Astrodynamics (AERO0024) The two­body problem Lamberto Dell'Elce Space Structures & Systems Lab (S3L)0 Outline The two­body problem µ rÜ = – r Equations of motion r 3 Resulting orbits 1 Outline The two­body problem µ rÜ = – r Equations of motion r 3 Resulting orbits 2 What is the two­body problem (or Kepler problem)? F 21 F m1 12 m2 Motion of two point masses due to their gravitational interaction 3 Two­body problem vs real world 4 What is the interest in the two­body problem? 5 Gravitational force of a point mass F 21 F m1 12 m2 r Norm: m1 m2 kF 12k = kF 21k = G r 2 Direction: • Along the line joining m1 and m2 • Directed toward the attractor 6 Gravitational constant 7 Gravitational parameter of a Celestial body 8 Satellite laser ranging 9 Satellites as bodies in free fall 10 Is point­mass a good approximation for Earth gravity? 11 Gravitational potential of a uniform sphere 12 Gravitational potential of a spherically­symmetric body 13 Outline The two­body problem µ rÜ = – r Equations of motion r 3 Resulting orbits 14 Dynamics of the two bodies 15 Motion of the center of mass 16 Equations of relative motion Assume m2 m1 17 Equations of relative motion 18 Integrals of motion: The angular momentum 19 Implication: Motion lies in a plane 20 Azimuth component of the velocity 21 Integrals of motion: The eccentricity vector 22 Relative trajectory 23 In summary 24 Outline The two­body problem µ rÜ = – r Equations of motion r 3 Resulting orbits 25 Conic sections in polar coordinates 26 Conic sections 27 Possible trajectories of the two­body problem 28 Digression: Energy of the orbit 29 Digression: Energy of the orbit at periapsis 30 Circular orbits (e = 0) 31 Orbital velocity 32 Orbital period 33 Two important quantities 34 Elliptic orbits (0 < e < 1) 35 Geometry of the elliptic orbit 36 Angular momentum and energy 37 Velocity on an elliptic orbit (vis­viva equation) 38 Kepler's second law 39 Kepler's third law 40 Parabolic orbits (e = 1) 41 Escape velocity 42 Hyperbolic orbits 43 Characteristic energy (c3) and excess of velocity 44 Delta II, Delta III, and Atlas IIIA 45 The two­body problem in a nutshell 46 What is missing? 47 Astrodynamics (AERO0024) The two­body problem Lamberto Dell'Elce Space Structures & Systems Lab (S3L)48 Astrodynamics (AERO0024) Exercise session 1 The two­body problem Lamberto Dell'Elce Space Structures & Systems Lab (S3L)49 Ex. 1: Geometry of the orbit Given: in1 Altitude first obs. in3 Altitude second obs. h2 = 852 [km] h1 = 1545 [km] in2 Anomaly first obs. in4 Anomaly second obs. θ2 = 58 [deg] θ1 = 126 [deg] Find: out1 Eccentricity out2 Perigee altitude out3 Semi­major axis 50 Schematic resolution r' r p θ out1 0.08164 [ ] a e a out2 595.5 [km] F' F out3 7593 [km] p = a 1 − e2 p p r(θ) = r = r(0 deg) = 1 + e cos θ p 1 + e r(θ) + r0(θ) = 2a 8θ 2 [0; 360] deg p r = r(180 deg) = a 1 − e 51 Ex. 2: GTO and GEO 52 Schematic resolution 53 Astrodynamics (AERO0024) Exercise session 1 The two­body problem Lamberto Dell'Elce Space Structures & Systems Lab (S3L)54.