dynamical modeling of the merger of the sagittarius dwarf ... - IOPscience

The Astrophysical Journal, 723:1057–1064, 2010 November 10 C 2010.

doi:10.1088/0004-637X/723/2/1057

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

DYNAMICAL MODELING OF THE MERGER OF THE SAGITTARIUS DWARF–MILKY WAY SYSTEM J. M. Myers1 , B. Snyder1,2 , M. Rusthoven2,3 , L.-S. The2 , and D. H. Hartmann2 1 Department of Physics and Astronomy, Francis Marion University, Florence, SC 29506, USA 2 Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0978, USA 3 Governor’s School for Science and Mathematics, Hartsville, SC 29536, USA Received 2008 December 17; accepted 2010 September 6; published 2010 October 18

ABSTRACT In hierarchical models of large-scale structure formation, most galaxies grow in clusters, where merging is an essential, if not dominant process. Understanding the merger process is greatly aided by examples in the vicinity of the Milky Way due to the availability of high quality positional and kinematic data. The Sagittarius Dwarf Galaxy at a distance of 25 kpc is a prime example. Detailed observations of stars in this dwarf galaxy and its associated tidal streams also provide a unique probe of the gravitational field of the Milky Way. To fully utilize this probe requires not only excellent observations but also accurate models of the merging event. We investigate the recent history of the Sagittarius Dwarf with the smoothed particle hydrodynamics/N-body GADGET-2 code, treating both the Milky Way and the dwarf galaxy as a particle system. In addition to considerations of particle number, gravitational softening, stability criteria such as Toomre Q parameter, and angular momentum transport, we add the distinction between static and dynamic potential differences as another aspect that deserves consideration. We find that the commonly employed approximation of a static Milky Way potential induces changes in the orbit corresponding to position changes on the sky of order 0.5◦ –1◦ . The weak dependence of the results on the live versus static Milky Way model justify the use of static potentials in situations which do not require very high positional accuracy. Key words: Galaxy: evolution – Galaxy: kinematics and dynamics – Galaxy: structure – galaxies: individual (Sagittarius Dwarf) – methods: numerical Online-only material: color figures

can provide high quality positional, kinematical, and chemical data, allowing for superb tests of galaxy models. While strong evidence for the buildup of massive galaxies via major merging events is present in surveys of the local universe (e.g., in the Sloan Digital Sky Survey (SDSS) data McIntosh et al. (2008) sampling z 0.2), some observations of massive star-forming galaxies at z ∼ 2 suggest that the baryonic mass assembly of galaxies may be dominated by a more continuous accretion process, rather than by major mergers (e.g., Genel et al. 2008). Ongoing tidal disruption of smaller galaxies in the gravitational field of our Galaxy leaves very extended and longlasting trails of stars, and this also has been established for an increasing sample of nearby systems. The large angular extent on the sky and very low surface brightness of galactic tidal streams constitutes a significant observational challenge, but their use as probes of the dark matter halo offers great incentives for resource allocation. For example, Helmi (2004b) studied the shape of the Milky Way’s halo utilizing the detailed observations of the Sagittarius Dwarf Galaxy (SDG). This system was discovered during a stellar abundance and kinematics survey of K and M stars in the galactic bulge by Ibata et al. (1995). In this work, we investigate certain aspects of merger modeling that bear on the interpretation of the observed positional and kinematical data of Sagittarius. To interpret the history of the collision with the Milky Way, especially the development of its associated tidal stream, the accuracy of merger calculations must be evaluated. A common approach in analyzing the kinematics, orbits, and survival of the SDG and its tidal streams, is to treat the dynamics of stars in the SDG as an N-body problem of a system moving in the static gravitational potential of the Milky Way (MW). Since this static MW approximation is adopted on an ad hoc basis (for simplicity and faster calculations), and also because the positional and kinematical data of stars in SDG are

1. INTRODUCTION Our understanding of galaxy formation in an expanding cold dark matter dominated universe has improved significantly through extensive surveys, semianalytic modeling, and detailed N-body/smoothed particle hydrodynamics (SPH) simulations of coupled gas–star–dark matter systems (e.g., Ellis 2000; Keel 2007; Okamoto 2008; Meisenheimer 2008). Initial density perturbations form dark matter halos through dissipationless collapse, and galaxies form within these structures after gas cooling processes become efficient. Subsequently larger structures grow by hierarchical merging of the dark matter halos. The good agreement in producing the observed galaxy distributions in magnitude, redshift, color, and the power spectrum of galaxy clustering supports this paradigm. Still, our understanding of how the present-day large-scale structure and galaxy morphologies formed, and our ability to model the relevant chain of events, is incomplete (e.g., Ellis 2000). Open questions include the missing satellite problem (Moore et al. 1999; Klypin et al. 1999) in the neighborhood of the Milky Way, and the angular momentum problem (e.g., Navarro & Benz 1991; Navarro & White 1994; Navarro et al. 1995). Producing realistic galaxy models requires a detailed treatment of supernova feedback in addition to the gravitational aspect of the problem. Extensive particle-hydro simulations indicate that the observed clustering on large scales can be reproduced within a ΛCDM cosmological framework, and that bound baryonic structures resembling present-day galaxies can be obtained (e.g., Springel 2007; Kobayashi et al. 2007). Many aspects of these simulations rely on simplified treatments of the “micro-physics,” i.e., star formation and supernova feedback (energetically and chemically). Useful guidance on this modeling task is provided by observations in the vicinity of the Milky Way. Tidal effects in ongoing galaxy mergers of the Milky Way with smaller satellite galaxies 1057

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much more accurate than those for other galactic collisional systems, we suspect that a dynamic MW may be required for correspondingly accurate calculations of the evolution of the SDG orbital path, and its tidal stream in particular. Our ultimate goal is the development of a SPH/N-body merger simulation that includes the hydrodynamics of the gaseous disk and feedback processes such as (merger-driven) star formation and supernovae. Toward this goal, the simulations presented here focus on the limitations of the assumption of a minor merger in a static potential. For the purpose of testing the accuracy of the static-potential assumption, we compare two simulations having static and dynamic potentials through N-body calculations. The paper is structured as follows. In Section 2, we present a brief review of previous simulations of the SDG–MW system reported in the literature. In Section 3, we describe our simulations with the GADGET-2 code (Springel et al. 2001; Springel 2005) to obtain the initial configurations of the Milky Way (Section 3.1), the Sagittarius Dwarf (Section 3.2), and the combined, merging system (Section 3.3). In Section 4, we show the resulting orbital paths, and its projection on the celestial sphere. In Section 5, we present position and velocity distributions obtained with the two different potentials for the Milky Way. These are the observables relevant for utilizing the dwarf as a probe of the galactic halo. Finally, in Section 6, we discuss our results and their implications for modeling the merger in detail. 2. PREVIOUS MODELING The establishment of Sagittarius’ orbital properties has enabled detailed numerical simulations of the merger, but the wide variety of techniques employed over the last two decades sometimes yields surprisingly different results. Since its discovery in 1995, the favored method for the study of the SDG–MW system is an N-body approach with a prescribed, static potential for the Milky Way (Johnston et al. 1995, 2005; Velazquez & White 1995; Ibata et al. 1997; Ibata & Lewis 1998; Helmi 2004a, 2004b; Law et al. 2005; Law & Majewski 2010). With the exception of Velazquez & White (1995), this potential generally combines a Miyamoto & Nagai (1975) disk, a Hernquist spheroid (or bulge), and a logarithmic halo. Similar values for the mass of the disk (1.0 × 1011 M ) and the bulge (3.4 × 1010 M ) are used except in the case of Velazquez & White (1995) where the disk, bulge, and halo masses are 5.5 × 1010 , 1.1 × 1010 , and 5.3 × 1011 M , respectively. In characterizing the logarithmic halo component, values for the velocity in the halo vary from 114 km s−1 (Johnston et al. 2005) to 131.5 km s−1 (Helmi 2004a). Only two groups employed an N-body representation for the Milky Way (Edelsohn & Elmegreen 1997; Gómez-Flechoso et al. 1999) in their study of the SDG–MW interactions. Gómez-Flechoso et al. (1999) use the Milky Way model from Fux (1997) which describes the Milky Way as a threecomponent system including an oblate nucleus-spheroid, double exponential disk, and oblate exponential dark matter halo. Their luminous component has a mass of 8.25 × 1010 M and a dark matter mass of 2.4 × 1011 M . Edelsohn & Elmegreen (1997) used a parallel hashed oct-tree N-body code for their study. Using a total mass of 2.5 × 1011 M for their MW, their system was designed to reproduce the MW properties summarized in Merrifield (1992). The disk component used a total of 40,000 particles and in their two cases the SDG system used 789 and 1589 particles. These simulations showed tidal streams should be present and gave predictions on the times SDG would cross

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through the MW disk. We will compare these times to those of our simulation in Section 4. A variety of different models have also been used for the SDG system, including King profiles (Ibata et al. 1997; Velazquez & White 1995), Plummer sphere models (Johnston et al. 1995, 2005), a spherical density distribution (Edelsohn & Elmegreen 1997), and most recently a spiral galaxy (Penarrubia et al. 2010). Values for scale length and mass vary substantially between these studies. Values ranging from 107 to 109 M have been used for the total mass of SDG in the literature. The question of dark matter in the SDG has been addressed by Gómez-Flechoso et al. (1999) and who also present an alternative to the dark matter dominated model for the dwarf. They conclude that the SDG has been orbiting the MW for a minimum of 5 Gyr and find a model without a dominant dark matter component can survive for up to 10 Gyr. 3. DYNAMICAL SIMULATION OF THE SYSTEM To study galaxy mergers in general, one must consider the collisionless dynamics of stars and dark matter, as well as the hydrodynamics of the gaseous components. Star formation and supernovae from massive stars is the dynamic driver of the composition and multi-component structure of the interstellar medium (ISM). All components are coupled gravitationally, and therefore one requires a numerical tool that can handle the Nbody aspects of galactic dynamics as well as gaseous flows, their cooling physics and feedback processes. GADGET-2 is specifically designed to evolve and to simulate such systems. However, in the study of the SDG–MW merger presented here, we do not include gas in our simulations. For the purpose of this work, star formation is not essential and the gas mass is not a major constituent of our Galaxy. For the particular problem of the SGD–MW merger, we thus do not apply the full power of GADGET-2, but treat the merger as a pure N-body problem. The initial systems for the Milky Way Galaxy and SDG were constructed using parameters adopted from the literature. The MW and the SDG systems were both evolved in isolation for 4 Gyr to allow each system to virialize. These systems were then combined and allowed to evolve jointly for 2.5 Gyr. In all of our simulations, we use 0.6 kpc as the value of the gravitational softening parameters for all particle types (bulge, disk, and halo). The model parameters used are described in the following sections. 3.1. Milky Way N-body simulations of disk galaxies require a sufficiently large number of particles to minimize two-body relaxation effects that thicken the disk artificially (Edelsohn & Elmegreen 1997; Widrow & Dubinski 2005). For the purpose of testing whether the predicted path of the SDG is determined accurately when a static potential for the MW is applied in comparison with the SDG path in a dynamic potential of the MW, we employ N-body calculations of the SDG–MW system in both cases. To treat the static gravitational potential of the MW, we do not employ analytical models for the potential, but instead fix the MW particles’ position while allowing the SDG particles to move in response to the total gravitational potential of the system. A similar method of implementing a fixed potential in GADGET was applied by Read et al. (2006). In the case of the dynamic potential of the MW, the MW particles are treated in the usual N-body fashion.

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3.1.1. Model Parameters

Several numerical models of the Milky Way have been constructed (e.g., Bahcall & Soneira 1980; Bahcall et al. 1982; Kuijken & Dubinski 1995; Fux 1997; Widrow & Dubinski 2005; Naab & Ostriker 2006; Just & Jahreiß 2010). The initial Milky Way model used in our simulations was constructed from the density profiles of Bahcall & Soneira (1980) and Bahcall et al. (1982). An updated version of this model based on star counts from Hipparcos and various optical and near-infrared surveys was developed by Robin et al. (2003), but the comparative nature of our study does not require such a refined model. The Bahcall model is based on star count data, providing a local scale height, luminosity function, and mass distribution that reproduce the observed rotation curve. The halo mass density has the form ∝ R−1.2 for radius 10 kpc R 30 kpc and ∝ R−1.7 at larger radius. The disk density distribution is an axisymmetric double exponential of the form ∝ e−z/(50 pc)−((r−8 kpc)/3.5 kpc) , and the bulge density distribution is such that it gives the projected surface brightness of de Vaucouleurs law. We perform Monte Carlo sampling to create the spatial distributions for the disk, halo, and bulge particles. For each randomly generated particle, we add seven mirror particles with respect to the planes of the Cartesian coordinate axes to place the center of mass of the system exactly at (x, y, z) = (0, 0, 0). In our simulations, the number of particles in the MW’s halo, disk, and bulge are 960,000, 320,000, and 32,000, respectively, representing a mass of 6.50 × 1011 M , 5.85 × 1010 M , and 2.21 × 1010 M . The halo and bulge components were given appropriate circular velocities, randomly directed in the tangent plane. Disk particles were given corresponding tangential velocities in the mid-plane of the MW. All velocities are determined by calculating the total gravitational forces on the particles (Binney & Tremaine 1987). Velocity dispersion is added to the circular velocity of disk particles to ensure stability against axisymmetric instabilities (Toomre 1964). The minimum radial and tangential velocity dispersion components according to Toomre (1964) are σr,min = 3.36Gμ/κ and σθ,min = κ/(2Ω) σr,min , respectively, where μ is the surface density, κ is the epicycle frequency, and Ω is the angular velocity (each quantity is determined locally). The radial and tangential components are multiplied by a velocity dispersion factor of 2.5 to avoid clumping and rapid growth of bar instabilities (e.g., Athanassoula & Sellwood 1986; Edelsohn & Elmegreen 1997; Khoperskov et al. 2003; Gabbasov et al. 2006). For each disk particle, we add to the purely circular streaming velocity random velocity components that are drawn from a normal distribution based on the positiondependent dispersions. 3.1.2. Evolution

Our initial MW is evolved for 4 Gyr as an isolated system allowing its particles to relax to a quasi-equilibrium (see Figure 1). During this time period, we observe many different features developing in the disk of the MW. Initially, the disk is very thin and by ∼117 Myr, a central clump and a ring have formed in the disk. The ring developed as the particles expand outward in the plane of the disk, and the central clump around the central bulge expands. The width of the ring becomes larger and by ∼332 Myr it has dissipated. Around 800 Myr, a bar feature and spiral arms start to appear for the first time. This spiral arm feature is transient in nature. The approximate time when the dynamical equilibrium of the disk is reached can also be recognized in the evolution of the

Figure 1. Snapshots of the simulated Milky Way particle distribution showing the bulge (light gray) and the disk (black) components for the 4 Gyr evolution. To reduce the figure file size, we exclude the halo component and only show 40,000 disk and 1600 bulge particles. (A color version of this figure is available in the online journal.)

z-component of the angular momentum of the MW system shown in Figure 2. Angular momentum is predominantly transferred from the disk to the halo and bulge which both started with zero angular momentum. The spheroidal components pick up a small amount of angular momentum reducing the dominant disk angular momentum by about 1%. The total angular momentum is well preserved and after 2.5 Gyr the coupling between the components has stabilized and the individual angular momentum remains constant. Figure 3 shows the evolution of the artificially defined radii containing a given fraction of the halo mass. The 10% mass radius is initially at 20.4 kpc, expanding slightly afterward. This radius exhibits fluctuations which dampen over a period of 2 Gyr, maintaining a relatively constant value of 22 kpc. The 50% and the 90% mass radii do not change by more than 0.3 kpc during the entire 4 Gyr evolution, demonstrating the near dynamical equilibrium of the halo component established by the initial configuration. One of the consequences of angular momentum transfer from the disk to the halo component is the resulting slightly oblate structure of the halo indicated by a difference of interparticle separation of ∼0.2 kpc with respect to the vertical and the planar directions. Thus, after 4 Gyr of evolution the interaction between the disk and the halo components leads to a slightly squeezed halo, as predicted in adiabatic models of galaxy formation (Blumenthal et al.

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Figure 2. z-component of the angular momentum evolution of the disk (solid line), the halo (long-dashed line), and the bulge (short-dashed line) components of the Milky Way, and the fraction of the total angular momentum change. After ∼2.5 Gyr evolution, the angular momentum components are quite steady. The z-component of the total angular remains constant throughout the evolution. Note that the unit of the angular momentum is written in the unit system where the gravitational constant G = 1, and the mass, length, and time units are set to 1.0 × 1011 M = 1, 1.0 kpc = 1, and 1.49 Myr = 1, respectively.

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1986) and confirmed in self-consistent equilibrium solutions to the coupled Poisson and collisionless Boltzmann equations (Widrow & Dubinski 2005). The vertical thickness of the disk (Figure 4) slowly grows over the course of the 4 Gyr evolution. The thickening of the disk is understood to be caused by the effect of two-body gravitational collisions (Ostriker & Peebles 1973; Binney & Tremaine 2008). The rate of the heating has been found to be affected by the value of the gravitational softening, the number of disk particles used in the simulation, and the efficiency of angular momentum transfer between the disk and the halo (Gabbasov et al. 2006). In an investigation of the effect of the number of particles used in the simulation, we performed a test calculation using a 10 times smaller number of particles and find the scale height of the disk particles at 3 Gyr is about twice as large.

The SDG model consists of a single spheroidal distribution based on a Plummer (1911) model with a scale radius (Plummer b parameter) of 1.0 kpc. In comparison, the simulations by Johnston et al. (1995, 2005) use scale radii of 0.6 kpc and 0.82 kpc, respectively. In our simulations, the number of particles used for the SDG is 200,000, corresponding to a total mass of 1 × 108 M . The initial particle distribution for the dwarf was created the same way as the MW halo. Each particle was given an appropriate circular velocity oriented randomly in the tangential plane. This initial model was then evolved for 4 Gyr. Similar to the case of the halo component of the MW, we find that the radii of the mass shells in the SDG system stabilize within 1 Gyr. We select the model at t = 2 Gyr as our initial dwarf model in the SDG–MW system. 3.3. The Combined System 3.3.1. Initial System

To create our simulation of the SDG–MW interaction, we adjoined the two separately evolved models into one system. The assumed initial position of the SDG relative to the center of the MW was determined by performing a simple two-body evolution to 2 Gyr in the past. Following Edelsohn & Elmegreen

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Table 1 Parameters for the SDG–MW System Parameter

Value

Reference

Heliocentric distance of SDG Galactocentric radial velocity of SDG Current position (l, b) Transverse proper motion of SDG Proper motion of SDG

25 kpc 172 km s−1 (+5.6, −14.09)◦ 250 km s−1 μl cos(b) = −0.47 mas μl = 2.25 mas yr−1 8.0 kpc 220 km s−1 (−10.0, 5.25, 7.17) km s−1

Ibata et al. (1994) Ibata et al. (1995), Ibata et al. (1997) SIMBAD (Wenger et al. 2000) Dinescu et al. (2005) Dinescu et al. (2005) Dinescu et al. (2005) Binney & Merrifield (1998) Binney & Merrifield (1998) Binney & Merrifield (1998)

Sun Galactocentric distance Local standard of rest rotation speed Sun’s peculiar velocity (U , V , W )

(1997), we used the present position, radial velocity, and proper motion of the SDG shown in Table 1 for the initial state of the two-body calculations. Reversing the direction of the current SDG velocity, we trace the orbit backward in time using Aarseth’s N-body code (Aarseth et al. 1974; Binney & Tremaine 1987). To minimize tidal effect of the MW on the dwarf at its initial position in the merged simulation, we select the starting coordinates near SDG’s farthest location from the MW center. This corresponds to a time of ∼1.2 Gyr in the past. Perhaps not surprisingly, given the large extent of the system on the sky, the position of the centroid of the SDG used by various authors can differ substantially (see Table 2 of Kunder & Chaboyer 2009 for a summary). We adopted the SIMBAD values (Wenger et al. 2000), and the references in Ibata et al. (1994), Ibata et al. (1995), Ibata & Irwin (1997), and Dinescu et al. (2005) for the radial velocity, heliocentric distance, and proper motion, respectively (see Table 1). At the present time, we place the Sun at x = 8 kpc, y = z = 0 kpc, and the SDG at x = −16.13 kpc, y = −2.37 kpc, and z = −6.09 kpc. After adjusting for the motion of the center of mass of the MW, in the Cartesian coordinate system where the center of mass of the MW is located at (0, 0, 0), the location of the SDG is at x = −35.91 kpc, y = −9.81 kpc, z = −33.87 kpc at ∼1.2 Gyr in the past. The corresponding initial velocity of the SDG is vx = −141.46 km s−1 , vy = −8.70 km s−1 , vz = +0.51 km s−1 . 3.3.2. SDG–MW Evolution: Two Cases

We calculate the orbital path of the SDG galaxy starting from its initial point ∼1.2 Gyr in the past. We set the initial position of the MW at the center of the Cartesian coordinate system and the SDG as described above in Section 3.3.1. We follow the trajectories of all particles with GADGET-2. For the goal of the current study, we created two main cases for simulations of the SDG–MW systems. The first case is the static case where all MW particles are fixed to their initial position and the SDG particles move in the gravitational potential of the MW and the dwarf particles. The second case is the dynamic case where all MW and SDG particles are dynamically evolved allowing the positions and velocities of all particles in the system to change dynamically according to the total force. In all of our comparisons of the two cases, we compare them at exactly the same evolved time. In our simulations, the number of particles in the MW’s halo, disk, and bulge are 960,000, 320,000, and 32,000, respectively, representing a mass of 6.50 × 1011 M , 5.85 × 1010 M , and 2.21 × 1010 M . Therefore, the total number of particles for the MW is 1,312,000, representing a mass of 7.306 × 1011 M . The number of particles of the dwarf is 200,000, corresponding to a total mass of 1 × 108 M .

Figure 5. Time series of MW and SDG particles (blue) from the start of the simulation to ∼2.0 Gyr (or ∼0.78 Gyr in the future). The label on the panels shows the time in units of Myr. The fifth panel (at 1230 Myr) is closest to the present time. To reduce the figure file size only 12,000 halo, 80,000 disk, and 800 bulge particles are displayed for the MW and 5000 particles for the SDG.

4. RESULTS The panels in Figure 5 show a series of snapshots in edge-on view of the MW and SDG system. The first panel at the topleft corner of the figure shows the particle distribution near the initial time of the system, which is about 1.2 Gyr in the past. The second panel at the top-right corner shows the configuration of the particles after the SDG made its first pass through the MW disk in the simulation around 0.49 Gyr after the simulation starts or about 0.73 Gyr in the past. The third and the fourth panels show the second time (or the last time before the present) the SDG crosses the MW disk around 0.73 Gyr in the simulation time or about 0.49 Gyr in the past, while

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Figure 6. Left: the snapshot of the simulated particle distribution showing a leading stream and trailing tail of SDG particles (blue dots) at the present time, with MW particles forming a disk pattern near the center of the coordinate system in an edge-on view. The box symbols follow the predicted path of SDG’s center of mass from the initial position to today’s position (light green circle) in the two-body calculation found in Section 3.3.1. To reduce the figure file size only 12,000 halo, 8000 disk, and 800 bulge MW particles and 5000 SDG particles are displayed. Right: distribution on the sky in an Aitoff projection with 5000 SDG particles displayed. (A color version of this figure is available in the online journal.)

Edelsohn & Elmegreen (1997) in their simulation case 1 show that the collision occurred ∼0.17 Gyr ago. After the SDG passes through the disk (the fourth panel at 1.1 Gyr), the SDG particles are spread significantly, producing a long leading stream and a trailing tail. The fifth panel at time 1.23 Gyr shows the particle distribution near today’s time. The sixth panel at time 1.37 Gyr shows the configuration after the SDG crosses through the MW disk 147 Myr in the future. During the time between the fifth and sixth panels, the centroid of the SDG crosses the MW disk at a time of 1.27 Gyr (about 50 Myr from today). In comparison, Edelsohn & Elmegreen (1997) found that the collision will happen 35 Myr in the future. The last two panels show the SDG system approaching the MW disk again in the future with the leading stream and trailing tail even longer than before. Considering the large differences in the crossing times mentioned above, we caution the reader to not overinterpret such differences as the initial conditions vary between authors. Figure 6(a) shows a snapshot of our simulation, taken near the present time, with SDG particles forming a leading stream and trailing tail, and MW particles near the center of the coordinate system. The position of the highest particle density of the SDG is near l = +7.◦ 7, b = −17.◦ 5. consistent with today position of SDG (l = +5.◦ 6, b = −14.◦ 09) in the SIMBAD database (Wenger et al. 2000). The box symbols show the path of the center of mass of SDG from its initial position to the present in the twobody simulation described in Section 3.3.1 showing quite good agreement with the GADGET-2 calculations. Figure 6(b) shows the SDG particle distribution in an Aitoff projection near the present time in the simulations of our two cases: the dynamic (blue dots) and static (red dots) of SDG in the SDG–MW system. This presentation indicates that both simulations yield almost identical results. However, closer inspection of the longitude and latitude distributions (see below) reveals differences on the degree scale, so that any interpretation of observations that require accuracies better than a degree must be approached with caution. 5. OBSERVABLES Ibata et al. (1995) in their SDG discovery paper utilized the observed radial Galactocentric velocity distribution for testing

their Galaxy model while Ibata et al. (1997) in their Figure 10 fit both the Galactocentric radial velocity distribution and the SDG heliocentric distance distribution as functions of latitude to determine the orbital parameters of the SDG. Following their approach, to demonstrate the different effects of a dynamic and static MW on the dynamics of SDG particles, we compare several observables of the SDG particles in the simulations relevant at the present time of 1.219 Gyr. Key observables are the distributions in longitude, latitude, heliocentric distance, and Galactocentric radial velocity. Figures 7(a) and (b) show the present-day longitude and latitude distributions of the SDG particles. The result for the static MW potential is shown by a solid line while the dynamic case is shown by a dashed line. The longitude distribution in the dynamic case (dashed line) had to be shifted by −0.◦ 18 to match the static case for easy comparison of their distributions. The inset shows both distributions without any shifting. Both longitude distributions have similar shapes except for the slightly higher density and narrower clump of the leading stream near l = 4◦ in the dynamic case relative to the static case. The differences between the resulting distributions are manifested in the overall shift (0.◦ 18) as well as some differences in the length of the leading tidal stream. Similarly, Figure 7(b) shows the SDG latitude distribution where the latitude in the dynamic case had to be shifted +1.◦ 2 to match the static case for comparing the shape of the distribution with the static case. The inset shows both latitude distributions without any shifting. Both latitude distributions have similar shapes near their peak distributions. Furthermore their leadingstream distributions are also very similar up to ∼15◦ . Figure 8(a) compares the SDG heliocentric distance distribution in the dynamic (dashed line) and the static (solid line) MW potential at the present time. For the purpose of comparing the distance distributions, we shift the dynamic distance distribution by −0.15 kpc so the peak location matches with the peak of the static case. The shape of the distributions is similar, except in the dynamic case the trailing tail (near a heliocentric distance of ∼20 kpc) is slightly narrower than in the static case. The inset shows the peak of the SDG heliocentric distance distribution at the present time without any shift. The peak of the distribution is ∼0.15 kpc farther from the Sun in the dynamic case than in the static case.

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Figure 7. Today’s SDG particle distribution in longitude (left) and latitude (right). The SDG longitude distribution in the dynamic case (dashed line) has been shifted −0.◦ 18 for comparing the shape of the distribution with the static case. The distributions look similar except the secondary clump near l = 4◦ in the dynamic case is slightly denser and narrower than in the static case. The inset expands the longitude axis near the peak of the longitude distribution and shows both distributions without any shift. The SDG latitude distribution in the dynamic case (dashed line) has been shifted +1.◦ 2 for comparison to the static case. The inset is unshifted but expands the axis near the peak. (A color version of this figure is available in the online journal.)

Figure 8. Left: SDG heliocentric distance distribution with the dynamic case (dashed line) shifted by −0.15 kpc for comparison to the static case (solid line). The inset shows the unshifted peak in the distributions. Right: today’s SDG velocity distribution relative to Galactic center in the dynamic case (dashed line) and static case (solid line). The difference between the two cases is in the velocity distribution of the leading stream near a velocity of 195 km s−1 . (A color version of this figure is available in the online journal.)

Comparing our SDG peak distribution with the predicted path in the two-body calculation (which used the kinematic data presented by Ibata et al. (1997) in their Figure 10), we find the peak of the SDG heliocentric distance distribution (in Figure 8(a)) is ∼1.4 kpc farther away. Several factors may contribute to this difference; the most significant factor being the substantial change in the MW potential from the one we use in the two-body calculation to obtain the SDG initial position and velocity. Figure 8(b) compares the Galactocentric radial velocity distribution of the SDG in the dynamic (dashed line) and static (solid line) MW potential. The figure shows that the shape and the location of the main peak are similar. However, there is a difference in the shape and the location of the secondary leading-stream velocity distribution near 190 km s−1 . The observed velocity distribution presented in Figure 7 of Ibata et al. (1997) shows a peak at 172 km s−1 and not extending beyond 200 km s−1 . In contrast, our calculated peak is at ∼160 km s−1

with a maximum at ∼230 km s−1 , in reasonable agreement with the observed values. 6. DISCUSSION AND CONCLUSIONS We consider two different numerical approaches to simulate the merger of the Sagittarius Dwarf with the Milky Way, referred to as dynamic and static cases. In both cases, we utilize the Nbody capabilities of GADGET-2. The most realistic case allows both galaxies to evolve under the influence of their combined potentials (the dynamic case) while most other simulations only consider N-body simulation of the dwarf in the presence of a static MW potential. To test this approximation, we carry out the second case (static) by simply fixing the positions of all MW particles. A comparison between these two simulations reveals the importance of the response of the MW to the merger event. The tidal effects on the dwarf in a static potential are not identical to those of the dynamic case. Given the fact that this

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merger event is in close proximity and thus provides excellent observational constraints led us to investigate this issue. Before building detailed models for comparison with observations, we investigate the relative changes induced by the common method in fixing the MW potential. The two cases presented in this work clearly show differences in all observables and in particular the positions. The magnitude of the positional differences ranges from 0.◦ 5 to 1◦ . Depending on the accuracy needed for comparison with a particular set of observations such differences may be significant. The weak dependence of the results on the live versus static MW model suggests that the use of static potentials is justified when very high positional accuracy is not required. Increasing observational efforts have dramatically improved our understanding of the large-scale structure of the tidal streams and reveal sub-structures (bifurcation) in the streams (e.g., Fellhauer et al. 2006; Yanny et al. 2009). These properties provide information on the halo of the MW but also the internal structure of the Sagittarius Dwarf. Thus, to utilize the SDG–MW merger as a probe it appears essential to use the most accurate merger simulations. Our simulations indicate that the static MW approximation should be replaced in favor of fully dynamic simulations if computational resources allow. Our simulations with GADGET-2 did not include a gas component in either of the two merging systems, which would be necessary for a study of merger induced star formation and subsequent wind and supernova driven feedback. GADGET-2 is in principle capable of treating all components with additional numerical routines to calculate star formation and supernova feedback. While in general the simulation of the merger of two distant galaxies does not require a very accurate treatment, the proximity of merging satellites with the MW offers the opportunity for deeper insight and thus would benefit from correspondingly accurate simulations. Our simulations as well as those of Gabbasov et al. (2006) demonstrate sensitivity to several aspects of the numerical studies of merger events. In addition to considerations of particle number, gravitational softening, stability criteria such as Toomre Q parameter, and angular momentum, we add the distinction between static and dynamic potential differences as another aspect that deserves consideration. We are grateful for the financial support of Francis Marion University and to Volker Springel for developing GADGET-2 and making it available to the public. Two previous Francis Marion students, Blane McCraken and Ashley G. Messick, also assisted during the very early stages of this project. We also acknowledge the generous support from Clemson CCIT, especially Jill Gemmill and Edward Duffy, who arranged the access to CPU time in Clemson’s Palmetto cluster and assisted in implementing GADGET-2. Last but not least, we express our gratitude to the anonymous referee for a constructive report on the manuscript. REFERENCES Aarseth, S. J., Henon, M., & Wielen, R. 1974, A&A, 37, 183 Athanassoula, E., & Sellwood, J. A. 1986, MNRAS, 221, 213 Bahcall, J. N., Schmidt, M., & Soneira, R. M. 1982, ApJ, 258, L23

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