N-Body/SPH simulations of induced star formation in dwarf ...lib.ugent.be/fulltxt/RUG01/002/061/248/RUG01... · GHENT UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF PHYSICS AND ASTRONOMY

GHENT UNIVERSITY

FACULTY OF SCIENCESDEPARTMENT OF PHYSICS AND ASTRONOMY

N-Body/SPH simulations of induced starformation in dwarf galaxies

AUTHOR: TINE LIBBRECHT

Supervisors:Annelies CLOET-OSSELAERDr. Mina KOLEVAJoeri SCHROYEN

Promotor:Prof. Dr. Sven DE RIJCKE

Master thesis in context of obtaining the academic degree of Master of Science in Physics and AstronomyJune 2013

”Les gens ont des etoiles qui ne sont pas les memes. Pour les uns, qui voyagent, lesetoiles sont des guides. Pour d’autres elles ne sont rien que de petites lumieres. Pourd’autres qui sont savants elles sont des problemes. Pour mon businessman elles etaient del’or. Mais toutes ces etoiles-la elles se taisent. Toi, tu auras des etoiles comme personnen’en a... ”- Antoine de Saint-Exupery - Le Petit Prince1

1Figures front page: Density evolution projected on the xy plane. This density evolution belongs to sim1058 and shows the density evolutionduring an extreme peak in the SFR and right afterwards. The extreme star formation has caused extreme feedback, and the gas is entirely blownapart by supernovae and stellar winds. Star formation is not possible for the next 2 Gyr. For more explanations, see Chapter 4.

Preface

While finishing this master thesis, it almost seems like a miracle that all these words are finally put onpaper and that it got somehow finished (even almost or not really quite in time). It is hard to decidewhere to start thanking people because I am very greatful to everyone that helped me along the processand every opportunity that came along the way.First of all, I want to thank professor Sven De Rijcke. Offering the vastly interesting subject and super-vising patiently, I always felt inspired after leaving his office. I am also greatful for his effort to hand in anFWO project for me, even when at the time, my thesis was rather unexisting still.Secondly, I don’t think I can thank Annelies enough. She has spent a tremendous amount of time, help-ing with software problems. Also, she answered my every (stupid) question, she gave advice and sheproofread several parts. The same is true for Mina and Joeri. The three of them have spent so muchtime and efforts helping me that I feel like my results are their results too.Also it has been a pleasure to work together with Robbert. It is a lot of fun to have someone to obsesswith over your subject. This way we have killed the mood at several parties (to great boredom of ourfriends).I am very greatful to my close friends and roommates Amelie De Muynck, who calculates faster thanher shadow, and Thomas Boelens, Maple wizard. They were always ready to help selflessly, like truefriends do.In the same context, my dear boyfriend Koen Hendrickx should get an award for living together with themost distracted person alive. Yes, driving left side of the road or using toxic plants in our dinner - thinkingit is laurel, I can be dangerous at times. He suppported me all year long and he always believed in me,even when I did not believe in myself. I am looking forward to move to Stockholm with him and start anew chapter of our lives.Finally, I am very greatful to my family. My dad Dirk Libbrecht, who has been enthusiastic and interestedin my thesis, my mum An Hautekiet for comforting me always, and my brother Robbe Libbrecht, whohas also written a thesis this year so we have “suffered” together.I enjoyed working on this thesis very much, I hope you enjoy reading it.

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ii PREFACE

Toelating tot bruikleen

De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van demasterproef te kopieren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van hetauteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden bij hetaanhalen van resultaten uit deze masterproef. Op datum van 11 juni 2013.

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iv TOELATING TOT BRUIKLEEN

Contents

Preface i

Toelating tot bruikleen iii

Nederlandse samenvatting vii

Introduction 1

1 Blue Compact Dwarfs 51.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Starbursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Evolution scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Feedback and instability 112.1 Physics of star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Components of the interstellar medium . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Jeans mass and Jeans radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.4 Induced star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Parameterization of star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Kennicutt-Schmidt law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Oscillations in star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 Hurwitz criterium for stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.4 Star formation systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Research hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Implementation of induced star formation in an N-body/SPH code 353.1 The code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Dwarf galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2 Collisionless N-Body simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.3 SPH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.4 Neighbor search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.5 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Additional physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.1 Star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.2 Star mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.3 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.4 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.5 Self-regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Implementation of induced star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 Star formation law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.2 Calculation of ρs,i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.3 Integration of the star formation law . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.4 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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4 Results and discussion 494.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.1 Reference dwarf galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 Behavior of the star formation rate and parameter values . . . . . . . . . . . . . . 534.1.3 Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.4 Influence of search radius and age criterium . . . . . . . . . . . . . . . . . . . . . 61

4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.1 Why a change in star formation rate? . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.2 Where does star formation occur? . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.3 Scaling relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Conclusions and outline 795.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

List of Abbreviations 81

Bibliography 86

Nederlandse samenvatting

Introductie

Ons universum wordt beschreven door een donkere energie en koude donkere materie kosmologie.Deze theorie verklaart het ontstaan van galaxieen aan de hand van hierarchische structuurvorming.Kleine dichtheidsperturbaties groeien en versmelten en geven zo het ontstaan aan dwerggalaxieen.Grote sterrenstelsels zoals onze Melkweg worden dan gevormd door de versmelting van vele kleinedwergstelsels.Dwerggalaxieen zijn kleine galaxieen met absolute magnitudes groter dan −18 en optische groottesvan ongeveer 1 kpc. Verschillende subtypes van dwergstelsels worden ingedeeld waaronder gasarmeelliptische dwergen (dEs), gasrijke onregelmatige dwergen (dIrrs) en blauwe compacte dwergen (BCDs).Deze laatste worden onder de loep genomen in deze thesis.Blauwe compacte dwergen vertonen extreem hoge stervormingstempo’s, starbursts genaamd. Tot opheden is er geen consensus over de oorzaak van deze starbursts. De oorzaak kan gezocht worden bijeen invloed van buitenaf (externe mechanismes) of invloed van binnenuit (interne mechanismes).Het is uiteindelijk de bedoeling te onderzoeken of BCDs kunnen gesimuleerd worden. Hiervoor focussenwe op interne mechanismes in de dwergstelsels, meer specifiek stervorming. De stervormingswet inde gebruikte code is een eenvoudige lineaire wet, de Kennicutt-Schmidt wet genoemd. Theoretischeberekeningen worden gebruikt om deze wet aan te passen en non-lineairiteit te introduceren. Hierdoorkunnen oscillaties in de stervorming ontstaan die mogelijks de starbursts in BCDs verklaren.

Blauwe compacte dwerggalaxieen

Verschillende vragen blijven overeind in verband met BCDs. De oorzaak van hun starbursts kan mogelijkverklaard worden door interne mechanismes. Een voorbeeld daarvan is de invloed van draaimoment.Zo is het aangetoond dat hoger draaimoment leidt tot een meer continue stervorming terwijl een lagerdraaimoment zorgt voor een sterk gepiekte stervorming.Externe mechanismes ter verklaring van BCDs zijn bijvoorbeeld het versmelten van twee dwergstelselswat leidt tot een intense stervormingsepisode.Ook evolutiescenario’s worden bestudeerd. Als de stervormingsepisode in een BCD stilvalt, hoe zaldit stelsel dan waargenomen en geıdentificeerd worden? De evolutie van een gasrijke BCD naar eengasarme dE vereist een verklaring voor het gasverlies. Gasverlies kan optreden als gevolg van externemechanismes zoals ram pressure stripping (het verliezen van gas ten gevolge van de druk veroorzaaktdoor het bewegen door een intergalactisch medium) of interacties met andere galaxieen of graviterendeobjecten. Toch blijft het op basis van andere eigenschappen zoals metalliciteit en draaimoment on-waarschijnlijk dat een BCD naar een dE zou evolueren. Hoogstwaarschijnlijk zullen stilgevallen BCDsals dIrrs geclassificeerd worden.

Feedback en instabiliteit

Om starbursts in BCDs te kunnen verklaren, is het nodig om een grondig inzicht in het verschijnselstervorming te verkrijgen. Stervorming is onderhevig aan vele feedback effecten. Stervorming vindtplaats in koele en dichte gebieden. Wanneer interstellair gas gekoeld wordt, kan het ineenstorten en eenster vormen. Pasgeboren sterren op hun beurt zenden stellaire winden uit of ontploffen in supernova’s.

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viii NEDERLANDSE SAMENVATTING

Hierdoor zal het interstellair medium verhitten en ioniseren. Het interstellair gas wordt in bellen uit elkaargeblazen en hierdoor zal stervorming stopgezet worden. Anderzijds zullen diezelfde stellaire winden ensupernova’s aanleiding geven tot het vormen van meer sterren. Aan de randen van de geblazen bellenzal het gas bijeen gedrukt worden en hoge dichtheden veroorzaken. Dichte gaswolken kunnen dan weerineenstorten tot sterren. Het fenomeen waarbij meer sterren gevormd worden door de aanwezigheid vansterren in de omgeving wordt geınduceerde stervorming genoemd. Een andere complicatie is dat viasupernova’s ook metalen vrijgelaten worden in het interstellaire medium die het koeltempo versnellen.Het is duidelijk dat veel verschillende effecten stervorming beınvloeden. Daardoor is dit een moeilijk tevoorspellen en te parametriseren grootheid.Berekeningen zijn ondernomen om de stabiliteit van stervormingssystemen te testen. Het blijkt dathet toevoegen van geınduceerde stervorming in een stervormingwet kan zorgen voor instabiliteit enoscillerend gedrag van het stervormingstempo. Kunnen deze oscillaties van het stervorminstempo,veroorzaakt door het inbrengen van een non-lineaire term in de stervormingswet, de verklaring zijn voorstarbursts in BCDs? Om deze vraag te onderzoeken wordt gebruik gemaakt van simulaties.

Implementatie van geınduceerde stervorming in een N-Body/SPH code

De N-Body/SPH code GADGET-2 wordt gebruikt om dwerggalaxien te simuleren. In deze code zijnslimme boomstructuren geımplementeerd om de berekening van de gravitationele interacties te ver-snellen. Diezelfde boomstructuren worden gebruikt om buurdeeltjes te zoeken, nodig om dichtheden teberekenen. Als men de parametrisatie van geınduceerde stervorming wil inbrengen in de stervormings-wet, is het namelijk nodig om de gasdichtheid en sterdichtheid van een deeltje te kennen.De uitbreiding van de stervormingswet zorgt er ook voor dat er geen analytische oplossing meer voorhan-den is. Daarom wordt een integratiealgoritme geımplementeerd: een vierde orde Runge-Kutta. Eenvergelijking werd gemaakt met een lineaire benadering, maar op grond van fysische argumenten wordtde Runge-Kutta methode verkozen.

Resultaten en bespreking

Eens de stervormingswet aangepast is, kunnen simulaties uitgevoerd worden. Het blijkt al snel dat denieuwe stervormingswet aanleiding geeft tot oscillaties in het stervormingstempo. Twee vrije parame-ters die verband houden met de parametrisatie van geınduceerde stervorming worden uitvoerig getestin een groot bereik. Er kan echter geen verband gevonden worden tussen parameterwaardes en kwa-litatief gedrag van het stervormingstempo. Met andere woorden, een ontaarding doet zich voor tussende parameterwaardes en het kwalitatief gedrag van de stervorming.Het blijkt dat galaxieen met een hoog stervormingstempo in de eerste 3 gigajaar van hun evolutie, laterhoogstwaarschijnlijk nog weinig sterren zullen vormen en geen oscillaties vertonen in hun stervormings-tempo. Galaxieen die daarentegen eerder bescheiden van start zijn gegaan en in de eerste 3 gigajaarminder sterren hebben gevormd, zullen daarna een grotere kans hebben om oscillaties te vertonen inhun stervormingstempo en zullen op het einde meer stermassa gevormd hebben.Het verloop van een stervormingsepisode gaat als volgt: gas stort ineen naar het centrum van hetgalaxie. Hoge dichtheden worden bereikt en stervorming begint. Tijdens een stervormingsepisode,zullen supernovae en stellaire winden ervoor zorgen dat de site van actieve stervorming zich weg ver-plaatst van het centrum van het galaxie naar grotere afstanden. Dit is de geınduceerde stervorming aanhet werk. Na een korte periode van stervorming zorgen de feedback effecten ervoor dat stervormingweer helemaal stilvalt.Verschillende schalingsrelaties worden bestudeerd en het blijkt dat de gesimuleerde galaxieen typischeschalingrelaties zoals de Faber-Jackson relatie reproduceren in vergelijking met observaties van dwerg-stelsels. Dit is een goede indicatie dat de gesimuleerde galaxieen fysisch zijn.De dwergstelsels die oscillaties vertonen, blijken helderder, blauwer en diffuser te zijn dan gesimuleerdegalaxieen zonder oscillerend gedrag. Dit is gemakkelijk te verklaren. De stelsels zijn helderder omdat zijmeer sterren gevormd hebben, blauwer omdat zij recenter sterren gevormd hebben, en diffuser omdat

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er meer feedback geweest is en stervorming daardoor van het centrum weg verschuift.Als we deze resultaten interpreteren in de context van het simuleren van BCDs, dan moeten we besluitendat onze simulaties geen BCDs voorstellen. De oscillaties in stervorming zijn niet intens genoeg om eenstarburst in een BCD te kunnen zijn. Ook zijn onze systemen niet compacter maar diffuser geworden,en hebben zij hogere metalliciteiten. Allemaal eigenschappen die in tegenspraak zijn met observatiesvan BCDs.Tenslotte werden mergers bestudeerd van dichte gaswolken en dwergstelsels. Deze mergers hadden alaangetoond tot starbursts te leiden, gebruik makend van een standaard stervormingswet. Onze nieuweaangepaste wet die rekening houdt met geınduceerde stervorming, lijkt de starbursts te versterken.

Conclusie en vooruitblik

Het aanpassen van de stervormingswet door rekening te houden met geınduceerde stervorming leidtniet tot het simuleren van BCDs. Het is echter wel mogelijk dat deze aangepaste wet zijn rol kan spelenin het simuleren van sterke starbursts bij merging. Dit moet in groter detail bestudeerd worden.

x NEDERLANDSE SAMENVATTING

Introduction

Cosmology

Our universe is believed to be described by a lambda cold dark matter ΛCDM cosmology. Dark energyΛ is responsable for 76% of the universe energy content and is an unknown energy component thatcauses the universe to expand in an accelerating way.Another 20% of the universe consists of unknown cold dark matter. Cold, because dark matter particlesare non-relativistic. Dark, because the particles are only able to interact gravitationally and not electro-magnetically. Therefore, it is impossible to observe dark matter, at least by means of electromagneticradiation.The remaining 4% of the universe consists of radiation and baryonic matter as we know it on our humbleEarth.Using ΛCDM cosmology, theories of structure formation in the young universe were developed. It turnsout that dark matter is necessary to explain large scale structure formation and the origin of galaxies.Structure in the universe is believed to originate hierarchically. Primarily, small density perturbationsarise. Pressureless dark matter clumps together in these density perturbations and dark matter halosare formed.After the decoupling of baryonic matter and radiation, baryons fall into the gravitational potentials of thedark matter halos. When the clumps of dark and baryonic matter are sufficiently large, gravity will haveincreased and will attract other clumps. These clumps will start moving, attracting one another, andmerge. When the formed structures of dark matter and baryons contain a sufficient amount of baryonicgas, they can start forming stars and dwarf galaxies are born. Large galaxies as for instance our MilkyWay are believed to be formed through the merging of dwarf galaxies.Cosmological structure formation has been simulated by for instance the Millennium simulation. Thissimulation clearly illustrates hierarchical structure formation, starting with small density perturbationsand ending up with large scale structures as galaxies and clusters, which are still in process of formingeven larger structures. Furthermore, simulations have been undertaken to follow hierarchical structureformation in detail via merger trees, where one keeps track of all merged progenitors that eventuallybecome galaxies.From the theory of structure formation, two important conclusions are drawn. First of all, without colddark matter, galaxies as we know them would never have been able to form. This is an important argu-ment in favor of the existance of dark matter. Secondly, the origin of dwarf galaxies and the fact that theyact as progenitors for large galaxies is an important motivation to study dwarf galaxies.

Dwarf galaxies

Now, knowing how dwarf galaxies have originated, we look into the definition and properties of dwarfgalaxies. They can be defined as small galaxies with absolute V-band magnitudes larger than −18 (notethat larger magnitudes correspond with fainter luminosities and that magnitudes vary logarithmically).Dwarf galaxies have sizes of only 1 to a few kpc. For comparison, the Milky Way has an absolutemagnitude of −21 and a size of 30 kpc.Another important property of dwarf galaxies is that they are dark matter dominated. They have largemass to luminosity ratios which implies that there is far more mass present than from starlight alonewould be estimated. The large mass to light ratios suggest that large amounts of dark matter arepresent in dwarf galaxies. Furthermore, in comparison to large galaxies, dwarf galaxies have a shallower

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2 INTRODUCTION

gravitational potential wells. Therefore, they tend to be more diffuse systems.Dwarf galaxies are typically divided into subcategories, for example:

• dwarf ellipticals (dEs) have absolute magnitudes between −14 and −18. They have old redstellar populations and contain almost no gas. Their name is derived from the fact that they haveelliptical isophotes. Dwarf ellipticals are the most common type of galaxies in the universe and aremostly found in cluster environments.

• dwarf spheroidals (dSphs) are the “little brothers” of the dwarf ellipticals. They share mostly thesame properties but have absolute magnitudes larger than −14.

• dwarf irregulars (dIrrs) could be described as the opposite of dSphs. They are situated in thesame magnitude range, but they contain a large amount of gas. Therefore, stars are being formedand the galaxy contains a young and blue stellar population. Dwarf irregulars are mainly found inthe field as opposed to dEs and dSphs populating clusters.

• blue compact dwarfs (BCDs) are galaxies showing extreme bursts of star formation. The youngpopulation of stars gives rise to its blue colors. The starburst is located centrally in the galaxyhence its compact appearance.

Many questions remain unanswered in regard to dwarf galaxies. Dwarf ellipticals and dwarf spheriodalshave old stellar populations. This implicates that for a certain point in their history, they should havecontained gas in order for star formation to be possible. Currently they are devoid of gas. By whichmeans did their gas amount disappear? Have dIrrs evolved to dEs by the loss of their gas and by whichmechanisms would this gas loss have occured? On the other hand, dEs also show fundamentally differ-ent properties compared to dIrrs as for instances larger metallicities and lower angular momenta. Thiscomplicates possible evolution scenarios.The case of BCDs is even more mysterious and provides the context of this master thesis. Up till thepresent, no explanation is confirmed for the origin of the observed starbursts in BCDs. Moreover, itis unclear whether these starbursts are cyclic phenomena or a single event. In Chapter 1, BCDs arediscussed into detail. The cause of the starbursts in BCDs can be searched for in external or internalmechanisms.External mechanisms are all influences on the dwarf galaxy originating from its environment and/ornearby objects. Examples of external mechanisms are interactions with an intercluster medium, tidalinteractions with other galaxies or black holes, mergers of galaxies, etc. A master thesis has been writ-ten by Verbeke [Verbeke, 2013], with the aim of simulating BCDs via mergers of dense gas clouds anddwarf galaxies.It is also possible that starbursts in BCDs can be explained by internal mechanisms. These includeall physical processes that occur in the galaxy without an influence of its environment. Properties asangular momentum, processes as star formation, gas cooling and heating, supernovae, etc. are all con-sidered as internal mechanisms. This master thesis investigates the origin of starbursts in BCDs byfocussing on the internal mechanism of star formation.

Star formation

In order to understand BCDs, it is necessary to obtain a good insight into star formation. What factorslead to an increased star formation rate and under which circumstances do (cyclic) starbursts occur?Gas density is a very important parameter in this regard. The star formation rate can be put proportionalto (a power of) the local gas density.But star formation is more complex. Numerous feedback effects - positive as well as negative - influencestar formation. For instance cooling of the interstellar gas will cause a gas collapse and provoke starformation. On the other hand, the newborn stars give rise to stellar winds and often supernovae, heatingand ionizing the interstellar medium. These conditions will halt star formation. It gets even more compli-cated when realizing that supernovae and stellar winds at the same time also trigger star formation. At

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the edges of the bubbles blown in the interstellar gas by supernovae, densities are increased and gascollapses to form stars. The process of star formation initiated by supernovae and stellar winds is calledinduced star formation.Another effect is caused by supernovae releasing metals in the interstellar medium, which will acceleratethe cooling rate, favoring star formation.The complex interplay between different feedback effects leads to self-regulation of star formation. Anincrease of the gas density for instance will not simply lead to higher star formation rates. Moreover, re-search has shown that self-regulation can lead to non-linear oscillations in the star formation rate. This isan interesting property in regard to gain insight in BCDs. Could these oscillations lead to increased starformation rates and hence partially or entirely explain the observed (maybe cyclic) starbursts in BCDs?In Chapter 2, star formation and all its feedback effects are discussed into detail. Furthermore, a closerlook is taken at self-regulation and oscillatory behavior of star formation.

Simulations

The final aim of this thesis is to investigate if it is possible to simulate BCDs. For the simulations, anN-Body/Smoothed Particle Hydrodynamics code is used, which is thoroughly discussed in Chapter 3.Findings from a literature study and theoretical calculations in Chapter 2 will be used to alter the starformation prescription in the code. This may increase the chances for oscillatory behavior to occur inthe star formation rate and cause starbursts in the simulated galaxies.The results of the simulations and a discussion is presented in Chapter 4. Enjoy!

4 INTRODUCTION

1Blue Compact Dwarfs

1.1 Definition

Blue compact dwarfs (BCDs) are dwarf galaxies with remarkably high star formation rates (SFRs), dom-inated by hot blue stars with possibly low metallicities. The neutral gas component as well as the areasof active star formation (SF) are centrally concentrated in BCDs, hence their compact appearance. Likeother dwarf galaxies, they have absolute magnitudes of MB ≥ −18 and optical sizes of about 1kpc.The possibly low metallicities of BCDs and the presence of recently formed stars, suggest at first sightthat BCDs are young systems. Near-infrared images however show that underlying star populationswith ages of a few Gyrs are present [James, 1994, Thuan, 1983]. The observed near-infrared fluxes aredue to the presence of K and M giants1, while the observed star formation episodes are younger than5 · 107 Gyr. This time span cannot contain the evolution of young blue stars to K and M giants. Hence aprevious period of star formation has occured before the presently observed star formation episode.Nowadays, it is generally assumed that the star formation episodes in BCDs happen in a transientand possibly even in a cyclic way. These star formation episodes with unusually high SFRs are calledstarbursts. Star formation rates in BCDs were estimated using Hα

2 observations in the optical. As anexample, the estimate for the SFR in BCDs by Sethuram [Sethuram, 2011] is of the order of 0.1− 1.0solar masses per year (M�/yr).

1.2 Starbursts

The cause of starbursts in BCDs is still an area of active research. As mentioned in the introduction, wecould look for the explanation of starbursts into external or internal mechanisms.

Internal mechanisms Van Zee et al. [van Zee et al., 2001] studied 6 BCDs through the observationof the 21 cm emission line of HI (for a discussion on the components of the interstellar medium, seePar. 2.1.1). The observed velocity fields suggest that the 6 BCDs have some rotational velocity. In spiteof these rotational velocities, the specific angular momenta (J/M) of BCDs are smaller than the angularmomenta of for instance low surface brightness dwarf irregulars (LSB dIrrs) because BCDs are more

1K and M giants are stars in an evolved life stage. Hydrogen burning has reached its end phase and the stars evolve away from the mainsequence towards red colors.

2Hα is the Balmer line corresponding with the transition in hydrogen from excitation state n = 3 to n = 2. The transition corresponds witha visual wavelength of λ = 656.28 nm. Hα radiation is an important tracer for star formation because young stars tend to ionize and excitetheir surrounding interstellar medium.

5

6 Chapter 1. Blue Compact Dwarfs

Figure 1.1: (a) Specific angular momenta of LSB dIrrs (open circles) and BCDs (closed circles) as a function of absolute B-band magnitudeMB. (b) Maximal rotational velocity of LSB dIrrs and BCDs as a function of MB. Figure from [van Zee et al., 2001].

compact objects. This is shown in Figure 1.1(a). The maximal rotational velocities of LSB dIrrs andBCDs do have comparable values, see Figure 1.1(b). For BCDs however, these maximal rotationalvelocities are reached within a smaller radius from the center than for LSB dIrrs, because BCDs arecompact. This means that BCDs have steeper rotation curves than LSB dIrrs.Toomre instability analysis [Toomre, 1964] balances gravity, rotational effects and thermal effects, to cal-culate the threshold gas density for star formation. This analysis leads to threshold densities that areproportional to the slope of the rotation curve [van Zee et al., 2001]. Steeper rotation curves implicatehigher threshold densities, hence BCDs have higher threshold densities for the onset of SF, thus it willtake a longer time before SF initiates. Therefore more “fuel” for SF is present when the threshold den-sity value is attained and intenser SF can occur. It is possible that this property leads to the observedstarbursts in BCDs. In this context it seems that starbursts are a logical implication of the compact ap-pearance and hence the lower specific angular momenta of the BCDs. The frequency of the consecutivestarbursts would then be regulated by the rate at which the gas in the central core reaches the thresholddensity and initiates SF until feedback effects (as for instance supernovae) redistribute the gas and stopor slow down SF.More recently, Schroyen et al. investigated the influence of rotation in simulated dwarf galaxies[Schroyen et al., 2011]. They found a significant influence of angular momentum on the appearance andevolution of dwarf galaxies, and hence they proposed angular momentum as the second most importantparameter (after total galaxy mass) for determing the behavior of dwarf galaxies. They compared be-tween three different types of dwarf galaxy simulations: a rotating model (which is rotationally flattened),a non-rotating spherical model and a non-rotating manually flattened model. These models showedfundamentally different star formation histories (SFHs). The rotating model showed a rather continuousSFH where periods of increased SF may occur but where the SFR never completely falls back to zero.This strongly contrasts with the non-rotating model that exhibits multiple episodes of relatively high SF;

1.3. Evolution scenarios 7

Figure 1.2: SFHs of a rotating model (red) and non-rotating spherical models (green). Figure from [Schroyen et al., 2011]. Full lines depictthe SFR. Dashed lines depict the total formed star mass.

in between the SFR always diminishes to approximately zero. Both SFHs of the rotating and non-rotatingmodel are shown in Figure 1.2. The model that does not rotate but is manually flattened, shows more orless the same behavior as the non-rotating spherical model. The differences in SF behavior are henceclearly due to the differences in angular momentum and not to flattening itself. In conclusion, lowerangular momenta of galaxies have an influence on the SFR and cause it to be more bursty in nature,however, these starbursts are not sufficiently intense to explain observed starbursts in BCDs.

External mechanisms To study and simulate starbursts in BCDs, Bekki [Bekki, 2008] used a differentapproach. His simulations of interacting and merging dwarf galaxies lead to starbursts. The most suc-cesfull model consists of two dwarf galaxies, of different mass and in a different life stage, that collideand merge. The inward movement of HI gas from the outer areas of the galaxy leads to a strongly in-creased star formation rate, consistent with the observed SFRs in BCDs. A blue compact core is formed,surrounded by the already existing stars from the two progenitor galaxies embedded in an HI cloud. Theobserved low metallicities of the young blue stars also have a natural explanation in the model of Bekki:the new stars are formed by extremely metal-poor gas from the outer regions of the progenitor galaxies.In conclusion, it is not yet clear whether the starbursts in BCDs are a consequence of internal or externalmechanisms, or a combination of both.

1.3 Evolution scenarios

The classification of a galaxy is based on its observational properties. Dwarf galaxies with extremestar formation rates and a central dense and blue core will be classified as BCDs. However, when thestarburst has come to an end, the question arises how the galaxy will be observed an classified. Will


a faded out BCD resemble a dwarf irregular, or rather a dwarf elliptical? Several scenarios have beenstudied for the evolution of BCDs to dIrrs and to dEs.An important parameter is the amount of neutral gas in the galaxy. BCDs and dIrrs are generally gas-rich with MHI ' 107M�, while dEs are generally gas-poor with MHI ' 105M� [van Zee et al., 2001].Evolution scenarios from BCDs to dEs should hence explain by what means a large quantity of gas candisappear from a galaxy.

Evolution of BCDs to dEs Gas removal can be considered in terms of internal processes or in terms ofexternal processes. Internal processes are for example the use of total gas amount by the stellar mass(this process is called depletion), supernovae (SNe) and stellar wind (SW) effects blowing out the ISM,...External processes are for instance tidal interactions or ram pressure stripping (explained below).In search for a process that removes gas from a galaxy, Dekel and Silk [Dekel and Silk, 1986] proposedin 1986 an internal mechanism in which a starburst can expel the interstellar medium (ISM). They cal-culated that the virial velocity of a galaxy should be smaller than the critical value of 100 km/s to allowglobal gas loss as a result of the kinetical energy of the starburst. This theory however has been falsifiedby more recent models: it seems to be harder than previously thought to remove the ISM by one singlestarburst. Another argument against this model is that multiple starbursts are seen to occur in one singledwarf galaxy so it seems even less plausible that only one starburst can expel the ISM.In fact, simulations of isolated dwarf galaxies have shown that all simulated galaxies retain a significantamount of their gas, in other words, gas cannot be expelled by internal mechanisms alone [Valcke, 2010].This is also supported by observations. Indeed, in (the center of) clusters, elliptical galaxies are far morecommon than in the field. Likely, the gas is removed in by processes that involving interactions with theintracluster medium or other galaxies (tidal interactions), hence by means of external processes.An interesting external mechanism for gas removal, is ram pressure stripping. This phenomenon oc-curs when a galaxy moves through hot intergalactic clouds of gas, common in clusters. This causesthe interstellar gas of the galaxy to be removed by the pressure of the intergalactic gas. Ram pressurestripping was proposed by Gunn en Gott [Gunn and Gott, 1972]. They studied the effect of infall of massin clusters and the influence on galaxy formation and properties of galaxies.Ram pressure Pr is given by

Pr = ρv (1.1)

where ρ is the gas density in the cluster and v the velocity of the infalling galaxy. Gunn and Gott cal-culated that in this way a galaxy could be stripped of its interstellar material when the density of theintergalactic medium exceeds 5 · 10−4 atomes/cm3.Modelling as well as observing galaxies subject to ram pressure stripping is still an area of active re-search. Multiple recent studies use (magneto)hydrodynamical models to simulate ram pressure strip-ping, e.g. [McCarthy et al., 2008, Shin and Ruszkowski, 2013, Ruszkowski et al., 2012]. Ram pressurestripping is also supported by observational evidence, e.g. [Jachym et al., 2013, Bernard et al., 2012].According to Van Zee et al. it is possible for BCDs originated in clusters to evolve to dEs - however notby the single process of ram pressure stripping alone. This statement is based on rotational arguments.The observed BCDs were selected to have properties similar to dEs. Therefore it would be likely thatthey evolve to dEs after the current starburst fades out. These six BCDs however each have a certainangular momentum while dEs generally do not rotate [Bender and Nieto, 1990]. Hence the evolution ofBCDs to dEs would require loss of angular momentum which is not possible by ram pressure stripping.Angular momentum can be transferred via merging or tidal interactions between galaxies. Via thesemore violent processes, maybe it could be possible for a BCD to evolve to a dE. Bekki [Bekki, 2008]studied this scenario. He simulated BCDs by means of the merging of two dwarf galaxies and studiedthe evolution scenarios. He concludes that it is highly unlikely that a BCD would evolve to a gas poor dEbecause the simulated BCD still contains a large amount of HI gas after the fade out of the starburst.

Evolution of BCDs to dIrrs Following from the above discussion, it is hence most probable that quies-cent BCDs will be classified in the class of the gas rich dIrrs. The question arises: “do all dIrrs show

1.3. Evolution scenarios 9

starbursts at a certain moment in there lifetime, or would BCDs be a true subclass of dIrrs with differentproperties throughout their lifetime?”Van Zee [van Zee, 2001] studied a large sample of dIrrs using UBV3 and Hα observations. By meansof properties as color gradient, stellar composition and surface brightness, they concluded that the lumi-nosities of dIrrs today can be explained by a continuous SFR without a need of one or more starburststo explain the observations. Certain observed dIrrs however, show properties that are very similar to theproperties of BCDs, as for instance a compact central gas core and similar color gradients to the onesof BCDs. But the observed galaxies are currently not going through a starburst phase. These dIrrs arethus probably examples of faded out BCDs.

3UBV are photometric filters, corresponding to ultraviolet, blue and visual wavelengths. The same filters correspond to V-band and B-bandmagnitude.


2Feedback and instability

2.1 Physics of star formation

This section studies the phenomenon of star formation into detail. Star formation takes place in cooland dense regions of molecular clouds. When a cloud collapses under its own gravity, thermonuclearreactions start taking place and a star is born. Depending on the properties of the star, it emits mass andenergy through supernovae events or stellar winds. This emission is called feedback. The interstellarmedium recieves feedback and is heated and blown appart. Hence feedback annuls the conditionsnecessary for star formation. On the other hand, the same feedback effects also increase the probabilityof stars being born in the approximity of the emitting star. This phenomenon of induced star formation isdeepened in Par. 2.1.4.It is clear that the birth of a star involves, and has an influence on different components of the interstellarmedium. These components are discussed in Par. 2.1.1. The Jeans criterium yields conditions for acollapse to occur and is derived in Par. 2.1.2. Furthermore, the proces of cooling of the interstellar gas,giving rise to star formation regions is explained in Par. 2.1.3.

2.1.1 Components of the interstellar medium

The interstellar medium (ISM) consists out of several components that are not necessarily spatiallyseparated, the distinction is made by their temperatures (paraphrased from [Baes, 2010]).

HII regions Hot ionized gas has temperatures of T ∼ 104 − 106 K, also known as HII regions. Mostlyobserved in areas of recent star formation, they obtained their high temperatures and state of ionisationfrom UV-light emitted by hot young stars. HII clouds are mainly observed by one of the Balmer lines inhydrogen which comes in existence when an electron undergoes a transition from ionisation state n = 3to n = 2 and emits a photon at a wavelength of 656.3 nm in the optical. This line is also called theHα-line and is widely used in astronomy to estimate SFRs.

HI regions Another component of the ISM is neutral atomic HI gas at much cooler temperatures ofT ∼ 10− 100 K. The only way to observe HI gas is via the 21 cm line using radio spectroscopy. This lineis due to a transition of the electron spin state relatively to the proton spin state. In the lowest energystate of the hydrogen atom, the spin of the proton and electron are antiparallel. A slightly excited stateexists when the spins are parallel. The transition between those two states is forbidden with a decay timeof 107 yr, however HI gas is so common in galaxies that the 21 cm line is easily observed. Moreover, this

11

12 Chapter 2. Feedback and instability

transition can be observed at all temperature scales since the requested temperature for this transitionis very small (T � 1 K).

Molecular clouds Further cooling of the HI gas leads to formation of molecular clouds, mainly consistingof H2 but also other elements as CO, HCN and CS. These clouds are extremely cold with temperaturesaround T ∼ 5− 30 K. Since H2 is a very poor emitter (see Par. 2.1.3), molecular clouds are detected bymillimeter radiation from rotational CO transitions.Molecular clouds also contain molecular interstellar dust, which are condensed solid molecular particlesof less than 0.1 ¯m. This dust usually has temperatures of around T ∼ 20− 30 K, therefore it radiates ininfrared and submillimeter wavelengths. Furthermore it acts as a katalysator on the formation of molec-ular hydrogen gas.Observations as well as simulations show that the very cold and dense molecular clouds are regions ofactive star formation. These regions are thus of particular interest in the research on star formation. Thedifferent components of the ISM interact in numerous ways and it is especially interesting to understandwhat processes lead to formation of molecular clouds and thus to star formation regions. A good insightin the properties of molecular clouds, yields information on the circumstances and conditions for starformation. Following paragraphs discuss this in more detail.

2.1.2 Jeans mass and Jeans radius

The density and temperature of a gas cloud is determined by the balance between gravity and thermalpressure. A spherical cloud is in hydrostatic equilibrium when

dPdr

= −Gρ(r)M(r)r2 , (2.1)

with P the internal pressure, ρ(r) the density at a radius r and M(r) the total gas mass enclosed in r. Fora gravitational collapse to occur, this hydrostatic equilibrium has to be disbalanced. This happens whenthe Jeans mass and Jeans radius are exceeded. To calculate the critical values for mass and radius(approach similar to [Perkins, 2003]), one starts from the free fall time. This is the time a small mass min the outer shell of a gas cloud needs to reach the center of the gas cloud when this cloud undergoesa pressureless gravitational collapse. When this small mass m at position r0 moves to a new position rcloser to the center of the cloud, the kinetic energy rises along 1

2 m(drdt )

2 while the change in potentialenergy equals GMm

r − GMmr0

. The free fall time tFF is hence calculated from

tFF =∫ dr

drdt

=∫ dr√

2GMr − 2GM

r0

. (2.2)

The integration of this expression (a goniometric substitution adjusts the boundaries from [r0, 0] to[−π

2 , 0]) yields

tFF =π

4

√2r3

0GM

=

√3π

32Gρ, (2.3)

where the relationship M = ρ 43 πr3

0 was used. The free fall time is now to be compared to the time ts for awave to travel the cloud at sound speed cs with ts = r0/cs. When tFF � ts, a density perturbation will beimmediately annihilated by the fast travelling pressure waves. However, when tFF � ts, the gas aroundthe pressure perturbation will already be collapsing before pressure waves can erase the perturbation.Under this circumstances a gravitational collapse can occur. In other words, for each cloud a criticallength scale and total mass exist and these are called the Jeans length λJ and Jeans mass MJ . Whenthe Jeans mass and length are exceeded, a collapse is able to take place. Jeans length is calculatedfrom

λJ = cstFF ' cs

√π

Gρ, (2.4)

2.1. Physics of star formation 13

where the sound speed can be relativistic or non-relativistic. In the case of non-relativistic particles thesound speed equals

c2s =

∂P∂ρ

=γkTm

, (2.5)

which yields a Jeans length of

λJ '√

5πkT3Gρm

, (2.6)

with γ = 5/3 in the case of neutral hydrogen. The Jeans mass MJ is then simply given by

MJ ' πρλ3J , (2.7)

and can thus be seen to be proportional to

MJ ∝ ρ−1/2T3/2. (2.8)

The equalities are not exact due to constant factors close to unity, depending on the mass distribution inthe cloud.The non-relativistic Jeans radius and mass can also be calculated by requesting equality of the gravi-tational potential energy Egrav ' GM2

r and the kinetic energy (according to the equipartition theorem)Ekin = 3

2MkT

m where m is the average mass of one gas particle. This method yields

rcrit '2MGm

3kT' 3

2

(kT

2πρGm

)1/2, (2.9)

where Eq. 2.7 and Eq. 2.6 were used. Isolating ρ yields

ρcrit '3

4πM2

(3kT2mG

)3. (2.10)

With the collapse of the cloud, kinetic energy is released in the form of heat. Eventually hydrostaticequilibrium is reached. Further collapse is only possible when heat is radiated away in the coolingprocess. The cooler the cloud gets, the more molecules are able to be formed.

2.1.3 Cooling

Different cooling mechanisms occur at different temperatures and densities in a molecular gas cloud.The two main types of cooling that take place in molecular clouds are cooling by dust grains and coolingby molecular line transitions. Discussion paraphrased according to [De Rijcke, 2011].

H2 In most density ranges, cooling by molecular line transitions is the commonest mechanism. Themost abundant molecule in molecular clouds is H2. However, this molecule does not radiate in the tem-perature ranges of molecular clouds. This is due to the quantummechanical structure of H2. Since H2is a homonuclear molecule, two possibilities for the electronic ground state exist, namely orthohydrogenin which the protonspins are parallel and parahydrogen with antiparallel protonspins. Protons as well aselectrons are fermions, hence the wavefunction for H2 needs to be antisymmetric for the exchange ofprotons and of electrons. Orthohydrogen has a symmetric wavefunction for the protonspins, hence theangular momentum wave function needs to be antisymmetric and thus has an odd parity. The oppositeis true for parahydrogen: its angular momentum wave function is symmetric and has an even parity. Re-garding transition probabilities, electric dipole transitions are in general the most likely to occur. For H2however, electric dipole transitions are not possible. Indeed, a dipole transition does not change parity,


Figure 2.1: The relevant molecular and atomic transitions for cooling of molecular clouds. Table taken from [Goldsmith and Langer, 1978].

hence only quadrupole transitions are possible (J : 3 → 1 for orthohydrogen and J : 2 → 0 for parahy-drogen). Probabilities for quadrupole transitions are very low and moreover, the excitation from a stateof J = 0 to J = 2 requires an energy of ∆E = k · 500 K. Hence in very cold regions, this excitation is noteven possible apart from its very low probability. In practice, the transition does not occur in molecularclouds.H2 also radiates in the infrared by vibrational transitions. However, this kind of excitations only occurs attemperatures of the order of 103 − 104 K, which is generally too high for molecular clouds. Furthermore,vibrational (and rotational) excitations mainly take place as a consequence of shock waves or a largequantity of high energy radiation, phenomena which are usually not present in molecular clouds.Finally, electronic transitions for H2 have emission and absorption lines in the ultraviolet (Lyman andWerner), but these excitations require high photon energies that do not occur in the temperature rangesof molecular clouds. To summarize all the above, the H2 gas in the temperature ranges of molecularclouds does not radiate, hence it does not provide cooling.

CO For molecular clouds at temperatures between 10 − 40 K and molecular densities of n(H2) <3 · 104 cm−3, the 12CO molecule, which is the second most abundant molecule after H2, is the dominantcoolant [Goldsmith and Langer, 1978]. CO does have a net dipole moment and no parity problems dooccur since CO is a heteronuclear molecule. For these reasons, CO does allow transitions in which∆J = ±1. Moreover, this rotational transition has an excitation temperature of only 5.5 K, meaning thatCO can be easily excitated by collisions with hydrogen molecules.

Other molecules and atoms For densities of n(H2) = 1 · 103 cm−3 and larger, other isotopic species ofCO, other molecules as for instance O2, H2O and hydrides, and atoms as CI provide a large percentageof the cooling [Goldsmith and Langer, 1978]. An overwiew of molecules and atoms that are important inregard to cooling is given in Figure 2.1.

Dust At high gas densities, cooling by dust grains is the dominant mechanism. Dust acts on the com-ponents of the ISM in different ways. As mentioned before, it stimulates the formation of H2 molecules.Dust gathers HI atoms at its surface which facilitates interaction to form H2 molecules. Furthermore, dustacts as a coolant as a consequence of its thermal radiation. Since it usually has temperatures of 20− 30K, it emits in the infrared. However, for the molecular gas to cool along with the dust, both these compo-nents need to be in thermal equilibrium, otherwise only the dust will cool while the gas remains heated.For dust and gas to be well-coupled, a density of n(H2) > 1.5 · 104 cm−3 [Goldsmith and Langer, 1978]


is necessary. Densities in molecular clouds vary between 1− 105cm−3 hence dust as a coolant is onlyimportant in the densest regions of the molecular cloud.

Collapse By means of the discussed cooling mechanisms, a gravitational collapse can occur withouta raise of temperature. This leads to further stimulation on formation of molecules, which then againfacilitates the cooling process. This provides a positive feedback mechanism for the collapse of thecloud and thus the formation of a star. To summarize, stars are formed in cold, dense and collapsingmolecular clouds.In galaxies, giant molecular clouds are observed with radii of the order of 100 pc. These giant molecularclouds are subdivided in very dense, cool regions of subcloud complexes with smaller radii of more orless 10 pc; this is called fragmentation. When the Jeans criteria are satisfied in a subcloud, a star isformed in this region.When the cloud has collapsed to a density of 1015 kg/m−3 and a radius of 1015 m [Perkins, 2003], ahydrostatic equilibrium is attained and a protostar is formed. Contraction slowly goes on while energyis radiated away. This process obeys the virial theorem which says that for a bound system of non-relativistic particles, the time averaged potential and kinetical energy of the system, respectivily < V >and < E >, are related by − < V >= 2 < E >. Hence with the decrease of the radius r of the cloud,the potential energy V = −GM

r decreases which means that − < V > and thus < E > increase. Allthis kinetic energy is radiated away.At a certain point, the emitted energy is of the order of 10 eV per hydrogen atom which means that disso-ciation of H2 molecules and ionization of H atoms can take place. Eventually the collapse is stopped bythe onset of thermonuclear reactions and the proton-proton chain or CNO chain start producing helium.

2.1.4 Induced star formation

Initial mass function Stars form in different spectral types and mass ranges (see Table 2.1). The pro-bability that a star will form with a certain mass M is calculated by an initial mass function ξ(M). Forinstance Eq. (2.11) was proposed by Salpeter [Salpeter, 1955]

ξ(M) ≈ 0.03(

MM�

)−1.35, (2.11)

for M between 0.4 and 10 M�. This result was obtained by fitting observed luminosity functions of mainsequence stars.In 1976, Silk published several articles discussing the temperature dependence of the initial mass func-tion. He expects from observations and calculations of the Jeans mass, initial mass function and astability analysis, that it is expected that high mass stars form in turbulent regions with higher tempera-tures (T ∼ 100 K) while lower mass stars form in somewhat less violent regions with lower temperatures(T ∼ 10 K) [Silk, 1977a, Silk, 1977b].More recently, results from research in regard to the environmental dependency of the initial mass func-tion suggest that massive stars are formed in regions of high surface density where radiative feed-back raises the temperature more effectively hence increasing the Jeans mass (MJ ∝ ρ−1/2T3/2)[Krumholz et al., 2010]. These results were obtained from simulations using a hydrodynamical numeri-cal model. The environmental dependency of the initial mass function is still a subject of active researchand discussion, but is also supported by observational hints, e.g. [Hsu et al., 2012, Hsu et al., 2013].Once a star with a certain mass M is formed, it exchanges energy with its environment in several ways.This released energy causes instability in the surrounding molecular cloud region and causes the birthof more stars. The process of stimulated star formation by the presence of other stars is called inducedstar formation or triggered star formation. Several forms of induced star formation exist and the way inwhich energy is released from a star and added to the ISM depends on the spectral type of the star andits current life stage.

OB associations OB associations are star clusters populated by mainly O and B stars, see Table 2.1.This massive and short-lived type of stars produce strong stellar winds during their entire lifetime which


spectral type mass (M�) temperature (K) radius (R�) luminosity (L�)O 60.0 50000 15.0 1400000B 18.0 28000 7.0 20000A 3.2 10000 2.5 80F 1.7 7400 1.3 6G 1.1 6000 1.1 1.2K 0.8 4900 0.9 0.4M 0.3 3000 0.4 0.04

Table 2.1: Spectral types of stars along the Harvard classification and their properties. Only applicable to main-sequence stars. Data from[Schombert, 2013].

Figure 2.2: The OB subgroup has emitted the ionization-shock front which gave rise to the HII region. Instability in the CPS layer causesthe birth of more OB stars. Their presence is indicated by the observated infrared and/or compact continuum sources and H2O and OHmasers. The direction of the magnetical field for the ideal case of maximal transmission of the ionization-shock front is indicated. Figure from[Elmegreen and Lada, 1977].

is of the order of 107 yr. Stellar winds consist of large fluxes of photons with wavelengths close to theLyman line. Therefore the radiation from stellar winds ionizes the hydrogen molecules from the molecularcloud surrounding the OB association and gives rise to expanding HII regions (see Par. 2.1.1), alsocalled Stromgren spheres. The interesting property of ionization fronts is that they trigger high massstar formation. This phenomenon was firstly discussed in the 1970’s by e.g. Elmegreen and Lada[Elmegreen and Lada, 1977]. They discussed the instability in the layer between the ionization front andthe preceeding shock front, see Figure 2.2. Combining observations and results from an analyticalmodel in which they calculated the instability of this cooled post-shock layer (CPS), they conclude thatafter approximately one and a half million years, a new OB subgroup will form. Indeed, as discussedbefore, massive stars tend to form in turbulent, dense regions of slightly higher temperature and theseconditions are present in the CPS. The new OB association is then called the second generation andalso emits stellar winds. This proces continues and sequential formation of OB associations takes place.Moreover, stellar winds are not the only feedback process originating from OB associations that returnsenergy to the ISM. The short lifetimes of the massive OB stars result after 107 yr in supernovae oftype SNe II when the degeneration pressure of the free electrons is not sufficient anymore to balancethe gravity of the fusioning star producing iron. In an instant, all electrons recombine with the protonsto form neutrons, the degeneration pressure disappears completely and the star collapses under its


Figure 2.3: Illustration of induced star formation in molecular clouds. When SNe and other feedback effects emmit shock waves in the ISM,new generations of stars are formed in spatially ordered subgroups. Illustration from [Brau, 2013].

own gravity. The result is an energy transfer to the ISM of the order of e51 erg by a massive releaseof neutrinos and a giant shock wave travelling through the ISM. During the adiabatic expansion of thisshell, which is called the Sedov-Taylor phase, the radius of the shell is given by [Sedov, 1958]

rs =

(E0

α(γ)ρ

)1/5t2/5, (2.12)

with E0 the released energy by the SN II, α(γ) a numerical factor obtained from imposing energy con-servation [Vandenbroucke et al., 2013], ρ is the density of the ISM and t the time interval for which rsis calculated. For the standard release of energy E0 = e51 erg, typical molecular cloud densities ofρ = 100 amu/cm3 and a time span of 104 yr, this results in a rough estimate for the characteristic radiusrs ∼ 10− 30 kpc. This can be interpreted as the influence sphere of an OB association or more generallythe characteristic length scale for induced star formation. A nice illustration of sequential induced starformation, is shown in Figure 2.3.Sequential formation of OB subgroup association is strongly supported by observational evidence. Sim-ply the spatial ordening of the OB subgroups in regard to their ages is a very good indication of thetheory. A chain of OB subgroups at similar distances lying along the galactic plane with monotonicallyincreasing age were observed by e.g. Ambartsumian [Ambartsumian, 1958] and Blaauw [Blaauw, 1958].

Supernovae As already discussed, SNe II are an important feedback effect in regard to induced starformation.In fact, supernovae in general (also of type SN Ia for instance) release energy in the ISM, and hencehave a large influence on star formation.Supernovae of type Ia occur only in binary systems. In most cases, one of the binary stars will eventu-ally become a white dwarf. When the compagnion star starts to blow out matter in an evolved life stage,under certain circumstances this matter will be transferred to the white dwarf in a Roche lobe. A SN Iaoccurs when the mass of the white dwarf transcends the Chandrasekhar limit of 1.4 M�.Whether the influence of SNe on star formation is mainly positive or negative is ambiguous. On the onehand, because SNe release a vast amount of energy, the surrounding gas is heated and ionized andgas densities are decreased which are unfavorable conditions for star formation. On the other hand, at


the edges of the pressure and ionization front, instability arises and star formation is provoked. Numer-ous studies have shown that SNe induce SF, for example a numerical study by Krebs and Hillebrandt[Krebs and Hillebrandt, 1983], who have used a 2-dimensional hydrodynamical model. Their resultsshowed that SNe can lead to a compression of a nearby molecular cloud (in a radius of more or less 20pc), if this cloud is not too far from the critical Jeans mass and if cooling is sufficiently efficient.Another important consequence of SNe is that they release metals in the ISM. Metallicity of the inter-stellar gas has a great influence on its cooling efficiency, see Par. 2.1.3.

T associations When stars in mass ranges smaller than 2 M� are formed and the proces of hydrogenburning is started, they go through a violent phase, called the T Tauri phase, before reaching the mainsequence stadium on the Hertzsprung-Russel diagram. These T Tauri stars posses a thick circumstellaraccretion disk consisting out of gas gradually falling onto the surface of the star, producing strong stellarwinds in the direction of the rotation axis of the protostar and losing a significant fraction of the stellarmass. These stellar winds ionize the surrounding molecular clouds and are hence forming expandingHII bubbles.In 1978, Norman and Silk [Norman and Silk, 1980] proposed a model showing that the presence of TTauri stars induces more (low mass) star formation. However, their model does require an initial highenergy trigger as for instance SNe or luminous OB stars. Once the T Tauri stars are formed, theyproduce stellar winds giving rise to ionized bubbles. These bubbles expand and tend to intersect. Atthe edges of these bubbles, clumpy cloud filaments are formed that coalesce and eventually collapse.This gives rise to the formation of new (low mass) stars. The entire process is illustrated in Figure2.4. Triggered star formation in context of T Tauri stars is also supported by more recent studies and

Figure 2.4: Illustration of T Tauri triggered star formation. Figure taken from [Norman and Silk, 1980].

observations. For instance Chauhan et al. [Chauhan et al., 2009] have observed near-infrared excessstars in HII regions and they found that these stars are aligned from the ionization source to the edgeof the HII cloud. Moreover, the ages of the aligned stars also gradiently increase towards the ionizationsource. These observations strongly support the hypothesis of sequential star formation originating froman ionization source.

Length and time scales We have shown from the above discussion that induced star formation originatesfrom a number of different processes. Therefore, it is surprising that length and time scales for all of these

2.2. Parameterization of star formation 19

processes are more or less similar.From the spatial separation of sequential OB and T associations, length scales of the order of 10− 30 pcare observed. For supernova induced star formation, we have calculated using the Sedov-Tayler radiusthat the characteristic radius of influence is about the same.Time scales can be estimated as the lifetimes of OB stars (∼ 107 yr) and the duration of the T Tauri stage(∼ 107 − 108 yr).

2.2 Parameterization of star formation

In order to model star formation and to study the behavior of star formation systems, the physical phe-nomena of Sect. 2.1, need to be parametrized.

2.2.1 Kennicutt-Schmidt law

In 1959, Schmidt proposed a star formation law [Schmidt, 1959], currently known as the Schmidt law,which states that the star formation rate in a galaxy is proportional to a power n of the local gas densityρg. To get this result, he assumes that the initial luminosity function of all main sequence stars ψ(Mv) istime independent, with Mv an absolute magnitude in the visual.The luminosity function φ(M)v, which yields the total observed luminosity of all stars for a certain mag-nitude Mv, is given by

φ(M)v =ψ(Mv)T(Mv)

T(gal). (2.13)

Here, ψ(Mv) is the initial luminosity function, T(Mv) is the lifetime for a star of magnitude Mv andT(gal) is the total galaxy age. Hence through observations, the initial luminosity function ψ(Mv) can beobtained.When now the assumption is made that the initial luminosity function is time independent, the rate ofstar formation equals

dN(Mv, t)dt

= ψ(Mv) f (t), (2.14)

with N(Mv, t) the number of stars with a magnitude Mv at time t, and f (t) is called the rate function.In order to obtain an analytical solution for the rate function f (t), Schmidt defined P as the ratio ofpresent gas density ρg(1) to initial mass density ρg(0)

P =ρg(1)ρg(0)

, (2.15)

with t = 1 the present and t = 0 the beginning of the era of star formation. He assumes that the numberof stars formed per time unit varies with a power of the gas density,

dNdt

= f (t)∑Mv

ψ(Mv) = Cρng(t). (2.16)

Evaluating this equation at t = 1 and t = 0 yields

f (1) = Pn f (0). (2.17)

For n = 1 the analytical solution for the rate function is given by

f (t) = f (0)e−t/τ , (2.18)

where τ is determined by

e−1/τ = P. (2.19)


This solution is obtained by realizing that f (t) is proportional to dρg(t)dt . For a detailed derivation, see

[Schmidt, 1959]. An analytical solution which we do not discuss, is available for n > 1.In general, the Schmidt law is rather presented in terms of gas density than in terms of the rate function.Hence to summarize, the SFR dN

dt is proportional to the rate function f (t) (see Eq. (2.16)), which is

proportional to dρg(t)dt , which is proportional to a power n of the gas density ρg. We obtain the expression

dρs

dt= −dρg

dt= c?ρn

g , (2.20)

where c? is a constant, ρs is the star density, and it is assumed that all lost gas mass is converted intostars, hence dρs

dt = −dρgdt . The analytical solution in terms of gas density for n = 1 equals

ρg(t) = ρg(0)e−c? tτ , (2.21)

with in this case

e−c? 1τ = P. (2.22)

To obtain a value for n, several observable quantities are calculated for different values of n, as forinstance the initial luminosity function, the rate of star formation, the number of white dwarfs, the abun-dance of helium, etc. Schmidt concluded that a value around n = 2 provided the best reproduction ofobservational data.The Schmidt law was used almost exclusively for decades to obtain estimates for SFRs in galaxies,although its validity was questioned several times. The problem with the Schmidt law was that severalstudies obtained very divergent values for n, going from n = 0 to n = 4, corresponding to differentconditions in disks. For example, very large values of n were found in the arms of spiral galaxies, evenwhen the gas density in those arms was similar to dense regions in smooth disks, where n is smaller.Uncertainty arose whether gas density was the only parameter that describes SFRs and even whethera universal SF law does exist at all.In 1988, Kennicutt published an article stating that SFRs not only depend on the gas density but alsoon the ratio of this density to a critical threshold density determined by dynamical properties of the diskof the galaxy [Kennicutt, 1989]. This gas density threshold takes into account gravitational instabilitiesin the disk. The Schmidt law is then valid for gas densities above the threshold value. In this regime,the Jeans mass is exceeded and the gas is gravitationally unstable. A power law with a shallow slopedescribes the SFR.Below the threshold density, almost no star formation takes place. In this regime, the dependency of theSFR to the gas density is approximately zero.In the transition zone with gas densities around the threshold value, a very steep dependency of the gasdensity is observed. Kennicutt states that non-linear dependencies of the SFR rate to the gas densitycould be explained by this effect. The three different regimes are shown in Figure 2.5 and it is clear thatthe Schmidt law does depend on the distance to the center of the disk, called the radius R.Kennicutt continued his research concerning the Schmidt law and in 1997, he proposed an alternativeSchmidt law, later referred to as the Kennicutt-Schmidt law, in which the SFR is indeed proportional to(a power of) the gas density, but is also weighted with the average orbital timescale, the dynamical time(shown in Figure 2.6). The dynamical time τ is defined as

τ =2π

σ=

2πRv(R)

, (2.23)

with σ the angular rotational velocity and v(R) the velocity in function of radius R. The dependency ofthe Kennicutt-Schmidt law of the dynamical time accounts for the previous findings that the Schmidt lawis dependent on the distance R to the center of the galaxy.The final proposition of Kennicutt for the star formation law equals

dρs

dt= −dρg

dt= c?

ρng

τ, (2.24)

which is called the Kennicutt-Schmidt law. Today, it is widely used to model SFRs in numerical simula-tions of galaxies.


Figure 2.5: SFR dependency of gas surface density. Three different regimes are showed. For densities below a threshold value, there is almostno dependence of the SFR on the gas surface density. For gas densities above the threshold value, a Schmidt law with modest power defines thedependency. In the transition zone around the threshold density, a very steep dependency is observed. From left to right, the curves correspondto smaller distances to the center of the galaxy, and higher values for the threshold density. Figure taken from [Kennicutt, 1989].

2.2.2 Oscillations in star formation

However usefull the Kennicutt-Schmidt law may be, it does not explicitly take into account the numer-ous feedback effects that have an influence on star formation and lead to self-regulation (a.k.a. self-organization). Molecular cooling, induced star formation, supernovae and other feedback effects canbe parameterized and put into equations to describe a star formation system, consisting out of severalcomponents, namely stars and gas (sometimes a distinction is made between molecular, atomic andionized gas). Cooling converts atomic gas into molecular gas, molecular gas can form stars, massivestars explode through SNe and return ionized gas and metals to the ISM. The entire mechanism can bedescribed by a set of equations and put into a dynamical system of the form

dXdt

= f (X). (2.25)

As shown by e.g. Bodifee and De Loore [Bodifee and De Loore, 1985], a star formation system de-scribed by Eq. (2.25), with different X being the components of the ISM and where effects of feedbackare explicitly accounted for, is self-regulating and can lead to oscillatory behavior of the variables X.Their system to describe star formation, makes use of four different components namely atomic gasmass A, molecular gas mass M, massive star mass S with stars of a mass > 20 M� and a reservoir forold and less massive stars R. The reason to seperate between S and R is to gather the stars S that havean important influence on their environment hence stars that trigger SF.A main difficulty is to parameterize molecular cooling. Cooling proceeds more efficiently with higherdensities of molecular and atomical gas, but is impeded by ionizing radiation of nearby massive stars.Therefore, Bodifee and De Loore proposed a cooling rate proportional to An2(M + αA)n3 S−n4 . Parame-ters n2, n3, and n4 are assumed to be between 1 and 3 and α expresses an efficiency ratio of molecularand atomic cooling, hence α has values between 0 and 1. The entire set of equations is given by


Figure 2.6: Obervations of SFRs in function of the ratio of gas surface density and dynamical timescale. Filled circles correspond to disksamples, open circles correspond to SFRs and gas densities in the center of the disk, squares represent starburst galaxies. [Kennicutt, 1998].

[Bodifee and De Loore, 1985]

dAdt

= K1S + K2S + K3Mn1 − K4 An2(M + αA)n3 S−n4 , (2.26)

dMdt

= K4 An2(M + αA)n3 S−n4 − K5SMn1 − K3Mn1 , (2.27)

dSdt

= K5SMn1 − K1S− K2S, (2.28)

dRdt

= K1S + K3Mn1 . (2.29)

In Eq. (2.26), we see that the amount of atomic gas decreases in time with an increased cooling raterepresented by K4 An2(M + αA)n3 S−n4 . This loss in atomic gas mass is converted into molecular gashence this term appears in Eq. (2.27) with an opposite sign. Molecular gas is subject to spontaneousstar formation described by a Schmidt law K3Mn1 as well as induced star formation parametrized by theterm K5SMn1 . Induced star formation is hence proportional to the density of (massive) stars and to apower of the molecular gas density. As described in Par. 2.1.4, massive stars induce the birth of othermassive stars. The loss of molecular gas mass that describes induced star formation is thus added tothe change in massive stellar mass, Eq. (2.28). Active massive stars S lose mass through stellar winds;this proces is parametrized by −K2S and this mass loss returns as atomic gas. There is also a fractionof the active stellar material S that is transformed to inactive stars R given by the term K1S. In the end,stellar rest products R are returned to the ISM as atomic gas.The parameterization of Eqn. (2.26) to (2.29) inexplicitly assumes that all stars originated by sponta-neous star formation, have masses less than 20 M�, and moreover that those stars are not activematerial that is able to induce star formation. However, there is no reason to assume that the spon-taneous birth of stars with masses > 20 M� is impossible, although it will be less likely. Furthermore,stars that have been formed with masses less than 20 M� go through the T Tauri phase and thus areable to induce star formation. Another striking feature of the system, is that the same power n1 for thedependency of the (molecular) gas density is used for spontaneous star formation as for induced starformation. In spite of these assumptions, the parameterization does provide a good description of thephenomenon of star formation and the interactions between the different components of the ISM that gowith it.Bodifee and De Loore studied the proposed system in great detail, more specifically the influence of all


introduced parameters on the systems behavior.Two main types of behavior were found. Depending on different values of the parameters, the systemis stable and evolves to a final stationary state, or it becomes unstable, bifurcates, and shows self-sustaining oscillations behaving like a limit cycle1. Interesting findings are that oscillations occur for asmall value of n1, and that the parameter K5 parameterizing induced star formation, does have a greatinfluence on the system behavior. For very small values of K5, initial oscillations are damped. Withincreasing K5, the amplitude of the oscillations enlarges until a limit cycle is obtained. Beyond a criticalvalue, the system evolves towards a stationary state without oscillations.Bodifee and De Loore conclude that self-regulation and oscillatory behavior is found in a four componentsystem where molecular synthesis and induced star formation are included.Similar systems were studied by Korchagin et al. [Korchagin et al., 1988]. They proposed 3 mecha-nisms that lead to the development of oscillations in the system, namely induced star formation, strongnon-linearity in feedback mechanisms, and additionally the presence of time delay.Time delay includes a time scale on which for instance stellar winds are ejected and stellar mass isreturned to the ISM in the form of gas. When the process of mass ejection is delayed with a time thatexceeds a critical value, the system can go into oscillatory behavior. However, they conclude that themechanisms of induced star formation together with non-linear cooling are sufficient to obtain oscillationsof the components of a system, shown by the following three component system

dSdt

= βSMn1 − αS, (2.30)

dMdt

= −βSMn1 + dMn2 A, (2.31)

dAdt

= Q + αrS− dMn2 A. (2.32)

(2.33)

S represents star mass, M is the molecular cloud mass and A is gas mass in the atomic phase. Theparameter β regulates induced star formation, α is the rate at which stellar material is removed (with afraction r put back into atomic gas). Parameter Q describes the accretion rate. Parameter d describesthe rate at which atomic gas is converted into molecular gas and vice versa. Note that this parame-terization of cooling is different to the previous system. Moreover in this system, no spontaneous starformation takes place. The system should be considered as a somewhat simplified model to isolatethe effects of induced star formation and cooling on the systems behavior. No mass conservation isimposed.Korchagin et al. studied the (in)stability of the system not only numerically but also analytically using theHurwitz criterium (see Par. 2.2.3). From this Hurwitz criterium, they found that instability in the systemand oscillatory behavior occured when n2 > n1, or in other words, when the non-linearity in molecularcloud production is stronger than in induced star formation. Numerically, undamped oscillations beha-ving like a limit cycle are found in the system.Much like Bodifee and De Loore, Korchagin et al. concluded that the explicit parameterization of (non-linear) feedback mechanisms leads to self-organization, and they propose induced star formation as(one of) the most important mechanism(s) in order to obtain oscillatory behavior in the system variables.The presence of oscillatory behavior in the star component is interesting in regard to investigating (cyclic)starbursts present in BCDs. Therefore, instability and oscillatory behavior in star formation systems arestudied in detail in Par. 2.2.4 where conditions for oscillatory behavior are set up.

2.2.3 Hurwitz criterium for stability

The Hurwitz criterium provides a test for the presence of (in)stability of a star formation system. Thecriterium will be used in Par. 2.2.4 to test which conditions need to be valid to obtain instability in asystem that may lead to oscillatory behavior.

1Limit cycle behavior occurs in non-linear dynamical systems and defines a closed trajectory in phase space. In this case, limit cyclebehavior corresponds to unlimited undamped oscillations in the star formation system components.


A non-linear system consisting out of a set of differential equations of the form

X(t) = f (X(t), Y(t), Z(t)),

Y(t) = g(X(t), Y(t), Z(t)),

Z(t) = h(X(t), Y(t), Z(t)),

(2.34)

is studied. The equilibrium solutions X0, Y0 en Z0 of the system can be found by putting the equationsof system (2.34) to zero:

0 = f (X(0), Y(0), Z(0)) = f (X0, Y0, Z0),0 = g(X(0), Y(0), Z(0)) = g(X0, Y0, Z0),0 = h(X(0), Y(0), Z(0)) = h(X0, Y0, Z0).

(2.35)

To perform stability analysis, the influence of perturbations eλt is studied: X(t) → X(t)eλt, Y(t) →Y(t)eλt and Z(t)→ Z(t)eλt, where λ are the eigenfrequencies of the system. It is clear that a system isonly stable when all eigenfrequencies have a negative real part and the perturbations are damped. Whenone of the eigenfrequencies has an imaginary part, oscillations will occur. Either damped or enhancedoscillations arise when there is a respectivily negative or positive real part as well as an imaginary partto the eigenfrequency. These eigenfrequencies are calculated by setting up the characteristic equationof the system through |J − Iλ| = 0. The Jacobian J is defined as

J =

∂ f∂X

∂ f∂Y

∂ f∂Z

∂g∂X

∂g∂Y

∂g∂Z

∂h∂X

∂h∂Y

∂h∂Z

X0,Y0,Z0

. (2.36)

The characteristic polynomial equals

P(λ) = a3λ3 + a2λ2 + a1λ + a0 = 0, (2.37)

where a3 > 0. To check whether the real parts of the three solutions for λ are negative, the Hurwitzcriterium can be used. A polynomial with only negative real parts is called Hurwitzian and is alwaysstable.To check under which conditions the system will show oscillations, a solution for the eigenfrequencyλ = iω is inserted in Eq. 2.37 and this yields

−iω3 − a2ω2 + ia1ω + a0 = 0. (2.38)

If this ω exist and is real, we know that the system has at least one imaginary eigenfrequency andoscillatory behavior can occur in the system.Splitting real and imaginary parts in Eq. (2.38) yields two equations for ω, namely

ω2 = a1, (2.39)

ω2 =a0

a2. (2.40)

For this ω to exist and to be real, both these expressions should obviously be equal and positive. Thisyields the conditions

a1a2 = a0, (2.41)a1 > 0, (2.42)a0

a2> 0. (2.43)

If Eq. (2.41) is satisfied, than Eq. (2.42) and Eq. (2.43) are identical. Hence for a purely imaginaryeigenfrequency to exist, instability to arise and oscillations to occur in the system, we need to check if


a1a2 = a0 and a1 > 0.In a two component system, purely imaginary eigenfrequencies do not exist, therefore we insert a par-tially imaginary eigenfrequency λ = γ+ iω in the two component characteristic equation λ2 + a1λ+ a0 =0. Instability arises when γ > 0. If there is also an imaginary part to the eigenfrequency (hence if ω isreal), oscillations will occur and will be amplified in the system.We obtain for the real and the imaginary part of the characteristic equation with inserted eigenfrequency

γ = − a1

2, (2.44)

ω =

√−a2

14

+ a0. (2.45)

For the system to be instable, it is clear that a1 < 0. If this is the case, we can also see that ω will be

real if a0 > − a21

4 and hence oscillations will occur. If a1 > 0, the system will always be stable.Generalisation of these calculations for an n-component system provides the Hurwitz criterium for sta-bility. When one of the Hurwitz conditions is violated, the system has at least one eigenfrequency thathas a positive real (and sometimes an imaginary) part. A characteristic polynomial which satisfies theconditions to be stable is called Hurwitzian.

Hurwitz criterium A polynomial P(λ) = anλn + an−1λn−1 + · · ·+ a1λ + a0 is Hurwitzian if and only ifthe determinants D1 up to and including Dn are positive [C., 2013]

D1 = an−1 > 0, (2.46)

D2 =

∣∣∣∣an−1 anan−3 an−2

∣∣∣∣ > 0, (2.47)

...

Dn−1 =

∣∣∣∣∣∣∣∣∣∣∣

an−1 an 0 · · · 0an−3 an−2 an−1 · · · 0an−5 an−4 an−3 · · · 0

......

.... . .

...0 0 0 · · · a1

∣∣∣∣∣∣∣∣∣∣∣> 0, (2.48)

Dn = a0Dn−1 > 0. (2.49)

For a system of third order, one obtains the following conditions for stability

a2 > 0, (2.50)a2a1 > a0a3, (2.51)

a0(a2a1 − a0a3) > 0. (2.52)

These are also called the Hurwitz conditions for stability. Instability is obtained when at least one of theHurwitz conditions is not valid. It should be noted that when one of the Hurwitz criteria is invalid, it is notsure whether oscillations will occur. It is also possible that only a negative real part to the eigenfrequencyis present without an imaginary part. In this case, perturbations will be enhanced and the system willexponentially move away from its equilibrium solution, without any presence of oscillatory behavior.


2.2.4 Star formation systems

The Hurwitz criterium can be used to study the behavior of different star formation systems. In thisparagraph, we focus on different systems with the phenomenon of induced star formation included andtry to determine conditions for which this system will or will not be stable and potentially show oscillations.

Induced SF with non-linearity in gas density We start simple, with a two component star formationsystems consisting of a component stellar mass S, and a component gas mass M. The first systemunder study is

S = −αS + βMnS,

M = αS− βMnS,(2.53)

with α the fraction of stars that perish and are transferred to molecular gas and β is the factor thatregulates induced star formation parameterised by βMnS. This is a vastly simplified system to study theinfluence of induced star formation on the systems behavior. Note that no spontaneous star formationtakes place in this system and that mass is conserved.First of all, equilibrium solutions are calculated. For the system in Eq. (2.53), an equilibrium solution M0exists for all values of S, namely

M0 =

(α

β

) 1n

. (2.54)

The Hurwitz conditions for a two component system with characteristic equation a2λ2 + a1λ + a0 equal

a1 > 0,a0a1 > 0.

(2.55)

For system (2.53) this yields

a1 = α− βMn0 + nβMn−1

0 S0 = nβ

(α

β

) n−1n

S0 > 0,

a0 = 0.

(2.56)

Since we know that n, α, β and S0 are all positive, the first Hurwitz criterium will always be fulfilled. In fact,because a0 = 0, the criterium a0a1 is violated. Still, we see from Eq. 2.44 that for a1 > 0 and a0 = 0, thevalue for ω will be imaginary and hence no imaginary eigenfrequency exists. This means that all rootshave negative real parts and a small perturbation will be damped and evolve towards a stable solution.To make the system somewhat physically more acceptable, we propose an accretion term Q that in-troduces a rate for which gas accretes and will be dense enough to start collapsing and form a star.Furthermore we drop mass conservation because only a fraction r of old star mass αS is returned tomolecular gas. We obtain hence a system

S = −αS + βMnS,

M = αrS− βMnS + Q.(2.57)

Equilibrium solutions M0 and S0 are given by

M0 =

(α

β

) 1n

, (2.58)

and

S0 =Q

α(1− r). (2.59)


Also an equilibrium value for S0 is present in this system. The changes brought into this system howeverdo not lead to instability. Indeed, the factors a1 and a0 are given by

a1 = α− βMn0 + nβMn−1

0 S0 = nβ

(α

β

) n−1n Q

α(1− r)> 0,

a0 = nβ

(α

β

) n−1n

Q > 0.

(2.60)

The Hurwitz conditions are always valid and no oscillatory behavior will be observed in this system.

Induced SF with non-linearity in star density Different results are obtained for a system with a power nover the star mass instead of the gas mass. Physically this parameterization makes less sense. It wouldbe more logical that star formation, even induced star formation, is more dependent on gas mass thanon stellar mass. However we define this system to study its (in)stability:

S = −αS + βMSn,

M = αS− βMSn,(2.61)

where mass is conserved. Equilibrium solutions M0 and S0 are related by

M0 =αS1−n

0β

. (2.62)

Determination of the characteristic polynomial for this system and inserting the equilibrium solutionsyields

a1 = α(1− n) + βSn0 ,

a0 = αβSn0 (1− n).

(2.63)

This is an interesting result. For n > 1, a1 can be negative, depending on the values of α, β and S0. Alsoa0 will defenitely be negative hence at least one of the Hurwitz conditions for stability is not valid. Indeed,even if a1 is positive, a0a1 will always be negative for a value of n > 1. In conclusion, non-linear inducedstar formation can lead to instability, perturbations will amplify instead of being damped and oscillatorybehavior of the systems variables S and M is a possibility.To study this system into more detail, the same extra physics as in system Eq. (2.57) is added, resultinginto

S = −αS + βMSn,

M = αrS− βMSn + Q.(2.64)

The equilibrium solutions of system (2.57) are

M0 =α

βS1−n

0 , (2.65)

and

S0 =Q

α(1− r). (2.66)

The values of

a1 = α(1− n) + β(

Qα(1−r)

)n, (2.67)

a0 = αβ(

Qα(1−r)

)n(1− n), (2.68)


are obtained.For n > 1, detailed function testing (using a python script) serves us with a negative value of a1 for alarge range of values of all parameters (going over 10−6 to 105 for α, Q and β and from 0 to 1 for r).Actually, what is important are not the absolute values of α, β and Q, but their relative values. Theaccretion term Q needs to be several orders of magnitude smaller than α and β for a1 to be negative.Indeed, when the accretion term dominates, we see from Eq. (2.67), that a1 remains positive.For the parameters α and β, it is important that they do not differ too much (generally not more thanone order of magnitude). Indeed, the term of induced star formation is balanced with the rate at whichstars have reached their life ending. When this balance is distorted too much, no equilibrium solution ispresent around which the oscillations can take place.

Inclusion of spontaneous SF Since we know that instability occurs for systems using a parameteriza-tion of induced star formation with a power over the star density βMSn, we would like to know underwhich circumstances a system using a parameterization of induced star formation with a power over gasdensity βMnS becomes instable. Additional physics is introduced into the system and the effect on thestability of the system is studied.There is a disadvantage to all the studied systems up till now: no spontaneous star formation is in-cluded. This also means that when initially no stars are present, no stars are able to be formed. Asimple Schmidt law γM is added to the systems in order to study its effect on the (in)stability of thesystem. For instance system (2.57) becomes

S = −αS + βMnS + γM,

M = αrS− βMnS + Q− γM,(2.69)

with γ a parameter with dimensions of time−1 to regulate spontaneous star formation.First of all, we study the case of n = 1. Equilibrum solutions are given by

S0 =Q

α(1− r), (2.70)

M0 =Qα

βQ + αγ(1− r). (2.71)

Via the characteristic equation of the system, values for a1 and a0 are calculated and equal

a1 = α− βQα

βQ + αγ(1− r)+

βQα(1− r)

+ γ, (2.72)

a0 = βQ + αγ(1− r) > 0. (2.73)

Since the system is getting more complicated, it is harder to see under which conditions the value ofa1 will get negative. A python script testing a1 for a large range of parameter values yields no negativevalues for a1 within the tested boundaries.We are curious whether this is the same for n = 2 in system (2.69). In this case equilibrium values are

S0 =Q

α(1− r), (2.74)

M0 =−γ +

√γ2 + 4βQ2

α(1−r)2

2βQα(1−r)

, (2.75)

and a0 and a1 equal

a1 = α− βM20 + 2βM0S0 + γ, (2.76)

a0 = 2αβM0S0(1− r) + αγ(1− r) > 0, (2.77)

where Eq. (2.74) and Eq. (2.75) should be inserted.Using the python script, no negative values for a1 are found. Up till now, no violation of the Hurwitz


conditions could be found for systems with only a power over gas density and no power over star den-sity in the induced star formation term. However, a description with a power over gas density would bepreferred since it is physically better defensible.

Time-delay According to Korchagin et al. [Korchagin et al., 1988], the presence of time-delay in combi-nation with induced star formation is a sufficient condition for the development of non-linear oscillationsin star formation systems. As discussed in Par. 2.2.2, time-delay introduces a time scale on which stellarwinds take place and return stellar mass to molecular gas. Physically, this makes sense because stellarwinds take place over time scales of O(107) yr. For old stars, it takes a delay for the old stellar materialαS to be transferred into molecular gas αrS: it needs to be cooled first (however cooling is not explicitlyparameterized in this system). The system under study with the inclusion of time-delay equals

S(t) = βS(t)M(t)− αS(t),

M(t) = −βS(t)M(t) + Q + αrS(t− T),(2.78)

with the time dependencies explicitly written down. The parameter T is the time of delay for old stellarmaterial to be transferred into molecular gas. The equilibrium solutions can be calculated as always byputting S and M to zero. This yields

S0 =Q

α(1− r), (2.79)

M0 =α

β. (2.80)

The calculation of the characteristic equation studies the effect of small perturbations eλt on the system,with λ the eigenfrequencies of the system. Up till now, the characteristic equation and the perturbationswere always calculated at t = 0, therefore the perturbations yielded eλ(t=0) = 1. Now, they have to beexplicitly taken into account for the calculation of the characteristic equation∣∣∣∣−α + βM− λ βS

αreλT − βM −βS− λ

∣∣∣∣ = 0, (2.81)

which yields

λ2 +βQ

α(1− r)λ +

βQ1− r

=rβQ1− r

eλT . (2.82)

Definition of the following dimensionless quantities

λ =α(1− r)

βQλ, (2.83)

E =α2(1− r)

βQ, (2.84)

T =βQ

α(1− r)T, (2.85)

yields a simplified characteristic equation

λ2 + λ + E = rEeλT . (2.86)

It is not trivial to determine the eigenfrequencies λ for this equation. As we know, if λ(T) = γ(T)+ iω(T)has a positive real part γ(T), the system is instable. If λ has only an imaginary part for a certain criticalvalue of Tc, it is sure that oscillations will be found in the system. In that case we have λ(Tc) = iω(Tc)and this leads to

−ω2 + iω + E = rEeiω+Tc . (2.87)


Decomposition into real and imaginary parts yields the expressions

sin(ωTc) =ω

rE, (2.88)

cos(ωTc) =1r

(1− ω2

E

). (2.89)

Via the Pythagorean trigonometric identity, ω is calculated and equals

ω2 = E− 12±√

14− E + E2r2. (2.90)

Since we have defined Tc as the value for which ˜λ(Tc) = iω(Tc), it is certain that ω has to be real for avalue Tc to exist. This yields constraints for Eq. (2.90)

14− E + E2r2 > 0, (2.91)

implicating

E >1 +√

1− r2

2r2 . (2.92)

We also have from Eq. (2.90) that

E >12

. (2.93)

Combining these expressions we have

E >1 +√

1− r2

2r2 >12

. (2.94)

Using Eq. (2.84), the final condition for a critical value Tc to exist and oscillations to occur is

α2(1− r)βQ

>1 +√

1− r2

2r2 >12

. (2.95)

For values of r < 1, the latter inequality is always satisfied. The first inequality is also easily fulfilledfor values of r close to 1. It is hence important that a large fraction of the stellar mass is returned asmolecular gas to obtain oscillations. Another important quantity is α, the rate at which old stellar materialis transferred to gas needs to be high enough for oscillations to occur. This term balances the systemand without a balance, no equilibrium solutions are present to oscillate around. The denominator βQneeds to be small enough hence when induced star formation is large, the gas accretion should be smalland vice versa.When it is sure that the inequality of Eq. (2.95) is satisfied, the value for Tc can be calculated from Eq.(2.88) and Eq. (2.89).Now, we repeat this calculations for a system with only spontaneous star formation and no induced starformation present:

S(t) = γM(t)− αS(t),

M(t) = −βM(t) + Q + αrS(t− T).(2.96)

With the same method of calculation as for system (2.78), we find a characteristic equation

λ2 + λ(α + γ) + αγ = αγreλT . (2.97)

2.2. Parameterization of star formation 31(1)(1)

Figure 2.7: Plot of the quantity αγ(α+γ)2 for a range of α = [10−5, 105] and γ = [10−5, 105]. We see that this quantity reaches its limit in 0.25.

In this case, dimensionless quantities are defined as

λ =λ

α + γ, (2.98)

E =αγ

(α + γ)2 , (2.99)

T = (α + γ)T, (2.100)

and this yields the same characteristic equation as Eq. (2.86):

λ2 + λ + E = rEe˜λT . (2.101)

For this equation, we have obtained the condition

E =αγ

(α + γ)2 >1 +√

1− r2

2r2 >12

, (2.102)

for the existence of a critical Tc for which an imaginary frequency exists and hence oscillations occur.However, this inequality can never be fulfilled. Indeed, plotting of the quantity αγ

(α+γ)2 yields values smaller

than 14 , see Figure 2.7. This is an important result: we obtain that oscillations can occur in systems with

induced star formation combined with time delay but not in systems with spontaneous star formationcombined with time delay.Korchagin et al. [Korchagin et al., 1988] did very similar calculations, but with a power of n = 2 over thegas density for spontaneous star formation γM2. They have obtained equal results and concluded thatinduced star formation is necessary (in combination with time delay) to observe oscillatory behavior inthe system.

Three component systems The two component systems up till now did not explicitly account for an im-portant phenomenon in regard to star formation, namely cooling. Indeed, molecular gas is cold whileejected stellar mass is hot and is hence rather in the atomic phase than in the molecular phase (or evenin the ionized phase but this would lead us too far).Therefore we take a closer look at the three component system, studied by Korchagin et al. [Korchagin et al., 1988]


and discussed in Par. 2.2.2.

S = βSMn1 − αS,

M = −βSMn1 + dMn2 A,

A = Q + αrS− dMn2 A,

(2.103)

with the atomic gas mass indicated as A. Cooling is parameterized by the term dMn2 A.Korchagin et al. [Korchagin et al., 1988] have calculated the characteristic equation for this system, andthe coefficients a2, a1 and a0 equal (as confirmed by our own calculations)

a2 =

[d(

α

β

)n2/n1

+Q

1− r

(β

α

)1/n1

(n1 − n2)

], (2.104)

a1 =Qn1

1− r

(β

α

)1/n1[

α + d(

α

β

)n2/n1]

, (2.105)

a0 = Qn1αd(

α

β

)(n2−1)/n1

. (2.106)

The Hurwitz conditions in three dimensions (see Eq. (2.50)) yield the conditions

0 <

[d(

α

β

) n2n1

+Q

1− r

(β

α

) 1n1(n1 − n2)

], (2.107)

(1− r)αd(

α

β

) n2n1

<

[d(

α

β

) n2n1

+Q

1− r

(β

α

) 1n1(n1 − n2)

] (α + d

(αβ

)n2/n1)

, (2.108)

for stability. Instability can occur for the case of n1 < n2, in other words, when the non-linearity in thecooling is stronger than for induced star formation, as discussed in Par. 2.2.2.Will the same behavior occur for systems with spontaneous star formation instead of induced star for-mation? We check for the system

S = γMn1 − αS,

M = −γMn1 + dMn2 A,

A = Q + αrS− dMn2 A,

(2.109)

with γMn1 representing spontaneous star formation.Equilibrium solutions are found to be

S0 =Q

α(1− r), (2.110)

M0 =

(Q

γ(1− r)

) 1n1

, (2.111)

A0 =γ

d

(Q

γ(1− r)

) n1−n2n1

. (2.112)

Calculation of the characteristic equation yields values for

a2 = α + γ

(Q

γ(1− r)

) n1−1n1

(n1 − n2) + d(

Qγ(1− r)

) n2n1

, (2.113)

a1 = n1dγ

(Q

γ(1− r)

) n1+n2−1n1

+ αγ

(Q

γ(1− r)

) n1−1n1

(n1 − n2) + αd(

Qγ(1− r)

) n2n1

, (2.114)

a0 = n1αγd(

Qγ(1− r)

) n1+n2−1n1

(1− r). (2.115)

2.3. Research hypothesis 33

Application of the Hurwitz conditions a2 > 0 and a2a1− a3a0 > 0 yields instability for the case of n1 < n2,the same condition as in the system (2.103) with induced star formation included instead of spontaneousstar formation. We conclude that instability in this systems is provided by the presence of cooling andnot by the presence of induced star formation.

Conclusion We have found several examples of systems where instability and hence oscillatory behav-ior can occur.First of all, including induced star formation with a power over the stellar density βMSn introduces insta-bility in star formation systems. However, this parameterization of induced star formation is not preferred:physically gas mass seems more important than stellar mass in regard to the probability of star forma-tion to take place. Therefore we test different systems with a parameterization of βMnS for induced starformation.Including spontaneous star formation in combination with induced star formation in a system will notincrease its chances of being instable.The presence of time delay is an important phenomenon in regard to instability. If it is combined with thepresence of induced star formation βMnS, instability arises and oscillatory behavior can be found in thesystem. A clear condition for the parameter values can be found for which an imaginary frequency existsat which small perturbations in the system will start to oscillate. Interestingly, when the phenomenon oftime delay is combined with the presence of spontaneous star formation instead of induced star forma-tion, no such imaginary frequencies exist and the system is stable for all parameter values.Finally, three component systems were studied that accounted for the phenomenon of cooling of theatomic gas to molecular gas. It was found that this system is instable when the non-linearity in star for-mation is smaller than the non-linearity in cooling, and this result is valid for the phenomenon of inducedstar formation as well as for spontaneous star formation.

2.3 Research hypothesis

In Chapter 1, the origin of starbursts in BCDs was discussed. Up till the present, a lot of questions inregard to starbursts are left unanswered, e.g. are starbursts in BCDs cyclic phenomena? Are starburststhe consequence of internal mechanisms (rotation, feedback, self-regulation,...), external mechanisms(gas infall, merging, tidal interactions,...), or perhaps a combination of both?Star formation was studied in detail in Chapter 2, where we looked into feedback mechanisms and thephenomenon of induced star formation. It is shown that the interplay of non-linear feedback mechanismsand in particular the presence of induced star formation, leads to self-regulation of star formation. Withincertain conditions, a self-regulating star formation system tends to show oscillations and limit cycle be-havior.Is it possible that these oscillations in star formation rate, obtained from taking into account non-linearfeedback and induced star formation, can be partially or entirely responsable for the presence of (cyclic)starbursts in BCDs?In search for the answer to this question, the phenomenon of induced star formation is implemented ina state of the art N-Body/SPH (Smoothed Particle Hydrodynamics) code that simulates dwarf galaxies.The currently used Kennicutt-Schmidt law is altered to account for induced star formation and the effecton the star formation rate is studied.


3Implementation of induced star formation in an

N-body/SPH code

3.1 The code

The used N-body/SPH code is GADGET-2, developped by Springel [Springel, 2005] and designed to fol-low the evolution of a self-gravitating N-body system. In the research group it is used to simulate dwarfgalaxies and it is constantly modified to improve the description of phenomena as feedback, cooling,star formation, etc.GADGET-2 is written in C, it is free, open source, and able to perform the simulation of a large varietyof phenomena on different space and time scales. Not only isolated (dwarf) galaxies can be modelledbut also interactions, mergers and even full cosmological simulations (for instance the Millennium simu-lation) have been undertaken with GADGET-2.

3.1.1 Dwarf galaxies

In order to simulate dwarf galaxies, initial conditions for the simulation need to be generated. Forour simulations, we make use of the E-models (which are the GADGET-2 equivalent C models from[Valcke et al., 2008]), representing galaxies with mass ranges going from dwarf spheroidals to the light-est dwarf ellipticals. Initially, only atomic gas and dark matter are present. The initial conditions arecosmologically motivated, meaning that they share properties with small structure formations found fromsimulations of merger trees. However, cosmology is not explicitely taken into account in the simulations.Gas is a collisional fluid, meaning the mean free path of a gas particle is very small in comparison to thecharacteristic length scale of a galaxy. Collisional fluids support shock and sound waves in the medium.The hydrodynamics of the fluid is taken into account with SPH. With the generation of initial conditions,one has to choose the density profile of the atomic gas. The gas is put into a pseudo-isothermal sphere,meaning that all gas particles are at a temperature of 104 K and has a density distribution of the form

ρg(r) =ρc

1 +(

rrc

)2 , (3.1)

with ρc the critical density threshold for star formation, see Par. 3.2.1, and rc a scale radius (for moreinformation, see [Schroyen et al., 2013] in preparation). The initial metallicity equals Z = 0.0001 Z� andexpressed the mass fraction of particles that are not hydrogen or helium.

35

36 Chapter 3. Implementation of induced star formation in an N-body/SPH code

Table 3.1: Properties of the initial conditions for an E07 model as used in Chapter 4.

Dwarf galaxy type E07Number of gas particles 100000Number of DM particles 100000Mg,i 262×106 M�MDM,i 1238×106 M�DM profile NFWInitial gas temperature 10000 KInitial metallicity 0.0001 Z�Feedback efficiency εFB 0.7Softening length ε 30 pcSF threshold density ρc 100 amu cm−3

Star formation parameter c? 0.25Rotation none

Cold dark matter interacts only gravitationally, hence collisionless dark matter (DM) is not describedby hydrodynamical equations. Therefore, the collisionless dark matter particles are treated as pointmasses. For the dark matter halo, the Navarro-Frenk-White (NFW) profile is adopted

ρDM(r) =ρs

rrs

(1 + r

rs

)2 , (3.2)

with rs and ρs characteristic values to define the profile (for more detailes, see [Cloet-Osselaer et al., 2012]).During the simulation, stars are formed. The mean free path length of a star is a lot larger than the char-acteristic length scale of galaxies, indeed the collision of two stars is a very rare event. Star particlesare collisionless and interact through gravity. The star particles in the simulation do not represent singlestars but are SSP (Single Stellar Population) particles that contain a large amount of stellar mass, rep-resenting a population of stars with the same age. The mass distribution of an SSP particle is describedby the initial mass function, see Par. 3.2.2.Also other parameters as rotation, number of particles (gas and DM), and criteria according to starformation and feedback are chosen when the initial conditions are generated. An overview of used pa-rameters is given in Table 3.1.According to [Valcke, 2010], starting from a certain lower limit for number of particles, the accuracy ofthe simulation results will not increase with an increasing number of particles. He advises to use at least100000 gas particles and 100000 DM particles to obtain good results.

3.1.2 Collisionless N-Body simulation

The collisionless fluids of dark matter and stars interact gravitationally via Newton’s law. For a userdefined number of N bodies, the gravitational force on one body with mass mi equals

Fi =N

∑j=1j 6=i

−Gmimj

r2 er, (3.3)

with r = |ri − rj|, er = r/r and G the gravitational constant.A first problem with this equation is that it is divergent for very small values of r. Therefore, softening isintroduced:

Fi =N

∑j=1j 6=i

− Gmimj

(r + ε)2 er, (3.4)

3.1. The code 37

Figure 3.1: The left figure shows the division of the problem domain into ocants, each octant containing one particle. The right figurerepresents the hierarchical tree structure with its nodes and subnodes pointing to the particles. Figure from [Liu and Liu, 2003].

with ε the gravitational softening length. GADGET-2 is provided with the possibility to use an adaptivesoftening kernel which is at its maximum value when r = 0. However in our simulation, a constantsoftening length is used, namely ε = 30 pc.Gravity is a force acting on an infinite range, each particle interacts with every other particle in the galaxy.This gives rise to a computational cost that scales with N2 for the calculation of the accelerations in onetime step. Without clever algorithms and approximations, the simulation of galaxies would be (nearly)impossible.In the case of GADGET-2, a tree algoritm with multipole expansion is used to decrease the computationalcost to O(N log N) [Springel, 2005]. A tree algoritm starts with dividing the problem domain into octants.In every octant, the mass distribution is checked and further subdivided in octants untill each octantcontains only one particle, see Figure 3.1. The problem domain exists now out of cubes of differentdimensions, each cube containing one particle. At the root of the hierarchical tree structure, the nodeswith all subnodes point to all particles in the node. To calculate the gravitational force on a target particlei, a certain node will be responsible for a gravitational force on the particle. When a multipole expansionof the node is considered a sufficiently accurate contribution to the total gravitational force, the expansionis used and no further descend into subnodes is necessary. GADGET-2 makes only use of monopolemoments to approximate the gravitational force contribution in a node. For nodes spatially nearby thetarget particle, further descend will be necessary to obtain an accurate calculation of the force. Theaccuracy can be entirely determined by the user: the further the descend level of the treewalk, the betterthe accuracy at the price of a higher computational cost. The set-up of the tree structure is not onlyvery useful for the calculation of the gravitational force, but also for the determination of the neighbors,in order to calculate the density of a gas particle (see Par. 3.1.3).

3.1.3 SPH

In SPH, the collisional fluid (in this case the gas particles) is described in a comoving frame of reference,which is the Langrangian approach. The following Navier-Stokes equations describe fluids and expressrespectively conservation of mass, also known as the continuum equation, see Eq. (3.5a), conservationof momentum, see Eq. (3.5b) and conservation of energy, see Eq. (3.5c)

dρ

dt= −ρ∇ · v, (3.5a)

dvdt

= −1ρ∇p +F , (3.5b)

dε

dt= − p

ρ∇ · v. (3.5c)


Figure 3.2: Cubic spline kernel for a smoothing length of h = 1.

Galaxies are self gravitating systems hence the external force F in Eq. 3.5b is expressed by Newton’slaw for N particles. The equation of motion for particle i becomes

dvidt

= − 1ρi∇pi −

N

∑j=1j 6=i

Gmj(xi − xj)

|xi − xj|3. (3.6)

Hence in order to solve this equation, for each particle the term ∇piρi

needs to be calculated. SPH codeshave a very specific approach for determining the density of a particle. They interprete each particle notas a point mass but rather as a smeared out distribution of mass that contributes to the density of thesurrounding particles. The density is then given by

ρi =N

∑j=1

mjWij, (3.7)

where N is the number of particles that contributes to the density of particle i, mj is the mass of eachcontributing particle, and Wij is the smoothing kernel which represents the mass distribution and hasdimensions 1

length3 . This method of calculating densities and other quantities in a discrite way is calledthe particle approximation.The smoothing kernel depends on the smoothing length h and on the distance between particle i and jhence

Wij = W(|xi − xj|, h). (3.8)

The smoothing length h determines the length scale on which the density distribution Wij is smeared outand hence influences other particles. In GADGET-2, the default smoothing Kernel function is the cubicspline kernel, shown in Figure 3.2, given by

Wij =8

πh3

{1 + 6u2(u− 1), 0 ≤ u < 1

22(1− u)3, 1

2 ≤ u ≤ 1(3.9)

with u =|xi−xj |

h . A great advantage of SPH is that the derivative of a certain quantity f (x) can becalculated in the particle approximation as follows

< ∇ · f (x) >= −N

∑j=1

mj

ρjf (xj) · ∇W(x− xj, h), (3.10)

3.1. The code 39

making use only of the derivative of the smoothing kernel.The length under the curve of the cubic spline kernel is called the support domain (in a 3D code, supportdomains are spherical) and its scope is determined by the smoothing length h. Therefore, the determi-nation of h should be handled with thought: when h is too small, the support domain of the particles willbe too small and quantities as pressure and density will be calculated inaccurately. On the other handwhen h is too large, local effects will be smoothened out. For a 3D code, an ideal number of neighboringparticles is around 57 [Liu and Liu, 2003]. To obtain a more or less constant number of neighbors, anadaptive smoothing length is necessary, hence each particle i has its own smoothing length hi. It is clearthat smoothing lengths in dense regions are smaller than in low density regions.A way of introducing an adaptive smoothing length, is to make use of the continuity equation to updatethe smoothing length along with variation of density

dhdt

= −1d

hρ

dρ

dt=

hd∇ · v, (3.11)

with d the number of dimensions, in this case 3. Indeed, with increasing density, dhdt is negative hence

the value of h decreases.GADGET-2 only uses Eq. (3.11) as an initial guess to calculate hi. The initial guess is iserted in anumerical method to solve the following equation [Springel, 2005]

4π

3h3

i ρi = Nsphm, (3.12)

with m the average mass of a particle and Nsph the ideal number of neighbors.Now, when every particle has its own adapted smoothing length hi and when for instance the densityof particle i is calculated, then the influence of particle j on particle i is determined by W(|xi − xj|, hi)whereas the influence of particle i on particle j is determined by W(|xj − xi|, hj). Since hi 6= hj, thecontribution is not symmetrical. This violates basic principles of physics and should be solved by asymmetrization procedure to obtain a new value for the smoothing length hij, for instance the average ofboth particles smoothing lengths hij =

12 (hi + hj) or as implemented in GADGET-2 , the maximum value

hij = max(hi, hj).In fact, all equations need to be symmetric for the exchange of two particles. The force on particle i fromparticle j has to be equal to the force on particle j resulting from particle i. A symmetrization procedureis applied to Eq. (3.5b) and results into [Springel, 2005]

dvidt

= −N

∑j=1

mj

[fi

Pi

ρ2i∇iWij(hi) + f j

Pj

ρjr∇jWij(hj)

], (3.13)

making use of the particle approximation and Eq. (3.10). The factors fi are defined by

fi =

(1 +

hi3ρi

∂ρi∂hi

)−1, (3.14)

and result from the fact that ρi depends on hi.

3.1.4 Neighbor search

As discussed in Par. 3.1.2, the problem domain is divided into octants with a hierarchical tree structure,each octant containing one particle. In order to compute the density ρi of a certain particle, one needs todetermine the neighboring particles contributing to this density. A cubic search domain around particle iis defined with sides 2hi. Now, when the treewalk is started at the root of the tree, it is checked whetherthe node has an overlap with the search domain, see particle i in Figure 3.1. If not, the nodes do nothave to be opened at all. If so, the treewalk is continued to a subnode and the process is repeated. Withthis method, the treewalk is restricited to the local area of particle i. The number of computations withthis method scales with O(N log N).


Figure 3.3: Panel (a) illustrates the scatter approach where the density of particle i is calculated by contributions of the support domains ofthe neighboring particles with their corresponding smoothing lengths hj. Panel (b) represents the gather approach: the density of particle i isdetermined by checking which particles are situated within the support domain of particle i determined by its own smoothing length hi . In thecase of the cubic spline kernel, the factor κ = 1. Figure from [Liu and Liu, 2003].

To determine the density ρi for a certain particle i, two different approaches exist. One can check withinthe smoothing length hi of the particle for the number of particles that are situated within hi, this is calledthe gather approach. Another method is the scatter approach, which checks whether the particle i issituated in the support domain of its surrounding particles. The difference between both approaches isshown in Figure 3.3. GADGET-2 makes use of the gather approach.

3.1.5 Integration

All ingredients are present now for running the code. Newtons law, Eq. (3.4) and the equations describ-ing the hydrodynamics Eq. (3.5a), (3.5b) and (3.5c), need to be integrated simultaneously. For systemsas galaxies, particles move on many different time scales. For example, a gas particle in a low-densityregion in the outer borders of a galaxy will not change its position and other properties as fast as aparticle in the high-density center of the galaxy. Therefore, adaptive timesteps are used. In GADGET-2,the Hamiltionan calculating the forces is split up in a long-range and a short-range part [Springel, 2005].The integration scheme used in GADGET-2 is the leapfrog method. The reason this scheme was chosenis that it behaves more stable in combination with individual time-steps per particle than for instance afourth order Runge-Kutta which is computationally more expensive [Springel, 2005].

3.2 Additional physics

GADGET-2 is a general N-Body/SPH code which can be adjusted to the specific needs of the user. In thiscase, for the simulation of dwarf galaxies, physical phenomena as cooling, feedback and star formationhave been added to the code by [Valcke et al., 2008].

3.2.1 Star formation

As discussed in Chapter 2, stars form in dense, cold and converging regions in molecular clouds whichobey the Jeans criteria. To select candidate particles to form stars, a loop is run over all gas particles

3.2. Additional physics 41

and the following criteria are checked

∇ · v ≤ 0, (3.15)ρg ≥ ρc = 100 amu cm−3, (3.16)T ≤ Tc = 15000 K. (3.17)

A critical temperature of Tc = 15000 K seems rather high in regard to the discussion of star formationin cool molecular clouds (see Par. 2.1). In fact, as will be shown later, almost all particles satisfythis temperature criterium and the selection of eligible gas particles for the formation of stars is almostentirely based on the gas density ρi of the particle. When a particle has collapsed to a density ofρi = 100 amu cm−3, it is assumed that this particle is sufficiently cool to form a star.In less recent codes, the star formation criteria used to be based on the Jeans criterium. However, thisapproach is unfavorable since it depends on the number of particles in the code and hence the resolutionof the simulation. We recall that the Jeans criterium is satisfied when the dynamical time τ is smallerthan the sound crossing time ts =

hcs

. Here, the free-fall time tFF discussed in Par. 2.1.2 is equal to thedynamical time τ discussed in Par. 2.2.1. In this case it is defined as

τ =1√

4πGρg. (3.18)

For the critical density ρc, the dynamical time is τ = 2.4 Myr. The sound crossing time is calculated with

the sound speed cs =√

γkTcm = 14 · 103 m/s = 14 kpc/Gyr. In dense regions of our simulations, a typical

smoothing length equals h = 30 pc. Using this, we obtain a sound crossing time ts = hcs

=∼ 2.1 Myr.For the SFC of Eq. (3.15), we are more or less at the critical value where τ = ts, and with a furtherincrease in density and decrease in temperature, the Jeans criterium will be definitely satisfied (becauseτ ∝ ρ−1/2 and ts ∝ T1/2). This result however, is only valid for our specific simulations with a resolution of100000 gas particles. For other resolutions, different values for hi will be present (because hi is adaptivein order to have a more or less constant number of neighbors) and this will yield different values for thesound crossing time ts. In conclusion, we should not interprete the SFC in terms of the Jeans instabilitycriterium, but simply on the fact that star formation takes place in cold, dense and converging regions.Once the star particles are selected satisfying the SFC, one needs to decide whether they will form astar or not. To make a prediction of this star formation probability, the Kennicutt-Schmidt law is used,discussed in Par. 2.2.1:

dρs

dt= −dρg

dt= c?

ρg

τ, (3.19)

with τ the dynamical time and a power of n = 1 over the gas density was chosen. This leads to theanalytical solution of

ρg(t) = ρg,0 exp(−c?

tτ

), (3.20)

where ρg,0 is the initial gas mass. The probability that the elected gas particle will form a star is equal tothe ratio of the “disappeared” gas mass to the total gas mass, assuming that all “disappeared” gas hasbeen transferred to stars, along Eq. (3.19). This probability p? is then given by

p? = 1− ρg

ρg,0= −∆ρg

ρg,0, (3.21)

with ∆ρg = ρg − ρg,0.For one timestep ∆t in the code, the solution for p? is given by

p? = 1− exp(−c?

∆tτ

). (3.22)


The integration step ∆t in GADGET-2 equals the systems (adaptive) timestep which is of the order ofmagnitude ∆t ∼ O(104) yr in star formation regions, while the dynamical time tc is of the order oftc ∼ O(106) yr. This implies that ∆t/τ ∼ O(10−2) and thus that Eq. (3.22) can be approximated by

p? = c?∆tτ

. (3.23)

However this approximation is not used in the present version of the code.It is clear that p? always has a value between 0 and 1. When now a random number r is calculated inthe [0, 1] interval, the candidate particle will form a star, or more precise an SSP particle, when p? > r.

3.2.2 Star mass

The newborn SSP particle has the same initial mass as its progenitory gas particle. For the case of theE-models, the total initial gas mass (see Table ??) is divided by the number of gas particles (in this case100000) hence the mass of one SSP particle is of the order of 103 M�.The Salpeter initial mass function (see Par. 2.1.4) is adopted to describe the distribution of star masswithin an SSP particle and takes the form

Φ(m)dm = Am−(1+x)dm, (3.24)

with Φ(m) = dN/dm and x = 1.35 as proposed by Salpeter. Parameter A is chosen by normalization:∫ mu

ml

Φ(m)dm = 1. (3.25)

For lower limit ml = 0.1 M� and upper limit mu = 60 M� the value of A = 0.06 is obtained [Valcke, 2010].Increasing the upper limit mu would increase the number of SNe II and hence the feedback to the ISM.The calculation of a certain average property of an SSP is then given by [Valcke, 2010]

=MSSP∫ mu

mlmΦ(m)dm

·∫ mB,u

mB,l

B(m)Φ(m)dm, (3.26)

with mB,u and mB,l the upper and lower limit for which the property B is applicable.

3.2.3 Feedback

As discussed in Par. 2.1.4, feedback to the ISM occurs via stellar winds (SW) originating from massiveOB stars and T-Tauri stars, and via SNe of type Ia and II.To calculate the amount of energy that will be released from SNe II events Etot,SNII, Eq. (3.26) is usedand this yields

Etot,SNII = ESNII ·MSSP∫ 60 M�

m0.1 M�mΦ(m)dm

·∫ 60 M�

8 M�Φ(m)dm, (3.27)

with the lower limit for a SN II event to occur is 8 M� and with the energy release of one SN II eventESNII = 1051 erg assumed to be mass independent. The result of the integral equals [Valcke, 2010]

Etot,SNII = 7.31 · 10−3MSSPESNIIM−1� . (3.28)

The calculation for Etot,SW is identical, replacing ESNII with ESW = 1050 erg.A feedback efficiency εFB is defined and set to 0.7. This feedback efficiency expresses the efficiency atwhich the ISM absorbs the emitted energy, hence the actual energy the ISM absorbs is only 70% of thetotal energy release. The feedback efficiency εFB is an important parameter in regard to star formation

3.2. Additional physics 43

and self-regulation. When it is too low, the ISM will be less heated and be less subject to shock waves.This results in unrealistic high SFRs.The effect of εFB on the SFR was studied by Cloet-Osselaer et al. [Cloet-Osselaer et al., 2012], andthe relation between εFB and the density threshold ρc for SF was investigated. It was concluded thata degeneracy is present between the two parameters. Several models were in good agreement withthe observations, for instance a model with parameters ρc = 6 amu cm−1, εFB = 0.7 and one withρc = 50 amu cm−1, εFB = 0.9. This arises the question whether the value of εFB = 0.7 for a criticaldensity threshold ρc = 100 amu cm−1 as used in our simulations is a good choice. Furthermore, sincefeedback is an important parameter in regard to induced star formation and self-regulation, it could beinteresting to study the effect of different values of the feedback efficiency when induced star formationis included in the star formation law. However, since star formation is a self-regulating phenomenon,a (modest) increase of the feedback efficiency will probably not change the qualitative behavior of thesystem.If the feedback efficiency is too small (for instance for εFB ∼ 0.1), an insufficient amount of energy isreleased in the ISM to locally blow the gas apart. Therefore, the simulated galaxies will have continuousstar formation rates that are too high. There is no source present for the gas to be blown apart and/orheated and self-regulation is eliminated.The total energy release for SNe type Ia is calculated analogously to Eq. (3.28), but is multiplied with afactor ASNIa = 0.15 because not all stars in the applied mass range (3 M� − 60 M�) undergo a SN Ia.Only stars in a binary system are eligible, see Par. 2.1.4.The feedback energy is not released instantaneously. OB stars emit stellar winds during their entirelifetimes of O(107) yr and moderate mass stars emit stellar winds during the T-Tauri stage which durationis of the same order of magnitude O(107) yr. The energy release of stellar winds to the ISM εFB · Etot,SWis homogeneously spread over this time span.In the same way the energy release is spread over the time span of SNe type II events that occur whenmassive stars have reached their life ending. This occurs between O(106)-O(107) yr. The precise timeranges in which the feedback energy is released, were calculated using a formula for the amount of timestars spend on the main-sequence. The values are given by [Valcke, 2010]

• SW: 0− 4.3 · 107 yr

• SNe II: 5.4 · 106 − 4.3 · 107 yr

• SNe Ia: 1.543 · 109 − 1.87 · 109 yr

We conclude that stars with an age larger than 1.87 · 109 yr will no longer give rise to stellar feedbackand hence will not induce star formation in the code.Another interesting aspect to this spreading of the emission of stellar feedback is that it shows that time-delay is included in the simulations: one of the aspects that give rise to instability and the presence ofoscillations in a star formation system, according to Korchagin et al. [Korchagin et al., 1988], see Par.2.2.2 and confirmed by or own calculations, see Par. 2.2.4.The feedback distribution makes use of the smoothing kernel hs of the star (stars are treated as pointmasses and are not described by a smoothing kernel; however they inherit the smoothing length of theirprogenitory gas particle). A gas particle j will recieve an amount ∆Ej from a star i that is in the proces ofgiving feedback. The quantity ∆Ej equals [Valcke, 2010]

∆Ej =EmjW(|~ri −~rj|, hs)

∑Nk=1 mkW(|~ri −~rj|, hs)

. (3.29)

During stellar feedback, not only energy is released but also a certain amount of mass is returned to theISM in the form of metals. The release of metals in the ISM is important since it has a major influenceon the cooling of gas, see Par. 2.1.3.Once an SSP particle is formed in the simulation, it will never die. The massive star fraction of theSSP particle, determined by the IMF, will give rise to feedback and therefore, the SSP particle willlose a fraction of its mass. In the E07 model, the SSP particles lose 11% of their mass by the meansof feedback. After the feedback has finished, the mass of the SSP particle remains constant. The


Figure 3.4: Cooling curves in function of temperature for different metallicities Z. The cooling rate Λ is calculated from interpolation betweentabulated values of these cooling curves. Data from [Maio et al., 2008, Sutherland and Dopita, 1993].

stellar population in the SSP particle will age and this will have an influence on observable quantities asluminosity and color of the SSP particle.

3.2.4 Cooling

Implementing cooling in an SPH code, requires an adjustment of the energy equation Eq. (3.5c)[Valcke, 2010]:

dε

dt= − p

ρ∇ · v +

Γ−Λρ

, (3.30)

with Γ a heating term, and Λ a cooling term. Heating results from feedback by SNe and SW, see Par.3.2.3. When a particle is heated during a timestep from feedback, it is not allowed to cool radiatively inthe same timestep.The code provides cooling for a temperature range of 10− 109 K. The chosen value Λ for one timestep isinterpolated from tabulated values for cooling, depending on temperature and metallicity Z. The coolingcurve implemented in the code are shown in Figure 3.4.Several regimes can be distinguished for the cooling curves in Figure 3.4 (paraphrased from [De Rijcke, 2011]).In the temperature range of 107− 109 K, all particles are fully ionized and Bremmstrahlung dominates thecooling process. Between temperatures of 104 and 107 K, the particles change their state from neutralto ionized. For a certain temperature range depending on the type of element, there is a contributionof recombination and line cooling which gives rise to peaks in the cooling curve. Below 104 K, no ionsand free electrons are present and only line cooling can contribute to the cooling process. Moreover, inthe temperature range of molecular clouds, hydrogen atoms cannot be excitated by collisions anymoreand do not contribute to the cooling curve. In this temperature range, cooling proceeds mainly throughmolecular line transitions, as discussed in Par. 2.1.3.

3.3. Implementation of induced star formation 45

3.2.5 Self-regulation

As a consequence of the implementation of stellar feedback (with time-delay) and cooling, the star for-mation in the simulations is self-regulating. The implemented cooling will cause the gas particles to cool,and gravitational and hydrodynamical interactions cause a collapse of the gas. Conditions are favorablefor star formation and the Kennicutt-Schmidt law decides how much stars are being formed, proportionalto the gas density.Once new stars are born, they initiate stellar feedback for a few Myr to Gyr. Energy is released whichcauses the interstellar gas to heat and to be blown apart. At the edges of the blown bubbles, the gas ispressed together and these high densities give rise to induced star formation. When a lot of stars areformed, the stellar feedback is strong and will eventually halt star formation and blow the ISM apart.During the feedback, metals have been released which influence cooling and will cause it to proceedfaster (see Figure 3.4). The cycle is complete now and the proces can start all over. This is an illus-tration of how self-regulation can lead to oscillations in star formation. One could hence wonder whyinduced star formation still needs to be included in the star formation law if the phenomenon is alreadypresent in the code.In simulations using a Kennicutt-Schmidt law, supernova explosions will indeed give rise to high densi-ties, but the prediction of the star formation rate p? only accounts for high densities. It does not reckonwith nearby (young) stars giving rise to stellar feedback. If we want to implement induced star formationlocally, the chances of forming a star should be higher for particles subject to feedback effects of nearbystars. Therefore, the adjusted star formation law Eq. (4.1) is proposed.Another motivation to alter the star formation law is that it increases the chances of instability in thestar formation system and oscillatory behavior of its components, as thoroughly discussed in Par. 2.2.4.In the context of understanding and simulating BCDs, oscillations in the star formation rate are favorable.

3.3 Implementation of induced star formation

3.3.1 Star formation law

Now we use results of the discussion in Par. 2.2.4 where conditions for instability and oscillatory behav-ior were set up, to choose a parameterization of induced star formation which we will implement in thenew star formation law. However, it is hard to simply make use of the drawn conclusions, since phenom-ena as feedback and cooling are explicitely included in the code and will have their influence, withouta specific parameterization of this phenomena present. For instance, the phenomenon of cooling wasstudied a the three component star formation system with a (simple) parameterization of dMn A, with Mthe total molecular gas mass, A the total atomic gas mass and d and n free parameters, see Eq. (2.103).However, as shown by Figure 3.4, a much more sophisticated cooling is present in the code. Is the usedparameterization of cooling dMn A even a remote description of the implemented cooling curves?Looking at dMn A dimensionally, we see that dMn should have dimensions of 1

time (see Eq. (2.30)) henceit could be interpreted as a cooling rate C(T, n), with n the number density of the gas. If we look at thecooling curve Λ(T) in Figure 3.4, it is clear that it depends only weakly on temperature in the range of102 − 104 K. For a cooling rate in function of density for a constant temperature, we have that C(n) ∝ n2,as proposed in literature by e.g. [Oppenheimer and Dalgarno, 1975]. Now, for a changing density n anda changing temperature (along T ∝ 1

n ), we have that C(n, T) ∝ n2 because cooling is not temperaturedependent in the range of 102 − 104 K. Hence the term dMn A would in this case be dM2 A. Of course,this is a very rough estimate only to show under which conditions the parameterization for cooling ofdMn A makes sense.If we assume that a three component system with inclusion of spontaneous star formation as in Eq.(2.109) is approximately a good description of the dwarf galaxies simulated with GADGET-2, the Hur-witz conditions discussed in Par. 2.2.4 yield that oscillations can occur when the non linearity in cloudformation (cooling) is stronger than in star formation. Hence if the cooling is indeed parameterized bydM2 A, than the power over the star formation law (Kennicutt-Schmidt law ) should be smaller than two.Currently, the Kennicutt-Schmidt law is implemented in the code as dρs

dt = c?ρgτ . We see that the Hurwitz


condition for stability is already violated in this system because the power over the star formation law isone and the power over cloud formation which is approximately two.In order to increase the chances for oscillatory behavior to occur, it does make sense to adjust the starformation law to account for induced star formation if time delay is present in the code. We know thattime delay is implemented, see Par. 3.2.3. We have shown in Par. 2.2.4 that instability arises in a twocomponent system when induced star formation is accounted for, and no oscillations occur when onlyspontaneous star formation is implemented. In this regard, we can use the parameterization βMnS ofinduced star formation and include this term in the currently used star formation law

dρs

dt= −dρg

dt= c?,1

ρg

τ+ c?,2ρn

gρs, (3.31)

with the induced star formation term c?,2ρngρs adjusted to the units and notation of the Kennicutt-Schmidt

law.This renewed star formation law is implemented in the code and tested for different values of c?,2 andn. Do oscillations occur with this parameterization of induced star formation and for which parametervalues? The results of the simulations are discussed in Chapt. 4.

3.3.2 Calculation of ρs,i

The star formation prescription law is used to make a prediction for the SFR, in order to decide whetherthe concerning gas particle that satisfies the SFC, let’s call it the target gas particle, will become a star.To implement Eq. (3.31) in GADGET-2, the first quantity needed is the star density ρs in the surroundingsof the target gas particle. Methods to calculate the gas density ρg are already present in the code fornumerous reasons, including for the integration of the originally used Kennicutt-Schmidt law. We canwork analogously to the gas density method, with a few modifications.In order to know the surrounding star density ρs,i of the gas particle i, the neighboring star particleshave to be known. This is done by a treewalk algorithm, described in Par. 3.1.4. However the sizeof the box needs to be adjusted. Indeed, we do not look for stars within the smoothing length of thetarget particle, but within the characteristic length for induced star formation to have an influence. Fromthe Sedov-Taylor radius and observations of spatial separation of T-Tauri and OB associations (see Par.2.1.4), we know that this characteristic length for induced star formation is of the order of 10− 30 pc.Therefore, we choose a search radius of radius = 50 pc.The treewalk algorithm is performed over star particles with a search cube of sides 2 · radius = 100 pc.As shown in Figure 3.1, it is checked whether the search cube around particle i overlaps with a treenodeoctant and if further descend is necessary. In this way, the number of total star neighbor particles iscounted and a list of all neighboring star particles is kept up. However, this algorithm yields all neighborsin a box with side 2 · radius, but since we are looking for neighbors within the characteristic radius, onlythe neighbor particles j with a distance |rj − ri| < radius are selected to contribute to the star densityρs,i of particle i.A loop is run over the list of neighboring star particles in order to calculate their contribution to ρs,i. Thecontribution of a star particle j to ρs,i equals

(ρs,i)j = mjW(|xj − xi|, radius), (3.32)

with W the cubic spline kernel.It is hence the case that star particles at a distance of radius = 50 pc and larger, do not contributeto the star density at all. For distances of 10− 30 pc as we proposed for the length scale of inducedstar formation (see Par. 2.1.4), the parameter u in the smoothing kernel function is 0.2− 0.6 and thecontribution of particles at this distance can be seen from Figure 3.2.

3.3.3 Integration of the star formation law

A definite advantage of the Kennicutt-Schmidt law is that it has an analytic solution. This is no longerthe case for our proposal for the star formation law in Eq. (3.31). Maple was used to check for analytical

3.3. Implementation of induced star formation 47

solutions, without a result. Therefore, Eq. (3.31) is integrated numerically. A fourth order Runge-Kuttamethod is chosen to perform the numerical integration, since this algorithm offers a high level of accuracyand is easy to implement.First of all, a driver function g(ρs, ρg, τ) is defined which is equal to the star formation law

g(ρs, ρg, τ) = −c?,1ρg

τ− c?,2ρn1

g ρs. (3.33)

The simulation timestep is of the order of ∆t = 104 yr or in code units ∆t = 10−5 Gyr. In order to make aprediction for ρs,i(t + ∆t), ∆t is subdivided in 100 smaller integration steps ts and a loop over this smallertimesteps is started. The values of ρg,i(t) and ρs,i(t) at time t are present (see Par. 3.3.2).The fourth order Runge-Kutta integration method makes use of four constants [Baes, 2011]

k1 = tsg(ρg, ρs, τ), (3.34)

k2 = tsg(ρg +k1

2, ρs −

k1

2, τ), (3.35)

k3 = tsg(ρg +k2

2, ρs −

k2

2, τ), (3.36)

k4 = tsg(ρg + k4, ρs − k4, τ). (3.37)

Since we know that dρsdt = −dρg

dt , the same constants k1 to k4 can be used for the decrease/increase ingas/star density but with an opposite sign.After one itegration step ts, the gas and star density become

ρg(t + ts) = ρg(t) +k1

6+

k2

3+

k3

3+

k4

6, (3.38)

and

ρs(t + ts) = ρs(t)−k1

6− k2

3− k3

3− k4

6. (3.39)

After 100 integration steps, values for ρg(t + ∆t) and ρs(t + ∆t) are obtained. The latter quantity isnot strictly necessary to estimate the star formation rate, but we need to keep up its value in the loop.Indeed, stars that are formed within this timespan of ∆t will have an influence on their neighboring gasparticles and increase the star formation rate.In the end, the probability p? that the target gas particle i becomes a star is calculated as before through

p? = 1− ρg,i(t + ∆t)ρg,i(t)

. (3.40)

The fourth order Runge-Kutta integration method does not seem to noticeably slow down GADGET-2,probably due to the relativily large integration time step ts. However a computationally more efficientapproximation was considered to obtain p? by a linear approximation. Since

p? = 1− ρg,i(t + ∆t)ρg,i(t)

= − ∆ρg

ρg(t), (3.41)

and if we approximate ∆ρg by

∆ρg '(−c?,1

ρg

τ− c?,2ρn1

g ρs

)∆t, (3.42)

a value for p? of

p? '(

c?,11τ+ c?,2ρn1−1

g ρs

)∆t, (3.43)


is obtained.The Runge-Kutta method and the linear approximation method were used in simulations with the exactsame initial conditions. The values obtained for p? were the same for two significant figures or more.Assuming that the implementation of the linear approximation is correct, this provides a good indicationfor the implementation of the Runge-Kutta method to be correct too. One would hence naively assumethat the linear approximation performs as well as the Runge-Kutta method and since it is more efficient,it should be preferred.However, by running multiple simulations with the linear approximation method, we noticed that forthe low mass models, the results were not different compared to the same models without inducedstar formation. Actually, some simulations where the linear approximation was applied did not showany difference in the star formation rate with a simulation where no induced star formation law wasimplemented and the original Kennicutt-Schmidt law was used. This was the case for light galaxies witha very small number of stars.This effect is completely logical. Light galaxies form less stars and in the case of a very small numberof stars, ρs(t) is (mostly) zero and the second term in Eq. (3.43) has no effect at all. In that case, onlythe Kennicutt-Schmidt law is accounted for. The problem can be overcome by simulating more massivegalaxies that have more star formation or to adjust the search radius in which star particles contribute tothe star density of a gas particle.We decided to simulate more heavy galaxies (E07 instead of E05) but the Runge-Kutta method was stillpreferred above the linear approximation based on physical arguments. Because, even when the stardensity ρs,i(t) is zero for a certain particle, a prediction for the star formation rate should include inducedstar formation for the stars that are formed in the time span of the prediction. This will never be possiblewith the linear approximation, as it performs no integration over time.To overcome the less computational efficiency of the fourth order Runge-Kutta, other faster numericalalgorithms could be considered for instance the leapfrog method or the second order Runge-Kutta.Also, one can decrease the number of integration time steps ts. However, in both cases the effect on theaccuracy should be studied.

3.3.4 Test

Now that we are convinced that the Runge-Kutta method performs well and is a good choice, we wantto test if the treewalk algorithm to calculate ρs,i is implemented correctly.This is done by writing an output file with all the relevant quantities for every timestep: the ID number ofthe target gas particle, the number of its star particle neighbors calculated by the treewalk algorithm andtheir ID numbers, the search radius, the masses, the smoothing kernel attribution to the star density, thestar density itself, etc.When the simulation is run, every 0.1 Gyr a snapshot is given as output. We can visualize and analyzethese snapshots with Hyplot, which is a program specifically designed [Valcke, 2010] to interpret theoutput of GADGET-2. For every snapshot, by means of particle IDs, loads of information can be checkedon the concerning particle: particle type, mass, smoothing length (for gas particles), position, velocity,age (for stars), etc.If the output file lists a star particle to be a neighbor of a certain gas particle, we can check in hyplotwhether these two particles are confirmed to be seperated by a distance less than 50 pc. Checking themass of the star particle, the smoothing kernel contribution can be verified by hand and all contributionsare put together in the quantity ρs,i.Of course, the proces of verifying by hand has been preformed numerous times on several simulationsand different snapshots for different particle IDs. However, it is only a random sample that has beenchecked. It cannot be used as hard proof that the method is correct. In conclusion, it is very likely thatthe treewalk algorithm to calculate ρs,i performes correctly.So, as we are convinced that induced star formation is included in a correct way in GADGET-2, we canstart simulations of dwarf galaxies. The results are discussed in Chap. 4.

4Results and discussion

4.1 Results

4.1.1 Reference dwarf galaxy

In Chapter 3, we have introduced a new star formation law

dρs

dt= −dρg

dt= c?,1

ρg

τ+ c?,2ρn

gρs, (4.1)

in the code that accounts for the phenomenon of induced star formation. To study its influence, a refer-ence simulation is needed, where the original Kennicutt-Schmidt law regulates star formation. When theinitial conditions of this reference simulation are exactly equal to a simulation using a new star formationlaw, the influence of the implemented induced star formation can be studied. Table 3.1 lists the usedparameters and properties to set up the initial condition file for the reference galaxy.Since we are studying oscillations in star formation rates, we will mostly be comparing star formationhistories of different simulations. Therefore, we should understand the star formation history of the refer-ence galaxy, shown in the top panel of Figure 4.1. The depicted SFH can be explained by means of theSFC of Eqs. (3.15). The number of particles exceeding the critical density of star formation ρc is calledNdens and is also depicted in Figure 4.1. Also Nentr and Nconv are shown, respectively the numberof particles with a temperature T < 15000 K satisfying the temperature criterium, and the number ofparticles satisfying ∇ · v < 0.The most striking feature in the SFH of this dwarf galaxy is that almost no star formation takes place after3 Gyr. Initially between 0 and 3 Gyr, the star formation rate is rather high and irregular peaks show upin the SFH. These peaks are due to the collapse of the initial state, a sphere with density distribution ofEq. (3.1) and a constant temperature of T = 10000 K. When the initial gas sphere collapses, the criticaldensity threshold ρc for star formation is exceeded by a sufficient amount of gas particles, see Ndens inFigure 4.1. Indeed, the density criterium is the most severe criterium for the onset of star formation anddecides almost exclusively whether a particle is eligble for star formation or not.From Figure 4.1, it is clear that the temperature criterium for star formation T < 15000 K is almostalways satisfied for all gas particles, which is logical when the initial gas temperature is 10000 K for allgas particles and when the cooling rate drops several orders of magnitude at a temperature of 10000 K(see Figure 3.4).The number of gas particles Nconv, shows us that the initial gas sphere collapses between 0− 0.1 Gyr.Afterwards, during the first 3 Gyr, Nconv increases right before a peak in the SFR. A collapse is nec-essary in order to increase the density preliminary to star formation. The new born stars give rise to

49

50 Chapter 4. Results and discussion

0.000

0.002

0.004

0.006

0.008

0.010

SF

R[M�/yr] reference sim1006

012345

Mst

ar[1

06M�

]

0

20

40

60

80

MgasB

O[%

]

0.0

0.1

0.2

0.3

0.4

Ndens[%

]

0

20

40

60

80

Nconv[%

]

95

96

97

98

99

Nentr

[%]

0 2 4 6 8 10 12

time[Gyr]

0.000.020.040.060.080.100.120.14

Nto

t[%

]

Figure 4.1: Top to bottom: star formation rate SFR, total star mass Mstar, fraction of the initial gas mass MgasBO currently blown out (at adistance > 30 kpc), number of gas particles Ndens exceeding ρc, number of gas particles Nconv satisfying ∇ · v < 0, number of gas particlesNentr with T < 15000K and finally number of gas particles Ntot satisfying all three SFC. These properties belong to the reference simulationsim1006 using a Kennicutt-Schmidt star formation law.

4.1. Results 51

(a) (b)

(c) (d)

Figure 4.2: Density evolution projected on the xy plane. Reference simulation sim1006 is shown at different times indicated in units of Gyr,following an increase and decrease in the SFR (compare times with the SFH in Figure 4.1). In (a) the density is still reasonably low and thegalaxy is collapsing. In (b) densities are high and some fragmentation is observable. In this high density fragmentated subclouds, particles areeligble for star formation and a peak in the star formation rate is observed. After more or less 0.1 Gyr, feedback effects of this new born starsshow up in (c). Bubbles are blown in the ISM and the gas of the dwarf galaxy is blown appart. This results into a broad low density gas cloudin (d) where the SFR has entirely dropped to zero.


(a) (b)

Figure 4.3: Density evolution projected on in the xy plane. The reference simulation sim1006 is followed at different times indicated in unitsof Gyr. Recently formed stars with ages < 10 Myr are shown as purple dots. In (a) the interstellar gas has collapsed to high densities andstar formation is possible in the fragmentated subclouds. Feedback effects are present in (b) giving rise to high densities near the edges of theblown bubbles in the interstellar gas. Induced star formation occurs as is seen from the recent stars formed in these high density regions.

feedback, in a time span of 0− 0.043 Gyr for SW and SN II and between 1.5− 1.9 Gyr for SN Ia. Hence,a large amount of the feedback by new born stars has already been done by the time of the next snap-shot after 0.1 Gyr. The effect of the stellar feedback is immediately shown in the decrease of the densityand a decrease in Nconv. Feedback blows bubbles in the ISM, as discussed in Par. 2.1.4. We see thatthe number of particles Ndens satisfying the density criterium and Nconv satisfying the convergerencecriterium decreases after a peak in the SFR.For simulations like ours with a high feedback efficiency (ε = 0.7) and a severe density threshold cri-terium (100 amu/cm3), periods with a continuously high SFR extending over more than a few tenths ofGyr are rare. We can also clearly follow the collapse before and the feedback effects after a peak in theSFR when the density is plotted for the galaxy, see Figure 4.2.So we have explained the origin of star formation in the first 3 Gyr of the evolution of the galaxy. But whydoes the star formation drop back afterwards?Star formation does not drop because of an insufficient amount of total gas mass available to form stars.Stars use gas mass when they are formed, but after 12 Gyr, the total gas mass has only decreased bymore or less 5%, hence 95% is still available for star formation.Star formation decreases strongly at 3 Gyr because the constant stellar feedback has caused the ISMto decrease in density in such amount that star formation becomes very hard. The intense peaks of starformation at times of 2 and 2.8 Gyr (see the SFH in Figure 4.1). The effect of feedback on the density ofthe ISM has already been illustrated in Figure 4.2.Another quantity we can look at to observe the effect of feedback is MgasBO in Figure 4.1. This quantityrepresents the amount of gas situated at very large distances from the galaxy center, out of a box withsides of 30 kpc. Between 4− 12 Gyr, 80% of the gas is situated at very large distances from the galaxycenter. This also makes it more plausable that high densities in the galaxy center are hard to achieveand star formation stops.An interesting remark is that the phenomenon of induced star formation is already present in the refer-ence simulation in a self consistent way. The simulated galaxies use a normal Kennicutt-Schmidt lawwithout the induced star formation term. However since stellar feedback is present in the code, starsare formed at the edges of bubbles in the ISM blown by supernovae and stellar winds. As discussed inPar. 2.1.4, these bubbles give rise to a high density cooled post shock layer at the edge of the bubblesand induces star formation. This effect is observable in Figure 4.3(b) where recently formed stars are

4.1. Results 53

Table 4.1: Simulations with the induced star formation law of Eq. (4.1) and the exact same initial conditions as reference simulation sim1006.The simulation numbers are indicated in bold if small peaks in the SFR occur (where sim1006 did have a neglegible SFR), and in red whenthese peaks show hints of periodic behavior and have a reasonably high amplitude.

n = 1 n = 1.5 n = 2 n = 2.5 n = 3 n = 3.5 n = 4sim1042 sim1020 sim1019 sim1015 sim1018 sim1040 sim1041

c?,2 = 104 c?,2 = 3.1 · 104 c?,2 = 105 c?,2 = 3.1 · 105 c?,2 = 106 c?,2 = 3.1 · 106 c?,2 = 107

plotted.Now, we have gained a good understanding of the behavior of the reference galaxy and the effect of theextra term for induced star formation in the star formation law can be studied.

4.1.2 Behavior of the star formation rate and parameter values

The adjusted star formation law Eq. (4.1) introduces two new parameters, namely n and c?,2.We recall that c?,2 has units of 1

time×densityn . Therefore, we tried to find a realistic value for c?,2 from di-

mensional arguments. In star formation regions, we find a typical dynamical time of the order of 10−3 Gyrand typical gas densities of the order of 0.1× 1010 M�Gpc−3, more or less equal to the critical densitythreshold for star formation. These characteristic values are used for the calculation of c?,2 for differentvalues of n. The results can be found in Table 4.1. Simulations with the exact same initial conditionsas the reference galaxy sim1006 (see Table 3.1) have been undertaken for different values of n with thecorresponding calculated values for c?,2.We based our choice for the values of n on the findings of Par. 2.2.4. We found that for a three com-ponent system with cooling included, the value of the power should be more or less 1 < n < 2. If weconsider the system with time delay, n can equal 1 or any other higher value in regard to obtaining os-cillatory behavior. Other arguments are that in literature, no powers higher than n = 4 have been foundfor the Schmidt law, hence this would be improbable to be the case for the parameterization of inducedstar formation.From Table 4.1, we see that several simulations indeed show oscillations in the SFR in comparison to thereference galaxy, hence there is a definite influence of the extra term introduced in the star formation law.

Oscillatory behavior First of all, when do we state that oscillations do occur? Let’s take a close look tothe SFHs of the simulations of Table 4.1 shown in Figure 4.4. For instance for sim1018, no peaks oroscillations in the SFR occur in the system. Between 4− 12 Gyr, there is some very modest and slowstar formation compared to the absolute absence of star formation of reference simulation sim1006, butthis is considered irrelevant in regard to the oscillatory behavior we try to obtain. Simulations with thistype of behavior are indicated in normal font in Table 4.1 and are later referred to as non-oscillatinggalaxies.Then for sim1015, a clear increase in the SFR rate is present but describing these peaks as obvious os-cillations would be an overstatement. The peaks are not very high but appear to be somehow periodic,although it is hard to say at sight. Simulation sim1015 and similar simulations are indicated in bold fontin tables and referred to as half-oscillating simulations.It gets harder to define oscillatory behavior when we look at sim1020. The height of the peaks is defi-nitely significant now. There seems to be a hint of periodic behavior but it is once more hard to defineat sight. For sim1019, we conclude that oscillatory behavior is present. The amplitude of the peaks inthe SFR is significant and even at sight one can say that the occurence of the peaks is periodic. Still,it seems that the oscillations are damped towards later times. Simulations that show peaks in the SFRwith a significant amplitude (comparable to the amplitude of the peaks in the SFR of the reference sim-ulation sim1006) and with a hint of periodic behavior are concluded to have oscillatory behavior. Thesimulation numbers of these galaxies are indicated in bold and red in tables and are referred to as oscil-lating galaxies.Further conclusions are that in all simulations where induced star formation is included, the total cumula-


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012345678

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ar[1

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]

Figure 4.4: Star formation histories and total cumulative star mass for different simulations with induced star formation included in the starformation law. Parameters for the simulations can be found in Table 4.1. The reference simulation sim1006 is always indicated for which aKennicutt-Schmidt law is implemented.

4.1. Results 55

Table 4.2: Tested values for n and c?,2 for simulations of dwarf galaxies with the exact same initial condition file as sim1006, tabulated inTable 3.1. The simulation numbers are indicated in bold if small peaks in the SFR occur (where sim1006 did have a neglegible SFR), and inred when these peaks show hints of periodic behavior and have a reasonably high amplitude.

n = 1 n = 1.5 n = 2 n = 2.5 n = 3 n = 3.5 n = 4c?,2 = 107 sim1070 sim1054 sim1056 sim1055 sim1071 sim1072 sim1041c?,2 = 106 sim1043 sim1021 sim1022 sim1023 sim1018 sim1038 sim1039c?,2 = 105 sim1044 sim1032 sim1019 sim1034 sim1035 sim1036 sim1037c?,2 = 104 sim1042 sim1048 sim1049 sim1050 sim1057 sim1052 sim1053c?,2 = 102 sim1045 sim1024 sim1025 sim1026 sim1027 sim1046 sim1047c?,2 = 10−2 sim1031 sim1030 sim 1029 sim1028

tive star mass is higher at the end time of the simulations than in the reference simulation. It also seemsto be that in simulations where peaks in the star formation occur, the initial star formation between 0− 3Gyr is lower than in simulations where no peaks occur.

Parameter influence Looking at Table 4.1, no correlation appears to be present between the power nand the qualitative behavior of the dwarf galaxy. Since we have found different types of qualitative be-havior, it is necessary to study the influence of both parameters n and c?,2 introduced via Eq. 4.1 for alarge range of parameter values. Values for n and c?,2 are tabulated in Table 4.2.For this large parameter range, we find different types of qualitative behavior, comparable to the typesof behavior shown in Figure 4.4.Some of the simulations show large peaks in the SFR in the first 3 Gyr of their existence, with compa-rable or larger heights than the peaks of the reference simulation sim1006 in the first 3 Gyr. In most ofthe cases, this results into a non-existing to very low SFR in the later stages of its evolution. This type ofbehavior is found for several examples in Table 4.2 for the non-bold and non-colored simulation numbers(the non-oscillating galaxies).For the half-oscillating galaxies (simulations numbers indicated in bold), we find a qualitative behaviorsimilar to the behavior shown in Figure 4.4(b) of simulation sim1015. We find modest peaks in the starformation rate. Sometimes a hint of periodic behavior is present.The most successful simulations have a number indicated in bold red. The initial star formation is mostlylower compared to the reference galaxy and no extreme peaks are present in the first 3 Gyr. This resultsinto oscillatory behavior in the SFR with a significant yet not striking amplitude. Indeed, the amplitudeof the peaks in the SFR of even the most successful simulations is never higher than the maximal am-plitude of the peaks in the SFR of the reference galaxy in the first 3 Gyr. On the other hand, timing isimportant and a difference in the SFR from 10−5 M�/yr to more than 10−3 M�/yr at the same time inthe evolution history of both galaxies is significant.To illustrate the main types of qualitative behavior, we refer to Figure 4.5 where it is easier to comparestar formation histories because the axes are put on the same scale. The three columns of Figure 4.5indicate the three types of qualitative behavior discussed above. On the other hand, one could alsointerprete the figure continuously from top to bottom and left to right. Indeed, it is hard to decide whenoscillations do occur or when absolutely no oscillatory behavior is observed. While changing parame-ters, all intermediate types of behavior can be found.Looking at Table 4.2, no relation seems to be present between qualitative behavior and the value for n.At most, one could say that n = 4 is not very successful in regard to obtaining oscillatory behavior, but itcould be a coincidence. No trend is found towards higher or lower values of n, at least not by the looksof Table 4.2.For the parameter of c?,2 it seems that very small values 10−2 will not lead to any oscillatory behav-ior. This is probably due to the fact that the term that introduces induced star formation in Eq. (4.1)will be too small. Indeed, looking at orders of magnitude for Eq. (4.1), we have that c?,1 ∼ 0.1,ρg ∼ 0.1× 1010 M�Gpc−3 in star formation regions and τ ∼ 10−3 Gyr. This yields an order of magnitudefor spontaneous star formation of about 1× 1010 M�Gyr−1 Gpc−3. For the induced star formation term,


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1019

Figure 4.5: Star formation histories for a large number of simulations with different values for n and c?,2, the exact values are listed inTable 4.2. Top left, the reference galaxy sim1006 is plotted with no induced star formation included. The figure can be interpreted in threecolumns to differ between the three qualitative types of behavior. In the left column, the non-oscillating galaxies are shown with no peaksand/or oscillatory behavior in the SFR after ∼ 3.5 Gyr. In the middle, modest peaks show up in the SFR with sometimes a hint of periodicbehavior. These are called the half-oscillating galaxies. The right column collects simulations which have been most successsfull in regard tooscillatory behavior: clear peaks in the SFR are present and oscillatory behavior is found. The figure can also be interpreted continuously fromtop to bottom and left to right.

making use of c?,2 = 10−2 (1010 ×M�)−nGpc3n Gyr−1, ρg ∼ 0.1× 1010 M�Gpc−3 and ρs ∼ 0.001×1010 M�Gpc−3, the induced star formation term is of the order of 10−6× 1010 M�Gpc−3 Gyr−1 for n = 1and even smaller for n > 1. The terms for spontaneous star formation and for induced star formation arehence different by 7 orders of magnitude which is apparently too large a difference to obtain an effectof the induced star formation term on the SFR. For the value of c?,2 = 102 (1010 ×M�)−n Gpc3n Gyr−1,spontaneous star formation and induced star formation differ by 3 orders of magnitude which shows tobe sufficient to have an effect of induced star formation on the SFR.For values of c?,2 between 102 − 107, there seems to be no trend or preferred value. In the detailed fourcomponent star formation system of Bodifee and De Loore [Bodifee and De Loore, 1985], discussed inPar. 2.2.2, it was found that the coefficient K5 that regulates induced star formation in the same way asc?,2, has a great influence on the systems behavior. Small values for K5 yield damped oscillations, largervalues yield oscillations behaving like a limit cycle, and too large values for K5 result in a stationarysolution. With some wishfull thinking, we could say that this is somehow similar to the three types ofqualitative behavior, shown in Figure 4.5. For simulations in the left column of Figure 4.5, one could saythat the star component has evolved to a more or less stationary state. For the second and third column,oscillations are obtained which are highly or modestly damped. However, the qualitative behavior that isfound in the implemented star formation system, does not seem to be directly related to values for c?,2.This can be seen from Table 4.2: no indication is present for more successsfull values of c?,2 or a trendtowards larger values of c?,2. Therefore, we would rather conclude that the systems qualitative behavior

4.1. Results 57

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SF

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r] 1058 1019

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012345678

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ar[1

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]

Figure 4.6: Star formation histories of sim1058 and sim1019. These two simulations have the exact same initial conditions and the exact sameparameter values n = 2 and c?,2 = 105 (1010 ×M�)−2 Gpc6 Gyr−1. The only difference is the random seed used to decide whether a gasparticle eligble for star formation will eventually become a star. It is clear that entirely different qualitative behavior of the SFR is obtained.

is chaotically and depends on random factors, more than on parameter values n and c?,2.Even literally, when the seed of the random generator is changed, used to determine if a target gasparticle will eventually become a star (see Par. 3.2.1), the qualitative behavior of the system can alsochange. This statement is illustrated by Figure 4.6. The qualitative behavior of the two simulations isentirely different while their parameter values and initial conditions are exactly the same. This proofsthat the star formation system implemented in the code is chaotical: it is nearly impossible to generallyrelate qualitative behavior to parameter values. As long as c?,2 is not too small, oscillatory behavior canoccur at any parameter values. Therefore we conclude that a degeneracy is found between the behaviorof the SFR and the parameter values. Multiple combinations of parameter values lead to the same typesof qualitative behavior.In order to further study the change in the systems behavior with a change of parameter values, we zoomin on simulation 1019, which we consider as a simulation with oscillatory behavior present in the SFRand differ the parameter values for n and c?,2 in a smaller range, see Table 4.3. It is noteworthy that a

Table 4.3: Zoom on successful oscillating simulation sim1019. Parameter values are varied in a smaller range to check whethe similarqualitative behavior is found for similar parameter values.

n = 1.9 n = 2 n = 2.1c?,2 = 1.2 · 105 sim1082c?,2 = 1.1 · 105 sim1083c?,2 = 105 sim1064 sim1019 sim1065c?,2 = 9 · 104 sim1061c?,2 = 8 · 104 sim1063

small change in the value of n will yield entirely different qualitative behavior, while small changes in thevalue of c?,2 yield more or less similar qualitative behavior. This suggests that the choice for parameter nis completely random, while for a given simulation and a given n, probably some more successful valuesand less successful values for c?,2 can be found, however one can never predict the type of behaviorfrom the parameter values.


Figure 4.7: Expected correlation between n and c?,2 (units depend on value of n), based on sim1019 with n = 2 and c?,2 = 105 (1010 ×M�)−2Gpc6 Gyr−1.

Correlation We also wondered if there would be any correlation between successful values of c?,2 andfor n. Table 4.2 does not suggest a correlation, however theoretically, we would expect one as we explainnow.Assume that for a simulation with certain initial conditions, we have found oscillatory behavior for asuccessful value of c?,2,osc and nosc. Looking at Eq. (4.1), this yields a certain contribution CISF =c?,2,oscρnosc

g ρs to the star formation law. In fact, if this contribution of induced star formation CISF wouldremain exactly equal for different values of c?,2,osc and nosc, the same simulation result with oscillatorybehavior would be obtained. If we assume that ρg and ρs are constants (this is not a too far fetchedassumption, especially since ρg ∼ ρc ∼ 0.1× 1010 M�Gpc−3 and ρs is mostly of the order of 10−4 ×1010 M�Gpc−3), then we find a correlation between c?,2,osc and nosc of

nosc =log(

CISFc?,2,oscρs

)log ρg

. (4.2)

If for a successful simulation, for instance sim1019, orders of magnitude in CISF are inserted, we obtainthe equation

nosc = −3 + log c?,2,osc, (4.3)

which is plotted in Figure 4.7. Since oscillations are obtained in simulation 1019, which is situated onthis line, we expect oscillations for other combinations of n and c?,2 on this line too. On the other hand,we have found that the simulation results are very sensitive to even the smallest change in parametervalues or any other change. Indeed, for the case of sim1019, the extra term of induced star formation inEq. (4.1) increases p? by more or less 1% compared to the dominant spontaneous star formation term.Nevertheless, this has a major influence on the star formation history and the qualitative behavior of theSFR. Therefore, the assumption that ρs and ρg are constants, will not be valid because small differencesin ρs and ρg of less than 1% can have a major influence. At most we can use this expected correlationas a guideline to choose parameter values that have a chance of being successful for a certain initialconditions.

4.1.3 Oscillations

Up till now, we have defined oscillatory behavior and the presence of oscillations at sight. It would benice to have a more quantitative benchmark to decide whether oscillations are present.Hence, what does define oscillatory behavior? Not only an amplitude, which is rather easy to determineat sight, but also a period. It is not easy to define at sight whether peaks in the SFR are periodic or have

4.1. Results 59

a more random character. Therefore, the SFHs can be Fourier transformed to look for periodic behavior.For a time dependent function f (t), in this case f (t)=SFR, we have that

f (t) =1

2π

∫ ∞

−∞F(iω)eiωtdω, (4.4)

and

F(iω) =∫ ∞

−∞f (t)e−iωtdt, (4.5)

with ω the frequency, easily converted to a period P with P = 2πω .

The results of the Fourier transformation of SFRs are shown in Figure 4.8 for simulations that areconsidered to have oscillatory behavior and in Figure 4.9 for simulations without oscillations. Since inall simulations, the SFR shows irregular peaks in the first 3 Gyr of the SFH, the Fourier transform is onlyapplied between 3.7− 12 Gyr. It should be noted that the choice of different time boundaries for theapplication of the Fourier transform can yield different results.From Figure 4.8, we see that in most cases, there is one frequency that dominates and there aresupplementary frequencies showing up. The periods of these frequencies are situated between 0.5− 2Gyr. When comparing the Fourier transform of the SFR with the SFH of the same simulations, the periodcalculated with the Fourier transform can most of the times be identified with oscillations showing up inthe SFR. For instance for sim1070, the main frequency yields a period of 0.7 Gyr, which can be identifiedwith the oscillations in the SFR of sim1070 in Figure 4.5.For sim1020, no period is dominant over others, which is no surprise when we look at the irregularbehavior of the SFR of sim1020 in Figure 4.4. In the Fourier transform of sim1052, we observe a sort ofregular pattern. Indeed, looking at the SFH of sim1052, a certain damped pattern in the SFR shows up.When the time boundaries are varied (for instance 4− 12 Gyr), the location of the peak for a calculatedperiod does not vary, but its amplitude can be different.For a proper comparison, we have also Fourier transformed the SFHs of simulations where no oscillatorybehavior was found, see Figure 4.9. The height of the peaks are naturally a lot smaller than in Figure4.8 because the SFR is a lot smaller too. We also see that for sim1037 en sim1040, some small peaksshow up. This could indicate the presence of highly damped oscillations. This results suggest, asconcluded before, a continuous transition from simulations with no oscillatory behavior to simulationswith manifest oscillations.The behavior of the Fourier transform of sim1058 is due to the fact that clearly no period can be foundfor a very large single peak, see Figure 4.6.The Fourier transformations of the SFHs do reveal some interesting aspects but it is still hard to use themas a quantitative benchmark for oscillations to be present. In some cases a period does show up, butthen the question remains: how high and how distinctive should the peak be. Hence whether oscillationsoccur or not is still to determine at sight, using the SFH and additionally the Fourier transforms.

4.1.4 Influence of search radius and age criterium

Search radius In fact, the in Par. 4.1.2 extensively discussed parameters c?,2 and n are not the onlyparameters newly introduced in the code by the new star formation law of Eq. 4.1.In order to compute the star density around a target particle, the search radius, called radius was in-troduced in Chapt. 3. Star particles within this search radius contribute to the star density of a targetparticle according to the cubic spline kernel. From calculations of the Sedov-Taylor radius in Par. 2.1.4and observations of spatial ordening of OB and T-Tauri associations, the search radius was chosen tobe radius = 50 pc. Other values for this search radius are defensible and therefore, different values forthe radius are tested and their influence on the SFR is studied.The SFHs are shown in Figure 4.10. One would naively expect that for small search radii, the SFHswould be similar to the SFH of the reference galaxy sim1006, because less neighboring particles will befound and the influence of induced star formation should be small. On the other hand, we already haveshown that SFHs are sensitive to even the smallest changes in parameter values and p?.Indeed, looking at Figure 4.10, we have found oscillatory behavior for a radius of radius = 20 pc,radius =


Figure 4.8: Fourier transform of several simulations considered to have oscillatory behavior present in their SFR. The associated SFHs areshown in Figure 4.4, Figure 4.5 and Figure 4.6. The Fourier transform is only applied between 3.7− 12 Gyr in the SFH of the simulations.

4.1. Results 61

Figure 4.9: Fourier transform of several simulations considered to have no oscillatory behavior present in their SFR. Some of the associatedSFHs are shown in Figure 4.5. The Fourier transform is only applied between 3.7− 12 Gyr in the SFH of the simulations.

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Figure 4.10: Star formation histories of different simulations with parameters n = 2 and c?,2 = 105 × (1010 ×M�)−2Gpc6 Gyr−1. Thereference simulation sim1006 is shown with no induced star formation present. In the other simulations, the search radius is varied for whichneighboring particles contribute to the star density of a certain target particle.


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]

Figure 4.11: Two simulations with the exact same parameter values n = 3.5 and c?,2 = 104 × (1010 ×M�)−3.5Gpc10.5 Gyr−1. In sim1052,with label “original”, no age criterium is defined, in sim1078 with label “age criterium”, only young stars are accounted for with an age < 1.87Gyr.

50 pc and radius = 80 but not for radius = 40 pc and not really (although discussable) for the case ofradius = 60. Hence, we cannot conclude that there is a trend towards higher or lower search radii. Thebehavior of the SFH is more dependent on the complex and unpredictable interplay between a higherstar formation probability p? and the feedback effects.It is also found that the larger the search radius, the more computational time the simulation takes. Forlarger search radii, the search domain overlaps with more tree nodes, hence more branches of the treeneed to be descended. The treewalk is the most expensive algorithm in the code, thus altering it canhave a large influence on the computational time.

Age criterium Another aspect which we have not accounted for up till now is the age criterium. InPar. 2.1.4, we have defined length and time scales for induced star formation to occur, and the timescale turned out to be of the order of O(107) − O(108) yr. The implemented feedback comes to anend at 1.87 Gyr (see Par. 3.2.3) when the SN of type Ia go off. Therefore, stars that are older than1.87 Gyr in the code, will never be able to induce star formation because they do not give rise to anyfeedback. Therefore, with the calculation of ρs, only star particles that are younger than 1.87 Gyr shouldbe included.If we implement this in the code, we can look at SFHs to see whether this age criterium for stars hasan influence, see Figure 4.11. As we can see, the age criterium has no influence in the first 4 Gyr inthe evolution of the galaxy. This is surprising because the criterium is set at < 1.87 Gyr hence we wouldexpect an influence starting 1.87 Gyr after the birth of the first star. The surprising agreement of bothsimulations is probably due the fact that induced star formation is already explicitely present in the code(see Figure 4.3) and star forming regions are hence located in the neighborhood of young stars.After 4 Gyr, the influence of the age criterium is starting to show up but no obvious qualitative differenceis found between the two simulations. In fact, the simulation with the age criterium applied formed slightlymore stars than the simulation without the age criterium. This also proofs that a decrease of the inducedstar formation term c?,2ρn

gρs can lead to an increase in the SFR and the total star mass.

4.2 Discussion

4.2.1 Why a change in star formation rate?

The inclusion of induced star formation in the star formation law has shown to result in different types ofqualitative behavior of the SFR, sometimes giving rise to oscillatory behavior. The effect of parameters nand c?,2 was discussed in detail, but no obvious link is present between parameter values and behaviorof the SFR. Hence for now, we only know that oscillations can occur, but we have not really found anexplanation for them. What exactly does happen when p? is increased by the extra term in the SF law?

4.2. Discussion 63

Why do some simulations show oscillatory behavior and others don’t?First of all, it is interesting to look at the star formation criteria. We take the example of sim1019, forwhich oscillatory behavior is observed, and the SFC are plotted in Figure 4.12. The comparison is madewith the reference galaxy sim1006 without induced star formation included.It is clear from the SFR, that during the first 3 Gyr of the evolution history, the SFR of sim1019 is moremodest while the SFR of sim1006 shows more violent peaks. One could wonder why this is the casebecause in fact, the probability for star formation p? is larger in sim1019 than in sim1006, due to theinduced star formation term in the star formation law.The answer is of course feedback effects. In sim1019 and sim1006, the initial star formation peak at∼ 0.1 Gyr is exactly the same. After that, stars are present in the galaxy hence the influence of inducedstar formation should be noticable. Indeed, we see at a time of∼ 0.3 Gyr, sim1019 already starts formingstars again, while sim1006 is still recovering from the first initial star formation. This early higher SFRof sim1019 results in a lot more feedback than in sim1006, and therefore, star formation proceeds moreslowly afterwards. The total cumulative star mass is lower for sim1019 in the first 3 Gyr and the effectsfrom feedback are also clear from the gas mass MgasBO that has been ejected. In sim1019, initiallymore feedback has been done, therefore more gas has been blown out, and the SFR slows down.After a period of 3 Gyr, reference galaxy sim1006 that has been forming more stars up till now, especiallyduring 2 violent starbursts at ∼ 2 Gyr and ∼ 2.8 Gyr. There has been giving rise to as high amounts offeedback that the density has decreased significantly and star formation is almost entirely stopped. Thisalso shows up in the SFC as discussed in Par. 4.1.1. Almost no particles satisfy the density criterium.For sim1019, an entirely different situation occurs. Due to the relatively more moderate SFR in the first3 Gyr of the evolution history, less feedback has been done, larger gas densities remain in the center ofthe galaxy, less gas is situated at large distances, and more gas is available to collapse, satisfy the SFC,and give rise to star formation. After each peak of SF, feedback is done and gas will be blown apart.Therefore, the peaks in SF are damped towards later times.As before, the temperature criterium proofs to be unimportant, almost all particles satisfy T < 15000 K.The convergence criterium can play a role, although it is a lot less severe than the density criterium.We also observe as before, that after peaks in the SFR, less particles satisfy the convergence criteriumbecause the gas is blown apart.Now, we can compare the oscillating simulation sim1019 with a non-oscillating simulation with one verylarge peak in the SFR, sim1058, see Figure 4.13. Both simulations make use of the altered star forma-tion law with the induced star formation term included.At the time of ∼ 2.6 Gyr, sim1058 does show a peak in the SFR after which the SFR is reduced toalmost zero for a time of ∼ 1 Gyr. Apparently, this has provided the galaxy with time to collapse to highdensities and give rise to an extreme peak in the SFR, giving rise to extreme feedback and blowing thegas apart. The feedback is as extreme that the gas only forms a more or less spherical gas cloud againwhen 2 Gyr have passed and hardly reach densities again that exceed the SF threshold. We can seethe feedback happening when the gas density is plotted, see Figure 4.14.It is also interesting that we observe a manifest low in the temperature criterium after the extreme star-burst has occured. Clearly, the gas has been heated by the extreme feedback effects.

4.2.2 Where does star formation occur?

In order to further study the properties of the galaxies that show oscillatory behavior, we investigatewhether star formation occurs rather centrally, or more spreaded over the dwarf galaxy. Therefore, wehave resimulated the successful simulation sim1019 but printing out snapshots every 0.01 Gyr insteadof every 0.1 Gyr. This allows us to zoom in on a star formation peak, see Figure 4.15.During the occurence of the SF peak, we look at the projected density on the xy plane in Figure 4.161.Right before the maximum in the SFR around 4.63 − 4.64 Gyr, the gas halo is still rather spherical.Therefore, the highest densities and hence star formation occur at radii smaller than 1 kpc in the centerof the galaxy. This is clear from the density-radius plot of Figure 4.16 because the threshold density isindicated with a dashed line at 100 amu cm−3. Particles that exceed this threshold are eligible for starformation and their distance to the center of the galaxy is easily read from Figure 4.16.

1An animated video of this figure is available at http://www.youtube.com/watch?v=zWy6tzvYhqg


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[%]

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]

Figure 4.12: Top to bottom: star formation rate SFR, total star mass Mstar, fraction of the initial gas mass currently blown out (at a distance> 30 kpc) MgasBO, number of gas particles Ndens exceeding ρc, number of gas particles Nconv satisfying ∇ · v < 0, number of gas particlesNentr with T < 15000K and finally number of gas particles Ntot that satisfy all three SFC. Oscillating galaxy sim1019 is compared with thereference simulation sim1006.

4.2. Discussion 65

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[%]

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]

Figure 4.13: Top to bottom: star formation rate SFR, total star mass Mstar, fraction of the initial gas mass currently blown out (at a distance> 30 kpc) MgasBO, number of gas particles Ndens exceeding ρc, number of gas particles Nconv satisfying ∇ · v < 0, number of gas particlesNentr with T < 15000K and finally number of gas particles Ntot that satisfy all three SFC. Oscillating and non-oscillating galaxies sim1019and sim1058 are compared. Both have a star formation law with induced star formation included.


Figure 4.14: Density evolution projected on the xy plane. This density evolution belongs to sim1058 and shows the density evolution duringan extreme peak in the SFR and right afterwards. It is clear that the extreme star formation has caused extreme feedback, and the gas is entirelyblown apart by supernovae and stellar winds. Star formation is not possible for the next 2 Gyr.

The largest part of the gas particles have temperatures around 104 K, because initially all gas particlesare at temperatures of T = 104 K. Another reason is that the cooling rate shows an extreme decreasearound 104 K (see Figure 3.4), hence particles that have been heated, cool rather easily to temperaturesof 104 K and much more inefficient to lower temperatures.Comparing the density-radius plots with the temperature-radius plots, we see that at the same radiiwhere little peaks in the density occur, lows in the temperature show up. The gas needs to be cold be-fore it is able to collapse and form stars. We can see that the hydrodynamic equations in the simulationsaccount for this condition, because the star formation regions and the cool regions occur at the sameradii. The fact that the critical temperature for star formation is very high (T < 15000 K) is hence not toomuch of a problem since high density regions proof to be cold enough to form stars.After the maximum value in the SFR has been reached, the first effects of supernovae become visible,around 4.66− 4.67 Gyr. Bubbles are blown in the ISM, and at the edges of these bubbles, the gas ispressed together and this high densities give rise to star formation. The induced star formation is veryobvious: the newly formed stars are situated exactly in these high density regions at the edges of theblown bubbles and the region of star formation has been shifted towards larger radii of more or less 1kpc, shown in Figure4.16.At times of 4.69− 4.7 Gyr, the feedback has been strong enough for the gas to be vastly blown apart andhigh density regions to disappear. The star formation peak has more or less come to an end, with little

4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90time[Gyr]

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Figure 4.15: Zoom on a peak in the star formation rate of sim1075, with the same parameters as sim1019: n = 2 and c?,2 = 105 × (1010 ×M�)−2Gpc6 Gyr−1. The green lines indicate the begin and end time of the density plots in Figure 4.16.

4.2. Discussion 67

Figure 4.16: Left: projected density plots on the xy plane. Time is shown in Gyr and zooms in on the star formation history of sim1075, seeFigure 4.15 on a star formation peak, between 4.63− 4.7 Gyr. Recently formed stars with ages < 0.01 Gyr, are plotted as purple dots.At the right of the projected density plots, the density and temperature are shown as a function of radius. This radius is the distance between theparticles location and the center of the galaxy (the center is calculated using the center of mass of the dark matter halo). The threshold densityfor star formation (nSF = 100 amu cm−3) is shown as a dashed line. We can see some gas particles exceeding the threshold density for starformation.


recrudescences of star formation around 4.74 and 4.76 Gyr.On the density-radius plots, it is clear that the gas particles move away from the center of the galaxywith time and the star formation shifts towards larger radii with it. For instance at 4.69 Gyr, very few gasparticles are located in the center of the galaxy (radii< 0.5 kpc). The few particles in the center have lowdensities of 10−2 amu cm−3 and are very hot with temperatures of T ∼ 106, from the recieved feedback.The observed behavior of star formation, starting with a collapse of the gas, induced by supernovaeand feedback and finally stopped by the gas being blown apart, is quite the same as the star formationbehavior in simulations without an altered star formation law. As discussed before, induced star for-mation is already present in a self-consistent way in the code. On the other hand, we need to realisethat without the altered star formation law, this observed peak in the star formation rate would not havebeen present. The reference galaxy using the Kennicutt-Schmidt law does not show star formation atevolution times later than 4 Gyr.

4.2.3 Scaling relations

Via observations of different (dwarf) galaxies, certain observed quantities of these galaxies have beenproven to show a strong correlation. When the correlated properties of these galaxies are plotted, allgalaxies are found on more or less one line/plane. If a simulated dwarf galaxy would be situated entirelyaway from this line/plane, this would be a strong indication for the simulation to be unphysical andprobably galaxies with these properties as simulated, do not exist.Several of these correlated quantities, called the scaling relations, are plotted in Figure 4.17. Evenwithout focussing on the details of the different scaling relations, we see that the simulated galaxies(colored dots) agree with the observations (grey dots).

Half light radius

Panel (a) of Figure 4.17 shows the half light radius in function of the absolute V-band magnitude MV (theV-band corresponds to a wavelength of λ = 550 nm). We recall that apparent magnitudes are alwaysdefined as a difference between apparent magnitude m1 and m2,

m1 −m2 = −2.5 log(

F1

F2

), (4.6)

with F1 and F2 the according fluxes of the observed objects. It is easy to show that the fluxes vary as

F2

F1' 2.51m2−m1 , (4.7)

hence when the magnitudes differ by a value of ∆m = 1, one of the object is 2.5 times brighter than theother.Using this to interprete Figure 4.17(a), we see that the simulated galaxies are spreaded over MV be-tween −11.3 and −12.3. This means that the maximum difference between the dimmest and brightestsimulated galaxies is maximal a factor 2.5.The figure has been plotted in more detail to check for a difference between the oscillating and the non-oscillating galaxies, see Figure 4.18(a). The simulated galaxies (almost) all show larger MV and largerlog(Re) than the reference galaxy. All simulations are still situated in the region with the observations,this means that the simulated galaxies are shifted along the scaling relation. The altered star forma-tion law has increased the amount of total star mass, hence it is logical that the absolute magnitudeincreases. There is also an indication present for the oscillating galaxies to be shifted more than thenon-oscillating galaxies. This effect is natural because the oscillating galaxies form more stars than thenon-oscillating galaxies.Is it also explainable that the half light radius Re increases? When plotting the half light radius of anoscillating galaxy, and the reference galaxy, see Figure 4.18(a), we see that the oscillating galaxy endsup with a larger half light radius at 12 Gyr than the reference galaxy. We know that the oscillating galaxyhas produced more stars and therefore has been subject to larger amounts of feedback. As shown in

4.2. Discussion 69

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c2]

f.)

Figure 4.17: Overview of different scaling relations. The simulations (colored dots) are shown and compared with the observa-tions (grey dots). The mapped proberties are measured at the end of the evolution time of the galaxy after 12 Gyr. (a) Halflight radius log(Re) as a function of the visual V-band absolute magnitude MV . (b) The fundamental plane FP substracted withthe half light radius log(Re) as a function of the B-band luminosity log(LB). (c) Color V-I as a function of the V-band magni-tude MV (V is in the visual and I in the infrared). (d) Metallicity [Fe/H] as a function of V-band magnitude MV . (e) The Ser-sic index n defines the shape of the luminosity profile. (f) Central surface brightness µ0 as a function of the V-band magnitude MV .The observational data are taken from [Jimenez et al., 2008, Burstein et al., 1997, de Rijcke et al., 2005, Geha, 2003, Grebel et al., 2003,Guzman et al., 2003, McConnachie and Irwin, 2006, Caldwell, 1999, Zucker et al., 2007, Irwin and Hatzidimitriou, 1995, Lianou et al., 2010,Mateo, 1998, Michielsen et al., 2007, Koyama et al., 2008, de Rijcke et al., 2009, Peterson and Caldwell, 1993, Sharina and Puzia, 2008]


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Re[

kp

c]

(b)

Figure 4.18: The left panel (a) shows a zoom on Figure 4.17a. Almost all simulated galaxies are shifted along the scaling relation towardslarger magnitudes MV and larger half light radii Re in comparison with the reference galaxy. The right image (b) shows the detailed starformation of an oscillating galaxy (sim1019) and the reference galaxy (sim1006). For the oscillating galaxy, a first hint of the negativecorrelation between Re and MV can be observed.

Par. 4.2.2, this feedback blows apart the gas and the regions of active star formation shift towards largerdistances from the center of the galaxy. Careful comparison of evolution of the SFR and the Re showsthat right before and during a peak in SFR, the half light radius shows a low and when the SFR dropsback to zero (due to feedback) the half light radius increases. Are these two quantities truly negativelycorrelated?To anwer this question, the correlation matrix of the two quantities SFR and Re is calculated for theoscillating galaxies, as well as for the non-oscillating galaxies and the reference simulation. The results,shown in Table 4.4 are remarkable. All the oscillating galaxies show a significant negative correlation rbetween the SFR and the half light radius Re, while all the non-oscillating galaxies show eather no or apositive correlation r.The difference in correlation is an indication for different physical mechanisms to be dominant. In theoscillating galaxies, the star formation peaks start with a more or less spherical galaxy that collapses tohigh densities. Star formation is concentrated centrally and the half light radius of the galaxy decreases.At the maximum and right after the peak in the SFR, feedback has shifted star formation towards largerdistances from the galaxy center and therefore, the half light radius increases.For the non-oscillating galaxies, looking at the SFH of the reference galaxy, we see that the half lightradius increases strongly in the first 3 Gyr of active star formation, since the first stars are being formed.Afterwards, the SFR almost completely drops down to zero and the half light radius remains more orless constant. When no active star formation is present, the half light radius will be rather determined bythe age of the stellar populations and their according distances from the galaxy center.

Table 4.4: Correlation r between the SFR and the half light radius Re of oscillating an dnon oscillating simulated galaxies. The referencegalaxy is shown too.

osc r non-osc rsim1020 -0.56 sim1054 0.42sim1036 -0.43 sim1045 0.03sim1019 -0.39 sim1037 0.19sim1070 -0.38 sim1040 0.08sim1052 -0.39 sim1030 0.14sim1036 -0.34 reference 0.42

4.2. Discussion 71

Fundamental plane

The fundamental plane represents an observed relation between the half light radius Re, mean effectivesurface brightness e and the central velocity dispersion σ of elliptical galaxies (no dEs). It is usuallyexpressed as

log Re = α log σc + β log e +γ, (4.8)

with α, β and γ fundamental constants equal to [Burstein et al., 1997]

α = 1.38 (4.9)β = −0.835 (4.10)γ = −0.629. (4.11)

The quantity plotted in panel (b) of Figure 4.17, depicts the relation

0 = α log σc + β log e +γ− log Re, (4.12)

as a function of B-band luminosity log LB. The elliptical galaxies that obey the fundamental planewill obviously be situated at zero. Dwarf galaxies are observed to deviate towards higher values.This can probably be explained by dwarf galaxies being more diffuse systems large elliptical galax-ies [Cloet-Osselaer et al., 2012]. Indeed, dwarf galaxies have shallower gravitational potentials and aremore easily blown apart by feedback in comparison to elliptical galaxies.Figure 4.17 (b), shows that the simulated dwarf galaxies are deviating from the fundamental plane to-wards positive values. This is in agreement with observations of dwarf galaxies. However, in comparisonwith the reference galaxy, it seems that the galaxies with an altered star formation law deviate less fromthe fundamental plane. Looking at specific values for the simulations, we see that oscillating galaxiestend to have smaller mean effective surface brightnesses e and larger Re (as discussed above) incomparison to the reference galaxy. Mainly the larger value of Re will cause the simulated galaxies tobe shifted towards zero, in accordance with the fundamental plane. However they still clearly agree withthe dwarf galaxy regime.

Color

Panel (c) of Figure 4.17 depicts V-I color index as a function of V-band magnitude. We recall that the

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I


non-oscreference

Figure 4.19: A zoom on Figure 4.17 panel (c). The simulated galaxies are shifted along the scaling relation towards larger magnitudes andbluer colors. Oscillating and half-oscillating galaxies seem to be shifted more than the non-oscillating galaxies in comparison to the referencegalaxy.

V-band is a visual band with a wavelength of λ = 550 nm, the I-band is situated at the transition betweenvisible light and the infrared at a wavelength of λ = 790 nm. Hence when the color index V-I increases,the galaxy shifts towards higher wavelengths. The scaling relation shows that galaxies with larger V-band magnitudes are bluer. This makes perfectly sense because young stars proof to be brighter and


bluer.The simulated galaxies seem to be shifted along this scaling relation in comparison to the referencegalaxy. Indeed, they have not only produced more star mass along their evolution history, but also starformation has occured more recently than in the reference galaxy. Hence the simulated galaxies arebrighter and bluer.It is also observed when zoomed in on this scaling relation, that the half-oscillating and the oscillatinggalaxies are shifted more towards this scaling relation than the non-oscillating galaxies, as one wouldexpect. This effect is shown in Figure 4.19.

Metallicity

Another important quantity in regard to galaxies is metallicity. Metallicity can be expressed in severalways. In panel (d) of Figure 4.17 the metallicity is depicted as [Fe/H] which is defined as

[Fe/H] = log(

NFe/NH

(NFe/NH)�

). (4.13)

Another way of describing metallicities is to consider mass fraction Z of particles that are no hydrogenor helium.The simulated galaxies agree with the observations. They show rather high metallicites and the refer-ence galaxy and the non-oscillating galaxies tend to be metallicity poorer than the oscillating galaxies,see Figure 4.20(a). This effect is once more explainable by the fact that the oscillating galaxies haveproduced more star mass, there has occured more feedback and therefore the ISM has been more en-riched with metals. We can see this enrichment of metals happening when we look at the SFH of an

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Z

(b)

Figure 4.20: Figure (a) shows a detail of panel (d) of Figure 4.17. The simulated galaxies have shifted along the scaling relation towardsbrighter magnitudes and metallicities. The oscillating and half-oscillating galaxies are generally shifted more than the non-oscillating galaxies.Figure (b) shows the SFH of an oscillating galaxy with the according metallicity Z.

oscillating galaxy, see Figure 4.20(b). The mass fraction of the metals is clearly increasing becausemore stars are being formed.

Sersic profile

The two last panels (e) and (f) of Figure 4.17 show properties of the Sersic profile. This profile describesthe surface brightness of a galaxy as a function of the radius. The fitted Sersic profile is of the form

I(R) = I0e−(R/Re)1n , (4.14)

4.2. Discussion 73

(a) (b)

Figure 4.21: Panel (a) shows Sersic profiles of the reference galaxy and several oscillating galaxies. The oscillating galaxies tend to have amore concave hence diffuse profile than the reference galaxy. Panel (b) shows the Sersic profile of an oscillating simulating at different timesbefore, during and after a starburst.

with I(R) the surface intensity as a function of radius R and with Re the half light radius. The central sur-face brightness µ0 shown in panel (f) of Figure 4.17 is the same as I0 but in units of mag/arcsec2.The parameter n depicted in panel (e) of Figure 4.17 is the Sersic index and defines the diffuse-ness/compactness of the profile. For values of n > 1, the surface brightness profile has a concaveshape and the surface brightness decreases sharply within small radii from the center. Values of n < 1describe a convex hence a rather diffuse profile that does not show a large decrease in surface bright-ness within small radii from the center.Looking at panel (e) of Figure 4.17, we see that the reference galaxy has a Sersic index of aroundn = 1. Most of the simulated galaxies (oscillating and non-oscillating) have a Sersic index around oneour smaller, although some of them have a larger index n. When the values for the central surfacebrightness µ0 in panel (f) of Figure 4.17 are compared with the reference galaxy, the simulated galaxies(oscillating and non-oscillating) show a clear increase in V-band magnitude as was already discussed.However no clear increase in central surface brightness µ0 is observed. This would mean that the galax-ies with the altered star formation law are somewhat more diffuse than the reference galaxy. We canlook at this in detail, plotting the Sersic profiles, see Figure 4.21(a). We see indeed that the oscillat-ing galaxies have a slightly more concave profile that does not decrease as fast with the radius as thereference galaxy. An explanation is not hard to find, the oscillating galaxies have been more subjectto feedback and the star formation regions have shifted towards somewhat larger distances from thegalaxy center, as discussed in Par. 4.2.2. Therefore the surface brightness profiles are more diffuse,compared to the profile of the reference galaxy. Indeed, the reference galaxy has hardly been subjectto any feedback during the last 8 Gyr. The oscillating galaxies being more diffuse, also agrees with ourfindings that the oscillating galaxies have larger half light radii than the reference galaxy.We also looked into the change of the surface brightness profile during a peak in the SFR, here called (alittle exagerated) a burst, shown in Figure 4.21(b). The early burst profile clearly shows to be straight oreven a little convex compared with the concave late burst and post burst profile. During the early birst,star formation is located in the center of the galaxy and this is translated to a higher value for µ0 and astraight to slightly convex surface brightness profile. A few tenths of Gyr later, feedback has caused thestar formation to shift towards larger radii, µ0 drops and the surface brightness profile gets a concaveshape.


4 5 6 7 8 9 10log10(LB)

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10(σ

)[k

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]

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oschalf-oscnon-oscreference

Figure 4.22: Faber-Jackson relation for observations, indicated as grey dots, and simulations, indicated as colored dots.

Faber-Jackson

The Faber-Jackson relation2 is somehow similar to the fundamental plane, but lacks the quantity Re. Itexpresses a relation between the B-band luminosity and the velocity dispersion σ of the form

LB ∝ σ4, (4.15)

for elliptical galaxies. We have plotted our simulations on the Faber-Jackson relation, shown in Figure4.22, and our simulations are found to agree with the observations.The oscillating galaxies are found to be more luminous than the reference galaxy, in agreement withabove findings. There does not seem to be a trend towards more or less central velocity dispersionalong the line of sight.

Classification

dSphs or dIrrs The observations used in panel (d) of Figure 4.17 are depicting several types of galax-ies, from dSphs and dEs over dSphs/dIrrs to dIrrs. If we use this information to plot the dSphs in adifferent color, see Figure 4.23, we can see that the simulated galaxies agree with the metallicities of dedSphs. On the other hand, we know from Figure 4.12 and Figure 4.13 that unlike dSphs, all simulatedgalaxies retain a significant amount of gas mass: around 20% of the initial amount within a box of 30 kpc(MgasBO). Therefore, they should be classified as dIrrs. Clearly an inconsistancy is present.It has been shown by Valcke [Valcke, 2010] that isolated galaxies as simulated, will always retain a sig-nificant amount of their gas because no tidal interactions and/or ram pressure stripping can occur. If wewould simulate galaxies with identical properties in a cluster environment, probably the gas would havebeen removed by tidal interactions or ram pressure stripping and the galaxies would resemble typicaldSphs.Simulating dIrrs is presently hard in the code because all simulated galaxies tend to have too high metal-licities to resemble dIrrs. The cause of this excess in metals is due to the overestimation of star formationin the current simulations. This leads to more stellar feedback and hence more enrichment of the ISMwith metals. Star formation is overestimated because presently, ionization is not taken into account inthe simulations. The UV radiation of young stars gives rise to Stromgren spheres in which star formationis impossible. However in the code, only a neutral gas component is present. Currently, efforts are made

2The Tully-Fisher relation is not applied to our set of simulations. It was decided that this relation is probably not relevant for non-rotatinggalaxies, since the Tully-Fisher relation makes use of the circular rotation velocity of galaxies.

4.2. Discussion 75

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H]

Observations dSphsObservations dIrrs, dEsosc

half-oscnon-oscreference

Figure 4.23: Metallicity in function of V band magnitude as in panel (d) of Figure 4.17. Dwarf spheroidals are indicated in grey, othergalaxies (dIrrs and dEs) are indicated in purple. We can see that the simulated galaxies agree with the metallicities of dSphs.

to account for this UV background radiation by [Vandenbroucke et al., 2013], and future work.The oscillating and non-oscillating simulations with the altered star formation law, all experience morestar formation than the reference galaxy. The problem with too high metallicities was already present,and our galaxies suffer even more because of their higher star formation rates.

BCDs Now we interprete our findings in context of Chapter 1, where we were looking for an explanationfor the occurence of BCDs. We can say that our simulations with the altered star formation law do notresemble BCDs and hence do not provide an explanation for their existence.We recall that BCDs are blue dwarf galaxies with extreme starburst going on in them. The star formationrates of BCDs are estimated to be 0.1− 1 M�/yr. We have discussed that BCDs have underlying olderpopulations hence BCDs are not young systems. If we would have simulated BCDs, we would observeextreme peaks in the SFR in the latest few Gyr of the SFH. This was not the case for any galaxy wehave simulated.The most extreme peak in the SFR in our simulations we have found, was in sim1058 at 4.1 Gyr wherethe SFR was of the order of 0.015 M�/yr, see Figure 4.6. The SFR at this peak is still one order ofmagnitude too small and the peak is situated rather early in the SFH, therefore, we would probably neverobserve this dwarf galaxy as such (unless we look at very high redshifts).BCDs are also compact systems as their name suggests. We, on the other hand, have found that ouroscillating galaxies are more diffuse and have larger half light radii than the reference galaxy using aKennicutt-Schmidt law. Altering the star formation law will defenitely not lead to more compact systems.Also BCDs are thought to have low metallicities. From Figure 4.23, we see that our simulation do nothave lower but higher metallicities than the reference galaxy.In conclusion, altering the star formation law as in Eq. (4.1) to include the effect of induced star formationwill not lead to the simulation of BCDs. It is however possible that the altered star formation law andthe effect of induced star formation will play its role in the explanation and simulation of BCDs whencombined with merging galaxies or interactions with gas environments. Therefore, we should have alook at interaction scenarios with induced star formation included in the star formation law.Another remark is that only one parameterization is tested for the inclusion of induced star formation.In Chapter 2, we have shown that other parameterizations could also lead to instability and oscillatorybehavior and it is hard to predict how the simulations would turn out using another parameterization forthe star formation law (for instance c?,2ρgρn

s with a power of the star density instead of over the gas den-sity). Naively speaking, it seems that another parameterization would not yield too much of a differencefor the qualitative behavior of the SFR. On the other hand, we have shown that small alterations of lessthan 1% to the chance p? of forming a star, can yield entirely different qualitative outcome of the SFH.


4.3 Mergers

To explain BCDs, there are two possible angles: explain the starbursts by internal mechanisms, or byexternal mechanisms.In this thesis, by altering the star formation law, we have studied internal mechanisms. However, oursimulations did not resemble BCDs hence we could not provide and explanation for their existence andthe origin of their starbursts.Meanwhile, research has been conducted to investigate external mechanisms, more specifically mergersof dense gas clouds and dwarf galaxies [Verbeke, 2013]. The results of these mergers suggest thatstarbursts are triggered by gas infall in a dwarf galaxy. Also rotation of the dwarf galaxy (host galaxy inthis context) is listed as an important parameter in regard to starbursts. When rotation of the host galaxyis too low, the infalling gas cloud cannot be slowed down sufficiently and the gas cloud passes through.On the other hand, when rotation of the host galaxy is too high, the gas cloud get sucked in completelyand is spreaded vastly over the galaxy. The triggered starbursts will therefore be less intense and lesscentered.Interesting results can be obtained from merging a gas cloud and a dwarf galaxy, using our proposedstar formation law with the induced star formation term instead of the standard Kennicutt-Schmidt law.Only a few exploring simulations have been undertaken. Their initial conditions are listed in Table 4.5 andtheir further properties in Table 4.6. We have opted for heavier galaxies of type E09 since these have

Table 4.5: Properties of the initial conditions for an E09 model used as host galaxies for the mergers with gas clouds.

Dwarf galaxy type E09Number of gas particles 200000Number of DM particles 200000Mg,i 525×106 M�MDM,i 2476×106 M�DM profile NFWInitial gas temperature 10000 KInitial metallicity 0.0001 Z�Feedback efficiency εFB 0.7Softening length ε 30 pcSF threshold density ρc 100 amu cm−3

Star formation parameter c?,1 0.25

been investigated by [Verbeke, 2013] and prooven to be successful host galaxies in order to achieve astarbursts. A disadvantage is that we are not sure which parameters for c?,2 and n to choose. Very brieftesting of some random parameter values has been undertaken, see Table 4.6 and Figure 4.24. In this

Table 4.6: Properties of simulations used as host galaxies for mergers.

simulation number rotation c?,2 nsim0017 5 km/s 0 0sim3003 5 km/s 104 2sim3006 5 km/s 105 2

context, sim0017 is our reference simulation, using a Kennicutt-Schmidt law. Figure 4.24 shows thatsim0017 already has oscillatory behavior in its star formation history. Simulations sim3003 and sim3006each use the altered star formation prescription with different parameter values, see Table 4.6.We see that sim3006 experiences little effect of the altered star formation law. The qualitative behaviorof the star formation rate seems similar and even the total star mass is the same as in the referencegalaxy after 12 Gyr. For sim3003, there is a clear influence of the altered star formation prescription.During the first 3 Gyr of the simulation, star formation has been more modest, therefore, more star masshas been formed by the end of the simulation.

4.3. Mergers 77

0.000

0.005

0.010

0.015

0.020

0.025

SF

R[M�/y

r] reference sim0017sim3006

sim3003

0 2 4 6 8 10 12time[Gyr]

0

10

20

30

40

Mst

ar[1

06M�

]

Figure 4.24: Comparison of a rotating simulation sim0017 using the Kennicutt-Schmidt law, and rotating simulations using the altered starformation law, see Table 4.6.

0.0000.0050.0100.0150.0200.0250.030

SF

R[M�/y

r] 30064001

40024003

0 2 4 6 8 10 12time[Gyr]

05

10152025303540

Mst

ar[1

06M�

]

Figure 4.25: Star formation history of host galaxy sim3006 using the altered star formation law. Gas clouds are falling in at different snapshottimes (seperated by 0.1 Gyr). A strong starburst is found for merger sim4002.

The fact that these galaxies rotate as opposed to previous simulations with the altered star formation,can have its influence. It has been shown that adding rotation in the simulations, leads to less burstyand more continuous star formation histories [Schroyen et al., 2011]. On the other hand, the fact thatsim3006 does not show many differences with sim0017 can also be a simple coincidence. Previousfindings (see Par. 4.1.2) have indicated three types of qualitative behavior, one of them being more orless similar to the reference simulation as what we have here.Simulation sim3006 is chosen to be the host galaxy. There is no obvious influence of the altered starformation prescription, but this can change when gas clouds are falling in.Gas clouds are generated with detailed properties as described in [Verbeke, 2013]. The gas clouds ofsim4001, sim4002 and sim4003 are equal, but they fall into the host galaxy at different times. Theyhave a density profile proportional to 1

r and their temperature is determined by their mass and density,because the cloud needs to be in virial equilibrium.The results of this gas infall in Figure 4.25, show that the timing at which the gas clouds fall in has amajor influence. In this case, sim4002 was the most successful in initiating an intense starburst. Thestarburst is at its maximum at 0.025M�/yr. Peaks with this maximum value have never been able tobe simulated using only the altered star formation without the merging gas cloud. It is clear that themerging gas cloud has triggered star formation. Moreover, the observed peak in the star formation ratebelongs among the most intense that have been found by Verbeke, where a Kennicutt-Schmidt law is


9.5 10.0 10.5 11.0 11.5 12.0time[Gyr]

0.000

0.005

0.010

0.015

0.020

0.025

0.030

SF

R[M�/y

r]

30064001

40024003

Figure 4.26: Detail of Figure 4.25. It is clear that the clouds fall in at different times. This small difference in time yields entirely differentqualitative behavior of the bursts.

used. Therefore, since we only tested three scenarios and already one of them is very successful, it isvery plausible that the altered star formation law enhances the starburst.Figure 4.26 zooms in on the different mergers. An intersting finding is that all mergers show oscillatorybehavior after they have fallen in. Simulations sim4001 and sim4003 give no rise to a high star formationpeak but form more stars in the end, see Figure 4.25. This is in accordance with previous conclusionsthat more modest star formation during a certain period of the evolution history, will in the end give riseto oscillations and a higher total star mass.In conclusion, mergers of gas clouds and dwarf galaxies using an altered star formation law lead tointense starbursts and should be investigated in greater detail in the context of explaining BCDs.

5Conclusions and outline

5.1 Conclusions

• Theoretical calculations were used to choose a parameterization of induced star formation. It wasshown that in simulations where time delay is included, induced star formation leads to instability,while spontaneous star formation does not have this effect. In a three component system wherecooling is included, any parameterization of star formation leads to oscillatory behavior. For oursimulations with cooling and time delay included, we opted for a parameterization of c?,2ρn

gρs. Thisterm was added to the Kennicutt-Schmidt star formation law.

• Alteration of the star formation law with an extra non-linear term accounting for induced star for-mation leads in several cases to oscillatory behavior of the star formation rate. However, we wereunable to relate qualitative behavior to parameter values of c?,2 and n. We can conclude that c?,2should be larger than 10−2 (1010×M�)−nGpc3n Gyr−1 to have an influence on the behavior of thestar formation rate. All tested values of n between 1 and 4 can give rise to oscillatory behavior. Inother words, a degeneracy is found between star formation behavior and parameter values.

• When the star formation rates of the oscillating simulations are Fourier transformed, in most casesa period shows up for the oscillations of the order of 1 Gyr.

• The star density around a gas particle ρs is calculated counting the stars in a certain search radius.No relation between the star formation behavior and the search radius can be found.

• When the age criterium is applied, that only takes into account young stars which are still in theprocess of giving feedback, a surprising accordance is found with simulations not using the agecriterium.

• Simulations that have a more modest star formation rate in the first 3 Gyr of their evolution, willafterwards show oscillatory behavior and form more stars, in comparison to simulations with aviolent start.

• A star formation episode starts centrally. The newborn stars give rise to feedback effects whichshift the star formation regions towards larger distances from the galaxy center. After more or lessa tenth of a Gyr, feedback effects halt star formation completely.

• The simulated galaxies turn out to reproduce different scaling relations when compared to obser-vations of dwarf galaxies. This is an indication for the simulations to be physically acceptable.

79

80 Chapter 5. Conclusions and outline

• The oscillating galaxies using the altered star formation law turn out to be brighter, bluer and morediffuse than the reference galaxy (which uses a Kennicutt-Schmidt law). This is easily explainedby the fact that oscillating galaxies have formed more stars, hence they are brighter. They haveexperienced star formation more recently, hence they are bluer. This larger star mass has givenrise to more feedback, shifting the location of star formation to larger radii, therefore, the oscillatinggalxies are more diffuse and have larger half light radii.

• The simulated galaxies all have large metallicities, according to the regime of dSphs. However,they retain a significant amount of their gas and should be classified as dIrrs. Clearly an inconsis-tency is present.

• Oscillations in the star formation rate are observed, however the star formation episodes are notsufficiently intense to represent BCDs. Moreover, the oscillating systems are more diffuse andhave higher metallicities in comparison to our reference simulation. These properties do not agreewith observed properties of BCDs.

• Merging dense gas clouds with dwarf galaxies, by using an altered star formation law, seems toyield definite intense starbursts. The starbursts are probably intensified by the effect of the alteredstar formation law.

5.2 Outline

• Only one parameterization of induced star formation is tested. It is fairly unpredictable what effectto expect from other parameterizations as for instance c?,2ρgρn

s .

• The star formation law can be extended even more with different terms. For instance a parameter-ization for cooling could be considered.

• The inititial mass function has an upper limit of 60 M�. It was already suggested by [Valcke, 2010]that it would be interesting to look at higher upper limits. This will enhance the number of su-pernovae and have a definite effect on induced star formation explicitely present in the code. Incombination with an altered star formation law, this could give interesting results. Also, environ-mental dependency of the initial mass function can be considered. This can lead to the simulationof for instance OB associations. On the other hand, a higher resolution will be necessary to do so.

• We showed that mergers combined with an altered star formation law probably leads to intensestarbursts. This can be investigated into more detail.

List of Abbreviations

ΛCDM Lambda Cold Dark Matter

BCD Blue Compact Dwarf

CPS Cooled Post-Shock Layer

dE Dwarf Elliptical

dIrr Dwarf Irregular

DM Dark Matter

dSph Dwarf Spheroidal

HSB High Surface Brightness

ISM Interstellar Medium

LSB Low Surface Brightness

NFW Navarro-Frenk-White profile for dark matter

SF Star Formation

SFR Star Formation Rate

SFR Star Formation Rate

SN Supernova

SPH Smoothed Particle Hydrodynamics

SSP Single Stellar Population

SW Stellar Winds

81

82 LIST OF ABBREVIATIONS

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