Quantifying stellar radial migration in an N-body simulation: blurring, churning, and the outer regions of galaxy discs
AAstronomy & Astrophysics manuscript no. 25612˙apnoref c (cid:13)
ESO 2018July 24, 2018
Quantifying stellar radial migration in an N-body simulation:blurring, churning, and the outer regions of galaxy discs
A. Halle , , P. Di Matteo , M. Haywood , and F. Combes Observatoire de Paris, GEPI, 5 place Jules Janssen 92190 Meudon, France Observatoire de Paris, LERMA, 61 Av. de l’Observatoire, 75014 Paris, France UPMC-CNRS, UMR7095, Institut dAstrophysique de Paris, 98 bis boulevard Arago, 75014 Paris, Francee-mail: [email protected]
ABSTRACT
Radial stellar migration in galactic discs has received much attention in studies of galactic dynamics and chemicalevolution, but remains a dynamical phenomenon that needs to be fully quantified. In this work, using a Tree-SPHsimulation of an Sb-type disc galaxy, we quantify the effects of blurring (epicyclic excursions) and churning (change ofguiding radius). We quantify migration (either blurring or churning) both in terms of flux (the number of migratorspassing at a given radius), and by estimating the population of migrators at a given radius at the end of the simulationcompared to non-migrators, but also by giving the distance over which the migration is effective at all radii. We confirmthat the corotation of the bar is the main source of migrators by churning in a bar-dominated galaxy, its intensitybeing directly linked to the episode of a strong bar, in the first 1-3 Gyr of the simulation. We show that within theouter Lindblad resonance (OLR), migration is strongly dominated by churning, while blurring gains progressively moreimportance towards the outer disc and at later times. Most importantly, we show that the OLR limits the exchange ofangular momentum, separating the disc in two distinct parts with minimal or null exchange, except in the transitionzone, which is delimited by the position of the OLR at the epoch of the formation of the bar, and at the final epoch. Wediscuss the consequences of these findings for our understanding of the structure of the Milky Way disc. Because theSun is situated slightly outside the OLR, we suggest that the solar vicinity may have experienced very limited churningfrom the inner disc.
Key words.
Galaxies: formation — Galaxies: evolution — Galaxies: spiral — Galaxies: structure — Galaxy: stellarcontent
1. Introduction
Radial stellar migration in the galactic discs has been at-tracting increasing attention, including some theoretical(Sellwood & Binney 2002; Minchev & Famaey 2010; Daniel& Wyse 2014) or numerical works (Brunetti et al. 2011;Minchev et al. 2011, 2012b; Di Matteo et al. 2013; Roˇskaret al. 2013; Vera-Ciro et al. 2014), and studies based onobservations of the stars of the Milky Way (e.g. Haywood2008; Sch¨onrich & Binney 2009; Yu et al. 2012).In galactic discs where non-axisymmetric potential per-turbations such as bars or spiral arms occur, stars can gainor lose angular momentum (Lynden-Bell & Kalnajs 1972,e.g.), leading to outward or inward migration, respectively.The result of this mechanism is that stars can be foundat a galactocentric radius differing significantly from theirbirth radius. Another reason for apparent radial migrationis simply the nature of orbits in axisymmetric or nearlyaxisymmetric potentials: stars oscillate radially around aguiding radius, the amplitude of the oscillations increas-ing with the radial velocity dispersion. Sellwood & Binney(2002) studied the impact of transient spiral arms on thechange in angular momentum of stars in N-body simula-tions. They found that stars nearly at corotation with aspiral pattern experience the largest changes in angular mo-mentum, larger than what can be experienced by stars at
Send offprint requests to : A. Halle the Lindblad resonances of the spiral pattern or in the restof the disc. This prominent effect of corotation was con-firmed, in particular by Roˇskar et al. (2012), in N-body sim-ulations exhibiting several spiral patterns amongst whichthe strongest one causes large changes in angular momen-tum for stars around its corotation radius. Other studieshave shown the influence of the bars on radial migration(Di Matteo et al. 2013; Brunetti et al. 2011; Kubryk et al.2013, e.g.). In the case where both a bar and spiral armsare present, the bar-spiral resonance overlap studied byMinchev & Famaey (2010); Minchev et al. (2011, 2012b)can produce some non-linear coupling that can generatelarger changes in angular momentum. In addition, whileangular momentum can occur at corotation without anyincrease in radial kinetic energy, resonance overlap is ex-pected to increase radial energy: stars that were originallyon nearly circular orbits can thus have significantly moreeccentric orbits once they have experienced a change in an-gular momentum that is due to a resonance overlap.As it redistributes stars over the disc of a galaxy, ra-dial migration has been invoked as a possible explanationto a number of unanswered questions of galactic and extra-galactic stellar populations: the upturn of the mean age inthe outer parts of discs (Roˇskar et al. (2008) and e.g Bakoset al. (2008); Zheng et al. (2014)), the formation of theGalactic thick disc (Sch¨onrich & Binney 2009; Loebmanet al. 2011), the dispersion in metallicity at a given age a r X i v : . [ a s t r o - ph . GA ] M a y . Halle et al.: Blurring, churning, and the outer regions of galaxy discs observed on stars in the solar vicinity (Haywood 2008;Sch¨onrich & Binney 2009).Each of these problems requires a different level of mi-grations. For instance, U-shaped age profiles would requirethat stars from the inner disc of a galaxy are massivelypresent in the outer regions. Similarly, shifting the meansolar-vicinity metallicity towards higher values, as has beensuggested to explain the flatness of the age-metallicity re-lation at the solar vicinity (Loebman et al. 2011), wouldrequire massive migration of stars to the solar radius.Explaining the range of metallicities in the solar vicinityis a more subtle problem because it may require only verylittle migration as it depends on the local (6-10 kpc) metal-licity gradient. If the gradient is steep, the presence of high-metallicity stars found at the solar radius may be explainedby blurring alone, without any significant churning. Thatthe gradient is steep is suggested by the data, which nowheavily indicate that the Sun is at the interface between aninner and an outer disc that each have different chemicalproperties, as advocated in Haywood et al. (2013), and con-firmed on more extended data by Nidever et al. (2014). Therecent results found for stars at the solar radius seem thusto dispute that strong migration by churning is necessaryto explain the metallicity distribution.A common prediction of N-body models of galaxy evo-lution is the significant migration of inner disc stars in theexternal parts of galaxy discs (see for example Roˇskar et al.(2008); Minchev et al. (2011, 2014)). In the case of the outerdisc of the Milky Way, which contains stars with guidingradii greater than ∼
10 kpc – some of them with pericen-tres small enough to reach the solar vicinity– there is noevidence that this extreme migration has ever occurred atany time in the last 10 Gyr. At every age, stars of theouter disc are more metal poor than inner disc stars andmore alpha-enhanced (Haywood et al. 2013), which indi-cates that the outer Galactic disc must have followed a dif-ferent chemical evolutionary path (Snaith et al. 2014). Evenwithout access to age information, large-scale spectroscopicsurveys like APOGEE are confirming the substantial differ-ence between the inner and the outer Milky Way disc, as isshown by the properties of their stars in the [ α /Fe]-[Fe/H]plane or by the change of the metallicity distribution func-tion with distance from the Galaxy centre (Anders et al.2014; Nidever et al. 2014). It is striking in this context thatregardless of the nature and formation of the outer disc ofthe Milky Way, stars from the inner thin disc have not beenable to significantly pollute the outer parts of the Galacticdisc, when models of radial migration would predict this asa common scenario. This also casts some doubts on the ideathat migration due to the present bar/spiral is at the originof the U-shaped age profiles observed in external galaxies,at least in bar-dominated systems where this inversion oc-curs outside of the outer Lindbland resonance (OLR) of thebar.However, it is possible that the first pattern thatoccurred in the galaxies had a lower speed than thecurrent speed and that the OLR was at larger radii, whichwould have extended the churning action at larger radii.During the mass assembly of galaxies, the concentrationof mass increases, and consequently the pattern speedincreases, implying a smaller OLR radius with time. Thesimulations of Bournaud & Combes (2002), which includedgas accretion, showed such an increase in the bar patternspeed that is due to a central mass accumulation that causes the increase of the azimuthal and radial frequen-cies, and subsequently an increase of the bar pattern speed.In addition, the characteristics of the thick disc of theMilky Way hardly require any significant migration, andthey can be well explained by formation from a disc thatis rich in gas and turbulent (Haywood et al. 2013; Lehnertet al. 2014; Haywood et al. 2015). On the theoretical side,the uncertainties are equally important, and althoughphenomenological radial migration is now implemented inseveral Galactic chemical evolution models (Sch¨onrich &Binney 2009; Kubryk et al. 2014a,b), there is still muchuncertainty about the importance of the role of radialmigration. For example, what fraction of stars are subjectto migration and on which distances? Is the migrationepisodic or continuous? How are the different parts ofthe disc affected by migration? While tentative answersto these questions have been given in past works, fromthe simple model of recurring spiral perturbations in anisolated disc in Sellwood & Binney (2002) to recent modelsin a cosmological context by Minchev et al. (2014), thedetailed quantitative effects of radial migration are stillsubject to debate.In this paper, we are interested in distinguishing be-tween the effects of ‘blurring’, or in other words, the radialmigration that is due to epicyclic excursions around a fixedguiding radius, and ‘churning’, which is the radial migrationthat is due to a change in this guiding radius (according tothe terminology of Sch¨onrich & Binney (2009)). In partic-ular, we seek to quantify the fraction of the stars involvedin the migration, the extent of the migration, the fractionof migrators at a given galactocentric radius, and the re-gions affected by the process. The study is performed usingan N-body+SPH simulation of an Sb-type galaxy that wasfirst presented in Halle & Combes (2013). A few results onradial migration for this simulations were presented in theAppendix of Di Matteo et al. (2014). In the present pa-per, we first briefly list the parameters of the simulationin Sect. 2. In Sect. 3 we study the density resonances, andin Sect. 4 we quantify the effects of blurring and churn-ing, their strength, and their extent. Section 5 discussesthe behaviour of migrators near the outer Lindblad reso-nance, and in particular its role as a barrier for migrators.Finally, the kinematic characteristics of migrating stars arediscussed in Sect. 6, and we conclude in Sect. 7.
2. Numerical simulation
We used one of the simulations presented in Halle &Combes (2013). We briefly summarize the main character-istics of the simulations. They were performed with theN-body SPH code Gadget-2 (Springel 2005) and includestochastic star formation following a Schmidt law, kineticfeedback from core-collapse supernovae and some detailedcooling of the gas down to 100 K, with the possibility ofincluding cooling by molecular hydrogen, whose local massfraction is computed based on a semi-analytic recipe fromKrumholz et al. (2008, 2009); McKee & Krumholz (2010).The simulated Sb type galaxy includes
2. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
Table 1.
Sb galaxy parameters
Sb M h M d M b M g r h a d h d r b a g h g [M (cid:12) ] [M (cid:12) ] [M (cid:12) ] [M (cid:12) ] [kpc] [kpc] [kpc] [kpc] [kpc] [kpc]1.7 10
12 5 0.5 1 11.8 0.2 – a stellar ( s ) and a gas ( g ) discs that both haveMiyamoto-Nagai density profiles: ρ s,g ( R, z ) = h s,g M g π × a s,g R + (cid:16) a s,g + 3 (cid:113) z + h s,g (cid:17) (cid:16) a s,g + (cid:113) z + h s,g (cid:17) (cid:18) R + (cid:16) a s,g + (cid:113) z + h s,g (cid:17) (cid:19) (cid:0) z + h s,g (cid:1) (1) – a stellar bulge ( b ) and a dark matter halo ( h ) withPlummer density profiles: ρ b,h ( r ) = 3 M b,h πr b,h (cid:18) r r b,h (cid:19) − . (2)The masses of the different components and density pro-files parameters are shown in Table 1. There are initially400 000 particles of each component with particle masses ofm DM , m (cid:63) and m g for the dark matter, stars, and gas, listedin Table 2. The softening length (cid:15) used in the gravitationalforce computation and the smoothing length h used for hy-drodynamics are also listed in Table 2. Particle velocitieswere assigned in the following way (see Halle & Combes(2013) for details): For gaseous and stellar disc particles,we computed circular velocities from gravitational acceler-ations and applied an analytic asymmetric drift correctionto obtain a more realistic profile. Radial velocity disper-sions were derived as σ r ( r ) = 3 . GQ Σ( r ) κ ( r ) , with κ ( r ) andΣ( r ) the epicyclic frequency and the surface density at ra-dius r , respectively, and adopting a Toomre parameter Q equal to 1. The azimuthal velocity dispersions σ θ ( r ) werecomputed from the radial velocity dispersions σ r ( r ) usingthe epicycle approximation, and the vertical velocity disper-sion σ z ( r ) was derived by assuming isothermal equilibriumfor the discs. For the spheroidal components, the veloc-ity dispersion is isotropic and was derived from the secondmoment of the Jeans equation. It can be noted here thatour initial conditions for the evolution of the disc are notrepresentative of clumpy discs observed at high-redshifts(e.g. Elmegreen et al. 2007). We focused on the effect ofa bar on the evolution of an already formed thin disc. Inclumpy discs with a chaotic evolution, resonant patternscausing significant churning are difficult to form because ofthe higher velocity dispersion, and the radial migration ismore likely to be some blurring. The uncertainty on themorphology and kinematics of galactic discs before bar for-mation is large. In addition, we did not take into accountany thick-disc component that may have contributed sig-nificantly to the total mass of the disc at the epoch of barformation in the Milky Way, for example (e.g. Haywoodet al. 2013; Di Matteo et al. 2014; Snaith et al. 2014). Thiskinematically hot component may have been less affected Table 2.
Resolution of the simulation (cid:15) h m DM m (cid:63) m g [pc] [M (cid:12) ] [M (cid:12) ] [M (cid:12) ]100 ≥ (cid:15)
10 3.7 10 by radial migration (Loebman et al. 2011; Brunetti et al.2011).The simulated time-span we used is 9 Gyr, which is sim-ilar to the estimated time-span during which the Milky Wayhas been evolving with no major mergers (Haywood et al.2013; Hammer et al. 2007, e.g.). Figure 1 shows the surfacedensity maps of different components of the disc (gas, allstars, and new stars formed during the simulation). A bar isformed during the first Gyr. In this simulation, the feedbackefficiency is 10% and there is no molecular hydrogen cool-ing. With these parameters, the surface density of gas andstars is relatively smooth because this feedback efficiencyand the absence of molecular hydrogen cooling prevent gasand stellar clumps from forming. We note that compared tomany previous studies (for example Minchev et al. (2011);Di Matteo et al. (2013)), the stellar and gaseous discs arevery extended (initial conditions were generated with aninitial cut at R = 36 kpc for both gas and stars). Thischoice also allows studying regions outside the OLR at alltimes, which in this model is located between 11 and 23 kpcfrom the centre (see next sections). The maximum extent ofdiscs is difficult to determine observationally, be it at highor low redshifts, which means that considering an extendeddisc is not necessarily a singular case. The recent study byvan Dokkum et al. (2014) showed, for example, that thedisc of M101, previously strongly underestimated, extendsto as much as 18 scale lengths in radius. Moreover, we herefocused rather on the effect of radial migration with respectto the location of the main resonances than on the abso-lute sizes. Figure 2 shows the azimuthally averaged surfacedensity of the simulated disc components as a function ofthe galactocentric radius. The old stellar disc dominates thesurface density at all radii throughout the simulated time-span. Its surface density profile is very stable, with onlymild departures from the initial conditions, which is due tothe presence of the bar and spiral arms formed during thesimulation.
3. Resonances in the disc
The disc forms a central bar that persists during the wholesimulated time-span. Transient spiral arms are also present.We determined the pattern speeds through a classic Fouriermethod: – At times spaced by a constant interval of 10 Myr, weperformed a spatial Fourier transform of the surfacedensity Σ(
R, θ ) of the stars to obtain the dominant az-
3. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
Gas Formed stars All stars t=1.0 Gyr l og ( Σ / [ M fl / p c ] ) Gas Formed stars All stars t=3.0 GyrGas Formed stars All stars t=5.0 GyrGas Formed stars All stars t=7.0 GyrGas Formed stars All stars t=9.0 Gyr
Fig. 1.
Surface density maps of the gas (left), stars formed during the simulation (middle), and all the stars (right).Face-on view boxes have a size of [80 kpc ×
80 kpc] and edge-on views a size of [80 kpc ×
20 kpc]. The colour scale,shown in the top right corner, is the same for all the views.
4. Halle et al.: Blurring, churning, and the outer regions of galaxy discs Ω [ k m / s / k p c ] Ω [ k m / s / k p c ] Fig. 3.
Spectrograms of the m = 2 Fourier mode. Top left plot: integration on 9 Gyr from 0.5 Gyr to 9.5 Gyr. Otherplots: integrations on 1 Gyr centred on the times specified in the plots. The black curves are Ω( R ) (solid), Ω( R ) − κ ( R )2(dot-dashed), Ω( R ) + κ ( R )2 (dashed). The horizontal line represents the estimate of the bar pattern speed. The verticallines represent the estimates of the bar ILR (blue), corotation (red), and OLR (green) radii. -2 -1 Σ [ M fl / p c ] Fig. 2.
Time evolution of the azimuthally averaged surfacedensity of the old stellar disc without the bulge (solid), theyoung stellar disc (dashed), and the gas disc (dot-dashed),as a function of the distance from the galaxy centre.imuthal modes in the different radial bins, S m ( R ) ∝ (cid:90) π Σ( R, θ ) e imθ d θ. (3) – We then performed a time Fourier analysis of the differ-ent modes in each radius bin to study their azimuthal speed, T m ( R, ω ) ∝ (cid:90) t f t i S m ( R ) e iωt d t. (4)In the spatial Fourier transforms, the m = 2 mode usu-ally dominates, which corresponds to π -periodic patterns:the central bar or two-armed spirals. The contours of theobtained power in the R -Ω plane for this mode are shownin Fig. 3 for a time Fourier-integration performed on thetotal time intervals of 9 Gyr (first panel) and of 1 Gyr cen-tred on 1, 3, 5, 7, and 9 Gyr. Ω = ωm , with m = 2 here, isthe pattern speed. The integration on 9 Gyr shows that thecontours at low radii, which correspond to the bar, fill a Ωrange from (cid:39)
10 to (cid:39)
30 km/s/kpc. The time interval of1 Gyr on which the time Fourier transform was then per-formed around specific times was chosen so as to optimisethe determination of the pattern speeds: for a given timeresolution, a too short integration time prevents correctlydetermining low pattern speeds and yields a poor resolu-tion in frequency, while a too large integration time leadsto a determination of overly averaged pattern speeds if theychange significantly during this time. Figure 3 shows thatthe bar slows down with time. The bar transfers angularmomentum to the rest of the disc and to the stellar bulgeand dark matter halo, as already discussed, among oth-ers, by Debattista & Sellwood (1998); Athanassoula (2002);Martinez-Valpuesta et al. (2006); Saha et al. (2012); Saha& Naab (2013); Di Matteo et al. (2014). The angular mo-mentum transfer between the different components of the
5. Halle et al.: Blurring, churning, and the outer regions of galaxy discs galaxy is detailed in Fig. 4. At the beginning of the sim-ulation, the angular momentum is contained in the discbecause the DM halo and stellar bulge have no initial rota-tion. In 9 Gyr, the disc loses 8% of its angular momentum,which is transferred mainly to the DM halo, while a slightfraction is given to the bulge.The contours of Fig. 3 show the main peaks of power inthe Ω- R plane. The bar (power peak at low radii) is clearlythe strongest π -periodic surface density perturbation with arigid-body rotation in the integration periods of 1 Gyr, butspiral arms (peaks at larger radii) are visible as well. For ex-ample, the panel corresponding to t = 3 Gyr indicates spi-ral patterns rotating at (cid:39)
18 km / s / kpc and (cid:39) / s / kpc.These resonant modes with pattern speeds such that someof the resonance radii coincide might be sustained by non-linear mode coupling (Tagger et al. 1987, e.g.). Some higher m -periodic features are also present ( m = 3 , L z [ × M fl k p c k m / s ] ∆ L z / L z t o t(t = ) DM haloOld stellar discDisc of gas+new starsTotal discBulgeTotal
Fig. 4.
Top: Time evolution of the vertical component ofthe angular momentum L z for different galaxy compo-nents. Bottom: Time variation of the fraction of the totalangular momentum contained in each galaxy component L z − L z t=0 L z tot t=0 .By determining the angular speeds corresponding tothe main contours, we obtained the radii at which starson circular orbits are in resonance with the main pat-terns. We computed the angular speed Ω from the poten-tial by Ω = 1 R d φ d R z =0 and the epicyclic frequency κ by κ = (cid:18) d φ d R + 3 R d φ d R (cid:19) z =0 . The corotation resonance (CR)occurs for Ω = Ω P , the inner Lindblad resonance (ILR) forΩ − Ω P = κ − Ω P = − κ A m at each time is shown inFig. 6. The bar strength increases until t = 3 Gyr , when itdrops before increasing again as a result of the angular mo-mentum exchange with the DM halo (Athanassoula 2002).The drop in strength corresponds to the start of the buck-ling of the bar. This vertical instability, leading to a thickX-shaped bar, can be seen in the edge-on views of the newstars and the whole stellar component of Fig. 1. R [ k p c ] ILRCROLR Ω [ k m / s / k p c ] Ω bar Fig. 5.
Time evolution of the ILR, CR, and OLR radii ofthe bar (left y-axis) and of the bar speed Ω (right y-axis).The various resonances are expected to locally affect theangular momentum of the stellar disc and thus to generatesome radial migration. We study next the radial migrationthat occurs between 1 and 9 Gyr of evolution (after theinitial building of the bar).
4. Blurring and churning
In this section, we quantify different types of radial stellarmigration. We examine – migration in terms of galactocentric radius (hereaftersimply radius), that is, the difference of radius betweentwo different times. Blurring and churning can bothcause a change in the galactocentric radius of a star,thus this definition captures both types of migratorsand can be considered as the overall migration experi-enced by disc stars. The comparison between radii attwo different times during disc evolution is widely usedin the literature, see for example Roˇskar et al. (2008);Brunetti et al. (2011); Loebman et al. (2011); Bird et al.(2012); Di Matteo et al. (2013); Kubryk et al. (2013);Minchev et al. (2012b, 2014).
6. Halle et al.: Blurring, churning, and the outer regions of galaxy discs m a x ( A / A ) R Fig. 6.
Time evolution of an estimate of the bar strengthfrom the m = 2 coefficient of the Fourier decompositionof the stellar surface density. The bar strengthens between0 and 1 Gyr before weakening when a buckling instabilityoccurs around 3 Gyr, and it then strengthens again. – migration in terms of guiding radius, or churning, thatis, the difference of the guiding radius between two dif-ferent times. This follows the nomenclature of Sch¨onrich& Binney (2009). Stars that increase the amplitude oftheir radial oscillations over time but do not changetheir guiding radius, that is, stars that only experienceblurring, are not considered as migrators according tothis definition.We recall that our model is not intended to reproduce ei-ther the characteristics of the Milky Way disc – as an ex-ample, the pattern speed of our simulated bar is lower thanthe pattern speed measured for the Galaxy (Gerhard 2011)and as a consequence, the main resonances are located atmuch larger distances from the centre than those measuredfor the Milky Way – or of any other specific galaxy. As aresult, the exact values of spatial or kinematic variationsexperienced by stars in the model are not directly applica-ble to any galaxy. However, the effects described below canbe considered as representative of those that any typicalbar-dominated galaxy probably experiences. A stellar radius oscillates around a guiding radius. In thecase of low eccentricities, it is possible to determine thisguiding radius by finding the radius R c at which a star hasthe same vertical component of the angular momentum anda circular trajectory: L z = R c v circ ( R c ), where v circ is thecircular velocity obtained from the potential. The ampli-tude of the radial oscillations around the guiding radius isexpected to scale as σ R κ , where σ R is the radial velocitydispersion, and σ R and the epicyclic frequency κ are bothfunctions of the radius R .The output snapshots of our simulation are separatedby 10 Myr, which is small enough to allow computing theguiding radius at any time t by a simple method: – determination of the relative minima and maxima of theoscillatory evolution of the radius. – use of a linear fit between the relative minima on theone hand and the relative maxima on the other to obtaina local minimum radius R min ( t ) and a local maximumradius R max ( t ) at time t . – definition of the guiding radius at t by the average ofthese two radii: (cid:104) R (cid:105) ( t ) = R min ( t ) + R max ( t )2 .Figure 7 shows examples of this determination of theguiding radius for two random stellar particles. The posi-tions of the particles are centred on the centre of mass ofthe whole galaxy at each time-step for this analysis. Thismethod directly provides the guiding radius and also theamplitude of the radial oscillations at any time t . R [ k p c ] Fig. 7.
Determination of the guiding radius for two randomstellar particles. Blue lines: galactocentric radii R ( t ). Redlines: R min ( t ). Green lines: R max ( t ). Black lines: guidingradii (cid:104) R (cid:105) ( t ) = R min ( t ) + R max ( t )2 . Our aim is to compare the overall population of migratingstars, by blurring and churning, with that of migrators bychurning alone. For this, we first examined the distributionsof the variations of radius (/guiding radius) as a functionof the initial radius (/guiding radius), for different time-intervals between 1 and 9 Gyr of evolution. The top setof plots of Fig. 8 shows the difference of radii of stellarparticles at final time t f and initial time t i as a function ofthe radius at t i , while the bottom set shows the difference ofguiding radii as a function of the guiding radius at t i , where t i and t f are given a range of values. Thus, in the top set ofplots the whole population of migrators is shown, whilst inthe bottom plots only migrators by churning are selected.The considered stellar components are the old stellar discand the new stars that are formed before t i . The RMS ofthe variations are indicated for each time interval.In each of the panels of Fig. 8, the shaded areas showthe span of the values of the ILR (blue), CR (red) and OLR(green) radii of the bar determined from Fig. 3 at t i and
7. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
Fig. 8.
Distributions of the variations of galactocentric radii (top half) and guiding radii (bottom half). The colour ofeach bin signifies the stellar mass in the bin and the colour scale shown at the bottom of the figure applies everywhere.The shaded areas represent the bar ILR (blue), CR (red), and OLR (green) radii variation in the time-span of each plot.The red vertical lines are the average of the bar CR radius during the time-span of a plot. The diagonal red lines helpidentify the migrators that cross the bar CR radius from lower radii (dot-shaded part of the schematic plot) or fromlarger radii (grid-shaded part). The diagonal green lines similarly help identify migration with respect to the OLR radiusat the end of the time interval.
8. Halle et al.: Blurring, churning, and the outer regions of galaxy discs t f . The vertical solid line is the average of the corotationradius and the diagonal line equation is y = R CR − R i forthe radii plots, y = R CR − (cid:104) R i (cid:105) for the guiding radii plots.The regions between the diagonal and vertical lines allowestimating the migrators that cross the corotation radiusby blurring+churning or by churning alone: – In the region corresponding to the dot-filled region ofthe schematic diagram, the stars in the radii plots have R i < R CR and R f > R CR , and those on the guidingradii plots have (cid:104) R i (cid:105) < R CR and (cid:104) R f (cid:105) > R CR . – In the region corresponding to the grid-filled region ofthe schematic diagram, the stars in the radii plots have R i > R CR and R f < R CR , and those on the guidingradii plots have (cid:104) R i (cid:105) > R CR and (cid:104) R f (cid:105) < R CR .Analogously, in each panel, the vertical dashed line showsthe position of the OLR at t = t f , and the dasheddiagonal line equation is y = R OLR − R i for the radiiplots, y = R OLR − (cid:104) R i (cid:105) for the guiding radii plots. Theregions between the diagonal and vertical dashed linesthus allow estimating the migrators that cross the OLR,at its final position in the time interval considered, byblurring+churning or by churning alone. We use thesecriteria in Sects. 4.3 and 5.The results presented in Fig. 8 can be summarised asfollows:1. As a general behaviour, the distribution for the wholepopulation and for migrators by churning alone showssome features that appear as diagonal structures. Thesefeatures are seeded by several resonances that are due tothe presence of the bar and of the spiral arms. The mostprominent diagonal features occur around the corota-tion of the bar, the main source of migrators in thestellar disc, as in Minchev & Famaey (2010); Minchevet al. (2011); Brunetti et al. (2011).2. Quantifying radial migration by means of the instan-taneous difference between radii at two different timesleads to overestimating churning both in terms of max-imum extent and RMS of the variations and in terms offraction of migrators (see further discussion in Fig. 9).As an example, the RMS values for the 1 to 3 Gyr time-interval is 2.5 kpc for the variations of radius, while it isonly 1.8 kpc for the variations of guiding radius (we notethat the absolute values are specific to our model andmay not be directly applicable to any specific galaxy,MW included). These differences are due to the radialexcursions that can cause a stellar particle to have aradius between (cid:104) R (cid:105) ( t ) − A ( t ) and (cid:104) R (cid:105) ( t ) + A ( t ), where A ( t ) = R max ( t ) − (cid:104) R (cid:105) ( t ) is the semi-amplitude of theradial oscillations at time t . The radius R ( t ) at time t can be expressed as R ( t ) = (cid:104) R (cid:105) ( t ) + r ( t ) , (5)where r ( t ) is in the interval [ − A ( t ) , A ( t )]. The vari-ation in radius R between time t i and t f is thus R ( t f ) − R ( t i ) = (cid:104) R (cid:105) ( t f ) − (cid:104) R (cid:105) ( t i ) + r ( t f ) − r ( t i ) , (6)and r ( t f ) − r ( t i ) is in the interval [ − ( A ( t f ) + A ( t i )) , A ( t f ) + A ( t i )]. Table 3.
RMS values of the variations of radius induced byblurring alone, at different times. See Fig. 8 for a compari-son with the variations obtained for the whole population ofmigrators and for those migrated by churning alone. Notethat these values are obtained with an isolated galactic discwith specific size parameters (possibly more extended thanthe Milky Way disc) and are not applicable to any specificgalaxy. t f = 3Gyr t f = 5Gyr t f = 7Gyr t f = 9Gyr t i = 1Gyr 1.73 kpc 1.73 kpc 1.68 kpc 1.95 kpc t i = 3Gyr 1.84 kpc 1.83 kpc 2.04 kpc t i = 5Gyr 1.90 kpc 2.05 kpc t i = 7Gyr 2.13 kpc
3. The migration in terms of change of radius (=blur-ring+churning) keeps approximately the same distri-bution and RMS value on time-intervals of the samelength, although the amplitude and RMS of the migra-tion in terms of change of guiding radius (=churningalone) are smaller at late times. For example, for time-intervals of 2 Gyr, the RMS value of the radius varia-tion only varies from 2 . . t = 1 to3 Gyr to 1.1 kpc between t = 5 and 7 Gyr or t = 7 and9 Gyr. This difference is due to the increase, with time,of the amplitude of radial oscillations, that is, blurring,as quantified in Table 3, where the RMS of the distri-bution is given for the same time intervals as shownin Fig. 8. In particular, from the comparison of the val-ues given in Table 3 with those of churning alone givenin the bottom panels of Fig. 8, it can be noted that inthe early phase of disc evolution, the spatial variationsinduced by churning overwhelm those due to blurring.This trend in reversed already after 3 Gyr of disc evolu-tion, when the extent of the radial variations by blurringdominates the variations induced by churning alone.4. The disc experiences its most intense phase of migra-tion by churning in the very early phase of its evolu-tion, when the stellar bar forms from an axisymmetricpotential and it is thin and strong. After the buckling in-stability and the formation of the boxy/peanut-shapedbulge at t ∼ The RMS values were evaluated as the quadratic differencebetween the RMS value of the overall population of migratorsand that migrated by churning alone. 9. Halle et al.: Blurring, churning, and the outer regions of galaxy discs x [kpc] M ( | ∆ R | > x ) / M d i s c All stars1 to 3 Gyr
10 5 0 5 10 x [kpc] M ( | ∆ R | > | x | ) / M d i s c Outward migrationInward migration x [kpc] M ( | ∆ R | > x ) / M ( R i ) R i = ± x [kpc] M ( | ∆ R | > | x | ) / M ( R i ) x [kpc] R i = ± x [kpc] x [kpc] R i = ± x [kpc] Fig. 9.
Migration between 1 and 3 Gyr. First line: Massfraction of stars that migrate by more than x kpc inwardsor outwards in terms of radius (solid line) or guiding radius(dashed line). Second line: Mass fraction of stars that mi-grate outwards (right half of the plot) or inwards (left part)by more than x kpc in terms of radius (solid) or guidingradius (dashed). Third and fourth lines: Same study, butonly for stars with a radius (solid lines) or a guiding radius(dashed lines) around R i .represent in the top panels of Fig. 9 the mass fraction ofthe stars that migrate by more than x kpc in terms of ra-dius or guiding radius from 1 to 3 Gyr. Figures 10 and 11then show the same fraction, but for stars migrating in thetime interval between 5 and 7 Gyr and over the whole in-terval 1-9 Gyr, respectively. Figure 9 for example showsthat between 1 and 3 Gyr, (cid:39)
40% of the ensemble of stars x [kpc] M ( | ∆ R | > x ) / M d i s c All starsAll stars5 to 7 Gyr
10 5 0 5 10 x [kpc] M ( | ∆ R | > | x | ) / M d i s c Outward migrationInward migration Outward migrationInward migration x [kpc] M ( | ∆ R | > x ) / M ( R i ) R i = ± x [kpc] M ( | ∆ R | > | x | ) / M ( R i ) x [kpc] R i = ± x [kpc] x [kpc] R i = ± x [kpc] Fig. 10.
Migration between 5 and 7 Gyr. First line: Massfraction of stars that migrate by more than x kpc inwardsor outwards in terms of radius (solid line) or guiding radius(dashed line). Second line: Mass fraction of stars that mi-grate outwards (right half of the plot) or inwards (left part)by more than x kpc in terms of radius (solid) or guidingradius (dashed). The yellow lines in these two plots are thecorresponding migration from 1 to 3 Gyr (identical to thecurves of Fig. 9). Third and fourth lines: Same study, butonly for stars with a radius (solid lines) or a guiding radius(dashed lines) around R i .(with mass-weighting accounting for the different masses ofstellar particles) change their instantaneous radius by morethan 2 kpc, while this fraction reduces to only (cid:39)
20% fora change of the guiding radius of the same amplitude. Theplotted fraction decreases more rapidly with x for the vari-
10. Halle et al.: Blurring, churning, and the outer regions of galaxy discs x [kpc] M ( | ∆ R | > x ) / M d i s c All starsAll stars1 to 9 Gyr
10 5 0 5 10 x [kpc] M ( | ∆ R | > | x | ) / M d i s c Outward migrationInward migration Outward migrationInward migration x [kpc] M ( | ∆ R | > x ) / M ( R i ) R i = ± x [kpc] M ( | ∆ R | > | x | ) / M ( R i ) x [kpc] R i = ± x [kpc] x [kpc] R i = ± x [kpc] x [kpc] R i = ± x [kpc] Fig. 11.
Migration between 1 and 9 Gyr. First line: Massfraction of stars that migrate by more than x kpc inwardsor outwards in terms of radius (solid line) or guiding radius(dashed line). Second line: Mass fraction of stars that mi-grate outwards (right half of the plot) or inwards (left part)by more than x kpc in terms of radius (solid) or guidingradius (dashed). The yellow lines in these two plots are thecorresponding migration from 1 to 3 Gyr (identical to thecurves of Fig. 9). Third and fourth lines: Same study, butonly for stars with a radius (solid lines) or a guiding radius(dashed lines) around R i .ation of guiding radius and it extends to a lower maximumvalue. The plots in the second row show the detail of theinward and outward migration. In the negative x left partof the plot, the curves represent the fraction of stars mi-grating inwards by more than | x | kpc, while in the rightpositive x part, the curves represent the fraction of starsmigrating outwards by more than x kpc. The curves arealmost symmetric with respect to the x = 0 vertical axis.The migration has a lower amplitude in terms of guiding ra- dius than in terms of radius for both inwards and outwardsmigration. The lower panels show the same study for starsin specific initial radius or guiding radius bins. More specif-ically, in these last two rows we quantify the stellar massmigrating of more than ∆ R (showing both the absolute andthe real value of the variation) from some initial radii R i ,normalising this stellar mass to the mass initially containedat R = R i . In particular, we chose to analyse the fraction ofmigrating stars with respect to three different initial radii,around R i = 5 kpc, R i = 10 kpc and R i = 20 kpc, cor-responding to radii between the ILR and the CR, the CRand the OLR, and outside the OLR, respectively. From in-spection of these plots, we can deduce the following:1. Over the time period 1-3 Gyr, the whole population ofmigrators moves limitedly with respect to its initial ra-dius, independently of the location of this initial radiuswith respect to the bar’s resonances. As an example, thefraction of stars that has a variation of its radius in ab-solute values greater than | ∆ R | = 4 kpc is about 10% at R i = 5 kpc, and 20% at R i = 10 kpc and R i = 20 kpc.These fractions decrease when considering stars that ex-perience churning, the fraction of stars that change theirguiding radii by more than 4 kpc is smaller than 10%for all values of the initial radius inspected.2. If one considers the sign of the displacement (bottompanels in Fig. 9) – with positive ∆ R corresponding tooutward migration and negative ∆ R to inward migra-tion – , one sees that migration is, in general, not sym-metric. For radii between the ILR and the CR, there isan excess of stars migrating outwards, ∆ R = − R i be-ing the largest possible extent of inwards migration formigrators originating at R = R i . Between the CR andthe OLR, there is an excess of inward migrators, bothin terms of blurring+churning and in terms of churningalone. We note in particular that outward migrators bychurning do not cross the OLR at t=3 Gyr, the fractionof stars with ∆ R > R i = 5 kpc). We discuss this importantfinding in more detail in the next section. Finally, mostof the stars that initially were located at R i = 20 kpc,thus outside the OLR between 1 and 3 Gyr, do not reachthe OLR, with migration mostly spatially redistributingstars that are ab initio in the outer disc. Moreover, thereis a slight excess of inward migrators by blurring in thisregion.The results found in Fig. 9 are confirmed when analysingmigration over the same time duration (2 Gyr), but for adifferent time interval (5-7 Gyr). Figure 10 shows the samestudy as Fig. 9, for the time period from 5 to 7 Gyr, withthe bottom panels detailing the migration for initial radiilocated again between the ILR and the CR ( R i =7 kpc),between the CR and the OLR ( R i =15 kpc), and beyondthe OLR ( R i =25 kpc). Because the bar slows down withrespect to earlier times, the initial radii R i are now moreexternal than those chosen for the analysis shown in Fig. 9.The striking difference between migration between 5 and7 Gyr, compared to the time interval 1-3 Gyr (Fig. 9), isthe even smaller change in guiding radius experienced by
11. Halle et al.: Blurring, churning, and the outer regions of galaxy discs the stars at these late times: there are no migrators bychurning with | ∆ R | > R i = 15 kpc at t i = 5 Gyr isabout 3.5 kpc, which is not enough to cross the OLR, whoselocation is at R OLR = 20 kpc at time t = 7 Gyr.It is not straightforward to identify the reason of thedecrease of the importance of churning with time that isobserved in our modelled galaxy. It may be due to a con-comitance of factors that may be difficult to separate. Thisdecrease with time can be explained by the evolution of thebar, and of its strength, the influence of other resonancesin the disc, and the kinetic state of the disc. As an exam-ple, Fig. 6 shows that after an abrupt decline, the bar takestime to regain strength in the final several Gyr. It is thennot as concentrated in the galactic plane (it has a peanutshape), and the tangential force it exerts on stars in theplane is thus weaker than at earlier times. In the periodfrom 1 Gyr to about 3 Gyr, there is also a phenomenonof resonance overlap that can increase the radial migration(see Minchev & Famaey (2010). The disc also gradually be-comes hotter (as we show in Sect. 6, see especially Figs. 16and 21), and stars thus become less responsive to potentialperturbations.Finally, when analysing migration over the whole timeinterval 1-9 Gyr, as done in Fig. 11, one can see that themaximum displacement of migrators has increased. This isbecause stars have had time to cross a larger portion of thedisc. For example, 5% of migrators by churning migratedby more than | ∆ R | = 6 kpc, whilst this fraction was nullbetween 1 and 3 Gyr. But when examining the details of mi-gration, it is still valid that the amount of the displacementdepends both on its sign (inward versus outward migra-tors) and on the location of the initial radius R i . We notein particular that the largest displacement of outwards mi-grators occurs in the region between the ILR and the CR( R i = 6 kpc and R i = 9 kpc, respectively), but that evenover such a long time interval, these outwards migrators donot cross the final OLR radius. Furthermore, blurring sig-nificantly contributes to shaping the outer disc – comparefor example the fraction of outwards migrators in the outerdisc ( R i = 30 kpc) by churning+blurring versus churningalone. We now attempt to determine which migration effect dom-inates at some radius (migration only due to epicyclic ex-cursions away from a guiding radius or a change of guidingradius), before computing the fraction of different types ofmigrators at a final radius.A quantification of a ‘migration flux’ as a function of theradius can be obtained by computing the mass fraction ofthe disc that crosses a given radius. We represent in the topplots of Fig. 12 as a function of the galactocentric radius R – the whole migration flux, that is, the mass fraction ofthe stellar disc that crosses R , the crossing of R beingquantified in terms of radius, that is: R i < R , R f >R from time t i to time t f for outward migration (solidlines) and R i > R , R f < R for inward migration (dashedlines) – the migration flux by churning alone , that is, the massfraction of the stellar disc that crosses R in terms ofradius and also of guiding radius: R i < R , R f > R and (cid:104) R i (cid:105) < R, (cid:104) R f (cid:105) > R for outward migration (solid lines)and R i > R , R f < R and (cid:104) R i (cid:105) > R, (cid:104) R f (cid:105) < R for inwardmigration (dashed lines).Both fluxes in the top panels of Fig. 12 are normalised tothe total stellar mass in the disc.The whole outward migration flux at the corotation ra-dius is thus the portion of stars that belongs to the dot-filledarea of the schematic diagram of Fig. 8 in the R f − R i vs R i plots, and the whole outward migration flux at a radius R different from the corotation radius is simply obtained byshifting the threshold radius and the diagonal line accord-ingly. Similarly, the whole inward migration corresponds tothe grid-filled area of the schematic diagram. The migra-tion flux by churning alone is obtained by selecting starsthat are both in the shaded zones of the radii plots and ofthe guiding radii plots (top and bottom panels in Fig. 8,respectively). The migration flux by churning alone is thusalways inferior to the whole migration flux because it isa fraction of it. The inner kpc inward migrators (both interms of change in their instantaneous radius or in theirguiding radius) are stars captured by the bar. Some othermuch smaller peaks are observed at large radii, correspond-ing to spiral patterns. However, in agreement with previousresults (see Introduction), the most significant part of themigration flux occurs near the corotation of the bar becausethe bar is the strongest potential perturbation in our sim-ulation and its corotation radius is located in a disc regionwith a high surface density. In discussing these plots, wewish to emphasize two points: – The finding that the corotation is the locus of thestrongest flux of migrators by churning can be partic-ularly appreciated considering the time intervals of dif-ferent lengths, for example comparing the time interval1-3 Gyr with those at 1-5 Gyr, 1-7 Gyr, and 1-9 Gyr.The corotation spans an increasingly larger radial ex-tent as the duration of the time interval grows. As aresult, the flux through the corotation region, which ismostly a thin spike around the CR for short time inter-vals (cf. 1-3 Gyr), transforms into a sort of large plateaufor longer time intervals, whose extent corresponds tothe spatial extent spanned by the corotation during thecorresponding time. – The dominant role of churning near corotation can alsobe appreciated in the bottom panels of Fig. 12, where weshow the fraction of stars migrated by churning with re-spect to the whole sample of migrators crossing a givenradius R . While more than 60% of migrators crossingthe corotation region are migrating by churning, thisfraction significantly decreases for other regions of thedisc. In particular, it can be appreciated that the frac-tion of migrators by churning decreases for later timesoutside corotation. This confirms the growing impor-tance of radial heating (i.e. blurring) as time increases(see also discussion in the previous section).
12. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
R [kpc] D i s c f r a c t i o n m i g r a t i n g t h r o u g h R Blurring+churning outward fluxBlurring+churning inward fluxChurning outward fluxChurning inward flux
R [kpc] C hu r n i n g / ( b l u rr i n g + c hu r n i n g ) f l u x e s Churning/(blurring+churning) outward fluxesChurning/(blurring+churning) inward fluxes
Fig. 12.
Top half: Mass fraction of the stellar disc crossing a radius R in terms of radius (apparent migration) or bothradius and guiding radius (true migration). Bottom half: Ratios of the true to apparent migrators. The shaded areasrepresent the bar ILR (blue), CR (red), and OLR (green) radii variation in the time-span of each plot.
13. Halle et al.: Blurring, churning, and the outer regions of galaxy discs R f [kpc] F r a c t i o n o f m i g r a t o r s a t R f R f -R i >2 kpcR f -R i < -2 kpcR f -R i >4 kpcR f -R i < -4 kpcR f -R i >6 kpcR f -R i < -6 kpc R f [kpc] F r a c t i o n o f m i g r a t o r s a t R f
Fraction of migrators as a function of radius (top half) or guiding radius (bottom half) at the end of the time-span specified in each plot. The shaded areas represent the bar ILR (blue), CR (red), and OLR (green) radii variation inthe time-span of each plot. Note that these values are obtained with an isolated galactic disc with specific size parameters(possibly more extended than the Milky Way disc) and are not applicable to any specific galaxy.
14. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
In the previous sections, we have quantified the mainsources of migration in the disc and the flux of migrat-ing stars as a function of distance from the galaxy centre.We now discuss where migrators are redistributed in thedisc, and in particular, how many migrators can be ex-pected at different radii throughout the disc. To this aim,we represent in Fig. 13 the fraction of stars of a radius binaround R f at t f that have migrated by more than n kpc( n =2, 4, and 6) in radius since t i (top half of the fig-ure) or whose guiding radius has changed by more than n kpc since t i (bottom half). These fractions peak on bothsides of the CR radius of the bar, consistently with themigrating fluxes of Fig. 12, which peak at the bar CR ra-dius: radii R f > R CR receive stars migrating outwards from R i < R CR , while radii R f < R CR receive stars migratinginwards from R i > R CR . However, as already discussed pre-viously (see also next section), inward and outward migra-tors cannot cross the whole disc, and in particular, outwardmigrators originated around the CR cannot reach the outer( R > R
OLR ) regions.At first glance, the fraction of the whole population ofmigrators (churning+blurring) seems very high at certainradii (see top panels in Fig. 13): as an example, in the timeinterval 1-9 Gyr, 40% of the stars in the CR region arestars that have reached this region by migrating by morethan 2 kpc from some inner radii by blurring or churning;in the OLR region between 40% and 50% of stars are out-ward 2 kpc–migrators; outside the OLR, the contribution ofthese stars to the local population increases nearly mono-tonically up to 80%. This high fraction is due to the ex-ponentially declining stellar surface density that causes in-wards migrators to easily constitute a large part of the starsat an external final radius. The population of more extrememigrators – those that have migrated by more than 6 kpcoutwards by blurring or churning – are still important con-tributors to the local (= at a given R f ) stellar population:the fraction has a local maximum in the OLR region, atradii < R OLR ( t = t f ), where it peaks at ∼ R f = R OLR , to finally increase again nearlymonotonically in the outer disc. However, when we examinethe contribution of migrators by churning alone at a givenradius (bottom panels of Fig. 13), we can notice some dif-ferent trends and absolute values. In particular, extreme(∆
R > f [kpc]0123456 σ [ k p c ]
RMS values of galactocentric and guiding radii at1 Gyr of stars with galactocentric radii in 2 kpc wide binsaround R at 9 Gyr. The vertical dashed lines, as in Fig. 14,are the CR (red) and OLR (green) radii at 9 Gyr. Thevertical solid green line is the OLR radius at 1 Gyr.Fig. 14 shows the distribution of guiding radii at 9 Gyrof stars whose radii are in different bins (shown in shadedcolours). The histograms, especially at large radii, tend topeak at smaller radii than the average radii of the shadedarea, exhibiting the well-known asymmetric drift effect. Thesecond row shows the distribution of guiding radii at 1 Gyrof the same stars (with guiding radii in the shaded bins at9 Gyr). Comparison of the top and middle row shows thatthe distributions of guiding radii change between the initialand final times. The amplitude of the distributions tend todecrease at all radial bins (this is also observed by Minchevet al. (2014)). The initial distributions of (cid:104) R i (cid:105) , skewed to-wards low radii, translate into distributions of (cid:104) R f (cid:105) that aremuch more symmetric. The differences between the distri-butions of initial and final guiding radii is evidence thatmigration by churning has occurred, and in particular, theincrease of guiding radii with time indicates a predomi-nance of outward migration. The bottom line shows thegalactocentric radii of the same stars at 1 Gyr. The starsclearly have migrated significantly. The distributions arevery similar to that of the middle line, indicating that nearthe beginning of the simulation, at t =1 Gyr, most of thestars are in circular orbits. We compare the distributions ofinitial radii and guiding radii more quantitatively in Fig. 15by representing the RMS value of the initial galactocentricradii or guiding radii of stars in bins of final radius (weuse overlapping bins of 2 kpc width as in Minchev et al.(2014)). As in Minchev et al. (2014), we see these RMS val-ues, estimates of the widths of the histograms of Fig. 14,are generally slightly higher for the initial galactocentricradii. The implications of Figs. 14 and 15 on the mixing ofthe disc are discussed in more detail in the next section.
5. OLR: a barrier for migrators
In the previous section, we have reported that migratorsby churning do not cross the OLR. In this section wedevelop this point further.As previously described, the bottom panels of Fig. 8 show
15. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
Distributions of final guiding radii (top), initial guiding radii (middle), and initial galactocentric radii (bottom)of stars whose final galactocentric radii belong to bins of 2 kpc width. Each curve corresponds to the closest same-colourshaded bin of final galactocentric radii. The time interval is from 1 Gyr to 9 Gyr. The vertical dashed lines are the CR(red) and OLR (green) radii at 9 Gyr. The vertical solid green line is the OLR radius at 1 Gyr. In the two bottom lines,the histograms corresponding to stars in radius bins outside the OLR radius at 9 Gyr have been plotted with thickerlines (see next Section). The stellar masses M in bins of each histogram are divided by the largest mass M max containedin a bin of the histogram.the variation ∆ R = (cid:104) R f (cid:105) − (cid:104) R i (cid:105) of guiding radii as a func-tion of their initial guiding radius R i in the time interval[ t i , t f ]. The dashed vertical line indicates the position of theOLR at t f , that is, at the end of the time interval underconsideration, and the diagonal dashed line the equation∆ R = R OLR ( t f ) − (cid:104) R i (cid:105) . It is evident from these figuresthat at all times, stars with (cid:104) R i (cid:105) < R OLR ( t i ) do not crossthe position of the OLR at the final time t f , these stars arealways below the line ∆ R = R OLR ( t f ) − (cid:104) R i (cid:105) . Migrators inthe OLR region in the time interval under consideration,that is, with R OLR ( t i ) ≤ R i ≤ R OLR ( t f ), can cross theOLR, but these are stars that at least at t = t i had guidingradii all beyond the initial OLR position R OLR ( t i ). Inother words, the only migrators by churning in a barredgalaxy at any given time that can be found beyond theOLR are stars migrating from a region between the OLRat the initial time of bar formation and the current OLRposition. The larger the variations of the pattern speedof the bar over time, and consequently, the larger the variations ∆ R OLR = R OLR ( t f ) − R OLR ( t i ) of the OLRposition, the larger will be the region of the inner discwhere migrators crossing the OLR can originate from. Inno case, however, can stars born at corotation cross thefinal OLR. Analogously, if we were to draw a diagonal lineof equation ∆ R = R OLR ( t i ) − (cid:104) R i (cid:105) in the bottom panels ofFig. 8, this line separating stars that can cross the initialposition of the OLR, we would find that no stars with (cid:104) R i (cid:105) ≥ R OLR ( t f ) would migrate inward crossing the radius R = R OLR ( t i ). This suggests that the OLR region, thatis, the region between the initial (i.e. at the epoch of barformation) and final (i.e. current) position of the OLRis a transition region for a disc galaxy, the only regionwhere migrators by churning can be exchanged betweenthe inner and the outer disc. But neither stars with (cid:104) R i (cid:105) ≤ R OLR ( t i ) can cross the OLR, nor, in the oppositesense, stars with (cid:104) R i (cid:105) ≥ R OLR ( t f ) can penetrate below theradius R i = R OLR ( t i ). Note that some more contaminationof the regions around the OLR can be caused by stars
16. Halle et al.: Blurring, churning, and the outer regions of galaxy discs migrating by blurring (top panels of Fig. 8), even if, ingeneral, these polluters originate in regions outside thosespanned by the CR. The middle row of Fig. 14 also showsthe very limited mixing between the regions inside theinitial OLR (at t =1 Gyr) and the regions outside thefinal OLR (at t =9 Gyr). There are no stars that migratedby churning from inside the initial OLR and, vice versa,inside the initial OLR, there are no stars coming fromoutside the final OLR. The lack of contamination of thedisc region outside the final OLR radius by the regionsinside the initial OLR radius can be seen from the extentof the distributions of the initial guiding radii of starsthat are beyond the final OLR (plotted as thicker lines).For example, none of the stars ending up at the finaltime between 24 and 26 kpc (that is, just outside theOLR at the final time t =9 Gyr) come from radii below R OLR ( t =1 Gyr)=11 kpc, whereas stars belonging to thebins centred between 5 and 21 kpc, inside R OLR ( t =9 Gyr),all have distributions of (cid:104) R i (cid:105) that can reach values aslow as 3 kpc. Figure 15 shows that RMS values of initialgalactocentric or guiding radii of stars in a final galacto-centric radius bin are monotonically increasing from theinnermost regions up to a radius inside the final OLR.After this maximum, the RMS values tend to decrease inthe outer disc, indicating again that there is no accumula-tion of migrators from the inner disc outside the final OLR.To our knowledge, this result has not been pointedout before, and it recalls the similar barriers encounteredby the gas in a barred galaxy (Schwarz 1981; Combes1988): similarly to stars, gas in the CR-OLR region canalso gain angular momentum to reach the OLR position,at most. Similarly to stars (Fig. 13, bottom panels), gasaccumulates in the OLR region, but differently from thestellar component, gas is then able to dissipate part of itsenergy, shocks and forms rings of concentrated materialat the OLR. Similarly, gas outside the OLR is not able topenetrate this resonance and access the inner region of thedisc, until the barrier is removed (because of a significantchange in the bar properties, see for example Combes(2011)).The results we presented are based on the analysis ofa single simulation, and one may have concerns about thegenerality of the findings discussed. It is difficult to comparewith other N-body works presented in the literature, eitherbecause they usually lack information about the position ofthe OLR at initial times and only give its average positionor position at the final time analysed or because the OLRis too close to the initial boundary of the simulated stellardiscs to make robust predictions (see for example Minchevet al. (2012b); Di Matteo et al. (2013)). However, an inter-esting comparison can be made with test particle simula-tions: in these experiments, the pattern speed of the mainasymmetry is kept fixed, and as a consequence, the posi-tion of the OLR does not change with time; the role playedby the OLR in these cases can be discussed. A comparisonwith the work by Minchev & Famaey (2010), for example,shows that when only a bar pattern is present in the disc(their Fig 2, left panels), the angular momentum exchangeinduced by the bar diminishes closer to the OLR and stopsvery clearly outside it, with a null angular momentum vari-ation at radii outside the OLR. Independently of the barstrength, their experiments confirm the robustness of our results: the angular momentum exchange is only allowedinside the OLR, and, in particular, the maximal variationof angular momentum produced at the CR (where most ofthe migrators by churning are generated) is not sufficient toallow those stars to cross the outer resonance. In the rightpanels of the same figure, a similar analysis is made for asingle spiral pattern. Unfortunately, the OLR is outside thex-range shown in these plots, and as a consequence, is notpossible to discuss where the exchange of angular momen-tum stops. This can be seen in the work by Sellwood &Binney (2002), however. Their constrained model, contain-ing a unique spiral pattern, shows that the OLR limits theexchange of angular momentum, with any variation stimu-lated at the OLR being extremely local and concerning avery limited number of stars. Of course, the response of adisc to asymmetric perturbations changes when many per-turbations of similar amplitudes are superimposed, as is thecase of several transient spiral patterns (Sellwood & Binney(2002), but also Roˇskar et al. (2008) and Vera-Ciro et al.(2014)) or of the overlap of a bar and a spiral pattern ofsimilar strengths (see, for example, Fig. 3 of Minchev &Famaey (2010)). But these cases are different from the casediscussed here, where a unique main pattern is present (thebar), with possibly an overlap with weak spiral arms, es-pecially in the first 2 Gyr of evolution of the system. Inthis case, most of the angular momentum exchange occursnear the corotation of the bar, far from the OLR, and thecontamination of the outer disc ( R > R
OLR ( t f ) with starsmigrated from the inner disc ( R < R
OLR ( t i ) is very weakor null. In this case of a growing bar, the stopping of an-gular momentum exchange at the OLR of the bar is more-over consistent with the mechanism of bar growth: barsstrengthen and grow by receiving angular momentum fromstars near their ILR and by giving angular momentum tostars near their OLR (Lynden-Bell & Kalnajs 1972). Thereis thus some mixing close to the OLR, but beyond the OLRregion there are no more resonant stars that may exchangeangular momentum with the bar.Why is this result important? For the Milky Way, atleast for two reasons:1. It may explain why the inner and the outer discs of theGalaxy have been able to maintain two different stellarpopulations over a time interval of ∼
10 Gyr, which prob-ably corresponds to the whole interval of secular evolu-tion experienced by the Galaxy. We recall here the re-sults of Haywood et al. (2013), who extensively showedand discussed that the chemical properties of outer-discstars ( (cid:104) R (cid:105) ≥
10 kpc) are substantially different fromthose of stars of the inner disc of similar ages. Theseresults have been confirmed by Anders et al. (2014) andNidever et al. (2014), who also showed that the chem-ical properties of the outer disc are significantly differ-ent from those of the inner disc (see also Bensby et al.(2012), for a high-resolution spectroscopy study of theinner and outer disc). The current position of the OLRin the Milky Way is estimated to be slightly inside thesolar radius by some observational or theoretical stud-ies (Dehnen 2000; Famaey et al. 2005; Minchev et al.2007). This may therefore explain why the Sun appearsto be in a transition region (Haywood et al. 2013) andwhy the outer disc does not show signatures of any sig-nificant pollution by stars originating in the inner disc( R ≤ −
17. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
2. Depending on the exact location of the Sun with respectto the OLR, the solar vicinity may have been stronglypolluted by stars migrating by churning from the innerdisc (see Fig. 13, bottom panels, for stars inside the finalOLR position), or not at all. If the Sun is outside theOLR position, as has been deduced by Dehnen (2000);Famaey et al. (2005); Minchev et al. (2007) and the baris the main source of asymmetric perturbations in theMilky Way, then the impact of churning at the solarvicinity may have been significantly overestimated (seeFig. 13, bottom panels, for stars beyond the OLR re-gion). In particular, stars that migrated by churning atthe solar vicinity may well originate in regions muchcloser to the solar radius than previously thought, andin this case, invoking substantial migration from thecorotation and the inner disc is not possible.This result may also lead to doubts about the interpreta-tion of U-turn in age profiles or inversion in colour-profilesfound in the outer disc of external galaxies. When these in-versions occur outside the OLR position for bar-dominatedgalaxies, it is difficult to explain them in terms of strongmigration from the inner disc, since, according to our re-sults, no strong migration from the corotation is expectedto contaminate the outer disc, unless another pattern withlower speed existed before the present one, with an OLRextending to outer radii.
6. Migration and cooling/heating
The effect of radial migration on the vertical structure ofdiscs has been studied in a number of works (Minchev et al.2012a; Solway et al. 2012; Roˇskar et al. 2013; Vera-Ciroet al. 2014, e.g.). The assumption that outward migrationwould help create a thick disc because of the higher verti-cal velocity dispersion of the outward migrators originatingfrom the inner hotter disc has been debated because of thevertical cooling outward migrators should undergo if thevertical action of a stellar orbit is conserved (Minchev et al.2012a; Solway et al. 2012; Roˇskar et al. 2013).We observe that the vertical velocity dispersion of thestellar disc slightly increases with time at all radii. In Fig. 16we plot the vertical velocity distribution of the old stellardisc, the stellar disc of stars formed during the simulation,and the gas disc. The velocity dispersion of the new stars,formed from the colder gas disc, is generally lower than thevelocity dispersion of the old stellar disc. The mass of newstars is only a small fraction of the total stellar mass (seeFig. 2), however, so star formation has no net cooling effecton the total stellar disc.We would like to know if the stars that are going tomigrate are a special sub-population in terms of kineticsof stars with the same initial guiding radius, whether themigration affects their kinetic state, and if the migratorsaffect the population of stars at their final guiding radius.Vera-Ciro et al. (2014) found a provenance bias of mi-grators in terms of kinetic state. They found that radialmigration driven by spiral arms (seeded in their simula-tions by massive perturbers) preferentially affect stars witha lower velocity dispersion than the average velocity disper-sion at the initial radius. We find a similar trend, as shownin Fig. 17, where the ratio of the vertical velocity disper-sion of stars in bins in (cid:104) R i (cid:105) and (cid:104) R f (cid:105) − (cid:104) R i (cid:105) (we only used σ z [ k m / s ] Fig. 16.
Time evolution of the vertical velocity dispersionof the old stellar disc without the bulge (solid), the youngstellar disc (dashed), and the gas disc (dotted-dashed).bins that contained at least the mass of ten old-disc stellarparticles) to the vertical velocity dispersion of all stars ina bin at (cid:104) R i (cid:105) (a column in the plots) is represented. Thestars migrating the most from an initial guiding radius tendto be colder in the z-direction than the average for all thestars at this initial guiding radius.We next investigated whether the migrators can be sub-ject to heating (or cooling) when they reach hotter (re-spectively colder) regions. In Fig. 18 we represent the dis-tribution of the ratio of the final vertical velocity disper-sion to the initial one for stars that have migrated by∆ R = (cid:104) R f (cid:105) − (cid:104) R i (cid:105) from their initial radius R i in a timeinterval [ t i , t f ]. We observe that, especially in the inner re-gions of the disc, that is, inside the OLR, outward migra-tors tend to lower their velocity dispersion, while inwardmigrators tend to increase it. The effect increases with theamplitude of the variation (it is more visible for the extremecases of migration).We now discuss how the final vertical velocity disper-sion of migrators compares to that of stars with the samefinal guiding radii (cid:104) R f (cid:105) . In Fig. 19 we represent the distri-bution of the ratio of the velocity dispersion of migrators ina bin in (cid:104) R f (cid:105) and (cid:104) R f (cid:105) − (cid:104) R i (cid:105) to the velocity dispersion ofall stars of guiding radius (cid:104) R f (cid:105) . We observe different trends,depending, among others, on the location of the radius R f under consideration. We start by discussing what occurs in-side the OLR at the final time. We observe that in the veryinner regions, stars migrating inwards to reach a guiding ra-dius (cid:104) R f (cid:105) have a higher velocity dispersion than the wholepopulation of stars at (cid:104) R f (cid:105) . These stars must be takingpart in the peanut-shaped bar with a high vertical veloc-ity dispersion. This agrees with the results reported by DiMatteo et al. (2014), who found that outside-in migratorsparticipating in a boxy/peanut shaped structure tend tobe kinematically hotter than in situ stars. At higher radii,however, the inward migrators tend to have lower velocitydispersions than the average at their final guiding radius.Outward migrators tend to have a slightly higher ( ∼ R variations, found between the CR and the
18. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
Ratio of vertical velocity dispersion of stars in bins in (cid:104) R i (cid:105) and (cid:104) R f (cid:105) − (cid:104) R i (cid:105) to the vertical velocity dispersion ofall stars in radial bin centred on (cid:104) R i (cid:105) . The shaded areas represent the bar ILR (blue), CR (red), and OLR (green) radiivariation in the time-span of each plot. The red vertical lines are the average of the bar CR radius during the time-spanof a plot.
Distribution of the ratio of final to initial vertical velocity dispersion of stars migrated by ∆ R = (cid:104) R f (cid:105) − (cid:104) R i (cid:105) from their initial guiding radius (cid:104) R i (cid:105) in the time interval [ t i , t f ]. The shaded areas represent the bar ILR (blue), CR (red),and OLR (green) radii variation in the time-span of each plot. The red vertical lines are the average of the bar CR radiusduring the time-span of a plot.OLR region. These extreme migrators can be significantlyhotter (up to 50% more) than the whole population at the same final guiding radius. We note that this effect is onlyevident when a selection on the amplitude of the migration
19. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
Distribution of the ratio of vertical velocity dispersion of stars in bins in (cid:104) R f (cid:105) and (cid:104) R f (cid:105) − (cid:104) R i (cid:105) to the verticalvelocity dispersion of all stars with guiding radii centred on the final value (cid:104) R f (cid:105) . The shaded areas represent the bar ILR(blue), CR (red), and OLR (green) radii variation in the time-span of each plot. The red vertical lines are the average ofthe bar CR radius during the time-span of a plot. σ z [ k m / s ] All starsNon R migratorsOutward R migratorsInward R migratorsNon
Fig. 20.
Vertical velocity dispersion at t = 3 Gyr at R of allstars, non-migrators, or migrators, that migrated by morethan 2 kpc in terms of radius or guiding radius from 1 to3 Gyr.∆ R is made. When the overall population of (inward oroutward) migrators is analysed as a whole, the final verti-cal velocity dispersions of migrators and non-migrators aremuch more similar, which is because apart from the ex-treme cases of migration, many stars that have migratedby small ∆ R tend to have a final kinematic similar to thatof the overall population. We can see this in Fig. 20. In thisfigure, we have plotted the vertical velocity dispersion of allstars as a function of radius at t = 3 Gyr and the velocity dispersion of inward and outward migrators that are in aradius bin and that have migrated by more than 2 kpc interms of radius or guiding radius since t = 1 Gyr, and thenon-migrators defined as the stars of the radius bin thathave not changed their radius or guiding radius by morethan 2 kpc. The inward migrators have a slightly lowervertical velocity dispersion, the outward migrators have aslightly higher velocity dispersion, but the total velocitydispersion is very close to the one of the non-migrators,indicating a weak effect of the overall migration on the lo-cal velocity dispersion. We insist, however, on the fact thatthe most extreme migrators, in particular the most extremeoutward migrators, can have dispersions significantly differ-ent (50%) from the average. While we agree with the resultsby Minchev et al. (2012a), therefore, that migration over-all contributes little to disc thickening, extreme migratorscan depart from the average small increase of the heatingfound in Fig. 20 and found also by Minchev et al. (2012a),contributing with significantly higher vertical velocity dis-persions to the whole sample of migrating stars that endup at the same final guiding radius. The radial velocity dispersion also increases on average withtime, as can be seen in Fig. 21. We can follow the amplitudeof the radial oscillations as well as the eccentricity of thetrajectories of the stellar particles defined as e = R max − R min R max + R min . (7)The eccentricity varies between 0 for circular orbitsand 1 for radial orbits. The mean eccentricity of the disc
20. Halle et al.: Blurring, churning, and the outer regions of galaxy discs σ R [ k m / s ] Fig. 21.
Time evolution of the radial velocity dispersionof the old stellar disc without the bulge (solid), the youngstellar disc (dashed), and the gas disc (dotted-dashed). A m p li t u d e [ k p c ] E cc e n t r i c i t y Fig. 22.
Time evolution of the amplitude of radial oscilla-tions (top) and eccentricity (bottom) with time.stars increases with time. This increase is mainly caused bythe growth of the bar: More and more stars are gradually
10 5 0 5 10
Change of eccentricity as a function of change ofguiding radius from 1 to 3 Gyr (top) and 5 to 7 Gyr (bot-tom). The colour scale represents the mass in each bin andis logarithmic. The solid black line is the average of thechange in eccentricity by bin of change of guiding radius(i.e. the average of the corresponding column).trapped by the bar and gain eccentricity as their orbits be-come elongated. The evolution of the amplitude of radialoscillations (2 A ( t ) where A ( t ) is the semi-amplitude intro-duced in Sect. 4.2) and eccentricity e as a function of radiuscan be seen in Fig. 22. The bar growth is visible from theincrease of eccentricity and amplitude with time in the in-ner kpc. The amplitude of radial oscillations increases onaverage with time at all radii, which is consistent with theincrease of radial velocity dispersion shown in Fig. 21 andwith the growing role of blurring with time, as discussedin Sect. 4.2. The eccentricity also slightly increases at allradii.It is interesting to investigate whether radial migrationchanges the radial amplitude and the eccentricity of thestars trajectories. In Fig. 23, we represent the change ineccentricity as a function of the change in guiding radiusbetween t = 1 and 3 Gyr in the top panel and between t = 5and 7 Gyr in the bottom panel. The distribution is broad.The majority of stars only slightly change their eccentric-ity, but on average (black line), stars migrating outwards
21. Halle et al.: Blurring, churning, and the outer regions of galaxy discs (with positive (cid:104) R f (cid:105) − (cid:104) R i (cid:105) ) tend to decrease their eccentric-ity while the opposite stands for stars migrating inwards.The inward effect increase of the eccentricity (especiallyvisible in the bottom panel of Fig. 23 from 5 to 7 Gyr) isagain dominated by the capture of stars by the central bar.The decrease of eccentricity does not necessarily mean thatthe orbits are circularised in terms of decrease in amplitudeof radial oscillations because of the denominator in the ex-pression of the eccentricity, which can increase in the caseof outwards migration. In Fig. 24 we therefore represent thevariation of the amplitude of radial oscillations and sepa-rate in the middle and right columns the disc into an ‘innerdisc’ and an ‘outer disc’ by a cut between the CR and OLRbar radii at 10 kpc for the first time-interval and at 15 kpcfor the second one, so that the effect of the bar growth onthe stellar orbits is not present in the ‘outer disc’ plots. Inthe inner-disc plot of the time-span from 5 to 7 Gyr, a cleartrend of an increase of the amplitude of the radial oscilla-tions is visible for inward migration, corresponding to starsmigrating inwards and contributing to the bar. All the dis-tributions are again broad, with the largest variations ofamplitude occurring for stars that do not migrate signifi-cantly. Stars that migrate outwards the most increase theirradial oscillations amplitude on average, but the possibleincrease is limited compared with the rest of the distribu-tion.
7. Conclusions
We have studied stellar radial migration in a simulation ofan Sb-type extended galactic disc. We confirmed the mainrole of corotations of the density resonances as seeds forradial migration in terms of both blurring and churningand observed a strong radial migration when there is a bar-spiral resonance overlap; this is consistent with the resultsof Minchev & Famaey (2010); Minchev et al. (2011).We have quantified the effects of blurring and churning.Migration defined as the simple difference in galactocentricradii between two times can lead to significantly overesti-mating the fraction of migrators by churning – and alsothe spatial extension of this migration. The intrinsic na-ture of the epicyclic orbits and the increase in the ampli-tude of radial oscillations with time – both effects beingincluded when migration is quantified by a change of in-stantaneous radius – can indeed give the impression of amuch stronger migration than that really experienced bystars in the galaxy. This is particularly true, for example,for the fraction of migrators in the outer disc of our simu-lated galaxy. When blurring is excluded, the fraction of ex-treme migrators contributing to the outer disc populationdecreases by a factor between 2 and 8 (compare Fig. 13,top and bottom panels, for the time interval 1 to 9 Gyr).Whilst the spatial extent covered by migrating stars in-creases with time, our simulation suggests that migratorsencounter barriers. In particular, stars migrating by churn-ing from corotation cannot cross the OLR, and vice versa,stars born beyond the OLR cannot reach the inner disc. TheOLR region – defined as the region between the position ofthe OLR at the epoch of bar formation, and at the finalepoch – is a transition region, the only region where somepollution between the inner and the outer disc is allowed.Even though our model is not intended to reproducethe Milky Way – the pattern speed of our simulated baris lower than the pattern speed measured for the Galaxy (Gerhard 2011), and in consequence, the main resonancesare located at much larger distances from the centre thanthose measured for the Milky Way, and the study assumesan initially already formed thin disc – we think that theprevious result might help understand the puzzling natureof the outer Galactic disc and its significantly different stel-lar populations. It has recently been shown that the MilkyWay outer disc followed a different chemical evolution his-tory than did the inner disc (Haywood et al. 2013; Snaithet al. 2014). Regardless of the formation mechanism of theouter regions of the Galaxy, stars there have been able toevolve independently of the inner disc. Our model suggeststhat the Galactic OLR and its location (estimated to belocated slightly inside the solar radius, see Dehnen (2000))have a major role in explaining this finding. The OLR in-deed acts as a barrier for gas (Combes 1988), but - as weshowed here - for stars as well, inhibiting migration fromthe inner to the outer disc and vice versa. This may explainwhy stars with inner thin disc chemistry are not observedin the outer disc (Haywood et al. 2013). This finding, if con-firmed with future studies, may also lead to doubts aboutthe interpretation of the U-shape in age profiles, or inver-sion in colour-profiles found in the outer disc of externalgalaxies. We do not yet know whether our Galaxy has theU-profile in stellar ages that is sometimes observed in exter-nal galaxies like M33. When these inversions are observedoutside the OLR position in barred galaxies, it is difficultto explain them in terms of strong migration from the in-ner disc, with the observed bar/spiral pattern speed, sincethey occur beyond their OLR. However, it is possible thatprevious bar/spiral waves have developed with lower pat-tern speeds, implying OLR and migration at larger radii.Another possible explanation is that the settlement of theGalactic disc in the outer regions permitted the formationof a significant number of old stars in situ, as proposed forthe Milky Way in Haywood et al. (2013) and Snaith et al.(2014).Finally we have analysed the kinematics of migratingstars.We confirmed the results found by Vera-Ciro et al. (2014)for spiral galaxies. Similarly to that case, also when migra-tion is mainly induced by a stellar bar, there is a provenancebias of migrators in terms of their kinetic state. The starsmigrating the most from an initial guiding radius tend tobe colder in the z-direction than the average of all the starsat the same initial guiding radius.We also confirmed the results by Minchev et al. (2012a): mi-gration contributes little to disc thickening, but we pointout, consistently with Minchev et al. (2012b), that there isa trend of increasing vertical velocity dispersion with theextent of migration: the most extreme outward migratorsthat end up at a given final radius tend to also have thehighest velocity dispersions when compared to the velocitydispersion of all the stars found at that final radius. Thus,while the overall effect of heating at a given radius is weak,we suggest that at a given radius, extreme outward migra-tors from the inner disc are identifiable as stars that havethe highest velocity dispersions among those measured forstars of the same age, at the same radius. We recall thatthis signature is different from that induced by mergers,which could heat the outer disc enough for extreme migra-tors from the central parts of the galaxy to have a coolingeffect on the outer disc (Minchev et al. 2014).
22. Halle et al.: Blurring, churning, and the outer regions of galaxy discs
10 5 0 5 10
10 5 0 5 10
Fig. 24.
Change of radial oscillation amplitude as a function of change of guiding radius from 1 to 3 Gyr (top) and 5 to7 Gyr (bottom). The colour scale represents the mass in each bin and is logarithmic. The solid black line is the averageof the change in amplitude by bin of change of guiding radius (i.e. the average of the corresponding column).Overall, our findings challenge the current view of theeffect that radial migration from the inner disc may have inthe outer regions of disc galaxies when only a main asym-metric pattern is present. This may fundamentally affectthe understanding of stellar populations in bar-dominatedgalaxies, which we will investigate in future studies.
Acknowledgements.
The authors wish to thank A. Gomez for hersupport, encouragements, and suggestions. This work also benefitedof several enriching discussions with D. Katz, M. D. Lehnert, andO. N. Snaith. AH thanks the Observatoire de Paris, which funded herwork through an ATER grant.
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