IEEE - Aerospace and Electronic Systems - February 2022 - 9

Yang et al.
Table 1.
Orbit Models Used in This Article
Satellite model Geometry-based shape
Constant mass
Reference frame
Earth gravity
field
Third-body
attraction
Relativistic
effect
Atmospheric
drag model
SRP
International Celestial
Reference Frame
GGM05S (100 x 100) model
[30] Tides corrections [26]
Sun, moon, and planets
ephemerides: JPL DE430
[31]
Post-Newtonian correction
[26]
Geometry-dependent
projected area
Various AMD models
Geometry-dependent
projected area
Shadow model: Earth
eclipse considered
Numerical
integrator
Implicit Runge-Kutta solver
RADAU II [32]
array drive angles are also provided for accurate effective
area calculation [see Figure 1(b) for the satellite geometry].
In this work, the COSMIC ephemerides will be used for
assessing the OP performance with various AMD models.
ORBIT PROPAGATOR
For spacecraft in near Earth's orbit, a Newton-Kepler system
is traditionally used to describe the orbit for the twobody
case. Precision orbit modeling, however, should take
into account additional gravitational and nongravitational
perturbations. In general, the accelerations acting on the
satellite consist of terms for the Earth's geopotential, the
third-body gravitational attraction of the sun, moon, and
other planets, the SRP and atmosphere drag on the spacecraft,
if no active orbital maneuver is performed. The
exact formulations for each term can be obtained from,
e.g., [26] and [27]. A modified version of the open-source
high precision orbit propagator [28] is used in this work,
which has been validated by authors using the general
mission analysis tool [29]. The dynamic models/parameters
used for orbit propagation are summarized in Table 1.
The motion of the satellite along with the time t is
modeled by the conventional orbit dynamics in the Cartesian
coordinates
d
dt
xðtÞ¼ fðt; xðtÞ;ppðtÞÞ ¼
FEBRUARY 2022
vðtÞ
aðt; rðtÞ;vvðtÞÞ
;
(1)
IEEE A&E SYSTEMS MAGAZINE
where the p is the orbital parameter vector, including the
SRP and drag coefficients (Cr and Cd). x comprises the
position r and v vectors in the international celestial reference
frame. a is the acceleration vector acting on the LEO
satellite and can be calculated by modeling the aforementioned
perturbative forces. The orbit can be propagated
using the integration as follows:
xðtÞ¼ xðt0Þþ
where t0 is the initial epoch.
SURFACE FORCES
A simple cannon ball assumption does not realistically
represent the shape and geometry of the COSMIC satellites.
Increased model fidelity can be obtained via summation
of the projected areas of different components.
Hence, the surface forces of drag and SRP can be computed
more precisely.
The acceleration due to aerodynamic drag is formulated
as [27]
adrag ¼
1
2
Cdr
Ad
m
jv vwjðv vwÞ
(3)
where Cd is the atmospheric drag coefficient. r is the local
AMD, Ad is the projected area in the instantaneous direction
of travel of the satellite, m is the mass of the satellite,
v is the velocity of the space object, and vw is the velocity
of the atmosphere. The wind velocity only considers the
corotation of the atmosphere and the horizontal winds are
neglected during geomagnetic quiet time.
The mass ofCOSMIC before launch was 61:14 kg [25].
This value will be used throughout all OP simulations, by
neglecting the mass decrease due to the propellant consumption.
The COSMIC satellite travels in a manner that the þX
direction in Figure 1 always aligns with the flight direction
[25]. Hence, the total area for surface forces calculation
can be formulated as
Atotal ¼ Amain þApanel
with
Apanel ¼ 2 p
Amain ¼ 1:034 0:132 ðm2Þ
0:9742
2
sin u ðm2Þ
(5)
where u is the rotational angle of the solar panel (see
Figure 1) in and out of the plane, whose thickness is
neglected. The u values in the body frame can be obtained
from CDAAC. The projected area for drag calculation is
then calculated via
Ad ¼ Atotalacos
vðv vwÞ
jvjjv vwj
:
(6)
9
(4)
Zt
t0
fðxx;pp; tÞdt
(2)
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