AY202a
Galaxies & Dynamics
Lecture 5:
Galactic Structure
Spirals
NFW
Calculate structure of CDM Halos
Isothermal over large range of radii
Shallower at the center
Steeper past virial radius
NFW 1997, ApJ 490, 493
“A Universal Density Profile from
Heirarchical Clustering”
ρ(r)
ρcrit
δc
= (r/r ) (1 + r/r )2
s
s
where rs is a scale radius, δc is a characteristic density and
ρcrit = 3H2/8πG is the critical density.
Disk Kinematics
Disk kinematics are driven
by differential rotation.
Generally the inner stuff
has a shorter rotation period
than the outer stuff.
Ω(r) = v(r)/r
Angular Rotation Rate
☼
GC
Differential Rotation
Disk Kinematics
Next, what is the distribution of stellar radial
velocities as a function of distance and of
galactic longitude?
l = longitude
☼
v☼
d
d = star dist
l
S
R☼ Solar dist
vr
R☼ 90+α *
α = angle
α
R
between vr
and l.o.s.
α
Thus the observed radial velocity of the star S
vS = vR cos(α) - v☼ sin(l)
by law of sines
sin(l)
R
so
vS = (
sin(90+α)
cos(α)
=
=
R☼
R☼
vR R☼
R
) sin(l)
- v☼ sin(l)
 vS = (Ω(R) - Ω☼) R☼ sin(l)
and the transverse velocity is
vT = (Ω(R) - Ω☼) R☼ cos(l) - Ω(R) d
so
0
0
0 < l < 90
vS
d
900 < l < 1800
If we look very locally, such that d << R
 (Ω(R) - Ω☼) ~ (dΩ/dR)
(R
R
)
☼
R☼
These expressions were expanded by J. Oort
who introduced two constants to describe the
local galactic velocity field:
The local SHEER
=A
The local VORTICITY = B
Oort’s Constants
A =(
-r dΩ )
1
B = - 2r
2
=
½
(
dr R☼
v(r) - dv(r) )
r
dr
R☼
d
v(r)
dv(r)
2
dr (r Ω(r)) = - ½ [ r + dr ]R☼
with these definitions
vS = A d sin(l)
vT = d (Acos(2l) + B)
Ω☼ = A – B
(ang speed at Sun)
Best Current Estimates:
From Hipparcos
A = 14.8 ± 0.8 km/s/kpc
B = -12.4 ± 0.6 km/s/kpc
Ω☼ = v/R = A-B = 27.2 km/s/kpc
independently we think
v☼~ 220 km/s, R☼~8.2 kpc
 Ω☼ = 26.8 km/s/kpc
Vertical Structure of the Disk
Now that you’ve seen how thin/thick the disk of
the MW is, how is it described?
Oort derived the relation describing vertical
structure, equating surface potential with kinetic
energy:
1
∂(ν v2) = - ∂Φ
ν
∂z
∂z
where ν = # density of stars, ρ = mass density
and v 2 = velocity dispersion
Adding Poisson’s Equation:
∂2Φ
= 4π G ρ
2
∂z
∂
1
∂(νv 2)

[ ν ∂ z ] = -4 πGρ
∂z
thus the derivatives of the stellar density
distribution and velocity dispersion w.r.t. z
provide a measure of the integrated mass
density in the disk, ρ0 = ρ(R☼,z=0) =
the Oort Limit
Current best estimate from Hipparcos +
dispersion measures
~ 0.15 ± 0.02 M☼/pc3
this is about twice the density of stars + gas +
dust actually measured in the Solar
neighborhood  disk dark matter exists.
Surface Mass Density
Also, we can calculate the integrated surface
mass density of the disk (same formalism):
+z
μ(z) = ∫ ρ(z’)dz’ =
-z
1
∂(ν v 2)
2πGν
∂z
which gives μ(700pc) ~ 75 M☼/pc2
the mean surface mass-density inside 700pc
Spiral Galaxy Structure
What gives Spiral Galaxies their appearance?
There are 2 main components (plus others less
visible)
Disk --- rotationally supported
--- thickness is a function of the local
vertical “pressure” vs gravity
Spiral Pattern --- Three models
Density Wave
Tidal Interactions
SPSF = self propagating star formation
Density Waves  Lin’s “Grand Design”
spirals (M81, M83)
Interaction Induced Spirals  Good Looking
spirals with Friends (M51)
Self Propagating Star Formation --detonation waves, SF driven
by SF,  “Flocculent”
Spirals
M81
Classic
Grand
Design
Spiral
Another GD Spiral
M51 Interacting System
Optical
Molecular Gas -CO
M33 A Flocculent System
NGC4414 another Flocculent S
Spiral Structure
Some Definitions:
Number of Arms = m, most spirals have
m=2, i.e. twofold symmetry
Arm Orientation:
Leading
rotation
Trailing
Density Wave Theory
Developed over many years by first Bertil
Lindblad, then C.C. Lin, then Frank Shu:
Quasi-stationary Spiral Structure Hypothesis
(spiral pattern changes only slowly w. time)
+
Density Wave Hypothesis
Pattern is a SF pattern driven by density change
Follow the Mass
Density Response of
Stellar Disk
Gravitational Field
due to Stars & Gas
+
Density Response of
Gaseous Disk
Total material
needed to maintain
the field
||
=
TOTAL RESPONSE
Consider the “Pattern Speed” ΩP of a fixed
pattern in a differentially rotating disk of
stars which are in closed orbits. As in very
early (and wrong) models of the Solar
system, closed elliptical stellar orbits can be
approximated by circular orbits + epicycles.
The Epicyclic Frequency, κ, can be calculated
from the disk rotation speed and the stellar
angular momentum:
(
dΩ2
)
κ2 = R dR + 4Ω2
Epicyclic Frequency
4Ω2)
d(r
κ2 = r-3
dr
∂2Φeff
=(
)
2
∂R
Angular momentum
In the rotating frame
Φeff = Φ(R,z) + LZ2/2R2
term
And since Ω(r) = L/r2

km/s/kpc

κ2 = r
+ 4Ω2
= 2 Ω (1
100
50
dΩ2
dr
-½
– A/B)
Ω + κ/2
Ω
Generally
ΩP = Ω –nκ/m
Ω – κ/2
5
r
10
Ω-κ
The features seen in the rotation curve and
overall pattern are then
(1) Pattern Speed ΩP
(2) Physical Rotation Ω(R)
(3) Co-rotation Radius Rc, ΩP = Ω(RC)
(4) Inner and Outer Lindblad Resonances
corresponding to closed orbits at the
pattern speed --ΩP
= Ω(R) ± κ/m
•
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 & 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))