IEEE - Aerospace and Electronic Systems - August 2021 - 8

Simulation Framework for Orbit Propagation and Space Trajectory Visualization
d0 ¼
3
2
n0
a0
00 ¼
k2
a2
2
ð3 cos 2i0 1Þ
ð1 e0
2Þ
n0
1 þ d0
00 ¼ðken0
00Þ
2
3
where k2 ¼ 1
2 J2aE
3
2
(5)
C4 ¼ 2n0ðq0 sÞ44a0b2
2k2
a0ð1 h2Þ
(6)
(7)
1:082616 103, ke ¼
2 (units of Earth radii), J2 ¼
p
ffiffiffiffiffiffiffiffiffi
GM
¼ 0:0743669161, G is the
universal gravitational constant, M is the mass of the Earth,
and aE is the equatorial radius of the Earth. Acceleration
due to atmosphere drag is the next variable that is considered
in initialization. The mathematical model of this term
is based on a density function from power law as in [22]
r ¼ r0
ðq0 sÞ4
ðr sÞ4 :
(8)
In this equation, r represents the distance between the
space vehicle and the Earth's center, q0 and s are the
parameters for the density function. The value of q0 is 120
km plus the radius of the Earth and the value of s is the
height of the spacecraft at perigee. In this model, if the
perigee altitude becomes less than 156 km, the value of
some of the parameters will be altered within the initialization.
So, the SatRec structure will be updated accordingly
u
¼ cos ði0Þ
¼
1
a0 s
b0 ¼ð1 e0
h ¼ a0e0
C2 ¼ðq0 sÞ44n0ð1 h2Þ7
2
e0h3Þþ
3
2
k2
ð1 h2Þ
ð þ u2Þð8 þ 24h2 þ 3h4Þ
1
2
3
2
a0ð1 þ h2 þ 4e0hþ
3
2
(13)
C1 ¼ BC2
C3 ¼
8
ðq0 sÞ45Að3;0Þn0aE sin i0
k2e0
(14)
(15)
2Þ
1
2
(11)
(12)
V_ ¼
"
a0
3k2u
2b0
4 þ
3k2
2ð4u2 19u3Þ
2a0
4b0
where k4 ¼3
8 J4aE
8 þ
5k4uð3 7u2Þ
2a0
4b0
8
4, and J4 ¼1:65597 106.
The secular coefficient and the long-period coefficient
for solar and lunar gravity are also initialized as the mentioned
variables are calculated. This computation includes
some additional terms if the orbital period of the space
vehicle is greater or equal to 225 min (DSCOM, DPPER,
DSINIT). Details are provided in [23]. Finally, when all of
the initial variables are computed, the orbit is propagated
once to initialize the states at epoch time.
IEEE A&E SYSTEMS MAGAZINE
AUGUST 2021
#
n0
(23)
(9)
(10)
v_ ¼
"
þ
3k2ð1 5u2Þ
2a0
2b0
4b0
4 þ
8
3k2
2ð7 114u2 þ 395u4Þ
16a0
4b0
5k4ð3 36u2 þ 49u4Þ
4a0
#
n0
8
(22)
C5 ¼2ðq0 sÞ44a0b2
D2 ¼ 4a0C1
D3 ¼ a02ð17a0 þ sÞC1
4
3
D4 ¼ a0
2
3
2
0ð1 h2Þ7
2
"
1 þ
11
4
þ ð1 u2Þð2h2 e0h e0h3Þ cos ð2v0Þ
3
4
"
0ð1 h2Þ7
2
( "
3ð1 3u2Þð1 þ h2 2e0h e0h3Þ
3
2
#)
(16)
hðh þ e0Þþ e0h3
#
(17)
(18)
3
(19)
23ð221a0 þ 31 sÞC1
where A3;0 ¼J3aE
4
(20)
3, and J3 ¼0:253881 105.
Zonal harmonics of the Earth are considered in initialization
as
M_ ¼
"
3k2ð1 þ 3u2Þ
2a0
2b0
3
þ
3k2
2ð13 78u2 þ 137u4Þ
16a0
4b0
7
#
n0
(21)
1
2
2hð1 þ e0hÞþ e0 þ h3
1
2
1
2
#

IEEE - Aerospace and Electronic Systems - August 2021

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