Stability of selfgravitating systems E.V.Polyachenko Institute of Astronomy RAS, Moscow October 23, 2013
Contents Homogeneous medium uid: Jeans instability stars: Jeans instability, Landau damping, inverse LD Disks (razor-thin) uid uniformly rotating sheet uid and stellar dierentially rotating disks (WKB) Instabilities for grand-design structures bar-mode instability swing amplication matrix methods bar-mode instability as a manifestation of inverse LD buckling instability Spheres stability of isotropic systems f (H) radial orbit instability in anisotropic systems f (H, L) gravitational loss cone instability
Introductory remarks What? Why? Build appropriate models Helps in understanding observations Discover obscured matter Understand the origin of spiral structure, bars, bulges, etc. Restrictions Linear theory Fluid (gas): no dissipation (Euler eq.), barotropic eq. of state Starts: CBE no collisions, t < t relax continuous DF f (x, v, t)
Homogenious systems Innite homogeneous media
Homogenious systems Innite uid medium Qualitative explanation R J vs 2 (1) Gρ 0 Approaches: modal and initial value problem A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a xed phase relation. Fluid: Jeans instability HD eq-s (continuity, Euler, barotropic) Poissoin eq. Jeans swindle ω 2 = v 2 s (k 2 k 2 J ), k2 J 4πGρ 0 v 2 s, λ J = πv 2 s Gρ 0 (2)
Homogenious systems Innite stellar medium Vlasov DR (1938, 1945) Landau DR (1946), Landau contour, Landau damping
Homogenious systems Innite stellar medium Vlasov DR (1938, 1945) Landau DR (1946), Landau contour, Landau damping
Homogenious systems Innite stellar medium Vlasov DR (1938, 1945) Landau DR (1946), Landau contour, Landau damping initial conditions
Homogenious systems Innite stellar medium Vlasov DR (1938, 1945) Landau DR (1946), Landau contour, Landau damping initial conditions Van Kampen modes (1955)
Homogenious systems Innite stellar medium K.M. Case (1959). Conclusion:... while the operator occurring in the linearized equation for... oscillations is not bounded and self-adjoint it does share many of the properties of such operators. In particular the eigenfunctions satisfy analogous completeness and orthogonality relations. The dierence is manifested in that the eigenfunctions must be understood in the sense of the theory of distributions (which is physically reasonable in this case) and that some of the eigenvalues can be complex.
Homogenious systems Innite stellar medium Linear Landau damping t γ 1, ( Φ 1 ) 1/2 γ /k (3)
Homogenious systems Innite stellar medium Linear Landau damping t γ 1, ( Φ 1 ) 1/2 γ /k (3) Non-Jeans Instability: inverse Landau damping (ILD) γ f 0 > 0 (4) u u=ω/k
Homogenious systems Innite stellar medium Linear Landau damping t γ 1, ( Φ 1 ) 1/2 γ /k (3) Non-Jeans Instability: inverse Landau damping (ILD) γ f 0 > 0 (4) u u=ω/k ILD is dierent from Jeans instability
Disks Disks
Disks Innite uid uniformly rotating sheet Rotating frame Ω HD eq-s for motion in a plane, `plane' pressure Jeans swindle perturbations exp(ikx i ωt) ω 2 = 4Ω 2 2πGΣ 0 k + k 2 v 2 s (5) Ωv s GΣ 0 > π 2 (6) Ωv s GΣ 0 > 1.06 (thick uid sheet) (7) Exact DR Instability: Jeans inst. suppressed on large scales by rotation
Disks Fluid and stellar dierentially rotating disks (WKB) Dierential rotation Ω = Ω(R) Gravitational coupling A.Kalnajs, C.C.Lin, A.Toomre 1964-1966: locally tightly wound spiral arms are plane waves decoupling Pitch angles for spiral galaxies of dierent Hubble types (Ma 2002)
Disks Fluid and stellar dierentially rotating disks (WKB) WKB appoximation: kr 1, d dr 1 R (8) Lindblad resonances Re ω m Ω p = Ω(R) ± 1 κ(r) m (9) WKB dispersion relation for uid disk (ω mω(r 0 )) 2 = κ 2 2πGΣ 0 k + v 2 s k2 (10) Lin-Shu-Kalnajs DR for stellar disk (ω mω(r 0 )) 2 = κ 2 2πGΣ 0 k F (11)
Disks Local stability of the disks Radial perturbations m = 0, WKB, local stability: uid: Q = κv s >1 πgσ 0 κσ stars: Q = >1 3.36GΣ 0 Nonradial perturbations m > 0, no WKB, local stability: uid: Q > 1.7 stars: Q > 3.15 (V.Polyachenko, EP, Strelnikov 1997):
Disks Bar-mode instability N-body simulation of bar-mode instability Start movie
Disks Swing amplication Swing amplication Spiral & bar formation CR region spirals high amplication bars low amplication bars lumpy structure (superposition of tightly wound leading and trailing spirals) Feedback through the center
Disks Bar as a normal mode of stellar disk Kuzmin-Toomre SG disk with retrograde orbits Q 1.5 Usual PM planar scheme Gravity softening r 1 (r 2 + ε 2 ) 1/2 Quiet start Athanassoula, Sellwood 1986 Visually, one can distinguish three stages Axisymmetric, t < 120 Bar formation, 120 < t < 150 Non-linear bar, t > 150
Disks Bar as a normal mode of stellar disk Fourier analysis and expansion in logarithmic spirals A m (p, t) = R max 2π Σ(r, ϕ, t) exp( i[mϕ + p ln r])r dr dϕ (12) 0 0 Pitch angle α = arctg(m/p), p is spirality (trailing p > 0, bars p = 0). Bisymmetric perturbations: m = 2. Jalali, Hunter, 2005 (JH05)
Disks Bar as a normal mode of stellar disk Exponential growth of the amplitude in the linear regime Γ(t) = max ln A 2 (p, t) p
Disks Bar as a normal mode of stellar disk Constant pattern speed in the linear regime Ω ε p = 1 m dφ dt
Disks Bar as a normal mode of stellar disk Extrapolation to ε 0 and comparison with JH05 ε 0.20 0.15 0.10 0.07 0.05 0 JH1 JH2 Ω p 0.229 0.267 0.314 0.344 0.366 0.428 0.445 0.294 γ 0.076 0.110 0.161 0.196 0.229 0.312 0.308 0.109
Disks Bar as a normal mode of stellar disk Thus, the detected mode is the most unstable mode found by JH05, i.e. the mode with the largest growth rate
Disks Bar as a normal mode of stellar disk Power spectrum spiralitypattern speed P s t (p, Ω p ) = 1 2π t+s Power spectrum radiuspattern speed R s t(r, Ω p ) = 1 2π t t+s t A 2 (p, t ) exp Γ(t ) eimω pt s H Ã 2 (R, t ) exp Γ(t ) eimω pt s H t (t )dt t (t )dt 2 2
Disks Bar as a normal mode of stellar disk Time evolution of spiralitypattern speed spectrum Start movie
Disks Matrix methods Kalnajs matrix method (1971, 1977) Linear matrix method (EP 2004, 2005) Petrov-Galerkin formulation (M.A.Jalali 2007)
Disks Bar-mode instability as a manifestation of inverse LD L m m n (1) f n (J) 2 dj f (n) 0 (J) Lynden-Bell & Kalnajs, 1972 = 0, f (n) 0 (J) n f 0 J 1 + m f 0 J 2
Disks Buckling instability Karlsrud et al. (1971) ω 2 = ν 2 + 2πGΣ k σ 2 k 2 ω 2 = κ 2 2πGΣ k + σ 2 k 2 (Bending) (WKB) Polyachenko & Shukhman (1977) ν = 0 3 α 2 < kh 1 α α σ σ z α c = 2.7 kh = 0.3 2h = R 10
Disks Buckling instability NGC 1381, NIR + isophotos Solar neighbourhood: α = σ/σ z 1.7 stability (Merritt & Sellwood 1994) bars thickening in thin disks thickening of thin dynamically hot stellar systems (elliptical galaxies)
Spheres Spheres
Spheres Stability of isotropic systems Antonov-Lebovitz th. All non-radial modes Y m (θ, ϕ), l > 0 of a l uid barotropic sphere with dp dρ > 0 are stable. Antonov 1-st law A stellar system having an ergodic DF f 0 (H 0 ) with f 0 (H 0) < 0 is stable if the uid barotropic sphere with the same equilibrium density distribution is stable. Antonov 2-nd law All non-radial modes of a stellar system having an ergodic equilibrium DF f 0 (H 0 ) with f 0 (H 0) < 0 are stable. Doremus-Feix-Baumann th. All radial modes of a stellar system with an ergodic DF f 0 (H 0 ) and f 0 (H 0) < 0 are stable.
Spheres Anisotropic systems Doremus-Feix-Baumann ext. All radial modes of a spherical stellar system with a DF f 0 (H 0, L) and f 0 / H 0 < 0 are stable. Non-radial perturbations: radial orbit instability (ROI) gravitational loss cone instability (glci)
Spheres Radial orbit instability Polyachenko & Shukhman (1972): anisotropic Jeans instability Zeldovich, Polyachenko, Fridman & Shukhman (1972) Antonov (1973): purely radial orbits Lynden-Bell (1979): precessing orbits, coalescence and capture Polyachenko & Shukhman (1981): Kalnajs m.m. for spheres, Russian stability criterium χ = 2T r T = 1.7 ± 0.25 Barnes (1985), Merritt & Aguilar (1985)... Perez et al. (1996), Maréchal & Perez (2009)
Spheres Radial orbit instability Kalnajs matrix method for spheres Numerical calculations
Spheres Radial orbit instability Kalnajs matrix method for spheres Numerical calculations
Spheres Radial orbit instability Kalnajs matrix method for spheres Numerical calculations Stability of Idlis models The rst determination of stability boundary, ζ = 2T r T
Spheres Radial orbit instability Precession of orbits and the moment of inertia φ = π+p(e)l+... p(e) <> 0 Ω pr = dφ dt = φ π T r (E) = p(e) T r (E) L d 2 φ dt 2 = p(e) T r (E) N I = T r(e) p(e) Ω pr > 0 growth of the perturbations (instability) Ω pr < 0 weakening of the perturbations (?)
Spheres Gravitational loss cone instability Plasma physics analogy SMBH Mirror trap Loss cone Type of the DF (nearly isotropic, strongly anisotropic) Geometry
Spheres Gravitational loss cone instability Disks are usually unstable Tremaine 2005 Spheres, monotonic DF stable Spheres, anisotropic DF with a loss cone unstable, l 3 EP, V.Polyachenko, Shukhman 2007-2008