Imprints of Zero-Age Velocity Dispersions and Dynamical Heating on the Age-Velocity dispersion Relation

2.1 MB·

by  Jun Kumamoto

Publ.Astron.Soc.Japan(2014)00(0),1–13 1
doi:10.1093/pasj/xxx000
Imprints of Zero-Age Velocity Dispersions and
Dynamical Heating on the Age-Velocity
dispersion Relation
7
1
0 Jun KUMAMOTO1,2, Junichi BABA2,3, and Takayuki R. SAITOH2
2
1 AstronomicalInstitute,TohokuUniversity,Aramaki,Aoba,Sendai,Miyagi980-8578,Japan.
n
a 2 Earth-LifeScienceInstitute,TokyoInstituteofTechnology,Ookayama,Meguro,Tokyo
J 152–8551,Japan.

3 3 ResearchCenterforSpaceandCosmicEvolution,EhimeUniversity,Bunkyo-cho,
1
Matsuyama,Ehime,790–8577,Japan.

A] ∗E-mail:[email protected]
G Receivedh13-Sep-2016i;Acceptedh09-Jan-2017i
.
h
Abstract
p
- Observations of stars in the the solar vicinity show a clear tendency for old stars to have
o
r larger velocity dispersions. This relation is called the age-velocity dispersion relation (AVR)
t
s and it is believed to provide insight into the heating history of the Milky Way galaxy. Here,
a
in order to investigate the origin of the AVR, we performed smoothed particle hydrodynamic
[
simulations of the self-gravitating multiphase gas disks in the static disk-halo potentials. Star
1
formationfromcoldanddensegasistakenintoaccount,andweanalyzetheevolutionofthese
v
8 star particles. We find that exponents of simulated AVR and the ratio of the radial to vertical
6 velocitydispersionareclosetotheobservedvalues. WealsofindthatthesimulatedAVRisnot
6
asimpleconsequenceofdynamicalheating. Theevolutiontracksofstarswithdifferentepochs
3
0 evolvegraduallyintheage-velocitydispersionplaneasaresultof: (1)thedecreaseinvelocity
. dispersion in star forming regions, and (2) the decrease in the number of cold/dense/gas as
1
0 scatteringsources. TheseresultssuggestthattheAVRinvolvesnotonlytheheatinghistoryof
7
astellardisk,butalsothehistoricalevolutionoftheISMinagalaxy.
1
: Keywords:Galaxies:kinematicsanddynamics—Method:numerical
v
i
X
r
a 1 Introduction asinglepower-lawfunctionforboththeradial(σ )andverti-
R
cal(σ )directions:
z
Itiswellknown thatthevelocitydispersion ofstarsintheso-
lar vicinity increases with stellar age. This is called the age– σ∝τβ, (1)
velocitydispersionrelation(hereafterAVR).Stro¨mberg(1946)
andSpitzer&Schwarzschild(1951)werethefirsttoinvestigate whereσ representsσR orσz andτ is theageofthestar. The
the association between stellar velocity dispersion and spec- power-lawindex,β,is0.3–0.5withcertainerrors(Seetable8
tral types, and then they discovered this relation. Subsequent of Sharma et al. 2014). The index for the radial direction is
observations (Nordstro¨m et al. 2004; Seabroke & Gilmore slightlysmallerthanthatfortheverticaldirection. However,it
2007; Holmberg et al. 2007; Holmberg et al. 2009; Aumer & isnotclearwhetherthisdifferenceisrealornot. σz/σR∼0.5
Binney 2009; Just & Jahreiß 2010; Sharma et al. 2014) con- has also been ascertained by observations (Dehnen & Binney
firmed the existence of the AVR with much larger samples. 1998).
According to these observations, this relation is well fitted by Somesecularheatingprocessesand/ortimeevolutioninthe
(cid:13)c 2014.AstronomicalSocietyofJapan.
2 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
disk play crucial roles in establishing the AVR (e.g., Wielen et al. (2012) carried out N-body/smoothed particle hydrody-
1977; Freeman & Bland-Hawthorn 2002; Bland-Hawthorn & namics (SPH) simulations with the mass and spatial resolu-
Gerhard 2016 for reviews), since the older stars tend to have tions of 7300 M⊙ and 155 pc. They showed that old stars
larger velocity dispersions. There are a number of scenar- thathavelargervelocitydispersionwerebornatlowerradiiand
ios for the origin of the AVR: heating by giant molecu- arekinematicallyhotterthanthosebornatalatertime. Martig
lar clouds (GMCs; Spitzer & Schwarzschild 1951; Spitzer et al. (2014) used adaptive mesh refinement code RAMSES
& Schwarzschild 1953; Kokubo & Ida 1992), by transient (Teyssier 2002) and they showed that AVR is not determined
spiral arms (Carlberg & Sellwood 1985; De Simone et al. bythestellarvelocitydispersionatbirthtime,butistheresult
2004),bythecombinationofGMCsandspiralarms(Carlberg of subsequent heating. Furthermore, House et al. (2011) in-
1987; Jenkins &Binney 1990), by halo black holes (Lacey& vestigatedthevariationsofAVRusingdifferentmodelingtech-
Ostriker1985;Ha¨nninen &Flynn2002),orbyminormergers niques. Grandetal.(2016)usedthestate-of-the-artsimulation
(Toth&Ostriker 1992;Walker etal.1996;Huang &Carlberg code,AREPO(Springel2010),fortheirzoom-insimulationsand
1997). Until now, there has been no consensus about the pri- reported thatthey obtained anAVRthat is consistent with the
mary source of the AVR. In table 1, we summarize the theo- observedAVR.Whileallstudiesshowedpower-law-likeAVRs,
reticalpredictionsofvelocitydispersionsinboththeradialand we note that their simulations are insufficient to disclose the
verticaldirections. Alloftheaboveprocessesarekeyfeatures originoftheAVRbecauseofthelimitednumericalresolutions
ofgalaxyformation.Thus,understandingtheoriginoftheAVR (theforceresolutionis≥100pc),whichislargerthanthethick-
isessentialinordertoadvanceourunderstandingofgalaxyfor- ness of the molecular gas disk (48–160 pc)in the Milky Way
mation. galaxy (Nakanishi & Sofue 2006). Some models did not di-
rectlysolvethelowtemperature(<100K)andhigh-densityre-
Here, wemainlyfocusonthecontribution ofGMCstothe ∼
gions(>100cm−3)thatmimicGMCs. Inaddition,theorigin
formation of theAVR. Someprevious studies investigated the ∼
oftheAVRisnotwellinvestigatedinthesestudies.
effectofgravitational scatteringbyGMCsonthevelocitydis-
persionsofstars.Kokubo&Ida(1992)calculatedstellarorbits
undertheinfluenceofthegravitationalpotentialofGMCs,and In this paper, we find out the origin of the AVR by us-
theyfoundthattheevolutionofvelocitydispersioncouldbedi- inghigh-resolutionN-body/SPHsimulations. Insteadofdoing
videdintotwophases. Intheearlyphase,theevolution ofthe the cosmological simulations, we adopt an idealized model in
velocity dispersionis proportional toexp(τ),whileinthelate which only gas and stars formedfrom cold and dense gas are
phase,itisproportionaltoτ0.25.Ha¨nninen&Flynn(2002)per- solved. The halo and (the main body of) the stellar disk are
formedN-bodysimulationsofthestellardiskinvolvingGMCs expressedbystaticpotentials. Thisapproximationallowsusto
modeled by spherical mass distributions of uniform density. use high mass and spacial resolutions, such as 6000 M⊙ and
Theyfoundthattheradialandverticalvelocitydispersionsare 10pc.Asatrade-off,wecannotincludethecontributionofthe
proportional to τ0.21 and τ0.26, respectively. This is consis- spiral arms at present, but this will constitute the next step of
tent with the prediction from the simplified model of Kokubo thisstudyandwewillinvestigateitinaforthcomingpaper.
& Ida (1992), but the expected values of velocity dispersions
aresomehowdifferentfromthoseobtained fromobservations. The structure of this paper is as follows. In §2, we de-
Hence, gravitational scattering by GMCs is not considered a scribetheinitialconditionsofoursimulationsandournumeri-
primary source of the AVR. However, the models used above calmethods. Wepresenttheevolutionofthesimulatedgalactic
showsignificantroomforimprovement. Forinstance,theydid diskandAVRsinour simulation in§3. Heating ratesarealso
nottakeintoaccountgasdynamics,theydidnotmodelGMCs discussedinthissection. Finally,in§4,weshowthesummary
asrealisticstructures,andtheydidnotconsiderstar-formation ofourresultsanditsimplications forthehistoryoftheforma-
processes. Aumeretal.(2016b)andAumeretal.(2016a)per- tionandevolutionoftheMilkyWaygalaxy.
formed controlled N-body simulations of disk galaxies to in-
vestigatethediskgrowth. Inthesesimulations, theystudy the
contribution ofGMCstothediskgrowthbyusingmassiveN-
bodyparticleswhichrepresentGMCs. Theyfoundthattheef- 2 Models and Methods
ficiencyofGMCheatingdepends onthefractionofdiskmass
residingintheGMC.Modelsthataremorerealisticareneces-
Our three dimensional (3D) N-body/SPH simulations con-
saryto conclude whether or not GMC heating is important in ductedinthisstudywereperformedusingtheASURA-2(Saitoh
theestablishmentoftheAVR.
&Makino2009;Saitoh&Makino2010). Weinvestigatedthe
Recently, some zoom-in cosmological simulations have AVR for stellar particles formed from SPH particles moving
beenusedtoderivenumericalestimationsoftheAVRs. Brook withinastatichaloanddiskpotential.
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 3
Table1. Velocitydispersionspredictedbythetheoreticalmodels.WesummarizethemodelsreferringtoAVRsanditsexponents.Note
that,inthecaseofminormergers,onlytheamountoftheincreaseofvelocitydispersionsareshown.
heatingsource velocitydispersionσ
Spitzer&Schwarzschild1953 GMC σ ∝τ1/3
mean
Kokubo&Ida1992 GMC σ ∝τ0.25,σ ∝τ0.25
R z
Ha¨nninen&Flynn2002 GMC σ ∝τ0.21,σ ∝τ0.26
R z
DeSimoneetal.2004 spiralarm σ ∝τ0.20-0.60
R
(dependingonspiralproperties)
Jenkins&Binney1990 GMC&spiralarm σ ∝τ0.5,σ ∝τ0.3
R z
Ha¨nninen&Flynn2002 haloblackhole σ ∝τ0.50,σ ∝τ0.50
R z
Walkeretal.1996 minormerger inceasebyδσ∼(10,8,8)kms−1atsolarradius
Huang&Carlberg1997 minormerger σ increasebyafew–dozens%
z
(dependingonsatellitemassandfallingangle)
2.1 GalaxyModel
In order to concentrate on the contribution of the clumpy in-
terstellar medium (ISM) to the stellar velocity dispersion, we
solve the evolution of the gas with high resolutions. For this
purpose,hereweassumethatthehaloandthemainbodyofthe
stellardiskaredescribedbyfixedpotentials. Thismodelingis
identicaltothatofSaitohetal.(2008).
We assume that the dark matter halo follows the Navarro-
Frenk-White(NFW)profile(Navarroetal.1997),whichisex-
pressedas
ρ
ρ (x)= c , (2)
halo x(1+x)2
r
x= , (3)
r
s
where ρ and r are the characteristic density and a scale ra-
c s
dius ofthis profile, respectively. Weassumethat ρ =4.87×
c
106M⊙kpc−3andrs=21.5kpc.
The potential of the stellar disk is described by the
Miyamoto-Nagaimodel(Miyamoto&Nagai1975):
Fig.1. Initialrotationalvelocityasafunctionofradius. hvφiistheaverage
ρ∗(R,z)= M∗z∗2 × aazwimiduththoaflv0e.1lockiptycotfointhiteialraSdPiaHlbpianrsti.cleslocatedineachradialbin.Weapply
4π
R∗R2+(R∗+3 z2+z∗2)(R∗+ z2+z∗2)2,(4)
[R2+(R∗+ pz2+z∗2)2]5/2(z2p+z∗2)3/2 2.2 Methods
whereM∗,R∗,andz∗arethpestellardiskmass,thescaleradius Wesolvetheevolutionofgasandstarparticlesinthehaloand
andthescaleheight,respectively. WeuseM∗=4×1010 M⊙, diskpotentialusingourN-body/SPHcode,ASURA-2(Saitoh&
R∗=3.5kpc,andz∗=400pc. Makino 2009; Saitoh & Makino 2010). The numerical tech-
Initially, the gas disk has simple exponential and Gaussian niqueisalmostthesameasthoseusedinabovepapers,butwith
distributions intheradialandverticaldirections. Thescalera- slightupdates. Wenowbrieflydescribeourmethod.
diusandthescaleheightofthegasdistributionaresetto7kpc The self-gravity among the gas and star particles is calcu-
and400pc,respectively. Figure1showstheaveragedrotation lated by the tree with the GRAvity PipE (GRAPE) method
curveofgasparticlesattheinitialconditions. Thetotalmassof (Makino 1991). The opening angle is set to 0.5 and only
thegasdiskis6×109M⊙,andweuse1millionSPHparticles monopole moment is considered. In order to accelerate the
toexpressthisgasdisk. Hence,themassofeachSPHparticle computation,weadoptthePhantom-GRAPElibrary(Tanikawa
is6000M⊙.Thegravitationalsofteninglengthissetto10pc. etal.2013),whichisasoftwareemulatorofGRAPE.
4 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
The gas dynamics are solved by using SPH (Lucy 1977; 3 Results
Gingold & Monaghan 1977; Monaghan 1992). The cubic-
WeanalyzetheAVRinthesimulationsandinvestigateitsevo-
splinekernelfunctionisutilized. Forthefirstspacederivative
lutionin§3.1. WethendiscusstheoriginsofAVRevolutionin
ofthekernel,Thomas&Couchman’smodificationisemployed
§3.2.
(Thomas&Couchman1992). Thenumberofneighboringpar-
Beforepresentingthedetails oftheAVRanditsorigin, we
ticlesforeachSPHparticleiskeptwithin32±2inthesupport
firstexplaintheglobalevolutionofourgalaxies.Figure2shows
radiusofthekernel.Theshockishandledwithanartificialvis-
the evolution of our simulated galaxies. The upper and lower
cositytermproposedbyMonaghan(1997).Theviscositycoef-
panels show overviews and close-up views of the gas density
ficientissetto1.0. WealsointroducetheBalsaralimiterinor-
onthe galactic plane (z=0). Fromtheclose-up views of the
dertoreducetheunwantedangularmomentumtransfer(Balsara
gas density, we can see rich inhomogeneous structures of the
1995).
ISM.Thesestructures areinduced by our modeling with high
Thetimeintegrationisconductedbyasecondorderscheme resolution(Saitohetal.2008).Fromt=1–3Gyr,thecontrast
(see appendix A in Saitoh & Makino 2016). For SPH parti- ofthegasdensitydecreasesbecauseofthegasconsumptiondue
cles, we use the FAST scheme so that we can accelerate the tostarformation. Thisimpliesthattheprimarysourceheating
time-integrationinsupernova(SN)-heatedregionsbyusingdif- thediskstarsdecreaseswithincreasingtimeinoursimulation.
ferenttime-stepsforgravitationalandhydrodynamicalinterac- Wediscusstherelationbetweenthecontrastofthegasdensity
tions(Saitoh&Makino2010)andthetime-steplimitersothat andheatingratein§3.2.2.
wecanfollowtheshockedregioncorrectly(Saitoh&Makino The high-density regions exceed the star formation thresh-
2009). olddensity (i.e. n >100cm−3)Therefore,starsareformed
H
inthesehigh-densityregionswhentheothertwostar-formation
Insimulationsinthispaper,weadopttheradiativecooling,
conditions are satisfied. Figure 3 shows the evolution of stel-
starformation,andtypeIIsupernovae(SNe)feedback. Thera-
lar surface density. The stellar surface density distribution
diative cooling of the gas is solved by assuming an optically
mapsshownostrongspiralarmsdevelopedbystellardynami-
thin cooling function, Λ(T,f ,G ), for a wide temperature
H2 0 calmechanismssuchastheswing-amplification(Toomre1981)
range of 20 K<T <108 K (Wada et al. 2009) For simplic-
and non-linear interactions between wakelets (Kumamoto &
ity, we fix the molecular hydrogen fraction at f =0.5, and
H2 Noguchi 2016). As the skeletal structure of the disk is con-
thefar-ultravioletintensityisnormalizedtothesolarneighbor-
structed of the smoothed and axisymmetric Miyamoto-Nagai
hood G = 1. We do not have to use the artificial pressure
0 potential, itis difficulttoenhance thespiralarmsandabarin
floor method, which is highlighted in Saitoh et al. (2006) and
thismodel;thus,ithasaflocculentstructure.
Robertson&Kravtsov(2008). 1 Starformationismodeledin
Figure4showsthestar-formationrateasafunctionoftime.
a probabilistic manner following the Schmidt law (e.g., Katz
Itischaracterizedbytwophases: theinitialrapidlyincreasing
1992; Saitohetal.2008). AnSPHparticlewhichsatisfies the
phase(t<100Myr),andthesucceedingexponentiallydecreas-
followingthreeconditions,(1)n >100cm−3;(2)T<100K;
H
ing phase. The decay time-scale of the star-formation rate is
and(3)∇·v<0,spawns astarparticlewiththeSalpeterini-
∼1Gyr. Thistimescaleisrathershortcomparedtotheesti-
tialmassfunction(Salpeter1955)andiswithinthemassrange
mationofthegalacticchemicalevolution(Larson1972;Yoshii
of0.1M⊙<m⋆<100M⊙. Asanearlystellarfeedback,the
etal.1996),asoursimulationdoesnotconsiderthesecularevo-
H -regionfeedbackusingaStromgrenvolumeapproach(Baba
II
lutionduetogasaccretionfromthehalo.Thisisalsothereason
etal.2017)isemployed.TheenergyfeedbackfromtypeIISNe
whywestopoursimulationatt=3Gyr,beforethegasisex-
isimplemented. EachSNreleases1051 erg,andthisenergyis
hausted. A model incorporating gas accretion will be used in
injected into the surrounding ISM. We adopt the probabilistic
ourforthcomingpapers.
mannerofOkamotoetal.(2008)toevaluatetheinjectiontime.
3.1 AVR
Figure5showstheradial,azimuthal,verticalandtotalAVRsin
our simulated galaxy at t=2 Gyr. In this AVR analysis, we
1Theideatointroducethepressurefloortosimulationsmightcomefromthe
onlyusethestarparticlesformedfromgas.Thereddotsdenote
numericalexperimentsshownbyBate&Burkert(1997).Theypointedout
forthefirsttimethatartificialfragmentationmightoccur,ifSPHsimulations theresultswhenonlystarswhosegalactocentricdistanceRwas
didnothavesufficientmassresolution. Afterthisindication,thepressure intherange8kpc<R<9kpcwereused,whereasthebluetri-
floorisintroduced,inparticulartoavoidartificialfragmentations.However, anglesrepresentthecasewhereallregionswereincluded. The
Hubberetal.(2006)concludethatundertheinsufficientmassresolution
velocitydispersionsasafunctionofstellarage,τ,arecalculated
case,thegrowthofgravitationalinstabilityissuppressedwhentheartificial
fragmentationdidnottakeplace. byusingthefollowingequations:
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 5
Fig.2. Thegasdensityonthegalacticplane(z=0)att=1,2,and3Gyr. Theupperpanelsshowaregionof−12kpc<x<12kpcand−12kpc<
y<12kpc.Thelowerpanelsdisplaythecloseupviewofapartofthediskregionof0.6kpc<x<5.4kpcand0.6kpc<y<5.4kpc.
Fig.3. Thestellarsurfacedensityatt=1,2,and3Gyr.Forthestellarsurfacedensity,weusethesumofstellarparticlesformedfromthegasparticlesand
staticsurfacedensitycalculatedbytheMiyamoto-Nagaimodel(Miyamoto&Nagai1975).
σR(τ)= hvR2iτl<τ<τh−hvRi2τl<τ<τh 1/2, (5) meantheaverageofstellarvR2 whoseageisbetweenτlandτh.
We apply the width of age bins in such a way that every bin
σφ(τ)= (cid:2)h(vφ−vφ(R))2iτl<τ<τh−hv(cid:3)φ−vφ(R)i2τl<τ<τh 1/2,(6)has equal numbers of stars (2000 stars in each bin). For the
distributions ofvelocity dispersions, thesamplingrangeeffect
σ (τ)=(cid:2)hv2i −hv i2 1/2, (cid:3)(7)
z z τl<τ<τh z τl<τ<τh isinsignificant.
σ (τ)=(cid:2) σ (τ)2+σ (τ)2+σ (τ)2(cid:3)1/2, (8)
tot R φ z
The solid lines in figure 5 represent the fitting results with
whereτ an(cid:2)dτ arethetwoedgesofth(cid:3)eagebin. hv2i a power-law function for stellar particles found between R=
l h R τl<τ<τh
6 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
Fig.5. Radial(topleft),vertical(topright),azimuthal(bottomleft)andtotal(bottomright)AVRsinoursimulatedgalaxyatt=2Gyr.Thereddotsindicatethe
AVRcalculatedwithstellarparticlesfoundbetween8kpcand9kpcfromthegalacticcenter. Thesolidlinesarethefittingresultswithapower-lawfunction
forthestellarparticlesinthisregion.Forthisprocedure,weonlyusestarparticlesolderthan0.5Gyr(ontherightsideofthedashedline).Thebluetriangles
indicatetheAVRcalculatedwithallstellarparticles.
8kpcand9kpc. Here,weadoptstarsolderthan0.5Gyr. The τ 0.36
σ =25.0 kms−1. (12)
functionalformsofthefittedage-radial,azimuthal,vertical,and tot 1Gyr
(cid:18) (cid:19)
totalvelocitydispersionrelationsare
Theexponentsoftheradial,azimuthal,vertical,andtotalAVR
0.37 andtheratiooftheradialtoverticalvelocitydispersionareclose
τ
σ =19.4 kms−1, (9)
R 1Gyr totheobservedvalues (see§1). Theradialtoverticalvelocity
(cid:18) (cid:19)
dispersionratiobecomes
0.33
τ
σφ=13.4 1Gyr kms−1, (10) σz =0.38 τ 0.22. (13)
(cid:18) (cid:19) σ 1Gyr
R
(cid:18) (cid:19)
0.59
σ =7.36 τ kms−1, (11) Observationally, the age dependence on this relation is not
z 1Gyr
(cid:18) (cid:19) found.Thisweakdependenceonτ inoursimulatedAVRagain
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 7
Fig.4. Star-formationrateasafunctionoftime.
Fig.6. σφ2/σR2 (red triangles) and κ2/4Ω2 (blue dots) as a function of
favorsobservations(Dehnen&Binney1998). galactocentricradiusatt=2Gyr. Thesetwoparametersareequalwhen
epicycleapproximationissatisfied.
Thebluetrianglesinfigure5showtheAVRcalculatedwith
all stellar samples. Here, we reapply the width of age bins in
such away that each bin has 10000 stars. Radial and vertical oursimulatedgalaxies,althoughtheyareregardedasaheating
AVRs are similar to the local AVRs shown by the red points sourceofradialvelocity dispersion(Carlberg1987;Jenkins&
inspiteofthedifferenceinthesamples.Thisagreementmeans Binney1990). Interestingly,withoutstrongspiralarms,wecan
thatradialandverticalvelocitydispersionsareroughlyconstant successfullyreproducetheobservedexponents ofnotonlythe
throughoutthediskregion. verticalAVR,but alsotheradialone. Wediscussthis pointin
On the other hand, azimuthal velocity dispersion with all §4.
stellarsamplesislargerthanthosewithlocalstarsintheouter Figure7showstheAVRsatthethreedifferentepochs. We
region(8kpc<R<9kpc). Thatisbecausetheazimuthalve- canseethatthesimulatedAVRsofthethreeepochsdonotover-
locity dispersion is determined by the epicycle approximation lap;thereisaclearindicationoftheevolutionoftheAVR.Both
(Binney&Tremaine2008), the initial velocity dispersions and heating efficiency evolve.
This isinconsistent with theconventional interpretation ofthe
σ2 κ2
φ = , (14) AVR, i.e., that it is the evolution track of stars on the age–
σ2 4Ω2
R velocitydispersionplane.
where κ and Ω are epicycle frequency and angular velocity, Infigure8,weshowtheevolutiontracksofstellargroupsof
respectively, and the ratio of κ and Ω changes depending on 26differentepochsontheage–velocitydispersionplanes.Each
thegalactocentricdistance. Figure6showsσφ2/σR2 andκ2/Ω2 grouphasadifferentinitialvelocitydispersionandheatingrate.
as a function of radius. There is a good agreement between Theslopeofeachevolutionsequenceislesssteepthanthosein
then,whichmeansthatstellarmotioniswelldescribedwiththe observations.However,whenwepickupvalues,forinstance,at
epicycleapproximation. Weseethatσφ2/σR2 ∼0.5intheouter t=2Gyrforallevolutionsequences(blacktrianglesinfigure
region (R>8 kpc) where the rotational velocity becomes al- 8),weobtainAVRsthatareconsistentwiththoseobtainedfrom
mostflat(asshowninfigure1),andthisisalsoconsistentwith observations. ThisrevealsthatAVRsareinconsistent withthe
thepredictionoftheepicycleapproximation.Intheinnerregion evolutiontracksofstarsontheage-velocitydispersionplane.
(R<8kpc), however, the rotational velocity is anincreasing
functionofR,andthenσ2/σ2 arelargerthanthoseintheouter
φ R
3.2 OriginsofsimulatedAVR
region.Here,σ doesnotdependonradiusasmentionedinthe
R
aboveparagraph,andso,σφ becomesadecreasingfunctionof In the previous section, we showed that the AVR is not just a
radiusaccordingtotheepicycleapproximation. Therefore,az- simple evolutionary track of stars on the age–velocity disper-
imuthalAVRwithallstellarsamplesdeviatesupwardfromthat sionplane.Inthissection,weinvestigatetwoimportantfactors
withlocalstarsbetween8kpcand9kpcasshowninfigure5. thataffecttheevolutionofstarsontheage–velocitydispersion
Hereafter, we discuss the radial and vertical AVR with all plane: theinitialvelocitydispersion(i.e. velocitydispersionat
stellarparticlesinthegalacticdisktoensuresufficientsamples thebirthtimeofstars)andtheheatingrateasafunctionofage.
andsimplification. The evolution of the initial velocity dispersion is discussed in
Itis worthnoting that strongspiralarmsdonot develop in §3.2.1,whilethatoftheheatingrateisdiscussedin§3.2.2.
8 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
Fig.7. Evolutiontracksofstellarage-velocitydispersionswithdifferentsampleswhosebirthepochist=1,2,and3Gyr. Differentsymbolsrepresent
differentgroups.
Fig.8. Dotsshowthetimeevolutionofstellarvelocitydispersion.Theleftandrightpanelsshowtheradialandverticalvelocitydispersion,respectively.Each
colorindicatesadifferentbirthtime.TheblacktrianglepointsshowtheAVRatt=2Gyr.
3.2.1 Zero-AgeVelocityDispersion thatsatisfythestar-formationcriteriaasafunctionoftime. We
First,wediscussthestellarvelocitydispersionatthebirthtime can see that both velocity dispersions monotonically decrease
ofstars. Wecallthis the“zero-agevelocity dispersion”(here- with time and they can be fitted by a power-law function of
after,ZAVD).Asshowninfigure8,theZAVDdecreaseswith t. However,theirpower-lawindicesaredifferent;theindexof
increasingage. Thisisoneofthekeyreasonsfortheinconsis- the ZAVD for the vertical direction is about twice as steep as
tencybetweentheAVRandtheevolutiontracks. thatfortheradialdirection. Gravitationalinteractions,radiative
cooling,starformationandfeedbackplaycrucialrolesindeter-
The ZAVD is strongly related to the velocity dispersion of
miningtheISMstructure,andtheseeffectsdependonthemass
thestar-forminggasbecauseitinheritsthevelocitydispersion.
ofthegasinthedisk. Therefore,itisnaturalthatthestatusof
Figure9showstheradialandverticalvelocitydispersionsofgas
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 9
Fig.9. Velocitydispersionofgasthatsatisfiesthestarformationcriteriaasa
functionoftime.Boththeradialandverticalvelocitydispersionsareshown.
Bluedotsrepresenttheradial-velocitydispersions, whilereddotsindicate
thevertical-velocitydispersions.Thelinesaretheresultsoftheleastsquare Fig.10. Verticalvelocitydispersionasafunctionoftime.Twenty-onegroups
fittingwithapower-lawfunction. havebeenselected.Eachsequencerepresentstheevolutionofeachgroup.
Solidcurvesareresultsofthefitting(seeEq.(17)).
theISMisrelatedtotheoriginoftheAVR.Inotherwords,the
AVRinvolvesthehistoricalevolutionoftheISMinagalaxy. Byintegratingthisequation,wefinallyhave
−0.33 0.40
t
3.2.2 DynamicalHeatingRate σ =11.8 A− kms−1, (17)
z 1Gyr
In the previous section, we discussed the ZAVD. Here, we " (cid:18) (cid:19) #
investigate the evolution of the dynamical heating rate (i.e. where A is a constant of an integral. The different sequence
dσ/dt).Oncethesetwofactorsareunderstood,wecanusethem found in figure 10 can be expressed by choosing a different
topredicttheentireevolutionoftheAVR. valueofA.
Figure 10 shows the evolutions of vertical velocity disper- In figure 2, the dense gas (n > 100 cm−3) has clumpy
H
sions, σz, as a function of simulation time. Each sequence is structures,andisexpectedtoplaytheroleofaheatingsource.
characterized by two evolution phases: the rapidly increasing Thisclumpyanddensegasisdecreasingwithsimulationtime.
phaseintheearlystageandtheasymptoticallyflatteningphase Wesuspectthatthedecreasingheatingratewithincreasingsim-
inthelatestage. Latertimeorhighervelocitydispersioncause ulation time may result from the decrease in the dense gas.
theheatingratetobelower. Hence,weshowthetimeevolutionofthedensegasmassM
dens
Figure11showstheverticalheatingrate(dσ /dt)asafunc- infigure12. Here,wedefinethegasabovethestarformation
z
tionoftandσ . Wechosefourrepresentative epochsandσ . thresholdasthedensegas.Weseethatthemassofthedensegas
z z
Solidlinesarethefittingresultwithpower-lawfunctionsofσ monotonicallydecreasesanditcanbefittedbyasimplepower-
z
andt. Wecanseethattheheatingrateobtainedbyoursimula- lawfunctionwiththepower-lawindexof−1.24. Interestingly,
tioniswellfittedbypower-lawfunctions. the power-law index is consistent with that for the simulation
Usingthefittingresultsshowninfigure11,wecandescribe timefoundinEq. (16). Thisimpliesthatthetimeevolutionof
theheatingrateasafunctionoftandσ : theheatingrateinoursimulationiscontrolledbythetotalmass
z
ofthedensegas.
dσ
z ∝t−ασ−β. (15)
dt z Our result can be reduced to that obtained in the previous
We apply this equation to sequences in figure10 and then we study where the contribution of gas is unchanged (Spitzer &
obtain the concrete values of α and β. With these values, we Schwarzschild1951;Spitzer&Schwarzschild1953;Kokubo&
have Ida1992;Ha¨nninen&Flynn2002).Considerthecaseinwhich
the dense gas mass does not evolve; this can be described by
dσ
dtz ∝t−1.3σz−1.5. (16) lettingα=0inEq.(15),andthenweobtain
10 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
Fig.11. Heatingratesasafunctionofσz(left)andasafunctionoftimet.Fourrepresentativeepochs(t=1.0,1.5,2.0and2.5Gyr)andfourrepresentative
valuesofσz(σz=5.0,6.0,7.0and8.0kms−1)areemployed.Thesolidlinesrepresenttheresultsoffittingwithpower-lawfunctions.
σ ∝(A−t)1/(β+1). (18)
z
This is consistent with the time evolution of velocity disper-
sionshownbyapreviouswork(e.g. Ha¨nninen&Flynn2002),
wherethemodelstreatedtheGMCsasatime-independentpo-
tential.
4 Discussion and Summary
OursimulationshowsthattheAVRisnottheevolutiontrackof
stars on the age–velocity dispersion plane. This result means
that the AVR can be reproduced even if the heating rate is
smaller than the slope of the AVR. The time evolution of the
stellarvelocitydispersiondependsuponthecombinationofthe
initialvelocitydispersionandtheheatingrate.Inparticular,the
AVRisstronglyaffectedbytheevolutionoftheZAVD.
The importance of the ZAVD was pointed out by Wielen
(1977); however they did not attempt to elucidate the details
becauseofthecomplicatednatureofgasdynamics. Recentnu-
mericalsimulations ofgalaxy formation alsodemonstrate that
the ZAVD has a crucial role in the establishment of the AVR
(Birdetal.2013;Grandetal.2016). Theobservations oftur-
bulent gas disk athigh redshift(Fo¨rsterSchreiber etal.2009)
Fig.12. Timeevolutionofgasmassthatismuchdenserthanthestarfor-
mationthresholddensity.Thesolidlineindicatestheresultofthefittingbya supportthepresenceoftheevolutionofZAVD.
power-lawfunctionanditsexponentis−1.24. TheotherkeyprocessforestablishingtheAVRisthesecular
heating duetodensegas. Thisisconsistent with Aumeretal.
(2016a),whoshowsthattheevolutionofheatinghistoryshapes
anAVR.Oursimulationcansuccessfullyderivethecontribution
ofthisprocess.Wepointoutthatthesufficientspatialresolution