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  6.674210 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  6.674210 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  6.674210 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] Earth 3.98600441 x 1014 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 reference    m2  m2 R2  F21 R2   r R 1   m1  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       m1R1  F12 m2 R2  F21 inertial frame of reference  m2 R2   r R1 m1

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       m1R1  F12 m2 R2  F21 inertial frame of reference  m R 2  2 R   G r R   1  m R  m R m 1 1 2 2 1 RG  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       m1R1  F12 m2 R2  F21 inertial frame of reference    m2  R 2 x time m R  m R  2 R  1 1 2 2 R  derivative G  G r m1  m2 R   1  m R  m R m 1 1 2 2 1 RG  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       m1R1  F12 m2 R2  F21 inertial frame of reference    m2  R 2 x time m R  m R  2 R  1 1 2 2 R  derivative G  G r m1  m2 R   center of mass is: 1  m1R1  m2 R2   • motionless m1 R  G R  0 • or motion is in m1  m2 G 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       m1R1  F12 m2 R2  F21

      m1R1  m2 R2 RG  m1  m2 center of mass is:   • motionless R  0 • or motion is in G 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       m1R1  F12 m2 R2  F21

      m1R1  m2 R2 RG  m1  m2 center of mass is:   • motionless R  0 • or motion is in G 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 m1R1  F12 m2 R2  F21 reference   k m2 R2      m1m2   m1m2  j r m1R1  G 3 r m2 R2  G 3 r  r r  R i 1 m1

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 m1R1  F12 m2 R2  F21 reference   k m2 R2    m m  m m   m R  G 1 2 r m R  G 1 2 r  j r 1 1 3 2 2 3  r modification of r R1 i gravitational parameter equations m1   Gm  m  1 2    if: m1  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    r r  0 – Newton's laws of motion r 3 inertial frame of reference    k m2 vector r defined in R2   inertial frame of reference    j r expressed in coord. system i jk  R i 1 m1     r  (x2  x1)i  (y2  y1) j  (z2  z1)k

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    r r  0 – Newton's laws of motion r 3 inertial frame of reference    k m2 R2 vector r  can be expressed in coord.   system , that rotates about  i1 j1k1  j k1 r inertial coord. system with instant   R1 angular velocity  and instant angular i   acceleration m1  i1 j1     r  (x21 )i1  (y21 ) j1  (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    r r  0 – Newton's laws of motion r 3 inertial frame of reference   k m2 R2    j k1 r              R r  rrel  r r  2vrel 1 i  2 x time derivative in i m1 j 1 1 inertial frame of reference     r  (x21 )i1  (y21 ) j1  (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    r r  0 – Newton's laws of motion r 3 inertial frame of   reference r  rrel      k m2 R2 if i 1 j 1 k 1 is not rotating coord.    system j k1 r              R r  rrel  r r  2vrel 1 i  2 x time derivative in i m1 j 1 1 inertial frame of reference     r  (x21 )i1  (y21 ) j1  (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    r r  0 – Newton's laws of motion r 3 inertial frame of   reference r  rrel relative acceleration of   moving (non-rotating) k m2 R2  frame of reference in     x21   3 x21 coord. components  j k1 r r  R1  i   y   y m1 21 3 21 i1 j1 r  z   z 21 r 3 21 + 6 initial conditions

Orbital Mechanics 24 Space for Education, Education for Space 1. The two body problem Angular momentum

• relative angular momentum of body m 2 per unit mass  r  1     h  r m2r r r m m 2 2  1 x time derivative dh         r r  r r  r r r dt

m1

trajectory

Orbital Mechanics 25 Space for Education, Education for Space 1. The two body problem Angular momentum

• relative angular momentum of body m 2 per unit mass  r  1     h  r m2r r r m m 2 2  1 x time derivative dh         r r  r r  r r r dt

    m r  3 r  0 1 r trajectory

Orbital Mechanics 26 Space for Education, Education for Space 1. The two body problem Angular momentum

• relative angular momentum of body m 2 per unit mass  r  1     h  r m2r r r m m 2 2  1 x time derivative dh         r r  r r  r r r dt      angular momentum dh  m  0 r  3 r  0 1 is conserved dt r trajectory

Orbital Mechanics 27 Space for Education, Education for Space 1. The two body problem Angular momentum

• relative angular momentum of body m 2 per unit mass  r velocity vector can be expressed as      vr  r  v  v  vr v m2 angular momentum can be expressed as           r h  r r  r v  r vr  r v

m1

trajectory

Orbital Mechanics 28 Space for Education, Education for Space 1. The two body problem Angular momentum

• relative angular momentum of body m 2 per unit mass  r velocity vector can be expressed as      vr  r  v  v  vr v m2 angular momentum can be expressed as           r h  r r  r v  r vr  r v  k • unit vector      1 • time invariant h  r v  rv k  hk m   1 1 1 h • magnitude of h  rv angular momentum trajectory • time invariant

Orbital Mechanics 29 Space for Education, Education for Space 1. The two body problem Angular momentum

• relative angular momentum of body m 2 per unit mass  r  v  r v m • Cartesian coord. 2 system  r   k • unit vector j      1 • time invariant 1 h  r v  rv k  hk m    1 1 1 h • magnitude of i1 angular momentum trajectory • time invariant

Orbital Mechanics 30 Space for Education, Education for Space 1. The two body problem Angular momentum

• relative angular momentum of body m 2 per unit mass  r  v  r v m • Cartesian coord. 2 system  d  • Polar coord. v  r   r system r dt    • unit vector k1  j ir       • time invariant j 1 h  r v  rv k  hk  m    1 1 1 h • magnitude of i1 angular momentum trajectory • time invariant

Orbital Mechanics 31 Space for Education, Education for Space 1. The two body problem Angular momentum

• relative angular momentum of body m 2 per unit mass  r      vr   2  h  r  r k1  r k1  hk1 v m • Cartesian coord. 2 system  d  • Polar coord. v  r   r system r dt    • unit vector k1  j ir       • time invariant j 1 h  r v  rv k  hk  m    1 1 1 h • magnitude of i1 angular momentum trajectory • time invariant

Orbital Mechanics 32 Space for Education, Education for Space 1. The two body problem Solution of problem • Equation of orbit    r r  0  r 3 r   cross product with h vr        v r h   3 r  h  r m • Cartesian coord. 2 system • Polar coord.  system r     j1 ir j m1  i1 trajectory

Orbital Mechanics 33 Space for Education, Education for Space 1. The two body problem Solution of problem    • Equation of orbit   2    r  rir h   r k1    r  3 r  0  r and h expressed r r by polar coordinates   cross product with h vr        v r h   3 r  h  r m • Cartesian coord. 2 system • Polar coord.  system r     j1 ir j m1  i1 trajectory

Orbital Mechanics 34 Space for Education, Education for Space 1. The two body problem Solution of problem    • Equation of orbit   2    r  rir h   r k1    r  3 r  0  r and h expressed r r by polar coordinates   cross product with h vr        v r h   3 r  h  r m • Cartesian coord. 2 system      • Polar coord. rh   j system r     j1 ir j m1  i1 trajectory

Orbital Mechanics 35 Space for Education, Education for Space 1. The two body problem Solution of problem    • Equation of orbit   2    r  rir h   r k1    r  3 r  0  r and h expressed r r by polar coordinates   cross product with h vr        v r h   3 r  h  r m • Cartesian coord. 2 system      • Polar coord. rh   j system r angular momentum is const. vector       j1 ir dh j  0 m1  dt i1 trajectory

Orbital Mechanics 36 Space for Education, Education for Space 1. The two body problem Solution of problem    • Equation of orbit   2    r  rir h   r k1    r  3 r  0  r and h expressed r r by polar coordinates   cross product with h vr        v r h   3 r  h  r m • Cartesian coord. 2 system      • Polar coord. rh   j system r angular momentum is const. vector         j1 ir dh d d  j  0 r h  j m1  dt dt dt i1 trajectory

Orbital Mechanics 37 Space for Education, Education for Space 1. The two body problem Solution of problem

• Equation of orbit d   d  r h  j  dt dt r  v  r v m • Cartesian coord. 2 system  • Polar coord. r • Unit vectors of polar system coord. system are not   constant vectors      j1 ir ir  cos i1 sin j1 j m1     i1 j  sin i1  cos j1 trajectory

Orbital Mechanics 38 Space for Education, Education for Space 1. The two body problem Solution of problem

• Equation of orbit d   d  r h  j  dt dt r  v  r v m • Cartesian coord. 2 system   di   • Polar coord. r • Unit vectors of polar r  sin i  cos j system coord. system are not d 1 1   constant vectors      j1 ir ir  cos i1 sin j1 j m1     i1 j  sin i1  cos j1 trajectory

Orbital Mechanics 39 Space for Education, Education for Space 1. The two body problem Solution of problem

• Equation of orbit d   d  r h  j  dt dt r  v  r  v  dir m2  j • Cartesian coord. d  system   di   • Polar coord. r • Unit vectors of polar r  sin i  cos j system coord. system are not d 1 1   constant vectors      j1 ir ir  cos i1 sin j1 j m1     i1 j  sin i1  cos j1 trajectory

Orbital Mechanics 40 Space for Education, Education for Space 1. The two body problem Solution of problem

• Equation of orbit d   d  •  is scalar – time r h  j  dt dt  invariant parameter r     v d d  r r h  ir  dt dt  v  dir m2  j • Cartesian coord. d  system   di   • Polar coord. r • Unit vectors of polar r  sin i  cos j system coord. system are not d 1 1   constant vectors      j1 ir ir  cos i1 sin j1 j m1     i1 j  sin i1  cos j1 trajectory

Orbital Mechanics 41 Space for Education, Education for Space 1. The two body problem Solution of problem

• Equation of orbit d   d  r h   j •  is scalar – time     invariant parameter dt dt r  d   d   v • e is integration  r r h  ir  dt dt constant, i.e. const. vector v    m  • Cartesian coord. 2 r h  ir  e system • Polar coord.  system r     j1 ir j m1  i1 trajectory

Orbital Mechanics 42 Space for Education, Education for Space 1. The two body problem Solution of problem

• Equation of orbit d   d  r h   j •  is scalar – time     invariant parameter dt dt r  d   d   v • e is integration  r r h  ir  v dt dt constant, i.e. const. vector      m2  r h   i  e • Cartesian coord. • dot product with r    r  system        • Polar coord. r r h  r ir  e system r     j1 ir j m1  i1 trajectory

Orbital Mechanics 43 Space for Education, Education for Space 1. The two body problem Solution of problem

• Equation of orbit d   d  r h   j •  is scalar – time     invariant parameter dt dt r  d   d   v • e is integration  r r h  ir  v dt dt constant, i.e. const. vector      m2  r h   i  e • Cartesian coord. • dot product with r    r  system       • Polar coord.  r  r h   r  i  e r    r        system • using a b c abc      r r h   r 1 ecos      j1 ir j m1  i1 trajectory

Orbital Mechanics 44 Space for Education, Education for Space 1. The two body problem Solution of problem

• Equation of orbit d   d  r h   j •  is scalar – time     invariant parameter dt dt r  d   d   v • e is integration  r r h  ir  v dt dt constant, i.e. const. vector      m2  r h   i  e • Cartesian coord. • dot product with r    r  system       • Polar coord.  r  r h   r  i  e r    r        system • using a b c abc      r rh   r 1 ecos   j i  j 1 r  2 m1  h 1 r  i1 r  • Scalar equation of orbit   trajectory  1 ecos  • e is eccentricity

Orbital Mechanics 45 Space for Education, Education for Space 1. The two body problem Solution of problem • Equation of orbit  r  v  r v h   v   1 ecos  m  • Cartesian coord. 2 r h system  • Polar coord. r system h  rv    j i  j 1 r  2 m1  h 1 r  i1 r  • Scalar equation of orbit   trajectory  1 ecos  • e is eccentricity

Orbital Mechanics 46 Space for Education, Education for Space 1. The two body problem Solution of problem • Equation of orbit  r  v  r v h   v   1 ecos  m  • Cartesian coord. 2 r h system  • Polar coord. r dr  system h  rv vr  r   esin   dt h  j i  j 1 r  2 m1  h 1 r  i1 r  • Scalar equation of orbit   trajectory  1 ecos  • e is eccentricity

Orbital Mechanics 47 Space for Education, Education for Space 1. The two body problem Energy law • energy of system written in inertial frame of

reference placed in center of mass Etot  Ek1  Ek2  Ep

inertial frame of reference m  2 r

m1

Orbital Mechanics 48 Space for Education, Education for Space 1. The two body problem Energy law • energy of system written in inertial frame of

reference placed in center of mass Etot  Ek1  Ek2  Ep

1 1 m m E  m v2  m v2 G 1 2 expressed by inertial inertial frame of tot 2 1 m1 2 2 m2 r motion reference m  2 r

m1

Orbital Mechanics 49 Space for Education, Education for Space 1. The two body problem Energy law • energy of system written in inertial frame of

reference placed in center of mass Etot  Ek1  Ek2  Ep

1 1 m m E  m v2  m v2 G 1 2 expressed by inertial inertial frame of tot 2 1 m1 2 2 m2 r motion reference m  2 r 1 m m m m E  1 2 v2 G 1 2 expressed by tot relative motion 2 m1  m2 r m1 reduced mass m1m2 of system m1  m2

Orbital Mechanics 50 Space for Education, Education for Space 1. The two body problem Energy law • energy of system written in inertial frame of

reference placed in center of mass Etot  Ek1  Ek2  Ep

1 1 m m E  m v2  m v2 G 1 2 expressed by inertial inertial frame of tot 2 1 m1 2 2 m2 r motion reference m  2 r 1 m m m m E  1 2 v2 G 1 2 expressed by tot relative motion 2 m1  m2 r specific orbital energy 2 m1 reduced mass m1m2 v  (total energy per unit of system    reduced mass) m1  m2 2 r vis viva equation

Orbital Mechanics 51 Space for Education, Education for Space 1. The two body problem Energy law • energy of system written in inertial frame of

reference placed in center of mass Etot  Ek1  Ek2  Ep

1 1 m m E  m v2  m v2 G 1 2 expressed by inertial inertial frame of tot 2 1 m1 2 2 m2 r motion reference m  2 r 1 m m m m E  1 2 v2 G 1 2 expressed by tot relative motion specific energy 2 m1  m2 r expressed by e specific orbital energy 2 m1 1  2 v  (total energy per unit 2    reduced mass)    2 1 e  2 h 2 r vis viva equation

Orbital Mechanics 52 Space for Education, Education for Space 1. The two body problem Trajectories • Shape of trajectory depends on eccentricity e • Equation of orbit is equation of conic sections: – circle e  0 • equation of orbit – ellipse 0  e 1 h2 1 r  – parabola e 1  1 ecos  – hyperbola e 1

h  0

Orbital Mechanics 53 Space for Education, Education for Space 1. The two body problem Trajectories • Circle: (bounded trajectory) e  0 • equation of h2 1 h2 r  r  orbit  1 ecos  

• speed of   v  v  1 ecos  v  motion h r r 2 r 2 • period T  T  r 3/ 2  / r 

•specific 1  2 1     1 e2     energy 2 h2 2 r

Orbital Mechanics 54 Space for Education, Education for Space 1. The two body problem Trajectories • Circle: (bounded trajectory) e  0  2 3/ 2 v  T  r r  central body circ. velocity circ. period [km/s] [min.] Earth 7.90 84.48 Moon 1.68 108.36 Mars 3.55 100.19 Sun (surface) 436.7 166.91 Sun (Earths) 29.78 5.26x105 •trajectories of satellite in different

altitude passed in time TEarth

Orbital Mechanics 55 Space for Education, Education for Space 1. The two body problem Trajectories • Circle: (bounded trajectory) e  0  2 3/ 2 v  T  r •trajectories of satellite in r  different altitude passed in in

time TEarth  84.48min. TGEO  86164 s

rGEO  42164km

vGEO  3.07km/s

Orbital Mechanics 56 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1

b – semiminor axis

empty focus

C - center a – semimajor axis F - focus P -periapsis A - apoapsis

Orbital Mechanics 57 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1 2 h 1 rA  rP r  e  P r  r  1 e rP 1 e A P h2 1  r  rA 1 e  1 ecos  2 h 1 b – semiminor axis rA   1 e  -true anomaly empty focus r

C - center a – semimajor axis F - focus P -periapsis A - apoapsis

Orbital Mechanics 58 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1 2 h2 1 h 1 r  a  P  1- e2 2  1 e h 1 2a  rP  rA r   1 ecos  2 h 1 b – semiminor axis rA   1 e  -true anomaly empty focus r

C - center a – semimajor axis F - focus P -periapsis A - apoapsis

Orbital Mechanics 59 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1

2 2a  rP  rA b  a 1-e  r  r 2 2 2 e  A P CF  ea b  a CF r  r A P b – semiminor axis

2CF  rA  rP  -true anomaly empty focus r

C - center a – semimajor axis F - focus P -periapsis A - apoapsis

Orbital Mechanics 60 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1 • eccentricity • flattening 2 2 2 a  b a  b e  f  1 1 e2 f  a 2 a e  0.1 f  0.1

Orbital Mechanics 61 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1 • eccentricity • flattening

usage: description usage: description of orbits of planet shape e  0.1 f  0.1

flattening of Earth: 1/ f  298.257 21.4 km diff. in radius

Orbital Mechanics 62 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1 • equation of h2 1 r  orbit  1 ecos 

• speed of v2   2 motion    v    a r 2 r

• period 2ab 2 3 T  T  a 2 h 

•specific 1  2 1  energy    1 e2     2 h2 2 a

Orbital Mechanics 63 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1 2 3 2 • ellipses with equal T  a  equal period and semimajor axis a : orbital energy 1     2 a

shape of location ellipses of orbits Orbital Mechanics 64 Space for Education, Education for Space 1. The two body problem Trajectories • Ellipse: (bounded trajectory) 0  e 1 2 3 2 • ellipses with equal T  a  equal period and semimajor axis a : orbital energy 1      2 2 a v    a r rp[km] vp[km/s] ra[km] va[km/s] 42164 3.07 42164 3.07 29514.8 4.19 54813.2 2.25 16865.6 6.15 67462.4 1.53 8432.8 9.22 75895.2 1.02

Orbital Mechanics 65 Space for Education, Education for Space 1. The two body problem Trajectories • Parabola: (open trajectory) e 1 • equation of h2 1 r  orbit  11cos 

• speed of v2  2 motion    v   - true anomaly 2 r r r •specific 1  2    1 e2 energy 2   2 h   0 F - focus P -periapsis

directrix

Orbital Mechanics 66 Space for Education, Education for Space 1. The two body problem Trajectories • Parabola: (open trajectory) e 1 1 2 hp  REarth v  10 r

central body esc. velocity [km/s] Earth 11.18 Moon 2.37 Mars 5.02 Sun (surface) 617.5 Sun (Earths) 42.12

Orbital Mechanics 67 Space for Education, Education for Space 1. The two body problem Trajectories • Parabola: (open trajectory) e 1 1 h  R •trajectories of satellite in p Earth 1 10 different h  R p 10 Earth

hp [km] (Earth) vp[km/s] 0 11.18 637.8 10.66 1275.6 10.20 1913.4 9.80 2551.2 9.44 3189.0 9.12 3826.8 8.83

Orbital Mechanics 68 Space for Education, Education for Space 1. The two body problem Trajectories • Parabola: (open trajectory) e 1 1 h  R •trajectories of satellite in p Earth 1 10 different h  R p 10 Earth

Orbital Mechanics 69 Space for Education, Education for Space 1. The two body problem Trajectories • Parabola: (open trajectory) e 1 1 h  R p 10 Earth

Orbital Mechanics 70 Space for Education, Education for Space 1. The two body problem Trajectories • Parabola: (open trajectory) e 1 1 h  R p 10 Earth

Orbital Mechanics 71 Space for Education, Education for Space 1. The two body problem Trajectories • Hyperbola: (open trajectory) e 1

asymptotes

a- semimajor axis vertex

empty F- focus focus

Orbital Mechanics 72 Space for Education, Education for Space 1. The two body problem Trajectories • Hyperbola: (open trajectory) e 1 • equation of h2 1 r  orbit  1 ecos  asymptotes • speed of 2 2  2 motion v  v   a- semimajor    a r axis 2 r vertex

 •specific   r energy 2a 2 empty 1  2 F- focus    1 e   focus 2 h2

Orbital Mechanics 73 Space for Education, Education for Space 1. The two body problem Trajectories • Hyperbola: (open trajectory) e 1 •trajectories of satellite with different e  0.1

• periapsis: rp  REarth

e vp[km/s] 1.1 11.45 1.2 11.72 1.3 11.98 1.4 12.24 1.5 12.49 1.6 12.74 1.7 12.99

Orbital Mechanics 74 Space for Education, Education for Space 1. The two body problem Trajectories • Hyperbola: (open trajectory) e 1 •trajectories of satellite with different e  0.1

• periapsis: rp  REarth

e vp[km/s] 1.1 11.45 1.2 11.72 1.3 11.98 1.4 12.24 1.5 12.49 1.6 12.74 1.7 12.99

Orbital Mechanics 75 Space for Education, Education for Space 1. The two body problem Trajectories • Hyperbola: (open trajectory) e 1 •trajectories of satellite with

different alt. hp  0.1REarth • eccentricity: e 1.1

hp vp[km/s]

0.1xREarth 11.45

0.2xREarth 10.92

0.3xREarth 10.45

0.4xREarth 10.04

0.5xREarth 9.68

0.6xREarth 9.35

0.7xREarth 9.05

Orbital Mechanics 76 Space for Education, Education for Space 1. The two body problem Trajectories • Hyperbola: (open trajectory) e 1 •trajectories of satellite with

different alt. hp  0.1REarth • eccentricity: e 1.1

hp vp[km/s]

0.1xREarth 11.45

0.2xREarth 10.92

0.3xREarth 10.45

0.4xREarth 10.04

0.5xREarth 9.68

0.6xREarth 9.35

0.7xREarth 9.05

Orbital Mechanics 77 Space for Education, Education for Space 1. The two body problem Time and position • Two cases can be investigated: – time as a function of position – position as a function of time • Only ellipse orbit is presented, but similar expressions can be derived for all trajectories

Orbital Mechanics 78 Space for Education, Education for Space 1. The two body problem Time and position • True, Mean and Eccentric anomalies

orbit

auxiliary circle

Orbital Mechanics 79 Space for Education, Education for Space 1. The two body problem Time and position • True, Mean and Eccentric anomalies location of satellite true anomaly 

orbit focus

auxiliary circle

Orbital Mechanics 80 Space for Education, Education for Space 1. The two body problem Time and position • True, Mean and Eccentric anomalies location of satellite true anomaly 

virtual location on circle with const. motion with the same period as satellite has

orbit M focus mean e anomaly

auxiliary circle

Orbital Mechanics 81 Space for Education, Education for Space 1. The two body problem Time and position • True, Mean and Eccentric anomalies projection of location of satellite true location on anomaly circle virtual location on circle eccentric with const. motion with anomaly  the same period as satellite has Ee

orbit M focus mean e anomaly

auxiliary circle

Orbital Mechanics 82 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position • using mean anomaly d h   r 2  r 2 dt

Orbital Mechanics 83 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position • using mean anomaly d h   r 2  r 2 dt

1 dt  r 2d h

Orbital Mechanics 84 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position • using mean anomaly d h   r 2  r 2 dt

1 dt  r 2d h

h2 1 r   1 ecos 

Orbital Mechanics 85 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position • using mean anomaly d h   r 2  r 2 dt

1 h3 d dt  r 2d dt  h  2 1 ecos 2

h2 1 r   1 ecos 

Orbital Mechanics 86 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position • using mean anomaly h3  d  2 d 2 t  t  h   r  r p 2  2 dt  0 1 ecos

1 h3 d dt  r 2d dt  h  2 1 ecos 2

h2 1 r   1 ecos 

Orbital Mechanics 87 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • using mean anomaly h3  d t  t  p 2  2  0 1 ecos t p  0

Orbital Mechanics 88 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • using mean anomaly h3  d t  t  p 2  2  0 1 ecos t p  0  d 1   1 e   1 e2 sin   2tan 1 tan   e   1 ecos 2 2 3/ 2  1 e 2  1 ecos 0   1 e     

Orbital Mechanics 89 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • using mean anomaly h3  d t  t  p 2  2  0 1 ecos t p  0  d 1   1 e   1 e2 sin   2tan 1 tan   e   1 ecos 2 2 3/ 2  1 e 2  1 ecos 0   1 e     

 d 1  M e  2 2 3/ 2 0 1 ecos 1 e 

Orbital Mechanics 90 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 M [rad] • using mean anomaly e h3 1 t  M e  2 2 3/ 2 1 e  e  0 (circle)

e  0.15

 [rad]   1 e   1 e2 sin  M  2tan 1 tan   e  e  1 e 2  1 ecos    

Orbital Mechanics 91 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • using mean anomaly e  0.576 M e [rad]

 [rad]

  1 e   1 e2 sin  M  2tan 1 tan   e  e  1 e 2  1 ecos    

Orbital Mechanics 92 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • using eccentric anomaly h3 1 t  2 3/ 2 M e  1 e2 

sin E Ee e

  1 e   1 e2 sin  M  2tan 1 tan   e  e  1 e 2  1 ecos    

Orbital Mechanics 93 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • using eccentric anomaly h3 1 • Kepler’s equation t  M e 2 2 3/ 2  1 e  M e  E esin E

sin E Ee e

  1 e   1 e2 sin  M  2tan 1 tan   e  e  1 e 2  1 ecos    

Orbital Mechanics 94 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 E [rad] • using eccentric anomaly e

e  0 (circle)

e  0.15

 [rad]  1 e   E  2tan 1 tan  e    1 e 2 

Orbital Mechanics 95 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • using eccentric anomaly E [rad] e  0.576

 [rad]

 1 e   E  2tan 1 tan  e    1 e 2 

Orbital Mechanics 96 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • defined 

 1 e   E  2tan 1 tan  e    1 e 2 

Orbital Mechanics 97 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • defined 

 1 e   E  2tan 1 tan  e    1 e 2 

M e  E esin E

Orbital Mechanics 98 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • defined 

 1 e   E  2tan 1 tan  e    1 e 2 

M e  E esin E

h3 1 t  2 3/ 2 M e  1 e2 

Orbital Mechanics 99 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • defined 

 1 e   E  2tan 1 tan  e    1 e 2 

M e  E esin E

h3 1 T t  M t  2 3/ 2 M e e  1 e2  2

Orbital Mechanics 100 Space for Education, Education for Space 1. The two body problem Time and position • Time as a function of position - ellipse 0  e 1 • defined 

 1 e   E  2tan 1 tan  e    1 e 2 

M e  nt

M e  E esin E 2 n  h3 1 T T t  M e t  M e • average angular 2 2 3/ 2 2  1 e  velocity

Orbital Mechanics 101 Space for Education, Education for Space 1. The two body problem Time and position • Position as a function of time - ellipse 0  e 1 • defined t

2 M  t e T

Orbital Mechanics 102 Space for Education, Education for Space 1. The two body problem Time and position • Position as a function of time - ellipse 0  e 1 • defined t

2 M  t e T

M e  Ee esin Ee 

Orbital Mechanics 103 Space for Education, Education for Space 1. The two body problem Time and position • Position as a function of time - ellipse 0  e 1 • defined t

2 M  t e T

M e  Ee esin Ee 

• E e must be computed numerically

Orbital Mechanics 104 Space for Education, Education for Space 1. The two body problem Time and position • Position as a function of time - ellipse 0  e 1 • defined t

2 • the problem of finding M e  t true anomaly for defined T time is called Kepler’s problem M  E esin E  e e e  1 e E    2 tan 1 tan e     1 e 2 

• E e must be computed numerically

Orbital Mechanics 105 Space for Education, Education for Space 2. Orbits in three dimensions

• Frame of reference • Earth-based systems • Orbital elements • Calculation of elements

Orbital Mechanics 106 Space for Education, Education for Space 2. Orbits in three dimensions Frame of reference • to describe orbits in three dimensions, the coordinate system in frame of reference must be defined • Newton laws are valid in inertial frame of reference • practically only pseudoinertial frame of reference can be considered • coordinate system is formed in considered frame of reference

Orbital Mechanics 107 Space for Education, Education for Space

2. Orbits in three dimensions Frame of reference • coord. system is defined by: • origin, fundamental plane and preferred direction • choice of frame of reference and subsequently coordinate system depends on considered trajectory: • Interplanetary trajectory – Interplanetary systems, e.g. Heliocentric coordinate system • Earth orbits – Earth-based systems

Orbital Mechanics 108 Space for Education, Education for Space 2. Orbits in three dimensions Earth-based systems • Geocentric Equatorial System (GES) - the most common system in astrodynamics • the center of coord. system is at Earth’s center • not-rotating coord. system • fundamental plane – Earth’s equator plane • axis X points towards the vernal equinox • axis Z extends through the North Pole

Orbital Mechanics 109 Space for Education, Education for Space 2. Orbits in three dimensions Earth-based systems • Geocentric Equatorial System (GES): • is often considered as Earth-Centered Inertial system (ECI) • ECI frame of reference is not fixed in space: • gravitational forces of planets – planetary precession • gravitational forces of Moon and Sun – luni-solar precession with period 26,000 years • combined effect – general precession • inclination of Moon – additional torque on Earth’s equatorial bulge – nutation with period 18,6 years • due to precession and nutation equinox is moving

Orbital Mechanics 110 Space for Education, Education for Space

2. Orbits in three dimensions Earth-based systems • Geocentric Equatorial System (GES): • for all precise applications, ECI must by defined on specific date • J2000 - commonly used ECI frame is defined with the Earth's Mean Equator and Equinox at 12:00 Terrestrial Time on 1 January 2000 • other Earth-based systems: • Earth-Centered, Earth-Fixed Coord. System – rotate with Earth • Perifocal Coord. System

Orbital Mechanics 111 Space for Education, Education for Space

2. Orbits in three dimensions Orbital elements • Location of the satellite: 1. the location of the orbital plane in defined coord. system of chosen frame of reference , i 2. the position of the elliptical orbit in this plane  3. the characteristics of ellipse e, h (or a) 4. the position of the moving satellite on the orbit  (or M)

Orbital Mechanics 112 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements • the goal is to determine  orbital elements from: • r  r i  r j  r k position vector  x  y  z  • velocity vector v  vxi  vy j  vz k

• both vectors are defined in GES at time t0

Orbital Mechanics 113 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements     r  rxi  ry j  rz k     v  vxi  vy j  vz k

  state vector r and v

Orbital Mechanics 114 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements     r  rxi  ry j  rz k     v  vxi  vy j  vz k

   i j k    h  r v  rx ry rz

  vx vy vz state vector r and v

1st element  h

Orbital Mechanics 115 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements     r  rxi  ry j  rz k     v  vxi  vy j  vz k     h  hxi  hy j  hz k  h  i  cos 1 z   h    state vector r and v

1st element  2nd element i h

Orbital Mechanics 116 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements     r  rxi  ry j  rz k     v  vxi  vy j  vz k     h  hxi  hy j  hz k

 1  2      e  v  r  r vv   r     3th element state vector r and v  e 1st element  2nd element i h

Orbital Mechanics 117 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements     r  rxi  ry j  rz k     v  vxi  vy j  vz k        i j k h  h i  h j  h k    x y z  n  k  h  0 0 1    e  exi  ey j  ez k hx hy hz   3th element state vector r and v  e 1st element   2nd element i h n vector of node line

Orbital Mechanics 118 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements     r  rxi  ry j  rz k     v  vxi  vy j  vz k     h  hxi  hy j  hz k    1 nx     cos   e  exi  ey j  ez k  n      n  nxi  ny j  0k   3th element state vector r and v  e 1st element   2nd element i h n  4th element

Orbital Mechanics 119 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements     r  rxi  ry j  rz k     v  vxi  vy j  vz k       h  hxi  hy j  hz k 1 n e        cos    e  exi  ey j  ez k  n e      n  nxi  ny j  0k   3th element state vector r and v  e  5th element 1st element   2nd element i h n  4th element

Orbital Mechanics 120 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements     r  rxi  ry j  rz k       v  vxi  vy j  vz k 1 e r        cos    e r h  hxi  hy j  hz k       e  exi  ey j  ez k 6th element      n  nxi  ny j  0k   3th element state vector r and v  e  5th element 1st element   2nd element i h n  4th element

Orbital Mechanics 121 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example:

rp  2004.75, 6174.08, 1567.56 km input parameters: v p  - 7.556, 1.581, 3.435 km/s

Orbital Mechanics 122 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: h  56430.1 km 2 / s i  28 e  0.196   45   30   0

rp  2004.75, 6174.08, 1567.56 km input parameters: v p  - 7.556, 1.581, 3.435 km/s

Orbital Mechanics 123 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: h  56430.1 km 2 / s Orbit in 2D view: i  28 e  0.196   45   30   0

rp  2004.75, 6174.08, 1567.56 km input parameters: v p  - 7.556, 1.581, 3.435 km/s

Orbital Mechanics 124 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: Orbit in 3D view:

rp  2004.75, 6174.08, 1567.56 km input parameters: v p  - 7.556, 1.581, 3.435 km/s

Orbital Mechanics 125 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: Orbit in 2D map: Orbit in 3D view:

rp  2004.75, 6174.08, 1567.56 km input parameters: v p  - 7.556, 1.581, 3.435 km/s

Orbital Mechanics 126 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: GEO Orbit in 2D map: Orbit in 3D view:

input parameters: GEO, circular orbit, i  2.5

Orbital Mechanics 127 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: GEO Detail view: Orbit in 2D map:

input parameters: GEO, circular orbit, i  2.5

Orbital Mechanics 128 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: GEO Detail view: Orbit in 2D map:

input parameters: GEO, e  0.01575 i  0

Orbital Mechanics 129 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: GEO Detail view: Orbit in 2D map:

input parameters: GEO, e  0.01575 i  2.5

Orbital Mechanics 130 Space for Education, Education for Space 2. Orbits in three dimensions Calculation of elements Example: Molnija Orbit in 2D map: Orbit in 3D view:

e  0.75 h  40089 km input parameters: a i  63.41hp  260km

Orbital Mechanics 131 Space for Education, Education for Space 3. Orbital perturbations

• Perturbing forces • Geopotential • Orbit propagation • Variation of parameters • Examples of orbits

Orbital Mechanics 132 Space for Education, Education for Space 3. Orbital perturbations Perturbing forces Orbits of Earth satellites are influenced by 2 facts: • The Earth is not exactly spherical and the mass distribution is not exactly spherically symmetric • The satellite feels other forces apart from the Earth’s attraction: • attractive forces due to other heavenly bodies • forces that can be globally categorized as frictional

All these influences are called perturbations

Orbital Mechanics 133 Space for Education, Education for Space 3. Orbital perturbations Perturbing forces

Perturbing forces

Conservative forces – Non-conservative forces – can be derived from cannot be derived from potential: potential – dissipative • flattening of the Earth forces: • Attraction of the Moon • atmospheric drag • Attraction of the Sun • radiation pressure • Attraction by other planets

Orbital Mechanics 134 Space for Education, Education for Space 3. Orbital perturbations Perturbing forces

Influence of perturbing forces expressed by accelerations: • GM – attraction of Earth (sphere shape) • J2 – flattening of the Earth (Earth ellipsoid) • J4, J6 – potential of Earth expressed by higher orders • Moon, Sun, Planets – their attraction

source: Capderou: Handbook of Satellite Orbits Capderou: Handbook Satellite of source:

Orbital Mechanics 135 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of single mass point:

position of masses m  potential energy of mass 2 in r gravitational field of mass

m1m2 E p  G m1 r

Orbital Mechanics 136 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of single mass point:

position of masses m  potential energy of mass 2 in r gravitational field of mass

m1m2 E p  G m1 r gravitational potential E  Ur   p  m2 r

Orbital Mechanics 137 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of single mass point:

position of masses m  potential energy of mass 2 in r gravitational field of mass

m1m2 E p  G m1 r gravitational potential equation of motion E p  Ur    expressed by potential m r 2  r  gradU 

Orbital Mechanics 138 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 1. approximation - sphere

position of masses dM  m2 d potential energy of mass in  gravitational field of dM r

dMm dE  G 2 p d M

Orbital Mechanics 139 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 1. approximation - sphere

position of masses dM  m2 d potential energy of mass in  gravitational field of dM r

dMm dE  G 2 p d M gravitational potential dE GdM dU   p  m2 d

Orbital Mechanics 140 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 1. approximation - sphere

position of masses dM  m2 d potential energy of mass in  gravitational field of dM r

dMm dE  G 2 p d M gravitational potential

dE p GdM dU    GdM m d U  dU  2   M M d

Orbital Mechanics 141 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 1. approximation - sphere

position of masses dM  m2 potential energy of mass in d equal potential as  gravitational field of dM r single mass potential GM  dMm2 Ur   dE p  G r r d M gravitational potential integration over sphere boundary dE p GdM dU    GdM m d U  dU  2   M M d

Orbital Mechanics 142 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 2. approximation - ellipsoid

position of masses dM  m2 d  r

  M Position of : • longitude  • latitude  • radius r

Orbital Mechanics 143 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 2. approximation - ellipsoid

position of masses dM Ellipsoid:  m2 • longitude d   • latitude r  • radius 

  M Position of : • longitude  • latitude  • radius r

Orbital Mechanics 144 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 2. approximation - ellipsoid

position of masses dM Ellipsoid:  m2 • longitude d   • latitude r  • radius 

  M  angle between and Position of : 2 • longitude      • latitude  d  r 1 2 cos    • radius r r  r 

Orbital Mechanics 145 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 2. approximation - ellipsoid

position of masses dM Ellipsoid:  m2 • longitude d   • latitude r  • radius 

  GdM U  M   angle between and M d Position of : 2 • longitude      • latitude  d  r 1 2 cos    • radius r r  r 

Orbital Mechanics 146 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 2. approximation - ellipsoid

position of masses using: dM  m2 • expansion of 1/d in terms d  of Legendre polynomials r • symmetric properties of ellipsoid

  GdM U  M  M d

Orbital Mechanics 147 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: 2. approximation - ellipsoid

position of masses using: dM  m2 • expansion of 1/d in terms d  of Legendre polynomials r • symmetric properties of ellipsoid

  GdM 2 U  M    R  3sin2  1  d Ur,,   Ur,   1  J  M  2  r   r  2  R is equatorial radius 1 3 J 2  2 I x  I z  J 1.082610 J 2 dimensionless coefficient MR 2 Orbital Mechanics 148 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees • potential is function of all 3 coordinates, i.e. Ur,, 

Orbital Mechanics 149 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees • potential is function of all 3 coordinates, i.e. Ur,, 

l    R   l  Ur,,     Clm cosm Slm sinmPlmsin  r l0  r  m0 

Orbital Mechanics 150 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees • potential is function of all 3 coordinates, i.e. Ur,,  parameters are obtained from precise observation of the motion of satellites

l    R   l  Ur,,     Clm cosm Slm sinmPlmsin  r l0  r  m0 

Orbital Mechanics 151 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees • potential is function of all 3 coordinates, i.e. Ur,,  parameters are obtained from precise observation of the motion of satellites

l    R   l  Ur,,     Clm cosm Slm sinmPlmsin  r l0  r  m0 

Legendre functions

Orbital Mechanics 152 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees • potential is function of all 3 coordinates, i.e. Ur,,  parameters are obtained from precise observation of the motion of satellites

l    R   l  Ur,,     Clm cosm Slm sinmPlmsin  r l0  r  m0  products Legendre functions sinmPlmsin  cosmPlmsin 

Orbital Mechanics 153 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees • potential is function of all 3 coordinates, i.e. Ur,,  parameters are obtained from precise observation of the motion of satellites

l    R   l  Ur,,     Clm cosm Slm sinmPlmsin  r l0  r  m0  products Legendre functions sinmPlmsin  im Hlm,   Plmsin  e cosmPlmsin  Complex functions called Spherical Harmonics

Orbital Mechanics 154 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees • potential is function of all 3 coordinates, i.e. Ur,,  parameters are obtained from precise observation of the motion of satellites

l    R   l  Ur,,     Clm cosm Slm sinmPlmsin  r l0  r  m0  • for m=0: , , are called zonal harmonics, Cl0  Jl Sl0  0 Hl0 , 

• for m=l: H ll   ,   are called sectoral harmonics

• all other functions H lm   ,   are called tesseral harmonics

Orbital Mechanics 155 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees m=0 m=1 m=2 m=3 l=0

Spherical Harmonics (SH)

l=1

Zonal Harmonics l=2

l=3

Orbital Mechanics 156 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees m=0 m=1 m=2 m=3 l=0

Spherical Harmonics (SH)

l=1 Sectoral

Harmonics

l=2

l=3

Orbital Mechanics 157 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees m=0 m=1 m=2 m=3 l=0

Spherical Harmonics (SH)

l=1

Tesseral Harmonics l=2

l=3

Orbital Mechanics 158 Space for Education, Education for Space 3. Orbital perturbations Geopotential Potential of Earth: expansion to higher degrees m=0 m=1 m=2 m=3 l=0

Spherical Harmonics (SH)

l=1 geopotential model of Earth is using coefficients in SH expansion, for example l=2 Goddard Earth Model 10b (GEM10b) is using 21x21 SH expansion

l=3

Orbital Mechanics 159 Space for Education, Education for Space 3. Orbital perturbations Orbit propagation • the goal is to solve equation of motion with initial conditions  r  gradU      r(t  0)  r0 and r(t  0)  r0 • potential U expresses influence of central acceleration and perturbative acceleration

U U0  R • for example, perturbative potential  R2 3sin2  1 U  R   J 0 r r 3 2 2

Orbital Mechanics 160 Space for Education, Education for Space 3. Orbital perturbations Orbit propagation • analytical methods: general perturbations • expresses modification of motion • enable to determine whether the eccentricity increases, the orbit begins to precess, and so on • numerical methods: special perturbations • one step methods – purely mathematical approach: Runge-Kuta • multistep methods – methods developed by astronomers to determine the motions of planets: Adams-Bashforth, Adams- Moulton • special methods design specially for artificial satellites

Orbital Mechanics 161 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion • Mathematical intro: diff. equation with right hand side dy  f ty  gt dt

Orbital Mechanics 162 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion • Mathematical intro: diff. equation with right hand side dy  f ty  gt dt

dy  f ty  0 dt homogenous equation

Orbital Mechanics 163 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion • Mathematical intro: diff. equation with right hand side dy  f ty  gt dt

dy dy  f ty  0   f tdt dt y homogenous equation

Orbital Mechanics 164 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion • Mathematical intro: diff. equation with right hand side dy  f ty  gt dt

dy dy  f t dt  f ty  0   f tdt y  ce dt y homogenous solution homogenous equation c – int. constant

Orbital Mechanics 165 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion • Mathematical intro: diff. equation with right hand side dy to obtain solution of eq.  f ty  gt with right hand side, we dt allow c to be function of t

dy dy  f t dt  f ty  0   f tdt y  ce dt y homogenous solution homogenous equation c – int. constant

Orbital Mechanics 166 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion • Mathematical intro: dc  f t dt diff. equation with e   gt right hand side dt dy to obtain solution of eq.  f ty  gt with right hand side, we dt allow c to be function of t

dy dy  f t dt  f ty  0   f tdt y  ce dt y homogenous solution homogenous equation c – int. constant

Orbital Mechanics 167 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion f t dt ct  C  gt e dt • Mathematical intro:  dc  f t dt diff. equation with e   gt right hand side dt dy to obtain solution of eq.  f ty  gt with right hand side, we dt allow c to be function of t

dy dy  f t dt  f ty  0   f tdt y  ce dt y homogenous solution homogenous equation c – int. constant

Orbital Mechanics 168 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Similar process can be applied to system of diff. eq. • diff. equation of motion can be written as system of equations  dr   v dt  dv    r  gradR dt r 3

Orbital Mechanics 169 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Similar process can be applied to system of diff. eq. • diff. equation of motion can be written as system of equations  dr     v solution r  rt, 6 constants dt without right  hand side   dv    r  gradR v  vt, 6 constants dt r 3

Orbital Mechanics 170 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Similar process can be applied to system of diff. eq. • diff. equation of motion can be written as system of equations  dr     v solution r  rt, 6 constants dt without right  hand side   dv    r  gradR v  vt, 6 constants dt r 3

6 int. constants are 6 orbital elements , i,,a,e,M

Orbital Mechanics 171 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Similar process can be applied to system of diff. eq. • diff. equation of motion can be written as system of equations  dr     v solution r  rt, 6 constants dt without right  hand side   dv    r  gradR v  vt, 6 constants dt r 3 variation of all 6 orbital elements 6 int. constants are 6 orbital elements  t, i t, t, , i,,a,e,M a t,e t, M t Orbital Mechanics 172 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Similar process can be applied to system of diff. eq. • diff. equation of motion can be written as system of equations  dr     v solution r  rt, 6 constants dt without right  hand side   dv    r  gradR v  vt, 6 constants dt r 3 variation of all 6 6 int. constants are 6 calculation of orbital elements orbital elements parameters  t, i t, t, , i,,a,e,M a t,e t, M t Orbital Mechanics 173 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Similar process can be applied to system of diff. eq. • diff. equation of motion can be written as system of equations d  1 R di 1 R cos i R     dt nabsini i dt nabsini  nabsini  dM 2 R b2 R da 2 R     dt na a na 4e e dt na M de b R b2 R d  cos i R b R       dt na3e  na 4e M dt nabsini i na3e e Lagrange’s planetary equations

Orbital Mechanics 174 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential must be expressed by orbital elements , i,,a,e,M R2 3sin2  1 R   J r 3 2 2

Orbital Mechanics 175 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential must be expressed by orbital elements , i,,a,e,M R2 3sin2  1 R   J 2 sin  sinisin   r 3 2

2 2. approximation - ellipsoid a1 e  r  R  R 1 ecos

Orbital Mechanics 176 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential must be expressed by orbital elements , i,,a,e,M R2 3sin2  1 R   J 2 sin  sinisin   r 3 2

2 2. approximation - ellipsoid a1 e  r  R  R 1 ecos • average value of R in one period T

1 T 1 2 R  Rdt  RdM T 0 2 0

Orbital Mechanics 177 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential must be expressed by orbital elements , i,,a,e,M R2 3sin2  1 R   J 2 sin  sinisin   r 3 2

2 2. approximation - ellipsoid a1 e  r  R  R 1 ecos • average value of R in one period T

2 1 T 1 2 1 R 2 R  Rdt  RdM R   3/ 2 J 2 3sin i  2 T 0 2 0 4 a3 1 e2 

Orbital Mechanics 178 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential can be decomposed into average (secular) and periodic part

R  Rs  Rp

Orbital Mechanics 179 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential can be decomposed into average (secular) and periodic part

R  Rs  Rp

average value in one period is zero

Orbital Mechanics 180 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential can be decomposed into average (secular) and periodic part

R  Rs  Rp

average value in one period is zero Rs  R

Orbital Mechanics 181 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential can be decomposed into average (secular) and periodic part

R  Rs  Rp

average value in one period is zero Rs  R • Replacing perturbative potential by its secular part

R Rs R

Orbital Mechanics 182 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential can be decomposed into average (secular) and periodic part

R  Rs  Rp

average value in one period is zero Rs  R • Replacing perturbative potential by its secular part 2 1 R 2 R Rs R R   3/ 2 J 2 3sin i  2 4 a3 1 e2 

Orbital Mechanics 183 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Perturbative potential can be decomposed into average (secular) and periodic part

R  Rs  Rp

average value in one period is zero Rs  R • Replacing perturbative potential by its secular part 2 1 R 2 R Rs R R   3/ 2 J 2 3sin i  2 4 a3 1 e2 

R  Ra,e,i

Orbital Mechanics 184 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Lagrange’s planetary equations R  Ra,e,i de  R R  di  R R  da  R   e ,   i ,   a  dt   M  dt     dt  M 

d   R     dt  i  dM  R R   M  ,  dt  a e  d   R R    ,  dt  i e  Orbital Mechanics 185 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Lagrange’s planetary equations R  Ra,e,i de  R R  di  R R  da  R   e ,   i ,   a  dt   M  dt     dt  M 

e, i, a are constants

d   R     dt  i  dM  R R   M  ,  dt  a e  d   R R    ,  dt  i e  Orbital Mechanics 186 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Lagrange’s planetary equations R  Ra,e,i de  R R  di  R R  da  R   e ,   i ,   a  dt   M  dt     dt  M 

e, i, a are constants

2 d   R  d  3  R        2 nJ 2   cos i dt  i  dt 21 e2   a  2 dM  R R  dM 3  R  2  M  ,   n  3/ 2 nJ 2   3cos i 1 dt  a e  dt 41 e2   a  2 d   R R  d  3  R  2   ,   2 nJ 2   5cos i 1 dt  i e  dt 41 e2   a  Orbital Mechanics 187 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Lagrange’s planetary equations

Kepler’s orbit

input parameters:

rp  2004.75, 6174.08, 1567.56 km

v p  - 7.556, 1.581, 3.435 km/s

Orbital Mechanics 188 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Lagrange’s planetary equations

Perturbed orbit Kepler’s orbit after 100 x T

only J2 is considered   32.9   53.9

input parameters:

rp  2004.75, 6174.08, 1567.56 km

v p  - 7.556, 1.581, 3.435 km/s

Orbital Mechanics 189 Space for Education, Education for Space 3. Orbital perturbations Variation of parameters • Numerical solution of orbital equations

Red color – perturbed orbit in specific time range Blue color – unperturbed Kepler’s orbit

t  0, 40T t  40T, 80T t  80T, 120T t  120T, 160T

Orbital Mechanics 190 Space for Education, Education for Space 3. Orbital perturbations Examples of orbits • Sun-synchronous orbits • Earth rotates counterclockwise around the Sun with angular velocity 0.986° per day • if satellite orbit rotates clockwise with the same angular velocity, position of orbit relative to the Sun will be still the a / R same 2 d  3  R    2 nJ 2   cos i dt 21 e2   a 

7 / 2 d  3   R    J 2   cos i dt 2 R3  a  i

Orbital Mechanics 191 Space for Education, Education for Space 3. Orbital perturbations Examples of orbits • Sun-synchronous orbits • Earth rotates counterclockwise around the Sun with angular velocity 0.986° per day • if satellite orbit rotates clockwise with the same angular velocity, position of orbit relative to the Sun will be still the same a / R

imin  95.6 for a  R

amax 12331 km for i 180

Operating S-s satellites: h  700 900 km orbit: circular or near circular

i

Orbital Mechanics 192 Space for Education, Education for Space 3. Orbital perturbations Examples of orbits • Sun-synchronous orbits Landsat – 4: i  99.07 a  7285.799km

Blue: Kepler’s orbit view from Sun Red: sun-synchronous orbit Orange: Sun and sun beam

Orbital Mechanics 193 Space for Education, Education for Space 3. Orbital perturbations Examples of orbits • Sun-synchronous orbits Landsat – 4: i  99.07 a  7285.799km

view from Earth

Orbital Mechanics 194 Space for Education, Education for Space 3. Orbital perturbations Examples of orbits • Sun-synchronous orbits Landsat – 4: i  99.07 a  7285.799km

Orbital Mechanics 195 Space for Education, Education for Space 4. Orbital maneuvers

• Impulsive maneuvers • Hohmann transfer • Non-Hohmann transfer • Plane change maneuvers

Orbital Mechanics 196 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • brief firings of rocket motors change the magnitude and direction of the velocity vector instantaneously • during an impulsive maneuver, the position of the spacecraft is considered to be fixed, only the velocity changes impulsive maneuver is an idealization • velocity increment is related to consumed propellant

 m  m   S 0 S  vS  ueLn   mS 0 

Orbital Mechanics 197 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • brief firings of rocket motors change the magnitude and direction of the velocity vector instantaneously • during an impulsive maneuver, the position of the spacecraft is considered to be fixed, only the velocity changes impulsive maneuver is an idealization • velocity increment is related to consumed propellant ue  I s g  m  m   S 0 S  vS  ueLn   mS 0 

Orbital Mechanics 198 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • brief firings of rocket motors change the magnitude and direction of the velocity vector instantaneously • during an impulsive maneuver, the position of the spacecraft is considered to be fixed, only the velocity changes impulsive maneuver is an idealization • velocity increment is related to consumed propellant ue  I s g v  S  m  m  m I g v  u Ln S 0 S  S  1 e S S e   m  mS 0  S 0

Orbital Mechanics 199 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • specific impulse characteristics

mS / m S0 [-] Propellant Specific impulse Is [s] cold gas 50

I s  50 s Monopropellant 230 hydrazine I  230 s s LOX/LH2 455

I s  455 s Ion propulsion >3000

v [m/s] ue  I s g S v  S  m  m  m I g v  u Ln S 0 S  S  1 e S S e   m  mS 0  S 0

Orbital Mechanics 200 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • impulse at periapsis rP  rEarth  300km

3 v  7.8km/s 2 vP P1 e  0.019 1 1 A P

rP ,vP1

vP

Orbital Mechanics 201 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • impulse at periapsis rP  rEarth  300km

3 v  7.8km/s 2 vP P1 e  0.019 1 1 A P

rP ,vP1 vP2  vP1  vP

vP

Orbital Mechanics 202 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • impulse at periapsis rP  rEarth  300km

3 v  7.8km/s 2 vP P1 e  0.019 1 1 A P

rP ,vP1 vP2  vP1  vP

vP h2  rPvP2

Orbital Mechanics 203 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • impulse at periapsis rP  rEarth  300km

3 v  7.8km/s 2 vP P1 e  0.019 1 1 A P

rP ,vP1 vP2  vP1  vP

vP h2  rPvP2 e2 new orbit

Orbital Mechanics 204 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • impulse at periapsis rP  rEarth  300km

3 v  7.8km/s 2 vP P1 e  0.019 1 1 A P vP3  2km/s vP2 1km/s

vP2  8.8km/s vP3  9.8km/s

e2  0.297 e3  0.609 rP ,vP1 rA3  12331.4km rA3  27482.1km vP2  vP1  vP

vP h2  rPvP2 e2 new orbit

Orbital Mechanics 205 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • impulse at apoapsis rP  rEarth  300km 5 vP2  8.8km/s

e2  0.297 3 2 v  0.4km/s 4 A3 A vA4  0.919km/s

vA5  2.km/s

vA

Orbital Mechanics 206 Space for Education, Education for Space 4. Orbital maneuvers Impulsive maneuvers • impulse at apoapsis rP  rEarth  300km 5 vP2  8.8km/s

e2  0.297 3 2 v  0.4km/s 4 A3 A vA4  0.919km/s

vA5  2.km/s

e  0.174 vA 3

e4  0 circle

e5  0.416 A  P

Orbital Mechanics 207 Space for Education, Education for Space 4. Orbital maneuvers Hohmann transfer • 2 impulse maneuvers circle orbit 1: circle orbit 2 - GEO: r  r 1000km r  r  42164km 2 1 Earth 2 GEO

v1   / r1 v2   / r1 HT vHT1 Hohmann transfer:

1 r1

r2

vHT2

Orbital Mechanics 208 Space for Education, Education for Space 4. Orbital maneuvers Hohmann transfer • 2 impulse maneuvers circle orbit 1: circle orbit 2 - GEO: r  r 1000km r  r  42164km 2 1 Earth 2 GEO

v1   / r1 v2   / r1 HT vHT1 Hohmann transfer:

1 r1 aHT hHT r2 eHT

vHT2

Orbital Mechanics 209 Space for Education, Education for Space 4. Orbital maneuvers Hohmann transfer • 2 impulse maneuvers circle orbit 1: circle orbit 2 - GEO: r  r 1000km r  r  42164km 2 1 Earth 2 GEO

v1   / r1 v2   / r1 HT vHT1 Hohmann transfer:

1 r1 aHT hHT r2 eHT

vHT2

vHT1  2.24km/s

vHT2 1.39km/s

Orbital Mechanics 210 Space for Education, Education for Space 4. Orbital maneuvers Non-Hohmann transfer

• 2 impulse maneuvers • transfer between 2 elliptical orbits • elliptical orbits are in the same plane and are coaxial • locations of points A and B are defined by true anomaly  A a  B

point B

orbit 1 point A

orbit 2

Orbital Mechanics 211 Space for Education, Education for Space 4. Orbital maneuvers Non-Hohmann transfer

• 2 impulse maneuvers • transfer between 2 elliptical orbits • elliptical orbits are in the same plane and are coaxial • locations of points A and B are defined by true anomaly  A a  B

transfer point B trajectory transfer trajectory

orbit 1 point A h2 1 h2 1 rA  rB   1 ecos A   1 ecos B  orbit 2

h, e

Orbital Mechanics 212 Space for Education, Education for Space 4. Orbital maneuvers Non-Hohmann transfer

• 2 impulse maneuvers • transfer between 2 elliptical orbits • elliptical orbits are in the same plane and are coaxial • locations of points A and B are defined by true anomaly  A a  B • transfer orbit h, e  point B vTB  vTB orbit 1 point A

orbit 2

Orbital Mechanics 213 Space for Education, Education for Space 4. Orbital maneuvers Non-Hohmann transfer

• 2 impulse maneuvers • transfer between 2 elliptical orbits • elliptical orbits are in the same plane and are coaxial • locations of points A and B are defined by true anomaly  A a  B • transfer orbit h, e  point B vTB   v2B vTB orbit 1 point A

orbit 2

Orbital Mechanics 214 Space for Education, Education for Space 4. Orbital maneuvers Non-Hohmann transfer

• 2 impulse maneuvers • transfer between 2 elliptical orbits • elliptical orbits are in the same plane and are coaxial • locations of points A and B are defined by true anomaly  A a  B • transfer orbit h, e  point B v TB   v vB  B   v2B vTB v2B orbit 1 point A

    orbit 2 vB  v2B  vTB .v2B vTB 

      vB  v2Bv2B  vTB vTB  2v2BvTB

Orbital Mechanics 215 Space for Education, Education for Space 4. Orbital maneuvers Non-Hohmann transfer

• 2 impulse maneuvers • transfer between 2 elliptical orbits   • elliptical orbits are in the same plane and are vTA vA coaxial  • locations of points A and B are defined by v 1A true anomaly  A a  B • transfer orbit h, e  point B v TB   v vB  B v2B orbit 1 point A

orbit 2

Orbital Mechanics 216 Space for Education, Education for Space 4. Orbital maneuvers Plane change maneuvers • single impulse maneuver • To change the orientation of a satellite's orbital plane, typically the inclination, the direction of the velocity vector has to be changed.

orbit 2

orbit 1

Orbital Mechanics 217 Space for Education, Education for Space 4. Orbital maneuvers Plane change maneuvers • single impulse maneuver • To change the orientation of a satellite's orbital plane, typically the inclination, the direction of the velocity vector has to be changed.

 v1 orbit 2

orbit 1

Orbital Mechanics 218 Space for Education, Education for Space 4. Orbital maneuvers Plane change maneuvers • single impulse maneuver • To change the orientation of a satellite's orbital plane, typically the inclination, the direction of the velocity vector has to be changed.

 v1 orbit 2  v2

orbit 1

Orbital Mechanics 219 Space for Education, Education for Space 4. Orbital maneuvers Plane change maneuvers • single impulse maneuver • To change the orientation of a satellite's orbital plane, typically the inclination, the direction of the velocity vector has to be changed.

    v  v2 v1 v1  orbit 2 v pure rotation  v2     is angle between the v  2 vsin  planes  2  orbit 1

Orbital Mechanics 220 Space for Education, Education for Space