Prometheus Asteroid Redirection MissionMission Design, Spacecraft Design, Orbital Dynamics Code

Development

Niklas Anthony

Space Engineering, masters level 2016

Luleå University of Technology Department of Computer Science, Electrical and Space Engineering

PROMETHEUS ASTEROID

REDIRECTION MISSION

Niklas Anthony nik.anthony@gmail.com

Abstract 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"

1

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 ................................................................................................................................. 17

ii. 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

i. Launch ............................................................................................................................................. 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

3

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].

5

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 Tsiolkovsky rocket equation was developed to easily determine a spacecraft’s 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.

9

(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 ∗ 𝑟3

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.

10

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) = 9.8

𝑚

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)) = 9.8

𝑚

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.

14

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.

It is estimated the total mass of the propulsion system will be 100 kg and use 24 kW of power.

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 = �� ∗ 9.8 𝑚

𝑠2 ∗ 3005 𝑠 → �� = 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) = �� (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) = �� (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 via spiral is the objects orbital velocity at the beginning, 7,616 m/s. This 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 = �� ∗ 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

18

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.

19

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)

(3 ∗ 108

7.145 ∗ 109)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 ∗ 1011

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

equipment and the antenna’s receiver to be kept very cool (see Thermal section).

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 ∗ 10−23𝑚2𝑘𝑔

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.

v. Capture

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:

��(𝑟𝑎𝑑

𝑠2) =

−2 ∗ ��(𝑟𝑎𝑑

𝑠 ) ∗ 𝑟(𝑚𝑠 )

𝑟 (𝑚)

��(𝑚

𝑠2) = �� (

𝑟𝑎𝑑

𝑠)

2

∗ 𝑟(𝑚) − 𝐺 ∗ 𝑀𝑠(𝑘𝑔)

𝑟 (𝑚)2

Where �� is angular acceleration, �� is radial acceleration, �� is angular velocity, commonly written as ω,

�� 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 ∗ 𝑡2

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:

��𝑡𝑜𝑡 = ∑−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

28

next position and velocity. Once the object’s new position and velocity is determined, the Earth’s and Moon’s

positions are updated for that same integration period, t. To keep things simple, Earth’s new position and

velocity use only the Sun’s gravity. Simple kinematic equation for position was used in each dimension,

iterated using the values calculated after running the loop once.

𝑥𝑖 = 𝑥𝑖−1 + 𝑣𝑥 ∗ 𝑡 + 1

2∗ 𝑎𝑥 ∗ 𝑡2

A separate function file was created, “stepxyz.m”, which does all the calculations. This file has 2 major

inputs, a matrix of the heliocentric x-y-z positions and velocities of the object, the Earth, and the Moon, and

a constant, t, which represents integration period. It has 6 outputs, the new heliocentric x-y-z position and

velocity vectors of the object after feeling the effects of the Earth’s, Moon’s, and Sun’s gravity. A second file,

“moveplanets.m”, which is similar to the previous code, was created to calculate the Earth’s and Moon’s next

position and velocity. In the calculations, a step size of 360 seconds was used to keep simulation times down.

The Moon has incredibly complex orbital parameters. Its semi-major axis grows by a few centimeters

every year, in addition to changing in magnitude every month. Its argument of perigee, relative to Earth,

completes one “orbit” every 8.85 years in a process called apsidal procession. By using only the effects of the

Sun and Earth on the Moon, it cannot be confirmed that all of these processes will emerge or be accounted

for.

iii. Thruster Integration

A separate function, “thruster.m”, was written to simulate the effects of the thrust from the

spacecraft. It imports 7 variables, the object’s position and velocity vectors plus integration time, t, and uses

them to orient an additional acceleration boost to one of 4 directions: Pro-/Retrograde, Normal/Anti-normal.

In order to simulate the object accelerating in the prograde direction, normalize the velocity inputs

and multiply it by the 1 N of acceleration it will feel.

Retrograde is done by negating the same calculation.

The normal direction is calculated by finding the cross product of the normalized velocity and position

vectors. Again, anti-normal is done by negating the “normal” operation.

As of now, the code does not simulate radial- or nadir-facing burns. At the time of code creation, the

spacecraft’s geometry did not allow for these types of burns at full thrust, so they were excluded,

more is written in the future work section.

When changing inclination, it is most efficient to thrust near the ascending and descending nodes. So

the code restricts normal/anti-normal burns to +/- π/8 radians of those nodes. The remaining time will be

spent manipulating the semi-major axis to either catch up with Earth or slow down enough for Earth to catch

it. Half the mission is spent slowing down/speeding the object up enough that by the end of the other half

negating what has been done, the object will be captured by the Earth-Moon system. Any sort of thrust

integration changes where the objects ascending and descending nodes are, which is why the code re-

simulates a full orbit of the object to find the new nodes, and the process is repeated.

Seen in Figure 7 below is a simulation of the spacecraft’s effects on the object. The red circle represents

the orbit of Earth. The green line begins below the circle, and where the simulation begins. The spacecraft is

firing retrograde until it reaches a point where it switches to anti-normal. The blue line represents the object’s

trajectory if the spacecraft continued firing retrograde, and the green curve represents the object’s changed

trajectory via inclination change. You see a drop in the blue line, which represents the point at which the

spacecraft switches from firing anti-normal to retrograde again. It continues firing retrograde until it

29

approaches the descending node, where it switches to firing normal. Again, the blue line represents the

unaffected path, and the green represents the altered one. In just one orbit, the inclination change is nearly

removed. The cycle of finding the next node, and designing a firing pattern around it is repeated until the

relative inclination reaches zero. The yellow dot is the Sun, the blue dot is Earth’s starting position, and the

brown dot is the object’s starting position.

Figure 7: Inclination Change Simulation

(Output of “motion.m” file for removing inclination and changing semi-major axis, heliocentric XYZ counter-

clockwise motion)

iv. Capture Mechanics

Small body capture is a phenomenon unexplainable by the two-body problem. This process can only

be observed in an n-body problem, in this case, n=4. The physics behind capture in this case is quite simple:

the Earth’s and Moon’s gravity will pull or hold the object into a stable orbit around the Earth. It is important

to note that the approach velocities of the object will be relatively slow. This means, absent other forces, an

encounter with Earth will be hyperbolic.

To begin, the code used only the Sun’s and Earth’s pull to create a semi-stable orbit, excluding lunar

gravitational effects. Below in Figure 8 is the output for one set of inputs over a period of around 230 days,

where the object enters the Earth’s sphere of influence (around one million kilometers) at the green square

and begins orbiting the Earth in the clockwise direction. At the end of the 230 days, the object is at the black

hexagon, where it appears to remain in orbit around the Earth. The code was run using the same heliocentric

Cartesian coordinates, but plotted to show a geo-centric motion. The blue dot represents Earth, not to scale.

30

Figure 8: Capture Mechanics Code

(Geocentric XY coordinates, clockwise motion)

In this case, the object approached the earth from behind and left (-x, -y, relative to the Earth), which

would represent an object returning from the L5 point. The gravity from the sun prevents the object from

falling deep into Earth’s gravity well, and escaping again. The orbits vary in parameters, but generally keep a

wide elliptical shape.

For objects returning from the L4 point, their initial pass by Earth will be quite close, and need to be

monitored carefully. Once the object emerges on the Sun-side, it will feel the capture effects of the Sun, and

find a stable orbit. It is vital that these input parameters be optimized, as the spacecraft will not be able to

impart a lot of delta-v during this phase, only around 1 m/s per 5 days. A new code was written to find an

approach velocity and position for capture into the Lunar L5 point. A matrix of input variables was created,

including incoming position (with a distance of 900,000 km, varying angles around Earth) and incoming

relative velocity. It was found, however, that this required an immense amount of time and processing

power, as the code had to simulate a year of orbits for each input set. It was calculated to take years on this

computer to run through all of the input sets. By eliminating any input set that went under a certain distance

from the Earth (Geostationary orbit distance) and any set that didn’t make one orbit, the time was reduced,

but would still take days on end to optimize.

31

6. Risk & Financing

a. Risk

There is risk for every single mission to space: deployment mechanisms need to function perfectly, as

there are no backups. The software on board must be thoroughly tested for any event, so it does not send

the spacecraft into an unstable spin or orbit. The communication systems need to work together to ensure

the spacecraft is always under control. All of the engines on the spacecraft, payload, propulsion, and RCS,

should be operating nominally for the entire mission duration.

The largest risk here is if the spacecraft runs out of fuel mid-mission and puts the object on a collision

trajectory with Earth. In this case, the object is very small, and would likely break up in the atmosphere. It is

a C-type asteroid, which we expect to be loosely packed with ice and hydrocarbons. If all 500,000 kg of the

object were travelling at escape velocity when it reached the Earth, around 11,000 m/s, its kinetic energy

would be on the scale of 30 TJ, or an equivalent of 7 kilotons of TNT.

𝐸𝑘 =1

2∗ 𝑚 ∗ 𝑣2 =

1

2∗ 500,000 ∗ 11,0002 = 3.03 ∗ 1013 𝐽

𝐸𝑘 = 3.03 ∗ 1013 𝐽 ∗1 𝑘𝑡 𝑇𝑁𝑇

4.184 ∗ 1012= 7.18 𝑘𝑡 𝑇𝑁𝑇

For comparison, the Chelyabinsk object was estimated to be around 12,500,000 kg, 20 times as

heavy as our object. It was also travelling over 19,000 m/s. Due to its entry trajectory in the Earth’s

atmosphere, it completely blew up in the atmosphere before causing any significant impact damage on the

ground.

In another scenario, the object is in a highly elliptical orbit around the Earth, where it could possibly

cross the orbits of geosynchronous satellites, or lower-orbiting satellites. Damages or destruction of

satellites would have to be paid for by the funding committee, and obviously a second mission must be sent

to fix the orbit of the object.

In a third scenario, the object could crash into the surface of the Moon. The mission would be

considered a failure, but the object could still eventually be accessed, if a future mission is designed to land

on the Moon and study the object.

It is also unknown how the exhaust plume will interact with the object. It could remain a plasma, and

charge the object to an incredibly large potential. The hollow cathode attached to the exhaust of both sets

of thrusters aims to reduce the total charge of the plume, but long term accumulation of exhaust has not

been studied.

The mission is over 23 years long, which increases the risks of component failure due to radiation and

solar panel malfunction due to micro-meteor impacts.

b. Financing

The 2012 Keck study [48] developed a cost estimate based on using NASA as the prime contractor

and set in the 2012 fiscal year, using technologies of TRL 6. Figures 10 describes the cost of each subsystem,

including development costs and contractor fees.

32

Figure 10: Spacecraft Design Costs [48]

DDT&E stands for Design, Development, Test, and Evaluation, which this mission seeks to nearly

eliminate by using state-of-the-art technology. The spacecraft itself is quoted as costing 336 million USD,

which can be assumed to have decreased over the past four years as well. The summary also does not

include fuel costs. Xenon gas can be bought for around $15/L. This mission requires 23,333 L of xenon,

which will come out to 350,000 USD. Higher quality xenon will cost more, but with a bulk purchase, would

be less per liter. It will be assumed in this case to be a maximum of 1 million USD. Seen below, in Table 5, is

the estimated costs of this spacecraft.

Table 5: Prometheus Spacecraft Costs

Subsystem Cost (Millions USD)

Payload 80

Command & Data Handling 40

Communications & Tracking 30

GNC 20

Electrical Power 200

Thermal 30

Structure & Mechanisms 60

Propulsion 150

Propellant 1

Subtotal 611

Integration & Tests 250

TOTAL 861

The figures above include development costs, as the design driver of the mission is to launch

as soon as possible, avoiding DDT&E. Payload and Propulsion have increased due to the number of

thrusters in each, which need a little more development. The cost of the spacecraft is then added to the

overall mission cost, which can be summed up in Figure 11 below.

33

Figure 11: Overhead and Operations Costs [48]

The Keck Institute used NASA as the primary contractor, not an open market, which is reflected in

the insight/oversight, phase A, and spacecraft costs. The launch vehicle will not be an Atlas V, but a Falcon

Heavy, which reduces the cost by almost 200 million USD. In Table 6 below, these factors are all included in

a price estimate for this mission.

Figure 12: Prometheus ARM Cost Estimates

Item Cost (millions USD) Description

Spacecraft Total 860 Off-the-shelf parts & non-singular contractor

Launch Vehicle 100 Falcon Heavy

Mission Ops/GDS 269 23 year mission + setup

Reserves 227 20% reserves

Xenon Fuel 1 23,333 L of high quality xenon

Total 1458

In addition to traditional funding, this mission should use “in-kind” commercial agreements, where a

member of a group will offer a product or service to the group calculated in monetary terms by the group’s

bylaws and how much the member contributed to the group. This can include testing facilities, launch

costs, product development, etc. This can greatly reduce cost of design and development. With this mission

being such a diversely-funded project, with companies, agencies, and universities from all over the world

contributing in some manner, the cost can be driven down quite extensively.

34

7. Conclusions

This report has four main conclusions:

1) Mission length needs to be reduced

2) Orbital Dynamics need to be optimized

3) NEOs need to be studied more

4) Mission is feasible

The mission length is over 24 years, including building and asteroid selection. There are a number of

ways to reduce this. Firstly, the escape phase is almost 9 years without Lunar assistance. By including the

Moon as a slingshot, both the escape phase and rendezvous phase can be reduced by around 20%.

Depending on the funding campaign, it might be more feasible to use chemical rockets to push the

spacecraft into an encounter with the Moon, to save 7 years of spiraling around and avoiding other

satellites.

The redirection phase can be reduced by either selecting a smaller object or developing new thruster

or energy generation technology. By selecting a smaller object, the entire mission’s fuel requirements go

down. The object will accelerate faster with the same amount of force, thus would require less fuel to get

from the Lagrange point to Earth. This would reduce the fuel costs for the rendezvous and escape phases as

well. The other solution is to develop technologies capable of putting out more thrust for the input power

and to generate electrical power without the use of heavy, degrading solar panels. As mentioned, the

VASIMR plasma engine is suggesting large amounts of thrust output for relatively low power consumption.

The Megaflex solar arrays could reduce the mass and size of the dry spacecraft. If nuclear fusion or fission

can be developed in space, perhaps the solar panels can be removed altogether.

The orbital dynamics code that was written was very simplistic, developed only for individual mission

phases, to get a rough estimate of the fuel costs. By integrating the phases and running optimization

software on the inbound and outbound trajectories, the fuel totals would be reduced by an estimated 30%.

The capture dynamics need to be studied over long periods, and the spacecraft needs to be either

cannibalized or sent to collide with the Moon or burn up in Earth’s atmosphere.

We know very little about NEOs less than 100 meters in diameter. We are constantly surprised by

objects whizzing by within Lunar distance, only being detected hours before-hand. The Chelyabinsk object

was only 20 meters across and caused a lot of property damage and injured thousands. The effects of a

larger object hitting the surface would be much greater. Companies like Planetary Resources and Deep

Space Industries have taken it upon themselves to start developing nanosatellites capable of scanning

down objects, which is a great start [49]. Not only is it vital we observe these objects, but it is also

necessary to develop the technology and missions capable of diverting potentially hazardous objects, to

ensure our safety.

And finally, the mission is feasible. With the combination of in-kind commercial agreements, reduced

cost of access to space, and the readiness of redirection technology, humanity should see the redirection of

its first object launching in the next five to ten years. This mission also serves as a standardized platform to

base other missions off of. If an object is detected that would require less total delta-v than this mission,

but in a completely different orbit, then the spacecraft would still be the same. The communication and

power generation requirements might differ slightly, but overall would be relatively the same. A smaller

object could be redirected from further away or a larger object that might be passing close by could both

be returned using this mission as a basis.

35

8. References

[1] R. McMillan, "SPACEWATCH® Project," University of Arizona, 3 June 2016. [Online]. Available:

http://spacewatch.lpl.arizona.edu/. [Accessed 30 June 2016].

[2] F. Crick and L. Orgel, "Directed Panspermia," Icarus, vol. 19, no. 3, pp. 341-346, 1973.

[3] H. H. Hsieh and D. Jewitt, "A Population of Comets in the Main Asteroid Belt," Science, vol. 312, no.

5773, pp. 561-563, 2006.

[4] M. E. Lawler, D. E. Brownlee, S. Temple and M. M. Wheelock, "Iron, magnesium, and silicon in dust

from Comet Halley," Icarus, vol. 80, no. 2, pp. 225-242, 1989.

[5] A. F. Witt, H. C. Gatos, M. Lichtensteiger, M. C. Lavine and C. J. Herman, "Crystal Growth and Steady‐

State Segregation under Zero Gravity: InSb," Journal of the Electrochemical Society, vol. 122, no. 2,

pp. 276-283, 1975.

[6] G. J. Consolmagno and M. J. Drake, "Composition and evolution of the eucrite parent body: evidence

from rare earth elements," Geochimica et Cosmochimica Acta, vol. 41, no. 9, pp. 1271-1282, 1977.

[7] D. Agle, "Radar Images Provide New Details on Halloween Asteroid," NASA JPL, 3 November 2015.

[Online]. Available: http://www.jpl.nasa.gov/news/news.php?feature=4763. [Accessed 7 July 2016].

[8] M. Connors, P. Wiegert and C. Veillet, "Earth’s Trojan asteroid," Nature, vol. 475, pp. 481-483, 2011.

[9] B. Zellner, "Asteroid taxonomy and the distribution of the compositional types," Asteroids, Vols.

A80-24551 08-91, pp. 783-806, 1979.

[10] D. Mendez, "VASIMR," Ad Astra Rocket Company, [Online]. Available:

http://www.adastrarocket.com/aarc/VASIMR. [Accessed 7 July 2016].

[11] O. ATK, "MegaFlex Solar Array - Fact Sheet," [Online]. Available: https://www.orbitalatk.com/space-

systems/space-components/solar-

arrays/docs/FS008_15_OA_3862%20MegaFlex%20Solar%20Array.pdf. [Accessed 7 July 2016].

[12] Chodas, Paul, "Potentially Hazardous Asteroids," NASA, 19 Feb 2016. [Online]. Available:

http://neo.jpl.nasa.gov/orbits/. [Accessed 20 Feb 2016].

[13] M. Freeman, "Mining helium-3 on the Moon for unlimited energy," EIR Science & Technology, pp.

24-29, 31 July 1987.

[14] B. Versteeg, Deep Space Industries, [Online]. Available:

https://tctechcrunch2011.files.wordpress.com/2016/05/harvestor.jpg?w=738. [Accessed 7 July

2016].

[15] M. Bazzocchi and R. Emami, "Comparative Analysis of Redirection Methods for Asteroid Resource

Exploitation," Acta Astronautica, p. 18, 2016.

[16] K. Tsiolkovsky, "Treastice on Outer Space Rocket Equations," 1903.

36

[17] N. Anthony, M. I. Nazarious and A. Walden, "Asteroid Redirect Mission Conceptual Design," Luleå

University of Technology, Luleå, 2016.

[18] J. J. Lissauer, "Timescales for planetary accretion and the structure of the protoplanetary disk,"

Icarus, vol. 69, no. 2, pp. 249-265, 1987.

[19] G. A. Krasinsky, E. V. Pitjeva, M. V. Vasilyev and E. I. Yagudina, "Hidden Mass in the Asteroid Belt,"

Icarus, vol. 158, no. 1, pp. 98-105, July 2002.

[20] P. Chodas, "Introduction: NASA Near-Earth Object Search Program," NASA, 11 December 2013.

[Online]. Available: http://neo.jpl.nasa.gov/programs/intro.html. [Accessed 7 July 2016].

[21] W. F. J. Bottke, A. Morbidelli, R. Jedicke, J.-M. Petit, H. F. Levison, P. Michel and T. S. Metcalfe,

"Debiased Orbital and Absolute Magnitude Distribution of the Near-Earth Objects," Icarus, vol. 156,

no. 2, pp. 399-433, 2002.

[22] I. Sample, "Scientists reveal the full power of the Chelyabinsk meteor explosion," The Guardian,

November 7, 2013.

[23] A. Mainzer, T. Grav, J. Bauer, J. Masiero, R. S. McMillan, R. M. Cutri, R. Walker, E. Wright, P.

Eisenhardt, D. J. Tholen, T. Spahr, R. Jedicke, L. Denneau, E. DeBaun, D. Elsbury, T. Gautier, S.

Gomillion, E. Hand, W. Mo, J. Watkins, A. Wilkins, G. L. Bryngelson, A. Del Pino Molina, S. Desai, M.

Gomez Camus, S. L. Hidalgo, I. Konstantopoulos, J. A. Larsen, C. Malezewski, M. A. Malkan, J.-C.

Mauduit, B. L. Mullan, E. W. Olszewski, J. Pforr, A. Saro, J. V. Scotti and L. H. Wasserman, "NEOWISE

OBSERVATIONS OF NEAR-EARTH OBJECTS: PRELIMINARY RESULTS," The Astrophysical Journal, vol.

743, no. 2, 2011.

[24] J. Nelson, "NASA's WISE Finds Earth's First Trojan Asteroid," NASA JPL, 27 July 2011. [Online].

Available: http://www.jpl.nasa.gov/news/news.php?feature=3080. [Accessed 4 March 2016].

[25] G. Abramson, "Celestia," University of New Mexico, 2015. [Online]. Available:

http://fisica.cab.cnea.gov.ar/estadistica/abramson/celestia/. [Accessed 7 July 2016].

[26] "Hall Effect Thrusters," Busek, 2016. [Online]. Available:

http://www.busek.com/technologies__hall.htm. [Accessed 7 July 2016].

[27] L. Davis and L. Filip, "How Long Does It Take to Develop and Launch Government Satellite Systems?,"

The Aerospace Corporation, El Segundo, CA, USA, 2015, March.

[28] P. Blau, "Delta IV," United Launch Alliance, 2015. [Online]. Available:

http://www.ulalaunch.com/Products_DeltaIV.aspx. [Accessed 7 July 2016].

[29] "Falcon Heavy," SpaceX, 2016. [Online]. Available: http://www.spacex.com/falcon-heavy. [Accessed

7 July 2016].

[30] SpaceX, "Falcon Heavy Capabilities and Services," Pasadena, CA, 2016.

[31] M. Adler, "Space," Stack Exchange, 12 March 2015. [Online]. Available:

http://space.stackexchange.com/questions/8420/general-guidelines-for-modeling-a-low-thrust-ion-

spiral. [Accessed 7 July 2016].

37

[32] P. Keaton, "Low Thrust Rocket Trajectories," Los Alamos National Laboratory, Los Alamos, NM, Nov

2002.

[33] J. Szabo, B. Pote, L. Bryne, S. Paintal, V. Hruby, R. Tedrake, G. Kolencik, C. Freeman and N. Gatsonis,

"Eight Kilowatt Hall Thruster System Characterization," International Electric Propulsion Conference,

Washington DC, 2013.

[34] NASA, "Dawn - Power On! Ion Propulsion System," NASA JPL, Pasadena, CA, 2016.

[35] S. Batteries, "Space," February 2016. [Online]. Available: http://www.saftbatteries.com/market-

solutions/space. [Accessed 7 July 2016].

[36] H. Zirin, "Solar Constant," Encyclopedia Brittanica.

[37] W. Shockley and H. J. Queisser, "Detailed Balance Limit of Efficiency of p‐n Junction Solar Cells,"

Journal of Applied Physics, vol. 32, no. 510, 1961.

[38] S. W. Benson, "Solar Power for Outer Planets Study," NASA Glenn RS, Cleveland, OH, Nov, 2007.

[39] "Rosetta FAQs".

[40] "International Space Station - Solar Arrays," NASA.

[41] "RDR 68 Momentum and Reaction Wheels (RDR)," Rockwell Collins, [Online]. Available:

https://www.rockwellcollins.com/Data/Products/Space_Components/Satellite_Stabilization_Wheel

s/RDR_68_Momentum_and_Reaction_Wheel.aspx. [Accessed 8 July 2016].

[42] B. Gouda, B. Fast and D. Simon, "Satellite Attitude Control," 7 July 2004. [Online]. Available:

http://academic.csuohio.edu/simond/pubs/Gouda04.pdf. [Accessed 8 July 2016].

[43] J. Fisher, "Monopropellant Rocket Engines," [Online]. Available:

http://www.rocket.com/files/aerojet/documents/Capabilities/PDFs/Monopropellant%20Data%20Sh

eets.pdf. [Accessed 7 July 2016].

[44] O. ATK, "Diaphragm Tanks Data Sheets - Sorted by Part Number," [Online]. Available:

http://www.psi-pci.com/Data_Sheet_Index_Diaphragm-PN.htm. [Accessed 8 July 2016].

[45] S. Slobin, "70-m Subnet Telecommunications Interfaces," NASA JPL, Pasadena, Aug 2013.

[46] D. Scuka, Tracking the Spacecraft Following a Comet, ESA, Aug, 2014.

[47] K. Burnett, "Kepler's equation and the Equation of Centre," 1 Novermber 1998. [Online]. Available:

http://www.stargazing.net/kepler/kepler.html. [Accessed 8 July 2016].

[48] Keck Institute for Space Studies, "Asteroid Retrieval Feasibility Study," NASA Jet Propulsion

Laboratory at California Institute of Technology, Pasadena, CA, USA, Apr 2012.

[49] "Prospector 1," Deep Space Industries, 2016. [Online]. Available:

http://deepspaceindustries.com/prospector-1/. [Accessed 11 September 2016].

[50] C. Wolff, Minimum Detectable Signal (MDS),

http://www.radartutorial.eu/09.receivers/rx51.en.html, 2016.

38

[51] "Xenon Propulsion System," Surrey Satellite Technology Ltd, [Online]. Available:

https://www.sstl.co.uk/Products/Subsystems/Propulsion-Systems/Xenon-Propulsion-System.

[Accessed 8 July 2016].

39

9. Appendix

Attached is the code used in this report: phase.m, motion.m, stepxyz.m, moveplanets.m, and thruster.m.

a. Phase.m

%Constants

au = 1.49598e11; %astronomical unit (m)

G = 6.67408e-11; %gravitational constant

Ms = 1.9891e30; %mass of sun (kg)

Me = 5.9722e24; %mass of earth (kg)

Mm = 7.3477e22; %mass of moon (kg)

m= 29.5e3;

t = 2.55*365*24*3600/1e6;

%1 create circular orbit

%2 fire retrograde for 1/2 year, fire prograde for 1/2 year

blep = zeros(4,1e6);

blep(1,1)=au;

blep(2,1)=0;

blep(3,1)=0;

blep(4,1)=sqrt(G*Ms/au)/au;

blep2 = zeros(4,1e6);

blep2(1,1)=au;

blep2(2,1)=0;

blep2(3,1)=0;

blep2(4,1)=sqrt(G*Ms/au)/au;

for i = 2:1e6

ar = blep(4,i-1)^2*blep(1,i-1) - G*Ms/blep(1,i-1)^2;

aph = -2*blep(4,i-1)*blep(2,i-1)/blep(1,i-1);

blep(1,i) = blep(1,i-1) + blep(2,i-1)*t + 0.5*ar*t^2;

blep(2,i) = blep(2,i-1) + ar*t;

blep(3,i) = blep(3,i-1) + blep(4,i-1)*t + 0.5*aph*t^2;

blep(4,i) = blep(4,i-1) + aph*t;

ar2 = blep2(4,i-1)^2*blep2(1,i-1) - G*Ms/blep2(1,i-1)^2;

aph2 = -2*blep2(4,i-1)*blep2(2,i-1)/blep2(1,i-1);

if i < (1e6)/2

aph2 = aph2 + 1/(m*blep2(1,i-1));

blep2(1,i) = blep2(1,i-1) + blep2(2,i-1)*t + 0.5*ar2*t^2;

blep2(2,i) = blep2(2,i-1) + ar2*t;

blep2(3,i) = blep2(3,i-1) + blep2(4,i-1)*t + 0.5*aph2*t^2;

blep2(4,i) = blep2(4,i-1) + aph2*t;

else

aph2 = aph2 - 1/(m*blep2(1,i-1));

blep2(1,i) = blep2(1,i-1) + blep2(2,i-1)*t + 0.5*ar2*t^2;

blep2(2,i) = blep2(2,i-1) + ar2*t;

blep2(3,i) = blep2(3,i-1) + blep2(4,i-1)*t + 0.5*aph2*t^2;

blep2(4,i) = blep2(4,i-1) + aph2*t;

end

end

40

b. Motion.m

au = 1.49598e11; %astronomical unit (m)

G = 6.67408e-11; %gravitational constant

Ms = 1.9891e30; %mass of sun (kg)

ESOI = 9e8; %earths sphere of influence, object will always start this far

Ve = sqrt(G*Ms/au);

rmin = 4.2e7;

rmax = 1e9;

va_or = zeros(2,1000); %valid orbs

%position

th1 = linspace(pi,(3*pi/2),10);

x_t = ESOI*cos(th1);

y_t = ESOI*sin(th1);

%velocity

vels = zeros(2,10,10);

th2 = linspace(0,(2*pi),10);

Vc = 0;

for q = 1:10

vels(1,q,:)=Vc*cos(th2);

vels(2,q,:)=Vc*sin(th2);

Vc = Vc + 50;

end

%start pos loop here (p)

for p = 1:10

%we should have 100 x-y positions and 10000 velocity options for each

%position

xe = au;

ye = 0;

dxe = 0;

dye = Ve;

%dxo = -Ve/80;

%dyo = Ve*1.01;

xo = xe + x_t(p);

yo = ye + y_t(p);

%start vel loops here (g,f)

for g = 1:10

for f = 1:10

dxo = dxe + vels(1,g,f);

dyo = dye + vels(2,g,f);

t=315;

p

g

f

ploot = zeros(1e5,8);

ploot(1,:)=[xo,yo,dxo,dyo,xe,ye,dxe,dye];

valid = 1;

for i = 2:1e5

ploot(i,:)=stepxy(ploot(i-1,:),t);

r=sqrt((ploot(i,1)-ploot(i,5))^2+(ploot(i,2)-ploot(i,6))^2);

if r > rmax || r < rmin

i=1e5;

valid = 0;

end

end

41

if valid == 1

%check if valid orbit

th(1)=atan2(yo,xo);

if th(1) < 0

th(1)=th(1)+2*pi;

end

n=0;

rm = 1e9;

for k = 2:1e5

th(k) = atan2((ploot(k,2)-ploot(k,6)),(ploot(k,1)-ploot(k,5))); %angle

r(k) = sqrt((ploot(k,1)-ploot(k,5))^2+(ploot(k,2)-ploot(k,6))^2);%dist

if th(k) < 0

th(k) = th(k) + 2*pi;

end

%check if completed orbit

if ((th(k) - th(1)) > 0) && ((th(k-1) - th(1)) < 0)

n=n+1;

end

if r(k) < rm

rm = r(k);

end

end

va_or(1,((p-1)*1000+(g-1)*10+f))=n;

va_or(2,((p-1)*1000+(g-1)*10+f))=rm;

end

end

end

end

c. Stepxyz.m

function [out] = stepxyz(body,t)

%body is 3x6 matrix of object,earth,moon positions and vectors in xyz

%t is time increment

au = 1.49598e11; %astronomical unit (m)

G = 6.67408e-11; %gravitational constant

Ms = 1.9891e30; %mass of sun (kg)

Me = 5.9722e24; %mass of earth (kg)

Mm = 7.3477e22; %mass of moon (kg)

%distances between object and sun,earth,moon (r^2=sqrt(x^2+y^2+z^2)

d_o_s = sqrt(body(1,1)^2+body(1,2)^2+body(1,3)^2);

d_o_e = sqrt((body(1,1)-body(2,1))^2+(body(1,2)-body(2,2))^2+(body(1,3)-

body(2,3))^2);

d_o_m = sqrt((body(1,1)-body(3,1))^2+(body(1,2)-body(3,2))^2+(body(1,3)-

body(3,3))^2);

%newtons gravitational law for each body in each direction

a_x = (-G*Ms*body(1,1)/(d_o_s)^3) + (-G*Me*(body(1,1)-body(2,1))/(d_o_e)^3) + (-

G*Mm*(body(1,1)-body(3,1))/(d_o_m)^3);

a_y = (-G*Ms*body(1,2)/(d_o_s)^3) + (-G*Me*(body(1,2)-body(2,2))/(d_o_e)^3) + (-

G*Mm*(body(1,2)-body(3,2))/(d_o_m)^3);

a_z = (-G*Ms*body(1,3)/(d_o_s)^3) + (-G*Me*(body(1,3)-body(2,3))/(d_o_e)^3) + (-

G*Mm*(body(1,3)-body(3,3))/(d_o_m)^3);

%kinematic eqtns

x_n = body(1,1) + body(1,4)*t + 0.5*a_x*t^2;

y_n = body(1,2) + body(1,5)*t + 0.5*a_y*t^2;

z_n = body(1,3) + body(1,6)*t + 0.5*a_z*t^2;

dx_n= body(1,4) + a_x*t;

42

dy_n= body(1,5) + a_y*t;

dz_n= body(1,6) + a_z*t;

ph_n = atan2(y_n,x_n);

if ph_n < 0

ph_n = ph_n + 2*pi;

end

out = [x_n y_n z_n dx_n dy_n dz_n ph_n];

end

d. Moveplanets.m

function [out] = moveplanets(body,t)

au = 1.49598e11; %astronomical unit (m)

G = 6.67408e-11; %gravitational constant

Ms = 1.9891e30; %mass of sun (kg)

Me = 5.9722e24; %mass of earth (kg)

%move moon via earth and sun

%distances between moon and sun/earth

d_m_s = sqrt(body(3,1)^2+body(3,2)^2+body(3,3)^2);

d_m_e = sqrt((body(3,1)-body(2,1))^2+(body(3,2)-body(2,2))^2+(body(3,3)-

body(2,3))^2);

%accelerations from newtons law

a_x = (-G*Ms*body(3,1)/(d_m_s)^3) + (-G*Me*(body(3,1)-body(2,1))/(d_m_e)^3);

a_y = (-G*Ms*body(3,2)/(d_m_s)^3) + (-G*Me*(body(3,2)-body(2,2))/(d_m_e)^3);

a_z = (-G*Ms*body(3,3)/(d_m_s)^3) + (-G*Me*(body(3,3)-body(2,3))/(d_m_e)^3);

%kinematics for moon

xmn = body(3,1) + body(3,4)*t + 0.5*a_x*t^2;

ymn = body(3,2) + body(3,5)*t + 0.5*a_y*t^2;

zmn = body(3,3) + body(3,6)*t + 0.5*a_z*t^2;

dxmn= body(3,4) + a_x*t;

dymn= body(3,5) + a_y*t;

dzmn= body(3,6) + a_z*t;

%move earth via sun

d_e_s = sqrt(body(2,1)^2+body(2,2)^2+body(2,3)^2);

a_x = (-G*Ms*body(2,1)/(d_m_s)^3);

a_y = (-G*Ms*body(2,2)/(d_m_s)^3);

a_z = (-G*Ms*body(2,3)/(d_m_s)^3);

xen = body(2,1) + body(2,4)*t + 0.5*a_x*t^2;

yen = body(2,2) + body(2,5)*t + 0.5*a_y*t^2;

zen = body(2,3) + body(2,6)*t + 0.5*a_z*t^2;

dxen= body(2,4) + a_x*t;

dyen= body(2,5) + a_y*t;

dzen= body(2,6) + a_z*t;

phe_n = atan2(yen,xen);

if phe_n < 0

phe_n = phe_n + 2*pi;

end

phm_n = atan2(ymn,xmn);

if phm_n < 0

phm_n = phm_n + 2*pi;

end

43

%output 2x6 matrix describing new pos' and vel's of earth and moon

out = [xen yen zen dxen dyen dzen phe_n; xmn ymn zmn dxmn dymn dzmn phm_n];

end

e. Thruster.m

function [out] = thruster(body,t,m,o)

a = 3/m;

rp = sqrt(body(1)^2+body(2)^2+body(3)^2); %magnitude of pos vector

rv = sqrt(body(4)^2+body(5)^2+body(6)^2); %magnitude of vel vector

%acceleration vector in prograde

ap = a*[body(4)/rv body(5)/rv body(6)/rv];

%accel vector in normal (cross product posxvel to get norm)

an = a*cross([body(1)/rp body(2)/rp body(3)/rp],[body(4)/rv body(5)/rv

body(6)/rv]);

%o acts as a selector for which thrust direction the s/c fires in

%if o = 1, prograde

if o == 1

out = [(body(1)+(0.5*ap(1)*t^2)) (body(2)+(0.5*ap(2)*t^2))

(body(3)+(0.5*ap(3)*t^2)) (body(4)+ap(1)*t) (body(5)+ap(2)*t) (body(6)+ap(3)*t)

0];

%if o = 2, retrograde

elseif o == 2

out = [(body(1)-(0.5*ap(1)*t^2)) (body(2)-(0.5*ap(2)*t^2)) (body(3)-

(0.5*ap(3)*t^2)) (body(4)-ap(1)*t) (body(5)-ap(2)*t) (body(6)-ap(3)*t) 0];

%if o = 3, normal

elseif o == 3

out = [(body(1)+(0.5*an(1)*t^2)) (body(2)+(0.5*an(2)*t^2))

(body(3)+(0.5*an(3)*t^2)) (body(4)+an(1)*t) (body(5)+an(2)*t) (body(6)+an(3)*t)

0];

%if o = 4, anti-normal

else

out = [(body(1)-(0.5*an(1)*t^2)) (body(2)-(0.5*an(2)*t^2)) (body(3)-

(0.5*an(3)*t^2)) (body(4)-an(1)*t) (body(5)-an(2)*t) (body(6)-an(3)*t) 0];

end

ph_n = atan2(out(2),out(1));

if ph_n < 0

ph_n = ph_n + 2*pi;

end

out(7)=ph_n;

end