Mon. Not. R. Astron. Soc.000, 1–13 (2011) Printed 5 April 2011 (MN LATEX style file v2.2)

Radial Mixing in Galactic Disks: The Effects of Disk Structure andSatellite Bombardment

Jonathan C. Bird⋆, Stelios Kazantzidis, & David H. WeinbergDepartment of Astronomy and the Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210

5 April 2011

ABSTRACTWe use a suite of numerical simulations to investigate the mechanisms and effects of radialmigration of stars in disk galaxies like the Milky Way (MW). Anisolated, collisionless stellardisk with a MW-like scale-height shows only the radial “blurring” expected from epicyclic or-bits. Reducing the disk thickness or adding gas to the disk substantially increases the level ofradial migration, induced by interaction with transient spiral arms and/or a central bar. We alsoexamine collisionless disks subjected to gravitational perturbations from a cosmologicallymotivated satellite accretion history. In the perturbed disk that best reproduces the observedproperties of the MW, 20% of stars that end up in the solar annulus 7 kpc < R < 9 kpcstarted atR < 6 kpc, and 7% started atR > 10 kpc. This level of migration would addconsiderable dispersion to the age-metallicity relation of solar neighborhood stars. In the iso-lated disk models, the probability of migration traces the disk’s radial mass profile, but inperturbed disks migration occurs preferentially at large radii, where the disk is more weaklybound. The orbital dynamics of migrating particles are alsodifferent in isolated and perturbeddisks: satellite perturbations drive particles to lower angular momentum for a given changein radius. Thus, satellite perturbations appear to be a distinct mechanism for inducing radialmigration, which can operate in concert with migration induced by bars and spiral structure.We investigate correlations between changes in radius and changes in orbital circularity orvertical energy, identifying signatures that might be usedto test models and distinguish radialmigration mechanisms in chemo-dynamical surveys of the MW disk.

Key words: methods: numerical; Galaxy: kinematics and dynamics; Galaxy: disc; galaxies:formation; galaxies:evolution

1 INTRODUCTION

Under the influence of gravity, disk galaxies are expected to assem-ble in an “inside-out” fashion: stars form first from high-densitygas in the central region of the galaxy where the potential is deep-est, and subsequently at increasing galacto-centric radii (e.g. Lar-son 1976). An immediate consequence of this formation scenario isthat stars born at the same time and in the same region of a galaxyshould have similar chemical compositions. However, modern ob-servations of stars in our Galaxy have challenged this tenet of tra-ditional galaxy formation theory. Most immediately, our own Sun’scomposition is substantially more metal rich than that of nearbysolar age stars (Wielen et al. 1996), and while the local interstellarmedium (ISM) is close to solar metallicity today, chemical evo-lution models predict a lower ISM metallicity five Gyr ago whenthe sun formed. More generally, the age-metallicity relationships(AMRs) of field and solar neighborhood stars are characterized byhigher dispersions than expected (Edvardsson et al. 1993; Nord-strom et al. 2004). In addition, simple chemical evolution models

⋆ E-mail:bird@astronomy.ohio-state.edu

predict many more low metallicity G-dwarfs in our region of thedisk compared to those observed, a discrepancy known as “the lo-cal G-dwarf problem” (van den Bergh 1962; Schmidt 1963). Evi-dence seemingly in contradiction to standard galaxy chemical evo-lution theory is not limited to our own Galaxy. Metallicity gradientsin disk galaxies are shallower than predicted by classical models(e.g., Magrini et al. 2007). Ferguson & Johnson (2001) and Fergu-son et al. (2007) find unexpectedly old stellar populations on nearlycircular orbits in the outskirts of M31 and M33, respectively. Theseperplexing observations cannot be readily explained within the con-fines of classic galaxy formation models.

Wielen et al. (1996) suggested that the Sun must have mi-grated to its current location from an initial position in the innerGalaxy; by extension, present day radii of many stars could be sig-nificantly different from their birth radii. One difficulty in estab-lishing radial migration as a common phenomenon lies in finding adynamical mechanism that can cause a substantial fraction of starsto migrate several kiloparsecs while retaining the observed approxi-mately circular orbits. To this end, Sellwood & Binney (2002, here-after SB02) proposed that stars could radially scatter when they arein corotational resonance with transient spiral waves. Their calcu-

2 Bird, Kazantzidis, & Weinberg

lations showed that this process could produce large stellar excur-sions while maintaining near-circular orbits in disks, but SB02 didnot estimate the overall efficiency of the process. In the present pa-per, we examine radial migration in simulations of disk galaxies.Our experiments include galactic disks evolved both in isolationand under the action of infalling satellites of the type expected inthe currently favored cold dark matter (CDM) paradigm of hier-archical structure formation (e.g., Peebles 1982; Blumenthal et al.1984). The latter set of experiments were presented in the studiesof Kazantzidis et al. (2008, hereafter K08) and Kazantzidis et al.(2009) and were utilized to investigate thegeneric dynamical andmorphological signatures of galactic disks subject to bombardmentby CDM substructure.

Inspired by SB02, several groups have recently investigatedthe potential role of radial mixing in the chemical and dynam-ical evolution of disk galaxies. Schonrich & Binney (2009) pre-sented the first chemical evolution model to incorporate radial mi-gration. The rate at which stars migrate via the SB02 mechanism isleft as a free parameter constrained by the metallicity distributionfunction (MDF) of solar neighborhood stars in the Geneva Copen-hagen Survey (GCS; Nordstrom et al. 2004). Their model suc-cessfully reproduced, within systematic uncertainties, the observedage-metallicity distribution of stars in the GCS (Holmberg et al.2007) and the observed correlation between tangential velocity andabundance pattern described by Haywood (2008). However, there ispartial degeneracy between the magnitude of radial migration andother parameters in the model such as star-formation rates and gasinflow characteristics. Furthermore, it is unclear whether the levelof migration required to fit the data is consistent with theoreticalexpectations .

Numerical simulations have confirmed the occurrence of ra-dial migration under a variety of conditions. Roskar et al. (2008a,b)studied the migration of stars in a simulation of an isolatedMilky Way (MW)-sized stellar disk formed from the cooling ofa pressure-supported gas cloud in a1012M⊙ dark matter halo. Intheir simulations, some older stars radially migrated to the outskirtsof the disk while maintaining nearly circular orbits, forming a pop-ulation akin to that observed in M31 and M33 (Ferguson & John-son 2001; Ferguson et al. 2007). Roskar et al. (2008b) found that∼ 50% of all stars in the solar neighborhood were not bornin situ;this is a natural explanation for the observed dispersion in the AMRand solar neighborhood metallicity distribution function (MDF).

More recently, Quillen et al. (2009) investigated radial migra-tion in a stellar disk perturbed by a low-mass (∼ 5 × 109 M⊙)orbiting satellite. Their numerical simulations integrated test par-ticle orbits in a static galactic potential and highlighted the factthat mergers and perturbations from satellite galaxies and subhaloscan induce stellar radial mixing. Although informative, test particlesimulations in a static isothermal potential may not capture all therelevant physics of the process of stellar radial migration in diskgalaxies. In our paper, we expand on the analysis of Quillen et al.(2009) by investigating radial migration using fully self-consistentnumerical simulations both with and without satellite bombard-ment.

There are now three established mechanisms by which starsmove away from their birth radii. The random motions of stars andtheir local encounters with structures such as molecular clouds dy-namically heat the disk and increase their epicyclic energy withtime (e.g. Spitzer & Schwarzschild 1953; Binney & Tremaine2008). While maintaining the same guiding center and angular mo-mentum, these stars can be spread over the galacto-centric radialrange defined by their pericenter and apocenter, “blurring” the disk.

The SB02 mechanism, in contrast, actually shifts the guiding cen-ter of stellar orbits. Stars scattered at corotational resonance withspiral waves can drift several kiloparsecs away from their initialradii while staying on nearly circular orbits. For any single spiralwave, SB02 predict that stars are scattered on each side of the coro-tational resonance, “churning” the contents of the disk1. Stars mayundergo several encounters with transient spiral waves throughouttheir lifetimes. The third pathway for radial migration is gravi-tational perturbations from orbiting satellites as discussed above.However, there is neither detailed theory describing the physics ofthis process nor predictions of the resulting stellar dynamics. In thiswork, we aim to characterize this radial migration process and com-pare its effect on the stellar disk with those of the other establishedmechanisms of radial migration.

Our investigation complements earlier and ongoing radial mi-gration studies. We perform a simulation campaign, including nu-merical experiments of isolated disk galaxies with different scaleheights and gas fractions, which in turn lead to different levels ofspiral structure. For the first time, we examine the effect of satel-lite bombardment on radial migration utilizing simulations wheregalactic disks are subjected to a cosmologically motivated satelliteaccretion history. Via a comparative approach, we determine howthe magnitude and efficiency of radial migration depend on inputphysics, establish correlations between orbital parameters and mi-gration, and present evidence that each of the three migration mech-anisms is distinct in the examined parameter space. These charac-teristics lead to possible observational signatures that may constrainthe relative importance of each migration mechanism in the MilkyWay.

2 METHODS

2.1 Isolated Disk Models

We employ the method of Widrow & Dubinski (2005) to con-struct numerical realizations of self-consistent, multicomponentdisk galaxies. These galaxy models consist of an exponential stel-lar disk, a Hernquist bulge (Hernquist 1990), and a Navarro et al.(1996, hereafter NFW) dark matter halo. They are characterized by15 free parameters that may be tuned to fit a wide range of ob-servational data for actual galaxies including the MW and M31.The Widrow & Dubinski (2005) models are derived from three-integral, composite distribution functions and thus represent self-consistent equilibrium solutions to the coupled Poisson and col-lisionless Boltzmann equations. Owing to their self-consistency,these galaxy models are ideally suited for investigating the complexdynamics involved in the process of radial mixing. The Widrow& Dubinski (2005) method has been recently used in a variety ofnumerical studies associated with instabilities in disk galaxies, in-cluding the dynamics of warps and bars (Dubinski & Chakrabarty2009; Dubinski et al. 2009), the gravitational interaction betweengalactic disks and infalling satellites (Gauthier et al. 2006; K08;Purcell et al. 2009; Kazantzidis et al. 2009), and the transforma-tion of disky dwarfs to dwarf spheroidal galaxies under the actionof tidal forces from a massive host (Kazantzidis et al. 2011). Werefer the reader to Widrow & Dubinski (2005) for an overview ofall relevant parameters and a detailed description of this technique.

For the majority of numerical experiments in the present

1 Blurring and churning are the terms proposed by Schonrich & Binney(2009) to describe these distinct aspects of radial migration.

Radial Mixing in Galactic Disks 3

study, we employ model “MWb” in Widrow & Dubinski (2005),which satisfies a broad range of observational constraints on theMW galaxy. Specifically, the stellar disk has a mass ofMdisk =3.53 × 1010 M⊙, a radial scale length ofRd = 2.82 kpc, anda sech2 scale height ofzd = 400 pc. We note that the adoptedvalue for the scale height is consistent with that inferred for theold, thin stellar disk of the MW (e.g., Kent et al. 1991; Juric et al.2008). The equivalent exponential scale height is approximately200 pc, but the sech2 vertical distribution is more accurate. Thebulge has a mass and a scale radius ofMb = 1.18 × 1010 M⊙

andab = 0.88 kpc, respectively. The NFW dark matter halo hasa tidal radius ofRh = 244.5 kpc to keep the total mass finite(K08), a mass ofMh = 7.35 × 1011 M⊙, and a scale radius ofrh = 8.82 kpc. The total circular velocity of the galaxy modelat the solar radius,R⊙ ≃ 8 kpc, is Vc(R⊙) = 234.1 km s−1,and the Toomre disk stability parameter is equal toQ = 2.2 atR = 2.5Rd. We note that direct numerical simulations of the evo-lution of model MWb in isolation confirm its stability against barformation for10 Gyr.

We wish to address the dependence of radial mixing in iso-lated disk galaxies upon initial disk thickness and the presence ofgas in the galactic disk. Because of their stronger self-gravity, thin-ner disks yield stronger and better defined spiral structure, so wemight expect them to cause enhanced radial mixing compared totheir thicker counterparts. Similarly, because radiative cooling inthe gas damps its random velocities, the presence of gas is as-sociated with long-lived spiral structure in disks (e.g., Carlberg& Freedman 1985), which may act to increase the amount of ra-dial mixing. Correspondingly, we initialize several additional diskgalaxy models.

The first modified galaxy model was constructed with thesame parameter set as MWb but with a scale height2 times smaller(zd = 200 pc). Except for disk thickness and vertical velocity dis-persion, all of the other gross properties of the three galactic com-ponents of this model are within a few percent of the correspondingones for MWb. Not surprisingly, being characterized by a strongerdisk self-gravity, this model develops a bar inside∼ 5 kpc at timet ∼ 1.2 Gyr. The second set of modified galaxy models are thesame as MWb except for the fact that a fractionfg of the mass ofthe initial stellar disk is replaced by gas. Thus, the resulting gaseouscomponent is constructed with the same initial density distributionas the stellar disk. In the present study, we employ two values forthe gas fraction,fg = 0.2 andfg = 0.4, and the adopted methodol-ogy will be described in detail in a forthcoming paper (Kazantzidiset al. 2011, in preparation). Briefly, the construction of the gas diskis done by assuming that its vertical structure is governed by hydro-static equilibrium and by computing the potential and the resultingforce field for the radially varying density structure of the gas com-ponent. By specifying the polytropic index and the mean molecularweight of the gas and using a tree structure for the potential cal-culation, the gas azimuthal streaming velocity is determined by thebalance between gravity and centrifugal and pressure support.

2.2 Perturbed Disk Models

In the standard CDM paradigm of hierarchical structure formation,galaxies grow via continuous accretion of smaller satellite systems.To investigate the effect of accretion events on radial mixing ingalactic disks, we have also analyzedN -body simulations of thegravitational interaction between a population of dark matter sub-halos and disk galaxies. These simulations were first presented byK08, and we summarize them briefly here.

K08 performed high-resolution, collisionlessN -body simula-tions to study the response of the galactic model MWb discussedabove to a typicalΛCDM-motivated satellite accretion history. Thespecific merger history was derived from cosmological simulationsof the formation of Galaxy-sized CDM halos and involved six darkmatter satellites with masses, orbital pericenters, and tidal radiiof 7.4 × 109 . Msat/M⊙ . 2 × 1010, rperi . 20 kpc, andrtid & 20 kpc, respectively, crossing the disk in the past∼ 8 Gyr.As a comparison, the Large Magellanic Cloud has an estimatedto-tal present mass of∼ 2 × 1010 M⊙ (e.g., Schommer et al. 1992;Mastropietro et al. 2005), so the masses of the perturbing dark mat-ter satellites correspond to the upper limit of the mass function ofobserved satellites in the Local Group. K08 modeled satellite im-pacts as a sequence of encounters. Specifically, starting with thefirst subhalo, they included subsequent systems at the epoch whenthey were recorded in the cosmological simulation. We note thatalthough K08 followed the accretion histories of host halos sincez ∼ 1, when time intervals between subhalo passages were largerthan the timescale needed for the disk to relax after the previousinteraction, the next satellite was introduced immediately after thedisk had settled from the previous encounter. Each satellite was re-moved from the simulation once it reached its maximum distancefrom the disk after crossing. This approach was dictated by the de-sire to minimize computational time and resulted in a total simu-lation time of∼ 2.5 Gyr instead of∼ 8 Gyr that would formallycorrespond toz = 1.

K08 and Kazantzidis et al. (2009) found that these accretionevents severely perturbed the galactic disk of model MWb withoutdestroying it and produced a wealth of distinctive morphologicaland dynamical signatures on its structure and kinematics. In thispaper, we will investigate the magnitude of radial mixing inducedin disk model MWb by these accretion events. As in the case of iso-lated disk models, it is worthwhile to examine the dependence ofradial mixing upon disk thickness. For this purpose, we employedthe thinner galaxy model with a scale height ofzd = 200 pc de-scribed above and repeated the satellite-disk encounter simulationsof K08.

2.3 Numerical Parameters

All collisionless numerical simulations discussed in this paper werecarried out with the multi-stepping, parallel, treeN -body codePKDGRAV (Stadel 2001). The hydrodynamical simulations wereperformed with the parallel TreeSPHN -body code GASOLINE(Wadsley et al. 2004). In the gasdynamical experiments, we includeatomic cooling for a primordial mixture of hydrogen and helium,star formation and (thermal) feedback from supernovae. Our starformation recipe follows that of Stinson et al. (2006), which isbased on that of Katz (1992). Gas particles in cold and dense re-gions which are simultaneously parts of converging flows spawnstar particles with a given efficiencyc⋆ at a rate proportional to thelocal dynamical time. Feedback from supernovae is treated usingthe blast-wave model described in Stinson et al. (2006), which isbased on the analytic treatment of blastwaves described in McKee& Ostriker (1977). In our particular applications, gas particles areeligible to form stars if their density exceeds 0.1 atoms/cm3 andtheir temperature drops belowTmax = 1.5 × 104 K, and the en-ergy deposited by each Type-II supernova into the surrounding gasis 4 × 1050 erg. We note that this choice of parameters and nu-merical techniques is shown to produce realistic disk galaxies incosmological simulations (Governato et al. 2007). Lastly, in an at-tempt to investigate the effect of star formation efficiency on radial

4 Bird, Kazantzidis, & Weinberg

Initial Disk Final Disk (Isolated) Final Disk (Perturbed)

Figure 1. Surface density maps of the stellar distributions of galactic disks with different initial scale-heightszd. Maps are presented only for the collisionlessexperiments and include the initial (left panels), final isolated (middle panels), and final perturbed disks (right panels). Each panel includes the face-on (bottompanels) and edge-on (upper panels) distributions of disk stars. Particles are color-coded ona logarithmic scale, with hues ranging from blue to white indicatingincreasing stellar density. Local density is calculated using an SPH smoothing kernel of32 neighbors. The density ranges from7×105 to 2×109M⊙/ kpc2.

mixing, for each of the gas fractionsfg above, we adopted twodifferent values forc⋆, namelyc⋆ = 0.05 (which reproduces theslope and normalization of the observed Schmidt law in isolateddisk galaxies) and a lower value ofc⋆ = 0.01.

All N -body realizations of the disk galaxy models containNd = 106 particles in the disk,Nb = 5 × 105 in the bulge, andNh = 2 × 106 in the dark matter halo. The gravitational soft-ening lengths for the three components were set toǫd = 50 pc,ǫb = 50 pc, andǫh = 100 pc, respectively. These are “equiva-lent Plummer” softenings; the force softening is a cubic spline. Inthe hydrodynamical simulations of isolated galaxies, gaseous diskswere represented withNg = 2 × 105 andNg = 4 × 105 for gasfractions offg = 0.2 andfg = 0.4, respectively. In these cases,the gravitational softening length for the gas particles was set toǫg = 50 pc. All numerical simulations of isolated and perturbeddisks were analyzed at2.5 Gyr. This time-scale corresponds to ap-proximately17.5 orbital times at the disk half-mass radius. Forevaluating the impact of accretion events, it is fairest to compareisolated and perturbed disks after the same amount of integration,but we may underestimate the total amount of mixing in the iso-

lated (and, to a lesser extent the perturbed) models. In the future,we plan to evolve selected simulations for the full8 Gyr intervalsincez = 1, but such simulations will require more than threetimes the computational resources used here.

Exchange of angular momentum between the infalling satel-lites and the disk tilt the disk plane and cause the disk center ofmass to drift from its initial position at the origin of the coordi-nate frame (K08; Kazantzidis et al. 2009). Therefore, we calculate∆r in the satellite-disk encounter experiments after removing theglobal displacement and tilt of the disk galaxy by determining theprincipal axes of the total disk inertia tensor and rotating the disksuch that this tensor is aligned with our original coordinate system.

3 RESULTS

Figure 1 presents surface density maps of the stellar distributionsof our four collisionless galactic disks with different initial scale-heightszd. Each simulation exhibits a distinctive combination ofmorphological features that could affect radial migration. The ini-

Radial Mixing in Galactic Disks 5

Figure 2. The distribution of radial shift (∆r= rf − ri) for all disk particles in each of the six isolated disk simulations.∆rmed and∆r80 specify themedian and80th percentile distance traveled, respectively. Gray values refer to the hatched histograms; black values are associated with the solid black linein each panel. We report the fraction of all particles in non-overlapping500 pc bins of∆r. The two collisionless simulations (left panel) illustratethe effectof disk structure on radial mixing, i.e., there is more radial migration in the smaller scale height (zd = 200 pc, black line) disk than in its thicker counterpart(zd = 400 pc, hatched histogram). The four hydrodynamical simulations arein the middle (fg = 20%) and right (fg = 40%) panels. In both of thesepanels, the gray, hatched histograms represent the simulations with higher star formation efficiency (c⋆= 0.05); those withc⋆= 0.01 are shown in black.Both increased gas fractions and smaller star formation efficiencies yield greater radial mixing.

tial smooth disks are unstable to spiral instabilities arising fromthe swing amplification of particle shot noise, an effect that leadsto emerging spiral structure as seen in the final isolated disks(e.g. Julian & Toomre 1966). Owing to its higher self-gravity, thezd = 200 pc disk (hereafter, disks with this initial scale height willbe referred to as “thin”) develops prominent spiral structure and astrong bar. Conversely, thezd = 400 pc disk (hereafter, disks withzd = 400 pc will be referred to as “thick”) has relatively little spi-ral structure and does not form a bar in isolation. We emphasizethat the “thick”, zd = 400 pc disk is the one in best agreementwith the old, thin disk of the MW, whilezd = 200 pc is too thin(see Section 2.1).

The radial and vertical morphology of the perturbed disks isdistinct from their isolated counterparts. Both perturbed disks de-velop bars and are characterized by prominent flaring and muchlarger scale heights compared to those of the isolated disks (K08;Kazantzidis et al. 2009). There is evidence that some spiral struc-ture evident in the isolated disks has been washed out in the simula-tions with substructure bombardment: within10 kpc of the galac-tic center, local enhancements of the stellar surface density evidentin the isolated disks are substantially more diffuse after the actionof the infalling satellites. Throughout this section, we will inves-tigate how the growth and dissipation of spiral structure affect ra-dial migration. While we have investigated similar maps in the fourhydrodynamical simulations, we do not show them here as theyare qualitatively similar to their collisionless counterparts. As ex-pected, however, the magnitude of spiral structure increases as thegas fraction rises. We note that the differential effect of gas on thestrength of spiral structure we find here is smaller than that reportedin previous numerical investigations of isolated disk galaxies (e.g.,Barnes & Hernquist 1996). This is mainly due to the effect of stel-

lar feedback, which causes the ISM in our simulations to becometurbulent and multi-phase (see also Stinson et al. 2006).

3.1 Radial Migration

We first investigate where particles move throughout each simula-tion. Figure 2 shows the distribution of∆r= rf − ri in the sixisolated disk simulations, whererf andri refer respectively to eachparticle’s final and initial projected distance from the galactic cen-ter. Each panel contains the∆r distribution, the median|∆r| (de-noted∆rmed), and80th percentile|∆r| (denoted∆r80) for two ofthe six isolated galaxies in our simulation suite.

The isolated, collisionless simulations (left panel) clearlydemonstrate that disk scale height and gas content affect the ra-dial migration process. The thin disk’s∆r distribution (black line)is considerably broader than the thick disk’s (gray hatch). Both∆rmed= 0.91 kpc and∆r80= 2.03 kpc of particles in the thindisk are approximately double those found in the thick disk.

The remaining panels of Figure 2 compare the∆r distribu-tions of the hydrodynamical simulations of our sample. We exam-ine four isolated disks with two initial gas fractions:fg = 20%(middle panel) andfg = 40% (right panel). The two histograms ineach panel represent star formation efficiencies (c⋆) of 1% (solidline) or 5% (gray, hatched histogram). Gas has a strong impact onparticle radial migration:∆rmed increases from0.47 kpc in thecollisionless case (all hydrodynamical simulations are of “thick”disks) to∼ 0.70 kpc when fg = 20% and∼ 0.80 kpc whenfg = 40%. The extent of radial migration is less dependent onstar formation efficiency; there are only minor differences amongsteach pair of histograms in the middle and right panels. However, inboth panels,∆rmed and∆r80 are highest whenc⋆= 1%. In thesefour hydrodynamical simulations, the time-averaged gas content of

6 Bird, Kazantzidis, & Weinberg

Figure 3. The ∆r distribution for all disk particles in the four collision-less simulations. Each histogram show the fraction of particles in non-overlapping500 pc bins of∆r. The isolated (hatched, gray histogram) andperturbed (black line)zd = 200 pc (left panel) andzd = 400 pc (rightpanel) disks are shown. The perturbed disks show greater dispersion in∆r

compared to isolated galaxies with the same geometry. The red histogramsdenote the expected∆r distribution from epicyclic motion alone, given theinitial orbital configurations of the particles in each disk. ∆rmed and∆r80

are labeled and color-coded for the three distributions in each panel.

the disk is directly correlated with the percentage of particles thatmigrate and their typical displacement relative to their formationradius.

Decreasing the scale height of the stellar disk, increasing thefraction of the initial disk mass in gas, and lowering the star-formation efficiency all increase the midplane density of the disk.This enhanced density increases the coherence and self-gravity ofspiral structure, supporting it against the dissipative nature of in-dividual particles’ random motions (as described by, e.g., Toomre1977). The correlation of spiral structure strength with the fractionof particles that migrate away from their birth radii and the medianmigratory distance traversed suggests that the SB02 mechanism,intimately linked with spiral waves, plays a major role in the mi-gration of particles in the isolated systems. The symmetric shapes(to within 0.4% about0 kpc) of the∆r distributions are also con-sistent with migration via the SB02 mechanism, though epicyclicmotion also has no preferred radial direction for particle movement.Resonances between the bar and spiral structure could also influ-ence the timescale for migration in the thin disk models (Minchev& Famaey 2010), but thezd = 400 pc models do not develop bars,regardless of whether they include gas.

Figure 3 compares the∆r distributions of the isolated colli-sionless disks to those of their perturbed counterparts. The∆rmed

and ∆r80 of each distribution are labeled and color-coded tomatch the corresponding histogram. In each panel, the most sharplypeaked histogram (red) shows the expected∆r distribution fromepicyclic motion alone, which we compute given the particle’s ini-tial phase space coordinates. We model the epicycle as a simpleharmonic oscillator about the particle’s guiding center radius (Rg)

Figure 4. The fraction of particles that migrate more than1.0 kpc (solidline),2.0 kpc (long dashed line), and3.0 kpc (dotted line) as a function offormation radius. Results are binned such that the migration probability iscalculated for particles in non-overlapping1.0 kpc wide annuli. Lines con-nect the migration fraction in each bin (x-coordinates are bin centers, from5.5 to 19.5 kpc). The gray regions are the radial mass profiles of the ini-tial disks normalized such that the total mass contained in thefirst annulusequals2/3 on this scale. The migration probability follows the mass distri-bution in the isolated disks but is anti-correlated with massin the perturbeddisks.

with an amplitude set by its initial radial energy (Er) (Chapter 3,Binney & Tremaine 2008). The energy associated with the circularcomponent of the particle’s orbit (Ecirc) is set byRg, which is theradius of a circular orbit with angular momentum equal to the mid-plane component of the particle’s angular momentum. The radialenergy of the orbit isEr = Etot −Ez −Ecirc whereEz is the ver-tical energy component andEtot is the total energy . AsEr sets theamplitude of the epicycle, we can solve for the position of the par-ticle as function of time with respect to its guiding center radius.We choose two random phases of the epicycle oscillation, desig-nating the first asri and the second asrf , and compute∆r. Theresulting distribution is shown in red. The shape of the distributiondoes not change if we change our random seed or average a largernumber of phases to obtainrf andri. For the isolated thick disk(right panel), the∆r distribution from full dynamical evolution isonly marginally broader than this epicyclic “blurring”. However,satellite bombardment broadens the∆r distribution dramatically,nearly doubling both∆rmed and∆r80.

For the thin disk, isolated evolution produces a much broader∆r distribution than epicyclic blurring, demonstrating the impactof the bar2 and spiral structure in this more unstable disk. In thiscase, satellite bombardment only slightly increases the width of the∆r distribution, despite the strong impact on disk structure that isvisually evident in Figure 1. We suspect that the small net differ-ence between these two histograms reflects a cancellation betweentwo competing effects of satellite bombardment. Accretion eventsheat the stellar disk and thereby suppress the development of spi-ral structure, thus reducing the level of SB02 migration. However,the accretion events also induce radial mixing directly via their dy-namical perturbations. The∆r histograms of both perturbed disks,

2 Bars can increase the eccentricity of particles, contributing to the “blur-ring” of the disk, and their potential resonance overlap with spiral structurecan induce migration (see Minchev & Famaey 2010).

Radial Mixing in Galactic Disks 7

Figure 5. Lindblad diagrams for all collisionless simulations. Each column shows the initial (left); final, isolated (middle); and final, perturbed (right) particleangular momentum projected along thez axis (Jz , kpc km s−1) vs. specific energy (E, km2 s−2). The red, yellow, and orange regions encompass68%,95%, and99%, respectively, of particles centered on the medianJz as a function ofE. Jz > 99% outliers are plotted as points. The green line denotesthe theoretical angular momentum - energy curve for circular orbits. Each panel includes a histogram illustrating the distribution of circularity (ǫ) for all diskparticles (see text for details). The isolated thin disk shows evolution towards slightly non-circular orbits. There is little change inǫ in the isolatedzd = 400 pc

case. Simulations with satellite bombardment show a pronounced shift towards less circular orbits. For reference, the total energy of midplane orbits at2, 5,10, and15 kpc from the galactic center is approximately−2.0, −1.6, −1.2, and−1.0 km2 s−2, respectively.

while still approximately symmetric, are noticeably more asym-metric than those of the isolated disks, with2% (3%) more particlesmoving outwards than inwards in the thin (thick) cases, comparedto 6 0.4% for the isolated models.

Figure 4 presents clear evidence that satellite-induced migra-tion is a distinct mechanism, acting in different environments fromeither epicyclic blurring or spiral-induced churning. In each panel,solid, dashed, and dotted lines show the fraction of particles at eachbirth radiusRform that migrate by more than|∆r| = 1.0, 2.0, or3.0 kpc, respectively. Shaded regions show the scaled radial sur-

face density profile of the initial disk. For both isolated disks, themigration probability decreases outwards and approximately tracesthe surface density profile. This behavior is similar to that assumedin the chemical evolution models of Schonrich & Binney (2009),who parametrized the probability that a star migrates as propor-tional to the mass surrounding it. However, for both perturbed disksthe probability of migration is flat or increasing outwards beyondRform= 10 kpc, where the disk potential weakens, and it definitelydoes not trace the mass distribution. To better understand this dif-ference between satellite- and spiral-induced radial mixing, we now

8 Bird, Kazantzidis, & Weinberg

investigate how changes in angular momentum and energy are cor-related with change in radius.

3.2 Orbital Characteristics

Total energy and angular momentum are the classic two-dimensional integrals of motion (Binney & Tremaine 2008). Fig-ure 5 shows Lindblad diagrams for the two initial and four finalstates of the collisionless simulations, plotting particle specific an-gular momentum projected along the axis of symmetry (Jz) versusspecific binding energy (E). Particles are grouped into 50 distinctlinear bins ofE. We calculate the medianJz in each bin. Red, yel-low, and orange regions connect the68th, 95th, and99th percentileJz intervals (centered on each medianJz), respectively, across allenergy bins. The1% most discrepant particles inJz for a givenEare plotted as individual points. Using each disk’s rotation curve,we plot Jz andE for circular orbits in the midplane of the disk(green line). Here,Jz = vc(r) × r , wherevc(r) is the circularvelocity at radiusr andE = 1

2v2

c (r)+Φ(r), whereΦ(r) is the po-tential in the disk midplane at radiusr. By definition, particles ona circular orbit have the maximumJz allowed given their energy.

By construction, particles are initially (left column of Fig-ure 5) on circular orbits with a small radial velocity component.The red region, falling close to the circular orbit curve in both initialdisks, confirms thatσvr

is small. Particles significantly displacedfrom the green curve in the initial states either have extremevr orare on highly inclined orbits (Jz is projected along thez axis). Dueto this inclination effect, the initial thick disk has a lower medianJz as a function ofE than the thin disk despite having the samevr

distribution.In the isolated thin disk (upper middle panel of Figure 5), the

range ofJz grows relative to the initial values at every energyE.The change is largest at lower energies. Recall from Figure 1 thatthe isolated thin disk develops a bar, which is composed of particleson radial orbits with relatively lowJz, in the most bound regionof the disk. Thus, this relatively large change towards less circu-lar orbits at low energies can be associated with bar formation. Incontrast, the distribution ofJz in the isolated thick disk is basicallyequivalent to its initial state. In the isolated thick disk there is nobar, relatively little spiral structure develops, and we find the leastradial migration among the four simulations. Ignoring the region ofthe Lindblad diagram influenced by members of the bar, the iso-lated disk diagrams suggest that the extensive radial migration seenin the thin disk and associated with guiding center modification de-creases the angular momentum of particles.

The two perturbed disks (right column of Figure 5) showlarger changes in their Lindblad diagrams. Bar formation in bothsimulations can explain the significantly lower medianJz at lowerenergies. At higher energies, corresponding to less bound particlesfurther out in the disk, both disks show a much larger dispersionin Jz than their isolated counterparts. Additionally, there is sub-structure in the Lindblad diagrams (groups of relatively lowJz ina narrow range of energy) not seen in the isolated disks. Satellitebombardment in the perturbed disk simulations has a qualitativelydiscernible impact on the angular momentum distribution of thedisk.

To quantify changes inJz, we introduce the circularity (ǫ)quantity (e.g., Abadi et al. 2003). For a particle of energyEi andangular momentumJi in the z-direction, we define circularity asǫ = Ji/Jcirc(Ei), the ratio ofJi to the specific angular momentumthe particle would have if it were on a circular orbit with energyEi

(obtained using the circular orbit curve). Circular orbits haveǫ = 1

Figure 6. Possible values of circularity as a function of eccentricity in thesolar annulus (7 > r > 9 kpc) of the zd = 400 pc disk. The top edgeof the dark region represents orbits in the midplane of the disk. The darkregion encompasses the95th percentile of particle circularity in the solarannulus of each simulation; the bottom edge of the light region correspondsto the99th percentile circularity.

and radial orbits haveǫ = 0. Negative circularities correspond toretrograde orbits. We note that this method is formally differentfrom that of Abadi et al. (2003). The circular orbit curve may notrepresent the highestJz for a givenE due to shot noise in the po-tential or gravitational potential asymmetries. Strictly defining themaximumJz in each energy bin (as in Abadi et al. (2003)) resultsin the same qualitative trends discussed later. Our method benefitsfrom the use of a mathematically constructed and reproducible ro-tation curve and systematically decreases∆ǫ= ǫf − ǫi at the0.01level. The inset of each panel in Figure 5 shows the circularity dis-tribution of each simulation.

Figure 6 plots the relation between circularity and eccentric-ity for orbits near the solar radius (8.0 kpc) in the zd = 400 pcdisk. Circularity is related to eccentricity via the particle’s radialand vertical energies as well as its orbital inclination (sinceJz isa projected quantity). Orbits in the midplane of the disk have thehighest circularity for a given eccentricity. Shaded regions in Fig-ure 6 show the95th and99th percentile circularity as a functionof eccentricity at the solar radius. For reference in interpreting Fig-ure 5 and subsequent figures, it is worth noting that changes of∼ 0.02 in ǫ typically correspond to quite noticeable changes in ec-centricity. The boundaries of these regions are not smooth due tobinning and small number statistics for initially high eccentricityparticles.

Returning to Figure 5, we see that the two isolated disks showmarkedly different evolution of their circularity distributions. Inthe isolated thin disk, the fraction of stars withǫ ∼ 1 (rightmostbin) drops from0.38 to 0.26, and the medianǫ drops from0.980to 0.955. Particles in the bar predominantly populate the newlyformed low circularity tail. In the thick isolated disk, on the otherhand, theǫ distribution is nearly identical to the initial disk’s, withmedianǫ dropping only0.002. The radial migration in the isolated

Radial Mixing in Galactic Disks 9

Figure 7. The initial and finalE, Jz pairs for ten randomly selected parti-cles in each simulation with initial energy−1.405 6 Ei 6 −1.395× 105

km2 s−2 and angular momentum1.645 6 Jz,i 6 1.655 × 103 kpc

km s−1 (large, filled square). For clarity we require that plotted particleslose at least0.05 × 105 km2 s−2 in energy during the simulation. Thefinal E,Jz pairs of the ten particles are plotted as open squares. Linescon-nect the initial and finalE, Jz points of each particle. TheE, Jz curvepopulated by circular orbits in the initial state of each simulation is indi-cated by the thick black curve. The dashed line is tangent to the circularorbit curve at the initial energy of all ten particles.

thick disk is not associated with systematic changes in angular mo-mentum, a further indication that epicyclic motion is the dominantmigration mechanism in this experiment.

The circularity distributions of the perturbed disks are demon-strably altered from their isolated counterparts but are similar toone another. The most common circularities are now in the range0.96 6 ǫ 6 0.98 in both disks, with a decrease forǫ> 0.98. Thethin disk starts with moreǫ ∼ 1 particles because of its smallerorbital inclinations, and its circularity distribution evolves morestrongly, with medianǫ dropping from0.980 to 0.936 vs. 0.963to 0.929 for the perturbed thick disk. Notably, the perturbed thinand thick disks evolve to similar∆r (Figure 3) andǫ distributionsdespite starting with different scale heights.

Figure 7 tracks the changes of selected individual particles inthe (E, Jz) space of the Lindblad diagram. WhileE andJz arenot individually conserved in the presence of a non-axisymmetricperturbation, the Jacobi invariantI = E − ΩbJz is, whereΩb

is the pattern speed of the perturbation, assumed to be static andsmall (SB02; Sellwood 2010). If∆I = 0, then∆Jz/∆E ≈ Ωb.The SB02 mechanism operates at the corotation resonance of thestar/particle and the spiral wave, requiring thatΩb = Ωrot. Thus,in the galaxy, particles should move parallel to the line that is tan-gent to the circular orbit curve at their binding energy prior to scat-tering. In other words, the SB02 mechanism requires that changesin energy be accompanied by changes in angular momentum thatpreserve the orbital shape, modulo differences in the slope of thecircular orbit curve over the range[Ei, Ef ] of a given particle.

Figure 7 zooms in on the area of the Lindblad diagram des-ignated by the two small boxes drawn on the initial state diagrams

Figure 8. Median change in circularity∆ǫ as a function of∆r for the entiredisk (open squares) and for particles withrf , ri > 4.0 kpc (filled squares).Particles are sorted into non-overlapping0.5 kpc bins of ∆r. Error barsmark the25th and75th percentile∆ǫ in each bin. Histograms at the bot-tom of each panel represent the relative fraction of particles in each bin forthe caserf , ri > 4.0 kpc.

in Figure 5. We randomly select ten particles within the same ini-tial E andJz range (filled, dark squares in Figure 7) and plot theirfinal E and Jz. We requireEf − Ei = ∆E < −0.05 × 105

km2 s−2 to ensure that the individual tracks are visible. In the iso-lated thin disk (top left), most particles move parallel to the cir-cular orbit curve tangent (dashed line); this relationship between∆Jz and∆E is consistent with that predicted by the SB02 mecha-nism. When particles lose energy in the isolated thick disk (bottomleft), their Jz typically remains closer to the circular orbit curvethan in the isolated thin disk. The changes in energy and angu-lar momentum are relatively small, indicating that guiding centersremain unchanged (Binney & Tremaine 2008). The randomly se-lected particles in both perturbed disks have slopes∆Jz/∆E thatare steeper than the slope of the tangent to the circular orbit curveat corotation. This distinct coupling of∆E and∆Jz, combinedwith our results concerning the migration probability as a functionof Rform (Section 3.1), offer compelling evidence that migrationin the perturbed disks is driven, at least in part, by a mechanismthat does not operate in the isolated systems. Figure 7 shows thatthe orbital characteristics of many particles in the perturbed disksare modified in a fashion inconsistent with either epicyclic motionor a single spiral wave scattering event.

We now examine the correlation between migration andchanges in orbital properties. Since the metallicity of star-forminggas increases with time and decreases with radius, any such corre-lations also imply observable correlations between the present or-bital parameters and metallicities of stars as a function of age andGalacto-centric radius. Figure 8 shows the median change in circu-larity, ∆ǫ = ǫf − ǫi, as a function of radial migration distance∆rin the four collisionless simulations (open squares). In the isolatedthick disk; changes in circularity are tiny (median|∆ǫ| 6 0.005)at any∆r; Figure 7 shows that particles in this simulation stayclose to the circular velocity curve even if they change energy. Inthe other three simulations, particles with negative∆r show sub-stantial drops in circularity (typical median∆ǫ < −0.06), whichare almost certainly associated with bar formation. The bars thatform in these three simulations increase the orbital eccentricities oftheir members. To focus on behavior in the disk proper, the filled

10 Bird, Kazantzidis, & Weinberg

Figure 9. Like Figure 8, but the change in maximum vertical displacement,∆zmax, is plotted in place of the change in circularity. Open squares showall disk particles, filled squares show those withrf , ri > 4 kpc, and errorbars mark25th and75th percentile at a given∆r.

squares in each panel show the median∆ǫ vs. ∆r for those par-ticles that start and end the simulation atr > 4 kpc, beyond theextent of the bar. Error bars mark the inter-quartile range (25th to75th percentile) at each∆r.

In the isolated thin disk, particles that migrate inwards (∆r<0) experience a modest decrease in circularity, stronger for morenegative∆r (and∆E), consistent with the tracks shown in Fig-ure 7. Particles that migrate outwards (∆r> 0) increase their en-ergy and typically traverse areas of the Lindblad diagram with rel-atively little change in the slope of the circular orbit curve. Fol-lowing the tangent to the circular orbit curve, particles will notsignificantly change their circularity in such a scenario (note therelative lack of low circularity particles atE > −1.5 km2 s−2 inFigure 5). The perturbed disks show a decrease in median circular-ity at every∆r, and a much wider inter-quartile range indicatinga greater range of orbital inclinations and eccentricities. Circular-ity drops more strongly for particles that have experienced strongradial migration, either inward or outward. The minimum in thesecurves is slightly offset to positive∆r because dynamical heat-ing slightly puffs up the disk radially, decreasing the potential ata given radius, thereby increasing its total energy, moving parti-cles to the right on the Lindblad diagram , and lowering the circu-larity for particles with∆r= 0. Figure 8 indicates that stars withanomalous chemistry for their age and current position should havepreferentially more eccentric orbits. The trend is smaller than theinter-quartile range, but similar in magnitude.

Schonrich & Binney (2009) and Loebman et al. (2010) discussthe possible role of radial migration in producing thick disks likethe ones observed in the Milky Way (Gilmore et al. 1989) and otheredge-on galaxies (Dalcanton & Bernstein 2002). Figure 9 is similarto Figure 8, but instead of∆ǫ it plots the change in vertical energy,quantified by the maximum distancezmax that a particle can reachfrom the disk plane. We computezmax approximately from eachparticle’s vertical velocity component assuming that the final po-

tential of each simulation is static, namelyzmax= |z| +v2

z

4πGΣ(r)

wherez is the vertical position of the particle at the final output,vz is the velocity along thez axis,G is the gravitational constant,andΣ(r) is the surface density of the disk at radiusr assumingall the mass of the disk is in the midplane. The change inzmax is

Figure 10. The distribution of formation radius (Rform, in kpc) for allparticles that are in the solar annulus (7 kpc 6 r 6 9 kpc) at the end ofeach simulation. Histograms report the fraction of solar annulus particlesemigrating from non-overlapping500 pc annuli inRform. Particles withinthe dashed lines remained in the solar annulus throughout thesimulation.Both thin disks and the perturbed thick disk show a broad range of Rform

at the solar annulus. The gray histogram in each panel is theRform distri-bution for those particles that are in the solar annulus and at large heightsabove the plane (1.0 < |z| < 1.5 kpc) in the final simulation output.

∆zmax= zmax,f − zmax,i where the subscriptsi andf refer to theinitial and final simulation snapshots, respectively.

The two isolated simulations show a shallow linear trend be-tween∆zmax and∆r (open squares for all particles; filled squaresfor those withri,rf> 4 kpc). When particles move outwards (in-wards) through the disk, they experience a weaker (stronger) grav-itational potential, thus increasing (decreasing)zmax. However,changes inzmax are small, less than300 pc even when we con-sider the quartile range at the extremes of∆r. The perturbed disks,by contrast, show larger changes in the medianzmax and a dramaticincrease in the inter-quartile range at all∆r. K08 show that the per-turbed400 pc disk develops a two-component vertical structure inquantitative agreement with the observed thin/thick disk structureof the MW. The perturbed200 pc disk increases its scale height(to ≈ 500 pc at the solar annulus), but it can still be described bya single component model. This vertical heating by satellite per-turbations is evident in Figure 9. Particles that have large positive∆r have the largest increase inzmax, which is plausibly a conse-quence of moving outwards to regions of lower disk surface densityand thus weaker vertical restoring force. For∆r < 0, the trend ofmedian∆zmax with ∆r is approximately flat, suggestion a cancel-lation between the effects of increasedΣ(r) at smallerr and directsatellite-induced heating of those particles with the largest excur-sions. Figure 9 implies that stars with high metallicity for their ageand present location should have preferentially largerzmax, thoughthe scatter is larger than the trend.

3.3 Solar Annulus

The solar neighborhood is easier to study than other regions ofthe Galaxy, since high-precision spectroscopy is easier for brighterstars and parallax and proper motion measurements are more ac-curate at smaller distances. Some of the most detailed chemo-dynamic surveys, such as the Geneva-Copenhagen Survey (Nord-strom et al. 2004) and the Radial Velocity Experiment (RAVE,

Radial Mixing in Galactic Disks 11

Figure 11. The change in circularity∆ǫ as a function of∆r for particlesending the four collisionless simulations in the solar annulus (7 kpc 6

rf 6 9 kpc). Particles are sorted into non-overlapping0.5 kpc bins of∆r. We plot the median (squares) and the25th and75th percentile (errorbars)∆ǫ in each bin. Histograms at the bottom of each panel represent therelative fraction of particles in each bin.

Steinmetz 2006) concentrate on the solar neighborhood. In this sec-tion, we repeat some of our earlier analysis specifically for starsthat reside in the solar annulus (7 kpc 6 rf 6 9 kpc) at the endof each simulation. This focus on the solar annulus also removesmuch of the impact of the bars that dominate evolution of the innerdisk (r < 3 kpc) in three of our simulations, though some particlesfrom the bar region can migrate as far as the solar radius, and reso-nances between the bar and spiral structure may increase migrationfrequency (Minchev & Famaey 2010).

Figure 10 shows the radius of formation (Rform) distributionof particles residing in the solar annulus, marked by the verticaldashed lines. Only the isolated thick disk simulation predicts a fi-nal solar annulus dominated by stars born in the solar annulus,with tails extending1–2 kpc on either side. The broad∆r dis-tributions of the other three simulations show that their stars mi-grate to the solar annulus from a wide range of formation radii.Tracking their similar full-disk∆r distributions (Figure 3), the so-lar annulusRform distributions of the isolated and perturbed thindisks are remarkably similar, despite the differences in migrationmechanisms discussed in Section 3.2. In the isolated thin disk,32%of solar annulus particles originated atRform6 6 kpc and4% atRform> 10 kpc. Corresponding numbers for the perturbed thindisk are36% and3%. TheRform distribution of the perturbed thickdisk is slightly narrower, but it still broad with respect to the iso-lated thick disk.

Figure 11 shows the correlations between∆ǫ and∆r for par-ticles that end in the solar annulus. Consistent with results for thefull disk (Figure 8), the solar annulus particles in the isolated thickdisk show no significant change in medianǫ regardless of∆r. Inthe isolated thin disk, the range of∆ǫ is much larger, with a mod-est decrease in medianǫ. The median∆ǫ drops at large positive∆rbecause particles move to less inclined orbits as they migrate out-wards and disk heating has slightly modified the galaxy’s circularvelocity curve, allowing outward moving particles to potentially in-crease circularity. In the perturbed thick disk, there is a strong andnearly linear trend between∆ǫ and∆r. Particles that migrated tothe solar annulus from the outer disk have experienced substantialdrops in circularity, while particles migrating from the inner disk

Figure 12. Like Figure 11, but the change in vertical displacement∆zmax

is plotted in place of the change in circularity.

show only modest decreases. The range of∆ǫ is large at all∆r.Results for the perturbed thin disk are intermediate between thoseof the isolated thin and perturbed thick disks: quasi-linear trends atlarge|∆r| but a flat plateau at intermediate|∆r|. This behavior isplausibly a consequence of two different mechanisms contributingto migration, with spiral-induced mixing dominating at intermedi-ate∆r and satellite-induced mixing dominating at the extremes.

Figure 12 shows the solar annulus correlations of∆zmax with∆r, analogous to Figure 8 for the full disk. For the two isolateddisks there is a clear linear trend of median∆zmax with ∆r asexpected from the arguments in Section 3.2: particles that migrateoutward move to a region of lower disk surface density, so if theirvertical velocities are not systematically changed by the radial mi-gration they will attain higherzmax. Both perturbed disks showsigns of the strong vertical heating induced by satellite accretion.The median∆zmax is > 0.2 kpc almost independent of∆r, andthe range of∆zmax is much larger than in the isolated disks. Sinceinwardly migrating particles experience a higher vertical potentialat rf thanri, they must experience more vertical heating than out-wardly migrating particles to keep the∆zmax trend flat.

Returning to Figure 10, gray histograms represent theRform

distributions of particles that end the simulations in the solar an-nulus at highz, 1.0 < |z| < 1.5 kpc. In the isolated thick diskand both perturbed disks, theRform distribution of highz particlesresembles that of all solar annulus particles. In these three models,therefore, selecting high-z particles does not isolate a populationwith atypical radial migration. In the isolated thin disk, on the otherhand, theRform distribution of high-z particles is strongly skewedtowards low formation radii, with a peak atRform= 5 kpc. Here,particles can only move to large heights above the plane when theyhave a large initial vertical velocity and migrate outwards so thatthey experience a weaker potential. Only0.1% of the solar annulusis at highz in the isolated thin disk (truly the tail of the initial verti-cal velocity dispersion); this number rises to3.2% and7.5% in thethin and thick perturbed disks, respectively. The radial migrationmechanisms in the perturbed disks ensure that even the highz pop-ulation of the solar annulus comes from a broad range ofRform.Measurements of the age-metallicity relation for high-z stars couldbe a valuable diagnostic for distinguishing models of radial migra-tion and vertical heating.

12 Bird, Kazantzidis, & Weinberg

4 SUMMARY AND DISCUSSION

Observations suggest that radial migration plays an important rolein the chemical evolution of the MW disk (Wielen et al. 1996;Schonrich & Binney 2009). SB02 described a mechanism by whichstars at corotational resonance with spiral waves can scatter, pro-ducing large changes in guiding center radius while keeping starson nearly circular orbits. Previous simulations have shown that mi-gration over several kiloparsecs can occur in isolated disks grownby smooth accretion (Roskar et al. 2008a) and that encounters withsatellites can also induce migration (Quillen et al. 2009). Here wehave carried out a systematic investigation of a variety of simula-tions to characterize the role of stellar disk properties, gas fractions,and satellite perturbations in producing radial migration. Most im-portantly, our suite of simulations includes experiments with a levelof satellite bombardment expected inΛCDM models of galaxy for-mation, which earlier investigations (K08, Kazantzidis et al. 2009)have shown to produce vertical and in-plane structure resemblingthat seen in the MW.

For disks evolved in isolation, the degree of migration cor-relates with the degree of disk self-gravity, susceptibility to barformation, and spiral structure. The collisionless stellar disk withzd = 400 pc, chosen to match that of the thin disk in the MW,has no bar and minimal spiral structure. It exhibits limited radialmigration; the median value of radial change|rf − ri| is ∆rmed=0.47 kpc, consistent with the level expected from epicyclic mo-tion (Figure 3). The collisionlesszd = 200 pc disk is much moreunstable, develops a bar and spiral structure (Figure 1), and hasa higher degree of radial migration, with∆rmed= 0.91 kpc and∆r80= 2.03 kpc. The presence of gas is a catalyst for radial mi-gration in thezd = 400 pc case. Both∆rmed and∆r80 increase inmodels with larger gas fractions or lower star formation efficiency(which consumes gas more slowly). The galactic bar strongly in-fluences, and perhaps dominates, the resultant individual particledynamics in the inner galaxy. Substantial radial migration, con-sistent with the SB02 mechanism or being satellite-induced, oc-curs in the outer portion of the disk in all of our collisionlessexperiments except for the isolatedzd = 400 pc disk. In thezd = 200 pc disk, stars that finish the simulation in the solar annu-lus (7 < R < 9 kpc) have∆rmed= 1.36 kpc, ∆r80= 2.68 kpc.

Adding satellite perturbations dramatically increases radialmigration in thezd = 400 pc disks, raising∆rmed from 0.47 kpcto 0.89 kpc globally and from0.60 kpc to 1.15 kpc in the solarannulus. Perturbations do not change the∆r distribution so drasti-cally in thezd = 200 pc disks, but there are other indications thatthe nature of radial migration is different. In the isolated galaxies,migrating particles in the outer disk follow tracks inE, Jz spacethat parallel the tangent to the local circular velocity curve, as ex-pected for the SB02 mechanism (see Section 3.2). However, in-wardly migrating particles in the perturbed disks lose more angularmomentum for a given change in energy, suggesting a more violentmigration mechanism. The radial distribution of migrating parti-cles presents an even clearer distinction, one with important impli-cations for chemical evolution models (Figure 4). In the isolateddisks, the probability that a particle with a formation radiusRform

undergoes significant migration (|∆r| > 1 kpc) is approximatelyproportional to the disk surface density atRform; i.e., migrationfollows mass. In the perturbed disks, the probability of migrationis flat or increasing with radius, so a much larger fraction of migra-tion comes from the tenuous outer disk, which is more susceptibleto dynamical heating from the satellites. Thus, satellite-induced mi-

gration appears to be physically distinct from the migration inducedby spiral waves (SB02) or bar resonances.

Satellite bombardment heats the disk both vertically and ra-dially. Compared to the isolated disks, the perturbed disks experi-ence a greater drop in median circularity (hence growth of medianeccentricity) during their evolution, and individual particles experi-ence a wider range of circularity changes. The circularity change ismoderately correlated with radial migration distance and direction,though the trend is smaller than the inter-quartile range at a given∆r (Figure 8). If we restrict our analysis to the solar annulus, thecorrelation between∆r and∆ǫ is more significant. In particular,particles in the solar annulus of thezd = 400 pc disk that migrate2–4 kpc inwards experience substantial drops in circularity, whilethose that migrate the same distance outwards nearly maintain theirinitial circularity (Figure 11). The large inward excursions from theouter disk, driven by satellite perturbations, systematically removeangular momentum from particle orbits.

As shown by K08, vertical heating by satellite bombardmentproduces (in the case of thezd = 400 pc disk) a two-componentvertical structure in quantitative agreement with the thin and thickdisk profiles observed in the MW. In the isolated disks, we findthe expected trend that as particles migrate outwards, they expe-rience a weaker vertical potential and increase their vertical en-ergy (characterized byzmax, the maximum distance from the planethat a particle can reach given its current location and velocity).However, in our simulations, this effect is not sufficient to producea second, “thick-disk” component in the isolated systems even ifthey have substantial radial migration. The satellite perturbationsincrease the median and range ofzmax considerably at all∆r. Theresulting overall trend of∆zmax with ∆r is fairly flat (especially ata fixedrf , such as the solar annulus, see Figures 9 and 12), suggest-ing that systems with low vertical velocity dispersion and subjectedto weaker potentials (as in the outer disk) are more susceptible tovertical heating (similar to the radial heating trends seen in Sec-tion 3.1). Particles withRform interior or exterior to a givenrf canbe found at at significant distances above the plane in the perturbeddisks, while increases inzmax are smaller in the isolated systemsand rely on particles moving outwards (Figure 10).

Our results confirm earlier findings that spiral structure de-velopment in isolated disks (Roskar et al. 2008a; Loebman et al.2010) and perturbations by satellites (Quillen et al. 2009) can pro-duce significant radial mixing of stellar populations while retainingreasonable orbital structure for disk stars. They strongly supportthe view (Wielen et al. 1996; Schonrich & Binney 2009) that ra-dial mixing is an essential ingredient in understanding the chemicalevolution of the MW and disk galaxies in general. While a combi-nation of metallicity and age can be used to estimate a star’s forma-tion radius given a chemical evolution model, the uncertainties inthis approach (including the difficulty of estimating ages for typicalstars in a spectroscopic survey) will make it difficult to reconstructformation radius distributions for observed stars, even in the solarneighborhood. However, our analysis provides theoretical guidancefor chemical evolution models that incorporate radial mixing (e.g.Schonrich & Binney 2009) and suggestions for the correlations onemight search for between chemical abundances and orbital proper-ties (though the predicted trends are fairly weak). We regard ourperturbedzd = 400 pc disk as the most relevant simulation, as itincludes the satellite accretion events expected inΛCDM and pro-duces a final vertical structure similar to that measured in the MW.In this simulation41% of particles in the7 kpc < R < 9 kpc so-lar annulus were “born” in that annulus,20% migrated there fromR < 6 kpc, and7% migrated there fromR > 10 kpc.

Radial Mixing in Galactic Disks 13

Radial mixing and orbital dynamics changes are sensitive toseveral different aspects of disk modeling, as shown by the vari-ety of our results. Robust predictions should therefore come fromcalculations that self-consistently include star formation, chemicalenrichment, gas accretion, and accretion events, and the interplayamong these elements. We will move in this direction with ourfuture simulations. Giant spectroscopic surveys such as SEGUE,RAVE, APOGEE, LAMOST, and HERMES offer an extraordinaryopportunity to unravel the formation history of the MW, and theyoffer an exciting new challenge to theoretical models of galaxy for-mation.

ACKNOWLEDGMENTS

We acknowledge useful discussions with Mario Abadi, JulianneDalcanton, Rok Roskar, Ralph Schonrich, Scott Gaudi, Tom Quinn,and James Wadsley. S.K. is funded by the Center for Cosmologyand Astro-Particle Physics (CCAPP) at The Ohio State University.This work was supported by NSF grants AST0707985 and AST1009505, and by an allocation of computing time from the OhioSupercomputer Center (http://www.osc.edu). This research madeuse of the NASA Astrophysics Data System. D.W. acknowledgesthe hospitality of the Institute for Advanced Study and the supportof an AMIAS membership during part of this work.

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