The Terminator Tether - Tethers Unlimited

98-3491APPLICATION OF THE TERMINATOR TETHER ELECTRODYNAMIC DRAGTECHNOLOGY TO THE DEORBIT OF CONSTELLATION SPACECRAFTRobert L. Forward, Robert P. HoytTethers Unlimited, Inc.8114 Pebble Ct., Clinton, WA 98236phone/fax: +1-206-306-0400Chauncey UphoffFortune-Eight Aerospace Industries, Inc.The Terminator Tether is a small, lightweight system that will use passiveelectrodynamic tether drag to rapidly deorbit spacecraft from low Earth orbit. Studies of theapplication of electrodynamic drag to the deorbit of constellation satellites indicates that theTerminator Tether offers significant mass savings compared to conventional rocket-baseddeorbit systems. Moreover, because it uses passive electrodynamic drag to achieve deorbit, i tcan deorbit the spacecraft even if the host has lost power and control functions. Numericalanalyses of the performance of the Terminator Tether indicate that a 5 to 10 km longconducting tether weighing only 2% of the host spacecraft mass can deorbit a typicalconstellation satellite within a few months. Although the tether increases the total collisioncross-sectional area of the satellite system during the deorbit phase, the electrodynamic dragis so many times greater than atmospheric drag at constellation altitudes that the tether canreduce the collisional Area-Time product for the satellite by several orders of magnitude. TheTerminator Tether thus can provide a low-cost and reliable method of mitigating the growthof debris in valuable constellation orbits.I . I N T R O D U C T I O NThis paper investigates the use of a highlysurvivable, conducting electrodynamic tether foruse as a “Terminator Tether” for removingunwanted Low-Earth-Orbit (LEO) spacecraftfrom orbit at the end their useful lives. 1 , 2 Whena spacecraft fails, or has completed its missionand is no longer wanted, the Terminator Tether,weighing a small fraction of the mass of the hostspacecraft, will be deployed. At both ends of thetether, a means of providing electrical contactwith the ambient plasma will be provided toenable current to be transmitted to and from theionospheric plasma. The electrodynamicinteraction of the conducting tether moving a torbital speeds across the Earth’s magnetic fieldwill induce current flow along the tether. Theresulting energy loss from the heat generated bythe current flowing through the ohmic resistancein the tether will remove energy from thespacecraft. Consequently, the orbital energy ofthe spacecraft will decay, causing it to deorbitfar more rapidly than it would due toatmospheric drag alone. Whereas a defunctspacecraft left in its orbit can take hundreds orthousands of years to deorbit due to atmosphericdrag, a spacecraft with a Terminator Tether canbe deorbited in weeks or months. The TerminatorTether thus is a low-mass means of reducing boththe risk of spacecraft fratricide and the amountof orbital space debris that must be coped with inthe future.In the first section of this paper, we willbegin by discussing the orbital debris problemmotivating the development of the TerminatorTether. We will then describe the basic conceptof electrodynamic tether drag, and review theresults of past experiments related to thisconcept. In the second section, we will developanalytical methods for predicting the effectivenessof Terminator Tether systems fordeorbiting spacecraft from various orbits. In thethird section, we will describe methods ofoptimizing the electrodynamic drag on thespacecraft, while concurrently stabilizing theelectrodynamic tether libration. In the fourthsection, we examine the effectiveness of theTerminator Tether for reducing the Area-Time-Product for orbital decay of LEO spacecraft, andcompare it to conventional deorbit methods.Finally, we describe two implementations of theTerminator Tether concept for reducing the LEOdebris environment.1Terminator Tether and Remora Remover are trademarks of Tethers Unlimited, Inc.Copyright © 1998 by Tethers Unlimited, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

98-3491( ) 2 (1)P m T v M B T=rdwhere r is the resistivity and d the density of theconducting tether material. This power isconverted into heat by the resistance of thetether and radiated away into space, extractingkinetic energy from the host spacecraft. For am T =10 kg tether of aluminum with resistivity ofr=27.4 nΩ-m and density d=2700 kg/m 3 , orbitingover the magnetic equator at an altitude of1000 km, at a velocity v M =6814 m/s relative tothe Earth's transverse magnetic field B T =20 µT,the power dissipated is P=2510 W! This energyloss in the form of heat must necessarily come outof the kinetic energy of the host spacecraft. For atypical example, a 1000 kg spacecraft in a1000 km high orbit subjected to an energy loss of2510 J/s from a 10 kg tether (1% the mass of thehost spacecraft) will be deorbited in a few weeks.Similar conclusions have been reached by manyothers, including members of the NASA/MSFCProSEDS team. 7 ,8,9,10SatelliteVelocity,TetherCurrentConductingHoytetherTerminator TetherDeployer/ControllerEarth's⊗ MagneticFieldElectrodynamicDrag ForceFigure 1. The Terminator Tether concept.Experimental Confirmations of Induced PowerLevelsPower levels of the magnitude estimated inthe previous paragraph have been measured in ar e a l orbital space experiment, the TSS-1Rmission carried out on the Shuttle Orbiter in1995. 11 In that experiment, a large Italianspacecraft, 1.6 m in diameter, was deployedupward from the Shuttle Orbiter at the end of aconducting copper wire tether covered withelectrical insulation. As the tether was slowlydeployed upwards, a series of measurements weremade of the open circuit voltage induced in thetether by its motion through the Earth's magneticfield. The voltage between the end of the tetherand the Orbiter ground varied from zero volts a tthe start to 3500 V when the amount of tetherdeployed approached its maximum length of20 km. Periodically, the end of the tether wasconnected either to one of two different electronguns, which supplied contact to the surroundingspace plasma, or to the Orbiter ground, whichproved to be a surprisingly good plasma contactorvia a combination of ion collection and secondaryelectron emission. The current flow through thetether was deliberately limited by controlcircuits and the current capacity of the electronguns, but power levels of 1800 W were reached.The tether was intended to have a fullydeployed length of 20 km, but at a deployedlength of 19.5 km, when about 3500 V was beinginduced at the end of the tether inside theOrbiter reel mechanism, a flaw in the insulationallowed an electrical spark to jump in anuncontrolled manner from the tether to theOrbiter ground. With no control circuits to keepthe current level down, the current flow jumped to1.1 A and the total power generated wasP=3850 W. Most of this energy went into theelectrical arc, which burned through the tether,causing it to break and halting the experiment.This experiment showed that large areas of bareconducting material, such as that provided by theItalian spacecraft and the Orbiter spacecraft, cancollect amperes of current, while thousands ofvolts of potential can be generated by sufficientlylong tethers moving at orbital speeds.Thus, both theory and experimental dataindicate that significant amounts of electrodynamicdrag force can be obtained from a lowmass conducting tether attached to a hostspacecraft, p r o v i d e d the ends of the conductor can3American Institute of Aeronautics and Astronautics

98-3491exchange sufficient numbers of electrons with thesurrounding space plasma.Experimental data from the TSS-1R dataalso produced the result that the efficiency of abare metal surface in “contacting” the spaceplasma is many times better than the standardtheory would predict. The 8 square meters of baresurface area of the Italian spacecraft weresufficient to collect the 1.1 A of electron current.Flight Demonstration of Electrodynamic DragBecause of the results of the TSS-1Rexperiment, and because of recent theoreticalstudies that indicate that a bare wire can easilycollect electrons, 12 Les Johnson, Nobie Stone,Chris Rupp, and others at NASA Marshall SpaceFlight Center have formed a team, whichincludes the present authors, which is embarkedon a new flight experiment. 10The experiment isscheduled for a piggy-back flight on a Delta IIlaunch of an AF Global Positioning Satellite inearly 2000. The goal of the experiment is todemonstrate that electrodynamic drag from awire moving at orbital speeds through theEarth's magnetic field will create a large enoughelectrodynamic drag force to deorbit the >1000 kgDelta II second stage in a few weeks. This isessentially a demonstration of the Loftus electrodynamicdrag deorbit concept and the first step inthe development of a Terminator Tether.The ProSEDS (Propulsion Small ExpendabletetherDeployer System) mission is presentlybaselined to use a 5 km long copper wire massing18 kg, a 20 km long nonconducting tether, and a25 kg ballast mass on the end of the tether. Thetotal of 25 km of tether length and the 25 kgballast mass on the end will provide enoughgradient force to keep the tether aligned near thezenith, so that the direction of the current in thetether is at right angles to both the direction ofthe spacecraft motion in the nominal EWdirection and the Earth's near-equatorialmagnetic field in the nominal NS direction.An important feature of the ProSEDSexperiment is that it is designed to be completelyself-powered. It uses a battery to initiatedeployment and to power up the plasmacontactor, but once current is flowing through thetether, some of the power is tapped off and usedto recharge the battery. The battery in turnpowers the current control electronics, thetelemetry system, and the plasma contactor. TheProSEDS mission will not be designed to allowground control changes in operation, primarilybecause of the increase in cost associated withthat option.Terminator TetherIn this paper we propose a commercializedversion of the ProSEDS experiment, which wouldconsist of a small, low-mass deployer/controllerpackage containing a large collecting area, shortlength, highly-survivable, multiline spacetether, such as a Hoytape mesh 13 made ofaluminum wire, as a “Terminator Tether” forupper stages and LEO spacecraft, especially theexpected multitude of LEO constellationsatellites and their upper stage launcher/dispensers. The Terminator Tether would bedeployed when the host vehicle is no longerworking or no longer wanted. The electrodynamicdrag from the Terminator Tether would rapidlyremove the unwanted vehicle from theconstellation orbit altitude and a few weeks latercomplete the deorbit of the host vehicle fromspace by burnup in the upper atmosphere of theEarth. For a Terminator Tether to be of maximumusefulness for constellation satellites, it would bedesirable to minimize the mass and the length ofthe tether, while at the same time maximizingthe electrodynamic drag force. A lower tethermass means more mass for revenue producingtransponders, while a shorter tether lengthmeans a lower collision cross-section Area-Time-Product during deorbit. Since the proposedTerminator Tether would autonomously maintaincontact with ground control during the deorbitphase, and ground control can control its rate ofdescent, a Terminator Tether can avoid the largerspacecraft with well-known and predictableorbits, thus decreasing the probability of acollision below that predicted using the Area-Time-Product alone.Electrodynamic Tether ConstraintsThe electrodynamic tether is assumed to bemade of a conducting metal, and have a length L,density d, resistivity r, and cross-sectional area Athat is constant along the length of the tether. Ifthe tether is a single round wire of diameter D,then the cross-sectional area is Α=πD 2 /4. Becauseof the micrometeorite and space debris hazard,however, it is likely the tether will be made upof redundantly interconnected multiple lineswhose individual cross-sectional areas add up toA. Given these assumptions, the tether mass is4American Institute of Aeronautics and Astronautics

98-3491then m T =dLA, while the end-to-end tetherresistance is R=rL/A=rdL 2 /m T .S p e c i f i c C o n d u c t i v i t y P a r a m e t e r : The choice ofthe metal conductor to be used in a space tether isdetermined by a combination of low resistivity(high conductivity) and low density, with cost,strength, and melting point as secondaryconsiderations for certain applications. Copperhas a resistivity r=17.0 nΩ-m, a densityd=8933 kg/m 3 , and a "specific conductivity" of1/rd=6,585 m 2 /Ω-kg. Aluminum has a resistivityr=27.4 nΩ-m, which is significantly greater thanthat of copper, but it has a much lower density ofd=2700 kg/m 3 . As a result, aluminum’s “specificconductivity” of 1/rd=13,500 m 2 /Ω-kg is twice theconductivity per unit mass of copper. Silver,because of its higher density and higher cost, isnot competitive as an electrodynamic spacetether even though its resistivity of 16.1 nΩ-m isslightly less than that of copper. An alternatecandidate material would be beryllium, with aresistivity r=32.5 nΩ-m, density d=1850 kg/m 3 ,and a specific conductivity of 1/rd=16,630 m 2 /Ωkg,slightly better than that of the much cheaperaluminum. Beryllium also has a higher meltingpoint at 1551 K than aluminum at 933 K, so someof its alloys may be a preferred material for someelectrodynamic applications despite its highermaterials cost. Unfortunately, despite decades ofmetallurgical research by the nuclear powerindustry, highly ductile alloys of beryllium havenot been found, so it is difficult to make it intowire. As a result, because of its high specificconductivity, low cost, and ready availability inductile wire form, we will assume for this paperthat the electrodynamic tether will be made ofaluminum wire.T y p i c a l R e s i s t a n c e V a l u e s : To be competitive,the mass of the tether needs to be a small fractionof the mass of the host spacecraft it is required todeorbit. Since a typical constellation satellitehas a mass of about 1000 kg, a typical TerminatorTether with a mass that is 2% of the hostspacecraft mass would consist of a deployer/controller package with a mass m D =10 kg,containing an aluminum tether with a massm T =10 kg with a volume of m T /d=LA=3.70x10 −3 m 3 .If this 10 kg of aluminum were formed into atether with a length of L=2 km and a crosssectionalarea of A=1.85 mm 2 , then the end-toendresistance of the tether would beR=rL/A=rdL 2 /m T =29.6 Ω. A longer tether wouldhave a proportionately smaller cross-sectionalarea and a higher resistance; for example, a 5 kmlong tether with the same mass would have aresistance of 185 Ω.I I .T ERMINATOR T ETHER A NALYSIS ANDO PTIMIZATIONI I . A . C h a n g i n g a S p a c e c r a f t O r b i t U s i n gE l e c t r o d y n a m i c T e t h e r P r o p u l s i o nTo determine the effectiveness of theTerminator Tether system for de-orbiting a spentspacecraft, we will now develop analytical toolsfor predicting the time required for a electrodynamictether to deorbit a spacecraft from aspecified altitude h and inclination i.Assumptions:C i r c u l a r O r b i tWe will assume that the spacecrafttrajectory is a nearly circular spiral which can,for each orbit, be approximated by a circular orbitwith radius r.T e t h e r O r i e n t a t i o nWe will assume that there is a balancebetween the electrodynamic drag on the tetherand the gradient forces on the tether whichcauses the tether to hang at an angle α from thelocal vertical, with the rotation in the directionopposite to the velocity vector. In reality,variations in the electrodynamic forces along thetether length will likely cause the tether to hangin a curved manner, and variations in the dragforce during an orbit will cause the tether tolibrate around some equilibrium point, but for thisanalysis we will assume that it hangs straight a tthe specified angle.The tether length vector can thus beexpressed asL = L rˆ cosα + vˆ sinα . (2)( )C u r r e n t C o l l e c t i o nWe will assume that the Terminator Tethersystem provides sufficient current contact withthe ionospheric plasma to transmit the fullcurrent possible between the tether and theambient plasma. Consequently, we will ignorethe limitations in the tether current level thatmay occur due to ionospheric plasma densityvariations between the day and night sides of theEarth.5American Institute of Aeronautics and Astronautics

98-3491ϕ ≈ 11.5°izλyspinaxismagneticaxisFigure 2. Tilted-dipole approximation to thegeomagnetic field.G e o m a g n e t i c F i e l d M o d e l :To first order, the Earth’s magnetic field canbe approximated by a magnetic dipole with themagnetic axis of the dipole tilted off from thespin axis by approximately ϕ=11.5°, asillustrated in Figure 2. For this analysis, we willignore the 436 km offset of the dipole center fromthe Earth’s center. At any given point, themagnetic field can be expressed as consisting oftwo components, a vertically-oriented component:B V = B E R E 3sin Λ (3)r3and a North-South oriented horizontalcomponent:B H = B E R E 3cos Λ (4)r3where B E =31 µT=0.31 gauss is the strength of themagnetic field on the magnetic equator at thesurface of the Earth, R E =6378 km is the radius ofthe Earth, r is the radial distance of the pointfrom the center of the Earth, and Λ is themagnetic latitude starting from the magneticequator.R e f e r e n c e F r a m eIn order to make the calculations tractable,we will perform the calculations in a referenceframe that is rotated so that it has its z axisaligned with the axis of the Earth’s magneticdipole. The inclination λ of the spacecraft orbitwith respect to the geomagnetic frame will varyxfrom λ=i-ϕ to λ=i+ϕ once a day as the Earthrotates. For the analysis in this paper we willneglect the slight variation of λ during a singleorbit, and average the effects of the rotation overmany orbits. For simplicity, we also choose theorientation of the reference frame so that theascending node of the orbit lies on the x axis.Expressed in Cartesian coordinates, thegeomagnetic field is given by⎡ ⎤B E R ⎢B =E 3 3xz / r2⎥⎢ 3yz / r2⎥r3⎢3z2/ r2− 1⎥⎣ ⎦⎡⎤B E R=E 3 3sin λsinθcosθ⎢2 ⎥r3 ⎢3sin λcos λsinθ⎥⎣⎢3sin2λsin2θ − 1⎦⎥, (5)In this rotated frame, the circular orbit of thespacecraft can be parameterized in terms of theangle θ around the orbit normal asand the velocity asmagneticaxisFigure 3. Spacecraft orbit in the referenceframe aligned with the magnetic axis.⎡x⎤⎡ cosθ⎤r =⎢ ⎥⎢y⎥= r⎢⎢cos λsinθ⎥⎥, (6)⎣⎢z⎦⎥⎣⎢sin λsinθ⎦⎥⎡ − sinθ⎤⎢⎥v = v o ⎢cosλcosθ⎥, (7)⎢⎥⎣⎢sinλcosθ⎦⎥6American Institute of Aeronautics and Astronautics

98-3491where the magnitude of the orbital velocity isgiven by:vo = G M E , (8)rwhere G=6.67x10 -11 m 3 /kg-s 2 is Newton'sgravitational constant, and M E =5.976x10 24kg isthe mass of the Earth.The motion of the tether across thegeomagnetic field induces an electric field in thereference frame moving with the tether 14E =− v× B. (9)Consequently, in the reference frame of the tetherthere is a voltage along the tetherV = E•L. (10)After some trigonometric yoga, Eq. 10 reduces toLV =B E RE 3 vocosαcosλr3= L B T vocosα . (11)where B T =B E R E 3 cosλ/r 3 =B H (Λ=λ) is thetangential component of the magnetic field a tright angles to both the velocity vector and thetether. 15If the Terminator Tether system provides ameans for the tether to make electrical contactwith the ambient space plasma, such as a hollowcathode plasma contactor, field emission device,or a bare wire anode, this voltage will cause acurrent to flow through the tether conductor. I fthe total resistance of the Terminator Tethersystem, including tether resistance, control circuitresistance, plasma contact resistance, andparasitic resistances, is R, the current in thetether will beI = V Lˆ . (12)RIf, as will be the case most of the time, theelectron current is leaving the space plasma andentering the tether along an appreciable lengthof the tether near the end, then Eq. 12 needs to bereplaced with an integral of the current along thelength of the tether.The reaction of this current with thegeomagnetic field will induce a Lorentz force onthe tether. Integrating this force along thelength of the tether, the net electrodynamic forceF E on the tether system isVFE= L( I× B) = ( L×B)RL vo= − 2B cos cos2E 2 R E 6 α λR r6L v= − 2B T 2 o cosαR. (13)The drag force F D on the tether is the componentof the electrodynamic force F E that is parallel tothe velocity vector v,FD= FE• vˆ = FEcosαL vo= − 2B cos2α cos2E 2 R E 6λR r6L v= − 2B o cos2T 2 αR. (14)Using Lagrange’s planetary equations and theassumption that the orbit is nearly circular, thetime rate of change of the orbital semi-major axisa can be found to be∂a− 2L2 B E 2 R E 6 cos2 α cos2 λ ⎛ 1 ⎞=⎜ ⎟ , (15)∂tMSR ⎝ a5 ⎠where M S is the total mass of the spacecraft(including the tether system), and cos 2 λ is theaverage of cos 2 λ as λ varies over one day due tothe rotation of the Earth:[ ]cos cos ( )cos2 1 ⎛6+ 2 2i+ 3 2 i−ϕ⎞λ =16⎜⎟ . (16)⎝ + 2cos 2φ+ 3cos [ 2( i + ϕ)] ⎠Taking the reciprocal of Eq. 15 andintegrating from the initial to the final orbitradius, we obtain an estimate of the total timerequired for a Terminator Tether to deorbit aspacecraft:M Ra∆tS6 final=a12L2 B E 2 R E 6 2 2 . (17)cos α cos λ a initialIt should be noted that if current can flow inonly one direction in the system, then thecalculation of cos 2 λ must be handleddifferently for orbits with inclinations greaterthan 78.5°. This is due to the fact that for suchhigh inclination orbits, the spin of the Earth will7American Institute of Aeronautics and Astronautics

98-3491rotate the magnetic dipole so that thespacecraft’s orbit will actually move in theretrograde direction relative to the magneticfield during a portion of the day. Consequently,the voltage will reverse direction for a part ofthe day.Deorbit Times for Constellation SatellitesTo illuminate the utility of using t h eTerminator Tether system to remove dead andobsolete satellites from useful orbital slots, wehave used the equations developed above toestimated the time required for a tether massingonly 2.5% of the total satellite mass to deorbitsatellites from typical Big- and Little-LEOconstellation orbits. In these calculations, wehave assumed that the control and parasiticresistances in the system equal half of the tetherresistance. Table 1 compares the predicted timerequired for a Terminator Tether to deorbit asatellite from the orbits used by several existingor planned constellation systems to the timerequired for the satellite to deorbit under t h einfluence of atmospheric drag alone. The timesfor atmospheric decay are rough estimates basedupon the assumption of a 10 m 2satellite crosssectionalarea, a drag coefficient of 2.0, and use ofthe 1977 Jaccia static atmosphere model 16 for themean exospheric temperatures (see Section III.A).As Table 1 shows, a Terminator Tether massingonly a very small percentage of the total systemmass can deorbit a satellite within a few weeksto a few months, many orders of magnitude fasterthan the satellite would deorbit due toatmospheric drag alone.It should be noted that the deorbit timeestimates given above are based upon atheoretical model that does not includeconsiderations for ionospheric plasma densityeffects. These effects may increase deorbit timesfor higher altitudes, where the plasma is moretenuous. More accurate calculations includingthese and other effects are currently underway.It also should be noted here that the truemeasure of the effectiveness of a deorbit methodis not just whether it reduces the orbital lifetimecompared to atmospheric drag decay, butwhether it reduces the product of the orbitallifetime times the collision cross-sectional areaof the spacecraft. This Area-Time-Productprovides a measure of the risk of the defunctspacecraft colliding with another spacecraftduring its orbital lifetime. In Section III, we willshow that the Terminator Tether cansignificantly reduce the Area-Time-Product formost LEO orbits.Inclination ChangeWhile Table 1 shows that a TerminatorTether system can rapidly deorbit a spacecraftfrom orbits with inclinations below about 75°, therate of altitude drop for a near-polar orbit is low.Because electrodynamic tether propulsion can beused to change the inclination of an orbit, 18 weexplored the possibility of decreasing the deorbittime for a polar satellite by modulating thetether current to reduce the orbit inclination, thusbringing the satellite into an orbit with a morefavorable interaction between the velocity andmagnetic field vectors. For a nearly polar orbit,however, the rate of inclination changeTable 1. Deorbit times from example constellation orbits using a Terminator Tether system with analuminum tether massing 2.5% of the spacecraft mass.ConstellationAltitude(km)Inclination(degrees)Deorbit Time,no TT(Derelict)8American Institute of Aeronautics and AstronauticsInitial OrbitDecay Rate(km/day)Deorbit Time,withTerminator TetherOrbcomm 1 775 45 100 years 44 11 daysOrbcomm 2 775 70 100 years 11.6 41 daysLEO One USA 950 50 600 years 32 18 daysGlobalStar 1390 52 9,000 years 22.3 37 daysSkybridge 1475 55 11,000 years 18.5 46 daysFaiSat 1000 66 800 years 13.5 45 daysIridium 780 86.4 100 years 2.1 17 7.5 months 17M-Star 1350 47 7000 years 27 28 daysCelestri 1400 48 9000 years 26 32 daysTeledesic 1350 ~85 7000 years 1.7 17 17 months 17Note: All spacecraft are assumed to have an effective drag cross section of 10 m 2 .

98-3491achievable by a passive tether system turns out tobe very small. The rate of inclination change isgiven by the rate of orbit precession caused by thenet electrodynamic torque T T on the orbit in thetransverse direction perpendicular to both theline of nodes (the x axis, in this case) and theorbit axis:∂iT= T , (18)∂tΩwhere Ω= r× p = M S ( r×v) is the angularmomentum of the satellite in its orbit. Thistorque results primarily from the out-of-planeforces on the tether when the satellite is near theequator; because the velocity vector for polarorbits is nearly parallel to the magnetic vectorwhen the satellite is at the equator, this force israther small. Averaging the torque on the orbitdue to the electrodynamic force given by Eq. 13over one orbit, we obtain∂iL2B2E 2 R α λ ⎛= E 6 cos sin 2 1 ⎞⎜ ⎟ . (19)∂t4MSR ⎝ r6 ⎠If we choose an orbit such as the one used by th eIridium constellation (780 km altitude, 86.4°inclination), the maximum inclination changethat a tether massing 2.5% of the spacecraft masscould cause would be only about 0.35° per year.Consequently, modulating the tether current tomaximize the inclination change rate will notresult in a significant improvement in the overalldeorbit rate.I I . B . M a x i m i z i n g E l e c t r o d y n a m i c D r a gBecause the electrodynamic forces actperpendicular to the tether, the tether will trailbehind the spacecraft. In fact, it is necessary forthe tether to hang at an angle behind vertical forthe electrodynamic forces to decelerate thespacecraft. The hang angle of the tether dependsupon the balance between the electrodynamicdrag force, which tends to pull the tether back,and the gradient force, which tends to restore thetether to a vertical orientation. Because thegradient force decreases as the tether librationincreases, if the electrodynamic drag is too large,this balance can become unstable, resulting in lossof control of the tether system. In this section, weanalyze the drag torque to gradient torquebalance and develop a means of not onlyoptimizing the electrodynamic drag but alsostabilizing the tether libration.Force and Torque Balance AnalysisWe will now calculate the forces and torqueson the tether and, using the fact that the electrodynamicand gradient torques on the tether mustbalance each other out to achieve a stable tetherorientation angle, calculate some optimum valuesfor some of the Terminator Tether parameters.E l e c t r o d y n a m i c F o r c e a n d T o r q u eAs discussed in previous sections, both theoryand experiment show that: provided theconducting tether is moved rapidly through theEarth's magnetic field in order to generate avoltage across it, and provided good contact ismade with the space plasma, we will have aconducting tether that has a current flowingthrough it. When a wire (moving or not) carryinga current I is embedded in a magnetic field B,there will be an electrodynamic force F Egenerated on each element of the wire. Theelectrodynamic force will be at right angles toboth the magnetic field vector and the lengthvector of the wire, with a magnitude given byEq. 13:L vFE=− B T2 2 o cosα . (20)RThe electrodynamic force is always at rightangles to the conductor, and stays at right anglesto the conductor as the angle α varies, as shown inFigure 4. Assuming that the electrodynamic dragforce is applied uniformly along the length of thetether, we can make the simplifying assumptionthat the integrated force is effectively appliedat right angles to the center of mass of the tetherat the point L/2 as shown in Figure 4. Theelectrodynamic torque on the tether is:T F L B2 L 3T v o cosαE = E =. (21)2 2RG r a d i e n t F o r c e s a n d T o r q u e sWhen a tether and its ballast end mass aredeployed from a host spacecraft, the gravitygradient force field of the Earth, combined withthe orbital centrifugal gradient force field willcause the tether to deploy either up or down fromthe host spacecraft, depending upon the directionin which the ballast mass is ejected from the hostvehicle. In the Terminator Tether system, thetether will be deployed below the hostspacecraft.In the absence of electrodynamic and atmosphericdrag, the equilibrium direction of the9American Institute of Aeronautics and Astronautics

98-3491V,Elocalverticalm sαT GTIm TF GT⊗ BT EF EF Dwhere the gradient field strengthΓ=ω 2 ο =v 2 o /r 2 =GM E /r 3 . The strength of the forcedepends not only on the ballast mass m B and thestrength of the gradient field 3Γ, but also theradial component of the distance of the ballastmass from the center of mass of the spacecraft,which is Lcos(α). As shown in Fig. 4, this forceacts in the vertical direction along the radiusvector leading from the ballast mass towards thecenter of the Earth. The component of thisgradient force that is at right angles to thetether, given by F GB sin α, will produce a torqueΤ GB on the tether that tends to restore the tethertoward the vertical, lessening the angle α.TGB = LFGB sinα = 3 Γ mBL2sinα cosα(23)The tether mass m T also contributes to thegradient force and torque. If we assume that thetether has a uniform cross section, then we canreplace the distributed mass of the tether withan equivalent point mass m T placed at the centerof mass of the tether, which is the point L/2along the tether, and a distance L/2 cos α in theradial direction. The gradient force due to t h etether mass is then:T GBm BFGT= 3 Γ mTL cosα (24)2F GBFigure 4. Gradient and electrodynamic forcesand torques on the tethered system.tether would be exactly along the vertical, sincethe combined gradient field is a maximum in thatdirection. Because the tether will generate asignificant amount of electromagnetic drag,however, the actual equilibrium position of thetether with respect to the local vertical will beat some angle α lagging behind the spacecraftmotion in the plane of the orbit, as shown inFigure 4. In the following analysis, we find thereis an optimum angle for α that produces thelargest electrodynamic drag force on the hostspacecraft, minimizing its deorbit time.The combined vertical gravity gradient andcentrifugal gradient field 3Γ acting on the ballastmass m B at the end of the tether of length L willproduce a gradient force F GB given by:FGB= 3 Γ mBL cosα , (22)While the gradient torque is:L 3TGT = F GT sinα = Γ m T L2cosα sinα(25)24The total gradient torque attempting to restorethe tether to its vertical orientation is then:TG = TGB + TGTm= 3 Γ⎛m TB +⎞L2(26)cosαsinα⎝ 4 ⎠It is important to notice the variation of the totalgradient torque as the tether angle α is varied.Since the gradient force is always in the radial orvertical direction, there is no torque on the tetherwhen the tether is vertical, as would be the casewhen there are no aerodynamic or electrodynamicdrag forces. Once the drag forces becomeimportant and start to apply torque to the tether,increasing the tether angle α, those drag torquescausing an increase in tether angle α will beopposed by a rising gradient torque which willattempt to decrease the tether angle. Thegradient torque reaches its maximum at α=45°,where sinα=cosα=0.707 and sinαcosα=0.50. Whenthis angle is reached, we are at a point of10American Institute of Aeronautics and Astronautics

98-3491rotational instability, for if there is a furtherincrease in the electrodynamic drag force due toan increase in magnetic field strength or plasmadensity, causing an increase in current flowthrough the tether and causing the angle α tobecome greater than 45°, the gradient torque,instead of growing stronger to counteract theincreased drag torque, will become weaker. Thetether will become unstable and the angle α willgo rapidly to 90°, where the drag force will alsodrop to near zero.To restore control to the tether angle if theinstability occurs, it will be necessary to turn offthe electrodynamic drag forces by shutting off thecurrent flow through the tether. The α=90°position for the tether and ballast mass is agravitationally unstable orientation. After atime, slight fluctuations in the gravity field willallow the gradient force to slowly take over andrestore the tether to the vertical orientation,which, unless it can be controlled in some way, isequally likely to be up or down. It would thereforebe desirable to maintain control of the tetherangle so as to avoid the tether angle getting intothe region of instability. To avoid this possibilityof tether instability, the ProSEDS missionplanners are planning on using a large ballastmass and a long non-conducting tether in order tokeep the gradient forces high. For TerminatorTether applications on commercial spacecraft,however, use of a large ballast mass will not beeconomically feasible. Consequently, theTerminator Tether will use feedback control onthe tether current to maximize the drag force andstabilize the tether dynamics.Torque Balance on a Stable TetherThe angle α of a stable tether is determinedby the balance between the electrodynamic torqueΤ E attempting to increase the angle α and thegradient torque Τ G attempting to decrease theangle α. Balance is achieved when the twotorques are equal:TE = TG = TGB + TGT(27)or, using Eq. 21 and Eq. 26:B2L3vTE= T o cosα2 R(28)m= 3 Γ⎛m TB +⎞L2sinαcosα= T⎝ ⎠G4Simplifying Eq. 28, we obtain a relationshipbetween the electrodynamic and gradientparameters of the tether that must hold if thetether is to be in a stable equilibrium state.B2T LvoRm= 6 Γ ⎛m TB +⎞⎝ 4 ⎠sinα (29)At first glance, it might seem that theoptimum angle for the tether would be 45°, sinceat that angle the gradient torque is largest andtherefore can counteract a larger electrodynamicdrag force, despite the fact that at 45° the tetheris at the onset of instability. The optimum angle,however, is that which maximizes thehorizontal or drag component of the electrodynamicforce that opposes the host spacecraftmotion, not the t o t a l electrodynamic force. Thishorizontal drag force is given by Eq. 14:B2L2vFD= FEcosα= − T o cos2αRB2T L2v=−o cos2αRC+ rdL2/ mT(30)This equation gives a maximum drag force for longtether length L, small tether resistance R andsmall tether angle α. But to maintain α near zerowhen there is a large drag force on the tetherrequires a large ballast mass or a very longtether. If a large ballast mass were available,such as might be obtained by cutting off a largeportion of the host vehicle (a solar panel, forexample), then this is a mode of operation whichcan allow the maximum electrodynamic force F Ethat is available to produce the maximum dragforce F D . If, however, the amount of drag forcethat can be applied to the tether is limited bytether instability, as it is in the NASA/MSFCProSEDS mission and the various TerminatorTether applications, then instead of looking a tthe electrodynamic limits to maximizing thedrag force F D , we want to look at the gradientlimits to maximizing the drag force. To do this,we use Eq. 29 in Eq. 30 to obtain:mFD=− 6 Γ L⎛m TB +⎞sinαcos 2 α (31)⎝ 4 ⎠This equation says that for maximum drag forceon the host spacecraft, you want long tetherlength L, as well as massive ballast mass m B andtether mass m T . The equation also states that asmall tether angle α (tether near vertical) is notoptimum. If a very large ballast mass isavailable then it is possible to operate with α at11American Institute of Aeronautics and Astronautics

98-3491a small angle and get the maximum drag forceavailable from the maximum electrodynamicforce made possible by the available environmentalparameters. More realistically, for anygiven ballast mass, it is better to operate thetether at the angle α that maximizes the dragforce. We can determine that optimum angle bysetting the partial derivative ∂F D /∂α=0 andsolving the resulting equation. When we do this,we find that the optimum angle for the tetherthat gives the maximum electrodynamic dragforce F D , while still keeping the tether torquesbalanced and under control, is α=arctan(0.707)=35.26˚. This angle is well below the angle of 45˚where tether instability sets in. With this angleselected, we obtain an equation for the maximumstable drag force of:mFD(max, α = 35.26 ° ) = − 6sinαcos2αΓL⎛m TB +⎞⎝ 4 ⎠(32)m=− 2.31 Γ L⎛m TB +⎞⎝ 4 ⎠The tether angle in a Terminator Tether willbe controlled by controlling the current throughthe tether to compensate for variations inmagnetic field strength and direction, plasmadensity (which affects the plasma resistance),and other factors, and thereby maintain thetether at an intermediate angle where both theelectrodynamic and gradient forces are at anappreciable level and balance each other. Thiscan be done in a number of ways, either by varyinga control resistor or inserting stepped values ofballast resistors in series with the resistance ofthe tether, or by periodically interrupting thecurrent through the tether to keep the averagecurrent at the desired value.There are many ways to generate the sensinginformation needed to provide the feedbacksignals to the tether current controller. Thesimplest is to measure the drag acceleration oneither the host spacecraft or the end mass with aset of accelerometers and maximize thedeceleration force in the direction opposite to thehost spacecraft motion. Another method wouldbe to measure the current in the tether, and,knowing the tether resistance and the amount ofcontrol resistance, calculate the power beingextracted and maximize that value. Alternatemethods would be to use GPS receivers at bothends of the tether to measure the angle of thetether or an optical position sensor to measure theposition of the ballast mass with respect to thehost spacecraft. These methods of controlling thedrag force or the tether angle should alsostabilize the tether oscillations that presentlyconcern the ProSEDS mission planners.E l e c t r o d y n a m i c D r a g F o r c e a n d P o w e r L e v e l sWe will now estimate the magnitude of theelectrodynamic drag force and power attainablefrom a Terminator Tether. If we assume theTerminator Tether is in orbit at an altitude of1000 km, where the gradient field Γ=0.99x10 -6 /s 2 ,and the electrodynamic tether has a length ofL=5 km, a mass of m T =10 kg, a ballast mass ofm B =10 kg, and a tether tilt angle α=35.26˚. thenthe gradient-force-limited maximum allowablestable drag force using Eq. 32 is F D =0.143 N. Thisis to be compared with the electrodynamic dragforce obtainable from the aluminum tethermoving at velocity v M =6814 m/s with respect tothe transverse magnetic field B T =20 µT. If weassume the control resistor R C =0 Ω, then themaximum available electrodynamic drag forceusing Eq. 30 is 0.246 N, which is more than th estable drag force of 0.143 N. The controlresistance must be increased to lower the currentflow through the tether and bring the electrodynamictorque down to a level where it willbalance the gradient torque and leave the tetherat the optimum angle to produce the stable dragforce level of 0.143 N.This maximum stable drag force F D =0.143 Nopposing the motion of the host spacecraft,assumed to be in an equatorial orbit with λ=0 anda velocity with respect to the magnetic fieldofv M =v o -ω E r cosλ=(7350-536) m/s=6814 m/s, isequivalent to a deceleration power of:P = FDvM= 975 W(33)Since, as pointed out in Eq. 1, the powergeneration capability of an electrodynamictether is proportional primarily to its mass, theTerminator Tether will be designed to have ahigh conductivity tether with enough mass toexceed the design power levels needed for anyparticular initial orbit and host vehicle. Thecurrent through the tether would then becontrolled at the gradient-limited maximumstable power level so as to maintain the tether a tthe optimum angle to give maximum stable drag.For example, the power level P that could begenerated and dissipated in an electrodynamictether can be obtained either by using Eq. 11 forthe voltage induced across the tether and12American Institute of Aeronautics and Astronautics

98-3491Product by several orders of magnitude. As aresult, the Terminator Tether system can greatlyreduce the risks of a decaying spacecraft collidingwith another spacecraft.As pointed out before, a well-designed TerminatorTether can lower the collision probabilityeven further than the blind chance probabilityimplied by the use of the Area-Time-Productcriteria, by using ground control of the rate ofdescent to avoid collision with the larger objectsin space with well-known and predictable orbits.Note that Fig. 5 is conservative in two ways.First, the assumed cross-sectional area of thetether is much larger than its neutral drag crosssection (the area presented to the “wind”) and,second, the power generated in the tether isassumed to be constant at values that areconsiderably less than those to be expected in therange of altitudes shown in Figure 5. Forexample, a 5 km, 13 kg tether whose resistance is143 ohms, would generate over 5000 watts at 622km altitude if it had perfect contact with the10 6plasma and if it were orbiting in the magneticequator. In these examples, we have assumedthat the same tether will generate only 2040watts throughout its descent from 622 to 250 kmaltitude, although, in the ideal case, the powerwould increase with decreasing altitude. Theseassumptions are based on the power levelsobserved in the TSS-1R electrodynamic tetherexperiment. These lower power levels arethought to have resulted from incomplete contactwith the plasma. As the technology matures, thehigher theoretical values may be possible. Theinduced power values used in the calculationspresented in Fig. 5 are the lower values, whichwe can be confident of, rather than the highertheoretical values.E f f e c t s o f O r b i t a l I n c l i n a t i o nFig. 5 also shows the effects of changing froman equatorial orbit to a polar orbit where theaverage values of inclination with respect to thegeomagnetic equator have been applied accordingto Eqn. 16.10 5T ∞ = 800 kelvinsT ∞ = 1100 kelvinsArea-Time Product (m 2 years)10000100010010T ∞ = 1400 kelvins5% Tether - 30 km x 1 cmMass = 65 kg; Power = 200 wattsInclination = 90°Inclination = 0°15% Tether - 5 km x 10 cmMass = 65 kg; Power = 10200 watts1% Tether - 5 km x 10 cmMass = 13 kg; Power = 2040 watts0.1300 500 700 900 1100 1300 1500Circular Orbit Altitude (km)Area-Time Product for 1300 kg SpacecraftNote: A∆t Scales Linearly with Spacecraft MassFigure 5. Area-Time Product vs. Initial Altitude for 1300 kg spacecraft at inclinations of 0° and90°. Upper curves show results for atmospheric drag alone at mean and extremes of exospherictemperature. Lower three curves show results for Terminator Tether systems.14American Institute of Aeronautics and Astronautics

98-3491Note that the area of the tether for the 90°orbit is only 300 m 2 while the area for the 0°examples is 500 m 2 . If all factors were equal inthe comparison, we should expect a factor ofabout 50 in A∆t as we change the inclination ofthe orbit from a daily average of 11°.5 to 82° , theequivalent average inclinations of the equatorialand polar orbits with respect to the geomagneticequator. We find at the 1500 km level anincrease in A∆t of a factor of about 30 from thelower tether curve of Fig. 5 to the upper tethercurve showing the decay of a polar satellite fromvarious altitudes at a fixed induced power levelof 200 watts. If the area of the polar orbit tetherwere 500 m 2 , the A∆t value at 1500 km would beabout 500 m 2 -years instead of 300 m 2 -years.Thus, we find that it is considerably moredifficult to bring down a polar orbiter than ones inthe low to middle inclinations. As a result, weexpect to require longer tethers at the higherinclinations in order to generate enough potentialdifference along the tether to permit good contactwith the ionospheric plasma and to provide astable current along the tether. We haveassumed, for the calculations shown in Fig. 5,that the tether angle α is always 0°. Thiscorresponds to a situation where the tether tipmass is large as would be the case if a large pieceof the spacecraft could be separated from themain spacecraft and used to hold the tether morenearly vertical.Nevertheless, the Area-Time products of thetether decay examples are still less than 1% ofthe neutral drag decay values, even for polarorbits. These high-inclination and/or highaltitudeorbits are typical of those underconsideration for the Teledesic, M-Star, andCelestri constellations. Reasonable estimates oftheir mass and area show them to have neutraldraglifetimes of several thousand years and A∆tvalues of 100,000 m 2 -years. 5% TerminatorTethers, made to be somewhat longer than forlower inclinations, can bring these spacecraftdown in one or two years with A∆t values of 300to 500 m 2 -years.Furthermore, it is important to note that thebuilt-in conservatism of using fixed power tethersyields a comfortable pad that can be realized i fwe design the tethers to operate at the maximumpower levels available at the lowest altitude inthe descent profile. The calculations and Figuresin this paper have been based on power levelsfixed at the highest altitude. Thus, the tetherdescent is not assumed to take advantage of theincreased power levels available as thespacecraft altitude decreases. Preliminarycalculations based on Eqn. 17 of this paperindicate that the Area-Time products and thedescent times can be decreased by a factor of 3simply by designing for the lower altitude powerlevels. This will be particularly helpful andeasy to do for the high-inclination orbits wherethe available induced power is of the order ofhundreds of watts rather than the kilowattlevels available at lower inclinations.I I I . B . C o m p a r i s o n w i t h S o l i d R o c k e t M o t o r sThe other conventional method of removing aspacecraft from a LEO orbit is to build into thespacecraft system a rocket mechanism capable ofdeorbiting the spacecraft. This method,however, requires that a significant fraction ofthe spacecraft’s launch mass be dedicated to thepropellant needed for deorbit.If a spacecraft manufacturer were to use arocket deorbit system, the design requirements forthe system will be more stringent than those forordinary spacecraft; the system must operateafter many years on-orbit and when some or a l lother components of the spacecraft have failed.Moreover, a rocket deorbit system must be capableof proper operation under many kinds ofanomalous situations, such as spacecraft tumblingdue to attitude control failure, offset of center ofmass, or lack of orbital position knowledge.Figure 6 shows the percent additional solidrocketpropellant mass required to drop aspacecraft from a circular orbit at the specifiedaltitude to a new orbit with a perigee of 200 km.At this altitude, atmospheric drag will remove atypical spacecraft from orbit in a few revolutions.The contours of constant stage propellant massfraction range from low values of 0.5 up to thevalues associated with the best solid motors(≈0.93) that can be built without adding anyextra hardware to the deorbit stage. Aneffective, independent stage to provide a retro∆V of 50-325 m/s will almost certainly have amass fraction on the order of 0.6 to 0.75. If thedeorbit stage is required to perform its ownattitude determination, the stage propellantmass fraction may be as low as 0.5.The figure shows that a solid-rocket deorbitsystem will require a mass allocation that is a15American Institute of Aeronautics and Astronautics

98-349130Stage Propellant Mass Fraction(Typical Range Shaded)0.50250.55Percent Increase in Orbital Mass20150.600.650.700.750.800.850.900.9510significant fraction of the spacecraft’s launchmass. For a spacecraft in a 1000 km orbit, adeorbit rocket system with a reasonablepropellant mass fraction of 0.7 will consumenearly 13% of the vehicle’s launch mass. ATerminator Tether system, however, can achievedeorbit of the spacecraft while requiring as littleas 2 to 5% of the launch mass. The mass savingsachieved with the Terminator Tether system canbe used for additional revenue-producingequipment or for additional station-keeping fuelto provide longer operational lifetimes.I V . A .I V . I M P L E M E N T A T I O NT HE T ERMINATOR T ETHER The standard Terminator Tether DeorbitSystem will be a small, lightweight, highlyautonomouspackage that is attached to the hostvehicle before launch. Only minimal electronicinterfacing will be required, likely limited toseveral circuits for monitoring the host status.The Terminator Tether unit will contain the5300 500 700 900 1100 1300 1500Circular Orbit Altitude (km)Figure 6. Conventional solid rocket motor deorbit system percent mass increase vs. altitude andstage propellant mass fraction (I sp =288 sec).tether and its deployer, an electronics controlpackage, a power conversion circuit to enable theelectronics to be powered by the tether current, athree-axis accelerometer, a radio receiver, asmall battery, a small solar panel to keep thebattery charged during the long dormant period,and an electron emission device such as a hollowcathode plasma contactor or a Spindt Cathode.D o r m a n t P h a s e : During the operational missionof the host spacecraft, the Terminator Tetherwill be dormant. Periodically, a timer circuitwill wake the system up and it will check thestatus of the host. Multiple, redundant methodsof observing host status will be used to preventany premature activation. The system will alsouse the radio receiver to listen for activationsignals from ground stations. If the host isdefunct, or if the system receives an authenticactivation signal, it will initiate the deploymentand deorbit sequence.D e p l o y m e n t : The tether could be deployedupwards from the host like the ProSEDS tether,16American Institute of Aeronautics and Astronautics

98-3491redundancy enables the Hoytether to withstandmany line cuts by small impactors while stillmaintaining the design load. As a result, it canprovide very high reliability of survival forperiods of years.V . C O N C L U S I O N SBy using electrodynamic drag to greatlyincrease the orbital decay rate of a spacecraft, aTerminator Tether system can remove unwantedobjects from LEO rapidly and safely. Thistechnology uses a passive interaction with theEarth’s magnetic field to generate drag, so nopropellant or input power is required.Consequently, the Terminator Tether can deorbiteven defunct spacecraft. Using an analyticalapproach, we have developed methods forpredicting Terminator Tether deorbit times fromvarious orbits. Using these methods, we haveshown that tether systems massing just 2 to 5% ofthe total spacecraft mass can deorbit a typicalcommunication satellite within several weeks ormonths, depending upon the initial orbit. It wasshown that high-inclination orbits are moredifficult to bring down than those in moderate tolow inclination orbits. The Area-time productsfor polar orbits are about 50 times larger than theequivalent system in an equatorial orbit. Thisstill leaves nearly 3 orders of magnitudeadvantage in Area-Time product over neutraldrag decay, even for polar orbits. The low massrequirements of a Terminator Tether systemmakes it highly advantageous compared to aconventional solid-rocket deorbit stage.Moreover, the drag enhancement provided by theelectrodynamic tether technique is so large thatthe total deorbit Area-Time-Product can bereduced by several orders of magnitude comparedto atmospheric drag alone, minimizing the longlivedorbital debris hazard created by aconstellation spacecraft after their end-of-life.In addition, we have developed a method ofoptimizing the electrodynamic drag on the tethersystem by controlling the tether hang angle. Thismethod also provides a simple method forstabilizing the tether libration.AcknowledgementsThis work was supported in part byNASA/MSFC SBIR contract NAS8-97025 andNASA/MSFC purchase order H-28540D.REFERENCES AND NOTES1. Loftus, J.P., JLoftus@ems.jsc.nasa.gov,personal communication via email Monday 10June 1996 15:50:10.2. Forward, R.L., "Electrodynamic Drag'Terminator Tether'", Appendix K, in FinalReport on NAS8-40690, 10 July 1996.3. Office of Science and Technology Policy: 1995,Interagency Report on Orbital Debris 1995,The National Science and Technology Council4. NASA Safety Standard: Guidelines andAssessment Procedures for Limiting OrbitalDebris. (NSS 1740.14, August 95).5. Space News, March 31-April 6, 1997, p 16.6. Space News, Aug 4-10, 1997, p 30.7. Estes, R., restes@mars.harvard.edu, personalcommunication via email Friday 31 May 199610:39:16.8. Loftus, J.P. JLoftus@ems.jsc.nasa.gov, personalcommunication via email Wednesday 19 June1996 08:20:39.9. Lorenzini, E., personal communication, 13March 1997.10. Johnson, L., "Propulsive Small ExpendableDeployer System Mission (ProSEDS)", OASTAdvanced Propulsion Workshop, JPL,Pasadena, CA, 20-22 May 1997.11. Gilchrist, B., personal communication, 12May 1997.12. Sanmartín, J.R., Martínez-Sánchez, M.,Ahedo, E., “Bare Wire Anodes forElectrodynamic Tethers,” J. Propulsion andPower, 7(3), pp. 353-360, 199313. Hoyt, R.P., Forward, R.L., "ElectrodynamicHoytethers for ISS and LEO Spacecraft",Appendix E of High Strength-To-WeightTapered Hoytether for LEO to GEO PayloadTransport, Final Report on NASA ContractNAS8-40690,10 July 1996.14. Note that in Eq. 9, the correct velocity vectorto use is the relative velocityvM=vo−ωEr cosλ between the orbitingspacecraft and the geomagnetic field, sincethe geomagnetic field rotates with the Earth18American Institute of Aeronautics and Astronautics

98-3491at the rate of ωE=2π rad/day. For anequatorial orbit at an altitude of 1000 km, thevelocity of the geomagnetic field is 0.536km/s or only 7% of the orbital velocity of7350 m/s. For nonequatorial orbits thedifference is even smaller. We will ignorethis small difference to keep the equationsmanageable.15. By a geometric coincidence, the transversemagnetic field BT and therefore voltage Vgiven by Eq. 11 are both essentially constantover the entire orbit, despite the fact thatthe horizontal magnetic field varies from amaximum at the magnetic equator BH(Λ=0)to a smaller value of BH(Λ=λ) at thenorthernmost portion of an orbit withgeomagnetic inclination λ. The variation inhorizontal magnetic field strength BH(Λ)with latitude Λ on the Earth and thevariation in the angle at which the velocityvector crosses BN, combine to produce aconstant transverse magnetic fieldBT=BH(Λ=λ) over the entire orbit.16. Jaccia, L.G., “Thermospheric Temperature,Density, and Composition: New Model,” SAOSpecial Report 375, Mar. 1977.17. The results for orbits with i>78.5 assume thatthe tether system can carry current in bothdirections; if the tether system is designed tocarry current in one direction only, the deorbittimes will be roughly twice as long, due toportions of the day when the orbit isretrograde to the magnetic dipole.18. Penzo, P.A., and Ammann, P.W., eds., TethersIn Space Handbook, 2nd Edition,NASA/OSF, 1989, p. 168-9.19. Hoyt, R.P., Forward, R.L., “The Hoytether: AFailsafe Multiline Space Tether Structure,”Tether Technical interchange Meeting,Huntsville, AL, Sept 9-10, 1997.19American Institute of Aeronautics and Astronautics