Phase Mixing

Figure5.4: Intheleftpanel,weanseetheinitialphasepointsat

p θ i = 0

^,^where

f = F

As the system evolvesthe phasepointsshear and thephase spaevolumegetsthinner and

oupies a larger regions. As more and more 'air' is mixed into the oarse-grained DF, it

dereases with time. The right panel depits a very late stage of the system, where the

ne-grainedDF

f

^onsists^ofinnitesimalthinlines. Atthisstage

F << f

^(see^also^Binney

& Tremaine (2008)).

Apartfromtheseexamples,thereamany otherauthors,who triedtogeta

desrip-tion of the nal state of a ollisionless relaxing system by using entropy arguments

(Nakamura, 2000; Stiavelli & Bertin, 1987). But reently Arad& Lynden-Bell (2005)

argued,thatallofthemhavelimitationsandtheyadditionallyshowthatthe

statistial-mehanialtheoriesofviolentrelaxationarenon-transitive. Thisnontransitivityyields

twodierentresults,ifasystemeitherundergoesoneviolentrelaxationproessatone

or two proesses of omparable magnitude. Finally, Arad & Lynden-Bell (2005)

on-lude, that the already mentioned inompleteness of violent relaxation (see 5.3.1) is

the mostimportantreasonfor theseshortomings. One way tooveromethe problems

is tond a useful evolution equationfor the oarse-grained DF.

Figure5.4 shows ashematirealization of this senario. The further evolution of

the system an be desribed by the ollisionless Boltzmann equation, whih implies

that the ne-grained DF

f

^stays ^onstant. ^Therefore ^the ^density ^of ^an innitesimal volume arounda phase pointsdoes not hange.

N

^pendulums,^eah^of^the ^same^length

L

^,^though^all^of^them ^have ^the^same^dynamial

properties. At the beginning, all of them have are swung bak by an angle

θ i=0,...,N

whihalllieinaverysmallinterval

∆θ << θ i

^. ^Now^we^dene^the^ne-grained^DF

f

^for

this system, whihis initiallythe same asthe oarse-grained DF

F

^. ^If^the ^pendulums

arereleasedtheyallhaveadierentangularveloity

θ ˙ i

^with^momenta

p θ i = l θ ˙ i

^,^i.e. ^the

pendulums with higherinitial

θ i

^have ^lower ^momenta ^ompared ^to^those ^with ^smaller

initialangles.

A marosopiobserver just an look ata ell of nite size and then he alulates

the oarse-grained distribution inthis ell. Initially

f = F

^,^but ^as^the ^system ^evolves,

the phase-spae volume winds up in to innitesimal thin laments (see Figs. 5.4).

Then the observer measures a muh smaller phase-spae density, beause now, a lot

of phase-spae around the measured phase point is not oupied but empty. Finally

the measured oarse-grained DF dereased a lot, ompared to the initialvalue. This

derease of

F

^, ^as^the ^pendulums ^get ^out ^of ^phase^is ^alled ^phase ^mixing.

CHAPTER 6

RELAXATION AND STRIPPING: THE

EVOLUTION OF SIZES AND DARK

MATTER FRACTIONS IN MAJOR

AND MINOR MERGERS OF

ELLIPTICAL GALAXIES

In this hapter we investigate ollisionless major and minor mergers of

spheroidalgalaxiesintheontextofreentobservationalinsightsonthe

stru-ture of ompat massive early-type galaxies at high redshift and their rapid

size evolution onosmologialtimesales. The simulationsare performedasa

seriesof mergerswith mass-ratiosof1:1and 1:10formodels representing pure

bulgesaswellas bulgesembeddedindarkmatterhalos. Formajor andminor

mergers, respetively, we identify and analyse two dierent proesses, violent

ralaxation and stripping, leading to size evolution and a hange of the dark

matterfration. Violent relaxation- whihis the dominant proess for major

mergers but not important for minor mergers - satters relatively more dark

matterpartilesthanbulge partilestoradii

r < r e

^. ^Stripping ⁱⁿ^minor

merg-ers assembles satellite bulge partilesat large radii inhalo dominatedregions

of the massive host. This eet strongly inreases the size of the bulge into

regions with higher dark matter frations. For a mass inrease of a fator of

two, strippingin minor mergersinreases the dark matterfration within the

eetive radius by 75 per ent whereas relaxation in one equal-mass merger

only leads to an inrease of 25 perent. Compared to simple one-omponent

virial estimates, the size evolution in minor mergers of bulges embedded in

massivedark matter halos are very eient. Ifsuh a two-omponent system

grows by minor mergersonly its size growth,

r ∝ M ^α

^, ^will^exeed ^the ^simple

theoretiallimitof

α = 2

^. ^Our ^results ^indiate^that ^minor^mergers^of ^galaxies

embedded inmassive darkmatter halos provideaninteresting mehanismfor

a rapid size growth and the formation of massive elliptialsystems with high

6.1 Introdution

Reent observations have revealed a populationof veryompat, massive (

≈ 10 ¹¹ M ⊙

⁾

and quiesent galaxies at z

∼

² ^with ^sizes ^of ^about

≈ 1kpc

^(Daddi ^et^al., ^2005; ^T^rujillo

et al., 2006; Longhetti et al., 2007; Toft et al., 2007; Zirm et al., 2007; Trujillo et al.,

2007;Zirm etal.,2007; Buitrago etal.,2008; vanDokkum et al., 2008;Cimattiet al.,

2008; Franxet al., 2008; Sarao et al., 2009; Damjanov et al., 2009; Bezanson et al.,

2009). Elliptial galaxies of a similar mass today are larger by a fator of 4 - 5 (van

der Wel et al.,2008) with atleast an order of magnitude lowereetive densities and

signiantly lower veloity dispersions than their high-redshift ounterparts (van der

Wel etal., 2005, 2008; Cappellariet al., 2009; Cenarro & Trujillo,2009;vanDokkum

et al., 2009; van de Sande et al., 2011). The measured small eetive radii are most

likelynotausedby observationallimitations,althoughthe lowdensity materialinthe

outer parts of distantgalaxies is diultto detet (Hopkins etal. 2009a). Their

lus-teringproperties,numberdensitiesand orepropertiesindiatethatthey are probably

the progenitors of the most massiveelliptialsand Brightest ClusterGalaxies (BCGs)

today (Hopkins etal.,2009a;Bezanson etal., 2009).

Possible formation senarios for suh ompat massive galaxies at redshifts

z ≈ 2 − 3

^inlude ^gas-rih ^major ^disk^mergers ^(W^uyts ^et^al.,^2010;^Bournaud ^et^al.,^2011),

aretion of satellites and gas, giant old gas ows diretly feeding the entral galaxy

inaosmologialsetting(Kere²etal.,2005;Naabetal.,2007,2009;Joungetal.,2009;

Dekeletal.,2009;Kere²etal.,2009;Oser etal.,2010)oraombinationofallofthese.

To explain the subsequent rapid size evolution dierent senarios have been proposed

(Fan et al., 2008; Hopkins et al., 2010; Fan et al., 2010). Frequent dissipationless

minor and major mergers, whih are also expeted in aosmologial ontext, seem to

be the most promising (Khohfar & Silk, 2006; De Luia et al., 2006; Guo & White,

2008; Hopkins et al., 2010). Minor mergers, in partiular, an redue the eetive

stellar densities, mildly redue the veloity dispersions, and rapidly inrease the sizes,

buildingupextendedstellarenvelopes(Naabetal.,2009;Bezansonetal.,2009;Hopkins

et al., 2010; Oser etal., 2010,see however Nipoti etal.,2009a). Dissipationlessmajor

mergerswillontributetomassgrowth,however,theirimpatontheevolutionofstellar

densities, veloity dispersions and sizes is weaker (Boylan-Kolhin et al.,2005; Nipoti

et al., 2009a). Observations and theoretial work also provide evidene that

early-typegalaxiesundergo onaverageonlyone majormergersineredshift

∼

²^(Bell^et^al.,

2006b;Khohfar&Silk,2006;Belletal.,2006a;Geneletal.,2008)whihwouldnotbe

suient toexplain theobserved evolution(Bezanson etal.,2009). Inaddition,major

mergers are highly stohasti and some galaxies should have experiened no major

mergeratall,and would thereforestillbeompat today. However, suh apopulation

ofgalaxieshasnotbeenfoundyet(Trujilloetal.,2009;Tayloretal.,2010). Simulations

ina fullyosmologial ontext support the importaneof numerous minormergersfor

the assembly of massive galaxies. They might initiallyform at higher redshift during

an early phase of in-situ star formation in the galaxy followed by a seond phase

dominatedbystellar aretion(dominatedbyminormerging)ontothe galaxy,driving

the size evolution (Naab et al., 2009; Oser et al., 2010). Diret observational and

irumstantial evidene has been reently presented in support of the minor merger

senario(van Dokkum etal., 2010;Trujilloetal., 2011).

Using the virialtheorem, Naab etal. (2009) and Bezanson et al.(2009) presented

a very simple way to estimate how sizes, densities and veloity dispersions of

one-omponent ollisionlesssystems evolve during mergers with dierent mass ratios.

A-ording to this simplied model assuming one-omponent systems on paraboli

or-bits, the aretion of looselybound material(minor mergers) results in asigniantly

stronger size inrease than predited for major mergers (Naabet al., 2009). With the

sameapproahBezansonetal.(2009)foundthateightsuessivemergersofmassratio

1:10 an lead to a size inrease of

∼

^5, ^whih ^orresponds ^to ^the ^observed ^dierene

between old ompat galaxies and today`s massive elliptials. Of ourse, this is only

valid for global system properties like the gravitational radii and total mean square

speeds. The simple modelisnot inludingviolentrelaxation eets like mass loss,

o-urring during the enounter or non-homology eets whih might hange observable

quantities.

Early papers on the interations of spheroidal galaxies already disussed many of

the abovementionedeetsusing N-bodysimulationsofone-omponentspherial

sys-tems. White (1978, 1979), who arried out one of the rst simulations of this kind,

already found that relaxation eets are very eient in equal-mass enounters and

ompletelyhangetheinternalstrutureofthenalremnants. Firstofalltheyontrat

in the entral regionsand build up diuse envelopes of stars (see alsoMiller& Smith

1980; Villumsen 1983; Farouki et al. 1983), whih leads to a break of homology. F

ur-thermore, equal-mass mergersderease populationgradients due tothe redistribution

of partiles in strong mixing proesses (White, 1980; Villumsen, 1983), whih breaks

down in unequal-mass mergers, whih even an enhane metalliity or olor gradient

(Villumsen, 1983). Farouki et al. (1983) also showed, that their multiple equal-mass

mergersnielyreovertheFaber-Jaksonrelation(Faber&Jakson1976,seealso

Se-tion 2.2) and that the veloity dispersion gets radially biased in the outer regions of

the newly developed extended envelope. However, they all just used one-omponent

models and therefore ould not investigate the inuene of the most massive part of

a real galaxy, whih is its dark matter halo. Naab & Trujillo (2006) and Hopkins

et al. (2009b) already showed, that more realisti galaxy models, where the bulge is

embedded ina darkmatter halo, an hange the size inrease.

Althoughdissipationlessminormergersingeneralare abletoinrease sizesand

de-rease veloity dispersions, it is not lear if this senario works quantitatively. Nipoti

et al. (2003), who are among the rst using spherial two-omponent models, argued

that dry major and minor mergers alone annot be the main mehanism for the

evo-lution of elliptial galaxies, beause their simulated merger remnants did not follow

the Faber-Jakson(Faber&Jakson1976)andKormendy relations(Kormendy1977),

although they stayed on the fundamentalplane. Nipotiet al. (2009a) found that dry

major and minor mergers an bring ompat early-type galaxies loser to the

funda-mental plane but the size inrease was too weak for the assumed merger hierarhies.

Furthermore, dissipationless major merging introdues a strong satter in the saling

relations,whihareobservationallyverytight. Finally,Nipotietal.(2009b)laimthat

early-type galaxies assemble only 50

%

^of ^their ^mass ^via ^dry ^merging ^from ^z

∼

² ^until

now and the expeted size growth of a fator of

∼

⁵ ^is ^hardly ^reprodued. ^However,

espeially in their minor merger sequenes, they use very ompat satellites, whih

mightunderpredit the eetive size growth.

Therearetwomainquestionsweaddressinthishapter. First,usinghighlyresolved

multiple equal-mass mergers, we investigate the impat of the massive dark matter

halo on the dynamis of the nal systems. Does it aet the entral regions and

an suh mergers really hange the entral dark matter fration, or is the inrease

just an artefat of the inreasing radius (Nipoti et al., 2009b). Seond, we revisit,

whether dissipationless minor mergers are really too weak to fully aount for the

observed evolution of ompat early-type galaxies(Nipotietal.,2003,2009a) and the

implied size growth. Usingmore realisti two-omponent models, we are able to draw

onlusions about the hange of internal struture for the galaxies in both merging

senarios.

This hapter is organized as follows. First, in setion 6.2 we give an overview of

our initialgalaxy setup and the employed numerialmethods, beforewe highlightthe

virial preditions in setion 6.3. In Setion 6.4 and 6.5we show the results for major

and minor mergers, respetively. Finally, we summarize and disuss our ndings in

setion 6.6.

Im Dokument Dissipationless merging and the evolution of early-type galaxies (Seite 57-63)

p θ i = 0

f = F

f

F << f

f

N

L

θ i=0,...,N

∆θ << θ i

f

F

θ ˙ i

p θ i = l θ ˙ i

θ i

f = F

F

r < r e

r ∝ M α

α = 2

≈ 10 11 M ⊙

∼

≈ 1kpc

z ≈ 2 − 3

∼

∼

%

∼

∼

r ∝ M ^α

≈ 10 ¹¹ M ⊙