Space Propulsion - IAP/TU Wien

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Space Propulsion Gravitational trajectories Assumption: conservative 1/r 2 force field F = − dU dr mμ = − 2 r V 2 2 μ − = ε = const. r conservation of total energy; ε is specific total energy; V 2 ⎛ = ⎜ ⎝ dr dt 2 ⎞ ⎟ ⎠ ⎛ + ⎜r ⎝ 2 dθ ⎞ ⎟ dt ⎠ magnitude 2 of velocity in polar coordinates (r, θ) μ − + r 1 2 ⎛ ⎜ ⎝ dr dt 2 ⎞ ⎟ ⎠ + 2 1 ⎛ dθ ⎞ ⎜r ⎟ 2 ⎝ dt ⎠ = ε 2 2 2 h ⎛ dr ⎞ h 2μ differential equ. of ⎜ ⎟ + − = ε 4 2 r ⎝ dθ ⎠ r r trajectory dr dt dr dθ dr ⎛ h = ⎜ dθ dt dθ ⎝ r = 2 ⎞ ⎟ ⎠ dθ V sinφ = = dt r h 2 r dθ = h / r 2 2μ h ε + − r r 2 2 dr subst. r=1/u dθ = − h ε + 2μ. u − h 2 u 2 du θ + C = −h ∫ du 2 ε + 2μ. u − h u 2 general solution
Space Propulsion Gravitational trajectories θ + C = −h ∫ du 2 ε + 2μ. u − h u 2 r = 1/u r = 1− 1+ h 2 2ε. h 2 μ / μ 2 .cos ( θ + C) when θ is counted from minimum r, then cos = -1 From geometry: p r = 1+ ε cosθ Is equation of conical section in polar coordinates (r,θ) when origin is in focal point; p is parameter and ε numerical excentricity of conic section; ε > 1 …hyperbola ε = 1 …parabola ε < 1 …ellipse ε = 0 … circle r = 1+ 1+ h 2 / μ 2ε. h 2 μ 2 .cosθ Trajectories under influence of gravity of the sun are conical sections with the sun in one focal point 1 st Kepler
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Page 17 and 18: Space Propulsion The orbit of a bod
Page 19 and 20: Space Propulsion Elliptical orbits
Page 21 and 22: Space Propulsion Tsiolkovsky equati
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Page 31 and 32: Space Propulsion m i = ρmL + ψ .
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Page 37 and 38: Space Propulsion Example 1: Simple
Page 39 and 40: Space Propulsion Hohmann transfer:
Page 41 and 42: Space Propulsion Velocity increase
Page 43 and 44: Space Propulsion Combined maneuver:
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Space Propulsion Attitude maneuvres
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Space Propulsion Thruster combinati
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Space Propulsion Attitude control t
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Space Propulsion Kinetics for rotat
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Space Propulsion one - axis maneuvr
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Space Propulsion Example 6: One-Axi
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Space Propulsion Example 7: Precess
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limit cycle cont‘d Space Propulsi
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Space Propulsion Example 8: Limit-C
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Space Propulsion dir. of ext. torqu
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Space Propulsion forced limit cycle
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eaction wheel, cont’d Space Propu
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eaction wheel, cont’d Space Propu
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Space Propulsion - IAP/TU Wien

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