Orbital Mechanics Vladimír Kutiš, Pavol Valko Space for Education, Education for Space ESA Contract No. 4000117400/16NL/NDe Specialized lectures Orbital Mechanics Space for Education, Education for Space Contents 1. The two body problem 2. Orbits in three dimensions 3. Orbital perturbations 4. Orbital maneuvers Orbital Mechanics 2 Space for Education, Education for Space 1. The two body problem • Motion in inertial frame • Relative motion • Angular momentum • Solution of problem • Energy law • Trajectories • Time and position Orbital Mechanics 3 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: – Newton’s law of gravitation position of masses gravitational forces m r 2 F 21 m1m2 F F21 F12 G 3 r m1 12 r universal gravitational 11 3 2 constant G 6.674210 m / kg s Orbital Mechanics 4 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: – Newton’s law of gravitation position of masses gravitational forces m r 2 F 21 m1m2 F F21 F12 G 3 r m1 12 r 1st time measured by Cavendish, 1798 universal gravitational 11 3 2 constant G 6.674210 m / kg s Orbital Mechanics 5 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation E G 1 2 p r conservative force can be expressed by potential energy position of masses gravitational forces m r 2 F 21 m1m2 F F21 F12 G 3 r m1 12 r 1st time measured by Cavendish, 1798 universal gravitational 11 3 2 constant G 6.674210 m / kg s Orbital Mechanics 6 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: – Newton’s law of gravitation Central body [m3/s2] 14 Earth 3.98600441 x 10 Moon 4.90279888 x 1012 Mars 4.2871 x 1013 m m Sun 1.327124 x 1020 F F G 1 2 r 21 12 r 3 can be measured with m considerable precision by F F 2 r astronomical observation 21 12 r 3 Orbital Mechanics 7 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: – Newton’s law of gravitation m m F F G 1 2 r 21 12 r 3 2 m r g G E g E 2 0 r rE z Orbital Mechanics 8 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: – Newton’s law of gravitation Orbital Mechanics 9 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: – Newton’s law of gravitation Orbital Mechanics 10 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion inertial frame of r m2 reference m2 R2 F21 R2 m1 R 1 m1R1 F12 Orbital Mechanics 11 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion inertial frame of r m2 reference m2 R2 F21 R2 m1 R 1 m1R1 F12 Orbital Mechanics 12 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion inertial frame of r m2 reference m2 R2 F21 R2 m1 R G R 1 m1R1 m2 R2 RG m1R1 F12 m1 m2 center of mass Orbital Mechanics 13 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion inertial frame of r m2 reference m2 R2 F21 R2 2 x time m1R1 m2 R2 m1 R R derivative G G m1 m2 R 1 m1R1 m2 R2 RG m1R1 F12 m1 m2 center of mass Orbital Mechanics 14 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion inertial frame of r m2 reference m2 R2 F21 R2 2 x time m1R1 m2 R2 m1 R R derivative G G m1 m2 R center of mass is: 1 m1R1 m2 R2 • motionless RmG R F • or motion is in 1 1 12 RG 0 m1 m2 straight line with center of mass constant velocity Orbital Mechanics 15 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion m2 R2 F21 m1R1 m2 R2 RG m1 m2 center of mass is: • motionless m R F • or motion is in 1 1 12 RG 0 straight line with constant velocity Orbital Mechanics 16 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion m2 R2 F21 m1R1 m2 R2 RG m1 m2 center of mass is: • motionless m R F • or motion is in 1 1 12 RG 0 straight line with constant velocity Orbital Mechanics 17 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion inertial frame of reference r m2 k m m m m R m R mG2 R2 1 F2 21r m R G 1 2 r m j 2 1 1 3 2 2 3 1 r r i R 1 m1R1 F12 Orbital Mechanics 18 Space for Education, Education for Space 1. The two body problem Motion in inertial frame • Two body problem can by defined by: m m – Newton’s law of gravitation F F G 1 2 r 21 12 r 3 – Newton's laws of motion inertial frame of reference r m2 k m m m m R m2 R2 1 F2 21 1 2 2 m1R1 G 3 r m2 R2 G 3 r m1 j r modification of r i equations R gravitational parameter 1 Gm1 m2 m R F 1 1if: m112 m2 r r 0 r 3 Gm1 Orbital Mechanics 19 Space for Education, Education for Space 1. The two body problem Relative motion • Two body problem can by defined by: – Newton’s law of gravitation – Newton's laws of motion inertial frame of reference k r m2 vector r defined in inertial frame of reference R m j 2 expressed in coord. system i jk 1 i R 1 r (x x )i (y y)j (z z )k 2 1 r2 1 r 0 2 1 r 3 Orbital Mechanics 20 Space for Education, Education for Space 1. The two body problem Relative motion • Two body problem can by defined by: – Newton’s law of gravitation – Newton's laws of motion inertial frame of reference r m2 k vector r can be expressed in coord. system , that rotates about R i1 j1k1 j k21 m1 inertial coord. system with instant angular velocity and instant angular i R acceleration i11 j1 r 3 r 0 r (x21 )i1 (y21 ) j1 r(z21 )k1 Orbital Mechanics 21 Space for Education, Education for Space 1. The two body problem Relative motion • Two body problem can by defined by: – Newton’s law of gravitation – Newton's laws of motion inertial frame of reference k r m2 R k21 m1 j r rrel r r 2vrel i 2 x time derivative in Ri1 j 1 1 inertial frame of reference r 3 r 0 r (x21 )i1 (y21 ) j1 r(z21 )k1 Orbital Mechanics 22 Space for Education, Education for Space 1. The two body problem Relative motion • Two body problem can by defined by: – Newton’s law of gravitation – Newton's laws of motion inertial frame of r r reference rel k r m2 if is not rotating coord. system R i1 j1k1 k21 m1 j r rrel r r 2vrel i 2 x time derivative in Ri1 j 1 1 inertial frame of reference r 3 r 0 r (x21 )i1 (y21 ) j1 r(z21 )k1 Orbital Mechanics 23 Space for Education, Education for Space 1. The two body problem Relative motion • Two body problem can by defined by: – Newton’s law of gravitation – Newton's laws of motion inertial frame of reference r rrel relative acceleration of moving (non-rotating) k r m2 frame of reference in x21 3 x21 coord.