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'* Carleton Universrty Ottawa, Canada KI S SB6
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The University Library
Magnetic Propulsion for Spîcecraft
by
Walter Gillies, B.Sc.(Hons.)
A thesis submitted to
the faculty of Graduate Studies and Research
in panial fulfillment of
the requirements for rhe degree of
Masrer of Engineering
Department of Mechanical and Aerospace Engineering
Ottawa-Carleton Institu te for
Mechanical and Aeros~ace Engineering
Carleton University
Ottawa, Ontario
August 30, 1996
O Copyright
1996 Walter Gillies
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Acknowledgments
Al1 men dream, but not equally. Those who dream by night in the dusty recesses of their mincis wake in the day to find chat it was vaniry, but the drearners of the day are dangerous men, for they may acr their dreams with open eyes, to make it possible.
T.E. Lawrence The Seven Pillars of Wisdom
Many thanks to my supervisors, Dr. Don Russell and Dr. Doug Staley, whose time and patience were taxed while 1 drearnt of elecnic spacecraft.
Walter Gillies
Abstract
Superconducting current loops are proposed as a rneans of spacecraft propulsion.
Magnetic interaction with the geomagnetic field is the propulsive source. The general
layout of the propulsion system is developed for both a single loop design and a triple-
loop design. Challenges related to the practical design and implementation of
superconducring lwps as a means of spacecraft propulsion are addressed. Future
potential for magnetic propulsion is explored, including interplanetary mvei, self-
launching capability and interstellm mvel. Levitation from eanh surface and
in terstellar flight do no t appear pracàcai because present-day materials are no t scrong
enough and superconducting currents are nor large enough. However, a single loop
used in a polar orbit does appear practical for orbit uansfers and interplanetary navel.
Table Of Contents:
Magneric Propulsion for Spacecrafi .............................................................................................. ii Acknow1edgrnent.r ......................................................................................................................... iv
......................................................................................................................................... A bsrracr Y
Table Of Contents: ....................................................................................................................... vi .......................................................................................................................... Table of Figures: ix
List Of Variables: ......................................................................................................................... xi
CHAPTER 1 : INTRODUCTION ..................................... .............................................................. 1
....................................................................................................... 1.1 Propulsion Requiremenrs 1 1 2 Magnetic Propulsion ............................................................................................................. 1
7 ..................... 1 3 The Superconductor Revolurion ........................................................................ ................................................................................................................ . I 4 Objecrive of Thesis 3
............................................................ .......................... CHAPTER 2 : LITERATURE REVIEW ,, 4
3.1 Origins ................................................................................................................................... 4 ..................................................... ............................... 2 2 Levirared Superconducrors ,.,. 4
......................................... 2 3 Early Magnetic Spacecraft ................................................................................................. 2.4 Presenr Magneric Spacecrafr 6
CHAPTER 3 : THE GEOMAGNETIC FIELD .............................................................................. 8
................................................... 3.0 General Description .... 8 ............................................................................................................ 3.1 D ipo le A pproximar ion -9
.. 3.2 Variarions in Ambienr Fiel& and Currents .................................................................... 12
..................................................................................................... CHAPTER 1 : LOOP DESIGN 13
3.1 Objective .............................................................................................................................. 13 .............................................................................. ............. 4 2 Magnetic moment of loop ... 13
4.3 NonuniJorm Field Force ............................................. .... ............................................................................................................................ 4.3 Single ioop 19
................................................................................................................................ 3.4.1 Thnisiers 22 .......................................................................................................... 4.4.2 Conml Momenr Gyros 23 ........................................................................................................ 4.4.3 Sin& Loop Applications 23
. ...................................................................................*................................... 4 5 Double b o p 25 4.6 Triple b o p .......................................................................................................................... 26
................ .............*.......................................*........*-.*........,.... 4.6.1 Triple Loop Refinernent ..,... 27 ........................................................................................................................ 1.6.2 Storing Energy 31
3.6.3 Crew Procecrion ...................................................................................................................... 32 ......... ....................................................................... 4.7 Magneric Propulsion Advanrages ..., 32
............................................................................................................... 4.8 Design Challenges 33
.................................................................................. CHAPTER 5 : LOOP REINFORCEMENT 35
............................................................................................................................ . 5 1 Objective ..35 5 3 Methodology ........................................................................................................................ 35
................................................................................................................ 5 3 Maximum Cwrenr 35 ............................................................................................................... 5.4 Self-Currenl Stress 36
5 5 Torques berween b o p s ........................................................................................................ 42 ................................................................................................ 5.6 Force fiom cwrenr in tether -44
................................................................................................................. 5.7 Terhers to b o p 1 46 ................................ 5.8 Rein forcernenr Marerial ......-....... 4 8 ...................................................................................................... 5.9 Size of Tethers ro Loop I 49
................................................. ............................................. 5.10 M a s of Terhers to b o p 1 ., 3 1 ............................................................................................................ 5.1 1 Secrion Modufus - 5 2
......................................................................................... 5.12 Tethers benveen loops 2 and 3 -54 .................................................................................... 5.1 3 Torque between Secondary Loops -36 .................................................................................... 5.14 Tethers Ber ween Secondary Loops 5 7
........................................................................................................ C . IS Seconduq b o p Mus 59 .................................................................................................. 5.15.1 Secondary-Secondary Force 60
..................................................................... ......................... 5.15.2 Secondary-Main Force .... 63 5.15.3 Total Shear and Bending Moment .........................................................
..................................................................................................... j -1 6 Srress on Cwved Beam -66 ................................................... ....................................... 5.1 7 Primary b o p Reinforcemenr ... 69
CHAPTER 6 : PERFORiMANCE ................................................................................................. -73
....................................................................................... .............................. 6.1 Overview ..., 73 ..................................................................................................................................... 6.2 Drag 73
6.2.1 Variations in Drag .................................................................................................................. 75 ........................................................................................................... 6.2.2 Drag in Orbit Rograrn: 76
................................................................................................................... 6 3 Attitude Conrrol 77 6.4 Gravity ................ .. ............................................................................................................ 77
.............................................................................................................. 63 Elecrrostatic Fiel iis 78 ............................................................................................................. 6.6 Oprimm b o p Size: 79
6.6.1 Orher Aurhors ......................................................................................................................... 79 ......................................................................................................... 6.6.2 LVagnetic Force on Loop 80
6.6.3 Net force equation .................................................................................................................. 84 6.6.3 Single Ring ........................................................................................................................... -84 6.6.5 Triple Ring ................. ,... .......................... ... ................ 86
................................................................................................ 6.6.6 Loop Paramcrcr Assumptions -90 ....................................................................................................... 6.7 Accelerarion Programr 91
....................................... ..... 6.7.1 Accelcration vs . lmp radius .. -92 ......................................................................... 6.8 Previous Researchers' Orbir Calculariuns -93
................................................................................................... 6.9 Orbit Trunsfer Program -95 6.9.1 Single Loop ......................................................... ... ................................................. -95 6.9.2 Triple Loop .......................................................................................................................... 103
.................................................................................... CHAPTER 7 : FUTURE DIRECTIONS 106
.................................................................................................... 7.1 Spaceflighl Challenges 106 ....................................................................................................... 7 2 hterplanerary Travel 1 0 6
........................................................................................................................... 7 .2 . I Solar Wind 1 O? ..................................................................................................................... 72.2 Magsail Orbits 107
....................................................................................................................... 7.2.3 Space Sailing 110
7.4 Interstellor Travel .............................................................................................................. 11 1 7.4 Self-hunching Capability ................................................................................................. 111
7.4.1 Triple Ring ................. ,.. ................................................................................................... 112 7-42 Single Loop .......................................................................................................................... 114 7.4.3 Radical thsumptions ............................................ -. ................................................................ 117
CHAPTER 8 : PROBLEMS TO ADDRESS .................................. .. ................................. 121
8.1 Orbiz Conrrol Barriers ....................................................................................................... 121 8.2 Absolure Barriers-Orbit Control ........................................................................................ 121
8.2.1 Superconductor Materials ..................................................................................................... 122 8.2.2 Power Conuol ...................................................................................................................... 126
8.3 Lesser Barriers-Orbit Conrrol ........................................................................................ 1 3 0 8.3.1 Transportacion and Deploymcnr ............................................................................................ 130 8.3.2 Geomagnetic Inductance ....................................................................................................... 131 8.3.3 Powcr Required: ................................................................................................................... 132 8.3.4 orbi^ rypc ............................. ......, ........................................................... 135 8.35 Single Loop Twnbling .......................................................................................................... 136
8.3 Barriers . Advmced Applications ...................................................................................... 137
CHAMTER Y : CONCLUSIONS ............... ....,.., ............................................................ 1 3 8
LIST OF WORKS CITED.... ....................................................................................... 1 3 9
APPENDIX A ............................ ............... ........................................................................... 133
tî.1 Strucrrcre ............................................................................................................................ 112 ................................................................................................... A 2 Single b o p Accelerarion 142
A 3 Triple Ring Acceleration ....................... ....,, ....................................................................... 144 A.3 Triple Ring Mars ................... ,. ............ ,. ....................................................................... 146 A.6 Single Loop Orbir Program I ........................................................................................... 147 A.7 Single Loop Orbis Program 2 ................... .,.. .............................................................. 150 A8 Triple b o p Orbit P rugram ...................... ..., ..................................................................... 152 A9 Single b o p Levirarion ................... ..., ................................................................................ 156
A.9.1 Al Reinforcement .................................................................................................................. 156 A.9.2 UHS Reinforcement .............................................................................................................. 159
A.10 Triple Loop Levitarion ...................... ........ ................................................................ 161 A.1 I Constanrs File ....................... .. ....................................................................................... 163 A12 Air Densip File ............................................................................................................ .... 164 A.13 Drag Force File ............................................................................................................... 165 A.IQ Magnetic Force File ......................................................................................................... 166 Al5 Orbir Calcularions File ..................................................................................................... 166 A16 Radial Force File ............................................................................................................. 168 A .I7 Single Loop Mass File ...................................................................................................... 168 A.18 Triple b o p Mass File ...................................................................................................... 169 A I 9 Flexibility Program .......................................................................................................... 174 A 2 0 hreurral Axis Calculat ions ................................................................................................. 176
APPENDIX B: ................................................................... ....................................................... 180
B.1 Orbit Coordinares .............................................................................................................. 180
Table of Figures:
........................... FIGURE 3.1 h d ~ G h m C MOMEXT OF C L W S T L W P (AFER WAKGS .;ES s (1986) . P . 304) 13 ................................................................................................ FIGURE 3.2 FIELD OF A MAGh'ETlC DIPOLE 15
.................................................................... FIGURE 3.3 S KGLE LOOP OF CLIRREhT I$ A MAGSETIC FIELD 19 FIGURE 3.3 SATE~LITE GRBIT AROUND THE EARTH SHOWWG REGIOSS OF MAGNETIC STABILITY ................. 21 FIGURE 3.5 TWO CURREST-CARRYKG t OOPS AT 90° TO EACH OTHER ....................... ,.. ................... 25
........................ . RGURE 4.6 TRIPLE LOOP OF SUf ERCOSDUCTLYG LOOPS WTii SFï .MA Gh'EnC MOMEST M 27 ............................... FIGURE 4.7 STABLLITY REGIOXS FOR MAGSETIC COirlTROL SHOWISG CIRCCLAR ORBIT 28
........... ..... RG~RE 3.8 MODIRED TRIPLE L W P , ViEW IS DIRECTIOK OF +Z M I S (TOWARDS EARTH).. ,... 30
FIGURE 5.1 S LPERCO.~'DUCT~,~;G .MA GXETiC EXERGY STORAGE SYSTEM (S MES) ........................................ 36 ............. FIGURE 5.2 OLTWGRDS TEiiSIOS ON THE .MA II; LOOP wHEPu' CARRYIKG CURREhT ...................... ., 37
FIGURE 5.3 VIEW OF CLIRRE~\T-CARRYIKG LOOP IS PLAST OF LOOP .......................................................... 38 FIGURE 5.4 VECTOR DIAGRAM SHOWIXG RELATIONSHIP BETWEEN OLWARD FûRCE AND TENSION ............. 40 FIGGRE 5.5 TORQLE ox LOOP 2 DüE TO CLRREhT 13 L W P 1. LOOKISG ALONG -X AXIS ............................... 33 RGLXE 5.6 LOOP M E A GIVING MAGXETIC FIELD FOR TETHER STRESS CALCLIATIOS ................................ 45
.............. FIGLRE 5.7 VIEW LOYG X AXIS TOWARDS TRIPLE RIKG, SHOWIXG TORQL'E OS SECOKDARY LOOP 46 ...................... FIGL~RE 5.8 DOGBLE TETHER ARRASCEMEXT FOR COXSECTiNG MAIX TO SECOSDARY LOOPS 47
FIGL'RE 5.9 T E ~ O S O S IFiHER ASD FORCE ARISISG FROM TORQUE OS SECOSDARY LOOP ........................ 39 RGL'RE 5.10 STRESS OX A LOADED METAL CYLINDER .............................................. .... .............. 52 FIGURE 5.1 1 TORQL'E OK LOOP 2 DbE TO THE FIELD OF L O P 1 XXD LOOP 3 RESPECnVELY ........................ 54 FIGLXE 5.12 NO POSSIBLE TETHER COSFIGLiRATiOSS TO COSTAIX THE SECOSDARY LOOPS ...................... 55 FIGURE 3-13 MAGSETIC MOiE\T OF LOOP 3 19 THE CE \. TER FIELD OF LOOP 2 ........................................... 56 FIGURE 5.13 FORCES DLE TO TETHERS BETWEEii SECOKDARY LûûPS ........................................................ 57 FIGURE 5.15 LOOP CABLE DIMElKSIOKS .................................................................................................... 59 FIGURE 5.16 F I ~ D L I M ~ OF LOOP 2 C C ; ' ~ S G LOOP 3 ............................................................................ 61 FIG~'RE 5.17 LOOP 2 IN MAGNFnC FIELD OF LOOP 1 ................................................................................. 63 FIGURE 5.18 C U R R E ~ IS LOOP 2 AT ASGLE T (THET,\ ) TO PRIMARY FIELD .................... .. .................. 61 F I G ~ R E 5- 19 FORCES ON LOOP QUr\RTER.SECTION. ................................................................................... 66 FIGURE 5-20 S ECI?OS OF LOOP WI11-I NEClKAL ASD CESTKOID AXIS ................. ... ................................ 67 FIGURE 5.21 TESSIOS DIRECnOSS O S THE MAIS LOOP TEMERS ............................................................... 69 FIGURE 5.32 TETHEK 'IE.iS1OS. COMPRESSIOS AND BESDISG VECTOR DItGRAM ............................. ,..,.. 70
................. R G ~ 5 -23 EXPASS ION AND COMPRESSIOX FûRCES ON 7ET)IERED, CURREST-CARRY IXG LOOP 71
FIGURE 6.1 DIRECIIOX OF MAGSETIC MOMEST FOR A cLIRRE~T LOOP ....................................................... 80 FIGLRE 6.2 SKGLE-LOOP DESIGN FOR MACSEIICALLY PROPEUED SPACECRAFT ..................................... 85 FIGbRE 6.3 FROh;TAL ViEW OFSPACECRAFT WITH TRIPLE RISG Ah'D WHERS ................... ,. .................... 87 FIGURE 6.4 LOOP TL'BE DIAMETER FOR VARIOCS WALL THICKhES' ...................................................... 88 FIGURE 6.5 TOTAL MASS VERSUS W A i i THICKXSS FOR TRIPLE RiSG ..................... ,. ............................. 89 FIGLW 6.6. TRIPLE RIXG WITH 8 EITIER-PAIR ARRA!GEME'\'T ...................... ,.. .................................. 89 FIGURE 6.7 SATELLITE ORBIT AROCND €ARTH SI 1OWSG STARLE TtIRLiST ARC ........................................... 94 RGtX 6.8. TRAXSFER FROM CIRCCLAR ORBlT RADIUS A TO ELLIPnCAL ORBIT PEiüGEE A, APOGEE B ...... 97 FIGLRE 6.9. DEFNTIOS OF APOGEE, PERlGEE, TRLE ASOMALY ASD SEM->LUOR U I S .................... .......... 98
...... FIGURE 6.10 SISGLE LOOP CHASGE IS APOCIEE W I m ORBIT NCMBER . SO DRAC; ............................. ,., 101
FIGURE 6.1 1 NO DRAC SISGLE LOOP CHAXGE IN DEtTA V PER ORBiT ....................................... ... 1 0 2 FIGURE 6- 12 SIMXE LOOP CHAXGE IN APOGEE WITH TIME . WITH DRAC .............................................. 1 0 3
FIGURE 7- 1 I,MP~ZSIVE H O H L ~ X N 'IRkYSFER. TYPICALLY USED BY ROCKETS ............................................ IO8 FIGURE 7.2 DIREC~OS OF MAGSAIL THRUST FROM SOUR WXD ........................................................ FIGUE 7.3 MAGSAILS DRAGGING AYD SAILKG ON THE SOLAR WIKD ....................................................... 110 FIGURE 7.4. SIKGLE SL'PERCOXDL'CnXG L0OPWITi-i GYROSCOPIC SHELL .................................................. 115 FIGURE 7.5 MASS AVD M A S ABLE IID LiR VS . C W h T FOR SINXE 100 M RADIUS LOOP ........................ 116
FIGURE 8.1 ~WERATURE. 'MAGh'EnC FIEiD AXD CL'RREhT DEPSSiïY LIMITS ~ V I L S O K . 1983) ................. -123 FIGURE 8.2 SI-MPLE HOMOPOLAR FLLX PUMP SHOWIXG PATH OF FLUX SPûT .......................................... 127 FIGURE 8.3 ELECTROXIC HOMOPOLAR FLLX P ~ W ................................................................................ 128 FIGURE 8.3 TRfiSFOR-MER-RECnFIER TYPE OF FLüX P W ...................................................................... 129
FIG~RE . 1 CLASSICAL ORBIT PARAUETERS FOR A S ELLPTICXL ORBU .......................................... .... ... 180
List Of Variables:
a = angle in radians a = flexibility factor comparing curved beam to smight 0 = CO-declinarion, angle fiom equatorial plane 0 = angle from geomagnetic nonh 8 = angle of arc subtended Q = allowable angular deviation from static equilibrium $ = right ascension, angle h m star Aries @ = area of flux (I = flux density (I = airnuth angle p = density of atmosphere
= elecuical conductiviry E = electromotive force sr: = Pi, 3.141592654 z = torque v = m e anomaly of oribt (see Appendix B) pd = density of alurninum
= allowed stress pl = mass per unit length po = pemeability of free space, 4nx 10" Tm/A o, = stress at outer surface, curved section O= = outer surface stress, straight section AV = change in velocity o, = yeild stress a = acceleration A = area, cross section a = semi-major axis A = total area of loop Ag = area of current loop perpendicular to 0 an = acceIeration due to drag A. = area of current loop perpendicular to r And = radial component of area A, = tangential acting part of loop area B = magnetic field B, = magnetic field at earth surface B,, Bw, B, = magnetic field in x, y and z directions BiPd = radial part of magnetic field B, = tangential part of magnetic field c = distance fiorn neutral axis CD = coefficient of spacecraft drag c, = distance to outer surface from centroid d = diameter of tether wire
da = differential element of area ds = differentid element of curve dt = differential torque e = eccenmcity of orbit E = specific energy of orbit F = force f = frequency FI? = force on loop 1 due to secondary loop FE = force on loop 2 due to other secondary loop Fc = force of compression Fdr% = force of drag Fm,,,, = magnetic force F,, = net force on satellite FT = force total Fm = force tangential to orbit Grad = gradient operator 1 = current 1 = moment of inertia 1, = current in loop 1 Iz = current 1 loop 2 1, = second moment of inertia of cross section L = inductance 1 = length (of wire, tether or loop) 11.1 = magneric moment m = mass M(x) = bending moment at point x Mm, = maximum bending moment G, = mass of superconducting material
= total mass of spacecraft N = number of turns in loop P = power r = radius from centre of eanh, vector from origin r = radius of loop RI = radius of loop 1, main loop R2 = raius of loop 2, secondary loop r~ = radius of orbit A rg = radius of orbit B RMy = radius of satellite body rcarh, rc = radius of earth R, = radius ro inner edge of loop % = radius ourer edge loop r, = radius of pengee r,, = radius of superconducting marerial cross section S = section moduIus T = orbital period
T = tension T = tension in tether t = thickness of reinforcement wall t = nme T = tirne period for acceleration T(0) = rhrust function, see equation 6-19 TIzY = tension from tether linking loop 1 and 2 in y plane TZzzzY = tension between seondary loops in y plane tu = torque between loops 2 and 3 ü = rnagnetic energy v = velocity V = velocity V = voltage VA = Alfven velocity x = distance along circumference of Ioop Y = dismace from neutral axis to cenuoidai axis z = distance fiom plane of loop
Chapter Introduction
1.1 Propulsion Requirements
Spacecraft require propulsion to attain, change and escape from orbit. Even
mainrainine an eanh orbit below about 600 km requires propulsion to counteract
atrnosphenc drag from the small traces of gas remaining. So far, propulsion forces
have k e n provided exclusively by rockets. Ropellant mass must be expended, with
the amount depending on the mission. The space station will require occasional
refueling by space shuttle for drag make-up thrusring. If a system could replace the
chernical thnisrin; and Save even one shuttle flight, one half billion dollars could be
saved. For a Mars mission, there may well be a need to shuttle payloads fiom low to
high eanh orbit so thar the trip can stan funher from eanh's gravity. A sysrem that
could repeatedly change orbit without need for fuel mass wouId k a great
advancemen t
1.2 Magnetic Propulsion
One possible replacernenr for rocker propulsion is the use of magnetic forces.
In low eanh orbit (LEO), by far the larges1 arnbient magnetic field is the geornagnenc
field, so it will be considered first. The eanh's magnetic field is similar to a fictional
bar magnet buned in the core of the eanh, tilted at 1 1.5 degrees from the axis of
rotation (Memll, 1983). It is well known rhat like poles of two bar masnets repel one
another. so one could postulate a very strong magnet lifting itself in the geomagnetic
field. It would also be necessary to find some means of controlling and stabilizing
such a magnet, for the situation would be as unstable as trying to balance one magnet
over another with like poles opposing. Not only would the levitated magnet tend to
nip over, it would also only be in equilibrium at one height, since there is no conuol
over magnetic field strength.
To control field strength, one could use a coi1 of current-carrying wire.
Current flow creates a magnetic field, and if the current flows in a circle, the field
resembles that of a bar magnet (Bleaney, 1976). One could change the current in this
loop to obtain the required magnetic force for propulsion. The large sire of the
required currents prevenrs the concept of superconducting loops fiom k i n g used for
spacecraft propulsion. Development of powerful superconducting materials is
addressing this problem.
1.3 The Superconductor Revolution
Superconductivity is the absence of any resistance to current flow. Current can
flow ui an isolated superconductor indefinitely wirh no Iosses. Prior to 1986,
superconductivity had only occurred in rnetals that had k e n cooled to near absolute
zero (Burns, 1992).
In 1986, G. Bednorz and A. Muller found superconductivity occumng in a
type of copper oxide at temperatures rhat, with funher research, reached over 80 K.
This is warrner than the boiling point of Iiquid nicrojen (= 63 K), which is cheaper to
buy than Canadian rnik (Burns, 1992). By 1994, temperatures of over 150 K for
superconductivity had been claimed (Moon, 1994). ïhese temperatures are much
above that of the space environment back,pund temperature of 2.7 K (Zubrin and
Andrews, 1991). The current densiries likewise increased by orders of magnitude, ro
the point whcre diin füms can carry on the order of 10' ~ / c m * (Moon, 1994), which
means a wire of one square centimeter area could cany a current of 10 million
amperes-
1.4 Objective of Thesis
The layout of magnetic propulsion systems utilizing both single and mple rings
of superconducring loops interacting with the eanh's magneric field will be developed.
Intemal and exremal stress on the loops will be calculated, as will the amount of
reinforcement needed. Simulation programs for borh the single and mple loop designs
will be run for situations with and without drag to estimate tirnes required for a
selected orbit transfer. Based on the expecred mass of the system for the forces
obtained, the practicality of replacing rockets with magnetic propulsion for launching
into and travelling through space will be judged. The future potential of
superconductin; magnetic-loop propulsion as a means of interplanetary and
interstellar travel will be outlined, along with calculations showing the barrien that
must be overcome to make these developmenrs pracrkal.
Chapter 2 : Literature Review
2.1 Origins
Meissner (1933) is genedly credited with the fvst comprehensive examinanon of
the phenomena of superconductivity, which is the complete absence of elecmcal resistance
in a material. Stamng to undentand the phenomena of superconductivity allowed study of
the possibility of magnetic flight because powerful elecnomagnets could now be made
which required no power input beyond that needed to start the current. However, any
system that uses anything but stable attraction forces--a self-levitating system, for
example--wouid need an attitude conrrol system. Careful design and control have resulted
in many artificially stabilized, levitated superconductors.
2.2 Levitated Superconductors
In the 1960's, efforts were made to create gyros from levitated, spinning
superconductors. The gyros were levitated by inducing a curren t in the superconductor,
which then repelled it fiom a magnet below. (Culver and Davis, 1957, Bourke, 1964 and
Harding, 1965) In the late 1960's and early 19703, magnetic plasma containment systems
were investigated for fusion reactors. The idea was ro levirate a superconducting current
ring that would generate a plasma containment field without need of physical connections
through the plasma (File et al., 1968). These ideas were not pracrical for production
because superconductors could not operare at high enough tempentures at the tirne.
2.3 Early Magnetic Spacecraft
Until 1986, the limits of current for superconductors compared to the size of the
magnetic fields and field gradients in space made the idea of magnetic propulsion
unfeasible. Magnetic acceleration for space travel has concentrated on using magnetic
guns, either to eject a probe at high velocity (Lernke, 1982) or to eject particles from the
spacecraft for thrust (Kom, 1995 and Mauidin, 1992).
Corliss (1960) considered a non-superconducting ring-shaped satellite cutting the
earrh's magneac lines of force as a means of generanng power and also as a means of
propulsion when a current was created in the ring. However, without a superconducting
ring, the iow thrust to power ratio made the idea unfeasible.
Engelberger ( 1964) proposed use of the geomagnetic field for propulsion.
Engelberger suggested that a superconducting hbSn wire, tens of kilometers in radius,
could be established in a polar orbit and that the current be controlled for propulsion.
Since the geomagnetic field changes direction as one orbits from pole <O pole, the forces
o n the loop would altemate with each half orbit. Engelberger's idea came before the
development of advanced superconductors.
In 1971, the Cjnited States Air Force (USAF) did a review of advanced spacecrafr
propulsion concepts, including magnetic propulsion. called Project Outgrowth. The
USAF team appears to have copied Engelberger's dia,mms and equations. The report
concluded that there was no significant value to magnetic propulsion with srare-of-the-an
superconductors in 197 1.
Alfien (1972) suggested using pulses of current dong a rigid tether to produce
forces for orbit msfe r . Wih a magnenc field in space of 5x10'~ Tesla and 2 Volt,km of
elecaic field, to get one Newton of force, one would need a spacecraft tens of LaIometen
long. Even so, Netzer and Kane (1994) wrore about the same idea, with linle change
excep t for improved superconduc tor materials.
2.4 Present Magnetic Spacecraft
The discovery of vastly improved superconductors by Bednoz and .Muller in 1986
Ied to new interest in the use of mapnetic forces for spaceaaft propulsion. Old ideas were
re-examined in the light of new superconductor capabihies.
Ackerman (1989) suggests using an arrangement of two superconducting loops,
one rotating at right angles ro the other, so as to provide stabilizing control when moving
throujh a magnetic field. Ackerman's concept is by far the rnost futunstic of any of the
invesrigators, considering currents of hundreds of millions of amperes thar allow self-
launching of the magneac craft in the geomagnetic field. However, the mass estirnates of
Ackerman's proposa1 do not consider the stresses on the structure created by the magnetic
currents.
Zubrin and Andrews ( 1990) proposed an almost idenrical design to Engelberger,
but with a completely different propulsive effect in mind. They used a h J S n
superconducring wire tens of kilometers in radius, but their propulsion came from
deflection of the interplanet- or interstellar plasma winds. For a 10-km radius loop at
earrh-distance from the Sun, this plasma effect would give an acceleration of 0.008 m/s2.
Their analysis included the mass for cooling and power generating equipment.
Vulpetti (1990) considered a "field-sail", with the same design as Zubnn (1990),
concentrating on intersrellar propulsion using the solar wind. Like Zubrin, Vulpetti did his
calcularions with a superconducting loop, but also considered a directed beam of particles
from an emitter at the starting point for added thnisr.
Zubrin (1991) considered the geomagnetic field interaction after a fiend (R.
Forward) suggested it to him. Geornagnetic field interactions cause forces an order of
magnitude greater than the solar forces for identical loops close to the eanh. Zubrin
noted that if the magneac sail attitude could be controlled, a force away from the eanh
could be obtained. Zubnn confined his anaiysis to the self-stabilized case, which means
that the magnetic moment of the loop was allowed to align itself with the earth's magnetic
field. Currents are applied when the loop is favorably positioned, but there is no attitude
control over the loop. Zubnn suggesrs that if superconductor curent densities were
increased 80 tirnes from their 199 1 values, his loop design would be capable of self-driven
orround launch to earth orbit, but mass estimates of mechanical reinforcernent at the new "
currents were nor included.
Chapter 3 :The Geomagnetic Field
3.0 Generai Description
The eanh has a magnetic field amund it. Fig. 3.1 shows the magnetic lin force.
Figure 3.1 Eanh with magnetic field lines (Tascione, 1988).
The source of the earth's magnetic field is not yet known with certainty, but the
most popular theory holds that convection movements in the core create a self-charging
dynamo, which means thar the movements of charge and mass in the molten ion core of
the emh create effects which sustain those same movements (Tascione, 1988)-
To simulate the movement of a satellite through the geomagnetic field, it is
necessary to calculate magnetic field strength at each orbit position. In practice, the
satellite control system will measure the field with on-board magnetometers. Simulation
of this situation requires redistic values of field strength. A simple mode1 of the
geomagnetic field will also enhance undentanding of the conrrol problem.
3.1 Dipole Approximation
In low eanh orbit , the geomagnetic field may be approximated as a dipole whose
magnetic moment is tilted at 11.5 a to the axis of the eanh's rotation.
Axis of
field
Figure 3.2 Geomagnetic field approximated as a dipole.
The rnagnetic nonh pole of the earth lies at about 79' N 6g0W (1989) and has an
eanh surface suength of 3.05~ IO-' Tesla, rhough both the position and field suength of
the pole Vary with tirne (Corliss, 1960). Field strength decreases as the inverse cube of
distance until one leaves the region of the eanh's magnetism (the rnagnetosphere) and
enters interplanetary space. The dipole approximation is good to within 20% of the
actual field at low eanh orbit distances (Memll, 1983). In polar spherical coordinates,
die magneric field strengrh, B, is @en as:
Where: B = magneric field strength r = distance from center of earth r, q =radius unit vector, tangential unit vecror, ïearh = eanh's radius B, = surface field of earth. 0 = CO-declination, angle from equatorial plane o = right ascension, angle from Aries, a star.
The tangential unit vector q is in the direction of angle 9. Sphencal coordinates r ,
Q and 0 are shown in Figure 3.3 below.
2-axis, rotational axis of earth
Figure 3.3 Spherical coordinates for earth frame.
Note that the magnetic field strength is symme~cal about the magneac z-axis in
figure 3.1. Thus, in equation 3.1, the field is given in two directions only. Note also that
the magnetic force on a dipole depends not on the field suength, bur rather on the gradient
of field suength.
3.2 Variations in Ambient Fields and Currents
Space is filled with ail sons of currents and elecmc fields (Tascione, 1988). The
earth. the sun and some orher planetary bodies have magnetic fields and surrounding
atmospheres that can cany current. These fields will make a difference to the forces on a
loop with large cumnts flowing through it. Not only that, the fields and currents change
with time according to a multitude of factors which are often unpredictable, such as solar
activity. Probably the actual implementation of magnenc propulsion would require a
merging of modeled and sarnpled magnenc field data. For orbiting spacecraft, the data
sampled from the previous orbit rnight be used ro predict the field strength expected on
the next orbit. The solution will most iikely nor be as simple as the geomagnetic reference
field approximation used in this thesis. However, the dipole approximation used will yield
accurate figures to be used in loop size and performance calculations when in the
preliminary design phase of a magnetically propelled vehicte.
Chapter 4 : Loop Design
4.1 Objective
It is proposed that a satellite in low eanh orbit will use a superconducting loop of
current to create forces for propulsion by interaction with the geomagnetic field. A loop
of current creates a magneric dipole that behaves in the same way as a smail magnet
placed in the field of a large magnet. The forces developed wiil depend on the magnetic
moment, which depends on the area enclosed by any given curren t. Therefore, a circular
loop is chosen as it will enclose the largest area for a given length of superconducting
wire. The objecrive is to find a suitable number of loops and a suitable arrangement of
those loops with respect to the spacecraft body.
4.2 Magnetic moment of loop
The definirion of magnetic moment for a current dismbuted about a curve (C) is
gïven as:
w here: M = magnetic moment of curve C 1 = current around curve C r = vector from origin to a point on curve C ds = differential element of curve C
For a curve that lies in a one plane, the integral pan of the expression above is
twice the area of the bop, so the expression for magnetic moment is:
r = radius of loop M = magnetic moment of loop 1 = current in loop
The magneric moment vector is in the direction shown in Figure 4.1.
magnetic moment vector, m
current loop +
Direction of Current, I
Figure 4.1 Magnetic moment of currenr loop (after Wangsness (1986), p. 304).
The magnetic field created by the loop is iike that of a mapetic dipole, as shown
in Figure 4.2 below:
Figure 4.2 Field of a magnetic dipole.
4.3 Nonuniform Field Force
The force (F) on a magnetic dipole moment (M) in a nonuniform field (B) is:
The simple expression above follows From the definition of magnetic induction:
where
F = force on curve C 1 = curent ds = elernent of curve C
B(r) = rnagnetic induction, or B field
r = vector from the origin to a point on the curve
Consider a small current loop in an extemd magnetic induction (B field), which
lies in the xy plane, k e thar of Figure 4.1. To make the integral easy, consider r in
azimuthal coordinates:
r = vector from origin to current loop a = radius of current loop @ = angle r makes wirh the x axis in the xy plane x, y = unit vectors in the x and y directions
The ds expression above is used, so that the integral term of 4-4 becomes:
The x, y and z subscripts on the B term indicate the parts of the magnetic field in
the x, y and z axis directions. Since the loop is small, the field does nor V a r y much over it,
so the magnetic field can be expanded in a power series about the origin. If the rnagnetic
field is expanded in a power series about the ongin and only the fust ternis retained, the x
cornponent of the equation above is found as:
The inte,mtion about a complete circuit means integratinj $ from O to 21r, which
means thar the (cos@, (sino) and (cospsing) terms are al1 zero, while the (cos29) tenn
inteptes to x. The x component of the force is found as:
The extemal magnetic field is assumed to be generated far away From the Ioop.
Thus, the field is uniform and V x Bo = O , which implies (6BJ8x) = (6BJ6z).
Remembenng that the magnetic moment only has a componenr in the z direction, the force
in the x direction can be rewritten as:
The forces in the y and z directions can be found by exactIy the same procedure as:
Combining the expressions for force in the three directions, one obrains the general
expression for force given at the beginning of Section 4.3. For a loop of area A and
current 1, the equarion for magnetic force in an extemal induction B may be writren as:
The power series expansion used above for the magnetic field can also be done in
sphencal coordinates. which are often convenient to use for spacecraft work. See Figure
3.3 for an explanation of spherical coordina tes in an eanh-centered geomagnenc reference
frame. A transformanon of equations 4.10 to 4.12 fiom (x,y,z) coordinates to the
spherical coordinates of the earth (r,Q,B) yields the following new expressions:
1 = currenr through loop r = radius from centre of geomagnetic field û = angle from geomagnetic North pole @ = azirnuth angle & = Area of current loop perpendicular to 0 vector Be = Component of magnehc field perpendicular to 9 vector A, = Area of current loop perpendicular to r vector Br = Component of magnetic field perpendicular to r vector
Notice rhat equation 4.16 gives the azimuthal force F, as O. This is a
consequence of using the magneric a i s of the eanh as a reference. If the reference used
was the axis of rotation of the eanh, an azimuthal force would appear due to the magnetic
field *gradient in the q direction. This magnetic force is simply a consequence of the
reference frame used. As seen in Figure 4.2, the magnetic Field is symmemc about the
plane of the magneric equator, implying that magneric potenrial is the same for dl anglesq
about the magnetic axis. In a more intuitive exarnple, one rnay think of the current loop
as a mal1 magnet aligning itself in the field of a much larger magnet. The small magner
would twist to align itself with the field lines of the large magnet and move towards the
nearest pole. There would be no force to make it revolve around the magnetic axis of the
larger magner.
This movement towards the nearest pole could be used in orbit by inducing current
in the loop during the times when the attraction force is in the desired direction.
However, there will always be a radial downward component to the aruactive force ( F r )
as well as the desired tangentid component (F , ).
To be able to obtain repulsive forces as well as attractive forces towards the
nearest pole, some rneans of conuolling the direction of the magneac moment vector of
the loop must be found. This is the sarne situation as orientanng a small magnet in the
field of a large magner. For instance, for a repelling force, the magnetic moment of the
small magnet would have to be opposite the magnetic field lines of the larger magnet,
which is the same as saying that Iike poles repel each other. Similarly, a current loop
could be onentated in the geornagneric field to obrain either repulsive or attractive forces.
1.4 Single loop
With a single b o p of current, control over the direction of the magnetic moment
vector implies use of an attitude connol system (see figure 4.3). To decide on a suitable
attitude control system, the disturbance torque on the loop must be known. The torque
on a current elzment in a magneric field is ziven in equarion 4.17.
Superconducting loop - - Current, l
B, Magnetic Field Lines Figure 4.3 Single loop of currenr in a magnetic field.
The [orque on the loop of Figure 4.3 is:
dt = differential torque dF = differential force
r = distance from the loop centre
Using previous equanons for force in Section 4.3, one can integrare around the current loop to obtain:
Using the relation A x ( B x C) = B(A C) - C(A B ) allows the term in square
brackets in the equation above to be written as:
Since 1 ds 1 = 1 dr 1 (an infinitesimai change in arc length is the sarne as an
infinitesimal change in radius) the second term above can be chançed CO :
Integrated. the expression above is equal to zero, since the inte,pI of a finite
scalar over a closed parh is always zero. This Ieaves only the fust term in equation 4.70.
Stokes's theorem is used ro transform the path inre-pi to a surface integral, so thar the
combination of equations 4.19 and 4.20 becomes:
da = differential area.
cis = differential element of curve Bo = magnetic field strength t = torque r = distance from centre to element k i n g integrated. I = current in amperes
Lf Bo is taken as constant, then V(r Bo ) = Bo and
If the surface area of the loop is A and the current is 1, then the magnetic moment,
M, equals IA and torque is given as:
At 700 km altitude, the geomagnetic field, Bo is given by the equations in Chaprer 3 as:
r = radius from earth centre B, = geomagnetic field snength at polar coordinates radius r and angle 0 B, = geornagetic field at eanh surface (r = earth radius), 3.05~10" Tesla average. r, = radius of the eanh, 6378 km. r. q = unit vectors in the r and 8 directions (see Figure 3.3)
For a 3 km loop with a currenr of 50 kA, the m i m u m torque at 200 km altitude is then:
This torque is for the case when the magnetic moment of the loop is 90° to the
aeomagnetic field. At O* or 180' to the field lines, the torque on the loop would be zero P
because the cross product is proportional to the sine of the angle, and sin(OO) = sin(180°)
= O. However, at the 180' position, any disturbance would cause the loop to flip over to
align inelf with the pomagnetic field in the same way that a srnûll bar magnet will flip
over when one arternpts to balance it over another magnet. To balance a single loop in the
unstable 180' position or to damp out oscillarions about the stable posiàon. some means
of attitude contro1 would have to be used.
4.4.1 Thrusters
As the loop orbits the earth. a continua1 change in attitude is needed since the
geomagnetic field direction is changing relative ro the moving satellite. Using thrusters
to conuol the attitude of the loop durin; the unstable part of the orbit is pointless as the
objective of using magnetic forces was to avoid thrust by mass expulsion and its attendent
fuel requiremenr. Large torques could be conuolled with large enough rockets and
enough fuel, but the mass of the fuel expended would be prohibitive for the type of long-
term mission for which a magnetically propelled vehicle would be besr suited.
4.4.2 Control Moment Gyros
Control moment Gyros are spinning wheels rnounted in a gimbal. A torque can be
created by tuming the gimbal axis. In effect, the spacecraft is turned around the ,op,
which tends to remain in a fixed orientation in space. Control moment gyros currently
used can give torques on the order of 500 N-m for short penods (Mauldin, 1993).
7 Torques on the order of 10 N-m, as found in Section 4.4, are much too great for
conventional gyro wheels. However, if a big enough gyro wheel were made, or if much
higher rotor speeds were possible, the torque could be made large enough. The torque is
directly proportionai to the mass of the spinning rotor and the square of the rotor radius of
gyration. Larger torques irnply propomonally larger mass, which is a problem for space
applications where mass is critical. However. the use of a gyro for a self-levirating single
loop (where attitude conuol is essential) will be briefly considered.
4.4.3 Single Loop Applications
Even if attitude control of a single loop is difficult, this does not mean that the
single loop concept is useless. Far from it: a single loop does not have the complexiry of
other options, such as double or mple loops and, when the magnetic field is pointed in the
same direction as the desired force, the loop can be used for useful thrust without any type
of attitude control system.
Predominantly Radial forces
Stable Thrust
\
Predominantly Radial Forces
Fi-me 4.4 SateIlite orbit around the eanh showing regions of mapnetic stability.
In Figure 4.1, the areas Iabeled "stable thrust arc region", are where the magnetic
moment of a single loop simply attempts to align itself with the local magnenc fieId. An
attractive force towards the nearest pole is present, which can be used to connol polar
orbits. No anitude control system is needed for this type of orbit correcrion, but some
means of damping out oscillations about the stable position would have to be found.
Calculations of performance of a single loop will be included in Chapter 6.
4.5 Double Loop
Some satellites, such as ISIS, launched in the 196OYs, use two circular cunent
loops at 90° to create magnetic dipoles needed for spin-up and planer atùtude control of
the satellite (Vigneron, 1972). The second bop, at 90' ro the first, allows the magnetic
moment vector to be placed anywhere within a plane by varying the currents ii the two
loops. The total magnetic moment vector is the sum of the vecton for the two loops.
moment Magnetic m 1 f \ k + o l Morne
Figure 4.5 Two current-carrying loops at 90' to each other
Two loops could place the magnetic moment vector anywhere in a plane, but this
is not good enough for complete attitude control because the satellite requires three-
dimensional magnetic moment pointing ability as it orbits through the geomagnetic field.
With a double loop, attitude conuol is only possible at the poinrs in orbit when the satellite
is favorably oriented with respect to the geomagnetic field. In other words, since the
magneac moment vector can only be controlled in a plane, one would have to wait until
diat plane is already in the correct attitude with respect to the geomagnetic field. Planar
p o i n ~ g ability is d l that is possible for a double loop, since the magnetic moment vector
is only controllable within a plane.
A possible solution is to make the entire spacecraft spin. Then the magnetic
moment could be placed where needed by pulsing its current at the times when the
orientation is correct. The loop would have to be charged and discharged in response to
conrroller commands. Charging and discharging the loop completely every few seconds
would be a challenge and probably a waste of power. Some capacitor system would have
to be found to store the charge. There would also be problems keepin; the spin constant,
since the magnetic torques would affect the spin rate.
For propulsion, the double loop concept does not have the simplicity and fieedom
from stresses that a single loop does. nor does it have m e attitude connol. It would be
better if the attitude control and propulsion sysrem had no moving pans and would work
regardlzss of satellite attitude. Performance of a double loop will not be invesngated.
4.6 Triple Loop
With an arrangement of three loops perpendicular to one another, the attitude of
the spacecraft is irrelevant for propulsion or attitude control because currents may be
varied to create a total magnetic moment vector in any direction.
Curre
Figure 4.6 Triple loop of superconducting loops with net magnetic moment , m-
The attitude and propulsion systems are al1 one, since the appropnate currents cm
yield both magnetic forces and torques. Beyond the current switching devices, there are
no actuators or other moving parts to the system. Power could be obtained via solar
panels or other long-terrn supplies such as on-board nuclear reactors. Since the loops are
superconducting, elecaical energy could be converted directly to gravitational or kinetic
energy. The only losses would be those associated with controlling the currenrs. The
mple Ioop will allow force to be applied in any direction, Save only those directions that
lead to areas of exactly the same magnetic potential.
4.6.1 Triple Loop Refinernent
The amount of current required to generate even small forces will be very large
compared to the current needed for attitude control. Therefore, during the parts of the
orbit where forces are king created, large torques tending to align the spacecraft with the
local field will arise unless the magnetic moment vecror is at O* or 180' to the local
magnetic field. This torque is the cross product of the magneuc moment and the magnetic
field, which is proponional to the sine of the angle between the two tems (which is zero
for 0" or 180°). This torque makes half of the thnist arc stable and half of the thmst arc
unstable. The remainder of the orbit is so close to the magnetic pole that only
predominan tl y radial forces could be created.
Predominantly
\ Radial forces ,
Unst Thru
Thrust
Unstable Thrust arc Reg ion
Predominantly Radial Forces
Figure 4.7 Stability regions for magnetic control showing circular orbit.
ï h e stable part of the thrust arc is the part of the orbit where the loop is king
ataacted to the nearest magnetic pole. With no attitude control system at all, a single loop
would align itself with the local magnetic field, which happens to be roughly tangennal to
the orbit parh. There would be oscillations about this stable position which would have ro
be d a m p d The unstable part of the thrust arc is when the magnetic moment of the
loop is king pushed away b m the nearest magnetic pole. A single loop without an
attitude conmi system would flip over 180' and try to move towards the nearest pole.
The triple loop design provides a means of stabilizing the unstable thrust arc
pomon of the orbit via a feedback connol system. The thmst arc cime can be doubled
compared to a single loop with no attitude control. The satellite will also be controllable
in attitude at any rime, which is useful for pointing payloads toward tarsets.
Upon reflection, one cm see that the arrangement of rhree equally-sized loops is
unnecessary. In the unstable pan of the arc, the lmps used for attitude control would nor
need to be nearly as large as the loop king used for tangential thrust i f the loop used for
rangenrial thrut is close ro 90' ro the local magneric field. This is posnble because the
torque on the main bop depends on the sine of the angle benveen its magnetic moment
and the local rnagneàc field. When the plane of the loop is 90' to the field, this implies
that the magnetic moment of the loop is at either O0 or 180' to the field. Since sin(OO) =
sin(180°) = 0, the torque is also zero for the main loop. On rhe other hand, the secondary
loops, used for attitude control, have magnetic moments at close to 90' to the field, where
the torque is a maximum for a given magnetic moment because sin(90°) = 1.
T Y axis (plan view)
Direction of Flight and x-axis
LI/ Magnetic field
Figure 4.8 Modifiecl triple lwp, view in direction of tz axis (towards eanh)
As long as the angle the rnagnetic moment of loop 1 makes with the magnetic fieid
is small, loops 2 and 3 can be much smailer than loop 1, while still counteracting the
torque in the unstable part of the orbit. The size ratio of loops 2 and 3 to loop 1 will be
set by the angle of deviation, $, at which designers wish the secondary loops to be able to
generate equd torque to the main lwp . This angle represents the maximum staric
deviation from unstable equilibrium that can be recovered by the secondary loops without
reducing the current in the main loop. To find the bop size ratio for an allowable staac
deviation of 4, the rorques are equated for both loops:
M = rnagnetic moment of loop, areaxcurrent t = torque on loop due to geomagnetic field B = geomagneric field Q = permissible staac deviation from unstable equilibrium
Assurning that the geomagnetic field is the sarne for both loops and that the loops
can carry the same current, the loop size ratio cm be found as follows:
sin Q ~2 = 4 j-- sin(9O0-+ )
Ri = Radius of loop 1, the main loop Rz = Radius of loop 2, a secondary loop <b = allowable static angle of deviation from unstable equilibrium
Should the main loop be disturbed beyond the point where the control loops can
recover it. the result is not catastrophic. One could simply wait for the satellite to fly into
the stable pan of the thnist arc, which aiways occurs imrnediately after the unstable part.
Altemanvely, the current in the main loop could be reduced so thar the smaller lwps
could re-establish the desired attitude.
4.6.2 Storing Energy
When no control forces or torques are required, current could be stopped by
introducing some resistance into the circuit, perhaps by heating the superconducror.
However, dissipating energy is wasteful, especially when the supply may be intermittent,
as with solar cells. For the 1rip1e loop, one solution is to have current dismbuted arnong
the loops to align the nec rnagnetic moment vector paralle1 ro the local magnetic field. A
control system would have to sense the local field direction, then use the correct
combination of the three mutually perpendicular magnetic moments to align the net
magnetic moment either direcdy with or directly against the local field. In this way,
energy could be stored without creating [orques. However, forces will be created
whenever there is a net rnagnetic moment present. Unwanted forces could be averaged to
zero by switching the magnetic moment vector exactly opposite the local magnetic field
after a short period. In any event, for the portion of the orbit on the dark side of the eanh,
or for a single loop design satellite designers will doubtlessly use some other form of
batteries, at least as a backup to the energy stored in the superconducting loops-
4.6.3 Crew Protection
For human crews, a benefit to using magnetic forces for propulsion is rhat charged
particles will be deflected by the fields present. Ions that are damaging to spaceaaft and
crew alike could be deflecred with the protection of a magnetic field. The effecrs of the
field itseff on biological organisms would have to be tested thoroughly, even though later
Chapters will show that the size of the main loop will be on the order of kilometers, which
makes the centre field only a few rimes stronpr than arnbient magnetic fields on eanh.
4.7 Magnetic Propulsion Advantages
Magnetic propulsion via superconducting ioops has the following advantages:
no moving pans to the basic propulsive system
crew protection from charged particles
very litde fuel used if nuclear powered, none if solar powered
sustained acceleration possible without large masses of fuel
Amtude conirol using a mple ring has the following specific advantages:
works in any spacecraft attitude
attitude control and propdsion are one system
To realize these advantages, there are numerous design challenges that must fmt
be overcome.
4.8 Design Challenges
To design a magneticdly-propelled spacecraft, one must find the stresses on the
structure for selected loop sizes and currents, then obrain mass estimates. Once the
physical charactenstics of mass and volume are known, magneac force and atmospheric
drag may be approximated. Work cm then be@n on an orbit control system and, for rhe
oiple loop arrangement, an attitude conuol system.
The orbit conuol system can be open-loop, with the desued impulse of thmst
king calculated based on the orbit correction required. The orbit correction will be
provided by the main loop. During the unsrable pan of the thrust arc, the attitude control
system on the mple loop will need to maintain the artitude so that the magnenc moment of
loop 1 is at 180' to the local magnetic field. The single l w p arrangement will only use the
stable pan: of the t h s t arc and allow itself to be aligned with the geomagnetic field.
Once the equations for drag force and thrust on the loop are found, a size for the
main loop can be selected based on the expected acceleration using reasonable currents
and materials limits. Reinforcement mass is a key concem for the triple loop, since to
obtain more magnetic force, one needs larger currents, which imply more reinforcernent
mass. At each loop size, the intemal and extemal stresses rnay be calculared so that a
mass can be found. based on suitable matenal strengths (Aluminurn alloy is used as an
example). Based on the expecred performance in terms of the mass of the propulsion
sy stem, strengths of materials and su perconductor performance required. the feasibility of
magnetic propulsion will be assessed.
Chapter 5 : Loop Reinforcement
5.1 Objective
As implied by the title, the objective of this chapter is to find expressions for the
thickness and diameter of loop mbing needed to support the stresses imposed on both
single and triple loop arrangements. Wirh these expressions, the mass for any main loop
radius rnay be found, along with the acceleration in a typical field. The expressions for
loop mass, gïven main loop radius, will be used for performance calculations in Chapter 6.
5.2 Methodology
A realistic maximum current d l be selected, based on examples frorn
superconducting magnetic energy storage (SMES) projects. Fiiinj the current will fix the
esrimates used for coolin; system and superconducting material mass per meter of loop.
The amount of reinforcement marenal uill be found by calcularing the stress on the loop
and using enough material to support that stress.
5.3 Maximum Current
Peck and Michels (1991) report on a 700 kA Superconducting Magnetic Energy
Storage (SMES) system, consisting of a NbTi superconductor srabilized with high purity
aluminum The idea is to store energy so that as demand varies in a town or indusmal
plant, the power production system does not have to Vary its output, but can run at a
constant, high efficiency rare. SMES also has applications for emergency power.
Superconductor
Magnetic field lines
Figure 5.1 Superconducting magnetic energy storage system (SMES)
Spadoni (1991) States that the maximum current in a superconducting loop is
limited by the voltage used for charging and the heat uansfer loss connecting to the room
temperature DC power input. Work is continuing on 100 kA to 1000 kA currents, but
only 50 kA has ken demonsuated by physical cable in useful lengths, and that was for a
NùTi low-temperature superconductor. The researchers M t e confidenrly about applying
high-temperature ceramic superconducrors to this application, but to be conservative, 50
kA will be selected as the current available.
5.3 Self-Current Stress
irrespective of extemal magnetic fields, there are several forces rhat must be
considered when current exists in the main loop. A radial force that puts tension on the
loop exists because the current in the loop creates a magnetic field. Every element of
current in the loop is suspended in the field of al1 the other current elements, and since the
magnetic force is a cross product of current direction and magneac field, the force is
outwards.
main loop Forces outwards
e e
satellite 1% Figure 5.2 Outwards tension on the main Ioop when carrying current.
The force on the wire is thus a tangentid tende force. To obtain the magnetic
field dong the axis of the loop, one uses the law of Biot and Savart:
Figure 5.3 View of current-carrying loop in plane of bop.
Along the axis of the loop, only the part of the field parallel to the z axis (dB 1 1 )
concributes to the field, since there is symmetry about the z axis.
p, 1 sin(90°) cosa (ds) B = 1 d ~ , = I d ~ c o s a = 1
bop bop ' 0 9 4n r2
R cosa = - =
R r 1
(R' +z2):
Combining and simplifying the above two expressions:
po IR ' B(z) = [z direction]
2 ( R ~ + z2)T
B(z) = magnetic field strength parallelro z mis, at distance z from loop plane & = 4nx 10" T d A , permeability of vacuum 1 = current in lmp creating field ds = differentiai element of loop length R = radius of loop r = distance from element of loop length to point P = = see Figure 5.3, angle in radians
Assurning z = O yields a very simple expression for the magnetic field snength at
the centre of a loop of current. This estirnate of field strength will be conservative, since
the value of z will be greater than zero for most calculations.
P o 1 B = - [z direction] 2R
The force on a lengh of current-carrying conductor in a magneric field is gïven by
the fundamental elecuodynamics equation of
1 = length of current conductor 1 = current in conductor B = magneac field F = force on conductor
Using the field B above and a circular currenr loop, the force per meter of loop is:
R = radius of current loop = permeability of free space, 4~x10-' Trn/A
F = force on current loop B = magnetic field suength 1 = Iength of wire, circumference in rhis case
The expression above is for the radial force on the entire loop due to its own
current To find the tension in the wire, Figure 5.4 is used.
Tension, T \ Angle
Figure 5.4 Vector dia-mm showing relationship between outward force and tension.
The Iength of arc enclosed by an angle 0 is RB. From rhe diagram, ii can be
deduced thar the outward force is reacted by the downward (in the diagram) part of the
tension vectors.
F = magnetic force per meter 1 = length of cable considered p,, = permeability of free space R = radius of loop 1 = current in loop 0 = angle of arc subtended by segment length 1.
Simplifying and taking the limit of this expression as 8+O, one obtains:
lim 1 0 II, ' 2 T=- -- 8 8 8+0 7 (frornsin,=-as0-tO) - 2
The nominal value for current is 50 kA.
- This is a conservative estirnate of tension, since the field experienced by the loop
will not be as strong as the centre field under maximum current conditions. 137 M Pa is
used as a
Area
typical allowable tensile stress for aluminum (See Section 5.8).
- - Tension
GA;
1570N Area = = 1.1 x 10"m2
137 x 106 ~a
w here:
= aUowed stress for alurninum
This area, quivalent to a solid wire with a radius of less than 2 mm, is the area
used when detiding on reinforcement for a single loop. However, the rriple loop will be
shown to require a larger amount of reinforcement due to other loads.
5.5 Torques between Loops
The net torque of al1 loops due to each other's fields is zero, as with any closed
system. However, there will be large torques between the loops, since their magnetic
moments are perpendicular to each other. To find the reinforcement needed to resist the
bending moments of one loop due to the field of another, the worst case scenaio of
maximum currents in both loops will be used.
\
y- axis
@ Magnetic field ou t of page
Curent in loop
r-y Torque on loop 2 with current as shown
Figure 5.5 Torque on loop 2 due to currenr in loop 1, looking along -x axis.
The torque between loops may be understood as ariàng because the magnetic
moment vectors of the loops try to align with each orher, just as a magner mes to align
with a magnetic field. This is the major cause of stress on the loops and tethers when the
mple-ring arrangement is used. As an example, consider a 500 meter radius main loop
and a 2 10 meter radius second- Ioop.
Moment = M ,, x B
Magnetic moment = Mm, = IT R:I~
Moment = 4.35 x 10' Nm ( 5.22)
This is the moment which acrs on the entire secondary loop, which will be
supponed with 2 tethers for each loop.
5.6 Force from current in tether
The tethers between lmps rnay carry current if they are used to uansfer current
between loops, or if the power source happens to be located at the spacecraft centre. The
tethers would then experience bending forces, as rhey would be carrying current through
the magnetic field of the loops. The question is: how strong must the attachment points
be for the tethers and the loop joints because of currents that may be carried in the tether?
If flux pumps are used to control the current in the loops, then the current in the
tethers to the flux pumps will much less than the loop currents'. probably 0.28. To be
extra conservarive, let us say that the magnetic field on the tether is equal to that which
would be produced by a loop one quaner the size of the main bop, rather than the centre
field of the main loop, which would be weaker.
I see chapter 7 section on flux pumps.
secondary loops \
Figure 5.6 Loop area giving magnetic field for tether stress c
The magnetic field, and the force on the tether due to the field are found as
follows:
P o 1 - CL, 1 -- P o 1 El= 2 (~adius) -
B = magnetic field snengrh 1 = current in Amperes RI = radius of loop 1
po = pemeability of k e space 1 = Length of current camier
CalcuIauons in the next few sections will show that this 6 N force is completely
insignificant compared to the forces that will arise due to the loops' interactions with each
others magnetic fields in the mple ioop arrangement. That is. the tension in the tethers
fiom resnaining nvisting between h p s wiLI be the determining factor for tether site in the
triple ring arrangement.
5.7 Tethers to Loop 1
The torques created between the loops wiil cause large stresses in the tethers.
Torque on seconda
Main loop loop
Z axis
Figure 5.7 View along x axis towards triple ring, showing torque on secondary loop.
If single tethers were used at A, B, C and D, then the torque, which acts as shown
for current in the main loop and'in the secondary loop fiom D to A, will cause bending
and torsion. There would be torsion dong tethers A and D and bending for tethers B and
C. For minimum mas, torsion is better than bending to handle a load, and tension is a
better way than both. To niinimize the amount of material needed, good engineering
design calls for as much of the load as possible to be held in tension. For this reason,
single tether designs can not compete with multi-tether designs because the mass of the
single terhers would be excessive. Below is a tether design that will allow most of the
load between the secondary and main loops to be held in tension:
Front view
Figure 5.8 Double tether arrangement for connecting main to secondary loops.
To visualize the loads on the terhers, imagine the main and secondary loops
twisting to align their magneac moments with each other. The tethers are assurned to be
flexible, so that no load is held in compression or bending. There are a total of 8 tethers
from the main loop to the secondary loops. There will also be 8 mialler tethers from the
secondary loops to the spacecraft cenne.
5.8 Reinforcement Material
For example purposes, consider the reinforcement rnatenal used in the tether and
loops to be aluminum alioy. In practice, the reinforcement may well be some other
material, but alumuiurn is a realisnc material to use for aerospace applicarions. The
sarnple material used will be alumùium alloy 2014-T6 with a safety factor of 3. From
Table 5.1 below, one can see that there are both suonger and weaker alloys of aluminum.
densi ty, kg/m3
Table 3.1 Yield and ultimate mess for types of aluminum.
The safety factor of 3 is included by dividing the yield stress by 3.
a,,, ultimate stress (failure) 70 MPa
aluminum type
Pure Al
0, 410MPa --- %owed - -
3 3 = 137 MPa
O,, yield stress (deformation) ,
20 MPa
137 MPa is the ailowable stress (adiowd) that will be used in subsequent
cakulations of the mass of reinforcement needed.
5.9 Size of Tethen to Loop L
The size, and thus the mass, of the tethers from the main loop to the secondary
ioops will be detemined by the allowable stress on the tether material and the tension to
which they are subjected. As an exarnple, consider the 500 m radius main loop and 310
meter radius secondary loops. The tension in each tether can be deduced from Figure 5.9.
Tangential Force
Radius 1,500 rn
Figure 5.9 Tension on tether and force arising from torque on secondary loop.
From Figure 5.9,
L
Tension = - cos e
moment F=
R,
Moment Tension =
(2 rethers)~, cos8
8 = angle at which tether pulls on secondary loop F = force at 90" to plane of secondary loop, caused by moment moment = moment in Nm for secondary loop in field of loop 1
RI, RI = radii of lwps 1 and 2
Notice that the moment is divided over the two tethers holding the loop. The
moment h m the interaction of loops 1 and 2 at maximum current was found previously.
The diameter of wire needed to resist the tension is found beiow assurning alurninum
2014-T6 is used with 137 MPa allowabIe stress, as calculated in Section 5.8.
4.35 x 10' Nm Tension = = 1124N
2 ( î 10 rn)(cos 22-80]
Tension Tension
d = diameter of tether Tension = tension on the tether due to secondary loop in field of main loop G ~ I , , ~ = allowed stress on duminum, taken as 137 MPa .
An alurninum wire of 3 mm diameter should thus be able to react the sratic forces
on the secondary loop with a safety factor of 3. Given the size of the srructure, however,
one would expect various vibrations that may well cause the forces to be iarger than those
experienced in the static situation. A detailed design calculation, including possible
vibrations, would increase the safety factor of 3 upwards.
5.10 Mass of Tethers to Loop 1
The mass of wire per rneter, given the diameter, is the volume of wire per meter
multiplied by the density of aluminum alloy. The sample wire for a 500 meter main loop is
used as an example.
pl = mass per meter of wire PA = density of aluminum 20WT6 alloy, 2800 kgm3 d = diameter of wire used for tether
There are 8 tethers from the main loop to the secondary loops, and 8 tethers from
the secondary loops to the spacecraft centre. Assurning the size of tethers is the same as
in Section 5.9, the total mass is found below.
This mass is for the tethers from the main loop to the secondary loops, and also
from the secondary loops to the spacecraft centre. However, this mass does not include
the tethers between secondary loops: they are a different size because the forces between
secondary loops are different from the forces between the main loop and a secondary loop.
Tethea between secondary loops are included in Section 5-12.
5.11 Section Modulus
ï h e concept of section modulus is used to determine the amount of reinforcement
that is needed for a given bending load on the loops.
S = section moduius M, = maximum bending moment G ~ I , ~ = allowable stress on material
The section modulus may be calculated from the moment of inenia and th
distance from the neutral axis. The neutral axis is the line dong which there is no stress.
Consider a solid, circular cylinder which is k i n g loaded by a bending moment:
Load direction
Neutra1 axis Forces
Compressive
Figure 5.10 Stress on a loaded metal cylinder.
It cm be seen that the maximum stresses are furthest h m the neutral axis. Let c
be the distance from neunai axis to the outer edge. Then the section modulus is given by:
S = section modulus 1 = moment of inenia about mis moment is acting c = distance from the neutral axis
Section moduli Vary greatly from one cross section to another, and the type of
cross section chosen will have a great effect on the mass of structure required to support a
given load. Section modulus gives a measure of the ability of a particular cross-section to
resist bending loads. Some examples of section moduli are given below:
solid circle
1 Type of cross section area section rnodulus
soiid square
Table 5.1 Examples of section moduli for various cross-sections.
n 6 d 3 S =
pipe, or hollow circle
L
By calculating the area of each cross secnon for the same section modulus, one can
see that a square requires less material than a circle to resist the same bending moment.
Likewise, a hollow cylinder requires less than both as a consequence of having nearly a l l
its material far from the neutral axis, so that al1 material is used to suppon stresses. A
hollow cylinder will be used for the riple loop arrangement since it is an efficienr way ro
x d2t s=- 1
d = diameter
t = thickness of shell
support bending mess. The single loop amangement can be made from a simple cable
with little regard for cross section resistance to bending because the largest stress for the
single-bop case will be tension.
5.12 Tethers between loops 2 and 3
The force neated by secondary loops is always directly against a tether when they
cany current in the field of loop 1. However, when current is present in borh secondary
loops, their magneric moments will tend to align with each other, creating torsion between
the loops that can not be supported by the tethers from the main loop to the secondary
loops.
Loop 2 T x axis
+ Y axis
Loop 2 torquew @ LOOP 2 torque due due to loop 1. to loop 3 field
Figure 5.1 1 Torque on loop 2 due to the field of loop 1 and loop 3 respectively.
For exarnple, if loop 2 has a moment clockwise as viewed above, tethers 1 and 3
would be tipht and 4 and 2 would be slack. If tethers 1 and 3 borh behave the same way
under load, loop 3 will feel no load at al1 from b o p 2, but will simply rotate clockwise by
whatever amount tether stretching will allow. The connecrion benveen the secondary
rings at the "top" and "bottom" of loop 2 (viewed above) does not have to support forces
on loops 2 or 3 due to the fieid of loop 1: the tethers take that load.
If loops 2 and 3 have current in them simultaneously, there will be a torque that
tends to line the loops up in the same plane with magnetic moments in the sarne direction.
This is the moment about the x-axis in the previous Figure. This moment could be resisted
either by making the loop connections and the loops themselves strong enough to resist
the load in bending, or by using another set of tethers connected to the main loop in the
XY plane, or by using a set of tethers between the secondary loops. The best opnon is the
one thar will use the least mass, so the "strong and rigici'' option is a non-smter since it is
much easier to cary loads in tension with a tether rather than in bending or torsion. As
for connections to the main loop or connections between secondary loops, it can be seen
that 8 tethers would be required for the main loop-to-secondary connection, while only 3
tethen could be used between the secondary loops.
Tethers between
Figure 5.12 Two possible tether configurations to contain the secondary loops.
Since the second configuration of 4 tethers requires much less tether length, ir wiU
be selected for initial calculations of tether mass,
5.13 Torque between Secondary Loops
In the worst case, both secondary loops would be operaung at full current, 50 kA.
For example purposes, the standard 210 m secondary loops. used previously, will be used
here. The field strength acting on the dipole moment of one loop is the centre field of rhe
other loop. From Figure 5.13, it can be seen that the magnetic moment of loop 2 acts at
the centre field point of loop 3.
- Field lir
loop3
Figure 5.13 Magnetic moment of loop 3 in the centre field of loop 2.
The torque between the loops in this situation would be:
tl) =Mm% x B
- P o 1 M,,=KR:I, B=- ZR,
P 1 V, I~R* 7 = K R: I (sin 9 0 ° ) i =
ZR, 2
This torque between secondary loops is larger than the torque expenenced by a
secondary loop due to the field of loop 1, since the current in al1 Ioops has the same
maximum and loop 1 is funher away h m the secondary Ioops than the secondary loops
are from each other. When estimating the size of the secondary loop reinforcement and
terhen between the secoridary loops, this torque is used.
5.14 Tethers Betrveen secondary Loops
Tension between secondary loops is directed at a 45' angle to the force at the
connection due to torque on the loop. The tension is found from Figure 5.13.
from torque Plane of Reaction Loop 2 Plane of in loop
45 degrees 100~ 3 Tension
Fispre 5.14 Forces due to tethen between secondary lwps.
The worst-case torque between secondary loops is as found by equation 5.45,
remembenng that two tethers absorb this torque.
M, Force = - ZR2
Force M, - Tension = - - cos0 2R,cosû
L04 x 106 Nrn Tension = -
2(210m)c0s45~
M- = maximum torque between secondary lwps R2 = radius of secondary loops 0 = angle tension acts to force
The area of the wire may be found from Section 5.9, using the new tension.
d = wire diamerer cdl,él. = allowable stress on aluminum alloy
A 6 mm wire diameter should be able to support the sratic stress. The length of
each tether is found by the Pythagorean theorern as the secondary loop radius multiplied
by d2. An estimate of mass is found for a 210 m secondary Ioop.
R2 = radius of secondary loop pAl = density of alurninum alloy used d = diarneter of tether
The mass calculation programs of Chapter 6 will change the thickness and length
of the tethen according to the caiculations used above.
5.15 Secondary Loop M a s
To fmd the mass needed for secondary Ioop reinforcement. the force at every point
along the loop is found. Integrating the force by the distance along the loop yields the
bending moment. From the moment, the amount of reinforcement
found. The total magneric force
secondary loop, the magnetic force
the geomagnetic field interactions.
found firs~
is the sum of the magnetic
from the primary loop, the
The thickness of the cable
material needed can be
forces from the other
self-induced tension and
(see Figure 5.15) will be
thickness
Figure 5.15 Loop cable dimensions.
To find an approximate mass for reinforcernent needed for a @ven stress, the
hollow circle section modulus from Section 5.1 1 is used. The thickness is set at a fixed
value, then the diameter needed at each point on the loop calculated. For simplicity, the
loop size is found by using the largest bending moment present, that between the magnetic
moments of different loops. This approximation ignores the self-induced loop tension and
any torsion that may develop. Following Bowes, Russell and Suter (1984). one obtains:
S = secrion modulus MW = maximum bending moment on loop, found previously odl, = allowable mess on material d = diarneter of loop tube reinforcement t = thickness of b o p reinforcement
The voIume of reinforcement can be found by a surnmation of aU the different
cross-sections multiplieci by the length of each cross-section. The cdculations can be
repeated for a range of different thickness values to arrive at an optimum thickness in
rems of reinforcement material mass. For the mass calculations, a quarter section of
secondary loop will be used. Each quaner section experiences the sarne loads and is
anchored by tethers at each end, so multiplying by four will yield the total mass.
5.1 5.1 Secondary-Secondary Force
The force on a current carrying wire in a rnagnenc field is
dF(x) = Idl x B (x)
dF(x) = magnetic force at point x x = distance dong arc of loop 1 = current dl = differential elemenr of length B ( x ) = rnagnenc field ar poinr x
plane of loop 2
Figure 5.16 Field lines of loop 2 cutting loop 3.
The magnetic field strength is approximated to be constant for any plane in z, with
z = O taken as the plane of loop 2. The magnetic field direction is not fixed, but due to
the symmeuy of the 2 secondary lwps, it is always at 90' to the current. The law of Biot-
Savart gives the field srrength at any z-plane.
B(z) = magnenc field strength = perrneability of free space
1 = current in secondary bop R2 = radius secondary loop z = distance from plane of loop
The starting and ending values of field strength are used to create a saaight-line
relationship between field strength and arc length over a quaner-loop. This approximation
is used to obtain a simple expression of field suength at any arc length x. This linear
approximation of field strength will have no error at each end of the quaner-section.
1 x2R2 - loop circumference = - 4 4
field end = PO P o IR: 3 - 5
field (x) = B(x) =
x = arc length, or Iength along b o p from crossing point
This field equation is used with the differential force expression below.
P It -=- ZR, 1
FZ2 = force on loop 2 due ro other secondary loop
This is the differential force on an element of loop arc length dx.
5.1 5.2 Secondary-Main Force
The main loop has a significantly larger radius than the secondary loops (see
Figure 5.17). The main loop field is thus approximated as king constant in direction with
respect to the secondary loops. In other words, the curvarure near the cenue of the field
is assumed to be negligible.
B Field from primary
Figure 5.17 Loop 2 in magnetic field of loop 1.
The magnetic field is rhen taken from the law of Biot-Savart.
P o 1 field start = - ZR,
field end = P. IR; 3
d ~ ~ , ( x ) = I dr sin@ ) B ( x )
FI2 = force on loop 1 due to current in loop 2 1 = current
B(x) = rnagneuc field at an: length distance x RI, Rz = radius of loops 1 and 2
Assumed B-Field of bop 1
Figure 5.18 Cunent in loop 2 ar angle t (theta) to primary field.
define:
A e = - (radians) R,
3
R1 (R; +R:)? a = (for simplicity of B(x))
Notice that x is the distance in arc length dong the quaner-section of loop.
5.15.3 Total Shear and Bending Moment
The rotal force on a secondary loop due to the orher secondary loop (Fz2) is:
L J
Force on loop 1 due to the field of loop 2 (FI*):
integrate by pans
1
a, 8 and other variables are the same as in Section 5.252
Total shear force is the sum of self-induced tension, primary-secondary force and
secondary-secondary force. For stanc loads, the total moment is the inte-ml of the shear
force by the distance dong the loop circumference.
M(x) = total moment at point x dong loop circumference TtZy = tension from tether linlànp loop 1 and 2 in y plane TzV = tension from rether linking secondary loops in y plane
Stress on the beam will be calculated using the moment found above. The free-
body diagram of the loop quaner-section below shows the won< case scenario for stress.
Magnetic forces
T22x --I
self-tension
tether 1 tension
Figure 5.19 Forces on Ioop quaner-section.
The mass obtained from these force equanons is an approximation only. Including
dynamic loads and torsion could well increase the rot al rein forcement m a s required.
Also, Figure 5.19 shows a straight bearn, while the Loop is actudly curved. The straight
beam approximation is jusufied in the next section.
5.16 Stress on Curved Beam
The maximum stress on a saaight beam under a bending moment is on the outside
edge of the beam and may be simply found (Bowes et al., 1984).
O, = stress in Pa. for straight beam outer surface
M = bending moment, force h e s distance c, = distance to outer edge from centroid axis 1 , = second moment of inenia of cross section
Since the loop quarter-section is curved rather than k i n g a straight beam, a
correction factor is applied when calculating the stress. From Bowes et al. (1984):
R = radius From centre loop to centmid of cross section Rn = radius £Yom centre loop to neutrai axis Y = distance from neutrai axis to centroidai axis A = area of loop cross section &, = outer radius of loop Ri = inner radius of loop c, = distance from centroid to outer wall
outer wall
-.
Figure 5.20 section of loop with neural and cenrroid axis.
The integrai in the neutrd axis equation above was solved numerically (see
Appendix A, program "flex.cppW and function "nacpp"). Beams which are s h
compared to their radius of curvature closely approximate snaight beams with regard to
stress under bending load. A measure of how accurate this approximation is may be
obtained by taking the ratio of the straight-bearn stress (equaeon 5.78) to the c w e d beam
stress (5.79) at either the inner or outer surface. If this ratio is close to 1, then the
approximation is good. This ratio is found by equation 5.81 in program "flex.cpp".
G, = stress ar outer surface, curved section o, = stress at outer surface, straight section 1 = second moment of inertia of cross section (see Figure 5.20 and equations above for other variables)
Another measure of the accuracy of the straight-beam approximation is a
"flexibility factor", explained in Bowes, Russell and S uter (1 984), equation 3-4 1.
Flexibility is a measure of resistance ro bending load. If the ratio of curved beam to
straight bearn flexibility is close to 1, then the approximation of using straight beam
bending equations is god .
- flexibility (curved) EYAR, a = -
flexibility (suaight) - - Ml
- a = flexibility of curved beam compared tu straight 1 = length dong cenaoidal axis of beam M = moment on beam (see Figure 5.20 and equations above for other variables)
The program "fiex-cpp" (Appendix A) found the ratio of curved bearn to smight
barn stress (the "stress factor") to be 0.9769 for the inner surface and the flexibiliry factor
to be 0.9761 for a 500 m radius main loop with a 0.80 m radius loop cross section of wall
thickness 1 cm. Thus, the saaight-beam equations used should yield an answer rhat only
differs from the curved beam equations by about 4%. Given that the bending equanons
are a rough approximation to begin with, the straight beam equations are acceptable.
5.17 Primary Loop Rein forcement
There is both bending and compression on the main loop due to the tether tension.
secondary loops rotation
front view side view
Figure 5-21 Tension directions on the main loop tethers
Current in the main loop will cause an expansion force, which will resist the
compressive effect of the tether. The compressive force will be compared to the current-
induced expansive force. The size of the radial inward and bending forces c m be found
Figure 5.22 and the tension from chapter 5.
Compression
Bending
Figure 5-22 Tether tension. compression and bending vector diagram
Fc = compression force T = tension 8 = found in Section 5.9, for example 500 m radius main loop
The expansion force will be calculared using the cenue field of the loop, which is
assumed constant in the plane of the loop. The pan of the expansion force per meter that
directly opposes tether compression will then be inteagrated over a half loop. A half loop is
selected since if al1 three loops had full current in them, there could be four tethers under
tension and a haif loop would be connected io al1 four. The same calculations as for the
secondary loop are used.
Force F ~ 'np , , P.I' - - - - - - - meter r2R1 a2R, 2 R ,
F = force 1 = current RI = radius of loop 1
= permeability of vacuum
The force opposing the tether tension is the ince,@ of the force in the y direction
in Figure 5.23.
Expansion forces
Tension direction
t
Figure 5.23 Expansion and compression forces on tethered, current-carryin; loop.
FT = total expansion force in y direction for bop quarter-section
Tension forces causing compression are far enough below (1041 N vs. 3 142 N)
expansion forces thar bending rather than buckling wil1 be used for reinforcement mass
estimates of the main loop. For the main loop, the mass per quaner section will be found
by calculating the mass needed to resist bending, as for the secondary loops. The
cornputer code using the above expressions for mass is found in Appendix A. The code
was nin with a 500 m radius main loop and 210 m radius secondary l w p s and the total
mass was found.
1 thickness of tube 1 tube cross-section radius, (m) 1 rnass triple ring]
Table 5.2 Mass of aiple ring, radius1 = 500.0 m, with varying tube wall thickness.
wall (m) 0.00 1 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.01 1 0.013 0.013 0.0 14
Table 5.3 clearly shows that the best mass is at the minimum practical thickness of
0.001 m. In theory, a very thin, large diameter tube would be best.
0.135 0.095 0.078 0.067 0.060 0.055 0.05 1 0.048 0 .O45 0,043 0-04 1 0.039 0.037 0.036
reinforcement (kg) 2235 1.2 3 1609.4 387 13.4 44702.4 49978.8 54749-0 59 135.7 632 18.7 07Û33.6 70680.7 741 30.5 77426.8 80588.4 83630.5
Chapter 6 : Performance
6.1 Overview
With the mass of the spacecraft and the magnetic force on it, one cm calculate
acceleration. With acceleration and a knowledge of how long thrust can be obtained each
orbit, one can calculate orbit transfer times and tirnes to escape from low earth orbit. In
this chapter, the performance at various loop radii for both triple and single loops will be
found. By irying many different loop radii with the equations developed for loop
reinforcement, drag and magnenc force, one can see how the acceleration is related to
loop size and thickness of reinforcement. Decisions can then be made about which way to
drive the design: either towards a very large loop with a mal1 current, or towards a
compact loop with a large current.
6.2 Drag
Ir is a complex problem ro accurately predict the amount of drag to be expected
for a given satellite shape and density of atmosphere. Wertz (1991) gives a general
expression for the acceleration due to atmospheric drag on a satellite:
p = amosphenc density CD= coefficient of drag z 2.3 A = cross sectional area m = rnass of satellite V = velocity with respect to the atmosphere
Echo 1, a large balloon satellite orbiting at 1600 h, expenenced a drag force
exceeding expected aerodynamic drag by a factor of 50. Drell et al. (1965) explained the
drag as an elecuomagnetic coupling between the vehicle and plasma (charged particles) of
the eanh7s magnetosphere. Momentum of the vehicle was uansferred to the plasma
Drell (1965) gives this drag as:
FD = force due to drag m = mass of satellite a = acceleration of satellite due to drag p = mass density of plasma V = vehicle velocity relative to plasma A = frontal area of vehicle Va = Alfven velocity, after D. Alfven B = geomagnetic field transverse to flight path.
In the magnetosphere, V g 100 km/s for most sarellite orbits, which have flight
paths nearly transverse to the geomagnetic field. The relative magnitudes of the
amospheric drag and the electromagnetic drag at 800 km will be compared below. The
plasma density is found from Tascione (1988) for 800 km altitude.
3 -3 Ppiasma = 1 O panides/crn = 4 . 4 ~ 1 0 - l ~ kg/m 3
7 A = 100 mu
V = 8 k m / s VAgven = 100 km/s for typical orbit
14 3 Pair = air density = 10- kglm at 800 km.
Alfven drag = (4.4 x 10-l6 kg/ rn3)(8x 103 m/ s)(100x 10) m/ s)(100m2)
Alfven drag = 35x IO" N
I Atmosphexic Drag = 7 ( 1 ~ - ' 4 kg/ m3)(22)(100rn2)(8x lo3 m/ s ) ~ -
Atmosphenc Drag = 7.0 x 10" N
Elecrromagnetic drag is thus half of aunospheric drag at 800 km for a flight path
transverse to the geomagnetic field. Further calculations show that as altitude increases,
elecuomagnetic drag becornes progressively Iarger than amospheric drag. However,
thoupht must be given to the type of orbit before one includes Alhen drag. The magnetic
field used in calcularing Alfven velocity is the field transverse to the flight path. Since a
magnetically propelled satellite would likely be in a polar orbit in order to navel along the
magnetic field lines, the Alfven velocity would be very small, since almost none of the field
is transverse to the flight path. Alfven drag will thus not be included in the calculations of
drag in the orbit simulanon program in this thesis. Detailed calculations of perfoxmance.
pnor to actually building a magnetically propelled spacecraft would certainly need to
consider Alfven drag. especially for orbits that do not follow geornagnetic field lines.
6.2.1 Variations in Drag
Any heating of the earth's atmosphere will produce densiry changes. Hearing
implies expansion of the atmosphere, which means that the amount of drag a satellite
experiences in low eanh orbit increases. Geomagnetic stoms, associated with solar
acavity, durnp large numbers of particles into the high-latitude atmosphere, causing one
type of heating. Most heating will be near the auroral zone (northem lights), so polar
orbiting satellites experience the greatest effect. Auroral substorms can also produce a
horizontal circulation pattern across the polar cap, with wind speeds of 1 to 2 km/s. For a
low-altitude polar orbiter moving at 6 or 7 km/s, the drag of the circulation can be
sigrufïcant (Tascione, 1988). As satellite altitude changes, as for any ellipacal orbit,
atmospheric density and drag will change as well.
6.2.2 Drag in Orbit Program:
Drag in the orbit program will be calculated using the atrnospheric drag formula
above. Atmospheric density will be taken as the maximum values at each altitude, as
given by Larson and Wenz (199 1 ), which means that the 1 l -year so1a.r cycle, associated
with drag increases, will be considered as active. The data table below will be included,
in expanded fom, as a look-up table (called density-cpp) in the program (see Appendix
A)*
Altitude, km above earth
surface 100
Atmosphenc Density, kg/m3
525x10"
Delta V to mainrain Altitude, rnean m/s per Y car
5.09~ 1 o6
Notice the third column above, which gives the delta-V, or change in velocity,
needed to maintain alritude for a spacecraft with a ballistic coefficient (a rneasun of mass
to surface area) of 100 kg/m2. Above about 600 km, the delta-V needed is so srna11 that
orbits usually Iast g-reater than 10 years, which is more than most satellite lifespans. In
other words, atmospheric drag is not significant above about 600 km unless the mission is
more than 10 years long or has very saingent orbit requirements.
6.3 Attitude Control
The attitude controller for the mple loop design controls the currents in loops 2
and 3 (the secondary loops) during the unstable parts of the thnist arc. Near the magnetic
poles, little tangential thnist can be produced, so the main loop would be available for
attitude control. The attitude controller of the mple l m p arrangement is assurned to keep
the main loop in equilibriurn during the unstable part of the thmsting arc in the orbit
transfer programs of this chapter.
6.4 Gravity
Gravity causes the satellite to accelerate towards the centre of the eanh. In Low
Eanh Orbit (LEO), the gravity of the moon, the sun and the other planets is so srnall
compared to that of the eanh that it is typically neated as a perturbation to the orbit.
Funhermore, most LEOs are nearly circular: otherwise they encounter too much
amosphere (or, worse yet, the earth surface) at perigee. For a circular orbit close to the
earth, the force balance for gravity is:
G universai gravitational consran t ml mass of satellite
mass of earth v velocity of satellite F force r radius frorn earth centre to satellite
Assuming a sphencal earthl, no pan of the gravity force is rangential to a circular
orbit, so gravity is no< explicidy included in calculations to find an opamum loop size for
tangentid acceleration. However, the effecr of gravity is irnplicitly included since if
atmospheric drag decreases the velocity, rhen the gavity force will cause an undesirable
radial movement towards the centre of the earth. This effect is the primary orbit
perturbation which the magnenc loop will correct and is the basic explanation of how the
Keplerian equauons used in the orbit rransfer programs of this chapter work. Equanon
6.6 will be used to find a velocity for circular orbits in the orbit transfer programs used in
this chapter. Velocity is needed in the calculations of atmosphenc drag.
6.5 Electrostatic Fields
Sometimes spacecraft become charged, building up a different electrostaric
potential from the surrounding plasma in the same way that mbber shoes on a polyester
- - - - -
1 Ignores uiaxiality, or oblate spheroid shape of earth.
mg c m build up a static charge on a person. On a satellite, a large charge could create
electrostatic forces. However, the c harge-buildu p phenomena has only ken observed on
geosy nchronous satellites. For low eanh orbir vehicles, charging is rare1 y observed
because low e m h orbits are within the ionosphere (Tascione, 1988).
The ionosphere is a region where the atmosphere is partially ionized due to
photodissociation. Photodissociation is the breaking apan of molecules by high-energy
solar radiation to form ions (hence, the ionosphere). An arbitrary upper lirnit for the
ionosphere was set at 2000 km, which means a low eanh orbit satellite will be within it
C O ~ M U ~ ~ Y (Tascione. 1988). In the presence of ions, an electrostatic field will cause
current to flow, neutralizing a charge imbalance (Corlis, 1960). Charge buildup and
electrostatic forces will not be included for this thesis.
6.6 Optimum Loop Size:
6.6.1 Other Authors
Zubnn (199 l), Vulpetti (1990) and Engelberger (1964) al1 propose usine
superconducting loops with radii on the order of tens of kilometers for spacecraft
propulsion. The reason for the large radii is that rhe maximum current that can be carried
by a superconducting wire is limited. If superconducting current allowed is small, the loop
size must be large for an appreciable magnetic moment. Thick wires presendy cary much
less current density and lose their superconducting state with much weaker magnetic
disturbances than thin wires or thin films. Present-day technology thus inclines the design
towards either a thin wire or a superconducting film spread over a large surface area, as
with a large-diameter hollow tube.
6.6.2 Magnetic Force on Loop
The magnetic force on a current loop is equal to the dot product of the magnetic
moment and the gradient of the local magneac field. (Chapter 4 develops this expression
for rnagnetic force) The magnetic moment vector is perpendicular to the plane of the
loop and equal in size to the current mulriplied by the area of the loop.
hl = magnetic moment vector B = geomagnetic field I = current in b o p Area = area of loop
Magnetic Moment, m
Figure 6.1 Direction of rnagnetic moment for a curent loop.
During the thrusting part of each orbit, the magnetic moment vector will be aligned
with the local geomagnetic field, which means only pan of the area of the main loop can
be used to increase satellite velocity unless the field happens to be pardel to the orbit.
The other part of the loop area will be creating a radial downwards force because the
geornagnetic field makes the loop tilt with respect to its flight path. This also means rhat
die ratio of "radial area" to "tangential area" is equal to the ratio o f the radial and
tangen rial magneac field components.
Amd = radial component of area A, = tangential component of area Bmd = radial component of magnetic field B, = tangential component of magnetic field
The radial and tangential areas squared and added equal the total area of the loop
squared. Combined with the magnetic field expressions given in Chapter 3, the radial and
rangendal areas may be found as a function of the angle from magnetic north.
Use the identiry cos% + sin2@ = 1, then use the sum of squares to replace A,.
A = total area enclosed by loop 0 = angle from magnetic north
The tangential force is then the tangential area multiplied by the tangential part of
the gradient in magnetic field, found in Chapter 3. The radial force will nor be
considered in the orbit control program, but rather the perigee will be kept constant. The
justification for this is that the radial force is small compared to the tangential force
because the thrust arc selected is mainly away from the magnetic poles, where the field is
vertical. Also, the effect of the radial inwards force dunng the thnist arc is offset by the
non-impulsive nature of the thrust, which would tend to make the pengee increase over
time in the absence of the radial inwards force. The maximum magneric force the loop can
create in the 8 , or rangential, direction to directly counter the drag is expressed following
Zubrin (1991). Force variarion is descnbed with a "thrust function", T(0), which scales
the magnenc force according to position frorn magnetic nonh.
sin 28 define: T(O ) =
2 (3 COS^ 0 + 115
B- = magnetic field at earth surface, 3 . 0 5 ~ 10-5 Tesla r d = radius of eanh, 6.378~10~ meters r = position from eanh cenne T(8) = thmst function, defined here. 8 = angle from magneric axis of earth I = curent carried in loop A, = area for tangential thmst, as found above- A = area of loop
Substituting for T(0) and the constants and multiplying through by r,., one obtains:
This is the expression for the force tangennal to the orbit which will be used in rhe
orbit simulation program. The thntst function, T@), highlights a shortcorning of
magnetic propulsion: over the magnetic pole (8 = 0') or over the magnetic equator (0 =
90°), no force can be created in the tangenaal direction if the emh's magnetic field is
assumed to be a dipole. Of course, the field is not a perfect dipole. There are always
some gradients at any point in the field, but as the dipole approximation suggesrs, certain
locations in orbit will need very large currents to generate the desired forces. Directly
over a magnetic pole, for instance, radial forces will be a maximum, but tangential forces
will be very small. The situation is like holding a small magner directly over the pole of a
larger magnet. The force on the small magnet is suaight down towards the pole. There
is no tangential force because the small magnet is already over the pole of the large
rnagnet.
The loop size question can be resolved with a specification of the acceleration
desired at a specific location. Toward 0 = 0°, the radius of the loop wiil tend toward
infinity as sin(28)jO. As 8- 45*, the radius necessary for a given magnetic force reaches
a minimum Once the altitude changes, the density of the atmosphere and the magnetic
field will change as well and the loop size required for a given accelerauon will change.
6.6.3 Net force equation
To find the best loop radius, one can find the besr tangential acceleration as loop
radius varies with constant current and constant drag per m2 of area. The net force on the
satellite is the rnagnenc force less the drag:
Rearranging equation 6- 1 , and using F = ma, the expression for armosphenc drag is:
p = aanospheric density CD = coefficient of drap A = spacecraft frontal area V = velocity of spacecraft
In the orbit pro-mm, the density of the atmosphere and velocity of the spacecraft
wiil change with position. Larger loop radii do not necessarily irnply geater accelerations
for al1 loop designs, since the increased thrust frorn a Iarger-radius current loop will be
offset by greater mass and larger loop area.
6.6.4 Single Ring
When shuttling payloads from low to high orbits, attitude control is not necessary
if only the stable pan of the thnist arc is used. Most of the single-loop reinforcement is
needed for self-induced tension rather than knding. Most of the tether force wiil be
opposed by the self-induced radial outwards force created by the current in rhe loop.
Figure 6.2 Single-loop design for magneticaily propelled spacecraft.
For a single loop design, the frontal area consists of the magneac lmp, the
satellite body and the tethers between the body and the loop. The area and mass
calculations for a single-loop design are straightforward when the loop is of unifom
thickness. The tension in the loop is found as described in Chapter 5, then the area of
reinforcement found for that tension. For the tethers, rhe same equations as for the triple
loop are used, except that the tension rem is much smaller because rhere are no bending
forces to restrain.
CL, 1' Tension = - 2
Tension Area,,, =
ak7w
Area-j, = area of loop reinforcement cross section Tension = loop tension caused by own current ~ 1 , = allowable stress on reinforcement materiai diameter = diarneter of loop cable
Since bending forces are not a significanr considerarion for the single loop, the
superconductor cable does not have to be a hollow tube, but can be a simple cable. This
rneans that the diarneter equaaon can be used for either the loop cable or the tether cables,
so long as the appropriate tension is used. For borh single and triple rings, the total mass
is:
mas,, = rnass,, + mas,, + mass,,, + mass,,, + mass,,
rnasstd = total mass of the spacecraft rnass,,t = mass of rein forcement material mStc,has = mass of the te thers mass,, = mass of superconductor used mas,, = mass of cwling, solar paneling and systems dependent on loop size massuy = mass of cenue spacecraft body, including payload
A size for the single loop will be selected based on self-accelerarion. The selected
loop will be used in the orbir uansfer program described in Section 6.7.
6.6.5 Triple Ring
The frontal area of the satellire is composed of the three loops, the tethers and the
main body area.
\ to main \
Figure 6.3 Frontal view of spacecraft with mple ring and rethers
Remembering that there are two main b o p tethers for each edge of the secondary
loops, the area of the spacecraft rhat meets atmosphenc drag is approximated by using a l l
the tethers. Tethers hidden behind the rethers shown in the Figure above are too far apan
to be considered as blocked h m drag. Lwps are symrnemcal about their quaner-
section, so loop 1 uses 4 urnes its quarter-sections area. However, loops 2 and 3 are
edge-on, so their area is approximated as 3 rimes quarter-section area, irnplyinp sorne
blocking of the miling edge of the loop, but not complete blocking. There are 1
secondas, loops. so rhe multiplier of the quaner-secnon area summanon is 6 .
Area =
Area = area of spacecraft for drag calculations
sections = number of sections used for numencal integration of area d, = diameter of section n ddcrl, drtrhcr~ = diameter of tethers 1 and 2 RI, R2 = radius of loops 1 and 2, on the order of hundreds of meten A body = fiontal area of spacecraft body
The prograrns in the thesis use the subroutine "mple-cpp" (see Appendix A) to
calculate the diameter of loop section for different thicknesses of reinforcement. As an
example, the tube diameter for various wall thicknesses is shown below for a 500 m main
loop radius with a 50 kA cument in both main and secondary loops. The methods of
Chapter 5 were used to arrive at the wall thickness values.
Tube Diameter vs Thickness
Wall t hickness (m)
Figure 6.4 Loop tube diameter for various wall thicknesses.
The mass of the triple ring will Vary with the thickness of tube reinforcement
needed. Figure 6.5 shows that the total spacecraft mass is least for the minimum
thickness, but this is not automan'cally the besr option. because minimum thickness means
mêwmum loop diameter and thus inaeased drag, as shown by Figure 6.4. See
"mple.cpp" in Appendix A for mass calculations.
Total spacecraft mass vs. wall thickness
Tube walI thickness (m)
Figure 6.5 Total mass versus wall rhickness for mple ring.
One possible way to reduce the amount of reinforcernent needed for the mple loop
is to use more tethers to resist the bending moments between the loops. For example, if
one wished to double the number of tethers, the arrangement below mipht be used.
tethers added
loop
Figure 6.6: Triple ring with 8 tether-pair arrangement.
The new arrangement with 8 tether-pain could be used in the acceleration
program with the appropnare new loads over each loop section. Consequently, a slight
reduction in rnass may be expected at the expense of complexity. However, further
calcularions wiU show that the Diple ring is so massive that even reducing its mass by an
order of magnitude would not make it cornpetitive with a single loop for either orbit
conaol or any of the other applications suggested.
6.6.6 Loop Parameter Assumptions
The minimum practical thickness of the hollow loop wall will be ser at 1 mm. 1
mm is selected because other parts of the spacecraft will need 10 be connected to the Ioops
at various points, creating stress concentration factors much above those considered for
die bending, tension and torsion. With very thin walls, tears will develop at attachment
points. AIso, the diameter of the hollow loop will become very large as the thickness
becomes very small, creating transport problems.
The area of the superconducting cable will be taken h m the superconducting
magnetic energy storage prototype cable mentioned previously, which had a radius of
about 0.013 rn for the complete 50 kA cable. For conserÿarisrn, the area of a round 0.02
m radius cable will be assumed. 5000 kg,/m3, the density of copper oxide (CuO), will be
used since virtually al1 high temperature superconductors are copper oxides. The mass of
solar panel, reflecting layen and other systems dependent on the size of the loop will be
set at 0.05 kg per meter of loop.
There are two acceleration programs, one for the single-loop mode1 and one for
the three-loop rnodel. Both programs are listed in Appendix A, the main difference
between them being the manner in which reinforcement mass is calculated (see Chapter 5).
A function caiculates the rnass of superconductor material, then adds the mass of solar
panel and reflecting (or cooling) material needed for any given loop radius. Total mass for
a loop is the reinforcement mas, the superconductor mass and the systems mass (solar
and cooIing).
G, = rnass of superconducting material r,, = radius of superconducting cable den,, = density of superconducting material (CuO, 5000 kg/m3) q- = solar and reflecting systems mass, taken as 0.05 kg/m of loop
i c me-", the solar and reflecting systern mass used, is defined per meter of loop,
so as loop size increases, the systems mass will increase proponionateiy. Frorn equarion
6.21. the tangential acceleration of the loop, considering atmospheric drag and rnagnenc
forces, is:
a = acceleration (tangential to orbit)
F, = magnetic force less drag p = density of atmosphere CD = coefficient of drag V = velocity of spacecraft 1 = current in loop RI = radius of loop 1
6.7.1 Acceleration vs. b o p radius
To find the best radius for the main loop, a number of different radii were aied
and the acceleration found at each. Table 6.7 shows the variation of spacecraft mass,
acceleration and area with increasing loop radius for a single loop spacecraft. The single
loop acceleration program was mn with drag calculated at an altirude of 300.0 km (the
sarne pengee as Zubnn and Andrews) and a current of 50 kA.
As the above table shows, single loop acceleration at a fixed current is best for the
largest radius possible. The increase in mass and drag wirh loop radius is more than
h p Radius, meters
, 600 1 200 i 800 3000
, 6000 , 9000
12000 15000 18000 2 1000 27000
Acceleration, rd?
0.0002 0-0005 0.00 10 0.0022 0,0057 0.0095 0.0 1 33 0.0 172
508.573
Table 6.2 Single loop accelerarion change with radius, including drap
Magneric force Newtons
O. 20 3 0.8 14 1.83 1 5.086
20.343 45.772 8 1.372 127,143
Mass, kg, total
1255 1510 1765 3275
855 30000 1 0.0370
183.086 249.20 1 41 1.944
Area meeting drap, rn2
66 82 98 132
13750
533 614 775
0.02 12 0.025 1 0.0330
3550 4825 6100 7375 8650 9925 12475
21 1 292 372 453
93
for by the increased magnetic force on a single loop. The next table shows
spacecraft mass, acceleration and area for various main loop radii of a triple-loop
spacecraft, again with a superconducring current of 50 kA and drag calculation for density
and velocity conditions at a 300 km circula orbit.
Radius main Acceleration, ms' Mass, kg bop, m 2
Table 6.3 Acceleration of mple ring for various radii, including drag.
From even a casual look at the tables, it is obvious that the triple b o p design wili
require massive reinforcement compared to the single loop. The large mass of the
reinforcement means that putting such a spacecraft into LE0 would be exrremely
expensive. Acceleration would be exuemely small because of the dra; of the structure and
large mas. For satellite orbit conuol, the mass of a the mple ring design is prohibitive.
6.8 Previous Researc hers' Orbit Calcula tions
Magnetic propulsion rnijht be used to escape frorn low eanh orbit (LEO), as well
as to change from one orbit to another. Zubnn and Andrews (1991) calculate
performance for a 3 1 km radius loop with a 50 kA cun-ent and a 14 ton payload. 5 tonnes
(5000 kg) are assurned for the superconducting loop ("magsail") and 1 tonne for power
and sysrems. Since their sample magsail consists of a single superconducting loop with no
attitude control, ir is only switched on for 8 = 20' co 0 = 80°, where 8 is the polar angle
frorn the magnetic pole. This range of angles represents the stable porrion of the thrust
arc.
stable part thrust arc 1
Figure 6.7 Satellite orbit around eanh showing stable thrust arc.
Zubnn and Andrews consider a transfer from a circufar 300 km orbit to a
geosynchronous polar elliptical orbit with a apogee (furthest point) of 71090 km and
peripe (closest point) of 636 km. The self-acceleration of Zubrin's sarnple ma@ is
0.0206 m/s2, which by cornputer integrarion (not shown) requires 160 orbits in 34 days to
meet the required AV (change in veiocity) for the uansfer. Zubrin and Andrews give the
foliowing flight times for their bop:
1 Orbit# 1 Time 1 Perigee 1 Apogee ( Velocity 1
Table 6.4: Orbit rransfer flight rimes, Zubrin and Andrews (1991).
O 60 120 -
The same orbital uansfer will be used in the single loop orbit program of this
thesis and the results compared to Zu bnn and Andrew 's table above.
6.9 Orbit Transfer Prograrns
6.9.1 Single Loop
O days 4.9
The orbit transfer program uses the acceleration values found in the single loop
acceleraaon program (see Appendix B) and basic orbital mechanics to find the rime and
number of orbits necessary to achieve a @en nansfer. Thrust is only used for the
stable p a n of the orbir, which is a quaner-section or less of the total orbit. Pengee
(closest point) is kept small and apogee (funhest point) steadily increases. The perigee is
kept close because Fe, rhe tangential magnetic thrust, falls off as (radius14, as noted in the
magnetic force section. The impulse per orbit is maximized by keeping perige small and
290 km 383
15 1 451
3 10 km 6076
7.7 km/s ,
8-75 21723 9-7 .
using as much of the dinisr arc as possible per orbit before the radial component of the
magnetic field becomes too large.
A factor of 0.9 is included because the magneac poles are rotatïng with the earth,
while the satellite orbit is not. The average loss in force was found as 0.9 by Zubrin and
Andrews (1991). The delta4 required for any panicular orbit nansfer is calculated using
Keplerian equations and assurning a Hohmann transfer (after Larson and Wem, 1992).
The urne under acceleraaon for a given delta-V is found fiorn elementary physics.
avg = r;\ + r5
3 -
AV = velocity needed for aansfer Gm. = gravitarional constant and mass of earth, 3.986~10'~ m3fZ r ~ , r ~ = radius of orbits A and B avg = average of orbit radius A and B V = final velocity V, = initial velocity a = acceleration t = time
The AV equation above is for a transfer from a cucular orbit of radius A to an
ellipacal orbit with peripe equal to radius A and apogee equal to radius B. The magsail
thrust arc is not impulsive, but neither is i t consrant thrusr, since the thrust portion is
always less than a quaner orbf for a single loop and less than a half-orbit for a triple loop.
The impulsive Hohmann m s f e r equauon for AV will be used as an approximation.
However, AV in the orbit program will assume gravity losses of 5% from the ideal AV for
the maneuver because of the semi-constant thrust aansfer instead of the impulsive, single
orbit Hohrnann rransfer. 5% is used because Zubrin and Andrews ( 199 1) cdculated a 5%
loss for their magnetic propulsion mnsfer.
Figure 6.8: Transfer from circula orbit radius A to ellipncal orbit perigee A. apogee B.
Since both air density and magnetic field change with altitude, it was necessary to
make the drag and mapneric force subroutines funcrions of altitude. For an elliptical
orbit, altitude above the earth is changing constantly during the thrust arc. To obtain an
accurate delta-V expected from a given thrust arc, the thmst arc was numerically
integrated. The velocity and total energy of the orbit were updated after each integration
sep . At the end of a t h s t arc, the orbit energy was used to update the shape and sire of
the next orbit Altitude for any angle from pengee for a given elliprical orbit was found by
using a combinanon of Newton's second law and his law of ,gravitation, which is the
position equation:
r = radius from earth center, see diagram below. G = gravitational constant M, = mass of earth r = time, used for second derivative
/< 2(a), a=semi-major ods 7
perige I< f
r-a, apogee distance
Figure 6.9: Definition of apogee, pense, m e anomaly and semi-major a is .
Notice that the eanh is one focus of the ellipse, while the other focus is twice the
semi-major axis away. For such an elliptical orbit, standard geomemc equarions were
used to lïnd position for any angle from perigee. Altitude was found by subnacting the
radius of the eanh from the position.
l + ecosv
a = semi-major mis of orbii, half distance between two foci of the ellipsz e = eccenaicity of orbit, measure of circularity v = m e anomaly, angle from perigee, radians, as in Figure 6.9 r, = radius at perigee, closest point to eanh in orbit
The acceleration time for each seDrnent of the thmst arc is found by using the
change in a quantity called the mean anomaly (M in equation 6.40) for a given change in
n e anornaly (angle h m perigee, see Figure 6.9) over the arc segment. The orbit conuol
program assumed an orbir with the perigee mid-way through the thrust arc and subdivided
the thrust arc into se,pents of equal change in m e anomaly. For each segment of thmst
arc, the time of flight was found by use of other canonical Keplerian orbit elemenrs. See
Appendk B for an explanaaon of Keplerian elements.
e + cosv E = cos-'
1 + ecosv -
.=('y )' t-t, = time of flight M = mean anomaly in radians E = eccentric anomal y e = eccentricity
v = m e anornaly q = mean motion G M, = gravitational constant and mass of eanh.
The magnetic force and drag force for a piven loop radius were found using the
altitude as found above. The acceleration urne for a segment is the time-of-flight. The
delta-V for a segment is the acceleration (total force minus drag divided by mass)
multiplied by the acceleration time. This delta-V was used to update the energy of the
orbit after the calculation for each segment-
v2 GM, &=- +-
2 r
E = energy of orbir per unit mass of spacecraft V = velocity, found by adding delta-V to previous velocity r = position from eanh center
At the end of a thrust arc, the energy of the orbit was used to find a new serni-
major axis for the next thrust arc. A new semi-major a i s automatically changed the
eccenmcity, mean anomaly, e c c e n ~ c anomaly and rnean motion, but the perigee was
assumed to be fixed. Fiiing the perigee is equivalent to assuming thar the thrust arc is an
impulsive delta-V inpur The size of the serni-major axis is a direct measure of the energy
of the orbit, so the new serni-major axis is found from Larson and Wenz (1992) as:
-GM, a=- ( 6.43)
2E
The orbit program was mn twice wirh the same size of loop as Zubnn and
Andrews (1991), first by neglecting drag, as they did. then including drag. The graphs
below show that, without drag, the orbits and tirne to achieve a transfer from a 300 km
circular orbit to an elliptical orbit with apogee 71090 km are almost identical to the table
of Z u b ~ and Andrews given previously: 155 orbits for the thesis pro,- vs. 16 1 for
Zubrin and Andrews. Note that Zubrin assumes a 11 ton payload and 6 tons for the rest
of the spacecraft. The mass esrimares in the rhesis show that a loop of the sarne size as
Zubnn would allow a 7 ton payload, and 13 tons for the rest of the spacecraft. Al1 the
same, the total m a s is made to be 20 cons, so that the results are comparable.
31.6 km Single Loop, No Drag
80000 1 -
- V ) Q i O b - - N ~ * , . Z ~ Z k G m LD
T T
Orbit number
Figure 6.10 Single loop change in apogee with orbit number, no drag.
As apogee increases. the speed of the spacecraft during the thnisting arc will also
increase. As a result, the delta-V possible during each thrusting arc will decrease even if
perigee is constant. This effect is shown in Figure 6.1 1.
31 -6 km Single Loop, No Drag
Figure 6.11 'JO drag single Ioop change in delta V per orbit.
Under identical conditions, the same prog= was run with drag. Not only is there
drag during the thnisting arc, reducing the delta-V available. but there is also drag for the
rest of the orbit. which is significanr unril the apogee is greater than about 600 km.
Figures 6.11 and 6.12 (below) l show that the acceleration of the single ring is snong
enough that drag does not significantly affect its performance for the @en uansfer.
Apogee vs. Orbit number
v O O O O O O CD CU v
Orbit number
Figure 6.12 Single loop change in apope with tirne, with drag.
The total urne for the orbit rransfer is 36 days with drag, 34.9 days without drag,
in excellent agreement with Zubnn and Andrew's estimate of 34.3 days without drag.
6.9.2 Triple Loop
A pro,mrn sirnilar to the single loop program with drag was run for the aiple loop.
Narurally, the area and mass caiculations were different, but the thrust arc was also
doubled, since the triple loop rnay use the unstable part of the thrust arc as well as the
stable pan. Since a 50 kA cumenr and a 31 km radius main loop were not sufficient to
counteract drag (see Table 6.5), the current and radius were changed in an effort to see if
any combination of current and loop size could make the h-iple ring work. However, ail
orbits decay from the start perigee of 300 km and go below 150 km in less than 2 days, no
marrer what current or radius is used.
Table 6.5 Triple ring, 3 1 km radius. mass 9,746,8 18 kg and current 50 kA.
1 orbit 1 days 1 delta V (m/s) 1 apogee, km 1
Table 6.6: Triple ring, radius 31 km, mass 97,147,734 kg and current 500 kA.
396 276 236 309 189 175
1 3 7 10 15 20
[ orbir 1 days 1 de1 V(rn/s) 1 apogee(krn) 1
1 oibit 1 da.yi6 1 del v 1 apo~;;~ 1 -1.19
0.19 -2.97 275
Table 6.7: Triple ring, radius 1 km, mass 565,734 kg, and current 500 kA.
0.06 0.19 0.44 0.63 0.94 1.25
1 3 7
-1-15 -2.93 -2.70 -2.69 -0.82 -0.81
0.06 0.19 0.44
-1.15 -2.94 -2.71
296 276 236
Table 6.8: Triple ring, radius I km, mass 59348 kg and current 50 kA.
Tables 6.6 to 6.8 show that larger currents do not yield better performance
because the mass and area of reinforcement increase to counterdct the larger torque
between loops thar larger currents bring. To see if the mass could be reduced. a sxnaller
radius loop (1 km) was used for the lasr two rables, but the result was sri11 a decay in orbit
below 150 km in less than two days.
Chapter 7 : Future Directions
7.1 Spaceflight Challenges
Travehg through space cosmic distances wiil take a long urne using rockets. Mars
would be a two year round u5p ushg rockets, and the size of the ship would require many
shuttle flights to assemble it in eanh orbit (Kumar, 1994). Fuel mass required on a rocket-
driven spacecraft increases exponentially with distance of the mission and mass of the payload.
For the Apollo rnoon missions, 2700 tonnes of mass on the ground were needed to lifi 40
tonnes of mass to the moon . For a 1 kg probe to be accelerated to 100 km/s, one would need
a 70 miIlion tonne rocker. This does not include any fuel for deceleration at destination, and
cerrainly not any return nip. Worse yet, 100 km/s is slow when one considen that a one-way
aip to the nearest star with planers (10 light years) would take 30 000 years (Mauldin. 1992).
Clearly, alternatives to rockets such as magneuc loops musr be investigared in order to uavel
cosmic dis tances.
7.2 Interplanetary Travel
Zubrin and Andrews (1991) describe the use of superconducting current loops, which
they cal1 magsails, for interplanetary travel. The mechanism for propulsion is deflection of
interplanetary plasma winds. Accelerations on the order of 0.01 r n ~ ' ~ are calculared for a
single loop of radius 30 km. The great advantages of a magsail are that multiple mps can be
made per mission and that trips are not subject to a launch window, as are Hohmann transfers.
Tangenaal force (or "ift") cm be generated by tuming the loop at an angle to the sola. wind,
hence the narne "magneuc sail" or "magsail".
7.21 Solar Wind
A flux of a few million charp i particles per cubic meter flows by the eanh from the
Sun. The eanh's magnetosphere deflecrs this plasma wind, protecnng Life on eanh from the
various ill effects of high-energy particles. The solar wind is visible when charged panicles
manage to penetrate die magnetosphere in the vicinity of the magneric poles, creating the
Nonhem Lights (Aurora Borealis) and Southern Lights (Aurora Ausnalis). Solar wind
velocity is 400-600 km/s, which sets an upper lirnit to the velocity a magsail can attain by
drag, or repulsion of the particle wind.
Since the solar wind consists of charged panicles. the number of parricles affected by
the magnetic field of the Ioop is much geater than those that pass through the loop area. In
fact, the area of effective total reflection of particles is calculated by Zubrin and Andrews as
75 urnes the area of the loop for a 50 kA current. The larger the cument for a given loop size,
the larger the area of effective total reflection.
7.2.2 Magsail Orbits
A mp from eanh-orbit distance from the Sun to the orbit of Mars is calculared by
Zubrin and Andrews (1991) to take 273 days for a 5-tonne magsail with 3 tonnes of
insulation, shielding and power equipment and 37.5 tonnes of payload. This is only sliphùy
longer than the Hohmann rransfer tirne (see Figure 7. l) , but i t should be remembered that the
estimates of this thesis suggesr that the payload mass estimates of Zubrin and Andrews may be
optirnistic, given the b o p reinforcement required.
1
Hohman
1 More thrust! 5' circularize needed \ /Thun+ I
\ Second \
Figure 7.1 Impulsive Hohmann transfer, typically used by rockers.
Once at the orbit radius of Mars, the magsail could circularize irs orbit and, while
remaining under power, stay in the orbit of Mars while traveling at a different velocity from
Mars. The magsail force is canceling some of the sun's attractive force. so the magsail can be
aaveling about 4.6 W s slower than Mars while at the same distance fkom rhe Sun. This
rneans that a magsail can launch at any time: there is no launch window as with an impulsive
Hohmann transfer because the magsail can wait for the destination planet to catch up with it
once at the same orbit distance from the Sun.
Figure 7.2 Direcaon of magsail thrust from solar wind.
In the reference frame of a nonrotating sun, the rnagsail orbit transfer would look
similar to a Hohmann transfer, except that at the destination orbit, the tangenàal speed would
be the sarne as at the initial orbit. If the initial orbit were at earth distance from the Sun, and
one warched the mapail from the eanh, it would appear to move radial outwards from the
sun to the new orbit.
The advantage of the rnagsail becomes even grearer after the cargo is delivered: now
the spacecraft can retum for another m p . The magneac field of the destination planet could
be used for an initial impulse out of orbit. If the next destination is in the direction of the solar
wind, then the rnagsail could proceed as previously. However, if the mission is to retum to
eanh for another load, then the rnagsail could either use the gravity of the Sun alone to return,
or do some sailing to speed up the process.
7.2.3 Space Sailing
All of the preceding discussion has assumed that the magsail does not change its
angular momentum about the Sun. No Lit is pnerated; the magsail simply reflects the charged
particles of the solar wind, like a sailboat traveling snaight downwind. To generate lift, or
"sail", one would need to deflect the solar wind at an angle.
I > Solar wind Direction
Loop 1 Loop 2 dragging
A Sailing
* Net drag force
t Lift 9
Drag force
Solar wind d Figure 7 -3 Magsails draggin; and sailing on the solar wind.
If lifi were to be used, the single loop proposed by Zubrin and Andrews would need
some form of attitude control. They suggest chan& the spacecraft centre of mass by
shifting the payload on a set of tethers, thus offsetring the centre of pressure from the centre
of mass and creating a [orque. Using lift allows trip times to be reduced and flexibiliry in
missions to be increased. For example, using lift means that it is not necessary for ellipacal
orbits to be used ro meet up wirh a cenain planet. The Mars mission ames in Zubrin and
Andrew's 1991 paper did not consider using planetary magnetic fields, but it is obvious that
the mvel times would be reduced using planetary magnetic fields as well as the solar wind.
Flexibility, large payload capability and multi-rrip missions make magnetic propulsion
extremely amactive for interplanetas, missions.
7.3 Interstellar Travel
The solar wind is about 500 km/s, so the maximum theoretical speed a magsail could
ever hope to attain from the solar wind using drag alone is 500 kmfs. Of course, as the
spacecraft aavels funher from the Sun, the density of the solar wind decreases, so that the
efficiency of solar wind propulsion must also decrease. It would be a fantastic feat to obtain
300 km/s from the solar wind. 1s 300 km/s fast or slow? In the case of interstellar travel, it is
slow. Desirable, "close" interstellar destinations are on the order of 30 light-years away.
Light navels at 300 000 k d s , so that a 300 km/s solar wind-powered spacecraft would rake
30, 000 years to reach the destination. Acceleration and deceleration times would add a few
thousand years, but g v e n average hurnan lifespans of =72 years. the point is moot.
7.4 Self-Launchino, Ca pability
Zubrin and Andrews (1991) claim thac 80 times the best cunent densities available
today would be needed to obtain self-launching capability from eanh surface. Ackerman
(1989) also suggests magnetic self-levitation. With extremely large currents available,
Ackerman assumes a Ioop size of less than 100 meters and a total mass on the order of 1000
kg. The estimates in b i s thesis show that very large cuments will not yield self-levitarion
because the reinforcemeni mass grows faster with current rhan the magnetic force.
7.4.1 Triple Ring
For hovering flight, a single ring with no amtude conuol is not adequate. To find if
self-levitanon is possible to achieve using a rriple ring arrangement, an iteraave procedure was
used in the program "U-cpp" (see Appendix A). A stamng radius was selecred, then the
mass that could be lifted at a starting current calculated and compared to the mass of the
spacecraft, including reinforcement. The current was then incremenred, the mass of the
spacecraft and the mass thar could be Iifted recakulated, and the difference between the mass
that could be lifted and the mass of the structure recorded. The current at which there was
the smallest difference berween the mass of the spacecraft and the mass that could be lified
was selected as the best current for that radius. Then the radius was increased and the
procedure repeated. The mass that could be lifted was found as follows.
Area = area of main loop curent = current in main loop grad-B = gradient in the magnetic fieid mass = mass that can be levitated gravity = 9.7 1 m/s2 at earth surface
The gradient in the magnetic field is needed in the radial direction, not the tangential,
as with the orbit progams. The denvarive of rhe magnetic field as given in Chapter 3 is:
B- = magnetic field at eanh surface r- = radius of earth r = distance from centre of earth 9 = angle h m magnetic nonh r, q = radial and tangentid unit vectors.
Notice that anywhere other than a magnetic pole, where 8 = O or x, there wiU be a
tangential force as well as a radial one. This means that the loop would have to tilt to fly
upwards at any point other than directly over a magnetic pole. See program
"mag-for-rad.cppW in Appendix A, where the condirions are assumed to be at eanh surface, at
a magnetic pole, for the details of the radial magnetic force caiculation. The results for the
niple ring are rabulated in Table 7.1 for currents from 4500 A to 1 000 000 A.
- - - - -
Table 7.1 Mass that can be lifted by magnetic force and spacecraft mass for mple ring.
radius1 (m) mass able to lift, total (kg)
I I
best mass (kg) current (A)
It h a . been shown in Chapter 6 that mple ring acceleration decreases when larger loop
radü are used. Smaller loop radii were used here to find if reinforcement mass at smaller sizes
would grow less quickly than magnetic lift. With saess as approxirnated in the "aiple.cpp"
routine (Appendix A), the mass of the aiple ring grows much faster than the lift force as
current is increased. The program selects the best current by finding the smallest difference
between the total mass and the total lift. In al1 cases, it is the smallest or starring current
which is selected.
The problem is that reinforcement mass increases more quickly with current than lifi,
so thar the triple ring, as proposed, will always be too massive to fly, regardless of the sire of
superconducting currents available. It would then appear that, using alurninum rein forcement
in the configuration of this thesis, the self-levitûaon postulated by Ackerman (1989) would
nor be possible. Some other configuration, or a reinforcement material that is much lighter
and stronger than aluminum, would have to be used.
7.4.2 Single Loop
Since the triple bop seems too massive for levitation, let us assume a single loop with
an attitude stabilizauon system such as a controllable-speed gyoscopic shell. Before
considering how such an attitude control system rnay work, consider the practicality of
achieving levitation. As in the mple-loop lifi promgam, the lifàng force and the total mass are
compared for many different loop radii and current sizes. The attitude control system is
included by assurning gyro mass equal a circular shell of aluminum one mm thick and of the
same area as the main loop (optirnistic, but sriil excessively massive, as shall be seen). Figure
7.4 shows the proposed design of a single loop gyro-stabilized superconducting magneric
I Spin axis
Side View Loop Plan View
Figure 7.1: Single superconducting loop with gyroscopic shell.
The single-loop lifting program (Appendix A, "1l.cpp") is the same as the mple-loop
program except that the mass caiculations are for the single, not the riple loop and the
s&g current is larger. As with the triple bop , the best current is selected as the current
where the difference between the rota1 mass and rhe mass that can be lifted is srnailest. Also
as with the triple loop, the best curent is the starting current since single loop reinforcement
rnass grows at a faster rate than magnetic force for eanh-surface conditions and alurninum
reinforcement,
To observe how the single loop mass changes with increased current ar a radius fixed
at 100 meters, see Figure 7.5.
Main loop radius (ml
50 100
Single Loop Mass & Lift vs. Current
Current (Ampetes)
Figure 7.5 Mass and mass able to lifr vs. current for single 100 m radius loop.
Total mass (kg)
2020 3039
Mass abIe to lift, Total (kg)
. 0.01149 0.04594
150 200 250 300 350
Curren t (A)
5OoOOO 500000
O. 10337 0- 18377
4059 5079 6099 7118 8138
500000 500000
400 450 500
Table 7.2 Single loop self-levitanon currenr selected for smallest lift-mass difference.
0,287 14 0.41 349 0.56280
500000 500000 500000 500000 500000 500000
9158 10178 11197
0.73508 0.93034 1.14857
The reinforcement mass increases aemendously with current, but the mass that it is
possible to lift increases so litde that the line appears to lie dong the X -axis in Figure 7.5. As
with the mpie loop, the mass of reinforcement gows much more quickly than the mass that
can be lifted With aluminum reinforcernent in the configuration selected, the single loop will
not self-Ievitate for any current value.
7.4.3 Radical Assumptions
Even if technoiogy evolves to allow very large superconducting currents, a
fundamental problem appears to be the mas of reinforcement needed to contain the forces
that those same currents induce. Aluminum is simply too massive for the craft to fly in the
configuration suggested. However, there are prospects for much stronger materials to be
used, as the data taken from Friedrich et al (1984) shows in Table 7.6.
- - - - - - 1 Yield or Fracture Suess, 1 density, g/cm3
Al Alloy, HS
Maraging Steel. Ni-Co- 1 3500 (fracture) 1 7.8
Steel, UHS
Ti Alloy, HS
Maraging Steel, wires
M Pa 550 (yield) 2.7
3000 (yield)
1500 (yield)
7.8
4.5
4000 (fracture) 7.9
7000 (fracture) 2.3
B - Fiber
S - Glass
C - Fiber 1 4378 (fracture)
4585 (fracture)
3500 (fracture)
1.7
3.6
Table 7.3 Yield and fracture stresses of high strengrh (HS) matenals.
KevIar 49
A1203 whisker
Some explanarion of the high srrength (HS) and Ultra-High Strength (UHS) yield and
fracture snengths listed above is in order. Yield stress is the stress at which a sample of
material under stress begins to permanently deform; it happens gradually. Fracture stress à
the stress at which a sample of the matenal would break. There is no m u a l elongarion of
such a sample in tension; the material is fine until the fracture stress, then it suddenly breaks
completely. This fracturing is a feature of most high strength materiais, and the stronger the
version of a panicular alloy or fiber, the more brittle it becomes. The last six materials on
the table are whiskers or fibers, which means that they can not be used directly for structure
3600 (fracture)
43 000 (fracture)
1.45
3.96
rnaterials, but are a volume fraction in composite rnaterials. T h e composite will typically have
less than half the swngth to density ratio of the whisker or fiber, often much less than half. It
is a great challenge to rnake pracacal structural materials with the srrengths in the table above.
The single-ring Iifnng program was modified to run with an M20i composite material
of fracture swngth 9000 M Pa and density 10 glcm3. The reduced snength and greater
density allow for the composite reinforcement of the whiskers. With a safety factor of 3, the
allowable sness was set at 3000 M Pa. See Appendix A, program "l2.cppy' for the pro,oram.
Table 7.4 Ultra-high strength reinforcement single-loop.
current (A)
500000 500000 500000
radius, main loop (m)
50 1 O0 150
- - - - - - - - - -
Total mass (kg). 1026 1053 1079
- - - -
Total mass able to lift, (kg)
0.0 1 149 0.04594 O. 1 0337
Table 7.5 Single loop of radius 100 m with UHS reinforcement
Tables 7.4 and 7.5 show clearly that even with ulua-high strength materials for
reinforcement. the single loop is still too massive to fiy. Increasing either current or loop
radius results in a greater m a s increase than the gain in mass that can be lifted. Fanrasticdly
strong materials or a design that somehow contains the self-induced stress would be needed
before magneric levitation from eanh surface could be considered as practical, even after
superconductors evolve to cany mega-arnp currents.
- -
mass ableto lift (kg) 0,0459
---- - - - - -
current , (A) 500000
- - - -- -
total mass (kg) 1053
Chapter 8 : Problems to Address
8.1 Orbit Control Barriers
Although superconductors exist that carry large enough currents to make magnetic
propulsion for orbit control appear possible with a single Ioop, there rernain numerous
practicd bamers that must be overcome. These barriers can be divided into absolute
barriers and lesser barriers. Absolute barrien are fundamental problems that must be
solved More development of magnetic propulsion can proceed. These include
superconductor limits, matenal fabrication and power system problems. Lesser bmien
are problems that are chailenging, but unlike the absolute barriers, they are almost certain
to be solved. Lesser barriers include nansponation and deplopent of the loops,
reinforcement of the tethers and conml of vibrations.
8.3 Absolute Barriersorbit Control
By far the greatest impediments to practicai use of magneric propulsion are the
limitations imposed by present-day superconductor technology. Until the crincal current
densiries, aïtical temperatures and crirical magnetic fields of superconductors improve,
the superconducting lwps will require insulation and cooling systems to function in the
near-eanh space environmenr. The mass of such cooling systems would almost cenainly
make rocket propulsion more practical for nearly al1 missions less than decades long.
Closely linked with superconductor limits is the problem of fabncating suitable materials
to support high-current superconducrivity in space w hile remaining flexible and resistant to
damage. Fmally, a pracacal means of conaolling the currents in the superconducting
loops will require a concentrated development effon.
8.2.1 Superconductor Materials
A superconductor is a perfect conductor. There is no intemal resistance to current
flow at al, which means thai once started, a current will persisr indefinitely if there are no
funher extemal interactions with it. Superconductors must be used because normal
conductors simply take too much power to make magnetic propulsion practical. For
example, most metailic conductors such as copper have conductivity in the range of 10'
dl-'. so the power for a typical current loop can be found as follows:
1 = cument, 50 kA 1 = length of wire, 3000 m o = conductivity, 1 o7 mR" A = area of wire, 3 mm2 = 3 x 1 0 ~ m'
A solar panel would have to use 1008 of the solar radiation passing though a 182
kmZ area ro supply this power (assuming 1376 ~ / m ' solar energy at eanh orbit). Such a
large power requirement makes magnetic propulsion feasible only with superconductors.
Prior to the 1980's, superconducrivity only occurred at remperatures close to
absolute zero and could only support very srnaIl currents in very weak magnetic fields. In
the 198OYs, a new class of ceramic superconductors were developed which allowed higher
tempemures, larger current densities and smnger
returning to the normal conducting stare. The praph
magnetic field suengths before
below shows the three Iunits on
superconducuvity: current density, magnetic field srrengrh and temperature. If any one of
these is too high, the superconducring effect is desrroyed and the matenal "goes normal".
Temperature 1
L/ Magnetic Field (T)
Figure 8.1 Temperature, magnetic field and current density Limits (Wilson, 1983).
Starring from absolute zero temperature with zero magnetic field and no currenr
density, one can desnoy the superconducting effect either by heating, increasing the
magnetic field, or increasing the currenr beyond the surface defined on the a i s above.
The required current densities and resistance ro extemal magnetic fields necessq for
magnetic propulsion have been shown in laboratory conditions on small samples.
Unforrunately, the superconducring propemes of a material change as one mes to scale up
the size of the wire. This is one of the primary reasons why superconducting power lines.
superconducting elecmc motors, or magnetic propulsion systems have not k e n produced
in quantity yet. To make the concept of magnetic propulsion cornrnercially viable, a
breakthrough in the size of superconducang current that it is possible to cary in a wire is
necessary.
So far, superconducting wins are composites of a superconductor and normal
conducting material. The superconducting state is stabilized by using rnany
superconducting filaments in a low resistivity conducting matrix. Srabilized means the
superconductor is protected against quenching. Quenching is a transition to nomal
conducting state, caused by some disturbance, either magnetic, themal or mechanical.
Even if the wire is operaang well below the critical temperature, magnetic field and
current density, a temporary disturbance will cause a flux jump, which means a temporary
disruption of the superconducring stare. Once resisrance exists, there is funher heating,
which funher destroys the superconducting effect, leading to more resistance and heating
until the superconducting current is desrroyed or iquenched".
One can stabilize the superconducror by fine subdivision of the superconducror
volume so that the heat capacity of the marenal limits the heating or by damping the flux
change rate by use of a highly conducting normal conductor (such as Cu or Al) to reduce
heat generation and allow heat removal. A problzm is that the highly conducting matenal
between the superconducting filaments can inrroduce new instabilines. Also, the induced
currents in the support mamix change with the wire length, which means that long lengths
of wire will have differenr performance than shon lengths. However, Barone et. al.
(1991) repon that over a certain critical length. superconducting properties are nearly
constant as the induced currents behave as if the wire is infinite.
Ceramic High-Temperature Super Conductors (HTSC) are greatly supenor to
previous Nb-Ti alloys in terms of critical current density, temperature and magnetic field.
But most HTSC compounds have failed in application because of low critical curent
density in buik fom, poor workability into mulàfüamentary wires and hiph cost. Only
NbTi and &Sn alloys, which usually operate at about 4.2 K, have k e n used in m e
large-scale indusmal applicaaons. Moreover, HTSC crirical parameters are not exactly
defined, especially in buk foms like wires. Criacal current and critical temperature tend
ro depend on the application rather chan being an absolute propeny of the material.
Initially, one rnay believe that a superconducting loop in space must have litde
problem with temperarure because background radiation remperarure in space is a mere
2.7 K. However, this temperature is for deep space, away from sunshine or planetaq
atmospheres. Since the thesis presumes that the magnetic forces are used ro counter
amospheric drag, the satellite is in contact with the atmosphere of the eanh. Al1 is not
lost, though, because the atmosphere is so thin at these altitudes that air temperature has
no appreciable effect o n a satellite. In fact, the temperature of the satellite is controlled by
the radiant heat of the sun and the reflectance of the satellite surface (King, 1987). If the
surface of the loop was a blackbody (a perfect absorber), it would hear up fast and
superconducring would be unanainable. On the other hand, a perfect reflector would
enable the loop to absorb none of the soiar radiation. Perf'ect blackbodies and perfect
reflecton do not exist, but it is obvious that the loop will need a coating or covering of the
closest thing to a perfect reflector that can be developed. At the same time, this coating
should allow heat to escape from the ring so that the ring may cool down after heatin; up
due ro resistance heating, should the superconducting effect be lost after too large a
current was inadvertendy suppiied for the conditions. The requirements for good
radiarion of heat and good rejection of incoming radiarion are connadictory, so design of
die insulation will be an interesting challenge in itself.
8.2.2 Power Control
Once power is produced by solar cells or other sources, it rnust be transferred to
current in the superconducting loops as required. One can not simply connect positive and
negative leads to the superconducting bop, since those leads would break the
superconducting path, as well as potentially creating heat problems. Instead, it is
necessary to use a device such as a flux pump.
A flux pump is a device where power can be introduced either mechanically or
electrically with ,mal1 heat input to a Iow temperature region. Once in the low
temperature region of the superconductor, the power is convened to a low-volt, high-
current supply. There are two general types of flux pump: the homopolar generator and
the mnsformer- rectifier system (Barone et al, 1991).
Plate (lead) n
net
Plan view
Q direction of rotation
O O Axis of rotation
magnet C 1 1 Plate
axis of rotation
Edge view
oil
-3
Figure 8.2 Simple homopolar flux pump showing path of flux spot
The homopolar generator consists in its simplest form of a magnet which can
rotate over a superconducting plate. The magneuc field generated is greater than the
critical field of the plate material, but is less than the criacal field of the superconducting
wire. The flux spot created by the magner quenches superconductivity locally on the
plate, but there is always a superconducting path through the plate. Rotation moves the
flux spot over the plate to the coil circuit. Since the coil circuit has a high critical field, the
wire is not quenched and the flux is left trapped in the circuit. Work is done moving the
magnet away from the coil circuit. There is also an elecrronic version of the homopolar
generator. In this case, a three phase current supplied to coils can make a rnoving flux
spot. Diodes ensure that the flux over the plate is of only one polariry.
plate flux spot movement
/[ 1 -
Figure 8.3 Elecrronic homopolar flux pump.
The voltage supplied by a flux pump is equal to the area of the flux spot swept by
Super -conductor coi1
the magner multiplied by the frequency of the sweep (Barone et al., 1991).
V = voltage f = frequency of sweep in Hz 9 = flux area swept by magnet
A problem wirh obtaining high voltages from a homopolar flux pump is thar
frequency is limited by materials to about 50 Hz. The rime to energize a coi1 is given as:
L/ T = - v ( 8.4)
T = tirne in seconds
L = inductance of the coil 1 = cunent in the coil V = voltage supplied
LI is in Volr-s and, for a given frequency, one may find the area of the flux spot
needed to energize a coil in a given time. For 1OOO Vs, one needs 17 minutes with 1 V if
the maximum flux is 1T and the area swept is 200 cm2. For larger power outputs, the
area and mass of the homopolar generator become undesirably high. The transformer-
rectifier system, an alternative to the homopolar generaror, is pictured below.
High current circuit
C 1 rectifier
1 transformer with super- conductor [
1 windings i
Superconductor coi1
rectifier I l
Figure 8.4 Transformer-rectifier type of flux pump.
The transformer core must be of material with low eddy currenr and hysteresis
losses, which are problems with superconductors which carry altemating current (AC).
The rectifiers are devices known as cryotrons, which means thar they cari be switched
from the superconducring to the normal stare by applying a magenc field. The resistance
in the normal state limits the reverse current to a mail fiaction of the forward current in
the superconducting state, and the current is thus rectifiai. AC is introduced into the
primary, low current circuit, which induces AC of opposite phase in the highcurrent
windings. M e n the voltage is, for example, negative in one secondary circuit, that
circuit's cryotron rectifier is made resisuve. The other cryoaon is made superconducting
and forward current flows. When the cunent alternates, the rectifiers are switched, and
current flows in only one direction in the high-currenr loop.
Using this arrangement. large power outputs are possible wirhour need for large
areas to pump flux over. However, using this type of flux pump will depend o n cryotron
development. Both zero field and zero voltage over the cryotron rectifier are needed to
make it superconducting, and some form of liquid coolin; (Nzcii or He(!)) is usually used.
This arrangement would be difficult to make robust and reliable enough for a space
applicaaon.
8.3 Lesser Barriers-Or bit Control
8.3.1 Transportation and Deployment
At the present rime, there is no launch capability for loops with diameten on the
order of 1 km. This irnplies that the superconducting loops must be assembled in orbit.
6 Flexible superconducting tape is now available. The tape can cary 1.3 x10 A per cm2 at
liquid nitrogen temperatures and is flexible enough to be bent around a pencil with only a
9% current loss. Los Alamos National Labs are now making 5 cm x 1 m saips that hold
their superconducting ability in 2 Telsa magnetic fields. Evidently, these srrips are
representative of longer lengths because the strips are not single crystals. Development of
thallium based superconductors is continuing. Thallium would allow superconductivity at
125 K, which is well above the temperature of space. so there is great hope that using
simple reflective insularion would allow superconductivity with only space as a cryocooler
(Scott, 1995). However, according to the Los Almos scientists, the technology will need
several years of development before evcn terresmal commercial applications are practical.
8.3.2 Geomagnetic Inductance
Inductance is a measure of how much magnetic flux is changing with tirne within a
conductor. Magnetic flux can change because the conductor moves through the magnetic
field. or because the magnetic field changes over tirne. Geomagnetic inductance is
inductance caused by movement through or change in the geomagnetic field. Since the
superconducting loops are continuously traveling through the geomagnetic field, there is a
voltage induced in thern.
E = electromotive force, volts 9 = magnetic flux density, Weber/mL v = velocity of loop B = geomagnetic field strength ds = element of loop curve length da = element of surface area v = velocity
The geomagnetic field is changing in a quasi-predictable fashion over thousands of
years and ais0 in an unpredictable fashion on timescales as short as a minute (Tascione,
1988). Both these types of changes would affect the aB/& terrn above, with the short
term changes, caused by solar activity, dominating. The arnbient magnetic field would
need to be continuously monitored in application, so that Ioop currenrs could be adjusted
as the geomagnetic field strength changes. The current conuoller wouId also have to
consider the induced emf caused by satellite movement through the geomagnetic field.
8.3.3 Power Required:
When current in a b o p is made to change, the flux through the loop will also
change. A self-induced, or back-emf, will be presenr when the current is changing
(Wangsness, 1986). The flux change creates an emf that opposes the current change. The
change in flux though the loop for a aven change in cument is related to the propemes of
the loop via the self-inductance, L, which follows directly from Faraday's Law.
E d r = self-induced electrornotive force (emf) L = self-inductance for Ioop 1 = current
A combination of the Biot-Savan Law and Maxwell's equations give self-
inductance as (Wangsness, 1986):
p ds, ads, 8 4a ape Rij
= penneability of free space Ra = distance from opposing elemenrs of current
ds = elments of cuve length L = self-inductance
For a circle, ds=RdB and Rij=2r, R=radius.
L =
L =
The self-inductance of each ring of current can be calculated as above, or by
c o n s i d e ~ g the definition of inductance with t h e assumption rhat the magneuc field in the
plane of a currenr-canying loop is constant.
L = inductance N = nurnber of tums in Iwp, 1 for case considered B = magnetic field inside the loop, as calculated in chapter 5. A = area of loop I = current in loop
It is "matifying that the results by two differenr methods agee. Whenever the
current in the loop is made to change, this self-inductance will determine how much
voltage is required for a given change in current. When the current is constant in a
superconducting h p , there is no voltage needed to maintain the current, so this seif-
inductance would only be considered dunng power-up and power-down of the loop.
The current in the superconducting loop can not be turned on permanendy. If it
was, the loop would not gain any net velocity over an orbit because half the attractive part
of the orbit would be agaiost the flight path and half in the same direction. For example,
if the orbit carried the loop from the South magnetic pole to the North magnetic pole, the
magnetic force h-om the South pole to the magneric equator would slow the satellite
down. There is approximately a 20' pan of the arc between the unstable and stable parts
of the thrust arc mentioned in chaprer 4 where the currenr would have to reverse direction
in the main loop for the mple ring or be tumed o n for the single ring. A realistic nominal
case to find the time available for a complete current change is 500 km altitude.
m3 where GM, = 3.986005 x 10'' -
s
,+= where R = R, + 500 km
T = orbital period V = orbital velocity, circular orbit R = radius of orbit from center of earth R, = radius of earth, 6378 140 meters GM, = universal ,gavitational constant and mass of eanh
Time for current change is 20°/3600 of the orbital period.
3
2x(6378140m+500~10b)~ zoo time to change = x- = 31%
360'
Thus, for a 94 minute orbit period (500 kn circular), the loop current must be
changed in about 5.3 minutes if both the stable and unstable parts of the thrust arc are
k i n g used. If a single-loop option is k i n g used, there will be more rime, but any current
in the loop before crossing the magnetic equator into the stable portion of the thrust arc is
working against the velocity of the spacecraft. The power required to charge the loop
rnay be deduced from the potential energy sstored in the loop and the cime, calculated
above, in which the Ioop musr be fdIy charged.
- Yower = -- '
A t 4(5.3 min)(60s / min)
Power = 3880 W
Elecmcal power on the order of 4 kW is thus required for a 500 m radius loop.
8.3.4 Orbit type
Nor al1 orbits can use magnenc propulsion effectively. Also, at certain points in
any orbit, no force can be created in the desired direction. This is because directly over
the magnetic poles, there can be no tangennal force on the loops. Al1 force would be
radial, either directly up or directly down. If it so happens that a tangential impulse is
needed near the poles, it will not be possible. Cenainly, there are always some tiny
gradients in the field, but the currents required to use them are astmnornicaily high. A
more practical solution is simply to wair an orbit or two until the ground uack of the
satellite has moved with respect to the magnetic pole. Nearly dl polar orbits have a
ground track that shifts substantially with time, so a favorable field orientation is bound to
occur somerime, but it is essential that it be considered when planning a mission.
8.3.5 Single Loop Tumbling
When the single loop has current in it, the magnetic moment aligns wirh the
geomagnetic field with libration (oscillations). Since the geomagnetic field changes
direcrion over the thrust arc, the loop will be tuming. With nothing to prevent it, the loop
will tumble for the remainder of the orbit until the current is turned on again for the
beginning of the next thnist arc, where the moment of the loop will rapidly realign wirh the
geomagnetic field. This realignment from a tumbling state will inrroduce new forces on
the structure thar have not k e n accounted for in the mass esrimates of reinforcement,
Some suategy ro limit turnbling and vibration of the single loop will have to be used to
stop it from vibrating or twisting apart as current is turned on and off. A possible solution
is to inrroduce a small current during pans of the arc where no thrust is required. This
srnall current would cause enough [orque to align with the local field, but any forces
created would be orders of magnitude smaller. In this way, the single Ioop could tum in a
slow and controlled rnanner throughout its orbit.
8.4 Barriers - Advanced Applications
To be used for in terplanetary rravel, in terstellar travel or self-launching flights,
magnetic propulsion requires that al1 of the above problems for orbit control be solved, but
on a larger scale. Since best b o p performance is for very large loop radii, interplanetary
or intentellar missions would require extremely large loop radii. Al1 the problems for
orbit control are an order of magnitude harder for interplanetary efforts, and two orders or
magnitude harder for inrerstellar, since interstellar missions would require a means of
accelerating faster than the 500 km/s solar wind. As for self-launching capability, one
would need not only fantastically large superconducting currents that do not quench in
their own fields, but also exnemely srrong reinforcement marenal thar is also much lighrer
than Alurninurn. Interstellar magnetic propulsion and self-launching capability require
stunning breakthroughs in superconductor current limits and marerials strengths before
even king considered for development-
Chapter 9 : Conclusions
Magnetic propulsion appears practical to develop for a single loop used for
satellite orbit control. The triple ring design appears too massive to use for orbit conuol
with aluminum as the structural material. Interplanetary navel using magnetic propulsion
should be practical at roughly the sarne rime as orbit transfer using rnagnetic propulsion.
Intentellar travel and self-levitation fiom the e h ' s surface using magnetic propulsion do
not appear possible with the design proposed and present day materials limits, even if
superconducnng currents increase by orders of magnitude.
The most efficient loop design for acceleration is a large-radius loop with a
relatively mal1 current. A transfer from a 300 km circula orbit to a 70, 000 km apogee
elliptical orbit would rake about 33 days with a 50 kA current in a 30 km radius loop of
the design proposed. Adding drag does not increase the rransfer time by more rhan about
1 day.
For shon duration missions, or missions thar require large delta-V in a shon time,
rockets would be more effective than magnenc saiis. The flexibility of using a rocket in
any orbit and in any direction is a geat advanrage. Magnetic propulsion, if it can be
developed in the single-loop fom proposed, appears better chan using rockets for missions
that lasr many years, or missions that require conrinual changes in velocity while in polar
orbit. For exarnple, a low-to-high polar orbit shuttle in preparation for a Mars mission
may well use rnagnetic propulsion, since the Ioops could continue making trips until the
solar cells deagade. Reusability means that magnetic propulsion has a future.
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Appendix A
A.1 Structure
Below are found C++ programs used to find mass, acceleration and orbit uansfer
times for both single and triple loops. Afrer the pro,pams themselves are various
subrouunes that are used as "include files" in dl progams.
A.2 Single Loop Acceleration
/* single loop acceleration program */ /* calculates acceleration of a single superconducting ioop for a range of radii */
double tether-area(doub1e radius 1 ) /* rerums area of tethers from loop to main body*/ ( return(43adius 1 *diamerer-tether); 1 /****t***************************************************~***** / main(void) { double radiusl; P radius of superconducting loop */
double mass-total; /* total mass of s p a c e d t */ double area-total; /* total mass of spacecraft meering drag */ double accel; /*acceieration of spacecraft */ double current; /* current in loop, amperes */ double dtitude; /* m above earth surface */ double velocity; /* velocity through atm for drag calc. */ double me-anom; /* me anomaly. assume pengee magnetic nonh */ double position; /* distance from center of eanh */ double bodymass; /*mass spacecraft body */
FILE *fp;
current = 50E3; radius 14.; /* start radius of main loop */ altitude = 300E3; /*alritude where thrust takes place */ position = altitude t radiuseanh; mie-anom = O. ; /*at 30 deg from mag nonh=perigee */ body-mass = 1000.;
Fprinti(fp,"Single loop with drag, alrirude %5.1 If km, current %5.1 If W. altitude/1000., current);
fprintf(fp, "radius accel mass area mag-force Li ");
for (n= 1 ; nd2; n++) ( radius 1 -dius 1 +6OO.;
rnass~total=loop~tnass(radius 1 ,current)+ùody~mass+tether~mass(radius 1); area-total = loop-area(radius 1)itether-ma(radius 1 )+bod y-area;
accel = (mabforce(radius 1 ,cunent,position,mie~anom)- .5*am~density(altitude)*Cd*pow(vel0~ity,2.)*(area-tota~)) /(mass-[oral);
fprintf(fp,"%5.0lf, 85.41f, %6.01f, %6.01f, ", radius I ,accel,mass~totaI,area~total);
r e m (O) ; 1
A 3 Triple Ring Acceleration
/* ProPm a3.cpp. Acceleratin of 3 ring arrangement finds acceleration at various lwp sizes for triple ring. estimates drag and mass increase with size
*/
#include cmath.h> #include cstdli b. h> #inchde <stdio.h> #include âcntI.h> #include csys\stat.h> #inchde <sning.h> h c l u d e <c:\wg\consr.cpp> /*cornmon constants between pro,pms */ #include <c:\wgWensity.cpp> /*file of atmospheric density */ #inchde <c:\wgbrb_calc.cpp> /* basic orbit calculations */ #inchde cc:\wgùnag-for.cpp> /*magnetic force routine */ #inchde <c:\w&riple.cpp> /* mass and area calculations triple loop +/ /************************************************************** / main (void)
double t hic kness,increment,radius 1 ,radius2; short n,m,loop; /* loop is a flag for loop 1 or 2 */ double accel; /* accelerarion of the loop */ double best-accel; /* maximum acceleration at given radius */ double area-total; /* total area meeting atmospheric drag +/ double mass-total; /* total mass of spacecraft */ double bestthick; /* thickness selected for best mass */ double best-mass; /* mass at best accelerarion */ double altitude; /* m above earth surface */ double position; Pdistance from eanh center to satellite position */ double me-anom; /* mie anomaly of satellite position */ double current; /* current in the loops */ double semimajor; /*semi-major axis of orbit */ double mass-body; /+ mass of spacecraft body */
FILE 'fp;
increment = 0.001 ; /* incremen t thickness by meters */
radius 1 = 0.0; /* start radius 1, m */ altitude = 8ûûE3; /* start at 800 km */ position = radius-earth+altitude; ûue-anom = O.; /*talce measurement at pengee */ semimajor = position; /*circular orbit */ mass-body = 2000. ; /* 2000 kg of payload, container */ curreot = 50E3;
fpnntf (@,"Triple ring with drag hW); fprintf(fp,"radiusl bat-accel best-mass thickness Li");
for (m=l;mc30;m++) /* different trys for radius 1 */
radius l =radius 1 +200.; thickness = 0.001; /* reset thickness for loop wall */ best-accel = 0.0; /* start acceIertion */
for(n=l;nc=lO;n+t) /* different nies for thickness */ I area-total= body-area +area_Ioops(radius 1 -0.2) +ares-te thersiradi u s 1 ,current) ;
accel = (magforce(radius 1 ,current,position,me-anom)- .5*aun4ensity(altirude)*Cd* pow(velocity(semi~major,posicion),22)k(area~total)) /(mass-to tal);
if (n == 1) /* stan best at first value */ ( best-acceI=accel: bat-thick = thickness; best-mass = mas-total; 1 ;
if (best-accel < accel) { best-accel=accel; /* finding best acceleration*/ best-thick = thickness; bestmass = mass-total; 1;
thickness = thickness + increment;
1 ; /* end thickness iteration */
fprind(fp," %6.01f, %9.81f, ", radiusl &est-accel);
A.4 Triple Ring M a s / *
Program m3.cpp mass of 3 ring arrangement. gïves mass of primary and secondas, loops for various thicknesses uses a standard loop size of 500 rn main and 210 m secondary uses Aluminum alloy reinforcement
"/ #inchde <math.h> #include cstdli b. h> #inchde <stdio.h> #inchde tio. h> #inchde <fcntl.h> #inchde <sysktat.h> #inchde tiostream.h> #indude a m n g . h > #inchde <c:\wg\ronst.cpp> // basic constants used in programs #indude <c:\wg\niple.cpp> // niple loop mass calculations /***********f***************************************************** / mai n(void)
I double thickness; /* wall of loop reinforcement */ double mas-body; /+ mass of spacecraft body in kg */ double increment; /*amount thickness increased each iteration */ double radiusl; /*radius of superconducun; loop */ doubIe current; /* max curent in al1 Ioops */ double mass-rein force; // mass of rein forcement double radius-tube; // radius of loop circular cross section
short n,loop; /* loop is a flag for loop 1 or 2 */
fp=fopen("m3.out", "w");
thickness = 0.001; /* stan thickness for loop wdl* / mass-body = 2000.; P kg */ incremenr = 0.001; /+ increment thickness by meters */ currenr = 5OE3; /*curent in loops */ radius1 = 500.; /* 500 meter main loop */
fprind(fp,"radiusl %S. llf Li",radiusl); fprintf(fp,"thickess tube-radius mass-reinf mass-to~al Li");
for(n=l;ndO;n++) /* different tries for thickness */ { mass-reinforce = mass-reinf(radius l .thickness,curren t);
radiris-tube = mass-rein force (Pi*2.*thickness*Pi*radius L W density-Alr3.);
fprintf(fp,"%5.3lf. B5.31f. %S.llf, %5.11f,Li", thickness,radius~tube,mass~reinforce, mass-rein force+mass~super(radius i )+ mass-system(radius l)+rnass~body+rnass~tether(radius 1 ,curren 1)):
thickness = thickness + incremenr; 1
fclose( fp); retum 0; 1
A.6 Single Loop Orbit Program 1
/* single loop orbit uansfer program, no drag included */ /* acceleration taken kom accel I .cpp */
#indude <rnath.h> #include <stdlib.h> #inchde estdio. h> #include dcntI.h> #inchde <syskat.h> #inchde <string. h> #inchde <c:\wgkonst.cpp> /*constants definition file */ #inchde <c:\wgùna~_for.cpp> /* magnetic force file */
#include cc:\wg\orb-calc.cpp> P basic orbit calculaaons file */ /************************************************************************/ main(void) { double radius 1 ; /* radius of superconducting loop */ double final-apogee; /* apogee new orbit */ double pengee; /* perigee of orbit in meters */ double apogee; /* apogee of orbir in meters */ double cunent; /* cunent in bop, amperes */ double thetal, theta2; /* stm and end angles of thrust arc */ double segments; /* how many segments t h s t arc divided into */ double position; /*vector from eanh center to position *I double me-anom; /* m e anomaly, radians */ double delta-me-anom; /* delta true anomaiy, segment of thmst arc */ double del-V; /* delta-velocity of satellite */ double semi-major; /* semi-major axis of orbir */ double accel-time: /* time for acceleraiion */ double accel; /* acclerarion of spacecraft */ double mass-total; /* total mass of spacecraft */ double time; /* time to complete mnsfer */
short orbit; /* number of orbit, multiple orbits for transfer */ short n; /* incremental counter * /
curent = SOE3; rddius 1 =3 1 -6E3; /* radius of main loop */ thetal = 2O.*Pi/l80.; /* sran thrusr arc */ theta2 = 8O.*Pi/180.; /* end thnist arc */ mass-total = 20E3; /* total mass. kg. of sarellire "1 del-V = 0.0; /* start delta V */ orbit = 0; perigee= 30OE3+radius_earth; /* circula starting orbi t */ apogee = perigee; /* stan simulation */ final-apogee = 7 109OE3+radius_eanh; I* ending apogee eliptical orbit */ segments=25.; /*divide thrusr arc into segmentse/ position = perigee+radius-eanh; /* start orbit */ delta-me-anom = (theta2-theta l)/segmenrs; semi-major = (pengee+apogee)R.; time 4.;
fprintf(fp,"Pengee %4.01f km loup radius %6.01f rn kW', (pengee-radius-eanh)/1000., radius 1 );
fprintf (@,"orbi t days del-V (rn/s) apogee(km)\n");
while (apogeedinal-apogee && orbitc 1000) /* if orbit gets too large, something is wrong */ { mie-mom = -(3O.*Pi/l80.); /* start thmst @ 80, perigee theta= j0 */
for (n= 1; ncsegments; n++) { position = semimajoe( 1 .-pow(eccent(pengee,semi~major),2.))/ (1 .+eccent(perigee.semi-major) *cos(tme-anom));
if (n==l) /* same accel time through 1 thmst arc */ accel~ume=(mean~anom(eccent(perigee.semi~ma~or) ,me-anom) - m e a n ~ a n o m ( e c c e n r ( p e r i g e e , s e m i ~ r n a j o r ) , ~ e d a n o m ) ) /mean-motion (semi-major); accel = (mag-force(radius 1 ,current,position,rrue_anom)) /mass-total;
del-V = del-V+accel*accel-time;
orbit = orbit + 1; time = time+period(semi-major);
fpnntf (fp," Qd, %5.21f, %4.21f, %8.01f, Li", orbit,timz,del-V,(apogee-radiusSearth)/lOOO.);
del-V = 0. ; /* reset to zero for next orbit */
} ; /* end while */
A.7 Single Loop Orbit Program 2
/* single loop orbit transfer program, same as orbl except with drag */ /* acceleration taken from accel 1 .cpp */
#indude <math.h> #indude Cstdlib. h> #incl ude atdio. h> #inchde dcntLh> #inchde ays\stat.h> #indude <smng.h> #indude <c:\wg\constcpp> /*constants definition file */ #inchde <c:\wg\densiry.cpp> /* atmospheric density file */ #inchde <c:\wg\drag.cpp> /* drag calculations * / #indude <c:\wghag-for.cpp> /* magnetic force calculations */ #inchde cc:\wgbrb-calc.cpp> /* basic orbit calculations */
main(void) { double radiusl; /* radius of superconducring loop */ double final-apogee; /* apogee new orbit */ double perigee; /* perigee of orbit in meten */ double apogee; /* apogee of orbit in merers */ double current; /* current in loop, amperes */ double thetal, theta2; /* start and end angles of thrust arc */ double thnistseg; /* how many segments thrust arc divided into */ double d r a ~ s e g ; /* how many segments drag arc divided inro */ double position; /*vector from eanh center to position */ double miepnom; /* m e anornaly, radians */ double delta-me-anom; P delta me anomaly. segment of thmst arc */ double del-V; /* delta-velocity of satellite */ double semi-major; /* semi-major a i s of orbit */ double accel-time; /* tirne for acceleration */ double accel; /* accleration of spacecraft */ double area-total; /* area for drag on spacecraft */ double mas-total; /* total mass of spacecraft */ double time; /* time to complere transfer */
shon orbir; /* number of orbit, multiple orbits for mnsfer */ short n; /* incremen ta1 counter */
fp=fopen("02.0utt'." w");
current = 50E3; radius l=3 1.6E3; /* radius of main loop */ theta 1 = 2O.*W180.; /* sstart thrust arc */ theta2 = 8O.*Pi/i80.; /* end thmst arc */ area-total = 885.; /*total area, m2, of satellite */ mas-total = 20E3; /* total mass, kg, of satellite */ dei-V = 0.0; /* srart delta V */ orbit = 0; perigee- 3ûUEJ+radius_earth; /* circular stamng orbit */ apogee = perigee; /* start simulation */ final-apogee = 7 1090E3+radius_eanh; /* ending apogee eliptical orbit */ thrust-seg=25.; /*divide thrust arc into segments */ position = perigee+radius-eanh; /* stan orbit */ delta-me-anom = (thete- theta 1 )/thru st-seg; semi-major = (perigee+apogee)/L; tirne =O.;
fprintf(fp,"Perigee %4.01f km loop radius %6.01f rn bb", (perigee-radius_eanh)/1000., radius 1);
fprintf (fpYworbit days del-V(m/s) apogee(km) W);
while (apogee<finalppogee && orbit<1000) P if orbit gets too large, something is wrong */
tmeanom = -(3O.*Pi/180.); /* sran thmsr @ 80, perigee rheta=50 */
for (n=l; ncthrust-seg; n++) ( position = semi-major*( 1 .-pow(eccenr(perigee,semi~major),2.))/ ( 1 .+eccen t(pengee,serniimajor) *cos(true-anom));
if (n==l) /* same accel time through 1 thnist arc */ accel~time=(mean~anom(eccen t(perigee,se-,mie-anom) -mean-anom(eccen t(pe~igee,semi~ma~or),ûueanom+del ta-true-anom)) /mean-motion(semi-major);
accel = (rna=_force(radius 1 ,current,posirion,rme_anom)- dragfome(position, area-total, velocity(semi~major,posirion))) /mas-total;
meanom =t.e-anom+deI ta-me-anom;
1 I* now the drag part of the orbit */ dragseg = (23"-(theta2-thetal))/deltatatnieeanom;
for (n=l; ndragseg; n++)
I posirion = semi-majoP(1 .-pow(eccenr(perigee.semi~major),2.))/ (1 .+ec~ent(perigee~se~~major) * cos(tme-anom));
/* same accel time through 1 thnist arc */ accel~time=(mean~anom(eccen t(perigee,serni-major) ,mie-anorn) -mean~anom(eccent(perigee,se~jor),~e~anom+delta~~e~anom)) /mean_rnotion(semi_major); accel =drag_force(position, area-total, velociry(semi-major,posirion)) /mass-total;
orbit = orbit + 1 ; time = time+penod(semi~major);
del-V = 0. ; /* reset to zero for next orbit */
}; /* end while */
A 8 Triple Loop Qrbit Program
/* orb3.cpp orbit calcularions for a mple-ring arrangement incl udes drag set the condition number from 1 to 4 for different current and
main loop radius conditions * /
#include <math. h> #inchde cstdli b- h> #inchde <stdio.h> #inchde <fcntl.h> #inchde <sysbtat.h> #inclde csaing.h> #include cc:\wgkonst.cpps /*cornmon constants between programs */ #inchde <c:\wgLlensity.cpp> /*file of aunosphenc densi- */ #inchde cc:\wgbrb-calc.cpp /* basic orbit calculations +/ #inchde <c:\wgLnagfor.cppz /*magnetic force routine */ #include <c:\wg\drag.cpp> /*drag force calcularions */ #include cc:\wg\mple.cpp> /* mass and area calculations mple loop */ /*************8***++;i(t*****+***I*g+*8*lice******************e***** 1
{ double radiusl; /* radius of superconductin; loop */ double final-apogee; /* apogee new orbir */ double pengee; /* pengee of orbit in meters */ double apogee; /* apogee of orbit in meters */ double current; /* current in loop, amperes */ double thetal, theta?; /* start and end angles of thrust arc */ double thnisr-seg; /* how many segments thrust arc divided inro */ double draeseg; /* how many segments drag arc divided into */ double position; /*vector h m earth center to position */ double trueanom; /* m e anomaly, radians */ double delta-me-anom; P delta mie anomaly, segment of thmst arc */ double del-V: /* delta-velocity of satellite +/ double serni-major; /* semi-major axis of orbir */ double accel-time; /* time for acceleration */ double accel; /* accleration of spacecraft Y double area-total; /* area for drag on spacecraft */ double mass-total; /* total mass of spscecraft */ double time; /* time to complete transfer */ double thickness; Pthickness of reinforcement wall */ double mass-body; /* mass of spacecraft body */ double termination-orbit; /*orbit to stop loop at */
shon orbit; /* number of orbit, multiple orbits for transfer */ shon n,m,loop: /* loop is a flag for loop 1 or 2 */ shon condition; /+ condition of loop currenrs and sizes */
*fp; mas-body = 2000. ; l* 2000 kg of payload, container */ thickness = -00 1; Ppractical lower limit rein forcement thickness */ theta 1 = 20-*Pi/180.; /* start thrust arc */ theta2 = 80-*Pif1 80.; /* end thrust arc */ del-V = 0.0; /* start delta V */ orbit = 0; perigee= 300E3+radius_eanh; /* circular sraning orbit */ apogee = perigee; /* stan simulanon */ final-apogee = 7 1090E3+radius-eanh; /* ending apogee eliptical orbit */ thnist-seg=25.; /*divide thmst an: into segments*/ position = perigee+radius-eanh; /* start orbir */ delta-mue-anom = (theta-thetal)/thrust-seg; semimajor = (pengee+apogee)/2.; terminationorbit=1000; /*teminates after this orbit if no success */ time =O-:
condition = 1; /* set the condition of loop radius and currenr */
if (condition= 1) { current = 50E3: radiusl=lûûO.; /* radius of main loop */ fp=fopen("o3t l .out","w"); 1 else if (condition=2) { current = 500E3; radius I =1000.; /* radius of main loop */ fp=fopen("o3t?.out"."w"): 1; if (condition=3) { current = 50E3; radius l=3 1000.; /* radius of main loop */ fp=fopen("o3t3.out","w"); 1 ;
if (condition4) { current = 500E3; radius 1 =3 1000.: /* radius of main loop */ fp=fopen("o3tl.out"," w"); 1 ;
area-total= bod y-area
fprintf(fp,"Perigee %4.01f km radius1 %5.01f mass %8.01f area %6.OIf Li". (perigee-radiusearth)/l 000.. radius l . mass-tota1,area-total);
fprintf(fp," currenr %6.OIf hW', current);
fprintf (@,"orbit days del-V(m/s) apogee(km) li");
while (apogeecfinal-apogee && orbircterminanon-orbit)
for (n= 1 ; ncthrust-seg; n++)
{ position = semi-major*( 1 .-pow(eccenr(perigee,semi_major),2.))/ ( l .+eccen t(pengee,semi-major) *cos(me-anom));
if (n==l) /* same accel time rhrouph 1 thmsr arc */ accel~time=(mean~anom(eccen~(perigee,semi_m,uue~anom) -mean~anorn(eccent(perigee.semi~ma~or),uue~anom+del ta-me-anom)) /mean~rnoaon(semi~major);
accel = (mag-force(radius 1 ,current,posi tion,mie_anom j- .5*atm~density(posi40n-radius~emh)*Cd* pow(velocity(semi~major,position),2.)*(area~total)) /(mass-total);
/* now the drag pan of the orbit */ cira;-seg = (3.*Pi-itheta2-theta 1 ))/delta-true-anom;
for (n=l; ncdrag-seg; nt+)
position = semi~major*(l.-pow(eccenr(perigee,semi~major),2.))/ (1 .+eccent(pengee,semi-major) *cos(mie-anom));
/* same accel time through 1 thrust arc */ accel~time=(mean~anom(eccent(perigee,semi~major) ,me-anom) - m e a n ~ a n o m ( e c c e n t ( p e r i g e e , s e m i ~ r n a j o r ) , ~ n o m ) ) /rnean_motion(semi~major);
accel =draagforce(position, area-total, veIocity(semi~major,posioon)) /mass-total;
if (orbit%5 ==O I I orbit < 25) { fpnntf (fp," Qd, %5.21f, %4.21f, %8.0lf, \nu, orbit.rime.de1-V,(apogee-radius-earth)/lOûû.); 1
if (apogee c radius-emh+ ljOE3) orbit=rerrninarion-orbit; /* orbit decayed inro earth */
del-V = 0. ; /* reset to zero for nexr orbit */
} ; /* end while */
A.9 Single Loop Levitation
A.9.1 Al Reinforcement
/ *
1 l xpp: single loop sel f-levitation prokgam
calculates mass and lifting ability of single superconducring loop for a range of radii. Shows mass change with current for 100 m loop assumes Aluminum alloy for reinforcement macrial. 2 files output: Il.out and m Lout
*/ #inchde <math.h> #indude es tdlib. h> #indude <stdio.h> #indude dcntl.h> #indude <sysbrar.h> #inchde <smng.h> #inchde <c:\wg\consr.cpp> #inchde <c:\wgkad-for.cp p> #inchde cc:\wgkingle.cpp> /O********Y*************b***************************************~**/
double system-mass (double radius 1) P finds mass of attitude control system for single loop stabilization by gyroscopic shell. Assumes a shell of the size of the loop made fiom Aluminum */ { renirn (Pi*pow(radius l.L)*.OO 1 *density-Al); 1
/*********~*******t****Jct*************************b***************** /
rnain(void)
double radius 1; /* radius of superconducting loop */ double rnass-[oral; /* total mass of spacecraft */ double current; /* current in loop, amperes */ double nue-anorn; /* nue anomaly, assume perigee magnetic north */ double position; /* distance from center of eanh */ double mass-Mt; /* mass rhar can be lifted by magnetic force */ double altitude; /* altitude above earth surface, m */ double theta; /*angle from magnetic nonh, radians */ double body-mass; /*mass spacecraft body */ double bat-mass-lift; /*point with least difference benveen mass-lift*/ double best-masstotal; /*and mass-total has "best" values */ double besr-current; /* current least difference lift, total mass */ double min-difc /* minimum difference between mass-lift and mass-total*/
FILE *fp, *fp2;
altitude = O.; /*altitude where thmst takes place */ position = altitude + radius-earth; theta = 0.; /* at a magnetic pole */ body_mass = 1000.;
fprintf(fp,"Single loop self-levitation current Li");
fpnntf(fp,"radius mass-total rnass-lift cument hW);
fprintf(fp2," Single loop radius 100 rn Li"); fpnntf(fp2," c m n t mass-total mass-lifth");
for (radius 1 =û.;radius 1 < 500.;) I radius 1 =radius 1 +50.; current = 0.; mass-lift=O.; /* stan the loop making ma mass-total= 1.; min-diff = 1E9; /* stan at high value so 1st value below replaces */
while (mas-lift < mass-total) { current = current +5E5;
if (min-diff > (mass-total-mass-Mt))
min-diff = mas-total-mass-lift; besr-rnass-lift = mass-lift; best-mass-to ta1 = mass-total; best-current = current; 1 ;
if (curent> 1 ES) break;
if (radius1 == 100. ) fprintf(fp2," %7.01f %7.01f WAlf Li", current. mass-tota1,mass-lift); }; /* end while loop */
fprintf(fp, "%S.Olf, %6.01fT %8.51f, %9.0lfT hW, radius l , b e s t ~ m a s s ~ t o t a l , b e ~ t ~ m a s s ~ l i f t , ~ n t);
} ; /* end for loop */ fclose (fp); return(0); 1
A.9.2 UHS Reinforcement
/* l2.cpp: single loop self-levitation program The same as Il-cpp, except for Ultra-high strength composite rein forcement instead of Aluminum
calculates mass and lifting ability of single superconducting b o p for a range of radii. Shows mass change with current for 100 m loop 2 files output: 12.our and m2.out */
#inchde <math. h> #include cstdlib. h> #include cstdio. h> #include dcntl.h> #include csys\stat.h> #inchde <string. h> #inchde <c:\wgkonst.cpp> #inchde cc:\wgù-ad-for.cp p> #indude <c:\wgkingle.cpp> /****************************************************************** / dou bIe system-mass (double radius 1) /* finds mass of attitude conuol sysrem for single loop stabilizarion by gyroscopic shell. Assumes a shell of the size of the loop made Erom Alurninum */ t r e m (Pi*pow(radius 1,2.)*.00 Pdensity-fiber); 1
/t***************ax***t****~~~iiciic**;i~;i~lic******************************* / rnain(void)
double radius1 ; /* radius of superconducting loop */ double mass-total; /* total mass of spacecraft */ double current; /* current in loop, amperes */ double mie-anom; /* m e anornaly, assume pengee magnetic north */ double position; /* distance from center of eanh */ double mass-lift; /* mass that cm be lified by magnetic force */
double altitude; /* altitude above earth surface, rn */ double theta; /*angle fiom magnetic north, radians */ double bod y-mass; P rnass s pacecraft body */ double bar-mass-lift; /*point with least difference benveen mass-liftc/ double best-masstotal; /*and mass-rotai has "best" values */ double best-current; /* current least difference lifi, total mass */ double min-diff; /* minimum difference be tween masslift and mas-to tal*/
altitude = O.; /+altitude where thmst takes place */ position = alritude + radius-eanh; theta = O.; /* at a magnetic pole */ body-mass = 1000.;
fprintf(fp,"Single loop self-levitauon current h");
fprintf(fp,"radius mass-total mass-lift currenr Y);
fprintf(fp7," Single loop radius 100 m Li"); fprintf(fp2," curren t mass-total rnass-Iifth");
for (radius 1 =O.;radius l< 500.;) { radius 1 =radius 1 +5O.; current = O-; mass-lift4.i /* start the loop making mass-lift less than mass-total*/ mass-total= 1 .; min-diff = 1E9; /* s t a n at high value so 1st value below replaces */
while (mass-li ft < mass-total) { current = current +SES;
mass-lift =fabsl(mag~for~rad(radius 1 ,currenr,theta,posihn)/9.8 1 );
mass~total=loop~mass(radius 1 ,cument,stress-allow-fiber,density_fiber)+ body-mass+tether-mass(radius 1 ,density-fi ber); + system-mass(radius 1 );
if (min-diff > (mas-total-mass-lift)) I
min-diff = mas-total-mass-lift; best-mass-lift = rnass-lift; bestmass-total= mass-rotal; best-curren t = curren t; 1;
if (cutreno 1 E8) break;
if (radius l = 100. ) fprintf(fp2," %7.01f %7.01f W.4K h", current, mass~toial,mass~lift); ) ; /* end w hile loop */
fprintf(fp, "%5.0lf, %6.01f, %8.51f, %9.01f, h", radius 1 .best~mass~totaI,best~mass~1ift9besrcurren t);
) ; /* end for loop */ fclose (fp); return(0); 1
A.10 Triple Loop Levitation
/* lift-cpp. mple ring. Fix radius. Fix altitude. calculates mass various currenrs can lift against graviry calculates mass of spacecraft for various currents (reinforce changes) objective: current for self-levitation, mass @ that current
*/ #inchde <math.h> #include <stdlib.h> #inchde <stdio,h> #inchde <fcntl.h> #inchde csysbtat. h> #inchde cstring.h> #indude <c:\wg\Eonst.cpp> /*cornmon constants berween pro,grams */ #inchde <c:\wg,lorb_calc.cpp> /* basic orbit calcuiations */ #inchde <c:\wg\rad-for.cpp> /* radial magnetic force routine */ #indude cc:\wgkriple.cpp> /* mass and area calculations triple loop */ /*************8~*********************************************** /
{ double thickness,radius 1 : /* thickness of loop radius 1 */ double mass-total; /* toral mass of spacecraft */ double best-thick; /* thickness selected for best mass */
double bat-mass; /* mass at best acceteration */ double altitude; /* m above earth surface */ double position; /*distance from eanh center to satellite position */ double current; /* current in the loops */ double mas-body; P mass of spacecraft body */ double mass-lift; /* mass that can be lifted */ double theta; /* angle h m magnetic nonh, radians */ double smallestdiff; /* smallest difference between lift and mass */ double best-current; /* current at smallest difference */ double bat-lift; /* mass that can be lifted at best curent*/ double increment; /* arnount to increment thickness by */
short n; /* counter for loop */
incrernent = 0.001 : /* increment thickness by meters */ mas-body = 1000.; /* kg of payload, container */ altitude = 0.; /* start km */ theta = O.; P start angle from magneac pole */ position = radiu s-earth+altitude; best-mass = mass-body; mass-lift = 0.;
fprintf(fp."Triple ring Lift hW); fprinafp, "radius 1 bat-lift best-mass bestcurrenr mass-lift mass Li" );
for (radius l=û.;radius l<?OO.;) ( current = 4E3; /* start current */ radius 1 -dius 1 -c IO.; smallestdiff = 777777.;
while (mass-liftebest-mass) { thickness = 0.001; /* reset thickness for loop wall */ current = current+500.;
for@= 1 ;n<= 1 O;n++) /* different mes for thickness */ { mass-total= massreinf(radius 1, thickness,current)+mass~super(mdius 1 )+
mass~system(radius 1 )+mass-bdy y;
thickness = thickness + increment;
if (n= 1) best-mass = mass-total;
if (best-mass > rnass-total) best-mass=mass-total;
); /* end thickness iteration */
if ((mass-total-mass-1ifr)c srnallest-diff) { bestlift=mass-lift; smallest-diff = mass-total-mass-lift; bestcurrent = current; 1 ;
if (currenu500000.) break; ); /* end of current while loop */
fprintf(fp,"87.01f, %7.4lf, %7.01f, %9.01f1 87.41f, 87.01f,Li", radius 1, best~lift,bes~lift+smallest~difflbest~curren t,
mass-lift, bat-mass );
1; /* end of radius iteration */
fclose(fp); return O; 1
A.11 Constants File
/* include file const-cpp contains constants common to al1 othcr programs of orbit controi and mass/acceleration calculations
*/
#de fine Pi 3.14 1 592654 #define radius-earth 6378E3 /* radius of eanh, m */ #define density-Alt800.0 /* density of aluminum alloy 2014-T6 */ #define perrn-free 4.0 + Pi * (pow10(-7)) /* pemeability of vacuum */
#define stress-allow-Al 137E6 /* Pa of mess allowed on Al */ #define GMe 3.986E14 P universal gravitational constant for eanh */ #define B-surface 3.05E-5 /*magnetic field at emh surfacc Tesla*/ #define Cd 2.2 /* coefficient of drag */ #define diameter-wire 0.003 /* diameter of loop wire */ #de fine diameter-tether 0.002 /* diarneter of techer wire */ #define radius-super 0.00 126 /*radius solid superconducnng wire */ #define density-super 5000. /*kg/m3 of CuO, basic HTSC marerial */ #define body-area 50. /* rn2 of spacecraft center body area */ #define deviation 10*W180 /* deviation fkorn unstable equilibnum */ #define density-fiber 1000. /* density of UHS composite */ #define suess-allow-fiber 3000E6 /* allowable stess on UHS composite +/ #define r-outer-start 0.05 // start value hollow circle radius ifdefine delta-r-outer 0.05 // incrernent radius outer
A.12 Air Density File
double am-densiry(doub1e alritude) /* density of atmosphere at any altitude given in m */ /* atrnospheric densities given are maximums. for peak solar cycle */ { double densi ty; altitude = altitude/1000.;
if (altitude>O. && altitude < 200.) density = (10.); else if (altitude>= 100. && alti tude < 1 50.) density = (6E-7); else if (altitude>= 150. && altitude < 200.) density = (2E-9); else if (al titude>=2OO. && altitude < 250.) density = (3E- 10); else if (alUtude>=250. && altitude < 300.) density = (1E- 10); else if (ahitude>=300. && altitude c 350.) density = (4E- 11); eke if (aItitude>=350. && altitude < 400.) density = (2E- 1 1 ); else if (aItitude>=4OO. && altitude < 450.) density = (IE- 11); else if (altitude=.4SO. && altitude < 500.) density = (5E- 12); else if (altitudez=500. && altitude < 550.)
densiry = (3E- 12); else if (altitude>=550. && altitude < 600.) density = (1 SE- 12);
else if (altitude>=600. && altitude < 650.) density = (8E-13);
else if (altitude>=650. && altitude < 700.) density = (4E- 13);
else if (altitude>=700. && altitude < 750.) density = ( U E - 13);
else if (altitude>=750. && altitude < 800.) density = ( 1 . 6 13);
else if (altiiude>=800. && altitude < 850.) densi ty = (9.4E- 14);
else if (altitude>=850, && altitude < 900.) density = (5.7E- 13);
else if (altitude>=900. && altitude < 950.) density = (3.5E- 14);
else if (altitude>=950. && altitude < 1000.) density = (2.2E- 14);
else if (altitude>= 1000. && altitude < 1250.) density = (1.4E- 14); else if (altitude>= 1350. && altitude c 1500.) density = (ME- 15);
else if (altitude>= 1500. && altitude c 2000.) density = (LIE-15); else if (altitude>2000.) density = (0.0);
return(densi ty); 1 /******************************************************/
A13 Drag Force File
/* calculates drag force on satellite of area-total at atmospheric density found at alti tude.*/
double altitude; /* altitude above e m h surface in km */
altitude = (position - radius-earth);
A.14 Magnetic Force File
double rnagforce(doub1e radius 1, double current, double position, double me-anom)
/* find the magnetic force on lwp of radius1 with current in it at altitude above the surface of the eanh and angle theta fiom magnetic north * /
double moment; /* magnetic moment of loop 1 */ double thrust; /* thnist function due position in geomagnetic field*/ double grad-B; /* --dient in geomagnetic field */ double theta; /* angle fiom magnetic north */
theta = 5O.*Pi/l80.-mie-anom; /* assume perigee center thrusr arc */
thrust =0.5*sin(2*theta)/sqrt(3*pow(cos(theta).2.)+1);
moment = current*Pi*pow(radius 1 3 ;
/* assume -9 loss for eanh rotation, poles moving */ retum(0.9*momen t*grad-B);
1
A15 Orbit Calculations File
/* basic orbit calcutation routines used by orbl, orbll, orb3.cpp */
double velocity (double semi-major, double position) /* rerums velocity at position in elliptical orbit of semi major axis */
retum (sqrt(2*GMe*( 1 ./position- 1 ./(2.*semi_major)))); 1 /***********************************%****%************************ / double delV-total(dou ble orbit-radiusA, double orbitradiusB)
/* r e t m s the required delta V for a transfer from an initial orbit with orbit-radius to final eliptical orbit with perigee A and apogee B
*/ { double semi-major; /* semi-major axis of orbit */ double delV ; /* delta V for transfer to eliptical orbit */
semi-major = (orbi t-radiusA+orbi t-radiusB)Q.; delV = sqrt(GMe)*abs(sqrt(2/orbit~radiusA- l/semi-major)- sqn( l/orbit-radiusA));
retum(delV); 1 /********************************************************************** / double thrust-arc(doub1e theta l ,double theta2,double pengee) /* finds length of thrust arc, approximated as a circle seamen[
with a radius of perige * 1
retum (pengee*abs(theta2-rhera 1 )):
1 /**************************************************************z******* / double vel-pengee(dou ble perigee,dou ble apogee) /* finds velocity at perigee for an elliptical orbit */ I double semimajor: /* semi-major axis of orbi t */
double eccent (double perigee, double semi-major) /* retums eccenticity given per-igee and semi major axis of orbit */ I retum ( 1 .-perigeehemi-major); 1 /********************************************************************** /
double meananom (double eccent, double medanom) /* retums mean anomaly given orbit eccennicity and me anomaly at point in question */
t double eccent-anom; /* eccentric anomaly for orbit */
eccen t-anom = (acos((eccen t+cos(rrueeanom))/ ( 1 .+eccen t*cos(true-anom))));
double period (double semi-major) /*period of orbit in days aven semi major axis*/
double period; /* period of orbit */ /* pend = sqrt(7.W *pow(semi-major.3 .)/GMe); */ period = pow((semimajor/33 1.2E3), 1.5)/(60. *N); retum(penod); 1 /********************************************************************** /
A16 Radial Force File
double mag-for-rad(double radius 1, double current, double theta, double position) /+ returns the magnetic force in the radia1 direction */ t double radial; /*radia1 force function of theca */
A.17 Single Loop M a s File / *
include file single.cpp: calculations used for mass of single loop used by a l .cpp and I l .cpp
* /
double area-reinf(doub1 curren t, double stress-allow )
/* area cross section of reinforcement needed in main loop to counter tension €rom current in the Ioop */
{ double tension; /* tension in loop due to own current */
double bop-rnass(double radius 1, double cunent, double stress-allow, double densikreinf)
double vol-super. /* volume of superconducring material*/ double vol-reinf; /* volume of reinforcemenr material */ double system-mass; /* mass of cooling, reflecting, power systems */
vol-reinf=2.*Pi*radius l ' a r e a ~ r e i n f ( c u ~ e s s ~ a l 1 o w ) ; vol-super=2.*Pi*radius 1 *Pi*pow(radius-super,2.); systern-mass = 2.*Pi*radiusl*O.005: /* sysrems */
double tether-mass(doub1e radius 1, double density-reinf) /* fin& mass of rethen from main loop to center body */ { retum(radius 1 *Pi*po~(diarneter-tether,2~)*densitv-reinf); /* 4 tethers cancei with factor of 4 denominator */ 1
/**************************************t************************ /
A.18 Triple Loop Mass File /* include file: tripie-cpp calculaaons associared with triple loop mass, area and loads
* / ........................................................... /
double radiusS(dou bls radius 1 ) P calculares radius loops 2 and 3 for given deviation and radius of 1 */ { retum (radius1 *sqrt(sin(Pi 1 2 -deviation)/sin(deviation))); 1
double tension 12 (double radiusl, double current) /* tension between bop1 and secondary loops */ { double angle; /*angle in radius that tether makes with loop 1 plane +/ double moment; /* in N-m that tether supports */
angle = atan(radius2(radius l)/radius 1); moment = Pi*pow(radius2(radius l),2-O)*pow(current,2.O)*perm-free/ (2.*radius 1);
retum (momenV(2. *radius2(radius 1 )*cos(angle))); 1 /*********************************************** / double area-tethers(dou ble radius I , double current) /* retums frontal area of tethers between the loops, used for drag*/ { double angle; P angle between tether and bop plane */ double moment; /* torque between loops due to current in loops */ double tension23; /* tension on rethers berween secondary loops */ double diameter: /* of rether */ double tetherl~ether2; /*frontal area of tethers 1 and 2 */
/* first for tethers fiom loop 1 to loops 2 and 3 */ diarnerer=sqn(4*rension 1 l(radius 1 ,current)/(Pi*sness~allow~Al)); tetherl = radius 1 *diamerer; /*fron ta1 area of tethers loop 1 to loops 7,3*/
/* now tethers between secondary loops */ angle = 45*Pi/180; /* secondaries are the same size */ moment = Pi*radius2(radius1)*pow(current,2.)*pem~free/(2); tension22=momenr/(2*radius2(radius I)*cos(angle)); diameter=sqrt(4*tension22/(Pi*sness~a~low~Al)); tether2 = sqn(?)*radius2(radius 1 )*diameter, /*frontal area rerhers 2,3*/
double del taX (double radius, double sections) /* gives length of an element of suaight beam used to represent quaner Ioop "/
retum (Pi*radius/(2.*sections));
double mass-super(doub1e radius 1 ) /* mass of superconducting material */ {
return((2*Pi*radius 1 +4*Pi*radius2(radius 1))*Pi*pow(radius~super,2.)* density-super);
double mas-system(doub1e radius 1 ) P mass of the solar, cooling and power disaibution systerns */ P mass of al1 systems assumed to be 0.05 kg/m */ { return((Pi*2*radius 1 +4*Pi*mdius2(radius 1 ))*0.05); 1 /***************************************************************** 1 double area-loops(dou ble radius 1, double r-outer) // returns the area of loops meeting atmosphenc drag // given radius main lwp and outer radius of loop cross section
I double areal, area2; /* areas of primary & secondary loops */
/* add main loop area to rwo secondary loops */
double force22(dou ble x,dou ble radius 1 ,double deltaX, double current) /* force on current elernent deltaX in secondary loop due to current
in other secondary loop */ { return(pow(current,2.)*pemIlIlfree*deltaX*( l.+2.*(1 .-pow(2.,- lS))*x/(Pi* radius2(radius 1 )))/(2.*radius2(radius 1 )));
1 /*************************************************************** / double force2l(doubIe x,double radius1 ,double delta)<, double current) /* force on current element deltaX in secondary loop due to cunent
in Ioop 1 */ ( retum(pow(current,2.)*pem~free*deltaXLsin(x/radus2(radius 1 ))*(l ./
double radius-outer(doub1e thickness, double moment) // returns outer radius of loop cross section // uses bending beam suaight-beam equations // thicknes of hollow circle cross section // moment on cross section // for a saaight beam: // radiusputer = sqrt(mornen t l/(stress~a1low~Al* thicknessw ) I retum (sqn(fabs(mornent/(sness~allow~A~*thickness*Pi)~)); 1 /*************************************************************** / double mass-prirnary(doub1e radius 1, double thickness,dou ble current)
// finds mass of the pnmary loop, given the radius and thickness // worst-case tension is assumed, with entire loop made with same // cross-section. Worst-case bending stess assumed over entire loop.
{ double mass; /* of a quaner section of main loop */ double moment 1; /* moment up to point x */
moment l = (tension 13(radius 1 ,~urrent)~cos(Pi/6.)- 1 OO.*perrn-free* pow(current,2.)/(2.*radius1))/(Pi*2.*radius 118.); // max. moment loop 1 due current in loop 2, less expansion // force from primary loop mass = Pik2,*radius-ou ter(thickness,moment 1 )*thickness *Pi*radiusl*2. * density-Al;
/* mass is for a straightened quaner-section of primary loop */
doubIe mass~second(doub1e radius 1, double thickness,double current)
/* finds mass of one secondary loop, given the radius of loop 1, the thickness of loop 2 reinforcement, the current in al1 loops (assumed to be max. and height of square cross-section of reinforcement */ {
double delta-m; /* differential m a s element */
double mass2; /* total mass element */ double x ; /* distance dong rirc of secondary loop quarter section */ double sections; /*number sections divided inro for integmtion*/ double moment2 /* moment at point x dong beam */ double shear-force; /*shear force sr any point x dong arc */ short n;
sections = 1000.0; /* number of pans qumer loop divided into */ mass3 = 0.0 ; /* stan m a s 2 */ moment2 = 0.; /* moment at end of beam */ x = 0.: /* sran value for distance dong beam */
for in= 1 ; n~sections; n++) { shearforce =force2 1 (x,radius 1 ,deltaX(radius2(radius 1 )
,sections), curren t) +force33(x,radius i ,deltaX(r;idiusS(ndiu.; 1 ),sections),current);
1; /* end for*/ /* rnass is for a swightened quaner-section of seconda. b o p */
re turn (4.*mass3); 1 /*********************************************************************/
mas-tether(doub1e radius 1 ,double current) /* finds mass of tethers from main to secondaries and between
secondary loops */
double angle; /* angle between tether and loop plane */ double moment: /* torque between loops due to current in loops */ double rension; /* tension on tether */ double diameter; /* of tether */ double mas-tether 1 .mass_tether2; /*m;~ss of tethers 1 ;tnd 2 */
/* f i s t for tethers from loop 1 to loops 1 and 3 */
angle = atan(radius?(radius l)/radius 1); moment = P i * p o w ( r a d i u s 2 ( r a d i u s l ) , 2 . 0 ) * p o w ( c u ~ e n ~ d i u s 1); tension=momenr/(2*radius2(radius 1 )*cos(angIe)); diameter=sqrt(4*tension/(Pi*s~ess~allow~AI)); mas-tetherl = (sqn(pow(radius 1,2.)+pow(radius?(radius 1),2.)) +sqrt@ow(radius2(radius 1),2.)+ 100.)) *densixAl* pow(diameter,2.)*Pi/4.; /* includes the body to secondary tethers. makes same diamerer +/
/* now tethers between secondary loops */ angle = 45*Pi/l8O; /* secondaries are the same àze */ moment = Pi*radius?(radius 1 )*pow(curren t,2.)*pemrnifiee/(3); tension=moment/(2*radius2(radius l)*cos(angle)); diameter=sqn(4* tension/(Pi*stress-a1 low-Al)); mass-tether2 = sqrt(2.)*radius2(radius I )*density_Al*Pi*diameter/4.;
double rnass-reinf (double radius l ,double chickness,dou ble current) /* total reinforcement mass is the sum of primary and secondary mass */
{
retum (mass-pnmary(radius 1 ,thickness,curren t)+2.*mass-second(radius 1. thickness,current));
A.19 Flexi bility Program
// flex-cpp finds flexibiliry of given curved beam compared to smight // calculates the neural axis position // assumes hollow circle cross section used for reinforcement
#inchde <iosueam.h> #inchde <smng.h> #inchde <c:\wg\consr.cpp> // basic constants used in program #indude <c:\wgLia.cpp> // neuual a i s calcularions
main(void) { double flexibility; // cornparkm to saaight beam flexibility double area; // area of loop cross section double second-moment; // of area about loop cenrer double r-outer; // outer radius of cross section double thickness; // thickness of loop reinforcernent double radius 1; // radius of pnmnry loop
FiLE *fp;
r-outer = -8; thickness = 0.0 1 ; radius 1 = 500.;
area=area-tu be(r-outer, thic kness);
second-moment =(pow(r-outer.4.)-pow((r-ou ter-thickness),4.))*Pi/l.;
flexibility = second~mornenr/(neutral~axis(r~outer,radius 1, thickness) *area*(radius 1 +r-outer));
fprintf(fp,"stress factor, inner %Wlf Li", ( 1 .-neutral_rixis(r-outer,radius 1 .thickness)/(r-outer-thickness))"
second-momen t/(neu tral_axis(r_outer,r3dis 1 .thickness) *nreri*ndiiis 1 ));
fpnntf(fp,"stress factor. ourer 8 S . M Li". ( 1 .+neutrai-axis(r-outer,radius 1, thickness )/r-ou ter)*
second~rnoment/(neut~lI~xis(r_outer,rdi~s 1 .thickness)*;ire;i*(rndiiis 1 +2. *router)));
fprintf(fp,"calciiiated area* 10% %7Slf Li", Pi*(pow( r-oiiter.2.) -pow((r-outer-thickness),2.))* 1 E6); fprintf(fp,"integrated area* 1W6 WSlf a r e 1 E6); fprintf(fp,"second moment* 10A6 B7.5lf \n ".second-moment* l E6): fpnntf(fp,"neuml mis %Mlf LiW. neuml-mis@-outer,ndius 1 .thickness));
fprintf (fp," flex factor % 8.7ifLi ", flexibility);
A.20 Neutra1 Axis Calculations
// nacpp neutral-axis // inctude file to calculate the neunal axis posinon from the cenmid // of the hollow loop used for reinforcemen t
double neutral-axis(doub1e router, double radius 1, double thickness)
// retums neutral axis position from centroid of cross section // performs integral for second rem of equation for neuml axis position // as found in curved beam equations from Bowes, Russell and Suter, p. 129
// r-outer is radius of outer surface of circular cross section // radius 1 is radius primary loop // thickness is thickness of cross sectional reinforcement
{ double area; // cross sectional area double delta-r: // increment in radius for inteption double delta-area; // increment of cross section area double r: // radius fiom center of primary loop double integral; // intemgal of area over radius for cross section double theta-outer: // angle from venical to point on outer surface double theta-inner; // angle to point on inner surface of cross section double length-outer; // horizontal length from center cross section double lenph-inner, // horizontal length to point inner surface
short sections; // number of sections divided into for integrarion shon n; // counter
sections = 5000.; area = 0.; del ta-area=O. ; inceaml = 0.;
// for radius 1 < r < radius 1 + thickness delta-r = thickness/sections; r = radiusl;
for (n=l; ncsections; n++) ( integral = integral + delta-area/r, the ta-outer = acos((r-r-ou ter-radius 1 )/r-ou ter); delta-area = fabs(S.*r-outer*sin(theta-outer)*delta-r); area = area + delta-area; r = r + delta-r; 1
// for radius l+thickness c r < radius l + P r o u t e r - 3*thickness
delta-r = (2.*r_outer-3.*thickness)/sections; r = radius l +thickness;
for (n= i ; ncsecrions; n++) ( integral = integral + delta_area/q if (fabs(-r+radius 1 +r-outer)/(r-outer-thickness) >= 1 .) ( theta-inner = 0.; cheta-outer = 0.; 1 else { theta-outer = acos((-r+radius 1 +r-ou ter)/r-ou ter); theta-inner = acos((-r+radius 1 +r-oucer)/(r-outer-thickness)): 1 ; lengh-ou~er = fabs(r-ou ter*sin(theta-ou ter)); length-inner = fabs((r-outer-thickness)*sin(theta-inner)); delta-area = (lengh-outer-length-inner)*Z.*delta-r. area= xea+ de 1 t a-area; r = r t delta-r; 1
// for radius 1+2*r-outer-thickness < r < radius 1 +?%-outer delta-r = thickness/sections;
// r = radius 1 +Pr-outer-thickness;
for (n=l; ncsections; n++)
in t e p l = integral + delta_area/r; theta-outer = acos((r-r-ou ter-radius 1 )/r-ou ter); delta-area = fabs(2.*r-outer*sin(thetaouter)*delta-r); area = area + delta-area; r = r + delta-r;
double area-tube (double r-outer, double thickness) // r e m s cross sectional area of hollow circle of outer radius // router and thickness { double area; // area of cross section double delta-r; // incremen t in radius for integration double delta-area; // increment of cross section area double r, // radius from center of primary bop double theta-outer; // angle from vertical to point on outer surface double theta-inner; // angle to point on inner surface of cross section double length-outer; // horizontal length from center cross section doubie lenghjnner, // horizontal length to point inner surface
short secnons; // number of sections divided into for intemgration shon n; // coun ter
sections = 5000.; area = 0.; del ta-area=O. ;
delta-r = thickness/sections; r = 0.;
for (n=l; ncsections; nt+) { theta-outer = acos((r-router)/r-outer); delta-area = fabs(2.*-outer*sin(thetaguter)*delta-r); area = area + delta-area; r = r + delta-r; 1
delta-r = (23-ou ter-2.*thickness)/sections;
for (n=l; n<sections: nt+) { if ( f a b s ( - - o u ter)/(r-ou ter-thickness) >= 1 .) { theta-inner = 0.; thetaouter = 0.; 1 else
( thetaouter = acos((-rtr-outer)/r-outer); theta-inner = acos((-r+r_outer)/(r-outer-thickness)); 1 ; lengthouter = fabs(r-outePsin(theta_outer));
delta-area = (lengh-ou ter-length_inner)*2.*delta_r; area= area+ delta-area; r = r + delta-r, 1
delta-r = t hickness/sections;
for (n=l; ncsections; n++) ( theta-ou ter = acos((r-r-outer)/r-outer); delta-area = fabs(2.*r-outePsin(theta-outer)*deIta-r); area = area + delta-area; r = r + delta-r; 1 return(area); 1
Appendix B:
B.1 Orbit Coordinats
The position and orbit of a satellite is defined by six parameten, a, e, i, R, o. and
a tirne variable, usually the tirne of perigee passage. These parameters are known as the
Keplerian orbit elemen ts
Line to perigee
a = semimajor mis
O = Right ascension of ascending node: angle from Aries, which is towards the son at vernal equinox, to the ascending node vector from the eanh cen ter eastward.
i = Inclination angle benveen equatonal and orbit planes, about the ascending node vec tor.
o = Argument of perigee: angle between the vecror from eanh cenier to the penfocus and the ascending node vector, in the orbit plane.
v = True anomaly
e = Eccentricity = d a
c = Distance from orbit center to apogee (funhest point).
The ascending node is the point where the orbit cuts the equatorial plane from
South to North, and the ascending node vector is the line from eanh center to the
ascending node. The semimajor axis and eccenmcity define the shape of the orbit. For a
circular orbit, e = O and o is not defined.