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6 Spiral Rotation
7 Spiral Arms
8 Barred Spirals
9 Dwarf Galaxies
10 Galaxy Luminosity
Basic components of Spirals reviewed:
 Disks: metal rich stars and ISM, nearly circular orbits
with little random motion, spiral patterns
 Bulge: metal poor to super-rich stars, high stellar
densities, mostly random motion – similar to
ellipticals
 Bar: present in 50 % of disk galaxies, long lived, flat,
linear distribution of stars
 Nucleus: central (<10pc) region of very high mass
density, massive black hole or starburst or nuclear
star cluster
 Stellar halo: very low surface brightness (few % of
the total light), metal poor stars, GCs, low-density hot
gas, little/no rotation
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 Dark halo: dominates mass (and gravitational
potential) outside ~10kpc, nature unknown?
Luminosity profiles: (1D):
 Exponential disk: I(r) = I0 exp(-r/rd)
o rd= disk scale length, typically ~2-6 kpc
o Light falls off sharply beyond Rmax ~3-5rd
 In the central regions, also see light from the bulge
o Bulge follows the r1/4-law, like ellipticals
Vertical disk structure:
The surface brightness perpendicular to the disk is also
described by a exponential or better by a sech law
 I(z) = I(0) exp(-|z|/z0)
 I(z) = 0.25 I(0) sech2 (-z/2z0) …… recall
sech(z)=2/[exp(z)+exp(-z)]
 z0 is the scale height of the vertical disk
Different populations have different scale heights. In the
Milky Way:
 Young stars & gas ~ 50pc
 Old thin disk ~ 300-400 pc (older stars, like
the sun)
 Thick disk ~1 – 1.5 kpc (older, metal-poor
stars)
Best interpretation of many of these is a trend in star
formation history
 Early type spirals formed most of their stars early on (used
up their gas, have older/redder stars)
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 Late type spirals have substantial on-going star-formation,
didn’t form as many stars early-on (and thus lots of gas left)
 Spirals are forming stars at a few Msun per year, and we know
that there is ~a few x 109 Msun of HI mass in a typical spiral
Rotation curves of other galaxies
On the left, a spiral galaxy image, with spiral arms delineated by
HII regions.
On the right, the spectral image: light from a narrow strip running
along the major axis of the galaxy has been spread into a
spectrum, between about 6500 and 6800 Angstroms.
The rotation of the galaxy is seen in the emission lines from H
alpha at 6563 Angstroms (the brightest line), as well as other
fainter lines in this region due to [NII]. HII regions appear reddish in
this image because of the prominence of the H alpha line in the
red region of the spectrum.
The dark features running vertically trace emission in the hydrogen
H-alpha line (left and stronger feature, rest wavelength 6563
Angstroms), and in nitrogen ([NII], right feature, rest wavelength
6583 Angstroms). These emission lines arise from hot gas
surrounding hot, young, massive O and B-type stars.
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We can measure rotation curves via:
HI mapping: 21cm emission from atomic hydrogen
Optical spectroscopy: Optical absorption lines from the stellar
component or Optical emission lines from hotter gas.
◦
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1. More luminous galaxies have higher rotation velocities,
2. Later type galaxies have slower rise in velocity.
The rotational velocity is constant,
m* Vrot2 / R = G M m* / R2
Vrot2 = G M / R
So that means M(R)  R.
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Goes beyond limits of stellar disks, which are showing an
exponential drop off in light (and thus mass) anyway!
Tully-Fisher Relation:
Tully & Fisher (1977) found that
L Vmax
where  ~ 4
The Tully-Fisher relation for spiral galaxies (and the FaberJackson relation for ellipticals), follow from the dynamics if we
assume constant mass-to-light ratio and surface brightness.
Plot the maximum circular velocity of spiral galaxies against their
luminosity in a given band:
….to find that L and Vmax are closely correlated
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Smallest scatter when L is measured in the red or the near-infrared
wavebands.
The Tully-Fisher relation reflects simple gravitational dynamics of a
disk galaxy:
Since m Vrot2 / R = G M m / R2 ,
the mass M is proportional to Vrot2 R
The observed flux or Luminosity L of a galaxy can be determined
photometrically
▪ e.g. simply integrating the surface brightness to determine the
total flux or luminosity from the galaxy.
◦ which is a function of the (visible) mass
(more massive - more stars - more emission)
Assuming all galaxies have the same M/L ratio and the same
surface brightness, then the relation L is proportional to Vrot4.
Why?
The Tully-Fisher relation reflects simple gravitational dynamics of a
disk galaxy. Estimate the luminosity and maximum circular velocity
of an exponential disk of stars.
Luminosity
Empirically, disk galaxies have an exponential surface brightness
profile:

I(R) = I(0) exp[-R / hR ]
…with central surface brightness I(0) a constant. Integrate
this across annuli to get the total luminosity:
L= 2R I(0) exp[R/ hR] dR
Can integrate this expression by parts, finding:

L I(0) 2hR 2
So, for constant central surface brightness, the luminosity
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scales with the square of the galaxy scale length. The relationship
is then predicted (taking R and hR to be linearly related).
For 2 distinct clusters, one a distance D further away:
7. Spiral Structure
The most obvious feature in a spiral galaxy is its spiral structure.
Here are M101 & M100 with ‘grand design’ spiral arms (type Sc):
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M101 is 9Mpc away;20x20 arcmin, mV= 7.9
M100 is 20Mpc away; 6x7 arcmin, mV= 9.3
Flocculent spiral: NGC 4414
Structure is made up from young, bright stars.

There are different types of spiral arms
o “Grand-Design” – two well-defined spiral arms (10%)
o Multiple-arm spirals (60%)
o Flocculent spirals – no well-defined arms at all, “ratty”
(30%)
Because disk galaxies rotate differentially, the orbital period is
an increasing function of radius R.
Thus if spiral arms were material features then differential rotation
would soon wind them up into very tightly-coiled spirals.
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The expected pitch angle of material arms in a spiral galaxy like
the Milky Way is only about 0.25 degrees. In fact, pitch angles
measured from photographs range from about 5 degrees for Sa
galaxies to 20 degrees for Sc galaxies.
The most likely implication is that spiral arms are not material
features.
First ingredient for producing spiral arms is differential
rotation. For galaxy with flat rotation curve:
V(R
(R)
= V/R
Angular velocity decreases with increasing radius
Any feature in the disk will be wrapped into a trailing spiral pattern
due to differential rotation.


Open spiral structure cannot be maintained in this way.
This problem is usually known as the winding dilemma
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In the 1950's it was thought that magnetic fields could be the
mysterious generators of spiral structure.
Conclusion: The stars in a spiral arm cannot always be the same
since spiral structure would wind up very tightly.
Though galaxy arms appear to be intact features, stars are actually
constantly moving in and out of them as they orbit the centre of the
Milky Way, like London commuters in a busy traffic circle.
Our own Sun might have once resided in a different arm. Since it
was formed more than 4 billion years ago, it has travelled around
the galaxy 16 times.
Open Questions:



Are spiral arms leading or trailing?
What is the nature of the arms?
A viable solution to this dilemma was finally sorted out by Lin
& Shu in 1963.
o Their solution was to assume that:
 Stars follow slightly elliptical orbits
 The orientations of these orbits are correlated:
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This arrangement produces a spiral density wave: spiral arms
are caused by a density perturbation that moves along at a speed
different from the speed of the objects within it. The density wave
resists the spiral’s tendency to wind up and causes a rigidly
rotating spiral pattern
Properties of spiral arms can be explained if they are not rotating
with the stars, but rather density waves:
• Spiral arms are locations where the stellar orbits
are such that stars are more densely packed.
• Gas is also compressed, possibly triggering star
formation and generating population of young stars.
• Arms rotate with a pattern speed which is not equal
to the circular velocity - i.e. long lived stars enter and
leave spiral arms repeatedly.
Material travels around undisturbed elliptical orbits, but sometimes
many orbits come close together, so the density increases.
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High densities also compress the magnetic fields, which produces
a maximum in the radio continuum emission in regions of highest
density.
So, bright stars should appear "down stream" from the peak in the
radio continuum emission.
This effect is, indeed, observed, and so the density wave theory is
vindicated!
In the inner parts of disks, stars are moving faster than the
pattern speed and overtake the density wave.
In the outer parts, stars move more slowly than the pattern
speed, and the spiral arms over take the stars

A remaining question is why orbits
themselves in correlated ellipses.
o the answer is self organization:
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arrange
This feedback loop can also generate the bars in SB
galaxies
Such a runaway process is called a dynamical instability
Note that this process only works if there is enough mass in the
disk for the perturbations to modify the gravitational field
In early-type spirals (Sa's) where most of the mass is in the bulge
not the disk, the instability will be partly suppressed.
This suppression explains the anti-correlation between bulge
size and strength of spiral structure.
Density Wave Theory:
1. Stars in the disk oscillate about their roughly circular orbits.
2. This yields the epicyclic frequency for radial oscillations:
2
= 4 2 + R d2/dR
3. Rigid body: (R) = constant, 
4. Keplerian:(R) = 1/R 3/2, 
5. Flat rotation; :(R) = 1/R, 
= 2
= 
= 2 1/2 
 < 2
So, typically
Therefore stars oscillate quite slowly around the circular orbit.
Resonances could occur between orbital and epicylic motions.
Suppose the pattern speed is
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p = 
(at the co-rotation radius
Rotate in a frame with the pattern speed:
Then, just see epicyclic motions, no orbital motion (subtracted off).
If p =  -/m
…the orbit is closed in the pattern rotating frame.
For a flat rotation curve with m = 2:
 -/m
= ( 1- 0.707)  = 0.293
However it is a function of R, so the spiral pattern is still wound up
in time!
Need: mutual gravitation attraction across the radii to balance the
winding tendency.
This produces an effective pattern speed almost independent of
radius.
Spiral arm pattern is amplified by resonances between the
epicyclic frequencies of the stars (frequency of deviations
from circular orbits) and the angular frequency of the spiral
pattern
Spiral waves can only grow between the inner and outer
Lindblad resonances where
p =  -/m
and p =  + /m
where  is the epicyclic frequency (frequency of radial
oscillations) and m is an integer (the number of spiral arms)
Why? Beyond the Outer: epicyclic frequency is too slow to
respond.
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Stars outside this region find that the periodic pull of the spiral is
faster than their epicyclic frequency, they don’t respond to the
spiral and the wave dissipates.
Resonance in narrow annulus can explain why 2 arm spirals are
more prominent.
Flocculent Spirals.
Note that density wave theory does not explain flocculent spirals.
Those can be explained by self-propagating star formation:
Star forming regions produce supernovae, which shocks the gas,
which triggers more star formation, etc, etc, etc
Differential rotation stretches out the regions of star formation into
trailing, fragmentary arms
No global symmetry (as observed)
Are spiral arms leading or trailing????
8. Barred Galaxies
e.g. NGC 1300:
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Half of all disk galaxies show a central bar which contains up
to 1/3 of the total light
Bars are almost as flat as surrounding disks.
S0 galaxies can have bars – a bar can persist in the absence
of gas
Bar patterns are not static, they rotate with a pattern speed,
but unlike spiral arms they are not density waves. Stars in
the bar stay in the bar.
The bar rotates as a unit in a rigidly rotating disk.
The asymmetric gravitational forces of a disk allow gas to
lose angular momentum (via shocks) compressing the gas
along the edge of the bar.
The gas loses energy
(dissipation) and moves closer to the center of the galaxy.
9. Dwarf Galaxies
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Dwarf Elliptical
Faint, M > -18, Low-luminosity: 106 – 1010 L
Low-mass: 107 – 1010 M
Small in size, ~few kpc
Often low surface brightness, so they are hard to find!
Why are dwarf galaxies important??
Majority of galaxies are dwarfs!! There are probably
lots of these, in the Local Group there are >30!
Dwarf galaxies may be remnants of galaxy formation
process: “proto-dwarf” gas clouds came together to
form larger galaxies (hierarchical formation)
Dwarf galaxies are currently being “absorbed” by larger
galaxies
Dwarf galaxies are relatively simple systems, not
merger products
Different types of dwarf galaxies
Dwarf ellipticals (dE): Note that these are structurally
very different from luminous E’s. Gas-poor, old stellar
population. Note that many dE’s have nuclei (dE,N).
Dwarf spheroidals (dSph): Gas-poor, diffuse
systems. Low luminosity (low surface brightness end
of dE’s.
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Dwarf irregulars (dIrr): Extreme end of late type
spirals. Active, on-going star-formation but low surface
brightness (like dSph’s). Gas-rich. Note that there are
no dwarf spirals!!
In the Local Group, we can study the resolved stellar
population (colour magnitude diagrams) to determine
the star formation histories of dwarf galaxies
Dwarf ellipticals are generally old (stars formed > 10
Gyr old), but some may have had more recent (a few
Gyr ago) weaker episodes of star formation
Dwarf irregulars tend to have quasi-continuous
star formation (perhaps interspersed with bursts).
Lower luminosity dIrr’s more likely to have a bursty
history
Environmental effects may play a role (e.g., tidal
stripping removing gas from dSph’s)
No two galaxies have the same star formation
history
Dwarfs do not contain dark matter…..however,
unusually:
 Dwarf Spheroidal, Leo I :
 Leo I
Low Surface brightness galaxies (LSB)
 Very difficult to detect!
 Need dedicated surveys
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 Recent automated CCD surveys suggest there may be
more LSB galaxies than all the other types of galaxy
put together
Peculiar Galaxies
 In particular, interacting galaxies
 Many cataloged by Arp in 1966
10. Galaxy Luminosity Function
Count the number of galaxies as a function of luminosity
(or absolute magnitude)
Useful for:
 Understanding galaxy formation (distribution by
luminosity implies distribution by mass – how many
galaxies of a given type and mass were formed
 Galaxy evolution models – either must reproduce
observed LFs (hierarchal formation models) or assume
them (and work backwards in time). Can also measure
evolution in LFs vs. redshift!
 Galaxy Properties
Schechter (1976) found that
 (L)dL = *(L/L*) exp{-L/L*}d(L/L*)
 (L)dL = number of galaxies per unit volume
with luminosities between L and L+dL
 Where L* = 1.9 x1010h72-2 Lsun is a characteristic
luminosity cutoff,  is the power-law slope at the faint
end, * is the normalization (# galaxies/Mpc3)
 This function is a power law for L< L* , but cuts off
rapidly for L > L*
 Usually measured in magnitude:
(M)dM =
(0.4 ln10)x * x 10 0.4(+1)(M*-M) x exp{-10 0.4(M*-M)}dM
* = 0.45 x10-2h723 Mpc-3
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Schechter Function by galaxy type and
environment
Field – dominated by Spirals, faint end dIrr
Clusters – many more E/S0 galaxies, faint end dE, more
dwarfs than in field
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Approximate Schechter values:
M* ~ -20.5 (in B), depends on H0
L* ~ 2 x 1010 L (~Milky Way)
 ~ -1 to –1.5 , often take -1 . 2 5
Normalization is uncertain!
Integrating the Luminosity Function
n* = 8 x 10-3 Mpc-3
L* = 1.4 x 1010 L

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…where L = 3.9 x 1033 erg s-1 is the Solar luminosity.
1. The Total Number of Galaxies:
If we integrate the Schechter function, we get the total
number of galaxies (per Mpc3), we find:
 N = ∫0 (L)dL = * L* (+1)
 Where  is the gamma function, (j+1)=j! when j is
an integer
 If  < -1, (+1) is undefined (!), and N is infinite!!
2. The Total Luminosity of Galaxies:
We can also integrate to find the total luminosity
Total luminosity = ∫0 L (L)dL = * L* (+2), which
diverges if  < -2
So the total amount of light is finite! (Phew!!)
Dominated by galaxies with L ~ L* for typical value of
.
3. Mass function of galaxies
For stars, measurements of the luminosity function can be
used to derive the Initial Mass Function (IMF).
For galaxies, this is more difficult:
• Mass to light ratio (M/L) of the stellar population
depends upon the star formation history of
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the galaxy.
• Image of the galaxy tells us nothing about the amount
and distribution of the dark matter.
More complex measurements are needed to try and get at the
mass function of galaxies.
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