Orbital MechanicsSpace for Education, Education for Space
Space for Education, Education for SpaceESA Contract No. 4000117400/16NL/NDe
Specialized lectures
Orbital Mechanics
Vladimír Kutiš, Pavol Valko
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
2. Orbits in three dimensions
3. Orbital perturbations
4. Orbital maneuvers
Contents
2
Orbital MechanicsSpace for Education, Education for Space
• Motion in inertial frame
• Relative motion
• Angular momentum
• Solution of problem
• Energy law
• Trajectories
• Time and position
1. The two body problem
3
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
Motion in inertial frame
4
rr
mmGFF
3
211221
position of masses gravitational forces
1m
2mr
21F
12F
2311 s kg/m 106742.6 Guniversal gravitational constant
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
Motion in inertial frame
5
rr
mmGFF
3
211221
position of masses gravitational forces
1m
2mr
21F
12F
2311 s kg/m 106742.6 Guniversal gravitational constant
1st time measured by Cavendish, 1798
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
Motion in inertial frame
6
rr
mmGFF
3
211221
position of masses gravitational forces
1m
2mr
21F
12F
2311 s kg/m 106742.6 Guniversal gravitational constant
conservative force can be expressed by potential energy
r
mmGE p
21
1st time measured by Cavendish, 1798
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
Motion in inertial frame
7
rr
mmGFF
3
211221
rr
mFF
3
21221
can be measured with considerable precision by astronomical observation
Central body [m3/s2]
Earth 3.98600441 x 1014
Moon 4.90279888 x 1012
Mars 4.2871 x 1013
Sun 1.327124 x 1020
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
Motion in inertial frame
8
rr
mmGFF
3
211221
2
02
zr
rg
r
mGg
E
EE
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
Motion in inertial frame
9
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
Motion in inertial frame
10
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
11
inertial frame of reference
1m
2m
r
2R
1R
rr
mmGFF
3
211221
2122 FRm
1211 FRm
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
12
inertial frame of reference
1m
2m
r
2R
1R
rr
mmGFF
3
211221
2122 FRm
1211 FRm
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
13
inertial frame of reference
1m
2m
r
2R
1R
rr
mmGFF
3
211221
2122 FRm
GR
center of mass
1211 FRm
21
2211
mm
RmRmRG
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
14
inertial frame of reference
1m
2m
r
2R
1R
rr
mmGFF
3
211221
2122 FRm
GR
center of mass21
2211
mm
RmRmRG
21
2211
mm
RmRmRG
2 x time derivative
1211 FRm
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
15
inertial frame of reference
1m
2m
r
2R
1R
rr
mmGFF
3
211221
2122 FRm
GR
center of mass21
2211
mm
RmRmRG
21
2211
mm
RmRmRG
2 x time derivative
0
GR
center of mass is:• motionless• or motion is in
straight line with constant velocity
1211 FRm
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
16
rr
mmGFF
3
211221
21
2211
mm
RmRmRG
0
GR
center of mass is:• motionless• or motion is in
straight line with constant velocity
2122 FRm
1211 FRm
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
17
rr
mmGFF
3
211221
21
2211
mm
RmRmRG
0
GR
center of mass is:• motionless• or motion is in
straight line with constant velocity
2122 FRm
1211 FRm
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
18
rr
mmGFF
3
211221
2122 FRm
1211 FRm
rr
mmGRm
3
2111 r
r
mmGRm
3
2122
inertial frame of reference
1m
2m
r
2R
1R
i
j
k
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Motion in inertial frame
19
rr
mmGFF
3
211221
2122 FRm
1211 FRm
rr
mmGRm
3
2111 r
r
mmGRm
3
2122
modification of equations
03
rr
r
21 mmG
1Gmif: 21 mm
inertial frame of reference
1m
2m
r
2R
1R
i
j
k
gravitational parameter
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Relative motion
20
03
rr
r
kzzjyyixxr
)()()( 121212
vector defined in inertial frame of referenceexpressed in coord. system
r
kji
inertial frame of reference
1m
2m
r
2R
1R
i
j
k
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Relative motion
21
03
rr
r
inertial frame of reference
1m
2m
r
2R
1R
i
j
k
1i
1j
1k
vector can be expressed in coord. system , that rotates about inertial coord. system with instant angular velocity and instant angular acceleration
r
111 kji
121121121 )()()( kzjyixr
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Relative motion
22
03
rr
r
121121121 )()()( kzjyixr
2 x time derivative in inertial frame of reference
relrel vrrrr
2
inertial frame of reference
1m
2m
r
2R
1R
i
j
k
1i
1j
1k
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Relative motion
23
03
rr
r
121121121 )()()( kzjyixr
2 x time derivative in inertial frame of reference
relrel vrrrr
2
if is not rotating coord. system
111 kji
relrr
inertial frame of reference
1m
2m
r
2R
1R
i
j
k
1i
1j
1k
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Two body problem can by defined by:
– Newton’s law of gravitation
– Newton's laws of motion
Relative motion
24
03
rr
r
relrr
relative acceleration of
moving (non-rotating) frame of reference in coord. components
21321
21321
21321
zr
z
yr
y
xr
x
+ 6 initial conditions
inertial frame of reference
1m
2m
r
2R
1R
i
j
k
1i
1j
1k
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• relative angular momentum of body per unit mass
Angular momentum
25
1m
2m
r
rrrmrm
h
2
2
1
rrrrrrdt
hd
1 x time derivative
r
2m
trajectory
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• relative angular momentum of body per unit mass
Angular momentum
26
rrrmrm
h
2
2
1
rrrrrrdt
hd
1 x time derivative
03
rr
r
1m
2m
r
r
trajectory
2m
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• relative angular momentum of body per unit mass
Angular momentum
27
rrrmrm
h
2
2
1
rrrrrrdt
hd
1 x time derivative
03
rr
r
0
dt
hdangular momentum is conserved1m
2m
r
r
trajectory
2m
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• relative angular momentum of body per unit mass
Angular momentum
28
angular momentum can be expressed as
velocity vector can be expressed as
rvvvr
vrvrvrrrh r
1m
2m
r
r
v rv
trajectory
2m
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• relative angular momentum of body per unit mass
Angular momentum
29
angular momentum can be expressed as
velocity vector can be expressed as
rvvvr
vrvrvrrrh r
11 khkrvvrh
1k
• unit vector• time invariant
h rvh
• magnitude of angular momentum • time invariant
1m
2m
r
r
v rv
trajectory
2m
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• relative angular momentum of body per unit mass
Angular momentum
30
11 khkrvvrh
1k
• unit vector• time invariant
h • magnitude of angular momentum • time invariant
1m
2m
r
r
trajectory
v rv
1i
1j
• Cartesian coord.system
2m
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• relative angular momentum of body per unit mass
Angular momentum
31
11 khkrvvrh
1k
• unit vector• time invariant
h
rrdt
dv
• magnitude of angular momentum • time invariant
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
2m
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• relative angular momentum of body per unit mass
Angular momentum
32
11 khkrvvrh
1k
• unit vector• time invariant
h
rrdt
dv
• magnitude of angular momentum • time invariant
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
11
2
1 khkrkrrh
2m
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
33
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
03
rr
r
cross product with h
hrr
hr
3
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
34
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
03
rr
r
cross product with h
hrr
hr
3
1
2 krh
rirr
and expressed by polar coordinatesr
h
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
35
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
03
rr
r
cross product with h
hrr
hr
3
1
2 krh
rirr
jhr
and expressed by polar coordinatesr
h
Orbital MechanicsSpace for Education, Education for Space
• Equation of orbit
angular momentum is const. vector
1. The two body problemSolution of problem
36
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
03
rr
r
cross product with h
hrr
hr
3
1
2 krh
rirr
jhr
0
dt
hd
and expressed by polar coordinatesr
h
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
37
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
03
rr
r
cross product with h
hrr
hr
3
1
2 krh
rirr
jhr
0
dt
hd
j
dt
dhr
dt
d
and expressed by polar coordinatesr
h
angular momentum is const. vector
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
38
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
11 sin cos jiir
j
dt
dhr
dt
d
11 cos sin jij
• Unit vectors of polar coord. system are not constant vectors
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
39
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
11 sin cos jiir
j
dt
dhr
dt
d
11 cos sin jid
id r
11 cos sin jij
• Unit vectors of polar coord. system are not constant vectors
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
40
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
11 sin cos jiir
j
dt
dhr
dt
d
11 cos sin jid
id r
11 cos sin jij
jd
id r
• Unit vectors of polar coord. system are not constant vectors
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
41
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
11 sin cos jiir
j
dt
dhr
dt
d
11 cos sin jid
id r
11 cos sin jij
jd
id r
ridt
dhr
dt
d
• Unit vectors of polar coord. system are not constant vectors
• is scalar – time invariant parameter
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
42
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
j
dt
dhr
dt
d
ridt
dhr
dt
d
eihr r
• is scalar – time invariant parameter
• is integration constant, i.e. const. vector
e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
43
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
j
dt
dhr
dt
d
ridt
dhr
dt
d
eihr r
eirhrr r
• is integration constant, i.e. const. vector
e
• dot product with r
• is scalar – time invariant parameter
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
44
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
j
dt
dhr
dt
d
ridt
dhr
dt
d
eihr r
eirhrr r
cos1 erhrr
• dot product with r
• is integration constant, i.e. const. vector
e
cbacba• using
• is scalar – time invariant parameter
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
45
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
j
dt
dhr
dt
d
ridt
dhr
dt
d
eihr r
eirhrr r
cos1 erhrr
cos1
1
2
e
hr
cbacba• using
• dot product with r
• is integration constant, i.e. const. vector
e
• is scalar – time invariant parameter
• Scalar equation of orbit• is eccentricity
r
e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
46
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
cos1
1
2
e
hr
rvh
cos1 ehr
hv
• Scalar equation of orbit• is eccentricity
r
e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Equation of orbitSolution of problem
47
1m
2m
r
r
trajectory
v rv
1i
1j
ri
j
• Cartesian coord.system
• Polar coord.system
cos1
1
2
e
hr
rvh
cos1 ehr
hv
sin ehdt
drrvr
• Scalar equation of orbit• is eccentricity
r
e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• energy of system written in inertial frame of reference placed in center of mass
Energy law
48
1m
2mr
inertial frame of reference
pkktot EEEE 21
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• energy of system written in inertial frame of reference placed in center of mass
Energy law
49
1m
2mr
inertial frame of reference
pkktot EEEE 21
r
mmGvmvmE mmtot
212
22
2
112
1
2
1 expressed by inertial
motion
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• energy of system written in inertial frame of reference placed in center of mass
Energy law
50
1m
2mr
inertial frame of reference
pkktot EEEE 21
r
mmGvmvmE mmtot
212
22
2
112
1
2
1
r
mmGv
mm
mmEtot
212
21
21
2
1
expressed by inertial motion
expressed by relative motion
21
21
mm
mm
reduced mass of system
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• energy of system written in inertial frame of reference placed in center of mass
Energy law
51
1m
2mr
inertial frame of reference
pkktot EEEE 21
r
mmGvmvmE mmtot
212
22
2
112
1
2
1
r
mmGv
mm
mmEtot
212
21
21
2
1
expressed by inertial motion
expressed by relative motion
21
21
mm
mm
reduced mass of system
r
v
2
2specific orbital energy (total energy per unit reduced mass)vis viva equation
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• energy of system written in inertial frame of reference placed in center of mass
Energy law
52
1m
2mr
inertial frame of reference
pkktot EEEE 21
r
mmGvmvmE mmtot
212
22
2
112
1
2
1
r
mmGv
mm
mmEtot
212
21
21
2
1
expressed by inertial motion
expressed by relative motion
r
v
2
2specific orbital energy (total energy per unit reduced mass)vis viva equation
specific energy expressed by
2
2
2
12
1e
h
e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Shape of trajectory depends on eccentricity
• Equation of orbit is equation of conic sections:
– circle
– ellipse
– parabola
– hyperbola
Trajectories
53
e
0e
10 e
1e
1e
cos1
1
2
e
hr
• equation of orbit
0e 10 e 1e 1e 0h
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Circle: (bounded trajectory)Trajectories
54
0e
cos1
1
2
e
hr
• equation of orbit
• speed ofmotion
• period
•specificenergy
2hr
r
cos1 eh
vv
rv
r
rT
/
2
2/32
rT
2
2
2
12
1e
h
r
2
1
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Circle: (bounded trajectory)Trajectories
55
0e
2/32rT
•trajectories of satellite in different altitude passed in time EarthT
central body circ. velocity[km/s]
circ. period [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
rv
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Circle: (bounded trajectory)Trajectories
56
0e
s 86164 GEOT
km42164GEOr
km/s07.3GEOv
•trajectories of satellite in different altitude passed in in time .min48.84EarthT
2/32rT
rv
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
57
10 e
a – semimajor axis
empty focus
b – semiminor axis
P -periapsisA - apoapsis
C - center
F - focus
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
58
10 e
cos1
1
2
e
hr
1
1
2
e
hrP
1
1
2
e
hrA
1
1
e
e
r
r
A
P
PA
PA
rr
rre
P -periapsisa – semimajor axis
F - focus
empty focus
b – semiminor axis
A - apoapsis
r
-true anomaly
C - center
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
59
10 e
cos1
1
2
e
hr
1
1
2
e
hrP
1
1
2
e
hrA
AP rra 2
2
2
-1
1
e
ha
P -periapsisa – semimajor axis
F - focus
empty focus
b – semiminor axis
A - apoapsis
r
-true anomaly
C - center
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
60
10 e
P -periapsisa – semimajor axis
F - focus
empty focus
b – semiminor axis
A - apoapsis
r
-true anomaly
AP rra 2 21 -eab
PA
PA
rr
rre
2 PA rrCF
C - center
eaCF 222 CFab
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
61
10 e• eccentricity • flattening
2
222
a
bae
a
baf
211 ef
1.0 e 1.0 f
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
62
10 e• eccentricity • flattening
1.0 e 1.0 f
usage: description of orbits
usage: description of planet shape
298.257 /1 f
flattening of Earth:
21.4 km diff. in radius
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
63
cos1
1
2
e
hr
• equation of orbit
• speed ofmotion
• period
•specificenergy 2
2
2
12
1e
h
a
2
1
10 e
2
32
aT
h
abT
2
r
v
2
2
rav
2
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
64
10 e
• ellipses with equal semimajor axis :a
a
2
1
2
32
aT
equal period andorbital energy
location of orbits
shape of ellipses
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Ellipse: (bounded trajectory)Trajectories
65
10 e
• ellipses with equal semimajor axis :a
a
2
1
2
32
aT
equal period andorbital energy
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
rav
2
Orbital MechanicsSpace for Education, Education for Space
• Parabola: (open trajectory)
- true anomaly
r
1. The two body problemTrajectories
66
1e
F - focus
directrix
P -periapsis
cos11
1
2
hr
• equation of orbit
• speed ofmotion
•specificenergy 2
2
2
12
1e
h
0
r
v
2
2
rv
2
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Parabola: (open trajectory)Trajectories
67
1e
central body esc. velocity[km/s]
Earth 11.18
Moon 2.37
Mars 5.02
Sun (surface) 617.5
Sun (Earths) 42.12
rv
2Earthp Rh
10
1
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Parabola: (open trajectory)Trajectories
68
1e•trajectories of satellite in different
Earthp Rh10
1
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
Earthp Rh10
1
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Parabola: (open trajectory)Trajectories
69
1e•trajectories of satellite in different
Earthp Rh10
1
Earthp Rh10
1 Earthp Rh
10
1
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Parabola: (open trajectory)Trajectories
70
1e
Earthp Rh10
1 Earthp Rh
10
1 Earthp Rh
10
1
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Parabola: (open trajectory)Trajectories
71
1eEarthp Rh
10
1
Earthp Rh10
1
Earthp Rh10
1
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Hyperbola: (open trajectory)Trajectories
72
1e
F- focusempty focus
asymptotes
vertex
a- semimajor axis
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Hyperbola: (open trajectory)Trajectories
73
1e
F- focusempty focus
asymptotes
vertex
a- semimajor axis
r
• equation of orbit
• speed ofmotion
•specificenergy
2
2
2
12
1e
h
r
v
2
2
cos1
1
2
e
hr
a2
rav
22
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Hyperbola: (open trajectory)Trajectories
74
1e•trajectories of satellite with different• periapsis:
1.0e
Earthp Rr
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 MechanicsSpace for Education, Education for Space
1. The two body problem
• Hyperbola: (open trajectory)Trajectories
75
1e•trajectories of satellite with different• periapsis:
1.0e
Earthp Rr
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 MechanicsSpace for Education, Education for Space
1. The two body problem
• Hyperbola: (open trajectory)Trajectories
76
1e•trajectories of satellite with different alt.• eccentricity: 1.1e
Earthp Rh 1.0
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 MechanicsSpace for Education, Education for Space
1. The two body problem
• Hyperbola: (open trajectory)Trajectories
77
1e•trajectories of satellite with different alt.• eccentricity: 1.1e
Earthp Rh 1.0
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 MechanicsSpace for Education, Education for Space
1. The two body problem
• 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
Time and position
78
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• True, Mean and Eccentric anomaliesTime and position
79
orbit
auxiliary circle
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• True, Mean and Eccentric anomaliesTime and position
80
orbit
auxiliary circle
location of satellite
true anomaly
focus
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• True, Mean and Eccentric anomaliesTime and position
81
orbit
auxiliary circle
eM
location of satellite
virtual location on circle with const. motion with the same period as satellite has
true anomaly
meananomaly
eM
focus
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• True, Mean and Eccentric anomaliesTime and position
82
orbit
auxiliary circle
eE
eM
location of satelliteprojection of location on circle
true anomaly
eccentricanomaly eE
focus
virtual location on circle with const. motion with the same period as satellite has
meananomaly
eM
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of positionTime and position
83
22 rdt
drh
• using mean anomaly
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of positionTime and position
84
22 rdt
drh
drh
dt 21
• using mean anomaly
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of positionTime and position
85
22 rdt
drh
drh
dt 21
cos1
1
2
e
hr
• using mean anomaly
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of positionTime and position
86
22 rdt
drh
drh
dt 21
cos1
1
2
e
hr
22
3
cos1
e
dhdt
• using mean anomaly
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of positionTime and position
87
22 rdt
drh
drh
dt 21
cos1
1
2
e
hr
22
3
cos1
e
dhdt
0
22
3
cos1 e
dhtt p
• using mean anomaly
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
88
0pt
0
22
3
cos1 e
dhtt p
10 e• using mean anomaly
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
89
0pt
0
22
3
cos1 e
dhtt p
10 e
cos1
sin1
2tan
1
1tan2
1
1
cos1
21
2/320
2 e
ee
e
e
ee
d
• using mean anomaly
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
90
0pt
0
22
3
cos1 e
dhtt p
10 e
cos1
sin1
2tan
1
1tan2
1
1
cos1
21
2/320
2 e
ee
e
e
ee
d
• using mean anomaly
eM
ee
d2/32
0
21
1
cos1
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
91
10 e• using mean anomaly
]rad[
]rad[ eM
(circle) 0e
15.0e
eM
e
ht
2/322
3
1
1
cos1
sin1
2tan
1
1tan2
21
e
ee
e
eM e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
92
10 e• using mean anomaly
]rad[
]rad[ eM
cos1
sin1
2tan
1
1tan2
21
e
ee
e
eM e
.5760e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
93
10 e
cos1
sin1
2tan
1
1tan2
21
e
ee
e
eM e
eM
e
ht
2/322
3
1
1
• using eccentric anomaly
eE eEsin
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
94
10 e
cos1
sin1
2tan
1
1tan2
21
e
ee
e
eM e
eM
e
ht
2/322
3
1
1
• using eccentric anomaly
eE eEsin
EeEM e sin
• Kepler’s equation
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
95
10 e
2tan
1
1tan2 1
e
eEe
• using eccentric anomaly
]rad[
]rad[ eE
(circle) 0e
15.0e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
96
10 e• using eccentric anomaly
]rad[
]rad[ E
2tan
1
1tan2 1
e
eEe
.5760e
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
97
10 e
2tan
1
1tan2 1
e
eEe
• defined
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
98
10 e
2tan
1
1tan2 1
e
eEe
• defined
EeEM e sin
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
99
10 e
2tan
1
1tan2 1
e
eEe
• defined
EeEM e sin
eM
e
ht
2/322
3
1
1
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
100
10 e
2tan
1
1tan2 1
e
eEe
• defined
EeEM e sin
eM
e
ht
2/322
3
1
1
eMT
t2
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Time as a function of position - ellipseTime and position
101
10 e
2tan
1
1tan2 1
e
eEe
• defined
EeEM e sin
eM
e
ht
2/322
3
1
1
eMT
t2
ntM e
Tn
2
• average angular velocity
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Position as a function of time - ellipseTime and position
102
tT
M e
2
t
10 e
• defined
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Position as a function of time - ellipseTime and position
103
eee EeEM sin
tT
M e
2
t
10 e
• defined
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Position as a function of time - ellipseTime and position
104
• must be computed numerically
eee EeEM sin
tT
M e
2
t
10 e
• defined
eE
Orbital MechanicsSpace for Education, Education for Space
1. The two body problem
• Position as a function of time - ellipseTime and position
105
• must be computed numerically
eee EeEM sin
tT
M e
2
t
10 e
• defined
• the problem of finding true anomaly for defined time is called Kepler’s problem
eE
2tan
1
1tan2 1 eE
e
e
Orbital MechanicsSpace for Education, Education for Space
• Frame of reference
• Earth-based systems
• Orbital elements
• Calculation of elements
2. Orbits in three dimensions
106
Orbital MechanicsSpace for Education, Education for Space
Frame of reference
107
2. Orbits in three dimensions
• 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 MechanicsSpace for Education, Education for Space
Frame of reference
108
2. Orbits in three dimensions
• 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 MechanicsSpace for Education, Education for Space
Earth-based systems
109
2. Orbits in three dimensions
• 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 MechanicsSpace for Education, Education for Space
Earth-based systems
110
2. Orbits in three dimensions
• 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 MechanicsSpace for Education, Education for Space
Earth-based systems
111
2. Orbits in three dimensions
• 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 MechanicsSpace for Education, Education for Space
Orbital elements
112
2. Orbits in three dimensions
• Location of the satellite:
1. the location of the orbital plane in defined coord. system of chosen frame of reference
2. the position of the elliptical orbit in this plane
3. the characteristics of ellipse
4. the position of the moving satellite on the orbit
i ,
)(or , ahe
)(or M
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
113
2. Orbits in three dimensions
• the goal is to determine orbital elements from:• position vector
• velocity vector
• both vectors are defined in GES at time
kvjvivv zyx
0t
krjrirr zyx
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
114
2. Orbits in three dimensions
vr
and
kvjvivv zyx
krjrirr zyx
state vector
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
115
2. Orbits in three dimensions
vr
and zyx
zyx
vvv
rrr
kji
vrh
kvjvivv zyx
krjrirr zyx
1st element
state vector
h
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
116
2. Orbits in three dimensions
vr
and
2nd element
h
hi z1cos
kvjvivv zyx
krjrirr zyx
1st element
state vector
khjhihh zyx
h
i
Orbital MechanicsSpace for Education, Education for Space
kvjvivv zyx
Calculation of elements
117
2. Orbits in three dimensions
h
vr
and
i
vvrr
rve
21
krjrirr zyx
khjhihh zyx
3th element
2nd element
1st element
state vector
e
Orbital MechanicsSpace for Education, Education for Space
khjhihh zyx
Calculation of elements
118
2. Orbits in three dimensions
h
vr
and e
i vector of node line
zyx hhh
kji
hkn 100
kvjvivv zyx
krjrirr zyx
kejeiee zyx
3th element
2nd element
1st element
state vector
n
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
119
2. Orbits in three dimensions
h
vr
and e
n
i 4th element
kvjvivv zyx
krjrirr zyx
khjhihh zyx
kejeiee zyx
kjninn yx
0
n
nx1cos
3th element
2nd element
1st element
state vector
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
120
2. Orbits in three dimensions
h
vr
and e
n
i
5th element
kvjvivv zyx
krjrirr zyx
khjhihh zyx
kejeiee zyx
kjninn yx
0
e
e
n
n
1cos
4th element
3th element
2nd element
1st element
state vector
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
121
2. Orbits in three dimensions
h
vr
and e
n
i
6th element
kvjvivv zyx
krjrirr zyx
khjhihh zyx
kejeiee zyx
kjninn yx
0
r
r
e
e
1cos
5th element
4th element
3th element
2nd element
1st element
state vector
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
122
2. Orbits in three dimensions
input parameters:
Example:
km/s 3.435 1.581, 7.556,-
km 1567.56 6174.08, 2004.75,
p
p
v
r
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
123
2. Orbits in three dimensions
input parameters:
0
30
45
196.0
28
s/km 56430.1 2
e
i
hExample:
km/s 3.435 1.581, 7.556,-
km 1567.56 6174.08, 2004.75,
p
p
v
r
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
124
2. Orbits in three dimensions
input parameters:
0
30
45
196.0
28
s/km 56430.1 2
e
i
hExample:
Orbit in 2D view:
km/s 3.435 1.581, 7.556,-
km 1567.56 6174.08, 2004.75,
p
p
v
r
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
125
2. Orbits in three dimensions
input parameters:
Example:
Orbit in 3D view:
km/s 3.435 1.581, 7.556,-
km 1567.56 6174.08, 2004.75,
p
p
v
r
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
126
2. Orbits in three dimensions
input parameters:
km/s 3.435 1.581, 7.556,-
km 1567.56 6174.08, 2004.75,
p
p
v
r
Example:
Orbit in 2D map: Orbit in 3D view:
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
127
2. Orbits in three dimensions
input parameters:
Example: GEO
Orbit in 2D map: Orbit in 3D view:
GEO, circular orbit,
2.5i
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
128
2. Orbits in three dimensions
input parameters:
Example: GEO
Orbit in 2D map:Detail view:
GEO, circular orbit,
2.5i
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
129
2. Orbits in three dimensions
input parameters:
Example: GEO
Orbit in 2D map:Detail view:
GEO,
0i
01575.0e
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
130
2. Orbits in three dimensions
input parameters:
Example: GEO
Orbit in 2D map:Detail view:
GEO,
5.2i
01575.0e
Orbital MechanicsSpace for Education, Education for Space
Calculation of elements
131
2. Orbits in three dimensions
input parameters:
Example: Molnija
Orbit in 2D map:
41.63i
75.0e
Orbit in 3D view:
km 40089ahkm260ph
Orbital MechanicsSpace for Education, Education for Space
• Perturbing forces
• Geopotential
• Orbit propagation
• Variation of parameters
• Examples of orbits
3. Orbital perturbations
132
Orbital MechanicsSpace for Education, Education for Space
Perturbing forces
133
3. Orbital perturbations
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 MechanicsSpace for Education, Education for Space
Perturbing forces
134
3. Orbital perturbations
Perturbing forces
Conservative forces –can be derived from potential:• flattening of the Earth• Attraction of the Moon• Attraction of the Sun• Attraction by other planets
Non-conservative forces –cannot be derived from potential – dissipative forces:• atmospheric drag• radiation pressure
Orbital MechanicsSpace for Education, Education for Space
Perturbing forces
135
3. Orbital perturbations
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
sou
rce:
Cap
de
rou
: H
and
bo
ok
of
Sate
llite
Orb
its
Orbital MechanicsSpace for Education, Education for Space
Geopotential
136
3. Orbital perturbations
Potential of single mass point:
position of masses
1m
r
r
mmGE p
21
potential energy of mass in gravitational field of mass
1m2m2m
Orbital MechanicsSpace for Education, Education for Space
Geopotential
137
3. Orbital perturbations
Potential of single mass point:
position of masses
1m
r
r
mmGE p
21
potential energy of mass in gravitational field of mass
1m2m
rm
ErU
p
2
gravitational potential
2m
Orbital MechanicsSpace for Education, Education for Space
Geopotential
138
3. Orbital perturbations
Potential of single mass point:
position of masses
1m
r
r
mmGE p
21
potential energy of mass in gravitational field of mass
1m2m
rm
ErU
p
2
gravitational potential
Ur grad
equation of motion expressed by potential
2m
Orbital MechanicsSpace for Education, Education for Space
Geopotential
139
3. Orbital perturbations
Potential of Earth: 1. approximation - sphere
position of masses
M
2m
d
dMmGdEp
2
potential energy of mass in gravitational field of dM
2mdM
d
r
Orbital MechanicsSpace for Education, Education for Space
Geopotential
140
3. Orbital perturbations
Potential of Earth: 1. approximation - sphere
position of masses
M
2m
d
dMmGdEp
2
potential energy of mass in gravitational field of dM
2m
gravitational potential
dM
d
r
d
GdM
m
dEdU
p
2
Orbital MechanicsSpace for Education, Education for Space
Geopotential
141
3. Orbital perturbations
Potential of Earth: 1. approximation - sphere
position of masses
M
2m
d
dMmGdEp
2
potential energy of mass in gravitational field of dM
2m
gravitational potential
MMd
GdMdUU
dM
d
r
d
GdM
m
dEdU
p
2
Orbital MechanicsSpace for Education, Education for Space
Geopotential
142
3. Orbital perturbations
Potential of Earth: 1. approximation - sphere
position of masses
M
2m
d
dMmGdEp
2
potential energy of mass in gravitational field of dM
2m
gravitational potential
MMd
GdMdUU
integration over sphere boundary
equal potential as single mass potential
rr
GMrU
dM
d
r
d
GdM
m
dEdU
p
2
Orbital MechanicsSpace for Education, Education for Space
Geopotential
143
3. Orbital perturbations
Potential of Earth: 2. approximation - ellipsoid
position of masses
M
2mdM
d
r
Position of :• longitude• latitude • radius r
2m
Orbital MechanicsSpace for Education, Education for Space
Geopotential
144
3. Orbital perturbations
Potential of Earth: 2. approximation - ellipsoid
position of masses
M
2mdM
dM
Ellipsoid:• longitude• latitude • radius
d
r
Position of :• longitude• latitude • radius r
2m
Orbital MechanicsSpace for Education, Education for Space
Geopotential
145
3. Orbital perturbations
Potential of Earth: 2. approximation - ellipsoid
position of masses
M
2mdM
dM
Ellipsoid:• longitude• latitude • radius
d
r
Position of :• longitude• latitude • radius r
2m
2
cos21
rrrd
angle between andr
d
Orbital MechanicsSpace for Education, Education for Space
Geopotential
146
3. Orbital perturbations
Potential of Earth: 2. approximation - ellipsoid
position of masses
M
2mdM
dM
Ellipsoid:• longitude• latitude • radius
d
r
Position of :• longitude• latitude • radius r
2m
2
cos21
rrrd
angle between andr
d
M
d
GdMU
Orbital MechanicsSpace for Education, Education for Space
Geopotential
147
3. Orbital perturbations
Potential of Earth: 2. approximation - ellipsoid
position of masses
M
2mdM
d
r
M
d
GdMU
using:• expansion of 1/d in terms of Legendre polynomials• symmetric properties of ellipsoid
Orbital MechanicsSpace for Education, Education for Space
Geopotential
148
3. Orbital perturbations
Potential of Earth: 2. approximation - ellipsoid
position of masses
M
2mdM
d
r
M
d
GdMU
using:• expansion of 1/d in terms of Legendre polynomials• symmetric properties of ellipsoid
2
1sin31,,,
2
2
2
Jr
R
rrUrU
is equatorial radiusRdimensionless coefficient2J zx II
MRJ
22
13
2 100826.1 J
Orbital MechanicsSpace for Education, Education for Space
Geopotential
149
3. Orbital perturbations
Potential of Earth: expansion to higher degrees
• potential is function of all 3 coordinates, i.e. ,,rU
Orbital MechanicsSpace for Education, Education for Space
Geopotential
150
3. Orbital perturbations
Potential of Earth: expansion to higher degrees
• potential is function of all 3 coordinates, i.e. ,,rU
l
m
lmlmlm
l
l
PmSmCr
R
rrU
00
sinsincos,,
Orbital MechanicsSpace for Education, Education for Space
Geopotential
151
3. Orbital perturbations
l
m
lmlmlm
l
l
PmSmCr
R
rrU
00
sinsincos,,
parameters are obtained from precise observation of the motion of satellites
Potential of Earth: expansion to higher degrees
• potential is function of all 3 coordinates, i.e. ,,rU
Orbital MechanicsSpace for Education, Education for Space
Geopotential
152
3. Orbital perturbations
l
m
lmlmlm
l
l
PmSmCr
R
rrU
00
sinsincos,,
Legendre functions
parameters are obtained from precise observation of the motion of satellites
Potential of Earth: expansion to higher degrees
• potential is function of all 3 coordinates, i.e. ,,rU
Orbital MechanicsSpace for Education, Education for Space
Geopotential
153
3. Orbital perturbations
l
m
lmlmlm
l
l
PmSmCr
R
rrU
00
sinsincos,,
Legendre functions sinsin lmPm
sincos lmPm
products
parameters are obtained from precise observation of the motion of satellites
Potential of Earth: expansion to higher degrees
• potential is function of all 3 coordinates, i.e. ,,rU
Orbital MechanicsSpace for Education, Education for Space
Geopotential
154
3. Orbital perturbations
l
m
lmlmlm
l
l
PmSmCr
R
rrU
00
sinsincos,,
Legendre functions sinsin lmPm
sincos lmPm
im
lmlm PH e sin,
products
Complex functions called Spherical Harmonics
parameters are obtained from precise observation of the motion of satellites
Potential of Earth: expansion to higher degrees
• potential is function of all 3 coordinates, i.e. ,,rU
Orbital MechanicsSpace for Education, Education for Space
Geopotential
155
3. Orbital perturbations
l
m
lmlmlm
l
l
PmSmCr
R
rrU
00
sinsincos,,
parameters are obtained from precise observation of the motion of satellites
• for m=0: , , are called zonal harmonics,
• for m=l: are called sectoral harmonics
• all other functions are called tesseral harmonics
00 lS ,0lH
,llH
,lmH
ll JC 0
Potential of Earth: expansion to higher degrees
• potential is function of all 3 coordinates, i.e. ,,rU
Orbital MechanicsSpace for Education, Education for Space
Geopotential
156
3. Orbital perturbations
Potential of Earth: expansion to higher degrees
ZonalHarmonics
l=0
l=1
l=2
l=3
m=0 m=1 m=2 m=3
Spherical Harmonics (SH)
Orbital MechanicsSpace for Education, Education for Space
Geopotential
157
3. Orbital perturbations
Potential of Earth: expansion to higher degrees
SectoralHarmonics
l=0
l=1
l=2
l=3
m=0 m=1 m=2 m=3
Spherical Harmonics (SH)
Orbital MechanicsSpace for Education, Education for Space
Geopotential
158
3. Orbital perturbations
Potential of Earth: expansion to higher degrees
TesseralHarmonics
l=0
l=1
l=2
l=3
m=0 m=1 m=2 m=3
Spherical Harmonics (SH)
Orbital MechanicsSpace for Education, Education for Space
Geopotential
159
3. Orbital perturbations
Potential of Earth: expansion to higher degrees
l=0
l=1
l=2
l=3
m=0 m=1 m=2 m=3
Spherical Harmonics (SH)
geopotential model of Earth is using coefficients in SH expansion, for example Goddard Earth Model 10b (GEM10b) is using 21x21 SH expansion
Orbital MechanicsSpace for Education, Education for Space
Orbit propagation
160
3. Orbital perturbations
• the goal is to solve equation of motion with initial conditions
• potential U expresses influence of central acceleration and perturbative acceleration
• for example, perturbative potential
00 )0( and )0(
grad
rtrrtr
Ur
RUU 0
rU
0
2
1sin3 2
23
2
J
r
RR
Orbital MechanicsSpace for Education, Education for Space
Orbit propagation
161
3. Orbital perturbations
• 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 MechanicsSpace for Education, Education for Space
Variation of parameters
162
3. Orbital perturbations
• Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion
• Mathematical intro:
tgytfdt
dy
diff. equation with right hand side
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
163
3. Orbital perturbations
• Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion
• Mathematical intro:
tgytfdt
dy
diff. equation with right hand side
0 ytfdt
dy
homogenous equation
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
164
3. Orbital perturbations
• Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion
• Mathematical intro:
tgytfdt
dy
diff. equation with right hand side
0 ytfdt
dy
homogenous equation
dttfy
dy
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
165
3. Orbital perturbations
• Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion
• Mathematical intro:
tgytfdt
dy
diff. equation with right hand side
0 ytfdt
dy
homogenous equation
dttfy
dy
dttf
cy ehomogenous solutionc – int. constant
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
166
3. Orbital perturbations
• Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion
• Mathematical intro:
tgytfdt
dy
diff. equation with right hand side
0 ytfdt
dy
homogenous equation
dttfy
dy
dttf
cy ehomogenous solutionc – int. constant
to obtain solution of eq. with right hand side, we allow c to be function of t
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
167
3. Orbital perturbations
• Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion
• Mathematical intro:
tgytfdt
dy
diff. equation with right hand side
0 ytfdt
dy
homogenous equation
dttfy
dy
dttf
cy ehomogenous solutionc – int. constant
to obtain solution of eq. with right hand side, we allow c to be function of t
tg
dt
dc dttfe
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
168
3. Orbital perturbations
• Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion
• Mathematical intro:
tgytfdt
dy
diff. equation with right hand side
0 ytfdt
dy
homogenous equation
dttfy
dy
dttf
cy ehomogenous solutionc – int. constant
to obtain solution of eq. with right hand side, we allow c to be function of t
tg
dt
dc dttfe
dttgCtc
dttfe
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
169
3. Orbital perturbations
• Similar process can be applied to system of diff. eq.
• diff. equation of motion can be written as system of equations
vdt
rd
Rrrdt
vdgrad
3
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
170
3. Orbital perturbations
• Similar process can be applied to system of diff. eq.
• diff. equation of motion can be written as system of equations
vdt
rd
Rrrdt
vdgrad
3
solution without right hand side
constants 6 ,trr
constants 6 ,tvv
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
171
3. Orbital perturbations
• Similar process can be applied to system of diff. eq.
• diff. equation of motion can be written as system of equations
vdt
rd
Rrrdt
vdgrad
3
solution without right hand side
constants 6 ,trr
constants 6 ,tvv
6 int. constants are 6 orbital elements
Meai ,,,, ,
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
172
3. Orbital perturbations
• Similar process can be applied to system of diff. eq.
• diff. equation of motion can be written as system of equations
vdt
rd
Rrrdt
vdgrad
3
solution without right hand side
constants 6 ,trr
constants 6 ,tvv
6 int. constants are 6 orbital elements
Meai ,,,, ,
variation of all 6 orbital elements
tMteta
ttit
, ,
, , ,
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
173
3. Orbital perturbations
• Similar process can be applied to system of diff. eq.
• diff. equation of motion can be written as system of equations
vdt
rd
Rrrdt
vdgrad
3
solution without right hand side
constants 6 ,trr
constants 6 ,tvv
6 int. constants are 6 orbital elements
Meai ,,,, ,
variation of all 6 orbital elements
tMteta
ttit
, ,
, , , calculation of parameters
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
174
3. Orbital perturbations
• Similar process can be applied to system of diff. eq.
• diff. equation of motion can be written as system of equations
i
R
inabdt
d
sin
1
R
inab
iR
inabdt
di
sin
cos
sin
1
e
R
ena
b
i
R
inab
i
dt
d
3sin
cos
M
R
nadt
da
2
M
R
ena
bR
ena
b
dt
de
4
2
3
e
R
ena
b
a
R
nadt
dM
4
22
Lagrange’s planetary equations
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
175
3. Orbital perturbations
• Perturbative potential must be expressed by orbital elements
2
1sin3 2
23
2
J
r
RR
Meai ,,,, ,
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
176
3. Orbital perturbations
• Perturbative potential must be expressed by orbital elements
2
1sin3 2
23
2
J
r
RR
Meai ,,,, ,
cos1
1 2
e
ear
sinsinsin i
RR
2. approximation - ellipsoid
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
177
3. Orbital perturbations
• Perturbative potential must be expressed by orbital elements
• average value of R in one period T
2
1sin3 2
23
2
J
r
RR
Meai ,,,, ,
cos1
1 2
e
ear
sinsinsin i
RR
2. approximation - ellipsoid
dMRdtRT
RT
2
00 2
11
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
178
3. Orbital perturbations
• Perturbative potential must be expressed by orbital elements
• average value of R in one period T
2
1sin3 2
23
2
J
r
RR
Meai ,,,, ,
cos1
1 2
e
ear
sinsinsin i
RR
2. approximation - ellipsoid
dMRdtRT
RT
2
00 2
11
2sin3
14
1 2
22/323
2
iJea
RR
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
179
3. Orbital perturbations
• Perturbative potential can be decomposed into average (secular) and periodic part
ps RRR
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
180
3. Orbital perturbations
• Perturbative potential can be decomposed into average (secular) and periodic part
ps RRR
average value in one period is zero
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
181
3. Orbital perturbations
• Perturbative potential can be decomposed into average (secular) and periodic part
ps RRR
RRs
average value in one period is zero
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
182
3. Orbital perturbations
• Perturbative potential can be decomposed into average (secular) and periodic part
• Replacing perturbative potential by its secular part
ps RRR
RRs
average value in one period is zero
RsR R
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
183
3. Orbital perturbations
• Perturbative potential can be decomposed into average (secular) and periodic part
• Replacing perturbative potential by its secular part
ps RRR
RRs
average value in one period is zero
RsR
2sin3
14
1 2
22/323
2
iJea
RR
R
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
184
3. Orbital perturbations
• Perturbative potential can be decomposed into average (secular) and periodic part
• Replacing perturbative potential by its secular part
ps RRR
RRs
average value in one period is zero
RsR
2sin3
14
1 2
22/323
2
iJea
RR
R
ieaRR ,,
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
185
3. Orbital perturbations
• Lagrange’s planetary equations ieaRR ,,
M
Ra
dt
da
M
RRe
dt
de,
RRi
dt
di,
i
R
dt
d
e
R
i
R
dt
d,
e
R
a
RM
dt
dM,
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
186
3. Orbital perturbations
• Lagrange’s planetary equations ieaRR ,,
M
Ra
dt
da
aie , , are constants
M
RRe
dt
de,
RRi
dt
di,
i
R
dt
d
e
R
i
R
dt
d,
e
R
a
RM
dt
dM,
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
187
3. Orbital perturbations
• Lagrange’s planetary equations ieaRR ,,
M
Ra
dt
da
aie , , are constants
M
RRe
dt
de,
RRi
dt
di,
i
R
dt
d
e
R
i
R
dt
d,
e
R
a
RM
dt
dM,
i
a
RnJ
edt
dcos
12
3 2
222
1cos3
14
3 2
2
22/32
i
a
RnJ
en
dt
dM
1cos5
14
3 2
2
222
i
a
RnJ
edt
d
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
188
3. Orbital perturbations
• Lagrange’s planetary equations
Kepler’s orbit
input parameters:
km/s 3.435 1.581, 7.556,-
km 1567.56 6174.08, 2004.75,
p
p
v
r
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
189
3. Orbital perturbations
• Lagrange’s planetary equations
Kepler’s orbitPerturbed orbitafter 100 x Tonly J2 is considered
9.53
9.32
input parameters:
km/s 3.435 1.581, 7.556,-
km 1567.56 6174.08, 2004.75,
p
p
v
r
Orbital MechanicsSpace for Education, Education for Space
Variation of parameters
190
3. Orbital perturbations
• Numerical solution of orbital equations
Red color – perturbed orbit in specific time rangeBlue color – unperturbed Kepler’s orbit
Tt 40 ,0 TTt 80 ,40 TTt 012 ,80 TTt 016 ,120
Orbital MechanicsSpace for Education, Education for Space
Examples of orbits
191
3. Orbital perturbations
• 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
i
a
RnJ
edt
dcos
12
3 2
222
ia
R
RJ
dt
dcos
2
3 2/7
32
i
Ra /
Orbital MechanicsSpace for Education, Education for Space
Examples of orbits
192
3. Orbital perturbations
• 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
i
Ra /
Rai for 6.95min
180for km 12331max ia
Operating S-s satellites:orbit: circular or near circular
km 900700 h
Orbital MechanicsSpace for Education, Education for Space
Examples of orbits
193
3. Orbital perturbations
• Sun-synchronous orbitsLandsat – 4:
km799.7285
07.99
a
i
Blue: Kepler’s orbitRed: sun-synchronous orbitOrange: Sun and sun beam
view from Sun
Orbital MechanicsSpace for Education, Education for Space
Examples of orbits
194
3. Orbital perturbations
• Sun-synchronous orbits
view from Earth
Landsat – 4:
km799.7285
07.99
a
i
Orbital MechanicsSpace for Education, Education for Space
Examples of orbits
195
3. Orbital perturbations
• Sun-synchronous orbitsLandsat – 4:
km799.7285
07.99
a
i
Orbital MechanicsSpace for Education, Education for Space
• Impulsive maneuvers
• Hohmann transfer
• Non-Hohmann transfer
• Plane change maneuvers
4. Orbital maneuvers
196
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
197
4. Orbital 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
0
0Ln S
SSeS
m
mmuv
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
198
4. Orbital 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
gIu se
0
0Ln S
SSeS
m
mmuv
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
199
4. Orbital 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
gIu se
0
0Ln S
SSeS
m
mmuv
gI
v
S
S S
S
m
m
0
e1
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
200
4. Orbital maneuvers
gIu se
0
0Ln S
SSeS
m
mmuv
gI
v
S
S S
S
m
m
0
e1
Propellant Specific impulse Is [s]
cold gas 50
Monopropellant hydrazine
230
LOX/LH2 455
Ion propulsion >3000
• specific impulse characteristics
[-] / 0SS mm
[m/s]Sv
s 50sI
s 455sI
s 230sI
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
201
4. Orbital maneuvers
• impulse at periapsis
1, PP vr
Pv
1
23
Pv
P
0.019
km/s8.7
km300
1
1
Earth
e
v
rr
P
P
A
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
202
4. Orbital maneuvers
• impulse at periapsis
1, PP vr
Pv
PPP vvv 12
1
23
Pv
P
0.019
km/s8.7
km300
1
1
Earth
e
v
rr
P
P
A
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
203
4. Orbital maneuvers
• impulse at periapsis
22 PPvrh
1, PP vr
Pv
PPP vvv 12
1
23
Pv
P
0.019
km/s8.7
km300
1
1
Earth
e
v
rr
P
P
A
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
204
4. Orbital maneuvers
• impulse at periapsis
22 PPvrh
1, PP vr
Pv
PPP vvv 12
1
23
Pv
P
0.019
km/s8.7
km300
1
1
Earth
e
v
rr
P
P
2e new orbit
A
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
205
4. Orbital maneuvers
• impulse at periapsis
22 PPvrh
1, PP vr
Pv
PPP vvv 12
1
23
Pv
P
0.019
km/s8.7
km300
1
1
Earth
e
v
rr
P
P
2e new orbit
km/s8.93 Pv
km/s23 Pvkm/s12 Pv
km/s8.82 Pv
0.609 3 e
27482.1km 3 Ar
0.297 2 e
12331.4km 3 Ar
A
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
206
4. Orbital maneuvers
• impulse at apoapsis
23
4
0.297
km/s8.8
km300
2
2
Earth
e
v
rr
P
P
5
Av
km/s4.03 Av
km/s919.04 Av
km/s.25 Av
A
Orbital MechanicsSpace for Education, Education for Space
Impulsive maneuvers
207
4. Orbital maneuvers
• impulse at apoapsis
23
4
0.297
km/s8.8
km300
2
2
Earth
e
v
rr
P
P
5
Av
km/s4.03 Av
km/s919.04 Av
km/s.25 Av
174.03 e
04 e
416.05 e PA
circle
A
Orbital MechanicsSpace for Education, Education for Space
Hohmann transfer
208
4. Orbital maneuvers
• 2 impulse maneuvers
1
2
HTHT1v
HT2v
11
Earth1
/
km1000
rv
rr
circle orbit 1:
12
GEO2
/
km42164
rv
rr
circle orbit 2 - GEO:
Hohmann transfer:
1r
2r
Orbital MechanicsSpace for Education, Education for Space
Hohmann transfer
209
4. Orbital maneuvers
• 2 impulse maneuvers
1
2
HTHT1v
HT2v
11
Earth1
/
km1000
rv
rr
circle orbit 1:
12
GEO2
/
km42164
rv
rr
circle orbit 2 - GEO:
Hohmann transfer:
HTa1r
2r HTeHTh
Orbital MechanicsSpace for Education, Education for Space
Hohmann transfer
210
4. Orbital maneuvers
• 2 impulse maneuvers
1
2
HTHT1v
HT2v
11
Earth1
/
km1000
rv
rr
circle orbit 1:
12
GEO2
/
km42164
rv
rr
circle orbit 2 - GEO:
Hohmann transfer:
HTa1r
2r HTeHTh
km/s24.2HT1 v
km/s39.1HT2 v
Orbital MechanicsSpace for Education, Education for Space
Non-Hohmann transfer
211
4. Orbital maneuvers
• 2 impulse maneuvers
orbit 1
orbit 2
point A
point B
A B
• 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
Orbital MechanicsSpace for Education, Education for Space
Non-Hohmann transfer
212
4. Orbital maneuvers
• 2 impulse maneuvers
orbit 1
orbit 2
point A
point B
A B
cos1
1
2
A
Ae
hr
cos1
1
2
B
Be
hr
transfertrajectory
transfer trajectory
eh ,
• 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
Orbital MechanicsSpace for Education, Education for Space
Non-Hohmann transfer
213
4. Orbital maneuvers
• 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• transfer orbit
orbit 1
orbit 2
point A
point B
A B
TBv
eh ,
TBv
Orbital MechanicsSpace for Education, Education for Space
Non-Hohmann transfer
214
4. Orbital maneuvers
• 2 impulse maneuvers
orbit 1
orbit 2
point A
point B
A B
TBv
Bv2
• 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• transfer orbit eh ,
TBv
Bv2
Orbital MechanicsSpace for Education, Education for Space
Non-Hohmann transfer
215
4. Orbital maneuvers
• 2 impulse maneuvers
orbit 1
orbit 2
point A
point B
A B
Bv2
Bv
TBv
Bv
TBBTBBB vvvvv
22 .
TBBTBTBBBB vvvvvvv
222 2
• 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• transfer orbit eh ,
TBv
Bv2
Orbital MechanicsSpace for Education, Education for Space
Non-Hohmann transfer
216
4. Orbital maneuvers
• 2 impulse maneuvers
orbit 1
orbit 2
point A
point B
Bv2
Bv
TBv
Bv
Av
TAv
Av1
A B
• 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• transfer orbit eh ,
Orbital MechanicsSpace for Education, Education for Space
Plane change maneuvers
217
4. Orbital maneuvers
• To change the orientation of a satellite's orbital plane, typically the inclination, the
direction of the velocity vector has to be changed.
• single impulse maneuver
orbit 1
orbit 2
Orbital MechanicsSpace for Education, Education for Space
Plane change maneuvers
218
4. Orbital 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 1
orbit 2
1v
Orbital MechanicsSpace for Education, Education for Space
Plane change maneuvers
219
4. Orbital maneuvers
• single impulse maneuver
2v
orbit 1
orbit 2
• To change the orientation of a satellite's orbital plane, typically the inclination, the
direction of the velocity vector has to be changed.
1v
Orbital MechanicsSpace for Education, Education for Space
Plane change maneuvers
220
4. Orbital maneuvers
• single impulse maneuver
2v
1v
orbit 1
orbit 2
• To change the orientation of a satellite's orbital plane, typically the inclination, the
direction of the velocity vector has to be changed.
v
v
12 vvv
pure rotation
2sin 2
vv
is angle between the planes