# CC Lin ```AY202a
Galaxies &amp; Dynamics
Lecture 6:
Galactic Structure, con’t
Spirals &amp; Density Waves
A Rotation Pattern with
Two Inner LB Resonances
ΩP
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 &amp; gas:
Assume that the gravitational potential is a
superposition of plane waves in the disk:
Φ (r,φ,t) =
-2πGμ
|K|
eiK(r,t)(r-r0)
uniformly rotating
sheet
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
-2πG
Φ(r,φ,t) = |K| H(r,t) e-i(mφ + f(r,t))
Differentiate and find
μ(r,φ,t) =
iK
2G
d
Φ(r,φ,t)
dr
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)
Note that
K &lt; 0 corresponds to Leading Arms
K&gt;0 “
“ “ Trailing “
and
i (mφ
– ωt) = i m(φ – ΩPt)
ΩP = ω/m
Response of the
motions of stars
or gas to nonaxisymetric
forces F1.
F1 is assumed to
be periodic in
time and angle.
•
N. Cretton
Density
Wave
Models
+
Bar
Potential
With a gas law:
a2 = dP/dρ ≈ dP/dμ
Sound speed ~ velocity dispersion
of the gas in equilibrium
We calculate a dispersion relation for the gas
(ω – mΩ)2 = κ2 - 2πGμ|K| + K2a2
ω2 = κ2 + K2a2 - 2πG|K|μ
F. Shu solved the special case of a flat
rotation curve, rΩ(r) = constant = v0
Mass Model
μ = v02/2πGr
 = √2 Ω

|K| = 4r
[1 &plusmn;
and the wavenumber
m
(1
–
r/r
)
]
0
√2
where r0 is the co-rotation radius
Inner and outer Lindblad resonances are at
r = ( 1 &plusmn; √2/m) r0
For m = 2, LR are at 0.293r0 + 1.707r0
m= 1, There is no inner LR
Response of the Gas depends on a 
5
μ/μ0
a = sound speed
||
128
1
32
in km/s
(Shu etal 1973)
8
t or φ
NB For an adiabatic shock, max μ/μ0 = 4 for =5/3
How does over density relate to SFR?
Schmidt-Kennicutt Law
ΣSFR = (2.5 &plusmn;0.7)x10-4
ΣGas
1.4 &plusmn; 0.15
( 1 M☼ pc-2 )
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-0.5. 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 = 0.017 ΣGas ΩG
There also appears to be a cutoff at low surface mass gas density:
Schmidt-Kennicutt Law
vs Elmegreen/Silk
Disk Stability
Toomre (1964) analyzed the stability of gas (and
stars) in disks to local gravitational instabilities.
Simply, gravitational collapse occurs if Q &lt; 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
Starburst Galaxies
Kennicutt ‘06
Normal Disks
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
&gt;&gt; 1 and is vigorous in regions where Q &lt;&lt; 1
1. Density waves are found between the ILR
and OLR
2. Stellar Rings form at Co-rotation and OLR
3. Bars inside CR, probably rotate at pattern
speed
4. Gas rings at ILR
For the MW ILR ~ 3 kpc,
CR ~ 14 kpc,
OLR ~ 20 kpc
Interaction induced
Spiral Structure = Tides
Based on Strong Empirical Evidence for star
formation induced by galaxy interactions
(Larson &amp; Tinsley 1978)
Models now “abundant” --- Toomre2 1970’s,
Barnes et al 1980’s, many more today.
Bars also act as drivers of density waves
2
Toomre
model for
the Antennae
Toomre2
galaxy.interaction.mpg
Bar Driven
Density
Wave
Self Propagating Star formation
Mueller &amp; Arnett 1976  Seiden &amp; Gerola
1978,  Elmegreens 1980’s+
based on galactic SF observations (e.g. Lada)
Seiden &amp; Gerola 1978
Spore
Galaxy Rotation Curves
MW HI
MW Rotation Curve
D. Clemens 1985
•
Zwicky’s Preface
```