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Dynamics of Collisionless Systems Summer Semester 2005, ETH Zürich Dynamics of Collisionless Systems Summer Semester 2005, ETH Zürich Frank C. van den Bosch Useful Information TEXTBOOK: Galactic Dynamics, Binney & Tremaine Princeton University Press Highly Recommended WEBPAGE: http://www.exp-astro.phys.ethz.ch/ vdbosch/galdyn.html LECTURES: Wed, 14.45-16.30, HPP H2. Lectures will be in English EXERSIZE CLASSES: to be determined HOMEWORK ASSIGNMENTS: every other week EXAM: Verbal (German possible), July/August 2005 GRADING: exam (2=3) plus homework assignments (1=3) TEACHER: Frank van den Bosch ([email protected]), HPT G6 SUBSTITUTE TEACHERS: Peder Norberg ([email protected]), HPF G3.1 Savvas Koushiappas ([email protected]), HPT G3 Outline Lecture 1: Introduction & General Overview Lecture 2: Cancelled Lecture 3: Potential Theory Lecture 4: Orbits I (Introduction to Orbit Theory) Lecture 5: Orbits II (Resonances) Lecture 6: Orbits III (Phase-Space Structure of Orbits) Lecture 7: Equilibrium Systems I (Jeans Equations) Lecture 8: Equilibrium Systems II (Jeans Theorem in Spherical Systems) Lecture 9: Equilibrium Systems III (Jeans Theorem in Spheroidal Systems) Lecture 10: Relaxation & Virialization (Violent Relaxation & Phase Mixing) Lecture 11: Wave Mechanics of Disks (Spiral Structure & Bars) Lecture 12: Collisions between Collisionless Systems (Dynamical Friction) Lecture 13: Kinetic Theory (Fokker-Planck Eq. & Core Collapse) Lecture 14: Cancelled Summary of Vector Calculus I A~ B~ = scalar = A~ B~ cos = A B (summation convention) · j j j j i i A~ B~ = vector = ~e A B (with the Levi-Civita Tensor) × ijk i j k ijk Useful to Remember A~ A~ = 0 × A~ B~ = B~ A~ × − × A~ (A~ B~ ) = 0 · × A~ (B~ C~ ) = B~ (C~ A~) = C~ (A~ B~ ) · × · × · × A~ (B~ C~ ) = B~ (A~ C~ ) C~ (A~ B~ ) × × · − · (A~ B~ ) (C~ D~ ) = (A~ C~ )(B~ D~ ) (A~ D~ )(B~ C~ ) × · × · · − · · ~ =vector operator= ( @ ; @ ; @ ) r @x @y @z ~ S = gradS = vector r ~ A~ = divA~ = scalar r · ~ A~ = curlA~ = vector r × Summary of Vector Calculus II 2 2 2 Laplacian: 2 = ~ ~ = scalar operator = @ + @ + @ r r · r @x2 @y2 @z2 2S = ~ (~ S) = scalar r r · r 2A~ = (~ ~ )A~ = vector r r · r ~ (~ A~) = 2A~ = vector r r · 6 r ~ (~ S) = 0 curl(gradS) = 0 r × r ~ (~ A~) = 0 div(curl A~) = 0 r · r × ~ (~ A~) = ~ (~ A~) 2A~ r × r × r r · − r ~ (ST ) = S ~ T + T ~ S r r r ~ (SA~) = S(~ A~) + A~ ~ S r · r · · r ~ (SA~) = S(~ A~) A~ ~ S r × r × − × r ~ (A~ B~ ) = B~ (~ A~) A~ (~ B~ ) r · × · r × − · r × Integral Theorems I Gradient Theorem: Let γ be a curve running from ~x0 to ~x1, d~l is the directed element of length along γ, and φ(~x) is a scalar field then: ~x1 ~x1 ~ φ d~l = dφ = φ(~x ) φ(~x ) r · 1 − 0 ~x0 ~x0 R R It follows that ~ φ d~l = 0 r · Divergence Theorem (Gauss’H Theorem): Let V be a 3D volume bounded by a 2D surface S, and let A~(~x) be a vector field, then: ~ A~ d3~x = A~ d2S~ V r · S · Curl Theorem (Stokes’R Theorem): LetRS be a 2D surface bounded by a 1D curve γ, and let A~(~x) be a vector field, then: (~ A~) d2S~ = A~ d~l S r × γ · R H Integral Theorems II NOTE: Since a conservative force F~ can always be written as the gradient of a scalar field φ, we have from the gradient theorem that F~ d~l = 0 · From the curl theorem we immediatelH y see that ~ F~ = 0 r × We immediately infer that a conservative force is curl free, and that the amount of work done (dW = F~ d~r) is independent of the path taken. · From the divergence theorem we infer that φ~ A~ d3~x = φA~ d2S~ A~ ~ φ d3x~ V r · S · − V · r R R R which is the three-dimensional analog of integration by parts u dv dx = d(uv) v du dx dx − dx R R R Curvi-Linear Coordinate Systems I In addition to the Cartesian coordinate system (x; y; z), we will often work with cylindrical (R; φ, z) or spherical (r; θ; φ) coordinate systems Let (q1; q2; q3) denote the coordinates of a point in an arbitrary coordinate system, defined by the metric tensor hij . The distance between (q1; q2; q3) and (q1 + dq1; q2 + dq2; q3 + dq3) is 2 ds = hij dqi dqj (summation convention) We will only consider orthogonal systems for which hij = 0 if i = j, so 2 2 2 6 that ds = hi dqi with @~x hi hii = ≡ j @qi j The differential vector is @~x @~x @~x d~x = dq1 + dq2 + dq3 @q1 @q2 @q3 The unit directional vectors are @~x @~x 1 @~x ~ei = = = @qi j @qi j hi @qi 3 so that d~x = hi dqi ~ei and d ~x = h1h2h3dq1dq2dq3. i P Curvi-Linear Coordinate Systems II The gradient: 1 @ ~ = ~ei r hi @qi The divergence: 1 @ @ @ ~ A~ = (h2h3A1) + (h3h1A2) + (h1h2A3) r · h1h2h3 @q1 @q2 @q3 h i The curl (only one component shown): 1 @ @ (~ A~)3 = (h2A2) (h1A1) r × h1h2 @q1 − @q2 h i The Laplacian: 1 @ h h @ @ h h @ @ h h @ 2 = 2 3 + 3 1 + 1 2 r h1h2h3 @q1 h1 @q1 @q2 h2 @q2 @q3 h3 @q3 h i Cylindrical Coordinates For cylindrical coordinates (R; φ, z) we have that x = R cos φ y = R sin φ z = z The scale factors of the metric are: hR = 1 hφ = R hz = 1 and the position vector is ~x = R~eR + z~ez Let A~ = AR~eR + Aφ~eφ + Az~ez an arbitrary vector, then A = A cos φ A sin φ R x − y A = A sin φ + A cos φ φ − x y Az = Az _ Velocity: ~v = R_ ~eR + R~eR + z_~ez = R_ ~eR + Rφ_~eφ + z_~ez Gradient & Laplacian: ~ A~ = 1 @ (RA ) + 1 @Aφ + @Az r · R @R R R @φ @z 2 2 2 = 1 @ R @ + 1 @ + @ r R @R @R R2 @φ2 @z2 Spherical Coordinates For spherical coordinates (r; θ; φ) we have that x = r sin θ cos φ y = r sin θ sin φ z = r cos θ The scale factors of the metric are: hr = 1 hθ = r hφ = r sin θ and the position vector is ~x = r~er Let A~ = Ar~er + Aθ~eθ + Aφ~eφ an arbitrary vector, then Ar = Ax sin θ cos φ + Ay sin θ sin φ + Az cos θ A = A cos θ cos φ + A cos θ sin φ A sin θ θ x y − z A = A sin φ + A cos φ φ − x y _ Velocity: ~v = r_~er + r~er = r_~er + rθ~_eθ + r sin θφ_~eφ Gradient & Laplacian: ~ A~ = 1 @ (r2A ) + 1 @ (sin θA ) + 1 @Aφ r · r2 @r r r sin θ @θ θ r sin θ @φ 2 2 = 1 @ r2 @ + 1 @ sin θ @φ + 1 @ r r2 @r @r r2 sin θ @θ @θ r2 sin2 θ @ 2 Introduction COLLISIONLESS DYNAMICS: The study of the motion of large numbers of point particles orbiting under the influence of their mutual self-gravity EAMPLES OF COLLISIONLESS SYSTEMS Galaxies (ellipticals & disk galaxies) N 106 1011 • ∼ − Globular clusters N 104 106 • ∼ − Galaxy clusters N 102 103 • ∼ − Cold Dark Matter haloes N 1050 • MAIN GOALS • Infer mass distribution from observed kinematics. Comparison with light distribution learn about dark matter and black holes ) • Understand observed structure of galaxies: 1. Galaxies formed this way learn about Galaxy Formation ) 2. Galaxies evolved this way learn about Stability of galaxies ) Newtonian Gravity Newton’s First Law: A body acted on by no forces moves with uniform velocity in a straight line Newton’s Second Law: ~ d~v dp~ (equation of motion) F = m dt = dt Newton’s Third Law: F~ = F~ (action = reaction) ij − ji m m Newton’s Law of Gravity: ~ i j Fij = G ~x ~x 3 (~xi ~xj ) − j i− j j − 11 2 2 9 1 2 1 G = 6:67 10− Nm kg− = 4:3 10− ( km s− ) M − Mpc × × Gravity is a conservative Force. This implies that: scalar field V (~x) (potential energy), so that F~ = ~ V (~x) • 9 −r The total energy E = 1 mv2 + V (~x) is conserved • 2 Gravity is a curl-free field: ~ F~ = 0 • r × Gravity is a central Force. This implies that: The moment about the center vanishes: ~r F~ = 0 • × ~ Angular momentum J~ = m~r ~v is conserved: ( dJ = ~r F~ = 0) • × dt × The Gravitational Potential Potential Energy: F~(~x) = ~ V (~x) −r Gravitational Potential: V (~x) Φ(~x) = m ~ Gravitational Field: ~g(~x) = F (~x) = ~ Φ(~x) m −r From now on F~ is the force per unit mass so that F~(~x) = ~ Φ(~x) −r For a point mass at : GM M x~0 Φ(~x) = ~x ~x − j − 0j 0 ρ(~x ) 3 For a density distribution : 0 ρ(~x) Φ(~x) = G ~x ~x d ~x0 − j − j R The density distribution ρ(~x) and gravitational potential Φ(~x) are related to each other by the Poisson Equation 2Φ = 4πGρ r For ρ = 0 this reduces to the Laplace equation: 2Φ = 0. r see B&T p.31 for derivation of Poisson Equation Gauss’s Theorem & Potential Theorem If we integrate the Poisson Equation, we obtain 4πG ρ d3x~ = 4πGM = 2Φ d3x~ = ~ Φd2~s V r S r R R R Gauss’s Theorem: ~ Φ d2~s = 4πGM S r Gauss’s Theorem statesRthat the integral of the normal component of the gravitational field [~g(~r) = ~ Φ] over any closed surface S is equal to 4πG r times the total mass enclosed by S. cf. Electrostatics: E~ ~n d2~s = Qint S · 0 R For a continuous density distribution ρ(~x) the total potential energy is: 1 3 W = 2 ρ(~x) Φ(~x) d ~x R (see B&T p.33 for derivation) NOTE: Here we follow B&T and use the symbol W instead of V .
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