## Crockett Grabbe

### Abstract

Electrical propulsion of rockets is developing potentially into the use of 3 different thrusters for future long-distance space missions that primarily involve plasma dynamics. Resarch is proposed on theoretical development of the plasma physics of these thruster, with particular focus on the VASIMIR Thruster. The development of electrical propulsion into plasma dynamic thruster has concentrated more on experimental research in that development, and there are important theoretical questions to be addressed to put a better foundation on the experimental work and reap the enhanced insight it can provide. The more important of these unanswered theoretical questions the VASIMIR development will be addressed and explored.

## Principles of Electrical Propulsion

The Rocket Equation was first derived by Tsytovich in 1903, and tells us much about the dynamics of rockets:

## Thrust= m dv/dt = -ve dm/dt [1]

which has the solution

## Δ v = ve ln(m1/m2) [2]

where m1 = mfuel + m2, with ve is the effective exhaustive velocity (e.g. chemical combustion, nozzle...) mfuel is the fuel spent in going from state 1 to state 2. There are 2 general types of thrust that have been developed:

### 1. Chemical thrust: ve ~ 1-5 km/s with T ~ 0.1 - 107 N

This is used for all launches from the Earth, where large enough acceleration to achieve a velocity increase of Δ v  11.2 km/s required to escape the gravitational field, means that m1 >> m2. Thus most of the rocket load has to be fuel.

### 2. Electrical thrust: ve ~ 15-80 km/s with T ~ 0.001 - 10 N

This thrust lets us achieve Δv ~ 100 km/s with far less mfuel, but over a much longer time. It is useful for distance in space, but not planetary launches or assists. The essential concepts of the 2 types of rocket thrust are illustrated in Figure 1, and a summary of the comparison of the 2 types of thrusts in given in Table 1.

A useful quantity often used by rocket engineers releated to this equation is specific impulse:

### Isp= ve/g [3]

To escape gravity in planetary launches, the thrust upward must exceed gravity's force downward:

so

### m2 < m1 exp[-(t2-t1)g/ve] [5]

Thus we can understand the specific impulse Isp as the thrust T divided by the time rate of propellant weight (as measured on the Earth's surface) delivery.

## Table 1: Comparison of Types of Propulsion

### Planned for future (e.g. human) missions to Mars

This demarcation between the 2 types of propulsion was appreciated in the early days of the space race. While the development of chemical propulsion was of high priority for getting rockets from the Earth's surface into space, there was also research put into the development of ion engines for travelling a distance into space once it was out there. [NYT, 1964] This led to the development of 2 types of electrostatic plasma engines discussed in the next section.

# Electrostatic Plasma Propulsion

The Gridded Electrostatic Ion Engine is shown in Figure 2. In this engine a neutral atomic propellant is inject into a hollow cathode, where electrons impact the atoms to create ions. The ions are electrostatically accelerated out of the rocket to create the thrust. Electron beams are also injected into the ions leaving the rocket in order to help neutralize them and create the electrical drag they would produce.

### Figure 2: Gridded Electrostatic Ion Engine [Wikipedia]

A Gridded Electrostatic Ion Engine experiment was placed in the Deep-Space 1 mission launched to Comet Borrelly in 1998 (its logo is shown in Figure 3). This mission had 12 advanced- technology tests, 1 of which was the solar-power Gridded Ion Thruster test called NSTAR. NSTAR used Xenon (Xe) to produce the ions, because its low ionization potential and high molecular were favorable for this application. NSTAR had an Isp of 1000 - 3000 s and a total power of 2.1 kW. The typical velocity produced for the Xe ions was about 35 km/s, which is about 10 times as large as the acceleration produced by chemical propulsion.

Deep Space 1 had a total payload of 374 kg. Of that 74 kg was Xenon for the test, or just under 20% of the payload. The test produced an accerlation of the probe speed of 4.3 km/s.

### Figure 3: Deep Space 1 Mission [NASA]

This Gridded Ion Thruster was also used on previously unplanned mission on the European Space Agency satellite Artemis because of a chemical propulsion error in its launch. As a result of the error, the was not in the geo-stationary orbit that had been intended. The Gridded Ion Thruster was used to correctly transfer it into a geo-stationary orbit.

Another electrostatic plasma engine is that of the Hall-Effect Thruster, as shown in Figure 4. This engine was first used in Soviet spacecraft in 1972 [REF??]. Here an axial electric field Ez (involving typically 300 V potential difference) is produced by an electron plasma at the end of the thruster instead of a grid, as well as a radial magnetic field Br. Br traps the electrons in the plasma in a Hall current by an E X Btheta force. On the other hand, the ions accelerated undergo very little trapping by the magnetic field and are acceleration out of the rocket by the elect, creating a thrust of around 0.04 - 0.6 N and an Isp of 1000 - 5000 s.

The European Space Agency (ESA) used a Hall thrust in the SMART-1 spacecraft which they launched to the moon in 2003. That thruster used a total of 58.8 kg of Xe in its thruster, which resulted a net velocity increase of 2.74 km/s.

A Hall-thrust experiment called HTEX was started at the Princeton Plasma Physics Labs (PPPL) in 1998 [REF??]. One of the objectives of that experiment was to analyze effects the thruster had on other experiments on the spacecraft, to determine how reliable such experiments are with the effects of that thruster taken into consideration. That is a serious issue that is undergoing examination for new thruster at several locations. For example, the issue of antenna performance on spacecraft with Hall thrusters was tackled initially by the development of a 3D geometric optics ray-tracing code for studying the effect of those plasma thrusters on communication satellite antennas.[Hallock et al, 2002].

# Electromagnetic Plasma Propulsion

There are basically 4 major types of electromagnetic plasma-propulsion thrusters currently under development for future spacecraft missions. These can be briefly described as follows.

In this thruster a radial current Jr and a magnetic field in the azimuthal direction Bθ drives ions to exhaust nozzle. An example is shown in Figure 5.

### Figure 5. CGI-rendered lithium Self-field MPD Thruster, marking the magnetic field force lines and direction of electric field [NASA Glenn Research Center]

This MPD thruster is analogous operation to rail gun, with concentric cathode & anode like the rails and the plasma ions like the cross-rail conductor that is accelerated in the rail gun. The thruster ionizes injected gas near front of cathode, and the resulting Jr X Bθ forces accelerates the ions out.

The MPD Thruster operates at a high temperature range between 2.5 and 25 eV and a specific impact Isp between 1500 and 6000 secs). Those are attractive features, but there are still 2 problems hindering the potential of this thruster. First, sources supplying 100s of kV are needed for to supply the energy needed for operation, and secondly, the electrodes do not last for a sufficient length of time. These are the 2 main barriers that must be overcome before it becomes an attractive option for spacecraft.

### 2. Pulse Induction Thruster (PIT)

In this thruster an azimuthal current Jθ and the magnetic field component Br in the radial direction drives ions to the exhaust nozzle. A picture of one at the company TRW is shown in Figure 6.

### Figure 6: Pulsed Inductive Thruster (PIT) [TRW, 2002]

In a PIT, ammonia (NH3) or argon gas is pulsed into the thruster in a thin layer. Capacitors provide a high-current pulse discharge through a large induction coil, which induces an Eθ parallel to it. That ionizes the injected gas & drives a current Jθ in it. The resulting force JθX Br accelerates plasma ions to exhaust.

Like the MHD thruster, the PIT thruster has a high temperature and high Isp. However, it does need a strong power supply to create the pulses discharged through the gas injected into the thruster.

### 3. Electrodeless Thruster

This thruster uses nonlinear pondermotive force of electromagnet waves to generate its power. WRITE UP THIS SECTION

### 4. VASIMR® Thruster

This thruster is a variant of MPD Thruster that was invented by Chang-Diaz, standing for Variable Specific Impact Magnetoplasma Rocket. A sketch of this engine is shown in Figure 7.

### Figure 7: VASIMR® Engine [Chang-Diaz, 2006]

In the VASIMR® engine, rocket propellant is fed into a tube surrounded by a helicon antenna, which ionizes the propellant to form a plasma. The plasma is also surrounded by magnetic coils, and is fed into a region with a radio-frequency (RF) booster antenna that is tuned to the ion-cyclotron resonance frequency (ICRF, which we will denote as fci) of the plasma. Considerable heating of the plasma occurs from the antenna, and the heated plasma rapidly expands out in a flare out the nozzle. The exhaust then undergoes detachment using reattachment of the electrons.

The VASIMIR Engine is undergoing considerable development at the Ad Astra Rocket Company in Houston, and will be the focus of the rest of this review.

## VASIMR® Plasma Propulsion

In VASIMR® propulsion, both the plasma ionization and the ion cyclotron heating are done by electromagnetic (helicon and ICRF) antennas. There are no electrodes as there are in most other types of thrusters, creating an advantage over them, This VASIMR® engine process basically consist of 4 stages, shown in the schematic diagram of Figure 8. The first is the ionizing phase, where gas is inject into the region surrounded by the helicon antenna to initially create a plasma. The second is the energizing phase, where the ICRF booster antenna rapidly turns the RF waves into heat energy for the plasma. Because of the intense pressure created by the heating, the plasma quickly accelerates into regions going out the nozzle where is can rapidly expand in the third stage. The substantial heating & nozzle acceleration on ions creates a large specific impact Isp, generally larger than what has been produced in the MPD and PIT thruster schemese discussed in the previous section. Finally, the fourth important phase is the detachment of the plasma ejected the nozzle from the spacecraft, which requires reattaching electrons to the ions, to rid them of electrostatic attaction back to the spacecraft.

### Figure 8: Schematic of the 4 stages of the VASIMR® engine [after Bering et al, 2009]

Experiments for the VASIMR® engine were done one the VX-10 device at the Johnson Space Center around 2005 (ee Figure 8). In those experiment the RF booster antenna for ICRF heating of the plasma was surrounded by superconducting magnet coils operating at 40 K, which produced a radial magnetic field component of Br=0.3 Teslas. (Even stronger magnets were developed there after 2005.) The ICRF is given by ωci = 2 π fci = Be/mi so fci~ 4.5 MHz for a hydrogen plasma. For an argon plasma this frequency would be about 110 KHz.

### Figure 10: VASIMR® system efficiency η, boost partition f, & thrust F (which we call T) [from Chang-Diaz, 2006]

A phenomenological plot of the system effeciency η , the boost partition f, and the thrust F as a function of the specific impulse Isp for plasmas of various elements and isotopes is shown in Figure 10. A phenomenological relationship between the thrust T and the specific impulse Isp for a given system efficiency η can be easily derived from Equs. [1] and [3]. The power of the thruster is:

so

### Isp = (P/g)(1/ηT) [6]

Taking P = 1 kW as the power driving the ICRF heater yields T = (102 kg-m/s)/ηIsp, and this relation is plotted in the graph in Figure 10 for the system efficiencies of several elements used to create the plasmas. Note in the graph taken for Figure 10, F was used to denote the thrust force (rather than T which has been used in this paper). The partition of power f going to the RF booster (instead of the ionization to create the plasma) is also plotted as a function of Isp.

A more well-development general theory of VASIMR® propulsion was initiated by Alexey and Breizman [2004]. In the theoretical analysis of VASIMR® 3 aspects were developed: (1) A first-principles model for the helicon plasma source (2) A nonlinear model for RF power deposition into the ion cyclotron frequency of the single-pass plasma flow (i.e. the plasma only passes the ICRF Heater Antenna once) (3) A discussion of the relevant plasma detachment mechanism. The last aspect awaited further development of the theory, which was done in a later paper.[Alexey and Breizmann, 2005] An ideal MHD analysis was carried out, but they noted a concern in that analysis was how well the ideal MHD assumption was -- whether it might be broken by anomolous resistivity created by high-frequency instabilities.

This problem of

The VX-10 experiment at the Johnson Spacecraft Center in Houston tested PICRF up to 1.444 kW, which from Eq. (6) yields an Isp ~ 12,000 (note η ~ 0.5 for all elements here). IS THIS CORRECT? The VX-50 experiment, subsequently conducted at the Johnson Spacecraft Center, tested the theoretical predictions for the ICRF heating using 2-4 MHz electromagnetic waves at powers up to 20 kW. [Bering et al, 2008]. The experiment measurements supported the theoretical predictions, noting that for a deuterium plasma, up to 80% efficiency absorption of the input power was achieved. They saw no evidence for power-limiting instabilities in the exhaust beam.

The Ad Astra Rocket Company was formed in 2005 as a contractor to Johnson Spaceflight Center. Their initial project was the VX-100 experiment. It was reported both from the earlier experimental studies on the VX-50 and in initial studies on the VX-100, the ICRF heating was single-pass and that no problems of nonlinear saturation were encountered.[{\it Bering et al}, 2009a]

Testing on the newer experiment VX-200 was initied at Ad Astra in October of 2008, with the helicon driven by a power Phelicon=30 kW, and the ICRF heat with power PICRF= 170 kW [{\it Bering et al}, 2009b]. An important focus of this experiment was putting this device in a space-vacuum simulation chamber, with the intent of testing it at conditions very similar to those that would be encountered in space.

The VASIMR engine package would be placed on a rocket as shown in Figure 11. It is expected that this engine will be tested on a rocket sometime in the near future. The continue success of laboratory experiments seems to make that quite feasible.

## Summary of Plasma Propulsion

Plasma propulsion can achieve much larger velocities in space with much fuel mass then chemical propulsion used to launch the rockets. These are ideal to use for long-distance travel in space, but of course they are not useful for launch from planets. Presently the most used plasma propulsion systems are the Gridded Ion Thrusters and Hall Thrusters. However future trends indicate that electromagnetic plasma thrusters will like as MPD, the PIT, and the VASIMR will in the future exceed the capacities of these electrostatic plasma thrusters. The VASIMR engine appears to hold the best prospects for future needs, likw sending humans to Mars, and sending spacecraft to far planets, comets, and interstellar medium.

### REFERENCES

Alexy V. Arefiev and Boris N. Breizman, "Theoretical Components of the VASIMR® Plasma Propulsion Concept," Physics of Plasmas, 11, 2942 (2004).

Alexy V. Arefiev and Boris N. Breizman, "Magnetohydrodynamic Scenario of Plasma Detachment in a Magnetic Nozzle," Physics of Plasmas, 12, 043504 (2005).

Edgar A. Bering III, Franklin R. Chang-Diaz, Jared P. Squire, M. Brukardt, Timothy W. Glover, Roger D. Bengston, V.T. Jacobson, Greg E. McCaskil, and Leonard D. Cassidy, "Electromagnetic Ion Cyclotron Resonance Heating in the VASIMR®," Advances for Space Research, 42, 192-205 (2008).

Edgar A. Bering III, Benjamin W. Longmier, Timothy W. Glover, Franklin R. Chang-Diaz, Jared P. Squire and Michaeo Brukardt, "High-Power Electric Propulsion Using VASIMR®: Results from Flight Prototype," 47th AIAA Aerospace Sciences Meeting, 254 (2009).

Edgar A. Bering III, Benjamin W. Longmier, Timothy W. Glover, Franklin R. Chang-Diaz, Jared P. Squire, Mark D. Carter, Leonard D. Cassidy, Chris S. Olsen, Greg E. McCaskil, and William J. Chancery, "VASIMR® VX-200: High-Power Electric Propulsion for Space Transportation Beyond LEO," 2009 AIAA Space Conference and Exposition, 6481 (2009).

Franklin R. Chang and J.L. Fisher, "A Supersonic Gas Target for a Bundle Divertor Plasma" Nuclear Fusion, 22, 1003 (1982).

Franklin R. Chang-Diaz, "Plasma Propulsion for Interplanetary Space Flight," Thin Solid Films 506-507, 449-453, 2006.

Deep Space 1 ref

Christopher A. Deline, Roger D. Bengtson, Boris N. Breizman, Mikhail R. Tushentsov, Jonathan E. Jones, D. Greg Chavers, Chris C. Dobson, and Branwen M. Schuettpelz, "Plume Detachment from a Magnetic Nozzle," Physics of Plasmas, 16, 033502 (2009).

Nat Fisch, HTEX refs at PPPL

Gridded Electrostatic Ion Engine refs

Hallock, Gary... "Impact Analysis of Hall Thrusters on Satellite Antenna Performance," Journal Spacecraft & Rockets 39, 115, Jan-Feb, 2002.

Andrew V. Ilin, Franklin R. Chang-Diaz, Boris N. Breizman and Mark Carter, "Modelling of Ion Cyclotron Waves Absorption in VASIMR®," Advances in Space Research, 42, 192-205 (2008).

Andrew V. Ilin, Franklin R. Chang-Diaz, Jared P. Squire, and Mark Carter, "Plasma Heating Simulation in the VASIMR® System," AIAA Aerospace Sciences Meeting, 0949 (2005).

NASA Glenn Research Center self-field MPD Thruster, online at http://www.grc.nasa.gov/WWW/RT/RT2000/5000/5430lapointe.html

"An Electric Rocket Passes Flight Test," New York Times, July 21, 1964.

Martin I. Panevsky and Roger D. Bengtson, "Characterization of the Resonant Electromagnetic Mode in Helicon Discharges," Physics of Plasmas, 11, 4196-205 (2004).

TRW photo of its PIT experiment, 2002. (TRW of Redondo Beach, CA has since become part of Grumman, Inc.)

Tsytovich, 1903