Post on 18-Feb-2018
Basics of Orbital Mechanics II
Modeling the Space Environment
Manuel Ruiz Delgado
European Masters in Aeronautics and SpaceE.T.S.I. Aeronauticos
Universidad Politecnica de Madrid
April 2008
Basics of Orbital Mechanics II – p. 1/24
Basics of Orbital Mechanics II
Keplerian and Perturbed Motion
Magnitude of the Perturbations
Special Perturbations all, numericalEncke’s MethodCowell’s Method
General Perturbations some, analytical, approximateOsculating OrbitVariation of ParametersLagrange Equations potentialGauss Equations potential & not potential
General Perturbations: Analytical approx/Semianalytical
Numerical Integration
Basics of Orbital Mechanics II – p. 2/24
Keplerian and Perturbed Motion
r = −G (M + m)r
|r|3︸ ︷︷ ︸
Kepler Problem
+P1
m1− P2
m2︸ ︷︷ ︸
Perturbation
rk =
ak︷ ︸︸ ︷
−G (M + m)rk
|rk|3
rp = −G (M + m)rp
|rp|3+ ap
rp
rk
m
M
Usually, |ap| ≪ |ak| ⇒ rp ≃ rk How small?
Basics of Orbital Mechanics II – p. 3/24
Perturbations (LEO)
1e−008
1e−006
0.0001
0.01
1
100
10000
1e+006
0 100 200 300 400 500 600 700 800 900
Acc
eler
atio
n (m
/s2 )
Height (km)
Accelerations of the Satellite (BC=50)
Shuttle
ISS
KeplerJ2
C22Sun
MoonDrag (low)
Drag (high)Prad
Basics of Orbital Mechanics II – p. 4/24
Perturbations (GEO)
1e−008
1e−006
0.0001
0.01
1
100
10000
1e+006
0 5000 10000 15000 20000 25000 30000 35000 40000
Acc
eler
atio
n (m
/s2 )
Height (km)
Accelerations of the Satellite (BC=50)
GEOGPS
KeplerJ2
C22Sun
MoonDrag (low)
Drag (high)Prad
Basics of Orbital Mechanics II – p. 5/24
Encke’s Method
kepl
eria
n
perturbed
rk
δr
r0
v0
Epoch
rp
M
Compute only the differenceδr
rk = −µrk
|rk|3rp = −µ
rp
|rp|3+ ap
δr = rp − rk |δr| ≪ |rp|
Basics of Orbital Mechanics II – p. 6/24
Encke’s Method
kepl
eria
n
perturbed
rk
δr
r0
v0
Epoch
rp
M
Compute only the differenceδr
rk = −µrk
|rk|3rp = −µ
rp
|rp|3+ ap
δr = rp − rk |δr| ≪ |rp|
δr = rp − rk = −µrp
|rp|3+ µ
rk
|rk|3+ ap =
Basics of Orbital Mechanics II – p. 6/24
Encke’s Method
kepl
eria
n
perturbed
rk
δr
r0
v0
Epoch
rp
M
Compute only the differenceδr
rk = −µrk
|rk|3rp = −µ
rp
|rp|3+ ap
δr = rp − rk |δr| ≪ |rp|
δr = rp − rk = −µrp
|rp|3+ µ
rk
|rk|3+ ap =
δr = − µ
|rk|3δr +
µ
|rk|3
(
1 − |rk|3
|rp|3
)
rp + ap
Basics of Orbital Mechanics II – p. 6/24
Encke’s Method
Aboutf(q), cf. Battin, p. 389 and 449
kepl
eria
n
perturbed
rk
δr
r0
v0
Epoch
rp
M
Compute only the differenceδr
rk = −µrk
|rk|3rp = −µ
rp
|rp|3+ ap
δr = rp − rk |δr| ≪ |rp|
δr = rp − rk = −µrp
|rp|3+ µ
rk
|rk|3+ ap =
δr = − µ
|rk|3δr +
µ
|rk|3
(
1 − |rk|3
|rp|3
)
rp + ap
1 − |rk|3
|rp|3= −f(q) = −q
3 + 3q + q2
1 + (1 + q)3
2
q =δr · (δr− 2rp)
rp · rp
Basics of Orbital Mechanics II – p. 6/24
Encke’s Method
Aboutf(q), cf. Battin, p. 389 and 449
kepl
eria
n
perturbed
rk
δr
r0
v0
Epoch
rp
M
Compute only the differenceδr
rk = −µrk
|rk|3rp = −µ
rp
|rp|3+ ap
δr = rp − rk |δr| ≪ |rp|
δr = rp − rk = −µrp
|rp|3+ µ
rk
|rk|3+ ap =
δr = − µ
|rk|3δr +
µ
|rk|3
(
1 − |rk|3
|rp|3
)
rp + ap
1 − |rk|3
|rp|3= −f(q) = −q
3 + 3q + q2
1 + (1 + q)3
2
q =δr · (δr− 2rp)
rp · rp
δr = − µ
|rk|3δr− µ
|rk|3f(q) rp + ap
Basics of Orbital Mechanics II – p. 6/24
Encke’s Method
Aboutf(q), cf. Battin, p. 389 and 449
kepl
eria
n
Epoch∣∣2
perturbed
rp
M
Compute only the differenceδr
rk = −µrk
|rk|3rp = −µ
rp
|rp|3+ ap
δr = rp − rk |δr| ≪ |rp|
δr = rp − rk = −µrp
|rp|3+ µ
rk
|rk|3+ ap =
δr = − µ
|rk|3δr +
µ
|rk|3
(
1 − |rk|3
|rp|3
)
rp + ap
1 − |rk|3
|rp|3= −f(q) = −q
3 + 3q + q2
1 + (1 + q)3
2
q =δr · (δr− 2rp)
rp · rp
δr = − µ
|rk|3δr− µ
|rk|3f(q) rp + ap
if δr ↑, rectify: δr = 0
rk
∣∣1→ rk
∣∣2
Basics of Orbital Mechanics II – p. 6/24
Loss of Precision
REAL*4 = Single-Precision = 6-7 DIGITS
REAL*8 = Double-Precision = 15-16 DIGITS
0.100000000000000 E+00+ 0.123456789012345 E-10
= 0.100000000000000 E+00+ 0.000000000012345 E+00
= 0.100000000012345 E+00
0.123456789012345 E+00- 0.123456789000000 E+00
= 0.000000000012345 E+00
= 0.123450000000000 E-10
Basics of Orbital Mechanics II – p. 7/24
Loss of Precision
1 − |rk|3
|rp|3
REAL*4 = Single-Precision = 6-7 DIGITS
REAL*8 = Double-Precision = 15-16 DIGITS
0.100000000000000 E+00+ 0.123456789012345 E-10
= 0.100000000000000 E+00+ 0.000000000012345 E+00
= 0.100000000012345 E+00
0.123456789012345 E+00- 0.123456789000000 E+00
= 0.000000000012345 E+00
= 0.123450000000000 E-10
Basics of Orbital Mechanics II – p. 7/24
Cowell’s Formulation
Direct numerical integration of the equations
ODE: r = −µr
|r|3+ ap (r, r, t)
IC: t0, r0, r0
r = r (t, t0, r0, r0)
x =
x
y
z
vx
vy
vz
x =
vx
vy
vz
x
y
z
=
vx
vy
vz
− µr3 x + ax
− µr3 y + ay
− µr3 z + az
x = f (x, t)
Basics of Orbital Mechanics II – p. 8/24
Osculating Orbit - Variation of Parameters
perturbed
M
r0
v0
Epoch
Satellite inr0, v0 atEpocht0
Followsperturbedtrajectoryrp(t)
Basics of Orbital Mechanics II – p. 9/24
Osculating Orbit - Variation of Parameters
ke
pler
ian
perturbed
M
r0
v0
Epoch
Satellite inr0, v0 atEpocht0
Followsperturbedtrajectoryrp(t)
Osculating Orbitat r0, v0:
TheKeplerianorbit followed by the satelliteif all perturbationsbecome zero from thispoint on.
Basics of Orbital Mechanics II – p. 9/24
Osculating Orbit - Variation of Parameters
ke
pler
ian
oscu
lati
ng
perturbed
rp(t)
M
r0
v0
Epoch
Satellite inr0, v0 atEpocht0
Followsperturbedtrajectoryrp(t)
Osculating Orbitat r0, v0:
TheKeplerianorbit followed by the satelliteif all perturbationsbecome zero from thispoint on.
Osculating orbit elements can be used ascoordinates
r0,v0 , t0 ⇒ i,Ω, ω, a, e, τ , θ, t0
rp(t),vp(t) , t ⇒ i(t),Ω(t), ω(t), a(t), e(t), τ(t) , θ(t), t
Basics of Orbital Mechanics II – p. 9/24
Variation of Parameters:Fast/Slow variables
θM , Æ
ω
FastVariables:
θ, M , Æ, t
r(t) ,v(t)
SlowVariables:
i, Ω, ω, a, e, τ (M0)
Basics of Orbital Mechanics II – p. 10/24
Variation of Parameters:Secular/Periodic
Secular
Secular+ Long periodic
Secular+ Long periodic+ Short periodic
“Short” ∼ Orbital period
Orb
italP
aram
eter
t
Basics of Orbital Mechanics II – p. 11/24
Variation of Parameters - Lagrange
Variation of Parameters:
r = −µr
|r|3+ ap
r = r (i(t),Ω(t), ω(t), a(t), e(t), t)
x = ⌊i,Ω, ω, a, e, τ⌋T
x = f (x, t)
Lagrange Planetary Equations: Conservative perturbations
ap = ∇R R(i,Ω, ω, a, e,M0) M0 = n τ
x = ⌊i,Ω, ω, a, e,M0⌋T
x = f (x,∇R)
Basics of Orbital Mechanics II – p. 12/24
Lagrange Planetary Equations
Singularities forlow eccentricity
or inclination
di
dt=
1
na2√
1 − e2 sin i
(
cos i∂R
∂ω− ∂R
∂Ω
)
dΩ
dt=
1
na2√
1 − e2 sin i
∂R
∂i
dω
dt=
√1 − e2
na2 e
∂R
∂e− cos i
na2√
1 − e2 sin i
∂R
∂i
da
dt=
2
na
∂R
∂M0
de
dt=
1 − e2
na2 e
∂R
∂M0−
√1 − e2
na2 e
∂R
∂ω
dM0
dt= −1 − e2
na2 e
∂R
∂e− 2
na
∂R
∂a
Basics of Orbital Mechanics II – p. 13/24
Lagrange Planetary Equations
Singularities forlow eccentricity
or inclination
di
dt=
1
na2√
1 − e2 sin i
(
cos i∂R
∂ω− ∂R
∂Ω
)
dΩ
dt=
1
na2√
1 − e2 sin i
∂R
∂i
dω
dt=
√1 − e2
na2 e
∂R
∂e− cos i
na2√
1 − e2 sin i
∂R
∂i
da
dt=
2
na
∂R
∂M0
de
dt=
1 − e2
na2 e
∂R
∂M0−
√1 − e2
na2 e
∂R
∂ω
dM0
dt= −1 − e2
na2 e
∂R
∂e− 2
na
∂R
∂a
(
M = n − 1−e2
na2 e∂R∂e − 2
na∂R∂a
∣∣a
)
Basics of Orbital Mechanics II – p. 13/24
Lagrange VOP: Kozai’s Method
Separate disturbing potentialR into constant/periodic, and orders ofmagnitude:R = R1 + R2 + R3 + R4
R1 =3
2
µ J2 R2E
a3
(1
3− 1
2sin2 i
)(1 − e2
)1/2R2 = 0
R3 =3
2
µ J3 R3E
a4sin i
(
1 − 5
4sin2 i
)
e(1 − e2
)−5/2
sin ω
R4 =3
2
µ J2 R2E
a3
(a
r
)3(
1
3− 1
2sin2 i
)[
1 −(r
a
)3 (1 − e2
)−3/2
]
+
+1
2sin2 i cos 2 (ν + ω)
Only gravitational perturbationsJ2 (flattening) andJ3 (pear-shape)are included.
Basics of Orbital Mechanics II – p. 14/24
Lagrange VOP: Kozai’s Method (secular)
di
dt=
3
8n J3
(RE
p
)3
cos i(4 − 5 sin2 i
)sin2 i cos ω
da
dt= 0
dΩ
dt= −3
2n J2
(RE
p
)2
cos i − 3
8n J3
(RE
p
)3(15 sin2 i − 4
)e cot i sin ω
dω
dt=
3
4n J2
(RE
p
)2(4 − 5 sin2 i
)+
3
8n J3
(RE
p
)3 [(4 − 5 sin2 i
)·
·(sin2 i − e2 cos2 i
)
e sin i+ 2 sin i
(13 − 15 sin2 i
)e
]
sin ω
de
dt= −3
8n J3
(RE
p
)3
sin i(4 − 5 sin2 i
) (1 − e2
)cos ω
dM
dt= n
[
1 +3
2J2
(RE
p
)2(
1 − 3
2sin2 i
)(1 − e2
)1/2]
−
− 3
8n J3
(RE
p
)3
sin i(4 − 5 sin2 i
) (1 − 4e2
)(1 − e2
)1/2
esin ω
Basics of Orbital Mechanics II – p. 15/24
Gauss Planetary Equations
Conservative and not conservative perturbations
Use the Orbital Frame forap
ap = ar ur + aθ uθ + az uz
Peric.
Sat.
hur
uθ
eω
Ω
θ
uN
i
i
x1
y1
z1
Basics of Orbital Mechanics II – p. 16/24
Gauss Planetary Equations
Singularities for loweccentricity or inclination
di
dt=
r cos θ
na2√
1 − e2az
dΩ
dt=
r sin θ
na2√
1 − e2 sin iaz
dω
dt=
√1 − e2
na e
[
− cos θ ar + sin θ
(
1 +r
p
)
aθ
]
− r cos i sin θ
h sin iaz
da
dt=
2
n√
1 − e2
(
e sin θ ar +p
raθ
)
de
dt=
√1 − e2
na
[
sin θ ar +
(
cos θ +e + cos θ
1 + e cos θ
)
aθ
]
dM0
dt=
1
na2 e[(p cos θ − 2er) ar − (p + r) sin θ aθ]
M =n+ b
ah e[(p cos θ−2re) ar−(p+r) sin θ aθ]
Basics of Orbital Mechanics II – p. 17/24
Numerical Methods: Euler
t
y
y0
y(t1)
y1
t0 t1
h
y = f(y, t)
y0 = y(t0)
y1 = y0 + f [y(t0), t0] · h. . .
yn = yn−1 + f [yn−1, t0 + (n − 1)h] · h. . .
Error = O(h2)
Basics of Orbital Mechanics II – p. 18/24
Numerical Methods: Midpoint
t
y
y0
y(t1)
t0 t1
h
y = f(y, t)
y0 = y(t0)
Basics of Orbital Mechanics II – p. 19/24
Numerical Methods: Midpoint
y1
t
y
y0
y(t1)
t0 t1
h
y = f(y, t)
y0 = y(t0)
y1 = y0 + f [y(t0), t0] · h/2
Basics of Orbital Mechanics II – p. 19/24
Numerical Methods: Midpoint
y1
t
y
y0
y(t1)
t0 t1
h
y = f(y, t)
y0 = y(t0)
y1 = y0 + f [y(t0), t0] · h/2
y1 = f [y1, t0 + h/2]
Basics of Orbital Mechanics II – p. 19/24
Numerical Methods: Midpoint
y1
y2
t
y
y0
y(t1)
t0 t1
h
y = f(y, t)
y0 = y(t0)
y1 = y0 + f [y(t0), t0] · h/2
y1 = f [y1, t0 + h/2]
y2 = y0 + y1 · h. . .
Error = O(h3)
Basics of Orbital Mechanics II – p. 19/24
Numerical Methods: Runge-Kutta 4
tn tn+1h
yn
y(tn+1)
y = f(y, t)
Basics of Orbital Mechanics II – p. 20/24
Numerical Methods: Runge-Kutta 4
y1
tn tn+1h
yn
y(tn+1)
y = f(y, t)
k1 = h f (yn, tn) y1 = yn + k1/2
Basics of Orbital Mechanics II – p. 20/24
Numerical Methods: Runge-Kutta 4
y1
y2
tn tn+1h
yn
y(tn+1)
y = f(y, t)
k1 = h f (yn, tn) y1 = yn + k1/2
k2 = h f (y1, tn + h/2) y2 = yn + k2/2
Basics of Orbital Mechanics II – p. 20/24
Numerical Methods: Runge-Kutta 4
y1
y2
y3
tn tn+1h
yn
y(tn+1)
y = f(y, t)
k1 = h f (yn, tn) y1 = yn + k1/2
k2 = h f (y1, tn + h/2) y2 = yn + k2/2
k3 = h f (y2, tn + h/2) y3 = yn + k3
Basics of Orbital Mechanics II – p. 20/24
Numerical Methods: Runge-Kutta 4
y1
y2
y3
y4
tn tn+1h
yn
y(tn+1)
y = f(y, t)
k1 = h f (yn, tn) y1 = yn + k1/2
k2 = h f (y1, tn + h/2) y2 = yn + k2/2
k3 = h f (y2, tn + h/2) y3 = yn + k3
k4 = h f (y3, tn + h) y4 = yn + k4
Basics of Orbital Mechanics II – p. 20/24
Numerical Methods: Runge-Kutta 4
y1
y2
y3
y4
yn+1
tn tn+1h
yn
y(tn+1)
y = f(y, t)
k1 = h f (yn, tn) y1 = yn + k1/2
k2 = h f (y1, tn + h/2) y2 = yn + k2/2
k3 = h f (y2, tn + h/2) y3 = yn + k3
k4 = h f (y3, tn + h) y4 = yn + k4
yn+1 = yn + k1
6+ k2
3+ k3
3+ k4
6
Error = O(h5)
Basics of Orbital Mechanics II – p. 20/24
Numerical Methods: Burlish-Stoer
tn tn+1h
yn
y = f(y, t), yn, tn
Basics of Orbital Mechanics II – p. 21/24
Numerical Methods: Burlish-Stoer
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
n = 2
n = 4n = 6
tn tn+1h
yn
y = f(y, t), yn, tn
Compute the intervalh with n stepshn , n =
k︷ ︸︸ ︷
2, 4, 6 . . .
Basics of Orbital Mechanics II – p. 21/24
Numerical Methods: Burlish-Stoer
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
n = 2
n = 4n = 6
tn tn+1h
yn
h2
h6
h4
0
y
y = f(y, t), yn, tn
Compute the intervalh with n stepshn , n =
k︷ ︸︸ ︷
2, 4, 6 . . .
Basics of Orbital Mechanics II – p. 21/24
Numerical Methods: Burlish-Stoer
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
n = 2
n = 4n = 6
tn tn+1h
yn
y(tn+1)
h2
h6
h4
0
y
Error =O(h2k+1
)
y = f(y, t), yn, tn
Compute the intervalh with n stepshn , n =
k︷ ︸︸ ︷
2, 4, 6 . . .
Polynomial extrapolation ton → ∞, h → 0
Basics of Orbital Mechanics II – p. 21/24
Adaptive Stepsize Control
Set a truncation errorǫ and stepsizeh
Give a step with a method of ordern
Repeat the step with ordern + 1
If the difference is> ǫ, decreaseh
If the difference is< ǫ, increaseh
Each section of the curve is integrated with the maximumhcompatible withǫ
This reduces the number of steps, but may require more derivativeevaluations
Basics of Orbital Mechanics II – p. 22/24
COWELL Program
Begin y = f(y, t)
Initializations
Input dataKB/File
ODE Integrator Call Int step Call Derivs
Compute elements
Compute Kepler
Save Data
INTTRAJ.DAT
OSCELEM.DAT
KEPTRAJ.DAT
Plot
End
aKep
agrav
a3Body
aDrag
aPrad...
Basics of Orbital Mechanics II – p. 23/24
ODE Integrator
Fixed Step
ti = ti−1 + ∆t
Dumb Integr Step Derivs
t = tf ?
Yes
No
Adaptive Stepsize
ti = ti−1 + ∆t
Adjust∆t
QS Integr Step Derivs
Error⋚ ǫ
t = tf ?
OK
Yes
> <No
Basics of Orbital Mechanics II – p. 24/24