Magnetic propulsion systems

Progress in Aerospace Sciences 37 (2001) 245}261

Magnetic propulsion systems Valentine Pulatov 58 Apartment, 11-A Street Seminarskaya, 65039 Odessa, Ukraine

Abstract Magnetic propulsion systems are based on the direct interaction of the vehicle's own magnetic "eld with the natural magnetic "eld, particularly the geomagnetic one, without using jet propulsion. Three such systems are reviewed in the order of their feasibility of automatic control over the thrust force vector. One of these magnetic propulsion systems permits partial control and is competitive with the electromagnetic or plasma rocket orbital microthrusters. The importance of the other two promising systems is to establish the main principles of magnetic propulsion. Their development depends on progress in solid-state physics. One of them may be able to have total control over the direction and modulus of the electrodynamic thrust force vector.  2001 Elsevier Science Ltd. All rights reserved.

Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Basic principle. . . . . . . . . . . . . . . . . . . . . . . . 1.2. Scope of this paper . . . . . . . . . . . . . . . . . . . . . 2. Simple magnetic propulsion system . . . . . . . . . . . . . . 2.1. Dipole force . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Example of design . . . . . . . . . . . . . . . . . . . . . 2.3. Thrust-to-mass ratio . . . . . . . . . . . . . . . . . . . . 3. Partially controlled system . . . . . . . . . . . . . . . . . . . 3.1. Cross force . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Partial control. . . . . . . . . . . . . . . . . . . . . . . . 3.3. Example of a rational scheme. . . . . . . . . . . . . . . 3.4. Comparison with the electromagnetic microthrusters . 4. Total control . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Electrodynamic force as the result of composition. . . 4.2. Arti"cial stability . . . . . . . . . . . . . . . . . . . . . . 4.3. Total control attainment. . . . . . . . . . . . . . . . . . 5. Summary and conclusions . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction 1.1. Basic principle Magnetic propulsion is based on the force interaction between the magnetic "eld of the #ight vehicle and the external natural magnetic "eld, e.g., the geomagnetic "eld. This external natural magnetic "eld plays the role of the so-called `applied "elda.

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The natural magnetic "elds are greatly extended. There is no near theoretical limit to the value of the interaction force. The greater the desired force of interaction, the farther away the magnetic "eld of the vehicle must be propagated. The propulsor of such a system may be an electromagnet. Its thrust force may be treated as the electrodynamic force of interaction of the electric current inside its conductors with the applied natural

0376-0421/00/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 6 - 0 4 2 1 ( 0 1 ) 0 0 0 0 6 - 9

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Nomenclature A B, B B  B " B "0 B "F B ,B ,B V W X B  e F, F F ,F ,F V W X F " F "0 F "F F ,F N O F  F  F ! H, H H  H # h i, j, k I J ¸ ¸  ¸  M, M M . M  m

area encircled by the turns of coil vector and modulus of the magnetic induction magnetic induction of the applied natural "eld, T dipole component of the magnetic induction of the applied "eld radial (vertical) component of the dipole component of the induction of the applied "eld de"ned by Eq. (9) tangential (horizontal) component of the dipole component of the induction of the applied "eld de"ned by Eq. (10) rectangular components of the magnetic induction vector of the applied "eld, Eqs. (1)}(3) magnetic induction inside the body of the screen charge of an electron, Eq. (20), C vector and modulus of the electrodynamic force rectangular components of the electrodynamic force, Eqs. (1)}(3) electrodynamic force on magnetic dipole (dipole force), N radial or vertical component of the dipole force de"ned by Eq. (17) tangential or horizontal component of the dipole force de"ned by Eq. (18) two centrally symmetric Ampere's forces on current-carrying loop or coil, Fig. 3 Ampere's force on the segment of the currentcarrying loop or coil without the screen Ampere's force on the segment of the currentcarrying loop or coil surrounded by the screen electrodynamic cross force vector and modulus of the magnetic intensity magnetic intensity of the applied natural "eld, A/m magnetic intensity of the distorted applied "eld outside of the screen de"ned by Eq. (22) or (26) height, m, km unit vectors electric current, A current density, A/cm length length of the segment without the screen of the circumferential coil length of the segment of the circumferential coil which is surrounded by the scrren vector and modulus of the magnetic moment magnetic moment of the dipole component of the magnetic "eld of the planet, A/m magnetic moment of the main force coil of the electromagnet of the magnetic propulsion system de"ned by formula (4) mass

N  N ,N   n

C

O o P R r r  r  t
C X, >, Z x, y, z

number of turns of the main force coil of the electromagnet of the magnetic propulsion system number of turns of the coils of the system of the arti"cial angular stabilization number of electrons per unit of volume, Eq. (20), cm\ origin of the spherical coordinate system in the center of the planet dipole origin of the polar coordinate system on the axial circumference of the ring coil arbitrary point of space, Fig. 1 radial distance in the spherical coordinate system, m, km radial distance in the polar or cylindrical coordinate system internal radius of the screen pipe external radius of the screen pipe time velocity of electrons de"ned by expression (20), cm/s axes in the rectangular coordinate system, Fig. 1 coordinates in the rectangular system

Greek letters  angle between the vectors of the magnetic induction and the magnetic moment  angular di!erence between  and  de"ned by Eq. (39)  ratio of the internal and external radii of the screen pipe de"ned by Eq. (24)  angular argument in the spherical coordinate system
angular argument in the polar or cylindrical coordinate system, deg  absolute magnetic permeability, 4;10\ H/m   absolute magnetic permeability of the body of  the screen
ratio of the magnetic permeabilities  and   de"ned by Eq. (25)   constant which depends on the screen design, de"ned by Eq. (23)  relation of the circumferential length to the diameter, 3.141592 magnetic conductance of the surrounding me+ dium de"ned by Eq. (38), H magnetic conductance of the ferromagnetic + screen material de"ned by Eq. (37)

electrodynamic torque on the main force coil  of the electromagnet of the propulsion system

,

electrodynamic torques on the coils of the   system of the arti"cial stabilization

angular argument in the spherical coordinate system I moment of inertia of the vehicle relative to the  axis in the plane of the main force coil

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"eld. Such a propulsion system can be described as `an electrodynamic systema [1]. The physical law of the electrodynamic interaction of the electric current with the applied magnetic "eld forms the basis for the force exerted on the #ight vehicle similar to the other physical laws which apply to other aircraft and aerospace vehicles. The values of the magnetic intensity or induction of the natural magnetic "elds are usually relatively constant in time and space. Consequently, the coil of the electromagnet of the magnetic propulsion system must be energized with direct electric current in order to obtain a continuous constant electrodynamic thrust force. In the process of the motion of the #ight vehicle with this propulsion system, the electric energy in the coil of the electromagnet of the propulsion system is evidently converted into the mechanical energy of the vehicle by direct action. Therefore one can forecast high economical operation and ecological purity for such a vehicle. Inasmuch as the applied magnetic "eld has no mass at rest, it cannot be considered as the support medium in the physical sense. The mechanical force of reaction of the magnetic propulsion system must be transmitted by the applied natural "eld, as some intermediary, to the physical body which has this magnetic "eld. In this case it is the body of the planet. Consequently, in the sense of thrust generation, the magnetic propulsion system could be similar to the means of air transport and territorial electric transport. For example, when in near-to-earth space, the magnetic propulsion system is `repellinga from the planet's body or it is `leaninga on this body through the interaction of the magnetic "eld of the electromagnet of the propulsion system with the geomagnetic "eld. Inasmuch as the mechanical force of reaction is a!orded by the support medium (the planet body), one avoids the complexity of a propulsion system with its own mass. Hence, a #ight vehicle with such a magnetic propulsion system is an object which retains constant mass and in this respect, it is distinct from a lot of other systems. The magnetic propulsion system di!ers fundamentally, for example, from the electromagnetic propulsors of the PPD and MPD types [2] which are categories of rocket engines. The qualities of the system appear attractive but its development and application will require the solution of some di$culties and shortcomings. 1.2. Scope of this paper The present state of modern technology allows the use of magnetic propulsion systems as orbital microthrusters in the near-to-earth space. The current thrust-to mass ratios of these systems are relatively low. Nevertheless, since they do not require propulsive mass, the vehicle mass is constant. Moreover, the thrust can be continuous with low electric consumption.

247

Sections 2}4 describe three possible magnetic propulsion systems. In each case, the physical principle of the action is "rst explained and proved; then, the thrust-tomass ratio is estimated, possible applications are considered and some relevant engineering proposals are mentioned. The "rst propulsion system uses only the dipole electrodynamic force. It is entitled `Simplea because there is no attempt to improve the design of the electromagnet of the propulsion system. It is impossible to change the direction of the thrust force vector in any point of space but its absolute value may be controlled. The United States' Patent of Engelberger [3] is an example of such a system and this is considered in detail below. The second system does not use the dipole force which may be neglected in this case. Rather, it uses the electrodynamic `cross forcea [4]. Its vector may have an arbitrary direction but in only one plane which is perpendicular to the direction of the applied natural "eld. Therefore, such a system is called `Partially Controlleda. Its thrust-to-mass ratio allows such a system to be used for continuous propulsion with low thrust in low-altitude equatorial orbits and to compete with microthrusters. Finally, Section 4 is devoted to `Total Controla. Its action is proved and explained in detail. This system allows arbitrary changes in the direction and modulus of the thrust electrodynamic force vector and so, we have reached the same possibilities for control as for other #ight vehicles. In the `Summary and Conclusionsa the promising possibilities of current propulsion systems are estimated. The paper is mostly concerned with the theoretical possibilities; there are many engineering proposals but it is not possible yet to assess which will ultimately emerge as the best because this will depend on the progress (rapid at present) in solid-state physics. The numerical examples quoted allow the propositions to be veri"ed and other engineering variants to be validated. Results are quoted mostly in SI units. 2. Simple magnetic propulsion system The electromagnet of such a magnetic propulsion system is the simplest that could be envisaged: it consists of only a current-carrying loop or coil. The thrust force in this case is provided simply by the dipole electrodynamic force. The formulation is easy to interpret and readily suitable for engineering calculations. 2.1. Dipole force In the rectangular coordinate system the components of the vector of the electrodynamic force F are B B B W #M X, V #M F "M W x X x V V x

(1)

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V. Pulatov / Progress in Aerospace Sciences 37 (2001) 245}261

B B B V #M W #M X, F "M W V y W y X y

(2)

B B B V #M W #M X, F "M X V z W z X z

(3)

in which B /x, etc. are partial derivatives of the rectanV gular components B , B , B of the vector of the magnetic V W X induction B of the applied "eld with respect to the axis  X, >, Z;M , M , M are also the rectangular compoV W X nents of the vector of the magnetic moment M of the  propulsion system. For example, if the coil of the electromagnet of the propulsion system has a number of turns N and a surrounded area A, then  M "N I A (4)    with the de"nition of its vector direction according to the right hand rule with respect to the direction of the electric current I [5].  In the case of the planetary magnetic applied "eld the spherical system of coordinates is convenient (Fig. 1). Then the origin O of the coordinate system is in the center of the planet magnetic dipole and the variable di!erentials are expressed in terms of latitude and longitudinal angles  and . They will be R, R, and R sin  where R"OP (Fig. 1). And then formulae (1)}(3) for the components of vector F in any point P of space will be transformed into B B B F #M (, 0 #M F "M F R ( R 0 0 R

(5)

B B B F #M (, 0 #M F "M F 0 R F R ( R

(6)

B B B 0 #M F #M ( . F "M ( 0 R sin 

F R sin 

( R sin 

(7) Here the vectors F, B and M are decomposed on   three mutually orthogonal directions, which are de"ned by the unit vectors i, j, k (Fig. 1). These vectors are also mutually orthogonal and therefore this coordinate system is orthogonal curvilinear. Each vector, for example, the electrodynamic force vector F"iF #jF #kF . (8) F ( 0 The planetary magnetic "elds are mainly of the dipole structure [6}8]. Therefore the problem may be treated as the interaction of the two dipoles: those of the planet and of the vehicle. Let us take into consideration only one component of the planetary magnetic induction B , namely its dipole  component B . The error of such a substitution consti" tutes about several percentage units and rises in the regions, where non-dipole component is high, as one

Fig. 1. The coordinate system.

could see on the maps of the geomagnetic "eld [7]. The resulting accuracy will be quite enough to obtain the necessary estimation of the numerical values of the planetary magnetic induction and electrodynamic force. The structure of the magnetic dipole "eld is axially symmetrical. It does not contain the longitudinal component of the magnetic induction, that is B "0 [9]. ( Then in the spherical system of coordinates the other two components [9] of the induction of the planetary dipole will be B " M (4R)\ 2 cos , (9) "0  . B " M (4R)\ sin , (10) "F  . in which  "4;10\ H/m is the absolute magnetic  permeability, M is the magnetic moment of the planet . dipole, which is 8.2;10 A m for the Earth [6], R is the distance from the center O of the planet magnetic dipole to the chosen point P. For example, if the height h above the terrestrial magnetic equator ("/2) is 250 km and equatorial radius of the terrestrial surface is R "6,378,245 m, then the distance R"6,628,245 m  and B "0, B "28;10\ T. "0 "F More correct results may be obtained, for example, by counting the location of the center and axis of the geomagnetic dipole with respect to geographic spherical coordinates [6]. The electrodynamic force may be calculated with the help of Eqs. (5)}(7) in any angular position of the vector of the vehicle magnetic moment M . But this dipole at  the same time is under the action of the electrodynamic torque

"M B sin , (11)    where  is the angle between vectors M and B [5]. For   this reason the coil of the electromagnet of the propulsion

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system may take up only one stable angular position with "0 when the vectors M and B coincide in their   direction. (In passing, it may be noted that this phenomenon is sometimes used for the orientation and oscillation damping of orbital vehicles.) Owing to this coincidence the relation of the components of the vector of the vehicle's magnetic moment M is the same as in (9), (10), that is  M /M "2 cos / sin  (12) 0 F and consequently M

0 and

"M (3 cos #1)\ 2 cos  

(13)

M "M (3 cos #1)\ sin . (14) F  In considering only the dipole component of the planetary magnetic "eld the expressions (5)}(7) may be simpli"ed. The third terms in the right hand side are equal to zero because of B "0. The "eld structure is axially ( symmetric and may be described by its form in only one surface which contains the vector of the magnetic moment M [9]. Expression (7) also disappears. As a result it  may be considered that the components of the dipole force B B "0 #M "F , F "M "0 0 R F R

(15)

B B "F , "0 #M F "M "F 0 R F R

(16)

where B and B are correspondingly the radial (verti"0 "F cal) and tangential (horizontal) components of the dipole component B of the planetary magnetic induction. " Substitution of expressions (9), (10) and (13), (14) into (15), (16) gives important formulae for the numerical estimations of the dipole electrodynamic force components F "!3 M M (4R)\(3 cos #1), (17) "0  .  F "!3 M M (8R)\(3 cos #1)\ sin 2. "F    (18) One can see that the dipole electrodynamic force decreases with height proportionally to the fourth power of the intercenter distance R. It is more rapid than decreasing to the second power of the gravitational force. So the low-altitude orbits are more e$cient. It is obvious also that the sign of the radial (vertical) component F of the dipole force does not depend on "0 the latitude angle . The vehicle's dipole is in a sole stable position always and it is attracted to the planet's dipole everywhere. At the same time the sign of the tangential (horizontal) component F of the dipole force is nega"F tive in the quadrants with odd numbers (see Fig. 2) and

Fig. 2. Examples of the composition of the components of the dipole force.

positive in the quadrants with even numbers, that is, it is an alternating one. The ratio of the components of the dipole force F /F " sin 2/2(3 cos #1) "F "0

(19)

and it does not depend on the vehicle's magnetic moment M . Consequently, the change in the value of M from   zero to the maximum only allows proportional control of the absolute value of this electrodynamic force. But its direction is constant at any point in space and depends only on the latitude angle . This ratio has the maximum and it is equal to 1/4 when the latitude angle "63.433, 116.573,2 . Typical situations of the direction and the absolute value of the dipole electrodynamic force of attraction of the planet's and the vehicle's dipoles in the "rst quadrant are illustrated in Fig. 2. The plane representation is reasonable because of the axial symmetry of the dipole magnetic "eld structure with respect to the axis Oz. The disposition of the vector M of the planet's geomagnetic . moment is such that the north geographic pole is at the top of the "gure together with the north geomagnetic pole (south magnetic pole). The numbers of quadrants are denoted by Roman numerals in Fig. 2. The radial and tangential components and resultant are illustrated at selected angles with a spacing of 153 in the same scale. In the second quadrant the disposition of

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V. Pulatov / Progress in Aerospace Sciences 37 (2001) 245}261

the components is symmetric to the "rst one with respect to the horizontal axis OX. In the fourth and third quadrants the disposition of the components is symmetric corresponding to that of the "rst and second quadrants with respect to the vertical axis OZ. 2.2. Example of design The United States' patent `Space Propulsion Systema of Engelberger [3] is based on using only a dipole electrodynamic force for the discontinuous thrust of an orbital vehicle. This patent surpasses earlier propositions, particularly from Cutler, Cherry, Hulliger and Engelberger himself as also the patents of the United States. At the same time it demonstrates possibilities of the simplest physical principle which does not contain any improvements just in the electrical equipment of the electromagnet of the propulsion system. As it was shown in the previous subsection, there is no longitudinal component of the dipole force, that is F "0. Therefore Engelberger considers his invention ( only suitable for use in polar orbits. Besides it is clear from the explanations of Fig. 2 and in the text that the quadrants, where the horizontal component F of the dipole force coincides with the direc"F tion of the orbital motion, are alternating with the quadrants, where this component is opposite to that direction. According to the Engelberger's proposition, to obtain some horizontal thrust, the electric current in the coil of the electromagnet of the propulsion system must be switched o! while the motion is taking place in the unfavorable quadrants and again must be switched on during the motion in the favorable quadrants. So it is evident that the horizontal thrust is discontinuous. In this connection the inventor had made an important proposition to equip the electromagnetic propulsion system with a `charge storage devicea*the electric capacitor. The main part of the speci"cation of his invention is devoted to the process of energy exchange between the winding of the coil of the electromagnet and the capacitor in the moments of switching on and o!. According to (11), the instability of the angular position with " theoretically follows from the negative sign of the derivative d /d in this position, while this sign is positive in the sole stable position with "0. In other words, the smallest magnet tends to rotate in the stable position to be attracted by the biggest one. But Engelberger places the current loop into the second angular position, which is unstable, with the electrodynamic moment equal to zero. He is also con"dent that this position may be continuously maintained by itself. Unfortunately, this is erroneous. For a continuous operation in the unstable zero angular position the electromagnet of the propulsion system must contain the special additional system of artixcial stabilization which is described in Section 4.2 of this article.

Although the geomagnetic "eld is a non-uniform one, this non-uniformity is very slight and therefore its derivatives in (1)}(3) or in (5)}(7) are relatively small. Consequently, the maximum of the possible value of the magnetic moment M is necessary to obtain a suitable  value of the dipole force F . In compliance with (4) the " values of the electric current I and the area A must be as  large as possible. Engelberger was the "rst to suggest that superconductivity should be used to obtain large values of the electric current for the orbital magnetic propulsion. For achieving a large value of the area A the inventor had suggested implementation by making the electromagnet of the propulsion system a kind of #exible superconducting cable. It must be delivered to the orbit in a compact stowed form. After orbiting this `cablea must be uncoiled and energized. It remains connected with the body of the spacecraft, where the energy source, cooling system, etc., are arranged. Then under the condition of weightlessness, the superconducting current-carrying loop must assume a circumferential form under the acting ponderomotive forces. Of course, it must take up the sole stable zero position at the same time. In Engelberger's numerical example the electric current I "4000 A, radius of the superconducting current  carrying loop 10,000 m, the height h"200,000 m (the corresponding radial distance for the equal-size sphere R"6,571,110 m). Consequently, the vehicle's magnetic moment from (4) M "1.257;10 A m. For the max imum ratio (19) the latitude angle "63.433 and then in accordance with (18) F "!5.245 N, in accordance "F with (17) F "!20.98 N. "0 This numerical result is obtained at the expense of the large size of the propulsion system and without taking into account the mass of the on-board equipment especially of the cryogenic one. Only the mass of the superconductor without the material of its matrix was taken into account. In the next subsection of this article an attempt is made to correct this de"ciency and to revise the estimation of the thrust-to-mass ratio. This would need considerable e!ort from a project group. 2.3. Thrust-to-mass ratio Let us follow Engelberger's idea that the electromagnet of the propulsion system consists of a superconducting ring coil without a core and it is delivered into the orbit in a stowed compact form. The estimation of the mass of the magnetic propulsion system would take into account the mass of the superconductor, the cryogenic system and the energy source. While attempting an approximate outline of such a design, one could see that a lot of the necessary data are absent in the literature or are antiquated. In that case the result obtained may be taken into consideration as a low limit or a demonstration of the order of magnitude only.

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Table 1 Estimation of the sizes of the layered structure of the refrigerated superconducting coil of the electromagnet for its total mass determination Layer

Superconductor Nb Sn  Matrix Al Insulation 10% mass Gaseous helium at 10}12 K and 10 Pa Distance bars, 10%

Diameter (mm) Intern. Extern.

0.0

14.0

Thickness (mm)

Cross-selectional area (cm)

Volume at the length, 100 m (m)

Mass density Mass (kg/m) (kg)

0.40

4.0;10\

8120

32.5

14.70

0.40

4.0;10\

2790

11.2 4.4

5.00 2.00

0.24

2.400

0.6

0.27

2.40

2.4;10\

1100

26.4

12.00

0.48 9.20 0.55

4.8;10\ 9.2;10\ 5.5;10\

2790 40.000 2790

13.4 3.7 15.4

6.10 1.70 7.00

5.70

5.7;10\

804.300

45.8

21.00

0.57

5.7;10\

1100

6.3

2.80

7.00

24.00 14.0

60.0

23.00

Pipe for helium 60.0 Thermal insulation 60.5 Pipe over insulation 69.5

60.5 69.5 70.0

0.25 4.50 0.25

Liquid nitrogen Distance bars, 10%

75.0

2.50

70.0

Pipe for nitrogen 75.0 Thermal insulation 75.5 Covering re#ecting pipe 109.5

Mass percentage (%)

75.5 109.5

0.25 17.00

0.59 49.40

5.9;10\ 0.494

2790 40.000

16.5 19.8

7.50 9.00

110.0

0.25

0.86

8.6;10\

2790

24.0

11.00

220.0

100.00

Total amount

The possible application in near space demands the minimum of energy consumption and mass. Special demands are also made on the reliability and life expectancy. The concept of maintainability loses its sense. The design of the magnetic propulsion system must take into account the achievements of the applied superconductivity. The multi-turn ring coil of the electromagnet of the propulsion system has an advantage over the uni-turn current-carrying loop. Then the lead-in wires must be accounted for when the current is less than the total ampere-turns of the coil; their overall dimensions could be less and they demand less energy expenditure for the refrigeration. The load matching with the power supply source could be simpler, the inductance is higher and the transient processes could be turned out more slowly. At the same time correct knowledge of the number of turns is not needed for "nding the total quantity of the wire. The total cross-sectional area of the superconductor with the matrix may be found according to the total number of ampere-turns. The estimation of the thrust-to-mass ratio here is based on the data of experimental superconducting cable

[10], assuming the ratio of the cross-sectional areas of the superconductor and matrix of 1 : 1 [11,12]. The superconducting coil with its current 40 kA has a design which is similar to this cable and it has a mass of 2,2 kg/m per unit length (Table 1). The multi-layer thermal insulation is assumed to be composed of aluminum foil and of the glass paper with the carbon "laments [13]. The total mass of the nitrogen and helium refrigerators [14,15] with the e$ciency and power consumption [15}17], which may be energized with the solar cells [18,19], may be estimated at about 375 kg. Then according to (4) magnetic moment M "31,831,000 A.m, the horizontal  component of the dipole force for the height of 250 km and the latitude angle 63,433 will be 4;10\ N from formula (18). Thrust-to-mass ratio consequently will be 1.1;10\ N/kg (or m/s). This estimation was made with data for the usual practical superconducting and cryogenic materials and technologies. The use of light materials for the purpose of the aerospace would give better results. The main reserve of increasing of the dipole force is connected with the low current density of the modern

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superconducting materials. Their concentration of superconducting electrons, for example in Nb Sn,  n "10 cm\ and it is about 1/100,000 of the number  of all the free current electrons. On the other hand, the average speed of electrons drift is
"J/en , (20)   where J"5;10 A/cm is the current density and e"1.6;10\ C is the charge of one electron. Then
"3.12;10 cm/s. The last value is approximately  1/1000 from the speed of the electron motion around the nucleus [9]. Therefore the total possible physical reserves of the current density in the superconductivity are large enough. The other branches of solid-state physics as, for example, the laser engineering or power semiconductor electronics have demonstrated a growth of several parameters a billion times during 15}20 years. So the dipole force used for magnetic propulsion is signi"cant. Besides, it is important for the total control over the resulting vector of the electrodynamic force (Section 4).

3. Partially controlled system The cause of the small value of the dipole force is the initial small value of the derivatives in (5)}(7). Compared with this the Ampere's force exerting on each meter of the superconductor with a current of 40,000 A at a height of 250 km is, for example, F "B I ) 1"28;10\ ) 4;10"1.12 N. (21)  "  It is 11,000 times more than the horizontal component of the dipole force exerted on the whole ring coil of the length of the axial circumference of 100 m. This fact is the basis of the following attempt to use the electrodynamic force for the magnetic propulsion which is described in this section. 3.1. Cross force Suppose the applied magnetic "eld is a uniform structure. Coil 1 of the circumferential form (Fig. 3) with the current I takes up the sole stable position with zero  electrodynamic torque when its plane is perpendicular to the applied "eld magnetic induction vector B . Two  arbitrary equal and symmetric about the center o segments are exerted by the equal and opposite Ampere's forces F and F because of the same induction, current N O and length values (Fig. 3). These forces are mutually balanced. The equality to zero of the summed electrodynamic force overall the circumference is evident and it follows also from (1)}(3) or (5)}(7) as the derivatives of the magnetic induction of such an applied "eld are equal to zero.

Fig. 3. The cross force arising.

Let some interference be introduced to change the interaction of some segment of the current-carrying loop with the uniform applied "eld. For example, some screen 2 of length ¸ (Fig. 3) covers this segment of the loop or  coil and a!ords decreased or increased interaction [4]. Then the equality and balance of the forces F and  F will be violated and the resulting di!erence of forces  F "F !F will be exerted. This electrodynamic force !   will not be of the dipole nature because of the assumption about the uniformity of the applied "eld. Its physical meaning is the di!erence in force between two opposing Ampere's forces. It is called `cross forcea because of its perpendicularity to the magnetic induction vector of the applied "eld in the sole stable angular position. Thus new possibilities may be opened. On one hand such an electrodynamic force could not depend on the derivatives of the magnetic induction of the applied natural "eld. Therefore it may not be extremely small. So a new opportunity of practical application could appear. It is considered in Section 3. On the other hand the principle of the total control of the electrodynamic force vector also becomes possible. It is considered in the concluding Section 4. An example of violating the balance of forces F and  F is the ferromagnetic `screeninga. The inverted com mas mean that the e!ect is di!erent from the usual screening which moderates the internal "eld. Moreover, it is the opposite one. In this case the uniform applied external "eld, which is imaged by the "eld lines (Fig. 4), undergoes some distortion near the ferromagnetic screen due to the greater quantity of its magnetic conductance in comparison with the magnetic conductance of the surrounding medium (see the numerical example for the expressions (37) and (38) in Section 3.3). The cross section of this screen in the form of a pipe is denoted by the circumference 2 in Fig. 4 and the cross section of the screened internal conductor with the current I is de noted by the internal circumference 1. The description of the external locally distorted applied "eld outside of the ferromagnetic screen [20] may

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Fig. 4. Schematic cross-section of the ferromagnetic screen 2 within the distorted applied "eld.

be represented in the form H "!H (1#r /r) cos
) k #   #H (1!r /r) sin
) i, (22)   where H is the vector of magnetic intensity of the dis# torted applied "eld consisting of two components: radial and tangential; radius r and angle
are the polar (cylindrical) coordinates with the origin o; i and k are the unit vectors; H is the magnetic intensity of the non-distorted  applied "eld; r is the external radius of the electromag netic screening pipe (Fig. 5);  is constant "[(1!)(1!
)][(1#
)!(1!
)]\

(23)

in which the ratio of the internal and external radii (Fig. 5) of the screening ferromagnetic pipe "r /r (24)   and the ratio of the magnetic permeabilities of the surrounding media and of the material with a ferromagnetic property of
" / . (25)   It is notable that the extent of distortion in (22) is inversely proportional to the second power of the distance and therefore it decreases more slowly than the magnetic induction of the dipole "eld in (9), (10). Further, the natural applied "eld is represented as the quasi-uniform one due to its very small non-uniformity because of low values of the derivatives of the magnetic induction in (1)}(3) or (5)}(7). Then expression (22) may be used for the computation and graphical construction of the corresponding "eld structure.

Fig. 5. Schematic cross-section of the pipe of the ferromagnetic screen.

Accurate evaluation of the cross force quantity through the computation of the resulting "eld structure, which shows the interaction of the applied distorted "eld and the circumferential "eld of the screened current I ,  may be obtained by the integration of the elementary electrodynamic forces over the entire boundary in the resulting "eld between the distorted external "eld and internal "eld of the current-carrying conductor itself. Authenticity of the solution may be veri"ed by the derivation of the formula of Ampere similarly. We shall use a simpli"ed method for the preliminary estimations of the value of the cross force of the described nature instead of the accurate treatment. First, the magnetic intensity of the distorted applied "eld is de"ned only in the plane, where the Ampere's force exerts, that is
"/2, and here H "H (1!r /r). #  

(26)

Second, the distances to the zero branch point of the resulting "eld structure are considered: r "N I /2H #   #

(27)

and r "N I /2H ,    

(28)

where N I are the ampere turns of the current-carrying   ring coil in Fig. 3, H and H are the magnetic intensities #  after and before the distortion.

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It follows from (26)}(28) that r !r r !r "0 #  #  and consequently

(29)

r "r /2#[(r /2)#r ]. (30) #    It may be seen that r 'r and the explanation of #  F 'F (Fig. 3) may be as follows:   The "eld of the `screeneda current propagates in the moderated distorted "eld on the greater distance r than # r of the non-screened one. Such an increasing propaga tion is usually the result of the proportional current increasing is may be seen from (27), (28). Besides, the boundary in the resulting "eld between the distorted external applied "eld and the internal "eld of the current is longer without a screen and the interaction takes place with the more distant lines of force. Therefore we shall consider that F /F "r /r "#[#(r /r )].      #   Supposing the Ampere's force is

(31)

F "B N I ¸ , (32) '     where B is the magnetic induction of the applied natural  "eld, and from (31) F " #[#(r /r )] B N I ¸ , (33)          we shall obtain the next expression for the desired crossforce estimation: F " [#(r /r )]! B N I ¸ . (34) !         The discrepancy of the calculations according to (34) with the data of the experiments constituted from 2% to 30% for the thin-walled ferromagnetic screens which are just suitable for the purpose of #ying. However, magnetic saturation of the screen material should be avoided as this would destroy the necessary distortion of the applied "eld. Therefore the radius of the screening pipe must be large enough in the case of the screening of the currents of large values. But it should be remembered that the centrally symmetric non-screened segment of the ring coil can also in#uence the magnetic saturation of the screen. The condition of the non-saturated state of the material of the screen demands `the low-currenta coil of the electromagnet of the propulsion system. It obstructs the use of `the high-currenta superconducting coil. The numerical example follows in Section 3.3. 3.2. Partial control It is clear from the consideration of Fig. 3 that screen 2 may be positioned in any place of the ring currentcarrying coil. The cross force F , which coincides with ! the force F in direction, may have any chosen arbitrary  angular direction in the plane which is perpendicular to

the magnetic induction vector B of the applied natural  "eld under the condition of the sole stable position with zero electrodynamic torque. At the same time the possible rotation of this ring coil of the electromagnet of the propulsion system around the central axis turns out to be possible while remaining in the sole stable plane. Hence there is axial instability. Both these circumstances demand automatic control for "xing any angle of orientation. This control may be called `partiala since the cross-force vector may have any direction but only in the stable plane, that is it remains perpendicular to the vector of the magnetic induction of the applied natural "eld. For the purpose of such control, the pipe of the screen may be designed as several cylindrical or thoroidal sections along the circumference. Their `screeninga action could be controlled by the di!erent magnetic biasing of each section. Thus the rotation is controlled and so any angle of turning may be arbitrarily chosen and "xed. There is another kind of sectionalization. The sections of the screen may also be designed either as separate parts with intervals or the parts may have di!erent thickness or magnetic biasing, etc., bounded along the generating line of the cylinder. Such a ferromagnetic screen con"guration is similar to a row of rectangles. The resulting asymmetric distortion of the applied natural "eld relative to the coil's plane could nevertheless provide some angular de#ection of the cross-force vector from this plane. So the partial control may be improved. But additional investigations are needed to quantify this e!ect. The choice of direction of the cross force vector may be made also in the same manner as in other #ight vehicles, i.e., by turning the body of the #ight vehicle around the axis which is parallel to the applied "eld direction. 3.3. Example of a rational scheme An important proposition about a rational design scheme for obtaining the maximal thrust may be made from the analysis of the semi-empirical formula (34). It may be represented taking into account (28) and (32) as F "[(F /2)#(2r ¸ B H )]!F /2, !      

(35)

in which F is the Ampere's force (32), r and ¸ are    correspondingly the external radius and the length of the pipe of the ferromagnetic screen, B and H are the   induction and the intensity of the applied magnetic "eld, and  is the constant determined according to (23). The graphic representation may be suggested as the triangle of forces (Fig. 6a) similar to the well-known triangles of resistances, currents, powers, etc. in electrical engineering. The analogous triangle of the ampere turns (Fig. 6b) may be obtained after the representation of

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Fig. 6. The triangles of forces (a) and of ampere-turns (b).

formula (34) with the addition of (28) as F "B ¸ [(N I /2)#(2r H )]!N I /2, !         (36) where N I are the ampere turns of the screened coil. It   may be seen that the growth of the cross force F de! mands decrease in the Ampere's force F and increase in  the second term of the expression between the rectangular brackets in (35) and (36). The last term in (35) is evidently connected with the magnetic pressure on the surface of the ferromagnetic screen from the side of the applied "eld because it contains the product of quantities of the magnetic intensity H and induction B . It de  pends mainly on the sizes of the ferromagnetic screen: the external radius r of its pipe and its length ¸ .   One can understand that the growth of the radius r on the one hand and the decrease in the ampere-turns  N I on the other hand result in lower values of the   magnetic intensity of the material of the ferromagnetic screen. The corresponding growth of its magnetic permeability helps to better in#uence the screen. But it is evident also that increase in the sizes r and ¸ must take   place without increasing the mass of the #ight vehicle. Consequently the thickness of the screen's wall must be minimized to the thickness either of the foil or the "lm. We shall continue the supposition that the external radius of the ferromagnetic screen r +r from (28). For   the numerical example the coil of the electromagnet of the propulsion system is supposed to be in the form of a rectangle 3;6 m (Fig. 7). One of these sides is surrounded by the cyclindrical screen of electrical sheet steel with external radius r "3 m, length ¸ "3 m and  

thickness  "5;10\ m. The high electric conductance  of the steel allows the use of the body of the screen instead of the screened part of the current-carrying loop or coil (it is shown by vertical dotted line in Fig. 7) that may complicate the design. The magnetic propulsor is schematically shown in Fig. 7 in the angular position which a!ords a continuous horizontal direction of the electrodynamic thrust on the equatorial orbit. The plane surrounded by the turns of the coil occupies the sole stable zero angular position which is perpendicular to the applied geomagnetic "eld direction. Maintaining this or any other chosen direction of the electrodynamic thrust force may be either owing to the aerodynamic stabilization due to the small air resistance or may be organized in accordance with the proposals in Section 3.2. The magnetic conductance of the two halves of the cyclindrical shell of the screen is "2  ¸ (r )\, +    

(37)

where  is the absolute magnetic permeability of the  material of the screen,  is its thickness, ¸ is its length,   and r is its internal radius. In the case of its  H "20 A/m, B "0.7 T,  "3.5;10\ H/m (electri   cal-sheet steel `380a) and "1;10\ H. The mag+ netic conductance of the same volume of space without the screen is " ¸ , +  

(38)

where  is the absolute magnetic permeability of the  surrounding medium. Here "3.77;10\ H, that is + 26.5 times less than that of the two halves of the screen.

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3.4. Comparison with the electromagnetic microthrusters

Fig. 7. Example of the coil 1 with the cylindrical ferromagnetic screen 2.

Therefore the distortion of the applied magnetic "eld near such a screen may be expected to be quite e!ective. There are technological di$culties because of the shelltype construction of this screen. But such a design is characteristic of magnetic propulsors in general. For the magnetic intensity in the body of the screen H "20 A/m and its external radius r "3 m and the   length ¸ "3 m we have the ampere-turns of the coil  N I "377 A. Therefore in that case the superconduc  tivity seems to be unnecessary. For the magnetic induction in the body of the screen B "0.7 T it will be from  (23) to (25) "8.5;10\. At an altitude of 250 km, where the magnetic induction of the applied "eld B "28;10\ T (see the example in Section 2.1), the " cross force in accordance with (34) is F "6.2;10\ N. ! The result is about 20% of the Ampere's force value F "3.17;10\ N obtained on the basis of (32). The  horizontal component of the dipole force in this numerical example according to the area 3;6 m will be 8.6;10\ N in compliance with (18), that is about 70,000 times less than the cross-force computed. Therefore it may be neglected in the case of the magnetic propulsion by the cross-force only according to the described variant. The non-linearity of the ferromagnetic material of the screen was neglected in this simple estimate. But it a!ects the result of summation of the magnetic #uxes of the applied "eld and of the screened current in the body of the thin-walled pipe of the screen because of their approximate equality under the condition of nonsaturation. In a more exact computation, this must be taken into account. The determining factor of the described method of obtaining the cross-force is the state of the magnetic saturation of the ferromagnetic screen. Any decrease in magnetic permeability by increasing the screened current leads to a lower e!ectiveness of the screen or demands an increase of its mass. So the current value is limited. The need to use superconductivity is essential for the dipole force method but in the case of the cross-force obtained by the method described the advantage of the superconductivity needs to be "rst justi"ed.

In the case of using the usual conduction the crosssectional area of the wire is determined by mass and energy loss. The smaller the cross-sectional area the lower is the mass of the conductor but the energy loss is greater because of the growth of ohmic resistance. We choose the value of the current density of 2 A/mm. Then for the ampere-turns value of 377 A the crosssectional area will be 187.5 mm. Supposing the length of one turn of the coil is (Fig. 7) 18 m for simplicity, we have the mass of aluminum wire 9.1 kg and the dissipated power 370 W. The ferromagnetic screen of the foil of the electrical steel of the described sizes will be of 22.5 kg. The mass of the solar cell [18,19] is estimated as 18.5 kg. The total mass is thus 50 kg. This value is brought up to 100 kg when allowance is made for insulation, wiring, automatic equipment, accumulator, details of construction, etc. Then for the cross-force value of 6.2;10\ N at a height of 250 km the thrust-to-mass ratio is 6;10\ N/kg. For comparison, the MPD Propulsor [2] has a thrust-to-mass ratio of 13 times more. But it operates in the pulse mode and it consumes 6 g/s of argon. The described magnetic propulsor does not expend a propulsive mass, hence its thrust may be continuous and long term. Consequently the total propulsive mass capacity for the entire period of operation of the MPD Propulsor must be taken into account for the comparison of the thrust-to-mass ratio. Then such a total thrust-to-mass ratio turns out to be also 6;10\ when the on-board capacity of argon is enough for 60 h of continuous operation. In other words, the engine total impulse of the magnetic propulsion system may be compared favorably with a reactive microthruster of any type owing to the longterm period of operation without the consumption of the propulsive mass. The ground test of the magnetic propulsion system is simple because of measurement of the electrodynamic force with one of the well-known methods, for example, with a torsion balance. The modulus and the direction of the geomagnetic inductance vector must be previously veri"ed experimentally [21] because of the distortion of this "eld inside a building and the de"cient detail of the maps of geomagnetic "eld. But applying only the cross force has one great disadvantage. Similar to the application of the dipole force, as the thrust force is for the polar orbits only (Section 2.2), the cross force may be used for equatorial orbits only because it acts in the perpendicular direction to the vector of the magnetic induction of the applied "eld. To obtain the possibility of the choice of any arbitrary orbit the total control over the electrodynamic force is necessary. This is described in Section 4.

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4. Total control Any arbitrary orbit-inclination angle may be chosen when the electrodynamic thrust force vector takes on also any arbitrary direction in any point of space. Then the direction of the electrodynamic thrust force vector could be maintained throughout the chosen orbit by the automatic control. A total control over this vector is also necessary to compete with existing microthrusters which have no such problem. The solution of this problem is not evident because of the uniqueness of the direction both of the magnetic induction vector of the applied "eld and of its gradient in any point of space. An easy-to-interpret principle of such a total control is described below. 4.1. Electrodynamic force as the result of composition The principle of the total control [22] may be based on the summation of two vector components: the dipole force and the cross force. The dipole force F (Fig. 8) acts on the current-carry" ing coil (it is not shown in "gure), which is in the only stable zero angular position. This dipole force is also in the sole direction in a given point of space. The modulus of the dipole force vector is directly proportional to the value of the current in the coil. It may take on any arbitrary value from zero to a maximum. It plays the role of one component of the resulting vector F of the electrodynamic force. This dipole component may be found by calculation from formulae (17) and (18). The cross force F plays the role of the second com! ponent. The last one may take on any arbitrary direction in the plane of the coil. Six forces of several directions are shown in Fig. 8. Any chosen direction and the necessary modulus of the cross force may be obtained, for example, through the dc magnetic biasing of the sectionalized ferromagnetic screen. The cross component may be found by calculation using formula (35) or (36). The example of the geometrical summation of two components F and F to obtain the resulting electrodynamic " ! force vector F is shown in Fig. 8. It is evident that the resulting vector F may take on any arbitrary direction and modulus value through the control over the modulus value of the dipole force F and over the direction and " modulus value of the cross force F . ! The numerical example in Section 2.3 considers the superconducting coil of the diameter of the axial circumference of 31.8 m with the ampere-turns of 40,000 A. We shall consider that all these ampere-turns are enclosed by the ferromagnetic cyclindrical pipe of a diameter of 30 m, with a length of 5 m and with a thickness of 3;10\ m. Then the magnetic intensity inside the body of this screen will be 424.4 A/m, the magnetic induction of the electrical steel will be 1.56 T, constant from (23) "9.5;10\ and consequently the cross force F "4.5;10\ N. The !

Fig. 8. Resulting electrodynamic force F as a vector sum of the dipole Component F and the cross component F in the sole " ! stable angular position.

last one is commensurable with the dipole component F "10;10\ N in this case. This numerical example is " designed for corroborating the possibility of obtaining a result by the composition of the vectors. Its aim is to demonstrate the physical and engineering reality of the concept. But the described methods of creation of the dipole and cross components turned out to be in contradiction because of the low e$ciency of the ferromagnetic screen under the condition of magnetic saturation and its high mass. This is not an inherent weakness of such a method because of the possibility of other engineering realizations. Besides such a supposed control provides a position of the resulting vector F in the lower hemisphere only. This arises from the fact that two dipoles attract. The upper hemisphere may be operative if the direction of the dipole force vector has reversed. It could correspond to the repulsion of two dipoles and it is considered in Sections 4.2 and 4.3. 4.2. Artixcial stability The electrodynamic torque, which acts on the currentcarrying coil, according to (11) always positions this coil in the sole stable zero angular position with "0 and with the same direction of the vectors M and B. The second position, when electrodynamic torque (11) is

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equal to zero, with " and with opposite directions of these two vectors is unstable. It was supposed to apply a system of three mutually perpendicular coils for control over the angular position of the #ight vehicle without the mechanical motion of any details [23]. A similar arrangement may be applied for another purpose: controlling the main force coil to remain in the unstable angular position with " [24]. This position corresponds to the repulsion of two dipoles of the planet and the vehicle. Two auxiliary `stabilizinga coils 3 and 4 (Fig. 9) are rigidly connected with the main force coil 1 in such a manner that their planes are mutually perpendicular to each other. The current-carrying force coil 1 continuously tends to recover from the unstable zero position, that is from the position with the electrodynamic torque which is either equal to zero or close to zero. Coil 1 turns around any axis 5 which is arbitrarily chosen in this example. The set of such possible axes lies in the plane which is perpendicular to the vector B , the magnetic  induction of the applied "eld. The plane of the force coil 1 coincides with this `zeroa plane at zero electrodynamic torque. Hence there is plane instability. (The smallest magnet tends to rotate in the stable angular position to be attracted by the biggest one.) When the plane of the main force coil 1 recovers from the zero plane, the currents I and I in the stabiliz  ing coils 3 and 4 must be increased automatically in such a relationship that their resulting magnetic moment M could be perpendicular to axis 5. This magnetic  moment is the geometrical sum of the moments M and  M . It must restore the counteractive electrodynamic  torque to the original state. Measuring the direction of the applied "eld under the condition of high intensity of the vehicle's own "eld is the main problem. But it could be treated by using a transducer to measure the angular acceleration. Let us estimate the current value and the mass of the stabilizing coils 3 and 4 relative to those of the main force coil 1. If the speed of the response of the system of the automatic stabilization is high enough then the quantities of deviation  of the angle  from the value  "!

"I N AB , (42)    

"I N AB . (43)     The resulting electrodynamic stabilizing torque is the vector sum

" # . (44)    It must be directed opposite with respect to the electrodynamic torque .  In accordance with the second Newton's law for the rotational motion we have

(39)

(40)

"J d/dt. (46)   Therefore in order to design the system of the arti"cial stabilization one may proceed from

Then according to (11) and (4) the early electrodynamic torque, which acts on the main force coil 1 with the current I and number of turns N encircling   the area A, in the applied "eld with the magnetic induction B will be 

+I N AB .    

The planes of the stabilizing coils 2 and 3 are perpendicular with respect to the main force coil 1 (Fig. 9). Therefore the sine of the angle of inclination in (11) is practically equal to one. Correspondingly their electrodynamic torques are

! "J d/dt, (45)    where d/dt is the angular acceleration and J is the  moment of inertia. In the case of "0 expression (45)  would be

are small and consequently sin ".

Fig. 9. Illustration of the principle of arti"cial stabilization.

(41)

'2 . (47)   The maximal load of the stabilizing coil is in accordance with (44) when the rotation of one stabilizing coil occurs around the axis of the other stabilizing coil. For example, if only coil 3 was developing a reaction then the current value in coil 4 would be equal to zero

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and according to (47)

'2 . (48)   The substitution of the electrodynamic torque expressions from (41) and (42) into the latest inequality (48) taking into account the equality of the areas A, which are encircled by the turns of the coils, gives N I /N I '2.     Similarly

(49)

N I /N I '2. (50)     It is evident that the higher the sensitivity of the transducer and more the speed of response of the automaticcontrol system the less the operating values of the angles  and of the ampere-turns N I or N I . Correspond    ingly, the mass of the stabilizing coils is less. For example, when (0.001 radian then the masses of these coils also are less by a factor of 10 than that of the main force coil. It is an important advantage of the described variant of the stabilization. The second advantage is the inductive decoupling of the stabilizing coils from each other and from the main force coil owing to the right angle between their axes and the equality to zero of their mutual inductance. When the current and the magnetic #ux in any coil change then the electromotive force is not induced in two other coils. The stablizing torque must conform to the condi tion d /dt'd /dt#d/dt(J d/dt). (51)    We make again the assumption about the action of the only one stabilizing coil ( " ). Substituting into (51)   the expression of from (42), we shall have  N AB dI /dt'2J d/dt. (52)     Inequalities (49), (50) and (52) are the basis for the design of the arti"cial stabilization system and they specify the requirements to the power supply source. There are two possible variants of the design scheme of the arti"cial stabilization system. We shall call them `the shell-type designa and `the skeleton designa. Both the constructions can be orbited in a stowed compact form and extended when in orbit. Then both the constructions must keep up their form and not distort when loaded. The binding element of the shell-type construction is the closed envelope of the "lm which is similar to balloon and which connects all three coils. Once in orbit, it may be "lled with some gas similar to `the Echoa satellites. Its advantage is in the tangential carrying of the distributed electrodynamic forces along the surface of the envelope. Its disadvantage is that it makes the system heavy. The mass of this envelope may be reduced by lending a lens form to the coils with the envelope (Fig. 10a). Then the quantity of area surrounded by stabilizing coil will be

Fig. 10. Variants of binding of the main force coil 1 and the stabilizing coils 3 and 4 by means of the envelope of the lens form (a) and of the telescopic rod 6 (b).

reduced and its ampere-turns value must be essentially increased. But these ampere-turns are so relatively small that the "nal gain, as the simple geometrical calculations show, is undoubted owing to more reduction in the surface and the mass of the envelope. The disadvantage of the shell-type construction is the large midship section. The base of the skeleton construction is the telescopic folding rod 6 (Fig. 10b) which stretches the stabilizing coils 3 and 4 into a rhombic form. Its advantage is the small midship section. But the disadvantage is the carrying of the distributed forces with the sign inversion similar to the wire of the aerial electric power line. If the sti!ness of the main force coil was not enough then some more similar rods in the coil's plane would be necessary in the skeleton design. 4.3. Total control attainment Arti"cial stability corresponds to the repulsion of both dipoles of planet one and the vehicle. The direction of the vector of the dipole force is opposite relative to its direction in the sole stable position. It may be characterized by the vector !F (Fig. 11). The composition of negative " dipole force !F with the arbitrary cross force F , as " ! two components, allows to obtain the resulting vector of the electrodynamic force F in the upper hemisphere

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V. Pulatov / Progress in Aerospace Sciences 37 (2001) 245}261

Fig. 12. Current-carrying loop in the non-uniform applied magnetic "eld.

elementary forces dF is not balanced and it quanti"es 8 the dipole force which is equal according to Purcell in this example

Fig. 11. Resulting electrodynamic force F as the vector sum of the reverse dipole component !F and the cross component " F in the arti"cially stable angular position. !

(Fig. 11). All the reasons are analogous to that in Section 4.1. But this hemisphere is more important than the lower one for the purpose of force provision. Both zero angular positions*the sole stable one and the arti"cially stable one*give total control over the vector F of the resulting electrodynamic force in the both hemispheres. Then any arbitrary angle of inclination of the orbit could be possible. But for each point of the orbit the variant of the composition of both components is characteristic. A continuous automatic control over the resulting vector of the electrodynamic force is necessary. The representation of the electrodynamic force as the vector sum of two components may be deepened with the easy-to-interpret explanation of the nature of the dipole force which was exempli"ed by Purcell [9]. Let the elementary electrodynamic force dF, which acts on the length d¸ of the loop with the current I, is perpendicular to the magnetic line of force which intersects this length d¸, that is perpendicular to the magnetic induction vector B (Fig. 12). The force dF may be resolved into the components dF and dF . These elementary forces may 0 8 be expressed as the Ampere's forces dF"BI d¸,

(53)

dF "B I d¸, 0 8

(54)

dF "B I d¸. 8 0

(55)

It is evident that the sum of the elementary forces dF 0 over all the loops is equal to zero. The sum of the

F "IA(grad B) . " 8

(56)

It is evident also that in the case of uniform magnetic "eld and of grad B"0, dF"dF and dF "0 (the situation 0 8 was discussed in Section 3.1). So the dipole force may be reduced to the Ampere's forces being exerted. The nature of the cross force is analogous as may be seen from Section 3.1: the cross force is some di!erence of the Ampere's forces. Thus the resultant*the totally controlled electrodynamic force*may be quanti"ed as the vector sum of all elementary Ampere's forces. Such a method is quite correct. But the totally controlled electrodynamic force as a vector sum of two controlled components is easier to understand. It is also quite correct. For example, when the cross force is about 20% of the corresponding Ampere's force (see Section 3.4) then the remaining 80% of it takes part in the resulting dipole force. One can see that the algorithm of the automatic total control may be based on any of the two described ways. The abstractions based on dividing one action into two di!erent actions are widely known. This is the electrodynamic torque and the electrodynamic force, the electric and magnetic "eld and the others. Such abstractions make easier the understanding of the total process but they may not be helpful when undertaking a detailed design. The ferromagnetic screening as a method of the cross force generation is suitable for magnetic microthrusters but it is a palliative one because of the impossibility of screening the currents of the large values because of the magnetic saturation in#uence. At the same time the concept of total control is based on a generalized physical principle which would have other engineering realizations which may di!er from what is implied in the present article. Then more powerful magnetic vehicles would be realized to a large extent.

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5. Summary and conclusions (1) Magnetic propulsion is obtained owing to direct electrodynamic interaction which is the most e$cient and pure power process at present. (2) Magnetic propulsion systems may be realized in three variants with the di!erent possibilities and features. Better designs, than those implied in the present article, are also possible. (3) Today's performance data of materials and engineering knowledge allow one to apply magnetic propulsion system as a kind of magnetic microthruster for orbital manoeuvring or balancing the air resistance when the thrust is admitted to be small but this is compensated by the continuous and stable value of the thrust. (4) A magnetic microthruster based on using the cross force may be tested and then applied on low-altitude equatorial orbits. (5) Use of the dipole force is de"ned by today's performance data of superconducting materials which are still under development and further growth is likely. This force is necessary as a component for total control over the resulting electrodynamic thrust force vector. (6) A total control over the electrodynamic force brings the magnetic propulsion to an equal position with the other aerospace means in the aspect of control. It allows use of orbits with an arbitrary angle of inclination. (7) This concept of total control appears encouraging as a way of operating magnetic #ight vehicles and thus realizing their advantages compared with other systems in terms of high speci"c impulse and ecological purity.

Acknowledgements I wish to express my appreciation to the Editor, Mr A.B. Haines, OBE, and the referee, Professor J.E. Allen, for their help in producing this paper.

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