1. AY202a Galaxies & Dynamics Lecture 6: Galactic Structure, con’t Spirals & Density Waves

2. A Rotation Pattern with Two Inner LB Resonances

3. Lindblad first noted that for n=1, m=2
(Ω – κ/2) is constant over a large range of radii such that ΩP = Ω – κ/2 and that a pattern could exist and be moderately stable.
C.C. Lin computed the response of stars & gas:
Assume that the gravitational potential is a superposition of plane waves in the disk:
Φ (r,φ,t) = eiK(r,t)(r-r0)
-2πGμ
|K|
uniformly rotating sheet

4. Φ(r,φ,t) = H(r,t) e-i(mφ + f(r,t))
Where K = wave number = 2π/λ
and μ = surface density
Now find a dispersion relation 
if μ(r,φ,t) = H(r,t) ei(mφ + f(r,t))
then
Φ(r,φ,t) = H(r,t) e-i(mφ + f(r,t))
-2πG
|K|

5. μ = μa(r) ei(mφ - ωt) Differentiate and find μ(r,φ,t) = Φ(r,φ,t) e.g.
These equations have solutions with a spiral like family of curves
m(φ – φ0) = Φ(r) – Φ(r0)
e.g.
μ = μa(r) ei(mφ - ωt) iK d dr 2G

6. ΩP = ω/m Note that K < 0 corresponds to Leading Arms
K > “ “ “ Trailing “
and
i (mφ – ωt) = i m(φ – ΩPt)
ΩP = ω/m

7. Response of the motions of stars or gas to non-axisymetric forces F1.
F1 is assumed to be periodic in time and angle.

9. N. Cretton

10. Density Wave Models + Bar Potential

11. We calculate a dispersion relation for the gas
With a gas law:
a2 = dP/dρ ≈ dP/dμ
We calculate a dispersion relation for the gas
(ω – mΩ)2 = κ2 - 2πGμ|K| + K2a2
ω2 = κ2 + K2a2 - 2πG|K|μ
Sound speed ~ velocity dispersion of the gas in equilibrium

12. F. Shu solved the special case of a flat rotation curve, rΩ(r) = constant = v0
Mass Model μ = v02/2πGr
 = √2 Ω and the wavenumber
|K| = [ 1 ± (1 – r/r0)]
where r0 is the co-rotation radius
Inner and outer Lindblad resonances are at
r = ( 1 ± √2/m) r0 m 4 r √2

13. Response of the Gas depends on a 
For m = 2, LR are at 0.293r r0
m= 1, There is no inner LR
Response of the Gas depends on a 
μ/μ0 5
a = sound speed
in km/s
(Shu etal 1973) ||
128 32 8 1
t or φ
NB For an adiabatic shock, max μ/μ0 = 4 for =5/3

14. How does over density relate to SFR? Schmidt-Kennicutt Law
ΣSFR = (2.5 ±0.7)x10-4 ( ) M☼/yr kpc-2
an exponent of ~1.5 is expected for self gravitating disks if SRF scales as the ratio of gas density to free fall time which
is proportional to ρ This lead Elmegreen and separately Silk to argue for an SFR law where the SFR is related to the gas density over the average orbital timescale:
ΣSFR = ΣGas ΩG
There also appears to be a cutoff at low surface mass gas density:
ΣGas
1.4 ± 0.15
1 M☼ pc-2

15. Schmidt-Kennicutt Law vs Elmegreen/Silk

16. Disk Stability
Toomre (1964) analyzed the stability of gas (and stars) in disks to local gravitational instabilities. Simply, gravitational collapse occurs if Q < 1.
For Gas Q = κ CS / (π G Σ)
For Stars Q = κ σR / (3.36 G Σ)
where Σ is again the local surface mass density,
κ is the local epicyclic frequency,
σR is the local stellar velocity dispersion,
and CS is the local sound speed

17. Starburst Galaxies
Kennicutt ‘06
Normal Disks

18. Kennicutt (1989) rephrased the Toomre argument in terms of a critical surface density, ΣC where
ΣC = α κ C / (π G)
Q = ΣC / ΣG
Where α is a dimensionless constant and
C is the velocity disperison of the gas, and
ΣG is the gas mass surface density.
For this definition of the Q parameter, as before, star formation is also suppressed in regions where Q >> 1 and is vigorous in regions where Q << 1

20. Some facts about spirals
Density waves are found between the ILR and OLR
Stellar Rings form at Co-rotation and OLR
Bars inside CR, probably rotate at pattern speed
Gas rings at ILR
For the MW ILR ~ 3 kpc,
CR ~ 14 kpc,
OLR ~ 20 kpc

21. Interaction induced Spiral Structure = Tides
Based on Strong Empirical Evidence for star formation induced by galaxy interactions (Larson & Tinsley 1978)
Models now “abundant” --- Toomre2 1970’s, Barnes et al 1980’s, many more today.
Bars also act as drivers of density waves

23. Toomre2
model for
the Antennae

25. Toomre2
galaxy.interaction.mpg

26. Bar Driven Density Wave

27. Self Propagating Star formation
Mueller & Arnett  Seiden & Gerola 1978,  Elmegreens 1980’s+
based on galactic SF observations (e.g. Lada)

28. Seiden & Gerola 1978
Spore

29. Galaxy Rotation Curves
MW HI

32. MW Rotation Curve
D. Clemens 1985

35. Zwicky’s Preface