, so only a very small force is transferred to the system.  The ion then moves towards the top of the engine and applies a positive force to the system as its z-axis velocity is reversed.  The ion’s z-axis velocity is almost constant inside the decelerator, so only a very small force is transferred to the system.  The ion then returns to the bottom after exiting the decelerator, which applies a negative force.  The velocity change for an ion traveling through a single cycle is small, however the beam guide contains over 3×1012 ions that complete over a half million round trip cycles each second.

Each ion applies a force aligned with the z-axis when the z-axis component of its velocity changes.  The z-axis velocity component of each ion as it completes a round trip is shown in Fig 6, starting from the bottom of the engine. The ion’s z-axis velocity is negative when it is moving in the inner core back towards the bottom of the engine because it is then traveling in the negative z-axis direction.  The ion’s absolute velocity in Fig 6 includes all three axis velocity components.  As the ion’s absolute velocity increases, its relativistic momentum increases non-linearly at a faster rate than the classical formula as predicted by the momentum equations in Table 1.

Regrettably, the accelerator section of the design defined in Table 2 requires 165 MW of power for continuous operation.  This power can also be approximated using the change in ion relativistic kinetic energy and the amount of time required to make this change.  The actual “wall plug” power will depend on many other factors.  Fortunately, power produced in the inner core’s decelerator is nearly equal to this amount.  If all the power collected in the decelerator could be applied to the accelerator, the design described in Table 2 would require less than 10 watts to offset momentum and radiation losses for all ions during operation.  Additional power would be required to generate control electric and magnetic fields inside the beam guide.  Harvesting power in the decelerator is not only important for improving engine efficiency – it also reduces the amount of heat that must be dissipated from the engine during operation.
The engine scales linearly with power.  100x the power and 100x the ion density, yields 100x the thrust without increasing the peak magnetic field.  Thrust is scaled by adjusting the ion density (number of ions inside the engine) and power, or ion mass and power – not the velocity of the ions inside the system.  The ion velocity range is set when the engine is designed because this range determines the radius of curvature in the beam guide.  Replacing ionized Helium with Xenon for example would increase ion mass and increase the available engine thrust.

[Figure: Fig. 6  Z-axis and Absolute Ion Velocity (Magnitude) During A Round Trip.]

The efficient transfer of energy from the inner to outer core is important if this engine is to become feasible.  An initial concept for harvesting energy in the decelerator is to use magnetic undulators tuned to produce RF fields that directly accelerate ions in the accelerator.  This approach transfers energy efficiently, avoiding parasitic losses typical with electrical conversion processes.  The helical design brings the accelerator and decelerator beam guide sections into proximity, reducing energy transfer path lengths.  Additional loops can be added to increase thrust or aid in thermal dissipation.  Ions travel in bunches and their position can be accurately controlled.  The author proposes to use this precision to position decelerating ions to reinforce accelerating ions in an adjacent section of the beam guide.
Ions in a synchrotron may have lifetimes of 10 hours or more – implying the design presented in Table 2 may require only 17.76 nanograms of ionized Helium for each year of continuous operation.  Although specific impulse is not strictly appropriate for this class of engine, this ion lifetime gives an equivalent specific impulse of 1.86×1017 seconds for the engine design defined in Table 2.  Ions in the beam guide could be replenished by injection from a secondary linear accelerator, a technique currently used in synchrotrons.  Ion lifetime may be increased by operating the engine continuously because this could gradually eliminate contamination in the beam guide.  It is also possible the engine could harvest helium from its surrounding environment or the power generation source.  Alpha particles can be produced during radioactive decay.
Modeling indicates the engine architecture will produce measurable torque around the z-axis if the radii of inner and outer cores are not equal.  This may be beneficial, and the design could be adapted by substituting electrons for ions in a compact cyclotron/microwave tube assembly to replace satellite reaction wheels.  A pair of engines with counter-rotating ions could produce and manage torque.