AIAA Simulations of Plasma Detachment in VASIMR

Transcription

1 AIAA Simulations of Plasma Detachment in VASIMR A.V. Ilin, F.R. Chang Díaz, J.P. Squire, A.G. Tarditi, ASPL, JSC / NASA, Houston, TX, B.N. Breizman, UT, Austin, TX, M.D. Carter, ORNL, Oak Ridge, TN 40th AIAA Aerosace Sciences Meeting & Exhibit 4-7 January 00 / Reno, NV For ermission to coy or to reublish, contact the coyright owner named on the first age. For AIAA-held coyright, write to AIAA, Permission Deartment, 80 Alexander Bell Drive, Suite 500, Reston, VA

2 SIMULATION OF PLASMA DETACHMENT IN VASIMR Andrew V. Ilin, Franklin R. Chang Díaz, Jared P. Squire, Alfonso G. Tarditi, Advanced Sace Proulsion Laboratory, JSC / NASA, Houston, TX, Boris N. Breizman, University of Texas at Austin, Austin, TX, and Mark D. Carter #, Oak Ridge National Laboratory, Oak Ridge, TN ABSTRACT Detachment of the lasma exhaust is an essential element of a magnetic nozzle oeration that requires exerimental, analytic and comuter simulation studies. The resent work is devoted to comuter simulations of the lasma detachment in the Variable Secific Imulse Magnetolasma Rocket. Both article simulation and MHD methods have been used and results are reorted in this aer. INTRODUCTION A magnetic nozzle lays two imortant roles in the oerations of the Variable Secific Imulse Magnetolasma Rocket (VASIMR),,3,4. First, the erendicular siral motion of ions is converted into the axial motion. Second, the lasma detaches from the thruster. Both effects are critical in roviding rocket thrust. The goal of mathematical simulations, reorted in the aer, is to hel understand the hysics of lasma detachment and to assist in the design of a thruster suitable for an actual flight test. Comuter simulations of lasmas are tyically done by using one, or a combination of three general techniques: fluid models, numerical solution of kinetic equations (Vlasov / Fokker-Planck), and article models. Recent growth in comuter seed and memory of has made the article descrition articularly attractive 5. Particle methods, both Particle-in-Cell (PIC) and direct simulation Monte-Carlo (DSMC) methods, have been effectively used for simulation of Pulsed Plasma Thrusters (PPT) 6, Hall Thrusters 7 as well as Ion Thrusters 8. In this aer we describe a article trajectory method 9, and comare it with other modeling techniques to rovide a quite accurate descrition for VASIMR lasma exhaust. The article code is used for demonstrating lasma detachment in VASIMR. The results of the article code are comared with results of MHD codes and with analytical considerations. PREVIOUS PLASMA DETACHMENT STUDIES Previous studies of the lasma detachment in a magnetic nozzle used simle models for the lasma flow. Kosmahl s 0 and Sercel s models involve calculation of trajectories of ions and electrons guiding centers for given vacuum magnetic field. They observe lasma detachment by analyzing how the ion velocities behave and how the ion trajectories cross magnetic field lines. The study by Sercel showed the imrovement of roulsive efficiency of the magnetic nozzle by introducing extra tuning electric coils, acting as magnetic lens. York s and Hooers s 3 model considers a single-fluid MHD flow for given magnetic field. The condition for detachment in Hooer s study is given by the scaling arameter G <,750, defined as G = (eb z r (z 0 )) / (m e m i (u(z 0 )) ). This work continues the revious studies of lasma detachment in the magnetic nozzle by considering additional simulation methods to analyze the detachment. MAGNETIC CONFIGURATION IN VASIMR The VASIMR system consists of three major magnetic cells, denoted as forward, central, and aft,,3,4. A magnet configuration examle (related to a 4 kw VASIMR thruster 4 concetual design) and the corresonding magnetic field rofile is shown in Figure. The forward end-cell rovides the injection of the neutral gas to be ionized by electromagnetic waves that are roduced by helicon antenna. In the central-cell, Research Scientist, Muniz Engineering, Inc. ilin@jsc.nasa.gov NASA Astronaut, ASPL Director Senior Research Scientist, Muniz Engineering, Inc. # Senior Research Scientist, SAIC Research Scientist, Institute for Fusion Studies Research Staff, Fusion Energy Division Coyright 00 by the, Inc. No coyright is asserted in the United States under Title 7, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the coyright claimed herein for Government Puroses. All other rights are reserved by the coyright owner.

3 the lasma is also electromagnetically heated by waves oerating near the Ion-Cyclotron Resonant Frequency (ICRF). The aft end-cell ensures that the lasma will efficiently detach from the magnetic field to rovide roulsion through a highly directed exhaust stream. This magnetic configuration allows the lasma exhaust to be guided and controlled over a wide range of lasma energies and densities. Figure : Geometry and magnetic field configuration for 4 kw VASIMR thruster. In Figure, the forward end-cell is located at z < 0.7 (m), the central cell at 0.7 < z < and the aft end-cell is at < z. In this aer we investigate the lasma exhaust in the aft end-cell in the domain < z < 5. The osition z 0 = (m) will be referred as the inlet of the exhaust. Currently, the VASIMR system is under develoment for a first sace flight exeriment using 4 kw of DC electric ower. In the future, several megawatt VASIMR thrusters will be considered for human interlanetary flights to Mars and beyond 5. This aer deals with exhaust lasma detachment for the 4 kw VASIMR thruster (VF-4), which is assumed to have the oerational arameters shown in Table. Oerating Definition/Tyical value arameter. Inut ower P = 4,000 (W). Power efficiency ε = 0.4 (fraction of the inut ower going into the thrust ower). 3. Secific imulse I s = 0,000 (sec). 4. Exhaust thrust velocity u = I s g = 0 5 (m/s) 5. Thrust force εp F = u = 0.9 (N). 6. Proellant rate F 6 m& = =.9 0 (kg/s). u 7. Proellant ion mass in terms of roton mass m i = m (Deuterium) 8. Exhaust ion energy miu Wi = =00 (ev) e 9. Radius of lasma at the exhaust inlet r (z 0 )=0.05 (m) 0. Directed energy of W z (z 0 ) = ½ W i = 50 (ev) the lume at the exhaust inlet. Electron temerature T e = 5 (ev) is assumed constant. Orthogonal ion T i = ½ W i = 50 (ev) temerature at the exhaust inlet 3. Average ion density m& at the exhaust inlet ni = = m πr ( z ) v ( z ) 4. Maximum ion density at the exhaust inlet i 0 8 (m -3 ) n max = n i = 0 8 (m -3 ), assuming arabolic density rofile n i (r, z 0 ) = n max ( r /r (z 0 )). Table : Definitions and tyical values of oerating arameters for 4 kw VASIMR thruster. Axial rofiles for basic lasma arameters are shown in Figure. The Debye length λ D is much less than the lasma radius r everywhere, therefore the lasma quasi-neutrality will hold excet in very localized sheath regions. Although collisional rocesses can lay an imortant role in the lasma source region, the exhaust can be assumed reasonably collisionless beyond the central section as long as the mean free ath λ mf for various collision rocesses is much larger than lasma characteristic length. While the electrons follow the magnetic field through most of the region of interest, the ion Larmor radius r i eventually becomes L larger than the magnetic field curvature radius. 0 0

4 increases downstream. When beta is greater than unity, the lasma has enough energy to stretch the magnetic field lines along the flow and thereby detach from the thruster. The condition β > also means that the flow velocity is greater than the Alfvén velocity, which shows that lasma detachment is essentially a transition from sub-alfvénic to suer-alfvénic flow. Similarly to what occurs in a suersonic flow created by the Laval nozzle, erturbations in the suer-alfvénic lume cannot be left in the sub-alfvénic region and leave the system together with the outgoing flow. Our numerical simulations confirm this behavior. PARTICLE TRAJECTORY SIMULATIONS OF THE PLASMA PLUME The magnetic and electric fields, lasma density and electric current density are assumed at steady state and having cylindrical symmetry, which leaves deendence only on radial and axial coordinates r and z. The magnetic field vector is a sum of the fields generated by the external coil and by the lasma current: B(r, z) = B 0 (r, z) + B (r, z). The electric field in the lasma lume is the ambiolar electric field: E(r,z) = E (r, z). Figure : Plasma density rofile (to) and scale lengths (bottom) of 4 kw VASIMR thruster. ANALYTIC ESTIMATES FOR PLASMA DETACHMENT A scaling analysis, also roosed by Dr. Roald Sagdeev (and reroduced below), is helful in understanding some of the hysics. In the VASIMR lasma exhaust, the magnetic flux is conserved. Thus, if B is the vacuum magnetic field and r is the radial coordinate lasma radius r, then B ~. r From article flux conservation, the lasma density n has the same deendence on r as B: n ~ B ~. r Assuming that ion kinetic energy is a constant in the lasma exhaust (when the axial ambiolar electric field is neglected): W ~ const, the lasma beta, defined through the total energy W, has the following deendence on lasma radius: nw β ~ ~ ~ r. B B This estimate shows that the ratio of lasma kinetic energy density to the magnetic field energy density The article simulation currently incororates five integrated models for calculation of these fields as shown in Figure 3. Figure 3: Mathematical simulation of lasma in VASIMR. First, the magnetostatic field roduced by the magnet coils, B 0 (r, z), is accurately generated. This field does not change throughout the remaining calculation. Second, a fully nonlinear article model, that is currently collisionless, calculates the ion ositions and velocities x i (t), v i (t) by solving the momentum equation. 3

5 The Lorentz force is calculated from the static and RF fields obtained from the Maxwell equation solvers in a three-dimensional sace. Third, the ion density, n i (r, z) and ion current density j i (r, z) are calculated from the ion ositions and velocities averaging over the gyro-motion, using a article to grid weighting. The resulting ion density is aroximately equal to the electron density in a quasineutral aroximation, and it is fed into the fourth ste. The fourth ste iterates over the revious models using n i (r, z) and a Boltzmann aroximation for the electron distribution to solve Poison s equation for the steady state lasma otential giving E (r, z). This field further modifies the lasma density and hence the RF couling. The steady state lasma current density, j i (r, z) becomes imortant in the exhaust region where B (r, z) can become significant comared with the fields from the magnet coils. Thus, the fifth and final ste calculates steady state magnetic field corrections caused by the lasma in the exhaust region. ) Particle Dynamics. The ion motion satisfies the following equation of motion: dvi mi = e( vi B + E ). (3) dt The single article trajectories are integrated from equation (3) with an adative time-scheme, which can quickly solve extensive article simulations for systems of hundreds of thousands of articles in a reasonable time (- hours), and without the need for a owerful suercomuter. The article calculation method is described in revious ublications 8,9,0. Figure 4 illustrates magnetic field lines and a tyical ion trajectory in the exhaust area of the VASIMR. From the ion trajectory observation, one can see the beginning of the article detachment from the magnetic field in the exhaust area with weak magnetic field. ) Magnetostatic Equations. The magnetostatic roblem is a steady-state case of two vector Maxwell equations: B = j B = A, () µ 0 0, where B 0 is the vacuum magnetic induction vector, µ is the magnetic ermeability, j 0 is the current density in electromagnets and A 0 is the magnetic vector otential. When modeling the VASIMR system, the assumtions of cylindrical symmetry and constant magnetic ermeability µ are valid. In that case, the magnetic vector otential A 0 (as well as current density vector j 0 ), written in the cylindrical coordinate system (r, φ, z), has only an azimuthal nonzero comonent: A = (0, A φ (r, z), 0) and the roblem () can be rewritten in the following form: Φ Φ r = µ rj0, () r r r z where Φ(r, z) = r A φ (r, z) is the magnetic flux. 0 0 Equation () is solved with a high level of accuracy using a finite difference scheme, which is solved by a fast iterative method, described in revious ublications 6, 7. Figure demonstrates the numerical solution for the vacuum magnetic field for a 4 kw VASIMR thruster. The calculated vacuum magnetic field B 0 is used in both article and MHD calculations. Figure 4. To: Magnetic field lines and test ion trajectory in the VASIMR. Middle: Total, axial and erendicular energies of the test ion. Bottom: Magnetic moment of the test ion trajectory. The ion detachment due to Larmor radius increase is observed. 4

6 The ion Larmor radius r L = m i v i / (q B) is rather small inside and near the thruster. However, it increases significantly down the flow, when the trajectory converges to a straight line. In the magnetic nozzle, the ion rotational energy converts into axial energy, which is demonstrated in the middle art of the Figure 4. Since the magnetic flux B r is a constant, where r is a lasma radius, the magnetic field goes down as fast as r -. The calculations show in the bottom of Figure 4, that the magnetic moment µ i = m i v i / ( B) is aroximately constant inside and near the thruster (0.7 < z <. m) but goes u in the exhaust area down the flow. It is accomanied with the erendicular velocity v going down as fast as r -. This makes the Larmor radius r L = µ i / (q v i ) increase faster than the magnetic moment. Note, that we moved the inlet osition for the articles, from z = m (B = 0.65 T) to z = 0.7 m (B = T). This allows us to have à refluid stabilization domain 0.3 m long. In that stabilization domain the magnetic moment was conserved and initial conditions for articles where taken as W = 0 ev, T = 80 ev at z = 0.7 m, to yield conditions W = 50 ev, T = 50 ev at z = m. This comutational ste gives a smoother calculated fluid variables of lasma density n, electric field E and magnetic field B for z > m. Figure 6 demonstrates the trajectory of a single electron with 5 ev energy. In contrast to the ions, the comlete electron attachment to the magnetic field is observed. During extensive article simulation, one needs to define an initial ion velocity distribution. Current simulation assume a Maxwell distribution of ion velocity before going into the nozzle at z = 0.7 m. Figure 5 demonstrates initial ion velocity distributions inside the VASIMR engine at z = 0.7 (B = T) and recalculated distribution at the boundary area with magnetic moment being conserved (z =. m, B = T). Figure 6: Magnetic field lines and test electron trajectory. The comlete electron attachment is observed. 3) Particle to Grid Weighting Figure 5: Initial ion velocity distributions at z = 0.7 m, B = T and at z =. m, B = T. The ion density n i is calculated by using a weighting method 5,0 for method of trajectories 9. With a given distribution for the initial osition and velocity vector, a large number (order of 0 5 ) of ion trajectories is 5

7 calculated. Every single trajectory is used to generate a number of articles distributed along it with equal time ste between them. Plasma density clouds with a certain weight and a size of the finite difference cells are roduced around each article oint, which after summation, became discrete ion density n i defined constant at each finite difference cell, using the following formula n ( X ) = w Q( X x, i j i j k ) k where X j is a osition of the j-cell, x k is a osition of k- article, w I is a article weight, Q(.) is a cloud density function. In our simulation, we used a continuous iece-wise-linear function with a suort equal shae of the j-cell. The article weight w i is calculated, such that it makes the grid density equal given value at given oint: n i ( X 0 ) = n. The ion current density j i is calculated by a technique, similar to that used to calculate the ion density in Figure. Namely, 0 i j X ) = ew Q( X x ) v. (4) i( j i j k k k kte ni ϕ = ln. (9) e n0 To avoid ln(0) calculations, the ion density can be adjusted by some small ositive constant. Equation (9) has to be solved in the loo with article simulations for the ions using an udated electric field. To achieve convergence in the self-consistent lasma electric field calculations, under-relaxation (daming) is needed for udating the electric otential: * ϕ new = τϕ + ( τ ) ϕ new with a relaxation arameter τ <. In ractical simulation, 0 iterations were enough to get a convergence with the relaxation arameter τ = 0.3. Figure 7 demonstrates the electric otential solution for the lasma system shown at Figure. The negative ambiolar electric otential in the exhaust area accelerates ions, which adds to the VASIMR erformance. 4) Electrostatic Equations. The electric field E can be calculated from the electric otential ϕ: E = ϕ. (5) In the system with cylindrical symmetry, it yields that E has only radial and axial nonzero comonents. The electric otential also satisfies the Poisson equation: ϕ = e( n i n e ), (6) where the right-hand side is a lasma charge density. To avoid calculation of the electron density function, the Boltzmann relation is used: eϕ ne = n0 ex, (7) kte where the bulk electron density n 0 is assumed equal the ion density at the lasma inlet (assuming that ϕ=0 there), and is a constant function along every magnetic field line. In the resent simulations the electron temerature T e is assumed constant. Due to the very small value of the Debye length for the studied lasma system: kt 4 λd = < 0 m, (8) 4πne the Poisson equation (6) can be simlified to the quasineutrality relation: n i = n e, which gives the following formula for the electric otential: Figure 7. To: Ion energy and temerature and electric otential in the 4 kw VASIMR thruster. Bottom: corresonding D contour-lot of the electric otential and magnetic field lines. The electric field is calculated self-consistently with the lasma density, shown in Figure. 5) Calculation of the internal lasma magnetic field To calculate the lasma magnetic field, the lasma current has to be derived first. In an axisymmetric system with cylindrical symmetry, this current is urely azimuthal. 6

8 The momentum balance equation ρ ( u ) u = j B () yields the following deendence of lasma current due to the curvature of the vacuum magnetic field in the exhaust: ( u ) u B j = ρ. () B The lasma current generates a lasma magnetic field that satisfies Amere s law: = j. (3) µ B The ratio of the lasma magnetic field to the vacuum magnetic field can be estimated by the following exression: B B µρu r r = β, (4) B a a where a is the magnetic field curvature. The article simulation results are consistent with this analytical estimate. The internal lasma magnetic field can be calculated using the same solver, as used for the vacuum magnetic field calculation. The only difference in this calculation is a current density source j. The calculation of B should be iterated with the calculation of lasma velocity and density. Figure 9: Plasma current density (diamagnetic and curvature) and corresonding magnetic field lines. As numerical exeriments show, the lasma-generated magnetic field in the area close to the thruster core is due to the diamagnetic effect and has an oosite direction to the vacuum magnetic field. In the exhaust area, when ions detach from the vacuum magnetic field, the lasma-generated field is due to the curvature current, which has the same direction. Although the value of beta reaches unity at z = 3m, the lasmagenerated field is still smaller than the vacuum magnetic field for the studied range of arameters. That is shown in the Figure 0. Figure 8 demonstrates the lasma current density due to the curvature of the magnetic field calculated by formula () and the magnetic field calculated by equation (3). In the lasma exhaust, the lasma magnetic field lines aroach the direction of the z-axis. Figure 8: Curvature current density and corresonding magnetic field lines. Note, that comlete lasma current density calculated from article trajectories by the formula (4) includes both magnetic field curvature and diamagnetic effects. The diamagnetic effect is essential only for high magnetic field in the area close to the thruster core and becomes negligible further away from it. It is demonstrated in Figure 9. Figure 0: Demonstrating lasma detachment due to increasing lasma beta and lasma magnetic field aroaching a constant. Plasma magnetic field B, calculated by the article code is correlated well with semianalitical disturbed field δb, defined from the formula (4). 6) Observation of lasma detachment During article simulations we have observed the following indications of lasma detachment in VASIMR: ) Axial ion energy W z aroaches a constant, which indicates that ion motion is not affected by the magnetic field, as shown in Figure 4. Thus, the velocity distribution function on the 7

9 axis becomes indeendent of z for large z. The examle of the velocity distribution function is demonstrated in Figure. Figure : D icture of lasma beta in the VASIMR exhaust. MHD SIMULATION WITH THE NIMROD CODE Figure : Calculated velocity distribution function using article simulation. The effect of magnetic nozzle is observed. ) Magnetic moment µ i increases (Figure 4). 3) Plasma β increases above. For the small ower sace-flight exeriment VASIMR configuration, lasma β goes above one at the distance of.8 m from the nozzle magnet. Figures 0 and demonstrate lasma β along axis z and in two-dimensional cross-section. 4) Plasma magnetic field is aroaching a constant in the lasma exhaust (Figures 8, 9, 0). As demonstrated in Figure 0, the ratio of lasma magnetic field B to the vacuum magnetic field B 0 goes u from order 4 at the nozzle magnet to order at m from the nozzle magnet. Even in the area of lasma β larger than one, the lasma magnetic field is still much less than vacuum magnetic field. Also, Figure 0 demonstrates that lasma magnetic field B, calculated by the article code correlates well with semianalitical disturbed field δb = (r / a ) B, defined from the formula (4). 5) Ion Larmor radius r L becomes greater than vacuum magnetic field line curvature a. A 3-D, MHD simulation tool is being develoed and validated to bring the resent theoretical analysis of the VASIMR nozzle/exhaust lasma region to a more accurate level. The simulation is able to reroduce quantitatively the exhaust lasma rofiles at a distance from the engine where the detachment takes lace. The code is based on an ugraded version of NIMROD, a code built for three-dimensional nonlinear fluid modeling of magnetized lasmas. This code is the result of a still ongoing multi-institutional effort suorted by the U.S. Deartment of Energy and it is available in the ublic domain. For this alication, the NIMROD has been ugraded to introduce the density equation in the model (removing the incomressible fluid aroximation of the current NIMROD release), and a rovision for oen-end boundary conditions and a lasma source term to simulate the exhaust flow. The alicable set of equations is: ) Maxwell Equations B = E, B = j t µ ) Continuity Equation ρ = ρu t 3) Momentum Equation u ρ + u u = ( I + P) + j B t 8

10 4) Energy Equation 3 + u = u P : u t 5) Ohm s Law (ideal MHD) E = u B The resent simulations use the viscosity coefficient η to obtain a closure for the stress tensor as P = η u. Preliminary tests have been erformed in a D (r-z) section of a cylindrical domain in D with the axis along the z direction. Oen-end boundary conditions are imosed, along both the longitudinal and the radial direction. A lasma source term is injecting lasma in a low-density background environment and over time a lasma ulse is formed and starts to roagate along the magnetic nozzle. The simulation shows the initial transient of the ulse formation before the equilibrium between the exhaust rate and the injection rate is reached. The results of the MHD codes are being used for validation of the article code results and to obtain a better understanding of the lasma detachment. As shown in Figure, the results of article simulation and MHD simulation on lasma beta analysis are very similar. CONCLUSION The described article simulations in VASIMR demonstrate lasma detachment from the magnetic nozzle. Reasonable agreement between MHD and article simulation is observed in lasma beta detachment analysis. The codes develoed so far are being validated in the VX-0 laboratory exeriment and assisting researchers in the design of a VASIMR flight demonstration exeriment. NOMENCLATURE A magnetic otential (Weber / m) a magnetic field curvature (m) B magnetic induction (0 Tesla) E electric field (Volt / m) e electron charge ( Coulomb) F thrust (0. 0. N) G dimensionless detachment scaling arameter g gravitational acceleration (9.8 m / s ) I s secific imulse ( s) I identity tensor j current density (0 0 5 Amere / m ) k Boltzmann constant ( J / K) m article mass (D ion: kg) m& roellant flow rate (0-6 kg / s) n lasma article density (0 0 8 m -3 ) P ower (4,000 Watt) lasma ressure (Pa) Q article cloud function r, φ, z cylindrical coordinates: radial (m), azimuthal (rad) and axial (m) r L Larmor radius (0 - m) r lasma radius ( m) t time (s) T e electron temerature ( 0 ev) T i ion temerature (0 00 ev) u exhaust (fluid) velocity ( m / s) v article velocity (ion: m / s) W energy (ev) w article cloud weight X cell osition (m) x i ion 3-D osition vector (m) β ratio of lasma kinetic ressure to the magnetic ressure (0 0) η viscosity (kg / (m s)) ε ower efficiency (0.4) Φ magnetic flux (Tesla m ) ϕ electric otential ( Volt) λ D Debye length (0-4 m) λ mf ion mean free ath ( m) µ magnetic ermeability ( Henry / m) µ i magnetic moment P stress tensor π ρ lasma mass density ( kg / m 3 ) τ relaxation arameter (0.3) Subscrits: 0 vacuum, inlet e electron i ion j cell k article L Larmor lasma, roton orthogonal to vacuum magnetic field B 0 arallel to vacuum magnetic field B 0 ACKNOWLEDGMENTS This research was sonsored by NASA L. Johnson Sace Center. REFERENCES. Chang Díaz, F. R., Research Status of The Variable Secific Imulse Magnetolasma Rocket, Proc. 39th Annual Meeting of the Division of Plasma Physics (Pittsburgh, PA, 997), Bulletin of APS, 4 (997) Chang Díaz, F. R., Research Status of The Variable Secific Imulse Magnetolasma 9

11 Rocket, Proceedings of Oen Systems July 7-3, 998, Novosibirsk, Russia, American Nuclear Society, Trans. of Fusion Technology, 35 (999) Chang Díaz, F. R., Squire, J. P., Ilin, A. V., et al. "The Develoment of the VASIMR Engine", Proceedings of International Conference on Electromagnetics in Advanced Alications (ICEAA99), Set. 3-7, 999, Torino, Italy, (999) Chang Díaz, F. R., Squire, J. P., Bering, E. A. III, George, J. A., Ilin, A. V., Petro, A. J., and Cassady, L., The VASIMR Engine Aroach to Solar System Exloration, Proceedings of 39 th AIAA Aerosace Sciences Meeting and Exhibit Jan. 8-, 00, Reno, NV, AIAA (00). 5. Birdsall, C. K., and Langdon, A. B., Plasma Physics Via Comuter Simulation, Inst. of Physics Publishing, Bristol UK, (995) Boyd, I. D., Keidar, M., and McKeon, W., Modeling of a Plasma Thruster From Plasma Generation To Plume Far Field, Proceedings of 35-th Joint Proulsion Conference, June 0-4, 999, Los Angeles, CA, AIAA (999) VanGilder, D. B., Keidar, M., and Boyd, I. D., Modeling Hall Thruster Plumes Using Particle Methods, Proceedings of 35-th Joint Proulsion Conference, June 0-4, 999, Los Angeles, CA, AIAA (999). 8. Wang, J., Anderson, J., and Polk, J., Three- Dimensional Particle Simulation of Ion Otics Plasma Flow, Proceedings of 34-th Joint Proulsion Conference, July 3-5, 998, Cleveland, OH, AIAA (998). 9. Ilin, V. P., Numerical Methods for Solving Problems in Electrohysics, (in Russian) Nauka, Moscow (985) Kosmahl, H. G., Three-Dimensional Acceleration Through Axisymmetric Diverging Magnetic Fields Based on Diole Moment Aroximation, NASA TN D-378, (967) 8.. Sercel, J. C., A Simle Model of Plasma Acceleration in a Magnetic Nozzle, AIAA , (990) 7.. York, T. M., Jacoby, B. A., and Mikellides, P., Plasma Flow Processes Within Magnetic Nozzle Configuration, J. of Proulsion and Power, 8 (99) Hooer, E. B., Plasma Detachment from a Magnetic Nozzle, J. of Proulsion and Power, 9 (993) Petro, A. J., Chang Díaz, F. R., Ilin, A. V., and Squire, J. P., Develoment of a Sace Station- Based Flight Exeriment for the VASIMR Magneto-Plasma Rocket, to be ublished in Proceedings of 40 th AIAA Aerosace Sciences Meeting and Exhibit Jan. 4-7, 00, Reno, NV, AIAA (00). 5. Chang Díaz, F. R., Braden, E., Johnson, I., Hsu, M. M., and Yang, T. F., Raid Mars Transits With Exhaust-Modulated Plasma Proulsion, NASA Technical Paer 3539, Houston, TX (995) Ilin, A. V., Chang Díaz, F. R., Gurieva, Y. L., and Il in, V. P., Accuracy Imrovement in Magnetic Field Modeling for an Axisymmetric Electromagnet, NASA Technical Paer , Houston, TX (000) Ilin, A. V., Bagheri, B., Scott, L. R., Briggs, J. M., and McCammon, J. A., Parallelization of Poisson- Boltzmann and Brownian Dynamics calculation, Parallel Comuting in Comutational Chemistry, ACB Books, Washington D.C., (995) Hockney, R. W., and Eastwood, J. W., Comuter Simulation Using Particles, Inst. of Physics Publishing, Bristol UK, (988). 9. Ilin, A. V., Chang Díaz, F. R., Squire, J. P., and Carter, M. D., Monte Carlo Particle Dynamics in a Variable Secific Imulse Magnetolasma Rocket, Proceedings of Oen Systems July 7-3, 998, Novosibirsk, Russia, American Nuclear Society, Trans. of Fusion Tech. 35 (999) Ilin, A. V., Chang Díaz, F. R., Squire, J. P., Breizman, B. N., and Carter, M. D., Particle Simulations of Plasma Heating in VASIMR, Proceedings of 36 th AIAA/ASME/SAE/ASEE Joint Proulsion Conference July 7-9, 000, Huntsville, AL, AIAA (000) 0.. Glasser, A. H., Sovinec, C. R., Nebel, R. A., Gianakon, T. A., Plimton, S. J., Chu, M. S., Schnack, D. D., and the NIMROD Team, "The NIMROD Code: a New Aroach to Numerical Plasma Physics," Plasma Phys. Control. Fusion 4, A747 (999). 0