Theoretical Physics

Electrostatic ion thrusters for space debris removal

Oscar Larsson, osclars@kth.seGustav Hedengren, ghedeng@kth.se

SA114X Degree Project in Engineering Physics, First LevelDepartment 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 preventcollisions. In this paper it is examined whether an electrostatic ion thruster canbe 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 numberof pieces by maneuvering between them. We conclude that the removal processseems possible although some improvements are needed. Maneuvering is costlybut despite conservative assumptions, we estimate that about 800 pieces can beremoved in one journey made by a satellite weighing ten tonnes of which nine arexenon.

Abstract

Den nuvarande nivån av rymdskrot är kritisk och åtgärder krävs för att förhindrakollisioner. I denna artikel undersöks det huruvida en elektrostatisk jonkanonär kraftfull nog för att bromsa skrot tillräckligt. Fortsättningsvis undersöks detom denna process är effectiv nog för att återupprepas för ett betydande antalbitar, inklusive manövrering bitarna emellan. Vi drar slutsatsen att processenverkar möjlig att genomföra även om vissa förbättringar behövs. Manövreringenär kostsam men trots konservativa antaganden uppskattar vi att ungefär 800 bitarkan tas ned under en resa av en satellit med vikt 10 ton varav nio ton är xenon.

Contents1 Introduction 2

2 Background 22.1 The space debris problem . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 The Dual-stage 4 grid ion thruster (DS4G) . . . . . . . . . . . . . . . . . 2

3 Problem 33.1 Required breaking, ∆v . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Beam divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Curvature due to Earth’s magnetic field . . . . . . . . . . . . . . . . . . 63.4 Maneuvering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 Xenon usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Results 94.1 Required ∆v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Aiming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Xenon usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Discussion 115.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Summary and Conclusions 12

7 Acknowledgements 12

Bibliography 13

8 Appendix 148.1 ∆v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.2 The curvatures independence of firing angle . . . . . . . . . . . . . . . . 14

1

1 IntroductionWe are two students doing a Degree pro-gram in engineering physics at the RoyalInstitute of Technology, KTH. This pa-per on the removal of space debris is ourbachelor’s thesis and the main part of ourbachelor’s project. The idea is entirely ourown but discussions have been held withour supervisor Prof. Christer Fuglesang.

We propose a system based on an electro-static ion thruster positioned on a trackingspacecraft. The thruster will fire xenonions at a piece of debris, causing it to losespeed and fall towards earth. We havechosen to focus on two key aspects:

• Is it possible to obtain enough thrustand firing-precision to bring downa single piece of debris using thismethod?

• Is the procedure, considering bothmaneuvering and firing, efficientenough to be extended to a largequantity of debris?

After a quick review of the background tothe problem in section 2, the breaking re-quirements for a single piece will be exam-ined in section 3.1. The following sections,3.2 and 3.3, handles the precision and rangeof the thruster whereas the maneuveringof the tracking satellite, as well as the re-quired fuel, will be dealt with in section3.4 and 3.5. Calculated results will be pre-sented in section 4 and discussed in section

5. At the very end an appendix containingcalculations is provided.

2 Background

2.1 The space debris problem

The European Space Agency, ESA, definesspace debris as

"...all the inactive, manmade objects, including frag-ments, that are orbitingEarth or reentering the atmo-sphere..."

and they warn how 10 km/s collisions withitems larger than around 10 cm could leadto complete destruction of an active space-craft [1]. It is shown in fig 1a) that impactvelocities of around 10 km/s are to be ex-pected if collision occurs. As of January2017, US Space Surveillance System, themain source of large debris information,has published cataloguing of around 19000 space objects larger than 5–10 cm, to-talling close to 8000 metric tons in Earthorbit.[2] The majority of debris is found inlow-earth-orbit and as seen in fig 1b), thehighest density is found around 800 km. Acollision will create new fragments that cancause new collisions in an avalanche effectknown as Kessler’s syndrome [3]. With ev-ery space mission potentially adding debrisand risking collision it is increasingly im-portant to deal with this issue.

2.2 The Dual-stage 4 grid ionthruster (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 MASTER2009code. The impact velocity and flux for 1–10 cm debris components calculated for theISS orbit for 2008, here fragments (red-line) include debris from satellite explosions orcollisions. (b) Recent evolution of 1–10 cm debris in low-earth-orbit altitudes.[4]

during the early 2000s, designed for longjourneys with large spacecrafts. The ideawas originally proposed by D.Fearn [5] andhas thereafter been tested for ESA by C.Bramanti and colleagues [6]. The thrusterwas not designed for the purpose of remov-ing space debris but possesses characteris-tics desired in order to do so.

An electrostatic thruster works by extract-ing ions from a plasma sheath and thenaccelerating them using an electrostaticpotential. The DS4G thruster has, as itsname suggests, two stages consisting of fourgrids instead of the conventional three grid-ded system. The extra grid allows the firststage to extract the ions from the plasmasheath before the second stage acceleratesthem. Stage 1, consisting of grid 1 and 2,has its potential limited to < 5 kV whilststage 2, grid 3 and 4, can have potentials upto 80 kV. This allows for less beam curva-ture/divergence and higher velocities than

conventional thrusters. Bramanti’s teamcalculated that a single 20 cm diameter 4-gridded ion thruster could operate at 250kW power to produce a 2.5 N thrust anda specific impulse of 19,300 s from Xenonpropellant using 30 kV beam potential and1 mm ion extraction grid separation [6].

3 ProblemThe first question is whether ions ex-hausted from a Dual-stage 4 Grid enginewill give enough thrust to slow down the de-bris to a lower orbit velocity where the at-mosphere can finish the process. The mag-nitude of the thrust as well as the abilityto focus it at the debris is of importance.Thereafter it will be examined whetherthe process is efficient enough so that areasonable payload of xenon could bringdown a significant number of debris. Bothparts are heavily intertwined as a more con-centrated beam would have a greater hit-

3

percentage, giving better breaking abilitiesas well as minimizing waste. Better preci-sion also allows for greater range, meaningless maneuvering and saving fuel.

3.1 Required breaking, ∆v

The targeted piece of debris is assumed tobe in a circular orbit at a specific altitudeand to satisfy the equation for specific or-bital energy

ε = εk + εp = v2 − µ

r=−µ2a

(1)

where v is the relative orbital speed, ris the distance from Earth’s center, µ =G(m + M) is the sum of the gravitationalbodies weighed with the gravitational con-stant, a is the semi-major axis and εk andεp are the kinetic and potential energies.Since µ and a are constants for two bodiesin a specific orbit, the total energy ε is con-stant over time and for a non-circular orbit,speed varies with radius. Thus, if a circu-lar orbit has its kinetic energy reduced at acertain point, the point will instead becomea low speed outer point of a smaller ellipticorbit (see figure 2). The speed difference,∆v, necessary to move from a certain cir-cular orbit to a desired elliptical orbit canbe obtained by defining the original and de-sired orbits, solving for v1 and v2 and there-after taking the difference (see Appendix8.1 for details). The formula becomes

Figure 2: Initial and resulting orbits afterbreak impulse. (Not to scale)

∆v =√µ

(√r2

r1(r1 + r2)−√

1

2r1

)(2)

where r1 is the original orbital radius andr2 is the smallest radius for the new ellip-tic orbit. Since Earth’s mass, M , is manytimes larger than the mass of any piece ofdebris the approximation µ ' GM is made.Thus it is obtained that,

∆v =√GM

(√r2

r1(r1 + r2)−√

1

2r1

)(3)

and ∆v does not depend on the mass ofthe debris, m. A graph for ∆v required toobtain a perigee altitude of 350 km as afunction of starting altitude can be foundas figure 5 in section 4.1

3.2 Beam divergence

One of the most important factors, bothconcerning range and efficiency, is the di-vergence of the beam. Coletti, Gessini andGabriel [7] has derived that the diffractionangle of the beam can be expressed as

4

α = 0.62 S

[P

P0

− 0.4r2

r1

Γ2

λ(1 + Γ)+ 0.53

r2

r1

− 1

]+

+ 0.31 S

(P

P0

)[1 +

t2t1

+ 0.35r2

r1

(λ+

d3 + t3 + t2d1

)(1 + 0.5Γ)−1.5

](4)

where Γ = (V2 − V3)/(V1 − V2) is the ra-tio between the acceleration and extractionvoltage, λ = d2/d1 is the ratio between thefirst and second stage grid gaps, ti, i =1, 2, 3, ri, i = 1, 2, are geometrical proper-ties of the tube and S = r1/d1. Given equa-tion 4 they could then calculate the optimalvalue of the perveance ratio P/P0 to allowfor zero beam divergence. The perveance,P , is a measurement of how the much thespace charge affects the beam’s motion. Itis a constant dependent on the geometricalproperties of the thruster. P0 is a constantwith value P0 = 4

9ε0(2e

m)1/2d2

1 where m isthe mass of the ions in the beam and e theelectronic charge.

P =2.335 · 10−6A

d2cg[1 + 1

µ(dcpdcg

)3/2]4/3[A/V3/2] (5)

where A is the cross-sectional area, dcg isthe distance between cathode and grid anddcp is the distance cathode to plates.

Perveance increases with increasing area ofthe tube and decreasing distance betweenthe plates and the grid. Coletti, Gessiniand Gabriel argue that there are two limi-tations on the minimal spacing between theplates. The first is the maximum electricfield that can be applied without causingarcing of the beam and the second is the

engineering limit. They further argue thata reasonable minimum value on the gridspacing is 0.5mm as opposed to the 1mmused by Bramanti [6]. Important to note isthat the diffraction angle of the beam canbe reduced to zero and it is from now on as-sumed that the beam is parallel when fired.

Sedlaček [8] calculated that given a per-veance P the beam diameter of an electronbeam at a distance z can be calculatedusing

2.09

√rbb− 1 =

z

b

√κP (6)

where b is the initial radius of the beam,κ = 1

2πε0η1/2= 3.034 · 104 [V3/2/A] and rb

is the beam radius which is defined as theposition of the outermost ion. The beamdiameter was calculated for an idealizedbeam only affected by the repulsive forcesof the electrons. For an optimal beam ofelectrons with the data measured by [6] thebeam radius is about 20 cm at a distanceof z = 10 m and about 7 m at a distanceof z = 100 m. The principle is the samefor a xenon beam. Despite the mass of thexenon ions being greater by a factor 105

compared to the electrons the spread of thebeam will increase compared to the elec-tron beam. This is due to the +8 chargeof the ions causing the space charge to be

5

many times greater. However, when firingions it is also necessary to fire equal chargeof the opposite sign to neutralize the satel-lite. Kaganovich et.al [9] suggest that it ispossible to neutralize a beam with the helpof electrons by 50-90% depending on howthe electrons are fired in comparison to thebeam.

3.3 Curvature due to Earth’smagnetic field

When firing charged particles in space, thecurvature due to Earth’s magnetic field hasto be accounted for. The curving force isdetermined by Lorentz’s formula,

~F = q~v× ~B. (7)

Considering an arbitrary xenon ion, its ve-locity can be split into two components,one parallel to, and one perpendicular tothe magnetic field lines, v‖ and v⊥ respec-tively. The perpendicular component willbe affected by equation 7 and form a circu-lar trajectory of radius rc according to therelation with the centrifugal force F = mv2

r,

rc =mv⊥qB

. (8)

On the contrary, the parallel componentwill not be affected by the magnetic fieldand thus remain in its original direction.The total velocity will therefore result in atrajectory the shape of a helix,

~r(t) = (rc cos(ωt),−rc sin(ωt), v‖t) (9)

where ω = qBm

and the z-axis is placed alongthe magnetic field lines which are approx-imated as straight and infinite. The rangeis limited in the radial direction by the di-ameter of the circle defined by equation

8, 2rc. This value can be maximized by let-ting v⊥ = v but will remain finite. Alongthe field lines however the range is infinite,although utilizing this range is not neces-sarly desirable. As shown above, despitezero beam divergence, spreading will stilloccur and limit us to a certain firing range,here denoted L. The helix shape gives

L2 = r2φ2 +v2‖r

2φ2

v⊥(10)

where φ is the curved angle around the z-axis (φ = 2π would mean orbiting the z-axis once). Rearranging and using equation8 yields

φ =LqB

mv(11)

meaning that for a fixed L, the curve of thetrajectory is independent of the ration be-tween v⊥ and v‖ as long as the beam is notparallel to the magnetic field (see appendix8.2 for details).

Continuing, the vector from thruster totarget is denoted as ~d. It is now clearthat the projection of ~d onto the xy-planeis dxy = 2rc| sin

(ωL2v

)| and that ~dz =

v‖L

v.

This yields

d2 = 4r2c sin2

(ωL

2v

)+

(v‖L

v

)2

=

=4m2v2 sin2 (β)

q2B2sin2

(qBL

2mv

)+ L2 cos2(β)

(12)

where β ∈ [0, π] is the firing angle of thebeam measured between ~v and z.

Considering instead the angle between ~dand z, θ ∈ [0, π], the distance is calculatedto be

6

d2 =L2

cos2(θ)

(1− ω2r2

c

v2

)=

L2

cos2(θ)

(1− v2

⊥v2

)(13)

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

β = arcsin

(√L2q2B2(1− cos2(θ))

4m2v2 cos2(θ) sin2( qBLmv

)− L2q2B2 cos2(θ) + L2q2B2

)(14)

The angle θ will be a function of time sincethere is a limit on the mass output persecond in the DS4G-thruster. To calculatethe exact direction, dependent on time, inwhich to fire will require a careful anal-ysis of the trajectory of both debris andsatellite and precise knowledge of the en-gineering limits of the engine. This is leftout in this report.

At impact, only the force-component per-pendicular to the trajectory of the debriswill contribute to retardation. Thus, amaximum range with the angle of inci-dence considered is of interest but this ishighly dependent of the trajectory of thetargeted debris. The criterion reads

~vXe · ~vdebrisvXevdebris

≥ cosα (15)

for allowed angle of attack α.

3.4 Maneuvering

Since the allowed firing distance is limited,the satellite must come close to the debris.When maneuvering satellites, two changes

are considered; the change of altitude andthe change of angle. A change of altitudeis relatively cheap in fuel and is performedby the Hohmann maneuver. The maneu-ver utilizes two velocity changes in order tochange altitude, the first to leave the cur-rent orbit and a second to exit the ellipticaltransfer orbit and enter a new circular or-bit.

∆v1 =

õ

r1

(√2r2

r1 + r2

− 1

)(16)

∆v2 =

õ

r2

(1−√

2r1

r1 + r2

)(17)

where r1 is the initial orbital altitude andr2 is the new orbital altitude.

Changing the angle, i, which is either theangle between the orbit and the equatoriallatitude, inclination, or the longitude an-gle, right ascension of the ascending node(RAAN), is significantly more costly. Thevelocity change is

∆vi = 2vinitial sin

(θ

2

)(18)

where θ is the absolute value of the dif-ference in angle, θ = |inew − iinitial|. See

7

figure 3 for illustration.

Figure 3: The inclination angle θ is markedin the image. RAAN is the correspondinglongitude.

Furthermore, due to the conservation ofmomentum, the satellite will accelerate it-self when firing ions at the debris. Consid-ering the firing process as a constant thrust,Fthrust, over time, t. The change in velocitybecomes

∆v =Fthrust · t

m(19)

where m is the mass of the spacecraft andconsidered constant during the relativelysmall impulse. This single change in veloc-ity will have similar effect as on the debris;an elliptical orbit will be created, howeverthis one being larger than the previous cir-cular one. Thus, equation 3 can be usedand solved for r2. The change in altitudeas a function of ∆v can be found in fig-ure 4. As equation 3 only depends on themass of Earth, this plot looks the same forboth satellite and debris. The difference isfound in equation 19 where msatellite couldbe 10, 000 times larger than mdebris.

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

3.5 Xenon usage

To estimate the consumption of xenon,both the xenon fired at debris as well as thexenon used for maneuvering have to be con-sidered. Using the specific impulse, Isp, aswell as the standard gravity, g0 = 9.80665m/s2, the thrusters mass-flow can be ex-pressed as

m =Fthrustg0 Isp

. (20)

Furthermore, the rocket equation uses thesame g0 and Isp when relating the velocitydifference to the fuel required to perform amaneuver

∆v = ve ln

(m0

mf

)= g0Isp ln

(m0

mf

). (21)

Here, ve is the exhaust velocity of the pro-pellant, m0 is the initial mass includingpropellant, and mf is the final mass of thesatellite.

An arbitrary maneuver when targeting apiece of debris includes both a change oforbital altitude, using the Hohmann ma-neuver, equations 16 and 17, and also an

8

adjustment of inclination angle, equation18. Denoting initial and final orbital radiias r1 and r2, the angle change θ and thesatellite mass m0, the xenon consumption,mXe is

mXe = m0

(1− e−(∆v1+∆v2+∆vi)/g0Isp

)(22)

using the rocket equation, (21), for each ofthe velocity changes.

Once in position, the cannon will fire untilsufficient momentum, ∆p, has been trans-fered. Assuming the thruster force, F, isconstant, using p = F , and introducing ahit-factor, fh (this represents the percent-age of the original thrust that makes it tothe target), the amount of xenon used is,

mXe−firing =Ft

g0Ispfh=

∆p

g0Ispfh. (23)

Adding equations yields the total fuel used.

4 Results

4.1 Required ∆v

In figure 5 the required velocity lossesare plotted for different starting altitudes,equation 3, within low earth orbit. In fig-ure 1 it is clear that the highest densityof debris is found, with a sharp peak, ataltitudes around 800 km corresponding to∆v = 86 m/s. The inner altitude h2 =350 km is chosen as an altitude where theatmosphere is dense enough to finish theprocess within a few months [10].

Figure 5: Required ∆v to achieve a lowerorbital radius r2 = R+3.5·105m, as a func-tion of initial altitude, h1 = r1 − R, whereR is Earth’s radius (see figure 2). The x-axis covers all LEO altitudes above 350 kmand the dotted line follows the altitude withgreatest debris density, 800 km.

4.2 Aiming

Using the fact that the magnetic fieldstrength at 800 km altitude has roughly 40nT [11] as its maximum near the poles andthe exhaust velocity is 200.000 m/s, a burstperpendicular to the field will, accordingto equation 8, allow for a curvature radiusof ∼ 850.000 m. Thus, the double radiimaximum range is irrelevant compared tothe range due to beam divergence. Fur-thermore, using equation 11 with L = 100meters, the curve will be about 0.007◦ re-gardless of firing angle. Thus the magneticfield can be neglected until longer range isachieved.

Assuming a 1% hit factor the allowed firingdistance for an electron beam, also assum-ing uniformly distributed space charge, isabout 40 m. Based on the data provided in3.2 it is assumed that the spreading of thexenon beam can be made lower than that ofan electron beam, since the mass is manymagnitudes greater and the space-charge

9

can be significantly neutralized.

Figure 6: The curvature radius due to themagnetic field, rc, as a function of differentfiring angles, β measured in radians.

4.3 Xenon usage

Figure 1 shows the peak density of debris asapproximately 10−6pieces/km3. Based onthis, an assumption of the distance neededto travel from piece to piece can be made.Based on the peak debris density it can becalculated that at 800 km altitude thereare approximately 650 pieces of debris perkm altitude of the size 1-10 cm in diameter.Most of the debris is concentrated aroundcertain orbits and it is assumed, based onfigure 7 that there are around 400 piecesof debris in the interval i = [90◦, 100◦].Also, according to figure 7 the debris isuniformly spread across the other angle,RAAN. Based on this, it is assumed thatthe average change in inclination angle re-quired to reach a piece is ≈ 3◦. This an-swers to an average mass output, usingequation 19, of the n:th maneuver beingmn ≈ 0.0013m0, where m0 is the currentmass of the spacecraft, needed to adjust

the orbit from one piece to another. Incomparison this is approximately the samemass needed to change altitude from 800to 850 km. It is noted that moving in theradial direction instead of the azimuthal ismore efficient. Therefore, the overall xenonusage might be reduced by first adjustingthe altitude to minimize θ. It is further-more not sufficient to be in the same orbitas the piece, the thruster also has to bewithin firing range. Precision maneuveringand a safety margin is added as a factor 2,bringing mass output to m = 0.0025m0.

Figure 7: Space debris dispersion of RAANfor different inclinations. Circa 900 piecescovered. [12]

The Xenon needed to bring down the de-bris is calculated according to 23 with thehit-factor, fh = 0.01, as argued in 3.2. Thevelocity of the debris at the altitude 800km is roughly 5 km/s and it is assumedthat the angle of attack presented in equa-tion 15 is never worse than 60◦ so that massneeded will increase by a factor 2 at most.Using these assumptions the mass outputper kilogram debris will be 1.07. A piece isassumed to weigh on average 1 kg and themass output for breaking will therefore be1.07 kg per piece.

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5 Discussion

Removing space debris with a focusedxenon-beam is beneficial in many ways assuggested by previous calculations. Firstof all, it does not require physical contactwith the debris (although decreasing fir-ing distance would certainly improve per-formance). Furthermore, there is alreadyexisting research and knowledge regardingxenon-fueled thrusters which is of essenceas modifications to existing equipment isneeded. The method is applicable on avariety of debris as it relies on the ba-sic principle momentum conservation (al-though breaking larger debris would requirea greater mass flow). Considering that thedebris is distributed mainly at certain or-bits, smart maneuvering might allow a re-duction in fuel consumption and so moredebris can be brought down.

5.1 Limitations

Tracking debris from Earth is today possi-ble using radar systems but maneuvering isnevertheless the main contributer to massoutput. Positioning will be the most de-manding part of the process.

Since the hit factor is low, even for shortfiring distances, the firing process takesabout 28 h (based on the current hit factorand distance) and so the satellite would berequired to follow the debris closely overa long time. The model used to calcu-late ∆v was based on the assumption thatthe momentum was transferred instanta-neously. For the current time required thisis clearly wrong and would result in exten-sive setbacks such as higher required ∆v.

This will also require additional maneuver-ing which has not been taken into accountwhen calculating xenon usage. In order toshorten the time needed the hit-factor andthe mass flow would have to be increased.A more focused burst would not only leadto less maneuvering but less fired xenonas well. If possible, it might be beneficialto position the satellite closer than 40 minitially, in order to increase the hit factorand therefore reduce both firing and ma-neuvering.

The assumptions made when deciding thehit factor should be considered to be con-servative. It was assumed that the space-charge was uniformly distributed over thecross sectional area of the beam. This isdefinitely a poor estimate considering howthe radius was defined. The beam will bedenser towards the center, but the exactdistribution remains unknown. Notwith-standing, beam spreading was assumedonly due to Coulomb forces and factorslike Maxwellian velocity distribution areneglected. This suggests that the spread-ing might not be too underestimated. Still,we believe losses to be smaller in real ap-plication.

5.2 Modifications

The DS4G has shown performance desir-able for the purpose of this report, mainlyin terms of high exhaust velocity and massoutput per second. It is important to keepin mind that the DS4G engine is not yetin use. It has however, in several experi-ments, proven to be of interest for futurespace missions. The modifications neededfor our use of the engine has not yet beendiscussed. Throughout this report it has

11

been assumed that it is possible to makethe beam completely parallel in accordancewith equation 4. However changing theperveance means changing the geometricalproperties of the thruster. If and how thisaffects performance has been left out andcalculations have been performed under theassumption that the effect on performanceis, or can be made, negligible.

An engineering improvement which wouldrapidly increase the hit factor is the neu-tralization of the beam. In theory (onceagain assuming only Coulomb forces affectthe spread) a neutralized beam would notspread, clearly increasing the allowed fir-ing distance. Neutralizing more than the90% suggested by [9] could greatly improveperformance, especially considering thatspreading depends on the distance firedsquared.

6 Summary and Conclu-sions

The results of this paper shows that itseems possible to bring down one piece ofdebris using an electrostatic ion thrusterprovided that certain improvements aremade. The calculations made in this report

has not been optimistic and most calcula-tions has been a worst-case scenario. De-spite this, if at a distance of 40 m from thedebris the mass output is approximately1.07 kg per kg debris. Even if the distanceneeds to be increased it would still be pos-sible to bring down a single piece of debrisdue to the high amount of xenon able tobring to space. It is further concluded thatthis method can be repeated in order tobring down multiple pieces of debris. Inthis worst case scenario it was estimatedthat, with 9 tonnes of xenon and a to-tal satellite weight of 10 tonnes, about 800pieces of debris with the average size of 1kg can be brought down using this method.A report by NASA [13] suggests that only5 pieces a year need to be removed in orderto keep the debris at a stable level whichhas been shown possible.

7 AcknowledgementsWe wish to give thanks to Dr. Nicko-lay Ivchenko and Prof. Martin Norgrenwho has answered many questions. Wealso wish to thank Anton Åkesson who hasaided in discussions regarding space flight.Lastly we wish to give our warmest grati-tude to our supervisor Prof. Christer Fu-glesang for invaluable guidance.

12

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[4] T.Ebisuzaki et al.,Demonstration designs for the remediation of space debris fromthe 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 InternationalAstronautical congress, Valencia, Spain 2006.

[7] Coletti, M & Gessini, P & Gabriel, S A 4-Gridded Ion Engine for High ImpulseSpace 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-EnergyIon 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 removalin 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

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8 Appendix

8.1 ∆v

The specific energy is

ε = v2 − µ

r=−µ2a

(24)

where ε is the total energy, a is the semi-major axis, r the radius, v the velocity andµ = G(M + m) is the sum of the masses weighed by gravitational constant. Definingouter circular orbit gives,

ε1 = v21 −

µ

r1

=−µ2a1

=⇒ v1 =

√−µ2r1

+µ

r1

=

õ

2r1

(25)

since a = r1. For the inner elliptical orbit 2a = r1 + r2 which yields,

ε2 = v22 −

µ

r1

=−µ2a2

=⇒ v2 =

√−µ

r1 + r2

+µ

r1

=

√r2µ

r1(r1 + r2). (26)

The difference is,

∆v = v1 − v2 =

õ

2r1

−√

r2µ

r1(r1 + r2). (27)

8.2 The curvatures independence of firing angle

A helix has the curve length

L2 = (rφ)2 +

(dz

dφφ

)2

(28)

where φ is the rotation angle. Since dzdφ

=v‖r

v⊥, rearranging gives

φ2 =L2

r2 +v2‖r

2

v2⊥

=L2v2

⊥r2(v2

‖ + v2⊥)

=L2v2

⊥r2v2

. (29)

Using equation 8, r = mv⊥qB

yields

φ =LqB

mv. (30)

which does not depend on aiming.

14