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
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)
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 ∀θ ∈ [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