From star-disc encounters to numerical solutions for a subset of the restricted three-body problem

3.5 MB·

by  Andreas Breslau

Astronomy&Astrophysicsmanuscriptno.AA-2015-26068 (cid:13)cESO2017
January5,2017
From star-disc encounters to numerical solutions for a subset of
the restricted three-body problem
AndreasBreslau,KirstenVincke,andSusannePfalzner
Max-Planck-InstitutfürRadioastronomie,AufdemHügel69,53121Bonn,Germany
e-mail:[email protected]
ABSTRACT
7
Variousastrophysicalprocessesareknown,wherethefly-byofamassiveobjectaffectsmatterinitiallysupportedagainstgravityby
1
rotation.Examplesareperturbationsofgalaxies,protoplanetarydiscsorplanetarysystems.Weapproximatesucheventsassubsetof
0
therestrictedthree-bodyproblembyconsideringonlyperturbationsofnon-interactinglow-massobjectsinitiallyoncircularKeplerian
2
orbits.Inthispaperwepresentanewparametrisationoftheinitialconditionsofthisproblem.Undercertainconditionstheinitial

n positionsofthelow-massobjectscanbespecifiedlargelyindependentoftheinitialpositionoftheperturber.Exploitingadditionally
a theknownscalingsoftheproblemreducestheparameterspaceofinitialconditionsforonespecificperturbationtotwodimensions.To
J thistwo-dimensionalinitialconditionspacewehaverelatedthefinalpropertiesoftheperturbedtrajectoriesofthelow-massobjects
3 fromournumericalsimulations.Thatway,mapsshowingtheeffectoftheperturbationonthelow-massobjectshavebeencreated,
whichprovideanewviewontheperturbationprocess.Comparingthemapsfordifferentmass-ratiosrevealsthattheperturbations
] bylow-andhigh-massperturbersaredominatedbydifferentphysicalprocesses.Theequal-masscaseisacomplicatedmixtureof
P theothertwocases.Sincethefinalpropertiesoftrajectorieswithsimilarinitialconditionsareusuallyalsosimilar,theresultsofthe
E limitednumberofintegratedtrajectoriescanbegeneralisedtothefullpresentedparameterspacebyinterpolation.Sinceourresults
. arealsouniquewithintheaccuracystrivedfor,theyconstitutegeneralnumericalsolutionsforthissubsetoftherestrictedthree-body
h
problem. As such, they can be used to predict the evolution of real physical problems by simple transformations like scaling and
p
withoutfurthersimulations.Possibleapplicationsaretheperturbationofprotoplanetarydiscsorplanetarysystemsbythefly-byof
-
anotherstar.Here,themapsallow,forexample,toquantifytheportionofunboundmaterialforanyperiastrondistancewithoutthe
o
needforfurthersimulations.
r
t
s Keywords. protoplanetarydiscs,planetsandsatellites:formation
a
[

11. Introduction 2005b) and neglect interactions of the disc material with itself,
v
like self-gravity or viscosity, in order to approximate all these
5
Various astrophysical processes are known where matter, ini- casesassubsetoftherestrictedthree-bodyproblem.
5
tially supported against gravity by rotation, is affected by the
8
Theinvestigationofthethree-bodyproblemhasalonghis-
gravitationalforceofapassing-bymassiveobject.Amongthem
0
torystartingwithNewton,Euler,LagrangeandJacobi(seee.g.
0are for example galaxy-galaxy encounters and perturbations of
Marchal1990;Valtonen&Karttunen2006;Musielak&Quarles
.protoplanetarydiscs,debrisdiscs,orplanetarysystemsbypass-
1 2014 and references therein). A well studied subset of the gen-
ing stars (see e.g. Toomre & Toomre 1972; Farouki & Shapiro
0 eral three-body problem is the perturbation of stable two-body
1981;Clarke&Pringle1993;Heller1995;Halletal.1996;Hall
7
systemsbyathirdbody.Thisproblemwasinvestigatedanalyti-
1997;Kobayashi&Ida2001;Melitaetal.2002;Levisonetal.
1
cally(e.g.Hut1983;Heggie&Hut1993)andnumerically(e.g.
:2004; Pfalzner et al. 2005b; Olczak et al. 2006; Scharwächter
vet al. 2007; Malmberg et al. 2011; Craig & Krumholz 2013; Valtonen & Heggie 1979; Hut & Bahcall 1983; Heggie & Hut
XiPunzoetal.2014).Someoftheseprocesseshappenondifferent 1993). The perturbation of low-mass matter due to a fly-by is
related to this problem. It was analytically approximated, for
scalesandmightinvolveadditionalforceslikee.g.viscousfric-
r example, by Ostriker (1994), Larwood (1997), and D’Onghia
ation or magnetic fields. Nevertheless, they all have in common
et al. (2010). However, to date no satisfying general analytical
that approximately at the closest approach of the perturber, the
descriptionofsuchperturbationshasbeenfound.Therefore,pre-
perturber’sgravitationalforcedominatestheevent.Therefore,it
viousstudiesinvestigatingtheinfluenceofanencounteronplan-
isverylikelythatatleastmatterwhichisclosetotheperturber
atpericentrepassageisinallcasesaffectedinasimilarway. etarysystemsortheglobalpropertiesofdiscs(e.g.energy,mass,
size)usuallyperformedstatisticalnumericalsimulationscover-
Forsimplicity,wefocushereoncaseswhererelativelylow- ing certain subsets of the large space of encounter parameters
massmatterorbitsamassiveobject(thehost)andisperturbedby (e.g.Halletal.1996;Hall1997;Pfalzneretal.2005b;Fregeau
anothermassiveobject(theperturber).Thiscanbe,forexample, et al. 2006; Steinhausen et al. 2012; Hao et al. 2013; Breslau
asingleplanet,oradiscofgasordebrisorbitingastarwhichis etal.2014;Li&Adams2015).
perturbedbyanotherstarflyingby.Incaseofadisc,wefollow
theapproachofseveralpreviousstudies,(e.g.Toomre&Toomre Alsobasedonnumericalsimulations,someapproacheswere
1972; Hall et al. 1996; Kobayashi & Ida 2001; Pfalzner et al. madetogeneralisetheeffectofsuchperturbations.Therein,usu-
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A&Aproofs:manuscriptno.AA-2015-26068
ally properties of the perturbed system were related to initial For simplicity of description, we consider a disc of mass-
properties. lessparticlesinitiallyorbitingcounter-clockwiseoncircularKe-
Clarke & Pringle (1993) and Hall et al. (1996) investigated plerian orbits in the x-y-plane around a host with mass M
h
the perturbation of protoplanetary discs in equal-mass encoun- in the centre of the coordinate system. These particles shall
ters and presented the probability for a disc particle to have a be perturbed by a perturber of mass M . The actual orbit of
p
certainfate(remaining bound,capturedbyperturber, unbound) the perturber depends on the mass-ratio of host and perturber,
depending on its initial distance to the host, r , normalised to m= M /M ,theeccentricityoftheorbit,e ,thepericentredis-
init p h p
the perturber’s periastron distance, r . 1 The perturber’s pe- tance,r ,andtheplaneoftheorbitrelativetotheorbitofthe
p,peri p,peri
riastron distance means here and in the following the distance particles.Wefurtherrestricttocaseswheretheorbitofthepar-
betweenhostandperturberatthemomentwhentheperturberis ticles and the orbit of the perturber are coplanar and prograde,
inthepericentreofitsorbit. thusreducingtheproblemtotwodimensions.Thelinefromthe
In a similar scenario, Hall et al. (1996) identified the parti- hosttothepericentreoftheperturber’sorbitdefinesthex-axisof
cleswhichsufferthelargestenergychangeduringanencounter ourreferencesystem.Theparticles’gravitationaleffectonhost
andlocatedthemwithinthedisc(seetheirFigures4a,5,and6). andperturberisassumedtobenegligible.
They found that shortly before periastron passage, these parti- Likeinsomepreviousstudies(e.g.Toomre&Toomre1972;
clesformedfourdistinctgroupswhichinteractwiththehostand Clarke&Pringle1993;Halletal.1996),wescalealllengthsof
perturber in different ways. However, since the disc is already theproblembysettingr =1.Theinitialradiusofparticlei’s
p,peri
deformed at the time of periastron passage a description of the orbit,r ,isthengiveninunitsofr .
i,init p,peri
spatialdistributionoftheseparticlesproveddifficult. Weassumetheperturbertopassthehostonaparabolicor-
Toomre & Toomre (1972) performed simulations of galaxy bit. Therefore, their distance is at t = −∞ and t = ∞ in princi-
perturbationswithnon-interactinglow-massdiscparticles.Since pleinfinite.Inthemomentofpericentrepassagethedistanceis
they modelled the gravitational potential of the involved galax- smallest,namelyr ,andtheperturber’sinfluenceonthehost
p,peri
iesbypointmasses,theirresultsarecomparabletosimulations isstrongest.Thegravitationalforceoftheperturberonthepar-
of low-mass discs around stars. Similar to Clarke & Pringle ticleisstrongestclosetopericentrepassage.Asitisimpossible
(1993)andHalletal.(1996),theyhaveshownthefates(remain- tocovertheperiodfromt = −∞tot = ∞inasimulation,this
ing bound, captured by perturber, unbound, forming tails and a factcanbeusedtorestricttheinvestigationoftheperturbationof
bridgebetweenhostandperturber)ofthediscparticles,butde- theparticle’sorbittoatimeperiodaroundpericentrepassage3.
pendingon“thepositionstheywouldhavereachedattimet=0 Theperturbedparticletrajectoriesobtainedforonespecificper-
intheabsenceofforcing”.With“absenceofforcing”theymean turberorbitconvergewithincreasinginitialseparationbetween
withouttheforcefromtheperturberontheparticlesandintheir perturberandhost.
setup,t = 0isthemomentoftheperturberspericentrepassage Due to this convergence, numerical investigations of the
(seetheirFigure15andthedescribingtext). perturbation process should start when the perturber’s influ-
Using a similar approach like Toomre & Toomre (1972), ence on the particles is still negligible. At that time (t = t )
init
we developed a new parametrisation of the initial conditions the perturber’s position shall be given in polar coordinates by
of the problem. For a given perturbation 2 the resulting two- (r ,θ ).Duetoabove’scondition,theperturberisthenfar
p,init p,init
dimensional initialcondition spaceallows creatingmaps of the outsidetheinitialparticleorbits,r >> r .Foraparabolic
p,init i,init
properties of the system, like e.g. final values. This new view orbit the time of flight from this initial position to pericentre is
on the problem gives a better understanding of the underlying givenby4
perturbationprocessitself.Sincethesolutionforonesetofini-
(cid:115)
tial conditions is unique, once obtained, numerical results can 2r3 (cid:16) (cid:17)
beeasilyappliedtodifferentphysicalproblemsbysimpletrans- t −t = p,peri E−e sinE (1)
peri init µ p
formations,likee.g.scaling,makingadditionalsimulationsun-
necessary. This reduces the computational effort of parameter
withthestandardgravitationalparameterforthetwo-bodyprob-
studiesinthiscontextsignificantly. lemµ=G(M +M )andtheeccentricanomaly
This paper focuses on the presentation of the new method h p
which is described in Sect. 2 along with the used numerical e −cosθ
scheme. In Sect. 3 some exemplary results of the new method E =arccos p p,init . (2)
1+e cosθ
areprovidedintheformoftheabovementionedmaps.Thelim- p p,init
itationsofthemethodalongwiththeapplicationtorealphysical
Since the particles orbit the host initially on circular Kep-
systemsarediscussedinSect.4.
lerian orbits with radii r , their initial angular velocities are
i,init
givenby
2. Method
(cid:115)
M
Ournewmethodisbasicallyarefinementoftheapproachused ω = G h , (3)
i,0 r3
byToomre&Toomre(1972)fortheirFig.15.Sincetheymen- i,init
tioned it only in passing and without complete description, we
willfirstexplaintheirapproachanddescribethenhowwedevel- whereG is the gravitational constant. The particles’ orbital pe-
opeditfurther. riods are Ti = 2π/ωi,0. Without the influence of the perturber,
1 Variableswiththeindex’p’denotepropertiesoftheperturberorits 3 Indeed, this fact is used in all numerical investigations of gravita-
orbit.Variableswithoutthisindexusuallyrefertotheparticle. tionalperturbationstorestrictthesimulationtime(e.g.Halletal.(1996)
2 A certain perturbation is here defined by the mass of the perturber starttheirinvestigationoftheperturbationofadiscwithaninitialdis-
relativetothemassofthehost,m,theeccentricity,e ,oftheperturber tanceofatleast10timesthediscradius).
p
orbitandtheorientationoftheplaneoftheperturberorbitrelativeto 4 The mathematical bases can be found for example in Bate et al.
theplaneoftheorbitinglow-massobject. (1971)orAppendix1.5ofKemble(2006).
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AndreasBreslauetal.:Fromstar-discencounterstonumericalsolutionsforasubsetoftherestrictedthree-bodyproblem
thepositionsoftheparticlesaregivenforalltimesbytheinitial (a)
radiusr andtheangle
i,init
θ(t)=θ +tω . (4)
i i,0 i,0
During the period, the perturber needs to reach pericentre,
t − t , a particle being initially, at t=t , at the position
peri init init
(θ ,r ), would move without the perturber’s influence to
i,init i,init
(θ + ω (t − t ), r ) 5. This is the position Toomre
i,init i,0 peri init i,init
& Toomre (1972) called “the position [it] would have reached
at time t = 0 in the absence of forcing”. Since the particle will
neverbeatthisposition,wewillcallitVirtualPericentrePosi-
tion (VPP) in the following. This position is obtained from the
initialpositionbythetransformation:
(cid:104) (cid:105)
θ = θ +ω (t −t ) mod 2π (5)
i,vpp i,init i,0 peri init
r =r . (6)
i,vpp i,init (b)
Figure 1 shows this exemplary for a disc of particles per-
turbedbyaperturberwithmass M = M .Theperturberpasses
p h
thehostonaparabolicorbitcomingfromthebottomoftheim-
age,passingpericentreat(1,0),andleavingatthetopoftheim-
age (see black line in Fig. 1a). The particles are coloured by
their final fates: blue particles remain bound to the host, green
particlesbecomecapturedbytheperturber,andredparticlesbe-
come unbound. Fig 1a shows the particles when the perturber
has nearly reached pericentre (perturber position indicated by
blackdotatthebottomoftheimage).Duetotheinfluenceofthe
perturber,thediscisalreadydeformedatthatmoment.
Figure1bshowsthesamefort=t .Duetothedifferential
init
rotationwithinthedisc,thepatternfromFig.1aappears“wound
up“. In contrast to Fig. 1a the disc is not yet deformed by the
influenceoftheperturber.
Figure1cshowstheparticlesattheirVPPs.Inthiscase,the
(c)
disc is neither deformed nor is the pattern from Fig. 1a wound
uplikeinFig.1b.
2.1. Newparametrisationoftheinitialconditions
Insteadofsettingupsimulationsbysamplingtheinitialpositions
of the particles (at t = t ) randomly and plotting the particles
init
laterattheVPPsasshownforexampleinFig.1c,weinvertthe
transformation given by Eq. (5) and Eq. (6) to parametrise the
initial conditions. This is analogue to the definition of the im-
pact parameter for the description of scattering processes, e.g.
of a test-particle in the field of a single attracting body. The
impact parameter describes at which distance the test-particle
would pass the target without being influenced. Similarly, we
define the initial particle positions by the positions where the
particles would be when the perturber is in pericentre without
the perturbation. For a given initial distance between host and
perturber,theinitialparticlepositionsareobtainedbysampling Fig. 1: Example of the Virtual Pericentre Positions for a per-
theVPP-spaceandtransformingthecoordinateswith turbeddisc.Theperturberhasthesamemassasthehostandthe
pericentre of the parabolic, prograde orbit is at (1,0). In a) and
(cid:104) (cid:105)
θ = θ −ω (t −t ) mod 2π (7) b)thediscisseenfaceonwhentheperturberhasnearlyreached
i,init i,vpp i,0 peri init
r =r . (8) pericentreandatt=tinit,respectively.Inc)theVPPsofthedisc
i,init i,vpp particlesareshown.Forfulldescriptionseetext.
2.2. Thenumericalsimulations
dividedintobinsandsampledwithparticles,eachinthecentre
ForoursimulationswesampledtheVPPspacespannedby x
vpp
of the respective bin. For the figures presented in Sect. 3, the
and y with mass-less tracer particles. Both dimensions were
vpp region x /r ∈ [−5;5], y /r ∈ [−5;5] was sampled
vpp p,peri vpp p,peri
t5heTpahritsicploeswitiilolnbeshaotutld=ntpoetribweitchotnhfeuspeedrtuwribthati(oθni(.tperi),ri(tperi)),where pweirtthur5b0e0r·w5i0th0mpaasrtsicMleps.=Tmhemmovaesssoonf athpearhaobsotliicsoMrbhit=(ep1,=th1e)
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with a pericentre of r =1. The trajectory of each particle 3.1. Low-massperturbation
p,peri
wasintegratedindividuallywiththeadaptiveLSODAintegrator
Figure2showsthemapsfortheregionwithr /r < 5for
fromtheODEPACKlibrary(Hindmarsh1983). vpp p,peri
themass-ratiom = 0.1.Thiscorrespondstotheperturbationof
Theinitialpositionoftheperturberwasdeterminedindivid- adiscwithinitialradiusofr =5r ,oraperturbationwith
uallyforeachparticle.Tofulfiltheclaimthattheinitialinfluence r =0.2r respectively.dTischepartpi,cpelreiswereorbitingthehost
oftheperturberisnegligible,theinitialdistanceoftheperturber p,peri disc
(indicatedbytheblackdotat(0,0))initiallycounterclockwise.
waschosensothattheforcefromtheperturberontotheparticle,
Theperturber’spericentreisat(0,1).
F ,wassmallrelativetotheforcefromthehost,F :
p h Figure2ashowstheparticlefates:particleswithVPPsinthe
blue region remain bound to the host, particles from the green
Fp/Fh <(cid:15). (9) region6arecapturedbytheperturber,andparticlesfromthered
regionbecomeunbound.TheblacklineinFig.2ashowstheorbit
This maximum initial influence of the perturber deter- of the perturber – the perturber came from the bottom edge of
mines the initial position of the perturber. For a particle with theimage(asindicatedbytheblackarrow),movedto(1,0),and
(cid:113)
r = x2 +y2 , the minimum initial distance between par- leftacrossthetopedge.Thewhiteregionsindicateparticlesfor
vpp vpp vpp whichtheaccuracyachievedwiththeusedintegrationmethodis
ticleandperturber,dp,followsfromEq.(9): notsufficientandwhicharethereforenotshown(seebelowfor
furtherexplanationandSec.4).
(cid:112)
d =r m/(cid:15). (10) It is often assumed that there is a simple radial dependence
p vpp
onwhetherparticlesremainboundornot.Fig.2showsthatthis
is certainly not the case, but a much more complicated depen-
Forpracticalreasons,theinitialdistancebetweenhostandper-
turberissettor = d +r .Fromr andtheeccentricity denceexists.OnecanseethatonlyparticleswithVPPsinanar-
p,init p vpp p,init row band along the approaching branch of the perturber’s orbit
oftheperturberorbit,e ,followsθ accordingto
p p,init (black line in Fig. 2a) become unbound. On the side of the de-
partingbranchoftheperturber’sorbit,particleswithVPPsfrom
r = rp,peri(1+ep) (11) abroaderregionbecomeunbound.Theonlyregionfromwhere
p,init 1+e cos(θ ) particlesarecapturedbytheperturberisclosetotheperturber’s
p p,init
pericentre (green area around (1,0)). Due to the low perturber
andthusthetimeuntilpericentrepassage,t −t ,withEq.(1) mass-ratio the region from where particles become unbound is
peri init
and (2). After determining the time until pericentre passage of relativelysmall.
theperturber,theinitialpositionoftheparticlewasobtainedac- AscanbeseeninFig.2a,theshapeoftheperturberorbitcan
cordingtoEq.(7)andEq.(8). not be used to explain e.g. the shape of the region from where
Tests have shown that for (cid:15) = 10−4 the particles are suffi- particlesbecomeunbound.Thereasonisthattheperturberorbit
shows all positions of the perturber for a time span while the
ciently “unperturbed” at the onset of the simulations. The ob-
mapshowsthefatesofparticleswhichareatacertainpositionat
tained accuracy will be discussed in Sect. 4. With these initial
acertainmoment.Evenundertheassumptionthattheparticles
conditionstheperturberisinitiallyatmuchlargerdistancesfrom
are not influenced by the perturber at all, particles “meeting”
the host than in most simulations of this kind performed so far
theperturberarenotshownneartheperturberorbitinthemap.
sinceinearlierstudiesthecomputationalresourcesnecessaryfor
Whentheperturberisatthecoordinates(r ,θ )atthetimet ,a
alargernumberofsuchsimulationswerenotavailable. 1 1 1
particlebeingexactlyatthispositionatthattimeisshowninthe
Duringthesimulationitwasregularlytestedwhetherthepar- (cid:113)
ticle had settled into a stable orbit. Here, possible stable, final mapat theposition(r ,θ + µ/r3(t −t )). Thesepositions
1 1 1 peri 1
orbits are either bound orbits (e ≤ 1) around host or perturber, are highlighted for all t with the black line in Fig. 2b and are
1
or unbound orbits (e > 1) with the centre of mass. A bound referredtointhefollowingastheperturber’s“interactionorbit”.
orbit with the centre of mass is only a transition state. An or- Figure 2b shows that the “interaction orbit” matches the
bit was accepted as stable when one of above’s conditions was shapeoftheregionfromwhereparticlesbecomeunboundmuch
fulfilledandtheorbitalelements(eccentricity,semi-majoraxis, better than the real perturber orbit. Comparing Fig. 2a and 2b,
argumentofperiapsis,...)calculatedfortherespectiveorbitdid one can also see that part of the white dots in Fig. 2a are very
not change more than 1 % between two tests. The duration be- closetotheinteractionorbit.ThismeansthatparticleswithVPPs
tweentwotestswasusually(tperi−tinit)/50.Whentheorbitwas intheseregionsaredirectlyonthewayoftheperturberandap-
stable,theintegrationoftheparticle’strajectorywasstopped. proachtheperturberveryclosely.Thisresultsinintegrationer-
rors,whytheseparticlesareexcludedfromtheresults.
Besidesthefatesoftheparticles,theshapesoftheirfinalor-
3. Results bits are important to describe the outcome of the perturbation
process.Theshapesoftheorbitsaredefinedbytheireccentrici-
Fromoursimulationswehavedeterminedthefatesoftheparti- tiesandsemi-majoraxes.Thesetwopropertiescanalsobetrans-
cletrajectories:whethertheparticlesremainboundtothehost, ferredintoenergyandangularmomentumtoobtainanotherof-
are captured by the perturber, or become completely unbound. tenusedviewontheperturbationprocess.
Additionally,thefinalorbitalelementsoftheparticleswerede-
Fortheparticleswhicharefinallyboundtohostorperturber
termined. They define the final orbits completely. These final the final eccentricities (e < 1) and semi-major axes relative to
properties can be presented as maps depending on the particles
theinitialradiioftheirorbitsareshowninFig.2candFig.2d.
VPPs. Because we want to present the new parametrisation of
Since the unbound particles are of minor importance for most
the initial conditions along with some consequences here, we
show only some of the final properties for three sample mass- 6 To improve the readability we will use “particles from ... region”
ratios. synonymousfor“particleswithVPPsin...region”.
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AndreasBreslauetal.:Fromstar-discencounterstonumericalsolutionsforasubsetoftherestrictedthree-bodyproblem
(a) (b)
(c) (d)
Fig. 2: Maps of the final properties of the particles depending on their VPPs for the region r /r < 5 for a mass-ration of
vpp p,peri
m=0.1.
a)andb)showwhethertheparticlesarefinallyboundtohostorperturber,orbecomeunbound.a)showsadditionallytheorbitof
theperturberthroughtheCartesianspaceandb)theperturber’s“interactionorbit”(seetextforexplanation).
c)showsthefinaleccentricitiesoftheboundparticlesandd)showsthefinalsemi-majoraxesoftheboundparticlesrelativetothe
radiioftheirinitialorbits.
applications,theirfinalorbitalelementscanbefoundinFig.A.1 agrees with the finding of Pfalzner et al. (2005a) that the cap-
intheappendix. turedparticlesareusuallyonverytightorbits.
Fortheshownmass-ratioonecangeneralisethat,ascouldbe
Thefinaleccentricitiesoftheparticleswhichremainbound
expected, the closer the VPP of a particle to the perturber’s in-
to the host can be considered as nearly symmetrical to the x-
teractionorbit(seeabove)thestrongeritisinfluenced:particles
axis (see Fig. 2c). Close to the region from where the particles
with VPPs very close to the interaction orbit become unbound,
becomeunbound(whiteregioninFig.2c),theeccentricitiesare
particles with VPPs far away from the interaction orbit remain
very high, up to e = 1. With increasing distance to the white
boundtothehostwithrelativelyloweccentricities.
region, the eccentricities decrease. Around the host, there is an
Sincetheparticleswereinitiallyrotatingcounter-clockwise,
approximatelyroundareawheretheparticlesarestilloncircular
theclearedarea(redareainFig.2a)mostlylagsbehindtheinter-
orbits,i.e.theeccentricitiesarecloseto0.
action orbit. Particles which move ahead 7 of the perturber are
Figure2dshowsthesemi-majoraxesoftheparticleswhich decelerated by its gravitational attraction. Those particles lose
are finally bound to host or perturber relative to the radius of part of their energy and/or angular momentum, and remain fi-
the initial particle orbit around the host. The particles from the nallyoftenonaneccentricorbit,boundstrongertothehostthan
greenarea(whichindicatesvaluesaround1)abovethenegative before. In contrast, particles which move behind the perturber
x-axis have final semi-major axes which are close to the ini- are accelerated. If the acceleration is strong enough they gain
tial ones. Only particles from a small region close to pericen- enough energy to reach escape velocity and become unbound.
tre (yellow to red region approximately around (0.8,-0.6) have The rear (in direction of rotation) borders of the red region in
finally larger semi-major axes than initially, these particles are Fig. 2a mark therefore those particles which gain enough en-
transportedoutwards.Particlesfromallotherregions,especially ergytoexactlyreachescapevelocity.Theparticlesbeyondthese
along the cleared area and those particles captured by the per-
turber,havefinallysmallersemi-majoraxes,sotheyaremoved 7 Ahead and behind in the following mean ahead and behind in the
inwards (cyan to blue regions). For the captured particles, this directionofrotation.
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(a) (b)
(c) (d)
Fig.3:LikeFig.2butforamass-ratioofm=20.0.
linesgainlessenergyandremainboundtothehostonorbitsof orbit. In case of a prograde encounter, the particles outside the
decreasing eccentricity with increasing distance from the lines. host’s orbit (see also Fig. 4) have a higher velocity relative to
Since particles ahead the interaction orbit are decelerated and theperturberduetotheirsuperimposedrotationaroundthehost.
remainboundtothehostwhileparticlesbehindareaccelerated Sincetheperturber’sattractionisweakeroutsidethehostsorbit,
and may become unbound, the perturber’s interaction orbit de- these particles become unbound. This is the case for the parti-
scribesapproximatelythefrontborder(indirectionofrotation) cles in the upper “half” of your parameter space. The removal
oftheclearedregion. of these particles can be described as “centrifugal clearing”.
The particles inside the host’s orbit have lower velocities rela-
3.2. High-massperturbation
We show next the extreme case of a perturber with a relative
massofm = 20.AscanbeseeninFig.3,theresultsdiffercon-
siderably from the previous case as now most of the material
fromtheinvestigatedparameterspaceisremoved,eitherbybe-
comingunboundorbybeingcapturedbytheperturber.Onlypar-
ticleswithVPPsinrelativelysmallregionsremainboundtothe
host(blueregionsinFig.3a).Theseregionsaremainlyaround
thehostandalongthenegativeandpositivex-axis.Interestingly,
onlyveryfewparticleswithVPPsbetweenr /r ≈0.3and
vpp p,peri
r /r ≈ 2 remain bound to the host. Most of the particles
vpp p,peri
from the upper half of the parameter space become unbound,
fromthelowerhalfmostparticlesarecapturedbytheperturber.
Figure. 3b shows that the “interaction orbit” (see Sec. 3.1)
doesnotmakemuchsenseforthiscase.Thisisaconsequenceof
differentdominatingphysicalprocessesfordifferentmass-ratios. Fig.4:Relativeorbitsincaseofahigh-massperturber.Thecen-
Incaseofahigh-massperturber(m(cid:29)1),theperturberrests tre of mass is indicated by the cross. The solid line depicts the
orbit of the host and the dashed line the orbit or the perturber.
closetothecentreofmasswhilethehostandtheparticlesmove
The grey circle denotes the low-mass material and the arrows
onawideorbit.Thedistancebetweenhostandperturber,andthe
showtherotationdirections.
relative velocity of the host is thereby defined by the parabolic
Articlenumber,page6of13
AndreasBreslauetal.:Fromstar-discencounterstonumericalsolutionsforasubsetoftherestrictedthree-bodyproblem
tivetotheperturberthannecessarytoovercometheperturber’s are finally bound to the host with relatively large semi-major
gravitationalattractionattheirpositions.Theseparticlesbecome axes.Theseregionsarecontinuationsoftheregionfromwhere
mainlycapturedbytheperturber. theparticlesare“centrifugallyremoved”.Here,theenergygain
Likethe“interactionorbit”helpedtoexplainthefinalprop- isjustnotenoughtoleavethegravitationalfieldofthehost.
ertiesoftheparticlesinthelow-massperturbationcase,thispro- Althoughanencounterbetweenstarsofequalmasswasof-
cesshelpstoexplainthemforthehigh-masscase.Especially,it tentakenasastandardcasetoinvestigatee.g.theeffectondisc
explainswhytheparticlesfromtheupper“half”ofourparame- properties(e.g.Clarke&Pringle1993;Halletal.1996)thefate
terspacearedifferentlyaffectedduringtheperturbationthanthe andfinalpropertiesoftheparticlesarenoteasytoexplain.This
particlesfromthelower“half”. isbecausethiscaseisratheracomplicatedmixtureoflow-and
Like in the low-mass case (see Sect. 3.1), particles from an high-massencounters.
approximatelycircularregionaroundthehosthavefinallyrela-
tivelyloweccentricities(e≈0,seeFig.3c).Incontrast,mostof
3.4. Finalorbitalelementsoftheboundparticles
theparticleswhicharecapturedbytheperturberhaveveryhigh
eccentricities. In the Figures 2, 3, and 5 it can be seen that there are large re-
Nearlyallparticleswhichremainboundtothehosthavefi- gions in the investigated parameter space where particles have
nally semi-major axes which are significantly smaller than the thesamefate.Furthermore,similarinitialconditionswithineach
radii of their initial orbits around the host (see Fig. 3d). Only oftheseregionsusuallyresultinsimilarfinalorbits,indicatedby
particleswhichalmostbecomecapturedbytheperturberorbe- thesmoothcolourgradientsintheimageswiththefinalorbital
comeunboundremainboundtothehostwithlargerfinalsemi- elements of the particles (e.g. Fig. 2c and 2d). This knowledge
majoraxes(e.g.theredregionat≈(3,0.8)to(5,0.8)inFig.3d). can be used to interpolate the final orbital elements for initial
Alsotheparticlescapturedbytheperturberhaveingeneralfinal conditionsbetweenthenumericallyintegratedpoints.
semimajor axes which are smaller than the radii of their initial Fortheinnerpartsoftheinvestigatedparameterspacewhere
orbits around the host. An exception are the particles from the allparticlesremainboundtothehost,thefinalorbitalelements
greenregionaround(4,1.5)inFig.3a. oftheparticlesfollowrelativelysmoothfunctionsoftheradius
Asignatureofthe“centrifugalclearing”canalsobeseenin andtheangle,θ .Asanexample,inFig.6thefinaleccentric-
vpp
thefinalperiapsideswiththecentreofmassoftheunboundpar- itiesareshownforsomesampleradii(r /r )asfunctionof
vpp p,peri
ticles (see Fig. A.1d in the appendix). The particles which are θ . In contrast to the data for Fig 2, 3, and 5, these data were
vpp
subject to the “centrifugal clearing” are ejected roughly in the obtainedfromsimulationswheretheparameterspacewassam-
direction they were moving shortly before periastron passage, pledinpolarcoordinates.Thedataarenotsmoothed.
whichisfromrighttoleft(seealsoFig.4).Theirhyperbolicor- Forafixedθ ,thefinaleccentricitiesarehigherforlarger
vpp
bitsoriginatebasicallyfromthepositionswheretheywereatthat initial radii, as could be expected. Around θ ≈ 0 the final
vpp
time. Therefore, the periapsides of these particles can approxi- eccentricitiesareincreasedrelativetothevaluesoutside−90 (cid:46)
mately be explained with the initial radii of their orbits around θ (cid:46) 90. Thevaluesseem tobea simplefunctionof θ and
vpp vpp
thehostplusthedistancebetweenhostandperturberatthemo- r /r .Butthederivationofthisdependencyliesbeyondthe
init p,peri
mentofejection,whichis≈1.Forexample,aparticlefrom(0,1) scopeofthispaper.
hasaninitialorbitalradiusof1andafinalperiapsisof≈2.
4. Discussion
3.3. Equal-massperturbation
We have presented a method to obtain general numerical solu-
Comparingtheresultsfortheequal-masscase(m = 1)withthe tionsfortheperturbationoflow-massobjectsoncircularKeple-
low-mass and the high-mass case, one can see (in Fig. 5) that rianorbitsaroundamassiveobjectbythefly-byofanothermas-
theequal-masscaseshowsfeaturesofbothothercases.Likein sive object. Like for all numerical solutions the quality of the
thelow-masscase,particlesfromabandalongtheapproaching resultsdependsonthechosenspacialandtemporalresolution.
branchoftheperturberorbitbecomeunbound.Duetothehigher The temporal resolution is limited by the used integrator.
perturbermassthebandisbroader.Therearborderofthisarea Thehere-usedLSODAintegratorisawell-establishedintegrator
depictsagaintheregionwhereparticlesgainexactlyescapeve- witherrorcontrolledtimestepsize.Asmaximumrelativeerror
locityandthefrontbordercanalsoapproximatelybedescribed oftheintegratedvalues,thebuild-indefaultof≈ 1.5·10−8 was
withtheperturber’sinteractionorbit(seeFig.5b). used.Sincethetrajectoryofeachparticlehasbeenintegratedin-
Thered,drop-shapedregionaround(-2,1)inFig.5ais“cen- dividually,theobtainedtrajectoriesformostparticlesaremuch
trifugally cleared” like in the high-mass case. Due to the lower more precise than e.g. those of Breslau et al. (2014). Only for
perturber mass the region is smaller. All other features, espe- some particles which approach one of the massive objects so
cially the red and green regions in the right half of Fig. 5a, are close that very small time steps are required, the accuracy was
caused by repeated interactions of the particles with host and deemedinsufficient.Therefore,particleswhichapproachoneof
perturber. thetwomassiveobjectscloserthan0.0001·r havebeenex-
p,peri
Incontrasttothelow-masscase,thefinaleccentricitiesand cludedfromtheshownimages(seeSect.3.1).
semi-majoraxesoftheparticles(seeFig.5cand5d)donotshow Other potentially critical factors for the accuracy of the re-
anysymmetry.However,thefindingthatclosetotheclearedre- sults are the duration of the simulation before (t −t ) and
peri init
gions, the final eccentricities are very high (e ≈ 1), still holds. after (t −t ) pericentre passage of the perturber. By vary-
end peri
Thenearlyunperturbedregionaroundthehostissmallerthanin ingtheinitialforceinfluenceoftheperturber(seealsoSect.2.2)
thelow-masscaseandbiggerthaninthehigh-masscase. andhencet −t ,weinvestigatedtheinfluenceofthestarting
peri init
Most particles which remain bound to the host or are cap- time on the results. We found an initial relative force influence
tured by the perturber have final semi-major axes, which are ofF /F (cid:46)10−4tobesufficientforourpurpose.
p h
smaller than the radii of their initial orbits. Only particles with Theresultsconvergealsofort −t →∞.Byintegrating
end peri
VPPsintheredregionsaround(-4,2)and(-1.5,2.5)inFig.5d theparticletrajectoriesindividuallyuntiltheorbitalelementsdo
Articlenumber,page7of13
A&Aproofs:manuscriptno.AA-2015-26068
(a) (b)
(c) (d)
Fig.5:LikeFig.2butforamass-ratioofm=1.0.
not change more than 1 % during a certain time span (see also ble for a period after the perturbation which is shorter than the
Sect.2.2)weensurethatourresultsarealsoconvergedwithre- viscous timescale. Depending on the strength of viscosity, dur-
specttot . ingthisperiodtheorbitalelementsofthematerialmightdeviate
end
Forcaseswheretheorbitsofthelow-massparticlesarenot considerablyfromourresultssincetheinfluenceoftheperturber
exclusivelydeterminedbythemassesofhostandperturber,e.g. isnotyetnegligibleandtheparticleshavenotyetsettledintosta-
duetoself-gravityand/orviscosity,ourresultscannotdirectly bleorbits.Afterthisshortperiod,theviscousevolutionleadsto
beapplied.Sinceinthesecasestheinitialangularvelocityofthe furtherchangeoftheorbitalelementswhichmakesacomparison
particlesdoesnolongerdependonlyonthedistancetoandthe difficult.Additionally,predictionsfromourresultsmightonlybe
massofthehost,theparticle’sVPPscannotbedeterminedan- possibleformaterialwhichdoesnotmovethroughregionswith
alytically.However,aslongasadditionalforcesaresmallcom- higherdensityduringandaftertheperturbation.
paredtothegravitationalforceoftheperturberaroundpericen-
tre,itisstillpossibletomakesomepredictionssincethephysical
processeswillbesimilar. 5. Applicationtoplanetarysystemsand
Forexample,forself-gravitating,non-viscousdiscsitcanbe non-viscousdiscs
expected that a low-mass perturber clears also only a narrow
band along its orbit, depending on the range of its force. The Theresultsoftherestrictedthree-bodyapproximationpresented
eccentricities of the remaining particles will also decline with here are directly applicable to stars surrounded by low-mass
increasingdistancetotheperturber’sinteractionorbit.Similarly, discs,eitherprotoplanetaryordebris,orplanetarysystemswhich
evenself-gravityandviscositycanonlyweakenbutnotprevent are disturbed by the fly-by of a second star. However, this ap-
the “centrifugal clearing” of the upper half of our parameter proachisonlyvalidifviscousforcescanbeneglected.Further-
space in case of a high-mass perturber (see also Sec. 3.2). Es- more,thematterhastobeoncircularorbitsbeforetheperturba-
pecially viscosity does not provide a strong enough additional tion.
attracting force. The repulsing effect of viscosity has also no For the application to discs it is important to note that the
impacthere,sincethematerialisremovedinadirectionwithout presenteddatasetscontaininformationwhathappenstodiscma-
othermaterial. terial for all encounters from penetrating (with r =0.2r
p,peri disc
Incaseofviscousdiscs,thegradualexchangeofenergyand ⇔ r /r = 5 ) up to very distant encounters. As already
disc p,peri
angularmomentumleadstoprocessesinthedisclikeaccretion pointedoutbyToomre&Toomre(1972),theresultsfordiffer-
anddiscspreading,evenwithoutaperturbation.Therefore,pre- entinitialdiscsizesarecontainedinourdatasetassubsets.When
dictions for viscous discs from our results may only be possi- investigating the effect of an encounter on a disc with a certain
Articlenumber,page8of13
AndreasBreslauetal.:Fromstar-discencounterstonumericalsolutionsforasubsetoftherestrictedthree-bodyproblem
(a) (a)
1.0
rvpp/rp,peri
0.8 0.56
0.52
y
ntricit0.6 0.46
e 0.41
c
c
e0.4
al
n
fi
0.2
0.0
-180 -90 0 90 180
θvpp[◦]
(b)
1.0
rvpp/rp,peri (b)
0.8 0.35
0.31
y
ntricit0.6 0.28
e 0.25
c
c
e0.4
al
n
fi
0.2
0.0
-180 -90 0 90 180
θvpp[◦]
(c)
1.0
rvpp/rp,peri
0.8 0.15
(c)
0.13
y
ntricit0.6 0.12
e 0.11
c
c
e0.4
al
n
fi
0.2
0.0
-180 -90 0 90 180
θvpp[◦]
Fig. 6: Final eccentricities of particles in the central re-
gion depending on the VPPs for the mass-ratios m=0.1(a),
m=1.0(b),andm=20.0(c).
size,onlytheradialsubsetwhichcorrespondstotherelativesize Fig.7:Sampleapplicationoftheparticlefatesfornon-viscous,
oftherespectivedischastobeconsidered. low-mass discs with different initial sizes perturbed by a per-
If e.g. a disc with an initial outer radius of 300 AU turberwithamass-rationofm=1.0anddifferentpericentredis-
is perturbed by a star with a periastron of 100 AU, tances.ThescaleisthesameasinFig.5forbettercomparison.
one would only consider the parameter space up to Forfulldescriptionseetext.
r /r =300AU/100AU=3. For the equal mass case
init p,peri
this is illustrated in Fig. 7a, which shows only the inner area
of Fig. 5a. Figure 7b shows the same results for a disc with
an initial outer radius of 150 AU perturbed by a star with a 7c shows the results for a disc with an initial outer radius of
periastronof50AU.Sincethewholeproblemscaleswithsize, 200AUperturbedbyastarwithaperiastronof100AU.Itcan
the result is exactly the same as for the previous case. Figure beseenthattheresultistheinnerpartofFig.7a.
Articlenumber,page9of13
A&Aproofs:manuscriptno.AA-2015-26068
simulationswheretheparameterspacewassampledinpolarco-
100
bound ordinates.
%] 80 unbound
[
ate captured 6. Summary
f
al 60
fin Inthiswork,wepresentedanefficientwaytogeneralisethere-
yfor 40 sults of a perturbation of low-mass bodies initially on circular
abilit Kotehpelremriaanssoivrebiotsbjaercotu.nIndcaonmtraassstivteoporbejveciotubsyintvheestfligya-btiyonosf,oaun-r
prob 20 newparametrisationmaintainsnotonlytheradialresolutionfor
the particles but also the angular resolution. By exploiting also
0
theotherknownscalingrelationsoftheproblem,theinitialcon-
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ditions for the low-mass bodies for one specific perturber orbit
rinit/rp,peri cannowbespecifiedwithjusttwoparameters.Wehavecreated
maps that relate the orbital properties of the low-mass bodies
Fig. 8: Probabilities for certain fates depending on the relative after the perturbation to this two dimensional initial condition
radius of a particle’s initial orbit (rinit/rp,peri) for the mass-ratio space. These allow to see the correlation between the particle’s
m=1.0.Seetextforfulldescription. initialpositionanditsfinalproperties.Thisgivesabetterunder-
standingoftheunderlyingdynamics.
Maps of the particle fates (whether they remain bound to
Above’sexampleillustratesalsothattheherepresentedway thehost,arecapturedbytheperturber,orbecomeunbound)and
to represent the results of a perturbation is independent of the somefinalorbitalelementsofperturbedparticleorbits(i.e.thefi-
initialdiscsizeasaparameter.Previousinvestigationsoftenper- naleccentricitiesandsemi-majoraxes)werepresentedforthree
formedsimulationsofperturbationsofdiscwithdifferentinitial sample mass-ratios. With these maps a detailed comparison of
sizes (e.g. Pfalzner et al. 2005b). But due to the inherent geo- theperturbationswithdifferentmass-ratiosispossible.Themaps
metrical scaling of the problem (see also Sec. 2), the obtained showthatalow-massperturber(Mp/Mh < 1)clearsbasicallya
information were redundant. Normalising the initial conditions “tunnel”alongitsorbitthroughtheparameterspace.Thecloser
withtheperturber’spericentredistance,asalreadydonebye.g. to the perturber orbit the higher the final eccentricities and the
Clarke&Pringle(1993),removesthisredundancy.Forthesim- lowerthesemi-majoraxesoftheparticleswhichremainbound
ulationsforaparameterstudythisisareductionbyonefreepa- tothehost.Forahigh-massperturber(Mp/Mh > 1)theeffectis
rameter–theinitialdiscsize.Wheninvestigatingtheeffectofan much more violent - only material from small regions remains
encounteronacertaindisc,thisparameterhastobeintroduced finallybound.Inthiscase,theremovalofparticlesfromthehost
again.Butthisismuchfasterthanperformingonesimulationfor isaconsequenceofthecentrifugalforceactingontheparticles
eachdiscsize. duringthehost’spericentrepassageonitsorbitaroundtheheav-
Ourmethodusestestparticlestodescribetheeffectofafly- ierperturber.Theequal-masscase(Mp/Mh = 1)isamixtureof
thelow-massandthehigh-masscase.
byonobjectswithVPPsinagivenarea.Howmuchmatterre-
Regionsinthepresentedmapswhereparticleshavethesame
sidesinitiallyinthisareadependsonthemassdistributionwithin
fate, show smooth gradients in the final orbital elements. As a
thedisc.Inordertoinvestigatethechangeinaphysicalquantity
consequence,thepresentednumericalresultsarenotonlyvalid
like e.g mass, angular momentum, or energy, one has therefore
for exactly the integrated initial conditions, but can be gener-
toattributemassestotheparticles.
alisedforthefullpresentedparameterspacebyinterpolation.
Furthermore, a real physical disc has also a vertical
Theresultsaredirectlyapplicabletoastrophysicalprocesses
mass/particledistributionwithacertainscaleheightwhichhas
like the perturbation of planetary orbits or protoplanetary discs
to be considered. The material which moves initially on orbits
bypassing-bystarswithouttheneedforadditionalsimulations.
which are inclined relative to the plane of the perturber orbit,
Thefatesofplanetsinanencounterandthefinalorbitalelements
is affected differently than in the here presented coplanar case.
can directly be obtained from the maps by scaling the initial
Theresultingdiscwouldthenbeacombinationoftheresultsfor
planetary orbit with the perturber’s periastron distance. When
thecoplanarcaseandtherespectiveinclinedcases.Typicalscale
applied to discs, our results are independent of the initial disc
heights of such discs correspond to orbital inclinations (cid:46) 10◦.
size,incontrasttomostpreviousworks.Theresultofthepertur-
Test have shown that for these inclinations the influence is of
bation of a disc with finite size can be obtained from our maps
minorimportance.
byconsideringonlytherespectiveradialsubset.Forthesimula-
In contrast to the application to discs, the post-encounter tionsofaparameterstudyinthiscontext,thisreductionbyone
properties of a planet can be directly obtained from the maps parameterreducesthenumericaleffortsignificantly.
by scaling the initial radius of the planetary orbit to the per- Even for the case of self-gravitating and viscous discs, the
turber’speriastron.Forexample,thefinalpropertiesofaplanet results might be applicable to some extent. Material which is
initially on an orbit with r = 100 AU around a star of mass removedfromthediscwithoutmovingthroughdenserregions,
Mh = 1 M(cid:12) perturbed by another star with mass Mp = 1 M(cid:12) likeincaseofahigh-massperturber,willmostlikelybehaveina
and a periastron of rp,peri = 200 AU can be found in the maps similarway.Incontrast,incaseofcrossingparticletrajectories,
at rvpp/rp,peri = 0.5. Since in practice, θvpp may be unknown, viscositywillchangetheresultsconsiderably.
it might be sufficient to obtain the probability for a certain re-
sult depending on r /r . This method was already applied
vpp p,peri
References
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arehigherresolvedandshowlessstatisticalscatter(seeFig.8).
Bate,R.R.,Mueller,D.D.,&White,J.E.1971,Fundamentalsofastrodynamics.
Like the data for Fig. 6, the data for Fig. 8 were obtained from Breslau,A.,Steinhausen,M.,Vincke,K.,&Pfalzner,S.2014,A&A,565,A130
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