The Milky Way in the Gaia Era The fine structure of the Galactic - - PowerPoint PPT Presentation

The Milky Way in the Gaia Era

The fine structure of the Galactic disc Jason L. Sanders

Institute of Astronomy University of Cambridge

49th Saas Fee School 2019

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 1 / 171

A naive model of a star-forming galaxy such as the Milky Way is an vertically-thick axisymmetric disc of point mass stars embedded in a (possibly triaxial) dark halo. At zeroth order, such a model provides a good representation of the outer parts of the Milky Way (away from the bar/bulge). However, such a model neglects

stars have varying properties such as mass, age, metallicity, other chemical abundances and non-axisymmetric features in discs.

Viewing the Galaxy in this richer space is incredibly important as it reveals the history and formation of the Galaxy as well as providing us with an understanding of what drives the evolution of the Galaxy. The non-axisymmetric features in the Milky Way drive much of its quiescient evolutionary properties. The purpose of this series of lectures is to go beyond the zeroth order representations of our Galaxy and inspect its chemical and kinematic intricacies.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 2 / 171

Outline of the lectures

This series of lectures will be split into two key components. Non-axisymmetric structure in the Galactic disc

The velocity field in the (extended) Solar neighbourhood

Oort constants, vertex deviation, breathing, bending, warping,

Moving groups in the Solar neighbourhood

Velocity and action/angle space,

Response of axisymmetric disc to non-axisymmetric perturbation, Radial migration, dynamical heating.

Chemical structure of the Galactic disc

Extracting properties from spectra,

Abundances, ages

The power of asteroseismology, Chemical evolution,

Sites of production, Single-zone, Multi-zone

Geometrically subdividing the disc (the thick disc), Chemically and chronologically dividing the disc, Inside-out formation, gas-flows, heating and flaring.

Naturally, many of the discussed topics are covered in Binney & Tremaine (2008) and Binney & Merrifield (1998). Also see Sellwood (2014) for a recent review that covers most of first half of lectures.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 3 / 171

Gaia data

Although this series of lecture is not on how to analyse Gaia data, I will use Gaia data to illustrate and motivate the points made. Therefore, here I give a brief overview of the Gaia 2nd data release that I will use. Full Gaia dataset – 1.3 billion parallaxes, proper motions, three photometric bands: broad G, blue GBP and red GRP. For a bright subset of (with GRVS < 14 mag – a narrow photometric band), there are radial velocities – this subset has full 6D information. There is no further spectroscopic information (e.g. effective temperature, surface gravity, metallicity, abundances) so we complement with large-scale spectroscopic surveys e.g. APOGEE, LAMOST, GALAH – using the catalogue of Sanders & Das (2018). We won’t worry about the systematics present in the Gaia data here as we are using the data for purely illustrative purposes.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 4 / 171

What does axisymmetry mean?

An axisymmetric disc is spatially defined by ρ(R, z) which we can consider as Σ(R) – a surface density profile – and ρ(z|R) – a vertical density profile. The velocity distribution is aligned with (R, φ) system – no cross-terms. There can only be non-zero mean azimuthal velocity, not radial Admits circular orbits.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 5 / 171

What do real galaxies look like?

Figure 1: M74: a nearby spiral galaxy. Descubre Foundation, Calar Alto Observatory, OAUV, DSA, V. Peris (OAUV), J. L. Lamadrid (CEFCA), J. Harvey (SSRO), S. Mazlin (SSRO), I. Rodriguez (PTeam), O. L. (PTeam), J. Conejero (PixInsight).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 6 / 171

What does Milky Way look like?

Figure 2: Mean radial velocity field in plane of the disc for Gaia RVS sample (Gaia Collaboration et al., 2018b).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 7 / 171

In this set of lectures we will discuss Stellar velocity structure in the solar neighbourhood (1,2), Characterising this using action variables (2) and Oort’s constants (3), Heating and the vertex deviation (4), Linear response of an axisymmetric disc to a perturbation (5), The effect of radial migration (6), resonances and phase-mixing (7), the bending, breathing and warping of the Galactic disc (8). The second half of these lectures will focus on the properties of stars and the subpopulations of the Galactic disc.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 8 / 171

Section 1 Non-axisymmetric structure 1: Local velocity structure I

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 9 / 171

Galactic Centre Star b

ℓ

Sun Solar rotation x, U z, W y, V North Galactic Pole φ Distance R₀ φ Peculiar motion μl μb vlos R Figure 3: Definition of coordinates use in these lectures

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 10 / 171

Solar neighbourhood UV plane

Figure 4: (U, V) plane for stars in the solar neighbourhood from Gaia Collaboration et al. (2018b).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 11 / 171

Solar neighbourhood UV plane

Figure 5: (U, V) plane for stars in the solar neighbourhood from Gaia Collaboration et al. (2018b) labelled with moving groups.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 12 / 171

Extended solar neighbourhood UV plane

Figure 6: (U, V) plane for stars in different spatial locations aroun the solar neighbourhood from Gaia Collaboration et al. (2018b).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 13 / 171

Why dynamical?

Originally believed to have a common birth origin (Eggen, 1965), but detailed photometry and spectroscopy shows mixture of ages and abundances in each moving group (e.g. Famaey et al., 2005; Bensby et al., 2007).

50 50

vR/kms−1

140 160 180 200 220 240 260 280

vφ/kms−1

τ < 2Gyr

50 50

vR/kms−1

3Gyr < τ < 6Gyr

50 50

vR/kms−1

τ > 8.5Gyr

Figure 7: ‘(U, V)’ plane separated by age using the catalogue of Sanders & Das (2018). The moving groups are present in all age bins. The ‘hotter’ moving groups (e.g. Hercules) are more visible in the older, hotter populations.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 14 / 171

Why dynamical?

Originally believed to have a common birth origin (Eggen, 1965), but detailed photometry and spectroscopy shows mixture of ages and abundances in each moving group (e.g. Famaey et al., 2005; Bensby et al., 2007).

50 50

vR/kms−1

140 160 180 200 220 240 260 280

vφ/kms−1

[Fe/H] > 0.2

50 50

vR/kms−1

0 < [Fe/H] < 0.2

50 50

vR/kms−1

[Fe/H] < −0.2

Figure 8: ‘(U, V)’ plane separated by metallicity using the catalogue of Sanders & Das (2018). The moving groups are present in all metallicity bins. Some moving groups (e.g. Hyades and Hercules) are more visible in the more metal-rich populations – this is likely a reflection of the disc metallicity gradient.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 15 / 171

Take-home messages

Local velocity distribution rich in non-axisymmetric substructure, ∼ 5 significant structures (Sirius, Hyades, Pleiades, Coma Berenices, Hercules), Moving groups contain stars with range of ages and metallicities – not common birth origin, dynamical origin. Structure varies with Galactic position. Horizontal streaks (transient structure).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 16 / 171

Section 2 Non-axisymmetric structure 2: Local velocity structure II

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 17 / 171

Action-angle variables

Actions J and angles θ (frequencies Ω = ∂H

∂J ).

Canonical set of coordinates obeying ˙ J = 0 and ˙ θ = Ω. Adiabatic invariants – invariant under slow potential changes. Action zero-points meaningful. In axisymmetry, Jφ is the conserved component of angular momentum, JR quantifies the radial extent of the orbit and Jz the vertical extent – not linked to a particular coordinate system (despite labels).

Figure 9: Cartoon of action-angle variables.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 18 / 171

Action-angle variables

Figure 10: Angle coordinates in epicyclic approximation from McMillan (2011) –

• scillations about the guiding centre are due to a combination of θR and θφ.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 19 / 171

Local transformation

Figure 11: Relationship between the actions and the velocities (vx, vy) in the solar

• neighbourhood. The left panel shows lines of constant Jφ and θφ (approximately

Cartesian grid) and the right panel lines of constant JR and θR (approximately polar grid) (from McMillan, 2011).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 20 / 171

Solar neighbourhood

1000 1250 1500 1750 2000 2250 2500

Jφ/kpckms−1

2 4 6 8 10 12 14

q

JR/kpckms−1

1 2 3 4 5 6

θR/rad

0.6 0.4 0.2 0.0 0.2 0.4 0.6

θφ/rad

Figure 12: Action-angle distributions of Gaia RVS stars.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 21 / 171

Selection effects

1000 1250 1500 1750 2000 2250 2500

Jφ/kpckms−1

2 4 6 8 10 12 14 q

JR/kpckms−1

1 2 3 4 5 6

θφ/rad

0.6 0.4 0.2 0.0 0.2 0.4 0.6

θR/rad

100 50 50 100

vR/kms−1

160 180 200 220 240

vφ/kms−1

RAVE-like sample within 400 pc

Figure 13: Action-angle distributions of samples from an axisymmetric model folded with a selection function like that of the RAVE survey.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 22 / 171

Substructures

Figure 14: Annotated action-angle distributions of Gaia RVS stars.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 23 / 171

Take-home messages

Action-angles are a useful coordinate system as obey simple equations

• f motion.

Locally, action-angles are a simple transformation of velocity (radial action = polar, azimuthal = Cartesian). Globally, actions compress 6D position and velocity structure. Selection volume imposes structure on action-angle distribution. Velocity substructures evident in action space.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 24 / 171

Section 3 Non-axisymmetric structure: Generalized Oort Constants

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 25 / 171

Generalized Oort constants

The velocities of stars in the solar neighbourhood can be decomposed into streaming motion and random velocities. For kinematically-cold disc stars, the streaming motion is a reflection of the nature of the closed

• rbits in the Galactic potential.

However, measurement of the streaming motion is difficult as the Sun corotates with the surrounding stars. Oort (1927) showed that large-scale asymmetries in the local velocity field could be related to the net streaming of stars and distinguished from the solar peculiar motion. Oort worked in the axisymmetric limit where net streaming is only possible azimuthally. However, the approach can be extended to non-axisymmetric motions (Milne, 1935; Chandrasekhar, 1942; Kuijken & Tremaine, 1991; Olling & Dehnen, 2003).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 26 / 171

Generalized Oort constants

We expand the planar streaming velocity field ¯ v in a Taylor series in the planar Cartesian coordinate system x as ¯ vi = Hijxj + O(x2). (1) Without loss of generality, we write H = K + C A − B A + B K − C

• .

(2) The four generalized Oort constants are the K: local divergence as ∇ · ¯ v = 2K, B: vorticity as ˆ ez · ∇ × ¯ v = 2B, A: azimuthal shear, C: radial shear. Despite the name, the Oort ‘constants’ can vary both with stellar type (Bovy, 2017) and time.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 27 / 171

Generalized Oort constants

In cylindrical polars, the velocities are vx = −vR cos φ + vφ sin φ, vy = vR sin φ + vφ cos φ, (3) and the derivatives evaluated at the Solar location are ∂ ∂x

• (R0,0) = − ∂

∂R

• (R0,0),

∂ ∂y

• (R0,0) = 1

R0 ∂ ∂φ

• (R0,0).

(4) Applying these relations to the Taylor expansion, we yield expressions for the Oort constants as 2A = ¯ vφ R − ∂¯ vφ ∂R − 1 R ∂¯ vR ∂φ , 2B = − ¯ vφ R − ∂¯ vφ ∂R + 1 R ∂¯ vR ∂φ , 2C = − ¯ vR R − 1 R ∂¯ vφ ∂φ + ∂¯ vR ∂R , 2K = ¯ vR R + 1 R ∂¯ vφ ∂φ + ∂¯ vR ∂R , (5) where all terms are evaluated at the Solar location. In the axisymmetric limit, ¯ vR = 0 and all φ-derivatives are zero so C = 0 and K = 0 and the familiar Oort constants are recovered.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 28 / 171

Generalized Oort constants

We can instead work in Galactic coordinates (ℓ, b). We assume the vertical streaming velocity ¯ vz is zero. We introduce the parallax ̟. Generalizing to 3D and setting the solar motion to 0 x = ̟−1 cos b cos ℓ, y = ̟−1 cos b sin ℓ, ¯ v|| = (¯ vx cos ℓ + ¯ vy sin ℓ) cos b, ¯ vℓ = (−¯ vx sin ℓ + ¯ vy cos ℓ) cos b, ¯ vb = −(¯ vx cos ℓ + ¯ vy sin ℓ) sin b. (6) Therefore, ¯ v||̟ = (K + C cos 2ℓ + A sin 2ℓ) cos2 b, ¯ µℓ = (B + A cos 2ℓ − C sin 2ℓ) cos2 b, ¯ µb = −(K + C cos 2ℓ + A sin 2ℓ) sin b cos b. (7)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 29 / 171

Accounting for the solar motion, v = (u, v, w), we find ¯ v||̟ = (K + C cos 2ℓ + A sin 2ℓ) cos2 b − u cos ℓ cos b − v sin ℓ cos b − w sin b, ¯ µℓ = (B + A cos 2ℓ − C sin 2ℓ) cos2 b − u sin ℓ − v cos ℓ, ¯ µb = −(K + C cos 2ℓ + A sin 2ℓ) sin b cos b + u cos ℓ sin b + v sin ℓ sin b − w cos b. (8) Note that the solar motion has a different angular dependence to the Oort constants.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 30 / 171

Measured values

K and C describe the non-axisymmetric nature of the local velocity

• distribution. However, a non-axisymmetric Galaxy can still produce K = 0

and C = 0 if the Sun is on a principal axis of an elliptical potential (Kuijken & Tremaine, 1994). From the parallaxes and proper motions of the first Gaia data release, Bovy (2017) found A = (15.3 ± 0.4)km s−1 kpc−1 and B = (−11.9 ± 0.4)km s−1 kpc−1 implying

vφ R = (A − B) = (27.2 ± 0.6)km s−1 kpc−1 and ∂vφ ∂R = −(A + B) = (−3.4 ± 0.6)km s−1 kpc−1.

Other determinations (e.g. Feast & Whitelock, 1997) use more extended samples of tracers (e.g. Cepheids) where higher-order velocity structure contributes.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 31 / 171

Data

100 200 300

Galactic longitude / deg

40 30 20 10 10 20 30 40

(v||, gc /cos2b)/kms−1 kpc−1

> 2mas, /σ > 3 0.6 < (GBP − GRP) < 0.7 (K + Asin2ℓ + Ccos2ℓ) K = (−0.27 ± 0.24), A = (15.63 ± 0.30) C = (−2.76 ± 0.32)

100 200 300

Galactic longitude / deg

40 30 20 10 10 20 30 40

(µℓ, gc/cos2b)/kms−1 kpc−1

(B + Acos2ℓ − Csin2ℓ) B = (−11.88 ± 0.21), A = (17.01 ± 0.28) C = (−5.14 ± 0.28)

100 200 300

Galactic longitude / deg

40 30 20 10 10 20 30 40

(µb, gc/(sinbcosb))/kms−1 kpc−1

−(K + Asin2ℓ + Ccos2ℓ) K = (−3.26 ± 0.36), A = (14.75 ± 0.47) C = (−3.35 ± 0.52)

Figure 15: Results from Gaia DR2 RVS sample: proper motions of stars with ̟ < 2 mas, ̟/σ̟ > 3 and 0.6 < (GBP − GRP) < 0.7. Left-panel shows the ‘proper-motion’ in the line-of-sight direction.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 32 / 171

Caveats

Asymmetric drift means azimuthal velocity of hotter populations is less than the local circular speed. This means A and B vary with the population and must be corrected to measure the underlying properties

• f the potential – use bluer, younger populations e.g. Cepheids (Feast &

Whitelock, 1997). We have neglected net streaming perpendicular to the Galactic plane as well as planar gradients in the vertical velocity. Typically samples of stars extend beyond the typical disc scale-height which is the scale on which vertical gradients are expected. Extension to planar gradients is simple though the contributions have angular terms cos ℓ, sin ℓ so are partially degenerate with the solar peculiar motion. However, such gradients are expected and measured (see warp). Olling & Dehnen (2003) discuss how incompleteness (through varying survey depth and extinction) gives rise to ‘mode-mixing’ which bias the Oort constants.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 33 / 171

Take-home messages

Generalized Oort constants characterize local velocity structure. Disentangle solar motion from collective motion of solar neighbourhood stars. A and B non-zero for axisymmetric galaxies. C and K zero for axisymmetric galaxies. Non-zero C and K from Gaia RVS data: velocity field not axisymmetric.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 34 / 171

Section 4 Non-axisymmetric structure 4: Heating and the vertex deviation

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 35 / 171

Velocity dispersion tensor

We have inspected the local mean velocity field. The Oort constants allow us to formalize the mean velocity structure. We will now look at the dispersion structure. The local velocity dispersion tensor characterised by six numbers: σ =   σRR σRφ σRz σRφ σφφ σφz σRz σφz σzz.   (9) In the axisymmetric case, σRφ = σφz = 0 (as potential axisymmetric). If the Galaxy is symmetric, z → −z then in the plane z = 0 σRz = 0. However, far from the plane this tilt term can be significant: σRz = 0. For inspecting the solar neighbourhood (z ≈ 0), we will consider the three diagonal terms: σRR, σφφ, σzz, and the cross-term σRφ (sometimes we will write σ2

R ≡ σRR = (vR − vR)2 for example).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 36 / 171

Oort ratio

In the axisymmetric case, radial and azimuthal in-plane oscillations are linked (via epicyclic theory). For small oscillations, we can write (Binney & Tremaine, 2008) v2

R = κ2(∆R)2,

(vφ − vc)2 = 4B2(∆R)2, (10) so σφφ σRR = − B A − B = κ2 4Ω2 . (11) This is known as the Oort ratio, which is 1

2 for a flat rotation curve. In practice,

the Oort ratio is quite different from 1

2 due to

rotation curve not completely flat, the asymmetric drift: hotter populations lag the circular speed due to the asymmetry in the number of stars from inside and outside, highly non-Gaussian vφ distribution – third order corrections are significant.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 37 / 171

Vertex deviation

Traditionally, the vertex deviation ℓv has been used to measure the angle of the major axis of the dispersion tensor relative to the radial tan 2ℓv = −2σRφ σRR − σφφ . (12) In the axisymmetric limit, ℓv = 0.

Figure 16: Definition of vertex deviation. The Galactic centre is off to the top of the diagram.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 38 / 171

Data

0.50 0.75 1.00 1.25 1.50 1.75 2.00

(GBP − GRP)

5.0 2.5 0.0 2.5 5.0 7.5 10.0

MG

Main sequence, > 2mas, /σ > 3

0.4 0.6 0.8 1.0 1.2 1.4

(GBP − GRP)

10 20 30 40

σi/kms−1

Parenago 10 Gyr 12.5 Gyr R φ z

0.4 0.6 0.8 1.0 1.2 1.4

(GBP − GRP)

5 10 15 20 25 30

lv/deg

tan2lv = −

2σRφ σRR − σφφ

lv, red = 8.73 ◦

Figure 17: Results from Gaia DR2 RVS sample: left panel shows the main sequence selection, middle panel the dispersions as a function of colour and right panel the vertex deviation.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 39 / 171

Stellar ages

2 4 6 8 10 12

τ/Gyr

8 16 24 32 40 48 56 64 72 80

σi/kms−1 σR σφ σz σtot

Figure 18: Age-velocity dispersion relation using stars within 300 pc from the catalogue of Sanders & Das (2018). Power-law slopes from top-to-bottom are 0.45, 0.43, 0.31 and 0.35.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 40 / 171

Ages with actions

Dispersion in action σJi ∝ σ2

i .

2 4 6 8 10 12

τ/Gyr

5 10 15 20 25 30 35 40 45 50

σJi/kpckms−1 σJR σJz

Figure 19: Age-action dispersion relation using stars within 300 pc from the catalogue

• f Sanders & Das (2018). Power-law slopes from top-to-bottom are 0.9 and 0.62.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 41 / 171

Beyond solar neighbourhood

τ/Gyr

10

2

2 × 10

1

3 × 10

1

4 × 10

1

6 × 10

1

σR/kms−1

10 10

1

τ/Gyr

2 × 10

1

3 × 10

1

4 × 10

1

6 × 10

1

σz/kms−1

4 6 8 10 12 14

Radius/kpc

4 6 8 10 12 14

Radius/kpc

Figure 20: Age-velocity dispersion relations beyond solar neighbourhood from Sanders & Das (2018).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 42 / 171

Dynamical heating

In an axisymmetric Galaxy, cold gas will settle onto circular orbits (hot

• rbits disallowed as gas is collisional).

Therefore, stars formed with initial velocity dispersion of their birth environment. Non-axisymmetries is the disc scatter stars from their initial orbit, slowly injecting energy into the orbit. Transient spiral arms are efficient in-plane scatterers (increase σRR and σφφ). Poor vertical heaters as ν ≫ m(Ω − Ωp). Fixed pattern spirals fix stars onto more radial orbits but the radial energy does not change over time (see Radial migration lecture.)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 43 / 171

Dynamical heating

Giant molecular clouds efficient at converting radial energy into vertical energy (increase σzz). The magnitude of the impulse from a molecular cloud scales with the interaction time ∆v ∝ v−1 giving the rate of change of vertical energy as dEz/dt ∝ n/v (n the number density of molecular clouds). If the molecular clouds are confined to a thin layer and the stars vertically

• scillate harmonically, n ∝ v−1 so dEz/dt ∝ 1/Ez giving Ez ∝ t1/2 so

σz ∝ t1/4. Difference between heating as a function of time and dispersion with age (as it depends on the strength of perturbations over time – e.g. Aumer et al. (2016)). The oldest stars do not fit into this picture. They may have formed hotter (in an early turbulent phase of the Galaxy).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 44 / 171

Take-home messages

Vertex deviation measures local non-axisymmetry. Dispersions different for different subpopulations. This is primarily the effect of different age populations. Older stars have undergone more heating. Sources of heating: spirals, molecular clouds. However, some populations could be born hot. Vertex deviation also function of age (dispersion).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 45 / 171

Section 5 Non-axisymmetric structure 5: Response to a perturbation

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 46 / 171

Response to a perturbation

In general, the non-axisymmetric effects discussed in previous lectures are

• small. They must be driven by non-axisymmetric perturbing potentials. For

instance, sources of non-axisymmetry in the Galaxy are the Galactic bar spiral arms satellite perturbations triaxial dark halo giant molecular clouds · · ·

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 47 / 171

Response to a perturbation

The response of an axisymmetric disc to a non-axisymmetric potential can be considered using perturbation theory. In general, there is a need for a full 3D theory for warm discs. Here we work with cold discs in 2D (R, φ). The equations for the system (collisionless Boltzmann / Jeans’ equations) are linearized assuming that the perturbation is small. Here we closely follow Kuijken & Tremaine (1991) and Binney & Tremaine (2008). All perturbative responses are 2π periodic and rotate at frequency ω so can be expressed in the form f(R, φ) =

• m∈N

fm(R)ei(mφ−ωt) (13) We will work with a single m angular wave-number. For spiral patterns m is the number of spiral arms. A bar has m = 2 as its dominant wave-number. More complex perturbations can be considered by summing the responses over m.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 48 / 171

We start with the collisionless Boltzmann equation for distribution function f in potential Φ (where we use the Einstein summation convention) ∂f ∂t + vi ∂f ∂xi − ∂f ∂vi ∂Φ ∂xi = 0. (14) We obtain three Jean’s equations by multiplying by 1, vj and vjvk respectively and integrating over all velocities d3v as ∂ρ ∂t + ∂(ρvi) ∂xi = 0 (1), ∂(ρvj) ∂t + ∂(ρvivj) ∂xi + ρ∂Φ ∂xj = 0 (2), ∂(ρvjvk) ∂t + ∂(ρvivjvk) ∂xi + vjρ ∂Φ ∂xk + vkρ∂Φ ∂xj = 0 (3). (15) The first and second equations can be combined to give ρ∂vj ∂t + ρvi ∂vj ∂xi = −ρ∂Φ ∂xj − ∂(ρσij) ∂xi . (16)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 49 / 171

We express our Jeans equations in terms of cylindrical polar coordinates (R, φ). This requires derivatives of vectors and tensors. Using the notation ˆ eij ≡ ˆ ei ⊗ ˆ ej and ˆ eijk ≡ ˆ ei ⊗ ˆ ej ⊗ ˆ ek, we have for vector v and tensor S ∇v = ∂vR ∂R ˆ eRR + 1 R ∂vR ∂φ − vφ

• ˆ

eRφ + ∂vφ ∂R ˆ eφR + 1 R ∂vφ ∂φ + vR

• ˆ

eφφ, (17) and ∇S =∂SRR ∂R ˆ eRRR + 1 R ∂SRR ∂φ − (SφR + SRφ)

• ˆ

eRRφ +∂SRφ ∂R ˆ eRφR + 1 R ∂SRφ ∂φ + (SRR − Sφφ)

• ˆ

eRφφ +∂SφR ∂R ˆ eφRR + 1 R ∂SφR ∂φ + (SRR − Sφφ)

• ˆ

eφRφ +∂Sφφ ∂R ˆ eφφR + 1 R ∂Sφφ ∂φ + (SRφ + SφR)

• ˆ

eφφφ. (18)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 50 / 171

We work in the regime where the terms ∂(ρσij)

∂xi

are negligible. That is that the

• rbits are cold. More precisely, the radial excursion of an orbit (max(|u|, σR)/κ

– for epicyclic frequency κ and radial velocity perturbation u) is shorter than the radial scale of the potential. In this case, the density cancels through the Jeans equation and the two equations in cylindrical coordinates are ∂vR ∂t + vR ∂vR ∂R + vφ R ∂vR ∂φ − v2

φ

R = −∂Φ ∂R , ∂vφ ∂t + vR ∂vφ ∂R + vφ R ∂vφ ∂φ + vRvφ R = − 1 R ∂Φ ∂φ (19) We linearize the equation as vR = ǫRe

• u(R)ei(mφ−ωt)

, vφ = vc + ǫRe

• v(R)ei(mφ−ωt)

, Φ = Φ0 + ǫRe

• Φ1(R)ei(mφ−ωt)

. (20)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 51 / 171

We obtain u = i ∆

• ˜

ω ∂Φ1 ∂R − 2mΩΦ1 R

• ,

v = 1 ∆

• 2B ∂Φ1

∂R − m˜ ω Φ1 R

• ,

(21) where Ω ≡ vc/R = 2(A − B), ˜ ω = ω − mΩ and ∆ = κ2 − ˜ ω2. These expressions are suitable for cold discs. More generally, the response for a warm disc is weaker and summarised by a reduction factor F – see Binney & Tremaine (2008) Appendix K for a calculation of F assuming a tightly-wound spiral perturbing a Schwarzschild distribution function. With the velocity perturbations computed, it is simple to use the continuity equation (Jeans equation (1)) to demonstrate the perturbation Σ1 to the unperturbed surface density Σ0 is i ˜ ωΣ1 = 1 R ∂(RuΣ0) ∂R + imΣ0 R v. (22) Now we go on to inspect the effect on the dispersions and vertex deviation.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 52 / 171

Kuijken & Tremaine (1991) show how the vertex deviation can be related to the velocity perturbations independent of the potential. We wish to combine the three Jeans equations retaining terms up to those proportional to σijk: σijk ≡ (vi − vi)(vj − vj)(vk − vk) = vivjvk − vivjvk − vjvivk − vkvivj + 2vivjvk. (23) We subtract vj times equation (1) from equation (2) and vjvk times equation (1) from equation (3) to yield ρ∂vj ∂t + ∂(ρvivj) ∂xi − vj ∂(ρvi) ∂xi + ρ∂Φ ∂xj = 0 (4), ρ∂(vjvk) ∂t + ∂(ρvivjvk) ∂xi − vjvk ∂(ρvi) ∂xi + vjρ ∂Φ ∂xk + vkρ∂Φ ∂xj = 0 (5). (24) Now we subtract equation (4) times (vj + vk) from equation (5) which eliminates the potential yielding ρ∂σjk ∂t + ∂(ρvivjvk) ∂xi − vjvk ∂(ρvi) ∂xi − vj ∂(ρvivk) ∂xi − vk ∂(ρvivj) ∂xi + 2vjvk ∂(ρvi) ∂xi = 0. (25) We can rewrite all derivatives of ρ as e.g. vjvk

∂(ρvi) ∂xi

= ∂(ρvjvk vi)

∂xi

− ρvi

∂(vjvk) ∂xi

.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 53 / 171

This yields ρ∂σjk ∂t + ∂(ρσijk) ∂xi +ρvi ∂(vjvk) ∂xi +ρvivk ∂vj ∂xi +ρvivj ∂vk ∂xi +2ρvi ∂(vjvk) ∂xi = 0. (26) We neglect the term with σijk and write vi ∂(vjvk) ∂xi + vi ∂(vjvk) ∂xi = vi ∂σjk ∂xi , (27) and vivk ∂vj ∂xi + vivj ∂vk ∂xi + vi ∂(vjvk) ∂xi = σik ∂vj ∂xi + σij ∂vk ∂xi . (28) This gives us the final equation ∂σjk ∂t = σik ∂vj ∂xi + σij ∂vk ∂xi + vi ∂σjk ∂xi . (29) This equation tells us the relationship between the dispersions and mean velocities independent of the potential. It is therefore appropriate for considering perturbations from both spirals and the bar.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 54 / 171

In terms of cylindrical polar coordinates, we obtain three equations ∂σRR ∂t + 2σRR ∂vR ∂R + 2σRφ R ∂vR ∂φ − 2vφ

• + ∂σRR

∂R vR + 1 R ∂σRR ∂φ vφ = 0, ∂σφφ ∂t + 2σRφ ∂vφ ∂R + 2σφφ R ∂vφ ∂φ + 2vφ

• + ∂σφφ

∂R vR + 1 R ∂σφφ ∂φ vφ = 0, ∂σRφ ∂t + σRR ∂vφ ∂R + σRφ 1 R ∂vφ ∂φ + vR R + ∂vR ∂R

• + σφφ

R ∂vR ∂φ − vφ

• + ∂σRφ

∂R vR + vφ R ∂σRφ ∂φ + (σRR − σφφ)

• = 0.

(30)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 55 / 171

We linearize the equation as σij = σ0

ij + ǫRe

• sij(R)ei(mφ−ωt)

, ℓv = ǫRe

• ℓ1ei(mφ−ωt)

. (31) For simplicity, we introduce the parameter α = −(A + B)/(A − B). This corresponds to the slope of a power-law rotation curve with vc ∝ Rα. Hence Ω ≡ vc/R = 2(A − B) and ∂vc/∂R = αΩ = −2(A + B). To leading order in ǫ we have (using the Oort ratio for the unperturbed dispersion ratio) ℓv ≈ − sRφ σ0

RR − σ0 φφ

= (A − B) A sRφ σ0

RR

(32) ℓ1 = 2 (1 − α)∆2

• − 2(α + 1)Ω ∂u

∂R − i ˜ ω ∂v ∂R + (α + 1)(2Ω + 1

2m˜

ω) u R + i[2m(α + 1)Ω + α˜ ω] v R

• + · · ·

(33) where ∆2 = 8(1 + α)Ω2 − ˜ ω2 = 4κ2 − ˜ ω2,

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 56 / 171

and for the dispersions sRR σ0

RR

= 1 ˜ ω

• − 2i ∂u

∂R + iu hσ − 2iΩ(1 − α)ℓ1

• ,

sφφ σ0

φφ

= 1 ˜ ω

• i

1 hσ − 2 R

• u + 2mv

R + 2iΩ(1 − α)ℓ1

• .

(34)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 57 / 171

Near a Lindblad resonance, ∆ ≈ 0 then v = −i ˜ ωu 2Ω , sRR ≪ sφφ = −2i ∂u

∂R

˜ ω σ0

φφ, ℓ1 =

1 (1 − α)Ω ∂u ∂R . (35) Radial compression (expansion) leads to positive (negative) vertex deviation. Radial and azimuthal velocities are π/2 out of phase. Near corotation, ˜ ω = 0 u = − imΦ1 (1 + α)ΩR , v = 1 2Ω ∂Φ1 ∂R , sRR σ0

RR

= sφφ σ0

φφ

= 1 ˜ ω

• − i ∂u

∂R + i 1 hσ − 1 R

• u + mv

R

• ,

ℓ1 = − 1 2(1 − α)Ω ∂u ∂R − u R − imv R

• = − C

2A. (36) Near corotation, the dispersion diverge but the axis ratio remains fixed to the unperturbed value. The vertex deviation can be related to the Oort constants giving ℓv ≈ 7◦ for the solar neighbourhood.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 58 / 171

Tightly-wound spiral

As gravity is a long-range force, perturbations in all parts of the system are

• coupled. It is often useful to work in the limit of a tightly-wound spiral as the

long-range forces are negligible and the perturbation acts only locally. In the tightly-wound limit, the derivatives of the potential are dominated by steep local gradients. A general plane wave surface density profile is Σ′ = Σ′

1ei(mφ−ωt)+ikR,

(37) where more generally k(R) but here we consider local perturbations. The corresponding plane-wave potential solution (∇2Φ1 = 4πGΣ′) is Φ1(R, φ, z, t) = Φ1,0ei(mφ−ωt)+ikR−|kz|, (38) where the z term is required to satisfy Σ(z) = 0 for z = 0 (eq. 5.161 Binney & Tremaine, 2008). The relationship between Σ′

1 and Φ1,0 is found by

4πGΣ′ = ǫ

−ǫ

dz ∂2Φ1 ∂z2 = ∂Φ1 ∂z ǫ

−ǫ = −2|k|Φ1.

(39)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 59 / 171

The tightly-wound approximation is |kR| ≫ 1 (k > 0 gives a trailing spiral, k < 0 leading). In this case, radial derivatives of the potential kΦ1(R, φ, t)/R dominate the potential Φ1(R, φ, t). The derived expressions simplify to u = − ˜ ωkΦ1 ∆ , v = ikΩ(1 + α)Φ1 ∆ , ℓ1 = 6ik2Ω˜ ω(1 + α)Φ1 ∆∆2(1 − α) (40) The radial velocity is in phase with the potential whilst the azimuthal velocity and vertex deviation are π/2 out of phase.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 60 / 171

Tightly-wound spiral

1 2 3 4 5 6

φ

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

R

20 15 10 5 5 10 15 20

ℓv/deg

Figure 21: Mean velocities (stream lines) and vertex deviation (colours) for a tightly wound spiral with Φ1 = −0.002Re(exp(20i ln R − 2i(φ − t))) in a galaxy with a flat rotation curve vc = 1. The grey lines are potential minima. Corotation is at R = 1. The spiral is trailing, moving to the right.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 61 / 171

Self-consistent waves

The perturbations are self-consistent if the driving potential equals the response potential. Returning to equation (22) we replace ∂(RuΣ0)/∂R = ikRuΣ0 as the derivative of u is dominated by the gradients of Φ1 in the tightly-wound approximation, and this term also dominates that which is ∝ v. Therefore, Σ1 = kuΣ0 ˜ ω = −k2Φ1 ∆ = 2πG|k| ∆ Σ′

1.

(41) For self-consistent waves, Σ1 = Σ′

1 so we get the dispersion relation

∆ = κ2 − (ω − mΩ)2 = 2πG|k|F, (42) where we have included the reduction factor F. Considering axisymmetric disturbances m = 0 we find cold discs F = 1 are unstable (i.e. there is some k that produces negative ω hence growing modes). Warmer discs F < 1 can be stable provided Toomre Q =

σRκ 3.36GΣ > 1.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 62 / 171

Take-home messages

We can compute the linear response of the local velocity field to a spiral perturbation for cold populations. But need numerical models to consider warm populations. Tightly-wound approximation allows for simpler analytic progress but the results are only indicative for real galaxies (which are not tightly-wound). Variation of ℓv with dispersion indicates small-scale perturbation responsible for local velocity distribution. Full 3D spiral theory is needed. This couples the radial oscillations to vertical oscillations (see Masset & Tagger (1997) and Monari et al. (2016)) and can give rise to bending/breathing modes (see lecture 8) but

• f lower amplitude than observed in the data – likely vertical oscillations

are due to Sgr.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 63 / 171

Section 6 Non-axisymmetric structure 6: Radial migration

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 64 / 171

We have discussed the heating of disc stars and the bulk effects on the velocity distribution from non-axisymmetric features. In a potential perturbation with pattern speed ω, the Jacobi energy is conserved: EJ = E − ωJφ. (43) Therefore, changes in angular momentum are related to changes in energy ∆E = ω∆Jφ = ΩR∆JR + Ωφ∆Jφ, (44) so ∆JR = ω − Ωφ ΩR ∆Jφ. (45) Therefore, at corotation (ω = Ωφ) a star can change its angular momentum without changing its radial action significantly – circular orbit to circular orbit. For fixed spiral patterns the stars will stay on cold orbits within the spiral

• potential. However, for transient spirals the star will move along the line of

constant Jacobi energy and then be deposited at a new radial action and angular momentum when the spiral disappears. This generically causes a heating when multiple spirals with different pattern speeds appear and

• disappear. A subset of stars around corotation of the spiral patterns will move

from circular orbit to circular orbit remaining radially cold.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 65 / 171

Figure 22: Energy-angular momentum diagram from Sellwood & Binney (2002). The solid curve is the set of circular orbits. All orbits more energetic than this are eccentric. A non-axisymmetric steadily-rotating perturbation permits the integral E − ΩpL corresponding to the dashed line. The circular orbits at corotation can be scattered to new circular orbits.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 66 / 171

Figure 23: Effect of a single transient spiral on the stellar angular momentum distribution (left panel all particles, right those with low energy, Sellwood & Binney (2002)). The dashed line corresponds to symmetric scattering about the corotation resonance (solid vertical line).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 67 / 171

Figure 24: Radial metallicity gradient from Cepheids (Genovali et al., 2014).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 68 / 171

2 4 6 8 10 12

τ/Gyr

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6

[Fe/H]

0.75 0.50 0.25 0.00 0.25 0.50

[Fe/H]

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Density (arb. units)

Within 250 pc

Cold All

Figure 25: Age-metallicity relation (left) and metallicity distributions split by dynamics at fixed angular momentum (equal to that of the Sun) using the catalogue of Sanders & Das (2018). Cold means JR < 1 kpc km s−1, Jz < 1 kpc km s−1 (which is dispersion ∼ 5 km s−1).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 69 / 171

A single transient spiral event produces a spread in angular momentum

• changes. Over the lifetime of the disc, many transient spiral events will scatter

the stars and will act as a diffusive process in angular momentum. Therefore, we can model with simple kernels (e.g. Gaussian) – e.g. Hayden et al. (2015).

Figure 26: Metallicity distributions for APOGEE stars in different radial bins from Hayden et al. (2015). Note the changing skewness with positive skew interior to the solar circle and negative skew exterior.

Appears from APOGEE data that migration strength is a function of location in disc – difficult to explain with heating alone.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 70 / 171

Take-home messages

The interaction of a single star with a transient non-axisymmetric perturbation generically causes heating. Around corotation, no heating is imparted. Star scattered from circular

• rbit to circular orbit.

Simulations demonstrate transient spirals cause significant migration. Observations of the local age-metallicity distributions suggest radial migration is significant. Dynamically cold populations share a similar metallicity spread to all stars. Modelled as a diffusive process – dependent on properties of disc.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 71 / 171

Section 7 Non-axisymmetric structure 7: Resonances and phase-mixing

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 72 / 171

Resonances

In the presence of a rotating non-axisymmetric Φ(R, φ) = Φ0(R)eim(φ−Ωpt) there are closed resonant orbits when the condition m(Ω − Ωp) = nκ (46) is satisfied. These resonant orbits correspond to cases when the radial frequency is commensurate with the azimuthal frequency in the frame rotating with the potential. This means a star will repeatedly encounter the same part

• f the non-axisymmetric potential.

Important cases when n = 0: corotation resonance, star circulates around fixed point in bar frame. n = ∓1: Inner/Outer Lindblad resonances. m = 2: a bar is dominated by the m = 2 mode, as is a two-armed spiral.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 73 / 171

Resonances

Figure 27: Resonant orbits in the corotating frame (Binney & Tremaine, 2008) – solid line is corotating, short-dashed inner Lindblad resonance and long-dashed outer.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 74 / 171

Resonances

0.0 2.5 5.0 7.5 10.0 12.5 15.0

R/kpc

50 100 150 200

Frequency/kms−1 kpc−1

Ω Ω − /2 Ω + /2 Bar Solar radius

Figure 28: Frequency combinations with radius in the Galactic potential of McMillan (2017). Solid curve corresponds to corotation, short-dashed outer Lindblad and long-dashed inner Lindblad. The blue vertical line is the solar radius and red horizontal line the approximate pattern speed of the bar (with the corotation radius shown as a dot).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 75 / 171

Resonances

Figure 29: Orbit structure near outer Lindblad resonance from Dehnen (2000) – orbits just inside and just outside the outer Lindblad resonance (outer dashed circle) have different orientation.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 76 / 171

Hercules OLR, corotation, spirals etc.

Features in the local velocity distribution produced by families of orbits on both sides of resonance. Dehnen (2000) attributed Hercules to the OLR of the bar. This requires a fast bar. More recent work attributes it e.g. to corotation of the bar (P´ erez-Villegas et al., 2017), m = 4 bar resonance (Hunt et al., 2018). Other moving groups can be considered as resonances from spirals (e.g. Hyades, Sellwood, 2010; McMillan, 2011). Overlapping of resonances gives rise to complex features.

Figure 30: (U, V) diagram from the models of Dehnen (2000) showing a Hercules-like feature.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 77 / 171

Phase-mixing

Bar is long-lived. But many perturbations come and go, e.g. transient spiral

• arms. The stars are then picked up on resonant orbits inducing structure into

the local velocity distribution, before being released again when the spiral

• disappears. The resonant structures will then phase-mix back into the

distribution. Phase-mixing is very simple in action-angle space. If a clump initially has an action spread ∆J0 and angle spread ∆θ0, then after some time t the distributions will be ∆J = ∆J0, ∆θ = ∆θ0 + ∆Ω0t ≈ ∆Ω0t, (47) where ∆Ω0 = ∂2H/∂2J · ∆J0 and we have introduced the Hessian matrix and Hamiltonian H. The final approximation holds if we consider large times. This shows us that phase-mixed stars in a given angle range have approximately the same actions.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 78 / 171

We can relate this to statements about the velocity distribution of phase-mixed populations by using the epicyclic approximation. A star with epicyclic amplitude X obeys (see Binney & Tremaine, 2008) R = Rg + X cos(κt + ψ), φ = Ωgt + φ0 − 2ΩgX κRg sin(κt + ψ), vφ(R) − vc(R) = − κ2 2Ωg X cos(κt + ψ) = κ2 2Ωg (Rg − R). (48) Therefore, selecting stars in a fixed angular range ∆φ ≪ 1 we select stars with approximately the same Ωg and hence κg. This implies that vφ and R are related via vφ = const. − κ2 2ΩR, (49) for un-phase-mixed material.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 79 / 171

UV Streaks

Figure 31: (U, V) planes from the models of De Simone et al. (2004) (bottom two rows, top rows Hipparcos data). Note the near-horizontal streaks.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 80 / 171

UV Streaks

Figure 32: (U, V) planes from the models of Hunt et al. (2018). Time increasing left-to-right, lifetime of spiral increasing top-to-bottom.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 81 / 171

vφ − R Streaks

7.50 7.75 8.00 8.25 8.50 8.75 9.00

R/kpc

180 200 220 240 260

vφ/kms−1

vφ = const. −

2

2ΩR Figure 33: vφ against R for Gaia RVS sample. Phase-mixing populations exhibit streaks in this plane with gradient κ2/2Ω – see Antoja et al. (2018).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 82 / 171

vφ − R Streaks

Figure 34: vφ against R for two models from Antoja et al. (2018). Left is the result of phase-mixing, right for a barred potential.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 83 / 171

Take-home messages

Non-axisymmetric potentials admit resonant orbits. These resonances can be trapping and surrounded by chaotic regions. Traditionally, Hercules has been associated OLR of a fast bar. Recent pattern speed measurements place solar neighbourhood between corotation and outer Lindblad resonance. Coupling of bar and spirals necessary. Transient spiral features produce multiple scattering events that phase-mix back in.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 84 / 171

Section 8 Non-axisymmetric structure 8: Bending, breathing and warping

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 85 / 171

HI warp

Figure 35: Warp structure in the HI gas (Kalberla & Kerp, 2009).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 86 / 171

Stellar warp

Figure 36: Warp structure in 2MASS star counts (Reyl´ e et al., 2009).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 87 / 171

Wobbly Galaxy

Figure 37: Vertical velocity against angular momentum for solar neighbourhood stars (Sch¨

• nrich & Dehnen, 2018).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 88 / 171

Wobbly Galaxy

Figure 38: Vertical density profile from SDSS (Widrow et al., 2012).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 89 / 171

Bending and breathing

Figure 39: Bending and breathing modes in disc.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 90 / 171

Vertical waves in disc

Toomre showed that in an infinite thin slab with surface density Σ and in-sheet dispersion σx admitted vertical bending waves satisfying the dispersion relation ω2 = 2πGΣ|k| − σ2

xk2.

(50) Here the gravity acts to stabilise the waves whilst the inertial term involving the dispersion destabilises. This second term is that responsible for the buckling instability and bar formation. However, for cold discs the bending waves are stable. This is in contrast to the density waves which are unstable. When embedded in an external potential, we must modify the dispersion relation to include the epicyclic frequency ω2 = ν2 + 2πGΣ|k| − σ2

xk2.

(51) This further stabilises the bending waves. Therefore, warps need to be excited by external perturbations. This relation holds for axisymmetric bending waves in a disc. It can be generalized to non-axisymmetric bending waves by ω → ω − mΩ although warps far from tightly-wound. See Sellwood (2013) and Binney & Tremaine (2008) for more.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 91 / 171

Bending and breathing

When a satellite passes near the disc, we can compute the instantaneous response using the impulse approximation. The satellite passes through the disc at an angle θ to the normal at speed vs. A stationary star located at (x, z) within the satellite’s orbital plane has impact parameter b = x cos θ + z sin θ. If ψ is the angle between the line connecting satellite and star and the vertical, and r the length of this line, the change in vertical velocity is ∆vz =

• azdt = 1

vs

• azdℓ = −GM

vs cos ψ r 2 dℓ = −GM vs ℓ cos θ − b sin θ (b2 + r 2)3/2 dℓ = 2GM bvs sin θ. (52) Therefore in the limit x ≫ z, the bending response is ∆vz,bend = 1

2(∆vz(x, z) + ∆vz(x, −z)

• = 2GM

vsx tan θ (53) and the breathing response is ∆vz,breath = ∆vz(x, z) − ∆vz(x, −z) = 2GMz vsx2 sin θ tan θ. (54) Both bending and breathing responses are excited by a satellite.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 92 / 171

Phase-space spiral

Figure 40: (z, vz) plane coloured by density (left), vR (middle) and vφ (right) (Antoja et al., 2018).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 93 / 171

Phase-space spiral

Figure 41: Vertical frequency of orbits as a function of radius coloured by their vertical amplitude (Antoja et al., 2018).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 94 / 171

Take-home messages

Large-scale Galactic warp in stars, gas and dust. Evidence of breathing, bending and warping in vertical kinematic field. Vertical waves produced by external perturbation. Phase-space spiral result of interaction with Sagittarius and subsequent phase-mixing.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 95 / 171

Section 9 Subpopulations of the disc 1: Introduction and spectroscopy

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 96 / 171

Introduction

Not all stars are the same. The only way to learn how the Galaxy formed is to sub-divide based on the stellar properties. Example 1: you have found a metal-poor star near the Sun. If formed in the Milky Way, it should be old, or has moved from a lower metallicity region of the Galaxy. But could be associated with an accreted dwarf galaxy/globular cluster? Example 2: the stars near the Sun at pericentre are more metal-rich than those at their apocentres. The Galaxy has a negative radial metallicity

• gradient. But how does this depend on when the stars were formed?

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 97 / 171

Introduction

In this set of lectures we will discuss measuring stellar properties with spectroscopy (1,3), asteroseismology (2,3) and photometry (3), Galactic chemical evolution (4,5), the geometric structure of our Galaxy and the thick disc (6), and the chemical/age structure of our Galaxy (7+).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 98 / 171

Gaia HR diagram

Figure 42: HR diagram using Gaia parallaxes and photometry (Gaia Collaboration et al., 2018a).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 99 / 171

Gaia + Spectroscopy HR diagram

Figure 43: Spectroscopic HR diagram using large spectroscopic surveys and Gaia

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 100 / 171

Gaia + Spectroscopy HR diagram

Figure 44: Spectroscopic HR diagram using large spectroscopic surveys and Gaia

• parallaxes. Left is raw, right is folding with stellar models (Sanders & Das, 2018).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 101 / 171

Spectroscopy

Typically a spectrum is characterized by three numbers: Teff temperature of star – hotter stars have broader lines. log g surface gravity – puffier stars (giants) have lower surface gravity and narrower lines (pressure broadening). [M/H] – more metals, deeper lines (but also depends on other stellar properties). Then need to find the abundances e.g. O, Na on a line-by-line basis (either atomic or molecular).

Figure 45: Sensitivity of lines to surface gravity (decreasing upwards).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 102 / 171

Spectroscopy: methods

Methods Stellar model fitting – generate libraries of artificial stellar spectra – requires accurate line-lists (locations of lines, strengths etc.) Data-driven approach (the Cannon) – use a training set (from stellar model fitting) to fit a parametric model to a spectrum pixel-by-pixel. Identifies features associated with certain elements but susceptible to measuring correlations.

Figure 46: Data-driven spectrum model from Ness et al. (2015). The top panel shows

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 103 / 171

Spectroscopy: deriving distances and extinctions

The luminosity of a star is linked to its spectroscopic properties. With apparent magnitudes we can convert luminosities derived from stellar models to absolute magnitudes. This complements Gaia parallaxes. For nearby stars Gaia provides accurate distances and constrains the stellar parameters. For distant stars spectrophotometric distances are superior. Also, extinction – stars with same spectra should have same colours (pair method).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 104 / 171

Take-home messages

Large-scale spectroscopic surveys delivering metallicities and abundances for millions of stars. Allows accurate characterization, distance and extinction estimation.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 105 / 171

Section 10 Subpopulations of the disc 2: Asteroseismology

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 106 / 171

The Sun oscillates

Figure 48: Radial velocity time series for Sun

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 107 / 171

The Sun oscillates

Figure 49: Power spectrum of radial velocity time series

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 108 / 171

Stars oscillate

p-modes are pressure-driven (convection in the stellar atmosphere shakes teh star driving modes, like a boiling pan of water), g-modes gravity driven (in giants).

Figure 50: Solar-like oscillation spectra for five stars with similar mass but decreasing

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 109 / 171

What do oscillations tell you?

Two key quantities to extract from spectra ∆ν – separation of peaks. This is a measure of the dynamical timescale in the star and hence scales like √ρ so ∆ν ∝

• M

R3 (55) νmax – the spectra are modulated by a Gaussian distribution. This tells us about the excitation and damping of the modes. The cut-off frequency νac tells us the maximum frequency modes can have to be reflected at the

• boundary. Using an isothermal approximation, νac = c/(4πH) where c is

the sound speed and H the scale-height. Hypothesising νac ∝ νmax we have νmax ∝ c H ∝ gT −1/2

eff

= MT −1/2

eff

R2 . (56) Using measurements of the Sun as an absolute calibrator, we can use ∆ν and νmax to measure the mass M and R provided we have effective temperatures from spectroscopy or photometry.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 110 / 171

Stars oscillate

Figure 51: Asteroseismic HR diagram (Yu et al., 2018).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 111 / 171

Mode mixing

g-modes are restricted to the core so in general difficult to see. In giants, the p-modes in the convective zone and g-modes in the core can couple producing ‘mixed modes’. These mixed modes are visible in the convective envelope and reflect the properties of the core. Can distinguish red giant stars from red clump stars that are core helium burning. Important as red clump are standard candles.

Figure 52: Mixed g mode period spacing against ∆ν. The red clump stars are the clump in the top left (Vrard et al., 2016).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 112 / 171

Take-home messages

Photometric and radial velocity oscillations give details on internal stellar structure. Radius and mass can be extracted (provided effective temperature) from two summary statistics: νmax and ∆ν. Mode mixing allows separation of red giant and red clump (useful standard candles).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 113 / 171

Section 11 Subpopulations of the disc 3: Stellar ages

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 114 / 171

Stellar age estimation: methods, Soderblom (2010)

Absolute ages Sun (radioactive decay) Kinematic ages from time of flight of cluster dispersal Model-dependent ages (for an evolved star mass tells you approximately age) Isochrones (e.g. blue stars young) Giant C & N abundances Ness et al. (2016), Martig et al. (2016a), Masseron & Gilmore (2015). H lines Bergemann et al. (2016). Gyrochronology Open clusters Kinematics/Dynamics Lithium depletion Chromospheric activity Binaries

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 115 / 171

Isochrone ages

Figure 53: Stellar models overlaid on spectroscopic data from Bergemann et al. (2016). At turn-off, stars can be distinguished in age both from spectroscopy and accurate distances, provided metallicities. On giant branch, surface gravity insufficient, need distances.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 116 / 171

Isochrone ages

Figure 54: Degeneracy between age and metallicity using solely photometry (Howes et al., 2018) – colour is log(probablity).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 117 / 171

Ages from [C/N]

Figure 55: CNO cycle. The proton capture onto 14N is slow so 14N builds up in the core.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 118 / 171

Ages from [C/N]

Figure 56: Distribution of elements in a solar metallicity star at the turn-off (Charbonnel & Lagarde, 2010).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 119 / 171

Ages from [C/N]

Figure 57: Correlation between asteroseismic mass and [C/N] from APOGEE. Models

• verplotted (Martig et al., 2016a).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 120 / 171

Take-home messages

Stellar ages most reliable for main-sequence turn-off Need accurate luminosities for giant stars Mass-dependent processing of carbon and nitrogen permit spectroscopic mass estimation.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 121 / 171

Section 12 Subpopulations of the disc 4: Chemical evolution I

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 122 / 171

Alpha Fe diagram

Figure 58: [α/H] against [Fe/H] (top) from complete sample of Fuhrmann (2011) – circle size correspond to ages.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 123 / 171

Fundamental equations

We assume axisymmetry and an infinitely-thin disc (see Matteucci, 2012). The star-forming gas naturally forms a very thin disc. However, gas returned from stellar evolution may initially enter a more vertically-extended disc (as the stars have been dynamically heated) but we assume the time-scale for the cooling of this gas back on to the thin gas disc is small so can be neglected (note some use a hot gas reservoir that feeds the cold star-forming gas disc). In this way, we reduce the problem to a single radial dimension, R. We define the gas surface density at radius R and time t as Σ(R, t). The evolution of Σ is given by ˙ Σ(R, t) = −Γ(R, t) + G(R, t) + I(R, t) + R(R, t), (57) where the first term gives the gas depleted by star formation, the second term the gas returned from stellar evolution, the third term the inflow of cold gas

• nto the disc and the final term the radial flow of gas in the disc. Each star

formation event produces stars that follow an initial mass function ξ(M) normalized here such that

• Mξ(M)dM = 1. This means ΓξdM is the total star

formation in solar masses in the interval dM.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 124 / 171

Fundamental equations

The evolution of the stellar density is given by ˙ Σ∗(R, t) = Γ(R, t) − G(R, t), (58) which we call the reduced star formation rate. In this way, equation (57) can be rewritten as ˙ Σ(R, t) + ˙ Σ∗(R, t) = I(R, t) + R(R, t) ≡ Σeff(R, t), (59) where ˙ Σeff(R, t) is the effective accretion rate. The evolution of chemical species i is given by ˙ Σi(R, t) = −Γ(R, t)Xi(R, t) + Gi(R, t) + I(R, t)Xi,I(R, t) + Ri(R, t), (60) where Xi gives the mass fraction of species i and Xi,I(R, t) describes the mass fraction of inflowing gas which we assume to be equal to the initial mass fraction Xi,I(R, t) = Xi(R, 0).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 125 / 171

Ingredients: yields

Figure 59: Stellar yields: top AGB, middle Type II SN (both coloured by mass) , bottom Type Ia.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 126 / 171

Ingredients: yields

Not all elements are independent. They reflect different processes. Importantly, alpha elements reflect alpha process which is nuclear fusion involving a He. Alpha elements are produced with even numbers of protons. Alpha elements and iron produced in SNII. In SNIa more iron per alpha. Enriched gas is returned to the ISM through supernovae (both Type Ia and Type II) and asymptotic giant branch (AGB) star winds. Enrichment from both Type II and AGB stars are similar in that metals are returned at the end of the life of a single star. The total gas return is given by G(R, t) =

• 1 − χ(R, t)
• GII+AGB(R, t) + GIa(R, t)
• ,

(61) where χ(R, t) gives the fraction of returned gas that flows off the disc (next lecture). For species i, the gas return per unit time is given by Gi,II+AGB(R, t) = t dt′ Mi,ej(M′(τ))ξ(M′(τ))Γ(R, t′)

• − dM′

dτ

• ,

(62) where Mi,ej is the mass of the ejecta of species i, the age τ = t − t′, and M′ is the mass of a star with lifetime τ (could also be function of metallicity Z). Need to know lifetimes of stars from stellar models. Roughly M ∝ τ −1/3.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 127 / 171

Ingredients: yields

Figure 60: Chemical evolution models for a selection of abundances using different yields (Andrews et al., 2017).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 128 / 171

Ingredients: IMF

Initial mass function (IMF) ξ(M) describes the mass distribution of stars formed in each star formation episode. Simple version is ξ(M) ∝ M−α with α = 2.35. Its universality is currently debated. Steeper mass functions produce more low mass stars relative to high. This produces fewer SN and hence less enrichment.

Figure 61: Illustration of varying the IMF (Andrews et al., 2017).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 129 / 171

Ingredients: SFR

Star formation rate is observed to be linked to the gas surface density (Kennicutt-Schmidt ˙ Σ⋆ ∝ Σ1.4). It is the star formation efficiency (star formation rate/gas mass) that affects the distribution of abundances as abundances are differential.

Figure 62: Illustration of varying the SFE (Andrews et al., 2017).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 130 / 171

Alpha-knee

Figure 63: Alpha-knee for different dwarf spheroidal galaxies – reflecting different star formation efficiency time-scales.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 131 / 171

Sausage merger

Figure 64: Sausage knee – evidence of an early major merger (Hayes et al., 2018).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 132 / 171

Sausage merger

Figure 65: Sausage knee – kinematic evidence of an early major merger.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 133 / 171

Ingredients: Supernovae delay times

We use the delay-time-distribution formalism for the Type Ia supernova rate proposed by Greggio (2005). In this formulation, the delay-time distribution D(τ) describes the rate of Type Ia supernovae from a single stellar population

• f age τ. Such a formulation does not assume any particular progenitor

scenario and allows for flexible combinations of several channels. The gas return per unit time for species i is Gi,Ia(R, t) = Mi,ej t dt′ D(τ)Γ(R, t′), (63) where the ejecta mass of species i is independent of metallicity and all Type Ia supernovae explode at the Chandrasekhar mass, Mch = 1.3745M. We adopt the analytic delay-time distribution fitted by Matteucci et al. (2006)to the results of Mannucci et al. (2006) as D(t) = AB mmax

mmin dM ξ(M)

• 101.46−50(log10 t−1.3)

0.04 < t < 0.085, 10−0.74−0.9(log10 t−0.3) 0.085 < t < 14, (64) where t is in Gyr and the lower and upper age limits correspond to stars with masses 8M and 0.8M respectively. AB gives the fraction of all stars that will form a Type Ia supernova.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 134 / 171

Ingredients: Supernovae delay times

Figure 66: Delay time distribution (Matteucci et al., 2006) – peaks for prompt and tardy.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 135 / 171

Ingredients: Supernovae delay times

Figure 67: Illustration of varying the decay time (Andrews et al., 2017).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 136 / 171

Closed box model: predictions

Figure 68: G dwarf problem: a closed box model produces too many metal-poor stars (Pagel, 1992).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 137 / 171

Closed box model: predictions

Figure 69: α − Fe diagrams for a closed box model (Andrews et al., 2017).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 138 / 171

Take-home messages

Ingredients for closed box chemical evolution model: IMF, SFR, Yields (SN, AGB) Steeper IMF, fewer SNII, lower [α/Fe] Higher specific SFR, knee at higher [Fe/H]. Evidence of a chemically-distinct accreted population in local halo distribution (Nissen & Schuster, 2010). Closed box produces too many metal-poor stars: G dwarf problem Similar formalism (yields, SNIa delay times, stellar ages) used in large (hydro) simulations.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 139 / 171

Section 13 Sub-populations of the disc 5: Chemical evolution II

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 140 / 171

Outflows

Figure 70: Galactic fountain in NGC 253: HI plumes

A simple ‘Galactic fountain’ model of χ(R, t) =

• χin,

R < RT, χout, R > RT. (65)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 141 / 171

Radial gas flows and inflow

Figure 71: High velocity clouds HI – accreting gas (Wakker et al., 2008).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 142 / 171

Radial gas flows and inflow

We follow the gas flow prescription given by Pezzulli & Fraternali (2016). The circular velocity of gas falling onto the Galactic disc lags the local circular velocity Vc(R, t) by α(R, t)Vc(R, t). By considering the conservation of mass and angular momentum, Pezzulli & Fraternali (2016) demonstrate that the inflow rate is given by I(R, t) = −µ(R, t) 2παR2 , (66) and the radial velocity of the gas is u(R, t) = µ(R, t) 2πRΣ , (67) such that R(R, t) = − 1 2πR ∂µ ∂R . (68) The radial mass flux is µ which can be chosen to reproduce a given star formation law.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 143 / 171

Radial gas flows and inflow

Figure 72: Radial mass flux and radial velocity for a toy model of gas accreted at angular momentum lag (Pezzulli & Fraternali, 2016).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 144 / 171

Radial migration

Radial migration affects both the stars and the gas in the Galaxy. Must use simulation-inspired recipes in chemical evolution codes. Non-axisymmetric potential perturbations scatter stars/gas in angular momentum. The gas resides on near-circular orbits so changes in angular momentum correspond to changes in circular radius. The kernel governing the transition rate from radius R′ to radius R over time t is given by K(R, R′, t) such that the surface density of species i after a time τ (ignoring other evolutionary effects) is Σi(R, t + τ) = 1 R

• dR′ R′Σi(R′, t)K(R, R′, τ).

(69) We can also consider a discretized version as in Sch¨

• nrich & Binney (2009).

Mi =p(i + 1 → i)Mi+1 + p(i − 1 → i)Mi−1 − p(i → i + 1)Mi − p(i → i − 1)Mi. (70) If p(j → i) ∝ Mi∆t angular momentum is conserved.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 145 / 171

Radial migration

Figure 73: Metallicity distributions from Sch¨

• nrich & Binney (2009). Blue without

radial migration and gas flows.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 146 / 171

Inside-out formation

More metals produced in the inner Galaxy (higher SFE, gas flows) so expect negative gradient in vφ against [Fe/H]. However, in metal-poor ‘thick disc’ populations positive gradient observed (e.g. Lee et al., 2011). Due to inside-out formation. Series of negative gradient populations sum to produce a positive gradient.

Figure 74: vφ against metallicity from the models of Sch¨

• nrich & McMillan (2017).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 147 / 171

Take-home messages

More advanced chemical evolution models account for gas flows (inflow,

• utflow and radial flow).

Radial migration also an important ingredient in describing metallicity spread. Inside-out formation necessary ingredient to explain increase in vφ with metallicity at low metallicity (in ‘thick disc’).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 148 / 171

Section 14 Sub-populations of the disc 6: Geometric thick disc

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 149 / 171

Thick disc

Figure 75: Gilmore & Reid (1983) South Pole density profile derived from photometric distances.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 150 / 171

Thick disc

Figure 76: Vertical density profile of upper main sequence stars with Gaia parallaxes. Gilmore & Reid (1983) in green, Juri´ c et al. (2008) in red.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 151 / 171

External galaxy thick discs

Figure 77: Edge-on view of NGC 522 which shows a thick disc (Comer´

• n et al., 2011).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 152 / 171

Kinematics of thick disc

Figure 78: The thick disc is kinematically hot (Casetti-Dinescu et al., 2011).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 153 / 171

Age and alpha of thick disc

Figure 79: The thick disc is primarily old and alpha-rich. We will see later a more complicated picture (Haywood et al., 2013).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 154 / 171

Thick disc formation scenarios?

Heating Major merger (extreme heating!) Upside-down Flaring Radial migration

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 155 / 171

Take-home messages

Milky Way has a geometric thick disc – large vertical scaleheight. Thick discs common in external galaxies Production of thick disc: last major merger, upside-down formation? Thick disc is old and metal-poor, and has hot kinematics. But this picture is somewhat historical based on local observations (next lecture...)

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 156 / 171

Section 15 Sub-populations of the disc 7: Chemical and chronological disc separation

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 157 / 171

Reconciling picture of thick disc

Figure 80: Radial density profile of SDSS main sequence stars from Juri´ c et al. (2008). Each set of points corresponds to a different Galactic height. The two dashed lines are exponentials with scales 3 kpc and 5 kpc. Note the ‘thick disc’ has a longish ∼ 4 kpc scalelength – this appears consistent with external galaxies.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 158 / 171

Reconciling picture of thick disc

Figure 81: Radial scale-radius against vertical scale-height for a set of ‘mono-abundance populations’ (narrow bins in [α/Fe] against [Fe/H]) using SEGUE from Bovy et al. (2012a). Note the α-rich disc (traditionally associated with the thick disc) has a short scalelength ∼ 2 kpc.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 159 / 171

Reconciling picture of thick disc

Figure 82: [α/Fe] against [Fe/H] distributions for a series of radial (left to right) and vertical (bottom to top) spatial bins for APOGEE data from Hayden et al. (2015). Note the radially truncated high [α/Fe] sequence, the increasing contribution of high [α/Fe] vertically and the presence of α-poor stars at high heights in the outer disc.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 160 / 171

Flaring of discs

Figure 83: Simulated mono-age populations from Minchev et al. (2015). Top panel shows their scale heights and bottom the radial distributions. The squares and triangles show the two exponential fit which produces a thin and thick disc over the whole radial range.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 161 / 171

Age gradient in geometric thick disc

Figure 84: Radial age gradient at a range of Galactic heights from Martig et al. (2016b). Evidence of inside-out formation and younger disc stars contributing to the

• uter geometric thick disc.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 162 / 171

The mono-abundance view: radial

Figure 85: Surface density profiles of mono-abundance populations from Bovy et al. (2016). All high [α/Fe] populations display exponential profile with ∼ 2 kpc scalelength. The low [α/Fe] bins exhibit broken exponential with peak moving outwards for lower

• metallicity. Evidence of metallicity gradient and radial migration.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 163 / 171

The mono-abundance view: vertical

Figure 86: Scaleheights of mono-abundance populations from Bovy et al. (2016). High [α/Fe] populations don’t flare, low [α/Fe] do.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 164 / 171

The mono-age view: radial

Mono-abundance populations difficult to interpret as relationship to age non-linear.

Figure 87: Radial profile split by age from Mackereth et al. (2017). Note broadening with age (radial migration and disc heating).

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 165 / 171

The mono-age view: vertical

Figure 88: Scaleheight split by age from Mackereth et al. (2017). Young disc significantly flared.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 166 / 171

Rethinking thick disc formation

Figure 89: Surface density against scaleheight for mono-age populations (Mackereth et al., 2017). Note the smooth distribution with no sign of a bimodality. Perhaps evidence of no violent formation event of thick disc.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 167 / 171

Chemical bimodality

However, there is a bimodality in chemical space. This is awkward to reconcile with the smooth spatial properties of the populations. The bimodality can possibly be explained by 1. tuned chemical evolution, 2. gas-rich merger,

• 3. clumpy star formation.

Figure 90: Solar neighbourhood [α/Fe] against [Fe/H] from Hayden et al. (2015). Note the bimodality, particularly at low [Fe/H].

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 168 / 171

Dynamical modelling of populations

What about velocities?

Figure 91: Vertical dispersions of mono-abundance populations from Bovy et al. (2012b). Note the run of σz with height seen previously is due to a sum of near isothermal populations.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 169 / 171

Dynamical modelling of populations

Bovy & Rix (2013) used the mono-abundance populations hz and σz distributions to constrain the Galactic potential. A quasi-isothermal f(J) model (see Binney lecture ‘Models from distribution functions’) is fitted for each MAP whilst simultaneously adjusting the potential Φ(x). In axisymmetry, Jz distribution specifies σz and vertical density profile, Jφ specifies radial surface density profile and JR distribution specifies σR and σφ. If we know the potential σz is fixed by vertical profile. σz can be used to measure potential. σR(R) must be constrained from the data (e.g. see ‘Dynamical heating and the vertex deviation’) lecture.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 170 / 171

Take-home messages

Geometric thick disc has long scalelength but α-rich disc has short scalelength. α-rich populations (at all ages) approximately exponential (scalelength 2 kpc). α-poor populations a series of broken profiles with an anticorrelation between peak radius and metallicity. Width broadens with age. α-rich populations truncated but α-poor populations flare. Geometric thick disc composed of sum of flaring populations. Smooth spatial distribution points to non-violent past formation. Likely the disc is formed through a combination of inside-out formation and outward radial migration. However, chemical bimodality still a puzzle: tuned chemical evolution, wet merger, clumpy star formation? With Gaia, the populations can be inspected dynamically.

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 171 / 171

References Andrews B. H., Weinberg D. H., Sch¨

• nrich R., Johnson J. A., 2017, ApJ, 835,

224 Antoja T., et al., 2018, Nature, 561, 360 Aumer M., Binney J., Sch¨

• nrich R., 2016, MNRAS, 462, 1697

Bensby T., Oey M. S., Feltzing S., Gustafsson B., 2007, ApJ, 655, L89 Bergemann M., et al., 2016, A&A, 594, A120 Binney J., Merrifield M., 1998, Galactic Astronomy Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press Bovy J., 2017, MNRAS, 468, L63 Bovy J., Rix H.-W., 2013, ApJ, 779, 115 Bovy J., Rix H.-W., Liu C., Hogg D. W., Beers T. C., Lee Y. S., 2012a, ApJ, 753, 148 Bovy J., Rix H.-W., Hogg D. W., Beers T. C., Lee Y. S., Zhang L., 2012b, ApJ, 755, 115 Bovy J., Rix H.-W., Schlafly E. F ., Nidever D. L., Holtzman J. A., Shetrone M., Beers T. C., 2016, ApJ, 823, 30 Casetti-Dinescu D. I., Girard T. M., Korchagin V. I., van Altena W. F ., 2011, ApJ, 728, 7

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 171 / 171

Chandrasekhar S., 1942, Principles of stellar dynamics Chaplin W. J., Miglio A., 2013, ARA&A, 51, 353 Charbonnel C., Lagarde N., 2010, A&A, 522, A10 Comer´

• n S., et al., 2011, ApJ, 741, 28

De Simone R., Wu X., Tremaine S., 2004, MNRAS, 350, 627 Dehnen W., 2000, AJ, 119, 800 Eggen O. J., 1965, Moving Groups of Stars. the University of Chicago Press,

• p. 111

Famaey B., Jorissen A., Luri X., Mayor M., Udry S., Dejonghe H., Turon C., 2005, A&A, 430, 165 Feast M., Whitelock P ., 1997, MNRAS, 291, 683 Fuhrmann K., 2011, MNRAS, 414, 2893 Gaia Collaboration et al., 2018a, A&A, 616, A10 Gaia Collaboration et al., 2018b, A&A, 616, A11 Genovali K., et al., 2014, A&A, 566, A37 Gilmore G., Reid N., 1983, MNRAS, 202, 1025 Greggio L., 2005, A&A, 441, 1055 Hayden M. R., et al., 2015, ApJ, 808, 132

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 171 / 171

Hayes C. R., et al., 2018, ApJ, 852, 49 Haywood M., Di Matteo P ., Lehnert M. D., Katz D., G´

• mez A., 2013, A&A,

560, A109 Howes L. M., Lindegren L., Feltzing S., Church R. P ., Bensby T., 2018, arXiv e-prints, Hunt J. A. S., Hong J., Bovy J., Kawata D., Grand R. J. J., 2018, MNRAS, 481, 3794 Juri´ c M., et al., 2008, ApJ, 673, 864 Kalberla P . M. W., Kerp J., 2009, ARA&A, 47, 27 Kuijken K., Tremaine S., 1991, in Sundelius B., ed., Dynamics of Disc

• Galaxies. p. 71

Kuijken K., Tremaine S., 1994, ApJ, 421, 178 Lee Y. S., et al., 2011, ApJ, 738, 187 Mackereth J. T., et al., 2017, MNRAS, 471, 3057 Mannucci F ., Della Valle M., Panagia N., 2006, MNRAS, 370, 773 Martig M., et al., 2016a, MNRAS, 456, 3655 Martig M., Minchev I., Ness M., Fouesneau M., Rix H.-W., 2016b, ApJ, 831, 139 Masseron T., Gilmore G., 2015, MNRAS, 453, 1855

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 171 / 171

Masset F ., Tagger M., 1997, A&A, 318, 747 Matteucci F ., 2012, Chemical Evolution of Galaxies. Springer Berlin Heidelberg, doi:10.1007/978-3-642-22491-1, https://doi.org/10.1007/978-3-642-22491-1 Matteucci F ., Panagia N., Pipino A., Mannucci F ., Recchi S., Della Valle M., 2006, MNRAS, 372, 265 McMillan P . J., 2011, MNRAS, 418, 1565 McMillan P . J., 2017, MNRAS, 465, 76 McMillan P . J., et al., 2018, MNRAS, 477, 5279 Milne E. A., 1935, MNRAS, 95, 560 Minchev I., Martig M., Streich D., Scannapieco C., de Jong R. S., Steinmetz M., 2015, ApJ, 804, L9 Monari G., Famaey B., Siebert A., 2016, MNRAS, 457, 2569 Ness M., Hogg D. W., Rix H.-W., Ho A. Y. Q., Zasowski G., 2015, ApJ, 808, 16 Ness M., Hogg D. W., Rix H.-W., Martig M., Pinsonneault M. H., Ho A. Y. Q., 2016, ApJ, 823, 114 Nissen P . E., Schuster W. J., 2010, A&A, 511, L10 Olling R. P ., Dehnen W., 2003, ApJ, 599, 275 Oort J. H., 1927, Bull. Astron. Inst. Netherlands, 3, 275

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 171 / 171

Pagel B. E. J., 1992, in Barbuy B., Renzini A., eds, IAU Symposium Vol. 149, The Stellar Populations of Galaxies. p. 133 P´ erez-Villegas A., Portail M., Wegg C., Gerhard O., 2017, ApJ, 840, L2 Pezzulli G., Fraternali F ., 2016, MNRAS, 455, 2308 Reyl´ e C., Marshall D. J., Robin A. C., Schultheis M., 2009, A&A, 495, 819 Sanders J. L., Das P ., 2018, MNRAS, 481, 4093 Sch¨

• nrich R., Binney J., 2009, MNRAS, 396, 203

Sch¨

• nrich R., Dehnen W., 2018, MNRAS, 478, 3809

Sch¨

• nrich R., McMillan P

. J., 2017, MNRAS, 467, 1154 Sellwood J. A., 2010, MNRAS, 409, 145 Sellwood J. A., 2013, Dynamics of Disks and Warps. p. 923, doi:10.1007/978-94-007-5612-0˙18 Sellwood J. A., 2014, Reviews of Modern Physics, 86, 1 Sellwood J. A., Binney J. J., 2002, MNRAS, 336, 785 Soderblom D. R., 2010, ARA&A, 48, 581 Vrard M., Mosser B., Samadi R., 2016, A&A, 588, A87 Wakker B. P ., York D. G., Wilhelm R., Barentine J. C., Richter P ., Beers T. C., Ivezi´ c ˇ Z., Howk J. C., 2008, ApJ, 672, 298

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 171 / 171

Widrow L. M., Gardner S., Yanny B., Dodelson S., Chen H.-Y., 2012, ApJ, 750, L41 Yu J., Huber D., Bedding T. R., Stello D., Hon M., Murphy S. J., Khanna S., 2018, ApJS, 236, 42

Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 171 / 171