Prometheus Asteroid Redirection Mission
Mission Design, Spacecraft Design, Orbital Dynamics Code
Development
Niklas Anthony
Space Engineering, masters level
2016
Luleå University of Technology
PROMETHEUS ASTEROID
REDIRECTION MISSION
Niklas Anthony
nik.anthony@gmail.comAbstract
This report will design a mission and spacecraft to redirect the first Near-Earth Object (NEO) to a stable orbit in the Earth-Moon system. The mission profile includes a soon-as-possible launch, spiral-out escape from the Earth-Moon system, rendezvous, ion beam redirection method, and decommissioning phases, each with accompanying orbital dynamics code written in Matlab. The spacecraft design will include power and mass budgets for each of the subsystems including power, thermal, communications, GNC, fuel, and thrusters. The orbital dynamics code is detailed in the final section of the report. DISCLAIMER: "This project has been funded with support from the European Commission. This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein." "Co-funded by the Erasmus+ Programme of the European Union"
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Table of Contents
Table of Contents ... 1 1. Executive Summary ... 3 2. Introduction ... 5 a. Reasoning ... 5 b. Redirection Method ... 6 c. Asteroid Selection ... 7 3. Mission Design ... 10 a. Overview ... 10 b. Mission Profile ... 10 i. Setup ... 10 ii. Launch ... 11 iii. Escape ... 11 iv. Rendezvous ... 12 v. Redirection ... 12 vi. Parking ... 13 vii. Decommissioning ... 13 4. Spacecraft Design ... 14 i. Overview ... 14 ii. Payload ... 14 iii. Propulsion ... 15 i. Propulsion Thrusters ... 15 ii. Fuel ... 16 iv. Power ... 17 i. Power Storage ... 17ii. Power Generation ... 18
v. Guidance, Navigation, and Control (GNC) ... 19
i. Reaction Wheels ... 19
ii. Reaction Control System (RCS) Thrusters ... 19
vi. Communications ... 21
i. Main Dish ... 21
ii. Dipole Antennae ... 22
iii. Telemetry, Tracking, and Command (TT&C) ... 22
vii. Thermal ... 22
5. Orbital Dynamics ... 24
a. Overview ... 24
2 ii. Escape ... 24 iii. Rendezvous ... 25 iv. Operations ... 25 v. Capture ... 25 b. Code Development ... 26 i. Importing Data... 27
ii. Orbital Motion ... 27
iii. Thruster Integration ... 28
iv. Capture Mechanics ... 29
6. Risk & Financing ... 31
a. Risk ... 31 b. Financing ... 31 7. Conclusions ... 34 8. References ... 35 9. Appendix ... 39 a. Phase.m ... 39 b. Motion.m ... 40 c. Stepxyz.m... 41 d. Moveplanets.m ... 42 e. Thruster.m ... 43
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1. Executive Summary
Asteroids are bodies of dirt, ice, rock, and metal that were not absorbed by the sun or planetoids in the early solar system. They populate the entire solar system, with most being located between Mars and Jupiter in the Main Asteroid Belt. The first Near-Earth Object (NEO) was discovered by the Spacewatch Project at the University of Arizona in 1989 on a CCD imager [1]. Since then, approximately fifteen thousand objects have been detected around Earth, ranging in size from a car to small cities, with the amount constantly increasing. The large objects are relatively easy to detect, but can still slip by the detectors, such as object 2015 TB 145, also known as the Halloween Asteroid, which was only detected as it passed by Earth at 487,000 km away. This object was around 600 m wide; compare that to the Chelyabinsk meteorite, which was only 20 m across, which wasn’t detected until it exploded over the small Russian town, causing property damage and injuring over a thousand people. It is therefore vital that these objects are monitored and studied.
Asteroids can tell us about the history of the solar system, including how it formed in the first place. Some objects have been found to contain organic compounds, such as the Murchison meteorite, which leads some scientists to suggest that life itself started because of these objects in an idea called panspermia [2]. Companies are also interested in studying NEOs as they could be a viable and valuable source of materials in orbit around Earth. Water ice is thought to be common among NEOs; this could provide a source of oxygen for life support and hydrogen for rocket fuel, which would not have to be launched from the surface of Earth anymore [3] . Silicon, which was found in the tail of Halley’s Comet, could be used in semiconductor production in space [4] [5]. NEOs also contain precious metals and rare-earth elements which could be valuable to buyers on the Earth’s surface [6]. They could also serve as shelter vehicles for astronauts travelling to other planets and moons around the solar system.
(Image 1: Object 2015 TB 145, the “Halloween Asteroid”, photo courtesy of NASA JPL) [7]
It is therefore, important to study these objects much closer. Since no NEO has been detected in direct orbit around Earth, one must be redirected there. This report will design a mission for a spacecraft to rendezvous with a NEO and redirect it to the Earth-Moon L5 Lagrange point, where the object will be further analyzed, studied, tested, and exploited. The report is split into three major sections: mission design,
4 spacecraft design, and orbital dynamics. The mission design involves determining what NEOs make ideal candidates for redirection, selecting the redirection method, and outlining the various phases of the mission profile. The spacecraft design section will involve determining the operating parameters of each subsystem, including power generation, propulsion method, fuel storage, thermal control, Guidance, Navigation, and Control (GNC), and communications systems. The orbital dynamics section will describe the motion of the spacecraft and object over time, including launch, escape from Earth, rendezvous, and redirection.
NEOs’ orbital parameters vary a lot, and most objects’ closest approach to Earth involve a high relative velocity, which requires a large momentum change in a short period of time to capture, which the spacecraft cannot output. It is therefore necessary to find a more stable source of NEOs. An object will be chosen from either the L4 or L5 points of the Earth-Sun system, as they will remain relatively stationary over time, and should be relatively abundant, as we see with Jupiter’s large Greek and Trojan Lagrange asteroid populations [8]. Due to current thrust output of ion beam engines, the object will be under 9 m in diameter, and will take 10 years to fully redirect. The desired composition will be a C-type asteroid, as it is the least dense and most abundant [9]. The mission will use the Moon and Sun to capture the object in a stable Earth orbit, preferably in the Earth-Moon L5 Lagrange point, as the Moon is in the best position to impart momentum on the object. The project is estimated to cost under 1 billion USD, which accounts for the spacecraft itself, launch services, ground station operations over the mission duration, and a 30% overhead. Two technologies would greatly improve this spacecraft, but have not reached a high enough technology readiness level high enough for consideration. These are the VASIMR plasma engine [10], for redirection method, and Orbital ATK’s Megaflex solar panel arrays [11].
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2. Introduction
a. Reasoning
Asteroids are remnants of the early solar system. Their composition and structure can give scientists more insight into how solar systems form and what ours looked like particularly. They contain a plethora of atoms and molecules, ranging from simple water ice to complex compounds to heavy metals. The idea of panspermia suggests that an asteroid brought life with it to early Earth, which begs further investigation [2]. Learning how an asteroid forms in terms of geometry in zero-gravity is useful for crystal growth and could be useful in the semiconductor industry [5]. It is also of utmost importance we learn how to redirect an asteroid in the aspect of planetary defense. NASA’s NEO program shows there are over 1600 potentially hazardous asteroids (PHAs) that we know of, with more being discovered almost every day. [12] From science fiction to science fact, we could use the remains of a hollowed-out asteroid as radiation protection for long-duration, deep-space missions between planets.
According to Planetary Resources, the potential income from asteroid redirection could be worth several trillion dollars. Companies like Planetary Resources and Deep Space Industries could use the materials the asteroid is made up of to sell for profit to the emerging space industry. Water ice and carbonaceous compounds can be turned into rocket fuel and life support materials, expensive commodities in space [3]. Helium ice could be used in fusion reactors and engines. [13] Rare earth metals and silicates can be used to grow extremely precise semiconductors and other technologies. Simple materials such as dirt, stone, and iron can be used to create massive structures and ships that would be impossible to build and launch from Earth.
(Image 2: Deep Space Industries aims to mine asteroids for their materials) [14]
This mission would be the first to ever move an object in the Solar System. It is therefore suggested that national space agencies, universities, and companies from all over the world come together to see this mission done, perhaps using in-kind commercial agreements. It will create a new age in interest and accessibility to the solar system that the entire world can take part in. In a period of global tensions, having a new frontier to explore and utilize would encourage cooperation between countries. There have been a few missions to asteroids and comets, but none have redirected them, unless you consider the Deep Impact mission to be an asteroid redirection mission. The latest mission OsirisREX seeks to bring back up to 2kg of
6 asteroid material from Bennu, and NASA’s ARM mission seeks to pick up a boulder from the surface of a larger asteroid.
This mission, Prometheus, will aim to move a free-floating boulder to the Moon’s L5 Lagrange point. As this would be the first redirection mission, it would serve as a proof-of-concept and test bed for further missions. The spacecraft will be using ion beam redirection, so it will use the state of the art ion engines and solar panel technology. It will also push the limits of station keeping and attitude control as the distance and firing angles needs to remain precise over several years. The major incentive of this mission will be the end result: having an easily-accessible asteroid from deep space in the Earth-Moon system. NASA and ESA want to send human explorers to asteroids, and this would be the perfect starting location to test relevant human habitation technologies. Having the asteroid so close to home will allow groups with smaller budgets, such as universities or private companies, to test their relevant asteroid technologies; such as composition analysis tools, landing and attachment gear, and mineral extraction methods. Future missions will most likely target larger asteroids and will most certainly use the technology developed form this missions result.
b. Redirection Method
There are many different ways an object can be moved in space. Some promising methods include ion beam, tugboat, gravity tractor, and laser sublimation. Ion beam involves aiming an ion-emitting thruster at an object and letting the exhaust plume impart force onto the object. Tugboat involves landing rockets on an object, and thrusting along the center of mass to move the object. Gravity tractor utilizes Newton’s Law of attraction to pull the asteroid towards the craft in a desired direction. Laser sublimation is the process of aiming a high-powered laser at the object, heating a pinpoint area, causing material to sputter off, creating a plume of dust and dirt, thus a momentum shift. There are many factors to consider for each method, such as amount of delta-v required, robustness, cost, power, and technology development. Each have their strengths and weaknesses [15].
Delta-v, or change in velocity, represents an absolute measure of orbital change; it is how much a spacecraft’s velocity will change, with reference to its initial velocity, when it fires its engines. The dry mass of the spacecraft, amount of fuel, and efficiency of the firing engine all contribute to a spacecraft’s total delta-v capability. The Tsiolkodelta-vsky rocket equation was dedelta-veloped to easily determine a spacecraft’s delta-delta-v, seen below, where g is the acceleration due to gravity at the surface of Earth (9.8 m/s2), used only for units
convention, ISP is the spacecraft engine’s specific impulse, and mi and mf are the spacecraft’s masses before and after firing the engine.
𝑑𝑉 (𝑚 𝑠) = 𝑔 ( 𝑚 𝑠2) ∗ 𝐼𝑆𝑃(𝑠) ∗ 𝑙𝑛 ( 𝑚𝑖(𝑘𝑔) 𝑚𝑓(𝑘𝑔)) [16]( 1 )
Robustness refers to how well a technology can adapt to objects of different size, mass, or spin rate. For instance, the tugboat method requires the spacecraft to attach and anchor itself to the object, which would be incredibly difficult on any object that is spinning relatively quickly, or an object made of a completely different material than the attachment method is designed for, whereas ion beam does not need to land or attach, so it can still operate on a wide range of objects.
In Michael Bazzocchi & Reza Emami’s “Comparative Analysis of Redirection Methods for Asteroid Resource Exploitation”, each method is analyzed in great detail according to these parameters, and suggested ion beam and tugboat to be the most promising choices, with ion beam being more robust, but less fuel-efficient, and vice-versa for tugboat. [15] Below, in Table 1, is a table outlining the final compilation of their work in the form of a table, where RM-01 represents “redirection method 1”, ion beam redirection.
7 RM-02 is “tugboat”, RM-03 is “gravity tractor”, RM-04 is “laser sublimation”, and RM-05 is “mass ejector”, a process of launching rocks off of the object to create an opposing force on the object.
Table 1: Comparison of Various Redirection Methods
(Table comparing redirection methods against 8 criteria, taken from Bazzocchi & Emami) [15]
Ion beam redirection was chosen because it scored better in the system cost and technology readiness sections. In previous work, different types of ion beam redirection were compared, including science and history, which found gridded ion thrusters and Hall Effect thrusters (HET) to be the two most promising technologies. [17] Currently, HETs show more promise in total thrust, but it appears more research is focusing on gridded ion thrusters. For instance, the Busek 20 kW HET can put out over 800 mN of thrust (at 15 kW) while the most powerful gridded tech, the NEXT thruster (used on the Dawn spacecraft) only puts out around 240 mN (at 7 kW). As mentioned, several agencies and institutes are researching how to make these Solar Electric Powered (SEP) thrusters more efficient, including using various ion propellants and exhaust plume focusing.
c. Asteroid Selection
According to the accretion disk theory, all the asteroids in our solar system were formed in the early solar system, just as planets and the sun were beginning to form. Most of the asteroids were consumed by the sun and planets, and certain resonant orbits of asteroids with large planets became unstable, and were ultimately consumed as well [18]. Most of the asteroids in our solar system exist between Mars & Jupiter in the Asteroid Belt, but we can observe others all over the solar system [19]. They range in composition from simple water ice to complex molecules to heavy metals, in shape from spherical to rubber ducks, in size from micrometers to kilometers, and in spin from stationary to several times per minute.
Many of the asteroids that have been detected so far have been as a result of the Lincoln Near-Earth Asteroid Research (LIENAR) program and the Wide-field Infrared Survey Explorer (WISE) mission from 2009 to 2011 and its continued use in the NEOWISE project (2013-present) [20]. Detecting small asteroids is very difficult: the data is still being reviewed and more discoveries are found almost every day. Objects under 20 m show up as bright, blurry dots, meaning shape cannot be determined. Other observation programs are being run by both NASA and ESA, primarily searching for large Potentially Hazardous Asteroids (PHAs). NASA’s Near-Earth Object (NEO) program lists several ground station observatories that are contributing to the search, and ESA’s NEO program utilizes professional and amateur observations from around the world; these include optical, infrared, and radar imaging. Seen below, in Figure 1, is a graph of the discovery statistics of NEOs by various platforms, taken from NASA’s website. Each point of data is the number discovered over a half-year period, not showing the total, but clearly showing that the discovery rate is increasing over time.
8 Figure 1: NEO discovery statistics by half-year.
(NEO discovery rate, taken from NASA’s NEO webpage) [20]
A majority of papers concerning asteroid detection only consider large asteroids, typically having over 100 meter diameters [21]. However both NASA’s and ESA’s catalogue of NEOs include objects much smaller than this. Very little concern has been given to the 10 meter boulder-sized objects which are more numerous and fly by us all the time, like the object that exploded over Chelyabinsk, Russia (20 m wide) which caused a lot of property damage and injured thousands of people when it exploded above the ground [22]. Had that object entered at a slightly different trajectory, it could have killed those of people.
NASA states they use various methods to characterize asteroids, including: radar measurements, ground- and space-based infrared observations, and long-arc high-precision astrometry. Obviously, with large objects it is relatively easy to map the surfaces and determine the composition, but for relatively small objects (under 100 m diameter), the data cannot be accurately measured [23]. It is possible to track small objects, but they are simply small dots of “light”. It is therefore necessary to get closer to small objects to determine these more advanced details about them.
This mission will select an asteroid from the existing archive of objects in the 5-10 m range and have a scout spacecraft fly to it to get detailed images, orbital parameters, and composition. One option is to select objects in Earth’s Lagrange 4-5 points, like Jupiter’s Trojans, seen below in Image 3. It is, however, very difficult to see these objects, as the angle between them and the sun is small enough that most results are washed out by sunlight [24]. However, the author of this paper hypothesizes that NEOs found in this region would be relatively simple to redirect back to Earth, as they share the same orbital size, and simply need to be shifted in phase. Objects have been detected in Jupiter’s, Mars’, and Neptune’s Lagrange points, thus it is hypothesized there are more NEOs in Earth’s Lagrange points as well [8].
Data about size, composition, and orbital parameters will be collected by the scout spacecrafts and sent back to Earth. Each of the objects that the probes detect will be added to NASA’s and ESA’s database of NEOs. From this updated database, an object will be chosen based on its easiness to redirect: nearest to 500,000 kg, C-type, orbital elements ideal for redirection. Once the object is selected, a command will be given to the probe to approach the selected object for detailed images and information about its composition, shape, spin rate, mass, etc.
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(Image 3: Objects discovered in Jupiter’s Lagrange points: The Trojans and The Greeks Image courtesy of Guillermo Abramson, University of New Mexico, March 2012) [25]
After developing some preliminary orbital dynamics code, Phase.m, explained in the Code Development section of the report, it was found that in order to change phase by 60 degrees (the angle between L4/L5 and Earth) in 10 years at 1N of constant thrust, the target object needs to be under 500 metric tons. In the case of L4, the spacecraft would need to slow the object down so Earth could catch up, and then speed it back up to re-align the orbits (firing prograde for 5 years, then switching to retrograde for the remaining 5). For L5 objects, the opposite firing pattern is used. This code did not take into account slight changes in inclination or eccentricity. The density of asteroids can range between 1380 kg/m3, typically
C-type (carbonaceous), to 5320 kg/m3 M-types (metallic) [19]. For simplicity’s sake, the mass of the object can
be converted to size using the equation for a perfect sphere with a density of 1400 kg/m3, even though the
asteroids will be misshapen in real life. This is very close to the C-type density, which is estimated to be the most numerous and least dense [21].
𝑚 = 𝜌 ∗ 𝑉 5 ∗ 105 𝑘𝑔 = 1400𝑘𝑔 𝑚3∗ 𝑉 𝑉 ≈ 357.1 𝑚3 𝑉𝑠𝑝ℎ= 4 3 ∗ 𝜋 ∗ 𝑟 3 𝑟𝑜𝑏𝑗 = √357.1 𝑚3∗ 3 4 ∗ 𝜋 3 ≈ 4.4 𝑚 This means we can search for objects roughly less than 9 m in diameter.
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3. Mission Design
a. Overview
This mission will have 5 design drivers:
1) Target a C-type asteroid, under 10m in diameter, in either the L4 or L5 points of the Sun-Earth system
2) 10 year maximum redirection phase length
3) Design a spacecraft capable of using ion beam redirection technology to apply a constant 1 N of force on the object
4) Park the object in the stable L5 point in the Earth-Moon system.
5) Soon-as-possible launch/minimal technological research or development
This mission will be a proof-of-concept mission, showing that a near-Earth object, NEO, can be redirected into the Earth-Moon system with ion beam technology. A C-type object is chosen due to its low density, thus large surface area, and general abundance [21]. Candidate objects must be smaller than 9 m, due to the low thrust capabilities of today’s technology. Investors in this mission will want their return as fast as possible, which is why the duration is limited to 10 years. 1 N of force has been determined to be the ideal thrust due to technological limitations. The largest thrust output of any ion thruster that was researched was 800 mN by the BHT20k Hall Effect Thruster, which hasn’t been tested in space yet [26]. Power generation technology with capabilities of 50 kW beyond traditional solar panels is not yet advanced enough either. Radioactive Thermo-Electric systems do not have the output power capability required for 1 N of thrust in both directions. Having the object in the L5 point around Earth will allow organizations to access, analyze, exploit, and test the object for a fraction of the cost of visiting a NEO in its natural orbit. This mission seeks to launch as soon as possible, which means technology development time is kept to a minimum, meaning construction and testing of the spacecraft can begin as soon as funding is acquired. Note that as time goes on, SEP technology will only become more powerful and efficient, so the 10-year mission duration figure can be shortened if thruster technology becomes more powerful.
b. Mission Profile
The mission profile can be broken down into 7 stages: setup, launch, escape, rendezvous, redirection, parking, and decommissioning.
i. Setup
The setup phase will include the design, development, and mission of the scout probes, which is estimated to take around one year to build and two years to rendezvous with the Lagrange points. While this is happening, the spacecraft will be assembled and tested, which is estimated to take around two years [27]. Very few objects have been detected in Earth’s L4 and L5 points, not because they are not numerous, but because detection methods are limited. Programs that try to view the L4/L5 points often are washed out by the Sun’s radiation, thus detailed information about mass, size, shape, and spin are near impossible to obtain [8]. Two cubesats will be sent to the L4/L5 points, where they will begin to analyze the environment, finding suitable targets for the mission. Once the ideal target is chosen, the cubesat will rendezvous with the object
11 to take detailed measurements to verify it as a feasible target. Once a target is identified, the spacecraft will be launched at a time that the Moon can be used as a gravity slingshot to rendezvous with the object.
ii. Launch
Since this will be one of the heaviest modern spacecraft ever flown, around 42,000 kg, a heavy launch service is required. This value is calculated in the Fuel section of the Propulsion subset in Spacecraft Design below. As of writing this, the rocket capable of lifting the most is the United Launch Alliance’s (ULA) Delta IV Heavy, which can launch payloads of just under 29,000 kg to low Earth orbit [28]. There are several companies/agencies developing vehicles capable of launching this mission’s spacecraft into LEO, such as SpaceX’s Falcon Heavy, due to have its first launch this year [29], or ESA’s Araine 6 or ULA’s Vulcan, scheduled to launch in 2020. SpaceX launches from two locations: California and Florida, both in the USA. The launch location for this mission would be closer to the equator, such as the Guiana Space Center in French Guiana. SpaceX quotes 90 million USD for a standard launch package of the Falcon Heavy into a 28o inclined orbit of
54,400 kg, however they say other options are available, and that does not factor in the reusability of their launchers, which will bring the price down [30]. SpaceX does not list its definition of low in LEO; the highest possible “LEO” orbit will be chosen, as to minimize fuel requirements of the spacecraft and the time spent in the escape phase.
iii. Escape
The propulsion set of ion engines will create a low-thrust, spiral-shaped escape profile, seen below in Figure 2. The maneuver can be thought of as an infinite number of Hohmann transfers, where the delta-v required for each maneuver is equal to the equivalent change in orbital velocity.
Figure 2: Spiral escape trajectory
(Space Stack Exchange, courtesy of Mark Adler, 2015) [31]
There were strange perturbations when using Phase.m to calculate the time required to escape, therefore a rule-of-thumb estimate for the total delta-v was used: the total delta-v for any spiral maneuver is the difference in speed between two orbital heights [32]. The initial height is the height after launch, which
12 should be around 500km. In this case, the destination orbit is an escape trajectory, where the orbital velocity is zero, relative to Earth, which means the difference between the orbital velocities is equal to the initial orbit’s velocity. In order to calculate the orbital velocity in a circular orbit at a height, R, the following equation is used:
𝑉(𝑅) = √𝐺∗𝑀
𝑅 ( 2 )
Where G is the gravitation constant, M is the mass of the parent body (in this case, the Earth) in kilograms, and R is the distance from the center of the parent body in meters to the spacecraft. A 500 km orbital height corresponds to an orbital velocity of 7,616.6 m/s. It is estimated to take nearly nine years to spiral out from LEO to an escape trajectory, calculated in the Fuel section of the Propulsion subsystem in Spacecraft Design below. This fails to include the lunar gravity assist, which would require an optimization of launch date and trajectory, but will save time and fuel. The sphere of influence (SOI) of the moon is estimated to be 66,000 km from its center. The height at which the spacecraft enters the Moon’s sphere of influence is simply the difference between the orbital height of the Moon and it’s sphere of influence.
ℎ𝑒,𝑚(𝑘𝑚) − 𝑆𝑂𝐼𝑚(𝑘𝑚) = ℎ𝑒,𝑠𝑐 (𝑘𝑚)
378,029 𝑘𝑚 − 66,100 𝑘𝑚 = 311,929 𝑘𝑚
Where he,sc is the height of the spacecraft from the surface of Earth, he,m is the height of the moon from the surface of Earth, and SOIm is the sphere of influence of the Moon. This value is inserted into Equation 2, which gives a value of 1,130.4 m/s. This means the spacecraft will only have to cancel out 6,482.2 m/s, and would be placed into an ideal rendezvous trajectory.
iv. Rendezvous
The Lagrange points sit 60 degrees ahead (L4) and behind (L5) Earth in its orbit. The code Phase.m was used to simulate this transfer, where the spacecraft is travelling at circular velocity at 1 AU to begin with, and ending at the same speed, but 60 degrees out of phase. The solution is to fire 14 months in prograde and then 14 months in retrograde, or vice versa, depending on whether the destination is L5, or L4, respectively. Since the object is in the L4 or L5 Lagrange point, it will be nearly in the plane of the ecliptic, and thus, for estimation purposes, it will be assumed no inclination change will be necessary. Equation 1 is used to determine the delta-v requirement of this phase of the mission. The spacecraft’s wet mass before and after this maneuver are calculated in the Fuel section of the Spacecraft Design section of the report, and the engine specific impulse is calculated in the Payload section of the report, see Table 2.
𝑑𝑉 (𝑚 𝑠) = 𝑔 ( 𝑚 𝑠2) ∗ 𝐼𝑆𝑃(𝑠) ∗ ln ( 𝑚𝑖 𝑚𝑓 ) = 9.8𝑚 𝑠2∗ 3005 𝑠 ∗ ln ( 32,952 𝑘𝑔 30,218 𝑘𝑔) = 2,551 𝑚 𝑠
Objects do not sit exactly in the Lagrange points, they orbit them in what are known as “halo” orbits. Some overhead in fuel estimations are used to account for any sort of trim burns required on approach to meet up exactly with the object in its halo orbit.
v. Redirection
Once the spacecraft is at the object, it will begin the redirection process. This involves pointing its payload thrusters at the object and pushing it in the appropriate direction. The mission was constrained to
13 have a 10-year redirection phase, targeting up to a 500 ton boulder. As the spacecraft is firing in both directions, its position will only change at the same rate the object is moving. A 500 ton object, experiencing a force of 1 Newton will accelerate at:
𝐹(𝑁) = 𝑚 (𝑘𝑔) ∗ 𝑎(𝑚 𝑠2)
1 𝑁 = 5 ∗ 105 𝑘𝑔 ∗ 𝑎
𝑎 = 2 𝜇𝑚 𝑠2
This acceleration value can be integrated over 10 years to find a net delta-v of: 𝑑𝑉 = 𝐴 ∗ 𝑡 = 2 ∗ 10−6𝑚
𝑠2 ∗ 10 𝑦𝑟𝑠 = 631
𝑚 𝑠
This is not an accurate depiction of the fuel costs, however. The spacecraft will be firing 6 engines at full thrust for 10 years, which would amount to a lot more equivalent delta-v than 631 m/s. To determine the equivalent delta-v, Equation 1 is again used. By the end of the redirection phase, the spacecraft will be devoid of fuel, leaving its dry mass. Again, the fuel costs and engine specific impulse are calculated in Spacecraft Design section of the report.
𝑑𝑉 (𝑚 𝑠) = 𝑔 ( 𝑚 𝑠2) ∗ 𝐼𝑆𝑃(𝑠) ∗ ln ( 𝑚𝑖(𝑘𝑔) 𝑚𝑓(𝑘𝑔) ) = 9.8𝑚 𝑠2∗ 3005 𝑠 ∗ ln ( 30,218 𝑘𝑔 8,800 𝑘𝑔) = 36,331 𝑚 𝑠
If all 6 thrusters were pointed in the same direction, and the total delta-v imparted in one quick impulse, the spacecraft would have an aphelion past Pluto.
vi. Parking
As the object is in one of Earth’s Lagrange Points, the return window will be unlimited as well. The spacecraft can begin its redirection phase at an ideal time such that the Moon will be in the perfect location for a gravity assist or capture upon entering the system. As the object will be travelling at nearly the same speed as Earth when it arrives, the orbit relative to Earth will be highly elliptical. This is where the Moon will be used as a third capture body, in addition to the Earth and Sun. The object will be placed in another halo orbit around the L5 point, where it will remain for study and exploitation. The L5 point was chosen as a point of easy access. The amount of fuel required to get there is the same as the Moon, but spacecraft visiting the L5 point will not have to expend the fuel that would have been used to enter the orbit of the Moon.
vii. Decommissioning
After the final trim and insertion burns are completed, the spacecraft will have completed its goal. The spacecraft and object will begin to experience an attractive force of gravity. The spacecraft must either dock with the object or move to another halo orbit, far enough away to avoid collision. Not only is the object worth studying, but the spacecraft itself could provide insight into long-term stays in the deep-space environment, including the effects of solar wind, radiation, and micrometeorites. The massive solar arrays could be utilized as a mothership power source, or its extra fuel removed for future cubesat missions.
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4. Spacecraft Design
i. Overview
It has been decided that this mission will consist of one spacecraft, not a constellation or formation of several. The relative size of the object compared to the spacecraft limits the operating space around it. The spacecraft will be roughly cube-shaped, approximately 4 m in each dimension. On two opposite sides, the two solar arrays will be affixed, which will reach out to 24 m, totaling around 55 m in width. On an open side, the payload thrusters will be situated: 4 total (3 in use, 1 spare) in an equilateral triangle shape, with the spare in the center. On the opposite side of that, the propulsion thrusters will be fixed in the same configuration as the payload thrusters. If any of the thrusters break down, the opposite side’s thrusters must fire in the same way, to eliminate any torque on the spacecraft. On one of the remaining two sides will be the long-rage communications dish, around 4 m in diameter, see calculates in Communications section below. It will lie flat against the spacecraft when stored, and open on a hinge once in space, able to be tilted to allow for more communication time with Earth, similar to Rosetta’s configuration. On the remaining free side the heat radiators and various sensors will be mounted.
The exact subsystems will be described below in the following sections: Payload, Power, Propulsion, GNC (Guidance, Navigation, and Control), Communications, and Thermal
ii. Payload
The primary “instrument” will be the ion beam source. A value of 1 Newton of exhaust thrust is required by the payload thrusters to complete its redirection phase in 10 years. This balances the use of highly powerful thrusters while keeping power requirements low enough to not require unfeasibly large solar panels. There are many different ion thruster technologies. Mentioned in the Redirection Method section of the report, gridded ion thrusters and Hall-effect thrusters are the two most well-developed and –tested types. To achieve 1 Newton of thrust, three Busek BHT8000 HETs clustered in a triangle, with one spare, will be used. The voltage of the thrusters can be varied to increase or decrease thrust and specific impulse. The higher the voltage, the higher the specific impulse, but the lower the thrust [26]. These results can be summed up in Table 2 below.
Table 2: Performance of the BHT8000 Hall Effect Thruster
Thrust (mN) Voltage (V) ISP (s)
449 (max) 400 2210
325 (min) 800 3060
333 (ideal) 774 3005
These values are for Xenon fuel, but it is also interesting to note that Busek has been testing their thrusters with varying types of propellant, such as other gasses like Iodine and Argon, as well as metals like Zinc and Magnesium. A large portion of thruster power consumption goes into ionizing the fuel, which is stored in a neutral state. Details of these tests on thrust output and power consumption are not presented for the BHT8000, therefore Xenon will be used as the propellant.
The BHT8000 has not been tested for a mission of this sort of duration. It is quoted for 1,000 kg of mass throughput, and the thrusters must be cooled to around 1000 K in order to last for longer than 10,000 hrs
15 [33]. Rated at a technology readiness level (TRL) of 5, its one of the few thrusters with a high enough thrust output for this mission at that high of a TRL.
Exhaust plume characteristics were measured, such as current density, and particle counts, but little has been tested on the shape of the plume. It is vital that the plume be focused enough to ensure all of the exhaust momentum is imparted onto the object, and not in a cloud around it.
The effects of gravity will also be a concern, as the spacecraft will be spending over 10 years at the object. The payload thrusters must output an additional thrust to maintain distance between the object and spacecraft:
𝐹 =𝐺∗𝑀∗𝑚
𝑟2 ( 3 )
6.674 ∗ 10−11∗ 500,000 𝑘𝑔 ∗ 29418𝑘𝑔
20 𝑚2 ≈ 2.5 𝑚𝑁
The secondary payload includes the sensor suite and add-ons, including sensors and cameras. In order to maintain optimum distance from the object, an altimeter is needed. Since the spacecraft will be operating within an estimated 20 m, a low-range photoelectric sensor will be used. Several accelerometers are needed to accurately describe the gravitational field around the object. In order to gather data about the object’s spin rate, shape, size, and composition, several cameras will be attached, measuring many wavelengths, including infrared, visible and ultraviolet. The images captured by these cameras will be sent back to Earth for the public to see and for scientists to study. The plasma ejected from the spacecraft would be interesting to study; a Langmuir probe, magnetometers, electron, and ion detectors should give an accurate depiction of the plasma environment around the object and how it changes over time during the mission. It is assumed that all of these sensors and instruments add up to 50 kg.
It is estimated the total mass of the payload subsystem will be 175 kg and use up to 30 kW of power, if everything is being used at once.
iii. Propulsion
i. Propulsion Thrusters
The primary propulsion thrusters must match the payload thrusters exactly, plus be able to generate enough thrust to keep up with the object that is accelerating away. Three Busek BHT8000 HETs firing simultaneously in the same configuration as the payload thrusters should create an equalizing force on the spacecraft. Since the object will be accelerating away from the spacecraft at a rate of 2 µm/s2, the
propulsion thrusters must output the following extra thrust: 𝐹 = 29,418 𝑘𝑔 ∗ 2 𝜇𝑚
𝑠2 = 58.8 𝑚𝑁
Overall, the spacecraft’s propulsion thrusters need to fire 56.3 mN (previous number minus the force of gravity) harder than the payload thrusters to keep up with the object, which is well within the operating parameters of the BHT8000 models. Seen below in Figure 3 is the nominal setup of the BHT 8000.
16 Figure 3: Nominal layout of the BHT 8000 propulsion panel.
(Figure taken from Busek’s IEPC paper) [33]
ii. Fuel
The spacecraft’s dry mass is estimated to be 8,800 kg by the time the mission is complete, from the fuel tank, solar panel, and overhead figures. The amount of fuel required can be calculated by analyzing the mission phases in reverse order. The last phase of the mission is the redirection phase, which requires the most fuel. The total fuel consumed can be calculated using the mass flow rate via a variation of the Tsiolkovsky equation: 𝑇(𝑁) = 𝑚 (𝑘𝑔 𝑠 ) ̇ ∗ 𝑔(𝑚 𝑠2) ∗ 𝐼𝑆𝑃(𝑠) 1 𝑁 = 𝑚̇ ∗ 9.8 𝑚 𝑠2 ∗ 3005 𝑠 → 𝑚̇ = 34 𝜇𝑔 𝑠
The spacecraft will be firing 6 thrusters, in both directions for 10 years, which corresponds to a total mass flow of:
𝑚𝑓(𝑘𝑔) = 𝑚̇ (
𝑘𝑔
𝑠 ) ∗ 𝑡(𝑠) = 34 ∗ 10
−9 𝑘𝑔
𝑠 ∗ 2 ∗ 10 𝑦𝑟𝑠 = 21,418 𝑘𝑔
Therefore, at the start of the redirection phase, the spacecraft’s mass is 30,218 kg. As mentioned, Phase.m was used to determine the length of the rendezvous phase. For the spacecraft to travel 60 degrees in phase to the L4 point, it takes 2.45 years, and the L5 takes 2.55 years. For L5, it spends 465 days firing prograde and the remaining 465 days retrograde, putting it nearly exactly at 1 AU, but 60 degrees behind the Earth. We can find how much fuel this would require by the same process as the redirection phase, only 3 thrusters will be used, instead of 6:
𝑚𝑓(𝑘𝑔) = 𝑚̇ (
𝑘𝑔
𝑠 ) ∗ 𝑡(𝑠) = 34 ∗ 10
−9 𝑘𝑔
𝑠 ∗ 2.55 𝑦𝑟𝑠 = 2,734 𝑘𝑔
And finally, the first phase is the spiral-out from Earth. As mentioned previously, the amount of delta-v required to escape delta-via spiral is the objects orbital delta-velocity at the beginning, 7,616 m/s. This delta-value was
17 inserted into Equation 1 to solve for the fuel required for this phase of the mission. The final mass is the mass of the spacecraft as it begins its rendezvous phase, which is the sum of the previous two phases: 32,952 kg.
𝑑𝑉 = 𝑔 ∗ 𝐼𝑆𝑃 ∗ ln (𝑚𝑖 𝑚𝑓 ) = 7,616 𝑚 𝑠 = 9.8 𝑚 𝑠2∗ 3005 𝑠 ∗ ln ( 𝑚𝑖 32,952 𝑘𝑔) → 𝑚𝑖 = 42,677 𝑘𝑔. The fuel required by this phase is the difference between the calculated value and the final mass:
42,677 − 32,952 = 9,725 𝑘𝑔
Up until this point, it was unknown how long the escape phase would take. With the fuel total and mass flow rate, the total duration can be calculated. The spacecraft will be using its three propulsion thrusters for this phase, so the equation becomes:
𝑚𝑓= 𝑚̇ ∗ 𝑡 = 9,725 𝑘𝑔 = 34
𝜇𝑔
𝑠 ∗ 𝑡 → 𝑡 = 9.07 𝑦𝑟𝑠 This duration was unexpectedly long, which will be discussed in the conclusion.
The total amount of fuel required for every phase of the mission is the mass calculated in the last step subtracted from the spacecraft’s dry mass, 33,877 kg. Since the object might need to change inclination by a few degrees, and for the mission to include a few trimming burns, an overhead is included, which brings up the fuel requirement to 35,000 kg, an extra 1,123 kg. For comparison, the Dawn spacecraft only had around 425 kg of xenon [34].
Using the density of compressed xenon to be around 1500 kg/m3, [34], the spacecraft needs a xenon
fuel tank with a capacity of 23.3 cubic meters, or 23,333 liters, roughly a 3.5 m-wide sphere, which has not been built by anyone yet. Orbital ATK has a 133 liter tank (80458-1) which weighs 20.4 kilograms. By scaling linearly at 6.52 L/kg, a 23,333 liter tank will weigh 3,579 kilograms. The total mass of the fuel subsystem will be 38,579 kg.
iv. Power
i. Power Storage
The spacecraft will need to operate in a hibernation mode until it can deploy its solar panels and become self-sustaining. An average launch into LEO takes 1 hour, and the deployment of the panels will take several more hours. Lightweight lithium batteries are capable of providing 170 Wh/kg (quoted from Saft Batteries VL51ES model) [35]. An estimated 1 kW is used for the communications, camera, and solar panel deployment mechanism over 5 hours. The following equations are used to determine the mass of the batteries required for the mission:
𝐸𝑡𝑜𝑡= 𝑃 ∗ 𝑡 = 1 𝑘𝑊 ∗ 5 ℎ = 5,000 𝑊ℎ 𝑚𝑏𝑎𝑡 = 𝐸𝑡𝑜𝑡 𝐸𝑘𝑔 =5,000 𝑊ℎ 170 𝑊ℎ𝑘𝑔 = 29.4 𝑘𝑔
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ii. Power Generation
The limiting factor in this design is the power generation. The solar constant is the flux of energy at a distance from the sun of 1 AU. It varies over time dependent on the solar cycle, but on average is valued at 1366 W/m2 [36]. The total energy flux of the sun can be found by integrating this constant over a sphere of
radius 1 AU. Using this value, the flux of energy per square meter can be determined for any distance from the sun. Since the spacecraft is operating between the Lagrange points and Earth, the maximum distance from the sun would only be around 1.1 AU, which corresponds to a power flux of 1129 W/m2. At an operating
efficiency of around 30% [37], the available power input is only 339 W/m2. NASA estimates a deep space
mission to Saturn of 11 years will degrade the panels by 15% [38], so an additional 20% overhead will be added to ensure full power requirements are met, equivalently reducing the input power to 271.2 W/m2.
This can be seen in the following equation where the power generation capability, Pi,min, is equal to the solar flux at 1.1 AU, Pt, multiplied by the efficiency of the solar panels, effpg, and the deterioration loss factor,
lossdet. 𝑃𝑖,𝑚𝑖𝑛 = 𝑃𝑡∗ 𝑒𝑓𝑓𝑝𝑔∗ 𝑙𝑜𝑠𝑠𝑑𝑒𝑡 ( 4 ) 1129 𝑊 𝑚2∗ 0.3 ∗ (1 − 0.2) = 271.2 𝑊 𝑚2
To operate six thrusters at the same time, plus the satellites other functions, 60 kW of total power is estimated to be needed at maximum consumption rate. The minimum power input of 271.2 W/m2 yields a
required surface area of 221.2 m2. The spacecraft will have two sets of panels, each the width of the
spacecraft itself, 4 meters, and a length of 27.7 m each. From tip-to-tip, the spacecraft will be around 60 meters. To get a sense of scale, the following table compares the solar panel sizes of Rosetta, this mission, and the International Space Station (ISS):
Table 3: Solar Panel Sizes
Mission Area (m2) Length (m) Width (m)
Rosetta [39] 64 14 2.29
Prometheus 221.2 27.7 4
ISS (one wing) [40]. 812 35 11.6
Orbital ATK’s MegaFlex solar panels promise large surface area and efficient solar cells, but are currently only at TRL of 6 [11]. Future missions are highly recommended to use this once they are tested in space. Other power generation methods were considered, such as fission or Radio-Thermal Generators (RTG). RTGs are useful for missions that require low power, and are far away from the sun, as the solar intensity begins to get very low. The Soviet Union tested nuclear power in space, but suffered several accidents with containment of nuclear materials over Earth’s surface, and the project was discontinued. Project Prometheus, the same name as this mission’s, was a program run by NASA to build a nuclear reactor in space, but was shut down in 2006 due to budget issues. It is unknown what TRL these technologies are currently at, but could prove to be a useful power generation tool in the future. At an estimated 20kg per 400 W generated, plus overhead for deployment and structures, the total mass of the power generation subsystem is 3,100 kg and can produce 60 kW of power.
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v. Guidance, Navigation, and Control (GNC)
There are satellites in geostationary orbit that use SEP for station-keeping and attitude control, which was a point of consideration. For tiny changes in attitude on a geostationary satellite, this is ideal. This mission, however, requires a heavy craft to be able to turn relatively quickly when transitioning between phases, and the ability to reposition itself around the asteroid. For this, custom reaction wheels will be used in conjunction with 24 reaction control thrusters.
i. Reaction Wheels
Rockwell Collins offers a complete reaction wheel assembly that will satisfy spacecraft up to 7 metric tons. A RDR-68-3 wheel and WDE-8-45 control unit will weigh 9.05 kg, capable of spinning up to 6000 rpm in both directions, producing 0.075 NM of torque and use around 90 W of power at nominal speed [41]. As this spacecraft is around six times as heavy, customized heavy reaction wheels need to be built and tested. By estimating linearly, this craft requires a 212.4 kg wheel and require 540 W of power. Four wheels will be installed, in a configuration seen below where 3 are perpendicular to each other, such that if one fails, the dimension is not lost, but severely crippled. The fourth is angled such that it is 45o between the other 3
wheels, see Figure 4 below. Assuming 3 are spinning at max speed at once (highly unlikely) a total of 1.62 kW of power is required.
Figure 4: Reaction Control Wheel Assembly (W1,W2,W3 primary, W4 extra)
Courtesy of: ISSL at Cleveland State University [42]
ii. Reaction Control System (RCS) Thrusters
Aerojet offers a wide range of RCS thrusters, of which the MR-106L monopropellant thruster was chosen, detailed in Figure 5’s schematic below. In order to rotate the craft in one axis, 4 thrusters on one face will be used, one in each corner. Each thruster’s axis will line in the plane of the face that it is on; each positioned such that the angle between the edge of the craft and the thrust axis will be 45 degrees; each are 90 degrees rotated clockwise from each other. Each face will have this configuration; six faces equals 3 degrees of rotation in each direction, controllable by firing 4 thrusters on a face simultaneously. If each face is roughly 4 m wide, the total torque will be 769.3 N-M of torque in any given axis/direction. In the rare event
20 that a thruster cannot fire due to obstruction by communication dish or solar panels, its opposite thruster on the face can be disabled, leaving only two to fire; the same result, but with only half the torque. The total power of one thruster will be assumed to be the sum of the 3 listed power consumptions of the heaters and valve, 41.7 W at full thrust. At any given time, 12 thrusters will be firing at once, any more would be negating applied torque.
Figure 5: MR-106L Reaction Control Thruster
Courtesy of: Aerojet Rocketdyne Monopropellant Engine Catalogue, 2006. [43]
In order to accurately control attitude, a series of accelerometers, gyroscopes, and sensors are needed. Four accelerometers, 3 primary, 1 spare, similar configuration to the reaction wheels, are needed to accurately determine angular acceleration, thus position and velocity as well. Four laser gyroscopes, 3 primary, 1 spare, will be used as reference. A sun sensor will be used to maintain maximum solar incidence. A star constellation tracker will be used for orientation.
A budget analysis of the Attitude Control Subsystem can be seen in Table 4 below: Table 4: Attitude Control Subsystem Budget
Mass (kg) Tot Mass (kg) Power (W) Tot Power (W)
Reaction Wheel 212.4 849.6 (4x) 540 1620 (3x)
MR-106L 0.59 14.16 (24x) 41.7 500.4 (12x)
Fuel Lines 2 48 - -
Control Sys - - 10(est) 10
N2H4 37.6 75.2 (2x) - -
N2H4 Tank * 5.6 11.2 - -
TOTAL 949.2 2130.4
Two of ATK Space System’s 80308 diaphragm tank, which is 419 x 508 mm and pill-shaped, will be used to store the hydrazine, which should correspond to around 75.2 kg of fuel [44]. Note, by selectively toggling the spacecraft’s ion thrusters, a small torque can be applied, which could be helpful for when long-duration, low-value changes in attitude are required. In total, the GNC subsystem will weigh 950 kg and use 2.1 kW of power.
21
vi. Communications
The spacecraft will use two sets of antennae: One large high-gain parabolic dish antenna for primary use with commands and telemetry and two smaller low-gain dipole antennae for emergency telemetry, health, and housekeeping data, one placed next to the high-gain antenna and the other placed on the opposite side of the craft, to cover every direction, regardless of spacecraft orientation.
As this will be a multi-national effort, both NASA’s Deep Space Network (DSN) and ESA’s large antennae will be used to communicate with the spacecraft from Earth. For calculation purposes, numbers and values are taken from NASA’s DSN, and is assumed to operate similarly to ESA’s. The DSN operates at both X-band (7145-7190 Mhz up, 8400-8450 Mhz down) and Ka-band (34200-34700 Mhz up, 31800-32300 Mhz down). Command signals are only available in the X-band from Earth. Telemetry is also available in that band, and will be the ultimate design choice; a specific frequency will be allocated by the International Telecommunications Union (ITU) [45].
i. Main Dish
Since the Lagrange points, the Sun, and Earth make up the three corners of an equilateral triangle, the maximum distance that the spacecraft will be from Earth will be 1 AU. For distances of 1 AU, a 4 m wide dish will be sufficient. Using an efficiency of 65% and operating frequency of 7145 Mhz uplink, the gain of the main dish is calculated:
𝐺𝑠𝑐,𝑢𝑝 = 4𝜋𝐴𝑒 𝜆2 ( 5 ) 4𝜋(𝜂𝜋(𝐷2)2) (𝑐𝑓)2 = 4𝜋(0.65𝜋(42)2) (7.145 ∗ 103 ∗ 108 9)2 = 2.187 ∗ 104= 47.65 𝑑𝐵
Using the free-space path loss equation, the strength of signals from Earth can be determined. Since the L4 and L5 points form an equilateral triangle with the object, Sun, and Earth as its corners, the FSPL equation will use 1 AU as its primary distance factor. (1 AU = 1.496x1011 m)
𝐹𝑆𝑃𝐿 = (4∗𝜋∗𝑑 𝜆 ) ( 6 ) (4 ∗ 𝜋 ∗ 1 ∗ 1.496 ∗ 10 11 0.042 ) 2= 2.004 ∗ 1027= 273.02 𝑑𝐵
NASA’s 70 m dishes in the DSN can transmit 20 kW at 7145 MHz with a gain of 73.15 dB.
𝑃(𝑑𝐵𝑊) = 10 ∗ 𝑙𝑜𝑔10( 𝑃(𝑊) 1𝑊 ) ( 7 ) 10 ∗ 𝑙𝑜𝑔10( 20000 𝑊 1 𝑊 ) = 43.01 𝑑𝐵𝑊 𝑃𝑟 = 𝑃𝑡+ 𝐺𝑡+ 𝐺𝑟− 𝐹𝑆𝑃𝐿 (𝑑𝐵) ( 8 ) 𝑃𝑟 = 43.01 𝑑𝐵𝑊 + 73.15 𝑑𝐵 + 47.65 𝑑𝐵 − 273.02 𝑑𝐵 = −109.21 𝑑𝐵𝑊 = 1.2 𝑝𝑊
The signal strength at the receiver is on the scale of picowatts (10-12) which requires highly sensitive
22
ii. Dipole Antennae
ESA’s 35 m antennae can pick up signals of strength -225.69 dBm [46]. The spacecraft’s emergency antennae need to meet this requirement, so the reverse of the equation above is used to determine how much power is needed to receive that signal at Earth. Dipole antennae have a standard gain of 2.15 dB. The receive gain of the 70 m DSN antennae are 74.55 dB at the downlink frequency of 8450 Mhz. The FSPL will increase by around 1.5 dB with this change in carrier frequency.
−𝑃𝑡= −𝑃𝑟+ 𝐺𝑡+ 𝐺𝑟− 𝐹𝑆𝑃𝐿
−𝑃𝑡 = −(−225.69 𝑑𝐵𝑚) + 2.15 𝑑𝐵 + 74.55 𝑑𝐵 − 274.52 𝑑𝐵 = −27.87 𝑑𝐵𝑚 = 1.6 𝜇𝑊
In an emergency shutdown mode, and the main antenna is blocked or behind the spacecraft, the dipole antennae will be transmitting at many orders of magnitude higher than microwatts.
The main dish will be the primary mass of this system, with the dipoles only being a few kilograms each. The total for this subsystem is 150 kg, drawing a maximum of 1 kW of power.
iii. Telemetry, Tracking, and Command (TT&C)
The DSN has download bit rate of between 10 bits per second (recommended 40 bps) to 10 Mbps [45]. Since this spacecraft’s power requirements are high during the redirection portion of the mission, a middle-ground bitrate will be used, around 1 Mbps. DSN’s large antennae could easily pick up this information.
The spacecraft will operate mostly autonomously, only receiving commands if something needs to be changed or a specific camera shot is requested. Otherwise, the spacecraft’s communication system will be used mostly for science data transmission, which depends on how many sensors and cameras are installed. It is important that any onboard data handling hardware should be able to withstand radiation exposure of 9 years within Earth’s magnetosphere, spending some time in the Van Allan belts on the spiral out, and 13 years in deep space.
vii. Thermal
The spacecraft will experience a wide range of temperature changes. At 0.9 AU, it will experience up to 1700 W/m2 and at 1.1 AU, it will experience as low as 1100 W/m2. When in Earth escape phase, it will also
experience higher incident power from Earth, both through black body radiation and reflection. Using an albedo of 0.31 and blackbody temp of 255 K, the spacecraft will experience an additional 410 W/m2 and 236
W/m2; combined with solar constant at 1 AU of 1370 W/m2 yields a max heat load of 2016 W/m2 [17].
The exterior of the spacecraft will need to be painted in either thermal control paints or multi-layer-insulation (MLI). In order to keep the electronics at a safe operating temperature of 290K, we will need an internal heater. The far side of the spacecraft will be specifically used for thermal radiators and star trackers.
The main dish antenna must be able to receive commands with signal strength of around 10-12 W. If a
signal-to-noise ratio of 10 is required, the thermal noise can be set to 10-13 W. The temperature that we need
to keep the antenna at can be found using the simple Nyquist equation. The assumed bandwidth of the noise will be 1/10 of the received signal frequency, 7145 Mhz.:
23 𝑃𝑛= 𝑘𝐵∗ 𝑇 ∗ ∆𝑓 → 10−13= 1.38 ∗ 10−23
𝑚2𝑘𝑔
𝑠2𝐾 ∗ 𝑇 ∗ 7.145 ∗ 108𝐻𝑧 = 101.4 𝐾
Room temperature is 300 K, so a system of heat syncs and pipes need to be installed to keep the antenna cool enough to receive commands. The heater will be the most power-consumptive unit, but will not be on all of the time. The paint, heater, and some pipes and heat syncs will weigh around 200 kg and consume up to 1 kW of power.
24
5. Orbital Dynamics
a. Overview
The exact orbital dynamics will be listed in a similar format to when they were mentioned in the mission overview section: Launch, Escape, Rendezvous, Operations, and Capture. After each portion of the mission profile is detailed, the paper will describe how the code was developed and works for the mission. Below in Figure 6 is an outline of the mission phases and durations.
Figure 6: Mission Phase Overview
(Self-Made)
i. Launch
The object sits in either the L4 or L5 point, which means the object will stay almost motionless relative to Earth over time. The L4 and L5 points sit in the plane of the ecliptic, which means 23.4 degrees of inclination will need to be removed during the launch, which will increase the cost of the launch. As this spacecraft is the heaviest ever launched, it would be logical to use the Earth’s rotation to its max potential, which requires a launch site close to the equator, such as French Guiana. The Falcon Heavy is quoted as being able to put 54,400 kg into “LEO”, but does not specify at what inclination or height exactly. For calculations, a 500 km launch height was used. The launch date will be chosen to be as soon as possible, such that when the object returns to the Earth-Moon system, it can utilize the Moon’s gravity to position itself into a stable orbit in the Lunar L5 point around the Earth.
ii. Escape
Many SEP spacecraft currently in geostationary orbit and beyond have used the technique called spiral out, where the payload thrusters are constantly firing for sometimes months on end. The idea is to constantly fire prograde, in order to constantly raise the periapsis. In order to maintain the prograde direction, the spacecraft will have to be spin-stabilized, completing one full rotation every orbit. At low altitudes, this rotation rate is relatively quick, but as it gets further away, the rate will decrease. Using a combination of the reaction wheels and RCS thrusters, this varying spin rate stabilization can be achieved. As calculated previously, the spiral out phase will take several years due to the small thrust-to-weight ratio.
25 The overall fuel requirement can be greatly reduced by using the Moon as a slingshot. By timing the launch correctly, the spacecraft can approach the Moon from behind, letting it pull the spacecraft faster and higher relative to Earth. Once the spacecraft enters the sphere of influence of the Moon, the spacecraft will have to switch from its spin-stabilized escape trajectory to a trajectory that will maximize the Oberth effect from the Moon. The Oberth effect essentially states that it is most energy efficient to change orbital height by firing engines when a spacecraft is nearest the parent body. For this reason, the spacecraft will be aiming to pass by the Moon at a very close distance, 50 km from the surface at periapsis. Again, by timing the launch and entry trajectory properly, the spacecraft can be put into an escape trajectory from Earth and save fuel.
iii. Rendezvous
From the escape trajectory, the spacecraft will begin its rendezvous maneuvers. A second change of attitude must be done in order to fire either prograde or retrograde relative to the Sun, depending on if the target object is in the L5 or L4 points, respectively. Again the spacecraft must be spin-stabilized, this time at a rate of one rotation per year nearly. Halfway through the rendezvous phase, the spacecraft must spin around 180o, changing to the opposite direction from what it was previously facing (from prograde to
retrograde or vice versa). Alternatively, the propulsion thrusters can be turned off and the payload thrusters turned on, eliminating the need to use the reaction wheels or thrusters. This will place the spacecraft 1 AU from the Sun and 60o out of phase relative to the Earth, exactly where the Lagrange points are. For example,
if the target is the L4 point, the spacecraft will fire retrograde for half of the phase, then switch to prograde for the other half. Firing retrograde lowers the periapsis on the opposite side of the orbit, decreasing the semi-major axis, and thus, orbital period, allowing the spacecraft to catch up to the L4 point. Switching back to prograde will increase the periapsis, increasing the semi-major axis back to 1 AU, where the target object is. Upon arrival at the Lagrange point, the spacecraft will have to reorient itself to reduce its velocity relative to the object. The spacecraft could either target the object itself for alignment, or the small scanner probe that was sent to analyze it.
iv. Operations
After the final insertion burns are completed, the spacecraft will begin the redirection phase, which will see the first use of the payload engines. Using a similar procedure to the rendezvous maneuver, the spacecraft pushes the object in the prograde/retrograde direction to change the phase and eventually return to the same 1 AU as the Earth. The object will most likely have a small amount of inclination, which needs to be removed in order to use the Moon’s gravity upon arrival. This is one of the factors in the asteroid selection phase.
When removing inclination, the thrusters will push the object in either the normal or anti-normal direction, relative to the object’s orbit, when the object passes by the descending node or ascending node, respectively. The spacecraft will slowly move from the prograde or retrograde vector such that when the object crosses the nodes, the spacecraft will be perpendicular to the orbital plane, then begin returning to its prograde or retrograde vector.
26 The capture method for either L4 or L5 objects will be very similar. Due to the nature of the redirection method, the object will be approaching the Earth-Moon system at relatively low speeds. From the L4 point, it will be approaching from a higher orbit than Earth, relative to the sun, and from the prograde direction. Objects approaching from the L5 point will have a lower orbit, and from the retrograde direction. Using both the Earth and the Moon’s gravity, the spacecraft will be able to position the object in the L4 position of the Moon, 60 degrees behind it in orbit.
b. Code Development
Modern orbital dynamics code has two simulation options: patched conics or integration. Patched conics is the concept that at any given time, only one massive body will have dominating gravitational effects on an object, and in simulation, when the object gets closer to another massive body, it will switch focus from the first to second massive body, excluding all others. This makes calculations and modelling pretty fast and simple, but does not quite represent reality. Integration adds up the effects of gravity from multiple sources on the object in a small time increment, then steps the simulation forward for that one increment. It is very time and processer consumptive, but gives a much more accurate and realistic simulation of orbital motion. For the full mission simulation, a large computer must be used to run the integration method to accurately portray the mission’s orbital paths. For testing purposes, however, individual scripts were written which use patched conics to better understand concepts and to keep simulation time reasonable.
A total of five programs were written, two main functions, and three sub functions. Phase.m has been repeatedly referenced, and shall now be explained. The user inputs the mass of the object or spacecraft, the force acting on that object, a step integration period, and the duration of the maneuver. The code then puts the object in a two-dimensional, circular orbit around the Sun at a radius of 1 AU. It then uses the following two equations to calculate the object’s angular and radial accelerations, respectively, in polar coordinates:
𝜃̈(𝑟𝑎𝑑 𝑠2 ) = −2 ∗ 𝜃̇(𝑟𝑎𝑑𝑠 ) ∗ 𝑟(𝑚𝑠 )̇ 𝑟 (𝑚) 𝑟̈(𝑚 𝑠2) = 𝜃̇ ( 𝑟𝑎𝑑 𝑠 ) 2 ∗ 𝑟(𝑚) − 𝐺 ∗ 𝑀𝑠(𝑘𝑔) 𝑟 (𝑚)2
Where 𝜃̈ is angular acceleration, 𝑟̈ is radial acceleration, 𝜃̇ is angular velocity, commonly written as ω, 𝑟̇ is radial velocity, r is the distance between the center of the Sun and the object, G is the gravitation constant, and Ms is the mass of the sun. These acceleration variables are derived from Kepler’s laws, and can be used with the object’s position and velocity values in the basic kinematic equations, seen below, to determine the object’s next position and velocity values, for both angle and distance.
𝑥𝑖 = 𝑥𝑖−1+ 𝑣𝑥∗ 𝑡 +
1
2∗ 𝑎𝑥∗ 𝑡
2
𝑣𝑖 = 𝑣𝑖−1+ 𝑎𝑥∗ 𝑡
An additional acceleration is added, as specified by the user in the prograde or retrograde direction, depending on which Lagrange point the user wants to end up at. Once the new position and velocity values are calculated, they are stored in an array and fed recursively into the same equations to simulate orbital motion. This acceleration is added for every integration period, until the integrated duration has passed one half of the total duration specified by the user. The acceleration is then applied in the opposite direction, eventually putting the object back out to 1 AU. The code then outputs the total phase the object has changed relative to its starting position.
27 To calculate the mass of the object, the user inputs 1 N of force, and a 10 year duration. The user then varies the mass of the object until the phase becomes at least 60 degrees. This is how a mass of 500 metric tons was calculated. Similarly, to calculate the duration of the rendezvous phase, the user inputs a force of 1 N and the mass of the spacecraft (fueled up). The user then varies the duration until the phase becomes at least 60 degrees for both the L4 and L5 points.
The other main function, motion.m, could theoretically simulate the entire mission, given enough processing power. It calls on the other three scripts: stepxyz.m, moveplanets.m, and thruster.m to simulate the effects of the spacecraft on the object’s orbital motion. The code is described in the following subsections, where data about the Earth, Moon, and object’s orbits are entered. The object’s motion is handled by stepxyz.m, and the Earth’s and Moon’s motion is handled by moveplanets.m. Finally, the spacecraft’s imparting force is simulated by thruster.m.
i. Importing Data
To begin, the object’s, Earth’s, and Moon’s positions and velocities at a given time are imported. NASA’s Solar System Dynamics database can give the x, y, and z position and velocity vectors of the Earth and Moon in heliocentric coordinates for any time between -3000 BC and 3000 AD. The object’s orbital parameters can be found in either NASA’s or ESA’s NEO database, given as Kepler parameters: semi-major axis, eccentricity, inclination, longitude of ascending node, argument of periapsis, and mean anomaly (a,e,i,Ω,ω,Mo). For consistency and ease of use, the objects Kepler parameters are converted to Cartesian using Matlab scripts. In order to calculate the Cartesian coordinates of the object, the eccentric anomaly via Kepler’s equation must be found:
𝑀𝐴 = 𝐸𝐴 − 𝑒 ∗ 𝑠𝑖𝑛(𝐸𝐴) [47]( 9 )
Mean anomaly, MA, is found by subtracting the sine of the eccentric anomaly, EA, multiplied by the eccentricity, e, from the eccentric anomaly itself. This equation has no numerical solutions, it is therefore suggested to use an iterative estimation approach, in this case, via Newton-Raphson estimates:
𝐸𝐴𝑖+1= 𝐸𝐴𝑖−
𝐸𝐴𝑖−(𝑀𝐴+𝑒∗𝑠𝑖𝑛(𝐸𝐴𝑖))
1−𝑒∗𝑐𝑜𝑠(𝐸𝐴𝑖) [47]( 10 )
Begin with an estimate for EA1 to be MA if the orbit is not highly eccentric. From eccentric anomaly the objects true anomaly is calculated, and thus, actual Cartesian coordinates.
ii. Orbital Motion
The orbital motion of the object, Earth, and Moon are written in a “motion.m” file, using the integration method. Since the objects are all in the same coordinate system, relative forces can be calculated by subtracting distances. The force acting on the object is simply the sum of all of the parent bodies: Sun, Earth, and Moon:
𝑎⃑𝑡𝑜𝑡= ∑
−𝐺 ∗ 𝑀𝑖
𝑅𝑖2 3
𝑖=1
Where i represents the object (Sun, Earth, Moon), Mi represents the parent body’s mass, and Ri represents the distance between the object and the parent body. This acceleration vector, in addition to a specified integration step period, t, is used to determine, using basic kinematic equations below, the object’s
