LOFAR on the Moon: Mission Configuration and Orbit Design

Transcription

1 LOFAR on the Moon: Mission Configuration and Orbit Design Maximizing the Payload Mass Using Chemical or Electrical Propulsion May 12, 2015 Aerospace Engineering - Space Exploration

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3 LOFAR on the Moon: Mission Configuration and Orbit Design Maximizing the Payload Mass Using Chemical or Electrical Propulsion For the degree of Master of Science in Aerospace Engineering at the Delft University of Technology ( ) Under the supervision of: Ir. R. Noomen Ir. H. Cruijssen Ir. L. Meijer May 12, 2015 Faculty of Aerospace Engineering Delft University of Technology

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5 Abstract Unperturbed observation of deep space radio waves is impossible to achieve from Earth but could be managed if the instruments were positioned in a place shielded from Earth, such as the far side of the Moon or the Shackleton crater, situated at its South Pole. In order to keep the costs of such a mission as low as possible, the VEGA launcher needs to be used. This MSc Thesis studies the possible mission designs and trajectories to land a minimum of 150 kg of payload in such a place. For the eight mission scenarios considered, the spacecraft can either use a Chemical Propulsion System (CPS) or an Electric Propulsion System (EPS) for the transfer to Low Lunar Orbit (LLO), the spacecraft can either circularize around the Moon in a polar orbit or in an orbit situated in the lunar orbital plane, and the relay can either be sent to the Lagrangian point L2 (L2) or be landed on the Moon. After preselecting the four feasible mission scenarios, a Matlab program was coded to simulate the trajectories flown from Low Earth Orbit (LEO) to LLO using either a CPS or an EPS, the trajectory from LLO to L2 using a CPS which is applicable to certain mission scenarios only, and the trajectory for the descent and landing also using a CPS. The inclination of the lunar orbital plane with respect to the launching site varying during the year, it was chosen to take the worst-case scenario for this report. The subsystem masses were then assessed to determine the payload mass that can be placed on the Moon. It was found that with the most successful CPS mission 34 kg of payload could be placed on the rim of the Shackleton crater, while 107 kg could be placed at that same place when using an EPS. For both cases, no relay module was sent to L2 and the spacecraft circularized around the Moon in a polar orbit. Even though the payload mass does not meet the 150 kg requirement, the reader should not forget that this study belongs to the preliminary design phase of the mission: he could therefore look into the recommendations for further work to alter several parameters that could change the outcome of this report.

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7 List of Acronyms ADS-NL AOCS CCW CPS CW DL EPS ESA GTO L2 LEO LLO LOFAR MSc OBC PCU SOI SPS TCS TTC Airbus Defence and Space Netherlands Attitude and Orbital Control System Counter-Clockwise Chemical Propulsion System Clockwise Descent and Landing Electric Propulsion System European Space Agency Geostationary Transfer Orbit Lagrangian point L2 Low Earth Orbit Low Lunar Orbit LOw Frequency ARray Master of Science On Board Computer Power Conditioning Unit Sphere Of Influence Stackable Platform System Thermal Control System Telemetry, Tracking and Command

8 iv List of Acronyms TU Delft VEGA Delft University of Technology Vettore Europeo di Generazione Avanzata (Advanced Generation European Carrier Rocket)

9 Table of Contents Abstract List of Acronyms i iii Preface Acknowledgments ix xi 1 General Introduction 1 2 The Mission Purposes of LOFAR on the Moon Mission outline VEGA launcher Earth-Moon transfer Landing the payload on the Moon Relay options Goal of this MSc thesis Optional mission scenarios Mission design considerations, assumptions and constraints Study of the different mission scenarios Summary Approach for this study Transfer orbits Spacecraft architecture Transfer orbit representation

10 vi Table of Contents 3 From LEO to LLO Using CPS Introduction Discretization of the VEGA performance chart Optimal parking orbit around the Earth Considerations Matlab solving Simplification of the problem: from 3D to 2D The in-plane three-body problem Equations of motion Explanation of the Matlab program Calculations: inputs and results Minimum value of V tot Results and conclusions From LEO to LLO Using EPS Introduction Procedure Transfer orbit Perturbations Acceleration provided by the engines Constraints and assumptions Available power Time constraint Transfer geometry and constraints Constraints on the engines Apogee raising The spiral out orbit Influences of thrust, specific impulse and altitude of parking orbit on the transfer time First engine selection: transfer time Second engine selection: mass at lunar altitude Time in Van Allen belts Lunar capture Results From LLO to the Lagrangian Point L Introduction Theorem of image trajectories Procedure Computations From L2 to LLO The image orbit Overall results and conclusions

11 Table of Contents vii 6 From LLO to Lunar Surface Introduction Descent Hovering and landing Overall results Best Mission Scenario Introduction Different mission scenarios Mass left in LLO Mission Scenarios C2, C3 and C Mission Scenario E Overview of transfer parameters Electrical stage Lander Results Conclusion and Recommendations General conclusion Recommendations for further work A Old Preface 97

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13 Preface [An unconventional preface was first written before being judged of unprofessional quality. This old preface can be found in Appendix A.] This Master of Science (MSc) thesis was carried out at the company Airbus Defence and Space Netherlands (ADS-NL), formerly called Dutch Space, situated in Leiden, The Netherlands. This MSc thesis is supervised by Ron Noomen, assistant professor at the disciplinary group Astrodynamics and Space Missions of the Aerospace Engineering faculty at Delft University of Technology (TU Delft), together with Henk Cruijssen, Systems Engineer at the Research and Development Projects department of ADS-NL and Lex Meijer, Lead Engineer of the Operations, Optical and Electrical Engineering department of ADS-NL.

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15 Acknowledgments I would like to take this opportunity to thank my TU Delft supervisor, Ir. R. Noomen for guiding me through this work, and more generally my Master of Science program. Being my Master coordinator, he helped me out in the completion of my studies from the first day until the end of my and challenged the results of my thesis all along its completion. I am also grateful to Ir. H. Cruijssen, systems engineer at Airbus Defence and Space Netherlands, who was supervising my thesis at the company. He offered me to work on this topic and led me through my work. Being in charge of the project, his contribution to my thesis was essential in many ways as he provided me with advice but also challenging questions and ideas. Finally, I would like to show my appreciation to my second company supervisor, Ir. L. Meijer, lead engineer at Airbus Defence and Space Netherlands, who helped me with orbit-related questions and enlightened me with his knowledge of the SMART-1 mission. Delft University of Technology May 12, 2015

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17 As I stand out here in the wonders of the unknown at Hadley, I sort of realize there s a fundamental truth to our nature. Man must explore. And this is exploration at its greatest. David Scott, Commander of Apollo 15 (July 31, 1971)

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19 Chapter 1 General Introduction This Master of Science (MSc) thesis is part of a bigger project from Airbus Defence and Space Netherlands (ADS-NL) called LOw Frequency ARray (LOFAR) on the Moon, which is a mission proposal for the European Space Agency (ESA). The main objective of the LOFAR on the Moon project is to deploy a network of radio telescopes on the Moon in such a place that the array would be protected from the electromagnetic radiation coming from the Earth. Indeed, the absence of electromagnetic pollution would allow unperturbed observation of deep-space radio waves and hence better analysis results than the ones that are currently obtained by radio telescope arrays on Earth. The research question giving birth to this MSc thesis is as follows: What mission configuration and associated trajectory would be the best option to land in a controlled manner a minimum of 150 kg of payload on a part of the Moon shielded from the Earth? The only restrictions are to make use of the VEGA launcher, to fly the trajectory using a propellant system, to guarantee information relay and to have the system operational on the Moon less than one year after launch. The meaning of best will be explained in the following chapter. The work of this MSc thesis is based on the previously completed Literature Survey entitled A Survey on Earth-Moon Transfer Orbits using Chemical or Electrical Propulsion [1]. In Chapter 2, the mission will be described completely, including the different mission scenarios that need to be investigated in this MSc thesis. Chapters 3 and 4 will focus on the transfer from Low Earth Orbit (LEO) to Low Lunar Orbit (LLO) using a chemical propulsion system or an electrical propulsion system respectively. Chapter 5 will deal with the transfer from LLO to the Lagrangian point L2, which could be an important part of the mission. In Chapter 6, the lunar descent and landing will be investigated. In Chapter 7, the different subsystem masses will be calculated and the best mission scenario will be determined. Chapter 8 will conclude this report and give some recommendations for further work.

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21 Chapter 2 The Mission The LOw Frequency ARray (LOFAR) on the Moon mission is a proposal of the company Airbus Defence and Space Netherlands (ADS-NL), situated in Leiden, The Netherlands, for the European Space Agency (ESA). This chapter aims at detailing the considerations for this mission while linking them to the work of this MSc thesis. After exposing the different purposes of LOFAR on the Moon, the outline of the mission will be explained, from launch to the landing of the payload on the Moon. Then, the research question will be detailed, highlighting the goal of this MSc thesis. Afterwards, the different mission scenarios will be listed, studied and preselected taking into account the assumptions and constraints of the work to be done. Finally, the approach followed for the work of this study will be described. 2-1 Purposes of LOFAR on the Moon The first main goal of this mission is to land a network of radio telescopes which would observe the deep space radio waves at a location on the Moon that is hidden from Earth. Being shielded from Earth and thus its radiation would allow non-polluted hence better observations of those waves than what can be achieved by such instruments on Earth or in orbit around Earth. The second main goal of this mission is to investigate the interplanetary capability of the European launcher VEGA (abbreviation for Vettore Europeo di Generazione Avanzata, which translates to Advanced Generation European Carrier Rocket). Most spacecraft flying a transfer orbit to the Moon have the advantage of being launched into a Geostationary Transfer Orbit (GTO), which cannot be achieved using VEGA. Indeed, the VEGA launcher can only deliver its payload into Low Earth Orbit (LEO). From such a parking orbit, it is more complicated and more propellant-demanding to perform interplanetary transfers such as Earth-Moon

22 4 The Mission transfers since the amount of velocity change that needs to be applied to the spacecraft is much greater than when the spacecraft is launched from a regular GTO. Thus using the VEGA launcher for this mission is a challenge in itself, and the outcome of this study will provide valuable information on the possibility for VEGA to be used for a broader range of missions than it is today. Figure 2-1: Illustration showing how the SPS can be mounted on top of each other [2] A secondary goal of this mission is to make use of the Stackable Platform System (SPS). The idea of the SPS is that several satellites or parts of a same satellite can directly be mounted on top of each other in the payload fairing for launch cost sharing. Figure 2-1 illustrates this principle. The SPS family is comprised of SPS-1, SPS-2, SPS-3 and SPS-4. The SPS product family is organized in an evolutionary fashion increasing in mass and complexity. The main load-carrying structure for the SPS product family is a cylinder that matches the payload adapter ring of the primary payload satellite (Figure 2-2). The SPS-2, 3 and 4 are also equipped with a means for separation from the launcher. The spacecraft could be equipped with an SPS-4 for the mother ship (responsible for the transfer), combined with for instance an SPS-3 for the lander. The way the LOFAR on the Moon mission will be orchestrated is presented in the following section.

23 2-2 Mission outline Mission outline Figure 2-2: SPS illustration showing the payload adapter ring [2] This section describes the different episodes of the mission and highlights their main parts, from launch to the landing of the payload on the Moon including the relay considerations VEGA launcher It is required for the mission to make use of the VEGA launcher. VEGA is a European rocket and is therefore launched from Kourou, in French Guiana. As seen on the VEGA performance chart (Figure 2-3), the launcher is only able to deliver its payload (the spacecraft) in LEO, and not in GTO. Therefore the starting point for the mission is one of the LEO possibilities that can be seen in this chart. The chart shows that the VEGA rocket is able to place around 2.3 tons of payload in LEO at an altitude of 300 km given that the rocket does not need to provide its payload with an extra inclination change. The in-orbit inclination with respect to the Earth would then be the latitude of the launching site (Kourou), which is 5.24 degrees [1]. Providing the payload delivered to LEO with a different inclination would cost additional propellant, therefore additional mass, and the bigger the inclination the more the propellant

24 6 The Mission Figure 2-3: VEGA Performance for circular orbits [3] used. This means that the inclination needed to be achieved by VEGA has a direct influence on the mass available for payload (the total spacecraft mass, also called wet spacecraft mass). Similarly, if the jettison of the spacecraft from the launch adapter occurs at a higher altitude, more propellant has to be used by the VEGA launcher and therefore less mass can be allocated to the payload of the launcher (the spacecraft itself). The VEGA performance chart represents the only data dealing with the relationship between parking orbit and payload mass that is available. It will therefore be used intensively during the rest of this study Earth-Moon transfer Once the spacecraft has been inserted into LEO, it will enter a transfer orbit to Low Lunar Orbit (LLO) using a propellant system to propel itself. The propellant system can either be chemical or electrical. The two propulsion systems will trigger different trajectories, leading to different quantities of propellant used. For instance, using a Chemical Propulsion System (CPS) allows for a fast transfer to LLO but requires a large amount of propellant while an Electric Propulsion System (EPS) would need less propellant, but the duration of the transfer would be much longer, in the order of several months. If a CPS is used, the transfer orbit will be flown using several impulsive shots as will be seen in Chapter 3. If however the trajectory is flown with an EPS, the transfer will be executed in two phases: apogee raising and lunar capture (which also includes the lowering of the orbit into LLO) as will be seen in Chapter 4.

25 2-2 Mission outline 7 The two propulsion systems will also trigger different structure designs, leading to different structural masses and requirements. In addition, the architectures will differ: for instance, the solar array size for EPS has a large influence on the system. Either way, the end of the transfer occurs when the spacecraft is orbiting in circular LLO at either one of the following altitudes: 100, 300, 500, 1000 or 2000 km. These LLO altitudes of circularization were chosen as a sample in order to investigate the consequences for propellant usage. In a further stage of the design it could be interesting to investigate a range of values around the best found LLO altitude Landing the payload on the Moon Once in LLO the spacecraft will brake and begin its descent to the Moon using a CPS. Indeed using an EPS is not an option here because of the very low acceleration provided by electrical engines. The spacecraft will then drop off the maximum amount of payload possible (including the instruments) on its surface. These instruments need to be protected from the Earth s electromagnetic radiation to be able to observe the deep space radio waves. Therefore they need to be placed such that they are shielded from Earth. As discussed in the literature survey, this leaves two options for the position of the payload on the Moon: either on the far side of the Moon or behind the rim of the Shackleton crater, situated on the South Pole of the Moon [1]: if the payload is landed on the far side of the Moon, the antennas can benefit from an unlimited field of view: the entire Universe can be observed. However half of the time and for more than 14 days straight the instruments are eclipsed from the sunlight, which complicates the design of the payload itself. if the payload is landed behind the rim of the Shackleton crater, the antennas can only observe the southern sky of the Universe. However most of the time sunlight would not be an issue and the instruments would benefit from a near-constant power input. Either way, since the payload is not able to see the Earth directly, it is not able to communicate with it. A relay has to be used in order to send the data gathered by the instruments back to the Earth. The different relay options are presented in the next subsection Relay options The instruments that will be deployed on the Moon will generate a large amount of data, which will need to be transmitted to the Earth. However, these instruments will be shielded from the Earth (a mission requirement), which means that the need for an extra means to relay information is present. There are several options to achieve information relay:

26 8 The Mission When in LLO, a relay module can be separated from the spacecraft and can be sent to orbit around the Lagrangian point L2. Communication between Earth and the payload would then be possible as the relay module would be able to see the Earth and the payload (if it is placed on the far side of the Moon) at all times. When in LLO, a relay satellite can be separated and keep orbiting around the Moon in LLO while the lander descends and lands on the surface. Communication between the payload and the relay would be possible but not throughout the entire orbit. The same holds for the Earth-satellite communication. In case the radio telescopes are placed behind the rim of the Shackleton crater, it would be possible to position the communication module on the rim of that crater. This module would then be able to communicate with both the instruments and the Earth. Furthermore direct exposure to sunlight would be possible most of the time [1]. Collecting sunlight with solar arrays would be easily doable, for instance using 360 degrees steerable solar arrays or a cylindrical or conical solar array. The arrays would be placed upright, perpendicular to the incoming sunlight in order to maximize their efficiency. As can be seen, a communication module is needed for this mission and its position highly depends on where on the Moon the payload is situated. 2-3 Goal of this MSc thesis In order to determine whether it is possible to meet the purposes of the mission, the following research question needs to be answered: What mission configuration and associated trajectory would be the best option to land in a controlled manner a minimum of 150 kg of payload on a part of the Moon shielded from the Earth? The only restrictions are to make use of the VEGA launcher, to fly the trajectory using a propellant system, to guarantee information relay and to have the system operational on the Moon less than one year after launch. As has been seen in Section 2-2, the LOFAR on the Moon mission can be achieved in different ways, depending on the parking orbit chosen around the Earth, on the propulsion system used, on the position of the payload on the Moon, on the information relay etc. Several mission scenarios can be considered, and for each many candidate orbit transfers can be flown. The goal of this MSc thesis is to find the combination that would answer the problem the best way (the best solution). For this MSc thesis, how good a solution is depends on 4 criteria: mass of payload delivered (as long as it is larger than 150 kg), cost, risk and time of achievement (as long as it is shorter than one year). This means that a solution will not be considered further if the payload mass delivered is smaller than 150 kg or if the time needed to deliver the payload on the Moon from launch is greater than one year. For every solution considered however, the higher the payload mass the better the solution; the lower the cost of the mission the better the solution; the lower the amount of risk the better the solution;

27 2-4 Optional mission scenarios 9 and finally, the smaller the time needed the better the solution. This MSc thesis will deal with the study of the conceivable mission scenarios as well as the computation of the different transfer orbits. The different mission scenarios will be detailed in the next section and the determination of the best solution will be made in Chapter Optional mission scenarios Mission design considerations, assumptions and constraints The transfer from LEO to lunar touchdown is carried out in steps: first, the spacecraft flies from LEO to LLO (at the chosen altitude). At this altitude, a module may detach to go orbiting around the L2 point if applicable. The (remaining part of the) spacecraft then descends and lands on the Moon at the chosen destination using a CPS. This (remaining part of the) spacecraft carrying the payload will be called lander for the rest of the study. For design considerations, the relay module will not be a satellite orbiting around the Moon in LLO. Indeed, first of all, most LLO are unstable orbits [4]. Secondly, a satellite orbiting the Moon would only see the payload (and hence be able to gather data from it) during less than half of its orbital period. On the other side of the Moon, it would send the data back to Earth. This set-up would raise requirements regarding the storage capacities of the payload and relay, as well as their transmission speeds. The possible places to land on the Moon are the far side and the Shackleton crater, situated on the South Pole [1]. If the payload is landed on the South Pole of the Moon, then as seen in Subsection 2-2-4, it would be positioned in such a way that an external relay would be able to see both the Earth and the payload shielded from Earth. In this case, no relay spacecraft around the L2 point is needed. If however the payload is landed on the far side of the Moon, then a relay needs to be sent to orbit around the L2 point. Once orbiting around L2, the module can see both the payload on the far side of the Moon and the Earth at all times and can relay the information in real time. The module going to L2 will be using a CPS. Indeed, going to L2 using an EPS is very complicated and will therefore not be covered in this MSc thesis [5]. No inclination change will be performed in LLO because of the propellant cost it would trigger. Therefore, if the spacecraft is circularized in LLO in the lunar orbital plane, the payload can only be landed on the far side of the Moon and a module has to be sent to L2 to relay the data. If however the spacecraft is orbiting the Moon in a polar LLO, the choice can be made to either land the payload on the South Pole, or to land it on the far side and to send a module around L2. There is indeed no point sending a module to L2 if the payload has to be landed on the rim of the Shackleton crater.

28 10 The Mission Finally, it is chosen for the spacecraft to limit the number of stages to two (for instance an EPS and a lander, or a CPS and a lander). Having three stages (for instance an EPS, a CPS and a lander) would trigger a too high structure mass which would not be beneficial for the payload mass Study of the different mission scenarios For the transfer, either a CPS or an EPS can be used. Both options have to be investigated. With either system, the spacecraft can get captured by the Moon and reach LLO in the Orbital Plane of the Moon or Polar LLO. Both options are considered here. Once in LLO, there are again two options: either a module separates from the spacecraft and travels towards L2 using a CPS then orbits around it in order to serve as relay satellite while the other part of the spacecraft descends and lands on the lunar surface, or the whole spacecraft descends and lands on the Moon. The Descent and Landing (DL) uses a CPS. Mission Scenario C1: CPS, circularization in LLO in the lunar orbital plane, no L2 Mission Scenario C1 considers the case for which the spacecraft is propelled by a CPS for the transfer to LLO. The transfer orbit and the circularization around the Moon take place in the same plane: the lunar orbital plane. No secondary module detaches from the main spacecraft to go orbiting around the L2 point. Since the circularization around the Moon is made in the lunar orbital plane, the spacecraft will land on the far side of the Moon. Since there will be no module orbiting around L2, it is impossible to transfer the collected data back to Earth. This mission scenario is a No Go. Mission Scenario C2: CPS, circularization in LLO in the lunar orbital plane, L2 Mission Scenario C2 considers the case for which the spacecraft is propelled by a CPS for the transfer to LLO. The transfer orbit and the circularization around the Moon take place in the same plane: the lunar orbital plane. A secondary module detaches from the main spacecraft and enters an orbit around the L2 point. Since the circularization around the Moon is made in the lunar orbital plane, the spacecraft will land on the far side of the Moon. The collected data will be transferred back to Earth by the module orbiting around L2. This mission scenario is a Go. Mission Scenario C3: CPS, circularization in polar LLO, no L2 Mission Scenario C3 considers the case for which the spacecraft is propelled by a CPS for the transfer to LLO. The transfer orbit takes place in a different plane than the lunar orbital

29 2-4 Optional mission scenarios 11 plane. After the transfer, the spacecraft enters a polar LLO. No secondary module detaches from the main spacecraft to go orbiting around the L2 point. Since the spacecraft orbits around the Moon in a polar orbit, it is possible to choose whether the payload will be landed on the far side of the Moon or on the South Pole of the Moon. However, landing the payload on the far side of the Moon would result in a No Go for the mission, because the data could not be transmitted back to Earth. It is thus preferred to choose the second landing option, which is the South Pole of the Moon. The data can easily be transmitted back to Earth. This mission scenario is a Go. Mission Scenario C4: CPS, circularization in polar LLO, L2 Mission Scenario C4 considers the case for which the spacecraft is propelled by a CPS for the transfer to LLO. The transfer orbit takes place in a different plane than the lunar orbital plane. After the transfer, the spacecraft enters a polar LLO. A secondary module detaches from the main spacecraft and enters an orbit around the L2 point. Since the spacecraft orbits around the Moon in a polar orbit, it is possible to choose whether the payload will be landed on the far side of the Moon or on the South Pole of the Moon. However, because a module is sent to the L2 point, there is no point landing the payload on the South Pole of the Moon, otherwise this mission scenario is the same as C3 but with less payload mass available (the mass that went to L2). The payload is therefore landed on the far side of the Moon and the data is relayed by the module orbiting around L2. This mission scenario is a Go. Mission Scenario E1: EPS, circularization in LLO in the lunar orbital plane, no L2 Mission Scenario E1 considers the case for which the spacecraft is propelled by an EPS for the transfer to LLO. The transfer orbit and the circularization around the Moon take place in the same plane: the lunar orbital plane. No module detaches from the main spacecraft to go orbiting around the L2 point. Since the circularization around the Moon is made in the lunar orbital plane, the spacecraft will land on the far side of the Moon. Since there will be no module orbiting around L2, it is impossible to transfer the collected data back to Earth. This mission scenario is a No Go. Mission Scenario E2: EPS, circularization in LLO in the lunar orbital plane, L2 Mission Scenario E2 considers the case for which the spacecraft is propelled by an EPS for the transfer to LLO. The transfer orbit and the circularization around the Moon take place in the same plane: the lunar orbital plane. A module detaches from the main spacecraft and

30 12 The Mission enters an orbit around the L2 point. The first part of the transfer is executed with the EPS. The DL phase has to be carried out with a second stage, using chemical propulsion. A third stage with a CPS is needed for the module to go to the L2 point. As seen before, having three stages is not acceptable due to design considerations. This mission scenario is a No Go. Mission Scenario E3: EPS, circularization in polar LLO, no L2 Mission Scenario E3 considers the case for which the spacecraft is propelled by an EPS for the transfer to LLO. The transfer orbit takes place in a different plane than the lunar orbital plane. After the transfer, the spacecraft enters a polar LLO. No secondary module detaches from the main spacecraft to go orbiting around the L2 point. Since the spacecraft orbits around the Moon in a polar orbit, it is possible to choose whether the payload will be landed on the far side of the Moon or on the South Pole of the Moon. However, landing the payload on the far side of the Moon would result in a No Go for the mission, because the data could not be transmitted back to Earth. It is thus preferred to choose the second landing option, which is the South Pole of the Moon. The data can easily be transmitted back to Earth. This mission scenario is a Go. Mission Scenario E4: EPS, circularization in polar LLO, L2 Mission Scenario E4 considers the case for which the spacecraft is propelled by an EPS for the transfer to LLO. The transfer orbit takes place in a different plane than the lunar orbital plane. After the transfer, the spacecraft enters a polar LLO. A secondary module detaches from the main spacecraft and enters an orbit around the L2 point. The first part of the transfer is executed with the EPS. The DL phase has to be carried out with a second stage, using chemical propulsion. A third stage with a CPS is needed for the module to go to the L2 point. As seen before, having three stages is not acceptable due to design considerations. This mission scenario is a No Go Summary Table 2-1 summarizes the different mission scenarios that are considered in this MSc thesis. Only Mission Scenarios C2, C3, C4 and E3 are considered. Indeed, as seen in the previous subsection, Mission Scenarios C1 and E1 were discarded because it would not be possible to transfer the collected data back to Earth and Mission Scenarios E2 and E4 were dismissed because the spacecraft would need to be composed of three stages, which would trigger a too high structure mass.

31 2-5 Approach for this study 13 Table 2-1: Summary of the different Mission Scenarios CPS EPS Orbital plane LLO Polar LLO Orbital plane LLO Polar LLO No L2 C1 = No C3 = Yes E1 = No E3 = Yes L2 C2 = Yes C4 = Yes E2 = No E4 = No 2-5 Approach for this study This section explains what approach will be followed for this study Transfer orbits Matlab will be used to compute the transfer orbits of this study. The goal is to determine the minimum amount of propellant mass that can be used for each leg of the transfer. The transfer orbits that will be computed are: in Chapter 3: from LEO to LLO circularization in the lunar orbital plane using a CPS, including determination of the best parking orbit (applicable to Mission Scenario C2). in Chapter 4: from LEO to LLO circularization in polar orbit using an EPS, including determination of the best parking orbit (applicable to Mission Scenario E3). in Chapter 5: from LLO to L2 in the lunar orbital plane (applicable to Mission Scenario C2). in Chapter 6: from LLO to lunar touchdown (applicable to every mission scenario). Note that the transfer orbits from LEO to LLO for Mission Scenarios C3 and C4 as well as the transfer orbit from LLO to L2 for Mission Scenario C4 will not be computed. Instead, the quantity of propellant needed for those transfers will be estimated in Chapter 7. In the end the best LLO altitude will also be determined Spacecraft architecture In this subsection the spacecraft architecture is detailed for all the mission scenarios considered in this study. The masses mentioned will be determined throughout this report if applicable.

32 14 The Mission Figure 2-4: Architecture of the spacecraft when using a CPS and going to L2 (Mission Scenarios C2 and C4) Mission Scenarios C2 and C4 Figure 2-4 shows the architecture of the spacecraft for Mission Scenarios C2 and C4 (if a CPS is used and if (part of) the spacecraft is going to orbit around L2). From bottom to top: The first block is composed of an SPS carrying an engine using chemical propulsion. The solar panels (in red) are attached to this block. The propellant mass is M p1 and the dry mass M d1, leading to a total mass of M 1 = M p1 + M d1 for the first block. This first block detaches and flies to L2. The second block corresponds to the lander. Chemical propulsion is also used for this stage. The solar panels for this stage are not included in this figure, but they are part of this block. Note that during the descent phase, the solar panels cannot be deployed due to the high deceleration undergone by the spacecraft. Therefore the DL phase must go on batteries. Only once landed on the Moon the solar array can be deployed. The propellant mass of this second block is M p2 and the dry mass M d2, which leads to a total mass of M 2 = M p2 + M d2.

33 2-5 Approach for this study 15 The third block corresponds to the payload that will be dropped off on the Moon surface. It has a mass of M payload. The total mass is M tot = M 1 + M 2 + M payload. Mission Scenario C3 Figure 2-5: Architecture of the spacecraft when using a CPS and not going to L2 (Mission Scenario C3) Figure 2-5 shows the architecture of the spacecraft for Mission Scenario C3 (if a CPS is used and if the spacecraft is not going to L2). From bottom to top: The first block is composed of an SPS carrying an engine using chemical propulsion. The solar panels (in red) are attached to this block. The propellant mass is M p1 and the dry mass M d1, leading to a total mass of M 1 = M p1 + M d1 for the first block. This first block also serves as the lander that will drop off the payload on the surface of the Moon. The second block corresponds to the payload that will be dropped off on the Moon surface. It has a mass of M payload. The total mass is M tot = M 1 + M payload.

34 16 The Mission Mission Scenario E3 Figure 2-6: Architecture of the spacecraft when using an EPS (Mission Scenario E3) Figure 2-6 shows the architecture of the spacecraft for the mission scenario E3 (if an EPS is used). From bottom to top: The first block is composed of an SPS carrying an engine using electrical propulsion. The solar panels (in red) are attached to this block. The propellant mass is M p1 and the dry mass M d1, leading to a total mass of M 1 = M p1 + M d1 for the first block. The second block corresponds to the lander. Chemical propulsion is used for this stage to facilitate the DL. The solar panels for this stage are not included in this figure, but they are part of this block. Note that during the descent phase, the solar panels cannot be deployed due to the high deceleration undergone by the spacecraft. Therefore the DL phase must go on batteries. Only once landed on the Moon the solar array can be deployed. The propellant mass is M p2 and the dry mass M d2, which leads to a total mass of M 2 = M p2 + M d2 for the second block. The third block corresponds to the payload that will be dropped off on the Moon surface. It has a mass of M payload.

35 2-5 Approach for this study 17 The total mass is M tot = M 1 + M 2 + M payload Transfer orbit representation All along this thesis report, the same sort of figures will be used to describe the orbit of the spacecraft from Earth to the Moon, from the Moon to the L2 point or any other transfer orbit. The figures are composed of three (or sometimes two) parts: the left-hand part, the right-hand part and the middle red rectangle. Figure 2-7 resembles the figures contained in this thesis report. As explained in the figure, the left-hand part plots the orbit of the spacecraft in the XY space, with the relevant bodies. The right-hand part shows the evolution (from left to right) of the distance to the Moon with respect to time, scaled to the distances of the left-hand part. The middle red rectangle shows the velocity of the spacecraft with respect to the center of the reference frame (in this case the Earth); it is only there for indication and is therefore not to scale. Figure 2-7: Figure example and explanations

36 18 The Mission

37 Chapter 3 From LEO to LLO Using CPS In this chapter the orbit from Low Earth Orbit (LEO) to Low Lunar Orbit (LLO) will be computed, provided that the spacecraft uses a Chemical Propulsion System (CPS). At the end of the chapter, the spacecraft will be orbiting around the Moon in LLO. Every piece of Matlab program developed and used for this chapter has been tested to certify that the results are correct, however those tests have not been included in this report. 3-1 Introduction This chapter deals with the transfer orbit from LEO to LLO using a CPS. As seen in the literature survey, CPS transfers are short (in the order of days) [1] and therefore for this section the time requirement of one year as seen in Chapter 2 will not be of interest. Furthermore, contrary to a spacecraft powered with an Electric Propulsion System (EPS), the propulsive energy does not come from the solar array. Eclipses can thus be ignored without risking a major design flaw. After performing a discretization of the VEGA performance chart, the optimized parking orbit around the Earth will be determined. Afterwards, it will be seen that the problem can be simplified into a two-dimensional problem. This in-plane three-body problem will then be solved for a specific set of parameters and the more general results will be shown in the last section. 3-2 Discretization of the VEGA performance chart The starting point of the transfer orbit from LEO to LLO is the parking orbit around the Earth. The VEGA performance chart (Figure 2-3) displays the possibilities for parking orbits

38 20 From LEO to LLO Using CPS Figure 3-1: Results of the linear interpolation performed using Matlab: the left-hand side displays the 3D plot showing the evolution of the payload mass depending on the inclination and the altitude; the right-hand side shows in a 2D plot the payload mass depending on the inclination for the same values of altitude than the VEGA performance chart. around the Earth, including the link between altitude, inclination and spacecraft mass. However Figure 2-3 does not give precise information on how much payload mass can be placed into orbit for altitudes that are not already present on the diagram, such as 1000 km for instance. Therefore a linear interpolation using Matlab needs to be made, which will result in a complete set of solutions (altitude, inclination, mass) over the full range of capabilities of the VEGA rocket. It is assumed that values outside the altitude range already present on the diagram (below 300 km or above 1500 km) cannot be reached by VEGA. Therefore the linear interpolation will be bound to the altitude range km. By taking a few points at different inclinations and belonging to the five altitude lines of the VEGA performance chart, it is possible to interpolate the behavior of the evolution of the altitude of the parking orbit with respect to the payload mass for a given orbit inclination. Doing a linear interpolation can be regarded as extracting the five altitude lines of the figure and completing the space in-between in a three-dimensional plot. The Matlab program relative to the linear interpolation is available on the separate CD. The outcome of the program is a complete 3D mapping of the payload mass that can be placed in LEO depending on the altitude (ranging from 300 to 1500 km) and the inclination (from 5 to 90 degrees). This mapping is illustrated by Figure 3-1. The left-hand side of the figure displays the 3D

39 3-3 Optimal parking orbit around the Earth 21 plot showing the evolution of the payload mass depending on the inclination and the altitude. It is printed with inclination values ranging only from 5 to 30 degrees to clarify the reading of the plot but the linear interpolation was performed until 90 degrees (indeed, the linear interpolation gets more precise if a broader range of values is used). The right-hand side of the figure shows in a 2D plot the payload mass depending on the inclination for the same values of altitude as the ones displayed in Figure 2-3 (300, 500, 700, 1200, 1500 km) for comparison. 3-3 Optimal parking orbit around the Earth In this section the optimal parking orbit around the Earth will be determined Considerations As has been seen in Subsection 2-2-1, the higher the altitude of the parking orbit around the Earth the lower the payload mass of the VEGA launcher (the spacecraft mass), and thus the lower the payload mass of the spacecraft brought to the Moon. The first consideration is therefore: Consideration 1: In order to maximize the payload mass on the Moon, the parking orbit should be as low as possible. However, this consideration is not necessarily correct. Indeed, it can be proven that for a satellite in circular orbit around the Earth, the closer to the Earth the more the energy required to raise its altitude to a certain value. For instance, to enter a Hohmann transfer orbit between the Earth and the Moon where the perigee would be the LEO at altitude r p and the apogee the altitude of the Moon r a, the following change in velocity (also called V ) has to be applied: V p = V cp ( 1 + e 1) (3-1) where V p is the change in velocity that the spacecraft needs to accomplish to go from LEO to the transfer orbit, and where V cp (the circular velocity at perigee altitude) and e (the eccentricity of the transfer orbit) are defined by: V cp = µe r p (3-2) e = r a r p r a + r p (3-3) and where µ E is the standard gravitational parameter of the Earth [6]: µ E = m 3 /s 2 (3-4)

40 22 From LEO to LLO Using CPS Combining Equations (3-1), (3-2) and (3-3) leads to: V p = µe 2ra ( 1) (3-5) r p r a + r p From Equation (3-5), it is easy to see that V p increases when r p decreases. In other words when the altitude of the parking orbit decreases, the change in velocity required to raise the altitude of the spacecraft and reach the Moon increases. A higher change in velocity is performed using more propellant, which reduces the dry mass (the total mass minus the propellant mass) of the spacecraft and so the payload mass on the Moon. The second consideration is therefore: Consideration 2: In order to maximize the payload mass on the Moon, the parking orbit should be as high as possible. The considerations 1 and 2 clearly contradict each other, which means that finding the altitude of the LEO maximizing the dry mass of the spacecraft is not trivial. In order to find this optimal parking orbit altitude, a brief Matlab program has been developed Matlab solving It is reminded that Tsiolkovsky s rocket equation links the V with other parameters of the spacecraft [7]: V = I sp g 0 ln( M total M dry ) (3-6) where I sp is the specific impulse (expressed in seconds), a fundamental characteristic of the engine, g 0 is the standard gravity (g m/s 2 ), and M total and M dry are the total mass and the dry mass of the spacecraft respectively, such that: M total = M dry + M prop (3-7) where M prop is the propellant mass needed for the change of V. Therefore [1]: M dry = M total exp ( V I sp g 0 ) (3-8) Based on Equation (3-8), the spacecraft dry mass (after transfer to the altitude of the Moon) is assessed for different parking orbit altitudes. A Hohmann transfer is performed from the LEO altitude considered to the altitude of the Moon (at the end of the transfer, the spacecraft is circularized around the Earth at the lunar altitude). This is aimed at giving a rough estimate of the quantity of V needed for the

41 3-3 Optimal parking orbit around the Earth 23 Figure 3-2: Spacecraft dry mass (in percent of its total mass and in kg) as a function of LEO altitude for a specific impulse of 100 (in blue, on the bottom), 300 (in green, in the middle) and 500 seconds (in red, on top). On the right hand side, for each color the top line corresponds to an inclination of the parking orbit of 5 degrees and the bottom one to an inclination of 30 degrees. transfer to the Moon to determine the best parking orbit around the Earth; these calculations are however not aimed at giving any preliminary information regarding the quantities of V needed for the mission. With this quantity of V and with Equation (3-8), the propellant mass of the spacecraft can be calculated, leading to the spacecraft dry mass. These calculations are performed directly in the Matlab program for the range of LEO altitudes considered in Section 3-2 as well as for different specific impulses. The Matlab code is available on the separate CD. The results of the code are shown in Figure 3-2. This figure displays on the left-hand side the dry mass of the spacecraft in percent of its total mass as a function of the altitude of the parking orbit around the Earth using three different values of specific impulse I sp : 100, 300 and 500 seconds. The blue color is used for I sp = 100 s (on the bottom of both plots), the green color for I sp = 300 s (in the middle of both plots) and the red color for I sp = 500 s (on top of both plots). It is assumed that a Hohmann transfer from LEO to a circular orbit at the same altitude as the Moon s is performed. As was predicted, for each value of specific impulse the higher the altitude of the parking orbit the higher the ratio dry mass over total mass. On the right-hand side of the figure is printed the dry mass of the spacecraft (in kg) with respect to the altitude of the parking orbit, once again for the 3 different values of specific impulse (the 3 different colors). This time the lines are doubled: for each color the top one corresponds to an inclination of the parking orbit of 5 degrees and the bottom one to an inclination of 30 degrees. It is supposed that no further change in inclination is required to reach the Moon. The important conclusion lies in the fact that the allowable dry mass of the spacecraft de-

42 24 From LEO to LLO Using CPS creases when the altitude of the parking orbit around the Earth increases. It can therefore be concluded that the altitude of the parking orbit maximizing the total payload mass that can be placed on the Moon is 300 km. Therefore consideration 1 dominates over consideration 2. For the continuation of the CPS study, it will be assumed that the VEGA rocket places its payload in a 300 km LEO. The inclination of the parking orbit still needs to be determined. 3-4 Simplification of the problem: from 3D to 2D At this altitude (300 km), VEGA is able to put 2300 kg of payload into orbit if the orbit has an inclination of 5 degrees (Figure 2-3). Increasing the inclination would lower the payload mass. The inclination change i that needs to be performed from the launch site to the inclination of the lunar orbit, by either the VEGA rocket or the spacecraft can be calculated as follows: Subsection showed that the latitude of the launch site with respect to the equator i Kourou Equator is 5.24 degrees (North of the equator). the inclination of the equator of the Earth with respect to the ecliptic plane i Equator Ecliptic is degrees [8]. as seen in the literature survey, the inclination of the plane containing the orbit of the Moon (called the lunar orbital plane) with respect to the ecliptic plane i Moon Ecliptic is 5.16 degrees on average [1]. Therefore under the assumption that the launch date cannot be chosen (only the time of the day at which the launch takes place can be chosen), the inclination change that needs to be performed is comprised between: i min = i Equator Ecliptic i Kourou Equator i Moon Ecliptic (3-9) and: i min = = deg (3-10) i max = i Equator Ecliptic + i Kourou Equator + i Moon Ecliptic (3-11) i max = = deg (3-12) To reach the Moon, the inclination change that needs to be performed by either the VEGA rocket or the spacecraft is thus somewhere between 13 and 34 degrees depending on the day of launch. Figure 3-3 illustrates the situation without including the latitude of the launch site. Thus, the inclination change required to be able to reach the Moon will always be smaller than 34 degrees. For such an inclination, VEGA can place 2200 kg of payload (the spacecraft) into orbit, as can be seen in Figure 2-3. Therefore assuming that the VEGA launcher performs that inclination change leads to a spacecraft mass 100 kg lower (at most) than if the spacecraft would simply be placed in

43 3-4 Simplification of the problem: from 3D to 2D 25 Figure 3-3: Range of inclination values between the Earth s equator and the Moon [9] LEO with a 5 degrees inclination. The cost η (in terms of spacecraft mass) of making this assumption will therefore always be smaller than: η = = 4.35% (3-13) Indeed, if the spacecraft would need to change the inclination from 5 degrees to the correct one, it would cost some propellant anyway. Since 100 kg is only 4.35 % of the 2300 kg, it can be concluded that for this study, it is acceptable to assume that these 100 kg are a reasonable cost for the inclination change. Furthermore, this worst-case scenario assumption significantly simplifies the subsequent calculations. Thus for the continuation of this chapter, it will be assumed that VEGA places 2200 kg of payload (the spacecraft) at the inclination needed to reach the Moon: the spacecraft will not need to operate any further inclination change. Therefore the problem becomes a 2D problem.

44 26 From LEO to LLO Using CPS 3-5 The in-plane three-body problem In this section the transfer from LEO to LLO is analyzed Equations of motion As seen in the literature survey, the Sphere Of Influence (SOI) of the Moon with respect to the Earth falls completely within the SOI of the Earth with respect to the Sun [1]. This means that for transfers from the Earth to the Moon, the gravitational attraction of the Sun is a perturbation. Figure 3-4 plots the gravitational accelerations of the Earth, the Moon and the Sun on the spacecraft as a function of position along the Earth-Moon axis. As can be seen in the figure, the Sun s attraction is an almost constant acceleration which is not dominant compared to the two other accelerations. Therefore for this study the solar gravitational attraction is neglected and because the solar gravitational attraction is the most significant perturbation for such a transfer orbit, all the other perturbations are also neglected and as a result the spacecraft is only influenced by the Earth s and the Moon s attractions. Figure 3-4: Accelerations of the Earth, the Moon and the Sun on the spacecraft as a function of position along the Earth-Moon axis The spacecraft, the Earth and the Moon are considered point masses. The spacecraft s mass is too small to be of any influence over the other bodies. However, the two bodies will exert the following forces on the spacecraft (SC): F Earth/SC = µ Earthm SC rearth/sc 3 r Earth/SC (3-14) F Moon/SC = µ Moonm SC rmoon/sc 3 r Moon/SC (3-15)

45 3-5 The in-plane three-body problem 27 where µ Earth and µ Moon are the standard gravitational parameters of the Earth and the Moon respectively, where m SC is the spacecraft mass and where r Earth/SC and r Moon/SC are the distance vectors from Earth to the spacecraft and from the Moon to the spacecraft respectively. From Newton s second law in an inertial reference frame: FSC = m SC a SC = m SC d V SC dt Incorporating Equations (3-14) and (3-15) into (3-16) gives: F Earth/SC + F Moon/SC = µ Earthm SC r 3 Earth/SC r Earth/SC µ Moonm SC dv rmoon/sc 3 r Moon/SC = m SC SC dt (3-16) (3-17) which simplifies to (in an inertial reference frame): d V SC dt = µ Earth rearth/sc 3 r Earth/SC µ Moon r 3 Moon/SC r Moon/SC (3-18) To consider the motion of the spacecraft in the geocentric referential, the attraction force of the Moon applied to the Earth has to be subtracted from the previous equation [8]. Therefore Equation (3-18) becomes: d V SC dt = µ Earth rearth/sc 3 r Earth/SC µ Moon r 3 Moon/SC r Moon/SC µ Moon r 3 Earth/Moon r Earth/Moon (3-19) Figure 3-5 plots the three terms of Equation (3-19) as a function of position along the Earth- Moon axis. It is clear that the attraction force of the Moon applied to the Earth can be neglected with respect to the two other (main) accelerations. Therefore for the rest of this study Equation (3-18) will be used to describe the motion of the spacecraft in the geocentric reference frame. Furthermore the Moon will be assumed in a circular orbit around the Earth. In the geocentric reference frame, in the plane containing the Earth, the Moon and the spacecraft (the lunar orbital plane), the coordinate x is defined as pointing towards a fixed direction: the direction of the ascending node of the lunar orbit. The y coordinate is defined so that the (x, y) system is a Cartesian coordinate system. In this system, the Earth has the coordinates: ( ) 0 E = (3-20) 0 The Moon s coordinates are: M = ( ) dem cos(θ M (t)) d EM sin(θ M (t)) with θ M the angle defined between the X axis and the (Earth Moon) line: θ M (t) = (3-21) 2π T Moon t + θ Minit (3-22)

46 28 From LEO to LLO Using CPS Figure 3-5: Accelerations undergone by the spacecraft as a function of position along the Earth- Moon axis in the geocentric reference frame Finally, the spacecraft has the following coordinates: SC = ( ) x y = V SC = (ẋ ) ẏ = d V SC dt (ẍ ) = ÿ (3-23) With these notations, one can express: r Earth/SC = (x 0) 2 + (y 0) 2 (3-24) and r Moon/SC = (x d EM cos(θ M (t))) 2 + (y d EM sin(θ M (t))) 2 (3-25) Combining Equations (3-20), (3-21) and (3-23) into Equation (3-18) leads to the following set of equations: ẍ = xµ Earth (x 2 + y 2 ) 3/2 µ Moon (x d EM cos(θ M (t))) ((x d EM cos(θ M (t))) 2 + (y d EM sin(θ M (t))) 2 ) 3/2 (3-26) ÿ = yµ Earth (x 2 + y 2 ) 3/2 µ Moon (y d EM sin(θ M (t))) ((x d EM cos(θ M (t))) 2 + (y d EM sin(θ M (t))) 2 ) 3/2 (3-27) Equations (3-26) and (3-27) are the equations of motion of the spacecraft in the simplified geocentric reference frame, where only the Moon and the Earth exert attraction on the spacecraft. They are the basis of the Matlab program that was developed in order to calculate the spacecraft s trajectory throughout the entire orbit, including changes in velocity. The Matlab code is available on the separate CD.

47 3-5 The in-plane three-body problem Explanation of the Matlab program This Matlab program takes into account the gravitational pulls of both the Earth and the Moon on the spacecraft. It does not take into account the gravitational force exerted by the Sun as well as all the other perturbations, as seen in Subsection Also from Subsection 3-5-1, the Moon is assumed to be in circular rotation around the Earth. The Matlab program solves the set of differential equations previously derived (Equations (3-26) and (3-27)) using the function Ode45 of Matlab. This function employs a variable step size Runge-Kutta integration method. Ode45 uses a 4th and 5th order pair of formulas for high accuracy [10]. It is possible to change the precision of the Ode45 Matlab function in the input part of the code. A higher accuracy leads to better results but to a higher computation time. Depending on the inputs given, the program will calculate the total amount of V needed to go from a circular orbit around the Earth to a circular orbit around the Moon. This V is called V tot. There are eight inputs to the program. The first six inputs of the program are: the altitude of LEO, which is 300 km as seen in Subsection the altitude of LLO (100, 300, 500, 1000 or 2000 km). the direction of LLO circularization: Clockwise (CW) or Counter-Clockwise (CCW). the angle α defining the direction of V 0 (the first V ) in the 2D plane of the orbit. the value of V 0. the initial angle that determines the position of the Moon with respect to the Earth in the orbital plane, called θ Minit. This angle goes along with the previous angle α, so that both can be changed to achieve the same result. Once the spacecraft is near the Moon, a second V, called V 1, needs to be applied to enter a Hohmann transfer orbit with apocenter the current altitude above the Moon and pericenter the altitude of the LLO. After half a period of this second transfer orbit, the spacecraft is at the required altitude above the lunar surface. At this point, another change in velocity, V 2, is automatically applied in order to circularize at this altitude. The total V is then calculated, and the previous calculations are done once again with different input values. The minimum allowed distance separating the spacecraft from the Moon when the second V is applied is the seventh input. One should bear in mind that allowing the spacecraft to be at a larger distance from the Moon when entering the second transfer orbit can save some propellant, however it also means that the Matlab model loses precision. Indeed, the gravitational pull of the Moon is less important when the spacecraft is far away from it and therefore the previously neglected perturbations play a more important role than if the spacecraft were closer to the Moon. The eighth input is the time when V 1 is applied to the spacecraft, named t V1. It is assumed

48 30 From LEO to LLO Using CPS that t = 0 corresponds to the moment when the spacecraft fires its engines to leave the LEO. Time t V1 has to be greater than the time needed to reach the minimum allowed distance to the Moon. Figure 3-6: Geometry of the transfer: the colors (red, then blue, then green) show the evolution of the orbits of the spacecraft and of the Moon when time is increasing. Figure 3-6 can be used to explain a bit more in detail the previous paragraph. The circles around t 1, t 2 and t 3 on the spacecraft s trajectory are an illustration of the seventh input (the minimum distance separating the spacecraft from the Moon when V 1 is applied). The times t 1, t 2 and t 3 correspond to the eighth input (the time when V 1 is applied to the spacecraft). At t = t 1, the spacecraft is too far from the Moon and no impulsive shot is applied. At times t = t 2, t = t 3 and any other time in between, the change in velocity V 1 needed to enter a Hohmann transfer orbit from the current position to LLO is calculated for either the CW or the CCW option, depending on the input chosen. Once at the pericenter of the transfer orbit, a third V is applied to circularize completely at this altitude. The total amount of V to circularize around the Moon V tot will vary depending on the time t chosen to apply V 1. The Matlab program computes all the V s and deduces the transfer requiring the smallest amount of V tot, which corresponds to the best strategy for the transfer. The two following parameters have a resolution: the direction of V 0 and the time t V1 when V 1 is applied to the spacecraft. These coefficients determine how precise the calculations are, but also how long the computations will take. Figures 3-7 and 3-8 show an example of CW circularization around the Moon at random times t 1 and t 2 > t 1. Similarly, Figures 3-9 and 3-10 show an example of CCW circularization around the Moon at random times t 1 and t 2 > t 1.

49 3-5 The in-plane three-body problem 31 Figure 3-7: Example of a CW circularization at time t 1 Figure 3-8: Example of a CW circularization at time t 2 > t 1 Once the whole range of input values has been investigated, the program keeps the option that requires the least amount of V, thus of propellant. The option found is then plotted. The program also displays all the parameters needed to be able to reconstruct the orbit, which

50 32 From LEO to LLO Using CPS Figure 3-9: Example of a CCW circularization at time t 1 Figure 3-10: Example of a CCW circularization at time t 2 > t 1 are: the total V needed, V 0 as well as its orientation in the XY-plane, V 1 and V 2 as well as the moment at which they are performed.

51 3-5 The in-plane three-body problem Calculations: inputs and results To allow a relatively small calculation time, the resolution of the direction of V 0 is chosen to be 1 degree, the one relative to the time when V 1 is applied is chosen to be 1 per 2.4 hours. In this subsection, one specific case will be investigated. The results for all cases can be found in Section 3-6. The input values for this case are taken as follows: altitude of the LEO: 300 km. altitude of the LLO: 300 km. direction of LLO circularization: CW. direction of V 0 : from 0 to 360 degrees (all the possibilities are investigated). initial angular position of the Moon in the XY-plane θ init : 180 degrees. value of V 0 : from 3083 m/s to 3164 m/s: this range of values corresponds to a range of Hohmann transfer orbits for which the apogee varies between 300,000 and 1 million km, according to Equation (3-5) (the Moon is situated 384,400 km away from the Earth [1]). minimum distance to the Moon when V 1 is applied: km (well within the SOI of the Moon, which has a radius of 66,183 km [1], so the lunar gravitational attraction can be considered the only force acting on the spacecraft). time when V 1 is applied: from 2 to 10 days. Indeed, starting from a LEO altitude of 300 km, the spacecraft needs more than 2 days to reach an altitude close enough to the Moon s; furthermore this range of values extends until 10 days to allow the spacecraft to circularize around the Moon on its way back from the apogee, in case it had (voluntarily) missed it during the first leg of its trajectory. The trajectory of the spacecraft is computed for 20 days after the first impulsive shot. After 8 hours of computation, the program ended up finding possible solutions. The best one has the following properties: direction of V 0 : 56 deg. V 0 = 3108 m/s. V 1 = 175 m/s is applied after t 1 = 5.50 days. V 2 = 535 m/s is applied after t 2 = t days = 6.84 days. The orbit resulting from the Matlab calculations is called optimized transfer orbit. This orbit can be seen in Figures 3-11, 3-12, 3-13 and 3-14, each figure corresponding to a different moment in time to show the evolution of the orbit. It can be seen that the spacecraft first passes the Moon, and only on its way back an impulsive shot is applied. In Figure 3-13 the

52 34 From LEO to LLO Using CPS Figure 3-11: Optimized transfer orbit at t = 7.00 days Figure 3-12: Optimized transfer orbit at t = 8.28 days spacecraft is already circularized around the Moon. It remains in LLO. After 20 days (Figure 3-14), the spacecraft is still orbiting the Moon and will remain there unless another impulsive

53 3-5 The in-plane three-body problem 35 Figure 3-13: Optimized transfer orbit at t = 9.81 days Figure 3-14: Optimized transfer orbit at t = days shot is applied.

54 36 From LEO to LLO Using CPS The total amount of V needed for this orbit is: V totmatlab = 3818 m/s (3-28) The values chosen for the resolutions being really coarse, the results of the calculations are not extremely precise. However, the value obtained for V tot is a first step. Increasing the resolutions would allow more possibilities to be investigated, resulting in a much bigger number of solutions. This is left as a recommendation for further work. Some of the solutions found could be better than the one found here and for that reason V tot to go from a LEO 300 km to a LLO 300 km is smaller than 3818 m/s. V tot < V totmatlab = 3818 m/s (3-29) This minimum value of V tot found by the program will be used for the rest of this study. However, it is interesting to determine the minimum value of V tot that the program could ever find. The next subsection will deal with this topic Minimum value of V tot The previous subsection gave a good estimation of the amount of V tot needed to go to LLO 300 km. With finer calculations (thus a longer calculation time), this value could probably be lowered. This subsection shows in a really simple way that this value will never go lower than 3083 m/s. While the distance Earth-Moon is 384,400 km [1], the SOI of the Moon is R SOIMoon = 66, 183 km [1]. This means that 300,000 km away from the Earth in the Moon direction, the spacecraft will experience more gravitational pull from the Earth than from the Moon. If no extra velocity change is applied, the spacecraft will never be able to approach the Moon at a smaller distance. Therefore by definition the amount of V required for this mission can never be smaller than the one needed to enter a Hohmann transfer orbit with perigee 300 km and apogee 300,000 km. This V, called V F irst maneuver, is calculated as follows [11]: V F irst maneuver = V HohmannP erigee = V CP ( 1 + e Hohmann 1) (3-30) where V CP is the circular velocity at the perigee and where e Hohmann is the eccentricity of the Hohmann transfer orbit. So: µe V F irst maneuver = ( 1 + r a r p 1) (3-31) r p r a + r p where r a and r p are the apogee and perigee distances respectively. Therefore: V F irst maneuver = µe 2(RE + h a ) ( 1) (3-32) R E + h p 2R E + h a + h p

55 3-6 Results and conclusions 37 where R E is the radius of the Earth and h a and h p are the altitudes of the apogee and perigee respectively. Equation 3-32 becomes: V F irst maneuver = ( 2( ) ) (3-33) V F irst maneuver = m/s (3-34) This proves that in any case V tot cannot be smaller than 3083 m/s. 3-6 Results and conclusions The amount of velocity change that is required to circularize 300 km above the lunar surface in a CW direction, starting from a LEO of altitude 300 km is (see Subsection 3-5-3): V tot = 3818 m/s (3-35) Doing the same calculations again for the different values of LLO altitude and for both CW and CCW circularization leads to the results found in Table 3-1. The resolutions were taken to be 1 degree for the direction of V 0 and 2.4 hours for the time when V 1 is applied. It is interesting to notice that the lower the circularization altitude, the more the amount of V needed for the transfer. Indeed, at lower altitudes, the circular velocities are higher, which leads to a greater amount of V needed to circularize at these altitudes, which translates into a greater total amount of V needed. Table 3-1: V tot from LEO to LLO with CW or CCW circularization Clockwise Counter Clockwise Altitude LLO (km) V tot (m/s) V tot (m/s)

56 38 From LEO to LLO Using CPS

57 Chapter 4 From LEO to LLO Using EPS In this chapter the orbit from Low Earth Orbit (LEO) to Low Lunar Orbit (LLO) will be computed, provided that the spacecraft uses an Electric Propulsion System (EPS). At the end of the chapter, the spacecraft will be orbiting around the Moon in polar LLO. Every piece of Matlab program developed and used for this chapter has been tested to certify that the results are correct, however those tests have not been included in this report. 4-1 Introduction As seen in the previous chapter, using a Chemical Propulsion System (CPS) allows the spacecraft to reach the Moon within a week. However, as seen in the literature survey, most of the spacecraft mass is constituted of propellant, and very little is left for the payload [1]. Using an EPS would decrease the amount of propellant needed, which would increase the total payload mass on the Moon. However, reaching the Moon would no longer be a matter of days but rather a matter of months, or even years. What are the actual savings in mass permitted by an EPS, and what are the drawbacks of such a system? How long is actually needed to circularize around the Moon starting from LEO? What electrical engine(s) would be best suited for this mission? These questions will be answered in this chapter. 4-2 Procedure In order to determine how much payload mass would be placed on the Moon using an EPS, the orbit of the spacecraft will be determined using a Matlab program developed for this purpose. In this section, the physical description of the transfer will be detailed. The Matlab

58 40 From LEO to LLO Using EPS program will make use of this description. This Matlab program is available on the separate CD Transfer orbit As seen in Chapter 2, for this chapter the spacecraft needs to circularize around the Moon in a polar orbit. The transfer will be performed in two steps, which are two successive two-body problems: a first step called apogee raising in which the spacecraft increases its distance to the Earth: during this phase the spacecraft is only influenced by the gravity of the Earth until it reaches the Sphere Of Influence (SOI) of the Moon. It is reminded that the radius of the SOI of the Moon is R SOIMoon = 66, 183 km [1]. a second step called lunar capture during which the spacecraft decreases its altitude with respect to the Moon and finishes in polar LLO: the only gravitational attraction influencing the spacecraft is the one of the Moon. The two steps of the transfer will be linked by the technique of patched conics. Even though this consideration is an approximation, it is relevant for early designs of transfer orbits [11]. In order to circularize around the Moon in a polar orbit, the apogee raising will take place in a different plane than the lunar orbital plane. For this first phase of the transfer it is assumed that the only gravitational acceleration acting on the spacecraft is the one of the Earth. It is also assumed that the problem is two-dimensional: the launcher will provide the spacecraft with the inclination needed to be in the orbital plane considered, represented in Figure 4-1. The goal of this first phase is to aim a little bit lower or higher in order to end up after the apogee raising with the Moon either above or below the spacecraft. At this point the spacecraft will be at the border of the SOI of the Moon. For simplification purposes, the altitude reached with respect to the Earth, even though slightly higher than the altitude of the Moon, will be called altitude of the Moon or lunar altitude for the rest of this study. Precise calculations concerning the plane of the apogee raising as well as the actual altitude reached will be done in Subsection At this moment, the second phase of the transfer will take place. The spacecraft being at the border of the SOI of the Moon, it is relevant to assume that the only gravitational acceleration acting on the spacecraft is the one of the Moon. With the Moon either above or below itself, the spacecraft will decrease its apocenter with respect to the Moon while remaining in polar orbit. In the end, a polar LLO will be reached. Figure 4-1 shows the complete transfer for the case for which the spacecraft aims lower than the Moon with respect to the lunar orbital plane. A similar figure could be drawn for the higher case.

59 4-2 Procedure 41 Figure 4-1: Apogee raising and lunar capture (case for which the spacecraft aims lower ) Perturbations Even though spacecraft propelled with EPS are really sensitive to perturbations due to their low thrust [1], for this study none of the perturbations is modeled by the program, including the Moon s gravitational attraction in the first leg of the transfer and the Earth s gravitational attraction in the second leg. Hence the gravity of the Moon/Earth (depending on the phase of the transfer), the Earth oblateness, the solar gravitational attraction, the gravitational attractions of the other planets of the solar system, the eccentricity of the lunar orbit, the solar radiation pressure and the atmospheric drag of the Earth are all assumed to be equal to zero. The reason is that it would be too computationally demanding to include those perturbations for the upcoming grid search. Indeed, other parameters would vary in the grid search (the position of the Moon at the beginning of the apogee raising, if only one had to be mentioned). The final result is really sensitive to the initial position of the Moon, therefore the position of the Moon would have to be varied with very small steps, which is computationally heavy. The same holds for the other perturbations. However, with a higher processing power these perturbations and especially the gravitational pull of the Moon/Earth could be included in the study. Of course, for the proper design of a mission these perturbations should be taken into account. Every single one of these perturbations influences the trajectory and the time needed to complete the transfer. But if one wants to design a first approximation of the transfer orbit in order to assess the time and propellant quantity needed, then these perturbations can be left aside. To compensate for the negligence of these perturbations, an extra 10% of the total propellant consumption will be allocated for guidance, as will be seen in Section 4-6. This is relevant because during the first-orbit estimations of SMART-1, the same amount of propellant was allocated to the guidance of the spacecraft to compensate for trajectory control, perturbations etc [1].

60 42 From LEO to LLO Using EPS Acceleration provided by the engines As seen in the literature survey [1], the thrust is assumed to be of constant magnitude. Furthermore, by applying the thrust tangentially to the trajectory, a maximum instantaneous rate of change of the orbital energy can be achieved [11]. Many studies have revealed that when the thrust is small compared to the acceleration due to gravity (which is the case for this study), a tangential thrust of constant magnitude is close to the optimum thrust profile [11]. If the thrust is applied along the velocity vector, the spacecraft will accelerate. If the thrust is applied in the opposite direction of the velocity, the spacecraft will brake. The spacecraft is regarded as a point of mass M(t). The starting condition is a circular LEO at altitude and inclination to be determined in Subsection At t=0, the mass M(0) is a function of the altitude and inclination of the LEO according to the VEGA performance chart (Figure 2-3). The exact value of M(0) depending on the LEO conditions will be computed in Subsection The Matlab program solves the orbital equations of motion using Ode45, like for the CPS. For the EPS though, the change in velocity of the spacecraft does not need to be known to compute the instantaneous acceleration and mass of the spacecraft at an instant t. The derivation of the formula for the acceleration provided by the propulsion system is as follows: a(t) = T M(t) (4-1) The instantaneous mass of the spacecraft can be rewritten as (in case of continuous and constant thrust): M(t) = M(0) ṁt (4-2) where ṁ is the mass flow of the engine, which is linked to the thrust by: Rewriting the mass flow, Equation (4-2) becomes: T = ṁi sp g 0 (4-3) M(t) = M(0) And finally incorporating Equation (4-4) into Equation (4-1): T t I sp g 0 (4-4) a(t) = T M(0) T t = I spg 0 T M(0) 1 T t M(0)I spg 0 (4-5) which can be rewritten as: a(t) = a 0 1 a 0t I spg 0 (4-6) This acceleration was incorporated in the Matlab program together with the orbital accelerations. All these accelerations determine the movement of the spacecraft.

61 4-3 Constraints and assumptions Constraints and assumptions In this section the different parameter values assumed for this study will be presented Available power The Stackable Platform System (SPS) can be regarded as a box of width and length equal to w SP S = m and of height h SP S = m. The SPS fits perfectly into the circular payload bay of the VEGA rocket. Indeed, as can be seen in Figure 4-2, the maximum diameter of the payload for the straight part of the fairing is d max = m, which corresponds to a square of width w max such that: d max = 2wmax 2 (4-7) which means that: w max = d max 2 = m > w SP S = m (4-8) Figure 4-2: Allowed payload volume for the spacecraft [3] On two opposite wings of the SPS, five solar panels will be folded (in red in Figure 2-6). For every wing of the SPS, a space of 70 mm needs to be saved for the mechanism holding the solar panels, plus a width of 30 mm per panel (22 mm of material and 8 mm of gap between

62 44 From LEO to LLO Using EPS panels). Note that the same 8 mm gap is needed between the SPS wing and the first panel. Therefore the total width added per side of the SPS is: w solar panels = = 220 mm = m (4-9) Figure 4-3 shows the geometry of the SPS as well as the solar panels as seen from above. The maximum length of a solar panel l max can be calculated as follows: which translates to: Thus: d 2 max = (w SP S + 2 w solar panels ) 2 + l 2 max (4-10) l max = d 2 max (w SP S + 2 w solar panels ) 2 (4-11) l max = m (4-12) Therefore the length of a solar panel is chosen to be l solar panel = m < l max. As seen in Figure 4-2, an extra margin is taken between the maximum allowed width of the payload and the width of the payload bay. Figure 4-3: Geometry of the SPS and solar panels in the VEGA payload module From Figure 2-6 it can be seen that the lander and the payload, the other parts of the spacecraft necessary for the mission, are mounted on top of the SPS. It is assumed that the height of a solar panel is equal to the maximum height allowed into the straight part of the payload module, namely h solar panel = m (Figure 4-2). It could be possible to attach some solar panels on the two remaining sides of the SPS. Those solar panels could either be deployed below and on top of the spacecraft, or they could be

63 4-3 Constraints and assumptions 45 attached to the solar panels already present on the two other sides of the satellite (hence the solar panels would be folded on both adjacent sides, doubling their length). However the momentum resulting from either configuration, the added mass for structure and solar panels as well as the trade-off resulting from this added mass (is it worth adding mass to shorten the transfer time?) make it an interesting but complicated topic that will not be treated in this MSc thesis. Hence only 2 sides of the spacecraft are used for the solar panels, and the total area of solar cells is: S = 2 5 l solar panel h solar panel = = m 2 (4-13) When dealing with solar cells, it is preferred to work with average conditions rather than End of Life conditions [12]. According to Airbus Defence and Space Netherlands (ADS-NL) specialists, the average power per square meter for a solar panel would be, for a typical AZUR SPACE GaAs (29 % Beginning of Life): u = 285 W/m 2 (4-14) This value of average power per unit area has been averaged for 1 to 2 years and takes into account the temperature differences from LEO to the Moon and the radiation damage from the Van Allen belts. It could be interesting to know the actual damage of the Van Allen belts on the efficiency of the solar array, especially at the end of the transfer (near the Moon). However this will be left as a recommendation for further work, as stated in Subsection The total power output (taking into account the degradation) from the solar panels is: P output = S.u = 9995 W (4-15) From this value has to be subtracted a total of 800 Watts (worst-case scenario) for the functioning of the bus (communication, attitude control, thermal control, etc.) [12]. Thus the power left is: P P CU = = 9195 W (4-16) The power P P CU is the power that goes into the Power Conditioning Unit (PCU). This power unit conditions the energy and delivers it continuously in an appropriate form during the overall mission [13]. The losses associated with the PCU are assumed to be 10% as a first approach. Therefore the power available for the propulsion system is: P P rop = 0.9 P P CU = = 8275 W (4-17) Due to the uncertainties of these calculations, the power for the propulsion system is approximated to: P P rop = 8250 W (4-18) Time constraint The spacecraft will carry a lander which will use a CPS to control the final Descent and Landing (DL) on the Moon. This propellant will not be used until the spacecraft has successfully entered an LLO around the Moon. In order to avoid engine malfunctioning during

64 46 From LEO to LLO Using EPS the DL, it is recommended to limit the complete transfer time to one year (from LEO to lunar touchdown in 365 days). Therefore it was decided to limit the transfer time of the apogee raising to 300 days. This leaves sufficient time for the spacecraft to get captured by the Moon, decrease its apocenter, descend and land on it and also allows some margin for potential coasting periods (trajectory arcs without thrusting) for accurate orbit insertions as well as some margin for system check-outs required before a new critical phase of the transfer commences. Furthermore, for this EPS transfer orbit, eclipses are ignored. When in an eclipse, the spacecraft does not have power coming from the solar array, hence cannot thrust. If the spacecraft does not thrust, no propellant is consumed during the eclipse phase. However, the eclipse phase is seen as a coasting period from the point that during that time, the spacecraft does not use its engines, thus does not modify the orbit it is flying. Eclipses hence simply increase the duration of the orbit transfer. As will be seen in the last section of this chapter, the final design allows for some margin in terms of mission duration: therefore, even if the eclipses were taken into account, the transfer time of one year would still be met Transfer geometry and constraints The geometry of the transfer can be seen in Figure 4-4. As explained in Section 4-2, the apogee raising takes place in a different plane than the lunar orbital plane (the spacecraft either aims higher or lower than the Moon). Once above or below the Moon, the spacecraft stops the thrusting that increases its apocenter with respect to the Earth. It will therefore remain at the same distance to the Earth, with its velocity vector directed orthogonally with respect to the Earth-spacecraft axis, and thus parallel to the velocity vector of the Moon (because the Moon is just above or below). At this point, the spacecraft will be at the border of the SOI of the Moon and will be captured by its gravity before decreasing its altitude in a two-dimensional transfer orbit orthogonal to the Earth-Moon axis. As seen in the literature survey, the obliquity of the Moon is 6.68 degrees [1]. For this EPS transfer it will be neglected (assumed 0) and the spacecraft will circularize around the Moon in polar orbit even though this polar orbit can be 6.68 degrees off. In Figure 4-4 it looks like the spacecraft crosses the SOI of the Moon before being exactly below it. Actually, this is because the figure is two-dimensional. When the apocenter of the spacecraft with respect to Earth is raising, the Moon is at a different place on its orbit and is therefore not just above the spacecraft. Even though not treated in this study, the geometry of the transfer is computed such that the spacecraft and the Moon are exactly aligned vertically (from the point of view of the lunar orbital plane) at the end of the apogee raising. It is reminded that the VEGA launcher is not able to place the spacecraft in a Geostationary Transfer Orbit (GTO). Instead, the spacecraft has to start from LEO, at an altitude varying from 300 to 1500 km. The lower the altitude the higher the mass of the spacecraft and therefore the more payload it can take on board (see Subsection 2-2-1). However, the lower

65 4-3 Constraints and assumptions 47 Figure 4-4: Geometry of the transfer and important parameters the altitude the longer the spacecraft needs to reach the altitude of the Moon. And with a highly restricting time constraint of 300 days for the apogee raising, even 100 kilometers at the altitude of the parking orbit can be the difference between an acceptable mission design and an unacceptable one. For simplification purposes, it is assumed that the correct inclination of the orbital plane is provided by the VEGA launcher. The problem can therefore be considered two-dimensional. Following the same logic as in Section 3-4, because the launch date has not been determined, the exact inclination change that needs to be performed by the VEGA launcher is unknown. Therefore the worst-case scenario is envisaged, which is when the inclination change that needs to be performed is maximal. From Section 3-4 it is reminded that the inclination change that needs to be performed from the launching site to the lunar orbital plane obeys: deg i Kourou Moon deg (4-19) For the apogee raising, the orbit of the spacecraft will occur in a different plane than the lunar orbital plane. More precisely, as seen in Figure 4-4, the angle between the orbital plane of the apogee raising and the lunar orbital plane is: Thus: i Moon apogee raising = arctan( R SOI Moon 66, 183 ) = arctan( d Earth Moon 384, 748 ) (4-20) i Moon apogee raising = 9.76 deg (4-21) Because it can be chosen to aim lower or higher than the Moon, the value of the angle i Moon apogee raising can either be added to or subtracted from the inclination change i Kourou Moon. As can be seen, subtracting this angle yields a lower value for the inclination change provided by VEGA. The worst-case scenario becomes: i Kourou apogee raisingmax = = deg (4-22)

66 48 From LEO to LLO Using EPS For the rest of the study it is assumed that to place the spacecraft into LEO, VEGA performs the inclination change of i Kourou apogee raising = 24 deg. The mass of the spacecraft M(0) at t=0 can therefore be determined. This mass M(0) depends on the altitude of the parking orbit around the Earth as can be seen in the VEGA performance chart (Figure 2-3) [3]. In order to limit the number of orbit designs to be considered, and hence limit the total computation time, only the 5 values of LEO present in the VEGA performance chart will be considered here. Therefore the mass of the spacecraft at t=0 is: M 300km (0) = 2250 kg (4-23) M 500km (0) = 2130 kg (4-24) M 700km (0) = 1980 kg (4-25) M 1200km (0) = 1670 kg (4-26) M 1500km (0) = 1480 kg (4-27) The distance d apogee raising that the spacecraft has to reach in order to be above or below the Moon can also be determined. Using Figure 4-4: d apogee raising = RSOI 2 Moon + d 2 Earth Moon = km (4-28) which is less than 6000 km more than the distance Earth-Moon Constraints on the engines As seen in the literature survey, electrical propulsion transfers are a matter of months or years [1]. One way to accelerate the transfer is to use an engine (or several) that can provide as much thrust as possible. However, the more thrust the engine can provide the more power it needs to function. As seen in Subsection 4-3-1, a maximum of 8250 W is available for propulsion. Since this mission would be European, the engine selected for the transfer has to be European. Therefore only European engines are considered in this study. Furthermore, for several reasons (design, cost, volume accommodation...) it was decided that a maximum of 4 engines could be mounted on the spacecraft. It means that as long as the power requirements are met, more engines can be added to the propulsion system to accelerate the transfer, with a maximum of 4 engines. Table 4-1 lists the engines having a thrust greater than 80 mn and a power requirement smaller than 8250 W that are manufactured in European companies. The companies looked at are the main manufacturers of electrical engines, namely Airbus Defence and Space, Thales and Snecma (from the Safran group). As can be seen from the table, some engines have a value of total impulse too small to be able to thrust continuously for one year. For instance,

67 4-3 Constraints and assumptions 49 the PPS-1350-E by Snecma is only able to thrust for 215 days. For this study however, the engines will be assumed to be of then-technology, meaning that it is supposed that in a few years (when the mission will take place) the upgraded versions of these engines will be able to thrust continuously for a year (or as long as needed for this study). Table 4-1: European Electric Engines [14] [15] [16] [17] [18] [19] Name Power Thrust Isp Mass Total Impulse Thrust Time (kw) (mn) (s) (kg) (MNs) (days) PPS-1350-G (Snecma) PPS-1350-E (Snecma) PPS-5000 (Snecma) ? TMA (Snecma) RIT-22 (Airbus Defence and Space) ??? RIT-XT (Airbus Defence and Space) ??? HEMP-T 3050 (Thales) ??? HEMP-T (Thales) ?? With an available power of 8250 W, several engines can be used together, as long as the number of engines does not exceed four. Table 4-2 lists, for each engine, the thrust and if applicable the tuned-down thrust provided by the maximum number of engines that can be used. Because for the PPS-1350-E, the PPS-5000, the RIT-22 and the RIT-XT, the input power is slightly over the maximum allowed, the "tuning-down" of these engines was considered. For the HEMP-T however, the input power of two engines would have almost been twice the maximum allowed and for that reason, the tuning-down of this engine was not considered. Tuning down an engine means not using it at its full range of capabilities. It is assumed that for values relatively close to 100% of the intended power input (second column of Table 4-1), thrust is a linear function of the power input. Therefore tuning down an engine means in this case using it with only 80 or 90% of the input power, resulting in a thrust equal to 80 or 90% of the nominal thrust. Furthermore it is assumed that the tuning down of an engine has no influence on the specific impulse of that engine. Table 4-2: Characteristics when maximum number of engines used Maximum number of engines Power Tuned-down Isp Thrust Tuned-down power thrust (kw) (kw) (s) (mn) (mn) PPS-1350-G (Snecma) x PPS-1350-E (Snecma) x PPS-5000 (Snecma) x TMA (Snecma) x RIT-22 (Airbus Defence and Space) x RIT-XT (Airbus Defence and Space) x HEMP-T 3050 (Thales) x HEMP-T (Thales) x

68 50 From LEO to LLO Using EPS 4-4 Apogee raising In this section, only the apogee raising (the transfer from the parking orbit to the lunar altitude) is computed. This means that the transfer time is the time needed from LEO to the altitude of the Moon, excluding the lunar capture and the altitude lowering. Furthermore, if any term such as mass left at the end of the transfer or mass left at the altitude of the Moon is encountered in this section, it will mean the mass left at the end of the apogee raising, before the lunar capture is started (the mass left refers to the total spacecraft mass minus the mass of the propellant already used for the transfer). At the end of the section, the spacecraft will be at the border of the SOI of the Moon, ready to start the lunar capture and to decrease its apocenter with respect to the Moon The spiral out orbit Figure 4-5 shows the orbit of the spacecraft as powered by an EPS. As seen in Subsection 4-2-3, the thrust is directed along the velocity vector in order to accelerate, hence to raise the apogee of the orbit. The orbit of the spacecraft is spiraling away or spiraling out from the central body, the Earth. The apogee is raised until the altitude of the Moon is reached. The Moon has deliberately been hidden to show that its gravity is not taken into account for this part of the transfer. For this figure, a random set of parameters was used (thrust, specific impulse, altitude of parking orbit). The influence of these parameters will be studied in the following subsection Influences of thrust, specific impulse and altitude of parking orbit on the transfer time In this subsection the influence of each one of the following parameters on the transfer time of the apogee raising is studied: thrust, specific impulse and LEO altitude. Grid search Assuming a constant thrust along the velocity vector all along the transfer from LEO to the altitude of the Moon, a grid search was made in order to determine the influence of the thrust magnitude, the specific impulse and the altitude of the parking orbit on the transfer time. The grid search was performed with the thrust varying from 200 to 600 mn and the specific impulse from 1500 to 4500 seconds, which corresponds approximately to the ranges of thrust and specific impulse that are suitable for our case (as was seen in Table 4-2). The altitude of the parking orbit was varied in accordance with the performances of the VEGA launcher as can be seen in the VEGA performance chart (Figure 2-3).

69 4-4 Apogee raising 51 Figure 4-5: Typical apogee raising of a spacecraft powered by EPS Figure 4-6 displays the time needed for the spacecraft to reach the altitude of the Moon as a function of these parameters. As explained in Section 4-2, the lunar gravitational attraction is neglected. This means that the gravity of the Moon could play an important role in the escape of the spacecraft from the Earth through lunar resonances, especially at the end of the apogee raising. Figure 4-6 displays then what can be called a "worst-case scenario", where the Moon does not help in the apogee raising. Therefore in real life, the transfer times are expected to be slightly shorter (in the order of a few days). The different colors represent the different LEO altitudes; the different symbols are for the specific impulses and the thrust is varied on the X-axis. On the Y-axis is plotted the time in days. In total 6 hours of computation were needed to obtain these results. As can be seen, when the thrust increases the transfer time decreases. This is logical since at any time t, thrust and acceleration are linked to the spacecraft mass by the following formula: a(t) = T M(t) (4-29) Therefore the acceleration increases when the thrust increases, leading to a faster transfer.

70 52 From LEO to LLO Using EPS Figure 4-6: Grid search: study of the influence of thrust, specific impulse and altitude of parking orbit on the transfer time from LEO to the lunar altitude. Furthermore, the transfer time is shorter when engines with smaller specific impulses are used. Indeed, even though the specific impulse needs to be maximized in order to save propellant, the smaller it is the greater the acceleration of the spacecraft is (see Equation (4-6)), resulting in a faster transfer. It can also be noticed that the transfer time is greatly influenced by the altitude of the parking orbit. More precisely, if a spacecraft A is placed at a high altitude (for instance 1500 km), it will take a certain time to reach the altitude of the Moon; placing another spacecraft B at a lower altitude (300 km for instance) means that it will first need to raise its altitude by 1200 km before being at the same starting point as spacecraft A. Conclusions on influence of parameters As can be seen from Figure 4-6, although thrust and specific impulse both depend on the engine, the absolute thrust has way more influence on the transfer time than the specific impulse. Therefore in order to meet the one-year transfer requirement, the absolute thrust level needs to be increased as much as possible. However engines with a higher thrust level also require a greater power input, and as has been derived in Subsection the power input for the propulsion system is limited to 8250 W. Another way to achieve a higher thrust would be to increase the number of engines used (as long as it is not greater than 4), but here again, the power input requirement of 8250 W is limiting. It should not be forgotten that non-technological requirements can also play a role in the mission design: engines are costly

71 4-4 Apogee raising 53 devices so every addition should be properly justified. To meet the apogee raising time requirement of 300 days, it is also possible to increase the altitude of the parking orbit. However, as seen in Subsection 4-3-3, this would trigger a lower initial spacecraft mass, which ultimately would lead to a lower payload mass on the surface of the Moon. The answer is not easy and an optimization needs to be made. The goal is to keep the configuration (engine, LEO altitude) meeting the time requirement of 300 days, that maximizes the mass left at the altitude of the Moon. This configuration will then be used for the lunar capture. The selection of the best configuration will be made in the next subsections First engine selection: transfer time Using the Matlab program, the transfer time to the lunar altitude was determined for every engine of Table 4-1. For every engine, several cases were taken into account: the case for which only one engine is used, the case for which two engines are used, etc. until the case for which the maximum number of engines (tuned-down if necessary) are used as determined in Table 4-2. For each case, the five altitudes of parking orbit allowed by the VEGA launcher were studied. It is recalled that the transfer time for the apogee raising is limited to 300 days, as seen in Subsection Figure 4-7: Transfer time from LEO to the lunar altitude with the European engines In Figure 4-7, the colors represent the different LEO altitudes and the symbols represent the different European engines. The different number of engines used (the different cases) can be

72 54 From LEO to LLO Using EPS found when moving along the X-axis, representing thrust. For instance, the stars at 280 mn represent the case for which 2 PPS-1350-E engines are used, the stars at 420 mn are for the case for which 3 PPS-1350-E engines are used and the stars at 462 mn are for the case where 4 of these engines are used (tuned-down). Therefore every symbol on the graph (every "dot") represents a given parking altitude, a specific engine and a given number of these engines. A total of 3.5 hours of Matlab computations was needed to obtain this plot. As can be seen, only some of the configurations for which the total thrust is above 300 mn qualify for the time limit. The RIT-22, the RIT-XT, the HEMP-T-3050 and the HEMP-T are therefore discarded for the rest of the study. The shortest transfer time is achieved with 2 PPS-5000 engines (tuned-down) starting from 1500 km altitude and is approximately 175 days. Only the engines that allow for a transfer shorter than 300 days are selected for the rest of the study, namely the PPS-1350-G, the PPS-1350-E, the PPS-5000 and the TMA, all four by Snecma. In addition, depending on the engine used, the spacecraft will not be able to be launched at any altitude but will have to respect the altitude(s) that meet(s) the maximum allowed transfer time of 300 days (see Figure 4-7). The configurations (engine(s) and LEO altitude) that qualify for the next selection can be found in Table 4-3. Table 4-3: Configurations qualifying for the second engine selection Name of engines Number of engines LEO altitudes (km) PPS-1350-G , 1500 PPS-1350-E 3 700, 1200, 1500 PPS-1350-E (tuned-down) 4 500, 700, 1200, 1500 PPS PPS-5000 (tuned-down) 2 300, 500, 700, 1200, 1500 TMA , Second engine selection: mass at lunar altitude The goal of this study is the maximization of the payload mass on the Moon. Besides, it is important to notice that the larger the mass at the end of the first part of the transfer, the larger the payload mass on the Moon. Thus for the engines and altitudes selected (so as long as the transfer time meets the requirements), the mass left at the lunar altitude needs to be maximized. The second selection phase therefore aims at choosing the combination of engine and LEO altitude that maximizes the mass left at the lunar altitude, allowing for the largest mass of payload. The mass left at the altitude of the Moon can be calculated as follows: M left = M wet M prop (4-30)

73 4-4 Apogee raising 55 where M wet is the total mass of the spacecraft when delivered in LEO by the VEGA launcher, so this value varies from 1480 kg and 2250 kg depending on the altitude; and where M prop is the propellant mass used until this altitude (it is different than the total propellant mass which includes for instance the propellant mass for the lunar capture and altitude lowering). The mass left at the end of the apogee raising can be determined using Equation (4-4). It is therefore: M left = M(t A ) = M(0) T t A I sp g 0 (4-31) where t A is the time at which the spacecraft has finished its apogee raising. Figure 4-8: Mass left at the altitude of the Moon for the selected engines and for different parking orbit altitudes Using the engines and corresponding LEO altitude(s) selected in the previous subsection, the mass left at the altitude of the Moon was determined using the Matlab program and is shown in Figure 4-8. As can be seen in the figure, only when the spacecraft is parked at 300, 500 or 700 km altitude is the mass left larger than 1300 kg. It is also interesting to notice that the mass left at the altitude of the Moon varies between less than 1000 kg and almost 1700 kg. The configuration leading to the highest mass left at the altitude of the Moon is when two tuned-down PPS-5000 are used and when the altitude of the parking orbit is 300 km. This configuration is therefore selected for the rest of this study. At the lunar altitude, the spacecraft: has spent 284 days raising its apogee. now has a mass of 1679 kg.

74 56 From LEO to LLO Using EPS now has an absolute velocity V SC = 1049 m/s. These results were obtained with Matlab. The velocity of the spacecraft is slightly higher than the circular velocity at the altitude of the spacecraft, which is: muearth V circapogee raising = = 1010 m/s (4-32) r apogee raising For that reason the spacecraft will not enter a circular orbit around the Earth if the thrust is stopped but rather a slightly elliptical orbit (ellipse resembling a circle). However, for this preliminary mission design this slight difference will be ignored and the velocity of the spacecraft will be assumed to be exactly the one needed to circularize around the Earth at the altitude of the apogee raising (in direction and intensity). Therefore the velocity of the spacecraft is V SC = 1049 m/s. Figure 4-9: Distance from the center of the Earth as a function of time Figure 4-9 displays the evolution of the apogee raising as a function of time for the chosen configuration. It is interesting to notice that due to the small acceleration of the EPS compared to the Earth gravitational attraction, the spacecraft takes more than 200 days to raise its apogee to 50,000 km while the (almost) 350,000 km remaining are achieved in less than half that time. The nature of this transfer brings the question to the time spent in the Van Allen belts, which could strongly damage important parts of the spacecraft Time in Van Allen belts As seen in the literature survey, the Van Allen belts are areas encircling the Earth in which there are relatively large numbers of high-energy charged particles. There are two belts: an

75 4-4 Apogee raising 57 inner belt which extends from 1000 to 6000 km above the Earth and an outer belt which starts at approximatively 10,000 km and finishes at 65,000 km above the Earth [1]. The greatest intensity of the Van Allen belts, hence its most damaging part for the spacecraft ranges from 14,500 to 19,000 km [1]. Therefore the time spent by the spacecraft in the Van Allen belts and more specifically this most-damaging area should remain below an acceptable level. Figure 4-10: Distance of the spacecraft from the center of the Earth with respect to time, starting from LEO In Figure 4-10 is plotted the altitude of the spacecraft with respect to time for the chosen configuration: the figure is a zoom-in of Figure 4-9. From this figure, it is possible to see that it takes nearly 240 days to exit the outer Van Allen belt, which is significant. However the 4500 km of the most damaging part of the Van Allen belts, corresponding roughly to the range 21,000-25,500 km on the figure (distance from the center of the Earth), are crossed by the spacecraft in approximately 20 days. The spacecraft will therefore spend less than a month in this most-damaging area, which is considered acceptable. Because of the extended amount of time spent in both Van Allen belts, the efficiency of the solar cells will be reduced over time due to severe radiation damage. Therefore the available power per unit area will decrease during the transfer. However, the average power provided by the solar cells had already been compensated for such a loss, as stated in Subsection Therefore no design changes result from this analysis. This consideration is however reflected on the mass of the solar array, as will be seen in Chapter 7. A thicker cover glass is used for radiation protection, leading to a heavier mass per area. It could be interesting to look into the actual damage of the Van Allen belts on the solar panels depending on the radiation intensity and to deduce the exact power per square meter as a function of time along the transfer. This is left as a recommendation for further work.

76 58 From LEO to LLO Using EPS 4-5 Lunar capture This section deals with the capture, orbit lowering and circularization of the spacecraft around the Moon. As seen in the previous section, at the end of the apogee raising the spacecraft is situated at the border of the SOI of the Moon, thus 66,183 km below the Moon. At this point, the gravitational pull of the Earth is ignored and only the lunar gravitational attraction is taken into account. The first step of the transfer (the apogee raising) is patched to the second part of the transfer (the orbit lowering studied in this section) with the technique of patched conics [1]. For this section only, the Y-axis is chosen to be pointing from the position of the spacecraft at the beginning of the orbit lowering phase to the Moon; the Z-axis is chosen to be pointing from the position of the spacecraft at the beginning of the orbit lowering phase to the Earth; the X-axis is chosen to be such that XYZ is a Cartesian coordinate system. The beginning of the lunar capture starts 284 days after the beginning of the apogee raising. At this moment, the spacecraft has a mass of 1679 kg. It is reminded that two tuned-down PPS-5000 are used as engines (see Subsection 4-4-4). Figure 4-11: Capture by the Moon with no thrust

77 4-5 Lunar capture 59 It is assumed that the velocity of the spacecraft with respect to the Earth is contained in the same plane as the velocity of the Moon, as seen in Subsection Therefore the second value is subtracted from the first one and the initial velocity of the spacecraft with respect to the Moon can be determined. Because the initial position of the spacecraft is also known, it is possible to draw the orbit that the spacecraft would fly around the Moon if no thrust were applied. This orbit is represented in Figure As can be seen, the spacecraft is captured by the Moon and the orbit flown is an ellipse with apocenter the initial position of the spacecraft (at the border of the SOI of the Moon) and with a pericenter situated more or less 8000 km above the lunar surface. In Figure 4-11 as well as in every other figure showing an orbit in this section, the orbit is seen from the Earth s point of view. Hence the orbital plane considered in the figure is moving around the Earth and its velocity vector is directed towards the -X axis. As seen in the literature survey, for the European mission SMART-1, the lunar capture occurred 60,000 km above the lunar surface and cost no propellant [1], which is exactly what happens in this case. Therefore it is relevant to think that the lunar capture for the LOFAR on the Moon mission is correct. Similar to the SMART-1 mission, the engines will be used in braking to lower the altitude of the spacecraft. Figure 4-12: Capture by the Moon if a constant braking thrust is applied Figure 4-12 shows the orbit of the spacecraft if a constant thrust opposite to the velocity vector is applied. As expected, the apocenter of the orbit is lowered. However, the pericenter is also lowered and during the second revolution, the spacecraft crashes on the surface of the Moon. Therefore it is important to consider raising the pericenter to avoid such a crash. Figure 4-13 shows what would happen if a constant thrust along the velocity vector would be applied. As can be seen, the pericenter is raised as compared to both previous figures.

78 60 From LEO to LLO Using EPS Figure 4-13: Capture by the Moon if a constant accelerating thrust is applied However, by keeping a constant thrust along the velocity vector, the velocity of the spacecraft and the altitude of the apocenter increase such that after a little bit more than half a revolution, the spacecraft escapes from the gravitational pull of the Moon. Twelve days after the beginning of this maneuver, the spacecraft is situated more than 150,000 km away from the Moon. It is therefore obvious that an optimization technique needs to be used in order to lower the spacecraft altitude using the minimum amount of propellant and in the shortest time possible. This optimization is left as a recommendation for further work. In this section, the lowering and circularization of the spacecraft will be executed in the following manner: During the first 20 days: if the distance to the Moon is greater than the semi-major axis of the instantaneous orbit flown by the spacecraft, then the spacecraft thrusts in the direction of its velocity to increase the altitude of the pericenter of the instantaneous orbit. if however the distance to the Moon is shorter than the semi-major axis of the instantaneous orbit flown by the spacecraft, then thrust is applied in the direction opposite to the velocity of the spacecraft in order to decrease the altitude of the apocenter of the instantaneous orbit. The semi-major axis of the instantaneous orbit flown by the spacecraft can be calculated as follows [8]: 1 a = 2 r V (4-33) 2 µ Moon

79 4-5 Lunar capture 61 with r the instantaneous distance to the center of the Moon and V the instantaneous velocity of the spacecraft with respect to the Moon. Figures 4-14 and 4-15 show the evolution of the spacecraft orbit after 13 and 20 days respectively. After 20 days, the orbit of the spacecraft is nearly circular (Figure 4-15). For the rest of this study, it will be assumed completely circular. Thus from this point on, constant thrust is applied in the direction opposite to the velocity to decrease the altitude of the spacecraft with respect to the Moon (the spacecraft is spiraling-in ) until the chosen LLO altitude is reached (100, 300, 500, 1000 or 2000 km). Figure 4-16 shows the complete circularization and lowering of the orbit of the spacecraft until an LLO altitude of 100 km is reached. Figure 4-14: Evolution of the circularization around the Moon in 13 days The lowering of the altitude of the spacecraft can also be seen in Figure 4-17, which plots the evolution of the distance to the surface of the Moon with respect to time, until an altitude of 100 km is reached. Once at this LLO altitude, the mass left of the spacecraft is 1566 kg days are necessary to achieve the orbit lowering to this specific altitude. Stopping the process at all chosen LLO altitudes leads to the results found in Table 4-4. As can be seen, less than 60 days are needed to circularize around the Moon at any chosen altitude. V circ refers to the circular velocity of the altitude considered. It is calculated as follows: V circ = µmoon r µ Moon = (4-34) R Moon + h LLO

80 62 From LEO to LLO Using EPS Figure 4-15: Evolution of the circularization around the Moon in 20 days Figure 4-16: Evolution of the complete circularization around the Moon with a LLO altitude of 100 km where R Moon is the radius of the Moon and h LLO the LLO altitude. In Table 4-4, the difference indicates the difference in percent between the spacecraft velocity and the circular velocity at the considered altitude. As can be seen, this difference is negligible

81 4-6 Results 63 Figure 4-17: Evolution of the distance of the spacecraft to the surface of the Moon for the complete orbit lowering until an LLO altitude of 100 km Table 4-4: Results of the lunar descent for the different LLO altitudes Altitude LLO (km) Mass left (kg) Transfer time (days) Velocity (m/s) V circ Difference (%) and is probably due to the approximation made earlier regarding the fact that after 20 days of circularization, the orbit was assumed completely circular. The spacecraft is now circularized around the Moon at any chosen altitude (100, 300, 500, 1000 or 2000 km). 4-6 Results As seen in Section 4-4, two tuned-down PPS-5000 are used for the EPS transfer. The altitude of the Earth parking orbit is chosen to be 300 km. After spending 284 days raising its apogee, the spacecraft has a mass of 1679 kg and is ready to start the lunar capture and orbit lowering. The spacecraft needs between 39.6 and 56.2 days to circularize at the chosen LLO altitude, and at this altitude its mass left is between 1566 and 1599 kg.

82 64 From LEO to LLO Using EPS Figure 4-18 shows the entire orbit of the spacecraft, from LEO at an altitude of 300 km to LLO at an altitude of 300 km. To simplify the reading and understanding of the orbit, Figure 4-18 is in a rotating frame (the Moon appears immobile with respect to the Earth). In theory, during the lunar capture and orbit lowering, the Moon would have the time to achieve one to two revolutions around the Earth. The cover of this MSc thesis also represents the same transfer orbit as Figure 4-18, but seen from different angles. Figure 4-18: Orbit transfer of the spacecraft propelled by an EPS The starting mass of the spacecraft (wet mass) is 2250 kg as seen previously in this chapter. This means that between 651 and 684 kg of propellant are needed for the complete transfer from LEO to LLO. As mentioned in Subsection 4-2-2, an extra 10% of propellant mass should be allocated for orbit control. After taking care of those extra 10% of propellant mass, the overall results for the EPS transfer can be found in Table 4-5. The mass left indicates the wet mass minus the propellant mass used to reach the LLO altitude. Table 4-5: Results of the EPS transfer for the different LLO altitudes Altitude LLO (km) Total propellant mass (kg) Mass left (kg) Total transfer time (days) As mentioned in Subsection 4-3-2, the eclipses were ignored for this transfer. This means that the actual transfer time will be longer than the one calculated in this chapter. However, depending on the LLO altitude chosen, the transfer is completed in a time ranging from 324 to 340 days, leaving between 25 and 41 days of margin for the mission requirement of one year. It is assumed that with this more or less one month margin, even if eclipses were taken into account, the time requirement of one year would still be met.

83 Chapter 5 From LLO to the Lagrangian Point L2 In this chapter, the transfer orbit to go from Low Lunar Orbit (LLO) to the Lagrangian point L2 will be designed. After an introduction, the theorem of image trajectories will be introduced: this theorem will shape the calculations of the chapter. Afterwards, the orbit from L2 to LLO will be computed, followed by the determination of the trajectory from LLO to L2 (the image orbit). In the last section the overall results will be computed. Every piece of Matlab program developed and used for this chapter has been tested to certify that the results are correct, however those tests have not been included in this report. 5-1 Introduction As seen in Chapter 2, this part of the study aims at determining the orbit needed to go orbiting around the L2 point, starting from LLO. For this study, the trajectory will be computed from LLO to L2 itself. Although not entirely correct, it is assumed that the amount of V and hence of propellant needed to go to L2 are the same as if the spacecraft enters an orbit around the Lagrangian point. The computation of the proper trajectory is left as a recommendation for further work. Furthermore, this study begins with the spacecraft orbiting in a circular LLO in the lunar orbital plane. By definition, the L2 point is situated in that same plane. Therefore the trajectory to be designed also lies in that plane. These considerations lead to a two-dimensional trajectory, which is less complicated to compute than a three-dimensional one. It is also assumed that a Chemical Propulsion System (CPS) is used for this part of the transfer. Indeed, going from LLO to L2 using an Electric Propulsion System (EPS) is very complicated and will therefore not be treated in this MSc thesis, as already discussed in Subsection

84 66 From LLO to the Lagrangian Point L2 5-2 Theorem of image trajectories The theorem of image trajectories was established in 1961 by A. Miele. It states that when it is assumed that the scope of the study is the restricted three-body problem in the Earth- Moon space (Earth, Moon, spacecraft), if a trajectory can be flown in that system then three image trajectories are also possible [20]. For this chapter, what is important to consider is the following part of the theorem: If a trajectory is feasible in the Earth-Moon space, then its image with respect to the Earth- Moon axis is feasible provided it is flown in opposite direction. For this chapter, the scope of the study is the trajectory computation from LLO to L2. Figures 5-1 and 5-2 illustrate with an appropriate example the way in which the theorem works for this case: if a trajectory from L2 to LLO is feasible, then its image orbit would be the trajectory from LLO to L2. Figure 5-1: Theorem of Image Trajectories: L2 to LLO and LLO to L2 trajectories as seen with respect to a fixed Earth-Moon axis.

85 5-3 Procedure 67 Figure 5-2: The theorem of Image Trajectories: L2 to LLO and LLO to L2 trajectories as seen in the inertial reference frame. 5-3 Procedure The use of the theorem of image trajectories makes it easier to design the transfer orbit to L2 using Matlab. Indeed, it is more difficult to design a transfer orbit to reach an unstable equilibrium point than to reach an orbit around a body with a strong gravitational pull (the Moon). Therefore, based on the theorem of image trajectories, it is possible to compute the trajectory from L2 to LLO and then to take its image with respect to the Earth-Moon space to obtain the required transfer orbit. However, this theorem does not provide any information regarding the quantity of V that is needed for the image orbit. A verification of the image orbit will therefore need to be performed, and the correct amount of V will be deduced from there. The procedure to compute the trajectory from L2 to LLO is as follows: Starting from L2, a first impulsive shot is applied, perpendicular to the Earth-Moon axis. It is called V L2transferrange. If at some point the distance to the surface of the Moon becomes equal to the required LLO altitude, a second impulsive shot is applied to circularize around the Moon in a Clockwise (CW) or Counter-Clockwise (CCW) direction, depending on what has been

86 68 From LLO to the Lagrangian Point L2 chosen.the time elapsed from the beginning is called t LLO. This impulsive shot is called V LLOtransfer. The spacecraft is now orbiting around the Moon at the required LLO altitude in a CW or CCW direction. After some time, the spacecraft is exactly between the Earth and the Moon (the Earth, the spacecraft and the Moon are aligned). This time elapsed between the second impulsive shot and the moment when the three bodies are aligned is called t aligned. At this point, the return trajectory, also called the image orbit can be computed: Starting from the end-position of the previous transfer orbit (spacecraft circularized at the required LLO altitude, exactly between the Earth and the Moon), nothing is done for the amount of time t wait = t aligned. At this moment (after waiting for t wait ), the spacecraft is positioned at the symmetric of the point of LLO circularization with respect to the Earth-Moon axis. A third impulsive shot is then applied, called V LLOimagerange : the orientation of this impulsive shot is the symmetric of the one of V LLOtransferrange with respect to the Earth-Moon axis but the value of the impulsive shot is varied in the range 0.95 V LLOtransferrange < V LLOimagerange < 1.05 V LLOtransferrange. The orbit is then integrated for a time t integ = t LLO. Afterwards, the distance to L2 is computed for every possible V LLOimagerange and the value for which the spacecraft is the closest to the L2 point is determined. This value is called V LLOimage. For the chosen value V LLOimage, the spacecraft is now at the L2 point. At this moment, another impulsive shot V L2image is applied to remain at L2. The value of the impulsive shot is V L2image = V L2transferrange but the direction is the opposite of the one of the first impulsive shot V L2transferrange. As suggested by its name, the first impulsive shot applied V L2transferrange can vary within a range of values. The previous calculations are therefore repeated for this whole range of values and in the end, the value minimizing the total quantity of V needed for the image trajectory V totalimage = V LLOimage + V L2image is determined. This value is called V L2transfer. The resulting values V LLOimage, V L2image and therefore V totalimage are the solution to the transfer orbit from LLO to L2 for the given LLO altitude and direction of circularization. This procedure is then repeated for every LLO altitude considered and for both directions of circularization: CW and CCW. Figure 5-3 displays the minimum distance to the center of the Moon as a function of V L2transferrange. As can be seen, for values between 130 and 375 m/s the spacecraft gets closer than approximately 3800 km, which corresponds to an altitude of 2000 km. The range of values investigated in this chapter is therefore: 130 m/s < V L2transferrange < 375 m/s. it is interesting to notice that for 500 m/s and higher, the closest distance to the Moon is achieved in the beginning, at the L2 point. The impulsive shot is too strong for the lunar

87 5-4 Computations 69 Figure 5-3: Minimum distance to the center of the Moon (in m) as a function of V L2transferrange (in m/s) gravitational attraction and the spacecraft exits the Sphere Of Influence (SOI) of the Moon. The Matlab program used for this transfer orbit is based on the one described in Section 3-5. It is available on the separate CD. 5-4 Computations As seen in the previous section, the transfer orbit will be computed in two steps: first, the transfer from L2 to LLO and then its image orbit (from LLO to L2) From L2 to LLO Starting from the L2 point, the spacecraft enters a trajectory that is supposed to finish in a circular orbit around the Moon at a given altitude (100, 300, 500, 1000 or 2000 km) and at a given direction of circularization (CW or CCW). Figure 5-4 shows the spacecraft at the L2 point, without any orbit control for 7 days. As can be seen, the distance to the Moon is very slowly decreasing (a drop is present in the plot on the right-hand side of the figure), which is believed to be due to Matlab s numerical approximations. Besides this slight imperfection and even though the L2 point is known to be unstable, it is clear that once the spacecraft is at the L2 point, it can remain there provided

88 70 From LLO to the Lagrangian Point L2 that it is equipped with an orbit control system which would counteract the instabilities of the orbit. Figure 5-5 displays the orbit transfer from L2 to LLO. In the figure, the specific example of a CCW circularization at 300 km altitude is taken. However, this figure is also characteristic of every other LLO circularization considered in this section. The time needed for the spacecraft to go from L2 to LLO is approximately 4 days, depending on the LLO altitude chosen (a few more hours are needed to circularize at an altitude closer to the Moon). Figure 5-4: Spacecraft at the L2 point for 7 days Figure 5-5: Trajectory from L2 to LLO with CCW circularization at 300 km altitude The results of the computations can be seen in Table 5-1. It is interesting to notice that the

89 5-4 Computations 71 lower the circularization altitude the more the V needed. This is mainly due to the higher circular velocity of lower orbits. Another interesting phenomenon is the fact that CCW circularization is more propellant efficient than CW circularization, which is likely to be due to the geometry of the three-body problem (the Moon rotates around the Earth in the CCW direction). Table 5-1: V s from L2 to LLO with CW or CCW circularization Clockwise Altitude LLO (km) V L2transfer (m/s) V LLOtransfer (m/s) V totaltransfer (m/s) Counter Clockwise Altitude LLO (km) V L2transfer (m/s) V LLOtransfer (m/s) V totaltransfer (m/s) The image orbit For this subsection, the theorem of image trajectories is used. For a given LLO altitude and direction of circularization, the spacecraft flies the image of the trajectory computed in the previous subsection with respect to the Earth-Moon axis. As seen in Section 5-3, the theorem of image trajectories tells nothing about the quantity of V needed for the image orbit. Therefore as explained in that same section, the first impulsive shot of the image orbit is varied from 95 to 105% of its original value (the transfer orbit) to determine the exact amount of V that is needed to place the spacecraft at the L2 point. However the second impulsive shot applied to remain at L2 has the same value as the original. Figure 5-6 shows the total orbit as computed by the Matlab program. Starting from L2, the spacecraft circularizes at LLO (in this case at 100 km altitude in a CCW direction). Once in LLO, it completes less than one revolution before going back to L2 and remaining there, as can be seen in the bottom part of the figure. It is important to notice that only the second leg of the transfer (from LLO to L2) is of interest for this study. The first leg is only here to make the computations easier. The results of the computations can be seen in Table 5-2. The next section will compare these results with the ones found for the original transfer.

90 72 From LLO to the Lagrangian Point L2 Figure 5-6: From L2 to LLO CCW circularization at 100 km altitude (less than 1 revolution) and back to L2. The plot above displays the orbit 8.06 days after the first impulsive shot; the plot below displays the orbit days after the first impulsive shot. Table 5-2: V s from LLO with CW or CCW circularization to L2 Clockwise Altitude LLO (km) V LLOimage (m/s) V L2image (m/s) V totalimage (m/s) Counter Clockwise Altitude LLO (km) V LLOimage (m/s) V L2image (m/s) V totalimage (m/s)

91 5-5 Overall results and conclusions Overall results and conclusions Table 5-3 compares the results obtained for the image orbit with the ones of the original transfer. For every case, the absolute difference is only a few meters per second. As can be seen, the root mean square of the difference of the V values is 0.49% for the CW case and 0.14% for the CCW. It is reminded that the root mean square of n values a i, i=1..n is calculated as follows: RMS = 1 n n a 2 i (5-1) i=1 Those differences between the total V needed for the original transfer and the one needed for the image orbit are probably due to the numerical approximations of Matlab, in which case it would mean that for this particular trajectory the theorem of image trajectories can be expanded as follows (with the same prerequisites as the ones of Section 5-2): If a trajectory is feasible in the Earth-Moon space, then its image with respect to the Earth- Moon axis is feasible provided it is flown in opposite direction and the amounts of V needed to fly both trajectories are equal. Table 5-3: Comparison of V totaltransfer (from L2 to LLO) and of V totalimage (from LLO to L2) Clockwise Counter Clockwise Altitude LLO (km) Difference in m/s Difference in % Difference in m/s Difference in % a 1 = b 1 = a 2 = b 2 = a 3 = b 3 = a 4 = b 4 = a 5 = b 5 = 0.18 RMS Therefore the total amount of V needed to go from LLO to L2, depending on the LLO altitude and direction of circularization can be seen in Table 5-4.

92 74 From LLO to the Lagrangian Point L2 Table 5-4: V total of the image orbit (from LLO with CW or CCW circularization to L2) Clockwise Counter Clockwise Altitude LLO (km) V totalimage (m/s) V totalimage (m/s)

93 Chapter 6 From LLO to Lunar Surface This chapter focuses of the descent and soft landing of the spacecraft on the Moon. It is supposed that a Chemical Propulsion System (CPS) is used. The starting point is a circular Low Lunar Orbit (LLO) at an altitude 100, 300, 500, 1000 or 2000 km. At the end of this chapter, the spacecraft is positioned on the surface of the Moon. Every piece of Matlab program developed and used for this chapter has been tested to certify that the results are correct, however those tests have not been included in this report. 6-1 Introduction As seen in the literature survey, the Moon is not a flat sphere and landing on it is therefore complicated. However for this chapter the Moon will be assumed a perfect sphere of radius 1737 km [1]. The descent and landing phase is split into two parts. The first part, the descent, lowers the altitude of the spacecraft until a few tens of meters (from 20 to 100) above the surface of the Moon. Because compared to the radius of the Moon these few meters are insignificant, it was chosen to simulate the descent until the surface of the Moon, which is 1737 km away from the center of the Moon. The results can be proven unaffected if the same calculations are made for an ending point 20 to 100 meters higher. For the second part, the landing, it is assumed that the spacecraft needs to find the best spot to land, hover and move towards it, then land. Starting from a few tens of meters above the lunar surface, this phase ends once the spacecraft has landed.

94 76 From LLO to Lunar Surface 6-2 Descent There are many ways for the spacecraft to descend from the LLO altitude to the surface of the Moon. In this section, only the simplest one will be investigated and the results will be computed from there. More complicated descent strategies could be investigated, especially dealing with landing accuracy and obstacle recognition and clearance. These strategies are left as a recommendation for further work. The spacecraft is orbiting at a distance d SC/Moon = R Moon + h LLO from the center of the Moon, where R Moon is the radius of the Moon (1737 km) and h LLO is the altitude of the LLO (100, 300, 500, 1000 or 2000 km). A first impulsive shot V 1 is applied in such a way that the spacecraft enters a Hohmann transfer orbit with an apocenter situated at the LLO altitude and a pericenter situated at a distance d peri R Moon such that the spacecraft would hit the surface of the Moon if no further change of velocity would be applied [11]: where V cllo = µ Moon d SC/Moon 2d peri V 1Hohmann = V cllo (1 ) (6-1) d peri + d SC/Moon is the circular velocity of the LLO altitude of the spacecraft. The minimum change in velocity V 1min concerns the limit case for which the pericenter of the Hohmann transfer would be exactly at the altitude of the Moon, namely d peri = R Moon. The change of velocity is therefore: 2R Moon V 1min = V cllo (1 ) (6-2) R Moon + d SC/Moon Similarly, the maximum change in velocity V 1max concerns the limit case for which the pericenter of the Hohmann transfer would be situated exactly at the center of the Moon. Therefore the change of velocity applied is equal to the circular velocity of the LLO altitude of interest: V 1max = lim V 2d peri c LLO (1 ) (6-3) d peri 0 d peri + d SC/Moon V 1max = V cllo (6-4) A grid search is performed for every value of V 1 ranging from V 1min to V 1max. For simplification purposes, V 1min and V 1max were rounded up to the nearest integer and the grid search was performed with a step-size of 1 m/s. The Matlab code is available on the separate CD. Table 6-1 shows the range of values for V 1 for the different LLO altitudes considered. Figure 6-1 shows the orbit of the spacecraft for the limit case V 1 = V 1min for a LLO altitude of 500 km. At the surface of the Moon, the velocity of the spacecraft is 1783 m/s. In the legend situated on top of the figure is displayed the time needed for the descent: this time is 3793 seconds, which is a little bit more than an hour. Similarly, Figure 6-2 shows the orbit of the spacecraft for the limit case V 1 = V 1max for a

95 6-2 Descent 77 Table 6-1: Range of values for V 1 Altitude LLO (km) V 1min (m/s) V 1max (m/s) LLO altitude of 500 km. At the surface of the Moon, the velocity of the spacecraft is 1123 m/s. For this case the time needed for the descent is 971 seconds (around 16 minutes). Figure 6-1: Orbit of the spacecraft for V 1 = V 1min, starting from a LLO altitude of 500 km Once at the surface of the Moon (or a few tens of meters above the surface of the Moon), a second impulsive shot V 2 is applied to cancel out the spacecraft velocity in every direction. Therefore at the lunar surface: V 2 = V spacecraft (6-5) The total amount of V for the descent, V totdescent = V 1 + V 2, is plotted as a function of V 1 ranging from V 1min to V 1max. The result can be seen In Figure 6-3 for the case for which the LLO altitude is 500 km. The optimized amount of V totdescent is reached for V 1 = V 1min. The same procedure is repeated for every LLO altitude considered. For every case, the value of V 1 leading to the minimum value of V totdescent is V 1 = V 1min. Therefore for the

96 78 From LLO to Lunar Surface Figure 6-2: Orbit of the spacecraft for V 1 = V 1max, starting from a LLO altitude of 500 km Figure 6-3: V totdescent as a function of V 1 for a LLO altitude of 500 km descent it is chosen to apply a first impulsive shot of value V 1 = V 1min, and a second impulsive shot at the surface of the Moon of value V 2 = V spacecraft. The results are summarized in Table 6-2. As can be seen from Figures 6-1 and 6-2, this descent phase takes an amount of time ranging from minutes to a little bit more than an hour. The time for the descent and landing is therefore not further investigated because it is considered of too little influence for the overall