Solar neighbourhood 0.6 14 0.4 12 J R / kpckms − 1 10 0.2 θ φ / rad 8 0.0 6 0.2 q 4 0.4 2 0.6 0 1000 1250 1500 1750 2000 2250 2500 0 1 2 3 4 5 6 J φ / kpckms − 1 θ R / 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 RAVE-like sample within 400 pc 0.6 14 240 0.4 12 J R / kpckms − 1 220 10 0.2 v φ / kms − 1 θ R / rad 8 0.0 200 6 0.2 180 q 4 0.4 2 160 0.6 0 1000 1250 1500 1750 2000 2250 2500 0 1 2 3 4 5 6 100 50 0 50 100 J φ / kpckms − 1 v R / kms − 1 θ φ / rad 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 of 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 orbits 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 v i = H ij x j + O ( x 2 ) . ¯ (1) Without loss of generality, we write � K + C � A − B H = . (2) A + B K − C The four generalized Oort constants are the K : local divergence as ∇ · ¯ v = 2 K , B : vorticity as ˆ e z · ∇ × ¯ v = 2 B , 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 v x = − v R cos φ + v φ sin φ, v y = v R sin φ + v φ cos φ, (3) and the derivatives evaluated at the Solar location are ∂ ( R 0 , 0 ) = − ∂ ∂ ( R 0 , 0 ) = 1 ∂ � � � � ( R 0 , 0 ) , ( R 0 , 0 ) . (4) � � � � ∂ x ∂ R ∂ y R 0 ∂φ � � � � Applying these relations to the Taylor expansion, we yield expressions for the Oort constants as 2 A = ¯ R − ∂ ¯ ∂ ¯ ∂φ , 2 B = − ¯ R − ∂ ¯ ∂ ¯ v φ ∂ R − 1 v φ v R v φ ∂ R + 1 v φ v R ∂φ , R R (5) 2 C = − ¯ ∂ ¯ ∂φ + ∂ ¯ ∂ R , 2 K = ¯ ∂ ¯ ∂φ + ∂ ¯ v R R − 1 v φ v R v R R + 1 v φ v R ∂ R , R R where all terms are evaluated at the Solar location. In the axisymmetric limit, ¯ v R = 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 ¯ v z 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 || = (¯ ¯ v x cos ℓ + ¯ v y sin ℓ ) cos b , (6) ¯ v ℓ = ( − ¯ v x sin ℓ + ¯ v y cos ℓ ) cos b , v b = − (¯ ¯ v x cos ℓ + ¯ v y sin ℓ ) sin b . Therefore, v || ̟ = ( K + C cos 2 ℓ + A sin 2 ℓ ) cos 2 b , ¯ µ ℓ = ( B + A cos 2 ℓ − C sin 2 ℓ ) cos 2 b , (7) ¯ µ b = − ( K + C cos 2 ℓ + A sin 2 ℓ ) sin b cos b . ¯ 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 2 ℓ + A sin 2 ℓ ) cos 2 b ¯ − u cos ℓ cos b − v sin ℓ cos b − w sin b , µ ℓ = ( B + A cos 2 ℓ − C sin 2 ℓ ) cos 2 b ¯ (8) − u sin ℓ − v cos ℓ, µ b = − ( K + C cos 2 ℓ + A sin 2 ℓ ) sin b cos b ¯ + u cos ℓ sin b + v sin ℓ sin b − w cos b . 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 R = ( A − B ) = ( 27 . 2 ± 0 . 6 ) km s − 1 kpc − 1 and v φ ∂ v φ ∂ 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 K = ( − 0 . 27 ± 0 . 24) , A = (15 . 63 ± 0 . 30) B = ( − 11 . 88 ± 0 . 21) , A = (17 . 01 ± 0 . 28) K = ( − 3 . 26 ± 0 . 36) , A = (14 . 75 ± 0 . 47) C = ( − 2 . 76 ± 0 . 32) C = ( − 5 . 14 ± 0 . 28) C = ( − 3 . 35 ± 0 . 52) 40 40 40 ( K + A sin2 ℓ + C cos2 ℓ ) ( B + A cos2 ℓ − C sin2 ℓ ) ( µ b, gc / (sin b cos b )) / kms − 1 kpc − 1 30 30 30 / cos 2 b ) / kms − 1 kpc − 1 ( µ ℓ, gc / cos 2 b ) / kms − 1 kpc − 1 20 20 20 10 10 10 0 0 0 10 10 10 20 20 20 ( v || , gc 30 30 30 > 2mas , /σ > 3 − ( K + A sin2 ℓ + C cos2 ℓ ) 0 . 6 < ( G BP − G RP ) < 0 . 7 40 40 40 0 100 200 300 0 100 200 300 0 100 200 300 Galactic longitude / deg Galactic longitude / deg Galactic longitude / deg Figure 15: Results from Gaia DR2 RVS sample: proper motions of stars with ̟ < 2 mas , ̟/σ ̟ > 3 and 0 . 6 < ( G BP − G RP ) < 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 of 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 RR σ R φ σ Rz σ = σ R φ σ φφ σ φ z (9) σ Rz σ φ z σ zz . In the axisymmetric case, σ R φ = σ φ z = 0 (as potential axisymmetric). If the Galaxy is symmetric, z → − z then in the plane z = 0 σ Rz = 0. However, far from the plane this tilt term can be significant: σ Rz � = 0. For inspecting the solar neighbourhood ( z ≈ 0), we will consider the three diagonal terms: σ RR , σ φφ , σ zz , and the cross-term σ R φ (sometimes we will R ≡ σ RR = ( v R − v R ) 2 for example). write σ 2 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) v 2 R = κ 2 (∆ R ) 2 , (10) ( v φ − v c ) 2 = 4 B 2 (∆ R ) 2 , so A − B = κ 2 σ φφ B = − 4 Ω 2 . (11) σ RR 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 v φ 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 ℓ v has been used to measure the angle of the major axis of the dispersion tensor relative to the radial − 2 σ R φ tan 2 ℓ v = . (12) σ RR − σ φφ In the axisymmetric limit, ℓ v = 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 Main sequence, > 2mas , /σ > 3 5.0 40 30 2 σ Rφ tan2 l v = − 2.5 25 σ RR − σ φφ R 30 0.0 20 σ i / kms − 1 φ l v / deg M G 2.5 20 15 z 5.0 10 10 10 Gyr 12.5 Gyr l v, red = 8 . 73 ◦ 7.5 5 Parenago 10.0 0 0 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 ( G BP − G RP ) ( G BP − G RP ) ( G BP − G RP ) 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 σ R σ φ σ z σ tot 80 72 64 56 48 40 32 σ i / kms − 1 24 16 8 2 4 6 8 10 12 τ/ Gyr 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 σ J i ∝ σ 2 i . σ JR σ Jz 50 45 40 35 30 25 20 15 10 σ Ji / kpckms − 1 5 2 4 6 8 10 12 τ/ Gyr Figure 19: Age-action dispersion relation using stars within 300 pc from the catalogue of 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 2 10 14 12 1 6 × 10 Radius / kpc σ R / kms − 1 10 1 4 × 10 8 1 6 3 × 10 4 1 2 × 10 14 τ/ Gyr 1 6 × 10 12 Radius / kpc σ z / kms − 1 1 4 × 10 10 1 3 × 10 8 1 2 × 10 6 4 0 1 10 10 τ/ Gyr 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 orbits 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 σ RR and σ φφ ). Poor vertical heaters as ν ≫ m (Ω − Ω p ) . 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 σ zz ). 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 d E z / d t ∝ n / v ( n the number density of molecular clouds). If the molecular clouds are confined to a thin layer and the stars vertically oscillate harmonically, n ∝ v − 1 so d E z / d t ∝ 1 / E z giving E z ∝ t 1 / 2 so σ z ∝ t 1 / 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 2 π periodic and rotate at frequency ω so can be expressed in the form � f m ( R ) e i ( m φ − ω t ) f ( R , φ ) = (13) m ∈ N 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 ∂ f − ∂ f ∂ Φ ∂ t + v i = 0 . (14) ∂ x i ∂ v i ∂ x i We obtain three Jean’s equations by multiplying by 1, v j and v j v k respectively and integrating over all velocities d 3 v as ∂ρ ∂ t + ∂ ( ρ v i ) = 0 ( 1 ) , ∂ x i ∂ ( ρ v j ) + ∂ ( ρ v i v j ) + ρ∂ Φ = 0 ( 2 ) , (15) ∂ t ∂ x i ∂ x j ∂ ( ρ v j v k ) + ∂ ( ρ v i v j v k ) + v j ρ ∂ Φ + v k ρ∂ Φ = 0 ( 3 ) . ∂ t ∂ x i ∂ x k ∂ x j The first and second equations can be combined to give ρ∂ v j ∂ v j = − ρ∂ Φ − ∂ ( ρσ ij ) ∂ t + ρ v i . (16) ∂ x i ∂ x j ∂ x i 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 e ij ≡ ˆ ˆ e i ⊗ ˆ e j and ˆ e ijk ≡ ˆ e i ⊗ ˆ e j ⊗ ˆ e k , we have for vector v and tensor S ∇ v = ∂ v R e RR + 1 � ∂ v R e R φ + ∂ v φ e φ R + 1 � ∂ v φ � � ∂ R ˆ ˆ ∂ R ˆ ˆ ∂φ − v φ ∂φ + v R e φφ , (17) R R and ∇ S = ∂ S RR e RRR + 1 � ∂ S RR � ∂ R ˆ ˆ − ( S φ R + S R φ ) e RR φ ∂φ R + ∂ S R φ e R φ R + 1 � ∂ S R φ � ∂ R ˆ ˆ + ( S RR − S φφ ) e R φφ R ∂φ (18) + ∂ S φ R e φ RR + 1 � ∂ S φ R � ∂ R ˆ ˆ + ( S RR − S φφ ) e φ R φ R ∂φ + ∂ S φφ e φφ R + 1 � ∂ S φφ � ∂ R ˆ ˆ + ( S R φ + S φ R ) e φφφ . R ∂φ Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 50 / 171
We work in the regime where the terms ∂ ( ρσ ij ) are negligible. That is that the ∂ x i orbits are cold. More precisely, the radial excursion of an orbit ( max ( | u | , σ R ) /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 ∂φ − v 2 ∂ v R ∂ v R ∂ R + v φ ∂ v R R = − ∂ Φ φ + v R ∂ R , ∂ t R (19) ∂ v φ ∂ v φ ∂ R + v φ ∂ v φ ∂φ + v R v φ = − 1 ∂ Φ + v R ∂ t R R R ∂φ We linearize the equation as � u ( R ) e i ( m φ − ω t ) � v R = ǫ Re , � v ( R ) e i ( m φ − ω t ) � v φ = v c + ǫ Re , (20) � Φ 1 ( R ) e i ( m φ − ω t ) � Φ = Φ 0 + ǫ Re . Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 51 / 171
We obtain u = i ω ∂ Φ 1 ∂ R − 2 m ΩΦ 1 � � ˜ , ∆ R (21) v = 1 2 B ∂ Φ 1 ω Φ 1 � � ∂ R − m ˜ , ∆ R ω = ω − m Ω and ∆ = κ 2 − ˜ ω 2 . These where Ω ≡ v c / R = 2 ( A − B ) , ˜ 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 Σ 1 to the unperturbed surface density Σ 0 is ω Σ 1 = 1 ∂ ( Ru Σ 0 ) + im Σ 0 i ˜ v . (22) R ∂ R R 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 σ ijk : σ ijk ≡ ( v i − v i )( v j − v j )( v k − v k ) = v i v j v k − v i v j v k − v j v i v k − v k v i v j + 2 v i v j v k . (23) We subtract v j times equation (1) from equation (2) and v j v k times equation (1) from equation (3) to yield ρ∂ v j ∂ t + ∂ ( ρ v i v j ) ∂ ( ρ v i ) + ρ∂ Φ − v j = 0 ( 4 ) , ∂ x i ∂ x i ∂ x j (24) ρ∂ ( v j v k ) + ∂ ( ρ v i v j v k ) ∂ ( ρ v i ) + v j ρ ∂ Φ + v k ρ∂ Φ − v j v k = 0 ( 5 ) . ∂ t ∂ x i ∂ x i ∂ x k ∂ x j Now we subtract equation (4) times ( v j + v k ) from equation (5) which eliminates the potential yielding ρ∂σ jk ∂ t + ∂ ( ρ v i v j v k ) ∂ ( ρ v i ) ∂ ( ρ v i v k ) ∂ ( ρ v i v j ) ∂ ( ρ v i ) − v j v k − v j − v k + 2 v j v k = 0 . ∂ x i ∂ x i ∂ x i ∂ x i ∂ x i (25) ∂ ( ρ v i ) = ∂ ( ρ v j v k v i ) ∂ ( v j v k ) We can rewrite all derivatives of ρ as e.g. v j v k − ρ v i . ∂ x i ∂ x i ∂ x i Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 53 / 171
This yields ρ∂σ jk ∂ t + ∂ ( ρσ ijk ) ∂ ( v j v k ) ∂ v j ∂ v k ∂ ( v j v k ) + ρ v i + ρ v i v k + ρ v i v j + 2 ρ v i = 0 . (26) ∂ x i ∂ x i ∂ x i ∂ x i ∂ x i We neglect the term with σ ijk and write ∂ ( v j v k ) ∂ ( v j v k ) ∂σ jk v i + v i = v i , (27) ∂ x i ∂ x i ∂ x i and ∂ v j ∂ v k ∂ ( v j v k ) ∂ v j ∂ v k v i v k + v i v j + v i = σ ik + σ ij . (28) ∂ x i ∂ x i ∂ x i ∂ x i ∂ x i This gives us the final equation ∂σ jk ∂ v j ∂ v k ∂σ jk = σ ik + σ ij + v i . (29) ∂ t ∂ x i ∂ x i ∂ x i 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 ∂σ RR ∂ v R ∂ R + 2 σ R φ � ∂ v R + ∂σ RR ∂ R v R + 1 ∂σ RR � + 2 σ RR ∂φ − 2 v φ ∂φ v φ = 0 , ∂ t R R ∂σ φφ ∂ v φ ∂ R + 2 σ φφ � ∂ v φ + ∂σ φφ ∂ R v R + 1 ∂σ φφ � + 2 σ R φ ∂φ + 2 v φ ∂φ v φ = 0 , ∂ t R R (30) � 1 ∂σ R φ ∂ v φ ∂ v φ ∂φ + v R R + ∂ v R + σ φφ � ∂ v R � � + σ RR ∂ R + σ R φ ∂φ − v φ ∂ t R ∂ R R + ∂σ R φ ∂ R v R + v φ � ∂σ R φ � + ( σ RR − σ φφ ) = 0 . R ∂φ Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 55 / 171
We linearize the equation as � s ij ( R ) e i ( m φ − ω t ) � σ ij = σ 0 ij + ǫ Re , (31) � ℓ 1 e i ( m φ − ω t ) � ℓ v = ǫ Re . For simplicity, we introduce the parameter α = − ( A + B ) / ( A − B ) . This corresponds to the slope of a power-law rotation curve with v c ∝ R α . Hence Ω ≡ v c / R = 2 ( A − B ) and ∂ v c /∂ R = α Ω = − 2 ( A + B ) . To leading order in ǫ we have (using the Oort ratio for the unperturbed dispersion ratio) s R φ s R φ = ( A − B ) ℓ v ≈ − (32) σ 0 RR − σ 0 σ 0 A φφ RR 2 − 2 ( α + 1 )Ω ∂ u ω ∂ v � ℓ 1 = ∂ R − i ˜ ( 1 − α )∆ 2 ∂ R (33) ω ) u ω ] v � + ( α + 1 )( 2 Ω + 1 2 m ˜ R + i [ 2 m ( α + 1 )Ω + α ˜ + · · · R where ∆ 2 = 8 ( 1 + α )Ω 2 − ˜ ω 2 = 4 κ 2 − ˜ ω 2 , Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 56 / 171
and for the dispersions s RR = 1 − 2 i ∂ u ∂ R + iu � � − 2 i Ω( 1 − α ) ℓ 1 , σ 0 ω ˜ h σ RR (34) � 1 s φφ = 1 − 2 u + 2 mv � � � i + 2 i Ω( 1 − α ) ℓ 1 . σ 0 ω ˜ h σ R R φφ Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 57 / 171
Near a Lindblad resonance , ∆ ≈ 0 then 2 Ω , s RR ≪ s φφ = − 2 i ∂ u v = − i ˜ ω u 1 ∂ u ∂ R σ 0 φφ , ℓ 1 = ∂ R . (35) ω ˜ ( 1 − α )Ω Radial compression (expansion) leads to positive (negative) vertex deviation. Radial and azimuthal velocities are π/ 2 out of phase. Near corotation , ˜ ω = 0 ( 1 + α )Ω R , v = 1 im Φ 1 ∂ Φ 1 u = − ∂ R , 2 Ω � 1 s RR = s φφ = 1 − i ∂ u − 1 u + mv � � � ∂ R + i , (36) σ 0 σ 0 ω ˜ h σ R R φφ RR � ∂ u 1 ∂ R − u R − imv = − C � ℓ 1 = − 2 A . 2 ( 1 − α )Ω R 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 ℓ v ≈ 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 ′ 1 e i ( m φ − ω t )+ ikR , (37) where more generally k ( R ) but here we consider local perturbations. The corresponding plane-wave potential solution ( ∇ 2 Φ 1 = 4 π G Σ ′ ) is Φ 1 ( R , φ, z , t ) = Φ 1 , 0 e i ( m φ − ω t )+ 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 Φ 1 , 0 is found by � ǫ d z ∂ 2 Φ 1 � ∂ Φ 1 � ǫ 4 π G Σ ′ = ∂ z 2 = − ǫ = − 2 | k | Φ 1 . (39) ∂ z − ǫ 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 k Φ 1 ( R , φ, t ) / R dominate the potential Φ 1 ( R , φ, t ) . The derived expressions simplify to , ℓ 1 = 6 ik 2 Ω˜ u = − ˜ ω k Φ 1 , v = ik Ω( 1 + α )Φ 1 ω ( 1 + α )Φ 1 (40) ∆ ∆ ∆∆ 2 ( 1 − α ) 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 1.4 20 1.3 15 1.2 10 1.1 5 ℓ v / deg 1.0 0 R 5 0.9 10 0.8 15 0.7 20 0.6 0 1 2 3 4 5 6 φ Figure 21: Mean velocities (stream lines) and vertex deviation (colours) for a tightly wound spiral with Φ 1 = − 0 . 002 Re ( exp ( 20 i ln R − 2 i ( φ − t ))) in a galaxy with a flat rotation curve v c = 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 ∂ ( Ru Σ 0 ) /∂ R = ikRu Σ 0 as the derivative of u is dominated by the gradients of Φ 1 in the tightly-wound approximation, and this term also dominates that which is ∝ v . Therefore, = − k 2 Φ 1 Σ 1 = ku Σ 0 = 2 π G | k | Σ ′ 1 . (41) ω ˜ ∆ ∆ For self-consistent waves, Σ 1 = Σ ′ 1 so we get the dispersion relation ∆ = κ 2 − ( ω − m Ω) 2 = 2 π G | 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 σ R κ be stable provided Toomre Q = 3 . 36 G Σ > 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 ℓ v 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 of 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: E J = E − ω J φ . (43) Therefore, changes in angular momentum are related to changes in energy ∆ E = ω ∆ J φ = Ω R ∆ J R + Ω φ ∆ J φ , (44) so ∆ J R = ω − Ω φ ∆ J φ . (45) Ω R Therefore, at corotation ( ω = Ω φ ) 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 − Ω p L 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
Within 250 pc 0.6 Cold 0.4 3.0 All 0.2 Density (arb. units) 2.5 0.0 2.0 [Fe / H] 0.2 1.5 0.4 1.0 0.6 0.5 0.8 1.0 0.0 2 4 6 8 10 12 0.75 0.50 0.25 0.00 0.25 0.50 τ/ Gyr [Fe / H] 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 J R < 1 kpc km s − 1 , J z < 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 orbit 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 ) = Φ 0 ( R ) e im ( φ − Ω p t ) there are closed resonant orbits when the condition m (Ω − Ω p ) = n κ (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 of 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 200 Frequency / kms − 1 kpc − 1 Solar radius 150 100 Ω + / 2 Bar 50 Ω Ω − / 2 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 R/ kpc 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 ∆ J 0 and angle spread ∆ θ 0 , then after some time t the distributions will be ∆ J = ∆ J 0 , ∆ θ = ∆ θ 0 + ∆Ω 0 t ≈ ∆Ω 0 t , (47) where ∆Ω 0 = ∂ 2 H /∂ 2 J · ∆ J 0 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 = R g + X cos ( κ t + ψ ) , φ = Ω g t + φ 0 − 2 Ω g X sin ( κ t + ψ ) , κ R g (48) v φ ( R ) − v c ( R ) = − κ 2 X cos ( κ t + ψ ) = κ 2 ( R g − R ) . 2 Ω g 2 Ω g Therefore, selecting stars in a fixed angular range ∆ φ ≪ 1 we select stars with approximately the same Ω g and hence κ g . This implies that v φ and R are related via v φ = const . − κ 2 2 Ω R , (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
v φ − R Streaks 260 240 v φ / kms − 1 220 2 v φ = const . − 2Ω R 200 180 7.50 7.75 8.00 8.25 8.50 8.75 9.00 R/ kpc Figure 33: v φ against R for Gaia RVS sample. Phase-mixing populations exhibit streaks in this plane with gradient κ 2 / 2 Ω – see Antoja et al. (2018). Jason Sanders The Milky Way in the Gaia Era 49th Saas Fee school 2019 82 / 171
v φ − R Streaks Figure 34: v φ 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¨ onrich & 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 σ x admitted vertical bending waves satisfying the dispersion relation ω 2 = 2 π G Σ | k | − σ 2 x k 2 . (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 ω 2 = ν 2 + 2 π G Σ | k | − σ 2 x k 2 . (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 − m Ω 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 v s . 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 � cos ψ � a z d t = 1 � a z d ℓ = − GM ∆ v z = d ℓ r 2 v s v s � ℓ cos θ − b sin θ (52) = − GM d ℓ = 2 GM sin θ. ( b 2 + r 2 ) 3 / 2 v s bv s Therefore in the limit x ≫ z , the bending response is = 2 GM � ∆ v z , bend = 1 2 (∆ v z ( x , z ) + ∆ v z ( x , − z ) v s x tan θ (53) and the breathing response is ∆ v z , breath = ∆ v z ( x , z ) − ∆ v z ( x , − z ) = 2 GMz v s x 2 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 , v z ) plane coloured by density (left), v R (middle) and v φ (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
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