1	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.
2	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., r^1/4 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
3	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 ~ σ^x where x ≈4 So, get dispersion from spectrum, get luminosity, and with magnitude get distance!
4	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.
5	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.
6	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 ~ L^0.2) Metallicity also?
7	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
8	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?
9	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: T_relax/t_c = (h/R)(N/(8logN)) – “collisionless” if large. For N = 10¹¹, the ratio of relaxation to crossing time is about 10⁸.
10	Spiral Galaxies -- Stability Small Scale Stability Must not satisfy the Jean’s criteria Large Scale Stability Rotational support
11	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 = ρ₀v_s², then perturbations with λ > v_s(G ρ₀)^-1/2 = λ_J are unstable. Basically, does the disturbance have time to cross in a freefall time, t_ff = (Gρ)^-1/2?If so, then pressure forces are negligible. The sound crossing time is r/v_s, which gives the λ_J criteria For galaxies, we can write an effective pressure in terms of the velocity dispersion: λ > σ(G ρ₀)^-1/2 = λ_J
12	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 > L_crit = 2Gμ/3Ω², so requires large scales. Order of magnitude, Ω²R = V²/R ~ GμR²/R² = Gμ, so L_crit ~ R
13	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 L_crit. Order of magnitude then, write Jean’s length in terms of surface density by equating free fall and crossing times: λ_J = σ²/Gμ = L_crit, or σ=Gμ/Ω A more rigorous calculation gives σ_r=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.
14	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.
15	Stellar Orbits κ=2Ω is an ellipse, for instance.
16	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.
17	Linblad Resonances There are resonances of interest between stellar orbits (Ω(r)) and spiral patterns (Ω_p).
18	Density Wave Theory Grand Design Spirals
19	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
20	Stochastic Spirals AKA “Self-sustaining Star Formation” = SSF or “Contagious” star formation.
21	Stochastic Spirals AKA “Self-sustaining Star Formation” = SSF or “Contagious” star formation.
22	Precession–a clue to Grand Design Spirals? Epicyclic orbits are basically precessing ellipses.
23	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.
24	Density Waves Barred or spiral kinematic waves = density waves.
25	Trailing and Leading Spirals Winding direction vs. sense of rotation.
26	How about the Continuous ISM?
27	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).
28	Excitation of Spiral Waves by Companion Galaxies
29	For Next Week Chapter 7 in Combes et al. on interacting galaxies. Start on Data Reduction and Analysis New problem set next week likely.