TITLE: Contribution to the Dynamics of a Solar Sail in the Earth - Sun
System (PhD Thesis)

AUTHOR:
Ariadna Farres
Departament de Matematica Aplicada i Analisi
Universitat de Barcelona
Gran Via 585, 08007 Barcelona, Spain

E-mail: ari@maia.ub.es

ABSTRACT:
Solar sails are a new concept of spacecraft propulsion that has more
adepts everyday. The idea is to provide a spacecraft with a large
membrane mirror such that the impact of the photons emitted by the Sun
and their further reflection produce momentum on it. For the moment
there has yet not been a successful deployment of a solar sail in
space, although lately there have been a couple of attempts: Cosmos 1
and NanoSail. Both missions failed before the spacecraft could get to
the nominal orbit, not being able to deploy the solar sail and test
the technology. In the last years, space agencies have started to
invest in this technology and it seems that at some point solar sails
will become a reality.

Studies on the use of a solar sails have been done in the past. One of
the reference books in the field is [McI99] where most of the studies
on the subject up to 1999 are summarised. The design of solar sails,
the force models for the different structures, the dynamics on
heliocentric or geocentric orbits are some of the subjects covered in
this book. Nevertheless, dynamical system tools have had a small
influence in this area.

The use of dynamical systems tools in astrodynamics is not new for the
UB-UPC Dynamical System group. Lots of studies have been made in the
past applying this tools to several astrodynamical problems [GLMS01a,
GLMS01b, GJMS01]. We propose to use similar ideas to navigate through
the Earth - Sun system with a solar sail.

One of the goals of this thesis is to study, in an extended way, the
natural dynamics of a solar sail in the Earth - Sun system. This is a
first step of a more ambitious project of designing strategies for
different kind of mission application such as, station keeping
strategies around equilibrium points, periodic orbits and invariant
tori, or consider using invariant manifold to go from one region on
the phase space to the other in a natural way.  Either using solar
sails or other type of low - thrust spacecraft propulsion, covering
most of this aspects within the framework of dynamical systems.

In Chapter 1 we review some of the known aspects on solar sails and
explain the model and problems that we want to face. We use the
Restricted Three Body Problem (RTBP) adding the solar radiation
pressure on the solar sail as a model and study some of its most
relevant dynamical properties.

It is well known [Sze67], that the Restricted Three Body Problem in
synodical coordinates has 5 equilibrium points, which correspond to
the position where the gravitational attraction of the two primaries,
Earth and Sun, compensate. When we consider the extra effect of the
solar sail, there is a 2D family of new equilibria parametrised by the
sail orientation. These artificially generated equilibrium points open
a wide new range of possible mission applications, the Geostorm
mission and the Polar Observer are two examples [McI99]. The Geostorm
is a mission concept where a modest sail is placed Sunwards of the
classical Earth - Sun L1 point. Then using a magnetometer to detect
the solar wind polarity enables to double the time of alert of a
conventional L1 Halo orbiter such as SOHO. The Polar Observer aims to
use an artificial equilibrium point displaced above the ecliptic
plane, high above one of the Earth’s poles. This would provide
constant real-time views of the polar latitudes for studding, for
instance, climate change.

Most of these equilibrium points are unstable, hence a station keeping
strategy is required if we want to maintain the solar sail close to
equilibria for a long time. In Chapter 2 we derive a station keeping
strategy using dynamics system tools. The idea is to understand the
variation of the phase space when the sail orientation is changed. We
can see that the linear dynamics around these equilibrium points is
closely approximated by a saddle × centre × centre motion. Hence when
the sail is close to the equilibrium point its trajectory will escape
along the unstable direction. If we change the sail orientation, the
fixed position varies slightly, and so do the stable and unstable
directions. We want to find a new sail orientation such that the
unstable direction of the new equilibrium brings the trajectory back
to a vicinity of the initial position. Furthermore, one must take into
account the centre projection of the motion, as this one can result of
an unbounded growth.  We have applied these strategies for the two
missions mentioned before maintaining the solar sail around the
desired equilibrium point up to 30 years. We have also tested the
robustness of our strategies including errors in the position and
velocity determination, as well as errors on the orientation of the
sail at each manoeuvre. We will discuss the effect of these errors on
the controllability of the solar sail.

Further on, we would like to extend these ideas to derive station
keeping strategies around periodic orbits. For this reason we need to
have a more complete understanding of the non-linear dynamics around
an equilibrium point, and how it varies when the sail orientation
changes. In this thesis we have focused on the motion in a close
neighbourhood of the displaced L1 equilibrium point for a solar sail,
called SL1.

We have developed numerical tools for the study of the bounded motion
close to SL1. These techniques are very general and can be applied
around other equilibrium points.

Due to the instability of the region, we cannot take arbitrary initial
conditions and integrate them numerically, as they would quickly
escape from the vicinity of the fixed point. For this reason, we
propose to perform the reduction to the centre manifold around the
different equilibrium points. We want to find a high order
approximation of the motion on the centre manifold and use it to
describe the motion on it. As the system is only Hamiltonian for a
small set of values of the sail orientation we cannot take advantage
of this as in [Jor99, JM99] where the motion around the collinear
points of the RTBP is discussed. Instead, we compute the power
expansion of the graph, y = v(x), of the centre manifold around an
equilibrium point up to high order [Sim90, Har08]. We are interested
in an efficient algorithm as we want to do the reduction to the centre
manifold for different sail orientations. In Chapter 3 we describe
this algorithm and also give some details on the implementation of an
efficient code. We also compare the efficiency of our algorithms with
the Lie series method, for the particular case of a sail oriented
perpendicular to the Sun - line, when the system is Hamiltonian.

Finally, in Chapter 4 we describe the dynamics around different
equilibrium points close to SL1 . We have computed the families of
periodic orbits by means of a continu- ation method. We have also used
the approximation to the centre manifold obtained in the previous
chapter to have a description of the periodic and quasi-periodic
motion in an extended neighbourhood of these equilibrium points.
Around each of the equilibrium points we find families of planar and
vertical periodic orbits related to the two frequencies defining the
centre motion. Families of Halo - type orbits can also be found. The
interaction between the two frequencies gives rise to families of
invariant tori.

At the end of this dissertation we summarise the main results and
point out some possible directions for future work.