And yet it moves: The dangers of artificially fixing the Milky Way center of mass in the presence of a massive Large Magellanic Cloud
Facundo A. Gómez, Gurtina Besla, Daniel D. Carpintero, Álvaro Villalobos, Brian W. O'Shea, Eric F. Bell
aa r X i v : . [ a s t r o - ph . GA ] F e b D RAFT VERSION J ULY
26, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
AND YET IT MOVES: THE DANGERS OF ARTIFICIALLY FIXING THE MILKY WAY CENTER OF MASS IN THEPRESENCE OF A MASSIVE LARGE MAGELLANIC CLOUD F ACUNDO
A. G
ÓMEZ
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USAInstitute for Cyber-Enabled Research, Michigan State University, East Lansing, MI 48824, USA andMax-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748, Garching, Germany G URTINA B ESLA
Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA D ANIEL
D. C
ARPINTERO
Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Argentina andInstituto de Astrofísica de La Plata, UNLP-Conicet La Plata, Argentina Á LVARO V ILLALOBOS
Astronomical Observatory of Trieste, via G.B. Tiepolo 11, I-34143 Trieste, Italy B RIAN
W. O’S
HEA
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USALyman Briggs College, Michigan State University, East Lansing, MI 48825, USAInstitute for Cyber-Enabled Research, Michigan State University, East Lansing, MI 48824, USA andJoint Institute for Nuclear Astrophysics
AND E RIC
F. B
ELL
Department of Astronomy, University of Michigan, 830 Dennison Bldg., 500 Church St., Ann Arbor, MI 48109, USA
Draft version July 26, 2018
ABSTRACTMotivated by recent studies suggesting that the Large Magellanic Cloud (LMC) could be significantly moremassive than previously thought, we explore whether the approximation of an inertial Galactocentric referenceframe is still valid in the presence of such a massive LMC. We find that previous estimates of the LMC’s orbitalperiod and apocentric distance derived assuming a fixed Milky Way are significantly shortened for modelswhere the Milky Way is allowed to move freely in response to the gravitational pull of the LMC. Holding otherparameters fixed, the fraction of models favoring first infall is reduced. Due to this interaction, the Milky Waycenter of mass within the inner 50 kpc can be significantly displaced in phase-space in a very short periodof time that ranges from 0.3 to 0.5 Gyr by as much as 30 kpc and 75 km/s. Furthermore, we show that thegravitational pull of the LMC and response of the Milky Way are likely to significantly affect the orbit andphase space distribution of tidal debris from the Sagittarius dwarf galaxy (Sgr). Such effects are larger thanprevious estimates based on the torque of the LMC alone. As a result, Sgr deposits debris in regions of the skythat are not aligned with the present-day Sgr orbital plane. In addition, we find that properly accounting forthe movement of the Milky Way around its common center of mass with the LMC significantly modifies theangular distance between apocenters and tilts its orbital pole, alleviating tensions between previous models andobservations. While these models are preliminary in nature, they highlight the central importance of accountingfor the mutual gravitational interaction between the MW and LMC when modeling the kinematics of objectsin the Milky Way and Local Group.
Subject headings: galaxies: formation – Galaxy: formation – Galaxy: halo – methods: analytical – methods:numerical – methods: statistical INTRODUCTION
Recently, a series of studies based on photometric, kine-matic and dynamical arguments have enhanced our currentunderstanding of the orbital history and mass of the Mag-ellanic Clouds system (see, e.g. Kallivayalil et al. 2013, andreferences therein). The results presented in these studies sug-gest that the Large Magellanic Cloud could be significantly
Electronic address: [email protected] more massive than previously thought. Besla et al. (2010,2012) showed that the observed irregular morphology and in-ternal kinematics of the Magellanic System (in both the gasand stellar components) are naturally explained by interac-tions between the Large and the Small Magellanic Clouds(LMC and SMC hereafter), rather than gravitational interac-tions with the Milky Way (MW hereafter). Kallivayalil et al.(2013, hereafter K13) showed that in order for the SMC to bebound to the LMC for periods as large as 2 Gyr (the estimated Gómez et al.age of the Magellanic stream) a LMC with a mass greaterthan 1 × M ⊙ is required. In addition, based on propermotion measurements obtained using the Hubble Space Tele-scope, Besla et al. (2007) and K13 showed that for such mas-sive LMC models, the Magellanic Clouds are likely to be ex-periencing their first infall towards the MWCould the acceleration of the inner regions of the MW in-duced by such a massive LMC be significant, even if it is ex-periencing its first pericenter passage? In binary stellar sys-tems, the two stars orbit about a common center of mass thatis often exterior to the more massive star. The MW+LMCsystem may be analogous, where the center of mass of thecombined system may be at a non-negligible distance fromthe Galactic center. A simple back-of-the-envelope calcula-tion suggests that this may indeed be the case. For example,assuming a MW model with a dark matter halo of viral mass M vir = 1 × M ⊙ , the mass of the MW enclosed withinthe LMC present-day position, R LMC ≈
50 kpc, is approxi-mately M ≈ - × M ⊙ . The LMC canonical modeladopted by K13, based on the requirement that the LMC andSMC have been a long-lived binary, assumes a total mass of M LMC = 1 . × M ⊙ . In this MW + LMC system, the or-bital barycenter could be displaced by as much as ≈
14 kpcfrom the Galactic center. The associated phase-space dis-placement of the MW with respect to its orbital barycentercould have a substantial impact on the inferred orbital prop-erties of satellite galaxies, including the LMC itself. In otherwords, such a massive satellite orbiting the MW at the presentday could pose a serious challenge to the commonly-adoptedassumption of an inertial Galactocentric reference frame.While understanding the motion of the Milky Way and itsneighbors is of relevance for many Local Group studies, adeep understanding of the expected response of the MW to thegravitational pull of such a massive LMC is urgently neededfor analyses based on orbital integration using present-dayphase-space coordinates as initial conditions. Furthermore,due to the extended nature of the MW stellar halo, not all starswill experience the same acceleration from the LMC. This dif-ferential acceleration could introduce observable signatureson the phase-space distribution of extended tidal streams,such as those associated with the Sagittarius dwarf galaxy(Sgr). The Sgr tidal tails span at least 300 ◦ across the sky(Ibata et al. 1997), and have been observed at Galactocen-tric distances as large as 100 kpc (e.g Majewski et al. 2003;Newberg et al. 2003; Ruhland et al. 2011; Drake et al. 2013).Indeed, Vera-Ciro & Helmi (2013) showed that the torque onSgr exerted by the LMC can introduce non-negligible pertur-bations to the orbit of Sgr and its distribution of debris. Theirwork, however, considered an spatially-fixed MW model, thusneglecting the dynamical response of the MW to the gravita-tional pull of the LMC.The aforementioned perturbations, associated with theplausible presence of a massive LMC, could even influencethe determination of the present-day Galactic mass distribu-tion. Multiple observational programs have provided, and willcontinue to provide, very accurate photometric, astrometric,and spectroscopic information for enormous samples of stars,not only in the Galactic disk but also in the more extendedstellar halo (see e.g. York et al. 2000; Perryman et al. 2001;Steinmetz et al. 2006; Keller et al. 2007; Yanny et al. 2009;Barden et al. 2010; Cui et al. 2012; Gilmore et al. 2012).During the last two decades several studies were devotedto the development and application of powerful theoreti- cal and statistical tools that could allow us to efficientlymine these observational data sets. An important goal inmany of these studies is to statistically infer the present-day Galactic mass distribution. It is customary for thesestudies to consider as input data dynamically young andextended stellar streams (e.g. Helmi 2004; Johnston et al.2005; Law & Majewski 2010; Koposov et al. 2010; Lux et al.2012; Sanders & Binney 2013b; Vera-Ciro & Helmi 2013;Bonaca et al. 2014; Deg & Widrow 2014; Gibbons et al.2014; Price-Whelan et al. 2014). The reason behind thischoice is simple: these types of spatially extended streamsare expected to approximately delineate the orbit of their cor-responding progenitors in phase-space (see Eyre & Binney2011; Sanders & Binney 2013a).For simplicity, in most of these works the MW’s mass dis-tribution has been assumed to be smooth and static, not onlystructurally but also spatially. Assuming a frozen-mass poten-tial may not strongly affect the results of these analyses. TheMW’s mass is not expected to have significantly evolved dur-ing the last 2 to 3 Gyr (e.g. Bullock & Johnston 2005), a dy-namical timescale that pertains to these studies. On the otherhand, the assumption that the MW can be regarded as an iner-tial frame has not been thoroughly tested. If one neglects thepresence of the LMC, the MW’s accretion activity can be re-garded as quiescent during this period of time. However, thedegree to which the presence of a massive LMC could sig-nificantly affect the statistically-inferred parameters that bestdescribe the Galactic potential remains to be studied.The dangers associated with artificially fixing the MWcenter of mass have been considered by several authorsin the past. One of the first works to explore this waspresented by White (1983). Using N -body simulations,this study showed that the orbital decay rate of a satel-lite galaxy is artificially enhanced by fixing the host cen-ter of mass. Current analytic prescriptions to model dy-namical friction are fine-tuned by calibrating against resultsof fully self-consistent N -body simulations (e.g., Cora et al.1997; Benson et al. 2004; Just & Peñarrubia 2005). Morerecently, Perryman et al. (2014) discussed the effects thatthe time evolution of the orientation of the disk angu-lar momentum vector with respect to an initial referenceframe could have on Gaia measurements. Such perturba-tions to the disk angular momentum could be caused by,e.g., the time-dependent accretion of gas (Shen & Sellwood2006; Roškar et al. 2010), the predicted tumbling of theGalactic dark matter halo (e.g. Bailin & Steinmetz 2005;Bryan & Cress 2007; Vera-Ciro et al. 2011) and by the tidalinteraction of a fairly massive LMC (Bekki 2012).In this work we revisit the problem of a non-inertial MWreference frame by modeling the interaction between the MW,the Sagittarius dwarf galaxy (Sgr hereafter), and a LMC thatis undergoing its first infall at the present day. We focus ouranalysis on two possible situations where the response of theMW to the gravitational pull of the LMC could induce sig-nificant perturbations: namely, the inferred orbit of the LMCabout the MW, and the orbit and tidal debris from the Sgrdwarf galaxy. To this end, we use a variety of different tech-niques to model the gravitational interaction between thesethree galaxies. In Section 2 we provide a justification for theLMC mass range explored in our experiments. In Section 3and 4 we use smooth analytic representations of the Galacticpotentials to characterize the significance of this perturbativeeffect on the orbital properties of both the LMC and Sgr. InSection 5 we use full N -body simulations to explore the con-nd yet it moves 3sequences of a non-inertial Galactocentric reference frame onthe phase-space distribution of the Sgr tidal debris. We con-clude and discuss our results in Section 6. THE MASS OF THE LMC
In this study we consider LMC mass models that range from3 × to 2 . × M ⊙ . We would like to stress that theconsidered LMC masses are meant to represent the total LMCinfall mass up to its virial radius, as opposed to the present-day observational constraint within its optical radius. In thissection we justify this mass range and explain why high-massLMC models are currently favored. We refer the reader toBesla (2014) for a more extended discussion.Our goal is to explore the effects of a massive LMC on theassumption that the MW can be considered an inertial frameof reference. The mass of the LMC is the dominant uncer-tainty in the orbital history of the Magellanic Clouds sincedynamical friction is proportional to its mass. Moreover, themass of the LMC also controls the orbit of the Small Magel-lanic Cloud, ultimately determining how long the two galaxieshave interacted with each other as a binary pair.Observationally, the total mass of the LMC is onlyconstrained within the optical radius. The LMC hasa well-defined rotation curve that peaks at V circ = 91 . ± M (8.7 kpc) = 1 . × M ⊙ , and fur-ther implies that the LMC is dark matter-dominated. The totalmass of the LMC may be much larger than this, depending onthe tidal radius.There is strong evidence that the stellar disk of the LMC ex-tends to 15 kpc (Majewski et al. 2009; Saha et al. 2010). If therotation curve stays flat to at least this distance then the totalmass enclosed is M (15 kpc) = V r / G ∼ × M ⊙ . Thisminimum value is consistent with LMC masses adopted bytraditional models of the orbital evolution of the MagellanicClouds (e.g. Murai & Fujimoto 1980; Gardiner & Noguchi1996). Note however that this estimate only takes into ac-count the total mass of the LMC within 15 kpc. Thus, it maysignificantly underestimate its total infall mass within its virialradius; this is the quantity of interest for this work.The total dynamical mass of the LMC at infall, up to itsvirial radius, can be estimated using its baryon fraction. Cur-rently, the LMC has a stellar mass of 2 . × M ⊙ and a gasmass of 5 . × M ⊙ . The baryonic mass of the LMC isthus M bar = 3 . × M ⊙ . Using the minimum total mass of M tot = 3 × M ⊙ , the baryon fraction of the LMC becomes M bar / M tot = 11%. This is much higher than the baryon fractionof disks in galaxies like the MW, which is of the order of 3-5%. In the shallower halo potentials of dwarf galaxies, stellarwinds should be more efficient, making baryon fractions evenlower, not higher.This analysis is further complicated if material has beenremoved from the LMC. Fox et al. (2014) have recently es-timated the total gas mass (HI and ionized gas) outside theMagellanic Clouds at 2 × ( d / M ⊙ , with d the dis-tance to the Magellanic stream. If half of this material camefrom the LMC, as suggested by Nidever et al. (2008), its ini-tial baryon fraction would be 14% – approaching the cosmicvalue. Note that the bulk of the Magellanic Stream likely re-sides at distances of order d =100 kpc, rather than 55 kpc, in which case the baryon fraction would increase to ∼ f bar ∼ - × M ⊙ . This highertotal mass is consistent with cosmological expectations fromhalo occupation models that relate a galaxy’s observed stel-lar mass to its halo mass. Using relations from Moster et al.(2013), the mean halo mass for a galaxy with a stellar mass of2 . × M ⊙ is 1 . × M ⊙ , implying a baryon fraction of f bar ∼ . × M ⊙ .The halo occupation model relations are primarily invokedto motivate initial conditions for a first infall model. As shownin K13 and later in this work, first infall models are obtainedin Milky Way-like hosts with a total mass ≈ × M ⊙ , re-gardless of the total LMC mass (within the range consideredhere). If the Clouds have only recently been accreted therehas not been enough time to severly truncate the LMC haloand, as a result, its current mass should approximately reflectits infall mass; i.e., the mass the LMC halo had upon firstcrossing the virial radius. Note as well that high-mass LMCmodels, > M ⊙ , are necessary in models of the forma-tion of the Magellanic Stream as they allow for a long-lived( ∼ ∼
130 km/s; high-mass LMCmodels (masses of order 10 M ⊙ ) are needed to explain howthe LMC can have held on to the SMC if it is moving at suchspeeds. This argument has been outlined in K13. Based ontheir parameter space search, and the requirement that theLMC and SMC have been a long-lived binary, we adopt acanonical mass model for the LMC = 1 . × M ⊙ .A very important uncertainty in the kind of analytic orbitalintegration schemes we employ in this study is the mass evo-lution of the LMC over time. The arguments laid out in thissection are for the required infall mass of the LMC. The sub-sequent mass loss incurred by the LMC as it orbits about theMW will necessarily cause significant modifications in the or-bits presented in the following section. A proper analysis ac-counting for this effect requires detailed N -body simulationsthat, in principle, are beyond the scope of the simple back-ward integration scheme presented here. Nonetheless, to vali-date our assumptions, we will compare the results from one ofthe MW+LMC mass combinations with those obtained withthe corresponding fully self-consistent N -body model. THE ORBIT OF THE LMC ABOUT THE MW
Our goal in this Section is to explore whether artificiallyfixing the MW center of mass could have significant implica-tions on the inferred orbital properties of the LMC. For thispurpose, we will integrate the orbits of different LMC modelsbackwards in time in MW-like hosts that are kept artificiallyfixed in space and that are also allowed to react to the gravita-tional pull of the LMC. In order to make a direct comparisonwith the results presented in K13 we will start by consider-ing smooth, analytic representations of the Galactic poten-tials. We will later compare our results with those obtainedfrom full N -body simulations. The models and methodologyare described in Section 3.1. Our results are presented on Sec-tion 3.2. Analytic models
Methodology
Gómez et al. G a l ac t o ce n t r i c r a d i u s [ kp c ] R vir MW = 1 x 10 LMC = 0.3 x 10 LMC = 0.5 x 10 LMC = 0.8 x 10 LMC = 1.0 x 10 LMC = 1.8 x 10 LMC = 2.5 x 10 G a l ac t o ce n t r i c r a d i u s [ kp c ] R vir MW = 1.5 x 10 Lookback time [Gyr] G a l ac t o ce n t r i c r a d i u s [ kp c ] R vir MW = 2 x 10 F IG . 1.— Time evolution of the galactocentric radius of different LMCmodels in three MW-like host potentials. Orbits are integrated backwards intime. Note that t = 0 Gyr corresponds to present-day. The different color-coded lines show the results obtained with different LMC models, as indi-cated in the top panel. From top to bottom, the MW models have M vir = 1 , . × M ⊙ , respectively. The dashed lines show the results obtained inMW models in which the center of mass has been artificially fixed. Solid linesshow the results obtained when the host is allowed to react to the gravitationalpull exerted by the LMC. Note the shorter LMC orbital periods obtained inthe latter case. The more massive the LMC, the larger the change in orbitalperiod. To follow the evolution of the gravitational interaction be-tween the MW and the LMC we used a symplectic leapfrogintegration scheme (Springel et al. 2001). Both the host andthe satellite are represented with analytic potentials; the cen-ter of each one follows the orbit that results from the accel-eration of the other. In practice, this is done by assigning tothe center of mass of each galaxy a mass-less tracer particle.The orbit of each tracer particle is determined by the smoothgravitational potential associated with the secondary galaxy. If, as in Section 4, a third galactic model is included, the orbitof each tracer particle will be determined by the smooth andnon-trivial potential associated with the overlapping densitydistributions of the two remaining galactic models. Note that,even though we use mass-less particles as phase-space trac-ers of the galactic centers of mass, we assign to each galac-tic model a spatially extended density distribution (see Sec-tion 3.1.2). Thus, at any given time, the acceleration exertedby the LMC on the MW (and vice versa) is computed by onlytaking into account the mass that is enclosed within a spherecentered on the LMC (and vice versa) of radius equal to thedistance between the two center of masses. In all cases the or-bits are integrated backwards from their present-day positionsand velocities.As in Besla et al. (2007) and K13, we ignore the mass evo-lution of the LMC owing to the MW’s tidal field. In ad-dition, we do not follow the time evolution of the mass orthe structural parameters of the MW potential. Since thepotentials considered are structurally frozen, there is no dy-namical friction exerted on the satellite galaxies. Therefore,we model this acceleration using an approximation of Chan-drasekhar’s dynamical friction formula (Chandrasekhar 1943;Binney & Tremaine 2008),d v d t = - π G M ρ ln Λ (cid:20)Z v v f ( v )d v (cid:21) v v , (1)where the subindex 1 refers to the galaxy causing the friction,the subindex 2 to the galaxy being decelerated, v is the rel-ative velocity of both interacting galaxies, M is the mass ofthe corresponding galaxy, ρ the mass density, f the distribu-tion function of velocities, G the gravitational constant and Λ is the Coulomb factor. For simplicity, in these experimentswe neglect the dynamical friction exerted by the LMC on theMW.Under the assumption of a Maxwellian velocity distributionand a constant background density field, it is possible to ap-proximate the integral in Equation 1 by: Z v v f ( v )d v ≈ (cid:18) erf( x ) - x √ π e - x (cid:19) , (2)where x = v / √ σ (Binney & Tremaine 2008). Here, σ isthe one-dimensional velocity dispersion of the host dark mat-ter halo. We adopt an analytic approximation of σ for anNFW profile as derived by Zentner & Bullock (2003). Fol-lowing Besla et al. (2007) and K13, for these experimentswe consider a value of the Coulomb factor that varies as afunction of the satellite’s galactocentric distance as describedby Hashimoto et al. (2003). The Hashimoto et al. (2003)Coulomb factor not only scales as a function of the satellite’sseparation to the host but also as a function of the satellite’sscale radius. As in K13, a fixed scale radius of 3 kpc is as-sumed in all cases. This may possibly overestimate the roleof dynamical friction in the orbital history of high mass LMCmodels. Detailed comparisons with N -body models are re-quired to properly estimate the degree of error, which will becomplicated by mass loss owing to MW tides and the pres-ence of the SMC. Such an analysis is beyond the scope of thisstudy. Nonetheless, as we will later show in this section, sucheffects are very small when considering LMC models that arecurrently undergoing their first pericenter passage. Moreover,our goal is to assess the effects of a non-inertial frame of refer-ence on the LMC’s orbit rather than to determine the exact or-bital history itself. As such, this methodology will sufficientlynd yet it moves 5 TABLE 1P
ARAMETERS OF THE
MW-
LIKE POTENTIAL USED IN OURSIMULATIONS . M vir R vir r s M disk r a r b M bulge c bulge
100 261 26 .
47 6 . . .
53 1 . . .
27 5 . . .
53 1 . . .
15 5 . . .
53 1 . . OTE . — Masses are in 10 M ⊙ and distances in kpc. The scaleradius for the Hernquist profile dark matter halos are obtained from r s through equation 9 TABLE 2P ARAMETERS OF THE
LMC
MODELS USED IN OURSIMULATIONS . M LMC [10 M ⊙ ] 3 5 8 10 18 25 r LMC [kpc] 8 11 14 15 20 22.5N
OTE . — These parameters are used for both Plummer andHernquist profiles. illustrate the general change in the orbits. If indeed dynamicalfriction is overestimated for high mass LMC models, their or-bits will be less eccentric backwards in time, augmenting theperturbative effects we illustrate here.
Galactic potentials
To model the MW potential we choose a three-component system, including a Miyamoto-Nagai disk(Miyamoto & Nagai 1975) Φ disk = - GM disk r R + (cid:16) r a + q Z + r (cid:17) , (3)a Hernquist bulge (Hernquist 1990), Φ bulge = - GM bulge r + r c , (4)and a Navarro, Frenk & White dark matter halo (Navarro et al.1996, hereafter, NFW) Φ halo = - GM vir r (cid:2) log(1 + c ) - c / (1 + c ) (cid:3) log (cid:18) + rr s (cid:19) . (5)Here, R and Z are the radial and vertical cylindrical coordi-nates and r is the radial spherical coordinate. The dark matter(DM) halo viral mass, M vir , is defined as the mass enclosedwithin the radius where the dark matter density is 360 timesthe average matter density (van der Marel et al. 2012). In allmodels the disk scale length and height, r a and r b , are keptfixed at 3.5 and 0.53 kpc, respectively. The bulge mass andscale radius, M bulge and r c , are also kept fixed at 10 M ⊙ and0.7 kpc, respectively. In addition, the NFW density profilesare truncated at the virial radius. The remaining parametersare allowed to vary in order to explore different models forthe MW potential. The adiabatic contraction of the dark mat-ter halo associated with the presence of a disk was taken intoaccount using the CONTRA code (Gnedin et al. 2004). Thevalues of the parameters for the different models are listed inTable 1. The circular velocity curve of these Galactic mod-els is shown in Figure 8 of K13. Note that in all cases thecircular velocity at the solar circle, R ⊙ ≈ . V ⊙ = 239 km/s (McMillan 2011). Note that a lowervalue of V ⊙ , e.g. V ⊙ = 218 ± Φ LMC = - GM LMC q r + r . (6)Following K13, a variety of LMC masses are explored,ranging from M LMC = 3 × to 2 . × M ⊙ . A de-tailed justification for this explored LMC mass range wasprovided in Section 2. The parameters that describe theseLMC models are listed in Table 2. Note that the scaleradius of each model is chosen such that the total masscontained within 9 kpc is ≈ . × M ⊙ , as indicatedby the LMC’s rotation curve (van der Marel et al. 2009;van der Marel & Kallivayalil 2014). An interacting MW + LMC model
The orbital properites of the LMC
To explore whether artificially fixing the MW’s center ofmass could have a significant effect on the inferred orbitalproperties of the LMC, we generate LMC-like orbits by in-tegrating the galaxies backwards in time from their presentday phase-space coordinates. The initial orbital conditionsfor all LMC models, in a Galactocentric reference frame, are( X , Y , Z ) LMC = ( - , - , -
28) kpc and ( v x , v y , v z ) LMC =( - , - , M vir = 1 , . , and2 × M ⊙ , respectively. As expected, our results are invery good agreement with those found by K13 (see Fig. 11 ofK13).In a MW model with M vir = 1 × M ⊙ (top panel of Fig-ure 1), the resulting LMC-like orbits show periods larger thana Hubble time, T ≈ .
73 Gyr, independent of the satellite’stotal mass. Increasing the mass of our MW models resultsin shorter LMC orbital periods. As a result, the less massiveLMC models start to show more than one pericenter passagewithin 8 Gyr. Note that, as discussed by K13, orbits withperiods, P , between T > P & T / < T / ≈ . M vir = 1 . × M ⊙ (mid-dle panel), LMC models with masses M LMC ≤ × M ⊙ have completed a full orbital period within 6.9 Gyr. For ourmost massive MW model ( M vir = 2 × M ⊙ , bottom panel) Gómez et al. O r b it a l P e r i od [ G y r] T / LMC mass [10 M ⊙ ] A po ce n t r i c d i s t a n ce [ kp c ] MW = 1.5 x 10 , freeMW = 1.5 x 10 , fixedMW = 2 x 10 , freeMW = 2 x 10 , fixed F IG . 2.— Top panel: Orbital period of LMC-like orbits as a function ofthe LMC total mass. The red and blue symbols indicate the period of orbitsintegrated in MW-like host with M vir = 1 . × M ⊙ , respectively.Open symbols show the orbital periods obtained when the MW is artificiallyfixed in space. Filled symbols show the results obtained when the MW isallowed to freely react to the gravitational pull exerted by the LMC. Theblack dashed line indicates half of the Hubble time, T /
2. Bottom panel: Asin the top panel, for the apocentric distance of the different LMC-like orbits.The red and blue dashed lines indicate the virial radius of the MW-like hostwith M vir = 1 . × M ⊙ , respectively. The orbital properties shownin this figure were obtained using the mean LMC’s velocity presented in K13. only the two most massive LMC models ( M LMC = 2 . . × M ⊙ ) exhibit orbital periods longer than 6.9 Gyr,in agreement with K13.The color-coded solid lines in Figure 1 show the orbits ofthe same LMC models, now in MW potentials that are al-lowed to freely react to the gravitational pull of the LMC. Itbecomes abundantly clear that nailing down the MW centerof mass has a very significant effect on the backward time in-tegrated orbits, particularly for the most massive LMC mod-els. In all cases, the orbital periods and apocentric distancesare significantly shorter. As the mass of the LMC becomeslarger, and thus more comparable to the MW mass enclosedwithin the LMC’s location, the two-body interaction becomesmore relevant. In other words, the more massive the LMCmodel, the more significant the changes in the resulting orbitsare. This can be inferred from the orbits shown in, e.g., themiddle panel of Figure 1 ( M vir = 1 × M ⊙ ). The acceler-ation experienced by the MW towards the LMC, and the cor-responding displacement of its center of mass, result in both ashorter LMC orbital period and a smaller apocentric distancein a Galactocentric reference frame.Note that this change in orbital period is not related tothe artificial enhancement of dynamical friction discussed by White (1983, W83). Using N -body simulations, W83 findsthat artificially fixing the host’s center of mass results in moreefficient dynamical friction than when the host is allowed toorbit. The reason for this behavior is attributed to the differ-ent global patterns excited by the orbiting satellite on the den-sity distribution of the host (for a detailed discussion aboutthis subject see Cora et al. 1997). As opposed to the N -bodymodels considered in W83, the galaxies in our analysis aremodeled through analytic and structurally frozen potentials.Thus, the perturbation of the satellite cannot generate wakesin the host’s density field. Changes shown in Figure 1 aremainly a reflection of the resulting orbits about the barycenterof the system. Note however that, due to the shortening of thesatellites’ orbital periods and apocentric distances, dynami-cal friction would act more efficiently in the free MW modelsthan in the fixed MW models.The orbits of the LMC models in a MW-like host with amass of 1 . × M ⊙ are shown in the middle panel of Fig-ure 1. Even in this more massive host, the effects of “freeing”the MW are still very significant. Now, all LMC models with M LMC ≤ × M ⊙ have completed a full orbit within 6 . × M ⊙ (bot-tom panel) all but the most massive LMC model have com-pleted a full orbit within 6.9 Gyr.We summarize and quantify the changes in our LMC-likeorbits in Figure 2. The top panel shows the orbital periods ofthe most recent orbit, obtained in both free (filled symbols)and fixed (open symbols) MW-like models. We focus onlyon those orbits that have completed at least one orbit aboutthe MW within a Hubble time. This figure clearly shows howdramatic the change on the orbital period can be, especiallyfor the most massive LMC models. For example, for a MW-like host with M vir = 1 . × M ⊙ and a LMC model with M LMC = 1 × M ⊙ , the period changes from 13 . ≈ T (fixed MW) to 6.8 Gyr ≈ T / R apo , as a function of LMC mass. Note that for the MW +LMC mass model combination discussed above, the apocen-ter goes from R apo ≈ . R vir to R apo ≈ . R vir . The changein the inferred orbital properties of our LMC-like models sug-gests that, even though a first-infall is still a very plausible sce-nario, the limiting LMC-MW mass combinations that couldhost a first infalling LMC are noticeably affected; it raises therequired minimum LMC mass and disfavors MW models with M vir ≥ . × M ⊙ . Phase-space displacement of the MW center of mass
We have illustrated that the presence of a massive LMC cansubstantially alter the orbital barycenter of the MW + LMCsystem even in a first infall scenario. It is thus interesting toquantitatively characterize the displacement of the MW centerof mass as function of time due to this gravtiational interac-tion.Before we move any forward, it is worth recalling that theinferred LMC’s orbital properties, quoted both here and inK13, do suffer from a number of simplifications. If the LMCorbits are significantly affected by this simplifications, thenthe estimated phase-space displacement of the MW center ofmass could also be significantly affected.On the one hand, as shown in K13, including a simplemodel for the time evolution of the MW potential would in-crease these orbital periods by . ≈ ±
19 km/s,will yield, in some cases, orbits with significantly larger pe-riods. Furthermore, we have neglected LMC mass loss dueto the tidal interaction with the MW. Note, however, that per-turbations on the inferred orbital properties due to LMC massloss are not expected to be significant in those cases where theLMC is clearly undergoing its first infall. On the other hand,the treatment of dynamical friction implemented here may beoverestimated, thus artificially increasing the orbital periods(see Section 3.1.1).A more accurate determination of the LMC’s orbital prop-erties as a function of LMC mass would require a full N -bodytreatment. Even though this is not the goal of this work, to ex-plore whether our approximations regarding dynamical fric-tion and LMC’s mass loss are valid, we have ran a full N -bodysimulations considering one of the MW+LMC mass combina-tions analyzed in this work. The MW-like host was modeledas a self-consistent three-component system consisting of aNFW dark matter halo, an exponential stellar disk, and a cen-tral bulge following a Hernquist profile. The LMC galaxy wasmodeled as a self-consistent Plummer sphere. The masses andparameters that specify each galactic component were chosento reproduce the analytic rigid representation of the galacticpotentials associated with a MW of M vir = 1 × M ⊙ and aLMC of total mass M LMC = 1 . × M ⊙ . These parametersare listed in Table 3. Initial positions and velocities for theLMC and MW centers of mass were obtained from the nu-merically integrated orbits using the analytic rigid potentials.The simulations were started at a lookback time equal to 2Gyr.In Figure 3 we show, with purple lines, the time evolutionof the position (top panel) and the velocity (bottom panel) ofthe MW center of mass, with respect to its present-day co-ordinates, due to the gravitational interaction with the LMCmodel M LMC = 1 . × M ⊙ . The solid line show the resultfrom the rigid analytic representation of the potentials whilethe dashed dotted line shows the result from for the fully N -body representation. Note the very good agreement betweenthe results obtained from the two different modelling tech-niques. The final phase-space displacement of the MW centerof mass in the N -body model is slightly smaller than whatwas obtained with the analytic rigid case. The differences are ≈ ≈ M vir = 1 × M ⊙ . Note that, for thislow-mass MW model, the LMC is on its first infall about the MW, regardless of the LMC mass .Changes in both position and velocity are very rapid andtake place primarily during the last ≈ . - . R LMC ≈
50 kpc, themass of the MW enclosed within a radius of R LMC becomessmaller. Assuming a MW model with M vir = 1 × M ⊙ ,at present day M ≈ . × M ⊙ becomes comparableto the mass of the LMC. Thus, the orbital barycenter of theMW + LMC system is significantly displaced from the MWcenter of mass. For example, for a LMC model with a totalmass M LMC = 1 . × M ⊙ , the orbital barycenter is locatedat ≈
14 kpc from the MW center of mass at the present epoch.In this model, as the LMC approaches its current location theMW is displaced by ≈
30 kpc and its velocity has changed by ≈
75 km/s in ≈ . THE ORBIT AND TIDAL DEBRIS FROM SGR
In this section we explore the implications of the motion ofthe MW around its center of mass with the LMC for the or-bit of the Sagittarius dwarf galaxy (Sgr) and its distributionof tidal debris. For this purpose, we will integrate Sgr-likeorbits in different scenarios. We will consider spatially “free"and “fixed" MW-like hosts, with and without the presence of aLMC-like satellite. As in Section 3, the experiments analyzedin this section will assume smooth and analytic representa-tions of the Galactic potentials. The models and methodol-ogy are described in Section 4.2. We briefly review the mainproperties of the Sgr stream and discuss previous attempts toconstrain the shape of the MW dark matter halo based on thestream’s phase-space distribution in Section 4.1. Our resultsare presented on Section 4.3. Note that, throughout this study,we are not interested in obtaining an orbit that could accu-rately reproduce the observed distribution of the Sgr tidal de-bris. Our goal is rather to simply characterize the significanceof artificially fixing the MW center of mass on Sgr-like or-bits. If the effect is significant, this justifies the incorporationof complete and realistic modeling of the LMC+MW interac-tion in future analyses of Sgr and other long stellar streams inthe halo of the MW.
The complex nature of the Sgr streams
As discussed in Section 1, the Sgr tidal stream and its rem-nant core have been used multiple times in the past to probethe mass distribution of the MW. The main reason behind thewide popularity of this satellite galaxy is the very large ra-dial and angular extent covered by its debris. The Sgr stellar Gómez et al. G a l ac t o ce n t r i c r a d i u s [ kp c ] MW = 1 x 10 LMC = 0.3 x 10 LMC = 0.5 x 10 LMC = 0.8 x 10 LMC = 1.0 x 10 LMC = 1.8 x 10 LMC = 2.5 x 10 Lookback time [Gyr] G a l ac t o ce n t r i c v e l o c it y [ k m / s ] MW = 1 x 10 F IG . 3.— Time evolution of the position and velocity of the MW center ofmass with respect to its position at t = 0 Gyr. The results are obtained fromsimulations where the MW model is allowed to react to the gravitational pullexerted by the LMC. Orbits are integrated backwards in time. Note that t = 0Gyr corresponds to present-day. The solid lines show the results obtainedwith a MW model with a dark matter halo of M vir = 1 × M ⊙ . Thedifferent colors indicate the results obtained with different LMC models, asindicated in the top right corner of the top panel. The most significant changesin both the position and the velocity of the MW center of mass take placeonly during the last 0.3 to 0.5 Gyr, the time at which both, the MW massenclosed within the LMC Galactocentric radius becomes comparable to thatof the LMC and the distance between both galaxies becomes short enough.For comparison, the dashed dotted line shows the displacement of the MWcenter of mass obtained from a fully live N -body simulations considering anLMC model with M LMC = 1 . × M ⊙ . Note the very good agreementbetween the results obtained with the rigid analytic potential and their fullylive N -body counterpart. stream spans at least 300 ◦ across the sky (Ibata et al. 1997),and observations suggest that it can be observed at Galac-tocentric distances as large as 100 kpc (e.g Majewski et al.2003; Newberg et al. 2003; Ruhland et al. 2011; Drake et al.2013). As discussed by Deg & Widrow (2014), the Sgr streamhas a very complicated structure, making it difficult to model.The mean orbital poles of the great circles that best fit debrisleading and trailing the Sgr core show a difference of ∼ ◦ (Johnston et al. 2005). Stars in the trailing and leading armsshow very different apocenters (Belokurov et al. 2014) and bi-furcations have been observed in both arms (Belokurov et al.2006; Koposov et al. 2012; Slater et al. 2013). In addition,Peñarrubia et al. (2010) showed that the phase-space configu-ration of the Sgr stream strongly depends on the structure ofthe progenitor.Given the complex nature of this stream, it is not surpris-ing that several previous studies have yielded contradictory F IG . 4.— Sgr-like and LMC orbits obtained in a free MW model with adark matter halo of 10 M ⊙ . For this example an LMC model with totalmass M LMC = 1 . × M ⊙ was considered. The orbits are illustrated in the XZ (top panel) and YZ (botom panel) Galactocentric planes, with Z pointingtowards the Galactic pole and X pointing towards the opposite direction ofthe Sun. The black and color coded lines indicate the Sgr-like and the LMCorbit, respectively. The color coding indicates the LMC’s galactocentric dis-tance. Solid and dashed black lines show the Sgr’s first and the second Gyr ofbackwards evolution, respectively. For clarity, the LMC orbit is only shownduring the first Gyr of backwards evolution. results with regards to the structure of the MW’s gravitationalpotential. For example, the previously mentioned tilt of theorbital plane can be reproduced with N -body simulations ifthe MW dark matter halo is modeled as a mildly oblate galac-tic component (Johnston et al. 2005). On the other hand, ra-dial velocity measurements of a sample of M giant stars fa-vor a prolate dark matter halo (Helmi 2004). Furthermore,Law & Majewski (2010) showed that a triaxial dark matterhalo model could reproduce the angular position, distance,and radial velocity constraints imposed by current wide-fieldsurveys of the Sgr stream. However, the results from thismodel are bound by a number of caveats. Firstly, the modelrequires the disk’s minor axis to be aligned with the interme-diate axis of the triaxial halo. As shown by Debattista et al.(2013), this configuration is extremely unstable. The problemcan be alleviated if the assumption of a disk-halo alignmentnd yet it moves 9 −20 0 20 40−40−2002040 X [kpc] Z [ kp c ] LMC = 1.8 x 10 MW = 1.0 x 10 MW free + SgrMW free + Sgr + LMCMW fixed + Sgr + LMC −20 0 20 40
X [kpc]
LMC = 1.8 x 10 MW = 1.5 x 10 −20 0 20 40 X [kpc]
LMC = 1.8 x 10 MW = 2.0 x 10 F IG . 5.— The different lines show Sgr-like orbits integrated for 2 Gyr in different models of the MW potential. The orbits are illustrated in the XZ Galactocentricplane, with Z pointing towards the Galactic pole and X pointing towards the opposite direction of the Sun. The LMC moves in a direction that is approximatelyperpendicular to this plane. Sgr orbits are integrated backwards in time. Thus, present-day positions are the same in all cases. Solid and dashed lines show thefirst and the second Gyr of backwards evolution, respectively. The blue lines show the Sgr-like orbits obtained in MW models that are allowed to react to anyexternal perturbation (Sgr + MW only). The green lines show the same orbits, now introducing a LMC model with a mass of M LMC = 1 . × M ⊙ , whichfollows the orbit described in Figure 1. The red lines show the corresponding Sgr orbit obtained after artificially fixing the MW center of mass, but including theperturbative effects of the LMC. From left to right, the different panels show the results obtained using MW models with masses M vir = 1, 1.5 and 2 × M ⊙ ,respectively. Note that the differences on the Sgr-like orbits between “free” (green line) and “fixed” (red line) MW models are even larger that those obtained inmodels with and without the LMC (green and blue lines). is relaxed when searching for a best fitting Galactic poten-tial (Deg & Widrow 2014). Secondly, the resulting axis ratiosare not compatible with expectations derived from cosmolog-ical simulations (e.g., see discussion by Vera-Ciro & Helmi2013, hereafter VCH13). Interestingly, VCH13 showed that ifthe gravitational field of the LMC is taken into account whencomputing the orbit of Sgr, the triaxial configuration of theMW-like dark matter halo can be brought to a more cosmo-logically plausible shape.The analysis presented in VCH13 shows that the torque onSgr exerted by the LMC can be as important as that of theMW’s dark matter halo, introducing non-negligible perturba-tions to the orbit of Sgr and its distribution of debris. Attemptsto reproduce the Sgr stream without a model for the LMC per-turbation will consequently force searches of the best fittingparameters that characterize the MW’s gravitational potentialto artificially adjust in order to account for this perturbation.While a very relevant conclusion, the work of VCH13 (aswell as many of the previously cited works) considered MWmodels that are fixed in phase-space. In addition, their re-sults were based on test particle simulations in which the Sgrstream is being significantly perturbed by the LMC over 3 to4 Gyr. As shown in Section 3, even relatively low-mass first-infall LMC models can significantly accelerate the MW innerregions in a very short period of time. It is thus likely that, ina Galactocentric reference frame (as opposed to a barycentricreference frame), the distribution of Sgr debris, which cov-ers a radial extension of ∼
100 kpc, will be significantly per-turbed due to the phase-space displacement of the MW centerof mass. To explore this, we integrate Sgr-like orbits in MWpotentials that are allowed to freely react to the gravitationalpull of the LMC.
Analytic models
Methodology
As in Section 3, to follow the gravitational interaction be-tween the MW, the LMC, and Sgr, we used a symplecticleapfrog integration scheme (Springel et al. 2001). The host and the two satellites are represented with analytic potentials;the center of each one follows the orbit that results from theacceleration of the other two. In all cases the orbits are inte-grated backwards in time from their present-day positions andvelocities. For simplicity, in this section we model the darkmatter halo of all galaxies with Hernquist profiles. This al-lows us to model the dynamical friction that each of our threegalaxies (LMC, Sgr, and MW) induces on the remaining twoby approximating the integral in Equation 1 as follows: Z v v f ( v )d v ≈ (cid:18) erf( x ) - x √ π e - x (cid:19) , (7)where x = 2 v p r / ( M G ). Here, r is the scale radius of thegalaxy causing the friction. The Coulomb factor is computedas Λ = rv GM ( r ) , (8)where r is the distance between the centers of the two galax-ies and M ( r ) is the mass of the galaxy being decelerated en-closed within r (Carpintero et al. 2013). Galactic models
We model the MW as a three component system. The maindifference with the MW model presented in Section 3.1.2 isthe profile of its DM halo. In this Section, all DM halos followa Hernquist profile (Eqn. 4). The parameters that describe ourMW models are listed in Table 1. The scale radius of theHernquist profile DM halos, r H , are obtained from the NFWscale radii listed in Table 1 following Springel et al. (2005), r H = r s r (cid:16) log(1 + c ) - c + c (cid:17) . (9)To smoothly model Sgr, we also choose a Hernquist pro-file. The model parameters are based on those presented byPurcell et al. (2011, hereafter P11). Our single component0 Gómez et al. −20 0 20 40−40−2002040 X [kpc] Z [ kp c ] LMC = 1.8 x 10 MW = 1.0 x 10 Θ fixapo Θ freeapo Θ fr e ea p o < Θ fixa p o MW free + Sgr + LMCMW fixed + Sgr + LMC F IG . 6.— Angular distance between two consecutive apocenters obtainedfrom Sgr-like orbits integrated in a free (green) and a fixed (red) MW model.In both cases, a model for the LMC is also included in the simulations. Forclarity, orbits are shown only during the first Gyr of backwards evolution.Note that allowing the MW model to react to the pull of its satellites, espe-cially the LMC, results in a significant decrease of Θ apo . model consists of a DM halo with a mass, prior to cross-ing the MW virial radius ( R vir ), of M Sgr = 10 M ⊙ . Asdescribed by P11, this large value of M Sgr prior to infallis obtained from a cosmological abundance matching argu-ment (Conroy & Wechsler 2009; Behroozi et al. 2010), basedon the present-day luminosity of the Sgr core and tidal de-bris. Lower mass models of Sgr are presented in Section 5.Note however that, as in P11 and references therein, thesatellite is initially launched 2 Gyr ago at a distance of 80kpc from the Galactic centre, traveling vertically at 80 kms - toward the North Galactic Pole. Thus its mass, at thispoint in time, is truncated at the instantaneous Jacobi radius r J ≈
30 kpc. This leaves a total bound mass (2 Gyr ago) of M ≈ . × M ⊙ , i.e., a factor of ∼ M vir . The scale length of theprofile is R Sgr = 13 kpc. In order to crudely account for Sgrgalaxy’s mass loss due to tidal interaction with the MW po-tential, we assume that its mass linearly varies during the 2Gyr of evolution between M and M Sgr = 10 M ⊙ (Law et al.2005; Purcell et al. 2011). A Hernquist profile is also usedto model the LMC. The parameters that specify each of ourLMC models are listed in Table 2. An analytic treatment of the MW + LMC + Sgr system
In this Section we characterize the significance of the per-turbative effects associated with a first-infall LMC on the or-bit of Sgr. In all experiments, the orbits of our three galaxiesare integrated backward in time for 2 Gyr from their present-day positions. To be consistent with the full N -body integra-tions that we analyze in Section 5, present-day initial con-ditions for our Sgr-like orbits are obtained as follows. Asdiscussed in Section 4.2.2, we first integrate our Sgr modelforward in time for 2 Gyr in a free MW model. The ini-tial conditions at this initial time are ( X , Y , Z ) Sgr = (80 , , Note that the equivalent NFW scale radius is 6.5 kpc (see Eq. 9). kpc and ( v x , v y , v z ) Sgr = (0 , ,
80) km/s. Note that, onlyin this first step, we neglect Sgr’s mass loss due to the tidalinteraction with the MW potential. The position and veloc-ity of the Sgr model at the final integration point (i.e., af-ter 2 Gyr of evolution), ( X , Y , Z ) Sgr = (0 , , -
3) kpc and( v x , v y , v z ) Sgr = (413 , , -
46) km/s, are then used as present-day initial conditions for the backward integration. As an ex-ample, we show in Figure 4 the resulting Sgr-like and LMCorbits obtained in a free MW model with a dark matter halo of10 M ⊙ . For this integration an LMC model with total mass M LMC = 1 . × M ⊙ was considered.In Figure 5, we compare the resulting backward integratedSgr-like orbits obtained with free and fixed MW models, withand without the LMC. The blue line in the left panel shows thebackward time integrated Galactocentric orbit of our Sgr-likesatellite in a MW model with a dark matter halo of 10 M ⊙ .The solid and dashed lines indicate the first and second Gyrof evolution, respectively. In this orbital integration, the MWcenter of mass is allowed to react to any external potential.However, the initial mass of Sgr, 10 M ⊙ , is very small andthus the orbital barycenter is approximately located at theMW center of mass.The green line in the same panel shows the orbit of Sgr ina Galactocentric reference frame, now including a model forthe LMC. As before, we allow the MW center of mass to reactto the pull of any external potential. For this experiment wehave chosen a LMC model with a total mass of M LMC = 1 . × M ⊙ that is experiencing its first pericentric passage at thepresent-day (see Section 3 and Table 2). This LMC modelrepresents the canonical model described by K13. Such amassive LMC is required to keep the LMC-SMC binary con-figuration for longer than 2 Gyr in MW models with masses ≤ . × M ⊙ (Gnedin et al. 2010; Boylan-Kolchin et al.2013; Kallivayalil et al. 2013; Piffl et al. 2014). It has alsobeen used in the past to successfully reproduce many of theobservable properties of the Magellanic stream (Besla et al.2012). A comparison of the blue and green lines shows thatthe perturbation on the Sgr orbit due to the LMC’s gravita-tional pull is indeed significant, as first suggested by VCH13.Note, however, that in our case we are considering a first in-fall scenario for the LMC, and so its perturbative effects haveonly operated over the past ∼ Θ apo . As shown in Figure 6, allowing the MW model to re-act to the pull of its satellites, especially the LMC, resultsin a significant decrease of Θ apo . A comparison between theSgr-like orbits in the “free” and the “fixed” MW models thatinclude the LMC yields a ∆Θ apo ≈ ◦ . Instead, as can beseen in the left panel of Figure 5, a comparison between theSgr orbits obtained in free MW models with (green line) andwithout the LMC (blue line) yields a smaller but still notice-nd yet it moves 11 −20 0 20 40−40−2002040 X [kpc] Z [ kp c ] LMC = 3 x 10 MW free + SgrMW free + Sgr + LMCMW fixed + Sgr + LMC −20 0 20 40
X [kpc]
LMC = 5 x 10 −20 0 20 40 X [kpc]
LMC = 8 x 10 −20 0 20 40 X [kpc]
LMC = 1 x 10 F IG . 7.— As in Figure 5, but for LMC models with different total masses. In each panel, the mass of the corresponding LMC model is indicated in the bottomright. In all cases, we consider a MW potential with mass M vir = 1 × M ⊙ . Note that perturbations in the Sgr-like orbits are noticeable in all simulations thatinclude LMC models with masses > × M ⊙ . able ∆Θ apo ≈ ◦ . Belokurov et al. (2014, hereafter B14) findsa Θ apo between the apocenters of the Sgr leading and trail-ing arms that is ∼ ◦ smaller than what is predicted for Sgrorbits in logarithmic fixed halos. Thus, taking into accounta free MW and a model of the LMC could at least partiallyexplain this observed smaller-than-predicted angular distancebetween the consecutive apocenters. Note that the magnitudeof ∆Θ apo strongly depends on the initial orbital conditions ofSgr, as well as on the mass of the three galaxies involved.The middle and right panels of Figure 5 show the same ex-periments, now in MW models with 1 . × M ⊙ ,respectively. Even in these more massive MW models theperturbation to the orbit of Sgr associated with fixing the MWcenter of mass is very significant, and again larger than thatobtained by the inclusion of the LMC torques on the Sgr orbitalone.In Figure 7 we explore the Sgr orbital history includingLMC models of different total masses. In all cases, the MWmodel contains a dark matter halo of 1 × M ⊙ . The leftpanel shows the results obtained with our least massive LMCmodel, M LMC = 3 × M ⊙ . In this case, perturbations to theorbit of Sgr are almost negligible, regardless of whether weinclude a model for the LMC or consider a free MW. How-ever, as we increase the mass of the LMC, the perturbationon the orbit of Sgr quickly becomes noticeable. For a LMCwith M LMC = 8 × M ⊙ (i.e., the mass used by VCH13), theperturbation is very clear. As before, the largest changes inthe orbital path of Sgr are obtained when we allow the MW toreact to the external gravitational potential of the LMC. N-BODY MODELS OF THE PHASE-SPACE DISTRIBUTION OFSGR-LIKE TIDAL DEBRIS
Thus far, we have explored the effects of allowing the MW’scenter of mass to respond to perturbations from its satelliteson the history of the Sgr dwarf galaxy’s orbit. However, suchanalytic arguments are insufficient to explore the significanceof such perturbations on the phase-space distribution of theSgr stellar stream. To explore this, we run a new set of ex-periments, now based on full N -body numerical simulationsfollowing the evolution of Sgr forward in time. We describethe models and methodology in Section 5.1, and present ourresults in Section 5.2. N-body models
Methodology
TABLE 3S
UMMARY OF THE SET - UP FOR THE N - BODY SIMULATIONS ANALYZED IN S ECTION Host
DM halo N part = 2 . × Virial mass 1 × [ M ⊙ ]Scale radius 26 . N part = 3 × Mass 6 . × [ M ⊙ ]Scale length 3.5 [kpc]Scale height 0.53 [kpc]Stellar bulge N part = 5 × Mass 1 × [ M ⊙ ]Scale radius 0.7 [kpc] Sgr Satellites
DM halo
Light Heavy N part = 2 . × Virial mass 0 . × × [ M ⊙ ]Scale radius 4 . . N part = 5 × Mass 6 . × . × [ M ⊙ ]Scale radius 0 .
85 0 .
85 [kpc]
LMC Satellite
Single spheroid N part = 2 × Mass 1 . × [ M ⊙ ]Scale radius 20 [kpc] The N -body systems are evolved using GADGET-2.0(Springel 2005), a well-documented, massively parallel Tree-SPH code. To construct self-consistent stable models of theMW, the LMC, and Sgr, we follow the procedure describedby Villalobos & Helmi (2008). In the following sections wedescribe the main properties of each galactic model. In gen-eral, the force softening is chosen to be a tenth of the meaninterparticle distance of each system, calculated using parti-cles located within a distance of ten scale length radii. In par-ticular, when dealing with Plummer models we compute oursoftening lengths as described by Athanassoula et al. (2000).Keeping a MW model fixed in a N -body simulation is sig-nificantly more challenging than in simulations with analytic,smooth galactic models. Having illustrated in Section 4.3 thatperturbations on the orbit and debris of Sgr when fixing theMW are quite significant, in what follows we will only con-2 Gómez et al. F IG . 8.— Present-day distribution of the simulated Sgr stream and core in different projections of phase-space. These distributions are obtained from thesimulations with the Heavy Sgr model. The black circle indicates the current location of the simulated Sgr remnant core. From top to bottom we show the stellarparticle distribution projected in right ascension versus declination, right ascension versus line-of-sight velocity with respect to the Galactic standard-of-rest(V los ), and right ascension versus heliocentric distance. The star particles are color coded according to the quantity indicated in the color bars. The panels onthe left show the results obtained after simulating the MW-Sgr interaction in isolation. The black squares show data from 2MASS M-giant stars (Majewski et al.2004). The panels on the right show the results obtained after including in the simulation a LMC model with total mass M LMC = 1 . × M ⊙ . Note thatsignificant perturbations to the phase-space distribution of Sgr debris are induced by the LMC. These perturbation are the result of both the torque exerted by theLMC on Sgr and the response of the MW to the LMC’s gravitational pull. sider free MW models with and without the presence of amassive LMC. Note that, due to the relatively low mass ofSgr, the pertubative effects associated with the phase-spacedisplacement of the MW center of mass in previous studiesthat have only considered the interaction between the MWand Sgr (i.e. disregarding the LMC) are negligible. Galactic models
We model the MW-like host as a self-consistent three-component system containing a NFW dark matter halo, anexponential stellar disk, and a central bulge following a Hern-quist profile. The dark matter halo has a total mass of M vir =10 M ⊙ , a scale radius r s = 26 .
47 kpc, and is initially adi-abatically contracted to model its response to the formationof a stellar disk in its central region (Blumenthal et al. 1986;Mo et al. 1998). The exponential disk has a total mass of6 . × M ⊙ and a scale length and height of 3.5 and 0.53kpc, respectively. For the bulge we assume a mass of 10 M ⊙ and a scale radius of 0.7 kpc. As previously discussed, the cir-cular velocity profile takes a value of ∼
239 km s - at ∼ . M LMC = 1 . × M ⊙ and a scale radius r LMC = 20kpc. Since the mass spreads out to infinity in Plummer mod-els, the density profile is initially truncated at the radius thatencloses 95% of the LMC’s total mass.Based on Purcell et al. (2011), the Sgr progenitor is self-consistently initialized with a NFW dark matter halo and aspheroidal stellar component that follows a Hernquist profile.Two different dark matter mass models are considered:1. a “Light” model with a DM halo mass of 10 . M ⊙ ;2. a “Heavy” model with a DM halo mass of 10 M ⊙ .The stellar components in both models have a totalmass of 6 . × M ⊙ and a scale radius of 0 .
85 kpc(Niederste-Ostholt et al. 2012). As discussed in Section 3.1.2(see also P11), the Sgr-like satellites are launched at a Galac-tocentric distance of 80 kpc from the Galactic centre in theplane of the MW disk, traveling vertically at 80 km s - to-wards the North Galactic Pole. To account for the mass lossthat would have occurred between the crossing of the MW’svirial radius and this “initial” location, the Sgr progenitorNFW mass profiles are initially truncated at the correspond-ing instantaneous Jacobi radius, r J ≈
30 and 23 kpc for theHeavy and Light Sgr, respectively.nd yet it moves 13Table 3 summarizes the values of all of the parameters thatdescribe our N -body models. An N-body treatment of the MW + LMC + Sgr system
The N -body models discussed in this section will build onthe previously-discussed analytic models to explore the im-pact of LMC perturbations in a first infall scenario on theSgr tidal debris. To this end, we compare simulations inwhich the LMC is included against others in which it is not.In what follows, all simulations include fully self-consistentthree-component MW models that are allowed to respond tothe gravitational pull of any external source. As described inSection 5.1.2 we consider Heavy and Light Sgr models, bothself-consistently initialized with spherical baryonic and darkmatter components. The Heavy and the Light Sgr are fol-lowed for ∼ . . M LMC = 1 . × M ⊙ . The initial conditions forthe LMC models were obtained by backward time integrationfrom its present-day location in a “free” MW scenario (seeSection 3) until 2.1 or 2.6 Gyrs ago, depending on the Sgrmodel. Note that, due to the slight overestimation of the roleof dynamical friction in the analytic calculations (see discus-sion in Section 3.2), the LMC initial conditions (ICs) were it-eratively calibrated by comparing the resulting LMC N -bodyorbits with their analytic counterpart. The goal of this exercisewas to obtain a set of ICs for the N -body simulations that, atpresent day, yields the correct phase-space coordinates. TheSgr dwarf galaxy is launched with the same initial conditionsas in the “LMC-less” simulations. Clearly, the addition of theLMC results in significant perturbations in the phase-spacedistribution of Sgr debris.The top panels of Figures 8 and 9 show the simulatedpresent-day Sgr stream and the remnant core projected in rightascension (RA) and declination. A comparison between sim-ulations with and without the LMC model reveals an interest-ing feature at RA ≈ ◦ . When the LMC is accounted for,tidal debris that otherwise would overlap when projected onthe sky are split into two distinguishable arms. This is true forboth Sgr models, suggesting that tidal material could also bedeposited in regions of the sky that are not delineated by thepresent-day Sgr orbital plane. Note that the two arms showboth opposite heliocentric distance gradients as a function ofRA (see bottom right panel in Figures 8 and 9) and oppositeline-of-sight velocities.The middle panels of Figures 8 and 9 show the projectedSgr distribution in RA versus Galactic standard-of-rest lineof sight velocity ( V los ) space. A quick comparison between the left and right panel reveals very significant changes inthe distribution of V los . In general, adding the LMC resultsin a much broader distribution at all RA. This can be moreclearly seen on Figure 10 where, as an example, we show the V los distributions of Heavy Sgr star particles located within250 ◦ < RA < ◦ . Recall that, as shown in Figure 7, theseperturbations are not just the result of the LMC torque on Sgr,but are also due to the self-consistent response of the MWto the LMC’s gravitational pull. The phase-space distributionof the Sgr debris obtained when the LMC is included in thesimulation results in a worse fit to the Majewski et al. (2004)data. However, in this work we have not attempted to find aset of initial conditions that could fit the Sgr debris in a sce-nario in which the LMC is included. Starting with differentinitial conditions for the Sgr orbit or a lower LMC mass couldplausibly bring the velocities into better agreement. Instead,our goal is simply to explore what perturbations are inducedand whether they are significant.Perturbations to the Sgr debris phase-space distribution canalso be observed in the bottom panels of Figures 8 and 9,where we show the projection onto RA versus heliocentricdistance space. It is clear that the addition of the LMC re-sulted in a significant spatial redistribution of Sgr debris. Notethat, independent of whether the LMC is included or not, starparticles in the leading and trailing arm can reach distancesof ∼
50 kpc (at RA ≈ ◦ ) and ∼
100 kpc (RA ≈ ◦ ), respectively (also, see Figure S4 from P11). These differentdistances are similar to the leading and trailing tail’s apoc-entric distances of the Sgr stream, as traced by B14. Theyfind R apolead ≈
48 kpc and R apotrail ≈
102 kpc, respectively. Thedifferent apocentric distances reached by the star particles inour simulations in the leading and trailing arms are merely aconsequence of considering a self-gravitating Sgr model (seeChoi et al. 2007; Gibbons et al. 2014).It is also interesting to explore whether the self-consistentaddition of the LMC could at least partially explain the ≈ ◦ difference between the mean orbital poles of the greatcircles associated with the debris leading and trailing Sgr(Johnston et al. 2005). In Figure 11 we show the time evolu-tion of the Heavy Sgr orbital angular momentum orientation, ˆ L . Since Sgr is launched in the X-Z plane, its angular mo-mentum initially points in the ˆ Y direction. The red line showsthe angular displacement of ˆ L with respect to ˆ Y in the LMC-less scenario. As expected from a polar orbit in an axisym-metric potential, the orientation of the angular momentum re-mains nearly constant and close to 0 ◦ at all times. The blackline shows the result obtained after adding the LMC model.Clearly, as the LMC approaches the MW, the Sgr orbital planestarts to tilt with respect to its initial orientation. This tiltingtakes place during the last 0.5 Gyr of evolution, in good agree-ment with the results shown in Figure 3. At present-day, theangular displacement of ˆ L is of approximately 9 ◦ , similar tothe value reported by Johnston et al. (2005). DISCUSSION AND CONCLUSIONS
In this work we have performed and analyzed a set of nu-merical simulations using smooth and N -body gravitationalpotentials. Our goal was to explore whether the approxima-tion of an inertial Galactocentric reference frame holds in thepresence of a relatively massive LMC that is experiencing itsfirst infall towards the MW. In a nutshell, if the LMC currently Similar results are obtained in a Galactocentric reference frame F IG . 9.— As in Figure 8, for the simulation with the Light Sgr model. Note again the significant perturbations to the phase-space distribution of Sgr debrisinduced by the LMC. −500 −250 0 250 50010 Vlos [km/s] N u m b e r c oun t F IG . 10.— Distribution of lines-of-sight-velocities, V los , of Heavy Sgr starparticles located within 250 ◦ < RA < ◦ . The red line shows the resultsobtained in a self-consistent N -body simulation of the interaction betweenthe MW and Sgr. The black line shows the results obtained when a model ofthe LMC is added to the simulation. has a total mass of at least 5 × M ⊙ , the answer is likelyto be no.To arrive to this conclusion, we have focused our effortson two possible situations where artificially fixing the MWcenter of mass could have a significant effect. Our first ob-vious choice was to explore the implications on the orbitcalculations of the LMC about the MW. Our results clearlyshow that the LMC’s orbital period and apocentric distanceare significantly shortened if we allow the MW to react tothe LMC’s gravitational pull. As the mass of the LMC be- Time [Gyr] O r b it a l p l a n e tilt [ d e g ] MW free + SgrMW free + Sgr + LMC F IG . 11.— Time evolution of the Heavy Sgr orbital angular momentumorientation with respect to its initial direction. The red line shows the resultsobtained in a self-consistent N -body simulation of the interaction betweenthe MW and the Sgr dwarf galaxy. The black line shows the results obtainedwhen a model of the LMC is self-consistently added to the simulation. comes larger, and thus more comparable to the MW mass en-closed within the LMC’s location, the two-body interactionbecomes more relevant. Thus, the more massive the LMC,the larger the changes on its orbital periods. The change inthe inferred orbital properties of our LMC-like models sug-gest that, even though a first-infall is still a very plausible sce-nario, the limiting LMC-MW mass combinations that couldhost a first-infalling LMC are noticeably affected; it raisesthe required minimum LMC mass and disfavors MW modelsnd yet it moves 15with M vir ≥ . × M ⊙ . A detailed dynamical analysis, in-cluding N -body models that can naturally account for the twobody interaction, the LMC’s tidal mass loss and dynamicalfriction, and a model for the time evolution of the MW po-tential, would be required to robustly characterize the orbitalhistory of the different LMC mass models. This is beyond thescope of the work presented in this paper.We have also characterized how the MW itself respondsto the gravitational pull of the LMC. We find that significantchanges in both the position and velocity of the MW center ofmass takes place only during the last 0 . - . M LMC = 1 . × M ⊙ , the orbitalbarycenter is located at ≈
14 kpc from the MW center of massat the present day. For this LMC model, the MW was dis-placed by ≈
30 kpc and its velocity changed by ≈
75 km/s inthis very short amount of time. Note that similar results wereobtained in simulations of the collision between Andromedaand its satellite galaxy M32 (Dierickx et al. 2014).Due to the extended nature of the MW stellar halo, not allstars are accelerated at the same rate by a massive satellite. Itis thus likely that this differential acceleration will have im-portant effects on the observable properties of extended stel-lar streams. For example, the distribution of Sgr debris, whichcovers a radial extension of ∼
100 kpc, could be significantlyperturbed due to the phase-space displacement of the MilkyWay center of mass, in addition to the perturbations associ-ated with the LMC torque (e.g. VCH13). To explore this, weintegrated Sgr-like orbits in MW potentials that are allowedto freely react to the gravitational pull of the LMC. We wouldlike to stress that in this work we have not attempted to finda set of ICs for the Sgr progenitor that could fit the Sgr de-bris in a scenario in which the LMC is included. The com-plexity behind the search for best-fitting ICs in a three-bodyproblem scenario using fully self-consistent models is beyondthe scope of this work. Instead, our goal is simply to explorewhether or not such perturbations are significant.Our analysis showed that, indeed, the presence of the LMCintroduced significant perturbations on Sgr-like orbits andtheir associated distribution of debris. We have confirmed pre-vious results presented by VCH13, where they showed thatthe torque on Sgr exerted by the LMC can introduce non-negligible perturbations on its orbit and distribution of debris.However, we find that the differences between the Sgr-likeorbits obtained in “free” and a “fixed” MW + LMC modelsare even larger that those obtained in “free” models with orwithout the LMC. Furthermore, we have shown that this per-turbation is significant even in the scenario where the LMC isundergoing its first pericenter passage. Attempts to reproducethe Sgr stream without a model for the LMC perturbation willthus force searches for the best-fitting parameters that charac-terize the MW gravitational potential to artificially adjust inorder to account for this perturbation.An example is the discrepancy discussed byBelokurov et al. (2014, B14) in the angular distance be-tween the inferred apocenters of the Sgr leading and trailingarms. Observations suggest that this angular distance issmaller than what is predicted in a fixed logarithmic poten-tial. Our analysis showed that the differences in the angulardistances between the last two Sgr’s orbital apocenters could at least be partially accounted for with both a free MW modeland the inclusion of the LMC.Another example is the ≈ ◦ difference between the meanorbital poles of the great circles associated with the debrisleading and trailing the Sgr core, reported by Johnston et al.(2005). We find in our simulations that, due to the gravita-tional pull exerted by the LMC, the Sgr orbital plane tilts withrespect to its initial orientation by ≈ ◦ during the last 0.5 Gyrof evolution. These are just two examples of peculiar charac-teristics of the Sgr debris that could be naturally and, at least,partially accounted for if a fully self-consistent model of theMW + LMC + Sgr interaction is considered.Interestingly, these results were obtained without the needfor a prolate/oblate model of the Galactic DM halo. To ac-curately quantify the significance of these perturbations, fullyself-consistent models of the MW + LMC + Sgr interactivesystem are required. Note that, for each combination of galac-tic models, a specially tailored set of initial orbital conditionsfor the LMC and Sgr will be required. We defer this analysisto a follow-up work.The orbit of the LMC about the MW and the orbital his-tory and phase-space distribution of Sgr debris are just twoexamples where perturbations induced by the MW + LMCinteraction could be significant. The inferred orbital prop-erties of other MW dwarfs, such as Carina, Fornax, Sculp-tor and Ursa Minor, obtained using present-day phase-spacecoordinates, could also be affected by such interaction if theLMC is massive enough (e.g. Pasetto et al. 2011; Angus et al.2011). Furthermore, using HST proper-motion measure-ments, van der Marel et al. (2012) estimated a radial velocityof M31 with respect to the MW of V rad , M31 = - . ± . V tan , M31 = 17 . σ confidence region V tan , M31 ≤ . . × M ⊙ , the velocityof the MW center of mass could have changed by as muchas 75 km/s in less than 0.5 Gyr. Decomposing this velocityinto a tangential and radial components toward M31 yields V rad , MW ≈
37 km/s and V tan , MW ≈
66 km/s. This suggests thatestimates of the Local Group mass based on timing argumentscould be affected by such a two-body interaction. In addi-tion, a significant fraction of the present-day relative velocityof M31 with respect to the Galactic centre could be associ-ated to the temporary Galactic displacement about its orbitalbarycenter, thus affecting the projected evolution of the MW+ M31 system.We are on the verge of the so-called
Gaia era. In additionto the very accurate phase-space catalogs that we are alreadymining,
Gaia is starting to collect phase-space information formany millions of stars. The high-quality data that will soonbecome available clearly calls for the development of modelsthat are as detailed as possible, and which include all knownsources of significant interactions. The results presented inthis work suggest that, if the LMC is as massive as suggestedby recent studies, to properly interpret this data it is essentialto consider in the analyses self-consistent MW + LMC modelsthat are allowed to freely react to their mutual gravitationalinteractions.FAG and BWO are supported through the NSF Office ofCyberinfrastructure by grant PHY-0941373 and by the Michi-gan State University Institute for Cyber-Enabled Research(iCER). BWO was supported in part by NSF grant PHY 08-22648: Physics Frontiers Center/Joint Institute for Nuclear6 Gómez et al.Astrophysics (JINA). DDC is supported by the UniversidadNacional de La Plata, Argentina, and the Instituto de As-trofísica de La Plata, UNLP-Conicet, Argentina. G.B. ac- knowledges support from NASA through Hubble Fellowshipgrant HST-HF-51284.01-A.is starting to collect phase-space information formany millions of stars. The high-quality data that will soonbecome available clearly calls for the development of modelsthat are as detailed as possible, and which include all knownsources of significant interactions. The results presented inthis work suggest that, if the LMC is as massive as suggestedby recent studies, to properly interpret this data it is essentialto consider in the analyses self-consistent MW + LMC modelsthat are allowed to freely react to their mutual gravitationalinteractions.FAG and BWO are supported through the NSF Office ofCyberinfrastructure by grant PHY-0941373 and by the Michi-gan State University Institute for Cyber-Enabled Research(iCER). BWO was supported in part by NSF grant PHY 08-22648: Physics Frontiers Center/Joint Institute for Nuclear6 Gómez et al.Astrophysics (JINA). DDC is supported by the UniversidadNacional de La Plata, Argentina, and the Instituto de As-trofísica de La Plata, UNLP-Conicet, Argentina. G.B. ac- knowledges support from NASA through Hubble Fellowshipgrant HST-HF-51284.01-A.