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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, 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=−˜ε−1

2(4)

for the EDromo formulation, where

˜ε=ε+V

is the total energy, i.e., the sum of the Keplerian orbital energy

ε=v2/2−µ/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 °

Ω116.640 939 804 248 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 ≥Ntess∆t,(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

a≥20 000 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

a−ai(km)

0

1

2

e×10−4

57

59

61

i(deg)

90

180

270

Ω(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

10−4

to

10−15

, 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

Ω

, 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

log10δΩ(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 Ntess∆t=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 10−4

to 10−15.

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 °

Ω49.351 °

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

r−RE<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

Ntess∆t=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

10−2

and

10−1

, 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

a−ai(km)

0.05

0.5

0.95

e

0

45

90

i(deg)

90

180

270

Ω(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

log10δΩ(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 Ntess∆t=40 d. The

results with THALASSA are obtained by decreasing the solver tolerance from 10−4to 10−15.

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 ×105km3s−2and µE/µM=1/81.45.

Particle Moon

x4.467 794 60 ×1053.844 000 0 ×105km

y0 0 km

Û

x0 0 km s−1

Û

y1.199 786 3 1.024 612 7 km s−1

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

1×10−12

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.

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