The dynamics of galaxy discs

John Magorrian

20 January 2018

We live inside a disc galaxy:

Discs are easier to see from outside. Here is M51:

Notice: satellite, spiral arms, dust, young stars, pitch angle.

Spiral arms are ubiquitous in galaxy discs: the flocculent M63

What are spiral arms?

← top-down view of disc.

Just differential rotation?No, winds up too quickly!

Can magnetic fields stop spirals winding up?No, spirals present in old disc stars.On large scales 1

2µ0B2 1

2ρ(∆v)2.

Lin & Shu (1966) proposed that spirals are density waves.

Outline1 An idealised model of a galactic disc2 perturbation theory (analogy w/ ugrad QM/EM)

disc is an elastic medium3 intro to dynamics of spiral arms

Understanding the stellar dynamics of discs

Simplifying assumptions:

1 Ignore magnetic fields, gas, cosmic rays, ...2 Disc is 2d, isolated,3 composed of identical stars,4 moving in their own mean-field potential Φ(x).5 These stars are eternal: no star formation or destruction.

Disc completely described by two functions H = 12v2 + Φ(x)

and f (x,v), coupled via

∂f∂t

+ [f ,H] = 0, ∇2Φ = 4πG∫

dv f︸ ︷︷ ︸=ρ(x)

.

This talk: What happens when we poke a disc that is indynamical (not thermal!) equilibrium?

Coordinates: angle–action variables

Unperturbed disc is in steady, axisymmetric state withHamiltonian H0(x,v) = 1

2v2 + Φ(R).

Integrals of motion (E ,L), or J = (Jr , Jφ).

Location along an orbit given by θ = (θr , θφ).These increase at a rate θ = Ω ≡ ∂H0

∂J = (κ,Ω).

Dynamical equilibrium requires phase-space DFf = f (E ,L) = f (J) (Jeans’ thm).

Epicycles(Hipparchus, 190-120 BC)

Almost-circular orbits (Jr ' 0) can be thought of as:1 an underlying circular orbit at some guiding center

radius Rg, with2 epicyclic oscillations superimposed.

Corresponding angularfrequencies Ω = ∂H/∂J:

1 θφ = Ω,2 θr = κ.

Jr ∝ κA2, where A = max(R − Rg)Local “temperature” ∼ 〈v2

R〉 ∼ κ2〈A2〉.

Perturbation theory

DF f and Hamiltonian H = 12v2 + Φ coupled by CBE+Poisson:

∂f∂t

+ [f ,H] = 0, ∇2Φ = 4πG∫

dv f .

Equilibrium (f0,H0) boring.

From t = 0 apply perturbation εΦp(x, t).Then

f = f0 + εf1 + · · · ,H = H0 + ε (Φp + Φs)︸ ︷︷ ︸

≡Φ1

+ · · · .

Cf electrostatics:

Φp ⇔ D,Φ1 ⇔ ε0E,

Φs = Φ1 − Φp

⇔ ε0E− D= −P.

Linearized evolution equations are

∂f1∂t

+ [f1,H0] = −[f0,Φ1], ∇2Φs = 4πG∫

dv f1.

Matrix mechanics(Kalnajs 1976)

We want to understand t > 0 behaviour of∂f1∂t

+ [f1,H0] = −[f0,Φ1], ∇2Φs = 4πG∫

dv f1.

Choose an orthogonal basis Φα(x) for Φ(x) and expand

Φp(x, t) =∑α

aα(t)Φα(x),

Φs(x, t) =∑α

bα(t)Φα(x).

Transform (x,v)→ (θ,J) and expand as Fourier series in θ.Fourier transform t → ω so that F (t) = 1

2π

∫dωF (ω)e−iωt .

Then

bα(ω)︸ ︷︷ ︸“P”

=∑β

Pαβ(ω)[aβ(ω) + bβ(ω)︸ ︷︷ ︸“E”

],

where Pαβ is polarization matrix (⇔ −χ in electrostatics).

Matrix mechanics: normal modes

We have just seen that

b(ω)︸︷︷︸response

= P(ω)[a(ω) + b(ω)︸ ︷︷ ︸stimulus+response

].

Isolated galaxy: stimulus Φp = 0, so a = 0.

Normal modes have b = P(ω)b.Find them by solving |P(ω)− I| = 0 for ω.

Time consuming numerical calculation. Difficult to gain insight.

WKB approximation

Gain insight from incomplete, approximate set of Φα(x).

Assume |kR| 1 in

Φ1(R, φ, t) = −2πG|k |

A(R)ei(kR+mφ−ωt)e−k |z|,

Σ1(R, φ, t) ' A(R)ei(kR+mφ−ωt).

Pattern speed Ωp = ω/m.Increasing |k | tightens.

Basis labels α = (m, k ,R): local approximation.

Then Pαβ(ω) is diagonal with (after some manipulation)

PmkR(ω) =2πGΣ|k |F

κ2 − (ω −mΩ)2 ,

where F ≤ 1 depends on disc DF and on (ω,m, k ,R).Self-consistent waves have PmkR(ω) = 1.

Whither that [κ2 −m2(Ωp − Ω)2] denominator?

Consider a star at guiding centre radius R, frequencies Ω, κ.To it, spiral perturbation comes by at rate m(Ωp − Ω).

If m(Ω− Ωp) = nκthis resonateswith radial motion:self sustaining!

Most important resonances for discs:corotation Ωp = Ω (i.e., n = 0).“Lindblad” resonances n = ±1.

Whither that [κ2 −m2(Ωp − Ω)2] denominator?

Consider a star at guiding centre radius R, frequencies Ω, κ.To it, spiral perturbation comes by at rate m(Ωp − Ω).

If m(Ω− Ωp) = nκthis resonateswith radial motion:self sustaining!

Most important resonances for discs:corotation Ωp = Ω (i.e., n = 0).“Lindblad” resonances n = ±1.

Wave propagation

Dispersion relation for waves Φ1 ∝ A(R)ei(kR+mφ−ωt), assuming|kR| 1, is

(ω −mΩ)2 = κ2 − 2πGΣ|k |F .

Take lukewarm disc. Apply spiral with m = 2 arms, Ωp ≡ ωm = 1.

Group velocity,

vg,R =∂ω(R, k)

∂k.

Collisionless dissipation as k →∞. Changes f (J), makes Jr ↑.Perturbation Φ1 lowers effective κ around ILR/OLR.

[But what about that k ∼ 0 stage?]

Beyond WKB: Swing amplification(Toomre 1981)

WKB incomplete. Full calculation of spiral unwinding gives:

Beyond WKB: Swing amplification(Toomre 1981: shear, shake and self-gravity)

Peak spiral unwinding rate almost matches κ for stars havingΩ ' Ωp: stars encouraged to join wave.

Discs are extremely responsive to perturbations.

Radial migration(Sellwood & Binney 2002; Ralph’s talk)

Spiral arms redistribute disc material. Here is one example.

Consider star (E ,L) perturbed by spiral Ωp.

E − ΩpL = const.⇒ Ωp∆L = ∆E

=∂E∂Jr

∆Jr +∂E∂L

∆L

= κ∆Jr + Ω∆L.So,

∆Jr =Ωp − Ω

κ∆L.

Keeping cool: spirals can shuffle disc stars in R while keepingthem on almost circular orbits!

Summary

Galaxies can never achieve thermal equilibrium.How do disc galaxies satisfy their urge to increase entropy?

local star-star encounters (as in globular clusters, 1010 yr);but stars “dressed” by polarisation wakes,with resonant absorption of collective oscillations acrossdisc

∼ 103x faster than local, undressed 2-body.

WIP, but modern surveys encouraging deeper thought.

Some important omissions1 nonlinear response: mode mixing.2 gas: star formation, molecular clouds, accordions.3 Only a single DF f : blind to different stellar populations.

Seeing DFs in “colour” is best way of probing galaxies’ longmemories.