Sinking Satellites and Tilting Disk Galaxies

Siclin Huang

.EL thesis submitted in

conformity with the requirements

for thc Degrce of Doctor of Philosoph y

Graduate Department of .-!.~tronomy

University of Toronto

@ Siqin Huang 1997

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Sinking Satellit es and Tilting Disk Galaxies

Siclin Huang Doctor of Philosophy. 1997

Department of Astronomi;. University of Toronto

Abstract

[ perforni fully self-consistent disk+halo+satellite Y-body simulations to investi-

gate the dynamical interaction between a disk galaxy and an infalling satellite. In

particitlar. 1 study the following three differerit dynamical responses of the disk to the

infalling satellite: tilting, warping. and t hickening. as well as the dynamical effects

of the parent gaiaxy on the infalling satellite: orbital decay and ticlal disroption.

The mode1 in this thesis is characterized with two cosmologically significant improve-

ments. First. the satellite starts at a distance more than three times of the radius

of the optical disk. This ensures a realistic interaction among the satellite. the disk.

and the halo in the course of the satellite infall. Secondly. evolution of the struc-

ture and velocity ellipsoid of the disk diie to interna1 heating is allowed. 1 study the

cornnionly arising case of satellites having density profiles comparable to that of the

parent galaxy in contrast to that of compact satellites considered in previous work.

1 find that a disk is mainly tilted rather than heated by infalling satellites. Satel-

lites of 10%. 20%. and 30% of the disk mass tilt the disk by angles of (2.9 f 0.3)'.

(6.3 k 0.1)'. and (10.6 f 0.2)'. respectively. However. only 3.4'3,. 6.9%. and 11.1% of

the orbital angiilar momentum is transferrecl to the parent galasy. The kinetic energy

associated with the vertical motion in the initial coordinate frarne of the disk is in-

creased by (6f 3)%, ( - 6 f 3)%. and (51 f 5)%. respectively. whereas the corresponding

thermal energy associated wit h the vertical random motion in the tilted coorclinate

frame is only increased by (4 f 3)%, (6 zk 2)%, and (10 f Z)%, respectively. 1 find t hat

satellites are mainly accreted ont0 the parent halo. Satellites having up to 20% of the

disk mass produce no observable thickening. whereas a satellite with 30% of the disk

mass produces lit tlc observable t hickening inside the half-mass radius of t he disk but

significant thickening beyond this radius. Hence, a high cosmological accretion rate

and thin disks can coexist if most infalling satellites have density profiles comparable

to that of the parent galaxy.

Acknowledgement s

It is a pleasure to thank -- supervisor. Ray C'arlherg. for introdiicing me to this

iriteresting project ancl proïiding insightful aclvice. 1 greatly appreîiate his patience.

support. anci tirne-

1 rvould like to thank my Ph.D. cornmittee rnembers. Sinion Lilly and Scott

Tremaine. for their encouragement and valuable suggestions. as well as Shogo In-

agaki For his comrnents and advice during t,he coiirse of this research.

I am ver - happy that 1 spent the last few pears in siich a friendly environrnent.

I received help and encouragement froni many people in the department. Thank

y011 all. Especially. 1 would like to thank Sang-Hee Kim for heiping me adjust to a

new environment when I arrived in Canada. Saridra Scott for her help and constant

encouragement. Pierre Grave1 and $larcin Sawicki for interest ing cliscussions and

suggestions. and Teresa Iiroeker for solving cornputer problems. Sandra also reacl

the whole thesis, and made many corrections and suggestions. I really appreciate her

tirne and effort.

1 acknowledge the Department of Astronomy and Lïniversity of Toronto for finan-

cial support.

1 thank my parents. Shuzhen and Lide. for allowing me to pursue a career accorcling

to my own choice, my sisters. Sifeng and Sifen. and brot her. Sijun. for t heir love and

care. Special thanks to Sifen for always providing support. and Sijun for beinp a

mode1 and inspiration. 1 would like to espress niy gratitude to my parents-in-law.

Maria and Fortunato. Without their help and generosity. 1 would not have been able

to complete my thesis in a timely manner. I thank rny little star. Cristella, for making

rny Iife full of joy and happiness. Finally. deepest gratitude to my husband, Giovanni.

for his suggestions and advice concerning this work and Iiis continued lriendship and

care.

Dedication

To Gioannni jor his ~ n c . o u r n g e r n e n t . s u p p o r t . and lorr.

Contents

Abst ract

... Acknowiedgements 111

Dedicat ion iv

1 Introduction 1

2 Numerical Methods 10

'3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

'3.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 N-body C'ode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 siirnmary . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . 19

3 The Evolution of an Isolated Galaxy 20

3.1 Introditction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . '20

3.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 I\;ineniatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . '21

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Satellite-Bar Interaction 33

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Infalling satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1

4.3 SIowdown of weak bars . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Disk Tilting and Heating 40

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Angular Moment uni Transfer . . . . . . . . . . . . . . . . . . . . . . -LU

.5 . 3 EnergyTransfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Summary 5 (1

6 Warping 52

6.1 Introdirction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.2 The Ring hloclel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.3 The Particle Mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7 Measurement of Disk Heating 60 . r . 1 Introdiiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 . i . 2 L:ert ical iieloci ty Dispersion . . . . . . . . . . . . . . . . . . . . . . . 62 . r . .3 Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8 Orbital Decay and Tidal Stripping of Satellites 68

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.2 Theoretical Esti~nates . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . I l

.. 8.4 Satellite Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . r r

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9 Orbital Decay Rate of Satellites and Galactic Accretion 81

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

9.2 SatelliteOrbitalDecay . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9.3 Galactic Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10 Discussion and Conciusion 92

10.1 Density of Satellites ancl Cialaxy Accretion Rate . . . . . . . . . . . . $12

10.2 Disk Tilting . Holmberg Effect . ancl Rotation of Stellar Halo . . . . . . 9:)

10.3 hlerger Rate and R = L CD1.I C'osmology . . . . . . . . . . . . . . . . 96

10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

1 0 5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A The Scale Length of the Galaxy 103

-1- 1 htroduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3

A.2 The scale length of the Cialaxy . . . . . . . . . . . . . . . . . . . . . . 104

X.3 Sumrnary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

References 110

List of Tables

. . . . . . . . . . . . . 3 . 1 The relation among Q . T, and rn,.~./n~pa.t.ci. 31

. . . . . . . . . . . . 9 . i Dynamical Friction Time and Galaxy . iccretion 86

... V l l l

List of Figures

Initial conditions imposed on the disk and the satellite . . . . . . . .

Surface clensity of Mode1 O . . . . . . . . . . . . . . . . . . . . . . . .

Histograms of particle distribution in vertical direction of 10 ring . .

Scale height of the asymptot ic exponent ial distribution versus radius

andtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Distributions of velocity dispersions. OR. a,: and a;. ancl of the Toomre

parameter. Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Evolution of the velocity dispersions . . . . . . . . . . . . . . . . . . .

Evolution of t h e amplitude of rn = 2 bar mode . . . . . . . . . . . . .

Evolution of an isolated disk and of a disk with a 30% clisk-niass infall

Aiigular rotation speed of disk and satellite as a function of radius and

t h e angular speed of bar for Mode1 3 . . . . . . . . . . . . . . . . . .

C'omparison amongst an isolated disk and a disturhed clisk vierved in

t he initial and t ilted coordinate frames . . . . . . . . . . . . . . . . .

Evolution of the disk. satellite. and total angular moment.um directions

.\ngular mornenta of the satellite. the disk. and the halo versus t ime .

Iiinetic energy associated with the vertical motion in the initial coor-

dinate frame k= and in the tilted coordinate frame . . . . . . . . . . .

Vertical velocity dispersion as a function of radius and time . . . . . .

fi . L Evolution of t lie tlisk and satell i te particles of LIodel 2 as i.iewcl in

t he . r z plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . 6.2 10 rings of Sfodel 3 projected on t h e r z plane

. . . . . . 6.3 Evoltition of t h e two Eulerian angles of Ring 6 and Ring 9

6.4 Evolution of t he angiilar momentum directions and t he traces of the

. . . . . . . . . . . . . . . . . positions for four typical disk particles

. r - 1 Densities of disk ancl halo versus radius . . . . . . . . . . . . . . . . .

7.2 Particles located on the original ring and t he tilted ring . . . . . . .

7.3 Evolution of t h e dispersions of 19 of Mode1 2 and Mode1 3 . . . . . . .

7 . 4 Evolution of t he thickness of di& . . . . . . . . . . . . . . . . . . . .

8.1 Orbital decay and mass stripping of satellites in Sfodel 4 . !dodel 3 . hIodet 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2 Mass inside t he half-mass radius of t h e sateilite versils t he distance

between t h e satellite and t he parent galaxy for Mode1 1 . SIoclel 2 . and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hiode1 3

8.3 Orbital decay of t he direct and retrograde orbital satellites as a function

of t ime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 Satellite particles of .\ Iode1 3 projected on bot h the .cg a n d r z planes 78

8.5 Distribution of t h e satellite particles of Mode1 3 in phase space ( I;,, . r ) 19

9.1 Sa te l l i t eorb i ta ldecay . . . . . . . . . . . . . . . . . . . . . . . . . . . $4

10.1 Evolution of t h e disk and satellite particles of Slodel 3 a n d Mode1 O as

. . . . . . . . . . . . . . . . . . . . . . . . . . viewecl in the r z plane 94

A.1 Observeci z-profile of 'iCX'424-L . . . . . . . . . . . . . . . . . . . . . 106

A.% Function f ( z / h z . R l h ) of Equation A.4 versus Rlh and b . . . . . . . 1 ~ 7 '

Chapter 1

Introduction

Galactic mergers. which mere once t hought to be interest ing but rare phenomeria.

are now considerecl to be one of the dominant processes governing the format ion ancl

evolution of galaxies. In any hierarchical cosmological model. merging is inevitable

because small mass pert tirbat ions coliapse hefore large ones and t h e evolu tion pro-

ceecls as a cascade of mergers froni smali to large scales. Therefore. most galaxies have

had major mergers during their formation and a large fraction of galaxies have hacl

minor mergers subsequently. In the colcl tlark matter (C'DM) model. most galaxies

accreted at least 10% of their niass over past 5 billion years (Bahcall S: Tremaine

1988. Kauffmann. White k C;iiiclercloni 1993. Lace- k Cole 1993). By ernpl-irig

the t~vo-point correlat ion function (Ci rot h k Peebles 1977) for the galaxy clist ributioii

and Schechter's luminosity function (Schechter 1976) for t h e galarx mass distribution.

with the assurnption that the mass-to-light ratio for galaxies is a constant. Tremaine

(1980) estimated the total mass accreted by a typical galasy. Tremaine assumed

that a spiral galaxy is surrounded tiy an extended isothermal halo, and the ciecay of

the satellite orbit is caused by clynamical friction arising from the halo. If a typical

galaxy is defined to be one having the luminosity L = 6.8 x 10'h-*Lo in Schechter's

luminosity function, where h = Ho/(lOOkm/s/Mpc)l then

tvhere rr represents the one-dimensional l-~locitj* dispersion of tlie halo. / is t he age

of the gnlasj-. and .Z denotes the ratio of the size of t h e parent galaxy to ttiat of a

t ~ * ~ i c a l sa tel lit,^. By assuming tlie reasoriahle valiies a = L4Okrnfs. .\l/ L = 6.11. / L . . &t = 1. h = 1. and in ;\ = 3. 1 ohtairi

ni , , , = 6 x l0?\1+. ( 1.2)

This is about IO%, of the disk niass of the 41ilk~- LYaj-. and it is also comparable to

the mass of the Large 4lagellanic C'loiicl ( liiinkel. Demers. k tru-in 1996).

Obsen-ational evidence for past and ongoing minor mergers between disks ancl

satellite galaxies is numerous. The recetit observat ional results from the Hiibble

Telescope show t hat some quasar act ivity is associated wit h minor rnergers ( Bahcall.

Iiirhakos. k Schneider 1995). A large fraction of disk galaxies are warped. which

may relate to recent accretion (Binney). and grancl design spiral structtire is often

associated wit h small close companions. such as 11.5 1 ( Toomre 198 1 ). The recent ly

discovered Sagittarius dwarf is at a distance of only 16 kpc from the center of the

Milky LVay It is a large. highly elongatecl dSph galaxy which appears to be in the

process of tidal disriiption (Ibata. C;ilriiore. k Irwin. 1995). Most galasies have a

niiniber of satellite companions (Zaritsky et al. 1993) at close orbits. for esample. t h e

Large and Small Magellanic Cloods (L5IC ' and S4IC) are a t distances of 31 SIpc and

63 Slpc respectively from the Sitn. These satellites. if located inside the dark matter

halo of their parent galasy. will inevitably spiral into the center of the parent galaxy

due to dynamical friction.

On the other hand, the stellar coniponent of a galactic disk is built up over a

period comparable to the Hubble time. The resolting stellar disk is so thin that the

vertical scale height is only about 10% of the radial scale length. Although the details

of vertical heat ing processes remai n cont roversial ( Wielen 1977. Lacey 1984. Car1 berg

1987. Binney k Lacey 1988), the vertical extent of the disk is probably consistent with

interna1 heating processes alone. The contribution from external sources of vertical

( ' H A PTER 1 . I.t'TR0 D C'C'Tl0.V

heat i r i~ riiiist thercfore rernain very srnall. Prrl-ioiis stiidies of the infall of a high

tlensit~. satellite orito a t hin clisk showvecl tliat t l i r clisk can 11e significarit lj- t tiickenecl

by a lu'% tlisk-niass satellite. Quinn k Goodnian I 1986. Iiereafter QG) first performed

siniiilat ioris of an infalling satellite ont0 a spiral e;ala?tj.. Their S- body simiilat ions

were niainlu focused on the orbital d e c - of rigicl body satellites. but the- noted

strong rlisk heating. Furt her ';-hoci' simulations by Quinn. Herncl~iist. k Fiillagar

( I W 3 . hereafter QHF) showecl that a thin clisk is thickenecl hy a factor of two at the

solar rieighhorhood hy a 10% clisk-mass infalling satellite. and they pointetl out that

the riiinor rnerser ma\. be the mechanisni for the formation of the thick disk of the

Galas'.. \+ru recent ly. L\-alker. Mihos. k Hernquist ( 1996. hereafter LVSIH ) performed

liigh cliiality. h i ly self-consistent simulations aricl presentecl accurate results for the

clisk heating and sinking time of t he satellite. They found that a 10% disk-mass

infalling satellite thickens a disk at the solar circle about 60%. and the sink tirne of

t h e satellite is - 1 Gyr. T6th S: Ostriker's ( 1992. hereafter TO) analytical calculation

showed that the Slilky \Vay cannot have accreted more tlian 4% of its mass inside

ttie solar circle in the last 5 Gyr. The? concl~icled that the high merging rate clerived

from the Cl = 1 Lniverse is incorrect.

Howe\-er. al1 t hese st iidies have treatecl t h e satellites as ha\-ing extremely high

central densiti-. so the satellites are relativelj. immiine to ticlal disruption. Since a

large fraction of the satellite mass reaches the ceriter of the disk. t remendous heating

in tlie clisk is inevitable. For example. in TO and QHF's model. the density profiles of

1 satellites are tlescribed by the Jaffe rnodel. p n -. Such a functional form leads

to an estremely high density at the centes of the satellite as compared to the density

of the parent galaxy. Therefore. the core of the satellite could survive and reach the

center of the disk. The infall of such a high density satellite ni- well represent the

infall of a compact dwarf galaxy, such as the infall of Y33 onto 313 1. but certainly does

not represent the infall of satellites ont0 the Milky Way. Observations show t hat most

of the dwarf galaxies in our Local Group have low central densities. The exception

is M32. which is sometimes called an underluminous normal ellipt ical. or compact

ellipt iral. Iiornienclj- ( L08.3) clemonstratrd t. hat elliptical galasies fa11 into a two-

faiiiily riassification. as first proposecl I>!. \ \ Ï r th k Gallagher ( 1984): higli Iiirninosity

normal rllipticals have a surface brightness that increases with decreasing Iiiniinosity

ending at 1132: while the dwarf secluence is cliaracterized by a surface hright ness

t hat tlecreases wi t li clecreasing luminosi t!.. SIost of the cornpanions of t he Mlky \.\.-a!

C;alaq- a r e low luniinosity and non-nocleatecl clwarf elliptical galasies (or. as thq-

are t rad it ionally called. clrvarf spheroidal galasies) escept t he LMC*. an irregiilar. and

the ShIC'. a tlwarf irregiilar. T h e surface briglitness of al1 t hese clwarf ellipt icals is

well descsihccl h>- the King mode1 (King 1966). with low central concentration or hy

the exponential law which is often usecl t o tlescribe the surface hriglitness of ciwarf

irregulars. Flirt herrnore. t he density profiles. clerirecl from t h e rotation ciirves of

several clwarf spirals. are well fitted by an isothermal density function. p x ~ / ( r f + r 2 ) .

with large core radius. r , (Moore 1994). The nearly constant central clensit ies of t hese

clwarf spirals a r e ver. simiiar to the King rnoclel profile. which is employecl to tlescribe

hot h t he satellite and lia10 density profiles in t his t hesis.

Furt hermore. observations show t hat accret ion ont0 the outer halo aritl outer disk

are common hot the fact tha t t he inner disk remairis thin suggests tliat infalling

satellites a r e ticlally tlisriipted before t hey reach the inner disk. There is acciimiilating

observational eividence showing t hat the outer stellar halo may forni by t he accretion

of small. metal-poor fragments like gas-rich dwarf galaxies. as proposecl by Searle Sr

Zinn ( 1978). Thex found t hat t h e chernical and orbital properties of t lie oiiter globular

clusters a r e decoiipled. Preston, Beers k Shectman (1994) discovered a population

of hlue meter-poor (BMP) stars in the solar neighborhood. Their ages a re > 3 Gyr.

they have [Fe/H] < - 1. and the- a r e kinematically intermediate between the rapidly

rotating disk and the siowly rotating halo. Since Preston. Beers 8c Shectman noted

that nearby satellites. like the Carina dSph galaxy, have significant intermediate-age.

met al-poor components, they suggested t hat t he galact ic BMP s ta rs may come from

sirnilar dwarf galaxies. Rodgers e t al. (1981) and Lance (1988) found young main

sequence 4 stars with [Ca lH] > -0.5 at heights up to 11 kpc from the galactic

plane. Since t hese .A stars are a kinemat ically unusual population. t h o - arguecl t hat

the formation of these stars is associatecl with the accretiori of somewliat riietal poor

gas. perhaps froni a clwarf galaxy. The recently discoi-erecl Sagittarius clwarf is ai a

tlist ance of only 16 kpc Froni the center of t h e Mlky \\'au. I t is a large. highly elongatetl

dSph galaxy which appears to be in the process of tidal tlisruption ( Ihata. C;ilnrore.

k i n . 1 ) Ot her accret ion eïitlence is metal-poor. ret rogratle mot-ing groups in

the field (Eggen 1979. .LIajeivski et al. 1994) and Young retrogracie globolar clusters

(Zinn 1993. van den Beryh 1993a.b. Da Costa S: r\rmanclroff 1995). Zaritsky ( 1995)

fountl evidence of recent accretion in the outer disks of nearby late-type galaxies II?-

estirnat ing the durat ion of steep ahuiitlance gradients. elevatecl rates of s tar format ion.

or outer disk nsynimetries. The fart t hat accret ion ont0 the outer clisk is common and

the inner disk apparently remains iindisturbed suggests that the infalling satellites are

tidallv disrupted hefore t hey reacli the inner disk. and therefore they are loir c l e n s i t ~

satellites.

Due to ticlal stripping. low clensity satellites are niainly accreted ont0 the halos

or outer clisks rather than the inner clisks. Therefore. the low accretion rate ont0 the

inner disk alone cannot I)e usecl to set the limit on the accretion rate of t h e galasy.

In order to set a limit o n the rate of galactic accretion. it is necessarj- to inclode

satetlite accretion ont0 not only the inner disk. but also the outer disk and the outcr

halo. Therefore. in this thesis. infalling satellites are introducecl inside the parent

halo at cosmologically appropriate distances from the edge of the disk. whereas in

previous simulations (IVSIH and QHF). satellites started at the edge of the tlisk. The

physical complication of starting satellites further away from the edge of the disk is

not the longer C'PU time required in the simulations. but rather the accomulatetl

disk internal heating the internal evolution of the disk The clisk mode1 in this thesis

is designed in such a way that the disk internal heating produces a velocity ellipsoid

similar to that observed in the Galaxy. Furthermore. by moving satellites further

away from the edge of the disk, 1 can study, in addition to the external heating of

the disk, other dynamical responses of the disks to infalling satellites. namely disk

C'El.-\ PTER 1 . 1:VTROD L'C'TIO.\-

tilting antl warping. Binney k )[a>- ( 1986) clemonstrated. b- ~ising straightforwarcl

test part icle orbital integrations. t liat disks are ext remely resilieiit i n the prcsencr r,f

a sloiv rotating externa1 torqoe: theu tilt iiearlj- as a unit. and i n extreme cases. t hej-

can be t urned -upside down" over time. Morve\-er. the important issues. such as t h e

rate of t ilting. clisk heating. antl clynaniical friction in the systeiri. are not tliscitssecl

in their stuc1~-. and these issues will be investigated in this thesis. 1 will show in full!-

self-consistent simulations t hat t hin tlisks are mainly t ilteci. rat her t han t hickenecl b>-

infalling loiv densit- satellites. Since clisks tilt towarcls the orbital plane of a satellite

on a direct orbit. the disk thickening in t his work is mucli decreased. compared wit h

that in previous work in which disk tilting is either not alloived (TO). o r ver- small

(QHF ancl LVMM) due to srnall init.ia1 separation between the disk and the satellite.

Galactic warps are very common and remain a long standing puzzle in galactic

dynamics. Atternpts to resolve this puzzle are summarizecl by Toomre (1983) and

Binney ( 1992). and the solutions are classifiecl into two groups: ivarps can either be

escited during the formation of an isolatecl galaxy or during the interaction of galaxies.

However. studies show that warps are damped in a time periocl much shorter than

the Hubble time (Nelson k Tremaine 1995a. Dubinski S- li~iijken 190.5). Thiis. the

lact that 50% of galaxies are warpecl coiilcl be an indication that disks are in fact

being subjected to the addition of considerable niisaligned angular momeiitum. Iri

t his t hesis. a disk subjected to t lie acldi t ion of rnisaligned angular momentum froni

an infaling satellite is investigated.

This thesis is divided into 10 chapters. In Chapter 2, the initial conditions for

the disk. halo. and satellite components. as weil as the relation among these three

components will be discussed. The tree code employed in the simulations of this

t hesis will be briefly described. The N-body simulation mode1 in t his work has sev-

eral features which are significantly different from those considerecl in previous work

(WMH, QG. QHF, and TO). First. the initial separation between satellites and the

parent galaxy is cosmologicaiiy realistic. This ensures a true dynamical interaction

between the satellite, the disk, and the halo. Secondly, interna1 evolution of the disk

is allo~vetl. Thirclly. infaliing satellites ha\-e tlensi ty profiles comparable t O t liat of the

parent galas!-. Finall~.. a smail softening Iength is employecl in the simulation so that

the rlisli is st ronglv self-gravitating. This niininiizes the coiipling between the vertical

motion of the satellite and the vertical oscillations of stars within the clisk. that is.

the clisk responds to the infalling satellite as a unit.

In C'liapter 3. 1 will discuss the evolution of the striictiire and kinematics of an

isolatecl galaxy model in order to carry out a cornparison between the galasy model

i r i t his t hesis ancl spiral galaxies. especially the hIiikj. Way. The resiilt ing clist ri bu-

tions of bot h the surface density ancl the velocity dispersions are exponential. The

measiired scale length is a constant. biit the scale height of the asymptotic espo-

nential clistribution increases with time as observed in t h e Milky Ii-a.. For t h e solar

neighborhood. the rneasured scale height and velocity dispersions for the motlel clisk

are similar to what is observed in the hlilky CC'ay. At a given location. the squared

veloci ty dispersions increase proportionally wi t h increasing tirne. The similarit!. of

the structirre and kinematics between the mode1 disk and the hfilkj- Way inclicates

that the model of the disk is dynamically reasonable. This model is usec! to further

sttidy the external heating of the clisk causecl bu infalling satellites.

In C'liapter 4. the interaction betiveen an infalling satellite ancl a weak bar which

forms in the disk ivill be investigated. I ivill show that the growth rate of the rn = 2

bar in a disk with an infalling satellite is somewhat higher than that in the same

clisk but without the infalling satellite. However. the growth rates of t h e rn = 2 bar

mode in both cases are very low due to the high halo-to-clisk mass ratio ernployed

in the model of this thesis. I will show that t lie pattern speed of a weak bar decays

very slowly compared with that of a strong bar studied in previoiis work. The outer

Lindblad resonance of a weak bar remains in the inner disk. whereas the satellite is

Iocated far from the resonance. There is no angular momentum transfer from the bar

to the outer disk and the satellite. The satellite orbital decay rate is not affected by

the formation of a weak bar in the disk.

The results on three major dynamical effects of sinking satellites on disk galaxies,

naniel- tilting. warping. and heating will be presentecl in Chapter 5 . C'hapter 6. and

C'liapter 7 respectively. First. 1 will clemonstrate. in C'hapter 5 . t hat a tlisk is niainly

tiltecl ratlier than heated by infalling satellites. Satellites of 10%. 20%. and 30% of

the tlisk niass tilt the disk by angles of (2.9 f 0.3)'. (6.3 k 0.1)'. and ( 10.6 f 0.2)'

respect ively. However. only 3.4%. 6.%. ancl 1 1.1% of the orbital angular moment iim

is t ransferrecl to the parent galaxy. T h e kinet ic energy associatecl wit h vertical mot ion

in the initial coordinate frame of the disk is increased by (6 * :3)%1. (26 k 3)R. and

( 5 i 5 .j)% respectively. whereas t h e corresponding thermal energ' associated wit h the

vertical random motion in the tilted coordinate frame is increased h>. only ( 4 f 3 ) % .

(6 k ?)Yl. and (10 f 2)% respectively. Next. 1 will show. in Chapter 6. that infalling

satellites can excite warps in the disk. However. the warps event~ially fade away diie

to the fact t hat both the inner disk and outer disk are tiltecl towarcls the sarne plane.

Finally. in C'hapter 7, the vertical velocity dispersion and t h e vertical scale height of

a disk in a local tilted coordinate frame are measurecl. Inside the half-mass radius of

the disk. the vertical heating of the disk measured with both methods is equivalent.

hoivever. out sicle t his radius. the vertical heating inferred from the vertical veloci ty

dispersion is greater than that determinecl from the vertical scale height. 1 will show

that the infall of a 20% disk-mass satellite causes heating near the center and eclge of

the disk. On t h e other hand. the infall of a 30%1 disk-niass satellite causes the entire

disk to be heated, with the outer regions of the disk heated much more than the inner

ones.

In C'hapter 8. two dynamical efFects of a parent galaxy on an infalling satellite

will be cliscussecl. One is satellite mass stripping which is caused by the tidal force

from the parent galax-. and t h e other is satellite orbital decay which results from

the dynamical friction exerted on the satellite by the parent galaxy. First. 1 will

show that the m a s loss of the satellite is roughly proportional to the halo mass

which the satellite traverses, if self-similar density profiles are employed for both t h e

satellite and the halo. Then, I will show that the orbital decay rate. obtained [rom

both theoretical estimates and simulation results, is constant because of the rnass

stripping of the infalling satellite. In the case of a solid satellite. the orhita1 ciecay

rate is inversely proportional to the distance between the ceriter of t h e satellite a n d

the center of the parent galas'.. The orbital tlecay rates for satellites on nead>- rirciilar

orbits. but clifferent orbital inclinations with respect to the parent tlisk plane. and

for satellites on direct anci retrograde orbits will be studied. In physical space. the

final shape of t h e ticlally stripped satellite particles resembles a warpecl toms. The

planes of t h e outermost and innermost parts of the torils are alignecl respectirelj-

with the planes of t h e initial and final orbits of the satellite. In phase space. the

satellite part icles are clist inguished from halo part icles by t heir high rot a t ional speecl

and clumpy distribution.

In Chapter 9. 1 will calculate. by employing both analytical approximations and

numerical integration niethocls. the orbital decay time of a satellite ivhich esperiences

tidal disruption as its elliptical orbit decays. The orbital decay times estimatecl with

both methods are similar. Therefore. based on the analytical result. 1 tvill calc~ilate

the gaiactic accretion for the case of low mass infalling satellites. In tlie calciila-

tion. 1 will follow Trernaine's ( 1980) method. However. I mil1 introduce the following

modifications: high peciiliar velocity cutoff. high niass cutoff. and non-constant mass-

to-light ratio for dwarf galaxies. I ivi l l show that for a median eccentricitj-. t = 0.3.

a typical galasy like our own 4lilky \Ca>- Calaxy has absorbed about :3O<i: of its

disli rnass in the form of infalling satellites. ;\ccorcling to results drawn froni the

simulations tliscussecl i n t his thesis. it can be accommodatecl without unacceptably

thickening the disk insicle the half-mass radius.

Finally. in Cliapter 10. the implications of the main results of this thesis on tlie

clensity of satellites. the Holmberg effect. t h e rotation of the stellar halo of the C' la 1 axy.

and the t heoretical nierger rate for the Cl = 1 C'DM cosmology will be discussecl. At

the end. some coiicluding remarks and suggestions for future work will be presented.

Chapter 2

Numerical Met hods

2.1 Introduction

The gravi tational interaction b~ tween a tlisk galasy and a satellite is invest igatecl

bot h analytically ancl nitmerically. ,-lnalytical treatments of the interactions niain1~-

use t hc perturbation t heory and the inipuise approximation. However. the pertiirba-

t ion t heory is generally rest rictecl to weak interactions ( Lj-clen-Bell 22 Kalnajs 1912.

C;oldreich k Tremaine 1979. Palnier k Papaloizou 1982. Palmer 1983). and tlir im-

piilse approximation is iiselul onl?. rvlien the satellite moves sufficiently rapidly witli

respect to the parent tlisk (hg~i i lar k White 198.5). In order to st~icly the clynaniical

responses of a disk to a n infalling satellite. naniel- disk tilting. warping. and thicken-

ing. as rvell as t o investigate t h e dynarnical effects of the parent clisk on the irifalling

satellite. such as the orbital clecay and tidal stripping of satellites. one must employ

nunierical simulations.

Simulating gravitational interactions between galaxies kvas initiated on an analog

computer by Holmberg (1941) who founcl that a close encounter caused observable

tidal distortion and resulted in capture due to orbital energy loss. Two decades

later. Pfleiderer Sr Siedentopf (1961) and Pfleiderer (1963) revisitecl the subject on

an electronic digital cornputer. Due t o the fact that a number of observecl galaxies

show signs of interaction (Arp 1966), many simulations of galaxy interactions were

performed after 1970. including those by kabushita (1971), Wright ( 1972), Toomre

k Toomre (1972). and Eneev. [ion-01. k Stinyaev (1973). .Aniongst these works.

the paper of Toomre k Toomre is rriost citetl becaiise they siiccessfii1l~- motleleci

several pairs of interact ing galaxies. siich as ;\rp29.5. SI5 1 + S C;C'.i 19.5. NC;C'-L038/39

and .\iC;C'-!676. T h e good agreement of the bridges and tails between their models and

observations showed that t h e gravitat ional interaction is responsible for the observed

niorphological distort ions. Since t h e late 1080's. simulating galas>- interactions has

become an active field dile to Loth the accumulation of merging evidence and the

theoretical support from hierarchical cosmological moclels. as well as t h e cleielopment

of hierarchical tree code in conjunct ion wi t ti availahle fast computers. In t liis chapter.

1 will first esplain. in Section 2.2. the detailecl initial conditions of my simulation

moclels. then. I rvill briefiy describe. in Section 1.3. the tree code ernployed in the

simulations.

2.2 Initial condit ions

.ksurning ttiat a dark matter halo is formecl at a redshift. z . by a spherical top

hat perturbation (Ciunn Sz Gott 1972) froni a spherical region of conioving radius.

ro. in current units. Narayari 5; White (1988) show tha t t h e mass. 41. the niean

density. p. and t h e velocity dispersion. o. of the dark mat te r halo are 31 = y p o r & p = l'ïBpo( 1 + z ) ~ . and o r ( 1 + = ) I l 2 ~~r~ respectively. In these expressions. pu and &

are respectively t h e critical density and t tie Hubble constant a t present. Therefore.

the size of the halo is rsize = (&A~/ . l /p ) ' /~ x ( ro / ( l + Thus. a t a given redshift.

the relation among the size, r,i=, , the m a s . and the velocity dispersion of the dark

matter halo is

r,i,, ! w / ~ . (2 .1)

In other words, al1 dark matter halos formed a t the same redshift z have the same

mean density and structure. However. since low mass halos are, on the average.

formed earlier. their mean densities are higher than those of high mass halos. Eisen-

stein & Loeb (1996) derived that the relation between hf and o is LM ;x a", with

o = 3.1 - 9.2 instead of 3.0 i i i Equation 2.1. Recentl!.. Savarro. Frenk. k \\\-hite

( I W i ) showd t h the high mass halo is less centrally concentratecl tlian the low

n i a s orle. Ilut they iridicatecl that their halos are too concentratecl to be consistent

wi th the halo parameters inferred from clrvarf irregulars. In this thesis. L assume

t hat the halos of t.he satellite ancl disk are formed a t t h e same reclshift. and I choose

self-siniilar clensity profiles For both halos. Since the total mass of most low density

dwarf galaxies is mainly contrihiited by dark matter halos. the density profile of the

satellite halo is callecl the satellite density profile in the folloming cliscussion.

Observations show t hat the observed mass-to-light ratio and relation hetiveen

himinosi ty ancl velocity dispersion from 17 clwarf elliptical galasies (6 are galact ic

dwarf spheroidals) are M I L x L-0-40*0-06 and L x o"-"'0-9 respectively ( Peterson

k C'aldwell 1993). This is in accord wit h the corresponding t heoretical predict ions

.\II L x L-0-37 and L x a's3 (Dekel k Silk 1986). T h e resulting relation between

mass onci velocity dispersion is .U x a3.3-3-4 . On the other hancl. since the observecl

relation between luminosity ancl core radius is L x R5.0*0-"Peterson & C'aldwell

1993). t h e resulting relation between mass and core radius is .LI x R3. Based on t hese

ol~serïecl relations among core radius. velocity dispersion. and mass. the folloiving

scaling relation. R x a x .LI I l 3 . is employed to scale models. In t h e mode1 scalecl

~~ncler t his relation, the central densities of both the sateliite and the host are similar.

t herefore. t his mode1 cannot be applied to study the infall of a compact dwarf.

As a reasonable approximation to a relaxed halo. 1 employ a srrongly concentratecl

King moclel with tlie dimensioiiless central potential I f C i = 8.0. which corresponds to

the central concentration (King 1966). c = log(r, /rh-) = 1.833. in which. r~ is tlie

and r, is the tidal radius. In Equation 2.2. po is the central density. and o o is the

central velocity dispersion.

1 choose the initial orbital parameters of satellites, such as the distances from

the center of the parent galaxy. the orbital eccentricities. etc.. to be as realistic as

possible. For example. a satellite starts at the apocentar i.+ of ari c = 0.2 elliptical

orbit. The radius r+ is chosen to be more than three tinies of the initial disk radius.

a significant increase relative to the values eniployed in previous work ( i \ -41H and

QHF). i r i which satellites started at the edge of the clisk. In my motlels. satellites are

initially set so far away froni the edge of the disk that they do not penetrnte the clisk

when t,hey pass the disk plane.

Prirnarily to create an ecpilibriurri disk. I choose one out of each eight halo part icles

to becorne disk particles. Since al1 halo particles ( i = 1. .V) are initially sortecl from

zero to the tidal racliiis. each chosen particle located at the spherical coorclinates.

r ( i ) . O(i). and o ( i ) is moved to t h e corresponcling half-mass racliris at the cylindrical

coordinates R(i) = r - ( i l 2 ) . o(i). and ~ ( i ) = O. The new clisk is embeddecl inside the

lialf-mass raclius of the haIo. and it has zero thickness. but. the t hickness increases

with increasing time. The total mass ratio of the halo to the tlisk is 7. and the mass

ratio inside t h e disk racliiis is 3.5. The newly formed disk has zero t hickness. but. the

t hickness increases wi t h increasing time.

Init iallj-. al1 disk particles have only rotational velocities. The rotat iorial velocity

of particle i. locatecl a t R ( i ) From the center of the clisk. is delined by rrat(i) =

1 R i ) ) R i ) [ii t his equation. .il( R ( i ) ) is the total mass contaiiied ivitliin the

radius R( i ) . The post-setup rotation curve of the disk. or the rotation velocity of a

sntellite versus the distance between the satellite and the disk. is sliown in Figure 2.1.

The resiilting disk and halo system has the initial dimensionless spin X = 0.023. which

is lower than t h e average value of 0.05-0.07' found in papers by Efstathiou k .Jones

(1979). Barnes S. Efstathiou (1985), and Warren et al. (1991). due to the non-rotating

halo and the liigh mass ratio of halo to disk in niy models.

Galactic clisks are usually treated as quasi-equilibriurn systems for periods much

shorter than the Hubble tirne. However, since I am studying the interaction between

a disk and an infalling satellite over a period comparable to the Hubble time, it is

necessary to consider the interna1 dynamical evolution of the disk. O bservational

C'H--1 PTER 2. '.bIERiC'.-\ L JIETHODS

eviclence of t tiis interrial cvol~it ion consists of the variation of the velocity dispersion

and the scale height froni the yoiingst to the olclest disk stars ( \Vielen 1977. C'arlher,

et al. 1985. Edvardsson et al. 9 ) . For esample. the relation hetween the velocitj-

dispersions o l disk stars ancl their ages can be describecl as CT x t". wliere ci is

approxirnately eclual to 0.5 (N'ielen 1977). There are man? interna1 heat ing sources

in the disk. bot it is widely accepted that the disk stars are rnainly heated by scattering

with giant molecular cloucls (Spitzer k Schwarzschild 19-53. Lace? 1984. Villumsen

198.5) and spiral arms (Barhanis k \Voltjer 1967. Sellwood Li: C'arlberg 1984. C'arlberg

S. Sellwood 198.5).

In N- body simulations. the t wo- body relaxation. due to the accumulation of many

small cleflections of t h e orhit of a particle arising from encounters with other particles.

increases the velocity dispersion o of a spherical system as 0 x (T/T,)O." (Huang.

Dubinski. k Carlberg 1993). In t h e relation. a is velocity dispersion. T is time and

T, is the two-body relaxation tinle which is defined as (Binney k Tremaine 1987)

in which .Y is the number of particles. and Tc is the crossing time. In a simulation

with the softenecl potential O,, = -Gmim,/(r?, + Ç ' ) ~ ' ~ . where c is the softenirig

lerigt h. the t wo- body relasat ion t inie is redefined as (Huang. Dubinski. Sc C'arlberg

1993)

(2.4)

in which R is the size of the system.

Hoivever. 1 will show. in C'hapter 3. that Equation 2.4 cannot be appliecl to a

differentially rotating disk because the disk responds to non-axisymmetric initial dis-

turbances in a remarkably spiral Iike and vigorous rnanner (Julian & Toomre 1966.

hereafter JT). Therefore. each disk particle can be equivalent to a spiral like clensity

wake in the disk, and the amplitude of the density wake is related to the Toomre

parameter Q. When Q is very low at the beginning of a simulation, t h e amplitude of

T,-tblc 2.1 : iC[oclcl Paranieters

the clensity wake is ver' high. The initial noise of the particle distribution through

the swing amplification (Toomre 198 1 ). inimediately develops into the visible spiral

structures. and the wakes of part icles becorne st rongly correlated. The st rongly cor-

related wakes heat the disk ver' rapidlj- in the first few disk rotations. .As Q increases

and the spiral structure fsdes away. the amplitude of clensity wakes decreases. ancl

the wakes of disk particles becorne uncorrelatecl or weaklp correlated due to the tran-

sient weak spiral structures. The interaction between these uncorrelated or weakly

correlated wakes perform as a cont irioits interna1 heating source in the simulatecl clisk.

Further cliscussion about the evoltition of the velosity ellipsoid of the disk is discussed

in C'hapter 3.

Table 2.1 Lists the values of the paranieters associated with a series of simulations.

for which I present results in this thesis. The number of particles. LV. is Listed in

column 2. In our standard models. the number of disk particles is chosen to be

10,000, and the number of halo particles. 70,000. The mass ratios of halo to disk.

iLIH/MD, and satellite to disk, i\.ls/MD, are listed in columns 3 and 4 respectively.

The parameters of the satellite orbit, such as the apocenter r+, the orbital inclination

Figure 2.1: Initial conditions imposed on the disk and the satellite. The initial position and the orbit of the satellite are shown in the top panel. whereas the rotation curve of the disk, or the rotation velocity of a satellite versus the distance between the satellite and the disk is shown in the bottom panel.

Table 2.2: Parameters in Standard [.'nits

Quantitj- Si-mbol \aliie Galas. mass JI = .\ID + -\lH 3.2 x 10"J l i Half-mnss radius of halo R 16 kpc

1,-elocit~. 1.'- = 262 krn/s Dyna~nical t ime t = R)L- fi x 10'yr Timestep At 2.4 11yr Softening lengt h e 400 pc

0. and the eccentricity t are listed in columns 5 . 6 and 7 respectively. The direction

of satellite rotation ~v+/c,. either direct (+) or retrograde (-1. is listecl in column 8.

In the previous fully self-consistent N-boclj- siniulat ion (LV.\I FI). halo part icles were

treated as heavier particles in order to provitle better sanipling of the clisk and satellite

coniponents. However. this resulted in an extra lieating source in the lighter particle

coniponents t h e to the scattering between light ancl heavy part icles. In t his t liesis. al1

particles in t h e disk. the halo. ancl the satellite have the same mass so t hat t here is no

extra Iieating caused by the scattering between the light aiitl Iieavj- part icles. In t lie

simulation. 1 ernploy mode1 units with the gravitation constant G' = 1. t h e total mass

of t lie parent galasu. including the niass of the disk and the halo .\ID + JIH = 1. ancl

the hall mass radius of the halo R = 1. For purposes of presentation. uriits are scaled

to facilitate cornparison with observed galaxies sucli t hat MD + JIH = 3.2 x 10" .Li 3

and R = 16 kpc. The resulting velocity unit is 1'62 km/s which leads to

l++=8.5k,, * 220 km/s (see Table 2.2).

2.3 N-body Code

The algorit hms for computing the potential of a systein of N part.icles can be divided

into "direct" and "field" methods. The former explicitly treats interactions between

individual particles while the latter does so only indirectly, through the contribution

of particles to t h e glol>al gravitational field. T h e -*direct- niethocl is flexible but has a

C'PL- cost per step. sraling as .Y" -4 variety of niethocls have heeri proposecl to reclucc

the cost of compiit in? the self-consistent potent ial. stich as test-particle niet tiods

(Schivarzschild 1979. Quinn 1984). the restricted t hree-hocl>- met liotl ( Pfleiderer k

Siedentopf 1961. Toomre Sr Toomre 1972). or semi-restrictecl Y-body codes (Lin k

Tremaine 1983. Quinn k C;ooclman 1986. Hernq~iist k [kinherg 1989). However. the

dynamics of interacting galasies will not be represented faithfully withoiit inclucling

self-gravit? ( Barnes II)Y8). Early self consistent met hods. known as gricl met hocls. are

not usefiil for stiicl~-ing interaction of galaxies (Sellwootl 1987. Hockney S- East wood

1988) because of t h e low efficience in increasing resolut ion. The recent clevelopment

of the *-hieratchical tree" method (Appel 1985. Jernigan 1985. Barnes k H u t 1986)

is the best technicliie for studj-ing the interaction of galaxies hecause i t provides high

resolution in high clensity regions and low resolution in lom density regions.

For al1 simulations presentecl in this thesis. 1 eniploy a --tree'. Y- body simulation

code which incl~icles cIiiaclruple correction in the cell-particle force ( Barnes k Hut 1986.

Hernquist 1987. Dubinski 1988). The tree code is an algorithm which sorts particles

in a 3-body sxstem into groups in a hierarchy of cubes. Each cube of particles is

suhclivided into eight subcubes wit h hall the length of the parent. The ciibe hierarchy

forms an octal tree chta structure which is iised to calculate the gravitational force

on a particle. The net force eserted on a particle by the rest of the particles in the

N-hocly system is calculated as the sum of t h e forces due to nearby particles and

distant groups in hierachical cubes. The force from a distant group is determined

from a quadruple orcler of expansion of the potential about the mass center of the

group. A group is considered clistant if s l d < B. in which d is the distance from the

particle to the mass center of the group. s is the size of the cube. and 0 is called

tolerance parameter which is the maximum allowed angle subtended by the cube as

seen from the particle. If a group is too close t o the particle. it is siibdivided into

eight cubic subgroups and the subgroups can be furt.her subdivided until the criterion

s l d < 0 is reached. A tree code can determine the force on a particle in a time of

orcler .\- log .\- corn parrd tvi t h .\" for a direct force calciilat ion.

The evolution of thr JJ-stem is followed 1,v a leapfiog integrator with a constant

tiniestep At = 0.0-1 (o r 2.1 SIyr in Table 2.2) and a tolerance paranieter O = 0.7. The

softening lengt h E is itsitally çhosen to be of order K/.\"/? in which R is the size of t lie

systern. 1 therefore select 6 = 0.025 (or 400 pc in Table 2 . 2 ) in the simiilations of this

thesis. A small softeniiig length is essential in order that tlie disk remains strongl?-

self-gravitating. 1 find that tlie resiilting conserlation of total energv is satisfiecl to

better than O.l-58% o v e r 4.000 tiniesteps or 9.6 Gyr.

2.4 Summary

The N-body simulation mode1 in tliis work lias several featiires which are significantly

different from t hose consitlered i n previous work (IL-SIH. QG. QHF. and TO). First.

the initial separation between satellites and the parent galasy is cosmologically realis-

tic. This enstires a triie clyriamicat interaction between the satellite. the clisk. ancl the

halo. Seconclly. the moclel disk is unconventionally constructed from a cosniological

perspect i1;e-e: the interna1 evolution of the disk is allowecl. Thirdly. the density of in-

falling satellites is cosniologically scaled. The infalling satellites have density prof les

comparable to tha t of the parent galaxy. Finally. a small softening length is ernployecl

in the simulation so t hat the disk is strongly self-gravitating. This minimizes the coii-

pling between the vertical motion of the satellite and the vertical oscillations of stars

within the disk. tliat is. the disk responds to the infalling satellite as a unit.

Chapter 3

The Evolution of an Isolated Galaxy

3.1 Introduction

Surface photomet ry of spiral systems ( Freeman 1970. De Jong 1996) shows t hat alniost

al1 of them have exponential surface brightness profiles ~ c ( R ) x esp( - R / h R ) . where

h R is called scale lengt h. The vertical light clist ribution of edge-on spiral galaxies ( vari

cler Iiruit Sr Searle 1981) can be ezpressed by t h e asymptotic exponential relation

L z x sech2(z/h,). correspontling to içotherrnal sheets (Spitzer 1942). In the above

relation. h z is called scale height. which increases with increasing age of the stellar

group and is believed to h e a constant as a function of disk radius. Howet-er. increasing

evidence shows that the scale height of disk galaxies may not be constant (Rohlfs k

LViemer 1982. Iïent. Dame. k Fazio 1991 (hereafter KDF). Filx S: Martinet 1994

( hereafter FM)) but incienses wit h increasing radius. In most N-body simiilations.

the initial particle distribution of a disk mode1 is given by the folloiving relation.

p( R. z ) x esp(- ~ / h ~ ) s e c h ~ ( z / . 2 l i , ) . wit h constant h R and h;. However. in this thesis.

the mode1 disk is unconventionally constructed from a cosmological perspective as

described in Chapter 2. I t is t herefore important to ensure tha t the prescription

employed for constructing the disk galaxy yields a galaxy similar to a present cl-

spiral galaxy in terms of its structure and kinematics.

The velocity dispersions of disk stars increase with increasing age (Spitzer &

Schwarzschild 1953, Wielen 197'7, Carlberg e t al. 1985, Edvardsson et al. 1993),

o x t U ivith a = 0.3 - 0.5. III t his thesis. the velocity dispersion of disk particles

iricreases witti increasing tinie as o x tO-' due priniarilu to t h e weakly correlatecl in-

teraction between particie ir-akes. [t is important to ensiire that the amoiint o l t h e

disk heating in the disk rnodel eniployed in tliis thesis is siniilar to the aniount of

internal heating in the Galaxy. In other words. the resultirig velocity ellipsoicl in the

disk moclel shoulcl be similar to what is observer1 in the Ga1a.u~-.

In this chapter. I disciiss the evolution of the structure and kinematics of an

isolateci galaxy (Slodel O ) in order to carry out a cornparison with spiral galaxies.

especially rvit h the LIilky CVa,-. First. in Section 3.2. 1 compare the clisk surfacc

density and the vertical distribution of disk particles of Mode1 O n i th those of spiral

galaxies. Then. in Section 3.3. 1 compare the three diniensional velocity clispersions V

of the model disk with those observed in the Milky \J7ay. The Toomre parameter Q

wliich characterizes the stabilitp of disks (Tooinre 1964) is also discussecl.

Structure

In the followiiig tliscussioii. tlie time clock is reset to zero after the disk evolves for

three disk rotation periods because wit hin this time period. the initial cold ancl zero

tliickness disk is rapidly heated by the newly fornied spiral arnis. The disk t hen enters

a slow evolution phase in which two-body relaxation is the major source of internat

heating. The clisk rotation period clefined at R = RfE. = 8.5 kpc is 2.4 x 10-r or 4

mode1 time units.

T h e initial surface density distribution of the disks is a projected King model.

which is very close to the exponential distribution. S ( R ) = E(O) exp(- Rlh). Fig-

ure 3.1 shows the surface density of Model O at t=O. 3.6. and 7.2 C+r. The surface

density distribution of the disk changes little. except at the center region R < 1 kpc,

where the surface density increases with time as shown in the figure. The resulting

surface density distribution is consistent with the observed surface brightness distri-

bution of disk galaxies: C(R) = C(0) e x p ( - R / h R ) . For t = 0, 3.6, and 7.2 Gyr, hR

Figure :3.i: Surface density of Moclel O at t = O (dash-dottecl lirie). t = 3.6 Gyr (dashed line). and t = 7.2 Gyr (solid line). The clotteci straight line represents an - exponential fit. wi th scale length. h R = 3.8f 0.1 kpc. of the surface density at t = 1.2 Gyr.

is 3.8 k 0.1. 3.7 k 0.1. and 3.8 k 0.1 kpc respectively. The observed scale length for

the Milky Way varies from 1.8 kpc to 6.0 kpc (see the review by K D F ) depending on

the data set (see discussion in Appendix A ) . However. the value of 3.5 kpc (de Vau-

couleurs k Pence 1978) is often used. 1 therefore conclude that the surface density

distribution of the disk mode1 employed in this thesis is similar to that observed in

t he Mi lkj. I\a>..

ln orcler to nieasiire t h e vertical distribution of disk particles. I clivided the disk

from R = O to R = -10 lipc into 10 rings with A R = 2 kpc. The histograms of particle -

clistrihution in t h e vertical clir~ction of 10 rings at t= 1 . 2 G y is shown in Figure 3.2.

The solid iines represent as'-mptotic esponential fit. .V(z) = .\;(0)sech'(z/2h,). ivhere

both :V(z = O ) decrrases and h z increases tvith increasing radius. Figure 3.2 shows

t hat the particle clist ribut ion in the vertical direction fits the asyrnpt,otic esponential

distribution ver>- ivell. Therefore. t he paraineter h= is a good clescription of the

t hickness of clisk.

Figure 3.3 shows t h e measurecl scale height and its standard deviat ion as a funct ion

of disk radius and time. The measured scale height is not a constant but increases

slowly with increasing disk radius: h=(R) = h,(O)esp(R/h). in which h is called the

radial variance of vert,ical scale height and equals 32 i 3. 20 f 1. and 21 h 1 kpc for

t = 0. 3.6. ancl 7 .2 C;yr respectivel-. For the Galaxy. IiDF proposed a non-constant

scale height mode1 in which the scale height is constant at a value hZsmi, insicle a

characterist ic radius R,,,, and increases linearly wit h radius for radii larger t han

R i The? fourid tliat the oiitwards increasing hl significantl- improvecl the fit of

their observation data. ancl the measured hR is 3.0 kpc insteacl of 2.7 kpc in constant l i :

niodel. Lnspired by I iDF ancl iising the similar model with hR = 3.0 kpc but oiitwards

increasing h: . FSI pointecl out that the model may fit, the pioneer 10 da ta as me11 as van

der Kruit's (1986) rnotlel rvith hR = 5 .5 kpc and constant A,. By employing the values

h(h-DF) of h ,,,,, and h,(R,:,) in IiDF's model a n d assuming that h z ( R = 0 ) = , = 165

pc and h,(RG3) = h,(R = O ) exp(R, , /h) = h , ( ~ , ~ ) ( " ~ ~ ) = 247 pc. ivith RGI: = 8.0

kpc, I obtain h = 20 kpc for the Galaxy. The estimated h is similar t o tha t measiired

in the disk model. Based on the exponential scale height model h2 x exp(Rl2Okpc)

derived from the disk model of this thesis, 1 explain how the constant h= assumption

causes the discrepancy among varies photometric determinations of scale length in

Appendix A.

The thickness of the disk. defined as h,(Ro), is 144. 447, a n d 614 pc for t = 0.

Figure 3.2: Histograms of particle distribution in the vertical direction of 10 rings. The solid lines represent asyniptotic exponential fit. The particle distribution in vertical direction fits the asy mptotic exponential distribut ion well.

Figure 3.3: Scale lieight of the asymptotiç exponential distribution versus radius and time. The measured scale height increases slowly with increasing disk radius: h,( R) = A,(O) e x p ( R / h ) - in which h is equal to 32 3 . 20 k 1. and 21 k 1 kpc for t = 0. 3.6. and 7.2 Gyr respectively.

3.6, and 7.2 C;yr respectively. I t is consistent with observation that the scale height of

different stellar groups increases from the youngest group to the oldest group (Blaauw

1965). For example, the scale heights measiired from O stars. F stars. and dIi and

dM stars are 50. 190, and 330 pc (Mihalas S: Binney 1981). Note that the exponential

scale height is equivalent to half of the asymptotic exponential scale height. Therefore,

Figure 3.4: Distributions of velocity dispersions. OR, on. and 0,. and of the Toomre parameter. Q at t=O (.). 3.6 (+), and 7.2 Gyr ( * ) respectiveiy The solid lines. which - are the least squares fit of the velocity dispersions at t = i .:! Gyr. show t hat. to a good approximation. t h e velocity dispersions decrease exponentially wi t h increasing disk radius.

the s t ruc ture of the disk mode1 in th i s thesis is similar to the structure of the Milky

Way.

C1H.4PTER 3. THE EI'OL I-TIO:\ O F -4-Y 1.50 L.4TED G.4L;ISk'

3.3 Kinemat ics

In t his section. 1 compare the Linerriatics of i he clisk rnoclel {vit h t hose obser~ecl in t he

i l a Figure 3.4 shows the velocitj- dispersions. OR. O,. <r,. and the Toornre

parameter. Q. as a fiiriction of disk radius and tirne for Mode1 O. The solici lines. -

which are the least scliiares fit of t tie velocity dispersions a t t = r -2 Gyr. stiow t hat to

a gootl approximation the velocity dispersions decrease exponentially wit h incrrasing

disk radius. Note that a t 7.2 C;yr. the velocity clispersicns in regions R < 5 kpc evoli-e

a w - froni the esponential trend. which is due to the e s t r a heating ca i i sd h!. tlie

scattering of disk particles witti the weak bar formed a t the center. The formation of

a weak bar is disciissed in the next cliapter.

Direct rneasurements of radial velocity dispersion of the Galaxy (Lewis cL- Fcee-

man 1989. hereafter LF) also show an esponential distribution. By assumin: 0; x

e s p ( - ~ / h g ) . LF estimated that h g of tlie Cialaxy is 4.4 kpc which is 47'7; longer

than the photometric measiirement of hk = 3.0 kpc ( K D F ) . S o t e tha t hAf and hR

represent respect ively the kinemat ic nieasurement and the photometric cleterniiria-

tion of scale length. Van der Krui t & Freeman ( 1986) also shoivecl t hat in SC;C 5-47

the rela~ion between hg and hR is: h R = 1.2hk (o r h g = 1.4hR after cleletin:, a

discrepant point). The cliscrepancy hetween h i and h g ma!- bc due to the assurnp-

tions of a constant scale height and of a constant rnass-to-light ratio. In the disk

rnodel of this thesis. hK rneasurecl from vertical velocity dispersion of disk particles.

a! x e x p ( - ~ / h R ) . is equal t o l .4hk. where hk is rneasured from the disk particle

surface density. Zd x exp(- RlhR). This is because in t h e clisk model. the rneasurecl

scale height increases exponentially with increasing disk radius: h, x c s p ( Rlh) . and

the mass-to-light ratio defined as the surface density ratio of both disk and halo par-

ticles to disk particles only. M I L = r d + h / y d O: e x p ( R / h ) . Therefore. 1 obtain the

following relation among h g , ha and h , l / h R = l / h k - 2 / h . Independent measiire-

ments of these three parameters from vertical velocity distribution, surface densi ty

distribution, and scale height distribution in the disk model of this thesis satisfy the

ahove relation. The observed h i f = 4.4 (LF) . h k = 3.0 kpc (KDF). and h = 20 kpr

estimat.ed frorii KDF-s data for t h e Cialas- also satisfy t h e above relation.

The Toonire parameter which characterizes t lie stabili ty of clisks ( Toomre 196~L )

is defined as

where K ( R ) is epicycle frequency which is derived from rotation velocity R ( R ) . ti =

R- + -ln2. OR and S d ( r ) represent respect ively radial velocity dispersion and r surface clensitj. of disks. The obserl-ed value of Q at R.? depends on the measurement

of local surface density Sd(R,a) . Q = 1.5 f 0.5 with Xi(R.,) = 80 f pc-'

(Bahcall 1984). and Q = 2.7 k 0.9 with Y d ( R.?.) = 45 i 9 . 1 f 0 . p ~ - 2 (Iiuijken k Gilmore

1989). For other spiral galaxies. i t is believecl that Q is a constant as a function of

disk radius (Sellwood 5: C'arlberg 1984. hlart inet 1988. Bot tema 1993). By assuniing

( ~ t l / L ) ~ = 2.0. Bottema (1993) fountl that Q = 2 - 2.5 for 12 spiral galaxies. This

result coincides with the general stability criterion for galaxies as derived in numerical

simulations i Athanassoula & Sellwood 1986). Q measured in the disk mode1 of this

thesis is approximately constant. however. the average value of Q. which is equai to

2.0. 2.3. ancl 2.9 at t = 0. 3.6, and 7.2 Gyr. is higlier than t h e ohserved value for the

Galaxy.

As 1 have discussed in Chapter 2. due to the fact that a differentially rotating clisk

responds to non-asisymmetric initial disturbances in a remarkably spiral like ancl

vigorous rnanner (JT) . each disk particle performs as a spiral like density wake in the

disk. Therefore? the main disk heating could be due to the interaction between density

wakes; rather than the two-body relaxation. By rneasiiring the relaxation time. the

disk heating rate. and the shape of the velocity ellipsoid. I will investigate whether

the main disk heating is due t o the interaction between the density wakes of disk

particles, or due to the uncorrelated orbi ta1 deflect ion between particles. Furt hermore.

by comparing the equivalent mass of density wakes measured in rny simulation with

t hat estimated by JT, 1 will shor; whetlier the interaction between the particle wakes

are correlated.

O 5 13

Time (Gyr)

Time (Gyr)

Figure 3.5: Evolution of the velocity dispersions. OR, a,. 0.. and the ratios of a,/aR and o,/04 at Re, = 8.5 kpc. On the top panel, t h e dashed lines represent a least squares fit to the data with a a t Il2.

First. 1 nieasure t iie relasation tinie froni t h e evolirtion of the velocity dispersions

at R. = S.-> kpc. The top panel of Figure 3.5 shows that the velocity dispersions.

OR. O,. 0:. and <T at R,; increase as the square root of time. This is consistent

with t h e age and \-elocrit' dispersion relation derived LL-ielen ( 1977) who assometl

t hat the local fl~ict~iations of the gravitational field are the cause of the evolution of

the stellar velocity clispersion. I define the relaxation time to be the time for the

kinetic energy associated with the random motion to double. At time t'. the square

of the ve1ocit~- dispersion is modified from its expression at time t accorcling to the

relation $ ( i f ) = d ( t ) + % ( t f - 1 ) . If 02(1') = 20"t)). then Tr = t' - t = 02(t)/s. Therefore. the relasation time clepends on time t . At t = 0. 3.6 and 7.2 Ci-r. t he

relaxai ion t inie is 4.7 Gyr. 8.3 Gyr. and 1 1.9 Gyr. respectivelj-. On the other hand.

the two-body relaxation time can be estimated from Equation 2.4. Csing R = 1

moclel radial unit. 3=80.000 particles. e = 0.025 model radial unit. ancl Tc = 1 model

time unit. I obtained from Equation 2.4 the two-body relaxation time TLr = 2704

niodel time units. or 162.2 Gyr. The fact that the relaxation time rneasored from

the velocity dispersion is much shorter t han the two-body relasation time estimatecl

from Equation 2.4 implies that the two-body relaxation is not the niain source of

heating in the disk. Instead. it is the interaction between particle wakes. Since the

aniplit~iclc of part icle wakes decreases mit h increasing Toonire parameter Q (JT). ancl

t hat Q itself incrcases wit h simulation time. the measured relasation t ime increases

with tinie. Table :3.1 lists the relaxation time measured at three different times and

the corresponding ~ a l u e s of Q.

Secondlu. 1 measure the disk heating rate and the shape of t h e velocity ellipsoid

at R.! = 8.5. If the disk is mainly heated by two-body relaxation. its heating rate

shoiild be isotropic: di . 5 dl . - 2 dt = 1 : 1 : 1. However, the rneasured heating rate 5 . A . - - du2 da?

is dt - di dt - 1 : 0.44 : 0.24. On the other hand, 1 find that the ratio of

o ~ / o & , measured from Figure 3.5 is 0.66, which is approximately equal to the ratio

of K/&Q = 0.67 measured from the disk rotation curve at Rc, = 8.5 kpc. The fact

that crR/od is determined by the rotation curve also shows that the disk heating is

Table 3.1: The relation among Q. Tr and m,..,~,/rn, ,,,,, 1,

not due to the two-body relaxation. hut clue to t h e disk dynamics. I also firid tha t

oz/n0 increases with the thickness of the disk. with a value of 0.73 at 10 Gyr.

1 conclticle from the relaxation time. t he rate of disk heating. ancl the s h n p ~ of t h e

\-elocity ellipsoid. t ha t the main sources of heating a r e not due to orbital deflections

between disk and halo particles or deflections between disk particles only. hut due

to t he interactions of particle wakes. This could ei ther be clue t o the interactions of

uncorrelatecl wakes of clisk particles or weakly correlated wakes in the lorn1 of weak

transient spiral arms. In order to investigate whet her t h e interact ion bet ween particle

wakes are correlated. I compare the equivalent mass of particle wakes computed in

my simulations with that calculatec~ by JT. T h e equivalent mass of particle wakes is

defined as rn,,k, = Trr/Trm,rt, , l , . The values of m,ak,/mp,rt, ,r, at different (Z a r e

listed in Table 3 . 1 . For Q = '1.0. m,,k, = 34.5mparticre. On the other hand. JT found

that a disk with a flat rotation curve and Q = 2.0 responds to a pert~irhative density

transform Do with a maximum density transform of 10.8 Do. Since the spatial clensitj-

a t a given location is proportional to the maximum densi ty transform. the equivalent

mass of density wakes is defined as rnWake zz DD,,,/Dorn,a,ticle = 10.8m,a,t,,r,. The

higher equivalent mass of particle wakes found in my simulations shows that particle

make interactions a re correlated in the form of weak transient spiral structures.

Finally. t he total amount of heating and the heating distribution in R. 0. and z

directions in t he simulations of this thesis is similar to those observed in the Galaxy.

For example, a t t = 5 Gyr, the measured velocity dispersions, a ~ . a,. and a,. are

40. 28. and 16 km/s. which are similar t o t he observed velocity dispersions. 39, 23,

ancl 20 kiii/s. of olcl disk stars in the solar neighborhood (\.L*ielen 1977). 1 t iierc~fort~

concl~icle on t his hasis t hat the riiiiriber of tlisk particles in the galaxy mode1 of t iiis

t hesis is acleqiiate for simulating t hiri clisk clyriarnics.

Summary

In surnrnarj-. the ïlist,ributioris of hoth siirface density and velocity clispersiona of

the disk rnoclel are esponential. nhicli are consistent wit h observecl distributions

for spiral galaxies. The scale lengt h rneasurecl from t lie clisk surface clensity ( '3 x

exp(- ~ / h A)) is shorter t han t hat nieasorecl from the t-ertical velocitj- dispersion

(a: x esp(- R/hAi)) . hf = 1 . 4 h i . This discrepnnc~ between photornetric and kinc-

matic measurernents. wliich is also ohservecl in t h e C;alasy ancl iIGC.5247. mat- be dile

to both the non-constant scale height ancl mass-to-light ratio. The vertical clistribu-

tion of particles lolloivs an asymptotic exponential clistribiition very well, however. the

scale height increases exponentially with clisk radius. and it increases proportionally

to the square root of time. 1 fincl that at R.:. it is equal to 144. 447. ancl 614 pc for

t = 0. 3.6. and 1.2 Gur. respectii-ely. which is coinciclent with the result ohtained from

O. F. and d!vi+dI\: stars. At a given location. the squared velocity dispersions in-

crease proportionally with increasing time. For the solar neighborhood. at 1 = 3 Gyr.

the velocity dispersions. oo. aricl O, are 40. 28. and 16 km/s respectivel-. whereas

the observecl resiilts are 39. 23: ancl 20 km/s (Wielen 197'7). The Toomre parameter

is approsimate constant. and the average value of Q is appro'tirnately eqiial to 2.0.

2.5, and 2.9 at t = 0. 3.6, and 7.2 Gyr. respectively. it is slightly higher than that

observed in the Galasy, Q=l..j (Bahcall 1984) and 2.7 (Iiuijken 9- CXmore 1989). or

in other spiral galaxies Q = 2 - 2.5 (Bottema 1993). The similarity of the structure

and kinematics between the disk model employed in this thesis ancl the Milky Way

indicates that the clisk model is dynamically reasonable. In the following chapters. 1

will employ t, his model to furtlier investigate the external heating of the disk caused

by infalling satellites.

Chapter 4

Satellite-Bar Interaction

4.1 Introduction

It is iviclely believed that bars in galaxies a re formed throiigh a global dynarnical in-

stability. discovered in Y-body simulations ( Hohl 5: Hockey 1969. Miller. Prendergast

S: Quirk 1970). Toomre (1981) proposed that the instability is caused by a swing-

amplified feed-back loop. and there are two niethocls to suppress the instability. One

is to reduce the gain of the swing-amplifier throiigh a high i-elocity dispersion in the

clisk or hY immersing the disk in a massive halo (Ostriker S. Peebles 1973). The other

is to suppress the feeclback through the center by adcling a bulge-like mass compo-

nent with a small core radius (Sellwood 1989). The method of irnmersing a disl- \ ' in a

massive halo is widely used to prevent bar instability in disks because the presence of

clark halos is inferred from the observed flat rotation ciirves of spiral galaxies (Robin

et al. 198.5. van Albatla et al. 1985, Kent 1987).

In the disk mode1 of this thesis. a high halo-to-disk mass ratio. .LfH/MD = 3.5 is

employetl in order to suppress the bar instability for a very long period. However. a

weak inner bar slowly develops in Mode1 O after the disk evolves for 25 disk rotations

as shown in Figure -1.2. lncreasing the number of particles can decrease the shot

noise caused by random distribution of part ides- which can consequent ly suppress

the formation of a bar for a longer period. However, for simulations run as long as 50

di& rotations in t his t hesis, very large N simulations are computationally expensive

aritl riot rirîrssar>- for t h e main piirpose of this thesis u-hicli is to stiicl~. the cl>-naniiral

responses of a disk to infalling satellites. iricliiding the iritrractiori betrveen infallinj

satellite ancl bar. The bar phenorrienon is comrnon arnorig the disk galaxies. incliicli ng

oui. oirri galas'.. De laucouleurs ( 196:3) founcl amorig 5)94 spirals that 3 1 % are S.A.

28% are SXB. and :37% a re SB. It is riot unrealistic that the disk moclel of this thesis

has a weak bar formeci a t t h e center.

It is important to consider the possible interaction between an infalling satellite

and the weak bar. This is because i f a satellite is iritrotliicrd into tlie system. t h e

satellite nia)- esci te a stronger bar in t lie disk t han t hat in t lie isolatecl clisk (At hanas-

s o d a 1996). Consequent ly. the stronger bar wodtl heat t h e inrier disk more t han the

weak bar woiild. Therefore. the infalling satellite could heat the oiiter disk clirectly by

depositing energy. ancl heat the inner disk inclirectly by esciting a stronger bar. O n

the other hancl. the satellite-bar interaction may affect t h e orbital decay rate of the

satellite. which may consequently alter the disk heating rate causecl by the satellite.

The purpose of this chapter is t o investigate the interaction hetwcen a weak bar

formecl in the disk and an infalling satellite in the stanclarci moclel of this thesis. hlore

gerieral satellite-bar interactions were disciissed hy Athanassoula ( 1996). T h e effect

of an infalling satellite on the forniation of a weak bar in tlie parent clisk is studiecl

in Section -4.2. and the effect of a weak bar on the orbital tlecay rate of an infalling

satellite is discussecl in Section 4.3. .A brief surnrnary is given in Section 4.4.

4.2 Infalling satellite

A tidal interaction with a n external perturber cati cause the formation of a bar in

the disk (Xoguchi 1987) o r can excite a stronger bar in the disk (Athanassoula 1996).

1 first compare the rn = % bar modes of an isolated disk and of a disk with a n

infalling satellite. Figure 4.1 shows the evolution of the amplitude. IA(m = 2)1, of

the bar mode for the isolated disk (Mode1 O, dashed line) and for the disk with a

30% di&-mass satellite infall (Mode1 3, solid line). Here 1 A(rn = 2)I is defined by

Time (Gyr )

Figure 4.1: Evolution of the amplitude of the nt = 2 bar mode for Mode1 3 (solid line) and Mode1 O (dashed line). The bar mode of Mode1 3 is stronger aiid formecl slightly earlier than that of Model 0.

IA(m = 2 ) ( = l / ~ \ l ( ~ ? C O S ( % B ~ ) ) ~ + (Sr sin(2dj))'. in which IV is the number of

particles inside the turnover radius because the bars in both models end inside t his

radius. The bar mode for Model 3 grows slightly earlier than that for Mode1 0: and

the resulting growth rate and the amplitude of the bar mode for Model 3 are slightly

higher than those for Model O. These results can be seen directly from Figure 4.2

which shows the evolution of the disk particles of Model O and Model 3. The bars in

i T-O , 7-20

Figure 4.2: Evoiution of an isolated disk. Model O ( top tmo rom) and of a disk wit h a 30% disk-mass infall, Mode1 3 (bottom two rows). Although weak bars formed in both models, the one in Model 3 is slightly stronger than the one in Mode1 O.

both models are weak but the one in hIodel 3 formed slightly earlier and is slightly

stronger than the one in Model O. This is because in Mode1 0, the m=2 bar mode

is triggered by the density fluctuation arising from random distribution of particles.

whereas in Model 3: both the density fluctuation and the infalling satellite are the

origin of the m=2 bar mode in the disk. 1 therefore conclude that an infalling salellite

can excite a slightly stronger bar in the parent disk than that in an isolated galaxy.

CVH.4PTER -!. S_-lTELLITE-B.4-1R l:VTER.-IC'TIO,Y

4.3 Slowdown of weak bars

It is believed that during the evolution of a barred galasy. angular rnornentum redis-

tribution among the bar. the clisk. and the halo. Ieads to a gradua1 slowdown of the

bar. Linear theor? estimates (Weinberg 1985. Herncpist Cc Weinberg 1992) predict

that a strong bar transfers a significant fraction of its angular momentiim to the halo

in only a few rotation periods. Similarly high slowclown rates were also found in the

numerical simulations of Iiernquist 9: Weinberg ( 1992) and Sellwood k Debat t ista

( 1996). Hoivever. Combes et al. (1990). Little Sc C'arlberg ( 199 1) . ancl Athanassoula

(1996) foiind in their simulations a lower slowclown rate: of orcler 30 to 40 initial bar

periods are needecl to slow down the bar by a factor of two. Obriouslp. the slowdown

rate of a bar depencls on the simulation rnodels. For esample. in Sellwood 5: Debat-

tista's rnodel. the gravitation of the disk is dominant. A strong bar forms quickly.

within three disk rotations. and it slows down a t a high rate. However. in one of

Athanwsoula's rnodels. the gravitation of the halo is dominant. The bar forms slowly

and slows down a t a low rate.

In the disk rnodel of this ttiesis. the bar forms extremely slowly clue to the high

niass ratio of the halo to disk eniployed in the moclel. Therefore. the bar slows down

at an'ert reme!y low rate. I estimate the bar pattern speecl. Rb. by performing Fourier

transform of I;L(t. rn = 2 ) 1. I found that the pattern speecl of bar decays linearly wit h

time. and the bar pattern speed of Mode1 3 decays sliglitly faster than that of Model

O. This is because the bar in Model 3 is slightly stronger than that in Mode1 O. It

is consistent wit h the results presented by At hanassoiila ( 1996). However. t lie decay

rates for both Model O and Model 3 are very low: rlQa/Ri,; equals 0.11 and 0.16

respect ively in 50 initial bar rotation periods.

Due to t h e slow decay of the pattern speed, t h e outer Lindblad resonance (OLR)

of the weak bar of Model 3, which is located a t the inner disk R < 10 kpc as shown

in Figure 4.3, has little outwards shifting. On the other hand. the infalling satellite

in Model 3 never reaclies the inrier disk because it is rapidly disrupted as soon as it

n 0 Q 3 \

C/)

\ E x u

rn \ 2 i-i c (3 C 0

c

Figure Rotation curve of disk and satellite and the angular speed of bar for Model 3 at t 4 . 0 Gyr ( top line) and 10.8 Gyr (bottom line). There is little coupling Setween the rotation of t h e bar and the rotation of the satellite.

enters the disk. Therefore, there is no direct angular momentum transfer from the

inner weak bar to infalling satellite. The satellite orbital decay rate is therefore not

affected by the formation of a weak bar a t the center of the disk.

4.4 Summary

In t lie stanclarcl modcl of t his thesis. an infalling satellite catises a slightly st ronqer

bar than that in an isolated ciisk. however. in both cases. the bars are weak and f o r ~ t i

i -er- slomlj-. in approximate 25 disk rotations. This is due to the high halo-to-tlisk

mass ratio ernployecl in the disk moclel. The pattern speed of the stronger bar in

Mode1 3 decays slightly faster than that of the weaker one in h[odel 0. however. t h e

pattern speeds of weak bars in both rnoclels clecay ver- slowly relative to those of

strong bars found in previous work. Since the OLR of the weak bar remains at t h e

inner disk. and the infalling satellite is located far from the OLR. there is no direct

angular momentum transfer from the bar to the infalling satellite. Therefore. the

satellite orbital clecay rate is not affected by the formation of a weak bar in the clisk.

In other words. disk heat ing caused by the infalling satellite is not iinclerestimatetl i r i

the moclels of t his t hesis.

Chapter 5

Disk Tilting and Heating

Introduction

As t h e orbit of an infalling satellite decays. part of the orbital energy and orbital

angular momentum is transferred into the disk and the halo. ancl the rest is carriecl

by the satellite remnants. The orhit of the infalling satellite and its host galactic disk

are usually not in the same plane. The resulting dynamical issues are what fraction

of the orbi ta1 energy associated ivi t h the vertical mot ion of the satellite is t ransferred

to the disk coherently: and what fraction is thermalized in the disk. In t h e former

case. the energy transfer learis to the tilting of the disk. whereas in the latter case.

i t causes the thickening of the disk. In Section 5.2. I corripute the disk tilting angle

tlirough the direction of the angular momentiim of disk. In Section 5.3. I calculate

the total kinetic energy associated with the vertical motion of the disk particles in

both the initial coordinate frame and the tiIted coordinate frame. so that. I will be

able to answer the question of whet her the disk is more likely to be tilted or t hickened

by an infalling satellite.

5.2 Angular Moment um Transfer

First, I compare a perturbed disk (Model 3) in the initial coordinate frame and in

the tilted coordinate frame with an isolated disk (Model O ) in the initial coordinate

Figure 5.1: Disk particles of Nodel O projected in +y, r z and y - planes (top row). Disk particles of Mode1 3 projected in both xy, $2 and y= planes (micldle row) and tilted x'yt,x'z'. y':' planes (bottom row) at T=144 time model units or :3G disk rotation periods.

frame. In Figure 5.1. the disk particles of Model O are projected o n the .ry. XI. and

yz planes (top panels), and the disk particles of Model 3 are projected on the xy.

xz, and y= planes (middle panels), as well as on the tilted x'y'. x'z'. and y':' planes

(bottom panels) a t T = 144 model time units or t = 8.64 Gyr. Here x. y, and : are

the initial coordinates, and x', y', and z' are tilted coordinates which are determined

Direction of Angular Mornenturn

Figure 5.2: Evolution of the disk (D) , satellite (S), and total (T) ang~ilar niornentum directions. x0 = Bo cos o. and y" = Bo sin d. After 40 disk rotations. the changes in the disk and satellite angular moment um directions are 10.6" and 9.6" respectively. and the direction drift of the total angular momentum is about 0.8'.

by the three principal ases of inertia of the tilted disk. Because the disk of Mode1 3

is tilted by the infalling satellite dong both the x and y axes. the disk viewed in the

initial coordinate frame as shown in the middle panels seems quite thickened by the

infalling satellite, however, the real thickening in the tilted coordinate frame is not

very large, as shown in the bottom panels.

In order to show ho\\. the infalling satellite gracliiall>- tilts the disk to the position

as shotvn in Figure 5.1. I plot i n Figure 5.2 the evolution of the angular momentuni

directiolis of the disk. the satellite. anci the whole system. For a given angular nia-

+

mentum L(B. O ) . its direction is plottecl as a point in polar coordinates ( O . O ) . In t h e

figure. xO = Bo cos o. and ,y0 = BOsin o. The changes of the angular mornentiim di-

rections of the clisk and the satellite in t he .r coordinate show t hat the angle hetween

the clisk plane and the sateliite orbital plane decreases due to the angular nionlentuni

transfer between t hem. Also it can be seen from the figure that t h e disk plane ancl

the satellite orbital plane are precessing in opposite directions because the disk ancl

the satellite exert an opposite torqiie on each other. I find tiiat after 40 disk rota-

tions. the directions of the disk and satellite angtilar momenta shifted 10.6" and 9.6'

respectively. During the same period. the drift of the direction of the total angular

momentuni, due to the accumulateci integration error. is very small. It is O.YO in -10

disk rotations. or on the average. 0.7" per timestep.

In order to follow t h e angiilar momentum transfer amonp the satellite. the clisk.

ancl the halo? I measore the relative changes in both the rnagrii tude and the direction

of the angular mornentum of each component. 1 first consider the results of a simple

case which involves the angiilar momentum transfer betmeen a disk and a haio in

an isolated galaxy. namelu hlodel 0. Since the initial rotation of the halo is nearly

zero. the clisk rotation angular momentum is almost the total angular momentum

of tlie whole systern. As the galas- evolves with time. the disk gradually transfers

its angular momentum to its surrounding halo at a steady. low rate of 0.1% per

rotation. By the end of tlie simulation, 40 disk rotations. due to the formation of the

weak bar in the center of the disk as discussed iri Chapter 4. the disk has transferred

4.6% of its angular momenturn to its surroundirig halo. Conservation of total angtilar

momentum is very good: there is no more than a 0.06% variation during the 40 disk

rotations.

When a satellite is accreted oiito the isolated galaxy, there is angular momentum

transfer among the satellite, the disk. and the halo. Figure 5.3 shows t h e evolution of

the angular monienta of the satellite. the clisk. ancl the tialo for hlotlel 1. hloclel 2 . and

Sloclcl 3 . The left ancl right panels show respectively the evoliit ion of the magnitudes

ancl the direct ion of the various angiilar monmita. C'onipared rvi t h the isolated clisk.

t lie disks with satellites transfer more of their angular niornenta to t heir halos due to

the st ronger bars esci ted by the satellites. The amount of t h e angular momentum loss

clepends on the strength of the bars. The disks afso absorb angular momenta from

their satellites. and the amount of angular momentum gain depencls on the masses

of the satellites. Xs a result. bj- the end of 40 disk rotations. the magnitudes of the

disk angulax momenta of 'ilodel 1. SIotlel 2. and Iloclel 3 have clecreasecl by 6.8%.

5.3%. and 6.2%. Although the bar in ~ l o d e l 2 is stronger t han t hat in Slodel 1. the

net angular momentum loss of the disk in Mode1 2 is less than that in hloclel 1. This

is because the disk in Model 2 absorbs sorne angular momentum from its satellite.

whereas the disk in &Iode1 1 absorbs no angular momenturn froni its satellite. which

is tidally disrupted before it reaches the clisk. The disk in Modei 3 absorbs more

angular momenturn froni its satellite than does the disk in !dodel 2. However. in this

case. the net loss of angular momentum is grenter due to the stronger bar. which

transfers more of its angiilar moment& to the surrounding halo.

Due to t h e large initial distances between satellites and disks. the ratios of the

disk angular rnomenta to t h e satellite ang~ilar niomenta are much larger than their

corresponding mass ratios. In Model 1. .LIodel 2. ancl .\[ode1 3 . the ratios of satellite

angular momentum to disk angular niomentum are 43%. 90%, and 130%. whereas the

corresponding satellite to disk mass ratios are only IO%, 20%. and 30%. The large

angular rnomenta of the satellites make ciisk tilting very easy. The disks of Mode1 1.

Model 2. and hiIodel 3 are tilted by angles of (2.9 rt0.3)". (6.3 k 0.1)". ancl (10.6it0.2)"

respect i vely.

Now, 1 discuss the evolution of the angular rnomenta of the satellites. Since the

d ynamical friction exerted on an infalling satellite is proport ional to the square of

the satellite rnass, a heavier satellite is subjected to a stronger dynamical friction.

This stronger dynamical friction, in turn. results in greater angular momentum loss

- ""F à

rime

Figure 5.3: Angular momenta of the satellite: the tlisk. and the halo versus time. The evolution of the angular momenta of the three components nf Model 1. Mode1 2. and Model 3 is shown from top to bottom. Most of the initial angular montenta of t h e satellites remain in the satellite remnants, and the halos absorb angular momentum from both the disks and the satellites.

2 5 E B . L O - . Y

T a - -

' . . ' - ' - - , " ' ..' , 8.(L(t).L(O)),,,k 0.:- -._ ........ m . e0(L(t).L(O))&,,,;t. . . . . . . - 0 . . '

9: . . - 4

Wb.

0.-

O -

front t lie satellite. For a rigid or point niass satellite. the orbital angtilar niomenturti

of the satellite is cornpletel>- transferred to the disk and halo system. Howc\-et-. for

a self-consistent low density satellite. most of the orbital angtilar niornentwn is kept

as rot.ational angular momentum of the satellite remnants. d u e to ticlal stripping.

Ir i the case of 10%. 20%. and 30% clisk-mass satellites. 1 find that hy the end of

t.he simulations they have lost respectively 3.-1%. 6.9%). and 11.1% of their angiilar

momenta. The majority of the angular mornenta of the satellites is carriecl hy t heir

remnants. The direct ions of the satellite ang~ilar momenta have changed respect iveil-

angles of 6.3". 7.9". and 9.6' relative to t heir initial directions in Model 1. SIoclel 2.

and blodel 3. .As for the halo angular momenturn. it reaches a fraction approxiriiately

equal to 10% of the total angular momenturn of the system for al1 three niodels. but in

the heavier satellite model. the halo angular momentum is slightly larger because t h e

halo absorbs more angular mornentum from the satellite. Clearly. the halo absorbs

angular momenturn very efficient 1- from bot h its satellite and its embedded disk.

5.3 Energy Transfer

The infalling satellite not only tilts t he disk through the transfer of angular mornen-

t u m between the satellite and the disk. but also heats the disk by depositing eiiergy.

In this thesis? I only investigate the disk heating in the vertical direction. Figure 5.4

shows the kinetic energy associatecl with the vertical motion of disk particles in the

initial coordinate frame. E;,. and in the tilted coordinate franie. k , ~ . for Mode1 1,

Model 2. and Model 3. Here the tilted coordinate frame is constructed by employing

the principal axes of the inertia tensor of the disk. I find that the 10%. 20%. ancl

305% disk-mass satellite in fa11 increases the thermal energy associated wi t h random

motion of disk particles in the tilted coordinate frames by only (4 f 3 ) % , (6 f 21% and

( 10 31 2 ) W respectively, as cornpared to the isolated model. Howewr. in coritradis-

tinction? the kinetic energy associated with the vertical motion of the disk particles

in the initial coordinate frames is increased bÿ ( 6 f 3)%. (26 f 3)%, and (51 f 5)X1

Figure 5.4: Kinetic energy associated with the vertical motion in the initial coordinate frarne k, and in the tilted coordinate frame as a function of time for. lrom top t o bottom' Mode1 1, Mode1 2. and Mode1 3. After a satellite which Lias 10%. 20% or 30% disk mass falls into the disk, the kinetic energy associated with the vertical random motion of the disk k,t is increased by only 4 f 3%, 6 f 2% or 10 k 2% respectively, as compared to a n isolated galaxy.

respect ivelj-. as coniparetl to the isolated mnclel. C'onseqiient lu. t lie clisks are mainlj-

tiltecl rat her tlian heated hy the infalling satellites.

In orcler to stiicly the clistribution of t lie thermal energ!. associat~cl rvit h the ran-

dom vertical niot ion of clisk part icles. 1 plot. in Figure 5.5. the evolu t ion of the vertical

velocity dispersion of the disks as a function of disk radius in the global tilted co-

ordinate Irame. The clisks are not heated uniformly as a fiinction of radius by the

infalling satellites. I fincl that both the outer and the inner regions of the clisks are

heated more than the other regions. The satellites heat the outer disks directly by

transferring energy to the outer disks and heat the inner disks indirectly bu exciting

slightly stronger bars. Note that a fraction of the heating of the iiiner disk shown in

Figiire 5.5 is artificial due to the mismatch between the inner local tiltecl coordinate

Frame ancl the one eniployed in the cornputation. As there is rvarping in the clisk. the

tilting angles of the inner and outer parts of the disk are different irom each ot her.

and they are also different from the tilting angle of the disk as measured in the tilted

coordinate frame. defined by the principal axes of the total inertia tensor. In fact. the

direction of the axes in the tilted coordinate frame is mainly fised by the oriter part

of the disk (inertia increases quadratically with distance). To explicitly verif? which

fraction of the thickcning of the inner part of the disk is caused by real thermal effects

and which part is an artifact associated wit h the choice of reference franie. i study

the vertical velocity clispersion in the local tilted coordinate frame in Chapter T .

The results of Figure 5.5 can be siimmarized as follows: the evoliition of the

vertical velocity dispersion in the disk subjected to the infall of a 10% disk-mass

satellite is almost the same as the one in the isolated galaxy. and there is no cletectable

thickening in the disk. A disk is only slightly thickened by the 20% clisk-mass satellite.

However. the infall of a 30% disk-mass satellite definitely causes detectable thickening

of the di&, especially in its outer region. Finally, in order to compare this work with

previous work (TO, QHF and WMH) that suggested that a dense 10% disk-mass

satellite is sufficient to thicken a thin disk, 1 show in the last panel of Figure 5.5 the

disk heating caused by a dense 10% disk-mass satellite (Mode1 10). The satellite is

Radius (kpc)

Radius (kpc)

Radius ( k ~ c )

5 1 O 15

Radius (kpc)

Figure 5.5: Vertical velocity dispersion as a function of radius and time for Model 1. Mode1 2, Model 3, and Model 10 (solid line) compared with that for Model O (dashed line). In each panel. from bottom to topo the distribution of the vertical velocity dispersion at t = 0, 3.6. and 7.2 Gyr is plotted. Only the 30% disk-mass satellite and the dense 10% disk-mas satellite can cause detectable thickening in the outer part of the disks.

iritrodocecl into the systeni at t h e eclge of ille disk. r-+ = 1.0. and its half-niass radius.

coniparecl wi th that of the satellite in Slodel 1. is clecr~asecl b!- >O%. 1 fincl that the

clisk heating caused bx the 10% clisk-rnass high densit?. satellite is e\-en larger than

t tint causecl by the 30% clisk-mass but low density satellite. Therefore. the densitj.

of an infalling satellite is a crucial parameter in determinina t h e amount of esternal

heating caused hy the infalling satellite.

5.4 Summary

Due to the angular momentum transfer betiveen the ciisks and their satellites. t he disks

in .\Iode1 L. 5Ioclel 2 and SIoclel 3. are tilted by angles of (2.9 f 0.3)". (6.3 i- O. 1)' and

( 10.6 41 0.2)' respectively by the end of 10 disk rotation periods. In the case of the

satellites. they have changecl respectively by angles of 6.3". 7.9' and 9.6' relative to

t heir initial direction in hfodel 1. Mode1 2 and Mode1 3 in the same period. .As for

the magnitude of these satellite angular momenta. they have lost respectively 3.4%.

6.9% and 1 1.1% of t heir angular nionlenta. The remaining angular momenta of t hose

satellites are left with their rernnants.

The answer to the qiiestion that 1 posed at the beginning of this chapter is that a

large fraction of the orbi ta1 energy associated wit 11 the vertical motion of the satellite

is aclclrd to the disk coherently and a small fraction is thermalized in the disk. A

10%. 20% and 30% disk-mass satellite infall increases the thermal energy associated

with random motion of disk particles in the tilted coordinate frames by only (4&3)%.

(6 IL 2 )% and (10 & 2)% respectively. as cornpared to the isolated model. However.

in contradistinction, the kinetic energy associated with the vertical motion of clisk

particles is increased respectively by (6 & 3)%. (26 f 3)% and (51 I 3% in the initial

coordinate fratlies z s compared to the isolated model. Therefore. there is more added

energy related to the tilting of the disk than that related to the t hickening of the disk.

The evolution of the vertical velocity dispersion in the disk with the infall of a

10% disk-mass satellite is almost the sarne as the one in the isolated galaxy so there

is no cletectahle thickeriing in the tlisk. The disk is slightl?. tliickeiietl by the 205

clisk-i-iiass satellite ancl t h e 30% clisk-rnass satellite infall clefiriitrl>- causes cletectahle

thickeninj in the disk. especiallj- at t h e outer part of t h e disk.

Chapter 6

Warping

6.1 Introduction

Calactic ivarps are ver- cornmon. For esample. al1 three spiral galaxies in the 10-

cal group are warpecl. ancl more than half of al1 disk galaxies appear to be ivarped

( Sanchez-Saavedra. Battaner k Florido 1990. Bosma 199 f ). Therefore, warps are

either long-lived or recently excited (excellent review papers a re written by Toomre

(1983). Binney ( 1990) ancl Nelson k Tremaine ( 1995b)). However. studies show that

warps are damped in a time period much shorter than the H ~ i b b l e t ime (Nelson k

Tremaine !995a. Dubinski XL Iiuijlien 109.5) due to the fact t hat short-wavelengt h

bending [raves are clampecl by mave-part icle resonances ( Hunter k Toomre 1969.

Weinberg 199 1. Yelson & Tremaine l995b), while long-wavelengt h hencling waws

are strongly dampetl by clynamical friction froni an oblate halo tha t is either non-

rotating or rotating in the same direction a s the disk (Nelson k Tremaine 1905a).

Therefore, warps are unlikely to be long-lived. and they must be excited recentlv.

Nelson & Tremaine (1995b) surnmarized t h e following four mechanisrns which can

excite warps: excitation by dynarnical friction from the halo, if the halo and clisk

rotate in the opposite direction (Nelson & Tremaine 1995a); gravitational noise from

halo ( Nelson & Tremaine 1995b); Coriolis force from a twisting halo ( Ostriker k

Binney 1989); ticlal fields from satellites (Burke 1975, Kerr 197.5, Hunter 8~ Toomre

1969, Bertin Sr Mark 1980. Lynden-Bell 1985, Weinberg 1995). In this chapter. t he

warp escited bu the tidal field of a satellite will be clisciissetl.

The simiilat ions in t liis t liesis clemorist rate t liat lvarps are esci tecl hj- a n infalling

satellite. I n Figure 6.1. 1 show t h e evolution of the clisk and satellite particles of

.\Iode1 2. projectecl on the .cz plane. Since the orbit of the satellite is inclinecl 30'

with respect to the disk. the angiilar momenturn transfer between t h e disk and the

satellite causes the outer part of the disk to tilt [aster than the inner part. Therefore.

a slight warp appears after T=fiO. As the system evolves in time. the inner part of the

disk is tilted in the same plane wi t hout precessing becaiise the gravitat ional influence

of the infalling satellite is small. Howeïer. the outermost part of the disk has a

greater tilting angle t han the inner part. and it is subject to a slow precession under

the torque imparted by the infalling satellite. The projected clisk appears slightly

thicker than it actually is in Figure 6.1. because the disk is not only tilted along the

x axis. but also along the y axis.

In this cliapter, 1 first investigate the kinematics of the warps in this chapter bu

employing two models: one. which I call the ring mode1 discussed in Section 6.2. in

which the disk is subdivided into rings and the evolution of these rings is studied: the

other. which 1 call the particle mcdel discussed in Section 6.:3. in which the motion

of a individual particle is followed. X brief siirnmary is given in Section 6.4.

6.2 The Ring Mode1

1 first construct a ring mode1 in order to investigate the kinematics of the warps. A

disk is subdivided into rings and the evolution of these rings is studied. 1 regroup disk

particles into 10 rings (i=O-9). The particles located between R(i ) and R(i + 1) form

Ring i: where R(i) is defined by R( i ) = 0.2 x 1.25'. The disk particles are divided

in this way so that there is a sufficient number of particles in each ring t o make a

reliable estimate of the moment of inertia tensor. For each Ring i: the principal axes

of its inertia tensor are used as local coordinate axes (xi, y:, 2:). In Figure 6.2, 1 plot

10 rings according to their new coordinates (x:, y:, z:) at T = 100 mode1 time units

Figure 6.1: Evolution of the disk and satellite particles of Mode1 2 as viewed in the s z

plane. The dominant halo particles are not plotted. Xote the slight warp for T > 60 model time units. or t > 3.6 Gyr. The size of the boxes is 6 model radial units. or 96 kpc.

Figure 6.2: 10 rings of Mode1 3 projected on the x i plane. .+' on each ring indicates the direction of the y!.

or t = 6 Gy. The figure shows that the larger the radius of the ring. the more it

is tilted. In other words. the warps can be seen very clearly in the figure. The *+'

symbol on each ring indicates the direction of the y: axis. The outer rings. Rings 7.

8, and 9, show a clear warp. and the directions of their yj axes indicate the trend of

precession.

1 then study the evolution of the warps by following the orientations of individual

rings. The orientation of ring i is specified by the two Eulerian angles Bi and $i.

Figure 6.3 shows the r\-olution of 0 ancl O for two typical rings. 1 select Ring 6 to

show the typical beha\-ior of inner rings and Ring 9 to represent the typical heliavior of

oiiter rings. I observe t hat Ring 6 is t iltecl wit h sonie nutation and wit liout precession.

which means that the inner part of the disk is gradually tilted in one direction tvithoiit

precession. On the other hancl. Ring 9 is tilted mhile undergoing some slow precession.

because the outer ring is tiltecl away from the rest of clisk and starts to precess iincler

the torque imparted its displacement. Since the inner disk is slowlp tiltecl towarcis

the outer disk. the warps can last for a very long tirne. As shoivn in Figure 6.1. t h e

warp caused by the satellite in blodel 2 lasts for more than 30 disk-rotation periods.

I conclude that infalling satellites can excite warps in the disk. However. the warps

eventually fade away due to the fact t hat bot h the inner disk and outer disk are ti!tecl

towards the same plane.

6.3 The Particle Mode1

.An alternative way to study the kinematics of warps is to follow the clirections of

orbital angular mornenta of individual disk particles which are located at different

radii. I have plotted in Figure 6.4 the evolution of the directions of the angular

momenta in polar coordinates ( O . O ) for four typical particles as well as the traces

of their positions. In t h e iipper left panel. 1 consider the case in which a part icle is

located in the disk. The motion of the particle is mainly circular with little inwards

shifting caused by the decreasing of rotation velocity ancl the increasing of random

veloci ty due to two-body relaxation. However. the direction of the angular moment um

shows clear evolution. The mean value of 0 increases wi th time, which means that

the orbit of the particle is graclually tilted by the infalling satellite. An increase of

the dispersion of 8 , with movement from the origin. indicates that the particle is

gradually heated due to two-body relaxation. There is no precession of the angular

rnomentum for such a particle. In the di&, the angular mornenturn of most particles

evolves in this way, which irnplies that most of the disk does not precess. This result

Figure 6.3: Evolution of the two Eulerian angles (p. tl) plot t.ed in the polar coordinates frame of Ring 6 and Ring 9. Ring 6 is tiltecl with some nutation. bu t does not undergo precession. Ring 9. however. precesses very slowly.

Figure 6.4: Ei~olution of the angular momentum directions ( O 0 . O") plotted in the polar coordinate frarne for four typical disk particles. The traces of their positions (r. 6') are also plotted in the polar coordinate frame, which are shown in the upper right corner of each panel. The symbol + indicates the initial positions and angular momentum directions of t h e particles. The simulation is run for 50 rotation periods.

is consistent ivith the conclusion of our ring mode1 study. which suggests that the

innermos t rings do not precess.

In the upper right panel of Figure 6.4, 1 consider the case of a particle which is

very close to the eclge of the clisk. The part icle orbits witli increasingl~. greater radiiis

around the center of t h e disk diie t o the increnient of angiilar nionientum raused

by the infalling satellite. .-\s t h e particle moves outivards. its mean orbital plane is

gradually tilted a w q from the rest of the disk because the direction of the angular

momentum absorbed From the satellite is not aligned tcith the direction of the initial

angular angular momentiim. Its angular moment um starts to precess iinder the torqiie

imparted by the infalling satellite. This result is also consistent tvitti the result ttiat

1 have described for the ring model. namely t hat the outerrnost rings precess.

The particle shown in the loaer right panel has a motion and an evolution of

the angular momentiim which is similar to the one depicted in the iipper left panel.

except that the inward motion and the standard deviation of O are larger in this case.

This follows the fact that since 6 is defined as cos-L(I~/ l ) . 8 becomes very large for

a particle with small 1:. as is the case in this example of a particle located near the

center of the disk.

Finally in the lower left panel. 1 consider a particle which is initiallx located at the

edge of the disk. I observe that its mean orbital plane is easily tilted by the inlalling

satellite and that the angular momentum precesses iinder the torque imparted hj- the

infalling satellite.

1 conclude. from the analysis of the results obtained with botli the ring ancl particle

models that the infalling satellite can excite warps in the disk. However t h e warps

eventually fade away due to the fact that both the inner disk and outer disk are

finally tilted towards the same plane. Therefore, the warps are unlikely to be long

lived phenornena. The fact that most galaxies are warped could be a n indicator t hat

disks are, in fact. being subjected to a recent infall of satellites, though the infalling

satellites may not be easily detected because of tidal disruption.

Chapter 7

Measurement of Disk Heating

7.1 Introduction

Since warping in the disk is excited by an infalling satellite, it is necessary t o consider

a local tilted coordinate systern in order to s tudy the disk thickening and its time

dependence. Traditionally. there are two met hods to rneasure the disk t hickening.

One is t o measure the scale height of the disk. h,, and the other is t o measure

the vertical velocity dispersion. a,. These tmo measurements are usually equivalent

because if the presence of a dark matter halo is ignored, the relation hetween hr and

O, is

o: 'X S ( R : +,. (1.1 )

where Z ( R , z ) is the surface density of the disk. This relation is derived from the

vertical component of the Jeans eqiiat ion,

in which Q is the potential field. and v is the space density of particles. On t h e left

hand side of the equation, -g = 4 ~ ~ ~ 5 1 , p(R1 2) = 41rGC(Rl 2). and on the right

hand side of the equation, k&(uo:) = of / h z under the assumption tha t cr, does

not Vary with z. By substituting the above two equations in Equation 7.2, I obtain

R o r r (kpc)

Figure 7.1: Densitics of disk and halo versus radius. The dashed line is a n exporiential fit of the disk density.

Relation 7.1.

However. Relation 7.1 may not he valid across the disk in a typical niodel of this

thesis becaiise t h e contribution from halo particles may not be ignored. especially

near the edge of the disk where the density of t h e halo is much Iarger than t hat of the

disk as shown in Figure 7.1. In the figure. the densities of the disk and halo versus

radiiis for Mode1 O a re plotted. By replacing v with v d + uh and still assuming that

o, does not Vary with z. I obtain that the right hand side of Equation 7.2 is equal

.? 1 .'r to lry-- vd+uh - ' Z (u.i + v h ) I f v i z ~ h . and s u d » Gvh. Relation 7.1 is still \-alid. but. i f 8) -

vd « v h riear the eclge of disk. Relation 7.1 is no longer valid.

I vi-il1 rneasure hoth h, and a: in this chapter. To measiirr either h , or a- is a two-

step procecliire: first. to determine the local tiltetl coordinate system ria the direction

of angular mornentum or the direction of the maximum principal asis of the inertia

moment: second. to measiire t h e t hickness in the determinecl local tilted coordinate

system. I will clevelop. in Section 7.2. an one-step methocl in which the vertical

veloci tj- dispersion is measured in the local t iltecl coordinate system. but rvi t hotit

compiiting the exact orientation of the tilted coordinate system. The thickness of

the disk measured from the traditional two-step method is also given in Section 7.3.

Finally. 1 will briefly surnmarize the results in Section 7.4.

7.2 Vertical Velocity Dispersion

First. 1 demonstrate in Figure 7.2 the one-step measuring method by iising a simple

example. 1 assume that some particles are uniformly distributed in a ring. as shown

in Figure ?a. and that their average velocities in spherical coordinates are = 0. - P,. ancl l;s = O. The velocity dispersions of the particles are a,,. a,, . and a,. The

direction of the angular mornentum Z(9;. oi) of part,icle i is plottecl in Figure 7 . 2 ~ as -.

a point ( O ; . oi j in polar coordinates. The plotted circles correspond to the standard

deviation of the direction of the angiilar mornentum. ad. of the particles located in the

ring. Here. O; is tlefined by the following equation ta.n(Oi ) = l,, / lo t = cle, l u m , . in which

ue, and r,, tlenote the vertical and circular velocities respectivelu. Since ( uc I / vn , 5 0.1 -

usually. 1 can write Bi zz tan(&) = us, lo,,. Therefore. 0; = C T ~ ~ / ~ ' + ~ ~ ~ v ~ / v ~ . --

hr thermore . since vi/ud e 0.01, and = 4.0, the second term is very small

in cornparison to the first term and can therefore be ignored. Finally. 1 obtain the

following relation, 0 6 zz o,,/q, which indicates that the standard deviation in 9 is

proport ional to the vertical velocity dispersion.

If the ring of particles is sirnply tilted by an infalling satellite by an angle 6

Figure 7.2: Particles located on the original ring ( a ) and the tilted ring ( b ) . The

directions. in polar coordinates ( O , q5): of the angiilar rnornenta L. of these particles are plot ted in t h e panels of c ancl cl.

as shown in

i is changed

Figure X b . the resulting 0 0 1 remains i ~ n c h a n ~ e d . -#

into 0; = 6 + 8: (Figure 7.2d). However. if the

though 8; of particle

particles in the ring

are also heated by the inhlling satellite. the value of 061 increases proportionally

wi th increasing a,;, where v i is the vertical velocity dispersion in t h e local tilted

coordinates. Therefore, 1 can obtain the increment of vertical velocity dispersion ir

the local tilted coordinates by measuring t h e increment of 06..

Final1~-. 1 compare the evolution of of llodel 2 and 1Iodel 3 with that of Slodel

O. to calculatr their relative incrernents of velocity dispersion causecl hy infalling

satellites in Figure 7.3. 1 find tliat the infall of the 20% disk-mass satellite mainly

causes Iieat ing near t tie center and edge of the disk. In t h e case of the infall of the

30% clisk-mass satellite. the vertical reiocity dispersion is increased i n the ont ire disk.

but the otiter regions of the disk are heated much more thaii the inner regions. I

find that. when averaged over radii greater tlian the half-mass radiiis. the velocity

dispersion of the clisk of Mode1 2 is increased hy 5%. whereas that of Mode1 3 is

increased significantly niore to 28%. For radii smaller than the half-mass radius. the

average increment in velocity dispersion of hlodel 3 is 6% whereas that of Mode1 2 is

4 XI 8

7.3 Thickness

In this section. 1 measure the thickness of disk in local tilted coordinate frames fol-

lowing a two-step procedure. First. the disk is divided into rings. and their local

t ilted coordinate frames are determined by the principal axes of t heir inert ia tensors.

The thickness of eacli ring is then measured in the local tilted coordinate frarne. The

disk is uniformly divideci into 10 rings from radius. R = 0.0 to R = 20 kpc. For eacli

ring. the principal ases of its inertia tensor are employd as i ts local tilted coordinate

axes. and the t hickness of the disk is measured in the local tilted coordinate frame.

The thickness of disk is defined by the scale height, h-. of asyrnptotic exponential

distribut.ion. sech2(z/h,). It is shown in Figure 3.2 that t h e particle distribution in

vertical direct ion fits the asymptot ic exponential distribut ion ver- well. Therefore.

h , is a good description of the thickness of disk.

1 plot. in Figure 7.4. the scale height, h- , versus radius, R. for Mode1 2 and Mode1

3 (solid lines) compared with that for Model O (dash lines) at T=O. 60. and 120 time

mode1 units. I find t hat when averaged over radii greater than the ha

the thickness of the disk of Model 2 is increased by 7% com~ared with

.If-mass radius,

that of Model

Figure 7.3: Evolution of the dispersions of û of Model 2 and Model 3 in cornparison with that of Model O. In each panel, t h e distributions of the dispersion of 8 at T = 0, 60, and 120 mode1 time units are plotted.

Figure 7.4: Disk scale height, h,, versus radius. R, for Model 2 and Model 3 (solid line) compared with that of Model O (dashed line) at T=O. 60, and 120 time mode1 units.

O. whereas t hat of Sloclel 9 is incrcasecl significantly more to 21%. For radii snialler

t hari t lie Iial f-mass radi LIS. the nveragecl increnient i ri the t hirkness of clisk of Sloctrl

3 relatii-e to t h a t of Slodel O is 12% whereas that of llociel 2 is orily 6%.

7.4 Summary

1 have nieasured the disk heating caused by infalling satellites in Motlei 2 and .\.Iode1

3 through hoth the increments of vertical velocity dispersion ancl scale height relatii-e

to those of 1Ioclel 0. 1 fincl in Mode1 '2 that the infa11 of the 20% disk-mass satellite

niainly causes heat ing near the center and edge of t h e disk. In the case of the infall of

the 30% disk-mass satellite in 4lodel 3 . the entire disk is heatetl. but the outer regioiis

of the disk are heated much more t han the inner regions. Inside t h e half-mass radius

of the disk. Relation 7.1 is approximately valid because the thickness increment is

approxiniately twice that of the increment of the vertical velocity dispersion. In that

region. although the densities of halo and disk are similar. the derivative of density

in vertical direction of the halo are much smaller t han t hat of the ctisk. and t herefore

is negligihle. Hoivever. outside the half-mass radius of disk. half of the thickness

increment is much less than the increment of vertical velocity dispersion. This is

because the density of the halo is larger than tliat of clisk. and Relation 7.1 is not

valicl.

Chapter 8

Orbital Decay and Tidal Stripping of

Satellites

8.1 Introduction

I have investigatecl the dynarnical responses of a disk to infalling satellites in previotis

chapters. In this chapter. 1 will s tudy the dynarnical effects of the parent galasy o n

infalling satellites. namely orbital decay and ticlal stripping. In the motlels of this

t hesis. the orbital decay of infalling satellites is mainly caosed by dynaniical fric-

tion against the dark rnatter halo because the satellites are rapidly disriipted when

the- enter the parent disk. However. in previous work (QG. QHF. WMH). dense

satellites are introduced at the edge of their parent disk. so the cl-namical friction

against the halo is not important. and t h e drag on the satellites is contributeci by

the disk. In those cases. the drag is due to the following three important mecha-

nisms (QG): dynamical friction against the disk, second-order pert urbat ive torque at

Lindblad resonances ( Lynden-Bell & Kalnajs 1972. Goldreich Sc Tremaine 1982). and

nonperturbative horseshoe orbits. A more detailed discussion can be found in QG.

In order t o understand the simulation results which a i l1 be presented in this

chapter, 1 will briefly estirnate. in Section 8.2. the orbital decay rate for a satellite

with deceasing mass due to tidal stripping. In Section 8.3, 1 will first present results

of orbital decay and mass stripping for satellites on nearly circular orbits but difFerent

orbital incliriations with respect to the parent clisk plane. I will t hen disciiss t lie resolts

for satellites Iiaving different niasses on an elliptical orbit. Finally. 1 will compare

t h e orbital clecay rates for a satellite on both direct and retrograde orhits. Ttir

distribution of satellite remnants in pliysical and phase space is studied in Section S.-!.

Finallj-. a brief surnmarj- is given in Section 8.5.

8.2 Theoretical Estimates

.As the orbit of a satellite decays inside the halo. the mass of t h e satellite clecreases

due t o tidal stripping. I will first show that the fraction of remaining mass of the

satellite. .II(r,)/:II,. is proportional t o the fraction of halo mass ivithin the location of

the satellite. :M(r)/Mh. i f the satellite and the parent galaxy have self-similar density

profiles. Basetl on this resolt and Chandrasekhar's formiila. 1 will show that the

orbital decaj- rate is a constant as a function of the distance between the center of

the satellite and the center of i ts parent galaxy. and it is proportional to the initial

mass of t h e satellite.

Since the density profile of the King moclel cannot be expressed explicitl-. I ilse

9 t lie following simi1a.r density profile as s substitute. p = =a;/(r2 + r i - ) for r 5 r,,;,

and p = O for r > rsizc. I assume t hat the tidal ra.dius of the satellite. r,. is clefined

by the Jacobi limit (Binney S- Tremaine 1987). I then have the relation between the

rnean density of the satellite insicle the tidal radius and the mean clensitv of the halo

inside r where the sateIlite is located.

By substituting p , ( r , ) = &aj(l - ( r ~ , s / r t ) tan- l ( rr / rK,s)) / r l R -&o:/r: for ri > >

i ~ , ~ and p h @ ) -&oi/r2 for r >> rK,h in the above equation, 1 obtain the relation

between r, and r, r, rr: (~ / f i ) (o , /o~) r . By using Equation 2.1 in the above relation.

1 have t h e following relation: r t / r , x ( l / f i ) ( r / r h ) ? where r, and r h are the sizes of

the satellite and the halo respectively. Since the ratio of t he remaining mass to the

total niass of the satellite is .If ( r - , )/.\l, - ( r-,Ir, j. and the ratio of the niass inside r

to the total mass of the halo is . \ l ( r . ) / J l h z ( r / r h ). I have the relation between t hese

ttvo mass ratios.

O 1 - ( ) / ( ) ( - ( / h j . Thus. as the orbit decays. the mass loss

of the satellite is rotighly proportional to the halo mass which the satellite traverses.

In reality. the mass loss of the satellite can be higher tlian our estimate because of

the eccentric orbit. tidal distortion arici evaporat ion of the sateiiite.

Chandrasekhar (1943) showed that a particle of mass -11, moving through a homo-

geneous background of individually much Iighter particles with an isotropic velocity

distribution suffers a drag force.

where c, is the speed of the satellite with respect to the mean velocity of the field.

ancl p ( < r,) is the total clensity of the field particles with speed less than us. The

parameter A = pma, /pmin . where p,,, is conventionally taken to be the hall-mass

radius of the field system. and p,,, is the larger of the two scales pg0 z G3ls / r ;

and the Iialf-mass radius of' the satellite ( White 1976). Detailed studies (Lin L

Tremaine 1983. White 1983. Tremaine k Weinberg 1981. Weinberg 1986. Bontekoe

& van Albada 1987, Zaritsky 5: White 1988) show that Equation 8.3 gives a reliable

estirnate of the rate of orbital decay. On the other hand. Weinberg (1989), Hernquist

S: Weinberg (1989), and Prugniel k Combes (1992) showed tliat Chandrasekhar's

formula may not adequatelj- represent the orbital decay of an extended satellite in

general due to self-gravitating of the halo and tidal deformation of the satellite. Since

I will discuss the relation among the orbital decay rate. the distance between the

satellite and parent galaxy? and the initial mass of t h e satellite: rather than calculate

accurate decay tirne, Equation 8.3 is employed in following discussion.

By assuming that the halo is an isothermal sphere wit h one dimensional velocity

C'H.4 PTER 8. ORBITAL DEC.4 k' A:YD TID.4 L STRIPPI.YC; O F S.4TELLITE.5

dispersion 0. and t h e satel1it.e follows a slowly clec-.ing circular orbit. Binney k

Treiiiairie ( 1986) obtainecl ( Equation 7-35)

whew L*, = fin. Equation Y.-! indicates that t he

proportional to t h e distance between the center of

orbit,al tlecay ra te $ is inversel!

the satellite and the center of its

parent galaxy. hoivever. it is proportional t o t h e mass of t h e satellite. In the moclels

of t his t hesis. since self-similar clensity profiles a r e employed for t h e satellite and the

halo. ticlal stripping iç unavoiclable. The mass of t h e satellite decreases as its orhit >

decays. as shown in Equation 8.2. By repiacing .Clh(r) - y in Equation (1.2. 1 obtain

the fol lowi ng equat ion

Eq~ ia t ion 8.4 t herefore can be rewri t ten as

Ecluation 8.6 shows that . due to the ticlal stripping as approsinmted for the niodel

of this thesis. the decay rate of the orbital radius is a constant as a fiinction of the

distance between t h e center of the satellite and the center of its parent galaxy. and

it is proportional t o t h e initial mass of the satellite.

8.3 Simulation Result s

I first st tidy the orbital decay rate of a satellite on a nearly circular orbit. However

the orbital inclination with respect to the parent galactic plane is 30". 60'. and 90'

for Mode1 4. Mode1 3, and Model 6 respectively. The satellites have 10% of the

disk mass and star t a t a distance of 2.5 mode1 radial units from t h e center of the

parent galaxy. Figure 8.1 shows orbital decay and mass stripping for Morlel 4 ( top) ,

Model j(middle), and Model 6 (bottom) respectively. In the figure. t he mass inside

Figure 8.1: Orbital decay and mass strippitig of satellites in Mode1 1 (top). Mode1 .i (middle), and Mode1 6 (bottom). The first column shows that the mass inside the initial half-mass radius of satellite decreases linearly with the decreasing distance between the center of the satellite and the center of the parent galaxy. The second column shows that the orbital decay rate is approximately constant. Hence. the mass loss rate is also approximately constant, as shown in the third column.

the i r i i t ial tialf-mass radius of t iie satellite is riornialized according to the follou-ing

relation. j = r r z ( ~ - ~ , ~ ~ ) / r n ~ , ~ , , ~ . For example. at 1 = 0. f = 0.5. Duc to ticlal stretching

ancl stripping. the center of mass ni- riot be located at the center of t he satellite. I

therefore redefine the density peak of the satellite as the center of t h e satellite. As

the orbit clecays. the satellite is gradiially disrupted. and the location of the density

peak of t h e satellite becomes lincertain. If the standard deviation of the position of

the clensitu peak becomes larger than the initial half-mass radius of the satellite. the

satellite is consiclerecl completely stripped.

In the first colurnn of Figure 8.1. I plot the distance between the center of the

satellite and the center of the parent disk. r. versus the mass insicle the i n i t in1 hnlf-

mass radius of t h e satellite. f . It shows that f decreases linearly with clecreasing

r . In the second column. r versus t is plotted. It shows that the orbital decay rate

is approximately constant. which is in accord with the prediction of Equation 8.6.

Finally. in the last colurnn. f versus t is plotted. The mass loss rate. g. is also

approximately constant. The satellites in t hese three models start at a distance of

2 . - mode1 radial units and are completely disrupted at a distance of approximately

1.6 model radial units. The stripped satellite remnants are clistributed in a n ann~ilar

ring. between 1.0 and 5.0 model radial units from the center of the parent galas'.

whereas the disk particles are distributed between O and 1.2 nioclel radial units from

the center. Since there is little interaction betweeri these satellites and the disks of

their parent galaxies. the orbital decay is independent of the orbital inclination.

Secondly. I investigate the orbital decay rate and mass stripping of a satellite on

an initially elliptical orbit with e = 0.2, but the satellite has 10%. 20%: and 30% of

the disk mass for 3Ioclel 1. Mode1 2. and Model 3 respectively. From top to bottom.

Figure 8.2 shows the mass inside the initial half-mass radius of the satellite versus

the distance between the center of the satellite and the center of the parent galaxy

for hlodel 1, Model 2, and Model 3 respectively. I find that during the same time

period. t h e apocenter r+ of the satellite orbit decreases by 1.0. 2.0. and 3.0 model

radial units for Model 1. Model 2, and Model 3 respectively. In other words, the

Figure 8.2: Mass inside the half-rnass radius of the satellite versus the distance be- tween the satellite and the parent galaxy for Model l. Model 2. and Model 3. T h e heavier t he satellite, the faster t h e orbit decays, and the faster t h e orbital eccentricity decreases.

C'H.4 PTER S. ORBIT-4 L DEC.4 1.' .-l.VD TID.4 L STRIPPIXC; OF S-ATEL LITES

orbital der-- rate nieasured a t the apocenter is proportional to the initial niass of

the satellite. On t h e other hand. the orbital decaj- rate measured at the pcricenter

r- is also proportional to the initial mass of the satellite. It is also consistent with

t h e precliction of Equation 8.6.

The eccentricity of the satellite orbit is defined bu e = ( r + - r-)/( r+ + r- ). I find

tliai in 1Ioclel 1. the final r+ and r- of the satellite orbit are 3.0 ancl 2.0 moclel taclia1

iinits respect irely. Therefore. the final eccent ricity the orbit is 0.2 ivhich remains

~inchanged [rom its initial value. On the other hand. in 1loclel :3. the evolut ion of

orbital eccentricity can be ciivided into two stages. One is a slow decaj- stage in the

halo where the orbital eccentricitx remains almost unchangecl. Ancl the other is a

fast decay stage in the disk where the orbital eccentricity rapidly clecreases to zero.

The fast clecay stage in the disk can be explaineci as following. First. it is clue to the

addit ional clynamical friction against the disk. Secondly. the eccent rici tj- decay rate.

according to Equation 9.21. is y x -+. For large r . the decay ra te is ver' small.

t he orbital eccent ricity is nearly constant. but. for small r. the eccent ricity decays

ver'. fast .

Finally. anot her interesting point is to compare the orhital clecay of direct (I lodel

Y ) and retrograde (Slodel 9) satellites. This is done in Figure 8.3. T h e sirnilar orbital

decay for both satellites show that the dynamical friction exerted on tlie satellite by

tlie halo is the major factor that causes tlie orhital decay of the satellites. Howeirer.

the fact that the retrograde orbit decays faster than the direct one suggests tliat the

distant interaction between the satellite and the galactic disk is not completely neg-

ligible. although a 10% disk-mass satellite is completely disrupted in t h e halo before

it enters the galactic disk. This is due to the fact that the Lindblad resonances of the

disk esert a perturbative torque on the satellite (QG). Lynden-Bell Sr Kalnajs (1972)

have shown that a uniformly rotating. perturbing potential exerts a negative torque

on inner Lindblad resonances and a positive torque on outer Lindblad resonances.

In other words. the torque on the satellite due to the inner Lindblad resonances is

positive and that due to the outer Lindblad resonances is negative (QG). Figure 4.3

C'HAPTER S. ORBIT.4 L DEC'A 1.- .-LW TIDA L S T R I P P L W OF SATEL L I TES

Figure 13.3: Orbital clecaj. of the direct and retrograde orbital satellites as a function of time. The retrograde orbit decays laster than the direct one.

shows that the outer Lindblad resonances are not present in the mode1 clisks. but that

t here are two inner Lindblad resonances. Therefore, the inner Lindblad resonances

exert positive torque on the satellite. This srnall positive torque increases the anpiilar

momentum of a direct orhit satellite resulting in a sloiver decay of the satellite orbit

thaii would otherwise occur if only dynamical friction was involved. On the other

hand, 1 find that for the satellite rnoving on retrograde orbit. the torque is opposite

to t h e angular rnomentum. This, as expected, decreases the angular momentum and

the orbit decays faster. The fact t hat the retrograde orbit tlcca>-5 faster t han the

direct one when the satellite is located outsicle the disk is opposite to wtiat is foiind

when the satellite is locatecl inside t lie tlisk ( Q G ) .

8.4 Satellite Remnants

1 show in Figure 8.4 the evolution of the satellite particles projected on the sy and .rz

planes for 'vlodel 3. -1s the orbit of the satellite deca~-s inside the halo. the satellite is

partial11 tidally stripped by the halo. Once the satellite remnant enters the disk. it

is rapidly disrupted by the tidal force due to the high density of t he disk. T h e sniall

core of the satellite survives for about 3.5 disk rotation periods or 8.4 Gyr. The final

shape of the tidally st rippecl satellite part icles resembles a warpecl t.orus. T h e planes

of the ootermost and innermost parts of the torus are aligned with the planes of the

initial and final orbits of the satellite.

I find that the 10% clisk-mass satellite of blodel 1 is completely stripped before it

enters the disk. Only 2% of the satellite particles are accreted onto the edge of the

disk. T h e 20% clisk-mass satellite of !vlodel 2 is completely stripped at the edge of tlie

disk. About 11% of the satellite particles are accreted ont0 the disk. and tlie particles

are located outside of the half-mass raclius of the disk. Finally. the 30% disk-nias

satellite of Slodel 3 is rapidly stripped when it enters the clense disk. Approximately.

18% of the satellite particles are accreted onto the disk. and the particles. as in the

previous case. are mostly located outsicle of the solar circle.

Figure 8.5 shows that the distribution of the satellite particles of Mode1 3 in phase

space ( C 5 0 t , r ) is very different from t hat of the halo particles. not only because of t heir

high rotation speeds. but also because of t heir nonuniform distribution. If the stellar

halo of a galaxy is produced by multiple satellite accretions. the stellar distribution

in phase space should be clurnpy. This is in accord with the recent observation of

halo stars (Majewski, Hawley k Munn 1996) which shows a high degree of d u m p i n g

in the U-V-W-[Fe/H] distributions.

Figure 8.4: Satellite particles of Mode1 3 projected on both the xy (top two rows) and xz (bottorn two rows) planes. The size of the boxes is 6 mode1 radial units or 96 kpc.

Figure 8.5: Distribution of the satellite particles of Mode1 3 in phase space (I/;,,.r). Only Iimited phase space is filled by the satellite particles.

CH.4 PTER 8. ORBITAL DEC.4Y- .1.VD TID--1L STRIPPI!VG OF S-ATELLITES

8.5 Summary

For the examples considered in this chapter. I ha\re shown tha t all 10% disk mass

satellites are completely t idal l - strippecl hefore the- penetrate the disks. Since the

orbi ta1 clecax is niainly caused by the dark niatter halo instead of the parent disk. the

orbital parameters. such as orbital inclination of the satellite relative to the parent

galactic disk and the direction of the satellite rotation relative to the direction of

disk rotation only play a small role in the satellite orbital deca-. The orbital decay

rate is constant for both circular ancl elliptical orbits. and it is proportional to t he

initial mass OF the satellite. The orbital eccentricity is nearly constant in the halo

but decreases rapidly in the disk. The mass inside the initial hall-mass radius of t he

satellite decreases with increasing time ancl clecreasing distance between the center of

the satellite and the center of the parent disk. Most of the tidally disrupted satellite

part icles are accreted ont0 the halo ancl the outer disk rat her t han onto the inner disk.

In physical space. the final shape of the tidally stripped satellit,e particles resembles

a ivarped torus. The planes of the outermost and innermost parts of the torus a re

aligned with the planes of the initial and final orhits of the satellite. In phase space.

t he satellite particles are distinguished from halo particles by t heir high rotational

speed and clumpy distribution.

Chapter 9

Orbital Decay Rate of Satellites and

Galactic Accretion

9.1 Introduction

Slost satellites orbit around their parent galasy on elliptical orbits. For esample the

L41C and SMC orbit around Our own Galaxy on elliptical orbits ( Murai % Fujimoto

1980. Lin S. Lynden-Bell 1982). It can be understood this way (Holrnberg 1940.

Tremaine 1980): when a small galasy has a close encounter wi th a big galaxy. the

small galasy will lose kinetic energy due to tidal dist ~trbance and may become a bound

satellite galaxy. i.e.. its parabolic orbit changes irito an ellipticai orbit with a rather

large eccent ricity. Then, because of the dynamitai friction. every subsequent passage

of the satellite galaxy t hrough the pericenter tends to decrcase the eccentricity. and

eventually. the satellite galaxy will spiral into the parent galaxy. It is obvious that

satellite galaxies spend nearly al1 their lifetime on elliptical orbits. However. in ana-

lyt ical calculat ions. t lieir orbits are usuaily t reated as circular for simplicity. Ariot her

simplification in traditional estimates of orbital decay rate is t h e assumption of a rigid

satellibe, which is subject to no tidal disruption. However, if both the satellite and

its host have self-similar density profiles. the rnass loss of t h e satellite is proportional

to the host mass which the satellite traverses. It is therefore necessary to calculate

the orbital decay rate of a decreasing mass satellite on an elliptical orbit.

1 ~ i . i l l first irii-estigate. in Section 9.2. t h e orbital tlecay rate of a tlecreasing m a s

satellite on ari ellipt ical orbit by eniploying analyt ical approximations arid direct

integrations. S~conclly. hased on the resulting orbital clecaj- rate. I rvill then. in

Section '3.3. calc~ilate the galactic accret ion rate bj- follorving Trernaine's est iniate.

bu t incliitle the following modifications: t idal disriipt ion of satellites. elliptical orbits.

non-constant rnass-to-light ratio for tlwarf galaxies. upper mass lirnit for satellites.

and high peculiar velocity cutoff. Finally. a hrief summary is given in Section 9.4.

9.2 Satellite Orbital Decay

The flatness of niany observed rotation curves of spiral galasies suggests that the

density distribution of a galactic halo is given by the following expression

ivhich holds for a singular isothermal sphere with circular velocity r, and velocity

dispersion O = r,/&. As a satellite of mass .I(,(r) orbits through t h e galasy. it is

suhject to dynamical friction

in which .Y = I * - \ ~ / U , . and A is the ratio of the size of the parent galaxy to that

of a satellite' and erf is the error function, erf(X) = 5 J$ e - X 2 d ~ . The angular

momentum loss of the satellite can therefore be writ.ten as

By substituting Equation 9.1 and Equation 9.2 in Equation 9.3. 1 obtain the following

eciuat ion.

For a circular orbit. .Y = I . L = LI = r,r. Equation 9.4 cari be rewritteri as

For a rigid satellite of .\.l,(r) = .II,. the dynamical friction time is

Hotvever. the clecay of an elliptical orbit is much more cornplicated than that of the

circular one because there is no explicit expression of as a function of r for

t h e elliptical orbit. I therefore have to use a few approximations in the following

calculation. The orbital time of a satellite is TOr6 5 5. and the ratio of - q r , ,

r $ GAI, -. 2 < 1. Since Torb < Tfrici a continuous orbital d e c q ( the solitl line in

Figure 9.1 ) can be treated approximately as a series discontinuotis ellipses ( the dash

line in Figure 9.1). Each of the ellipses is characterized by their apocenter r,. r+/.

... On each ellipse. there are the approximate conservation of angular momentum

per unit mass. L+/:ll ,(r+) s L_/ .LL(r- ) , i-e., r+tl+ x r-r- . ancl the approsimate

conservation of energy per unit mass E+/.bl,(r+) z E-/.ll,(r-). Le.. $ P : +c< ln ( r+) z

Ir: + t?: ln(r-) . in which the expression for potential @ ( r ) = cz ln ( r ) is employecl. 1

t hen obtain the following relation. 5 = Q=& 2e In '+r in ivhich e is the eccentricity 1-0 '

defined by e = . For a given e. the ratio of 2 is listed in column 2 of Table 9.1.

On the other hand. 1 assume that the eccentricity e is a constant as the orbit decays.

This assumption is discussed later in this section.

Since the satellite velocity. o , \ ~ ~ is not constant on elliptical orbits. 1 calculate the

average angda r moment urn loss rate on each ellipse according to following equat iono

r i- ' cir

Figure 9.1: Satellite orbital decay. The solicl line shows clliptical orbital decay of a satellite. and the dash line is the approximate orbital decay. The circular orbit decay referred to in this thesis is shown by the dash-dot line.

Substituting Equation 9.4 in Equation 9.7. 1 obtain

On a given ellipse, using the approsimate conservation of energy per unit mass

E z E+. I obtain that the square of the satellite velocity at any distance r from the

CH-4PTER 9. ORBIT,-11, DEC.41' R.-\TE OF .S.-\TELLITES .4ND G.4L.AC'TIC' ,-1 C'C'R ETIOLV

ancl the square of the radial velocity of the satellite is piven bu the following expression

r+ 2 "+] . ) + ? I n -

'le L - E I* r (9. IO)

.\ssuming that infalling satellites have clensity profiles similar to that of the host.

ancl according to Equation 8.2. I obtain

Combining Equation 9.11. Equation 9.9. and Equation 9.10 into t h e integration Equa-

tion 9.8, I then have

where !(el is listed in colurnn 3 of Table 9.1. The tlynamical friction time is

Equation 9.13 can also 'De rewritten as

Assurning I n n = 3.0. (7, = 220 km/s. :\l,/Mh = 0.1. a n d r = 50 kpc. 1 obtain

in column 4 of Table 9.1: the dynarnicai friction time ta,, for satellites on elliptical

orbits with e = O to 0.9.

The orbital decay time ratio between ail elliptical orbit and a circiilar orbit is

tellip/tcir = j(O)/ f ( e ) , with r+ = r,,, For a typical elliptical orbit, r+ oc/&

i.e.. r = 0.3, t h e dynamical friction time is 78% of that for the circular one. 1 have

C'HA P T E R 9. ORBIT.4 L DEC'A k- R.4TE OF SATEL LITES .-LW (3.4 L A C T I C ' .-i C'T'R F,'TlO!\i

Table 9.1: Dynarnical Friction Time and Galas! Accret ion

ernployed r+ and v+ to describe the satellite orbit because in my simulations. satellites

are introduced at a distance r+ [rom a parent gaiaxu with a tangential velocity r+.

In terms of mean radius. a. of an ellipse. Equation 9.13 can be rewritten as

The value of / ( ~ ) / ( 1 + e ) varies from 0.428 to 0.389 for e = O to 0.9 (Table 0 . 1 ) . It

shows that most of the elliptical orbits can be simplifiecl as circular ones w i th less

than 10% error.

In t he above calculation, for simplicity. 1 assume that the orbital eccentricity is

constant as an orbit decays. I discuss here the decay rate of an orbital eccentricity

by employing t h e first order perturbation theory. 1 rewrite the dynamical friction

in which &Y) = [ e r f ( ~ ) - $$c"~]/.Y~. In plane polar coordinates ( r , ~ ? ) . the equation

of motion for a satellite located a t (r .4~) is

By assumirig t* , = i-. ancl rcv = L*, + L*,.. ivhere bot h t*, and r:,-. are sniall per t~ i rha t ions .

Equat,ion 9-17 can b e rewritten into

.tf, ln 11 in which = c , / r . k = f iAl , , dg(.Y). and t h e second

neglec t ed in t h e ecjuat ion.

. order terms. rz. a n d c,c*,.. are

Since a n d X. change ve- slowly with t ime. t hey a re treated as constants when

1 solve Equat ion S. 18. T h e solutions for t h e equation a re

2 cc k d tv, = A esp ( -k t ) cos(-&t + a.) - Pd2 + p

and

In which .4 a n d L- are constant. and they depend on t h e initial conditions ïo. c*o. and

.At a n d c-' a re also constant and can be determined by t h e knowleclge of .-l a n d L*. For a

circiilar ocbit. .A. -4'. c*. and Q' are equal t o zero. The first terms in both Equat ion 9.1'3

and Equation 9.20 show hom an elliptical orbi t clecays towards a circular one.

Integrating Ecluation 9.19. I obtain t hâ t r ( t ) = ~ e i i ~ ( - k t ) s i n ( ? f i d l + U T ) + r . In

which r is t h e i n tegrat ion of t he second t e rm in Equation 9.19. and i t varies very slowly

comparecl with r l ( t ) . Orbital eccentricity is t hen defined as e = ( r + - r - ) / ( r + + r - ) =

2Bexpi-kt) /r . a n d

For large r - . the tlecay rate is ver>- small. and the orbital eccentr ic i t~ is nearly constant.

however. for small r. the assumpt ion of constant eccent ricity for ellipt ical orbital clec-.

is no longer valid.

For a given set of 1.0. e ? ~ . tu and c.0. the orbital decay time of a satellite cari

be calculated by integrating Equation S. LY numerically. .Assume ln !\ = 3.0. -11 =

2.0 x 10'~.11,. for ro = JO kpc. c.0 = 0.0. ïo = 0.0. and &*O = c+/ro. the numerically

integratecl cljmamical friction time t i r L t is listed in column 5 of Table 9.1. Comparing

the +-namical friction time ta,, in column 4 with tint in column 5 . 1 fincl tliat the

clynamical friction times calculated from both the analytical approximation and di-

rection integration are consistent. escept those of high eccentricities e = 0.8 and 0.9.

This is because for low e. $ .x e O. and the constant eccentricity assumption in the

analytical approximation is valid. however. for high e. the constant eccentricity as-

sumpt ion in the analytical approximation is no longer valid. In general. the analytical

approximation predicts a very good approximation of orbital decay time. Therefore.

based on the analytical approximation. 1 ivill estimate the galactic accretion in the

following section.

9.3 Galactic Accretion

Employing the two-point correlation function for galaxy distribution Groth S: Pee-

bles 1971) and Schechter's (1976) luminosity function for galaxy mass distribution.

Tremaine ( 1980) estimated the total accretion of a typical galaxy.

in which the number density of satellites of mass MS. at a distance r from a given

galaxy is n(r, iMs)diM, = no(iLl,)d!Cl,(r/ro)-7. Based on Schechter's luminosity fiinc-

t ion arid coristant inass-to-light ratio. t lit. field riensit>- of satellites is ! t u ( .lis )d.\I, =

n-(.\Is/.\I- ) - " esp(-.IIs/.1.I.)d(,ll,/.ll.). in wliirti n = 1-25. n. = 1.2 x l ~ - ' h ' l ~ ~ c - ~ .

ancl L. = 1.O x LO'OL.; , . r ( t w ) is cleriveci froni Ecliiation 9.6 by ernploj-in:, t j r , , = i /Ho.

r.(t ) = J-Hor~c,. Equatiori 9.23 is t hen rewrit ten as

whicli is Eq~iatioii 7-31 of Binnev Sr Tremaine (1986). For ro = :3h-'SLpc. vc = 220.

Ho = lOOh km/s/hIpc. J I I L = 12h.\li / L (. . Ecluation 9.24 gives .\I,,/.l% = 0.16.

In this section. 1 calculate the galactic accretion rate by folloiving Tremaine's

( 19230) estimate. but include t h e following nioclificat ions. The first modification is to

incliide tidal disruption and elliptical orbits of satellites. By assuming that satellites

have clensity profiles comparable to that of the host. I obtain r ( t H ) from Equation 9.13.

Secondly. some nearby galaxies have rather high relative peculiar velocities. and

they are not likely to merge with each 0 t h . The relative line-of-sight peculiar velocitj-

distribution from CfAi (Davis tk Peebles 1983) and Cf.\'+SSRS'L redshift surveys

(4Iarkze et al. 1995) is

In which 1.; is the relative line-of-sight pecuiiar velocity. and 012 is the rins relative

line-of-sigh velocity Davis SL Peebles found that 012 = 340 f 10 kni/s. and Markze

et al. measured ai* = 540 180 km/s. btit o 1 2 drops to 295 f 99 km/s if al1 Abell

clusters with richness ciass R 2 1 are removed. For the case cliscussed in this thesis,

a reasonable value for a12 is 300 krn/s. At a given distance from a typical galaxy,

nearby galaxies on elliptical orbits are likely to merge. The upper Iimit for the relative

peculiar velocity is /pi2 + V: + Iri2 = a u c . '4

Therefore. the fraction of the galaxies which

which corresponds to a parabolic orbit.

are going to merge is

IrJ + I ~ I ) / ( ~ ~ ~ d l $ c f C i l K ,

in whicti 1;' + 1,; + I-1' 5 2c-f. and I ha\-e assiinied isotropie rms relative velocitj-

(T, = og = 06 = cl2. The integration gives P = :35%. where (;- = 220 km/s is i i s~d.

Due to higli relative peciiliar velocitj- cutoff. orily 35% of galaxies are likely to nierse.

Ttiirrllj-. non-constant mass-to-light ratio for dwarf ellipticals is emplo~ecl. I as-

sume L / L . = (M/.\.I. )y3. which is based on the relation :CIIL x L-'-' for dwarf

ellipt.icaIs (Dekel k Silk 1986. Peterson k C'altlwell 1993). Then the mass distribu-

tion of satellites becomes

Finally. I integrate the satellite mass from O to 0.3.11. rather than from O to

x in Tremaine's estimate because I only consider the infall of low niass satellites.

Therefore. a typical galaxy accretion is

Combining al1 the modifications into the integrat ion Equat ion 9.08. I obtain

.Assuming that r?, = 22Okm/s. Ha = 100 km/s/Slpc. ro = 3.0 Mpc. and 1nA = 3 are

reasonahle. I obtain the final result MaCc/M. = O.O0h(f) . where h ( e ) = (f (e)/f(o))'.'.

which is listed in the last column of Table 9.1. varies from 1 to '3. Furthermore.

- ( f ) ( C f ) For JId;3k/ . \ I . = O. 1. -\focc--lIdlsl- = O.Zh(e) . ibfacc/~bIddsk -

For a median value of e=O..j. the accretion rate is -Ilacc/i\ld;sk z 30%. According

to the simulations in this thesis. t h e infall of a 30% disk-mass satellite can ~ r o d u c e

observable thickening only beyond the half-mass radius of the parent disk.

1 calculate the orbital d e c q tinir of a satellite which esperiences ticlal clisriiption as

its ellipt ical orbit decays. bu employing both analvtical approximation and numerical

integration niethods. For a median eccentricity. e = k.3. the orbital decay time is

approxirnately .;O% shorter than tha t of the circular one if r+(t = 0 ) = r ( t = 0).

The orbital decay times estirnated by hot h metliods are similar. t herefore. based on

the analytical result. I calculate the galactic accretion for the case t hat the infalling

satellites have low masses. In estimating the galactic accretion. I follow Tremaine's

( 1980) met hocl. but incliide the following modifications: high peculiar velocity cutoff.

tiigh mass cutoff. and non-constant mass-to-liglit ratio for tlwarf galaxies. In t his

case. the \ A u e for the accretion ra te varies between 13% ancl 23% of Tremaine's

estimate. For a median eccentricity. E = 0.5. a typical galaxy. like our own SIilky

U*ay C;alaxy. has absorbed about 3 % of i ts own mass. or 30% of its d isk mass. if the

disk niass is assiimecl t o be 105% of t h e total mass. in the form of low mass infalling

satellites. Accorclinp to the results drawn from the simulations in this thesis. it can

be acconirnodatecl without unacceptable disk thickening wit hin t h e half-mass radius

of the disk.

Chapter 10

Discussion and Conclusion

10.1 Density of Satellites and Galaxy Accretion

Rate

The infall of a high density satellite on its parent clisk is of interest because the hi&

clensity satellite coulcl produce some observed features in the di&. sirch as the for-

mation of the thick clisk nricl the bulge. On the other hand. a low density satellite.

which is tidally disrupted before it reaches the inner disk. procluces little observable

eviclence in the inner disk. and it has heen neglected in the past. However. Zarit-

sky (1996) found evidence of recent accretion in the outer tlisks of nearhy late-type

galaxies by estimating the durat ion of steep abundance gradients. elevated rates of

star formation. and outer ciisk asymmet.ries. The fact that accretion orito the outer

disks is common. ancl the inner clisks remain undisturbetl suggests that the infallirig

satellites are tidally disruptecl before the. reach the inner disk. therefore. they are

low density satellites. This is not surprising since al1 but one of low mass galaxies i r i

our locai group are low density galaxies. Therefore. accretion of Iow density satellites

appears t o be more common than that of high density satellites.

By iising the thickness and the Toomre parameter Q of the Galasy at the solar

circle, TO estimatecl an upper limit for the disk accretion rate. In their model. density

profiles of satellites are described by the Jaffe Model, p oc A. Such funct ional

forms lead to an estremely tiigli densitu at ttie center of the satellite in comparisori to

the density of the parent galaxj.. Tlierefor~. the core of such a satellite cotild sur\-ive

and reach the center of the disk. For csaniple. in the tivo cases considerecl bu TO.

84% and -KI%. of the satellite n i a s respectively. cannot be tidally stripped: it reaches

the center. and tremendous heatirig in the tlisk is tinavoidable. Therefore. a low disk

accretion rate is required. TO Ftirther suggested a iow clensity iiniverse in which the

galaxy accretion is suppressed at the current epoch. Mowewr. I foiind that a low

tlensity satellite is mainly accreted ont0 the outer halo anci the outer tlisk. Therefore.

1 argile t hat first. the disk accretion rate is much smaller than the galaxy accretion

rate. and secondly. that the accret ion rate nt the inner disk is much smaller t han t hat

a t the outer disk. Therefore. the t hinness and coldness of the inner disk can be used

to set a limit on the special high deiisity satellite infall rate. but certainly not on the

general satellite infall rate. In the case of satellites having densitj- profiles siniilar to

that of the host galaxy. a high galaxy accretion rate and thin disks can coexist.

10.2 Disk Tilting, Holmberg Effect, and Rotation

of Stellar Halo

In the course of a search for cornpanions of nearby bright spiral galaxies, HoImberg

( 1969) found t hat essentially none of the physical cornpanions were wit hin 30' of the

major axis of ttie parent disk. QG discussetl a possible explanation for this effect ancl

attempted to explain it by employing the relation between the orbital decay rate and

the orbital inclination with respect to the plane of the parent disk. However. their

simulation results predicted a very weak effect. Here. I find that disk tilting caused

by an infalling satellite can produce a somewhat stronger effect.

I have shown that disks respond to infalling satellites primarily by tilting. 1 found

t h a disk tilts towards the orbital plane of a direct satellite (Mode1 3) and away

from the orbital plane of a retrograde satellite (Model 9 ) 7 as shown in Figure 10.1.

Model 9 has the sarne mode1 parameters as those of Model 3. except for the rotational

Figure 10.1: Evolution of the disk and satellite particles of Mode1 3 (top two rows) and Mode1 9 (bottorn two rows) as viewed in the .rz plane. The size of the boxes is 6 mode1 radial units. or 96 kpc. The disk tilts towards the orbital plane of the direct satellite and away from the orbital plane of the retrograde satellite. Note that due to disk precession. the disk appears much thicker than it actually is.

direction of its satellite. In other words. as the orbit of the direct satellite deca~-S. the

orbital inclination with respect to the plane of the disk decreases. On the ot her hand.

as the retrograde orbit decays. the orbital inclination with respect to the plane of the

disk increases. Since the orbital decay of distant satellites is caused by the cl>-naniical

friction arising from the halo. the decay rate does not depend on the orbital direction

and inclination. Therefore. my simulation results imply that a rather unilorm spatial

distribution of distant satellites is expectecl unless the halo rotates significantly. This

is in accord with the observational result of Zaritsky et al. (1993). However. the

orbital clecay of close satellites is mainly caused hy the clynamical friction arising

from the disk. Therefore. low inclination and direct orbits decq- faster than high

inclination and retrograde ones. As a resiilt. there should he a net escess of siirviving

close satellites on high inclination and retrograde orbits. This is supported bp the

satellite sample of Zaritsky et al. ( 1993) but a larger sample is needed to further

confirm our suggestion. In their sample. 1 found tha t there are five satellites locateci

nt distances less than 20 kpc from their parent galaxies. Three of thern have unknown

directions of orbit rotation, and two of t hem have retrograde orbits wit h 59' and 60"

inclinations. In QG's calculation. al1 satellite orbits are assumed to be direct. which

implies that only the relation betiveen orbital decay rate and orbital inclination is

counted as the cause for the Holmberg effect. However. the more important relation

between orbital decay rate and otbit rotation direction of the satellite is negiected.

Usually. the decay tirne of a retrograde orbit is a few times longer than that of a

direct one. Therefore. my simulation results on disk tilting' suggest a much stronger

anisotropy of satellite distribution than that found by QG.

Remnants of tidally disrupted satellites may become part of the stellar halo. Searle

S. Zinn (1978) proposed that the stellar halo formed by the accretion of small. metal-

poor fragments like gas-rich dwarf galaxies. because they found tha t the chemical

and orbital properties of the outer globular clusters are decoupled. Direct evidence

for present and past accretion (Freeman 1996a) is: the Sgr drvarf galaxy (Ibata et

al. 1994): young main sequence A stars with [Ca/H]> -0.5 at heights up to 11 kpc

from t h e galactic plane (Rodgers et al. 1981. Lance LSYP): blue metal-poor rriaiii

secluence stars witti [Fe/H]< - 1 at age> 3 C$r (Preston. Beers. k Shectman 199-4 ):

rnetal-poor. retrograde nioving groiips in the field (Eggen 1979. hlajewski et al. 1994):

Young retrograde globuiar clusters (Z inn 1993. van den Bergli 1993a.h. Da Costa k

Armandroff 1904). The mean rotation of the metal-poor halo near the galactic plane

is direct. and at a large height from t h e plane is retrograde (Carne' 1996). This

phenonienon can be explained by disk tilt ing due to infalling satellites. Our siniulat ion

results show that a disk tilts towards the orbital plane of a direct satellite. but away

from the orbital plane of a retrograde satellite. This implies that the remnants of a

direct satellite are located near t he disk plane. whereas the remnants of a retrograde

satellite are distributed away from the disk plane. If part of the stellar halo consists

of remnants from many infaliing satellites which were assumed uniforrnly distributed.

my simulation results suggest that there should be a net excess of retrograde remnants

ïi-ith high orbital inclinations ancl a net excess of direct remnants with low orbit.al

inclinations.

10.3 Merger Rate and R = 1 CDM Cosmology

In any hierarchical cosmological model. merging is inevi table because srnall mass

perturbations collapse before large ones and the evolution proceeds as a cascade of

mergers from srnall to large scales. Most galaxies have had major mergers cluring

their formation and a large fraction of galaxies have had minor mergers during their

evolution. CJnder the cold dark mat ter (CDM) model, most galaxies accretecl at least

10% of their mass over past 5 billion years (Bahcall Sr Tremaine 1988. Frenk et al.

1988. Carlberg S. Couchman 1989, Kauffmann. White Pc Guiderdoni 1993. Lacey S:

Cole 1993). However, studies showed that a thin disk can be damaged by an infall

of 10% disk-rnass. high density satellite (TO. QHF. WMH). TO therefore suggested

that the theoretical merger rate for R = 1 C'DM cosmology is too high.

By employing Carlberg's ( 1990a. b) formula for galaxy merger rate. TO obtained.

For O = 1 C'DI1 cosmology. the probability of a itierger between a nornial L. galas!.

wit h one having a mass .\;I = .Hlz x 1 0 ' ~ . \ l ; hetween redshift zi ancl z = 0.

ancl t h e probabi1it~- of at least one merger back

The- found that the probability of at least one merger between the Milky Wa?- and

a satellite of -\lrr = . l fb /Rb = 0.029 for .\b = 2 x 10?\II ancl Rb = 0.07 after the Sun

[vas horn at z, = 0.33 ( R = 1 and flo = 50) is 90%. However. they found in their

calculation. that no more than 4% of t h e Galaxy's mass can haive accreted inside the

solar circle in the last 5 Gyr. The? t herefore concl~ide t hat Carlberg's formula gives

too high a merger rate. and R = 1 CDSL cosmology is incorrect.

In the simulations of this thesis. an infalling satellite having up to 20% of the

disk mass produces little observable tliickening in the disk. A 30% clisk-mass satellite

produces little observable thickening inside the half-mass radius of the disk but great

thickening beyond the half-mass radius. The satellite mass for Mode1 1. Mode1 2 .

and Model 3 is 4 x I O ~ . L I , ~ . 8 x 109.UI. and 12 x 10".\.Iz, respectively. According to

Equation 10.2. the probability of at least one merger hetween the M l k y Kay and

a satellite liaving a mass equal to that of Mode1 1. Model 2. and Mode1 3 is 71%.

47%1 and 38% respectively. In other mords, 38% of spiral galaxies have merged ivith

a 30% disk-mass satellite in last 5 Gyr. and in these galaxies, disk thickening can be

observed beyond tlieir half-mass radii. T herefore. according to the simulation results

presented in tliis t hesis. the theoretical merger ra te for R = 1 C'DM cosmology is not

unreasonable.

I loiincl that our thin tlisks primarilx responcl to infalling satellites hy tilting. A

clisk tilts towards the orbital plane of a direct satellite and away froni the orbital

plane of a ret rogra.de satellite. Disk heating in t h e vertical direction. or eqiiivalently

clisk t hickening. is Iess in the models of this thesis relative to that computed in

other rnodels. In these latter cases. the disk tilting is either not aliowed (TO) or

very small due to the limited initial separation between the satellite and the disk

(QHF. LC'MH). Satellites with 10%. 20%. and 30% of the clisk mass tilt the disk by

angles of (2.9 f 0.3)'. (6.3 0.1)'. and (10.6 rt 0.2)" respectively. although only 3.4%.

6.9%. and 11.1% of the orbital angular niomentum of these satellites is transferrecl

to the corresponding parent galaxy. In these cases. the kinetic energ? associated

with vertical motion in the initial coordinate frarne of the the disk is increasecl by

(6 I R ) % . (26 f 3 ) % . and (51 f. .5)% respectively. whereas the corresponcling thermal

energy associated with the random vertical motion in the tilted coordinate frame is

increasecl by only (4 f 3)%. (6 f 2)%. and (10 i~ 2)'X respectively.

1 found that in the case of satellites having tlensities comparable to that of the

parent galaxy. a 10% disk-mass satellite is completely disrupted by tidal force before

it enters the galactic disk. Since 98% of the satellite mass is accreted onto the halo.

the darnage to the disk due to such an infalling satellite is not observable. A 20%

tlisk-mass satellite is cornpletely tidally disrupted at the edge of the disk. and 11% of

the satellite mass is accreted ont0 the outer disk. Hence. only the fraction of the disk

outside the half-mass radius is directly heated by t h e infalling satellite. Finally. a

30% disk-mass satellite is completely disrupted in the outer disk. AIthough only 18%

of the satellite mass is accreied ont0 the disk. the disk beyond the half-mass radius

is significantly heated by the satellite. Both 20% and 30% disk-mass satellites cause

a slightly stronger bar in the disk than that in an isolated di&, therefore. the disk

inside the half-mass radius is slightly heated indirectly by these infalling satellites.

1 concluded that the accretion onto the halo is more common than that ont0 the

disk. and the accretiori ont0 the otiter ciisk is more comnion than that ont0 t h e inrier

clisk. Thrret'ore. a high cosmological infall rate is cornpatilde with the esistence of

t hiri inrier disks. g i ~ e n our result t hat the infall of ION- density satellites causes lit tle

disttirbance to the inner disk.

I ha\-e shown that an infalling satellite can excite warps in the disk. However the

warps eventually fade aw- clue to the fact that both the inner disli ancl outer clisk

are tilted towards the same plane. Therefore. the warps are unli kelv to be long lived

pheriomena. The fact that most galaxies a re warped could be a n indicator that disks

are in fact being subjected to a recent infall of satellites. thougli the infalling satellites

may not he easily detectecl because of tidal disruption. The results of disk warping

presented in this thesir cd:: rcpresent the warps in stellar disks.

1 founcl tha t an infalling satellite causes a somewhat stronger bar than that lorrned

in an isolated disk. However. in bot h cases. the bars are weak and evolve very slowly.

taking approximatelv 25 disk rotations to form. This is clue to the high halo-to-disk

rnass ratio employed in the models of this thesis. The pattern speed of weak bars

d e c q s very slowly relative to that of strong bars in previous work ( LVeinberg 1985.

Hernqiiist k Weinberg 1991. Sellwood S: Debattista 1996). Sirice. in rny model. the

OLR of the wenk bar remains at the inner disk. and the infalling satellite is located

far from t h e OLR. there is no direct angular rnomentum transfer from the bar to

the irifalling satellite. Therefore. the satellite orbital clecay rate is not affected by

the formation of a weak bar in the disk. In other worcls. clisk heating caused by the

infalling satellite is not underestimated in the simulations of this thesis. The results

obtained from my simulations do not represent the dynamical interaction betrveen a

strong bar and an infalling satellite.

Since self-sirnilar density profiles are employed for both the satellite and the halo

in my models, the satellite mass decreases proportionally to the decreasing distance

between the center of the satellite and t h e center of the parent galaxy. In this case.

1 founcl that the orbital decay rate is constant for both circular and elliptical or-

bits. whereas the orbital clecay rate for a solid satellite in previoiis work (Binney 9-

C'H.4 PTER 10. DISC'CiSSIO;V .4,YD C'OiYC'L C ' S I 0 3

Tremaine 1986) increases ivith decreasing distarice between the center of the satellite

and the center of t,he parent galnx>-. $ x - j. The orbital decay rate is proportional

to the initial mass of the satellite. For elliptical orbits. the decay of the orbital ecccn-

tricit!- is very slow in the halo but decreases rapidly in the disk. Slost of the tidall>-

clisrupted satellite particles are accretecl onto the halo and the outer disk rather than

ont0 the inner disk. In physical space. the final shape of the tidally strippecl satellite

part icles resembles a warpecl torils. The planes of the outermost and innermost parts

of the torus are aiigned with the planes of the initial and final orbits of the satellite.

In phase space. satellite particles are distinguished from halo particles bu their high

rot,ational speed and clumpy clistribiition. This is in accord mith the recent observa-

tion of halo stars (Majewski. Hawley 9- Munn 1996) which shows a high clegree of

clumping in the U-V- W-[Fe/H] distributions.

1 calculated the orbi ta1 decay time of a satellite which experiences tidal disruption

as its eliiptical orbit decays by employing both analytical approximations and niimer-

ical integration methods. For a median eccentricity. e = 0.5. the orbital decaY time

is approximately 50% shorter than that for the circular orbit i f r+(t = 0) = r ( t = 0).

Orbital decay times estimated with both methods are similar. Therefore. based on

the analytical result, 1 calculated the galactic accretion for the case of Ion- m a s

satellites. In estimating the galactic accretion. 1 followed Tremaine's (1980) method.

However. I introduced the following modifications: high peculiar velocity cutoff. high

mass cutoff. and non-constant rnass-to-light ratio for dwarf galaxies. In this case, the

value for the accretion rate which 1 obtained represents 13% to 25% of Treniaine's

est imate. For a median eccent ricity, e = 0.5. a typical galaxy like our own SIilky Way

Galaxy has absorbed about 30% of its disk mass in the form of low rnass infalling

satellites. According to the simulation results discussed in this thesis. such an amoiint

of accretion can be accommodated without unacceptable disk thickening within the

half-mass radius.

Tlie interaction between a clisk galas>- and ari infalling satellite can trigger star for-

mation in both t h e disk and the satellite. However. star formation canriot be stticlietl

in the piirely stellar models of t his t hesis. An important refinernent to simulatioris

of t his kind rvoulcl be to construct moclels ~vhich explicitly incorporate gas dynarnics.

Hernciuist and 5Iihos ( 199.5) studied mergers between gas rich disks ancl stellar clwarf

galasies with high densities and found that a large fraction of the gas is driren into

the inner regions of the disk. The high clensity gas may cause starburst in the center

of the disk. On the other hand. Zaritsky (1996) found elcvated rates of s ta r formation

in the oiiter disks of nearby late-type galaxies. which may result [rom the infall of

lorv densit- gas rich satellites. In future models which esplicitly inclucle a gas com-

ponent. it tvould b e interesting to stuc1:- the gas remnants of the infalling satellites.

This r n q explain s ta r formation in the ooter disks of nearbv late-type galaxies. as

well as provide an explanation for t h e formation of the halo peculiar A stars with

[Ca/H]> -0.5 a t heights iip to 11 kpc from the galactic plane (Rodgers et al. 198L.

Lance 1988).

L:sually. Lrarps observed in gas disks are stronger than those observed i t i stellar

disks. 1 have performed some simulations in mhich gas dissipation is modeled bu

applying a friction in the radial direction on disk particles. These preliminary simu-

lation results show tha t both spiral structures and warps last longer in models which

include ttiis cooling. However. in order to truly understand warps in both gas disks

and stellar disks excited by infalling satellites. it would be necessary t o use models

incorporating gas dynarnics. It is also very interesting to study tlie disk tilting rate

of both the gas disk and the stellar clisk.

Optical sitrface photornetry of harred spiral galaxies reveal tha t there are two

types of bars: large bars tend to have a nearly constant surface brightness (flat bar).

whereas smaller bars tend to have a clecreasing surface brightness with a scale length

similar to the disk (exponential bar) (Elmegreen 1996). By carrying ou t a statistical

s t itdy of t lie distribiition of galaxies in cli fferent environnients. Elmegreen. Bellin.

,I- Elniegreen ( 1990) foond a strong link bet rveeri close interactions and Aat bars.

However. siich a link cannot be investigatecl in t h e moclels OF t his t hesis becaiise o n 1 ~ -

weak bars are fornied <lue to t h e fact tha t a higli halo-to-clisk mass ratio is eniplo-ed.

1 suggest that it m a - be interesting study the link between close interactions and Aat

bar in Future moclels in which a low halo-to-clisk rnass ratio is employed.

Appendix A

The Scale Length of the Galaxy

A.1 Introduction

The space emissivity profile for the Galasy. u( R. 2). can be written in the following

forni

v(R. z ) = p ( ~ ) exp(- ~ / h ~ ) s e c h ~ ( z / h , ) / ( 2 h , ) . ( A . 1)

rvhere K and z are cylinclrical coordinates. / r ( O ) is the central surface brightriess. and

h R and h z are called scale lengt h and scale height respect ively. Tliere are photomet ric

and kinematic methods to determine the scale length of the Galasy. Hoive\-er. the

measured scale length based on both methods shows a ver? large spread. frorn 1.8 kpc

to 6 kpc (see review of the photometric determinations bu I iDF and t h e review of the

kinemat ic est imates by FM). For photometric determiriat ions. it was believed t hat

the measured scale lengt h depends on t h e wavelengt h of ohservat ions: hR = 2 - 3 kpc

for around 2.2 pm. 3.5-5.5 kpc for optical bands. and 4.5-6 kpc for IRAS OH/IR stars.

However. LTBV photometry towards the anti-center down to a magnitude mi. = 2.5

led Robin et al. (1992) to reach a value hR = 2.5 kpc. compatible with the near IR

values. Therefore. the measured value of h R depends not oniy on the wavelength.

but also on the location of the observed sources. For observations carried out at a

wavelength of approximate 2.2 Pm. sources are located at low latitude 161 < 10". On

the other hand, the OH/IR stars of IRAS observations are located at Ibl > 2'. In

P P E A. THE SCALE L E - W T H OF T H E GALAXY

opt ical bands. the location of sources varies: for example. i = 179'. h = 2.5' ( Robin et

al. 1 !)!IL>) alid 1 hl > ?O0 (van der Iiriii t 1986). FM raised the issue t liat t lie discrepancy

betwei.een the various photometric determinations of h R of the C;alasy ma' arise froni

the assiiniption of a constant scale heiglit. The constant scale Iieiglit which is inferred

from surface photometry on t wo eclge-on spiral galaxies NGC-I'L44 ancl NGC'5907 ( van

der Iiruit k Searle 1981 ) ma)- not he valid for the Galaxy.

K D F proposed a non-constant scale tieight model for the Cklaxy. In t heir motlel.

the scale height is constant with a value hmin inside a charecteristic radius R,,,,. and

increases linearly wit h raclius for radii larger than R,,,. Fnder such an assuniption

for the radial dependence of h,. IiDF found that the description of t heir observat ional

data \vas significantly improved. They inferred a value h = 3.0 kpc in cornparison to

the smaller h R = 2.7 kpc inferred on the basis of a constant h z assumption. Inspired

by the mode1 of K D F and using a similar model with hR = 3.0 kpc but outwards

increasing h,. PSI founcl that their non-constant scale height model coolcl fit the

pioneer 10 t h t a as well as van der Iiruit's (1986) model with h R = 3.5 kpc and

constant h,. Therefore. the assumption of a constant h , for the Galasy ma? not be

well-founded. The purpose of this appendix is to show how the discrepancy hetween

photometric determinations of h R for t h e Galaxy arises from the assumption of a

constant scale height.

A.2 The scale length of the Galaxy

By replacing h , in Equation A . l bu the lorm hz(0)exp(R/h) disctissed in Chapter 3

of this thesis. 1 obtain the following equation

1 introduce t h e identity

where f( R / h . z / h , ( O ) ) is a furictiori of R / h and z/h,(O) defined hy t h e follo~ving

Then Ecliiation A.? can be rewrittm as

ivhich in form is very similar to Equation A.1. By comparing t hese two equations. 1

obtain the following relation between t.wo measorecl scale lengths hgn3' and hR.

In Ecl~iation A.6. i f h >> hR. the effect of a non-constant scale heipht can l x

ignored. For example. in NGC4244. my estimate of h is 3R,,, >> hR. .Alttioagh

tlie non-constant f i : model fits tlie observed z-profile of NGCX24-L slightly better than

constant h , model. the latter is a very good approximation as shown in Figure A. 1.

Hov;ever. in the Galaxy. according to the estimate based on IiDF's data given in

Chapter 3. h - R,,, = 20 kpc. therefore. the effect of a non-constant scale height

can not be ignored in this case. in niy model. i f the average value of f ( R l h . z / h , ( O ) is

equal to 1. then the effect of a non-constant scale height is not noticeable. Mowever. i f

the average value of f ( R/ h. z / h , ( û ) departs froni uni ty. t hen hRnst is no longer equal

to h R . 1 find that if f (R /h . : /h , (O)) > 1. hgnsL > hR1 and for f ( R / h . - / h Z ( 0 ) < 1.

h z n s L < hR. Tlierefore. the Eunction f (R /h . z/h,(O) gauges whether hgnst is over-

estimateci or underestimated. By choosing values h = 20 kpc. h,(O) = 200 pc. and

Ro = 8 kpc. 1 plot in Figure A.2 f ( R l h . =/h , (O)) as a function of Rlh and the galactic

latitude b. I also calculate the averaged value of f ( : t h S . Rlh) for some simple cases

in Table A.1: from R = 2 kpc to 8 kpc in 1 = 0" direction: from 8 kpc to 18.3 kpc in

1 = 90": and from 8 kpc to 20 kpc in 1 = 180" direction. These data will be used in

t h e following discussion concerning the discrepancy that exists between the different

Figure A l : Observed z-profile of NGC-1244 (taken from van der Iiruit SL Searle 1981) and constant h , fitting (dashed lines) and non-constant h,(R) = h , ( O ) e x p ( R / h ) fit- ting (solid lines). The non-constant h- fi ts the data only slightly better that constant h, in NCC4244.

observations of hR.

Determinations of hR, based on 2.2 pm observations of hR are quite short: 2.0 kpc

(Jones et al. 1981) and 3.0 kpc (Eaton et al. 1984. I iDF 1991). In al1 cases. sources

are located at low latitude. Therefore. the average value of f (Rlh . z / h z ( O ) ) is smaller

than 1. Hence, hYst is smaller tliat h R. In the first two cases. stars located near b = O

Figure A.?: Function f ( 4 h 2 , R l h ) of Equation A.4 versus Rlh for different values of 6 . where 6 varies from O" to IO0.

are counted. and / ( R / h , z / h z ( O ) ) z 0.0. KDF averaged the surface brightness profiles

in four cuts of constant latitude covering the range 161 < 1' ( J ( R / h , z / h z ( 0 ) ) < 0.14)

l0 < 161 < 2" ( f ( R / h , r / h : ( O ) ) < 0.55). Sa < Ibl < 5' ( f (R /h . z / h , (O) ) < 1.26). and

-ia < 161 < 10" ( / ( R / h , r / h , ( û ) ) < 1.58). Since the total averaged f ( R l h , z /h,(O)) is

less than 1. hgnSt < hR. which is consistent with KDF's resiilts tha t h T s t = 2.7 kpc

and hR = G.0 kpc. Therefore. under t h e assurnption of a constant scale height, the

2.%pm photometric determinations of h R are very low.

Table .A . [ : f ( R / h . z / h , ( O ) )

On t h e o ther hand. hR determined from IRAS s ta rs counts are iocated nt t h e

high end of values: 4.5 kpc (Habing 1988) and 6.0 kpc ( Rowan-Robinson S: Chester

1987). This is due to the fact t h a t the sources a r e Iocated between 2' < b < 10".

10" < [ < :JO0 (Rowan-Robinson S: Chester (1987) a n d 1-5' < 1 < 180". but mainly

1.5' < 1 < 90° (Habing 1988). In both cases. j ( R / h . z / h , ( O ) ) > 1.0. therefore.

hsonst R > h ~ -

C:\f-B photometry towards the anti-center (1 = l79O. b = 2 . 5 O ) lecl Robin et al.

(1992) to t h e es t imate hgnSt = 2.3 kpc. By using f (R/h .z /h , (O)) = 0.44 (1 =

179". b = 2.5") in Equation A.6. 1 obtain h R = 2.7 kpc. which is very close t o the

value h R = 2.8 kpc obtained by Robin et al. (1996) based on t h e ttiick disk model.

but rvithout t h e assumption of a constant h z .

Van der Kruit (1986) analyzed t he surface briglitness of t h e Galact ic background in

the pioneer 10 background starl ight experiment and found tha t the surface brightness

ratio to be

SB( - . 90) /SB(180 ,0) = h , / h R = 1f11.0 (A.:)

By choosing h, = 325 pc for old dwarfs in the di&, van der h u i t obtained h~ = 5.5

kpc. If t lie scale height is not a constant. Eqiiat ion .-1.7 can be writ teri as

in which h,(O) can be calculated from the following equation. h z ( 0 ) esp( Ro/h ) = 32.5

pc. Bu assuming h = 20 kpc and Ro = 8.5 kpc. I determine h=(O) = 212 pc. irhich

yields. when substituteci in Equation h.8. a value h R = 4.4 kpc. It is consistent with

F.\I's conclusion that a mode1 with shorter hR. but raclially increasing h -. ma!- fit the

pioneer 10 data as well as van der Kruit's (1986) model with hgngL = 5.5 kpc and a

constant h,.

A.3 Summary

1 have shown that the assumption of a constant scale height for the C;alas~. may be

t he reason for the discrepancy which has emerged among the different photometric

determinations of the scale lengt h of the C;aIaq. Bq- using a non-constant scale height

model. hl ( R) = h:(O) esp( R l h ) . whicli is derived on the basis of my numerical simu-

lat ions. 1 have demonst rated t hat the scale lengt h deterinined under the assumpt ion

of a constant scale height can ei t her be underest imated or overest iniated. depending

on the location of the sources. In general. the scale height determinecl by - 2 4 r m

observations is underestimated. ivliereas that inferred irom IRAS OH/IR star counts

is overestimated. The optical measurements of the scale length can either he under-

estimated (Robin et al. 1992) or overestimated (van der Kruit 1986) depending on

the location of the sources. Future observational d a t a reduction. based on rny non-

constant scale height model, h,(R) = h,(O) exp(R/h) . perhaps result in a measured

scale length which weakly depends on or may even be independent of the wavelength

of observations.

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