Electrostatic Ion Thrusters for Space Debris Removal

Theoretical Physics

Electrostatic ion thrusters for space debris removal

Oscar Larsson, [email protected] Gustav Hedengren, [email protected] SA114X Degree Project in Engineering Physics, First Level Department of Theoretical Physics Royal Institute of Technology (KTH) Supervisor: Christer Fuglesang October 28, 2018 Abstract The current levels of space debris are critical and actions are needed to prevent collisions. In this paper it is examined whether an electrostatic ion thruster can be powerful enough to slow down the debris in a sufficient manner. Furthermore, it is looked into whether the process can be repeated for a significant number of pieces by maneuvering between them. We conclude that the removal process seems possible although some improvements are needed. Maneuvering is costly but despite conservative assumptions, we estimate that about 800 pieces can be removed in one journey made by a satellite weighing ten tonnes of which nine are xenon. Abstract Den nuvarande nivån av rymdskrot är kritisk och åtgärder krävs för att förhindra kollisioner. I denna artikel undersöks det huruvida en elektrostatisk jonkanon är kraftfull nog för att bromsa skrot tillräckligt. Fortsättningsvis undersöks det om denna process är effectiv nog för att återupprepas för ett betydande antal bitar, inklusive manövrering bitarna emellan. Vi drar slutsatsen att processen verkar möjlig att genomföra även om vissa förbättringar behövs. Manövreringen är kostsam men trots konservativa antaganden uppskattar vi att ungefär 800 bitar kan tas ned under en resa av en satellit med vikt 10 ton varav nio ton är xenon. Contents

1 Introduction 2

2 Background 2 2.1 The space debris problem ...... 2 2.2 The Dual-stage 4 grid ion thruster (DS4G) ...... 2

3 Problem 3 3.1 Required breaking, ∆v ...... 4 3.2 Beam divergence ...... 4 3.3 Curvature due to Earth’s magnetic ﬁeld ...... 6 3.4 Maneuvering ...... 7 3.5 Xenon usage ...... 8

4 Results 9 4.1 Required ∆v ...... 9 4.2 Aiming ...... 9 4.3 Xenon usage ...... 10

5 Discussion 11 5.1 Limitations ...... 11 5.2 Modiﬁcations ...... 11

6 Summary and Conclusions 12

7 Acknowledgements 12

Bibliography 13

8 Appendix 14 8.1 ∆v ...... 14 8.2 The curvatures independence of ﬁring angle ...... 14

1 1 Introduction 5. At the very end an appendix containing calculations is provided. We are two students doing a Degree pro- gram in engineering physics at the Royal Institute of Technology, KTH. This pa- 2 Background per on the removal of space debris is our bachelor’s thesis and the main part of our 2.1 The space debris problem bachelor’s project. The idea is entirely our The European Space Agency, ESA, defines own but discussions have been held with space debris as our supervisor Prof. Christer Fuglesang. "...all the inactive, man We propose a system based on an electro- made objects, including frag- static ion thruster positioned on a tracking ments, that are orbiting spacecraft. The thruster will fire xenon Earth or reentering the atmo- ions at a piece of debris, causing it to lose sphere..." speed and fall towards earth. We have chosen to focus on two key aspects: and they warn how 10 km/s collisions with items larger than around 10 cm could lead • Is it possible to obtain enough thrust to complete destruction of an active space- and firing-precision to bring down craft [1]. It is shown in fig 1a) that impact a single piece of debris using this velocities of around 10 km/s are to be ex- method? pected if collision occurs. As of January • Is the procedure, considering both 2017, US Space Surveillance System, the maneuvering and firing, efficient main source of large debris information, enough to be extended to a large has published cataloguing of around 19 quantity of debris? 000 space objects larger than 5–10 cm, to- talling close to 8000 metric tons in Earth After a quick review of the background to orbit.[2] The majority of debris is found in the problem in section 2, the breaking re- low-earth-orbit and as seen in fig 1b), the quirements for a single piece will be exam- highest density is found around 800 km. A ined in section 3.1. The following sections, collision will create new fragments that can 3.2 and 3.3, handles the precision and range cause new collisions in an avalanche effect of the thruster whereas the maneuvering known as Kessler’s syndrome [3]. With ev- of the tracking satellite, as well as the re- ery space mission potentially adding debris quired fuel, will be dealt with in section and risking collision it is increasingly im- 3.4 and 3.5. Calculated results will be pre- portant to deal with this issue. sented in section 4 and discussed in section

2.2 The Dual-stage 4 grid ion thruster (DS4G) The DS4G is a powerful concept engine, developed by the European space agency 2 Figure 1: a) Results from simulations for debris 1-10cm performed with MASTER2009 code. The impact velocity and ﬂux for 1–10 cm debris components calculated for the ISS orbit for 2008, here fragments (red-line) include debris from satellite explosions or collisions. (b) Recent evolution of 1–10 cm debris in low-earth-orbit altitudes.[4] during the early 2000s, designed for long conventional thrusters. Bramanti’s team journeys with large spacecrafts. The idea calculated that a single 20 cm diameter 4- was originally proposed by D.Fearn [5] and gridded ion thruster could operate at 250 has thereafter been tested for ESA by C. kW power to produce a 2.5 N thrust and Bramanti and colleagues [6]. The thruster a speciﬁc impulse of 19,300 s from Xenon was not designed for the purpose of remov- propellant using 30 kV beam potential and ing space debris but possesses characteris- 1 mm ion extraction grid separation [6]. tics desired in order to do so.

An electrostatic thruster works by extract- 3 Problem ing ions from a plasma sheath and then accelerating them using an electrostatic The first question is whether ions ex- potential. The DS4G thruster has, as its hausted from a Dual-stage 4 Grid engine name suggests, two stages consisting of four will give enough thrust to slow down the de- grids instead of the conventional three grid- bris to a lower orbit velocity where the at- ded system. The extra grid allows the first mosphere can finish the process. The mag- stage to extract the ions from the plasma nitude of the thrust as well as the ability sheath before the second stage accelerates to focus it at the debris is of importance. them. Stage 1, consisting of grid 1 and 2, Thereafter it will be examined whether has its potential limited to < 5 kV whilst the process is efficient enough so that a stage 2, grid 3 and 4, can have potentials up reasonable payload of xenon could bring to 80 kV. This allows for less beam curva- down a significant number of debris. Both ture/divergence and higher velocities than parts are heavily intertwined as a more concentrated beam would have a greater hit-

3 percentage, giving better breaking abilities as well as minimizing waste. Better precision also allows for greater range, meaning less maneuvering and saving fuel.

3.1 Required breaking, ∆v Figure 2: Initial and resulting orbits after The targeted piece of debris is assumed to break impulse. (Not to scale) be in a circular orbit at a speciﬁc altitude and to satisfy the equation for speciﬁc orbital energy √ r r r 1 ∆v = µ 2 − (2) r1(r1 + r2) 2r1

2 µ −µ where r1 is the original orbital radius and = k + p = v − = (1) r 2a r2 is the smallest radius for the new elliptic orbit. Since Earth’s mass, M, is many times larger than the mass of any piece of where v is the relative orbital speed, r debris the approximation µ ' GM is made. is the distance from Earth’s center, µ = Thus it is obtained that, G(m + M) is the sum of the gravitational bodies weighed with the gravitational con- r stant, a is the semi-major axis and k and √ r r 1 ∆v = GM 2 − (3) p are the kinetic and potential energies. r1(r1 + r2) 2r1 Since µ and a are constants for two bodies in a specific orbit, the total energy is con- and ∆v does not depend on the mass of stant over time and for a non-circular orbit, the debris, m. A graph for ∆v required to speed varies with radius. Thus, if a circu- obtain a perigee altitude of 350 km as a lar orbit has its kinetic energy reduced at a function of starting altitude can be found certain point, the point will instead become as figure 5 in section 4.1 a low speed outer point of a smaller elliptic orbit (see figure 2). The speed difference, 3.2 Beam divergence ∆v, necessary to move from a certain circular orbit to a desired elliptical orbit can One of the most important factors, both be obtained by defining the original and de- concerning range and efficiency, is the di- sired orbits, solving for v1 and v2 and there- vergence of the beam. Coletti, Gessini and after taking the difference (see Appendix Gabriel [7] has derived that the diffraction 8.1 for details). The formula becomes angle of the beam can be expressed as

4 P r Γ2 r α = 0.62 S − 0.4 2 + 0.53 2 − 1 + P0 r1 λ(1 + Γ) r1

P t r d + t + t + 0.31 S 1 + 2 + 0.35 2 λ + 3 3 2 (1 + 0.5Γ)−1.5 (4) P0 t1 r1 d1

where Γ = (V2 − V3)/(V1 − V2) is the ra- engineering limit. They further argue that tio between the acceleration and extraction a reasonable minimum value on the grid voltage, λ = d2/d1 is the ratio between the spacing is 0.5mm as opposed to the 1mm first and second stage grid gaps, ti, i = used by Bramanti [6]. Important to note is 1, 2, 3, ri, i = 1, 2, are geometrical proper- that the diffraction angle of the beam can ties of the tube and S = r1/d1. Given equa- be reduced to zero and it is from now on as- tion 4 they could then calculate the optimal sumed that the beam is parallel when fired. value of the perveance ratio P/P0 to allow for zero beam divergence. The perveance, Sedlaček [8] calculated that given a per- P , is a measurement of how the much the veance P the beam diameter of an electron space charge affects the beam’s motion. It beam at a distance z can be calculated is a constant dependent on the geometrical using properties of the thruster. P0 is a constant r 4 2e 1/2 2 rb z√ with value P0 = 9 0( m ) d1 where m is 2.09 − 1 = κP (6) the mass of the ions in the beam and e the b b electronic charge. where b is the initial radius of the beam, 1 4 3/2 κ = 1/2 = 3.034 · 10 [V /A] and rb 2π0η 2.335 · 10−6A is the beam radius which is defined as the P = [A/V3/2] (5) d2 [1 + 1 ( dcp )3/2]4/3 position of the outermost ion. The beam cg µ dcg diameter was calculated for an idealized where A is the cross-sectional area, dcg is beam only affected by the repulsive forces the distance between cathode and grid and of the electrons. For an optimal beam of dcp is the distance cathode to plates. electrons with the data measured by [6] the beam radius is about 20 cm at a distance Perveance increases with increasing area of of z = 10 m and about 7 m at a distance the tube and decreasing distance between of z = 100 m. The principle is the same the plates and the grid. Coletti, Gessini for a xenon beam. Despite the mass of the and Gabriel argue that there are two limi- xenon ions being greater by a factor 105 tations on the minimal spacing between the compared to the electrons the spread of the plates. The first is the maximum electric beam will increase compared to the elec- field that can be applied without causing tron beam. This is due to the +8 charge arcing of the beam and the second is the of the ions causing the space charge to be

5 many times greater. However, when firing 8, 2rc. This value can be maximized by let- ions it is also necessary to fire equal charge ting v⊥ = v but will remain finite. Along of the opposite sign to neutralize the satel- the field lines however the range is infinite, lite. Kaganovich et.al [9] suggest that it is although utilizing this range is not neces- possible to neutralize a beam with the help sarly desirable. As shown above, despite of electrons by 50-90% depending on how zero beam divergence, spreading will still the electrons are fired in comparison to the occur and limit us to a certain firing range, beam. here denoted L. The helix shape gives

2 2 2 3.3 Curvature due to Earth’s vkr φ L2 = r2φ2 + (10) magnetic field v⊥ where φ is the curved angle around the z- When firing charged particles in space, the axis (φ = 2π would mean orbiting the z- curvature due to Earth’s magnetic field has axis once). Rearranging and using equation to be accounted for. The curving force is 8 yields determined by Lorentz’s formula, LqB φ = (11) F~ = q~v × B.~ (7) mv Considering an arbitrary xenon ion, its ve- meaning that for a fixed L, the curve of the locity can be split into two components, trajectory is independent of the ration be- one parallel to, and one perpendicular to tween v⊥ and vk as long as the beam is not parallel to the magnetic field (see appendix the magnetic field lines, vk and v⊥ respec- tively. The perpendicular component will 8.2 for details). be affected by equation 7 and form a circu- Continuing, the vector from thruster to lar trajectory of radius rc according to the 2 ~ relation with the centrifugal force F = mv , target is denoted as d. It is now clear r that the projection of d~ onto the xy-plane mv is d = 2r | sin ωL | and that d~ = vkL . r = ⊥ . (8) xy c 2v z v c qB This yields On the contrary, the parallel component will not be affected by the magnetic field ωL v L2 d2 = 4r2 sin2 + k = and thus remain in its original direction. c 2v v The total velocity will therefore result in a 4m2v2 sin2 (β) qBL = sin2 + L2 cos2(β) trajectory the shape of a helix, q2B2 2mv (12) ~r(t) = (rc cos(ωt), −rc sin(ωt), vkt) (9) where β ∈ [0, π] is the firing angle of the qB beam measured between ~v and zˆ. where ω = m and the z-axis is placed along the magnetic field lines which are approx- imated as straight and infinite. The range Considering instead the angle between d~ is limited in the radial direction by the di- and zˆ, θ ∈ [0, π], the distance is calculated ameter of the circle defined by equation to be

6 L2 ω2r2 L2 v2 d2 = 1 − c = 1 − ⊥ (13) cos2(θ) v2 cos2(θ) v2

Equating (12) and (13) allows for an expression of the ﬁring angle as a function of the position of the debris to be derived.

s ! L2q2B2(1 − cos2(θ)) β = arcsin (14) 2 2 2 2 qBL 2 2 2 2 2 2 2 4m v cos (θ) sin ( mv ) − L q B cos (θ) + L q B

The angle θ will be a function of time since are considered; the change of altitude and there is a limit on the mass output per the change of angle. A change of altitude second in the DS4G-thruster. To calculate is relatively cheap in fuel and is performed the exact direction, dependent on time, in by the Hohmann maneuver. The maneu- which to ﬁre will require a careful anal- ver utilizes two velocity changes in order to ysis of the trajectory of both debris and change altitude, the ﬁrst to leave the cur- satellite and precise knowledge of the en- rent orbit and a second to exit the elliptical gineering limits of the engine. This is left transfer orbit and enter a new circular or- out in this report. bit.

r r At impact, only the force-component per- µ 2r2 pendicular to the trajectory of the debris ∆v1 = − 1 (16) r1 r1 + r2 will contribute to retardation. Thus, a r r maximum range with the angle of inci- µ 2r1 ∆v2 = 1 − (17) dence considered is of interest but this is r2 r1 + r2 highly dependent of the trajectory of the where r is the initial orbital altitude and targeted debris. The criterion reads 1 r2 is the new orbital altitude.

~v · ~v Changing the angle, i, which is either the Xe debris ≥ cos α (15) vXevdebris angle between the orbit and the equatorial latitude, inclination, or the longitude an- for allowed angle of attack α. gle, right ascension of the ascending node (RAAN), is signiﬁcantly more costly. The velocity change is 3.4 Maneuvering θ ∆vi = 2vinitial sin (18) Since the allowed ﬁring distance is limited, 2 the satellite must come close to the debris. where θ is the absolute value of the dif- When maneuvering satellites, two changes ference in angle, θ = |inew − iinitial|. See

7 ﬁgure 3 for illustration.

Figure 4: Change in orbital altitude, ∆h as a function of changed velocity. Initial altitude h1 = 800 000 [m].

3.5 Xenon usage Figure 3: The inclination angle θ is marked in the image. RAAN is the corresponding To estimate the consumption of xenon, longitude. both the xenon fired at debris as well as the xenon used for maneuvering have to be con- Furthermore, due to the conservation of sidered. Using the specific impulse, Isp, as momentum, the satellite will accelerate it- well as the standard gravity, g0 = 9.80665 2 self when firing ions at the debris. Consid- m/s , the thrusters mass-flow can be ex- ering the firing process as a constant thrust, pressed as Fthrust, over time, t. The change in velocity becomes F m˙ = thrust . (20) g0 Isp F · t Furthermore, the rocket equation uses the ∆v = thrust (19) m same g0 and Isp when relating the velocity difference to the fuel required to perform a where m is the mass of the spacecraft and maneuver considered constant during the relatively small impulse. This single change in veloc- m0 m0 ity will have similar effect as on the debris; ∆v = ve ln = g0Isp ln . (21) an elliptical orbit will be created, however mf mf this one being larger than the previous cir- Here, ve is the exhaust velocity of the pro- cular one. Thus, equation 3 can be used pellant, m0 is the initial mass including and solved for r2. The change in altitude propellant, and mf is the final mass of the as a function of ∆v can be found in fig- satellite. ure 4. As equation 3 only depends on the mass of Earth, this plot looks the same for An arbitrary maneuver when targeting a both satellite and debris. The difference is piece of debris includes both a change of found in equation 19 where msatellite could orbital altitude, using the Hohmann ma- be 10, 000 times larger than mdebris. neuver, equations 16 and 17, and also an

8 adjustment of inclination angle, equation 18. Denoting initial and ﬁnal orbital radii as r1 and r2, the angle change θ and the satellite mass m0, the xenon consumption, mXe is

−(∆v1+∆v2+∆vi)/g0Isp mXe = m0 1 − e (22) Figure 5: Required ∆v to achieve a lower 5 using the rocket equation, (21), for each of orbital radius r2 = R+3.5·10 m, as a func- the velocity changes. tion of initial altitude, h1 = r1 − R, where R is Earth’s radius (see figure 2). The x- Once in position, the cannon will fire until axis covers all LEO altitudes above 350 km sufficient momentum, ∆p, has been trans- and the dotted line follows the altitude with fered. Assuming the thruster force, F, is greatest debris density, 800 km. constant, using p˙ = F , and introducing a hit-factor, fh (this represents the percentage of the original thrust that makes it to 4.2 Aiming the target), the amount of xenon used is, Using the fact that the magnetic field strength at 800 km altitude has roughly 40 nT [11] as its maximum near the poles and F t ∆p mXe−firing = = . (23) the exhaust velocity is 200.000 m/s, a burst g0Ispfh g0Ispfh perpendicular to the field will, according to equation 8, allow for a curvature radius Adding equations yields the total fuel used. of ∼ 850.000 m. Thus, the double radii maximum range is irrelevant compared to the range due to beam divergence. Fur- 4 Results thermore, using equation 11 with L = 100 meters, the curve will be about 0.007◦ re- 4.1 Required ∆v gardless of firing angle. Thus the magnetic field can be neglected until longer range is In figure 5 the required velocity losses achieved. are plotted for different starting altitudes, equation 3, within low earth orbit. In fig- Assuming a 1% hit factor the allowed firing ure 1 it is clear that the highest density distance for an electron beam, also assum- of debris is found, with a sharp peak, at ing uniformly distributed space charge, is altitudes around 800 km corresponding to about 40 m. Based on the data provided in ∆v = 86 m/s. The inner altitude h2 = 3.2 it is assumed that the spreading of the 350 km is chosen as an altitude where the xenon beam can be made lower than that of atmosphere is dense enough to finish the an electron beam, since the mass is many process within a few months [10]. magnitudes greater and the space-charge

9 can be significantly neutralized. the orbit from one piece to another. In comparison this is approximately the same mass needed to change altitude from 800 to 850 km. It is noted that moving in the radial direction instead of the azimuthal is more efficient. Therefore, the overall xenon usage might be reduced by first adjusting the altitude to minimize θ. It is furthermore not sufficient to be in the same orbit as the piece, the thruster also has to be within firing range. Precision maneuvering and a safety margin is added as a factor 2, bringing mass output to m = 0.0025m0.

Figure 6: The curvature radius due to the magnetic field, rc, as a function of different firing angles, β measured in radians.

4.3 Xenon usage Figure 1 shows the peak density of debris as approximately 10−6pieces/km3. Based on this, an assumption of the distance needed to travel from piece to piece can be made. Based on the peak debris density it can be Figure 7: Space debris dispersion of RAAN calculated that at 800 km altitude there for different inclinations. Circa 900 pieces are approximately 650 pieces of debris per covered. [12] km altitude of the size 1-10 cm in diameter. Most of the debris is concentrated around The Xenon needed to bring down the de- certain orbits and it is assumed, based on bris is calculated according to 23 with the figure 7 that there are around 400 pieces hit-factor, fh = 0.01, as argued in 3.2. The of debris in the interval i = [90◦, 100◦]. velocity of the debris at the altitude 800 Also, according to figure 7 the debris is km is roughly 5 km/s and it is assumed uniformly spread across the other angle, that the angle of attack presented in equa- RAAN. Based on this, it is assumed that tion 15 is never worse than 60◦ so that mass the average change in inclination angle re- needed will increase by a factor 2 at most. quired to reach a piece is ≈ 3◦. This an- Using these assumptions the mass output swers to an average mass output, using per kilogram debris will be 1.07. A piece is equation 19, of the n:th maneuver being assumed to weigh on average 1 kg and the mn ≈ 0.0013m0, where m0 is the current mass output for breaking will therefore be mass of the spacecraft, needed to adjust 1.07 kg per piece.

10 5 Discussion This will also require additional maneuvering which has not been taken into account Removing space debris with a focused when calculating xenon usage. In order to xenon-beam is beneficial in many ways as shorten the time needed the hit-factor and suggested by previous calculations. First the mass flow would have to be increased. of all, it does not require physical contact A more focused burst would not only lead with the debris (although decreasing fir- to less maneuvering but less fired xenon ing distance would certainly improve per- as well. If possible, it might be beneficial formance). Furthermore, there is already to position the satellite closer than 40 m existing research and knowledge regarding initially, in order to increase the hit factor xenon-fueled thrusters which is of essence and therefore reduce both firing and ma- as modifications to existing equipment is neuvering. needed. The method is applicable on a variety of debris as it relies on the ba- The assumptions made when deciding the sic principle momentum conservation (al- hit factor should be considered to be con- though breaking larger debris would require servative. It was assumed that the space- a greater mass flow). Considering that the charge was uniformly distributed over the debris is distributed mainly at certain or- cross sectional area of the beam. This is bits, smart maneuvering might allow a re- definitely a poor estimate considering how duction in fuel consumption and so more the radius was defined. The beam will be debris can be brought down. denser towards the center, but the exact distribution remains unknown. Notwith- standing, beam spreading was assumed only due to Coulomb forces and factors 5.1 Limitations like Maxwellian velocity distribution are neglected. This suggests that the spread- Tracking debris from Earth is today possi- ing might not be too underestimated. Still, ble using radar systems but maneuvering is we believe losses to be smaller in real ap- nevertheless the main contributer to mass plication. output. Positioning will be the most de- manding part of the process. 5.2 Modifications Since the hit factor is low, even for short firing distances, the firing process takes The DS4G has shown performance desir- about 28 h (based on the current hit factor able for the purpose of this report, mainly and distance) and so the satellite would be in terms of high exhaust velocity and mass required to follow the debris closely over output per second. It is important to keep a long time. The model used to calcu- in mind that the DS4G engine is not yet late ∆v was based on the assumption that in use. It has however, in several experi- the momentum was transferred instanta- ments, proven to be of interest for future neously. For the current time required this space missions. The modifications needed is clearly wrong and would result in exten- for our use of the engine has not yet been sive setbacks such as higher required ∆v. discussed. Throughout this report it has

11 been assumed that it is possible to make has not been optimistic and most calcula- the beam completely parallel in accordance tions has been a worst-case scenario. De- with equation 4. However changing the spite this, if at a distance of 40 m from the perveance means changing the geometrical debris the mass output is approximately properties of the thruster. If and how this 1.07 kg per kg debris. Even if the distance affects performance has been left out and needs to be increased it would still be pos- calculations have been performed under the sible to bring down a single piece of debris assumption that the effect on performance due to the high amount of xenon able to is, or can be made, negligible. bring to space. It is further concluded that this method can be repeated in order to An engineering improvement which would bring down multiple pieces of debris. In rapidly increase the hit factor is the neu- this worst case scenario it was estimated tralization of the beam. In theory (once that, with 9 tonnes of xenon and a to- again assuming only Coulomb forces affect tal satellite weight of 10 tonnes, about 800 the spread) a neutralized beam would not pieces of debris with the average size of 1 spread, clearly increasing the allowed fir- kg can be brought down using this method. ing distance. Neutralizing more than the A report by NASA [13] suggests that only 90% suggested by [9] could greatly improve 5 pieces a year need to be removed in order performance, especially considering that to keep the debris at a stable level which spreading depends on the distance fired has been shown possible. squared. 7 Acknowledgements

6 Summary and Conclu- We wish to give thanks to Dr. Nicko- sions lay Ivchenko and Prof. Martin Norgren who has answered many questions. We The results of this paper shows that it also wish to thank Anton Åkesson who has seems possible to bring down one piece of aided in discussions regarding space ﬂight. debris using an electrostatic ion thruster Lastly we wish to give our warmest grati- provided that certain improvements are tude to our supervisor Prof. Christer Fu- made. The calculations made in this report glesang for invaluable guidance.

12 References

[1] European Space Agency, Space debris: The ESA approach, 2 (2017).

[2] National Space Agency, Orbital Debris, Quarterly news, vol.22, 10, (2018)

[3] Donald J. Kessler and Burton G. Cour-Palais (1978). "Collision Frequency of Arti- ﬁcial Satellites: The Creation of a Debris Belt". Journal of Geophysical Research. 83: 2637–2646.

[4] T.Ebisuzaki et al.,Demonstration designs for the remediation of space debris from the International Space Station, Acta Astronautica, vol 112, 102-113, (2015)

[5] Fearn, D G, The use of ion thrusters for orbit raising, J Brit Interplan Soc, 33, 129-137, (1980).

[6] Bramanti, C et.al The innovatice DUAL-STAGE 4-GRID ion thruster concept – Theory and experimental results IAC-06-C4.4.7, Presented at 57th International Astronautical congress, Valencia, Spain 2006.

[7] Coletti, M & Gessini, P & Gabriel, S A 4-Gridded Ion Engine for High Impulse Space Mission (2010)

[8] Sedlaček, Miroslav. Electron Physics of Vacuum and Gaseous Devices, Royal In- stitute of Technology (1995)

[9] Kaganovich, I.D. et.al. (2010). Physics of Neutralization of Intense High-Energy Ion Beam Pulses by Electrons. Physics of Plasmas.056703

[10] Khan, Michael War der Weiterbetrieb von Envisat fahrlässig? (2012). https://scilogs.spektrum.de/go-for-launch/war-der-weiterbetrieb-von-envisat- fahrl-ssig/ Accessed 2018-04-16.

[11] National Center for Environmental Information Geomagnetic calculator https: //www.ngdc.noaa.gov/geomag-web/#igrfwmm (2018-05-07)

[12] Olympio J.T., Frouvelle N. Space debris selection and optimal guidance for removal in the SSO with low-thrust propulsion. Acta Astronautica, vol 99 (2014)

[13] J.C. Liou et.al. (2010). The Kessler Syndrome: Implications to Future Space Op- erations.AAS 10-016

13 8 Appendix

8.1 ∆v The speciﬁc energy is

µ −µ = v2 − = (24) r 2a where is the total energy, a is the semi-major axis, r the radius, v the velocity and µ = G(M + m) is the sum of the masses weighed by gravitational constant. Deﬁning outer circular orbit gives,

r r 2 µ −µ −µ µ µ 1 = v1 − = =⇒ v1 = + = (25) r1 2a1 2r1 r1 2r1 since a = r1. For the inner elliptical orbit 2a = r1 + r2 which yields,

r r 2 µ −µ −µ µ r2µ 2 = v2 − = =⇒ v2 = + = . (26) r1 2a2 r1 + r2 r1 r1(r1 + r2) The diﬀerence is,

r r µ r2µ ∆v = v1 − v2 = − . (27) 2r1 r1(r1 + r2)

8.2 The curvatures independence of ﬁring angle A helix has the curve length dz 2 L2 = (rφ)2 + φ (28) dφ where φ is the rotation angle. Since dz = vkr , rearranging gives dφ v⊥

L2 L2v2 L2v2 φ2 = = ⊥ = ⊥ . (29) v2r2 2 2 2 2 2 2 k r (v + v⊥) r v r + 2 k v⊥

mv⊥ Using equation 8, r = qB yields

LqB φ = . (30) mv which does not depend on aiming.