Orbital dynamics in the tidally-perturbed Kepler and Schwarzschild systems. Sam R. Dolan
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1 Orbital dynamics in the tidally-perturbed Kepler and Schwarzschild systems Sam R. Dolan All Scales, Nottingham, 24th Aug 2015 Sam Dolan (Sheffield) Perturbed dynamics Nottingham 1 / 67
2 work in collaboration with Robert Blaga West University of Timisoara, Romania. (in progress) Sam Dolan (Sheffield) Perturbed dynamics Nottingham 2 / 67
3 Inspirations: (1) The Lidov-Kozai resonance Modern Celestial Mechanics (A Morbedilli, 2011) [An early] Soviet artificial satellite, initially put on a quasi-polar orbit around the Earth, crashed to the ground after a few days because of the Kozai resonance generated by the joint Lunar and Solar perturbations forced its eccentricity to increase so much that the perigee became smaller than the Earth s radius. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 3 / 67
4 Inspirations: (2) Asteroid belts & resonances P Deuar Sam Dolan (Sheffield) Perturbed dynamics Nottingham 4 / 67
5 Inspirations: (3) Black holes, separatrices & chaos L r The black hole and the pea (Cornish & Frankel, 1997) Keplerian systems are impervious to small perturbations. In contrast, black hole space-times are at the edge of chaos, just waiting for the proverbial butterfly to flap its wings. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 5 / 67
6 Overview 1 Introduction Hamiltonian methods Delaunay variables 2 Perturbed Kepler Examples: l = 2 intrinsic & extrinsic Kozai-Lidov resonance Poincaré sections & chaos 3 Perturbed Coulomb 4 Perturbed Schwarzschild Kozai resonance in relativity Tidally-perturbed black hole Chaos 5 Discussion Sam Dolan (Sheffield) Perturbed dynamics Nottingham 6 / 67
7 A simple model Newtonian gravity in vacuum Laplace s equation: 2 V = 0 Expand axisymmetric static potential in multipoles: V ɛ = l=1 V = M r + V ɛ, [ a l (t) r l + b l (t) r (l+1)] P l (cos θ) Extrinsic and intrinsic perturbations of Keplerian potential. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 7 / 67
8 A simple model I will consider two quadrupolar examples: 1 intrinsic bulge : V ɛ = ɛr 3 P 2 (cos θ) 2 extrinsic tidal distortion : V ɛ = ɛr 2 P 2 (cos θ) Small spheroidal perturbation: { ɛ > 0 : oblate ( squashed ) ɛ < 0 : prolate ( stretched ) Sam Dolan (Sheffield) Perturbed dynamics Nottingham 8 / 67
9 Orbital elements Image credit: Lasunncty Sam Dolan (Sheffield) Perturbed dynamics Nottingham 9 / 67
10 Keplerian dynamics Angular momentum L = r p is conserved Laplace-Runge-Lenz (eccentricity) vector e is also conserved: e = 1 M p L ˆr, e L = 0 Five conserved quantities {E, L, L z } and {Ω, ω} Super-integrable system {E, L, L z } may be replaced by geometric quantities: Semi-major axis: a = M/( 2E) Eccentricity: e 2 = 1 L 2 /a Inclination: cos i = L z /L. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 10 / 67
11 Hamiltonian methods Lagrangian L(q, q; t). Euler-Lagrange equations L q = d dt Hamiltonian H(q, p; t) where p L q and Hamilton s equations H = p q L ( ) L q. q = H p, ṗ = H q. Poisson bracket: {A, B} = A B q p A p B q Sam Dolan (Sheffield) Perturbed dynamics Nottingham 11 / 67
12 Hamiltonian methods: properties 1 Hamiltonian flow: F = F (q, p; t). df dt = F F q + q p ṗ + F t = F H q p F H p q + F t F = {F, H} + F t 2 Conservation of H: dh dt = H t Autonomous Hamiltonian H = H(q, p) H is conserved. 3 Conservation of phase-space volume δv (Liouville s theorem): 1 δv ( where F = ( q, ṗ) = H p, H q dδv dt ) = F F = 0. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 12 / 67
13 Canonical transformations Canonical transformation (restricted) A change of variables q Q(q, p), p P(q, p) such that Hamilton s equations keep their form: Q = H P, Ṗ = H Q Hamilton s equations re-expressed in symplectic form: ẋ = A H ( ) ( ) q 0 I x, where x = and A = p I 0 ( ) ( ) Q Q/ q Q/ p Let X = P. Then Ẋ = Jẋ where J = P/ q P/ p Ẋ = JA H x = ( JAJ t) H X Transformation x X is canonical if and only if the Jacobian preserves symplectic form: JAJ t = A. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 13 / 67
14 Generating functions Canonical transformations can be effected by generating functions, e.g., W (q, P ), p = W q, Q = W P For autonomous Hamiltonians, a generating function W is defined by the (restricted) Hamiltonian-Jacobi equation: ( H q, W ) = E q i.e., non-linear ODE for Hamilton s characteristic function W. If the H-J equation has (at least) n 1 constants of integration then the system is integrable. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 14 / 67
15 Kepler Hamiltonian Consider Keplerian Hamiltonian in spherical coordinates: H 0 = 1 2 p2 M r ( (p 2r + 1r2 = 1 2 p 2 θ + p2 φ sin 2 θ )) M r, where p r = ṙ, p θ = r 2 θ, pφ = r 2 sin 2 θ φ 3 constants: L z = p φ, L = p 2 θ + csc2 θp 2 φ and E = H 0. H-J equation is separable: W = W r (r) + W θ (θ) + W φ (φ) 1 2 (W r) [ (W 2r 2 θ ) 2 + csc 2 θ(w φ )2] M r = E Sam Dolan (Sheffield) Perturbed dynamics Nottingham 15 / 67
16 Kepler Hamiltonian Separability leads to p φ = W φ = L z p θ = W θ = L 2 csc 2 θl 2 z p r = W r = ( 2E + 2M/r L 2 /r 2) 1/2 W (q, P) generates a canonical transformation (q, p) (Q, P). p = W q, Q = W P We can now choose the new momenta P i to be the constants α i {E, L, L z }. Or we can choose new momenta to be some functions of these, P = P(α i ). Sam Dolan (Sheffield) Perturbed dynamics Nottingham 16 / 67
17 Kepler Hamiltonian in action-angle variables Choose the momenta to be the action variables defined by: J i 1 p i dq i = 1 W i (q i ; {α j })dq i 2π 2π Then J i = J i (α j ) : no dependence on q j. The conjugate angle variables w i are defined via W by w i = W J i J i (α j ) can be inverted to give α j (J i ), and thus the new Hamiltonian is simply H = E(J i ) Thus J i = E = 0, w i ẇ i = E J i ˆΩ i = const. J i = const. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 17 / 67
18 Kepler Hamiltonian in action-angle variables In the Kepler case, the action integrals have simple closed-form solutions: J φ = 1 L z dφ = L z 2π J θ = 1 L 2π 2 csc 2 θl 2 zdθ = L L z J r = 1 2E M + 2M/r L 2π 2 /r 2 dr = L + 2E M E(J) = 2(J φ + J θ + J φ ) 2 Clearly, the r, θ and φ frequencies are degenerate (equal) Sam Dolan (Sheffield) Perturbed dynamics Nottingham 18 / 67
19 Kepler Action-Angle variables Now introduce alternative action-angle variables: J 1 = J φ, J 2 = J φ + J θ, J 3 = J φ + J θ + J r, M J 1 = L z, J 2 = L, J 3 = 2E H = M 2J3 2. with frequencies 0, 0 and ˆΩ Kepler = ( 2E) 3/2 / M = M 1/2 /a 3/2, w 1 = W L z = Ω w 2 = W L = ω w 3 = W J 3 = M. Here M is the mean anomaly, where M = u e sin u and 1 e tan(u/2) = 1+e tan(ν/2) and u & ν are eccentric & true anomalies Sam Dolan (Sheffield) Perturbed dynamics Nottingham 19 / 67
20 Delaunay variables Delaunay variables are also action-angle variables: J 1 = L z = L cos i, w 1 = Ω, ˆΩ1 = 0, J 2 = L = Ma 1 e 2, w 2 = ω, ˆΩ2 = 0, J 3 = Ma, w 3 = M, ˆΩ3 = M 1/2 /a 3/2. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 20 / 67
21 The perturbed Kepler system Sam Dolan (Sheffield) Perturbed dynamics Nottingham 21 / 67
22 Example #1: Intrinsic quadrupole r 3 Earth has a equatorial bulge, like an oblate spheroid: This causes precession of the equinoxes Moon experiences a quadrupolar perturbation: H ɛ = ɛr 3 P 2 (cos θ). Insert the geometric relation cos θ = sin i sin(ν + ω), H ɛ = 1 4 ɛr 3 (( 1 3 cos 2 i ) 3 sin 2 i cos 2(ν + ω) ) One fast variable w 3 = M t and five slow variables J 1, J 2, J 3, w 1, w 2 Sam Dolan (Sheffield) Perturbed dynamics Nottingham 22 / 67
23 Example #1: Intrinsic quadrupole r 3 Adiabatic approximation Average H ɛ over one cycle of the fast angle w 3 : H ɛ = 1 2π H ɛ dw 3 2π 0 Insert unperturbed (Kepler) values and evaluate integral to get secular Hamiltonian at O(ɛ) H ɛ is an adiabatic invariant Q. Why can we average at the level of the Hamiltonian? A. Because the period T depends only on semi-major axis, so ω = 1 T T 0 H ɛ L dt = L 1 T T 0 H ɛ dt = H ɛ L Sam Dolan (Sheffield) Perturbed dynamics Nottingham 23 / 67
24 Example #1: Intrinsic quadrupole r 3 H ɛ = 1 T = = T 0 H ɛ dt = 1 LT 2π 2π 0 r 2 H ɛ dν ɛ 4L 3 (1 + e cos ν) ( 1 3 cos 2 i 3 sin 2 i cos 2(ν + ω) ) dν T 0 ɛ ( 1 3 cos 2 4L 3 a 3/2 i ) Here we insert the unperturbed relations before evaluating the integral, r = L e cos ν, ν = L r 2, T = 2πa3/2. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 24 / 67
25 Example #1: Intrinsic quadrupole r 3 H ɛ = ɛ 4L 3 a 3/2 ( 1 3 cos 2 i ), cos i = L z /L Applying Hamilton s equations, Ω = H ɛ 3π cos i = ɛ L z T a 2 (1 e 2 ) 2, ω = H ɛ L = ɛ3π(5 cos2 i 1) 2T a 2 (1 e 2 ) 2 ȧ = ė = i = 0, since H ɛ does not depend on ω or Ω. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 25 / 67
26 Example #1: Intrinsic quadrupole r 3 H ɛ = ɛ 4L 3 a 3/2 ( 1 3 cos 2 i ), cos i = L z /L 3π cos i Ω = ɛ a 2 (1 e 2 ) 2, ω = ɛ3π(5 cos2 i 1) 2a 2 (1 e 2 ) 2 Shape and inclination of ellipse is fixed. Orbital plane precesses around symmetry axis at fixed rate. Periastron will advance if i < i c and retreat if i > i c where cos i c = 1 5 i c = At critical inclination i = i c, the r and θ motions are in 1:1 resonance. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 26 / 67
27 Example #2: Extrinsic quadrupole r 2 H ɛ = 1 LT = ɛl7 2T = ɛa2 8 2π 0 2π H ɛ = ɛr 2 P 2 (cos θ) r 2 H ɛ dν 3 sin 2 i sin 2 (ν + ω) 1 0 (1 + e cos ν) 4 dν ( (2 + 3e 2 )(3 cos 2 i 1) + 15e 2 sin 2 i cos(2ω) ) In this case, H ɛ depends on ω too Hence L and i will evolve. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 27 / 67
28 Example #2: Extrinsic quadrupole r 2 Let L = a l, L z = a l z e 2 = 1 l 2 H ɛ = ɛa2 8l 2 ( (5 3l 2 )(3l 2 z l 2 ) + 15(1 l 2 )(l 2 l 2 z) cos(2ω) ) Hamilton s equations Ω = 1 a H ɛ l z, ω = 1 a H ɛ l, l = 1 a H ɛ ω : Ω = 3πɛa3 l z 2l 2 ( 3l (1 l 2 ) cos(2ω) ) ω = 3πɛa3 2l 3 ( 5l 2 z l 4 + 5(l 4 l 2 z) cos(2ω) ) l = 15πɛa3 2l 2 (1 l 2 )(l 2 lz) 2 sin(2ω). Sam Dolan (Sheffield) Perturbed dynamics Nottingham 28 / 67
29 Example #2: Extrinsic quadrupole r 2 Simple 2D dynamical system in l and ω : ω = 3πɛa3 2l 3 ( 5l 2 z l 4 + 5(l 4 l 2 z) cos(2ω) ) l = 15πɛa3 2l 2 (1 l 2 )(l 2 lz) 2 sin(2ω). We may seek fixed points & classify using the Jacobian. A saddle point at l = 1, cos(2ω) = (1 5l2 z ) 5(1 l 2 z) if cos i < 3/5 an unstable inclined circular orbit. Saddle point is associated with a homoclinic orbit (separatrix) Centre at ω = π/2 (or ω = 3π/2) and 3l 4 = 5lz, 2 i.e. 3 cos i = 1 e 2 5 Sam Dolan (Sheffield) Perturbed dynamics Nottingham 29 / 67
30 Example #2: Phase diagram of Kozai resonance 1 - e 2 1 Separatrix between oscillation and libration Separatrix exists if l z < 3/5 Recall l z is conserved : 0.4 l z = 1 e 2 cos i π 4 π 2 3 π 4 π ω Eccentricity can increase significantly Sam Dolan (Sheffield) Perturbed dynamics Nottingham 30 / 67
31 Example #2: Unstable circular orbit θ 3 π 4 π 2 Polar axes Nearly-circular orbit at 3 cos i = 5 becomes highly eccentric π r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 31 / 67
32 Example #2: Unstable circular orbit 1.0 z x 2 + y 2 Same trajectory on Cartesian axes Sam Dolan (Sheffield) Perturbed dynamics Nottingham 32 / 67
33 Example #2: Surface of section ϵ = r p_r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 33 / 67
34 Example #2: Surface of section ϵ = r p_r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 34 / 67
35 Example #2: Surface of section ϵ = r p_r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 35 / 67
36 Example #2: Surface of section ϵ = r p_r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 36 / 67
37 Example #2: Surface of section ϵ = r p_r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 37 / 67
38 Example #2: Surface of section ϵ = r p_r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 38 / 67
39 The perturbed Coulomb system Sam Dolan (Sheffield) Perturbed dynamics Nottingham 39 / 67
40 Perturbed Coulomb problem Chemists can measure the electronic structure of the hydrogen atom in external fields: cf. the quadratic Stark effect. H = 1 2 p2 + V, V = 1 r + V 1 Librational and vibrational modes found in experiments. The Generalized Van de Waals potential is V 1 = γ ( β 2 z 2 + x 2 + y 2) Our extrinsic quadrupole case is γ = 1 2 ɛ and β2 = 2 The GVdW system is integrable for β 2 = 1 4, 1 and 4. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 40 / 67
41 Hydrogen atom in GVdW potential Alhassid et al. (1987): The weakly-perturbed system has an adiabatic invariant Λ = (4 β 2 )e 2 + 5(β 2 1)e 2 z where e is eccentricity (LRL) vector. The adiabatic invariant Λ is equivalent to the averaged Hamiltonian H ɛ Sam Dolan (Sheffield) Perturbed dynamics Nottingham 41 / 67
42 Hydrogen atom in GVdW potential Elipe & Ferrer (1994): types of motion depend on β 2 and σ = l z (recall l z = L z / a = 1 e 2 cos i). Below is the bifurcation diagram: 1 σ II V III IV I Kepler quadrupole case : β 2 = 2 Three integrable cases : β 2 = 1/4, 1 and 4. β 2 Sam Dolan (Sheffield) Perturbed dynamics Nottingham 42 / 67
43 The perturbed Schwarzschild system Sam Dolan (Sheffield) Perturbed dynamics Nottingham 43 / 67
44 Some motivation Black holes admit unstable circular timelike geodesics i.e. homoclinic orbits i.e., there is a separatrix between absorption and scattering Cornish & Frankel (1997) described how, under perturbation, the separatrix acquires a fractal structure, like this: L Q. What happens to the Kozai separatrix? r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 44 / 67
45 Kozai resonance: re-considered GR generates anomalous precession (e.g. Mercury s orbit). Other perturbers (planets) also produce additional precession. Suppose : ω = A ( 1 e 2 5 cos 2 i 5(1 e 2 cos 2 i) cos(2ω) + ɛ 1 δω 1 ) l = B(1 l 2 )(l 2 l 2 z) sin(2ω). Possible fixed points ( ω = 0 = l) to consider: 1 l = 1 (circular) 2 ω = π/2, 3π/2 3 ω = 0, π. δω 1 = 0 (1) & (2) exist if i > i c, giving a saddle point and a centre, respectively. Case (3) does not exist. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 45 / 67
46 Kozai resonance: re-considered ω = A ( 1 e 2 5 cos 2 i 5(1 e 2 cos 2 i) cos(2ω) + ɛ 1 δω 1 ) l = B(1 l 2 )(l 2 l 2 z) sin(2ω). Cases: (2) ω = π/2, 3π/2 (3) ω = 0, π. Case (2) exists if : cos i = 3 5 (1 e2 ) ɛ 1 δω This resonance will disappear as δω/ɛ increases. Case (3) exists if : ɛ 1 δω 1 = 4(1 e 2 ) i.e. a fixed orbit with a 90 -rotated orientation arises, with pericenters in the ecliptic. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 46 / 67
47 Perturbed Schwarzschild system Geodesic motion: Lagrangian / Hamiltonian formulation: L = 1 2 g µνẋ µ ẋ ν H = 1 2 gµν p µ p ν. We studied a model of a tidally-perturbed BH : R Moeckel, A Nonintegrable Model in General Relativity, in Commun. Math. Phys. 150, 415 (1992). Perturbed metric: g tt = f ( 1 ɛr 2 fp 2 (cos θ) ) 1 ( g rr = 1 + ɛr 2 fp 2 (cos θ) ) f ( ( g θθ = r ɛr 2 1 2M ) ) r 2 P 2 (cos θ) g φφ = sin 2 θg θθ. where f = 1 2M/r. Sam Dolan (Sheffield) Perturbed dynamics Nottingham 47 / 67
48 Geodesics in perturbed spacetime Significant recent literature on evolving geodesics on bumpy black holes See e.g. W Schmidt, CQG (2002) Pound & Poisson, PRD 77, (2008) Hughes & Vigeland, PRD 81, (2010) Gair, Flanagan, Drasco, Hinderer, Babak, PRD 83, (2011) Geyer, MSc thesis (2013) Methods : 1 Method of Osculating Orbits 2 Determination of fundamental frequencies / action-angles 3 Direct integration Sam Dolan (Sheffield) Perturbed dynamics Nottingham 48 / 67
49 Geodesics in perturbed spacetime θ Contours of effective potential V eff (r, θ) = 0 where 1 2 (g rrṙ 2 + g θθ θ2 ) = V eff r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 49 / 67
50 Geodesics in perturbed spacetime π 3 π 4 π 2 Unperturbed Schwarzschild π 4 ϵ = Sam Dolan (Sheffield) Perturbed dynamics Nottingham 50 / 67
51 Geodesics in perturbed spacetime π 3 π 4 π 2 Rotational π 4 ϵ = Sam Dolan (Sheffield) Perturbed dynamics Nottingham 51 / 67
52 Geodesics in perturbed spacetime π 3 π 4 π 2 Librational π 4 ϵ = Sam Dolan (Sheffield) Perturbed dynamics Nottingham 52 / 67
53 Geodesics in perturbed spacetime How does the geodesic structure change as we progress into the strong-field regime? Apply a rescaling to effect this change : r r/λ H λh ɛ λ 3 ɛ L z L z /λ 1/2 Let s examine prolate case ɛ > 0 ( stretched along one axis) Sam Dolan (Sheffield) Perturbed dynamics Nottingham 53 / 67
54 Perturbed Schwarzschild: Surface of Section ϵ = r p r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 54 / 67
55 Perturbed Schwarzschild: Surface of Section ϵ = = θ = = r p r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 55 / 67
56 Perturbed Schwarzschild: Surface of Section ϵ = r p r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 56 / 67
57 Perturbed Schwarzschild: Surface of Section ϵ = r p r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 57 / 67
58 Perturbed Schwarzschild: Bifurcations & chaos Now let s try increasing the strength of the perturbation Sam Dolan (Sheffield) Perturbed dynamics Nottingham 58 / 67
59 Perturbed Schwarzschild: Bifurcations & chaos Sam Dolan (Sheffield) Perturbed dynamics Nottingham 59 / 67
60 Perturbed Schwarzschild: Bifurcations & chaos Sam Dolan (Sheffield) Perturbed dynamics Nottingham 60 / 67
61 Perturbed Schwarzschild: Bifurcations & chaos Sam Dolan (Sheffield) Perturbed dynamics Nottingham 61 / 67
62 Perturbed Schwarzschild: Bifurcations & chaos Sam Dolan (Sheffield) Perturbed dynamics Nottingham 62 / 67
63 Perturbed Schwarzschild: Bifurcations & chaos Sam Dolan (Sheffield) Perturbed dynamics Nottingham 63 / 67
64 Perturbed Schwarzschild: Bifurcations & chaos Sam Dolan (Sheffield) Perturbed dynamics Nottingham 64 / 67
65 Perturbed Schwarzschild: Bifurcations & chaos Sam Dolan (Sheffield) Perturbed dynamics Nottingham 65 / 67
66 Sam Dolan (Sheffield) Perturbed dynamics Nottingham 66 / 67
67 Summary Small perturbations can have large cumulative effects As the Kepler system is degenerate (super-integrable), small perturbations can impose new structure (e.g. separatrices) The Kozai resonance is altered (ɛ > 0) or reduced (ɛ < 0) by relativistic effects Chaos is fascinating, beautiful... and potentially relevant (e.g. Hyperion; chaotic mixing) ϵ = r p_r Sam Dolan (Sheffield) Perturbed dynamics Nottingham 67 / 67
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