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THALASSA: a fast orbit propagator for near-Earth and
cislunar space
Davide Amato∗†
University of Arizona, Tucson, AZ, 85721, USA
Technical University of Madrid, Madrid, 28040, Spain
Aaron J. Rosengren
University of Arizona, Tucson, AZ, 85721, USA
Claudio Bombardelli§
Technical University of Madrid, Madrid, 28040, Spain
THALASSA is a numerical propagation code for orbits in the Earth-Moon system, integrat-
ing either unregularized or regularized equations of motion through a multi-step, variable
step-size and order numerical solver. It includes a trajectory splitting algorithm that switches
the primary body and the associated coordinate system along the integration to efficiently
propagate cis- and translunar trajectories. THALASSA is one order of magnitude slower than
the semi-analytical propagator STELA in the long-term integration of a quasi-circular MEO,
but it is capable of achieving a significantly higher accuracy. When a HEO is integrated
instead, THALASSA accurately computes the trajectory when employing regularized formula-
tions, whereas STELA fails to reproduce the qualitative behaviour due to the approximations
involved in the averaging process and in the related analytical developments. The trajectory
splitting algorithm is tested through the integration of a periodic orbit in the Earth-Moon
Circular Restricted Three-Body Problem (CR3BP). Using the algorithm in conjunction with
the Kustaanheimo-Stiefel regularized formulation increases the accuracy by four orders of
magnitude for the same number of right-hand side evaluations and integration steps. If the dy-
namics are significantly perturbed, or in the presence of resonances, integrating non-averaged,
regularized equations of motion is a recommended alternative to semi-analytical techniques.
Nomenclature
d= distance from the Moon
l0= reference unit of length
n0= reference unit of frequency
s= fictitious time
t= time
L= order of lunisolar expansions
Ntess = minimum period of the retained resonant harmonics as a multiple of integration time step
RHill = Hill radius of the Moon
Rsw = threshold distance for the switch of coordinate systems
T= orbital period
V= perturbing potential
α= order of the Sundman transformation
δr= position error
ε= Keplerian orbital energy
Postdoctoral Research Associate, University of Arizona, Department of Aerospace & Mechanical Engineering, 1130 N. Mountain Ave, Tucson,
85721, AZ, USA.
Ph.D. candidate, Technical University of Madrid, School of Aerospace Engineering, Plaza Cardenal Cisneros 3, Madrid, 28040, Spain.
Assistant Professor, Department of Aerospace & Mechanical Engineering, 1130 N. Mountain Ave, Tucson, 85721, AZ, USA.
§Associate Professor, School of Aerospace Engineering, Plaza Cardenal Cisneros 3, Madrid, 28040, Spain.
˜ε= total orbital energy, sum of the Keplerian orbital energy and of the perturbing potential
µ= gravitational parameter
τtess = minimum period of the retained resonant harmonics
t= integration time step
r= position vector
v= velocity vector
u= KS position vector
y= state vector
F= total perturbing acceleration
P= non-conservative perturbing acceleration
λ= EDromo state vector
Subscripts
p= perigee
syn = synodic
E= Earth
M= Moon
I. Introduction
Thanks to a synergy of technological developments in satellite miniaturization and mass production, and in launcher
reusability, the cost of access to space is rapidly decreasing. As a consequence, the population of Resident Space Objects
(RSOs) is expected to rise significantly in the coming years. Although existing Space Situational Awareness (SSA) assets
are steadily increasing the number of cataloged RSOs, future improvements will be needed to tackle this scenario. In the
US Space Surveillance Network (SSN) these improvements will materialize in added capabilities such as the Lockheed
Space Fence,
1
whereas Europe is also bolstering its own independent SSA resources through the ESA SSA program.
By decreasing the detectability size threshold and providing more extensive coverage, next-generation systems for
space surveillance will considerably increase both the quantity and the quality of observations. Such observations
will need to be processed in order to gain a comprehensive understanding of the near-Earth space environment. This
translates in the need for propagating the RSO catalog for short- and mid-term conjunction analyses, and for long-term
debris population studies. Also, a new approach to spacecraft end-of-life disposal, in which lunisolar perturbations
are exploited to achieve disposal orbits with reduced propellant consumption, has arisen recently.
2–4
In order to find
appropriate dynamical configurations leading to disposal orbits, it is necessary to generate maps of the near-Earth
and cislunar space numerically through long-term ensemble propagations for thousands or millions of different initial
conditions.
Classically, long-term orbit propagation has relied heavily on semi-analytical techniques employing the method of
averaging.
5–11
In this approach, the equations giving the rates of change of osculating orbital elements are averaged
either analytically or numerically.
12;13
The resulting averaged equations describe the evolution of mean orbital elements,
which are only subject to long-periodic and secular behaviors that (ideally) describe well the trajectory on long time
spans. The osculating elements can often be recovered by adding short-periodic terms to the mean orbital elements.
Special care is required when the mean motion is commensurable with that of one of the perturbers, i.e. in the case
of mean motion resonances. In this case, terms which would only generate short-periodic oscillations otherwise have to
be retained in the perturbing potential as not to miss important contributions to to the long-term behavior. Also, highly
elliptical orbits are challenging to propagate semi-analytically as the most significant perturbations change during one
orbital period, requiring comprehensive, high-fidelity physical models. A novel semi-analytical technique employing
Lie transforms has recently been developed to tackle this specific problem.14
Contemporaneously with the development of semi-analytical techniques in the 1960s and 1970s, extensive efforts
were being devoted to improve numerical integration through the process of regularization, i.e., the elimination of
singularities from the equations of motion through analytical procedures. Stiefel and Scheifele accomplished the full
regularization and linearization of the two-body problem by applying the spinor formalism originally developed by
Kustaanheimo. In their seminal work,
15
they provide regularized sets of coordinates and orbital elements which have
consistently displayed excellent numerical performances. 16
URL:
http://www.esa.int/Our_Activities/Operations/Space_Situational_Awareness/About_SSA
, last visited: October 18
th
,
2017.
2
It is common in astrodynamics to encounter situations in which the two-body problem is weakly perturbed. In this
case, it is possible to further improve numerical integration by applying Variation Of Parameters (VOP) techniques
along with regularization. Such a strategy is followed in the VOP method presented in Reference [
15
], and by Burdet
and Sperling.
17
This approach has also been followed for the development of the family of Dromo formulations of the
perturbed two-body problem in the last decade.18–22
The numerical integration of regularized equations of motion constitutes a possible alternative to semi-analytical
techniques for the long-term propagation of orbits in the Earth-Moon system. Whereas the efficiency of semi-analytical
techniques is unquestionable for LEOs and MEOs, their accuracy is intrinsically limited due to the approximations
involved in the averaging process and in the necessary analytical developments, such as the expansion in Legendre
series of the third-body potentials.
23
Moreover, semi-analytical techniques might fail in the presence of mean motion
resonances and of strong perturbations by the Moon. Our goal in this work is to study and quantify the performance
of regularized formulations in the propagation of orbits of particular operational interest, and taking semi-analytical
techniques as a reference when possible. Ultimately, we hope to find an optimal compromise between the efficiency
of semi-analytical techniques and the accuracy and reliability accorded by the integration of regularized equations of
motion.
The numerical code that we developed for this study,
THALASSA
, includes the Cowell, Kustaanheimo-Stiefel (KS),
15
and EDromo
20
formulations. The equations of motion are integrated by the adaptive multi-step numerical solver
LSODAR
, which includes dense output and event location capabilities. It includes a mid-accuracy physical model
which takes into account a perturbing geopotential up to degree and order
15
, gravitational perturbations from the Sun
and the Moon, air drag from several atmospheric models, and solar radiation pressure according to the cannonball
model.
THALASSA
efficiently handles strong lunar gravitational perturbations through a trajectory splitting algorithm,
24
according to which the trajectory is integrated in different coordinate systems depending on the primary body. We made
the Fortran source code available in a public repository as to enhance the reproducibility of results and as to provide an
aid for the implementation.
The performance of
THALASSA
is evaluated on three orbit propagation test cases: a MEO similar to those followed
by the Galileo constellation, the HEO of the proposed Simbol-X space observatory,
25
and a periodic orbit in the planar
Earth-Moon Circular Restricted Three-Body Problem (CR3BP). In the first two test cases, we compare the performance
of
THALASSA
with respect to that of the “Semi-Analytic Tool for End of Life Analysis” (
STELA
), a highly efficient
semi-analytical orbit propagator developed by CNES.
The paper is structured in the following way. In the next section, we describe the main characteristics of
THALASSA
.
The MEO and HEO test cases are presented in section III, where the performance of
THALASSA
is compared with that
of
STELA
. In section IV, we take into account the propagation of a periodic orbit in the planar CR3BP. We summarize
the conclusions of the study and outline future improvements in section V.
II. Description of THALASSA
In the following, we expound on the main characteristics of the
THALASSA
code. Particular attention is devoted to
the characteristics of the available sets of equations of motion, and of the numerical solver. Moreover, we describe the
trajectory splitting algorithm allowing the efficient integration of orbits in the Earth-Moon system, and the implemented
physical model.
A. Equations of motion
The choice of the equations of motion to be numerically integrated, that is the particular formulation of the perturbed
two-body problem, can substantially affect performance. The most straightforward choice involves the equations for the
perturbed two-body problem in Cartesian coordinates,
Ü
r=µ
r3r+F,(1)
where Fincludes both potential and non-conservative perturbations:
F=V
r
+P.(2)
URL: https://computation.llnl.gov/casc/odepack/, last visited: November 28th, 2017.
URL: https://logiciels.cnes.fr/fr/node/36?type=desc, last visited: October 31st, 2017.
3
This approach, denoted as the Cowell formulation, is simple and reliable, but suffers from a series of drawbacks.
First of all, the presence of the 1
/r2
singularity implies the amplification of the numerical error after close approaches
to the primary body. Moreover, the solutions of Equation 1 are unstable in the sense of Lyapunov even for Keplerian
motion (that is when
F=0
). These aspects can be significantly ameliorated by eliminating the singularity through
the analytical procedure of regularization. This is accomplished through a change of the independent variable to the
fictitious time sthrough the generalized Sundman transformation:
dt
ds
=f(r,v,s)rα,f>0.(3)
Using the fictitious time as the independent variable in a numerical integration implies the presence of an “analytical
step size regulation,” since equal steps in fictitious time
s
correspond to steps in physical time
t
which are the smaller the
closer to collision (
r
0). By changing also the dependent variables and performing further analytical manipulations,
it is possible one obtains regular (i.e., singularity-free) equations of motion which are also Lyapunov-stable for Keplerian
motion. Since numerical error can be seen as a small deviation from a reference trajectory, the stabilization of Keplerian
motion implies that its growth during the integration is reduced.15;24
In most situations of interest the perturbation
F
in Equation 1 is weak, that is
Fr 2/µ
1. In this case, Variation of
Parameters (VOP) methods can be used to derive sets of orbital elements which are slowly varying. The most immediate
example is the set of classical orbital elements
(a,e,i,, ω, M)
which, however, suffers from singularities for
e=
0and
i=
0. By applying VOP methods to the solution of regularized equations of motion, it is possible to obtain sets of
orbital elements which are free from these singularities.
In addition to the Cowell formulation, the code includes the Kustaanheimo-Stiefel (KS) and the EDromo regularized
formulations of dynamics.
15;20
They use the fictitious time
s
defined in Equation 3 with
α=
1and either
f=
1(for the
KS formulation) or:
f=˜ε1
2(4)
for the EDromo formulation, where
˜ε=ε+V
is the total energy, i.e., the sum of the Keplerian orbital energy
ε=v2/2µ/rand the perturbing potential V.
The KS formulation expresses the state of the particle through a 4-dimensional position vector
u
and its
s
-derivative
u0
. Their evolution is described by a second-order linear equation for
u00
, which has to be integrated along with the rate
of change of the total energy
˜ε
. In total, one has to integrate 10 first-order scalar equations, as one more is needed to
recover the value of physical time.
The EDromo formulation is based on a VOP method in which the state is expressed through a set
λ
of 7 orbital
elements, whose evolution is described by first-order equations. Two of them are the projections of the eccentricity vector
on an “intermediate” frame, one is a function of the total orbital energy, and the remaining four are the components of a
quaternion describing the orientation of the intermediate frame with respect to the inertial. This set of elements is only
valid for ˜ε < 0, although analogous formulations exist for ˜ε > 0.21;22
In both formulations, physical time is recovered by integrating either Equation 3 or the equation for a time element.
The latter choice often presents numerical advantages, since the variation of the time element is smooth for weak
perturbations, unlike
t
. Physical time is then computed from the time element through algebraic relations at each step
of the integration. A time element is denominated either constant or linear, according to its behavior for Keplerian
motion. The KS formulation is endowed with a constant time element (provided that
˜ε <
0), whereas the EDromo
formulation considers either a constant or a linear time element. We did not implement the constant time element of the
KS formulation in THALASSA as to treat all types of orbits.
All the equations are non-dimensionalized with reference units of distance
l0
and of frequency
n0
. The first is the
initial distance from the primary body, whereas the latter is n0=qµ/l3
0.
Exhaustive descriptions of the KS and EDromo formulations and of their derivation can be found in References [
15
]
and [20].
B. Numerical solver
THALASSA
uses the Livermore Solver for Ordinary Differential equations with Automatic Root-finding (
LSODAR
),
an implicit multi-step solver with variable step-size and order.
26
To advance the integration, it employs either the
Adams-Bashforth-Moulton (ABM) numerical scheme or backward differentiation formulas (BDF) depending on whether
the ODE is detected to be stiff. The maximum orders used in the integration are 12 and 5 for the ABM and BDF
4
methods, respectively. The accuracy of integration is adjusted by assigning absolute and relative tolerances on the local
truncation error, which we set to the same value. The only exception is the absolute tolerance assigned to the EDromo
constant time element, which is always fixed to 1for numerical stability reasons.
The choice of the solver has been mostly driven by the peculiarities involved in the integration of regularized
equations of motion. In many situations, it is necessary to obtain the value of the solution at a prescribed value of the
physical time t. Since tis a dependent variable, one first has to solve the event location problem
t(s) − t=0.(5)
The root
s
of Equation 5 must be found during the integration of the equations of motion through an iterative process.
The value of the state vector at the prescribed time
y(t)=y(t(s))
is then computed by interpolating the numerical
solution.
LSODAR
implements a root-finding capability,
27
along with a dense output algorithm which interpolates the
solution between the variable-sized integration steps at no loss of accuracy.
C. Trajectory splitting algorithm
Close encounters with the Moon introduce strong perturbations in the equations of motion. The resulting oscillations
in the solution are detrimental to the accuracy of integration of the regularized equations of motion. To avoid this
situation,
THALASSA
implements a trajectory splitting algorithm. The numerical integration is carried out in a geocentric
coordinate system (E,x,y,z)if the distance from the Moon dis greater than a user-assigned threshold Rsw,
d>Rsw,(6)
and in a selenocentric system
(M,X,Y,Z)
otherwise. The directions of the geocentric and selenocentric axes are the
same in both coordinate systems, and are aligned with those of the International Celestial Reference Frame (ICRF). In
this way, we split the trajectory into a sequence of arcs with different primary bodies according to Equation 6. Before
starting the integration of each arc, the reference units for non-dimensionalization of the equations are updated with the
values of the initial distance from the primary body. In each of the arcs, the magnitude of the perturbation
F
(which is
proportional to 1/d2) is significantly reduced, increasing numerical efficiency.
The algorithm, which is conceptually similar to the patched conics approach commonly used in the preliminary
design of interplanetary trajectories, was originally developed for the accurate integration of close encounters by
Near-Earth Asteroids.24 In this work, it has been extended to handle arbitrarily long sequences of close encounters.
D. Physical model
As
THALASSA
was originally developed for a comparative study with respect to the
STELA
orbit propagator, its
physical model closely resembles that of the latter. In particular,
THALASSA
takes into account the EGM-GOC-2
geopotential model with degree and order up to 15, gravitational perturbations from the Moon and the Sun, solar
radiation pressure (without considering eclipses), and atmospheric drag. The non-spherical geopotential is implemented
as a conservative perturbation for the KS and EDromo formulations. The ephemerides for the Sun and the Moon are
either computed from analytical formulas,
28§
or from JPL Developmental Ephemerides (DE) which are read through
the SPICE toolkit.
29
Four atmospheric models are included: a piecewise exponential model, the US 1976 Standard
Atmosphere, the Jacchia 1977 model, and the NRLMSISE-00 model.
In the test cases shown in the following section, we only consider perturbations stemming from a 2
×
2geopotential
and gravitational perturbations due to the Sun and the Moon, whose ephemerides are computed from analytical formulas.
Additional investigations including atmospheric drag are left for a forthcoming work.
E. Code availability
We made available the Fortran source code of
THALASSA
on the public repository
https://gitlab.com/
souvlaki/thalassa
, as to enhance the reproducibility of results and its implementation. It has been compiled
and tested using the
gfortran 5.4.0
compiler on Ubuntu and Mint Linux distributions. The code requires minor
modifications to run in Windows environments or with different Fortran compilers. The public repository also includes
Python 2 scripts that can be used to execute propagation batches with THALASSA.
§The subroutine for the computation of the analytical ephemerides was kindly provided by F. Deleflie.
5
Table 1 Initial Modified Julian Date and osculating orbital elements for the Galileo test case.
Initial value Unit
MJD 58 474.7433
a29 601.310 447 014 60 km
e0.0001
i56 °
116.640 939 804 248 0 °
ω0°
M0°
III. Performance for high-altitude orbits
The computational time and the accuracy obtained from propagations with
THALASSA
are compared to those obtained
with
STELA
3.1.1.
STELA
integrates singly-averaged equations of motion for a set of mean equinoctial elements
30
in
physical time using a fixed-step Runge-Kutta scheme of 6
th
order, and it takes into account a wide range of perturbations.
Its physical model was partly re-implemented in
THALASSA
(subsection II.D), as to avoid any spurious results in the
evaluation of numerical error. The conversion between osculating and mean elements is handled in
STELA
by adding or
subtracting short-periodic terms, which are computed taking into account only the
J2
and lunisolar perturbations; the
initial conditions in the mean orbital elements are also computed in this way.
STELA
requires the user to fine-tune several parameters in order to obtain the best propagation performance. First of
all, the accuracy of the numerical integration can be changed by varying the fixed time step
t
of the numerical scheme.
Resonant tesseral harmonics in the expansion of the geopotential are retained and averaged if
τtess Ntesst,(7)
where
τtess
is the period of each of the harmonics, and
Ntess
can be chosen by the user. Also the order
L
of the expansions
of the lunar and solar perturbing potentials can be selected from values between
2
and
5
. We disabled precession and
nutation effects for these tests.
We remark that
STELA
and
THALASSA
are implemented in Java and Fortran, respectively. Calculations performed in
the latter language generally display lower computational times; the amount by which the computational time differs
depends on several factors such as the specific algorithm, the system characteristics, and the particular versions of the
Fortran compiler and the JRE/JDK. Benchmarks of generic scientific computing and astrodynamics applications show
that Java codes can be up to one order of magnitude slower, although this depends on the particular application.
31;32
The increase in CPU time of Java algorithms with respect to Fortran can be estimated conservatively at 5 times for the
numerical tests shown here.
We examine the performance of
THALASSA
in the propagation of high-altitude orbits, that is with a semi-major axis
a20 000 km
. We take into account two test cases whose initial conditions reflect operational orbits of real or planned
missions, i.e., Galileo and Simbol-X. Each of these initial conditions is propagated for 200 years or until a re-entry at a
height of
80 km
is detected. The physical model takes into account a 2
×
2geopotential and perturbing accelerations
from the Moon and the Sun considered as point masses.
In each of the test cases, we compute a reference solution in quadruple precision using the Cowell formulation in
THALASSA
together with a very strict solver tolerance. The reference solution is used to check that the trajectory is
qualitatively reproduced at lower accuracies using both
STELA
and
THALASSA
, and to quantify errors on the osculating
state. The workstation used for the tests has a quad-core Intel i7-4771 processor at
3.50 GHz
,
32 GB
of DDR3 RAM,
and runs Linux Mint 17.3.
A. Galileo
Initial conditions corresponding to a Galileo-like orbit are displayed in Table 1. The evolutions of the reference
osculating orbital elements and of the osculating and mean orbital elements from a
STELA
propagation are shown in
6
-4
-2
0
aai(km)
0
1
2
e×104
57
59
61
i(deg)
90
180
270
(deg)
90
180
270
ω(deg)
0 50 100 150 200
t(yr)
0
90
180
270
M(deg)
THALASSA STELA (osc) STELA (mean)
Fig. 1 Orbital elements as a function of time for the propagation of the Galileo initial conditions in Table 1 for
200 years. The curves “THALASSA” and “STELA (osc)” refer to the osculating orbital elements obtained in the
reference propagation with THALASSA, and with the STELA propagation for L=
4
,Ntess =
0
.
4
and t=120 h,
which exhibit a significant overlap. The curve “STELA (mean)” refers to the mean orbital elements obtained
from the STELA propagation. The semi-major axis is plotted as a difference from the initial osculating value.
7
Figure 1 for 200 years of propagation. The latter is obtained by setting
L=
4,
Ntess =
0
.
4, and
t=120 h
. A value of
Ntess =
0
.
4implies that only tesseral harmonics with a period greater than
2 d
are retained in the geopotential. Both
orbit propagators reproduce the qualitative characteristics of the orbit well, which is quasi-circular during the whole
integration span. The osculating orbital elements obtained with the reference and the
STELA
propagation overlap. This
suggests that the recovery of the short-periodic terms due to
J2
and lunisolar perturbations is sufficient to adequately
describe the evolution of the osculating trajectory, in this case.
A more encompassing picture of the numerical performance of
THALASSA
can be built by performing batch
propagations of the initial conditions in Table 1. We repeat the 200-year propagation with
THALASSA
for solver
tolerances decreasing from
104
to
1015
, and with
STELA
for
t
decreasing from
12 d
to
0.5 d
. In the latter code, we set
L=
4
,Ntess =
0
.
4. Figure 2 shows the CPU time as a function of the errors on the orbital elements. These are computed
as absolute differences with respect to the values obtained with the
THALASSA
reference propagation, whereas the CPU
time is the average on 3 runs of the same propagation. We display the results for all of the formulations implemented
in
THALASSA
. When the EDromo formulation is used, we propagate the physical time by integrating the Sundman
transformation, or by employing a constant or linear time element.
The semi-analytical approach used by
STELA
is the most efficient for moderate levels of accuracy. In fact,
STELA
shows a minimum error in osculating semi-major axis on the order of
100 m
, with computational times between
2 s
and
15 s
. This error reduces to
1 cm
when using the EDromo formulation, albeit with a computational time that is
75
to
100
times higher when taking into account the different languages of implementation. This aspect is relevant, as it is
important to achieve a high accuracy in the computation of the semi-major axis (equivalently, the orbital energy or
period) to constrain the long-term growth of the numerical error in the along-track direction.
The performance of
STELA
is relatively insensitive to the value of the integration time step. This is because the
equations of motion are necessarily approximated in a semi-analytical approach. The first step of an analytical averaging
process involves the expansion of the right-hand side of the osculating equations of motion in power series of the small
parameters attaining to the perturbations. As to compute explicit expressions for the right-hand sides of the averaged
equations of motion, these series must be truncated and then averaged over one period while assuming some simplifying
hypotheses (usually, that the mean orbital elements are constant during the averaging operation). We denominate as
averaging error the contribution to the integration error due to these simplifications. As a consequence, the choice of
the solver tolerance (or time step) has a limited impact on the accuracy attainable with semi-analytical methods, as
these parameters do not affect the averaging error. This is quite evident in the errors on
i
and
, which are completely
dominated by the averaging error. We remark that the right-hand side of the mean equations of motion must be truncated
even if the averaging operation is performed through a numerical scheme,
12;13
as is usually the case for non-conservative
perturbations. Thus, orbit propagations will be affected by averaging error even in this approach.
Figure 2 shows that the regularized formulations implemented in
THALASSA
give rise to a better numerical efficiency
with respect to the Cowell formulation. In particular, EDromo allows to obtain values of the eccentricity which are
four orders of magnitude more accurate than Cowell and KS. While these initial conditions do not lead to a growth
in eccentricity sufficient for a re-entry in 200 years, this could happen for sets of initial conditions which are “close
enough”.
33
In this respect the accurate computation of the eccentricity is particularly relevant, as it drives the radius of
perigee and thus the possibility of interaction with the dense layers of the atmosphere.
Overall, the best performance with
THALASSA
is obtained by using the EDromo formulation, whose elements
vary quite slowly for a weakly-perturbed, quasi-circular orbit. However, even switching from the Cowell to the KS
formulation results in an increase in efficiency. Since angles at the origin are doubled when transforming from Cartesian
to KS coordinates, the orbital frequency of the KS solution is half that of the solution in Cartesian coordinates. This
results in a smoother right-hand side, which is more amenable to numerical integration.
For large tolerances (right part of the plots in Figure 2) the accumulation of truncation error induces numerical
instability in the solution which leads to the complete loss of information on the orbit. Note that it is necessary to assign
a very small tolerance to avoid this instability when using the Cowell formulation; this leads to correspondingly higher
computational times. For very small tolerances, the best attainable accuracy is limited by the accumulation of round-off
error in double-precision arithmetic.
We remark that, when taking into account the fivefold increase in CPU time due to the Java implementation of
STELA
,
THALASSA
is about 20 times slower for the same accuracy of
1 km
in semi-major axis. It is only possible to
improve accuracy by choosing a smaller tolerance in
THALASSA
, whereas
STELA
is limited in this respect due to the
reasons describe above.
8
-6 -4 -2 0 2 4
log10δa(km)
0
10
20
30
40
50
60
CPU time (s)
-8 -6 -4 -2 0
log10δe
-11 -9 -7 -5 -3 -1
log10δi(deg)
0
10
20
30
40
50
60
CPU time (s)
-7 -5 -3 -1 1 3
log10δ(deg)
-5 -3 -1 1 3
log10δu(deg)
0
10
20
30
40
50
60
CPU time (s)
Cow EDr(t) EDr(c) EDr(l) KS STELA
Fig. 2 CPU time as a function of the errors on the osculating orbital elements after 200 years for the Galileo
test case. The CPU time is measured as an average of the ones measured in 3 runs of both codes, and it does not
take into account differences due to the implementation in different languages. The points are obtained with
the formulations included in the THALASSA code (in black), and with STELA (in red). The label “Cow” refers to
the Cowell formulation, whereas the labels “EDr(t)”, “EDr(c)” and “EDr(l)” refer to the EDromo formulation
in which the physical time is propagated through the Sundman transformation, a constant time element, and a
linear time element, respectively. The label “KS” refers to the Kustaanheimo-Stiefel formulation. The “STELA
data series refers to propagations performed with L=
4
and Ntesst=2 d and integration time steps decreasing
from 12 d to 0.5 d, whereas the results for THALASSA are obtained by decreasing the solver tolerance from 104
to 1015.
9
Table 2 Initial MJD and osculating orbital elements for the Simbol-X test case.
Initial value Unit
MJD 56 664.863 368 054 20
a106 247.136 454 000 0 km
e0.751 73
i5.2789 °
49.351 °
ω180 °
M0°
B. Simbol-X
We consider the orbit of Simbol-X, an X-ray space telescope that was proposed by ASI and CNES.
25
This test case
was also used as a challenging problem for the semi-analytical propagator developed in Reference [
14
], who provide
the initial conditions displayed in Table 2. These correspond to a highly elliptical orbit with a large semi-major axis,
resulting in a 7:1mean motion resonance with the Moon. Due to the large semi-major axis and to the presence of the
resonance, we expect lunisolar perturbations to play a rather important role. These characteristics make it an interesting
test case for orbit propagation software.
The evolution of the mean and osculating orbital elements for the
THALASSA
reference propagation and for two
propagations with
STELA
are displayed in Figure 3. The latter are performed by setting the order
L
of the expansion of
the lunar and solar perturbing potentials to either
4
or
5
. All the propagations are stopped as soon as the condition
rRE<80 km is satisfied, which signals a re-entry. This takes place after 80 years for the reference propagation and
the STELA propagation with L=4, and after 82 years for the one with L=5.
The frequency of the long-periodic oscillations in the eccentricity differs significantly between the
STELA
and
reference propagations, and the remaining orbital elements start diverging already after 14 years. Moreover, the
trajectories computed for different values of
L
display different qualitative behaviors, showing that
L
has a significant
impact on the propagation. Additional tests, which we do not present here, have also evidenced that smaller values of
L
lead to trajectories shifting even further from the reference.
In fact, in the averaging approach used by
STELA
one assumes that the orbital elements of the third bodies are
constant during one orbital period, which in this case is of about
4 d
. This approximation is problematic, as the Moon
moves by about
53°
along its orbit during one orbital period of the spacecraft. Also,
STELA
averages over the critical
effects of the 7:1mean motion resonance with the Moon, thus missing long-periodic behaviors associated with it. All
these issues stem from the very high value of the ratio of the semi-major axis of the spacecraft to that of the Moon,
(a/aM) ≈
0
.
3. The time scale of the orbital period is relatively close to that of the lunar perturbations, which is an
unfavorable situation for a semi-analytical method.
Analogously to subsection III.A, we perform batch propagations of the initial conditions in Table 2 with both codes.
Differently from the Galileo test case, the
STELA
time step is varied between
40 d
and
1 d
; we use larger time steps as to
accommodate the larger orbital period. We consider two propagation batches for
STELA
corresponding to
L=
4
,
5, and
set the minimum period of the tesseral harmonics retained in the potential to
Ntesst=40 d
. Since the re-entry date
changes depending on the code used and on
L
, we consider the errors in the orbital elements and the corresponding
CPU times at an epoch 75 years from the initial, when no computed trajectory has re-entered yet.
The CPU time as a function of the errors on the osculating orbital elements is displayed in Figure 4. As in the
Galileo test case, decreasing the
STELA
time step does not significantly improve the accuracy of the propagation. Even
considering a larger
L
does not affect the numerical error at 75 years significantly, but it does lead to a better qualitative
reproduction of the reference trajectory, as in Figure 4. Note that
STELA
commits an error in eccentricity between
102
and
101
, which corresponds to an error in
rp
between
103km
and
104km
. As stated in the previous section, such a
significant uncertainty in
rp
implies that the potential interaction of the spacecraft with the atmosphere cannot be reliably
estimated.
Large errors in mean anomaly for
STELA
and Cowell imply that the position of the spacecraft is already lost after 75
10
-500
0
500
aai(km)
0.05
0.5
0.95
e
0
45
90
i(deg)
90
180
270
(deg)
90
180
270
ω(deg)
0 10 20 30 40 50 60 70 80
t(yr)
90
180
270
M(deg)
THALASSA STELA,lM= 4 (mean) lM= 5 (mean) lM= 4 (osc)
Fig. 3 Orbital elements as a function of time for the propagation of the Simbol-X initial conditions in Table 2 for
83 years. The curve “THALASSA” refers to the osculating orbital elements obtained in the reference propagation.
The curves “STELA,L=
4
(mean)”, “L=
4
(osc)” and “L=
5
(mean)” refer to the mean (osculating) orbital
elements obtained from STELA with the reported values of the truncation order of the lunisolar expansions L
and with t=48 h,Ntess =
20
. The re-entry dates of the STELA propagations differ with respect to the reference
by up to two years.
11
-4 -2 0 2 4
log10δa(km)
0
2
4
6
8
10
12
14
CPU time (s)
-6 -4 -2 0
log10δe
-7 -5 -3 -1
log10δi(deg)
0
2
4
6
8
10
12
14
CPU time (s)
-5 -3 -1 1 3
log10δ(deg)
-5 -3 -1 1
log10δω(deg)
0
2
4
6
8
10
12
14
CPU time (s)
-2 0 2
log10δM(deg)
Cow EDr(t) EDr(c) EDr(l) KS STELA,lM= 5 STELA,lM= 4
Fig. 4 CPU time as a function of the errors on the osculating orbital elements after 75 years of propagation for
the Simbol-X test case. The CPU time is measured as an average of the ones measured in 3 runs of both codes,
and it does not take into account differences due to the implementation in different languages. The points are
obtained with the formulations included in the THALASSA code (black points), and with STELA (red points). The
labels for the non-averaged formulations are the same as in Figure 2. The propagations with STELA are reported
for L=
4
and L=
5
, and integration time steps decreasing from 40 d to 1 d; in both cases Ntesst=40 d. The
results with THALASSA are obtained by decreasing the solver tolerance from 104to 1015.
12
Table 3 Initial position and velocity for the test particle and the Moon in the geocentric coordinate system,
derived from the dimensionless initial conditions in the synodic coordinate system in Reference [35] with
µE=3.986 004 41 ×105km3s2and µE/µM=1/81.45.
Particle Moon
x4.467 794 60 ×1053.844 000 0 ×105km
y0 0 km
Û
x0 0 km s1
Û
y1.199 786 3 1.024 612 7 km s1
years of integration. In fact, the Cowell formulation exhibits the worst performance among the non-averaged methods.
The integration in physical time involves larger accumulated round-off and truncation errors, which limit the accuracy
attainable with the Cowell formulation to moderate values. In contrast, regularized formulations improve the accuracy of
up to four orders of magnitude. EDromo and KS are equally efficient, since regularized coordinates and elements evolve
on similar time scales for such a high orbit. However, EDromo provides up to one order of magnitude improvement in
accuracy.
We remark that the numerical solver used for all the formulation is adaptive. If the step-size and integration order
had been fixed, the advantage in performance of the regularized formulations over Cowell would only increase. Most
importantly, the regularized formulations in
THALASSA
reproduce the qualitative behaviour of the trajectory faithfully
over the entire propagation span, unlike the semi-analytical approach used in STELA.
IV. Performance for a translunar orbit
The last of the numerical tests proposed in the present work is executed on a periodic orbit in the planar Moon-Earth
CR3BP. We consider the Moon to be moving on a circular orbit about the Earth with mean motion
nM=p(µE+µM)/aM
,
according to the assumptions of the CR3BP and differently from the tests performed in the previous section.
The initial position and velocity in the geocentric coordinate system are displayed in Table 3 for both the orbiter and
the Moon. These correspond to the dimensionless initial conditions for a periodic orbit originally found by Davidson,
34
which are provided in the synodic coordinate system. The orbit is close to a 4 : 13 commensurability with that of the
Moon, as the value of its synodic period
Tsyn =127.92 d
implies 4
Tsyn
13
TM
. The trajectory is displayed in Figure 5
in both the geocentric and synodic coordinate systems, for one synodic period. The orbit is very unstable, as changes
of less than
1 %
in the mass ratio may lead to a completely different dynamical behavior.
35
Moreover, we performed
numerical tests evidencing that relative changes on the order of
1×1012
in the initial conditions imply a loss of the
periodicity for
t>
3
Tsyn
. This intrinsic instability makes propagating this orbit quite a challenging test for numerical
propagators.
We built a reference trajectory by propagating in quadruple precision and with a very strict tolerance the initial
conditions in Table 3. As to guarantee the existence of the periodic orbit, we only consider the gravitational acceleration
from the Earth and the Moon in the equations of motion. We set the integration interval to 1
Tsyn
since the amplification
of the numerical error makes its measurement difficult at the end of larger intervals.
The total orbital energy with respect to the Moon
˜εM
changes sign during the propagation.This prevents the
integration of the trajectory with the EDromo formulation using the trajectory splitting algorithm, if no checks on the
sign of the orbital energy are performed. It is possible to devise a switch criterion analogous to Equation 6, but imposing
a relative or absolute threshold on orbital energy rather than distance. This will be considered for further investigation, as
it is out of the scope of the present work. One could still be able to integrate the trajectory with the EDromo formulation
in the geocentric coordinate system without applying trajectory splitting, as the total orbital energy with respect to
the Earth is
˜εE<
0always. However, the integration of such a strongly perturbed trajectory (see Figure 5a to verify
the significantly non-Keplerian character of the orbit) with a VOP method would result in a very poor performance.
Therefore, we omit results obtained with the EDromo formulation altogether. Also, it is not possible to propagate this
The synodic coordinate system
(CG, ξ, η)
has its origin in the center of gravity, its
η
-axis directed towards the Moon, and its
η
-axis directed
along the Moon’s velocity. It rotates with respect to the geocentric coordinate system with angular velocity nM.
13
-1 -0.5 0 0.5 1
x(aM)
-1
-0.5
0
0.5
1
y(aM)
(a) Geocentric. (b) Synodic.
Fig. 5 Periodic orbit displayed in the geocentric (left panel) and synodic (right panel) coordinate systems, for
a duration of 1 synodic period Tsyn =127.92 d and in units of lunar semi-major axis aM=384 400 km. The
Earth and the Moon are displayed to scale as gray circles. The Moon’s orbit is displayed with a gray line in
the left panel. The initial conditions (Table 3) correspond to the points (x0,y0)=(
1
.
162
,
0
),(ξ0, η0)=(
1
.
150
,
0
).
The trajectory is plotted in black where the distance from the Moon is greater than Rsw =
1
.
1
RHill, and in red
otherwise. The dashed red circle is centered on the Moon and is of radius Rsw. Compare the right panel with
Fig. 9.24(b) of Reference [35].
orbit with
STELA
, due to the physical model corresponding to the CR3BP not being implemented and to the intrinsic
limitations of semi-analytical approaches in the integration of three-body problems.
As in previous sections, we repeat the propagation in double precision with several values of the tolerance, using
either the Cowell or KS formulations. Each of the propagations is performed twice. In a first instance, we keep the
coordinate system selenocentric during the whole integration period. Subsequently, we enable the trajectory splitting
algorithm described in subsection II.C, setting
Rsw =1.1RHill =67 914 km,(8)
where
RHill =61 740 km
is the Hill radius of the Moon. This value of
Rsw
in terms of Hill radii has been found to give
optimal performances in past studies using the trajectory splitting algorithm.
24
In Figure 5, parts of the trajectory in
which Equation 6 is satisfied are displayed in black.
The average computational time on
6
runs, number of right-hand side evaluations, and integration steps taken as a
function of the position error with respect to the reference trajectory
δr
are displayed in Figure 6. The error is measured
at the end of the 1
Tsyn
propagation span. We choose this particular metric for the numerical error rather than one based
on prime integrals, such as the Jacobi constant, since the latter can still show small values for high values of the position
error. The tests are performed on a dual-core Intel i7-7500U at
2.70 GHz
machine with
16 GB
DDR3 RAM, running
Ubuntu 16.04.
Figure 6a shows that the computational time is higher when using the trajectory splitting algorithm due to the
overheads associated with event location, the switch of coordinate systems, and the re-initialization of the integration at
each switch. Also, the KS formulation exhibits a slightly higher computational time since its right-hand side involves
more algebraic computations.
In practical scenarios in which more complex physical models are considered, the main computational burden
resides in the evaluation of the perturbation routines. In this case, the computational cost is better conveyed by the
number of right-hand side evaluations and of integration steps taken (Figures 6b and 6c). According to these metrics,
the KS formulation together with the trajectory splitting algorithm increases the position accuracy by four orders of
magnitude. In fact, splitting the trajectory results in a smoother evolution of the KS coordinates and of the total energy.
On the other hand, it does not affect neither the number of right-hand side evaluations nor the number of steps for the
Cowell formulation. Changing the coordinate system implies moving the largest gravitational acceleration in or out of
14
(a) CPU time. (b) Right-hand side evaluations. (c) Steps taken.
Fig. 6 CPU time (left panel), right-hand side evaluations (middle panel), and steps taken (right panel) as a
function of the position error measured after 1 synodic period Tsyn =127.92 d of the periodic orbit with initial
conditions displayed in Table 3. The data series “Cow” and “KS” denote propagations performed with the
Cowell and Kustaanheimo-Stiefel formulations, respectively. Red markers are relative to propagations in which
the trajectory is split according to the algorithm in subsection II.C, whereas black markers are obtained by
keeping the coordinate system selenocentric during the whole integration.
the term
F
in Equation 1, however the equations of motion formally stay the same and the change has no numerical
consequence except when the contribution of round-off error dominates. In this circumstance, some significant digits
may indeed be lost in the computation of
F
when the trajectory is not split. Moreover, the numerical values of
l0
and
n0
should be updated to reduce the accumulation of round-off error; this is only done when the trajectory is split.
Examination of Figure 6 reveals that trajectory splitting increases the maximum accuracy achievable by two orders
of magnitude for both formulations. This is significant, as a smaller accumulation of numerical error implies that the
main characteristics of the orbit will be preserved for longer integration spans; equivalently, the onset of the numerical
instabilities degrading the results is delayed.
V. Conclusions and outlook
A. Concluding remarks
The code
THALASSA
numerically propagates orbits in the Earth-Moon system by integrating either unregularized or
regularized equations of motion (the latter using the Kustaanheimo-Stiefel and EDromo formulations) with a multi-step,
variable step-size and order numerical solver. Cis- and translunar orbits are efficiently integrated by adopting a trajectory
splitting algorithm that switches the primary body (and the associated coordinate system) between the Earth and the
Moon along the propagation.
THALASSA
includes perturbations from a 15
×
15 geopotential, air drag from several
atmospheric models, gravitational accelerations from the Sun and the Moon, and attitude-independent solar radiation
pressure. Solar and lunar ephemerides are obtained either from analytical formulas or from reading JPL DE ephemerides.
We compare
THALASSA
against the semi-analytical orbit code
STELA
in the propagation of MEO and HEO orbits.
STELA
is most efficient for moderate levels of accuracy in the MEO case, showing a minimum error in osculating
semi-major axis on the order of
100 m
after 200 years of propagation, with a CPU time from
2 s
to
15 s
.
THALASSA
achieves the same accuracy with a computational time
15
to
20
times higher. In contrast to
STELA
, it is possible to
achieve centimeter-accuracy in the semi-major axis by decreasing the solver tolerance.
For the HEO test case, the trajectory computed by
STELA
diverges from the reference already after 14 years of
propagation. The approximations involved in the averaging process and the presence of a 7:1mean-motion resonance
with the Moon imply that the dynamics cannot be accurately followed by
STELA
. The position along the orbit after 75
years of propagation is only recovered by
THALASSA
using regularized formulations, whereas it is lost with the Cowell
formulation.
A semi-analytical approach is undoubtedly more efficient than the numerical integration of the non-averaged
equations for dynamics which are “regular enough”, short orbital periods, and moderate accuracy requirements. In these
cases, numerical methods based on regularized formulations can provide similarly accurate results in a computational
time that, while longer, may still be acceptable depending on the application. However, the robustness afforded by such
15
methods also allows the accurate integration of highly elliptic, strongly perturbed orbits, with long orbital periods, and
in the presence of resonances.
Finally, we evaluate the performance of
THALASSA
in the propagation of a periodic orbit in the Earth-Moon CR3BP,
with the additional purpose of validating the trajectory splitting algorithm. We find that by using the Kustaanheimo-Stiefel
formulation along with trajectory splitting the accuracy increases by four orders of magnitude with respect to the Cowell
formulation, after one synodic period and for the same number of integration steps and right-hand side evaluations.
B. Outlook
Further work is needed to investigate additional improvements to
THALASSA
. We will take into account additional
criteria for switching coordinate systems and primary bodies in the trajectory splitting algorithm. For instance, one
could set a threshold on the local truncation error generated by the gravitational attraction of the perturbing body. In
fact, its estimates are already available during the integration due to the need of adapting both step-size and order of the
solver. Operating directly on the local truncation error (rather than on distance) would mitigate its accumulation at the
end of the integration.
Currently, the gravitational perturbations from the perturbing bodies are implemented as non-conservative accel-
erations in regularized formulations. It might be possible to further enhance performance by implementing them as
time-dependent perturbing potentials instead, especially when the periods of the perturbing bodies are much smaller
than that of the particle being propagated.
As to make trajectory splitting viable for the propagation of realistic translunar trajectories, the definition of the
selenocentric coordinate system will be made consistent with the 2009 IAU recommendations;
36
similarly, precession
and nutation of the geocentric coordinate system will be implemented for consistency with the ICRF.
Acknowledgments
The development of the project was started during the research visit of D. Amato at the Aristotle University of
Thessaloniki (Greece). D. Amato thanks K. Tsiganis for his supervision during the visit, and J.F. San-Juan, M. Lara, and
D. Hautesserres for helpful discussions on semi-analytical techniques. D. Amato gratefully acknowledges the assistance
of F. Deleflie for the implementation of the analytical ephemerides.
References
[1]
Koltiska, M. G., Du, H., Prochoda, D., and Kelly, K., “AN/FSY-3 Space Fence System Support of Conjunction Assessment,”
2016 AMOS Conference Proceedings, Maui Economic Development Board, 2016.
[2]
Colombo, C., Letizia, F., Alessi, E. M., and Landgraf, M., “End-of-life Earth re-entry for highly elliptical orbits: the INTEGRAL
mission,” Advances in the Astronautical Sciences, Vol. 152, 2014, pp. 26–30. Proceedings of the AAS/AIAA Spaceflight
Mechanics Meeting, January 2014, Santa Fe, NM, USA.
[3]
Colombo, C., “Long-Term Evolution of Highly-Elliptical Orbits: Luni-solar perturbation effects for stability and re-entry,
Advances in Space Research, Vol. 155, 2015, pp. 3117–3140. Proceedings of the AAS/AIAA Spaceflight Mechanics Meeting,
January 2015, Williamsburg, VA, USA.
[4]
Rosengren, A. J., Amato, D., Daquin, J., and Gkolias, I., “The dynamical placement of satellite constellations and designing
for demise,Proceedings of the 9
th
International Workshop on Satellite Constellations and Formation Flying, University of
Colorado, Boulder, CO, USA, 2017.
[5]
Cefola, P. J., Long, A. C., and Holloway, G., “The long-term prediction of artificial satellite orbits,” AIAA 12
th
Aerospace
Sciences Meeting, American Institute of Aeronautics and Astronautics, Washington, D. C., USA, 1974. Paper 74-170.
[6]
Velez, C. E., and Fuchs, A. J., “A review of averaging techniques and their application to orbit determination systems,AIAA
12
th
Aerospace Sciences Meeting, American Institute of Aeronautics and Astronautics, Washington, D. C., USA, 1974. Paper
74-171.
[7]
McClain, W. D., “A recursively formulated first-order semianalytic artificial satellite theory based on the generalized method
of averaging. Volume 1: The generalized method of averaging applied to the artificial satellite problem,” Tech. Rep. NASA
CR-156782, NASA Goddard Space Flight Center, Greenbelt, MD, USA, 1977.
16
[8]
McClain, W. D., “A recursively formulated first-order semianalytic artificial satellite theory based on the generalized method of
averaging. Volume 2: The explicit development of the first-order averaged equations of motion for the nonspherical gravitational
and nonresonant third-body perturbations,” Tech. Rep. NASA CR-156783, NASA Goddard Space Flight Center, Greenbelt,
MD, USA, 1978.
[9]
Danielson, D. A., Sagovac, C. P., Neta, B., and Early, L. W., “Semianalytic Satellite Theory,” Tech. Rep. ADA276836,
Department of Mathematics, Naval Postgraduate School, Monterey, CA, USA, 1995.
[10]
Métris, G., and Exertier, P., “Semi-analytical theory of the mean orbital motion,” Astronomy and Astrophysics, Vol. 294, 1995,
pp. 278–286.
[11]
Cefola, P. J., Folcik, Z., Di-Costanzo, R., Bernard, N., Setty, S., and San-Juan, J. F., “Revisiting the DSST standalone orbit
propagator,Advances in the Astronautical Sciences, Vol. 152, American Astronautical Society and American Institute of
Aeronautics and Astronautics, Univelt, Inc., Santa Fe, NM, USA, 2014.
[12] Uphoff, C., “Numerical averaging in orbit prediction.” AIAA Journal, Vol. 11, 1973, pp. 1512–1516. doi:10.2514/3.50620.
[13]
Ely, T. A., “Mean element propagations using numerical averaging,The Journal of the Astronautical Sciences, Vol. 61, No. 3,
2014, pp. 275–304. doi:10.1007/s40295- 014-0020-2.
[14]
Lara, M., San-Juan, J., and Hautesserres, D., “HEOSAT: a mean elements orbit propagator program for highly elliptical orbits,”
CEAS Space Journal, 2017. doi:10.1007/s12567-017- 0152-x, online first.
[15]
Stiefel, E. L., and Scheifele, G., Linear and Regular Celestial Mechanics, Springer-Verlag, Berlin, Germany, 1971. doi:
10.1007/978-3- 642-65027- 7, URL
http://link.springer.com/10.1007/978-3- 642-65027- 7{%}5Cnhttp://d-nb.
info/458247197.
[16]
Janin, G., “Accurate computation of highly eccentric satellite orbits,” Celestial Mechanics, Vol. 10, 1974, pp. 451–467.
doi:10.1007/BF01229121.
[17]
Bond, V. R., and Allman, M. C., Modern Astrodynamics: Fundamentals and Perturbation Methods, Princeton University Press,
Princeton, NJ, USA, 1996.
[18]
Peláez, J., Hedo, J. M., and Rodríguez de Andrés, P., “A special perturbation method in orbital dynamics,” Celestial Mechanics
and Dynamical Astronomy, Vol. 97, 2007, pp. 131–150. doi:10.1007/s10569- 006-9056- 3.
[19]
Baù, G., Bombardelli, C., and Peláez, J., “A new set of integrals of motion to propagate the perturbed two-body problem,
Celestial Mechanics and Dynamical Astronomy, Vol. 116, 2013, pp. 53–78. doi:10.1007/s10569- 013-9475- x.
[20]
Baù, G., Bombardelli, C., Peláez, J., and Lorenzini, E., “Non-singular orbital elements for special perturbations in the two-body
problem,” Monthly Notices of the Royal Astronomical Society, Vol. 454, 2015, pp. 2890–2908. doi:10.1093/mnras/stv2106.
[21]
Roa, J., and Peláez, J., “Orbit propagation in Minkowskian geometry,Celestial Mechanics and Dynamical Astronomy, Vol.
123, 2015, pp. 13–43. doi:10.1007/s10569- 015-9627-2.
[22]
Baù, G., Amato, D., Bombardelli, C., and Milani, C., “New orbital elements for accurate propagation in the Solar System,
Proceedings of the 6th International Conference on Astrodynamics Tools and Techniques (ICATT), European Space Agency,
Darmstadt, Germany, 2016.
[23]
Musen, P., “On the Long-Period Lunisolar Effect in the Motion of the Artificial Satellite,” Journal of Geophysical Research,
Vol. 66, 1961, pp. 1659–1665. doi:10.1029/JZ066i006p01659.
[24]
Amato, D., “Advanced orbit propagation methods applied to asteroids and space debris,” Ph.d. thesis, School of Aerospace
Engineering, Technical University of Madrid, Madrid, Spain, July 2017.
[25]
Ferrando, P., Arnaud, M., Briel, U., Cavazutti, E., Clédassou, R., Counil, J. L., Fiore, F., Giommi, P., Goldwurm, A., Lamarle,
O., Laurent, P., Lebrun, F., Malaguti, G., Mereghetti, S., Micela, G., Pareschi, G., Piermaria, M., Roques, J. P., and Tagliaferri,
G., “The Simbol-X mission,” Memorie della Società Astronomica Italiana, Vol. 79, 2008, pp. 19–25.
[26]
Hindmarsh, A. C., “ODEPACK, a systematized collection of ODE solvers,” Scientific Computing, IMACS Transactions
on Scientific Computation, Vol. 1, edited by R. S. Stepleman, M. Carver, R. Peskin, W. F. Ames, and R. Vichnevetsky,
North-Holland, 1983, pp. 55–64.
[27]
Hiebert, K. L., and Shampine, L. F., “Implicitly defined output points for solutions of ODEs,” Tech. Rep. SAND80-0180,
Sandia Laboratories, Albuquerque, NM, USA, 1980.
17
[28] Meeus, J., Astronomical Algorithms, Willmann-Bell, Inc., Richmond, VA, USA, 1998.
[29]
Acton, C. H., “Ancillary data services of NASAs Navigation and Ancillary Information Facility,” Planetary and Space Science,
Vol. 44, 1996, pp. 65–70. doi:10.1016/0032-0633(95)00107- 7.
[30]
CNES, “
STELA
3.1.1 User Manual,” Tech. rep., CNES, Dec 2016. URL
https://logiciels.cnes.fr/sites/default/
files/Stela-User- Manual_4.pdf.
[31]
Bull, J. M., Smith, L. A., Pottage, L., and Freeman, R., “Benchmarking Java against C and Fortran for scientific applications,
Proceedings of the 2001 joint ACM-ISCOPE conference on Java Grande, Association for Computing Machinery, Palo Alto,
CA, USA, 2001, pp. 97–105.
[32]
Eichhorn, H., Reiner, A., Cano, J. L., and F., M., “A Comparative Study of Programming Languages for Next-Generation
Astrodynamics Systems,Proceedings of the 6
th
International Conference on Astrodynamics Tools and Techniques (ICATT),
European Space Agency, Darmstadt, Germany, 2016.
[33]
Gkolias, I., Daquin, J., Gachet, F., and Rosengren, A. J., “From Order to Chaos in Earth Satellite Orbits,” The Astronomical
Journal, Vol. 152, 2016, 119. doi:10.3847/0004-6256/152/5/119.
[34]
Davidson, M. C., “Numerical examples of transition orbits in the restricted three body problem,” Astronautica Acta, Vol. 10,
1964, p. 308.
[35] Szebehely, V., Theory of orbits. The restricted problem of three bodies, Academic Press, New York, NY, USA, 1967.
[36]
Archinal, B. A., A’Hearn, M. F., Bowell, E., Conrad, A., Consolmagno, G. J., Courtin, R., Fukushima, T., Hestroffer, D., Hilton,
J. L., Krasinsky, G. A., Neumann, G., Oberst, J., Seidelmann, P. K., Stooke, P., Tholen, D. J., Thomas, P. C., and Williams, I. P.,
“Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2009,” Celestial Mechanics and
Dynamical Astronomy, Vol. 109, 2011, pp. 101–135. doi:10.1007/s10569- 010-9320- 4.
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... A sophisticated numerical solver, named LSODAR (Livermore Solver for Ordinary Differential equations with Automatic Root-finding), has been included to integrate the differential equations of motion. Some results using a preliminary version of THALASSA for cis-and translunar orbits have been shown in Amato et al. (2018). ...
... We consider the orbit of the proposed Simbol-X mission as a test case representative of a high-altitude HEO. The initial conditions in Table 6 were used to benchmark the performance of the semi-analytical propagator by Lara et al. (2017), and for the study performed on THALASSA in Amato et al. (2018). The large values of eccentricity and semimajor axis make this orbit a challenging test case for both semi-analytical and numerical methods. ...
... This leads to an error in the lifetime estimation of 92 years, since the condition on the instantaneous radius of perigee does not correspond to a physical reentry. For l M < 5, discrepancies with respect to the reference solutions are even more relevant; however, we omit the corresponding trajectories here as the interested reader can find these results in Amato et al. (2018). ...
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... This is a very demanding process in terms of CPU time (the computation of MiSO initial conditions for an individual constellation plane can take a few hours with an Intel Core processor i7-4790 @ 3.6 GHz), where the use of a very efficient orbit propagator is paramount. All numerical propagations were performed using the THALASSA orbit propagator [16,17]. All MiSO orbits initial conditions derived in this work are reported in Appendix B for reproducibility purposes. ...
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