Space Debris: from LEO to GEO Anne LEMAITRE naXys - University of Namur
Plan Space debris problematic Forces Gravitational resonances Solar radiation pressure (SRP) Shadowing effects Lunisolar resonances Numerical integrations Chaos Atmospheric drag Other aspects : rotation, Yarkovsky, synthetic population Post-doc : Deleflie and Casanova, and Phd : Valk, Delsate, Hubaux, Petit and Murawiecka Anne LEMAITRE Space debris
Number of debris chart-historical-debris-growth.jpg 572 × 369 pixels Anne LEMAITRE Space debris
Number of debris Anne LEMAITRE Space debris
Chinese explosion : Fengyun 2007 New York Times Explosion of the chinese satellite Fengyun FY-1C on January 11, 2007 Anne LEMAITRE Space debris
Recent collision : Cosmos - Iridium 2009 Collision • Iridium 33 (active American telecommunication satellite) • Cosmos 2251 (non active military Russian satellite) • Date : February 10, 2009 • Speed : 11.7 km/second Anne LEMAITRE Space debris
What are Orbital Space Debris? Definition Orbital debris refers to material on orbit resulting from space missions but no longer serving any function. Launch vehicle upper stages Abandoned satellites Lens caps Momentum flywheels Core of nuclear reactors Objects breakup Paint flakes Solid-fuel fragments Anne LEMAITRE Space debris
Current debris population There are about 18 000 objects larger than 10 cm TLE Catalogue About 350 000 objects larger than 1 cm More than 3 × 10 8 objects larger than 1 mm Catalogued objects (NASA) 6 % Operational spacecrafts 24% Non-operational spacecrafts 17% Upper stages of rockets 13% Mission related debris 40% Debris mostly generated by explosions & collisions Anne LEMAITRE Space debris
Computer generated images Figure: LEO image Figure: GEO image
LEO-MEO-GEO EARTH ENVIRONMENT resonance between the orbital period of the satellite and the rotation of the Earth = gravitational resonance RES 1:2 RES 1:1 LEO MEO GEO 400 km 20 000 km 36 000 km METEOSAT ISS GPS Anne LEMAITRE Space debris
Rossi et al (2005) Number of debris LEO 3 PEAKS IN DENSITY MOST POPULAR ORBITS − 3 10 − 4 10 MEO GEO > 1 mm − 5 10 − 6 10 Density [objects/km 3 ] − 7 10 − 8 10 > 1 cm − 9 10 − 10 10 > 10 cm − 11 10 − 12 10 0 0.5 1 1.5 2 2.5 3 3.5 4 Altitude [km] 4 x 10 Rossi et al., Proceedings of the IAU Colloquium, No. 197, 2005 and MASTER 2009 Anne LEMAITRE Space debris
Problematic situation Situation and solutions Size (r) Characteristics Protection Number cumulative effects r < 0.01 cm not necessary surface erosion significant damages 0.01 < r < 1 cm armor plating 170 000 000 objects perforation 1 < r < 10 cm important damages no solution 670 000 objects catastrophic events r > 10 cm manoeuvres < 20 000 objects catalogued (TLE) Anne LEMAITRE Space debris
Analogy with natural bodies Natural and artificial objects Natural Artificial Debris existing orbits chosen orbits existing orbits no control control no control long times short times long times model and huge numerical model and observations integrations observations stability precision stability Anne LEMAITRE Space debris
Natural cleaning Long term dynamics ESTIMATION OF LIFETIMES FOR USUAL OBJECTS 300 km 1 month 400 km 1 year 500 km 10 years 700 km 50 years 900 km 1 century 1200 km 1 millennium Anne LEMAITRE Space debris
The forces for MEO and GEO Order of magnitude of the perturbations 10 -2 GEO GM 10 -4 J 2 10 -6 A/m 40 m 2 /kg A/m 10 m 2 /kg Acceleration [km/s 2 ] 10 -8 A/m 1 m 2 /kg Sun Moon J 22 10 -10 A/m 0.01 m 2 /kg J 3 10 -12 10 -14 Jupiter 10 -16 10000 15000 20000 25000 30000 35000 40000 45000 50000 Distance from the Earth’s center [km] Anne LEMAITRE Space debris
The forces First contribution : forces Dynamics of a debris = Keplerian orbit around the Earth + rotation of the Earth + shape of the Earth (geopotential - J 2 ) + third body perturbations (Moon and Sun) + solar radiation pressure + shadowing effects + atmospheric drag (LEO) : cleaner Anne LEMAITRE Space debris
The Hamiltonian formulation Hamiltonian formalism H deb (v, Λ ,r, θ ) = H kep (v,r) + H rot ( Λ ) + H geo (r, θ ) + H 3b (r) + H srp (r) Anne LEMAITRE Space debris
The geopotential � ρ ( r p ) U ( r ) = µ � r − r p � dV , µ = G m ⊕ V x = r cos φ cos λ x p = r p cos φ p cos λ p = r cos φ sin λ = r p cos φ p sin λ p y y p z = r sin φ z p = r p sin φ p � R e � n n ∞ � � U ( r , λ, φ ) = − µ P m n ( sin φ )( C nm cos m λ + S nm sin m λ ) r r n = 0 m = 0 R e : the equatorial Earth’s radius � � r p � n 2 − δ 0 m ( n − m )! P m C nm = n ( sin φ p ) cos ( m λ p ) ρ ( r p ) dV M ⊕ ( n + m )! R e V � r p � n � 2 − δ 0 m ( n − m )! P m S nm = n ( sin φ p ) sin ( m λ p ) ρ ( r p ) dV M ⊕ ( n + m )! R e V Anne LEMAITRE Space debris
The geopotential J 2 = − C 20 = 2 C − B − A B − A and C 22 = 2 M ⊕ R 2 4 M ⊕ R 2 e e � R e � n n ∞ � � U ( r , λ, φ ) = − µ r + µ P m n ( sin φ ) J nm cos m ( λ − λ nm ) r r n = 2 m = 0 C nm = − J nm cos ( m λ nm ) S nm = − J nm sin ( m λ nm ) � C 2 nm + S 2 J nm = nm � − S nm � m λ nm = . arctan − C nm Anne LEMAITRE Space debris
The geopotential: Kaula formulation � R e � n ∞ n n + ∞ � � � � U = − µ µ r − F nmp ( i ) G npq ( e ) S nmpq (Ω , ω, M , θ ) a a n = 2 m = 0 p = 0 q = −∞ � + C nm � n − m even S nmpq (Ω , ω, M , θ ) = cos Θ nmpq (Ω , ω, M , θ ) − S nm n − m odd � � n − m even + S nm + sin Θ nmpq (Ω , ω, M , θ ) + C nm n − m odd Kaula gravitational argument , θ the sidereal time : Θ nmpq (Ω , ω, M , θ ) = ( n − 2 p ) ω + ( n − 2 p + q ) M + m (Ω − θ ) Anne LEMAITRE Space debris
The luni-solar perturbations The acceleration : � � r − r i r i ¨ r = − µ i � r − r i � 3 + . � r i � 3 The potential (i=1 for the Sun, i=2 for the Moon): � � � r − r i � − � r . r i � 1 R i = µ i . � r i � 3 � r � n � R i = µ i P n ( cos ψ ) r i r i n ≥ 2 r i the geocentric distance ψ the geocentric angle between the third body and the satellite P n the Legendre polynomial of degree n . Anne LEMAITRE Space debris
The luni-solar perturbations The three components ( x , y , z ) of the position vector r expressed in Keplerian elements ( a , e , i , Ω , ω, f ) The Cartesian coordinates X i , Y i and Z i of the unit vector pointing towards the third body. Usual developments of f and r a in series of e , sin i 2 and M � a � n � � | l | + 2 j 3 + ∞ � � R i = µ i sin i A ( n ) k , l , j 1 , j 2 , j 3 ( X i , Y i , Z i ) e | k | + 2 j 2 cos Φ r i r i 2 n = 2 k , l , j 1 , j 2 , j 3 Φ = j 1 λ + j 2 ̟ + j 3 Ω , λ = M + ω + Ω , ̟ = ω + M Anne LEMAITRE Space debris
Poincaré variables Delaunay canonical momenta associated with λ , ̟ and Ω : � � L = √ µ a , µ a ( 1 − e 2 ) , µ a ( 1 − e 2 ) cos i G = H = Non singular Delaunay elements, keeping L and λ : P = L − G p = − ω − Ω Q = G − H q = − Ω Poincaré variables : √ √ x 1 = 2 P sin p x 4 = 2 P cos p √ √ x 2 = x 5 = 2 Q sin q 2 Q cos q x 3 = λ = M + Ω + ω x 6 = L Anne LEMAITRE Space debris
Dimensionless Poincaré variables � � 2 P 2 Q U = V = L L � � 1 1 − U 2 2 = U − 1 1 8 U 3 − 128 U 5 + O ( U 7 ) e = U 4 � � − 1 1 − U 2 2 sin i 2 = V + 1 4 VU 2 + 3 32 VU 4 + O ( U 6 ) 2 = V 2 Non canonical dimensionless cartesian coordinates ξ 1 = U sin p η 1 = U cos p ξ 2 = V sin q η 2 = V cos q Anne LEMAITRE Space debris
Hamiltonian nmax 2 R ( n ) H 2 b + ˙ � � H pot = θ Λ + pot + H i n = 2 i = 1 nmax Nn µ 2 1 A ( n ) ( ξ 1 , η 1 , ξ 2 , η 2 ) B ( n ) 2 L 2 + ˙ � � = − θ Λ + ( λ, θ ) j j L 2 n + 2 n = 2 j = 1 nmax Nn 2 L 2 n C ( n ) ( ξ 1 , η 1 , ξ 2 , η 2 , X i , Y i , Z i ) D ( n ) � � � + ( λ ) r n + 1 j j i = 1 n = 2 j = 1 i Dynamical system ξ i = 1 ∂ H η i = − 1 ∂ H ˙ ˙ i = 1 , 2 L ∂η i L ∂ξ i � 2 � 2 � � λ = ∂ H ∂ L − 1 ∂ H ∂ H L = − ∂ H ˙ ˙ ξ i + η i 2 L ∂ξ i ∂η i ∂λ i = 1 i = 1 Anne LEMAITRE Space debris
Semi-analytical averaged method Use of a series manipulator λ θ ξ 1 η 1 ξ 2 η 2 L X Y Z r X ⊙ Y ⊙ Z ⊙ r ⊙ Coefficient cos (0 0) (0 0 0 0 -6 0 0 0 0 0 0 0 0) 0.12386619D-04 cos (0 0) (0 0 0 2 -6 0 0 0 0 0 0 0 0) -0.18579928D-04 cos (0 0) (0 0 0 4 -6 0 0 0 0 0 0 0 0) 0.46449822D-05 Averaging process over the fast variable : λ Semi-analytical averaged solution Anne LEMAITRE Space debris
Number of terms Perturbation Number of terms n -order expansion ξ i 1 1 η i 2 i 3 2 η i 4 1 ξ 2 with i 1 + i 2 + i 3 + i 4 ≤ n n = 2 n = 4 n = 6 n = 8 Geopotential H J 2 5 15 31 53 (33) (145) (410) (895) External Body - Sun & Moon up to degree 2 27 86 197 390 (205) (836) (2374) (5480) up to degree 3 73 250 611 1227 (645) (2642) (7854) (18380) See also STELA (Deleflie - CNRS) Anne LEMAITRE Space debris
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