Plasma Thruster Using Momentum Exchange in ...

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Plasma Thruster Using Momentum Exchange in Crossed Magnetic Fields

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Abstract— We examine the creation and design of a plasma thruster based on the converting of radial and circumferential momentum of the plasma flow into the axial direction. The acceleration effect is formed by the azimuthal electric field which comes about as a variation of an external, axial magnetic flux. Simultaneously, the azimuthal electron and ion flows interact with the radial magnetic field via the Lorentz force. Owing to this process, the transformation of azimuthal momentum transfers into axial momentum. As a result, the electron and ion flows are accelerated in one axial direction. Estimations specifying the admissible range of parameters of this device and the plasma flow are derived. Index Terms— Accelerators, Hall effect devices, magnetic fields, plasma engines.

I. I NTRODUCTION

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LASMA thrusters have an undeniable advantage over the conventional chemical jet or rocket engines [1]–[4]. This follows directly from the Tsiolkovsky formula: M0 V f = Vex ln (1) Mf

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by the magnetic field. Here, the electrons would form spiral trajectories about the coil. By pulsating, the electrons enter into the central region and when the field dies, the electrons due to inertia should continue in the same direction along the axis to generate thrust. In this case for a plasma thruster, the thrust generated by electric propulsion systems is only limited by the amount of electric power and a small propellant that can be supplied (see [1], [2]). However, all these devices are characterized by a low specific impulse that comes about because of a low density of neutralizing the plasma flow. An increase in specific impulse increases the mass efficiency while also decreasing the thrust thereby increasing trip travel time. Therefore, one option of the problem is to increase the density of the plasma flow at a higher velocity. In the present piece, we shall focus on the plasma thrusters with the crossed fields accelerating plasma ions. As a rule, in such devices (so-called Hall thrusters) would use the strong magnetic field perpendicular to the flow and the axial electric field to accelerate an ion component. The magnetic field impedes the counter flow of electrons in the accelerating field which does away with the space-charge limitation. This restricts the flow and thrust of ion engines, i.e., acceleration of ion flow in this region come about in quasi-neutral conditions and there is no limitation brought about by the space charge (see [3], [4]). This feature makes the Hall thrusters being one of the most efficient types of plasma devices among plasma accelerators. For comparison, the efficiency of an ionic electrostatic accelerator such as gridded ion thrusters is limited by the Langmuir law. However, it should be borne in mind that although the Hall thrusters are mechanically simpler, physics of Hall thrusters are more complex and they achieve slightly less electric efficiency and specific impulse than ion thrusters. So, as a first step, it will be useful to consider the basic physical features of the existing Hall thrusters.

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and Paul A. Murad

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Alexander R. Karimov

where V f is the final velocity of the rocket, M0 is the initial total mass, including propellant, M f is the final total mass of spacecraft without propellant, and Vex is the effective exhaust velocity. As is seen from (1), to have a large mass fraction for the delivered payload, the specific impulse of the thruster must be on the order of the total Vex . However, the effective exhaust velocity of the jet engine is limited by the enthalpy of the used fuel. Such a limit is absent in plasma thrusters that make it more efficient to obtain the high-velocity exhaust velocities at a low mass fraction compared to chemical engines. In a simplistic approach, if you have a single coil with an electric current, a magnetic field is created. The approach looks into this where electrons pass through the coil and move in a direction similar to the central axis of the coil. If this is steady state, electrons would follow the closed lines of force created Manuscript received August 7, 2017; revised October 11, 2017 and December 26, 2017; accepted February 20, 2018. The review of this paper was arranged by Senior Editor A. I. Smolyakov. (Corresponding author: Alexander R. Karimov.). A. R. Karimov is with the Institute for High Temperatures RAS, 127412 Moscow, Russia and also with the Department of Electrophysical Facilities, National Research Nuclear University MEPhI, 115409 Moscow, Russia (e-mail: [email protected]). P. A. Murad is with Morningstar Applied Physics, LLC, Vienna, VA 22182 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPS.2018.2810779

II. O PERATIONAL P RINCIPLES OF H ALL T HRUSTERS In Hall thrusters, the orthogonal electric E and magnetic B fields are used to produce thrust by the ion component created in the plasma (see [2], [3]). As we have said, an electron component is assumed to be fixed in the axial direction. A quasi-equilibrium situation may exist in a pressure-less plasma possessing high electron conductivity σ when it is possible to allow σ → ∞ (i.e., the frictional force between electrons and ions is negligible and the corresponding term in the equation of motion for electrons and ions is absent).

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Fig. 1.

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The plasma parameters in the acceleration zone are defined by the following equations: 1 e E + vi × B ∂t vi + (vi · ∇)vi = (2) mi c 1 E = − ve × B (3) c where m i is the ion mass, vi is the ion fluid velocity, ve is the electron fluid velocity, and c is the velocity of light. Here E and B represent a superposition of external and internal electric and magnetic fields, respectively. As is seen from relation (3), the electric field is able to accelerate ions that exist in the plasma only due to B when ve = vi . In this case, the electron density n e is approximately equal to the ion density n i . The external magnetic field is assumed to be large enough to magnetize electrons but not to impact ions [2], [3]

Schematic of cylindrical Hall thruster [6].

In fact, the electron fluid in these thrusters acts as an immobile (in the axial direction) neutral medium so long as condition (4) is satisfied. It should be noted that the effect of the ion acceleration process is limited by the charge polarization owing to the generation of an internal electric field having the order ∼ n e λ D , where λ D is the Debye length. One may try to go to the limits of the magnetized electrons approximation if both ion and the electron flows move in one or the same direction. Such acceleration may occur if plasma flow is in the crossed magnetic fields varied with time and space. Typically, this idea is similar to an end-Hall thruster and a cylindrical type Hall thruster where the acceleration is in the axial direction. The generation of an electric field brings about the interaction between the electron flow and a magnetic field (see [7], [8]). At the same time, this system provides thrust without the need for additional current and space-charge neutralization. It is also worth noting that the similar mechanism of acceleration occurs with a coaxial plasma accelerator [9], [10] or a magnetoplasmadynamic thruster [11], [12]. The typical principle of cylindrical Hall thruster is depicted schematically in Fig. 2. This thruster distinguishes from conventional annular and end-Hall thrusters by using a cylindrical configuration with an enhanced radial component using a cusptype magnetic field. Such a magnetic field configuration can be created with the help of an electromagnet coil [7], [13] or permanent magnets [14]. Unlike conventional Hall thrusters, cylindrical hall thrusters do not have the center stem that provides a radial magnetic field. Similar to the conventional annular type Hall thruster, there is a closed E × B electron drift and acceleration of nonmagnetized ions by the electric field defined by (3). Here the radial component of the magnetic field crossed with the azimuthal electron current produces thrust. At the same time, the electrons are not confined to an axial position, and they bounce over an axial region. This can be considered as an electrostatic viewer on the acceleration mechanism from an electromagnetic point of view. There is another interpretation. The azimuthal Hall current interacts with an applied radial magnetic field and

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Fig. 2.

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Schematic of stationary plasma thruster [6].

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r Le L r Li

(4)

where r Le and r Li are the Larmor radius for electrons and ions, respectively, and L is a length of the self-consistent accelerating layer. If these requirements are fulfilled, and we can always find such external magnetic field to satisfy (4), then the space charge of the ions is compensated by a small mobility of electrons across the magnetic field. As a result, the electrons can experience the rotation in the azimuthal direction with a drift velocity V E = (E × B/B 2 )c, and ions may move only in the axial direction. Thus, radial magnetic field confines electrons which compensate the space charge of the flow, while an axial electric field accelerates ions. This idea is for stationary plasma thrusters where the thruster includes an anode layer in which the electrical discharge has an E×B configuration where the external magnetic field is radial and perpendicular to the axial electric field (see [2]–[5]). A schematic of this thruster is shown in Fig. 1. The size of the acceleration zone in the axial direction of stationary plasma thrusters is acceptable to be greater in thrust performance than the one in thrusters with an anode layer. Nevertheless, these thrusters are in accordance with the principle of this operation and the parameters achieved.

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Fig. 5.

Schematic of electron and ion acceleration region.

One may use microwave discharge in a waveguide for the plasma production or the initiation of discharge in a low pressure gas may be propelled by microwave radiation with a stochastically jumping phase (see [17]–[19]). Although the first option is more technically simple but a breakdown power minimum for microwaves with a stochastically jumping phase depends weakly on a working gas pressure that is caused by the anomalous nature of collisionless electron heating [18]. Such feature allows extending the discharge existence region in a direction of over pressures. Therefore, it would be useful to combine both methods in the source worked out. Nevertheless, in both cases the length of the discharging zone must be designed to minimize the ionization mean free path localizing the ionization of the working gas into the ionization volume. In the present scheme, the permanent magnet system creates a stationary radial magnetic field and the solenoid feeds through a low-frequency alternating current by creating predominantly an axial magnetic field that varies in time. As a result, the configuration of the crossed magnetic fields presented in Fig. 5 uses a combination of external fields to produce an axisymmetric magnetic field

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Fig. 3. Schematic of end-Hall thruster [15]. Here jϕ is the sum of electron jeϕ and ion jiϕ current fluxes.

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KARIMOV AND MURAD: PLASMA THRUSTER USING MOMENTUM EXCHANGE IN CROSSED MAGNETIC FIELDS

Fig. 4. Design of device on the base [16]. Here a and b are the internal and external radiuses of the magnetic system; L is the length of the self-consistent accelerating layer; and δ is the length of plasma source.

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accelerates the plasma axially through the Lorentz force. Such a mechanism arises naturally in the end-Hall thruster (see Fig. 3). Unlike stationary plasma thrusters and cylindrical Hall thrusters, there is no neutralizer in this scheme but the size of the acceleration zone is extremely small and is defined by the configuration of the magnetic field at the output of the magnetic coil (see [14]). Thus with this appearance, the magnetic coils in these devices create a configuration where there is both a radial and axial component of the magnetic field.

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III. D ESIGN OF P LASMA ACCELERATOR

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This acceleration mechanism for the rotating plasma flow in the external magnetic fields can be strengthened in the schematic of a thruster concept sketched in Fig. 4. This is based on the design described in [16]. Here, a gas fuel is injected into the system through the gas inlet. Then, the gas flux enters the zone of plasma source where the gas is ionized to create a plasma flux that would be used to generate thrust. In the present schematic, we have only drawn the ionization zone without discussing the physical mechanism and construction of the plasma source since it requires a separate discussion and we hope to return to the issue in our forthcoming studies. Now, we only state some general considerations about the plasma source which may be used in this schematic.

B0 = Br0 er + Bz0 ez

(5)

where Br0 = Br0 (r ) and Bz0 = Bz0 (t) are some known, independent functions having the following properties. To prevent the influence of the border effects of the solenoid, we should put L b, here L is the solenoid length coinciding with length of the self-consistent accelerating layer, and b is the external radius of the magnetic system, therefore, we can neglect the axial dependence of B0 (see Fig. 4). Then taking into account, the lack of divergence for external magnetic field B0 creating by coil and permanent magnet system, one can come to the conclusion that Br0 (r ) = B0 /r for a ≤ r ≤ b, where B0 is some constant defining the characteristic field of magnetic system which we have used for further estimations. However, it should be noted that such a distribution is suitable only for estimates. In reality, the distribution of the magnetic field in the this region will be more complicate since one should take into account the geometry of permanent magnet and coil, and this issue required a separate examination. Besides, we neglect the influence of diamagnetic plasma currents on the full magnetic field B so that B ≈ B0 .

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Such a diamagnetic effect is negligibly small if β=

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where Te is the electron temperature, and Ti is the ion temperature (for convenience, here and below, all temperatures are written in energy units). This ratio is a consequence of maintaining the total pressure in the plasma which is composed of gas dynamic and magnetic pressures [3], [20]. It implies that we ought to have deal with a cold, sufficiently rarefied plasma. Assuming a b, where a is the internal radius of the magnetic system, from an equation of induction written in integral form for this field B0 c

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it follows that the external electric field has only an azimuthal component: r E ϕ0 ≈ − ∂t Bz0 (7) 2c which rotates electron jeϕ and ion jiϕ current fluxes in different directions. The azimuthal electron and ion flows interact with the radial magnetic Br0 via the Lorentz force. This leads to an axial acceleration of electrons and ions in the same axial direction since the resulting forces jeϕ × Br0 and ion jiϕ ×Br0 are also directed in the same direction. Owing to this process, the transformation of azimuthal momentum transfers into axial momentum. As a result, the electron and ion flows are accelerated in one axial direction sketched in Fig. 3. Thus, one can accelerate the electron and ion flows in the axial direction by accelerating them in the azimuth direction by a vortex electric field that may be generated by varying the axial magnetic field in time [24], [25]. Besides, in this case, the direction of the plasma inlet coincides with the direction of the Ampere force that arises in a given configuration of magnetic fields and the axial fluxes of electrons and ions. Moreover, such a configuration of external magnetic fields may increase the transfer processes of energy/momentum when the oscillation energy in the radial mode tends to be transferred to other degrees of freedom of the rotating plasma flows. The discussion of this idea was presented in [21]–[23] where for spatially homogeneous flows with n e = n e (t) and n i = n i (t) it was shown that the radial component does not vary with time, whereas the axial component changes under the action of the azimuthal electric field. In this case, the energy/momentum exchange comes about because of a local violation of quasi-neutrality that leads to the generation of local time-dependent azimuthal electric fields, which provide the redistribution of energy/momentum in space and time. So, one can expect a similar behavior in the general case of inhomogeneous flows in time-dependent external magnetic fields but this requires special discussion.

where is the constant defining the signal period. It is obvious that in this case, the magnetic field Bz0 (t) will repeat the form of the signal I0 (t). On the other hand, we can present I0 (t) in a formal Fourier series as ∞ π 2 2πl (−1)l+1 1 t + cos t . (8) I0 (t) = cos 2

π 4l 2 − 1

IV. T HRUST H ANDLING However, as is seen from (7) for a harmonic function, Bz0 = Bz0 (t) is considered as very weak to start creating the plasma acceleration which can occur only for a quarter period

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As is seen from (8), for l > 1 the amplitude of the upper harmonics is negligible. Therefore, we may describe the signal I0 (t) with just a few lower harmonics. For example, if we are restricted by the crude case when l = 1 and we may use the bifilar coil with a parallel winding of wires, to make the crude form of the current I0 (t) in this coil. Certainly, the waveform can be approximated to the form of I0 (t) if we use trifilar, tetraphilic, pentafilar, etc., coils which are able to pass higher harmonics but this issue is not considered in the framework of this paper. Nevertheless, it is useful to touch on the connection of the value with the solenoid length L and the characteristic velocity of plasma injection to the solenoid from the inlet. As a characteristic estimate of this velocity, we take the thermal velocity of electrons v T e = (Te /m e )1/2 . It is clear that for the time interval /2 when I0 = 0, there is no acceleration of plasma flux by external field Bz0 (t). We can neglect any processes in this interval if during time /2 the flux flies a negligible distance in comparison with the length L, i.e., /2 meets the inequality

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(6)

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n e Te + n i Ti 1 B 2 /8π

when ∂t Bz0 ≤ 0. It implies that we have to use some electronic unit to form the signal Bz0 (t) waveform with the required phase. Although this point is purely a technical problem that depends on the amplitude and frequency of the original signal, we show that in principle, the problem is solvable. Let the current creating the axial magnetic flux in the solenoid be

π t , 0 < t < /2 cos

I0 (t) = 0,

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L/v T e .

(9)

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In this case, we can use a continuous plasma feed from plasma source to create a continuous thrust. As the plasma flux accelerates, the condition (9) will be violated so this relation determines only the initial parameters of the solenoid and the characteristic velocity of plasma injection. In opposite case, when ∼ L/v T e we have to pass to the discrete gas inlet mode not to waste fuel.

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V. A DMISSIBLE PARAMETERS OF ACCELERATOR

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In order to determine the parameters of this acceleration scheme, we ought to consider the generation within the electric and magnetic fields in the rotating plasma flow propagating in the external fields (5). However, even in the case of spatially uniform but nonstationary external magnetic fields, this is a complex problem [24]. So, we are restricted by crude estimations through considering the dynamics of electrons and ions caused by an external magnetic field (5).

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KARIMOV AND MURAD: PLASMA THRUSTER USING MOMENTUM EXCHANGE IN CROSSED MAGNETIC FIELDS

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If we take the value v e = (Te /m e as a characteristic velocity of electron flow, and v i = (Te /m i )1/2 as a characteristic velocity of the ion flow, then we can rewrite the condition of (10) in the following form: (Te m e )1/2 (Te m i )1/2 a< < < b. eBz0 eBz0

Wi πb2 δ = W R πb2 L + 2πbL Jw

(10)

(11)

Our model presumes the absence of collisions between particles of the plasma medium. Such a situation wittingly occurs if

λ (b − a)2 + L 2 . (12) Here λ = 1/(σ0 n e ) is the mean free path of the plasma flow, where σ0 is the effective cross section. It is worth noting that the present condition may be weakened by selecting the largest value of L and b − a as a characteristic scale of the plasma flow. As a characteristic estimate, we take σ0 = πa02 , where a0 is the Bohr radius. Taking into account this value of σ0 , we arrive at

(b − a)2 + L 2 n e 1016 . (13)

by using (14)–(16) we arrive at σe 4π n e n e 1 m e n c r0 Ui ξ + 2+ exp(−Ui /Te ) σc Te 3 nn nc π m i nn b (17) where σc = πr02 and n c = 1/r03 . Thus, relationships (13) and (17) makes it possible to draw a tentative conclusion about the range of the parameters for which our assumption about collisionless plasma is valid. Also, we give up the generation of the magnetic field of the plasma flow (i.e., the magnetic field brought about by the movement of plasma electrons and ions) that would reduce this effect. It implies that we can neglect and ignore the spatial dependence of this magnetic field. According to the estimation [24], such an assumption is valid for b2 n e 1013.

Moreover, from (10) and (12), it follows that the cyclotron frequency ωsc for the s-component exceeds the collision frequency νcoll between the charged particle of the s-kind with particles of other kinds (i.e., the Hall parameter h s = ωsc /νcoll 1) and the medium is considered to be a magnetized plasma. In this case, all transport properties of plasma are defined by the magnetic field and the medium becomes anisotropic. Also it should be borne in mind that such approximation remains valid until the particle production processes in the plasma source will dominate over the processes of particle losses. To obtain a rough estimation, we restrict our consideration of the losses processes by the bulk three-body electron recombination and escape of particles to the accelerator walls. For impact ionization by thermal electrons in a quasi-neutral plasma, we can write the rate of particle production as [17] Ui 3 exp(−Ui /Te ) Wi ≈ n e n n v e σe 2 + (14) 2 Te where Ui is the ionization potential, σe is the characteristic ionization cross section, and n n is the density of neutral gas. The rate of bulk recombination can be written [17] W R ≈ 2π 2r05 v e n 3e

(15)

where r0 = 2e2 /3Te is the Coulomb radius. According to [20], the electron flux to the wall can be estimated as 1 Jw ≈ v i n e . (16) 2 Presume that the length of plasma source δ is more less than the length of acceleration zone L and hence we may set δ δ ≈ = ξ. δ+L L

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a < r Le < r Li < b.

Then, from the requirement that the rate of particle production is much larger than that of particle loss

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In order to escape the ions and electrons entering into the accelerator chamber, instead of (4), we must put

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(18)

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However, even with these parameters, they are beyond the range of condition in (18) and one can expect to result in some additional thrust effects.

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VI. S PECULATING P ROSPECTS

It is also worth noting that there may be a lot of technical implementations for this idea. Here, we discussed the schematic based on the geometry presented in [16]. However, instead of an electromagnet here, we propose to use a permanent magnet to create the constant radial magnetic field (see Fig. 4). Also, we do not discuss the electronic method for the formation of signal Bz0 (t) of the required form. One can say that this is a solvable issue from the technical point of view but the solution depends on both the amplitude and frequency of the original signal. Furthermore, this problem originates from the framework of this paper and we hope to discuss this question later. Here, we have considered the design where the plasma source is located in the zone of gas inlet system (see Fig. 4). However, there is another schematic when the process of plasma production can be achieved by a radial alternating electric field to propel a plasma discharge directly in the acceleration zone. From this technical point of view, such radial excitement is simpler than other methods for propelling plasma discharge. Herewith, we can expect that the used configuration of external magnetic fields may increase the transfer processes of energy/momentum when the oscillation energy in the radial mode tends to be transferred to other degrees of freedom induced by the rotating plasma flows [21]–[25]. However, in this case, one should take into account the coupling of ionization processes and nonlinear plasma oscillations [18], [26]–[30]. This may be used with a transformation of energy/momentum radial oscillations directing into the axial direction by the plasma inhomogeneity with nonlinear coupling among the electron and ion flow components and oscillations [22], [28], [31].

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The design of a plasma thruster that accelerates a plasma flow in the crossed magnetic fields [see (5)] uses a timevarying axial magnetic field Bz0 (t)ez . This magnetic field provides an azimuthal acceleration of the plasma flow, and a permanent radial magnetic field Br0 (t)er provides the transformation of the azimuthal momentum into the axial momentum. Namely, the acceleration effect is caused by the azimuthal electric field which comes from the variation of an external magnetic flux (7). Besides, it is possible to go beyond the approach of quasi-neutrality. These processes can lead to the generation of a local azimuthal time-dependent electric field which may accelerate the rotation of the electron and ion fluids in different directions. As previously mentioned, the interactions of these flows with the external magnetic field propel the axial acceleration of the flow. The presented relationships (11)–(18) define the range of allowable parameters of the device. In fact, the proposed design is a hybrid of induction accelerator and end-Hall thruster. ACKNOWLEDGMENT The authors would like to thank S. I. Krasheninnikov and V. A. Kurnaev for helpful discussions and their interest in this paper. R EFERENCES

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[17] P. Yu Raizer, Gas Discharge Physics. New York, NY, USA: Springer, 1991. [18] F. F. Chen, “Plasma ionization by helicon waves,” Plasma Phys. Control. Fusion, vol. 33, no. 4, pp. 339–364, 1991. [19] M. Lieberman and A. Lichtenberg, Principles of Plasma Discharges and Materials Processing. Hoboken, NJ, USA: Wiley, 2005. [20] F. F. Chen, Introduction to Plasma Physics and Controlled Fusion. New York, NY, USA: Plenum, 1984. [21] A. R. Karimov and S. M. Godin, “Coupled radial-azimuthal oscillations in twirling cylindrical plasmas,” Phys. Scripta, vol. 80, no. 3, p. 035503, Aug. 2009. [22] A. R. Karimov, L. Stenflo, and M. Y. Yu, “Coupled flows and oscillations in asymmetric rotating plasmas,” Phys. Plasmas, vol. 16, no. 10, p. 102303, Oct. 2009. [23] A. R. Karimov, M. Y. Yu, and L. Stenflo, “Asymmetric oscillatory expansion of a cylindrical plasma,” J. Plasma Phys., vol. 79, no. 6, pp. 1007–1009, Dec. 2013. [24] A. R. Karimov and P. A. Murad, “Acceleration of rotating plasma flows in crossed magnetic fields,” IEEE Trans. Plasma Sci., vol. 45, no. 7, pp. 1710–1716, Jul. 2017. [25] A. R. Karimov and P. A. Murad, “The processes of energy/momentum exchange for rotating plasma flows in crossed external fields,” J. Space Exploration, vol. 6, no. 1, pp. 113–126, Jan. 2017. [26] F. F. Chen and R. W. Boswell, “Helicons-the past decade,” IEEE Trans. Plasma Sci., vol. 25, no. 6, pp. 1245–1257, Dec. 1997. [27] A. N. Kozlov, “Ionization and recombination kinetics in a plasma accelerator channel,” Fluid Dyn., vol. 35, no. 5, pp. 784–790, Sep. 2000. [28] A. R. Karimov, M. Y. Yu, and L. Stenflo, “Large quasineutral electron velocity oscillations in radial expansion of an ionizing plasma,” Phys. Plasmas, vol. 19, no. 9, p. 092118, Sep. 2012. [29] A. R. Karimov and V. A. Shcheglov, “Nonlinear Langmuir oscillations under nonequilibrium conditions,” Phys. Plasmas, vol. 7, no. 3, pp. 1050–1052, Feb. 2000. [30] M. Mirzaie, B. Shokri, and A. A. Rukhadze, “Analysis of overcritical plasma production by strong microwave fields,” Plasma Phys. Controlled Fusion, vol. 49, no. 8, pp. 1335–1345, Jul. 2007. [31] A. R. Karimov, M. Y. Yu, and L. Stenflo, “Flow oscillations in radial expansion of an inhomogeneous plasma layer,” Phys. Lett. A, vol. 375, no. 27, pp. 2629–2636, Jul. 2011.

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VII. C ONCLUSION

[1] G. P. Sutton and O. Biblarz, Rocket Propulsion Elements. New York, NY, USA: Wiley, 2010. [2] A. I. Morozov and V. V. Savelyev, “Fundamentals of stationary plasma thruster theory,” in Reviews of Plasma Physics, vol. 21, B. B. Kadomtsev and V. D. Shafranov, Eds. New York, NY, USA: Consultants Bureau, 2000, pp. 203–291. [3] D. M. Goebel and I. Katz, Fundamentals of Electric Propulsion: Ion and Hall Thrusters. Hoboken, NJ, USA: Wiley, 2008. [4] R. G. Jahn, Physics of Electric Propulsion. New York, NY, USA: McGraw-Hill, 1968. [5] V. Kim, “Main physical features and processes determining the performance of stationary plasma thrusters,” J. Propulsion Power, vol. 14, no. 5, pp. 736–743, Sep. 1998. [6] S. McGrail and S. Parker, “Preliminary design of a laboratory cylindrical Hall-effect thruster,” B.S. thesis, California Polytech. State Univ., San Luis Obispo, CA, USA, 2012 [7] A. Smirnov, Y. Raitses, and N. J. Fisch, “Experimental and theoretical studies of cylindrical Hall thrusters,” Phys. Plasmas, vol. 14, no. 5, p. 057106, May 2007. [8] C. L. Ellison, Y. Raitses, and N. J. Fisch, “Cross-field electron transport induced by a rotating spoke in a cylindrical Hall thruster,” Phys. Plasmas, vol. 19, no. 1, p. 013503, Jan. 2012. [9] A. N. Kozlov, “Basis of the quasi-steady plasma accelerator theory in the presence of a longitudinal magnetic field,” J. Plasma Phys., vol. 74, no. 2, pp. 261–286, Apr. 2008. [10] A. N. Kozlov, “Two-fluid magnetohydrodynamic model of plasma flows in a quasi-steadystate accelerator with a longitudinal magnetic field,” J. Appl. Mech. Tech. Phys., vol. 50, no. 3, pp. 396–405, May 2009. [11] R. G. Jahn and E. Y. Choueiri, “Electric propulsion,” in Encyclopedia of Physical Science and Technology, vol. 5, 3rd ed. New York, NY, USA: Academic, 2002, pp. 125–141. [12] E. Choueiri, “Scaling of thrust in self-field magnetoplasmadynamic thrusters,” J. Propulsion Power, vol. 14, no. 5, pp. 744–753, Sep. 1998. [13] Y. Raitses and N. J. Fisch, “Parametric investigations of a nonconventional Hall thruster,” Phys. Plasmas, vol. 8, p. 2579, May 2001. [14] Y. Raitses, E. Merino, and N. J. Fisch, “Cylindrical Hall thrusters with permanent magnets,” J. Appl. Phys., vol. 108, no. 11, p. 093307, Nov. 2010. [15] S. D. Grishin, L. V. Leskov, and H. P. Kozlov, Electric Rocket Engines. Moskow, Russia: Mechanical Engineering, 1975. [16] V. P. Kim, “Design features and operating procedures in advanced Morozov’s stationary plasma thrusters," Tech. Phys., vol. 60, no. 3, pp. 362–375, Mar. 2015.

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Alexander R. Karimov received the Ph.D. and Doctor of Science degrees in physics and mathematics from Moscow Engineering Physical Institute, Moscow, Russia, in 1994 and 1999, respectively. He is currently a Professor with the Department of Electrophysical Installations, Moscow Engineering Physical Institute. Since 1996, he has been a Staff Member of Institute for High Temperatures Russian Academy of Sciences, Moscow, where he is a Leading Researcher with the Division for MagnetoPlasma Aerodynamics and MHD Energy Conversion. His current research interests include theoretical laser, beam and plasma physics, hydrodynamics, and biophysics.

Paul A. Murad has over 25 years of public service as a Senior Technology Analyst with the Department of Defense looking at game-changing technology as well as defining future U.S. satellite systems for the next 20 years. He started several technical conferences to include the First High-Frequency Gravity Wave Meeting in 2003 as well as five STAIF Conference that covered new space propulsion technology, energy devices, and communications issues. With 18 years in the private sector service, he was with two Aerospace Corporations, Bendix, Teterboro, NJ, USA, and AAI Corporation, Cockeysville, MD, USA, in an executive capacity. He was involved in the Apollo, the NERVA Nuclear Rocket Engine, the Space Shuttle and numerous tactical and strategic missile systems as well as the Navy High Energy Laser project. He is currently the Chief Executive Officer with the Morningstar Applied Physics, LLC, Vienna, VA, USA. His current research interests include electrodynamics, hydrodynamics related to solving closed-form Navier–Stokes equations as well as tornado phenomenology, and astrophysics. Mr. Murad is an Associate Fellow of the AIAA.

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