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Dynamics of Collisionless Systems Summer Semester 2005, ETH Zürich

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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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    HELLENICHELLENIC ASTRONOMICALASTRONOMICAL SOCIETYSOCIETY PROCEEDPROCEEDINGSINGS 6th HELLENIC ASTRONOMICAL CONFERENCE Penteli, Athens, Greece, 15-17 September 2003 Sun and Heliosphere Stars & Stellar Systems Extragalactic Astronomy High Energy Astrophysics Cosmology, Relativity & Relativistic Astrophysics Galactic Dynamics & Chaotic Dynamical Systems Infrastructure of Astronomy in Greece History & Teaching of Astronomy Edited by PAUL G. LASKARIDES ATHENS March 2004 Proceedings of the 6th ASTRONOMICAL CONFERENCE of Hel.A.S. PUBLISHED BY The Hellenic Astronomical Society PRINTED BY The Editing Office of the University of Athens, Greece All rights reserved © 2004 Hellenic Astronomical Society No part of this material, protected by this copyright notice, may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the editor, or the President of Hel.A.S. Hellenic Astronomical Society e-mail: [email protected] ISBN 960-88092-0-7 Hellenic Astronomical Society (Hel.A.S.) Proceedings of the 6th ASTRONOMICAL CONFERENCE 15 – 17 September 2003, Penteli, Athens, Greece Edited by Paul G. Laskarides Athens 2004 COUNCIL OF Hel.A.S. Laskarides Paul, University of Athens (Chairman) Antonopoulou Eugenia, University of Athens (Vice Chairman) Tsinganos Kanaris, University of Athens (Secretary) Theodossiou Efstratios, University of Athens (Treasurer) Plionis Manolis, I.A.A. National Observatory of Athens Hatzidimitriou Despina, University of Crete Moussas Xenophon, University of Athens Scientific Organizing Committee P. Laskarides (Chairman, Univ.of Athens), S. Persides (Thessaloniki U.), C. Goudis (IAA- NOA & Patras U.), N. Kylafis (Crete U.), K. Tsinganos (Athens U.), I. Daglis (NOA Athens), K. Kokkotas(Thessaloniki U.), A.
  • N-Body Simulation of a Clumpy Torus: Application to Active Galactic Nuclei

    N-Body Simulation of a Clumpy Torus: Application to Active Galactic Nuclei

    Mon. Not. R. Astron. Soc. 000, 1–?? (2012) Printed 10 августа 2021 г. (MN LaTEX style file v2.2) N-body simulation of a clumpy torus: application to active galactic nuclei E.Yu. Bannikova1,2⋆, V.G. Vakulik1,2† and A.V. Sergeev1,2 1 Institute of Radio Astronomy of the National Academy of Sciences of Ukraine, Krasnoznamennaya 4, 61022 Kharkov, Ukraine 2 Karazin Kharkov National University, Svobody Sq. 4, 61022 Kharkov, Ukraine Accepted 2012 April 25 . Received 2012 April 12 ; in original form 2012 January 18 ABSTRACT The gravitational properties of a torus are investigated. It is shown that a torus can be formed from test particles orbiting in the gravitational field of a central mass. In this case, a toroidal distribution is achieved because of the significant spread of inclinations and eccentricities of the orbits. To investigate the self-gravity of the torus we consider the N-body problem for a torus located in the gravitational field of a central mass. It is shown that in the equilibrium state the cross-section of the torus is oval with a Gaussian density distribution. The dependence of the obscuring efficiency on torus inclination is found. Key words: gravitation - galaxies: active -galaxies: nuclei - galaxies: Seyfert 1 INTRODUCTION warm component is 3 x 4pc (Raban et al. 2009). Thus, the observations confirmed a substantial geometrical thickness Obscuring dusty tori are important structural elements of of the torus: the minor-to-major radius ratio r is about Active Galactic Nuclei (AGN). In the framework of the 0 0.7. It is suggested that the hot component discovered by unified scheme, the differences in the AGN are explained Jaffe et al.