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Prole Shape Evolution

For along period,the surfae brightness prolesof elliptialgalaxies have been tted

bythedeVauouleurs

r

1/4

prole(deVauouleurs,1948). Butmorereentworkshows,

that the urvature of the light proles seems to be very important as it orrelates to

otherobserved propertiesofelliptialgalaxies,suhasthe eetiveradius

r

e

,the total luminosityandthe stellarmass(Caonetal.,1993;Nipotietal.,2003;Naab&Trujillo,

2006;Kormendyetal.,2009). Therefore weuse the Sersi

r

1/n

(Sersi,1968)proleto

tthe surfae brightness proles of our simulations. The formulaan be writtenas

I(r) =

I

e

·10

−bn((r/re)

1/n−1)

,

(7.1)

wherethe three free parametersare halfof the total luminosity

I

e

, the eetiveradius

r

e

and the so alled Sersi index

n

, whih gives the shape of the prole. The fator

b

n

, whih only depends on

n

, is hosen suh that the eetive radius

r

e

enloses half of the total luminosity. For the expeted range of Sersi indies, this fator an be

approximated by the relation

b

n

= 0.868n−0.142

(Caon et al., 1993). In the ase of

n= 4

, Eq. 7.1 redues to the de Vauouleurs

r

1/4

law.

In ordertoget abetteromparisontoreent observationsofelliptialgalaxies(e.g.

Trujillo et al. 2004; Kormendy et al. 2009) we onvert the projeted surfae densities

ofsetion7.4toaV-bandsurfae brightness. Assumingaonstantmass-to-lightratio,

the radial luminosity prole an be writtenas

L(r) = Σ(r)·10

−MV/2.5

,

(7.2)

where

M

V

= 7.1973

is the absolute magnitude of a star in the V-band at a distane of

10pc

, a stellar age of

10

10

yr and lose to solar metalliity

Z

= 0.02

(see Bruzual & Charlot 2003).

Σ(r)

is the projeted surfae density of the previous setion and the V-band magnitude an be alulated,

µ

V

(r) =−2.5·logL(r) + 21.5721,

(7.3) where

21.5721

is just a fator to onvert surfae density to mag/arse

2

. As we want

tot

µ

V

with a Sersi funtionwe have totakethe logarithmof Eq. 7.1 whih yields

µ(r) =−2.5 logI(r) =µ

e

+c

n

[(r/r

e

)

1/n

−1],

(7.4) with

c

n

= 2.5·b

n

and

µ

e

=−2.5 logI

e

.

Figure 7.5: Top panels: Surfae brightness proles

µ

V

(r)

of the two-omponent major (1:1) and minor merger (1:5) generations plotted against the radius. The blak symbols

depit the initial Hernquist prole, the blue symbols the prole of the rst remnant and

the redsymbolsthe nal proles. Theoverplotteddashed linesin theorresponding olors

show the best tting Sersi funtion, whih yields an inreasing Sersi index

n

with eah

subsequent merger generation. The tting range starts at

0.02·r

e

(

r

e

is the eetive radius of Fig. 7.2) and ends at either

10·r

e

or a limiting surfae brightness of

m

V

=

27mag/arcsec

2

, whih results in residuals

∆µ <

0.2mag/arcsec

2

(small panels below).

As the proles show an artiial ore like struture (given by the initial onditions), the

residuals inrease for very small radii and the tted eetive radii

r

e,f it

(thin lines at the bottom)aresmallerthan

r

e

ofFig. 7.1(aordingarrows). Bottompanels: Thesameasin the fourtop panels,forthe1:10 minor mergerwith thediuse satellite(Sat 1:10,left) and

for the full set of 10 generations with a ompat satellite (Sat 1:10) and head-on orbits

(right). Inthe rst ase, the nal surfaebrightness proleshows a prominent kink,whih

results in an unrealisti high Sersi index

n

∼

20

for the best t. The minor mergers of ompat satellites, show more reasonable results, but the Sersi index saturates rapidly at

Figure 7.5 shows the Sersi ts to the surfae brightness proles of the equal-

mass mergers(1:1, top left),the 1:5 (top right) and 1:10 minor mergers with angular

momentum (bottom left) and the head-on senario of Fig. 7.4 (bottom right). We

hose a ttingrange, sothat weget a goodt tothe mainparts of the prole(

>95%

along the major axis). If we start at

0.02·r

e

and either go out to more than

10·r

e

or to a limiting surfae brightness of

m

V

= 27

mag/arse

−2

, whih is the limit of

reentobservations(Trujilloetal.,2004;Kormendyetal.,2009),the residualsarevery

small(

∆µ <0.2

mag/arse

2

), exeptinthe innermostregions,where theproleshave

a ore like struture. Looking at the proles of the initial Hernquist spheres (blak

irles in all panels) we an see that we get a shape parameter of

n

= 3.9

, whih almost resembles the de Vauouleurs prole (

n

= 4

). As expeted, the Hernquist sphere isavery goodapproximationof the de Vauouleurs

r

1/4

lawoveralarge radial

range and has a ore prole in the innermost region (see also Naab & Trujillo 2006).

Due to the ore, the tted Sersi prole overestimates the entral surfae brightness

whih leads to a tted eetive radius

r

e,f it

(narrow vertial lines at the bottom of eahsurfaebrightnesspanel),whihisslightlysmalleromparedtothe'real'eetive

radius

r

e

(orrespondingarrows)ofFig. 7.1. Thisamountof'artiialextra-light'from the tted prole aounts for the disrepany between these two radii, for all shown

mergersenarios.

Intheaseof1:1mergersoftwo-omponentmodels(topleftpanel,Fig7.5),wean

see that theprole shapebarely hanges forthe remnants. Therefore, the Sersi index

shows only asmallinrease (see alsoblak solid line, Fig. 7.6)ompared tothe added

stellar mass

M

∗

,whihis not enoughtoexplain the veryhigh numbers, observers nd for large ore elliptial galaxies (

n∼

10

, see Caon etal. 1993; Kormendy et al.2009). This is a onsequene of the violent mergingproess, where the material assembles at

all radii and the surfae brightness gets shifted nearly parallel to higher values, but

does not signiantlyhangethe slopeof the prole.

This piture hanges dramatially if we go to higher initial mass ratios, where

the merging proess beomes dierent. In minor mergers violent relaxation does not

aet the host galaxy (see Chapter 6), just the in-falling satellites. The latter one

instantaneous feels the deep potential well of the host at losest approah and suers

strongly from rapid potential utuations. Furthermore dynamial frition and tidal

stripping get more prominent, as the satellites are more loosely bound than the host

galaxy and the tidal fores are strong enough to strip a big amount of the satellite's

material(see also Setion5.2).

Therefore, inthe aseof minormergers, mostofthe areted materialassembles at

larger radii of the galaxy and the entral regions are hardly aeted (see also setion

7.4). Regarding the surfae brightnesses of the 1:5 minor mergers (top right panel,

Fig. 7.5), this implies, that the urvature, measured by the Sersi index

n

, hanges rapidly with eah further generation (see also red solid line, Fig. 7.6). Already after

the rst generation with a mass inrease of a fator of

∼

1.2

we get a Sersi index of

n >7

and the nalremnanthas aslopeof

n= 9.5

, whihperfetlyliesinthe rangeof observations (Caonetal.,1993;Kormendyetal.,2009). Theorrespondingbulgeonly

Figure 7.6: Evolutionofthe Sersi indiesforallmerger generationsofFig. 7.3, exept

the 1:10 senario, whih yields unrealisti ts (see bottom left, Fig. 7.5). As equal-mass

mergers do not signiantly hange the slopes of the surfae brightness the Sersi index

after one generation is

n

∼5−6

(blak lines). For the bulge+halo minor mergers with a mass ratio of 1:5 (red solid line), the slope inreases rapidly for the rst two generations

before it onverges to a nal value of

n

∼

9.5

. The 1:10 head-on minor mergers with a ompat satellite (green solid line) show the same trend, i.e. an initial fast inrease of

n

for the rst generations before it onverges to a value of

n

∼7−8

. For ompletness, the dashed lines show the bulge only simulations, where the urvature for the minor mergers

senario(reddashedline,Fig. 7.6)showsthesametrend,andyieldsanalSersiindex

of

n= 7.8

,buttheoverallevolutionismuhmoreeientwithtwo-omponentmodels. This again indiates, that the mass ratio and darkmatter halo are very important,as

they inrease the eet of dynamial frition and tidal stripping in a way, that the

areted stellar mass assembles at the 'right' regions of the host galaxy, and leads to

the observed prole shapesof ore elliptials.

Ifwefurtherinreasetheinitialmassratioto1:10(bottompanels,Fig. 7.5),wean

seethatthe Sersiindexgetsunrealistilarge

(n >20)

forthesenariowiththe diuse satellite(leftpanel). Asthe satelliteisonlyweaklybound,itloosesallitsmassatvery

large radii, develops a kink at a radius of

r

≈

4

kp and the best tting Sersi prole yields a very high urvature. This piture improves, if we take more bound satellites

(Sat 1:10). Then the mass assembles more smoothly outside a radius of

r

≈

1.5

kp but still produes an extended outer envelope (right panel, Fig. 7.5). Although the

evolution of the proles look very reasonable, the Sersi index onverges ata value of

n≈7−8

(solid green line, Fig. 7.6), whih an be explainedwith the very high mass ratio of the nal generations (the lastgeneration has a mass ratio of 1:19).

Altogether we an say that a massive dark matter halo enhanes the eet of dy-

namial frition and tidal stripping. Considering satellites, whih are not too weakly

bound,itisthe maindrivertoaretetheluminousmatteratthe 'right'regions. Then

we alsoget reasonable resultsfor the evolution of the Sersi index of

n

∼8−10

(Fig.

7.6). In the ase of equal-mass mergers the eet of violent relaxation and mixing is

moredominantand doesnothange theslopeof thesurfae brightness prolesandwe

only get amildinrease afterone generation

n

∼5−6

(Fig. 7.6).

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