6 Spiral Rotation 7 Spiral Arms

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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

  • Dark halo: dominates mass (and gravitational potential) outside ~10kpc, nature unknown?

Luminosity profiles: (1D):

  • Exponential disk: I(r) = I0 exp(-r/rd)

    • rd= disk scale length, typically ~2-6 kpc

    • Light falls off sharply beyond Rmax ~3-5rd

    • 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)

  • 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.

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. ◦

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) µ R.

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

  • Lµ Vmaxa where a ~ 4

The Tully-Fisher relation for spiral galaxies (and the Faber-Jackson 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

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.


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.


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

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):

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

    • Grand-Design” – two well-defined spiral arms (10%)

    • Multiple-arm spirals (60%)

    • 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.

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) = constant

(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

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.

    • Their solution was to assume that:

      • Stars follow slightly elliptical orbits

      • The orientations of these orbits are correlated:

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.

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 arrange themselves in correlated ellipses.

    • the answer is self organization:

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:

k2 = 4 2 + R d2/dR

  1. Rigid body: (R) = constant, k = 2 

  2. Keplerian:(R) = 1/R 3/2, k = 

  3. Flat rotation; :(R) = 1/R, k = 2 1/2

So, typically k < 2

Therefore stars oscillate quite slowly around the circular orbit. Resonances could occur between orbital and epicylic motions.

Suppose the pattern speed is

Wp = W

(at the co-rotation radius

Rotate in a frame with the pattern speed:

Then, just see epicyclic motions, no orbital motion (subtracted off).

If Wp = W -k/m

…the orbit is closed in the pattern rotating frame.

For a flat rotation curve with m = 2:

W -k/m = ( 1- 0.707) W = 0.293W

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

Wp = W -k/m and Wp = W + k/m

where k 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.

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:

  • 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

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.

    • 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

    • Recent automated CCD surveys suggest there may be more LSB galaxies than all the other types of galaxy put together

  • Peculiar Galaxies

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

    • F(L)dL = F*(L/L*)a exp{-L/L*}d(L/L*)

    • F(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, a is the power-law slope at the faint end, F* 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:

F(M)dM =

(0.4 ln10)x F* x 10 0.4(a+1)(M*-M) x exp{-10 0.4(M*-M)}dM

F* = 0.45 x10-2h723 Mpc-3

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

Approximate Schechter values:

  • M* ~ -20.5 (in B), depends on H0

  • L* ~ 2 x 1010 L¤ (~Milky Way)

  • a ~ -1 to –1.5, often take -1.25

  • Normalization is uncertain!

Integrating the Luminosity Function

n* = 8 x 10-3 Mpc-3

L* = 1.4 x 1010 L¤


…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¥ F(L)dL = F* L* G(a+1)

    • Where G is the gamma function, G(j+1)=j! when j is an integer

    • If a < -1, G(a+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 F(L)dL = F* L* G(a+2), which diverges if a < -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

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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