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天X物理 Galactic Dynamics: the Tale of Spirals

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天X物理 Galactic Dynamics: the Tale of Spirals &Κ£ Galactic Dynamics: The Tale of Spirals μͬŦ ɖT1Z ̇Zɋ ģ\1ZΟǾ£εĭZĤɊI Outline ZAn introduction of galaxies ZSpiral density wave theory ZNumerical simulations 1pc =3.26 lyrs A part of universe Observations Hubble Tuning-fork Diagram Typical Early-Type galaxies IC 4296(E0) NGC 4365(E3) NGC4564(E6) NGC 4623(E7) NGC 4251(S0) NGC 4340(SB0) Typical Barred Spirals NGC 175 (SBab) NGC 1300 (SBb) NGC 2525(SBc) Disk structures: Bar, Spiral & Ring Bars & Spirals Barred Spiral Galaxy NGC 1300 Bars & Spirals Barred Spiral Galaxy NGC 7479 Type SBb Galaxy in Pegasus with several distant companions A long bar-shaped structure and spiral arms that are not symmetrical are visible in its central region. Bars & Spirals Barred Spiral Galaxy NGC 151 Spiral Structures flocculent grand-design Spiral Structures 9Bh9vxvTv hyBhyhvr 8r hyTv hyv7h rqBhyhIB8 "%$ 8r hySrtvs7h rqBhyh#" # Introduction to wavelet method Wavelet method provides us a multi-scale analysis. It decomposes data at some certain scale into different planes Wj , j=0,1,2,3,4,5,… . When we say wavelet planes 1 to 4, meaning W1+W2+W3+W4. The following is a numerical example: ++ = Wavelet NGC1667(Sc) NGC4321(Sc) NGC 1068 WFPC2 - Wavelet NICMOS - Wavelet What we have observed for galaxies? Bar Spirals (waves) Bars, bars, bars, Oh, bars ZHow important a bar in spiral galaxy? ZWhat is a bar consist of? Bh9vxv6pvr@yyvvphyBhyhH'& ShvsurBh9vxvH'& Rotation Curve of Galaxies NGC 2403 Milky Way Sppyiral Density Wave Theor y Spiral phenomena Spiral phenomena occur in nature, art and galaxies. One-arm spiral θ − Φ(r) = const. Spiral ⊗Miracle Stairs⊘⊛⊘⊛᳜ᅇ᳜ᅇ ๝ᇒ๝ᇒ⦸⦸⦸ٶŀଇť่ٶŀଇť่ٶʘǞƿᣦʘǞƿᣦ⊗⊗⊗่ŀଇť Ἔ̊Ἔ̊⊛⊛⊛ɐвྛ;CୈୡᳶƲɐвྛ;CୈୡᳶƲ ⊗Loretto⊘⊘෱෱෱⊛⊛⊛;Cୈ෱;Cୈ෱ Ἔ̊ɐвྛᾣɐᾣɦᷤྡ ἜἜ⊗⊗Miracle Stairs⊘( x = a cos(t) y = asin(t) z = bt NGC 1300 Waves, waves, waves, Oh! waves What’s the difference between the fllfollow ing waves ? Bridge wave (twist) Coffee wave (material wave) ᅍ඿ḃάື⌒(Starry Night) Thomas Kinkade Paintings spring ុែ Zoom in (ុែ) NGC 1300 The Governing Equations-1 r z θ r = (r,θ, z) v = (R,Θ, Z) The Governing Equations-2 For stars ∂Ψ ∂Ψ Θ ∂Ψ Θ2 ∂Ψ + R + + (a + ) ∂t ∂ r r ∂ ϑ r r ∂R Θ ∂Ψ + (aθ − ) = 0 r ∂Θ where ∂Φ 1 ∂Φ a = − , a = − r ∂ θ ∂θ r z=0 r z=0 The Governing Equations-3 ∂ 2Φ ∂Φ ∂ 2Φ ∂ 2Φ + 1 + 1 + ∂r 2 r ∂r r 2 ∂θ 2 ∂z 2 = π (σ +σ )δ 4 G S g (z) σ = Ψ Θ S mS OO dRd The Governing Equations-4 σ The surface density g must be determined from the gas dynamic equations, which are ∂ρ 1 ∂ 1 ∂ + ()ρru + ()ρv = 0 ∂t r ∂r r ∂θ ∂u ∂u v ∂u v2 1 ∂P ∂Φ + u + − = − − ∂t ∂r r ∂θ r ρ ∂r ∂r ∂v ∂v v ∂v vu 1 1 ∂P 1 ∂Φ + u + + = − − ∂t ∂r r ∂θ r ρ r ∂θ r ∂θ The Govering Equations-5 In the above equations, ρ is the volume density ∞ σ = ρ = ρ g O dz h −∞ which expresses the surface density as the product of average volume density times scale height. The pressure is related to density and temperature. For adiabatic process P = P () ρ. If we are considering polytropic gas, γ P = Kρ γ where K is a constant and is the polytropic index. Spiral Waves The full problem involves both stellar and gas components of the system. This is too complicate. For simplicity, we will just use gas-dynamic equations to describe the system. In other words, we will treat a fluid of stars. In doing so, we will lose some physics near the resonances, but the overall behavior will still be the same. Spiral Waves Since spiral structure is the basic problem for all disc galaxies, we must look for solutions of spiral forms. In general, ~ − mθ + Φ(r) = const. represents a spiral system. When m=2, we have a two- arm spiral system. The pitch angle α between the spiral and an intersecting circle may be determined by Δr tanα = rΔα Pitch Angle The pitch angle α between α the curve and an intersecting Δr circle may be determined by rΔα Δr tanα = rΔα When the pitch angle fixed, there exists logarithmic spiral structures. Spiral Waves ~ By − m θ + Φ ( r ) = const . , we have Δ ~ ' r m − mΔθ + Φ (r)Δr = 0, = ~ Δθ Φ' (r) Δr ~ m tanα = , Φ'(r) = rΔα r tanθ If α is constant, then ~ m Φ(r) = ln()r + C tanα It implies that the spiral is a logarithmic spiral. Spiral Waves Spiral Waves Leading and Trailing. If the galactic rotation is in the increasing in azimuthal direction, the spiral is leading whenever the pitch angle is positive, and the spiral is trailing whenever the pitch angle is negative. Spiral Waves Waves. In order to resolve the Winding Dilemma, the spirals must be waves, swirling around the center in constant angular speed. Thus, we must consider a form []− + Φ~ () e i ϖ t m θ r ϖ = Ω Ω When m p with p the pattern angular speed, the swirling rate of the wave pattern around the center. In a frame rotating at Ω , p we have − ()θ − Ω + Φ~ ()= − ψ + Φ~ () m p t r m r . Spiral Waves The form becomes []− ψ + Φ~ () e i m r The spirals are stationary in the rotating frame. The ppyerturbation density ρ ≈ ρ ()− ψ + Φ~ () 1 ˆ 1 cos m r The density crests are located on ~ − m ψ + Φ ()r = ± 2 n π . Material waves? I m p o s si b Winding dilemma problem l e Spiral Waves Winding Dilemma. Flat rotation curve 225 Ω = = 26.5 km / s − kpc 0 8.5 225 Ω = = 53 km / s − kpc 4.25 Life of a galaxy is about 10^(10) years. Spiral will wind up tightly within a couple of revolutions. Yet all spiral galaxies, including Milky Way, remain a fairly open pattern. Where do bars, spirals and waves live? Disk, Disk, Disk, Oh!, Disk Disk 200 pc 40 kpc Disks Disk observed from various angles Perturbation to the gasdynamic equations The system of equation in gas-dynamics approach are: ∂ σ 1 ∂ 1 ∂ + ()σ ru + ()σ v = 0 ∂ t r ∂ r r ∂ θ ∂ u ∂ u v ∂ u v 2 1 ∂ P ∂ Φ + u + − = − − ∂ t ∂ r r ∂ θ r σ ∂ r ∂ r ∂ v ∂ v v ∂ v vu 1 1 ∂ P 1 ∂ Φ + u + + = − − ∂ t ∂ r r ∂ θ r σ r ∂ θ r ∂ θ P = K σ r ∂ 2 Φ 1 ∂ Φ 1 ∂ 2 Φ ∂ 2 Φ + + + = 4π G σδ ()z . ∂ r 2 r ∂ r r 2 ∂ θ 2 ∂ z 2 Perturbation to the gasdynamic equations The basic state (stationary) is = = Ω ()σ = σ ()Φ = Φ ( ) u 0 , v r r , 0 r , 0 r , z 1 ∂ P ∂ Φ − r 2 Ω = − 0 − 0 σ ∂ r ∂ r 0 z = 0 ∂ P ∂ σ 0 = a 2 ∂ r ∂ r 1 ∂ Φ ∂ 2 Φ ∂ 2 Φ 0 + 0 + 0 = 4π G σ δ ()z r ∂ r ∂ r 2 ∂ z 2 0 Perturbation to the gasdynamic equations The perturbation state (stationary) is σ = σ + σ = + 0 1 , u 0 u , = Ω + Φ = Φ + Φ v r v 1 , 0 1 ∂σ 1 ∂ 1 ∂ 1 + ()σ ru + ()σ v + rΩ σ = 0 ∂t r ∂r 0 r ∂θ 0 1 1 ∂u ∂u a 2 ∂σ σ ∂σ ∂Φ + Ω − 2Ω v = − 1 + a 2 1 0 − 1 ∂ ∂ 1 σ ∂ σ 2 ∂ ∂ t r 0 r 0 r r ∂v d (rΩ ) ∂v a 2 1 ∂σ 1 ∂Φ 1 + u + Ω + Ω u = − 1 − 1 ∂ ∂θ σ ∂θ ∂θ t dr 0 r r ∂ 2 Φ 1 ∂Φ 1 ∂ 2 Φ ∂ 2 Φ + 1 + 1 + 1 = 4πG σ δ ()z . ∂r 2 r ∂r r 2 ∂θ 2 ∂z 2 1 Perturbation to the gasdynamic equations The spiral form is introduced as ~ ()σ Φ = (σ Φˆ ) i []ϖ t − m θ + Φ ()r 1 , u , v 1 , 1 ˆ , uˆ , vˆ , e ~ Φ ()r = O k ( r ) dr ~ Φ ' (r ) = k is the radial wave number. Perturbation to the gasdynamic equations Equation of continuity: uˆ σ iϖσˆ + σ + ikσ uˆ + 0 ()− im vˆ + Ω ()− im σˆ = 0 0 r 0 r σˆ m (1) ()ϖ − mΩ + ()k − i / r uˆ − vˆ = 0 σ 0 r Equation of motion a 2 C dσ S iϖ uˆ − im Ω uˆ − 2Ω vˆ = − D iσˆk + σˆ 0 T − rk Φˆ σ 0 E dr U C k 1 dσ S (2) i()ϖ − m Ω uˆ − 2Ω vˆ = − a 2 D i + 0 Tσ − ik Φˆ D σ σ T E 0 0 dr U κ 2 a 2 im (3) uˆ + i()ϖ − m Ω vˆ = − im σˆ + Φˆ Ω σ 2 0 r Perturbation to the gasdynamic equations r d κ 2 = 4Ω 2 (1+ Ω) 2Ω dr Perturbation to the gasdynamic equations Poisson equation: d 2 Φˆ ik m 2 (4) − k 2 Φˆ − Φˆ − Φˆ = 4πGσˆδ ()z dz 2 r r 2 m Tightly wound spirals: kr >> 1 >> 1 tan α σˆ (5 ) (ϖ − m Ω ) + kuˆ = 0 σ 0 σˆ ()6 i (ϖ − m Ω )uˆ − 2Ω vˆ = −ia 2 k − ik Φˆ σ 0 κ 2 ()7 uˆ + i()ϖ − m Ω vˆ = 0 2Ω d 2 Φˆ ()8 − k 2 Φˆ = 4πG σˆδ ()z dz 2 Perturbation to the gasdynamic equations These are called the asymptotic equations for large kr >> 1 The approach is also referred as WKBJ analysis, in which we consider ( ) A(r )e if r Where A is a slow varying function of r and f( r) is a rapidly varying function of r.
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