Feasibility study of the
implementation of a space
sunshade near the first Lagrangian point
MARÍA GARCÍA DE HERREROS MICIANO
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
implementation of a space sunshade near the first
Lagrangian point
MARÍA GARCÍA DE HERREROS MICIANO
Master in Aerospace Engineering Date: July 15, 2020
Supervisor: Christer Fuglesang Examiner: Christer Fuglesang School of Engineering Sciences
Swedish title: Möjlighetsstudie av solparasoll i rymden nära Lagrangepunt L1
Abstract
The lack of strong measures to avoid the possible fatal consequences of global warming is pushing researchers to look for other alternatives such as geoengi- neering. Within geoengineering, this study focuses on the space based solar radiation management methods. More precisely, the project evaluates the fea- sibility of implementing a space sun shade near the first Lagrangian point in the Sun-Earth system within a thirty year period time from now. The study is structured in three main blocks: spacecraft configuration, trajectory definition and launch. An analysis looking at the minimum cost system was carried out, starting with the definition of the mass and size of spacecraft. Furthermore, an optimization of the trajectory was developed in order to minimize the travel time to the vicinity of the Lagrangian point. The shades will be formed by swarms of 10 000 m2 solar sails that will cover an area of 6.3 × 1012m2 with a total mass of around 5.7 × 1010kg. The sails will be injected into a LEO and will start a trajectory to the vicinity of the first Lagrangian point that will take around 2.3 years. The total cost of the project is approximated to be 10 trillion dollars. The mission appears to be feasible from a technological point of view, with some development needed in the attitude control subsystem. The main challenge will be the launch of all the spacecraft. A space mission of this dimensions has never been attempted before so it will require a big advance from the launch vehicle industry.
Sammanfattning
Bristen på åtgärder för att undvika de konsekvenser som den globala uppvär- mingen leder till, har drivit forskare att leta efter alternativa lösningar, varav geoengineering är en av dem. Denna studie fokuserar på rymdbaserade strål- hanteringsmetoder, mer specifikt på hur huruvida implementationer av solpa- rasoller nära Lagrangepunkten L1 i sol-jord-systemet är möjlig eller ej. Studi- en är strukturerad i tre huvudsakliga block: rymdskeppskonfiguration, banade- finition och uppskjutning. Med målet att minimera kostnaderna, definierades rymdskeppets utforming, massa och storlek. Vidare så, optimerades vägen till närheten av L1 med avseende på att minimera tiden. Solparasollerna kom- mer vara placerade i svärmar med en area på 10 000 m2 vardera, totalt kom- mer solparasollerna att täcka en yta av 6.3 × 1012m2 med en total massa på 5.7 × 1010kg. Solparasollerna kommer skjutas upp till LEO och därefter star- ta sin resa till närheten av L1, vilket kommer ta cirka 2.3 år. Totala kostanden för projektet uppskattas till 10 billioner dollar. Efter genomförd studie visades projektet vara genomförbart sett från en teknisk synvinkel, men vidare studi- er behövs göras för att utveckla och fastställa styrsystemet. Huvudutmaningen kommer att vara uppskjutningen av rymdskeppen, då det kräver stora framsteg och utveckling inom rymdindustrin.
1 Introduction 1
1.1 Current Climate Situation and Policies . . . 1
1.2 Geoengineering the Climate . . . 3
1.3 Literature Review on Space Sun Shades . . . 4
1.4 Present Work . . . 5
2 Methodology 7 2.1 Initial Assumptions . . . 7
2.2 Launch . . . 8
2.2.1 Launcher Selection and Assumptions . . . 8
2.2.2 Target orbit . . . 9
2.3 Trajectory . . . 11
2.3.1 Reference Frames . . . 11
2.3.2 Solar Sail Dynamics . . . 12
2.3.3 New Equilibrium Point . . . 15
2.3.4 Escape Trajectory Optimization . . . 15
2.3.5 Trajectory to Sub-L1 Optimization . . . 16
2.4 Spacecraft Configuration . . . 18
2.4.1 Total Mass and Size Study . . . 19
2.4.2 Mass Budget . . . 20
2.4.3 Control . . . 21
3 Results 24 3.1 Spacecraft Configuration . . . 24
3.1.1 Total Mass and Size . . . 24
3.1.2 Mass Budget . . . 28
3.1.3 Control . . . 29
3.2 Launch . . . 31
3.2.1 Launcher Selection . . . 31
V
3.2.2 Launch in Numbers . . . 32
3.2.3 Target Orbit Definition . . . 33
3.3 Final Trajectory . . . 33
3.3.1 Escape Trajectory . . . 34
3.3.2 Travel to Sub-L1 . . . 35
4 Cost Analysis 40 4.1 Launch Cost . . . 40
4.2 Spacecraft Cost . . . 41
5 Discussion 43 5.1 Spacecraft . . . 43
5.2 Launch . . . 46
5.2.1 Launch in Numbers . . . 46
5.2.2 Environmental Impact . . . 47
5.3 Trajectory Results . . . 49
5.4 Cost . . . 50
6 Conclusions 52
Introduction
The emissions of greenhouse gases (GHG) have been changing the planet for decades but it was not until fifteen years ago, with the Kyoto Protocol, that climate change started to get attention from governments. Since then, the im- portance of it has been growing. However, up to now actions against it have not been taken with the urgency that the problem requires in order to avoid possible fatal consequences [1] [2]. This is pushing researchers to look for alternatives away from the reduction of greenhouse gases emissions, which as time passes seems harder to achieve on time and requires the commitment of the whole world [3] [4].
1.1 Current Climate Situation and Policies
Since the pre-industrial age, the world’s climate has changed significantly be- cause of the emission of GHG, but there has been a great acceleration of these changes in the past fifty years. The main contribution to the emissions is fossil carbon dioxide (CO2), which mainly comes form energy and industrial use.
This explains the acceleration during the industrial era and specially in the last decades.
After 2010, GHG emissions have been growing at a rate of 1.5 per cent per year and the peak of these emissions does not seem like it is going to take place any time soon [2]. Every year that this peak is delayed translates into a larger rise of the global mean temperature by the end of the century. Different scenarios are considered by the United Nations–sponsored Intergovernmental Panel on Climate Change concluding that in order to stabilize the mean global temperature around 2◦C above pre-industrial levels by 2100, this peak should
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take place between 2020 and 2050 [5].
By 2017 temperature had already risen 1◦C [5] and as a consequence global warming effects are already observable on the planet, although these will be more intense in the following decades [6]. Some of the most important are the rise of sea level as a result of the melting ice, the increase of extreme weather events such as droughts, heavy rainfalls or heatwaves, the extinction of plant and animal species, the reduction of crop fields and the increase of wildfires [7]. All these phenomena translate in large costs for society (increase in mor- tality, consequences in human health) and economy (damage of infrastructure, agriculture, tourism and energy sectors).
The most recent international agreement regarding climate change was the Paris Agreement (2016), ratified by 187 countries as of 2019. Here, all signa- tories compromised to: “holding the increase in the global average tempera- ture to well below 2◦C above pre-industrial levels and pursuing efforts to limit the temperature increase to 1.5◦C" [8]. Therefore, the upper limit that should not be trespassed is nowadays set to 2◦C. It must be kept in mind that the 0.5◦C difference between both limits means a significant increase in the intensity of the global warming effects mentioned above [5].
With the current climate policies and the ones expected to be implemented in the following years, by the end of the century temperature is predicted to rise between 2.6◦C and 3.7◦C, depending on the compliance with these poli- cies [1]. Thus, it seems clear that stronger measures need to be taken to reduce GHG emissions in order to achieve the defined goals. To meet the 1.5◦C limit, global CO2 emissions would need to reach zero by 2050 and keep decreasing afterwards. This means that emissions would need to start dropping immedi- ately at a faster rate than ever and once zero emissions were reached, carbon removal techniques would have to be implemented [9]. The chances of achiev- ing these objectives seem remote, specially keeping in mind that, in order to do so, an international response needs to be coordinated on a global level. Oth- erwise, despite all the climate policies that are being implemented in a lot of countries, the decrease of emissions in these will not be enough to offset the increase in others [2].
1.2 Geoengineering the Climate
As it has been described before, historically the dominant approach to fight climate change has been the reduction of GHG emissions. While the achieve- ment of the necessary levels of emissions in time is each year further away from reaching the goal, other alternatives are being considered in addition to these reductions. One of the options is so-called geoengineering, which consists on deliberately modifying Earth’s environment in order to counteract climate change. As addressed in Geoengineering the climate. Science, governance and uncertainty [3], these measures are still highly controversial but they are alternatives that can provide help, mitigating both short-term and long-term global warming effects. Despite being an important alternative to be consid- ered, it must be acknowledged that geoengineering methods should not be seen as a solution but as part of a larger set of measures to fight climate change.
Geoengineering techniques can be divided in two different groups: Carbon Dioxide Removal (CDR) and Solar Radiation Management (SRM). The first group aims to reduce the level of carbon dioxide in the atmosphere, while SRM techniques reduce the net incoming solar radiation received at the surface of the planet by deflecting it before reaching Earth or increasing the reflectivity of the planet surface or its atmosphere. Inside the second group, it is possible to find different techniques, such as the release of stratospheric aerosols, sur- face albedo enhancement or space based techniques. Although space based proposals are not seen as affordable given their large cost and necessary tech- nology development, they have two important advantages compared to Earth based techniques: they do not require the modification of the Earth surface or atmosphere and they have a more uniform effect [3]. Furthermore, they would be the most cost effective if long term geoengineering is necessary [3].
The implementation of the space based proposals aims to reduce the solar radiation reaching Earth. Theoretical calculations [10] point out that, in order to be able to offset the effects caused by a doubling of the carbon dioxide con- tent in the atmosphere (compared to pre-industrial levels) the solar radiation would need to be reduced 1.7% . This would be equivalent to reducing the average global temperature by approximately 2◦C [11].
Table 1.1: Literature review.
Authors Year Concept Total mass (tonnes)
Mautner [12] 1991 Solar screen orbiting the Earth 108− 109
McInnes [17] 2002 Reflecting discs in L1 2.6 × 108
Absorbing discs in L1 5.2 × 107
Pearson et al. [13] 2002 Particle rings orbiting the Earth 5 × 109 Spacecraft rings orbiting the Earth 2.3 × 1012
Angel [15] 2006 Cloud of spacecraft near L1 2.3 × 1012
Struck [14] 2007 Dust clouds at stable lunar Lagrange points 2.1 × 1011 Sánchez & McInnes [16] 2015 Discs in orbit near L1 1.4 × 107
1.3 Literature Review on Space Sun Shades
Among SRM geoengineering methods, space based techniques aim to avoid part of the solar radiation from reaching Earth’s atmosphere. The implemen- tation of these techniques has been discussed numerous times in the literature, considering a wide range of alternatives that can be found in Table 1.1. The creation of reflective rings in orbits around Earth has been mentioned by dif- ferent authors as Mautner [12] and Pearson et al. [13], but these techniques would create a large orbital debris hazard and they could affect how light is perceived from the surface of the planet [11]. Other options considered are related to the use of stable points in space to place these reflectors, both in the Moon-Earth system (Struck [14]) and the Sun-Earth system (Angel [15], Sánchez and McInnes [16], McInnes [17]). Out of these, the most effective method so far appears to be the use of the L1 point in the Earth-Sun system [11], since the use of the Earth-Moon system stable points would only allow to reduce solar radiation for a certain period of time each month. On the con- trary, the use of the first Lagrange point, which is located in the Sun-Earth line, would reduce solar radiation the whole time the reflector is in place.
The type of reflectors used to avoid sunlight from reaching Earth has also been discussed as well as where they should be manufactured. Struck [14], Pearson et al. [13] and Mautner [12] considered the use of clouds of dust par- ticles that should be obtained from asteroids or the moon surface. Other and more recent authors consider different kinds of spacecraft, with various levels of control over their movement. Angel [15] proposes the use of a large number of small and light vehicles placed in a random cloud with little control, mean- while Sánchez and McInnes [16] present two discs in a certain controlled orbit close to the L1 point. As pointed out by McInnes [17], the large discs proposed would need to be manufactured in space, while a smaller spacecrafts could be
manufactured on Earth, as Angel [15] describes.
Comparing the mass to make these studies a reality, as it can be seen in Table 1.1, the lightest option proposed is the most recent one [16], but the mass of the system is not always a driver in this case. For those proposals built in space, the mass is not a constraining factor. Although a new constraining factor appears here, which is the necessary advance in technology to be able to mass produce in space, which so far is not even contemplated.
1.4 Present Work
The aim of this project is to study the feasibility of the realization of space based geoengineering ideas in the near future. Considering the studies de- scribed in section 1.3, among all these methods, the alternative which is ap- parently more effective is the deployment of large discs in the vicinity of the first Lagrange point.
One of the problems faced by the most common space based techniques proposed is that, even though a uniform reduction of the solar radiation would decrease the global mean temperature, the effects at a regional level would change depending on the latitude [3]. More precisely, as studied in [16], there would be a cooling of latitudes close to the equator and a warming of the poles.
This is the reason why the most recent study regarding this type of geoengi- neering option [16] studies the optimal configuration of the shades in order to minimize the differences. The final result of the paper written by Sánchez and McInnes [16] leads to the deployment of two discs orbiting around the sub-L1 point (new point resulting of the equilibrium of gravitational forces and solar radiation pressure). From now on, it will be assumed that this is the config- uration of the shades to be deployed. The area and exact orbit of the discs are to be defined further in the project once the features of the spacecraft are described.
Since the goal is to find the best mean to accomplish the deployment of the shades in the following decades, the approach is focused on their trajec- tory to the vicinity of the first Lagrange point, the features of the spacecraft used regarding control and layout, and the launch of these spacecraft into or- bit. The technologies contemplated in the project already exist or are being developed inside the space sector, with technology readiness levels between four and nine. As any other space mission, and specially given the dimensions
of this project that can be anticipated going back to the literature study, the leading driver during the study is the cost of the mission, which in space usu- ally translates into mass. On a second level it is also considered the time of the mission. The reason for the lesser importance given to the time is that the period in which the shades should be put into place is not specifically defined.
The final goal of the shade is to avoid surpassing the 2◦C global mean temperature rise in order to avoid severe consequences [6]. Therefore, the time limit for the project must be defined based on this goal. In the worst case scenario studied in [1], which corresponds with a world where emissions keep rising till 2100, the 2◦C limit would be surpassed by 2050. Looking at these results, it was decided to set this as the time limit when the shades should be in position and operating by this year.
The first step in the study of the trajectories is the definition of the propul- sion system used in the spacecraft to travel to the Lagrange point. It is assumed that in the near future the production of large structures in space would not be possible, therefore the discs would be manufactured on Earth. As a conse- quence, the spacecraft size needs to be modified so it can be launched from the Earth’s surface into space. Later on, once the discs are in place, the control strategy that they will follow will be described. Next, the general layout of the spacecraft will be addressed, focusing on the materials, the mass budget, the size and the control techniques used, as well as the control strategy. To close the thesis, a discussion and a conclusion will be carried out. Here the results of the project will be commented, as well as its possible chances to become a reality and its social effects.
Methodology
The feasibility evaluation of the deployment of a space sun shade in the vicinity of the first Lagrangian point requires the analysis of certain areas, common in any space mission. In the methodology chapter, the different approaches used during the process are presented.
2.1 Initial Assumptions
In order to enable the selection of the trajectory for the mission there were some aspects that needed to be clarified. As already mentioned, assuming that in a near future the human race will not have access to the manufacturing of large structures in space, it will not be feasible to launch the two large discs described in [16] from Earth. For this reason, instead of using just discs, it was decided that each one of the two shades would be composed by a certain num- ber of smaller spacecraft, which can be put into orbit with an existing launch vehicle (or one under development). As an initial approximation, considering the total area described in [16] and that one spacecraft could have an area of 400 m2, the number of spacecraft orbiting the vicinity of the L1 point would be around 16 250 million.
Related to the travel to the sub-L1 point, the best propulsion system had to be chosen. Two options were considered at the very beginning of the research:
electric propulsion and solar sailing. No other systems were taken into account because they would require to carry large amounts of propellant and thus they would greatly increase the cost of the mission. Electric propulsion is a type of low thrust space propulsion that uses electrical power to accelerate a propel- lant and create thrust. It has a really high specific impulse which, compared to
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chemical propulsion, allows to create a low continuous thrust for long periods of time and with small amounts of propellant. Solar sails allow spacecraft to use solar pressure to propel themselves through space, without the need of any kind of propellant. The attitude of the sail surface controls the thrust vector, allowing the vehicle to travel in different directions.
The final goal of the spacecraft once in place is to cover a large area with the lowest weight possible, which gives as a result a vehicle with a design similar to a solar sail [16], even if this sail has never being considered to travel through space in previous studies. Considering this and the fact that solar sailing does not require any kind of propellant (which leads to a reduction in the launch mass compared to any other propulsion system), it was decided that the propulsion system best suited for the mission was solar sailing.
2.2 Launch
As outlined previously, it was assumed that the spacecraft will be launched from Earth. Consequently, it is necessary to study the best options to put such a large mass in orbit, regarding the type of vehicle and the orbit that this vehicle needs to reach. In addition, the launch site selection must also be examined.
2.2.1 Launcher Selection and Assumptions
The selection of the launch vehicle is crucial to define the final cost of the mis- sion, representing between fifteen and twenty-five percent of the total budget in the development of a regular space mission [18]. Moreover, in this project the launch percentage is expected to be even higher because of the large mass of the system. Therefore the driver choosing the launch vehicle for the shade was the price per kilogram.
The approximate mass expected for the system, using as reference the most optimistic of the studies treated previously, is around 1.4 × 107tonnes [16].
Considering that historically the launcher with the largest payload capacity has been the Saturn V, which stopped working in 1973 and had a capacity of 140 tonnes, the number of launches necessary to put the shade in space will be around 105. Since the second driver during the project was time, it was considered that the reduction of the number of launches needed would mean a reduction of the time it takes to put them in orbit. This together with the fact that the price must be as low as possible, points towards the search of the
Table 2.1: Launcher considered during the study.
Launcher Payload to LEO (kg) Cost ($/kg) Situation
Starship [19] 100 000 + 201 Under development Long March 9 [20] 140 000 - Under development SLS Block 2 [21] 130 000 8 0002 Under development SLS Block 1B [21] 105 000 8 0003 Under development Yenisei [22] 88 000 - 115 000 - Under development Falcon Heavy [23] 63 000 1 430 Operational
1 Cost for SpaceX, no information about the costumer cost yet. [24]
2 Cost estimated in 2019. [21]
3 Cost estimated in 2019. [21]
cheapest launch vehicle on the market and the one with the most payload ca- pacity, to reduce both price and number of launches needed.
The heaviest rocket currently operational is the Falcon Heavy from the company SpaceX [23], which is able to carry around 64 tonnes to Low Earth Orbit (LEO), but there are several launchers under development with larger payload capacities expected to be operational in the next decade. Regarding the cost, as it can be seen in the Table 2.1, the cheapest is the Starship devel- oped by SpaceX, although it must be pointed out that official costs have not been released about this launcher. The main reason for its lower price, when compared with the rest of heavy vehicles selected in Table 2.1, is its complete reusability.
2.2.2 Target orbit
Before starting the trip to the vicinity of the Lagrangian point the spacecraft must escape Earth’s gravitational sphere of influence. To do so, there are two different options to consider: the launcher injects the spacecraft directly in an escape trajectory or the spacecraft is injected in a parking orbit, from where it will escape the gravitational field using its own propulsion system.
The payload capacities in Table 2.1 are referred to LEO. If the spacecraft is to be injected in an escape trajectory, the capacity for these launchers is considerably reduced. Thus, the number of launches needed would increase as well as the cost per kilogram of each one of these launches. Consequently
Figure 2.1: Space debris distribution with altitude and inclination in 2020.
Data obtained from ESA MASTER 2009.
it was decided that the best option was to use a parking orbit from where the spacecraft would travel by its own means of propulsion. The next step was to select the best parking orbit. The main propulsion system of the spacecraft is a solar sail. Solar sails have a large area, which means that for low Earth orbits the atmospheric drag generated by them can not be neglected. For altitudes higher than 1000 km the acceleration generated by the solar pressure on the sail is higher than the one generated by the drag [25]. If this was not the case, the solar sail will never be able to escape, therefore, altitudes below 1000 km were dismissed. Moreover, it must be taken into account that solar sails are large areas made of extremely thin materials, which makes them especially sensible to any kind of space debris. In figure 2.1 is shown how the space debris is distributed in different orbits. Based on this, it was decided that to minimize the possibilities of collision, the spacecraft should be injected at an altitude higher than 2000 km . From here, the solar sail would be deployed and the escape trajectory would start. It must be mentioned that although the space debris in altitudes higher than 2000 km is much fewer than in lower altitudes, the solar sail may face some problems when crossing the geostationary orbits zone.
The selection of the inclination of the parking orbit was driven by: the possibility of cost reduction by using the Earth’s rotation boost and the launch window available. The reduction can be achieved by taking advantage of the Earth’s rotation when launching, which gives a boost to the vehicle and allows to carry more payload than without it. This boost increases if the vehicle is launched from a space port located close to the equator and to take the most advantage out of it, the launch azimuth must be close to 90◦. Taking into account these two characteristics the most convenient inclination for the orbit, regarding the cost reduction, must be close to zero. The reason for it can be deduced from Equation 2.1 which relates the latitude (Φ), launch azimuth (β) and inclination (i).
cos i = cosΦ cos β (2.1)
Given that the number of launches required to take the shade to space has been calculated to be around 105, it will be necessary to carry out several launches a day and therefore multiple space sports around the globe must be operating.
To make sure that in all of them the launch window is as wide as possible, the best is to place the spacecraft in an equatorial orbit. This type of orbits has a constant launch window when launching from equatorial locations.
2.3 Trajectory
This section describes the study of the trajectory of the spacecraft once it is released in the correct target orbit. It was decided to divide it into two phases, since the dynamics of the vehicle are different in each one of them. First the spacecraft will carry out an escape trajectory to exit Earth’s sphere of influence by just using the solar sail. Once outside of the planet gravitational field, the sail will start its way to the equilibrium point close to Sun-Earth L1 point. Both phases were optimized in order to reduce the travel time, since the propellant is not a driver in the case of solar sails.
2.3.1 Reference Frames
For each one of the phases different reference frames are used to study the dynamics of the vehicle, being a total of three different reference frames:
• Earth Centered Inertial (ECI)
• Circular Restricted 3 Body Problem (CR3BP)
• Earth Centered Sun Pointing (ECSP)
The ECI reference frame is an inertial frame centered on the Earth that does not rotate with respect to the stars. The x-axis points towards the vernal equinox, the z-axis coincides with the Earth’s rotational axis and the y-axis completes the right-handed orthogonal frame with the first two axes such as ˆ
y = ˆz × ˆx.
The CR3BP center is placed in the center of mass of the three body system, which in this case is placed on the Earth-Sun line. The x-axis points towards the Earth, the z-axis is perpendicular to the ecliptic plane and the y-axis com- pletes the right-handed orthogonal set just as before.
Finally, the ECSP is an Earth centered system that moves with Earth rota- tion around the Sun. Its x-axis points towards the Sun, the z-axis is perpendic- ular to the ecliptic plane and the y-axis completes the right-handed orthogonal set.
2.3.2 Solar Sail Dynamics
This project assumes ideal solar sails, which means that the force generated by the solar pressure on the sail is perpendicular to its surface ( ˆn being the vector in this direction). The acceleration that the solar radiation pressure induces in the sail is described as: [26]
~as= βµsun
r2s (ˆrs· ˆn)2nˆ (2.2) where µsun = 1.327 × 1020m3s−2is the gravitational constant of the Sun, ˆrs is the unit Sun-sail vector, rs is the Sun-sail distance and ˆn is the normal to the sail surface in the direction of the force. The parameter β is the solar sail lightness parameter, which is the ratio between the maximum acceleration of the vehicle due to the Solar Radiation Pressure (SRP) and the gravity of the Sun. This value is a function of the spacecraft areal density σ (in g m−2) and the optical properties of the sail Q. Being Q = 1 a perfectly reflecting surface and Q = 0 a surface that does not change the direction of the photons at all [16].
β = 1.53 · Q
σ (2.3)
The value 1.53 represents the critical loading parameter in g m−2. It is the mass to area ratio that a sail, oriented perpendicular to the sun line, should have to generate a force equal and opposite to the solar gravitational force.
The thrust vector (ˆn) can be defined using two angles that describe the orientation of the solar sail with respect the CR3BP axes: the cone angle α that corresponds to the angle between the normal to the surface and the x-axis and the clock angle δ which defines the angle between the normal vector and the y-z plane. In terms of these angles, the normal vector is described with the following expression.
ˆ
n = cos α, sin α cos δ, sin α sin δ
(2.4) Two-Body Problem Equations
The two body problem was solved when studying the escape trajectory of the solar sail, while the gravitational force of Earth was the primary force acting on it. The equations of motion of the spacecraft are written in the ECI reference frame.
~r = −¨ µ
|~r|3 · ~r + ~as (2.5)
where r represents the position of the spacecraft and~r its acceleration.¨ Three-Body Problem Equations
Once the solar sail is outside the sphere of influence of Earth the trajectory be- comes a three-body problem, where the gravity of the Sun also must be taken into account. To solve this new problem, first the state vector of the sail in the last position of the two body solution was transformed from the ECI coordi- nates to the CR3BP reference frame. Starting at this point, the equations of motion for the three-body problem (expressed in the CR3BP reference frame) were used to solve the movement of the spacecraft.
Given the large dimensions of the variables that these equations were go- ing to work with, it was decided to use dimensionless variables. In order to do so, new units were introduced. The unit length chosen was the distance between the Sun and the Earth (1AU); the unit of mass was defined as the sum of the mass of these two bodies such as msun + mearth = 1; the mass ratio µ = mearth/(msun+mearth) and finally, the time unit selected was 1/ω, where
Figure 2.2: Circular Restricted Three Body Problem schema [27]
.
ω is the angular velocity of the Earth around the Sun.
In this reference frame and with the units mentioned, the motion of the solar sail can be described as:
~r + 2ω × ˙¨ ~r = ~as− ∇U (2.6) where U is representing the effective gravitational potential, which can be writ- ten as:
U = −x2+ y2
2 − (1 − µ
|~r1| + µ
|~r2|) (2.7)
where the position vectors ~r1and ~r2, that can be seen in Figure2.2, are defined as ~r1 = x + µ y z and ~r2 = x − (1 − µ) y z. It must be stressed that this potential represents not only the gravitational potential but also the centripetal acceleration, included in the equation with the first term. To use the solar sail acceleration with these new dimensionless units it needs to be rewritten as:
~as = β1 − µ
(~r1)2 (ˆr1· ˆn)2nˆ (2.8) where ˆr1 and ˆn are unit vectors.
2.3.3 New Equilibrium Point
The first Lagrangian point is a location in space, laying on the Sun-Earth line, where the gravitational forces of the two bodies and the centrifugal force of the orbital motion of the third body create a stable location. Thus, a smaller mass placed in this point remains in the same relative position. When working with solar sails a new force needs to be taken into account, the solar radiation pressure. As a result, the point where a third mass is in equilibrium changes sunwards from the classical location, which is located around 1.5 × 106km away from Earth (1/100 of the total distance between the two bodies).
To figure the new position the new force equilibrium needs to be com- puted so the total acceleration is equal to zero. At this point the sail will be perpendicular to the sunlight, the normal to the surface will be ˆn =1 0 0.
Considering this and the fact that the point will be on the x-axis, the equation to find the location of the new equilibrium point (xe) is Equation 2.9 [16].
γ5−(3−µ)γ4+(3−2µ)γ3+(1−2µ−(β +1)(1−µ))γ2+2µγ −µ = 0 (2.9) with γ = xe − (1 − µ). As it can be seen from the equation, it is possible to change the position of the equilibrium point by selecting a certain lightness parameter for the sail. This variation will be used later on to study the area and total mass variation depending on the sail features.
2.3.4 Escape Trajectory Optimization
Solar sailing provides the spacecraft with very low acceleration, which leads to that an orbital maneuver takes a long period of time. Therefore, the goal of the escape trajectory optimization is to minimize its time.
The energy per unit mass of a body orbiting around a planet is defined as:
E = 1
2~vT~v + U (~r) (2.10) where U (t) is the potential energy per unit mass and it has a negative value.
The energy of the body remains constant and negative as long as it stays in orbit. To enable the body to escape the gravitational field, the energy must be above zero, which can be achieved by increasing its speed.
The strategy followed to achieve the escape consists in maximizing the in- stantaneous rate of increase of orbital energy. This approach does not lead necessarily to the minimum time solution, but it has been shown that for the order of magnitude of solar sail acceleration the solution obtained is near min- imum time [28]. The instantaneous rate of increase of the orbital energy is defined in Equation 2.11 [28].
dE
dt = ˙~vT~v − ~gcT
~v = ~asT
~
v (2.11)
It can be deduced that in order to maximize it, the component of the sail ac- celeration along the velocity vector must be maximized in each point of the trajectory. The optimization problem was solved by Coverstone and Prussing [28] in the ECSP reference frame resulting in the following control law for the normal of the sail:
nx = − |vy|
qvy2+ ξ2(vy2+ vz2) (2.12)
ny = ξnx (2.13)
nz = vz
vyny (2.14)
where the variable ξ is described as:
ξ =
−3vxvy− vyq
9vx2+ 8(vy2+ v2z)
4(vy2+ vz2) (2.15)
Based on this result, to compute the escape trajectory of the sail the equa- tions of motion 2.5 defined in section 2.3.2 were solved. In order to do so, everything was transformed to ECI coordinates. The simulation was stopped once the total energy of the sail reached zero, starting at this exact point the second phase of the trajectory outside the sphere of influence of Earth.
2.3.5 Trajectory to Sub-L1 Optimization
The minimum-time solar sail trajectory is an optimal control problem, which can be solved by two different methods: the indirect and the direct. For this project the direct approach was selected, as suggested in [29]. The direct meth- ods transform the optimal control problem in a parameter optimization prob- lem. In order to do so, the control variables are discretized by dividing the
trajectory in a certain number of segments. In the paper mentioned [29], it was demonstrated that with few segments a near minimum-time solution can be accomplished. During the study it was assumed that the changes between different angles were instantaneous, which must be taken into account when reading the final results.
Problem Definition
Any trajectory optimization problem can be described as a system of state variables (x(t)) and control variables (u(t)). The optimal control problem is generally defined as:
J (x, u) = Z tf
t0
f (t, x(t), u(t))dt (2.16)
˙x = f (t, x(t), u(t)), t ∈ [t0, tf] (2.17)
r(x(t0), x(tf)) = 0 or ≥ 0 (2.18)
g(t, u(t)) ≥ 0, t ∈ [t0, tf] (2.19) where J is the objective or cost function, ˙x are the state equations, r represents the boundary conditions (initial and final) and g stands for the path constraints defined during the trajectory. The function f represents any function depen- dant of those variables. Next each one of the functions selected to define the problem under study are presented.
• State and control variables. The state variables in this case are the po- sition and velocity coordinates of the spacecraft. The control variables are the two angles that define the normal of the solar sail, the clock angle and cone angle, which are defined in subsection 2.3.2.
• Cost function. The cost function is the function to be optimized, which here corresponds with the final time of the trajectory.
J = tf (2.20)
• State equations. These are equations that define the system dynamics, therefore the equations of motion 2.6 defined in section 2.3.2.
• Boundary conditions are usually defined as initial and final points of the trajectory. The initial point is defined by the final point of the escape trajectory, while the final point is the new equilibrium point in the Sun- Earth line. In order to facilitate the optimization process, the final point values were allowed to have a relative error of 10−5.
• Path constraints are restrictions imposed to the variables that must be respected throughout the complete trajectory, both nonlinear constraints and upper or lower bounds.
Regarding the upper and lower bounds, none were defined for the state and control variables. The only bounds determined for the optimization were defined for the final time (objective function), setting the lower bound to zero and testing the upper bound in order to get a solution.
One nonlinear constraint was set, in order to make sure that the reflective side of the sail is pointing towards the sun all the time.
ˆ
r · ˆn ≥ 0 (2.21)
Multiple Shooting Parametrization Method
As it was mentioned before, the optimal control problem was solved using a direct method that allows to treat the optimal control problem as an opti- mization problem by discretizing the control variables. A certain number of segments is defined to divide the trajectory, which was chosen to be ten. For each one of these segments, the cone and clock angles have a constant value.
These parameters are the optimization variables that the optimizer will change in the search of a solution. Thus, there are 10 × 2 optimization variables, two per segment. The optimization was developed using the SNOPT optimization software, which uses a sequential quadratic programming (SQP) algorithm.
The software was used on MATLAB, using the interface developed by Gill and Wong [30].
2.4 Spacecraft Configuration
Initially, the scope of this project did not cover the definition of the spacecraft features but, after observing how the results of its dynamics depend on it, it was decided to developed a rough description of the layout of the spacecraft.
The areas in which the description is more precise are the ones related directly with the performance of the solar sail, such as the sail area, the total mass of the spacecraft and the sail material.
2.4.1 Total Mass and Size Study
The main driver during the whole project was the cost and therefore the to- tal mass. This value depends on the total area (A) needed for the shade and the areal density of each one of the spacecraft (σ) that create this shade. This density can change during the mission because some of the elements can be jettisoned. For this study it was decided that this density would stay constant and therefore when examining the literature only initial densities were consid- ered. The total mass Mtotalthen becomes:
Mtotal= A · σ (2.22)
The total area depends on where the new equilibrium point is located and thus, if one looks at Equation 2.9, it depends on the lightness parameter of the sail (β). This parameter is subjected to the areal density and the reflectivity of the surface facing the Sun. How the area changes with the position of the new equilibrium point can be seen in Equation 2.23, where, keeping in mind that the shades are defined to be discs, the radius needed to reduce the solar insolation a certain value ∆S is defined [16].
Rshade = Rsun
dshade
dsun
r∆S
S (2.23)
where dsunand dshade are, respectively, the distances of the Sun and the solar sail from Earth, Rsunis the Sun’s radius and ∆SS represents the percentage that the solar radiation needs to be decreased, which, as already mentioned in sec- tion 1.2, in this case has a value of 1.7% [11].
Considering these dependencies, it is possible to study how the total mass varies for different values of reflectivity and areal density. To orient the study and make the final decisions later on, some dimensions of existing solar sails or other project ideas were used as a reference; these can be found in Table 2.2.
Table 2.2: Solar sail dimensions from the literature.
Solar sail Density (g m−2) Lightness parameter (β) Total area (m2)
IKAROS [31] 1550 0.001 200
Light Sail 2 [32] 156 0.01 32
Sunjammer [27] 150 0.0363 1 200
Heliostorm mission [33] 14.8 ( 91) 0.0379 10 000
Angel Sun Shade [15] 4 0.0153 1
1 Density without payload.
The lower and upper limits for the areal density and the reflectivity were defined based on the latest solar sail technologies found. Although Angel [15]
assumed that the areal density of the spacecraft could reach 4 g m−2, the sim- plicity of the spacecraft described in his paper does not match the one under study in this report and therefore this value was considered too low. Looking at Table 2.2 and dismissing this option, the lowest areal density of the solar sail (considering the total mass of the spacecraft) was set to be 9 g m−2. Further- more, Angel [15] discussed the minimum reflectivity achievable in a material, reaching a solution of Q ∼ 0.04, which was used as the lower bound for this variable. The outcome of the study for different combination of these two parameters can be found later on in the results section 3.1.1.
2.4.2 Mass Budget
Solar sails have been deeply studied for decades but so far the only solar sails that have reached space (IKAROS [31] and Light Sail [32] ) have been test sails, with the goal of demonstrating certain solar sail related technologies.
As a consequence, although these vehicles have proven that these kind of tech- nologies can work, the mass budget of the spacecraft were not representative of a real solar sail, since they carried other propulsion systems as back up, therefore did not need a low areal density to get acceptable accelerations. Fur- thermore, all the spacecraft studied carried payloads to develop scientific mea- surements that in the case under study are not needed.
Table 2.3: Mass budget study in percentage of the total spacecraft mass and TRL.
Spacecraft
Interstellar Heliopause Probe [34]
Heliostorm Mission [33]
Light Sail 2 [32]
IKAROS [31]
NASA study [35]
TRL 2 5+ 7 7 2
Solar sail
assembly 57 48 - - 64
Thermal 4 1 - - 1
AOCS 5 11 - - 7
Power 12 5 - - 5
Structure &
mechanisms 10 8 - - 20
CDH 7 15 - - 2
Payload 5 33 - - 5
The mass budget of the solar sails considered during the project can be found in Table 2.3. In the first row it is given the Technology Readiness Level (TRL). As seen in the table, the information regarding the two solar sails with the highest TRL could not be found. Instead, several projects with no experi- mental validation or small prototype tests were used as reference.
The final target of this study was to create a realistic mass budget for the spacecraft so, once defined the total mass expected, the mass assigned for each subsystem could be approximated. Hence, it would be possible to examine if those masses were realistic with the technologies available nowadays and the ones expected to be developed in the next decades.
2.4.3 Control
Control Subsystem
The solar sail must change its orientation constantly during its trajectory to the equilibrium point because it defines the direction of the thrust vector. Fur- thermore, once in the sub-L1 point, the optimal movement for the shade does not exactly follow a natural orbit, so the spacecraft will require regular attitude control of the clock and cone angles to describe the desired orbit. Therefore the spacecraft needs to have an active attitude determination and control sys- tem.
Concerning the attitude control, after selecting solar sailing as the main propulsion system in order to eliminate the use of propellant, it makes sense to follow the same reasoning in the control subsystem. Besides, the large mo- ment of inertia of the sail means large active control torques which translate in large amounts of propellant if thrusters were to be used, or large reaction wheels [36]. Thus, the use of conventional control techniques could only be considered as backup. Consequently, it was decided that the best option was to use solar radiation pressure to control its attitude. There are two main types of techniques for attitude control using solar radiation pressure.
• Adjustment of the position of the center of pressure.
• Adjustment of the position of the center of mass.
There are two different techniques to change the position of the center of pressure. The first technique consists of changing the reflectivity of the sail,
demonstrated by IKAROS [31] which was the first solar sail to reach success- fully interplanetary space (therefore its TRL is 7). This is achieved by using Reflectivity Control Devices placed at the edges of the sail membrane that are able to generate a torque by changing its reflectivity with electricity. In this control method the torques are constrained by the reflectivity modulation and the Sun angle, resulting in a limitation of the angle change in certain positions.
Furthermore, it needs to be kept in mind that these reflective devices are placed on the sail, affecting not only its orientation but also the total force generated by it.
The second technique considered is one that has been widely studied in various projects [27] but has never been tested in space, and it is the use of tip mounted vanes. These vanes are installed at the tips of the booms in or- der to have a large momentum arm that allows the vanes to have small areas.
Each vane has two degrees of freedom, allowing a three-axis attitude control over the spacecraft. The main problem that this kind of method presents is the complexity of the configuration and the deployment of the sail module.
Unlike center of pressure methods, techniques related with the displace- ment of the center of mass are usually completely independent from the solar sail assembly, which is an advantage since the performance of the sail is not af- fected. Nevertheless, they have one important drawback, they add more mass to the spacecraft. Some examples of these methods are the distributed mass method, which moves small masses along the booms or the bus, or the use of a gimbaled boom with a tip mass. These methods can only create pitch and yaw control because its movements can not create a torque in the perpendicu- lar direction of the sail, thus they need to be used together with other methods in order to achieve a 3-axis attitude control.
In regard to attitude determination, the sensors must be capable of working in the vicinity of the Lagrangian point. Because of this reason, magnetometers can not be used, since their utilization is usually limited to a maximum altitude of 6000 km. Earth horizon detectors are also discarded since this type of sen- sors needs to be used in an orbit around the Earth. Apart from this limitation, among the rest of sensors available on the market any could be used, however, the final sensors will be chosen looking for the minimum mass and minimum cost.
Control Strategy
Once each one of the spacecraft reaches the vicinity of the equilibrium point, they need to be arranged in two different groups to deploy the shadows as stated in [16], defined as two different discs. Each one of the discs must follow a cer- tain orbit with their respective control laws. Thus, it is necessary to control a large group of spacecraft in order to achieve a certain geometry in space. The use of groups of satellites with a certain formation with one mission has been lately introduced in the space community, mainly with observation satellites.
Angel[15] already tackled this problem and he proposed a cloud of ran- domly placed spacecraft, completely autonomous. In his proposal each space- craft must make sure that its facing the Sun and stay inside the cloud envelope, but the position inside of the cloud is unknown. The main reason for this strat- egy selection was to avoid the need of communication systems and complex station keeping requirements. Considering that in the case under study these elements are already implemented in the spacecraft in order to travel to that point using the solar sail, the use of a randomly position cloud loses most of its advantages, leading into the search of new control strategies for orbit for- mation.
The group of spacecraft must orbit the equilibrium point with a certain shape formation. This operation needs to be developed autonomously by the spacecraft, since trying to control from the ground such a large number (more than 100) of vehicles is not feasible [37]. A solution for this problem is the use of swarm strategies, which aim to find a global group organization with- out the presence of a centralized control that allows to reach a certain goal.
This is a strategy already being studied in observation satellites, since it allows to use smaller and simpler satellites to image the whole planet [38]. Swarm behaviour can be simulated implementing four basic rules in each of the in- dividuals: avoid collision (maintain a safe distance from each other), remain grouped (avoid isolation), align to the neighbor and reach the final goal. These laws allow the spacecraft to autonomously control its position with respect to the rest of the group. Each spacecraft performs this swarm maintenance ma- neuvers using a certain number of the closest satellites as a reference [39].
Results
As it was already mentioned previously in this report, the main goal of the project is to study the feasibility of placing a space sun shade near the first Lagrangian point. To do so, the most characteristic features of the spacecraft, the optimal trajectory and the launch have to be defined. In this section the final results for each one of these aspects obtained from the methods described before are presented.
3.1 Spacecraft Configuration
After the study developed in subsection 2.4 the final features of the spacecraft could be selected. Next, each one of the aspects treated previously, such as size and mass of the spacecraft, mass budget and controls systems, are finally defined. These results enabled to compute the optimal trajectory and choose the launch option later on.
3.1.1 Total Mass and Size
The total mass of the system depends on the areal density and the total area, as explained in subsection 2.4.1. At the same time, the total area depends on where the shade is placed and thus where the equilibrium point for the sail is located. As it was defined in Equation 2.9, the location of this point changes with the lightness parameter, which is defined by the reflectivity and the areal density of the sail. Therefore, it can be concluded that the total mass of the system changes with the reflectivity of the sail’s material and the areal density of the spacecraft. In order to find the combination which results in the mini- mum mass, this dependency was studied obtaining the graph in figure 3.1.
24
Figure 3.1: Variation of the total mass of the system with the areal density, for a certain set of values of the reflectivity parameter Q. Q has a value of 0.05 for the bottom line and 1 for the one located at the top of the graph.
Figure 3.1 shows the variation of the mass with the density and the reflec- tivity. In the graph it is also possible to find star points that show the lowest mass achievable for each reflectivity. It can be seen that depending on the value of the reflectivity, the density of the spacecraft for the minimum mass changes.
For reflectivities between 0.05 and 0.2 the minimum mass corresponds with the minimum density, whereas for reflectivities higher the minimum mass is located in higher values for the density.
This takes place because the density affects the total mass in two differ- ent ways: directly by itself and indirectly through the area by changing the equilibrium point position. But this equilibrium point position also changes with the reflectivity. The total mass is proportional to the density and the area.
So that, when the density increases, the mass tends to behave the same way but the area decreases, creating the opposite effect and decreasing the mass.
As the reflectivity increases, the equilibrium point moves towards the Sun, in- creasing the area needed. For reflectivities smaller than 0.2 the most important factor when looking for the minimum mass is the direct effect of the density
and accordingly, the minimum mass point is located in the minimum density.
When the reflectivity reaches 0.2 and the equilibrium point moves closer to the Sun, the area becomes more important when looking for the minimum mass.
This forces the minimum mass point to move to higher densities, in order to move the equilibrium point closer to the Earth and therefore decrease the area needed.
Based on these results, the minimum mass is obtained by choosing the min- imum areal density and the minimum reflectivity, giving a final mass of 28.5 million tonnes. However, there is something else that needs to be kept in mind and that is the performance of the solar sail. This performance is represented by the lightness parameter (β), whose values can be seen for each one of the minimum mass points in Figure 3.2. The higher the value of β, the solar sail is able to generate a greater acceleration and thus has a better performance.
Consequently, the combination that results in the minimum mass corresponds as well with the worst performance of the sail.
It can be noted that the lightness parameter corresponding with the mini- mum mass points increases with Q, until it stabilizes at a value of 0.035 from Q ∼ 0.2 on. In view of these values, if a combination different from the one resulting in the minimum mass is to be considered in order to achieve a better sail performance, it would be reasonable to choose the first point with a light- ness parameter of 0.035, since it is the one with the lowest mass inside this group. As a result, two options were under study for the definition of the fi- nal characteristics of the solar sail. The first corresponding with the minimum mass overall and the second being the one that, inside the group of minimum mass points for each reflectivity, has a lowest mass and a lightness parameter of 0.035. The definition of both points can be found in table 3.1.
Table 3.1: Options considered for the spacecraft definition.
Case 1 Case 2
Density (g m−2) 9 9
Reflectivity 0.05 0.2
Lightness parameter (β) 0.0085 0.035 Total mass (kg) 2.85 × 1010 5.66 × 1010 Escape trajectory time (days) 1 792 582
Trajectory to L1 time (days) - 317
Figure 3.2: Lightness parameter variation with the areal density, for a certain set of values of the reflectivity parameter Q. Q has a value of 0.05 for the bottom line and 1 for the one located at the top of the graph.
Between these two options it was clear that in order to achieve the lowest cost, the minimum mass point had to be selected. Nevertheless, given the low performance of it, it was considered that a trajectory analysis of this option was necessary in order to check if the sail could travel to the vicinity of the Lagrangian point in a reasonable time. Thereafter, the trajectory analysis for each one of the options was carried out. The results regarding the time that it took for each one of the sails to describe the desired trajectory can be found in Table 3.1. This table shows that for the minimum mass case it was not possible to find an optimal trajectory to the Lagrangian point, resulting in the elimina- tion of this option.
The number of spacecraft depends on the sail area of each one of them.
This area was decided taking into account the areal density selected and the values for previous solar sail projects proposed. It had to be considered that the sail would be deployed so its size is not limited by the size of the payload fairing of the launcher. The area selected needed to be large enough so that the mass of the spacecraft was reasonable. If the area was 10 m2, the total mass would need to be 90 g, which considering the need of a deployment mechanism and the rest of subsystems (such as structure, control, thermal or communication), could be hard to achieve with the current technologies. Keeping this in mind and based on past solar sail projects, which can be found in Table 2.2, the final size of the sail was set to be 10 000 m2. Table 3.2 shows the final definition of each one of the spacecraft that forms the shade.
Table 3.2: Final definition of the solar sail main features.
Density (g m−2) 9
Reflectivity 0.2
Lightness parameter 0.035 Area of the solar sail (m2) 10 000 Mass for one spacecraft (kg) 90
Total mass of the system (kg) 5.66 × 1010 Total area of the system (m2) 6.31 × 1012
3.1.2 Mass Budget
Previously the total mass for the spacecraft was set to be 90 kg in order to achieve the desired areal density. The goal in this section was to define the mass budget expected for each one of the spacecraft subsystems. Once done,
it could be examined if the mass assigned for each subsystem, given a 90 kg spacecraft, could be achieved with current technologies. This will be discussed later on in section 5.1.
Table 3.3: Mass budget study in percentage of the total spacecraft mass.
Subsystem Percentage
Expected weight considering sys- tem margin (kg)
Expected weight (kg)
Structure 20 12.6 18
Thermal 5 3.15 4.5
Propulsion (sail
module) 50 31.5 45
AOCS 5 3.15 4.5
Power 10 6.3 9
Mechanisms 2 1.3 1.8
CDH and teleme-
try 8 5.0 7.2
Total 100 63 90
In Table 3.3 three columns can be found. The first one represents the per- centage of the total mass that was assigned to that subsystem. The second column shows the maximum mass that the respective subsystem should have during this first phase of the design in order to comply with the mass target.
Lastly, the third column shows the weight that the final design of the subsystem should have, taking into account that this mass will grow as the design evolves [40].The margin used to calculate the difference between these two columns was 30%, considering that the design process is in the System Requirements Review phase [40].
3.1.3 Control
The trajectory that the solar sail needs to describe requires continuous thrust vector steering maneuvers and, as a consequence, the spacecraft must have an active three-axis attitude control system. Taking this necessity into account, as well as the technology readiness level and the benefits and drawbacks of each option, all the control techniques considered in subsection 2.4.3 were evalu- ated.
Change of the center of mass techniques were discarded for two reasons:
the limit to two-axis control, with its consequent need of an extra control method, and their higher mass when compared with change of center of pres- sure techniques. Among change of the center of pressure approaches it was de- cided to select Reflectivity Control Devices (RCD) as the main option, which uses changes in the reflectivity in different areas of the sail to create a torque, as it can be seen in Figure 3.3. This kind of method can change the total force created by the sail when changing the reflectivity, which could mean a loss of efficiency. Nevertheless, it is the only one that has reached a TRL of seven so far [36], since the rest of methods are just concepts and have never been tested.
Figure 3.3: Schematic representation of how the Reflectivity Control Device works [41].
Once every solar sail is in orbit, in order to achieve the two discs configu- ration described in [16] the spacecraft will use a swarm behaviour as control strategy. As explained in 2.4.3, in this kind of control scheme, each vehicle performs autonomously its own maneuvers based on the position of the rest of vehicles next to it following the rules defined in the section mentioned. The first three rules are common for every swarm, but the last one, called the goal
rule, is the one that defines the final objective of the swarm. In the case under study, the final goal is the movement of the two discs formed by the swarm in the correct orbit. The spacecraft would be divided beforehand in two dif- ferent groups. Once in the vicinity of the Lagrangian point, each solar sail would take its corresponding position relative to the rest of its group. At the same time, each one of the groups would be moving in the optimal orbit, pre- serving the same geometry at every moment thanks to the control laws. In principle, for the problem in hand, it was considered that in the next decades the large number of spacecraft would not entail any issue when implementing this behaviour. If this was not the case, the two swarms defined before could be divided in smaller ones, which would have a leader spacecraft that would be in charge of forming another swarm with the rest of the smaller groups.
3.2 Launch
Before starting the trajectory described in the project, the spacecraft must reach space from Earth’s surface. In this section the selection of the vehicle selected to carry out this task is presented, as well as the number of launches necessary to transport the total mass and the target orbit of these launches.
3.2.1 Launcher Selection
After the analysis of the current and near-future launch vehicle market de- veloped in subsection 2.2.1, it was decided that the lowest cost for the entire mission would correspond to the use of the Starship as main launcher. Even though the final customer price of this vehicle is still unknown, given the ini- tial approximations for the manufacturer cost and that currently the cheapest operational rocket belongs to SpaceX, it was decided to consider it as the best option. Furthermore, the selection of a reusable rocket has two extra advan- tages. The first being that the environmental impact of each launch will be much lower since the vehicle does not need to be built from scratch each time.
The second extra advantage is also related with this lack of necessity of build- ing one rocket per launch, which means that the launch frequency will not depend on the manufacturing rate of the vehicle but only in the refurbishment time, which should be lower.
