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AY202a Galaxies & Dynamics Lecture 6: Galactic Structure, con’t Spirals & Density Waves. A Rotation Pattern with Two Inner LB Resonances. Ω P. Lindblad first noted that for n=1, m=2
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AY202a Galaxies & DynamicsLecture 6: Galactic Structure, con’tSpirals & Density Waves
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) = eiK(r,t)(r-r0) -2πGμ |K| 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 Φ(r,φ,t) = H(r,t) e-i(mφ + f(r,t)) -2πG |K|
iK d Differentiate and find μ(r,φ,t) = Φ(r,φ,t) 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) dr 2G
Note that K < 0 corresponds to Leading Arms K > 0 “ “ “ Trailing “ and i (mφ – ωt) = im(φ – ΩPt) ΩP = ω/m
Response of the motions of stars or gas to non-axisymetric forces F1. F1 is assumed to be periodic in time and angle.
Sound speed ~ velocity dispersion of the gas in equilibrium With a gas law: a2 = dP/dρ ≈ dP/dμ 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 Ωand the wavenumber • |K| = [ 1 ± (1 – r/r0)] where r0 is the co-rotation radius Inner and outer Lindblad resonances are at r = ( 1 ± √2/m) r0 m 4 r √2
For m = 2, LR are at 0.293r0 + 1.707r0 m= 1, There is no inner LR Response of the Gas depends on a μ/μ0 5 a = sound speed in km/s (Shu etal 1973) || 128 32 8 1 t or φ NB Foran adiabatic shock, max μ/μ0 = 4 for =5/3
ΣGas 1.4 ± 0.15 How does over density relate to SFR? Schmidt-Kennicutt Law ΣSFR = (2.5 ±0.7)x10-4 ( ) 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: 1 M☼ pc-2
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 = κσR / (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 = ακ 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 • Density waves are found between the ILR and OLR • Stellar Rings form at Co-rotation and OLR • Bars inside CR, probably rotate at pattern speed • 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
Toomre2 model for the Antennae
Toomre2 galaxy.interaction.mpg
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
