Angular Momentum Changes Due to Direct Impact Accretion in a Simplified Binary System

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by  Jeremy F. Sepinsky

Angular Momentum Changes Due to Direct Impact
Accretion in a Simplified Binary System
Jeremy F. Sepinsky∗, Bart Willems†, Vassiliki Kalogera† and Frederic A. Rasio†
1 ∗DepartmentofPhysics,TheUniversityofScranton,Scranton,PA18510
1 †DepartmentofPhysicsandAstronomy,NorthwesternUniversity,2145SheridanRoad,Evanston,IL60208
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Abstract.
n
Wemodelacircularmass-transferringbinarysystemtocalculatetheexchangeofangularmomentumbetweenstellarspins
a
andtheorbitduetodirectimpactofthemasstransferstreamontothesurfaceoftheaccretor.Wesimulatemasstransferby
J
calculatingtheballisticmotionofapoint massejectedfromtheL point of thedonor star,conserving thetotallinearand
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3 angularmomentumofthesystem,andtreatingthestarsasuniformdensitysphereswithmainsequenceradiideterminedby
their masses. Weshow that,contrary toprevious assumptions intheliterature,directimpact does not always act asasink

] oforbitalangularmomentumandmayinfactincreaseitbyfacilitatingthetransferofangularmomentumfromthespinof
R
thedonortotheorbit.Here,weshowanexampleoftheexchangeofangularmomentum,aswellasameasureoftheorbital
S angularmomentumchangesforavarietyofbinarystarsystemswithmainsequencecomponents.
.
h Keywords: Stars:Binaries:Close,MassLoss,DirectImpact,AngularMomentumChanges
p PACS: 98.10+z,97.10.Gz,97.10Kc
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t INTRODUCTION
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[ Inclosebinarysystems,masstransferisalwaysaccompaniedbyexchangeofangularmomentum.Incaseswherethe

masstransferstreamdirectlyimpactstheaccretor,itiscommonlyassumedthatanyangularmomentumcarriedbythe
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v massistransferredentirelytothespinoftheaccretor,therebyremovingitfromtheorbit[e.g.1,2,3].Furthermore,
7 manyofthesamestudiesassumetheangularmomentumaddedtotheaccretor’sspinisidenticaltotheorbitalangular
6 momentumthe transferred mass had at ejection from the donor,neglecting changes due to gravitationalinteraction
1 withthebinary.
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Inordertoassess thefateofthese systems,accuratecalculationsoftheangularmomentumexchangeareneeded.
.
9 For example, calculationsof the systemic mass loss in Algol binariesdepend strongly upon the rotation rate of the
0 accretor[3],whilethelikelihoodthatdoublewhitedwarfswillbedriventocoalescerequiresknowledgeoftheorbital
0 angular moment losses [2, 1]. In this paper, we briefly present our preliminary results showing a violation of these
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standardassumptionsaboutorbitalangularmomentumtransportinasimplifiedbinarysystem.
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CALCULATIONS
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a
We considera circular binarysystem consisting of two mainsequencestars with masses M and M , radiiR and
D A D
R [4],anduniformrotationratesW andW ,withthesubscriptsDandAcorrespondingtothedonorandaccretor,
A D A
respectively.We let W be the Keplerianorbitalangularvelocity at the periastronof the orbit. We treat the stars as
K
uniformdensitysphereswithinertialconstantsk =k =2/5,andchoosetheorbitalseparationasuchthatthevolume
D A
equivalentradiusoftheeffectiveRochelobe[5]isequaltoR .
D
To modelthe response of the system to mass loss, we eject a single particle of mass M ≪M ,M from the L
P D A 1
point of the donor star [5, 6] with a velocity equal to the vector sum of the orbital velocity at the L point and the
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rotationalvelocityofthedonorstaratthatpoint.Thethree-bodysystemisthenevolvedvianumericalintegrationof
theNewtonianequationsofmotionuntilM impactsthesurfaceoftheaccretor.Duringbothaccretionandejection,
P
thelinearandangularmomentaofthesystemareconserved.Fordetailsofthecalculation,seeSepinskyetal.[7].To
modelcontinuousmass transfer in the circular orbit,we assume that, for sufficientlysmall M , the specific angular
P
momentumtransferredbyasingleejectedparticleisidenticaltothattransferredbyacontinuousmasstransferstream
whichfollowsthesametrajectory.
FIGURE1. a) Thechange inthespin(top) andorbital (bottom) angular momenta betweenejectionand accretionof asingle
ballisticparticleasafractionofthetotalangularmomentumforasystemwithMD=10M⊙,MA=1M⊙,andW D=0.9W K.b)
Thechangeintheorbitalangularmomentum(D J )ofthebinaryorbitperunitaccretedmass(M )forasystemwithW =W ,
orb P A K
MA=1.0⊙,andW D= fDW K.Notethatwecanseeeitheranincrease(D Jorb>0)oradecrease(D Jorb<0)intheorbitalangular
momentumofthesystemdependingontheorbitalparameters.Forcomparison,thesolidlineshowsthechangeinorbitalangular
momentumassumingalloftheejectedparticle’sinitialorbitalangularmomentumisdepositedintothespinoftheaccretor.
RESULTS &CONCLUSIONS
In Figure 1a, we show the change in the spin (top) and orbital (bottom) angularmomenta for a binary system with
theparametersgiveninthecaption.Massisejectedatt =0,andaccretesatt =0.216P whererapidjumpsinthe
orb
momentaoccurduetotheirconservationattheinstantaneousinelasticejectionandaccretion.Inthiscase,nearlyall
the orbital angular momentum of the particle is transferred to the orbital angular momentum of the accretor, while
onlyasmallamountistransferredtoitsspin.
In Figure 1b we show the change in the orbital angular momentum per unit M for a large number of binary
P
systems. We see that the orbital angular momentum of nearly all the tested systems increases due to direct impact
accretion. During the ejection of M , conservation of momentum dictates a loss of spin angular momentum of the
P
donor, increasing the orbital angular momentum of the particle. Upon accretion, a portion of this is added to the
accretor’sspin while the rest is addedto the orbit. These results are characteristicallydifferentfrom those obtained
by the standard assumption of decreasing the orbital angular momentum by the specific angular momentum of the
particle atthe L point(J ) and addingitentirely to the spin of the accretor.The solid line in Figure 1bshows the
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changeintheorbitalangularmomentumfollowingthisperscription.Thismayhavealargeeffectonthepredictions
ofthesurvivabilityofanyclassofsystemwhichundergoesadirectimpactmasstransferphaseduringthecourseof
itsevolution,e.g.,doublewhitedwarfs,Algolbinaries,cataclysmicvariables,etc.
Infuturework,wewillexplorethisintriguingtrendinmoredetail,applyingtheresultstosystemssuchasdouble
whitedwarfswherethechangeinorbitalangularmomentumiscriticaltoassessingthestabilityofmasstransfer.
TheauthorswouldliketothankChristopherDeloye,PaulGroot,andTomMarshforusefuldiscussions.
REFERENCES
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