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OPTIMAL DEFLECTION OF NEAR-EARTH OBJECTS THROUGH A KINETIC IMPACTOR
PERFORMING GRAVITY ASSIST
OCTOBER 3, 2018
Candidate:
Lorenzo Bolsi
858611
Advisor:
Prof. Camilla Colombo
POLITECNICO DI MILANO
FACULTY OF INDUSTRIAL ENGINEERING
DEPARTMENT OF AEROSPACE SCIENCE AND TECHNOLOGY
MASTER COURSE IN SPACE ENGINEERING
A.A. 2017 - 2018
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Aknowledgments – Ringraziamenti
First of all, I would like to acknowledge Camilla Colombo, my supervising professor,
for her precious teachings and suggestions during all my work. Her guide has
strongly helped me during the accomplishment of my work and allowed me to
achieve this important target of my life. Her motivation has been inspirational for
me.
During my years at University I had the fortune to meet awesome people with
whom I could share the joy and the pain of Politecnico. To Mathieu, Cao, Mary,
Glu, Fra and Ubi, my space companions during the master’s courses; to Xhenis and
Claudio, my side during the bachelor, with whom I shared unforgettable
moments; to Bavetta, Lorenzo and Donald Trumpet, for sharing the beautiful
“Aula Tesi” with me during the last year: a big “thank you”.
During my years in Milano I was lucky to share my life with great people. My thanks
therefore go to Salo and Sgarris (it was difficult to bear you to be honest), to Lollo
and Antonio (the craziest combination of roommates I have ever had), to Cecio
and Samu, to Nico, Otta, Andre, Alice, Giova, Barazz and Barta.
A huge hug to the crew form Motta Baluffi, which have accompanied me before
and during these years. It’s thanks to you (or it’s your fault) if I have become the
person I am today. My thanks to Biazzo, Jhonny, Steppo, Nazgul, Sara, Fede, Mozza
and all the others. “Puoi togliere il ragazzo da Motta Baluffi, ma non Motta Baluffi
dal ragazzo” (semicit.)
It’s impossible to forget The Group, with whom I shared intense and unforgettable
months in Japan. Thanks to Bryan, Robin, Ecem, Iffa, Jade, Toey, Dean, Eser,
Miriam, Kim, LiFeng, Stian, Ville.
This incredible path that brought me to this incredible goal would have never been
possible without my family. Your support throughout all of these years pushed me
to always improve. That’s why I want to devote this success to you.
Grazie mamma e grazie papà, per supportarmi e sopportarmi in tutto e per tutto
(anche se a volte sono i che devo sopportare voi).
Grazie Roby, per essere il mio modello da seguire e grazie Vale per essertelo preso,
perché anche lui sa essere bello pesante.
Grazie a zia Rita, Eva e Mirko, zia Donata e zio Beppe, zio Paolo, zio Giorgio e zia
Marta e tutti i miei bellissimi cugini per il vostro supporto e per credere sempre in
me.
Infine, un immenso grazie alla nonna Lina e al nonno Giuseppe che sono stati in
questi anni, di gran lunga, i miei fan numero uno.
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Table of contents
List of figures.............................................................................................................. 6
List of tables ............................................................................................................... 9
Abstract ................................................................................................................... 11
1. Introduction...................................................................................................... 13
1.1. Background ............................................................................................... 13
1.2. Deflection methods ................................................................................... 14
1.3. Past and future missions ............................................................................ 16
1.4. Thesis objectives........................................................................................ 17
2. Theory of asteroid deflection ............................................................................. 18
2.1. Deflection model ....................................................................................... 18
2.2. Mission design – Direct hit.......................................................................... 23
2.3. Orbital Mechanics ...................................................................................... 25
2.3.1. Interplanetary transfers ...................................................................... 25
2.3.2. Fly-by................................................................................................. 25
2.4. Mission design – Gravity assist scenario (not powered) ................................ 27
3. Test cases results .............................................................................................. 29
3.1. Direct Hit................................................................................................... 29
3.2. Single Gravity Assist ................................................................................... 36
3.2.1. Earth gravity assist.............................................................................. 36
3.2.2. Mars and Venus gravity assists ............................................................ 40
4. Improved test cases results ................................................................................ 44
4.1. Multi-revolution Lambert model ................................................................. 44
4.1.1. Theory ............................................................................................... 44
4.1.2. Simulation.......................................................................................... 45
4.2. Powered gravity assist................................................................................ 49
5. Analysis of the test case solutions ...................................................................... 54
5.1. Variation of time window ........................................................................... 54
5.2. Spacecraft’s initial mass and asteroid’s mass ............................................... 59
5.2.1. Relation between spacecraft’s initial mass and achievable deflection .... 59
5.2.2. Relation between asteroid’s mass and achievable deflection ................ 59
6. Deflection efficiency against the whole NEOs population..................................... 61
6.1. Generation of a population of asteroids ...................................................... 61
6.2. Creation of a synthetic population of asteroids............................................ 67
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6.2.1. Theory ............................................................................................... 67
6.2.2. Simulation.......................................................................................... 70
7. Conclusions and future works ............................................................................ 84
8. Bibliography...................................................................................................... 86
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List of figures
Figure 2.1 - Orbits of Earth and Asteroid and their positions at the MOID..................... 19
Figure 2.2 – Distance vectors at the MOID (non-deviated NEO in orange, deviated NEO in
red).......................................................................................................................... 20
Figure 2.3 - Explanation of angles. [35]....................................................................... 23
Figure 2.4 - Geometry of the hyperbola. [36] .............................................................. 26
Figure 3.1 – Pareto front for Direct Hit scenario - Colombo Phd thesis [17]. In the
colormap the achievable deflection. .......................................................................... 31
Figure 3.2 - Pareto-front for direct hit scenario - Present work. In colormap the
achievable deflection. ............................................................................................... 32
Figure 3.3 - Delta-v given to the asteroid VS Achievable distance for Direct hit scenario.
In colourmap the achievable deflection scaled on the Earth radius. ............................. 33
Figure 3.4 - In-plane angle VS out-of-plane angle between impactor and asteroid's
velocities for direct hit scenario. In colourmap the achievable deflection scaled on
Earth's radius. .......................................................................................................... 33
Figure 3.5 - Departure times (years before MOID) VS achievable distance for direct hit
scenario. In colourmap the TOF of the transfer........................................................... 34
Figure 3.6 - Departure time (years before MOID) VS Mass of the spacecraft at deflection
for direct hit scenario. In colourmap the TOF for the transfer. ..................................... 35
Figure 3.7 - Direct hit deflection mission. ................................................................... 35
Figure 3.8 - Pareto-front for Earth's gravity assist scenario. In colourmap the achievable
deflection................................................................................................................. 37
Figure 3.9 - Delta-v given to the asteroid VS achievable distance for Earth gravity assist
scenario. In colourmap the achievable deflection scaled on Earth's radius. .................. 37
Figure 3.10 - In-plane angle VS Out-of-plane angle between impactor and asteroid's
velocities for Earth gravity assist scenario. In colourmap the achievable deflection scaled
on Earth's radius. ...................................................................................................... 38
Figure 3.11 - Departure time (years before MOID) VS achievable distance for Earth
gravity assist scenario. In colourmap the sum of the TOF for the two interplanetary
transfers. ................................................................................................................. 39
Figure 3.12 - Departure time (years before MOID) VS mass of the spacecraft at deflection
for Earth gravity assist scenario. In colourmap the sum of the TOF for the two
interplanetary transfers. ........................................................................................... 39
Figure 3.13 - Earth gravity assist deflection mission. ................................................... 40
Figure 3.14 - Mars gravity assist deflection mission..................................................... 41
Figure 3.15 - Venus gravity assist deflection mission. .................................................. 41
Figure 3.16 - Comparison between the pareto-fronts of the 4 scenarios. Blue for Earth
gravity assist; Black for direct hit; green for Venus gravity assist; red for Mars gravity
assist........................................................................................................................ 42
Figure 3.17 -Comparison of Delta-v given to the asteroid VS achievable distance for the 4
scenarios. Blue for Earth gravity assist; Black for direct hit; Green for Venus gravity
assist; red for Mars gravity assist. .............................................................................. 42
Figure 4.1 - An example of possible Lambert’s transfers in the plot semi -major axis VS
transfer time ............................................................................................................ 45
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Figure 4.2 - Comparison of pareto-fronts for Mars gravity assist's missions. Blue for
Multi-revolution case and two rounds of optimisations; red for multi -revolution and one
round of optimisation; Black for zero-revolution......................................................... 46
Figure 4.3 - Comparison of delta-v given to asteroid VS achievable distance in Mars
gravity assist's mission. Blue for multi-revolution and two rounds of optimisation; red for
multi-revolution and one round of optimisation; Black for zero-revolution. ................. 47
Figure 4.4 - Comparison of pareto-fronts for Venus gravity assist's mission. Blue for
multi-revolution, two rounds of optimisation with 400 individuals; red for multi -
revolution, two rounds of optimisation with 200 individuals; black for zero-revolution. 48
Figure 4.5 - Comparison of delta-v given to asteroid VS achievable distance for Venus
gravity assist's mission. Blue for multi-revolution, two rounds of optimisation with 400
individuals; red for multi-revolution, two rounds of optimisation with 200 individuals;
black for zero-revolution. .......................................................................................... 48
Figure 4.6 - Pareto-front for Earth powered gravity assist's mission. ............................ 50
Figure 4.7 - Delta-v given to asteroid VS achievable distance for Earth powered gravity
assist's mission. ........................................................................................................ 50
Figure 4.8 – Pareto front for Mars powered gravity assist's mission. ............................ 51
Figure 4.9 - Delta-v given to asteroid VS achievable distance for Mars powered gravity
assist's mission. ........................................................................................................ 51
Figure 4.10 – Pareto front for Venus powered gravity assist's mission.......................... 52
Figure 4.11 - Delta-v given to asteroid VS achievable distance for Venus powered gravity
assist's mission. ........................................................................................................ 52
Figure 5.1 - Polar plot of delta-v given to the asteroid. Modulus and direction are shown.
................................................................................................................................ 54
Figure 5.2 - Polar plot of optimal delta-v given to asteroid for Earth gravity assist’s
mission. time window varying from 10 years to 1 year. ............................................... 55
Figure 5.3 - Polar plot of optimal delta-v given to asteroid for Earth gravity assist's
mission. Difference between high value or low values for time window. ...................... 56
Figure 5.4 - Comparison of pareto-fronts for Earth gravity assist's mission with varying
time window. ........................................................................................................... 57
Figure 5.5 - Comparison between delta-v given to asteroid VS achievable distance for
varying time window. ............................................................................................... 58
Figure 5.6 - Comparison between departure time (years before MOID) VS achievable
distance for varying time window. ............................................................................. 58
Figure 5.7 – Achievable deflection VS asteroid mass for Earth gravity assist’s mission ... 60
Figure 5.8 - Delta-v given to asteroid VS mass of asteroid for Earth gravity assist's
mission .................................................................................................................... 60
Figure 6.1 - Test run of the population generator: semi-major axis VS eccentricity........ 61
Figure 6.2 - Test run of the population generator: semi-major axis VS inclination ......... 62
Figure 6.3 – Test run of the poulation generator: eccentricity VS inclination................. 62
Figure 6.4 - Test run of the population generator: periapsis VS apoapsis ...................... 63
Figure 6.5 – MOID VS number of objects. Different colours for different categories of
asteroids .................................................................................................................. 63
Figure 6.6 - test run of the population generator: semi-major axis VS eccentricity.
Filtering action on pericentre and apocentre .............................................................. 64
Figure 6.7 - Test run of the population generator: semi-major axis VS inclination.
Filtering action on pericentre and apocentre. ............................................................. 65
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Figure 6.8 - Test run of the population generator: eccentricity VS inclination. Filtering
action on pericentre and apocentre ........................................................................... 65
Figure 6.9 - Test run of the population generator: periapsis VS apoapsis. Filtering action
on pericentre and apocentre ..................................................................................... 66
Figure 6.10 - Test run of the population generator: MOID VS number of objects. Filtering
action on pericentre and apocentre. .......................................................................... 66
Figure 6.11 - Visual representation of Earth's orbital plane and asteroid's orbital plane.68
Figure 6.12 - Simulation of deflection of the synthetic population. Blue for best
deflection through Earth gravity assist; Black for best deflection through Direct hit;
Green for best deflection through Venus gravity assist ............................................... 71
Figure 6.13 - Direct hit on asteroid - Deflection achieved. ........................................... 72
Figure 6.14 - Direct hit on asteroids population - Delta-v given to asteroid. .................. 72
Figure 6.15 - Direct hit on asteroids population - Distance from perigee at the deflection.
................................................................................................................................ 73
Figure 6.16 - Direct hit on asteroid population - in plane angle of deflection. ............... 73
Figure 6.17 - Direct hit on asteroid population - Out of plane angle of defelection ........ 74
Figure 6.18 - Earth gravity assist mission on asteroid population – Deflection............... 75
Figure 6.19 - Earth gravity assist mission on asteroid population - Delta-v given to
asteroid. .................................................................................................................. 75
Figure 6.20 - Earth gravity assist mission on asteroid population - Distance from perigee
at deflection. ............................................................................................................ 76
Figure 6.21 - Earth gravity assist mission on asteroid population - in-plane angle of
deflection................................................................................................................. 76
Figure 6.22 - Earth gravity assist mission on asteroid population - out-of-plane angle of
deflection................................................................................................................. 77
Figure 6.23 - Mars gravity assist mission on asteroid population - Deflection................ 78
Figure 6.24 - Mars gravity assist mission on asteroid population – delta-v given to
asteroid ................................................................................................................... 78
Figure 6.25 - Mars gravity assist mission on asteroid population – distance to perigee.. 79
Figure 6.26 - Mars gravity assist mission on asteroid population - in-plane angle .......... 79
Figure 6.27 - Mars gravity assist mission on asteroid population - out-of-plane angle ... 80
Figure 6.28 - Venus gravity assist mission on asteroid population - Deflection .............. 80
Figure 6.29 - Venus gravity assist mission on asteroid population - Delta-v given to
asteroid. .................................................................................................................. 81
Figure 6.30 - Venus gravity assist mission on asteroid population - Distance from perigee
at deflection. ............................................................................................................ 81
Figure 6.31 - Venus gravity assist mission on asteroid population - In-plane angle at
deflection................................................................................................................. 82
Figure 6.32 - Venus gravity assist mission on asteroid population - Out-of-plane angle at
deflection................................................................................................................. 82
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List of tables
Table 3.1 - Orbital parameters of 2010RF12. .............................................................. 29
Table 3.2 - Parameters for the mission. ...................................................................... 29
Table 3.3 - Bounds for the design variables - Direct Hit scenario. ................................. 30
Table 3.4 - Optimization parameters. ......................................................................... 30
Table 3.5 - Bounds for the design variables - Single gravity assist scenario.................... 36
Table 4.1 - Optimisation parameters for multi-revolution cases. .................................. 47
Table 4.2 - Bounds for ‖𝛥𝑣0‖ .................................................................................... 49
Table 4.3 - Optimisation parameters for powered gravity assist's missions cases .......... 49
Table 6.1 - Set of parameters used for the simulation ................................................. 70
Table 6.2 - Bounds for the design variables. Asteroid population case.......................... 70
Table 6.3 - Optimisation parameters. Asteroid population case ................................... 71
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Abstract
Un gran numero di corpi celesti minori popola il nostro Sistema Solare; alcuni di questi
vengono classificati come Near Earth Objects (NEO), corpi celesti la cui orbita si trova
vicino o addirittura interseca quella del nostro pianeta. La maggior parte di questi è
costituita da asteroidi (NEA), tra cui è possibile evidenziare i Potentially Hazardous
Asteroids (PHA), così chiamati poiché si ritiene che siano una minaccia per la Terra. Una
moltitudine di studi sui diversi aspetti che caratterizzano questo problema sono stati
affrontati dalla comunità scientifica, per giungere alla conclusione che la soluzione più
comune per affrontare il potenziale impatto è che l'asteroide venga deviato attraverso un
kinetic impactor, in modo tale che la sua orbita venga modificata modificata in misura tale
da essere ad una distanza di sicurezza con la Terra, così da non rappresentare più una
minaccia. La presente tesi espande i lavori passati su questo soggetto, includendo nella
geometria della missione del kinetic impactor non solo un singolo arco direttamente dalla
Terra all'asteroide, ma anche l'assist gravitazionale di tre pianeti: Terra, Marte e Venere.
Sarà analizzato inizialmente il caso reale di un singolo asteroide, trovando la migliore
metodologia di deviazione. Quindi la tecnica verrà sfruttata per analizzare gli effetti
qualitativi che ha su una nuvola di asteroidi, creata in modo tale da compre ndere ogni
possibile asteroide.
A very large number of minor celestial bodies populates our Solar System; some of these
are classified as Near Earth Objects (NEO), celestial bodies whose orbit lies close to or
even intersects our planet’s. The greater part of these are asteroids (NEA) and some of
these are classified as Potentially Hazardous Asteroids (PHA), called in this way because
they are believed to be a threat for Earth. A multitude of studies on the different aspects
that characterise this problem have been carried by the scientific community, finding that
the most common solution to face a potential impact situation is the deflection of
incoming asteroids through kinetic impactor, in such a way that their encounter with the
Earth is avoided or modified to an extent that it does not pose a threat. The present
dissertation expands the previous works on this subject, by including in the geometry of
the mission taken by the kinetic impactor not only a single branch from the Earth to the
asteroid, but also the swingby of three planets: Earth, Mars and Venus. It will be analysed
first a real case of a single asteroid, finding the best deflection methodology. Then the
technique will be exploited to analyse the qualitative effect it has on a cloud of asteroid,
capable to include every single possible asteroid.
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1. Introduction
1.1. Background
Near Earth Objects (NEO) are those celestial bodies whose orbit lies close to or even
intersects our planet’s, a few of which are believed to pose a potential threat to Earth.
Because of their hazardous nature, the presence of these celestial bodies in our Solar
System has caught the eye of both the public and the scientific community and the
concern has grown over the past decades. A multitude of studies on the different aspects
that characterise this problem have been carried out.
Our Solar System is populated by rocky and icy celestial bodies, known as asteroids and
comets. These latter are originated in the outer Solar System, and their agglomeration
process, around 4.6 billion years ago, is thought to be the reason of the birth of the giant
outer planets (Jupiter, Saturn, Uranus and Neptune). At the same way, inner rocky planets
(Mercury, Venus, Earth and Mars) are supposed to be born thanks to the agglomeration
of the asteroids present in this warmer region. Comets are still present in the outer region
of the Solar System, while asteroids are still located in the inner, mainly in the space
between Mars and Jupiter, forming the so called main asteroid belt or inner asteroid belt
(there is indeed an outer asteroid belt, called Kuiper belt). These objects are the remnant
and leftovers from this process of agglomeration, and therefore have a crucial scientific
value, since they may carry with them the explanations of the birth of planets and life [1].
Because of this in the last years both comets and asteroids have gained a lot of interest
among the scientific community, leading to have some scientific missions devoted to
reach and study these celestial bodies. Among these there are past and future missions,
like JAXA’s Hayabusa [2] and Hayabusa 2 [3], ESA’s Rosetta [4], NASA’s Deep Impact [5],
and AIDA, born from the collaboration between NASA (DART) [6] and ESA (AIM) [7].
Not only has this scientific importance attracted the interest of scientists, but also the
danger of a potential impact of an asteroid or comet with our planet. There are, in fact,
asteroids and comets that dive deep into the inner Solar System, sometimes getting close
to the Earth. These objects are classified as NEOs (Near Earth Objects), and they are
defined as comets and asteroids with a perihelion distance smaller than 1.3 AU [8]. Among
these the greatest part is represented by NEAs, the Near Earth Asteroids. It is possible to
furthermore differentiate these bodies into subcategories, to distinguish them: Atiras
asteroids are those NEAs having a semi-major axis smaller than 1 Astronomical Unit (AU)
and with a perihelion distance smaller than 0.983 AU, meaning that their orbit is
completely contained within Earth’s orbit; Atens are those NEAs with a semi -major axis
smaller than 1 AU but with a perihelion distance larger than 0.983 AU, having therefore
their orbit crossing Earth’s orbit; Apollos asteroids are those Earth-crossing NEAs with a
semi-major axis larger than 1 AU and perihelion distance smaller than 1.017 AU; Amors
are those NEAs whose orbit is exterior to Earth’s but interior to Mars’, with a semi -major
axis larger than 1 AU and perihelion distance between 1.017 AU and 1.3 AU; finally, we
define the Potentially Hazardous Asteroids (PHAs): these are those NEAs whose Minimum
Orbit Intersection Distance (MOID) with the Earth is 0.05 AU or less and whose absolute
magnitude (H) is 22.0 or brighter [1]. The MOID is defined as the minimum distance
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between the osculating orbits of two bodies. In this work, the MOID is always intended as
the minimum geometrical distance between the Earth and the asteroid, but it is used also
to define the location along the asteroid’s orbit when the minimum distance occurs. Over
than 600,000 asteroids are known as today, and more than 16,000 are classified as NEOs,
around the 10% of which can be defined PHA [9]. These are the most dangerous objects
orbiting in the Solar System, because an impact with them cannot be excluded. An impact
has in fact a low probability to happen, but not null. Also in recent past, in fact, we have
examples about how the asteroids can pose a serious threat on our planet and our lives.
A collision of epic proportion caused the extinction of the dinosaurs around 65 million
years ago, leaving behind the gargantuan crater of Chicxulub in Yucatan [10]. But also in
the recent past we have evidences of the destructive power of asteroids. The morning of
the 15th of February 2013 a house-sized asteroid entered the atmosphere over
Chelyabinsk, Russia, at over eleven miles per second and blew apart 14 miles above the
ground, releasing a power of around 4440,000 tons of TNT. Some buildings were damaged
and over 1,600 people injured in the blast, mostly due to broken glass [11]. In the next
future, a notable case of risk is identified with the asteroid 2010RF12, a PHA discovered
during a close approach in 2010, which will return close to our planet several times in the
next century. The encounter with the largest probability to end up in an impact is
predicted around the 6th of September 2095. The nominal value of this close approach is
of 1.209 · 10−4 𝐴𝑈 [12].
The concern generated by this problem has raised the interest of the major space agencies,
which have established units devoted to the observation of NEOs, as well as proposing
strategies to face an imminent impact, were it to be expected. Furthermore, international
efforts have come to exist, involving both the cooperation between the agencies
themselves and the institution of the UN-mandated Space Mission Planning Advisory
Group (SMPAG) [13], in order to coordinate the work aimed at preparing for such a threat.
On the other hand, asteroids can also be seen as resources to support space
industrialisation, as they appear to be the least expensive source of needed raw materials.
In a not so far future it will be possible to exploit asteroids as mines, to obtain precious
metals [14].
1.2. Deflection methods
The impact with a large asteroid is, therefore, a serious threat to our society and for this
reason various techniques have been developed to deflect an asteroid in a collision course
with Earth. These techniques have been studied [15] [16] [17] and we classify them here,
highlighting their main particularities.
The first method is the Nuclear Interceptor. It exploits the detonation of a nuclear
explosive to impart the necessary momentum variation and to deflect the selected
asteroid. This method is the most efficient, since the energy density of nuclear explosives
is huge, making possible to deflect also very big asteroids, that other methods cannot
handle. It becomes important here, the position of the detonation, because a detonation
on or below the surface could lead to a fragmentation of the asteroid instead of a
deflection [17]. Moreover, the use of nuclear explosions in space is banned by the Outer
Space Treaty [18]
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The spacecraft Propulsion method is a simple idea which involves attaching a spacecraft
to the selected asteroid and thrusting it to achieve the momentum exchange needed for
the deflection. The momentum exchange can be instantaneous, through a traditional
chemical rocket propulsion, or low-varying, using a low-thrust engine, such as ion engine.
Low thrust methods are, in this case, more efficient due to a much higher specific impulse
of these kind of engines, meaning that it would be sufficient much less fuel to obtain the
same momentum exchange as chemical propulsion, but the time required to achieve it
would be much higher. The drawback for this type of methods is that any rotation of the
asteroid would make the engine difficult to operate in the right direction all the time.
Either the rotation has to be controlled or the engine can operate only while pointing in
the correct direction.
The solar collector method involves collecting energy from the sun and using it to vaporise
part of the asteroid surface, resulting in a small thrust on the asteroid. The vaporisation
can be obtained through the use of a mirrored surface to direct Sun’s light onto the
surface, or energy can be first collected through solar panels and then used to power a
laser focused on the asteroid. Once again, the rotation of the asteroid could create
problems, since the area selected would be in the focus of the beam for less time,
resulting in a smaller outgassing effect [19].
The Mass Driver method [20] involves excavating material from the asteroid’s surface and
then propelling it using an electromagnetic rail gun. This method cannot operate
constantly, because of a possible rotation of the asteroid, but also because the material
to be propelled needs to be collected first. The efficiency of this method, though, is high,
since the propellant is found in situ, and doesn’t need to be carried by a spacecraft,
reducing difficulty and costs.
The Gravity Tractor method [21] exploits the Gravitational force between the asteroid and
the spacecraft. This force exerts a continuous pull on the asteroid, resulting in the desired
deflection. The main advantage of this method ids that the force directly acts on the
centre of mass of the celestial body, making irrelevant the rotation of the asteroid. On the
other hand, the thruster of the spacecraft needs always to be directed towards an empty
region of space, so that the exhausts do not interact with the asteroid itself.
The laser ablation method [22] is a process that consists in removing material from
asteroid’s surface by irradiating it with a laser beam or with the focusing of the Sun’s
radiation through a system of mirrors and lenses, in order to produce thrust through the
ejection of vapours from the surface.
The ion beaming method [23] that is based on displacing an asteroid by mean of a quasi-
neutral ion beam generated by a nearby spacecraft.
The Kinetic Impactor method [24] is for sure the simplest method listed here. In fact, it
involves a collision between the spacecraft and the asteroid to impart the desired
momentum exchange and to deflect the asteroid. The impact can be modelled as a
completely inelastic collision, and therefore the change in the velocity of the asteroid can
be immediately computed.
In general, impulsive techniques produce a larger deflection and are more easily
implementable. Low-thrust techniques would require a too large time window for
asteroids of relevant mass and size [17].
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For the scope of the present dissertation, only the kinetic impact method will be analysed,
since it has the highest technology readiness level due to its simplicity [25]. For these
reasons some studies have been carried on about the kinetic impactor.
NEOShield-2 project [26], continuing from previous NEOShield-1 project, is an
international project that assesses the threat posed by Near Earth Objects and looks at
the best possible solutions for dealing with a big asteroid or comet on a collision path with
our planet. The effort is led from Airbus Defence and Space, with Germany space agency
(DLR) as project coordinator. This project emphasizes on two aspects: the first aspect
focuses on technological development of essential techniques of guidance, control and
navigation (GNC) in the vicinity of celestial bodies. This is made through hitting the
asteroid with a kinetic impactor and observing it before and after for orbit determination
and monitoring; the second aspect focuses on refining NEOs characterisation.
Further studies have been carried in [27], in which it is studied the way orbital parameters
change due to an impact. In [15] a comparative assessment of different deflection’s
strategies for NEOs is presented. In [28] the deflection is studied to avoid the potential
threat and at the same time to ensure the increase of the MOID in future encounters.
1.3. Past and future missions
Some missions aimed to study asteroids and comets’ composition have already been
listed. Other missions, though, have been planned. Their scope is to prove the
effectiveness of the kinetic impactor method, to validate it also from a practical point of
view. The first mission that goes in this direction is a joint effort of NASA and ESA. The
name of the mission is Asteroid Impact & Deflection Assessment Mission (AIDA), still
under development [29]. A close-Earth encounter of the binary asteroid 65803 Didymos
(1996 GT) in October 2022 provides the optimal target for such a mission, allowing an
impact on Didymos’ secondary body to change its orbital period around the primary by a
measurable amount – as seen both from ground observatories and from rendezvous
spacecraft. AIDA mission is composed by two independent spacecraft: an asteroid
impactor, provided by NASA, called Double Asteroid Redirection Test (DART) [6], and an
asteroid rendezvous spacecraft, provided by ESA, called Asteroid Impact Mission (AIM)
[29]. DART will be launched between December 2020 and May 2021, and its duty will be
hitting the secondary body of the asteroid Didymos, Didymoon. AIM was scheduled for
launch in October 2020 and will be tasked with closely observing the impact, together
with ground-based telescopes. AIM will analyse not only the change in the orbital period
of Didymoon, but also the structure of the body after the impact and the ejecta plume.
All of these data will be exploited in future if a real threat scenario will occur. AIDA mission
replaced the Don Quijote Mission, an ESA prototype mission, which also f eatured two
spacecraft: an impactor and an orbiter [30].
The AIM spacecraft has been recently replaced by Hera [31], and will be the ESA
contribution to AIDA mission.
Most of the missions studied so far for the deflection of an asteroid through kinetic
impactor follow a direct hit strategy, meaning that the spacecraft designed for the
deflection perform an interplanetary transfer from Earth directly to the asteroid. In [32] a
gravity assist manoeuvre is included but only for a restricted pool of asteroids.
17
We can find interplanetary transfers from Earth to the asteroid, including gravity assist in
[33], in which the trajectory is optimised to maximise the number of visited asteroids of
the Atira group using the minimum propellant consumption. Anyway, this is not a
deflection mission, but a rendez-vous mission.
1.4. Thesis objectives
This thesis aims to find an optimal way to deflect any possible asteroid, using the kinetic
impactor strategy and exploiting the gravity assist of one of the following planets: Earth,
Mars or Venus. The spacecraft can also perform a deep space manoeuvre during its
interplanetary transfers. Moreover, also the case with a complete revolution of the
spacecraft during the Lambert’s arc will be introduced. The final aim of the mission will
be to maximise the deflection of the asteroid, minimising the initial mass of the spacecraft,
following the model built in [34]. This model is taken as starting point for the present work,
that wants to improve it thanks to some reasonings that will be presented later. It
becomes of primary importance of this work, therefore, the development of new
algorithms able to accomplish this duty.
Initially, a test case with an actual asteroid will be analysed. 2010RF12 asteroid’s threat
will be faced, first through a direct hit scenario and then exploiting the gravity assists of
the three planets. Once proved the effectiveness of the method, the same will be
repeated on the whole family of Near Earth Asteroids, created using a software provided
by the European Space Agency (ESA), the Near-Earth Object Population Observation
Program (NEOPOP). Then, following the model buil t up in [35], a synthetic population of
NEOs is created, comprehensive of all the spectrum of orbital parameters possible. Every
single node of this cloud is tagged with its probability to exist, according to the real
population created by NEOPOP, and is then deflected using the same method as before.
The effects are analysed, for each of the 4 missions (direct hit and the three gravity assists),
finding some characteristics useful to understand the best deflection strategy.
18
2. Theory of asteroid deflection
2.1. Deflection model
A way to identify the hazard of a Near Earth Object (NEO), is to compute its Minimum
Orbit Intersection Distance (MOID). MOID, as already said, is defined as the minimum
distance between the osculating orbits of two bodies. When the MOID is smaller, the
hazard is higher because a close approach is more likely to happen and, therefore, also an
impact if the two bodies are in phase. The MOID is not a constant, because ephemerides
of NEOs change in time, due to planetary perturbations [17]. In any case, the objective of
this work is to prove the effectiveness of the deflection through a kinetic impactor, with
gravity assists included in the spacecraft trajectory to intersect the NEO. Therefore, the
variation of the MOID in time is not taken in consideration, and so the resonance of the
asteroid’s orbit is not evaluated (as done in [36]). In fact, the deflection here is designed
in order to maximise the distance between the two bodies at the MOID.
When a real case scenario is envisaged (e.g., 2010RF12 in this case), the above-mentioned
assumptions are exploited, and the following procedure is used: since 2010RF12 has a
close approach foreseen in the end of 2095, the date chosen as starting time for the
search of the close approach is the 31st of January 2095. Then, the MOID is computed
numerically, by finding the minimum distance between the 2 orbits, given the orbital
parameters of the NEO and the Earth, and therefore the true anomalies of the two bodies
at MOID are found. These correspond to a time, that is computed through the Keplerian
law of planetary motion, starting from the previously selected starting time. Of course,
these two times will be different, because the two bodies will reach the MOID in different
instants, as it is shown in Figure 2.1.
19
FIGURE 2.1 - ORBITS OF EARTH AND ASTEROID AND THEIR POSITIONS AT THE MOID.
Once again, the purpose of the thesis is to prove the effectiveness of the deflection
manoeuvre. Therefore, the orbital parameters are changed in order to have the two
bodies at the MOID in the same instant. For this reason, the time at which MOID happens
is selected to be equal to the time at which Earth is at MOID, and consequently the
ephemerides of NEO are properly modified to create a synthetic case. In such a way, the
worst-case scenario is faced, giving more importance to this work, that, if able to solve
this case, will be even more able to solve “easier” cases. The choice to keep the real
Earth’s orbital parameters and to change those of the asteroid and not vice versa (as was
done in [34]) is due to the fact that also gravity assists have to be taken in account in this
study. It is important, therefore, to maintain unvaried the phasing between the planets.
It is much easier, indeed, to modify only the ephemerides of the asteroid with respect to
change all those of the bodies of the Solar System.
The objective is to maximise the deflection of the asteroid at the MOID, by applying a
deviation through the impact of the spacecraft with the asteroid. This impact acts as an
impulse, that perturbs the asteroid’s orbit. The impulse can be considered quasi-
instantaneous and the new orbit is considered to be proximal to the unperturbed one.
In this work the deflection model proposed by Vasile and Colombo is used [32], and is now explained. Let 𝑎 , 𝑒 , 𝑖 , Ω and ω be respectively the semi-major axis, eccentricity, inclination, right ascension of the ascending node and pericentre’s anomaly of the nominal orbit of the NEO (i.e. the unperturbed orbit). Let now be 𝜃𝑀𝑂𝐼𝐷 the true anomaly of the NEO at MOID along the nominal orbit as previously defined, and 𝜃𝑀𝑂𝐼𝐷
∗ = 𝜃𝑀𝑂𝐼𝐷 + ω the corresponding argument of latitude. We can now write the effect of the deviation
20
caused by the kinetic impactor, by using the proximal motion equations [37], obtaining in such a way the displacement with respect to the unperturbed position at the same epoch:
𝛿𝑠𝑟 ≈ 𝑟𝑀𝑂𝐼𝐷
𝑎𝛿𝑎 +
𝑎 𝑒 sin 𝜃𝑀𝑂𝐼𝐷
𝜂𝛿𝑀 − 𝑎 cos𝜃𝑀𝑂𝐼𝐷 𝛿𝑒
𝛿𝑠𝜃 ≈ 𝑟𝑀𝑂𝐼𝐷
𝜂3(1 + 𝑒 cos𝜃𝑀𝑂𝐼𝐷)2𝛿𝑀 + 𝑟𝑀𝑂𝐼𝐷 𝛿ω +
+ 𝑟𝑀𝑂𝐼𝐷 sin 𝜃𝑀𝑂𝐼𝐷
𝜂2(2 + 𝑒 cos𝜃𝑀𝑂𝐼𝐷)𝛿𝑒 + 𝑟𝑀𝑂𝐼𝐷 cos𝑖 𝛿Ω
𝛿𝑠ℎ ≈ 𝑟𝑀𝑂𝐼𝐷(sin 𝜃𝑀𝑂𝐼𝐷
∗ 𝛿𝑖 − cos𝜃𝑀𝑂𝐼𝐷∗ sin 𝑖 𝛿Ω )
EQUATION 2.1
The so computed 𝛿𝑠𝑟, 𝛿𝑠𝜃 and 𝛿𝑠ℎ represent the displacements in the radial, transversal
and perpendicular-to-the-orbit-plane directions. 𝑟𝑀𝑂𝐼𝐷 represents the distance of the
asteroid from the main body (i.e. the sun) at the MOID epoch, and 𝜂 = √1 − 𝑒2.
Two assumptions need to be made, to allow the proximal motion equations to be valid:
first, the deflection imparted to the asteroid 𝛿𝑟 = ‖𝛿𝒓‖, where 𝛿𝒓 = [𝛿𝑠𝑟 𝛿𝑠𝜃 𝛿𝑠ℎ]𝑇,
must be small with respect to the orbit radius 𝑟𝑀𝑂𝐼𝐷, that is 𝛿𝑟 ≪ 𝑟𝑀𝑂𝐼𝐷 [37] . Moreover,
because of the 𝜂 term, we are constrained to work with elliptical orbits ( 𝑒 < 1 ),
assumption that is valid for the case of NEOs but not for some comets.
In Figure 2.2 it is shown a schematic view of Earth, unperturbed asteroid and deflected
asteroid.
FIGURE 2.2 – DISTANCE VECTORS AT THE MOID (NON-DEVIATED NEO IN ORANGE, DEVIATED NEO IN
RED).
Following [32], the computation of the variation of the orbital parameters is made
through Gauss’ planetary equations [38], assuming that the variation in the velocity of the
asteroid 𝛿𝒗, caused by the impact, is instantaneous. The latter is decomposed as follows:
𝛿𝒗 = [𝛿𝑣𝑡 𝛿𝑣𝑛 𝛿𝑣ℎ]𝑇 , where 𝛿𝑣𝑡 and 𝛿𝑣𝑛 are the components in the plane of the osculating orbit, in tangential and normal directions respectively, while 𝛿𝑣ℎ is the
component perpendicular to the plane.
21
The Gauss’ planetary equations assume the form:
𝛿𝑎 =2 𝑎2 𝑣𝑑
𝜇𝑆𝑢𝑛𝛿𝑣𝑡
𝛿𝑒 = 1
𝑣𝑑[2(𝑒 + cos𝜃𝑑 )𝛿𝑣𝑡 −
𝑟𝑑
𝑎sin 𝜃𝑑 𝛿𝑣𝑛 ]
𝛿𝑖 = 𝑟𝑑 cos𝜃𝑑
∗
ℎ𝛿𝑣ℎ
𝛿Ω = 𝑟𝑑 cos𝜃𝑑
∗
ℎ sin 𝑖𝛿𝑣ℎ
𝛿𝜔 = 1
𝑒 𝑣𝑑[2 sin 𝜃𝑑 𝛿𝑣𝑡 + (2𝑒 +
𝑟𝑑
𝑎cos𝜃𝑑) 𝛿𝑣𝑛] −
𝑟𝑑 sin 𝜃𝑑∗ cos𝑖
ℎ sin 𝑖𝛿𝑣ℎ
𝛿𝑀𝑡𝑑= −
𝑏
𝑒 𝑎 𝑣𝑑
[2 (1 + 𝑒2 𝑟𝑑
𝑝)sin 𝜃𝑑 𝛿𝑣𝑡 +
𝑟𝑑
𝑎cos𝜃𝑑 𝛿𝑣𝑛]
EQUATION 2.2
where 𝑟𝑑 and 𝑣𝑑 are the norm respectively of the orbital radius and velocity at the point
in which the deflection is applied, ℎ is the angular momentum, 𝑝 the semilatus rectum, 𝑏
the semi-minor axis, 𝜇𝑆𝑢𝑛 the gravitational constant of the Sun, 𝜃𝑑 the true anomaly of
the asteroid at the deflection and 𝜃𝑑∗ = 𝜃𝑑 + 𝜔.
Another consideration has to be made: the so computed variation in the mean anomaly
M takes in account only the instantaneous change of geometry at the deflection time 𝑡𝑑.
However, the semi-major axis is changed too, and consequently the mean motion
𝑛 changes, provoking a second contribution to the variation in the mean anomaly [17]:
𝛿𝑀𝑛 = 𝛿𝑛(𝑡𝑀𝑂𝐼𝐷 − 𝑡𝑑) = 𝛿𝑛 𝛥𝑡
𝛿𝑛 = √𝜇𝑆𝑢𝑛
(𝑎 + 𝛿𝑎) 3 − √𝜇𝑆𝑢𝑛
𝑎3
𝛥𝑡 = (𝑡𝑀𝑂𝐼𝐷 − 𝑡𝑑)
EQUATION 2.3
𝛥𝑡 is named time-to-MOID. Finally, we can write the total variation of the mean anomaly
by adding the two above mentioned terms: 𝛿𝑀 = 𝛿𝑀𝑡𝑑+ 𝛿𝑀𝑛 [39].
22
Eventually, the way of computing 𝛿𝒗 is shown. The velocity of the spacecraft at the end
of the last Lambert’s arc is known and the velocity of the asteroid at the deflection time
is known as well. The spacecraft hitting the asteroid is modelled as a completely inelastic
collision. The collision is seen from a reference frame attached to the asteroid. Therefore,
the spacecraft, of mass 𝑚𝑆𝐶 computed through the Tsiolkowsky’s equation [40] , has an
impact velocity 𝜟𝒗, given by the difference between its absolute velocity and that of the
NEO. On the other hand, the asteroid is seen as still in this frame. After the collision, the
NEO is moving with the velocity 𝛿𝒗 previously defined and has a mass augmented by that
of the spacecraft. That said, it is possible to write the equation: 𝑚𝑆𝐶 𝜟𝒗 = (𝑚𝑆𝐶 +
𝑚𝑁𝐸𝑂) 𝛿𝒗. By rearranging the terms, we obtain:
𝛿𝒗 = 𝛽 𝑚𝑆𝐶
(𝑚𝑆𝐶 + 𝑚𝑁𝐸𝑂) 𝜟𝒗
EQUATION 2.4
and
𝑚𝑆𝐶 = 𝑚𝑆𝐶0 ∗ 𝑒−(
𝛥𝑣𝑚𝑎𝑛𝐼𝑠𝑝 𝑔0
)
EQUATION 2.5
where 𝑚𝑁𝐸𝑂 is the mass of the NEO, 𝛥𝑣𝑚𝑎𝑛 is the sum of the impulses given by the
spacecraft during the manoeuvres in the interplanetary transfer, 𝐼𝑠𝑝 is the specific
impulse of the engines, 𝑔0 is the standard gravity.
𝛽 is the momentum enhancement factor. This scalar value can be either equal or larger
than 1. It is 1 in the ideal case (like in this work), when neither the impactor nor any ejecta
escape during the collision, and so only the momentum of the impactor is transferred to
the asteroid. It is larger than 1 when the real case is studied. In fact, when the ejecta
produced are fast enough to escape the asteroid’s gravitational well, laboratory
experiments and simulations suggest that 𝛽 may be substantially larger depending on the
impact as well as the regolith, surface and subsurface properties of the target [41]. The
main factor that determining the momentum enhancement factor can be seen in porosity
and strength of the target’s material, besides from the influence of impact velocities and
masses of the colliding objects, their size, etc. [42]. The knowledge of the momentum
enhancement factor is one of the main unknown when planning a NEO deflection mission,
indeed is one of the target of the study of the currently planned NEO study missions
described in Section 1.3 [29].
It needs to be noticed that the 𝛿𝒗 obtained through the Equation 2.4 is in cartesian
coordinates. It has therefore first to be rotated to obtain along the tangential, normal and
perpendicular-to-orbit-plane components, exploited in Equation 2.2.
In such a way we have that the deflection 𝛿𝑟 is function of the variation of the orbital
parameters through the proximal motion equations (Equation 2.1), and the variation of
23
the orbital parameters is dependent on the variation of velocity given to the asteroid 𝛿𝒗,
through the Gauss planetary equation (Equation 2.2). This last 𝛿𝒗 is found thanks to
Equation 2.4, and it relies on 𝜟𝒗 that we can found from the geometry of the mission. The
problem is therefore in a closed form.
2.2. Mission design – Direct hit
The strategy used in this work follows [34] and is presented in the following
It is necessary now to define the design variables, that will uniquely determine the
geometry of the mission. These quantities, then, have to be analysed by the optimiser
that will find the best set, by minimising a previously selected objective function (this can
be in principle both a single-objective or a multi-objective optimisation function).
The first mission analysed is a direct hit with a deep space manoeuvre during the
interplanetary transfer.
The spacecraft leaves Earth’ Sphere of Influence (SOI) at time 𝑡0.
The escape velocity of the spacecraft from Earth’s SOI is identified by a vector of three
quantities: ‖𝛥𝑣0‖, 𝛼𝛥𝑣0 and 𝛿𝛥𝑣0
. ‖𝛥𝑣0‖ is the norm of the excess velocity. 𝛼𝛥𝑣0 is the
angle that identifies the in-plane direction of the excess velocity with respect to the
tangential direction of the heliocentric velocity of the Earth. 𝛿𝛥𝑣0 is the angle that
identifies the out-of-plane direction of the excess velocity with respect to the heliocentric
velocity of the Earth. These angles are shown in Figure 2.3.
FIGURE 2.3 - EXPLANATION OF ANGLES. [35]
After this phase, the spacecraft starts its interplanetary transfer towards the asteroid. At
time 𝑡𝐷𝑆𝑀1 engines are activated and the spacecraft modifies its orbit.
Finally, at time 𝑡𝑑 the deflection happens, following the model presented in Section 2.1.
For this problem it is necessary to set 7 design variables:
24
𝒙 = 𝛼0 𝛼1 𝑇𝑜𝐹1 ‖𝛥𝑣0‖ 𝛼𝛥𝑣0 𝛿𝛥𝑣0
𝑚𝑆𝐶0
EQUATION 2.6
𝛼0 is a parameter to select the starting time 𝑡0. This is computed as follows:
𝑡0 = 𝑡𝑖𝑛𝑖𝑡 + (𝑡𝑀𝑂𝐼𝐷 – 𝑡𝑖𝑛𝑖𝑡 − ∑ 𝑇𝑜𝐹𝑖
2
𝑖=1
) · 𝛼0
where 𝑡𝑖𝑛𝑖𝑡 is the beginning of the time window in which the spacecraft can start its
mission, 𝑡𝑀𝑂𝐼𝐷 is the time at which MOID happens and 𝑇𝑜𝐹𝑖 is the time of flight of the i-
th branch of the interplanetary transfer (for a direct hit there is only one branch, and
therefore only one time of flight; the equation is written in this way to include also the
single-flyby case, presented in Section 2.4) The initial time is found by setting a warning
time. This warning time is defined as a window of time, precedent to the MOID, in which
it is possible to perform the transfer and the deflection mission:
𝑡𝑖𝑛𝑖𝑡 = 𝑡𝑀𝑂𝐼𝐷 − 𝑤𝑎𝑟𝑛𝑖𝑛𝑔𝑇𝑖𝑚𝑒
𝛼0 is bound between the values 0 and 1. When it is 0, 𝑡0 becomes equal to 𝑡𝑖𝑛𝑖𝑡, meaning
that the spacecraft starts as soon as the asteroid is detected. When it is 1, 𝑡0 becomes
equal to 𝑡𝑀𝑂𝐼𝐷 minus the sum of the times of flight of the branches, meaning that the
spacecraft hits the asteroid at the MOID.
𝛼1 is a parameter to select the time at which the deep space manoeuvre is performed. It
is bound between the values 0 and 1, so that it identifies at which moment of the
interplanetary transfer the impulse is given:
𝑡𝐷𝑆𝑀1 = 𝑡0 + 𝛼1 · 𝑇𝑜𝐹1
𝑚𝑆𝐶0 is the initial mass of the spacecraft.
Then, also the objective function has to be selected. For the first analysis, a double-
objective function is selected, composed by two conflicting quantities:
𝑱 = −(𝑟𝑝 − 𝑟𝑝0) 𝑚𝑆𝐶0
𝑟𝑝 represents the minimum distance of the asteroid from the Earth at the MOID after the
deflection, while 𝑟𝑝0 is that without deflection. In other words, 𝑟𝑝 represents the perigee
of the close approach hyperbola between the NEO and the Earth.
Since the optimiser works by minimising the objective function, a minus is added to the
first quantity, so doing the solution searched are those with maximum deflection and
minimum spacecraft initial mass.
25
2.3. Orbital Mechanics
2.3.1. Interplanetary transfers
The interplanetary transfer taken by the kinetic impactor is divided in two branches.
The spacecraft exits from a planet sphere of influence with a known velocity and moves
in the space according to the propagation of a Keplerian orbit, assuming the hypothesis
of the restricted 2 body problem [43]. This model ends when the deep space manoeuvre
is made. From this point on the spacecraft moves along a Lambert arc going to the next
planet in a given time.
2.3.2. Fly-by
2.3.2.1. Geometry of the hyperbola
The model of the hyperbolic transfer follows [38] Every time an object in the Solar System,
such as a comet or an asteroid, passes close to a planet, its heliocentric trajectory is
modified. This is due to the gravitational pull of the planet playing an important role in
determining the small body’s orbit when performing a fly-by. In fact, a region of space in
which the attraction of a body dominates over the one of the Sun can be determined; it
is known as the Sphere of Influence (SOI).
The present section details some general properties concerning the fly-by of a small body
around a planet. The formulas featured in this chapter will be used throughout the
following steps of this dissertation to obtain some of the equations necessary for the work.
Considering the classic two-body problem relation, that will be exploited during all the
work,
𝑟 = 𝑝
1 + 𝑒 𝑐𝑜𝑠𝜃
EQUATION 2.7
which expresses the distance of the minor body with respect to the major one in the
perifocal reference frame, we can compute the direction of the asymptotes of a
hyperbolic flyby with respect to the perigee direction as
𝜃∞ = arccos(−1
𝑒)
EQUATION 2.8
where 𝑟 is the distance between the two bodies, 𝑝 and 𝑒 are the hyperbolic orbit’s semi-
latum rectum and eccentricity respectively and 𝜃 is the true anomaly (𝜃∞ is its value when
the distance goes to infinity).
26
The other involved angles will therefore be
β = π − 𝜃∞
δ = 2 𝜃∞ − π = arcsin (1
𝑒)
𝛿 is the turn angle, which provides information circa the degree of rotation during the fly-
by, whereas 𝛽 is the complementary angle to 𝜃∞ [44].
From Figure 2.4, which expresses the fly-by geometry, we can derive
FIGURE 2.4 - GEOMETRY OF THE HYPERBOLA. [36]
𝑏 = 𝑎 tan β
𝛥 = (𝑟𝑝 + 𝑎) sin β
By introducing the formulation of 𝛽 derived above, we obtain the impact parameter (also
known as the aiming radius) [38] as
𝑏 = 𝛥 = 𝑎 √𝑒2 − 1
where is the semi-major axis of the hyperbola after a sign change (that explains the hat
symbol) and 𝑟𝑝 is the pericentre radius.
The energy conservation equation applied for the distance going to infinity can be used
to obtain the modulus of the planetocentric velocity of an object leaving (or entering into)
a planet’s SOI on a hyperbolic trajectory
v∞ = √μ𝑃
𝑎
where μ𝑃 is the gravitational constant of the planet.
Therefore, given the impact parameter and the planetocentric velocity of an object, it is
possible to compute the pericentre radius as:
27
r𝑝 = √μ𝑃
v∞4 + 𝑏2 −
μ𝑃
v∞2
This formula is used also to find the distance of the asteroid from Earth before and after
the impact with the kinetic impactor, so that the deflection achieved can be computed as
a mere difference [45].
2.3.2.2. Powered gravity assist
During a powered gravity assist, with the powered manoeuvre happening at the
pericentre, it is important to notice that
δ = 𝜃∞1 + 𝜃∞2 − π
where 𝜃∞1 and 𝜃∞2 are the true anomalies at infinite distance before and after the
manoeuvre.
The procedure to follow is to find the pericentre velocity, given the initial infinite absolute
velocity and the pericentre radius. This is then increased of the value of the impulse given
𝛥𝑣𝑃𝑂𝑊. It is now possible to compute 𝜃∞1, 𝜃∞2 and the final infinite absolute velocity, that is changed. This is possible, exploiting the following equation, relative to hyperbolic
orbits:
𝑣𝑝 = √2 μ𝑃
r𝑝 −
μ𝑃
𝑎
𝑣∞ = √− μ𝑃
𝑎
𝑣𝑝2 = 𝑣𝑝1 + 𝛥𝑣𝑃𝑂𝑊
This set of equations, together with Equation 2.8, allows me to define the infinite velocity
of the spacecraft after the powered manoeuvre both in modulus and in direction.
Knowing also the plane on which the gravity assist is performed (in both powered or not-
powered gravity assist) closes the problem. This plane is identified by introducing an angle
𝛾 that will be defined in Section 2.4
2.4. Mission design – Gravity assist scenario (not powered)
To improve the results obtained in [34], also the possibility to perform a gravity assist
around one of the above mentioned planets (Earth, Mars, Venus) is analysed.
To do so, it is necessary to implement the design variables, to include some quantities to
identify the gravity assist. In fact, the spacecraft, at the end of the first interplanetary
transfer will now perform a swing-by around a planet. This is identified by two quantities:
𝛾2 is the first parameter used to identify the flyby around the second planet. It is an angle
that identifies the plane on which the swing-by happens. The procedure is the following:
28
the normal vector to the plane containing the incoming velocity of the spacecraft relative
to the flyby planet and the planet’s velocity is computed. Then, this vector is rotated
around the relative incoming velocity of the spacecraft of an angle 𝛾2. The vector obtained
is the normal vector to the plane on which the swing-by happens.
𝑟𝑝2 is the second parameter used to identify the flyby around the second planet. This
quantity, multiplied by the radius of the swing-by planet, determines the hyperbola’s
pericentre radius.
After the fly-by, the spacecraft starts for a second interplanetary transfer and, after having
performed a deep space manoeuvre at 𝑡𝐷𝑆𝑀2, it arrives and hits the asteroid.
For this problem it is necessary to set 11 design variables:
𝒙 = 𝛼0 𝛼1 𝑇𝑜𝐹1 𝛾2 𝑟𝑝2 𝛼2 𝑇𝑜𝐹2 ‖𝛥𝑣0‖ 𝛼𝛥𝑣0 𝛿𝛥𝑣0
𝑚𝑆𝐶0
EQUATION 2.9
𝛼1 is a parameter to select the time at which the second deep space manoeuvre is
performed. It is bound between the values 0 and 1, so that it identifies at which moment
of the interplanetary transfer the impulse is given:
𝑡𝐷𝑆𝑀2 = 𝑡0 + 𝑇𝑜𝐹1 + 𝛼2 · 𝑇𝑜𝐹2
where 𝑇𝑜𝐹2 is the time of flight on the second interplanetary branch.
29
3. Test cases results
3.1. Direct Hit
Initially the problem of the deflection of the 2010RF12 is faced through a direct hit
scenario. In Tab.3.1 we show the orbital parameter of the asteroid. [12]
Semi-major axis
Eccentricity Inclination Right ascension of ascending node
Argument of the periapsis
1.58· 108𝑘𝑚
0.187 0.911 𝑑𝑒𝑔 162 𝑑𝑒𝑔 267 𝑑𝑒𝑔
TABLE 3.1 - ORBITAL PARAMETERS OF 2010RF12.
The set of design variable, following Section 2.2, is defined in Equation 2.6.
This type of mission has already been analysed in [34], although some variations, that will
be explained later in this chapter, have been carried to obtain better solutions.
The other parameters set for the mission are shown in Tab.3.2
𝑤𝑎𝑟𝑛𝑖𝑛𝑔𝑇𝑖𝑚𝑒 10 𝑦𝑒𝑎𝑟𝑠
𝛥𝑣𝑙𝑎𝑢𝑛𝑐ℎ 1 𝑘𝑚/𝑠
𝐼𝑠𝑝 300 𝑠
𝐷𝑁𝐸𝑂 100 𝑚
𝜌𝑁𝐸𝑂 2600 𝑘𝑔/𝑚3
𝛽 1
TABLE 3.2 - PARAMETERS FOR THE MISSION.
𝛥𝑣𝑙𝑎𝑢𝑛𝑐ℎ and 𝐼𝑠𝑝 are parameters to define the launcher properties: the first is the
maximum excess velocity of the spacecraft from the Earth provided by the launcher, the
second the specific impulse. These values are more than feasible, according to [4].
𝐷𝑁𝐸𝑂 and 𝜌𝑁𝐸𝑂 are parameters to define the physical properties of the asteroid. From
these values, taken from [46] it is possible to calculate the mass of the asteroid to be
deflected, assuming the shape of it being a sphere:
𝑚𝑁𝐸𝑂 =1
6 𝜋 𝜌𝑁𝐸𝑂 𝐷𝑁𝐸𝑂
3
Finally, assuming 𝛽 = 1 means that the impact is modelled as an ideal totally inelastic
impact.
30
After these parameters are set, it is necessary to define the bounds of the design variables,
creating a domain in which the global evolutionary algorithm for the optimization has to
work to find the best solutions. The upper and lower bound for each one of the design
variables are shown in Tab.3.3.
Variable 𝛼0 𝛼1 𝑇𝑜𝐹1 ‖𝛥𝑣0 ‖ 𝛼𝛥𝑣0 𝛿𝛥𝑣0
𝑚𝑆𝐶0
Lower bound
0 0 0.01 𝑃𝑚𝑎𝑥 0 𝑘𝑚/𝑠 −𝜋 𝑟𝑎𝑑 −𝜋/2 𝑟𝑎𝑑
300 𝑘𝑔
Upper bound
0.99 1 4 𝑃𝑚𝑎𝑥 3 𝛥𝑣𝑙𝑎𝑢𝑛𝑐ℎ +𝜋 𝑟𝑎𝑑 +𝜋/2 𝑟𝑎𝑑
8000 𝑘𝑔
TABLE 3.3 - BOUNDS FOR THE DESIGN VARIABLES - DIRECT HIT SCENARIO.
𝛼0 and 𝛼1 bounds have already been explained in Section 2.2 and Section 2.4. 𝑃𝑚𝑎𝑥 is
defined as the maximum orbital period between the celestial bodies that are taken in
account during the simulation. In this case, therefore, it is the maximum period between
Earth and 2010RF12’s orbital periods. 𝑇𝑜𝐹1’s bounds are selected in order to allow the
simulation to find the real optimal solution, and for this reason the time window is that
wide. Here we find a variation with respect to the simulation carried in [34], where the
𝑇𝑜𝐹1 was bound between 0.01 𝑃𝑚𝑎𝑥 and 1.1 𝑃𝑚𝑎𝑥. This limitation could, in fact, restrict
too much the interplanetary transfer. ‖𝛥𝑣0‖, that is the excess velocity of the kinetic
impactor with respect to Earth’s velocity, is bound according to feasible values [4]. This
value is different from 𝛥𝑣𝑙𝑎𝑢𝑛𝑐ℎ, that is the maximum excess velocity furnished by the
launcher. So, for example, if the simulation finds that the best solution requires an excess
velocity of 2 𝑘𝑚/𝑠, this means that 1 𝑘𝑚/𝑠 will be furnished by the launcher and 1 𝑘𝑚/𝑠
needs to be furnished by using the on-board fuel. 𝛼𝛥𝑣0 and 𝛿𝛥𝑣0
bounds are selected to
not exclude any direction. 𝑚𝑆𝐶0 is bound between two feasible values. It is important to
notice that the initial mass is intended as the mass of the spacecraft after the launch.
Now that design parameters and variables have been set, it is necessary to define the
characteristics of the optimization.
The optimiser used is the Evolutionary Predictive Interval Computation (EPIC) algorithm,
developed by Vasile [47]. This works by performing two simulations: in the first, it explores
the domain, dividing it in sub-domains and searching for those “zones” of the domain that
have better solutions; in the second, it converges to the best solutions. The convergence
is highly influenced by the selection of the parameters of the optimisation. Since EPIC is
an evolutionary algorithm, it is necessary to define the number of function evaluations to
perform in every simulation and the number of individuals that analyse the domain i n
which it is possible to find the solution. These two parameters have been chosen as the
tuning parameters, to reach a better solution. In [34], these parameters were selected
according to domain definition. The values are shown in Tab.3.4, and that brought to the
solution shown in Figure 3.1.
Case Function evaluations Number of individuals
Colombo [17] 100,000 100
Present work 500,000 200 TABLE 3.4 - OPTIMIZATION PARAMETERS.
31
It is important now to define the Pareto front. The Pareto front is a set of optimal solutions
for a multi-objective function. It can be defined when the objectives are conflicting
between each other (like in our case). A solution belongs to the pareto front (is pareto-
optimal) when none of the objectives can be improved in value without degrading some
of the other objectives. In our case, it can be depicted through a 2D plot, having the
objectives as axes.
It is shown in Figure 3.1 the Pareto front of the direct hit scenario, according to the
parameters used in [17].
In the present work, anyway, we select to have a wider domain, due to the selection of a
wider window in the time of flight of the interplanetary transfer. The values shown in
Tab.3.4 are selected to reach the new solution of the problem.
It is possible to see, in Figure 3.2 that the new solution gives better results with respect
to that obtained in [34], as expected, thanks to the possibility to have a longer
interplanetary transfer. In fact, we see that with the same initial mass, we can now obtain
larger values in the achievable deflection. The Pareto front shows that the two objective
functions are conflicting, as expected. In fact, decreasing the initial mass brings to have a
smaller achievable deflection, and vice versa. Moreover, it is important to highlight the
FIGURE 3.1 – PARETO FRONT FOR DIRECT HIT SCENARIO - COLOMBO PHD THESIS [17]. IN THE COLORMAP
THE ACHIEVABLE DEFLECTION.
32
linearity that links the two quantities. This relation will be explained in the next chapter,
after that also the fly-by cases will be analysed.
FIGURE 3.2 - PARETO-FRONT FOR DIRECT HIT SCENARIO - PRESENT WORK. IN COLORMAP THE
ACHIEVABLE DEFLECTION.
In the figures from 3.3 to 3.7 we can see other characteristics of this kind of transfer.
In Figure 3.3 we can see the distance we can achieve thanks to the deflection versus the
modulus of the variation of velocity given to the asteroid. Also in this case, it is possible
to notice a linearity in the graphic, meaning that the two quantities have a direct
proportionality relationship.
In Figure 3.4 we can see the direction with which the kinetic impactor hits the asteroid.
Two considerations can be made. The first is that the in-plane angle between asteroid and
spacecraft takes values between −30 𝑑𝑒𝑔 and 30 𝑑𝑒𝑔. This means that the best way the
spacecraft can arrive and hit the asteroid is more or less tangentially. The second is that
the out-of-plane angle between the spacecraft and the asteroid is almost zero. This can
be easily explained looking at 2010RF12’s inclination in Tab.2.1. In fact, this asteroid has
an almost null inclination, as well as Earth, making easier for the spacecraft to reach this
new orbit on an almost coplanar orbit (for graphical depiction of angles, see Figure 2.3)
33
FIGURE 3.3 - DELTA-V GIVEN TO THE ASTEROID VS ACHIEVABLE DISTANCE FOR DIRECT HIT SCENARIO. IN COLOURMAP THE ACHIEVABLE DEFLECTION SCALED ON THE EARTH RADIUS.
FIGURE 3.4 - IN-PLANE ANGLE VS OUT-OF-PLANE ANGLE BETWEEN IMPACTOR AND ASTEROID'S
VELOCITIES FOR DIRECT HIT SCENARIO. IN COLOURMAP THE ACHIEVABLE DEFLECTION SCALED ON
EARTH'S RADIUS.
34
It is possible to notice in Figure 3.5 and Figure 3.6 the fact that the departure time happens
more or less 9.5 years before the predicted MOID. This means that, in order to achieve
the best possible deflection, it is necessary to leave as soon as possible. Given a time
window of 10 years in fact, the spacecraft waits only around 6 months to leave Earth.
Finally, in Fig.3.7 we can see the geometry of one of the trajectories found by the
optimizer, that shows us the orbits of Earth, asteroid and the interplanetary manoeuvres.
It is evident also from this plot that the spacecraft hits the asteroid almost tangentially.
FIGURE 3.5 - DEPARTURE TIMES (YEARS BEFORE MOID) VS ACHIEVABLE DISTANCE FOR DIRECT HIT
SCENARIO. IN COLOURMAP THE TOF OF THE TRANSFER.
35
FIGURE 3.6 - DEPARTURE TIME (YEARS BEFORE MOID) VS MASS OF THE SPACECRAFT AT DEFLECTION
FOR DIRECT HIT SCENARIO. IN COLOURMAP THE TOF FOR THE TRANSFER.
FIGURE 3.7 - DIRECT HIT DEFLECTION MISSION.
36
3.2. Single Gravity Assist
Once proved the effectiveness of the optimization method, the simulation was repeated,
including the gravity assists. The three solutions so obtained show the effect of the swing-
by of each one of the three above mentioned planets on the achievable deflection.
The parameters shown in Tab.3.2 are kept unvaried also in these simulations, since they
just define the launcher’s, asteroid’s and impact’s properties.
The bounds for the design variables shown in Tab.3.3 are also kept unvaried, but of course
it is necessary to define a new set of bounds, for the set of variables that was not present
during the direct hit scenario, defined in Equation 2.9. These are shown in Tab.3.5.
Variable 𝛾2 𝑟𝑝2 𝛼2 𝑇𝑜𝐹2
Lower bound −𝜋 𝑟𝑎𝑑 1.1 0 0.01 𝑃𝑚𝑎𝑥 Upper bound +𝜋 𝑟𝑎𝑑 66.0 1 4 𝑃𝑚𝑎𝑥
TABLE 3.5 - BOUNDS FOR THE DESIGN VARIABLES - SINGLE GRAVITY ASSIST SCENARIO.
𝛾2 is chosen in order to allow the swing-by to happen in every possible plane. 𝑟𝑝2 has his
minimum dictated by a small but feasible value, while his maximum is put at 10 times the
geo-stationary distance (42,168 km). 𝛼2 and 𝑇𝑜𝐹2 follow the same reasoning as 𝛼1 and
𝑇𝑜𝐹1 as explained in Section 3.1.
The same parameters as those of the direct hit scenario have been chosen for the
simulation (Tab.3.4). This allows the optimiser to get the best solution and makes possible
to compare the two methods starting from the same data and assumptions.
In the next paragraph the results for deflection missions via flybys of the Earth, Mars and
Venus, respectively, are presented.
3.2.1. Earth gravity assist
In Figure 3.8 we can see the Pareto front of Earth gravity assist scenario. Two
considerations can be made: first, the linearity is preserved, meaning that the linear
relation between initial mass of the spacecraft and achievable deflection exists no matter
the geometry of the mission; the second is that this mission allows to get greater
deflections with respect to the direct hit scenario, highlighting the usefulness of this new
method. This can in fact be seen as a possibility to get a better deflection using the same
initial mass, or also, to have the same deflection with a smaller initial mass (meaning a
cheaper mission).
The linearity is respected also in the relation between variation of velocity given to the
asteroid and achievable deflection (Fig.3.9). It is also possible to notice that, the variation
of velocity given to the asteroid is higher in this case, meaning that this new geometry of
the mission allows the spacecraft to reach the asteroid with a larger relative velocity or a
larger mass, looking at Equation 2.4.
37
FIGURE 3.8 - PARETO-FRONT FOR EARTH'S GRAVITY ASSIST SCENARIO. IN COLOURMAP THE
ACHIEVABLE DEFLECTION.
FIGURE 3.9 - DELTA-V GIVEN TO THE ASTEROID VS ACHIEVABLE DISTANCE FOR EARTH GRAVITY ASSIST
SCENARIO. IN COLOURMAP THE ACHIEVABLE DEFLECTION SCALED ON EARTH'S RADIUS.
38
In Fig.3.10 we can see the direction of the impact. It is immediate to make the same
considerations as the direct hit scenario. In fact, the in-plane angle between spacecraft’s
velocity and asteroid’s velocity is bound between −30 𝑑𝑒𝑔 and 10 𝑑𝑒𝑔, confirming the
fact that the best way to hit the NEO is tangentially as repeated in Section 3.1. At the same
way, out-of-plane angle between the two velocity takes small values (no more than
10 𝑑𝑒𝑔), confirming that the interplanetary transfer happens on plane almost coplanar
to those of Earth and asteroid.
FIGURE 3.10 - IN-PLANE ANGLE VS OUT-OF-PLANE ANGLE BETWEEN IMPACTOR AND ASTEROID'S
VELOCITIES FOR EARTH GRAVITY ASSIST SCENARIO. IN COLOURMAP THE ACHIEVABLE DEFLECTION
SCALED ON EARTH'S RADIUS.
Once again, looking at Fig.3.11 and Fig.3.12, it is evident that it is better to hit the
spacecraft as soon as possible. In fact, we can see that most of the solutions found have
their departure time 10 years before the expected MOID, i.e the launch happens
immediately, at the beginning of the time window. Moreover, exploiting the gravity assist
allows the spacecraft to use less fuel and in fact it reaches the asteroid with a larger mass.
The mission’s orbits are presented in Fig.3.13
39
FIGURE 3.11 - DEPARTURE TIME (YEARS BEFORE MOID) VS ACHIEVABLE DISTANCE FOR EARTH
GRAVITY ASSIST SCENARIO. IN COLOURMAP THE SUM OF THE TOF FOR THE TWO INTERPLANETARY
TRANSFERS.
FIGURE 3.12 - DEPARTURE TIME (YEARS BEFORE MOID) VS MASS OF THE SPACECRAFT AT DEFLECTION
FOR EARTH GRAVITY ASSIST SCENARIO. IN COLOURMAP THE SUM OF THE TOF FOR THE TWO
INTERPLANETARY TRANSFERS.
40
FIGURE 3.13 - EARTH GRAVITY ASSIST DEFLECTION MISSION.
3.2.2. Mars and Venus gravity assists
The procedure is now repeated for both Mars and Venus. The same values, both for design
parameters and optimiser parameters, are kept.
In Fig.3.14 and Fig.3.15 we can see some examples of optimal deflection trajectories,
including Mars gravity assist and Venus gravity assist.
In Fig.3.16 we can see the comparison between the Pareto fronts of all the four above-
mentioned strategies and in Fig.3.17 the slopes of the distances achievable versus the
variation of velocity given to the asteroid.
41
FIGURE 3.14 - MARS GRAVITY ASSIST DEFLECTION MISSION.
FIGURE 3.15 - VENUS GRAVITY ASSIST DEFLECTION MISSION.
42
FIGURE 3.16 - COMPARISON BETWEEN THE PARETO-FRONTS OF THE 4 SCENARIOS. BLUE FOR EARTH
GRAVITY ASSIST; BLACK FOR DIRECT HIT; GREEN FOR VENUS GRAVITY ASSIST; RED FOR MARS GRAVITY
ASSIST.
FIGURE 3.17 -COMPARISON OF DELTA-V GIVEN TO THE ASTEROID VS ACHIEVABLE DISTANCE FOR THE 4
SCENARIOS. BLUE FOR EARTH GRAVITY ASSIST; BLACK FOR DIRECT HIT; GREEN FOR VENUS GRAVITY
ASSIST; RED FOR MARS GRAVITY ASSIST.
43
Some considerations can be carried on. Once again, we see that the linearity relations are
respected. Anyway, the deflections achievable are much smaller in the cases including
Mars and Venus gravity assists.
For Mars and Venus, we can see that also the variation of velocity given to the asteroid
have low values, although the direction remains almost tangent. This means that the bad
deflection achievable is due to the timing of deflection, and therefore to a bad phasing
between the planets and the asteroid. Future works need to perform the same analysis
in a different launch window, when the phasing of the planets is good.
In all cases, the trend is to have the departure time as soon as possible in the time window
of 10 years.
In conclusion, it is immediate saying that direct hit scenario and Earth’s swing-by furnish
better solutions with respect to Mars and Venus gravity assists. For this reason, in the next
chapter it will be discussed how to improve the effectiveness of these two missions, by
varying the geometry and the design of the trajectories.
44
4. Improved test cases results
4.1. Multi-revolution Lambert model
4.1.1. Theory
The first modification at the mission affects the Lambert’s arc of the interplanetary
transfers. In fact, up to now the Lambert’s arc has been design as direct, meaning that the
path of the spacecraft along the orbit was shorter than the orbit itself. In other words, the
spacecraft was allowed to reach the next planet for the gravity assist or the asteroid for
the impact staying on its orbit for less than one period. But, according to [48], given a
certain 𝑇𝑜𝐹, a starting point (point A) and an arrival point (point B; these 3 conditions are
the conditions necessary to solve the Lambert Problem), it exists a number 𝑁𝑚𝑎𝑥 ≥ 0
such that in this exact 𝑇𝑜𝐹 the spacecraft starts from points A and arrives to point B
performing 𝑁𝑚𝑎𝑥 complete revolutions of its orbit. It is possible, therefore, to have at
least one solution of the Lambert problem performing 𝑁 ≤ 𝑁𝑚𝑎𝑥 complete orbit’s
revolutions. Moreover, for every 𝑁 ≥ 1, there are two solutions solving the Lambert
problem: one is given by an orbit with high eccentricity and low energy and one is given
by an orbit with low eccentricity and high energy. This is not valid for the zero-revolution
case, which has only one solution for a fixed time of flight (Fig.4.1 [48]).
Exploiting this theory, we can include in the simulation a single -revolution Lambert’s arc
(more than one revolution would be too cumbersome and time consuming) and see if this
can improve the effectiveness of the deflection. Since in the design of the mission there
are two interplanetary transfers (or one in the case of direct hit scenario), we can have 9
different designs for the same mission (one option for the zero-revolution and two for the
single-revolution Lambert’s arc, repeated for both the interplanetary transfers)
45
FIGURE 4.1 - AN EXAMPLE OF POSSIBLE LAMBERT’S TRANSFERS IN THE PLOT SEMI-MAJOR AXIS VS
TRANSFER TIME
Including this new technique in the design of the complete mission could affect the
effectiveness of the simulation. In order to get the proper solution, we should
exponentially increase the number of the individuals and the number of function
evaluations inside the optimising function. Anyway, this is not a good idea, because it
would increase also the simulation time exponentially.
The strategy here followed to overcome this problem is the space pruning. The term space
pruning refers to all techniques that allow reduction of search space focussing the
optimization in smaller areas where the optimal solutions are to be found. The output of
a typical pruning process is a set of hyperrectangles contained in the original search space
where, according to some criteria, good solutions are expected [49]. There are a large
number of criteria that can be adopted to define such regions and that are dependent on
the particular problem instance considered. The way that is implemented here is the
following: the optimisation is first ran with the same parameters as it has been done till
now. The solution obtained will be used as a starting point for a further optimization.
Basically, the first optimisation is used to decide how many revolutions it is better to
perform for each interplanetary transfer of each mission. After that, a second
optimisation is launched to find the final solution. This last optimisation has now the
number of revolutions for each transfer fixed and can therefore work without further
problems.
4.1.2. Simulation
The results of this new method are shown in this section.
Both for direct hit and Earth’s gravity assist, the first round of optimisation shows that the
best way to project the mission is to keep the zero-revolutions transfers. Therefore, for
these two options we cannot improve the achievable deflection.
46
We start and see improvements for the Mars and Venus gravity assist’s missions.
For Mars, we obtain that the best way to design the mission is the following: i n the first
interplanetary transfer (from Earth to Mars), the best solution is to perform a complete
revolution along the orbit with high energy (low eccentricity), while in the second transfer
(from Mars to the asteroid) it is better to not perform any revolution. The improvements
so obtained are visible in Figure 4.2 and Figure 4.3
FIGURE 4.2 - COMPARISON OF PARETO-FRONTS FOR MARS GRAVITY ASSIST'S MISSIONS. BLUE FOR
MULTI-REVOLUTION CASE AND TWO ROUNDS OF OPTIMISATIONS; RED FOR MULTI-REVOLUTION AND
ONE ROUND OF OPTIMISATION; BLACK FOR ZERO-REVOLUTION.
47
FIGURE 4.3 - COMPARISON OF DELTA-V GIVEN TO ASTEROID VS ACHIEVABLE DISTANCE IN MARS
GRAVITY ASSIST'S MISSION. BLUE FOR MULTI-REVOLUTION AND TWO ROUNDS OF OPTIMISATION; RED
FOR MULTI-REVOLUTION AND ONE ROUND OF OPTIMISATION; BLACK FOR ZERO-REVOLUTION.
It is evident how this solution allows reaching much higher distances than the standard
scenario. Moreover, in Fig.4.2 we can notice that the space pruning procedure (i.e. a
second round of optimisation) is necessary to reach the best solution (i.e. better
achievable deflections).
The procedure is repeated for Venus. In this case, in the first interplanetary transfer the
best way is to perform a complete revolution along the small energy orbit (high
eccentricity), while for the second, once again, the zero-revolution. The new results are
shown in Fig.4.3 and Fig.4.4. For this case a clarification needs to be made: the parameters
of optimisation, as kept till now, are no longer sufficient to converge to the optimal
solution. Therefore, a new set of parameters is chosen and shown in Tab.4.1.
Case Function evaluations Number of individuals
Zero-revolution cases 500,000 200
Multi-revolution (Mars) 500,000 200
Multi-revolution
(Venus)
1,000,000 400
TABLE 4.1 - OPTIMISATION PARAMETERS FOR MULTI-REVOLUTION CASES.
The results obtained show a slight improvement in the achievable deflection, highlighting
that, anyway, the multiple-revolutions in Lambert’s arcs is a strategy that must be taken
in account.
48
FIGURE 4.4 - COMPARISON OF PARETO-FRONTS FOR VENUS GRAVITY ASSIST'S MISSION. BLUE FOR
MULTI-REVOLUTION, TWO ROUNDS OF OPTIMISATION WITH 400 INDIVIDUALS; RED FOR MULTI-REVOLUTION, TWO ROUNDS OF OPTIMISATION WITH 200 INDIVIDUALS; BLACK FOR ZERO-REVOLUTION.
FIGURE 4.5 - COMPARISON OF DELTA-V GIVEN TO ASTEROID VS ACHIEVABLE DISTANCE FOR VENUS
GRAVITY ASSIST'S MISSION. BLUE FOR MULTI-REVOLUTION, TWO ROUNDS OF OPTIMISATION WITH 400
INDIVIDUALS; RED FOR MULTI-REVOLUTION, TWO ROUNDS OF OPTIMISATION WITH 200 INDIVIDUALS; BLACK FOR ZERO-REVOLUTION.
49
4.2. Powered gravity assist
Up to this point, the mission has exploited at maximum one gravity assist of one of the
three rocky planets selected (Earth, Mars and Venus). This gravity assist is designed as a
not powered swing-by, meaning that spacecraft is supposed to follow the hyperbolic orbit
around the planet with no interference. One way to improve the effectiveness of the
deflection is to add a manoeuvre to the geometry of the mission, with an ignition of the
engines of the spacecraft at the pericentre of the fly-by, as presented in Section 2.3.2.2.
This also affects the design variable vector, by augmenting its dimension of 1, making it
to become:
𝒙 = 𝛼0 𝛼1 𝑇𝑜𝐹1 𝛾2 𝑟𝑝2 𝛥𝑣𝑃𝑂𝑊 𝛼2 𝑇𝑜𝐹2 ‖𝛥𝑣0‖ 𝛼𝛥𝑣0 𝛿𝛥𝑣0
𝑚𝑆𝐶0
The dimension of the design variable becomes now 12.
The bounds for this new variable are set as in Tab.4.2 taking the same values as the
bounds for ‖𝛥𝑣0‖:
Upper bound 0 𝑘𝑚/𝑠
Lower bound 3 𝑘𝑚/𝑠
TABLE 4.2 - BOUNDS FOR ‖𝜟𝒗𝟎‖
The presence of a new variable can be seen as a problem from the computational point
of view, because, as already seen, a wider domain could require a larger computational
time. For this reason, the solutions obtained so far are analysed, in order to see if there
are regions of the domains in which a solution is never present.
It is found that, for all the cases taken in account so far, both 𝑇𝑜𝐹1 and 𝑇𝑜𝐹2 are never
bigger than 2 · 𝑃𝑚𝑎𝑥 . The assumption made in Tab.3.3 and Tab.3.5 can therefore be
changed, shrinking the window for the times of flight. All the other variables’ bounds are
kept unvaried.
With this new domain, it is possible to exploit the parameters in Tab.4.3 to reach a proper
solution.
Case Function evaluations Number of individuals
Powered gravity-assist cases
500,000 200
TABLE 4.3 - OPTIMISATION PARAMETERS FOR POWERED GRAVITY ASSIST'S MISSION CASES
The results are shown in the following figures. In particular, Fig.4.6 and Fig.4.7 refer to
Earth, Fig.4.8 and Fig.4.9 refer to Mars, Fig.4.10 and Fig.4.11 refer to Venus.
50
FIGURE 4.6 - PARETO-FRONT FOR EARTH POWERED GRAVITY ASSIST'S MISSION.
FIGURE 4.7 - DELTA-V GIVEN TO ASTEROID VS ACHIEVABLE DISTANCE FOR EARTH POWERED GRAVITY
ASSIST'S MISSION.
51
FIGURE 4.8 – PARETO FRONT FOR MARS POWERED GRAVITY ASSIST'S MISSION.
FIGURE 4.9 - DELTA-V GIVEN TO ASTEROID VS ACHIEVABLE DISTANCE FOR MARS POWERED GRAVITY
ASSIST'S MISSION.
52
FIGURE 4.10 – PARETO FRONT FOR VENUS POWERED GRAVITY ASSIST'S MISSION.
FIGURE 4.11 - DELTA-V GIVEN TO ASTEROID VS ACHIEVABLE DISTANCE FOR VENUS POWERED GRAVITY
ASSIST'S MISSION.
53
Also in this case, we can say that the new methodology shows slight improvements with
respect to the standard scenarios.
We can state that both the multi-revolution methodology and the powered gravity assist
methodology have to be taken in account because they improve the quality of the results.
54
5. Analysis of the test case solutions
After this first round of simulations, optimizations and solutions obtained, it is possible to
highlight some common characteristics of the various missions.
It is first analysed the deflection action and, in particular, the best deflection’s direction,
and the way it is affected by a change in the time window. After that it will be showed and
explained the relation between the achievable deflection and the initial mass of the
spacecraft (we have already highlighted the linearity relation between these two
quantities in Chapter 3 and Chapter 4). Finally, also a relation between the achievable
deflection and the mass of the asteroid will be showed.
5.1. Variation of time window
We have already underlined that the best way to deflect the asteroid is by hitting it along
the tangential direction (see Sections 3.1 and 3.2). This fact is also more evident in the
polar graphic in Fig.5.1, (relative to Earth’s gravity assist) where the optimal delta-v given
to the asteroid is shown on a polar plot.
FIGURE 5.1 - POLAR PLOT OF DELTA-V GIVEN TO THE ASTEROID. MODULUS AND DIRECTION ARE
SHOWN.
55
Every solution hits the asteroid with an impact angle lower than 30 𝑑𝑒𝑔. This fact was
visible also in Fig.3.4 and Fig.3.10.
Things change when the time window assigned to the problem change. In particular, we
see relevant changes when the time window becomes really small (one or two years). In
these cases, the trend is to hit the asteroid perpendicularly. This can be seen in Figure 5.2
and Figure 5.3. In all the cases from 10 to 3 years of warning time the deflection’s direction
is almost tangential. It becomes almost perpendicular for 1 and 2 years of warning time.
FIGURE 5.2 - POLAR PLOT OF OPTIMAL DELTA-V GIVEN TO ASTEROID FOR EARTH GRAVITY ASSIST’S
MISSION. TIME WINDOW VARYING FROM 10 YEARS TO 1 YEAR.
56
FIGURE 5.3 - POLAR PLOT OF OPTIMAL DELTA-V GIVEN TO ASTEROID FOR EARTH GRAVITY ASSIST'S
MISSION. DIFFERENCE BETWEEN HIGH VALUE OR LOW VALUES FOR TIME WINDOW.
This fact is not surprising. In fact, we have that if 𝛥𝑡 < 𝑇𝑁𝐸𝑂, the best way to deflect an
asteroid is along the normal direction, while if 𝛥𝑡 > 𝑇𝑁𝐸𝑂 , the best deflection’s direction
is the tangential (Sections 3.1 and 3.2). 𝛥𝑡 is the time-to-MOID, previously defined in
Equation 2.3.
Basically, when the time-to-MOID is low, it is better to shift asteroid’s orbit as soon as
possible, although not in most efficient way (normal direction), while when the time-to-
MOID is big enough, there is no need to immediately shift asteroid’s orbit and therefore
the most efficient deflection strategy can be followed (tangential direction) [32].
Figure 5.2 confirms this statement (as well as Figure 5.3 that highlights only the cases with
time window of 10 years, 2 years and 1 year).
Other considerations can be carried, looking at the effects of the variation of the time
window, as shown in Figure 5.4, Figure 5.5 and Figure 5.6.
In Figure 5.4 the Pareto fronts of the same mission with varying time window are
presented. The increasing steepness of the lines show that, decreasing the time window
reduces the maximum deflection available. As expected, a larger time window means a
larger deflection. For example, given an initial mass of the spacecraft of 5000 kg, a time
window of 10 years allows a deflection of about 4000 km, while a time window of 5 years
allows only to deflect the asteroid of less than 1000 km.
57
FIGURE 5.4 - COMPARISON OF PARETO-FRONTS FOR EARTH GRAVITY ASSIST'S MISSION WITH VARYING
TIME WINDOW.
Figure 5.5 confirms this last statement, highlighting the variation of velocity imparted to
the asteroid.
Finally, in Figure 5.6, we can see that it is always better to choose a departure time at
the beginning of the time window, as already found in Section 3.
58
FIGURE 5.5 - COMPARISON BETWEEN DELTA-V GIVEN TO ASTEROID VS ACHIEVABLE DISTANCE FOR
VARYING TIME WINDOW.
FIGURE 5.6 - COMPARISON BETWEEN DEPARTURE TIME (YEARS BEFORE MOID) VS ACHIEVABLE
DISTANCE FOR VARYING TIME WINDOW.
59
5.2. Spacecraft’s initial mass and asteroid’s mass
5.2.1. Relation between spacecraft’s initial mass and achievable deflection
The linearity between the initial mass of the spacecraft and the achievable deflection is
evident from all the Pareto fronts shown so far independently from the strategy adopted,
the number of revolutions in the Lambert’s arcs or the time window selected. This relation
is confirmed also from a theoretical point of view. Looking at the Equations 2.1, 2.2, 2.4
and 2.5 it is possible to show this linearity, by making some assumptions. The first is that
𝑚𝑆𝐶 ≪ 𝑚𝑁𝐸𝑂 . This assumption (more than acceptable), allows to rewrite the
denominator in Equation 2.4: 𝑚𝑆𝐶 + 𝑚𝑁𝐸𝑂 ≈ 𝑚𝑁𝐸𝑂. Moreover, since in the cases we
have faced and we will face the time window is 10 years, both 𝛿𝒗 and 𝜟𝒗 can be identified
with their tangential component (that is the dominating component), making the final
equation a scalar equation:
𝑟𝑝 = 𝐾 · 𝑚𝑆𝐶0
𝑚𝑁𝐸𝑂 · 𝛿𝑣𝑡
EQUATION 5.1
where 𝐾 is a term that depends on the type of mission adopted. The linearity is confirmed
both for initial mass of the spacecraft and for the variation of velocity given to the asteroid,
and this fact is very important. In fact, we can now reduce the design variables vector, by
eliminating 𝑚𝑆𝐶0. From now, we will set 𝑚𝑆𝐶0 = 1000 𝑘𝑔, knowing that it is possible to
recover solutions with other initial masses thanks to the linearity relation. Moreover, it is
possible to reduce the optimizer function to a single-objective function, including only the
deflection, reducing therefore the computational time.
𝐽 = −(𝑟𝑝 − 𝑟𝑝0)
5.2.2. Relation between asteroid’s mass and achievable deflection
Equation 5.1 also implies an inverse proportionality between the achievable deflection
and the mass of the asteroid. With the same reasoning as before, we can run every
simulation from now on assuming the mass of the asteroid to be 𝑚𝑁𝐸𝑂 = 1.36 · 109 𝑘𝑔
(i.e. the mass of a spherical asteroid with a diameter of 200 𝑚 and a homogeneous
density as shown in Tab.3.2), knowing that it is possible to recover solutions for asteroids
with different masses [35].
The inverse proportionality is evident in Figures 5.7 and 5.8.
60
FIGURE 5.7 – ACHIEVABLE DEFLECTION VS ASTEROID MASS FOR EARTH GRAVITY ASSIST’S MISSION
FIGURE 5.8 - DELTA-V GIVEN TO ASTEROID VS MASS OF ASTEROID FOR EARTH GRAVITY ASSIST'S
MISSION
61
6. Deflection efficiency against the whole NEOs population
6.1. Generation of a population of asteroids
In order to create a global model able to analyse every possible asteroid and the best way
to deflect it before an impact happens (where the impacts are yielded to happen at the
same predefined epoch), it is in principle necessary to create a synthetic population of
asteroid, comprehensive of all the possible orbital parameters.
Before doing this, a real set of asteroids is produced, thanks to the software released by
ESA, NEOPOP [50]. This software produces a set of orbital parameters of asteroids,
differentiating them according to different characteristics (MOID, category of NEO, size…),
according to the date in which the software is supposed to make his research. For our
case, the MOID is predefined to happen at the same 𝑡𝑀𝑂𝐼𝐷 as used since now. At this time,
both the Earth and the asteroids are defined to be at the same point. For this reason, the
software computes his research at this 𝑡𝑀𝑂𝐼𝐷.
A filter on the dimensions of the asteroid is activated, with the diameter constrained to
take values between 50m and 200m, because in this range there are the asteroids that
have a probability to hit our planet of about 1 every 100 years (a relevant probability
therefore), and their effect are dangerous for human life [35]. The results obtained are
shown in Figures from 6.1 to 6.5. There have been found 231,336 objects.
FIGURE 6.1 - TEST RUN OF THE POPULATION GENERATOR: SEMI-MAJOR AXIS VS ECCENTRICITY.
62
FIGURE 6.2 - TEST RUN OF THE POPULATION GENERATOR: SEMI-MAJOR AXIS VS INCLINATION.
FIGURE 6.3 – TEST RUN OF THE POPULATION GENERATOR: ECCENTRICITY VS INCLINATION.
63
FIGURE 6.4 - TEST RUN OF THE POPULATION GENERATOR: PERIAPSIS VS APOAPSIS.
FIGURE 6.5 – MOID VS NUMBER OF OBJECTS. DIFFERENT COLOURS FOR DIFFERENT CATEGORIES OF
ASTEROIDS.
Figures from 6.1 to 6.3 show the distribution of this population in the orbital parameters’
space. A high concentration of asteroid is found around values of semi-major axis of 2.5
AU and eccentricity 0.5
In Figure 6.4 the same population is depicted on a periapsis VS apoapsis graphic.
64
Finally, in Figure 6.5, the concentration of NEOs related to the MOID, showing a peak at
around 0.3 AU (therefore, at a safe distance).
But of course, not all of these asteroids are interesting for our discussion. In fact, those
that are of greater interest for us are the PHA. This means that the asteroids that have to
be analysed are those with a pericentre radius smaller than 1AU, and an apocentre radius
higher than 1AU. This condition is necessary, but not sufficient to have an intersection
with Earth orbit.
With this new filtering action, we have the final number of 112,471 asteroids, distributed
as shown in the Figures from 6.6 to 6.10.
FIGURE 6.6 - TEST RUN OF THE POPULATION GENERATOR: SEMI-MAJOR AXIS VS ECCENTRICITY. FILTERING ACTION ON PERICENTRE AND APOCENTRE.
65
FIGURE 6.7 - TEST RUN OF THE POPULATION GENERATOR: SEMI-MAJOR AXIS VS INCLINATION. FILTERING ACTION ON PERICENTRE AND APOCENTRE.
FIGURE 6.8 - TEST RUN OF THE POPULATION GENERATOR: ECCENTRICITY VS INCLINATION. FILTERING
ACTION ON PERICENTRE AND APOCENTRE
66
FIGURE 6.9 - TEST RUN OF THE POPULATION GENERATOR: PERIAPSIS VS APOAPSIS. FILTERING ACTION
ON PERICENTRE AND APOCENTRE.
FIGURE 6.10 - TEST RUN OF THE POPULATION GENERATOR: MOID VS NUMBER OF OBJECTS. FILTERING
ACTION ON PERICENTRE AND APOCENTRE.
67
In Figures from 6.6 to 6.8 we can see the effect of the filtering action on pericentre and
apocentre: the figures are now sharp due to this restriction (especially Figure 6.6). Also,
the region where there was a peak of asteroids in the un-filtered case (Figures from 6.1
to 6.5) is now excluded, meaning that it couldn’t have a close encounter with Earth.
In Figure 6.9 the filtering action is crystal clear. In fact, regions where the pericentre is
bigger than 1 AU or where the apocentre is smaller than 1 AU are empty.
Finally, also in Figure 6.10, we can see that less asteroids are left, and the peak is shifted
towards value of MOID of about 0.1 AU.
The advantage of using this software is that it gives you a lot of characteristics about the
population, included the probability of every single asteroid to collide with Earth.
6.2. Creation of a synthetic population of asteroids
6.2.1. Theory
After that the real population of asteroid is generated, we need to create a synthetic cloud
of asteroids, defined by a grid of homogeneously distributed points in a 3-dimensional
space, formed by the orbital parameters 𝑎 , 𝑒 , 𝑖 .
Some assumptions are made, in order to simplify the problem, following [35].
First of all, as already stated in Section 6.1, all the points with pericentre higher than 1 AU
or apocentre smaller than 1 AU are discarded. This constraint is necessary but not
sufficient to say that each one of those points yields an impact with Earth.
A further assumption is made to simplify the problem. Earth is assumed to be circular with
radius equal to 1 AU. With this assumption it is straight-forward to compute both Ω𝑖𝑚𝑝𝑎𝑐𝑡
and ω𝑖𝑚𝑝𝑎𝑐𝑡.
As shown in Figure 6.11 from [35], Ω𝑖𝑚𝑝𝑎𝑐𝑡 is defined by the position of the Earth at the
fixed epoch at which the virtual impact is set, while ω𝑖𝑚𝑝𝑎𝑐𝑡 has two possible
configurations (±ω𝑖𝑚𝑝𝑎𝑐𝑡).
In fact, once fixed the virtual impact in a position along Earth’s orbit (the position is
irrelevant since Earth’s orbit is assumed to be circular), these two quantities can be
recovered.
Due to the circularity of Earth’s orbit, Ω𝑖𝑚𝑝𝑎𝑐𝑡 can be chosen arbitrarily, and it is therefore
set to zero.
ω𝑖𝑚𝑝𝑎𝑐𝑡 can be found applying Equation 2.7, substituting at cos 𝜃 ,
cos (𝜃𝑖𝑚𝑝𝑎𝑐𝑡 + ω𝑖𝑚𝑝𝑎𝑐𝑡)
68
FIGURE 6.11 - VISUAL REPRESENTATION OF EARTH'S ORBITAL PLANE AND ASTEROID'S ORBITAL PLANE.
As can be seen in all of the images provided by NEOPOP software, there are region of
space much more densely populated with NEOs than others. There are, for example, many
more low-inclination objects than high-inclination. On the other hand, not only the NEO
population density is important when considering the impact frequency; the impact
geometry also plays an important role.
The relative frequency of each virtual impactor is given by means of two multiplying
factors; first, the NEO orbital distribution that defines the actual asteroid probability
density, and second, the collision probability of a given set of orbital parameters, which
assesses the likelihood of an impact for a given object.
The first term is the NEO orbital distribution and it is found thanks to the NEOPOP
software.
The probability of collision is found exploiting some reasonings. First, as already stated,
the pericentre of the impactor must be smaller than 1 AU and the apocentre larger than
1 AU [35]. This is not sufficient. In fact, only a limited set of ω yields an impact, only a
single Ω yields an impact, and also only a single 𝑀 yields an impact. In this section, we find
this probability of collision.
We first need to compute the maximum MOID that allows an Earth collision. For the latter,
the Earth’s gravity needs to be accounted, because an asteroid close to Earth will
essentially follow a hyperbolic trajectory with the Earth at its focus. A hyperbolic factor ε:
ε = 𝑟𝑎
𝑟𝑝 = √1 +
2 𝜇𝐸
𝑟𝑝𝑣2∞
accounts then for the curvature that the object’s trajectory would experience during the Earth approach. If we assume that the maximum distance at which collision happens is equal to the radius of Earth, we need to have a distance between Earth’s orbit and the asteroid smaller than 𝑀𝑂𝐼𝐷𝐸 to have an impact, where
𝑀𝑂𝐼𝐷𝐸 = 𝑟𝐸√1 + 2 𝜇𝐸
𝑟𝐸𝑣2∞
69
Another assumption needs to be made: both the right ascension of the ascending node
and the argument of periapsis are uniformly distributed in NEO space. We can assume
therefore that every value of these two quantities is equally possible. The same reasoning
is used for 𝑀.
We have that for ω = ω𝑖𝑚𝑝𝑎𝑐𝑡 , 𝑀𝑂𝐼𝐷 = 0 . All the values of ω such that 𝑀𝑂𝐼𝐷 <
𝑀𝑂𝐼𝐷𝐸 are also valid to have an impact.
It is possible to find [35] that the probability of having an argument of perigee ω such that
the impact can occur is
𝑔1(𝑎, 𝑒, 𝑖) = 2 · 𝛥𝜔
𝜋
where
𝛥𝜔 = 𝑀𝑂𝐼𝐷𝐸√1
(sin 𝑖)2 + (tan 𝛾)2
and
tan 𝛾 =𝑝
√𝑒2 − ( 𝑝 − 1 )2
being 𝛾 the flight path angle.
This probability identifies the possibility of an orbit to have a 𝑀𝑂𝐼𝐷 < 𝑀𝑂𝐼𝐷𝐸. But this is
not sufficient to yield an impact. In fact, also the fact that both the two bodies have to
pass at that point in the same instant needs to be verified. This is translated in the
following equation [35]:
𝑔2(𝑎, 𝑒, 𝑖) = 𝑙𝑚𝑎𝑥
4𝜋
and
𝑙𝑚𝑎𝑥 = 𝑀𝑂𝐼𝐷𝐸𝑣∞
√(𝑣∞2 − 𝜇𝑠𝑢𝑛(√𝑝cos𝑖 − 1)
2)
The relative frequency of the impactors can be found by multiplying 𝑔1, 𝑔2 and NEO
density distribution. For every point of the asteroid’s cloud it is possible to in tegrate this
value along the 𝛥𝑎 𝑥 𝛥𝑒 𝑥 𝛥𝑖 cube.
70
6.2.2. Simulation
The simulation to deflect a population of asteroids is here presented.
The assumption made in Section 6.2.1 are implemented to create a synthetic population
of asteroids. Moreover, according to the Figures from 6.6 to 6.10, it is possible to create
some bounds on the orbital parameters: 𝑎 is bound between 0.05 AU and 3 AU; 𝑒 is
bound between 0 and 1; the inclination 𝑖 is bound between 0 deg and 90 deg.
Moreover, the 𝑡𝑀𝑂𝐼𝐷 selected is the same as that used in Section 2, 3, 4 and 5, because it
could be selected arbitrarily. The important thing is that the ephemerides of the Earth and
of the population of asteroids are such that at this 𝑡𝑀𝑂𝐼𝐷 they both are at MOID. For Mars
and Venus the real ephemerides are used instead.
The simulation is repeated 4 times: 1 for the direct hit scenario, 3 for the gravity assist’s
missions. The multi-revolution method and the powered gravity assist are taken in
consideration. The assumption made for the simulations are shown in Tab.6.1.
𝑚𝑁𝐸𝑂 1.36 ∗ 109 𝑘𝑔
𝑚𝑆𝐶0 1000 𝑘𝑔
𝑤𝑎𝑟𝑛𝑖𝑛𝑔𝑇𝑖𝑚𝑒 10 𝑦𝑒𝑎𝑟𝑠
TABLE 6.1 - SET OF PARAMETERS USED FOR THE SIMULATION
we have to redefine the design variable as:
𝒙 = 𝛼0 𝛼1 𝑇𝑜𝐹1 𝛾2 𝑟𝑝2 𝛥𝑣𝑃𝑂𝑊 𝛼2 𝑇𝑜𝐹2 ‖𝛥𝑣0‖ 𝛼𝛥𝑣0 𝛿𝛥𝑣0
𝑚𝑆𝐶0
and so, also the bounds are redefined in Tab.6.2 as:
Variable 𝛼0 𝛼1 𝑇𝑜𝐹1 𝛾2 𝑟𝑝2 𝛥𝑣𝑀𝐴𝑁
Lower bound
0 0 0.01 𝑃𝑚𝑎𝑥 −𝜋 𝑟𝑎𝑑 1.1 0 𝑘𝑚/𝑠
Upper
bound
0.99 1 2 𝑃𝑚𝑎𝑥 +𝜋 𝑟𝑎𝑑 66.0 3 𝑘𝑚/𝑠
Variable 𝛼2 𝑇𝑜𝐹2 ‖𝛥𝑣0 ‖ 𝛼𝛥𝑣0 𝛿𝛥𝑣0
Lower bound
0 0.01 𝑃𝑚𝑎𝑥 0 𝑘𝑚/𝑠 −𝜋 𝑟𝑎𝑑 −𝜋/2 𝑟𝑎𝑑
Upper bound
1 2 𝑃𝑚𝑎𝑥 3 𝛥𝑣𝑙𝑎𝑢𝑛𝑐ℎ +𝜋 𝑟𝑎𝑑 +𝜋/2 𝑟𝑎𝑑
TABLE 6.2 - BOUNDS FOR THE DESIGN VARIABLES. ASTEROID POPULATION CASE
71
Optimisation parameters are shown in Tab.6.3
Case Function evaluations Number of individuals
Multiple asteroids 500,000 200 TABLE 6.3 - OPTIMISATION PARAMETERS. ASTEROID POPULATION CASE
The first result evident from simulation is that the best deflection is achievable through
Earth Gravity assist in most of the cases. It is in some other cases better to exploit direct
hit or Venus gravity assist for low eccentricity orbits (Figure 6.12).
FIGURE 6.12 - SIMULATION OF DEFLECTION OF THE SYNTHETIC POPULATION. BLUE FOR BEST
DEFLECTION THROUGH EARTH GRAVITY ASSIST; BLACK FOR BEST DEFLECTION THROUGH DIRECT HIT; GREEN FOR BEST DEFLECTION THROUGH VENUS GRAVITY ASSIST.
Some characteristics of the single cases are now analysed and highlighted.
72
6.2.2.1. Direct hit
Figures from 6.13 to 6.17 show the main characteristics of the direct hit scenario.
FIGURE 6.13 - DIRECT HIT ON ASTEROIDS POPULATION - DEFLECTION ACHIEVED.
FIGURE 6.14 - DIRECT HIT ON ASTEROIDS POPULATION - DELTA-V GIVEN TO ASTEROID.
73
FIGURE 6.15 - DIRECT HIT ON ASTEROIDS POPULATION - DISTANCE FROM PERIGEE AT THE DEFLECTION.
FIGURE 6.16 - DIRECT HIT ON ASTEROIDS POPULATION - IN PLANE ANGLE OF DEFLECTION.
74
FIGURE 6.17 - DIRECT HIT ON ASTEROIDS POPULATION - OUT OF PLANE ANGLE OF DEFLECTION.
From Figure 6.13 and 6.17 it is evident how the deflection achievable and the modulus of
the variation of velocity given to the asteroids have a similar behaviour (as expected), and
they both increase with the semi-major axis. Orbits with low semi-major axis are therefore
the most difficult to deflect. This result was found also in [35].
Figure 6.15 show that in most cases the deflection happens near to the apogee, while
some of the asteroids with semi-major axis close to but bigger than 1 AU have the
deflection happening at their perigee.
In Figures 6.16 and 6.17 we can see the direction of deflection. For what concerns the in-
plane angle, it is preferred to hit the asteroids almost tangentially (as already found also
in Section 3), and it is better to break the asteroid instead of accelerating it (angles near
to 180 deg). For what concern the out-of-plane angle, it increases with the inclination,
also confirming what expected (and stated in Section 3).
75
6.2.2.2. Earth flyby
The analysis is repeated for Earth flyby case. The results are plotted in Figures from 6.18
FIGURE 6.18 - EARTH GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION – DEFLECTION.
FIGURE 6.19 - EARTH GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - DELTA-V GIVEN TO
ASTEROID.
76
FIGURE 6.20 - EARTH GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - DISTANCE FROM PERIGEE
AT DEFLECTION.
FIGURE 6.21 - EARTH GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - IN-PLANE ANGLE OF
DEFLECTION.
77
FIGURE 6.22 - EARTH GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - OUT-OF-PLANE ANGLE OF
DEFLECTION.
Most of the comments made in Section 6.2.2.1 are still valid here. In fact, in Figures 6.18
and 6.19 it is possible to see how the achievable deflection and the modulus of the delta-
v given to the asteroid increases both with semi-major axis and inclination.
In Figure 6.20 we notice that for highly eccentric orbits, the deflection tends to happen at
the apogee, while decreasing the eccentricity the deflection tends to move towards the
perigee. Once again for orbits with semimajor axis close to, but bigger than 1 AU, the
deflection happens almost at the perigee.
For Figures 6.21 and 6.22 the reasonings are the same as Section 6.2.2.1. The impactor
tends to hit the asteroid always almost tangentially by breaking it more than accelerating
it. The out-of-plane angle increases with the inclination.
It must be noticed that the values of variation of velocity given to the asteroid are much
higher here, as well as the deflection reached.
78
6.2.2.3. Mars and Venus gravity assists
In this last section the gravity assists of Mars and Venus are presented. Figures from
6.23 to 6.27 refer to Mars, those from 6.28 to 6.32 refer to Venus.
FIGURE 6.23 - MARS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION – DEFLECTION.
FIGURE 6.24 - MARS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION – DELTA-V GIVEN TO
ASTEROID.
79
FIGURE 6.25 - MARS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION – DISTANCE TO PERIGEE.
FIGURE 6.26 - MARS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - IN-PLANE ANGLE.
80
FIGURE 6.27 - MARS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - OUT-OF-PLANE ANGLE.
FIGURE 6.28 - VENUS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION – DEFLECTION.
81
FIGURE 6.29 - VENUS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - DELTA-V GIVEN TO
ASTEROID.
FIGURE 6.30 - VENUS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - DISTANCE FROM PERIGEE
AT DEFLECTION.
82
FIGURE 6.31 - VENUS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - IN-PLANE ANGLE AT
DEFLECTION.
FIGURE 6.32 - VENUS GRAVITY ASSIST MISSION ON ASTEROIDS POPULATION - OUT-OF-PLANE ANGLE
AT DEFLECTION.
83
Once again we see that the best deflection are reachable for orbits with high semimajor
axis, as well as best variation of velocities given to the asteroids (Figures 6.23, 6.24, 6.28
and 6.29), although these values are in general worse than the Earth’s gravity assist
scenario.
Out-of-plane angles increase their values with inclination (Figures 6.27 and 6.32), as
expected.
It is curious the behaviour shown in Figure 6.31, where the in-plane-angle for Venus
gravity assist mission is bound between values of 180 deg and 110 deg, differently from
all the others cases, meaning that it always breaks the asteroid.
Figure 6.25, can be divided in two zones: the zone of shallow crossers (those asteroids in
the lower part of the V-shaped asteroid’s cloud), that have a deflection happening near
to the pericentre, and deep crossers (asteroids in the inner part of the V-shaped asteroid’s
cloud) that have their deflection happening close to the apogee.
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7. Conclusions and future works
The effect of the inclusion of a gravity assist in a mission aimed to deflect an asteroid
through kinetic impactor is here presented. In fact, gravity assists from Earth, Venus and
Mars have been analysed first on a test case (2010RF12), and then on a population of
asteroids comprehensive of all the spectrum of possible orbital parameters.
It is possible to say that the best solution, in most of the analysed cases, is represented by
the inclusion of a gravity assist of Earth inside the trajectory performed by the kinetic
impactor.
This fact can be seen as a possibility to achieve best deflections of asteroids with the same
initial mass or the possibility to achieve the same required deflection with a smaller mass
(meaning a cheaper mission) than all the other cases.
Anyway, the fact that the gravity assists of Venus and Mars had not the same performance
as the gravity assist of Earth must not be seen as a definitive result. In fact, choosing a
different time window for the deflection can boost the performances of these two cases,
just because also the phasing is changed. This topic must be extended and studied in
future works.
Other ways to improve the effectiveness of all the above-mentioned cases can be found
by tuning some of the missions’ parameters. For example, increasing the time window to
values of 20 years is expected to improve the efficacy of the methods.
Also, we have seen that it is always possible to better the concept of the mission. In fact,
the inclusion of a single-revolution Lambert arc or a powered gravity assist manoeuvre
had the result to improve also the achievable deflection of every single approach (direct
hit or gravity assist approach). Following this philosophy, it is possible to improve more
and more the efficiency of the method, for example including multiple revolution
(intended as more than one) in the Lambert’s arc or including more than one planet’s
gravity assist to the mission. Of course, it must be taken in account that increasing the
complexity of the mission results also in much more complex computations and very high
computational times.
The algorithm developed in this work has been used only to the aim of deflecting an
asteroid to a maximum distance. Anyway, by simply changing the optimiser function 𝐽 we
can use the same algorithm to analyse a different concept of mission. For example, by
minimising the difference of the velocity between the asteroid and the kinetic impactor
we can create and analyse a mission of rendez-vous.
85
86
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