Astr 5460     Wed., Sep. 29, 2004
   This week: Finish Fundamental Plane
Spiral Structure
Follow-up WIRO Observing trip
Spiral Structure
(Ch. 5, Combes et al., parts)
  Unless noted, all figs and eqs from Combes et al or Longair.

Observing Project: Reductions+
Raw data to one accessible directory
Copies of logs to everyone
Everyone needs to do basic reductions:
Bias subtraction
Flat fielding
Combination of images and cosmic ray rejection
Flux calibration/image zero points
“Basic” Analyses (on several ellipticals and spirals)
Integrated magnitudes and colors, preferably with errors
Surface Brightness Profiles and fits (e.g., r1/4 law)
“Pretty” three color pictures (presentation is important)
Additional Analyses (should try several of these)
Quantitatively measuring “diskiness/boxiness” of ellipticals (tricky)
Ellipse position angle, ellipticity as function of radius
Inclination angle of an intermediate spiral galaxy
Colors of bulge vs. dust lane across bulge
Colors of spiral arms/H II regions
Anything else clever you can think of – be creative!
Everyone should write up results as a paper using LaTeX with all the five standard parts of a scientific article

Fundamental Plane
Luminosities, surface brightness, central velocity distribution, (and others), are correlated, hence the term “fundamental plane.”  Ellipticals populate a plane in parameter space.  BIG area of research – very useful tool and helps us understand galaxies.
Faber & Jackson (1976) is a classic in this area (you might want to look up and read this one):
L ~ σx where x ≈4
So, get dispersion from spectrum, get luminosity, and with magnitude get distance!

Fundamental Plane
Dressler et al. (1987) include all three of the plane parameters and find a tight relationship:
Can also substitute in a new variable Dn (a diameter chosen to match a surface brightness) which incorporates L and Σ.
Can get distances then to various accuracies.

Fundamental Plane
Three views of the relationship from Inger Jorgensen et al. (1997).  The top is “face-on” and the other two views are projections.
As the relationship involves Luminosity it is a distance indicator.
Physical origin of interest for understanding early type galaxies.

Fundamental Plane
Physical origin:
If Virial Theorem applies, then the FP means that M/L ratio depends on the three variables.
The orientation of the plane in parameter space implies that M/L depends on the mass (M/L ~ L0.2)
Metallicity also?

Spiral Structure of Galaxies
Discuss basics only – too many topics to cover this semester.
Will be covered in more detail in “Stars and the Milky Way” next semester

Spiral Galaxies
Already discussed many observational properties (e.g., frequency, types, luminosity functions, rotation curves, light profiles, masses, etc.).
Now want to spend some time discussing their most notable feature – the spiral arms!  It hasn’t been such an easy thing to understand.  Why not?

Spiral Galaxies
Opening considerations:
Disk Stability Issues.  Most of the mass of the galaxy is in a dark halo, and stars are treated as particles in the overall potential, so should disks even be stable?
Note that nearby interactions are unimportant:  Trelax/tc = (h/R)(N/(8logN)) – “collisionless” if large.  For N = 1011, the ratio of relaxation to crossing time is about 108.

Spiral Galaxies -- Stability
Small Scale Stability
Must not satisfy the Jean’s criteria
Large Scale Stability
Rotational support

Spiral Galaxies -- Stability
Jean’s criteria governs “small scale” collapse due to self-gravitation
Treated as stability vs. perturbation in an infinite homogeneous medium in equilibrium
For fluid with a pressure P = ρ0vs2, then perturbations with λ > vs(G ρ0)-1/2 = λJ are unstable.
Basically, does the disturbance have time to cross in a freefall time, tff = (Gρ)-1/2?  If so, then pressure forces are negligible.
The sound crossing time is r/vs, which gives the λJ criteria
For galaxies, we can write an effective pressure in terms of the velocity dispersion: λ > σ(G ρ0)-1/2 = λJ

Spiral Galaxies -- Stability
Stability due to Rotation – large scales
See Toomre (1964) for details
Considered axisymmetric perterbations
Consider small region of size L on a disk with surface density μ (mass then is ~μL2), rotating at Ω, distance d from the center.  Imagine a perturbation that locally increases the surface density.
Angular momentum conserved, so the angular velocity must also increase, creating a centrifugal force in the rotating frame.  If this force can send the mass back to its original position, system is stable.
Stability:  L > Lcrit = 2Gμ/3Ω2, so requires large scales.  Order of magnitude, Ω2R = V2/R ~ GμR2/R2 = Gμ, so Lcrit ~ R

Spiral Galaxies -- Stability
Stability on all scales
Combine the criteria for large and small scales
Need a minimum velocity dispersion to make critical Jean’s length equal to Lcrit.
Order of magnitude then, write Jean’s length in terms of surface density by equating free fall and crossing times: λJ = σ2/Gμ = Lcrit, or σ=Gμ/Ω
A more rigorous calculation gives σr=3.36Gμ/κ, where κ is the epicyclic frequency (see below)
Ratio of observed radial dispersion to critical dispersion is called Q, and stability requires Q > 1.
Near the Sun, Q is between 1 and 2.

Stellar Orbits
First-order Epicyclic theory (yes, epicycles!)
Just an overview here for context
Work with nearly circular orbits in a flattened axisymmetric gravitational potential.  Write down the equations of motion in polar coordinates, using a Taylor series.
Basically just looking at the linear deviation from circular orbits.  You wind up with equations that solve for the radial excursions and the precession of the orbit in terms of “epicycles” of frequency κ.

Stellar Orbits
κ=2Ω is an ellipse, for instance.

Stellar Orbits
Thinking of rotation curves, the core shows fixed rotation (Ω = constant, V=Ωr, and κ=2Ω)
Differential rotation where V = constant (Ω=V0/r; then κ=√2Ω)
Ratio of κ/Ω stays between 1 and 2 for realistic potentials.

Linblad Resonances
There are resonances of interest between stellar orbits (Ω(r)) and spiral patterns (Ωp).

Density Wave Theory
Grand Design Spirals

Winding Problem
What’s up with these spiral patterns anyway?
Spiral flat rotation curves
Differential rotation
Implication for the persistence of a pattern
Let’s work it out

Stochastic Spirals
AKA “Self-sustaining Star Formation” = SSF or “Contagious” star formation.

Stochastic Spirals
AKA “Self-sustaining Star Formation” = SSF or “Contagious” star formation.

Precession–a clue to Grand Design Spirals?
Epicyclic orbits are basically precessing ellipses.

Precession–a clue to Grand Design Spirals?
Linblad: rate of precession of orbits Ω – κ/2 almost constant with r, so a 2-armed spiral wave would rotate without deformation.

Density Waves
Barred or spiral kinematic waves = density waves.

Trailing and Leading Spirals
Winding direction vs. sense of rotation.

How about the Continuous ISM?

How about the clouds?
Flow of molecular clouds in a spiral potential, treated as a 1D mechanical oscillator.  Clouds accumulate in the potential wells (spiral arms).

Excitation of Spiral Waves by Companion Galaxies

For Next Week
Chapter 7 in Combes et al. on interacting galaxies.
Start on Data Reduction and Analysis
New problem set next week likely.