D RAFT VERSION M AY 28, 2020 Typeset using L A TEX twocolumn style in AASTeX63 N-body Simulations of the Milky Way-Gaia Enceladus Merger G RAHAM AD A RCHIBALD ABSTRACT The formation of the Milky Way has long been credited to the collapse of a gigantic gas cloud leading to the spiral-armed disk we observe. New data has lead to an model involving a collision between the young Milky Way and a dwarf galaxy 8-11 billion years ago. While most studies explore this using massive, low particle resolution ab-initio simulations, this project demonstrates the advantages of a more controllable, high resolution n-body simulation resulting in an orbital anisotropy parameter of β = 0 41 Keywords: Galactic Formation — Gaia Sausage — N-body Simulation 1. INTRODUCTION Within the field of astrophysics, the formation of galaxies, and especially our home Milky Way, has long been a topic of interest, with many hoping to uncover the true mechanism of formation by which our celestial homes came to be. Early theories suggested the gradual accretion of mass through the monolithic collapse of gigantic gas clouds, with conserva- tion of angular momentum accounting for the mutation of these clouds into the thin spinning disks we currently observe (Eggen et al. 1962). Since the second data release from the Gaia observatory (Helmi et al. 2018) and the observation of acute anisotropy amongst high metallicity stars within the galactic centre (Be- lokurov et al. 2018), the true mechanism of formation of the Milky Way has been the subject of heavy debate. Several hy- potheses for the origin of the modern Milky Way have since emerged, though the most likely seems to be a significant merger event between the young Milky Way (hereby referred to as proto Milky Way ) and a massive dwarf galaxy known as the Gaia Enceledus (hereby referred to as Gaia ) (Myeong et al. 2018) between 8 and 11 billion years ago, resulting in the deposition of stars characterized by their highly radial orbital trajectories which travel past the galactic centre with high radial and low tangential velocities relative to the ma- jority of stars composing the galactic disc. The stars remnant of this proposed merger are now known as the Gaia Sausage for their characteristic sausage shape when viewed in veloc- ity space, as can be seen in Figure 1. Most research investigating this candidate as the potential true formation model have used ab initio simulations such as ‘ Evolution and Assembly of GaLaxies and their Environ- ments ’ (EAGLE) (Bignone et al. 2019) which produce thou- sands of Milky Way type galaxies, which can then be tapered for those sharing similar properties to the fine structure of the modern Milky Way. Figure 1. Disk and sausage features in velocity space. The high radial component of the Gaia stars name the feature ‘sausage’ (Be- lokurov 2018) Very recently, Bignone et. al used the DR2 data in conjunc- tion with the largest EAGLE simulation to date in order to identify Milky Way and Gaia analogs from a massive sample of ab initio systems. From the approximately 10000 Milky Way type galaxies formed in this simulation, they identi- fied an ideal candidate using quantities such as mass, radius, and orbital anisotropy within the region of the hypothesized merger deposition to select the system from the catalogue of candidates. Once this system was identified, they presented data regarding the dynamics of the collision, as well as pre and post merger quantities for both the Gaia and proto Milky Way, as displayed in Figure 2. 2 Figure 2. Table one from (Bignone et al. 2019) describing proper- ties of the Milky Way and Gaia analogs pre merger ( z = 1 7 ) and post merger ( z = 0) . These parameters are central to this project for creating the proto Milky Way and Gaia models. Though their paper has suggested this merger as a likely candidate for the galactic formation, they comment: ‘we expect that a higher resolution, zoom-in simulation of this system will allow for a more detailed exploration of these processes (Bignone et al. 2019). The aforementioned quote forms the basis of this thesis project; exploring the validity of the merger using a more finely controlled, high resolution n- body simulation. Compared to a massive ab-initio simulation such as EAGLE, which produces thousands of galactic for- mations, an n-body simulation provides much more control over specific parameters describing the system components as well as collision dynamics. Furthermore, an n-body sim- ulation allows for vastly increased particle resolution while conserving computational power and time. By making use of pre merger proto Milky Way and Gaia analogs, we hope to time-evolve the collision model and investigate the resultant system with optimized parameters. 2. METHODS Investigation of the validity of the merger between the proto Milky Way and the Gaia begins with selection of an appropriate ab initio model from which the n-body system is based. Our model uses the proto Milky Way and Gaia analogs found using the largest EAGLE simulation to date (Bignone et al. 2019, Table One) which found a single can- didate from a population of more than 10,000 Milky Way analogs based parameters such as mass, size, and central- ized, high orbital anisotropy. The values from this simula- tion, along with some derived values such as concentration factor from the Navarro–Frenk–White profile (Navarro et al. 1996) are input into GalactICS (Deg et al. 2019) as the foun- dation for the n-body simulation. GalactICS files are then used as input for the n-body simulation performed by Gadget in our case. The simulation produces snapshots at specified time-evolution intervals which are then analysed to investi- gate the validity of the models with quantities uch as radial density and orbital anisotropy. Beginning with GalactICS, we now examine each step of the methodology more closely. 2.1. GalactICS GalactICS is a program used for creating initial condi- tions of an n-body simulation (Deg et al. 2019). In this project, GalactICS functionality is central to creating pre- merger Milky Way and Gaia analogs which are to be used as the basis for the merger simulations. GalactICS uses three main files for a system composed of a disk and halo, in.disk, in.halo, and in.dbh. The role of the disk and halo files are to specify the particle count for each component, in our case, we have used a 5:1 ratio of proto Milky Way to Gaia particles in order for the mass of each systems particles to be approxi- mately equal. More crucial to the specifications of the desired simulation is the in.dbh file which describes the virial mass, radius, thickness, and truncations of the halo and disk com- ponents. The explicit input data used in the final simulation can be seen in the appendix 4. This is the crucial point in which the parameters of the ab-initio model (Bignone et al. 2019) are input as our initial conditions. GalactICS directly creates ASCII files for galactic components; disk and halo in our case. We next create the disk and halo components for each system by using the make disk and make halo com- mands. With these ASCII files, we prepare the simulation parameters with the GalCombine function, which takes as input the specified systems including their components, in our case, this consists of the proto Milky Way and Gaia sys- tems each with their disk and halo components. At this stage, we implement the initial positional and rotational alignments of the systems as are consistent with our ab-initio model (Bignone et al. 2019). Specifically, we place the Gaia at x, y, z coordinates [ − 85 , 47 , − 6] kpc while also flipping the Gaia such that it’s spin is counter rotating with respect to the proto Milky Way. This achieves a collision very similar to that of the ab-initio model in terms of collision and inclina- tion angles. After executing this program, we need only to convert the output into a Gadget compatible format by using the output file as input to our ascii2gadget command, bringing us to the next portion of our methodology. 2.2. Gadget Gadget (GAlaxies with Dark matter and Gas intEracT) is a program for cosmological simulations of structure forma- tion (Springel 2005) which we use to time-evolve the n-body analogs. Using the Gadget compatible file created in the pre- vious section Sec. 2.1, we need only specify a few parame- ters for the execution of the simulation using Gadget2 such as total simulation time and time between snapshots. In our case, we are using a total simulation time of 10 Gyr in order to mimic the merger timeline as used in the reference pa- per (Bignone et al. 2019). Using a snapshot interval of 0.05 3 Gyr, we run the simulation through openmpi and the Gadget2 command. Upon completion of the simulation, we now pos- sess our requisite snapshots to be used for the data analysis. 2.3. Pynbody Pynbody is a Python analysis package for n-body astro- physical simulations (Pontzen et al. 2013). It is central to this stage of the project as each snapshot requires pre-processing in the form of a coordinate transformation. After partitioning the snapshots into separate arrays of Milky Way and Gaia particles, we rotate and shift the particles of each snapshot into an appropriate frame of reference. Specifically, in or- der to analyze each snapshot from the frame of reference of the stationary Milky Way, we correct each snapshot with two rotations such that the disk lies flush in the xy plane. We proceed by centering the system about the centre of mass of the proto Milky Way disk according to the shrinking sphere model (Power et al. 2003), as well as correcting for the cen- tre of velocity of the proto Milky Way using vel center and center functions built in to the analysis module of Pynbody specifically for these n-body simulations. Now that we have a simple correction algorithm, we can apply this to each snapshot of interest and proceed with more specific analyses. We begin with a visual representation of the system evolution using a radial density plot. Using a com- bination of the matplotlib ‘histogram2d’ and ‘pcolormesh’ functions, we are able to visually examine the dynamics of the merger event while confirming that our centre of mass and rotational corrections are working as expected (Hunter 2007). Another way to visually examine the dynamics of the merger and the evolution of the system over time is through the use of velocity space corner plots (Foreman-Mackey 2016). These plots, comprised of radial, azimuthal, and polar velocities in our case, display the velocity distributions of the systems, al- lowing us to investigate the data for key features, such as the formation of the characteristic ‘sausage’ shape of Gaia stars in the v r v φ frame. Finally, in order to more quantitatively display the evolution of the characteristic parameters of each system, we plot various velocities of both the Milky Way and Gaia systems per time step (0.05 Gyr) over the entirety of the 10 Gyr evolution in order to view the effects of the merger event during the dynamic period of the collision and as the system settles towards a steady state. Finally, we can numer- ically compare our resultant system to the expected galaxy we observe today using the anisotropy parameter β . This pa- rameter, described by β = 1 − σ 2 θ + σ 2 φ 2 σ 2 r , (1) describes the orbital dynamics of stars using the σ i veloc- ity dispersion in spherical coordinates with values β = 1 corresponding to a completely tangential orbit and β = 0 representing an isotropic orbit. 3. RESULTS AND DISCUSSION With our simulation complete, we can begin to investigate the result of this n-body simulation. Initially, we look at a simple visual representation of the evolution of the system in the form of a radial density plot. Figure 3. Radial density plots for time evolution between 0.45 and 0.8 Gyr. In this stage, the Gaia makes several passes through the Milky Way system as they merge. When looking into these plots, we must consider that the systems are corrected to the center of mass, center of velocity, and transformed rotationally such that the proto Milky Way remains centred at the origin as well as being flush with the xy plane. This first set of plots in Figure 3 illustrate the colli- sion as the Gaia strikes the proto Milky Way from a slight in- clination of approximately 7 ◦ as the systems collide dynami- cally. Next, we examine late stage evolution of the system as 4 the two bodies merge and settle into a visually constant state beginning around 2 Gyr in Figure 4. Figure 4. Radial density plots for time evolution between 1.1 and 10.0 Gyr. During this stage, the system experiences no visual sep- aration of the two incident bodies and settles into a more constant state beginning around 2 Gyr. During this stage of the time-evolution, we observe what appears to be a single system with the later stages of evo- lution ceasing any visual indication that this was once two galaxies. Following this representation of the distribution of stars from early stages of the pre collision system to the end state system we observe at 10 Gyr, we can investigate some properties more crucial to our study by exploring the velocity distributions of these systems. 300 150 0 150 300 v 300 150 0 150 v r 300 150 0 150 300 v 300 150 0 150 300 v 300 150 0 150 300 v GE Velocity Triangles T = 0 Gyr 400 200 0 200 400 v 300 150 0 150 300 v r 150 0 150 300 v 400 200 0 200 400 v 150 0 150 300 v MW Velocity Triangles T = 0 Gyr Figure 5. Corner plots of the proto Milky Way and Gaia at 0 Gyr, the initial snapshot of the simulation. 5 The corner plots such as those in Figure 5 illustrate the dis- tributions of spherical coordinate velocities in both the proto Milky Way and Gaia systems with each plot displaying six subplots containing each velocity component on both the x and y axes of a 2d histogram aside from the v i vs v i plots which are 1d histograms. As expected, we begin with two systems with largely isotropic orbits as seen in the v r v φ plots. Proceeding into a snapshot within the most dynamic portion of the time-evolution, we can clearly observe the chaotic mo- tion of the Gaia stars as described by their orbital velocities. 0 200 400 600 800 v 400 0 400 800 v r 600 300 0 300 600 v 0 200 400 600 800 v 600 300 0 300 600 v GE Velocity Triangles T = 0.5 Gyr 400 200 0 200 400 v 400 200 0 200 400 v r 400 200 0 200 400 v 400 200 0 200 400 v 400 200 0 200 400 v MW Velocity Triangles T = 0.5 Gyr Figure 6. Corner plots of the proto Milky Way and Gaia at 0.5 Gyr. While we are still yet to observe the formation of any sausage like features within the v r v φ quadrant of the Gaia plots, these plots are not without merit. One interesting char- acteristic to note is the relatively large effects of the merger on the Gaia compared to the proto Milky Way; attributable to the mass difference between the systems. This mass delta aids us in understanding how the present Milky Way appears unperturbed aside from this deposition of old stars. Prior to looking at the late time-evolution snapshots in velocity space, we can zoom out in order to take a numerical look at what is really happening as these systems interact. To do this, we can plot various velocities for each system across the entirety of the simulation. 100 150 200 250 v 2 r 1/2 Merger Velocities (km/s) 0 50 100 150 v 50 75 100 125 ( v v ) 2 1/2 0 2 4 6 8 10 Time (Gyr) 80 100 120 140 v 2 1/2 MW GE Figure 7. Velocities of the proto Milky Way (blue) and Gaia (red). From top to bottom, we have root mean square of v r , mean of v φ , standard deviation of v φ , and root mean square of v θ 6 Figure 7 presents us with a clear picture of where each systems stars are tending towards as the simulation evolves. We see that in the early stages, especially in the range of 0 − 2 Gyr, the systems experience significant changes due to the dynamics of the merger. As the time progresses, most of the values tend to level out as the system evolves into a more steady state. Some strange results include the dispar- ity of some velocity parameters to the expected values jux- taposed with the accuracy of some others. Specifically, the values for v r of both systems are very similar to the values of ∼ 200 km/s for the Gaia and ∼ 120 km/s for the proto Milky Way while the circular velocity for the proto Milky Way as described by the blue data in the second to top frame is much lower than we expect which, rather than approaching 50 km/s, should be clustered much closer to ∼ 180 km/s. As we hoped, we note that the proto Milky Way and Gaia sys- tems have inverted and rather drastic contrast in terms of v r and v φ values. This suggests the formation of highly radial orbits amongst the stars composing the Gaia Sausage which we hoped to observe. We can try and confirm this finding by looking into later stages of the velocity distribution evolution, most aptly represented by the now familiar corner plots. Moving two billion years into the future, after the system has settled from a two component system into a single galaxy as we saw in Figure 4, we can clearly the formation of the sausage shaped feature as shown in the v r v φ quadrant of Fig- ure 8. We can quantitatively describe each of these systems in terms of their orbital anisotropy as described by equation 1. Using the velocity dispersions, we obtain β = − 0 10 for the proto Milky Way, and β = 0 44 for the Gaia. These results agree with the expected values as they describe anisotropic orbits for the Gaia stars as well as circular, isotropic orbits for the proto Milky Ways component. In terms of the mag- nitude of anisotropy for the Gaia stars, we obtain a β value comparable to that of the 101 final Milky Way candidates from the reference paper (Bignone et al. 2019), though it is significantly less than the β = 0 76 possessed by their final candidate, the parameters of which this study are based. My thoughts on the reasoning behind this lie mainly in my inability to incorporate specific initial parameters such as the precise distribution of stellar mass amongst stars within vary- ing ranges of circularity described by M ?, as can be seen in Figure 2 with circularity = L z /L z, max being characterized by the average and maximum angular momentum, L z along the main axis of rotation for stars of a given binding energy E Nevertheless, we proceed to the final corner plot represent- ing the velocity distribution of the system after 10 Gyr of time-evolution in Figure 9. 500 250 0 250 500 v 500 250 0 250 500 v r 500 250 0 250 500 v 500 250 0 250 500 v 500 250 0 250 500 v GE Velocity Triangles T = 2.5 Gyr 500 250 0 250 500 v 400 200 0 200 400 v r 400 200 0 200 400 v 500 250 0 250 500 v 400 200 0 200 400 v MW Velocity Triangles T = 2.5 Gyr Figure 8. Corner plots of the proto Milky Way and Gaia at 2.5 Gyr. The plots for the proto Milky Way are characterised by isotropic distributions of velocities with gaussian distributions of the 1d his- tograms. The Gaia stars are characterized by anisotropic velocity distributions, especially within the v r v φ quadrant. We also note the much sharper, leptokurtic distributions of the v φ and v θ 1d his- tograms. 7 600 300 0 300 600 v 500 250 0 250 500 v r 500 250 0 250 500 v 600 300 0 300 600 v 500 250 0 250 500 v GE Velocity Triangles T = 10.0 Gyr 500 250 0 250 500 v 400 200 0 200 400 v r 400 200 0 200 400 v 500 250 0 250 500 v 400 200 0 200 400 v MW Velocity Triangles T = 10.0 Gyr Figure 9. Corner plots of the proto Milky Way and Gaia at 10 Gyr, the final snapshot of the simulation. Very similar to Figure 8, as the system appears to settle into a steady state with little change over the previous 8 Gyr of evolution. 400 200 0 200 400 radial motion of MW, km/s 400 200 0 200 400 circular motion, km/s Disk 400 200 0 200 400 radial motion of GE, km/s Gaia Sausage Motions of Stars at 10 Gyr Figure 10. Isolated plots of the v r (radial velocity) and v φ (circular velocity) from the data used in Figure 9. The labels and colors used in this plot are selected to draw comparison to Figure 1 8 Looking at Figure 9, we note the similarity to 8 as the ve- locity distribution composing the systems seem to relax into these states relatively quickly once the dynamic interaction ceases at ∼ 2 Gyr. Lastly, we return to the question of the ability of the n-body simulation to reproduce the anisotropic feature we observe in the present Milky Way by isolating the velocity space plots of v r against v φ at the final stage of evolution an can plainly see the drastic difference in radial and circular velocity possessed by stars from the proto Milky Way and those belonging to the Gaia system. Comparing this to Figure 1, we note the radial dominated orbits of the Gaia stars, with similar distributions of velocities along the v φ axis representing circular motion in both the positive and negative directions. For the proto Milky Way component, we see that the majority of stars exist with small radial orbits and favor heavily the positive direction of circular motion, as most stars tend to orbit in a single direction. Quantitatively, we can once again measure the orbital anisotropy of both components of the now modern Milky Way. Again using Equation 1, we find that for the proto Milky Way component, we have a value of β = − 0 09 , and for the Gaia component, we find β = 0 41 Once again, the anisotropy parameter for the Gaia component is signifi- cantly less than the value of ∼ 0 76 we observe in the present day Milky Way. Although this is not the exact result we de- sire, the nature of the n-body simulation allows for massive amounts of control over minute details of both the initial sys- tem and the manner in which they interact over time. Further- more, as discussed in the Introduction 1, n-body simulation allows for much higher resolution for a given amount of com- putational time and power compared to a massive ab-initio simulation. In our case, even though we were not able to opti- mize our parameters to achieve the desired orbital anisotropy, we obtained a system of 300 , 000 stellar particles with a bary- onic particle mass of 3 33 × 10 4 M compared to the baryonic particle mass of 1 81 × 10 6 M (Bignone et al. 2019), while still only using a few days of computational time. While the particle resolution is certainly an advantage of the n-body simulation, I believe it should be increased only once the ini- tial parameters have achieved more accurate anisotropy val- ues through the modification of current parameters as well as the inclusion of more subtle parameters which may alter the development of the system such as star formation rates. Limitations of this project are mostly related to issues ob- taining full functionality of the merger simulations and thus a lack of fine tuning for the parameters of the collision as well as initial conditions of the proto Milky Way and Gaia components. The majority of issues regarding simulation functionality relate to use of the Gadget software due to the sheer complexity and breadth of potential uses for the soft- ware. With experience and through long periods of trial and error, the procedure of obtaining snapshots from initial sys- tem parameters can be done in a matter of hours (depending on simulation time), learning the entire process takes count- less hours and often requires the aid of those who are more experienced with the tools at hand. For this reason, I was un- able to adhere as strictly to my proposed timeline as I would have hoped and spent much more time obtaining full func- tionality of the methodology than expected. The result of these setbacks were that my initial estimations of the initial parameters for the GalactICS and Gadget parameter files re- mained largely unchanged . With hindsight, a more efficient strategy for completing this project would involve developing my analysis tools earlier so that I could investigate the simu- lation snapshots prior to having the functionality to simulate the full time evolution. This would have have encouraged my to question and adjust my initial conditions more frequently which would hopefully have produced more accurate results. With this in mind, my propositions for future direction, and perhaps a future student’s thesis project, could begin with the state I have been able to achieve in my work, and spend much more available time optimizing the parameters of the initial systems as well as the specific conditions of the collision. Nevertheless, I must reiterate that I believe this project has demonstrated the wonderful ability and efficiency of an n- body simulation to produce and analyse the collision respon- sible for the formation of our Milky Way. 4. CONCLUSION The formation of the Milky Way has long been attributed to the continuous accretion of debris through the collapse of gi- gantic gas clouds (Eggen et al. 1962). Recent studies, as well as new data release (Helmi et al. 2018) suggest that the true mechanism of formation could be explained by a massive collision between the Milky Way and a dwarf galaxy known as the Gaia Enceladus (Belokurov et al. 2018; Myeong et al. 2018). Most research into this hypothesized merger has made use of massive, relatively low resolution ab-initio simulations which create huge numbers of candidate galaxies which can then filtered for an ideal candidate and be decomposed in order to investigate the properties of their constituent parts. This research thesis attempts to simulate this hypothesized merger using a high resolution n-body simulation which of- fers much greater control over fine system parameters than an ab-initio , while also providing the ability for massive par- ticle resolution scaling at a much lower computational cost than an ab-initio model. Though my progress encountered several delays due to compatibility between programs, as well as a relatively steep learning curve regarding several astrophysical concepts, I was able to create a model which produced the sausage like feature quantitatively describable by an anisotropy parameter of β = 0 41 which we observe in our modern Milky Way 10, with particle resolution approximately two orders of magni- 9 tude greater than that of the paper on which my initial condi- tions were based. While this study has demonstrated the potential for a n- body simulation to investigate the origins of the Milky Way, there is certainly much more which can be done to ad- vance this research in terms of optimization. I would sug- gest that these results, in conjunction with optimization of the initial system parameters, beginning with mass distribu- tion amongst regions of varying circularity as discussed in (Bignone et al. 2019), be further explored in order to obtain a more accurate, high resolution model which advances the field of astrophysics towards the truth of the formation of our Milky Way. ACKNOWLEDGEMENTS I would like to thank Dr. Lawrence Widrow for the su- pervision of this research thesis, as well as providing me with the opportunity to work on such an interesting and new study within the field of astrophysics. Dr. Nathan Deg has also been hugely important to the progression of this project as well as aiding in my understanding and ability regarding functionality of each component of the methodology. Dr. Ja- cob Bauer has also provided me with some crucial analysis techniques which allowed me to proceed through a portion of the research which could have derailed a majority of the project. Lastly, I would thank Dr. Alex Wright for organiz- ing the PHYS 590 course this year and providing students with an approachable and engaging view into the world of research. REFERENCES Belokurov, V. 2018, Velocities of the Milky Way and the Sausage stars in Gaia. https://people.ast.cam.ac.uk/ ∼ vasily/gaia sausage/info.html Belokurov, V., Erkal, D., Evans, N. W., Koposov, S. E., & Deason, A. J. 2018, MNRAS, 478, 611, doi: 10.1093/mnras/sty982 Bignone, L. A., Helmi, A., & Tissera, P. B. 2019, The Astrophysical Journal, 883, L5, doi: 10.3847/2041-8213/ab3e0e Deg, N., Widrow, L. 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F., Jenkins, A., et al. 2003, Monthly Notices of the Royal Astronomical Society, 338, 14–34, doi: 10.1046/j.1365-8711.2003.05925.x Springel, V. 2005, Monthly Notices of the Royal Astronomical Society, 364, 1105–1134, doi: 10.1111/j.1365-2966.2005.09655.x 10 APPENDIX GalactICS in.dbh values Gaia z = 1 7 y 53.2 3.2 3.8 6. 1. 3. y 1.33 1.5 3. 0.2 2. 0. 0. n n n n 0.05 10000 10 GalactICS in.dbh values proto MW z = 1 7 y 113. 4.7 19. 15. 1. 3. y 6.45 2.1 4. 0.2 2. 0. 0. n n n n 0.05 10000 10