Emergence of a stellar cusp by a dark matter cusp in a low-mass compact ultra-faint dwarf galaxy

2.4 MB·

by  Shigeki Inoue

Mon.Not.R.Astron.Soc.000,1–??(2014) Printed10February2017 (MNLATEXstylefilev2.2)
Emergence of a stellar cusp by a dark matter cusp in a
low-mass compact ultra-faint dwarf galaxy
7
1 Shigeki Inoue⋆
0
2
Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The Universityof Tokyo, Chiba 277-8583, Japan

b Department of Physics, School of Science, The Universityof Tokyo, Bunkyo, Tokyo113-0033, Japan
e
F

9 10February2017

]
A
ABSTRACT
G Recent observations have been discovering new ultra-faint dwarf galaxies as small as
∼ 20 pc in half-light radius and ∼ 3 km s−1 in line-of-sight velocity dispersion. In
.
h thesegalaxies,dynamicalfrictiononastaragainstdarkmattercanbe significantand
p
alter their stellar density distribution. The effect can strongly depend on a central
-
o density profile of darkmatter,i.e. cusp orcore.In this study, I performcomputations
r using a classical and a modern analytic formulae and N-body simulations to study
t
s howdynamicalfrictionchangesastellardensityprofileandhowdifferentitisbetween
a a cuspy and a cored dark matter haloes. This study shows that, if a dark matter
[
halo has a cusp, dynamical friction can cause shrivelling instability which results in

3 emergenceofastellarcuspinthecentralregion <∼2pc.Ontheotherhand,ifithasa
v constant-densitycore,dynamicalfrictionissignificantlyweakeranddoesnotgenerate
9 a stellar cusp evenif the galaxyhas the same line-of-sightvelocity dispersion.In such
5 a compact andlow-mass galaxy,since the shrivelling instability by dynamicalfriction
4 is inevitable if it has a dark matter cusp, absence of a stellar cusp implies that the
7 galaxy has a dark-matter core. I expect that this could be used to diagnose a dark
0
matter density profile in these compact ultra-faint dwarf galaxies.
.
1
Key words: instabilities – methods: numerical – methods: analytical – galaxies:
0
dwarf – galaxies: kinematics and dynamics.
7
1
:
v
i
X
1 INTRODUCTION turn a cusp into a core by flattening the inner slopes of
r the primordial DM density profiles in dwarf galaxies as
a Dhaavreklomnagttbeeren(DdMeb)atedde.nsTithyeorpertoicfialelsstiundiedswsaurcfhgaaslacxoiess- massive as MDM ∼ 1010–1011 M⊙ (e.g. Governato et al.
2010; Pontzen & Governato 2012; Ogiya & Mori 2014;
mological N-body simulations have demonstrated that
El-Badry et al.2016;Di Cintio et al.2017).Ifthelattersce-
DM density increases toward the galactic centre inde-
narioisthecase,sincedynamicalmassesofsomeultra-faint
pendent of a halo mass (e.g. Dubinski& Carlberg 1991;
dwarfgalaxies(UFDs)inthelocalgrouphavebeenobserved
Navarro et al.1997;Klypin et al.2001;Springel et al.2008;
to be significantly smaller than the mass threshold above
Ishiyama et al. 2013).Ontheotherhand,observationspro-
which the baryonic effect is influential to their central DM
posedthatdwarfgalaxiesseemtohavenearlyconstantden-
densities, they could be expected to preserve the primor-
sities of DM at their central regions (‘cusp/core problem’,
dial DM density profiles which may be cuspy. Accordingly,
e.g. Gilmore et al. 2007; Oh et al. 2011; Hayashi & Chiba
it is interesting to try to determine DM density profiles of
2012).1
suchlow-massUFDs.Itis,however,stillimpossibletoknow
As a possible solution, if DM haloes consist of warm
whethertheirDMhaloeshavecuspsorcoresbecauseofonly
or self-interacting particles, all dwarf galaxies are ex-
a handful of stars observable by spectroscopy to measure
pected to have central DM cores. It has also been pro-
their line-of-sight velocities (LOSVs) and model their DM
posed, alternatively, that (recursive) baryonic feedback can
haloes. Hence, it is worthwhile looking for an alternative
method to deduce which typeof DM thelow-mass galaxies
have, cusp or core. For example, Pen˜arrubia et al. (2016)
⋆ E-mail:[email protected] have proposed a method using a fraction of wide binaries
1 Lowsurfacebrightnessgalaxieshavealsobeenobservedtohave which can be disrupted by tidal force depending on their
DMcores(deBlok2010,andreferencestherein)althoughIdonot
DMpotential in UFDs.
discussthesegalaxies.
2 S. Inoue
The smallest UFDs are as tiny as Rh < 30 pc in where ρDM,0 and rs are scale density and radius, and γ is
half-light radius and L 102−3 L in lumi∼nosity (e.g. aninnerdensityslope.IassumeacuspyDMhalotoberep-
⊙
∼
Willman et al. 2005; Belokurov et al. 2009; Laevens et al. resented by setting γ = 1, which corresponds to the Hern-
2015; Martin et al. 2016; Homma et al. 2016; Simon et al. quist model profile (Hernquist 1990), and a cored DM halo
2016) although current observations still cannot reject the is represented by γ =0. Local velocity dispersion of DM is
possibility that some of them are extended globular clus- generally computed by solving Jeans equation,
ttmehraasyt.bdReyencmaeamnrtiglcyian,laHflrleyircnteaiffonendcte(izDve(F2o)0n1o6nt)haehsattismaraenasacglaaylietniscotafldlyar1dk0ismGcuaystrsteeindr σD2M(r)= ρD1M Zr∞ρDMGMrD′M2(r′) dr′, (2)
∼
Draco II — observed physicalproperties of which areRh = where G is the gravitational constant, and MDM(r) is mass
129.9+−862p.c1,kbmrigsh−t1nemsseaMsuvre=d−w2it.h9i±n0R.8h,(LMOaSrVtindiestpearl.si2o0n16σ)h—= obfarDyoMnecnocmlopsoendenwtitihsinnerg.liHgiebrlee,aInadsstuhmatevtehlaotcigtyravdiitsytriobfua-
±
byadoptingtheChandrasekharDFformula(Chandrasekhar tion is isotropic. The analytic solutions of σDM for γ = 0
1943) to his singular isothermal DM halo model. His result and1canbefoundinDehnen(1993)andHernquist(1990).
impliesthatDFagainst DMcouldsignificantlychangestel- This study discusses how the DF against DM affects
lardistribution in UFDsmorecompact and/orless massive stellar distribution and how different it is between cuspy
than Draco II, which will be discovered by future observa- and cored DM haloes. I use a Plummer’s model for stellar
tions. distribution of a compact UFD,
The effect of DF strongly depends on the DM den-
sity profile. It has been known that DF drag force becomes 3M r2 −52
ρ (r)= ⋆ 1+ , (3)
significantly weaker in cored density distribution than in ⋆ 4πr3 (cid:18) r2(cid:19)
⋆ ⋆
cuspy one, once a massive particle enters the core (e.g.
whereM andr arethetotalmassandscaleradiusofstars.
Hernandez& Gilmore 1998). Studies using N-body sim- ⋆ ⋆
Thismodelhasacoreofstarsinr r wherethedensityis
ulations have demonstrated that drag force by DF does ≪ ⋆
cease practically in a constant-density core, probably by nearly constant. Two-dimensional half-light radius Rh and
non-lineareffects (e.g. Goerdt et al. 2006;Read et al. 2006; integrated mass Mh inside Rh are obtained by assuming a
constant mass-to-luminosity ratio and integrating equation
Inoue 2009, 2011; Arca-Sedda& Capuzzo-Dolcetta 2014;
Petts et al.2015,2016).Therefore,ifanextremelylow-mass (3).Inthis study,I assume r⋆ =20 pc(Rh =r⋆ in aPlum-
mer’s model). Luminosity-weighted LOSVdispersion inside
UFD has a constant-density core of DM, DF could be too
weak to affect the stellar distribution. On the other hand, Rh is given as
if such a UFD has a DM cusp, DF against DM could be 4π ∞ Rh
stitornon,gsuecnhoausghemtoergmenakcee oafltaersatteiollnasr ctuospitsorstfeolrlmaratdiiosntriobfua- σh2 = Mh Z0 dzZ0 ρ⋆σ⋆2(R′,z)R′ dR′, (4)
nucleus cluster as a remnant of stars fallen into the galac- wherestellar velocity dispersion σ⋆ is computedfrom equa-
ticcentre.Currentobservationsoflow-mass compactUFDs tion (2) in which ρ⋆ is substituted for ρDM. Since stellar
are limited to a close distance of d< 30 kpc from the sun gravity is now assumed to be negligible, setting σh gives
because of their faintness. At this d∼istance, each star in a ρDM,0 whentheotherparametersinequation(1)arefixed:2
UsmFeDarcinagnibsesmreasolllevredthsainnctehethmeetaynpisceaplasrizaetioonf oobfssetravrastieovneanl 1σ.h25∝anρdD3M.0,0.kImnsw−h1a.3tfFoilglo.w1si,llIusdtirsactuesssrtahdeiatwlporcoafisleessooffσσhDMto,
at the galactic centres. Therefore, theexpected stellar cusp circular velocities vcirc GMDM(r)/r and σ⋆ normalised
≡
andthenucleusclusterwouldbeobserved asadensegroup by σh in my cuspy and copred halo models with rs =125 pc
of stars at thegalactic centreif it exists. and 1 kpc.
Thisstudyaddressestheeffect ofDFbyDMonstellar
distribution in an extremely low-mass and compact UFD
2.1 Analytic formulae of dynamical friction
and focus on how different it is between cuspy and cored
DMdensity profiles. In Section 2, I perform analytical esti- 2.1.1 The Chandrasekhar formula
mation of stellar shrivelling due to DF based on a classical
IconsiderDFagainstDMonastar.TheChandrasekharDF
andamodernDFformulae.InSection3,IperformN-body
formula underMaxwellian velocity distribution4 is given as
simulations resolving every single star and demonstrate the
same as the analytical estimation presented in Section 2.
Finally, I present discussion and summary of this work in 2 When σh =1.5 kms−1, the values of ρDM,0 inthe cored DM
Section 4. models are 9.8, 7.0, 5.8 and 5.1×10−1 M⊙ pc−3 for rs = 125,
250, 500 pc and 1 kpc. Those inthe cuspy models are 1.1, 0.45,
0.20and0.096×10−1 M⊙ pc−3,respectively.
3 The cusp and the core models of equation (1) have the fi-
2 ANALYSIS USING DYNAMICAL FRICTION nite total masses, Mcusp,tot = 2πρDM,0rs3 and Mcore,tot =
FORMULAE (4/3)πρDM,0rs3, respectively. When σh = 1.5 kms−1, for rs =
125pc,thetotalmassesofcuspyandcoredhaloesareMcusp,tot=
Inthisstudy,Idescribeadensityprofileof aDMhalobya 1.3×106 M⊙ and Mcore,tot = 8.0×106 M⊙. For rs = 1 kpc,
Dehnen model (Dehnen1993), Mcusp,tot=6.0×107 M⊙ andMcore,tot =2.2×109 M⊙.
4 Although Chandrasekhar (1943) has also proposed more gen-
ρDM(r)= rγ(ρrD+M,r0sr)s44−γ, (1) reerfaelrfotormeqsuoaftaionnal(y5t)icasDCFhnaontdrraelsyeiknhgaronfoMrmauxlwaeilnl dthisitsripbaupteior.n, I
Emergence of a stellar cusp by dynamical friction 3
Figure2.Leftpanel:dragforcebyDFcomputedwithequation(5)withlnΛ=15normalisedbym2 inthehalomodelofequation(1).
⋆
TheblueandredlinesindicatetheresultsinthecuspyandthecoredDMmodels,respectively. Right panel:sameastheleftpanelbut
computed withequation (6)inthe caseof σh =1.5kms−1 and m⋆ =0.5M⊙.Theredarrowsindicatetidal-stallingradii,rTS,inthe
coredDMhaloes.Inthecuspyhaloes,rTS<0.1pc.
locityvectorv⋆.Byassumingacircularorbit,i.e.v⋆ =vcirc,
equation (5) can be solved with σDM from equation (2).
In the case considered here, equation (5) is independent of
ρDM,0 (i.e. σh) since ρDM, v⋆2 and σD2M are proportional to
ρDM,0althoughtheDFtimescale, m⋆σh/FDF,dependson
∼
ρDM,0. In addition, because theparentheses in equation (5)
is independentof m⋆, FDF/m2⋆ is independentof m⋆.
The left panel of Fig. 2 shows the Chandrasekhar DF
force of equation (5) with lnΛ = 15 normalised by m2 in
⋆
thecuspy and the cored haloes with various rs. The Figure
indicates that the strength of DF is remarkably different
between the cuspy and the cored DM haloes; DF increases
monotonically towards the centre in a DM cusp (the blue
lines), whereas it is approximately constant or gently de-
creases in a core (the red lines). The behaviour of DF is
almost independent of rs in the cuspy haloes, whereas DF
in a core becomes weaker when rs is larger. Although the
difference of DF inside rs between the cusp and the core
becomes smaller with decreasing rs, it is still quite large in
r <10 pcevenin thecase ofrs =125 pc.Thismeansthat,
if∼a stellar component of a UFD is deeply embedded in a
Figure1.RadialprofilesoflocalvelocitydispersionsofDMand
stars, and circular velocities in the cuspy (left) and the cored DM halo (i.e. Rh ≪ rs), one can expect that DF strongly
(right) halo models with rs =125 pc (top) and 1 kpc (bottom). depends on a density profile of DM and could change the
Theleftandrightordinatesarefortheleftandrightpanels.The stellar distribution if the compact UFD has a cusp of DM.
profilesarenormalisedbyσh. Moreover, in a cuspy halo, a star undergoing DF migrates
intoaninnerradius, at which DFis evenstronger(see Sec-
tion 4.1).
FDF=−4πlnΛGv22ρDMm2⋆ (cid:20)erf(X)− √2Xπ exp(−X2)(cid:21), (5)
⋆
where m and v are a mass and a velocity of a star, 2.1.2 Petts et al. formula
⋆ ⋆
X ≡v⋆/(√2σDM),andΛisaparameter,whosepropervalue The Chandrasekhar formula of equation (5) is, however,
is still under debate (e.g. Arca-Sedda& Capuzzo-Dolcetta
basedonseveralassumptionssuchastheMaxwellian veloc-
2014; Just & Pen˜arrubia 2005; Petts et al. 2015, 2016)5.
ity distribution and the invariable parameter of Λ. There-
The direction of FDF is presumed tobe opposite totheve- fore, improved formulae have been invented by previous
studies.Recently,Petts et al. (2016) proposed more sophis-
5 Basically,Λisdefinedtobearatiobetweentheminimumand ticated DF modelling based of the general Chandrasekhar
formula(equations25and26inChandrasekhar1943),which
maximum impact parameters of two-body gravitational interac-
tion, i.e., Λ ≡ bmax/bmin, where bmin ∼ Gm⋆/v⋆2, bmax ∼ rs in uses a distribution function instead of the Maxwellian dis-
the classical formula (Chandrasekhar 1943; Binney&Tremaine tribution and takes high-velocity encounters into account.
2008). They demonstrated that their improved DF model can re-
4 S. Inoue
produceorbitsofinfalling particlesin cuspyandcoredden-
sity fieldsbetterthan theclassical formula. Theyformulate
DFforce6 as
FDF=−2π2G2vρ⋆2DMm2⋆ Z0vescJ(vDM)f(vDM)vDM dvDM,(6)
v⋆+vDM v2 v2 b2 V4
J = 1+ ⋆− DM log 1+ max dV,(7)
Z|v⋆−vDM|(cid:18) V2 (cid:19) (cid:18) G2m2⋆ (cid:19)
where f(vDM) represents a distribution function of DM,
which is defined so that 4π f(vDM)vD2M dvDM = 1, and
escapevelocityvesc =√ 2Φ,RandV correspondstorelative
−
velocityofencounter.Inequation(7),themaximumimpact
parameter
ρDM(r)
bmax =min , r . (8)
(cid:18)dρDM/dr (cid:19)
The value of ρDM/(dρDM/dr) can be taken as the distance
withinwhichthedensityfieldcanbeconsideredtobehomo-
geneous (Just & Pen˜arrubia 2005; Just et al. 2011). How-
ever, since it can diverge in a constant-densitycore, bmax is
limited to be 6 r (Pettset al. 2015). Equation (7) can be
solved analytically (see AppendixA).
Inaddition,Petts et al.(2016)alsointroduceda‘tidal-
stalling’ radius, rTS, at which the tidal radius of a massive
particle is equalto its orbital radius. The tidal radius is
Figure 3. Time-evolution of the stellar surface density profiles
rt = Ω2 Gdm2Φ⋆/dr2, (9) ainndthσehcu=sp1y.5ankmd tsh−e1c.oIrnedthDeMtopdeannsditybomtotodmelpwainthelsr,se=qu1a2t5iopncs
− (5)and(6)areusedtomodelDFeffect.Inthetoppanel,results
whereΩ2 =GMDM/r3.Theyargued thatDFceases within at t = 4 and 7 Gyr in the cored halo are not shown. The black
the tidal-stalling radius because of non-linear effects, and solid line indicates the initial state of the profile. The ordinates
showed that the radius of rTS matches well results of N- are normalised by Σh ≡ M⋆/(2πR2h). The density peaks at the
body simulations (a radius of DF cessation in a core, see centre reach log(Σ⋆/Σh)=2.2 and 2.1at t=10 Gyr inthe top
andbottom panels.
Section1).InthePetts et al.DFmodel,FDF =0inr<rTS
although equation (6) still returnsa non-zero value.
The right panel of Fig. 2 shows the Petts et al. DF
force of equation (6) normalised by m2. Unlike the Chan- stars is given by Eddington’s formula (Binney & Tremaine
⋆
drasekharformula,nowFDF/m2⋆ weaklydependsonσh and 2008) with isotropy. I do not take into account mutual in-
m⋆. Here, I assume σh = 1.5 km s−1 and m⋆ = 0.5 M⊙, teractionsbetweenthestars,thereforetheresultisindepen-
however the results hardly change between σh = 1.5 and dent of thenumberof stars. For thesake of statistics, I use
3.0 km s−1. Radii of rTS become about 1.4 times smaller a random sample of ten million stars in each run. While
whenσh =3.0km s−1.IntherightpanelofFig.2,although integrating their orbits with respect to time, the stars are
the differences between the cuspy and the cored haloes are decelerated byDFrepresented bytheanalytic formulae ev-
still quite large, it is remarkable that the Petts et al. for- ery timestep. I assume m⋆ = 0.5 M⊙ as a typical mass of
mula predicts DF significantly stronger than the classical a star as old as 10 Gyr (Kroupa 2002; Maraston 2005).
∼
formula in the central regions of the cored haloes (‘super- The analytic DF is considered to work until a star reaches
Chandrasekhar DF’, Read et al. 2006; Goerdt et al. 2006; the radius rlimit at which MDM(rlimit) = m⋆. In my mod-
Zelnikov & Kuskov 2016; Petts et al. 2016). On the other els, rlimit 0.1 and 0.5 pc in the cuspy and the cored DM
≃
hand,theDFinthecuspyhaloesissimilartothatgivenby haloes.Whenastarentersrlimit withavelocityslowerthan
theChandrasekhar formula. vcirc|r=rlimit, the star is stopped there and considered to
be fallen into the galactic centre by DF. When the Chan-
drasekhar formula is applied, I set lnΛ = 15. When the
2.2 Orbital integration with the formulae Petts et al. formula is applied, DF ceases within rTS (i.e.
Using the models and the DF formulae described above, I FDF = 0). I use a second-order leap-frog integrator for the
orbital computations with a constant and shared timestep
performorbitalintegrationofstarsunderthepotentialgiven
bytheDMdistributionofequation(1).Withtheinitialspa- of ∆t = 0.01×rlimit/vcirc|r=rlimit. I confirmed the conver-
gence of my results with respect to ∆t. Orbits of stars are
tial distribution of equation (3),thevelocity distribution of
time-integrateduntilt=10Gyr,andIobtainstellarsurface
densities as functions of radius in the runs.
6 Pettsetal. (2016) proposed two models of DF: ‘P16’ and Fig. 3shows theresults oftheorbital integration using
‘P16f’.IusetheirP16fmodelinthisstudysincetheyconcluded the Chandrasekhar (top) and Petts et al. (bottom) formu-
thatP16fismoreaccuratethanP16. lae in the cuspy and the cored haloes with rs = 125 pc
Emergence of a stellar cusp by dynamical friction 5
and σh = 1.5 km s−1. Although the stellar surface density realisticmodels,IperformN-bodysimulationsinwhichthe
is nearly constant in R< 5 pc in the initial state (black modelsarefullyself-consistent,andDFdragforcenaturally
line), the density profiles∼are significantly steepened after arises as mutualinteractions between particles.
t 4 Gyr in the cuspy DM halo (blue lines). In addition,
∼
sharpstellarcuspslikenucleusclustersemergeatthegalac-
tic centres, which have five and four per cents of the total 3.1 Settings
numberofstellarparticleswithinR<0.5pcinthetopand
The initial conditions of my N-body simulations are the
the bottom panels. The cusps mainly consist of stars fallen
sameastheDMandstellarmodels(equation1and3)with
into the centres by DF. The steepened stellar distribution
profilesarenearlyexponentialoutsidethestellarcusps.Be- theparametersusedinFig.3:rs =125pc,σh =1.5km s−1
(ignoring stellar potential) and b=20 pc. The total stellar
cause the Chandrasekhar and the Petts et al. formulae are
massissettoM =2500M ,andamassofasinglestellar
not significantly different (Fig. 2), the results of Fig. 3 are ⋆ ⊙
particle is m = 0.5 M , i.e. the number of stellar parti-
similar in the cuspy halo. These results corroborate the ex- ⋆ ⊙
cles is N = 5000. A stellar particle has a softening length
pectation that,asHernandez(2016)proposed,astellar dis- ⋆
of ǫ = 0.1 pc. Velocity distribution is given by Edding-
tributioninalow-masscompactUFDcanbeaffectedbyDF ⋆
ton’s formula taking into account the total potential of the
against DM if it has a cusp.
DMand thestars. AlthoughtheactualLOSVdispersion of
If a DM halo has a core, however, the DF approxi-
mated by the analytic formulae is significantly less efficient starsinsideRh isslightlyhigherthan1.5km s−1 becauseof
self-gravity of the stars, theincrease is only a few per cent.
to steepen the stellar profile (the red lines), in spite of the
Althougheverysinglestarisresolvedwithapoint-masspar-
same σh meaning similar DM masses within Rh. When the ticle, the interactions between stars in the simulations are
Chandrasekharformulaisapplied(thetoppanel),thestellar
still collisionless (see AppendixB).
densityhardlychangesevenatt=10 GyrinthecoredDM
halo. Although the Petts et al. formula (the bottom panel) DF can arise if m⋆ ≫ mDM, where mDM is a mass of
a DM particle in simulations. Since m = 0.5 M in my
steepens the stellar density profile more than the classical ⋆ ⊙
formula, the density slope is clearly shallower than that in simulations, mDM should be < 0.05 M⊙. Achieving such a
high resolution requires appro∼ximately 2.6 and 16.0 107
thecuspyDM,andastellarcuspdoesnotform.Nostarsare ×
particlesforthecuspyandthecored DMhaloes.Tolighten
fallen intothecentrebyeitherDFmodellings. Theabsence
theheavyburdenoftheN-bodycomputations,Iemployan
of the stellar cusp is due to the week DF in the DM core
orbit-dependentrefinementmethod for a multi-mass spher-
andtheDFcessation assumedinr<rTS inthePetts et al. icalmodelproposedbyZemp et al.(2008).Thismethoddi-
model.
vides a DM halo into i shells and the central sphere (the
Fig. 4 shows the same results but for different settings
zerothshell).Basically,eachshellisresolvedintoDMparti-
faonrdrsσhan(di.eσ.hh.iTghheereρffDecMt,0o)f,DbFutbtehceomsteesepweenaiknegrfoofralasrugrefracres c(sleesewTiatbhleea1c)h.AmfatsesrraesssoigluntiinognamDDMM,ipaanrtdicsloeftinentihnegil-etnhgsthheǫlil
densityprofileandformationofastellarcuspbyaDMcusp
itsinitialpositionandvelocityandcomputingitspericentre
can be seen even when rs =1 kpc and σh =3.0 km s−1. In distanceinthefixedpotential,ifthepericentreintrudesinto
the case of cored DM haloes, on the other hand, the stellar
tdheenssiatmyepraosfiinlesthaerecuaslpmyohstaliontmacotdeelvse.nInththouegchasσehofatnhdercsoraerde tphaertiinclneesrwji-tthhtshheelml,atshsempDaMrt,ijclaenidstshpelitsoifntteonimngDMle,ni/gmthDǫMj.,7j
Thesplitparticlesaredistributedonrandompositionswhile
halowith σh =3.0 km s−1 andrs=125pc,thePetts et al. keeping the initial radius of the parent particle, and direc-
formulapredictsweaksteepening,butastellarcuspdoesnot
tions of their tangential velocities are randomly reassigned
emerge.Thus,significanceoftheDFeffectstronglydepends
while keeping the initial radial velocity and the kinematic
on whetherthe DMhalo has a cusp or a core even if thers energy of the parent particle. This refinement method can,
and σh are the same. The most noticeable difference is the by a substantial factor, reduce the computational run time
emergence of a stellar cusp in a DMcusp.
by decreasing the number of DM particles in outer regions
The total mass of the nucleus remnants consisting of
that are not important to this study, while preventing the
starsfallenintothecentrecandependnotonlyonDMden-
outer particles with larger masses from entering the inner-
sity but also stellar distribution. If Rh is larger, stars have most region resolved with the smallest particle mass. After
moreextendeddistribution,thereforeDFtimescalebecomes
therefinement,1.53and6.88 107 particlesarerequiredto
longer on average. Thus, a larger Rh leads a smaller frac- represent the cuspyand thec×ored DMhaloes.
tion of stars to fall into the centre by DF. As a result, a
IuseasimulationcodeASURA(Saitoh et al.2008,2009;
less prominent stellar cusp would form in such an extended
Saitoh & Makino 2009, 2010, 2013),8 in which a symmet-
galaxy.
ric form of a Plummer softening kernel (Saitoh & Makino
2012),aparalleltreemethodwithancomputationalacceler-
atorGRAPE(GRAvityPipE,Makino2004;Tanikawa et al.
3 N-BODY SIMULATIONS 2013) and the second-order leap-frog integrator with indi-
vidual timesteps are used. The number of stellar particles,
AsIshowedinSection2,theanalyticformulaeareusefulto
estimate themagnitude ofDF.The formulae, however,still
ignore non-linear effects. For example, they assume DF as 7 Therefore, the mass ratio mDM,i/mDM,j has to be a natural
a corrective effect of two-body interactions and donot take
number.
into account orbital periodicity of particles or reaction of 8 ASURA is an N-body/smoothed particle hydrodynamics (SPH)
fieldparticles.ToaddressfurthertheeffectofDFusingmore codealthoughthisstudyonlyusestheN-bodypart.
6 S. Inoue
Figure4.SameasFig.3butforhalomodelsdifferentinrsandσh.Inthetoppanels,thedensitypeaksatthecentrereachlog(Σ⋆/Σh)=
1.6,1.6and1.1att=10Gyrfromlefttoright.Inthebottompanels,thedensitypeaksarelog(Σ⋆/Σh)=2.1,1.6and1.6,respectively.
Table 1. The multi-shell structures for the refinement method Fig.5showsthestackingsofstellarsurfacedensitypro-
in my N-body simulations. From left to right, numbers of the files in the cuspy (blue) and the cored (red) DM haloes at
shells,radialrangesoftheshells,basicmassresolutionsandsoft- t=4,7and10Gyr.Theshadedregionsindicatetheranges
ening lengths of the DM particles in the shells. By the orbit- of upper and lower 1σ-deviations of the stackings. In the
based method for refinement, not all particles in the i-th shell DM cusp, the stellar density profile clearly demonstrates
have mDM,i although the central sphere (i = 0) consists of the theemergence of a stellar cusp after t=7 Gyr; the density
finest-resolutionparticlesofmDM,0. slope becomesremarkably steeperin R< 2pcthan thatin
R> 2 pc. On the other hand, such a st∼ellar cusp does not
em∼erge in the cored DM halo although the outer density
i-thshell range mDM,i [M⊙] ǫi [pc] slopeofthestarsinR> 2pcissimilartothatinthecuspy
DMhalo,whichisnear∼lyexponentialwithradius.Fromthis
0 r<rs 0.05 0.1
result,it can beseen that aDMcuspcan generate astellar
1 rs<r<2rs 0.1 0.14
2 2rs<r<3rs 0.2 0.2 cuspbyDFevenifthestellardensityprofileisinitiallyflat.
3 3rs<r<4rs 0.4 0.28 IntheN-bodysimulations, thestellarcuspshavemassesof
4 r>4rs 0.8 0.4 33.8+−45..53 M⊙ within R<2 pc,which correspondsto1.4 per
cent of the total stellar mass. Additionally, the difference
between thetwocases issignificant in spiteof thesame σh,
which meansthatthetwohalo models areconsidered tobe
N =5000, inmy simulations may betoosmall toobtain a
⋆ similar in observations. The emergence and the absence of
statistically certaindensityprofile.Toreinforcethispoint,I
stellar cusps in the cuspy and cored DM are approximately
perform ten runswith thesame initial condition but differ-
consistent with theresults of my orbital integration models
ent random-numberseeds.
using theanalytic DF formulae (Fig. 3 and 4).
3.2 Results
3.2.1 Evolution of the stellar density profiles Fig. 6 shows the evolution of Rh and σh during the
N-body simulations. Stellar half-mass radii Rh decrease
Iobtainthreesurfacedensityprofilesobservedfromperpen- slightly; by 0.5 and 1 pc in the cored and cuspy DM.
≃
dicular angles for each of the ten runs. Then, I compute a LOSV dispersions σh with in Rh are almost constant even
stackingofthethirtyprofilesofstellarsurfacedensityatthe aftertheemergenceofthestellarcusps.ThismeansthatDF
same time t for each case of the cuspy and the cored halo. isnoteffectiveformostofstarsaroundthehalf-massradius
Thecentreofthestellardistributionisdefinedtobetheme- although the central regions in r Rh are significantly af-
≪
dian position among all stellar particles in each snapshot. fected.
Emergence of a stellar cusp by dynamical friction 7
Figure6.Time-evolutionofRh(top)andσh(bottom)intheN-
bodysimulations.Thesolidlinesindicatethemedianvaluesofthe
stackingsofthethirtystellarprofiles,andtheshadedregionsare
therangesof±1σ-deviations.Thethindashedlineisanexample
chosenrandomlyineachDMmodel.
cuspy halo is still cuspy or cored after the creation of the
stellar cusp.
Fig. 7 shows DM density profiles in my N-body simu-
lations of thecuspyhalo model, in which I makea stacking
of theten runs. Here, the halo centreis defined to be a po-
sition of the particle that has the highest DM density in
each snapshot. I use a method like SPH to compute the lo-
cal DM densities for the centering; a cubic spline kernel is
appliedto128 neighbouringDMparticles. TheFigureindi-
Figure5.Time-evolutionofstellarsurfacedensityprofilesinthe cates that the DM cusp in the initial state is significantly
N-bodysimulations.Theblueandredcolourscorrespondtothe weakenedinr < 1pcatt=4Gyr(orange).Eventually,the
runsofthecuspyandthecoredDMhalomodels.Thesolidlines initialDMcusp∼isturnedintoacoreextendingtor 2pcat
≃
indicate the median values of the stackings of the thirty profiles t=10Gyr(green).ThisresultmeansthatthecentralDMis
(seethemaintext),andtheshadedregionsaretherangesofupper kinematically heatedbyinfallingstars, andtheDMdensity
andlower1σ-deviations.Thethindashedlinesisanexampleofa is decreased in r < 2 pc. The size of this region where DM
singleprofilechosen randomlyineach timeandDMmodel.The is affected is cons∼istent with the size of the stellar cusp in
blacksolidlinesaretheinitialstateofthestellardensityprofiles.
theN-bodysimulations (Fig. 5). Sincethesoftening length
The leftand rightordinate indicate surfacedensities normalised
by Σh ≡ M⋆/(2πR2h) = 0.99 M⊙ pc−2 and number densities of ofDMparticlesisǫ0 =0.1pc,thepeaksofDMdensitiesat
stars,respectively. r 0.1 pcmay betransient fluctuation.
≃
From the consistency of the sizes between the stellar
cusps and the DM cores created, the size of a stellar cusp
3.2.2 Evolution of the DM density profiles may be regulated by a DM density profile flattened by in-
falling stars. If this is the case, a larger number of stars
As I showed above, DM exerts DF on stars and can cause
falling into the centre can create a larger DM core and a
a low-mass compact UFD to have a stellar cusp if it has
broader stellar cusp. In the central region of a cuspy DM
a cuspy DM halo. On the other hand, the DM particles
can be kinematically heated by the stars spiraling into the halo,thesignificanceofDFbasicallydependsonσh9.Inad-
dition,thenumberofstarsinthecentralregionwherestars
centre, as the reaction of DF. Previous studies have shown
canreachthecentrebyDFwithin 10Gyrdependsonthe
that a DM cusp can be disrupted or made shallower by ∼
objects spiraling into the centre (e.g. Goerdt et al. 2010; initial stellar distribution, i.e. M⋆ and Rh.
Cole et al. 2011; Arca-Sedda& Capuzzo-Dolcetta 2017).
Inoue& Saitoh (2011) also demonstrated that a DM cusp
disruptedbyinfallingobjectscanberevivedifacentralrem-
nantoftheinfallingobjectsissufficientlymassive.Hence,it
is also interesting to look into evolution of the dark matter 9 FDF is almostindependent fromrs inthe Petts etal. formula
density profiles in my N-body simulations, i.e. whether the forcuspyhaloes(Fig.2).
8 S. Inoue
havealarge corewithasizeof rs > 100pc.IdiscussUFDs
in current observations on this poi∼nt in Section 4.3.
4.2 The analytic formulae vs N-body simulations
It is interesting to compare the results of the analytic
formulae with those of N-body simulations although it
is not the main purpose of this study. Early studies us-
ing numerical simulations have discussed that the Chan-
drasekhar formula assuming Maxwellian velocity distribu-
tion and a invariant Λ can give a quite accurate estimate
of DF force in various cases (e.g. Lin & Tremaine 1983;
Bontekoe & van Albada 1987). It was also reported, how-
ever, that the formula can be inaccurate in some specific
cases; analyses and N-body simulations have shown that
DF can be enhanced around a constant-density core, and
Figure 7. Profiles of spatial DM densities inthe N-body simu-
then suppressed in the core (e.g. Goerdt et al. 2006; Inoue
lationsofthecuspyhalomodelatt=4,7and10Gyr.Thesolid
linesindicatethemedianvaluesofthestackedprofilesofDM,and 2009; Goerdt et al. 2010; Arca-Sedda& Capuzzo-Dolcetta
theshadedregionscover±1σ-deviationsamongthetenruns.The 2014; Petts et al. 2015, 2016). These phenomena are incon-
thin dashed line is an example of a single run chosen randomly sistentwithpredictionsbythesimplifiedChandrasekharfor-
ineachtime.Theblacklinedelineates theanalyticmodelofthe mula. Various physical mechanisms of the deviation from
DMdistribution(equation1withγ=1). the analytic formula have been proposed: orbital reso-
nance between a massive and field particles (Inoue 2011;
Zelnikov & Kuskov2016),coherentvelocityfieldamongpar-
4 DISCUSSION AND SUMMARY ticles (Read et al. 2006), a non-Maxwellian velocity dis-
tribution (Silva et al. 2016; Petts et al. 2015; Just et al.
4.1 Summary and interpretation of the results
2011),decreaseoflow-velocityparticles(Antonini& Merritt
As I showed in Section 2 and 3, DF on stars against DM 2012; Petts et al. 2016; Dosopoulou & Antonini 2016) and
canlargelyalterthestellardensitydistributionifthegalaxy inhomogeneity of background density and a variable Λ
hascuspyDMdistributionandacompactstellarcomponent (Just & Pen˜arrubia 2005).
havingRh 20pcandσh <3km s−1.Themostimportant In the case of a DM cusp, the two analytic formulae
resultobtai≃nedfrommyN∼-bodysimulationsisthattheDF predict similar DF force in Fig. 2, and the results of my
by the DM cusp can arouse emergence of a stellar cusp in orbital integration models are qualitatively consistent with
the galactic centre <2 pc. On the other hand, if a DM theN-bodysimulations.However,thestellarcuspinmyN-
halohasacore,DFis∼notefficienttogeneratesuchastellar body simulations have the size of R 2 pc, and it could
≃
cusp, in spite of the same σh, although stellar density can be attributed to the weakened DM cusp shown in Fig. 7,
beaffected and increase slightly in a wide range < 10 pc. where DF is weakened. In addition, the DM density centre
The results mentioned above can be explain∼ed by the isnotnecessarily befixedontothestellarcentreinthesim-
differences of density and velocity dispersion between a ulations, and the slippage of the centres can broaden the
cuspy and a cored DM haloes. According to the analytic stellar cusp. Therefore, the broadness of the stellar cusp in
formulae, DFbecomes stronger when abackground density thesimulationsdoesnotnecessarily meaninaccuracyofthe
ishigherandavelocity dispersion islower. Inacuspyhalo, DF modellings. However, the stellar density slopes outside
DMdensityincreasestowardthecentre,andvelocitydisper- the stellar cusp is steeper in the orbital integration mod-
sion decreases (see Fig. 1), therefore DF becomes stronger els. Moreover, in my N-body simulations, the total mass
towards the centre. In this case, orbital shrinkage by DF of the stellar cusps within R = 2 pc are about four times
brings a star to an inner region where DF is even stronger: smallerthanthosepredictedbytheanalyticDFmodellings.
‘DF shrivelling instability’ (Hernandez 2016). On the other On these points, both analytic formulae would be overesti-
hand, in a cored halo, density and velocity dispersion are mating DFin thecuspy haloes.
nearlyconstantinthecentralregion,thereforeDFdragforce InthecaseofaDMcore,ontheotherhand,DFcannot
isapproximatelyindependentofradius;itactuallydecreases generateastellarcuspineithertheorbitalintegrationorthe
gently towardsthecentre(Fig. 2).This meansthat acored N-bodymodels.However,thereisadifferenceworthyofspe-
halo is relatively stable against theDF shrivelling of stars. cial mention: the simplified Chandrasekhar formula hardly
Interpretationoftheaboveresultsshouldbeconsidered changesthestellardensityslopes,whereastheN-bodysim-
carefully. It should be noted that presence of a stellar cusp ulations show significant increase of the stellar densities in
in a low-mass compact UFD is not necessarily evidence to a wide range of R< 10 pc (Fig. 5). This result shows that
prove a DM cusp. It is because we do not know the initial the Chandrasekhar∼formula assuming Maxwellian distribu-
conditionofthestellardensity;agalaxycancreateastellar tionandainvariableΛunderestimatesDFinthecoredhalo
cusp at its birth even if its DM halo has a core. It could be despitethattheclassicalformulaoverestimatesinthecuspy
said, however, that it is inevitable to have a stellar cusp if halo. It is noteworthy that using the Petts et al. formula
a low-mass compact UFD has a DM cusp. In other words, withtheirtidal-stallingmodelcandramaticallyimprovethe
if a low-mass compact UFD is observed to have no stellar reproducibility of DFeffect in thecored haloes, andthere-
cuspandbeoldenough,itsuggeststhatitsDMhalowould sult of the stellar density profile is almost consistent with
Emergence of a stellar cusp by dynamical friction 9
the N-body simulations (see the bottom panels of Fig. 3 from asimple stellar population model of Maraston (2005),
and 5). Thus, the DF modelling proposed by Petts et al. thestellarmassofDracoIIisapproximatelylog(M /M )=
⋆ ⊙
(2016)seemstobemoreaccuratethanthesimplifiedChan- 3.1–3.9. AttheHeliocentricdistanceofDracoII,20 3kpc
±
drasekhar formula although it may not be perfect yet in a (Laevenset al. 2015), the size of a stellar cusp expected
DM cusp. Although it is beyond the scope of this study to from my N-body simulations, 2 pc, corresponds to
≃ ≃
investigate the physical reasons of the differences between 0.3 arcmin. Unfortunately, the cusp region is smaller than
the analytic formulae and my N-body simulations, I con- thesizeoftheinnermostbinofastellarsurfacedensitypro-
siderthattheN-bodysimulationswouldbephysicallymore fileshowninfig.3ofLaevenset al.(2015),thereforeitmight
credible than theanalytic models. be still challenging for the current observations to detect a
stellarcuspinDracoIIevenifitispresent.Inaddition,the
observed numberof stars belonging toDraco II maybetoo
4.3 Comparison with observations small toexcavatea stellar cusp in thecurrentobservations.
Although DMdensity profilesof UFDsmay differfrom one
Here, I discuss the validity of my models of low-mass com-
another even if they have the same sizes and LOSV dis-
pact UFDs and the results in comparison with current ob-
persions, it could improve statistics for proving absence of
servations. First, it is still very difficult or impossible to
stellarcuspstomakeastackinglikeFig.5amongUFDshav-
determinemasses and sizes of DM haloes of UFDswith ac-
curacy in current observations. I have to note, therefore, ingsimilar andsufficiently small Rh and σh.Because of the
faintnessofUFDsassmallasDracoII,observationsarelim-
that the parameters in my DM halo models might be arbi-
itedtotheclosedistancefromthesolarsystem:d< 30kpc.
trary.Recentobservationalstudieshavearguedthatgalaxies
Itcouldbeexpected,however,thatfutureobserva∼tionswill
havetheuniversalDMsurfacedensity,µDM ≡ρDM,0rs,over explore vasterregions to discover such faint UFDs.
quite a wide range of luminosity when they are assumed to
havecoredDMhaloes (Spanoet al.2008;Hayashi & Chiba
2015; Kormendy & Freeman 2016).10 Donato et al. (2009)
and Kormendy & Freeman (2016) derived µDM = 140+−8300 ACKNOWLEDGMENTS
and 70 4 M pc−2 from their galaxy samples including
⊙
± The author thanks the referee for his/her useful comments
some satellite galaxies of the Milky Way. Although it has
that helped improve the article greatly, and Takayuki R.
to be noted that they assumed different models for their
SaitohforkindlyprovidingthesimulationcodeASURA.This
cored haloes, my cored DM models of equation (1) have
µDM = 122 and 176 M⊙ pc−2 for rs = 125 and 250 pc, re- study was supported by World Premier International Re-
spectively, when σh = 1.5 km s−1. Accordingly, my cored searchCenterInitiative(WPI),MEXT,JapanandCREST,
halo model with rs = 125–250 pc and σh = 1.5 km s−1 JST. The numerical computations presented in this paper
werecarriedoutonCrayXC30atCenterforComputational
would be consistent with the observed universality of µDM
Astrophysics,NationalAstronomicalObservatoryofJapan.
if it is extrapolated toextremely low-mass galaxies.
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