Injection and propagation of a nonrelativistic electron beam and spacecraft charging

Two-dimensional numerical simulations have been carried out in order to study the injection and propagation of a nonrelativistic electron beam from a spacecraft into a fully ionized plasma in a magnetic field. Contrary to the earlier results in one dimension, a high-density electron beam whose density is comparable to the ambient density can propagate into a plasma. A strong radial electric field resulting from the net charges in the beam causes the beam electrons to spread radially, reducing the beam density. When the injection current exceeds the return current, significant charging of the spacecraft is observed along with the acceleration of the ambient electrons back to the spacecraft. As a result, a beam current much larger than the thermal return current can be injected into space. Recent data on the electron beam injection from the Spacelab 1 (SEPAC) are discussed.


I. Introduction
A number of artificial particle beam experiments have been carried out in the last decade using sounding rockets and artificial satellites (Grandal, 1982). Most of such active experiments are carried out in order to study the properties of the ionospheric and magnetospheric plasmas and magnetic field.
In more recent years, as the power of the electron beam becomes large, such artificial beams can be used to actively modify the ambient space plasmas.
One such example is the SEPAC experiment carried out on board the Spacelab 1 in which an intense electron beam was injected into the ionosphere in an attempt to produce artificial auroral discharges. (Kawashima, 1982;Obayashi et al., 1984).
As the beam electrons become more energetic, it is very important to understand the interaction between beam particles and the ambient plasma, including spacecraft charging, beam-plasma instability, enhanced ionization of the surrounding gas, and radiation generation. These processes collectively determine the propagation of the injected beam into the ambient space plasma.
There are a number of one-dimensional studies on the injection and propagation of a nonrelativistic electron beam into a plasma (Okuda et al., 1987;Okuda and Kan, 1987). It was found that the electron beam can propagate into an ambient plasma freely so long as the beam density is much less than the ambient density.
In particular, the injection current can exceed the return current to the spacecraft carried by the thermal electrons in an ambient plasma. This is because the electric field p-ising from the beam injection accelerates the ambient electrons toward the spacecraft (Okuda e_t al., 1987). A strong beam-plasma instability is found to accelerate the beam and the ambient electrons beyond the initial beam speed thereby forming a high energy tail. A presence of such a high energy electron population can lead to the generation of an anomalous conductivity where th« classical mean-free path after the beam-plasma instability may be longer than the mean-free path of the original beam electrons (Okuda et al., 1985).
When the beam density becomes comparable to the ambient density, beam propagation is prohibited due to the reflection of the injected beam electrons, causing a charging of a spacecraft.
These results found in one-dimensional simulation remain valid only in the presence of a strong axial magnetic field preventing the beam electrons from spreading radially. In two and three dimensions where the radial spread of an electron is allowed, the one-dimensional results at high beam densities may have to be modified, especially when the ambient magnetic field is not strong enough. Even in a completely neutralized beam, it is pointed out that the beam electrons suffer from radial diffusion much larger than the diffusion of the ambient electrons (Okuda, 1987). This is due to the resonance of the beam electrons with the waves generated as a result of beam-plasma instability. The electrons experience a D.C. electric field, u -kj v 0 * 0, resulting in a large radial excursion given by eg x B/B 2 drift, and the diffusion coefficient may be estimated by (o E»/B')v" , where E" is the amplitude of the azimuthal electric field and Y is the growth rate of the beam-plasma instability.
When the beam is not neutralized, such as in the case of beam injection from a spacecraft into the ionosphere, effects of non-neutrality and spacecraft charging must be included to determine the beam propagation and beam diffusion.
In this paper, we present results obtained from two-dimensional numerical plasma simulations in a uniform ambient magnetic field. Following the introduction, the simulation model is presented in Sec. II. The results of simulations on the propagation of the beam electrons and spacecraft charging are given in Sec. III. In Sec. IV, data from the SEPAC experiments are compared with the results of the numerical simulations.

II. Simulation Model
Consider an electron beam injected into space from a spacecraft. The beam electrons are injected along a magnetic field at a constant rate in time at a given energy and density into a fully ionized ambient plasma. A spacecraft is assumed to be a perfect conductor located at y < 0 as shown in Fig. 1. The ambient plasma is initially uniform and has Maxwellian velocity distributions for both ions and electrons with T = T,, We shall assume that the spacecraft is much larger than the dimension of an electron beam so that the spacecraft extends from x = 0 to x = L . An electron beam with the width L b is injected from a locali2ed area of a spacecraft into the ambient plasma which is charge neutral before injection.
The injection of an electron beam and the flux of the ambient electrons and ions into the spacecraft determine the charging. As in the onedimensional simulation models, we shall assume the ambient electrons and ions impinging upon the spacecraft are lost from the simulations via recombination. The same is true for the beam electrons which come back to the spacecraft after reflection. Ambient particles are reflected at y = L so as to maintain the initial uniform plasma density (Okuda et al., 1987). -Periodic boundary conditions are employed at x = 0 and x = L boundaries. On the surface of the spacecraft, the electrostatic potential 4>(x, y = 0) = constant holds since the spacecraft is assumed a perfect conductor. At y = L , we shall assume <j>(x, y -L ) = 0 since the location is sufficiently away from the beam and the spacecraft for most of the time. This is an approximate boundary condition which is valid only for L = -=; however, the choice of the other boundary conditions at y : L such as 3»(s, y = L )/3x = 0 which assume that the electric field lines are parallel to y at y = L did not significantly change the results of the simulations. In the simulations, the potential is solved in the following manner: since *{x, y) is assumed periodic in x, stx, For the k x -0 mode, we shall apply the same boundary condition as in our one-dimensional model (Okuda et al., 1987) so that 4. (y-L ) -a<v (y=L )/3y = 0. For two-dimensional k * 0 modes, *k fy = 0) = 9r, (y=L v ) -0 are used. In this way the spacecraft potential * 3 = 4(x, y=0) can be determined uniquely.
Ill. Results of Simulations tte shall first consider a case where the beam density n^ is equal to the average ambient density n Q (Case 1). The beam velocity is 10 times the thermal speed of the ambient electrons. In this case, strong reflection of the beam electrons has been observed in the one-dimensional simulation (Okuda and Kan, 1987). A 64 (L x ) x 512 (L ) spatial grid was used for most of the simulation results reported here except for Case 5 where a 128 x 512 grid is used. The beam radius L b = 8 & where A is the grid size was chosen. There are k electrons and ions per grid for the ambient plasma in the simulations.
The Debye length of the ambient electrons and ions is chosen equal to A. We assume that the ambient magnetic field is relatively weak so that u = 22 e is chosen, which may be realized at low ionospheric altitudes. Here u and ft g are the ambient electron plasma frequency and electron gyrofrequency, defined b y ""pe = (4nn 0 e 2 /m e ) 1/2 and a e = eB 0 /m e c.
Since the ambient electrons are assumed to be a Maxwellian distribution at t = 0 given by the thermal current to the spacecraft across the y = 0 plane due to the ambient electrons is given by, in the absence of an electric field, (2*)* so that |I + /I_| = v/Vg = (T i /T e ) 1/2 (nig/m^1 72 « 1. Typically, 1 & n l = 1 and nig/m^^ = 1/100 have been chosen for the simulations. In tne absence of beam electrons, thr electron and ion fluxes muse be equal in order to maintain charge neutrality. As a result, an ambipolar electric field will be generated which then forces the two fluxes to be the same. It is well known that the resultant flow speed is the ion sound velocity. In the presence of beam electrons, particle fluxes given by Eqs. (2) and (3) must be modified since a much stronger electric field is generated in a plasma, as we see later.
The injection current to the ambient plasma from the spacecraft is given by In this Case 1, |I_/lJ = (v e /v 0 ). (L x /L b )/(2it) 1/ld = 0,32 so that the injection current is much larger than the thermal current. Mote, however, the thermal currents defined by Eqs.
(2) and (3) are obtained in the absence of an elec tric field. As we shall see, the injection of beam electrons will generate a large electric field which can significantly modify the return current. Beam-electron phase for space, y-v , shown in Fig. 2(c), shows familiar holes and vortices characteristic of beam-plasma instability seen in onedimensional simulations (Okuda et al., 1987). The phase space, y-v v , of the ambient electrons in Fig. 2(d) shows that a fraction of the ambient electrons are accelerated beyond the beam energy as a result of trapping. These observations agree with the results of one-dimensional simulations except that, in one-dimension, beam propagation is possible only when the beam density is substantially smaller than the ambient density (Okuda and Kan, 1987). In two-dimensions, beam electrons spread radially as shown in Fig.   2(a) which reduces the beam density substantially below the ambient density.
In' this respect, there is good agreement between one-and two-dimensional simulations. where v is the average beam speed in the plasma, v is substantially smaller than the initial injection speed.
The spatially averaged electrostatic potential ? shown in Fig. 3(b) at u e t = 20 decreases slowly toward the spacecraft from $(y = L ) = 0 until y = 10OA, where the potential structure begins to show an oscillatory structure.
The potential of the spacecraft at this moment is negative, however, and the spacecraft potential has been observed to fluctuate in time suggesting the spacecraft charging at this beam density is negligible.
Ambient electron velocity distribution shown in Fig. 3(c) indicates a presence of accelerated electrons toward the spacecraft, v < 0, as well as in the direction of beam injection, v v > 0. It is clear that there are more electrons accelerated toward the spacecraft than trapped by the beam-plasma instability. Therefore, the flux carried by the ambient electrons accelerated toward the spacecraft (v < 0) is larger than that given by Eq.
(2) which assumes a thermal Maxwellian distribution without a net drift.
Beam electron velocity distribution shown in Fig. 3(d) indicates a rapid spread ir. velocity space. High energy electrons beyond the injection energy is seen; however, most of the beam electrons are decelerated via beam-plasma instability.
At a later time, w^t = 50, we found the beam can propagate more or less pe freely at this density as shown in Fig. 4. Beam electrons shown in Fig. 4(a) indicate that they propagate into the ambient plasma without reflection back to the spacecraft. As seen at an early time, radial spread reduces the beam density below the ambient density so that the spacecraft charging remains small.
Beam phase-space, y-v ,, shown in Fig is much easier to move ions across the magnetic field than to move electrons. In fact, ions may be treated as unmagnetized so that their motion across the magnetic field is primarily controlled by the radial electric field. Figure 5 summarizes the total number of particles in the plasma.  which is normalized by the total injection current in the system. We observe that the beam current and the ambient current more or less cancel out each other leaving little net current I = I + I b as shown in Fig. 6(c)." Note the beam current increases at a rate about one-half of the injection rate given by the broken line in Fig. 6(a) suggesting the average beam speed in a plasma is v Q /2. This agrees with the beam electron velocity distribution shown in Fig.   2(d) and also explains that the wavelength of the phase-space modulation in Fig. 4(b) is about TTV 0 /ID The spacecraft potential <t> s shewn in Fig. 6(d) oscillates at a frequency near ID and does not show any net potential in agreement with une-dimensional results at low beam density (Okuda and Kan, 1987).
In the following, we shall study the dependence of the beam propagation on beam density, beam energy, magnetic field strength, and the surface areas of the spacecraft, respectively.
(A) Increasing the beam der.sity (Case 2) When the beam density ic increased to n^/r\ Q = 2 while keeping all the other paramaters the same (Case 2), we found that a significant fraction of beam electrons is reflected back to the spacecraft. beam density (Okuda et al., 1987). This is also confirmed from the electrostatic potential contours shown in Fig. 8(a) and the averaged potential 5(y), Fig. 8 Fig. 10(b). The total number of electrons in the plasma shown in Fig. 10(c) and the total number of ions in Fig. 10(d) show that the number of the electrons is slightly more than the number of the ions, suggesting the spacecraft is charged positive in this case.
Figures 11(a) and 11{b) show the current carried by the ambient electrons I, averaged over the plasma region and the spacecraft potential. Again, we a note the absolute value of the return current is smaller than the injection current shown by the broken line, The beam current more or less cancels the ambient current so that the net current is found to oscillate around zero.
The spacecraft potential shown in Fig. 11(b) oscillates around the mean positive potential of e* s /T e -40, which is about 40? of the injection energy.
We have also run a simulation in which the beam velocity is increased by a factor 2 so that v Q /v e = 20 while keeping the beam density twice that of the ambient density as in Case 2. We found that the spacecraft potential remains positive at around e* g /T g -200 confirming that the spacecraft potential increases with the injection energy, provided the. beam density remains sufficiently high relative to the ambient plasma .derfsi'ty. This appears to have been observed in the SEPAC data as will be discussed in Sec. IV.
(B) Increasing the beam energy (Case 3) We shall now study what happens when the beam energy is increased. We use the same parameters as in Case 1 except for the beam velocity, which is now increased to v /v = 20 (Case 3). The injection current defined by Eq.
(4) remains the same as in Case 2 so that it is twice that of Case 1. This amount of injection current was too large compared with the thermal return current so that about one-half of the beam electrons is reflected back to the spacecraft in Case 2.
fls we shall see, the results of the present case are different from those in Case 2 despite the fact that the injection currents are the same for both cases. Instead, we find that the beam electrons manage to propagate into the plasma without reflection, similar to the first case. The results of the simulations at u t = 25 are shown in Fig. 12 for, (a) the locations of the beam electrons, (b) the ambient electron phase space, y-v y , (c) the potential contours and, (d) the averaged potential i(y). It is clear from Fig. 12(a) that the beam electrons can propagate into the ambient plasma wich little reflection so that the beam-plasma instability develops as seen in the phase space of the ambient electrons in Fig. 12(b) where trapped particle vortices are clearly seen. The electrostatic potential contours show a presence of regular patterns determined by the size of the beam radius and the wavelength of the beam-plasma instability as seen in Fig. 12(c).
The spacecraft charged up positively at this moment as seen in Fig. 12(d) with a wavy structure away from the spacecraft characteristic to the beamplasma instability. These results of simulation are similar to 'what we have r seen in Case 1 rather than the results found in Case 2. These results are in.
agreement with the one-dimensional results where the injection ourrent much larger than the thermal return current can be injected only when the beam velocity is increased while keeping the beam density low (Okuda et_al., 1987). In this Case 3, the injection current I Q is 6.25 times the thermal return current, |I Q /I_| = 6.25. Beam electron velocity distribution along the magnetic field indicates the presence of a beam toward the spacecraft similar to that found in Case 1. These electrons represent an enhanced return flow of the ambient electrons.
The total number of beam electrons in a plasma increases linearly in time at the injection rate as shown in Fig. 13(a) while the number of ambient particles decreases in response to the injected electrons as shown in Fig. 13(b). The total number of electrons stays about the same as shown in When the ambient magnetic field is increased to fl e^W p e = 2 from n e /a> = 0.5 in Case 1, we may expect the electron motion will become one-dimensional since the motion across the magnetic field becomes prohibited. This does not mean, however, the whole dynamics can be approximated by a one-dimensional model since the width of the electron beam remains small in comparison with the spacecraft or the propagation length in the direction of magnetic field.
Electric field lines can bulge out of the beam electrons in two-dimensions which weakens the electric field along the magnetic field. As a result, the reflection of the beam electrons is reduced in two-dimensions compared with the one-dimensional simulations. This makes beam propagation possible even though the radial expansion of the beam is much smaller compared with Case 1 when the magnetic field is weak.
Using the same parameters as in Case 1, the ambient magnetic field is increased to 3 /u = 2 which may be relevant to the ionospheric plasma at an altitude of a few hundred km. The results shown in Fig. 15 are for (a) the beam electron spatial distribution in the (x-y) plane, (b) the ambient electron spatial distribution, (c) the beam electron phase-space, y-v", and (d) the ambient electron phase space, y-v", at u n _t = 50. Compared with Figs.
2, 4, 7, 9, and 12 which correspond to a weaker magnetic field, it is clear that the radial expansion of the beam and the ambient electrons are much smaller in trie present rjase. The beam electrons, in fact, propagate along the magnetic field without appreciable radial expansion as shown in Fig. 15(a).
The ambient electrons shown in Fig. 15 The electrostatic potential contours shown in Fig, 16(a) confirm that the structure of the electric field is similar to that in Case 1 so that the electrostatic reflection of the beam electrons back to the spacecraft is not much stronger here. The ambient electron velocity distribution along the magnetic field shown in Fig. 16(b) is again similar to Fig. 4(c) indicating a presence of the enhanced return flux to the spacecraft and the accelerated electrons in the direction of beam propagation. It is interesting to observe that the ion density becomes peaked at the center of the system as shown in Fig. 16(c). This is because the electrons cannot diffuse across the magnetic field in two-dimensional simulation where the cE x B/B drift is in the z-direction (which is the ignorable coordinate). Ions are attracted by the electrons since the ion mobility across the magnetic field is much larger.
The total electron density is almost identical to the ion density shown in Fig. 16(c). Beam electrons spread out more along the magnetic field as shown in Fig.16(d) where the energy of the accelerated electrons exceed the beam injection energy significantly. This is similar to the earlier onedimensional results where the high energy electrons can attain an energy twice that of the injection energy (Okuda et al., 1985). Eqs. (2) and (4), the return current carried by the ambient electrons is proportional to the surface area so that if the area is increased relative to the beam radius, it should be possible to inject a beam with higher density without significant reflection. If the beam density is too high compared with the ambient density, on the other hand, the local electric field may be strong enough to reflect the beam electrons back to the spacecraft as in the case of one-dimensional simulations (Okuda and Kan, 1987).
In Case 5, the simulation system is increased in x-direction by a factor of 2 from the other cases so that L x = 128A. We shall assume the beam density is twice the ambient density and the beam injection speed v /v = 10 so that the total injection current is the same as that of Case 2 where strong reflection of the injected beam to the spacecraft was found. According to Eq.
(2), the return current must increase by a factor of 2 in this case, so that a smaller amount of charging of the spacecraft may be expected to take place.
Results of the simulations in this case are shown in Fig. 18 for (a) the locations of the beam electrons in (x-y) space, (b) beam-electron phase space in (y-v v ) space, (c) electrostatic potential contour, and (d) the spacecraft potential e* s /T e at u De t = 50, Beam electrons propagate into the ambient plasma without significant reflection as shown in Fig. 18(a) and the phase space indicates a presence of trapped particles in Fig.18(b). Note the rapid radial expansion of the beam electrons near the injection point.
Electrostatic potential contours also indicate a coherent structure associated with the beam propagation, and the spacecraft potential, e$ /T , oscillates in time, although its mean value is positive, e* /T -25, suggesting that the spacecraft is charged up to the potential equal to one-quarter of the beam energy.
Time history of the total number of beam and ambient electrons and their currents are shown in Fig. 19. The number of the beam electrons increases linearly in time suggesting no reflection as shown in Fig. 19(a) until the end of the simulation when a small fraction of beam electrons is reflected. The ambient electrons decrease at the injection rate as shown in Fig. 19(b). The beam current, in the plasma increases at a rate roughly one-third of the injection current, which is more or less cancelled oy the return current carried by the ambient electrons to the spacecraft as shown in Fig. 19(c) and (d).
The results of this" simulation confirm that the increase of the conducting area of the spacecraft helps increase the injection current by drawing more return current to the spacecraft from the surrounding plasma.

IV. Discussion
The results of the two-dimensional simulations of the injection of an electron beam into a plasma have been presented in which the beam electron -19density is comparable to or larger than the ambient electron density and the injection current is larger than the thermal return current carried by the ambient electrons. In contrast to the earlier results in one-dimensional simulations, an electron beam can propagate into the ambient plasma when the density is comparable to or larger than the ambient density. This propagation is attained partly because of the radial spread of the beam electrons after injection due to the radial electric field and partly due to the finite radius of the beam which causes the bulging of the electric field along the beam injection.
Spacecraft charging remains negligible even when the injection current exceeds the thermal return current so long as the beam density is not much larger than the ambient density. This result agrees with the one-dimensional simulations where the electric field generated as a result of beam injection accelerates the ambient electrons toward the spacecraft, thereby cancelling the injection current. The resultant spacecraft charging remains small.
In order to increase the injection current, it is more effective to increase the beam energy rather than to increase the beam density (Cases 2 and 3). This is consistent with the results found in one-dimensional simulations.
When the size of the spacecraft is increased relative to the size of the beam, we found that the beam current can be increased while no reflection of the ambient electrons takes place (Case 5). This is in agreement with Eq. (2) in which the total return current increases with the conducting area of the spacecraft. The beam current in a plasma is generally smaller than the injection current for all cases. High energy electrons are found in both beam and ambient phase-space which contributes to the enhancement of the classical mean-free path as in one-dimensional results (Okuda et al., 1985).
The results of the present two-dimensional simulations may be used to analyze data of space experiments with particle accelerators (SEPAC). The SEPAC experiments involved injection of an electron beam into the ionosphere using an electron beam accelerator {Obayashi et al., 1984). It was found that the spacecraft potential increased with the injection current reaching the accelerator voltage of 7.5 keV for a beam current of 100 mA. P-syond that point, the voltage stayed constant, suggesting the beam electrons were not able to leave the spacecraft.
The ambient ionospheric electron density near the 200 km altitude is n < 5 x 10' cnT^ and the temperature T = 2Q00°K so that the maximum thermal return current would be en Sv I = ° T < 50 mA , (6) C2TT) 1 " where S = 60 m is the conducting area on Spacelab 1, Therefore, we find the beam injection currents of 100 mA and beyond are significantly larger than the maximum thermal return current so that significant charging should take place. This is in good agreement with the results of numerical simulation, suggesting the spacecraft potential must approach the accelerator potential.
The beam density is found from where a is the beam cross section, a = 12 cm 2 at the end of the accelerator but it becomes larger as the beam electrons move out of the spacecraft.
Assuming a = 200 car at the injection point, n. = 10 6 cm -3 , o which is about twice the ambient ionospheric density corresponding to Case 2 where a significant charging of the spacecraft and beam reflection take place. Realizing that the electron plasma frequency is f e = 6 x 10° em << f ce , the beam electrons should be able to expand radially once injected into the ionosphere as seen in our simulations for a weak magnetic field.
We should emphasize that while the simulation results reported here are, in general, consistent with the data from SEPAC, the ionosphere is far more complex than the simulation model used here. For example, the ionosphere is a weakly ionized plasma so that the neutral gas density is much larger than the plasma density. Even a small amount of additional ionization of the neutral gas due to high energy electrons can modify the ambient density, thereby modifying the spacecraft potential.
While we have studied only the injection of the electron beam alonp" the ambient magnetic field, it is important to study the oblique injection since the injection angle in real space experiments cannot be controlled as the spacecraft moves across the magnetic field. Furthermore, at. a higher altitude of the ionosphere, the effect of the ambient magnetic field becomes important since the beam electrons cannot expand freely as seen in Case 4, In such a case beam rotation around the ambient magnetic field and the associated plasma instability is important in determining the propagation and the stability of the beam. Such a process can be studied by using a three-dimensional simulation model with elongated grid (Cheng andOkuda, 1977, Okuda et al.. 1979). These results will be reported elsewhere.   shown by the broken line in Fig. 10(a).    Vy/V. Fig. 16