Dynamical friction for supersonic motion in a homogeneous gaseous medium

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by  Daniel Thun

Astronomy&Astrophysicsmanuscriptno.df ©ESO2016
January29,2016
Dynamical friction for supersonic motion
in a homogeneous gaseous medium
DanielThun,RolfKuiper,FranziskaSchmidt,andWilhelmKley
Institutfu¨rAstronomieundAstrophysik,Universita¨tTu¨bingen,AufderMorgenstelle10,D-72076Tu¨bingen,Germany
e-mail:{daniel.thun@, rolf.kuiper@, fr.schmidt@student, wilhelm.kley@}uni-tuebingen.de
Received;
6 ABSTRACT
1
0 Context. Thesupersonicmotionofgravitatingobjectsthroughagaseousambientmediumconstitutesaclassicalprobleminthe-
oreticalastrophysics.Itsapplicationcoversabroadrangeofobjectsandscalesfromplanetesimals,planets,allkindofstars,upto
2
galaxiesandblackholes.Especiallythedynamicalfriction,causedbytheformingwakebehindtheobject,playsanimportantrole

n forthedynamicsofthesystem.Tocalculatethedynamicalfrictionforaparticularsystem,standardformulae,basedonlineartheory
a areoftenused.
J Aims. Itisourgoaltocheckthegeneralvalidityoftheseformulaeandprovidesuitableexpressionsforthedynamicalfrictionacting

8 onthemovingobject,basedonthebasicphysicalparametersoftheproblem,namelythemass,radius,andvelocityoftheperturber,
2 thegasmassdensity,soundspeed,andadiabaticindexofthegaseousmedium,andthesizeoftheformingwakeorinteractiontime,
respectively.

] Methods. Weperformdedicatedsequencesofhighresolutionnumericalstudiesofrigidbodiesmovingsupersonicallythroughaho-
A
mogeneousambientmedium,andcalculatethetotaldragactingontheobject,whichisthesumofgravitationalandhydrodynamical
G drag.Westudycaseswithoutgravitywithpurelyhydrodynamicaldrag,aswellasgravitatingobjects.Invariousnumericalexperi-
ments,wedeterminethedragforceactingonthemovingbodyanditsdependenceonthebasicphysicalparametersoftheproblem,as
.
h givenabove.Fromthefinalequilibriumstateofthesimulations,wecomputeforgravitatingobjectsthedynamicalfrictionbydirect
p numericalintegrationofthegravitationalpullactingontheembeddedobject.
- Results. Thenumericalexperimentsconfirmtheknownscalinglawsforthedependenceofthedynamicalfrictiononthebasicphys-
o ical parameters as derived in earlier semi-analytical studies. As a new important result we find that the shock’s stand-off distance
r isrevealedastheminimumspatialinteractionscaleofdynamicalfriction.Belowthisradius,thegassettlesintoahydrostaticstate,
t
s which–duetoitssphericalsymmetry–causesnonetgravitationalpullontothemovingbody.Finally,wederiveananalyticestimate
a forthestand-offdistancethatcanusedconvenientlyincalculatingthedynamicalfrictionforce.
[
Keywords.General–Gravitation–Shockwaves–Waves–Hydrodynamics–Methods:numerical
1
v
9
9 1. Introduction Itsastrophysicalapplicationcoversawiderangeoffields,as
7 pointed out recently by Lee & Stahler (2014). In the common
7 The motion of gravitating objects through a gaseous ambient envelope phase, the decay time of the secondary companion is
0 medium is a classical problem in theoretical astrophysics. Of driven by dynamical friction (Ricker & Taam 2008, 2012), as
. special interest is the force acting on the object as it deter- well as the important survival rate of planets around evolved
1
0 minestypicalevolutionorequilibrationtimescales.Thisforceis stars (Villaver & Livio 2009). Further applications include the
6 causedbytwomainmechanisms,thepurelyhydrodynamicdrag, settling of massive stars in molecular clouds (Chavarr´ıa et al.
1 andbydynamicalfriction.Evenwithoutgravity,theflowofthe 2010),thedragonastarintheaccretionflowaroundblackholes
: gas will be perturbed by the body because it acts as an obsta- (Narayan 2000), the coalescence of black holes (Armitage &
v cleandblocksthegasinfrontofit.Thiscausesanexchangeof Natarajan2005),andthemigrationofplanetesimals(Grishin&
i
X momentum between the embedded body and the surroundings, Perets 2015). In the framework of planet evolution, dynamical
r leadingtoahydrodynamicdragforceontheobject.Onearththis friction plays a role in the change of planetary inclination due
a givesforexamplerisetotheairresistanceofallflyingobjects,a to the interaction with the disk (Rein 2012; Teyssandier et al.
dragforcethattendstoopposethemotionofthebody.Inanas- 2013). The dynamical interactions of evolved stars that move
trophysicalsetting,foramovingmassivebodyitsgravitational through the interstellar medium (ISM) determine the observa-
attractionontothesurroundingleadstotheformationofawake tionalpropertiesofthedustintheenvelope(Slavinetal.2004;
ofhigher-than-averagedensitybehindthemovingobject,and,as vanMarleetal.2011;Meyeretal.2014a,b,2015).
aresult,toagravitationalpullofthedenseregionontothebody.
Thisadditionalforceisagaindirectedoppositetothemotionof This incomplete list of applications shows that the phe-
thebodyandleadstoaslowdown.Thisforceisusuallycalled nomenonofdynamicalfrictioniswidespreadandofgeneralim-
gravitational drag force or dynamical friction (Chandrasekhar portance. But still, thereis no satisfying derivation of the force
1943; Ostriker 1999). Hence, the total drag force acting on a on the object, despite many years of research (see e.g. Lee &
gravitating body is the sum of the hydrodynamic drag and dy- Stahler2014).Inthisstudy,wetacklethisimportantproblemof
namicalfriction. dynamical friction via direct numerical modeling. We perform
1
DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
hydrodynamical simulations of a gravitating body moving su- infrontoftheobjectwhere(additional)energycanbedissipated.
personicallythroughahomogeneousmediumandextractthedy- UsingthelinearizedfluidequationsRuderman&Spiegel(1971)
namicalfrictionontheobject.Weanalyzeindetailthescalingof andRephaeli&Salpeter(1980)deriveacorrectiontotheabove
thedragwithimportantphysicalparameteroftheproblemsuch equationthatdependsontheMachnumberoftheobject.
asMachnumberandmassoftheobjectandothers,andcompare In an important work, Ostriker (1999) extended the above
thistoexistingformulaefordynamicalfriction. analysisandstudiedthetime-dependentdynamicalfrictionforce
The paper is organized as follows: In Sect. 2, we summa- acting on a massive object embedded into a gaseous medium
rizethestatusonformulaefordynamicalfriction.InSect.3,we in the sub- and supersonic regime. Her analysis resulted in an
describethehydrodynamicsequationsandthephysicalandnu- expressionforthecoefficientC ,verysimilartothatofRephaeli
A
mericalsetupofthesimulations.InSect.4,wepresentcompar- &Salpeter(1980),whichincludesatime-dependentmaximum
ison simulations to laboratory experiments for non-gravitating radius, s ∼ V t,wheretisthetimetheobjecthasinteracted
max ∞
objects, moving at supersonic speed through air. In Sect. 5, we withthegas.Heranalysisisbasedonlinearperturbationtheory
presentthesimulationsofgravitatingobjectsforvariousdiffer- andusefultostudytime-dependentphenomena.Theexpression
ent physical parameter and discuss their implication on the dy- forthesupersoniccasereads
namical friction. In Sect. 6, we present a new formula for the
dragforce.InSect.7,wegiveabriefsummaryofthestudy. CA =lnVs∞t (cid:32)MM2−2 1(cid:33)1/2 , (4)
min
2. Analyticalestimatesofdynamicalfriction where M = V /c denotes the Mach number of the problem.
∞ ∞
A first account of dynamical friction has been given by Obviously,ifwesetsmax =V∞t,thenEqs.(2)and(4)agreewith
Chandrasekhar(1943)whocalculatedthedynamicalfrictionof eachotherinthelimitoflargeM,becauseforhighlysupersonic
astarembeddedinastellarcluster.Hisresultswerelateradapted flows pressure effects become less important. Here, we are not
to the motion of stars through the ISM or galaxies through the interested in the time-dependent process but will focus on the
intergalacticmedium.Here,theambientgasistypicallytreated finalstationarystate.
as a static hydrodynamic continuum which is transversed by a Inadditiontotheseanalyticestimatestherehavebeenmany
movingbodyofmassMandthevelocityV .Usingtheimpulse numerical studies of moving gravitating objects embedded in a
∞
approximation, where the moving mass perturbs the surround- gas.ThefirsttoaddressthisissuenumericallywasHunt(1971)
ingsonlyforafinitetime,Ruderman&Spiegel(1971)derived who used a special shock fitting method to calculate the flow
ageneralformulaforthedynamicalfrictionforceF actingon aroundthebodyandtheaccretionrateontoit.Later,Shimaetal.
DF
themassM: (1985)werethefirsttousemoderntypefluiddynamicalmeth-
(cid:32)GM(cid:33)2 odsonthisproblemandperformedaxisymmetricsimulationsin
FDF =4πρ∞ V CA. (1) sphericalpolarcoordinates.Intheirnowclassicpapertheystud-
∞
iedthedragforceandtheaccretionrateontotheobjectasafunc-
In Eq. (1), ρ∞ denotes the ambient constant density, G is the tionofthevelocityandfoundveryroughqualitativeagreement
gravitational constant, and CA stands for the “Coulomb loga- with Eqs. (1) and (2), when using for smin the inner radius of
rithm” the computational domain. Most of the subsequent simulations
CA =ln(smax/smin) , (2) dealt with open inner boundary conditions and focused on the
massaccretionrateontotheobject.Anintroductorysummaryto
where s and s correspondtotheminimumandmaximum
min max this so called Bondi-Hoyle-Littleton accretion process is given
impact parameter of the interaction. A very similar expression
inthereviewarticlebyEdgar(2004).Inthispaperwewillnot
forthedragonastarmovinginazerotemperaturemediumhas
studyaccretionontotheobjectandstudyrigidbodiesonly.The
beenderivedbyBondi&Hoyle(1944)whoanalyzedthechange
mainfocuswilllieontheaccuratecomputationofthedynami-
inmomentumbythegasthatiscollectedinaverythinlinebe-
calfrictionand–asapre-requisite–onthedeterminationofthe
hind the object. Additionally, they derived relations for the gas
minimumintegrationradiuss .
accretionrateontothebody. min
The validity of the formula by Ostriker has been demon-
To apply Eq. (1) to various physical problems, the relevant
stratedbySa´nchez-Salcedo&Brandenburg(2001)whoshowed
length scales s and s need to be known. Typically, for
min max agreement for moderate Mach numbers in non-linear, isother-
s the maximum extension of the medium, or a characteris-
max mal simulations. However, they used a Plummer-type potential
tic (global) length scale is assumed. For the minimum length,
with a smoothing length much larger than R . Later, Kim &
s ,thesituationisnotsoclearandnogeneralacceptedrecipe A
min Kim (2009) reanalyzed dynamical friction for extended bodies
exists.Often,eitherthephysicalradiusRoftheobjectischosen
through numerical simulations. However, they did not consider
or, in case the body becomes very small, s ≈ max(R,R ).
min A objects with a more or less rigid surface (like stars) but used
HereR standsfortheaccretionradius
A again a Plummer-type potential with a smoothing length, and
2GM hence their results may be applicable more to galaxies moving
R = . (3)
A V2 in the intergalactic medium. They compare their results to the
∞
formulabyOstriker(1999)andprovideanewfittingformulain-
Recently,Canto´ etal.(2011)derivedanapproximationfor s volving R . They used an axisymmetric cylindrical coordinate
min A
usingballisticorbittheory,whichdoesnotaccountforthepres- system,whichsuffersintermsoflimitedresolutionclosetothe
sure, however. Obviously, the question of the correct minimum center.
radiusoftheobjectisstillnotclarifiedandremainsambiguous. Starting with Ruffert (1994) and Ruffert & Arnett (1994)
WhileEq.(1)wasderivedforapressurelessmedium(dust), there have been many simulations considering the three-
somecorrectionshavetobeappliedincasepressureeffectsplay dimensionalflowaroundanaccretingobject.Aninterestingfea-
arole.Inthesupersoniccase,withV > c ,wherec denotes turediscoveredalreadyinthesefirststudiesistheonsetofunsta-
∞ ∞ ∞
thesoundspeedoftheunperturbedmedium,abowshockforms bleflowwhenbreakingtheaxialsymmetry,aneffectthatismost
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DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
pronouncedinplanartwo-dimensionalflows,thatarenonphys- densityofthegas.Theaccelerationduetoexternalforcesa is
ext
ical however. A comprehensive summary of two-dimensional givenherebythegravitationalattractionofthemovingobject
(2D) and three-dimensional (3D) simulations carried out has
GM
beengivenbyFoglizzoetal.(2005).Theypointoutthattheori- a =− e (10)
ginoftheinstability,i.e.underwhatconditionsitisexpectedand ext r2 r
its effect on the drag force, are still not understood. In this pa-
where r is the distance from the center of the object to the po-
perwewillavoidthequestionofinstabilityandfocusonpurely
sition under consideration. We close the equations of hydrody-
axisymmetricflows.
namicswiththeidealgasequationofstate
ItwillbeusefultocomparethedragformulaEq.(1)tostan-
dard hydrodynamic drag formula. From dimensional analyses p=(γ−1)e . (11)
th
(Landau&Lifshitz1966)onefindsforthehydrodynamicaldrag
forceactingonanobjectwithcrosssectionA Gaspressure,density,andtemperaturearerelatedvia
k
F = 1C ρ V2 A. (5) p= B ρT, (12)
hydro 2 D ∞ ∞ µmH
The dimensionless drag coefficient C contains the details of where µ is the mean molecular weight, kB the Boltzmann con-
D
stantandm themassofthehydrogenatom.Thesoundspeedof
how the body and the medium physically interact with each H
other, e.g. through surface effects and the dependence on lam- thegasisgivenby
(cid:114)
inarvs.turbulentflows.Ifweusetheaccretionradiustosetthe c = γp. (13)
interactioncrosssectionA=πR2,theratioofthehydrodynami- ∞ ρ
A
caldragandthedynamicalfrictionisgivenby:
In our simulations we use a constant value of the adiabatic ex-
ponent γ. We study the influence of different γ-values on the
F C
hydro = D (6) dynamicalfrictionindedicatedparameterseries,seeSect.5.7.
F 2C
DF A Fromthegivenvaluesofthedensityρ andpressure p of
∞ ∞
theinitiallyhomogeneousmedium,itsadiabaticindexγ,andthe
In this study, we determine the dependence of dynamical
speedV ofthemovingobject,theassociatedMachnumberof
friction acting on a rigid, gravitating body moving supersoni- ∞
thesystemisgivenas
callythroughahomogeneousmediumonthebasicphysicalpa-
rametersofthesystem.First,wedemonstratethatthedynamical M=V /c . (14)
∞ ∞
frictionscaleslikeEq.(1)andwederiveananalyticrelationfor
the up-to-now undetermined minimum length scale s of the
min
problem, and finally give a convenient expression for the dy- 3.3. Numerics
namicalfrictioningeneral.
Simulationsarecarriedoutusingtheopen-sourcecodePLUTO
(Mignone et al. 2007), version 4. Part of the simulations were
carried out with a new in-house developed CUDA version for
3. Physicsandnumerics
usage on GPUs. We use a Runge-Kutta time stepping scheme
3.1. Problemsetup of second order with a second order reconstruction of states
in space, a van-Leer limiter in the reconstruction step, and a
We model gravitating, spherical objects with a given mass M
Harten-Lax-vanLeersolverfortheRiemann-Problem.
and radius R moving with supersonic speed V through a ho-
∞ Equations(7)to(9)aresolvedintheco-movingframeofthe
mogeneous medium, that is characterized by its mass density
object,i.e.theobjectisatrestinthemodelingframeandtheini-
ρ ,pressure p ,andtemperatureT orsoundspeedc ,respec-
∞ ∞ ∞ ∞ tialvelocityofthesurroundinghomogeneousgascorrespondsto
tively. As initial condition, the object is placed instantaneously
thephysicalspeedofthesolidbody.Weusea2Dsphericalgrid
into the ambient medium and the evolution of the gaseous sys- (r,θ) assuming axial symmetry. The object is fixed at the ori-
tem is followed using time-dependent hydrodynamical simula-
ginofthecoordinatesystemandthegasflowsintothenegative
tions. After the system reaches a steady state we determine the z-direction. The solid surface of the moving spherical object is
gravitationalpullofthegasontotheobject(thedynamicalfric-
representedasboundaryconditionsattheradialinnerboundary
tion)aswellasthehydrodynamicdrag.
ofthecomputationaldomain.Here,weusereflectingboundary
conditions, corresponding to a non-accreting object. The com-
putationaldomainextendsintheradialdirectionfromtheradius
3.2. Equations
R of the object up to R = 100or1000 R. The symmetry
domain
WestudythemotionoftheidealgasbysolvingtheEulerequa- axis of the domain is aligned with the trajectory of the object.
tions Inthepolardirection,thecomputationaldomainextendsfrom0
toπusingagridspacinguniforminangle.Toobtainhighreso-
∂ρ
+∇·(ρu)=0 (7) lutionattheinterestingareaaroundtheobject,andtoensurean
∂t approximatelyquadraticgridspacingintheradialandthepolar
∂ρu+∇·(ρu⊗u)+∇p=ρa (8) directionofeachgridcell,weusealogarithmicgridspacingin
∂t ext theradialdirection.
∂e +∇·[(e+p)u]=ρu·a . (9) At the outer radial boundary Rdomain, we set the boundary
∂t ext conditions according to the gas flow of the surrounding: In the
upperhemisphere(forθ ∈ [0,π/2]),weimplementedaninflow
Here, ρ denotes the gas mass density, u the velocity, p the gas boundarycondition,i.e.allghostcellsaresettotheunperturbed
pressure,ande=e +e thetotal(kineticandthermal)energy valuesofthesurroundingmedium.Inthelowerhemisphere(for
kin th
3
DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
θ ∈ [π/2,π]),weimplementedazero-gradientboundarycondi- thefinalforcebalanceas
tionsothattheout-flowingmaterialcanleavethecomputational
domain without reflections. As already mentioned, we perform Fipnrs+FDF+Fotruat+Fopurst =0. (17)
simulationsinaxialsymmetryandthereforeatθ=0(positivez-
axis)andatθ=π(negativez-axis)wemakeuseofaxisymmetric We calculate all these forces for our simulations and evaluate
theimportanceofthedifferentcontributions.Eq.(17)showsthat
boundaryconditions.
Simulations with a size of the body of R = 0.1R are per- thedragactingonanobjectcanbeeitherevaluatedattheinner
(cid:12)
boundary(plusgravity)orsolelyattheouterboundary.
formedonagridconsistingof700×480gridcells,simulations
with a smaller size of the body of R = 0.01R are performed
(cid:12)
onagridconsistingof860×400gridcells.Wepresentacorre-
4. Comparisontolaboratoryexperiments
spondingconvergencestudyindetailinappendixC.
In computational astrophysics, it is only rarely possible to
test the numerical algorithms against laboratory experiments.
3.4. Assumptionsandsimplifications
However, in the case of the problem on hand – a sphere mov-
The numerical experiments are performed to determine the dy- ing supersonically through a gaseous medium – experimental
namicalfrictioninageneralastrophysicalcontext.Hence,wedo dataisavailablefornon-gravitatingmovingbodies.Tocheckthe
nottakeintoaccounteffects,whichdependonaspecificsystem validity of our numerical ansatz and prove the accuracy of the
under investigation, such as radiative heating and cooling. The code,includingsetup,boundaryconditions,andthecalculation
flowisassumedtobeinviscid.Self-gravityofthegasisnottaken ofthedragforce,weperformcomparisonsimulationsofabody
intoaccount.Effectsofmagneticfieldsarenotinvestigated.The movingsupersonicallythroughahomogeneousgaseousmedium
long-evolution problem of a time-dependent slow-down of the (air), which can be compared to existing data from laboratory
moving body by the acting dynamical friction is not included experiments,namelybyvanDyke(1982)andBillig(1967).
inthisstudy,seeSa´nchez-Salcedo&Brandenburg(2001)fora
discussion on this issue. The formula for dynamical friction is
4.1. Morphology
derived within the co-moving frame of the moving body. The
object is treated as a rigid body where no accretion through its Theblack-and-whitephotographyinvanDyke(1982),Fig.266,
surface is allowed. The medium is considered to behave as an displaystheresultofalaboratoryexperimentofasphericalnon-
idealgas. gravitating body moving with supersonic velocity through air.
We present the right half of the original image in Fig. 1, bot-
tompanel.Themorphologyofthesystemischaracterizedbya
3.5. Calculationofthedragforce
standingshockfrontaheadofthemovingsphere,aclearlyvisi-
bleshockboundarybetweentheunperturbedandtheperturbed
Thetotaldragforceactingonanobjectmovingthroughanam-
gaseousmedium,andalow-densityregionpastthemovingob-
bientmediumcanbecalculatedfromthemomentumbalancein
ject.
thefinalequilibriumstate(Landau&Lifshitz1966).Inourcase,
the object is moving along the z-direction and we have to con- We model the same experiment numerically within our
siderthez-componentofEq.(8)whichreadsinequilibrium framework described in the previous section. Here, we switch
off the gravitational force of the moving body and use an adia-
∂p batic exponent for air of γ = 1.4. The body is moving into the
∇·(ρuzu)+ ∂z −ρaz =0, (15) positive z-direction; actually, simulations are performed in the
co-moving frame of the body, hence, the gas flow is initialized
wherea denotesthez-componentofthegravitationalaccelera- intothenegativez-direction.Thehomogeneousgasisinitialized
z
tionduetothemovingobject.Integratingoverthewholevolume withaconstantdensityρ∞.TheMachnumberoftheflowissetto
of the computational domain we can convert the first two parts M=V∞/c∞ =1.53.Inthewholecomputationaldomaintheini-
intosurfaceintegralsandobtain tialconditionisgivenbytheunperturbedflow.Atthestartofthe
simulation the rigid sphere is embedded and perturbs the flow.
(cid:90) (cid:90) (cid:90)
(cid:2)(ρu u)+pn(cid:3)·df+ ρa dV+ (cid:2)(ρu u)+pn(cid:3)·df =0, Werunthemodeluntilanequilibriumstatehasbeenreached.
z z z InFig.1,upperpanel,wedisplaythemorphologyofthegas
in Vol out
(16) densityaroundthesphereinthenumericalexperiment.Clearly
wheredf isthesurfaceelementanddVthevolumeelement,and visible is the bow shock in front of the sphere, where the gas
thesubscriptinandoutrefertotheinnerandouterboundaryof flow changes from supersonic to subsonic velocities, the shock
thecomputationaldomain.ThefirsttermofEq.(16)represents frontdividingtheperturbedfromtheunperturbedgas,aswellas
thetotalhydrodynamicalforceactingonthesurfaceofthebody the low-density region behind the moving body. As expected,
which is the sum of a momentum transport, Ftra, through the the gas density reaches a maximum directly in front of the
in
body’s surface (e.g. for porous objects or open, accreting bod- sphere. A visual comparison with the experimental data from
ies)andapressureforce, Fprs,actingdirectlyonthebody.The van Dyke (1982), Fig. 266, as presented in Fig. 1, shows ex-
in
second term is the dynamical friction force on the body, F , cellent agreement between the numerical experiment and the
DF
which is obtained by integrating over the whole volume of the laboratory experiment in terms of their morphological charac-
domain. The sum of these two contributions must be balanced teristics.Furthermore,themagnitudeofthedensityjumpatthe
by the corresponding momentum transport and pressure terms shockfrontfromthenumericalsimulationagreesverywellwith
at the outer boundary, Ftra and Fprs, respectively. In our case, the well-known analytical Rankine-Hugoniot jump conditions.
out out
thefirstsurfaceintegralinEq.(16)istakenatthesurfaceofthe In contrast to the axisymmetric and inviscid numerical result,
movingobject,heretheradius,R,ofthesphericalbody,andthe thelaboratoryexperimentshowsthatthelow-densityregionpast
lastsurfaceintegralattheouterboundaryofthedomain,R . themovingobjectissubjecttoweakturbulence,thatcannotbe
domain
Inthecaseofanimpermeablerigidbody,Ftra =0,andweobtain capturedintheidealizedsimulationsetting.
in
4
DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
fromthecenterofthesphere,asafunctionofM.TheR de-
SO
1.7
Thisstudy
VanDyke&Gordon(1959)
1.6
Billig(1967)
1.5
R 1.4
/
O
S
R 1.3
1.2
1.1
1.0
2 4 6 8 10 12 14
M
Fig.2. Quantitativecomparisonoftheshock’sstand-offdistance
asfunctionofMachnumber.Thesolidblacklinedenotesthere-
lationship derived from laboratory experiments as given in the
main text. Small red dots represent results from numerical ex-
perimentsforthesamesetup.Bluedotsdenoteearlynumerical
resultsbyvanDyke&Gordon(1959).
creaseswithincreasingMandapproachesasymptoticallyinthe
limitofhighlysupersonicflowavalueofR →1.14.Thisout-
SO
comeofthenumericalexperimentscanbedirectlyandquantita-
tivelycomparedtothelaboratoryexperimentsbyBillig(1967),
using a derived fit to the laboratory data given by R /R (cid:39)
SO
1+0.143exp(3.24/M),whereRdenotesthegeometricalradius
of the moving sphere. The results from the numerical and lab-
oratory experiments are quantitatively in very good agreement,
as shownin Fig. 2.They areconsistent with eachother at both
extremes, weak shocks (M = 1.53) as well as strong shocks
Fig.1. Visualization of a comparison simulation result of a (M → ∞).Intheregimeofmoderateshocks(M ∼ 2...4),the
sphere embedded in air (γ = 1.4) flow at a Mach number numerical experiments show slightly larger stand-off distances
M=1.53.Thehomogeneousgasarrivesfromthetopandflows
than the laboratory experiments. Additionally, in the numerical
inthenegativez-direction.Shownisthedensitydistributionfor
framework, the position of the shock front is associated with a
thefinalequilibriumstateinthecentralpartofthecomputational
certaingridcell,morespecificallyitscenter;nofurtherinterpo-
domainaroundthesphere.
lationhasbeenapplied.Hence,thevisiblestepsinthenumerical
Upper panel: Numerical results, black lines denote iso-density
datavisualizethefinitespatialresolutionofthenumericalexper-
contours.
iments.
Bottompanel:LaboratorydatafromvanDyke(1982),Fig.266.
4.3. Thehydrodynamicaldragforce
Thefollowingcomparisonchecksfortheshockfrontphysics InFig.3wedisplaythecontributionsofthedifferentforcesact-
anditsdependenceontheMachnumber.
ingonthegasandthesphere.Shownarethepressureforceact-
ingonthebody,Fprs,togetherwiththepressureandmomentum
in
fluxattheouterboundary.AstobeexpectedfromEq.(17)the
4.2. Thestand-offdistanceoftheshockfront
sumofall3forcecontributions, F addsuptozero.Theerror
tot
As we will point out later, the shock’s stand-off distance, R , for the total force is slightly increasing for higher Mach num-
SO
plays a crucial role in determining the dynamical friction on a bersbutisalwaysbelow2%forallsimulations.Inthiscasethe
gravitating moving body. Here, we measure R from the cen- hydrodynamic drag acting on the body, given by Fprs, is nega-
SO in
terofthemovingsphericalobject.Totestthedependenceofthe tivewhichindicatesaforceopposingthedirectionofmotionof
shock front on the problem’s Mach number, M, we compute a thebody.Thestandarddragforce,asgiveninEq.(5),depends
sequence of models for the same setup as in the previous sec- quadratically on the velocity V of the object. Our numerical
∞
tion but with a variety of different inflow velocities V . Fig. 2 resultsclearlyconfirmthisexpectedquadraticscalingwithV ,
∞ ∞
shows the resulting stand-off distance of the shock, measured seeaddedsolidlineinFig.3.
5
DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
# M[M ] R[R ] M c [kms−1] γ R [R ] Figures
Jup (cid:12) ∞ domain (cid:12)
Fiducialsimulation
F 60 0.01 4.0 58 5/3 10 Figs.A.1+A.2
Fiducialsimulationwithlargerdomainsize
C 0.01 25 Fig.9
Variationofthemassoftheperturber
M1 10... 50 0.1 Fig.8+16
M2 1...120 0.01 Figs.7+8+10+16
Variationofthevelocityoftheperturber
V1 0.1 3.5...20.0 49 1.2 Fig.12+13+15
V2 0.01 2.2...10.9 53 1.4 Figs.12+13+15+16
V3 0.1 2.0... 9.0 58 5/3 Figs.4+8+16
V4 0.01 2.0...50.0 58 5/3 Figs.5+7+8+11+12+13+15+16+17
V5 0.01 2.0...20.0 63 2.0 Figs.12+13+15+16
V6 0.1 2.0...10.0 77 3.0 Figs.12+13+15
V7 0.1 2.0...10.0 89 4.0 Figs.12+13+15
Variationofthesoundspeedofthemedium
S 0.01 1.6...12.6 18.3...183 5/3 Figs.8+16
Variationoftheadiabaticindexofthemedium
G 0.1 4.4 53 1.2...6.0 Fig.14+15
Table 1. Overview of series of simulations performed. Each row represents a series of individual simulations, mostly varying a
singlebasicsetupparameter.Thecolumnsdenotefromlefttorightthelabelofthesimulationseries,themass,theradius,andthe
velocity of the perturber, the soundspeed of the gaseous medium, the adiabatic index of the medium, and the outer radius of the
computationaldomain.Thelastcolumngivesthefigurenumbersassociatedwiththedataprocessingofthesimulationseries.The
bodyvelocityisgiveninunitsofMach.Ifthevalueofaparameterisnotexplicitlyspecified,thevaluefromthefiducialsimulation,
giveninthetoprow,isused.
ing body is produced very similar to the morphology found in
Fipnrs thelaboratoryexperimentsforthenon-gravitatingbodies.Fig.4
4000 Fopurts showsthesteadystatesolutionforthegasmassdensityandthe
Foturat velocityfieldforoneofoursimulations.Infrontoftheobjecta
Ftot
2000 2 bowshockformswherethematerialisdeceleratedfromsuper-
∝M
s] sonic to subsonic speeds. After passing the shock front, matter
t
ni closetotheobjectsettlesintoahydrostaticenvelope.Themajor
U
e 0 differencetothenon-gravitatingcaseisthatbehindthemoving
d
o object a wake of higher-than-average density instead of lower
C
[ densityisformed,whichisadirectconsequenceofthegravita-
F
2000 tionalattractionofthebody.Inturn,thiswakeofhigherdensity
−
yieldsagravitationallypullontotheobject,whichslowsitdown.
Thisisthephenomenonofdynamicalfriction.
4000 AsshownaboveinEq.(17)thetotaldragactingonagravi-
−
tatingrigidbodyisthesumofthehydrodynamicdrag(pressure
0 10 20 30 40 50 60 forceonthebody, Fprs)andthedynamicalfriction, F .Inour
in DF
M case,behindtheshockfrontasphericalhydrostaticshellforms
aroundtheobjectsuchthatthetotalpressureforceontheobject
isnegligible.Hence,thetotaldragontheobjectisgivensolely
Fig.3. Theforcecontributionsintheequilibriumstateaccording
bydynamicalfriction,andinthefollowingweshallconcentrate
to Eq. (17) for the non-gravitating, rigid sphere in air, given as
afunctionoftheMachnumber. Fprs denotesthepressureforce onthispartonly,seeFig.6below.ForveryhighMachnumbers
in the separation of the shock from the object’s surface becomes
measured at the surface of the sphere, Fprs the pressure force
out verysmallsuchthatnohydrostaticenvelopecanform,andpres-
measured at the outer boundary of the computational domain, sureeffectswillbecomeimportantagain.
and Ftrs the momentum transport through the outer boundary.
out The dynamical friction of a body moving with supersonic
Thesumofthesethreeforcesisdenotedby F .Thecurvede-
tot speed through a gaseous homogeneous medium denotes a well
notesaparabolawithF ∝M2.
defined problem, which involves a manageable amount of de-
pendencies. In the course of this section, we first give an
overview of the relevant problem parameters and how they are
5. Simulationsofdynamicalfriction
expectedtoimpacttheamountofdynamicalfriction(Sect.5.1).
Now we turn to astrophysical applications and consider the Afterwards,wepresentourrealizedsimulationseries,eachded-
motion of a gravitating body through a homogeneous gaseous icatedtoinvestigatetheimpactofaspecificparameter,anddis-
medium.Forsupersonicspeedsashockfrontaheadofthemov- cuss the simulation results in terms of scaling laws (Sects. 5.5
6
DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
thegasmassdensityρ andsoundspeedc astheindependent
∞ ∞
variables;themassdensityentersdirectlytheforcetermofthe
gravitational pull of the wake onto the moving body and the
soundspeedofthemediumsetstheMachnumberM = V /c
∞ ∞
oftheshock.
The thermodynamics of the gas is controlled by its caloric
and thermal equations of state Eqs. (11) and (12) with the adi-
abatic index γ as the only free parameter. Most important for
theshockphysics,theadiabaticindexcontrolshowthegaseous
mediumreactsincaseofcompressionandexpansion.
While the object is moving through the gaseous medium, a
wakeofhigherdensitywillformbehindtheobject,whichgrows
in time. For a known interaction time t, the extent of the wake
isconstrainedby s = V t.Wediscussthedependenceofthe
max ∞
dynamicalfrictiononthemaximumextentofthewakeinmore
detailinSect.5.4.
Summarized,thedynamicalfrictionofamassivebodymov-
ingsupersonicallythroughagaseoushomogeneousmediumde-
pendsonlyonthemassofthebody M,thevelocityofthebody
V ,itsgeometricalradiusR,themassdensityρ ofthegas,its
∞ ∞
Fig.4. Density and velocity structure of the final quasi- soundspeedc ,theadiabaticindexγ,andthemaximumextent
∞
stationary state around a gravitating object moving with su- ofthewakes .
max
personicmotionthroughagaseoushomogeneousmedium.The
InRuderman&Spiegel(1971),theauthorsgiveanexpres-
bodyismovingtowardsthepositivez-direction.Densityiscolor-
sion for the dynamical friction as given in Eq. (1). In this for-
codedandblacklinesdenoteiso-densitycontours.Velocitiesare
mula, the dynamical friction depends additionally on the mini-
givenasarrows,scaledbythespeed.
mumlengthscales oftheinteraction,aresultofthespatialin-
min
tegrationlimitsoftheanalyticalderivation.Fromtheanalysisof
therelevantparametersabove,itfollowsthattheparameter s
min
to 5.7). Most importantly, the simulations’ outcome reveal the is actually not a free parameter, but has to be a function of the
stand-off distance as the minimum spatial scale of the forming parametersgivenabove.InSect.5.3,wecomputetheminimum
anisotropic density structure around the moving body. We an- lengthscaleoftheinteractionfromournumericalsolutions,as-
alyze and discuss this new aspect in-depth and derive a semi- sociatethislengthscalewiththeshock’sstand-offdistance,and
analyticalrelationbetweenthestand-offdistanceandtheaccre- deriveitsdependenceontherelevantproblemparameters.
tion radius from fits to the numerical data in Sect. 5.3. Finally, Inthefollowing,wepresentthenumericalexperimentsper-
wecombinethesefindingstoderiveaconvenientsemi-analytical formed. The dependence of the dynamical friction on each of
expressionforthedynamicalfriction(Sect.6). the relevant parameters is determined in a single or multiple
Wewouldliketopointoutagainthatwedonotassumeana dedicated simulation series. Table 1 gives an overview of the
priorivalidityoftheclassicdragformulaasstatedinEq.(1).Our simulation series and their physical and numerical parameters.
procedure is to check the scaling of each parameter in Eq. (1) Due to the scale-freedom in the equations of the problem, the
individually and confirm the existence of the logarithmic term. gas mass density can be chosen arbitrarily, we use a value of
After this we demonstrate that the minimum distance smin in ρ∞ = 1.510−4gcm−3 inallsimulationsperformed.Inthetable,
Eq. (2) is closely related to the stand-off distance of the shock thevelocityofthebodyisgivinginunitsofMach.Simulations
fromtheobject.Wethenpresentanewformulaforcalculating with varying soundspeed of the medium (series “S”) use the
sminfromthebasicphysicalparameteroftheproblem. samevelocityoftheperturber,henceyielddifferentMachnum-
bers as well. Simulations with varying adiabatic index of the
medium (series “G”) use varying initial values of the gas pres-
5.1. Therelevantproblemparameters
sure to keep the soundspeed the same in all simulations of the
Firstofall,themovingobjectischaracterizedbyitsmassMand series.
velocity V . The impact of these parameters on the dynamical
∞
frictionisanalyzedanddiscussedinSects.5.5and5.6.Forhigh
5.2. Thegasmassdensity
enoughmassesorsmallenoughvelocities,thesmallestscaleof
interaction with the gas is set by the gravity of the moving ob-
Thedynamicalfrictionshouldscalelinearlywiththedensityof
jectratherthanitsgeometricalradius.Inthesecases,themoving
theenvironment:
body can also be treated as a point mass, as it is usually done
insemi-analyticalapproaches,e.g.inthederivationbyOstriker FDF ∝ρ∞ (18)
(1999). Its geometrical radius R becomes important for the in-
teraction with the surrounding gas in cases of either low mass Thisscalingbehaviorisadirectconsequenceofthefactthatthe
orhighvelocities,respectively.Weinvestigatetheimpactofthe total dynamical friction is given by the sum of all the gravita-
geometricalradiusinSect.5.3. tional pulls from the environment onto the body. Each gravita-
Thegaseoushomogeneousmediumischaracterizedbytwo tional pull scales linearly with the density of the environment.
ofthefourquantities:massdensityρ ,pressurep ,temperature Additionally, the hydrodynamical equations (see Sect. 3.2) are
∞ ∞
T ,orsoundspeedc .Theremainingtwoarethendetermined scale-free in density, and, hence, the flow morphology is inde-
∞ ∞
accordingtoEqs.(12)and(13).Inthefollowing,wewillchoose pendentontheinitialdensityofthemedium.
7
DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
5.3. Theminimumspatialscaleofinteraction
0.2 RSO
Thetotaldynamicalfrictionactingonthemovingbodyisgiven
bythespatialintegraloverthegravitationalpullofthesurround-
f 0.0
ing gaseous medium. According to Eqs. (1) and (2), it is ex-
pectedtoscalewiththevalueofaminimumspatialinteraction
scalesminaccordingtotheCoulomblogarithm: −0.2 M=2.00
100 101 102 103
F ∝ln(s /s ) (19) r/R
DF max min
Below this minimum interaction scale, the gaseous medium is 0.10
assumed to exert no net gravitational pull on the body. What RSO
0.05
actuallysetstheminimuminteractionscaleisanopendiscussion
intheliterature. f 0.00
0.05
− =4.00
5.3.1. Aspatialforceanalysis 0.10 M
− 100 101 102 103
In our study, the approach of direct numerical experiments al- r/R
lows us to properly determine the minimum spatial interaction
scaleincaseofgaseousmediaquantitatively.Inordertoderive 0.02
the impact of each spatial scale individually, we first calculate RSO
0.01
thegravitationaldragforceactingontheobjectfromthegasin-
side a specific shell at radius r with a radial shell thickness of f 0.00
i
ri+1/2−ri−1/2: −0.01 =8.00
f(r)=−(cid:90) GM ρ(x)eˆ dV, (20) −0.02100M 101 102 103
i r2 r r/R
S(ri)
wherethevolumeofshellS(r)isgivenby
i
(cid:90) π(cid:90) ri+1/2 Fig.5. Fractionsofthegravitationaldragforce f(ri)tothetotal
S(r)=2π r2sin(θ)drdθ. (21) gravitational drag force acting on the moving object as a func-
i
0 ri−1/2 tion of distance to the object. The vertical solid lines mark the
stand-offdistanceR oftheshockaheadoftheobject,andthe
Thetotalgravitationalforceactingontheobjectisgivenbythe SO
dashedverticallinesdenotetheaccretionradiusR accordingto
sumoverallshells: A
Eq. (3). Results are shown for simulations with three different
(cid:88)
Fnumerical = f(r). (22) MachnumbersMaslabeledinthebottomleftcorners.
DF i
i
Even though the above Eq. (20) is written in vector form, only symmetry, which actually results into a small gravitational ac-
its z-component is non-zero due to the axial symmetry of the celeration instead of a drag. But this effect remains negligible
problem.Thenetgravitationaldragwillleadtoaslow-downof comparedtothetotaldynamicalfrictionandisonlymarginally
thebody,whichismovingalongthepositivez-direction. visible in Fig. 5 by eye for the case of highest Mach number;
In Fig. 5, we show the fractions of the gravitational force here,thefractionofthedynamicalfrictionforceclosetobutstill
f(ri)asafunctionofradiusordistancetothemovingbody,re- belowthestand-offdistancehasapositivesign,indicatingposi-
spectively. The spatial analysis of the force is shown for three tiveaccelerationofthebody.
simulations with different Mach numbers of M = 2,4, and8. Clearly, the shells inside the sphere around the object with
SimulationparametersaregiveninTable1,series“V4”. radiusR donotcontributetothetotaldynamicalfrictionforce,
The spatial analysis clearly reveals the so-called stand-off becauseSaOroundtheobjectasphericallysymmetricalhydrostatic
distance RSO of the shock front as the searched-for minimum envelope forms. This is confirmed in Fig. 6 where we plot the
spatialinteractionscalesmin: individual contributions to the total drag force on the spherical
body, as a function of Mach number. As shown, the contribu-
smin =RSO. (23) tionofthepressureforce,Fprs,actingontheobjectisnegligible.
in
The associated bow shock forming in front of the body is vis- From this we can conclude that i) to calculate the drag acting
ible in the density morphology depicted in Fig. 4. The shock on the object it is sufficient to consider the dynamical friction
frontR denotestheradius,wheretheflowchangesfromsuper- alone, and that ii) the relevant quantity for the minimum inter-
SO
sonictosubsonicvelocities.Moreover,thedensitymorphology action scale smin is given by RSO. Additional simulations with
around the body is characterized by the formation of a hydro- fixed Mach number M but different object masses M give the
static envelope extending up to R . This hydrostatic envelope sameresultforthespatialanalysisoftheforce.
SO
isveryclosetosphericalsymmetry,asalsodepictedinFig.A.1.
Hence,althoughthisregionmarksthehighestgasmassdensity,
5.3.2. Aconvenientexpressionforthestand-offdistance
and in principle might have the strongest gravitational impact
duetoitscloseness,thenetgravitationalpullwithinthishydro- Itistheaimofthisstudytoderiveageneralexpressionforthe
static envelope (r < R ) turns out to be negligible. If at all, dynamical friction, that allows to compute the acting gravita-
SO
thestrongercompressionintheforwarddirectionofthetrajec- tional drag from the relevant problem parameters without the
toryofthemovingbodyyieldsaslightdeviationfromspherical need of direct numerical simulations. As revealed in the previ-
8
DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
200
Fipnrs Numericalexperiments
150 FFooptuurratts 102 Fit:a·Mb,b=−2.054
Fnumerical
DF
s] 100 Ftot
t
ni
U R
ode 50 R/SO101
C
[
F 0
50
−
100 100
− 2 4 6 8 10 100 101 102
M M
Fig.6. The force contributions for the equilibrium state ac- 102
Numericalexperiments
cording to Eq. (17) for the gravitating, rigid sphere, given as
afunctionoftheMachnumber. Fprs denotesthepressureforce Fit:a·Mb,b=1.018
in
measured at the surface of the sphere, Fnumerical denotes the to-
DF
taldynamicalfrictioncalculatedaccordingtoEq.(22), Fprs the
out
pressure force measured at the outer boundary of the compu-
R
tational domain, and Fotrust the momentum transport through the /O101
outerboundary.Thesumofthese4forcesisdenotedbyFtot. RS
ous section, such an expression for the dynamical friction will
includeascalingwith
F ∝ln(s /R ). (24)
DF max SO
In the next step of our investigation, we derive an expression 100 100 101 102
for the stand-off distance, which is based on the basic problem M/M
Jup
parameters only. Therefore, we make use of the apparent rela-
tionshipofthestand-offdistanceR totheaccretionradiusR ,
SO A Fig.7. Correlation of the shock’s stand-off distance R with
seeEq.(3)forthedefinitionoftheaccretionradius. SO
objectvelocityorMachnumberMrespectively(toppanel)and
Fig. 5 seems to indicate that R is directly proportional to
SO withobjectmassM(bottompanel).Solidlinesdenotelinearfits
the accretion radius R . Since R scales as ∝ M/V2 we check
thesescalinglawsfortAhestand-offAdistanceaswell.W∞eusease- inthelog-logplane.
riesofsimulationswherewevarysystematicallythemass M of
theobjectanditsvelocityV∞.Simulationparametersaregiven medium.DetailedsimulationparametersaregiveninTable1.To
in Table 1, series “M2” and “V4”. The results are displayed in findausefulgeneralexpressionforthestand-offdistanceR we
SO
Fig. 7. Both panels show least-square fit lines that support the utilizethenon-linearityparameter
factthatR ∝ M/M2.ForlargerMthestand-offdistancebe-
SO
comessmalleraccordingtoR ∝ M−2 asdisplayedinthetop GM 1 M2 R
SO η= = A , (26)
panel of Fig. 7. For fixed M, lowering the object mass yields (M2−1)c2R 2 M2−1 R
a smaller stand-off distance in agreement with R ∝ M as ∞
SO
showninthebottompanelofFig.7.Thisoutcomeconfirmsthe as introduced by Kim & Kim (2009). In Fig. 8 we show the
scaling laws as long as the geometrical extent of the body R is stand-off distance R as well as the ratio of R /R as func-
SO SO A
smallcomparedtoR .ForhighMachshocksandforlowbody tion of the non-linearity parameter η. As seen in the top panel,
SO
masses, respectively, the stand-off distance cannot decrease to for η > 1 (larger M, lower M) we find the scaling R /R ∝ η,
SO
smaller radii and approaches the size of the body R instead. In with an unknown proportional constant close to unity as indi-
allothercases,R isdirectlyproportionaltoR : catedbytheaddeddashedline.Thisisingoodagreementwith
SO A
the results of Kim & Kim (2009). The largest deviation from
R ∝R . (25)
SO A thistrendisseenintherightmostgreentriangle,whichbelongs
Animportantdifferencebetweenbothradiiisgivenbythefact to a simulation in the weakly supersonic regime M = 1.25. A
thatthestand-offdistancedependsonthethermodynamicsofthe spatial analysis of the final configuration of this model shows
system.Asaconsequence,westudytheimpactoftheadiabatic that behind the shock front and around the object the gaseous
indexγonthestand-offdistanceandthefinaldynamicalfriction mediumisnotinhydrostaticequilibrium–asforthesupersonic
terminSect.5.7below. cases – but displays small residual motions that did not disap-
Toinvestigatetheproportionalitybetweenthetworadiifur- peareveninlong-termruns.So,weexpectdeviationsfromthe
ther, we perform additional simulations varying the mass, ra- following relation (27) for non-supersonic flows, a regime that
dius,andvelocityoftheobjectaswellasthesoundspeedofthe is not part of the present study and will have to be explored in
9
DanielThunetal.:Dynamicalfrictionforsupersonicmotioninahomogeneousgaseousmedium
The bottom panel of Fig. 8 just confirms for completeness
103 againtheproportionalitybetweenthestand-offdistanceandthe
Variationofmach(R=0.01) accretionradiusfordifferentMachnumbers,objectmasses,and
Variationofmass(R=0.01) soundspeedsintheregimeR (cid:29)R.Wecheckedinadditionthat
SO
Variationofmach(R=0.1) achangeinthesizeRofthemovingbodydoesnotinfluencethe
Variationofmass(R=0.1) stand-offdistanceaslongasRremainssubstantiallysmallerthan
102 VRaSOri/aRtio=nηofsoundspeed(R=0.01) RSO.Hence,thecurvesforR=0.1andR=0.01areidentical.
R
/
O
RS 5.4. Themaximumextentofthewake
101
Themaximumextentofthewakepastthebodydeterminesthe
size of the perturbed region, which will contribute to the total
dynamicalfrictionviaitsgravitationalpull.InEq.(1),themaxi-
mumextentofthewakeentersastheouterradiusoftheintegral
100 s intheCoulomb-LogarithmC :
10−1 100 101 102 max A
η
F ∝ln(s /R ). (28)
DF max SO
101
Variationofmach(R=0.01)
Variationofmass(R=0.01)
For an infinite medium, the size of the wake s is given
Variationofmach(R=0.1) max
bythedurationofthemovementoftheperturbingobject.Fora
Variationofmass(R=0.1)
Variationofsoundspeed(R=0.01) bodystartingitsjourneyatt=0withaconstantvelocityV∞,the
time-dependentsizeofthewakes atalatertimetisgivenby
max
RA thedistancetravelledsmax(t)=V∞t.
/
O The relation (28) implies that the dynamical friction in-
S
R
creases as long as the body is moving through the medium,
100
leading to a larger and larger wake. In principal, the dynami-
calfrictionseemstoapproachaninfinitelyhighforceforanin-
finitelylongtravelingobject.Inpractice,thedynamicalfriction
cannotgrowinfinitely,evennotforaninfinitelylargemedium,
because the dynamical friction acting on the body leads to a
10−1 100 101 102 slow-down of the object on a dynamical timescale of the gas-
η
body interaction (Sa´nchez-Salcedo & Brandenburg 2001). The
dynamicaltimescaleofdynamicalfrictionisgivenastheratioof
Fig.8. Correlationoftheshock’sstand-offdistanceR withthe theinitialmomentumofthemovingobjectandtheactingforce
SO
non-linearityparameterη,asgiveninEq.(26),forvariousmodel tDF = MV∞/FDF. Hence, the velocity of the perturber mono-
sequences.Inthetoppanel,R isnormalizedbytheobjectra- tonicallydecreasesintimeuntilthebodyisatrest.Asaconse-
SO
dius R. For η > 1, a dashed line is superimposed, representing quence,thedynamicalfrictionfirstincreases(duetoagrowing
therelationR /R = η.Inthebottompanel,R isnormalized extentofthewake)uptoamaximumvalueanddecreasesafter-
SO SO
bythecorrespondingaccretionradius. wards(duetotheslow-downofthebody).
Here, we model the evolution of the system on timescales
muchsmallerthanthetimescalefortheslow-downofthebody
futurestudies.Forcomparison,wedenotethattherightmostred
toextracttheactingdynamicalfrictionasafunctionoftherel-
square belongs to a simulation with M = 2, but even larger η
evant initial system parameters. As a next step, we check the
than the green triangle run, and matches the found relationship
scaling of the dynamical friction force with the extent of the
muchbetter.
wake. We compute the resulting dynamical friction by numer-
For η < 1 (larger M, lower M) we enter the regime where
icallyintegratingthegravitationalpulloftheformingwakeonto
thestand-offdistanceislimitedbythesizeoftheobjectR,and thebodyuptodifferentradiiwithinthecomputationaldomain.
therefore the ratio R /R remains constant. For this regime of
SO As a result, we obtain the individual contributions of the wake
lowη,Kim&Kim(2009)findadifferentscalingduetothefact
ateachradius.Forfurtheranalysis,weextendedthesizeofthe
thattheyusedasmoothedgravitationalpotentialtomodeltheir computationaldomaintoR =25R (modelC),othersimu-
domain (cid:12)
perturberinsteadofthesolidsurfaceusedinourstudy.Usingthe
lationparametercorrespondtothefiducialcase(modelF)given
definitionofη(Eq.(26)),theproportionalityR /R∝ηimplies
SO inTable1.
1 M2 AsshowninFig.9,thescalingofthenumericallydetermined
RSO ∝ 2 M2−1 RA forRSO (cid:29)R. (27) dragforcewiththeextentofthewakefollowstheexpectedlog-
arithmic relation. In the following, simulations make use of an
Relation (27) represents a suitable expression for the stand-off arbitrary size of the computational domain, which just should
distance.Aspointedoutabove,theproportionalityconstantfor bemuchlargerthanthestand-offdistanceoftheshock.Thefi-
thesimulationsperformedsofarseemstobeveryclosetounity. nallyactingdynamicalfrictioncanthenbedeterminedusingthe
But the relation still lacks the effect of different equations of scaling relation Eq. (28). We use an outer radius of the com-
state, i.e. any dependence on the adiabatic index γ. We investi- putationaldomainofeither100Ror1000R,dependingonthe
gatethisdependenceindetailinthefollowingsection. specifiedradiusRofthemovingbody.
10