Basics of Orbital Mechanics II Modeling the Space Environment Manuel Ruiz Delgado European Masters in Aeronautics and Space E.T.S.I. Aerona´uticos Universidad Polite´cnica de Madrid April 2008 BasicsofOrbitalMechanicsII–p.1/24 Basics of Orbital Mechanics II Keplerian and Perturbed Motion Magnitude of the Perturbations Special Perturbations all, numerical Encke’s Method Cowell’s Method General Perturbations some, analytical, approximate Osculating Orbit Variation of Parameters Lagrange Equations potential Gauss Equations potential & not potential General Perturbations: Analytical approx/Semianalytical Numerical Integration BasicsofOrbitalMechanicsII–p.2/24 Keplerian and Perturbed Motion r P P r 1 2 ¨ = G (M + m) + − r 3 m − m 1 2 | | Perturbation Kepler Problem | {z } | {z } m a k r r p r k ¨ = G (M + m) k − r 3 r z }| k { | | M k r r p a ¨ = G (M + m) + p p − r 3 p | | a a r r Usually, ¨ ¨ How small? p k p k | | ≪ | | ⇒ ≃ BasicsofOrbitalMechanicsII–p.3/24 Perturbations (LEO) Accelerations of the Satellite (BC=50) 1e+006 Kepler J 2 C 10000 Shuttle 22 Sun Moon Drag (low) 100 Drag (high) ) 2 P s rad / m ( 1 ISS n o i t a r 0.01 e l e c c A 0.0001 1e−006 1e−008 0 100 200 300 400 500 600 700 800 900 Height (km) BasicsofOrbitalMechanicsII–p.4/24 Perturbations (GEO) Accelerations of the Satellite (BC=50) 1e+006 Kepler J 2 C 10000 22 Sun Moon Drag (low) 100 Drag (high) ) 2 P s rad / m ( 1 n o i t a r 0.01 e GPS l e c GEO c A 0.0001 1e−006 1e−008 0 5000 10000 15000 20000 25000 30000 35000 40000 Height (km) BasicsofOrbitalMechanicsII–p.5/24 Encke’s Method r Compute only the difference δ p e r t u r b r r e r k r p a d ¨ = µ ¨ = µ + k p p − r 3 − r 3 δr k p | | | | r r r r r r δ = δ p k p p − | | ≪ | | r v k M 0 r 0 Epoch n a i r e l p e k BasicsofOrbitalMechanicsII–p.6/24 Encke’s Method r Compute only the difference δ p e r t u r b r r e r k r p a d ¨ = µ ¨ = µ + k p p − r 3 − r 3 δr k p | | | | r r r r r r δ = δ p k p p − | | ≪ | | r r r r r r p k a k v δ¨ = ¨ ¨ = µ + µ + = M 0 p k p − − r 3 r 3 p k r | | | | 0 Epoch n a i r e l p e k BasicsofOrbitalMechanicsII–p.6/24 Encke’s Method r Compute only the difference δ p e r t u r b r r e r k r p a d ¨ = µ ¨ = µ + k p p − r 3 − r 3 δr k p | | | | r r r r r r δ = δ p k p p − | | ≪ | | r r r r r r p k a k v δ¨ = ¨ ¨ = µ + µ + = M 0 p k p − − r 3 r 3 p k r | | | | 0 Epoch n r 3 µ µ a i r r k r a r δ¨ = δ + 1 | | + e l p p p − r 3 r 3 − r 3 e ! k k k p | | | | | | BasicsofOrbitalMechanicsII–p.6/24 Encke’s Method r Compute only the difference δ p e r t u r b r r e r k r p a d ¨ = µ ¨ = µ + k p p − r 3 − r 3 δr k p | | | | r r r r r r δ = δ p k p p − | | ≪ | | r r r r r r p k a k v δ¨ = ¨ ¨ = µ + µ + = M 0 p k p − − r 3 r 3 p k r | | | | 0 Epoch n r 3 µ µ a i r r k r a r δ¨ = δ + 1 | | + e l p p p − r 3 r 3 − r 3 e ! k k k p | | | | | | r 3 2 r r r 3 + 3q + q δ (δ 2 ) k p 1 | | = f(q) = q q = · − − r 3 − − 3 r r 1 + (1 + q) 2 p p p · | | About f(q), cf. Battin, p. 389 and 449 BasicsofOrbitalMechanicsII–p.6/24 Encke’s Method r Compute only the difference δ p e r t u r b r r e r k r p a d ¨ = µ ¨ = µ + k p p − r 3 − r 3 δr k p | | | | r r r r r r δ = δ p k p p − | | ≪ | | r r r r r r p k a k v δ¨ = ¨ ¨ = µ + µ + = M 0 p k p − − r 3 r 3 p k r | | | | 0 Epoch n r 3 µ µ a i r r k r a r δ¨ = δ + 1 | | + e l p p p − r 3 r 3 − r 3 e ! k k k p | | | | | | r 3 2 r r r 3 + 3q + q δ (δ 2 ) k p 1 | | = f(q) = q q = · − − r 3 − − 3 r r 1 + (1 + q) 2 p p p · | | µ µ r r r a δ¨ = δ f(q) + p p − r 3 − r 3 k k | | | | About f(q), cf. Battin, p. 389 and 449 BasicsofOrbitalMechanicsII–p.6/24