Formation and Evolution of the Disk System of the Milky Way: [alpha/Fe] Ratios and Kinematics of the SEGUE G-Dwarf Sample

Abstract

We employ measurements of the [alpha/Fe] ratio derived from low-resolution
(R~2000) spectra of 17,277 G-type dwarfs from the SEGUE survey to separate them
into likely thin- and thick-disk subsamples. Both subsamples exhibit strong
gradients of orbital rotational velocity with metallicity, of opposite signs,
-20 to -30 km/s/dex for the thin-disk and +40 to +50 km/s/dex for the
thick-disk population. The rotational velocity is uncorrelated with
Galactocentric distance for the thin-disk subsample, and exhibits a small trend
for the thick-disk subsample. The rotational velocity decreases with distance
from the plane for both disk components, with similar slopes (-9.0 {\pm} 1.0
km/s/kpc). Thick-disk stars exhibit a strong trend of orbital eccentricity with
metallicity (about -0.2/dex), while the eccentricity does not change with
metallicity for the thin-disk subsample. The eccentricity is almost independent
of Galactocentric radius for the thin-disk population, while a marginal
gradient of the eccentricity with radius exists for the thick-disk population.
Both subsamples possess similar positive gradients of eccentricity with
distance from the Galactic plane. The shapes of the eccentricity distributions
for the thin- and thick-disk populations are independent of distance from the
plane, and include no significant numbers of stars with eccentricity above 0.6.
Among several contemporary models of disk evolution we consider, radial
migration appears to have played an important role in the evolution of the
thin-disk population, but possibly less so for the thick disk, relative to the
gas-rich merger or disk heating scenarios. We emphasize that more physically
realistic models and simulations need to be constructed in order to carry out
the detailed quantitative comparisons that our new data enable.

Full text

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arXiv:1104.3114v1 [astro-ph.GA] 15 Apr 2011
Submitted to ApJ on April 15, 2011
Formation and Evolution of the Disk System of the Milky Way:
[α/Fe] Ratios and Kinematics of the SEGUE G-Dwarf Sample
Young Sun Lee1, Timothy C. Beers1, Deokkeun An2, Željko Ivezi´ c3, Andreas Just4,
Constance M. Rockosi5, Heather L. Morrison6, Jennifer A. Johnson7, Ralph Schönrich8,
Jonathan Bird7, Brian Yanny9, Paul Harding6, and Helio J. Rocha-Pinto10,11
ABSTRACT
We employmeasurementsofthe[α/Fe] ratiofromlow-resolution(R∼2000)spectraof17,500G-type
dwarfs included in SDSS Data Release 8, selected using simple and well-understood selection criteria, to
separate them into likely thin- and thick-disk subsamples. This classification, based on chemistry, is
strongly motivated by the bi-modal distribution of stars in the [α/Fe] vs. [Fe/H] diagram. The resulting
subsamples allow, for the first time, investigations of the kinematic behavior of thin- and thick-disk stars
as a functionof metallicityand positionup to distances of 3 kpcfromthe Galactic plane. Both subsamples
exhibit strong gradients of orbital rotational velocity with metallicity, but with opposite signs (−20 to −30
km s−1dex−1for the thin-disk population, and +40 to +50 km s−1dex−1for the thick-disk population).
We find that the rotational velocity decreases with the distance from the plane for both disk components,
with similar slopes (10 km s−1kpc−1), and a nearly constant difference in the mean rotational velocity
of about 30 km s−1. The mean rotational velocity is uncorrelated with Galactocentric distance for the
thin-disk subsample, and exhibits only a marginally significant correlation for the thick-disk subsample.
Thick-disk stars exhibit a very strong trend of orbital eccentricity with metallicity (−0.2 dex−1), while
the eccentricity does not change with metallicity for the thin-disk subsample. The eccentricity is almost
independent of Galactocentric radius for the thin-disk stars, while a marginal gradient of the eccentricity
with distance exists for the thick-disk population. Both subsamples possess similar trends of increasing
eccentricity with distance from the Galactic plane, with a constant difference of about 0.1. The shapes of
the overall distributions of orbital eccentricity for the thin- and thick-disk populations are quite different
from one another, independent of distance from the plane; neither subsample has significant numbers of
1Department of Physics & Astronomy and JINA (Joint Institute for Nuclear Astrophysics), Michigan State University, East Lansing, MI 48824,
USA; lee@pa.msu.edu, beers@pa.msu.edu
2Department of Science Education, Ewha Womans University, Seoul 120-750, Republic of Korea
3Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195-1580, USA
4Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg (ZAH), Mönchhofstr. 12-14, 69120 Heidelberg, Germany
5UCO/Lick Observatory, Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA
6Department of Astronomy, Case Western Reserve University, Cleveland, OH 44106, USA
7Department of Astronomy, Ohio State University, Columbus, OH 43210, USA
8Max-Planck-Institute für Astrophysik, Karl-Schwarzschild-Str. 1, D-85741, Garching, Germany
9Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
10Universidade Federal do Rio de Janeiro, Observatório do Valongo, Lad. Pedro Antônio 43, 20080-090 Rio de Janeiro, Brazil
11Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, 20921-400 Rio de Janeiro, Brazil

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starswith eccentricityabove0.6. Theseobservationalresults providestrongnewconstraintsonmodelsfor
theformationandevolutionoftheMilkyWay’s disksystem. Forexample,theobserveddependenceofthe
mean rotational velocity on metallicity for thin-disk stars is inconsistent with predictions from classical
localchemicalevolutionmodels. We also considerthe predictionsof severalcontemporarymodels ofdisk
evolution, such as radial migration, gas-rich mergers, disk heating, and pure accretion models. We find
that radial migration appears to have played an important role in the evolution of the thin-disk population,
but possibly less so, relative to the gas-rich merger or disk heating scenarios, for the thick disk. Pure
accretion models appear to be ruled out by the observed distribution of eccentricities for thick-disk stars.
We emphasize that more physically realistic models, and simulations that probe a greater range of disk
formation scenarios, need to be constructed in order to carry out the detailed quantitative comparisons
that our new data enable.
Subjectheadings: Galaxy: disk—Galaxy: formation—Galaxy: kinematicsanddynamics—Galaxy: struc-
ture
1.Introduction
The Milky Way’s thick disk, first identified from fits of the vertical density profile of stars with a mix of expo-
nential functions (Yoshii 1982; Gilmore & Reid 1983), differs in many ways from the thin disk, e.g., in its kinematics
and chemical abundances.
The scale height of the thick disk is about 1 kpc, while that of the thin disk is ∼ 0.3 kpc. Typical thick-disk stars
have generally lower net orbital rotational velocities with larger velocity dispersions (Majewski 1993; Chiba & Beers
2000; Robin et al. 2003; Soubiran et al. 2003; Parker et al. 2004; Wyse et al. 2006), possess higher [α/Fe] ratios12,
and are older and more metal-poor than typical thin-disk stars (Bensby et al. 2003, 2005; Reddy et al. 2006, 2010;
Fuhrmann 2008; Haywood 2008).
Their higher [α/Fe] ratios and the older ages imply that thick-disk stars were born earlier than most thin-disk
stars, in an environment of rapid star formation, and that they have likely had more time to experience dynamical
heating and secular processes such as scattering by perturbations in the disk. As a result of the multiple complex
processes that thick-disk stars may have experienced during their lifetimes, consensus on the nature of the formation
and evolution of the thick disk has yet to be reached.
The currently discussed mechanisms for thick-disk formation can be broadly divided into two groups – violent
origin and secular evolution. Among the models involvingviolent origin, the heating scenario (e.g., Quinn et al. 1993;
Kazantzidis et al. 2008) posits that the thick disk results from a pre-existing thin disk that has been dynamically
heated by satellite mergers. In their simulations of this process, Villalobos & Helmi (2008) found that on the order
of 10–20% of the stars in the thickened disk component were accreted from satellites, the rest being heated thin-disk
stars. The accretion origin of the thick disk (e.g., Abadi et al. 2003) invokes the hypothesis that thick-disk stars were
predominantly formed in dwarf-like galaxies, which were then directly assimilated into the thick disk from orbits that
reached near the Galactic disk plane. Abadi et al. (2003) predicted that over 70% of thick-disk stars were accreted
from such disrupted galaxies. The third model among the violent origin class is that thick-disk stars may have formed
in situ through chaotic mergers of gas-rich systems, prompting simultaneous early star formation before and during
the mergers (Brook et al. 2004, 2005, 2007), and that thin-disk stars formed after the merger events settled down.
12The [α/Fe] ratio is often represented by an average of the [Mg/Fe], [Si/Fe], [Ca/Fe], and [Ti/Fe] ratios, which we adopt in this paper as well.

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Secular evolution by disk heating was first conceived by Spitzer & Schwarzschild (1953), who demonstrated that
encounters with molecular clouds could increase the velocity dispersion of late type, old stars. Barbanis & Woltjer
(1967) also showed that spiral structures might be the cause of the larger velocity dispersion of older stars in the
solar neighborhood. These ideas had been further developed by several studies (e.g., Fuchs 2001, and references
therein). Although challenged by Jenkins (1992), disk heating by secular processes have recently regained attention,
both observationally and theoretically, as possible thick-disk formation scenarios.
Indeed,recent theoreticalstudies andsimulations (Schönrich& Binney2009a,2009b; Loebmanet al. 2010)have
suggested that the thick disk might not require a violent origin, but rather could have formed by cumulative secular
processes associated with the radial migration of stars. Accordingto the migrationtheories (Sellwood & Binney 2002;
Roškar et al. 2008a), stars in the Galactic disk can radially move from the inner (outer) to the outer (inner) regions
due to resonant scattering by transient spiral structure. Based on their simulations, Minchev & Famaey (2010) also
suggestedthat long-livedspiral structures, interactingwith a centralbar, couldbe responsiblefor the radial movements
of stars in a disk galaxy.
These proposed models predict various trends between the kinematic parameters and chemical abundances of
disk-system stars, as well as between their kinematics and spatial distributions. For example, Schönrich & Binney
(2008b) suggested that local, relatively metal-rich thin-disk stars, formed in the inner part of the disk and moved
outward, while local metal-poor thin-disk stars were born in the outer disk and migrated inward to the solar radius,
retaining information on the kinematic differences between the two populations. Thus, there should exist a gradient in
the variation of rotational velocity with metallicity; evidence for such a behavior has been claimed observationally by
Haywood (2008). Models of disk heating via satellite mergers (Villalobos et al. 2010) result in proposed relationships
between rotational velocity and Galactocentric distance and distance from the Galactic plane. Gas-rich mergermodels
(Brook et al. 2007) also predict a gradient of rotational velocity with Galactocentric radius for disk stars near the solar
radius.
Sales et al. (2009)proposed that the distribution of orbital eccentricities for nearby thick-disk stars could be used
to provide constraints on the various suggested formation models. A number of recent papers also have employed
this framework to study possible origins of the thick disk, based on data from several large spectroscopic surveys.
For example, Wilson et al. (2011) have explored data from the RAdial Velocity Experiment (RAVE; Steinmetz et al.
2006), while Dierickx et al. (2010) used data from the seventh public release of the Sloan Digital Sky Survey (SDSS
DR7; York et al. 2000; Abazajian et al. 2009). The study of Casetti-Dinescu et al. (2011) combined RAVE data with
newly available proper motions from the fourth release of the Southern Proper Motion Catalog (SPM4; T. Girard et
al., in preparation). We discuss their analyses and conclusions further below.
Most previous observational studies that have sought to test the various correlations predicted by the models
mentionedabovehaveusedmethodsofassigningindividualstarstomembershipinthethin-andthick-diskpopulations
(based on a given star’s location or kinematics) that introduce manifest biases that can confound interpretations (as
previously noted by Schönrich & Binney 2009b and Loebman et al. 2010). As the chemical signatures of a star are
substantially less variable properties than its spatial position or velocities over its lifetime, it is instead desirable to
classify disk stars into their likely components according to their chemistry.
Among the various chemical abundance ratios that might be explored for this purpose, the [α/Fe] ratios appear
particularly useful. These ratios can be relatively easily measured (as described below), and have been proven to well
separate thick-disk stars from thin-disk stars. It is known that, at least in the solar neighborhood (where essentially
all previous studies have been conducted), thick-disk stars are on average enhanced in their [α/Fe] ratios by +0.2 to
+0.3 dex relative to their thin-disk counterparts at a given [Fe/H]. Local kinematically selected thin- and thick-disk
samples based on probabilistic membership assignments have confirmed this enhancement of [α/Fe] (e.g., Bensby et

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al. 2003; Reddy et al. 2006). Fuhrmann (1998, 2008) also demonstrated that dwarfs in his volume-limited sample
could be clearly separated into two populations as a function of [Fe/H] – one associated with high [Mg/Fe] and the
other with low [Mg/Fe]. The elemental abundance patterns of the stars with low [Mg/Fe] ratios and high [Mg/Fe]
ratios are very similar to the kinematically selected thin- and thick-disk samples. Finally, [α/Fe] ratios also provide
valuable information on the timescales and intensities of star formation in the populations involved.
In this study we make use of the first set of [α/Fe] ratios obtained for a large sample of low-resolution(R∼2000)
spectra from the Sloan Extension for Galactic Understandingand Exploration(SEGUE; Yanny et al. 2009). As shown
by Lee et al. (2011), for stars with SDSS/SEGUE spectra of signal-to-noise (S/N) ratios greater than 20, and with
temperatures in the range 4500 K ≤ Teff≤ 7000 K, one can estimate [α/Fe] with an accuracy of better than 0.1 dex.
This enables a chemical separation of the disk system into likely thin- and thick-disk populations. In this paper we
exploretheobservedcorrelationsofrotationalvelocityandorbitaleccentricitywithmetallicity,Galactocentricdistance
and distance from the Galactic plane, as well as the orbital eccentricity distributions for the individual populations,
and compare with the predictions of the radial migration, gas-rich merger, and dynamical heating models. Since we
believe that direct quantitative comparisons with the predictions made by various models (or simulations) mentioned
in this study are somewhat premature, we emphasize the more qualitative aspects of these comparisons.
This paper is outlined as follows. In Section 2 we present the G-dwarf sample from SEGUE, describe various
cuts imposed on the sample to obtain a refined disk dwarf sample, and discuss the calculations used to derive their
space motions and orbital eccentricities. Section 3 describes how we assign membership of the stars into either the
thin-or thick-diskpopulations. Results fromourG-dwarfsampleand discussionofcomparisonsof ourresults with the
predictions of various contemporary disk formation and evolution scenarios follow in Sections 4 and 5, respectively.
A summary and our conclusions follow in Section 6.
2.Selection of Local Dwarf Stars
2.1.The SEGUE G-dwarf Sample
Our initial sample comprises low-resolution (R ∼2000) spectra of ∼63,000 stars from SDSS Data Release 8
(DR8; Aihara et al. 2011), obtained during the SEGUE sub-survey, which were originally targeted as G-dwarf candi-
dates (withcolors andmagnitudesin therange0.48<(g−r)0< 0.55and r0< 20.2). As a result ofthe simple sampling
function, this dataset is expected to be relatively unbiased with respect to chemistry, and completely unbiased with
respect to kinematics. In order to obtain a subsample of disk stars with the most reliably estimated physical quantities
we apply several additional cuts.
First, we exclude stars lacking informationon their stellar parameters (effective temperature, Teff, surface gravity,
log g, and metallicity, [Fe/H]), radial velocities, or proper motions. The stellar atmospheric parameters were deter-
mined by the most recent version of the SEGUE Stellar Parameter Pipeline (SSPP; Lee et al. 2008a, 2008b; Allende
Prieto et al. 2008; Smolinski et al. 2011); typical external errors in these estimates are 180 K in Teff, 0.24 dex in
log g, and 0.23 dex in [Fe/H] (Smolinski et al. 2011). It has been shown that shifts in the SSPP-derived estimates of
[Fe/H] and [α/Fe] caused by the presence of unrecognized spectroscopic binaries are generally small (Schlesinger et
al. 2010). Although the typical uncertainty of the radial velocity varies with the S/N ratio of a spectrum, it is less than
5 km s−1for the greatmajorityofstars in oursample. Propermotioninformationwas obtainedbasedon the procedures
described by Munn et al. (2004); the systematic error noted by Munn et al. (2008) has been corrected (final typical
errors are 3–4 mas yr−1). In this regard, see also Bond et al. (2010), who investigated the systematic errors in Munn et
al. (2008) by comparison with the expected null proper motions of SDSS quasars.

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Distances to individual stars were estimated using a calibrated set of stellar isochrones (An et al. 2009a), fol-
lowing the prescription in An et al. (2009b). After correcting photometry for dust extinction, main-sequence fitting
was performedsimultaneouslyon three differentcolor-magnitudediagrams (CMDs), with r as a luminosityindex, and
g−r, g−i, and g−z as color indices, respectively. We adopted an SSPP-derived [Fe/H] in the distance estimation for
each star, and fixed a stellar age and [α/Fe] of the model at a given [Fe/H], assuming a linear relationship between
[Fe/H] and these quantities (see An et al. 2009b). Distance estimates obtained using [α/Fe] from this assumption may
not be internally consistent with analyses based on the SSPP-determined [α/Fe], but even a ∼0.1 dex difference in
[α/Fe] has a negligible impact on the derived distances (∼0.01 mag in distance modulus). We also limited models in
the fitting to log g ≥ 4.2 to minimize possible distance bias from stellar age effects near the main-sequence turn-off.
An inter-comparison of results from the three CMDs suggests that the internal error in the distance modulus is ∼0.1
mag; an additional ∼0.1 mag error is expected from the combined errors in age, [Fe/H], [α/Fe], and E(B−V). This
suggests that the associated distance-modulus error is ∼0.14 mag for individual stars. The effects of binarity are more
difficult to quantify, and are not included in this error estimate (see An et al. 2007; Sesar et al. 2008).
The [α/Fe] ratio is derived following the procedures described by Lee et al. (2011). Briefly summarizing, Lee
et al. first generated a grid of synthetic spectra, covering 4000 K ≤ Teff≤ 8000 K in steps of 250 K, 0.0 ≤ log g ≤
5.0 in steps of 0.2 dex, −4.0 ≤ [Fe/H] ≤ +0.4 in steps of 0.2 dex, and −0.1 ≤ [α/Fe] ≤ +0.6, in steps of 0.1 dex, then
determined[α/Fe] by searchingthe grid fora synthetic spectrumthat best matches a givenSDSS/SEGUE spectrum(in
regions that are most influenced by [α/Fe]). By comparing with a set of moderately high-resolution (R = 15,000) and
medium-resolution (R = 6000) spectra of SDSS/SEGUE stars, they demonstrated the ability to measure [α/Fe] from
SDSS/SEGUE spectra (with S/N > 20) with uncertainties less than 0.1 dex, for stars with atmospheric parameters in
the range Teff= [4500, 7000] K, log g = [1.5, 5.0], and [Fe/H] = [−1.4, +0.3], over the full range of [α/Fe] considered.
For stars with [Fe/H] < −1.4, slightly higher S/N was required to achieve this precision (S/N > 25).
In order to assemble a local dwarf sample, we only include stars with distances, d, less than 3 kpc from the Sun,
and with log g ≥ 4.2. These cuts ensure that we are selecting likely dwarfs from which we can obtain accurate space
motions(i.e., that do notsuffer fromsevere degradationdueto propagationof propermotionerrorsat largerdistances).
In order to perform a confident separation of the thin- and thick-disk populations on the basis of [α/Fe], we further
requirethat the spectra of the dwarf stars includedin our analysis have S/N ≥30. This conservativecut on S/N ensures
not only high quality estimates of [Fe/H] and [α/Fe], but also that our program stars have small errors in estimated
radial velocity (less than 5 km s−1).
2.2. Calculations of Space Motions and Orbital Eccentricity
With information on the distances, radial velocities, and proper motions for our program stars in hand, we then
derive the U, V, W space velocity components. We apply (U,V,W)⊙= (11.10,12.24,7.25) km s−1(Schönrich et al.
2010) to adjust for the solar peculiar motions with respect to the Local Standard of Rest (LSR). For the purpose of our
analysis, we also make use of the rotational velocity around the Galactic center in a cylindrical coordinate system,Vφ,
calculated assuming R⊙= 8.0 kpc and VLSR= 220 km s−1. The Galactocentric distance projected onto the Galactic
plane, R, and the vertical distance from the Galactic plane, |Z|, are also obtained. In addition, by adoption of an
analytic Stäckel-type gravitational potential (which includes a flattened, oblate disk and a spherically-shaped massive
dark halo; see Chiba & Beers 2000), we compute rapo(rperi), the maximum (minimum) distance from the Galactic
center that a star reaches during its orbit, as well as the orbital eccentricity, e, defined as (rapo−rperi) /(rapo+rperi).
Errors in the derived kinematics and orbital parameters for each star due to propagation of the errors in the observed
quantities (mostly dominated by distance and proper motion errors) are determined by 1000 realizations of a Monte
Carlo simulation.

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We nextremovestars fromoursample withderivedrotationalvelocitiesrelativeto theGalactic centerless thanVφ
= +50 km s−1, with [Fe/H] ≤ −1.2, and located outside the range 7 < R < 10 kpc, in order to minimize contamination
from the halo and outer-disk components.
Finally, we perform a simple check on the likely remaining halo contamination in our sample following the
prescription of Bensby et al. (2003). For calculation of the approximate disk and halo star fractions we adopt the local
stellar densities, velocity dispersions in U, V, and W, and the asymmetric drifts listed in their Table 1, assuming the
space velocities of the thin-disk, thick-disk, and halo stars are distributed as Gaussians. Based on these probability
distributions, we reject stars that have greater likelihood of belonging to the halo than to the disk system. This check
removes only about 60 additional stars from the sample, showing that the above selection criteria for thin- and thick-
disk stars are quite reasonable. We also experimented with the application of slightly different scale heights for
describing the variation of halo stellar densities with |Z|, but the above result appears quite robust. Note, however,
that these various cuts do not necessarily eliminate contamination by members of the so-called metal-weak thick-disk
(MWTD) population,which Carollo et al. (2010)have shown exhibits metallicities in the range −1.7 < [Fe/H] <−0.7,
and a prograde rotation ofVφ∼ +100 to +150 km s−1.
Summarizing the criteria used for our sample selection, surviving program stars satisfy d < 3 kpc, log g ≥ 4.2,
S/N ≥ 30, Vφ> +50 km s−1, [Fe/H] > −1.2, 7 < R < 10 kpc, and possess greater probability of belonging to the
disk system than to the halo. The surviving sample from the above cuts numbers ∼17,500 stars. Figure 1 shows the
distributions of [Fe/H], Vφ, |Z|, and R for the final dwarf sample (solid lines), before and after further division based
on the derived [α/Fe] ratios into the thin- and thick-disk populations, as described below.
3.Division of the Sample on [α/Fe] into Thin- and Thick-Disk Populations
As mentioned previously, since a stellar population’s kinematics and spatial distributions can be modified over
time (especially in the disk system), while a (dwarf) star’s atmospheric chemical abundance is essentially invariant
(except in unusual circumstances, such as binary mass transfer from an evolved companion), we make use of the
estimated [α/Fe] ratio as a reference to separate the thin- and thick-disk populations.
For the purposeof the present analysis, our dwarf sample is split into likely thin-disk (with low [α/Fe]) and thick-
disk (with high [α/Fe]) populations, based on the following scheme:
I) For stars with [Fe/H] ≥ −0.8
• thin disk, if [α/Fe] < −0.08·[Fe/H] + 0.15
• thick disk, if [α/Fe] > −0.08·[Fe/H] + 0.25
II) For stars with [Fe/H] < −0.8
• thin disk, if [α/Fe] < +0.214
• thick disk, if [α/Fe] > +0.314
This division into the thin- and thick-disk populations is devised based on examination of the distribution of
number densities in the [α/Fe] vs. [Fe/H] plane, shown in Figure 2. Note how well the populations appear to separate
above and below the solid line in this figure, which is our adopted fiducial.

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The dashed lines located ±0.05 dex in [α/Fe] above and below the fiducial solid line in Figure 2 indicate the
dividing points for the high-[α/Fe] (thick-disk) and low-[α/Fe] (thin-disk) stars. Note that this leaves a gap of 0.1 dex
in [α/Fe] between the thin- and thick-disk dividing lines. This choice serves to reduce the number of misclassified
stars that may arise from observationalerrors in their measured[α/Fe]. The dashed red line in Figure 1 shows the thin-
disk subsample, whereas the dotted-dash blue line is for the thick-disk subsample, classified by the dividing schemes
described above. From this figure, one can roughly read off the ranges and peak values of the estimated and derived
parameters for each subsample.
To check on the efficacy of the chemical separation of the disk populations through use of the [α/Fe] ratio, we
have investigatedthe variation of theU, V, W velocity dispersions of our sample with [α/Fe]. It is well knownthat the
dispersion of each velocity componentincreases with distance from the Galactic plane (as well as on age, on average).
In any event, the thick-disk population exhibits substantially higher dispersions than the thin-disk counterpart. Figure
3 shows the derived velocity dispersions of our sample as a function of [α/Fe]. It is readily apparent that, up to around
[α/Fe] = +0.2, the dispersion of each velocity component increases moderately. Above [α/Fe] = +0.2 the gradients of
the velocity dispersions with [α/Fe] become somewhat steeper. Above [α/Fe] = +0.3, the magnitude of each velocity
dispersion is larger by about 10 km s−1than for [α/Fe] < +0.2. As our thin-disk stars mostly have [α/Fe] < +0.2 and
thick-disk stars possess [α/Fe] > +0.3, Figure 3 kinematically confirms that the division by [α/Fe] into the thin- and
thick-disk populations is quite robust.
One may still be concerned about remaining biases in our initial sample selection, based on (g−r)0color, due to
its small, but non-zero, metallicity sensitivity. If our selected sample favors metal-poor over metal-rich stars, this bias
might produce misleading correlations between the parameters we are seeking to understand. For example, at least for
the thick-disk population, previous studies have indicated that the observedstellar orbital rotational velocity decreases
with declining metallicity. Thus, if biases have increased the relative numbers of metal-poor stars in the thick-disk
subsample, the overall distribution of Vφwill be shifted to lower rotational velocity. However, it should be kept in
mind that, because our sample does not suffer from kinematic bias, any correlations that we are seeking between
kinematics and chemical abundances will not be affected by any potential metallicity bias, as long as the correlations
are derived from ranges of R and |Z| that are sufficiently small that the correlations remain roughly constant over the
regions considered.
For the thick disk, there is some existing evidence for the lack of a metallicity gradient with distance above the
Galactic plane (e.g., Gilmore et al. 1995), or at most for only a small one, on the order of 0.1–0.2 dex kpc−1(Ivezi´ c
et al. 2008). So, if our thick-disk sample does not suffer from a significant metallicity bias, the shape of the observed
metallicity distribution functions (MDFs) at different heights should remain roughly constant. Figure 4 displays the
observed MDFs for both the thin- and thick-disk subsamples in different bins of |Z| distance. From inspection, the
relative numbers of metal-rich stars in the thick-disk subsample do not grossly change with different cuts on height
abovethe plane,so a significantmetallicitybias does notappearto exist forthis population. Quantitatively,the fraction
of the stars with [Fe/H] < −0.6 for the thick-disk subsample is 0.51, 0.51, 0.58, and 0.64 from the first to the fourth
panel (a resulting metallicity gradient with |Z| of −0.055 dex kpc−1), consistent with the expectation from previous
work.
However, this seems not to be the case for the thin-disk subsample. At heights above |Z| = 1.5 kpc (already many
thin-disk scale heights above the plane), we notice that more metal-rich stars have dropped out of the distribution,
compared to the upper three panels. This is quantitatively confirmed by examination of the fraction of stars with
[Fe/H] < −0.2 (0.44, 0.46, 0.58, and 0.71 from the first to the fourth panel). This may be a natural consequence of
selecting the sample without consideration of the |Z| distance, since at greater heights thick-disk stars are expected to
dominate. In other words, some of the stars in the metal-poor tail of the thin-disk subsample may in reality belong
to the thick disk, but they have been misclassified as thin-disk stars due to errors in the estimated [α/Fe]. Indeed,

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considering the distribution of [α/Fe] for the stars with [Fe/H] < −0.3 and |Z| > 1.5 in the thin-disk subsample, many
of the stars have [α/Fe] > +0.15. Thus, moderately α-enhanced stars at this height may mostly belong to the thick
disk rather than to the thin disk, with a much lower probability of old thin-disk membership.
Simple calculations confirm the above argument. According to Lee et al. (2011), the error in [α/Fe] at S/N = 30
is about 0.08 dex. If we assume this is a reasonable estimate of a 1σ error in [α/Fe] for all stars with |Z| > 1.5 kpc,
and perturb the measured [α/Fe] values by this amount,the total numberof stars classified as membersof the thin-disk
component falls to 41, substantially smaller than 203 that are claimed to be present. Thus, it is valid (within statistical
fluctuations) to say that the low-[α/Fe] metal-poor thin-disk stars in this |Z| distance region are likely spurious, and
are found at roughly the expected level of contamination. As the potential bias also depends on the age distribution
of our sample, which is not known at present, it is difficult to quantify it further. However, as the narrow color range
applied to originally select the G-dwarfs for spectroscopic follow-up also preferentially selects certain age ranges on
the main sequence, we might expectthat this bias might contribute at some level to the observedtrends (e.g., rotational
velocity versus metallicity) that we seeking to understand. Nevertheless, as the total number of thin-disk stars in this
most distant region is rather small, we expect the impact of such stars on our analysis to be minimal.
4. Results of the Observations
In this section we use our local G-dwarf sample to examine the observed gradients ofVφwith [Fe/H], R, and |Z|,
as well as trends of e with [Fe/H], R, and |Z| for the thin-disk and thick-disk populations as identified above.
4.1.Correlations between Rotational Velocity and Metallicity
The top panel of Figure 5 shows a color-coded distribution of Vφin the [α/Fe] vs. [Fe/H] plane for our G-dwarf
sample. Detailed examination of this panel (as well as Figure 2) reveals a metal-poor tail for the low-[α/Fe] stars
(< +0.2), which we associate with the thin disk, extending down to [Fe/H] = −0.7. This already implies that the thin
disk may not be well-described by a single metal-rich population with a peak around [Fe/H] = −0.2. We also notice
from this panel that a higher rotational velocity is observed in the region of the metal-poor thin disk ([α/Fe] < +0.2
and [Fe/H] < −0.3), suggesting a negative trend of Vφwith [Fe/H]. In contrast, the high-[α/Fe] stars (> +0.3, which
we associate with the thick disk) apparently exhibit a strong positive trend of Vφwith [Fe/H]. We investigate these
trends quantitatively below.
The bottompanel of Figure 5 displays the distribution of mean orbital radii (Rmean) in the [α/Fe] vs. [Fe/H] plane.
It is clear that the stars we associate with the thick-disk population exhibit smaller mean orbital radii than those with
associated with the thin disk. In addition, the metal-poor thin-disk stars possess larger mean orbital radii than the
dominant metal-rich thin-disk stars.
Looking at the results from other observational work, a recent study by Navarro et al. (2010) obtained a slightly
different result for their thin-disk subsample. These authors found little or no correlation between Vφand [Fe/H] for
their thin-disk stars (defined by [Fe/H] > −0.7 and [α/Fe] < +0.2), although the subset of their thin-disk subsample
with availableEuabundances(sothat potentialthick-diskorhalostars couldberejected)exhibitsaverysimilarpattern
to that we identify here. One should also keep in mind the possibility of effects from selection biases in their sample,
as it was based on an assembly of stars that included kinematically selected targets.
Haywood (2008) separated thin-disk stars with [Mg/Fe] < +0.2 from thick-disk stars with [Mg/Fe] > +0.2 in
the spectroscopic sample of Soubiran & Girard (2005), and found an increasing trend of the mean orbital radii with

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decreasing metallicity for the thin-disk population, along with a decreasing trend of mean radii with decreasing metal-
licity for the thick-disk population (see his Figure 3). He claimed that this tendency resulted from stars that migrated
from the inner and outer disk. These behaviors are qualitatively in very good agreement with our findings.
Rocha-Pinto et al. (2006) also reported a similar behavior between the mean orbital radii and the chemical
abundancesintheir volume-completesample of325late-typedwarfs. Theirprimaryresults were that, as the difference
in the distance between the mean orbital radius and the solar radius increases, the abundances of Fe, Na, Si, Ca, Ni,
and Ba all decrease. This relationshipbetween the chemical abundancesand the mean orbital radii could be accounted
for by radial displacements of the stars involved.
It is quite remarkable that all of the observed behaviors of the mean orbital radii from our G-dwarf sample agree
so well with several previous observational studies (based on much smaller samples).
Figure 6 indicates that there exists a clear gradient of Vφwith [Fe/H] at any given |Z| distance, for both the
thin-disk subsample (top panel) and the thick-disk subsample(bottom panel). Figure 7 displays the observedgradients
at different heights above the plane for both subsamples. Similar slopes of Vφfor both the low- and high-[α/Fe] stars
are obtained for the various slices in |Z| distance, although the slope of the thick-disk subsample becomes shallower
at larger distance (fourth panel), and slightly steeper for the thin-disk subsample (which only includes 203 stars). Due
to the small number of thin-disk stars in the vertical region 1.5 ≤ |Z| < 3.0 kpc, the slope and its error are obtained
by a linear fit to all stars, without binning in [Fe/H]. If significant contamination of our thick-disk subsample by
unrecognized MWTD stars were present, we might expect the slope of the correlation of Vφwith [Fe/H] to increase
with distance above the plane, due to the greater velocity lag, larger scale height, and lower metallities of the MWTD
component compared with that of the canonical thick disk (Carollo et al. 2010). That is, at larger distances from the
planeand at lower [Fe/H], the meanVφwould be expectedto be lower than it would be fora pristinethick-disksample.
We see no evidence for steepening of the gradient in Figure 7.
A gradient of −20 to −30 km s−1dex−1, on average, is shown to exist for the thin-disk subsample, and a strong
gradient of +45 to +50 km s−1dex−1for the thick-disk subsample. This result for the thick-disk population agrees
with the claim of Spagna et al. (2010), who derived a similar slope using F-, G-, and K-type dwarfs from SDSS DR7.
However, this finding clearly contradicts the results of Ivezi´ c et al (2008), who found little correlation betweenVφand
[Fe/H]. We discuss a possible resolution to this discrepancy in the Appendix.
Inordertocheckhowuncertaintiesintheparameters,Vφ, [Fe/H],[α/Fe], and|Z| affectourderivedgradientsofVφ
over |Z| for both the thin- and thick-disk subsamples, we have performed a simple Monte Carlo experiment. Random
changes in each parameter within 1σ of the estimated error of each parameter were applied to 1000 realizations of
each subsample (over the full range in |Z|), from which we obtained average gradients of −19.6 km s−1dex−1for the
thin-disk subsample and +46.0 km s−1dex−1for the thick-disk subsample, in good agreement with those shown in the
bottom panel of Figure 7. This indicates that our derived gradients ofVφover [Fe/H] are not grossly affected by errors
in the derived parameters.
4.2.Rotational Velocity Gradients with Distance from the Galactic Center and Galactic Plane
Figure8 shows the overalltrends ofrotationalvelocitywith distancefromthe Galactic center(toppanel)andwith
verticaldistancefromtheplane(bottompanel)forthethin-disk(blackdots) andthick-disk(opensquares)populations.
Inspection of the top panel of this figure indicates only a negligible rotational velocity gradient for the thin-disk
subsample (only −0.2 km s−1kpc−1), consistent with a flat rotation curve in the solar neighborhood. The asymmetric
drift is about 10 km s−1at the solar radius, as found by previous work (e.g., Soubiran et al. 2003). A small gradient of

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−5.6 km s−1kpc−1is found for the thick-disk subsample, which lags the VLSRby ∼40 km s−1, not far from the lag of
51 km s−1obtained by Soubiran et al. (2003). Note that even if we include in the analysis the stars with 0 < Vφ< 50
km s−1that were eliminated in our original selection, we obtain very similar asymmetric drifts and gradients for the
thin- and thick-disk populations.
The bottom panel of Figure 8 shows that the gradients of Vφwith |Z| distance are very similar (about −10.0 km
s−1kpc−1) for both the thin- and thick-disk subsamples. The difference in Vφ(the velocity lag) for the high-[α/Fe]
stars relative to the low-[α/Fe] stars is almost constant, ∼30 km s−1at any given |Z| distance. This again suggests that
contamination from MWTD stars is not a major issue for our thick-disk subsample.
Comparing with other recent studies, the vertical gradient of Vφwith |Z| for our thick-disk subsample, −10.8 km
s−1kpc−1, is smaller than that obtained by Casetti-Dinescu et al. (2011), −25 km s−1kpc−1, based on ∼4400 red clump
metal-rich thick-disk stars covering the metallicity range −0.6 < [Fe/H] < +0.5, that of Ivezi´ c et al. (2008) from their
SDSS sample (−29 km s−1kpc−1), that of Girard et al. (2006), who derived a gradient of −30 km s−1kpc−1from a
sample of about 1200 red giants located in the range |Z| = 1–4 kpc, and as obtained by Chiba & Beers (2000; −30 km
s−1kpc−1) for the subset of their non-kinematicallyselected stars in the metallicity range −0.8≤ [Fe/H] ≤ −0.6 within
2 kpc of the Galactic plane. Even if we cut our thick-disk subsample to include only stars with [Fe/H] > −0.6, we
obtain a slope of −8.6 km s−1kpc−1, consistent, within 3σ, with that derived from the subsample without a metallicity
restriction.
It is interesting to note that, if we consider our entire thin- and thick-disk subsamples with |Z| >1.0 kpc together,
we find a vertical gradient of −21.2 km s−1kpc−1, in better agreement with the previous studies. That is, the derived
vertical gradient of the rotational velocity becomes substantially steeper when the stars are not divided according to
their [α/Fe] ratios. We conclude that accurate determination of the vertical gradient of Vφwith |Z| for the thick disk
requires application of a chemical separation criterion (other than simply [Fe/H]) to isolate the various components.
Application of our simple Monte Carlo experiment with 1000 realizations of the subsamples yielded average
radial gradients of Vφwith R of −0.1 km s−1kpc−1and −4.0 km s−1kpc−1, and vertical gradients with |Z| of −9.6 km
s−1kpc−1and −10.0 km s−1kpc−1for the thin- and thick-disk subsamples, respectively. This confirms again that our
computed gradients ofVφare not strongly affected by errors in the parameters involved, as these values are very close
to the listed in Figure 8.
4.3. Correlations of Stellar Orbital Eccentricities with Metallicity, Distance from the Galactic Center, and
Height Above the Galactic Plane
Figure 9 shows trends of orbital eccentricities (e) for the G-dwarf sample, as a function of [Fe/H], R, and |Z|,
from the top to bottom panel, respectively. The black dots denote our thin-disk subsample, while the open squares
indicate the thick-disk subsample.
One outstanding feature from inspection of the three panels is that the overall distribution of the orbital eccentric-
ities for the thick-disk stars is easily separable from that for the thin-disk population. The top panel suggests that the
trend of the eccentricities for the thin-disk stars is independentof metallicity, i.e., an almost flat trend of e with [Fe/H],
indicative of a very narrow distribution of eccentricity, with a peak around e ∼ 0.14. On the other hand, the trend of
e for the thick-disk subsample increases as the metallicity decreases. Around [Fe/H] = −1.0 it flattens out, implying
a rather broad distribution of eccentricities. A slope of −0.2 dex−1is obtained from a least-squares fit to the averaged
points.
The second panel also shows several interesting features. As in the top panel, there is not much correlation

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between e and R for the thin-disk subsample, although the behavior trends slightly higher below R = 7.5 kpc. The
thick-disk stars generally exhibit an increasing trend of e with increasing R. The eccentricity distributions for the thin-
and thick-disk populations merge at R ∼ 7.0 kpc.
The bottom panel shows that the eccentricities for both low- and high-[α/Fe] stars increase on average the farther
away they are from the Galactic plane. In addition, similar to Figure 8, it is also noticed that the difference (about 0.1)
in the eccentricity between the low-[α/Fe] and high-[α/Fe] subsamples is constant at any given |Z|.
A simple Monte Carlo experiment with 1000 realizations of the subsamples also reveals that the derived trends
of the eccentricities with [Fe/H], R, and |Z| above are not strongly affected by errors in the parameters involved, as
the computed gradients of the eccentricities are within 3σ from those values listed in Table 1, which quantitatively
summarizes various correlations discussed in this section for the two subsamples.
5.Qualitative Comparisons with Predictions of Contemporary Models of Disk Formation
At present, it may be unwise to rely too strongly on the present predictions of the suggested thick-disk formation
models. Thisfollowsbecause,eventhoughtheyareabletoreproducesomeaspectsoftheMilkyWay’s disksystem, the
predicted properties are limited by large uncertainties with their treatment of star formation, the dynamical interaction
of presumed satellites with the disk, unavoidable numerical effects, and the myriad of assumptions that are required
in their construction. Thus, in this section, we compare our observational findings only with qualitative expectations
from the published radial migration, gas-rich merger,and disk heating models. It is our expectationthat, as the models
and simulations improve, these comparisons will increasingly be able to discriminate between the relative importance
of the various formation scenarios.
5.1. Correlations between Rotational Velocity and Metallicity
According to the radial migration models (Sellwood & Binney 2002; Roškar et al. 2008a; Schönrich & Binney
2009a; Minchev & Famaey 2010), the (presumably) metal-poor stars of the thin disk (which includes young, low-
[α/Fe] stars) that were born in the outer disk move inward to the solar neighborhood, while the (presumably) metal-
rich stars that formed in the inner disk migrate outward into the solar neighborhood(as the inner region of the disk has
a higher stellar and gas density, and is rapidly chemically enriched, most of the stars should be metal rich).
Schönrich & Binney (2009a) suggested that this radial movement can occur by two mechanisms: “blurring” and
“churning”. Blurringrefers to the increase of eccentricities overtime at a similar angular momentumdue to scattering,
e.g., on giant molecular clouds. Churning is mostly triggered by resonant scattering at co-rotation due to transient
spiral density waves, which transfers stars from inner (or outer) disk regions into the solar vicinity by changing their
angular momenta without alteration of their orbital circularity (hence eccentricities). These authors suggested that
churning is the dominant process by which stars in the inner disk migrate out to the solar annulus, thus providing
greater heterogeneity in the abundance and velocity distributions among solar neighborhoodstars.
The consequence of incomplete mixing from blurring and churning is that the metal-rich stars in the thin disk
possess relatively lower rotational velocities (Vφ), while the metal-poor stars have higherVφ. Thus, the expectation is
that there should exist a trend of Vφwith [Fe/H] among (at least) the thin-disk stars. Schönrich & Binney (2009a,b)
indeed predicted a significant downtrend of Vφwith [Fe/H] for the low [α/Fe] stars, due to incomplete mixing for
younger stars. This prediction was confirmed by the later N-body models of Loebman et al. (2010), who employed
slightly different treatments of radial mixing and star formation in their simulated disks from Schönrich & Binney

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(2009a,b), but found a gradient of −19.7 km s−1dex−1for younger stars (identified with the thin-disk component with
low [α/Fe]) in the solar neighborhood(7 < R < 9 kpc and 0.5 < |Z| < 1 kpc).
Our observed gradient of Vφwith [Fe/H] for the thin-disk stars in the range 0.5 < |Z| < 1.0 kpc (−24.1 km
s−1dex−1) is not far from the estimate of −20 km s−1dex−1obtained by Loebman et al. (2010) for their simulated
sample of young, low-[α/Fe] stars in their transition zone, which covers the same interval in height above the plane.
Note that the scale of their [α/Fe] determinations and ours are slightly different, and they also employed the predicted
oxygen abundance ratio as a proxy for [α/Fe], rather than the averages employed in our estimates. It appears that an
overall velocity gradient of −20 to −30 km s−1dex−1with metallicity for the thin-disk subsample qualitatively agrees
well with the expectations from the radial migration models.
The extended tail of low-[α/Fe] metal-poor stars observed in Figures 1, 2, 4, and 5 can be also explained by the
radial migration scenario. Roškar et al. (2008b) found that their simulated disk stars (when allowed to mix radially)
exhibited a MDF more like the observations by Holmberg et al. (2007) than that obtained from an in situ sample
without radial migration. These authors concluded that radial migration was the likely cause of the broader MDF,
which is also supported by our present data.
It is noteworthy that our observed negative gradient of Vφwith [Fe/H] for the thin-disk stars (−20 to −30 km
s−1dex−1) stands in contradiction to expectations from traditional local evolution models in the solar neighborhood
(without allowing for mixing or migration of stars), which predict a positive slope of Vφwith [Fe/H]. According to
these models, the stars that were born early in the history of star formationin the thin disk are expected to be relatively
metal-poor. These old metal-poor thin-disk stars should have experienced more perturbations, such as from variations
in the Galactic potential over time. As a result, such stars are expectedto exhibit slower rotational velocities and larger
velocitydispersions than the younger,moremetal-rich thin-diskstars. This inevitablyleads to the expectedproduction
of a positive gradient ofVφwith [Fe/H], which we clearly do not find.
When considering the radial migration models for the thick disk the case differs somewhat, in particular due to
the much older ages of these stars. According to these models, it is expected that most of the thick-disk stars that exist
in the solar neighborhoodtoday were born with high velocity dispersion in the inner portion of the Galaxy, in regions
of higher local density, at a time when the metallicity of the ISM was relatively low and the α-abundance ratios were
high. As they migrated outward over time, the lower gravitational restoring force of the local disk allowed these stars
to exploreorbitsreachinghigheraboveor belowthe plane. Relativelyfew thick-diskstars are thoughtto havemigrated
inward from the outer disk region. These old stars had more time to experience mixing of their orbits; in the case of
complete mixing for these older stars, one might expect little or no trends between rotational velocity and metallicity.
Schönrich & Binney (2009a,b) did not make predictions of velocity trends with metallicity, on the grounds that
insufficient knowledge of the earliest phases of disk formation exists to constrain expectations for such a potential
gradient (i.e., unknowninitial conditions). However,Loebmanet al. (2010)reportedfrom their simulation an insignif-
icant gradient of +1.4 km s−1dex−1for these older stars (> 7 Gyr, which generally matched the observed properties
of the thick-disk component, e.g., high [α/Fe] ratios). Even though the migration strength in their simulation induced
substantial mixing, the process was still incomplete. Thus, it would allow for the conservation of significant veloc-
ity/metallicity trends. It should be mentioned, however, that their model was not specifically intended to match the
properties of the Milky Way.
In any case, the small or absent predicted correlations between Vφand [Fe/H] for the high-[α/Fe] stars from the
migration models are in contrast to our determination of a steep gradient of +40 to +50 km s−1dex−1for the observed
high-[α/Fe] stars we associate with the thick disk, as shown in Figure 7. Thus, this trend of Vφwith [Fe/H] for the
thick-disk subsample can provide a useful constraint to the radial migration models mentioned above.

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In summary,the observedcorrelationsbetweenVφand[Fe/H] forourlow-[α/Fe] (thin-disk)stars can be naturally
explained by the radial migration of stars from the outer disk (more metal-poor stars) and from the inner disk (more
metal-rich stars) into the solar vicinity, as predicted by the migration models. As explained by Schönrich & Binney
(2009b), such a velocity gradient arises from the interplay between the churning and the blurring processes. The
behavior of the high-[α/Fe] (thick-disk) stars is rather different (exhibiting a much steeper gradient) than expected
from the simulated high-[α/Fe] stars of Loebman et al. (2010), although there remains the uncertainty of how well
thick-disk stars are represented in these models, and how well the models match the actual history of the Milky Way.
It appears that stellar radial migration may have played an important role in the evolution of the thin disk, but, based
on the information available from the current radial migration models and simulations, it is difficult to ascertain the
relative importance of radial migration for the formation and/or evolution of the thick disk.
5.2.Rotational Velocity Gradients with Distance from the Galactic Center and Galactic Plane
The gas-rich merger model of Brook et al. (2007) predicts a correlation between Vφand R for stars in the disk
system. According to their simulations (especially their Figure 5) there should exist a detectable velocity gradient for
the thin disk in the region of the solar neighborhood (R = 7–10 kpc). This differs from our null gradient for the thin-
disk subsample. Their simulations also indicate a negligible gradient for their thick-disk stars (which they refer to as
“merger stars”), which is at least qualitatively in line with our small value of −5.6 km s−1kpc−1. However, it should be
kept in mind that, as Richard et al. (2010) demonstrated in their various gas-rich merger simulations, the initial orbital
parameters of the mergers strongly affect the final kinematics and structures of the resulting disk populations. Brook
et al. (2007) performed a simulation with a particular set of parameters to produce their disk systems, which may not
necessarily match those of the Galaxy. For example, one rather large difference between this particular simulation
and our results is that, while we find a difference in velocity lag of about 30 km s−1between our thin and thick-disk
subsamples, the N-body prediction calls for a difference of over 150 km s−1. Additional simulations of this process,
better matched to the nature of the Milky Way, would clearly be useful to compare our results with.
The dynamical heating of a pre-existing thin disk, as modeled by the simulations from Villalobos et al. (2010)
also predicts gradients ofVφwith respect to both R and |Z|. Looking at their Figure 14, the thickened-disk component
exhibits a very weak trend ofVφwith R for low initial orbital inclination of the merging satellite, while the correlation
between the two quantities becomes stronger as the incidence angle is increased. Concerning the gradient of Vφwith
|Z|, it is evident in their Figure 14 that the vertical gradient of Vφis much shallower at higher orbital inclination.
Thus, roughly speaking, our radial gradient of Vφfor the thick-disk subsample agrees better with that expected for
low orbital inclination of the merging satellite, but our vertical gradient is better matched by mergers with high orbital
inclination. This may indicate that the compromise case of intermediate orbital inclination (i = 30◦) best describes
our observed results, a possibility also considered by Villalobos et al. (2010). In any event, the comparisons of our
thick-disk subsample with this particular model prediction imply that if the heating scenario played a major role in the
formation of the thick disk, the initial orbital inclination of the merging satellite could not have been too small or too
large. Of course, it is also possible that multiple satellite mergers may have been involved, which complicates these
simple comparisons with a single merger.
In summary, comparisons of our data with the gas-rich merger model from Brook et al. (2007) suggest that,
while this model may not explain the lack of a rotational velocity gradient with Galactocentric distance for the thin
disk, it does account for that observed for the thick disk. However, the much larger difference in the velocity lag than
our finding between the thin- and thick-disk stars remains to be resolved. The model of thin-disk heating by mergers
of Villalobos et al. (2010) qualitatively agrees with the expected kinematic features of our thick-disk subsample,
assuming that the merging satellite has an intermediate orbital inclination.

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In the previous section, which considered correlations between Vφand [Fe/H], our results for the thin-disk pop-
ulation were shown to be in qualitative accord with predictions of the radial migration models, while those for our
thick-disk population might not be. In order to be confident of the implications of this result, one would like to com-
pare with the predictions from more refined radial migration models that better reproduce the observed properties of
the thick-disk population. On the other hand, the relationship between Vφwith R and |Z| for the high-[α/Fe] stars
agrees better with the predictions of the gas-rich merger and thin-disk heating models that we have considered here.
Taken as a whole, the presently available comparisons of the various observed gradients suggest that the thick disk
may have formedfrom either the mergers of gas-richsystems or the heating of a pre-existingthin disk by mergers, and
has been little influenced by the secular process of stellar migration, while radial migration may well have strongly
affected the evolution of the thin disk.
5.3. Distribution of Stellar Orbital Eccentricities
Sales et al. (2009)demonstratedthat the orbitaleccentricities ofa stellar populationcouldalso be used as a toolto
probe the formation and evolution mechanisms of the disk system. In particular, taken at face value (and recognizing
that their summary only pertains to a limited set of model parameters and histories), their Figure 3 suggests that
radial migration models (e.g., Roškar et al. 2008a) generate symmetric distributions of stellar eccentricities with
rather narrow widths, while the gas-rich merger models (e.g., Brook et al. 2004, 2005) produce distributions that are
skewed toward higher eccentricity with larger widths. The accretion models (e.g., Abadi et al. 2003) distribute the
eccentricities rather broadly over a wide range. For the disk heating scenario (e.g., Villalobos & Helmi 2008), there is
a similarity of the eccentricity distribution with that of the merger model for e < 0.6, but there exists a secondary peak
at high eccentricity (e ∼ 0.8). Generally, they found that violent models such as disk heating and accretion generated
a distribution of stellar orbital eccentricities spanning a large range, with secondary peaks at higher eccentricity, or
at least with rather broad distributions of high eccentricity stars. By contrast, the smooth transition models, such as
radialmigrationor in situ star formationfromgas-richmergersproduceddistributionsdominatedbylower eccentricity
orbits covering relatively narrower ranges.
Several studies have compared the above expectations from these models to observed distributions of orbital
eccentricities for thick-disk stars in the solar neighborhood. Wilson et al. (2011), for example, investigated the
eccentricity distribution of a sample of thick-disk stars from RAVE. They concluded that their observed distribution,
which peaked at low eccentricity and exhibited a lack of high eccentricity stars, disfavored the pure accretion model
of Abadi et al. (2003), and was most consistent with the predictions of gas-rich merger models. Dierickx et al. (2010)
carried out a similar test, using a large sample of dwarfs from SDSS DR7, and suggested that their sample favored the
gas-rich merger scenario as well. Casetti-Dinescu et al. (2011) performed an analysis using a sample of ∼4400 red
clump thick-disk stars from RAVE Data Release 2 (Zwitter et al. 2008) with available proper motions from SPM4.
Their comparison of the derived orbital eccentricity distribution with model predictions supported the gas-rich merger
scenario, or possibly the minor merger heating model (arguing that the expected secondary peak at high eccentricity
couldbe avoided, dependingon the initial orbital configurationof the mergingsatellite(s)). Indeed, a recentsimulation
studybyDiMatteoetal. (2011)showedthat,withtheadoptionofaparticularset ofinitialconditions(a1:10massratio
and direct orbit of a presumed single interacting satellite), the disk heating model could also produce the distribution
of eccentricities observed by Wilson et al. (2011) and Dierickx et al. (2010) without creating a secondary peak at high
eccentricity, confirming that the heating model may also be a viable mechanism for thick-disk formation.
It is noteworthy that the various observational studies, based on different samples, with different distance esti-
mates, and adopting different models for the Milky Way potential, all produce similar eccentricity distributions for the
thick-disk population – a broad peak at low eccentricity and a lack of high eccentricity stars. Considering all of the

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studies mentionedabove, the favored mechanisms for thick-disk formationare likely to be either (or both) the gas-rich
mergers model or the thin-disk heating by minor mergers scenario, at least when considering only stellar orbital ec-
centricities as a probe. All of these studies rejected the pure accretion model of thick-disk formation (as advocated by
Abadi et al. 2003).
Unlike the previous observational studies mentioned above, which selected thick-disk stars mostly on the basis
of spatial extent, we have selected a subsample of likely thick-disk stars based on their measured [α/Fe], as described
in Section 3. We now compare the eccentricity distribution of our thick-disk subsample with expectations from each
model cited in Sales et al. (2009). The left-hand set of panels of Figure 10 displays the normalized distributions of
eccentricityforthelow-[α/Fe](topleft)andhigh-[α/Fe](topright)populations. Eachdistributioninthetoptwopanels
is restricted to different slices on distance from the Galactic plane, as listed in the figure legend. The e distribution of
the entire G-dwarf sample (without splits based on [α/Fe]), divided into regions that should emphasize the thin- and
thick-disk regions, is shown in the bottom left and bottom right panels, respectively.
The eccentricity distributions of the thin-disk subsample peak at much less than e = 0.2, with narrow widths,
and apparently include very few high eccentricity stars (e > 0.4) for the two |Z| regions shown in the top left panel.
In contrast, the distributions for the thick-disk subsample shown in the top right panel peak at e ∼0.2, and exhibit
extended tails of higher eccentricities up to e ∼0.8; there remains a relative lack of high eccentricity stars (e > 0.6).
Although we find that the relative frequency of the high-e stars increases a bit at larger |Z| distance (red dashed line)
for both subsamples, the distributions otherwise do not change significantly. This again confirms that the population
separation based on [α/Fe] appears to work quite well. The eccentricity distributions for the full sample of G-dwarf
stars exhibits some rather interesting features. Even at large |Z| distance (0.8–2.4 kpc, bottom right), where the thick-
disk stars should dominate, the eccentricity does not appear similar to that of the thick-disk subsample separated by
[α/Fe] (top right panel) in either range of |Z| distance; the peak and the width do not match. This underscores once
more that, for the purpose of the selection of thick-disk stellar samples, purely spatial separations are insufficient.
Comparing with the published model predictions in Sales et al. (2009), as shown in the right-hand panels of
Figure 10, the relative shortage of high eccentricity stars and the absence of the secondary peak at high e ∼0.8 in
our observed distribution exclude the accretion origin and the disk heating model for the thick disk. Although the
distribution expected from the radial migration models provides a viable description of stars in the low eccentricity
region,it fails to capturethe observedhigh eccentricitytail of the thick-diskstars. The skewed distributionof observed
eccentricities toward higher values is not well-represented by the radial migration predictions, which exhibit a more
Gaussian-like shape (lower left panel of the right-hand panels of Figure 10). It should be noted, however, that an
alternative radial migration model by Schönrich & Binney (2009a,b) indicates the presence of a peak eccentricity
between 0.1 and 0.2, with an extended tail towards high eccentricities, which is consistent with the shape of the
observed e distribution of our thick-disk sample. Hence, we must be cautious in drawing firm conclusions on the
formation mechanisms of the thick disk due to their apparent sensitivity to details of the models and simulations.
Solely based on comparisons with the predictions in the published models from Sales et al. (2009), it seems that our
eccentricity distribution most closely resembles that predicted from the gas-rich merger scenario.
The eccentricity distribution of our thick-disk subsample differs little from the disk heating model of the Di Mat-
teo et al. (2011)simulation. As identified by this simulation (and also mentionedin the discussions of the observations
of Casetti-Dinescu et al. 2011 and Wilson et al. 2011), the secondary peak or high eccentricity region (e > 0.6) in
the disk heating model is mostly occupied by the accreted stars (which retain the initial orbital characteristics of the
merging satellite). In addition, depending on the initial conditions (especially the orbital inclination of the interacting
satellite) of the simulation, the high eccentricity secondary peak may (or may not) be seen in the predicted distribution
of the eccentricities. In particular, the small satellite mass (1:10 mass ratio) in the Di Matteo et al. simulation would
likely not contribute large numbers of stars to the solar neighborhood.