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AY202a Galaxies & Dynamics Lecture 6: Galactic Structure, con’t Spirals & Density Waves A Rotation Pattern with Two Inner LB Resonances ΩP Lindblad first noted that for n=1, m=2 (Ω – κ/2) is constant over a large range of radii such that ΩP = Ω – κ/2 and that a pattern could exist and be moderately stable. C.C. Lin computed the response of stars & gas: Assume that the gravitational potential is a superposition of plane waves in the disk: Φ (r,φ,t) = -2πGμ |K| eiK(r,t)(r-r0) uniformly rotating sheet Where K = wave number = 2π/λ and μ = surface density Now find a dispersion relation if μ(r,φ,t) = H(r,t) ei(mφ + f(r,t)) then -2πG Φ(r,φ,t) = |K| H(r,t) e-i(mφ + f(r,t)) Differentiate and find μ(r,φ,t) = iK 2G d Φ(r,φ,t) dr These equations have solutions with a spiral like family of curves m(φ – φ0) = Φ(r) – Φ(r0) e.g. μ = μa(r) ei(mφ - ωt) Note that K < 0 corresponds to Leading Arms K>0 “ “ “ Trailing “ and i (mφ – ωt) = i m(φ – ΩPt) ΩP = ω/m Response of the motions of stars or gas to nonaxisymetric forces F1. F1 is assumed to be periodic in time and angle. • N. Cretton Density Wave Models + Bar Potential With a gas law: a2 = dP/dρ ≈ dP/dμ Sound speed ~ velocity dispersion of the gas in equilibrium We calculate a dispersion relation for the gas (ω – mΩ)2 = κ2 - 2πGμ|K| + K2a2 ω2 = κ2 + K2a2 - 2πG|K|μ F. Shu solved the special case of a flat rotation curve, rΩ(r) = constant = v0 Mass Model μ = v02/2πGr = √2 Ω |K| = 4r [1 ± and the wavenumber m (1 – r/r ) ] 0 √2 where r0 is the co-rotation radius Inner and outer Lindblad resonances are at r = ( 1 ± √2/m) r0 For m = 2, LR are at 0.293r0 + 1.707r0 m= 1, There is no inner LR Response of the Gas depends on a 5 μ/μ0 a = sound speed || 128 1 32 in km/s (Shu etal 1973) 8 t or φ NB For an adiabatic shock, max μ/μ0 = 4 for =5/3 How does over density relate to SFR? Schmidt-Kennicutt Law ΣSFR = (2.5 ±0.7)x10-4 ΣGas 1.4 ± 0.15 ( 1 M☼ pc-2 ) M☼/yr kpc-2 an exponent of ~1.5 is expected for self gravitating disks if SRF scales as the ratio of gas density to free fall time which is proportional to ρ-0.5. This lead Elmegreen and separately Silk to argue for an SFR law where the SFR is related to the gas density over the average orbital timescale: ΣSFR = 0.017 ΣGas ΩG There also appears to be a cutoff at low surface mass gas density: Schmidt-Kennicutt Law vs Elmegreen/Silk Disk Stability Toomre (1964) analyzed the stability of gas (and stars) in disks to local gravitational instabilities. Simply, gravitational collapse occurs if Q < 1. For Gas Q = κ CS / (π G Σ) For Stars Q = κ σR / (3.36 G Σ) where Σ is again the local surface mass density, κ is the local epicyclic frequency, σR is the local stellar velocity dispersion, and CS is the local sound speed Starburst Galaxies Kennicutt ‘06 Normal Disks Kennicutt (1989) rephrased the Toomre argument in terms of a critical surface density, ΣC where ΣC = α κ C / (π G) Q = ΣC / ΣG Where α is a dimensionless constant and C is the velocity disperison of the gas, and ΣG is the gas mass surface density. For this definition of the Q parameter, as before, star formation is also suppressed in regions where Q >> 1 and is vigorous in regions where Q << 1 Some facts about spirals 1. Density waves are found between the ILR and OLR 2. Stellar Rings form at Co-rotation and OLR 3. Bars inside CR, probably rotate at pattern speed 4. Gas rings at ILR For the MW ILR ~ 3 kpc, CR ~ 14 kpc, OLR ~ 20 kpc Interaction induced Spiral Structure = Tides Based on Strong Empirical Evidence for star formation induced by galaxy interactions (Larson & Tinsley 1978) Models now “abundant” --- Toomre2 1970’s, Barnes et al 1980’s, many more today. Bars also act as drivers of density waves 2 Toomre model for the Antennae Toomre2 galaxy.interaction.mpg Bar Driven Density Wave Self Propagating Star formation Mueller & Arnett 1976 Seiden & Gerola 1978, Elmegreens 1980’s+ based on galactic SF observations (e.g. Lada) Seiden & Gerola 1978 Spore Galaxy Rotation Curves MW HI MW Rotation Curve D. Clemens 1985 • Zwicky’s Preface