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 and Overview

• 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 :
  Circular Orbits : Simple kinematics in a differentially rotating disk
  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) Circular Rotation: Oort's Constants

• A simple derivation of their functional forms is given here: [image]

  For the radial and tangential velocities, we find:

  (6.1a) (6.1b)

  Where Oort's constants A and B are given by (R₀ = solar radius):

  (6.2a) (6.2b)

  Oort's A expresses local shear, while
  Oort's B expresses local vorticity, ie local rotation:   _loc  =  × V   (curl V)

• Here are some examples of apparent stellar motions for several different rotation curves: [image]

• Current best (Hipparcos) estimates for Oort's A and B are (B&M p642): [image]
      A = 14.8 ± 0.8 km/s/kpc and
      B = -12.4 ± 0.6 km/s/kpc

  Notice their dimensions are velocity gradients which are also frequencies
  E.g. using psm units: A = 0.0148 km/s/pc 0.014 Myr^-1 = 14.8 Gyr^-1

• From their definition some interesting properties of the MW rotation can be measured:

  (6.3a) (6.3b)

  The first of these confirms that the rotation curve is fairly flat near the sun (gently rising).

  The second yields an orbital period for the sun:
      P(R₀) = 2 / (R₀) = 2 / 27.2 Gyr^-1 = 0.23 Gyr = 230 Myr

  And when coupled with an estimate for the galactocentric distance, R₀, yields an orbital velocity:
      V_c(R₀) = 218 (R₀ / 8 kpc) km/s
  Which agrees fairly well with radio VLBI measurements of from proper motion of Sgr A^*
      V_c(R₀) = 241 (R₀ / 8 kpc) km/s

• This analysis assumes the sun and stars are all on circular orbits.
  In truth, this is only approximately true: the stars are in fact perturbed from circular orbits.
  This kind of motion can be analyzed using the concept of epicycles.

(3) 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

• Here's an intuitive way to understand the origin of the epicyclic motion: [image]
  • 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

• An equivalent description considers the coriolis force in a rotating frame: [image]

• For a Keplerian potential, the orbit and epicycle frequencies are the same, _g = _g
  The full orbit is closed: we have an offcentered Keplerian ellipse. [image]
  However, in general _g and are different so orbits don't close.......
  Unless they are observed from frame rotating at _g - ½ :
    orbits then appear closed ellipses, centered on galactic center.

• At this point, lets briefly anticipate the relevance of epicycles for spiral arms:

  • For epicycle phases which vary systematically with radius:
    nested elliptical orbits may crowd in a spiral pattern
    this is called a kinematic density wave [image]
  • 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

• Now let's return to the epicycles, and derive their characteristics :

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

    (6.4)

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

    (6.5)

• 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 :

  (6.6)

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

  (6.7)

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

  (6.8)

• For Milky Way disk near the sun, ² = 4 G _o, which gives ~ 0.072 Myr^-1,
  vertical oscillation period 2 / ~ 87 Myr ~ 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

  (6.9)

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

  (6.10)

• However, since L_z = R² (dot), then this can also be written as

  (6.11)

  Where the effective potential, _eff, allows us to describe the radial motion in 1-D form [image]
  Typically, _eff has a minimum, rising steeply at small R and slowly at large R
  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

  (6.12)

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

  (6.13)

  This gives SHM about the guiding radius  

  (6.14)

  with frequency , where

  (6.15)

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

  (6.16)

  Integration gives :

  (6.17)

• 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

  (6.18)

  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

  (6.19)

  Some properties of this motion are:

  • elliptical epicycle with radial/azimuthal axis ratio = / 2
  • epicyclic motion is retrograde w.r.t. orbit (c.f. Ptolemy's were prograde)

  • For Keplerian potential: R^-3/2 we get =
        epicycle axis ratio 2:1 (cf Ptolemy's were 1:1 circles)
        full orbit is closed ellipse, centered at the ellipse focus [image]
    
  • For flat rotation curve: R^-1 we get = 2
  • For solid body rotation: = const (harmonic potential [example]):
        = 2 giving circular epicycles and closed oval orbits
    

  • In general,   <     <   2   so   >  
        epicycle completed before rotation
        from inertial frame, orbits don't close, but regress

We can express the epicyclic and orbital frequencies at the solar radius, ₀, ₀, in terms of Oort's constants:

(6.20)

(6.21)

• The ratio ₀ / ₀ 1.3
      Solar neighborhood stars make 1.3 epicyclic rotations per orbit. [image]

• The ratio ₀ / 2 ₀ 0.7
      Epicycles have radial/azimuthal extent of ~0.7
      Stars with R_g = R₀ have velocity dispersions _R / = ₀ / 2₀    0.7
      However, at R₀, velocity dispersions are in fact _R / = 2₀ / ₀    1.5
      this is because stars found at R₀ tend to have R_g < R₀ (more stars at smaller radii)

• Epicycle sizes are / , so for _R ~ 30 km/s, we find ~ 1 kpc excursions
  Similarly, for _z ~ 30 km/s and 0.096 Myr^-1 we find vertical excursions ~ 300 pc.
  

• For the sun, z ~ 40pc and W ~ 7 km/s, so we expect modest vertical excursions.
  An amusing (speculative) theory is that disk crossings coincide with terrestrial craterings and/or extinctions
  The higher stellar densities perturb the Oort comet cloud, causing more impacts (!): [image]

(4) Resonances

• As we shall see (§ 5 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, and their response builds up
  

• 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
  The angular frequency of encountering each arm in a 2 armed spiral is 2(_p - _*)
  So the condition for resonance is:
  2(_p - _*) = ±   or   _p - _* = ± / 2     (± / m for an m armed spiral)

• There are two classes of resonance:
  _p - _* = - / 2   :   Inner Lindblad Resonance   (stars move past pattern)
  _p - _* = + / 2   :  Outer Lindblad Resonance   (pattern moves past stars)

• To establish the radii of these resonances, one needs to know: [image]
  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
  (there can also be 0,1,2 inner Lindblad resonances)

• Density waves: (see below)
  Can only survive inbetween the ILR and OLR (where we find arms)
  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:
  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 (as found)
  

• Gas driven inwards to ILR and outwards to ILR
  often find gas rings/disks/starformation near ILR

• 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

(5) Density Waves

• In general, orbits are not closed (~1-2 epicycles per orbit)

• However, they are closed in a frame which rotates at _g - ½ [image]
  In this frame, the orbit is a closed ellipse with the Galactic center at the center.

• Consider set of orbits whose epicyclic phases vary monotonically with radius
  (i.e. PA of ellipses rotates with increasing R)
  Simple orbit crowding will generate a two arm spiral pattern.
  This is called a kinematic density wave [image]

• If - ½ is roughly independent of radius (roughly true for flat rotation curve):
    the pattern is fixed and rotates at _p = _g - ½ = pattern speed
  In fact, for V_c = const, pattern speed varies slowly with R, so spiral slowly winds up

• 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) Sketch of Approach

• The generation of kinematic spirals assumes axisymmetric potential
  However, orbit crowding yields a non-axisymmetric spiral perturbation

  Star (and gas) orbits are 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)
  Considers waves propagating in a differentially rotating disk
  Derive a dispersion relation: = f(k) with phase velocity, /k, and group velocity, d/dk.

  Look for Quasi-Stationary Steady State solution (QSSS).

  (ii) Results

• Solutions are found with _p = - (d/dk) × ½ which is ~ independent of R
    Pitch angles (R) ~ const, yielding logarithmic spirals
    Waves survive between ILR and OLR
    Waves are absorbed at ILR.
    Waves weaker in disks with higher velocity dispersion
        need cold component to be replenished via star-formation (c.f. S0 disks don't have arms).

• Gas response:
    Non-linear, leading to collisions/shocks above a threshold response [image].
    Gas runs into itself (c.f. traffic jams) creating narrow gas features (as observed)
    Predict velocity streaming in vicinity of arms, roughly as found: [image]
    Geometry of density wave & strength of shock depend on central concentration & V_c
  
  explains correlation of pitch angle with bulge/disk ratio
  explains lack of dwarf-spirals: need threshold V_c to form disk and arms.

It is unclear how often the QSSS density wave theory is applicable: (see [o-link])
Some galaxies have spiral arms within the region where V_c r (solid body):
    arms don't wind up in this region.
Many galaxies are flocculent, with no clear density wave pattern (see § 6 below)
Many galaxies have an alternative source of density wave (see next)

There are two other obvious sources of density waves, both are m = 2.

(i) Tides from companions

Tidal field of passing neighbor creates a strong m = 2 perturbation: [image]
This drives a strong kinematic density wave.
Self gravity enhances this perturbation.
However, these spirals are transient.

(ii) Bars and Oval Distortions

Bars are another source of m = 2 perturbations [images]
Sanders & Huntley (1976) find even weak bars can generates strong spiral arms
However, the mechanism needs viscosity (ie gas dissipation) to work.
Oval distortions may have a similar (though weaker) effect.

(6) Disk Instabilities and Their Amplification

For the significant number of flocculent spirals, a different mechanism may be at work [image]
This involves the tendency of stars and gas within a disk to clump up gravitationally and form stars
These clumps then get pulled into spiral arm fragments by differential rotation

• When are self-gravitating disks vulnerable to local gravitational instabilities ?
  

  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 instability: Q < 1 where Q / (3 G )
  Where is the stellar velocity dispersion and is the local surface density

• Here's a simplified derivation based on a 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   [image]
    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

  (6.22)

  [A similar relation for gravitational stability for a gas disk is: Q     V_s / 3 G   >   1 ]

• Factors which promote gravitational instability (i.e. promote spiral structure) are:
  low stellar velocity dispersion
  high surface mass density
  low epicyclic frequencies (and/or low local rotation, i.e. low |B|).
  
  
• Solar neighborhood : ~ 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 [image]
• 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

(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 central regions emerge as leading waves [image]
  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 [image]
  Indeed, it is difficult to devise stable disk models even with Q > 1
  Reality of this bar instability has been verified using analytic methods
  

• 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.

• Early work (Hohl 1971, Ostriker & Peebles 1973) noted the severity of the bar instability
  As you might expect, 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 highly unstable to bar formation!

• What might suppress the bar instability in real galaxies?
  There are several possible mechanisms:
  • Put mass in a dark halo: this acts like a high dispersion component
    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 yields an (R) with an ILR