Basics of Orbital Mechanics II - OCW UPM

44 Pages·2009·0.41 MB·English

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

n
o
i
t
a
r
0.01
e
GPS
l
e
c
GEO
c
A
0.0001
1e−006
1e−008
0 5000 10000 15000 20000 25000 30000 35000 40000
Height (km)
BasicsofOrbitalMechanicsII–p.5/24
Encke’s Method
r
Compute only the difference δ p
e
r
t
u
r
b
r r
e
r k r p a d
¨ = µ ¨ = µ +
k p p
− r 3 − r 3 δr
k p
| | | |
r r r r r r
δ = δ
p k p p
− | | ≪ | |
r
v
k
M 0
r
0 Epoch
n
a
i
r
e
l
p
e
k
BasicsofOrbitalMechanicsII–p.6/24
Encke’s Method
r
Compute only the difference δ p
e
r
t
u
r
b
r r
e
r k r p a d
¨ = µ ¨ = µ +
k p p
− r 3 − r 3 δr
k p
| | | |
r r r r r r
δ = δ
p k p p
− | | ≪ | |
r r r
r r r p k a k v
δ¨ = ¨ ¨ = µ + µ + =
M 0
p k p
− − r 3 r 3
p k r
| | | | 0 Epoch
n
a
i
r
e
l
p
e
k
BasicsofOrbitalMechanicsII–p.6/24
Encke’s Method
r
Compute only the difference δ p
e
r
t
u
r
b
r r
e
r k r p a d
¨ = µ ¨ = µ +
k p p
− r 3 − r 3 δr
k p
| | | |
r r r r r r
δ = δ
p k p p
− | | ≪ | |
r r r
r r r p k a k v
δ¨ = ¨ ¨ = µ + µ + =
M 0
p k p
− − r 3 r 3
p k r
| | | | 0 Epoch
n
r 3
µ µ a
i
r r k r a r
δ¨ = δ + 1 | | + e
l
p p p
− r 3 r 3 − r 3 e
! k
k k p
| | | | | |
BasicsofOrbitalMechanicsII–p.6/24
Encke’s Method
r
Compute only the difference δ p
e
r
t
u
r
b
r r
e
r k r p a d
¨ = µ ¨ = µ +
k p p
− r 3 − r 3 δr
k p
| | | |
r r r r r r
δ = δ
p k p p
− | | ≪ | |
r r r
r r r p k a k v
δ¨ = ¨ ¨ = µ + µ + =
M 0
p k p
− − r 3 r 3
p k r
| | | | 0 Epoch
n
r 3
µ µ a
i
r r k r a r
δ¨ = δ + 1 | | + e
l
p p p
− r 3 r 3 − r 3 e
! k
k k p
| | | | | |
r 3 2 r r r
3 + 3q + q δ (δ 2 )
k p
1 | | = f(q) = q q = · −
− r 3 − − 3 r r
1 + (1 + q)
2 p p
p
·
| |
About f(q), cf. Battin, p. 389 and 449
BasicsofOrbitalMechanicsII–p.6/24
Encke’s Method
r
Compute only the difference δ p
e
r
t
u
r
b
r r
e
r k r p a d
¨ = µ ¨ = µ +
k p p
− r 3 − r 3 δr
k p
| | | |
r r r r r r
δ = δ
p k p p
− | | ≪ | |
r r r
r r r p k a k v
δ¨ = ¨ ¨ = µ + µ + =
M 0
p k p
− − r 3 r 3
p k r
| | | | 0 Epoch
n
r 3
µ µ a
i
r r k r a r
δ¨ = δ + 1 | | + e
l
p p p
− r 3 r 3 − r 3 e
! k
k k p
| | | | | |
r 3 2 r r r
3 + 3q + q δ (δ 2 )
k p
1 | | = f(q) = q q = · −
− r 3 − − 3 r r
1 + (1 + q)
2 p p
p
·
| |
µ µ
r r r a
δ¨ = δ f(q) +
p p
− r 3 − r 3
k k
| | | |
About f(q), cf. Battin, p. 389 and 449
BasicsofOrbitalMechanicsII–p.6/24