The Fundamental Scaling Relations of Elliptical Galaxies

21 Pages·2005·0.53 MB·English

DRAFTVERSIONNOVEMBER3,2005
PreprinttypesetusingLATEXstyleemulateapjv.9/08/03
THEFUNDAMENTALSCALINGRELATIONSOFELLIPTICALGALAXIES
BRANTROBERTSON1,5,THOMASJ.COX1,LARSHERNQUIST1,MARIJNFRANX2,
PHILIPF.HOPKINS1,PAULMARTINI3,VOLKERSPRINGEL4
DraftversionNovember3,2005
ABSTRACT
Weexaminethefundamentalscalingrelationsofellipticalgalaxiesformedthroughmergers. Usinghundreds
ofsimulationstojudgetheimpactofprogenitorgalaxypropertiesonthepropertiesofmergerremnants,wefind
thatgasdissipationprovidesanimportantcontributiontotiltintheFundamentalPlanerelation.Dissipationless
5 mergersofdisksproduceremnantsthatoccupyaplanesimilartothatdelineatedbythevirialrelation. Asthe
0 gascontentofprogenitordiskgalaxiesisincreased,thetiltoftheresultingFundamentalPlanerelationincreases
0 and the slope of the R - M relation steepens. For gas fractions f > 30%, the simulated Fundamental
e ⋆ gas
2 Planescalings(R ∝σ1.55I- 0.82)approachthoseobservedintheK-band(R ∝σ1.53I- 0.79). Thedissipationless
e e e e
v merging of spheroidal galaxies and the re-merging of disk galaxy remnants roughly maintain the tilt of the
o Fundamental Plane occupied by the progenitor ellipticals, approximately independent of the orbital energy
N orangularmomentum. Drymergingofspheroidalsystemsatredshiftsz<1isthenexpectedtomaintainthe
stellar-massFundamentalPlanerelationsimprintedbygas-richmergingduringtheepochofrapidspheroidand

1 supermassiveblackholegrowthatredshiftsz≈1- 3. Inoursimulations,feedbackfromsupermassiveblack
hole growth has only a minor influence on the stellar-mass scaling relations of spheroidalgalaxies, but may

1 playaroleinmaintainingtheobservedFundamentalPlanetiltatopticalwavelengthsbysuppressingresidual
v starformationinmergerremnants.
We estimate that ≈ 40- 100% of the Fundamental Plane tilt induced by structural properties, as opposed
3
to stellar population effects, owes to trends in the central total-to-stellar mass ratio M /M produced by
5 total ⋆
dissipation. Gascoolingallowsforanincreaseincentralstellarphase-spacedensityrelativetodissipationless
0
1 mergers, therebydecreasingthe centralMtotal/M⋆. Lower mass systemsobtain greaterphase-spacedensities
1 thanhighermasssystems,producingagalaxymass-dependentcentralMtotal/M⋆andacorrespondingtiltinthe
5 FundamentalPlane. Weaccountforthesetrendsintheimportanceofdissipationwithgalaxymassintermsof
0 theinefficientcoolingofcollisionallyheatedgasinmassivehalosanddynamicallyvaryinggasconsumption
/ timescalesinsmallersystems.
h
p Subjectheadings:galaxies:formation–galaxies:evolution
-
o
r 1. INTRODUCTION Faberetal.1987,hereafterthe“virialscaling”).
t
s Elliptical galaxies represent a fascinating combination of WhiletheobservationaldeterminationoftheFPwasorig-
a inally motivated as a precise distance indicator to improve
complexity and regularity. A leading theory for the origin
:
v of early-type galaxies is based on mergers of disk galax- upon the previously known luminosity (L) – velocity dis-
i ies (Toomre&Toomre 1972; Toomre 1977) and likely in- persion(σ) relation (Faber&Jackson 1976), the importance
X
of the FP scalings and its correspondingly small scatter for
volvesgasdissipation,starformation,andsupermassiveblack
r theories of elliptical galaxy formation was also realized.
holefeedback(Barnes1992;Barnes&Hernquist1992,1996;
a The first observationally determined FP scalings (α∼1.3-
Mihos&Hernquist1994,1996;DiMatteoetal.2005)inad-
1.4,β∼0.8- 0.9,atopticalwavelengths;Dressleretal.1987;
dition to stellar dynamics. Despite their complex origins,
Djorgovski&Davis1987)differedfromthevirialscaling,in-
early-typegalaxiesobeyaregularsetofscalingrelationsthat
dicating a “tilt” relative to the expectation for homologous
connecttheirphotometricandkinematicproperties,mostno-
systems. TheFPtiltimpliedthatthemass–to–lightratioM/L
tably the relation between effective radius R , central stellar
e likelyvariesasafunctionofgalaxymassorluminosityas
velocitydispersionσ, andaveragecentralsurfacebrightness
DIejkonrgoowvnskais&thDeaFvuinsd1a9m8e7n)talPlane(FP;Dressleretal.1987; M ∝Lγ, (2)
L
R ∝σαI- β. (1) withintherangeγ≈1/5- 1/4.Faberetal.(1987)notedthat
e e deviationsfromtheFPcanbeinducedbyM/Lvarianceow-
ing to e.g. metallicity or age trends in stellar populations,
The virial theorem can be used to calculate this relation
dynamical or structural properties, and the relative distribu-
for homologous systems, which gives α = 2, β = 1 (e.g.,
tion of darkand baryonicmatter. In principle, each of these
1 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cam- effectsmayalsointroduceasystematictiltintotheobserved
bridge,MA02138,USA. FPiftheyvaryasafunctionofellipticalgalaxymass.
2LeidenObservatory,P.O.Box9513,NL-2300RALeiden,Netherlands. Thepurposeofthecurrentpaperistogaugetheimportance
3 TheOhioState University, Department ofAstronomy, 140West18th of various contributionsto the tilt in the observed FP in the
Ave.,Columbus,OH43210,USA.
context of the scenario where elliptical galaxies form from
4 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1,
85740GarchingbeiMünchen,Germany. mergers. Using hundreds of simulated galaxy mergers that
[email protected] includethephysicsofgascooling,starformation,supernova
2 Robertsonetal.
feedback, and black hole accretion and feedback, we deter- iesdependprimarilyongalaxyluminosity. Faber&Jackson
mine that gas dissipation may significantly contribute to the (1976)establishedtherelationbetweenluminosityLandve-
tilt of the observed FP, in addition to tilt induced by M/L locitydispersionσ,providingfurtherevidencethatellipticals
trends from stellar populations. We propose that elliptical follow a regular sequence as a functionof mass. Kormendy
galaxies initially form in gas-rich mergers from disk galaxy (1977)showedthatthesurfacebrightnessesandeffectiveradii
progenitorswhosegasfractionsexceed f ∼30%,andshow of ellipticals correlate with galaxy luminosity and with one
gas
thattheseremnantsdisplaysubstantialFPtilt. another(seealsoBinggelietal.1984).
Weconnecttheoriginofthistilttothecentralstellarphase- Importantearlyindicationsthatasecondparameter(inad-
spacedensityofthe remnants. Insmallmasssystemswhere dition to mass) governs the properties of ellipticals came
dissipationismostimportant,thestellarphase-spacedensity with the Terlevichetal. (1981) and Tonry&Davis (1981)
ofremnantsincreasessubstantiallyinhigh-gasfractionmerg- work that implied a correlation of L- σ and absorption-line
ers. Thecentralstellar phase-spacedensityinmassive ellip- strength–luminosityrelation(Mg - L)residuals.Whilethese
2
ticals remains similar in mergerswith varying gas fractions, findingswerelatercontradicted(e.g.Dressler1984), thatel-
withtheirstarsonaverageobtaininglowerphase-spaceden- lipticalgalaxieswerenotaone-parameterfamilyremainedan
sities thanreachedin smallersystems. Thismass-dependent importantpossibility.
phase-space density trend translates into a mass-dependent The discovery of the FP (Dressleretal. 1987;
trend in the ratio of total mass to stellar mass M /M in Djorgovski&Davis 1987) definitively revealed that el-
total ⋆
the central regions of ellipticals and a corresponding tilt in liptical galaxy properties are set by at least two parameters.
the FP. We then explain this mass-dependent importance of Specifically,ellipticalswerefoundtoobeyarelationbetween
dissipationintermsoftheinefficientcoolingofcollisionally R , σ, and I as given by Equation (1), with less than half
e e
heated gas in massive ellipticals and the dynamically vary- thescatteroftheFaber&Jackson(1976)L- σ relation. The
inggasconsumptiontimescaleofsmallersystems. Itisinter- small scatter of the FP was immediately noticed, implying
esting,andpossiblysignificant,thatthegasfractionrequired that the process of elliptical galaxy formationmust result in
to reproduce the observed tilt in the FP is, as discussed by a veryregular mass-sequence. Faberetal. (1987) notedthat
Hernquist(1993) similar to thatneededformergersofdisks mass-to-light ratio (M/L) variations can influence observa-
toyieldthehighcentralphasespacedensitiesofellipticals. tions of the FP by inducing tilt relative to the plane defined
In accord with other work (e.g., Capelatoetal. 1995; by the virial relation. Much of the subsequent work on the
Dantasetal. 2003; Nipotietal. 2003; Boylan-Kolchinetal. FP has centered aroundpossible causes of M/L variation or
2005), we find that subsequent dissipationless merging be- otheroriginsfortiltingtheFPrelativetothevirialplane.
tween spheroidal galaxies roughly maintains the FP tilt. Numerous subsequent observations verified and im-
Moreover,we findthata singlegenerationof dissipationless proved the FP relation for ellipticals (Luceyetal.
re-mergingofremnantswillinducescatterintheM - σrela- 1991a,b; deCarvalho&Djorgovski 1992; Benderetal.
BH
tionbutwillnotdestroythecorrelation,asspheroidalgalaxy 1992; Jorgensenetal. 1992; Guzmanetal. 1993;
mergers also do not dramatically alter the R - M relation Jorgensenetal. 1993; Sagliaetal. 1993; Benderetal. 1994;
e ⋆
when viewed as a mass-sequence. Possible dry mergingbe- Prugniel&Simien 1994; Pahreetal. 1995; Jorgensenetal.
tweenspheroidalgalaxiesatredshiftsz<1asindicatedbyre- 1996; Prugniel&Simien 1996; Busarelloetal. 1997;
centobservations(Belletal.2005;vanDokkum2005)isthus Graham&Colless 1997; Benderetal. 1998; Pahreetal.
notexpectedtodestroytightFPorM - σrelationsgenerated 1998b,a; Mobasheretal. 1999; Kronawitteretal.
BH
duringspheroidformationthroughgas-richmergersathigher 2000; Gerhardetal. 2001; Bernardietal. 2003a;
redshifts. Padmanabhanetal. 2004; Wooetal. 2004; Cappellarietal.
Thispaperisorganizedasfollows. Wereviewtheobserva- 2005)andextendedtheFPdeterminationstohigherredshifts
tionalandtheoreticalworkonellipticalgalaxyformationand (Franx 1993; vanDokkum&Franx 1996; Kelsonetal.
scalingrelationsin§2.Wepresentourmethodologyin§3and 1997; Schadeetal. 1997; vanDokkumetal. 1998;
ourresultsin§4. Wediscusstheimplicationsofourworkin Jørgensenetal. 1999; Treuetal. 1999; Kelsonetal.
§6andsummarizeandconcludein§7. Throughout,weadopt 2000a,b,c; Kochaneketal. 2000; Treuetal. 2001;
aflatΛCDMuniversewithΩM=0.3,ΩΛ=0.7,Ωb=0.04,and vanDokkum&Franx 2001; vanDokkumetal. 2001;
aHubbleparameterH =100hkms- 1Mpc- 1withh=0.7. Treuetal. 2002; Gebhardtetal. 2003; Rusinetal. 2003;
0
vanDokkum&Stanford 2003; vanDokkum&Ellis 2003;
2. REVIEWOFELLIPTICALGALAXYSCALINGLAWS
vanderWeletal.2004). Whilethespecificdetailsmayvary,
Scaling laws describing the regularity of the properties of
theseworksgenerallyconcludethat
ellipticalgalaxieshavebeenknownsinceatleastFish(1964),
whoreportedarelationbetweentheirpotentialenergyW and • Atightfundamentalplanebetweentheellipticalgalaxy
mass M⋆ asW ∝M⋆3/2. Even earlier, deVaucouleurs(1948, propertiesRe,σ,Ie exists,andextendsinsomeformto
hereafterdV)haddemonstratedthatellipticalsgenerallyhave atleastredshiftz∼1.
asurfacebrightnessprofilelogI(r)∝r1/4,atleastoversome
range in radius r, and implications of the dV profile for the • Ellipticalsare old, with formationredshiftsz>1, and
mass-dependentpropertiesofellipticalswererecognizedbe- their color evolution, which controls the FP normal-
foretheFish(1964)paper(e.g.,Poveda1958). ization,isroughlyconsistentwithpassiveevolutionof
Sandage(1972)foundacolor–magnituderelationforVirgo theirstellarpopulations.
and Coma cluster ellipticals (see also Stebbins&Whitford
1952;deVaucouleurs1961). Faber(1973)discoveredasim- • Some of the FP tilt must originate from the changein
ilar color–magnitudetrend in LocalGroup and cluster ellip- stellarpopulationM/Lwithgalaxymass,buttheextent
ticals,aswellasanabsorption-linestrength–magnituderela- towhichstellarpopulationscontributetoM/L-induced
tion, and suggested that these properties of elliptical galax- FPtiltisdebated.
GasDissipation&theFP 3
Ofspecificinteresttomodelersaretheobservationsthatmade starsanddarkmatter. Thesecondcategory,termed“dissipa-
definite statements about the influence of structural or kine- tional” mergers, consists of simulations that account for gas
maticnonhomologyonM/LandtheFPtilt,especiallythose cooling,starformation,thephysicsoftheinterstellarmedium,
that conclude directly that these nonhomologies are either andsupernovafeedback.Werefertothethirdcategoryofcal-
unimportant or of minor significance (Gerhardetal. 2001; culationsas“full-model”simulationswhich,inadditiontothe
Cappellarietal. 2005), significant (e.g. Padmanabhanetal. processes accounted for in the “dissipational” mergers, also
2004), or of possible but as yet not fully determined impor- include supermassive black hole growth and feedback. All
tance(e.g.Pahreetal.1998b). threecategoriesofdisk–diskmergersarederivedfromastan-
Moreover, elliptical galaxy properties related to the dard set of galaxy models to enable a direct comparison of
FP or its projections have been extensively observed. physicalprocesses.
These include studies of the photometric profiles (e.g. To perform our numerical simulations, we utilize the
deVaucouleurs 1948; Sersic 1968; Kormendy&Illingworth GADGET2 code (Springel 2005). GADGET2 uses the
1982; Burkert 1993; Caonetal. 1993), internal kine- smoothed particle hydrodynamics (SPH) formalism (Lucy
matic structure (e.g. Binney 1978; Daviesetal. 1983; 1977; Gingold&Monaghan1977) in its entropy-conserving
Davies&Birkinshaw 1988; Benderetal. 1994), metallic- formulation(Springel&Hernquist2002)tocalculatethedy-
ity (deCarvalho&Djorgovski 1992; Benderetal. 1993, namical evolution of the gas and a tree-based method to
1996; Bernardietal. 1998, 2003b), and the M - σ rela- computegravitationalforcesbetweenparticles(Barnes&Hut
BH
tion (e.g. Gebhardtetal. 2000; Ferrarese&Merritt 2000; 1986).
Tremaineetal. 2002). An important, related property is Allprogenitorgalaxieswerecreatedusingthemethodsde-
the size-stellar mass relation (Shenetal. 2003; Trujilloetal. scribedinSpringeletal.(2005),whichallowforthegenera-
2004;Trujillo&Aguerri2004;McIntoshetal.2005),thatin- tion of stable equilibrium galaxy models. Each galaxy con-
dicates a power-law correlation between some characteristic tains an extended dark matter halo, and may also consist of
galaxysizeandthestellarmassorluminosity. a stellar disk, gaseous disk, stellar bulge, and a supermas-
Interpreting these observations has been the focus of sive black hole particle. The collisionless components of
various theoretical efforts (see, e.g. Barnes&Hernquist thegalaxymodelsarerequiredtosatisfytheJeansequations
1992, for a description of early results). Notably, sim- whilethestructureofthegascomponentisdeterminedbythe
ulations of the formation of ellipticals and their prop- equationofhydrostaticequilibriumandanintegralconstraint
erties in the context of the merger hypothesis have onthesurfacemassdensity.
been performed using a variety of progenitor mod-
els including spheroidal (White 1979; Capelatoetal. 3.1. DissipationlessDiskSimulations
1995; Dantasetal. 2003; González-García&vanAlbada
Thedissipationlessdiskprogenitorsconsistofanexponen-
2003; Nipotietal. 2003), disk (Toomre&Toomre 1972;
tial stellar disk embedded in a dark matter halo with virial
Gerhard 1981; Farouki&Shapiro 1982; Faroukietal. 1983; velocities in the range V = 80- 500 km s- 1. We initial-
Barnes&Hernquist 1991; Barnes 1992; Hernquist 1992, vir
ize the size of the disk according to the Moetal. (1998)
1993;Hernquistetal. 1993; Mihos&Hernquist1994, 1996;
formalism for dissipational disk galaxy formation (see also
Hibbard&vanGorkom 1996; Bekki 1998; Dubinski 1998;
Fall&Efstathiou1980;Blumenthaletal.1986)assumingthe
Naabetal.1999;Naab&Burkert2001;Aceves&Velázquez
disk contains a fraction of the total galaxy angular mo-
2005), and cosmologicalsystems (e.g. Aarseth&Fall 1980;
mentum equal to its mass fraction, which we set to m ≡
Sáizetal. 2004). Analytical models of elliptical galaxies d
M /M =0.041to matchthe MilkyWay-like modelused
have also been formulated (Hernquist 1990; Ciotti 1991; disk vir
bySpringeletal. (2005). Thediskscalelengthr isthende-
Ciottietal. 1996; Ciotti 1996; Ciotti&Bertin 1999), aiding d
terminedbythe galaxyspin λandthe darkmatterhalocon-
the interpretation of both the observational and simulation
centrationC . We adoptλ=0.033,whichisnearthemode
results. In what follows, we combine features of many of vir
of the redshift- and mass- independent distribution of dark
these previous theoretical endeavors by simulating mergers
matter halo spins measured in cosmological N-body simu-
between dissipational and dissipationless disk galaxies,
lations (Vitvitskaetal. 2002). For the Navarro-Frenk-White
spheroidal systems, and merger remnants to determine their
haloconcentrationC (Navarroetal.1997,NFW),weadopt
fundamentalscalingrelations. vir
themass-andredshift-dependentdarkmatterhaloconcentra-
tionsmeasuredbyBullocketal.(2001)
3. METHODOLOGY
Galaxymergingcombinesmuchcomplexphysics, includ- M - 0.13
ing e.g. the collisionless dynamicsof dark matter and stars, C (M ,z)≈9 vir (1+z)- 1, (3)
vir vir
(cid:18)M (cid:19)
gasdissipation,starformation,andfeedbackfromsupernovae coll,0
apnridncbilpalcekbheoelsesgernotiwalthf.orWdheitleermeaicnhinogfmtheersgeerprreomcensasnetspmroapyeirn- whereMcoll,0∼8×1012h- 1M⊙ isthelinearcollapsemassat
redshift z=0. In all cases, the stellar disk scaleheight h =
ties, their relative importancehas notbeen fullyestablished. d
0.2r , similar to the MilkyWay (c.f. Siegeletal. 2002). We
Bycomparingthestructureofmergerremnantsinsimulations d
model the dark matter halo with a Hernquist (1990) density
which systematically include or exclude various processes,
profileoftheform
we attempt to both test the merger hypothesis and identify
themostimportantphysicalmechanisms. M a
ρ (r)= DM , (4)
Tothisend,weperformasetofhundredsofsimulationsof h 2π r(r+a)3
galaxymergers.Oursuiteconsistsofthreecategoriesofdisk–
diskmergersandoneclassofspheroid-spheroidmergers.The wherethescalelengtha(C )mapstheHernquist(1990)pro-
vir
firstcategoryofsimulations,refereedtoas“dissipationless,” file parametersto the appropriateNFW haloparameters(for
includesonlysimulationsofdisksconsistingofcollisionless details,seeSpringeletal.2005).
4 Robertsonetal.
TABLE1. GALAXYMERGERS
Model Progenitor Redshift GasFraction ISMPressurization PericentricSeparation #ofSimulations
Vvir[kms- 1] z fgas qEOS rperi
“Full-Model”SimulationswithBlackHoles
Local 80,115,160,226,320,500 0 0.4,0.8 0.25,1.0 2rd 24
Intermediate- z 80,115,160,226,320,500 2,3 0.4,0.8 0.25,1.0 2rd 48
High- z 115,160,226,320,500 6 0.4,0.8 0.25,1.0 2rd 20
HaloConcentrations 160 0 0.4 1.0 2rd 5
DiskOrientation 160 0 0.4 1.0 Table2 14
OrbitalConfiguration 160 0 0.4 1.0 Table2 18
DissipationalSimulations
Local 80,115,160,226,320,500 0 0.4,0.8 0.25,1.0 2rd 24
Intermediate- z 80,115,160,226,320,500 2,3 0.4,0.8 0.25,1.0 2rd 48
High- z 115,160,226,320,500 6 0.4,0.8 0.25,1.0 2rd 20
fgasRuns 80,115,160,226,320,500 0 0.01,0.025,0.05 0.25,1.0 2rd 72
0.1,0.2,0.4
DissipationlessSimulations
Local 80,115,160,226,320,500 0 0.0 – 2rd 6
Intermediate- z 80,115,160,226,320,500 2,3 0.0 – 2rd 12
High- z 80,115,160,226,320,500 6 0.0 – 2rd 6
WideOrbit 80,115,160,226,320,500 0 0.0 – 0.4Rvir 6
WideOrbit,Int.- z 80,115,160,226,320,500 2,3 0.0 – 0.4Rvir 12
WideOrbit,High- z 80,115,160,226,320,500 6 0.0 – 0.4Rvir 6
Bulge,Local 80,115,160,226,320,500 0 0.0 – 2rd 6
Bulge,Intermediate- z 80,115,160,226,320,500 2,3 0.0 – 2rd 12
Bulge,High- z 80,115,160,226,320,500 6 0.0 – 2rd 6
High-Res 80,115,160,226,320,500 0 0.0 – 2rd 6
SpheroidSimulations
Local 80,115,160,226,320,500 0 0.0 – 0.025Rvir 6
Wide/EllipticalOrbit 80,115,160,226,320,500 0 0.0 – 0.4Rvir 6
Full-ModelRe-mergers 80,115,160,226 0 0.4 0.25 0.05Rvir 4
FollowingRobertsonetal.(2005b),wescaletheprogenitor mergerwithawiderparabolicorbitincreasedtor =0.4R
peri vir
galaxypropertiestoapproximatethestructureofdiskgalax- tojudgetheeffectofincreasedorbitalangularmomentumon
ies appropriate for redshifts z=0, 2, 3, and 6. Varying the the dissipationless merger remnants. In addition, we repeat
progenitorgalaxiesinthismannerenablesustodeterminethe eachnearly-radialmergerwithbulgecomponentsincludedin
impactofredshift-dependentgalaxypropertiesonthescaling the galaxies with mass fraction m ≡ M /M = 0.1367
b bulge vir
lawsofellipticals. KeepingthevirialvelocityV fixedwith to match the Milky Way-like model used in Springeletal.
vir
redshift,wescaletheprogenitorvirialmassandvirialradius (2005). WemodelthebulgeswithaHernquist(1990)density
usingtherelations profile form (see Equation 4) where we set the bulge scale-
lengthb=0.2r . Eachbulgecontains20,000particles, with
M = Vv3ir (5) the number ofddisk particles reduced to 60,000 to maintain
vir 10GH(z) thesamemassresolution. Finally,were-runallthepuredisk
simulationsatredshiftz=0withhigherresolutiondissipation-
V
R = vir , (6) lessmodelswith 180,000darkmatterparticlesand120,000
vir
10H(z) disk particles to examine issues related to numericalresolu-
tion. While we discuss these tests in more detail in §4, we
whereH(z) isthe Hubbleparameter. To suitablyresolvethe
noteherethatthelargesetofdissipationlesssimulationspro-
forcesbetweenparticlesinmodelsofhigher-redshiftsystems,
wereducethegravitationalsmoothingby(1+z)- 1. Thehalo ducesresultsveryconsistentwiththerestrictedsetofhigher
resolutionruns. Inallweperform78dissipationlesssimula-
concentrationsalsovarywithredshiftandmassaccordingto
tions,andweprovideacompletelistinginTable(1).
Equation(3)andthediskscalelengthsdecreasewithredshift
through their dependenceon R andC . We note that the
vir vir 3.2. DissipationalDiskSimulations
redshift-dependenceofdiskscalelengthsagreeswellwiththe
distributionofdiskscalelengthsseenouttoredshiftz≈1(e.g. To gauge the impact of dissipational gas physics on the
Ravindranathetal.2004;Bardenetal.2005). properties of merger remnants, we perform a suite of disk
At each redshift, we consider three separate types of dis- galaxymergersthat includegas cooling, star formation, and
sipationless disk mergers. First, we examine equal mass supernovafeedback. Thedissipationaldisk progenitorscon-
mergers of pure disk galaxies on prograde-prograde copla- tain exponential gaseous and stellar disks and Hernquist
nar parabolicorbitswith the pericentricpassage distance set (1990)darkmatterhalos,withtheirdisksizesdeterminedby
to r = 2r . The galaxies each contain 60,000 dark mat- theMoetal.(1998)formalismasdescribedin§3.1. Thever-
peri d
terand80,000stellardiskparticles. Second,werepeateach ticalstructureofthegaseousdisksaredeterminedbyaninte-
GasDissipation&theFP 5
gralconstraintfromthesurfacemassdensityandtherequire-
mentofhydrostaticequilibriumwithinthegalaxypotential. TABLE2. ORBITALVARIATIONS
The thermal properties of the gas are determined us-
ing the multiphase interstellar medium (ISM) model of DiskOrientations
Springel&Hernquist (2003). Star formation is prescribed Models [dθe1g] [dφe1g] [dθe2g] [dφe2g] [h-r1pekripc]
in the manner of Springel&Hernquist (2003), constrained
to approximate the Schmidt (1959) law for disk galax- b 180 0 0 0 5.0
c 180 0 180 0 5.0
ies as measured by Kennicutt (1998), including a density
d 90 0 0 0 5.0
threshold. Below this threshold the gas is modeled as a e 30 60 - 30 45 5.0
single-phase medium which is not star-forming. Dense gas f 60 60 150 0 5.0
above the threshold is modeled as a hybrid of cold, dense g 150 0 - 30 45 5.0
h 0 0 0 0 5.0
cloudsembeddedin a hot, diffuse mediumas envisionedby
i 0 0 71 30 5.0
McKee&Ostriker(1977). Thetemperatureofthehotphase j - 109 90 71 90 5.0
issetbysupernovafeedbackandtheefficiencyofcloudevap- k - 109 - 30 71 - 30 5.0
l - 109 30 180 0 5.0
oration,andhasanenergyperunitmassthatfarexceedsthe
m 0 0 71 90 5.0
coldphase. Eventhoughmostofthegasbymassiscold,the n - 109 - 30 71 30 5.0
hightemperatureofthehotphasemorethancompensatesfor o - 109 30 71 - 30 5.0
its small mass fraction, acting to pressurize the star-forming p - 109 90 180 0 5.0
gas,andleadingtoaneffectiveequationofstateP (ρ)thatis
eff OrbitalConfigurations
stifferthanisothermal(foranumericalfit,seeRobertsonetal.
e1 30 60 - 30 45 2.5
2004).ThemultiphasemodelofSpringel&Hernquist(2003)
e2 30 60 - 30 45 10.0
has been generalized by Springeletal. (2005) to allow for e3 30 60 - 30 45 15.0
an effective equation of state parameter q that linearly e4 30 60 - 30 45 20.0
EOS
interpolates between an isothermal gas (q = 0) and the e5 30 60 - 30 45 40.0
EOS e6 30 60 - 30 45 30.0
fully-pressurized multiphase ISM model (q = 1). In-
EOS h1 0 0 0 0 2.5
creasing qEOS improves the dynamical stability of the gas h2 0 0 0 0 10.0
and can prevent Toomre (1964) instability even in gas-rich h3 0 0 0 0 15.0
systems (Springel&Hernquist 2003; Robertsonetal. 2004; h4 0 0 0 0 20.0
h5 0 0 0 0 40.0
Springel&Hernquist2005;Robertsonetal.2005a).
h6 0 0 0 0 30.0
Forourdissipationalmodels, we re-runtheVvir=80- 500 k1 - 109 - 30 71 - 30 2.5
kms- 1 purediskmergersimulationsfrom§3.1withtwogas k2 - 109 - 30 71 - 30 10.0
fractionsof f =0.4,0.8,eachwithtwoequationofstatepa- k3 - 109 - 30 71 - 30 15.0
gas k4 - 109 - 30 71 - 30 20.0
rametersqEOS=0.25,1.0,atredshiftsz=0,2,3,and6. Each k5 - 109 - 30 71 - 30 40.0
progenitor galaxy has 60,000 dark matter particles, 40,000 k6 - 109 - 30 71 - 30 30.0
stellar disk particles, and 40,000gas particles. The systems
are merged on prograde-prograde parabolic coplanar orbits
withr =2r .Foranequationofstateparameterq =0.25
peri d EOS tion and feedback should preserve the power-law scaling of
we also systematicallyvarythe gasfractionofz=0 progen-
theM - σrelationbetweenredshiftsz=0- 6.
itors using f = 0.01, 0.025, 0.05, 0.1, 0.2 and 0.4. Our BH
gas Thefull-modelsimulationsaugmentthepurediskdissipa-
dissipationalsimulationcategoryhasatotalof164runs,with
tionalmergersimulationsfrom§3.2withsupermassiveblack
thecompletelistofsimulationsprovidedinTable(1).
holegrowthasdescribedabove. Eachprogenitorgalaxyhas
3.3. Full-ModelDiskSimulations 40,000stellardiskparticlesand40,000gasparticles,andare
mergedon prograde-progradeparaboliccoplanarorbitswith
Our full-modelcategory simulations include the complete
r =2r . The modelsare calculated forV =80- 500km
physical model presented in Springeletal. (2005), account- peri d vir
ing for gas cooling, star formation, supernova feedback, s- 1 galaxies at z= 0, 2, and 3 and Vvir = 115- 500 km s- 1
the Springel&Hernquist (2003) ISM model described in galaxies at z=6. Simulations are performed with gas frac-
§3.2, and a prescription for supermassive black hole growth tionsof fgas=0.4,0.8eachwithequationofstateparameters
and feedback. The supermassive black holes are included ofqEOS=0.25,1.0. Furthermore,werun14variationsofthe
as “sink” particles, with seed masses of 105h- 1M⊙. The Vvir=160kms- 1progenitormergerwherewechangethedisk
black holes are allowed to grow according to spherical orientationaccordingtothemethodofBarnes(1992)tochar-
Bondi-Hoyle-Lyttleton accretion (Hoyle&Lyttleton 1939; acterizetheeffectsofdiskalignment(seeTable2),andrunan
Bondi&Hoyle 1944;Bondi1952). The mass accretionrate additional 18 simulations where for 3 different orientations
˙ wevarythepericentricpassagedistance. Wealsorunasetof
M is determined from the density and sound speed of the
5 additionalsimulationsof theV =160km s- 1 halo where
gas near the black hole. A fraction ǫ =0.1 of the accretion vir
f
wesimulatedarkmatterhaloconcentrationsofC =5,7,9,
rate is radiatively released, of which a fraction η =0.05 vir
therm
11,and13. Inall,weperformatotalof129full-modelsim-
is coupled as thermal feedback into gas within an SPH ker-
ulationsandacompletelistoftheserunsisprovidedinTable
nel smoothing length of the black hole. The strength of the
(1).
thermalcouplingiscomparabletothethermalfeedbackcou-
pling of supernova energy used in cosmologicalsimulations
(e.g. Abadietal. 2003), and reproducesthe M - σ relation 3.4. SpheroidSimulations
BH
observedlocally(DiMatteoetal.2005). Usingthesamefull- Toexplorethepropertiesofremnantsformedbythemerg-
model simulations presented here, Robertsonetal. (2005b) ing of spheroidal systems, we also perform a suite of equal
havealso demonstratedthatthismodelforblackholeaccre- mass spheroid-spheroid mergers. The spheroid progenitors
6 Robertsonetal.
consist of stellar spheroids embedded in dark matter halos, The mean-squaredscatter about the direct best-fit plane can
both having Hernquist (1990) profiles. We assume a stellar thenbecharacterizedbythequantity
massfractionof f =0.05andtheconcentrationsofthehalos
are adjusted to ac⋆count for the mass-dependence measured ∆2 = σI2IσR2RσV2V- σI2IσR4V- σR2RσI4V (14)
in cosmological simulations (see Section 3.1). The sizes of (cid:10) (cid:11) (cid:0) - σ2 σ4 +2σ2 σ2 σ2
the stellar spheroids are set to follow the Shenetal. (2003) VV IR IR IV RV
Re- M⋆relationformassivegalaxiesas × σI2IσV2V- σI4V - 1(cid:1).
(cid:0) (cid:1)
R =4.16 M⋆ 0.56kpc, (7) Whenappropriatewecomparethequantity ∆2 1/2 withthe
e (cid:18)1011M⊙(cid:19) scatterdeterminedfromobservationalsampl(cid:10)es. (cid:11)
TheremnantpropertiesarealsocomparedtotheR - M re-
(see also Boylan-Kolchinetal. 2005). We vary the circular e ⋆
velocity of the halos between V = 80- 500 km s- 1. The lation,whichhasbeenmeasuredobservationallyintheSDSS
vir (Shenetal. 2003) and may be representedby the power-law
spheroidgalaxiesmergeoneithernearly-radialparabolicor-
form
bits with r =0.025R or wide elliptical orbitswith ellip-
peri vir R ∝Mµ. (15)
ticityǫ=0.5andr =0.4R . Eachspheroidprogenitorhas e ⋆
peri vir
1,200,000dark matter and 80,000stellar particles. A com- The R - M relation allowsfor a usefulcomparisonof rem-
e ⋆
pletelistofthespheroid–spheroidmergersisprovidedinTa- nant sizes for simulations with differing angular momenta,
ble(1). progenitorredshifts,ISMphysics,orgasdissipation.Inaddi-
Todeterminetheimpactsubsequentre-mergingbetweenel- tion, we find that the relative locationof remnantsin the FP
lipticalsformedfromdiskgalaxymergersmighthaveonthe can often be related to the impact of different physical pro-
structural properties of the remnants, we also re-merge disk cessesontheeffectiveradii. TheR - M relationalsoserves
e ⋆
galaxyremnantsfromthez=0full-modelsimulations.These asausefulcalibrationforourmethodtomeasuretheFPprop-
remnantsaremergedonparabolicorbitswithrperi=0.05Rvir, ertiesoftheremnants. Asmentionedin§3.4,asubsetofour
whereRvir isthe virialradiusoftheoriginaldiskprogenitor. simulationsuite involvesthe mergingof equilibriummodels
Were-mergeremnantswithprogenitorgalaxycircularveloc- of spheroidsinitialized to satisfy the Shenetal. (2003) rela-
itiesintherangeVvir=80- 226kms- 1. Acompletelistofthe tion. Asdiscussedinfurtherdetailin§4.4,theanalysistech-
re-mergersimulationsisprovidedinTable(1). nique used to measure the FP properties of remnants accu-
ratelyrecoverstheShenetal.(2003)relationwhenappliedto
3.5. Analysis thespheroidmodelprogenitorsandaffirmsourabilitytode-
Eachsimulationisevolveduntilthemergeriscompleteand terminesimulatedremnantpropertieswithreasonablefidelity.
theremnantsarefullyrelaxed,requiringintegrationsoftypi-
cally2-4Gyr. Theremnantsarethenkinematicallyanalyzed 3.6. ComparisonwithObservations
by measuringthe half-massstellar effectiveradiiR , the av-
e WeadopttheapproachofplottingtheFPpropertiesofsim-
erageone-dimensionalvelocitydispersionσ withina circu- ulatedremnantsintheR - σ2I- 1 virialplanecoordinatesys-
larapertureofradiusR ,andtheaveragestellarsurfacemass e e
e tem.ThetiltoftheFPrelativetothevirialplanecanbequan-
densityI measuredwithin R as I ≡M (r<R )/πR2. The
e e e ⋆ e e tifiedthroughthepower-lawrelation
quantities R , σ, and I are averaged over 100 random sight
e e
linestotheremnant. R ∝ σ2I- 1 λ, (16)
Once the FP parameters R , σ, and I are determined, we e e
e e (cid:0) (cid:1)
employthedirectfittingmethodofBernardietal.(2003a)to where λ=1 indicatesan alignmentof the FP with the virial
determinethe best-fit FP scalings. The directfitting method plane. Whenappropriate,wemeasurethisestimateoftheFP
seekstominimize tilt relative to the virial plane by linear least-squares fitting.
Ourchoiceof coordinatesystemsisnotunique,andalterna-
∆=logR - αlogσ- βlogI - δ, (8)
e e tiverepresentationsoftheFPincludetheκ-spacecoordinate
whereαandβaretheFPscalingindicesdefinedbyEquation system(Benderetal.1992)orthebest-fitFPcoordinatesde-
(1). Theminimizationof∆requires terminedbyobservationsinvariouspassbands. Theprimary
advantage of choosing the virial plane coordinate system is
(σ2σ2 - σ2 σ2 ) the easily determined tilt, which provides a gauge of possi-
α= II RV IR IV (9)
(σ2σ2 - σ4) blevariationsinthecentraltotal-to-stellarmassMtotal/M⋆ or
II VV II
kinematicnonhomologyofremnantsas a functionofgalaxy
(σ2 σ2 - σ2 σ2 ) mass.
β= VV IR RV IV (10)
(σ2σ2 - σ4 ) For our purposes, we choose not to use stellar population
II VV IV synthesistocomparewithdeterminationsoftheFPinoptical
δ=hlogR i- αhlogσi- βhlogI i, (11) passbands. ObservationsindicatethattrendsintheM/Lratio
e e
owingtostellarpopulationeffects(e.g. ageormetallicity)as
wheretheaveragehlogXiovertheNdatasamplesconsidered a function of galaxy mass or luminosity will contribute sig-
isdefinedas
nificantlytotheFPtilt,especiallyatshort-wavelengths(fora
hlogXi≡ logXi/N (12) recentresultonthisissue,seeCappellarietal.2005). Obser-
Xi vationshavealsodeterminedthatellipticalgalaxiesaretypi-
cally old and their stellar populations redden passively with
andtheco-variantdispersionisdefinedas
time (e.g. Benderetal. 1996; vanDokkum&Franx 1996).
σ2 = logX - hlogXi logY - hlogYi /N. (13) These constraints imply that to properly recover the short-
XY i i
wavelengthphotometricFP,preciseinformationonthestellar
Xi (cid:0) (cid:1)(cid:0) (cid:1)
GasDissipation&theFP 7
FIG.1.— FundamentalPlane(FP)relationproducedbythemergingofdis- FIG. 2.— EffectiveradiusRe–stellarmassM⋆relationproducedbythe
sipationlessdiskgalaxymodelsappropriateforredshiftsz=0(black),z=2 mergingofdissipationlessdiskgalaxymodelsappropriateforredshiftsz=0
(red),z=3(blue),andz=6(green)onnearlyradial, parabolicorbits. All (black),z=2(red),z=3(blue),andz=6(green)onnearlyradial,parabolic
modelsincludedarkmatterhalos. Thedissipationlessmergingofpuredisk orbits.Allmodelsincludedarkmatterhalos.Thedissipationlessmergingof
models(solidtriangles)anddiskmodelswithbulges(solidcircles)produce purediskmodels(solidtriangles)anddiskmodelswithbulges(solidcircles)
similarFPrelationsnearlyparalleltotheplanedefinedbythevirialrelation. produceRe- M⋆relationsshallowerthanthatmeasuredformassivegalaxies
Increasingtheangularmomentumoftheorbitbylengtheningthepericentric intheSloanDigitalSkySurvey(Shenetal.2003).Forcomparison,thebest
passagedistanceoftheorbitproducesanoffsetintheFPbyincreasingthe least-squaresfittotheRe- M⋆relationofpurediskmergerremnantsisplotted
effectiveradiusoftheremnants(opencircles),butthesystemsstillobtaina (dottedline).
FPscalingsimilartothevirialplane. Selecthigherresolutionrunsclosely
follow theFPdelineated bytheirlowerresolutioncounter-parts (opendia-
monds). Forcomparison, the best least-squares fitto the FP of pure disk
mergerremnantsisplotted(dottedline).
calpassbandluminosities(e.g.Bruzual&Charlot2003),and
typicalK-bandmass-to-lightratiosarewithin≈35%ofunity
for stellar populations formed at redshifts 0.75 < z < 5.0.
age, metallicity, and formation-redshiftdistribution of ellip- Thesepropertiesmakecomparisonsbetweenthestellar-mass
tical galaxies as a function of stellar mass at z=0 must be and near-IR FP scalings somewhat more sensible than com-
obtained. We note that knowing only either the formation- parisons with shorter-wavelength photometric FP scalings,
redshift (e.g. the redshift of the last major gas-rich merger) thoughnotideal.
or mean stellar age of elliptical galaxies may not be suffi- Inprinciple,amorestraight-forwardcomparisonwouldbe
cient to determine the photometric FP of the entire ellipti- tousehigh-resolutioncosmologicalsimulationsofgalaxyfor-
cal galaxy population. Recent surveys indicate that ellipti- mation to probe the fundamental plane with simultaneously
calstypicallyundergoa majordissipationlessmergeratred- accountingformetallicityandstellarageeffects. Previousat-
shiftsz<1(Belletal.2005;vanDokkum2005),and,inprin- temptstocompareresultsofcosmologicalsimulationsforthe
ciple, such mergers may induce galaxy mass-dependent tilt scalinglawsofellipticalgalaxieswithobservationshavebeen
fromstructuraleffectsthataredisjointfromM/Leffectsfrom made(e.g.Sáizetal.2004),butwithspatialresolutionsome
stellarpopulationsthatcharacteristicallypredatethoseevents. ≈20 times worse than the isolated merger simulations pre-
Clearly, applying stellar population synthesis models to el- sentedhere.Suchspatialresolutionislargerthantheeffective
liptical galaxies produced in individual galaxy merger sim- radiiofevenmoderately-sized(M ≥21)early-typegalaxies.
r
ulationsto producesimulated photometricFP scalings with- However,themetallicitiesandagesofthestellarpopulations
out attempting to correct for the cosmologically-determined ofgalaxiesproducedin cosmologicalsimulationscan be es-
propertiesoftherealellipticalpopulationislikelytoonaive. timated throughouttheir formation, and the results of stellar
Furthermore,comparingdirectlystellar-massFPscalingsde- populationsynthesismodelingwouldthereforebeeasytoin-
termined from simulations with short-wavelength (e.g. B- terpret.
bandorSloang-band)photometricFPscalingsshouldbeper- We mention here that semi-analytic techniques could
formedwithextremecautionastheshort-wavelengthFPmay be combined with the results of high-resolution merger
have additionalsourcesof tilt notpresentin the stellar-mass simulations in an attempt to account for the redshift-
FP. dependent formation of the elliptical galaxy population.
With these concerns in mind, our stellar-mass FP results For example, Robertsonetal. (2005a) used the results of
will be comparedwith the near-infrared(IR) FP determined Hopkinsetal. (2005a), who inferred a redshift-dependent
by Pahreetal. (1998b). While, as Pahreetal. (1998b) note, black hole mass function from the quasar luminosity func-
tilt owing to stellar population effects may still be present tion, to determine the influence of the redshift-dependent
and is not tightly constrained, structural or dynamical non- formation times of elliptical galaxies on the M - σ rela-
BH
homologies can contribute significantly to the FP scaling in tion. Using the results from Robertsonetal. (2005a) and
thenear-IR.TheK-bandmagnitudesofstellarpopulationsof Hopkinsetal. (2005a), Hopkinsetal. (2005b) combinedthe
agivenagearelessinfluencedbymetallicitythantheiropti- redshift-dependentpropertiesofgalaxyremnants,theM - σ
BH
8 Robertsonetal.
FIG. 3.— FundamentalPlane(FP)relationproducedbythemergingof FIG. 4.— EffectiveradiusRe–stellarmassM⋆relationproducedbythe
gas-richdiskgalaxieswithdarkmatterhalos,starformationandsupernova mergingofgas-richdiskgalaxieswithdarkmatterhalos,starformationand
feedback.Shownareremnantsproducedbymergersappropriateforredshifts supernovafeedback. Shownareremnantsproducedbymergersappropriate
z=0(blackcircles), z=2(reddiamonds),z=3(bluetriangles), andz=6 forredshiftsz=0(blackcircles),z=2(reddiamonds),z=3(bluetriangles),
(greensquares)withnearlyradial,parabolicorbits. Thedissipationalmerg- andz=6(greensquares)withnearlyradial, parabolicorbits. Thedissipa-
ingofpurediskmodelsproducesaFPnearlyparalleltotheobservedinfrared tionalmergingofpurediskmodelsproducesaRe- M⋆relationroughlypar-
FP(Pahreetal.1998b)andisalmostindependentoftheredshiftscalingsof alleltothatmeasuredformassivegalaxiesintheSloanDigitalSkySurvey
theprogenitorsystems. Forcomparison,thebestleast-squaresfittotheFP (Shenetal.2003). Forcomparison,thebestleast-squaresfittotheRe- M⋆
delineatedbytheremnantsisplotted(solidline). relationsdelineatedbythedissipationalsimulations(solidline)anddissipa-
tionlesssimulations(dottedline)areplotted.
TABLE3. BEST-FITSCALINGS
1 shows the location of dissipationless merger remnants in
Models FundamentalPlane Re- M⋆ the virial coordinate system, and plots remnants from pro-
α β λ ∆2 µ
genitors appropriate for various redshifts. The best-fit FP
(cid:10) (cid:11)
Dissipationless 2.00 1.01 1.00 0.018 0.45 scalings produced by pure disk systems (solid triangles) are
Dissipational 1.58 0.80 0.80 0.065 0.57 α=2.00, β =1.01, with almost no tilt relative to the virial
Full-Model 1.55 0.82 0.79 0.062 0.57
plane (λ = 1.00). Individually, progenitors at each of the
GasFraction fgasRuns foursimulatedredshiftsproducemergerremnantsalsoclosely
alignedwiththevirialplane(λ=0.97- 1.03).Includingstel-
fgas=0.01 1.81 0.75 0.97 0.009 0.41
fgas=0.025 2.11 0.74 0.96 0.011 0.41 larbulgesintheprogenitors(solidcircles)producesasimilar
fgas=0.05 2.07 0.64 0.95 0.011 0.41 FPscaling(α=1.95,β=0.98;λ=0.97). Amoresubstantial
fgas=0.1 2.01 0.61 0.92 0.014 0.42 changeintheFPisrealizedbyincreasingthepericentricpas-
fgas=0.2 1.89 1.20 0.89 0.024 0.44 sagedistanceoftheencountersfrom2r to0.4R (opencir-
fgas=0.4 1.64 1.07 0.83 0.033 0.51 d vir
cles), whichcorrespondinglyincreasesthe total angularmo-
mentaofthemergingsystems. Remnantsproducedinthedis-
sipationlesswideorbitmergerstypicallyhavelargereffective
radii, inducing an offset of ∆logR ∼- 0.4 in the FP. How-
relation,andtheredshift-dependentblackholemassfunction e
ever,thewideorbitFPscalings(α=1.97,β=1.04;λ=0.97)
tomodeltheevolutionofthered-galaxyluminosityfunction
are still very similar to both the nearly radial orbit FP and
and color-magnitude relation. A combination of the results
virialscalings. Increasingthenumberofdarkmatterparticles
from Hopkinsetal. (2005b) and the stellar-mass FP relation
per halo to 180,000 in the progenitors(open diamonds) has
presented here could be used to account for the effects of
verylittleeffectontheresultingFP,suggestingourresultsare
color-magnitudeevolutiononthe photometricFP, butwould
notstronglyinfluencedbyournumericalresolution.
likelyinvolveotherassumptionsbeyondthoseemployedhere The R - M relation produced by merging dissipationless
andwedefersuchanalysisforfuturework. e ⋆
disk progenitors, shown in Figure 2, reflects the trends ap-
parent in the FP those mergers generate and additional fea-
4. RESULTS
turesowingtoredshift-dependentprogenitorproperties. The
Below,wepresenttheFPandR - M relationsforthedis-
e ⋆ merging of pure disk galaxies (solid triangles) appropriate
sipationless,dissipational,full-model,andspheroidalmerger
for various redshifts generates remnants with a shallower
simulations.ForeachFPandRe- M⋆relation,welistthebest mean R - M relation (µ≈0.45, solid line) than that mea-
e ⋆
fitscalingsinTable3.
sured in late-type galaxies in the Sloan Digital Sky Survey
(Shenetal. 2003, µ≈0.56). The remnants also systemati-
4.1. DissipationlessDiskSimulations
callydecreasein effectiveradiuswith redshift,reflectingthe
Themergingof thedissipationlessdisksdescribedin §3.1 decrease in progenitor disk scalelength and pericentric pas-
produces a FP relation similar to the virial scaling. Figure sagedistance. Widerorbitmergers(opencircles)withlarger
GasDissipation&theFP 9
FIG. 5.— Fundamental Plane (FP) relation produced by the merging FIG. 6.— Effective radius Re – stellar mass M⋆ relation produced by
ofgas-richdiskgalaxies withdarkmatterhalos, starformation, supernova themergingofgas-richdiskgalaxieswithdarkmatterhalos,starformation,
feedback,andaprescriptionforfeedbackfromaccretingsupermassiveblack supernovafeedback, andaprescription forfeedback fromaccreting super-
holes. Shownareremnants produced bymergers appropriate forredshifts massiveblackholes. Shownareremnantsproducedbymergersappropriate
z=0(blackcircles), z=2(reddiamonds),z=3(bluetriangles), andz=6 forredshiftsz=0(blackcircles),z=2(reddiamonds),z=3(bluetriangles),
(green squares) with nearly radial, parabolic orbits. The merging of pure andz=6(greensquares)withnearlyradial,parabolicorbits.Alsoplottedare
diskgalaxiesusingourfullphysicalmodelproducesaFPnearlyparallelto remnants produced byvarying the system angular momentum through the
theobservedinfraredFP(Pahreetal.1998b)andnearlycoincidentwiththe initialdiskorientation(purplepentagons)andpericentricpassagedistances
FPproducedbysimilarsimulationswithoutblackholes. TheFPrelationis (orangehexagons)forasinglepairofprogenitormodels,whichproducesa
roughlyindependentoftheredshiftscalingsoftheprogenitorsystemsandthe spreadintheremnanteffectiveradius.Themergingofpurediskgalaxiesus-
locationofremnantswithinFPisfairlyinsensitivetoalargevarietyofdisk ingourfullphysicalmodelproducesaRe- M⋆relationroughlyparalleltothat
orientations(purplepentagons)andorbitalconfigurations(orangehexagons), measuredformassivegalaxiesintheSloanDigitalSkySurvey(Shenetal.
aschangesintheeffectiveradiusarecompensatedbychangesinthevelocity 2003),thoughwithanoffset.TheresultantRe- M⋆relationshiftsdownward
dispersionandsurfacemassdensity. Forcomparison,thebestleast-squares with the redshift of the progenitor systems as the remnants decrease with
fittotheFPdelineatedbytheremnantsisplotted(solidline). size. Forcomparison,thebestleast-squaresfittotheRe- M⋆relationdelin-
eatedbytheseremnants(solidline),aswellasthedissipationlessdiskmodel
Re- M⋆relation(dottedline)andtheShenetal.(2003)relation(dashedline)
areplotted.
angularmomentaproduceremnantswithlargereffectiveradii
whilegalaxiescontainingbulges(solidcircles),andtherefore
lessspecificangularmomenta,produceremnantswithsmaller
effectiveradii. Theseresultsareconsistentwithexpectations galaxiesvarysubstantiallywithredshift. Combined,varying
fordissipationlesssystemswheretheenergyandangularmo- thegasfractionabove f >0.4,changingthe ISMpressur-
gas
menta of the model system are manifestly conserved. As a ization dramatically, and scaling the progenitor systems for
finalnoteforthedissipationlessruns,increasingthenumeri- redshiftsz=0- 6 produceonly a small amountof scatter in
calresolutionofthedarkmatterby3×(opendiamonds)gives thedissipationalmodelFP( ∆2 1/2=0.065).
consistentresults,suggestingthatartificialheatingofthestel- The Re- M⋆ relation of th(cid:10)e re(cid:11)mnants, shown in Figure 4,
larcomponentbydiscretenesseffectsinthedarkmatterhalo ismorestronglyinfluencedby therangeof progenitorprop-
isnotstronglyinfluencingthestructureoftheremnants. erties. Thesmallerprogenitorgalaxiesappropriateforhigher
redshiftsproducesmallerremnants,whiletheless-pressurized
4.2. DissipationalDiskSimulations
ISM models also decrease the effective radii of the rem-
Themergingofgas-richdiskgalaxyprogenitorsincluding nants.ThedissipationalmodelR - M relationissignificantly
e ⋆
dissipationresultsinFPandR - M relationsthatdiffersub- steeper (solid line, µ= 0.57) than the relation produced by
e ⋆
stantiallyfromtheanalogousrelationsproducedbythemerg- thedissipationlessmergingofdiskgalaxies(dottedline, µ=
ing of dissipationless disks. Figure 3 shows the FP relation 0.45),andcompareswellwiththerelationmeasuredforlate-
generatedby the dissipationalmodeldisk galaxiesappropri- type galaxies in the SDSS (Shenetal. 2003). The dissipa-
ately scaled for various redshifts. The dissipational model tionalsimulationsproduceanR - M relationthathasalower
e ⋆
FP displays a tilt relative to the virial plane (λ=0.8), with R normalizationthantheShenetal.(2003)relation,butsub-
e
a scaling (α=1.58, β =0.80) similar to the near-IR photo- sequent re-mergingand a cosmologically-representativedis-
metricFP(Pahreetal.1998b,α=1.53,β=0.79). Thedissi- tributionoforbitswilllikelydecreasethediscrepancybyin-
pationalmodelFPincludesremnantsproducedfromprogen- creasing R at a given stellar mass. Since these the dissipa-
e
itorsthatvarybyafactor2×ingasfraction(f =0.4,0.8), tionless remnants lie above the Shenetal. (2003) relation,
gas
includingeitherstrongly(q =1.0)orweakly(q =0.25) re-merging and higher angular momentum orbits will only
EOS EOS
pressurized ISM equations-of-state. For gas-rich systems furtherincreasethisdiscrepancywithrealellipticalgalaxies.
(f ≥0.4),thegasfractionandISMphysicsoftheprogeni- While a proper accounting for the distribution of formation
gas
torshaveonlyaslighteffectontheFP.Theredshift-dependent redshifts for elliptical galaxies would alter the R - M rela-
e ⋆
propertiesofprogenitorsystemshavelittleeffectonthesim- tionbyweightingtheremnantsunequally,themergersofdisk
ulated FP plane, even as the structuralpropertiesof the disk galaxieswheredissipationisimportantwillneverthelesspro-
10 Robertsonetal.
FIG. 7.— FundamentalPlane(FP)relationproducedbythemergingof FIG. 8.— Effective radius Re –stellar massM⋆ relations produced by
spheroidalgalaxymodels. UsingHernquist(1990)stellarspheroidsmodels themergingofspheroidalgalaxymodels. ThemergersofHernquist(1990)
withdarkmatterhalosasinitialconditions(bluetriangles),theremnantspro- stellarspheroidsinitialconditionswithdarkmatterhalos(bluetriangles),ini-
ducedbydissipationlessspheroidmergersappropriateforredshiftz=0,with tialized to the Shenetal. (2003)relation (dashed line), with nearly radial,
nearlyradial,parabolicorbits(blackcircles)andwide,ellipticalorbitswith parabolic orbits (black circles) and wide, elliptical orbits with circularity
circularityǫ=0.5(reddiamonds)arecalculated.Theremnantsofspheroidal ǫ=0.5 (red diamonds) are simulated. The remnants of spheroidal merg-
mergersproduceaFPrelation similartotheirprogenitor systems, roughly ersproduceaRe- M⋆relationslightlyshallowerthantheobservedrelation,
independentoftheorbitalenergyorangularmomentum.Alsoshownarese- roughlyindependentoftheorbitalenergyorangularmomentum.Alsoshown
lectspheroidalremnantsfromgas-richdiskgalaxymergersimulationsthat arespheroidal remnants fromgas-richdiskgalaxy mergersimulations that
includestarformation,supernovafeedback,andaprescriptionforfeedback includestarformation,supernovafeedback,andaprescriptionforfeedback
fromaccretingsupermassiveblackholes(orangesquares). Thediskgalaxy fromaccretingsupermassiveblackholes(orangesquares). Thediskgalaxy
mergerremnants occupy a FPrelation similar tothat observed in infrared mergerremnantsoccupyaRe- M⋆relationwiththesamescalingasobserved
observations(solidline). There-mergingofthesediskgalaxyremnantson inSDSS(dashedline).There-mergingofthesediskgalaxymergerremnants
nearlyradial, parabolic orbits (greencircles), furtherdemonstrates thatthe onnearlyradial,parabolicorbits(greencircles)producesasimilarRe- M⋆
mergingofspheroidalremnantsproducesaFPsimilartothatoccupiedbythe relationasthedissipationless mergingofspheroidalgalaxies. Forcompar-
progenitorsystems(e.g.Capelatoetal.1995;Dantasetal.2003;Nipotietal. ison,thebestleast-squaresfittotheRe- M⋆ relationdelineatedbythedis-
2003;Boylan-Kolchinetal.2005).Forcomparison,thebestleast-squaresfit sipationlessspheroidalmergerremnants(dashed-dottedline),theShenetal.
totheFPdelineatedbythedissipationlessdiskmergerremnants(dottedline) (2003)relation,andtherelationproducedbysimulationsusingthefullphys-
andspheroidalmergerremnants(dash-dottedline)isplotted. icalmodel(solidline)areplotted.
ducea R - M relationthat is steeper than thatproducedby
e ⋆
dissipationlessmerging.
4.3. Full-ModelDiskSimulations
Introducingthe effectsofblackholefeedbackthroughthe
full-model simulations produces a set of fundamental scal-
ing relations similar to those in dissipational mergers with-
out black holes. The fundamental plane produced by the
full-model simulations yields nearly the same FP scalings
(α=1.55,β=0.82)andtilt(λ=0.79)asthedissipationalsim-
ulations(see§4.2),andissimilartotheobservednear-IRscal-
ings(Pahreetal.1998b). Thesesimulatedremnantsexhibita
scatter abouttheir mean FP of ∆2 1/2 =0.062, comparable
to boththe observedscatter int(cid:10)he F(cid:11)Pat opticalwavelengths
(e.g.Bernardietal.2003a)andthatproducedbythedissipa-
tionalsimulations.Blackholefeedbackcausesthefull-model
remnants to be slightly larger than the dissipational model
remnantsasfeedback-drivenwindsremovegaseousmaterial
fromtheinnermostregionsoftheremnantsthatwouldother-
wisecontributetothecentralstellarcontent. Figure5shows FIG. 9.— BlackholemassMBH –stellarvelocitydispersionrelationσ
thefull-modelremnantsproducedbymergersappropriatefor relationmeasurementsforgalaxiesproducedbythere-mergingofdiskgalaxy
variousredshiftsonnearlyradial,parabolicorbits.Theresul- mergerremnants that obey the local MBH- σ relation (e.g., Tremaineetal.
2002, solidline). Whilethe number ofre-mergers examined is limited, a
tantFPrelationisroughlyindependentoftheredshiftscalings
singlegeneration ofdissipationless mergers after theinitial formative gas-
oftheprogenitorsystemsandthelocationofremnantswithin richmergers that generate the MBH- σ relation is notexpected tostrongly
theFPisfairlyinsensitivetotheorientationsofthedisks(pur- altertheobservedMBH- σrelationbutmaybeasourceofscatter.
plepentagons,seeTable2)andorbitalconfiguration(orange