Dynamics of a Geostationary Satellite Clément Gazzino

Home , Geostationary orbit, Geosynchronous orbit, LAGEOS, True longitude

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Clément Gazzino. Dynamics of a Geostationary Satellite. [Research Report] Rapport LAAS n° 17432, LAAS-CNRS. 2017. hal-01644934v2

HAL Id: hal-01644934 https://hal.archives-ouvertes.fr/hal-01644934v2 Submitted on 4 Dec 2017

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Dynamics of a Geostationary Satellite

Technical Report

Clément Gazzino November 22, 2017

CONTENTS 1 Introduction4

2 Orbital Mechanics4 2.1 Keplerian Motion...... 4 2.1.1 Dynamics Equations for the Keplerian Motion...... 4 2.1.2 Keplerian Trajectory...... 5 2.1.3 The Geostationary Orbit...... 6 2.2 Orbital Perturbations...... 6 2.2.1 Gravitational Attraction of a non Spherical Earth...... 6 2.2.2 Sun and Moon Disturbing Gravitational Potential...... 8 2.2.3 Sun Radiation Pressure (SRP)...... 9 2.2.4 Summary...... 9

3 State Representation for the Spacecraft Motion 10 3.1 Cartesian State Representation...... 10 3.2 Orbital Elements State Representation...... 11 3.2.1 Classical Orbital Elements...... 11 3.2.2 Éléments orbitaux équinoxiaux...... 14

4 Free Evolution Equations 15 4.1 Keplerian Motion...... 16 4.2 Non Keplerian Motion...... 17 4.2.1 Osculating Orbits...... 17 4.2.2 Lagrange Equations...... 18 4.3 Linearized Evolution Equation...... 20 4.3.1 Linearization Point...... 20 4.3.2 Derivative of the State Vector at Order 1...... 20

5 Computation of the Matrices for the Linearized Dynamics 21 5.1 Keplerian Part...... 21 5.2 Non Keplerian Earth Gravitational Part...... 22 5.2.1 Introduction...... 22 5.2.2 Coeﬃcient C20 ...... 24 5.2.3 Coeﬃcient C21 ...... 25

1 5.2.4 Coefficient S21 ...... 26 5.2.5 Coefficient C22 ...... 27 5.2.6 Coefficient S22 ...... 28 5.2.7 Coefficient C30 ...... 30 5.2.8 Coefficient C31 ...... 31 5.2.9 Coefficient S31 ...... 32 5.2.10 Coefficient C32 ...... 33 5.2.11 Coefficient S32 ...... 34 5.2.12 Coefficient C33 ...... 35 5.2.13 Coefficient S33 ...... 37 5.3 Sun and Moon Gravitational Attractions...... 38 5.3.1 Sun and Moon Positions...... 38 5.3.2 Relative Dynamics Matrices...... 39 5.4 Solar Radiation Pressure...... 43

Appendix 47

A Reference Frames 47 A.1 Geocentric Inertial Reference Frame...... 47 A.2 Rotating Geocentric Reference Frame...... 48 A.3 Local Orbital Frame...... 50 A.4 Equinoctial Reference Frame...... 51

B Conversion Formulas with the Classical Orbital Elements 55 B.1 Keplerian Motion Integrals...... 55 B.2 Anomalies Transformations...... 56 B.3 Computation of the Orbital Elements from the Cartesian Positions and Ve- locities...... 57 B.4 Computation of the Cartesian Positions and Velocities from the Classical Orbital Elements...... 59 B.4.1 Computation of the Position...... 59 B.4.2 Computation of the Velocity...... 59

C Conversion Formulas with the Equinoctial Orbital Elements 60 C.1 Deﬁnition of the Equinoctial Orbital Elements from the Classical Ones.. 60 C.2 Computation of the Classical Orbital Elements from the Equinoctial Ones 61 C.3 Conversion from the Cartesian Position and Velocity to the Equinoctial Orbital Elements...... 64 C.4 Conversion from the Equinoctial Orbital Elements to Cartesian Position and Velocity...... 67

D Approximation Methods for Solving the Kepler Equation 69 D.1 Newton Algorithm...... 70 D.1.1 Kepler Equation Expressed in E ...... 70 D.1.2 Kepler Equation Expressed in ν ...... 71 D.2 Hull’s Method...... 71 D.2.1 With the Equations (223) and (224)...... 71 D.2.2 With the Equation (225)...... 72 D.2.3 Kepler Equation with the Equinoctial Orbital Elements...... 72

2 E Physical Parameters 74

3 1 INTRODUCTION The study of the dynamics of a body around the Earth began with the study of the gravitation equation attributed to Newton and the three Kepler laws for the motion around a spherical body. This motion is called keplerian motion. For space applications as the station keeping of a geostationary satellite, this keplerian motion is not enough to describe the spacecraft trajectory. Orbital disturbances have to be added in the description of the motion, namely the fact that Earth is not a spherical body, that the Sun and the Moon create a gravitational attraction on the spacecraft and that the Sun radiation pressure change the spacecraft trajectory. From the study of the keplerian dynamics and the orbital perturbations, a dynamical model of the spacecraft is set up. The state vector can consist either in the cartesian positions and velocities or the orbital elements. As for station keeping, the satellite stay in the vicinity of its station keeping point, it is possible to linearize the dynamics with respect to the gap between the actual position of the spacecraft and the station keeping one. This document is organized as follows. The ﬁrst section describes the physics of the keplerian motion and the orbital perturbations disturbing this keplerian motion. The second section focuses on the state representation of the spacecraft ﬂying around the Earth on the quasi-geostationary orbit. The following section derives the non linear and the linearized equation of motion and the last section computed the linearized dynamics matrices for each orbital perturbation.

2 ORBITAL MECHANICS

2.1 KEPLERIAN MOTION In this section, the equations for the keplerian motion are recalled (see for instance the references [Battin, 1999], [Sidi, 1997] ou [Vallado, 1997]) and some notations used through this document are presented.

2.1.1 DYNAMICS EQUATIONS FOR THE KEPLERIAN MOTION As a first approximation, the Earth can be considered as a spherical body. A spacecraft flying in the gravitational field of the Earth undergoes a force whose expression is given by the Newton gravitational law:

m msat ~r F~g = G ⊕ , (1) − r2 r with: • the gravitational constant, G • m the Earth mass, ⊕ • msat the satellite mass, • ~r the Earth-satellite radius vector, • r the norm of this vector.

4 The standard geocentric gravitational parameter is deﬁned by µ = m . ⊕ G ⊕ According to the Newton second law, the variation of the spacecraft momentum is equal to the sum of the external forces. As the spacecraft mass is supposed to remain constant, the equations of motion in the inertial geocentric frame supposed to be galilean (see the Figure8 page 47) are given by:

d2~r µ ~r

= ⊕ (2) dt2 − r2 r G B This diﬀerential equation describes the motion of the spacecraft and of the Earth around the center of mass of the system {Earth-spacecraft}. As the spacecraft mass is very small with respect to the Earth mass, the Earth-spacecraft center of mass center of mass is supposed to be the center of the Earth. Therefore, the Equation (2) describes the motion of the spacecraft around the Earth.

2.1.2 KEPLERIAN TRAJECTORY In order to solve the Equation (2), integrals of the motion are first derived. The specific angular momentum is defined by: ~h = ~r ~v, (3) × and is constant. As the angular momentum is perpendicular to the position and velocity vectors, the plane defined by the initial position and the initial velocity is constant. Therefore, the spacecraft trajectory lies in a plane. The solution of the Equation (2) is a conic. With a polar parametrisation, the equation of this conic is given by: a(1 e2) r = − , (4) 1 + e cos(ν) where a is the semi-major axis of the conic, e its eccentricity and ν the angular parameter. a and e describe the shape of this conic. In particular, e defines the conic type: • if 0 6 e < 1, the trajectory is an ellipse (or a circle in the case e = 0), • if e = 1, the conic is a parabola, • if e > 1, the conic is an hyperbola. The trajectory is bounded and the motion is periodic in the first case only. Therefore, in the sequel, only trajectories with eccentricities strictly smaller than 1 will be studied. The semi-minor axis b is also used and is defined by: b = a√1 e2. (5) − In the case e < 1, the trajectory is closed and the orbital period is defined as: v u u a3 T = 2πt . (6) µ ⊕ The mean motion is defined as: 2π rµ n = = ⊕ . (7) T a3 Conversion formulas between the position and the velocities on one hand and the parameters defining the spacecraft trajectory on the other hand are given in the AppendixB page 55.

5 2.1.3 THE GEOSTATIONARY ORBIT For spacecraft orbiting the Earth, the orbits are sorted in the following categories : • Low Earth Orbits (LEO), for an altitude smaller than 800 km (a < 7178 km), • Mid Earth Orbits (MEO), for an altitude between 800 km and 30 000 km (7178 km < a < 36378 km), • Geosynchronous Orbits, for which the orbital period of the spacecraft is equal to the rotation period of the Earth, • Geostationary Earth Orbits (GEO) for which the spacecraft stays over the same point of the Earth. The GEO orbit is a particular geosynchronous orbit with a zero inclination and a zero eccentricity. The GEO semi-major axis can be computed thanks to the Equation (6):

! 1 µ T 2 3 a = ⊕ ⊕ . (8) 4π2 With the sidereal Earth rotation period T = 86164 s and the geocentric standard ⊕ gravitational parameter µ = 3.986.1014 m3/s2, it follows a = 42158 km. ⊕

2.2 ORBITAL PERTURBATIONS The orbitals perturbations are all the forces, except the Earth gravitational one, that a spacecraft undergoes. Among this forces, the references [Shrivastava, 1978], [Valorge, 1995] and [Vallado, 1997] introduce the following ones relevant for a spacecraft on a GEO orbit: • the part of the Earth gravitational potential produced by the non spherical distribu- tion of the mass, • the Sun and and the Moon gravitational attractions, • the solar radiation pressure: photons emitted by the Sun are absorbed and re-emitted by the spacecraft, what induces a variation of its momentum. The following subsections describe the modeling of the disturbing accelerations and potentials for a spacecraft orbiting the Earth on a GEO orbit.

2.2.1 GRAVITATIONAL ATTRACTIONOFANON SPHERICAL EARTH If the Earth is supposed to be a perfect sphere, the gravitational potential is expressed as: µ (r, ϕ, λ) = ⊕ . (9) E − r However, the Earth is not a spherical body. Therefore, adopting the expression given in the reference [Vallado, 1997], the gravitational potential is given by:

Z dm(rP , ϕP , λP ) Z dm(rP , ϕP , λP ) p,pot(r, ϕ, λ) = = , (10) r 2 E G Ω(Terre) ~r ~rP G Ω(Terre) rP rP k − k r 1 cos Λ + 2 − r r

with ~r the vector from the Earth center to the potential computation point, ~rP the vector from the Earth center to a point of the Earth whose mass is dm and Λ the angle between these two vectors (see Figure1).

6 ~uZG

~r ~rP −

~rP Λ ϕP

~uYG λP

~uXG

Figure 1 – Angle Λ between the direction of the point in which the gravitational potential is computed and the direction of the current integration point (scheme from [Vallado, 1997]).

2 rP Assuming that 2 1, the integrand can be expanded in series of Legendre polynomi- r als: l 1 X∞ r r = Pl(cos Λ), (11) r2 rP P l=0 rP 1 cos Λ + 2 − r r where Pl is the Legendre polynomial of degree l given by: l 2 l l j 1 d (x 1) 1 X ( 1) (2l 2j)! l 2j Pl(x) = − = − − x − . (12) 2ll! dxl 2l j!(l j)!(l 2j)! j=0 − − The gravitational potential of a non spherical Earth is thus: l l µ X∞ X r p,pot(r, ϕ, λ) = ⊕ ⊕ Pl,m(sin ϕ) Clm cos(mλ) + Slm sin(mλ) , (13) E r l=0 m=0 r where the Pl,m are the associated Legendre functions: l+m 2 l 1 2 m d (x 1) Pl,m(x) = (1 x ) 2 − , (14) 2ll! − dxl+m and the Clm and Slm are the coeﬃcients of the decomposition of the potential in spherical harmonics. The gravitational potential can be rewritten as the sum of the keplerian gravitational potential and a disturbing gravitational potential (see for instance the references [Soop, 1994, Montenbruck and Gill, 2000, Sidi, 1997, Losa, 2007]). This potential is a function of the radius, the latitude and the longitude expanded on the spherical basis of the associated Legendre functions: µ p, (r, ϕ, λ) = ⊕ + P (r, ϕ, λ) E pot r U l l (15) µ µ X∞ X r = ⊕ + ⊕ ⊕ Pl,m(sin ϕ) Clm cos(mλ) + Slm sin(mλ) . r r l=0 m=0 r

In some textbooks, the Jl coeﬃcients are used instead of the Cl0 ones. These coeﬃcients are linked by Jl = Cl . − 0

7 2.2.2 SUNAND MOON DISTURBING GRAVITATIONAL POTENTIAL The Sun and the Moon create a gravitational force acting on a spacecraft orbiting the Earth. These attraction forces tend to change the spacecraft orbital plane due to the fact that the Sun and the Moon orbital planes are inclined with respect to the Earth equator. Thus these gravitational forces act on the orbital plane of the spacecraft. Denoting: • µ et µ the Sun and Moon gravitational parameters respectively,

• ~r et ~r $the position vectors of the Sun and the Moon respectively in the geocentric

inertial reference frame, with r and r their norms (see Figure2), $ the acceleration acting on the spacecraft due$ to the gravitational attraction of the Sun and the Moon is given by:   µ " ~r ~r ~r # ~r ~r ~r

~ap, = ~r + µ − 3 + µ  − 3  , (16) − r3 ~r ~r 3 − r ~r $ ~r 3 − r$ $ k − k $ k − k and is usually expanded with the two ﬁrst Legendre polynomials:$ $   µ µ " 3 # µ 3

~ap, ~r + 3 2 (~r ~r)~r ~r + 3  2 (~r ~r)~r ~r . (17) ≈ − r3 r r · − r$ r · − $ $ $ $ $

Soleil Lune

~r ~r −

~r ~r ~r − $

~r ~r $

Terre

Figure 2 – Illustration des vecteurs ~r et ~r .

$ 8 The Sun and Moon gravitational attraction potential is derived from the Equation (16) (see the reference [Cot, 1984] and [Campan et al., 1995]):

1 ~r ~r ! 1 ~r ~r ! p, = µ · µ · , E ~r ~r − ~r 3 − ~r ~r − ~r $3 $ k − k k k $ k − k k k  (18) µ r !2 3(~r ~r )2 1 µ$ r $!2 3(~r ~r )2 1 1 + · −  + 1 + · −  . ≈ r r 2 r r 2 $ $ $ $ 2.2.3 SUN RADIATION PRESSURE (SRP) The radiations emitted by the Sun create a pressure on the spacecraft because of the ab- sorption and emission of photons. According to [Soop, 1994] and [Montenbruck and Gill, 2000], 6 2 the mean solar power received per surface unit in the vicinity of the Earth is 1.4 10− kW/m . This solar power creates a mean pressure on a surface in the Earth-Sun direction P = 6 2 4.56 10− N/m . In order to compute the value of the force on the satellite from the pressure, it is mandatory to know the area projected on the spacecraft-Sun direction. Several models exist for the computation of this projected area. The most common one is the cannonball model that model the spacecraft as a sphere (see the reference [Lucchesi, 2001]). If S is the projected area in the Earth-Sun direction, c the radiation pressure coeﬃcient, ~usS the unit vector in the spacecraft-Sun direction and % a parameter being 0 if the satellite is in the shadow of the Earth and 1 otherwise, the acceleration created by the radiation pressure is written as: SP c ~a = % ~usS (19) SRP m Neglecting the eclipses and assuming that the spacecraft-Sun and Earth-Sun directions are almost the same, it is possible to deﬁne the pseudo-potential function for the Sun radiation pressure: SP c p,SRP(r, ϕ, λ) = % ~r ~r , (20) E m k − k

More complex models exist (see [McMahon and Scheeres, 2010] and the references therein) and take into account the attitude of the spacecraft for instance for a more precise computation of the eﬀect of the radiation pressure.

2.2.4 SUMMARY In the reference [Valorge, 1995], the relative effects of the disturbing forces with respect to the central gravitational force are compared for a LEO satellite and a GEO satellite. These results are summarized in the Table1. The J2 effect is in both cases 100 times as strong as the other disturbing effects. For GEO satellite, the atmospheric drag is completely negligible.

9 Disturbing force LEO GEO Central gravitational force 1 1 2 5 J2 eﬀect 10− 10− 4 7 Non spherical potential (except J2) 10− 10− 7 5 7 4 Sun and Moon attraction 10− 10− 10− 10− 13 ∼ 7 13 ∼ 7 Sun radiation pressure 10− 10− 10− 10− 9 ∼ 7 ∼ Atmospheric drag 10− 10− 0 ∼ Table 1 – Comparison between orbital disturbing with respect to the central gravitational force for a LEO and a GEO spacecraft.

3 STATE REPRESENTATION FOR THE SPACECRAFT MOTION Through this document, the system is a GEO spacecraft supposed to be a point mass that evolves in the Earth gravitational ﬁeld with the orbital disturbing forces introduced in the previous section. Six parameters or generalized coordinates are mandatory for the description of the satellite motion. These six parameters form the state vector of the spacecraft. Several state representations can be used. In the sequel, the state vector is written with the cartesian positions and velocities (cartesian parameters), then with the classical or equinoctial orbital elements. The reference [Hintz, 2008] introduce other orbital elements sets that can be used to describe the spacecraft motion.

3.1 CARTESIAN STATE REPRESENTATION The positions and the velocities of the spacecraft in the geocentric inertial reference frame as depicted in the Figure8 page 47 are:

x   ~r = y = x~uXG + y~uYG + z~uZG , (21) z G R   x˙ d~r   ~v = = y˙ =x~u ˙ XG +y~u ˙ YG +z~u ˙ ZG . (22) dt G z˙ R G R The cartesian state vector reads thus as: x(t)   y(t)   z(t) x (t) =   . (23) cart   x˙(t)   y˙(t) z˙(t)

10 3.2 ORBITAL ELEMENTS STATE REPRESENTATION

3.2.1 CLASSICAL ORBITAL ELEMENTS In the case where the spacecraft ﬂies in a central body gravitation ﬁeld, the trajectory is a conic. If the trajectory is closed, it is an ellipse or a circle. In that case, the trajectory can be represented by the following parameters: two parameters for the shape of the ellipse: • the semi-major axis a, the eccentricity e, three parameters for the orientation of the ellipse in space: • the inclination i of the orbit with respect to the ecliptic plane, the right ascension of the ascending node Ω : angle in the equatorial plane between the vernal equinox and the intersection direction between the orbital and the equatorial planes in the ascending direction, the perigee argument ω : angle between the ascending node and the perigee, a parameter called anomaly describing the position of the spacecraft on its orbit • (time-varying parameter). These six parameters are called classical orbital elements or keplerian orbital elements (see for instance the reference [Battin, 1999]). The physical meaning of these orbital elements is depicted on the Figure3. The mean motion of the spacecraft on its orbit n is usually associated to these parameters: rµ n = ⊕ , (24) a3

11 ~uZG

Descending node ν Perigee

Equatorial plane

G ω ~uYG

Ω i

Apogee Apsis line Ascending node

~uXG Nodes line Orbital plane

Figure 3 – Classical orbital elements

Three anomalies are classically handled in order to describe the spacecraft position on its orbit: • the true anomaly ν, • the eccentric anomaly E, • the mean anomaly M. The true anomaly is the angle between the perigee direction and the direction of the radius ~r. The eccentric anomaly is an angle deﬁned as illustrated by the Figure4. The sine, cosine and tangent of the eccentric anomaly are related to the sine, consine and tangent of the true anomaly thanks to the following relations:

sin ν√1 e2 sin E = − , (25a) 1 + e cos ν e + cos ν cos E = , (25b) 1 + e cos ν s E 1 e ν tan = − tan . (25c) 2 1 + e 2

12 b r E ν a ae

Figure 4 – True and eccentric anomalies

The mean anomaly corresponds to the position of a ﬁctitious spacecraft with a constant velocity on a circular orbit with a period equal to the one of the real spacecraft. The mean anomaly satisﬁes thus the following equality:

M(t) = M(t ) + n(t t ). (26) 0 − 0 The eccentric anomaly and the mean one are related through the Kepler equation:

E e sin E = M. (27) − This transcendental equation has no simple solutions. An approximated solution can be computed with an iterative process (Newton method). In the case where the eccentricity is small, the reference [Hull, 2003] gives a method for computing an approximate solution. This method is described in the AppendixD page 69. A state vector is created with the classical orbital elements:  a     e     i  x (t) =   , (28) COE    Ω     ω  ν(t) ou E(t) ou M(t)

13 and in the sequel, we will use the mean anomaly M as the time-varying parameter. The state vector is thus:  a     e     i  x (t) =   . (29) COE    Ω     ω  M(t) Conversion formulas between the cartesian parameters and the classical orbital elements can be found in the AppendixB page 55.

3.2.2 ÉLÉMENTSORBITAUXÉQUINOXIAUX Two singular cases may arise with the classical orbital elements: • when the orbit is circular, the eccentricity is zero and the perigee argument ω is not defined (because the perigee is not defined), • when the orbit lies in the equatorial plane, the inclination is zero and the right ascension of the ascending node is not defined (because the ascending node is not defined). In order to overcome these difficulties, the equinoctial orbital elements are introduced:   a   e e ω  x = cos( + Ω)   ey = e sin(ω + Ω) (30)  ix = tan(i/2) cos(Ω)    iy = tan(i/2) sin(Ω)   equinoctial anomaly

(ex, ey) is called the eccentricity vector and (ix, iy) the inclination vector. In the references [Battin, 1999] and [Cefola, 1972], the equinoctial orbital elements are deﬁned with switched ex and ey as well as switched ix and iy. Moreover, the inclination vector is sometimes deﬁned using alternative forms:   ix = sin(i/2) cos(Ω), ix = sin(i) cos(Ω), or (31) iy = sin(i/2) sin(Ω), iy = sin(i) sin(Ω).

The anomalies for the equinoctial elements are usually called longitudes (see [Battin, 1999] for instance): true longitude L, mean longitude l and eccentric longitude K. For clarity reasons and in order not to mix these anomalies with the geographical longitude, the time-varying parameters are renamed and written as:

true longitude : L true equinoctial anomaly : νQ = Ω + ω + ν, • → mean longitude : l mean equinoctial anomaly : MQ = Ω + ω + M, • → eccentric longitude : K eccentric equinoctial anomaly : EQ = Ω + ω + E. • →

14 Three other angular time-varying parameter called true longitude `νΘ, mean longitude `MΘ and eccentric longitude `EΘ are deﬁned by:

`ν = Ω + ω + ν Θ = νQ Θ, Θ − − `M = Ω + ω + M Θ = MQ Θ, (32) Θ − − `E = Ω + ω + E Θ = EQ Θ. Θ − − Θ(t) is the right ascension of the Greenwich meridian and is deﬁned by the Equation (129), page 49. We choose to replace the sixth parameter of the equinoctial orbital elements state vector by the mean longitude lMΘ in order to form the following state vector:

 a     ex     ey  x (t) =   . (33) EOE    ix     iy  `MΘ(t)

The anomalies deﬁned in this section are summarized in the Table2.

True equinoctial anomaly νQ = Ω + ω + ν Mean equinoctial anomaly MQ = Ω + ω + M Eccentric equinoctial anomaly EQ = Ω + ω + E Geographical longitude λ True longitude `ν = νQ Θ Θ − Mean longitude `M = MQ Θ Θ − Eccentric longitude `E = EQ Θ Θ − Table 2 – Equinoctial anomalies and longitudes.

4 FREE EVOLUTION EQUATIONS The previous sections described the environmental disturbing forces acting on the spacecraft and the state vector that will be used for the description of the motion. In this section, the dynamics equations will be derived for a satellite undergoing the Earth central gravitational attraction as well as the disturbing forces: the non spherical Earth gravitational ﬁeld, the Sun and the Moon gravitational forces and the force from the sun radiation pressure. The acceleration created by the spacecraft thrusters is not taken into account in this section.

15 4.1 KEPLERIAN MOTION When the Earth central gravitational attraction force is acting on the spacecraft, the non linear dynamics equation reads:  d~r   = ~v,  dt  G R (34)  d~v ~r   = µ .  dt − ⊕ ~r 3 G R || || It can be rewritten as:

 x˙     y˙     z˙       x  dxcart  µ   ⊕ 2 2 23 = − √x + y + z  . (35) dt    y   µ   3 − ⊕ √x2 + y2 + z2     z   µ  − ⊕ √x2 + y2 + z23

With the classical orbital elements, the anomaly is the only time-varying parameter. The evolution of the systems is thus given by:

 0     0     0  dxCOE   =   = K(x ). (36)  0  COE dt    0   r µ  n =  a3

The keplerian dynamics can be also expressed using the equinoctial orbital elements. Using the transformation between the classical orbital elements and the equinoctial ones xEOE = xEOE(xCOE, t), the keplerian dynamics reads:

dx ∂x dx ∂x EOE = EOE COE COE, dt ∂xCOE dt − ∂t ∂xEOE dxCOE (37) = ωT , ∂xCOE dt − = K(x ) ωT , EOE − with  0     0     0  ωT =   , (38)    0     0  ω ⊕ 16 ∂x and EOE the jacobian transformation matrix between the classical and the equinoctial ∂xCOE orbital elements: 1 0 0 0 0 0  ex  0 q 0 ey ey 0  2 2   ex + ey − −     ey  0 q 0 ex ex 0  2 2   ex + ey  ∂xEOE   =  2 2  . (39)  ix(1 + ix + iy)  ∂xCOE 0 0 q iy 0 0  2 2 −   2 ix + iy   2 2   iy(1 + i + i )   x y  0 0 q ix 0 0  2 2   2 ix + iy  0 0 0 1 1 1

The dynamics equation for the equinoctial state vector reads thus:  0     0     0  dxEOE   =   = K(x ) ωT . (40)  0  EOE dt   −  0  r   µ  ⊕ ω a3 − ⊕

4.2 NON KEPLERIAN MOTION The non keplerian motion refers to the motion of the spacecraft taking into account all the disturbing forces described in the section 2.2.

4.2.1 OSCULATING ORBITS When disturbing forces act on the spacecraft, the transformation between the cartesian positions and velocities and the classical orbital elements is time-dependent. At each time, a set of orbital elements can be defined. These elements are thus time-varying and are called osculating orbital elements, defining an osculating trajectory. This trajectory is the one the satellite would follow if the orbital disturbances were instantly removed. The osculating ellipse is thus tangent to the spacecraft trajectory, but their curvature are different. The state vectors in terms of the classical or equinoctial orbital elements is thus:  a(t)   a(t)       e(t)   ex(t)       i(t)   ey(t)  x (t) =   et x (t) =   . (41) COE   EOE    Ω(t)   ix(t)       ω(t)   iy(t)  M(t) `MΘ(t) The aim of this section is to derive the dynamics equation: dx = f(x,~ap, t), (42) dt

17 with ~ap the disturbing accelerations, i.e. the accelerations produced by the forces except the Earth central gravitational one. All these disturbing forces are deriving from a potential function. Therefore:

dx = f(x, p, t), (43) dt E where p is the disturbing potential. E With the Lagrange perturbation technique (see the reference [Zarrouati, 1987]), it is possible to derive the dynamics equation for the state vector expressed in terms of the orbital elements with respect to the perturbing potential, assuming that the perturbations are small with respect to the central acceleration term.

4.2.2 LAGRANGE EQUATIONS The Lagrange equation expresses the evolution of the state vector written with orbital elements for a spacecraft undergoing small perturbations deriving from a potential. Expressing this gradient with respect to the orbital elements, it comes:

!T ∂ p ~ap = E . (44) ∂xCOE

The motion equation is thus rewritten as:

!T dxCOE ∂ p = K(xCOE) + LCOE(xCOE, t) E , (45) dt ∂xCOE

where the ﬁrst term is the keplerian part and the second term is the eﬀect of the disturbing forces.

LCOE(xCOE, t) is the Lagrange matrix deﬁned in the reference [Zarrouati, 1987] by:

LCOE(xCOE, t) =  2  0 0 0 0 0 na  √1 e2 −e2 1   0 0 0 0 −2 −2   na e na e   0 0 0 1 cos i 0   na2√1 e2 sin i na2√1 e2 sin i   1 − − −  . (46)  0 0 0 0 0   − na2√1 e2 sin i   √1 e2 cos−i   0 −2 0 0 0   na e na2√1 e2 sin i  2 −1 e2 − na na−2e 0 0 0 0

The equation of motion (45) is then transformed in order to express the evolution of

18 the state vector in terms of the equinoctial orbital elements:

dx ∂x ! dx ∂x EOE = EOE COE COE, dt ∂xCOE dt − ∂t ! !T ∂xEOE ∂ p = K(xEOE) ωT + LCOE(xCOE, t) E , − ∂xCOE ∂xCOE ! !T ∂xEOE ∂ p ∂xEOE = K(xEOE) ωT + LCOE(xCOE, t) E , (47) − ∂xCOE ∂xEOE ∂xCOE ! !T !T ∂xEOE ∂xEOE ∂ p = K(xEOE) ωT + LCOE(xCOE, t) E , − ∂xCOE ∂xCOE ∂xEOE !T ∂ p = K(xEOE) ωT + LEOE(xEOE, t) E . − ∂xEOE The Lagrange matrix with the equinoctial orbital elements is: ! !T ∂xEOE ∂xEOE LEOE(xEOE, t) = LCOE(xCOE, t) . (48) ∂xCOE ∂xCOE The Lagrange matrix transformed with the equinoctial orbital elements is computed with the jacobian matrix of the transformation between the classical and the equinoctial 2 2 orbital elements (39) and the transformations of the section 3.2.2. Denoting p = 1 + ix + iy and recalling: rµ n = , a b = a√1 e2, (49) 2 2 − 2 e = ex + ey, the Lagrange matrix reads:

LEOE(xEOE, t) =  2   0 0 0 0 0   −na     b eyixp eyiyp exb   0 0   3 2   na 2nab 2nab na (a + b)    b exixp exiyp eyb   0 0   3 2   −na − 2nab − 2nab na (a + b) . (50)  2   eyixp exixp p ixp   0 0     − 2nab 2nab 4nab 2nab   e i p e i p p2 i p   y y x y y   0 0   − 2nab 2nab −4nab 2nab   2 exb eyb ixp iyp   0  na −na2(a + b) −na2(a + b) −2nab −2nab In the sequel, the equinoctial orbital elements will be used. Therefore, the index EOE will be omitted and the dynamics equation reads: real !T d x X ∂ pi = K(x) ωT + L(x, t) E = fL(x, t). (51) dt − perturbation i ∂x

The fL function encompass the Earth central acceleration term as well as the orbital disturbances.

19 4.3 LINEARIZED EVOLUTION EQUATION In the case of station keeping, the spacecraft has to stay in the vicinity of the station keeping position. Hence, the distance between the spacecraft and the station keeping point is very small with respect to the distance to the center of the Earth. It is thus possible to linearized the spacecraft dynamics with respect to the ideal trajectory of the station keeping point evolving on the geostationary orbit.

4.3.1 LINEARIZATION POINT The nominal station keeping position is a point deﬁned by the state vector:   ask    0     0  xsk =   , (52)    0     0  `MΘ,sk

with ask = 42165.765 km and `MΘ,mp the station keeping mean longitude. This point is a ﬁctitious point on a geostationary orbit and evolves following an unperturbed keplerian orbit:

dkepx

x˙ sk = dt x=xsk (53) = 0 by hypothesis. The gap between the station keeping position and the spacecraft position is written:

∆x = x xsk. (54) − ∆x(t) is at each time the gap between the ﬁctitious position of a point that would evolve on a geostationary unperturbed keplerian orbit and the real position of the spacecraft following the equation of motion: drealx x˙ = . (55) dt We assume that this gap is small enough, so that it is possible to develop the dynamics equation at the order 1 in ∆x.

4.3.2 DERIVATIVE OF THE STATE VECTOR AT ORDER 1 The dynamics of the gap between the real orbit and the ﬁctive one linearized with respect to ∆x is computed as:

d∆x drealx dkepx

= dt dt − dt x=xsk drealx = dt ! drealx ∂ drealx

= + ∆x dt ∂x dt x=xsk x=xsk

∂

= fL(xsk, t) + (fL(x(t), t)) ∆x (56) ∂x x=xsk Identifying:

∂ A(t) = (fL(x(t), t)) , (57a) ∂x x=xmp

D(t) = fL(xmp, t), (57b)

it is possible to recover the time-varying linear system:

d∆x = [A(t)]∆x + D(t). (58) dt

5 COMPUTATION OF THE MATRICES FOR THE LIN- EARIZED DYNAMICS This section aims at computing the A(t) and D(t) matrices for each of the disturbing forces described in Section 2.2 according to the formulas given by the Equation (57). As the motion has been linearized, the matrix A(t) and the vector D(t) can be decomposed in a keplerian and a disturbing parts:

A(t) = AK (t) + A˜(t), (59a)

D(t) = DK (t) + D˜(t). (59b)

Moreover, the term corresponding to the orbital disturbances is further decomposed in one term for each perturbation:

A˜(t) = A˜ (t) + A˜ (t) + A˜ (t) + A˜SRP(t), (60a) ⊕ D˜(t) = D˜ (t) + D˜ (t) + D˜ $(t) + D˜ SRP(t), (60b) ⊕ $ where A˜ and D˜ denote the effect of the non spherical Earth gravitational potential, A˜ ⊕ ⊕ and D˜ the effect of the gravitational attraction of the Sun, A˜ and D˜ the gravitational ˜ ˜ attraction of the Moon, and ASRP and DSRP the effect of the Sun$ radiation$ pressure. The physical parameters needed for the numerical computation of dynamics are summarized in the AppendixE page 74.

5.1 KEPLERIAN PART The non linear keplerian equation of motion is given by the Equation (40). Linearizing this equation according to the formulas given by the Equation (57) leads to:

 0 0 0 0 0 0    0 0 0 0 0 0    0 0 0 0 0 0     AK (t) =  0 0 0 0 0 0 , (61a)    0 0 0 0 0 0  s   3 µ   ⊕  5 0 0 0 0 0 −2 ask

21  0     0     0      DK (t) =  0  . (61b)    0  s   µ   ⊕  3 ω ask − ⊕

5.2 NON KEPLERIAN EARTH GRAVITATIONAL PART

5.2.1 INTRODUCTION As stated in the section 2.2.1, the gravitational potential of a non spherical Earth can be expanded in spherical harmonics. The potential is ﬁrst expressed in terms of the geographical position (radius, latitude and longitude). This geographical position is then converted in the cartesian position in the geocentric inertial reference frame, and ﬁnally in the equinoctial orbital elements. Following the process written in the reference [Losa, 2007], the Earth gravitational potential is expressed in terms of the radius, latitude and longitude of the spacecraft E⊕ up to the degree and order 3 as:

2 µ R 2 = ⊕ ⊕ C20 3 sin ϕ 1 E⊕ 2r3 − µ R2 + ⊕ ⊕ 3 cos ϕ sin ϕ C cos λ + S sin λ r3 21 21 2 µ R 2 + ⊕ ⊕ 3 cos ϕ C cos(2λ) + S sin 2λ r3 22 22 3 µ R 2 + ⊕ ⊕ C sin ϕ 5 sin ϕ 3 (62) 2r4 30 − 3 µ R 2 + ⊕ ⊕ cos ϕ 15 sin ϕ 3 C cos λ + S λ 2r4 − 31 31 sin 3 µ R 2 + ⊕ ⊕ 15 sin ϕ cos ϕ C cos(2λ) + S sin 2λ r4 32 32 3 µ R 3 + ⊕ ⊕ 15 cos ϕ C cos(3λ) + S sin 3λ , r4 33 33

with R the radius of the Earth and Clm and Slm the coeﬃcient of the Earth gravita- ⊕ tional spherical harmonics. Values for these coeﬃcients can be found for instance in the table D-1 of the reference [Vallado, 1997]. Using the conversion formulas (130) between the geographical position and the position in the geocentric inertial reference frame, the gravitational potential of the Earth is written as:

= ,C20 + ,C21 + ,S21 + ,C22 + ,S22 E⊕ E⊕ E⊕ E⊕ E⊕ E⊕ + ,C30 + ,C31 + ,S31 + ,C32 + ,S32 + ,C33 + ,S33, (63) E⊕ E⊕ E⊕ E⊕ E⊕ E⊕ E⊕ with:

2 2 2 2 µ R C20 2z x y ,C20 = ⊕ ⊕ − − , (64a) E⊕ 2 √x2 + y2 + z25

22 2 z(x cos Θ + y sin Θ) ,C21 = 3µ R C21 5 , (64b) E⊕ ⊕ ⊕ √x2 + y2 + z2

2 z(y cos Θ x sin Θ) ,S21 = 3µ R S21 − 5 , (64c) E⊕ ⊕ ⊕ √x2 + y2 + z2 2 2 2 (x y ) cos(2Θ) + 2xy sin 2Θ ,C22 = 3µ R C22 − 5 , (64d) E⊕ ⊕ ⊕ √x2 + y2 + z2 2 2 2 (y y ) sin(2Θ) + 2xy cos(2Θ) ,S22 = 3µ R S22 − 5 , (64e) E⊕ ⊕ ⊕ √x2 + y2 + z2 3 2 2 2 µ R C30 z(2z 3x 3y ) ,C30 = ⊕ ⊕ − − , (64f) E⊕ 2 √x2 + y2 + z27 3 2 2 2 µ R C31 (12z 3x 3y )(x cos Θ + y sin Θ) ,C31 = ⊕ ⊕ − − , (64g) E⊕ 2 √x2 + y2 + z27 3 2 2 2 µ R S31 (12z 3x 3y )(y cos Θ x sin Θ) ,S31 = ⊕ ⊕ − − − , (64h) E⊕ 2 √x2 + y2 + z27 2 2 3 z[(x y ) cos(2Θ) + 2xy sin(2Θ)] ,C32 = 15µ R C32 − 7 , (64i) E⊕ ⊕ ⊕ √x2 + y2 + z2 2 2 3 z[(y x ) sin(2Θ) + 2xy cos(2Θ)] ,S32 = 15µ R S32 − 7 , (64j) E⊕ ⊕ ⊕ √x2 + y2 + z2 2 2 2 3 (x cos Θ + y sin Θ)[x + y 4(y cos Θ x sin Θ) ] ,C33 = 15µ R C33 − 7 − , (64k) E⊕ ⊕ ⊕ √x2 + y2 + z2 2 2 2 3 (y cos Θ x sin Θ)[x + y 4(x cos Θ + y sin Θ) ] ,S33 = 15µ R S33 − − 7 . (64l) E⊕ ⊕ ⊕ √x2 + y2 + z2

Using the conversion formulas (216) between the cartesian position in the geocentric inertial reference frame and the equinoctial orbital elements, the Earth gravitational potential is expressed in terms of the equinoctial orbital elements. In the potential , the contributions of each harmonic are summed together. Therefore, E⊕ by linearity of the derivation, the matrix A˜ (t) and the vector D˜ (t) for the overall Earth ⊕ ⊕ gravitational potential are the sum of the linearization of each component of the spherical expansion. In the next subsections, the matrix A˜ and the vector D˜ for each harmonic ⊕ ⊕ are derived. In the sequel, the following notations will be used:

κ(t) = `MΘ + Θ(t), κsk(t) = `MΘ,sk + Θ(t),

cκ = cos κ(t), sκ = sin κ(t),

cκ,sk = cos κsk(t), sκ,sk = sin κsk(t),

c2κ,sk = cos(2κsk(t)), s2κ,sk = sin(2κsk(t)),

cκ+Θ = cos(κ(t) + Θ(t)), sκ+Θ = sin(κ(t) + Θ(t)),

c2κ+2Θ = cos(2κ(t) + 2Θ(t)), s2κ+2Θ = sin(2κ(t) + 2Θ(t)), (65)

c`,sk = cos `MΘ,sk, s`,sk = sin `MΘ,sk,

c2`,sk = cos(2`MΘ,sk), s2`,sk = sin(2`MΘ,sk),

c3`,sk = cos(3`MΘ,sk), s3`,sk = sin(3`MΘ,sk),

c2Θ = cos(2Θ(t)), s2Θ = sin(2Θ(t)),

c3Θ = cos(3Θ(t)), s3Θ = sin(3Θ(t)).

23 5.2.2 COEFFICIENT C20

The C20 term of the Earth potential is: α20 2 ,C20 = 3 sin ϕ 1 , E⊕ r3 − 2z2 x2 y2 = α20 − − , √x2 + y2 + z25 3 h 2 2 2 2 2 2 2 i (1 + excκ + eysκ) (ix + iy) + 2ix(6cκ 5) + 2iy(1 6cκ) + 1 + 24cκsκixiy = α − − , 20 a3(1 e2 e2)3(1 + i2 + i2)2 − x − y x y (66) 2 µ⊕R⊕C20 with α20 = 2 . The Equation (57) leads to the dynamic matrix:

 12 13  0 A ,C20 A ,C20 0 0 0  21 22⊕ 23⊕ 26  A ,C20 A ,C20 A ,C20 0 0 A ,C20  31⊕ 32⊕ 33⊕ 36⊕  α20 A ,C20 A ,C20 A ,C20 0 0 A ,C20 A˜ ,C =  ⊕ ⊕ ⊕ ⊕  , (67) 20 4  44 45  ⊕ ask√µ ask  0 0 0 A ,C20 A ,C20 0  ⊕  54⊕ 55⊕   0 0 0 A ,C20 A ,C20 0  61 62 63 ⊕ ⊕ A ,C20 A ,C20 A ,C20 0 0 0 ⊕ ⊕ ⊕ with:

21 21sκ,sk 31 21cκ,sk A ,C20 = ,A ,C20 = , (68a) ⊕ 2 ⊕ − 2 61 12 2 A ,C20 = 21,A ,C20 = 6asksκ,sk, (68b) ⊕ − ⊕ − 22 32 2 A ,C20 = 6askcκ,sksκ,sk,A ,C20 = 6ask(1 + cκ,sk), (68c) ⊕ − ⊕ 62 39askcκ,sk 13 2 A ,C20 = ,A ,C20 = 6askcκ,sk, (68d) ⊕ 2 ⊕ 23 2 33 A ,C20 = 6ask(1 + sκ,sk),A ,C20 = 6askcκ,sksκ,sk, (68e) ⊕ − ⊕ 63 39asksκ,sk 44 A ,C20 = ,A ,C20 = 6askcκ,sksκ,sk, (68f) ⊕ 2 ⊕ − 54 2 45 2 A ,C20 = 6asksκ,sk,A ,C20 = 6askcκ,sk, (68g) ⊕ − ⊕ 55 26 A ,C20 = 6askcκ,sksκ,sk,A ,C20 = 3askcκ,sk, (68h) ⊕ ⊕ − 36 A ,C20 = 3asksκ,sk, (68i) ⊕ − and:  0     3sκ,sk −  α20  3cκ,sk  D˜ ,C =   (69) 20 3   ⊕ ask√µ ask  0  ⊕    0  6

24 5.2.3 COEFFICIENT C21

The C21 term of the Earth potential is:

α21 ,C21 = cos ϕ sin ϕ cos λ, E⊕ r3 z(x cos Θ + y sin Θ) = α21 , √x2 y2 z25 + + (70) 3 (1 + excκ + eysκ) (iycκ ixsκ) = 2α − 21 a3(1 e2 e2)3(1 + i2 + i2)2 − x − y x y h 2 2 i cκ (i i ) + 2sκ ixiy + cos(`M ) , × +Θ x − y +Θ Θ 2 with α21 = 3µ R C21. ⊕ ⊕ The Equation (57) leads to the dynamic matrix:

 14 15  0 0 0 A ,C21 A ,C21 0  ⊕24 ⊕25   0 0 0 A ,C21 A ,C21 0   ⊕34 ⊕35  α21  0 0 0 A ,C21 A ,C21 0  A˜ ,C =  ⊕ ⊕  , (71) 21 4  41 42 43 46  ⊕ ask√µ ask A ,C21 A ,C21 A ,C21 0 0 A ,C21 ⊕  51⊕ 52⊕ 53⊕ ⊕56  A ,C21 A ,C21 A ,C21 0 0 A ,C21 ⊕ ⊕ ⊕ 64 65 ⊕ 0 0 0 A ,C21 A ,C21 0 ⊕ ⊕

41 7cκ,skc`,sk A ,C21 = , (72a) ⊕ − 4 51 7sκ,skc`,sk A ,C21 = , (72b) ⊕ − 4 2 42 3askcκ,skc`,sk A ,C21 = , (72c) ⊕ 2 52 3askcκ,sksκ,skc`,sk A ,C21 = , (72d) ⊕ 2 43 3askcκ,sksκ,skc`,sk A ,C21 = , (72e) ⊕ 2 2 53 3asksκ,skc`,sk A ,C21 = , (72f) ⊕ 2 14 2 A ,C21 = 4ask cos(2κsk Θ), (72g) ⊕ − 24 2 A ,C21 = 6asksκ,skc`,sk, (72h) ⊕ − 34 A ,C21 = 6askcκ,sksκ,skc`,sk, (72i) ⊕ 64 A ,C21 = 13asksκ,skc`,sk, (72j) ⊕ 15 2 A ,C21 = 4ask sin(2κsk Θ), (72k) ⊕ − 25 A ,C21 = 6askcκ,sksκ,skc`,sk, (72l) ⊕ 35 2 A ,C21 = 6askcκ,skc`,sk, (72m) ⊕ − 65 A ,C21 = 13askcκ,skc`,sk, (72n) ⊕ − 46 ask sin(2κsk Θ) A ,C21 = − , (72o) ⊕ − 2 56 ask cos(2κsk Θ) A ,C21 = − , (72p) ⊕ 2

25 and:  0     0    α21c`,sk  0  D˜ ,C =   (73) 21 3   ⊕ 2ask√µ ask cκ,sk ⊕   sκ,sk 0

5.2.4 COEFFICIENT S21

The S21 term of the Earth potential is:

β21 ,S21 = cos ϕ sin ϕ sin λ, E⊕ r3 z(y cos Θ x sin Θ) = β21 − , 2 2 25 √x + y + z (74) 3 (1 + excκ + eysκ) (iycκ ixsκ) = 2β − 21 a3(1 e2 e2)3(1 + i2 + i2)2 − x − y x y h 2 2 i sκ (i i ) + 2cκ ixiy + sin(`M ) , × +Θ y − x +Θ Θ 2 with β21 = 3µ R S21. ⊕ ⊕ The Equation (57) leads to the dynamic matrix:

 14 15  0 0 0 A ,S21 A ,S21 0  24⊕ 25⊕   0 0 0 A ,S21 A ,S21 0   34⊕ 35⊕  β21  0 0 0 A ,S21 A ,S21 0  A˜ ,S =  ⊕ ⊕  , (75) 21 4  41 42 43 46  ⊕ ask√µ ask A ,S21 A ,S21 A ,S21 0 0 A ,S21 ⊕  51⊕ 52⊕ ⊕53 56⊕  A ,S21 A ,S21 A ,S21 0 0 A ,S21 ⊕ ⊕ ⊕ 64 65 ⊕ 0 0 0 A ,S21 A ,S21 0 ⊕ ⊕

41 7cκ,sks`,sk A ,S21 = , (76a) ⊕ − 4 51 7sκ,sks`,sk A ,S21 = , (76b) ⊕ − 4 2 42 3askcκ,sks`,sk A ,S21 = , (76c) ⊕ 2 52 3askcκ,sksκ,sks`,sk A ,S21 = , (76d) ⊕ 2 43 3askcκ,sksκ,sks`,sk A ,S21 = , (76e) ⊕ 2 2 53 3asksκ,sks`,sk A ,S21 = , (76f) ⊕ 2 14 2 A ,S21 = 4ask sin(2κsk Θ), (76g) ⊕ − 24 2 A ,S21 = 6asksκ,sks`,sk, (76h) ⊕ − 34 A ,S21 = 6askcκ,sksκ,sks`,sk, (76i) ⊕ 64 A ,S21 = 13asksκ,sks`,sk, (76j) ⊕ 15 2 A ,S21 = 4ask cos(2κsk Θ), (76k) ⊕ − − 26 25 A ,S21 = 6askcκ,sksκ,sks`,sk, (76l) ⊕ 35 2 A ,S21 = 6askcκ,sks`,sk, (76m) ⊕ − 65 A ,S21 = 13askcκ,sks`,sk, (76n) ⊕ − 46 ask cos(2κsk Θ) A ,S21 = − , (76o) ⊕ 2 56 ask sin(2κsk Θ) A ,S21 = − , (76p) ⊕ 2 and:  0     0    β21s`,sk  0  D˜ ,S =   (77) 21 3   ⊕ 2ask√µ ask cκ,sk ⊕   sκ,sk 0

5.2.5 COEFFICIENT C22

The C22 term of the Earth potential is:

,S22 E⊕ α = 22 cos2 ϕ cos(2λ), r3 2 2 (x y )c2Θ + 2xys2Θ = α22 − , √x2 + y2 + z25 3 (1 + excκ + eysκ) = α 22 a3(1 e2 e2)3(1 + i2 + i2)2 − x − y x y h 4 4 2 2 2 2 i c κ ( i i + 6i i ) + (4s κ ixiy 2c )(i i ) c `,sk 4s ixiy , × 2 +2Θ − x − y x y 2 +2Θ − 2Θ y − x − 2 − 2Θ (78) 2 with α22 = 3µ R C22. ⊕ ⊕ The Equation (57) leads to the dynamic matrix:

 11 12 13 16  A ,C22 A ,C22 A ,C22 0 0 A ,C22  21⊕ 22⊕ 23⊕ 26⊕  A ,C22 A ,C22 A ,C22 0 0 A ,C22  31⊕ 32⊕ 33⊕ 36⊕  α22 A ,C22 A ,C22 A ,C22 0 0 A ,C22 A˜ ,C =  ⊕ ⊕ ⊕ ⊕  , (79) 22 4  44 45  ⊕ ask√µ ask  0 0 0 A ,C22 A ,C22 0  ⊕  54⊕ 55⊕   0 0 0 A ,C22 A ,C22 0  61 62 63 ⊕ ⊕ 66 A ,C22 A ,C22 A ,C22 0 0 A ,C22 ⊕ ⊕ ⊕ ⊕

11 A ,C22 = 10s2`,skask, (80a) ⊕ − 21 21sκ,skc2`,sk A ,C22 = , (80b) ⊕ − 2 31 21cκ,skc2`,sk A ,C22 = , (80c) ⊕ 2 61 A ,C22 = 21c2`,sk, (80d) ⊕ − 12 2 h 2 2 i A ,C22 = 6ask sκ,skc2Θ(6cκ,sk 1) + 2s2Θcκ,sk(2 3cκ,sk) , (80e) ⊕ − − −

27 22 h 2 4 2 i A ,C22 = ask 4c2Θcκ,sksκ,sk(3cκ,sk 2) + s2Θ( 12cκ,sk + 14cκ,sk 1) , (80f) ⊕ − − − 32 2 A ,C22 = 6ask(1 + cκ,sk)c2`,sk, (80g) ⊕ − 62 39askcκ,skc2`,sk A ,C22 = , (80h) ⊕ − 2 13 2 h 2 2 i A ,C22 = 6ask 2sκ,sks2Θ(3cκ,sk 2) + c2Θcκ,sk(6cκ,sk 5) , (80i) ⊕ − − − 23 2 A ,C22 = 6ask(1 + sκ,sk)c2`,sk, (80j) ⊕ 33 h 2 4 2 i A ,C22 = ask 4cκ,sksκ,skc2Θ(3cκ,sk 1) + s2Θ( 12cκ,sk + 10cκ,sk + 1) , (80k) ⊕ − − − 63 39asksκ,skc2`,sk A ,C22 = , (80l) ⊕ − 2 44 A ,C22 = 2askcκ,sk sin(κsk 2Θ), (80m) ⊕ − − 54 A ,C22 = 2asksκ,sk sin(κsk 2Θ), (80n) ⊕ − − 45 A ,C22 = 2askcκ,sk cos(κsk 2Θ), (80o) ⊕ − − 55 A ,C22 = 2asksκ,sk cos(κsk 2Θ), (80p) ⊕ − − 16 2 A ,C22 = 8askc2`,sk, (80q) ⊕ 26 h 2 2 i A ,C22 = 3ask 2s2Θsκ,sk(3cκ,sk 1) + c2Θcκ,sk(6cκ,sk 5) , (80r) ⊕ − − 36 h 2 2 i A ,C22 = 3ask c2Θsκ,sk(6cκ,sk 1) + 2s2Θcκ,sk(2 3cκ,sk) , (80s) ⊕ − − 66 A ,C22 = 12asks2`,sk, (80t) ⊕ and:   4s2`,skask    3sκ,skc2`,sk    α22  3cκ,skc2`,sk D˜ ,C =   (81) 22 3 −  ⊕ ask√µ ask  0  ⊕    0  6c `,sk − 2

5.2.6 COEFFICIENT S22

The S22 term of the Earth potential is:

,S22 ⊕ E 2 = β22 cos ϕ cos(2λ), (x2 y2) cos(2Θ) + 2xy sin 2Θ = β22 − , √x2 + y2 + z25 3 (1 + excκ + eysκ) = β 22 a3(1 e2 e2)3(1 + i2 + i2)2 − x − y x y h 4 4 2 2 2 2 i s κ ( i i + 6i i ) + (4c κ ixiy 2s )(i i ) c `,sk + 4c ixiy , × 2 +2Θ − x − y x y 2 +2Θ − 2Θ x − y − 2 2Θ (82) 2 with β22 = 3µ R S22. ⊕ ⊕

28 The Equation (57) leads to the dynamic matrix:

 11 12 13 16  A ,S22 A ,S22 A ,S22 0 0 A ,S22  21⊕ 22⊕ 23⊕ 26⊕  A ,S22 A ,S22 A ,S22 0 0 A ,S22  31⊕ 32⊕ 33⊕ 36⊕  β22 A ,S22 A ,S22 A ,S22 0 0 A ,S22 A˜ ,S =  ⊕ ⊕ ⊕ ⊕  , (83) 22 4  44 45  ⊕ ask√µ ask  0 0 0 A ,S22 A ,S22 0  ⊕  54⊕ 55⊕   0 0 0 A ,S22 A ,S22 0  61 62 63 ⊕ ⊕ 66 A ,S22 A ,S22 A ,S22 0 0 A ,S22 ⊕ ⊕ ⊕ ⊕

11 A ,S22 = 10c2`,skask, (84a) ⊕ 21 21sκ,sks2`,sk A ,S22 = , (84b) ⊕ − 2 31 21cκ,sks2`,sk A ,S22 = , (84c) ⊕ 2 61 A ,S22 = 21s2`,sk, (84d) ⊕ − 12 2 h 2 2 i A ,S22 = 6ask sκ,sks2Θ(6cκ,sk 1) + 2c2Θcκ,sk(3cκ,sk 2) , (84e) ⊕ − − − 22 h 2 4 2 i A ,S22 = ask 4cκ,sksκ,sks2Θ(3cκ,sk 2) + c2Θ(12cκ,sk 14cκ,sk + 1) , (84f) ⊕ − − − 32 2 A ,S22 = 6ask(1 + cκ,sk)s2`,sk, (84g) ⊕ − 62 39askcκ,sks2`,sk A ,S22 = , (84h) ⊕ − 2 13 2 h 2 2 i A ,S22 = 6ask 2sκ,skc2Θ(3cκ,sk 1) + s2Θcκ,sk(5 6cκ,sk) , (84i) ⊕ − − − 23 2 A ,S22 = 6ask(1 + sκ,sk)s2`,sk, (84j) ⊕ 33 h 2 4 2 i A ,S22 = ask 4cκ,sksκ,sks2Θ(3cκ,sk 1) + c2Θ(12cκ,sk 10cκ,sk 1) , (84k) ⊕ − − − 63 39asksκ,sks2`,sk A ,S22 = , (84l) ⊕ − 2 44 A ,S22 = 2askcκ,sk cos(κsk 2Θ), (84m) ⊕ − 54 A ,S22 = 2asksκ,sk cos(κsk 2Θ), (84n) ⊕ − 45 A ,S22 = 2askcκ,sk sin(κsk 2Θ), (84o) ⊕ − − 55 A ,S22 = 2asksκ,sk sin(κsk 2Θ), (84p) ⊕ − − 16 2 A ,S22 = 8asks2`,sk, (84q) ⊕ 26 h 2 2 i A ,S22 = 3ask 2c2Θsκ,sk(3cκ,sk 1) + s2Θcκ,sk(5 6cκ,sk) , (84r) ⊕ − − 36 h 2 2 i A ,S22 = 3ask s2Θcκ,sk(6cκ,sk 1) + 2c2Θ(3cκ,sk 2) , (84s) ⊕ − − − 66 A ,S22 = 12askc2`,sk, (84t) ⊕ − and:   4c2`,skask  −   3sκ,sks2`,sk    β22  3cκ,sks2`,sk D˜ ,S =   (85) 22 3 −  ⊕ ask√µ ask  0  ⊕    0  6s `,sk − 2

29 5.2.7 COEFFICIENT C30

The C30 term of the Earth potential is:

,C30 E⊕ α = 30 C sin ϕ 5 sin2 ϕ 3 , r4 30 − z(2z2 3x2 3y2) = α30 − − , √x2 + y2 + z27 (86) 4 (1 + excκ + eysκ) (iycκ ixsκ) = 2α − − 30 a4(1 e2 e2)4(1 + i2 + i2)3 − x − y x y h 2 2 2 2 2 2 2 i 3(i + i ) + 2i (3 10c ) + 2i (10c 7) + 3 + 40cκsκixiy , × x y y − κ x κ − 3 µ R C30 with α = ⊕ ⊕ . 30 2 The Equation (57) leads to the dynamic matrix:

 14 15  0 0 0 A ,C30 A ,C30 0  ⊕24 ⊕25   0 0 0 A ,C30 A ,C30 0   ⊕34 ⊕35  α30  0 0 0 A ,C30 A ,C30 0  A˜ ,C =  ⊕ ⊕  , (87) 30 5  41 42 43 46  ⊕ ask√µ ask A ,C30 A ,C30 A ,C30 0 0 A ,C30 ⊕  51⊕ 52⊕ 53⊕ ⊕56  A ,C30 A ,C30 A ,C30 0 0 A ,C30 ⊕ ⊕ ⊕ 64 65 ⊕ 0 0 0 A ,C30 A ,C30 0 ⊕ ⊕

41 27cκ,sk A ,C30 = , (88a) ⊕ 4 51 27sκ,sk A ,C30 = , (88b) ⊕ 4 42 2 A ,C30 = 6askcκ,sk, (88c) ⊕ − 52 A ,C30 = 6askcκ,sksκ,sk, (88d) ⊕ − 43 A ,C30 = 6askcκ,sksκ,sk, (88e) ⊕ − 53 2 A ,C30 = 6asksκ,sk, (88f) ⊕ − 14 2 A ,C30 = 12askcκ,sk, (88g) ⊕ − 24 2 A ,C30 = 24asksκ,sk, (88h) ⊕ 34 A ,C30 = 24askcκ,sksκ,sk, (88i) ⊕ − 64 A ,C30 = 51asksκ,sk, (88j) ⊕ − 15 2 A ,C30 = 12asksκ,sk, (88k) ⊕ − 25 A ,C30 = 24askcκ,sksκ,sk, (88l) ⊕ − 35 2 A ,C30 = 24askcκ,sk, (88m) ⊕ 65 A ,C30 = 51askcκ,sk, (88n) ⊕ 46 3asksκ,sk A ,C30 = , (88o) ⊕ 2 56 3askcκ,sk A ,C30 = , (88p) ⊕ − 2

30 and:  0     0    3α30  0  D˜ ,C =   (89) 30 4   ⊕ −2ask√µ ask cκ,sk ⊕   sκ,sk 0

5.2.8 COEFFICIENT C31

The C31 term of the Earth potential is:

,C31 E⊕ α = 31 cos ϕ 15 sin2 ϕ 3 cos λ, r4 − (12z2 3x2 3y2)(x cos Θ + y sin Θ) = α31 − − , √x2 + y2 + z27 (90) 4 (1 + excκ + eysκ) = 3α − 31 a4(1 e2 e2)4(1 + i2 + i2)3 − x − y x y h 2 2 i cκ (i i ) + 2sκ ixiy + cos(`M ) × +Θ x − y +Θ Θ h 2 2 2 2 2 2 2 i (i + i ) + 2i (10c 9) + 2i (1 10c ) + 1 + 40cκsκixiy , × x y x κ − y − κ 3 µ R C31 with α = 3 ⊕ ⊕ . 31 2 The Equation (57) leads to the dynamic matrix:

 11 12 13 16  A ,C31 A ,C31 A ,C31 0 0 A ,C31  21⊕ 22⊕ 23⊕ 26⊕  A ,C31 A ,C31 A ,C31 0 0 A ,C31  31⊕ 32⊕ 33⊕ 36⊕  α31 A ,C31 A ,C31 A ,C31 0 0 A ,C31 A˜ ,C =  ⊕ ⊕ ⊕ ⊕  , (91) 31 5  44 45  ⊕ ask√µ ask  0 0 0 A ,C31 A ,C31 0  ⊕  54⊕ 55⊕   0 0 0 A ,C31 A ,C31 0  61 62 63 ⊕ ⊕ 66 A ,C31 A ,C31 A ,C31 0 0 A ,C31 ⊕ ⊕ ⊕ ⊕

11 A ,C31 = 21asks`,sk, (92a) ⊕ 21 A ,C31 = 54sκ,skc`,sk, (92b) ⊕ 31 A ,C31 = 54cκ,skc`,sk, (92c) ⊕ − 61 A ,C31 = 108c`,sk, (92d) ⊕ 12 2 A ,C31 = 24ask sin(2κsk Θ), (92e) ⊕ − − 22 h 2 i A ,C31 = 3ask s`,sk(24cκ,sk 1) + 24cκ,sk sin Θ , (92f) ⊕ − 32 2 A ,C31 = 12ask(3cκ,sk + 2)c`,sk, (92g) ⊕ 62 A ,C31 = 102askcκ,skc`,sk, (92h) ⊕ 13 2 A ,C31 = 24ask cos(2κsk Θ), (92i) ⊕ − 23 2 A ,C31 = 12ask(3cκ,sk 5)c`,sk, (92j) ⊕ − 33 h 2 i A ,C31 = 3ask s`,sk(24cκ,sk + 1) + 24cκ,sk sin Θ , (92k) ⊕ 31 63 A ,C31 = 102asksκ,skc`,sk, (92l) ⊕ 44 A ,C31 = 3askcκ,sk [10cκ,sks`,sk + 11 sin Θ] , (92m) ⊕ − 54 A ,C31 = 3asksκ,sk [10sκ,sks`,sk + sin Θ] , (92n) ⊕ − 45 A ,C31 = 3askcκ,sk [10cκ,skc`,sk + cos Θ] , (92o) ⊕ 55 A ,C31 = 3asksκ,sk [10sκ,sks`,sk 11 cos Θ] , (92p) ⊕ − − 16 2 A ,C31 = 6askc`,sk, (92q) ⊕ − 26 A ,C31 = 12ask cos(2κsk Θ), (92r) ⊕ − − 36 A ,C31 = 12ask sin(2κsk Θ), (92s) ⊕ − − 66 A ,C31 = 24asks`,sk, (92t) ⊕ − and:   6s`,skask  −   12sκ,skc`,sk −  α31  12cκ,skc`,sk  D˜ ,C =   (93) 31 4   ⊕ ask√µ ask  0  ⊕    0  24c`,sk

5.2.9 COEFFICIENT S31

The S31 term of the Earth potential is:

,S31 E⊕ β = 31 cos ϕ 15 sin2 ϕ 3 cos λ, r4 − (12z2 3x2 3y2)(x cos Θ + y sin Θ) = β31 − − , √ 2 2 27 x + y + z (94) 4 (1 + excκ + eysκ) = 3β − 31 a4(1 e2 e2)4(1 + i2 + i2)3 − x − y x y h 2 2 i sκ (i i ) + 2cκ ixiy sin(`M ) × +Θ y − x +Θ − Θ h 2 2 2 2 2 2 2 i (i + i ) + 2i (10c 9) + 2i (1 10c ) + 1 + 40cκsκixiy , × x y x κ − y − κ 3 µ R S31 with β = 3 ⊕ ⊕ . 31 2 The Equation (57) leads to the dynamic matrix:

 11 12 13 16  A ,S31 A ,S31 A ,S31 0 0 A ,S31  21⊕ 22⊕ 23⊕ 26⊕  A ,S31 A ,S31 A ,S31 0 0 A ,S31  31⊕ 32⊕ 33⊕ 36⊕  β31 A ,S31 A ,S31 A ,S31 0 0 A ,S31 A˜ ,S =  ⊕ ⊕ ⊕ ⊕  , (95) 31 5  44 45  ⊕ ask√µ ask  0 0 0 A ,S31 A ,S31 0  ⊕  54⊕ 55⊕   0 0 0 A ,S31 A ,S31 0  61 62 63 ⊕ ⊕ 66 A ,S31 A ,S31 A ,S31 0 0 A ,S31 ⊕ ⊕ ⊕ ⊕

11 A ,S31 = 21asks`,sk, (96a) ⊕ − 21 A ,S31 = 54sκ,sks`,sk, (96b) ⊕ 32 31 A ,S31 = 54cκ,sks`,sk, (96c) ⊕ − 61 A ,S31 = 108s`,sk, (96d) ⊕ 12 2 A ,S31 = 24ask cos(2κsk Θ), (96e) ⊕ − 22 h 2 i A ,S31 = 3ask c`,sk(24cκ,sk 1) 24cκ,sk cos Θ , (96f) ⊕ − − − 32 2 A ,S31 = 12ask(3cκ,sk + 2)s`,sk, (96g) ⊕ 62 A ,S31 = 102askcκ,sks`,sk, (96h) ⊕ 13 2 A ,S31 = 24ask sin(2κsk Θ), (96i) ⊕ − 23 2 A ,S31 = 12ask(3cκ,sk 5)s`,sk, (96j) ⊕ − 33 h 2 i A ,S31 = 3ask c`,sk(24cκ,sk + 1) 24cκ,sk cos Θ , (96k) ⊕ − − 63 A ,S31 = 102asksκ,sks`,sk, (96l) ⊕ 44 A ,S31 = 3askcκ,sk [10cκ,skc`,sk 11 cos Θ] , (96m) ⊕ − 54 A ,S31 = 3asksκ,sk [10sκ,sks`,sk + cos Θ] , (96n) ⊕ − 45 A ,S31 = 3askcκ,sk [10cκ,sks`,sk sin Θ] , (96o) ⊕ − 55 A ,S31 = 3asksκ,sk [10sκ,skc`,sk 11 sin Θ] , (96p) ⊕ − 16 2 A ,S31 = 6asks`,sk, (96q) ⊕ − 26 A ,S31 = 12ask sin(2κsk Θ), (96r) ⊕ − − 36 A ,S31 = 12ask cos(2κsk Θ), (96s) ⊕ − 66 A ,S31 = 24askc`,sk, (96t) ⊕ and:   6c`,skask    12sκ,sks`,sk −  β31  12cκ,sks`,sk  D˜ ,S =   (97) 31 4   ⊕ ask√µ ask  0  ⊕    0  24s`,sk

5.2.10 COEFFICIENT C32

The C32 term of the Earth potential is:

,C32 E⊕ α = 32 sin ϕ cos2 ϕ cos(2λ), r4 (98) z[(x2 y2) cos(2Θ) + 2xy sin(2Θ)] = α32 − , √x2 + y2 + z27 3 with β32 = 15µ R S32. ⊕ ⊕ The Equation (57) leads to the dynamic matrix:  14 15  0 0 0 A ,C32 A ,C32 0  ⊕24 ⊕25   0 0 0 A ,C32 A ,C32 0   ⊕34 ⊕35  α32  0 0 0 A ,C32 A ,C32 0  A˜ ,C =  ⊕ ⊕  , (99) 32 5  41 42 43 46  ⊕ ask√µ ask A ,C32 A ,C32 A ,C32 0 0 A ,C32 ⊕  51⊕ 52⊕ 53⊕ ⊕56  A ,C32 A ,C32 A ,C32 0 0 A ,C32 ⊕ ⊕ ⊕ 64 65 ⊕ 0 0 0 A ,C32 A ,C32 0 ⊕ ⊕ 33 41 9cκ,sks2`,sk A ,C32 = , (100a) ⊕ − 4 51 9sκ,sks2`,sk A ,C32 = , (100b) ⊕ − 4 42 2 A ,C32 = 2askcκ,sks2`,sk, (100c) ⊕ 52 A ,C32 = 2askcκ,sksκ,skc2`,sk, (100d) ⊕ 43 A ,C32 = 2askcκ,sksκ,sks2`,sk, (100e) ⊕ 53 2 A ,C32 = 2asksκ,sks2`,sk, (100f) ⊕ 14 2 h 2 2 i A ,C32 = 4ask 2c2Θsκ,sk(3cκ,sk 1) + s2Θcκ,sk(5 6cκ,sk) , (100g) ⊕ − − 24 2 A ,C32 = 8asksκ,sks2`,sk, (100h) ⊕ − 34 A ,C32 = 8askcκ,sksκ,sks2`,sk, (100i) ⊕ 64 A ,C32 = 17asksκ,sks2`,sk, (100j) ⊕ 15 2 h 2 2 i A ,C32 = 4ask s2Θsκ,sk(6cκ,sk 1) + 2c2Θcκ,sk(3cκ,sk 2) , (100k) ⊕ − − − 25 A ,C32 = 8askcκ,sksκ,sks2`,sk, (100l) ⊕ 35 2 A ,C32 = 8askcκ,sks2`,sk, (100m) ⊕ − 65 A ,C32 = 17askcκ,sks2`,sk, (100n) ⊕ − 46 ask h 2 2 i A ,C32 = s2Θsκ,sk(6cκ,sk 1) + 2c2Θcκ,sk(3cκ,sk 2) , (100o) ⊕ 2 − − 56 ask h 2 2 i A ,C32 = 2c2Θsκ,sk(3cκ,sk 1) + s2Θcκ,sk(5 6cκ,sk) , (100p) ⊕ 2 − − and:  0     0    α32s2`,sk  0  D˜ ,C =   (101) 32 4   ⊕ 2ask√µ ask cκ,sk ⊕   sκ,sk 0

5.2.11 COEFFICIENT S32

The S32 term of the Earth potential is:

,S32 E⊕ β = 32 sin ϕ cos2 ϕ sin(2λ), r4 (102) z[(y2 x2) sin(2Θ) + 2xy cos(2Θ)] = β32 − , √x2 + y2 + z27

3 with β32 = 15µ R C32. ⊕ ⊕

34 The Equation (57) leads to the dynamic matrix:

 14 15  0 0 0 A ,S32 A ,S32 0  24⊕ 25⊕   0 0 0 A ,S32 A ,S32 0   34⊕ 35⊕  β32  0 0 0 A ,S32 A ,S32 0  A˜ ,S =  ⊕ ⊕  , (103) 32 5  41 42 43 46  ⊕ ask√µ ask A ,S32 A ,S32 A ,S32 0 0 A ,S32 ⊕  51⊕ 52⊕ 53⊕ 56⊕  A ,S32 A ,S32 A ,S32 0 0 A ,S32 ⊕ ⊕ ⊕ 64 65 ⊕ 0 0 0 A ,S32 A ,S32 0 ⊕ ⊕

41 9cκ,skc2`,sk A ,S32 = , (104a) ⊕ − 4 51 9sκ,skc2`,sk A ,S32 = , (104b) ⊕ − 4 42 2 A ,S32 = 2askcκ,skc2`,sk, (104c) ⊕ 52 A ,S32 = 2askcκ,sksκ,skc2`,sk, (104d) ⊕ 43 A ,S32 = 2askcκ,sksκ,skc2`,sk, (104e) ⊕ 53 2 A ,S32 = 2asksκ,skc2`,sk, (104f) ⊕ 14 2 h 2 2 i A ,S32 = 4ask 2s2Θsκ,sk(3cκ,sk 1) + c2Θcκ,sk(6cκ,sk 5) , (104g) ⊕ − − 24 2 A ,S32 = 8asksκ,skc2`,sk, (104h) ⊕ − 34 A ,S32 = 8askcκ,sksκ,skc2`,sk, (104i) ⊕ 64 A ,S32 = 17asksκ,skc2`,sk, (104j) ⊕ 15 2 h 2 2 i A ,S32 = 4ask c2Θsκ,sk(6cκ,sk 1) + 2s2Θcκ,sk(2 3cκ,sk) , (104k) ⊕ − − 25 A ,S32 = 8askcκ,sksκ,skc2`,sk, (104l) ⊕ 35 2 A ,S32 = 8askcκ,skc2`,sk, (104m) ⊕ − 65 A ,S32 = 17askcκ,skc2`,sk, (104n) ⊕ − 46 ask h 2 2 i A ,S32 = c2Θsκ,sk(6cκ,sk 1) + 2s2Θcκ,sk(2 3cκ,sk) , (104o) ⊕ 2 − − 56 ask h 2 2 i A ,S32 = 2s2Θsκ,sk(3cκ,sk 1) + c2Θcκ,sk(6cκ,sk 5) , (104p) ⊕ 2 − − and:  0     0    β32c2`,sk  0  D˜ ,S =   (105) 32 4   ⊕ 2ask√µ ask cκ,sk ⊕   sκ,sk 0

5.2.12 COEFFICIENT C33

The C33 term of the Earth potential is:

,C33 E⊕ α = 33 cos3 ϕ cos(3λ), r4 (106) (x cos Θ + y sin Θ)[x2 + y2 4(y cos Θ x sin Θ)2] = α33 − − , √x2 + y2 + z27

35 2 with α33 = 15µ R C33. ⊕ ⊕ The Equation (57) leads to the dynamic matrix:

 11 12 13 16  A ,C33 A ,C33 A ,C33 0 0 A ,C33  21⊕ 22⊕ 23⊕ 26⊕  A ,C33 A ,C33 A ,C33 0 0 A ,C33  ⊕31 32⊕ 33⊕ 36⊕  α33 A ,C33 A ,C33 A ,C33 0 0 A ,C33 A˜ ,C =  ⊕ ⊕ ⊕ ⊕  , (107) 33 4  44 45  ⊕ ask√µ ask  0 0 0 A ,C33 A ,C33 0  ⊕  54⊕ 55⊕   0 0 0 A ,C33 A ,C33 0  61 62 63 ⊕ ⊕ 66 A ,C33 A ,C33 A ,C33 0 0 A ,C33 ⊕ ⊕ ⊕ ⊕

11 A ,C33 = 21s3`,skask, (108a) ⊕ − 21 A ,C33 = 18sκ,skc3`,sk, (108b) ⊕ − 31 A ,C33 = 18cκ,skc3`,sk, (108c) ⊕ 61 A ,C33 = 36c3`,sk, (108d) ⊕ − 12 2 h 2 4 2 i A ,C33 = 8ask 2c3Θcκ,sksκ,sk(8cκ,sk 3) + s3Θ( 16cκ,sk + 14cκ,sk 1) , (108e) ⊕ − − − 22 3ask h 4 2 4 2 i A ,C33 = c3Θsκ,sk(32cκ,sk 28cκ,sk + 1) + cκ,sks3Θ( 32cκ,sk + 44cκ,sk 1)) , (108f) ⊕ 2 − − − 32 h 4 2 2 2 i A ,C33 = 4ask s3Θsκ,sk(12cκ,sk + 5cκ,sk 2) + c3Θcκ,sk(12cκ,sk cκ,sk 6) , (108g) ⊕ − − − 62 A ,C33 = 34askcκ,skc3`,sk, (108h) ⊕ 13 2 h 2 4 2 i A ,C33 = 8ask 2s3Θcκ,sksκ,sk(8cκ,sk 5) + c3Θ(16cκ,sk 18cκ,sk + 3) , (108i) ⊕ − − − 23 h 4 2 2 2 i A ,C33 = 4ask s3Θsκ,sk(12cκ,sk 23cκ,sk + 5) + c3Θcκ,sk(12cκ,sk 29cκ,sk + 15) , (108j) ⊕ − − 33 h 4 2 4 2 i A ,C33 = 3ask c3Θsκ,sk( 32cκ,sk + 20cκ,sk + 1) + cκ,sks3Θ(32cκ,sk 36cκ,sk + 5)) , (108k) ⊕ − − 63 A ,C33 = 34asksκ,skc3`,sk, (108l) ⊕ − 44 A ,C33 = 3askcκ,sk sin(2κsk 3Θ), (108m) ⊕ − − 54 A ,C33 = 3asksκ,sk sin(2κsk 3Θ), (108n) ⊕ − − 45 A ,C33 = 3askcκ,sk cos(2κsk 3Θ), (108o) ⊕ − − 55 A ,C33 = 3asksκ,sk cos(2κsk 3Θ), (108p) ⊕ − − 16 2 A ,C33 = 18askc3`,sk, (108q) ⊕ − 26 h 2 4 2 i A ,C33 = 4ask 2s3Θcκ,sksκ,sk(8cκ,sk 5) + c3Θ(16cκ,sk 18cκ,sk + 3) , (108r) ⊕ − − 36 h 2 4 2 i A ,C33 = 4ask 2c3Θcκ,sksκ,sk(8cκ,sk 3) + s3Θ( 16cκ,sk + 14cκ,sk 1) , (108s) ⊕ − − − 66 A ,C33 = 24asks3`,sk (108t) ⊕ and:   6s3`,skask    3sκ,skc3`,sk    α33  3cκ,skc3`,sk D˜ ,C =   (109) 33 3 −  ⊕ ask√µ ask  0  ⊕    0  8c `,sk − 3

36 5.2.13 COEFFICIENT S33

The S33 term of the Earth potential is:

,S33 E⊕ β = 33 cos3 ϕ cos(3λ), r4 (110) (x cos Θ + y sin Θ)[x2 + y2 4(y cos Θ x sin Θ)2] = β33 − − , √x2 + y2 + z27

2 with β33 = 15µ R S33. ⊕ ⊕ The Equation (57) leads to the dynamic matrix:

 11 12 13 16  A ,S33 A ,S33 A ,S33 0 0 A ,S33  21⊕ 22⊕ 23⊕ 26⊕  A ,S33 A ,S33 A ,S33 0 0 A ,S33  31⊕ 32⊕ 33⊕ 36⊕  β33 A ,S33 A ,S33 A ,S33 0 0 A ,S33 A˜ ,S =  ⊕ ⊕ ⊕ ⊕  , (111) 33 4  44 45  ⊕ ask√µ ask  0 0 0 A ,S33 A ,S33 0  ⊕  54⊕ 55⊕   0 0 0 A ,S33 A ,S33 0  61 62 63 ⊕ ⊕ 66 A ,S33 A ,S33 A ,S33 0 0 A ,S33 ⊕ ⊕ ⊕ ⊕

11 A ,S33 = 21c3`,skask, (112a) ⊕ 21 A ,S33 = 18sκ,sks3`,sk, (112b) ⊕ 31 A ,S33 = 18cκ,skc3`,sk, (112c) ⊕ 61 A ,S33 = 36s3`,sk, (112d) ⊕ − 12 2 h 2 4 2 i A ,S33 = 8ask 2s3Θcκ,sksκ,sk(8cκ,sk 3) + c3Θ(16cκ,sk 14cκ,sk + 8) , (112e) ⊕ − − 22 h 4 2 4 2 i A ,S33 = 3ask s3Θsκ,sk(32cκ,sk 28cκ,sk + 1) + cκ,skc3Θ(32cκ,sk 44cκ,sk + 11)) , (112f) ⊕ − − 32 h 4 2 2 2 i A ,S33 = 4ask c3Θsκ,sk(12cκ,sk + 5cκ,sk 2) + s3Θcκ,sk( 12cκ,sk + cκ,sk + 6) , (112g) ⊕ − − 62 A ,S33 = 34askcκ,sks3`,sk, (112h) ⊕ 13 2 h 2 4 2 i A ,S33 = 8ask 2c3Θcκ,sksκ,sk(8cκ,sk 5) + s3Θ( 16cκ,sk + 18cκ,sk 3) , (112i) ⊕ − − − 23 h 4 2 2 2 i A ,S33 = 4ask c3Θsκ,sk(12cκ,sk 23cκ,sk + 5) + c3Θsκ,sk( 12cκ,sk + 29cκ,sk 15) , (112j) ⊕ − − − 33 3ask h 4 2 4 2 i A ,S33 = s3Θsκ,sk( 32cκ,sk + 20cκ,sk + 1) + cκ,skc3Θ( 32cκ,sk + 36cκ,sk 5)) , ⊕ 2 − − − (112k) 63 A ,S33 = 34asksκ,sks3`,sk, (112l) ⊕ − 44 A ,S33 = 3askcκ,sk cos(2κsk 3Θ), (112m) ⊕ − − 54 A ,S33 = 3asksκ,sk cos(2κsk 3Θ), (112n) ⊕ − − 45 A ,S33 = 3askcκ,sk sin(2κsk 3Θ), (112o) ⊕ − 55 A ,S33 = 3asksκ,sk sin(2κsk 3Θ), (112p) ⊕ − 16 2 A ,S33 = 18asks3`,sk, (112q) ⊕ − 26 h 2 4 2 i A ,S33 = 4ask 2c3Θcκ,sksκ,sk(8cκ,sk 5) + s3Θ( 16cκ,sk + 18cκ,sk 3) , (112r) ⊕ − − − − 36 h 2 4 2 i A ,S33 = 4ask 2s3Θcκ,sksκ,sk(8cκ,sk 3) + c3Θ(16cκ,sk 14cκ,sk + 1) , (112s) ⊕ − − 66 A ,S33 = 24askc3`,sk, (112t) ⊕ 37 and:   6s3`,sk    4asksκ,sks3`,sk −  β33  4askcκ,sks3`,sk  D˜ ,S =   (113) 33 5   ⊕ ask√µ ask  0  ⊕    0  8asks3`,sk

5.3 SUNAND MOON GRAVITATIONAL ATTRACTIONS As presented the Section 2.2.2, the gravitational potential of the Sun and the Moon atraction is often expanded in Legendre polynomials (see for instance the thesis [Losa, 2007]). For the derivation of the linearized matrix, we choose not to make this approximation and to use the exact expression of the potential. The expression of the relative dynamics matrices derived in this subsection are valid for any disturbing body whose gravitational potential acts on the spacecraft orbiting the Earth on the GEO orbit. For this study, these disturbing bodies are only the Sun and the Moon.

5.3.1 SUNAND MOON POSITIONS The positions of the Sun and the Moon are involved in their gravitational potential. The ephemeris have been retrieved from the NASA JPL’s Horizons System (see [Giorgini and JPL Solar System Dynamics Group, 2005]). The Figures5 and6 display the Sun and Moon cartesian position in the geocentric inertial reference frame from January 1st, 2034 to January 1st, 2039.

108 1.6 ·

1.4

1.2 x y z 1

0.8

0.6

0.4

0.2

0.2 positions− (km) 0.4 − 0.6 − 0.8 − 1 − 1.2 − 1.4 − 1.6 − 0 100 200 300 400 500 600 700 800 900 1,0001,1001,2001,3001,4001,5001,6001,7001,8001,9002,000 time (days)

Figure 5 – Cartesian position of the Sun in the geocentric inertial reference frame from January 1st, 2034 to January 1st, 2039.

38 105 6 · x y z$ 5 $ $

0 positions (km)

1 −

2 −

3 −

4 −

5 − 0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800 time (days)

Figure 6 – Cartesian position of the Moon in the geocentric inertial reference frame from January 1st, 2034 to January 1st, 2039.

The radial position of these bodies are written: q 2 2 2 rP = ~rP = x + y + z for P = or P = . (114) k k P P P $ The notations presented in the Equation (65) are still valid here. The distance between the station keeping point (in the equatorial plane) and the disturbing bodies is written: q 2 2 2 r P = x P + y P + zP , (115) ⊕ ⊕ ⊕ with: x P (t) = xP (t) ask cos κsk, ⊕ − with P = or P = , (116) y P (t) = yP (t) ask sin κsk, ⊕ − $ and illustrated on the Figure7.

5.3.2 RELATIVE DYNAMICS MATRICES The gravitational potential of the Sun and the Moon are expressed in terms of the cartesian position of the sapcecraft and of the disturbing body in the geocentric inertial reference frame. As the chosen state vector is made of the equinoctial orbital elements, the cartesian position is transformed into the equinoctial orbital elements thanks to the formulas given by the Equation (216). The positions of the Sun and the Moon are interpolated with the values from the Horizon ephemeris generator. The matrices A˜ and A˜ are written:

s $ ˜ ask h ˜iji AP = µP AP , with P = or P = . (117) µ i=1,...,6, j=1,...,6 ⊕ $ 39 ~uY G P

y P ⊕

x P ⊕ ask

`MΘ,sk

~uXG Equatorial Plane

Figure 7 – Illustration of x P and y P for P = or P = . ⊕ ⊕ $

The expressions for each coeﬃcient are:

" ! # 11 1 2ask xP sκ,sk yP cκ,sk Ap = 3 x P sκ,sk y P cκ,sk 3 + 5 −3 , (118a) ⊕ − ⊕ rP r P − rP " ⊕ sκ,sk 1 x P cκ,sk + y P sκ,sk 2ask 1 xP cκ,sk + yP sκ,sk 21 ⊕ ⊕ Ap = 3 − + 3 ask −2 r P 2 rP ⊕ 2 # (x P cκ,sk + y P sκ,sk) ⊕ ⊕ 3ask 5 , (118b) − r P " ⊕ cκ,sk 1 x P cκ,sk + y P sκ,sk 2ask 1 xP cκ,sk + yP sκ,sk 31 ⊕ ⊕ Ap = 3 − 3 ask 2 r P − 2 rP ⊕ 2 # (x P cκ,sk + y P sκ,sk) ⊕ ⊕ +3ask 5 , (118c) r P " ⊕ # cκ,skzP 1 1 x P cκ,sk + y P sκ,sk 41 ⊕ ⊕ Ap = 3 3 + 6ask 5 , (118d) 4ask r P − rP r P " ⊕ ⊕ # sκ,skzP 1 1 x P cκ,sk + y P sκ,sk 51 ⊕ ⊕ Ap = 3 3 + 6ask 5 , (118e) 4ask r P − rP r P ⊕ ⊕ x P cκ,sk + y P sκ,sk 2ask xP cκ,sk + yP sκ,sk 61 ⊕ ⊕ Ap = 3 − 3 r P − rP ⊕ 2 (x P cκ,sk + y P sκ,sk) ⊕ ⊕ + 2ask 5 , (118f) r P ⊕

40 " x P s2κ,sk y P c2κ,sk xP s2κ,sk yP c2κ,sk 12 ⊕ ⊕ Ap = 2ask −3 + −3 − r P rP ⊕ # (x P sκ,sk y P cκ,sk)(x P cκ,sk + y P sκ,sk) ⊕ ⊕ ⊕ ⊕ 3askcκ,sk − 5 , (118g) − r P ⊕ 2(x P cκ,sk + y P sκ,sk) ask x P sκ,sk y P cκ,sk 22 ⊕ ⊕ ⊕ ⊕ Ap = cκ,sksκ,sk 3 − −3 r P − 2r P ⊕ ⊕ 4cκ,sksκ,sk(xP cκ,sk + yP sκ,sk) + xP sκ,sk yP cκ,sk + − 3 − rP 2 (x P cκ,sk + y P sκ,sk) ⊕ ⊕ + 3cκ,sksκ,sk 5 , (118h) r P ⊕ 2 2 2 2sκ,sk(x P cκ,sk + y P sκ,sk) + askcκ,sk (xP cκ,sk + yP sκ,sk) 32 ⊕ ⊕ 2 Ap = 3 3cκ,skask 5 r P − r P ⊕ ⊕ x P cκ,sk + y P sκ,sk 2 ⊕ ⊕ 2sκ,sk 3 , (118i) − rP 2 " # cκ,skzP 1 1 x P cκ,sk + y P sκ,sk 42 ⊕ ⊕ Ap = 3 + 3 3ask 5 , (118j) 2 −r P rP − r P ⊕" ⊕ # cκ,sksκ,skzP 1 1 x P cκ,sk + y P sκ,sk 52 ⊕ ⊕ Ap = 3 + 3 3ask 5 , (118k) 2 −r P rP − r P " ⊕ ⊕ 3(x P cκ,sk + y P sκ,sk) + 4ask 3 xP cκ,sk + yP sκ,sk 62 ⊕ ⊕ Ap = cκ,sk − 3 + 3 r P 2 rP ⊕ 2 # (x P cκ,sk + y P sκ,sk) ⊕ ⊕ 6ask 5 , (118l) − r P " ⊕ x P c2κ,sk + y P s2κ,sk xP s2κ,sk yP c2κ,sk 13 ⊕ ⊕ Ap = 2ask 3 + −3 − − r P rP ⊕ # (x P sκ,sk y P cκ,sk)(x P cκ,sk + y P sκ,sk) ⊕ ⊕ ⊕ ⊕ +3askcκ,sk − 5 , (118m) r P ⊕ 2 2 2 2cκ,sk(x P cκ,sk + y P sκ,sk) + asksκ,sk (xP cκ,sk + yP sκ,sk) 23 ⊕ ⊕ 2 Ap = 3 + 3sκ,skask 5 − r P r P ⊕ ⊕ x P cκ,sk + y P sκ,sk 2 ⊕ ⊕ + 2cκ,sk 3 , (118n) rP 2(x P cκ,sk + y P sκ,sk) ask x P sκ,sk y P cκ,sk 33 ⊕ ⊕ ⊕ ⊕ Ap = cκ,sksκ,sk 3 − −3 − r P − 2r P ⊕ ⊕ 4cκ,sksκ,sk(xP cκ,sk + yP sκ,sk) + xP sκ,sk yP cκ,sk + 3 − 2rP 2 (x P cκ,sk + y P sκ,sk) ⊕ ⊕ + 3cκ,sksκ,sk 5 , (118o) r P " ⊕ # cκ,sksκ,skzP 1 1 x P cκ,sk + y P sκ,sk 43 ⊕ ⊕ Ap = 3 + 3 3ask 5 , (118p) 2 −r P rP − r P ⊕ ⊕ 2 " # sκ,skzP 1 1 x P cκ,sk + y P sκ,sk 53 ⊕ ⊕ Ap = 3 + 3 3ask 5 , (118q) 2 −r P rP − r P " ⊕ ⊕ 3(x P cκ,sk + y P sκ,sk) + 4ask 3 xP cκ,sk + yP sκ,sk 63 ⊕ ⊕ Ap = sκ,sk − 3 + 3 r P 2 rP ⊕

41 2 # (x P cκ,sk + y P sκ,sk) ⊕ ⊕ 6ask 5 , (118r) − r P " ⊕ # cκ,sk cκ,sk x P sκ,sk y P cκ,sk 14 ⊕ ⊕ Ap = 4askzP 3 + 3 + 3asksκ,sk −5 , (118s) − − r P rP r P " ⊕ ⊕ # 1 1 x P cκ,sk + y P sκ,sk 24 2 ⊕ ⊕ Ap = 2zP sκ,sk 3 3 + 3ask 5 , (118t) r P − rP r P ⊕ " ⊕ # 1 1 x P cκ,sk + y P sκ,sk 34 ⊕ ⊕ Ap = 2zP sκ,skcκ,sk 3 3 + 3ask 5 , (118u) − r P − rP r P ⊕ ⊕ " 2 # y yP 3asksκ,skzP 44 Ap = cκ,sk 3 3 5 , (118v) r P − rP − r P ⊕ ⊕ " 2 # 54 yP yP 3asksκ,skzP Ap = sκ,sk 3 3 5 , (118w) r P − rP − r P ⊕" ⊕ # 1 1 x P cκ,sk + y P sκ,sk 64 ⊕ ⊕ Ap = 3sκ,skzP 3 + 3 4ask 5 , (118x) −r P rP − r P " ⊕ ⊕ # sκ,sk sκ,sk x P sκ,sk y P cκ,sk 15 ⊕ ⊕ Ap = 4askzP 3 + 3 3askcκ,sk −5 , (118y) − − r P rP − r P ⊕" ⊕ # 1 1 x P cκ,sk + y P sκ,sk 25 ⊕ ⊕ Ap = 2zP sκ,skcκ,sk 3 3 + 3ask 5 , (118z) − r P − rP r P " ⊕ ⊕ # 1 1 x P cκ,sk + y P sκ,sk 35 2 ⊕ ⊕ Ap = 2zP cκ,sk 3 3 + 3ask 5 , (118aa) r P − rP r P ⊕ ⊕ " 2 # 45 xP xP 3askcκ,skzP Ap = cκ,sk 3 + 3 + 5 , (118ab) −r P rP r P ⊕ ⊕ " 2 # x xP 3askcκ,skzP 55 Ap = sκ,sk 3 + 3 + 5 , (118ac) −r P rP r P "⊕ ⊕ # 1 1 x P cκ,sk + y P sκ,sk 65 ⊕ ⊕ Ap = 3cκ,skzP 3 3 + 4ask 5 , (118ad) r P − rP r P " ⊕ ⊕ x P cκ,sk + y P sκ,sk + ask xP cκ,sk + yP sκ,sk 16 ⊕ ⊕ Ap = 2ask 3 + 3 − r P rP ⊕ 2 # (x P sκ,sk y P cκ,sk) ⊕ ⊕ +3ask −5 , (118ae) r P ⊕ x P c2κ,sk + y P s2κ,sk xP c2κ,sk + yP s2κ,sk 26 ⊕ ⊕ Ap = 3 + 3 − r P rP ⊕ (x P sκ,sk y P cκ,sk)(x P cκ,sk + y P sκ,sk) ⊕ ⊕ ⊕ ⊕ + 3asksκ,sk − 5 , (118af) r P ⊕ x P s2κ,sk y P c2κ,sk xP s2κ,sk yP c2κ,sk 36 ⊕ ⊕ Ap = −3 + −3 − r P rP ⊕ (x P sκ,sk y P cκ,sk)(x P cκ,sk + y P sκ,sk) ⊕ ⊕ ⊕ ⊕ + 3askcκ,sk − 5 , (118ag) r P " ⊕ # zP sκ,sk sκ,sk x P sκ,sk y P cκ,sk 46 ⊕ ⊕ Ap = 3 + 3 3askcκ,sk −5 , (118ah) 2 − r P rP − r P " ⊕ ⊕ # zP cκ,sk cκ,sk x P sκ,sk y P cκ,sk 56 ⊕ ⊕ Ap = 3 + 3 + 3asksκ,sk −5 , (118ai) − 2 − r P rP r P ⊕ ⊕

42 " x P sκ,sk y P cκ,sk xP cκ,sk + yP sκ,sk 66 ⊕ ⊕ Ap = 2 −3 + 3 − r P rP ⊕ # (x P sκ,sk y P cκ,sk)(x P cκ,sk + y P sκ,sk) ⊕ ⊕ ⊕ ⊕ 3ask − 5 , (118aj) − r P ⊕ for P = or P = .

The vectors D˜ $and D˜ are written:

s$ a h i ˜ sk ˜ i DP = µP DP , with P = or P = . (119) µ i=1,...,6 ⊕ $ The expressions for each coefficient are: " # x P sκ,sk y P cκ,sk xP sκ,sk yP cκ,sk D˜ 1 a ⊕ ⊕ , P = 2 sk −3 −3 (120a) r P − rP " ⊕ # x P cκ,sk + y P sκ,sk xP cκ,sk + yP sκ,sk ˜ 2 ⊕ ⊕ DP = sκ,sk 3 3 , (120b) − r P − rP " ⊕ # x P cκ,sk + y P sκ,sk xP cκ,sk + yP sκ,sk ˜ 3 ⊕ ⊕ DP = cκ,sk 3 3 , (120c) r P − rP " ⊕ # ˜ 4 cκ,skzP 1 1 DP = 3 3 , (120d) 2 r P − rP " ⊕ # ˜ 5 sκ,skzP 1 1 DP = 3 3 , (120e) 2 r P − rP " ⊕ # x P cκ,sk + y P sκ,sk xP cκ,sk + yP sκ,sk ˜ 6 ⊕ ⊕ DP = 2 3 3 . (120f) r P − rP ⊕ 5.4 SOLAR RADIATION PRESSURE With the notations used for the previous computations of the linearized dynamics, the SRP pseudo-potential function defined in the Section 2.2.3 is written as: q 2 2 2 SRP = α (x x) + (y y) + (z z) , (121) E − − − with α = %P Sc (see the definition of these quantities in the Section 2.2.3). The Sun m cartesian position in the geocentric inertial reference frame is interpolated with the data retrieved from the Horizon ephemeris system and the cartesian position pf the spacecraft is transformed into the osculating equinoctial orbital elements with the formulas given by the Equation 216.

The matrix A˜SRP is written: s ask h ij i A˜SRP = α A˜ . (122) µ PRS i=1,...,6, j=1,...,6 ⊕ The expressions for each coeﬃcient are:

h 2 i (x sκ,sk y cκ,sk) 3z + x (3x askcκ,sk) + y (3y asksκ,sk) A˜11 = ⊕ − ⊕ ⊕ ⊕ − , (123a) PRS − r3 ⊕

43 " 2 # sκ,sk x cκ,sk + y sκ,sk 2ask (x cκ,sk + y sκ,sk) ˜21 ⊕ ⊕ ⊕ ⊕ APRS = − + 2 , (123b) r 2ask r ⊕ ⊕ " 2 # cκ,sk x cκ,sk + y sκ,sk 2ask (x cκ,sk + y sκ,sk) ˜31 ⊕ ⊕ ⊕ ⊕ APRS = − + 2 , (123c) − r 2ask r ⊕ " # ⊕ cκ,skz 1 x cκ,sk + y sκ,sk ˜41 ⊕ ⊕ APRS = + 2 , (123d) − 2r 2ask r ⊕ " ⊕ # sκ,skz 1 x cκ,sk + y sκ,sk ˜51 ⊕ ⊕ APRS = + 2 , (123e) − 2r 2ask r ⊕ ⊕ 2 x cκ,sk + y sκ,sk 2ask (x cκ,sk + y sκ,sk) ˜61 ⊕ ⊕ ⊕ ⊕ APRS = − 2 2 , (123f) − askr − r ⊕ ⊕ ˜12 2ask APRS = [x s2κ,sk y c2κ,sk r ⊕ − ⊕ ⊕ # (x sκ,sk y cκ,sk)(x cκ,sk + y sκ,sk) +cκ,skask ⊕ − ⊕ ⊕ ⊕ , (123g) r2 ⊕ ˜22 1 APRS = cκ,sksκ,sk ask 2(x cκ,sk + y sκ,sk) r − ⊕ ⊕ ⊕ 2 # 1 (x cκ,sk + y sκ,sk) + (x sκ,sk y cκ,sk) askcκ,sksκ,sk ⊕ ⊕ , (123h) 2 ⊕ − ⊕ − r2 ⊕ 2 2 2 32 2sκ,sk(x cκ,sk + y sκ,sk) + cκ,skask 2 (x cκ,sk + y sκ,sk) A˜ = ⊕ ⊕ + askc ⊕ ⊕ , (123i) PRS − r κ,sk r3 ⊕ ⊕ 2 " # 42 cκ,skz x cκ,sk + y sκ,sk A˜ = ⊕ 1 + ask ⊕ ⊕ , (123j) PRS 2r r2 ⊕ " ⊕ # 52 cκ,sksκ,skz x cκ,sk + y sκ,sk A˜ = ⊕ 1 + ask ⊕ ⊕ , (123k) PRS 2r r2 ⊕ ⊕ " 2 # ˜62 cκ,sk 3 (x cκ,sk + y sκ,sk) APRS = (x cκ,sk + y sκ,sk) 2ask + 2ask ⊕ ⊕ , (123l) r 2 ⊕ ⊕ − r2 ⊕ ⊕ ˜13 2ask APRS = [ (x c2κ,sk + y s2κ,sk) r − ⊕ ⊕ ⊕ # (x sκ,sk y cκ,sk)(x cκ,sk + y sκ,sk) +sκ,skask ⊕ − ⊕ ⊕ ⊕ , (123m) r2 ⊕ 2 2 2 23 2cκ,sk(x cκ,sk + y sκ,sk) + sκ,skask 2 (x cκ,sk + y sκ,sk) A˜ = ⊕ ⊕ asks ⊕ ⊕ , (123n) PRS r − κ,sk r3 ⊕ ⊕ ˜33 1 APRS = cκ,sksκ,sk ask 2(x cκ,sk + y sκ,sk) r − − ⊕ ⊕ ⊕ 2 # 1 (x cκ,sk + y sκ,sk) + (x sκ,sk y cκ,sk) + askcκ,sksκ,sk ⊕ ⊕ , (123o) 2 ⊕ − ⊕ r2 " # ⊕ 43 cκ,sksκ,skz x cκ,sk + y sκ,sk A˜ = ⊕ 1 + ask ⊕ ⊕ , (123p) PRS 2r r2 ⊕ ⊕ 2 " # 53 sκ,skz x cκ,sk + y sκ,sk A˜ = ⊕ 1 + ask ⊕ ⊕ , (123q) PRS 2r r2 ⊕ ⊕ " 2 # ˜63 sκ,sk 3 (x cκ,sk + y sκ,sk) APRS = (x cκ,sk + y sκ,sk) 2ask + 2ask ⊕ ⊕ , (123r) r 2 ⊕ ⊕ − r2 ⊕ ⊕

44 " # 14 4askz x sκ,sk y cκ,sk A˜ = cκ,sk + sκ,skask ⊕ − ⊕ , (123s) PRS r − r2 ⊕ ⊕ 2 " # 24 2sκ,skz x cκ,sk + y sκ,sk A˜ = 1 + ask ⊕ ⊕ , (123t) PRS − r r2 ⊕ " ⊕ # 34 2sκ,skcκ,skz x cκ,sk + y sκ,sk A˜ = 1 + ask ⊕ ⊕ , (123u) PRS r r2 ⊕ ⊕ " 2 # 44 cκ,sk y asksκ,skz A˜ = + , (123v) PRS r −r r2 ⊕ ⊕ ⊕ " 2 # 54 sκ,sk y asksκ,skz A˜ = + , (123w) PRS r −r r2 ⊕ " ⊕ ⊕ # 64 sκ,skz x cκ,sk + y sκ,sk A˜ = 3 + 4ask ⊕ ⊕ , (123x) PRS r r2 ⊕ " ⊕ # 15 4askz x sκ,sk y cκ,sk A˜ = sκ,sk + cκ,skask ⊕ − ⊕ , (123y) PRS − r r2 ⊕ " ⊕ # 25 2sκ,skcκ,skz x cκ,sk + y sκ,sk A˜ = 1 + ask ⊕ ⊕ , (123z) PRS r r2 ⊕ ⊕ 2 " # 35 2sκ,skz x cκ,sk + y sκ,sk A˜ = 1 + ask ⊕ ⊕ , (123aa) PRS − r r2 ⊕ ⊕ " 2 # 45 cκ,sk x askcκ,skz A˜ = , (123ab) PRS r r − r2 ⊕ ⊕ ⊕ " 2 # 55 sκ,sk x askcκ,skz A˜ = + , (123ac) PRS r r r2 ⊕ ⊕ " ⊕ # 65 cκ,skz x cκ,sk + y sκ,sk A˜ = 3 + 4ask ⊕ ⊕ , (123ad) PRS − r r2 ⊕ ⊕ " 2 # ˜16 2ask (x sκ,sk y cκ,sk) APRS = (x cκ,sk + y sκ,sk) + ask ⊕ − ⊕ , (123ae) r − ⊕ ⊕ r2 ⊕ ⊕ x c2κ,sk + y s2κ,sk A˜26 = ⊕ ⊕ PRS r ⊕ (x sκ,sk y cκ,sk)(x cκ,sk + y sκ,sk) asksκ,sk ⊕ − ⊕ ⊕ ⊕ , (123af) − r3 ⊕ x s2κ,sk y c2κ,sk A˜36 = ⊕ − ⊕ PRS r ⊕ (x sκ,sk y cκ,sk)(x cκ,sk + y sκ,sk) + askcκ,sk ⊕ − ⊕ ⊕ ⊕ , (123ag) r3 " ⊕ # 46 z x sκ,sk y cκ,sk A˜ = sκ,sk + askcκ,sk ⊕ − ⊕ , (123ah) PRS 2r r2 ⊕ " ⊕ # 56 z x sκ,sk y cκ,sk A˜ = cκ,sk + asksκ,sk ⊕ − ⊕ , (123ai) PRS 2r − r2 ⊕ ⊕ ˜66 2 APRS = [x sκ,sk y cκ,sk r ⊕ − ⊕ ⊕ # (x sκ,sk y cκ,sk)(x cκ,sk + y sκ,sk) ask ⊕ − ⊕ ⊕ ⊕ . (123aj) − r3 ⊕

45 The vector D˜ SRP is written: s ask h i i D˜ SRP = α D˜ . (124) µ PRS i=1,...,6 ⊕ The expressions for each coeﬃcient are:

2ask(x sκ,sk y cκ,sk) D˜ 1 = ⊕ − ⊕ , (125a) PRS − r ⊕ 2 x cκ,sk + y sκ,sk D˜ = sκ,sk ⊕ ⊕ , (125b) PRS r ⊕ 3 x cκ,sk + y sκ,sk D˜ = cκ,sk ⊕ ⊕ , (125c) PRS − r ⊕ cκ,skz D˜ 4 = , (125d) PRS − 2r ⊕ sκ,skz D˜ 5 = , (125e) PRS − 2r ⊕ x cκ,sk + y sκ,sk ask D˜ 6 = 2 ⊕ ⊕ − . (125f) PRS − r ⊕

46 APPENDIX

AREFERENCE FRAMES

A.1 GEOCENTRIC INERTIAL REFERENCE FRAME The reference frame for the cartesian positions and velocities is the Geocentric Inertial Reference Frame. A deﬁnition of this reference frame can be found in [Vallado, 1997] named Geocentric Equatorial Coordinate System or Earth Center Inertial. The origin is located at the center of the Earth and the axis are deﬁned by:

• the axis (G, ~uZG ) in the direction of the Earth North pole,

• the axis (G, ~uXG ) in the direction of the intersection between the equatorial plane and the ecliptic plane (vernal equinox),

• the axis (G, ~uYG ) completes the basis,

and are illustrated on the Figure8. The reference frame (G, ~uX , ~uY , ~uZ ) is denoted G G G RG and the associated basis . BG

Sun ~uZG Ecliptic plane Earth rotation

Earth

G Equatorial plane ~uYG

~uXG Vernal equinox

Figure 8 – Geocentric inertial reference frame

A point in space is represented by its three cartesian coordinates x, y and z or its three spherical coordinates r, α (right ascension) and δ (declination). The transformations between the cartesian and spherical coordinates are illustrated on the Figure9 and given by:

x = r cos δ cos α, (126a)

47 y = r cos δ sin α, (126b) z = r sin δ. (126c)

The inverse relationships are:

q r = x2 + y2z2, (127a) y α = atan , (127b) x z ! δ = atan , (127c) √x2 + y2

where α is chosen such that α [ 90◦, 90◦] for x > 0 and α [90◦, 270◦] for x < 0. ∈ − ∈

~uZG

δ G

~uYG

x α y

~uXG Equatorial plane

Figure 9 – Right ascension α and declination δ of a point in space

A.2 ROTATING GEOCENTRIC REFERENCE FRAME The rotating geocentric reference frame is named by the reference [Vallado, 1997] Earth Centered Earth Fixed reference frame. This is a reference frame whose origin is located at the center of the Earth and whose axis are deﬁned by: • an axis in the direction of the Earth North pole, • an axis in the direction of the Greenwich meridian, • an axis that completes the basis. This reference frame is illustrated on the Figure 10.

48 ~uZG

G ~uYG

Θ λ α Greenwich meridian

~uXG

Figure 10 – Rotating Geocentric Reference Frame

In this reference frame, a point is deﬁned by its radius r, its geographical longitude λ and its geographical latitude ϕ such that:

λ = α Θ(t), (128a) − ϕ = δ. (128b)

Θ(t) is the right ascension of the Greenwich meridian (angle between the axis (G, ~uXG ) of the Figure8 and the Greewich meridian). It is thus the Earth rotation angle at time t and is sometimes called sidereal time. Assuming that the Earth rotation rate ω is ⊕ constant, Θ(t) veriﬁes: Θ(t) = Θ(t0) + ω (t t0). (129) ⊕ − If [x y z]T denotes the position of a spacecraft in the geocentric inertial reference frame, its radius, latitude and longitude are computed with the following conversion formulas: q r = x2 + y2 + z2, (130a) x cos Θ(t) + y sin Θ(t) cos λ = , (130b) √x2 + y2 y cos Θ(t) x sin Θ(t) sin λ = − , (130c) √x2 + y2 √x2 + y2 cos ϕ = , (130d) √x2 + y2 + z2 z sin ϕ = . (130e) √x2 + y2 + z2

49 A.3 LOCAL ORBITAL FRAME It is possible to deﬁne a coordinates system linked to the position of the spacecraft on its orbit. This reference frame is labeled RTN or RSW as in [Vallado, 1997]. The axis are deﬁned by:

• the axis ~uN is in the direction of the angular momentum ~h,

• the axis ~uR is in the direction Earth-spacecraft,

• the axis ~uT completes the basis,

and are illustrated on the Figure 11. The reference frame (S, ~uR, ~uT , ~uN ) is denoted ROL and the associated basis . BOL

~uZG

~uN ~uT

~uR S

G ~uYG

~uXG

Figure 11 – Local orbital frame

The local orbital frame is deﬁned from the inertial geocentric reference frame thanks to three rotations (see the Figure 12):

• rotation about the axis (G, ~uZG ) with an angle Ω, • rotation about the axis (G, ~un) with an angle i,

• rotation about the axis (G, ~uh) with an angle ω + ν,

where ~un is a unit vector in the direction of the intersection between the equatorial plan and the orbital plane, and ~uh is a unit vector perpendicular to the orbit plane. The Figure 13 depicts these rotations in three dimensions and the Figure 14 the associated plane rotations. If ~σ is a vector whose coordinates are:   σx   ~σ = σy , (131) σz G B in the geocentric inertial frame and:   σr   ~σ = σt , (132) σt OL B 50 ~uZ G ~uT ~uN ~uR ~u ~uv h S

r ν ~um i ~ue Equatorial plane i Ω G

ω ~uYG

Ω i

~un ~uXG

Orbital plane

Figure 12 – Local orbital frame.

in the local orbital frame, the coordinates transformation is given by:           σx cos Ω sin Ω 0 1 0 0 cos(ω + ν) sin(ω + ν) 0 σr − − σy = sin Ω cos Ω 0 0 cos i sin i sin(ω + ν) cos(ω + ν) 0 σt ,      −      σz 0 0 1 0 sin i cos i 0 0 1 σt G OL B B cos (ω + ν) cos Ω cos i sin (ω + ν) sin Ω  − = cos (ω + ν) sin Ω + cos i sin (ω + ν) cos Ω sin i sin (ω + ν)    cos i cos (ω + ν) sin Ω sin (ω + ν) cos Ω sin i sin Ω σr − − cos i cos (ω + ν) cos Ω sin (ω + ν) sin Ω sin i cos Ω σt . − −    sin i cos (ω + ν) cos i σt OL B (133) It is also possible to deﬁne a local orbital frame from the trajectory tangent vector and the angular momentum vector, denoted NTW . When the orbit is circular, the RTN and NTW are the same.

A.4 EQUINOCTIAL REFERENCE FRAME

The equinoctial reference frame = (~uh, ~up, ~uq) is deﬁned by: BEQX • a rotation of the geocentric inertial reference frame about the axis (G, ~uZG ) with an angle Ω,

• a rotation about the new (G, ~un) axis (pointing in the ascending node direction) with an angle i,

51 Ω ~uZ ~uh ~uZG

Ω i

~uT ~uv ω + ν

i ~um

Ω

Image created by L. Sofía Urbina, LAAS-CNRS

~uYG

~uXG Ω ω + ν

i ~uR

~un

Figure 13 – Three dimensions rotation between the geocentric inertial reference frame and the local orbital frame.

~uYG ~uZG ~uv ~um ~uh ~uf ~uR

~un ~uv ~uT ~ue Ω i ω ν ν Ω i ω

~uXG ~um ~un ~uZG ~un ~uh ~uN Figure 14 – Planar rotations between the geocentric inertial reference frame and the local orbital frame.

52 • a rotation about the axis (G, ~uh) with an angle Ω, − and is denoted = (G, ~uh, ~up, ~uq). REQX The Figure 15 shows the equinoctial reference frame, the Figure 16 the three dimen- sional rotation and the Figure 17 the planar rotations for the transformations between the geocentric inertial reference frame and the equinoctial reference frame.

~uZG ~uv ~u h S ~uq r i ~um Ω Plan équatorial i Ω G

~uYG

Ω i Ω

~un ~uXG

~u Plan orbital p

Figure 15 – Equinoctial reference frame.

If ~σ is a vector whose coordinates are:   σx   ~σ = σy , (134) σz G B in the inertial geocentric reference frame and:   σp   ~σ = σq  , (135) σh EQX B in the equinoctial reference frame, the transformations are written:

53 Ω

~uh ~uZG

Ω i

~uq ~uv

Ω

i ~um

Ω

Image created by L. Sofía Urbina, LAAS-CNRS

~uYG

~uXG Ω Ω

i ~up

~un

Figure 16 – Three dimensions rotation between the geocentric reference frame and the equinoctial reference frame.

~uYG ~uZG ~uq ~um ~uh ~uv

~un ~uv ~un Ω i Ω

Ω i Ω ~uXG ~um ~up

~uZG ~un ~uh Figure 17 – Planar rotations between the geocentric reference frame and the equinoctial reference frame.

54           σx cos Ω sin Ω 0 1 0 0 cos Ω sin Ω 0 σp − σy =  sin Ω cos Ω 0 0 cos i sin i sin Ω cos Ω 0 σq  ,   −   −      σz 0 0 1 0 sin i cos i 0 0 1 σh G EQX B B  2 2    cos i sin Ω + cos Ω cos Ω sin Ω (1 cos i) sin i sin Ω σp 2 − 2 − = cos Ω sin Ω (1 cos i) sin Ω + cos i cos Ω sin i cos Ω  σq  .  −    sin i sin Ω sin i cos Ω cos i σh EQX − B (136)

BCONVERSION FORMULASWITHTHE CLASSICAL ORBITAL ELEMENTS This section presents the conversion formulas between the cartesian positions and velocities of a spacecraft given in the inertial geocentric reference frame and the classical orbital elements. This formulas are derived from [Battin, 1999], [Vallado, 1997] and [Lyon, 2004].

The position vector is: x   ~r = y , (137) z G B and its norm is: q r = x2 + y2 + z2. (138) The velocity vector is: x˙  ˙   ~v = ~r = y˙ , (139) z˙ G B and its norm is: q v = x˙ 2 +y ˙2 +z ˙2. (140)

B.1 KEPLERIAN MOTION INTEGRALS From the keplerian motion dynamics equation, some motion integrals can be derived. The speciﬁc angular momentum:     yz˙ zy˙ hx − ~h = ~r ~v = zx˙ xz˙ = hy , (141) ×  −    xy˙ yx˙ hz G G − R R is constant and its norm is: q 2 2 2 h = hx +y +hz, q (142) = (yz˙ zy˙)2 + (zx˙ xz˙)2 + (xy˙ yx˙)2. − − −

55 The speciﬁc angular momentum is perpendicular to the position and the velocities of the spacecraft and is a constant vector. Therefore, the spacecraft trajectory lies in a plane called orbital plane. The unit vector perpendicular to this plane is:   hx 1   ~uh = q hy , (143) h2 +2 +h2 x y z hz G R

The total energy conservation gives the following equation:

µ v2 µ = . (144) −2a 2 − r The eccentricity vector is deﬁned by:

~v ~h ~r ~e = × , (145) µ − r

and is in the direction Earth-perigee. ~ue is the unit vector in this direction. the norm e of the eccentricity vector is the eccentricity of the ellipse.

B.2 ANOMALIES TRANSFORMATIONS Le calcul de l’anomalie excentrique à partir de la l’anomalie vraie est eﬀectué selon : The eccentric anomaly is computed from the true on with the following relationships:

sin ν√1 e2 sin E = − , (146a) 1 + e cos ν e + cos ν cos E = , (146b) 1 + e cos ν s E 1 e ν tan = − tan . (146c) 2 1 + e 2

The Equations (146) are inverted as:

sin E√1 e2 sin ν = − , (147a) − e cos E 1 e cos E − cos ν = − , (147b) e cos E 1 s − ν 1 + e E tan = tan . (147c) 2 1 e 2 − The mean anomaly is computed through the Kepler equation:

M = E e sin E. (148) −

56 B.3 COMPUTATION OF THE ORBITAL ELEMENTSFROMTHE CARTESIAN POSITIONSAND VELOCITIES The equation (144) leads to: 1 a = . (149) 2 x˙ 2 +y ˙2 +z ˙2 √x2 + y2 + z2 − µ

The equation (145) leads to: v u !2 !2 !2 u z hx y˙ hy x˙ y hz x˙ hx z˙ x hy z˙ hz y˙ e = t + − + + − + + − , (150) r µ r µ r µ

and the vector ~ue is the unit vector in the direction of ~e. L’inclinaison est l’angle entre le plan de l’équateur et le plan de l’orbite, ou encore

l’angle entre le vecteur moment cinétique et le vecteur ~uZG En notant ~uh le vecteur uni- taire perpendiculaire au plan de l’orbite, c’est-à-dire dans la direction du vecteur moment cinétique ~h, il vient : the inclination os the angle between the equator and the orbital planes, as well as the

angle between the angular momentum and the ~uZG vectors. Denoting ~uh the unit vector in the angular momentum direction perpendicular to the orbital plane, the cosine of the inclination is: hz cos i = ~uZ ~uh = . (151) G · h As i [0, π], the Equation (151) can be inverted as: ∈ ! hz i = arccos (152) h

The angle Ω is the angle between the direction (G, ~uXG ) and the direction of the vector ~n, intersection of the orbital and the equatorial planes. ~n veriﬁes thus:   hy − ~n = ~uZ ~h =  hx  , (153) G ×   0 G R with the unit vector:  h  ~n y 1 −  ~un = = q hx . (154) ~n 2 2   hx + hy 0 k k G R The cosine of Ω is: hy cos Ω = ~uX ~un = . (155) G q 2 2 · − hx + hy As Ω [0, 2π], sin Ω has to be computed in order to recover the value of Ω. Deﬁning: ∈   hx 1 −  ~um = ~uZG ~un = q hy , (156) × 2 2 −  hx + hy 0 G B 57 as depicted on the Figures 13 and 14, the sine of the right ascension of the ascending node is: hx sin Ω = ~uX ~um = (157) G q 2 2 − · hx + hy

As ω is the angle between ~n et ~e, it comes : y hz x˙ hx z˙ x hy z˙ hz y˙ hx r + −µ + hy r + −µ cos ω = ~ue ~un = − (158) q 2 2 · e hy + hx

As ω [0, 2π], sin ω has to be computed. With the deﬁnition of ~uf : ∈

~uf = ~uh ~ue, (159) × depicted on the Figures 13 and 14, sin ω is computed as:

sin ω = ~ue ~uv · " ! !# 1 z hx y˙ hy x˙ y hz x˙ hx z˙ = hy hy + − + hz + − q 2 2 he hx + hy − r µ r µ (160) " ! !#! x hy z˙ hz y˙ z hx y˙ hy x˙ hx hz + − + hy + − − − r µ r µ

The local orbital frame (~uR, ~uT , ~uN ) is deﬁned such that: ~r ~uR = , r ~uZ = ~uh, (161) ~r ~uT = ~uZ ~uR = ~uh . × × r

The true anomalie ν is the angle between the vector ~e and the vector ~uR. From this, it is possible to write: ~r cos ν = ~ue ~uR = ~ue · · r " ! ! !# (162) 1 x hy z˙ hz y˙ y hz x˙ hx z˙ z hx y˙ hy x˙ = x + − + y + − + z + − −er r µ r µ r µ

As ν [0, 2π], sin ν has to be computed: ∈ ~r ! sin ν = ~ue ~uT = ~ue ~uZ ~uR = ~ue ~uh , − · − · × − · × r " ! ! 1 x hy z˙ hz y˙ y hz x˙ hx z˙ = hyz hzy + − + hzx hxz + − (163) rhe − r µ − r µ !# z hx y˙ hy x˙ + hxy hyx + − . − r µ

From the value of ν computed through the Equations (162) and (163), the eccentric anomaly is computed with the Equation (146) and the mean anomaly through the Kepler Equation.

58 B.4 COMPUTATION OF THE CARTESIAN POSITIONSAND VE- LOCITIESFROMTHE CLASSICAL ORBITAL ELEMENTS

B.4.1 COMPUTATION OF THE POSITION In the local orbital frame, the position vector components read: r   ~r = 0 . (164) 0 OL B Applying the rotation formula (133), it follows:

x cos (ω + ν) cos (Ω) cos (i) sin (ω + ν) sin (Ω)    −  ~r = y = r cos (ω + ν) sin (Ω) + cos (i) sin (ω + ν) cos (Ω) . (165) z sin (i) sin (ω + ν) G G R R Replacing r by its expression in terms of the classical orbital elements, the cartesian position components in the geocentric inertial reference frame are: a(1 e2) x = − (cos (ω + ν) cos (Ω) cos (i) sin (ω + ν) sin (Ω)) (166a) 1 + e cos ν −

a(1 e2) y = − (cos (ω + ν) sin (Ω) + cos (i) sin (ω + ν) cos (Ω)) (166b) 1 + e cos ν a(1 e2) z = − (sin (i) sin (ω + ν)) (166c) 1 + e cos ν

B.4.2 COMPUTATION OF THE VELOCITY The velocity in the inertial geocentric reference frame is computed by derivating the position on the inertial geocentric reference frame:

d~r

~v = (167) dt G R The total time derivative is decomposed in the partial derivatives with respect to the orbital elements: ∂~r ∂~r ∂~r ∂~r ∂~r ∂~r ~v = a˙ + e˙ + i˙ + Ω˙ + ω˙ + ν.˙ (168) ∂a ∂e ∂i ∂Ω ∂ω ∂ν Or, les éléments orbitaux classiques définissent une ellipse osculatrice, c’est-à-dire l’ellipse que le satellite suivrait dans le cas où les perturbations disparaissaient instantanément. Ainsi, les éléments orbitaux ci-dessus sont à considérer dans le cas képlérien, et seul ν est à dérivée temporelle non-nulle. On peut alors réécrire : As the orbital elements define an osculating ellipse that is tangent to the trajectory, the velocity of the spacecraft on its real trajectory is the velocity of the spacecraft if it would fly on the osculating ellipse. Therefor, the classical elements are to be considered in the keplerian case, and all the elements have a zero time derivative except the anomaly: ∂~r ~v = ν,˙ (169) ∂ν

59 with the time derivative of the true anomaly given by the reference [Battin, 1999]: q µa (1 e2) (1 + e cos (ν))2 ν˙ = − . (170) a2(1 e2)2 − Therefore, the cartesian velocity in the inertial geocentric reference frame is given by:

 s µ x˙ =  e sin ν sin(ω + ν) + (1 + e cos ν) cos(ω + ν) sin Ω cos i a(1 e2) −  + (1 + e cos ν) sin(ω + ν) e sin ν cos(ω + ν)  (171a) −

 s µ x˙ =  (1 + e cos ν) sin(ω + ν) + e sin ν sin(ω + ν) cos Ω a(1 e2) −  (1 + e cos ν) cos(ω + ν) + e sin ν sin(ω + ν) cos Ω cos i (171b) −

s µ z˙ = e sin ν sin(ω + ν) sin i + (1 + e cos ν) cos(ω + ν) sin i (171c) a(1 e2) − CCONVERSION FORMULASWITHTHE EQUINOC- TIAL ORBITAL ELEMENTS

C.1 DEFINITION OF THE EQUINOCTIAL ORBITAL ELEMENTS FROMTHE CLASSICAL ONES La déﬁnition des éléments orbitaux équinoxiaux présentée dans la section 3.2.2 page 14 est rappelée ici : The deﬁnition of the equinoctial orbital elements is recalled here:   a   e e ω  x = cos( + Ω)   ey = e sin(ω + Ω) (172)  ix = tan(i/2) cos(Ω)    iy = tan(i/2) sin(Ω)   equinoctial anomaly

Three anomalies are used

• true equinoctial anomaly: νQ = Ω + ω + ν ;

• mean equinoctial anomaly: MQ = Ω + ω + M ;

• eccentric equinoctial anomaly: EQ = Ω + ω + E

60 C.2 COMPUTATION OF THE CLASSICAL ORBITAL ELEMENTS FROMTHE EQUINOCTIAL ONES The eccentricity is computed as the norm of the eccentricity vector: q 2 2 e = ex + ey. (173)

The deﬁnition of the inclination vector leads to: i tan2 = i2 + i2. (174) 2 x y Hence: i 1 tan2 1 i2 i2 cos i = − 2 = − x − y , (175) 2 i 2 2 1 + ix + iy 1 + tan 2 and: i q 2 2 2 tan 2 ix + iy sin i = 2 = . (176) 2 i 2 2 1 + ix + iy 1 + tan 2 As i [0, π], the inclination reads: ∈ 1 i2 i2 ! i − x − y . = arccos 2 2 (177) 1 + ix + iy The right ascension of the ascending node is computed from the eccentricity vector:

ix cos Ω = , (178a) q 2 2 ix + iy

iy sin Ω = . (178b) q 2 2 ix + iy

If the inclination is zero, ix and iy are also zero and Ω is not deﬁned. It comes from the eccentricity vector:

ex cos(Ω + ω) = , (179a) q 2 2 ex + ey ey sin(Ω + ω) = , (179b) q 2 2 ex + ey then with trigonometric expansions:

ex cos Ω cos ω sin Ω sin ω = , (180a) q 2 2 − ex + ey ey sin Ω cos ω + cos Ω sin ω = . (180b) q 2 2 ex + ey

The system (180) of unknowns cos ω and sin ω has only one solution because the deter- minant is cos2 Ω + sin2 Ω = 1. The solution is:

ex cos Ω + ey sin Ω exix + eyiy cos ω = = , (181a) q 2 2 q 2 2q 2 2 ex + ey ex + ey ix + iy

61 ey cos Ω ex sin Ω eyix exiy sin ω = − = − . (181b) q 2 2 q 2 2q 2 2 ex + ey ex + ey ix + iy

If the eccentricity is zero, ex and ey are also zero and ω is not deﬁned. Expanding the deﬁnition of the true equinoctial anomaly leads to:

cos νQ = cos(Ω + ω) cos ν sin(Ω + ω) sin ν, − ex ey = sin ν cos ν, (182a) q 2 2 q 2 2 ex + ey − ex + ey

sin νQ = sin(Ω + ω) cos ν + cos(Ω + ω) sin ν, ey ex = cos ν + sin ν. (182b) q 2 2 q 2 2 ex + ey ex + ey

Hence, if the eccentricity is not zero:

ex cos νQ + ey sin νQ cos ν = , (183a) q 2 2 ex + ey

ex sin νQ ey cos νQ sin ν = − . (183b) q 2 2 ex + ey

If the eccentricity is zero but the inclination is not zero, it is possible to deﬁne the position of the spacecraft on its orbit by the angle ω + ν. The deﬁnition of the true equinoctial anomaly leads to:

cos νQ = cos Ω cos(ω + ν) sin Ω sin(ω + ν), − ix iy = cos(ω + ν) sin(ω + ν), (184a) q 2 2 q 2 2 ix + iy − ix + iy

sin νQ = sin Ω cos(ω + ν) + cos Ω sin(ω + ν),

iy ix = cos(ω + ν) + sin(ω + ν). (184b) q 2 2 q 2 2 ix + iy ix + iy (184c)

The solution of this system is:

ix cos νQ + iy sin νQ cos(ω + ν) = , (185a) q 2 2 ix + iy

ix sin νQ iy cos νQ sin(ω + ν) = − . (185b) q 2 2 ix + iy

Doing the same calculations with the mean and the eccentric anomalies leads to:

ex cos EQ + ey sin EQ cos E = , (186a) q 2 2 ex + ey

ex sin EQ ey cos EQ sin E = − . (186b) q 2 2 ex + ey

62 ix cos EQ + iy sin EQ cos(ω + E) = , (187a) q 2 2 ix + iy

ix sin EQ iy cos EQ sin(ω + E) = − . (187b) q 2 2 ix + iy

ex cos MQ + ey sin MQ cos M = , (188a) q 2 2 ex + ey

ex sin MQ ey cos MQ sin M = − . (188b) q 2 2 ex + ey

ix cos MQ + iy sin MQ cos(ω + M) = , (189a) q 2 2 ix + iy

ix sin MQ iy cos MQ sin(ω + M) = − . (189b) q 2 2 ix + iy

Using the Equation (183) in the Equation (146) leads to:

ex cos νQ + ey sin νQ e + q 2 2 2 2 ex + ey ex + ey + ex cos νQ + ey sin νQ cos E = = q , (190a) ex cos νQ + ey sin νQ 2 2 1 + e ex + ey(1 + ex cos νQ + ey sin νQ) q 2 2 ex + ey

ex sin νQ ey cos νQ q 2 2 − 1 ex ey q 2 2 q 2 2 ex + ey − − (ex sin νQ ey cos νQ) 1 ex ey sin E = = q − − − . (190b) ex cos νQ + ey sin νQ 2 2 1 + e ex + ey(1 + ex cos νQ + ey sin νQ) q 2 2 ex + ey

Identifying cos E et sin E with the (186), it follows:

2 2 ex + ey + ex cos νQ + ey sin νQ ex cos EQ + ey sin EQ = , (191a) 1 + ex cos νQ + ey sin νQ q 2 2 (ex sin νQ ey cos νQ) 1 ex ey ex sin EQ ey cos EQ = − − − , (191b) − 1 + ex cos νQ + ey sin νQ and: 2 2 q 2 2 ex ex + ey + ex cos νQ + ey sin νQ ey(ex sin νQ ey cos νQ) 1 ex ey E − − − − , cos Q = 2 2 (ex + ey)(1 + ex cos νQ + ey sin νQ) (192a) q 2 2 2 2 ex(ex sin νQ ey cos νQ) 1 ex ey + ey ex + ey + ex cos νQ + ey sin νQ E − − − . sin Q = 2 2 (ex + ey)(1 + ex cos νQ + ey sin νQ) (192b)

63 Using the Equation (186) in the Equation (147) leads to:

ex cos EQ + ey sin EQ e q − 2 2 2 2 ex + ey ex + ey ex cos EQ ey sin EQ cos ν = = q − − , (193a) ex cos EQ + ey sin EQ 2 2 e 1 ex + ey(ex cos EQ + ey sin EQ 1) q 2 2 − ex + ey −

ex sin EQ ey cos EQ − √1 e2 q 2 2 q 2 2 ex + ey − (ex sin EQ ey cos EQ) 1 ex ey sin ν = = q − − − . (193b) − ex cos EQ + ey sin EQ − 2 2 e ex + ey(ex cos EQ + ey sin EQ 1) q 2 2 − ex + ey

Identifying cos ν and sin ν with the equation (183), it follows:

2 2 ex + ey ex cos EQ ey sin EQ ex cos νQ + ey sin νQ = − − , (194a) ex cos EQ + ey sin EQ 1 − q 2 2 (ex sin EQ ey cos EQ) 1 ex ey ex sin νQ ey cos νQ = − − − , (194b) − − ex cos EQ + ey sin EQ 1 − and: 2 2 q 2 2 ex ex + ey ex cos EQ ey sin EQ + ey(ex sin EQ ey cos EQ) 1 ex ey ν − − − − − , cos Q = 2 2 (e + e )(ex cos EQ + ey sin EQ 1) x y − (195a) q 2 2 2 2 ex(ex sin EQ ey cos EQ) 1 ex ey + ey ex + ey ex cos EQ ey sin EQ ν − − − − − − . sin Q = 2 2 (e + e )(ex cos EQ + ey sin EQ 1) x y − (195b)

The mean equinoctial anomaly is computed from the eccentric one through the Kepler equation expressed by means of the equinoctial orbital elements:

EQ + ey cos EQ ex sin EQ = MQ. (196) −

C.3 CONVERSIONFROMTHE CARTESIAN POSITIONAND VE- LOCITYTOTHE EQUINOCTIAL ORBITAL ELEMENTS As for the classical orbital elements, the Equation (144) gives the semi-major axis as: 1 a = (197) 2 x˙ 2 +y ˙2 +z ˙2 √x2 + y2 + z2 − µ

The Equation (136) gave the transformation between the coordinates in the equinoctial and the inertial geocentric reference frames in terms of the classical orbital elements. Using

64 the deﬁnition of the equinoctial orbital elements, the transformation (136) is rewritten as:

   2 2    σ 1 + i i 2ixiy 2iy σ x 1 x − y − p σ   i i i2 i2 i  σ   y = 2 2  2 x y 1 x + y 2 x   q  (198) 1 + ix + iy − 2 2 σz 2iy 2ix 1 i i σh G x y EQX B − − − B This transformation matrix gives the coordinates of the equinoctial basis in the inertial geocentric one. In particular:   2iy 1 − ~u  i  , h = 2 2  2 x  (199) 1 + ix + iy 2 2 1 ix iy G − − B which allows to write the coordinates of the vector ~uh in the inertial geocentric reference frame:

2iy u ~u ~u , hx = h XG = −2 2 (200a) · 1 + ix + iy 2ix u ~u ~u , hy = h YG = 2 2 (200b) · 1 + ix + iy 1 i2 i2 u ~u ~u − x − y . hz = h ZG = 2 2 (200c) · 1 + ix + iy

With the Equation 199, the inclination vector is computed as:

uhy hy ix = = , (201a) 1 + uhz hz + h

uhx hx iy = = , (201b) −1 + uhz −hz + h where h is the norm of the vector ~h. With the Equation (198), the basis vectors are written:

1 + i2 i2 1 x − y ~u  i i  , p = 2 2  2 x y  (202a) 1 + ix + iy 2iy − G  B 2ixiy 1 ~u  i2 i2 . q = 2 2 1 + x y (202b) 1 + ix + iy − 2ix G B Combining the Figures 14 and 17, Ω + ω appears to be the angle between the vectors ~up and ~ue on one hand and ~uq and ~uf on the other hand (see the Figure 18). Hence: cos(Ω + ω) = ~ue ~up, sin(Ω + ω) = ~ue ~uq, (203a) · · and the components of the eccentricity vector are:

ex = e~ue ~up = ~e ~up, (204a) · · ey = e~ue ~uq = ~e ~uq, (204b) · ·

65 ~up ~uR ~uv

~ue

~uf

Ω ν ω ~un ω ~uT ν Ω ~up

~uh

~uN

Figure 18 – Rotations planes dans le plan orbital autour du vecteur ~uh.

and:

" ! 2 2 ! 1 x hy z˙ hz y˙ hy hx ex = 2 h2 + − 1 + hx y r µ h h 2 h h 2 1 + 2 + 2 − ( z + ) − ( z + ) (hz+h) (hz+h) z hx y˙ hy x˙ y hz x˙ hx z˙  2hx r + −µ 2hxhy r + −µ + 2  (205a) hz + h − (hz + h)

" ! 2 2 ! 1 y hz x˙ hx z˙ hy hx ey = 2 h2 + − 1 hx y r µ h h 2 h h 2 1 + 2 + 2 − − ( z + ) − ( z + ) (hz+h) (hz+h) z hx y˙ hy x˙ x hy z˙ hz y˙  2hy r + −µ 2hxhy r + −µ + 2  (205b) hz + h − (hz + h)

From the Figure 18, the true anomaly appears to be νQ = Ω + ω + ν the angle between the vectors ~up and ~uR on one hand, and ~uq and ~uT on the other hand. Therefore:

cos νQ = ~uR ~up, (206a) · sin νQ = ~uR ~uq, (206b) · and:

" 2 2 ! 1 hy hx cos νQ = 2 h2 x 1 + hx y h h 2 h h 2 r 1 + 2 + 2 ( z + ) − ( z + ) (hz+h) (hz+h) # 2hxz 2hxhyy + 2 , (207a) −hz + h (hz + h)

66 " 2 2 ! 1 hy hx sin νQ = 2 h2 y 1 hx y h h 2 h h 2 r 1 + 2 + 2 − ( z + ) − ( z + ) (hz+h) (hz+h) # 2hyz 2hxhyx + + 2 . (207b) hz + h (hz + h)

C.4 CONVERSIONFROMTHE EQUINOCTIAL ORBITAL ELE- MENTSTO CARTESIAN POSITIONAND VELOCITY The transformation of the equinoctial orbital elements to the cartesian position and velocity in the inertial geocentric reference frame uses the expression of the position vector in the equinoctial frame and the rotation matrix 136. The radius is expressed in terms of the equinoctial orbital elements as (see the reference [McClain, 1977]): a(1 e2 e2) r = − x − y , (208) 1 + ex cos νQ + ey sin νQ This expression of r uses the true equinoctial anomaly. As the sixth parameter of the state vector is the mean longitude one, it is mandatory to transform νQ in `MΘ. To this end, the Kepler equation expressed in equinoctial orbital elements is solved with the approximated method of the AppendixD:

νQ = Ω + ω + ν, = Ω + ω + M + 2e sin(M), (209) = MQ + 2e sin(M),

= `MΘ + Θ + 2e sin(M).

At the order 0 in eccentricity:

cos νQ = cos MQ = cos(`MΘ + Θ), (210a)

sin νQ = sin MQ = sin(`MΘ + Θ), (210b)

and the radius reads then: a(1 e2 e2) r = − x − y . (211) 1 + ex cos(`MΘ + Θ) + ey sin(`MΘ + Θ) The radius is expressed by means of the eccentric equinoctial anomaly as:

r = a(1 ex cos EQ ey sin EQ), (212) − − According to the Figure 18:

~uR = cos(Ω + ω + ν)~up + sin(Ω + ω + ν)~uq, (213) = cos(`MΘ + Θ)~up + sin(`MΘ + Θ)~uq.

Therefore :  `  a(1 e2 e2) cos( MΘ + Θ) x y   ~r = − − sin(`MΘ + Θ) . (214) 1 + ex cos(`M + Θ) + ey sin(`M + Θ) Θ Θ 0 EQX B 67 The coordinates of the position vector in the equinoctial reference frame are transformed in the inertial geocentric one with the rotation matrix (198), what leads to:

x   ~r = y z G B a(1 e2 e2) 1 − x − y = 2 2 (215) 1 + ex cos(`MΘ + Θ) + ey sin(`MΘ + Θ) 1 + ix + iy  2 2    1 + ix iy 2ixiy 2iy cos(`MΘ + Θ)  − 2 2 −     2ixiy 1 ix + iy 2ix  sin(`MΘ + Θ) , × − 2 2 2iy 2ix 1 i i 0 x y EQX − − − B and thus: 2 2 2 2 a(1 e e ) (1 + i i ) cos(`M + Θ) + 2ixiy sin(`M + Θ) − x − y x − y Θ Θ x = , (216a) 2 2 1 + ex cos(`MΘ + Θ) + ey sin(`MΘ + Θ) 1 + ix + iy

2 2 2 2 a(1 e e ) 2ixiy cos(`M + Θ) + (1 i + i ) sin(`M + Θ) − x − y Θ − x y Θ y = , (216b) 2 2 1 + ex cos(`MΘ + Θ) + ey sin(`MΘ + Θ) 1 + ix + iy

2 2 a(1 e e ) 2iy cos(`M + Θ) 2ix sin(`M + Θ) − x − y Θ − Θ z = . (216c) 2 2 1 + ex cos(`MΘ + Θ) + ey sin(`MΘ + Θ) 1 + ix + iy The velocity in the inertial geocentric reference frame is computed by derivation of the position in the inertial reference frame:

d~r

~v = (217) dt G R The total time derivative is decomposed in the partial derivatives with respect to the equinoctial orbital elements:

∂~r ∂~r ∂~r ∂~r ∂~r ∂~r d(`MΘ + Θ) ~v = a˙ + e˙x + + e˙y + i˙x + i˙y + . (218) ∂a ∂ex ∂ey ∂ix ∂iy ∂(`MΘ + Θ) dt

However `MΘ+Θ = MQ, the velocity reads:

∂~r ∂~r ∂~r ∂~r ∂~r ∂~r ~v = a˙ + e˙x + + e˙y + i˙x + i˙y + M˙ Q. (219) ∂a ∂ex ∂ey ∂ix ∂iy ∂MQ

Following the argumentation developed for the computation of the velocity with the classical orbital elements, the velocity is computed as:

∂~r ~v = M˙ Q, (220) ∂MQ

68 with r µ M˙ Q = n = . (221) a3 Hence :

r µ a(1 e2 e2) x − x − y ˙ = 3 2 a 2 2 1 + ex cos(`MΘ + Θ) + ey sin(`MΘ + Θ) 1 + ix + iy  2 2  (1+i i ) sin(`M +Θ)+2ixiy cos(`M +Θ) 1+ex cos(`M +Θ)+ey sin(`M +Θ) × − x − y Θ Θ Θ Θ  2 2 ey cos(`M + Θ) ex sin(`M + Θ) (1 + i i ) cos(`M + Θ) + 2ixiy sin(`M + Θ) , − Θ − Θ x − y Θ Θ (222a)

r µ a(1 e2 e2) y − x − y ˙ = 3 2 a 2 2 1 + ex cos(`MΘ + Θ) + ey sin(`MΘ + Θ) 1 + ix + iy  2 2  (1 i +i ) cos(`M +Θ) 2ixiy sin(`M +Θ) 1+ex cos(`M +Θ)+ey sin(`M +Θ) × − x y Θ − Θ Θ Θ  2 2 ey cos(`M + Θ) ex sin(`M + Θ) (1 i + i ) sin(`M + Θ) + 2ixiy cos(`M + Θ) , − Θ − Θ − x y Θ Θ (222b)

r µ a(1 e2 e2) z − x − y ˙ = 3 2 a 2 2 1 + ex cos(`MΘ + Θ) + ey sin(`MΘ + Θ) 1 + ix + iy   2ix cos(`M + Θ) 2iy sin(`M + Θ) 1 + ex cos(`M + Θ) + ey sin(`M + Θ) × − Θ − Θ Θ Θ  ey cos(`M + Θ) ex sin(`M + Θ) 2ix sin(`M + Θ) + 2iy cos(`M + Θ) . − Θ − Θ − Θ Θ (222c)

DAPPROXIMATION METHODSFOR SOLVING THE KEPLER EQUATION With the state space representation using the orbital elements, the sixth parameter is one of the three anomalies: the true one ν, the mean one M or the eccentric one E. Through this document, the mean anomaly is used for the simplicity of its time derivative. However, the Lagrange matrix is usually expressed in terms of the true anomaly. It is therefore mandatory to compute the former from the latter. This can be done in two ways:

69 • solving the Kepler equation: E e sin E = M, (223) − in order to compute E from M and computing ν with the formula: s  1 + e E ν = 2 arctan  tan  ; (224) 1 e 2 − • solving the Kepler equation directly in true anomaly: s   s  1 e ν 1 e ν 2 arctan  − tan  e sin 2 arctan  − tan  = M. (225) 1 + e 2 − 1 + e 2

As the Kepler equation can not be solved analytically, approximations techniques have to be used. The following sections present some of these techniques.

D.1 NEWTON ALGORITHM

D.1.1 KEPLER EQUATION EXPRESSEDIN E The Newton algorithm is an iterative method aimed at ﬁnding an approximation of the zero of the equation: f(x) = 0, (226)

from a ﬁrst guess of the solution x0. A sequence of zeros (xn)n N converging towards the ∈ solution x˜ is built. Each approximate solution xn+1 is computed from the previous one xn according to: f(xn) xn+1 = xn . (227) − f 0(xn)

In the case of the Kepler equation expressed in E, the algorithm reads:

En e sin En M En+1 = En − − (228) − 1 e cos Em − Once E is found, ν is computed with the Equation (224). A way to make the algorithm faster is to choose a starting point close to the actual solution. In the reference [Battin, 1999], a choice for the starting point is proposed. If the function f is monotonic on the interval [x1, x2] such that f(x1)f(x2) 6 0, then a good starting point may the the intersection between the abscissa axis and the line passing through (x1, f(x1)) and (x2, f(x2)). For the Kepler equation, the function f : E E e sin E M is increasing, and 7→ − − [Battin, 1999] suggests the following evaluation points: f(M) = e sin M 0, M [0, π], − 6 ∀ ∈ f(M + e) = e(1 sin(M + e)) 0, M. − > ∀ The initial point is thus the intersection with the abscissa line (for M [0, π]): ∈ e sin M E = M + . (229) 0 1 sin(M + E) + sin M −

70 D.1.2 KEPLER EQUATION EXPRESSEDIN ν The Kepler equation expressed in ν can be solved directly with the Newton algorithm using the functions: v   v  u u u1 e ν u1 e ν f(ν) = 2 arctan t − tan  e sin 2 arctan t − tan  M, (230a)  1 + e 2 −   1 + e 2 − s e ν!   v  1 2 u − 1 + tan u1 e ν 1 + e 2 f 0(ν) = 1 e cos 2 arctan t − tan  . , (230b)  −   1 + e 2 1 e ν 1 + − tan2 1 + e 2 and : f(νn) νn+1 = νn . (231) − f 0(νn)

D.2 HULL’S METHOD

D.2.1 WITHTHE EQUATIONS (223) AND (224) The Equation (223) is a transcendental equation with e as a small parameter as the orbit is almost a geostationary one. the reference [Hull, 2003] proposes a technique to solve the Kepler equation. Considering the funcion:

f :(E; e) f(E; e) = E e sin E M, (232) 7→ − − the equation to be solved is: f(E; e) = 0, (233) for E as unknown and e close to zero. The solution E of this equation is a function of e. ∗ As e is close to 0, the solution can be expanded power of e:

1 2 E = E0 + E1e + E2e ∗ 2 (234) = E0 + ∆E.

The function f(E; e) = f(E0 + ∆E; e) is then expanded with respect to (E0; 0) up to the order 2. The following notations will be used in the sequel:

∂f(E; e) f(E ; 0) f, E fe, 0 −→ ∂e |( 0;0) −→ ∂f(E; e) ∂2f(E; e) E fE, E fee, ∂E |( 0;0) −→ ∂e2 |( 0;0) −→ ∂2f(E; e) ∂2f(E; e) E fEE, E feE, ∂E2 |( 0;0) −→ ∂e∂E |( 0;0) −→

71 f(E; e) = f(E0 + ∆E; e),

1 2 1 2 = f + fE∆E + fee + fEE∆E + feee + feEe∆E, 2 2 (235) 1 2 1 2 2 1 2 2 = f + fE E e + E e + fee + fEEE e + feee + feEE e . 1 2 2 2 1 2 1

Identifying the coeﬃcients for each order of e leads to the following equations:

 f = 0,    fEE + fe = 0, 1 (236)   1 1 2 1  fEE + fEEE + fee + feEE = 0. 2 2 2 1 2 1 Solving this previous system leads to:   E0 = M,  E1 = sin(M), (237)   E2 = sin(2M),

and then: 1 E = M + e sin M + e2 sin(2M). (238) hull 2 From this point, the Equation (224) gives ν.

D.2.2 WITHTHE EQUATION (225) The Hull’s method is applied to compute an approximate solution of the transcendental Equation (225). The handled function is: v   v  u u u1 e ν u1 e ν f(ν; e) = 2 arctan t − tan  e sin 2 arctan t − tan  M. (239)  1 + e 2 −   1 + e 2 −

With similar developments as (245), the true anomaly reads: 5 ν = M + 2e sin M + e2 sin(2M). (240) 2

D.2.3 KEPLER EQUATION WITH THE EQUINOCTIAL ORBITAL ELEMENTS The Kepler equation in terms of the equinoctial orbital elements reads:

EQ + ey cos EQ ex sin EQ = `M + Θ(t). (241) − Θ

For a quasi-circular orbit, the eccentricity is very small and this transcendental equation can be solved in EQ with the Hull’s method applied to the function:

f :(EQ; ex, ey) f(EQ; ex, ey) = EQ + ey cos EQ ex sin EQ `M + Θ(t). (242) 7→ − − Θ 72 The solution EQ of the equation: ∗

f(EQ; ex, ey) = 0, (243) with EQ unknown and ex and ey small can be expanded in power series of ex and ey:

1 2 1 2 EQ = EQ0 + EQxex + EQyey + EQxxex + EQyyey + EQxyexey, ∗ 2 2 (244)

= EQ0 + ∆EQ.

f(EQ; ex, ey) = f(EQ0 + ∆EQ; ex, ey) is then expanded with respect to (EQ0; 0, 0) up to the order 2. For clarity reasons, the following notations will be used:

∂f(EQ; ex, ey) f E , f, f , ( Q0; 0 0) (EQ0;0,0) ex −→ ∂ex | −→ ∂f(EQ; ex, ey) ∂f(EQ; ex, ey) f , f , (EQ0;0,0) ey (EQ0;0,0) EQ ∂ey | −→ ∂EQ | −→ 2 2 ∂ f(EQ; ex, ey) ∂ f(EQ; ex, ey) E , fe e , E , fe e , 2 ( Q0;0 0) x x 2 ( Q0;0 0) y y ∂ex | −→ ∂ey | −→ 2 2 ∂ f(EQ; ex, ey) ∂ f(EQ; ex, ey) f , f , 2 (EQ0;0,0) EQEQ (EQ0;0,0) exey ∂EQ | −→ ∂ex∂ey | −→ 2 2 ∂ f(EQ; ex, ey) ∂ f(EQ; ex, ey) f , f . (EQ0;0,0) exEQ (EQ0;0,0) eyEQ ∂ex∂EQ | −→ ∂ey∂EQ | −→

f(EQ; ex, ey) = f(EQ0 + ∆EQ; ex, ey),

= f + fEQ ∆EQ + fex ex + fey ey+

1 2 1 2 1 2 + fE E ∆E + fe e e + fe e e 2 Q Q Q 2 x x x 2 y y y

+ fEQex ∆EQex + fEQey ∆EQey + fexey exey, 1 2 1 2 = f + fE EQ ex + EQ ey + EQ e + EQ e + EQ exey + fe ex + fe ey Q x y 2 xx x 2 yy y xy x y 1 2 2 2 1 2 1 2 + fE E EQ e + EQ e + 2EQ EQ exey + fe e e + fe e e 2 Q Q x x y y x y 2 x x x 2 y y y + fEQex EQxex + EQyey ex + fEQey EQxex + EQyey ey + fexey exey. (245) Identifying the coeﬃcients of the several orders of ex and ey so that the Equation (243) holds leads to the following system:

 f = 0,    fE EQ + fe = 0,  Q x x   fEQ EQ + fey = 0,  y  1 1 2 1 (246) fE EQ + fE E EQ + fe e + fE e EQ = 0,  Q xx Q Q x x x Q x x  2 2 2   1 1 2 1  fEQ EQ + fEQEQ EQ + feyey + fEQey EQ = 0,  2 yy 2 y 2 y   fEQEQ EQxEQy + fEQex EQy + fEQey EQx + fexey + fEQ EQxy = 0.

73 Solving this system leads to:   EQ = `MΘ + Θ(t),  0   EQ = sin(`MΘ + Θ(t)),  x   EQy = cos(`MΘ + Θ(t)), − (247)  EQ = cos(`M + Θ(t)) sin(`M + Θ(t)),  xx Θ Θ   EQ = cos(`MΘ + Θ(t)) sin(`MΘ + Θ(t)),  yy −  2 2  EQ = cos (`M + Θ(t)) sin (`M + Θ(t)). xy Θ − Θ Hence, the eccentric equinoctial anomaly is expressed in terms of the mean longitude as:

EQ = `MΘ + Θ(t) + ex sin(`MΘ + Θ(t)) ey cos(`MΘ + Θ(t)) − ! ! `MΘ + Θ(t) 2 2 `MΘ + Θ(t) + sin (e e ) + cos exey. (248) 2 x − y 2

At the order 0 in ex and ey, we get:

cos EQ = cos(`MΘ + Θ(t)), (249a)

sin EQ = sin(`MΘ + Θ(t)). (249b)

EPHYSICAL PARAMETERS This appendix gives numerical values for the physical parameters involved for the computation of the orbit of a geostationary satellite. In the sequel, the unit d stands for "day".

The physical and orbital parameters of the Earth are: • geocentric gravitational parameter: µ = 3.986 105 km3/s2 = 2.9755 1015 km3/d2, ⊕ 5 • rotation rate: ω = 7.2921 10− rad/s = 6.3004 rad/d, ⊕ • radius: r = 6378.137 km ⊕ • coeﬃcient of the spherical decomposition of the Earth gravitational ﬁeld: see the Table3 (these values have been taken from the reference [Vallado, 1997]) • sidereal angle Θ(t): using the computation algorithm from [Vallado, 1997], for Jan- st uary 1 , 2034, the value of the sidereal angle is: Θ0 = 1.7579 rad. If t denotes the elapsed time since January 1st, 2034, the sidereal angle at time t is computed as: Θ(t) = Θ0 + ω t. ⊕ • geostationary semi-major axis: ask = 42165.8 km. The gravitational parameter of the Sun is:

µ = 1.32712 1011 km3/s2 = 9.9069 1020 km3/d2. (250)

The gravitational parameter of the Moon is:

µ = 4.9028 104 km3/s2 = 3.6599 1014 km3/d2. (251) $ 74 degree l order m Clm Slm 3 2 0 1.083 10− − 10 × 9 2 1 2.414 10− 1.543 10− − 6 7 2 2 1.574 10− 9.038 10− − 6 3 0 2.532 10− 6 × 7 3 1 2.191 10− 2.687 10− 7 7 3 2 3.089 10− 2.115 10− − 7 3 3 1.006 10−7 1.972 10−

Table 3 – Clm and Slm coeﬃcients for the spherical harmonics decomposition of the Earth gravitational potential (values taken from [Vallado, 1997]). The means that for order 0, × the Sl0 coeﬃcient does not exist.

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