IEEE - Aerospace and Electronic Systems - August 2021 - 9

Shirazi et al.
UPDATEANDITERATION
After the initialization, the simulation goes through a loop
in time domain in which the spacecraft trajectory is propagated
with the constant variables calculated previously.
The updating steps consider short-period and long-period
effects of solar and lunar gravity, resonance effects of the
Earth's gravity, and atmospheric drag effects.
Considering these updates, propagation yields the
updated parameters in SatRec in each iteration. Then, the
vectors representing the unit orientations are computed
and the state vectors will be obtained. Knowing the state
vectors, the rest of the orbital characteristics can be calculated
accordingly (RV2ELEMENTS)as rk;
propagation models and provides a base for modeling
much more complicated trajectories.
As in HPOP, the equations of acceleration in (1) of a
space vehicle are computed in the inertial reference frame
as
~g ¼~gg þ~gng
(30)
where ~gg is the sum of the accelerations due to gravitational
forces, affecting the spacecraft motion other than
the typical term for the Earth's gravity ðm=r3~rÞ in (1),
and ~gng
is the sum of the nongravitational forces, which
r_k;r _fk;uk;ik,
and Vk in each iteration. Unit orientation vectors, as ~U
and ~V are determined as
U~¼ Msin uk þNcos uk
V~¼ Mcos ukNsin uk
where
M~ ¼
(24)
(25)
2
4
N~ ¼
2
4
sinVk cos ik
cosVk cos ik
sin ik
cosVk
sinVk
3
5:
3
5
are acted on the space vehicle surface areas.
Obviously, the main difference between this model
(26)
and the two-body model is that the perturbation term is
not assumed to be zero ð~g 6¼ 0Þ. The main challenge in
HPOP is the precise calculation of different perturbation
terms. For the sake of brevity, details of these terms are
omitted here and the reader is urged to refer to the references
in the following sections.
The first group of terms in HPOP is the gravitational
perturbations. The acceleration due to gravitational forces
in this term can be expressed as follows:
~gg ¼~ggeo þ~gst þ~got þ~grd þ~gn þ~grel
where ~ggeo is the term describing the orbital perturbations
(27)
caused by the Earth's geopotential, ~gst is the perturbation
caused by solid Earth tides, ~got is the perturbation caused
by the ocean tides, ~grd is the effect of rotational deformation,
~gn
is the gravitational effect of other giant masses,
Then, the position and velocity vectors are calculated
by
~r ¼ rk ~U
~v ¼ _rk ~Uþ r _fk ~V:
(28)
(29)
The simplified general perturbations consider secular
effects of J2, J4, and long-periodic effects of J3, and
short-periodic effects of J2, along with atmospheric drag.
This propagation method is very popular due to its widespread
application for various kinds of missions. More
detailed description of the SGP4 model as applied for the
generation ofNORAD 2-line elements is provided in [19].
HIGH-PRECISION ORBIT PROPAGATOR (HPOP)
The HPOP model consists of a propagation process in
which the general orbital perturbations along with the
gravitational forces due to other planets (N-body problem)
are taken into account. As a result, this model provides
more accurate prediction in comparison to other
AUGUST 2021
such as sun, moon, and planets, and ~grel is the perturbations
caused by the general relativity.
The gradient of the potential function U that satisfies
the Laplace equation, r2U ¼ 0, represents the Earth's
gravitational attraction. The resulting perturbation acceleration
can be specified as follows:
rU ¼rðUs þ DUst þ DUot þ DUrdÞ
¼~ggeo þ~gst þ~got þ~grd:
(32)
(33)
In this equation, Us denotes the potential caused by
solid-body mass distribution. The effect of solid-body
tides is represented by DUst, DUot denotes the effect of
potential changes due to the ocean tides, and DUrd is the
effect of the rotational deformations.
As for Us, spherical harmonic expansion is usually utilized
to express this variable, with respect to the bodyfixed
reference frame [24]. Also, since a nonrigid elastic
body is the real formation of the Earth, its mass distribution
varies with a nonuniform pattern. The solid Earth
tides that affect the variation of ðDUsdÞ are commonly
expressed by external geopotentials as described in [25].
Moreover, details regarding the computation of oceanic
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IEEE - Aerospace and Electronic Systems - August 2021

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