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16.346 Astrodynamics
Fall 2008
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Lecture 30
Effect of J 2 on a Satellite Orbit of the Earth #10.6
Variational Equations using the Disturbing Function
Recall the variational equatioms
∂s
∂s d α
ds
0
=
+
= Fs +
ad
dt
∂t
∂ α dt
=⇒
∂s d α
0
=
ad
∂ α dt
which we can write as
T
∂r dα
=0
∂α dt
or
T
T
∂r
∂r
∂v dα
=
ad
∂α
∂α dt
∂α
If we use the gradient of the disturbing function R for the disturbing acceleration
∂r dα
=0
∂α dt
∂v dα
= ad
∂α dt
∂v
∂α
adT =
then we have
so that
∂r
∂α
T
∂r
ad =
∂α
T ∂R
∂r
∂R
∂r
T
∂R ∂r
=
∂r ∂ α
T
∂R
=
∂α
T
T
T
∂v
∂R
∂v
∂r dα
=
−
∂α
∂α
∂α dt
∂α
Lagrange Matrix L
∂r
∂α
T
Therefore, the variational equation using the matrix L is
T
∂R
dα
=
L
dt
∂α
The Lagrange Matrix is skew-symmetric, i.e., L = −L T . Because of the skew-symmetry,
there are only 15 elements to calculate and only 6 of these are different from zero.
Lagrange’s Planetary Equations
1
dΩ
∂R
=
dt
nab sin i ∂i
1
cos i ∂R
di
∂R
=−
+
dt
nab sin i ∂Ω nab sin i ∂ω
cos i ∂R
b ∂R
dω
=−
+ 3
dt
nab sin i ∂i
na e ∂e
16.346 Astrodynamics
2 ∂R
da
=
dt
na ∂λ
b ∂R
b2 ∂R
de
=− 3
+ 4
dt
na e ∂ω
na e ∂λ
2 ∂R
b2 ∂R
dλ
=−
− 4
dt
na ∂a
na e ∂e
Lecture 30
Effect of the J2 Term on Satellite Orbits
Gravitational potential function of the earth
∞
Gm Gm req k
V (r, φ) =
−
Jk
Pk (cos φ)
r
r
r
k=2
=R
where the angle φ is the colatitude with cos φ = ir · iz and
ir =
[cos Ω cos(ω + f ) − sin Ω sin(ω + f ) cos i] ix
+ [sin Ω cos(ω + f ) + cos Ω sin(ω + f ) cos i] iy
(8.92)
Problem 3–21
+ sin(ω + f ) sin i iz
Hence cos φ = sin(ω + f ) sin i so that
2
GmJ2 req
(1 + e cos f )3 [3 sin2 (ω + f ) sin2 i − 1] + O[(req /r)3 ]
2p3
2π
1
R=
R dM
where
dM = n dt and r2 df = h dt
2π 0
2π
2
µJ2 req
n 2
1
2
=
Rr df =
3 (2 − 3 sin i)
3
2
2π 0 h
4a (1 − e ) 2
R=−
Averaged Variational Equations
1
dΩ
∂R
=
dt
nab sin i ∂i
di
=0
dt
cos i ∂R
b ∂R
dω
=−
+ 3
dt
nab sin i ∂i
na e ∂e
For the Earth
da
=0
dt
de
=0
dt
2 ∂R
b2 ∂R
dλ
=−
− 4
dt
na ∂a
na e ∂e
Problem 10–12
r 3.5
3 req 2
dΩ
eq
= − J2
n cos i = −9.96
(1 − e2 )−2 cos i degrees/day
dt
2
p
a
r 3.5
3 req 2
dω
eq
= J2
n(5 cos2 i − 1) = 5.0
(1 − e2 )−2 (5 cos2 i − 1) degrees/day
dt
4
p
a
Coefficients of the earth’s gravitational potential ( ×106 )
J2 = 1, 082.28 ± 0.03
J3 = −2.3 ± 0.2
J4 = −2.12 ± 0.05
16.346 Astrodynamics
J5 = −0.2 ± 0.1
J6 = 1.0 ± 0.8
Lecture 30