POLITECNICO DI MILANO · 2019. 11. 13. · anim o a cultivar la curiosidad y la creatividad. Julia, la cual me ha apoyado y animado a perseguir mis suenos,~ haciendo cada d a m as

POLITECNICO DI MILANO

DIPARTIMENTO DI SCIENZE E TECNOLOGIE AEROSPAZIALI

Master Degree Programme: Space Engineering

Master Thesis

STABLE ATTITUDE ORBITS IN

SOLAR RADIATION AND DRAG

DOMINATED REGIONS

Antonio Jose Garcıa SalcedoMatricola 897603

Supervised by:

Prof. Camilla Colombo

Co-supervised by:

Dr. Narcıs Miguel

Academic year: 2018-2019

Copyright© July 2019 by Antonio Jose Garcıa Salcedo. All rights reserved.

This content is original, written by the Author, Antonio Jose Garcıa Salcedo. All the non-

originals information, taken from previous works, are specified and recorded in the Bibliogra-

phy. When referring to this work, full bibliographic details must be given, i.e. Garcıa-Salcedo

Antonio J., “Stable attitude orbits in solar radiation and drag dominated regions”. 2019, Po-

litecnico di Milano, Faculty of Industrial Engineering, Department of Aerospace Science and

Technologies, Master in Space Engineering, Supervisor: Camilla Colombo, Co-supervisor:

Narcıs Miguel

Declaration of authorship

I, Antonio Jose Garcıa Salcedo, declare that this thesis titled, “Stable attitude orbits in solar

radiation and drag dominated regions” and the work presented in it are my own. I confirm

that:

• This work was done wholly while in candidature for a research M.sc. degree in Space

Engineering at Politecnico di Milano.

• Where any part of this thesis has previously been submitted for a master or any other

qualification at this University or any other institution, this has been clearly stated.

• Where I have consulted the published work of others, this is always clearly attributed.

• Where I have quoted from the work of others, the source is always given. With the

exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I have made clear

exactly what was done by others and what I have contributed myself.

Signed:

Date:

AJGARSAL

Máquina de escribir

25th July 2019

Acknowledgements

This work supposes the closure of my academical life, providing me with the key to open the

next door: my career.

It would not have been possible without my thesis supervisor Prof. Camilla Colombo and my

co-supervisor Dr. Narcıs Miguel, they have encouraged me to be curious and feel fascinated

about space field throughout the numerous events they organise (MeetMeTonight, NASA

Space Apps Challenge and SpaceDebris Day). They gave me the opportunity of contributing

to cutting-edge COMPASS project by working on this thesis, and share knowledge and

invaluable time to guide me until the last day. Thanks for everything.

A big thank-you to Alvaro Romero. He had been one of the closest people I had the last

months which makes me admire him as an engineer and college, but also as a fiend. He gave

me the chance of taking part in challenging projects, getting me closer to the engineering

world and pushing me into continuous learning. He is a great source of inspiration and

professionalism. Thanks for showing me the actual meaning of excellence.

The reader would allow me to switch to my mother tongue. Some people deserve it.

Todo esto ha sido posible gracias al apoyo incondicional de las personas que considero mi

familia. Soy consciente del gran esfuerzo tanto emocional como economico que me han

brindado lo cual me hace sentir privilegiado y por lo que estare eternamente agradecido.

Pilar, mi madre, que me ha ensenado el valor de la resiliencia y la importancia de seguir

adelante a pesar de las adversidades. Antonio, mi padre, el cual ha confiado siempre en

mi y no ha permitido que me falte de nada. Mi gran hermano Francisco, que siempre me

animo a cultivar la curiosidad y la creatividad. Julia, la cual me ha apoyado y animado a

perseguir mis suenos, haciendo cada dıa mas ameno y siendo una gran fuente de inspiracion

y motivacion. Mis titos Fali y Antonio, que me han tratado como un hijo mas. Toni, por

su esfuerzo para hacerme sentir en casa. A mis titas Encarna y Ani, y especialmente a mis

primos hermanos, por ensenarme el verdadero significado de la fraternidad.

I also want to thank my flatmates, friends and colleges Jose Luis J. and Arnab D. It has been

a pleasure having the opportunity to interact each other every day and share so much with

you.

A big thanks to my colleges and friends Alvaro D., Amador G., Gerardo A., Jason C., Juan

Vicente F., Jose Marıa B., Marıa del Carmen A., and Luminita B., Gianluca M. and Vittorio

S. We have shared great moments during our academic years.

Abstract

The growth of satellites orbiting the Earth gives rise to the need for investigating disposal

strategies for space vehicles to keep operative orbits safe for future space missions. Recently,

several studies have been focused on designing end-of-life missions. Technology development

has enabled the use of solar sails to control light spacecrafts. Solar sail shape with auto-

stabilising dynamical properties supposes an exceptional and attractive option to explore in

order to passively deorbit a vehicle with a minimum power cost. This work explores the stable

dynamics of a spacecraft with a simplified version of a pyramidal shape solar sail. In the last

stage of the orbit lowering, the attitude is affected by disturbances due to Earth oblateness

effect, solar radiation pressure, and atmospheric drag. A sensitivity analysis on the geometric

parameters of the sail and initial parameters of the orbit is performed. In particular, the

transition region between solar radiation pressure dominated region and atmospheric drag

dominated region is analysed. In this environment, spacecraft configurations and initial orbit

parameters that make the satellite remains stable until the very end of the re-entry are found.

These results could serve as a guideline for more accurate analyses, extending the work to

three-dimension study with high-fidelity models would suppose a step for this technology

readiness level, contributing in this way to the development of end-of-life mission designs.

Sommario

Il crescente numero di satelliti in orbita attorno alla Terra da origine alla necessita di in-

vestigare le strategie di smaltimento per i veicoli spaziali, per mantenere le orbite operative

sicure per le future missioni spaziali. Recentemente, diversi studi si sono concentrati sulla

progettazione di missioni di fine vita. Lo sviluppo della tecnologia ha reso possibile l’uso di

vele solari per il controllo di veicoli spaziali leggeri. La forma delle vele solari con proprieta

dinamiche auto-stabilizzanti diventa un’opzione eccezionale e attraente da esplorare per lo

smaltimento passivo di un veicolo con un consumo energetico minimo. Questo lavoro esplora

le dinamiche stabili di un veicolo spaziale attraverso una versione semplificata di una vela

solare con forma piramidale. Nell’ultima fase della caduta verso la terra, l’assetto influenzato

da disturbi dovuti all’effetto dello schiacciamento della Terra, alla pressione della radiazione

solare e alla resistenza atmosferica. Viene eseguita un’analisi di sensibilit sui parametri geo-

metrici della vela e i parametri iniziali dell’orbita. Un’analisi della regione di transizione tra

la regione dominata dalla pressione di radiazione solare e la regione dominata dalla resistenza

atmosferica e inclusa. In questo ambiente, si trovano confgurazioni di satelliti e parametri

orbitali iniziali in cui il veicolo cade sulla Terra in modo stabile. Questi risultati potrebbero

servire come base per analisi piu accurate, l’estensione di questo lavoro al caso tridimension-

ale tramite modelli ad alta fedelta farebbe avanzare il livello di prontezza di questo tipo di

tecnologia. Contribuirebbe inoltre allo sviluppo di una vera progettazione delle missioni di

fine vita.

Contents

1 Introduction 18

1.1 Space debris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Overview of passive mitigation technologies . . . . . . . . . . . . . . . . . . . 22

1.2.1 Area-augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2.1.1 Drag-augmentation device . . . . . . . . . . . . . . . . . . . 23

1.2.1.2 Solar radiation pressure augmented device . . . . . . . . . . 23

1.2.1.3 Deployable structures . . . . . . . . . . . . . . . . . . . . . 25

1.2.2 Electrodynamic tethers . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2.3 Mechanical tethers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2.4 Thrust propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2.5 Natural perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3 State of the art of solar sailing . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.3.1 The quasi-rhombic pyramid sail . . . . . . . . . . . . . . . . . . . . . 29

1.4 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.5 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Theoretical modelling 33

2.1 Spacecraft configuration. Geometry of the sail. . . . . . . . . . . . . . . . . . 33

2.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.1 Attitude dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.2 Solar radiation pressure . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.3 Gravity gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.4 Atmospheric drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Orbit dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.1 Cartesian propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3.2 Gauss propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Orbit perturbations 43

7

8 Contents

3.1 SRP-dominated region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 SRP force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.2 SRP torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.3 Orbit evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Drag-dominated region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.1 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 Drag torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.3 Orbit Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Numerical results 65

4.1 General simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Sensitivity analysis: α and d . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Sensitivity analysis: e and ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1 120 - 1000 km of altitude . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . 74

4.3.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.2 Above 1000 km of altitude . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.2.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . 79

4.3.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Conclusions and future work 84

Appendix 93

A Computation of Shadowed Area 95

List of Figures

1.1 Space debris scenario [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Evolution of the tracked and published space object population and its com-

position by object class [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3 Evolution of the tracked and published space object population and its com-

position by orbit class [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 The quasi-rhombic pyramid concept [4]. . . . . . . . . . . . . . . . . . . . . . 30

2.1 Sketch of the sail structure in body-frame. a) 3D sail model. b) x-y projection. 34

2.2 Relative position of the sail with respect to the bus (Fb top view). a) d = 0

m, b) d < 0 m, c) d > 0 m, d) d > 0 m. . . . . . . . . . . . . . . . . . . . . . 35

2.3 Value of d so that the bus in the sail tip. . . . . . . . . . . . . . . . . . . . . 35

3.1 Spacecraft orientation with respect to the Sun. Definition of λ. . . . . . . . . 45

3.2 ϕSRP definition. a) SRP regions. b) SRP regions including shadow in one

panel, w′ definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 SRP force in Fb as a function of the sail aperture angle α and Sun-spacecraft

orientation. ϕSRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 MSRP,3 as a function of the Sun-Spacecraft orientation . . . . . . . . . . . . 48

3.5 Bifurcation curves. a) d-α plane. b) η-α plane. . . . . . . . . . . . . . . . . . 49

3.6 a) Bifurcation curve evolution for different values of sail reflectance η in d-α

plane. b) Bifurcation curve evolution for different d in η-α plane. . . . . . . . 49

3.7 Phase space of system dynamics when only the SRP perturbation is present

for a circular orbit. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c) α = 30°,d = 0 m. d) α = 30°, d = 1.5 m. . . . . . . . . . . . . . . . . . . . . . . . . 50

3.8 Poincare section Σ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

9

10 List of Figures

3.9 Poincare maps for different spacecraft configurations under SRP, gravity gra-

dient and J2 perturbations with an initial orbit of h0 = 1500 km, ω = 0° and

e = 0. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c) α = 30°, d = 0 m. d)

α = 30°, d = 1.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.10 Poincare maps for different spacecraft configurations under SRP, gravity gra-

dient and J2 perturbations with an initial orbit of h0 = 1500 km, ω = 0° and

e = 0.1. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c) α = 30°, d = 0 m. d)

α = 30°, d = 1.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.11 Position of the spacecraft with respect to the relative velocity vector. Defini-

tion of δ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.12 ϕdrag definition. Left: general drag regions. a) Drag regions. b) Drag regions

including shadow in one panel, w′ definition. . . . . . . . . . . . . . . . . . . 54

3.13 Density profile. a) Decimal scale in x axis and y axis. b) Decimal scale in y

axis, logarithmic scale in x axis. . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.14 Drag force term fff in Fb as a function of the sail aperture angle α and relative

velocity orientation with respect to the aircraft ϕdrag. . . . . . . . . . . . . . 56

3.15 Phase space of system dynamics when only the atmospheric drag perturbation

is present for a circular orbit and h0 = 750 km. a) α = 45°, d = 0 m. b)

α = 45°, d = 1.5 m. c) α = 30°, d = 0 m. d) α = 30°, d = 1.5 m. . . . . . . . 57

3.16 Poincare maps for different spacecraft configurations under atmospheric drag,

gravity gradient and J2 perturbations with an initial orbit of h0 = 750 km,

ω = 0° and e = 0. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c) α = 30°,d = 0 m. d) α = 30°, d = 1.5 m. . . . . . . . . . . . . . . . . . . . . . . . . 58

3.17 Poincare maps for different spacecraft configurations under atmospheric drag,


ω = 0° and e = 0.1. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c) α = 30°,d = 0 m. d) α = 30°, d = 1.5 m. . . . . . . . . . . . . . . . . . . . . . . . . 59

3.18 Poincare map for different spacecraft configurations under atmospheric drag,


ω = 0° and e = 0. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c) α = 30°,d = 0 m. d) α = 30°, d = 1.5 m. . . . . . . . . . . . . . . . . . . . . . . . . 60

3.19 Orbit evolution until deorbit in drag-dominated region for three different initial

spacecraft attitude with respect to relative velocity direction, α = 30°, d = 0

m h0 = 600 km, ω = 0° and e = 0. a) Phase space. b) altitude evolution. . . 61

3.20 Model validation. Comparison between Cartesian propagation and Gauss

propagation in transition region. . . . . . . . . . . . . . . . . . . . . . . . . . 62

List of Figures 11


propagation in SRP-dominated region. . . . . . . . . . . . . . . . . . . . . . 62


propagation in drag-dominated region. . . . . . . . . . . . . . . . . . . . . . 63

4.1 Initial spacecraft attitude with respect to the Sun and the relative velocity

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Simulation results as a function of α, d and e for a spacecraft going through

transition region. a) Deorbiting time in days. b) Time in days in which the

spacecraft starts to tumble. c) Tumbling altitude in km. . . . . . . . . . . . 68

4.3 Orbit evolution for a α = 60°, d = 0 m spacecraft, e = 0. a) Altitude and

semi-major axis evolution. b) Attitude evolution. . . . . . . . . . . . . . . . 69

4.4 Orbit evolution for a α = 45°, d = 1.5 m spacecraft, e = 0.01. a) Altitude and


4.5 Sensitivity analysis with respect to α and d. a) Deorbiting time in days. b)

Portion of time that the spacecraft remains stable. c) Time until the spacecraft

starts to tumble in days. d) Altitude in which the spacecraft starts to tumble

in km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.6 Orbit evolution for a α = 30° , d = 1.5 m spacecraft, e = 0. a) Altitude and


4.7 Sensitivity analysis with respect e and ω, a = RE + 1000 km, As/m = 2

m2/kg. a) Deorbiting time in days. b) Portion of time that the spacecraft

remains stable. c) Time until the spacecraft starts to tumble in days. d)

Altitude in which the spacecraft starts to tumble in km. . . . . . . . . . . . 75









4.10 Orbit evolution for a α = 45°, d = 0 m, As/m= 4 m2/kg spacecraft, e = 0.0273

and ω = 135°. a) Altitude and semi-major axis evolution. b) Eccentricity

evolution. c) Attitude evolution. . . . . . . . . . . . . . . . . . . . . . . . . . 78

12 List of Figures

4.11 Sensitivity analysis with respect to e and ω, h0 = 1000 km, As/m = 4 m2/kg.

a) Deorbiting time in days. b) Portion of time that the spacecraft remains

stable. c) Time until the spacecraft starts to tumble in days. d) Altitude in

which the spacecraft starts to tumble in km. . . . . . . . . . . . . . . . . . . 80

4.12 Sensitivity analysis with respect to e and ω, h0 = 1000 km, As/m = 10 m2/kg.

a) Deorbiting time in days. b) Portion of time that the spacecraft remains

stable. c) Time until the spacecraft starts to tumble in days. d) Altitude in

which the spacecraft starts to tumble in km. . . . . . . . . . . . . . . . . . . 81

4.13 Initial scenario for λ0 = ω0 = 90°. . . . . . . . . . . . . . . . . . . . . . . . . 81

4.14 Orbit evolution for a α = 45°, d = 0 m, As/m = 10 m2/kg spacecraft, e =

0.1046 and ω = 90°, within a 20-years window. a) Altitude and semi-major

axis evolution. b) Eccentricity evolution. c) Attitude evolution. . . . . . . . 82

4.15 Orbit evolution for a α = 45°, d = 0 m, As/m = 10 m2/kg spacecraft, e =

0.1046 and ω = 270°, within a 20-years window. a) Altitude and semi-major

axis evolution. b) Eccentricity evolution. c) Attitude evolution. . . . . . . . 83

A.1 Geometry of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

A.2 Line intersection problem in auxiliary axes x’ y’. . . . . . . . . . . . . . . . . 96

List of Tables

3.1 Spacecraft parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1 General Simulation. Parameters. . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Sensitivity Analysis: sail configuration. Simulation Parameters. . . . . . . . 71

4.3 Sensitivity Analysis: e and ω, h0 = 120− 1000 km. Simulation Parameters. . 74

4.4 Sensitivity Analysis: e and ω, h0 = 1000 km. Simulation Parameters. . . . . 79

13

Nomenclature

α Aperture angle of the sail deg

δ Relative velocity pointing direction with respect to drag in ECI deg

η Sail reflectance −

γ1, γ2, γ3 Earth-spacecraft direction cosines −

λ Sun-pointing direction in ECI deg

Fb Body frame −

Ft−n−h Tangential-normal frame −

µ Earth’s gravitational parameter m3/s2

ν1, ν2, ν3 Relative velocity direction cosines −

Ω Orbit right ascension of the ascending node deg

ω Orbit argument of the perigee deg

Φ Angular velocity of the spacecraft rad/s

π+ Top panel of the sail −

π− Bottom panel of the sail −

aaa Spacecraft acceleration vector km/s2

aaadrag Atmospheric drag acceleration km/s2

14

Nomenclature 15

aaaJ2 J2 acceleration km/s2

aaaSRP Solar radiation pressure acceleration km/s2

nnn± Panel sail normal −

rrr Spacecraft position vector km

uuuE Earth-spacecraft direction g

uuurel Spacecraft relative velocity direction with respect to atmosphere −

uuus Sun-Earth direction deg

vvv Spacecraft velocity vector km/s

Σ Poincare section −

σ1, σ2, σ3 Sun-Earth direction cosines −

θ Orbit true anomaly deg

ϕ Spacecraft attitude in ECI frame deg

ω ω + Ω deg

ϕdrag Relative velocity direction in body-frame deg

ϕSRP Solar radiation pressure direction in body-frame deg

a Orbit semi-major axis km

Asp Sail panel area m2

CD Spacecraft drag coefficient −

d Distance between mass centres of the sail and the bus m

e Orbit eccentricity −

Fdrag Atmospheric drag force N

FSRP Solar radiation pressure force N

16 Nomenclature

G Universal gravitational constant m3/s2kg

h Sail panel height m

Ix,b, Iy,b, Iz,b Bus moments of inertia kgm2

Ix,sc, Iy,sc, Iz,sc Spacecraft moments of inertia kgm2

J2 Gravitational perturbation −

mb Bus mass kg

mE Mass of the Earth kg

ms Sail mass kg

Mdrag Atmospheric drag torque Nm

MGG Gravity gradient torque Nm

MSRP Solar radiation pressure torque Nm

n Orbit mean motion rad/s

n Sun’s mean motion rad/s

PSR Solar radiation pressure N/m2

RE Earth’s mean equatorial radius km

rp Perigee radius km

w Sail panel width m

Glossary

ESA European Space Agency

LEO Low Earth Orbit

MEO Medium Earth orbit

GEO Geosynchronous Equatorial

SRP Solar Radiation Pressure

QRP Quasi-Rhombic Pyramid

ECI Earth-Centered Inercial

RAAN Righ Ascension of the Ascending Node

IADC Inter Agency Debris Coordination Committee

COMPASS Control for Orbit Manoeuvring through Perturbations for Application to

Space Systems

Chapter 1

Introduction

In the middle of the fifties, Sputnik I, the world’s first artificial satellite, was launched into

space. That supposed the beginning of satellite era. Since that moment, more than 8950 [1]

satellites have been launched into the space providing useful technologies for society. The

development of space sector in last sixty years has been such that nowadays people are used

to hearing news about launchers and satellites every now and then, there is a space laboratory

in orbit and the biggest countries and communities have their own global positioning satellite

systems.

The number of satellites which are placed in orbit increases every year. The biggest countries

in the world have their own operating satellite constellations to provide positioning and

communication services among others. Thanks to technological improvement, space is more

accessible than ever. Small and light spacecraft have been proven to be a viable alternative

with large applicability. In this way, private companies start to see space as a potential

market.

Recently, satellite mega-constellations have been proposed. Satellite start-up OneWeb at-

tempts to place 900 small satellites into orbit to make broadband internet connections more

accessible [5]. On May 23 of 2019, SpaceX has just launched 60 internet satellites. This sup-

poses the beginning for Starlink project, the broadband constellation which has been granted

approval to scatter 12000 satellites through low Earth and very low Earth orbit [6]. Other

firms, such as Telesat and LeoSat, have similar, smaller-scale projects [7, 8].

The result of this scenario is that the region of space closer to the Earth is every day more

crowded. At the beginning of 2019 [1], about 8950 satellites have been placed into Earth

18

1.1. Space debris 19

orbit by nearly 5450 successfully rocket launches since the start of the space age. Approx-

imately 5000 of these satellites are still on space but only about 1950 are still operating.

Eventually, spacecraft can suffer collisions and explosions resulting in a wide generation of

new moving parts which cannot be controlled, which compromises the safety of the operating

satellites. There are more than 22300 debris objects regularly tracked by the Space Surveil-

lance Network and maintained in their catalogue [1]. In January 2019, the estimated number

of break-ups, explosions, collisions or events resulting in fragmentation is larger than 500.

Statistical models predict about 34000 objects in orbit larger than 10 cm, 900000 objects

ranging between 1 cm to 10 cm and 128 million objects from 1 mm to 1 cm [1].

Figure 1.1: Space debris scenario [1].

1.1 Space debris

According to the European Space Agency (ESA), space debris is defined as all non-functional,

human-made objects, including fragments and elements thereof, in Earth orbit or re-entering

into Earth atmosphere. When the satellite lifespan is over, it generally remains in orbit

without being operational. The growth of objects through space also increases the probability

of collisions. Collisions give rise to the appearance of a large number of objects of different

sizes, shapes, etc. that form a cloud. Some of these objects are unfeasible to be tracked.

Other events such as explosions can give rise to similar phenomena. When these objects

considerably grow, they generate a cascade effect in which debris create new debris. The

problem is commonly labelled as ”Kessler syndrome”, also called ”collisional cascading” [9]. It

may suppose the destruction of many operating satellites and can be triggered when reaching

a critical debris population density. Consequently, debris mitigation must be implemented.

20 Chapter 1. Introduction

The typical collision speed is about 10 km/s in Low Earth Orbit (LEO) [10]. At such

speed, debris larger than 1 cm could disable an active satellite or could cause the break-up

of a satellite or rocket body. The complete destruction of a spacecraft and the consequent

generation of a debris cloud can be triggered by the impact of debris larger than 10 cm.

Major contributions to the debris population are the result of a Chinese anti-satellite test

targeting the Feng Yun-1C in January 2007 [11]. This fact increased the trackable space

object population by 34%. Later, in February 2009, the first-ever accidental in-orbit collision

took place between the American communication satellite, Iridium-33 and a Russian military

Kosmos-2251 satellite [12]. Both were destroyed, generating more than 2300 trackable frag-

ments which added another 17% [2]. Both phenomena can be observed in Fig. 1.2 and 1.3,

where the count evolution by object type and object orbit respectively are illustrated with

respect to time. In addition, India shot down one of its satellites with a missile in March

2019 [13]. However, no enough data on the produced debris cloud is available since the event

is recent.

Figure 1.2: Evolution of the tracked and published space object population and

its composition by object class [2].

The catalogued objects can be divided into non-operational satellites 24%, rests of launchers

18%, leftovers from satellites and rocket bodies explosions, remnants from anti-satellite test

and debris fragments from other minor sources [2]. Nowadays, there is a rate of 70-90 launches

a year and it is increasing [14]. Each launch injects 30 or more satellites. Historical rates of

four to five break-ups per year can be assumed and consequently, the objects in space are

expected to increase. It supposes an increase of debris density and catastrophic collisions

1.1. Space debris 21

Figure 1.3: Evolution of the tracked and published space object population and

its composition by orbit class [3].

probability which could enable the collisional cascading initiation.

Many efforts are being taken to reduce space objects increase. The most effective strategy

is focused on reducing the space debris growth rate through in-orbit explosions or collision

prevention. The former can be achieved by passivation, which consists of exhausting the

active sources that can originate the explosion of the satellite at the end of their operational

life while the latter can be achieved by implementing collision avoidance manoeuvres during

their operational life.

End-of-life disposal strategies have been proposed to remove objects from the denser popu-

lated regions. In this way, satellites and orbital stages are suggested to reenter into Earth

atmosphere within 25 years of mission completion when their altitude is below 2000 km, in

LEO region. For operating setups in Geosynchronous Equatorial Orbit (GEO) region are

encouraged to change their orbit to a disposal or graveyard orbit at the end of their working

life. These are some of the requirements described in the Space Debris Mitigation Guidelines

[15] published by the Inter Agency Debris Coordination Committee (IADC).

To meet the space debris mitigation requirements, spacecraft self demise capabilities should

be considered for either controlled and uncontrolled space vehicles during the design phase,

this philosophy is known as ”design for demise”. In this way, the space system will disintegrate

in a desired way during re-entry avoiding damages to mankind or property on Earth.

Some projects have received economic support from an international network. This is the


case of Control for Orbit Manoeuvring through Perturbations for Application to Space Sys-

tems (COMPASS) [16], within the European Research Council of the Horizon 2020 program,

the project has the ultimate goal of researching the possibilities to harness the orbit pertur-

bations to optimise the space mission efficiency. In this framework, space debris disposals by

exploiting orbit perturbations are investigated.

1.2 Overview of passive mitigation technologies

Many disposal technologies can be used for passive mitigation at the satellite operational end

of life. The goal is to achieve spacecraft deorbit or a graveyard injection depending on the

region the vehicle is located. In this way, area-augmentation devices may be employed to

harness the atmospheric drag and, consequently to reduce the orbit perigee altitude allowing

the spacecraft re-entry. Fully autonomous chemical engine systems can provide ∆v to orbit

change. Thermal and communication subsystems may be used to guide the disabled space-

craft. Space tethers offer the opportunity to exploit the Earth magnetic field. Moreover,

solar sailing and electric propulsion have been recently investigated and proved for the small

satellites deorbiting in Medium Earth Orbit (MEO) and LEO regions [17, 18]. Mitigation

strategies should cover all possible spacecraft size and orbit region and go ahead of possible

future technologies such as micro and nano-satellites. Therefore, an assessment and compar-

ison of the efficiency of each strategy for different spacecraft characteristics and operation

orbit should be performed [19, 20].

1.2.1 Area-augmentation

Area-augmentation devices for passive deorbit is an attractive option in the end-of-life design.

The basic principle is based on a depleted flat sail that increases the area exposed to the

sunlight or atmospheric drag. As a consequence, the perturbation effect is enhanced and used

to orbit change. In the innermost orbiting regions, altitude up to 900-1000 km, solar sailing

devices represent the most efficient technology to deorbit for small and medium satellites. In

the region between 800 and 2000 km of altitude, where the atmospheric drag is not strong

enough to drive the spacecraft towards Earth, solar sailing becomes the best option to be

exploited. Under 800 km, atmospheric drag strength dominates above SRP force, and drag-

augmentation devices can be employed to achieve spacecraft re-entry.

1.2. Overview of passive mitigation technologies 23

1.2.1.1 Drag-augmentation device

In drag-dominated regimes, below 800 km of altitude, the required area-to-mass ratio to de-

orbit depends on the semi-major axis and it increases exponentially when altitude increases.

When the area increases, so does the aerodynamic resistance for low orbits, making the

spacecraft to decay. Main solutions are focused on increasing the ballistic coefficient, which

is determined by the spacecraft mass, size and shape, by using deployable or inflatable struc-

tures. Drag augmentation is obtained either by means of a spherical envelope or by means of

a gossamer structure. Drag enhancement was found to be the most mass-efficient method for

25-year deorbit of a satellite orbiting below 900 km [21] and 700 km [22] by using inflatable

balloons. Roberts et al. [23] found that faster deorbiting time is achieved for a lower initial

angle of attack with respect to the velocity flow, larger devices oscillate more slowly and are

less sensitive to density variations, and the deorbit time is dependant from the initial hour

angle. End-of-life disposal technologies were compared in [24], where gossamer structures

were found to be the best suited from a maximum allowable end-of-life mass, allowing en-

tirely passive operational mode. A full dynamics model considering the deformation of the

sail was studied in [25] for gossamer structures, where the resulting torques due to the elastic

behaviour of the sail were not negligible. Visagie et al. [26] studied the collision risk using

a deployable drag-sail to deorbit space debris. They used the area-time-product to perform

a comparative analysis and showed that collision risk can be reduced to less than 10% of

the non-mitigation scenario. The implementation of drag augmentation devices in CubeSats

placed at 600 km were addressed in [27], where the spacecraft was found to be capable of

entering and maintaining a low-drag configuration for five days while utilising a portion of

the available angular momentum to counterbalancing environmental disturbances. In 2017,

InflateSail [28], a 3U CubeSat equipped with a deployable drag sail, was launch into a 505

km of altitude. 72 days later, the spacecraft successfully return to the Earth, becoming the

first European demonstration of drag-sail deorbiting.

1.2.1.2 Solar radiation pressure augmented device

In Solar Radiation Pressure (SRP) dominated region, above 1000 km of altitude, the area-

to-mass ratio requirements for deorbiting strongly depends on orbit semi-major axis and

inclination [17, 18]. A solar sail is a reflective and deployable structure capable of allowing

a passive deorbiting from LEO and low MEO. It is a light assemblage which expands in

space increasing significantly the satellite area. Acceleration due to solar radiation pressure

and atmospheric drag are consequently enhanced. Solar sails are very similar to drag sails


in terms of structure and materials, nevertheless, the strategies to achieve deorbiting are

different in nature. Two methodologies can be employed for deorbiting with solar sail [19]:

• Inward deorbiting: the deorbiting is obtained by spiralling inward on a circular orbit.

• Outward deorbiting: the deorbiting is achieved by decreasing the semi-major axis of

the osculating ellipse, when eccentricity increases, the perigee radius decreases and the

goal is to make it be inside the atmosphere. Once the perigee radius passes through the

atmosphere, the drag acts reducing the orbit energy, thus, decreasing the semi-major

axis and eccentricity.

A reflective coating is used to augment solar radiation pressure acceleration exerted in the

sail, enabling deorbiting from high orbits in LEO and low orbits in MEO. Inward and outward

deorbiting can be accomplished by two possible strategies depending on the way that the sail

is controlled.

• Active attitude control: when the sail attitude is controlled along the vehicle trajectory

• Passive attitude control: when no active control is employed and the pursued effect is

achieved in long-term evolution. It is only possible for some spacecraft depending on

their area-to-mass ratio, otherwise, active control must be used.

1. Active attitude control strategy

A solar sailing strategy was proposed to deorbit via active attitude control. It con-

sists of changing the sail attitude twice per orbit [29]. When the sail normal points

the sunlight, the effect of solar radiation pressure results in decelerating the spacecraft.

Hence, the spacecraft energy is reduced. The strategy is based on facing the Sun when

the vehicle moves towards the star and minimising the area facing the sunlight when it

moves backwards to the Earth-Sun direction. An alternative semi active - modulating

strategy was proposed in [30], where the sail is activated depending on the Sun-perigee

angle with the purpose of increasing the long-term eccentricity evolution. The sail at-

titude is changed every 6 months on average, allowing the spacecraft to deorbit with

lower required area-to-mass ratio. The solutions present a remarkable drawback, active

control requires energy during satellite non-operational lifetime. It can become an un-

feasible option for small and light spacecraft with limiting power budget. Active solar

sailing can be achieved by adjusting sail orientation twice per revolution around the

Earth. Flat and pyramidal sail are available and proven technologies to be used with

active attitude control.


2. Passive attitude control strategy

Passive attitude control supposes a great alternative for the cost issue. A novel and

counter-intuitive solution was presented by Luking [17, 18], that consist of the sail is

kept to be always facing the Sun under SRP perturbation. The resulting long-term

effect is that eccentricity increases until the perigee radius reaches the Earth’s atmo-

sphere. Deorbiting is achieved in two phases. The first one is SRP-dominated, the main

effect is the eccentricity increment while the semi-major axis remains almost constant.

The second phase starts when the perigee radius enters inside the atmosphere. The

atmospheric drag decelerates the spacecraft and the vehicle experiments a naturally

inward deorbiting.

1.2.1.3 Deployable structures

To intensify SRP and drag effect on the spacecraft, the area has to be increased. Several solu-

tions have been proposed considering stored volume-to-mass ratio, reflectivity and dynamical

stability [31, 32].

1. Inflatable reflective balloon

There are available technologies which operate in high altitudes with no active con-

trol required. Sphere geometry is beneficial from a low-mass volume point of view, and

offers a reflective coefficient of 1 in all directions [33]. It is a perfect sail safe config-

uration with a constant cross-area with respect to SRP perturbation. However, this

technology has a lower reflectivity coefficient than other solutions and the process of

deployment is complex as the resin needs to solidify [32].

2. Flat sail

A flat sail is the simplest solution with a benefit from a low-mass volume. It is proven

and there are being used at this moment. They can be employed at high altitudes and

adapted to all spacecraft size. The reflective coefficient varies whether the sail faces

the Sun or not with a maximum of 2. It requires a complex deployment process [32].

The deorbiting can be obtained in two ways when using flat sail:

• Active attitude control: when inward deorbiting is aimed.


• No active control: in this case, the sail is not controlled, usually, the spacecraft

motion becomes tumbling.

3. Pyramidal Sail

The pyramidal sail is an evolution of flat sail with auto-stabilising properties depending

on the sail aperture angle and the centre of mass-centre of pressure offset. It represents

the best volume-mass ratio [31], and the technology is being developed and tested [34].

The reflective coefficient can vary from 1 to 2 depending on the angle of the sail. The

effective spacecraft area-to-mass ratio can be adjusted by modifying the sail angle. A

pyramidal sail with variable angle was proposed [4, 35] to achieve active control sailing.

4. Sail with reflective changing properties

The conventional SRP augmented devices strongly depend on the sail-Sun orientation.

To enhance potential applications a novel shape change with variable optical properties

technology has been proposed [36]. Electrochromic coatings can change colour by the

application of an electric stimulus which alters the oxidation state of the material, it

permits the electroactive material to be employed to counteract the gravity gradient

torques. The technology allows changing the sail shape profile to vary the reflectivity

coefficient. In this way, the forces and torques can be controlled by only changing the

coefficient. Electrochromic applications were proposed to control a swarm high area-

to-mass ratio spacecraft [37, 38]. Electrochromic control device for solar sail was first

proven in space with IKAROS mission [39]. The reflectance control device consisted of

a flexible multi-layer sheet whit liquid crystal encapsulated whose optical reflectance

properties change when an electrical voltage is applied. The spin axis direction was

changed via SRP by synchronising ON/OFF the device with the spinning phase, thus,

without fuel consumption. More recently, an inflatable balloon with colour-change

capabilities has been recently proposed [40].

1.2.2 Electrodynamic tethers

Electrodynamic tethers harness the Lorenz force by means of the interaction between a

current flowing inside the tether and Earth electromagnetic field. The electric current can be

provided by onboard systems or can be passively achieved by collecting free electrons from the

ionosphere. The current circulation is guaranteed in both cases by an electron emitter, which


discharges the electron back into the ionosphere [41]. Since no fuel is required, electrodynamic

tethers present an advantage in terms of mass efficiency in comparison with other propulsive

solutions. A high reduction of deorbiting time with respect natural atmospheric decay could

be achieved. Nevertheless, these tethers suppose an increment of spacecraft frontal area that

increases the collision probability.

1.2.3 Mechanical tethers

Mechanical tethers do not interact with the atmosphere. However, it does with other bodies

by means of momentum exchange. The YES2 experiment [42] was aimed to prove this

technology. The re-entry capsule was achieved by exchanging momentum from the capsule

to the satellite platform. The mission was successfully accomplished and mechanical tethers

were shown to be used for end of life disposal for small satellites.

1.2.4 Thrust propulsion

Thrust propulsion methods have been required from the beginning of space missions. They

have been used with the objective of reentering manned spacecraft among others. Thrust

propulsion devices are capable of producing a spacecraft acceleration by ejecting stored ma-

terial at high speed. Thrust can be generated in two different ways. It can be performed by

ejecting a large quantity of mass flow, which is the basic principle of chemical propulsion.

The other way consists of ejecting low mass but at very high speed, which is the working

principle of electric propulsion. On the one hand, chemical propulsion allows impulsive and

fast orbit change for end-of-life manoeuvres with a high mass penalty. On the other hand,

electric propulsion allows slow transfers. Traditional propulsive systems are not optimised

to be used in the satellite non-operational life, which increases the duration, complexity and

cost of the end-of-life operations. However, some companies such as D-ORBIT has developed

propulsive systems to be used in deorbiting schemes.

1.2.5 Natural perturbations

Earth oblateness and luni-solar perturbation effect was exploited for designing disposal strate-

gies of satellites placed in highly elliptical orbit [43, 44, 45, 46, 47], MEO [48, 49, 50]. End of


life disposals exploiting natural perturbations are under study nowadays within the COM-

PASS project [16], this document aims to contribute in this framework.

1.3 State of the art of solar sailing

Solar sailing has become popular in the last decade thanks to its wide variety of applications,

including displaced geostationary orbits [51], polar loitering when used together with solar

electric propulsion [52], orbit raising from low Earth orbit [53], inclination change [54] and

deorbiting [55], [29].

In 2010, the world’s first interplanetary solar sail spacecraft IKAROS was launched by Japan

Aerospace Exploration Agency (JAXA). It supposed a milestone in solar sail technology

since it demonstrated solar sail propulsion in interplanetary space [39]. In 2011, NASA’s

Nanosail-D2 mission was successfully launched after the failure of Nanosail-D mission [56]

due to the launching rocket crash. It supposed NASA’s first-ever solar sail deployment

in LEO. The mission showed the feasibility of sail deployment and its usage to deorbit a

spacecraft exploiting the atmospheric drag. In 2015, LightSail 1 was launched as a preliminary

technology demonstrator for a CubeSat spacecraft, the spacecraft successfully reentered the

atmosphere. It supposed the first mission of the LightSail project developed by The Planetary

Society to demonstrate controlled solar sailing for CubeSats in LEO. Following this project,

LightSail 2 has just been launched on 25 June 2019, the spacecraft is placed in a circular

orbit at 720 km of altitude and aims to raise its apogee and orbital energy following sail

deployment.

Lucking et al. [17] showed the feasibility of using solar sails to deorbit. The study was

first inspired in a planar model that was studied to find the needed area-to-mass ratio to

accomplish deorbiting within 5 years. The required area was found to depend only on semi-

major axis, eccentricity and the angle between the perigee and the solar radiation direction.

The work was extended to inclined circular and high eccentric orbits, where inclination

became a key-parameter for area requirement. Some regions where 25-years deorbit is possible

with less than 10 m2/kg were identified. In particular, spacecraft placed in Molniya orbits

could achieve deorbiting with a required 1 m2/kg. The possibilities of deorbiting circular

orbits were studied in [18], where the eccentricity evolution and the deorbiting time were

also subjects matter. The research focused on Sun-synchronous orbits and SRP-augmented

deorbiting was proved to be an effective method to passively deorbit spacecraft.

1.3. State of the art of solar sailing 29

Many studies have been carried out during the last years resulting in a variety of methods

and strategies to be used in end-of-life disposal [20, 57, 19]. Lately, many researching lines

contribute to this field. The coupled dynamics of a flexible sail, its deployment and space

vehicle were analysed in LEO [58, 59] where control manoeuvres were also considered. Natural

highways for end-of-life solutions in terms of required time and initial dynamical configuration

were studied [60] and possibilities of upper-stages [61] and nanosatellites [62] deorbiting were

analysed. Kelly et al. [63] proposed the TugSat mission, a spacecraft able to remove space

debris from GEO belt with a minimum power cost by means of solar sailing. The sail

orientation is optimised and controlled by rotating the sail perpendicular to the incoming

sunlight. Colombo et al. [64] studied the sail requirements in term of sail size achievable with

current technologies considering the increment of collision risk as a consequence of increasing

the spacecraft area by deploying the sail.

1.3.1 The quasi-rhombic pyramid sail

A novel variable-geometry, illustrated in Fig. 1.4, was proposed by Ceriotti et al. [4]. The

possibilities of the quasi-rhombic pyramid sail were studied to harness the solar radiation

pressure. Quasi-Rhombic Pyramid (QRP) geometry offers self-stabilising properties and

allows to modify the area facing the sunlight by controlling the flare angles of the sail. As

a result, strategies to obtain orbit raising were found using this technology for CubeSat

like spacecraft above 1000 km. The effect of gravity gradient torques and eclipses on this

technology were studied in [35, 65]. It was found that altitude adjustment can be obtained

by varying the sail angles. The possibilities of counteracting the negative effects of eclipses

and gravity gradient torques by means of spinning and ring damper were also analysed, and

stability regions of the sail were found where CubeSats in GEO can achieve orbit altitude

increase with no propellant consumption and no active attitude control after the proper

spinning is obtained. The concept was found to be unfeasible for orbits below 10000 km due

to eclipses and gravity gradient torques.

A recent work by researches in Politecnico di Milano [66] was focused on the dynamics of

uncontrolled spacecraft. A simplified version of the QRP solar sail was studied in planar

motion. Stable attitude dynamics close to the sun-pointing orientation were found in SRP-

dominated regions. Similarities between drag and SRP forces were identified, regions of stable

attitude were detected in the vicinity of the tangent-to-orbit attitude in drag-dominated

region. Drag force was found to passively stabilise a spacecraft for altitudes below 800 km.


Figure 1.4: The quasi-rhombic pyramid concept [4].

1.4 Scope of the thesis

Within a deorbiting scenario, the space vehicle flies from SRP-dominated to drag-dominated

regions before re-entering into the Earth, consequently, altitudes where spacecraft is under

the coupled effect of both phenomena must be crossed. Since these directions differ along

the satellite trajectory, uncontrolled vehicles probably end tumbling. This project is aimed

to extend the use of QRP solar sail and its stabilising properties in these regions inside of a

passive deorbiting scheme for end-of-life missions. Following the approach in [66], it explores

the behaviour of different spacecraft in regions, where SRP and drag are the dominant forces,

establishing the relation between the spacecraft attitude dynamics with the geometry of the

sail and initial orbit parameters.

As the first step to study the possibilities of achieving a passive deorbiting with QRP tech-

nology, a reduced sail model is analysed from SRP-dominated to drag-dominated regions. A

sail geometry is proposed, consisting of two square panels attached by one side and with a

certain aperture angle. It is connected to the spacecraft bus by means of a boom.

Only planar orbits are considered as a preliminary analysis, the space vehicle moves in orbit

which lies in the ecliptic plane where the tilt of Earth is neglected. The effect of eclipses are

not considered. Solar radiation pressure, atmospheric drag, J2 and gravity gradient pertur-

bations are modelled, the attitude dynamics is reduced to the study of the third component

of the torque in body-frame, restricting in this way the planar motion. Cartesian and Gauss

formulations are presented to propagate the orbital dynamics.

The evolution of the perturbation forces and torques with respect to the sail orientation is

analysed. Poincare maps are provided to study the spacecraft stability and stable regions

close to the Sun-pointing vector and tangent to orbit direction are found in SRP and drag-

1.5. Structure of the thesis 31

dominated regions respectively.

Once the spacecraft stability is understood in SRP and drag-dominated regions, a series of

simulations are performed to study the deorbiting vehicle dynamics going through a region

where SRP and drag forces are present. Results are provided in terms of deorbiting time and

the portion of time that the spacecraft remains passively stable. Sensitivity analyses with

respect to the spacecraft configuration parameters and initial orbital parameters are carried

out. As a result, the most favourable combination of variables is identified to achieve stable

deorbiting within the minimum possible duration.

The work focuses on the study of the spacecraft deorbiting capabilities using a two-panel solar

sail. The novelty of this work is the identification of the trajectory phases of a deorbiting

satellite through a transition region where SRP and drag forces are the main perturbations,

and how the sail geometry (aperture angle and centre of mass-centre of pressure offset) and

initial orbit (eccentricity, semi-major axis and Sun-perigee angle) influence the deorbiting

time and spacecraft stability. Which may serve as a guideline for future research with the

aim of designing control laws for solar and drag sail and improving end-of-life missions.

1.5 Structure of the thesis

The mathematical definition of the problem is given in Chapter 2. First, the spacecraft struc-

ture, composed of two square panel sail and the bus is presented and modelled in Section

2.1. Secondly, the state equation composed by the attitude dynamics and the non-Keplerian

planar motion is written for a spacecraft under the effect of solar radiation pressure, atmo-

spheric drag, gravity gradient and the gravitational perturbation J2 in Section 2.2. Finally,

the ordinary differential equation system to solve the state equation is introduced in 2.3.

The SRP and drag forces and torques depend on the orientation of the sail with respect the

Sun-pointing and tangent-to-orbit directions respectively. The system stability, depending

on the spacecraft attitude with respect the perturbation direction is analysed in Chapter

3 in SRP-dominated and drag-dominated regimes separately. The analysis is performed

for four different spacecraft configurations and different eccentricities, the coupled effects of

the dominant force with J2 and gravity gradient effects are investigated. The results are

summarised and discussed in Section 3.4.

A transition region where the spacecraft moves in a domain under the coupled effects of


SRP and drag perturbation is analysed in Chapter 4. A general simulation is performed

in Section 4.1, where the features that characterise a satellite through the transition region

are identified. Sensitivity analyses are carried out to understand the influence of the sail

configuration parameters, Section 4.2, and the initial orbit parameters, Section 4.3.

Finally, in Chapter 5, a summary of results and conclusions are presented and future work

lines of research are proposed.

Chapter 2

Theoretical modelling

2.1 Spacecraft configuration. Geometry of the sail.

The sail geometry selected for this work is the one previously studied in [66]. Two panels

sail is considered as a reduction of the quasi-rhombic pyramid concept to avoid out-of-plane

forces. The panels are denoted as π+ for the upper panel and π− for the lower one. The

sail size is fully determined by its height h and width w. The resulting area of the sail

is Asail = 2hw with a total mass of ms. In Fig. 2.1a) the sail model is depicted in the

spacecraft body-frame Fb. The sail configuration is determined by means of α and d. The

first parameter refers to the aperture angle of the sail while the second refers to the distance

between the sail and bus mass centres, which is an alternative way to measure the centre

of mass-centre of pressure offset. In Fig. 2.1b) the sail is sketched in the x-y plane of the

body-frame, the green circle represents the position of the bus.

A parametrization of the sail is written in the spacecraft body-frame Fb, whose base is defined

as iiix,y,z. Cylindrical coordinates are considered, thus, the panels can be represented as

Sail : π+∪π−, π± =

(aux− r cosα,±r sinα, z)> : r ∈ [0, w] , z ∈[−h/2, h/2

], (2.1)

where r = x2 + y2, and aux (see Eq. 2.4) is a free parameter that is chosen so that the

spacecraft mass centre is located at the origin of Fb.

33

34 Chapter 2. Theoretical modelling

Figure 2.1: Sketch of the sail structure in body-frame. a) 3D sail model. b) x-y

projection.

The normal of the panels and the sail centre of mass are computed as follows

nnn± = (sinα,± sinα, 0)> , (2.2)

rrrs =

(aux− 1

2w cos , 0, 0

)>, (2.3)

being nnn the sail normal, and the subindices denoting the panel at which refers. rrrs is the sail

centre of mass.

As long as the bus mass centre lies on the x-body axis, the principal axes of inertia of the sail

are parallel to Fb. Its position is assumed to be located at rrrs + (d, 0, 0)>. As stated before,

aux is chosen to place the spacecraft mass centre in the origin, so,

aux =1

2w cosα− d mb

ms +mb

, (2.4)

where mb and ms are the bus and sail masses respectively.

In Fig. 2.2, it can be seen the relative position of the sail with respect to the bus for different

values of d. For d > 0 m the bus can be placed inside, in the tip or in front of the sail

structure.

The value of d that makes the bus to be in the tip of the sail depends on the aperture angle,

in Fig. 2.3, it is depicted this relationship for a 9.2 × 9.2 m2 square sail of 3.6 kg of weight

attached to a 100 kg bus. It can be noted that in a flat sail (α = 90°) the bus is placed on

the sail tip when d = 0 m, and if the sail has a zero aperture angle the requested distance

between mass centres is half of the panel side.

The spacecraft structure is composed by the bus and the sail. The principal axes of the

2.1. Spacecraft configuration. Geometry of the sail. 35

Figure 2.2: Relative position of the sail with respect to the bus (Fb top view). a)

d = 0 m, b) d < 0 m, c) d > 0 m, d) d > 0 m.

Figure 2.3: Value of d so that the bus in the sail tip.

bus are assumed to be parallel to the ones of Fb, thus, the moments of inertia of the whole

spacecraft are computed using the parallel axes theorem. Thus,

Ix,sc = Ix,b +h2ms

6; Iy,sc = Iy,b +

h2ms

6+D(α, d); Iz,sc = Iz,b +D(α, d) (2.5)

D(α, d) =1

6msw

2 cos2 α +d2m2

b (mb + 2ms)

(mb +ms)2 , (2.6)

where Ix,sc, Iy,sc, Iz,sc are the satellite moments of inertia, Ix,b, Iy,b, Iz,b are the bus moments


of inertia, and D is an inertial parameter. It can be seen that d plays a key role in the inertia

moments along the y and z axes, the larger |d|, the larger the inertia moments are. Therefore,

it is directly related to attitude dynamics and gravity gradient torque.

2.2 Mathematical model

Building on the previous contribution such as Miguel et al. [66], the planar orbit and attitude

dynamics considered form a coupled system of differential equations in R6: spacecraft orien-

tation and angular velocity in z axis for attitude and the evolution of the planar position and

velocity. The spacecraft trajectory evolution can be propagated either by means of position

and velocity or by means of orbital elements.

Since a planar motion is studied, the rotation dynamics of the spacecraft can be fully ex-

plained by using a single Euler angle. Therefore, the system reads

Iz,scϕ = M3 or

ϕ = Φ

Φ = M3/Iz,sc(2.7)

being ϕ ∈ [0, 2π) the Euler angle of the rotation around z-axis, that is, the spacecraft altitude,

which also represents the relative position between body and ECI frames, Φ is the rotational

angular velocity, Iz,sc is the third spacecraft inertia moment (see 2.5), and M3 refers to the

sum of the torques produced by the perturbations acting on the vehicle along z-direction.

The state vector of the complete problem can be written by means of Cartesian coordinates

[ϕ Φ rrr vvv]> (2.8)

where rrr = (rX , rY )> and vvv = (vX , vY )> are the planar spacecraft position and velocity in

ECI frame. The state vector can be also expressed by means of orbital parameters, see Eq.

2.9. Since a planar orbital dynamics is considered, where the orbit remains in the ecliptic

plane and the tilt of the Earth is neglected, the spacecraft motion is fully defined by the

semi-major axis a, eccentricity e, the summation of the argument of perigee ω and the Right

Ascension of the Ascending Node (RAAN) Ω, and the true anomaly θ. It should be noted

that Ω is considered since J2 perturbation results in precession of the node.

[ϕ Φ a e ω + Ω θ]> (2.9)

2.2. Mathematical model 37

2.2.1 Attitude dynamics

The external forces, which produce torque and act on the spacecraft, drive the spacecraft

attitude dynamics. In this work, radiation pressure, atmospheric drag and gravity gradient

effects are considered. Therefore, the total torque exerted on the spacecraft is the summation

of the torques of these perturbations MMM = MMMSRP +MMMdrag +MMMGG.

To provide explicit expressions of the considered torques, the Sun-Earth, Earth-spacecraft

vectors and relative velocity of the spacecraft with respect to the atmosphere are written in

Fb as a unitary vectors

uuus = σ1iiix + σ2iiiy + σ3iiiz; rrrs = rsuuus (2.10a)

uuuE = γ1iiix + γ2iiiy + γ3iiiz; rrrE = ruuuE (2.10b)

uuurel = ν1iiix + ν2iiiy + ν3iiiz; vvvrel = vreluuurel (2.10c)

where uuus is the Sun-Earth direction vector, uuuE is the Earth-spacecraft direction vector, and

uuurel is the relative velocity direction vector. σ21 + σ2

2 + σ23 = γ21 + γ22 + γ23 = ν21 + ν22 + ν23 = 1

are the direction cosines, rrrs is the position of the Sun with respect to the Earth and rs the

Sun-Earth distance. rrrE refers to the position of the spacecraft with respect to the Earth, and

r is the distance between the spacecraft and the Earth. Finally, vvvrel represents the relative

velocity vector and vrel is the spacecraft relative velocity with respect to the atmosphere.

Since the sail geometry is selected to avoid out-of-plane forces, only rotations around the

z-axis are of interest. In what follows, the forces and torques produced by SRP, atmospheric

drag and gravity gradient are analysed and formulated.

2.2.2 Solar radiation pressure

Solar radiation pressure is the pressure exerted upon the spacecraft surface due to the ex-

change of momentum between the vehicle and the electromagnetic field emitted by the Sun.

This includes the electromagnetic radiation which is absorbed o reflected. Consequently, the

sail area, the orientation of the sail with respect to the Sun and the reflective properties of

the sail play a key role in the contribution due to the SRP phenomenon.

The SRP force exerted in each panel of the sail can be computed as [67]

FFF±SRP = −pSRAsp (nnn± · uuus)(2η (nnn± · uuus)nnn± + (1− η)uuus

), (2.11)


being FFF SRP the solar radiation force, pSR = 4.56× 10−6 N/m2 is the solar pressure at 1 AU,

which is considered to be constant, η ∈ (0, 1) is the dimensionless reflectance of the sail, Aspis the sail panel area, and nnn is the sail normal. It should be noted that the symbol ± refers

to either panel of the sail π+ or π−.

The torque due to SRP is the sum of contribution of both panels, thereby,MMM±SRP = rrr±∧FFF±SRP .

Its component around z-axis can be written as

M±SRP,3 =

Aspmb +ms

pSR2

(k1,1σ1σ2 ± k2,0σ2

1 ± k0,2σ22

), (2.12)

where MSRP,3 is the third component of the solar radiation torque, and

k1,1 (η) = sinα[2dmb

(2η cos (2α) + η + 1

)+ w (mb +ms)

(cosα− η cos (3α)

)],(2.13a)

k2,0 (η) = sin2 α[4dηmb cosα + w (mb +ms)

(1− η cos (2α)

)], (2.13b)

k0,2 (η) = cosα[2dmb

(η cos (2α + 1

)+ ηw

(mb +ms) sinα sin (2α

)], (2.13c)

it can be noted that the coefficients k1,1, k2,0 and k0,2 depend on the bus and sail masses, the

aperture angle of the sail α, the sail width w, the distance between the bus and sail mass

centres d as well as the sail reflectance η.

Once the parameters of the bus and the sail are selected (α, d, ms, mb and η), the force and

torque due to SRP exerted on the spacecraft only depend on its orientation with respect to

the Sun (uuus), it is studied in depth in Section 3.1.

2.2.3 Gravity gradient

A spacecraft can be modelled as a structure integrated with different separated point masses.

If some parts are closer to the Earth than others when the vehicle moves on its orbit, the

gravity force exerted on these parts is different. As a consequence, the difference in force

produces a torque that rotates the spacecraft.

The spacecraft under study can be considered as two point masses, bus and sail that have

masses of a different order of magnitude. Thus, the torque produced by the gravity gradient

must be considered. The rotation of asymmetrical bodies affected by a torque due to gravity

gradient can be written as [67]

MMMGG =3µ

r3uuuE ∧ IIISCuuuE, (2.14)

2.2. Mathematical model 39

being MMMGG the gravity gradient torque, µ = GmE = 3.986× 1014 m3/s2 is the gravitational

parameter of the Earth, r is the distance between the spacecraft and the Earth, IIISC =

diag(Ix,sc, Iy,sc, Iz,sc

)and uuuE is the spacecraft-Earth direction, see Eq. 2.10b). The third

component of the torque can be computed as

MGG,3 =3µ

r3(Iy,sc − Ix,sc

)γ1γ2, (2.15)

it can be noted that for the case in which the bus is symmetrical Eq. 2.15 is reduced to

MGG,3 = −3µ

r3D(α, d)γ1γ2, (2.16)

The gravity gradient depends on the planet parameters (µ and rE), aperture angle of the

sail (α) and the distance between mass centres (d) through the inertial parameter D, see Eq.

2.5, as well as the orientation of the spacecraft with respect to the Earth (uuuE). In terms of

the orbital elements, the Earth-spacecraft direction can be written as

uuuE =(− cos (θ + w − ϕ),− sin (θ + w − ϕ), 0

)T. (2.17)

2.2.4 Atmospheric drag

The motion of an object within a gas flow displaces the gas particles resulting in a deceleration

penalty. This phenomenon is generally named drag. The atmosphere is a gaseous medium,

therefore, a vehicle moving inside experiments a drag force which depends on the medium

properties and the vehicle shape.

In this work, the area of the bus is neglected since it is considered to be small in comparison

with the sail area. Consequently, the spacecraft front area is the one of the sail. The force

due to atmospheric drag can be decomposed as the summation of the drag force in each

panel, which can be written as [67]

FFF±drag = −1

2ρv2relCDAsp (nnn± · uuurel)uuurel, (2.18)

being FFF drag the atmospheric drag force, ρ is the atmospheric density, CD ∈ (1.5, 2.5) is

the empirically determined dimensionless drag coefficient, and uuurel is the relative velocity

direction vector (see Eq. 2.10c).

The atmospheric drag torque is computed as MMM±drag = rrr± ∧ FFF±drag, whose third component

can be computed as following

M±drag,3 =

Aspmb +ms

ρv2relCD4

(k′1,1ν1ν2 ± k′2,0ν21 ± k′0,2ν22

), (2.19)


where

k′1,1 = sinα[2dmb + w (mb +ms) cosα

]. (2.20a)

k′2,0 = w (mb +ms) sin2 α, (2.20b)

k′0,2 = 2dmb cosα, (2.20c)

the coefficients k′1,1, k′2,0 and k′0,2 depend on the bus and sail masses, the aperture angle of

the sail α, the sail width w and the distance between the bus and sail mass centres d. It

can be noted that Eq. 2.13a and Eq. 2.20 are related: k′1,1 = k1,1 (0), k′2,0 = k2,0 (0) and

k′0,2 = k0,2 (0), that is, the drag coefficients are the ones for the SRP case when the sail

reflectance is zero.

The total torque depends on the medium properties through the density, the bus and sail

geometry (α, ms, mb), shape (CD) and configuration (d), as well as the orientation of the sail

with respect to the relative velocity direction (uuurel). This is analysed in depth in Section 3.2.

2.3 Orbit dynamics

The orbit evolution can be computed through the equations of motion over time in Cartesian

coordinates or by propagating the orbit parameters. The two methods are presented here

as they can be used to check if they are correctly implemented by the time to perform

simulations. Three perturbations are considered in this work, SRP, atmospheric drag and

gravitational perturbation due to the Earth oblateness. The acceleration due to the SRP and

atmospheric drag phenomena can be easily obtained from the force formulation, Eq. 2.11

and Eq. 2.18, by simply dividing by the spacecraft mass, which is the summation of the sail

and bus masses, thus

aaaSRP =FFF SRP

mb +ms

, (2.21)

aaadrag =FFF drag

mb +ms

, (2.22)

where aaaSRP is the solar radiation acceleration and aaadrag is the atmospheric drag acceleration.

The acceleration due to the Gravitational perturbation can be computed from [68], the

2.3. Orbit dynamics 41

formulation is reduced to the planar case and it reads

aJ2,1 = −3

2J2µ

r2

(RE

r

)2(x

r

), (2.23a)

aJ2,2 = −3

2J2µ

r2

(RE

r

)2(y

r

), (2.23b)

being aJ2 the acceleration due to J2 perturbation, and the subindex denoting the vector

component, J2 = 1.082× 10−3 is the dominating term of spherical harmonics, µ is the Earth

gravitational parameter, RE is the Earth radius, r is the distance between the spacecraft and

the Earth and x and y are the coordinates of the spacecraft position in the reference frame.

2.3.1 Cartesian propagation

The spacecraft motion is described by means of Cartesian coordinates. The time evolution

of state vector, Eq. 2.8, is computed through the following ODE system

dϕ

dt= Φ, (2.24a)

dΦ

dt=

1

Iz,sc

(MGG,3 +MSRP,3 +Mdrag,3

), (2.24b)

drrr

dt= vvv, (2.24c)

dvvv

dt= − µ

r3rrr + aaaSRP + aaadrag + aaaJ2, (2.24d)

where ϕ is the spacecraft yaw angle, Φ is the angular velocity in z-body axis, MGG,3, MSRP,3

and Mdrag,3 are the torques that cause rotation around the z-body axis (Eq. 2.16, 2.12, 2.19),

Iz,sc is the moment of inertia on the z-body axis, rrr is the two dimensional position in ECI

frame and r refers to its module, vvv is the two dimensional spacecraft velocity in ECI frame

and finally aaaSRP , aaadrag and aaaJ2 are the accelerations due to SRP, atmospheric drag and J2perturbations in ECI frame.

Once the evolution is computed the orbital elements can be obtained through conversion at

each time step. See algorithm 4.2 in Curtis [68] p. 197-199.

2.3.2 Gauss propagation

The evolution of the state vector, Eq. 2.9, is computed through Gauss variational equations.

Since the atmospheric drag is present, it is convenient to consider the orbit perturbations


in the tangential-normal frame Ft-n-h. Its orthonormal base is denoted as iiit,n,h, and it is

composed by a parallel component to the tangential velocity of the motion iiit, the third

component is oriented perpendicular to the orbit iiih, and the missing one is obtained as

iiin = iiih × iiii. It can be noted that iiih ‖ iiiz ‖ iiiZ since the motion is planar. The variational

equations read [69]

dϕ

dt= Φ, (2.25a)

dΦ

dt=

1

Iz,sc

(MGG,3 +MSRP,3 +Mdrag,3

), (2.25b)

da

dt=

2a2v

µat, (2.25c)

de

dt=

1

v

[2 (e+ cos θ) at −

r

asin θan

], (2.25d)

dω

dt=

1

ev

[2 sin θat +

(2e+

r

acos θ

)an

], (2.25e)

dθ

dt=

h

r2− dω

dt, (2.25f)

where a, e, ω = ω+Ω and θ are the semi-major axis, eccentricity, sum of argument of perigee

and RAAN and the true anomaly of the orbit. at, an are the components of the acceleration

on Ft-n-h, the acceleration is computed translating aaa = aaaSRP + aaadrag + aaaJ2 from ECI to

tangential-normal frame. The Earth gravitational parameter is µ, the spacecraft distance

from the Earth and its velocity are denoted as r and v respectively. Finally, h = na√

1− e2where n =

√µ/a3 is the mean motion.

Chapter 3

Orbit perturbations

This section aims to extend the work done in [66] where stable attitude regions were found

in SRP and drag dominated regions in the cases that the sail is close to the Sun-pointing

vector and spacecraft relative velocity with respect to the atmosphere vector respectively. A

stable attitude region is referred to when there is an oscillatory motion either around the

sunlight direction or around the relative velocity vector. Those can be seen as closed lines in

ϕ, Φ plots. Here, the spacecraft dynamics is analysed considering also the scenarios where

the spacecraft attitude is not close to these directions. The study of the attitude dynamics

of a spacecraft using two panels sails in SRP and drag dominated regions allows analysing

the vehicle behaviour in a transition region where both effects are comparable.

The spacecraft parameters are chosen to be technologically feasible [64]. The considered

bus and sail parameters are listed in Table 3.1. The sail area-to-mass ratio can be easily

computed Asail/ (mb +ms) ≈ 1.6 m2/kg.

Table 3.1: Spacecraft parameters.

Parameter Symbol Value Unit

Bus mass mb 100 kg

Sail mass ms 3.6 kg

Sail panel height h 9.2 m

Sail panel width w 9.2 m

Sail reflectance η 0.8 -

Drag coefficient CD 2.1 -

43

44 Chapter 3. Orbit perturbations

Since the results depend strongly on the physical parameters α and d, two values are selected

to study the spacecraft dynamics dependence with the aperture angle of the sail, and two

values of the distance between sail and bus mass centres with the aim of providing informa-

tion about how does the gravity gradient effect modifies the spacecraft dynamics. Thus, 4

different spacecraft are proposed to be studied: the possible combination of two values of

each parameter, α = 30, 45° and d = 0, 1.5 m.

The orbit stability is analysed through Poincare maps [70]. They were first introduced by

H. Poincare, allowing to reduce the study of the continuous-time system to the study of

an associated discrete-time system (map). A transversal section to the flux is defined, the

map is composed by the intersection points of the flux with respect to the section. The

correspondence between these points can be interpreted as:

• A simple periodic orbit of the dynamical system is translated to a unique fix point in

the Poincare section.

• A quasi-periodic orbit trajectory becomes a closed curve.

• A chaotic orbit motion leads to erratically distributed points.

3.1 SRP-dominated region

The SRP effect on the spacecraft depends on its attitude with respect to the Sun, see Eq.

2.11. In Fig. 3.1 it can be seen the relative position of the Sun, illustrated as a yellow sphere,

and the spacecraft with respect to the ECI frame. The angle of the Sun-pointing vector is

denoted as λ. The spacecraft yaw angle ϕ also represents the relative position between ECI

and body frames since the problem is planar.

The spacecraft orientation with respect to the sunlight direction is denoted as ϕSRP and the

Sun-spacecraft vector as uuus, they can be computed as

ϕSRP = ϕ− λ, (3.1)

uuus =(− cos (−ϕSRP ),− sin (−ϕSRP ), 0

)>. (3.2)

It is assumed that λ = n, which is the Sun mean motion, that is, the apparent motion of

the Sun around the Earth is circular with constant angular velocity, being n = 2π365×24×3600 .

Therefore, the evolution of the Sun position over time can be easily recovered as λ (t) =

λ0 + nt, where λ0 is the initial Sun direction in ECI.

3.1. SRP-dominated region 45

Figure 3.1: Spacecraft orientation with respect to the Sun. Definition of λ.

The torque and acceleration due to the structure under consideration depend on the number

of panels that face sunlight, and how they are oriented with respect to it. Namely, the

torque/acceleration can be represented by splitting the possible sail orientation, ϕSRP =

[0, 2π), into 6 regimes. The problem is sketched in Fig. 3.2, where the regimes are separated

by means of dashed lines. In Fig. 3.2b), the case in which only a portion of the sail panel is

lighted by the sunlight is illustrated.

Figure 3.2: ϕSRP definition. a) SRP regions. b) SRP regions including shadow

in one panel, w′ definition.

It can be observed that there are some orientations in which one panel partially or totally

overshadows the other, when |ϕSRP | ∈ (α, π − α). The regions are defined depending on the

number of panels and their side (front or back) that are exposed to the sunlight.


3.1.1 SRP force

There are two special cases where only a portion of one sail panel back is exposed to the

sunlight, when ϕSRP ∈(π/2, π − α

)and ϕSRP ∈

(α− π,−π/2

). In these cases the portion

of the sail panel area is denoted as A′sp and it can be computed as explained in Appendix A.

The SRP force formulation for each regime reads

FFF SRP = −FFF ′+SRP +FFF−SRP if ϕSRP ∈(π2, π − α

),

FFF SRP = FFF−SRP if ϕSRP ∈(α, π

2

),

FFF SRP = FFF+SRP +FFF−SRP if ϕSRP ∈ (−α, α) ,

FFF SRP = FFF+SRP if ϕSRP ∈

(−π

2,−α

),

FFF SRP = FFF+SRP −FFF

′−SRP if ϕSRP ∈

(−π + α,−π

2

),

FFF SRP = −FFF+SRP −FFF

−SRP otherwise,

(3.3)

where FFF±SRP is the SRP force described in Eq. 2.11, the superscript refers to the top panel

π+ or to the bottom panel π−. The negative sign before the force summand indicates the fact

that the sail panel receives sunlight from behind, so the normal to the surface is −nnn± instead

of nnn±. Finally, the ′ sign is used to note that some of the sail panels are partially shadowed

by themselves, therefore its contribution is reduced since the lighted area is smaller. The

solar radiation force with a reduced area con be computed as

FFF ′SRP± = −pSRA′sp (nnn± · uuus)

(2η (nnn± · uuus)nnn± + (1− η)uuus

), (3.4)

A′sp = hw′, (3.5)

being w′ the width of the sail panel portion which is lighted.

To better understand how the force varies depending on the regimes, its evolution in body

frame with respect to the Sun-spacecraft orientation is depicted for different α values in Fig.

3.3.

In Fig. 3.3, it can be observed that the SRP force distribution in the x axis is symmetrical

since the sail is symmetric, the bigger is the aperture angle, the bigger is the deceleration

force when the sail faces the Sun. When the Sun is located at the spacecraft back, the solar

pressure acts as a propulsive force (Fx > 0). The distribution along y axis is anti-symmetrical,

when the Sun located at y > 0 side, ϕSRP ∈ (0,−π), Fy is found to be negative. The opposite

behaviour occurs when it is located at the negative side of the y-body axis.


Figure 3.3: SRP force in Fb as a function of the sail aperture angle α and

Sun-spacecraft orientation. ϕSRP .

3.1.2 SRP torque

Since SRP force depends on the Sun-spacecraft orientation, it does the SRP torque. It can

be computed as

MSRP,3 = −M ′+SRP,3 +M−

SRP,3 if ϕSRP ∈(π2, π − α

),

MSRP,3 = M−SRP,3 if ϕSRP ∈

(α, π

2

),

MSRP,3 = M+SRP,3 +M−

SRP,3 if ϕSRP ∈ (−α, α) ,

MSRP,3 = M+SRP,3 if ϕSRP ∈

(−π

2,−α

),

MSRP,3 = M+SRP,3 −M

′−SRP,3 if ϕSRP ∈

(−π + α,−π

2

),

MSRP,3 = −M+SRP,3 −M

−SRP,3 otherwise,

(3.6)

being M±SRP,3 the SRP torque described in Eq. 2.12, the superscript refers to the top panel

π+ or to the bottom panel π−. The ′ sign is used to note that some of the sail panels is

partially shadowed by themselves, therefore its contribution is computed for a reduced area

A′sp. Finally, the negative sign before the force summand indicates the fact that the sail panel

receives sunlight from behind, so the normal to the surface is −nnn± instead of nnn±.

The SRP torque, Eq. 2.12, is the force that defines the attitude evolution, see Eq.2.8a) and

2.8b). As a consequence, one can expect that the dynamical behaviour also depends on the

Sun-spacecraft orientation, a mathematical analysis of this dependence was studied in [55].


The total torque can be written solely as a function of ϕSRP while the other dependencies

can be seen as physical parameter. It can be noted that the attitude dynamics is written as

Eq. 2.7 so the system equilibria are the points such that Φ = 0, MSRP,3 = 0, and as MSRP,3

does not depend on Φ, finding equilibria reduces to finding zeros of MSRP,3. The SRP torque

evolution with respect to ϕSRP is depicted in Fig. 3.4 for a sail with η = 0.25 and d = 0, and

for three different aperture angles, α1 = 10°, α2 = 19.91° and α3 = 30°.

Figure 3.4: MSRP,3 as a function of the Sun-Spacecraft orientation

From Fig. 3.4, one can guess that the number of zeros of MSRP,3 are at least 2, (0, π) and

(0, 0), and there can be, either 2, 4 or 6 equilibria, depending on the value of the parameters

η, α and d. These are an example for which there are 6, 4 and 2 equilibria respectively.

Consequently, three scenarios can be found:

• 2 equilibria: 1 stable equilibria at ϕSRP = 0°and 1 unstable equilibrium point at ϕSRP= 180 deg.

• 4 equilibria: 2 bifurcation points appear where MSRP,3 is tangent to the horizontal axis.

• 6 equilibria: two pairs of stable-unstable equilibria that bifurcate from the previous

bifurcation points.

It should be pointed out that the value of α2 is a numerically approximated value. These

scenarios depend on the three parameters previously cited. Fixing one parameter, the study

is simplified and the regions which define the three scenarios can be represented. These cases

are presented in Fig. 3.5a) when the sail reflectance is fixed and in Fig. 3.5b) when d is

fixed. A bifurcation line, representing the set of points where there are exactly 4 equilibria,

separates the region of 6 equilibria with respect to the 2 equilibria.

To understand the evolution of the bifurcation curves when the fixed parameter varies, some


Figure 3.5: Bifurcation curves. a) d-α plane. b) η-α plane.

bifurcation curves for fixed sail reflectance are represented, in d-α plane, in Fig. 3.6a) for η

values ranging from 0 to 1. In Fig. 3.6b) the bifurcation curves are illustrated in η-α plane

for fixed d values.

Figure 3.6: a) Bifurcation curve evolution for different values of sail reflectance η

in d-α plane. b) Bifurcation curve evolution for different d in η-α plane.

In Fig. 3.6a), it can be seen that when η increases, the 2 equilibria region grows toward

d < 0 m and α > 45° (southeast side). It is worth noting that for d > 0.34 m the 2 equilibria

scenario is always present for all values of η and d. In Fig. 3.6b) a wider 2 equilibria region is

observed for higer sail reflectance. When d decreases, this region becomes smaller and moves

toward large η and α values (northest side).

In the case under study, the sail reflectance is fixed, see Tab. 3.1, and the corresponding

bifurcation curve is the one presented in Fig. 3.5a). It can be seen that for positive values

of d, the 2 equilibria scenario is always present. This is the most favourable case from a

stability point of view since the stability around ϕSRP = 0 is pursued, and the existence of


bifurcation or saddle points reduces the stable region size centred at this point.

The phase space of the system considering only the SRP perturbation is presented in Fig. 3.7

for different combinations of α and d, indicated as title in each of the panels. Dashed lines

are depicted at |ϕSRP | = α to represent the discontinuity in the regime centred at ϕSRP = 0.

The illustrated lines are obtained by propagating circular orbits under the effect of SRP

perturbation for different initial Sun-spacecraft orientations. Circular orbits are considered

since the only purpose is to illustrate the spacecraft attitude dynamics.

Figure 3.7: Phase space of system dynamics when only the SRP perturbation is

present for a circular orbit. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c)

α = 30°, d = 0 m. d) α = 30°, d = 1.5 m.

A pendulum shape is obtained for the 2 equilibria scenario. Configurations with low sail

aperture angle of the sail and higher distance between sail and bus mass centres present

smaller angular velocity ranges as it can be seen in Fig. 3.7d) while larger angular velocities

are obtained for higher α and low d values, which can be observed in Fig. 3.7a)

3.1.3 Orbit evolution

Once the system dynamics is understood for circular orbits under the effect of SRP, a more

realistic case should be addressed. The spacecraft dynamics under the SRP, gravity gradient

and J2 perturbations are analysed for circular and low-eccentric orbits. To see the spacecraft


dynamics evolution a Poincare map of an orbit propagated over 250 periods is presented.

The Poincare section Σ, illustrated in Fig. 3.8, is defined as

Σ =

(x, y) ∈ R2 | x > 0, y = 0, (3.7)

in this way, the orbit dynamics of an Earth orbiting spacecraft in ECI is always transversal

to Σ, and hence, the Poincae map is well defined.

Figure 3.8: Poincare section Σ.

Poincae maps are depicted for the four different spacecraft configurations under study from

an initial orbit of h0 = 1500 km, ω = 0°, e = 0, and λ0 = 90° in Fig. 3.9. The same variables

are represented for the same scenario but with e = 0.1 in Fig. 3.10. The illustrated points

represent the value of the variables when the vehicle pass trough the Poincae section, see Eq.

3.7. Each colour correspond to a different initial condition, being the blue ones the cases in

which the initial spacecraft attitude is closer to the Sun-pointing vector.

The effect of the gravity gradient can be observed in Fig. 3.9b) and Fig. 3.9d), when

d increases, makes the spacecraft dynamics becomes more chaotic. That is, the spacecraft

attitude oscillates more in the vicinity of the Sun-pointing direction, achieving ϕSRP > ϕSRP,0along its motion. If the spacecraft gets closer to the Earth, the gravity gradient torque

grows, and its effect could compromise the stability properties under consideration. The

same behaviour is found for small aperture angle of the sail in Fig. 3.9c) and Fig. 3.9d).

Smaller angular velocity values are obtained for higher α and lower d.

When eccentricity increases, the points on the maps spread out as it can be noticed in Fig.

3.10, which is the mean difference with respect to the circular case, Fig. 3.9. The vehicle

attitude becomes more unpredictable especially for non-zero d as the gravity gradient effect

becomes stronger. Stable regions are referred when the spacecraft dynamics remains inside a

close ϕSRP region, the most favourable scenario from this point of view is found for circular

orbits and small values of ϕSRP,0 and d.


Figure 3.9: Poincare maps for different spacecraft configurations under SRP,


ω = 0° and e = 0. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c) α = 30°, d = 0

m. d) α = 30°, d = 1.5 m.

Figure 3.10: Poincare maps for different spacecraft configurations under SRP,


ω = 0° and e = 0.1. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c) α = 30°,d = 0 m. d) α = 30°, d = 1.5 m.

3.2. Drag-dominated region 53

As eccentricity increase, different results can be obtained depending on the initial position of

the Sun λ0 and ω0. Moreover, this effect is more relevant when eclipses are considered and

the atmospheric drag perturbation is present. This study is proposed for future work.

3.2 Drag-dominated region

The atmospheric drag effect on the spacecraft depends on its attitude with respect to the

relative velocity to the atmosphere since the sail shape provides a varying frontal area when

the spacecraft attitude varies. In Fig. 3.11, the general scenario is sketched, where the

direction of the relative velocity of the vehicle with respect to the atmosphere is defined as

δ. The spacecraft attitude is denoted as ϕ, which also represents the angle between ECI and

body frames.

Figure 3.11: Position of the spacecraft with respect to the relative velocity

vector. Definition of δ.

Similarly as done with SRP in Eq. 3.1 and Eq. 3.2, the direction of the spacecraft relative

velocity with respect to the atmosphere is denoted as ϕdrag, therefore, the relative velocity

direction in body frame can be computed as

ϕdrag = ϕ− δ, (3.8)

uuurel =(cos (−ϕdrag), sin (−ϕdrag), 0

)>, (3.9)

In Fig. 3.12, the problem is sketched in body-frame. Since the area contributing to the drag

force depends on ϕdrag, it also does the atmospheric drag, hence, similarly as described for


the SRP in Section 3.1, the domain of possible sail orientation can be split into 6 regimes.

These regimes are illustrated in the sketch by means of dashed lines. In Fig. 3.12b), the case

in which only a portion of the panel back generates drag is depicted.

Figure 3.12: ϕdrag definition. Left: general drag regions. a) Drag regions. b)

Drag regions including shadow in one panel, w′ definition.

It can be noticed that for some orientations some of the panels is partially or totally shadowed

by the other, that is∣∣ϕdrag∣∣ ∈ (α, π − α). Therefore, the regions are defined depending on the

number of panels and their side (front or back) that face uuurel. In Fig. 3.12b), it is illustrated

the case in which only a portion of one panel back generates drag.

3.2.1 Drag force

It can be noted that the regions are the same found when only SRP is acting on the sail.

Again, there are two conditions in which only a part of the sail panel contributes when when

ϕdrag ∈(π/2, π − α

)and ϕdrag ∈

(α− π,−π/2

). The drag force exerted in the spacecraft is

defined in the following intervals

FFF drag = −FFF ′+drag +FFF−drag if ϕdrag ∈(π2, π − α

),

FFF drag = FFF−drag if ϕdrag ∈(α, π

2

),

FFF drag = FFF+drag +FFF−drag if ϕdrag ∈ (−α, α) ,

FFF drag = FFF+drag if ϕdrag ∈

(−π

2,−α

),

FFF drag = FFF+drag −FFF

′−drag if ϕdrag ∈

(−π + α,−π

2

),

FFF drag = −FFF+drag −FFF

−drag otherwise,

(3.10)

where FFF±drag is the drag force described in Eq. 2.18, the upper superscript refers to the top

panel π+ to the bottom panel π−. The negative sign before the force summand indicates the


fact that the sail panel back normal points to relative velocity direction, so the normal to the

surface is −nnn± instead of nnn±. Finally, the ′ sign is used to note that some of the sail panels is


A′sp, see Appendix A. The drag force with a reduced area can be computed as

FFF ′drag± = −1

2ρv2relCDA

′sp (nnn± · uuurel)uuurel, (3.11)

A′sp = hw′. (3.12)

Drag force depends on the spacecraft altitude through the density. The atmosphere density

depends on the altitude and the temperature, therefore, it can change during time. In this

work the Exponential Atmosphere Model [71] Table 8-4 is used. The density profile in altitude

is illustrated for decimal and logarithmic scales in Fig. 3.13.

Figure 3.13: Density profile. a) Decimal scale in x axis and y axis. b) Decimal

scale in y axis, logarithmic scale in x axis.

Drag force can be expressed as the product of the density, relative velocity magnitude and a

force term which depends on the spacecraft orientation

FFF drag = ρv2relfff(ϕdrag

), (3.13)

fff(ϕdrag

)= −1

2CDAsp (nnn · uuurel)uuurel, (3.14)

being fff a parameter which only depends on the vehicle relative orientation with respect to

atmospheric drag once the spacecraft parameters CD and Asp, introduced in 3.1, are fixed.

To see the evolution of the drag force when ϕdrag varies, the force term fff is depicted in body

frame with respect to the relative velocity direction in Fig. 3.14.


Figure 3.14: Drag force term fff in Fb as a function of the sail aperture angle α

and relative velocity orientation with respect to the aircraft ϕdrag.

In Fig. 3.14, it can be observed that in the same way that the SRP force, the distribution in

the x-body axis is symmetrical since the sail is symmetric in this axis. The distribution in y-

body axis is anti-symmetrical. When the vehicle moves toward positive y, ϕdrag ∈ (0°,−180°),the drag decelerates the vehicle (fy < 0) and vice versa.

3.2.2 Drag torque

Since drag force depends on the relative velocity angle, so does the drag torque. The same

regions are found for this effect, which can be formulated as a function of the different regimes

as

Mdrag,3 = −M ′+drag,3 +M−

drag,3 if ϕdrag ∈(π2, π − α

),

Mdrag,3 = M−drag,3 if ϕdrag ∈

(α, π

2

),

Mdrag,3 = M+drag,3 +M−

drag,3 if ϕdrag ∈ (−α, α) ,

Mdrag,3 = M+drag,3 if ϕdrag ∈

(−π

2,−α

),

Mdrag,3 = M+drag,3 −M

′−drag,3 if ϕdrag ∈

(−π + α,−π

2

),

Mdrag,3 = −M+drag,3 −M

−drag,3 otherwise,

(3.15)

where M±drag,3 is the drag torque described in Eq. 2.19, the superscript refers to the top panel

π+ or to the bottom panel π−. The ′ sign is used to note that some of the sail panels is



A′sp. Finally, the negative sign before the force summand indicates the fact that the sail back

points to the relative velocity direction, so the normal to the surface is −nnn± instead of nnn±.

The system can be understood in the same way that in the SRP case, Eq. 3.15, since the

term which depends on the Sun-spacecraft direction is equivalent to the one depending on

relative velocity direction in the drag torque, see Eq. 2.20. Thus, the three scenarios found

in Section 3.1 can appear. In the drag-dominated region, one of the three driven parameters

is fixed, η, as a consequence, the bifurcation curve is reduced to the one in which the sail

reflectance is zero, represented in Fig. 3.6a) in blue. Three scenarios for d > 0 m can be

observed depending on α. Hence, the dynamics phase space can be divided into one or three

stable regions depending on the considered spacecraft configurations with the parameters

introduced in Table 3.1, existing only the 2 equilibria scenario for d > 0.3422 m.

The phase space of the system considering only the atmospheric drag perturbation is pre-

sented in Fig. 3.15 for different combinations of α and d, indicated as title in each of the

panels. Dashed lines are depicted at∣∣ϕdrag∣∣ = α to represent the discontinuity in the regime

centred at ϕdrag = 0. The illustrated lines are obtained by propagating circular orbits for

different initial spacecraft orientations with respect to the relative velocity vector. Circular

orbits are considered since the only purpose is to illustrate the spacecraft attitude dynamics.

Figure 3.15: Phase space of system dynamics when only the atmospheric drag

perturbation is present for a circular orbit and h0 = 750 km. a) α = 45°, d = 0

m. b) α = 45°, d = 1.5 m. c) α = 30°, d = 0 m. d) α = 30°, d = 1.5 m.


Two scenarios are present depending on d, as can be seen in Fig. 3.15. 6 equilibria scenario

is obtained for d = 0 m cases, and the 2 equilibria one when d = 1.5 m. Large angular

velocities can be observed when α increases and d decreases.

3.2.3 Orbit Evolution

The spacecraft dynamics is studied in drag-dominated region when gravity gradient and J2perturbation are also present. To see the spacecraft dynamics evolution, Poincare maps of

orbits propagated over 250 periods are depicted. The Poincare section is the same defined

in SRP section, see Eq. 3.7 and Fig. 3.8. These maps are illustrated in Fig. 3.16 for the

spacecraft configuration under study, from an initial orbit of h0 = 750 km, ω = 0° and for

e = 0. The same variables are represented for the same scenario but with e = 0.1 in Fig.

3.17. The illustrated points represent the variables value when the vehicle pass trough the

Poincare section. Each colour correspond to a different initial condition, being the blue ones

the cases in which the initial spacecraft attitude is closer to the relative velocity vector.

Figure 3.16: Poincare maps for different spacecraft configurations under

atmospheric drag, gravity gradient and J2 perturbations with an initial orbit of

h0 = 750 km, ω = 0° and e = 0. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c)

α = 30°, d = 0 m. d) α = 30°, d = 1.5 m.

It can be seen in Fig. 3.16 that when d is small, the system dynamics shows three stable

regions, where the attitude oscillates around a stable point, as can be expected from Fig.


3.15. When d increases the gravity gradient torque becomes the dominant force, and two

stable equilibrium points can be observed when d = 1.5 m. It is worth noting that the dots

do not describe the theoretical lines obtained in Fig. 3.15, they oscillate around these lines

as a consequence of J2 and gravity gradient effects.

Figure 3.17: Poincare maps for different spacecraft configurations under


h0 = 750 km, ω = 0° and e = 0.1. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m.

c) α = 30°, d = 0 m. d) α = 30°, d = 1.5 m.

If the eccentricity increases, the change in altitude during the same period lead to chaotic

and unpredictable attitude dynamics (points spread in all the region), as it can be seen in

Fig. 3.17. Only the non-centred equilibrium regions of the gravity gradient torque remain,

what suggests that, in the drag-dominated region, small values of d and α in low eccentric

orbits ensure that the vehicle remains in an attitude stable region around the relative velocity

vector.

With the purpose of understanding the attitude behaviour when atmospheric drag torque is

stronger than gravity gradient torque, a simulation is performed closer to the Earth. In this

way, Poincare maps are provided in Fig. 3.18, from an initial orbit of h0 = 600 km, ω = 0°and for e = 0.

In this case, the atmospheric drag torque is bigger than the gravity gradient torque when

d = 1.5 m, and the stable regions obtained for the ideal case in Fig. 3.15 can be visualised,

that is, 2 equilibria when d = 1.5 m and 6 equilibria when d = 0 m. The orbit perturbations


Figure 3.18: Poincare map for different spacecraft configurations under


h0 = 600 km, ω = 0° and e = 0. a) α = 45°, d = 0 m. b) α = 45°, d = 1.5 m. c)

α = 30°, d = 0 m. d) α = 30°, d = 1.5 m.

make the points to change between the ideal curves, that is, what before seemed to be an

oscillating state with constant amplitude, see Fig. 3.15, now has variable amplitude. The

gravity gradient torque intensifies this effect.

With the purpose of verifying the regions previously identified, three circular orbits are

propagated from 600 km until deorbit (120 km) with different initial altitude with respect

to the relative velocity direction. In Fig. 3.19, the phase space of the vehicle attitude and

the altitude evolution can be observed for a α = 30 °and d = 0 m spacecraft. The initial

spacecraft attitude, ϕdrag,0, is chosen to place the dynamics in the three different regions

observed in Fig. 3.18c).

In Fig. 3.19a), it can be seen that depending on the initial value of ϕdrag, the spacecraft

attitude oscillates around ϕdrag ≈ 120° (red dots), and ϕdrag = 0° (blue dots). When ϕdrag,0 =

145° (yellow dots), spacecraft attitude oscillates around the origin although it can reach any

value (yellow dots spread horizontally), larger angular velocities are observed for this case.

In Fig. 3.19b), the deorbiting time is found to be slightly shorter when the initial spacecraft

orientation is closer to the relative velocity vector, which lead to narrow stable regions (red

and blue curves).

3.3. Model validation 61

Figure 3.19: Orbit evolution until deorbit in drag-dominated region for three

different initial spacecraft attitude with respect to relative velocity direction,

α = 30°, d = 0 m h0 = 600 km, ω = 0° and e = 0. a) Phase space. b) altitude

evolution.

3.3 Model validation

As explained in Section 2.3, there are two possibilities to propagate the orbit, computing the

evolution of the position and velocity vector by means of Cartesian propagation or calculating

the planar orbit parameters through Gauss propagation. Both methods are implemented and

compared to verify the model. In this way, semi-major axis, eccentricity, ω, true anomaly

altitude and spacecraft attitude evolution computed by means of Cartesian and Gauss prop-

agation during 100 orbits periods are depicted in Fig. 3.20. The initial orbit parameters

are randomly chosen, h0 = 650 km, e = 0.001, ω = 208°, the initial position of the Sun is

λ0 = 90°, and the spacecraft configuration is α = 45°, d = 0.5 m.

The predicted orbit evolution is the same in both cases at the beginning as it can be seen in

Fig. 3.20, at time ≈ 2 days, differences between both methods appear, which can be easily

visualised in semi-major axis and attitude evolution. The selected initial altitude is placed in

a region where SRP and atmospheric drag forces are equivalent and the differences observed

can be caused by the stiffness of the problem when the spacecraft is tilting.

With the aim of ensuring that the vehicle is placed in either SRP-dominated or drag-

dominated regimes, the same simulation is performed with an initial altitude of 1000 km

and 500 km respectively. To compare the results, the evolution of the same parameters over

100 periods are depicted in Fig. 3.21 for the SRP-dominated case and Fig. 3.22 when the

atmospheric drag is the dominant force.


Figure 3.20: Model validation. Comparison between Cartesian propagation and

Gauss propagation in transition region.


Gauss propagation in SRP-dominated region.

A perfect matching is obtained in SRP and drag dominated regions, as it can be observed in

Fig. 3.21 and Fig. 3.22, what suggests that the attitude prediction difference in the transition

region, where SRP and atmospheric drag perturbations are of a similar order of magnitude,

is likely to be due to the problem stiffness.

Cartesian propagation method is found to be more rigid when the problem tolerances vary.

Moreover, it does not present singularities for circular orbits. Once both methods are vali-

3.4. Final remarks 63


Gauss propagation in drag-dominated region.

dated, one is selected to perform the simulations avoiding extra computational costs, there-

fore, the Cartesian method is selected due to the cited advantages.

3.4 Final remarks

Depending on the values of η, α and d, three qualitatively different scenarios have been

identified characterised by the existence of either 6, 4 or 2 equilibria found in Φ = 0 rad/s

curve. The sail reflectance is fixed in both regimes, in SRP regime it is fixed by technology

development to η = 0.8, while the drag regime corresponds to the η = 0 case. Once one

parameter is fixed, the values of d and α determine which dynamic scenario describes the

spacecraft motion.

SRP-dominated regime was studied in previous work [66], the spacecraft dynamic was found

to be stable for d > dmin, being dmin < 0 m. The stable region was found in (−π + α, π + α)

since only the orientations in which one or two panels face the Sun were considered. How-

ever, in this document, all the possible orientation of the sail with respect the Sun-pointing

direction have been taken into account, ϕSRP ∈ (0, 2π]. Bifurcation curves and phase space

graphics show that only the 2 equilibria scenario appear when d ≥ 0 m. This scenario is

characterised for one stable point at the origin and one saddle point placed at π. As a

consequence, the stable region is expanded when all possible the orientations of the sail are


considered.

Stabilising properties in drag-dominated regime were also studied in [66], and narrow stable

regions were found around the relative velocity direction. Here, the study has been extended

and all possible orientation of the sail with respect to the relative velocity direction, ϕdrag ∈(0, 2π]. It has been found that three possible scenarios can describe vehicle dynamics when

d ≥ 0 m. For lower d values, 6 equilibria scenario is present, consisting of 1 stable point

on the origin, 1 saddle point in π, and symmetrical saddle-stable pair. For larger d values,

the 2 equilibria scenario, which is previously described, appear. The intermedium scenario

is similar to the last one but adding two symmetrical bifurcation points. However, it is

unlikely to appear in reality since a slight variation of some parameter translates the attitude

dynamics into either 2 or 6 equilibria scenario. In any case, a wider stable region with respect

to the previous research is found in this regime.

The same gravity gradient disturbing dynamic effect found in [66] has been observed when d

increases, that is, the spacecraft attitude oscillates more in the vicinity of the perturbation

direction. The effect becomes more critical in drag regime since the vehicle is closer to the

Earth surface. Finally, when eccentricity increases, the changes in altitude along the vehicle

trajectory increases the chaotic motion leading to unpredictable attitude dynamics, especially

in drag-dominated regime.

Chapter 4

Numerical results

Once spacecraft dynamics has been studied in the SRP-dominated region, Section 3.1, and

the drag-dominated region, Section 3.2, a depth study in a transition region, where both

effects SRP and atmospheric drag are present, is carried out. Stable regions have been found

in the previous section when either SRP or atmospheric drag is the dominant force. For

high altitudes, when SRP is the governing force, stable regions are identified around the Sun-

pointing direction. In the drag-dominated region, the stable domains are the ones around

the spacecraft relative velocity direction. Since these forces and torques act in a different di-

rection, in regions where their magnitude is comparable, one expects uncontrolled spacecraft

with sails becomes eventually tumbling. Gravity gradient torque and J2 perturbation also

are disturbances that can take spacecraft out of stable oscillation mode.

As reported before, for a fixed bus configuration, the attitude of the spacecraft along its

trajectory depends on several parameters: initial orbit (semi-major axis a, eccentricity e,

argument of perigee ω), geometry of the spacecraft (α, d, As/m) and the initial position of

the Sun (λ0). Computational simulations have been performed to understand the spacecraft

behaviour in the transition region and to know the influence of the driving parameters. Tum-

bling conditions are considered when the sail is not pointing neither the Sun-spacecraft nor the

relative velocity directions, which is numerically quantified as |ϕSRP | > 90°and∣∣ϕdrag∣∣ > 90.

The work aims to find a stable attitude orbits in the transition from SRP to drag-dominated

regions to apply in deorbiting schemes at the end of the satellite operational life. In this

framework, the deorbiting time, and the spacecraft attitude along its trajectory are of special

interest. Therefore, the simulated results are provided in terms of deorbiting time, simulation

time in which the spacecraft starts to tumble and the altitude in which this event occurs,

65

66 Chapter 4. Numerical results

as well as the percentage of the trajectory that the vehicle remains in stable attitude. Some

representative cases are illustrated in terms of the altitude, semi-major axis, and eccentricity

evolution. Along with the relative spacecraft orientation with respect to the Sun-pointing

direction ϕSRP and the relative velocity direction ϕdrag, allowing an easy visualisation of the

tumbling state.

4.1 General simulation

A general simulation is performed to understand how the vehicle behaves in a transition

region from SRP-dominated to drag-dominated regimes. As a first step, the sensitivity with

respect to α and d in quasi-circular orbits is analysed. The spacecraft stability has been

studied for these parameters and stable regions under the presence of a dominant force (SRP

or atmospheric drag) were provided in previous sections.

4.1.1 Simulation parameters

The chosen simulation parameters are those in Table 4.1. Three significative values between

totally closed and flat sail configuration of the sail aperture angle are selected, and three

values of the distance between the bus and sail mass centres are chosen to understand the

gravity gradient torque influence during the transition. Therefore, 9 different spacecraft

are simulated. Three different eccentricities are selected to see the effect when the orbit

becomes elliptic. An initial altitude of 850 km is taken since this region is considered as

SRP-dominated where the atmospheric drag is also present but with lower strength, the

initial spacecraft position is selected in such way that φ = λ = δ, allowing in this way the

dominant force to drive the vehicle dynamics.

The initial spacecraft attitude with respect to the Sun and the relative velocity directions

is illustrated in Fig. 4.1. The Sun and the Earth are represented as yellow and blue circles

respectively. The ECI frame axes are denoted as X and Y , while the x and y axis refer to

the body frame. The initial attitude is defined as ϕSRP = 0° and ϕdrag = 0°. Therefore, the

initial motion is driven by the dominant force at the initial altitude. The relative position of

the Sun with respect to the orbit perigee is the same in all the simulated cases since λ0 and

ω0 are fixed.

4.1. General simulation 67

Table 4.1: General Simulation. Parameters.

Parameter Unit Value

α deg 30 45 60

d m 0 1.5 3

As/m kg/m2 1.68

e - 0 0.001 0.01

h0 km 850

λ0 deg 90

ω0 deg 0

Figure 4.1: Initial spacecraft attitude with respect to the Sun and the relative

velocity direction.

4.1.2 Results

To understand the deorbiting and stabilising spacecraft capabilities, the deorbiting time, and

the time and altitude in which the vehicle starts to tumble are shown in Fig. 4.2. Tumbling

conditions are considered when neither ϕSRP ∈ (−90°, 90°) nor ϕdrag ∈ (−90°, 90°). In Fig.

4.2a), the deorbiting time is observed to range from 1200 and 3600 days. All the cases end

tumbling as it can be seen in Fig. 4.2b) and Fig. 4.2c), however, when d is small (left

side) the tumbling motion starts at lower altitudes, and not at the simulation beginning as a

consequence of lower gravity gradient torque as shown in Chapter 3. When α = 45, 60° and

d = 0 m (northwest side of each panel), the deorbiting times are smaller where only a small

portion of this time is under tumbling conditions at the end of the spacecraft trajectory.


Figure 4.2: Simulation results as a function of α, d and e for a spacecraft going

through transition region. a) Deorbiting time in days. b) Time in days in which

the spacecraft starts to tumble. c) Tumbling altitude in km.

To better understand the stability evolution of the cases that allow smaller deorbiting times,

spacecraft attitude and semi-major axis evolution when α = 60° and d = 0 m in a circular

orbit are represented in Fig. 4.3a). The spacecraft attitude is illustrated in Fig. 4.3b) by

means of the relative orientation with respect to the Sun-pointing and the relative velocity

vectors, ϕSRP and ϕdrag respectively. The vehicle is in tumbling state when |ϕSRP | > 90°and

∣∣ϕdrag∣∣ > 90°, that is, when blue and red points are located up or down the stable region

delimited by two dashed lines. In the beginning, the vehicle remains in SRP-stable region,

blue dots close to zero, which last almost the simulated time. It enters in tumbling, blue

and red points out of boundary dashed limits, at the end of the deorbiting (time ≈ 1150

days) when swiftly falls to the Earth surface. This last deorbiting stage is characterised for a

fast altitude decay where the atmospheric drag becomes the dominant force, pushing the sail

4.1. General simulation 69

to point towards the relative velocity vector, as it can be observed, red points agglomerate

around zero while blue points spread out before re-entering.

Figure 4.3: Orbit evolution for a α = 60°, d = 0 m spacecraft, e = 0. a) Altitude

and semi-major axis evolution. b) Attitude evolution.

To investigate the spacecraft behaviour in the cases where it starts to tumble earlier, the

case in which α = 45° and d = 1.5 m in a low-eccentric orbit, e = 0.01 is selected as a

representative case. Its altitude, semi-major axis and attitude evolution are depicted in Fig.

4.4. In the beginning, the spacecraft moves in SRP stable region although it oscillates more

than the previous case, Fig. 4.3b), around the Sun-pointing vector. The tumbling region is

observed from early (time ≈ 100 days), where blue and red points surpass the dashed lines.

Later, it can be noted some regions where the spacecraft stabilises with the Sun-pointing

vector (time ∈ (300, 700) days and time around 1700 days), however, they do not remain for

a long time. This case is considered a chaotic motion since once the tumbling state starts,

the spacecraft behaviour becomes unpredictable.

It is worth noting that between 400 and 600 km of altitude, the strength of drag force sig-

nificantly increases. Consequently, the orbit energy decreases faster, which can be identified

throughout the high reduction of semi-major axis and altitude as it can be seen in Fig. 4.3a)

and Fig. 4.4a) close to the simulation ending. Since the satellite is closer to the Earth surface,

atmospheric drag becomes the dominant force and the vehicle tries to stabilise around the

relative velocity direction. If the drag stabilisation is achieved, it does not last for long as it

can be noticed in Fig. 4.3b).


Figure 4.4: Orbit evolution for a α = 45°, d = 1.5 m spacecraft, e = 0.01. a)

Altitude and semi-major axis evolution. b) Attitude evolution.

4.2 Sensitivity analysis: α and d

In Chapter 3, the sail configuration parameters, aperture angle of the sail α and the cen-

tre of mass-centre of pressure offset throughout d, were found to be driven parameters for

the dynamical spacecraft behaviour when either SRP or atmospheric drag is the dominant

force. Moreover, in the previous section, the transition region where both effects, SRP and

atmospheric drag, are present they were also found to be influencing parameters. These

parameters can be easily changed along the spacecraft motion by means of on-board mecha-

nisms that allow to open/close the sail panels and extend/retract the boom that connects the

sail with the bus. To better understand their influence, a sensitivity analysis is performed.

4.2.1 Simulation parameters

8 different values of α are selected between totally closed and flat sail configuration, α = 0, 90°respectively. 7 values of d ranging from 0 to 3 m are selected, since negative values were found

to play against stability in [66] and larger values would considerably enhance the negative

stability effect due to gravity gradient torque, in total 56 spacecraft to be simulated. The

parameters can be found in Table 4.2, where an initial circular orbit, which represents the

most favourable scenario from a stable point of view as shown in Section 3, with an initial

4.2. Sensitivity analysis: α and d 71

altitude in SRP-dominated region, where drag effect is still low. It should be noted that all

cases are simulated for the same relative position of the Sun with respect to the orbit perigee

since λ0 and ω0 are fixed.

Table 4.2: Sensitivity Analysis: sail configuration. Simulation Parameters.


α deg 10:10:80

d m 0:0.5:3

As/m m2/kg 1.68

e - 0

h0 km 850

λ0 deg 90

ω0 deg 0

4.2.2 Results

A preliminary analysis suggests that tumbling motion cannot be avoided, therefore, the

target orbits are those where stable oscillations around any of the two directions (sunlight or

relative velocity) are maintained for the longer time span. To investigate and compare the

deorbiting capabilities of the different simulated satellites, the deorbiting time, the portion

of the trajectory that the spacecraft remains stable, the simulation time at which the vehicle

starts to tumble, and the initial altitude of this event are depicted in Fig. 4.5. Each grid

vortex represents a spacecraft whose α and d values are written in y and x axes respectively.

As previously stated, the tumbling state is considered when the sail points neither Sun-

pointing nor the relative velocity vectors.

In Fig. 4.5a), it can be seen that the deorbiting time is found to range between 1000 and

3500 days, and for larger α and smaller d (northwest side), the deorbiting times are found to

be the smallest. According to Fig. 4.5b), the spacecraft configurations in which α > 40°and

d < 1.5 m (yellow boxes), more than 90% of the time the satellite attitude dynamics remains

stable. In Fig. 4.5c), it can be observed that for small α and large d (southeast side of

the map) the tumbling motion starts from the beginning of the simulation. In Fig. 4.5d),

it can be seen that the altitude at which the spacecraft becomes tumbling ranges from 600

to 850 km, showing that tumbling motion cannot be avoided, the lower values (blue boxes)

correspond to the cases in which the spacecraft remains more time in stable mode.


Figure 4.5: Sensitivity analysis with respect to α and d. a) Deorbiting time in

days. b) Portion of time that the spacecraft remains stable. c) Time until the

spacecraft starts to tumble in days. d) Altitude in which the spacecraft starts to

tumble in km.

When the spacecraft attitude follows the Sun-pointing direction, the effective area of the sail

contributing to the drag force changes in time since the Sun-pointing and relative velocity

directions evolve in a different way. When the tumbling motion starts, the average effective

area in the drag contribution can increase. As a consequence, the tumbling motion enhance

its effect and the satellite decays faster to the Earth’s surface. This behaviour occurs in some

cases, for instance, when α = 30°, d = 1.5 m from a circular orbit. With the aim of observing

this event, the altitude and semi-major axis evolution of the case are depicted in Fig. 4.6a)

as well as the relative orientation of the sail with respect to sunlight and atmospheric drag

throughout ϕSRP and ϕdrag respectively, in Fig. 4.6b).

It can be observed that the spacecraft starts with a stable motion around the Sun-pointing

vector, blue dots are close to the origin, and the stability with respect to the atmospheric

drag perturbation is not achieved. A slight concentration of blue points can be seen around

ϕdrag = ±100° as a result of gravity gradient torque as observed in Fig. 3.16. The altitude

and semi-major axis remain almost constant, almost horizontal lines in Fig. 4.6a), until time

≈ 2300 days, where tumbling state starts, |ϕSRP | > 90° and∣∣ϕdrag∣∣ > 90°. It can be noted

that the slope of the altitude and semi-major axis evolution lines decreases before the fast

decay at the very end due to the increase of average area contributing to atmospheric drag

in tumbling state. Before the simulation ends, the red points agglomerate showing the drag

4.3. Sensitivity analysis: e and ω 73

Figure 4.6: Orbit evolution for a α = 30° , d = 1.5 m spacecraft, e = 0. a)

Altitude and semi-major axis evolution. b) Attitude evolution.

stabilisation effect.

4.3 Sensitivity analysis: e and ω

The deorbiting possibilities by means of solar sail were found to be dependant on the orbit

parameters such as eccentricity, semi-major axis and the relative position of the sun with

respect the orbit perigee [17, 18]. In this section, the influence of the eccentricity and Sun-

perigee angle on the region where SRP and atmospheric drag perturbations are present is

analysed.

4.3.1 120 - 1000 km of altitude

Since target orbits are those in which the spacecraft is placed at altitudes in which the

atmospheric drag is relevant, a simulation with a fixed semi-major axis a = 1000 +RE km is

performed, low-eccentricities are selected in such way that the initial altitude is below 1000

km.


4.3.1.1 Simulation parameters

The effect of orbit eccentricity and the relative direction of the sunlight with respect to the

argument of perigee is analysed for a fixed spacecraft configuration, α = 45° and d = 0 m,

at initial altitudes ranging from 122 to 911 km to test the effect of the initial atmospheric

drag magnitude. The simulation parameters can be found in Table 4.3, since a is fixed, 8

low-eccentricity values are selected to cover the initial altitudes mentioned. The initial Sun-

pointing direction λ0 is the same in all the cases, thus, different values of the Sun-perigee

angle are guaranteed through the selection of 8 different values of ω0. In total, 64 initial

scenarios are simulated. The analysis is performed for three different values of the sail size,

represented as the nominal area-to-mass ratio, however, it should be noted that these values

are different from the effective area-to-mass ratio contributing to either SRP or drag forces

since it depends on the sail orientation with respect the relative perturbation direction.

Table 4.3: Sensitivity Analysis: e and ω, h0 = 120− 1000 km. Simulation Parameters.


α deg 45

d m 0

As/m m2/kg 2,4,10

e - 0.12:0.0153:0.12

a0 km 1000 + RE

λ0 deg 90

ω0 deg 0:45:360

4.3.1.2 Results

To understand and compare the deorbiting capabilities of the different simulated scenarios,

in Fig. 4.7, four colour maps representing the deorbiting time (panel a), the percentage of the

trajectory that the spacecraft moves in stable conditions (panel b), simulation time in which

the tumbling motion starts (panel c) and the altitude of this event (panel d) are depicted

for a square sail panel of h = w = 10.18 m, which results in As/m = 2 m2/kg. Each grid

vortex represents one different case and its corresponding eccentricity and initial argument

of the perigee values are found in the y and x axes respectively. To better understanding of

the influence of increasing the sail size, the same results are presented for a h = w = 14.39

m, As/m = 4 m2/kg sail in Fig. 4.8 and for a h = w = 22.76 m, As/m = 10 m2/kg sail in

Fig. 4.9.


Figure 4.7: Sensitivity analysis with respect e and ω, a = RE + 1000 km, As/m

= 2 m2/kg. a) Deorbiting time in days. b) Portion of time that the spacecraft


Altitude in which the spacecraft starts to tumble in km.

In Fig. 4.7a), it can be seen that the deorbiting times range from almost 0 to 5000 days

depending on the initial eccentricity, which is a direct consequence of the difference between

initial altitudes. In Fig. 4.7b) and Fig. 4.7c), it can be noticed that a major portion of

the trajectory the satellite remains stable for low eccentricities (lower rows). In most of the

cases, the altitude in which the chaotic motion starts is around 600-800 km. When e ranges

from 0.06 to 0.12 the vehicle is placed at altitude 600-120 km in the orbit perigee whilst

the orbit apogee ranges between 1443 and 1885 km above Earth’s surface. Thus, during two

consecutive passages to the orbit perigee, the vehicle moves from drag-dominated to SRP-

dominated throughout a transition region in its way to the apogee, entering in tumbling

state from the beginning. Most of the initial tumbling altitudes are located around 600 and

900 km as it can be observed in Fig. 4.7d). Only for the portion of the cases which higher

eccentricities (upper rows), the initial altitude is located under 500 km. Some differences in

the initial tumbling altitude are observed with respect to ω0, although meaningful effect in

deorbiting times and the stable portion of the trajectory is obtained when w0 varies.

In Fig. 4.8a), the deorbiting times range from almost 0 to 2500 days as a direct consequence

that different eccentricities lead to different initial altitudes. In Fig. 4.8b) and 4.8c), it can

be noticed that lower eccentricities (lower rows) allow the spacecraft to remain more time

stable, for e = 0.05, the stable portion of the trajectory varies from 10% to 90% depending






on the ω0. Most of the initial altitudes range from 400 to 850 km as it can be seen in Fig.

4.8d), higher altitudes are found for larger eccentricity vales and ω0 = 0° while the lower

altitudes are found for the same eccentricities but a different initial argument of perigee.

In Fig. 4.9a), the deorbiting times range from almost 0 to 1000 days, where the lower values

are due to higher eccentricities leading to lower initial altitudes. It can be noted that for

constant eccentricity value, slightly lower deorbiting times are found for ω0 ≈ 90°. In Fig.

4.9b) and 4.9c) larger stable portion of the trajectory are found for lower eccentricities (lower

rows), when e = 0.05 a remarkable sensitivity with respect to the initial argument of perigee

is observed. Most of the initial tumbling altitudes are located between 300 and 750 km as

it can be seen in Fig. 4.9d), where the sensitivity with respect to ω0 observed in previous

figure is also present.

It should be noticed that when the sail area increases, the deorbiting time decreases. In Fig.

4.8 where the As/m is doubled with respect to the cases illustrated in Fig. 4.7, the deorbiting

times are found to be reduced almost the half. In Fig. 4.9 the nominal area-to-mass ratio

is increased 5 times with respect to the cases in Fig. 4.7, resulting in around 1/5 deorbiting

times. This suggests an almost indirect relationship between nominal area-to-mass ratio with

respect to the deorbiting time. It also can be noted that although no meaningful effect is






obtained with respect to ω0, larger sails are more sensitive to this parameter.

Regarding the relative position of the Sun with respect the orbit perigee, which is illustrated

through the sensitivity of ω0 since λ0 is fixed, no meaningful differences can be noticed with

respect to the deorbiting time and percentage that the spacecraft remains stable in its tra-

jectory. This parameter is relevant in the outward deorbiting first phase when the main

purpose is to increase eccentricity until reaching the atmosphere. However, the cases under

study have already reached the altitudes in which the atmospheric drag is present, there-

fore, from the beginning the spacecraft moves inside the second phase of outward deorbiting

strategy. The orbit energy and eccentricity decrease characterise this phase, which can be

derived from the semi-major axis and altitude reduction. To illustrate this phenomenon,

the altitude and semi-major axis, eccentricity and relative orientation of the spacecraft with

respect to sunlight and atmospheric drag direction are depicted in Fig. 4.10 for e = 0.0273

and ω = 135° case.

In Fig. 4.10a), the orbit apogee notably decreases (blue line upper limit) whilst the perigee

slightly does (blue line lower limit), before reaching the final part characterised for a fast

decay into the Earth’s surface. The orbit circularises since the eccentricity decreases as it

can be seen in Fig. 4.10b). It can be observed in Fig. 4.10c) that the spacecraft remain


Figure 4.10: Orbit evolution for a α = 45°, d = 0 m, As/m = 4 m2/kg

spacecraft, e = 0.0273 and ω = 135°. a) Altitude and semi-major axis evolution.

b) Eccentricity evolution. c) Attitude evolution.

stable around the sunlight pointing direction (blue dots close to the origin) until time 1550

days, where the tumbling motion starts (blue and red dots spread) and last until it reaches

the 120 km of altitude.

4.3.2 Above 1000 km of altitude

It has been seen that the last stage of outward deorbiting can be achieved within 14-year

(≈ 5000 days) window when the orbit perigee is placed below 1000 km of altitude. At this

distance from the Earth’s surface, the atmospheric drag contribution is low with respect

to SRP and the requested deorbiting time for spacecraft placed at higher altitudes can be

prohibitive from a design for demise strategy point of view. Eccentric orbits with apogee

altitudes above 1000 km spend long times in SRP-dominated regions and the relative position

of the orbit perigee with respect to the Sun can enhance the SRP contribution in lowering the

perigee altitude by increasing the eccentricity [17, 18]. In this way, a simulation is performed

to study the deorbiting capabilities inside a 20-years window of a spacecraft from an initial

altitude of 1000 km.


4.3.2.1 Simulation parameters

The simulation parameters are those in Table 4.4. The initial altitude of the spacecraft is

set to be at 1000 km where the atmospheric drag starts to influence the orbit dynamics, that

is, at the beginning of the second phase of outward deorbiting strategy. When a spacecraft

reaches this phase, it moves in an eccentric orbit, therefore low-eccentric orbits are analysed,

and 8 eccentricity values are selected, making the semi-major axis value to range from 7468

to 8384 km. Since the initial Sun-pointing direction λ0 is fixed, the variation of Sun-perigee

angle is studied selecting 8 different values of ω0. In total, 64 different initial scenarios. The

analysis is performed for a fixed spacecraft configuration and for two different values of the

sail size, represented as the nominal area-to-mass ratio, however, it should be noted that

these values are different from the effective area-to-mass ratio contributing to either SRP or

drag forces since it depends on the sail orientation with respect to the relative perturbation

direction.

Table 4.4: Sensitivity Analysis: e and ω, h0 = 1000 km. Simulation Parameters.


α deg 45

d m 0

As/m m2/kg 4,10

e - 0.12:0.0153:0.12

h0 km 1000

λ0 deg 90

ω0 deg 0:45:360

4.3.2.2 Results

To observe the deorbiting capabilities, simulation results are provided in four colour maps

for different values of e and ω0, y and x axes respectively in Fig. 4.11, showing the deorbiting

time (panel a), the percentage of the trajectory that the spacecraft moves in stable conditions

(panel b), and time and altitude of the tumbling initiation (panel c and d respectively). Only

the cases able to deorbit within a 20-years window are represented for a spacecraft whose sail

is composed by two square panels of h = w = 14.39 m, resulting in As/m = 4 m2/kg. With

the aim of observing the effect of increasing the sail size, the same results are depicted in Fig.

4.12 but for a spacecraft with different sail size, h = w = 22.76 m, resulting in As/m = 10

m2/kg.


Figure 4.11: Sensitivity analysis with respect to e and ω, h0 = 1000 km, As/m




Fig. 4.11a) shows that low-eccentric orbits (lower rows) allow smaller deorbiting times, and

when eccentricity increases, the deorbiting time can reach values larger than the 20-years time

constrain. All the cases capable of deorbiting in this time, allow the spacecraft to remain

stable more than 95% of its trajectory as it can be seen in Fig. 4.11b). For e = 0.05, the

re-entry within the time constraint is achieved for ω0 ∈ (0°, 180°). According to Fig. 4.11d),

the initial altitude at which the vehicle starts to tumble ranges from 620 to 740 km.

When the sail area increases, the deorbit can be achieved for more eccentric orbits as it can

be observed in Fig. 4.12. Lower deorbiting times are found for more circular orbits (lower

rows) in Fig. 4.12a). All the cases allow that the spacecraft remains stable more than 95%

of the time and the tumbling altitude ranges from 600 to 900 km, as it can be noticed in Fig.

4.12b) and Fig. 4.12d) respectively.

It can be observed that the relative direction of the argument of perigee with respect the

sunlight can compromise the deorbiting feasibility inside a fixed time window for more eccen-

tric orbits. In Fig. 4.12, it is worth noting that ω around 90° represents the most favourable

cases since deorbiting inside the time constraint window is achieved faster and for higher

eccentricities. To better understand what is going behind these cases, the initial orbit and

the Sun position are sketched in 4.13. Since the Sun is placed at λ0 = 90°, the spacecraft


Figure 4.12: Sensitivity analysis with respect to e and ω, h0 = 1000 km, As/m




faces the Sun when increasing its speed toward its perigee. In this situation, the SRP acts as

deceleration force, reducing the orbit energy. When the spacecraft moves towards the apogee,

the SRP accelerates the vehicle. As a consequence, the orbit circularisation is enhanced and

fast deorbit is achieved.

Figure 4.13: Initial scenario for λ0 = ω0 = 90°.


A representative case in which deorbiting is achieved is compared with a case in which it is not

achieved inside the time constrain window. Both cases are selected to have the same initial

parameters except the initial argument of perigee. In this way, in Fig. 4.14, the altitude

and semi-major axis (panel a), eccentricity (panel b) and the relative spacecraft orientation

with respect to SRP and drag perturbations (panel c) are depicted for a α = 45°, d = 0 m

and As/m = 10 m2/kg spacecraft in an initial orbit of e = 0.1046 and ω0 = 90°. The same

variables are presented in Fig. 4.15 for the same spacecraft but different argument of perigee,

ω0 = 270°.


spacecraft, e = 0.1046 and ω = 90°, within a 20-years window. a) Altitude and

semi-major axis evolution. b) Eccentricity evolution. c) Attitude evolution.

In Fig. 4.14a), the apogee altitude (blue line upper limit) decreases while the perigee altitude

(blue line lower limit) remains almost constant until the very end where the spacecraft swiftly

falls into the Earth’s surface. In Fig. 4.14b) the orbit circularisation is observed due to the

eccentricity long-term decreasing effect. In Fig. 4.14c), it is observed that the vehicle remains

stable around sunlight direction (blue dots inside stable region denoted by dashed lines) until

time ≈ 3400 days where tumbling state starts.

The circularisation process can be observed also in Fig. 4.15a) and Fig. 4.15b), however, it

is slower than the previous case and the eccentricity is not able to reach lower values, which

allow the re-entry, inside the 20-year window. The perigee altitudes persist about constant



spacecraft, e = 0.1046 and ω = 270°, within a 20-years window. a) Altitude and

semi-major axis evolution. b) Eccentricity evolution. c) Attitude evolution.

during the simulated time. In Fig. 4.15c) it can be noted that the spacecraft remains stable

around SRP perturbation.

In previous figures, it has been seen that when the initial orbit perigee is located at the

opposite side from the Sun, ω0 ≈ 270°, the deorbiting process becomes slower and, in some

cases, unfeasible for the selected time constraint.

Chapter 5

Conclusions and future work

The deorbiting possibilities by means of sailing have been studied with a special focus on a

transition region where both SRP and atmospheric drag perturbations have a relevant effect

on the dynamics. A simple spacecraft design consisting in 2 reflective panels and a bus is

considered, only two parameters drive the spacecraft configuration: the aperture angle of the

sail α and centre of mass-centre of pressure offset, that is measured through the distance

between the bus and sail masses, d. A simplified planar model of the attitude dynamics has

been established considering SRP, atmospheric drag and gravity gradient torques.

Spacecraft attitude dynamics has been studied in SRP and drag dominated regions. Follow-

ing the previous research [66], where only the cases in which one or two panels point the

perturbation direction with their frontal area as the back part of the sail was considered to

be black and hence non-reflective, all the possible directions, including the ones in which

one or the two sail panel backs contribute to the perturbation effect, are considered. The

dynamical system that consists of assuming only SRP/drag torque in rotation around the

body z-axis is found to have a pendulum-like structure that has, either 2,4 or 6 equilibria,

depending on η, α and d.

The sail reflectance is fixed according to the available sail technology [64] and a sensitivity

analysis with respect α and d shows.

• SRP-dominated region: Attitude stability has been observed to behave following the

Sun-pointing vector. For d ≥ 0 m, the SRP torque system has 2 equilibria. Theoreti-

cally, the spacecraft remains with a pendulum-like dynamics line whatever the initial

orientation with respect to the Sun is. However, gravity gradient and J2 effects can be

84

85

strong enough to take the spacecraft out of oscillatory motion, for large d and close to

Earth, the stability becomes chaotic and unpredictable. The effect intensifies as the

orbit eccentricity increases.

• Drag-dominated region: The three possible scenarios can be found depending on d and

α values. For d close to zero, the stability phase space is represented by 6 equilibria, it is

divided into one stable region around the relative velocity direction and two stable and

symmetrical sidelobes. As d increases, the sidelobes reduces and at a certain point, the

2 equilibria scenario appear, resulting in a wide stable region. Nevertheless, the larger

d is, the larger the gravity gradient torque becomes, destabilising the spacecraft. The

change in altitude suffered in elliptic orbits leads the vehicle to displace with chaotic

and tumbling motion.

Spacecraft scenarios moving through the regions where both perturbations are present have

been simulated. A transition region where the coupled effect of SRP and atmospheric drag

makes the spacecraft to end tumbling has been found in most of the cases. The spacecraft

starts moving in SRP-dominated region, the auto-stabilising properties remain until reaching

600-800 km of altitude where the atmospheric drag becomes stronger. As a consequence, the

vehicle enters what seems to be an unpredictable tumbling state that lasts until it is very

close to the Earth’s surface. The last deorbiting stage is characterised for a fast decay, the

atmospheric drag becomes the dominant force and the spacecraft starts to stabilise around

the relative velocity direction.

A sensitivity analysis suggests that satellite configuration placed in quasi-circular orbits with

large α and small d are able to passively deorbit with a stable attitude dynamics in the

vicinity of the Sun-pointing vector without any power cost, reducing the tumbling state to

the end of its trajectory. Therefore, a high percentage of the deorbiting time in which the

space vehicle remains stable without active control can be achieved. In some cases, the

effective average area contributing to the drag force increases when the spacecraft starts to

tumble, enhancing the deorbiting process. Hence, leading to uncontrolled tumbling motion

at some point of the orbit may reduce the deorbiting time. A trade-off between spacecraft

stability and deorbiting time must be performed for these cases.

Low-eccentric orbits are beneficial from a stable point of view since the small altitude vari-

ation between perigee and apogees allow the spacecraft remaining in the same orbit regime.

Otherwise, the spacecraft goes through SRP-dominated, transition and drag-dominated re-

gions in each orbit period, which make it to end tumbling. Deorbiting is achieved for initial

altitudes ranging from 120 to 1000 km when e < 0.12, for α = 45 °and d = 0 m spacecraft.

86 Chapter 5. Conclusions and future work

Orbits with initial low eccentricities spend most of the deorbiting time stable with respect

to the sunlight direction.

A simulated case from higher altitudes shows that low-eccentric orbits allow a stable re-

entry and the relative position between the Sun and the orbit perigee plays a key role in the

deorbiting time, having a relevant influence for eccentric orbits above 1000 km of altitude.

Since the sunlight can accelerate the circularisation process, orbits whose perigee point toward

the Sun can achieve lower deorbiting times. Remarkable differences have been found when the

initial argument of perigee changes, making deorbit unfeasible under certain time constraints.

The spacecraft stability has been analysed in planar orbits where the tilt of the Earth’s

rotational axis is neglected, and the effect of eclipses is not considered. Results from low-

eccentric orbits placed at LEO and low MEO altitudes show that 2 panel solar sail can be

employed to deorbit a satellite placed in quasi-circular planar orbits. The spacecraft remains

in stable motion until the end of the trajectory, which starts tumbling. If stable motion is

required during the whole trajectory, active control must be employed. Results suggest that

the control can be reduced to the transition region, reducing in this way the power cost.

A number of future lines of research emerge from this work:

• The study of orbit lowering from altitudes considering the simple-shape solar sail and

extend it to non-ideal conditions, that is, taking into consideration the eclipses effect

and high elliptic orbits.

• The study of control laws by means of varying sail aperture angle and distance between

sail and bus mass centres. Thus, the chaotic transition region in terms of stability may

be reduced or avoided with low power cost.

• The implementation of 3D QRP for non-planar orbits through the transition region.

• A further study of the spacecraft dynamics coupled with the dynamics of the sail,

considering deployment and sail flexibility under the SRP, atmospheric drag and gravity

gradient effect.

This thesis is a part of the COMPASS project: ”Control for orbit manoeuvring by surfing

through orbit perturbations” (Grant agreement No 679086). This project is a European

Research Council (ERC) funded project under the European Unions Horizon 2020 research.

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Appendix A

94

Appendix A

Computation of Shadowed Area

The effective spacecraft area depends on the relative orientation of the sail with respect to

the Sun-spacecraft or relative velocity directions. In some orientation regions, only a portion

of the panel width w′ contributes to the effective area since it is shadowed by the other panel.

In this section, the algorithm to compute the partial width is addressed. A general sketch of

the geometry of the problem is illustrated in Fig. A.1, where the green line represents the

Sun-spacecraft or relative velocity direction, ψ is the angle between the perturbing direction

and the vertical.

Figure A.1: Geometry of the problem

The problem can be reduced to a line intersection problem, which is illustrated in Fig. A.2.

The dashed line represents the satellite symmetric axis, the shadowed panel is represented in

95

96 Appendix A. Computation of Shadowed Area

blue and the direction of the considered perturbation in the new reference system in yellow.

Figure A.2: Line intersection problem in auxiliary axes x’ y’.

The angle between the perturbation direction and the y-body axis is computed as

ψ = |ϕ| − π

2. (A.1)

Three auxiliary distances are employed, a, b and c. They can be calculated as

a = w cosα (A.2a)

b = w sinα (A.2b)

c = 2b tanhψ (A.2c)

where w is the total panel width, that is, the blue line distance in Fig. A.2.

The two intersecting straight lines are:

Blue Line : ax′ − by′ = 0 (A.3)

Y ellow Line :c

2bx′ + y′ = c (A.4)

The intersection point is computed, being h its y′ coordinate.

h =c

1 + c2a

(A.5)

97

Finally, w′ can be obtained from h′ and the area is computed as A′sp = hw′.

w′ =h

sin(π2− α

) (A.6)

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