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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.
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
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,
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.
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
Classically, long-term orbit propagation has relied heavily on semi-analytical techniques employing the method of
In this approach, the equations giving the rates of change of osculating orbital elements are averaged
either analytically or numerically.
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,
they provide regularized sets of coordinates and orbital elements which have
consistently displayed excellent numerical performances. 16
, last visited: October 18
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 [
], and by Burdet
and Sperling.
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.
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
The numerical code that we developed for this study,
, includes the Cowell, Kustaanheimo-Stiefel (KS),
and EDromo
formulations. The equations of motion are integrated by the adaptive multi-step numerical solver
, 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
, gravitational perturbations from the Sun
and the Moon, air drag from several atmospheric models, and solar radiation pressure according to the cannonball
efficiently handles strong lunar gravitational perturbations through a trajectory splitting algorithm,
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
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,
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
with respect to that of the “Semi-Analytic Tool for End of Life Analysis” (
), 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
The MEO and HEO test cases are presented in section III, where the performance of
is compared with that
. 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
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,
where Fincludes both potential and non-conservative perturbations:
URL:, last visited: November 28th, 2017.
URL:, last visited: October 31st, 2017.
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
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
). 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:
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
correspond to steps in physical time
which are the smaller the
closer to collision (
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
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
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.
They use the fictitious time
defined in Equation 3 with
1and either
1(for the
KS formulation) or:
for the EDromo formulation, where
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
and its
. Their evolution is described by a second-order linear equation for
, which has to be integrated along with the rate
of change of the total energy
. 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
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
. 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
and of frequency
. The first is the
initial distance from the primary body, whereas the latter is n0=qµ/l3
Exhaustive descriptions of the KS and EDromo formulations and of their derivation can be found in References [
and [20].
B. Numerical solver
uses the Livermore Solver for Ordinary Differential equations with Automatic Root-finding (
an implicit multi-step solver with variable step-size and order.
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
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
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
is then computed by interpolating the numerical
implements a root-finding capability,
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
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,
and in a selenocentric system
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
(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
was originally developed for a comparative study with respect to the
orbit propagator, its
physical model closely resembles that of the latter. In particular,
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,
or from JPL Developmental Ephemerides (DE) which are read through
the SPICE toolkit.
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
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
on the public repository
, 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.
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
i56 °
116.640 939 804 248 0 °
III. Performance for high-altitude orbits
The computational time and the accuracy obtained from propagations with
are compared to those obtained
integrates singly-averaged equations of motion for a set of mean equinoctial elements
physical time using a fixed-step Runge-Kutta scheme of 6
order, and it takes into account a wide range of perturbations.
Its physical model was partly re-implemented in
(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
by adding or
subtracting short-periodic terms, which are computed taking into account only the
and lunisolar perturbations; the
initial conditions in the mean orbital elements are also computed in this way.
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
of the numerical scheme.
Resonant tesseral harmonics in the expansion of the geopotential are retained and averaged if
τtess Ntesst,(7)
is the period of each of the harmonics, and
can be chosen by the user. Also the order
of the expansions
of the lunar and solar perturbing potentials can be selected from values between
. We disabled precession and
nutation effects for these tests.
We remark that
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.
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
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
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
, 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
propagation are shown in
0 50 100 150 200
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=
,Ntess =
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.
Figure 1 for 200 years of propagation. The latter is obtained by setting
Ntess =
4, and
t=120 h
. A value of
Ntess =
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
propagation overlap. This
suggests that the recovery of the short-periodic terms due to
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
can be built by performing batch
propagations of the initial conditions in Table 1. We repeat the 200-year propagation with
for solver
tolerances decreasing from
, and with
decreasing from
12 d
0.5 d
. In the latter code, we set
,Ntess =
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
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
. 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
is the most efficient for moderate levels of accuracy. In fact,
shows a minimum error in osculating semi-major axis on the order of
100 m
, with computational times between
2 s
15 s
. This error reduces to
1 cm
when using the EDromo formulation, albeit with a computational time that is
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
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
, 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,
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
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
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
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
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
, whereas
is limited in this respect due to the
reasons describe above.
-6 -4 -2 0 2 4
CPU time (s)
-8 -6 -4 -2 0
-11 -9 -7 -5 -3 -1
CPU time (s)
-7 -5 -3 -1 1 3
-5 -3 -1 1 3
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=
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.
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 °
B. Simbol-X
We consider the orbit of Simbol-X, an X-ray space telescope that was proposed by ASI and CNES.
This test case
was also used as a challenging problem for the semi-analytical propagator developed in Reference [
], 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
reference propagation and for two
propagations with
are displayed in Figure 3. The latter are performed by setting the order
of the expansion of
the lunar and solar perturbing potentials to either
. 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
reference propagations, and the remaining orbital elements start diverging already after 14 years. Moreover, the
trajectories computed for different values of
display different qualitative behaviors, showing that
has a significant
impact on the propagation. Additional tests, which we do not present here, have also evidenced that smaller values of
lead to trajectories shifting even further from the reference.
In fact, in the averaging approach used by
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
along its orbit during one orbital period of the spacecraft. Also,
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) ≈
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
time step is varied between
40 d
1 d
; we use larger time steps as to
accommodate the larger orbital period. We consider two propagation batches for
corresponding to
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
, 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
time step does not significantly improve the accuracy of the propagation. Even
considering a larger
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
commits an error in eccentricity between
, which corresponds to an error in
. As stated in the previous section, such a
significant uncertainty in
implies that the potential interaction of the spacecraft with the atmosphere cannot be reliably
Large errors in mean anomaly for
and Cowell imply that the position of the spacecraft is already lost after 75
0 10 20 30 40 50 60 70 80
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=
(mean)”, “L=
(osc)” and “L=
(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 =
. The re-entry dates of the STELA propagations differ with respect to the reference
by up to two years.
-4 -2 0 2 4
CPU time (s)
-6 -4 -2 0
-7 -5 -3 -1
CPU time (s)
-5 -3 -1 1 3
-5 -3 -1 1
CPU time (s)
-2 0 2
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=
and L=
, 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.
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
x0 0 km s1
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
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
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
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,
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
. 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.
Moreover, we performed
numerical tests evidencing that relative changes on the order of
in the initial conditions imply a loss of the
periodicity for
. This intrinsic instability makes propagating this orbit quite a challenging test for numerical
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
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
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
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
-axis directed towards the Moon, and its
-axis directed
along the Moon’s velocity. It rotates with respect to the geocentric coordinate system with angular velocity nM.
-1 -0.5 0 0.5 1
(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)=(
),(ξ0, η0)=(
The trajectory is plotted in black where the distance from the Moon is greater than Rsw =
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
, 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)
RHill =61 740 km
is the Hill radius of the Moon. This value of
in terms of Hill radii has been found to give
optimal performances in past studies using the trajectory splitting algorithm.
In Figure 5, parts of the trajectory in
which Equation 6 is satisfied are displayed in black.
The average computational time on
runs, number of right-hand side evaluations, and integration steps taken as a
function of the position error with respect to the reference trajectory
are displayed in Figure 6. The error is measured
at the end of the 1
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
(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
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
when the trajectory is not split. Moreover, the numerical values of
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
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.
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
against the semi-analytical orbit code
in the propagation of MEO and HEO orbits.
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
15 s
achieves the same accuracy with a computational time
times higher. In contrast to
, 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
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
. The position along the orbit after 75
years of propagation is only recovered by
using regularized formulations, whereas it is lost with the Cowell
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
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
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
. 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;
similarly, precession
and nutation of the geocentric coordinate system will be implemented for consistency with the ICRF.
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.
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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 paper is concerned with the comparison of semi-analytical and non-averaged propagation methods for Earth satellite orbits. We analyse the total integration error for semi-analytical methods and propose a novel decomposition into dynamical, model truncation, short-periodic, and numerical error components. The first three are attributable to distinct approximations required by the method of averaging, which fundamentally limit the attainable accuracy. In contrast, numerical error, the only component present in non-averaged methods, can be significantly mitigated by employing adaptive numerical algorithms and regularized formulations of the equations of motion. We present a collection of non-averaged methods based on the integration of existing regularized formulations of the equations of motion through an adaptive solver. We implemented the collection in the orbit propagation code THALASSA, which we make publicly available, and we compared the non-averaged methods to the semi-analytical method implemented in the orbit propagation tool STELA through numerical tests involving long-term propagations (on the order of decades) of LEO, GTO, and high-altitude HEO orbits. For the test cases considered, regularized non-averaged methods were found to be up to two times slower than semi-analytical for the LEO orbit, to have comparable speed for the GTO, and to be ten times as fast for the HEO (for the same accuracy). We show for the first time that efficient implementations of non-averaged regularized formulations of the equations of motion, and especially of non-singular element methods, are attractive candidates for the long-term study of high-altitude and highly elliptical Earth satellite orbits.
... 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. ...
With the upcoming launch of large constellations of satellites in the low-Earth orbit (LEO) region, it will become important to organize the physical space occupied by the different operating satellites in order to minimize critical conjunctions and avoid collisions. This paper introduces the definition of space occupancy as the domain occupied by an individual satellite as it moves along its nominal orbit under the effects of environmental perturbations throughout a given interval of time. After showing that space occupancy for the zonal problem is intimately linked to the concept of frozen orbits and proper eccentricity, frozen orbit initial conditions are provided in osculating element space and a frozen-orbit polar equation is obtained to describe the space occupancy region in closed analytical form. Next, the problem of minimizing space occupancy is analyzed in a realistic model including tesseral harmonics, third-body perturbations, solar radiation pressure, and drag. The corresponding initial conditions, leading to minimum space occupancy (MiSO) orbits, are obtained numerically for a set of representative configurations in LEO. The implications for the use of MiSO orbits to optimize the design of mega-constellations are discussed.
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STORM is a general semi-analytical propagation code developed at Airbus Defence and Space and originally designed to perform long-term Mars orbit simulations, for mission analysis purposes. Since the 1960’s, humans have sent 18 spacecraft to study the Red Planet, 8 of which are active as of February 2021. While space-debris surveillance and mitigation have been gathering momentum in the past decades in the case of Earth, the issue of artificial objects left in orbit around Mars is another cause for concern in view of the Committee of Space Research’s Planetary Protection Policy. In that perspective, STORM is introduced as a potential tool for assessing the compliance of spacecraft with orbital lifetime requirements. The programme makes it possible to study the evolution of orbits over several decades by propagating a set of mean equinoctial orbital elements, with a comprehensive set of gravitational and non-gravitational perturbations. Particular emphasis is laid on atmospheric entry cases as the programme is interfaced with the European Mars Climate Database, which is the current best knowledge of the state of the Martian atmosphere. This work is focused on the semi-analytical propagator’s implementation and its actual use for predicting entry dates. A sensitivity analysis is conducted to ascertain the influence of atmosphere modelling and initial conditions, providing insight into how to account for uncertainties in a probabilistic way.
Full thesis available at The instabilities associated with the solution of the two-body problem are eliminated by time and space transformation, using fictitious time and fundamental features of Hopf fibration, famously known as Kustaanheimo-Stiefel (KS) transformation. High degree of precision can be achieved in computation of state vector with the usage of orbital frequency based on total energy. The differential equations are regular everywhere thereby avoiding the collision singularity unlike the non-linear Kepler problem and smoothed for eccentric orbits as generalized eccentric anomaly is used as the independent variable. KS regular elements equations are a very powerful method for obtaining analytical and numerical solutions in orbital motion with complex perturbations model. A modified analytical solution for the motion of spacecraft with the effect of J2, J3 and J4 in terms of KS regular and Keplerian elements is developed. Extensive comparative studies with previously developed analytical solutions alongside numerical integration revealed that the new hybrid theory can achieve good level of accuracy. The relative errors are of the order of 10^-7 in position and velocity. A new singularity-free analytical solution in terms of KS regular elements with the effect of luni-solar perturbations is developed. Extensive numerical and observed value comparison tests revealed high accuracy in position and velocity computations. The simulated relative errors are of the order ~10^-6. Seasonal variations is not only observed in the physical orbital parameters but also in the analytical solution accuracy which is revealed through the constants present in the mathematical model. Few applications using medium-fidelity numerical scheme using KS regular elements (KSROP) are studied. The characterization of the perigee oscillation and its advantage in improving the mission performance by the effect of solar gravity effect is studied. The Sun azimuth angle plays a dominant role in the dynamics of the perigee oscillation and thereby leading to typify the launch vehicle performance. Preliminary simulations for trans-lunar injection revealed considerable mass savings can be done by using natural perturbations as orbit-phasing propulsion. The usage of response surface methodology with genetic algorithm utilizing KSROP is a good numerical scheme for debris re-entry prediction from highly elliptical orbits. The average absolute relative percentage error yielded for the tested cases is found to be less than 5% when compared with real debris re-entry time.
We aim to provide satellite operators and researchers with an efficient means for evaluating and mitigating collision risk during the design process of mega-constellations. We first introduce a novel algorithm for conjunction prediction that relies on large-scale numerical simulations and uses a sequence of filters to greatly reduce its computational expense. We then use this brute-force algorithm to establish baselines of endogenous (intra-constellation), or self-induced, conjunction events for the FCC-reported designs of the OneWeb LEO and SpaceX Starlink mega-constellations. We demonstrate how these deterministic results can be used to validate more computationally efficient, stochastic techniques for close-encounter prediction by adopting a new probabilistic approach from Solar-System dynamics as a simple test case. Finally, we demonstrate how our methodology can be applied during the design phase of large constellations by investigating Minimum Space Occupancy (MiSO) orbits, a generalization of classical frozen orbits that holistically account for the perturbed-Keplerian dynamics of the Earth-satellite-Moon-Sun system. The results indicate that the adoption of MiSO orbital configurations of the proposed mega-constellations can significantly reduce the risk of endogenous collisions with nearly indistinguishable adjustments to the nominal orbital elements of the constellation satellites.
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Asteroids and space debris pose relevant menaces to civilization, both on ground and in space. Simultaneously, they present a number of common engineering and scientific challenges that must be tackled in the realm of Space Situational Awareness (SSA). As to improve current and future SSA technologies, robust and efficient orbit propagation methods are required. The main goal of the present thesis is to demonstrate that regularized formulations of dynamics entail significant advantages in the most demanding orbit propagation problems for asteroids and space debris. Regularized formulations are obtained by eliminating the 1/r^2 singularity in Newtonian equations of motion through an analytical procedure. The resulting regularized equations exhibit an excellent numerical performance. In this thesis, we consider the Kustaanheimo-Stiefel formulation and several methods of the \emph{Dromo} family, which represent the trajectory through a set of non-classical orbital elements. In the first part, we focus on the orbit propagation of planetary close encounters, taking into account several test cases. As scenarios of relevant practical importance, we propagate resonant returns of several fictitious asteroids and measure the error in the b-plane coordinates. To generalize the results, we carry out large-scale simulations in the Circular, Restricted Three-Body Problem by means of a bi-dimensional parametrization. We analyse the case of the asteroid (99942) Apophis, devoting particular attention to the amplification of the numerical error consequent to its deep close encounter in 2029. The second part is dedicated to the long-term prediction of Earth satellite orbits. We compare regularized formulations to a semi-analytical method based on equinoctial elements for several orbital regimes and perturbations. The parameters affecting the semi-analytical propagation efficiency are fine-tuned by analysing the different contributions to the integration error, which also gives insight on the limits of applicability of semi-analytical methods. Regularized formulations compare very favourably for highly elliptical and super-synchronous orbits, which encourages their application to lifetime analyses and numerical explorations of the cislunar space. Applications to asteroid impact avoidance are presented in the third part. We show the results of a geographical deflection of the fictitious asteroid 2015PDC obtained with an Ion Beam Shepherd spacecraft. Finally, we perform a systematic study of the potential resonant returns of the fictitious asteroid 2017PDC after its deflection by a nuclear device.
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The algorithms used in the construction of a semi-analytical propagator for the long-term propagation of highly elliptical orbits (HEO) are described. The software propagates mean elements and include the main gravitational and non-gravitational effects that may affect common HEO orbits, as, for instance, geostationary transfer orbits or Molniya orbits. Comparisons with numerical integration show that it provides good results even in extreme orbital configurations, as the case of SymbolX.
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We consider Earth satellite orbits in the range of semi-major axes where the perturbing effects of Earth's oblateness and lunisolar gravity are of comparable order. This range covers the medium-Earth orbits (MEO) of the Global Navigation Satellite Systems and the geosynchronous orbits (GEO) of the communication satellites. We recall a secular and quadrupolar model, based on the Milankovitch vector formulation of perturbation theory, which governs the long-term orbital evolution subject to the predominant gravitational interactions. We study the global dynamics of this two-and-a-half degrees of freedom Hamiltonian system by means of the fast Lyapunov indicator (FLI), used in a statistical sense. Specifically, we characterize the degree of chaoticity of the action space using angles-averaged normalized FLI maps, thereby overcoming the angle dependencies of the conventional stability maps. Emphasis is placed upon the phase-space structures near secular resonances which are of first importance to the space debris community. We confirm and quantify the transition from order to chaos in MEO, stemming from the critical inclinations, and find that highly inclined GEO orbits are particularly unstable. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors, and, from a mathematical perspective, have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.
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The goal of the Draper Semi-analytical Satellite Theory (DSST) Standalone Orbit Propagator is to provide the same algorithms as in the GTDS orbit determination system implementation of the DSST, without GTDS's overhead. However, this goal has not been achieved. The 1984 DSST Standalone included complete models for the mean element motion but truncated models for the short-periodic motion. The 1997 update included the short-periodic terms due to tesseral linear combinations and lunar-solar point masses, 50 x 50 geopotential, and J2000 coordinates. However, the 1997 version did not demonstrate the expected improved accuracy. Three projects undertaken by the authors since 2010 have led to the discovery of additional bugs in the DSST Standalone which are now resolved.
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Seven spatial elements and a time element are proposed as the state variables of a new special perturbation method for the two-body problem. The new elements hold for zero eccentricity and inclination and for negative values of the total energy. They are developed by combining a spatial transformation into projective coordinates (as in the Burdet–Ferrandiz regularization) with a time transformation in which the exponent of the orbital radius is equal to one instead of two (as commonly done in the literature). By following this approach we discover a new linearisation of the two-body problem, from which the orbital elements can be generated by the variation of parameters method. The geometrical significance of the spatial quantities is revealed by a new intermediate frame which differs from a local vertical local horizontal frame by one rotation in the instantaneous orbital plane. Four elements parametrize the attitude in space of this frame, which in turn defines the orientation of the orbital plane and fixes the departure direction for the longitude of the propagated body. The remaining three elements determine the motion along the radial unit vector and the orbital longitude. The performance of the method, tested using a series of benchmark orbit propagation scenarios, is extremely good when compared to several regularized formulations, some of which have been modified and improved here for the first time.
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The geometry of hyperbolic orbits suggests that Minkowskian geometry, and not Euclidean, may provide the most adequate description of the motion. This idea is explored in order to derive a new regularized formulation for propagating arbitrarily perturbed hyperbolic orbits. The mathematical foundations underlying Minkowski space-time \(\mathbb {M}\) are exploited to describe hyperbolic orbits. Hypercomplex numbers are introduced to define the rotations, vectors, and metrics in the problem: the evolution of the eccentricity vector is described on the Minkowski plane \(\mathbb {R}_1^2 \) in terms of hyperbolic numbers, and the orbital plane is described on the inertial reference using quaternions. A set of eight orbital elements is introduced, namely a time-element, the components of the eccentricity vector in \(\mathbb {R}_1^2 \), the semimajor axis, and the components of the quaternion defining the orbital plane. The resulting formulation provides a deep insight into the geometry of hyperbolic orbits. The performance of the formulation in long-term propagations is studied. The orbits of four hyperbolic comets are integrated and the accuracy of the solution is compared to other regularized formulations. The resulting formulation improves the stability of the integration process and it is not affected by the perihelion passage. It provides a level of accuracy that may not be reached by the compared formulations, at the cost of increasing the computational time.
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The long-term evolution characteristics and stability of an orbit are well characterized using a mean element propagation of the perturbed two body variational equations of motion. The averaging process eliminates short period terms leaving only secular and long period effects. In this study, a non-traditional approach is taken that averages the variational equations using adaptive numerical techniques and then numerically integrating the resulting equations of motion. Doing this avoids the Fourier series expansions and truncations required by the traditional analytic methods. The resultant numerical techniques can be easily adapted to propagations at most solar system bodies.
This paper is concerned with orbits in the restricted three body problem which describe essentially elliptic motion with respect to one of the heavy masses, then at some later time transfer to essentially elliptic motion with respect to the other heavy mass. All results presented are numerical, meaning the orbits have been found by searching for appropriate initial values. Even though the orbits selected for presentation are restricted to special initial values, the results indicate the large variety possible in the general case. New periodic orbits were found which have the general feature described above and others which are very near the terminal orbits of such families. The possible uses of such orbits for space flight are large in number, including lunar and inter-planetary missions.