Orbital Mechanics Project Laboratory
POLITECNICO DI MILANO
DEPARTMENT OF AEROSPACE SCIENCE AND TECHNOLOGY
MSc. Space Engineering
Academic Year 2021/2022
Authors: Group 2259
Francesco Calabrò
(10726829,220859)
Matteo Cannarile
(10651999,226304)
Domingos Savio
(10880091,221090)
Augusto Lima
(10916450,219888)
Professor:
Camilla Colombo
January 2023
1
Contents
List of symbols
1 Interplanetary Explorer Mission
1.1 Mission Requirements . . . . .
1.2 Design Process . . . . . . . . .
1.3 Interplanetary Trajectory . . .
1.4 Local Minimum Optimization .
1.5 Heliocentric Trajectory . . . . .
1.6 Flyby . . . . . . . . . . . . . .
3
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4
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2 Planetary Explorer Mission
2.1 Nominal orbit . . . . . . . . . . . . . . . . .
2.2 Ground track . . . . . . . . . . . . . . . . .
2.2.1 Unperturbed nominal orbit . . . . .
2.2.2 Unperturbed repeating ground track
2.2.3 Perturbed ground track . . . . . . .
2.3 Assigned perturbations . . . . . . . . . . . .
2.4 Orbit Propagation . . . . . . . . . . . . . .
2.5 Keplerian Elements Plot . . . . . . . . . . .
2.6 Evolution of the orbit representation . . . .
2.7 Filtering . . . . . . . . . . . . . . . . . . . .
2.8 Comparison with real data . . . . . . . . . .
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9
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List of symbols
a
aJ2
aM oon
ar
as
aw
AU
α
∆V
e
f
G.A.
h
i
Jn
ϕ
λ
µ
Ω
ω
p
ϕ
Pn
r
T
RAAN
Rf +ω
RE
Ri
RΩ
Semi-major axis of an orbit
Perturbing acceleration due J2 ef f ect
Perturbing acceleration due Moon effect
Radial acceleration
Transversal acceleration
Out of plane acceleration
Astronomical Unit
Right ascension
Change in velocity
Eccentricity of an orbit
True anomaly
Gravity Assist
Specific angular momentum
Inclination of an orbit
Zonal harmonics perturbing effect
Latitude
Longitude
Gravitational constant of Earth
Right ascension of the ascending node
Argument of perigee
Semi-latus rectum
Coelevation
Shaping polynomials
Orbit radius
Period
Right ascension of the ascending node
Matrix rotation of angle f + ω
Radius of Earth
Matrix rotation of angle i
Matrix rotation of angle Ω
3
1
Interplanetary Explorer Mission
As part of the mission analysis, a feasibility study is carried out to analyze a potential
Interplanetary Explorer mission, departing from Earth, performing a powered gravity assist on
a designated planet and arriving at a NEO asteroid.
1.1
Mission Requirements
The proposed mission has the following requirements:
• Departure Planet: Earth;
• Flyby Planet: Saturn;
• Arrival Asteroid: Near-Earth Object 73.
In addition, the dates for departure and arrival are also constrained.
• Earliest Departure: 02/05/2033;
• Latest Arrival: 29/10/2067.
It is also important to mention that the trajectories are calculated through the method of
patched conics, without considering planetary departure or insertion.
Lastly, the figure of merit for the mission is the total cost, such that the solution presented
in this work is based on the minimum viable total ∆V .
1.2
Design Process
As a preliminary analysis, a time window for departure and arrival is chosen. For that, the
trajectory is divided into two steps, the first being the departure from Earth and arrival at
Saturn for the flyby, and the second being the departure from Saturn and arrival at the NEO
73. To acquire a benchmark of how long a flight takes for each step, the orbits are assumed
to be coplanar and circular and the Hohmann transfer time of flight is then calculated. Using
the obtained results, tof1 for the first step and tof2 for the second, the initial choice for the
time windows are set. Equations (1), (2) and (3) demonstrate each window, in which dpt0
represents the earliest departure date and arvf the latest arrival date.
departurewindow = [dpt0 , arvf − 1.1(tof1 + tof2 )]
(1)
f lybywindow = [dpt0 + 0.9(tof1 ), arvf − 1.1(tof2 )]
(2)
arrivalwindow = [dpt0 + 0.9(tof1 ) + km(tof2 ), arvf ]
(3)
As seen in the previous equations, the departure window from Earth is considered to range from
the earliest departure date to the latest arrival minus the sum of tof1 and tof 2 multiplied by
a factor of 110%. The flyby window in Saturn is given by the earliest departure date summed
with 90% of tof1 and by difference between the latest arrival date and tof2 multiplied by the
same factor of 110% . Finally, the arrival window at the asteroid ranges from the flyby first
possible arrival plus 90% of tof2 until the latest arrival date. The factors of 110% and 90%
are used to consider a more conservative range of dates with respect to the Hohmann transfer
time of flight.
1.3
Interplanetary Trajectory
To calculate the trajectory, the flight is divided into two Lambert arcs that are connected
through a powered gravity assist manoeuvre applied at the perigee of the hyperbola. It is
important to mention that the minimum radius of perigee for the flyby hyperbola is another
constraint for the calculation, and is given by Saturn’s radius.
Since the problem has a total of three degrees of freedom (departure, flyby and arrival
times), a nested loop over each time window is implemented, in which the costs in ∆v for
4
both Lambert arcs and for the flyby manoeuvre are calculated, as well as the total ∆V for the
mission.
After finding the costs for the chosen time windows, the dates for the minimum total cost
of the mission are found, as shown in Table 1.
Mission Step
Departure
Fly-By
Arrival
Year
2039
2042
2046
Month
Nov
Dec
Jun
Day
26
20
06
Hour
17:35:15
07:40:20
12:35:53
∆v[km/s]
12.132
0.27247
17.054
Table 1: Dates and ∆vs for the Minimum Mission Cost
1.4
Local Minimum Optimization
The minimum ∆V data acquired in the nested loop is now used as initial condition to find
an optimal local minimum using MATLAB’s function fminunc. Table 2 compares the total
minimum cost before and after the optimization, and Table 3 shows the dates for the new local
minimum.
Simulation
Before Optimization
After Optimization
∆v[km/s]
29.459
28.03
Table 2: Minimum ∆V Comparison
Mission Step
Departure
Fly-By
Arrival
Year
2039
2043
2046
Month
Dec
Apr
Jun
Day
04
17
13
Hour
19:59:08
17:09:42
18:33:35
Table 3: Local Minimum Dates after Optimization
Figures 1 show, respectively, the Porkchop Plots for the first step of the mission, from Earth
to Saturn, and for the second one, from Saturn to NEO-73.
Considering that the Synodic Period of Earth and Saturn is Tsyn1 = 1.0357 year, one can
clearly see a pattern that repeats itself after this time on the image on the left. Also, minimum
values around 12 km/s are found, which are values close to the results obtained (see Table 5).
However, as for the figure on the right, since the synodic period between Saturn and NEO73 is Tsyn2 = 29.6381 years, it is not possible to detect any clear pattern for the timespan
considered for the mission. Even though, minimum values around 10 km/s are found, although
none of them were related to the global optimal solution.
5
Figure 1: Porkchop for interplanetary missions from Earth to Saturn (left) and from Saturn
to NEO-73 (right)
In order to further localize optimal conditions for the mission, the interplanetary trajectory
is again solved, but using progressively smaller time windows based on the optimal dates. In
this case, a range of 1 day for each optimal date is chosen and the trajectory and costs are
again calculated. The results gives us optimal dates within the much smaller established range,
as shown in Tables 4 and 5.
Mission Step
Departure
Fly-By
Arrival
Year
2039
2043
2046
Month
Dec
Apr
Jun
Day
04
17
13
Hour
20:28:31
05:24:24
19:02:58
Table 4: Optimal Dates
∆v[km/s]
11.555
0.60431
15.852
28.03
Departure
Flyby
Arrival
Total
Table 5: ∆V Cost for each main maneuver
6
1.5
Heliocentric Trajectory
Figure 2: Heliocentric Trajectory alongside Earth’s and Saturn’s Orbits
Figure 2 demonstrates the heliocentric path carried by the spacecraft alongside the orbits of
the concerned planets. The trajectory is composed by three manoeuvres. Firstly, the spacecraft
performs a manoeuvre to go into a highly elliptic orbit from Earth to Saturn. Then, arriving
at the Saturn’s sphere of influence, a powered gravity assist is performed, sending, lastly, the
spacecraft into another highly elliptical and much inclined orbit in the direction of the desired
destination, NEO 73 asteroid. The orbital parameters for the transfer orbits are shown in
Table 6.
Transfer
Earth-Saturn
Saturn-NEO 73
a [km]·109
1.4046
1.1846
e [-]
0.89509
0.9613
i [deg]
4.9725
36.029
Ω [deg]
72.526
221.47
ω [deg]
357.08
171.64
Table 6: Orbital parameters for the transfer orbits
1.6
Flyby
A powered gravity assist is performed at Saturn in order to give a boost to the spacecraft
into entering an orbit in the direction of the asteroid. The vehicle comes as close as 36016
km to the planet, gaining a total ∆V of 16.367 km/s and remaining 2882 hours inside the
planet’s sphere of influence. The powered gravity assist manoeuvre has a ∆V cost of 0.61948
km/s, which is only 3.8% of the velocity gained with the flyby. The incoming and outgoing
hyperbolas are shown in figure 3. The respective orbital elements are given in Table 7.
Figure 3: Gravity assist trajectory, showing the Incoming(
7
) and Outgoing(
) hyperbolas.
Orbit
Entry Hyperbola
Exit Hyperbola
a [km]
−4.2829 · 105
−3.0166 · 105
e [-]
1.2201
1.3124
i [deg]
173.48
173.48
Ω [deg]
198.94
198.94
ω [deg]
332.87
332.87
Table 7: Orbital Parameters of entry and exit hyperbolas
It is important to note that, due to Saturn’s mass particles from a wide variety of sizes are
orbiting the planet both in and out its rings. In addition, Saturn has a great number of moons
that may have to be considered as a spacecraft approaches the planet. Therefore, a further
study on Saturn’s neighborhood would be welcomed for the mission success.
8
2
Planetary Explorer Mission
For the second assignment, the study of the orbit for an Earth observation satellite is done.
A nominal orbit is determined as starting pointing for the analysis, from which its ground
track must be determined for unperturbed and perturbed conditions. A new orbit is proposed
in order to obtain a repeating ground track path.
Moreover, the models and numerical methods are evaluated and compared with real data
to verify their reliability.
2.1
Nominal orbit
For the Earth observation orbit, a nominal orbit was assigned, described by its orbital
elements shown in table 8. The values of Ω and ω were chosen by the group.
For the propagation of the orbits, it was considered that the initial true anomaly and the
Greenwich meridian angle with respect to the γ direction were zero. To compute the Moon
disturbance, it was considered that the satellite started its operation at May 2nd , 2033, as the
earliest departure of the 1st Assignment.
Semi-major axis [km]
Eccentricity [-]
Radius of perigee [km]
Radius of apogee [km]
Semi-latus rectum [km]
26297
0.5860
10886
41707
17267
Inclination [◦ ]
RAAN [◦ ]
Argument of perigee [◦ ]
Angular Momentum [km2 /s]
Period [h]
71.5846
0
15
82960.956
11.7888
Table 8: Characteristics of the assigned orbit
The unperturbed nominal orbit is shown in the figure 4 in the Earth-centered inertial
reference frame.
Figure 4: Nominal orbit of the satellite in Earth-centered inertial frame
2.2
Ground track
To compute the ground track of the satellite, it is first necessary to propagate the satellite’s
orbit. The motion of the spacecraft is assumed to be a perturbed two body problem in Cartesian
coordinates according to the theory discussed by Curtis [1], described by the equation (4):
r̈ = −
µ
r + aperturbation
r3
(4)
which is solved using MATLAB’s ode113 function, with a relative tolerance of 1e-13 and absolute tolerance of 1e-14.
9
From the Cartesian coordinates, the equivalent longitude and latitude of the spacecraft over
Earth’s surface are calculated, described by the set of equations (5).

ϕ = δ= sin−1


(z/r)



cos−1 x/r
;y>0
cos(δ)
α=
x/r

2π − cos−1
;y≤0

cos(δ)



λ = α − (θG (t0 ) + ωE · (t − t0 ))
2.2.1
(5)
Unperturbed nominal orbit
The first required analysis of the ground track is for the nominal orbit considering an
unperturbed situation, i.e., the perturbing acceleration in equation (4) is considered to be null.
The ground track was propagated for a period of 1 orbit of the satellite, 1 day and 10 days, as
shown in figure 5.
(a)
(b)
(c)
Figure 5: Ground track of the unperturbed nominal orbit. Ground track path (
point ( ), Ending point ( ).
2.2.2
), Starting
Unperturbed repeating ground track
It is also required that the orbit results in a repeating ground track with a ratio of 2:1 (for
every 2 orbits of the spacecraft, Earth must have performed 1 full revolution around its axis).
10
Therefore, the period of the repeating ground track orbit is computed by equation (6):
1
TEarth = 11.9681 h
(6)
2
For an unperturbed orbit, the period is only a function of the semi-major axis and can be
calculated through equation (7) to get the desired repeating ground track.
s
a3
T = 2π
→ a = 26563km
(7)
µ
Trepeating =
The other orbital parameters are kept from the nominal orbit and are shown in table 9.
Semi-major axis [km]
Eccentricity [-]
Radius of perigee [km]
Radius of apogee [km]
Semi-latus rectum [km]
26563
0.5860
10997
42129
17441
Inclination [◦ ]
RAAN [◦ ]
Argument of perigee [◦ ]
Angular Momentum [km2 /s]
Period [h]
71.5846
0
15
83380
11.9681
Table 9: Characteristics of the assigned orbit
From figure 6, the behaviour of the new orbit is shown and confirms the repeating ground
track with ratio 2:1, as the starting and ending points are at the same position after any
multiple of 2 orbits.
(a)
(b)
Figure 6: Ground track of the unperturbed repeating orbit. Ground track path (
point ( ), Ending point ( )
2.2.3
), Starting
Perturbed ground track
To have a more accurate prediction of the behaviour of the satellite orbiting the Earth, it’s
necessary to introduce disturbances. For this project, only the J2 effect and Moon disturbance
were taken into account, and are described in further detail in section 2.3.
Figure 7 shows the differences between the unperturbed and perturbed orbits for both the
nominal and the repeating orbits from sections 2.2.1 and 2.2.2.
From figure 7, it is noticeable that the perturbations have a light impact on the ground
track path of the satellite, and that it can not be neglected for the mission analysis. Both the
nominal and repeating orbits get out of phase slightly with relation to the unperturbed case.
Moreover, the repeating ground track orbit proposed in section 2.2.2 does not work with
the presence of perturbations. The method used to obtain the repeating orbit assumes that the
period and the semi-major axis of the orbit are constant which is not the case for a perturbed
motion.
11
(a)
(b)
Figure 7: Ground track of the perturbed obits. Ground track of unperturbed ( ), Starting
point ( ), Ending point ( ), Ground track of perturbed ( ), Starting point ( ), Ending
point ( )
2.3
Assigned perturbations
Two orbit perturbations are considered, J2 effect and Moon perturbation.
The first contribution consists of a perturbing potential on top of the central gravity field of
the Earth. TO model it, a zonal harmonic potential is used, function of the geocentric distance
r and of the coelevation ϕ.
n
∞
X
µ
RE
R(r, ϕ) = (−1 +
Jn
Pn (cosϕ))
r
r
n=2
(8)
In figure 8a zonal harmonics are represented. Only the first term of this summation is considered, the term correlated to J2 . This term is strictly correlated to Earth oblateness. The
perturbing acceleration due to J2 effect is the following:
2
2
2
2
x
z
y
z
z
z
3J2 µRE
aJ2 =
5
−
1
î
+
5
−
1
ĵ
+
5
−
3
k̂
(9)
r4
r
r2
r
r2
r
r2
Due to this perturbation semi-major axis a, eccentricity e and inclination i should remain
unperturbed. On the other hand, right ascension of ascending node Ω, argument of pericenter
ω and true anomaly f should be perturbed.
The second perturbation consists of the effect of Moon acting on orbit. For the modelling,
instead of using a three body problem, a two body problem is considered taking into account the
force of the moon as a perturbing acceleration. This perturbing acceleration can be calculated
from equation (10):
!
rm/s
rm
aM oon = µM oon
− 3
(10)
3
rm/s
rm
where rm/s is the vector that goes from the S/C to the moon and rm is the vector that goes
from Earth to Moon. In figure 8bthese vectors are shown. Moon’s position is directly taken
from ephemerides in order to have less uncertainty as possible on moon position. Principal
effects consist in a change of eccentricity e, inclination i and argument of pericenter ω.
12
(a) Earth zonal harmonics
(b) Main bodies
Figure 8: Earth zonal harmonics [4] and main bodies [1].
In figure 9 [2] all possible perturbations are shown. It can be observed that the two main
sources of perturbations are J2 effect (C2,0 in figure) and moon perturbation, indicating that
the calculations here represent a good model.
Figure 9: Orbit perturbations
2.4
Orbit Propagation
Orbits can be propagated in two main ways: using Cartesian coordinates, through Newton’s equations of motion (equations (4)), or Keplerian elements, through Gauss’ planetary
equations (11) to (16). The latter one is presented in Radial-Transversal-Out-of-plane reference frame.
da
2a2
p
=
(esinf ar + as )
dt
h
r
de
1
= (psinf ar + ((p + r)cosf + re)as )
dt
h
rcos(f + ω)
di
=
aw
dt
h
rsin(f + ω)
dΩ
=
aw
dt
hsini
dω
1
rsin(f + ω)cosi
=
(−pcosf ar + (p + r)sinf as ) −
aw
dt
he
hsini
df
h
1
= 2+
(pcosf ar − (p + r)sinf as )
dt
r
eh
(11)
(12)
(13)
(14)
(15)
(16)
The terms ar , as , aw are components of acceleration in RSW frame, h is the angular
momentum, and p is the semi-latus rectum.
13
In the case that the perturbations cannot be directly expressed in the RSW frame, as is
the case for the moon perturbation, it is possible to transform the perturbing accelerations
from Cartesian to RSW. First, a rotation of Ω around the third axis of the inertial frame is
performed, then a rotation of i around the first axis of the rotated frame, and finally a rotation
of an angle f + ω around the third axis of the last frame. The three rotation matrices are
shown in equation (17).

cos Ω
RΩ = − sin Ω
0
2.5
sin Ω
cos Ω
0


0
1
0 Ri = 0
1
0
0
cos i
− sin i


0
cos f + ω
sin i  Rf +ω = − sin f + ω
cos i
0
sin f + ω
cos f + ω
0

0
0
1
(17)
Keplerian Elements Plot
Figures 10 through 15 show the Keplerian elements obtained through the integration of
the equation of Gauss planetary equations and the relative error between both methods of
integration. A time window of two hundred orbital revolutions or 180 days was considered to
observe both the long-period and secular effects of the perturbations. A larger time window
could not be taken into account due to the eventual reentry of the spacecraft.
(a) Keplerian element
(b) Relative error
Figure 10: Semi-major axis a
(a) Keplerian element
(b) Relative error
Figure 11: Eccentricity e
(a) Keplerian element
(b) Relative error
Figure 12: Inclination i
14
(a) Keplerian element
(b) Relative error
Figure 13: RAAN Ω
(a) Keplerian element
(b) Relative error
Figure 14: Argument of pericenter ω
(a) Keplerian element
(b) Relative error
Figure 15: True anomaly f
It is possible to distinguish a long-periodic behaviour and a short-periodic behaviour by
looking at the evolution of orbital elements. It is very clear for eccentricity e and inclination
i. Semi-major axis presents both short-periodic and long-periodic behaviour. Regarding the
right ascension of ascending node Ω, argument of pericenter ω and true anomaly f , the shortperiodic behaviour is less visible but still present.
From figures 10b to 15b of relative error between Gauss’ resolution and Cartesian resolution,
it can be observed that the two methods are equivalent if the precision of the two is compared.
2.6
Evolution of the orbit representation
To give a better understanding of the orbit behaviour, Figure 16 shows how the orbit evolves
in time.
15
Figure 16: Orbit evolution representation. Initial orbit(-). Colorbar on the right is used in
order to let the reader understand evolution of the orbit.The initial position of spacecraft is
(⋆) and the final position of spacecraft is(⋆).
From figure 16 it is possible to observe the change of orbital elements. In particular shape
(semi-major axis a and eccentricity e) does not seem to change considerably. Orbit shows a
decrease in inclination i and a negative rotation of argument of pericenter ω. Change of right
ascension of ascending node Ω is not clearly visible.
2.7
Filtering
To see how the perturbations generate behaviours with different frequencies filtering of
results is performed. Figures 17 to 19 show the filtered results.
(a) Filtered semi-major axis a
(b) Filtered eccentricity e
Figure 17: Filtered semi-major axis a and eccentricity e: (
behaviour and ( ) secular effect.
16
) unfiltered, (
) long term
(a) Filtered inclination i
(b) Filtered RAAN Ω:
Figure 18: Filtered inclination i and RAAN Ω: (
and ( ) secular effect.
(a) Filtered argument of pericenter ω
) unfiltered, (
) long term behaviour
(b) Filtered true anomaly f
Figure 19: Filtered argument of pericenteromega and true anomaly f : (
long term behaviour and ( ) secular effect.
) unfiltered, (
)
As discussed in section 2.3, long-periodic behaviour is related to moon perturbation and
short-periodic perturbation is related to J2 effect. This is due to the fact that J2 perturbation
is related to spacecraft orbit whose period is much lower than the Moon orbit period.
Two different filters are used: the filter used to remove short-term behaviour has a cut-off
frequency of 100 days−1 , instead, the filter used to remove long-term behaviour has a variable
cut-off frequency as a function of parameters we want to filter.
Filtering of the results can show better the components related to two sources of disturbance. According to theory, it is possible to see that semi-major axis a does not change
secularly; instead eccentricity e is changing secularly and it will be clear for a larger time window. As expected inclination i, RAAN Ω and argument of pericenter ω are changing secularly.
2.8
Comparison with real data
To evaluate the precision of the methods used for the assignment, the models were compared
with real data from Atlas 5 Centaur, NORAD Catalogue Number is 29056.
This choice was motivated because the rocket body orbit’s is similar to the nominal orbit
from section 2.1, with a similar eccentricity and semi-major axis. Moreover, the altitude of the
orbit is such that the main disturbances acting on the body are J2 and Moon effects, as can
be seen in figure 8b. Near the perigee, other zonal harmonics effects will have the same order
of magnitude as Moon disturbances.
The data was collected from Space-Trak [3], with initial time 23th February, 2010, 07h28min32s
until 2nd December, 2020, 09h30min30s.
To compute the orbit propagation, the initial orbital elements were those collected from
Space-Track at the initial time, shown on table 10. To propagate the orbit, Gauss’ planetary
equations in RSW frame were solved numerically using ode113. A low-pass filter of frequency
0.9780 day −1 was used to get the mean value every two orbits, considering the initial period.
17
Semi-major axis [km]
Eccentricity [-]
Altitude of perigee [km]
Altitude of apogee [km]
Semi-latus rectum [km]
26936
0.5374
6089
35042
19156
Inclination [◦ ]
RAAN [◦ ]
Argument of perigee [◦ ]
Angular Momentum [km2 /s]
Period [h]
24.6709
344.0999
184.4477
87382
12.2212
Table 10: Characteristics of NORAD 29056 initial orbit
2.6955
104
Semi-major axis
Eccentricity
0.545
2.695
e [-]
a [km]
0.54
2.6945
0.535
2.694
2.6935
0.53
4000
4500
5000
5500
6000
6500
7000
7500
4000
4500
5000
Time[MJD2000]
25
5500
6000
6500
7000
7500
6500
7000
7500
6500
7000
7500
Time[MJD2000]
inclination
RA of ascending node
200
0
[deg]
i [deg]
24.5
24
23.5
-200
23
22.5
-400
4000
4500
5000
5500
6000
6500
7000
7500
4000
4500
5000
Time[MJD2000]
1000
5500
6000
Time[MJD2000]
Argument of perigee
4
106
True anomaly
3
600
f [deg]
[deg]
800
400
2
200
1
0
4000
4500
5000
5500
6000
6500
7000
7500
4000
Time[MJD2000]
4500
5000
5500
6000
Time[MJD2000]
Figure 20: Comparison between real data ( ), numerical propagation (
filtered ( ) of the orbit of rocket body NORAD 29056.
) and numerical
From figure 20, we notice that the numerical approach is capable of estimating the orbit
with good reliability as the trends of both results are very similar. The differences seen on
each plot were expected, as only J2 and Moon perturbations were taken into account for the
numerical model.
There’s a great difference in the short-term behaviour between numerical simulation and
the real data, particularly for semi-major axis and eccentricity. When filtering the short-term
oscillations, better results are obtained for the semi-major axis, but no significant changes are
seen for the other Keplerian elements.
18
References
[1]
H. D. Curtis. Orbital Mechanics for Engineering Students. 3 edition. Butterworth-Heinemann,
2014. isbn: 13 978-0-08-097747-8.
[2]
T. G. R. Reid. ORBITAL DIVERSITY FOR GLOBAL NAVIGATION SATELLITE SYSTEMS. Stanford University, 2017.
[3] Space-Track.org. https://www.space-track.org/.
[4]
P. M. B. Waswa. Analysis and Control of Space Systems Dynamics via Floquet Theory,
Normal Forms and Center Manifold Reduction. Arizona State University, 2019.
19