Thesis Guillermo Kardolus

Evaluation of mass loss in the simulation of

stellar clusters using a new multiphysics softwareenvironment

Guillermo Kardolus

August 25, 2010

Keywords: AMUSE, stellar dynamics, stellar evolution, stellar clusters, masssegregation

Supervisors:Dr. I. PelupessyProf. dr. S. F. Portegies Zwart

Mastercoordinator: Prof. dr. R.A.M.J Wijers

Astronomical Institute ”Anton Pannekoek”Astronomy and AstrophysicsUniversity of Amsterdam

Leiden ObservatoryComputational AstrophysicsLeiden University

Preface

This thesis covers the work done during my master’s research project as partof the fulfillment of the master’s degree in astronomy and astrophysics (O-profile) at the University of Amsterdam. This research project is done underthe supervision of Prof. dr. S. F. Portegies Zwart and Dr. I. Pelupessy atthe Leiden Observatory, and Prof. dr. R. A. M. J. Wijers at the AstronomicalInstitute ”Anton Pannekoek” (September 2009 - August 2010).

Abstract

In this research project stellar evolution and stellar dynamics are combined inthe simulation of stellar clusters and binary systems. The numerical approx-imations are done using a fairly new software framework written in Python.This framework is called AMUSE, an abbreviation of Astrophysical MUltipur-pose Software Environment. In contrary to the monolithic approach, where asoftware package that first covers one astrophysical domain is extended to alsocover another domain, AMUSE uses a modular approach. This means thatexisting packages, which are well-tested within their own domain, can be com-bined. AMUSE handles the communication and unit conversion between thesedifferent modules.The phiGRAPE and Hermite0 direct n-body integrators were combined with afast, fitting formulae based, stellar evolution program called Single Star Evolu-tion (SSE). The programs currently interact as followed: first a dynamical stepis taken, at which the n-body integrator synchronizes all the particles to thesame time. This step is then followed by an instant step of stellar evolution:the stellar evolution program sends an updated list of masses, based on the newglobal time, to the stellar dynamics program. If this method is accurate, de-creasing time steps should at some point converge to the same result. It shouldbe noted that the mass, lost during the stellar evolution of the individual stars,is assumed to leave the cluster instantly without interaction with the individualstars. The cluster is further idealized: the gas from the molecular cloud fromwhich the cluster originated is also assumed to be gone at the zero age mainsequence.However, before running these cluster simulations, the AMUSE framework hasto be tested first due to the fact that it is fairly new; a computer program isbuilt which computes various diagnostics in order to perform a qualitative com-parison with well-known results from earlier papers. After these tests, stellardynamics and stellar evolution are first combined in binary systems, which haveanalytical solutions. The binary simulations are done using the AMUSE Two-body module, which uses the generalized Kepler formulas to solve the two-bodyproblem for any orbit.Various expected results are observed using AMUSE and will be describedbriefly hereafter. The core collapse for a Plummer sphere occurs around tenhalf-mass relaxation times for one-component clusters, and is accelerated pro-portional to the low-mass component over the high-mass component in two-component simulations. Furthermore, there is indeed a sudden increase of bi-naries around core collapse.When mass loss is incorporated, the cluster expands and dissolves when 60 per-cent of the mass is taken away; this is indeed expected for a Plummer sphere.

iv

Finally, the cluster can remain in virial equilibrium when the mass loss occursin a time that is slow compared to the crossing time. When the mass loss is fast,this results in a steep increase of the number of escapers, and thus an increaseof the virial ratio of the cluster as a whole.It was concluded that AMUSE is suitable to perform an investigation of thetime stepping. However, when reproducible results are desired, the phiGRAPEmodule in GPU mode is not suitable: unexpected per run differences occurwhen simulations are performed with the exact same initial conditions. For thisreason the direct Hermite0 integrator was used in future cluster simulations.The binary simulations showed that, when incorporating adiabatic (Jeans mode)mass loss, results are more accurate when the mass loss occurs in a more (orbital-wise) symmetrical manner. This can be concluded from the fact that simula-tions are closer to the (analytically obtained) adiabatic value when more stepsare done, and from test runs in which the time steps are scaled with the orbitalperiod.In cluster simulations the results are closer to the adiabatic values as well —at least to a lower value of the virial ratio — when the time steps are shorter.However, this exact adiabatic value is unknown because there are no analyticalsolutions. When the time steps are shorter they appear to converge to the sameresult and it appears that smaller time steps are needed when the cluster has ashorter crossing time.

Popular summary

i Summary in English

Stars, including our own sun, are born in stellar clusters. Such a cluster isa multi-physical environment, in which various astrophysical domains interact.Examples of these domains are: stellar dynamics, stellar evolution, radiationtransfer, and hydrodynamics. In this research project stellar dynamics and stel-lar evolution evolution are combined in the simulation of stellar clusters andbinary systems. In dynamical simulations the velocities and positions of thestars have to be approximated corresponding to a certain time, and in stellarevolution the equations describing the stellar structure have to approximatedover time as well; both these domains are best studied with numerical approxi-mations using a computer. When stellar evolution is incorporated into the dy-namical simulation of a stellar cluster or binary system, this effectively meansthat the stars lose mass over time.Mass loss can be incorporated into the dynamical evolution of a stellar clusterby extending an existing program, that first covers only one of the domains, insuch a way that it also covers the other astrophysical domain. Such a monolithicapproach has many disadvantages: the resulting packages are often large, sufferfrom bugs, have sections of dead code, and are rarely documented. AMUSE,short for Astrophysical MUltipurpose Software Environment, was created to ad-dress these problems using a modular approach: existing, and thus well-tested,packages are combined into a larger framework written in Python.In AMUSE, the combination of the two astrophysical domains is done as fol-lowed: first a dynamical simulation is done over a specific time step, followed byan instant step of stellar evolution. The program responsible for stellar evolutionspecifies new particle masses and passes them to the stellar dynamics program.The stellar dynamics program is then ready to take a new time step. AMUSEhandles the communication and unit conversion between these programs.The question is whether this method is accurate. When this method is accu-rate, decreasing time steps should at some point converge to the same results.This, and the general effect of different time step sizes, is first investigated inbinary systems, which have analytical solutions. Research is then carried on incluster simulations. But because AMUSE is fairly new software, it has to betested first: a computer program is built which computes various diagnostics inorder to perform a qualitative comparison with well-known results from earlierresearch.In cluster simulations where all masses are equal (one-component simulations),as well as in two-component simulations, with and without mass loss, results are

vi I. SUMMARY IN ENGLISH

in line with what is expected. However, when reproducible results are desired,the stellar dynamics program, which runs on the GPU, is not suitable: unex-pected per run differences occur when simulations are performed with the exactsame initial conditions. For this reason another program, running on the CPUinstead, was used in future cluster simulations. This is unfortunate because theGPU is specialized in running vector calculations, and is therefore very fast inperforming stellar dynamical computations. The CPU is many orders slower inperforming these computations.The binary simulations showed that, for a mass loss that is supposed to beevenly spread over the orbit of the mass-losing star, results are more accuratewhen the mass loss occurs in a more (orbital-wise) symmetrical manner. Thiscan be concluded from the fact that simulations are closer to the analytical val-ues when more steps are done, and from test runs in which the time steps arescaled with the orbital period.In cluster simulations the results are more accurate as well when the time stepsare shorter. However, this exact analytical value is unknown because such so-lutions do not exist. When the time steps are shorter they appear to convergeto the same result and it seems that smaller time steps are needed when theclusters are more compact.

II. SAMENVATTING IN HET NEDERLANDS (DUTCH) vii

ii Samenvatting in het Nederlands (Dutch)

Sterren, inclusief onze zon, worden geboren in sterrenhopen. Een dergelijke ster-renhoop is een omgeving waarin verschillende astrofysische domeinen interactiemet elkaar hebben. Voorbeelden van deze domeinen zijn: dynamica, sterevolu-tie, stralingsoverdracht, en hydrodynamica. In dit onderzoek worden dynamicaen sterevolutie gecombineerd in de simulatie van sterhopen en dubbelsterren.In dynamische benaderingen worden de posities en snelheden van de sterrenbenaderd op een bepaald tijdstip, en in sterevolutie worden de structuurvergeli-jkingen die de ster beschrijven ontwikkeld over de tijd; beide domeinen zijn hetbest te bestuderen doormiddel van numerieke benaderingen met een computer.Wanneer sterevolutie wordt geıntegreerd in een simulatie die de dynamica vaneen cluster beschrijft, betekent dit effectief dat de individuele sterren massa ver-liezen na verloop van tijd.Massa verlies kan worden geıntegreerd in de dynamische evolutie van een sterren-hoop door een bestaand programma, dat voorheen maar een domein beschreef,uit te breiden met het andere fysische domein. Een dergelijke monolithischebenadering kent veel nadelen: de resulterende programma’s zijn vaak groot, be-vatten regelmatig fouten en stukken ongebruikte code, en zijn bovendien vaaknauwelijks gedocumenteerd. AMUSE, een afkorting voor ”Astrophysical MUlti-purpose Software Environment”, is ontwikkeld om deze problemen te adresserendoor middel van een modulaire aanpak: bestaande programma’s, die uitgebreidgetest zijn binnen het eigen domein, worden gecombineerd binnen een Pythonframework.In AMUSE worden de twee astrofysische domeinen als volgt gecombineerd: eerstwordt er een dynamische tijdstap genomen over een specifieke tijdstap, gevolgddoor een sterevolutie stap die instantaan plaatsvindt. Het programma dat desterevolutie verzorgt berekent nieuwe massa’s en geeft deze door aan het pro-gramma dat de dynamische simulaties verzorgt. Het dynamica programma isdan klaar om de volgende tijdstap te simuleren. AMUSE regelt de communi-catie en rekent eenheden om tussen de verschillende programma’s.Het is de vraag of deze methode een juiste representatie van de werkelijkheidgeeft. Wanneer dit inderdaad het geval is, dan zouden kleinere tijdstappen vanafeen bepaalde waarde met elkaar moeten convergeren tot hetzelfde resultaat. Hetvoorgaande, samen met de invloed van de tijdstap in het algemeen op simulatiesmet dynamica en sterevolutie, is eerst onderzocht in dubbelstersystemen. Hetvoordeel is dat dubbelsterren analytische oplossingen hebben in tegenstellingtot sterhopen. Vervolgens wordt het onderzoek verlegd naar de simulatie vansterhopen. Echter, AMUSE is vrij nieuwe software en zal daarom eerst getestmoeten worden: er wordt een computerprogramma geprogrammeerd dat ver-scheidene diagnostiek berekeningen uitvoert. Vervolgens kan deze diagnostiekworden vergeleken met bekende resultaten uit eerder onderzoek.In simulaties van sterhopen waarin alle sterren dezelfde massa hebben, maar ookin simulaties waarin twee verschillende massa componenten voorkomen, wordenresultaten verkregen die inderdaad in de lijn der verwachting liggen. Dit geldtzowel voor simulaties zonder, als simulaties met massa verlies. Echter, wan-neer reproduceerbare resultaten gewenst zijn, is het ster dynamica programmadat op de grafische kaart draait niet geschikt: niet verwachtte per-simulatie-verschillen worden waargenomen wanneer de simulaties worden uitgevoerd metexact dezelfde begincondities. Om deze rede is een ander programma, dat op de

viii II. SAMENVATTING IN HET NEDERLANDS (DUTCH)

CPU draait, gebruikt voor de rest van de cluster simulaties in dit onderzoek. Ditis erg jammer, omdat de grafische kaart gespecialiseerd is in vector berekeningenis deze significant sneller in het uitvoeren van ster dynamica berekeningen.De dubbelster simulaties lieten zien dat, wanneer massa verlies in een gelijkemate plaatsvindt over de baan van de ster, resultaten nauwkeuriger zijn alsde massa op een meer symmetrische manier (kijkend naar de baan) wordt ver-wijderd. Dit kan worden opgemaakt uit het feit dat simulaties dichter bij deanalytische waarde lagen wanneer er meer stappen werden genomen. Het kanook worden opgemaakt door resultaten te bekijken van test simulaties waarinde tijdstap wordt geschaald met de periode van de ster die massa verliest.Ook in cluster simulaties zijn de resultaten nauwkeuriger wanneer de tijdstapkleiner is. Hoewel de analytische waarde in dit geval niet kan worden berekenddoordat deze simpelweg niet bestaat. Op een gegeven moment blijkt het datkorte tijdstappen inderdaad convergeren naar dezelfde resultaten. Het lijkt eropdat kleinere tijdstappen nodig zijn wanneer het cluster compacter is.

Contents

Preface i

Abstract iii

Popular summary vi Summary in English . . . . . . . . . . . . . . . . . . . . . . . . . vii Samenvatting in het Nederlands (Dutch) . . . . . . . . . . . . . . vii

1 Introduction 11.1 Computational astrophysics . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Cluster evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Brief overview of the relevant literature . . . . . . . . . . 61.3 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Methods 112.1 Multi-physical software . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 AMUSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Stellar dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Hermite integration scheme . . . . . . . . . . . . . . . . . 142.2.2 Tree codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.3 Plummer model . . . . . . . . . . . . . . . . . . . . . . . . 162.2.4 Salpeter initial mass function . . . . . . . . . . . . . . . . 172.2.5 Special purpose hardware . . . . . . . . . . . . . . . . . . 182.2.6 Universal variable formulation . . . . . . . . . . . . . . . 20

2.3 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Timescales of stellar evolution . . . . . . . . . . . . . . . . 212.3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Mass loss (Single star evolution) . . . . . . . . . . . . . . 232.3.4 Mass loss II (Binary evolution) . . . . . . . . . . . . . . . 252.3.5 Fast models: Single Star Evolution (SSE) . . . . . . . . . 26

3 Tests and diagnostics 293.1 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 One-component simulations . . . . . . . . . . . . . . . . . 333.2.2 Two-component simulations . . . . . . . . . . . . . . . . . 373.2.3 Simulations incorporating mass loss . . . . . . . . . . . . 40

x CONTENTS

4 New results: binary systems 474.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.1 WR runs . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 AGB runs . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.1.3 Test runs . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 New results: cluster simulations 635.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Discussion, Conclusion, and Recommendations 67

A Code snippets 691.1 Cluster simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 691.2 Binary simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

B Parameters 73

Bibliography 75

Chapter 1

Introduction

Observations suggest that stars are formed in giant molecular clouds with amass of ∼ 105M� and a typical size of ∼ 10 pc (Pols, 2007). The molecu-lar cloud is in pressure equilibrium with its surroundings; star formation startswhen this equilibrium is disturbed, for instance, by the perturbation caused bya supernova or a collision with another cloud. The condition for the molecu-lar cloud to be stable against such perturbations is given by the Jeans mass(MJ . 105M� · (T/100K)1.5n−0.5, where n is the number density in cm−3).When a part of the cloud violates this criterium it will undergo a free-fall col-lapse. With the density going up, the Jeans mass within the collapsing cloudgoes down, leading to fragmentation. The fragmentation results in stars beingformed in clusters.Such stellar clusters are fascinating objects. Our own sun was probably formedin a cluster, so an understanding of cluster evolution can tell us something aboutour own origin. More detailed information about cluster evolution and a briefoverview of some of the relevant literature is given in Chapter 1.2.Over time the structure of the individual stars will change due to stellar evolu-tion, while the cluster as a whole also evolves dynamically. Both these astrophys-ical domains are best studied with numerical approximations using a computer.Stellar dynamics requires numerical approximations when more than two parti-cles are involved. Furthermore, the stellar structure equations are too complexto be solved analytically, and there are not enough detailed observations tostudy the impact of mass loss, by stellar evolution, on the dynamical clusterevolution. Using a computer to investigate these problems means that this re-search project belongs to the field of computational astrophysics. In Chapter1.1 computational astrophysics is described in more detail and is also put intoa historic perspective.This research project focusses on the most realistic method to combine stellarevolution and stellar dynamics: life stellar evolution is combined with dynam-ical n-body simulations. Existing, and thus well tested, packages of the twoastrophysical domains are connected in a modular software framework calledAMUSE1 (See Chapter 2.1).The time step for stellar evolution to take place can have a significant impacton the dynamical evolution of a stellar system. This can, for example, be seen

1AMUSE is an abbreviation of Astrophysical MUltipurpose Software Environment

2 CHAPTER 1. INTRODUCTION

in binary evolution, where the binary dissolves if half or more the mass is lostinstantly, while the system can never dissolve when mass is lost in a adiabaticmanner (see also Chapter 2.2.4). This research project investigates the influenceof the time step in the simulation of stellar clusters. In AMUSE, first a dynam-ical step is taken over a certain step dt, followed by an instant step of stellarevolution. Is this method accurate, which time steps converge to qualitativelythe same results, and how does this depend on the mass function and the initialdensity of the stellar cluster? The investigation of the time step will start withbinaries, which have analytical solutions (Chapter 4), and will then be extendedto stellar clusters (Chapter 5).However, because AMUSE is fairly new (developed since 2006), it has to betested first. This will be done by the reproduction of some of the results foundin earlier papers. A qualitative comparison, and the diagnostics used to analyzethe stellar cluster, are described in Chapter 3.In Chapter 2 it is described how the stellar cluster was initiated and evolved overtime. In this project the initialization of the particles (setting masses, velocity,and position) is done using a Plummer sphere (Chapter 2.2.3). In simulationswhere a more realistic mass distribution is desired, a Salpeter mass function isused (Chapter 2.2.4) to set the masses.Stellar dynamical simulations are done with the hermite0 and phiGRAPE soft-ware packages. The latter was initially designed to run on the special purposehardware called GRAPE, but was redisigned to work on a GPU (Chapter 2.2.5).Both these integrators follow the Hermite integration scheme (Chapter 2.2.1).The SSE package (Chapter 2.3.2) was used to perform the life stellar evolution.

It is concluded that not all the dynamical packages are suitable in case re-producible results are desired, and that the time step size can have a significanteffect on the dynamical simulations, of binaries as well as stellar clusters, thatincorporate stellar evolution. The differences are most noticeable when theclusters are compact, and in binary simulations with high mass-loss rates.

1.1 Computational astrophysics

In computational astrophysics computers are used to solve astrophysical prob-lems numerically. It is an interdisciplinary science comprising observationalastronomy, theoretical physics, and computer science. Numerical astrophysicsis applied in the simulation of n-body systems, fluid dynamics, structure andevolution of stars, and radiation transfer; among other things.Although numerical solutions are only approximations, very complex problemscan be solved numerically and with a high degree of accuracy. Computationalastrophysics offers a way to test a theory. Observed phenomena such as collisionsbetween galaxies, the behavior of young stellar clusters, and binary evolutioncan be reproduced using a computer; in many ways computational astrophysicslies between observation and theory.

An interesting implementation of computational astrophysics is the numericalsolution to the n-body problem: the problem of predicting position and velocityof a set of n gravitationally interacting particles. The differential equations de-scribing the motion of the n particles can only be solved analytically for n = 2,

1.1. COMPUTATIONAL ASTROPHYSICS 3

and in certain special cases for n = 3 (e.g. the Lagrange points). Solving ahigher order system seemed impossible and in 1885 a price was announced fora solution to the n-body problem. The problem could not be solved, it turnedout that general solutions of order n ≥ 3 can only be approximated numerically.The current situation is that special purpose supercomputers can simulate inthe order of 106 particles within a reasonable accuracy (Harfst et al., 2007). Seealso Chapter 2.2.

1.1.1 History

The first electromechanical computers were used by the military in World WarII to break codes. The first electronic general-purpose computer, ENIAC2, wasdesigned to compute artillery firing tables, although it was first used for compu-tations regarding the hydrogen bomb. The army funded this $500.000 computer,which became operational in 1946. The ENIAC had a remarkable size of 60 m3

and most of the programming was done by six women using patch cables andswitches. The ENIAC had a record uptime of five days and was taken out ofproduction in 1955. One of its main shortcomings was the low amount of mem-ory to store programs, IBM cards had to be used as input instead.With the first computers came the first extensive use of numerical models, suchas the Monte Carlo method. The use of a computer to study numerical modelswas quickly adopted in astrophysics. In the 1950s the first astrophysical codeswere constructed to describe stellar structure (Heger et al., 2000), for example,in 1956 a computer was used for the direct integration of the stellar structureequations while fitting the boundary conditions (Haselgrove and Hoyle, 1956).Different factors contributed to the fast development of computer science. Theinvention of high-level programming languages made it evident for scientiststo write numerical methods. Most important were compiled languages such asFortran (1953) and C (1972), object orientated languages such as C++ (1983)and Java (1994) and interpreted languages, for instance, Perl (1987) and Python(1989). Another important factor was the development of software aimed at col-laboration, ie. Apache’s Subversion and comment generators such as Doxygenand Sphinx. Results could also be viewed in higher quality with the develop-ment of graphical plotting programs, for example, gnuplot and matplotlib, andvisualization software such as OpenGL.Perhaps the most important factor was the exponential growth of hardwareperformance over time: the CPU transistor count (see Figure 1.1), hard drivecapacity, graphical cards, monitors, and memory. With this fast developmentof computers — desktop machines as well as special purpose supercomputers— simulations became more complex and gave a more accurate representationof reality. This ultimately lead to an increase in popularity of computationalastrophysics (see for instance Figure 1.2, where the exponential growth of sci-entific papers about n-body simulations is plotted).

2Electronic Numerical Integrator and Computer


Figure 1.1: Intel’s co-founder Gordon E. Moore described a trend in the tran-sistor count on CPUs in a 1965 paper. The count doubled every two years andMoore predicted that this trend would continue for at least ten years (Figurefrom Intel, inc.).

Figure 1.2: The occurrence of the quoted text ”n-body simulation” in abstractsof arXiv e-prints. Data for this graph was obtained using the Harvard abstractserver. This graph illustrates the growing popularity of numerical methods inastrophysics.

1.2. CLUSTER EVOLUTION 5

1.2 Cluster evolution

In this section the properties of stellar clusters in general are described, followedby an overview of the relevant literature, the most important results, how theywere obtained, and finally their similarities and differences will be discussed.

Globular clusters are relatively clean realizations of the classical n-body prob-lem; they contain little gas or dust, they are relatively isolated in space, andthe stars are approximately in the same stage of evolution (McMillan, 2003).Furthermore, the stars are relatively old, which weakens the effect of stellarevolution on the dynamical evolution of the globular clusters. Open clusters are”dirty” in the way that they do include gas and dust, are influenced by stellarevolution of massive stars, and are affected by the galactic tidal field (McMil-lan, 2003). It could be argued that globular clusters are very rare and are only”clean” because they survived the ”dirty” processes that destroyed most of theirsiblings (McMillan, 2003).In the Milky Way there are in the order of 102 known globular clusters (Harris,1996). The majority of galaxy clusters is located within 10 kpc of the galacticcenter, but they are also found in the outer parts of the halo. Globular clustersare the oldest objects in the galaxy; they consist of mainly old stars with thesame, low, metallicity and contain almost no gas and dust. The stars in theseclusters are at the same stage in stellar evolution, which can be an indicationthat the stars have been formed at the same time. It is however unknown howglobular clusters form. The masses of the Milky Way clusters range from 103M�to 2.2 ·106M�, which leads (based on the luminosity function) to a typical massof 2 · 105M� (Portegies Zwart et al., 2010).Open clusters are formed in giant molecular clouds, they are loosely bound andbecome disrupted by close encounters, both internally and with other clusters orclouds (Karttunen et al., 2003); the clusters typically survive for ∼ 108 yr. Thestars roughly have the same age and metallicity, and their radiation pressureeventually removes the gas remaining from the molecular cloud from which thecluster was formed.

In this research project stellar clusters, consisting of ∼ 103 stars, are simulated.Alternations take place in the presence of an initial mass function and/or stellarevolution. Assumptions are made to secure more cost efficient simulations; thecluster is considered to be isolated, thus ignoring tidal interactions and gravi-tational shocks, and, in case of stellar evolution, the gas is instantly removedwithout interaction with the stars in the cluster. Furthermore, the particles allhave zero radius, thus preventing collisions and mergers, are all assumed to bezero age main sequence stars at the start of the simulation, and there is no gasleft from the molecular cloud from which the cluster originated.Although these specific simulations are not intended to mimic the evolution of areal stellar cluster, globular clusters are expected to behave more like the simu-lations in which stellar evolution is turned off, while open clusters are expectedto show a similar behavior to the situation in which stellar evolution does takeplace. The distinction between these two type of clusters is based on their sizeand age; globular clusters consist of & 105, primarily old, stars, and the moreloosely bound open clusters typically contain ∼ 103, mostly young, stars.


1.2.1 Brief overview of the relevant literature

Various processes are of importance when studying the numerical outcome of(isolated) cluster simulations: relaxation, equipartition, core collapse, mass seg-regation, binary formation, and mass loss. This section describes these clusterevolution processes briefly.Remnants of the dynamical process called core collapse, the core size becoming(formally) zero and the density going to infinity, are observed as black holes inthe center of globular clusters, for instance in the M15 cluster (Gurkan et al.,2004).In single-component systems, containing only equal mass stars, core collapseis initially driven by the system its tension towards thermal equilibrium. Athermal velocity distribution is obtained by small changes of the particles ve-locity due to two-body interactions; a phenomenon called relaxation (Khalisiet al., 2007). The relaxation time is the time needed to deflect the direction ofa star’s movement by 90 percent relative to its orbit (Khalisi et al., 2007). Av-erage relaxation times for clusters range from 107 to 1010 years (Spitzer, 1987);the dynamical evolution of a cluster is comparable to the relaxation time. Asingle-component system reaches dynamical equilibrium on a crossing timescale:tcr = (GM/R3

vir)−1/2, where M is the total mass, G the gravitational constant,

and Rvir the virial radius: the radius within which the cluster is in virial equi-librium (McMillan, 2003). Thermal evolution takes place on the relaxationtimescale, half mass relaxation time is given by trh ∼ (N/8 log Λ) tcr, where Nis the number of particles, and Λ ∼ 0.1N (Spitzer, 1987).The system, however, cannot reach thermal equilibrium because of the finite es-cape velocity (Quinlan, 1996); when stars escape the core must compensate forthe lost energy by contracting. Because of its contraction the core heats up andenergy is transfered to the surrounding stars. This causes further contraction,generating more heat inside the core. This instability is called the gravothermalcatastrophe (Quinlan, 1996). When the system is near core collapse, the physicsbecomes more complex: massive stars in the core can evolve, stars can merge,and binaries can form inside the core (Quinlan, 1996).Various numerical methods exist for the dynamical simulation of a stellar clus-ter: direct or tree code n-body integrators, direct or Monte Carlo solutions tothe Fokker-Planck equation, and gaseous models. The core collapse is oftenassumed to take place between 12 to 19 half-mass relaxation times (Quinlan,1996). More specific, for a Plummer sphere, the well-known value of & 10trh isoften used (Gurkan et al., 2004).If point masses are used the mergers and collisions can be ignored. Several pro-cesses are left to be discussed, including: binary formation, mass segregation,and stellar evolution. When binaries harden — become more strongly bound —because of interaction with a third body, they can kick this star out of the core;the binding energy of the core becomes smaller, causing the core to re-expandin order to reestablish dynamical equilibrium (Makino, 1996). In this way hardbinaries, with a higher binding energy than the average kinetic energy of theparticles in the cluster, can stop core collapse.When different masses are initialized using a simple two-component mass func-tion or a power law mass distribution, a process called mass segregation willbecome important. On their way to energy equipartition, more massive starswill give some of their kinetic energy to stars with lower masses. This causes

1.2. CLUSTER EVOLUTION 7

the latter to gain velocity, and causes their orbits to widen. The former will losesome of their kinetic energy, and will sink towards the center of the cluster; en-ergy is thus transported outwards. Larger differences between the masses allowa more efficient way of energy transport to the outer regions, and thus speed upthe core collapse. As the mass segregation is in progress, the density of the coreincreases, leading to a shorter relaxation time. This leads to an increase in theenergy transfer rate (Gurkan et al., 2004). When the higher-mass stars sank tothe center, and the less-massive stars moved to the outer regions, the cluster issaid to be segregated (Vesperini et al., 2009). Observational evidence for masssegregation is found in, for example, the Trapezium cluster of Orion and in theyoung open cluster NGC 6231 (Khalisi et al., 2007).Some clusters seem to have segregated faster, based on their age, than predictedby their segregation timescale. This lead to a new theory of primordial masssegregation: massive stars are more likely to form in the center of star formingregions (Vesperini et al., 2009). Mass segregation has a strong effect on thedynamical evolution of a cluster; the timescale for core collapse is drasticallyshortened.Empirical studies of the outcome of n-body simulations show that the timescale,of a two- component system, of mass segregation scales with 1/µ, where µ is themass of the high mass component divided by the mass of the low mass compo-nent (Fregeau et al., 2002). The timescale for core collapse is also proportionalto 1/µ: tcc,µ ∝ 1

µ tcc,1, which is in line with simple theoretical arguments (Khal-isi et al., 2007).When a more realistic initial mass function (IMF) is used, ie. the Salpeter IMF(see also Chapter 2.1.4), core collapse occurs after . 0.1 trh(0), where trh(0)is the initial half mass relaxation time (Gurkan et al., 2004). This means thatdynamical evolution can be accelerated roughly by a factor 100 due to mass seg-regation. For a star of mass m and a mean cluster mass < m > the segregationtimescale is tseg ∼ tR < m > /m, where tR is the relaxation time (McMillan,2003).In a semi-analytical paper, Hills (1980) shows that a cluster expands whenmass is removed and dissociates if more than half the mass is removed within acrossing time, hereafter referred to as fast mass loss. After a fast removal of gas,the system has to find a new radius that satisfies the virial theorem for the newenergy. In case of slow mass loss, long compared to the crossing time, the systemremains in virial equilibrium and no amount of mass loss can dissociate the sys-tem; this type of mass loss is called adiabatic mass loss. Using the virial theoremfor an ideal gas (T0/V0 = −0.5, where T0 = M0 < V 2

0 > /2 is the kinetic energyand V0 = −GM2

0 /(2R0) the potential energy) to obtain the velocity dispersion(< V 2

0 >= GM0/(2R0)) and by comparison with the energy after fast mass loss(E = −GM2/(4R)) Hills found that R/R0 = (M0 − ∆M)/(2[M0/2 − ∆M ]).The equation for adiabatic mass loss is obtained by replacing ∆M with −dmand (R/R0) by [(R + dr)/R]. After separation and integration the result is:R/R0 = [M0/(M0 −∆M)]. These results are similar to those found in binarysystems: Jeans mode mass loss in case of the adiabatic variant, and supernovamass loss in case of the fast removal of gas. See also Chapter 2.3.4.The work of Hills was refined by Boily and Kroupa (2003b), hereafter BK, whoshowed, both semi-analytical and with n-body simulations, that dissociationalso depends on the initial mass function; up to 50 percent of the stars may


remain bound when the fraction of mass remaining in the cluster (ε) is smallerthan 1/2. BK find that Hills argument holds for clusters in virial equilibriumthat undergo a sudden mass loss. But Hills’ paper fails to explain the obser-vational evidence of bound stellar clusters from which 70 percent of the mass,in the form of the molecular cloud, was removed from the system. They arguethat a stellar cluster will survive mass loss of more than 60 percent if the stellarvelocity distribution favors stars with low velocities. This is the case for a massdistribution with a massive core and high-binding energy. They found that, incase of a Plummer distribution, the fraction of bound stars drops very rapidlyaround ε = 0.44 and after ε . 0.4 no stars remain bound.The papers by BK and Hills were intended for mass loss by the removal of thegas from which the cluster originated, but might as well be applied to simula-tions in which the molecular cloud is already removed at t = 0. In this case thesystem would dissolve if roughly half or more of the mass is lost by the evolutionof massive stars. This means that mass loss, and the possible dissociation thatgoes with it, is only important during the early stages of dynamical evolutiondue to the short lifetime of massive stars. Mass loss can significantly slow downcore collapse, or prevent core collapse from taking place. In segregated clustersmass loss by stellar evolution causes a stronger expansion than for unsegregatedclusters; strongly segregated clusters may therefore dissolve rapidly (Vesperiniet al., 2009).

What follows is a brief description of the methods and results in importantpapers regarding the combination of stellar dynamics and stellar evolution. Ap-plegate (1986) argued that, if the mass distribution allows enough massivestars, the combination of mass segregation and stellar evolution will dissolvethe cluster before core collapse. A cluster without massive stars will undergo arapid core collapse instead. More specific, Applegate found that mass functionswith a steeper slope than N(m) ∝ m−2 collapse rapidly; note that the SalpeterIMF has a slope of −2.35. Applegate used a relatively low upper mass cutoff:15 mass groups were made ranging from 4M� to 0.354M�.Chernoff and Weinberg (1990), hereafter referred to as CW, used a more so-phisticated method to confirm and refine Applegate’s findings. CW used aFokker-Planck equation with a simplified description of stellar evolution, a ta-ble of initial and final masses from which they linearly interpolated the initialmasses, to simulate initially unsegregated models initialized according to theKing model. They used initial stellar masses ranging from 0.4M� to 15M� dis-tributed with the power law slopes 1.5, 2.5, and 3.5. They find that ”mass lossduring the first 5 · 109 yr is sufficiently strong to disrupt weakly bound clusterswith a Salpeter IMF”.Fukushige and Heggie (1995), hereafter FH, used the same table for stellarevolution to investigate the evolution of the same King models and power lawslopes for the IMF. The difference is that FH used a second-order n-body inte-grator with predictor-corrector scheme and shared, constant, time step (see alsoChapter 2.2). The lost mass is assumed to leave the cluster without interaction.Their results are qualitatively in agreement with CW, although FH do obtainlonger, sometimes in the order of a magnitude, lifetimes for disrupted systems.Portegies Zwart et al. (1998) used a more sophisticated treatment of stellardynamics, with a fourth-order Hermite integrator with individual time steps(see also Chapter 2.2) running on a GRAPE-4 cluster (see also Chapter 2.2.5),

1.3. PROBLEM DEFINITION 9

and stellar evolution, with fitting formula based on detailed models by Eggle-ton. Qualitative agreement is made with the FH model, but the papers arein disagreement about the lifetime of dissolving clusters: the FH model has ashorter lifetime by roughly a factor 20.Vesperini et al. (2009) use the starlab package, an all in on multi-physicspackage, to investigate the influence of mass segregation on stellar dynamicscombined with stellar evolution. Mass is extracted with scaled Plummer mod-els; the scale radius is set smaller than the radius of the original cluster. Sub-tracting mass from a segregated cluster leads to earlier expansion than in thenon-segregated case; mass segregation thus has a destructive impact on thedynamical evolution of a stellar cluster.

1.3 Problem definition

Simulations of stellar clusters that incorporate stellar evolution usually use all-in-one software packages. In general the stellar dynamics code is extended toinclude stellar evolution. In this research project a new, modular based, softwareframework called AMUSE will be used to combine these different astrophysicaldomains (see also Chapter 2.1.1). AMUSE is fairly new; the incorporation ofstellar evolution has not yet been fully tested. First, dynamical evolution takesplace over a certain time step ∆t, after which the particles are synchronized (bythe n-body integrator), followed by an instant step of stellar evolution. Thisprocess repeats itself until t = tend. The question is whether stellar clustersconverge to, qualitatively, the same evolutionary outcome for different timesteps. This could be tested for segregated and unsegregated stellar clustersbecause of the significant impact on the dynamical evolution imposed by suchsegregation. Furthermore, another variable that needs to be tested is the densityof the cluster; a higher density speeds up the evolution, and a different time stepmight be required for an accurate simulation.Because the dynamical evolution of stellar clusters is very complex, a binarysystem will be studied first. The development of the semi major axis, andeccentricity, over time will be compared in models with different mass loss ratesand time steps, varying from adiabatic limit, to the limit in which the time stepis much larger than the binary period. Particularly interesting could be the casein which the mass loss takes place with the same step as the binary its period.Around core collapse, the number of binaries near the center of a stellar clustersuddenly increases. The influence of the time step on binary evolution cantherefore also be important for cluster evolution. There also might be a certain,useful, analogy between binary simulations and cluster simulations.However, because AMUSE is fairly new, the software framework needs to betested first: a qualitative comparison with well known results from the past isneeded.

Chapter 2

Methods

The simulations in this research project include stellar dynamics combined withstellar evolution. The main program used for stellar dynamics was the phi-GRAPE program on the GPU. This direct n-body integrator follows the fourthorder Hermite predictor-corrector integration scheme with blocked time steps.Furthermore, BHTree was used as a computationally cheap method to test thecalculations of various diagnostics. Both methods, as well as special purposehardware, are explained in Chapters 2.2.1 and 2.2.2 respectively. However, be-fore the n-body simulations can take place, the cluster first has to be initialized.This is done using the Plummer sphere (Chapter 2.2.3) and the Salpeter initialmass function (2.2.4). Special purpose hardware, the GPU, is used to calculateparts of the Hermite integration scheme. Because phiGRAPE was originallydesigned to run on a GRAPE cluster, both types of hardware are discussed inChapter 2.2.5.Mass loss is also investigated in binary systems. The binary simulations weredone using the twobody program. Twobody uses analytical formulae based onthe extension of Kepler’s equations in order to apply them to non-elliptic orbits.The twobody program is further explained in Chapter 2.2.6.The Single Star Evolution (SSE) program by Onno Pols is used to simulatestellar evolution. SSE is based on fitting functions of detailed models. First aquick overview of stellar evolution, and the assumptions that can be made tospeed up numerical methods, are given, and then the SSE program is explainedin more detail (Chapter 2.3.5).Mass loss plays a big role in this research project, both in binary systems and instellar clusters. In Chapter 2.3.3 and 2.3.4 it is explained where (evolutionarywise), and on which timescale, the mass loss occurs.Finally, multi-physics software is needed to study the problem defined in Chap-ter 1.3. The modular based AMUSE framework was chosen and will be furtherexplained in Chapter 2.1.1.

2.1 Multi-physical software

The Universe is a multi-physical environment in which astrophysical problemsoccur on multiple scales. Stellar dynamics, radiation transfer, stellar evolution,and hydrodynamics all have to be considered in a realistic simulation of, for

12 CHAPTER 2. METHODS

example, a stellar cluster. Scales may vary from 104 m and 10−3 s to 1020 mand 1017 s (Portegies Zwart et al., 2009).Different approaches can be considered when incorporating such a multi-scale,and multi-physics, environment into a numerical model. First there is the mono-lithic approach, a program that first simulates one part of astrophysics, for in-stance stellar dynamics, is extended to cover another field, for example stellardynamics. This approach has proven to have several disadvantages: resultingpackages are often large, prone to errors, suffer from bugs, are rarely docu-mented, have sections of dead code, and lack homogeneity (Portegies Zwartet al., 2009).The software package AMUSE1 was created to address these problems. It usesa modular approach: existing, and therefore well documented and tested, pack-ages are wrapped in an interface layer and combined into a larger framework(Portegies Zwart et al., 2009). The AMUSE package will be described furtherdetail in the following section.A modular approach has various advantages. Different packages from within thesame domain can be combined with packages from other astrophysical domains,and also new modules can be incorporated, without having to understand theframework in detail. The barrier to use the modular framework is therefore alsolow, because less understanding of the framework is required compared to themonolithic approach.

2.1.1 AMUSE

Figure 2.1: AMUSE is designedto work in 3 layers. (Figurefrom amusecode.org)

AMUSE was first developed during variousMODEST2 workshops and the first lines ofcode were written in 2006. Current develop-ment is done at the Leiden Observatory andfunding is provided by a NOVA grant.The AMUSE architecture is based on a 3 layerdesign (see Figure 2.10): a user script layer,an AMUSE code layer, and a legacy codelayer. Each layer adds functionality to a lowerlayer. The legacy code layer consists of theexisting astrophysical software packages andhas a built in functionality to communicateamongst these packages. The AMUSE codelayer provides an object orientated interfaceto the legacy layer and offers extra function-ality in the form of modules, such as, unitconversion, file handling, initialization of stel-lar dynamical code through a Plummer sphereand/or a Salpeter mass function. The extrafunctionality also include various functions to, for instance to compute the ki-netic energy and the potential. The communication between the AMUSE layerand the legacy layer is based on the MPI3 framework, a widely used communi-cation protocol for parallel computing. Finally, the user script layer consists of

1AMUSE is an abbreviation of Astrophysical MUltipurpose Software Environment2MODEST is short for MOdeling DEnse STellar systems.3MPI is an abbreviation of Message Passing Interface

2.2. STELLAR DYNAMICS 13

one or more Python scripts which evaluate certain astrophysical problems.

module reference lang. brief descriptionbhtree Barnes and Hut (1986) C++ Barnes-Hut tree codehermite0 Hut et al. (1995) C++ Direct n-body integrator

with shared, but variable,time step.

phiGRAPE Harfst et al. (2007) F77 Direct n-body integratorwith individual, blocked,timestep. Specialized to runon a GRAPE cluster.

twobody Bate et al. (1971) Python Semi analytical code basedon Kepler’s laws

smallN C++ Direct integrator for fewbody systems.

sse Hurley et al. (2000) F77 Stellar evolution based onanalytical formulas fitted todetailed models.

bse Hurley et al. (2002) F77 Binary evolution algorithm.evtwin Eggleton (2006) F77 Detailed stellar evolution

model.fi Pelupessy et al. (2004) F90 Parallel code for galaxy sim-

ulations.capreole F90 Grid hydrodynamics code

by Garrelt Mellema.

Table 2.1: Modules currently implemented in AMUSE.

2.2 Stellar dynamics

To predict the position and velocity of a set of n particles at any given time(the n-body problem), based on Newton’s law of universal gravity, the equationof motion needs to be solved with a given initial position and velocity

ai = G∑j 6=i

mjrijr3ij

i = 1, 2, ..., n (2.1)

where ai is the acceleration of particle i, G the gravitational constant (which isset to one in n-body units, together with the initial total mass), mj the massof particle j, and rij the distance between particles i and j.There are two popular types of numerical methods to approximate this equation.One involves direct summation and the computing speed scales with N2, whereN is the number of particles; this method is used for dense clusters and cansimulate up to ∼ 106 particles on special purpose hardware (Portegies Zwartet al., 2009). The other method uses a tree code and scales with N logN . Thismethod can be used for a larger number of particles, but is significantly lessaccurate than the direct method (Spurzem, 1999).


2.2.1 Hermite integration scheme

In case of direct summation a popular integration method, due to the rela-tively small number of scalar operations, is the fourth-order Hermite predictor-corrector integrator with block time steps (Makino and Aarseth, 1992). Eachparticle has its own position (ri), velocity (vi), acceleration (ai), jerk (ai), time(ti), and timestep (∆ti). The integration proceeds along the following steps

(1) The initial time steps can be set using a simple formula

∆ti = ηs|ai||ai|

(2.2)

Where ηs = 0.01 usually gives sufficient accuracy (Harfst et al., 2007).

(2) The global time t is set to the time corresponding to the particle withthe minimum time ti + ∆ti.

(3) The position and velocity of all the particles are predicted using a Taylorexpansion

rp,j = rj + (t− tj)vj +(t− tj)2

2aj +

(t− tj)3

6aj (2.3)

vp,j = vj + (t− tj)aj +(t− tj)2

2aj (2.4)

(4) With the predicted velocity and position, the acceleration and jerk of particlei at global time t is calculated using

ai = G∑j 6=i

mjrij

(r2ij + ε2)3/2(2.5)

ai = G∑j 6=i

mj

[vij

(r2ij + ε2)3/2+

3(vij · rij)rij(r2ij + ε2)5/2

](2.6)

where ε is the softening parameter. This parameter is used to prevent the ac-celeration and jerk from going to infinity. This parameter should have a smallvalue to keep the simulations realistic.

(5) The position and velocity are corrected with the higher order Taylor ex-pansion

ri = rp,i +∆t4i24

a(2)0,i +

∆t5i120

a(3)0,i (2.7)

vi = vp,i +∆t4i

6a(2)

0,i +∆t5i24

a(3)0,i (2.8)

a(2)0,i =

−6(a0,i − a1,i)−∆ti(4a0,i + 2a1,i)∆t2i

(2.9)

a(3)0,i =

12(a0,i − a1,i) + 6∆ti(a0,i + a1,i)∆t3i

(2.10)

where a(2)0,i and a(3)

0,i are the second and third time derivative of the acceleration atti, respectively. These are obtained by inversion of the Taylor expansion of a0,i.


The quantities a(2)1,i and a(3)

1,i are the time derivatives at ti+∆ti, and ∆ti = t−ti.

(6) Recalculate the timestep using the standard formula (Aarseth, 1985).

∆ti =

√√√√η|a1,i||a(2)

1,i |+ |a1,i|2

|a1,i||a(3)1,i |+ |a

(2)1,i |2

(2.11)

In case the simulation runs on a GRAPE cluster, the particles should be groupedusing a block time step in favor of a better performance (Harfst et al., 2007).The timestep is then replaced by ∆ti,b = (1/2)n, where n is chosen with thefollowing condition (

12

)n≤ ∆ti ≤

(12

)n−1

(2.12)

(7) Start over from step (2)

When diagnostics of the cluster are calculated the particles have to be syn-chronized to global time t. To do this the position and velocity of all particleshave to be predicted in order a(3). Since this was already calculated during thetime step calculation, all that needs to be done is storing these values.

2.2.2 Tree codes

Figure 2.2: Tree code: splitting the cells(two dimensions).

In the tree code method a cell is fitaround all the particles. This rootnode contains more than one particleand is further split into 8 child nodes.This operation is recursive and is re-peated until all particles have theirown cell (the leaf nodes), as is illus-trated in Figure 1.3.The force calculation proceeds as fol-lows: if the size of a cell is given by land the distance to the center of massof the cell is given by d, then the parti-cles are grouped when l/d < θ. Hereθ is the opening angle, and is oftenset to a value around unity (Aarseth,2003). When l/d > θ the cell is splitinto eight cells, recursively repeateduntil all particles are included.A tree code reduces the complexityfrom the direct summation methodO(N2) to O(N logN). This comes at a price, the tree code method is sig-nificantly less accurate then the direct integration method (Spurzem, 1999).The accuracy can be improved by reducing the value of θ, but when limθ→0 thecomplexity increased to O(N2), with additional costs for memory storage.


2.2.3 Plummer model

H.C. Plummer found a density distribution to fit the arrangement of observedglobular clusters (Plummer, 1911). This density distribution was later calledthe Plummer model and is often used in computer models to obtain an initialdensity distribution of a stellar cluster. Although the model is not representingany actual class of stellar clusters accurately, it is a frequently used test case(Harfst et al., 2007).

Figure 2.3: A 10,000 particle Plummer sphere: random positioned particles, ofequal mass, in compliance with Plummer’s density profile. Left panel: densityprofile. With increasing distance from the core, the density eventually drops asr−5. Right panel: particle position.

The Plummer density distribution is found by solving the analytical n = 5 caseof the Lane-Emden equation (2.14). This equation is used when the equation ofhydrostatic equilibrium (2.13) is solved for a polytrope, a star with an equationof state of P = Kρ(n+1)/n, where P is the pressure, ρ the density, and n thepolytropic index:

dP

dr= −GMρ

r2(2.13)

d

dr

{Kr2(n+ 1)nρ(n−1)/n

dρ

dr

}= −4r2Gρ (2.14)

With the boundary conditions ρ(0) = ρc and ρ(R) = 0 and by rewriting theLane-Emden equation in dimensionless form, the equation of hydrostatic equilib-rium can be solved — either numerical or analytical. Rewriting in dimensionlessform is done with ρ = λθn and r = αζ, where

α =[

(n+ 1)Kλ1/n−1

4πG

]1/2Substituting into the Lane-Emden equation gives

1ζ2

d

dζ

(ζ2 dθ

dε

)+ θn = 0 (2.15)


The n = 5 solution is given by

θ =1√

1 + 13ζ

2(2.16)

It can be shown that the total mass has a finite value (Aarseth, 2008), whichgives the following density distribution (the Plummer model) and correspondingpotential:

ρ(r) =(

3M4πa3

)(1 +

r2

a2

)−5/2

(2.17)

Φ(r) = − GM√r2 + a2

(2.18)

Where a is the Plummer radius: a parameter which determines the size of thecore.

2.2.4 Salpeter initial mass function

An initial mass function can be used to assign masses to the particles involvedin a n-body simulation. Based on observations it turned out to be convenientto express the mass distribution of a stellar cluster in the form of a power law:

M−α ∝ N(M)dM (2.19)

Using the observed luminosity function of solar neighborhood main sequencestars, the parameter α was determined to be 2.35 (Salpeter, 1955). The massesare restricted to a range between 0.1M� and 100M�, a lower mass star willbecome degenerate before it can ignite hydrogen, while a higher mass star wouldexceed the Eddington luminosity.

Figure 2.4: Left panel: Number of particles vs. mass using 10,000 particleswith random masses fitting the Salpeter mass function. Right panel: the result-ing mass distribution obtained by applying the Salpeter mass function to thePlummer sphere in the previous section.


2.2.5 Special purpose hardware

Figure 2.5: Basic concept of theGRAPE-1 and GRAPE-3 special pur-pose computers (Makino et al., 1997).(Figure from J. Makino)

The GRAPE4 system was designed toaccelerate the expensive O(N2) directsummation force calculations. Thefirst GRAPE was built at the TokyoUniversity in 1989, and had a perfor-mance of 240 MFLOPS5 at single pre-cision, while being more cost efficientthan general purpose computers withthe same performance (Makino et al.,1997). Its successor, GRAPE-2, ranat 40 MFLOPS, but at double pre-cision. Subsequently, the odd num-ber GRAPEs have single precision,while the even numbered GRAPEshave double precision. The GRAPE-3 was the first with a custom designedgravity chip and ran at a speed of 15 GFLOPS (Makino et al., 1997).The basic concept is that a general purpose computer sends the mass and posi-tion to the GRAPE, where the acceleration (1.5), jerk (1.6), and potential arecalculated on a force computation pipeline6. The results are then send back tothe host computer.The GRAPE-4 was completed in 1995, it included a predictor pipeline in orderto do Hermite individual time step calculations. It was the first computer tobreak 1 TFLOPS. In 2001 the GRAPE-6, with more pipelines and processorchips then its predecessor, was completed and its performance peaked at 64TFLOPS (Makino et al., 2003).

Figure 2.6: The four-cabinet GRAPE-4 system. (Image from J. Makino)

4GRAPE is an acronym for GRAvity PipE5FLOPS is an acronym for FLoating point Operations Per Second.6A pipeline is a series of data processing elements in which the output of one element is

the input for the next.


A GPU7 is designed to do vector calculations and is therefore, in theory, verysuitable for force calculations. In 2007 NVIDIA released CUDA8: a parallelcomputing architecture that can be used to write software for the GPU with arange of standard high-level programming languages. CUDA was used to writethe Sapporo library in order to run gravitational n-body simulations on a GPU.This library is very similar to the GRAPE-6 library and therefore programs thatwere previously running on a GRAPE-6 system could run on the GPU withoutchanging the source code (Gaburov et al., 2009).With a GRAPE-6 or GPU the (individual, blocked time step) integration pro-ceeds as followed:

(1) The system is initialized on the CPU9: each particle gets its own mass,position, velocity, acceleration, jerk, time, and time step.

(2) The CPU produces a list of particles to be integrated at the current time step.

(3) The CPU predicts the position and velocity of one of these particles andsends the result to the GPU/GRAPE-6, where the new time is stored in a reg-ister.

(4) The GPU/GRAPE-6 predicts the position and velocity of all other par-ticles at that specific time.

(5) The GPU/GRAPE-6 computes the force from all the other particles andsends the result to the CPU. Steps (3)-(5) are repeated until all particles in thelist are done.

(6) The global time is updated and the procedure is repeated from step (2)

Figure 2.7: The GeForce GT340 PCI express card by NVIDIA. Force and pre-dictor calculations are done on the GPU, located under the cooler. (Image fromNVIDIA, inc)

7GPU is short for Graphics Processing Unit.8CUDA is an acronym for Compute Unified Device Architecture.9CPU stands for Central Processing Unit.


2.2.6 Universal variable formulation

The universal variable formulation is a method to solve Kepler’s two-body prob-lem for elliptic, hyperbolic, and parabolic orbits. The twobody program, usedfor the binary simulations, incorporates the analytical formulae based on thismethod. At any time t, velocity (v(t)) and position (r(t)) need to be calculatedgiven the specific initial conditions v(t0) and r(t0). This is done by solving theequation of motion:

∂2r∂t2

+µ

r3r = 0, (2.20)

where µ ≡ G(m1 +m2). It is then convenient to introduce the universal variables, which fulfills the following equation (Danby, 1992):

∂s

∂t=

1r

(2.21)

Substituting the universal variable into the equation of motion gives the follow-ing equation.

∂2r∂s2

+µ

ar = −P, (2.22)

where P is a constant and a is the semi major axis. Deriving both sides to d/dsgives a differential equation with the following set of solutions (Danby, 1992).

t− t0 = r0 s c1(αs2) + r0∂r0∂t

s2 c2(αs2) + µs3 c3(αs2), (2.23)

where α = µ/a and the function cn is called the Stumpff function (Danby,1992): cn(x) =

∑i(−1)ixi/(n+2i)!. The value of s at time t can now be solved

numerically. Position and velocity at time t can then also be found:

r = r0f(s) + v0g(s) (2.24)v = r0f(s) + v0g(s) (2.25)

where the functions f(s) and g(s) are given by (Danby, 1992)

f(s) = 1− µ

r0s2 c2(αs2) (2.26)

g(s) = t− t0 − µs3 c3(αs2) (2.27)

f = − µ

rr0s c1(αs2) (2.28)

g = 1− µ

rs2 c2(αs2) (2.29)

2.3. STELLAR EVOLUTION 21

2.3 Stellar evolution

To determine the evolution of a single star, the following set of 4+N differentialequations (Pols, 2007) need to be solved simultaneously for a certain givenequation of state (P = P (ρ, T,Xi)).

Mass conservation:∂r

∂m=

14πr2ρ

(2.30)

Hydrodynamic changes:∂P

∂m= − Gm

4πr2− 1

4πr2∂2r

∂t(2.31)

Thermal changes:∂l

∂m= εnuc − εν − T

∂s

∂t(2.32)

Energy transport:∂T

∂m= − Gm

4πr4T

P∇ (2.33)

with ∇ =

{∇rad = 3κ

16πacGlPmT 4 , if ∇rad ≤ ∇ad

∇ad + ∆∇, if ∇rad > ∇ad

Composition changes:∂Xi

∂t=

mi

ρ

(∂ni∂t

)nuc

[+ mixing terms] (2.34)

with i = 1, 2, 3, ... N

Where r is the radius, m the mass, ρ the density, P the pressure, G the grav-itational constant, t the time, l the luminosity, εnuc the energy generated bynuclear reactions, εν the energy losses caused by neutrinos, T the temperature,s the entropy, ∇ the temperature gradient in either the energy transport byradiation or by convection, κ the opacity, a the radiation constant, c the speedof light, ∆∇ the superadiabaticity of the temperature gradient, Xi the fractionof element i, mi the mass of element i, ni the number density of element i, andthe ’mixing terms’ represent the redistribution of the composition in convectiveregions (Pols, 2007).

P, s, κ,∇rad,∆∇, εnuc, εν , and the reaction rate ∂ni

∂t can all be expressed as func-tions of ρ, T , and Xi (Pols, 2007). This means there are 4+N unknown variablesleft: r, l, ρ, T , and Xi, which can all be written as function of the independentvariables m and t. Therefore each of these variables has to be set at the bound-aries m = 0 and m = M . Initialization of the variables is needed at t = t0.

2.3.1 Timescales of stellar evolution

Hydrodynamical changes, disruption of the balance between gravity and pres-sure (gas and radiation), of star’s structure occur on the dynamical timescale(τdyn). This timescale is the same as the time it would take for a mass shell, ondistance R, to reach the center when radiation and gas pressure are suddenlyremoved. The speed at which the shell moves inwards is approximately thesame as the free-fall velocity. This results in the following expression for thedynamical timescale

τdyn ≈√

R3

GM(2.35)


The dynamical timescale of the Sun is around 5000 seconds.Changes in the thermal structure of a star occur on the so called Kelvin-Helmholtz timescale (τKH). This time is the same as a star its lifetime in case allenergy is being produced by contraction. With the virial theorem, the relationbetween internal and gravitational energy, for an ideal gas (Eint/Egr = −1/2),the thermal timescale can be expressed as

τKH =EintL

=Egr2L≈ GM2

2RL(2.36)

The thermal timescale of the Sun is approximately 15 million years.Changes to a star its composition occur on the nuclear timescale (τnuc), whichis the same as the lifetime of a star if all energy produced would come fromnuclear reactions.

τnuc =EnucL

= AMc2

L, (2.37)

where A is determined by the fraction of available fuel that is converted intoenergy. The thermal timescale of the Sun is in the order of 1010 years (Pols,2007). In conclusion, it can be stated that: τnuc � τKH � τdyn.

Significant mass loss usually occurs on the thermal and dynamical timescale,and is therefore rather abrupt compared to the nuclear, main sequence (MS),timescale. See also Chapter 2.3.3 and 2.3.5.

2.3.2 Assumptions

Detailed stellar structure programs try to solve the above set of stellar evolu-tion equations (2.20 to 2.24) with as few assumptions as possible. However, suchcalculations are computational expensive, certainly when a lot of stars have tobe evolved at the same time, as in the evolution of the individual stars in astellar cluster. Various assumptions, to accelerate these expensive equations,are described below.MS stars are close to thermal equilibrium (TE), because the energy loss on thesurface, the luminosity, is compensated by the energy generated by the nuclearfusion in the core. MS stars are also in hydrostatic equilibrium (HE), if theywere not this would result in a dramatic evolution: an implosion or explosion onthe dynamical timescale. In case of HE and TE the time derivatives in equation(1.21) and (1.22) vanish. Therefore the only time derivative left is the one thatdescribing the composition changes, which occur on the nuclear timescale.

The mechanical equations (1.20 and 1.21) can be decoupled in case of a poly-tropic equation of state: P = Kργ , where K and γ = 1+1/n are constants. Forthese polytropes the equation of hydrostatic equilibrium can be approximatedby the numerical solutions to the Lane-Emden equation (1.14). Degeneratestars — neutron stars (n = 3) and white dwarfs (n = 3/2) — comply to such atemperature independent equation of state. In this case K is a fixed parameter(Kippenhahn and Weigert, 1994). If a MS star is approximated by an Ideal gasequation of state, then modeling can be done using a n = 3 polytrope, where K,in this particular situation, is a free parameter (Kippenhahn and Weigert, 1994).


Another considerable simplification of the stellar evolution equations is the con-cept of homology. When two stars, scaled to the same radius, have the samemass profile, they are said to be homologous:

r1(x)R1

=r2(x)R2

, where x =m1

M1=m2

M2(2.38)

This offers the opportunity to describe one star using detailed stellar models,while all the homologous stars can be described using analytical scaling rela-tions. However, the conditions for homology are limited, see also Chapter 2.3.5.

Further simplifications can be made, for instance, the computational expen-sive mixing terms can be ignored. The same can be done with the neutrinolosses during MS, because these losses only become important in the late stagesof evolution (Pols, 2007). The superadiabaticity, the difference between the ac-tual temperature gradient and the adiabatic temperature gradient in convectiveregions, can also be ignored; it can be shown that only a tiny superadiabatic-ity is needed to transport heat in convective regions (Pols, 2007). Finally, theopacity, in the energy transport equation, can be approximated with a powerlaw: κ = κ0ρ

aT b, where a = 1 and b = −3.5 in case of Kramer’s opacity law(Pols, 2007).

2.3.3 Mass loss (Single star evolution)

Mass loss can have a substantial impact on stellar evolution, especially for mas-sive stars (M & 15M�), where mass loss by stellar wind is important duringall stages of evolution. The mechanisms responsible for mass loss are not wellunderstood, and can therefore impose a considerable uncertainty on the stellarevolution models.Various occurrences of mass loss will be described in the following section at thehand of evolutionary tracks of a low, and an intermediate-mass star.

Evolutionary tracks can be constructed using either detailed models or modelsusing fitting formulae (Chapter 2.2.5). The evolutionary tracks of low mass stars(< 2M�) are significantly different from that of high (> 8M�) and intermediate-mass stars. See also Figure 2.9.During the MS (Figure 2.9, points A to B) all stars evolve towards a higher lumi-nosity. Low mass stars also evolve towards a higher temperature, while high, andintermediate, mass stars evolve towards a lower temperature. Once the hydrogenin the core is exhausted, at point B, the core starts contracting. High and inter-mediate mass stars with a convective, well mixed, core suddenly find themselveswithout fuel for hydrogen fusion and start contracting on a thermal timescale.These stars therefore show a hook feature. When the temperature in the shell ishigh enough to ignite hydrogen, the hydrogen shell burning starts. This transi-tion is smooth for low mass stars. During hydrogen shell burning stars move, atalmost constant luminosity, towards a lower temperature on a thermal timescale.During this phase the shell keeps on adding mass to the core, which is still con-tracting. While the core contracts, the outer layers expand, thus making theshell act like a mirror. The outer layers cool, and the convective region grows.When stars have a deep convective envelope they move in a direction of higher


Figure 2.8: Evolutionary tracks of a 5M� (left panel) and a 1M� star (rightpanel) using a solar-like metallicity (Z = 0.02). See text for more information.(Figure from Onno Pols)

luminosity along the Hayashi line: an almost vertical (and thus temperature in-dependent) line, representing the location of fully convective stars in the HRD.

Figure 2.9: Various regionsin the Hertzsprung-Russell Di-agram (HRD).

It is during this particular phase, where thestars are close to the Red Giant Branch(RGB), that significant mass loss occurs inlow mass stars. See also Table 2.1.Along the RGB the radius of the envelopeincreases further, making it loosely bound.It therefore becomes easier to remove massby radiation (Pols, 2007). The rate of massloss during the RGB is often calculated withthe empirical law by Reimers (Kudritzki andReimers, 1978):

M = −4 · 10−13 ηL

L�

R

R�

M�M

M�/yr (2.39)

where η is often set to a value of order unity.Helium core burning starts when the tempera-ture of the core reaches the appropriate value.Low mass stars ignite helium, in a degeneratecore, in a runaway process called the heliumflash (point F). Intermediate and high massstars ignite helium in a non degenerate core,resulting in a loop on the Horizontal Branch (HB). When the hydrogen in thecore is exhausted, the process of contraction occurs again, but this time heliumis ignited in a shell. The stars cool along the Asymptotic Giant Branch (AGB),where both helium and hydrogen shell burning occur at the same time. Thestars in Figure 2.7 are on the AGB from point H to J. This is another phase ofsignificant mass loss. See also Table 2.1.


During the AGB phase thermal pulses in radius and luminosity occur due tothe instability between the helium and hydrogen shell burning. It is believedthat these pulses, together with the radiation pressure on dust particles in theatmosphere, are the source of the mass loss (Pols, 2007). The observational-determined mass loss rates vary from ∼ 10−7M�/yr to ∼ 10−4M�/yr.

For massive stars (M & 15M�) mass loss caused by stellar wind becomes im-portant in all evolutionary phases (Pols, 2007). Stars with M & 30M� have amass loss timescale that is shorter than the nuclear timescale, therefore massloss has a substantial impact on their evolution (Pols, 2007). These stars havean evolution in which mass loss will remove the envelope before the Hayashi lineis reached (in which case they would be classified as a Red Super Giant). Formassive stars, mass loss caused by the stellar-wind mechanism, can be approx-imated with the following empirical relation by De Jager (Nieuwenhuijzen andde Jager, 1990):

log(−M) ≈ −8.16 + 1.77 log(L

L�

)− 1.68 log

(Teff

K

)M�/yr (2.40)

The two empirical relations above indicate that, in general, mass loss (duringall evolutionary phases) becomes more important at increasing stellar mass.

2.3.4 Mass loss II (Binary evolution)

Mass transfer in a binary system can take place in a conservative (conservedmass and momentum) or non-conservative (mass and momentum loss) manner.Observations show evidence for both cases of mass transfer. Three cases of massloss will be described in further detail: mass loss through a fast stellar wind fromthe donor star, mass loss by the ejection of matter from the accreting star inthe form of a jet, and supernova mass loss.In case of a circular orbit, the orbital angular momentum of a binary system isgiven by the following equation (Verbunt, 2007)

J2 = GM2

1M22

M1 +M2a (2.41)

Where a is the semi major axis. A generalized equation for the case of masstransfer can be obtained by differentiating the above equation.

2J

J=a

a+ 2

Md

Md+ 2

Ma

Ma− Md + Ma

Md +Ma(2.42)

Where Ma is the mass of the accretor, and Md the mass of the donor. In caseof conservative mass transfer J = 0 and Ma = −Md and hence

a

a= 2

(Md

Ma− 1)Md

Md(2.43)

This means that the orbit will shrink as long the donor has a higher mass thanthe accretor and will expand otherwise. In case of non-conservative mass trans-fer only a fraction of the matter is absorbed by the accretor: Ma = −βMd. If γ


times the specific angular momentum of the binary is the specific angular mo-mentum loss due to mass loss (hloss = γh), then it can be shown that (Verbunt,2007)

a

a= −2

Md

Md

{1− βMd

Ma− (1− β)(γ +

12

)Md

Md +Ma

}(2.44)

In case of Jeans mode mass loss, driven by a fast stellar wind, the mass fromthe donor leaves the binary system without interaction with the accretor. Thismeans that hloss = a2

dω, where ad = aMa/(Md + Ma). It can be shown that inthis case γ = Ma/Md and finally

a

a= − Md

Ma +Md(2.45)

Where Md < 0 implies that a > 0, and hence the orbit expands during Jeansmode mass loss.When the accretor cannot accept the mass from the donor, because of a spin-upor a large Lacc, the excess mass will be ejected. This happens usually in theform of a jet. In this case the mass leaves the system with the specific angularmomentum of the accretor (hloss = a2

aω) and it turns out that

a

a=M

M

{2M(Md −Ma)−MdMa

MdMa

}(2.46)

It can then be shown that the orbit shrinks as long as Md/Ma & 1.28.

In a binary system a supernova can take place if a white dwarf is pushed overthe Chandrasekhar limit by mass transfer. When it is assumed that the massloss from the supernova is instantaneous, occurs in a circular orbit, and positionand velocity are the same after the explosion as they were before the explosion,then the orbital position where the supernova takes place is the new periastronof the post-supernova orbit. The post-supernova eccentricity is then given by

e =∆M

M1 +M2 −∆M(2.47)

Which means that the system will become unbound (e > 1) when more thanhalf the mass is lost during the explosion.

2.3.5 Fast models: Single Star Evolution (SSE)

In order to combine stellar evolution with the n-body simulation of a stellarcluster, a fast program is desired. SSE is such a program, it offers a fast ap-proximation of the evolution of stars and an accuracy within 5 % of that ofdetailed models (Hurley et al., 2000). The SSE program will be described infurther detail below.SSE uses formulas, as function of time, metallicity, and mass, to approximate theevolution of core mass, radius, and luminosity from the zero age main sequenceto the remnant stages (Hurley et al., 2000). The formulas are fitted based on agrid of evolutionary tracks, which were constructed using detailed — incorpo-rating convective overshooting — stellar models. The tracks are determined for


a stellar mass range from 0.5M� to 50M� and metallicities ranging from 0.0001to 0.03. First the mass function, with a specific metallicity, is fitted to get anidea of the functional form (Hurley et al., 2000), and then the fitted function isextended to a function of both mass and metallicity. Fitting functions are madefor different masses and different stages of stellar evolution. Mass division isbased on the maximum initial mass for which the helium flash takes place, andthe maximum initial mass for which helium ignites on the first giant branch.Formulas are constructed for the main sequence, Hertzsprung gap, first giantbranch, core helium burning, and asymptotic giant branch (Hurley et al., 2000).Various empirical relations are used to incorporate mass loss in the SSE pro-gram: Reimers’ formula for mass loss on the GB and beyond, de Jager’s formulafor the mass loss of massive stars throughout their entire evolution, and otherempirical relations for Wolf-Rayet stars and luminous blue variables.

Figure 2.10: Mass loss fraction over time (in Myr) of stars with various initialmasses and a solar-like metallicity (0.02). The numbers represent the initialmass in M�. Stars with a higher mass lose a larger fraction of mass on ashorter timescale. An exception is the case where there is a lot of mass loss dueto stellar wind, which happens during the evolution of the 50M� and 90M� star.These results were generated using the Single Star Evolution (SSE) program byOnno Pols.


Stellar

type

O-

50M�

O-

20M�

B-

10M�

A-

2M�

F-

1.2M�

G-

1.0M�

K-

0.6M�

M-

0.2M�

Evolu

tionphase

Tim

eM

assT

ime

Mass

Tim

eM

assT

ime

Mass

Tim

eM

assT

ime

Mass

Tim

eM

assT

ime

Mass

Main

sequenceStar

0.050.0

0.020.0

0.010.0

0.02.00

0.01.20

0.01.00

0.00.60

0.00.20

Hertzsprung

Gap

4.341.5

8.819.2

24.59.90

1.16·10

32.00

5.62·10

31.20

11.0·10

31.00

75.7·10

30.60

914·10

30.20

Giant

Branch

--

--

24.59.89

1.17·10

32.00

5.91·10

31.20

11.6·10

31.00

79.7·10

30.60

962·10

30.20

Core

Helium

Burning

4.341.4

8.819.2

24.59.89

1.20·10

32.00

6.38·10

31.05

12.3·10

30.76

--

--

First

AG

B-

-9.8

10.127.4

9.421.49

·103

1.986.51

·103

1.0112.46·10

30.72

--

--

SecondA

GB

--

--

--

1.49·10

31.95

6.51·10

30.96

12.5·10

30.59

--

--

Naked

Helium

MS

4.915.7

--

--

--

--

--

--

--

Naked

Helium

HG

4.911.9

--

--

--

--

--

--

--

Helium

WD

--

--

--

--

--

--

82.0·10

30.41

1.08·10

60.18

Carbon/O

xygenW

D-

--

--

-1.50

·103

0.646.51

·103

0.5412.5

·103

0.52-

--

-N

eutronStar

--

9.82.33

27.51.37

--

--

--

--

--

Black

Hole

4.911.7

--

--

--

--

--

--

--

Table

2.2:T

hem

ass(in

M�

)of

varioustype

stars,w

ithdifferent

initialm

asses,evolves

differentover

time

(inM

yr).R

esultsw

eregenerated

with

SSEusing

thesam

em

etallicityas

inF

igure1.9

Chapter 3

Tests and diagnostics

3.1 Diagnostics

To analyze the dynamical evolution of a stellar cluster there are a range ofdiagnostics that can be calculated. But first the cluster has to be initialized;velocities, masses, and positions are set using a Plummer sphere, then the stellardynamics package is defined. In these simulations PhiGRAPE for the GPU waschosen, because it is a fast, direct, n-body integrator. After the softening andtime step parameters are set, the stellar evolution package is defined. In thesesimulations SSE was chosen because it is fast and accurate within 5 percent ofdetailed models. Then the radius of the particles is set to zero to prevent thecomputational expensive collisions. The initial masses are set using the Salpetermass function. After the particles are shifted to the center of mass the diagnos-tics at t = 0 are calculated. Code snippets can be found in Appendix A.

Diagnostics can be compared and separated if computations are done beforeand after the stellar evolution step. For instance, the energy difference causedby evolution can be separated from the energy difference caused by the n-bodyintegrator.The AMUSE framework has built in functions to pass data from the AMUSElayer to the legacy layer, ie:

from_model_to_gravity.copy_attributes(["mass"])

and the other way around, ie:

from_gravity_to_model.copy()

The PhiGRAPE program uses individual, blocked, time steps; the time step of0.25 Myr is therefore a maximum time step, where the particles are synced tothe same time. Synchronization makes it possible to compute diagnostics andto set new masses based on the outcome of stellar evolution.PhiGRAPE and SSE are well tested within their own domain, the values ofthe phiGRAPE time step and the SSE parameters are the same as the valuessuggested by the programmers. A suitable value of the softening parameter wasdetermined by variation of the value over multiple n-body simulations. Withhigh values (ε ≥ 0.01 pc) the core of the stellar cluster did not collapse, and with

30 CHAPTER 3. TESTS AND DIAGNOSTICS

low values (ie. ε = 0) the program became slow because of the computationalexpensive close encounters.

What follows is a list of diagnostics with a brief explanation of how they werecomputed.

Time: The time is calculated in both SI and n-body units. Conversion ton-body units is done using the built in function:

convert_nbody.to_nbody( )

Energy: The energy error is a used as an indication for the accuracy of the n-body integrator. Conserved energy does not necessarily mean that the programis accurately representing reality, but a high energy error, on the other hand,does indicate that something is wrong.Before and after the evolution step the total energy is calculated and storedin different variables. By adding up values, energy differences are obtained forboth the evolution and the dynamical steps. A total energy error is computedby comparison with the initial energy, while the total error caused by the n-bodyintegrator is obtained by summing the per step values. The energy is obtainedusing the built in functions:

gravity.kinetic_energy + gravity.potential_energy

Mass: Total and per step information about the total mass of the cluster, andthe mass lost by evolution, is obtained by a simple loop through all the particlesin python:

totalMass = 0.0for x in particles:totalMass += x.mass.value_in(units.MSun)

T/V: A stable cluster obeys the virial theorem of an ideal gas in hydrostaticequilibrium: T/V = −0.5, where T is the kinetic energy, and V the potentialenergy. The fraction T/V therefore says something about the stability of thecluster. The built in functions are used to obtain kinetic and potential energy.

Half-mass radius: The half-mass radius gives an idea of the size of the cluster,and is later used to determine the number of stars that escaped from the cluster.The calculation is done using built in python functions. First all particles aresorted by their distance from the center of mass, then the masses are summeduntil half-mass:

stack = []

for x in particles:stackSize = len(stack)

stack.append([stackSize])

3.1. DIAGNOSTICS 31

stack[stackSize].append(x.distance)stack[stackSize].append(x.mass)

stack.sort(key=lambda x: x[1])

tmpMass = 0.0for x in range (0, stackSize):tmpMass += stack[x][2]if tmpMass / totalMass >= 0.5:RHalf = stack[x][1]

Escapers: The number of escapers can be calculated when also the total en-ergy of an individual particle is appended to the above stack. If a particle hasa positive energy and has a distance of more than 5 ·R1/2, it will be consideredto be unbound from the stellar cluster. A built in function for the potential ofa particle — potential() — was used in the computation of the energy.

Center of mass: In order to get accurate Lagrangian radii and core den-sity (see below), the escapers were not included in the calculation of the centerof mass. After a recalculation of the total mass, the center of mass is obtainedby the summation, weighted by particle mass, of the particle positions.

Lagrangian radii: These are the radii of various percentages of the totalmass, also of the escaped stars, as they develop over time; Lagrangian radii arean important tool to study the dynamical evolution of a stellar cluster. Themass fractions that were chosen are 0.5, 1, 5, 10, 25, 50, 75, and 90 percent.First the particles are shifted to the center of mass, and then sorted by thedistance from the center. Then it is just a manner of walking outside in a loopover all the particles.

Binaries: Binaries are formed mostly in the center of a stellar cluster andcan stop the core collapse. See also Chapter X. Binaries are found by computa-tion of the eccentricity; two stars are a binary when 0 ≤ e ≤ 1. The eccentricityis calculated with the following equations

e2 = 1− l2

G(M1 +M2)a, where l ≡ L

µ(3.1)

a = −G(M1 +M2)2ε

, where ε ≡ Ebin

µ(3.2)

Where l is the specific angular momentum, L the angular momentum, a thesemi major axis, µ the reduced mass, ε the specific orbital energy, G the gravi-tational constant, and M1 and M2 the masses.It is computational expensive to compute the eccentricity for each pair of starsof the stellar cluster. To speed up this calculation the stars were first sorted bytheir distance from the center of mass. Then the eccentricity of each star withits two closest ”neighbors” was calculated. These stars do not necessarily haveto be neighbors, because they might as well be at opposite sides of the cluster.By the comparison with two stars, however, the chances are large that all thebinaries will be, in fact, found. First the semi major axis is calculated. If it is


smaller than 0.1 pc the eccentricity is calculated.Three types of binaries are distinguished: hard, soft, and close binaries. If the(absolute) potential energy of the binary is larger than the average kinetic en-ergy of the particles in the cluster, the stars are considered to be hard binaries.The others are labeled soft binaries: binaries that are relatively easy to be dis-rupted by the other stars in the cluster. When the semi major axis is within 10percent of the softening parameter the binary, hard or soft, was considered tobe a close binary.The velocities, positions, and stellar masses were appended to a stack in orderto perform these calculations.

Core density: The core density is often defined as the density of the 5 starsclosest to the center of mass. When plotted against time it can be used todetermine the core collapse.The core density is determined in a similar fashion as the Lagrangian radii; thestars are sorted by their distance from the center of mass, then a loop is donefrom the innermost particle to the fifth particle. The distance from the latter isused to compute the core density.

Mass distribution: Each step the mass is plotted against the radius. Com-putation is done in a similar fashion as the Lagrangian radii.

Density distribution: Also plotted per step. Computation is similar to thecomputation of the core density.

Positions: Plotting the per step positions of the particles, the binaries, andthe center of mass can be very handy, especially if an obvious problem is en-countered in the numerical output. The phiGRAPE program has a OpenGLinterface which can be used to view the positions in 3D during runtime. But itcannot hurt to output plots as well, they can be put in a video, enabling theresearcher to fast forward and rewind through the simulation of a stellar cluster.

Core collapse: The moment of core collapse is numerically determined bytaking the maximum core density at which there are at least two hard binaries.The approximate time at which core collapse takes place is also determined priorto the first time step. First the virial radius (rv) — within the virial radius thesystem is in virial equilibrium — is calculated using rv = GM2/(−2V ), whereG is the gravitational constant, M is the total mass, and V is the potential en-ergy. Using the virial radius and the formulas from Chapter 1.2.1, the crossingtime, half mass relaxation time, and moment of core collapse are approximatedrespectively.

3.2 Tests

In the following section an isolated, n = 1024, system is tested against wellknown results. Isolation means ignoring gravity shocks and tidal interactionfrom the parent galaxy. All the gas from the molecular cloud is assumed to begone, and mass lost by stellar evolution does not interact with any of the stars,the mass is instantly removed instead. The particles all have zero radius, thus

3.2. TESTS 33

preventing collisions and mergers. Simulations were initialized with the samePlummer sphere. Variations occur in initial mass function (IMF) and in the oc-curance/incorporation of stellar evolution. Parameters of the relevant AMUSEmodules are listed in Appendix B.

The energy error is an indication for whether the simulations are accurate.A low energy error does not necessarily mean that the model is accurate, but ahigh error does indicate that something is wrong. The accuracy can also be de-termined by comparison with other models from well known papers, preferablyusing a completely different method.

3.2.1 One-component simulations

First simulations were done to determine if the maximum time step for phi-GRAPE converges to a similar evolution. Checking whether the time step con-verges is necessary because in a later stage the time step will be varied in thecase where stellar dynamics and evolution are combined. This time step is alsothe time over which the particles are synchronized in order to compute diag-nostics. Prior to the first time step the core collapse was approximated to takeplace at & 218 Myr for this specific set of 1024 particles, initialized using aPlummer sphere, with stellar masses of 0.345M� each.Although runs with identical initial conditions showed qualitatively very sim-ilar dynamical evolutions, they did show unexpected differences in the exactmoment of core collapse. Therefore multiple runs were done over which an av-erage was calculated. Per time step an average over 20 runs was calculated.The difference are probably due to the phiGRAPE GPU mode.The different time step runs show a qualitative similar evolution of the coreradius (Figure 3.1), although they seem to disagree on the moment of core col-lapse. In case ∆t = 0.25 Myr and ∆t = 0.125 Myr the core collapses at around190 Myr, similar to the expected value based on the well-known semi-analyticalformulae. Other diagnostics also appear very similar, ie. the number of escapers(Figure 3.3) and the central density (Figure 3.2). However, in case of a 0.5 Myrthe time step appears to take place quicker. Therefore 0.25 Myr was chosen asa usable time step for future simulations.

Qualitatively the simulations are in line with the well known theory, stars es-cape from the clusters (Figure 3.3), leading to contraction of the core while theouter layers expand (Figure 3.4). The gravothermal catastrophe, core collapse,takes place around the expected value of 200 Myr. Core collapse is eventuallystopped by the increase in hard binaries (Figure 3.5), which were verified tobe located very close to the center of mass. The system remains close to virialequilibrium (Figure 3.6), and the error eventually increases (Figure 3.7) due toclose encounters, especially close binaries with a semi major axis comparable tothe softening parameter, around the center of mass.


Figure 3.1: The core radius vs. time for different time steps, being 0.5, 0.25, and0.125 Myr from top to bottom. The core collapse is highlighted with a verticalline in the bottom two panels.

Figure 3.2: The central density vs. time for the same runs as in Figure 3.1.

3.2. TESTS 35

Figure 3.3: The number of escapers over time for the three time steps.

Figure 3.4: Lagrangian radii over time with ∆t = 0.25 Myr. The outer layersexpand while the core contracts.


Figure 3.5: Absolute number of binaries (hard, soft, and close) over time with∆t = 0.25 Myr.

Figure 3.6: Kinetic/Potential energy over time with ∆t = 0.25 Myr.

3.2. TESTS 37

Figure 3.7: Energy error over time with ∆t = 0.25 Myr. The error goes up withthe increase of close binaries.

3.2.2 Two-component simulations

This section describes the dynamical evolution of two component clusters; clus-ters containing two different masses. The following definitions are used in thissection: µ = mh/ml, where mh is the mass of the high mass stars, and ml themass of the low mass stars. Furthermore, q = Nh/Ncl, where Nh is the totalnumber of the high mass stars, and Ncl the total number of stars in the cluster.Every simulation started with the same, n = 1024, particle positions and veloc-ities as in the previous section. In each run a different, random, set of particleswas chosen to have a higher mass. Simulations were averaged over 20 runs andthe time step was set to 0.25 Myr.

Two-component systems have been extensively studied by Khalisi et al. (2007).The core radius (Figure 3.8) shows similar behavior to the radii found in thatpaper: the core collapse is accelerated by roughly a factor of 1/µ, and becomesless deep at higher µ. The latter is especially clear in the case where µ = 3.The acceleration of the dynamical evolution can also be seen by the number ofescapers (Figure 3.9), and hard binaries (Figure 3.10); these increase suddenlyaround core collapse, just as with the one-component runs. Also similar to theone-component simulations is the energy error (Figure 3.11), for which it wasverified that the increase was due to close binaries.Finally the fraction q was altered, and the result shows what is expected: ahigher number of high mass particles means that heat is more efficiently trans-ported outwards (Chapter 1.2.1), meaning that the core has to contract fasterto stay in thermal equilibrium.


Figure 3.8: Development of the core radius over time with q = 0.1 and threedifferent values for µ: 1, 2, and 3.

Figure 3.9: The number of escapers over time for the same runs that were usedfor the previous graph.

3.2. TESTS 39

Figure 3.10: The number of hard binaries over time for the same runs that wereused for the previous graph.

Figure 3.11: The energy errors over time for the same runs that were used forthe previous graph.


Figure 3.12: Development of core radius over time for three different fractionsq: 0, 0.1, and 0.25. The value of µ is kept at 2.

3.2.3 Simulations incorporating mass loss

In the first runs a fraction of the mass, of the same one-component n = 1024cluster, was taken away instantly to verify whether the cluster dissolves; for aPlummer sphere this should happen around Γ . 0.4, where Γ is the fraction ofthe mass left after mass loss (Boily and Kroupa, 2003a). The core radius indeedexpands strongly at Γ . 0.4 (Figure 3.13) and the cluster is unbound at thisspecific value of Γ according to the virial ratio (Figure 3.14).As predicted by Hills (1980) the cluster can remain in virial equilibrium whenthe mass loss is slow (Figure 3.16), divided over equal steps from 0 to 50 Myr,compared to the crossing time, which is ∼ 0.8 Myr for this particular cluster.When the mass loss is fast, the cluster has to find a new radius that fits thevirial theorem; in this case mass loss has a stronger disruptive effect on thedynamical evolution (Figure 3.15). The outcome is that the cluster has becomemore loosely bound, the virial ratio is closer to unity and the system suffersfrom more escapers (Figure 3.17).In the last set of runs the same Plummer sphere is used, but this time onerandom star has been given a 100 times higher mass. Runs with and withoutstellar evolution are compared. Figure 3.19 shows that the heavy star sankto either the core or at least close to the core when significant mass loss, dueto stellar evolution, becomes important. A fraction of 0.06 is lost, which, asexpected, slows down core collapse significantly (Figure 3.18), although thesame system does seem to recover to approximately the same virial ratio (Figure3.20). During the time where the system was more loosely bound, due to stellarevolution, more stars were able to escape compared to the runs without stellar

3.2. TESTS 41

evolution (Figure 3.21).

Figure 3.13: Core radius over time. At t = 50 Myr a fraction of the mass of allthe stars was taken away. Γ is the fraction that is left.

Figure 3.14: Virial ratios over time corresponding to the previous runs. Whenthe ratio > 1 the cluster is considered to be unbound.


Figure 3.15: Core radius over time. All stars lose half their mass, but in onecase this is done in equal steps from 0 to 50 Myr, and in the other case it isremoved suddenly at t = 50 Myr.

Figure 3.16: Virial ratios over time corresponding to the previous runs; thecluster can remain in virial equilibrium when the mass loss is slow compared tothe crossing time.

3.2. TESTS 43

Figure 3.17: The number of escapers over time corresponding to the previousruns.

Figure 3.18: Core radius over time obtained using the same Plummer sphere asin the previous section, but with one random star given a 100 times higher mass.In one of the simulations stellar evolution with SSE was turned on; resultingevolutionary stages of the high mass star are displayed.


Figure 3.19: Core density over time for the same runs as the previous graph.

Figure 3.20: Virial ratios over time for the same runs as the previous graph.

3.2. TESTS 45

Figure 3.21: Escapers over time for the same runs as the previous graph.

Chapter 4

New results: binarysystems

Binary systems are suitable to investigate the influence of time stepping be-cause they can be described analytically. Results from these simulations can bemeaningful for stellar cluster simulations, where binaries are formed close to thecenter of mass.A spherical, stellar-wind-driven, adiabatic (Jeans mode), mass loss is simulatedwhile alternations occur in the size of the time steps, mass loss rate, the eccen-tricity at t = 0, and the mass loss time span. The mass is assumed to leave thebinary system without interaction with the accreting star, and tidal interactionsare ignored.Three different scenarios are investigated. The first scenario includes a 50M�Wolf-Rayet (WR) star with a 2M� companion, with the WR star in its core he-lium burning phase. The mass loss rate is varied between 10−5 to 10−4M�/yr.The second scenario includes a 0.1M� and a 2M� star on the Asymptotic Gi-ant Branch (AGB); in this case the mass-loss rate is set between 10−7 and10−6M�/yr. Finally, a test scenario is used to investigate time stepping in theorder of the orbital period: a 2.0M� with a 1.0M� companion is simulatedwith an unrealistic high mass-loss rate ranging from 10−4 to 10−3M�/yr. Inthese test runs two types of time steps are used: a variable time step that scaleswith the binary period, meaning that in every orbit mass is being removed atthe same orbital location (angular-wise, see Figure 4.1), and a non-scaling timestep. In the latter, mass is removed on different orbital locations due to thechanging period.In all simulations the initial period is set to 1 year. Eccentricity, masses, andperiod are then used to determine the velocities and positions of the stars inthe x-y plane (see the code snippets in Appendix A).Results are compared to Jeans’ analytical value of a(T )/a(0) = M(0)/M(T )for adiabatic mass loss, where a and M are semi major axis and mass, andT is the time at which the simulation ended. This relation is obtained whenthe eccentricity is set to zero, but is similar to the relation for eccentric or-bits (Vanbeveren, 1998). The analytical value for the final eccentricity, in caseof the instant removal of mass from a circular orbit, is plotted as well usinge = [M(T )−M(0)]/M(T ) (see Chapter 2.3.3).

48 CHAPTER 4. NEW RESULTS: BINARY SYSTEMS

The AMUSE module twobody uses analytical formulae to calculate the positionand velocity of the binary stars over time, and is therefore a suitable package toperform accurate simulations. The used source code is described in AppendixA.

4.1 Results

The eccentricity and semi major axis, in an initially circular orbit, are in generalrelatively closer to the expected adiabatic values when the total number of stepsis increased. This happens because the chances are increased that mass is takenaway in a more symmetric manner if more steps are taken. The differences withthe adiabatic values increase with increasing mass-loss rate; this means thattime step size should decrease with increasing mass-loss rate in order to remainclose to the adiabatic variant. The above observations can be seen in case ofthe initially circular WR system (Figure 4.2 to 4.4 and 4.8 to 4.10) as well asin the AGB system (Figure 4.14 to 4.16 and 4.20 to 4.22).However, taking more steps does not necessarily mean the eccentricity andsemi major axis end up closer to the adiabatic value. For instance, at M =10−4M�/yr in Figure 4.2, the time step of 10.000 year is closer to the adi-abatic value (e = 0) than the 8.000 year time step. This can be explainedby the fact that, apparently, the 10.000 year steps are more symmetricallyspread over the mass-losing star its orbit than the 8.000 year steps. Thiswas further investigated in the test runs. When the time steps scale withthe period, a larger difference with the adiabatic value occurs because massis taken away at only one orbital point (the integer steps in Figure 4.26).

Figure 4.1: The mass-losing star is ini-tially in a circular orbit around the cen-ter of mass. Mass loss occurs at point 2and 3, and initially leads to an increasein eccentricity at point 2: e = ∆M/Mf .At point 3 mass loss circularizes the or-bit again.

The half integer steps mean that themass was taken away at opposite or-bital points, leading to a near adia-batic outcome (Figure 4.1). Up toM = 7 · 104 (in the test case - Figure4.26), taking the same mass away atonce (the analytical line in 4.26 indi-cating the upper value of instant massremoval in a circular orbit), wouldhave been closer to the analyticalvalue of e = 0 than when mass istaken away in integer steps. In otherwords, a single 150 yr step would havebeen a more accurate representationof the reality then 150 scaled steps of1 period.When the same initial steps are notscaled (Figure 4.27), the results forthe integer steps are closer to the an-alytical value. This is due to thechanging period: the semi major axisincreases, and Kepler’s 3rd law im-plies that the period must decrease.The changing period causes a more

4.1. RESULTS 49

symmetric mass loss when the time steps are fixed.

In an eccentric orbit (in these simulations e = 0.9) mass loss should have acircularizing effect, because more time is spend around the apastron (wherethe velocity is lower than in the periastron). This means more mass shouldbe lost around the apastron as well. Higher mass-loss rates should lead to amore circular orbit. In Figure 4.5 the eccentricity decreases between 10−5 and2 · 10−5M�/yr. At higher rates, and also in longer simulations (4.6 and 4.7),the eccentricity shows a somewhat chaotic development. This is likely due tothe fact that mass loss at periastron can have a strong disruptive effect, as canbe seen in Figure 4.19. In this simulation the binary system is disrupted whenthe time step is 100.000 years with a mass-loss rate of 10−6M�/yr.

4.1.1 WR runs

Figure 4.2: Eccentricity vs. mass-loss rate with T = 0.05 Myr, e(0) = 0.0,Md(0) = 50 M�, Ma = 2.0 M�, and P = 1 yr. The legend shows the step sizein years. All other simulations in this subsection describe stars with the sameinitial stellar masses and period.


Figure 4.3: Eccentricity vs. mass-loss rate with T = 0.1 Myr, and e(0) = 0.0


4.1. RESULTS 51





Figure 4.8: Semi major axis vs. mass-loss rate with T = 0.05 Myr, and e(0) =0.0

4.1. RESULTS 53

Figure 4.9: SMA vs. mass-loss rate with T = 0.1 Myr, and e(0) = 0.0


4.1. RESULTS 55



4.1.2 AGB runs

Figure 4.14: Eccentricity vs. mass-loss rate with T = 0.5 Myr, e(0) = 0.0,Md(0) = 2 M�, Ma = 0.1 M�, and P = 1 yr. The legend shows the step sizein years. All other simulations in this subsection describe stars with the sameinitial stellar masses and period.


4.1. RESULTS 57



4.1. RESULTS 59



4.1. RESULTS 61




4.1.3 Test runs

Figure 4.26: Eccentricity vs. mass-loss rate with T = 150 yr, e(0) = 0.0,Md(0) = 2 M�, Ma = 1 M�, and P = 1 yr. The legend shows the step size inyears. The time steps are scaled with the period.

Figure 4.27: The same run, but in this case the time steps are not scaled.

Chapter 5

New results: clustersimulations

The Hermite0 and SSE modules were combined for the direct n-body integrationand stellar evolution respectively. The parameters (Appendix B), and diagnos-tics, are the same as for the test runs in Chapter 3.A cluster with 1024 particles is initialized with a Plummer sphere and a SalpeterIMF. The Salpeter IMF included an 83M� star which is done evolving in 4.1Myr (see Table 5.1). All the other stars hardly evolve during the first 30 Myr(the time span chosen for the simulations). For different sets of runs the Plum-mer sphere is rescaled to different radii; clusters with a smaller radius have afaster dynamical evolution (because of the shorter crossing time), and thereforemight have different time steps which converge to the same result. The Plum-mer sphere in Figure 5.1 has a half mass radius of 0.56 pc and the massive starreaches the center after 3.5 Myr. In 5.2 the half mass radius is 1.12 pc and themassive star does not reach the center before the end of its evolution. Finally,in 5.3 the Plummer sphere was rescaled to a half mass radius of 0.4 pc and themassive star reaches the center in 2.8 Myr. The maximum energy errors, causedby the direct n-body integrator, lay between −0.8% and +0.3%.

Evolution phase Time (Myr) Mass (M�)Main sequence Star 0.0 83.230Hertzsprung Gap 3.6 55.946Naked Helium MS 3.7 22.305Naked Helium HG 4.1 14.391Black Hole 4.1 14.260

Table 5.1: The evolutionary phases of the 83M� star. This table was generatedusing SSE.

64 CHAPTER 5. NEW RESULTS: CLUSTER SIMULATIONS

5.1 Results

If mass loss occurs, the cluster responds by an increase of its radius on the dy-namical timescale. This is done to recover dynamical equilibrium. If mass lossis fast compared to the dynamical (crossing) timescale, the system is temporar-ily out of dynamical equilibrium and stars are able to escape. This causes adecrease in the cluster’s potential energy, and an increase in the kinetic energy,because the distance with the escaping stars increases and their velocities canhave a high value. Thus an increase in the virial ratio (|T/V |), of the cluster as awhole, is expected. It was verified that the number of escapers indeed increasedwith increasing virial ratio

The 0.005, 0.01, and 0.05 Myr time steps in Figure 5.1 converge to the sameresult. The 1.0 Myr time step is very close to these time steps as well, whilethe 0.1 and 0.25 Myr step clearly have a higher virial ratio. As with the binarysimulations, results are in general closer to the adiabatic value when the timestep is smaller (see also Figure 5.3).

By comparison of 5.1, 5.2, and 5.3 it seems that the time step size makes moredifference when the cluster is more tightly bound. In these simulations the dy-namical timescale decreases with increasing cluster density. This means that thehigh mass star segregates faster in 5.3, where the density is high. The stellarevolution of the high mass star will thus have more impact on the dynamicalevolution in this case. The time steps in 5.2 all converge to the same virial ratio,which is probably due to the fact that the high mass star has not segregatedyet before the end of its evolution. The cluster quickly dissolves because it ismore loosely bound.

5.1. RESULTS 65

Figure 5.1: Virial ratio vs time. Initial half mass radius: r1/2(0) = 0.56 pc.Initial crossing time: tcr(0) = 0.82 Myr. The legend displays the various timesteps in Myr.

Figure 5.2: Virial ratio vs time, r1/2(0) = 1.12 pc, tcr(0) = 2.31 Myr.

66 CHAPTER 5. NEW RESULTS: CLUSTER SIMULATIONS

Figure 5.3: Virial ratio vs time, r1/2(0) = 0.4 pc, tcr(0) = 0.53 Myr.

Chapter 6

Discussion, Conclusion, andRecommendations

By the qualitative comparison with earlier research in Chapter 3 it can beconcluded that AMUSE is a suitable framework for the dynamical simulationof stellar clusters, either with or without mass loss. The timescales for corecollapse in the dynamical simulations were in line with the expected values:& 10 half-mass relaxation times (trh) for one-component clusters, acceleratedby ml/mh (the masses of the low and high-mass component) for two componentclusters, and . 0.1trh for clusters with a Salpeter mass function. Furthermore,the number of binaries, close to the center of mass, does indeed suddenly in-crease around core collapse.When mass loss was incorporated in the dynamical simulation, it was observedthat the clusters expand, and can dissolve if roughly more than half the massis removed within the crossing timescale; in earlier research this was indeed theobtained approximate value for Plummer spheres. As expected, the cluster canremain in dynamical equilibrium when the mass loss is slow compared to thecrossing time; this means that there are less escapers and the system remainsaround the virial ratio of 0.5. From Chapter 3 it can also be concluded that thephiGRAPE module is not suitable if reproducible results are desired, becausethere are per run differences when the runs are performed under the exact sameinitial conditions. This is unfortunate and should be further investigated; thephiGRAPE module, in GPU mode, is significantly faster than other direct n-body integrators on the CPU.

From the binary simulations in Chapter 4 it can be concluded that, in gen-eral and in an initially circular orbit, adiabatic mass loss simulations are moreaccurate with decreasing time steps; smaller time steps are usually more sym-metrically spread over the orbit of the mass-losing star. Similar, if the timespan for mass loss in the simulations is increased, the final eccentricity and semimajor axis are closer to the analytical values as well. Furthermore, it can beconcluded that the time step should decrease with increasing mass-loss rate, inorder to keep the results close to the analytical values. If the time steps arescaled with the period, and in the order of the binary period, a large differencewith the analytical value occurs when mass is always taken away at the same

68 CHAPTER 6. DISCUSSION, CONCLUSION, AND RECOMMENDATIONS

orbital point (the periastron). Choosing two opposite orbital points for massloss with a scaled time step is enough to get a result close to the adiabaticvalue. When, in the same simulation, the time step does not scale with theperiod, results are in general close to the adiabatic value, because the changingperiod causes mass loss to occur in a more symmetrical manner. This couldbe investigated in more detail by a graph describing the eccentricity versus thenumber of steps per period. The orbital points for the mass loss to occur couldbe randomized using a Monte Carlo method and the upper and lower limitscould be found by purposely stepping in the most symmetrical and most asym-metrical manner.In an eccentric orbit (with e = 0.9) the evolution seems to be more chaotic:stepping in the periastron can have a disruptive effect on the binary. Massloss should circularize the orbit, which can only be observed at low mass-lossrates. Therefore it can be concluded that the time step in eccentric orbits (withe = 0.9) should be smaller than in initially circular orbits.

When stellar evolution and stellar dynamics are combined in cluster simulations(Chapter 5), it seems that smaller time steps, for the particle synchronizationof the Hermite integrator followed by an instant step of stellar evolution, areneeded when the dynamical time scale of the cluster decreases. This is possiblydue to the fact that a shorter dynamical timescale causes the cluster to segregatefaster, and therefore mass loss has a more disruptive effect on these clusters.However, the clusters in these particular simulations were only describing onespecific Salpeter initial mass function (which included a massive star of 80M�),while positions and velocities were initially set using a Plummer sphere.As with binary simulations, it appears that the results are closer to the adia-batic variant when the time step for stellar evolution is smaller, although it isunsure what the exact adiabatic value for the virial ratio is, because there areno analytical solutions at hand.In the cluster simulations with a half mass radius (r1/2) of 1.12 pc, all testedtime steps between 0.5 and 0.1 Myr converged to the same virial ratio. In thesimulations with r1/2 = 0.56 pc, the lowest tested time steps (5000 to 50.000 yr)converged. Surprisingly, the time step of 1.0 Myr converged with these shorttime steps as well. In analogy to the binary runs, it seems that there are certaintime steps which result in an unexpected outcome. When r1/2 = 0.4 pc, nonof the time steps seem to converge and it should be investigated if smaller timesteps do converge.Further research is needed to investigate the time stepping in stellar clusters.This could be done by averaging over the outcome of a set of various Salpeterinitial mass functions. It could also be interesting to investigate the limits inwhich the massive star(s) are initially segregated, and thus close to the centerof mass, or in the outer layers of the stellar cluster.

Appendix A

Code snippets

1.1 Cluster simulation

In AMUSE, a simplified code for the combination of the phiGRAPE stellardynamics package and the SSE stellar evolution package may look as followed.Note that initialization is done using a Plummer Sphere and a Salpeter massfunction.

particles = MakePlummerModel(number_of_stars);

gravity = PhiGRAPE(mode="gpu")gravity.initialize_code()gravity.parameters.timestep_parameter=0.01gravity.parameters.initial_timestep_parameter=0.01gravity.parameters.epsilon_squared = 0.000001 | units.parsec ** 2

stellar_evolution = SSE()stellar_evolution.initialize_module_with_default_parameters()

particles.radius = 0.0 | units.RSun

particles.mass = salpeter_masses

move_particles_to_center_of_mass(particles)

+---------- Diagnostic calculations at t = 0 ----------+

while time < end_time:

time += 0.25 | units.Myrgravity.evolve_model(time)from_gravity_to_model.copy()

+-------- Diagnostic calculations at t = time --------+

stellar_evolution.evolve_model(time)

70 APPENDIX A. CODE SNIPPETS

from_stellar_evolution_to_model.copy_attributes(["mass", "radius"])from_model_to_gravity.copy_attributes(["mass"])

move_particles_to_center_of_mass(particles)

+-------- Diagnostic calculations at t = time --------+

1.2 Binary simulation

In the AMUSE framework, the (simplified) source code for one specific massloss rate and time step looks as followed

import numpy

from amuse.legacy.twobody import twobody

from amuse.support.units import unitsfrom amuse.support.data import corefrom amuse.support.units import constantsfrom amuse.support.units import nbody_system

def binary(m1 = 50.0 | units.MSun,\m2 = 2.0 | units.MSun,\r1 = 0.0 | units.RSun,\r2 = 0.0 | units.RSun,\

period = 1.0 | units.yr,\ecc = 0.0):

convert_nbody = nbody_system.nbody_to_si(1.0 | units.MSun, 1.0 | units.AU)

mu=constants.G*(m1+m2)

# semi major axisa=(period**2/(4.0*numpy.pi**2)*mu)**(1./3.)

# apastronra=a*(1.0+ecc)

# angular momentumh=(mu*a*(1.0-ecc**2))**0.5

# vel at aphelionva=h/ra

x1=m2/(m1+m2)*ray1=z1=0.*x1vy1=-m2/(m1+m2)*vavx1=vz1=0.*vy1

x2=-m1/(m1+m2)*ra

1.2. BINARY SIMULATION 71

y2=z2=0.*x2vy2=m1/(m1+m2)*vavx2=vz2=0.*vy2

parts=core.Particles(2)

parts[0].mass=m1parts[0].radius=r1parts[0].x=x1

[.. etc ..]

gravity = twobody.TwoBody(convert_nbody)gravity.particles.add_particles(parts)

#set initial timestepdt = period/10.0

#set mass loss ratemdot = 10.0e-6 | units.MSun / units.yr

time = 0.0 | units.yr

while time < end_time:time += dtgravity.particles[0].mass = m1 - Mdot * dtgravity.evolve_model(time)

[ .. recalculate reduced mass, semi major axisangular momentum, eccentricity, period, step size .. ]

if __name__=="__main__":binary()

72 APPENDIX A. CODE SNIPPETS

Appendix B

Parameters

The test runs in Chapter 3 were done using PhiGRAPE in GPU mode, and thecluster simulations in Chapter 5 were done using the Hermite0 module. When-ever stellar evolution is involved the same SSE parameters are used.

PhiGRAPE parameters:time step parameter: 0.01initial time step parameter: 0.01softening parameter: 10−3 pc

Hermite0 parameters:softening parameter: 10−3 pc

SSE parameters:Metallicity: 0.02Reimers mass loss coefficient: 0.5Helium star mass loss factor: 1.0SN kick speed dispersion: 190.0 km/sWD IFMR flag: 0WD cooling flag: 1BH kick flag: 0NS mass flag: 1Maximum NS mass: 3.0 M�Fractional time step 1: 0.02Fractional time step 2: 0.01Fractional time step 3: 0.05

74 APPENDIX B. PARAMETERS

Bibliography

S. J. Aarseth. Direct methods for N-body simulations. In J. U. Brackbill &B. I. Cohen, editor, Multiple time scales, p. 377 - 418, pages 377–418, 1985.

S. J. Aarseth. Gravitational N-Body Simulations. Cambridge University Press,2003.

S. J. Aarseth. The Cambridge N-Body lectures. Cambridge University Press,2008.

J. H. Applegate. Dynamical effects of stellar evolution in globular clusters. ApJ,301:132–144, February 1986. doi: 10.1086/163881.

J. Barnes and P. Hut. A hierarchical O(N log N) force-calculation algorithm.Nature, 324:446–449, December 1986. doi: 10.1038/324446a0.

R. R. Bate, D. D. Mueller, and J. E. White. Fundamentals of astrodynamics.1971.

C. M. Boily and P. Kroupa. The impact of mass loss on star cluster for-mation - I. Analytical results. MNRAS, 338:665–672, January 2003a. doi:10.1046/j.1365-8711.2003.06076.x.

C. M. Boily and P. Kroupa. The impact of mass loss on star cluster formation- II. Numerical N-body integration and further applications. MNRAS, 338:673–686, January 2003b. doi: 10.1046/j.1365-8711.2003.06101.x.

D. F. Chernoff and M. D. Weinberg. Evolution of globular clusters in the Galaxy.ApJ, 351:121–156, March 1990. doi: 10.1086/168451.

J. M. A. Danby. Fundamentals of celestial mechanics. 1992.

P. Eggleton. Evolutionary Processes in Binary and Multiple Stars. January2006.

J. M. Fregeau, K. J. Joshi, S. F. Portegies Zwart, and F. A. Rasio. Mass Segrega-tion in Globular Clusters. ApJ, 570:171–183, May 2002. doi: 10.1086/339576.

T. Fukushige and D. C. Heggie. Pre-collapse evolution of galactic globularclusters. MNRAS, 276:206–218, September 1995.

E. Gaburov, S. Harfst, and S. Portegies Zwart. SAPPORO: A way to turn yourgraphics cards into a GRAPE-6. New Astronomy, 14:630–637, October 2009.doi: 10.1016/j.newast.2009.03.002.

76 BIBLIOGRAPHY

M. A. Gurkan, M. Freitag, and F. A. Rasio. Formation of Massive Black Holesin Dense Star Clusters. I. Mass Segregation and Core Collapse. ApJ, 604:632–652, April 2004. doi: 10.1086/381968.

S. Harfst, A. Gualandris, D. Merritt, R. Spurzem, S. Portegies Zwart, andP. Berczik. Performance analysis of direct N-body algorithms on special-purpose supercomputers. New Astronomy, 12:357–377, July 2007. doi:10.1016/j.newast.2006.11.003.

W. E. Harris. A Catalog of Parameters for Globular Clusters in the Milky Way.AJ, 112:1487–+, October 1996. doi: 10.1086/118116.

C. B. Haselgrove and F. Hoyle. A mathematical discussion of the problem ofstellar evolution, with reference to the use of an automatic digital computer.MNRAS, 116:515–+, 1956.

A. Heger, N. Langer, and S. E. Woosley. Presupernova Evolution of RotatingMassive Stars. I. Numerical Method and Evolution of the Internal StellarStructure. ApJ, 528:368–396, January 2000. doi: 10.1086/308158.

J. G. Hills. The effect of mass loss on the dynamical evolution of a stellarsystem - Analytic approximations. ApJ, 235:986–991, February 1980. doi:10.1086/157703.

J. R. Hurley, O. R. Pols, and C. A. Tout. Comprehensive analytic formulae forstellar evolution as a function of mass and metallicity. MNRAS, 315:543–569,July 2000. doi: 10.1046/j.1365-8711.2000.03426.x.

J. R. Hurley, C. A. Tout, and O. R. Pols. Evolution of binary stars and theeffect of tides on binary populations. MNRAS, 329:897–928, February 2002.doi: 10.1046/j.1365-8711.2002.05038.x.

P. Hut, J. Makino, and S. McMillan. Building a better leapfrog. ApJ, 443:L93–L96, April 1995. doi: 10.1086/187844.

H. Karttunen, P. Kroeger, H. Oja, M. Poutanen, and K. J. Donner. Fundamentalastronomy. 2003.

E. Khalisi, P. Amaro-Seoane, and R. Spurzem. A comprehensive NBODY studyof mass segregation in star clusters: energy equipartition and escape. MNRAS,374:703–720, January 2007. doi: 10.1111/j.1365-2966.2006.11184.x.

R. Kippenhahn and A. Weigert. Stellar Structure and Evolution. Springer-Verlag Berlin Heidenberg, 1994.

R. P. Kudritzki and D. Reimers. On the absolute scale of mass-loss in redgiants. II. Circumstellar absorption lines in the spectrum of alpha Sco B andmass-loss of alpha Sco A. A&A, 70:227–239, November 1978.

J. Makino. Postcollapse Evolution of Globular Clusters. ApJ, 471:796–+,November 1996. doi: 10.1086/178007.

J. Makino and S. J. Aarseth. On a Hermite integrator with Ahmad-Cohenscheme for gravitational many-body problems. PASJ, 44:141–151, April 1992.

BIBLIOGRAPHY 77

J. Makino, M. Taiji, T. Ebisuzaki, and D. Sugimoto. GRAPE-4: A MassivelyParallel Special-Purpose Computer for Collisional N-Body Simulations. ApJ,480:432–+, May 1997. doi: 10.1086/303972.

J. Makino, T. Fukushige, M. Koga, and K. Namura. GRAPE-6: Massively-Parallel Special-Purpose Computer for Astrophysical Particle Simulations.PASJ, 55:1163–1187, December 2003.

S. L. W. McMillan. Star Cluster Simulations Including Stellar Evolution. InJ. Makino & P. Hut, editor, Astrophysical Supercomputing using Particle Sim-ulations, volume 208 of IAU Symposium, pages 131–+, 2003.

H. Nieuwenhuijzen and C. de Jager. Parametrization of stellar rates of massloss as functions of the fundamental stellar parameters M, L, and R. A&A,231:134–136, May 1990.

F. I. Pelupessy, P. P. van der Werf, and V. Icke. Periodic bursts of star for-mation in irregular galaxies. A&A, 422:55–64, July 2004. doi: 10.1051/0004-6361:20047071.

H. C. Plummer. On the problem of distribution in globular star clusters. MN-RAS, 71:460–470, March 1911.

O.R. Pols. Stellar Evolution syllabus. September 2007.

S. Portegies Zwart, S. McMillan, S. Harfst, D. Groen, M. Fujii, B. O. Nu-allain, E. Glebbeek, D. Heggie, J. Lombardi, P. Hut, V. Angelou, S. Banerjee,H. Belkus, T. Fragos, J. Fregeau, E. Gaburov, R. Izzard, M. Juric, S. Justham,A. Sottoriva, P. Teuben, J. van Bever, O. Yaron, and M. Zemp. A multiphysicsand multiscale software environment for modeling astrophysical systems. NewAstronomy, 14:369–378, May 2009. doi: 10.1016/j.newast.2008.10.006.

S. Portegies Zwart, S. McMillan, and M. Gieles. Young massive star clusters.ArXiv e-prints, February 2010.

S. F. Portegies Zwart, P. Hut, J. Makino, and S. L. W. McMillan. On thedissolution of evolving star clusters. A&A, 337:363–371, September 1998.

G. D. Quinlan. The time-scale for core collapse in spherical star clusters. NewAstronomy, 1:255–270, November 1996. doi: 10.1016/S1384-1076(96)00018-8.

E. E. Salpeter. The Luminosity Function and Stellar Evolution. ApJ, 121:161–+, January 1955. doi: 10.1086/145971.

L. Spitzer. Dynamical evolution of globular clusters. 1987.

R. Spurzem. Direct N-body Simulations. Journal of Computational and AppliedMathematics, 109:407–432, September 1999.

D. Vanbeveren. The Brightest Binaries. 1998.

F. Verbunt. Binaries syllabus. September 2007.

E. Vesperini, S. L. W. McMillan, and S. Portegies Zwart. Effects of PrimordialMass Segregation on the Dynamical Evolution of Star Clusters. ApJ, 698:615–622, June 2009. doi: 10.1088/0004-637X/698/1/615.

Index

AMUSE, 1, 2, 9, 12, 29, 33, 48, 69, 70AMUSE modules

BHTree, 11Hermite0, 2PhiGRAPE, 2, 11, 29, 32SSE, 26, 29Twobody, 20

Asymptotic Giant Branch, 24

BinariesHard binaries, 32Soft binaries, 32

Core collapse, 6, 7–9, 31–33, 37, 40Core density, 32Core radius, 33, 37, 40Crossing time, 6CUDA, 19

Empirical mass loss formulaeDe Jager, 25Reimers, 24

ENIAC, 3Equipartition, 6

GPU, 2, 19GRAPE, 2, 15, 18

Half mass relaxation time, 6Hermite integration scheme, 14Hertzsprung-Russell Diagram, 24Homology, 23Horizontal Branch, 24

Jeans mass, 1Jeans mode mass loss, 7

Lagrangian radii, 31, 32, 35

Mass segregation, 7MODEST, 12

Plummer model, 16

Primordial mass segregation, 7

Red Giant Branch, 24Relaxation, 6

Salpeter initial mass function, 7, 17Sapporo library, 19Segregation timescale, 7Softening parameter, 14, 29, 73Stellar clusters

Globular clusters, 5Open clusters, 5

Stellar evolution timescaleDynamical timescale, 21, 22Nuclear timescale, 22Thermal timescale, 22, 23

Stellar structure equations, 16

Tree code, 15

Universal variable formulation, 20

Virial radius, 6, 32Virial theorem, 7

Thesis Guillermo Kardolus

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