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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.