Download N-Body/SPH simulations of induced star formation in dwarf galaxies

G HENT U NIVERSITY
FACULTY OF S CIENCES
D EPARTMENT OF P HYSICS AND A STRONOMY
N-Body/SPH simulations of induced star
formation in dwarf galaxies
AUTHOR : T INE L IBBRECHT
Supervisors:
Annelies C LOET-O SSELAER
Dr. Mina KOLEVA
Joeri S CHROYEN
Promotor:
Prof. Dr. Sven D E R IJCKE
Master thesis in context of obtaining the academic degree of Master of Science in Physics and Astronomy
June 2013
”Les gens ont des étoiles qui ne sont pas les mêmes. Pour les uns, qui voyagent, les
étoiles sont des guides. Pour d’autres elles ne sont rien que de petites lumières. Pour
d’autres qui sont savants elles sont des problèmes. Pour mon businessman elles étaient de
l’or. Mais toutes ces étoiles-là elles se taisent. Toi, tu auras des étoiles comme personne
n’en a... ”- Antoine de Saint-Exupéry - Le Petit Prince1
1 Figures front page: Density evolution projected on the xy plane. This density evolution belongs to sim1058 and shows the density evolution
during an extreme peak in the SFR and right afterwards. The extreme star formation has caused extreme feedback, and the gas is entirely blown
apart 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 on
paper and that it got somehow finished (even almost or not really quite in time). It is hard to decide
where to start thanking people because I am very greatful to everyone that helped me along the process
and every opportunity that came along the way.
First of all, I want to thank professor Sven De Rijcke. Offering the vastly interesting subject and supervising patiently, I always felt inspired after leaving his office. I am also greatful for his effort to hand in an
FWO 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, helping with software problems. Also, she answered my every (stupid) question, she gave advice and she
proofread several parts. The same is true for Mina and Joeri. The three of them have spent so much
time 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 obsess
with over your subject. This way we have killed the mood at several parties (to great boredom of our
friends).
I am very greatful to my close friends and roommates Amélie De Muynck, who calculates faster than
her shadow, and Thomas Boelens, Maple wizard. They were always ready to help selflessly, like true
friends do.
In the same context, my dear boyfriend Koen Hendrickx should get an award for living together with the
most distracted person alive. Yes, driving left side of the road or using toxic plants in our dinner - thinking
it 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 a
new chapter of our lives.
Finally, I am very greatful to my family. My dad Dirk Libbrecht, who has been enthusiastic and interested
in my thesis, my mum An Hautekiet for comforting me always, and my brother Robbe Libbrecht, who
has 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.
i
ii
PREFACE
Toelating tot bruikleen
De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van de
masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het
auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden bij het
aanhalen van resultaten uit deze masterproef. Op datum van 11 juni 2013.
iii
iv
TOELATING TOT BRUIKLEEN
Contents
Preface
i
Toelating tot bruikleen
iii
Nederlandse samenvatting
vii
Introduction
1
1 Blue Compact Dwarfs
1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Starbursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Evolution scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
5
7
2 Feedback and instability
2.1 Physics of star formation . . . . . . . . . . . . .
2.1.1 Components of the interstellar medium
2.1.2 Jeans mass and Jeans radius . . . . . .
2.1.3 Cooling . . . . . . . . . . . . . . . . . .
2.1.4 Induced star formation . . . . . . . . . .
2.2 Parameterization of star formation . . . . . . .
2.2.1 Kennicutt-Schmidt law . . . . . . . . . .
2.2.2 Oscillations in star formation . . . . . .
2.2.3 Hurwitz criterium for stability . . . . . .
2.2.4 Star formation systems . . . . . . . . .
2.3 Research hypothesis . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11
11
11
12
13
15
19
19
21
23
26
33
3 Implementation of induced star formation in an N-body/SPH code
3.1 The code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Dwarf galaxies . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Collisionless N-Body simulation . . . . . . . . . . . . . . .
3.1.3 SPH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Neighbor search . . . . . . . . . . . . . . . . . . . . . . .
3.1.5 Integration . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Additional physics . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Star formation . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Star mass . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Self-regulation . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Implementation of induced star formation . . . . . . . . . . . . .
3.3.1 Star formation law . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Calculation of ρs,i . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Integration of the star formation law . . . . . . . . . . . .
3.3.4 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
35
35
35
36
37
39
40
40
40
42
42
44
45
45
45
46
46
48
v
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
vi
4 Results and discussion
4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Reference dwarf galaxy . . . . . . . . . . . . . . . . . . .
4.1.2 Behavior of the star formation rate and parameter values
4.1.3 Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Influence of search radius and age criterium . . . . . . .
4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Why a change in star formation rate? . . . . . . . . . . .
4.2.2 Where does star formation occur? . . . . . . . . . . . . .
4.2.3 Scaling relations . . . . . . . . . . . . . . . . . . . . . . .
4.3 Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS
.
.
.
.
.
.
.
.
.
.
49
49
49
53
58
61
62
62
63
68
75
5 Conclusions and outline
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
79
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 galaxieën aan de hand van hiërarchische structuurvorming.
Kleine dichtheidsperturbaties groeien en versmelten en geven zo het ontstaan aan dwerggalaxieën.
Grote sterrenstelsels zoals onze Melkweg worden dan gevormd door de versmelting van vele kleine
dwergstelsels.
Dwerggalaxieën zijn kleine galaxieën met absolute magnitudes groter dan −18 en optische groottes
van ongeveer 1 kpc. Verschillende subtypes van dwergstelsels worden ingedeeld waaronder gasarme
elliptische 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 op
heden is er geen consensus over de oorzaak van deze starbursts. De oorzaak kan gezocht worden bij
een invloed van buitenaf (externe mechanismes) of invloed van binnenuit (interne mechanismes).
Het is uiteindelijk de bedoeling te onderzoeken of BCDs kunnen gesimuleerd worden. Hiervoor focussen
we op interne mechanismes in de dwergstelsels, meer specifiek stervorming. De stervormingswet in
de gebruikte code is een eenvoudige lineaire wet, de Kennicutt-Schmidt wet genoemd. Theoretische
berekeningen worden gebruikt om deze wet aan te passen en non-lineairiteit te introduceren. Hierdoor
kunnen oscillaties in de stervorming ontstaan die mogelijks de starbursts in BCDs verklaren.
Blauwe compacte dwerggalaxieën
Verschillende vragen blijven overeind in verband met BCDs. De oorzaak van hun starbursts kan mogelijk
verklaard 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 lager
draaimoment zorgt voor een sterk gepiekte stervorming.
Externe mechanismes ter verklaring van BCDs zijn bijvoorbeeld het versmelten van twee dwergstelsels
wat leidt tot een intense stervormingsepisode.
Ook evolutiescenario’s worden bestudeerd. Als de stervormingsepisode in een BCD stilvalt, hoe zal
dit stelsel dan waargenomen en geı̈dentificeerd worden? De evolutie van een gasrijke BCD naar een
gasarme dE vereist een verklaring voor het gasverlies. Gasverlies kan optreden als gevolg van externe
mechanismes zoals ram pressure stripping (het verliezen van gas ten gevolge van de druk veroorzaakt
door het bewegen door een intergalactisch medium) of interacties met andere galaxieën of graviterende
objecten. Toch blijft het op basis van andere eigenschappen zoals metalliciteit en draaimoment onwaarschijnlijk dat een BCD naar een dE zou evolueren. Hoogstwaarschijnlijk zullen stilgevallen BCDs
als dIrrs geclassificeerd worden.
Feedback en instabiliteit
Om starbursts in BCDs te kunnen verklaren, is het nodig om een grondig inzicht in het verschijnsel
stervorming te verkrijgen. Stervorming is onderhevig aan vele feedback effecten. Stervorming vindt
plaats in koele en dichte gebieden. Wanneer interstellair gas gekoeld wordt, kan het ineenstorten en een
ster vormen. Pasgeboren sterren op hun beurt zenden stellaire winden uit of ontploffen in supernova’s.
vii
viii
NEDERLANDSE SAMENVATTING
Hierdoor zal het interstellair medium verhitten en ioniseren. Het interstellair gas wordt in bellen uit elkaar
geblazen en hierdoor zal stervorming stopgezet worden. Anderzijds zullen diezelfde stellaire winden en
supernova’s aanleiding geven tot het vormen van meer sterren. Aan de randen van de geblazen bellen
zal het gas bijeen gedrukt worden en hoge dichtheden veroorzaken. Dichte gaswolken kunnen dan weer
ineenstorten tot sterren. Het fenomeen waarbij meer sterren gevormd worden door de aanwezigheid van
sterren in de omgeving wordt geı̈nduceerde stervorming genoemd. Een andere complicatie is dat via
supernova’s ook metalen vrijgelaten worden in het interstellaire medium die het koeltempo versnellen.
Het is duidelijk dat veel verschillende effecten stervorming beı̈nvloeden. Daardoor is dit een moeilijk te
voorspellen en te parametriseren grootheid.
Berekeningen zijn ondernomen om de stabiliteit van stervormingssystemen te testen. Het blijkt dat
het toevoegen van geı̈nduceerde stervorming in een stervormingwet kan zorgen voor instabiliteit en
oscillerend 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 voor
starbursts in BCDs? Om deze vraag te onderzoeken wordt gebruik gemaakt van simulaties.
Implementatie van geı̈nduceerde stervorming in een N-Body/SPH code
De N-Body/SPH code GADGET-2 wordt gebruikt om dwerggalaxiën te simuleren. In deze code zijn
slimme boomstructuren geı̈mplementeerd om de berekening van de gravitationele interacties te versnellen. Diezelfde boomstructuren worden gebruikt om buurdeeltjes te zoeken, nodig om dichtheden te
berekenen. Als men de parametrisatie van geı̈nduceerde stervorming wil inbrengen in de stervormingswet, 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 voorhanden is. Daarom wordt een integratiealgoritme geı̈mplementeerd: een vierde orde Runge-Kutta. Een
vergelijking werd gemaakt met een lineaire benadering, maar op grond van fysische argumenten wordt
de Runge-Kutta methode verkozen.
Resultaten en bespreking
Eens de stervormingswet aangepast is, kunnen simulaties uitgevoerd worden. Het blijkt al snel dat de
nieuwe stervormingswet aanleiding geeft tot oscillaties in het stervormingstempo. Twee vrije parameters die verband houden met de parametrisatie van geı̈nduceerde stervorming worden uitvoerig getest
in een groot bereik. Er kan echter geen verband gevonden worden tussen parameterwaardes en kwalitatief gedrag van het stervormingstempo. Met andere woorden, een ontaarding doet zich voor tussen
de parameterwaardes en het kwalitatief gedrag van de stervorming.
Het blijkt dat galaxieën met een hoog stervormingstempo in de eerste 3 gigajaar van hun evolutie, later
hoogstwaarschijnlijk nog weinig sterren zullen vormen en geen oscillaties vertonen in hun stervormingstempo. Galaxieën die daarentegen eerder bescheiden van start zijn gegaan en in de eerste 3 gigajaar
minder sterren hebben gevormd, zullen daarna een grotere kans hebben om oscillaties te vertonen in
hun 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 het
galaxie. 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 verplaatst van het centrum van het galaxie naar grotere afstanden. Dit is de geı̈nduceerde stervorming aan
het werk. Na een korte periode van stervorming zorgen de feedback effecten ervoor dat stervorming
weer helemaal stilvalt.
Verschillende schalingsrelaties worden bestudeerd en het blijkt dat de gesimuleerde galaxieën typische
schalingrelaties zoals de Faber-Jackson relatie reproduceren in vergelijking met observaties van dwergstelsels. Dit is een goede indicatie dat de gesimuleerde galaxieën fysisch zijn.
De dwergstelsels die oscillaties vertonen, blijken helderder, blauwer en diffuser te zijn dan gesimuleerde
galaxieën zonder oscillerend gedrag. Dit is gemakkelijk te verklaren. De stelsels zijn helderder omdat zij
meer sterren gevormd hebben, blauwer omdat zij recenter sterren gevormd hebben, en diffuser omdat
ix
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 besluiten
dat onze simulaties geen BCDs voorstellen. De oscillaties in stervorming zijn niet intens genoeg om een
starburst 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 observaties
van BCDs.
Tenslotte werden mergers bestudeerd van dichte gaswolken en dwergstelsels. Deze mergers hadden al
aangetoond tot starbursts te leiden, gebruik makend van een standaard stervormingswet. Onze nieuwe
aangepaste wet die rekening houdt met geı̈nduceerde stervorming, lijkt de starbursts te versterken.
Conclusie en vooruitblik
Het aanpassen van de stervormingswet door rekening te houden met geı̈nduceerde stervorming leidt
niet tot het simuleren van BCDs. Het is echter wel mogelijk dat deze aangepaste wet zijn rol kan spelen
in 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 energy
Λ is responsable for 76% of the universe energy content and is an unknown energy component that
causes the universe to expand in an accelerating way.
Another 20% of the universe consists of unknown cold dark matter. Cold, because dark matter particles
are non-relativistic. Dark, because the particles are only able to interact gravitationally and not electromagnetically. Therefore, it is impossible to observe dark matter, at least by means of electromagnetic
radiation.
The remaining 4% of the universe consists of radiation and baryonic matter as we know it on our humble
Earth.
Using ΛCDM cosmology, theories of structure formation in the young universe were developed. It turns
out 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 perturbations
arise. Pressureless dark matter clumps together in these density perturbations and dark matter halos
are formed.
After the decoupling of baryonic matter and radiation, baryons fall into the gravitational potentials of the
dark matter halos. When the clumps of dark and baryonic matter are sufficiently large, gravity will have
increased and will attract other clumps. These clumps will start moving, attracting one another, and
merge. When the formed structures of dark matter and baryons contain a sufficient amount of baryonic
gas, they can start forming stars and dwarf galaxies are born. Large galaxies as for instance our Milky
Way are believed to be formed through the merging of dwarf galaxies.
Cosmological structure formation has been simulated by for instance the Millennium simulation. This
simulation clearly illustrates hierarchical structure formation, starting with small density perturbations
and ending up with large scale structures as galaxies and clusters, which are still in process of forming
even larger structures. Furthermore, simulations have been undertaken to follow hierarchical structure
formation in detail via merger trees, where one keeps track of all merged progenitors that eventually
become galaxies.
From the theory of structure formation, two important conclusions are drawn. First of all, without cold
dark matter, galaxies as we know them would never have been able to form. This is an important argument in favor of the existance of dark matter. Secondly, the origin of dwarf galaxies and the fact that they
act 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 dwarf
galaxies. They can be defined as small galaxies with absolute V-band magnitudes larger than −18 (note
that 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 absolute
magnitude of −21 and a size of 30 kpc.
Another important property of dwarf galaxies is that they are dark matter dominated. They have large
mass to luminosity ratios which implies that there is far more mass present than from starlight alone
would be estimated. The large mass to light ratios suggest that large amounts of dark matter are
present in dwarf galaxies. Furthermore, in comparison to large galaxies, dwarf galaxies have a shallower
1
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 red
stellar populations and contain almost no gas. Their name is derived from the fact that they have
elliptical isophotes. Dwarf ellipticals are the most common type of galaxies in the universe and are
mostly found in cluster environments.
• dwarf spheroidals (dSphs) are the “little brothers” of the dwarf ellipticals. They share mostly the
same properties but have absolute magnitudes larger than −14.
• dwarf irregulars (dIrrs) could be described as the opposite of dSphs. They are situated in the
same magnitude range, but they contain a large amount of gas. Therefore, stars are being formed
and the galaxy contains a young and blue stellar population. Dwarf irregulars are mainly found in
the field as opposed to dEs and dSphs populating clusters.
• blue compact dwarfs (BCDs) are galaxies showing extreme bursts of star formation. The young
population of stars gives rise to its blue colors. The starburst is located centrally in the galaxy
hence its compact appearance.
Many questions remain unanswered in regard to dwarf galaxies. Dwarf ellipticals and dwarf spheriodals
have old stellar populations. This implicates that for a certain point in their history, they should have
contained gas in order for star formation to be possible. Currently they are devoid of gas. By which
means did their gas amount disappear? Have dIrrs evolved to dEs by the loss of their gas and by which
mechanisms would this gas loss have occured? On the other hand, dEs also show fundamentally different properties compared to dIrrs as for instances larger metallicities and lower angular momenta. This
complicates possible evolution scenarios.
The case of BCDs is even more mysterious and provides the context of this master thesis. Up till the
present, no explanation is confirmed for the origin of the observed starbursts in BCDs. Moreover, it
is unclear whether these starbursts are cyclic phenomena or a single event. In Chapter 1, BCDs are
discussed into detail. The cause of the starbursts in BCDs can be searched for in external or internal
mechanisms.
External mechanisms are all influences on the dwarf galaxy originating from its environment and/or
nearby objects. Examples of external mechanisms are interactions with an intercluster medium, tidal
interactions with other galaxies or black holes, mergers of galaxies, etc. A master thesis has been written by Verbeke [Verbeke, 2013], with the aim of simulating BCDs via mergers of dense gas clouds and
dwarf galaxies.
It is also possible that starbursts in BCDs can be explained by internal mechanisms. These include
all physical processes that occur in the galaxy without an influence of its environment. Properties as
angular momentum, processes as star formation, gas cooling and heating, supernovae, etc. are all considered as internal mechanisms. This master thesis investigates the origin of starbursts in BCDs by
focussing 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 factors
lead 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 proportional
to (a power of) the local gas density.
But star formation is more complex. Numerous feedback effects - positive as well as negative - influence
star formation. For instance cooling of the interstellar gas will cause a gas collapse and provoke star
formation. On the other hand, the newborn stars give rise to stellar winds and often supernovae, heating
and ionizing the interstellar medium. These conditions will halt star formation. It gets even more complicated when realizing that supernovae and stellar winds at the same time also trigger star formation. At
3
the edges of the bubbles blown in the interstellar gas by supernovae, densities are increased and gas
collapses to form stars. The process of star formation initiated by supernovae and stellar winds is called
induced star formation.
Another effect is caused by supernovae releasing metals in the interstellar medium, which will accelerate
the cooling rate, favoring star formation.
The complex interplay between different feedback effects leads to self-regulation of star formation. An
increase of the gas density for instance will not simply lead to higher star formation rates. Moreover, research has shown that self-regulation can lead to non-linear oscillations in the star formation rate. This is
an interesting property in regard to gain insight in BCDs. Could these oscillations lead to increased star
formation 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 closer
look 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, an
N-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 star
formation prescription in the code. This may increase the chances for oscillatory behavior to occur in
the 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
1
Blue Compact Dwarfs
1.1
Definition
Blue compact dwarfs (BCDs) are dwarf galaxies with remarkably high star formation rates (SFRs), dominated by hot blue stars with possibly low metallicities. The neutral gas component as well as the areas
of active star formation (SF) are centrally concentrated in BCDs, hence their compact appearance. Like
other 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 sight
that BCDs are young systems. Near-infrared images however show that underlying star populations
with ages of a few Gyrs are present [James, 1994, Thuan, 1983]. The observed near-infrared fluxes are
due to the presence of K and M giants1 , while the observed star formation episodes are younger than
5 · 107 Gyr. This time span cannot contain the evolution of young blue stars to K and M giants. Hence a
previous 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 transient
and possibly even in a cyclic way. These star formation episodes with unusually high SFRs are called
starbursts. Star formation rates in BCDs were estimated using Hα 2 observations in the optical. As an
example, the estimate for the SFR in BCDs by Sethuram [Sethuram, 2011] is of the order of 0.1 − 1.0
solar 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, we
could 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 observation
of the 21 cm emission line of HI (for a discussion on the components of the interstellar medium, see
Par. 2.1.1). The observed velocity fields suggest that the 6 BCDs have some rotational velocity. In spite
of these rotational velocities, the specific angular momenta (J/M) of BCDs are smaller than the angular
momenta 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 main
sequence towards red colors.
2 H is the Balmer line corresponding with the transition in hydrogen from excitation state n = 3 to n = 2. The transition corresponds with
α
a visual wavelength of λ = 656.28 nm. Hα radiation is an important tracer for star formation because young stars tend to ionize and excite
their 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 magnitude
MB . (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 and
BCDs do have comparable values, see Figure 1.1(b). For BCDs however, these maximal rotational
velocities are reached within a smaller radius from the center than for LSB dIrrs, because BCDs are
compact. This means that BCDs have steeper rotation curves than LSB dIrrs.
Toomre instability analysis [Toomre, 1964] balances gravity, rotational effects and thermal effects, to calculate the threshold gas density for star formation. This analysis leads to threshold densities that are
proportional to the slope of the rotation curve [van Zee et al., 2001]. Steeper rotation curves implicate
higher threshold densities, hence BCDs have higher threshold densities for the onset of SF, thus it will
take a longer time before SF initiates. Therefore more “fuel” for SF is present when the threshold density value is attained and intenser SF can occur. It is possible that this property leads to the observed
starbursts in BCDs. In this context it seems that starbursts are a logical implication of the compact appearance and hence the lower specific angular momenta of the BCDs. The frequency of the consecutive
starbursts would then be regulated by the rate at which the gas in the central core reaches the threshold
density and initiates SF until feedback effects (as for instance supernovae) redistribute the gas and stop
or 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 and
evolution of dwarf galaxies, and hence they proposed angular momentum as the second most important
parameter (after total galaxy mass) for determing the behavior of dwarf galaxies. They compared between 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 showed
fundamentally different star formation histories (SFHs). The rotating model showed a rather continuous
SFH 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 depict
the 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-rotating
model are shown in Figure 1.2. The model that does not rotate but is manually flattened, shows more or
less the same behavior as the non-rotating spherical model. The differences in SF behavior are hence
clearly due to the differences in angular momentum and not to flattening itself. In conclusion, lower
angular 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 different
approach. His simulations of interacting and merging dwarf galaxies lead to starbursts. The most succesfull model consists of two dwarf galaxies, of different mass and in a different life stage, that collide
and merge. The inward movement of HI gas from the outer areas of the galaxy leads to a strongly increased 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. The
observed 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 external
mechanisms, or a combination of both.
1.3
Evolution scenarios
The classification of a galaxy is based on its observational properties. Dwarf galaxies with extreme
star formation rates and a central dense and blue core will be classified as BCDs. However, when the
starburst has come to an end, the question arises how the galaxy will be observed an classified. Will
8
Chapter 1. Blue Compact Dwarfs
a faded out BCD resemble a dwarf irregular, or rather a dwarf elliptical? Several scenarios have been
studied 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 gasrich with MHI ' 107 M , while dEs are generally gas-poor with MHI ' 105 M [van Zee et al., 2001].
Evolution scenarios from BCDs to dEs should hence explain by what means a large quantity of gas can
disappear from a galaxy.
Evolution of BCDs to dEs Gas removal can be considered in terms of internal processes or in terms of
external 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] proposed
in 1986 an internal mechanism in which a starburst can expel the interstellar medium (ISM). They calculated that the virial velocity of a galaxy should be smaller than the critical value of 100 km/s to allow
global gas loss as a result of the kinetical energy of the starburst. This theory however has been falsified
by more recent models: it seems to be harder than previously thought to remove the ISM by one single
starburst. Another argument against this model is that multiple starbursts are seen to occur in one single
dwarf 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 significant
amount 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 more
common than in the field. Likely, the gas is removed in by processes that involving interactions with the
intracluster 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 occurs when a galaxy moves through hot intergalactic clouds of gas, common in clusters. This causes
the interstellar gas of the galaxy to be removed by the pressure of the intergalactic gas. Ram pressure
stripping was proposed by Gunn en Gott [Gunn and Gott, 1972]. They studied the effect of infall of mass
in 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 calculated that in this way a galaxy could be stripped of its interstellar material when the density of the
intergalactic medium exceeds 5 · 10−4 atomes/cm3 .
Modelling as well as observing galaxies subject to ram pressure stripping is still an area of active research. Multiple recent studies use (magneto)hydrodynamical models to simulate ram pressure stripping, e.g. [McCarthy et al., 2008, Shin and Ruszkowski, 2013, Ruszkowski et al., 2012]. Ram pressure
stripping 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 not
by 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 that
they evolve to dEs after the current starburst fades out. These six BCDs however each have a certain
angular momentum while dEs generally do not rotate [Bender and Nieto, 1990]. Hence the evolution of
BCDs 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 these
more 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 studied
the evolution scenarios. He concludes that it is highly unlikely that a BCD would evolve to a gas poor dE
because 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 quiescent 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 different
properties throughout their lifetime?”
Van Zee [van Zee, 2001] studied a large sample of dIrrs using UBV3 and Hα observations. By means
of properties as color gradient, stellar composition and surface brightness, they concluded that the luminosities of dIrrs today can be explained by a continuous SFR without a need of one or more starbursts
to explain the observations. Certain observed dIrrs however, show properties that are very similar to the
properties of BCDs, as for instance a compact central gas core and similar color gradients to the ones
of BCDs. But the observed galaxies are currently not going through a starburst phase. These dIrrs are
thus probably examples of faded out BCDs.
3 UBV are photometric filters, corresponding to ultraviolet, blue and visual wavelengths. The same filters correspond to V-band and B-band
magnitude.
10
Chapter 1. Blue Compact Dwarfs
2
Feedback and instability
2.1
Physics of star formation
This section studies the phenomenon of star formation into detail. Star formation takes place in cool
and dense regions of molecular clouds. When a cloud collapses under its own gravity, thermonuclear
reactions start taking place and a star is born. Depending on the properties of the star, it emits mass and
energy through supernovae events or stellar winds. This emission is called feedback. The interstellar
medium recieves feedback and is heated and blown appart. Hence feedback annuls the conditions
necessary for star formation. On the other hand, the same feedback effects also increase the probability
of stars being born in the approximity of the emitting star. This phenomenon of induced star formation is
deepened in Par. 2.1.4.
It is clear that the birth of a star involves, and has an influence on different components of the interstellar
medium. These components are discussed in Par. 2.1.1. The Jeans criterium yields conditions for a
collapse 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 spatially
separated, 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. Mostly
observed in areas of recent star formation, they obtained their high temperatures and state of ionisation
from UV-light emitted by hot young stars. HII clouds are mainly observed by one of the Balmer lines in
hydrogen which comes in existence when an electron undergoes a transition from ionisation state n = 3
to n = 2 and emits a photon at a wavelength of 656.3 nm in the optical. This line is also called the
Hα-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 of
T ∼ 10 − 100 K. The only way to observe HI gas is via the 21 cm line using radio spectroscopy. This line
is due to a transition of the electron spin state relatively to the proton spin state. In the lowest energy
state of the hydrogen atom, the spin of the proton and electron are antiparallel. A slightly excited state
exists when the spins are parallel. The transition between those two states is forbidden with a decay time
of 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 transition
is very small (T 1 K).
Molecular clouds Further cooling of the HI gas leads to formation of molecular clouds, mainly consisting
of H2 but also other elements as CO, HCN and CS. These clouds are extremely cold with temperatures
around T ∼ 5 − 30 K. Since H2 is a very poor emitter (see Par. 2.1.3), molecular clouds are detected by
millimeter radiation from rotational CO transitions.
Molecular clouds also contain molecular interstellar dust, which are condensed solid molecular particles
of less than 0.1 ¯m. This dust usually has temperatures of around T ∼ 20 − 30 K, therefore it radiates in
infrared and submillimeter wavelengths. Furthermore it acts as a katalysator on the formation of molecular hydrogen gas.
Observations as well as simulations show that the very cold and dense molecular clouds are regions of
active star formation. These regions are thus of particular interest in the research on star formation. The
different components of the ISM interact in numerous ways and it is especially interesting to understand
what processes lead to formation of molecular clouds and thus to star formation regions. A good insight
in the properties of molecular clouds, yields information on the circumstances and conditions for star
formation. 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 thermal
pressure. A spherical cloud is in hydrostatic equilibrium when
Gρ(r ) M(r )
dP
,
=−
dr
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. For
a gravitational collapse to occur, this hydrostatic equilibrium has to be disbalanced. This happens when
the 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 m
in the outer shell of a gas cloud needs to reach the center of the gas cloud when this cloud undergoes
a pressureless gravitational collapse. When this small mass m at position r0 moves to a new position r
2
closer to the center of the cloud, the kinetic energy rises along 12 m( dr
dt ) while the change in potential
− GMm
energy equals GMm
r
r0 . The free fall time tFF is hence calculated from
tFF =
Z
dr
dr
dt
=
dr
Z
q
2GM
r
−
2GM
r0
.
(2.2)
The integration of this expression (a goniometric substitution adjusts the boundaries from [r0 , 0] to
[− π2 , 0]) yields
s
s
2r03
π
3π
tFF =
=
,
(2.3)
4 GM
32Gρ
where the relationship M = ρ 43 πr03 was used. The free fall time is now to be compared to the time ts for a
wave to travel the cloud at sound speed cs with ts = r0 /cs . When tFF ts , a density perturbation will be
immediately annihilated by the fast travelling pressure waves. However, when tFF ts , the gas around
the 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 critical
length scale and total mass exist and these are called the Jeans length λ J and Jeans mass M J . When
the Jeans mass and length are exceeded, a collapse is able to take place. Jeans length is calculated
from
r
π
λ J = cs tFF ' cs
,
(2.4)
Gρ
2.1. Physics of star formation
13
where the sound speed can be relativistic or non-relativistic. In the case of non-relativistic particles the
sound speed equals
c2s =
γkT
∂P
=
,
∂ρ
m
(2.5)
which yields a Jeans length of
s
5πkT
λJ '
,
3Gρm
(2.6)
with γ = 5/3 in the case of neutral hydrogen. The Jeans mass M J is then simply given by
M J ' πρλ3J ,
(2.7)
and can thus be seen to be proportional to
M J ∝ ρ−1/2 T 3/2 .
(2.8)
The equalities are not exact due to constant factors close to unity, depending on the mass distribution in
the cloud.
The non-relativistic Jeans radius and mass can also be calculated by requesting equality of the gravi2
and the kinetic energy (according to the equipartition theorem)
tational potential energy Egrav ' GM
r
3 MkT
Ekin = 2 m where m is the average mass of one gas particle. This method yields
rcrit
2MGm
3
'
'
3kT
2
kT
2πρGm
1/2
,
(2.9)
where Eq. 2.7 and Eq. 2.6 were used. Isolating ρ yields
ρcrit '
3
4πM2
3kT
2mG
3
.
(2.10)
With the collapse of the cloud, kinetic energy is released in the form of heat. Eventually hydrostatic
equilibrium is reached. Further collapse is only possible when heat is radiated away in the cooling
process. 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 cooling
by molecular line transitions. Discussion paraphrased according to [De Rijcke, 2011].
H2 In most density ranges, cooling by molecular line transitions is the commonest mechanism. The
most abundant molecule in molecular clouds is H2 . However, this molecule does not radiate in the temperature ranges of molecular clouds. This is due to the quantummechanical structure of H2 . Since H2
is a homonuclear molecule, two possibilities for the electronic ground state exist, namely orthohydrogen
in which the protonspins are parallel and parahydrogen with antiparallel protonspins. Protons as well as
electrons are fermions, hence the wavefunction for H2 needs to be antisymmetric for the exchange of
protons and of electrons. Orthohydrogen has a symmetric wavefunction for the protonspins, hence the
angular momentum wave function needs to be antisymmetric and thus has an odd parity. The opposite
is true for parahydrogen: its angular momentum wave function is symmetric and has an even parity. Regarding transition probabilities, electric dipole transitions are in general the most likely to occur. For H2
however, electric dipole transitions are not possible. Indeed, a dipole transition does not change parity,
14
Chapter 2. Feedback and instability
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 parahydrogen). Probabilities for quadrupole transitions are very low and moreover, the excitation from a state
of J = 0 to J = 2 requires an energy of ∆E = k · 500 K. Hence in very cold regions, this excitation is not
even possible apart from its very low probability. In practice, the transition does not occur in molecular
clouds.
H2 also radiates in the infrared by vibrational transitions. However, this kind of excitations only occurs at
temperatures 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 large
quantity 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 and
Werner), but these excitations require high photon energies that do not occur in the temperature ranges
of molecular clouds. To summarize all the above, the H2 gas in the temperature ranges of molecular
clouds 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 12 CO molecule, which is the second most abundant molecule after H2 , is the dominant
coolant [Goldsmith and Langer, 1978]. CO does have a net dipole moment and no parity problems do
occur 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 that
CO 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 of
CO, other molecules as for instance O2 , H2 O and hydrides, and atoms as CI provide a large percentage
of the cooling [Goldsmith and Langer, 1978]. An overwiew of molecules and atoms that are important in
regard 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 components 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, dust
acts as a coolant as a consequence of its thermal radiation. Since it usually has temperatures of 20 − 30
K, it emits in the infrared. However, for the molecular gas to cool along with the dust, both these components 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]
2.1. Physics of star formation
15
is necessary. Densities in molecular clouds vary between 1 − 105 cm−3 hence dust as a coolant is only
important in the densest regions of the molecular cloud.
Collapse By means of the discussed cooling mechanisms, a gravitational collapse can occur without
a raise of temperature. This leads to further stimulation on formation of molecules, which then again
facilitates the cooling process. This provides a positive feedback mechanism for the collapse of the
cloud and thus the formation of a star. To summarize, stars are formed in cold, dense and collapsing
molecular clouds.
In galaxies, giant molecular clouds are observed with radii of the order of 100 pc. These giant molecular
clouds are subdivided in very dense, cool regions of subcloud complexes with smaller radii of more or
less 10 pc; this is called fragmentation. When the Jeans criteria are satisfied in a subcloud, a star is
formed in this region.
When the cloud has collapsed to a density of 1015 kg/m−3 and a radius of 1015 m [Perkins, 2003], a
hydrostatic equilibrium is attained and a protostar is formed. Contraction slowly goes on while energy
is radiated away. This process obeys the virial theorem which says that for a bound system of nonrelativistic 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. All
this 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 dissociation of H2 molecules and ionization of H atoms can take place. Eventually the collapse is stopped by
the 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 probability that a star will form with a certain mass M is calculated by an initial mass function ξ ( M). For
instance Eq. (2.11) was proposed by Salpeter [Salpeter, 1955]
ξ ( M ) ≈ 0.03
M
M
−1.35
,
(2.11)
for M between 0.4 and 10 M . This result was obtained by fitting observed luminosity functions of main
sequence stars.
In 1976, Silk published several articles discussing the temperature dependence of the initial mass function. He expects from observations and calculations of the Jeans mass, initial mass function and a
stability analysis, that it is expected that high mass stars form in turbulent regions with higher temperatures (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 function suggest that massive stars are formed in regions of high surface density where radiative feedback raises the temperature more effectively hence increasing the Jeans mass (M J ∝ ρ−1/2 T 3/2 )
[Krumholz et al., 2010]. These results were obtained from simulations using a hydrodynamical numerical model. The environmental dependency of the initial mass function is still a subject of active research
and 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 birth
of more stars. The process of stimulated star formation by the presence of other stars is called induced
star formation or triggered star formation. Several forms of induced star formation exist and the way in
which energy is released from a star and added to the ISM depends on the spectral type of the star and
its 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
16
Chapter 2. Feedback and instability
spectral type
O
B
A
F
G
K
M
mass (M )
60.0
18.0
3.2
1.7
1.1
0.8
0.3
temperature (K)
50000
28000
10000
7400
6000
4900
3000
radius (R )
15.0
7.0
2.5
1.3
1.1
0.9
0.4
luminosity (L )
1400000
20000
80
6
1.2
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 causes
the birth of more OB stars. Their presence is indicated by the observated infrared and/or compact continuum sources and H2 O and OH
masers. 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 the
Lyman line. Therefore the radiation from stellar winds ionizes the hydrogen molecules from the molecular
cloud surrounding the OB association and gives rise to expanding HII regions (see Par. 2.1.1), also
called Strömgren spheres. The interesting property of ionization fronts is that they trigger high mass
star 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 and
the preceeding shock front, see Figure 2.2. Combining observations and results from an analytical
model in which they calculated the instability of this cooled post-shock layer (CPS), they conclude that
after approximately one and a half million years, a new OB subgroup will form. Indeed, as discussed
before, massive stars tend to form in turbulent, dense regions of slightly higher temperature and these
conditions are present in the CPS. The new OB association is then called the second generation and
also 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 returns
energy to the ISM. The short lifetimes of the massive OB stars result after 107 yr in supernovae of
type SNe II when the degeneration pressure of the free electrons is not sufficient anymore to balance
the gravity of the fusioning star producing iron. In an instant, all electrons recombine with the protons
to form neutrons, the degeneration pressure disappears completely and the star collapses under its
2.1. Physics of star formation
17
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 release
of neutrinos and a giant shock wave travelling through the ISM. During the adiabatic expansion of this
shell, which is called the Sedov-Taylor phase, the radius of the shell is given by [Sedov, 1958]
rs =
E0
α(γ)ρ
1/5
t2/5 ,
(2.12)
with E0 the released energy by the SN II, α(γ) a numerical factor obtained from imposing energy conservation [Vandenbroucke et al., 2013], ρ is the density of the ISM and t the time interval for which rs
is calculated. For the standard release of energy E0 = e51 erg, typical molecular cloud densities of
ρ = 100 amu/cm3 and a time span of 104 yr, this results in a rough estimate for the characteristic radius
rs ∼ 10 − 30 kpc. This can be interpreted as the influence sphere of an OB association or more generally
the characteristic length scale for induced star formation. A nice illustration of sequential induced star
formation, is shown in Figure 2.3.
Sequential formation of OB subgroup association is strongly supported by observational evidence. Simply the spatial ordening of the OB subgroups in regard to their ages is a very good indication of the
theory. A chain of OB subgroups at similar distances lying along the galactic plane with monotonically
increasing 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 star
formation.
In fact, supernovae in general (also of type SN Ia for instance) release energy in the ISM, and hence
have a large influence on star formation.
Supernovae of type Ia occur only in binary systems. In most cases, one of the binary stars will eventually 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 Ia
occurs 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 one
hand, because SNe release a vast amount of energy, the surrounding gas is heated and ionized and
gas densities are decreased which are unfavorable conditions for star formation. On the other hand, at
18
Chapter 2. Feedback and instability
the edges of the pressure and ionization front, instability arises and star formation is provoked. Numerous 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 results
showed that SNe can lead to a compression of a nearby molecular cloud (in a radius of more or less 20
pc), 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 interstellar 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 hydrogen
burning is started, they go through a violent phase, called the T Tauri phase, before reaching the main
sequence stadium on the Hertzsprung-Russel diagram. These T Tauri stars posses a thick circumstellar
accretion disk consisting out of gas gradually falling onto the surface of the star, producing strong stellar
winds in the direction of the rotation axis of the protostar and losing a significant fraction of the stellar
mass. These stellar winds ionize the surrounding molecular clouds and are hence forming expanding
HII bubbles.
In 1978, Norman and Silk [Norman and Silk, 1980] proposed a model showing that the presence of T
Tauri stars induces more (low mass) star formation. However, their model does require an initial high
energy trigger as for instance SNe or luminous OB stars. Once the T Tauri stars are formed, they
produce stellar winds giving rise to ionized bubbles. These bubbles expand and tend to intersect. At
the 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 Figure
2.4. 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 excess
stars in HII regions and they found that these stars are aligned from the ionization source to the edge
of the HII cloud. Moreover, the ages of the aligned stars also gradiently increase towards the ionization
source. These observations strongly support the hypothesis of sequential star formation originating from
an ionization source.
Length and time scales We have shown from the above discussion that induced star formation originates
from 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 pc
are observed. For supernova induced star formation, we have calculated using the Sedov-Tayler radius
that 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 phenomena 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 density
ρ g . To get this result, he assumes that the initial luminosity function of all main sequence stars ψ( Mv ) is
time 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 magnitude 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 and
T (gal) is the total galaxy age. Hence through observations, the initial luminosity function ψ( Mv ) can be
obtained.
When now the assumption is made that the initial luminosity function is time independent, the rate of
star formation equals
dN ( Mv , t)
= ψ ( Mv ) f ( t ) ,
dt
(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 of
present gas density ρ g (1) to initial mass density ρ g (0)
P=
ρ g (1)
,
ρ g (0)
(2.15)
with t = 1 the present and t = 0 the beginning of the era of star formation. He assumes that the number
of stars formed per time unit varies with a power of the gas density,
dN
= f (t) ∑ ψ( Mv ) = Cρng (t).
dt
M
(2.16)
v
Evaluating this equation at t = 1 and t = 0 yields
f (1) = P n 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)
20
Chapter 2. Feedback and instability
dρ (t)
g
This solution is obtained by realizing that f (t) is proportional to 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
dρ g (t)
dt ,
which is proportional to a power n of the gas density ρ g . We obtain the expression
dρ g
dρs
=−
= c? ρng ,
(2.20)
dt
dt
where c? is a constant, ρs is the star density, and it is assumed that all lost gas mass is converted into
stars, hence
dρs
dt
=−
ρ g ( t ) = ρ g (0) e
dρ g
dt .
−c? τt
,
The analytical solution in terms of gas density for n = 1 equals
(2.21)
with in this case
1
e−c? τ = P.
(2.22)
To obtain a value for n, several observable quantities are calculated for different values of n, as for
instance the initial luminosity function, the rate of star formation, the number of white dwarfs, the abundance of helium, etc. Schmidt concluded that a value around n = 2 provided the best reproduction of
observational 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 several
studies obtained very divergent values for n, going from n = 0 to n = 4, corresponding to different
conditions in disks. For example, very large values of n were found in the arms of spiral galaxies, even
when 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 whether
a universal SF law does exist at all.
In 1988, Kennicutt published an article stating that SFRs not only depend on the gas density but also
on the ratio of this density to a critical threshold density determined by dynamical properties of the disk
of the galaxy [Kennicutt, 1989]. This gas density threshold takes into account gravitational instabilities
in 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 slope
describes the SFR.
Below the threshold density, almost no star formation takes place. In this regime, the dependency of the
SFR to the gas density is approximately zero.
In the transition zone with gas densities around the threshold value, a very steep dependency of the gas
density is observed. Kennicutt states that non-linear dependencies of the SFR rate to the gas density
could be explained by this effect. The three different regimes are shown in Figure 2.5 and it is clear that
the 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 alternative
Schmidt 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
2πR
=
,
σ
v( R)
(2.23)
with σ the angular rotational velocity and v( R) the velocity in function of radius R. The dependency of
the Kennicutt-Schmidt law of the dynamical time accounts for the previous findings that the Schmidt law
is dependent on the distance R to the center of the galaxy.
The final proposition of Kennicutt for the star formation law equals
ρng
dρ g
dρs
=−
= c? ,
(2.24)
dt
dt
τ
which is called the Kennicutt-Schmidt law. Today, it is widely used to model SFRs in numerical simulations of galaxies.
2.2. Parameterization of star formation
21
Figure 2.5: SFR dependency of gas surface density. Three different regimes are showed. For densities below a threshold value, there is almost
no dependence of the SFR on the gas surface density. For gas densities above the threshold value, a Schmidt law with modest power defines the
dependency. In the transition zone around the threshold density, a very steep dependency is observed. From left to right, the curves correspond
to 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 numerous feedback effects that have an influence on star formation and lead to self-regulation (a.k.a. selforganization). Molecular cooling, induced star formation, supernovae and other feedback effects can
be parameterized and put into equations to describe a star formation system, consisting out of several
components, namely stars and gas (sometimes a distinction is made between molecular, atomic and
ionized gas). Cooling converts atomic gas into molecular gas, molecular gas can form stars, massive
stars explode through SNe and return ionized gas and metals to the ISM. The entire mechanism can be
described by a set of equations and put into a dynamical system of the form
dX
= f ( X ).
dt
(2.25)
As shown by e.g. Bodifee and De Loore [Bodifee and De Loore, 1985], a star formation system described by Eq. (2.25), with different X being the components of the ISM and where effects of feedback
are 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 gas
mass A, molecular gas mass M, massive star mass S with stars of a mass > 20 M and a reservoir for
old and less massive stars R. The reason to seperate between S and R is to gather the stars S that have
an important influence on their environment hence stars that trigger SF.
A main difficulty is to parameterize molecular cooling. Cooling proceeds more efficiently with higher
densities 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 . Parameters n2 , n3 , and n4 are assumed to be between 1 and 3 and α expresses an efficiency ratio of molecular
and atomic cooling, hence α has values between 0 and 1. The entire set of equations is given by
22
Chapter 2. Feedback and instability
Figure 2.6: Obervations of SFRs in function of the ratio of gas surface density and dynamical timescale. Filled circles correspond to disk
samples, 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]
dA
= K1 S + K2 S + K3 Mn1 − K4 An2 ( M + αA)n3 S−n4 ,
(2.26)
dt
dM
= K4 An2 ( M + αA)n3 S−n4 − K5 SMn1 − K3 Mn1 ,
(2.27)
dt
dS
= K5 SMn1 − K1 S − K2 S,
(2.28)
dt
dR
= K1 S + K3 M n1 .
(2.29)
dt
In Eq. (2.26), we see that the amount of atomic gas decreases in time with an increased cooling rate
represented by K4 An2 ( M + αA)n3 S−n4 . This loss in atomic gas mass is converted into molecular gas
hence this term appears in Eq. (2.27) with an opposite sign. Molecular gas is subject to spontaneous
star formation described by a Schmidt law K3 Mn1 as well as induced star formation parametrized by the
term K5 SMn1 . Induced star formation is hence proportional to the density of (massive) stars and to a
power of the molecular gas density. As described in Par. 2.1.4, massive stars induce the birth of other
massive stars. The loss of molecular gas mass that describes induced star formation is thus added to
the change in massive stellar mass, Eq. (2.28). Active massive stars S lose mass through stellar winds;
this proces is parametrized by −K2 S and this mass loss returns as atomic gas. There is also a fraction
of the active stellar material S that is transformed to inactive stars R given by the term K1 S. 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 spontaneous star formation, have masses less than 20 M , and moreover that those stars are not active
material that is able to induce star formation. However, there is no reason to assume that the spontaneous 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 are
able to induce star formation. Another striking feature of the system, is that the same power n1 for the
dependency of the (molecular) gas density is used for spontaneous star formation as for induced star
formation. In spite of these assumptions, the parameterization does provide a good description of the
phenomenon of star formation and the interactions between the different components of the ISM that go
with it.
Bodifee and De Loore studied the proposed system in great detail, more specifically the influence of all
2.2. Parameterization of star formation
23
introduced parameters on the systems behavior.
Two main types of behavior were found. Depending on different values of the parameters, the system
is stable and evolves to a final stationary state, or it becomes unstable, bifurcates, and shows selfsustaining oscillations behaving like a limit cycle1 . Interesting findings are that oscillations occur for a
small value of n1 , and that the parameter K5 parameterizing induced star formation, does have a great
influence on the system behavior. For very small values of K5 , initial oscillations are damped. With
increasing K5 , the amplitude of the oscillations enlarges until a limit cycle is obtained. Beyond a critical
value, 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 component
system where molecular synthesis and induced star formation are included.
Similar systems were studied by Korchagin et al. [Korchagin et al., 1988]. They proposed 3 mechanisms that lead to the development of oscillations in the system, namely induced star formation, strong
non-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 is
returned to the ISM in the form of gas. When the process of mass ejection is delayed with a time that
exceeds a critical value, the system can go into oscillatory behavior. However, they conclude that the
mechanisms of induced star formation together with non-linear cooling are sufficient to obtain oscillations
of the components of a system, shown by the following three component system
dS
dt
dM
dt
dA
dt
=
βSMn1 − αS,
(2.30)
= − βSMn1 + dMn2 A,
(2.31)
= 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. The
parameter β regulates induced star formation, α is the rate at which stellar material is removed (with a
fraction r put back into atomic gas). Parameter Q describes the accretion rate. Parameter d describes
the rate at which atomic gas is converted into molecular gas and vice versa. Note that this parameterization of cooling is different to the previous system. Moreover in this system, no spontaneous star
formation takes place. The system should be considered as a somewhat simplified model to isolate
the effects of induced star formation and cooling on the systems behavior. No mass conservation is
imposed.
Korchagin et al. studied the (in)stability of the system not only numerically but also analytically using the
Hurwitz criterium (see Par. 2.2.3). From this Hurwitz criterium, they found that instability in the system
and oscillatory behavior occured when n2 > n1 , or in other words, when the non-linearity in molecular
cloud production is stronger than in induced star formation. Numerically, undamped oscillations behaving like a limit cycle are found in the system.
Much like Bodifee and De Loore, Korchagin et al. concluded that the explicit parameterization of (nonlinear) 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 are
studied 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. The
criterium will be used in Par. 2.2.4 to test which conditions need to be valid to obtain instability in a
system that may lead to oscillatory behavior.
1 Limit cycle behavior occurs in non-linear dynamical systems and defines a closed trajectory in phase space. In this case, limit cycle
behavior corresponds to unlimited undamped oscillations in the star formation system components.
24
Chapter 2. Feedback and instability
A non-linear system consisting out of a set of differential equations of the form
Ẋ (t) = f ( X (t), Y (t), Z (t)),
(2.34)
Ẏ (t) = g( X (t), Y (t), Z (t)),
Ż (t) = h( X (t), Y (t), Z (t)),
is studied. The equilibrium solutions X0 , Y0 en Z0 of the system can be found by putting the equations
of 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 ),
(2.35)
0 = h( X (0), Y (0), Z (0)) = h( X0 , Y0 , Z0 ).
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 is
only stable when all eigenfrequencies have a negative real part and the perturbations are damped. When
one of the eigenfrequencies has an imaginary part, oscillations will occur. Either damped or enhanced
oscillations arise when there is a respectivily negative or positive real part as well as an imaginary part
to the eigenfrequency. These eigenfrequencies are calculated by setting up the characteristic equation
of the system through | J − Iλ| = 0. The Jacobian J is defined as
 ∂f ∂f ∂f 
 ∂X

J =  ∂g
 ∂X
∂Y
∂Z
∂g
∂Y
∂g 

∂Z 
∂h
∂X
∂h
∂Y
∂h
∂Z

.
(2.36)
X0 ,Y0 ,Z0
The characteristic polynomial equals
P(λ) = a3 λ3 + a2 λ2 + a1 λ + a0 = 0,
(2.37)
where a3 > 0. To check whether the real parts of the three solutions for λ are negative, the Hurwitz
criterium can be used. A polynomial with only negative real parts is called Hurwitzian and is always
stable.
To check under which conditions the system will show oscillations, a solution for the eigenfrequency
λ = iω is inserted in Eq. 2.37 and this yields
−iω 3 − a2 ω 2 + ia1 ω + a0 = 0.
(2.38)
If this ω exist and is real, we know that the system has at least one imaginary eigenfrequency and
oscillatory behavior can occur in the system.
Splitting real and imaginary parts in Eq. (2.38) yields two equations for ω, namely
ω2
ω
2
= a1 ,
a0
=
.
a2
(2.39)
(2.40)
For this ω to exist and to be real, both these expressions should obviously be equal and positive. This
yields the conditions
a1 a2
a1
a0
a2
= a0 ,
> 0,
(2.41)
> 0.
(2.43)
(2.42)
If Eq. (2.41) is satisfied, than Eq. (2.42) and Eq. (2.43) are identical. Hence for a purely imaginary
eigenfrequency to exist, instability to arise and oscillations to occur in the system, we need to check if
2.2. Parameterization of star formation
25
a1 a2 = a0 and a1 > 0.
In a two component system, purely imaginary eigenfrequencies do not exist, therefore we insert a partially imaginary eigenfrequency λ = γ + iω in the two component characteristic equation λ2 + a1 λ + a0 =
0. Instability arises when γ > 0. If there is also an imaginary part to the eigenfrequency (hence if ω is
real), 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
ω
a
= − 1,
s2
− a21
=
+ a0 .
4
(2.44)
(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
a2
real if a0 > − 41 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 stability. When one of the Hurwitz conditions is violated, the system has at least one eigenfrequency that
has a positive real (and sometimes an imaginary) part. A characteristic polynomial which satisfies the
conditions to be stable is called Hurwitzian.
Hurwitz criterium A polynomial P(λ) = an λn + an−1 λn−1 + · · · + a1 λ + a0 is Hurwitzian if and only if
the determinants D1 up to and including Dn are positive [C., 2013]
D1 = an−1 > 0,
(2.46)
a
D2 = n−1
a n −3
(2.47)
an > 0,
a n −2 ..
.
Dn − 1
a n −1
a n −3
= a n −5
..
.
0
an
0
a n −2
a n −4
..
.
a n −1
a n −3
..
.
0
0
···
···
···
..
.
···
0 0 0 > 0,
.. . a1 Dn = a0 Dn−1 > 0.
(2.48)
(2.49)
For a system of third order, one obtains the following conditions for stability
> 0,
a2 a1 > a0 a3 ,
a0 ( a2 a1 − a0 a3 ) > 0.
a2
(2.50)
(2.51)
(2.52)
These are also called the Hurwitz conditions for stability. Instability is obtained when at least one of the
Hurwitz conditions is not valid. It should be noted that when one of the Hurwitz criteria is invalid, it is not
sure whether oscillations will occur. It is also possible that only a negative real part to the eigenfrequency
is present without an imaginary part. In this case, perturbations will be enhanced and the system will
exponentially move away from its equilibrium solution, without any presence of oscillatory behavior.
26
2.2.4
Chapter 2. Feedback and instability
Star formation systems
The Hurwitz criterium can be used to study the behavior of different star formation systems. In this
paragraph, we focus on different systems with the phenomenon of induced star formation included and
try 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 formation
systems consisting of a component stellar mass S, and a component gas mass M. The first system
under study is
Ṡ = −αS + βMn S,
(2.53)
Ṁ = αS − βMn S,
with α the fraction of stars that perish and are transferred to molecular gas and β is the factor that
regulates induced star formation parameterised by βMn S. This is a vastly simplified system to study the
influence of induced star formation on the systems behavior. Note that no spontaneous star formation
takes 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 M0
exists for all values of S, namely
M0 =
1
α n
.
β
(2.54)
The Hurwitz conditions for a two component system with characteristic equation a2 λ2 + a1 λ + a0 equal
a1 > 0,
a0 a1 > 0.
(2.55)
For system (2.53) this yields
a1 = α − βM0n + nβM0n−1 S0 = nβ
n −1
α n
S0 > 0,
β
(2.56)
a0 = 0.
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 a0 a1 is violated. Still, we see from Eq. 2.44 that for a1 > 0 and a0 = 0, the
value for ω will be imaginary and hence no imaginary eigenfrequency exists. This means that all roots
have 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 introduces 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 to
molecular gas. We obtain hence a system
Ṡ = −αS + βMn S,
Ṁ = αrS − βMn S + Q.
(2.57)
Equilibrium solutions M0 and S0 are given by
1
α n
M0 =
,
β
(2.58)
Q
.
α (1 − r )
(2.59)
and
S0 =
2.2. Parameterization of star formation
27
Also an equilibrium value for S0 is present in this system. The changes brought into this system however
do not lead to instability. Indeed, the factors a1 and a0 are given by
a1 = α − βM0n + nβM0n−1 S0 = nβ
n −1
α n
Q > 0.
a0 = nβ
β
n −1
Q
α n
> 0,
β
α (1 − r )
(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 n
over the star mass instead of the gas mass. Physically this parameterization makes less sense. It would
be more logical that star formation, even induced star formation, is more dependent on gas mass than
on stellar mass. However we define this system to study its (in)stability:
Ṡ = −αS + βMSn ,
Ṁ = αS − βMSn ,
(2.61)
where mass is conserved. Equilibrium solutions M0 and S0 are related by
M0 =
αS01−n
.
β
(2.62)
Determination of the characteristic polynomial for this system and inserting the equilibrium solutions
yields
a1 = α(1 − n) + βS0n ,
a0 = αβS0n (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 . Also
a0 will defenitely be negative hence at least one of the Hurwitz conditions for stability is not valid. Indeed,
even if a1 is positive, a0 a1 will always be negative for a value of n > 1. In conclusion, non-linear induced
star formation can lead to instability, perturbations will amplify instead of being damped and oscillatory
behavior 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, resulting
into
Ṡ = −αS + βMSn ,
Ṁ = αrS − βMSn + Q.
(2.64)
The equilibrium solutions of system (2.57) are
α 1− n
S
,
β 0
(2.65)
Q
.
α (1 − r )
(2.66)
M0 =
and
S0 =
The values of
a1
a0
n
= α(1 − n) + β α(1Q−r) ,
n
= αβ α(1Q−r) (1 − n),
(2.67)
(2.68)
28
Chapter 2. Feedback and instability
are obtained.
For n > 1, detailed function testing (using a python script) serves us with a negative value of a1 for a
large 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. The
accretion 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 than
one order of magnitude). Indeed, the term of induced star formation is balanced with the rate at which
stars have reached their life ending. When this balance is distorted too much, no equilibrium solution is
present around which the oscillations can take place.
Inclusion of spontaneous SF Since we know that instability occurs for systems using a parameterization of induced star formation with a power over the star density βMSn , we would like to know under
which circumstances a system using a parameterization of induced star formation with a power over gas
density βMn S becomes instable. Additional physics is introduced into the system and the effect on the
stability of the system is studied.
There is a disadvantage to all the studied systems up till now: no spontaneous star formation is included. This also means that when initially no stars are present, no stars are able to be formed. A
simple Schmidt law γM is added to the systems in order to study its effect on the (in)stability of the
system. For instance system (2.57) becomes
Ṡ = −αS + βMn S + γM,
(2.69)
Ṁ = αrS − βMn S + Q − γM,
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
=
M0
=
Q
,
α (1 − r )
Qα
.
βQ + αγ(1 − r )
(2.70)
(2.71)
Via the characteristic equation of the system, values for a1 and a0 are calculated and equal
βQα
βQ
+
+ γ,
βQ + αγ(1 − r ) α(1 − r )
βQ + αγ(1 − r ) > 0.
a1
= α−
(2.72)
a0
=
(2.73)
Since the system is getting more complicated, it is harder to see under which conditions the value of
a1 will get negative. A python script testing a1 for a large range of parameter values yields no negative
values 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
M0
=
=
Q
,
α (1 − r )
r
− γ + γ2 +
2βQ
α (1−r )
(2.74)
4βQ2
α (1−r )2
,
(2.75)
and a0 and a1 equal
a1
a0
= α − βM02 + 2βM0 S0 + γ,
= 2αβM0 S0 (1 − r ) + αγ(1 − r ) > 0,
(2.76)
(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
2.2. Parameterization of star formation
29
conditions could be found for systems with only a power over gas density and no power over star density in the induced star formation term. However, a description with a power over gas density would be
preferred since it is physically better defensible.
Time-delay According to Korchagin et al. [Korchagin et al., 1988], the presence of time-delay in combination with induced star formation is a sufficient condition for the development of non-linear oscillations
in star formation systems. As discussed in Par. 2.2.2, time-delay introduces a time scale on which stellar
winds take place and return stellar mass to molecular gas. Physically, this makes sense because stellar
winds take place over time scales of O(107 ) yr. For old stars, it takes a delay for the old stellar material
αS to be transferred into molecular gas αrS: it needs to be cooled first (however cooling is not explicitly
parameterized in this system). The system under study with the inclusion of time-delay equals
Ṡ(t) = βS(t) M (t) − αS(t),
Ṁ(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 stellar
material to be transferred into molecular gas. The equilibrium solutions can be calculated as always by
putting Ṡ and Ṁ to zero. This yields
S0
=
M0
=
Q
,
α (1 − r )
α
.
β
(2.79)
(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 perturbations
were always calculated at t = 0, therefore the perturbations yielded eλ(t=0) = 1. Now, they have to be
explicitly taken into account for the calculation of the characteristic equation
−α + βM − λ
βS (2.81)
αreλT − βM − βS − λ = 0,
which yields
λ2 +
βQ
βQ
rβQ λT
λ+
=
e .
α (1 − r )
1−r
1−r
(2.82)
Definition of the following dimensionless quantities
λ̃
=
E
=
T̃
=
α (1 − r )
λ,
βQ
α2 (1 − r )
,
βQ
βQ
T,
α (1 − r )
(2.83)
(2.84)
(2.85)
yields a simplified characteristic equation
λ̃2 + λ̃ + E = rEeλ̃T̃ .
(2.86)
It is not trivial to determine the eigenfrequencies λ̃ for this equation. As we know, if λ̃( T ) = γ( T ) + iω ( T )
has a positive real part γ( T ), the system is instable. If λ̃ has only an imaginary part for a certain critical
value of Tc , it is sure that oscillations will be found in the system. In that case we have λ̃( Tc ) = iω ( Tc )
and this leads to
−ω 2 + iω + E = rEeiω +Tc .
(2.87)
30
Chapter 2. Feedback and instability
Decomposition into real and imaginary parts yields the expressions
sin(ωTc )
=
cos(ωTc )
=
ω
,
rE
1
ω2
1−
.
r
E
(2.88)
(2.89)
Via the Pythagorean trigonometric identity, ω is calculated and equals
1
ω = E− ±
2
2
r
1
− E + E2 r 2 .
4
(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 a
value Tc to exist. This yields constraints for Eq. (2.90)
1
− E + E2 r2 > 0,
4
(2.91)
implicating
E>
1+
√
1 − r2
.
2r2
(2.92)
We also have from Eq. (2.90) that
E>
1
.
2
(2.93)
Combining these expressions we have
E>
1+
√
1
1 − r2
> .
2
2r2
(2.94)
Using Eq. (2.84), the final condition for a critical value Tc to exist and oscillations to occur is
√
α2 (1 − r )
1 + 1 − r2
1
>
> .
2
βQ
2
2r
(2.95)
For values of r < 1, the latter inequality is always satisfied. The first inequality is also easily fulfilled
for values of r close to 1. It is hence important that a large fraction of the stellar mass is returned as
molecular gas to obtain oscillations. Another important quantity is α, the rate at which old stellar material
is transferred to gas needs to be high enough for oscillations to occur. This term balances the system
and without a balance, no equilibrium solutions are present to oscillate around. The denominator βQ
needs to be small enough hence when induced star formation is large, the gas accretion should be small
and 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 star
formation present:
Ṡ(t) = γM(t) − αS(t),
Ṁ(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
Figure 2.7: Plot of the quantity
αγ
( α + γ )2
(1)
31
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
(2.98)
E
=
λ
,
α+γ
αγ
,
( α + γ )2
T̃
= (α + γ) T,
(2.100)
λ̃
=
(2.99)
and this yields the same characteristic equation as Eq. (2.86):
˜
λ̃2 + λ̃ + E = rEeλT̃ .
For this equation, we have obtained the condition
√
αγ
1 + 1 − r2
1
E=
>
> ,
2
( α + γ )2
2r2
(2.101)
(2.102)
for the existence of a critical Tc for which an imaginary frequency exists and hence oscillations occur.
αγ
However, this inequality can never be fulfilled. Indeed, plotting of the quantity (α+
yields values smaller
γ )2
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 formation
combined with time delay.
Korchagin et al. [Korchagin et al., 1988] did very similar calculations, but with a power of n = 2 over the
gas density for spontaneous star formation γM2 . They have obtained equal results and concluded that
induced star formation is necessary (in combination with time delay) to observe oscillatory behavior in
the system.
Three component systems The two component systems up till now did not explicitly account for an important phenomenon in regard to star formation, namely cooling. Indeed, molecular gas is cold while
ejected stellar mass is hot and is hence rather in the atomic phase than in the molecular phase (or even
in 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]
32
Chapter 2. Feedback and instability
and discussed in Par. 2.2.2.
Ṡ = βSMn1 − αS,
Ṁ = − βSMn1 + dMn2 A,
(2.103)
n2
Ȧ = Q + αrS − dM A,
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, and
the coefficients a2 , a1 and a0 equal (as confirmed by our own calculations)
" #
1/n1
α n2 /n1
Q
β
a2 =
d
+
( n1 − n2 ) ,
(2.104)
β
1−r α
"
n2 /n1 #
Qn1 β 1/n1
α
a1 =
α+d
,
(2.105)
1−r α
β
(n2 −1)/n1
α
.
(2.106)
a0 = Qn1 αd
β
The Hurwitz conditions in three dimensions (see Eq. (2.50)) yield the conditions
" n2
#
1
α n1
Q
β n1
0< d
+
( n1 − n2 )
,
β
1−r α
" n2
# n2
1
n2 /n1 α n1
α n1
Q
β n1
(1 − r )αd
< d
+
( n1 − n2 )
α + d αβ
,
β
β
1−r α
(2.107)
(2.108)
for stability. Instability can occur for the case of n1 < n2 , in other words, when the non-linearity in the
cooling 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 formation? We check for the system
Ṡ = γMn1 − αS,
Ṁ = −γMn1 + dMn2 A,
(2.109)
n2
Ȧ = Q + αrS − dM A,
with γMn1 representing spontaneous star formation.
Equilibrium solutions are found to be
S0
=
M0
=
A0
=
Q
,
α (1 − r )
1
n1
Q
,
γ (1 − r )
n1 − n2
n1
γ
Q
.
d γ (1 − r )
(2.110)
(2.111)
(2.112)
Calculation of the characteristic equation yields values for
n1 −1
n2
n1
n1
Q
Q
α+γ
( n1 − n2 ) + d
,
γ (1 − r )
γ (1 − r )
n1 + n2 −1
n1 −1
n2
n1
n1
n1
Q
Q
Q
n1 dγ
+ αγ
(n1 − n2 ) + αd
,
γ (1 − r )
γ (1 − r )
γ (1 − r )
n1 + n2 −1
n1
Q
n1 αγd
(1 − r ).
γ (1 − r )
a2
=
a1
=
a0
=
(2.113)
(2.114)
(2.115)
2.3. Research hypothesis
33
Application of the Hurwitz conditions a2 > 0 and a2 a1 − a3 a0 > 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 spontaneous
star formation. We conclude that instability in this systems is provided by the presence of cooling and
not by the presence of induced star formation.
Conclusion We have found several examples of systems where instability and hence oscillatory behavior can occur.
First of all, including induced star formation with a power over the stellar density βMSn introduces instability 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 formation to take place. Therefore we test different systems with a parameterization of βMn S for induced star
formation.
Including spontaneous star formation in combination with induced star formation in a system will not
increase its chances of being instable.
The presence of time delay is an important phenomenon in regard to instability. If it is combined with the
presence of induced star formation βMn S, instability arises and oscillatory behavior can be found in the
system. A clear condition for the parameter values can be found for which an imaginary frequency exists
at which small perturbations in the system will start to oscillate. Interestingly, when the phenomenon of
time delay is combined with the presence of spontaneous star formation instead of induced star formation, 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 the
atomic gas to molecular gas. It was found that this system is instable when the non-linearity in star formation is smaller than the non-linearity in cooling, and this result is valid for the phenomenon of induced
star 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 in
regard to starbursts are left unanswered, e.g. are starbursts in BCDs cyclic phenomena? Are starbursts
the 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 the
phenomenon of induced star formation. It is shown that the interplay of non-linear feedback mechanisms
and in particular the presence of induced star formation, leads to self-regulation of star formation. Within
certain conditions, a self-regulating star formation system tends to show oscillations and limit cycle behavior.
Is it possible that these oscillations in star formation rate, obtained from taking into account non-linear
feedback 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 in
a 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 effect
on the star formation rate is studied.
34
Chapter 2. Feedback and instability
3
Implementation 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 follow the evolution of a self-gravitating N-body system. In the research group it is used to simulate dwarf
galaxies 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 variety
of phenomena on different space and time scales. Not only isolated (dwarf) galaxies can be modelled
but also interactions, mergers and even full cosmological simulations (for instance the Millennium simulation) 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. For
our 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 lightest dwarf ellipticals. Initially, only atomic gas and dark matter are present. The initial conditions are
cosmologically motivated, meaning that they share properties with small structure formations found from
simulations 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 the
characteristic 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
ρ g (r ) =
1+
ρc
2 ,
(3.1)
r
rc
with ρc the critical density threshold for star formation, see Par. 3.2.1, and rc a scale radius (for more
information, see [Schroyen et al., 2013] in preparation). The initial metallicity equals Z = 0.0001 Z and
expressed 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
Number of gas particles
Number of DM particles
Mg,i
MDM,i
DM profile
Initial gas temperature
Initial metallicity
Feedback efficiency eFB
Softening length e
SF threshold density ρc
Star formation parameter c?
Rotation
E07
100000
100000
262×106 M
1238×106 M
NFW
10000 K
0.0001 Z
0.7
30 pc
100 amu cm−3
0.25
none
Cold dark matter interacts only gravitationally, hence collisionless dark matter (DM) is not described
by hydrodynamical equations. Therefore, the collisionless dark matter particles are treated as point
masses. For the dark matter halo, the Navarro-Frenk-White (NFW) profile is adopted
ρ DM (r ) =
ρs
r
rs
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 characteristic length scale of galaxies, indeed the collision of two stars is a very rare event. Star particles
are collisionless and interact through gravity. The star particles in the simulation do not represent single
stars but are SSP (Single Stellar Population) particles that contain a large amount of stellar mass, representing a population of stars with the same age. The mass distribution of an SSP particle is described
by the initial mass function, see Par. 3.2.2.
Also other parameters as rotation, number of particles (gas and DM), and criteria according to star
formation and feedback are chosen when the initial conditions are generated. An overview of used parameters is given in Table 3.1.
According to [Valcke, 2010], starting from a certain lower limit for number of particles, the accuracy of
the simulation results will not increase with an increasing number of particles. He advises to use at least
100000 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 user
defined number of N bodies, the gravitational force on one body with mass mi equals
N
Fi =
∑−
j =1
j 6 =i
Gmi m j
er ,
r2
(3.3)
with r = |ri − r j |, 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 is
introduced:
N
Fi =
Gmi m j
∑ − ( r + e ) 2 er ,
j =1
j 6 =i
(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 figure
represents the hierarchical tree structure with its nodes and subnodes pointing to the particles. Figure from [Liu and Liu, 2003].
with e the gravitational softening length. GADGET-2 is provided with the possibility to use an adaptive
softening kernel which is at its maximum value when r = 0. However in our simulation, a constant
softening length is used, namely e = 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 N 2 for the calculation of the accelerations in one
time 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 computational
cost 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 octant
contains only one particle, see Figure 3.1. The problem domain exists now out of cubes of different
dimensions, each cube containing one particle. At the root of the hierarchical tree structure, the nodes
with all subnodes point to all particles in the node. To calculate the gravitational force on a target particle
i, a certain node will be responsible for a gravitational force on the particle. When a multipole expansion
of the node is considered a sufficiently accurate contribution to the total gravitational force, the expansion
is used and no further descend into subnodes is necessary. GADGET-2 makes only use of monopole
moments to approximate the gravitational force contribution in a node. For nodes spatially nearby the
target particle, further descend will be necessary to obtain an accurate calculation of the force. The
accuracy can be entirely determined by the user: the further the descend level of the treewalk, the better
the accuracy at the price of a higher computational cost. The set-up of the tree structure is not only
very 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 express
respectively conservation of mass, also known as the continuum equation, see Eq. (3.5a), conservation
of momentum, see Eq. (3.5b) and conservation of energy, see Eq. (3.5c)
dρ
= −ρ∇ · v,
dt
dv
1
= − ∇p + F ,
dt
ρ
de
p
= − ∇ · v.
dt
ρ
(3.5a)
(3.5b)
(3.5c)
38
Chapter 3. Implementation of induced star formation in an N-body/SPH code
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’s
law for N particles. The equation of motion for particle i becomes
N Gm (x − x )
dvi
1
j
i
j
= − ∇ pi − ∑
.
3
dt
ρi
|
x
−
x
|
i
j
j =1
(3.6)
j 6 =i
∇p
Hence in order to solve this equation, for each particle the term ρ i needs to be calculated. SPH codes
i
have a very specific approach for determining the density of a particle. They interprete each particle not
as a point mass but rather as a smeared out distribution of mass that contributes to the density of the
surrounding particles. The density is then given by
N
ρi =
∑ m j Wij ,
(3.7)
j =1
where N is the number of particles that contributes to the density of particle i, m j is the mass of each
contributing particle, and Wij is the smoothing kernel which represents the mass distribution and has
1
dimensions length
3 . This method of calculating densities and other quantities in a discrite way is called
the particle approximation.
The smoothing kernel depends on the smoothing length h and on the distance between particle i and j
hence
Wij = W (|xi − x j |, h).
(3.8)
The smoothing length h determines the length scale on which the density distribution Wij is smeared out
and hence influences other particles. In GADGET-2, the default smoothing Kernel function is the cubic
spline kernel, shown in Figure 3.2, given by
(
1 + 6u2 (u − 1), 0 ≤ u < 12
8
Wij =
(3.9)
1
πh3 2(1 − u)3 ,
2 ≤u≤1
|x −x |
with u = i h j . A great advantage of SPH is that the derivative of a certain quantity f (x) can be
calculated in the particle approximation as follows
N
mj
f ( x j ) · ∇W ( x − x j , h ) ,
ρ
j =1 j
< ∇ · f (x) >= − ∑
(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, support
domains are spherical) and its scope is determined by the smoothing length h. Therefore, the determination of h should be handled with thought: when h is too small, the support domain of the particles will
be too small and quantities as pressure and density will be calculated inaccurately. On the other hand
when h is too large, local effects will be smoothened out. For a 3D code, an ideal number of neighboring
particles is around 57 [Liu and Liu, 2003]. To obtain a more or less constant number of neighbors, an
adaptive smoothing length is necessary, hence each particle i has its own smoothing length hi . It is clear
that 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 update
the smoothing length along with variation of density
1 h dρ
h
dh
=−
= ∇ · v,
dt
d ρ dt
d
(3.11)
with d the number of dimensions, in this case 3. Indeed, with increasing density, dh
dt 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 a
numerical method to solve the following equation [Springel, 2005]
4π 3
h ρ = Nsph m,
3 i i
(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 density
of particle i is calculated, then the influence of particle j on particle i is determined by W (|xi − x j |, hi )
whereas the influence of particle i on particle j is determined by W (|x j − xi |, h j ). Since hi 6= h j , the
contribution is not symmetrical. This violates basic principles of physics and should be solved by a
symmetrization procedure to obtain a new value for the smoothing length hij , for instance the average of
both particles smoothing lengths hij = 21 (hi + h j ) or as implemented in GADGET-2 , the maximum value
hij = max(hi , h j ).
In fact, all equations need to be symmetric for the exchange of two particles. The force on particle i from
particle j has to be equal to the force on particle j resulting from particle i. A symmetrization procedure
is applied to Eq. (3.5b) and results into [Springel, 2005]
"
#
N
Pj
dvi
Pi
= − ∑ m j f i 2 ∇i Wij (hi ) + f j ∇ j Wij (h j ) ,
(3.13)
dt
ρjr
ρi
j =1
making use of the particle approximation and Eq. (3.10). The factors f i are defined by
fi =
1+
hi ∂ρi
3ρi ∂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 to
determine the neighboring particles contributing to this density. A cubic search domain around particle i
is defined with sides 2hi . Now, when the treewalk is started at the root of the tree, it is checked whether
the node has an overlap with the search domain, see particle i in Figure 3.1. If not, the nodes do not
have to be opened at all. If so, the treewalk is continued to a subnode and the process is repeated. With
this method, the treewalk is restricited to the local area of particle i. The number of computations with
this method scales with O( N log N ).
40
Chapter 3. Implementation of induced star formation in an N-body/SPH code
Figure 3.3: Panel (a) illustrates the scatter approach where the density of particle i is calculated by contributions of the support domains of
the neighboring particles with their corresponding smoothing lengths h j . Panel (b) represents the gather approach: the density of particle i is
determined by checking which particles are situated within the support domain of particle i determined by its own smoothing length hi . In the
case 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 within
the smoothing length hi of the particle for the number of particles that are situated within hi , this is called
the gather approach. Another method is the scatter approach, which checks whether the particle i is
situated in the support domain of its surrounding particles. The difference between both approaches is
shown 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 describing the hydrodynamics Eq. (3.5a), (3.5b) and (3.5c), need to be integrated simultaneously. For systems
as galaxies, particles move on many different time scales. For example, a gas particle in a low-density
region in the outer borders of a galaxy will not change its position and other properties as fast as a
particle 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 chosen
is that it behaves more stable in combination with individual time-steps per particle than for instance a
fourth 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 this
case, for the simulation of dwarf galaxies, physical phenomena as cooling, feedback and star formation
have 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 which
obey 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
ρg
T
≤ 0,
≥ ρc = 100 amu cm−3 ,
≤ Tc = 15000 K.
(3.15)
(3.16)
(3.17)
A critical temperature of Tc = 15000 K seems rather high in regard to the discussion of star formation
in cool molecular clouds (see Par. 2.1). In fact, as will be shown later, almost all particles satisfy
this temperature criterium and the selection of eligible gas particles for the formation of stars is almost
entirely based on the gas density ρi of the particle. When a particle has collapsed to a density of
ρi = 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, this
approach is unfavorable since it depends on the number of particles in the code and hence the resolution
of the simulation. We recall that the Jeans criterium is satisfied when the dynamical time τ is smaller
than the sound crossing time ts = chs . Here, the free-fall time tFF discussed in Par. 2.1.2 is equal to the
dynamical time τ discussed in Par. 2.2.1. In this case it is defined as
τ= p
1
.
4πGρ g
(3.18)
For the critical density q
ρc , the dynamical time is τ = 2.4 Myr. The sound crossing time is calculated with
the sound speed cs =
= 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 = chs =∼ 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 further
increase in density and decrease in temperature, the Jeans criterium will be definitely satisfied (because
τ ∝ ρ−1/2 and ts ∝ T 1/2 ). This result however, is only valid for our specific simulations with a resolution of
100000 gas particles. For other resolutions, different values for hi will be present (because hi is adaptive
in order to have a more or less constant number of neighbors) and this will yield different values for the
sound crossing time ts . In conclusion, we should not interprete the SFC in terms of the Jeans instability
criterium, 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 a
star or not. To make a prediction of this star formation probability, the Kennicutt-Schmidt law is used,
discussed in Par. 2.2.1:
γkTc
m
dρ g
ρg
dρs
=−
= c? ,
dt
dt
τ
(3.19)
with τ the dynamical time and a power of n = 1 over the gas density was chosen. This leads to the
analytical solution of
t
ρ g (t) = ρ g,0 exp −c?
,
(3.20)
τ
where ρ g,0 is the initial gas mass. The probability that the elected gas particle will form a star is equal to
the ratio of the “disappeared” gas mass to the total gas mass, assuming that all “disappeared” gas has
been transferred to stars, along Eq. (3.19). This probability p? is then given by
p? = 1 −
ρg
∆ρ g
=−
,
ρ g,0
ρ g,0
(3.21)
with ∆ρ g = ρ g − ρ g,0 .
For one timestep ∆t in the code, the solution for p? is given by
∆t
p? = 1 − exp −c?
τ
.
(3.22)
42
Chapter 3. Implementation of induced star formation in an N-body/SPH code
The integration step ∆t in GADGET-2 equals the systems (adaptive) timestep which is of the order of
magnitude ∆t ∼ O(104 ) yr in star formation regions, while the dynamical time tc is of the order of
tc ∼ O(106 ) yr. This implies that ∆t/τ ∼ O(10−2 ) and thus that Eq. (3.22) can be approximated by
p? = c?
∆t
.
τ
(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 in
the [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 the
E-models, the total initial gas mass (see Table ??) is divided by the number of gas particles (in this case
100000) 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 mass
within 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:
Z mu
ml
(3.25)
Φ(m)dm = 1.
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 < B > of an SSP is then given by [Valcke, 2010]
< B >= R mu
ml
MSSP
·
mΦ(m)dm
Z m B,u
m B,l
B(m)Φ(m)dm,
(3.26)
with m B,u and m B,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 massive
OB 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 used
and this yields
Etot,SNII = ESNII · R 60 M
MSSP
m0.1 M
mΦ(m)dm
·
Z 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 event
ESNII = 1051 erg assumed to be mass independent. The result of the integral equals [Valcke, 2010]
1
Etot,SNII = 7.31 · 10−3 MSSP ESNII M−
.
(3.28)
The calculation for Etot,SW is identical, replacing ESNII with ESW = 1050 erg.
A feedback efficiency eFB is defined and set to 0.7. This feedback efficiency expresses the efficiency at
which the ISM absorbs the emitted energy, hence the actual energy the ISM absorbs is only 70% of the
total energy release. The feedback efficiency eFB 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 eFB on the SFR was studied by Cloet-Osselaer et al. [Cloet-Osselaer et al., 2012], and
the relation between eFB and the density threshold ρc for SF was investigated. It was concluded that
a degeneracy is present between the two parameters. Several models were in good agreement with
the observations, for instance a model with parameters ρc = 6 amu cm−1 , eFB = 0.7 and one with
ρc = 50 amu cm−1 , eFB = 0.9. This arises the question whether the value of eFB = 0.7 for a critical
density threshold ρc = 100 amu cm−1 as used in our simulations is a good choice. Furthermore, since
feedback is an important parameter in regard to induced star formation and self-regulation, it could be
interesting to study the effect of different values of the feedback efficiency when induced star formation
is 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 the
system.
If the feedback efficiency is too small (for instance for eFB ∼ 0.1), an insufficient amount of energy is
released in the ISM to locally blow the gas apart. Therefore, the simulated galaxies will have continuous
star formation rates that are too high. There is no source present for the gas to be blown apart and/or
heated and self-regulation is eliminated.
The total energy release for SNe type Ia is calculated analogously to Eq. (3.28), but is multiplied with a
factor 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 entire
lifetimes of O(107 ) yr and moderate mass stars emit stellar winds during the T-Tauri stage which duration
is of the same order of magnitude O(107 ) yr. The energy release of stellar winds to the ISM e FB · Etot,SW
is 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 when
massive stars have reached their life ending. This occurs between O(106 )-O(107 ) yr. The precise time
ranges in which the feedback energy is released, were calculated using a formula for the amount of time
stars 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 feedback
and 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 timedelay is included in the simulations: one of the aspects that give rise to instability and the presence of
oscillations 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 point
masses and are not described by a smoothing kernel; however they inherit the smoothing length of their
progenitory gas particle). A gas particle j will recieve an amount ∆Ej from a star i that is in the proces of
giving feedback. The quantity ∆Ej equals [Valcke, 2010]
∆Ej =
Em j W (|~ri − ~r j |, hs )
∑kN=1
mk W (|~ri − ~r j |, hs )
.
(3.29)
During stellar feedback, not only energy is released but also a certain amount of mass is returned to the
ISM in the form of metals. The release of metals in the ISM is important since it has a major influence
on 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 the
SSP particle, determined by the IMF, will give rise to feedback and therefore, the SSP particle will
lose a fraction of its mass. In the E07 model, the SSP particles lose 11% of their mass by the means
of feedback. After the feedback has finished, the mass of the SSP particle remains constant. The
44
Chapter 3. Implementation of induced star formation in an N-body/SPH code
Figure 3.4: Cooling curves in function of temperature for different metallicities Z. The cooling rate Λ is calculated from interpolation between
tabulated 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 as
luminosity 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]:
de
p
Γ−Λ
= − ∇·v+
,
dt
ρ
ρ
(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 in
the same timestep.
The code provides cooling for a temperature range of 10 − 109 K. The chosen value Λ for one timestep is
interpolated from tabulated values for cooling, depending on temperature and metallicity Z. The cooling
curve 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 the
cooling process. Between temperatures of 104 and 107 K, the particles change their state from neutral
to ionized. For a certain temperature range depending on the type of element, there is a contribution
of recombination and line cooling which gives rise to peaks in the cooling curve. Below 104 K, no ions
and free electrons are present and only line cooling can contribute to the cooling process. Moreover, in
the temperature range of molecular clouds, hydrogen atoms cannot be excitated by collisions anymore
and do not contribute to the cooling curve. In this temperature range, cooling proceeds mainly through
molecular line transitions, as discussed in Par. 2.1.3.
3.3. Implementation of induced star formation
3.2.5
45
Self-regulation
As a consequence of the implementation of stellar feedback (with time-delay) and cooling, the star formation 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 favorable
for star formation and the Kennicutt-Schmidt law decides how much stars are being formed, proportional
to the gas density.
Once new stars are born, they initiate stellar feedback for a few Myr to Gyr. Energy is released which
causes the interstellar gas to heat and to be blown apart. At the edges of the blown bubbles, the gas is
pressed together and these high densities give rise to induced star formation. When a lot of stars are
formed, 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 proceed
faster (see Figure 3.4). The cycle is complete now and the proces can start all over. This is an illustration of how self-regulation can lead to oscillations in star formation. One could hence wonder why
induced star formation still needs to be included in the star formation law if the phenomenon is already
present in the code.
In simulations using a Kennicutt-Schmidt law, supernova explosions will indeed give rise to high densities, but the prediction of the star formation rate p? only accounts for high densities. It does not reckon
with nearby (young) stars giving rise to stellar feedback. If we want to implement induced star formation
locally, the chances of forming a star should be higher for particles subject to feedback effects of nearby
stars. 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 the
star 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
3.3.1
Implementation of induced star formation
Star formation law
Now we use results of the discussion in Par. 2.2.4 where conditions for instability and oscillatory behavior were set up, to choose a parameterization of induced star formation which we will implement in the
new star formation law. However, it is hard to simply make use of the drawn conclusions, since phenomena as feedback and cooling are explicitely included in the code and will have their influence, without
a specific parameterization of this phenomena present. For instance, the phenomenon of cooling was
studied a the three component star formation system with a (simple) parameterization of dMn A, with M
the 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 used
parameterization of cooling dMn A even a remote description of the implemented cooling curves?
1
(see Eq. (2.30)) hence
Looking at dMn A dimensionally, we see that dMn should have dimensions of time
it could be interpreted as a cooling rate C ( T, n), with n the number density of the gas. If we look at the
cooling curve Λ( T ) in Figure 3.4, it is clear that it depends only weakly on temperature in the range of
102 − 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 and
a changing temperature (along T ∝ n1 ), we have that C (n, T ) ∝ n2 because cooling is not temperature
dependent 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 of
dMn 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 Hurwitz conditions discussed in Par. 2.2.4 yield that oscillations can occur when the non linearity in cloud
formation (cooling) is stronger than in star formation. Hence if the cooling is indeed parameterized by
dM2 A, than the power over the star formation law (Kennicutt-Schmidt law ) should be smaller than two.
ρ
dρ
Currently, the Kennicutt-Schmidt law is implemented in the code as dts = c? τg . We see that the Hurwitz
46
Chapter 3. Implementation of induced star formation in an N-body/SPH code
condition for stability is already violated in this system because the power over the star formation law is
one 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 star
formation law to account for induced star formation if time delay is present in the code. We know that
time delay is implemented, see Par. 3.2.3. We have shown in Par. 2.2.4 that instability arises in a two
component system when induced star formation is accounted for, and no oscillations occur when only
spontaneous star formation is implemented. In this regard, we can use the parameterization βMn S of
induced star formation and include this term in the currently used star formation law
ρg
dρ g
dρs
=−
= c?,1 + c?,2 ρng ρs ,
dt
dt
τ
(3.31)
with the induced star formation term c?,2 ρng ρs 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 and
n. Do oscillations occur with this parameterization of induced star formation and for which parameter
values? 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 whether
the 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 surroundings
of the target gas particle. Methods to calculate the gas density ρ g are already present in the code for
numerous reasons, including for the integration of the originally used Kennicutt-Schmidt law. We can
work 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 particles
have to be known. This is done by a treewalk algorithm, described in Par. 3.1.4. However the size
of the box needs to be adjusted. Indeed, we do not look for stars within the smoothing length of the
target particle, but within the characteristic length for induced star formation to have an influence. From
the 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 treenode
octant and if further descend is necessary. In this way, the number of total star neighbor particles is
counted and a list of all neighboring star particles is kept up. However, this algorithm yields all neighbors
in a box with side 2 · radius, but since we are looking for neighbors within the characteristic radius, only
the neighbor particles j with a distance |r j − ri | < radius are selected to contribute to the star density
ρs,i of particle i.
A loop is run over the list of neighboring star particles in order to calculate their contribution to ρs,i . The
contribution of a star particle j to ρs,i equals
(ρs,i ) j = m j W (|x j − 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 contribute
to the star density at all. For distances of 10 − 30 pc as we proposed for the length scale of induced
star formation (see Par. 2.1.4), the parameter u in the smoothing kernel function is 0.2 − 0.6 and the
contribution 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 longer
the 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-Kutta
method is chosen to perform the numerical integration, since this algorithm offers a high level of accuracy
and is easy to implement.
First of all, a driver function g(ρs , ρ g , τ ) is defined which is equal to the star formation law
g(ρs , ρ g , τ ) = −c?,1
ρg
− c?,2 ρng 1 ρs .
τ
(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 a
prediction for ρs,i (t + ∆t), ∆t is subdivided in 100 smaller integration steps ts and a loop over this smaller
timesteps is started. The values of ρ g,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
k2
k3
k4
= t s g ( ρ g , ρ s , τ ),
k
k
= t s g ( ρ g + 1 , ρ s − 1 , τ ),
2
2
k2
k2
= t s g ( ρ g + , ρ s − , τ ),
2
2
= t s g ( ρ g + k 4 , ρ s − k 4 , τ ).
(3.34)
(3.35)
(3.36)
(3.37)
dρ
dρ
Since we know that dts = − dtg , the same constants k1 to k4 can be used for the decrease/increase in
gas/star density but with an opposite sign.
After one itegration step ts , the gas and star density become
ρ g (t + ts ) = ρ g (t) +
k1
k
k
k
+ 2 + 3 + 4,
6
3
3
6
(3.38)
ρs (t + ts ) = ρs (t) −
k
k
k
k1
− 2 − 3 − 4.
6
3
3
6
(3.39)
and
After 100 integration steps, values for ρ g (t + ∆t) and ρs (t + ∆t) are obtained. The latter quantity is
not 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 gas
particles 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 −
ρ g,i (t + ∆t)
.
ρ g,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 efficient
approximation was considered to obtain p? by a linear approximation. Since
p? = 1 −
ρ g,i (t + ∆t)
∆ρ g
=−
,
ρ g,i (t)
ρ g (t)
(3.41)
and if we approximate ∆ρ g by
ρg
n
∆ρ g ' −c?,1 − c?,2 ρ g 1 ρs ∆t,
τ
a value for p? of
1
n1 −1
p? ' c?,1 + c?,2 ρ g ρs ∆t,
τ
(3.42)
(3.43)
48
Chapter 3. Implementation of induced star formation in an N-body/SPH code
is obtained.
The Runge-Kutta method and the linear approximation method were used in simulations with the exact
same 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 indication
for the implementation of the Runge-Kutta method to be correct too. One would hence naively assume
that 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 for
the low mass models, the results were not different compared to the same models without induced
star formation. Actually, some simulations where the linear approximation was applied did not show
any difference in the star formation rate with a simulation where no induced star formation law was
implemented and the original Kennicutt-Schmidt law was used. This was the case for light galaxies with
a very small number of stars.
This effect is completely logical. Light galaxies form less stars and in the case of a very small number
of stars, ρs (t) is (mostly) zero and the second term in Eq. (3.43) has no effect at all. In that case, only
the Kennicutt-Schmidt law is accounted for. The problem can be overcome by simulating more massive
galaxies that have more star formation or to adjust the search radius in which star particles contribute to
the star density of a gas particle.
We decided to simulate more heavy galaxies (E07 instead of E05) but the Runge-Kutta method was still
preferred above the linear approximation based on physical arguments. Because, even when the star
density ρs,i (t) is zero for a certain particle, a prediction for the star formation rate should include induced
star formation for the stars that are formed in the time span of the prediction. This will never be possible
with the linear approximation, as it performs no integration over time.
To overcome the less computational efficiency of the fourth order Runge-Kutta, other faster numerical
algorithms 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 the
accuracy 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 want
to 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 of
the target gas particle, the number of its star particle neighbors calculated by the treewalk algorithm and
their ID numbers, the search radius, the masses, the smoothing kernel attribution to the star density, the
star density itself, etc.
When the simulation is run, every 0.1 Gyr a snapshot is given as output. We can visualize and analyze
these snapshots with Hyplot, which is a program specifically designed [Valcke, 2010] to interpret the
output of GADGET-2. For every snapshot, by means of particle IDs, loads of information can be checked
on 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 hyplot
whether these two particles are confirmed to be seperated by a distance less than 50 pc. Checking the
mass of the star particle, the smoothing kernel contribution can be verified by hand and all contributions
are put together in the quantity ρs,i .
Of course, the proces of verifying by hand has been preformed numerous times on several simulations
and different snapshots for different particle IDs. However, it is only a random sample that has been
checked. It cannot be used as hard proof that the method is correct. In conclusion, it is very likely that
the 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 can
start simulations of dwarf galaxies. The results are discussed in Chap. 4.
4
Results and discussion
4.1
4.1.1
Results
Reference dwarf galaxy
In Chapter 3, we have introduced a new star formation law
dρ g
ρg
dρs
=−
= c?,1 + c?,2 ρng ρs ,
dt
dt
τ
(4.1)
in the code that accounts for the phenomenon of induced star formation. To study its influence, a reference simulation is needed, where the original Kennicutt-Schmidt law regulates star formation. When the
initial conditions of this reference simulation are exactly equal to a simulation using a new star formation
law, the influence of the implemented induced star formation can be studied. Table 3.1 lists the used
parameters 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 formation
histories of different simulations. Therefore, we should understand the star formation history of the reference galaxy, shown in the top panel of Figure 4.1. The depicted SFH can be explained by means of the
SFC of Eqs. (3.15). The number of particles exceeding the critical density of star formation ρc is called
Ndens and is also depicted in Figure 4.1. Also Nentr and Nconv are shown, respectively the number
of particles with a temperature T < 15000 K satisfying the temperature criterium, and the number of
particles satisfying ∇ · v < 0.
The most striking feature in the SFH of this dwarf galaxy is that almost no star formation takes place after
3 Gyr. Initially between 0 and 3 Gyr, the star formation rate is rather high and irregular peaks show up
in the SFH. These peaks are due to the collapse of the initial state, a sphere with density distribution of
Eq. (3.1) and a constant temperature of T = 10000 K. When the initial gas sphere collapses, the critical
density threshold ρc for star formation is exceeded by a sufficient amount of gas particles, see Ndens in
Figure 4.1. Indeed, the density criterium is the most severe criterium for the onset of star formation and
decides 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 almost
always satisfied for all gas particles, which is logical when the initial gas temperature is 10000 K for all
gas 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 necessary in order to increase the density preliminary to star formation. The new born stars give rise to
49
Chapter 4. Results and discussion
0.010
0.008
0.006
0.004
0.002
0.000
5
4
3
2
1
0
80
60
40
20
0
0.4
0.3
0.2
0.1
0.0
80
60
40
20
0
99
98
97
96
95
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
reference sim1006
Ntot [%]
Nentr [%]
Nconv [%]
Ndens [%]
MgasBO[%]
6
Mstar [10 M ]
SFR[M /yr]
50
0
2
4
6
time[Gyr]
8
10
12
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 a
distance > 30 kpc), number of gas particles Ndens exceeding ρc , number of gas particles Nconv satisfying ∇ · v < 0, number of gas particles
Nentr with T < 15000K and finally number of gas particles Ntot satisfying all three SFC. These properties belong to the reference simulation
sim1006 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 the
galaxy is collapsing. In (b) densities are high and some fragmentation is observable. In this high density fragmentated subclouds, particles are
eligble 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 stars
show 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 cloud
in (d) where the SFR has entirely dropped to zero.
52
Chapter 4. Results and discussion
(a)
(b)
Figure 4.3: Density evolution projected on in the xy plane. The reference simulation sim1006 is followed at different times indicated in units
of Gyr. Recently formed stars with ages < 10 Myr are shown as purple dots. In (a) the interstellar gas has collapsed to high densities and
star formation is possible in the fragmentated subclouds. Feedback effects are present in (b) giving rise to high densities near the edges of the
blown 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 snapshot after 0.1 Gyr. The effect of the stellar feedback is immediately shown in the decrease of the density
and a decrease in Nconv . Feedback blows bubbles in the ISM, as discussed in Par. 2.1.4. We see that
the number of particles Ndens satisfying the density criterium and Nconv satisfying the convergerence
criterium decreases after a peak in the SFR.
For simulations like ours with a high feedback efficiency (e = 0.7) and a severe density threshold criterium (100 amu/cm3 ), periods with a continuously high SFR extending over more than a few tenths of
Gyr are rare. We can also clearly follow the collapse before and the feedback effects after a peak in the
SFR 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 why
does 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 by
more 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 ISM
to decrease in density in such amount that star formation becomes very hard. The intense peaks of star
formation at times of 2 and 2.8 Gyr (see the SFH in Figure 4.1). The effect of feedback on the density of
the 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 quantity
represents the amount of gas situated at very large distances from the galaxy center, out of a box with
sides of 30 kpc. Between 4 − 12 Gyr, 80% of the gas is situated at very large distances from the galaxy
center. This also makes it more plausable that high densities in the galaxy center are hard to achieve
and star formation stops.
An interesting remark is that the phenomenon of induced star formation is already present in the reference simulation in a self consistent way. The simulated galaxies use a normal Kennicutt-Schmidt law
without the induced star formation term. However since stellar feedback is present in the code, stars
are formed at the edges of bubbles in the ISM blown by supernovae and stellar winds. As discussed in
Par. 2.1.4, these bubbles give rise to a high density cooled post shock layer at the edge of the bubbles
and 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 when
these peaks show hints of periodic behavior and have a reasonably high amplitude.
n=1
sim1042
c?,2 = 104
n = 1.5
sim1020
c?,2 = 3.1 · 104
n=2
sim1019
c?,2 = 105
n = 2.5
sim1015
c?,2 = 3.1 · 105
n=3
sim1018
c?,2 = 106
n = 3.5
sim1040
c?,2 = 3.1 · 106
n=4
sim1041
c?,2 = 107
plotted.
Now, we have gained a good understanding of the behavior of the reference galaxy and the effect of the
extra 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 .
1
We recall that c?,2 has units of time×density
n . 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 Gyr
and typical gas densities of the order of 0.1 × 1010 M Gpc−3 , more or less equal to the critical density
threshold for star formation. These characteristic values are used for the calculation of c?,2 for different
values of n. The results can be found in Table 4.1. Simulations with the exact same initial conditions
as the reference galaxy sim1006 (see Table 3.1) have been undertaken for different values of n with the
corresponding 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 component system with cooling included, the value of the power should be more or less 1 < n < 2. If we
consider the system with time delay, n can equal 1 or any other higher value in regard to obtaining oscillatory behavior. Other arguments are that in literature, no powers higher than n = 4 have been found
for the Schmidt law, hence this would be improbable to be the case for the parameterization of induced
star formation.
From Table 4.1, we see that several simulations indeed show oscillations in the SFR in comparison to the
reference 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 to
the SFHs of the simulations of Table 4.1 shown in Figure 4.4. For instance for sim1018, no peaks or
oscillations in the SFR occur in the system. Between 4 − 12 Gyr, there is some very modest and slow
star formation compared to the absolute absence of star formation of reference simulation sim1006, but
this is considered irrelevant in regard to the oscillatory behavior we try to obtain. Simulations with this
type of behavior are indicated in normal font in Table 4.1 and are later referred to as non-oscillating
galaxies.
Then for sim1015, a clear increase in the SFR rate is present but describing these peaks as obvious oscillations 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 font
in 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 definitely significant now. There seems to be a hint of periodic behavior but it is once more hard to define
at sight. For sim1019, we conclude that oscillatory behavior is present. The amplitude of the peaks in
the 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 SFR
with a significant amplitude (comparable to the amplitude of the peaks in the SFR of the reference simulation sim1006) and with a hint of periodic behavior are concluded to have oscillatory behavior. The
simulation numbers of these galaxies are indicated in bold and red in tables and are referred to as oscillating galaxies.
Further conclusions are that in all simulations where induced star formation is included, the total cumula-
54
Chapter 4. Results and discussion
1015
1006
Mstar[106 M ]
0
2
4
6
time[Gyr]
8
10
1020
12
0.010
0.008
0.006
0.004
0.002
0.000
8
7
6
5
4
3
2
1
0
0
2
4
6
time[Gyr]
8
10
1019
12
1006
6
Mstar[106 M ]
1006
Mstar[10 M ]
SFR[M /yr]
0.010
0.008
0.006
0.004
0.002
0.000
7
6
5
4
3
2
1
0
0.010
0.008
0.006
0.004
0.002
0.000
7
6
5
4
3
2
1
0
SFR[M /yr]
6
Mstar[10 M ]
1006
SFR[M /yr]
1018
SFR[M /yr]
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
7
6
5
4
3
2
1
0
0
2
4
6
time[Gyr]
8
10
12
0
2
4
6
time[Gyr]
8
10
12
Figure 4.4: Star formation histories and total cumulative star mass for different simulations with induced star formation included in the star
formation law. Parameters for the simulations can be found in Table 4.1. The reference simulation sim1006 is always indicated for which a
Kennicutt-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 in
Table 3.1. The simulation numbers are indicated in bold if small peaks in the SFR occur (where sim1006 did have a neglegible SFR), and in
red when these peaks show hints of periodic behavior and have a reasonably high amplitude.
c?,2
c?,2
c?,2
c?,2
c?,2
c?,2
107
=
= 106
= 105
= 104
= 102
= 10−2
n=1
sim1070
sim1043
sim1044
sim1042
sim1045
n = 1.5
sim1054
sim1021
sim1032
sim1048
sim1024
sim1031
n=2
sim1056
sim1022
sim1019
sim1049
sim1025
sim1030
n = 2.5
sim1055
sim1023
sim1034
sim1050
sim1026
sim 1029
n=3
sim1071
sim1018
sim1035
sim1057
sim1027
sim1028
n = 3.5
sim1072
sim1038
sim1036
sim1052
sim1046
n=4
sim1041
sim1039
sim1037
sim1053
sim1047
tive star mass is higher at the end time of the simulations than in the reference simulation. It also seems
to be that in simulations where peaks in the star formation occur, the initial star formation between 0 − 3
Gyr 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 n
and the qualitative behavior of the dwarf galaxy. Since we have found different types of qualitative behavior, it is necessary to study the influence of both parameters n and c?,2 introduced via Eq. 4.1 for a
large 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 types
of 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 comparable or larger heights than the peaks of the reference simulation sim1006 in the first 3 Gyr. In most of
the cases, this results into a non-existing to very low SFR in the later stages of its evolution. This type of
behavior 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 behavior
similar to the behavior shown in Figure 4.4(b) of simulation sim1015. We find modest peaks in the star
formation 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 mostly
lower compared to the reference galaxy and no extreme peaks are present in the first 3 Gyr. This results
into oscillatory behavior in the SFR with a significant yet not striking amplitude. Indeed, the amplitude
of the peaks in the SFR of even the most successful simulations is never higher than the maximal amplitude of the peaks in the SFR of the reference galaxy in the first 3 Gyr. On the other hand, timing is
important and a difference in the SFR from 10−5 M /yr to more than 10−3 M /yr at the same time in
the 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 compare
star formation histories because the axes are put on the same scale. The three columns of Figure 4.5
indicate the three types of qualitative behavior discussed above. On the other hand, one could also
interprete the figure continuously from top to bottom and left to right. Indeed, it is hard to decide when
oscillations do occur or when absolutely no oscillatory behavior is observed. While changing parameters, 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 it
could be a coincidence. No trend is found towards higher or lower values of n, at least not by the looks
of Table 4.2.
For the parameter of c?,2 it seems that very small values 10−2 will not lead to any oscillatory behavior. 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,
ρ g ∼ 0.1 × 1010 M Gpc−3 in star formation regions and τ ∼ 10−3 Gyr. This yields an order of magnitude
for spontaneous star formation of about 1 × 1010 M Gyr−1 Gpc−3 . For the induced star formation term,
SFR[M /yr]
SFR[M /yr]
SFR[M /yr]
SFR[M /yr]
56
Chapter 4. Results and discussion
0.012
0.010
0.008
0.006
0.004
0.002
0.000
0.010
0.008
0.006
0.004
0.002
0.000
0.010
0.008
0.006
0.004
0.002
0.000
0.010
0.008
0.006
0.004
0.002
0.000
0
2
4
6
8
time[Gyr]
1006
1054
1034
1070
1045
1048
1036
1037
1046
1052
1040
1056
1019
10
0
2
4
6
8
time[Gyr]
10
0
2
4
6
8
time[Gyr]
10
12
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 in
Table 4.2. Top left, the reference galaxy sim1006 is plotted with no induced star formation included. The figure can be interpreted in three
columns to differ between the three qualitative types of behavior. In the left column, the non-oscillating galaxies are shown with no peaks
and/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 periodic
behavior. These are called the half-oscillating galaxies. The right column collects simulations which have been most successsfull in regard to
oscillatory behavior: clear peaks in the SFR are present and oscillatory behavior is found. The figure can also be interpreted continuously from
top to bottom and left to right.
making use of c?,2 = 10−2 (1010 × M )−n Gpc3n Gyr−1 , ρ g ∼ 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 = 1
and even smaller for n > 1. The terms for spontaneous star formation and for induced star formation are
hence different by 7 orders of magnitude which is apparently too large a difference to obtain an effect
of 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 to
be 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 four
component star formation system of Bodifee and De Loore [Bodifee and De Loore, 1985], discussed in
Par. 2.2.2, it was found that the coefficient K5 that regulates induced star formation in the same way as
c?,2 , has a great influence on the systems behavior. Small values for K5 yield damped oscillations, larger
values yield oscillations behaving like a limit cycle, and too large values for K5 result in a stationary
solution. With some wishfull thinking, we could say that this is somehow similar to the three types of
qualitative behavior, shown in Figure 4.5. For simulations in the left column of Figure 4.5, one could say
that 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 is
found 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 trend
towards larger values of c?,2 . Therefore, we would rather conclude that the systems qualitative behavior
4.1. Results
57
1058
1019
Mstar[106 M ]
SFR[M /yr]
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
8
7
6
5
4
3
2
1
0
0
2
4
6
time[Gyr]
8
10
12
Figure 4.6: Star formation histories of sim1058 and sim1019. These two simulations have the exact same initial conditions and the exact same
parameter 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 gas
particle 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 gas
particle will eventually become a star (see Par. 3.2.1), the qualitative behavior of the system can also
change. This statement is illustrated by Figure 4.6. The qualitative behavior of the two simulations is
entirely different while their parameter values and initial conditions are exactly the same. This proofs
that the star formation system implemented in the code is chaotical: it is nearly impossible to generally
relate qualitative behavior to parameter values. As long as c?,2 is not too small, oscillatory behavior can
occur at any parameter values. Therefore we conclude that a degeneracy is found between the behavior
of the SFR and the parameter values. Multiple combinations of parameter values lead to the same types
of qualitative behavior.
In order to further study the change in the systems behavior with a change of parameter values, we zoom
in on simulation 1019, which we consider as a simulation with oscillatory behavior present in the SFR
and 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 similar
qualitative behavior is found for similar parameter values.
n = 1.9
c?,2
c?,2
c?,2
c?,2
c?,2
= 1.2 · 105
= 1.1 · 105
= 105
= 9 · 104
= 8 · 104
sim1064
n=2
sim1082
sim1083
sim1019
sim1061
sim1063
n = 2.1
sim1065
small change in the value of n will yield entirely different qualitative behavior, while small changes in the
value of c?,2 yield more or less similar qualitative behavior. This suggests that the choice for parameter n
is completely random, while for a given simulation and a given n, probably some more successful values
and less successful values for c?,2 can be found, however one can never predict the type of behavior
from the parameter values.
58
Chapter 4. Results and discussion
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 )−2 Gpc6 Gyr−1 .
Correlation We also wondered if there would be any correlation between successful values of c?,2 and
for n. Table 4.2 does not suggest a correlation, however theoretically, we would expect one as we explain
now.
Assume that for a simulation with certain initial conditions, we have found oscillatory behavior for a
successful value of c?,2,osc and nosc . Looking at Eq. (4.1), this yields a certain contribution CISF =
c?,2,osc ρng osc ρs to the star formation law. In fact, if this contribution of induced star formation CISF would
remain exactly equal for different values of c?,2,osc and nosc , the same simulation result with oscillatory
behavior would be obtained. If we assume that ρ g and ρs are constants (this is not a too far fetched
assumption, especially since ρ g ∼ 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
CISF
c?,2,osc ρs
log ρ g
.
(4.2)
If for a successful simulation, for instance sim1019, orders of magnitude in CISF are inserted, we obtain
the 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 on
this 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 parameter
values or any other change. Indeed, for the case of sim1019, the extra term of induced star formation in
Eq. (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 the
SFR. Therefore, the assumption that ρs and ρ g are constants, will not be valid because small differences
in ρs and ρ g of less than 1% can have a major influence. At most we can use this expected correlation
as a guideline to choose parameter values that have a chance of being successful for a certain initial
conditions.
4.1.3
Oscillations
Up till now, we have defined oscillatory behavior and the presence of oscillations at sight. It would be
nice 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 determine
at 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
Z ∞
−∞
F (iω )eiωt dω,
(4.4)
and
F (iω ) =
Z ∞
−∞
f (t)e−iωt dt,
(4.5)
with ω the frequency, easily converted to a period P with P = 2π
ω .
The results of the Fourier transformation of SFRs are shown in Figure 4.8 for simulations that are
considered to have oscillatory behavior and in Figure 4.9 for simulations without oscillations. Since in
all simulations, the SFR shows irregular peaks in the first 3 Gyr of the SFH, the Fourier transform is only
applied between 3.7 − 12 Gyr. It should be noted that the choice of different time boundaries for the
application 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 are
supplementary frequencies showing up. The periods of these frequencies are situated between 0.5 − 2
Gyr. When comparing the Fourier transform of the SFR with the SFH of the same simulations, the period
calculated with the Fourier transform can most of the times be identified with oscillations showing up in
the SFR. For instance for sim1070, the main frequency yields a period of 0.7 Gyr, which can be identified
with 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 irregular
behavior of the SFR of sim1020 in Figure 4.4. In the Fourier transform of sim1052, we observe a sort of
regular 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 calculated
period does not vary, but its amplitude can be different.
For a proper comparison, we have also Fourier transformed the SFHs of simulations where no oscillatory
behavior was found, see Figure 4.9. The height of the peaks are naturally a lot smaller than in Figure
4.8 because the SFR is a lot smaller too. We also see that for sim1037 en sim1040, some small peaks
show up. This could indicate the presence of highly damped oscillations. This results suggest, as
concluded before, a continuous transition from simulations with no oscillatory behavior to simulations
with manifest oscillations.
The behavior of the Fourier transform of sim1058 is due to the fact that clearly no period can be found
for 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 them
as a quantitative benchmark for oscillations to be present. In some cases a period does show up, but
then the question remains: how high and how distinctive should the peak be. Hence whether oscillations
occur 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 only
parameters 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 introduced in Chapt. 3. Star particles within this search radius contribute to the star density of a target
particle according to the cubic spline kernel. From calculations of the Sedov-Taylor radius in Par. 2.1.4
and observations of spatial ordening of OB and T-Tauri associations, the search radius was chosen to
be radius = 50 pc. Other values for this search radius are defensible and therefore, different values for
the 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 SFHs
would be similar to the SFH of the reference galaxy sim1006, because less neighboring particles will be
found and the influence of induced star formation should be small. On the other hand, we already have
shown 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 =
60
Chapter 4. Results and discussion
Figure 4.8: Fourier transform of several simulations considered to have oscillatory behavior present in their SFR. The associated SFHs are
shown 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 associated
SFHs are shown in Figure 4.5. The Fourier transform is only applied between 3.7 − 12 Gyr in the SFH of the simulations.
SFR[M /yr]
0.010
reference sim1006
radius=20pc
radius=40pc
radius=50pc
radius=60pc
radius=80pc
0.008
0.006
0.004
0.002
SFR[M /yr]
0.000
0.010
0.008
0.006
0.004
0.002
SFR[M /yr]
0.000
0.010
0.008
0.006
0.004
0.002
0.000
0
2
4
6
time[Gyr]
8
10
0
2
4
6
time[Gyr]
8
10
12
Figure 4.10: Star formation histories of different simulations with parameters n = 2 and c?,2 = 105 × (1010 × M )−2 Gpc6 Gyr−1 . The
reference simulation sim1006 is shown with no induced star formation present. In the other simulations, the search radius is varied for which
neighboring particles contribute to the star density of a certain target particle.
62
Chapter 4. Results and discussion
age criterium
original
6
Mstar[10 M ]
SFR[M /yr]
0.006
0.005
0.004
0.003
0.002
0.001
0.000
8
6
4
2
0
0
2
4
6
time[Gyr]
8
10
12
Figure 4.11: Two simulations with the exact same parameter values n = 3.5 and c?,2 = 104 × (1010 × M )−3.5 Gpc10.5 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 < 1.87
Gyr.
50 pc and radius = 80 but not for radius = 40 pc and not really (although discussable) for the case of
radius = 60. Hence, we cannot conclude that there is a trend towards higher or lower search radii. The
behavior of the SFH is more dependent on the complex and unpredictable interplay between a higher
star formation probability p? and the feedback effects.
It is also found that the larger the search radius, the more computational time the simulation takes. For
larger search radii, the search domain overlaps with more tree nodes, hence more branches of the tree
need to be descended. The treewalk is the most expensive algorithm in the code, thus altering it can
have a large influence on the computational time.
Age criterium Another aspect which we have not accounted for up till now is the age criterium. In
Par. 2.1.4, we have defined length and time scales for induced star formation to occur, and the time
scale turned out to be of the order of O(107 ) − O(108 ) yr. The implemented feedback comes to an
end at 1.87 Gyr (see Par. 3.2.3) when the SN of type Ia go off. Therefore, stars that are older than
1.87 Gyr in the code, will never be able to induce star formation because they do not give rise to any
feedback. Therefore, with the calculation of ρs , only star particles that are younger than 1.87 Gyr should
be included.
If we implement this in the code, we can look at SFHs to see whether this age criterium for stars has
an influence, see Figure 4.11. As we can see, the age criterium has no influence in the first 4 Gyr in
the evolution of the galaxy. This is surprising because the criterium is set at < 1.87 Gyr hence we would
expect an influence starting 1.87 Gyr after the birth of the first star. The surprising agreement of both
simulations 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 difference
is found between the two simulations. In fact, the simulation with the age criterium applied formed slightly
more stars than the simulation without the age criterium. This also proofs that a decrease of the induced
star formation term c?,2 ρng ρs can lead to an increase in the SFR and the total star mass.
4.2
4.2.1
Discussion
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 of
qualitative behavior of the SFR, sometimes giving rise to oscillatory behavior. The effect of parameters n
and c?,2 was discussed in detail, but no obvious link is present between parameter values and behavior
of the SFR. Hence for now, we only know that oscillations can occur, but we have not really found an
explanation 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, for
which oscillatory behavior is observed, and the SFC are plotted in Figure 4.12. The comparison is made
with 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 more
modest while the SFR of sim1006 shows more violent peaks. One could wonder why this is the case
because in fact, the probability for star formation p? is larger in sim1019 than in sim1006, due to the
induced 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 induced
star formation should be noticable. Indeed, we see at a time of ∼ 0.3 Gyr, sim1019 already starts forming
stars again, while sim1006 is still recovering from the first initial star formation. This early higher SFR
of sim1019 results in a lot more feedback than in sim1006, and therefore, star formation proceeds more
slowly afterwards. The total cumulative star mass is lower for sim1019 in the first 3 Gyr and the effects
from feedback are also clear from the gas mass MgasBO that has been ejected. In sim1019, initially
more 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, especially
during 2 violent starbursts at ∼ 2 Gyr and ∼ 2.8 Gyr. There has been giving rise to as high amounts of
feedback that the density has decreased significantly and star formation is almost entirely stopped. This
also 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 first
3 Gyr of the evolution history, less feedback has been done, larger gas densities remain in the center of
the 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 criterium
because the gas is blown apart.
Now, we can compare the oscillating simulation sim1019 with a non-oscillating simulation with one very
large peak in the SFR, sim1058, see Figure 4.13. Both simulations make use of the altered star formation 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 to
almost zero for a time of ∼ 1 Gyr. Apparently, this has provided the galaxy with time to collapse to high
densities and give rise to an extreme peak in the SFR, giving rise to extreme feedback and blowing the
gas apart. The feedback is as extreme that the gas only forms a more or less spherical gas cloud again
when 2 Gyr have passed and hardly reach densities again that exceed the SF threshold. We can see
the 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 starburst 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 investigate
whether star formation occurs rather centrally, or more spreaded over the dwarf galaxy. Therefore, we
have resimulated the successful simulation sim1019 but printing out snapshots every 0.01 Gyr instead
of 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 center
of the galaxy. This is clear from the density-radius plot of Figure 4.16 because the threshold density is
indicated with a dashed line at 100 amu cm−3 . Particles that exceed this threshold are eligible for star
formation and their distance to the center of the galaxy is easily read from Figure 4.16.
1 An
animated video of this figure is available at http://www.youtube.com/watch?v=zWy6tzvYhqg
Chapter 4. Results and discussion
0.010
0.008
0.006
0.004
0.002
0.000
8
7
6
5
4
3
2
1
0
80
60
40
20
0
osc sim1019
reference sim1006
0.15
0.10
0.05
0.00
90
80
70
60
50
40
30
20
10
99.5
99.0
98.5
98.0
97.5
97.0
96.5
96.0
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
Ntot [%]
Nentr [%]
Nconv [%]
Ndens [%]
MgasBO[%]
6
Mstar [10 M ]
SFR[M /yr]
64
0
2
4
6
time[Gyr]
8
10
12
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 particles
Nentr with T < 15000K and finally number of gas particles Ntot that satisfy all three SFC. Oscillating galaxy sim1019 is compared with the
reference simulation sim1006.
65
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
8
7
6
5
4
3
2
1
0
80
60
40
20
0
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
80
60
40
20
0
non-osc sim1058
osc sim1019
Ntot [%]
Nentr [%]
Nconv [%]
Ndens [%]
MgasBO[%]
6
Mstar [10 M ]
SFR[M /yr]
4.2. Discussion
95
90
85
80
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
2
4
6
time[Gyr]
8
10
12
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 particles
Nentr with T < 15000K and finally number of gas particles Ntot that satisfy all three SFC. Oscillating and non-oscillating galaxies sim1019
and sim1058 are compared. Both have a star formation law with induced star formation included.
66
Chapter 4. Results and discussion
Figure 4.14: Density evolution projected on the xy plane. This density evolution belongs to sim1058 and shows the density evolution during
an extreme peak in the SFR and right afterwards. It is clear that the extreme star formation has caused extreme feedback, and the gas is entirely
blown 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 particles
are at temperatures of T = 104 K. Another reason is that the cooling rate shows an extreme decrease
around 104 K (see Figure 3.4), hence particles that have been heated, cool rather easily to temperatures
of 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 radii
where little peaks in the density occur, lows in the temperature show up. The gas needs to be cold before it is able to collapse and form stars. We can see that the hydrodynamic equations in the simulations
account for this condition, because the star formation regions and the cool regions occur at the same
radii. The fact that the critical temperature for star formation is very high (T < 15000 K) is hence not too
much 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 is
pressed together and this high densities give rise to star formation. The induced star formation is very
obvious: the newly formed stars are situated exactly in these high density regions at the edges of the
blown bubbles and the region of star formation has been shifted towards larger radii of more or less 1
kpc, shown in Figure 4.16.
At times of 4.69 − 4.7 Gyr, the feedback has been strong enough for the gas to be vastly blown apart and
high density regions to disappear. The star formation peak has more or less come to an end, with little
1075
SFR[M /yr]
0.012
0.010
0.008
0.006
0.004
0.002
0.000
4.50
4.55
4.60
4.65 4.70 4.75
time[Gyr]
4.80
4.85
4.90
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 )−2 Gpc6 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, see
Figure 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 the
particles location and the center of the galaxy (the center is calculated using the center of mass of the dark matter halo). The threshold density
for star formation (nSF = 100 amu cm−3 ) is shown as a dashed line. We can see some gas particles exceeding the threshold density for star
formation.
68
Chapter 4. Results and discussion
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 galaxy
with time and the star formation shifts towards larger radii with it. For instance at 4.69 Gyr, very few gas
particles are located in the center of the galaxy (radii< 0.5 kpc). The few particles in the center have low
densities 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 supernovae
and feedback and finally stopped by the gas being blown apart, is quite the same as the star formation
behavior in simulations without an altered star formation law. As discussed before, induced star formation is already present in a self-consistent way in the code. On the other hand, we need to realise
that without the altered star formation law, this observed peak in the star formation rate would not have
been present. The reference galaxy using the Kennicutt-Schmidt law does not show star formation at
evolution times later than 4 Gyr.
4.2.3
Scaling relations
Via observations of different (dwarf) galaxies, certain observed quantities of these galaxies have been
proven to show a strong correlation. When the correlated properties of these galaxies are plotted, all
galaxies are found on more or less one line/plane. If a simulated dwarf galaxy would be situated entirely
away from this line/plane, this would be a strong indication for the simulation to be unphysical and
probably galaxies with these properties as simulated, do not exist.
Several of these correlated quantities, called the scaling relations, are plotted in Figure 4.17. Even
without 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 (the
V-band corresponds to a wavelength of λ = 550 nm). We recall that apparent magnitudes are always
defined as a difference between apparent magnitude m1 and m2 ,
F1
,
(4.6)
m1 − m2 = −2.5 log
F2
with F1 and F2 the according fluxes of the observed objects. It is easy to show that the fluxes vary as
F2
' 2.51m2 −m1 ,
F1
(4.7)
hence when the magnitudes differ by a value of ∆m = 1, one of the object is 2.5 times brighter than the
other.
Using this to interprete Figure 4.17(a), we see that the simulated galaxies are spreaded over MV between −11.3 and −12.3. This means that the maximum difference between the dimmest and brightest
simulated 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 nonoscillating galaxies, see Figure 4.18(a). The simulated galaxies (almost) all show larger MV and larger
log( 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 formation law has increased the amount of total star mass, hence it is logical that the absolute magnitude
increases. There is also an indication present for the oscillating galaxies to be shifted more than the
non-oscillating galaxies. This effect is natural because the oscillating galaxies form more stars than the
non-oscillating galaxies.
Is it also explainable that the half light radius Re increases? When plotting the half light radius of an
oscillating galaxy, and the reference galaxy, see Figure 4.18(a), we see that the oscillating galaxy ends
up with a larger half light radius at 12 Gyr than the reference galaxy. We know that the oscillating galaxy
has produced more stars and therefore has been subject to larger amounts of feedback. As shown in
4.2. Discussion
69
1.5
1.5
FP − log10 (Re )
log10 (Re ) [kpc]
1.0
2.0
non-osc
reference
Observations
osc
half-osc
0.5
0.0
−0.5
0.5
0.0
a.)
−1.0
−8
−10
−12
−14
MV
−16
−18
b.)
−0.5
−20
1.2
−1.0
1.1
−1.5
V−I
[Fe/H]
−0.5
6
8
log10 (LB )[L ]
10
12
−2.0
−3.0
0.8
c.)
−8
−10
−12
−14
MV
−16
−18
−20
−3.5
−4.0
10
4
d.)
−8
−10
−12
−14
MV
−16
−18
12
14
µ0 [mag/arcsec2 ]
3
2
1
16
18
20
22
24
0
e.)
−1
4
−2.5
0.9
0.7
2
0.0
1.3
1.0
n
1.0
−8
−10
−12
−14
MV
−16
−18
−20
26
28
f.)
−8
−10
−12
−14
MV
−16
−18
−20
Figure 4.17: Overview of different scaling relations. The simulations (colored dots) are shown and compared with the observations (grey dots). The mapped proberties are measured at the end of the evolution time of the galaxy after 12 Gyr. (a) Half
light radius log( Re ) as a function of the visual V-band absolute magnitude MV . (b) The fundamental plane FP substracted with
the half light radius log( Re ) as a function of the B-band luminosity log( L B ). (c) Color V-I as a function of the V-band magnitude 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 Sersic 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 [Jiménez et al., 2008, Burstein et al., 1997, de Rijcke et al., 2005, Geha, 2003, Grebel et al., 2003,
Guzmán 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]
70
Chapter 4. Results and discussion
non-osc
reference
−0.3
−0.4
0.010
0.008
0.006
0.004
0.002
0.000
0.5
0.4
0.3
0.2
0.1
0.0
osc
reference
Re[kpc]
log10 (Re ) [kpc]
−0.2
Observations
osc
half-osc
SFR[M /yr]
−0.1
−0.5
−11.0−11.2−11.4−11.6−11.8−12.0−12.2−12.4
MV
0
2
4
(a)
6
time[Gyr]
8
10
12
(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 towards
larger magnitudes MV and larger half light radii Re in comparison with the reference galaxy. The right image (b) shows the detailed star
formation of an oscillating galaxy (sim1019) and the reference galaxy (sim1006). For the oscillating galaxy, a first hint of the negative
correlation 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 larger
distances from the center of the galaxy. Careful comparison of evolution of the SFR and the Re shows
that right before and during a peak in SFR, the half light radius shows a low and when the SFR drops
back to zero (due to feedback) the half light radius increases. Are these two quantities truly negatively
correlated?
To anwer this question, the correlation matrix of the two quantities SFR and Re is calculated for the
oscillating 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 r
between the SFR and the half light radius Re , while all the non-oscillating galaxies show eather no or a
positive correlation r.
The difference in correlation is an indication for different physical mechanisms to be dominant. In the
oscillating galaxies, the star formation peaks start with a more or less spherical galaxy that collapses to
high 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 larger
distances 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 light
radius 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 or
less constant. When no active star formation is present, the half light radius will be rather determined by
the 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 reference
galaxy is shown too.
osc
sim1020
sim1036
sim1019
sim1070
sim1052
sim1036
r
-0.56
-0.43
-0.39
-0.38
-0.39
-0.34
non-osc
sim1054
sim1045
sim1037
sim1040
sim1030
reference
r
0.42
0.03
0.19
0.08
0.14
0.42
4.2. Discussion
71
Fundamental plane
The fundamental plane represents an observed relation between the half light radius Re , mean effective
surface brightness < I >e and the central velocity dispersion σ of elliptical galaxies (no dEs). It is usually
expressed as
(4.8)
log Re = α log σc + β log < I >e +γ,
with α, β and γ fundamental constants equal to [Burstein et al., 1997]
(4.9)
= 1.38
β = −0.835
γ = −0.629.
(4.10)
0 = α log σc + β log < I >e +γ − log Re ,
(4.12)
α
(4.11)
The quantity plotted in panel (b) of Figure 4.17, depicts the relation
as a function of B-band luminosity log L B . The elliptical galaxies that obey the fundamental plane
will 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 galaxies [Cloet-Osselaer et al., 2012]. Indeed, dwarf galaxies have shallower gravitational potentials and are
more 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 towards positive values. This is in agreement with observations of dwarf galaxies. However, in comparison
with the reference galaxy, it seems that the galaxies with an altered star formation law deviate less from
the fundamental plane. Looking at specific values for the simulations, we see that oscillating galaxies
tend to have smaller mean effective surface brightnesses < I >e and larger Re (as discussed above) in
comparison to the reference galaxy. Mainly the larger value of Re will cause the simulated galaxies to
be shifted towards zero, in accordance with the fundamental plane. However they still clearly agree with
the 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
0.90
0.88
Observations
osc
half-osc
non-osc
reference
V−I
0.86
0.84
0.82
0.80
−10.0−10.5−11.0−11.5−12.0−12.5−13.0
MV
Figure 4.19: A zoom on Figure 4.17 panel (c). The simulated galaxies are shifted along the scaling relation towards larger magnitudes and
bluer colors. Oscillating and half-oscillating galaxies seem to be shifted more than the non-oscillating galaxies in comparison to the reference
galaxy.
V-band is a visual band with a wavelength of λ = 550 nm, the I-band is situated at the transition between
visible 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 Vband magnitudes are bluer. This makes perfectly sense because young stars proof to be brighter and
72
Chapter 4. Results and discussion
bluer.
The simulated galaxies seem to be shifted along this scaling relation in comparison to the reference
galaxy. Indeed, they have not only produced more star mass along their evolution history, but also star
formation has occured more recently than in the reference galaxy. Hence the simulated galaxies are
brighter and bluer.
It is also observed when zoomed in on this scaling relation, that the half-oscillating and the oscillating
galaxies are shifted more towards this scaling relation than the non-oscillating galaxies, as one would
expect. This effect is shown in Figure 4.19.
Metallicity
Another important quantity in regard to galaxies is metallicity. Metallicity can be expressed in several
ways. In panel (d) of Figure 4.17 the metallicity is depicted as [Fe/H] which is defined as
NFe /NH
.
(4.13)
[Fe/H] = log
( NFe /NH )
Another way of describing metallicities is to consider mass fraction Z of particles that are no hydrogen
or helium.
The simulated galaxies agree with the observations. They show rather high metallicites and the reference 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 have
produced more star mass, there has occured more feedback and therefore the ISM has been more enriched with metals. We can see this enrichment of metals happening when we look at the SFH of an
−1.1
[Fe/H]
−1.2
Observations
osc
half-osc
non-osc
reference
−1.3
−1.4
−1.5
Z
−1.6
−1.7
−1.8
−10.5 −11.0 −11.5 −12.0 −12.5 −13.0
MV
(a)
0.010
0.008
0.006
0.004
0.002
0.000
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
0.0000
osc
SFR[M /yr]
−1.0
0
2
4
6
time[Gyr]
8
reference
10
12
(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 towards
brighter 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 because
more stars are being formed.
Sérsic profile
The two last panels (e) and (f) of Figure 4.17 show properties of the Sérsic profile. This profile describes
the surface brightness of a galaxy as a function of the radius. The fitted Sérsic profile is of the form
1
n
I ( R) = I0 e−( R/Re ) ,
(4.14)
4.2. Discussion
73
(a)
(b)
Figure 4.21: Panel (a) shows Sérsic profiles of the reference galaxy and several oscillating galaxies. The oscillating galaxies tend to have a
more concave hence diffuse profile than the reference galaxy. Panel (b) shows the Sérsic profile of an oscillating simulating at different times
before, 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 surface 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 Sérsic index and defines the diffuseness/compactness of the profile. For values of n > 1, the surface brightness profile has a concave
shape and the surface brightness decreases sharply within small radii from the center. Values of n < 1
describe a convex hence a rather diffuse profile that does not show a large decrease in surface brightness within small radii from the center.
Looking at panel (e) of Figure 4.17, we see that the reference galaxy has a Sérsic index of around
n = 1. Most of the simulated galaxies (oscillating and non-oscillating) have a Sérsic index around one
our smaller, although some of them have a larger index n. When the values for the central surface
brightness 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 galaxies with the altered star formation law are somewhat more diffuse than the reference galaxy. We can
look at this in detail, plotting the Sérsic profiles, see Figure 4.21(a). We see indeed that the oscillating galaxies have a slightly more concave profile that does not decrease as fast with the radius as the
reference galaxy. An explanation is not hard to find, the oscillating galaxies have been more subject
to feedback and the star formation regions have shifted towards somewhat larger distances from the
galaxy 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 subject
to any feedback during the last 8 Gyr. The oscillating galaxies being more diffuse, also agrees with our
findings 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 (a
little exagerated) a burst, shown in Figure 4.21(b). The early burst profile clearly shows to be straight or
even 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 a
straight to slightly convex surface brightness profile. A few tenths of Gyr later, feedback has caused the
star formation to shift towards larger radii, µ0 drops and the surface brightness profile gets a concave
shape.
74
Chapter 4. Results and discussion
Observations
osc
half-osc
non-osc
reference
2.0
log10 (σ) [km/s]
1.8
1.6
1.4
1.2
1.0
0.8
4
5
6
7
8
log10 (LB )
9
10
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 . It
expresses a relation between the B-band luminosity and the velocity dispersion σ of the form
L B ∝ σ4 ,
(4.15)
for elliptical galaxies. We have plotted our simulations on the Faber-Jackson relation, shown in Figure
4.22, 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 with
above findings. There does not seem to be a trend towards more or less central velocity dispersion
along the line of sight.
Classification
dSphs or dIrrs The observations used in panel (d) of Figure 4.17 are depicting several types of galaxies, from dSphs and dEs over dSphs/dIrrs to dIrrs. If we use this information to plot the dSphs in a
different color, see Figure 4.23, we can see that the simulated galaxies agree with the metallicities of de
dSphs. On the other hand, we know from Figure 4.12 and Figure 4.13 that unlike dSphs, all simulated
galaxies 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 significant amount of their gas because no tidal interactions and/or ram pressure stripping can occur. If we
would simulate galaxies with identical properties in a cluster environment, probably the gas would have
been removed by tidal interactions or ram pressure stripping and the galaxies would resemble typical
dSphs.
Simulating dIrrs is presently hard in the code because all simulated galaxies tend to have too high metallicities to resemble dIrrs. The cause of this excess in metals is due to the overestimation of star formation
in the current simulations. This leads to more stellar feedback and hence more enrichment of the ISM
with metals. Star formation is overestimated because presently, ionization is not taken into account in
the simulations. The UV radiation of young stars gives rise to Strömgren spheres in which star formation
is impossible. However in the code, only a neutral gas component is present. Currently, efforts are made
2 The Tully-Fisher relation is not applied to our set of simulations. It was decided that this relation is probably not relevant for non-rotating
galaxies, since the Tully-Fisher relation makes use of the circular rotation velocity of galaxies.
4.2. Discussion
75
2
1
Observations dSphs
Observations dIrrs, dEs
osc
half-osc
non-osc
reference
[Fe/H]
0
−1
−2
−3
−4
−8 −10 −12 −14 −16 −18 −20
MV
Figure 4.23: Metallicity in function of V band magnitude as in panel (d) of Figure 4.17. Dwarf spheroidals are indicated in grey, other
galaxies (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 more
star 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 explanation
for the occurence of BCDs. We can say that our simulations with the altered star formation law do not
resemble 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 formation
rates of BCDs are estimated to be 0.1 − 1 M /yr. We have discussed that BCDs have underlying older
populations hence BCDs are not young systems. If we would have simulated BCDs, we would observe
extreme peaks in the SFR in the latest few Gyr of the SFH. This was not the case for any galaxy we
have simulated.
The most extreme peak in the SFR in our simulations we have found, was in sim1058 at 4.1 Gyr where
the SFR was of the order of 0.015 M /yr, see Figure 4.6. The SFR at this peak is still one order of
magnitude too small and the peak is situated rather early in the SFH, therefore, we would probably never
observe 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 our
oscillating galaxies are more diffuse and have larger half light radii than the reference galaxy using a
Kennicutt-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 not
have 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 formation
will not lead to the simulation of BCDs. It is however possible that the altered star formation law and
the effect of induced star formation will play its role in the explanation and simulation of BCDs when
combined with merging galaxies or interactions with gas environments. Therefore, we should have a
look 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 oscillatory
behavior and it is hard to predict how the simulations would turn out using another parameterization for
the star formation law (for instance c?,2 ρ g ρns with a power of the star density instead of over the gas density). Naively speaking, it seems that another parameterization would not yield too much of a difference
for the qualitative behavior of the SFR. On the other hand, we have shown that small alterations of less
than 1% to the chance p? of forming a star, can yield entirely different qualitative outcome of the SFH.
76
4.3
Chapter 4. Results and discussion
Mergers
To explain BCDs, there are two possible angles: explain the starbursts by internal mechanisms, or by
external mechanisms.
In this thesis, by altering the star formation law, we have studied internal mechanisms. However, our
simulations did not resemble BCDs hence we could not provide and explanation for their existence and
the origin of their starbursts.
Meanwhile, research has been conducted to investigate external mechanisms, more specifically mergers
of dense gas clouds and dwarf galaxies [Verbeke, 2013]. The results of these mergers suggest that
starbursts are triggered by gas infall in a dwarf galaxy. Also rotation of the dwarf galaxy (host galaxy in
this context) is listed as an important parameter in regard to starbursts. When rotation of the host galaxy
is 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 completely
and is spreaded vastly over the galaxy. The triggered starbursts will therefore be less intense and less
centered.
Interesting results can be obtained from merging a gas cloud and a dwarf galaxy, using our proposed
star 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 and
their 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
Number of gas particles
Number of DM particles
Mg,i
MDM,i
DM profile
Initial gas temperature
Initial metallicity
Feedback efficiency eFB
Softening length e
SF threshold density ρc
Star formation parameter c?,1
E09
200000
200000
525×106 M
2476×106 M
NFW
10000 K
0.0001 Z
0.7
30 pc
100 amu cm−3
0.25
been investigated by [Verbeke, 2013] and prooven to be successful host galaxies in order to achieve a
starbursts. A disadvantage is that we are not sure which parameters for c?,2 and n to choose. Very brief
testing 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
sim0017
sim3003
sim3006
rotation
5 km/s
5 km/s
5 km/s
c?,2
0
104
105
n
0
2
2
context, sim0017 is our reference simulation, using a Kennicutt-Schmidt law. Figure 4.24 shows that
sim0017 already has oscillatory behavior in its star formation history. Simulations sim3003 and sim3006
each 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 behavior
of the star formation rate seems similar and even the total star mass is the same as in the reference
galaxy 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 mass
has been formed by the end of the simulation.
4.3. Mergers
77
reference sim0017
sim3006
sim3003
Mstar[106 M ]
SFR[M /yr]
0.025
0.020
0.015
0.010
0.005
0.000
40
30
20
10
0
0
2
4
6
time[Gyr]
8
10
12
Figure 4.24: Comparison of a rotating simulation sim0017 using the Kennicutt-Schmidt law, and rotating simulations using the altered star
formation law, see Table 4.6.
3006
4001
4002
4003
Mstar[106 M ]
SFR[M /yr]
0.030
0.025
0.020
0.015
0.010
0.005
0.000
40
35
30
25
20
15
10
5
0
0
2
4
6
time[Gyr]
8
10
12
Figure 4.25: Star formation history of host galaxy sim3006 using the altered star formation law. Gas clouds are falling in at different snapshot
times (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 bursty
and more continuous star formation histories [Schroyen et al., 2011]. On the other hand, the fact that
sim3006 does not show many differences with sim0017 can also be a simple coincidence. Previous
findings (see Par. 4.1.2) have indicated three types of qualitative behavior, one of them being more or
less 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 star
formation 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 of
sim4001, sim4002 and sim4003 are equal, but they fall into the host galaxy at different times. They
have a density profile proportional to 1r 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 a
major influence. In this case, sim4002 was the most successful in initiating an intense starburst. The
starburst is at its maximum at 0.025M /yr. Peaks with this maximum value have never been able to
be simulated using only the altered star formation without the merging gas cloud. It is clear that the
merging gas cloud has triggered star formation. Moreover, the observed peak in the star formation rate
belongs among the most intense that have been found by Verbeke, where a Kennicutt-Schmidt law is
78
Chapter 4. Results and discussion
3006
4001
SFR[M /yr]
0.030
0.025
0.020
0.015
0.010
0.005
0.000
9.5
10.0
10.5
11.0
time[Gyr]
11.5
4002
4003
12.0
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 different
qualitative behavior of the bursts.
used. Therefore, since we only tested three scenarios and already one of them is very successful, it is
very 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 oscillatory
behavior after they have fallen in. Simulations sim4001 and sim4003 give no rise to a high star formation
peak but form more stars in the end, see Figure 4.25. This is in accordance with previous conclusions
that more modest star formation during a certain period of the evolution history, will in the end give rise
to oscillations and a higher total star mass.
In conclusion, mergers of gas clouds and dwarf galaxies using an altered star formation law lead to
intense starbursts and should be investigated in greater detail in the context of explaining BCDs.
5
Conclusions and outline
5.1
Conclusions
• Theoretical calculations were used to choose a parameterization of induced star formation. It was
shown 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 where
cooling is included, any parameterization of star formation leads to oscillatory behavior. For our
simulations with cooling and time delay included, we opted for a parameterization of c?,2 ρng ρs . This
term 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 formation leads in several cases to oscillatory behavior of the star formation rate. However, we were
unable to relate qualitative behavior to parameter values of c?,2 and n. We can conclude that c?,2
should be larger than 10−2 (1010 × M )−n Gpc3n Gyr−1 to have an influence on the behavior of the
star formation rate. All tested values of n between 1 and 4 can give rise to oscillatory behavior. In
other 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 cases
a 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 the
process of giving feedback, a surprising accordance is found with simulations not using the age
criterium.
• Simulations that have a more modest star formation rate in the first 3 Gyr of their evolution, will
afterwards show oscillatory behavior and form more stars, in comparison to simulations with a
violent start.
• A star formation episode starts centrally. The newborn stars give rise to feedback effects which
shift the star formation regions towards larger distances from the galaxy center. After more or less
a tenth of a Gyr, feedback effects halt star formation completely.
• The simulated galaxies turn out to reproduce different scaling relations when compared to observations 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 more
diffuse than the reference galaxy (which uses a Kennicutt-Schmidt law). This is easily explained
by the fact that oscillating galaxies have formed more stars, hence they are brighter. They have
experienced star formation more recently, hence they are bluer. This larger star mass has given
rise to more feedback, shifting the location of star formation to larger radii, therefore, the oscillating
galxies 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 inconsistency is present.
• Oscillations in the star formation rate are observed, however the star formation episodes are not
sufficiently intense to represent BCDs. Moreover, the oscillating systems are more diffuse and
have higher metallicities in comparison to our reference simulation. These properties do not agree
with observed properties of BCDs.
• Merging dense gas clouds with dwarf galaxies, by using an altered star formation law, seems to
yield definite intense starbursts. The starbursts are probably intensified by the effect of the altered
star formation law.
5.2
Outline
• Only one parameterization of induced star formation is tested. It is fairly unpredictable what effect
to expect from other parameterizations as for instance c?,2 ρ g ρns .
• The star formation law can be extended even more with different terms. For instance a parameterization 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 supernovae and have a definite effect on induced star formation explicitely present in the code. In
combination with an altered star formation law, this could give interesting results. Also, environmental dependency of the initial mass function can be considered. This can lead to the simulation
of 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 intense
starbursts. 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
Bibliography
[Ambartsumian, 1958] Ambartsumian, V. A. (1958). On the problem of the mechanism of the origin of
stars in stellar associations. Rev. Mod. Phys., 30:944–948.
[Baes, 2010] Baes, M. (2010). Galactische sterrenkunde. Ghent University.
[Baes, 2011] Baes, M. (2011). Astrophysical Simulations. Ghent University.
[Bekki, 2008] Bekki, K. (2008). Formation of blue compact dwarf galaxies from merging and interacting
gas-rich dwarfs. Monthly Notices of the Royal Astronomical Society, 388:L10–L14.
[Bender and Nieto, 1990] Bender, R. and Nieto, J.-L. (1990). Internal kinematics of low-luminosity ellipsoidal galaxies. Astronomy and Astrophysics, 239:97–112.
[Bernard et al., 2012] Bernard, E. J., Ferguson, A. M. N., Barker, M. K., Irwin, M. J., Jablonka, P., and
Arimoto, N. (2012). A deep, wide-field study of Holmberg II with Suprime-Cam: evidence for ram
pressure stripping. Monthly Notices of the Royal Astronomical Society, 426:3490–3500.
[Blaauw, 1958] Blaauw, A. (1958). Stellar associations. Vatican Observatory, 11:405.
[Bodifee and De Loore, 1985] Bodifee, G. and De Loore, C. (1985). Oscillations in star formation and
contents of a molecular cloud complex. Astronomy and Astrophysics, 142:297–315.
[Brau, 2013] Brau, J. E. (2013). Ast122: The birth and death of stars.
[Burstein et al., 1997] Burstein, D., Bender, R., Faber, S., and Nolthenius, R. (1997). Global Relationships Among the Physical Properties of Stellar Systems. Astronomical Journal, 114:1365.
[C., 2013] C., S. (2013). Course on dynamics of multidisplicinary and controlled systems.
[Caldwell, 1999] Caldwell, N. (1999). Surface Brightness Profiles of Three New Dwarf Spheroidal Companions to M31. Astronomical Journal, 118:1230–1234.
[Chauhan et al., 2009] Chauhan, N., Pandey, A. K., Ogura, K., Ojha, D. K., Bhatt, B. C., Ghosh, S. K.,
and Rawat, P. S. (2009). Triggered star formation and evolution of T-Tauri stars in and around brightrimmed clouds. Monthly Notices of the Royal Astronomical Society, 396:964–983.
[Cloet-Osselaer et al., 2012] Cloet-Osselaer, A., De Rijcke, S., Schroyen, J., and Dury, V. (2012). The
degeneracy between star formation parameters in dwarf galaxy simulations and the Mstar -Mhalo relation. Monthly Notices of the Royal Astronomical Society, 423:735–745.
[De Rijcke, 2011] De Rijcke, S. (2011). Fysica van galaxieën. Ghent University.
[de Rijcke et al., 2005] de Rijcke, S., Michielsen, D., Dejonghe, H., Zeilinger, W. W., and Hau, G. K. T.
(2005). Formation and evolution of dwarf elliptical galaxies. I. Structural and kinematical properties.
Astronomy and Astrophysics, 438:491–505.
[de Rijcke et al., 2009] de Rijcke, S., Penny, S. J., Conselice, C. J., Valcke, S., and Held, E. V. (2009).
Hubble Space Telescope survey of the Perseus cluster - II. Photometric scaling relations in different
environments. Monthly Notices of the Royal Astronomical Society, 393:798–807.
[Dekel and Silk, 1986] Dekel, A. and Silk, J. (1986). The origin of dwarf galaxies, cold dark matter, and
biased galaxy formation. Astrophysical Journal, 303:39–55.
83
84
BIBLIOGRAPHY
[Elmegreen and Lada, 1977] Elmegreen, B. G. and Lada, C. J. (1977). Sequential formation of subgroups in OB associations. Astrophysical Journal, 214:725–741.
[Geha, 2003] Geha, M. (2003). Internal dynamics, structure and formation of dwarf elliptical galaxies.
PhD thesis, UNIVERSITY OF CALIFORNIA, SANTA CRUZ.
[Goldsmith and Langer, 1978] Goldsmith, P. F. and Langer, W. D. (1978). Molecular cooling and thermal
balance of dense interstellar clouds. Astrophysical Journal, 222:881–895.
[Grebel et al., 2003] Grebel, E. K., Gallagher, III, J. S., and Harbeck, D. (2003). The Progenitors of
Dwarf Spheroidal Galaxies. Astronomical Journal, 125:1926–1939.
[Gunn and Gott, 1972] Gunn, J. E. and Gott, III, J. R. (1972). On the Infall of Matter Into Clusters of
Galaxies and Some Effects on Their Evolution. Astrophysical Journal, 176:1.
[Guzmán et al., 2003] Guzmán, R., Graham, A. W., Matković, A., Vass, I., Gorgas, J., and Cardiel, N.
(2003). The Fundamental Properties of Dwarf Elliptical Galaxies in Clusters. In Perez, E., Gonzalez Delgado, R. M., and Tenorio-Tagle, G., editors, Star Formation Through Time, volume 297 of
Astronomical Society of the Pacific Conference Series, page 271.
[Hsu et al., 2012] Hsu, W.-H., Hartmann, L., Allen, L., Hernández, J., Megeath, S. T., Mosby, G., Tobin,
J. J., and Espaillat, C. (2012). The Low-mass Stellar Population in L1641: Evidence for Environmental
Dependence of the Stellar Initial Mass Function. Astrophysical Journal, 752:59.
[Hsu et al., 2013] Hsu, W.-H., Hartmann, L., Allen, L., Hernández, J., Megeath, S. T., Tobin, J. J., and
Ingleby, L. (2013). Evidence for Environmental Dependence of the Upper Stellar Initial Mass Function
in Orion A. Astrophysical Journal, 764:114.
[Irwin and Hatzidimitriou, 1995] Irwin, M. and Hatzidimitriou, D. (1995). Structural parameters for the
Galactic dwarf spheroidals. Monthly Notices of the Royal Astronomical Society, 277:1354–1378.
[Jachym et al., 2013] Jachym, P., Kenney, J. D. P., Ruzicka, A., Sun, M., Combes, F., and Palous, J.
(2013). Search for cold and hot gas in the ram pressure stripped Virgo dwarf galaxy IC3418. ArXiv
e-prints.
[James, 1994] James, P. A. (1994). Near Infrared Imaging of Dwarf Ellipticals Irregulars and Blue Compact Galaxies in the Virgo Cluster. Monthly Notices of the Royal Astronomical Society, 269:176.
[Jiménez et al., 2008] Jiménez, N., Smith Castelli, A. V., Cora, S. A., and Bassino, L. P. (2008). The
Color-Magnitude Relation of Cluster Galaxies: Observations and Model Predictions. Boletin de la
Asociacion Argentina de Astronomia La Plata Argentina, 51:263–266.
[Kennicutt, 1989] Kennicutt, Jr., R. C. (1989). The star formation law in galactic disks. Astrophysical
Journal, 344:685–703.
[Kennicutt, 1998] Kennicutt, Jr., R. C. (1998). The Global Schmidt Law in Star-forming Galaxies. Astrophysical Journal, 498:541.
[Korchagin et al., 1988] Korchagin, V. I., Korchagin, P. I., and Riabtsev, A. D. (1988). Nonlinear oscillations in star-forming systems. Astrophysics and Space Science, 141:55–64.
[Koyama et al., 2008] Koyama, H., Nagashima, M., Kakehata, T., and Yoshii, Y. (2008). Dynamical
response to supernova-induced gas removal in spiral galaxies with dark matter halo. Monthly Notices
of the Royal Astronomical Society, 389:237–249.
[Krebs and Hillebrandt, 1983] Krebs, J. and Hillebrandt, W. (1983). The interaction of supernova shockfronts and nearby interstellar clouds. Astronomy and Astrophysics, 128:411–419.
[Krumholz et al., 2010] Krumholz, M. R., Cunningham, A. J., Klein, R. I., and McKee, C. F. (2010).
Radiation Feedback, Fragmentation, and the Environmental Dependence of the Initial Mass Function.
Astrophysical Journal, 713:1120–1133.
BIBLIOGRAPHY
85
[Lianou et al., 2010] Lianou, S., Grebel, E. K., and Koch, A. (2010). Dwarf spheroidals in the M 81 group
- Metallicity distribution functions and population gradients. Astronomy and Astrophysics, 521:A43.
[Liu and Liu, 2003] Liu, G. R. and Liu, M. B. (2003). Smoothed Particle Hydrodynamics: A Meshfree
Particle Method. World Scientific Publishing Co. Pte. Ltd.
[Maio et al., 2008] Maio, U., Ciardi, B., Dolag, K., and Tornatore, L. (2008). Cooling in Primordial Structure Formation. In O’Shea, B. W. and Heger, A., editors, First Stars III, volume 990 of American
Institute of Physics Conference Series, pages 33–35.
[Mateo, 1998] Mateo, M. L. (1998). Dwarf Galaxies of the Local Group. Annual Review of Astronomy
and Astrophysics, 36:435–506.
[McCarthy et al., 2008] McCarthy, I. G., Frenk, C. S., Font, A. S., Lacey, C. G., Bower, R. G., Mitchell,
N. L., Balogh, M. L., and Theuns, T. (2008). Ram pressure stripping the hot gaseous haloes of
galaxies in groups and clusters. Monthly Notices of the Royal Astronomical Society, 383:593–605.
[McConnachie and Irwin, 2006] McConnachie, A. W. and Irwin, M. J. (2006). Structural properties of the
M31 dwarf spheroidal galaxies. Monthly Notices of the Royal Astronomical Society, 365:1263–1276.
[Michielsen et al., 2007] Michielsen, D., Koleva, M., Prugniel, P., Zeilinger, W. W., De Rijcke, S., Dejonghe, H., Pasquali, A., Ferreras, I., and Debattista, V. P. (2007). Toward a Solution for the Ca II
Triplet Puzzle: Results from Dwarf Elliptical Galaxies. Astrophysical Journal Letters, 670:L101–L104.
[Norman and Silk, 1980] Norman, C. and Silk, J. (1980). Clumpy molecular clouds - A dynamic model
self-consistently regulated by T Tauri star formation. Astrophysical Journal, 238:158–174.
[Oppenheimer and Dalgarno, 1975] Oppenheimer, M. and Dalgarno, A. (1975). The formation of carbon monoxide and the thermal balance in interstellar clouds. Astrophysical Journal, 200:419–425.
[Perkins, 2003] Perkins, D. (2003). Particle astrophysics. Oxford University Press.
[Peterson and Caldwell, 1993] Peterson, R. C. and Caldwell, N. (1993). Stellar velocity dispersions of
dwarf elliptical galaxies. Astronomical Journal, 105:1411–1419.
[Ruszkowski et al., 2012] Ruszkowski, M., Bruggen, M., Lee, D., and Shin, M.-S. (2012). The Flying
Spaghetti Monster: Impact of magnetic fields on ram pressure stripping in disk galaxies. ArXiv eprints.
[Salpeter, 1955] Salpeter, E. E. (1955). The Luminosity Function and Stellar Evolution. Astrophysical
Journal, 121:161.
[Schmidt, 1959] Schmidt, M. (1959). The Rate of Star Formation. Astrophysical Journal, 129:243.
[Schombert, 2013] Schombert, J. (2013). Ast122: The birth, death and destruction of stars.
[Schroyen et al., 2011] Schroyen, J., de Rijcke, S., Valcke, S., Cloet-Osselaer, A., and Dejonghe, H.
(2011). Simulations of the formation and evolution of isolated dwarf galaxies - II. Angular momentum
as a second parameter. Monthly Notices of the Royal Astronomical Society, 416:601–617.
[Sedov, 1958] Sedov, L. I. (1958). Examples of gas motion and certain hypotheses on the mechanism
of stellar outbursts. Rev. Mod. Phys., 30:1077–1079.
[Sethuram, 2011] Sethuram, R. (2011). Star formation in blue compact dwarf (BCD) galaxies. In Astronomical Society of India Conference Series, volume 3 of Astronomical Society of India Conference
Series, page 145.
[Sharina and Puzia, 2008] Sharina, M. E. and Puzia, T. H. (2008). VLT spectroscopy of globular clusters
in dwarf galaxies. In Guainazzi, M. and Osuna, P., editors, Astronomical Spectroscopy and Virtual
Observatory, page 57.
86
BIBLIOGRAPHY
[Shin and Ruszkowski, 2013] Shin, M.-S. and Ruszkowski, M. (2013). Ram pressure stripping in elliptical galaxies - I. The impact of the interstellar medium turbulence. Monthly Notices of the Royal
Astronomical Society, 428:804–814.
[Silk, 1977a] Silk, J. (1977a). On the fragmentation of cosmic gas clouds. II - Opacity-limited star formation. Astrophysical Journal, 214:152–160.
[Silk, 1977b] Silk, J. (1977b). On the fragmentation of cosmic gas clouds. III - The initial stellar mass
function. Astrophysical Journal, 214:718–724.
[Springel, 2005] Springel, V. (2005). The cosmological simulation code GADGET-2. Monthly Notices of
the Royal Astronomical Society, 364:1105–1134.
[Sutherland and Dopita, 1993] Sutherland, R. S. and Dopita, M. A. (1993). Cooling functions for lowdensity astrophysical plasmas. Astrophysical Journal Supplement Series, 88:253–327.
[Thuan, 1983] Thuan, T. X. (1983). Blue compact dwarf galaxies. II - Near-infrared studies and stellar
populations. Astrophysical Journal, 268:667–682.
[Toomre, 1964] Toomre, A. (1964). On the gravitational stability of a disk of stars. Astrophysical Journal,
139:1217–1238.
[Valcke, 2010] Valcke, S. (2010). Simulations of isolated and interacting dwarf galaxies. PhD thesis,
Ghent University.
[Valcke et al., 2008] Valcke, S., de Rijcke, S., and Dejonghe, H. (2008). Simulations of the formation and
evolution of isolated dwarf galaxies. Monthly Notices of the Royal Astronomical Society, 389:1111–
1126.
[van Zee, 2001] van Zee, L. (2001). The Evolutionary Status of Isolated Dwarf Irregular Galaxies. II.
Star Formation Histories and Gas Depletion. Astronomical Journal, 121:2003–2019.
[van Zee et al., 2001] van Zee, L., Salzer, J. J., and Skillman, E. D. (2001). Kinematic Constraints on
Evolutionary Scenarios for Blue Compact Dwarf Galaxies. I. Neutral Gas Dynamics. Astronomical
Journal, 122:121–139.
[Vandenbroucke et al., 2013] Vandenbroucke, B., De Rijcke, S., Schroyen, J., and Jachowicz, N. (2013).
Physics of a partially ionized gas relevant to galaxy formation simulations – the ionization potential
energy reservoir. ArXiv e-prints.
[Verbeke, 2013] Verbeke, R. (2013). Starbursts in Blue Compact Dwarf Galaxies triggered by gas inflow.
Master’s thesis, Ghent University, Belgium.
[Zucker et al., 2007] Zucker, D. B., Kniazev, A. Y., Martı́nez-Delgado, D., Bell, E. F., Rix, H.-W., Grebel,
E. K., Holtzman, J. A., Walterbos, R. A. M., Rockosi, C. M., York, D. G., Barentine, J. C., Brewington,
H., Brinkmann, J., Harvanek, M., Kleinman, S. J., Krzesinski, J., Long, D., Neilsen, Jr., E. H., Nitta,
A., and Snedden, S. A. (2007). Andromeda X, a New Dwarf Spheroidal Satellite of M31: Photometry.
Astrophysical Journal Letters, 659:L21–L24.