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Faculty of Sciences Department of Physics and Astronomy A numerical study of recent tidal interactions between dwarf galaxies Lars Bonne Supervisor: Prof. Dr. S. De Rijcke Dissertation advisor: R. Verbeke Dissertation submitted to obtain the academic degree of Master in Physics and Astronomy Academic year 2015-2016 Contents Acknowledgement 1 Introduction 1.1 The ΛCDM model . . . . . . 1.2 Dwarf Galaxies . . . . . . . . 1.2.1 Classification . . . . . 1.3 Tidal interactions . . . . . . . 1.4 Blue Compact Dwarfs . . . . 1.4.1 classification of BCDs 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Simulating and analysing dwarf galaxies 2.1 N -body/SPH simulations . . . . . . . . 2.2 Dwarf Galaxies and Initial Conditions . . 2.2.1 Included physics . . . . . . . . . 2.2.2 gogoIC, Kuzkut and Ganic . . . . 2.2.3 Initial conditions . . . . . . . . . 2.3 Interaction of dwarf galaxies . . . . . . . 2.3.1 Different interaction models . . . 2.3.2 The models . . . . . . . . . . . . 2.3.3 Initial velocity . . . . . . . . . . . 2.3.4 Amount of simulations . . . . . . 2.4 Hyplot . . . . . . . . . . . . . . . . . . . 2.4.1 Recognising the galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results 3.1 Star formation history . . . . . . . . . . . . . 3.1.1 Burst factors in the interacting models 3.1.2 Prograde vs retrograde . . . . . . . . . 3.1.3 Duration of the bursts . . . . . . . . . 3.2 Velocity dispersion . . . . . . . . . . . . . . . 3.3 The gas mass . . . . . . . . . . . . . . . . . . 3.4 B-I . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Gas distribution and gas flow . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 7 9 10 11 12 . . . . . . . . . . . . 13 13 18 18 23 24 36 38 41 43 45 46 46 . . . . . . . . 49 49 51 57 57 58 62 67 73 3.6 3.7 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-band surface brightness profile . . . . . . . . . . . . . . . . . . . . . . 4 Discussion 4.1 Comparison with merger tree . . . . 4.2 Gas flow and burst duration . . . . . 4.3 Moment of the burst . . . . . . . . . 4.4 Classification of the bursting galaxies 4.5 Metallicity . . . . . . . . . . . . . . . 4.6 Effect of tidal stripping and bursts on 4.7 The amount of bursts . . . . . . . . . . . . . . . . . . . . . . . . later . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 83 89 89 92 95 96 97 104 104 5 Conclusion 107 6 Nederlandstalige Samenvatting 109 Appendices 111 4 Acknowledgement First of all I would like to thank prof. De Rijcke for several reasons. One of them is providing this thesis subject and giving me the possibility to work on this topic. For me, it has been an interesting year discovering the world of dwarf galaxies and their tidal interactions. Further I would like to thank him for useful remarks during the year and making it possible for me to attend the workshop on Computational Solar and Astrophysical Modeling in September. It provided me with an intensive training in astrophysical simulations, which saved me a lot of time when fixing problems over the entire year such that I had more time to focus on the results. When talking about fixing problems, it is impossible for me not to thank Robbert. Helping me out over and over again when I got stuck trying to fix a problem. Further I also want to thank Robbert for answering all my questions, giving useful suggestions, insights and providing me with several scripts making it a lot easier for me to perform analysis of more complicated galaxy properties. I would also like to thank my friends and girlfriend for reminding me that there is more than finishing a dissertation. Lastly I want to thank my family for making it possible for me to study here in Ghent and showing interest in what I’m doing. In specific I want to mention my parents, giving a teenager the freedom of choosing his study of interest starting from the age of 13 till this moment. Even when he is ignoring the entire time advice from almost every high school teacher he had. 5 Chapter 1 Introduction 1.1 The ΛCDM model Trying to understand the evolution of our universe has led to the so called ΛCDM model, which is also known as the standard model of cosmology by now. The name refers to the two main current constituents of the universe according to this model. The Λ refers to a cosmological constant, associated with the unknown dark energy. And the CDM stands for cold dark matter, where the preposition cold implies that the dark matter was not relativistic at the moment it abandoned equilibrium. The cosmoligical constant was first introduced by Einstein to be able to derive a static universe from his equations. After his static universe lost all credibility when Edwin Hubble observed the expansion of the universe (Hubble, 1929), the cosmological constant was eventually reestablished to explain the observed accelerating expansion of the universe. This was discovered by looking at distant type Ia supernovae (Riess et al., 1998; Perlmutter et al., 1999). These supernovae occur when a white dwarf in a binary system superseeds the Chandrasekhar mass due to mass transfer in this binary system. In the ΛCDM model, because of the fact that the dark matter is supposed to be cold, the first dark matter halos that will be formed are the smallest ones. This implies that dwarf galaxies will be the first structures formed in the early universe. Merging of these smaller halos will lead to the formation of bigger structures. 1.2 Dwarf Galaxies As the name already indicates, a dwarf galaxy is a galaxy which is considerably smaller than some well known galaxies in our local group such as our own galaxy, M31 and M33. A dwarf galaxy has a diameter of a few kiloparsec (kpc) while the larger galaxies have diameters that can be expressed in units of 10 kpc. The presence of a dark matter halo that embeds the system is a requirement proposed for a while now (e.g. Mateo, 1998). This requirement would exclude globular clusters since it is believed that these systems do not have a dark matter halo. Apart from the 7 dark matter halo, efforts have been done to obtain a definition for dwarf galaxies. An often used definition was proposed by Tammann (1994). This definition was based on magnitudes in certain bands. Namely galaxies fainter than MV = -17 and MB = -16. The fact that dwarf galaxies are small makes them very interesting study objects. The first reason for this brings us back to the early stages of our universe when considering the ΛCDM model since in this model dwarf galaxies are the first structures that were formed. Because of their early formation and their small scale they are good study material for influences in the early universe, e.g. population III stars, the onset of the cosmic ultraviolet background (UVB),... Although dwarf galaxies should be the first systems that are formed, they produced some conflicting results with the CDM model that was very successful at large scales. Some of these successes were the large scale structure of galaxies (Davis et al., 1985) and the anisotropy in the cosmic microwave background (Peebles, 1987). On the other hand, one of the well known problems posed by dwarf galaxies is the ’missing satellites’ problem (Klypin et al., 1999; Moore et al., 1999). Since observationally there were less satellite dwarf galaxies found around major galaxies than the amount of dark matter substructures that were predicted by cosmological simulations. It was already proposed then that the UVB and supernova feedback could suppress the formation of these dwarf galaxies (e.g. Efstathiou, 1992; Haardt & Madau, 1996), such that this issue might be resolved. This missing dwarf problem, together with some other problems, mostly seems to be resolved now in simulations of Local Group environments due to ram pressure stripping, UVB and supernova feedback (Sawala et al., 2016). Another problem related to dwarf galaxies in this context is the existence of faint gasrich dwarf galaxies with rotational velocities of the order of 15 km/s, such as Leo P, Leo T and Pisces A (e.g. Ryan-Weber et al., 2008; Giovanelli et al., 2013; Tollerud et al., 2015), since more than 90 % of the dwarf galaxies with such rotational velocities are predicted to be dark halos (Sawala et al., 2014). This can be expected since previously mentioned effects remove gas from these low mass dwarf galaxies. However Tollerud et al. (2015) found significantly more faint gas-rich dwarf galaxies compared to what would be expected from cosmological simulations. This issue could be resolved by including the effects of population III stars in merger tree simulations of dwarf galaxy formation (Verbeke, Vandenbroucke & De Rijcke, 2015). The strong UV radiation coming from these population III stars delays the star formation in low mass dwarf galaxies, such that faint gas-rich dwarf galaxies can be found. As should be explained by now, it is very difficult for dwarf galaxies to keep most of their initial amount of baryons contained. This is in contrast with the more massive galaxies for which this seems less difficult. Thus creating a qualitative distinction of dwarf galaxies. An explanation for this is the fact that dwarf galaxies have a much more 8 shallow gravitational potential than the more massive galaxies. Due to this shallow gravitational potential the UVB, supernovae,... eject large amounts of baryons out of the dwarf galaxies (Tolstoy, Hill & Tosi, 2009). Due to the fact that they do not succeed in containing most of their initial gas, dwarf galaxies are heavily dark matter dominated. This attracted the attention from particle physicists. They hope to detect the annihilation of dark matter in dwarf galaxies, more specifically in dwarf spheroidal galaxies since they are even more dark matter dominated. A thing one would almost forget when talking about dwarf galaxies due to their low luminosity, is that they are more abundant in the universe than massive galaxies such that they certainly deserve the necessary attention to be well understood. Many observations of dwarf galaxies have led to their classification into different types (Tolstoy, Hill & Tosi, 2009). This classification will be shortly discussed in the following part. 1.2.1 Classification - Dwarf ellipticals (dE): These dwarf galaxies don’t have a lot of gas left and barely form any stars at the moment. They are mostly found in clusters of galaxies. Their magnitudes in the blue band are in the interval: -18 < MB < -14. As can be seen, this is not completely in agreement with the definition by Tammann (1994). However as indicated as well before, there is no unique general definition for dwarf galaxies. Another characteristic of dwarf elliptics are their smooth elliptical isophotes. - Dwarf Spheroidals (dSph): There is basically not much difference with dwarf ellipticals. The main distinction is based on the magnitude, where dwarf spheroidals are constrained by: -14 < MB < -8. Such that they are fainter than dwarf ellipticals. So they are less massive and more dark matter dominated. Dwarf ellipticals and spheroidals together are also classified as early type dwarf galaxies. - Dwarf irregulars (dIrr): In contrast to the two previous types, dwarf irregulars are still forming stars. They are also flattened by rotation (Côté, Carignan & Freeman, 2000) and are usually found in isolation in contrast to early type dwarf galaxies (e.g. Côté et al., 2009). They are classified as irregulars due to their isophotes that are not as smooth as the early type galaxies. - Blue compact dwarfs (BCD): Just like dwarf irregulars, they have irregular isophotes. However there is a difference with dwarf irregulars since they are generally bluer, more compact and have a higher central surface brightness (Papaderos et al., 1996a; Salzer & Norton, 1999). Together with the dwarf irregulars, they are referred to as late type dwarf galaxies. These BCDs will be discussed in more detail later in the introduction, since relations with tidal interactions are proposed. 9 - Transition type dwarfs (TTD): They are still forming stars at a rather low rate and possess properties of both early and late type dwarf galaxies. As their name suggests, they are expected to connect the early and late type dwarf galaxies. It was shown by Koleva et al. (2013) that with continued gas removal, these dwarf galaxies might evolve to dwarf ellipticals. - Ultra-faint dwarfs(uFd): Dwarf galaxies of this type have been observed rather recently. These dwarf galaxies have a very low metallicity (Kirby et al., 2008) and are strongly dark matter dominated (Salvadori & Ferrara, 2009; Bromm & Yoshida, 2011). It is assumed that they only had very few early star formation. - Ultra compact dwarfs (UCD): These are dwarf galaxies with very small effective radii and they are extremily faint (Penny et al., 2014). Compared to uFd, they are compacter and less dark matter dominated. The formation of these galaxies is still poorly understood. - Tidal dwarfs: Lastly one could mention this type of dwarf galaxy. These are formed in the tidal tails of interacting galaxies, which leads to the fact that they don’t possess a lot of dark matter. Because they are formed in tidal tails, they are formed out of enriched gas. Therefore these dwarf galaxies generally have high metallicities (Duc et al., 2000; Weilbacher, Duc & Fritze-v. Alvensleben, 2003; Sweet et al., 2014). 1.3 Tidal interactions In this thesis, we will specifically look at tidal interactions of dwarf galaxies. The interest in astronomy for tidal interactions between galaxies is certainly not new. This comes to no surprise, since probably most galaxies are affected somewhere in their evolution by interaction with other galaxies (e.g. Toomre, 1977). It was the Swiss astronomer Fritz Zwicky, also known for e.g. the first proposal of unseen dark matter when observing the Coma cluster, who first proposed that large extended stuctures seen in galaxies were caused by tidal interactions (Zwicky, 1956). This was certainly not directly accepted as an explanation, e.g. Gold & Hoyle (1959) suggested that electromagnetic interactions would contribute as well. That gravitational interaction could create these narrow extended structures was proved afterwards by several investigations (e.g. Pfleiderer & Siedentopf, 1961; Toomre & Toomre, 1972). These calculations were performed using simplified assumptions, yet demonstrated that close passages of galaxies can create bridges and tails. These bridges and tails are the most dominant features of tidally interacting galaxies. Since many dwarf galaxies are found near massive galaxies, studying this tidal interaction could be interesting to see the effect on the evolution of the dwarf galaxies (e.g. Mayer et al., 2001b, 2006; Klimentowski et al., 2009; Valcke, 2010). 10 These simulations demonstrated that the presence of a Milky Way-like dark matter halo has an influence on the evolution of the dwarf galaxies. In all studies tidal stripping and tidal tails were observed. These interactions also led to star formation bursts in the dwarf galaxies. And there was stated by Mayer et al. (2001a) that dwarf galaxies in the Milky Way potential were transformed from low surface brightness dwarfs to dSphs and from high surface brightness dwarfs to dEs. In the context of dwarf galaxy interactions, mergers of dwarf galaxies have been studied (e.g. Bekki, 2008; Cloet-Osselaer et al., 2014; Starkenburg, Helmi & Sales, 2016). However, the study of tidal interactions without eventual merger has never been done in great detail. Only one numerical study was found mentioning non-merger tidal interactions of dwarf galaxies (Bekki, 2008). This study of tidal interactions was far from a comprehensive study since there was mainly focussed on dwarf galaxy mergers triggering a star burst. So only tidal interactions with a low eccentricity of e = 1.1 were discussed, which is extremely close to a merging orbit. However, observational studies by Noeske et al. (2001) indicate that a wide range of eccentricities are possible as could be expected. This will be demonstrated in more detail in the next chapter. The study of tidal interactions between dwarf galaxies reaches some interesting questions. Are tidal tails and bridges formed just like during the tidal interaction of massive galaxies? Can these tidal interactions cause tidal stripping? Already for a long while observational studies indicate that some BCDs are accompanied by other low mass halos (Brinks & Klein, 1988; Brinks, 1990; Campos-Aguilar & Moles, 1991; Noeske et al., 2001). This was recently also confirmed with high resolution data from the Hubble Space telescope (Lelli, Verheijen & Fraternali, 2014). So are these dwarf galaxy companions possible triggers for the starburst occuring in these BCDs?,... The lack of numerical study of these tidal interactions and the questions this poses was the ground for the topic of this thesis. 1.4 Blue Compact Dwarfs The definition for BCDs is not always the same, it mostly depends on the author. But they generally include a few properties: it has a small optical size of maximally a few kpc, it has a small magnitude compared to massive galaxies and a large fraction of the light is emitted in the blue part of the spectrum caused by a starburst (e.g. Thuan & Martin, 1981; van Zee, Skillman & Salzer, 1998). Sevaral properties of these BCDs have been found by having a more detailed look at these objects. First of all, these galaxies appeared to contain old stars as well (e.g. Loose & Thuan, 1986; Kunth, Maurogordato & Vigroux, 1988; Papaderos et al., 1996a), implying it has a host galaxy. Surface brightness profiles of these BCDs have been created, showing that these galaxies have a higher central surface brightness and a smaller scale length (Papaderos et al., 1996b; Salzer & Norton, 1999). 11 Many investigations found a large central concentration of gas (e.g. Taylor et al., 1994, 1995; van Zee, Skillman & Salzer, 1998). Another very interesting observation is that BCDs can have very steep rotation curves, implying that there is a higher central density of dark matter (van Zee, Salzer & Skillman, 2001; Lelli et al., 2012a,b). It is also generally found that BCDs have lower gas metallicities than dIrrs (e.g. Izotov, Thuan & Lipovetsky, 1994; Izotov & Thuan, 1999; Hunter & Hoffman, 1999). Tidal interactions are not the only triggering mechanism proposed for the creation of BCDs. Mergers of dwarf galaxies in specific have been suggested as well to trigger a starburst (Östlin et al., 2001). This mechanism was confirmed in numerical simulations (Bekki, 2008; Cloet-Osselaer et al., 2014; Starkenburg, Helmi & Sales, 2016). Another mechanism that was proposed is an inspiralling gas clump (Elmegreen, Zhang & Hunter, 2012; Sánchez Almeida et al., 2015). This inspiralling gas cloud could lead to a higher central concentration of dark matter, stars and gas, which could cause a long-lived star burst. This was further supported by Koleva et al. (2014). They noted very irregular stellar kinematics in observed BCDs, which suggested that BCDs are experiencing a form of interactions even when they are found in isolation. This mechanism was numerically confirmed by Verbeke et al. (2014). So a couple of mechanisms have already been confirmed both observationally and numerically. For the tidal interactions between dwarf galaxies not involved in a merger process, this still has to be confirmed. Even though Bekki (2008) indicates that this mechanism should work as well, it was never studied in full detail. Based on the differences observed in BCDs, a classification scheme was proposed by Loose & Thuan (1986). This classification scheme will be shortly discussed below. 1.4.1 classification of BCDs - i0: This type of BCD seems to have no host galaxies. This would imply that they are young objects experiencing a first star formation burst. - nE: These BCDs are experiencing a nucleated burst in a host galaxy which has regular elliptical outer isophotes. - iE: Just like the previous type, this BCD has regular elliptical outer isophotes. But the burst occuring in the host galaxy has an irregular shape. - iI: This are BCDs that have an irregular burst in an irregular host galaxy. This class contains two further subdivisions. a) iI C: They have elongated hosts where the star formation is happening somewhere near the side of the galaxy. The C refers to the cometary shape of the isophotes. b) iI M: In this case the starbursting dwarf galaxy shows clear signs of a merger. 12 Chapter 2 Simulating and analysing dwarf galaxies Apart from observing dwarf galaxies, there are other ways to study these galaxies. A very powerful tool to study dwarf galaxies is N -body/SPH simulations. One can implement relevant physical processes in the code and study the effects of these processes on the evolution of a dwarf galaxy and compare this with observations. The small scale of dwarf galaxies is a particular advantage in the case of N -body simulations since this small scale allows a very high resolution. A resolution that is by far not reachable for the larger and more familiar galaxies with the same amount of resources. This again shows that dwarf galaxies are very interesting objects to study since they allow that the physical processes that govern galaxy evolution can be studied with a high resolution even when limited resources are available. In this chapter we will first discuss N -body simulations in general. A technique that is used to study a wide variety of problems in astrophysics and many other applications not directly related to astrophysics, such as ballistics and oceanography. In the second section we will discuss its use in specific for simulating dwarf galaxies. The initial conditions used for dwarf galaxies will be discussed in this section as well. The third section will talk about how the interaction of the dwarf galaxies is initiated. The final section in this chapter will shortly discuss the analysis tool HYPLOT that was used to analyse the data obtained from simulations. 2.1 N -body/SPH simulations In the simulations performed in this thesis, N -body systems were considered. To integrate these systems over time, an extended version of the freely available Gadget-2 code (Springel, 2005) was used. The extra physics included in this code will be discussed in the following part of this chapter. In this section the N -body techniques used and smoothed particle hydrodynamics (SPH) will be discussed in some more detail. The galaxies in our simulations consisted of dark matter, gas and stellar particles. All 13 the included particles interact gravitationally. For the gas particles, the pressure they experience has to be included as well. To calculate the gravitational force, the code is restricted to the non-relativistic case. In this case, one could determine the gravitational force on a particle by summing all the gravitational forces coming from the N-1 other particles. N X Gmj (xi − xj ) ẍi = − | xi − xj |3 j=1,j6=i (2.1) When a close encounter happens, this can lead to extreme accelerations of the particles because the denominator in 2.1 becomes very small. To solve this problem, an extra term was added to the denominator of the potential to avoid such problems, this is called gravitational softening. ẍi = − N X j=1,j6=i Gmj (xi − xj ) | xi − xj |2 )3/2 (2 + (2.2) This equation is then used to calculate the acceleration coming from gravitational interaction. The effect of gravitational softening on the acceleration of particles is plotted in figure 2.1. Gravitational softening brings some disadvantages since it can give rise to biases at small scales. However it avoids the problem of extreme accelerations at close encounter requiring a very small time step. It also quickly gets very close to the standard gravitational law at larger distances. In the case of our simulations, = 13 pc was used. When simply applying equation 2.2 to all particles in the simulation, the simulation time will scale with N2 . It is however possible to improve on this. This is done by using a hierarchical multipole expansion e.g. in the form of a tree algorithm. This allows to have a scaling like N log(N) instead of N2 . The tree is constructed by dividing the first node which contains all particles into 8 cubes with equal size and sides half the length of the original cube. This is division is done untill the moment that the cube contains one particle. The cubes that don’t possess any particles are left out. This is schematically shown for two dimensions in figure 2.2. Forces are calculated by walking through the tree, starting at the root node. Then it is checked if the the multipole expansion of the node provides an accurate enough force. If the multipole expansion provides an accurate enough force, the walk through that branch is stopped. If it is not accurate enough, the daughter nodes of this branch are called and checked. This multipole expansion leads to an approximate force, but the error on this force can be controlled by changing the criterion to call the daughter nodes. The criterion is given by equation: GM l 2 <α|α| (2.3) r2 r 14 Figure 2.1: The effect on the acceleration at close encouters when including gravitational softening can be clearly seen in this plot. This plot was taken from the course on Astrophysical Simulations of prof. Baes (Ghent University). On this plot the acceleration is plotted as a function of the seperation between the two particles. Where M is the mass in the node, l the size of the node and r the distance between the particle and the center of mass of the node. | α | is the acceleration at the previous timestep and α is the tolerance parameter. However it was shown that errors can become large when standard opening criterions are used (Salmon & Warren, 1994). This happens when the distance to the nearest particle in the node becomes very small. To protect against these errors, a second opening criterion is added to the code. | rk − ck |< 0.6l (k ∈ {1, 2, 3}) (2.4) Where r is the particle coordinate and c is the geometric centre of the node. There are several possibilities to evolve the system in time such as the Runge-Kutta integration, Gragg-Bulirsch-Stoer integration,... In the Gadget-2 code, a leapfrog integration was used. This integration scheme is a symplectic integration scheme, which means that it preserves the Hamiltonian structure over the time integration. Integration schemes such as Runge-Kutta do not preserve the Hamiltonian structure. Using a symplectic integration scheme leads to an extraordinary stabilitiy compared to e.g. Runge-Kutta integration. This is shown by Springel (2005), where Runge-Kutta integration was compared with leapfrog integration for an elliptic orbit with high eccentricity, as can be seen in figure 2.3. The leapfrog integration shows a precession, while the Runge-Kutta integration shows a drift in orbital energy. It has to be noted that the fourth order Runge-Kutta integration requires several force computations for one time step while leapfrog integration only requires one force computation per time step. This 15 Figure 2.2: This figure schematically shows the construction of a particle tree in two dimensions. This figure was taken from http://http.cs.berkeley.edu/∼demmel/cs267/lecture26/lecture26.html significantly reduces the computational cost while keeping a strong stability in the time integration. The timestep used in Gadget-2 depends on the gravitational softening , an accuracy parameter η and the magnitude of the acceleration | α | of the particle. The timestep is given by: h 2η 1/2 i (2.5) ∆t = min ∆tmax , |α| Where ∆tmax ≈ 0.02 Gyr is the maximal timestep allowed in our simulations. For simulating the behaviour of the gas, smoothed particle hydrodynamics (SPH) is used. SPH is a Lagrangian approach for solving the hydrodynamics equations of a system. A Lagrangian method has the specific property that it uses co-moving spatial coordinates of a fluid element. This is in contrast to the Eulerian method which follows the state of a system at a fixed coordinate x. In SPH, the fluid elements are represented by a limited amount of gas ’particles’. Then the Lagrangian momentum equation has to be be solved for all these particles. N X 1 Gmj (xi − xj ) ẍi = − ∇pi − ρi | xi − xj |3 j=1,j6=i (2.6) Where pressure experienced by the hydrodynamical particles is the only difference in this equation compared to the equation for the collisionless particles in the simulations. This pressure is coming from the collisions on a microscopic level. The effect of this is macroscopically represented by the pressure gradient. 16 Figure 2.3: The plot on the left shows the leapfrog evolution of the orbit, showing a precession of the orbit. The plot on the right shows the Runge-Kutta evolution of the orbit, showing a drift in the orbital energy. These plots were taken from Springel (2005) Of great importance when working with SPH is the density estimate. In the Gadget-2 code this is done using N X ρi = mj W (r ij , hi ) (2.7) j=1 Where r ij = r i - r j , hi is the smoothing length of the particle and W(r ij , hi ) is the SPH smoothing kernel. In the case of the Gadget-2 code, the smoothing kernel is given by 2 3 r 1 − 6 h + 6 hr , 0 ≤ hr ≤ 12 , 8 3 (2.8) W (r, h) = r 1 3 2 1− h , < hr ≤ 1, πh 2 r 0 >1 h This smoothing kernel was proposed by Monaghan & Lattanzio (1985). The smoothing length hi of these particles is defined in Gadget-2 such that each kernel contains a constant mass for the estimated density. This is constrained by 4π 3 h ρi = NSP H m (2.9) 3 i Where m is the average particle mass and NSP H the number of smoothing neighbours. The first term in the Lagrange momentum equation used in the Gadget-2 code was derived by Springel & Hernquist (2002): " # N X 1 Pi Pj ∇pi = mj fi 2 ∇i Wij (hi ) + fj 2 ∇i Wij (hj ) (2.10) ρi ρi ρj j=1 17 This equation conserves energy and entropy. Wij (h) stands for W(| r i − r j |, h) and the particle pressure Pi = Ai ργi with Ai the entropy of the particle and γ the adiabatic index. The coefficients fi are defined by hi ∂ρi fi = 1 + (2.11) 3ρi ∂hi To capture shocks that occur in the gas, an artificial friction term was added to the Gadget-2 code. N X (2.12) ẍi,visc = − mj Πij ∇i W ij j=1 Where Πij ≤ 0 is non-zero when particles approach each other in space and W ij is the average of the two kernels Wij (hi ) and Wij (hj ). This viscosity creates entropy, transforming kinetic energy irreversibly into heat. 2.2 Dwarf Galaxies and Initial Conditions As mentioned in the previous section, the N -body/SPH code that is used for the simulations is the freely available Gadget-2 code Springel (2005). This code was extended with many relevant physical processes that govern (dwarf) galaxy evolution. These effects include e.g. star formation, the heating and cooling of gas, chemical enrichment, feedback from stars and supernovae, the ionisation state of the gas, the cosmic ultra-violet background (UVB) and the formation of population III stars in non enriched gas (Valcke, de Rijcke & Dejonghe, 2008; Valcke et al., 2010; Schroyen et al., 2011; Cloet-Osselaer et al., 2012; Schroyen et al., 2013; De Rijcke et al., 2013; Vandenbroucke et al., 2013; Cloet-Osselaer et al., 2014; Verbeke, Vandenbroucke & De Rijcke, 2015; Verbeke et al., 2014; Vandenbroucke, Verbeke & De Rijcke, 2016). 2.2.1 Included physics Stellar particles It is well known that collapsing gas clouds in a galaxy can lead to star formation under certain conditions. To model this, star formation criteria have been included in the simulation code (Valcke, de Rijcke & Dejonghe, 2008). These criteria are given by ∇·v<0 (2.13) T < Tcrit = 15000K (2.14) ρg > ρcrit (2.15) 18 These criteria are checked every time step for all gas particles. Equation 2.13 is called the convergence criterion, which demands that the local gas flow is collapsing. The second equation 2.14 is called the temperature criterion. This demands a minimum temperature for the gas particle before it can be converted into a stellar particle. In our simulations, Tcrit = 15 000 K is the critical temperature used for star formation (Stinson et al., 2007). Apart from the gas being sufficiently cold, there is also need for a high gas density to form stars. This is included using the last criterion, namely the density criterion, with ρcrit the minimal density required. In our simulations, ρcrit = 100 amu cm−1 is used. This has been shown to be a good prescription for star formation (Governato et al., 2010; Schroyen et al., 2013). Generally speaking, this is the most stringent condition because it is rather unlikely that gas clouds reach such high densities when they are not collapsing or if they have T > Tcrit . If all these conditions are satisfied, the gas particle is allowed to transform into a stellar particle. The star formation is implemented in the code using a Schmidt law (Schmidt, 1959). dρg ρg dρ? (2.16) =− = c? dt dt tg Where ρ? is the stellar density, ρg the gas density, c? the star formation efficiency and tg is the dynamical time scale given by tg = p 1 4πGρg (2.17) It was shown that c? only has a minor effect on the star formation history and that models using a value around 0.2 for c? give dwarf galaxies that look most like observed dwarf galaxies (Stinson et al., 2006; Revaz et al., 2009). In the case of our simulations, c? = 0.25 is used (Cloet-Osselaer et al., 2014). When a gas particle is converted into a stellar particle, this stellar particle needs to be given certain properties. The stellar particle is assumed to represent a stellar population of a certain mass. This stellar population then has the same metallicity and age. These stars are modelled using an initial mass function, in the case of our simulations the one of Chabrier (2003) is used. This function is compared with other initial mass functions in figure 2.4. These newly formed stellar particles have a strong impact on the interstellar medium (ISM) of the dwarf galaxy. This happens through thermal feedback associated with type II supernovae and stellar winds coming from young heavy stars. It is assumed that these 43 young stars inject 1050 ergs M−1 J M−1 (or 10 ) into the interstellar medium via stellar winds. The type II supernovae will inject 1051 ergs M−1 into the ISM. The effect of type Ia supernovae on the galaxy is included as well. Since type Ia supernovae originate from white dwarfs accumulating mass in a binary, a delay is included for this energy to be injected in the ISM. To model this, a gaussian distribution is used with the mean at 4 Gyr and a standard deviation of 0.8 Gyr (Strolger et al., 2004). This distribution is cut off at a standard deviation of 3σ. 19 Figure 2.4: A plot of the Chabrier initial mass function compared with other initial mass functions that can be used (Mattsson, 2010). The green initial mass function is the initial mass function used in Mattsson (2010). The parameters given above only indicate the amount of energy released by these events. However, this doesn’t say that much about its eventual effect on the interstellar medium. To model this, one has to specify how easily this feedback can be absorbed by the ISM. In the case of our simulations, an absorption efficiency of 0.7 is used (Cloet-Osselaer et al., 2012). The feedback is distributed over the nearby gas particles using the SPH smoothing kernel. In the simulations, the effect of feedback on a gas particle results in the gas particle not being able to cool down radiatively during that time step. Apart from feedback, the supernovae produced by these stellar particles also enrich the dwarf galaxy with metals. In the case of these two types of supernovae included, one only has to follow the abundances of Mg (magnesium) and Fe (iron). Following the evolution of these two elements, one can deduce the entire chemical composition of the galaxy (De Rijcke et al., 2013). Population III stars In 1944 Baade, who was working at the Mount Wilson Observatory at that time, devided the observed stars in the universe in two types of stars (Baade, 1944). Namely population I and population II stars. This division was based on the difference in metallicity of these stars. Where population I stars have a higher metallicity than population II stars. It was found that there was a correlation between the age of the stars and their metallicity. The older stars seemed to have a lower metallicity. This led to the hypothesis of the so 20 called population III stars that were formed in the early universe. These stars that form in the early universe have been a topic of research for a while. First investigations of these stars include e.g. Ezer & Cameron (1971). They would have very low metallicity since they were formed out of unenriched gas. Although theoretical research about these objects has been done already, no stars with such a low metallicity have been observed so far. This is not really surprising since it was shown in simulations that there is no formation of population III stars at low redshift (Verbeke, Vandenbroucke & De Rijcke, 2015). In figure 2.5 a plot is shown of the metallicity of the observed stars. This plot clearly shows that one can expect that observations of nearby dwarf galaxies will not detect any low metallicity stars. Since no population III stars have been observed to date, the only possible input for these stars is based on theoretical studies. This leads to some uncertainty whether the effects of these stars in the simulations represent the reality. Yet it was shown by Verbeke, Vandenbroucke & De Rijcke (2015) that the inclusion of these population III stars can reproduce the most important observed quantities of gas rich dwarf galaxies. There are some very important differences between the properties of the population III stars and the stars forming out of enriched gas. First of all, they have a very different initial mass function. The initial mass function used, can be found in figure 2.6. The initial stellar masses vary between 0.7 M and 300 M . The initial mass function is clearly different from the Chabrier initial mass function, which contains a lot of low mass stars. On the other hand the initial mass function used for population III stars contains much more heavy stars (Susa, Hasegawa & Tominaga, 2014), this can be seen when comparing figures 2.4 and 2.6 . The assumption that population III stars form out of unenriched gas implies that one has to put constraints on the metallicity of the gas that can form these population III stars. When a stellar particle is formed in our simulations, it is assumed to be a population III stellar particle if [Fe/H] < -5 for that particle. One can ask whether this cut-off is at the ideal metallicity and probably the transition between population II and population III stars will be smoother than the transition used in our simulations. However not that much is known about population III stars for the moment. And it is shown that simulations with this cut-off for population III stars are able to produce dwarf galaxies that strongly resemble observed dwarf galaxies (Verbeke, Vandenbroucke & De Rijcke, 2015). Since these population III stars are clearly different from the population II stars, one can expect that their effect on the environment will be different as well. For example, the population III stars inject much more thermal energy into the galaxy. Their type II supernovae would inject about 4 times as much energy in the ISM as type II supernovae from the population II stars. The difference in energy injected by the young stars is even 21 Figure 2.5: This plot shows three different observations or theoretical observations of the fraction of stars with a certain metallicity. The histogram is a real observation of LeoI, the distribution that is rather nicely following this diagram is an observation of a simulation that includes population III stars. It shows as well that no low metallicity stars will be observed. The other graph is also an observation of a simulation, but this time without population III stars included. This plot was taken from Verbeke, Vandenbroucke & De Rijcke (2015) larger. It is assumed that the energy injected by young population III stars is about 40 times higher than the energy coming from young population II stars (Heger & Woosley, 2010). The stellar winds and supernovae give lots of stellar mass back to the interstellar medium. In the simulations a value of 45 % is used. The supernovae of population III stars obviously give rise to enrichement of the interstellar medium. It is assumed that the amount of Fe and Mg released is only 10 % of the chemical yield for these elements coming from type II supernovae of population II stars (Heger & Woosley, 2010). This yield has also been estimated based on observations of very low metallicity stars, since these stars contain signatures from this yield (Nomoto, Kobayashi & Tominaga, 2013). Cooling curves For the cooling and heating curves of the gas, the calculations performed by De Rijcke et al. (2013) were used. In this case the curves only depend on 5 parameters: temperature, density, redshift, [Fe/H] and [Mg/Fe]. Their calculations were done covering a large interval of physical scales: densities up to 100 amu cm−3 , tempuratures ranging from 10 K to 109 K and including effects up to redshift z = 15. The ionisation of 14 elements is determined for this range of physics including radiative and collisional ionization coming from the cosmic ultraviolet background and an interstellar radiation field, and charge transfer reactions. These elements include e.g.: Fe, Mg, C, O, N, Ni,... It concerns the elements that are mostly produced and/or released during type Ia and type II supernovae, and by intermediate mass stars. A simplified model was used for the 22 Figure 2.6: The green function is a fit through the data points obtained by Susa, Hasegawa & Tominaga (2014). This function is used as the initial mass function for the population III stars. It can be seen that the initial masses vary between ± 1 M and 300 M . This figure was taken from Verbeke, Vandenbroucke & De Rijcke (2015) enrichment of the interstellar medium. In this model a distinction was made between the fast contributions, coming from type II supernovae and young stars, and the slow contributions, coming from type Ia supernovae and less massive stars. A five dimensional interpolater was implemented by De Rijcke et al. (2013), allowing a fast interpolation of the tabulated cooling and heating curves in the simulations. The fact that only the Fe and Mg abundances need to be followed, reduces the memory requirements. Another effect taken into account in the simulations is that part of the UVB and supernova energy injected in the gas leads to ionisation of this gas. Due to this ionisation, the gas will be heated less since part of the energy is used to ionise this gas. This was implemented by Vandenbroucke et al. (2013). 2.2.2 gogoIC, Kuzkut and Ganic To set up the initial conditions, one uses gogoIC. This program calls two other programs, namely Kuzkut and Ganic. Kuzkut sets up the initial conditions for the dark matter 23 halo of the galaxy using a Monte Carlo sampling method. The second program that is called is Ganic. This programs adds gas particles to the previously created dark matter halo, which is done as well using a Monte Carlo sampling method. Although a dark matter halo is not required for Ganic to create a cloud of gas particles, but this feature will not be necessary for this project. Many properties of the initial gas cloud can be set, e.g. it is possible to give these gas particles an initial rotation. The initial rotation can be generated based on different rotation profiles, e.g. a constant rotation profile, a linearly increasing rotation velocity when moving away from the galaxy center or the rotation velocity following an arctangential profile. This initial rotation can have a strong influence on the galaxy observed in the current universe as will be explained later. This influence will be discussed later when we go more into detail on the initial conditions. Other properties of the gas cloud that can be set, are e.g. the initial temperature and metallicity of the gas. The gogoIC program can be used in two different ways. The first one is using a graphical user interface. This graphical user interface is a very easy way to set up the initial conditions, one simply has to select the desired dark matter halo, initial dark matter and gas mass, feedback efficiency, critical density for star formation,... Using this graphical user interface is however limited since it does not contain the latest updates. To really make use of the full possibilities of gogoIC, one has to write an inputfile and then start gogoIC using ’GOGOICmultiple.py’. An interesting possibility using the inputfile is that one can generate several initial condition files at the same time. This is particulary useful when working with a merger tree, since in this case lots of initial condition files have to be generated. Using the graphical user interface the generation of initial conditions has to be done one by one. Concerning our simulations, only the initial conditions of one simulation was set up with the graphical user interface. This is mentioned here because it will be shown later that there was something peculiar noticed about these initial conditions. This peculiarity will be discussed later in this chapter. 2.2.3 Initial conditions Cosmological parameters When creating the initial conditions for the dwarf galaxies, there has to be decided at which redshift one will set these initial conditions. This redshift will obviously be the starting redshift for the simulation. A large initial redshift was chosen, namely z = 13.5. From this redshift on, the created initial halo will evolve in a cosmological setting. The cosmological setting is the ΛCDM model, the current standard model of cosmology. This implies a flat universe currently dominated by the cosmological constant Λ. A flat universe implies that the current total critical density Ω0 is equal to 1. The critical density of the so called dark energy ΩΛ is 0.7274. The universe being flat implies that ΩM = 0.2726, where matter consists of cold dark matter and baryonic matter. In the case of our simulations ΩDM = 0.2250 and ΩBaryon = 0.0476. The Hubble constant is given by H0 = 70.4 km s−1 Mpc−1 . These values are in agreement with the conclusions of 24 WMAP-9 (Hinshaw et al., 2013). The total evolution starting from z = 13.5 in this cosmological setting takes 13.4 Gyr to arrive at z = 0. In the parameterfile of every simulation, one can set after which time interval a new snapshot of the simulation has to be saved. These snapshots contain all the information that can be used to analyse the simulations. Making this time interval smaller, one can study the time evolution of the galaxies with a higher resolution. Although using the birthtime parameter that is given for every stellar particle, it is possible to achieve a higher time resolution in the star formation rate without increasing the amount of snapshots per time interval. However to study other parameters of the galaxy with higher time resolution such as the density profile, the metallicity,... more snapshots are necessary. The masses of the dark matter halos It was found by many investigations both theoretical (Valcke, de Rijcke & Dejonghe, 2008; Revaz et al., 2009; Sawala et al., 2010) and observational (Tolstoy, Hill & Tosi, 2009; Mateo, 1998, and references therein) that the initial mass has a major influence on the evolution of dwarf galaxies. Figure 2.7 shows the difference in star formation depending on the initial mass of a dark matter halo. This figure was obtained from the investigation by Valcke, de Rijcke & Dejonghe (2008). This implies that these simulations were by far not as accurate as current simulations. Yet the plot clearly shows an effect of the initial mass on the star formation. This effect is obviously not limited to star formation alone, it will also have its effect on luminosity, metallicity,... The effect of the initial mass can be noted in our simulations as well. Figure 2.8 shows the difference in star formation depending on the mass of the galaxies in our simulations. The two star formation histories at the top of this figure are coming from two low mass dwarf galaxies, which have a different initial rotation of the gas cloud. By far the largest contribution to the mass of the galaxy is coming from the dark matter halo. This comes to no surprise since it is generally accepted that there is way more dark matter than baryonic matter in our universe. Since the initial gas mass of the galaxy is based on the mass of the dark matter halo in our simulations, one can say that the choice of the initial dark matter mass will have a major effect on the evolution of the galaxy. Since the initial mass of the dark matter halo and the gas cloud are related and due to the fact that the dark matter mass is much more dominant. When speaking about a galaxy with a certain mass, we mean the initial dark matter mass, unless it is specified differently. For the eventual study of the tidal interactions, galaxies with 3 different masses were used. Namely with masses: 1010 M , 2·1010 M and 4·1010 M . The magnitude in the V-band of the most heavy model is close to MV = -17 and in the B-band close to MB = -16. Which is an accepted rough working definition for dwarf galaxies proposed by Tammann (1994). This can be seen in figure 2.9 where the evolution of the B-band 25 Figure 2.7: This plot shows the effect of the initial mass on the star formation history of the galaxy. It contains 9 different mass models where C01 is the lightest model and C09 the heaviest model. An important remark for noticing the effect of the initial mass is the difference in scale on the y-axis. This plot was taken from Valcke, de Rijcke & Dejonghe (2008) and V-band magnitude are plotted for the 3 different masses used for the study of tidal interactions. It should be noted that the gas cloud of the plotted low mass galaxy (1010 M ) has an initial rotation with vmax = 5 km/s. This note is important since two types of low mass galaxies are used in the study of tidal interaction. One where the initial gas cloud has no rotation and one where the maximal rotation velocity is 5 km/s. These two galaxies have some different properties as will be discussed later. However, models with these three masses are not the only models that were run in isolation. Two models with an even lower mass of 2.5·109 M and 5·109 M were run as well. These simulations proved useful, since they provided a lower limit for the mass of the models that can be used for the main goals of this thesis. One of the main initial goals of this thesis was to study if tidal interactions of dwarf galaxies could trigger starbursts in these dwarf galaxies. So it is necessary that the galaxies still contain enough neutral hydrogen such that they can form a significant amount of stars. As can be seen from figure 2.10, these galaxies barely (in case of 5·109 M ) or don’t (in case of 2.5·109 26 SFR [M /yr] SFR [M /yr] SFR [M /yr] SFR [M /yr] 0.010 0.008 0.006 0.004 0.002 0.000 0.008 0.006 0.004 0.002 0.000 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.25 0.20 0.15 0.10 0.05 0.00 sim5 sim6 sim30 sim59 2 4 6 8 10 12 t [Gyr] Figure 2.8: This plot shows the star formation history of the galaxies used for the tidal interactions. sim5 and sim6 are the low mass galaxies. Sim30 is the medium mass galaxy and sim59 is the high mass galaxy. They show a clear difference in the rate of star formation. M ) form any stars after an initial burst. As can be noted from figure 2.10 as well, these galaxies barely possess any neutral hydrogen near z = 0. Thus excluding a peak in the star formation if no gas is captured e.g. by an inspiraling gas cloud. The capture of gas from tidal interactions is rather unlikely for these galaxies since they don’t have a lot of gas in their environment and neither possess enough mass to strip gas from other galaxies and capture this gas. The fact that this is unlikely can be noted later, where it can be seen that the low mass galaxies used in the interacting models don’t easily catch a lot of gas from the other galaxy. To be sure that this lack of remaining gas and star formation is no mass resolution effect, a second simulation was done to check if this could be an issue. Certainly for the galaxy with a halo mass of 5 · 109 M this could be interesting, since very rare star formation is noted. However this led to the same results. After an initial star formation peak both models almost completely stop forming stars. More information on the mass resolution can be found later in this section about the initial conditions. These low mass galaxies were run with the gas cloud having an initial rotation. This is 27 −18 −17 MB −16 −15 −14 −13 −12 −11 −10 −18 −17 MV −16 −15 −14 −13 −12 −11 low mass medium mass 0 2 4 6 t [Gyr] 8 10 high mass 12 Figure 2.9: This figure shows the evolution of MV and MB for three models with different masses as a function of time. The low mass model, is the galaxy with a high initial rotation velocity of the gas cloud (see later). done because simulations where the gas cloud has an initial rotation show a higher star formation rate as was noted by Schroyen et al. (2013) and also seen in our simulations. This implies that the low mass of the galaxies is probably the most important reason for the lack of star formation. The observation of this lack of star formation is in accordance with more realistic simulations involving a merger history by Verbeke, Vandenbroucke & De Rijcke (2015). They noted a lack of star formation for dwarf galaxies with these masses as well. To make life easier, we will use short notations to describe the galaxies used in the simulations of tidal interaction. There are four types of galaxies used in the simulations of tidal interaction. Namely a low mass dwarf galaxy (1010 M ) where the gas cloud has no initial rotation. To this dwarf galaxy will be referred with ”DG1”. A second type of low mass galaxy (1010 M ) was used. But this time the gas cloud was given an initial rotation with a maximal velocity of 5 km/s. To this dwarf galaxy will be referred with ”DG2”. The medium mass dwarf galaxy (2 · 1010 M ) used in the tidal interaction simulations, has a gas cloud that was given an initial rotation with a maximal velocity of 2.5 km/s. To this dwarf galaxy will be referred with ”DG3”. The high mass dwarf galaxy (4 · 1010 M ) has a gas cloud that was given an initial rotation with a maximal velocity of 2.5 km/s. To this dwarf galaxy will be referred with ”DG4”. This is also summarised in table 2.1. 28 0.030 0.025 0.020 0.015 0.010 0.005 0.000 5 4 3 2 1 0 −1 −2 −3 −4 10 −53 0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 −1 −2 −3 −4 10 −53 102 101 100 10−1 10−2 10−3 10−4 SFR [M /yr] Mstar [106 M ] [Fe/H]B 102 101 100 10−1 sim9 (high resolution) MHI [106 M ] MHI [106 M ] [Fe/H]B Mstar [106 M ] SFR [M /yr] sim21(high resolution) 0 2 4 6 t [Gyr] 8 10 12 0 2 4 6 t [Gyr] 8 10 12 Figure 2.10: The plot on the left shows a high mass resolution simulation of a galaxy where the dark matter halo has a mass of 5·109 M . It can be clearly seen that the star formation is almost completely shut down after 8 Gyr. On the right a galaxy is plotted with a dark matter halo mass of 2.5·109 M . This is also coming from a simulation with a higher mass resolution. Density profile of the dark matter halo The initial density profile used for all galaxies simulated in isolation is an NFW profile, named after Navarro, Frenk and White, which is cusped (Navarro, Frenk & White, 1997). This was proposed based on dark matter profiles obtained from large scale cosmological simulations that only included dark matter. On the other hand, observations indicate that dark matter profiles should have a core instead of a cusp (de Blok, 2010; Salucci et al., 2012). The way to resolve this, is to include baryonic matter in the simulations. It was found that through the gravitational interaction of the dark matter halo with the baryonic matter that the halo transformed from being cusped to having a core (Read & Gilmore, 2005; Governato et al., 2010; Cloet-Osselaer et al., 2012; Teyssier et al., 2013; Brooks & Zolotov, 2014). We use the assumption that in the early stages of the universe this baryonic effect did not really take place yet, such that the NFW profile is acceptable to use for the initial conditions. The NFW-profile is given by ρDM (r) = ρs (1 + rDM,s r r )2 rDM,s (r < rDM,max ) (2.18) This equation is used to generate the initial conditions for the dark matter halo. In this equation rDM,s is the scale length and ρs is a characteristic density for the dark matter halo. RDM,max is the radius where the profile is cut off. By now, generalised NFW profiles have been proposed based on high resolution cosmological simulations. For low mass and high redshift halos such a generalised NFW profile was proposed by Cen et al. (2004). For these halos, the density profile is dependent on 29 the halo mass and redshift. This generalised NFW profile is not necessary for the dwarf galaxies used in the tidal interactions, since it only have to be used for a halo mass under 109 M . The generalised NFW profile was used when constructing the merger tree simulations, to compare with the galaxies obtained in isolation. This is necessary since in a merger tree many low mass halos exist. And it was shown in the thesis by Cloet-Osselaer (2015) that using a generalised NFW profile can have a significant effect on the early evolution of these halos. In figure 2.11 the difference between a generalised NFW and an Figure 2.11: The initial density profiles for the dark matter halo and the gas cloud in Cloet-Osselaer (2015). These simulations where done with a halo mass of 0.9·109 M . NFW profile can be seen, where the generalised NFW profile has a higher central density. The creation of a generalised NFW profile is based on equation ρDM = ( rrs )α (1 ρ0 + rrs )4−2α (r < rmax ) (2.19) Where α and rs are dependent on the mass and redshift of the halo and rmax is the cutoff radius. Values for α and rs can be found in Cen et al. (2004). They obtained these values by fitting a statistical distribution to many values obtained for these parameters 30 from different dark matter halos. To obtain the values needed for different masses in the simulations the following function was used f (M, z) = a(1 + z)b ( M )(1+z)c 7 10 M (2.20) This function was fitted through their obtained data points for every parameter (CloetOsselaer, 2015). The gas cloud Not only a dark matter halo has to be created. As mentioned before a gas cloud has to be added to the dark matter halo. The mass of this gas cloud is then fixed by the mass of the dark matter halo. This is based on the ratio of baryonic and dark matter mass in the universe. The initial density profile of the initial gas cloud is a pseudo-isothermal profile (Revaz et al., 2009) ρg (r) = ρ0 2 1 + rr2 (r < rg,max ) (2.21) g In this equation ρ0 is the central gas density, rg is the scale length determined as in Schroyen et al. (2013) and rg,max is the radius where the gas cloud is cut off. Since it is determined as in Schroyen et al. (2013), the density of the gas cloud and the dark matter halo are related as can be seen from figure 2.11. The density difference of the gas cloud depending on the type of dark matter density profile will have its effect on the initial star formation. This was shown by Cloet-Osselaer (2015), where it was found that the higher central gas density for the generalised NFW profile led to a stronger initial star formation peak. It is possible to give the gas cloud an initial rotation. This initial rotation has a crucial influence on the galaxy observed at z = 0 and its evolution (Schroyen et al., 2011). It is also possible to have some different initial rotation profiles for the gas cloud. Namely a uniform rotation profile, a linearly increasing rotation velocity or the velocity following an arctangential profile as mentioned already before. Building on previous work done by Schroyen et al. (2013), the arctangential profile was chosen if an initial rotation was given to the gas cloud. vrot (r) = r 2 vmax arctan( ) π rrot,s (2.22) Where rrot,s is the scale radius and vmax the maximal rotational velocity. This profile is certainly favorable above the constant rotation profile, because this profile would give the gas particles at the center of the initial galaxy an unnatural high rotation speed. The influence of initial rotation can be seen in our simulations, namely in figures 2.12 and 2.13. First of all a more quiet start in the galaxy with an initial rotation is expected. 31 0.010 0.008 0.006 0.004 0.002 0.000 12 10 8 6 4 2 0 −1 −2 −3 −4 10 −53 0.010 0.008 0.006 0.004 0.002 0.000 25 20 15 10 5 0 −1 −2 −3 −4 10 −53 SFR [M /yr] Mstar [106 M ] [Fe/H]B 102 102 101 100 vinit,max = 5 km/s MHI [106 M ] MHI [106 M ] [Fe/H]B Mstar [106 M ] SFR [M /yr] vinit,max = 0 km/s 2 4 6 8 10 101 12 t [Gyr] 2 4 6 8 10 12 t [Gyr] Figure 2.12: The evolution of the star formation rate, stellar mass, metallicity and neutral hydrogen mass for the two low mass dwarf galaxies. On the left, the dwarf galaxy without initial rotation is plotted and on the right the dwarf galaxy with initial rotation. This can be explained by the fact that the initial rotation provides a barrier against immediate collapse of the gas. The initial rotation makes the gas spiral in, which makes the star formation ignite more quietly (Schroyen et al., 2011). However this won’t be compared here because of reasons explained later in this section. The first Gyr of the galaxy evolution is also not plotted in figure 2.12 for aesthetic reasons due to the way higher star formation in the first year of evolution. It can be seen that there is much less neutral hydrogen expelled from the galaxy with initial rotation. Because there is lots of neutral hydrogen continuously present in this galaxy, it is possible to have no significant breaks in the star formation history. This is clearly not the case in the galaxy without initial rotation. The galaxies with a higher mass, namely with 2·1010 M and 4·1010 M , were given an initial rotation as well, and don’t show any breaks in the star formation either. This can be seen in figure 2.8. One could argue that this could be because of their deeper gravitational potential such that not as much gas can escape. This might be part of the story, yet the initial rotation helps as well igniting the star formation less burstlike such that mass is not expelled in large amounts as would happen without initial rotation. The maximal initial rotation that was given to these more massive dwarf galaxies is 2.5 km/s. In Schroyen et al. (2011) it was shown that rotation has a strong effect on the star formation history. It was noted that dwarf galaxies with higher rotation show a less burstlike star formation history and have lower stellar mass. In our low mass galaxy with initial rotation there is a slight burstlike behaviour on top of the continuous star formation. This burstlike behaviour on top are statistical fluctuations in the star formation due to the formation of small clumps on top of the smeared out profile seen in figure 2.14. This is not really the case in the galaxy without initial rotation, where only 32 MB MV MI −16 −14 −12 −10 −8 −6 −4 −2 −160 −14 −12 −10 −8 −6 −4 −2 −16 −14 −12 −10 −8 −6 −4 −2 vrot = 0 km/s 0 2 4 6 t [Gyr] 8 10 vrot = 5 km/s 12 Figure 2.13: The evolution of the BVI magnitudes for the low mass dwarf galaxies with initial and no initial rotation of the gas cloud. It can be seen that the galaxy with initial rotation of the gas cloud is brighter in all bands. the burstlike behaviour is observed. The continuous star formation in the galaxy with initial rotation occurs because the gas is smeared out. This is not the case for the galaxy without initial rotation where the gas is not smeared out as smoothly, but collapses into clumps as seen in figure 2.14. Because of this, the star formation is more dependent on the occasional collapse of the gas cloud in this galaxy. The lower stellar mass with higher rotation is not observed in our simulations, as can be seen in figure 2.12. Just like in Schroyen et al. (2013) a higher stellar mass is seen when the gas cloud has an initial rotation. It has to be noted that in Schroyen et al. (2011) a uniform gas profile was used instead of a pseudo-isothermal gas profile. An explanation can be proposed then for this higher star formation. Since the pseudo-isothermal profile is more compact in the center and the rotational velocity is lower at the center due to the arctangential profile, it might be that the gas feels the centrifugal barrier less and that this leads to higher star formation. Further this could lead to a more burstlike star formation history of the galaxy, which can be observed in the star formation history 33 of our galaxy with initial rotation. This could be due to the fact that the initial star formation is stronger. This higher initial star formation could lead to slight disturbances on the smeared out galaxy. These clumps of higher density on top of the smeared out gas can be seen in figure 2.14. The higher star formation over the history of the galaxy with initial rotation obviously leads to a higher stellar mass and slightly higher metallicity, as can be seen in figure 2.12. In figure 2.13, a difference in the magnitudes between the two low mass galaxies is clearly visible. The difference in magnitude is around 1 mag most of the time. Where the galaxy with the higher brightness in all bands is as expected the galaxy with initial rotation, since it has a higher star formation rate and a higher stellar mass. Other parameters of the initial gas cloud are the same for all galaxies simulated. The gas cloud was given an initial temperature of 104 K (Cloet-Osselaer et al., 2014) and zero initial metallicity such that the gas was unenriched at the start of the simulations. While checking the correctness of the simulations, an odd initial density profile of the gas was found for the DG1 model as seen in figure 2.15. The initial conditions of DG1 were created with the graphical user interface of gogoIC. When creating exactly the same initial conditions with the inputfile, a completely different density profile for the gas was obtained as can be seen as well in figure 2.15. The initial dark matter density profiles are exactly the same up to some statistical scatter. One could think that different initial gas profiles were used. But a thorough search showed that they were both created using an pseudo-isothermal density profile and that the other initial conditions were the same as well. A search through the code that creates these initial conditions was done to find a reason for this strange behaviour. A multiplicative difference of 0.01 was found for the central density when creating a pseudo-isothermal profile. It is difficult to say that this is ’the’ reason for this odd behaviour. First of all because the central density of the gas should not be higher than the central density of the dark matter, even with a multiplicative difference. Secondly, the initial density profile of the gas created with the graphical user interface seems to be divergent. This is not the case for a pseudo-isothermal density profile as can be seen from equation 2.21. It also has to be noted that it is very weird that such a multiplicative difference for the central density was found while no direct reason seemed to be available for this. This doesn’t necessarily say that there is no reason for this difference, but this reason was not found. So a completely satisfying answer for this problem was not found. Since all tidal interactions with a DG1 galaxy are performed using the galaxy created with the graphical user interface, one obviously has to wonder if this gives similar results at the moment the tidal interactions will be studied. The tidal interactions were studied starting after ± 10.5 Gyr of evolution. To do this, several properties are studied over the evolution of the two galaxies. These properties are shown in figure 2.16. Looking at these plots one can see that there is a difference in star formation rate up to ± 9 Gyr. 34 Figure 2.14: Plotted on the left plot is a snapshot of the gas density in the low mass galaxy without initial rotation. The plot on the right shows the more smeared out low mass galaxy with initial rotation. This galaxy also shows some higher gas densities, yet it does not consist of two clear clumps like the plot on the left. The gas density in general is more smeared out because of the rotation. Afterwards the star formation rate gets very similar. This higher early star formation obviously leads to more stars in the galaxy at z = 0. However this difference is ± 20 %, which is way less than when comparing the effect of initial rotation. It was found that when the star formation becomes similar, the amount of neutral hydrogen is also at the same level in both simulations. The galaxy created with the inputfile had more neutral hydrogen up to ± 8 Gyr of evolution. But the higher star formation and thus more feedback leads to a slow exhaustion of this extra neutral hydrogen. Also the magnitudes in the different bands are very close to each other, although generally the DG1 created with the inputfile is slightly brighter because there are more stars created in total. The main difference can be seen when looking at the initial star formation shown in figure 2.17. The gas density with the strong central peak has a very strong initial burst compared to the pseudo-isothermal profile created with the inputfile. This comes to no surprise seeing the high central density, since the star formation has a density criterion. Mass resolution The amount of particles used for the simulations in isolation of the galaxies depends on the mass of the dwarf galaxy. This was done such that every galaxy was simulated with the same mass resolution. For the low mass galaxies with a dark matter halo of 1010 M , 2·105 particles were used to model the dark matter halo. This implies that the mass of every dark matter particle in the simulation is 5·104 M . For modelling the gas cloud, 2·105 particles were used as well in the low mass galaxies. As mentioned before, the amount of gas in the simulation is based on the ratio of the baryonic matter over the 35 104 DM (GUI) gas (GUI) DM (inputfile) gas (inputfile) ρDM [106 M /kpc3 ] 103 102 101 100 10−1 10−1 100 101 r [kpc] Figure 2.15: The density profiles of the dark matter and the gas for DG1 when created with the GUI or with the inputfile. dark matter in the universe for which a value of 0.2115 is used. This implies a total gas mass of 2.115·109 M in the initial galaxy. This leads to a mass resolution of 1.0575·104 M for the gas in the simulations. To obtain the same mass resolution for the medium and high mass galaxies used in the tidal interactions, they were simulated with respectively 4·105 and 8·105 dark matter and gas particles. The very low mass galaxies simulated, with masses of 2.5·109 M and 5·109 M , were given the same resolution as well. This implies that 5·104 and 105 particles were used to simulate the galaxies with respectively a dark matter halo of 2.5·109 M and 5·109 M . Although simulations with this mass resolution should be good enough for galaxies in isolation (e.g. Schroyen et al., 2013; Verbeke et al., 2014; Vandenbroucke, Verbeke & De Rijcke, 2016), a second simulation was run for the galaxies with a mass of 2.5·109 M and 5·109 M with respectively 105 and 2 · 105 particles. This was done since very occasionally a gas particle was transformed into a stellar particle for the galaxy with a halo of 5·109 M . So looking at this with a higher mass resolution was interesting since simulations of this kind of galaxies do not require a lot of resources. The simulations with higher resolution gave qualitatively and quantitatively the same results. Confirming that the used resolution will be good. 2.3 Interaction of dwarf galaxies To study the effects of tidal interactions of dwarf galaxies, two dwarf galaxies that were simulated in isolation were put on a trajectory passing each other. The starting distance 36 0.010 0.008 0.006 0.004 0.002 0.000 12 10 8 6 4 2 0 −1 −2 −3 −4 10 −53 0.010 0.008 0.006 0.004 0.002 0.000 16 14 12 10 8 6 4 2 0 −1 −2 −3 −4 10 −53 SFR [M /yr] Mstar [106 M ] [Fe/H]B 102 102 101 100 inputfile MHI [106 M ] MHI [106 M ] [Fe/H]B Mstar [106 M ] SFR [M /yr] GUI 101 2 4 6 8 10 100 12 2 t [Gyr] 4 6 8 10 12 t [Gyr] Figure 2.16: The evolution of DG1 constructed with the graphical user interface on the left and the one constructed with the inputfile on the right. This plot starts after one year of evolution, since the high initial star formation makes it difficult to create plots that nicely show all properties. between these galaxies was taken large enough such that tidal effects of both galaxies on each other should be almost neglectable. So the initial conditions were constructed such that the galaxies were initially seperated by at least 100 kpc. The specific models used will be discussed later in this section. Putting the galaxies together is a sort of balancing exercise. Since on one hand you would like to have them starting as far as possible from each other, such that no tidal effects are present at the moment these galaxies are put together. On the other hand if you put these galaxies too far from each other. You spend a lot of computing time just evolving galaxies that are in good approximation still evolving in isolation. To obtain an idea if the initial seperation of more than 100 kpc is big enough, let’s first have a look at the magnitude of tidal interaction. Where the tidal acceleration experienced by a particle is given by atidal = 2GMsource ∆r R3 (2.23) Msource is the mass of the source causing the tidal force, R is the distance between the center of mass of the object experiencing tidal forces and the source, and ∆r is the distance from the center of mass for the particle experiencing tidal accelaration. It can be noted from this equation that tidal force is an R−3 law. Such that this force drops extremely quick when increasing the distance of the source. 100 kpc is many times the size of dwarf galaxies which have a radius of only few kpc. This implies that ∆r R3 should make equation 2.23 relatively small for dwarf galaxies when R is higher than 100 kpc. 37 0.10 0.10 GUI SFR (M /yr) 0.08 SFR (M /yr) 0.08 0.06 0.06 0.04 0.04 0.02 0.00 inputfile 0.02 0.2 0.4 0.6 time (Gyr) 0.8 0.00 1.0 0.2 0.4 0.6 time (Gyr) 0.8 1.0 Figure 2.17: The star formation in the first Gyr of evolution of the two DG1 dwarf galaxies. On the left is the DG1 created with the graphical user interface. On the right is the DG1 created with the inputfile. 2.3.1 Different interaction models Trying to achieve a systematic study of tidal interactions of dwarf galaxies, one has to cover a large parameter space. A first parameter one can vary is the mass of the interacting galaxies. As was shown in the section on the mass of the dark matter halo, this can cover a large range of masses that lead to a different evolution of these galaxies. However it was found that for initial halo masses lower than 5·109 M there is absolutely no star formation and barely any gas present in the halo. It was also noted that 5·109 M was the absolute minimum for anything happening in the halo, but that even this mass was too low for expecting effects in tidal interactions with other dwarf galaxies. Halos with initial masses of 1010 M were the lowest mass halos that had a significant star formation over their evolution and were able to keep a sufficient reservoir of gas that could be affected by tidal interactions. The neutral hydrogen mass is of the order of 107 M . It thus seems fair to use this halo mass for the low mass halos in tidal interactions. As was noted earlier, the halo with initial mass of 4·1010 M almost exactly satisfies the limits of a dwarf galaxy that were proposed by Tammann (1994). So using the galaxy obtained with this halo mass should be a good reference for a high mass dwarf galaxy. The third halo mass used in the study of the tidal interactions has 2·1010 M as initial mass. This is the dubbel of the low mass and half of the high mass galaxy, such that it could be used as a medium mass dwarf galaxy. Not 3 different galaxies in isolation are used for studying the effects of tidal interactions, but 4 as mentioned earlier. As demonstrated in Schroyen et al. (2011), the initial rotation can have a big influence on the evolution and eventual galaxy observed. One 38 could expect that the effects of tidal interactions on these different galaxies would be different as well. This is why tidal interactions on galaxies with the same initial halo mass but with different initial rotation of the gas cloud are studied as well. The initial mass and rotation are the two main parameters that influence the evolution of a galaxy. So the division of the used galaxies is based on these two parameters. In table 2.1 the main parameters of the galaxies run in isolation are given together with the name they are referred to. One could wonder why only the effect of different initial rotation for name initial mass of the dark matter halo initial rotation of the gas cloud DG1 1010 M vinit = 0 km/s 10 10 M vinit = 5 km/s DG2 2 · 1010 M vinit = 2.5 km/s DG3 DG4 4 · 1010 M vinit = 2.5 km/s Table 2.1: This table contains the initial mass of the dark matter halo and the initial rotation of the gas cloud of the dwarf galaxies simulated in isolation that are used in the tidal interaction simulations. low mass galaxies is studied. This is purely based on the computational time needed for the more massive galaxies. Simulating the evolution of these galaxies is very costly and simulating the interactions with these heavier galaxies is even more costly. Such that it would require more resources to do the same study for the more massive galaxies. It could be questioned if this is sufficient to study the effect of rotation. But it was shown in Schroyen et al. (2011) that initial rotation leads to quantitatively and qualitatively different dwarf galaxies. Such that we should be able to study the general effects only related to the initial rotation in the tidal interactions. As one could expect, the main parameters of the galaxies are not the only parameters that matter in tidal interactions. The effects could depend on the galaxy it is interacting with. The effect of a more massive or a less massive galaxy could be completely different. Furthermore one might expect that it could depend on the distance at the moment of closest approach. The distance of closest approach already leads to a large parameter space one could cover. But it might also depend on the eccentricity of the orbit. One could try to reduce the size of this parameter space that has to be covered. For example when looking at relatively close encounters one can avoid very high eccentricities since this would require extremely high relative velocities. Since we are looking at tidal interactions, a lower limit can be put on the eccentricity. If the eccentricity is lower than 1, the 2 galaxies form a bound gravitational system such that they should merge after a while. Obviously the galaxies in this bound system will also experience tidal forces. Yet it seems applicable to study these tidal forces in the context of the merging process and not in the context of tidal interactions of two dwarf galaxies (Bekki, 2008; Cloet-Osselaer et al., 2014; Starkenburg, Helmi & Sales, 2016). In practice this lower 39 limit can be pushed up even a little more. This is possible because of the interaction of the gas clouds and the fact that we are considering N-body simulations, which could lead to interactions with eccentricity just above 1 eventually leading to a merger. Such that in our case a minimal eccentricity of 1.5 is used to study the tidal interactions. The maximal eccentricity used is 3.5 such that a rather wide range of interaction paths are covered. This can be seen in figure 2.18 where the paths for different eccentricities are plotted. These plotted trajectories are idealisations of the true trajectories that would be followed because of effects associated with the kinematics of the gas and the fact that N-body systems are involved instead of point sources. Yet it gives a good visualisation of the followed trajectories. An earlier study of the effects of a gas cloud falling in by Verbeke et al. (2014) showed great dependence on the way the gas cloud was falling in on the galaxy relative to its rotation. It was shown that when the gas cloud fell in along a retrograde path a strong burst in star formation was easily triggered. Yet a prograde path did not easily lead to a strong burst in the star formation rate. This is a different study, but seeing this result it is certainly worth to study this effect in tidal interactions. Figure 2.19 shows the concept of a retrograde orbit. This is an orbit where the first galaxy passes the other galaxy in the direction opposite to the rotation of that galaxy. A prograde orbit is then a path where the relative motion is in the same direction as the rotation of the object. In comparison to the study by Verbeke et al. (2014) the definition of prograde and retrograde is a bit more difficult since there are two galaxies involved and DG1 has no initial rotation and no clearly defined rotation in the x-y plane. So it does not necessarily mean that when a prograde interaction is chosen that this is prograde for both galaxies when they have a rotation. So it has to be checked that all galaxies with a rotation have the same rotation sense in the x-y plane. If they all have the same sense of rotation, this would not pose a problem. Since in that case the interaction would be e.g. prograde for both galaxies. If the two galaxies have opposite rotation sense, one galaxy will pass on a prograde trajectory while the other galaxy will pass on a retrograde trajectory. In case of our simulations this does not pose a problem since the rotation of all galaxies with a rotation is in the same sense, such that prograde and retrograde are properly defined. In the case DG1 is involved, it is put on a retrograde or prograde orbit of the other galaxy. When two DG1 type dwarf galaxies are interacting, this is hard to define. So when a prograde orbit is chosen, a track is used that would be prograde for one of the galaxies with initial rotation. But for these galaxies no difference between prograde and retrograde should be expected. The paragraphs above give an overview of the different parameters considered for the tidal interactions. To study the effects in a more systematic way, the simulations were divided over 7 models. These 7 models were based on the characteristics of the galaxies in isolation that were used in the interaction. In these seven models, simulations were 40 Figure 2.18: Plotted here are three orbits, with eccentricities ranging from 1.5 to 3.5, showing the range of paths that are covered in our simulations. run with a different orbit and different values for the pericenter distance and eccentricity. These 7 models were used to have a clear view on the effect of interaction with different galaxies. In the coming part, the specific properties of these 7 different models will be discussed. 2.3.2 The models In table 2.2, the specific properties of the interaction models are displayed. As was explained before, the galaxies used in our simulations depend on two parameters. Namely mass of the dark matter halo and initial velocity of the gas cloud. It are these 2 parameters of the 2 galaxies that are listed in this table. The first three interaction models only consider low mass galaxies. These are mainly used to study a possible effect of the initial rotation of a galaxy in the interaction. So the effect on the galaxy with initial rotation is studied when it is interacting with a galaxy without initial rotation but as well when it is interacting with a galaxy with initial rotation of the same mass. The same is done for the low mass galaxy without initial rotation. The following three models , so I4, I5 and I6, introduce a galaxy with a higher mass, referred to as medium mass galaxy or DG3. The gas cloud of this galaxy was given an initial rotation of 2.5 km/s. These interactions make it possible to study the interaction 41 Figure 2.19: This plot shows a retrograde orbit of the red dot. Changing the direction of rotation would lead to a prograde orbit of the red dot. This image was taken from https://en.wikipedia.org/wiki/Retrograde and prograde motion name I1 I2 I3 I4 I5 I6 I7 Mgal1 (M ) 1010 1010 1010 2 · 1010 2 · 1010 2 · 1010 4 · 1010 vinit,gal1 (km/s) 5.0 5.0 0 2.5 2.5 2.5 2.5 Mgal2 (M ) vinit,gal2 (km/s) 1010 0 1010 5.0 1010 0 10 10 5 1010 0 10 2 · 10 2.5 2 · 1010 2.5 Table 2.2: This table contains the mass of the dark matter halo and the initial rotation of the gas cloud for both galaxies that are used in the interaction models. of galaxies with a different mass. Studying the interaction of galaxies with different masses is certainly important since one can expect that this could lead to different effects. This expectation can be purely based on the higher mass leading to a higher gravitational and tidal force exerted on the other galaxy. Again both low mass galaxies are put in interaction with this medium mass galaxy to study the effect of their initial rotational velocity. These two low mass galaxies will also be able to point to possible effects only dependent on the mass of the halo. Not only is it possible to study the effects on a low mass dwarf galaxy when it is interacting with a heavier galaxy. Looking at the more massive galaxy can also tell us the effect of interacting with less massive galaxies. Model I6 looks at the interaction of two medium mass dwarf galaxies, such that this can be compared with the interactions with a low mass dwarf galaxy. 42 To make the study of the tidal interactions more complete, interactions with very heavy dwarf galaxies would be interesting. This brings us to model I7. More than the models before we are faced with computationally very demanding simulations, which strongly limits the study of these interactions. This is why only one model of interactions with a high mass dwarf galaxy is studied. Otherwise sufficient statistics on this model would be impossible, and a lack of statistics would make the study of these models rather useless. The question then is: which model to study? One could study the effect of the tidal interaction of a high mass galaxy and a low mass galaxy. But to some extent this was already achieved by models I4 and I5. It thus seemed more interesting to study the interaction of the medium mass dwarf galaxy with the high mass dwarf galaxy. This would make the study of mass dependence of the tidal interactions more complete. Certainly since the effect of rotation is already studied by the previous models. It could be argued that taking the medium mass and high mass dwarf galaxies would be far more computationally demanding than the interaction with low mass galaxies, since this would include more particles in the simulation. However the most important reason for the higher computational cost is dominantly that much more particles are in high density areas of these galaxies. Since the time step is dependent on the density, this leads to much more computations that have to be done. This was confirmed by noticing that the simulation times of the galaxies in isolation with different masses and different amount of particles clearly didn’t follow the N logN relation that only takes the amount of particles (N) into account. However some remarks should be made on this. First of all the difference in mass would be bigger in case the low mass galaxies were used, such that the galaxies might need to start with a larger distance between each other. Leading to the need for evolving this simulation longer and thus increasing the amount of computations needed. Secondly from the simulations of the galaxies in isolation it could be expected that the dominant computational cost of these models would be the high mass dwarf galaxy. As noted before, it not only has most particles but it also requires that more timesteps have to be calculated. This increases the amount of computational time needed significantly only because of the presence of the high mass galaxy. Taking into account that interactions with the low mass galaxy might need to evolve longer, these simulations might grow as costly as interactions with the medium mass galaxy. It is thus fair to say that computational costs should not be used as an argument not to study the interaction with medium mass dwarf galaxies. 2.3.3 Initial velocity When the two dwarf galaxies are put together such that they can interact, they are also given an initial velocity towards each other. This initial velocity is based on the trajectory of the galaxies and their masses. This mass was determined by looking at the 43 mass within a certain radius around the center of the galaxies. For DG1, DG2 and DG3 a radius of 20 kpc was used, for DG4 a radius of 30 kpc was used. These were rather good radii, since even the dark matter denstiy at these radii was very low. Trajectories with eccentricities between 1.5 and 3.5 were taken as mentioned before, since trajectories with a higher eccentricity seem rather unplausible. The initial velocities generated vary between minimally ± 60 km/s for the interactions of low mass dwarf galaxies and a maximum of ± 145 km/s for the interaction of medium and heavy dwarf galaxies. On itself these initial velocities don’t say that much since they are dependent on the initial distance of the interacting system. But this can be used for a comparison with observational data. For this we have a look at the observational data obtained by Noeske et al. (2001). They determined the recession velocity difference relative to us for dwarf galaxies within a projected distance of ± 100 kpc of one another. They found these recession velocity differences could reach up to 250 km/s but with the larger amount of velocity differences being way lower. The distribution can be seen in figure 2.20, which was taken from their publication. Looking at our initial velocities, they seem to be in the same interval as these observational measurements. The higher velocities in the observations could be explained because the above mentioned speeds in our simulations are only the initial relative speeds, thus the speeds at a distance of about 100 kpc and higher. But when the galaxies approach each other they will accelerate which will lead to higher relative velocities. To have an estimate of the relative velocity at the moment of closest approach, conservation of energy can be used. This is done using equation = v2 µ − 2 r (2.24) Where is the specific orbital energy of the system. µ is given by G(M1 + M2 ) with G the gravitational constant and M1 & M2 the two masses of the objects, in this case the galaxies. The orbital velocity of the two objects is given by v and the r gives the radial distance. This formula is used to have an estimate of the maximal relative velocity that occured in our simulations. To achieve this, formula 2.24 was applied to the system with the highest initial velocity. The relative velocity of these galaxies was then determined when their radial distance is 12 kpc. This is the radial distance of the two galaxies at the moment of closest approach. The calculations gave a maximal relative velocity of 213 km/s. This of the same order of magnitude of the maximal recession velocity differences proposed by Noeske et al. (2001). When having a detailed look at the velocity distribution in figure 2.20, one can see that there is also a significant fraction of galaxies with recession velocity differences smaller than 50 km/s. This is not really a huge problem. First of all, figure 2.20 only shows the recession velocity difference which is not always a good representation of the real relative velocity, such that the effective relative velocity might be higher. A second 44 Figure 2.20: Important is the continuous line in this plot which shows the observational difference in recession velocity of the dwarf galaxies that are within a projected distance of ± 100 kpc of each other. remark is, that they were only looking at dwarf galaxies near each other. However it is perfectly possible that some of these galaxies near each other will eventually merge, but merging galaxies was not part of our study. So in conclusion, the fact that the relative velocities in our simulations are roughly in the same interval as observational recession velocity differences gives confidence that realistic tidal interactions of dwarf galaxies are studied. 2.3.4 Amount of simulations As stated before a huge parameter space has to be covered when studying these tidal interactions. This leads to the requirement of a large amount of simulations for an indepth study. To achieve this, 140 different simulations of interactions were performed. This in not including some simulations that were done a second time but with a higher time resolution, which allows a deeper study of these simulations. These 140 simulations were rather equally distributed over the 7 interaction models. Only for model I7 less simulations were run due to the high computational cost. For model I7 only 10 simulations were run, where for the other models ± 20 simulations were done. In these specific models about half of the simulations were done with prograde trajectories and the other half with retrograde trajectories. Thus allowing a study of these 45 effects. All the initial conditions of the performed tidal interactions are listed in appendix 6. 2.4 Hyplot To analyse the data obtained from the simulations, scripts were written in python. These scripts made use of the analysis tool called Hyplot. This tool is publically available 1 and is designed for the analysis of N-body/SPH simulations. Very specifically for datafiles coming from the Gadget-2 simulation code. It can directly give quantities of galaxies evolved in isolation. This analysis tool is constructed using three programming languages. Namely C++, Fortran and Python. The C++ and Fortran code are used for reading the data files and performing calculations. While Python is mainly used for plotting and scripting (Valcke, 2010; Schroyen, 2013). As noted already, it is rather easy using HYPLOT to plot any basic quantities of a galaxy evolving in isolation. When working with two galaxies in the simulation, one has to take care that you know on which galaxy you are centered and thus taking data from. This extra care one has to take when more galaxies are involved will be discussed a little deeper in the coming part. 2.4.1 Recognising the galaxy HYPLOT has the built in function rcom(). Several arguments can be given to this function depending on what you want, but we won’t go into detail about this. This function doesn’t just calculate the center of mass in the data file but finds the center of mass of a galaxy. It can then put this center of mass at the origin, thus shifting the data. It can also give you the position of the center of mass with respect to the origin. But this feature was not really used. It is also possible to choose on which type of particle you want to center. You can look for the center of mass for the stars, gas or dark matter. It is also possible to find the center of mass based on all three types of particles. In most cases of the analysis, the centering is done based on the stellar particles. Since observationally, light coming from stars are a general way to identify a galaxy. Since this function will center on a galaxy in the data file, two questions arise: On which galaxy did it center? And secondly, how can you center on the other galaxy? Let’s start with answering the second question. After you centered on the first galaxy, you calculate the physical quantities of this galaxy. This is done assuming the galaxy is contained within a certain radius. When you have calculated the properties, you want to center on the second galaxy. This is done by putting the calculation of this physical quantity in a loop over the amount of galaxies 1 https://sourceforge.net/projects/hyplot/ 46 in the data, which is two in our case. Important to achieve recentering, is to use the reset() function which resets certain flags. This makes sure that using rcom() again will actually recenter the data. Otherwise the data will just stay at the same position since rcom() function gets the signal that it has already centered and will not try to recenter. You can expect that just using reset() and in the second part of the loop rcom() again, will lead to centering on the galaxy it already centered on the first time. This can be expected since this galaxy should still be the preferred center of mass. This can be avoided by removing this galaxy from the data. This is done by applying limits on the data based on the radius. When this galaxy is removed from the data, you can reuse rcom() for centering on the second galaxy. This procedure can be extended for as many galaxies as there are in the data. However one has to take care when two galaxies are very close that you don’t remove both galaxies or part of the second galaxy based on the radius. The previous part answered the question on how to recenter on the second galaxy in the data. But still you don’t know on which galaxy you centered during these processes. In the case of galaxies with different masses it will generally center first on the most massive galaxy and afterwards on the lower mass galaxy. However you want to be sure about this and in the case of two galaxies with almost the same mass you can’t really have an idea on which galaxy there was centered first. To know on which galaxy you centered, you can use the identities of the particles. The particles are namely identified by a number, so each particle in the simulations has a number. And maybe most importantly, this number stays the same over the entire lifetime of this particle. To define what is the first and what is the second galaxy, we look at the first datafile of this interaction. For which we perform the centering procedure explained above. But instead of calculating physical properties for the galaxy centered on, the id numbers of 100 stellar particles in both of these galaxies are stored in two different lists. Then when looking at the snapshots of interest, there is looped over the two galaxies to center on both of them. However except for calculating the physical properties of this galaxy, a list of all the id numbers of the stellar particles in the galaxy is made as well. It is checked for all these id numbers if they can be found in one of the two arrays that contain the reference id numbers. After this is done for all stellar particles, one could expect that this galaxy would contain 100 stars of a certain list and thus the galaxy is identified. This could be end of story, but since galaxies are dynamic enviroments it is possible that one or a couple of these stellar particles get ejected out of the galaxy. So a good criterion to identify this galaxy would be that the galaxy contains at least 70 or 80 of these reference stars. Since normally not that many stars are lost over the time we are looking at these galaxies, even when the galaxy is tidally interacting. 47 So to plot physical properties over the evolution of time for both galaxies, two arrays were defined such that each array contained the physical property of the same galaxy over its entire evolution. The allocation to these arrays was based on the identification explained above. 48 Chapter 3 Results In this chapter results obtained using the analysis tools will be presented and shortly discussed. Since the first objective of this thesis was to study if tidal interactions could trigger starbursts in dwarf galaxies, the first results to be discussed will be about the star formation rate over the evolution of the interacting galaxies. Afterwards other properties of these dwarf galaxies will be discussed as well. Such as the density profiles at certain points in the evolution, the evolution of the velocity dispersion, the gas mass available in the galaxy,... It will be checked if certain events in the evolution of these galaxies can be related and if certain events can be explained by other properties seen at the same time. This won’t be always discussed in full detail since a more in depth study will be done in the next chapter. 3.1 Star formation history By several observational studies it was shown that there are dwarf galaxies that show an extaordinary star formation who are accompanied by other dwarf galaxies (Brinks & Klein, 1988; Brinks, 1990; Campos-Aguilar & Moles, 1991; Noeske et al., 2001). For this reason it is of great interest to have a look at our simulations and see if these tidal interactions are able to trigger remarkable bursts in these dwarf galaxies. It has to be noted that it is not necessarily expected that both galaxies show star formation bursts and certainly not at the same time. Since these studies note that in many cases it is very hard to see the compagnon. When speaking of starbursts it would be nice to have a definition to characterise this burst. This is not that easy, since the star formation in the galaxies clearly varies over the evolution of the galaxy as can be seen from figure 2.8. So a star formation rate that would be characterised as a burst after 12 Gyr of evolution would not necessarily be characterised as a burst after 3 or 4 Gyr. This is why we use the definition proposed by Verbeke et al. (2014) to determine the burst factor of a certain star formation peak. 49 We can do this since both studies are about bursts at late stages of the dwarf galaxy evolution. Their definition only takes into account the star formation over the last 3 Gyr of the host, so the galaxy in isolation. The star formation rate at the peak is then divided by the average star formation rate of the host over the last 3 Gyr. This is expressed by equation (3.1) b = SF Rpeak /SF Rhost Where ’b’ is the burst factor. In their paper they also characterise these bursts based on the burst factor. When the burst factor of the peak is higher than 5, they characterise it as a burst. When the burst factor at the peak is larger than 10, they characterise it as a strong burst. This definition was used for all galaxy models in their simulations. This definition will not be used in this thesis. Why this definition is not used will be shortly discussed in the coming part. As already explained and you might have seen from figure 2.8, the star formation history is highly dependent on the dwarf galaxy model that is used. Some models clearly have a more burst like behaviour than other models, certainly after 10 Gyr of evolution. To show this, we apply definition 3.1 to the galaxies in isolation. We only consider the star formation history after 10 Gyr of evolution, since none of the tidal interactions in our models take place before 10 Gyr of evolution. The results are shown in table 3.1. name burst factor DG1 4.19 DG2 2.84 DG3 1.61 DG4 2.24 SF R (M /yr) 4.2 · 10−4 1.3 · 10−3 1.2 · 10−2 4.3 · 10−2 Table 3.1: This table contains the burst factors of the galaxies in isolation. Based on these numbers it could be claimed that there is not really a line in the burst factors. But when looking in great detail to figure 2.8. It can be seen that there is a small star formation peak in DG4 just after 10 Gyr of evolution. When looking at the star formation history of DG3, it can be noted that there is a slightly higher star formation rate just before 10 Gyr of evolution. Comparing this with the peak in DG4, one could expect that peaks in DG3 just before 10 Gyr of evolution will have a similar burst factor. This was checked by including 500 Myr extra in the analysis. In that case a burst factor of 2.13 is found for the medium mass galaxy (DG3) in isolation. This burst factor probably gives a more accurate representation for the medium mass galaxy. In conclusion, it can seen that there is a difference in what can be classified as a burst in each galaxy. Another reason why not necessarily the same definitions have to be used is the fact that new physical processes are included in the code used since the publication by Verbeke 50 et al. (2014). First of all an ultraviolet background is included and secondly, population III stars are included in the code used for this investigation. It is already shown in publications that this has a significant influence on the galaxy evolution (Verbeke, Vandenbroucke & De Rijcke, 2015; Vandenbroucke, Verbeke & De Rijcke, 2016). As a last argument, in this thesis the star formation rate is determined by looking at time intervals of 50 Myr. In each time interval the star formation rate is then determined. As mentioned in the previous chapter, this star formation rate can also be determined with other time resolutions. We will show later that this can lead to different burst factors. This is another reason why the definition of a burst and a strong burst by Verbeke et al. (2014) can not directly be applied to our simulations, since it is not immediatly clear if the same time interval is used in this paper. 3.1.1 Burst factors in the interacting models I1 All models had simulations where significantly higher burst factors were seen compared to the same galaxy in isolation. This burst factor was dependent on the galaxy as stated above. Looking at model I1, some remarkable bursts were noted in DG1. In 4 of the 23 simulations of this model, a burst factor higher than 6 was obtained. With a maximal burst factor of 6.99. As can be seen from figure 3.1, a burst factor bigger than 6 is certainly noteworthy for this type of galaxy. Such that we will use this burst factor to define a burst in DG1. In the case of DG2, the burst factors are not that high in the I1 model. This can be seen from figure 3.2, which shows the highest star formation peak in this model for DG2. This star formation rate is not that much higher than the maximal star formation rate in the model in isolation and neither remarkable enough to be classified as a burst. In the two plots mentioned above, the peak in the star formation rate happens shortly after the point of closest encounter. This closest approach occurs in all models after 11.4 or 11.5 Gyr of evolution. The bursts in the models with the low mass galaxies generally occur around this time. But for the moment we won’t go into detail about this. The moments that the bursts occur will be discussed in more detail in the following chapter. I2 In this model, two low mass galaxies with initial rotation are interacting with each other. Based on the maximal burst factor of the galaxy in isolation, it is not expected that these galaxies will reach burst factors as high as DG1. However higher star formation peaks are obtained in this model. The maximal burst factor obtained in these galaxies is 4.42, 51 Figure 3.1: This plot contains the comparison of the star formation rate of the galaxies in isolation and the two galaxies in an I1 interacting model. On the left DG2 is plotted, on the right DG1 is plotted. The burst factor at the peak of the galaxy on the right is 6.87, the maximal burst factor in the galaxy on the left is 3.09. The vertical line indicates the moment of closest approach. Figure 3.2: The plot on the left shows the highest star formation peak of the galaxy with initial rotation in the I1 model. The burst factor of this peak is 3.86. The vertical line indicates the moment of closest approach. 52 which is shown in figure 3.3. Over all the 22 simulations in total of this model, there are 3 simulations where a galaxy shows a peak with a burst factor higher than 4. The peak with a burst factor of 4.42 is strong compared to the star formation in isolation and in general any peak with a burst factor higher than 4 seems to be rather remarkable for this type of galaxy. This is why peaks, in low mass galaxies with initial rotation, that have a burst factor higher than 4 will be referred to as bursts in this thesis. I3 In this model, two low mass dwarf galaxies without initial rotation are tidally interacting. As mentioned before, a burst factor of 6 will be used to classify a peak as a burst. Of the 21 simulations done, 4 contain a galaxy with a burst factor higher than 6. The maximal burst factor that was obtained is 7.43. So it seems that it is possible to trigger significant star bursts in DG1 with galaxies of similar mass irrespective of their initial rotation. Based on model I1, this does not seem to be the case for DG2. This is not such a big surprise since the initial rotation of DG2 provides a certain stability against collapse of the gas (Schroyen et al., 2011). That in model I2 bursts occur might be attributed to the fact that DG2 creates a higher tidal force than DG1 since more gas and stars are contained within the galaxy. Although it could still be a statistical fluctuation due to the small amount of bursts in this type of galaxy. But the large difference in maximal burst factor for DG2 when comparing I1 and I2 doesn’t really support this last remark. I4 This is the first model that also includes something else than only low mass galaxies, namely DG3. In this specific model this medium mass dwarf galaxy is interacting with the low mass dwarf galaxy with initial rotation. For the low mass galaxies in this model, in 3 of the 23 simulations a peak with a burst factor higher than 4 is encountered which is earlier defined as a burst in this type of galaxy. The maximal burst factor that was obtained is 4.27. All these bursts occur when the distance at closest approach is larger than 20 kpc, which is rather remarkable. It could be that DG2 gets disturbed too much when it has a closer approach such that a burst can not occur. However no conclusive proof was found for this when looking at other properties of the galaxy. Also noteworthy peaks appear in the star formation history of the medium mass dwarf galaxies during these interactions as shown in figure 3.4. The highest peak during the interaction is remarkably higher than the star formation rate of the galaxy in isolation. But also very noticable and rather surprising is the peak in this galaxy only 100 Myr after the two galaxies were put together. To directly point at ’the’ explanation for this is quite hard. Since the two galaxies are put together and this peak does not show up in the galaxy in isolation it can be argued that this is caused by the second galaxy. However the small galaxy being able to trigger a burstlike event in the medium mass galaxy seems rather unlikely at that moment due to the small tidal forces then. Most likely this peak should not be classified as a burst taking into account what was mentioned 53 Figure 3.3: The peak in the left plot has a burst factor of 4.42 and the peak in the right plot has a burst factor of 3.76. The vertical line indicates the moment of closest approach. Figure 3.4: These star formation histories come from an I4 type interaction simulation. Both galaxies are displaying strong star formation peaks, with a maximum of 2.57 for DG3 and 4.12 for DG2. The vertical line indicates the moment of closest approach. 54 earlier. Namely that just before 10 Gyr this medium mass dwarf galaxy still has a slightly higher star formation rate where the highest peaks would give burst factors up to ± 2.1, although this is still lower than the burst factor of the early peak which is 2.31. It has to be noted that this was the maximal burst factor seen that early in the evolution of DG3 for all interaction simulations that were run. A burst factor has to be defined to classify a specific peak as a burst in the medium mass dwarf galaxies. We will use a burst factor of 2.5 to classify a peak as a burst in this thesis. Using this definition only one of the 23 simulations has a medium mass dwarf galaxy showing a burst. This burst has a burst factor of 2.57. I5 In this interaction model, the medium mass dwarf galaxy is interacting again with a low mass dwarf galaxy. But this time the low mass dwarf galaxy without initial rotation of the gas cloud. Based on the previously defined burst factors, there are bursts in both types of galaxies. In 2 of the 20 simulations of this model a burst occurs in the low mass dwarf galaxies, with the maximum burst factor being 6.36. Also in 2 of the 20 simulations, a burst occurs in the medium mass dwarf galaxy. It has to be noted that these 2 simulations are different simulations than the 2 with a burst in the low mass dwarf galaxy. The maximal burst factor obtained in a medium mass dwarf galaxy is 2.72. The star formation history of the simulation with the highest burst factor for the medium mass dwarf galaxy is plotted in figure 3.5. As can be seen, this peak is way higher than all other star formation. The peak occurs ± 150 Myr before the moment of closest approach. So in this case it seems very plausible that the tidal interaction with the other dwarf galaxy triggers the star formation burst. I6 This model studies the interaction of two medium mass dwarf galaxies. In 5 of the 21 simulations, a burst in at least one of the two galaxies is detected. Where 3.15 is the maximal burst factor that was obtained. It also has to be noted that in one of the simulations both galaxies had a burst factor higher than 2.5. The star formation histories for this simulation are plotted in figure 3.6. Also intersting to note is that this simulation contains the two star formation peaks with the highest burst factor of this model, namely 3.15 and 2.81. Remarkably these peaks occur at completely different moments. One happens directly after closest approach and the one in the other galaxy about 1 Gyr after closest approach. I7 In this model only very few tidal interactions were studied. But still a star formation peak was noticed in the medium mass dwarf galaxy. This peak had a burst factor of 55 Figure 3.5: This plot shows the strong star formation that is triggered in the medium mass dwarf galaxy because of tidal interactions with a low mass dwarf galaxy. The vertical line indicates the moment of closest approach. Figure 3.6: Plot of the star formation rate where both medium mass dwarf galaxies have a burst over their evolution. The vertical line indicates the moment of closest approach. 56 2.64, such that it is classified as a burst in our study. In the case of the high mass dwarf galaxy, no really significant peaks were observed. The maximal star formation peak in the interacting models still had a lower star formation rate than the small peak in the galaxy in isolation just after 10 Gyr of evolution. To classify a peak as a burst in DG4, a burst factor of 2.5 seems a fair requirement. Yet this is by far not reached in the performed simulations, so one can not speak of a burst in these galaxies. This doesn’t mean that no burst can occur since only 10 simulations with a DG4 were run. But no peaks with a burstfactor higher than 2.2 were seen, which indicates that it might be rather difficult to trigger a burst in DG4 with a less massive dwarf galaxy. 3.1.2 Prograde vs retrograde Looking at the orbit when a burst occurs, there is no unambiguous dependence on the type of orbit found. For DG1, 6 burst were noted when it was on a prograde orbit and 4 were noted when it was on a retrograde orbit. Indicating that the type of orbit is of no importance for this dwarf galaxy model, as was expected. For DG2, 4 galaxies had a burst when they were in a prograde orbit and 2 galaxies with a burst were found with a retrograde orbit. For DG3, 3 galaxies had a burst when they had a prograde orbit and 7 galaxies with a burst were found in a retrograde orbit. Seeing these results it is not easy to point at a specific orbit that more easily causes a burst in a dwarf galaxy. The small spread seen could be caused by rather poor statistics in the amount that could be classified as a burst. When taking the statistics for DG2 and DG3 together, 7 galaxies were in a prograde orbit and 9 were in a retrograde orbit. This doesn’t point to a dependence on the type of orbit for the dwarf galaxies to have a burst. This is in contrast with the observations of a gas cloud falling in, where a strong dependence on the type of orbit was found Verbeke et al. (2014). This doesn’t have to be completely surprising, since in the case of our simulations the burst is triggered using a different mechanism. A dependence on the eccentricity wasn’t found either. The orbits that had a burst, had eccentricities ranging between 1.8 and 3. That no bursts were found on an orbit with an eccentricity higher than 3, doesn’t really say much since very few interactions with an eccentricity higher than 3 were run. 3.1.3 Duration of the bursts As already mentioned, it is possible to determine star formation rates in different time intervals since the birthtime is stored. Looking at this might be interesting since it could shed more light on e.g. the duration of bursts. Using the 50 Myr time interval to determine the star formation rate didn’t really allow to determine the duration of the bursts, since in almost every case the peak consisted of only one time interval. To study the duration of these bursts, there was zoomed in on the bursts in time and using the 57 birthtime of the stars it was tried to determine the duration of the burst. In case of DG1, these bursts are of very short duration in time with a maximal duration of the order of ± 10 Myr as can been seen in figure 3.7. When looking more into detail to the gas flow at the moment of a burst in a DG1 dwarf galaxy we will try to explain why this happens. The fact that this burst is of such a short duration leads to a very strong sharp peak in the star formation history. When looking at the bursts of DG2 with a higher time resolution, a very different behavious is observed. The bursts occur during a longer time interval, going up to 50 Myr like in figure 3.8. Such that it is not one sharp peak, but the burst consists of an increased star formation for a short while. In the case of the burst in figure 3.8 the burst seems to be split into two by a very short interval of 10 Myr where no star formation occurs. In the case of DG3, two different types of burst were found when looking with a higher time resolution. This is shown in figure 3.9. The first type of burst can take up to 80 Myr, which has a strongly increased star formation over this time. This looks a lot like the bursts seen in DG2 type galaxies. On the other hand, bursts with one sharp peak have been found as well. These look more like the peaks observed in DG1 type dwarf galaxies. A possible explanation for this behaviour could be the initial rotation of the dwarf galaxy, since DG3 has a maximal initial rotation velocity of 2.5 km/s. This is exactly between the initial rotation velocities of the two low mass dwarf galaxies. 3.2 Velocity dispersion In the previous section it was shown that significant star formation peaks can occur while two dwarf galaxies are tidally interacting. Due to the large feedback coming from the young stars, one can expect that this will have a visible impact on the dwarf galaxies. First we will look for its effect on the velocity dispersion in the galaxy. Where both the velocity dispersion of the gas and the stars will be considered. When looking at the effect of feedback, it can also be checked if the tidal interactions have any effect on the velocity dispersion in the dwarf galaxies. To determine the velocity dispersion, the following formula is used r σx2 + σy2 + σz2 (3.2) σ= 3 P 2 Where σx2 = N1 N i=1 (vx,i - vx,av ) . This is calculated, taking into account all stellar and gas particles within a radius of 5 kpc. Such that N is the amount of gas or stellar particles within a 5 kpc radius from the galaxy center. For DG1, there is a strong peak in the velocity dispersion of the gas after a burst. This can be seen in figure 3.10. The effect on the stellar velocity dispersion is hardly or not noticable. The velocity dispersion can show peaks up to about twice the average 58 Figure 3.7: A high time resolution of a burst in a DG1 dwarf galaxy. The time resolution in this image is 5 Myr. Figure 3.8: A higher time resolution of a burst in a DG2 dwarf galaxy. It is plotted with a time resolution of 10 Myr. 59 Figure 3.9: The two types of bursts seen in the medium mass galaxy. These plots were made with a time resolution of 10 Myr. velocity dispersion of the galaxy. The time for the gas to recover to the normal velocity dispersion for these galaxies is about 50 to 100 Myr. Also clearly noticable in these dwarf galaxies, is the fact that the average velocity dispersion is significantly higher for the stars than for the gas. It can also be seen that the velocity dispersion of the gas has a peaked time evolution, where the peaks are related to small star formation seen in the galaxy. This is caused by the lack of initial rotation of the gas cloud. For DG2, also a peak in the velocity disperion of the gas is noted after a star formation burst. However a clear difference with the other low mass galaxy can be noted in figure 3.11. The peak in the velocity dispersion of the gas is not much higher than the average velocity dispersion of the gas. Certainly when comparing with the velocity dispersion peaks obtained for DG1. As can be seen from the evolution of the velocity dispersion in figure 3.11, it takes the gas about 50 to 100 Myr as well to come back to the original velocity dispersion. Noticable compared to the velocity dispersion of DG1 is the fact that the velocity dispersion of the gas and the stars are very similar, which happens because the average velocity dispersion of the gas is higher. The average velocity dispersion of the stars in the two low mass models is basically the same. An example of the velocity disperion of DG3 while interacting is shown in figure 3.12. As could be expected for a more massive dwarf galaxy, the average velocity dispersion is higher compared to DG1 and DG2 (e.g. Valcke, de Rijcke & Dejonghe, 2008; Schroyen et al., 2011). There is still a higher velocity dispersion of the gas when a starburst occurs, but it can not be characterised as a real strong peak. This is not really surprising because there is more gas in this galaxy. Such that the feedback coming from the young stars is more localised in the galaxy and doesn’t have a strong impact on all the gas in the galaxy. The time needed to relax to the average velocity disperion takes ± 50 Myr 60 Figure 3.10: The velocity dispersion of the stars and the gas is plotted together with the star formation rate (black line) for DG1, clearly showing a peak just after the burst. Figure 3.11: The evolution of the velocity dispersion over time for the gas and stars in DG2. This is plotted together with the star formation (black line.) 61 to 100 Myr, so rather similar to the low mass dwarf galaxies. As stated at the start of this section about the velocity dispersion, it would be very interesting to have a look at any effects on the velocity dispersion coming from the tidal interaction. When having a quick look through the results, immediatly some very strong and rather unnatural peaks in the stellar and gas velocity dispersion are observed at the moment of closest approach such as in figure 3.13. These peaks can be noted when galaxies are passing very close to each other. So close that the two galaxies are partly passing through each other at moment of closest approach. In that case part of the second galaxy is taken into account for the determination of the velocity dispersion in the first galaxy. This can lead to false peaks in the velocity dispersion. Normally this effect can be noticed by looking at the second galaxy in the interaction, where a similar effect is seen. Figure 3.14 shows the velocity dispersion of the galaxy which is interacting with the galaxy for which the velocity dispersion is displayed in figure 3.13. It can be seen that the velocity dispersion of the gas in this galaxy shows a strong sharp peak at the moment of closest approach as well. This peak is not because part of the first galaxy is included but because part of this second galaxy is cut out of the data when recentering to the second galaxy. Finally it has to be noted that a clear signature of this false peak, is the sharpness of it. It is only lasts for 50 Myr, afterwards the galaxies are seperated far enough such that this effect does not occur anymore. So when trying to the determine if there is an effect of tidal interaction on the velocity dispersion. There has to be taken care that no part of the other galaxy is included in the data. Some effects in the velocity dispersion were noted around closest approach for which the above remark didn’t provide an explanation. An example is shown in figure 3.15, in that case a strong increase in the velocity dispersion of the gas is noted for ± 400 Myr. There is also an increase for the stellar velocity dispersion but less significant however. A couple of times a similar behaviour was noted. Very remarkable is that all these events occured in DG1 of the I5 interaction model. For DG2 and DG3 this increase is never significantly seen. Why this happens, we hope to answer in the coming sections. Finally it should be noted that for DG4 no significant increase in velocity dispersion was found over their evolution. This is not really surprising since increase in velocity dispersion was generally found when a burst occurred, which did not happen in DG4. 3.3 The gas mass Apart from an effect on the velocity dispersion, one could also expect that the burst in star formation will have an effect on the amount of gas in the galaxy. This would be caused by the feedback ejecting gas out of the galaxy. To have a look at this effect, plots were produced showing the evolution of the amount of gas within a 5 kpc radius of the center of the galaxy. Again there has to be taken care when analysing the results that part of the other galaxy is not included in the gas 62 Figure 3.12: The velocity dispersion for DG3 (including a star burst) in an interacting model. Figure 3.13: An enormous peak in the velocity dispersion at the moment of closest approach in one of the galaxies at closest approach. The vertical line indicates the moment of closest approach. 63 Figure 3.14: An enormous peak in the velocity dispersion of the second galaxy at the moment of closest approach for a very close encounter. The vertical line indicates the moment of closest approach. Figure 3.15: Around the moment of closest approach, there is a strong increase for the velocity dispersion of the gas for ± 400 Myr. The increase in velocity dispersion for the stars is less strong but noticable as well. The vertical line indicates the moment of closest approach. 64 mass of the galaxy when analysing these results. This will only happen at the moment of closest encounter just like for the velocity dispersion, such that this effect was easily noticed. In this context, there was very regularly an increase in gas mass inside the galaxy when the two galaxies were getting close to closest approach. This can be seen in e.g. figures 3.16 and 3.18. One could try to attribute this to gas originally belonging to or hanging around the other galaxy that is now found in the galaxy we are looking at. However there are some odd things about this when one has a more detailed look. First of all, this effect is also seen when galaxies are passing by at more than 20 kpc. This is a very large distance to note such an effect, since only extremely rare gas can be found at such distances. Secondly looking at figures 3.16 and 3.18 it can be seen that this effect can be rather big. A lot of gas seems to show up in the galaxies up to ± 107 M even for low mass galaxies. This is a rather high gas mass. These amounts are even found for very large distances of closest approach. So based on the identity numbers of the gas particles, the amount of gas originally belonging to the other galaxy and now found in the galaxy near closest approach could be determined. It turned out that this was by far not close to the increase seen in the gas mass. The values for this comparison are given in appendix 6. This is only done for the low mass galaxies that had a burst since this burst generally occurred near the moment of highest mass accretion such that this might be related. The values for the low mass galaxies that don’t experience a burst are not given there, but for these galaxies similar gas accretion was found near closest approach as can be seen in figure 3.17. For the medium mass galaxies this was not done since the gas mass behaviour is generally more complex such that it is difficult to define this increase and the bursts in DG3 generally don’t occur near closest approach. But this will be discussed in more detail in the next chapter. So to compare this increase in gas mass with the accreted mass from the other galaxy, the gas increase was determined by comparing the maximal gas mass near closest approach with the moment the gas mass starts to increase monotonically towards this maximum. Never more than 43 % of the mass increase is coming from the other galaxy. And generally speaking about 80 - 90% of the gas increase is not coming from the other galaxy in the interaction. It has to be noted that after closest approach all galaxies seem to lose a lot of gas as well. One reason for this can be the starbursts that are sometimes initiated by this gas capture of the galaxy. Another reason that seems rather plausible is tidal stripping after closest approach which we hope to confirm by looking at the gas flow later. Certainly in figure 3.17 this seems the most plausible explanation for the gas lost. What can be seen as well in figure 3.18 is the effect of feedback when a large amount of stars are formed. This clearly leads to a sharp decrease in gas content of the galaxy. In figure 3.16, this effect is less clear since one could expect that tidal stripping is starting at that moment as well. The complete effect of feedback is however dependent on the 65 Figure 3.16: In black, the evolution of the gas mass in a DG1 and in red the evolution of the star formation rate. The moment of closest approach is indicated by the vertical line. Figure 3.17: The evolution of the amount of gas (shown in black) in a DG1 that is not experiencing a burst. This can be seen from the star formation history (shown in red). The moment of closest approach is indicated by the vertical line. Figure 3.18: In black, the evolution of the gas mass in a DG2 and in red the evolution of the star formation rate. The moment of closest approach is indicated by the vertical line. 66 mass of the galaxy. Where in the low mass galaxies the gas lost after a strong burst or tidal stripping is generally never retrieved, the gas lost by the medium mass galaxy can be partly retrieved after a while as in figure 3.19. For the high mass dwarf galaxies nothing different should be expected since they have an even deeper gravitational potential. However for the bursts there is no direct confirmation since no burst was found in the massive galaxy. The recapture of gas can be rather significant for DG3, such that a sequence of burst like events can be triggered as can be seen in figure 3.19. Since the gas is generally never retrieved for low mass dwarf galaxies, it is rather difficult for low mass galaxies to achieve a sequence of burst like events. Although not necessarily impossible, sometimes when the galaxy seems to stop losing gas after the burst a smaller burstlike event can be triggered even though the amount of gas in the galaxy does not seem to increase just before this. This could be because this gas never left the 5 kpc radius and is at that moment collapsing somewhere in the galaxy. 3.4 B-I At the moment of the burst, it can be expected that the galaxy will look bluer because of the young stars that were created recently. In the I-band on the other hand one can mostly see old stars. Such that a B-I map can show where the young stars will be found in the galaxy. Making use of the B-band and I-band magnitude of the stellar particles, maps of the galaxies are plotted at specific moments during the evolution. To produce B-I maps that look more like what would be seen observationally, the luminosity in both bands of the stellar particles is smeared out using a gaussian since stellar particles are unresolved during observations. In our case, the particles are smeared out with a standard deviation of 80 pc. This value for smoothing is chosen since it delivers a smooth B-I map, but also allows to identify the B-I substructure in the galaxy. From these B-I maps, it is then possible to obtain a B-I colour profile. This B-I profile can be deduced from this map using a technique similar to isophotal integration (Papaderos et al., 1996b; Micheva et al., 2013a,b; Verbeke et al., 2014). This isophotal integration is done in steps of 0.01 for the B-I values to obtain a smooth profile. The integration is done by looking for each B-I value to the cartesian grid cells of the B-I map. For each of these B-I values, the number of grid cells are determined that are bluer than this value. The area of these grid cells is then added together. Knowing the total area which is bluer, this is converted to a radius of a circle with the same area. It has to be noted that this radius is not the physical distance from the center of the galaxy. In figure 3.20, a B-I map is plotted of a DG1 before, after and at the moment of the burst. Before the burst, almost no blue spots are noticable in the galaxy since it barely forms any stars. When the strong burst occurs, it seems to happen near the center close to one dominant peak. Afterwards the peaks get less blue and are spread a bit more over the galaxy. In DG1, the burst doesn’t necessarily happen at the center of the galaxy, but 67 Figure 3.19: In black, the evolution of the gas mass in a DG3 and in red the evolution of the star formation rate. The moment of closest approach is indicated by the vertical line. Figure 3.20: The B-I map of a DG1 before (t = 11.76 Gyr), during (t = 11.82 Gyr) and after (t = 11.85 Gyr, t = 11.97 Gyr) the burst. 68 it takes place very concentrated as shown in figure 3.21. This is probably a consequence of a large amount of gas collapsing towards a small area in the galaxy. If starbursts occur at two different points, these points are rather close to each other. A difference between DG1 and DG2 can be seen in the B-I maps at the moment of a star burst. Where DG1 has one or two spots of strong star formation in the galaxy, the star formation in DG2 seems to be more distributed over the entire galaxy. An example of this is shown in figure 3.22. Where at the moment of the burst the star forming regions are distributed over the galaxy, indicating smaller clouds are collapsing to form stars around the same time. This might be a good explanation why the bursts in DG2 seem to last longer in time than DG1, where the burst is one strong and short peak. Due to the strong feedback on the large collapsing cloud in DG1, one can expect that star formation is immediatly shut down after the first stars are formed. Where in DG2 feedback of young stars formed in one cloud will not stop the star formation in another area of the galaxy. The distribution of peaks in the B-I map of DG3 during a burst is rather similar to the one of DG2. One can see several star forming areas distributed over the galaxy in figure 3.23. This is not really surprising when looking at Schroyen et al. (2011), which states that the star formation is more smeared out over the galaxy when the gas cloud has an initial rotation. As mentioned before, B-I profiles can be constructed from these maps. Figure 3.24 shows such a B-I profile before, during and after the burst of a DG1. It can be seen that there is a strong blue peak during the burst. However, this peak is not more compact than at other stages in the galaxy evolution. So during a burst in this type of galaxy it seems like the star forming area is larger in comparison with no burst. Except for the peak for R < 0.5 kpc, the profile is extremely flat. Indicating that star formation in this type of galaxy occurs very concentrated. This is in agreement with the fact that the star formation is dependent on the occasional collapse of the gas cloud in a certain region. When looking at B-I profiles of DG2 in figure 3.25, it can be noted that these profiles are slightly different from the B-I profiles of DG1. They are bluer up to a larger radius, normally around 1 kpc. This is in agreement with the fact that the star formation happens continuously and is smeared out over the galaxy. A strong blue peak is also noticable during the burst just like in DG1. During a burst in a DG3, the B-I profiles also show a blue peak as can be expected, this is shown in figure 3.26. Compared to the low mass galaxies, the B-I profile has a smooth evolution up to higher radii than 3 kpc. These higher radii are very dominant in the infrared, implying that very old stars are found there. In general it can be noted that the bursts in the galaxies show a blue peak on the background of older stars. The B-I magnitudes obtained during bursts are of the order of 0.6, with the strongest peaks going down to 0.5. Although these values are dependent on the radius used for the smoothing. 69 Figure 3.21: The B-I map of a DG1 at the moment of the star burst. On the left the galaxy is plotted edge one and on the right face on. Figure 3.22: The B-I map before (t = 11.47 Gyr), during (t = 11.53 Gyr) and after (t = 11.55 Gyr, t = 11.61 Gyr) the burst of a DG2. 70 Figure 3.23: The B-I map of a DG3 before (t = 11.47 Gyr), during (t = 11.59 Gyr) and after (t = 11.61 Gyr, t = 11.7 Gyr) the burst. 71 Figure 3.24: The B-I profile of a DG1 before (red), during (blue) and after (green) a burst. Figure 3.25: The B-I profile of a DG2 before (red), during (blue) and after (green) a burst. 72 Figure 3.26: The B-I profile of a DG3 before (red), during (blue) and after (green) a burst 3.5 Gas distribution and gas flow In the context of tidal interactions, the gas distribution in and around the dwarf galaxies can be studied for many purposes. It was noted in the B-I maps that the starbursts in different types of dwarf galaxies show different signatures. One could expect that this will be reflected in the gas distrution of these galaxies. Studying the tidal interaction of galaxies, many authors noted already long ago the existance of tidal tails around these galaxies (e.g. Pfleiderer & Siedentopf, 1961; Toomre & Toomre, 1972). One might expect a similar behaviour for the interaction between dwarf galaxies. While studying the velocity dispersion of the gas earlier in this thesis, it was noted that in some galaxies the gas has a significant increase in velocity dispersion at the moment of closest approach. One can hope that the distribution of the gas might teach something about what happens at that moment. A last thing one can look for, is the effect of feedback from a strong burst on the gas in the galaxy. Since the study of the velocity dispersion clearly showed strong peaks immediatly after the burst. First let’s have a look at the gas distribution in the galaxies when a burst occurs. In figure 3.27 an example of a burst in a DG1 is shown. Just before the burst, the gas becomes extremely concentrated in a small area where the burst will occur. The feedback from the recently formed stars has an immediate impact on the concentrated gas clump, which get disrupted. Leading to a strong increase in spread of the gas for ± 100 Myr and an irregular distribution of the gas inside the galaxy. This is perfectly in agreement with the observations of the velocity dispersion. Where it was noted that there is a strong increase in velocity dispersion of the gas up to 100 Myr 73 Figure 3.27: A plot of the gas column density in a DG1. It is shown before (t = 11.60), during (t = 11.65) and after (t = 11.70) the burst. The yellow dots show the positions of the stars formed in the previous 50 Myr. after the burst. Afterwards, the gas will start to get more concentrated again and show a distribution similar to the normal observed distribution in this type of galaxies. In the galaxy with initial rotation, it appeared that the burst occurs more spread out over the galaxy based on the B-I maps in figure 3.22. To confirm this, it is necessary that the gas is more spread out over the galaxy at the moment of burst. However one also expect strong clumps showing up in this galaxy where the new stars will be formed. These strong clumps might be formed because of the generally higher gas mass in the galaxy at the moment a burst occurs. This is also what is seen in these galaxies. Not one highly concentrated clump, but several high density clumps at different places in the galaxy are observed just before and during the star burst in figure 3.28. This could give an explanation for the lower peak in the velocity dispersion. Since in DG1 the burst has a huge impact coming from one star formation peak on almost all gas in the galaxy that is collapsing towards a small area. On the other hand in DG2, the burst occurs in smaller clumps where the star formation is a bit spread over time. This gives rise to less strong feedback on the gas involved such that the peak in the velocity dispersion is smaller. Even though not all the gas is collapsing to one clump, it can be noted from figure 3.28 that just before and during the burst the gas gets very compact for this type of dwarf galaxy while still having an irregular clumpy gas distribution. This was also seen in observations (e.g. van Zee, Skillman & Salzer, 1998). It was also noted that the bursts show wider peaks in DG2 when comparing with DG1, where the star formation is completely shut down after a burst. This can also be seen in figure 3.28, where it is shown that it is possible that directly after the first stars are formed that other clumps can occur at different places in the galaxy. This can lead to extra stars formed and wider star formation peaks. After this star formation peak the gas gets strongly dispersed as well for ± 100 Myr, although less strong than in DG1. This is again in accordance with the previously ob74 Figure 3.28: A plot of the gas column density in a DG2. It is shown before (t = 11.45), during (t = 11.50) and after (t = 11.55) the burst. The yellow dots show the positions of the stars formed in the previous 50 Myr. tained behaviour of the velocity dispersion of the gas in this type of galaxy, which is shown in figure 3.11. Where peaks in the velocity dispersion after a star burst are less strong than in DG1 and also lasted up to ± 100 Myr. A DG3 galaxy also shows compact gas clumps rather spread out over the galaxy as was expected. And again the feedback from the young stars has a less significant impact on the interstellar medium as shown in figure 3.29. In the case of these dwarf galaxies, not all gas in the galaxy seems to be involved in the starburst. This will be a reason for the less strong peak in the velocity dispersion. One would also like to have an explanation for the increase in velocity dispersion in the low mass dwarf galaxies in model I5 at the moment of closest encounter. When looking at figure 3.30, it gets clear quickly why there is such an increase in velocity dispersion of the gas. The gas of the low mass dwarf galaxy forms a large tidal tail when passing by the medium mass dwarf galaxy. Thus strongly increasing the velocity dispersion of the gas within a radius of 5 kpc. Also something looking like a bridge is formed near closest approach. The medium mass galaxy doesn’t really form a tidal tail, but something close to a small tidal tail is formed as well in this system. This can be seen in figure 3.31. However this happens at radii of 5 kpc and higher, such that this can’t be noted in the velocity dispersion. Since the formation of this small structure behind the medium mass dwarf galaxy was not seen in the velocity dispersion, it is certainly worth having a look if the formation of a tidal tail is also possible for galaxies other than DG1. First we will have a look if tidal tails can be formed during the tidal interaction of DG3 and DG2. In this case a bridge is formed between the galaxies and tidal tails are observed in both galaxies 3.32. This even occurs when both galaxies have rather large 75 Figure 3.29: A plot of the gas column density in a DG2. It is shown before (t = 11.30), during (t = 11.35) and after (t = 11.40) the burst. The yellow dots show the positions of the stars formed in the previous 50 Myr. Figure 3.30: The construction of a tidal tail near closest approach of a DG1. The final box zooms in on the small galaxy, showing the effect within 5 kpc. 76 Figure 3.31: In DG3, something close to a small tidal tail is formed after the passage of the low mass galaxy. Figure 3.32: The formation of a bridge and tidal tails in an I4 model. 77 distances at closest approach. Even in simulations where the closest encounter had a seperation of 20 kpc a bridge and tidal tails were observed. Since bridges are seen, one would expect that a significant amount of gas can be transferred. The effect of these tidal tails is not significantly seen in the velocity disperion of the DG2 and DG3 dwarf galaxies since the formation of these tails mainly starts at the outskirts of the galaxy. So at radii generally higher than 5 kpc. There also seems to be no link between a burst occuring and no tidal tails for DG2, since tidal tails were also observed in simulations where a burst occurred. When looking at the models with only low mass galaxies, generally no tidal tails or bridges are seen. With the exception of encounters closer than 10 kpc in models where a DG2 is included. Then small tidal tails or bridges can be observed. This seems to indicate that for dwarf galaxies with a low mass it is hard to give rise to tidal tails in the other galaxy. This seems rather contradictory with the study of the I4 model where tidal tails were observed after closest encounter in both dwarf galaxies. But it has to be noted since the DG3 has an initial rotation, the gas is spread out. Such that tidal tails could be formed from this gravitationally less bound gas. The tidal tail of the DG3 is not necessarily completely caused by the gravitational perturbation coming from DG2. But a contribution could also come from gas originally belonging to DG2 that is now falling in on DG3 and so enlarging the tail. This can be seen from figure 3.33, where it is shown that a large part of the gas in the tail originally belonged to DG2. Looking at the gas distribution when two DG3 galaxies are interacting in figure 3.34, it can be seen that tidal tails and bridges are formed. Only when going to tidal interactions with a closest approach at ± 25 kpc no tidal tails are visible anymore. Implying that these medium mass dwarf galaxies seem much stronger in the creation of tidal tails than the low mass dwarf galaxies. For the I7 interacting model this is rather similar, where tidal tails can also be formed when the closest approach is larger than 20 kpc. Tidal tails can be noted in the high mass dwarf galaxy as well. Regarding the effect of rotation on the formation of tidal tails and bridges, it looks like an initial rotation more easily gives rise to tidal tails and bridges. This is not completely surprising since this initial rotation gives rise to a more smeared out dwarf galaxy. Because of this, more gas is available that is continuously not that tightly bound to the galaxy. In that case, the tidal acceleration will have a bigger effect on this gas leading to formation of bridges and tidal tails. An extra reason can be found when looking at the equation of tidal acceleration 2.23. When gas is found at higher radii, there are more particles with a higher ∆r such that the tidal acceleration is bigger for these particles. Seeing the bridges formed between tidally interacting dwarf galaxies, it seems very plausible that the dwarf galaxies might capture gas originally belonging to the other galaxy. That this is possible can be seen in figure 3.35, where two DG3 dwarf galaxies are shown 78 Figure 3.33: On the left, the gas particles of the interacting system are shown. The particles belonging to different galaxies are in a different color. It shows a strong concentration of gas particles coming from the second galaxy in the tidal tail. Figure 3.34: The tidal interaction between to DG3 galaxies, showing the formation of a bridge and tidal tails. 79 ± 1.5 Gyr after the closest approach. Both dwarf galaxies contain gas originally belonging to the other galaxy, although the amount of gas transferred is rather low. In the case that low mass galaxies are involved in the interaction this effect is generally even smaller as can be seen in figure 3.36, which shows this property for a DG2 interacting with a DG1. 3.6 Density profiles The plots of the gas distribution show a more compact and higher density of the gas just before and during the burst. The feedback coming from the young stars then disperses the interstellar medium. Quantifying this effect is done by looking at the density profiles. These density profiles are constucted by determining the density of the different particles in spherically symmetric shells around the center of the galaxy. For the gas, it is already pretty clear what will happen around the moment of the burst. But what can we expect for the dark matter and stellar density, will they also become more compact? The theoretical investigation by Verbeke et al. (2014) about starbursting dwarf galaxies found a slightly more compact and a higher central stellar and dark matter density when a starburst occurred. This compactification was however way less than for the gas. This phenomenon could be explained by the deeper gravitational potential created by the compact gas cloud. Since in their investigation gas clouds of 107 M were captured by the system, the gas could have a significant effect on the gravitational potential. In the case of our simulations it does not necessarily have to be that big. Although gas was tunneled into the galaxy, it was not necessarily 107 M . It is also not known if gas hanging around the galaxy is tunneled into the galaxy when a gas cloud is captured. So in our simulations one can not immediatly expect that this effect will be significant when the gas becomes very compact. At the moment of burst in a DG1, the gas density shows a compact strong peak. The stellar and dark matter particles get a slight density peak as well at the center as can be seen in figure 3.37. When the feedback coming from the newly formed stars had its impact on the gas, this shows a way flatter and dispersed gas profile. This has its effect as well on the dark matter and stars for which the density at the center of the galaxy has a small drop. This is caused by the slightly shallower gravitational potential when the gas is dispersed. In the case of DG2 and DG3 a central peak in the stellar and dark matter density profile can be observed as well when a burst occurs. In contrast to DG1 a compactification is noted sometimes as well, although this is not always very significant. This can be seen in figure 3.38 where the stellar density is slightly more compact, but the dark matter density profile does not appear more compact in this case. It can be seen that the dark matter density profile still has a higher central density, which is oberved for all simulatutions. The density profiles of DG4 are not studied in this section since no starbursts were noted in this galaxy. Such that there was not immediatly a reason why one would study the 80 Figure 3.35: The gas particles of two DG3 dwarf galaxies that interacted with each other about 1.5 Gyr earlier. Showing that some gas from the other galaxy can be captured. Figure 3.36: The gas particles of a DG2 (left) and a DG1 (right) that interacted with each other about 1.5 Gyr earlier. Showing that generally very few gas is captured when low mass dwarf galaxies are interacting. 81 ρstar [106 M /kpc3 ] 10−1 ρgas [106 M /kpc3 ] 101 10−32 10 101 100 10−1 10−2 10−32 10 t = 11.75Gyr t = 11.8Gyr t = 11.9Gyr 100 ρDM [106 M /kpc3 ] 10−2 101 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 R [kpc] Figure 3.37: The stellar, gas and dark matter density profile of a DG1 before (t = 11.75 Gyr), during (t = 11.8 Gyr) and after (t = 11.9 Gyr) the burst. 82 density profiles of these galaxies in detail. This higher central density observed for all particles, will give rise to steeper rotation curves which were found in BCDs by van Zee, Salzer & Skillman (2001). The feedback coming from the formed stars disperses the gas. This leads to a drop in stellar and dark matter density and flattening of the rotation profiles (Verbeke et al., 2014). Such that this steep rotation profile is not a permanent property of galaxies hosting a starburst. 3.7 B-band surface brightness profile In the previous sections of this chapter, the galaxies having a burst show properties that are also observationally seen in dwarf galaxies that are experiencing a starburst. Seeing these results it would be interesting to look at the surface brightness profiles in the B-band. It is namely noted by observational studies that the surface brightness profiles have a smaller scale length in dwarf galaxies experiencing a starburst (e.g. Papaderos et al., 1996b; Salzer & Norton, 1999). To construct these profiles, again isophotal integration is used. And just like for the B-I maps produced earlier, the luminosity coming from the stellar particles is smeared out using a gaussian with a standard deviation of 80 pc. First of all, looking at the surface brightness of the DG1 type dwarf galaxies, a central peak is observed. But the profile doesn’t get significantly more compact when going to higher radii like in figure 3.39. This could be expected from the stellar density profiles discussed in the previous section. Just like in the B-I profiles at radii higher than 0.5 kpc, the profile is independent on a burst happening or not. This implies that the I-band magnitude won’t show a drop in surface brightness at higher radii as well. Since the host galaxy does not seem to be more compact, these bursts do not completely satisfy all the properties generally found for BCDs. The surface brightness profile in the case of a DG2 type dwarf galaxy is somewhat different. It does show a peak within a small radius similar to the DG1 type dwarf galaxies. But in contrast to the DG1 dwarf galaxies showing a burst, the DG2 dwarf galaxies their surface brightness shows a drop at higher radii like in figure 3.40 such that surface brightness profiles are more compact. This is in agreement with the observations of the density profile. When looking at the B-I profiles in figure 3.25, there is no difference at higher radii. This would imply that the surface brightness profile in the I-band should be more compact as well. To test this, the surface brightness profile was also plotted in the I-band. This is shown in figure 3.41 where the I-band surface brightness profile is also more compact as could be expected from the B-I profile. The peak is slightly less strong, but this comes to no surprise since the recently formed stars barely contribute to the I-band luminosity, so this can only be caused by a higher central stellar density. During the bursts in the DG3 type dwarf galaxies, two types of surface brightness profiles are observed. These two profiles are shown in figures 3.42 and 3.43. The first type shows a strong central peak in the surface brightness profile, but it doesn’t get more 83 t = 12.05Gyr t = 12.15Gyr t = 12.3Gyr 101 100 10−12 10 101 100 10−1 10−2 10−32 10 ρDM [106 M /kpc3 ] ρgas [106 M /kpc3 ] ρstar [106 M /kpc3 ] 102 101 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 R [kpc] Figure 3.38: The stellar, gas and dark matter density profile of a DG3 before (t = 12.05 Gyr), during (t = 12.15 Gyr) and after (t = 12.3 Gyr) the burst. Figure 3.39: The surface brightness profile in the B-band of a DG1 before (t = 11.76 Gyr), during (t = 11.82 Gyr) and after (t = 11.91 Gyr) the burst. 84 Figure 3.40: The surface brightness profile in the B-band of a DG2 before (t = 11.5 Gyr), during (t = 11.61 Gyr) and after (t = 11.79 Gyr) the burst. Figure 3.41: The surface brightness profile in the I-band of a DG2 before (t = 11.5 Gyr), during (t = 11.61 Gyr) and after (t = 11.79 Gyr) the burst. 85 compact at higher radii like the DG1 dwarf galaxies. The second type shows a wider but less strong peak and a lower surface brightness for higher radii, a bit like the DG2 dwarf galaxies. Both profile types are seen a couple of times, implying that none of these two profiles are one time events. 86 Figure 3.42: The surface brightness profile in the B-band of a DG3 before (t = 12.0 Gyr), during (t = 12.15 Gyr) and after (t = 12.3 Gyr) the burst. Figure 3.43: The surface brightness profile in the B-band of a DG3 before (t = 11.2 Gyr), during (t = 11.35 Gyr) and after (t = 11.5 Gyr) the burst. 87 Chapter 4 Discussion In this chapter, certain results will be discussed in more detail. For some of these more in depth discussions, simulations of merger trees will be used as well. Because of this, the simulations involving a merger tree will be discussed in the first section. There will also be attempted to characterise the bursting galaxies using isophotal maps. 4.1 Comparison with merger tree Although no merger tree was used when studying the tidal interaction of the dwarf galaxies, two simulations were run with a merger history. This was done to have a comparison of our simulations that were run in isolation with these merger tree dwarf galaxies. These two simulations were given the same initial mass, namely 2·1010 M , and were simulated using 300 000 gas and dark matter particles. The mass was chosen such that at the end of the galaxy evolution star formation would be still observed. It has to be noted that a lower mass resolution is used than for the simulations in isolation and the study by Verbeke, Vandenbroucke & De Rijcke (2015). This might pose some resolution effects for the low mass halos in the merger tree, certainly the low resolution for the gas particles is not ideal. But the final results show what could be expected. Namely gas-rich and low metallicity dwarf galaxies were found. Also similar effects were seen as in Vandenbroucke, Verbeke & De Rijcke (2016) where they used a higher resolution simulation performed for Verbeke, Vandenbroucke & De Rijcke (2015) to do a comparison with their simulations in isolation. This gives confidence that this resolution was not a major problem. To construct the merger history, a modified version of the GALFORM algorithm was used (Cole et al., 2000; Parkinson, Cole & Helly, 2008), which is based on the extended Press-Schechter algorithm (Press & Schechter, 1974; Bond et al., 1991; Lacey & Cole, 1993). This constructs the density threshold for a halo to become virialised, based on a Gaussian distribution of density fluctuations using a cosmological spherical collapse. For a given halo mass and redshift it creates a conditional mass function of the progenitors at a higher redshift. Then using a Monte-Carlo technique, the merger history is con89 structed based on the final mass of the galaxy. The modification by Parkinson, Cole & Helly (2008) was done to fit the conditional mass function of the Millenium Simulation (Springel et al., 2005). These merger tree simulations were performed in the same cosmological setting as the simulations in isolation, starting at a redshift of z = 13.5, and 20 redshift intervals were used for the merger processes. A minimal halo mass of 0.5·108 M was taken for the halos involved in the merger process. The two merger trees run for this thesis had a main difference. The first merger tree (MT1) has a late merger involving two rather massive halos and the other merger tree (MT2) that was run consists of many low mass halos that are merged over time with a more massive halo. These two merger trees are shown in figure 4.1. First of all, the star formation in both merger trees is compared. In both cases only the most massive halo can have star formation before the last merger in the tree, except for the initial star formation although this is not shown in the plots. The magnitude of the star formation in the two trees is however very different. MT2 shows on average a higher star formation in the last 2 Gyr of evolution than MT1 as illustrated in figure 4.2. It also has a slightly higher stellar mass and more neutral hydrogen at z = 0. The metallicity is almost exactly the same. These results are compared with DG1, DG2 and DG3 in the last 2 Gyr of their evolution, where DG1 and DG2 are shown in figure 2.12. Some effects of a merger history are immediatly visible compared to simulations in isolation. First of all, there is a difference in the star formation. The star formation of MT1 is around the same magnitude as the star formation in DG1 and slightly lower than the the star formation in DG2. The star formation of MT2 is of similar magnitude as the star formation in DG2. However since MT1 and MT2 have a double initial mass compared to DG1 and DG2, this implies that a higher initial mass is necessary for the merger trees to obtain a similar current magnitude of star formation as in isolation. When comparing the star formation with DG3, which has the same initial mass as MT1 and MT2, it was found that the star formation at z = 0 is way lower when the merger tree is involved. This is not really surprising since the merger tree consists of smaller halos, where star formation is immediatly shut down after the initial burst for most of these protogalaxies. Because of this, the star formation is mainly achieved by the protogalaxy with the highest mass. These many low mass halos which don’t form any stars after the initial burst also explain the lower stellar mass. Their stellar mass is even lower than in DG1. This lower star formation and stellar mass obviously leads to a lower metallicity for the merger trees. The amount of neutral hydrogen in the galaxy when considering the merger tree shows an interesting property. Although the merger trees retain less neutral hydrogen than DG3, they retain more neutral hydrogen than the low mass galaxies. Since the star formation in these mergers is similar to the low mass galaxies, this could imply that higher burst factors can occur when these galaxies are tidally interacting than in the case of interaction between dwarf galaxies simulated in isolation. 90 Figure 4.1: The visualisation of the two merger trees that were run as a function of the lookback time. 91 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 8.0 7.5 7.0 6.5 6.0 5.5 5.0 −1 −2 −3 −4 10 −52 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 11 10 9 8 7 6 5 −1 −2 −3 −4 10 −52 SFR [M /yr] Mstar [106 M ] 101 100 11.5 MT1 [Fe/H]B MHI [106 M ] MHI [106 M ] [Fe/H]B Mstar [106 M ] SFR [M /yr] MT1 12.0 12.5 t [Gyr] 101 100 10−1 11.5 13.0 12.0 12.5 t [Gyr] 13.0 Figure 4.2: The comparison of the star formation, stellar mass, metallicity and neutral hydrogen mass of the two merger trees in the last 2 Gyr of their evolution. In work performed by Cloet-Osselaer et al. (2014) a comparison between isolated models and merger trees was made as well. They also found a higher stellar mass for the isolated models at z = 0. However, they found a lower metallicity for most of the models in isolation. This is not in agreement with what we find. This happens since in their study, a lot of non-population III stars are formed in the initial burst leading to a high stellar mass. This strong initial burst is followed by a low star formation rate at later stages of the evolution, thus leading to a low metallicy. However recently, population III stars have been included in the code (Verbeke, Vandenbroucke & De Rijcke, 2015). In the case of a merger tree, it was shown that these population III stars delay the star formation in the galaxy. The effects of population III stars might be less strong in an isolated case, which leads to a stronger star formation history and thus a higher metallicity. This higher star formation rate in isolated models was very recently also found by Vandenbroucke, Verbeke & De Rijcke (2016) when population III stars were included. 4.2 Gas flow and burst duration In the previous chapter it was noted that depending on the type of galaxy, the bursts that occured were different. In the case of DG1 a very sharp peak was observed for only ± 10 Myr. Since this star formation happened in maximally two rather close compact clumps, it could be expected that feedback from the recently formed stars will shut this star formation down in the collapsing cloud. The prescriptions for the feedback from young stars can be found in Valcke, de Rijcke & Dejonghe (2008). The distribution of feedback from stellar winds and type II supernovae over time was based on the main 92 sequence liftime of stars with a certain mass David, Forman & Jones (1990): log τ (m) = 10 − 3.42log m + 0.88(log m)2 (4.1) Where m is expressed in solar masses. When applying the appropriate boundary masses for type II supernovae to this formula, which are 8 M and 70 M , one gets the following time intervals: - Stellar winds: 0 - 4.3 · 107 yr - Type II supernovae: 4.8 · 106 - 4.3 · 107 yr This provides a satisfying explanation for the short burst since stellar winds and supernovae from the most massive stars had their impact already. In the case two clumps form stars, this implies that these clumps have to start forming stars within ± 5 Myr time difference. Otherwise two distinct peaks would be observed. That this happens is not that surprising since these compact clumps are rather close to each other and most likely involved in the same collapse of the gas. This can be seen from the gas distribution maps in figure 3.27. In the case of DG2, this feedback from freshly formed stars can not directly explain the duration of the bursts, which last up to 50 Myr. Since after ± 40 Myr all feedback coming from massive stars has already been send into the ISM. Taking the gas flow into account explains the duration nicely, since many compact clumps over the entire galaxy can be seen in figure 3.28 just before the star formation is starting. Such that every clump probably only experiences star formation for about 10 Myr just like in the DG1 type galaxies. But since many clumps are collapsing over the galaxy, these short bursts together form one longer burst. This could also explain the short drop in star formation for 10 Myr in the middle of the burst, where due to a statistical fluctuation no clumps are collapsing. In the case of one or two clumps collapsing this drop would be very difficult to explain. If one clump was collapsing, the feedback shutting down the star formation would stop the star formation for longer than 10 Myr because the feedback of the last formed stars would still be too strong. In that case the gas can’t collapse in the same region. Two clumps are unlikely as well since not that much more gas is contained, compared to a DG1 dwarf galaxy. Such that two regions forming stars would have been shut down after 10 Myr like in the DG1 type dwarf galaxies. This would at best lead to two shorter peaks in the star formation. In DG3, two types of bursts are seen similar to the two types just discussed. This is not completely surprising since the initial rotation velocity was taken in the middle. However in all the galaxies where a burst occurred several clumps appeared in the gas maps. These gas maps are a bit misleading and it is important to look where te stars actually form, this is illustrated in figure 4.3. This reveals that in case of a sharp peak, most of the stars are formed in one small area, while for the wider star formation peak the recently formed stars appear more distributed over the galaxy just like what is seen 93 Figure 4.3: The gas map on the left shows the recently formed stars in a long burst. The gas map on the right shows the recently formed stars when a sharp burst occured. in figure 3.28 for a burst in DG2. In the case of the sharp peak, there are some stars formed elsewhere in the galaxy. This can be explained because the peak is not isolated like for DG1, but there is continuous star formation for the dwarf galaxy of this mass. So the other stars formed are a sort of continuous star formation background for the burst, which is caused by the collapse of one massive cloud. A correlation with the observed surface brightness profiles is found as well. The profile with the strong central peak that is not getting more compact has one or two compact areas where a lot of stars are formed. On the other hand, the more compact profile has a star formation spread over the galaxy. This result is again completely in accordance with the observations in DG1 and DG2. This short peak could also offer an explanation why no compact profile is observed in that case. Since the peak is extremely short, the gas becomes compact only for a very short time because it is immediatly dispersed again. This gives only a very short time for the other matter in the galaxy to get more compact, such that the galaxy might not have enough time to get significantly more compact. Although this could provide an explanation, it could also just be the time resolution used. If the compactification only happens for a short while, it might be that this is not captured by the snapshots that were saved. Although the higher resolution simulations using a time difference of ± 30 Myr per snapshot could not capture any compactification either, giving small support to the suggestion that the galaxy doesn’t really have the time to get significantly more compact. Since this is rather noteworthy, we will try to obtain a rough estimate of the time scale 94 needed for the galaxy to become more compact in the hope to learn something from this. For this, we will make use of the dynamical time scale given by equation 2.17. But ρg is changed by the density of dark matter and stellar particles. To define ρ we will use the simple assumption: 3M ρ= (4.2) 4πR3 Where M is the stellar and dark matter mass contained within radius R. This total stellar and dark matter mass was determined within three different radii. Namely 1 kpc, 3 kpc and 5 kpc. This was done for several snapshots of the galaxy in isolation to have a descent view on the usual mass within these radii. For DG1 and DG2 almost exactly the same masses were found within these radii, respectively: 1.5 · 108 M , 109 M and 2 · 109 M . For DG3 these masses were higher: 1.7 · 108 M , 1.75 · 109 M and 3.5 · 109 M . The values obtained for tdyn in DG1 and DG2 are respectively: ± 25 Myr, 40 Myr and 55 Myr. For DG3, this is respectively ± 20 Myr, 30 Myr and 40 Myr. Since these values are determined on very simple grounds one should not pay too much attention to these values. But they do provide an explanation why always a higher central density is obtained but not always a more compact profile at higher radii. This happens because the dynamical time scale is significantly smaller for the center of the galaxy than at higher radii. Since the higher central density is always seen, this again seems to support the explanation of the short burst. To conclude this section it can be stated that two types of bursts appear to be possible in tidal interactions. And the type of burst observed depends on the gas dynamics in the dwarf galaxy. 4.3 Moment of the burst In the introduction it was already stated that tidal interactions are considered as a possible trigger for starbursts because a companion dwarf galaxy is sometimes seen close to a starbursting dwarf galaxy. So it is interesting to ask when the bursts in our galaxies occur. Such that it can be checked if the galaxies are actually still close to each other when the burst occurs. A first remarkable feature when going through the moments of burst compared to the moment of closest approach is that ± 80 % of the bursts occur after the moment of closest approach. The results on the gas mass inside the galaxy point to an explanation for this. As was shown there, a significant amount of gas hanging around the galaxy gets tunneled into the dwarf galaxies when they are approaching each other due to the tidal interaction. This generally reaches its maximum very close to closest approach. This gas tunneling can then trigger a burst during this tunneling or afterwards. When the galaxy has a burst before closest approach, the galaxy was already accreting a significant amount of gas earlier, as is shown in figure 4.4. This accretion of the surrounding gas before a burst obviously agrees with the observation that dwarf galaxies experiencing a starburst are gas rich with a high central concentration (e.g. Taylor et al., 1994, 1995; 95 Figure 4.4: The gas mass evolution of a DG2 type dwarf galaxie having a burst before closest approach. The moment of closest approach is indicated by the vertical line. van Zee, Skillman & Salzer, 1998). When the burst occurs is also dependent on the type of galaxy, this can be seen from the data which is added in appendix 6. For the low mass galaxies, the burst almost always occurs within ± 250 Myr around the closest encounter. While it appears rather rare that a burst occurs within 250 Myr around the moment of closest approach in DG3. Bursts in these dwarf galaxies can easily occur more than 1 Gyr after closest approach. If a burst for the low mass galaxies occurs, the gas capture seems to initiate a burst rather quickly if one occurs. Such that in observations a companion will be found close to the starburst galaxy. For the more massive galaxies this immediate burst is not necessarily the case, such that the galaxies will easily have a separation of 100 kpc at the moment of a burst. This is still in agreement with recent observations by Lelli, Verheijen & Fraternali (2014) who found possible perturbers within a projected distance of 200 kpc. So not all perturbers are necessarily very close to the bursting galaxy. This delay that can be found in DG3 can be related to the recapture of gas generally somewhere after closest approach. So again these bursts can be related to the tunneling of gas into the galaxy, but this time it can also be an indirect consequence of the gas accretion due to tidal interactions. 4.4 Classification of the bursting galaxies Trying to classify the burst galaxies, we will use isophotal maps in the B-band. These isophotal maps are very useful since they give a great view on how the galaxy would be observed and the substructure it has. To create these isophotal maps, smoothing of the luminosity from the stellar particles is used with a standard deviation of 80 pc. This is exactly the same as for the other luminosity maps and profiles created. Because this resolution allows to see a lot of structure, also isophotal maps were produced with a standard deviation of 160 pc. This makes it easier to see the coarser structure of the 96 bursting dwarf galaxy, without being distracted by all the substructure. First the results for bursts in DG1 will be discussed. Both irregular and nucleated bursts are seen in this dwarf galaxy type, which is shown in figure 4.5. In both cases the outer regions are not that elliptical. When looking at the coarser map, they look more elliptical but not completely. The high resolution isophotal map shows that for the irregular burst two clumps are seen which are rather of center. In the coarser map, the irregular burst stays of center and slightly irregular. When having a look at the classification scheme in the introduction, there is not really a place to put the nucleated bursts with rather irregular outer isophotes. However it was already shown that the bursts in this dwarf galaxy type do not show all properties seen in BCDs. This might be because the gas cloud having absolutely no initial rotation could be a slightly extreme case to study, such that obtained results might not be optimal. For the DG2 dwarf galaxies, the burst is always irregular as could be expected from e.g. the B-I and density maps. An example of this is shown in figure 4.6. When looking at the coarser maps, the outer regions of DG2 galaxies appear rather elliptical. The inner regions on the other hand still show an irregular shape. Even in the high resolution map the outer region appears quite elliptical. Generally, the outer isophotes seem to be more elliptical in DG2 than DG1 as can be seen for example in figures 4.5 and 4.6. But it is hard to confirm this quantitatively. For the DG3 dwarf galaxies, rather irregular bursts are expected as well due to the initial rotation. This is shown in the face on isophotal map in figure 4.8. The outer regions in the coarse map are quite elliptical, but in the high resolution maps there is still a lot of scatter on these elliptical shapes. When looking at the edge on isophotal maps, iI C like burst galaxies can be seen like in figure 4.8. These comet like edge on isophotal maps were not seen for DG2. This could be because not that much bursts were seen and since not all DG3 burst galaxies show this comet like profile. However there is also some difference between the two bursts. DG3 can have a burst in one dominant area in contrast with DG2, such that a comet like profile can be formed more easily in DG3. In DG2 the burst always happens spread out over the galaxy, which is also reflected in a more spread burst in the edge on isophotal map in figure 4.7. 4.5 Metallicity The oxygen abundance of the ionised gas has been determined for the bursting dwarf galaxies like it would be determined observationally. Such that it could be compared with observational studies (e.g. Izotov, Thuan & Lipovetsky, 1994; Amorı́n et al., 2014). They observed BCDs with respectively 7.37 < 12 + log(O/H) < 8.04 and 7.5 < 12 + log(O/H) < 8.3. When looking at the values obtained from our simulations, which are tabulated in appendix 6. One can see that that the oxygen abundances are on the high side for the low mass dwarf galaxies, ± 8.1 for DG1 and ± 8.2 for DG2, and even too high for DG3, ± 8.5, 97 Figure 4.5: On top a nucleated burst is shown while below an irregular burst is shown for DG1. The two different resolutions are plotted. 98 Figure 4.6: A typical burst for DG2 is plotted, which is always irregular. This is still visible in the lower resolution isophotal map. Figure 4.7: The edge on view of a DG2 burst galaxy. A rather irregular burst profile can be seen, certainly in the high resolution map. 99 Figure 4.8: On top one can see the face on rather irregular burst. Below the rather comet like behaviour of the edge on isophotal map. 100 compared with observations. However one does not immediatly expect that this would be in the same region as the observations based on the section about the merger tree simulations. It was shown there that merger galaxies with a certain star formation had a significantly lower metallicity than the models in isolation with a similar star formation rate. However, the determination of these metallicities was very basic and not how it would be observed in reality. Because of this, the oxygen abundance of these two merger trees was determined as well. This would give a better view on the actual magnitude of the oxygen abundances observed. The oxygen abundances of the merger trees at z = 0, are given in 6. These are around 7.6. Seeing this, one could expect that the metallicities for the simulations in isolation are on the high side such that the medium mass dwarf galaxies might fit in the observed range as well when using a merger tree. Having a detailed look at the study by Verbeke, Vandenbroucke & De Rijcke (2015), one of the only results that did not completely agree with observations was the oxygen abundance. This abundance was somewhat on the high side as can be seen in figure 4.9 which was taken from their paper. This should also be the case in our merger tree simulations, since the same code was used. So if this small issue would be resolved, one could expect that simulations with merger trees would lead to even slightly lower metallicities. The discussion so far only looked at the magnitude of the observed metallicities. However it was found that BCDs have a metallicity that is generally lower than dIrrs (e.g. Izotov, Thuan & Lipovetsky, 1994; Izotov & Thuan, 1999; Hunter & Hoffman, 1999). This was not adressed so far in this discussion. When a low metallicity cloud is falling in on the galaxy, obtaining a lower gas metallicity for the BCD is obviously no problem. However in the case of tidal interactions it is not immediatly clear if a lower metallicity will be obtained when a burst will occur. We would like to check if this drop in metallicity is also possible with tidal interactions. Since the burst is triggered by the capture from surrounding gas, one could explain this by the gas falling in having a lower metallicity. When having a look at Schroyen et al. (2013) a metallicity gradient was found for simulated dwarf galaxies. They found long lived metallicity gradients in simulated dwarf galaxies even when they had an initial rotation. This metallicity profile was however based on stars, and we are dominantly interested in the gradient of the oxygen abundance in the gas. So a metallicity profile for this was constructed. It has to be noted that the determination of the oxygen abundance up to now was based on the oxygen abundance in ionised gas. However to construct the profile it was not restricted to ionised gas, but all gas was used to have a better view on the general oxygen abundance profile in the gas. The bins for plotting these profiles were taken large (in steps of 5 kpc) to have no jumps in the profile due to a lack of gas at a certain radius. Although this are very large bins, this is not really a big problem since we are interested in the coarse oxygen abundance profile at higher radii. These profiles were constructed for the host galaxies in isolation near times where generally bursts occured in the interacting models. 101 Figure 4.9: The oxygen abundance obtianed by Verbeke, Vandenbroucke & De Rijcke (2015) in their simulations compared with observations. Their oxygen abundance is slightly on the high side. When looking at these profiles in figures 4.10 and 4.11, it can be seen that the oxygen abundance profile is dropping at higher radii. Certainly at radii higher than 10 kpc. The behaviour is rather smooth for DG2 and DG3, seen in figure 4.11. For DG1 there is a bit more scatter on these results, which is shown in figure 4.10. Seeing these profiles, it seems possible that when gas is falling in on the galaxy from radii higher than 5 kpc that this could lead to lower oxygen abundances of the gas. However one can expect that this effect will by far not be as strong as a low metallicity gas cloud falling in, since the gas hanging around the galaxy is still enriched. To compare the gas oxygen abundance of a burst with the oxygen abundance when no burst happens is very complicated to do without biases. Since there is an evolution over time in the metallicity due to the star formation. Such that it is very difficult to find a statistically significant and representative sample of snapshots from the performed simulations to compare with the galaxies experiencing a burst. And since we can expect that the effect won’t be that big in our simulations it might be difficult to get conclusive results. It could be argued that one can try to study the evolution of the oxygen abundance in a galaxy experiencing a burst. However since the gas falls in over a large time interval, it should be difficult to see a significant drop in the evolution because the star formation is continuously enriching the gas and the metallicity is not much lower at higher radii. Although there is a drop at higher radii (r > 10 kpc) in the oxygen abundance, one can ask if this gas will fall in on the galaxy. If this doesn’t happen there won’t be a drop in the oxygen abundance. This would pose a problem for this model to create BCDs since they generally have a lower oxygen abundance. However the results discussed so far on the oxygen abundance gradient only considered the models in isolation. When looking at our merger tree simulations, which should be more realistic, they seem to offer 102 Figure 4.10: The course oxygen abundance profile of the gas in DG1 in isolation at different times during the evolution. Figure 4.11: The course oxygen abundance profile of the gas in DG3 in isolation at different times during the evolution. 103 a solution for this problem. As shown in figure 4.12 for MT2, the merger simulations seem to show a strong drop in oxygen abundance towards higher radii. It has to be noted that this is slightly less strong for MT1, but still way stronger than the simulations in isolation. 4.6 Effect of tidal stripping and bursts on later star formation It was noted from the gas density maps and the evolution of the gas mass in the galaxy that tidal tails can be formed and that there is some tidal stripping. Although significant amounts of gas can be stripped from the galaxy, it is certainly not able to strip enough gas from the galaxy to stop the star formation. This can be seen from the star formation histories e.g. in figure 3.16. Even when bursts occur, triggered by tidal interactions, the star formation maximally stops for 50-100 Myr in the DG1 model which is most sensitive to the starbursts. For DG2 and DG3 a stop in star formation for 50 Myr is not significantly seen in the star formation histories with a time interval of 50 Myr. In the higher resolution star formation histories, sometimes intervals after the burst of up to 20 Myr without star formation can be seen for DG2 e.g. in figure 3.8. In DG3 a break in the star formation is not really seen after a burst. 4.7 The amount of bursts Over all simulations, 26 galaxies were having a burst based on the definitions from the previous chapter. But it seems rather fair that one should not pay too much attention to this number. These definitions were proposed by looking at the burst factor in many simulations and comparing them with the star formation in these simulations and the simulations in isolation of the host galaxy. However the burst factors might be defined differently by another person so they are still slightly arbitrary, but this has been the case for all studies so far both theoretical and observational, since there is no unique definition for a starbursting dwarf galaxy. It was shown that galaxies with these burst factors generally show properties that are also seen observationally in galaxies experiencing a burst, which gives confidence that the definitions used are quite good. Although one should not pay too much attention to the number of bursts, one can try to have an idea on how easy or difficult it is to trigger a starburst using tidal interactions. 140 simulations were considered, which implies that 280 galaxies experiencing tidal interaction with another dwarf galaxy were studied. This implies that ± 10 % of the galaxies experienced a burst. Again a lot of arguments can be proposed that this number is not representative, it is also not claimed that this number is representative. Some of these arguments might be: - The definition of a burst is always slightly arbitrary. 104 Figure 4.12: The coarse oxygen abundance profile of the gas in MT2 at different times during the evolution. - It was attempted to study a representative range of orbits. But it is hard to put an upper limit for the radius of closest approach on what can be considered as tidally interacting dwarf galaxies. So it was just attempted to make this range as large as possible. - Other environmental influences were not included. But seeing the amount of galaxies that did not experience a burst, it is fair to say that tidal interaction is not a very efficient way to trigger a burst in a dwarf galaxy. This is also not necessary since not that many starbursts are seen in dwarf galaxies. The main reason why they are easily spotted is because of the higher luminosity caused by the recently formed stars. That it is not so easy to initiate a starburst in a dwarf galaxy was also noted by Verbeke et al. (2014), where only the most massive gas clouds could trigger a star burst rather easily. 105 Chapter 5 Conclusion In this thesis we numerically studied a wide range of recent tidal interactions between dwarf galaxies, attempting to understand the impact of these interactions on the evolution of dwarf galaxies. It was found that tidal tails and bridges can be formed if the galaxies passed each other sufficiently close. But to create these structures easily, a high enough mass of the galaxies is favorable. A dependence on the initial rotation of the gas cloud was found for this creation as well. Studying the amount of gas inside the galaxy, effects related to tidal interactions were found. Before the moment of closest approach it turned out that the tidal interaction generally led to the tunneling of surrounding gas into the galaxy. After closest approach tidal stripping of significant amounts of gas was observed. It was found that the accumulation of gas before closest approach and the recapture of gas after tidal stripping could initiate a starburst which again ejected a lot of gas out of the galaxy. The recapture of this gas could initiate a sequence of burstlike events. Finally it should be noted that the gas ejected out of the galaxy, both by tidal stripping and feedback, was never sufficient to create a star formation break longer than 100 Myr. Two types of bursts were found, depending on the dwarf galaxy dynamics. The first type of burst involved the collapse of maximally two massive clumps very close to each other, leading to a very short burst of maximally 10 Myr. The second type involved the collapse of several clumps at different places in the galaxy spread over a time interval of 40-80 Myr. Both bursts led to different features in e.g. the density profile and the surface brightness profile. It was found that the galaxies experiencing the second burst type show many properties that are observationally also seen in BCDs e.g. they are more compact, they are gas rich,... The property that could pose a problem for this model seems to be the possible lack of a metallicity drop when the burst occurs. However it was shown that more realistic simulations involving a merger tree could resolve this issue. 107 In the case of the first burst type, no compactification of the host galaxy was found. This is not necessarily a huge problem since generally BCDs are more compact. And it might be that a slight compactification is not noted in our simulations because of a too big time gap between the snapshots. It could also be that this type of burst is over abundant in our simulations due to the low mass dwarf galaxy without initial rotation, which is possibly not an ideal way to get rather realistic dwarf galaxies. 108 Chapter 6 Nederlandstalige Samenvatting Al geruime tijd zijn er observationele indicaties dat getijdenwerking tussen dwerggalaxieën een mogelijke oorzaak kan zijn voor een verhoogde stervorming die soms wordt waargenomen in dwerggalaxieën. Andere mogelijke oorzaken voor deze verhoogde stervorming, zoals het invangen van een gaswolk en het mergen van dwerggalaxieën is reeds numeriek aangetoond door verschillende studies. Getijdenwerking tussen dwerggalaxieën en het mogelijk effect hiervan op de stervorming is echter nog nooit diepgaand bestudeerd met computersimulaties. Daarom werd in deze thesis een uitgebreide numerieke studie gedaan van deze getijdenwerking. Hiervoor werd de interactie tussen verschillende dwerggalaxieën op verschillende relatieve trajecten bestudeert. Dit leverde enkele significante resultaten op. Ten eerste werd soms de vorming van bruggen en langwerpige structuren waargenomen rond de dichtste nadering van de twee galaxieën. Om deze structuren te vormen waren de dwerggalaxieën bij voorkeur voldoende massief en was de onderlinge afstand bij dichtste nadering van belang. Voor de vorming van deze structuren werd ook nog een andere afhankelijkheid gevonden, gerelateerd aan de initiële rotatie van de gaswolk. De getijdenwerking tussen de dwerggalaxieën had ook een effect op de hoeveelheid gas in de galaxieën. Als de galaxieën elkaar naderden, werd een stijging van de hoeveelheid gas in de galaxieën waargenomen. Dit was afkomstig van gas dat rond de galaxie hangt. Na dichtste naderen werd er dan weer een significant verlies van gas waargenomen. Al werd bij de meer massieve dwerggalaxieën een deel van het tijdelijk verloren gas na een tijdje weer terug ingevangen. Deze twee manieren van gas vangst konden tot een verhoogde stervorming leiden in de dwerggalaxien. De verhoogde stervorming kon ook grote hoeveelheden gas uit de dwerggalaxieën verwijderen. Het verlies van gas kon nooit tot periodes langer dan 100 Myr zonder stervorming leiden. Het deels terug invangen van het gas kon soms wel opnieuw tot een verhoogde stervorming leiden. Twee soorten verhoogde stervorming werden waargenomen, die bleken afhankelijk te zijn 109 van de interne dynamica van de dwerggalaxieën. Het eerste type burst, was een zeer korte burst van maximaal 10 Myr. Het tweede type burst bestond over een tijdsinterval van ± 40-80 Myr. Er was een duidelijk verschil merkbaar in hoe deze bursts tot stand kwamen. Het eerste type burst bestond uit maximaal twee massieve gaswolken die heel dicht bij elkaar sterren beginnen te vormen. Het tweede type burst bestond uit verschillende gaswolken die gespreid over de galaxieën sterren beginnen te vormen. Deze stervorming is iets meer gespreid in de tijd. Deze verschillende bursts leiden ook tot verschillende observaties, bijvoorbeeld in het oppervlaktehelderheid profiel. In het geval van het tweede type burst, hadden de dwerggalaxieën veel eigenschappen die waargenomen zijn in BCDs: het compacter zijn van de galaxie, hogere gashoeveelheid,... Alleen de nodige lagere metalliciteit leek voor een mogelijk probleem te zorgen. Meer realistische simulaties, gebruikmakende van een merger geschiedenis van de dwerggalaxieën, zouden dit probleem kunnen verhelpen. Het eerste type burst was niet echt compacter, dit hoeft niet noodzakelijk een probleem te zijn. Aangezien over het algemeen dwerggalaxieën met een verhoogde stervorming compacter zijn. Het kan ook zijn dat in onze studie dit soort burst overbenadrukt wordt door het bestuderen van het nogal extreme model zonder initiële rotatie van de gaswolk. En het kan misschien zijn dat het compacter worden van de galaxie niet gemerkt wordt door een gebrekkige tijdsresolutie in het wegschrijven van de snapshots. 110 Appendices 111 Appendix A pro vs retro tinit (Gyr) tp (Gyr) rp (kpc) prograde 9.95 1.0 15 9.95 1.0 15 retrograde prograde 10.45 1.0 5 10.45 1.0 7 retrograde retrograde 10.45 1.0 7 10.45 1.0 5 retrograde prograde 10.45 1.0 6 10.45 0.8 8 prograde prograde 10.45 0.9 7 10.45 1.0 7 prograde retrograde 10.45 1.0 9 retrograde 10.45 1.0 17 10.45 0.9 16 prograde 10.45 0.9 11 prograde prograde 10.45 0.9 12 retrograde 10.45 0.9 10 10.45 0.9 6 prograde 10.45 0.9 13 retrograde 10.45 0.9 17 prograde retrograde 10.45 0.9 18 prograde 10.45 1.0 19 retrograde 10.45 0.9 19 e 2 2 2 1.6 3 2 1.9 2.6 3 2 2 2 3 2.2 3 2.5 3.4 3 2.5 2.8 2 3 Table 1: The initial conditions of the I1 model interactions. Where tinit is the starting time of the interaction, tp the time after which the closest approach is achieved, rp the distance at closest approach and e the eccentricity. 112 pro vs retro tinit (Gyr) tp (Gyr) rp (kpc) prograde 10.45 1.0 7 prograde 10.45 0.8 5 10.45 0.9 6 retrograde retrograde 10.45 0.9 5 10.45 0.9 11 retrograde prograde 10.45 1.0 9 10.95 0.8 10 retrograde retrograde 10.45 0.9 12 10.45 1.0 11 prograde prograde 10.45 1.0 13 10.45 1.0 10 retrograde prograde 10.45 1.0 12 10.45 1.0 15 prograde retrograde 10.45 0.9 16 retrograde 10.45 1.0 18 retrograde 10.45 0.9 11 prograde 10.45 0.9 11 prograde 10.45 0.9 10 10.45 1.0 9 retrograde prograde 10.45 0.9 17 10.45 0.9 15 retrograde prograde 10.45 0.9 18 e 2.5 2.5 2 1.7 2 2 3 2.5 2 3 2 2 2 3 2 2.6 3.1 2.5 2 2.3 3 3 Table 2: The initial conditions of the I2 model interactions. Where tinit is the starting time of the interaction, tp the time after which the closest approach is achieved, rp the distance at closest approach and e the eccentricity. 113 pro vs retro tinit (Gyr) tp (Gyr) rp (kpc) retrograde 10.45 0.9 6 retrograde 10.45 1.0 10 10.45 1.0 9 prograde prograde 10.45 0.9 8 10.45 0.9 5 prograde retrograde 10.45 0.9 5 10.45 1.0 7 retrograde retrograde 10.45 1.0 4 10.45 1.0 8 prograde prograde 10.45 0.9 6 10.45 1.0 7 prograde retrograde 10.45 1.0 9 10.45 0.9 6 retrograde prograde 10.45 0.9 6 prograde 10.45 0.9 13 retrograde 10.45 0.9 12 retrograde 10.45 0.9 5 retrograde 10.45 1.0 8 10.45 1.0 7 retrograde prograde 10.45 1.0 7 10.45 1.0 12 prograde e 1.9 2 2 2 2 1.7 3 3 3 3 2 2 3 2.4 2.8 2.1 3 2 2 1.5 2 Table 3: The initial conditions of the I3 model interactions. Where tinit is the starting time of the interaction, tp the time after which the closest approach is achieved, rp the distance at closest approach and e the eccentricity. 114 pro vs retro tinit (Gyr) tp (Gyr) rp (kpc) retrograde 10.45 1.0 9 prograde 10.45 0.9 12 10.95 0.8 13 retrograde prograde 10.45 1.0 20 10.45 1.0 16 retrograde prograde 10.45 0.9 16 10.45 0.9 9 prograde retrograde 10.45 1.0 8 10.45 0.9 11 retrograde prograde 10.45 1.0 11 10.45 1.0 17 prograde retrograde 10.45 0.9 18 10.45 1.0 19 retrograde prograde 10.45 0.9 14 prograde 10.45 0.9 21 retrograde 10.45 0.9 22 prograde 10.45 1.0 10 prograde 10.45 1.0 22 10.45 0.9 23 retrograde retrograde 10.45 0.9 24 10.45 0.9 23 prograde retrograde 10.45 1.0 25 e 3 2.5 3 2 2.5 3 2.5 2 3 2 2 3 1.6 3 2.4 2.7 2 2.1 3 2.4 2.9 2 Table 4: The initial conditions of the I4 model interactions. Where tinit is the starting time of the interaction, tp the time after which the closest approach is achieved, rp the distance at closest approach and e the eccentricity. 115 pro vs retro tinit (Gyr) tp (Gyr) rp (kpc) retrograde 10.45 0.8 8 retrograde 10.45 1.0 10 10.45 1.0 9 prograde retrograde 10.45 0.9 16 10.45 0.9 18 prograde retrograde 10.45 0.9 14 10.45 0.9 8 retrograde prograde 10.45 0.9 12 10.45 1.0 11 retrograde prograde 10.45 1.0 11 10.45 1.0 13 prograde prograde 10.45 0.9 17 10.45 1.0 18 retrograde prograde 10.45 0.9 14 retrograde 10.45 0.9 13 prograde 10.45 1.0 19 retrograde 10.45 1.0 18 retrograde 10.45 1.0 12 10.45 1.0 12 prograde prograde 10.45 1.0 10 10.45 1.0 15 retrograde retrograde 10.45 0.9 21 e 1.7 2 2.5 3.5 3 3 2.5 3 2 2 2 3 2 3 3 1.9 2.1 1.8 1.8 2.3 2.2 2.6 Table 5: The initial conditions of the I5 model interactions. Where tinit is the starting time of the interaction, tp the time after which the closest approach is achieved, rp the distance at closest approach and e the eccentricity. 116 pro vs retro tinit (Gyr) tp (Gyr) rp (kpc) prograde 10.45 1.0 15 prograde 10.45 0.9 13 10.45 1.0 16 retrograde retrograde 10.45 0.9 15 10.45 1.0 10 retrograde prograde 10.45 1.0 11 10.45 0.9 12 retrograde prograde 10.45 0.9 20 10.45 1.0 19 retrograde prograde 10.45 1.0 17 10.45 1.0 20 retrograde prograde 10.45 1.0 14 10.45 0.9 23 prograde retrograde 10.45 0.9 13 retrograde 10.45 0.9 22 prograde 10.45 0.9 18 prograde 10.45 0.9 22 retrograde 10.45 0.9 14 10.45 0.9 24 retrograde prograde 10.45 1.0 24 10.45 0.9 25 retrograde e 2.5 3.5 2 3 2 2 3 3 2 2 2 2 2.1 2.6 2.8 3 2.6 2.2 2.5 2.1 2.3 Table 6: The initial conditions of the I6 model interactions. Where tinit is the starting time of the interaction, tp the time after which the closest approach is achieved, rp the distance at closest approach and e the eccentricity. 117 pro vs retro tinit (Gyr) tp (Gyr) rp (kpc) prograde 10.45 1.0 15 retrograde 10.45 0.9 13 10.45 0.9 17 prograde retrograde 10.45 1.0 18 10.45 0.9 20 retrograde prograde 10.45 0.9 22 10.45 0.9 12 prograde retrograde 10.45 1.0 24 10.45 1.0 20 prograde retrograde 10.45 1.0 27 e 2 2.5 2.8 2 2.7 2.2 2.6 2.1 2.3 2.2 Table 7: The initial conditions of the I7 model interactions. Where tinit is the starting time of the interaction, tp the time after which the closest approach is achieved, rp the distance at closest approach and e the eccentricity. Appendix B 200 Myr DG1 -100 Myr DG2 -250 Myr DG1 0 Myr DG3 300 Myr DG2 200 Myr DG2 1.1 Gyr DG1 -150 Myr DG1 150 Myr DG3 400 Myr DG2 Table 8: The times of the bursts in DG1, when the moment of closest approach is taken as the zero point. Added directly below the time is the galaxy type it is interacting with. 150 Myr DG2 150 Myr DG2 1.5 Gyr DG2 -150 Myr DG3 50 Myr DG3 250 Myr DG3 Table 9: The times of the bursts in DG2, when the moment of closest approach is taken as the zero point. Added directly below the time is the galaxy type it is interacting with. 118 1.1 Gyr DG3 700 Myr DG3 1.25 Gyr DG3 150 Myr DG3 750 Myr DG2 1 Gyr DG3 -200 Myr DG5 1 Gyr DG5 1.5 Gyr DG30 0 Myr DG4 Table 10: The times of the bursts in DG3, when the moment of closest approach is taken as the zero point. Added directly below the time is the galaxy type it is interacting with. Appendix C 12 + log(O/H) [Fe/H] 12 + log(O/H) [Fe/H] 8.12 8.07 8.16 8.03 8.09 -1.51 -1.58 -1.49 -1.56 -1.53 8.10 8.08 8.11 8.11 8.06 -1.54 -1.57 -1.57 -1.50 -1.52 Table 11: The 12 + log(O/H) and [Fe/H] values for DG1 experiencing a burst. 12 + log(O/H) [Fe/H] 8.23 8.24 8.20 8.19 8.25 8.24 -1.45 -1.46 -1.47 -1.47 -1.46 -1.46 Table 12: The 12 + log(O/H) and [Fe/H] values for DG2 experiencing a burst. 12 + log(O/H) [Fe/H] 12 + log(O/H) [Fe/H] 8.58 8.55 8.56 8.52 8.59 -1.22 -1.24 -1.24 -1.30 -1.19 8.55 8.53 8.56 8.57 8.52 -1.24 -1.28 -1.22 -1.23 -1.30 Table 13: The 12 + log(O/H) and [Fe/H] values for DG3 experiencing a burst. model MT1 MT2 12 + log(O/H) [Fe/H] 7.51 -1.73 7.66 -1.66 Table 14: The 12 + log(O/H) and [Fe/H] values for the merger tree simulations. 119 appendix D 0.94 0.4 2.46 1.25 1.60 3.14 2.38 1.46 2.12 0.34 0.0 0.06 0.0 0.70 0.30 1.16 0.04 0.12 Table 15: On top the increase in gas mass in the galaxy around closest approach for the DG1 dwarf galaxies having a burst. 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