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Millis 2018 Grant NNX17AE81G_for_CR.docx Air Force Research Lab (AFRL).
Introduction

Chemical, nuclear, and electric propulsion systems produce thrust by accelerating and expelling propellants.  Deep space travel is often a trade-off between thrust and large propellant storage tanks that eventually limit performance. The objective of this paper is to introduce and examine a unique engine that uses a closed-cycle propellant.  This in-
1 Manager, Office of Science and Technology.

| (d) Left |  | Right |
| --- | --- | --- |
|  |  |  |
|  |  |  |

(a)
(c)
(b)

[Figure: Fig. 1 Thought Experiment.]

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[Figure: Figure 1:  Halo Orbit Families]

The NRHOs are set apart from the rest of the halo orbits by their favorable stability characteristics [4] [5] [6]. They exist at the end of the halo orbit range with the lowest perilune radii [4].  Viewed in the rotating frame, “the NRHOs are characterized by an elongated shape that resembles an ellipse” [5], but they cannot be approximated as two-body Keplerian orbits.

In general, NRHOs were selected as the orbit for Gateway because they represent an advantageous staging orbit in which to aggregate resources for a variety of potential mission objectives.  They have neutral stability characteristics, and therefore low orbit maintenance costs [4] [6] [7].  These orbits are accessible to Orion within the capability of the existing Service Module [8] [9], and they offer access to the lunar surface with relatively short trip times of about 0.5 days [8].  Cargo can be delivered to the NRHO via four-body Ballistic Lunar Transfers (BLTs) with very low ∆V requirements [9] [10].  It is possible to depart for interplanetary destinations from the NRHO [11].  Also, they provide extended periods of communications coverage to one of the lunar poles, with only short blackouts [8] [12].
Introduction

The Gateway program has selected a Near-Rectilinear Halo Orbit (NRHO) of the L2 southern family as its operational orbit [1].  In order to facilitate mission analysis for Gateway and associated programs utilizing this orbit, a representative reference trajectory 15 years in duration has been developed.  The purpose of this white paper is to review the properties of this NRHO reference trajectory.

Background:  Selection of NRHO as Gateway Orbit
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David M. Burns, Ph.D.1 NASA, Marshall Space Flight Center, Alabama, 35812, USA
A new concept for in-space propulsion is proposed in which propellant is not ejected from the engine, but instead is captured to create a nearly infinite specific impulse.  The engine accelerates ions confined in a closed loop to relativistic speeds, and slightly varies their velocity to change their momentum.  The engine then moves the ions back and forth along the direction of travel to produce thrust.  This in-space engine is intended to be used for long-term satellite station-keeping without refueling or to propel spacecraft across interstellar distances.  The engine has no moving parts other than ions traveling in a closed-loop vacuum line, trapped inside electric and magnetic fields.
Nomenclature

%c = velocity represented as a percentage of the velocity of light in a vacuum ∆v = change in velocity 100x = one hundred times the original value a = acceleration atm = one standard atmosphere of pressure c = velocity of light in a vacuum: 299,792,458 meters/second cm = centimeter cm3 = cubic centimeter F = force He = Helium kg = kilogram km/s = kilometers/second L = length lbf = pound (force) m = mass if used in an equation, or meter if used to denote length m/s = meters/second mm = millimeter: 10-3 meters MW = megawatt: 106 watts (power) N = Newton (force) N⋅m = Newton-meter (torque) ns = nanosecond: 10-9 seconds 𝜌 = momentum (kg⋅m/s) RF = radio frequency electromagnetic radiation s = second T = Tesla (magnetic flux density) t = time v = velocity vz = z-axis velocity component, which is aligned with the engine’s direction of thrust W = Watt (power) 𝛾 = Lorentz factor, an element of Special Relativity defined in Table 1
because relationships for momentum, force, and kinetic energy become more complex.
The ring’s momentum is not constant during every point inside the local box-ring system in this thought
experiment.  Momentum is defined as the product of mass and velocity, and the ring’s mass is not constant.  The ring
requires an external action or force to change its mass, so the thought experiment does not have enough information
to evaluate conservation of momentum. If the box and ring continue to accelerate from perspective of an external
observer, then the local system’s total momentum will increase.  Momentum may be conserved in the global sense of
the box-ring system and the surrounding environment.  Adding or subtracting mass distorts the surrounding spacetime environment and requires an external intervention to the local box-ring system.

These influences may create a balancing force on the surrounding environment to the acceleration produced by the box and ring, which will conserve momentum at the global level.  A method for changing the ring’s momentum using external power is presented in the next section, and global conservation of momentum will be revisited in section IV.
Implementing Relativistic Momentum Change Using a Particle Accelerator

The proposed engine replaces the solid ring in Fig 1 with a ring of ions rotating in a plane orthogonal to the engine’s thrust vector as depicted in Fig 2.   The engine varies the rotational velocity of the ions to change their momentum as the ring is moved back and forth inside the engine.  To prevent ion velocity changes from negatively impacting the engine’s momentum, the radius of rotation for the ions in the ring increases as the ions accelerate and decreases when the ions decelerate.  The total number of ions in the engine does not change.  The ion’s z-axis velocity component is denoted by vz and is relative to the inside of the engine. Relationships for momentum, force and kinetic energy are presented in Table 1.   Relativistic formulas can be approximated by the classical formulas when an object’s velocity is much less than the speed of light [2,3].  𝛾 is the Lorentz factor, an element of Special Relativity [2].  The momentum of each ion (along with other properties, such as length and time) depends on its absolute velocity.  The ion’s momentum increases faster as a function of its absolute velocity using the relativistic instead of the classical formula because 𝛾≥1.
Top Bottom X 𝑣Z
Y
Z

[Figure: Fig. 2  Diagram of Proposed Engine Operation.]

[Table: Table 1  Classical and Relativistic Dynamics.]

|  | Classical | Relativistic | Definitions |
| --- | --- | --- | --- |
| Momentum | 𝝆 = 𝒎𝒗 | 𝟏 𝜸= 𝝆 = 𝜸𝒎𝒗 using 𝒗𝟐 𝟏− 𝒄𝟐 | 𝒂 Acceleration 𝑭 Force 𝑲𝑬 Kinetic Energy 𝜸 Lorentz Factor 𝝆 Momentum 𝒎 Resting Mass 𝒄 Speed of Light 𝒕 Time 𝒗 Velocity |
| Force | 𝒅𝝆 𝑭= 𝒅𝒕 | 𝒅𝝆∗ * Must account for 𝑭= 𝒅𝒕 direction of travel |  |
| Force | 𝑭= 𝒎𝒂 | 𝑭 = 𝜸𝟑𝒎𝒂 Parallel 𝑭 = 𝜸𝒎𝒂 Perpendicular |  |
| Kinetic Energy | 𝟏 𝑲𝑬 = 𝒎𝒗𝟐 𝟐 | 𝑲𝑬= 𝒎𝒄𝟐 𝜸−𝟏 |  |

|  | Classical | Relativistic | Definitions |
| --- | --- | --- | --- |
| Momentum | 𝝆 = 𝒎𝒗 | 𝟏 𝜸= 𝝆 = 𝜸𝒎𝒗 using 𝒗𝟐 𝟏− 𝒄𝟐 | 𝒂 Acceleration 𝑭 Force 𝑲𝑬 Kinetic Energy 𝜸 Lorentz Factor 𝝆 Momentum 𝒎 Resting Mass 𝒄 Speed of Light 𝒕 Time 𝒗 Velocity |
| Force | 𝒅𝝆 𝑭= 𝒅𝒕 | 𝒅𝝆∗ * Must account for 𝑭= 𝒅𝒕 direction of travel |  |
| Force | 𝑭= 𝒎𝒂 | 𝑭 = 𝜸𝟑𝒎𝒂 Parallel 𝑭 = 𝜸𝒎𝒂 Perpendicular |  |

| Kinetic Energy | 𝟏 𝑲𝑬 = 𝒎𝒗𝟐 𝟐 | 𝑲𝑬= 𝒎𝒄𝟐 𝜸−𝟏 |  | to its direction of travel (proportional to 𝛾).

This will be a key relationship that will determine the efficiency of the proposed engine.  Relativistic kinetic energy also increases faster than the classical definition as an object’s velocity approaches the speed of light.
If an ion rotating in a circle is moved along an axis perpendicular to its plane of rotation, the resulting motion is a helix.  Instead of mechanically guiding a fixed ion ring back and forth, a single closed-loop helically shaped beam guide can be used to transport the ions along the optimum path inside the engine. The beam guide is a coiled circular particle accelerator that uses electric and magnetic fields to accelerate and guide ions inside a closed and sealed tubular vacuum line.
The beam guide is designed to hold the ion’s z-axis component constant while its absolute velocity is increased or decreased as needed. Fig 3 demonstrates how an ion’s z-axis velocity component can be held constant while its absolute velocity increases.  The radius of curvature increases in proportion to its absolute velocity, so the ion has a longer distance to travel in the x-y plane as it moves upwards along the +z-axis.  The engine changes the momentum of ions by increasing and decreasing their rotational velocity around the z-axis, which is parallel to the engine’s direction of travel.  This allows the ion’s absolute velocity to be altered without imparting force along the engine’s direction of travel.
The Helical Engine has two concentric helical beam guide “cores.”  Ions travel toward the top of the engine in an outer core with a constant positive z-axis velocity and increasing absolute velocity. This corresponds to the action in Fig 1(a) and is accomplished by accelerating ions using strong electric fields and increasing the radius of rotation.  Ions return to the bottom of the engine in an inner core, corresponding to the action in Fig 1(c), with a constant negative z-axis velocity and decreasing absolute velocity.  This is accomplished by decelerating the ion’s rotational velocity using an undulator [4] or similar device to lower the ion’s kinetic energy and decrease the radius of rotation. The direction of rotation is the same for both cores.  Even if the engine is traveling at a relativistic speed, the system will accelerate because ions are changing momentum according to their own reference frame inside the engine under the General Theory of Relativity.

[Figure: Fig. 3  Increasing Ion Absolute Speed with a Constant]
An example design is presented in Fig 4(a), where lines represent the path of a single closed-loop beam guide. The green line in the center section of each core identifies the accelerator (outer core) or decelerator (inner core).  The radius of rotation changes as needed in the accelerator and decelerator sections to keep the z-axis velocity component constant.  The ion’s velocity on the z-axis is reversed in the blue lines at each end so ions can exchange momentum with the engine and the inner and outer cores can connect.  Ion absolute velocity (also referred to as the total magnitude of the velocity) is constant in the blue sections of the beam guide.
The beam guide’s helical path mimics the path an ion rotating in a ring follows, if the ring is rapidly moved from end to end.  Ions travel towards the top of the engine in the outer core in Fig 4(b), which has a radius of curvature that increases in the accelerator region (green line).  The beam guide radius of curvature then decreases so it can connect with the inner core.  Ions return to the bottom of the engine in Fig 4(c), which has a radius of curvature that decreases in the decelerator region (green line).  Ions travel in “bunches” in synchrotrons instead of being equally distributed throughout the volume.  It could be possible to set the radius of curvature to the same range for the inner and outer cores, and manipulate the travel time of bunches to avoid collisions.

[Figure: Fig. 4  Side and Top Down Views of a Helical Engine Beam Guide.  Ions Accelerate or Decelerate in Green]
| Status | Version | Change | Description | Release |
| --- | --- | --- | --- | --- |
|  | No. | No. |  | Date |
| Released Initial Release 4/12/17 Revision Totally rewritten, Revision A, CR ESD-0167-X 12/19/18 A NOTE: Updates to this document, as released by numbered changes (Change XXX), are identified by a black bar on the right margin. |  |  |  |  |
Thrust can be improved by designing the beam guide to operate with higher ion velocity, however the required accelerator power and magnetic field also increase.  A strong magnetic field is required to bend the ion’s direction of travel so that it remains in the center of the beam guide and does not impact the walls of the vacuum chamber.  The accelerator applies an in-line (parallel) force to accelerate ions, which scales proportionally to 𝛾3.  Since the beam guide is curved, most of the momentum exchange at each end of the engine is perpendicular to the ion’s direction of travel unless the ion is traveling at an extremely high velocity.  As a result, most of the momentum transfer forces in the top and bottom bends are proportional to 𝛾 until the ion velocity approaches the speed of light.  When ion velocity approaches the speed of light, the 𝛾3 momentum term in the ends of the engine begins to dominate.  Fig 7 shows the effect of increasing the minimum velocity using a constant input power of 165 MW.  Thrust efficiency increases dramatically for ion velocities at or above 99.99 %c, however the peak required magnetic field also increases rapidly. For comparison, the Large Hadron Collider can accelerate ions to 99.9999991 %c with a beam guide circumference of 27 kilometers and magnetic fields that exceed 8 T [5].
If most of the power in the decelerator can be recovered, then higher average ion velocities may improve overall system efficiency.  The total length of the beam guide decreases as the minimum ion velocity increases for a given applied power level because the range of the radius of curvature in the accelerator/decelerator sections decreases. Future designs could alter the ends of the engine to make momentum transfer more of a direct reflection (parallel force).  Additional beam guides could be “woven” into the architecture shown in Fig 4 to increase thrust.

[Figure: Fig. 7 Thrust and Peak Magnetic Field for a Constant Applied Power of 165 MW.]
The high required power and low resulting thrust seem to imply the principle of conservation of momentum is present, however the engine theoretically continues to accelerate if it has power.  There are several ways momentum can be conserved from a global perspective.
The process of increasing and decreasing the ion’s relativistic momentum along with producing the required power may distribute the momentum change to the surrounding environment external to the engine.  Some process must be used to generate the required power, and this process could conserve momentum in a global sense.  This is difficult to quantify, given the current abstract method for modeling the engine’s performance.
The ions inside the engine may accumulate an offsetting momentum.  For example, the ions may develop spin during each trip around the engine.  This spin could eventually lower or even halt engine thrust.  Ion spin is dependent on the specific beam guide implementation design and is not currently modelled in RMTM.  This is an area where a more detailed model or testing would be very useful.  It may be possible to mitigate ion spin by using a pair of counterrotating Helical Engines that share a single beam guide.  This technique could prevent ion spin from accumulating during subsequent round trips through the engine.  A pair of counter-rotating Helical Engines could also be used to manage torque.
Another possibility for momentum conservation is beam guide emissions.  The beam guide requires strong magnetic fields to bend the ion’s direction of travel and high electric fields to alter the ion’s velocity.  Ions traveling in the beam guide produce x-ray emissions during each circuit.  These fields and emissions may radiate energy to the surrounding area that would balance the accelerating engine.
Anchoring the Relativistic Momentum Transfer Model with relevant test data is essential to continuing the research.  Modeling the momentum of charged particles moving at relativistic velocities around an arbitrarily curved path is complex and the magnitudes of velocity and resulting forces span many orders of magnitude.  Small round-off errors could propagate into larger errors during each simulation.  RMTM has been evaluated using basic checks, such as calculating consistent thrust near boundary conditions; generating zero net thrust for non-helical architectures; and providing stable predictions for a wide range of simulation step sizes.  However, RMTM needs to be validated and verified through the testing of helical beam guide shapes in a synchrotron.
The Helical Engine design can be improved based on RMTM simulations.  The design currently uses a tapered power distribution profile in the accelerator and decelerator sections.  This causes ripples, which account for small negative forces in the accelerator and decelerator as listed in Table 2.  If power was applied and removed using a more optimum technique, then there would be less z-axis force losses during transitions in and out of these sections.  The engine design could also be improved so that more of the momentum exchange at each end results in parallel instead of perpendicular force based on the ion’s direction of travel.

This could greatly increase engine power efficiency and improve operation at higher minimum ion velocity.  Replacing ions with electrons would greatly decrease engine thrust, but would simplify construction, reduce costs, lower power requirements, and validate the concept.