Whittle : EXTRAGALACTIC ASTRONOMY

 
       
 

1 : Preliminaries	6 :   Dynamics I	11 : Star Formation	16 : Cosmology
2 : Morphology	7 :   Ellipticals	12 : Interactions	17 : Structure Growth
3 : Surveys	8 :   Dynamics II	13 : Groups & Clusters	18 : Galaxy Formation
4 : Lum. Functions	9 :   Gas & Dust	14 : Nuclei & BHs	19 : Reionization & IGM
5 : Spirals	10 : Populations	15 : AGNs & Quasars	20 : Dark Matter

(1) Introduction

• We now begin a study of Stellar Dynamics : the theory of gravitational many-body systems.
  This subject is mature, extensive and can be highly sophisticated
  Fortunately for us, we don't really need (and don't have the time for) a detailed treatment
  Instead, my aim will be to :
  • present the major themes in broad outline
  • extract some important analytic results
  • develop our intuitive understanding of each topic
  • connect each topic to our observational knowledge of galaxies

• Since the subject is so weighty, it is sensible to spread it out a bit :
  
  Topic 6 : considers disk dynamics and the formation of bars and spirals
  Topic 8 : covers the meat of the subject : ways to understand the general 3-D system
  Topic 12 : considers dynamical friction, encounters and mergers.
  Topic 13 : discusses how black holes can affect galaxy structure
  

  By the end we will have covered almost all of B&T while omitting much of the detail.

• So lets now begin with disk dynamics :
  The path for this topic takes us through a number of interesting themes :
  epicycles : perturbations of simple circular orbits
  resonances : when epicycles synchronize with the passage of disk patterns
  density waves : a self-consistent response to coherent epicyclic motion
  instabilities : when disks begin to clump up under self-gravity
  amplification : when this clumping grows to form bars and spirals.

(2) Epicycles

• Disk stars have approximately circular orbits with small deviations :
  As with the Ptolemaic system : star orbits can be described by superposition of :
  • circular orbit along guiding center (=deferent), radius R_g, angular velocity _g
  • smaller elliptical epicycle, angular velocity , retrograde
• consider gravity/centrifugal balance & conservation of AM :
  • consider star at guiding center (GC) and give it a kick radially outwards
    
  • conserving AM = mrv, since r increases, v decreases
      w.r.t. the guiding center, the star moves backwards
    
  • consider the new balance between gravity and centrifugal forces :
    under AM conservation,  F_centrifugal  v²/ r   r^-3   while F_grav falls more slowly than r^-2
      at larger radii F_grav > F_centrifugal and the star gets pulled back inwards
    
  • as it falls in, r decreases so v increases and the star moves forward relative to the GC
    
  • but now F_centrifugal > F_grav and the star moves outwards again
    

      the cycle repeats, and we have a small retrograde epicycle [ images ]
    (An equivalent description considers the coriolis force in a rotating frame)
    

• In general _g and are different so orbits don't close
• However, when observed from frame rotating at _g - ½ :
    orbits are closed ellipses, centered on guiding center

• For epicycle phases which vary systematically with radius :
  nested elliptical orbits may crowd in a spiral pattern
  this is called a kinematic density wave
• Orbits are in turn perturbed by the spiral density pattern,
  this modifies the simple epicycles
• Self-consistent solution is a density wave
  gas reponse causes shocks and star formation visible spiral arms [ images ]

• Consider a smooth axisymmetric flattened mass distiribution with potential (R,z)
  Since we have no azimuthal forces AM is conserved, and we have :

  (6.1)

• Separating the Equations of motion into components in cylindrical coordinates (R, ,z) :

  (6.2)

• Take first the z motion about the plane z=0
  since the disk is symmetric about z=0, then the z-force at z=0 is zero :

  (6.3)

  consider small motion above and below the plane,
  we simply expand the z-force linearly for small z :

  (6.4)

• This gives Simple Harmonic Motion (SHM), with frequency where ² = (² / z²)_z=0 :

  (6.5)

• For Milky Way disk near the sun, ² = 4 G _o, which gives :
  vertical oscillation period 2 / ~ 6.5 x 10⁷ yr ~ 1/3 circular period, .

• First consider the circular guiding orbit (R=const) :
  it has radius R_g, circular velocity V_c and angular velocity _g defined by

  (6.6)

  For non-circular orbits, the radial acceleration is given by (centrifugal - gravity) :

  (6.7)

• However, notice that this can also be written as

  (6.8)

  Where the effective potential, _eff, allows us to describe the radial motion in 1-D form
  Typically, _eff has a minimum, rising steeply at small R and slowly at large R [ images ]
  This inner steep term imposes an angular momentum (or centrifugal) barrier
  At the minimum in _eff, we recover the circular guiding orbit of radius R_g

  (6.9)

• Other orbits will oscillate in radius, about R_g.
  Consider the potential at R = R_g + x

  (6.10)

  This gives SHM about the guiding radius  

  (6.11)

  with frequency , where

  (6.12)

• Since L_z = R_g² _g = R² = const, changes in R yield changes in     (recall = (dot) )

  (6.13)

  Integration gives :

  (6.14)

• Thus, (t) follows the guiding center with small amplitude SHM superposed.
  Taking the y-axis in the forward direction with origin on the guiding center, we have

  (6.15)

  the oscillation of frequency is the same as in x, but out of phase by 90°
  

  Taken together, (and setting the initial phase ₀ = 0), we have

  • x = X cos(t)
  • y = -(2/) X sin(t)
  • elliptical epicycle with radial/azimuthal axis ratio = / 2
  • epicyclic motion is retrograde w.r.t. orbit (cf Ptolemy's were prograde)
  • for stars of same R_g, velocity ellipsoid _R / = / 2
  • however, at given R, velocity ellipsoid _R / = 2 /
  • for Keplerian : R^-3/2 we get =
    closed ellipse, centered at the ellipse focus
    axis ratio 2:1 (cf Ptolemy's were 1:1 circles)
  • For flat rotation curve R^-1 we get = sqrt(2)
  • for solid body rotation = const   we get = 2
  • in general,   <     <   2 so
      >   and epicycle completed before rotation
  • Here is a model of small amplitude epicyclic motion near the sun.

It is useful to re-express and in terms of Oort's constants A and B :
[a brief derivation and review of Oort's constants can be found here].

Oort's A expresses local shear
it is derived from radial velocities V_r / d = A sin(2l) where l=longitude :

(6.16a)

Current best (Hipparcos) estimates for Oort's A is 14.8 ± 0.8 km/s/kpc (B&M p642)

Oort's B expresses local vorticity, ie local rotation, × V
it is derived from proper motions V_t / d = A cos(2l) + B

(6.16b)

Current best estimate for Oort's B is -12.4 ± 0.6 km/s/kpc

Together these give :

(6.17a)

(6.17b)

(6.17c)

• Stars in the solar neighborhood make 1.3 epicyclic excursions per orbit.
• From disk mass models, we have a z-force law shown here
• for peculiar velocity _R ~ _z ~ 30 km/s we have :
  epicycle sizes ~ / ~ 1 kpc
  vertical excursions ~ / ~ 500 pc
  (SHM might not be quite valid for vertical motion)
• for the Sun, W = 7 km/s (z direction), giving the excursions shown here
  Note the sun passes through the disk plane every ~42 Myr
  there is evidence that species extinction and cratering has a similar period (~36Myr)
  tidal perturbations to the Oort cloud may be responsible : (simulation)

(3) Resonances

• As we shall see (§ 4 below) there can be regions of enhanced stellar density in the disk
  these can have the shape of spiral and bar patterns
  these patterns are neither stationary nor move with the disk stars,
  instead they move at some intermediate angular velocity, _p, called the pattern speed
  the interaction of these patterns with the epicyclic motion can lead to resonances :

• First consider a star which orbits at the pattern speed
  we have _* = _p   or   _p - _* = 0
• such stars experience a persistant non-axisymmetric perturbation
  their response can build up (epicycle phase rotates)
  (technically, this is a resonance to azimuthal epicycle perturbation, with natural frequency 0)

• Consider stars which complete exactly 1 epicycle between the passage of each arm
• their interaction with the spiral arm is resonant
  epicyclic amplitude is amplified and wave propagation is strongly modified
• where do such resonances occur in the galaxy ?
  the angular frequency of the star w.r.t. the pattern is _p - _*
  there are two cases :
  -ve if _* > _p ; star moves past arms
  +ve if _p > _* ; arms move past stars
  angular frequency of encountering each arm in an m armed spiral is m(_p - _*)
  so condition for resonance is :
  m(_p - _*) = ±   or   _p - _* = ± / m
  (notice that _p - _* = -½ is a special case of a two armed spiral with _p < _*)
• there are two classes of resonance :
  _p - _* = - / m   :   Inner Lindblad Resonance
  _p - _* = + / m   :   Outer Lindblad Resonance
• to establish the radii of these resonances, one needs to know :
  the pattern speed : _p
  the rotation curve, V_c(R), which gives (R) and (R)
  depending on V_c(R) and _p there may be 0,1,2,.... resonances [ images ]
  (note : there can be 0,1,2 inner Lindblad resonances)

• density waves can only survive inbetween the ILR and OLR
  the forcing function must be slower than to sustain the wave
• density waves cannot pass an ILR
  they are absorbed, like waves on a beach
  important in allowing/preventing propagation across the disk through the center
• orbit shapes change across the resonances :
  forcing below/above natural frequency yields in/out phase response :
  outside ILR, x₁ orbits parallel to bar
  inside ILR, x₂ orbits perpendicular to bar
  outside CR orbits again perpendicular
  cf near nuclear orthogonal bars are common
  cf bars don't extend beyond CR, stop close to it
  bars probably rotate with pattern speed _p ~ (R=CR)
• expect stellar rings to form at CR and OLR
  inner and outer rings do seem to correspond to expected CR/OLR locations
• gas driven inwards to ILR and outwards to ILR
  expect (and find) gas rings/disks/starformation near ILR
• gas only runs into arm at R < CR
  expect dust lanes and star formation ridges to fade out close to CR
• note that for the Milky Way, estimates are (for m=2) :
  ILR at ~3kpc, CR at ~14kpc, and OLR at ~20kpc, with _p ~ 15 km/s/kpc

(4) Density Waves

• In general, orbits are not closed (~1-2 epicycles per orbit)
  
• However, consider a frame which rotates at _g - ½
  • at _g we move with the guide center
    we witness just the epicycle with no orbital motion
  • now go slower by ½ ;
    after 1 epicycle, star moved through half an orbit
    after second epicycle back to start
  • orbit is closed in this frame
  • orbit is ellipse with GC at center
  • in general, - n/m will close after m radial oscillations [ images ]
• Now consider set of orbits whose epicyclic phases vary monotonically with radius
  (ie PA of ellipses rotates with increasing R)
• simple orbit crowding will generate a two arm spiral pattern [image].
• roughly, if - ½ ~ const for all radii, then pattern ~ fixed (in rotating frame)
  in non-rotating frame, pattern rotates at _p = _g - ½ = pattern speed
  for observed V(R), _g - ½ is fairly constant, so pattern is quite long lived
• for ~flat rotation curve, however, pattern speed does vary slowly with R, so spiral winds up
  but slower than "material" windup by factor ~ _p / ~ 0.3
  some improvement, but still not adequate
• However, including self-gravity can yield a different _p which is almost independent of radius
  this is a result from "density wave theory", to which we now turn :

(i) Motivation and Sketch of Approach

The generation of kinematic spirals assumes axisymmetric potential
However, orbit crowding yields a non-axisymmetric spiral perturbation
Star and gas orbits modified by the spiral perturbation
Their new orbits define a new surface density and associated potential
We need to look for a self consistent solution : response to input potential gives same potential
The analysis is difficult (Lin & Shu 1964, 1966 and much subsequent work)

Look for QSSS (Quasi Stationary Sprial Structure) -- ie _p(R) ~ const
Study response of epicycles to periodic forcing function (as they pass the spiral arms)
This is re-cast as waves propagating in a differentially rotating disk
Analytic methods only work for tightly wound spirals (WKB methods)
Derive a dispersion relation : w = f(k) giving


phase velocity = w / k and group velocity dw / dk
(w=frequency, k=wavenumber)
difficulties are encountered near the ILR, OLR, CR, but they are tractable

(ii) Results

• QSSS solutions are found with _p = - (dw/dk) × ½ ~ independent of R
• pitch angles (R) ~ const, yielding logarithmic spirals
• waves sustained if forcing frequency m|_p - (R)| is slower than (R)
  this occurs between ILR and OLR
  note : this region is larger for m=2 than m=4, so two spirals most common
• waves are absorbed at ILR
• wave response is reduced for disks with higher velocity dispersion
  probably need a "cold" component to be continually replenished
• response of gas is non-linear (collisions/shocks) above threshold [ images ] :
  k²F / |² - w²| > 1
  where spiral potential is (R,) = F cos(R + m)  with w = forcing frequency = m( - _p)
• gas runs into itself (cf traffic jams) :
  expect narrow gas features --- as observed
• predict velocity streaming in vicinity of arms :
  calculations (when tweaked) seem to fit well, cf HI study of M81 : [ images ]
• explains the fundamental correlation between Bulge/Disk ratio and Pitch angle in Spirals :
  Roberts, Roberts & Shu (1975) :
  show geometry of density wave and strength of shock depend on :
  (a) galaxy mass/size (~V_rot), and
  (b) central concentration.
  This explains :
  (a) correlation of pitch angle with B/D ratio
  (b) at given stage, pitch angle increases with lower luminosity gals
  (c) absence of dwarf spirals since in low mass galaxies the spiral
  shock is weaker/absent, leading to dwarf irregulars.

  (iii) Remaining Problems with QSSS

  despite much theoretical effort, still unclear if Lin-Shu QSSS important for real galaxies

• astrophysical complexities :
  cold gas only 2-4% of volume (hot gas won't respond)
  ... something else here ?
• unclear whether strength of many observed arms can be achieved
• passage through arms increases dispersion (and therefore Q)
  this will quench the arms and halt the process
  unclear whether creation of new cold component is sufficient to maintain process
• Careful study of ~50 galaxies (Kormendy & Norman) reveal few which may need QSSS :
  • Many galaxies have other causes of density waves (eg tides, bars, ovals -- see below)
  • Many (most) galaxies are flocculent with no obvious global spiral pattern
  • Yet other galaxies (eg late Sc) have large solid body rotation region
    coherent material arms can exist here --- no need for density wave.

There are two other obvious sources of density waves (both m=2)

(i) Tides

• tidal field of passing neighbor creates a strong m=2 perturbation
• early simulations (eg Toomre & Toomre 1972) nicely show strong outer arms
  arms did not, however, go to center (viewgraph)
• later simulations (Toomre 1981) included self-gravity and did better [ images ]
• nowadays, many simulations reveal a strong deep m=2 spiral
• note, however, these spirals are transient

  (ii) Bars and Oval Distortions

• bars are another source of m=2 perturbations [ images ]
• Sanders & Huntley (1976) found a weak bar generates strong spiral arms
• However, needs viscosity (ie dissipation)
  removing the viscosity stops the arm formation
• conclusion : spiral arms only need a bar and dissipation to form

(5) Disk Instabilities and Their Amplification

When are self-gravitating disks vulnerable to local gravitational instabilities ?
Such instabilities, together with differential rotation, may generate spiral structure

• Instabilities can arise from a competition between :
  • gravity causing overdense regions to collapse
  • stellar dispersion which inhibits the collapse
  • angular momentum which inhibits the collapse
• Toomre (1964) found the conditions for instabilities : Q < 1 where
  for gas disk : Q = V_s / ( G )
  for stellar disk : Q = _R / (3.36 G )
  where V_s is the sound speed and is the local surface density

Lets look briefly at the derivation :

(i) Modified Jeans Analysis

Consider overdense region radius R in a non-rotating disk
• the collapse time is t_coll ~ R / V where V ~ gravitational velocity ~ (G M / R)^½
  so t_coll   ~   R / (GM / R)^½ ~ (R³ / GM)^½   ~   (R / G )^½   ( is surface density)
  the time for stars to escape the region is : t_esc   ~   R /   ( is dispersion)
  so collapse occurs if t_coll   <   t_esc   ie (R / G )^½ <   R /

    The critical size for stability due to dispersion is therefore   :   R_J < ² / G



• Now consider a rotating disk
  
  the local angular velocity is Oort's constant B
  the region is stable if F_centrifugal   >   F_gravity
  in this case   R B²   >   GM / R² = G
  

    The critical size for stability due to rotation is therefore   :   R_rot   >   G / B²

• Combining these : the disk is unstable in the range   R_J   <   R   <   R_rot
  and therefore the disk is locally stable if R_J   >   R_rot
  ie   ² / G   >   G / B²   or   B / G   >   1
  recall that B = ² / 4 and ~ 1-2 so B ~ / 3
  

The final condition for disk stability is therefore : Q     / 3 G   >   1

(ii) Q from Disk Dispersion Relations

consider the time development of wave perturbations in a disk :       exp i (k . r - t)
Solve the Collisionless Boltzmann Equation and extract the dispersion relation :

(6.18a)

(6.18b)

where V_s = sound speed, and F is a "reduction factor" (see B&T § 6.2d)
the disk is unstable if ² < 0 for all k, since exp (-i t) = exp ( t) with real
This yields for stability :

Q_gas     V_s / G   >   1
Q_{stars     R} / 3.36 G   >   1
• Factors which promote gravitational instability are :
  low stellar velocity dispersion _R
  high surface mass density
  low epicyclic frequencies
  
  
• Note : even for Q > 1, disks may still be vulnerable to global instabilities (modes)
  
• Solar neighborhood : _R ~ 30 km/s ; ~ 50 M pc^-2 ; ~ 36 km/s/kpc
  these give Q ~ 1.4 and so the MW disk is locally stable near the sun

Some circumstances allow a powerful amplification of spiral patterns (eg Toomre 1981)

(i) The Swing Amplifier

If we have a leading spiral density wave, then

• differential rotation will gradually rotate it into a trailing spiral wave [ images ]
• the rotation of the pattern is retrograde
• the timescale for rotation is ~
  epicyclic motion approximately follows the arm
  long perturbation duration so epicycle amplified
  the emerging trailing pattern is strongly amplified
• amplification gain depends on Q and (radial wavelength of patern) [ images ]
  maximum ~10² when ~ 1.5 _crit, where
  _crit = 4 G / ² = shortest normally stabilized by rotation
  for > 2 _crit less effective; stabilized by rotation
• just disk noise has all present
  ~ 1.5 _crit amplified [ images ]
• as usual, subsequent disk heating raises Q and suppresses the amplification mechanism

  This swing amplification is thought to be very important

  (ii) Feedback for the Amplifier

  For this to work, we need a source of leading spiral waves
  however, these are not normally generated in a rotating disk
  Instead, look for feedback : trailing waves converted into leading waves.
  • waves reflected from outer edge experience 180° phase shift (trailing leading)
    unlikely to operate in real galaxies, edges too soft
  • trailing waves passing through the center emerge as leading waves [ images ]
    this can only occur when we have no ILR (which blocks wave passage)

  Swing amplification with feedback is probably very important in maintaining strong sprial structure.

• N-Body simulations of disks seem to form bars remarkably easily
  Indeed, it is difficult to devise stable disk models even with Q > 1
  Reality of this bar instability has been verified using analytic methods
  Here is an N-Body simulation showing bar formation :     [image]
      multi-arm spirals form first, then two arms, then a bar
      the bar lasts until the end of the simulation
  the instability can be suppressed with high dispersion (as for spiral density waves)
  

• Swing Amplification helps explain the instability :
  recall : leading waves are strongly amplified into trailing ones
  nothing happens unless there is a source of leading waves
  trailing waves pass through center and emerge as leading
  hence feedback keeps the amplifier going
  bar grows quickly.

• recall, waves cannot pass an ILR where - _b = - /2
  so for feedback to work : need _b > - /2 everywhere
  simulations verify this and "confirm" central tunneling as a source of feedback

• Equivalent description involves epicyclic response to forcing function
  if - _b < - /2 then the forcing frequency is greater than the epicyclic frequency
  response is out of phase by
  feeds x₁ orbits which extend along the bar
  bar can grow by accululating stars in x₁ orbits

  However, if ILR exists, - _b < /2
  response is in phase
  favours x₂ orbits which are perpendicular to the bar
  weakens the bar
  Likewise, outside co-rotation, - _b < /2
  x₁ orbits again suppressed while x₂ orbits reinforced
  explains why bars rarely extend beyond CR

• Early work (Hohl 1971, Ostriker & Peebles 1973) noted the severity of the bar instability
  as usual, increasing stellar dispersion can calm the instability
  they found disks were stabilized against bar formation for KE() / KE(rot) > 5
  Note that for the MW disk near the sun, KE()/KE(rot) ~ 0.15
  so our disk should be massively unstable to bar formation !

• What might suppress the bar instability in real galaxies ?
  there are several likely mechanisms :
  • put mass in a dark halo : this acts like a high dispersion component
    probably more of historical interest :
    back in 1973, any evidence for a dark halo was promoted
    however : dynamics of the inner regions are not influenced much by the halo
    the halo may nevertheless help stabilize disks at larger radii
  • achieve the same by having a high dispersion bulge or inner disk
    ingoing waves are damped before they pass though the center : cuts feedback
  • an ILR will shield the center and cut the feedback :
    a large central bulge mass will yield an (R) with an ILR

Recall that many spirals do not have strong density waves [image]
these may have a different cause of spiral arms :
patchy star formation smeared by differential rotation
dynamic process : arms come and go

Two processes are discussed :

(i) Local Disk Instability

for disks with Q_gas 1, gravity clumping will occur
for Q_gas ~ 1 the most unstable wavelength is ~2 G _gas/² ~ 200 pc
expected feature scale size ~ 0.2 kpc, as observed
this may be self-regulating : SN 'heats' gas; increases _gas and Q_gas stability

(ii) Self-Propagating Star Formation

Idea introduced by Mueller and Arnett (1976) :
SN & winds from star formation trigger further adjacent star formation
region is stretched by differential rotation spiral like patterns
for coherent spiral detonation wave, can yield fixed pattern
Problems : models need fine (~10%) tuning to get good arms