Basic Stellar Dynamics James Binney University of Oxford Saas Fee, - - PowerPoint PPT Presentation

Basic Stellar Dynamics

James Binney

University of Oxford

Saas Fee, January 2019

Equilibria & orbits Actions Actions & Jeans theorem Computing actions DFs for spheroids Secular evolution Spiral structure

Basic principle of stellar dynamics

We identify

◮ A steady state

◮ a high degree of symmetry (often axisymmetry)

Basic principle of stellar dynamics

We identify

◮ A steady state

◮ a high degree of symmetry (often axisymmetry)

◮ Fluctuations drive the system through a succession of

equilibria

Equilibrium models

◮ Most orbits in Φ(R, z) prove quasiperiodic:

x(t) =

n Xn cos(n · Ω t)

Equilibrium models

◮ Most orbits in Φ(R, z) prove quasiperiodic:

x(t) =

n Xn cos(n · Ω t) ◮ ⇒ ∃ 3 constants of motion I(x, v) = const on orbit

Equilibrium models

◮ Most orbits in Φ(R, z) prove quasiperiodic:

x(t) =

n Xn cos(n · Ω t) ◮ ⇒ ∃ 3 constants of motion I(x, v) = const on orbit ◮ E, Lz are 2 consts but there must exist a third (Arnold

1978)

Equilibrium models

◮ Most orbits in Φ(R, z) prove quasiperiodic:

x(t) =

n Xn cos(n · Ω t) ◮ ⇒ ∃ 3 constants of motion I(x, v) = const on orbit ◮ E, Lz are 2 consts but there must exist a third (Arnold

1978)

◮ 3 constraints Ii = const confine orbit to 3-surface in 6d

phase space.

◮ surface proves to be a 3-torus (Arnold 1978) ◮ A 3-torus is a room with opposite walls identified and

ceiling identified with the floor

Equilibrium models

◮ Most orbits in Φ(R, z) prove quasiperiodic:

x(t) =

n Xn cos(n · Ω t) ◮ ⇒ ∃ 3 constants of motion I(x, v) = const on orbit ◮ E, Lz are 2 consts but there must exist a third (Arnold

1978)

◮ 3 constraints Ii = const confine orbit to 3-surface in 6d

phase space.

◮ surface proves to be a 3-torus (Arnold 1978) ◮ A 3-torus is a room with opposite walls identified and

ceiling identified with the floor

◮ Since g(I1, I2, I3) = const we have choices

Actions

◮ Best to use actions Jr, Jz, and Jφ = Lz which are line

integrals Ji = 1 2π

• γi

p · dq around closed paths γi around the 3-torus

◮ In the room analogy, γ1 runs from a point on the front wall

to the corresponding point on the back wall, γ2 runs between points on the left & right walls, γ3 runs between floor and ceiling

◮ They have simple physical interpretations

◮ Jr quantifies radial oscillations ◮ Jz quantifies oscillations above & below the equatorial plane ◮ Jφ = Lz is cpt of angular momentum along symmetry axis

Properties of actions

◮ Adiabatic invariance: if Φ evolves slowly (accreting gas),

• rbit & its torus evolve but its Ji = const

Properties of actions

◮ Adiabatic invariance: if Φ evolves slowly (accreting gas),

• rbit & its torus evolve but its Ji = const

◮ Ji can be complemented by canonically conjugate

coordinate θi so (θ, J) form set of canonical coordinates

◮ So d3xd3v = d3θd3J ◮ Range θi = (0, 2π): x(θ1 + 2π, θ2, θ3) = x(θ1, θ2, θ3) ◮ ⇒

• rbits d3xd3v = (2π)3

d3J

◮ 3d action space a true map of 6d phase space

AA coords

◮ Hamilton’s eqs are trivial

0 = ˙ J = −∂H ∂θ ⇒ H(J)

◮ So

˙

θ = ∂H

∂J = Ω(J) (consts)

◮ ⇒ θ(t) = θ(0) + Ωt (quasi-periodicity)

Jeans theorem

◮ Let f(x, v) d3xd3v be probability that a randomly chosen

particle lies near (x, v)

Jeans theorem

◮ Let f(x, v) d3xd3v be probability that a randomly chosen

particle lies near (x, v)

◮ Liouville:

df dt = 0 ⇒ f is const of motion

◮ ⇒ f(J)

Jeans theorem

◮ Let f(x, v) d3xd3v be probability that a randomly chosen

particle lies near (x, v)

◮ Liouville:

df dt = 0 ⇒ f is const of motion

◮ ⇒ f(J)

◮ Porbits =

• d3xd3v f(x, v) = (2π)3

d3J f(J)

Jeans theorem

◮ Let f(x, v) d3xd3v be probability that a randomly chosen

particle lies near (x, v)

◮ Liouville:

df dt = 0 ⇒ f is const of motion

◮ ⇒ f(J)

◮ Porbits =

• d3xd3v f(x, v) = (2π)3

d3J f(J)

◮ Galaxy is an assembly of orbits ↔ points in 3d J-space

Jeans theorem

◮ Let f(x, v) d3xd3v be probability that a randomly chosen

particle lies near (x, v)

◮ Liouville:

df dt = 0 ⇒ f is const of motion

◮ ⇒ f(J)

◮ Porbits =

• d3xd3v f(x, v) = (2π)3

d3J f(J)

◮ Galaxy is an assembly of orbits ↔ points in 3d J-space ◮ (2π)3f(J) = density of stars in J-space ◮ Mass = (2π)3m

• d3J f(J) follows immediately from f(J)

◮ Equilibrium Galaxy comprises fα(J) for each population α

Computing actions

For decades the use of actions limited by difficulty of computing them

◮ Now θ(x, v) and J(θ, v) from St¨

ackel Fudge (Binney 2012)

◮ x(θ, J) and v(θ, J) from Torus Mapper (Binney &

McMillan 2016)

◮ State-of-the-art Python/C++ implementations in AGAMA

(Vasiliev 2018)

Computing actions

For decades the use of actions limited by difficulty of computing them

◮ Now θ(x, v) and J(θ, v) from St¨

ackel Fudge (Binney 2012)

◮ x(θ, J) and v(θ, J) from Torus Mapper (Binney &

McMillan 2016)

◮ State-of-the-art Python/C++ implementations in AGAMA

(Vasiliev 2018)

◮ St¨

ackel Fudge related to older technology of fitting St¨ ackel Φ (Dejonghe & de Zeeuw 1988, Sanders 2012) but more accurate (Vasiliev 2018)

◮ St¨

ackel Fudge has been extended to triaxial Φ in absence of figure rotation (Sanders & Binney 2015) but not to case of figure rotation

Computing actions

For decades the use of actions limited by difficulty of computing them

◮ Now θ(x, v) and J(θ, v) from St¨

ackel Fudge (Binney 2012)

◮ x(θ, J) and v(θ, J) from Torus Mapper (Binney &

McMillan 2016)

◮ State-of-the-art Python/C++ implementations in AGAMA

(Vasiliev 2018)

◮ St¨

ackel Fudge related to older technology of fitting St¨ ackel Φ (Dejonghe & de Zeeuw 1988, Sanders 2012) but more accurate (Vasiliev 2018)

◮ St¨

ackel Fudge has been extended to triaxial Φ in absence of figure rotation (Sanders & Binney 2015) but not to case of figure rotation

◮ TM extended to rotating bars by p-theory (Binney 2018) ◮ TM is a systematic approximation, but SF is not;

unfortunately x(θ, J) less useful than J(x, v)

Models from analytic DFs

Role of models

◮ A model interprets observations in terms of what’s out

there

◮ Discovering what’s there must preceed deducing how it got

there

◮ So we need an apparatus that fits models to data

independent of a creation myth

What must be included

◮ Bulge ◮ α-rich and α-poor discs ◮ stellar halo (impacts kinematics) ◮ dark halo (impacts dynamics) ◮ gas disc

From DF to observables

◮ Choose fα(J) for each component ◮ Guess Φ(x) ◮ Use St¨

ackel Fudge to evaluate ρ =

α

• d3v f(J) on a grid

in x

◮ Solve for Φ(x) and re-determine ρ(x) ◮ Converges in ≤ 5 cycles (5 min on my laptop) ◮ Hack your code from AGAMA

example self consistent model.cpp

What you can do with the model

◮ MC sample initial conditions for N-body model

◮ Model starts uniquely close to equilibrium ◮ Ideal for study of perturbations: spirals, warps, . . .

◮ Predict v-distribution of DM particles for direct-detection

expts

◮ Cross-section for detection expected to be speed-dependent

◮ MC sample with survey selection function to make mock

catalogue

◮ Compute at x (i) star counts and (ii) stellar v-distributions

for surveys

◮ Use f, Φ to understand dis-equilibria (streams, moving

groups, Antoja spiral)

Principles of picking a DF

◮ Each component has its own DF ◮ DF has to arrange stars in 3d J-space

f(Jr, Jz, Jφ)    Jr ↔ σR, σφ Jz ↔ σz Jφ ↔ Σ(R), vφ(R)

◮ Only 2 independent dispersions ◮ In case of disc

◮ σz, z0 and ρDM(R, 0) coupled ◮ f = f+ + f− and only f+ contributes to ρ

Spheroids: Posti DFs

◮ NFW, Hernquist, etc models generated by 2-power DFs

(Posti + 2015) ρ(r) = ρ0 (r/r0)α(1 + r/r0)β−α ↔ f(J) = M0 J3 [1 + J0/h(J)]a [1 + g(J)/J0]b with a = (6 − α)/(4 − α) b = 2β − 3 and g, h(J) = Jr + kφ|Jφ| + kzJz

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5

log10 r

−10 −8 −6 −4 −2 2 4 6 8

log10 ρ

f(J) Hernquist Hernquist

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5

log10 r

−6 −4 −2 2 4 6

log10 ρ

f(J) NFW NFW

Discs: quasi-isothermal DF

◮ Introduced by Binney 2010 & Binney & McMillan 2011 ◮ Motivated by Schwarzschild DF (epicycle approx J = H/Ω)

E = Ec + ER + Ez Ec = J2

φ

2R2

c

+ Φ(Rc), ER = κJr, Ez = νJz fSchw(Jφ, Er, Ez) = fφ(Rc) exp(−Er/σ2

R) exp(−Ez/σ2 z) ◮ Quasi-isothermal DF is

f(J) = fφ(Rc) exp(−κJr/σ2

R) exp(−νJz/σ2 z)

with κ(Rc) and ν(Rc)

Discs: quasi-isothermal DF

◮ Quasi-isothermal DF does a superb job in extended solar

nhd (e.g. Binney + 2014)

◮ But

◮ Behaves inappropriately when Jφ → 0 through Rc → 0 ◮ Undefined for some Φ(x) because κ2 can go -ve

◮ These flaws apparent when solving for self-consistent Φ

Discs: exponential DFs

◮ Basic principle: nothing in the DF should reference Φ ◮ Appearance of Rc, κ, ν in q-i DF violates this principle ◮ Instead try (Binney & Vasilev in prep)

f(J, τ) = fφ(Jφ)fr(Jφ, Jr, τ)fz(Jφ, Jz, τ) fφ(Jφ) ≡

• M

(2π)3 Jd J2

φ0 e−Jd/Jφ0

Jφ > 0 eJvJφ/J2

r0fφ(0)

Jφ < 0 fi = Jv Js pi 1 Ji0αi exp

• −

Jv Js pi Ji Ji0αi

• (i = r, z).

Jv ≡

• J2 + J2

v0,

Jd ≡

• J2 + J2

d0,

• J ≡ |Jφ|+krJr+kzJz

αi(τ) ≡ τ + τ1 1 + τ1 2βi (age-σ relation)

Secular evolution

◮ Thus far models dynamically static ◮ Stars are moved from orbit to orbit by fluctuations in Φ(x)

Secular evolution

◮ Thus far models dynamically static ◮ Stars are moved from orbit to orbit by fluctuations in Φ(x) ◮ Fluctuations contributed by bar, spirals, halo structures,

molecular clouds,..

Secular evolution

◮ Thus far models dynamically static ◮ Stars are moved from orbit to orbit by fluctuations in Φ(x) ◮ Fluctuations contributed by bar, spirals, halo structures,

molecular clouds,..

◮ Disc stars (also bulge stars?) born near plane on ∼circular

• rbits

Secular evolution

◮ Thus far models dynamically static ◮ Stars are moved from orbit to orbit by fluctuations in Φ(x) ◮ Fluctuations contributed by bar, spirals, halo structures,

molecular clouds,..

◮ Disc stars (also bulge stars?) born near plane on ∼circular

• rbits

◮ In J-space stars form along line Jr = Jz = 0

◮ Diffuse to higher Jr (growing eccentricity) ◮ Diffuse to higher Jz (growing inclination) ◮ Diffuse to different Lz = Jφ (radial migration)

Casagrande et al 2011

So older stars are radially & vertically hotter (Casagrande + 2011)

20 30 40 50 60 70 80 90 100 2 4 6 8 10 12 s (km/s) age (Gyrs)

BASTI, stars with good ages, [Fe/H]> -0.8, V > -150 km/s BASTI, stars with good ages, [Fe/H] > -0.5 BASTI, stars with good ages, no cuts Padova, stars with good ages, [Fe/H]> -0.8, V > -150 (km/s) BASTI, all stars, [Fe/H]> -0.8, V > -150 (km/s)

Heating/migration

◮ Radial migration and in-plane heating 2 aspects of 1

phenomenon

◮ Vertical heating (disc thickening) reflects a largely

independent process for which GMCs are important (Aumer+ 2016)

Heating/migration

◮ Radial migration and in-plane heating 2 aspects of 1

phenomenon

◮ Vertical heating (disc thickening) reflects a largely

independent process for which GMCs are important (Aumer+ 2016)

◮ Heating is irreversible

Heating/migration

◮ Radial migration and in-plane heating 2 aspects of 1

phenomenon

◮ Vertical heating (disc thickening) reflects a largely

independent process for which GMCs are important (Aumer+ 2016)

◮ Heating is irreversible

◮ Thin cold discs only possible in quiescent environment

◮ Spirals are by definition galaxies with thin, cold discs ◮ Gas an essential feature of any spiral galaxy

◮ Star formation keeps adding cool stars (Sellwood &

Carlberg 1984)

Heating/migration

◮ Radial migration and in-plane heating 2 aspects of 1

phenomenon

◮ Vertical heating (disc thickening) reflects a largely

independent process for which GMCs are important (Aumer+ 2016)

◮ Heating is irreversible

◮ Thin cold discs only possible in quiescent environment

◮ Spirals are by definition galaxies with thin, cold discs ◮ Gas an essential feature of any spiral galaxy

◮ Star formation keeps adding cool stars (Sellwood &

Carlberg 1984)

◮ S0s are galaxies with little cool gas & hotter, thicker discs

Spiral structure

◮ Stars in a razor-thin axisymmetric disc oscillate radially at

frequency κ = limJr→0 Ωr while circulating with frequency Ω = limJr→0 Ωφ

◮ In inertial frame orbits form rosettes that can be

considered (centred) ellipses with major axes that rotate with angular frequency Ωp = Ω − κ/2

Lindblad’s idea

◮ Lindblad posited a systematic twist of principal axes,

which creates spiral arms by orbit crowding

Lindblad’s idea

◮ Lindblad posited a systematic twist of principal axes,

which creates spiral arms by orbit crowding

◮ Problem: Ωp decreases outwards −

→ winding up of spiral

Lindblad’s idea

◮ Lindblad posited a systematic twist of principal axes,

which creates spiral arms by orbit crowding

◮ Problem: Ωp decreases outwards −

→ winding up of spiral

◮ Solution? Gravitational torques generated by misalignment

• f axes speeds precession of outer orbits & slows precession
• f inner orbits?

Lindblad’s idea

◮ Lindblad posited a systematic twist of principal axes,

which creates spiral arms by orbit crowding

◮ Problem: Ωp decreases outwards −

→ winding up of spiral

◮ Solution? Gravitational torques generated by misalignment

• f axes speeds precession of outer orbits & slows precession
• f inner orbits?

◮ leads to angular-momentum flux through disc ◮ In any gravitationally bound system entropy is increased

by moving L out and M in

WKB theory

◮ Self-gravity makes disc an elastic medium through which

density waves propagate

◮ WKB technique yields analytic theory for tightly wound

waves

◮ Assume Σ = Σ0 + Σ1ei[kR+m(φ−Ωp)t] ◮ Σ1 and k evolve as Σ0(R) changes ◮ Local relation Φ1 = −2πGΣ1/|k| holds between perturbed

density & potential

Lin-Shu-Kalnajs theory

◮ Posit Φ = Φ0 + Φ1ei[kR+m(φ−Ωpt)] ◮ Compute Σ1dyn dynamically generated by Φ1 ◮ Equate to Σ1grav = −|k|Φ1/2πG needed to generate Φ1 ◮ Then have the Lin-Shu-Kalnajs dispersion relation

Lin-Shu-Kalnajs dispersion relation

m2(Ωp − Ω)2 = κ2

• 1 − F |k|

kcrit

• kcrit ≡

κ2 2πGΣ F(s, χ) = 1 − s2 sin πs π dt e−χ(1+cos t) sin st sin t s = mΩp − Ω κ χ ≡

• k2 +

2mΩ κR 2 σ2 κ2

◮ s is perceived frequency ◮ √χ ∼ k × typical epicycle radius ◮ Implicit k(Ωp):

Lin-Shu-Kalnajs dispersion relation

m2(Ωp − Ω)2 = κ2

• 1 − F |k|

kcrit

• kcrit ≡

κ2 2πGΣ F(s, χ) = 1 − s2 sin πs π dt e−χ(1+cos t) sin st sin t s = mΩp − Ω κ χ ≡

• k2 +

2mΩ κR 2 σ2 κ2

◮ s is perceived frequency ◮ √χ ∼ k × typical epicycle radius ◮ Implicit k(Ωp): ◮ 0 ≤ F ≤ 1; F(s, 0) = 1 (ice-cold disc)

Lin-Shu-Kalnajs dispersion relation

m2(Ωp − Ω)2 = κ2

• 1 − F |k|

kcrit

• kcrit ≡

κ2 2πGΣ F(s, χ) = 1 − s2 sin πs π dt e−χ(1+cos t) sin st sin t s = mΩp − Ω κ χ ≡

• k2 +

2mΩ κR 2 σ2 κ2

◮ s is perceived frequency ◮ √χ ∼ k × typical epicycle radius ◮ Implicit k(Ωp): ◮ 0 ≤ F ≤ 1; F(s, 0) = 1 (ice-cold disc) ◮ In ice-cold disc

m2(Ωp − Ω)2 = κ2(1 − |k|/kcrit) so stable only for long waves (|k| < kcrit)

Lin-Shu-Kalnajs dispersion relation

m2(Ωp − Ω)2 = κ2

• 1 − F |k|

kcrit

• kcrit ≡

κ2 2πGΣ F(s, χ) = 1 − s2 sin πs π dt e−χ(1+cos t) sin st sin t s = mΩp − Ω κ χ ≡

• k2 +

2mΩ κR 2 σ2 κ2

◮ s is perceived frequency ◮ √χ ∼ k × typical epicycle radius ◮ Implicit k(Ωp): ◮ 0 ≤ F ≤ 1; F(s, 0) = 1 (ice-cold disc) ◮ In ice-cold disc

m2(Ωp − Ω)2 = κ2(1 − |k|/kcrit) so stable only for long waves (|k| < kcrit)

◮ In hotter disc F < 1 and stable also for shorter waves

◮ In warm disc can’t make right side as small as left near CR

Lin-Shu-Kalnajs dispersion relation

Lin-Shu-Kalnajs dispersion relation

Group velocity

◮ Short leading waves move from LR towards CR ◮ k decreases as they move - spiral unwinds ◮ Near “forbidden” zone around CR, WKB approx fails

Putting it together

◮ WKB theory is misleading in putting an exclusion zone

around CR

◮ CR is where the key action takes place ◮ illustrated by modes in N-body models

◮ Important points:

◮ Amplification of waves as leading → trailing (shearing

sheet)

◮ resonant absorption at LRs

Resonant absorption

◮ At ILR stars pushed along curve Ω − Ωp = κ/2

Balescu-Lenard eq N-body

◮ Narrow region of J-space with modified f partially reflects

trailing waves moving to new LR

◮ Reflected portion re-amplified ◮ Eventually all E absorbed at new LR ◮ New LR then marked by stronger modification to f ◮ Hence new LR a more highly silvered mirror ◮ Over time more and more wave action reflected for

re-amplification

Bar/bulge

◮ O(1) spiral structure soon leads to formation of a bar ◮ Initially the bar is a planar item

Bar buckling

◮ If it becomes strong enough, it will suddenly buckle

Portail+ (2017)

◮ Only centre of bar thickens

◮ Bulge/bar ◮ Long thin bar

◮ Buckling weakens bar

Bulge/bar modelling (Portail+ 2017)

BRAVA, ARGOS & OGLE surveys probed dynamics of bulge/bar

Portail+ 2017

BRAVA ARGOS

Portail+ 2017

φbar = 28◦ Ωp = 39 ± 3.5 km s−1 kpc−1 Mbar = 1.3 ± 0.1 × 1010 M⊙ MDM = 17 ± 2%

Chemodynamics of the bulge

◮ There’s a vertical [Fe/H] gradient in bulge ◮ Ness+ (2013) identified 5 [Fe/H] cpts with different spatial

distributions

◮ [Fe/H]> −0.5 boxy/peanut bulge (split RC on minor axis) ◮ [Fe/H]< −0.5 thick disc ◮ No mixing by merger since bar buckled

Chemodynamical bulge model

◮ Portail+ (2017) modelled bulge kinematics by augmented

M2M using ARGOS & APOGEE data

◮ Stars with [Fe/H]> −0.5 are strongly barred ◮ Less metal-rich stars have strong kinematic-chemical

correlations

◮ At r > 1 kpc metal-poor stars ∼ thick disc ◮ At smaller radii there’s an extra concentratio of metal-poor

stars

Portail+ 2017

Model-data comparisons: Portail+2017