A Plasma Thruster Based on Screw-Pinch Physics

Abstract

This research paper provides a conceptualization of a new type of plasma thruster based on screw-pinch physics and on the magnetic mirror concept. The article proposes a method to size a screw-pinch with a non-uniform axial magnetic field as a plasma thruster and to estimate its propulsive performance. The results obtained show that the plasma thruster is suitable for space missions inside the Earth’s sphere of influence and for space transportation of small satellites.

1. Introduction

This research work involves the study of a magnetic configuration based on the pinch concept [1], which has been extensively studied in the past [2] both for plasma generation [3] and for fusion research [4]. The z-pinch [2,3] is a plasma electric discharge established between two electrodes which is unstable in its basic configuration and can be stabilized with an external axial magnetic field to obtain the magnetic configuration known as the screw-pinch [2]. With respect to the stabilized magnetic configuration of the screw-pinch with a uniform axial magnetic field, it is possible to space the axial coils to create a non-uniform axial magnetic field, assuming a topology similar to that of a magnetic mirror where plasma-charged particles are partially confined. The axial end loss of charged particles of this magnetic configuration can generate a plasma jet, acting as a plasma thruster [5]. The effect of the magnetic field gradient on the plasma thruster performance [6] and the conceptualization of a plasma jet emitted from a magnetic mirror [7] has already been examined in the literature. In this work, a novel concept of a plasma thruster is examined that combines the physics of a magnetic mirror with its plasma jet and an electric arc as a plasma source. Plasma thrusters are a category of electric propulsion systems of interest in the space industry because of their high specific impulse, which allows them to carry out space missions with low propellant consumption. Their ionization mechanism, based on electron-neutral collision, makes this technology consolidated at the laboratory level and reliable for the design of new space propulsion systems. The current limitation of this category of thrusters is the low thrust they can exert due to the low mass flow rate (between mg/s and μg/s) of the charged particles, which are typically at low pressure (a few Pa) in the ionization chamber and do not have enough momentum to reach a thrust level of Newtons. However, even with this low thrust, such plasma thrusters can find good applications in satellite space missions for attitude control or for orbital maneuvers and also for space missions in the Earth’s sphere of influence where thrust levels in the range of 1 mN–1 N are enough to transport low-mass payloads. This paper describes the working principle of a plasma thruster based on screw-pinch physics and the magnetic mirror concept, deriving the performances that can be achieved in terms of thrust and power efficiency of such a system and its scalability.

2. The Screw-Pinch Configuration with Non-Uniform Axial Magnetic Field

The screw-pinch is a magnetohydrodynamic (MHD) equilibrium configuration of confined plasma that is obtained by stabilizing a plasma axial discharge with an axial magnetic field that is generated by surrounding angular coils. The geometry of this magnetic configuration is a plasma column with a helical magnetic field and axial and azimuthal plasma current components [1]. The axial plasma current is generated by the voltage breakdown between two electrodes positioned on opposite sides of a magnetic configuration, after which a plasma arc is generated. When the stabilizing axial magnetic field of the screw-pinch is generated by a Helmholtz couple, if we increase the axial distance of the coils, we obtain a non-uniform axial magnetic field with a maximum magnetic field at the coil axial position and a minimum magnetic field in the middle of the magnetic configuration. Thus, along the axis of the screw-pinch, there is a variation of the magnetic field, as illustrated in Figure 1, which is a typical topology of a magnetic mirror. This can be useful for controlling the axial confinement of the plasma and therefore the axial end loss of the magnetic configuration, which is relevant for thrust generation. Therefore, this plasma thruster configuration is characterized by a superposition of an arc discharge and a magnetic mirror. The presence of an axial electric field in the magnetic configuration changes the charged particles’ dynamic from that of a simple magnetic mirror, because the electrons will always have a drift velocity in the arc discharge that is proportional to the axial electric field through the electron mobility.

Magnetic mirrors are characterized by a continuous end loss of charged particles, which, due to their high axial kinetic energy, cannot be trapped inside a magnetic mirror and are emitted through a loss cone. This loss can be estimated using the conservation of the adiabatic invariant ($\mu$) and the charged particles’ kinetic energy (K) in the magnetic field, as stated in the following system of equations:

$$\left\{\begin{array}{c}\mu ={\displaystyle \frac{K_{\perp }}{B}}=constant\hfill \\ K=K_{\perp }+K_{\Vert }=constant\hfill \end{array}\right.$$

(1)

where the subscripts ‖ and ⊥ indicate the directions parallel and perpendicular to the magnetic field line, respectively, and B is the modulus of the magnetic field vector. When a charged particle is immersed in a non-uniform magnetic field, such as that of a magnetic mirror, its kinetic energy can be written as the following equation:

$$K={\displaystyle \frac{1}{2}}m{(v_{\Vert }^{\prime }+v_{{\Vert }_{drift}})}^2+\mu B=constant$$

(2)

where m is the mass of the charged particles. When the magnetic field varies between a minimum and a maximum value, there is also a variation of the parallel velocity components between a maximum and a minimum. In particular, when the magnetic field is minimum $B_{min}$, the parallel velocity is maximum, while on the contrary, when the magnetic field is maximum $B_{max}$, the parallel velocity is minimum. In Equation (2), the parallel velocity $v_{\Vert }$ is divided into a variable component $v_{\Vert }^{\prime }$ along the magnetic field and a constant component $v_{{\Vert }_{drift}}$ that is proportional to the axial electric field. Therefore, when a charged particle is immersed in a magnetic mirror configuration, it is confined in this non-uniform magnetic field, traveling back and forth between the two extremes of the magnetic mirror [8], as it happens for the charged particles confined in the Earth’s magnetosphere. From the magnetic mirror theory, it is known that there is a loss cone defined in the velocity space, and all the particles that have the velocity vector contained in this loss cone are lost from the magnetic mirror. The particle loss computed from this theory is enhanced due to collisions and kinetic instabilities in the plasma. The angle that defines the loss cone is called the pitch angle, which is the angle between the velocity vector and the axial magnetic field, and it is defined by the ratio between the perpendicular velocity and the parallel velocity of the particle. This ratio can be obtained from the previous system of equations (Equation (1)) between the conservation of the adiabatic invariant and the conservation of the kinetic energy, with the electron drift velocity defined as

$$v_{{\Vert }_{drift}}={\mu }_e{E}_{{\Vert }_p}$$

(3)

where $E_{{\Vert }_p}=V_p/L_p$ is the axial electric field associated with the plasma discharge and ${\mu }_e$ is the electron mobility computed as

$${\mu }_e={\eta }_s{n}_{e}e$$

(4)

where ${\eta }_s$ is the plasma resistivity, $n_e$ is the electron density and e is the electron charge. The loss cone for our magnetic configuration can therefore be computed as

$$tan{\theta }_{cone}={\displaystyle \frac{v_{\perp }}{v_{\Vert }}}=\sqrt{{\displaystyle \frac{B_{min}}{B_{max}-B_{min}+\lambda }}}$$

(5)

where $\lambda$ is a corrective factor due to the presence of the axial electric field of the plasma discharge, which is obtained from the introduction of the electron drift velocity (Equation (3)) in Equations (1) and (2), and it is defined as

$$\lambda ={\displaystyle \frac{n_e^2{e}^2{m}_e{\eta }_s^2{V}_p^2}{2\mu L_p^2}}$$

(6)

where $m_e$ is the electron mass, $V_p$ is the plasma potential drop of the plasma arc, $\mu$ is the adiabatic invariant of the magnetic field, and $L_p$ is the plasma length. Therefore, electrons are continuously lost from the screw-pinch through the center of the annular anode (Figure 1) and have a pitch angle $\theta <{\theta }_{cone}$. This physical property of a magnetic mirror can be exploited in a plasma thruster because the continuous axial loss of electrons in the anodic region can generate thrust by setting up an ambipolar electric field. which in turn accelerates the ions generating thrust [9].

3. Plasma Thruster Design

A plasma thruster fed by a screw-pinch magnetic configuration has an ionization chamber with a plasma discharge established between two electrodes. The electric discharge considered here is a plasma arc that is controlled in current intensity [3]. Since the jet power is proportional to the plasma temperature and to the plasma current, which in turn is proportional to the plasma temperature by means of Equations (13) and (14), the arc regime is considered in this study, where the electron temperatures of the plasma are around 1 eV and plasma currents are of the order of 10 A. The plasma arc is modeled as a DC electric circuit that is fed with a power supply that can sustain the formation of the electric arc, assuming that the total potential drop ($V_{tot}$) at the plasma arc is computed as the sum of the potential drop at the cathode ($V_c$), the potential drop at the central plasma column ($V_p$), and the potential drop at the anode ($V_a$) [3]. The cathode potential drop is computed according to the following equation:

$$V_c=V_{ion}+V_{therm}$$

(7)

where $V_{ion}$ is the ionization potential of the gas that takes into account the energy required for the ionization, and $V_{therm}$ is the thermionic potential of the cathode material (here, we consider electrodes of tungsten) that takes into account the energy required to extract an electron from the cathode. The potential drop of the central plasma column is computed according to the Ohm law:

$$V_p=RI={\eta }_s{\displaystyle \frac{d}{A_p}}I$$

(8)

where R is the plasma resistance and I is the plasma current. The plasma resistance is computed from the plasma resistivity ${\eta }_s$, given by the Spitzer formula [10], assuming that all the species are singly ionized, where d is the plasma length approximately equal to the distance between the electrodes and $A_p$ is the plasma cross-section. Finally, the anode potential drop is computed assuming that the sum of the cathode potential drop and the plasma potential drop ($V_c+V_p$) is approximately 80% of the total potential drop. The total input power to the electric arc is computed by multiplying this total potential drop on the arc for the plasma current, while the jet power, which is the output power of the thruster, can be estimated by assuming that it is a fraction of the convective loss power [11] to the anode, and it is computed by the following expression:

$$P_{jet}=\gamma P_{conv.anode}=\gamma {\displaystyle \frac{5}{2}}{J}_{z_p}{T}_e{A}_p$$

(9)

where $P_{jet}$ is measured in Watts, $J_z$ is the axial plasma current density expressed in A/m², $T_e$ is the electron temperature expressed in eV, and $A_p$ is the plasma cross-section. The proportionality factor $\gamma$ between the convective power and the jet power is computed from the solid angle of the loss cone, which is defined as

$$\mathsf{\Omega }=2\pi (1-cos\left({\theta }_{cone}\right))$$

(10)

and therefore, the factor $\gamma$ is defined by the ratio between the solid angle $\mathsf{\Omega }$ and the solid angle described by a semisphere, and it has the following expression:

$$\gamma ={\displaystyle \frac{\mathsf{\Omega }}{2\pi }}h=(1-cos\left({\theta }_{cone}\right))h$$

(11)

where ${\theta }_{cone}$ is the loss cone of the magnetic mirror already defined in Equation (5) and $h=h_1{h}_2$ is a correction factor of the axial loss of charged particles. $h_1$ models the increase in the lost particles due to collisions and kinetic microinstabilities of the plasma caused by anisotropy of the velocity distribution function, and it is assumed to be equal to 10 [12]. $h_2$ is the correction of the particle loss due to the real geometry of the plasma thruster (Figure 1). An annular anode allows the axial loss of charged particles but also sets up an electric field that deviates the particles towards its surface where the current path closes. This deviation of the charged particles towards the anode limits the axial loss and therefore the thrust of the plasma thruster. The correction factor to the loss cone $h_2$ is computed assuming that it is a function of the probability of losing an electron from a loss cone with a deviating electric field. First, the probability that an electron is deviated from the axial direction to the solid surface of the anode is assumed to be proportional to the ratio between the perpendicular ($E_{\perp }=\Delta V/L_2$) and the parallel ($E_{\Vert }=\Delta V/L_1$) electric field associated with the anodic region of the discharge. Then, the complementary probability that an electron can be lost ($h_2$) is computed as

$$h_2=1-{\displaystyle \frac{E_{\perp }}{E_{\Vert }}}=1-{\displaystyle \frac{L_1}{L_2}}\delta (L_2-L_1)$$

(12)

where $\delta (L_2-L_1)$ is a step function with the probability $h_2$ always defined between 0 and 1, $L_1$ is an axial length measured between the center of the magnetic mirror exit coil and the center of the annular anode, and $L_2$ is the radius of the annular anode, as illustrated in Figure 1. The jet power, as indicated in Equation (9), is a linear function of the electron temperature that is around 1 eV for an arc discharge. The electron temperature $T_e$ in this model is computed from the perfect gas state equation applied to the plasma, which is as follows:

$$T_e={\displaystyle \frac{p_e}{k_B{n}_e}}$$

(13)

where $p_e$ is the electron pressure, $n_e$ is the electron density, and $k_B$ is the Boltzmann constant. The equilibrium electron pressure on the axis of the arc discharge is computed, for a constant radial profile of the axial current density ($J_z$), from the analytical solution of the radial pressure profile for a z-pinch [3], and it has the following expression:

$$p_e={\displaystyle \frac{{\mu }_0{I}^2}{4{\pi }^2{a}_p^2}}={\displaystyle \frac{{\mu }_{0}I}{4\pi }}{J}_{z_p}$$

(14)

where $a_p$ (0.018 m) is the plasma radius and $J_{z_p}$ is the plasma axial current density, which is computed from the cathode axial current density $J_{z_c}$, assuming that the net current I in the plasma arc is constant, and it is computed as

$$J_{z_p}={\displaystyle \frac{J_{z_c}{A}_c}{A_p}}$$

(15)

where $A_c$ is the cathode cross-section (computed from Equation (17)) assuming a cylindrical cathode, and $A_p=\pi a_p^2$ is the plasma cross-section, also assuming cylindrical plasma. The cathode axial current density $J_{z_c}$ is computed from the Richardson law [13] that describes the thermoionic effect of a conductor when it is heated:

$$J_{z_c}=A_G{T}^2{e}^{-W/k_{B}T}$$

(16)

where $A_G$ = 0.6009 × 10⁶ A m⁻² K⁻² is a parameter that depends on the material properties to transmit the electron current, T is the cathode temperature, and $W=5$ eV is the extraction work of the cathode material (here tungsten). This equation allows us to compute the current density emitted from the cathode given the cathode temperature and its properties, and from this, it is possible to compute the cathode cross-section as

$$A_c={\displaystyle \frac{I}{J_{z_c}}}$$

(17)

The cathode cross-section should be compatible with the plasma cross-section and the plasma thruster size. We have chosen tungsten as the material for the cathode because it has the highest melting point among the pure elements. The material temperature must be controlled by a cooling system.

Finally, from the jet loss power, assuming that the ions are lost from the confined plasma through the loss cone at the sonic velocity (Bohm velocity), as stated in the Bohm criterion [14], we obtain the following expression for a thermal plasma:

$$v_{jet}=\sqrt{{\displaystyle \frac{e(T_e+T_i)}{m_i}}}$$

(18)

where e is the electron charge, $T_e$ is the electron temperature expressed in eV, $T_i$ is the ion temperature expressed in eV, and $m_i$ is the ion mass in kg; it can be computed the thrust exercised by the plasma thruster as

$$F={\displaystyle \frac{P_{jet}}{v_{jet}}}$$

(19)

Once the input power and the jet power of the plasma thruster are defined, it is possible to define the propulsive efficiency of the plasma thruster as follows:

$$\eta ={\displaystyle \frac{P_{jet}}{P_{in}}}$$

(20)

4. Results

We report here the main results obtained for the design of a plasma thruster based on a screw-pinch with a non-uniform axial magnetic field, with variation of the thruster propellants and the thruster geometry. The results have been obtained with the model described in the previous section that combines the physics of a magnetic nozzle with an arc discharge to compute the performance of the plasma thruster. The acceleration mechanism on which the model is based, from which the plasma jet velocity and the thrust are computed, is based only on the jet associated with the magnetic mirror concept [7] and not on the magnetic gradient effect (magnetic nozzle) [6] that would further increase the plasma thruster performance when considered. The inputs provided to the model for the estimation of the thruster performance are reported in the following

Table 1. Plasma parameters for the plasma thruster design.

Input	Value	Units
$n_e$	1.00 × ${10}^{18}$	particles/m3
$T_e$	1.59	eV
I	50.00	A
$B_{min}$	0.01, 0.03, 0.05, 0.08	T
$B_{max}$	0.10	T
$Z_i$	1.00	-
$h_1$	10.00	-
$h_2$	2.00, 5.00	-

, with some values specified for $B_{min}$ and also for $h_2$, with which a sensitivity analysis is performed.

Figure 2 shows the evolution of the input power required to feed the plasma thruster, with variation of the propellant species and the distance between the electrodes, according to the following equation:

$$P_{in}=V_{tot}I$$

(21)

assuming that this is the total input power provided to the plasma (here, we considered the power required to feed other on-board systems to be negligible). The results show that the input power increases with the plasma thruster’s axial size for a given plasma current (here fixed at 50 A) and it is very different from species to species, where hydrogen has the minimum and helium has the maximum input power required. Hydrogen and nitrogen seem to be the best propellants to choose for this type of plasma thruster, due to their lower input power required.

The thrust computed with variation of the plasma species (Figure 3) is low; indeed, this plasma thruster can be classified as a low-thrust propulsion system with a thrust level of the order of tens of millinewtons that increases with the atomic number of the propellant species. This increase in the thrust with the atomic number is due to the lower sonic velocity (Bohm velocity) of the more massive charged particles for a given jet power (here assumed equal to 33.95 W), and therefore, from the thrust point of view, the higher the ion mass, the higher the achievable thrust. The low thrust level makes this plasma thruster suitable for the transportation of low-weight payloads or for the attitude control of small satellites.

The efficiency of the plasma thruster is reported in Figure 4; it decreases with its axial size for a given current $I=50$ A, and it is computed considering only the acceleration of the particles inside the loss cone. Of course, this efficiency can be increased by considering a magnetic nozzle, which can further accelerate the charged particles, and increasing the thrust of the plasma thruster. From the efficiency profiles, it can be stated that the best propellant species to choose are still hydrogen and nitrogen, the ones with the lowest ionization energies. Therefore, the size of the thruster for a given mission must be selected in order to maximize its efficiency or equivalently to minimize its energy consumption.

Moreover, a sensitivity study for hydrogen gas has been performed on the thrust level given by variation of the magnetic field ratio (Figure 5) and by variation of the anode size (Figure 6). These are two parameters that can be modified in the design of the plasma thruster to achieve a controlled thrust. In particular, the thrust can be changed to a factor 7, changing the axial magnetic field ratio. The variation in the anode size also affects the thrust level of the plasma thruster, and in this case, an increase in the thrust can be obtained, moving the anode closer to the exit coil of the magnetic mirror and further from the axis. The results on the performance of the plasma thruster based on screw-pinch physics, obtained considering only the sonic particle jet generated from the magnetic mirror (without using a magnetic nozzle), showed that with a feeding power of the order of 1–2 kW, thrust levels of the order of 1–10 mN can be achieved. From the sensitivity analysis performed for hydrogen on the magnetic field gradient (Figure 5), it has been demonstrated that with this method, the thrust can be increased of a factor 10. In this way, the screw-pinch thruster can achieve a thrust-to-power ratio comparable to or even higher than that of the Hall thruster (0.05 N/kW [15]). In comparison to the Hall thruster, the non-uniform screw-pinch plasma thruster has a simpler design and its size is comparable to that of the Hall thruster (length of 10 cm and diameter of 6 cm with the cathode not included, as reported for the HT100 thruster [16]), making it a good alternative for the same missions. Assuming an increasing factor for the thrust of 10 for all the species (for the case with $B_{max}/B_{min}=1.25$),

Table 2. Thrust- to-power ratio for different propellants, with $B_{max}/B_{min}=1.25$ and $d=0.10$ m.

Species	${\mathit{H}}_2$	$\mathbf{He}$	${\mathit{N}}_2$	$\mathbf{Ar}$
$F/P_{in}$ (N/kW)	0.0169	0.0217	0.0603	0.096
$P_{in}$ (kW)	1.241	1.929	1.297	1.376

reports the corresponding thrust-to-power ratio, showing that argon would be the propellant with the highest thrust for a given feeding power, even if it is the one with the lowest specific impulse as computed in Figure 7. From the difference in thrust level and the specific impulse that determines the propellant consumption, it can be stated that argon could be preferred for mission profiles requiring high thrust for “fast” satellite motion (collision avoidance) while hydrogen would be preferable for lower but longer thrust mission profiles (orbit rising or attitude control).

5. Conclusions

We studied a new plasma thruster based on screw-pinch physics and the magnetic mirror concept. An arc discharge generates a thermal plasma column (with $T_i\approx T_e$) and a magnetic mirror topology allows us to control the axial loss of charged particles from the anode side of the magnetic configuration that can generate a thrust when integrated in a propulsive system. In the developed model, we considered the real geometry of the plasma thruster, made with an annular anode, also taking into account the correction due to the effect of an orthogonal electric field acting on the charged particle axial loss of the magnetic mirror. A thrust variation can be obtained during the thruster operations by changing the current that flows in the coils of the magnetic mirror and therefore changing the magnetic field intensity, obtaining a variable-thrust plasma thruster. The thrust level of the plasma thruster, controllable by changing the current in the coils, is in the range of tens of millinewtons, and therefore suitable for the transportation of small satellites and for attitude control systems. However, this thruster has a low efficiency that might be improved with a further acceleration system (magnetic nozzle), while the required input power increases with the axial size of the engine but remains around a few kilowatts for an arc regime with a current of around 50 A. All these features make this new concept of a plasma thruster suitable for operations in space.