Spiral Galaxies: Disk Kinematics
 As in all spiral galaxies, everything in our
Galaxy orbits around the Galactic Center (GC)
 The disk rotates with differential rotation 
material closer to the center travels on faster
orbits (takes less time to make one full orbit)
 Similar to the motion of the planets orbiting
the Sun
 Motions of stars and gas in disks can be used
to trace the mass distribution of our Galaxy
and other galaxies.
M(R) = 0R (r) dV
Motion at radius R depends only on M(R)
Objects responding to M(R) behave as if M(R) is centrally concentrated
For an object with mass m at R,
gravity balance acceleration of circular motion
M
GM(R)m/R2 = mv2/R
M(R) = v(R)2R/G
v
Measure v(R) and get M(R)
R
m
Let (R) = v(R)/R, then
M(R) = (R)2R3/G
v(R) or (R) is the rotation curve of the galaxy
Differential galactic rotation produces
Doppler shifts in emission lines from
gas in the Galactic disk
To determine galaxy rotation curve,
define Galactic Coordinates
b = galactic latitude in degrees
above/below Galactic disk
l = galactic longitude in degrees from
GC
The Sun (and most stars) are on slightly perturbed orbits (deviate slightly
from perfect circular motion). Must determine Sun motion wrt pure circular
motion by looking at average motions of all stars in the Sun’s vicinity.
Local Standard of Rest (LSR) - reference frame for
measuring velocities in the Galaxy. This would be the
position of the Sun if its motion were completely governed
by orbital motion around the Galaxy
V = Vy (velocity in direction of galaxy rotation)
U = Vx (velocity towards GC) = 10 km/s
W = Vz (velocity towards NGP) = 7.2 km/s
•V depends on color – stellar pops
•asymmetric drift – mean Vrot of stellar
pop lags behind LSR more and more
with increasing σ of stars.
•Comparing sun to this reference frame
produces V which grows with lag
•Fit to data gives at S2 = 0, V = 5.2 km/s
With respect to the LSR, the Sun is moving at about
~15.4 km/s towards l=53 degrees b=23 degrees and lies
about 10-20 pc above the Galactic plane. Use this to correct
doppler shift measurements to get radial velocity of object wrt LSR.
Position and Velocity of LSR in Galaxy
(adopted in 1985 by IAU; based on globular cluster positions)
Ro = 8.5 kpc
Vo = 220 km/s
Other values:
8 kpc and 200 km/s (SG 2.3 and Eisenhauer et al. 2003)
8.4 +/- 0.6 kpc and 254 +/- 16 km/s (Reid et al. 2009)
To map out vr throughout Galaxy,
divide the Galaxy into quadrants
based on value of galactic longitude.
Quad I (l<90) - looking to material closest to GC,
[(R) - 0] gets larger and vr increases. At point
of closest approach (subcentral point) vr is at
maximum for that los and then continues to
decrease to Sun’s orbit. Beyond Sun’s orbit, vr
becomes negative and increases in absolute
value.
Quad II (90<l<180) - all los pass through orbits
outside of the Sun’s. No maximum vr but
absolute values increase with d.
Quad III (180<l<270) - similar to Quad II but
opposite signs.
Quad IV (l>270) - similar to Quad I except reverse
signs.
Determination of Rotation Curve of
the Milky Way
 Determined from 21-cm line observations
 Assume circular orbits and that there is at
least some Hydrogen all along any given lineof-sight
 Especially important to have measure of gas
at subcentral point
•Find maximum shift of 21-cm line
along given los
•Assign that Doppler shift to
material at the subcentral point
(closest approach to GC)
•Rmin = Ro sin l
•(Rmin) = [vmax/(Ro sin l)] + o
•By studying los longitude values
from 0 to 90 degrees, Rmin will
range from 0 to Ro
Limitations
•No gas at subcentral point
•Non-circular orbits
•At Rmin = 0 and at Ro, difficult to
measure curve (small Doppler
velocity hard to determine)
Since there is no maximum Doppler shift for los directed away from GC,
rotation curve beyond Ro is more difficult – must measure velocity and
distance of material independently
Molecular Clouds
• velocity from radial velocities of CO emission
• distance from stars exciting clouds (spectroscopic parallax)
From tangent point method
with Vo= 200 km/s
Compares HI gas slightly
above and below plane
Rotation speed of outer
Galaxy
Vo = 200 km/s and 220 km/s
Xin and Zheng (2013) using Reid et al. LSR values
Spiral Galaxies: gas motions in galactic disks
•Disks are dominated by ordered motion – rotation
•Vmax = 50 – 400 km/s (most between 150-300 km/s)
•σ (gas) = 5 – 10 km/s
•σ (stars) = 5 – 50 km/s
-compare with Ellipticals σ = 50 – 500 km/s
Gas measured from Hα – ionized gas from disk HII regions
HI (21cm) – atomic gas – allows us to “see” beyond disk stellar edge
CO (2.6mm) – molecular gas (CO used to estimate H2)
HI in disks is optically thin – little absorption so mass ≈ intensity
*though in MW HI distribution can be hindered by dust in disk
*emission becomes optically thick (mass ≠ intensity)
Deep HI maps detect ~1019 H atoms/cm2 or 0.1M/pc2
Example: NGC 7331 contains 1.1 x 1010M HI gas
Ratio of galaxy mass in gas increases with later Hubble type
Mgas = MHI + MH2
Mdyn = dynamical estimate of total mass
Spiral galaxies vary in the amount of molecular to neutral Hydrogen
(50% to 10%)
Distribution of HI in external galaxies
•centers of disk galaxies are generally gas poor, gas is piled in a ring
several kpc out (also seen in MW and M31)
•more gas than stars in outer regions (little SF)
HI gas in NGC 5033 (Sc)
HI surface density vs radius
HI gas in NGC 7331
Example: Andromeda
HI - 21cm
FIR (SF & dust)
•CO displays sharp drop with radius
•Traces spiral arms
•CO more associated with arms than HI
which permeates galaxy (except in center)
•CO velocity map shows rotation
CO - 2.6micron
Optical
Compare to Distribution of HI and H2 gas in the Milky Way
•HI gas - mass does not decrease very quickly (mass interior to Ro = 109
Msun and outside Ro is 2x109 Msun)
•H2 gas - falls off rapidly (109 Msun inside Ro and 5x108 Msun outside)
•Feature in H2 called “molecular ring”
Correlations with molecular gas content in galaxies
Radio continuum
Molecular gas (CO emission line strength)
• tight correlation between molecular gas content, radio continuum emission,
and IR luminosity
Radio continuum
mid-IR luminosity
• Mid-IR from UV photons (from stars) converted to IR via dust
• Radio continuum from 1) free-free radiation in hot ionized gas and 2) synchrotron
from SNe remnants
 cool gas, SF, SNe – all related to the formation of massive stars in galactic disks
Spiral Galaxy Velocity Fields
At radius R, assume a gas cloud follows near-circular path with speed V(R)
Measure Vr (radial velocity) doppler shift
Velocity at galaxy center is Vsys, the systemic velocity
Observe a disk in pure circular motion at inclination i
For a star or gas cloud at radius R and azimuth ϕ, the radial velocity is
Vr(R,i) = Vsys + V(R) sini cosϕ
Contours of constant Vr connect points with the same value
V(R)cosϕ  spider diagram
A
B
for disk inclined 30 degrees
lines of constant Vr-Vsys
radius R/Rd
•line AB is the kinematic major axis – deviates furthest from Vsys
•center line (minor axis) is close to Vsys – all motion tangential
•Near center, lines are almost parallel to center line since V(R) goes
as R (solid body)
•Further out, where rotation speed becomes almost constant, lines
bend toward radial direction
Distorted Spider Diagrams: Kinematic Warps
•most galaxy spider diagrams show deviations from pure circular motion
•explained by warps in disk – tilted ring model
•HI maps of edge-on disks also reveal warps
M83 – model and data for less-inclined disk warp
Edge-on warp
Distorted Spider Diagrams: Oval Orbits
gas moves in oval (elliptical) rather than circular motions - effects on velocity fields:
•kinematic axes not perpendicular
•kinematic and photometric minor axes not aligned
•kinematic major axis close to line-of-nodes (line through galaxy’s center that
lies in sky plane and galaxy’s equatorial plane)
line of nodes
disks inclined by 1 degree (almost face-on)
left panel – photometric major axis at 70 degrees from LON
right panel – photometric major axis at 20 degrees from LON
Spiral Galaxy Rotation Curves:
evidence for Dark Matter
V(R) is of fundamental importance for determining M(<R) since
V2(R)= GM(<R)/R  V ~ sqrt(M/R)
For NGC7331, compare observed rotation curve with what we expect if the
galaxy mass is entirely in stars and gas
•assume stellar (bulge/disk) density ~ R-band light and typical M/L
•assume gas surface density (disk) is 1.4 x HI intensity
•adjust M/L so that gas and stars in disk account for as much of the galaxy’s rotation as
possible  maximum disk model
75% of
mass is in
a Dark
Matter Halo
Rotation curve should begin to fall at R ~ 20 kpc if only stars/gas important.
Van Albada et al (1985)
Dark Matter vs. Hubble type
Sa – Sb: longer scale lengths, more rapid rotation
~50% DM needed
V(R) climbs steeply  more mass close to center
Sd – Sm: shorter scale lengths, slower rotation
~80 – 90% DM needed
V(R) climbs more gradually  lacks central concentration
What mass distribution is required for a flat rotation curve?
for spherical symmetry

if V is constant, M(R) ~ R
Mo
r(a+r)2

where V=Vmax at large R
Density falls as R-2 for a flat rotation curve (slower than an exponential decrease)
What is the distribution of the Dark Matter Halo?
N-body simulations of gravitational clustering suggest
NFW law
where ρo and a are free parameters (Navarro, Frank & White 1997)
but fits can be made w/very different values of free parameters depending on the
adopted M/L for disk & bulge
Trends with
Luminosity
t
isk
Galaxies with larger L
have larger Vmax
Galaxies with larger L
have shorter radii of
solid-body rotation
DM
Fraction of DM inside
optical radius
increases with
decreasing Vmax
2-component
maximum
disk models
DM is less
concentrated than the
luminous matter
Kinematics from single emission line
Quick way to get Vmax  measure width of 21-cm HI emission w/single
dish radio telescope
W
Double-horned profile
results from flat rotation
curve
W ≈ 2 Vmax sin i
Measurements like this for
many galaxies revealed...
Tully-Fisher relation
L ~ Vmax4
Distance Indicator!
brighter galaxy  more massive  faster rotation
Usually use red or IR light for L to minimize effects of recent SF
Tully Fisher
In I-band
LI/4x1010LI, = (Vmax/200km/s)4
In H-band
LH/3x1010LH, = (Vmax/196km/s)3.8
Galaxies in Ursa Major group based on HI global profile
W/sin i = 2Vmax. Open circles are LSB galaxies.
TF can be dynamically understood without DM. But, since
Vmax comes largely from DM and L from luminous matter,
we see “coordination” between DM and LM.
Determining Vmax for distant galaxies to look for evolution in TF relation
Using optical light to get rotation curves
HST (ACS)
HST ACS
[OII]
model
residual
[OII]
model
residual
Tully-Fisher Evolution
• Comparing TFR at
0.9 < z < 1.4 to
Nearby Field Galaxy
Survey sample
(Kannappan et al.
2002)
• Offset (assumes no
slope change):
-1.7 mag at z~1.1
•
rotating disks 1.7
mag brighter at
<z>=1.1 than locally
(Metevier et al 2006)
Disk Galaxies: Gas Metallicity trend with Radius
Measure strength of emission lines in HII regions relative to H lines (e.g. Hβ)
O+ (3727Å), O2+ (5007Å), N+ (6583Å), S+ (6724Å) and correct for reddening and
underlying stellar absorption
 strength of both O+ and N+ fall w/radius; N+ falls faster
Interstellar abundances of
metals in disk galaxies
declines w/galactic radius
Since N is a secondary
element, its abundance
should be [N/H] ~ 2[O/H]
Observed [N/H] ~ 1.5[O/H]
Central [O/H] values
increase with galaxy
magnitude and Vmax (due to
TF; Zaritsky et al 1994)
Dors and Copetti (2005) for M101
Disk Galaxies: Gas Metallicity trend with mass
• Metallicity higher for more massive galaxies – MZ relation
• Metallicity lower at higher redshift at the same mass
• Metallicity at z~2.3 is less
than local galaxies at the
same mass and further from
local relation at lower
masses (red circles)
• Also observed at z~1 (blue
circles)
• evidence for evolutionary
downsizing  higher mass
galaxies evolve onto the
local MZ relation at earlier
times.
Henry et al. 2013
Why the decline of metallicity with radius?
Consider closed-box model of chemical evolution (BM 5.3.1)
Z = -p ln fgas
Z = Mheavy elements/Mgas
p = nucleosynthesis yield from stars
fgas = fraction of mass density in gas
gas fraction lower near center of Sa/Sb galaxies
 metallicity decreases w/radius
(trend prediction is correct but not quantitative)
Closed-box model predicts constant p (yield), but to match observed
trends, p must vary throughout disk – larger at small R and high Z
What else may be happening?
•gas drifts inward – carries metals inward
•gas replaced at large radii by accretion of metal-poor gas from IGM
Disk Galaxies: stellar ages and metallicities
Using old, giant branch
stars, determine stellar
metallicity vs radius.
Decrease in metallicity
with R until 15 kpc, then
flat. Is this the halo?
Probably not since halo
[Fe/H] is usually lower
metallicity ([Fe/H] ~ -1.5).
What is the reason for the
flattening of the relation at
large R?
Vlajic et al 2009
Accretion scenario...
Milky Way disk: stellar and cluster metallicities
•Galactic metallicity
decreases with radius and
levels beyond R =10 – 12
kpc
• α-elements produced in
Type II SNe are enhanced
at large R (more recent SF)
blue open and green = open clusters
red = field stars
black plus = Cepheid variables
Carney et al (2005)
•consistent with progressive
growth of the Galactic disk
with time and episodic
enrichment by Type II SNe
(e.g. recent enrichment
caused by enhanced SF
from minor mergers)
Milky Way disk: metallicity with vertical height
•[Fe/H] for 200,000 F,G stars in sdss
•metallicity distribution of the halo
component is Gaussian with mean
[Fe/H] = −1.46
•disk metallicity distribution is nonGaussian, with small scatter
•median smoothly decreasing with
distance from the plane from −0.6 at
500 pc to −0.8 beyond several kpc
Hayden et al. 2015
Ivezic et al (2008)
Spiral Galaxies: Spiral Structure
M81
Spiral Galaxies: Spiral Structure
•Almost all giant galaxies with gas in their disks display spiral structure
•Arms are blue  regions of active star formation
•Both ionized (Hα) and cool gas (HI) concentrate in arms
Shape of m-armed spiral
Function f describes tightness of spiral (large df/dr = closely wrapped arms)
Pitch angle  angle between arms and tangent to circle at R
angle i
Sa: i = 5 degrees
Sc: i = 10 to 30 degrees
•Spiral arms are almost always trailing –
tips point in direction opposite rotation
•Spiral shape not caused by galaxy’s
differential rotation
Spiral Structure – Winding Dilemma
For MW @ location of the Sun, V(R) = 200 km/s and R = 8 kpc
What is pitch angle? Recall Ω(R) = V/R and V is const w/R
Cot i ~ 200/8 * t

i = 2 degrees x (1 Gyr/t)
Spiral arms structure would be too tightly wound after ~1 Gyr due to
differential rotation - Eventually spiral structure is “smeared out”.
Density Waves in Spiral Galaxies
Spiral arms are caused by the compression of gas as it orbits the
Galactic center and encounters density waves that are essentially
stationary. Compression of gas causes stars to form which we see
as spiral arms.
•Cloud approaches arm at a relative
speed of ~100km/s.
•Arm acts as gravitational well, slowing
down the cloud.
•Arm will alter orbits of gas/stars, causing
them to move along arm briefly.
•Compresses HI gas and gathers small
MCs to form GMCs.
•GMCs produce O&B stars.
•Stellar radiation disrupts the clouds.
Self-propagating Star Formation
Grand design spirals are best
explained via the density
wave theory for spiral
structure.
Flocculant spirals may be
dominated by selfpropagating star formation,
resulting in more temporary
patterns, drawn into spirallike shapes due to the
rotation of the galaxy.
The formation of stars drives the waves –
shock waves from the later evolution of
stars creates denser regions where new
stars are created.
Flocculant spiral M33
How is the spiral pattern sustained?
Self-propagating SF  flocculant spirals
Kinematic spiral – stellar orbits not quite circular, eccentric pattern
arranged in a particular order
 Let stars in disk be on slightly eccentric paths
• Guiding center moves
uniformly at speed Ω(R)
so that its azimuth
ϕgc = Ω(Rg)t
• Oscillations in and out
described by epicyclic
motion
R = Rg + x = Rg + X cos(κt + ψ)
guiding center
Set stars with guiding centers spread around circle at Rg with ψ=2Φgc(0)
 stars lie on an oval with long axis at Φ=0.
At later time = t, Φgc(t) = Φgc(0) + Ωt and stellar radii now
R = Rg + X cos(κt + 2[Φgc(t) - Ωt])
R = Rg + X cos([2Ω-κ]t - 2Φgc(t))
Long axis now points along direction where
(2Ω-κ)t – 2Φ = 0 or Φ = (Ω-κ/2)t  Φ = Ωpt
where Ωp is the pattern speed
Pattern speed slower than stellar speed
Pattern of stars returns to original state after t=2π/Ωp
Star speeds allow them to return to original state after t=2π/Ω
Thus a 2-armed spiral can be made from a set of nested ovals of
stars with different guiding center radii Rg
For an m-armed spiral, set ψ = mΦgc(0) and Ωp = Ω - κ/m
Note: Ωp varies with Rg and therefore the spiral pattern will also wind up
with time, but slower than wind up due to stellar velocities.
 Ωp/Ω ~ 0.3 for flat rotation curve.
Density wave theory is based on the idea that mutual
gravitational attraction of stars and gas clouds at different R can
offset the spiral’s tendency to wind-up.
produces pattern which rotates rigidly within the galaxy disk.
How does the spiral pattern emerge?
Something has to "seed" the
perturbation (initial non-axisymmetry
or galaxy encounters). Then selfgravity of the disk will amplify the
perturbation and make it grow.
http://burro.cwru.edu/Academics/Astr222/
Galaxies/Spiral/spiral.html
Density inhomogeneities orbiting
within disks.
Using high-resolution N-body
simulations with mass
concentrations similar to GMCs
Induces spiral arms through
swing amplification
(D’Onghia et al. 2013)