Sect. 2) and measured how much material is 1 http://amusecode.org.
MNRAS000,1–20(2002)
4 J´ılkov´a et al.
upinsuchwaythattheamplitudeofthegravitationalforce
fromtheparentstarM atthedistanceofouteredgeofthe
2
disc(200aufromthestarofmassM ,seebelow)istentimes
2
largerthentheamplitudeofthegravitationalforcefromthe
other star with mass M . Such an initial separation is suf-
1
ficiently large and the influence of the star initially without
the disc on the planetesimals is negligible. We tested that
increasingtheinitialseparationuptowheretheforceatthe
outerdiscedgefromstarM is1%oftheonefromstarM
1 2
(asusedbyBreslauetal.2014)doesnotchangetheresults
(inagreementwithClarke&Pringle1993;Halletal.1996).
Weintegratetheencounteruntiltheseparationbetweenthe
two stars again reaches the initial value. Hall et al. (1996)
pointed out that most of the interactions between the disc
particlesandtheperturbertakesplaceshortlyaftertheclos-
estapproach.Wetestedandconfirmedthatourintegration
timeissufficientforthecapturedparticlestosettletotheir
final state.
Figure 1. A doodle of the initial conditions of the encounter
PlanetesimalsinitiallyorbitstarM inaflatdisc(that
simulation. The red bullet indicates star M1 which is initially is we do not consider any vertical profi2le). Unless specified
withoutthedisc,whilethebluebulletindicatesstarM2 andits otherwise,weuse104 discparticles.Byincreasingthenum-
discisshownbythelightgreenannulus.Thecoordinatesystem
ber of particles (up to 5·104) in several simulations, we
is centered on star M2 and the reference plane xy is defined by
the disc. The stars are moving on a parabolic orbit which is in- testedthatthe results do notchange for higher resolutions.
dicatedbythefullredline.Thepericenteroftheencounterqenc For several specific encounters, we run simulations with six
is indicated by the dashed red line. The plane of the encounter different values of the random seed. We estimate the error
is inclined by the inclination angle ienc around the x-axis. The of our results by the standard deviations of the quantities
argument of pericenter ωenc is measured in the encounter plane westudybelow(suchastheradiiofthetransferregionand
betweenthex-axisandthedirectiontothepericenter.Thedoodle thetransferefficiency,Sect.3),whicharesmallerthan2per
isnotscaled.
cent for all the studied cases.
Theinitialradiusandphaseoftheplanetesimalsinthe
to or stronger that of the other star and the assumption of discareuniformlydistributed,whichcorrespondstothesur-
ourhybridmethod,thatitcanbeconsideredasaperturba- facenumberdensity∝1/r,wherer istheradialdistanceof
tion,isnotfulfilled.Ourapproachisappropriateonlyforthe the particles from the parent star measured in the plane of
particles whose dynamics during the later stages of the en- thedisc.Suchaprofileisoftenusedtomodelprotoplanetary
counteraredominatedbythestarinitiallywithoutdisc,the discs (e.g. Pelupessy & Portegies Zwart 2013; Steinhausen
transferred particles, on which we focus here. This approxi- et al. 2012, and references therein) and is supported by ob-
mateapproachallowsustocarryoutmanyfastsimulations servations(e.g.Andrews&Williams2007).Sincethediscis
and map parameter space of the encounters systematically. represented by zero-mass particles, it is possible to re-scale
the surface number density profile in post-processing of the
simulations to represent different initial surface mass den-
2.2 Initial conditions
sity radial profiles. Each particle can be considered with a
We set the mass of the star without the disc, M1, to 1M(cid:12) weight given by its initial radius so that the surface radial
in all our simulations. We systematically varied the mass profileisaspecificfunctionofr(similarlyasinSteinhausen
of the star initially with disc M , the pericentre of the en- et al. 2012). We discuss the role of the initial disc surface
2
counter q , and the eccentricity of the encounter e , in density in Sect. 3.7. We set the radial extent of the disc to
enc enc
the ranges of 0.1–2M(cid:12), 200–500au, and 1.001–4.5, respec- 30–200au.Suchchoiceisconsistentwithdiscsizestypically
tively. These values correspond to encounters that occur in observedinclusteredenvironments(seeforexampleVincke
star clusters and associations (for example Lestrade et al. et al. 2015, and references therein). Unless specified other-
2011, and references therein). Here we restrict to mass ra- wise, the planetesimals are initialized on circular orbits.
tios M /M from 0.5 to 10. We note that encounters with Formostofourinitialconditions,thediscandtheorbit
2 1
highermassratios(uptofewhundred)arealsoexpectedin ofthetwostarsareinthesameplaneandthez-components
clustered environment (for example Olczak et al. 2010). oftheirangularmomentumhavethesamedirection.Sucha
We further run simulations with different mutual in- coplanar prograde case results in the most violent encoun-
clination of the encounter plane and the plane of the disc ters and with the highest number of transferred particles.
(whichistheplaneofreference,seeFig.1),i ,anddiffer- We carried out 1000 of such simulations and the grid of en-
enc
ent argument of periastron of the encounter, ω . Because counter parameters is given in the 1st section of Table 1.
enc
the disc is axisymmetric (see below), the third angle defin- To estimate the effect of the general geometry, we varied
ing the mutual geometry of the orbit of the two stars and the relative inclination of the plane of the encounter with
the disc—the longitude of ascending node—does not play respecttothedisc,ienc,andalsotheargumentofperiastron
arole.TheinitialconditionsareillustratedinFig.1andall oftheorbitofthetwostars,ωenc.Wespecifytheencounter
the considered parameters values are specified in Table 1. parameters in the 2nd and 3rd sections of the Table 1.
The initial separation of the encountering stars is set To estimate the effect of eccentricity of the planetesi-
MNRAS000,1–20(2002)
Transfer of debris during stellar encounters 5
Table 1. Initial conditions for encounter simulations. The val- ienc=0◦,eenc=1.0
200
ues of qenc, M2, and eenc on first section define a grid of 1000 500
runs.Further,therunswithienc andωenc variedarelisted.The 180
last section describes the cases when the eccentricity of the disc 450
particles,edisc,wasvaried. 400 150 160
350 125 140
gridparameters:
qenc [au] 200,220,240,260,280,300,350,400,450,500 au] 120[au]
eMen2c [M(cid:12)] 10..100,10,.21,.205.3,,10.5.4,,10.7.55,,02..705,,21.5.0,,31.0.2,53,.51,.54,.02,.04.5 q[enc 238000 100 100rtr,min
ienc [◦] 0
ωenc [◦] 90 260 75 80
edisc 0 240
60
varyingienc: 220 50
qenc [au] 200,300 200 40
M2 [M ] 0.1,0.5,1.0
eenc (cid:12) 1.001,1.5,3.0 0.1 0.2 0.3 0.4 0.5 0.8 1.0 2.0
ienc [◦] 0–180,instepsof15 M2/M1
ωenc [◦] 90
edisc 0
Figure 2. Minimal disc radius from where the particles can be
varyingωenc: transferred rtr,min for coplanar prograde (ienc = 0◦) parabolic
qenc [au] 200 (eenc =1.0) encounters. The horizontal axis shows the mass ra-
M2 [M ] 0.1 tioofstarinitiallywithtowithoutdiscM2/M1,theverticalaxis
eenc (cid:12) 1.0 showsthepericentreoftheencounterqenc.Notethatbothhori-
ienc [◦] 30,60,90,120 zontalandverticalaxesarelogarithmic.Thecolorscalemapsthe
ωenc [◦] 0–180,instepsof15 minimaltransferradiusrtr,min.Thecontourlevelsareinau.
edisc 0
varyingedisc: 100 particles are transferred (48 out of our 1000 coplanar
qenc [au] 200,300,400
prograde simulations result in 1–99 transferred particles).
M2 [M ] 0.1,0.5,1.0,1.5
eenc (cid:12) 1.001,1.5,3.0 Wefirstfocusonadescriptionoftheresultsofthesys-
ienc [◦] 0 tematic grid parameter space study (encounter parameters
ωenc [◦] 90 listed in the first part of Table 1, Sects. 3.1–3.3) and later
edisc 0–0.05,0–0.1 on the cases with a more general geometry (Sects. 3.4 and
3.5),thecasewithaneccentricdisc(Sect.3.6),andwealso
considertheroleofthesurfacedensityofthedisc(Sect.3.7)
mals orbits, e , we run simulations where the eccentrici-
disc
tiesarerandomlyselectedfromauniformdistributionfrom 3.1 Transfer region
0 to e =0.05 or 0.1 (see last section of Table 1).
disc,max The fate of a particle after the encounter is determined
by its orbit in the initial disc around the parent star M .
2
We identify a minimal disc radius from where the particles
can be transferred to M . We call this radius r and
1 tr,min
3 RESULTS
we show that it generally corresponds to the radius from
In each encounter experiment, we follow the disc particles where the particles can be unbound from the parent star
thataretransferredfromstarM2 tostarM1 (initiallywith M2,run,min.ItwasalreadypointedoutbyKobayashi&Ida
and without disc, respectively). We calculate the orbital el- (2001) that run,min depends on the parameters of the en-
ementsofthediscparticleswithrespecttobothstars—the counter—the mass ratio M1/M2, pericentre qenc, and the
semimajor axis, as, and eccentricity, es, inclination, is, and eccentricity eenc—as
argument of periastron, ωs (see Sect. 3.4.1 for the discus- r ≈α[(1+M /M )(1+e )](−1/3)q , (1)
un,min 1 2 enc enc
siononthechoiceofthereferenceplane);wheretheindexs
identifiesthestar,thatiss=1,2.Weidentifythecaptured where α ≈ 0.3 or 0.5 for a prograde or retrograde en-
particlesasthoseboundonlytothestarM attheendofthe counter,respectively.Ourresultsshowsimilartrendsandin
1
simulations. At the end of most of our simulations, a small Sect.4,weprovideadetaileddescriptionofaderivationofan
fraction of the particles (typically less than 5%) are bound approximate analytic formula for r (M /M ,q ,e )
tr,min 1 2 enc enc
tobothstars.Byincreasingtheintegrationtime(morethan and compare it with the results from our simulations in
fivetimes)andusingfullN-bodysimulations,wetestedthat Sect. 4.4.
these particles generally become bound only to star M or In Fig. 2, we show the dependence of r on the
2 tr,min
unbound from the system and do not change the charac- mass ratio of the encounter stars and the pericentre of the
teristics of the captured population. To avoid low number encounter for the coplanar prograde parabolic encounters
statistics we consider only the simulations in which at least (i.e. for which the highest number of particles is trans-
MNRAS000,1–20(2002)
6 J´ılkov´a et al.
M2/M1=0.5,ienc=0◦ qenc[au]: ienc=0◦,eenc=3.5 200
200.0 500
160 220.0 450 180
240.0
260.0
140 280.0 400 160
300.0
u] 120 350.0 350 140
r[atr,min 100 445050000...000 q[au]enc 300 175 120r[au]tr,max
80 226800 10 125 150 100
0
240 75
80
60 220
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 200 60
eenc 0.1 0.2 0.3 0.4 0.5 0.8 1.0 2.0
M2/M1
Figure 3. Minimal transfer radius, rtr,min, for encounters with
massratioM2/M1=0.5anddifferentpericentresqenc asafunc- Figure 4. Maximal disc radius from where the particles can be
tion of eccentricity, eenc. Here, rtr,min for the parabolic orbits transferred,rtr,max,forcoplanarprograde(ienc=0◦)encounters
(eenc = 1.0) correspond to the values mapped in Fig. 2 for the witheenc =3.5.Thehorizontalaxisshowsthemassratioofthe
massratioM2/M1=0.5fixedonthehorizontalaxis. stars M2/M1, the vertical axis shows the pericentre of the en-
counter.Notethatbothhorizontalandverticalaxesarelogarith-
mic.Thecolorscalemapsthertr,max.Here,200auistheinitial
ferred). The initial radial extent of the disc is 30–200au outeredgeofthediscandthereforealowerlimitofrtr,max.The
contourlevelsareinau.
(Sect.2.2)whichalsosetsthelimitsonr .Forthecases
tr,min
with large pericentres (qenc ∼>450au) and high mass ratios
(M2/M1 ∼>1.25),rtr,min isclose toorlargerthantheinitial 0.5
disc extent of 200au; for smaller pericentres (qenc ∼<200au) M2/M1=0.75,qenc=240au,eenc= 3.5
andlowmassratios(M2/M1 ∼<0.1),rtr,min isdecreasingup M2/M1=0.75,qenc=240au,eenc= 1.0
tothelowerdiscsizelimitof30au.Fig.2demonstratesthe
0.4
dependencyofr onthepericentreoftheencounterq
tr,min enc
and the mass ratio M /M and in Fig. 3, we show that the
2 1
minimal transfer radius r depends only weakly on the
tr,min 0.3
encountereccentricityeenc (inagreementwithKobayashi& ni
Ida 2001). /ni
For faster hyperbolic encounters (with eccentricities ntr
0.2
eenc ∼>2.5), we also identify a maximal radius from up to
where the particles are transferred to star M , r . For
1 tr,max
encounterswitheenc∼<2.5,theparticlesaretransferredfrom 0.1
uptotheouteredgeofthediscof200auwhichcorresponds
to the lower limit on r . In Fig. 4, we show r for
tr,max tr,max
coplanar prograde encounters with e = 3.5. As expected
enc 0.0
and similarly to rtr,min, rtr,max also increases with larger 40 60 80 100 120 140 160 180 200
mass ratios and larger pericentres. r [au]
ini
In Fig. 5, we show the radial distribution of the rela-
tive number of transferred particles for the encounters with
Figure 5. Radial distribution of the relative number of trans-
M /M = 0.75 and q = 240au and for two different en-
2 1 enc
ferred particles for two coplanar prograde encounters with the
countereccentricitiese =1.0and3.5,whicharealsoplot-
enc samemassratio(M2/M1=0.75)andpericentre(qenc=240au)
ted in Figs. 2 and 4. In the case of the slower, parabolic
and different eccentricities. The horizontal axis shows the initial
encounter, more particles are transferred from a wider ra-
discradiusoftheparticlesrini.Thedistributioniscalculatedin
dialrange,wheretheouteredgeisoutsidetheinitialextent equidistantradialbinsof10au.Theratioofthetransferred(ntr)
of the disc (of 200au, see red diamonds and dashed line in to the initial (nini) number of particles in each bin is shown on
Fig.5)Thefasterencounterresultsinfewertransferredpar- theverticalaxis.Bluebulletsconnectedbysolidlineandreddi-
ticles and r <200au. amonds connected by dashed line correspond to the encounter
tr,max
We note that in about 25 simulations (out of the total eccentricityof1.0and3.5,respectively.
1000)asmallnumberofoutlierparticles(thisisalways ∼<5
particles, which is never more than 5% of the total number
of the captured particles) are transferred from outside the
initial disc region limited by r and r . This is a
tr,min tr,max
MNRAS000,1–20(2002)
Transfer of debris during stellar encounters 7
200 boundto,fraction: 1.0
M2,0.74
M1,0.06
0.8
100 M1andM2,0.02
unbound,0.18
[au] 0 /nini 0.6
y X
n 0.4
100
−
0.2
200
− 0.0
200 100 0 100 200 40 60 80 100 120 140 160 180 200
− −
x[au]
r [au]
ini
Figure 6. Initial distribution of the disc for different fate of the particles for the encounter with M2/M1 = 0.75, qenc = 240au, and
eenc=3.5.Left:Theplaneofthediscwiththecoordinatesinannon-inertialreferenceframecenteredonthestarM2 (markedbyblack
bulletinthecentre).Diskparticlesarecolor-codedaccordingtotheirfinalfateaftertheencounterasindicatedinthelegendwherethe
fractionforeachoptionisalsogiven.Right:Relativedistributionsoftheinitialdiscradiusoftheparticlesofdifferentfinalstatus.The
distributions are calculated in equidistant radial bins of 10au. The ratio of the number of particles (nX, where X stands for bound to
M2 or/andM1 orunboundfromthesystem)totheinitial(nini)numberofparticlesineachbinisshownontheverticalaxis.
the particles in the right panel. Most of the particles, 74%
0.14 4.50 oftheinitial104,stayboundtotheparentstar;18%areun-
bound from the system and lost into interstellar space; 6%
0.12 4.00 are transferred to star M1; and 2% are left bound to both
of the stars. The small number of particles that are bound
n
/rtr,mi 0.10 3.50 twohobloethradstiaarlsex(tsehnotwonfitnhevdioilsect;)thisesieniptiaarltlyiclsepsrweaildltoyvpeircatlhlye
)
run,min 0.08 3.00eenc Aensdc-uapnubnebsoeuenndfrfroommtthheebslyusetermadoiarlbdoiusntrdibountliyontoinstaFrigM. 52.,
− the transferred particles (shown in dark blue) are initially
0.06 2.50
min enclosed in between two radii. Similarly, the particles that
r(tr, 0.04 2.00 are eventually unbound from the system (shown in green)
are limited by a minimal radius, called r , that is very
1.75 un,min
similar to the minimal radius of the transferred particles
0.02 1.50
r . In Fig. 7, we show the relative difference between
1.25 tr,min
r and r (the later is always smaller than the for-
0.00 1.00 tr,min un,min
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 mer). The difference between the two minimal radii is less
ntr than8%ofrtr,min foralltheencounterwitheenc ∼<3.5,and
less than 15% for the cases with higher eccentricities. This
mightresultfromthefasterencountersgenerallyproducing
Figure7.Relativedifferencebetweentheminimaltransferradius
a smaller number of transferred particles.
rtr,min,andtheminimalradiusoftheunboundparticlesrun,min,
asafunctionoftherelativenumberoftransferredparticlesntrfor
the1000encountersoftheparametersgrid.Coplanarencounters
3.2 Transfer efficiency
are plotted and color-coded by the eccentricity eenc (regardless
thepericenterqenc andmassM2). To measure the efficiency of the transfer we follow the ra-
tio of the number of transferred particles and the number
of particles initially located in the original disc within the
resultoftheapproximatehybridapproachwhenintegrating range r –r . We call this quantity the transfer effi-
tr,min tr,max
the orbits of the disc particles (see Sect. 2.1) and we tested ciency, µ . It is important to keep in mind that as showed
tr
the method correcting for the outliers by comparing with in Sect. 3.1, for a substantial part of the studied parameter
the N-body simulations. space, the maximal disc radius of the transferred particles
The initial position in the disc for the particles of dif- r is larger than the considered outer edge of the disc
tr,max
ferent final fate is showed in Fig. 6, for the encounter with of 200au. The complete transfer region of transfers is not
M /M = 0.75, q = 240au, and e = 3.5 (same en- covered for these cases.
2 1 enc enc
counter as already shown in Fig. 5 in the red distribution). InFigs.8and 9,wepresentµ forthepericentresq ,
tr enc
We show the initial disc aroundstar M color codedby the andmassratiosM /M ,oftheencounterforfixedeccentric-
2 2 1
fate of the particles after the encounter in the left panel of ities e of 1.0 and 3.5, respectively. We mark the encoun-
enc
Fig.6;andtheradialdistributionsoftherelativenumberof ters for which the transfer region is not completely covered
MNRAS000,1–20(2002)
8 J´ılkov´a et al.
ienc=0◦,eenc=1.0 0.44 M2/M1=0.5,ienc=0◦ eenc:
500 1.0
1.2
0.40 0.35
450 1.5
1.8
400 0.36 0.30 2.0
2.5
3.0
u] 350 0.32 µtr 0.25 34..50
q[aenc 300 0.28µtr 0.20 4.5
280
0.24 0.15
260
240 0.20 0.10
220
200 250 300 350 400 450 500
0.16
200 qenc[au]
0.1 0.2 0.3 0.4 0.5 0.8 1.0 2.0 Figure 10.Dependencyoftheefficiencyofmasstransferµtr on
M2/M1 thepericentreoftheencounterqenc.Theencountersarecoplanar
and prograde encounters with mass ratio M2/M1 = 0.5. Lines
ofdifferentcolorscorrespondtodifferentencountereccentricities,
Figure 8. Efficiency of mass transfer µtr for coplanar prograde eenc,asindicatedtotheright.Bulletsdepicttheencounterswith
parabolic encounters. The horizontal axis shows the mass ratio
completely covered transfer region, while crosses the encounters
M2/M1, the vertical axis shows the pericentre of the encounter withrtr,max>200au.
qenc. Note that both horizontal and vertical axes are logarith-
mic.Thecolorscalemapstheµtr.Theredcrossesmarkthebins
wherertr,max>200auandthetransferregionisnotcompletely
covered—whichisthecaseforallencountershere. 0.30 M2/M1=0.5,ienc=0◦ eenc:
1.0
1.2
ienc=0◦,eenc=3.5 0.25 1.5
0.225 1.8
500
2.0
0.20
450 0.210 2.5
3.0
400 0.195 ntr 0.15 3.5
4.0
4.5
350 0.180 0.10
u]
q[aenc300 0.165µtr 0.05
280 0.150
260
0.00
240 0.135 200 250 300 350 400 450 500
qenc[au]
220 0.120
200 Figure11.Dependencyoftherelativenumberofthetransferred
0.1 0.2 0.3 0.4 0.5 0.8 1.0 2.0 0.105 particlesntronthepericentreoftheencounterqencformassratio
M2/M1 M2/M1 =0.5, and different eccentricities eenc. See Fig. 10 for a
detaileddescription.
Figure 9. Efficiency of mass transfer µtr for coplanar prograde
encounters with an eccentricity of 3.5. See Fig. 8 for a detailed
description. inner and outer disc edges are closer to the values of rtr,min
and r , respectively), we tested that this result is not
tr,max
resolutiondependent.Toincreasetheresolutioninthemass
(i.e.,r >200au)bytheredcross.Regardlessofthein- ratioM /M ofourgrid,wealsorunadditionalsimulations
tr,max 2 1
completeness of the data, lower mass ratios result in higher with M2/M1 = 0.125, 0.15, and 0.175M(cid:12) for the encoun-
transfer efficiencies. This is consistent with the conclusion terswithe =3.5,andwefindthatthetransferefficiency
enc
of Pfalzner et al. (2005a, note that they defined the mass µ is indeed lowest for the M /M =0.1 and smoothly in-
tr 2 1
ratiooftheencounteringstarinversetotheoneusedhere). creasing up to M /M =0.2. The transfer efficiency is gen-
2 1
However, for eccentric orbits (Fig. 9), the encounters with erally higher for the parabolic encounters (µ = 0.1–0.23
tr
thelowestconsideredmassratioM /M =0.1,havealower for e =3.5 in Fig. 9 while µ =0.15–0.45 for e =3.5
2 1 enc tr enc
transfer efficiency than encounters with M /M =0.2. The in Fig. 8) as was already noted by Pfalzner et al. (2005b).
2 1
samefeatureispresentforallhigherencountereccentricities Regardingthedependenceofµ onthepericentreofthe
tr
eenc∼>3.0.Byincreasingthenumberofthediscparticles(to encounter qenc, Figs. 8 and Fig. 9 indicate that the transfer
5·104)andbydecreasingtheinitialdiscextent(sothatthe ratio is maximal for a particular value of q which is the
enc
MNRAS000,1–20(2002)
Transfer of debris during stellar encounters 9
eenc=1.0,ienc=0◦ M2/M1: ienc=0◦,eenc=1.0 200
0.40 0.1 500 175
0.35 00..23 450 150 180
0.30 00..45 400 125 160
0.8 140
0.25 1.0 350
ntr 0.20 112...250 q[au]enc300 100 110200a[au]1,min
0.15 280
0.10 260 75 80
240
0.05 60
220 50
0.00 200 250 300 350 400 450 500 200 40
qenc[au] 0.1 0.2 0.3 0.4 0.5 0.8 1.0 2.0
M2/M1
Figure12.Dependencyoftherelativenumberofthetransferred
particlesntr onthepericentreoftheencounterqenc forcoplanar Figure 13. Minimal semimajor axis of the transferred parti-
prograde parabolic encounters for different mass ratio M2/M1. cles for the coplanar prograde parabolic encounters (eenc = 1.0,
SimilarlytoFig.10,linesofdifferentcolorscorrespondtodifferent ienc =0◦).ThemassratioM2/M1 andthepericentreoftheen-
mass ratios M2/M1, as indicated to the right. Here, points are counter qenc is increasing along the horizontal axis and vertical
indicatedbycrossesbecausethetransferregionisnotcompletely axis,respectively.Notethatbothhorizontalandverticalaxesare
coveredforanyofthedisplayedsimulations(i.e.rtr,max>200au; logarithmic.Thecolorscalemapstheminimalsemimajoraxisof
seeFig.10). theorbitstransferredaroundthestarM1.Thecontourlevelsare
inau.
1000 M2/M1=0.5,ienc=0◦ eenc:
1.0
samefordifferentmassratios—260aufortheparabolicen- 1.2
counters(Fig.8)and∼350–400aufore =3.5(Fig.8).In 1.5
enc 800
Fig. 10, we show the dependency of µ on q for encoun- 1.8
tr enc
2.0
terswithafixedmassratiobutarangeineccentricities.The
2.5
transfer efficiency is higher at larger pericentre for higher u] 600 3.0
a
valuesofe .Regardlessofthepericentre,thetransfereffi- [ 3.5
ciency is geenncerally smaller for more eccentric encounters. a1,min 400 44..05
Inpreviousstudies,thenumberoftransferredparticles
was followed rather than its ratio to the initial population
inthetransferregion,thatisthanwhatwecallthetransfer 200
efficiency µ . In Fig. 11 we plot the number of transferred
tr
particlesasthefractionofthetotalnumberofdiscparticles,
n . Fig. 11 has the same setup as Fig. 10, where the mass 0
tr 200 250 300 350 400 450 500
ratio is fixed and the dependence on the encounter pericen- qenc[au]
treisshownfordifferenteccentricities.Mostoftheprevious
work focused on the case of a coplanar prograde parabolic Figure 14.Minimalsemi-majoraxisofthetransferredparticles
orbit which leads the most efficient transfer for any given a1,min as a function of the pericentre of the encounter qenc for
mass ratio and pericentre. The number of transferred par- differenteccentricitieseenc ofcoplanarprogradeencounters.The
ticles n decreases almost linearly with q in this case in mass ratio M2/M1 is 0.5 for all the depicted encounters. Lines
tr enc
agreement with Pfalzner et al. (2005a). This is further con- ofdifferentcolorsshowa1,min(qenc,M2/M1=0.5,eenc)forfixed
firmedinFig.12,whereweplotn forallthecoplanarpro- eenc asindicatedontherightsideoftheplot.
tr
grade parabolic encounters for different M /M . The linear
2 1
decreasehasanapproximatelyconstantslopeirrespectiveof
3.3 Orbits of the transferred planetesimals
the masses of the stars, while is generally lower for higher
massratios(thatisforlargemassesofthestarinitiallywith The transferred particles represent a specific population in
the disc, M ). orbit around their new host M . In this section, we analyze
2 1
Thenumberoftransferredparticlesaswellasthetrans- the orbits of particles transferred in coplanar prograde en-
fer efficiency can in principle depend on the surface num- counters (i = 0◦), these are described in Sects. 3.1 and
enc
ber density profile of the disc particles. As mentioned in 3.2. Orbits of particles transferred during encounters with
Sect. 2.2, we adopted an initial uniform distribution in r non-zero inclination of the disc and the plane of the en-
which corresponds to the surface density ∝1/r. To address counter, i , are described in Sects. 3.4 and 3.5.
enc
this effect, we consider different surface density profiles in In Fig. 13, we show the minimal semimajor axis of
Sect. 3.7. the transferred particles a (which corresponds to the
1,min
MNRAS000,1–20(2002)
10 J´ılkov´a et al.
1.0 M2/M1=0.5,eenc=1.0,ienc=0◦ 0.54 1.0 qenc=300au,eenc=1.0,ienc=0◦ 1.0 qenc=300au,M2/M1=0.5,ienc=0◦
0.48 0.64
0.48
0.8 0.8 0.42 0.8 0.56
0.42
0.36 0.48
0.36
0.6 0.6 0.30 0.6 0.40
0.30
e1 ntr e1 0.24ntr e1 0.32ntr
0.24
0.4 0.4 0.4
0.18 0.18 0.24
0.2 0.12 0.2 0.12 0.2 0.16
0.06 0.06 0.08
0.0 0.0 0.0 0.00
200220240260280300350400450500 0.1 0.2 0.3 0.4 0.5 0.8 1.0 1.2 1.5 2.0 1.0 1.2 1.5 1.8 2.0 2.5 3.0 3.5 4.0 4.5
qenc[AU] M2/M1 eenc
Figure 15.Distributionsoftheeccentricityofthetransferredparticlese1 forcoplanarprogradeencounters(ienc=0◦).Thethreeplots
show eccentricity distributions for different encounter pericentre qenc (Left), mass ratio M2/M1 (Middle), and encounter eccentricity
eenc (Right). Left: Eccentricity distributions for encounters with M2/M1 = 0.5, eenc = 1.0 (parabolic encounters), and qenc varying
alongthehorizontalaxis.Forclarity,thescaleofthehorizontalaxisisarbitrary—valuesofpericentreareequidistantlydistributedover
the horizontal axis. The vertical axis shows the eccentricity of the transferred particles e1 in 10 equidistant bins. The color scale maps
the relative number of transferred particles in each eccentricity bin as measured for individual encounters with different qenc. The red
symbols correspond to the median value of e1. Bullets depict the encounters with completely covered transfer region, while crosses the
encounterswithrtr,max >200au(seeSect.3.1and,e.g.,Fig.10).Middle:Eccentricitydistributionsforencounterswithqenc =300au,
eenc =1.0 (parabolic encounters), and different M2/M1 along the horizontal axis. Right: Eccentricity distributions for encounters with
qenc =300au, M2/M1 =0.5, and different eenc along the horizontal axis. The distribution at the qenc =300 on the horizontal axis of
theLeftplot,M2/M1=0.5oftheMiddleplot,andeenc=1.0oftheRightone,isthesame.
torfanthsfeermreadssorrbaittiowMith/Mthe,manindimtahleepneerrigcye)ntares aoffuthnecteionn- 1.0 qenc=500au,ienc=0◦ M2/M1:
2 1 0.1
counterq fortheparabolicprogradecoplanarencounters
enc 0.2
0.9
(eenc =1.0,ienc =0◦).Thereisacleartrend—asexpected, 0.3
forlargerpericentreandsmallermassratio,thelargera . 0.4
In Fig. 14, we show a as a function of q for diff1e,rmeinnt 0.8 0.5
1,min enc 0.8
e and fixed M /M . The minimal semimajor axis of the
enc 2 1 0.7 1.0
transferredorbitsa1,minisalinearfunctionofthepericentre med
of the encounter qenc and the coefficient of the proportion- e1, 0.6
ality depends on the eccentricity of the encounter e —
enc
largervaluesofe resultinsteeperincreaseofa with
enc 1,min 0.5
q .
enc
0.4
Mostofthetransferredparticlesareoneccentricorbits
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
(Pfalzner et al. 2005b). In Fig. 15, we show the eccentricity
e
distributions of the transferred particles for coplanar pro- enc
grade encounters. Similarly to Pfalzner et al. (2005b, their
Figure 16. Median value of the eccentricity of the transferred
Figure7,Bottom),fortheparabolicencountersthecaptured
particles e1,med as a function of the encounter eccentricity eenc
particles moveoneccentricorbitswithe1 ∼>0.8 irrespective for coplanar prograde encounters with pericentre qenc = 500au.
of the pericentre qenc and the mass ratio M2/M1 (Fig. 15, The lines of different colors correspond to encounters with stars
Left and Middle). The median eccentricity is generally de- ofdifferentmassratiosM2/M1,asindicatedontheright.Bullets
creasing with q (red crosses in Fig. 15, Left) and increas- depict the encounters with completely covered transfer region,
enc
ing with M /M (red crosses in Fig. 15, Middle). For the whilecrossestheencounterswithrtr,max>200au(seeSect.3.1).
2 1
parabolic encounters (e = 1.0, Fig. 15, Left and Mid-
enc
dle), a small fraction of the captured particles (about 5%
generallylowerthan0.5iftheencounterpericentreislarge,
oenrcloesusn)taerlsso(heavel>ow1e.0c)cesnhtorwicnitieins,Fe1ig∼<. 105.2,.RFiogrhht,ypsoemrbeoloicf qenc ∼>350au.
enc
thetransferredparticleshaverelativelyloweccentricity.For
fixed q and M /M , the median value of e is decreasing
enc 2 1 1 3.4 Inclination of the encounter plane
withtheencountereccentricityuptoe ∼2.5,andincreas-
enc
ing again for more eccentric encounters. We find a similar Toexplorehowthetransferefficiencyandthecharacteristics
dependenceirrespectiveoftheencounterpericentreq and ofthecapturedpopulationdependontheinclinationofthe
enc
massratioM /M .InFig.16,weshowthemedianvalueof discplanewithrespecttotheplaneoftheencounteri we
2 1 enc
thetransferredparticles’eccentricitye asafunctionof carried out simulations with i in the range 0–180◦ (see
1,med enc
e for encounters with pericentre q = 500au and dif- Table 1, section varying i ).
enc enc enc
ferent mass ratio M /M . We find that the median of e is In Fig. 17, we show the dependency of the minimal
2 1 1
MNRAS000,1–20(2002)