ORBITAL DYNAMICS OF SPACE NUCLEAR PROPULSION
SYSTEMS
by
LARA SCHOEFFLER
Submitted in partial fulfillment of the requirements for the degree of
Master of Science
Department of Mechanical and Aerospace Engineering
CASE WESTERN RESERVE UNIVERSITY
May 2021
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the thesis of
Lara Schoeffler candidate for the degree of Master of Science*
Committee Chair
Dr. Paul J. Barnhart
Committee Member
Dr. Yasuhiro Kamotani
Committee Member
Dr. Sunniva Collins
Committee Member
Dr. James Gilland
Date of Defense
3/19/2021
*We also certify that written approval has been obtained
for any proprietary material contained therein.
2
TABLE OF CONTENTS
TABLES ...... 5
FIGURES ...... 7
NOMENCLATURE ...... 9
ABSTRACT ...... 10
I. INTRODUCTION ...... 11
II. METHOD OF ANALYSIS ...... 15
A. Derivation of Equations ...... 15
B. Numerical Methodology Capabilities ...... 28 1) Orbits with No Applied Thrust ...... 29 2) Comparisons of Numerical and Exact Results...... 40 3) Outbound Circular Spiral Orbit with Applied Thrust ...... 44 4) Inbound Circular Spiral Orbit with Applied Thrust ...... 46 5) Radial Thrust Orbit ...... 48
C. Test Cases ...... 51 1) Test Case One: Circumferential Earth Escape ...... 51 2) Test Case Two: Tangential Earth Escape ...... 55 3) Test Case Three: NASA Dawn Mission ...... 59 4) Test Case Four: Jupiter Transit ...... 64
III. MISSION ANALYSIS ...... 71
A. Electric Propulsion ...... 71
B. Nuclear Thermal Propulsion ...... 86
C. Nuclear Fusion Propulsion ...... 102
IV. RESULTS AND DISCUSSION ...... 133
V. CONCLUSION ...... 139
VI. FUTURE WORK ...... 143
APPENDIX A: DERIVATION OF THE TWO-BODY PROBLEM ...... 145
3
APPENDIX B: INITIAL CONDITIONS FOR MISSION ANALYSIS CALCULATIONS ...... 156
APPENDIX C: MATLAB CODE FOR ORBITAL DYNAMICS CALCULATIONS ...... 164
BIBLIOGRAPHY ...... 169
4
TABLES
Table 1: Initial conditions for a circular orbit with no thrust ...... 31 Table 2: Initial conditions for an elliptical orbit with low eccentricity and no thrust...... 33 Table 3: Initial conditions for an elliptical orbit with high eccentricity and no thrust ..... 34 Table 4: Initial conditions for a parabolic orbit with no thrust ...... 37 Table 5: Initial conditions for a hyperbolic orbit with no thrust ...... 38 Table 6: Initial conditions for an elliptic orbit with no thrust ...... 41 Table 7: Orbit characteristics for exact radius calculations ...... 42 Table 8: Initial conditions for an outbound spiral orbit with applied thrust ...... 44 Table 9: Initial conditions for an inbound spiral orbit with applied thrust ...... 47 Table 10: Initial conditions for a radial thrust orbit ...... 49 Table 11: Initial conditions for circumferential, circular Earth escape orbit ...... 52 Table 12: Comparison of circumferential escape trajectories ...... 52 Table 13: Comparison of dimensional and non-dimensional orbit characteristics ...... 54 Table 14: Initial conditions for tangential, circular Earth escape orbit ...... 55 Table 15: Comparison of tangential escape trajectories ...... 56 Table 16: Comparison of dimensional and non-dimensional orbit characteristics ...... 58 Table 17: Dawn mission spacecraft characteristics [4]...... 61 Table 18: Initial conditions for Mars trajectory ...... 62 Table 19: Comparison of Mars trajectories ...... 63 Table 20: Known Jupiter mission characteristics [5] ...... 67 Table 21: Assumed Jupiter mission characteristics ...... 68 Table 22: Initial conditions for Jupiter trajectory ...... 68 Table 23: Comparison of Jupiter trajectories ...... 69 Table 24: Performance characteristics of conceptual nuclear electric propulsion system 75 Table 25: NEP flight characteristics after interplanetary flight and arrival at Mars ...... 76 Table 26: NEP flight characteristics for Mars arrival flight ...... 78 Table 27: NEP Mars escape flight leg of mission...... 79 Table 28: NEP flight characteristics after interplanetary flight and arrival at Earth ...... 81 Table 29: NEP flight characteristics for Earth arrival flight ...... 83 Table 30: Final mission characteristics for NEP mission to Mars ...... 85 Table 31: Performance characteristics of conceptual nuclear thermal propulsion system 88 Table 32: NTP Earth escape flight leg of Mars mission ...... 89 Table 33: NTP spacecraft flight characteristics upon reaching the edge of Earth's sphere of influence ...... 91 Table 34: NTP flight characteristics after interplanetary flight and arrival at Mars ...... 92 Table 35: NTP flight characteristics for Mars arrival flight ...... 94 Table 36: NTP Mars escape flight leg of mission...... 95 Table 37: NTP spacecraft flight characteristics upon reaching the edge of Mars’ sphere of influence ...... 97 Table 38: NTP flight characteristics after interplanetary flight and arrival at Earth ...... 98 Table 39: NTP flight characteristics for Earth arrival flight ...... 100 Table 40: Mission characteristics for NTP mission to Mars...... 101 Table 41: Performance characteristics of conceptual fusion propulsion system [14] .... 103 Table 42: Nuclear fusion Earth escape flight leg of Mars mission ...... 105
5
Table 43: Nuclear fusion spacecraft flight characteristics upon reaching the edge of Earth's sphere of influence ...... 107 Table 44: Nuclear fusion flight characteristics after interplanetary flight and arrival at Mars ...... 108 Table 45: Nuclear fusion flight characteristics for Mars arrival flight ...... 110 Table 46: Nuclear fusion Mars escape flight leg of mission ...... 111 Table 47: Nuclear fusion spacecraft flight characteristics upon reaching the edge of Mars’ sphere of influence ...... 113 Table 48: Nuclear fusion flight characteristics after interplanetary flight and arrival at Earth ...... 114 Table 49: Nuclear fusion flight characteristics for Earth arrival flight ...... 116 Table 50: Mission characteristics for a nuclear fusion mission to Mars ...... 118 Table 51: Nuclear fusion Earth escape flight leg of Jupiter mission ...... 119 Table 52: Nuclear fusion spacecraft flight characteristics upon reaching the edge of Earth's sphere of influence ...... 121 Table 53: Nuclear fusion flight characteristics after interplanetary flight and arrival at Jupiter ...... 122 Table 54: Nuclear fusion flight characteristics for Jupiter arrival flight ...... 124 Table 55: Nuclear fusion Jupiter escape flight leg of the mission ...... 125 Table 56: Nuclear fusion spacecraft flight characteristics upon reaching the edge of Jupiter’s sphere of influence ...... 127 Table 57: Nuclear fusion flight characteristics after interplanetary flight and arrival at Earth ...... 128 Table 58: Nuclear fusion flight characteristics for Earth arrival flight ...... 130 Table 59: Mission characteristics for a nuclear fusion flight to Jupiter ...... 131 Table 60: Comparison of Mars mission characteristics for the various propulsion technologies ...... 138 Table 61: Initial conditions for a NEP Earth-Mars flight ...... 156 Table 62: Initial conditions for a NEP Mars escape trajectory flight ...... 156 Table 63: Initial conditions for a NEP Mars-Earth flight ...... 157 Table 64: Initial conditions for nuclear thermal propulsion Earth escape flight ...... 157 Table 65: Initial conditions for a NTP Earth-Mars flight ...... 158 Table 66: Initial conditions for a NTP Mars escape trajectory flight ...... 158 Table 67: Initial conditions for a NTP Mars-Earth flight ...... 159 Table 68: Initial conditions for a nuclear fusion Earth escape flight ...... 159 Table 69: Initial conditions for a nuclear fusion Earth-Mars flight ...... 160 Table 70: Initial conditions for a nuclear fusion Mars escape trajectory flight ...... 160 Table 71: Initial conditions for a nuclear fusion Mars-Earth flight ...... 161 Table 72: Initial conditions for a nuclear fusion Earth escape flight ...... 161 Table 73: Initial conditions for a nuclear fusion Earth-Jupiter flight ...... 162 Table 74: Initial conditions for a nuclear fusion Jupiter escape trajectory flight ...... 162 Table 75: Initial conditions for a nuclear fusion Jupiter-Earth flight ...... 163
6
FIGURES
Figure 1: Two-Body problem ...... 16 Figure 2: Spacecraft thrust vectoring ...... 20 Figure 3: Tangential thrust ...... 21 Figure 4: Zero thrust orbit ...... 30 Figure 5: Circular orbit with no applied thrust ...... 32 Figure 6: Elliptical orbit with low eccentricity and no applied thrust ...... 34 Figure 7: Elliptical orbit with higher eccentricity and no applied thrust ...... 36 Figure 8: Parabolic trajectory with no applied thrust...... 38 Figure 9: Hyperbolic trajectory with no applied thrust ...... 39 Figure 10: Overlay of various orbit trajectories ...... 40 Figure 11: Elliptic orbit with 0.5 eccentricity ...... 41 Figure 12: Maximum error as a function of step size ...... 43 Figure 13: Outbound circular spiral orbit ...... 45 Figure 14: Outbound circular spiral orbit ...... 46 Figure 15: Inbound circular orbit initial conditions ...... 47 Figure 16: Inbound circular spiral orbit ...... 48 Figure 17: Thrust applied exclusively in radial direction ...... 49 Figure 18: Constant radial thrust orbit ...... 50 Figure 19: Circular Earth escape orbit with constant circumferential thrust ...... 53 Figure 20: Circular Earth escape orbit with constant tangential thrust ...... 57 Figure 21: Dawn mission trajectory [4] ...... 60 Figure 22: Low-thrust mission trajectory to Mars ...... 64 Figure 23: Mission trajectory to Jupiter [5] ...... 66 Figure 24: Mission trajectory to Jupiter including periods of acceleration, deceleration, and coasting ...... 70 Figure 25: Schematic of an ion thruster [6] ...... 73 Figure 26: NEXT Thruster ...... 74 Figure 27: NEP Mars arrival flight ...... 77 Figure 28: NEP Mars escape flight ...... 80 Figure 29: NEP Elliptical Earth arrival orbit ...... 82 Figure 30: NEP elliptical high-Earth orbit arrival flight ...... 83 Figure 31: Earth-Mars roundtrip mission for a NEP spacecraft ...... 85 Figure 32: Nuclear thermal rocket engine [10] ...... 87 Figure 33: Earth escape trajectory using a nuclear thermal propulsion system ...... 90 Figure 34: NTP spacecraft flight path to the edge of Earth's sphere of influence ...... 91 Figure 35: NTP Mars arrival flight ...... 93 Figure 36: NTP Mars escape flight ...... 96 Figure 37: NTP spacecraft flight path to the edge of Mars' sphere of influence ...... 97 Figure 38: NTP Earth arrival flight ...... 99 Figure 39: Earth-Mars roundtrip mission for a NTP spacecraft ...... 101 Figure 40: Imploding liner nuclear fusion propulsion concept [14] ...... 103 Figure 41: Earth escape trajectory using a nuclear fusion propulsion system ...... 106 Figure 42: Nuclear fusion spacecraft flight path to the edge of Earth's sphere of influence ...... 107
7
Figure 43: Nuclear fusion Mars arrival flight ...... 109 Figure 44: Nuclear fusion Mars escape trajectory ...... 112 Figure 45: Nuclear fusion spacecraft flight path to the edge of Mars' sphere of influence ...... 113 Figure 46: Nuclear fusion Earth arrival flight...... 116 Figure 47: Earth-Mars roundtrip mission for a nuclear fusion spacecraft ...... 118 Figure 48: Earth escape trajectory using a nuclear fusion propulsion system ...... 120 Figure 49: Nuclear fusion spacecraft flight path to the edge of Earth's sphere of influence ...... 121 Figure 50: Nuclear fusion Jupiter arrival flight ...... 123 Figure 51: Nuclear fusion Jupiter escape trajectory ...... 126 Figure 52: Nuclear fusion spacecraft flight path to the edge of Jupiter’s sphere of influence ...... 127 Figure 53: Nuclear fusion Earth arrival flight...... 129 Figure 54: Earth-Jupiter roundtrip mission for a nuclear fusion spacecraft ...... 131 Figure 55: Two masses in an inertial reference frame ...... 145 Figure 56: Gravitational force in the Two-Body Problem ...... 146 Figure 57: Keppler's Second Law ...... 151
8
NOMENCLATURE
푎 semi-major axis 푎푟 component of acceleration in the radial direction 푎휃 component of acceleration in the circumferential direction 퐶3 characteristic energy 푒 orbital eccentricity 퐸 error 𝑔 gravitational constant 퐺 universal gravitational constant ℎ step size in MATLAB calculation 퐼sp specific impulse 푚1 mass of the central force body 푚2 mass of the orbiting body 푚̇ 푓 propellant mass flow rate 푟 radius between 푚1 and 푚2 푟exact exact orbital radius calculated with the solution to the Two-Body Problem 푟0 initial radius between 푚1 and 푚2 푡 time 푇 thrust force 푇푟 component of thrust force in the radial direction 푇휃 component of thrust force in the circumferential direction 푉 velocity of 푚2 푉esc escape velocity
휃 angle swept out as 푚2 orbits 푚1 휇 gravitational parameter 휈 non-dimensional thrust force 휈푟 component of non-dimensional thrust force in the radial direction 휈휃 component of non-dimensional thrust force in the circumferential direction 휌 non-dimensional radius 휏 non-dimensional time 휙 thrust force angle
9
Orbital Dynamics of Space Nuclear Propulsion Systems
Abstract
by
LARA SCHOEFFLER
Space nuclear propulsion systems comprising low and continuous thrust propulsion technology are of critical importance to deep space exploration. This analysis presents numerical calculations of orbital trajectories of three propulsion systems: nuclear electric, nuclear thermal, and nuclear fusion. The mission analysis utilizes existing characteristics, like thrust, from reference material to determine if a two-year roundtrip mission to Mars is feasible with any of these propulsion systems. Ultimately, the nuclear thermal propulsion and nuclear fusion propulsion missions are able to meet the rapid mission time requirement with roundtrips of 422 days and 630 days, respectively. The
Mars surface stays are short: 30 days with nuclear thermal propulsion and 200 days with a nuclear fusion spacecraft. This study makes a recommendation for the technology best suited for a near-term crewed Mars mission while also advocating the need for additional design and study on both the nuclear thermal and conceptual nuclear fusion systems.
10
I. INTRODUCTION
On July 20, 1969, the world paused to watch Neil Armstrong set foot on the surface of the moon, his single step realizing the culmination of an unprecedented engineering effort to send men where none had gone before. The Apollo missions developed and embraced new technologies to solve extraordinary engineering challenges and achieve a goal considered to be impossible in preceding decades. The Apollo program is one of humankind’s most ambitious undertakings but represents only the first small step in human exploration of the solar system. NASA’s Artemis program is laying the foundation for the next generation of human space exploration by returning humans to the Moon by 2024. A permanent human presence on the Moon provides NASA the opportunity to test new equipment and technologies that could be used to send astronauts to Mars. A crewed mission to Mars would likely embrace different propulsion technology than used by the Apollo missions, which were propelled to the Moon by chemical rockets, including the massive Saturn V. A crewed mission to Mars would attempt to keep the duration of the roundtrip journey to two years; current chemical propulsion systems require optimal planetary alignment of Earth and Mars to facilitate a transfer, causing extended mission times and long surface stays. Space nuclear propulsion technology could likely achieve the two-year mission duration and maximize the length of surface stays. However, all crewed missions to-date have employed chemical propulsion systems, so any space nuclear propulsion system requires significant development and testing.
11
Chemical rockets are attractive for crewed missions because they create extremely high thrust levels and achieve adequate spacecraft acceleration in only a couple minutes.
However, even in this limited burn time, these rockets consume massive quantities of propellant. While the thrust capabilities of chemical rockets makes them attractive for deep space missions, their low specific impulse values make it exceedingly difficult, if not impossible, to transport enough fuel to achieve a roundtrip mission within the two- year window. Additionally, because chemical rockets must complete rapid engine burns in order to conserve propellant, they are limited in the orbital maneuvers and trajectories the spacecraft is capable of performing. These limitations make it advantageous to analyze other propulsion technologies (both conceptual and in-development) as alternatives for crewed space missions beyond cislunar space.
The analysis utilizes various propulsion technologies in orbital dynamics calculations to demonstrate mission feasibility, as well as identify the technologies best suited for human missions to other planets. It aims to identify the systems most suited for rapid, fuel efficient deep space missions and provide a comparison between the capabilities as demonstrated by the orbital dynamics calculations. The technologies in question are nuclear electric propulsion, nuclear thermal propulsion, and nuclear fusion propulsion. The characteristics of each as described in the literature are applied when calculating a roundtrip trajectory of a two-year crewed Mars mission. Note that the focus is on applying the propulsion system characteristics (thrust, propellant flow rate, etc.) as defined in existing literature, not on seeking to devise improvements to said propulsion systems.
12
It is likely that not all of the propulsion systems described here will be sufficient for a rapid round-trip mission to Mars. For example, while nuclear electric propulsion is highly fuel efficient and capable for providing thrust for thousands of hours, its exceedingly low thrust output may make it a better candidate for robotic missions where mission duration is not a critical factor. Conversely, nuclear thermal propulsion systems are capable of high thrust and are more fuel efficient than traditional chemical systems.
These attributes make it likely that a nuclear thermal propulsion system would be sufficient for a round trip Mars mission. However, nuclear thermal rockets are significantly less fuel efficient than a nuclear fusion propulsion rocket and may not be a good candidate for deep space missions. Nuclear fusion propulsion is the least developed of the propulsion concepts described here but could be the ideal system for rapid human missions to a variety of locations around the Solar System.
The following section is the Method of Analysis, which describes the mathematics of low, continuous thrust orbits. It includes a derivation of the governing equations stemming from the Two-Body Problem. After the Method of Analysis, is the
Numerical Methodology Capabilities section which describes the ability of the MATLAB code to create various orbital trajectories and shows the flexibility of the code to respond to an assortment of inputs. Next, are four test cases, which are used to verify that the
MATLAB code works appropriately. The test cases are a combination of low, continuous thrust orbit examples from the literature, as well as a real-world NASA mission. The following section titled Mission Analysis, details hypothetical round-trip missions using each propulsion technology: nuclear electric, nuclear thermal, nuclear fusion. In each
13
mission analysis, the low, continuous thrust orbit governing equations are applied when calculating and plotting the trajectories.
14
II. Method of Analysis
The methodology is a numerical system designed to calculate and plot orbital trajectories of spacecraft as they move with respect to various central force bodies; the equations of motion derived from the Two-Body Problem are the basis for the numerical method. These equations are then modified to include terms for thrust, propellant mass flow rate, and thrust angle; these characteristics allow the method to calculate unique trajectories for spacecraft with a variety of propulsion systems and capabilities. This section details the derivation of the equations of motion and their capability to plot various types of orbits. The section ends with a study of four test cases that serve to prove the flexibility and accuracy of the numerical methodology.
A. Derivation of Equations
Low or continuous thrust orbits are governed by a set of equations derived from the Two-Body Problem. The Two-Body Problem describes the motion of two objects in space; it assumes the only forces acting on a body arise from the other body under consideration. This model is used to predict the orbits of a variety of astronomical entities including planets, stars, and spacecraft. In this analysis, the Two-Body Problem provides the foundation for a set of equations that describe the continuous thrust trajectory of a spacecraft. The derivation of the Two-Body Problem (as shown in Appendix A) results in the following equation [1]:
푑2푟⃗ 휇푟⃗ + = 0 (1) 푑푡2 푟3
15
The Two-Body Problem is represented pictorially in Figure 1. This figure shows the more massive of the two bodies, 푚1, as the central force body; the less massive body,
푚2, moves in orbit around 푚1 with velocity, 푉. The distance between the centers of the bodies is radius, 푟. The radius vector sweeps through an angle, 휃, as 푚2 orbits the central force body.
Figure 1: Two-Body problem
The first term in the Two-Body Problem, 푑2푟⃗⁄푑푡2, is the relative acceleration between the two bodies; the second term, 휇푟⃗⁄푟3, represents the acceleration due to gravity, where the gravitational parameter, 휇, is a function of the universal gravitation constant, 퐺, and the masses of the bodies, 푚1 and 푚2.
휇 = 퐺(푚1 + 푚2) (2)
However, in this analysis 푚2, which is assumed to be the mass of the spacecraft, is negligible when compared with the large mass of the central force body, so 푚2 is neglected as shown in Equation 3.
휇 = 퐺푚1 (3)
16
Returning to Equation 1, it is necessary to expand this equation into radial and circumferential components for use later on. First, the relative acceleration vector,
푑2푟⃗⁄푑푡2, is expanded into its component form. This expansion begins with the position vector as shown in Equation 4.
푟⃗ = 푟푟̂ (4)
The first derivative of the previous equation yields
푑푟⃗ 푑푟 푑푟̂ = 푟̂ + 푟 (5) 푑푡 푑푡 푑푡 where the first derivative of 푟̂ is expanded using the cross product of angular velocity with 푟̂.
푑푟̂ 푑휃 = 휃̂ (6) 푑푡 푑푡
Now, Equation 6 is substituted into Equation 5. The resulting equation is the velocity vector, as a function of radial velocity, 푑푟⁄푑푡, in the 푟̂ direction and circumferential velocity, 푑휃⁄푑푡, in the 휃̂ direction.
푑푟⃗ 푑푟 푑휃 = 푟̂ + 푟 휃̂ (7) 푑푡 푑푡 푑푡
The acceleration vector is determined by obtaining the second derivative of Equation 7.
푑2푟⃗ 푑2푟 푑푟 푑푟̂ 푑푟 푑휃 푑2휃 푑휃 푑휃̂ = 푟̂ + + 휃̂ + 푟 휃̂ + 푟 (8) 푑푡2 푑푡2 푑푡 푑푡 푑푡 푑푡 푑푡2 푑푡 푑푡
Substituting Equation 6 into Equation 8 reduces the equation to the following:
푑2푟⃗ 푑2푟 푑푟 푑휃 푑2휃 푑휃 푑휃̂ = 푟̂ + 2 휃̂ + 푟 휃̂ + 푟 (9) 푑푡2 푑푡2 푑푡 푑푡 푑푡2 푑푡 푑푡
In order to simplify Equation 9 further, 푑휃̂⁄푑푡 can be expanded using the cross product of angular velocity with 휃̂.
17
푑휃̂ 푑휃 = − 푟̂ (10) 푑푡 푑푡
After substituting Equation 10 into Equation 9, the simplified relative acceleration term is shown below. As shown in Equation 11, the radial and circumferential acceleration terms are grouped by position vectors 푟̂ and 휃̂, respectively.
푑2푟⃗ 푑2푟 푑휃 2 푑푟 푑휃 푑2휃 = ( − 푟 ( ) ) 푟̂ + (2 + 푟 ) 휃̂ (11) 푑푡2 푑푡2 푑푡 푑푡 푑푡 푑푡2
After expanding the relative acceleration term in Equation 1, the second piece of the equation, 휇푟⃗⁄푟3, can be expanded into its component terms. This simpler analysis requires only that Equation 4, the expression for the radius vector, be substituted into the gravitational acceleration term as shown.
휇푟⃗ 휇 = 푟̂ (12) 푟3 푟2
The derivation of the Two-Body Problem is rewritten as the sum of Equations 11 and 12.
푑2푟 푑휃 2 휇 푑푟 푑휃 푑2휃 ( − 푟 ( ) + ) 푟̂ + (2 + 푟 ) 휃̂ = 0 (13) 푑푡2 푑푡 푟2 푑푡 푑푡 푑푡2
Now that both pieces of the Two-Body Problem have been broken into component terms, Equation 13 is rewritten as a set of two equations. The radial and circumferential components of acceleration are represented in Equations 14 and 15, respectively.
푑2푟 푑휃 2 휇 − 푟 ( ) + = 0 (14) 푑푡2 푑푡 푟2
푑푟 푑휃 푑2휃 2 + 푟 = 0 (15) 푑푡 푑푡 푑푡2
In the previous two equations, the angular velocity of the spacecraft is 푑휃⁄푑푡, and the radial velocity of the spacecraft is 푑푟⁄푑푡. The term 휇⁄푟2 represents the acceleration
18
of the spacecraft due to the gravitational force of the central force body. The radial and angular acceleration of the spacecraft are described by 푑2푟⁄푑푡2 and 푑2휃⁄푑푡2, respectively.
The Two-Body Problem, as described above, assumes no forces, other than the gravitational attraction between the two bodies, are applied to the spacecraft. In this scenario, the spacecraft will continue to orbit the central force body with no deviation in trajectory; the introduction of another force is required to change the spacecraft’s trajectory. Equations 14 and 15 represent Newton’s first law of motion: Every body continues in its state of rest, or in uniform motion in a straight line, unless compelled to change that state by forces acting upon it [2]. The applied force required to change the spacecraft’s orbit is thrust, which is generated by the propulsion system of the spacecraft.
The spacecraft propulsion system (chemical, electrical, or nuclear) accelerates the propellant through the engine; the propellant exits the engine, creating a thrust force applied to the spacecraft in a direction directly opposite to the exiting propellant. This thrust force is a direct application of Newton’s third law of motion: To every action there is an equal and opposite reaction; that is, the mutual forces of two bodies acting upon each other are equal in magnitude and opposite in direction [2].
19
Figure 2: Spacecraft thrust vectoring
Figure 2 shows the thrust force, 푇, applied to the spacecraft at an angle, 휙, from the radius vector, where 휙 is the thrust angle. Thrust vectoring can be accomplished during flight by changing the thrust angle; this is necessary to maintain a flight path or perform a variety of orbital maneuvers. The thrust angle can be specified as a constant value during a period of time; for example, a thrust angle of 90° results in circumferential thrust only with no radial component of thrust. However, it is also possible to specify a thrust angle that provides a tangential thrust force; the thrust angle is varied constantly with time to ensure that the thrust force remains colinear with the velocity vector, 푉⃗⃗, as shown in Figure 3.
20
Figure 3: Tangential thrust
Figure 3 also describes the radial velocity, 푟̇, and the circumferential velocity, 푟휃̇, of the spacecraft. The tangential thrust angle is calculated by first recognizing the following relationship
푟휃̇ tan 휙 = (16) 푟̇
Because Equation 16 is dependent on the radial and circumferential velocity of the spacecraft, the thrust angle will change along the trajectory of the flight path. The instantaneous thrust angle can be calculated as shown:
푟휃̇ if 푟̇ > 0 then 휙 = tan−1 ( ) (17) 푟̇
푟휃̇ if 푟̇ < 0 then 휙 = tan−1 ( ) + 휋 (18) 푟̇
휋 if 푟̇ = 0 then 휙 = (19) 2
21
The addition of a thrust force to the spacecraft changes Equations 14 and 15.
Now, the sum of the radial and circumferential acceleration terms is no longer zero, meaning the spacecraft is moving under the influence of an applied thrust.
푑2푟 푑휃 2 휇 − 푟 ( ) + = 푎 (20) 푑푡2 푑푡 푟2 푟
푑푟 푑휃 푑2휃 2 + 푟 = 푎 (21) 푑푡 푑푡 푑푡2 휃
The acceleration terms, 푎푟 and 푎휃, are functions of thrust, 푇, and spacecraft mass,
푚2. The thrust force applied to the spacecraft is expanded into radial and circumferential components; these forces are represented in Figure 2 as 푇푟 and 푇휃 and are determined using the thrust angle.
푇푟 = 푇 cos 휙 (22)
푇휃 = 푇 sin 휙 (23)
Using Newton’s second law, the radial and circumferential acceleration due to thrust can be described as follows
푇푟 푎푟 = (24) 푚2
푇휃 푎휃 = (25) 푚2
Now, Equations 24 and 25 are substituted into Equations 20 and 21. The velocity and angular momentum of the spacecraft are changing with time due to the applied thrust force.
푑2푟 푑휃 2 휇 푇 푟 2 − 푟 ( ) + 2 = (26) 푑푡 푑푡 푟 푚2
푑푟 푑휃 푑2휃 푇 휃 2 + 푟 2 = (27) 푑푡 푑푡 푑푡 푚2
22
The previous two equations can be rewritten as follows after multiplying both sides by 푚2
푑2푟 푑휃 2 휇 푚 ( − 푟 ( ) + ) = 푇 (28) 2 푑푡2 푑푡 푟2 푟
푑푟 푑휃 푑2휃 푚 (2 + 푟 ) = 푇 (29) 2 푑푡 푑푡 푑푡2 휃
It should be noted that Equations 28 and 29 are representations of Newton’s second law of motion: The time rate of change of linear momentum of a body is proportional to the force acting upon it and occurs in the direction in which the force acts
[2]. In both equations, the force, 푇, is equal to the spacecraft mass, 푚2, multiplied by the acceleration terms described previously.
The equations of motion, Equations 26 and 27, are converted to non-dimensional forms by utilizing non-dimensional quantities for orbit radius, time, and thrust. Non- dimensional orbit radius, 휌, is expressed in terms of the initial orbit radius, 푟0, as shown in Figure 1.
푟 휌 = (30) 푟0
Non-dimensional time, 휏, is expressed in terms of initial orbit radius, as well as the gravitational parameter of the central force body:
휇 휏 = 푡√ 3 (31) 푟0
The spacecraft thrust is also expressed in non-dimensional terms; the term, 휈, is a function of spacecraft mass and the gravitational acceleration at the initial orbit radius, 𝑔.
Neither the mass nor the thrust of the spacecraft is constant during the flight; mass
23
decreases as the flight progresses and propellant is expended; the mass change calculation is discussed later in this section. Additionally, the thrust level of the spacecraft can change during the course of the flight to facilitate acceleration, deceleration, or orbital maneuvering.
푇 휈 = (32) 𝑔푚2
The previous equation is the non-dimensional total thrust and can be expressed as a pair of equations with radial and circumferential components
푇푟 휈푟 = (33) 𝑔푚2
푇휃 휈휃 = (34) 𝑔푚2 where 𝑔 is defined by the initial orbit radius and the gravitational parameter of the central force body.
휇 𝑔 = 2 (35) 푟0
The radial and circumferential thrust expressions, Equations 22 and 23, are substituted into the non-dimensional thrust equations, Equations 33 and 34, showing the dependence on the thrust angle.
푇 cos 휙 휈푟 = (36) 𝑔푚2
푇 sin 휙 휈휃 = (37) 𝑔푚2
Finally, Equation 32 is solved for 푇 and substituted into Equations 36 and 37.
휈푟 = 휈 cos 휙 (38)
휈휃 = 휈 sin 휙 (39)
24
The previous two equations are the non-dimensional radial, 푟̂, and circumferential, 휃̂, components of the total thrust.
As mentioned previously, the spacecraft mass does not remain constant during the entirety of the flight. Because the spacecraft can experience variable thrust levels, the mass flow rate of the propellant, 푚̇ 푓, varies. Note, however, that the mass of the spacecraft remains constant during periods of coasting when no thrust is applied to the craft. The following equation describes the change in spacecraft mass during a period of time, Δ푡, which is a dimensional quantity.
푚2 = 푚2 − 푚̇ 푓∆푡 (40)
The mass flow rate of the propellant, with units of kg⁄s, is multiplied by the timestep; this quantity is subtracted from the spacecraft mass to give a new mass value.
The non-dimensional time, 휏, is converted back to a dimensional quantity before ∆푡 is calculated; the conversion of non-dimensional quantities to dimensional quantities is discussed later in this section.
Now that all the non-dimensional quantities for orbit radius, time, and thrust have been described, they can be substituted into the equations of motion describing the trajectory of the spacecraft. The equation of motion in the radial direction, Equation 26, is converted to a non-dimensional form using Equations 30, 31, 33, and 35.
푑2휌 푑휃 2 1 − 휌 ( ) + − 휈 = 0 (41) 푑휏2 푑휏 휌2 푟
Similarly, the equation of motion in the circumferential direction, Equation 23, is converted to a non-dimensional form using Equations 30, 31, 34, and 35.
푑2휃 2 푑휌 푑휃 휈 + − 휃 = 0 (42) 푑휏2 휌 푑휏 푑휏 휌
25
These equations are non-dimensional expressions of the Two-Body Problem with an applied thrust.
This coupled set of non-linear, ordinary differential equations is then reduced to a set of first-order differential equations in order to solve them. First, Equations 41 and 42 are rewritten in a simplified format as shown.
1 휌′′ = 휌휃′2 − + 휈 (43) 휌2 푟
2 휈 휃′′ = − 휌′휃′ + 휃 (44) 휌 휌
Then, they are reduced to a set of four first-order differential equations:
푑휌′ 1 = 휌휃′2 − + 휈 (45) 푑휏 휌2 푟
푑휌 = 휌′ (46) 푑휏
푑휃′ 2 휈 = − 휌′휃′ + 휃 (47) 푑휏 휌 휌
푑휃 = 휃′ (48) 푑휏
The previous four differential equations are solved using a fourth-order Runge-
Kutta method. The method determines 휌(휏), 휌′(휏), 휃(휏), and 휃′(휏), where 휌(휏) and 휃(휏) describe the trajectory of the spacecraft with time as it spirals outward from a planet.
After Equations 45-48 are solved non-dimensionally, the quantities 휌, 휌′, 휃, and
휃′ are converted to a dimensional format. The dimensional radius quantity is calculated by rearranging Equation 30 as shown.
푟 = 휌푟0 (49)
Similarly, the dimensional time value is calculated by rearranging Equation 31 as shown.
26
푟3 푡 = 휏√ 0 (50) 휇
The non-dimensional radial velocity term, 휌′, is converted to a dimensional quantity beginning with Equation 46. Substituting Equations 30 and 31 into Equation 46 yields the following
푑(푟⁄푟 ) 푟3 휌′ = 0 √ 0 (51) 푑푡 휇
After rearranging Equation 51, the following dimensional quantity represents the radial velocity of the spacecraft.
′ 휇 푟̇ = 푟0휌 √ 3 (52) 푟0
The non-dimensional angular velocity term, 휃′, is converted to a dimensional quantity in a similar fashion. Equation 31 is substituted into Equation 48.
휇 ̇ ′ 휃 = 휃 √ 3 (53) 푟0
The spacecraft velocity vector along the orbit is calculated using the dimensional values for radius, radial velocity, and angular velocity, as shown:
푉⃗⃗ = 푟̇푟̂ + 푟휃̇휃̂ (54)
The scalar quantity for spacecraft velocity follows:
푉 = √푟̇2 + 푟2휃̇ 2 (55)
The velocity of the spacecraft is an important parameter to track, as it is used to calculate the characteristic energy of the spacecraft. The characteristic energy, 퐶3, is the
27
measure of the excess kinetic energy over that which is required to escape the attraction of the central force body [1]. The velocity of the spacecraft is used to calculate 퐶3.
2휇 퐶 = 푉2 − (56) 3 푟
If 퐶3 is negative, the spacecraft continues in an elliptical orbit around the central force body and does not have enough energy to reach escape velocity. Escape velocity is the velocity required for the spacecraft to escape the gravitational attraction of the central force body. When 퐶3 equals zero, the spacecraft has reached escape velocity; a positive
퐶3 value means that the spacecraft has energy exceeding that which is needed to reach escape velocity and is moving on a hyperbolic escape trajectory. The escape velocity of a spacecraft is dependent on the central force body and the orbit radius. In the following equation, escape velocity 푉esc is calculated using the gravitational parameter of the central force body, as well as the orbit radius.
2휇 푉 = √ (57) esc 푟
The result of this equation provides the velocity required to reach escape speed at any given orbit radius. As the spacecraft’s orbit radius increases, the velocity required to escape decreases. The analysis described in this section is performed numerically in
MATLAB; the methodology is described in Appendix C.
B. Numerical Methodology Capabilities
The method described in the previous section is capable of producing a variety of orbital trajectories including circular, elliptical, parabolic and hyperbolic orbits. Unique
28
trajectories are defined by sets of initial conditions, which describe the position and velocity of the spacecraft at the beginning of the analysis; additionally, various spacecraft capabilities including thrust, thrust angle, and propellant flow rate are required for the analysis. The MATLAB program must be supplied with initial conditions for the
′ ′ following variables: 휌, 휌 , 휃, 휃 , 푇, 푚̇ 푓, and 휙. These quantities are determined dimensionally and then converted to non-dimensional quantities with the techniques discussed in the previous section. Distinctive sets of initial conditions result in different orbit trajectories. Several types of orbit scenarios are discussed below, including orbits with no applied thrust, outbound and inbound orbits with constant applied thrust, and orbits with constant thrust in the radial direction. The section describing orbits with no applied thrust includes circular, elliptic, parabolic, and hyperbolic examples stemming from the Two-Body Problem equations derived previously. Additionally, this section discusses a comparison of error between a numerically derived solution and the exact calculation determined with the solution to the Two-Body Problem in order to prove the accuracy of the numerical methodology. Note that the following analyses are performed with constant mass and thrust levels.
1) Orbits with No Applied Thrust: The movement of a body with respect to a central force body is governed by the Two-Body Problem equations of motion as described previously. Equations 14 and 15 describe the motion of a body in orbits of various eccentricities with no applied thrust force. The eccentricity of an orbit refers to how much the orbit deviates from a circular orbit, which has an eccentricity, 푒 = 0. The equation for eccentricity as a function of radius, angular velocity, angle and the gravitation parameter
29
is derived in Appendix A. All examples in the section assume Earth as the central force body. Additionally, all examples are completed with no applied thrust and constant spacecraft mass. Figure 4 shows the movement of 푚2 at constant velocity with respect to the central force body, 푚1.
Figure 4: Zero thrust orbit
For a circular orbit, the following initial conditions (in dimensional format) may be assumed from inspection of Figure 4:
푟 = 푟0 (58)
푟̇ = 0 (59)
휃 = 0 (60)
The initial spacecraft orbit radius is 푟0 as described in Figure 4. Note that the starting angle swept out by the radius vector can vary depending on the flight trajectory.
For this circular orbit, the initial angle, 휃, from the initial radius vector is zero degrees.
Additionally, the velocity of the spacecraft at the start of a circular orbit is wholly in the circumferential direction; therefore, the radial velocity of the spacecraft is zero as shown
30
in Equation 59. The initial angular velocity, 휃̇, of the spacecraft is calculated using
Equation 61 as derived in Appendix A and shown below
휇 휃̇ = √ (1 + 푒 cos 휃) (61) 푟3
However, because the MATLAB program uses non-dimensional equations of motion, the initial conditions must be converted to non-dimensional forms. The non- dimensional expression for angular velocity is calculated by substituting Equation 61 into
Equation 53.
휃′ = √1 + 푒 cos 휃 (62)
Using the expressions for non-dimensional terms derived previously, the non- dimensional initial conditions are as follows in Table 1:
Table 1: Initial conditions for a circular orbit with no thrust
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1
Thrust Angle (휙) n/a
Thrust (휈) 0
Propellant Mass Flow Rate (푚̇ 푓) 0
The following figure shows a circular orbit with no applied thrust. The spacecraft continues in an orbit of constant radius around the central force body unless an outside force, such as thrust, is introduced.
31
Figure 5: Circular orbit with no applied thrust
For an elliptical orbit, the initial conditions for radius, radial velocity, and angle are identical for those of a circular orbit. However, an elliptical orbit is more eccentric than a circular orbit and has an eccentricity value 0 < 푒 < 1. The closer 푒 is to one, the more eccentric the orbit becomes. For example, choosing a value of 푒 close to zero such as 푒 = 0.0001 results in an elliptical orbit that appears almost circular. The non- dimensional angular velocity is calculated using Equation 62. The initial conditions for this orbit are described in Table 2.
32
Table 2: Initial conditions for an elliptical orbit with low eccentricity and no thrust
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1.00005
Thrust Angle (휙) n/a
Thrust (휈) 0
Propellant Mass Flow Rate (푚̇ 푓) 0
Figure 6 shows elliptical orbit with low eccentricity. The spacecraft orbiting Earth begins at the same point as in the previous scenario, but travels on an elliptic path and reaches its apocenter at an angle of 180°; the spacecraft continues along this trajectory until reaching pericenter again at 0°.
33
Figure 6: Elliptical orbit with low eccentricity and no applied thrust
It is possible to create a highly elliptical orbit by choosing a value of 푒 close to one such as 푒 = 0.9. The initial conditions are shown below and are identical to those for the previous elliptical orbit with the exception of a larger non-dimensional angular velocity term.
Table 3: Initial conditions for an elliptical orbit with high eccentricity and no thrust
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
34
Angular Velocity (휃′) 1.378
Thrust Angle (휙) n/a
Thrust (휈) 0
Propellant Mass Flow Rate (푚̇ 푓) 0
This orbit is shown in Figure 7 below. As in the low-eccentricity example, the spacecraft reaches apocenter at 180°; however, the radius at apocenter in Figure 7 is significantly larger than that in the low-eccentricity example due to the higher starting
35
angular velocity. Without the addition of an outside force, such as thrust, the spacecraft will continue in this elliptical orbit forever.
Figure 7: Elliptical orbit with higher eccentricity and no applied thrust
A parabolic orbit is an orbit in which the spacecraft is not travelling in a bound path around the central force body. In this case, the spacecraft is either travelling away from a central force body on an escape path or travelling toward a central force body on a capture trajectory. Parabolic orbits have an eccentricity value 푒 = 1.0. This example orbit has the same initial conditions for radius, radial velocity, and angle as the previous three examples as shown in Table 4. The angular velocity term is higher than the previous example due to the higher eccentricity value.
36
Table 4: Initial conditions for a parabolic orbit with no thrust
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) √2
Thrust Angle (휙) n/a
Thrust (휈) 0
Propellant Mass Flow Rate (푚̇ 푓) 0
The parabolic orbit shown in Figure 8 is an escape orbit; the spacecraft will continue on this orbit trajectory indefinitely unless acted upon by an outside force.
37
Figure 8: Parabolic trajectory with no applied thrust
A spacecraft on a hyperbolic trajectory is travelling on an escape trajectory; hyperbolic trajectories have an orbital eccentricity 푒 > 1.0. This example orbit has the same initial conditions for radius, radial velocity, and angle as the previous four examples as shown in Table 5. However, due to the higher eccentricity, the orbit exhibits a higher angular velocity. This example assumes an eccentricity value 푒 = 1.6.
Table 5: Initial conditions for a hyperbolic orbit with no thrust
Radius (휌) 1
Radial Velocity (휌′) 0
38
Angle (휃) 0
Angular Velocity (휃′) 1.612
Thrust Angle (휙) n/a
Thrust (휈) 0
Propellant Mass Flow Rate (푚̇ 푓) 0
Figure 9 shows a spacecraft on a hyperbolic trajectory away from Earth; the spacecraft has enough speed to escape the Earth’s gravitational attraction and continue on away from the planet.
Figure 9: Hyperbolic trajectory with no applied thrust
39
For the purpose of comparison, Figure 10 overlays the circular, elliptical, parabolic, and hyperbolic orbit examples discussed in this section. Each orbit starts from the same point (due to the initial conditions) around the same central force body.
Figure 10: Overlay of various orbit trajectories
2) Comparisons of Numerical and Exact Results: The examples in the former section were solved numerically using the previously developed methodology. To confirm the accuracy of the numerical calculations, the results are compared with exact calculations completed using the solution to the Two-Body Problem (with no applied thrust), which is derived in Appendix A. The orbit used for this analysis is an elliptical orbit with eccentricity 푒 = 0.5 as shown in Figure 11.
40
Figure 11: Elliptic orbit with 0.5 eccentricity
As with the previous examples, the initial conditions must be described in non- dimensional terms. Table 6 describes the initial conditions used in this analysis.
Table 6: Initial conditions for an elliptic orbit with no thrust
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1.225
Thrust Angle (휙) n/a
41
Thrust (휈) 0
Propellant Mass Flow Rate (푚̇ 푓) 0
The exact solution to the Two-Body Problem for elliptical orbits relates the orbit radius to the orbit angle, the orbit eccentricity, and the semimajor axis, 푎, of the orbit. As derived in Appendix A, the equation for calculating the exact radius, 푟exact, of an orbit is shown below.
푎(1 − 푒2) 푟 = (63) exact 1 + 푒 cos 휃
The analysis begins with calculating the semimajor axis of the elliptical orbit using the initial orbit radius, eccentricity, and starting angle of the orbit. The assumed initial radius, 푟0, is 6,700 km with an angle of 0°. The semimajor axis is calculated by rearranging Equation 63 as shown.
푟 (1 + 푒 cos 휃) 푎 = 0 (64) 1 − 푒2
The semimajor axis, initial radius, and eccentricity are summarized in Table 7.
Table 7: Orbit characteristics for exact radius calculations
Semimajor Axis (푎) 13,400 km
Initial Radius (푟0) 6,700 km
Eccentricity (푒) 0.5
The numerical methodology calculates the elliptical orbit shown in Figure 11 and provides an output file containing orbit angles and corresponding radius values. The exact radius value is calculated using Equation 63 and the various orbit angles provided
42
by the numerical method. These two radius values are used to calculate the error, 퐸, as shown in the following equation.
푟 − 푟exact 퐸 = (65) 푟exact
The maximum error calculated for the elliptical radius described in this section is dependent on the step size, ℎ, used in the MATLAB program. The relationship between maximum error and step size is shown in Figure 12 below.
Figure 12: Maximum error as a function of step size
The value of maximum error 6.32 × 10-4 ≤ 퐸 ≤ 7.48 × 10-4 varies with step size
0.0001 ≤ ℎ ≤ 0.1. The maximum error does not change much and appears to be insensitive to changing step size. The numerical analysis can be run successfully with any
43
step size shown here; however, the smaller step sizes result in unacceptably long processing times.
3) Outbound Circular Spiral Orbit with Applied Thrust: The previous sections discussed various types of orbit trajectories derived from the Two-Body Problem with no applied thrust. The circular orbit can be modified by applying a thrust to the spacecraft; the addition of a thrust force causes the spacecraft to deviate from a circular orbit and begin to spiral outward away from the central force body. As with the previous examples, the analysis must begin with a set of initial conditions; the initial radius, radial velocity, angle, and angular velocity were derived in the previous section and are replicated in
Table 8.
The remaining initial conditions are thrust, mass flow rate, and thrust angle. This example was performed with constant spacecraft mass while assuming constant circumferential thrust. The non-dimensional thrust quantity, 휈, is assumed for the purposes of illustrating a spiral orbit.
Table 8: Initial conditions for an outbound spiral orbit with applied thrust
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1
Thrust Angle (휙) 90°
Thrust (휈) 0.005
44
Propellant Mass Flow Rate (푚̇ 푓) 0
The initial conditions in Table 8 produce the circular spiral orbit shown in Figure
13. The spacecraft spirals out from Earth at constant thrust; at a certain point, the engines are shut off, thrust goes to zero, and the spacecraft continues coasting in orbit around the
Earth. The spacecraft is unable to reach escape velocity. The circular mark on Figure 13 indicates the point at which thrust was reduced to zero, and the spacecraft begins to coast.
Figure 13: Outbound circular spiral orbit
Figure 14 is another example of an orbit with the initial conditions described in this section; in this trajectory, the spacecraft continues thrusting in a circular orbit for a
45
longer time period before coasting, resulting in a much larger orbit around Earth. The spacecraft’s engines are shut off before reaching escape velocity, resulting in a coasting orbit that is elliptical in shape.
Figure 14: Outbound circular spiral orbit
4) Inbound Circular Spiral Orbit with Applied Thrust: The previous section describes a spacecraft employing circumferential thrust to spiral out from a central force body. It is also possible for a spacecraft to employ an inbound spiral orbit when arriving at another body. The initial conditions for non-dimensional radius, radial acceleration, angular velocity, and angle are the same as those used for the circular outward spiral orbit
46
example as documented in Table 9. In this example, the thrust angle is different from the ninety degree circumferential thrust employed previously.
Figure 15: Inbound circular orbit initial conditions
In Figure 15, the applied thrust force, 푇, is directly opposite the velocity vector,
푉, of the spacecraft; the thrust angle is 270°.
Table 9: Initial conditions for an inbound spiral orbit with applied thrust
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1
Thrust Angle (휙) 270°
Thrust (휈) 0.005
Propellant Mass Flow Rate (푚̇ 푓) 0
47
The thrust acts as a retarding force on the velocity of the spacecraft and causes it to decelerate; the spacecraft is drawn inward by the gravitational attraction to the central force body, resulting in the circular spiral orbit shown in Figure 16.
Figure 16: Inbound circular spiral orbit
The spacecraft continues to spiral inward until the thrust force is no longer applied; at this point, the spacecraft continues to coast in a circular orbit around the central force body. The beginning of the coasting orbit is indicated by the circular mark on Figure 16.
5) Radial Thrust Orbit: In the circular orbit examples, the thrust force is applied circumferentially to the spacecraft to create a circular spiral orbit trajectory. If a thrust is
48
applied only in the radial direction, as shown in Figure 17, the angular velocity of the spacecraft does not change, and the spacecraft maintains a circular orbit around the central force body.
Figure 17: Thrust applied exclusively in radial direction
In this situation, the initial conditions required for the MATLAB program are the same as those required for the circular spiral orbit, as shown in Table 10. However, the thrust angle is 0° in the case of constant radial thrust, as opposed to a thrust angle of 90° for circumferential thrust.
Table 10: Initial conditions for a radial thrust orbit
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1
49
Thrust Angle (휙) 0
Thrust (휈) 0.005
Propellant Mass Flow Rate (푚̇ 푓) 0
Figure 18 shows the constant circular orbit of a spacecraft experiencing constant radial thrust only. The spacecraft maintains a circular orbit and is unable to move towards or away from the central force body.
Figure 18: Constant radial thrust orbit
50
C. Test Cases
After developing the methodology, four test analyses are performed to confirm
MATLAB code functionality. The test cases are selected to test a variety of code capabilities and are listed in order of increasing complexity. The first two cases are comparisons with hypothetical, continuous thrust, Earth escape trajectories using circumferential and tangential thrust, respectively, as described by Irving [3]. Both of these cases assume constant spacecraft mass for the duration of the analysis. The third test case is a comparison with the NASA Dawn mission, which sent an unmanned, low thrust, solar electric propulsion vehicle to visit Ceres and Vesta in the asteroid belt. The fourth and final test case is a comparison with a theoretical deep space mission to Jupiter.
These four analyses serve to prove that the code and methodology are robust and capable of accurate trajectory mapping activities.
1) Test Case One: Circumferential Earth Escape: The first test case describes a hypothetical spacecraft travelling on an outbound circular spiral orbit around Earth; the spacecraft thrusts at a constant level until reaching escape velocity. This spacecraft, as described by Irving [3], uses constant circumferential thrust, which requires a 90 thrust angle; he defines a non-dimensional thrust value, 휈, of 0.005. He chooses an initial orbit altitude of 322 kilometers. Note that this analysis is performed with constant spacecraft mass and constant thrust; there are no periods of coasting. The initial conditions for this analysis were defined in section II.B.3, which describes the initial radius, radial velocity, angular velocity, angle, and thrust angle for a spacecraft on an outbound circular orbit trajectory. The Irving analysis is performed non-dimensionally, so all initial conditions
51
and trajectory data are presented non-dimensionally, as well. The pertinent initial conditions are summarized below in Table 11.
Table 11: Initial conditions for circumferential, circular Earth escape orbit
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.005
Propellant Mass Flow Rate (푚푓) 0
Table 12 compares the escape trajectory characteristics calculated using the numerical methodology with the analysis presented by Irving. The radius and angle variables describe the position of the spacecraft in relation to the Earth when the spacecraft reaches escape velocity; the time variable describes the amount of time required for the spacecraft to reach escape velocity while undergoing constant circumferential thrust. Unfortunately, Irving does not provide values for the position of the spacecraft when it reaches escape velocity.
Table 12: Comparison of circumferential escape trajectories
Irving [3] Numerical Methodology
Radius (휌) Not provided 12.056
52
Angle (휃) Not provided 46.67
Time (휏) 159.8 159.84
The non-dimensional time derived by Irving matches the value calculated with the methodology described in this paper; additionally, the escape orbit shown in Figure 19 matches the trajectory plotted by Irving. The numerical solution of the Two-Body
Problem is capable of accurately replicating the results produced by Irving. Figure 19 shows the outbound spiral orbit with constant circumferential thrust.
Figure 19: Circular Earth escape orbit with constant circumferential thrust
Per Table 12, the spacecraft reaches escape velocity at a non-dimensional radius 휌 of 12.056. The initial radius is calculated by adding the 321.87 km orbit radius with the
53
radius of the Earth, 6,378 km, resulting in an initial orbit radius 푟0 of 6,699.87 km. Then, the radius at which escape velocity is achieved is determined by substituting the values for 휌 and 푟0 into Equation 49, resulting in a radius 푟 of 80,775 km. Now, the time required to reach escape velocity is calculated using Equation 50. Substituting 푟0, the gravitational parameter for Earth (398,600 km3⁄s2), and the non-dimensional time into
Equation 50 results in a flight time of 38.56 hours before reaching escape velocity. The dimensional angle, angular velocity, and radial velocity are calculated using equations derived in the previous sections. Table 13 summarizes the non-dimensional and dimensional characteristics of the orbit trajectory at the escape point as calculated by the numerical methodology.
Table 13: Comparison of dimensional and non-dimensional orbit characteristics
Non-Dimensional Dimensional
Radius 12.0562 80,775 km
Radial Velocity 0.2174 1.677 km⁄s
Angle 0.8145 46.67
Angular Velocity 2.857 × 10-2 3.289 × 10-5 rad⁄s
It is also possible to calculate the escape velocity of the spacecraft; this quantity is also used to confirm that the numerical methodology is accurate. Per Equation 57, the velocity required to escape an orbit of radius 80,775 km around Earth is 3.142 km⁄s.
Using Equation 55 to calculate velocity, the spacecraft velocity at the moment of escape
54
is 3.142 km⁄s. The calculated quantity matches the escape velocity determined using the
MATLAB program.
2) Test Case Two: Tangential Earth Escape: The second test case also describes a hypothetical spacecraft travelling on an outbound circular spiral orbit around Earth; as with the first test case, Irving uses a constant, non-dimensional thrust, 휈, of 0.005 [3].
However, this test case involves an added layer of complexity; the spacecraft experiences a constant tangential thrust force, which requires the use of a variable thrust angle, 휙. The thrust angle varies to match the direction of the velocity vector, resulting in tangential thrust. The tangential thrust angle is constantly changing to match the velocity vector, as opposed to the previous test case, which assumed a constant ninety-degree thrust angle.
Note that this analysis is also performed with constant spacecraft mass and constant thrust; there are no periods of coasting. The initial conditions for this analysis were defined in section II.B.2, which describes the initial radius, radial velocity, angular velocity, angle, and thrust angle for a spacecraft on an outbound circular orbit trajectory.
The Irving analysis is performed non-dimensionally, so all initial conditions and trajectory data are presented non-dimensionally, as well. The pertinent initial conditions are summarized in Table 14.
Table 14: Initial conditions for tangential, circular Earth escape orbit
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0
55
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.005
Propellant Mass Flow Rate (푚푓) 0
Table 15 compares the escape trajectory characteristics calculated using the methodology described in this paper with the analysis presented by Irving. The radius and angle variables describe the position of the spacecraft in relation to the Earth when the spacecraft reaches escape velocity; the time variable describes the amount of time required for the spacecraft to reach escape velocity while undergoing constant circumferential thrust. Unfortunately, Irving does not provide values for the position of the spacecraft when it reaches escape velocity.
Table 15: Comparison of tangential escape trajectories
Irving [3] Numerical Methodology
Radius (휌) Not provided 12.427
Angle (휃) Not provided 0.4644 (26.61)
Time (휏) 157.1 157.1
The non-dimensional time derived by Irving matches the value calculated with the methodology described in this paper; additionally, the escape orbit shown in Figure 20 matches the trajectory plotted by Irving. The numerical solution of the Two-Body
56
Problem is capable of accurately replicating the results produced by Irving. Figure 20 shows the outbound spiral orbit with constant tangential thrust.
Figure 20: Circular Earth escape orbit with constant tangential thrust
Per Table 15, the spacecraft reaches escape velocity at a non-dimensional radius 휌 of 12.427. The initial orbit radius is 6,699.87 km as calculated in the previous test case.
Then, the radius at which escape velocity is achieved is determined by substituting the values for 휌 and 푟0 into Equation 49, resulting in a radius 푟 of 83,255.9 km. Now, the time required to reach escape velocity is calculated using Equation 50. Substituting 푟0, the gravitational parameter for Earth, and the non-dimensional time into Equation 50 results in a flight time of 37.91 hours before reaching escape velocity. The dimensional
57
angle, angular velocity, and radial velocity are calculated using equations derived in the previous sections. Table 16 summarizes the non-dimensional and dimensional characteristics of the orbit trajectory at the escape point as calculated by the numerical methodology.
Table 16: Comparison of dimensional and non-dimensional orbit characteristics
Non-Dimensional Dimensional
Radius 12.4265 83,255.9 km
Radial Velocity 0.253 1.951 km⁄s
Angle 0.4644 26.61
Angular Velocity 2.506 ×10-2 2.885 × 10-5 rad⁄s
It is also possible to calculate the escape velocity of the spacecraft and confirm that the numerical methodology is accurate. Per Equation 57, the velocity required to escape an orbit of radius 83,255.9 km around Earth is 3.095 km⁄s. Using Equation 55 to calculate velocity, the spacecraft velocity at the moment of escape is 3.095 km⁄s. The calculated quantity matches the escape velocity determined using the MATLAB program.
This test case demonstrates that a tangential Earth escape can be achieved more rapidly than a circumferential one under the same thrust conditions. The spacecraft using constant, tangential thrust was able to reach escape velocity in 37.91 hours, as opposed to the constant, circumferential thrust option which took 38.56 hours. The more rapid flight of the tangential thrust trajectory results in reduced propellant usage and more rapid transit times when compared with the circumferential trajectory. For this reason, it is
58
generally more desirable to perform a continuous thrust orbital transit with a tangential thrust angle.
3) Test Case Three: NASA Dawn Mission: The Dawn Mission is a NASA mission
(2007-2018) that used an unmanned spacecraft powered by a solar-electric propulsion system. Dawn launched from Earth on September 27, 2007 and used a low thrust solar- ion propulsion system with xenon propellant. The Dawn spacecraft has three ion engines, each capable of 19-91 mN of thrust; one engine was active at a time with the others being redundant. This low thrust level is several orders of magnitude lower than traditional chemical rockets and is comparable to the force generated by dropping a single piece of paper. At maximum thrust, the ion engine consumed 3.25 mg⁄s of xenon propellant.
Dawn left Earth with 425 kg of propellant; the total spacecraft mass at launch was
1,172.1 kg. The engines provided constant thrust with intermittent coasting periods for communications or maintenance. After numerous periods of thrusting and coasting,
Dawn arrived at Mars on February 17, 2009. After a Mars gravity assist, Dawn continued on to Vesta, the second largest body in the asteroid belt. It arrived at Vesta on July 16,
2011 and remained in orbit around Vesta for 18 months. After completing its scientific mission at Vesta, Dawn flew to Ceres, a dwarf planet in the asteroid belt, arriving on
March 5, 2015. The spacecraft remains in orbit around Ceres after the conclusion of its mission in 2018. Dawn was the first spacecraft to orbit two destinations beyond Earth, as well as the first to orbit a dwarf planet.
This test case analyzes the first leg of the Dawn mission’s flight: the trip from
Earth to Mars. The trajectory of the Dawn mission is illustrated in Figure 21. The black
59
line segments indicate periods when Dawn coasted, while the blue indicate periods of thrusting. Additionally, the orbits of Earth and Mars are described by green and red lines, respectively.
Figure 21: Dawn mission trajectory [4]
Dawn launched from Earth on a Delta II rocket; it achieved Earth escape speed
(퐶3 = 0) and continued to coast away from Earth for 81 days until December 17, 2007 when it activated its engines to begin its interplanetary journey [4]. During the next 318 days, the Dawn spacecraft continued its flight to Mars; this transit time included periods of both thrusting and coasting. Dawn utilized its thrusters for 270 days (85%) of the total trip time and expended 72.1 kilograms of Xenon propellant [4]. This 318-day period was
60
followed by 109 days of coasting before rendezvous with Mars. All orbital dynamics calculations take place with the Sun as the central force body. Using the Earth departure date of September 27, 2007 and the HORIZONS Ephemerides, the radius from the Sun to the Earth is determined to be 1.003 AU, where one astronomical unit (AU) equals
1.49597871 x 108 km. The radius between Mars and the Sun on the Mars arrival date of
February 17, 2009 is determined to be 1.409 AU. Additionally, Mars has an orbital speed of 26.017 km⁄s on the date of spacecraft arrival. Unlike previous analyses, the initial orbit radius of the spacecraft does not start at 0 but begins at 2 as estimated from Figure
21. Table 17 lists Dawn spacecraft characteristics pertinent to the analysis.
Table 17: Dawn mission spacecraft characteristics [4]
Initial Mass 1,172.1 kg
Mass at Mars Arrival 1,100 kg
Maximum Thrust 91 mN
Propellant Mass Flow Rate 3.25 mg⁄s
Trip Time 508 days
Thrust Time 270 days
Coast Time 238 days
Radius: Earth to Sun (9/27/2007) 1.003 AU
Radius: Mars to Sun (2/17/2009) 1.409 AU
Angle: Earth (9/27/2007) 2
Angle: Mars (2/17/2009) 300
Velocity of Mars (2/17/2009) 26.017 km⁄s
61
There are a number of parameters required for this analysis that are not provided in [4]. First, the thrust angle 휙, which is the angle between the radius vector and the applied thrust force is not provided. In order to most closely replicate Dawn’s trajectory, this test case assumes a tangential thrust angle for the all periods of thrusting during the flight from Earth to Mars. Additionally, this analysis assumes periods of intermittent coasting and thrusting at maximum power.
The eccentricity of the orbit is not provided; this analysis assumes an elliptical orbit initially with eccentricity 푒 = 0.194. Because the initial orbit trajectory is elliptical, the angular velocity of the spacecraft is non-zero. From Equation 62, the non-dimensional angular velocity is calculated to be 휃′ = 1.093. The non-dimensional initial radius, angular velocity, and thrust angle are identical to those described in section II.C.1. As indicated in Table 17, the initial angle is estimated to be 2. Finally, the non-dimensional thrust, 휈, is zero initially because the spacecraft spends the first 81 days of flight coasting away from Earth. Similarly, the initial propellant mass flow rate is zero because no propellant is consumed while the spacecraft coasts. The initial conditions are summarized in Table 18.
Table 18: Initial conditions for Mars trajectory
Radius (휌) 1
Radial Velocity (휌′) 0
Angle (휃) 0.035
Angular Velocity (휃′) 1.093
Thrust Angle (휙) 90
62
Thrust (휈) 0
Propellant Mass Flow Rate (푚̇ 푓) 0 kg⁄s
Table 19 compares the Mars trajectory characteristics calculated using the numerical methodology with the Dawn mission. The radius and angle variables describe the position of the spacecraft in relation to the Sun when the spacecraft reaches Mars; the mass variable describes the spacecraft mass when it has consumed enough propellant to arrive at Mars.
Table 19: Comparison of Mars trajectories
Dawn [4] Numerical Methodology
Radius (푟) 1.4091 AU 1.409 AU
Angle (휃) ≈ 5.236 (300) 5.249 (300.72)
Mass (푚) 1,100 kg 1,100 kg
Velocity (푉) 26.017 km⁄s 25.587 km⁄s
The numerical solution of the Two-Body Problem is capable of accurately replicating the position and velocity of the Dawn spacecraft with the assumptions discussed previously. Figure 22 shows the outbound flight to Mars with periods of intermittent thrusting and coasting.
63
Figure 22: Low-thrust mission trajectory to Mars
4) Test Case Four: Jupiter Transit: The final test case is an analysis of a hypothetical mission to Jupiter in [5]. The paper discusses five categories of candidates for exploration and study: Primitive Bodies, Inner Solar System, Mars, Giant Planets,
Large Satellites [5]. The inner solar system (Mercury and Venus), as well as Mars, were not part of the study because their close proximity to Earth result in less significant flight times and propulsion system challenges [5]. However, the giant planets (Jupiter, Saturn,
Uranus, and Neptune) and large satellites, such as Pluto, represent a significant challenge.
These bodies are far from Earth and trip times with current propulsion system technology are prohibitively long for crewed missions due to health concerns caused by radiation exposure and a zero gravity environment. The study conducted in [5] analyzes crewed
64
missions to the planets Jupiter, Saturn, Neptune, Uranus, and Pluto using a nuclear electric propulsion system. This test case attempts to replicate the Jupiter mission in [5].
There are a number of assumptions used in [5] when describing the Jupiter mission. The outbound trip time is limited to 1.8 years due to concerns with radiation exposure and a zero gravity environment. The trip time is broken into periods of thrusting and coasting, with an initial thrusting acceleration phase followed by a period when the spacecraft is coasting. After coasting, the spacecraft begins thrusting to decelerate before arriving at the target system. The total mission length and thrust time is provided in [5], but the authors do not provide the breakdown between acceleration and deceleration time.
The total thrust time is one year, with 0.8 years of coasting time [5]. The maximum thrust level is provided and all thrusting takes place with the maximum available thrust.
However, the authors do not provide the thrust angle used to calculate the orbit trajectory.
The analysis performed in [5] yields Figure 23, where the green line indicates
Earth’s orbit around the sun. The solid blue line indicates periods of spacecraft thrusting, and the dotted blue line indicates coasting. The red line indicates the movement of Jupiter during spacecraft transit.
65
Figure 23: Mission trajectory to Jupiter [5]
The outbound flight leaves from Earth, as shown in Figure 23 where the blue and green lines intersect, and the mathematical analysis begins when 퐶3 equals zero and the spacecraft has escaped Earth’s gravity [5]; all orbital dynamics calculations take place with the Sun as the central force body. Using the Earth departure date of July 28, 2050 and the HORIZONS Ephemerides, the radius from the Sun to the Earth is determined to be 1.015 AU. The radius between Jupiter and the Sun on Jupiter arrival date of May 23,
2052 is determined to be 5.448 AU. Additionally, Jupiter has an orbital speed of 12.461 km⁄s on the date of spacecraft arrival. Unlike previous analyses, the initial orbit radius of the spacecraft does not start at 0 but begins at 305 as estimated from Figure 23.
66
The spacecraft launch mass is 20,000,000 kg, which includes 15,918,368 kg of propellant [5]. The final mass upon arrival at Jupiter is 4,081,632 kg [5]. Propellant mass flow rate is calculated by dividing the propellant mass by the total thrust time yielding a propellant mass flow rate of 0.505 kg⁄s at maximum thrust. The propellant mass flow rate is constant during periods of maximum thrust [5]. Table 20 reflects the spacecraft and mission characteristics provided in [5] and discussed above. All other parameters were allowed to vary freely in [5] to provide an “optimized” mission trajectory.
Table 20: Known Jupiter mission characteristics [5]
Initial Mass 20,000,000 kg
Final Mass 4,081,632 kg
Maximum Thrust 9,105 N
Propellant Mass Flow Rate 0.505 kg/s
Trip Time 1.8 years
Thrust Time 1.0 years
Radius: Earth to Sun (7/28/2050) 1.015 AU
Radius: Jupiter to Sun (5/23/2052) 5.448 AU
Angle: Earth (7/28/2050) 305
Angle: Jupiter (5/23/2052) 185
Velocity of Jupiter (5/23/2052) 12.461 km/s
There are a number of parameters required for this analysis that are not provided in [5]. First, the thrust angle 휙, which is the angle between the radius vector and the
67
applied thrust force is not provided. In order to most closely replicate the Jupiter trajectory in [5], this test case assumes a tangential thrust angle for the acceleration portion of the flight and a constant 165.2 thrust angle during the deceleration leg. Next, the total thrust time is known, but the breakdown between acceleration and deceleration time is unknown. This test case breaks the total thrust time into an acceleration time of
0.683 years and 0.317 years of deceleration time. These assumptions are compiled in
Table 21.
Table 21: Assumed Jupiter mission characteristics
Acceleration Thrust Angle Tangential
Deceleration Thrust Angle 165.2
Acceleration Time 0.683 years
Deceleration Time 0.317 years
Additionally, [5] does not provide information on the eccentricity of the acceleration orbit; this analysis assumes zero eccentricity. Because the initial orbit trajectory is circular, four of the initial conditions are duplicated from previous examples.
The non-dimensional initial radius, radial velocity, angular velocity, and thrust angle are identical to those described in section II.C.1. As indicated in Table 22, the initial angle is estimated to be 305. Finally, the non-dimensional thrust, 휈, is calculated per Equation
32.
Table 22: Initial conditions for Jupiter trajectory
Radius (휌) 1
Radial Velocity (휌′) 0
68
Angle (휃) 5.323
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.079
Propellant Mass Flow Rate (푚̇ 푓) 0.505 kg⁄s
Table 23 compares the Jupiter trajectory characteristics calculated using the numerical methodology with the analysis presented in [5]. The radius and angle variables describe the position of the spacecraft in relation to the Sun when the spacecraft reaches
Jupiter; the mass variable describes the spacecraft mass when it has exhausted its propellant upon reaching Jupiter.
Table 23: Comparison of Jupiter trajectories
McNutt [5] Numerical Methodology
Radius (푟) 5.448 AU 5.457 AU
Angle (휃) ≈3.229 (185) 3.217 (184.32)
Mass (푚) 4,081,632 kg 4,080,900 kg
Velocity (푉) 12.461 km⁄s 12.460 km⁄s
The numerical solution of the Two-Body Problem is capable of accurately replicating the position and velocity of the spacecraft produced in [5] with the assumptions listed in Table 21 and Table 22. Figure 24 shows the outbound flight to
Jupiter with an initial period of constant, tangential thrust followed by a period of
69
coasting. After coasting, the spacecraft begins to thrust again at a 165.2 thrust angle in order to decelerate before arriving at Jupiter.
Figure 24: Mission trajectory to Jupiter including periods of acceleration, deceleration, and coasting
70
III. MISSION ANALYSIS
The low or continuous thrust trajectory methodology described above can be applied to various propulsion systems for the purposes of mission design and analysis. Of great interest currently is a human mission to Mars. Due to radiation exposure and other health concerns, the mission length is limited to two years, meaning the spacecraft and its crew must fly to Mars, complete their mission, and return two Earth in a two-year period.
The various propulsion technologies described in the subsequent sections include nuclear electric propulsion, nuclear thermal propulsion, and nuclear fusion propulsion. Each section includes a Mars mission trajectory calculated using the thrust capabilities of the propulsion system under consideration. Additionally, the nuclear fusion section includes a human mission to Jupiter to highlight the advanced capabilities of this technology and its potential for missions deeper into the solar system.
A. Electric Propulsion
Electric propulsion (EP) is a type of propulsion that creates thrust by using electrical power to accelerate a propellant by magnetic or electrical means. Electric propulsion systems are widely used on spacecraft and satellites today. The Dawn mission is a notable example of a deep space mission powered by solar electric propulsion.
Electric propulsion systems are attractive for deep space travel because of their high Isp values and their ability to thrust continuously for long periods of time. However, electric thrusters are only capable of providing low amounts of thrust and are limited by their electric power source (solar or nuclear). The two styles of electric thrusters can be categorized as electrostatic and electromagnetic. Electrostatic thrusters, such as Ion and
71
Hall Effect thrusters, use an applied electric field to accelerate ions and create thrust, whereas electromagnetic thrusters, such as Magnetoplasmadynamic (MPD) and Pulsed
Inductive Thrusters (PIT), use magnetic fields to accelerate plasma to appropriate exhaust velocities [6]. More information about the functionality of Hall Effect, MPD, and PIT thrusters may be found in [6].
Ion thrusters use conducting grids to apply an electric field to accelerate ions to high exhaust velocities [6]. However, the propellant, which is typically xenon, begins with no charge; the propellant is bombarded with electrons generated by the hollow cathode in order to generate plasma. The plasma passes downstream through a positively charged grid (called the screen grid) and enters a region between the screen grid and accelerator grid. The accelerator grid is negatively charged and attracts the plasma, which passes through the small holes of the grid where they are discharged and create thrust. A schematic of an ion thruster is found in Figure 25.
72
Figure 25: Schematic of an ion thruster [6]
Electrostatic thrusters have been successfully used on several deep space missions, including Deep Space-1 and the Dawn mission, as previously mentioned. Both of these spacecraft used the NASA Solar Electric Propulsion Technology Applications
Readiness (NSTAR) thruster. The NASA Evolutionary Xenon Thruster (NEXT) system is an evolution of the proven NSTAR thruster and provides higher power, thrust, and specific impulse [7]. The NEXT thruster can be throttled from 25-235 mN of thrust at power levels ranging from 0.64-7.33 kW, with corresponding specific impulse values varying from 1,395-4,155 seconds [7]. The NEXT thruster has a mass of 13.5 kg and is only 36 cm in diameter [8] making it significantly smaller than traditional chemical
73
rockets. Figure 26 shows a NEXT ion thruster. The small size and mass of the NEXT thruster make it possible to easily include multiple thrusters on a spacecraft; this is critical because the low thrust capability of the NEXT thruster would necessitate that multiple engines be utilized on a crewed spacecraft. The higher power needs of multiple
NEXT engines can best be met through the use of a nuclear reactor.
Figure 26: NEXT Thruster
Electric propulsion systems are attractive for use on spacecraft because they are exceptionally fuel efficient and can be used continuously for thousands of hours.
However, the exceedingly low thrust capability of electric propulsion engines cause long transit times, which are not ideal for crewed missions. To date, electric propulsion systems have proven sufficient for use on unmanned, long-term missions. Electric propulsion systems have been reliably operated for thousands of hours both in a lab setting and in space. Currently, the NEXT thruster has achieved Technology Readiness
74
Level (TRL) 6 [6], meaning a system model or prototype has been demonstrated in a relevant ground or space environment [9].
This section will analyze a human mission to Mars utilizing a NEP system with eight NEXT engines in order to determine if the technology is capable of meeting the two year mission requirement. The analysis assumes all eight engines will be operated at maximum thrust, providing a total of 1.88 N of thrust. The propellant mass flow rate is calculated using specific impulse, average thrust, and the acceleration due to gravity as shown in the following equation:
퐹 푚̇ 푓 = (66) 𝑔0퐼푠푝
The propellant mass flow rate for this conceptual system is 4.6 × 10-5 kg⁄s. The thrust and propellant mass flow rate characteristics of this conceptual nuclear electric propulsion system are summarized in Table 24.
Table 24: Performance characteristics of conceptual nuclear electric propulsion system
Maximum Total Thrust 1.88 N
Specific Impulse 4,155 s
Propellant Mass Flow Rate 4.6 × 10-5 kg⁄s
The trajectory evaluated in this section, using the NEP characteristics described above, is a round trip flight to Mars of a spacecraft with a 50,000 kg starting mass. The first leg of the mission requires the spacecraft to escape Earth’s gravity and begin its interplanetary flight to Mars. However, due to the mass of the spacecraft and the minimal available thrust, the spacecraft is not capable of escaping Earth’s gravity in a timely
75
fashion using its electric thrusters. The spacecraft would orbit the Earth hundreds of time over the course of years before reaching escape velocity. Therefore, it is assumed that the spacecraft both launches from Earth and escapes Earth’s gravity using conventional chemical rockets. The low-thrust analysis will begin after the spacecraft reaches Earth- escape velocity and begins its interplanetary journey to Mars. At this point, the nuclear electric propulsion system kicks in and begins to provide maximum thrust. The trajectory is plotted using a Sun-centric calculation. All non-dimensional initial conditions used in the MATLAB code to calculate various trajectories may be found in Appendix B.
The spacecraft does not maintain constant thrust for the duration of the interplanetary trip but uses periods of thrusting and coasting. At the start of the trip, the spacecraft thrusts at 1.88 N and a tangential thrust angle; it maintains this thrust level for
98.38 days before stopping the engine and beginning to coast. The spacecraft coasts for another 37.04 days before arriving at the edge of Mars’ sphere of influence (SOI). The total interplanetary trip time is 135.42 days. The flight characteristics and trajectory are shown in Table 25 and Figure 31, respectively.
Table 25: NEP flight characteristics after interplanetary flight and arrival at Mars
Arrival Radius 2.275 108 km
Arrival Angle 1.937 rad (110.98)
Propellant Consumption 391 kg
Arrival Mass 49,609 kg
Arrival Velocity 24.26 km⁄s
Total Thrust Time 98.38 days
76
Coast Time 37.04 days
Total Trip Time 135.42 days
The spacecraft arrives in Mars’ sphere of influence and needs to continue thrusting in order to settle into a stable, circular orbit around the planet. As shown in
Figure 27, the spacecraft orbits Mars in a highly elliptical orbit as it spirals in towards the planet until settling in a stable, circular orbit at an altitude of 200 miles. The elliptical orbit reduces the time required to reach the 200-mile orbit.
Figure 27: NEP Mars arrival flight
77
The flight from the edge of Mars’ sphere of influence occurs at constant, tangential thrust with a starting velocity of 24.58 km⁄s. Over the course of 193.84 days, the spacecraft expends 770 kg of propellant. The time required to reach a stable, circular orbit around Mars is longer than the time required to fly from Earth because Mars’ gravitational force is now the primary force acting on the spacecraft at a closer distance.
After nearly eight revolutions around Mars, the spacecraft settles into a low-altitude circular orbit. It is worth nothing that this arrival flight time could be reduced if the main spacecraft were left in a high-altitude orbit and a smaller craft detached before flying to the low-altitude orbit. The flight characteristics for this portion of the mission are summarized in Table 26.
Table 26: NEP flight characteristics for Mars arrival flight
Starting Radius (edge of SOI) 593,120 km
Starting Angle 4.96 rad (284.46)
Propellant Consumption 770 kg
Arrival Mass 48,839 kg
Starting Velocity 24.58 km⁄s
Total Thrust/Flight Time 193.84 days
The mission can only spend 30 days on the surface of Mars. As stated in the test case section, radiation exposure and health concerns currently limit a crewed space mission to two years, which includes the time spent in situ and the transit time to and from Earth. The red line in Figure 31 shows the movement of Mars around the Sun during this mission. Note that the orbit of Mars is approximated as a circle at 1.523 AU.
78
After completing a one month mission on Mars, the spacecraft begins its journey back to
Earth. From a circular orbit, the spacecraft begins to accelerate at maximum thrust with a tangential thrust angle. It spirals out from Mars in a highly elliptical orbit until reaching escape velocity. The spacecraft reaches escape velocity in only 127.57 days using 507 kg of propellant. It is able to escape Mars’ gravity moving at 24.57 km⁄s. Table 27 contains the flight time and propellant usage of the Mars escape leg of the Mars mission.
Table 27: NEP Mars escape flight leg of mission
Propellant Consumption 507 kg
Final Mass 48,332 kg
Escape Velocity 24.57 km⁄s
Total Thrust/Flight Time 127.57 days
Figure 28 below shows the elliptical orbit employed by the spacecraft as it spirals out of Martian orbit and begins its trip back to Earth.
79
Figure 28: NEP Mars escape flight
After leaving Mars’ gravitational sphere of influence, the Sun again becomes the primary gravitational force acting on the spacecraft as it travels back to Earth. As with the previous analyses, the spacecraft utilizes maximum thrust during the interplanetary journey.
The spacecraft does not maintain constant thrust for the duration of the interplanetary trip but again uses periods of thrusting and coasting. At the start of the trip, the spacecraft thrusts at 1.88 N and a tangential thrust angle; it maintains this thrust level for 88.07 days before stopping the engine and coasting for 162.04 days. After coasting for over five months, the spacecraft begins to accelerate at maximum thrust and a tangential thrust angle for 138.89 days in order to arrive at the edge of Earth’s SOI. The
80
total interplanetary trip time is 389 days. A shown in Figure 31, the long return trip is due to the position of Earth with respect to Mars when the return journey begins. The spacecraft has to cover a farther distance before intercepting the Earth. The flight characteristics and trajectory are shown in Table 28 and Figure 31, respectively.
Table 28: NEP flight characteristics after interplanetary flight and arrival at Earth
Mars Departure Radius 2.281 × 108 km⁄s
Mars Departure Angle 5.155 rad (295.36)
Mars Departure Velocity 22.633 km⁄s
Earth Arrival Radius 1.496 108 km
Earth Arrival Angle 2.884 rad (165.241)
Propellant Consumption 902 kg
Earth Arrival Mass 47,430 kg
Earth Arrival Velocity 21.75 km⁄s
Total Thrust Time 226.96 days
Coast Time 162.04 days
Total Trip Time 389 days
The spacecraft arrives in Earth’s sphere of influence and needs to continue thrusting in order to settle into a stable, circular orbit around the planet. As discussed previously, the spacecraft does not have the thrust capabilities to spiral in to low-Earth orbit in a reasonable time frame. Even is the spacecraft used a highly elliptical orbit as
81
shown in Figure 29, it would still take 318 days of travel before arriving in low-Earth orbit.
Figure 29: NEP Elliptical Earth arrival orbit
Alternatively, the spacecraft can fly to a high-Earth orbit and rendezvous with a spacecraft propelled by traditional chemical rockets. This spacecraft can return the astronauts and payload to Earth in a matter of days, eliminating nearly a year of transit time as the spacecraft spirals to low-Earth orbit. As shown in Figure 30, the NEP spacecraft follows a highly elliptical flight path to high-Earth orbit to rendezvous with a second spacecraft.
82
Figure 30: NEP elliptical high-Earth orbit arrival flight
The flight from the edge of Earth’s sphere of influence begins with the spacecraft flying at 30.47 km⁄s and constant tangential thrust. After 9.44 days at maximum thrust, the spacecraft arrives in high-Earth orbit at an altitude of 35,786 km and expends only 38 kg of propellant. The details of the Earth arrival flight are summarized in Table 29 below.
At this point, the spacecraft can dock with another spacecraft, and the astronauts can be returned to Earth on the second spacecraft using conventional chemical rockets.
Table 29: NEP flight characteristics for Earth arrival flight
Starting Radius (edge of SOI) 925,870 km
Starting Angle 2.81 rad (161)
83
Propellant Consumption 38 kg
Arrival Mass 47,430 kg
Starting Velocity 30.47 km⁄s
Total Thrust Time 9.44 days
Total Coast Time 0 days
Total Flight Time 9.44 days
Assuming a high-Earth orbit rendezvous, the roundtrip mission to Mars described in this section has a total duration of 885.27 days, which includes a one-month dwell time at Mars and fails to come in under the maximum mission duration of two years. The spacecraft consumes 2,570 kg of propellant during the course of the mission. Figure 31 shows the Mars and Earth transits, as well as the movement of each planet during the course of the mission. Table 30 summarizes the final mission characteristics. The total thrust time during the course of this mission is 656.19 days, which far exceeds the capabilities of traditional chemical rockets and highlights the fuel efficiency of the propulsion technology.
84
Figure 31: Earth-Mars roundtrip mission for a NEP spacecraft
Table 30: Final mission characteristics for NEP mission to Mars
Total Propellant Consumption 2,570 kg
Final Mass 47,430 kg
Total Thrust Time 656.19 days
Total Coast Time 229.08 days
Mars Stay 30 days
Total Mission Duration Time 885.27 days
85
B. Nuclear Thermal Propulsion
Nuclear thermal propulsion (NTP) is a type of propulsion system using a nuclear thermal rocket (NTR). Nuclear thermal rockets were developed and tested from 1955-
1972 as part of the Rover and Nuclear Engine for Rocket Vehicle Applications (NERVA) programs. The Rover/NERVA programs built and ground tested twenty rockets, which demonstrated thrust levels from 25-250 klbf and specific impulse (Isp) values of approximately 900 seconds [10]. The demonstrated specific impulse values, which are twice those of chemical rockets, make nuclear thermal propulsion an attractive candidate for a crewed Mars mission. Additionally, nuclear thermal rockets achieved TRL ~5-6 during the Rover/NERVA programs [11].
A nuclear thermal rocket derives energy from the fission of Uranium-235 atoms, as opposed to traditional chemical rockets which use chemical combustion [10]. The
NTR uses a fission reactor core to generate the hundreds of megawatts of thermal power required to heat the liquid hydrogen propellant to high exhaust temperatures for rocket thrust [10]. The liquid hydrogen propellant flows from the turbopumps around the nozzle and reactor housing, cooling them and recovering heat; the heated propellant is expanded through the turbines, passes through the shielding and enters the reactor. The hydrogen flows through the coolant channels in the reactor core’s fuel elements, where it absorbs energy from the fission of U-235 atoms [10]; the now superheated propellant is expanded out through the nozzle, generating thrust. Figure 32 shows a NERVA nuclear thermal rocket engine.
86
Figure 32: Nuclear thermal rocket engine [10]
The expander-cycle NTR can be reconfigured for “bimodal” operation, meaning the rocket engine can be used for both thrust and power production [10]. Electric power can be generated for crew life support, communications, and other spacecraft operations.
When the spacecraft is producing thrust to perform orbital maneuvers, the reactor produces the megawatts of thermal power required to heat the propellant to the appropriate exhaust temperatures. However, when the spacecraft enters the coast phase of the mission, the control drums reduce the reactor power production to a lower level; the energy generated during this reduced-power phase is captured using a closed gas loop and fed through a Brayton cycle to generate electricity.
The bimodal nuclear thermal rocket engine used in the Mars mission trajectory described in this section produces a maximum of 25 klbf (111,206 N) with a 911 second specific impulse and a propellant flow rate of 12.45 kg⁄s [11]. The low specific impulse value necessitates short engine burn times to conserve propellant; three engines are required to effectively accelerate the spacecraft on its interplanetary journey. The rocket engine is 6.5 m long and has a thrust-to-weight ratio of ~5.5 [11]; the engine has a mass
87
of approximately 2060 kg. Table 31 lists the cumulative thrust and propellant flow rate values of three nuclear thermal rocket engines.
Table 31: Performance characteristics of conceptual nuclear thermal propulsion system
Maximum Total Thrust 333,618 N
Specific Impulse 911 s
Propellant Mass Flow Rate 37.35 kg⁄s
Nuclear thermal propulsion systems are ideal for use on spacecraft because they offer high power and a more fuel efficient means of accelerating a spacecraft than chemical rockets. As shown in Table 31, nuclear thermal rockets offer high thrusts, as well as specific impulses double those of traditional chemical rockets. Although nuclear thermal rockets have higher specific impulse values, they still require large quantities of propellant, which increase the system mass and cause slower transit times.
The trajectory evaluated in this section, using the nuclear thermal rocket characteristics described above, is a round trip flight to Mars. All MATLAB code initial conditions are summarized in Appendix B. After launching from Earth, the 250,000 kg spacecraft orbits Earth 200 miles above the surface; it leaves this low-Earth orbit using its nuclear thermal propulsion system at maximum thrust. All three spacecraft engines thrust for 36.48 minutes at a tangential thrust angle in order to reach an escape velocity of
38.278 km⁄s. After reaching escape velocity, the spacecraft’s engines shut off. The spacecraft consumes 81,760 kg of propellant during this leg of the flight. Table 32 details the fight time and propellant usage of the Earth escape leg of the Mars mission.
88
Table 32: NTP Earth escape flight leg of Mars mission
Propellant Consumption 81,760 kg
Final Mass 168,240 kg
Escape Velocity 38.278 km⁄s
Flight/Thrust Time 36.48 minutes
Escape Radius 10,913 km
Figure 33 below shows the trajectory of the spacecraft as it begins its journey away from Earth; the spacecraft does not circle the Earth during the escape leg of the flight.
89
Figure 33: Earth escape trajectory using a nuclear thermal propulsion system
After reaching escape velocity, the spacecraft flies to the edge of Earth’s gravitational sphere of influence at which point the Sun becomes the primary gravitational influence on the spacecraft. Figure 34 shows the spacecraft’s trajectory as it reaches the edge of Earth’s SOI. The spacecraft engines shut off after reaching Earth escape velocity, and it coasts to the edge of the SOI.
90
Figure 34: NTP spacecraft flight path to the edge of Earth's sphere of influence
The specific flight characteristics for this portion of the journey are listed in Table
33. The spacecraft has traveled a total of 6.765 days since leaving low Earth orbit; it expends no propellant after reaching escape velocity.
Table 33: NTP spacecraft flight characteristics upon reaching the edge of Earth's sphere of influence
Propellant Consumption 0 kg
Final Mass 168,240 kg
Velocity 30.88 km⁄s
Flight Time (incl. escape time) 6.765 days
91
Radius 925,290 km
At this point, the spacecraft leaves the Earth’s gravitational influence and begins its interplanetary flight; the next leg of the journey is the flight from Earth to Mars. The trajectory is plotted using a Sun-centric calculation. As with the Earth escape trajectory, the spacecraft utilizes three engines with a combined thrust of 333,618 N. The spacecraft does not maintain constant thrust for the duration of the interplanetary trip but uses periods of thrusting and coasting. At the start of the trip, the spacecraft accelerates under maximum thrust and a tangential thrust angle; it maintains this thrust level for 17 minutes before stopping the engines and beginning to coast. The spacecraft coasts for 131.933 days, which is the majority of the interplanetary flight duration. After traveling for over four months, the spacecraft arrives at the edge of Mars’ SOI. The flight characteristics and trajectory are shown in Table 34 and Figure 39, respectively.
Table 34: NTP flight characteristics after interplanetary flight and arrival at Mars
Arrival Radius 2.278 108 km
Arrival Angle 1.891 rad (108.35)
Propellant Consumption 37,780 kg
Arrival Mass 130,460 kg
Arrival Velocity 24.398 km⁄s
Total Thrust Time 17 minutes
Coast Time 131.933 days
Total Trip Time 131.945 days
92
The spacecraft arrives in Mars’ sphere of influence and needs to continue thrusting in order to settle into a stable, circular orbit around the planet. As shown in
Figure 35, the spacecraft flies in a straight trajectory until being captured by Mars’ gravity. It spirals in towards the planet until settling in a stable, circular orbit at an altitude of 200 miles.
Figure 35: NTP Mars arrival flight
The flight from the edge of Mars’ sphere of influence (577,000 km) occurs at constant, tangential thrust with a starting velocity of 24.878 km⁄s. The spacecraft coasts in towards the planet for 7.962 days before the spacecraft’s three engines begin to provide thrust. The three engines operate for 8.67 minutes until the spacecraft settles into
93
a circular orbit at an altitude of 200 miles. Over the course of 7.969 days, the spacecraft expends 19,420 kg of propellant. Table 35 summarizes the Mars arrival flight characteristics.
Table 35: NTP flight characteristics for Mars arrival flight
Starting Radius (edge of SOI) 577,630 km
Starting Angle 6.038 rad (345.98)
Propellant Consumption 19,420 kg
Arrival Mass 110,040 kg
Starting Velocity 24.878 km⁄s
Total Flight Time 7.969 days
The spacecraft stays in orbit around Mars for thirty days. As stated in the test case section, radiation exposure and health concerns currently limit a human space mission to two years, which includes the time spent in situ and the transit time to and from Earth.
The red line in Figure 39 shows the movement of Mars around the Sun during this mission. Note that the orbit of Mars is approximated as a circle at 1.523 AU. After completing a month-long mission on Mars, the spacecraft begins its journey back to
Earth. From a circular orbit, the spacecraft begins to accelerate at maximum thrust with tangential thrust angle. It spirals out from Mars until reaching escape velocity. The spacecraft reaches escape velocity after 7.25 minutes while all three engines are operating at full power and expend 16,240 kg of propellant. It is able to escape Mars’ gravity
94
moving at 28.95 km⁄s. Table 36 contains the flight time and propellant usage of the
Earth escape leg of the Mars mission.
Table 36: NTP Mars escape flight leg of mission
Propellant Consumption 16,240 kg
Final Mass 94,800 kg
Escape Velocity 28.95 km⁄s
Flight Time 7.25 minutes
Escape Radius 3,803 km
Figure 36 shows the spacecraft’s flight path as it escapes Mars’ gravity.
95
Figure 36: NTP Mars escape flight
After reaching escape the velocity, the spacecraft flies to the edge of Mars’ gravitational sphere of influence at which point the Sun becomes the primary gravitational influence on the spacecraft. Figure 37 shows the spacecraft’s trajectory as it reaches the edge of Mars’ SOI. The spacecraft stops thrusting after reaching escape velocity and coasts to the edge of Mars’ SOI.
96
Figure 37: NTP spacecraft flight path to the edge of Mars' sphere of influence
The specific flight characteristics for this portion of the journey are listed in Table
37. The spacecraft has traveled a total of 10.38 days since leaving Martian orbit; it continues to accelerate and expend propellant as shown below.
Table 37: NTP spacecraft flight characteristics upon reaching the edge of Mars’ sphere of influence
Propellant Consumption 16,240 kg
Final Mass 94,800 kg
Velocity 24.67 km⁄s
Flight Time (incl. escape time) 10.38 days
97
Radius 577,450 km
After leaving Mars’ gravitational sphere of influence, the Sun again becomes the primary gravitational force acting on the spacecraft as it travels back to Earth. As with the previous analyses, the spacecraft utilizes maximum thrust during the interplanetary journey. The spacecraft does not maintain constant thrust for the duration of the interplanetary trip but again uses periods of thrusting and coasting. At the start of the trip, the spacecraft coasts away from Mars for 120.09 days. When it engages its propulsion system, the spacecraft thrusts at 333,618 N and a tangential angle; it maintains this thrust level for 20.17 minutes before stopping the engine and beginning to coast for the remaining 109.90 days of the flight until it arrives at the edge of Earth’s SOI. The total interplanetary trip time is 230 days. The flight characteristics and trajectory are shown in
Table 38 and Figure 39, respectively.
Table 38: NTP flight characteristics after interplanetary flight and arrival at Earth
Mars Departure Radius 2.279 108 km⁄s
Mars Departure Angle 2.330 rad (133.50)
Mars Departure Velocity 23.956 km⁄s
Earth Arrival Radius 1.496 108 km
Earth Arrival Angle 0.947 rad (54.26)
Earth Arrival Velocity 25.92 km⁄s
Propellant Consumption 45,338 kg
Earth Arrival Mass 49,462 kg
98
Total Thrust Time 20.17 minutes
Coast Time 229.99 days
Total Trip Time 230 days
The spacecraft arrives in Earth’s sphere of influence and needs to continue thrusting in order to settle into a stable, circular orbit around the planet. As shown in
Figure 38, the spacecraft flies in a straight trajectory until being captured by Earth’s gravity. It spirals in towards the planet until settling in a stable, circular orbit at an altitude of 200 miles.
Figure 38: NTP Earth arrival flight
99
The flight from the edge of Earth’s sphere of influence begins with the spacecraft coasting at 31.44 km⁄s and zero thrust. After 5.09 days, the spacecraft begins to accelerate at maximum thrust for 6.83 minutes. Over the course of this portion of the flight, the spacecraft expends 15,313 kg of propellant and reaches a circular low Earth orbit at an altitude of 200 miles.
Table 39: NTP flight characteristics for Earth arrival flight
Starting Radius (edge of SOI) 925,760 km
Starting Angle 5.461 rad (312.89)
Propellant Consumption 15,313 kg
Arrival Mass 34,149 kg
Starting Velocity 31.44 km⁄s
Total Thrust Time 6.83 minutes
Total Coast Time 5.09 days
Total Flight Time 5.09 days
The roundtrip mission to Mars described in this section has a total duration of
422.15 days, which includes 30 days of dwell time at Mars and comes in under the maximum mission duration of two years. The spacecraft consumes 215,851 kg of propellant during the total thrust time of 87.73 minutes. Figure 39 shows the Mars and
Earth transits, as well as the movement of each planet during the course of the mission.
Table 40 summarizes the final mission characteristics.
100
Figure 39: Earth-Mars roundtrip mission for a NTP spacecraft
Table 40: Mission characteristics for NTP mission to Mars
Total Propellant Consumption 215,851 kg
Final Mass 34,149 kg
Total Thrust Time 87.73 minutes
Total Coast Time 390.69 days
Mars Stay 30 days
Total Mission Duration Time 422.15 days
101
C. Nuclear Fusion Propulsion
Nuclear fusion systems are conceptual and highly-energetic options for both terrestrial energy production and in-space propulsion systems. Nuclear fusion is a process by which energy is generated from the fusion of two lighter atomic nuclei; the fusion process results in a heavier nucleus formed during an exothermic reaction [12]. The fusion process differs drastically from nuclear fission, which splits heavy atomic nuclei into two lighter nuclei during an exothermic reaction [13]. The energy captured from either a nuclear fission reaction or a nuclear fusion reaction can be used to generate electricity. However, fusion has the advantage of greatly reduced radioactivity, no danger of a runaway reaction, and minimal nuclear waste [12]. Terrestrial nuclear fusion for power production has been an area of research and development for decades; however, no design has yet been capable of producing a net-positive power output [12].
One promising space nuclear fusion propulsion system uses the concept of pulsed fusion. In this type of fusion system, a small pellet of deuterium-tritium (D-T) fuel is compressed; the outer layer of the pellet explodes and generates an implosion that compresses the inner pellet material to conditions where fusion can occur [12]. The energy from the pellet implosion heats the hydrogen fuel surrounding the pellet, and the heated fuel is expanded through a nozzle, generating thrust. Prior concepts, including
Gevaltig and VISTA, proposed the use of laser ignited fusion systems to initiate the compression of the pellet [14]. A more recent concept uses a magnetic field to compress the D-T fuel and initiate fusion [14]. See Figure 40 for a schematic of this fusion concept.
The thrust level of a pulsed fusion propulsion system is controlled by the frequency with which pellets are compressed; a higher pulse repetition frequency results in higher thrust
102
levels. The rapid pellet ignition rate is used to calculate an average or close-to-continuous thrust level.
Figure 40: Imploding liner nuclear fusion propulsion concept [14]
Figure 40 gives a basic overview of the pulsed fusion implosion process.
However, it does not provide information pertaining to the estimated size or mass of the system, as nuclear fusion propulsion system designs remain largely conceptual.
Using a pulse repetition rate of 10 Hz, a specific impulse and average thrust are calculated in [14]; these values are recorded in Table 41. The propellant mass flow rate is calculated with Equation 66 using specific impulse, average thrust, and the acceleration due to gravity. The propellant mass flow rate for this conceptual system is 0.025 kg⁄s.
Table 41: Performance characteristics of conceptual fusion propulsion system [14]
Maximum Thrust 7,800 N
Specific Impulse 32,200 s
Propellant Mass Flow Rate 0.025 kg⁄s
103
Nuclear fusion propulsion systems are ideal for use on spacecraft because they offer an efficient means of accelerating a spacecraft. As recorded in Table 41, proposed fusion engine concepts have high specific impulses, which minimize fuel usage. Fusion engines could also operate continuously for days or weeks, allowing the spacecraft to accelerate to higher speeds during an interplanetary transfer, as well as drastically reducing the transfer time. This constant, long-term thrust capability differs substantially from traditional chemical rockets. Although chemical rockets create higher thrust than a nuclear fusion system, they have low specific impulse values of less than 500 seconds typically [15]. The low specific impulse requires the spacecraft to carry an enormous quantity of fuel; this prohibits the use of chemical rockets for roundtrip missions to Mars and beyond without some method of refueling mid-mission. For example, the space shuttle main engines each produced 470,000 lb (2,090 kN) of thrust in a vacuum [15] and had an Isp of only 452 seconds [16]. Using Equation 66, the propellant mass flow rate was
471.5 kg⁄s. This massive propellant consumption rate is not sustainable if chemical rockets were used in the two missions described below. Both consider rapid missions requiring long periods of constant thrust using nuclear fusion propulsion.
The first trajectory, using the nuclear fusion engine characteristics described above, is a round trip flight to Mars. After launching from Earth, the 350,000 kg spacecraft orbits Earth 200 miles above the surface; it leaves this low-Earth orbit using its nuclear fusion engine at maximum thrust. All non-dimensional initial conditions used in the MATLAB code are recorded in Appendix B. The spacecraft spirals out from low-
Earth orbit at constant, maximum thrust and a tangential thrust angle. After 3.25 days of
104
continuous thrust, the spacecraft reaches a velocity of 32.34 km⁄s and is able to escape
Earth’s gravity. It has consumed 6,930 kg of propellant during this leg of the mission.
Table 42 contains the flight time and propellant usage of the Earth escape leg of the Mars mission.
Table 42: Nuclear fusion Earth escape flight leg of Mars mission
Propellant Consumption 6,930 kg
Final Mass 343,070 kg
Escape Velocity 32.34 km⁄s
Flight Time 3.25 days
Escape Radius 116,560 km
Figure 41 below shows the trajectory of the spacecraft as it begins its journey away from Earth; the plot shows the spiral orbit ending when the spacecraft reaches escape velocity after sixteen revolutions around the Earth.
105
Figure 41: Earth escape trajectory using a nuclear fusion propulsion system
After reaching escape velocity, the spacecraft flies to the edge of Earth’s gravitational sphere of influence, at which point the Sun becomes the primary gravitational influence on the spacecraft. Earth’s SOI is 925,000 km. Figure 42 shows the spacecraft’s trajectory as it reaches the edge of Earth’s SOI. The spacecraft engines shut off after reaching Earth escape velocity, and it coasts to the edge of the SOI.
106
Figure 42: Nuclear fusion spacecraft flight path to the edge of Earth's sphere of influence
The specific flight characteristics for this portion of the journey are listed in Table
43. The spacecraft has traveled a total of 11.25 days since leaving low Earth orbit; it expends no propellant after reaching escape velocity.
Table 43: Nuclear fusion spacecraft flight characteristics upon reaching the edge of Earth's sphere of influence
Propellant Consumption 0 kg
Final Mass 343,070 kg
Velocity 30.65 km⁄s
Flight Time (incl. escape time) 11.25 days
107
Radius 925,430 km
At this point, the spacecraft leaves the Earth’s gravitational influence and begins its interplanetary flight; the next leg of the journey is the flight from Earth to Mars. The trajectory is plotted using a Sun-centric calculation. As with the Earth escape trajectory, the spacecraft utilizes its maximum thrust of 7,800 N. The spacecraft does not maintain constant thrust for the duration of the interplanetary trip but uses periods of thrusting and coasting. At the start of the trip, the spacecraft thrusts at 7800 N and a tangential thrust angle; it maintains this thrust level for 7.70 days before stopping the engine and beginning to coast. The spacecraft coasts for 52.37 days, which is the majority of the interplanetary flight duration. After coasting for almost two months, the spacecraft needs to decelerate in order to intercept Mars. The spacecraft thrusts at maximum thrust and a constant 206 thrust angle for 4.75 days in order to decelerate and arrive at the edge of
Mars’ SOI. This period of deceleration brings the total interplanetary trip time to 64.82 days. The flight characteristics and trajectory are shown in Table 44 and Figure 47, respectively.
Table 44: Nuclear fusion flight characteristics after interplanetary flight and arrival at Mars
Arrival Radius 2.277 108 km
Arrival Angle 1.262 rad (70.26)
Propellant Consumption 26,490 kg
Arrival Mass 316,580 kg
108
Arrival Velocity 29.398 km⁄s
Total Thrust Time 12.45 days
Coast Time 52.37 days
Total Trip Time 64.82 days
The spacecraft arrives in Mars’ sphere of influence and needs to continue thrusting in order to settle into a stable, circular orbit around the planet. In Figure 43, the spacecraft flies in a straight trajectory until being captured by Mars’ gravity. It spirals in towards the planet until settling in a stable, circular orbit at an altitude of 200 miles.
Figure 43: Nuclear fusion Mars arrival flight
109
The flight from the edge of Mars’ sphere of influence (577,000 km) occurs at constant, tangential thrust with a starting velocity of 29.516 km⁄s. Over the course of 3.2 days, the spacecraft expends 6,830 kg of propellant. These flight characteristics are recorded in Table 45.
Table 45: Nuclear fusion flight characteristics for Mars arrival flight
Starting Radius (edge of SOI) 575,480 km
Starting Angle 5.06 rad (289.98)
Propellant Consumption 6,830 kg
Arrival Mass 309,750 kg
Starting Velocity 29.516 km⁄s
Total Thrust/Flight Time 3.2 days
The spacecraft stays in orbit around Mars for 200 days. Radiation exposure and health concerns currently limit a human space mission to two years, which includes the time spent in situ and the transit time to and from Earth. The red line in Figure 47 shows the movement of Mars around the Sun during this mission. Note that the orbit of Mars is approximated as a circle at 1.523 AU. After completing the Mars mission, the spacecraft begins its journey back to Earth. From a circular orbit, the spacecraft begins to accelerate at maximum thrust with tangential thrust angle. It spirals out from Mars until reaching escape velocity. The spacecraft reaches escape velocity in only 1.17 days using 2,500 kg
110
of propellant. It is able to escape Mars’ gravity moving at 25.74 km⁄s. Table 46 contains the flight time and propellant usage of the Earth escape leg of the Mars mission.
Table 46: Nuclear fusion Mars escape flight leg of mission
Propellant Consumption 2,500 kg
Final Mass 307,250 kg
Escape Velocity 25.74 km⁄s
Flight Time 1.17 days
Escape Radius 36,107 km
Figure 44 shows the spacecraft’s spiral flight path as it orbits Mars five and a half times before reaching escape velocity.
111
Figure 44: Nuclear fusion Mars escape trajectory
After reaching escape the velocity, the spacecraft flies to the edge of Mars’ gravitational sphere of influence at which point the Sun becomes the primary gravitational influence on the spacecraft. Figure 45 shows the spacecraft’s trajectory as it reaches the edge of Mars’ SOI. The spacecraft continues to maintain maximum thrust and a tangential thrust angle as it flies to the edge of Mars’ SOI.
112
Figure 45: Nuclear fusion spacecraft flight path to the edge of Mars' sphere of influence
The specific flight characteristics for this portion of the journey are listed in Table
47. The spacecraft has traveled a total of 3.16 days since leaving Martian orbit; it continues to accelerate and expend propellant as shown below.
Table 47: Nuclear fusion spacecraft flight characteristics upon reaching the edge of Mars’ sphere of influence
Propellant Consumption 4,240 kg
Final Mass 303,010 kg
Velocity 29.58 km⁄s
Flight Time (incl. escape time) 3.16 days
113
Radius 577,280 km
After leaving Mars’ gravitational sphere of influence, the Sun again becomes the primary gravitational force acting on the spacecraft as it travels back to Earth. As with the previous analyses, the spacecraft utilizes maximum thrust during the interplanetary journey. The spacecraft does not maintain constant thrust for the duration of the interplanetary trip but again uses periods of thrusting and coasting. At the start of the trip, the spacecraft coasts for 4.82 days before turning on its engine and beginning to thrust at
7,800 N and a constant 170 thrust angle; the spacecraft maintains this thrust level for
13.89 days before stopping the engine and beginning to coast. The spacecraft coasts for another 31.42 days. After a month spent coasting, the spacecraft again begins to accelerate under maximum thrust; the spacecraft maintains maximum thrust and a tangential thrust angle for 11.52 days. The spacecraft coasts for the next 275.35 days before arriving at the edge of Earth’s SOI. The total interplanetary flight time is 337 days.
The flight characteristics and trajectory are shown in Table 48 and Figure 47, respectively. Note that the return flight to Earth includes a closer approach to the Sun
(~0.7 AU) than the outbound flight to Mars.
Table 48: Nuclear fusion flight characteristics after interplanetary flight and arrival at Earth
Mars Departure Radius 2.284 108 km⁄s
Mars Departure Angle 3.098 rad (177.502)
Mars Departure Velocity 29.269 km⁄s
114
Earth Arrival Radius 1.496 108 km
Earth Arrival Angle 4.107 rad (235.30)
Propellant Consumption 54,210 kg
Earth Arrival Mass 248,800 kg
Earth Arrival Velocity 27.19 km⁄s
Total Thrust Time 25.41 days
Coast Time 311.59 days
Total Trip Time 337 days
The spacecraft arrives in Earth’s sphere of influence and needs to continue thrusting in order to settle into a stable, circular orbit around the planet. As shown in
Figure 46, the spacecraft flies in a straight trajectory until being captured by Earth’s gravity. It spirals in towards the planet until settling in a stable, circular orbit at an altitude of 200 miles.
115
Figure 46: Nuclear fusion Earth arrival flight
The flight from the edge of Earth’s sphere of influence begins with the spacecraft coasting at 30.59 km⁄s and zero thrust. After 8.39 days, the spacecraft begins to accelerate at maximum thrust for 2.26 days. Over the course of 10.65 days, the spacecraft expends 4,820 kg of propellant and reaches a circular low Earth orbit at an altitude of 200 miles after orbiting the Earth nearly thirteen times. See Table 49 for a summary of the
Earth arrival flight characteristics described here.
Table 49: Nuclear fusion flight characteristics for Earth arrival flight
Starting Radius (edge of SOI) 925,510 km
Starting Angle 5.693 rad (326.201)
116
Propellant Consumption 4,950 kg
LEO Arrival Mass 243,980 kg
Starting Velocity 30.59 km⁄s
Total Thrust Time 2.26 days
Total Coast Time 8.39 days
Total Flight Time 10.65 days
The roundtrip mission to Mars described in this section has a total duration of
630.08 days, which includes a 200-day dwell time at Mars and comes in under the maximum mission duration of two years. The spacecraft consumes 106,020 kg of propellant during the course of the mission. Figure 47 shows the Mars and Earth transits, as well as the movement of each planet during the course of the mission. Table 50 summarizes the final mission characteristics. The total thrust time during the course of this mission is 49.73 days, which far exceeds the capabilities of both chemical and nuclear thermal rockets and enables a rapid transit time between planets.
117
Figure 47: Earth-Mars roundtrip mission for a nuclear fusion spacecraft
Table 50: Mission characteristics for a nuclear fusion mission to Mars
Total Propellant Consumption 106,020 kg
Final Mass 243,980 kg
Total Thrust Time 49.73 days
Total Coast Time 380.35 days
Mars Stay 200 days
Total Mission Duration Time 630.08 days
The second trajectory analysis uses the nuclear fusion engine characteristics described above for a round trip flight to Jupiter. This analysis serves to emphasize the usefulness of this propulsion technology for crewed deep space missions beyond Mars.
118
As with the Mars analysis, this mission has a two-year time limit. The analysis begins when the spacecraft leaves a 200-mile altitude low-Earth orbit using its nuclear fusion engine at maximum thrust. As noted previously, the non-dimensional initial conditions for the MATLAB code are recorded in Appendix B. The 500,000 kg spacecraft spirals out from low-Earth orbit at constant, maximum thrust and a tangential thrust angle. After
4.73 days of continuous thrust, the spacecraft reaches a velocity of 32.12 km⁄s and is able to escape Earth’s gravity. The more massive Jupiter-bound spacecraft takes more than a day longer than the Mars mission spacecraft to reach escape velocity. It has consumed 10,090 kg of propellant during this leg of the mission. Table 51 contains the flight time and propellant usage of the Earth escape leg of the Mars mission.
Table 51: Nuclear fusion Earth escape flight leg of Jupiter mission
Propellant Consumption 10,090 kg
Final Mass 489,910 kg
Escape Velocity 32.12 km⁄s
Flight Time 4.73 days
Escape Radius 139,200 km
Figure 48 below shows the trajectory of the spacecraft as it begins its journey away from Earth; the plot shows the spiral orbit ending when the spacecraft reaches escape velocity after twenty-three revolutions around the Earth. This spacecraft takes longer to accelerate to escape velocity when compared with the Mars spacecraft. It also has to make more revolutions around the Earth and reaches escape velocity at a greater distance from Earth because of its larger mass.
119
Figure 48: Earth escape trajectory using a nuclear fusion propulsion system
After reaching escape velocity, the spacecraft continues to fly to the edge of
Earth’s SOI at which point the Sun again becomes the primary gravitational influence on the spacecraft. Figure 49 shows the spacecraft’s trajectory as it reaches the edge of
Earth’s SOI. After spiraling out from Earth, the flight path becomes straight as the spacecraft engines shut off soon after reaching Earth escape velocity; it coasts to the edge of Earth’s SOI.
120
Figure 49: Nuclear fusion spacecraft flight path to the edge of Earth's sphere of influence
The specific flight characteristics for this portion of the journey are listed in Table
52. The spacecraft has traveled a total of 12.41 days since leaving low Earth orbit; it expends minimal propellant after reaching escape velocity.
Table 52: Nuclear fusion spacecraft flight characteristics upon reaching the edge of Earth's sphere of influence
Propellant Consumption 40 kg
Final Mass 489,870 kg
Velocity 30.71 km⁄s
Flight Time (incl. escape time) 12.41 days
121
Radius 925,420 km
The spacecraft now leaves the Earth’s gravitational influence and begins its interplanetary flight to Jupiter. This flight path is determined using a Sun-centric calculation. As with the Earth escape trajectory, the spacecraft utilizes its maximum thrust of 7,800 N. The spacecraft uses periods of thrusting and coasting during this interplanetary trip as well. At the start of the trip, the spacecraft thrusts at 7,800 N and a tangential thrust angle; it maintains this thrust level for 27.199 days before stopping the engine and beginning to coast. The spacecraft coasts for 134.14 days, which is the remainder of the interplanetary flight duration. During this flight, there is no need for the spacecraft to decelerate upon reaching the edge of Jupiter’s SOI; because of the large size of Jupiter’s SOI, it can decelerate an adequate amount as it begins to spiral in to a stable, circular orbit. The spacecraft arrives at Jupiter after a total of 161.34 days. The flight characteristics and trajectory are shown in Table 53 and Figure 54, respectively.
Table 53: Nuclear fusion flight characteristics after interplanetary flight and arrival at Jupiter
Arrival Radius 7.785 108 km
Arrival Angle 1.743 rad (99.68)
Propellant Consumption 57,970 kg
Arrival Mass 431,900 kg
Arrival Velocity 57.97 km⁄s
Total Thrust Time 27.199 days
122
Coast Time 134.14 days
Total Trip Time 161.34 days
The spacecraft arrives in Jupiter’s sphere of influence and needs to continue thrusting in order to settle into a stable, circular orbit around the planet. As shown in
Figure 50, the spacecraft flies in a straight trajectory until being captured by Jupiter’s gravity. It spirals in towards the planet until settling in a stable, circular orbit at an altitude of 200 miles. The spacecraft completes over 54 orbits around Jupiter during its deceleration flight.
Figure 50: Nuclear fusion Jupiter arrival flight
123
The spacecraft flies at constant, tangential thrust beginning at the edge of Jupiter’s sphere of influence (48,200,000 km) with a starting velocity of 57.64 km⁄s. After 42.76 days, the spacecraft expends 90,270 kg of propellant and arrives in orbit around the planet at a 200-mile altitude.
Table 54: Nuclear fusion flight characteristics for Jupiter arrival flight
Starting Radius (edge of SOI) 4.827 107 km
Starting Angle 5.181 rad (296.87)
Propellant Consumption 90,270 kg
Arrival Mass 341,630 kg
Starting Velocity 57.64 km⁄s
Total Thrust/Flight Time 42.76 days
The spacecraft stays in orbit around Jupiter for 120 days; the reduced duration when compared to the Mars mission is imposed by the two-year mission length requirement. As stated previously, radiation exposure and health concerns currently limit a crewed space mission to two years, which includes the time spent in situ and the transit time to and from Earth. The red line in Figure 54 shows the movement of Jupiter around the Sun during this mission. Note that the orbit of Jupiter is approximated as a circle at
5.2 AU. After completing a four-month mission at Jupiter, the spacecraft begins its journey back to Earth. From a circular orbit, the spacecraft begins to accelerate at maximum thrust with tangential thrust angle. It spirals out from Jupiter until reaching escape velocity. The spacecraft reaches escape velocity after 17.21 days using 36,740 kg
124
of propellant. It is able to escape Jupiter’s gravity moving at 24.44 km⁄s. Table 55 contains the flight time and propellant usage of the Jupiter escape leg of the mission.
Table 55: Nuclear fusion Jupiter escape flight leg of the mission
Propellant Consumption 36,740 kg
Final Mass 304,890 kg
Escape Velocity 24.44 km⁄s
Flight Time 17.21 days
Escape Radius 1,959,800 km
Figure 51 shows the spacecraft’s spiral flight path as it orbits Jupiter forty-two times before reaching escape velocity. When compared to the Mars mission, the lengthy flight time before reaching escape velocity is due to Jupiter’s massive size.
125
Figure 51: Nuclear fusion Jupiter escape trajectory
After reaching escape the velocity, the spacecraft flies to the edge of Jupiter’s gravitational sphere of influence at which point the Sun becomes the primary gravitational influence on the spacecraft. Figure 52 shows the spacecraft’s trajectory as it reaches the edge of Jupiter’s SOI. The spacecraft continues to maintain maximum thrust and a tangential thrust angle for 16.35 days before shutting down the engine and coasting to the edge of Jupiter’s SOI.
126
Figure 52: Nuclear fusion spacecraft flight path to the edge of Jupiter’s sphere of influence
The specific flight characteristics for this portion of the journey are listed in Table
56. The spacecraft has traveled a total of 3.16 days since leaving Jovian orbit; it continues to accelerate and expend propellant as shown below.
Table 56: Nuclear fusion spacecraft flight characteristics upon reaching the edge of Jupiter’s sphere of influence
Propellant Consumption 71,630 kg
Final Mass 270,000 kg
Velocity 58.66 km⁄s
127
Flight Time (incl. escape time) 36.03 days
Radius 4.824 107 km
The spacecraft begins its interplanetary journey back to Earth after leaving
Jupiter’s gravitational sphere of influence; The Sun again becomes the primary gravitational force acting on the spacecraft. As with the previous analyses, the spacecraft utilizes maximum thrust during the interplanetary journey. As with the previous interplanetary flights, the spacecraft does not maintain constant thrust for the duration of the trip. At the start of the flight, the spacecraft coasts for 6.06 days before turning on its engine at maximum thrust; it operates at constant thrust and a tangential thrust angle for
7.82 days. The spacecraft then coasts for 220.38 days, which is the majority of the interplanetary flight duration. After the period of coasting, the spacecraft needs to decelerate in order to intercept Earth. The spacecraft thrusts at maximum thrust and a constant 219.12 thrust angle for 11.1 days in order to decelerate and arrive at the edge of
Earth’s SOI. The total interplanetary trip time is 245.36 days. The flight characteristics and trajectory are shown in Table 57 and Figure 54, respectively.
Table 57: Nuclear fusion flight characteristics after interplanetary flight and arrival at Earth
Arrival Radius 1.496 108 km
Arrival Angle 4.107 rad (235.30)
Propellant Consumption 40,230 kg
Arrival Mass 229,770 kg
128
Arrival Velocity 29.72 km⁄s
Total Thrust Time 18.92 days
Coast Time 226.44 days
Total Trip Time 245.36 days
After arriving in Earth’s sphere of influence, the spacecraft needs to continue thrusting in order to settle into a stable, circular orbit around the planet. As shown in
Figure 53, the spacecraft flies in a straight trajectory until being captured by Earth’s gravity. It spirals in towards the planet, orbiting twelve times until settling in a stable, circular orbit at an altitude of 200 miles.
Figure 53: Nuclear fusion Earth arrival flight
129
The spacecraft coasts from the edge of Earth’s SOI at 30.66 km⁄s; after 7.96 days, the spacecraft begins to accelerate at maximum thrust for 2.08 days. Over the course of 10.04 days, the spacecraft expends 4,440 kg of propellant and reaches a circular low Earth orbit at an altitude of 200 miles after orbiting the Earth twelve times. Table 58 details the Earth arrival flight details.
Table 58: Nuclear fusion flight characteristics for Earth arrival flight
Starting Radius (edge of SOI) 925,430 km
Starting Angle 2.19 rad (125.2)
Propellant Consumption 4,440 kg
Arrival Mass 225,330 kg
Starting Velocity 30.66 km⁄s
Total Thrust Time 2.08 days
Total Coast Time 7.96 days
Total Flight Time 10.04 days
The roundtrip mission to Jupiter described in this section has a total length of
627.94 days, which includes four months of dwell time at Jupiter and comes in under the maximum mission duration of two years. Figure 54 shows the Jupiter and Earth transits, as well as the movement of each planet during the course of the mission. Table 59 summarizes the final mission characteristics.
130
Figure 54: Earth-Jupiter roundtrip mission for a nuclear fusion spacecraft
Table 59: Mission characteristics for a nuclear fusion flight to Jupiter
Total Propellant Consumption 274,670 kg
Final Mass 225,330 kg
Total Thrust Time 129.26 days
Total Coast Time 378.68 days
Total Mission Duration Time 627.94 days
The spacecraft consumes 274,670 kg of propellant during the course of the mission. The total thrust time during the course of this mission is 129.26 days, which far
131
exceeds the capabilities of traditional chemical rockets and enables a rapid transit time between planets.
132
IV. RESULTS AND DISCUSSION
Nuclear electric propulsion using NEXT Ion engines is a propulsion technology capable of transporting a mission to Mars. A human mission to Mars could be accomplished using a spacecraft sporting eight NEXT engines producing a cumulative
1.88 N of thrust; the mission would take over two years with only thirty days spent on
Mars. The technology is incredibly fuel efficient, requiring only 2,570 kg of propellant to complete this mission even though the electric engines provide thrust for a total of 656 days. However, the low available thrust necessitates a small spacecraft with minimal payload; this restriction could necessitate additional launches to deliver equipment to
Martian orbit or the planet’s surface.
The NEP technology places some limitations on the spacecraft’s orbital dynamic maneuvers. The ion engines produce such minimal thrust, that it is not advantageous to use them to escape Earth’s gravity. It would take over a year at maximum thrust for the spacecraft to slowly spiral out from low-Earth orbit; it is far more feasible to utilize traditional chemical rockets to accelerate the spacecraft to escape velocity before the electric engines are engaged for the interplanetary flight. This problem is also apparent when the spacecraft arrives in Mars’ SOI; the arrival flight is characterized by an elliptical, spiral orbit that deposits the spacecraft in a stable, circular orbit around the planet after several months. Upon departing Martian orbit, the spacecraft must again spiral slowly out until reaching escape velocity.
The interplanetary return flight to Earth is significantly longer than the Earth departure flight due to the movement of both Mars and Earth in the months since the mission began. The spacecraft must travel around the sun to catch up with Earth, which is
133
moving faster than Mars; the spacecraft does not approach closer than 1 AU from the sun during the flight. The NEXT propelled spacecraft is able to transport humans to Mars; it is unable to achieve the Mars roundtrip mission in the two year time window under the constraints and assumptions used in this analysis.
The thrust-limited nuclear electric propulsion system stands in sharp contrast to the next propulsion technology analyzed in the Mission Analysis section: nuclear thermal propulsion. Nuclear thermal propulsion systems use nuclear thermal rockets to generate much higher thrust levels than ion engines. This technology was tested extensively in the
1950’s and 1960’s, but was discontinued in favor of chemical rockets. Nuclear thermal rockets are capable of transporting a crewed mission to Mars. The spacecraft used in the
NTP analysis would require a trio of nuclear thermal rockets producing a cumulative
333,618 N of thrust and using 215,851 kg of propellant. The mission could be achieved within the two-year timeframe but still only allow a month spent on Mars. The vast majority of the mission would be consumed by interplanetary travel. The technology is certainly not lacking in thrust, but is hindered by high fuel usage (12.45 kg⁄s per engine).
The trajectories used to accomplish a roundtrip Mars mission using nuclear thermal rockets are limited by the propellant usage of the technology. The spacecraft is able to use nuclear thermal rockets to escape Earth’s gravity; the high thrust engines are able to accelerate the craft to escape velocity without spiraling out from low Earth orbit.
The Earth-escape flight is the longest consecutive engine burn of the mission at over 36 minutes. The interplanetary flight to Mars is short, less than five months, and characterized by a short engine burn (17 minutes) followed by over four months of
134
coasting. Upon arrival at Mars, the high thrust engines are able to achieve a circular Mars orbit in days rather than months by using short engine burns and periods of coasting.
The mission is only able to achieve a thirty-day Martian surface stay before beginning the return journey to Earth. This is the longest portion of the mission and accounts for more than half of the total mission duration. As with the NEP mission, the spacecraft must make nearly a full trip around the sun to catch up with the faster-moving
Earth. However, this return trajectory is characterized by its closer approach to the sun, flying inside the Earth’s orbital radius to 0.6-0.7 AU. This is typical of “opposition-class” missions, which are characterized by short mission durations, surface stays, and outbound transits, as well as long inbound transits; a two-year mission limit necessitates an opposition-class mission for a spacecraft propelled by nuclear thermal rockets. As the surface stay lengthens, it becomes more difficult to complete the inbound flight within the two-year window without using a prohibitive quantity of propellant. Nuclear thermal propulsion is capable of supporting a human roundtrip mission to Mars within the two- year limit.
The final propulsion technology under consideration is nuclear fusion propulsion.
Nuclear fusion rockets are a conceptual propulsion technology capable of high thrusts, high ISP values, and long thrust times. The nuclear fusion rocket described here is capable of 7800 N of thrust with a propellant mass flow rate of only 0.025 kg⁄s. A spacecraft employing one fusion rocket is able to complete a roundtrip human mission to Mars within the two-year timeframe, while achieving a 200-day stay on Mars. The spacecraft consumes 106,020 kg of propellant over the course of the mission.
135
The spacecraft, propelled by a nuclear fusion rocket, is able to escape Earth’s gravity in several days while accelerating under maximum thrust. The nuclear fusion engine is able to maintain maximum thrust levels for days or weeks at a time because of its low rate of fuel consumption. This continuous thrust feature allows the spacecraft to make the journey to Mars in only two months. The flight to Mars is characterized by an acceleration burn at the beginning of the flight and a deceleration engine burn upon arrival at Mars; the spacecraft engine provides continuous thrust for 12 days total. Upon arrival at Mars, the nuclear fusion rocket is able to achieve a circular Mars orbit in days by providing continuous thrust. After completing a 200-day mission on Mars, the mission begins its return journey.
The return voyage to Earth is the longest portion of this mission at over eleven months. As described with the other two Mars missions, the extended duration of the return transit is due to the movement of Mars and the Earth during the course of the mission. The spacecraft is forced to circumnavigate the sun, approaching to an orbital radius of 0.7 AU, while attempting to catch the Earth. The nuclear fusion rocket creates maximum thrust for 25 days of the total 337 day transit time. This trajectory for a roundtrip Mars flight could also be categorized as opposition-class mission, although the surface stay is longer than is typically described. Nuclear fusion propulsion is capable of transporting a human mission to Mars in the required time frame.
A mission to Jupiter was included in the nuclear fusion analysis section to demonstrate the feasibility of using this technology for missions beyond Mars. A spacecraft propelled by a nuclear fusion rocket is able to make a roundtrip journey to
Jupiter (including a 120-day stay in Jovian orbit) in 627.94 days; this timeline meets the
136
required mission time limit. The fusion rocket engine produces thrust for a total of 129.26 days and consumes 274,670 kg of propellant over the course of the mission. The interplanetary flights are characterized by minimal curvature caused by extended periods of high thrust, which allow the spacecraft to take the most direct route to Jupiter.
Each propulsion technology has advantages that make it attractive. NEP systems have extremely low propellant mass flow rates, making them attractive for use on deep space missions requiring conservative fuel use. Electric thrusters are also the furthest along in development of the three technologies. However, the exceedingly low thrust levels make it difficult to transport humans in an expedient fashion. Additionally, the thrust levels limit the mass of the spacecraft, reducing the payload delivery capability of the NEP system. A crewed Mars mission would likely require one or more prior additional missions (possibly unmanned) to send equipment to the Martian surface. The roundtrip Mars mission described in the Mission Analysis section only delivers a 49,839 kg spacecraft to Martian orbit. In comparison, the NTP spacecraft has a mass of 110,040 kg upon arrival at Mars, with a starting mass of 250,000 kg. The high thrust levels of nuclear thermal rockets are capable of transporting a massive spacecraft within a more practical timetable when compared with NEP. NTP systems are still limited by propellant use, however, and are unable to complete long stays on Mars, while also meeting the mission time limit. Nuclear thermal rockets were extensively tested decades ago and were proven as a viable option for space flight; unfortunately, testing was halted in the 1970s.
Nuclear fusion propulsion has the advantage of both high thrust and specific impulse. The technology is theoretically capable of providing continuous thrust for weeks while consuming less propellant than a NTP system. The roundtrip fusion Mars mission
137
consumes less than half of the propellant of the NTP mission, while achieving a longer
Martian stay and meeting the two-year mission limit. Of the three propulsion systems, the fusion rocket delivers the largest payload to Mars in the least amount of time.
Additionally, the fusion rocket-propelled spacecraft is capable of making a roundtrip journey to Jupiter, emphasizing its promise for rapid transit on deep space missions. The largest drawback to nuclear fusion propulsion currently is the theoretical nature of the technology. The propulsion technology is conceptual and thus not available for near-term
Mars or deep space missions, human or unmanned. Table 60 compares the Mars mission characteristics for the three propulsion technologies.
Table 60: Comparison of Mars mission characteristics for the various propulsion technologies
Mars Payload Mission Propellant Launch Stay Delivery to Duration Consumption Mass Duration Mars
Nuclear Electric 885.27 50,000 kg 48,839 kg 30 days 2,570 kg Propulsion days
Nuclear Thermal 422.15 250,000 kg 110,040 kg 30 days 215,851 kg Propulsion days
Nuclear Fusion 630.08 350,000 kg 309,750 kg 200 days 106,020 kg Propulsion days
138
V. CONCLUSION
The various propulsion technologies discussed here are appropriate for different types of missions. The analysis of a roundtrip NEP mission to Mars reinforces the conclusion that nuclear electric propulsion is not ideal for interplanetary crewed missions.
While incredibly fuel efficient, the low thrust levels are not sufficient to transport human beings within the required two-year window. Additionally, it necessitates a miniscule mission window at Mars, as well as minimal payload delivery. The low thrust capability of nuclear electric propulsion is ill-suited for human missions with tight timelines.
However, additional analysis could be performed to optimize the number of electric thrusters used on a spacecraft of this size, potentially decreasing the transit times and making NEP technology a more appealing candidate for a crewed Mars mission.
Additionally, the time required for a NEP-propelled spacecraft to spiral-in to Mars could be reduced by assuming a rendezvous with another vehicle propelled by higher thrust technology for the final approach to Mars. That being said, it is an advantageous near- term option for long-lasting robotic missions to deep space. The development of high thrust electric propulsion systems could change this conclusion, however, and make nuclear electric propulsion systems more attractive for human missions.
Nuclear thermal propulsion is a promising technology capable of transporting crewed missions to Mars. Nuclear thermal rockets have high thrust levels and are capable of hitting the two-year deadline for a roundtrip human mission to Mars. When compared to nuclear fusion, NTR propulsion technology has the advantage of extensive testing and development. However, it has been decades since nuclear thermal rockets were built and tested; another design cycle should be completed to verify the results obtained during the
139
Rover and NERVA projects. Additionally, more work is required to design and test a bimodal nuclear thermal rocket system, so that the nuclear reactor can serve as a source of electricity, as well as thrust. Nuclear thermal rockets are currently the best option for completing an expedient crewed mission to Mars. However, nuclear thermal propulsion is not ideal for deep space travel to destinations beyond Mars; NTP systems require large quantities of propellant, which makes it difficult to complete a roundtrip mission to a deep space destination.
Nuclear thermal rockets could possibly be utilized for other deep space missions if technological advancements increased the duration of crewed missions. Astronaut health concerns caused by extended exposure to cosmic radiation and a zero gravity environment dictate the tight mission timeline of two years; improvements in radiation shielding or the introduction of rotational artificial gravity on spacecraft could mitigate these concerns and allow extended missions to destinations beyond Mars. Extended missions to Mars would be characterized as “fast-conjunction” missions, which typically have long stays on Mars and short transits; these types of Mars missions are generally longer than two years. However, even if mission duration limits were expanded, nuclear thermal rockets are still limited by short burn times and high propellant usage.
Nuclear fusion propulsion is the best choice for human deep space exploration; the technology offers high thrust and low propellant usage. It can provide thrust for days or weeks at a time, resulting in rapid transits and increased stays at the intended destinations. As demonstrated previously, nuclear fusion propulsion can easily achieve a two year roundtrip mission to Mars; it can even complete a mission to Jupiter in this
140
timeframe. Improvements to radiation shielding and the introduction of artificial gravity as mentioned above would increase mission times and extend dwell times.
Unfortunately, nuclear fusion propulsion systems are largely theoretical at this time and are not an option for a human Mars mission in the coming decade. More development is required to make the technology a reality. Design studies and prototyping are required to more accurately judge the performance of a nuclear fusion propulsion system. Additionally, the design study should flesh out the structure of the propulsion system, as well as provide estimates of size and mass. Data gathered from prototype testing could be used to fine-tune and update the mission trajectories to Mars and Jupiter, giving a more accurate picture of the flight path of a nuclear fusion spacecraft.
Future human missions to Mars and beyond necessitate advanced propulsion systems with more capability than traditional chemical rockets. Nuclear thermal rockets, while the best choice for a near-term mission to Mars, are only a stepping stone along the path to deep space human exploration. The development of a nuclear fusion rocket would substantially change space exploration and allow crewed missions to destinations around the solar system, previously only visited by robotic spacecraft. Embracing improved propulsion technologies is the giant leap required for ushering in a new era of spaceflight and exploring the far reaches of the solar system.
141
142
VI. FUTURE WORK
This numerical methodology, based on equations derived from the Two-Body
Problem, could be further enhanced to provide increased usability and a more nuanced analysis. The current MATLAB code requires the user to manually modify the input variables to achieve the desired outcome. While this can effectively generate orbital trajectories, it is time-consuming to manually vary input parameters and run the code numerous times to achieve a desired trajectory. The methodology could be greatly improved with the addition of features to provide automated trajectory mapping. The user could specify both the initial parameters and the desired trajectory end points including radius and angle. The code could then vary other parameters, such as thrust, in order to achieve the appropriate flight path. The addition of automation to this MATLAB code would greatly improve its usability and allow a more rapid analysis of potential deep space missions.
The trajectory analyses for the NEP, NTP, and nuclear fusion missions could also be enhanced by modifying the numerical methodology to analyze slight plane changes of several degrees. The current methodology performs a two-dimensional analysis and assumes all movement occurs in the same plane. In reality, the movement of spacecraft around the solar system involves plane changes and requires a more complex analysis to map those trajectories. However, the addition of this feature to the code may not make much difference when analyzing mission trajectories; the plane changes are slight and might not have much effect on propellant usage or mission times.
Finally, the automated code as described previously could be applied to identify promising applications for space nuclear propulsion systems. The code is simpler and
143
more user-friendly than many low, continuous thrust trajectory codes and can be rapidly applied to a large number of potential missions. The automated code would be a powerful tool for easily weeding out the impractical missions and identifying those most suited for specific propulsion technologies.
144
APPENDIX A
DERIVATION OF THE TWO-BODY PROBLEM
The Two-Body Problem is a method by which to determine the motion of two bodies due to their mutual gravitation attraction alone; no other forces or bodies are considered during the course of the analysis. The celestial bodies under consideration can include stars, planets, and spacecraft. The derivation begins with Figure 55, which shows two bodies of masses 푚1 and 푚2, respectively, in inertial reference frame 푋푌푍; the reference frame is fixed with respect to fixed starts.
Figure 55: Two masses in an inertial reference frame
As shown, the body with mass 푚1 is the more massive of the two bodies. The position vector of mass 푚1 is 푟⃗1, and the position vector of mass 푚2 is 푟⃗2. The vector 푟⃗ is the relative position of 푚2 with respect to 푚1. The center of mass of the system, 퐶푀, is
145
located on 푟⃗; the position vector 푅⃗⃗ describes the position of the center of mass with respect to the origin, 푂.
Figure 56: Gravitational force in the Two-Body Problem
Each of the two bodies is acted upon by the gravitation attraction of the other;
Figure 56 shows these forces. 퐹⃗1 is the force exerted on 푚1 by 푚2, and 퐹⃗2 is the force exerted on 푚2 by 푚1. The forces are equal and opposite.
퐹⃗1 = −퐹⃗2 (67)
The derivation of the Two-Body Problem begins with Newton’s Second Law:
푑2푟⃗ 퐹⃗ = 푚 (68) 푑푡2
The only force under consideration is the gravitational attraction between the two masses, 퐹⃗푔. This force is defined by Newton’s Law of Gravity where 푟 is the magnitude of the position vector and 퐺 is the universal gravitational constant:
퐺푚 푚 퐹⃗ = 1 2 (69) 푔 푟2
146
Because gravity is the only force exerted on the bodies, the force, 퐹⃗푔, is equal to the forces exerted on each body, 퐹⃗1 and 퐹⃗2. The force acts along the position vector between the two bodies in the 푟̂ direction. Setting 퐹⃗1 and −퐹⃗2 equal to 퐹⃗푔 results in the following equations:
퐺푚 푚 퐹⃗ = 1 2 푟̂ (70) 1 푟2 퐺푚 푚 퐹⃗ = − 1 2 푟̂ (71) 2 푟2
The position vector 푟⃗ is equal to the magnitude of the position vector in the 푟̂ direction.
푟⃗ = 푟푟̂ (72)
Now, the unit vector 푟̂ is replaced in Equations 70 and 71 using Equation 72.
퐺푚 푚 퐹⃗ = 1 2 푟⃗ (73) 1 푟3
퐺푚 푚 퐹⃗ = − 1 2 푟⃗ (74) 2 푟3
Equations 73 and 74 are representations of Newton’s Second Law and can be set equal to Equation 68.
푑2푟⃗ 퐺푚 1 = 2 푟⃗ (75) 푑푡2 푟3
푑2푟⃗ 퐺푚 2 = − 1 푟⃗ (76) 푑푡2 푟3
2 2 In Equation 75, 푑 푟⃗1⁄푑푡 is the acceleration of 푚1 relative to the inertial
2 2 reference frame. Similarly, 푑 푟⃗2⁄푑푡 in Equation 76 is the acceleration of body 푚2 relative to the inertial reference frame. Subtracting Equation 75 from Equation 76 results in Equation 77, which is the Two-Body Problem.
147
푑2(푟⃗ − 푟⃗ ) 퐺(푚 + 푚 ) 2 1 = − 1 2 푟⃗ (77) 푑푡2 푟3
The Two-Body Problem in Equation 77 can be further simplified by recognizing that the position of 푚2 relative to 푚1 is equal to the difference between the absolute positions of both bodies as shown.
푟⃗ = 푟⃗2 − 푟⃗1 (78)
Additionally, the masses of both bodies can be combined and represented by 푀.
푀 = 푚1 + 푚2 (79)
Simplifying Equation 77 using Equations 78 and 79 results in the following
푑2푟⃗ 퐺푀 = − 푟⃗ (80) 푑푡2 푟3
Finally, the term 퐺푀 can be replaced with 휇, the gravitational parameter, as introduced in the Methods of Analysis section.
푑2푟⃗ 휇 + 푟⃗ = 0 (81) 푑푡2 푟3
The previous equation is the Two-Body Problem; this equation is the foundation of the equations derived in the Methods of Analysis section that are used to describe the motion of a spacecraft.
The Two-Body Problem equation is also used to derive Kepler’s Second Law and the orbit equation. Kepler’s Second Law states that equal areas are swept out by the position vector, 푟⃗, in equal times, meaning that bodies closer to the sun travel faster than those farther away [1]. This law is a function of radius and angular velocity, resulting in angular momentum. The orbit equation, which defines the path of 푚2 around 푚1, is a function of angular momentum as defined by Kepler’s Second Law [1]. The derivation
148
for Kepler’s Second Law begins by crossing the position vector with the Two-Body
Problem equation.
푑2푟⃗ 휇 푟⃗ × ( + 푟⃗) = 0 (82) 푑푡2 푟3
Because a vector crossed into itself is zero, the second term in Equation 81 is eliminated. The position vector crossed into the radial acceleration term is equal to zero as shown in Equation 83.
푑2푟⃗ 푟⃗ × = 0 (83) 푑푡2
The left hand side of Equation 84 is the position vector crossed into a force, which is a moment, or the derivative of the angular momentum, as shown on the right hand side.
푑2푟⃗ 푑 푑푟⃗ 푚 (푟⃗ × ) = 푚 (푟⃗ × ) (84) 푑푡2 푑푡 푑푡
푑 푑푟⃗ (푟⃗ × ) = 0 (85) 푑푡 푑푡
The angular momentum vector, ℎ⃗⃗, is obtained by integrating Equation 85, resulting in Equation 86.
푑푟⃗ ℎ⃗⃗ = 푟⃗ × (86) 푑푡
The angular momentum of the body is obtained by crossing the position vector, 푟⃗, with the velocity vector, 푑푟⃗⁄푑푡. The angular momentum vector is expanded into its component form in order to find a scalar expression. First, the radial velocity vector is written as the derivative of the product of radius, 푟, and unit vector, 푟̂, as shown in
Equation 87.
푑푟⃗ 푑 = (푟푟̂) (87) 푑푡 푑푡
149
The derivative of the velocity vector is calculated using the product rule with the results shown in Equation 88
푑푟⃗ 푑푟 푑푟̂ = 푟̂ + 푟 (88) 푑푡 푑푡 푑푡 where the first derivative of 푟̂ is expanded using the cross product of angular velocity with 푟̂.
푑푟̂ 푑휃 = 휃̂ (89) 푑푡 푑푡
After substituting Equation 89 into Equation 88, the velocity vector equation is represented as a function of radial velocity, 푑푡⁄푑푡, in the 푟̂ direction and circumferential velocity, 푑휃⁄푑푡, in the 휃̂ direction.
푑푟⃗ 푑푟 푑휃 = 푟̂ + 푟 휃̂ (90) 푑푡 푑푡 푑푡
Now, the velocity vector expression is substituted into Equation 85, the expression for the angular momentum vector.
푑푟 푑휃 ℎ⃗⃗ = 푟푟̂ × ( 푟̂ + 푟 휃̂) (91) 푑푡 푑푡
Crossing the 푟̂ unit vectors eliminates the first term in Equation 91, leaving a scalar expression for angular momentum as shown in Equation 92.
푑휃 ℎ = 푟2 (92) 푑푡
The angular momentum, ℎ, is the product of the radius squared and the circumferential velocity, 푑휃⁄푑푡.
Kepler’s Second Law is represented pictorially in Figure 57 below. The shaded section of the figure represents the area swept out by the position vector as 푚2 moves in an elliptical orbit around 푚1.
150
Figure 57: Keppler's Second Law
During the time interval, 푑푡, the position vector sweeps out an area, 푑퐴, as shown in Figure 3. The shaded section is approximated as a triangle for the purposes of calculating Kepler’s Second Law. The area is calculated as shown in Equation 93 by multiplying half the base by the height of the triangle.
푑퐴 푟2휃̇ = (93) 푑푡 2
The expression for angular momentum is substituted into Equation 93 yielding the following equation
푑퐴 ℎ = (94) 푑푡 2
The previous equation is Kepler’s Second Law.
Returning to the Two-Body Problem, Equation 81 as derived previously can be solved to yield the orbit equation. Solving this equation begins with the position vector as shown in Equation 95.
푟⃗ = 푟푟̂ (95)
151
The first derivative of the previous equation yields
푑푟⃗ 푑푟 푑푟̂ = 푟̂ + 푟 (96) 푑푡 푑푡 푑푡 where the first derivative of 푟̂ is expanded using the cross product of angular velocity with 푟̂.
푑푟̂ 푑휃 = 휃̂ (97) 푑푡 푑푡
Now, Equation 97 is substituted into Equation 96. The resulting equation is the velocity vector, as a function of radial velocity, 푑푟⁄푑푡, in the 푟̂ direction and circumferential velocity, 푑휃⁄푑푡, in the 휃̂ direction.
푑푟⃗ 푑푟 푑휃 = 푟̂ + 푟 휃̂ (98) 푑푡 푑푡 푑푡
The acceleration vector is determined by obtaining the second derivative of Equation 98.
푑2푟⃗ 푑2푟 푑푟 푑푟̂ 푑푟 푑휃 푑2휃 푑휃 푑휃̂ = 푟̂ + + 휃̂ + 푟 휃̂ + 푟 (99) 푑푡2 푑푡2 푑푡 푑푡 푑푡 푑푡 푑푡2 푑푡 푑푡
Substituting Equation 97 into Equation 98 reduces the equation to the following:
푑2푟⃗ 푑2푟 푑푟 푑휃 푑2휃 푑휃 푑휃̂ = 푟̂ + 2 휃̂ + 푟 휃̂ + 푟 (100) 푑푡2 푑푡2 푑푡 푑푡 푑푡2 푑푡 푑푡
In order to simplify Equation 100 further, 푑휃̂⁄푑푡 can be expanded using the cross product of angular velocity with 휃̂.
푑휃̂ 푑휃 = − 푟̂ (101) 푑푡 푑푡
After substituting Equation 101 into Equation 100, the simplified relative acceleration term is shown below. As shown in Equation 102, the radial and circumferential acceleration terms are grouped by position vectors 푟̂ and 휃̂, respectively.
152
푑2푟⃗ 푑2푟 푑휃 2 푑푟 푑휃 푑2휃 = ( − 푟 ( ) ) 푟̂ + (2 + 푟 ) 휃̂ (102) 푑푡2 푑푡2 푑푡 푑푡 푑푡 푑푡2
Now, Equation 102 is substituted into Equation 81.
푑2푟 푑휃 2 휇 푑푟 푑휃 푑2휃 ( − 푟 ( ) + ) 푟̂ + (2 + 푟 ) 휃̂ = 0 (103) 푑푡2 푑푡 ℎ2 푑푡 푑푡 푑푡2
Equation 103 is broken down into its radial and circumferential components as shown.
푑2푟 푑휃 2 휇 − 푟 ( ) + = 0 (104) 푑푡2 푑푡 푟2
푑푟 푑휃 푑2휃 2 + 푟 = 0 (105) 푑푡 푑푡 푑푡2
Equation 105, the circumferential component of the equation, can be rewritten as follows
1 푑 푑휃 (푟2 ) = 0 (106) 푟 푑푡 푑푡
Now, using the derivation for angular momentum, Equation 106 can also be rewritten.
푑휃 ℎ = 푟2 (107) 푑푡
Equation 107 is substituted into the radial component of the Two-Body Problem,
Equation 104 as shown.
푑2푟 ℎ2 휇 − + = 0 (108) 푑푡2 푟3 푟2
In order to solve this differential equation, first make a substitution for 푟.
1 푢 = (109) 푟
Using the chain rule, the derivative of Equation 109 follows.
푑푟 1 푑푢 푑휃 = − (110) 푑푡 푢2 푑휃 푑푡
153
Substitute the angular momentum expression for 푑휃⁄푑푡 in Equation 109. Additionally,
Equation 109 is substituted into Equation 110.
푑푟 푑푢 = −ℎ (111) 푑푡 푑휃
Finally, the chain rule is applied again to Equation 111 to calculate the second derivative of 푑푟⁄푑푡.
푑2푟 ℎ2 푑2푢 = − (112) 푑푡2 푟2 푑휃2
Equation 112 is substituted into Equation 107 yielding Two-Body Problem differential
Equation 113 that can be solved.
푑2푢 휇 + 푢 − = 0 (113) 푑휃2 ℎ2
The solution to the Two-Body Problem is the sum of the complementary and particular solutions of the equation as shown in the following equation.
푢(휃) = 푢퐶(휃) + 푢푃(휃) (114)
The complementary solution is shown below and includes two constants of integration, 퐴 and 휔.
푢퐶(휃) = 퐴 cos(휃 − 휔) (115)
The particular solution is shown in Equation 116. 휇 푢 (휃) = (116) 푃 ℎ2
The previous derivation yields the following solution. 휇 푢(휃) = + 퐴푐표푠(휃 + 휔) (117) ℎ2
Substituting for 푢 results in the following equation
154
(ℎ2⁄휇) 푟 = (118) 퐴ℎ2 ( ) (1 + 휇 cos 휃 + 휔 )
The previous equation represents the radius of the orbit of 푚2 with respect to 푚1 as a function of the angular momentum, ℎ, the gravitational parameter, 휇, and the angle,
휃. The quantity 퐴ℎ2⁄휇 is represented by the variable, 푒.
퐴ℎ2 푒 = (119) 휇
In the previous equation, 푒 is the eccentricity of the orbit. This dimensionless parameter indicates whether the orbit is circular, elliptical, parabolic, or hyperbolic in nature. Substitution Equation 119 into Equation 118 results in Equation 120.
푟3휃̇ 2 − 휇 푒 = (120) 휇 cos(휃)
155
APPENDIX B
INITIAL CONDITIONS FOR MISSION ANALYSIS CALCULATIONS
Table 61: Initial conditions for a NEP Earth-Mars flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1.15
Thrust Angle (휙) Tangential
Thrust (휈) 0.0064
-5 Propellant Mass Flow Rate (푚푓) 4.6 × 10 kg⁄s
Spacecraft Mass (푚) 50,000 kg
Table 62: Initial conditions for a NEP Mars escape trajectory flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 1.4 rad (80.21)
Angular Velocity (휃′) 1.405
Thrust Angle (휙) Tangential
Thrust (휈) 0.0064
-5 Propellant Mass Flow Rate (푚푓) 4.6 × 10 kg⁄s
Spacecraft Mass (푚) 48,839 kg
156
Table 63: Initial conditions for a NEP Mars-Earth flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 2.884 rad (165.241)
Angular Velocity (휃′) -1.103
Thrust Angle (휙) 90
Thrust (휈) 0.0064
-5 Propellant Mass Flow Rate (푚푓) 4.6 × 10 kg⁄s
Spacecraft Mass (푚) 48,332 kg
Table 64: Initial conditions for nuclear thermal propulsion Earth escape flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 3.7 rad (211.99)
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.136
Propellant Mass Flow Rate (푚푓) 37.35 kg⁄s
Spacecraft Mass (푚) 250,000 kg
157
Table 65: Initial conditions for a NTP Earth-Mars flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1.08
Thrust Angle (휙) 90
Thrust (휈) 0.136
Propellant Mass Flow Rate (푚푓) 37.35 kg⁄s
Spacecraft Mass (푚) 168,240 kg
Table 66: Initial conditions for a NTP Mars escape trajectory flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 4.27 rad (244.65)
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.136
Propellant Mass Flow Rate (푚푓) 37.35 kg⁄s
Spacecraft Mass (푚) 110,040 kg
158
Table 67: Initial conditions for a NTP Mars-Earth flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 0.947 rad (54.26)
Angular Velocity (휃′) -0.872
Thrust Angle (휙) 90
Thrust (휈) 0.132
Propellant Mass Flow Rate (푚푓) 37.35 kg⁄s
Spacecraft Mass (푚) 94,800 kg
Table 68: Initial conditions for a nuclear fusion Earth escape flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 0.69 rad (39.53)
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.003
Propellant Mass Flow Rate (푚푓) 0.025 kg⁄s
Spacecraft Mass (푚) 350,000 kg
159
Table 69: Initial conditions for a nuclear fusion Earth-Mars flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.003
Propellant Mass Flow Rate (푚푓) 0.025 kg⁄s
Spacecraft Mass (푚) 343,060 kg
Table 70: Initial conditions for a nuclear fusion Mars escape trajectory flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 2.0 rad (114.59)
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.003
Propellant Mass Flow Rate (푚푓) 0.025 kg⁄s
Spacecraft Mass (푚) 309,750 kg
160
Table 71: Initial conditions for a nuclear fusion Mars-Earth flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 4.087 rad (234.17)
Angular Velocity (휃′) -0.913
Thrust Angle (휙) 90
Thrust (휈) 0.003
Propellant Mass Flow Rate (푚푓) 0.025 kg⁄s
Spacecraft Mass (푚) 303,010 kg
Table 72: Initial conditions for a nuclear fusion Earth escape flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 2.20 rad (126.05)
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.003
Propellant Mass Flow Rate (푚푓) 0.025 kg⁄s
Spacecraft Mass (푚) 500,000 kg
161
Table 73: Initial conditions for a nuclear fusion Earth-Jupiter flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 0
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.003
Propellant Mass Flow Rate (푚푓) 0.025 kg⁄s
Spacecraft Mass (푚) 489,870 kg
Table 74: Initial conditions for a nuclear fusion Jupiter escape trajectory flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 4.95 rad (283.61)
Angular Velocity (휃′) 1
Thrust Angle (휙) 90
Thrust (휈) 0.003
Propellant Mass Flow Rate (푚푓) 0.025 kg⁄s
Spacecraft Mass (푚) 341,630 kg
162
Table 75: Initial conditions for a nuclear fusion Jupiter-Earth flight
Radius (휌) 1
Radial Acceleration (휌′) 0
Angle (휃) 4.62 rad (264.52)
Angular Velocity (휃′) -1.0
Thrust Angle (휙) 90
Thrust (휈) 0.003
Propellant Mass Flow Rate (푚푓) 0.025 kg⁄s
Spacecraft Mass (푚) 270,000 kg
163
APPENDIX C
MATLAB CODE FOR ORBITAL DYNAMICS CALCULATIONS
%------% low_thrust - low/continuous thrust orbit solution (using rk4) %------clear all; clc; disp('low_thrust'); fnmin=input('enter input file -> ','s'); fin=fopen(fnmin); data=input('enter data set file -> ','s'); fnmout=input('enter output file -> ','s'); fout=fopen(fnmout,'w'); disp(' '); %...read the input file, write to display m=fscanf(fin,'%g',[1,1]); s=fgetl(fin); fprintf(1,'%g %s\n',m,s); mu=fscanf(fin,'%g',[1,1]); s=fgetl(fin); fprintf(1,'%g %s\n',mu,s); ri=fscanf(fin,'%g',[1,1]); s=fgetl(fin); fprintf(1,'%g %s\n',ri,s); re=fscanf(fin,'%g',[1,1]); s=fgetl(fin); fprintf(1,'%g %s\n',re,s); ge=fscanf(fin,'%g',[1,1]); s=fgetl(fin); fprintf(1,'%g %s\n',ge,s); t1=fscanf(fin,'%g',[1,1]); s=fgetl(fin); fprintf(1,'%g %s\n',t1,s); t2=fscanf(fin,'%g',[1,1]); s=fgetl(fin); fprintf(1,'%g %s\n',t2,s); h=fscanf(fin,'%g',[1,1]); s=fgetl(fin); fprintf(1,'%g %s\n',h,s); iprt=fscanf(fin,'%hd',[1,1]); s=fgetl(fin); fprintf(1,'%hd %s\n',iprt,s); n=fscanf(fin,'%hd',[1,1]); s=fgetl(fin); fprintf(1,'%hd %s\n',n,s); tt=fscanf(fin,'%hd',[1,1]); s=fgetl(fin); fprintf(1,'%hd %s\n',tt,s); for i=1:n yv=fscanf(fin,'%g',[1,1]); s=fgetl(fin); %read in y initial conditions fprintf(1,'%g %s\n',yv,s); y(i)=yv; %write to display end
%set up column headers for output file fprintf(fout,'%11.4s','Time'); fprintf(fout,'%11.3s','Tau'); fprintf(fout,'%11.6s','Thrust'); fprintf(fout,'%11.12s','Thrust Angle'); fprintf(fout,'%11.5s','dr/dt'); fprintf(fout,'%11.8s','dRho/dt '); fprintf(fout,'%11.6s','Radius'); fprintf(fout,'%11.3s','Rho'); fprintf(fout,'%11.9s','dTheta/dt'); fprintf(fout,'%11.9s','dTheta/dt'); fprintf(fout,'%11.5s','Theta'); fprintf(fout,'%11.4s','Mass'); fprintf(fout,'%11.8s','Velocity'); fprintf(fout,'\r\n'); fprintf(fout,'%11.3s','(s)'); fprintf(fout,'%11.1s',' '); fprintf(fout,'%11.1s','N'); fprintf(fout,'%11.7s','Degrees');
164
fprintf(fout,'%11.6s','(km/s)'); fprintf(fout,'%11.1s',' '); fprintf(fout,'%11.4s','(km)'); fprintf(fout,'%11.1s',' '); fprintf(fout,'%11.5s','(1/s)'); fprintf(fout,'%11.1s',' '); fprintf(fout,'%11.5s','(rad)'); fprintf(fout,'%11.4s','(kg)'); fprintf(fout,'%11.5s','(km/s)'); fprintf(fout,'\r\n'); x1=t1; %map dimensional start time to non-dimensional start time variable r0=ri+re; %initial radius, km g=ge*(re^2)/(r0^2); %gravitational acceleration at initial orbit x2=nondim_t(t2,r0,mu); %non-dimensional time
Input=readmatrix(data,'Sheet',1); %read data from excel file, sheet 1 d=Input(:,1); p=numel(d); rr=2; r1=Input(1,1); r2=Input(rr,1); mf=Input(1,3); %mass flow rate at time zero
T=Input(1,2); %thrust at time zero phi_input=Input(1,4); %read in thrust angle at time zero phi=thrust_angle(phi_input,y); nu=T/(m*g); nuc=nu*sin(phi); %circumferential thrust component nur=nu*cos(phi); %radial thrust component t=dim_t(x1,r0,mu); drdt=dim_drdt(r0,y,mu); r=dim_r(y,r0); dthetadt=dim_dthetadt(r0,y,mu); th=y(4); V=(drdt^2+(r^2)*(dthetadt^2))^(1/2); %spacecraft velocity phi_d=phi*180/pi;
%write initial conditions to output file fprintf(fout,'%11.3e',t); fprintf(fout,'%11.3e',x1); fprintf(fout,'%11.3e',T); fprintf(fout,'%11.5e',phi_d); fprintf(fout,'%11.3e',drdt); fprintf(fout,'%11.3e',y(1)); fprintf(fout,'%11.3e',r); fprintf(fout,'%11.3e',y(2)); fprintf(fout,'%11.3e',dthetadt); fprintf(fout,'%11.3e',y(3)); fprintf(fout,'%11.3e',y(4)); fprintf(fout,'%11.3e',m); fprintf(fout,'%11.3e',V); fprintf(fout,'\r\n');
165
x=x1; ii=0; nn=fix((x2-x1)/h); %number of steps drhodt(1)=y(1); rho(1)=y(2); dtheta(1)=y(3); theta(1)=y(4); b(1)=nu; %h1=nondim_t(h,r0,mu) for jn=1:nn ii=ii+1;
xp=x+h;
t=dim_t(xp,r0,mu); dt=t-dim_t(x,r0,mu); %time change from last step
%check excel data - change thrust, mass flow rate with time if t>=r1 && t else T=Input(rr,2); mf=Input(rr,3); phi_input=Input(rr,4); phi=thrust_angle(phi_input,y); rr=rr+1; if rr>p break end r1=r2; r2=Input(rr,1); end %calculate mass change m=m-mf*dt; nu=T/(m*g); nur=nu*cos(phi); %radial thrust nuc=nu*sin(phi); %circumferential thrust dydx=derivs(x,y,nur,nuc); yout=rk4(y,dydx,n,x,h,nur,nuc); drdt=dim_drdt(r0,y,mu); r=dim_r(y,r0); dthetadt=dim_dthetadt(r0,y,mu); th=y(4); V=((drdt^2)+(r^2)*(dthetadt^2))^(1/2); phi_d=phi*180/pi; 166 if (ii==iprt) fprintf(fout,'%11.5e',t); fprintf(fout,'%11.5e',xp); fprintf(fout,'%11.3e',T); fprintf(fout,'%11.5e',phi_d); fprintf(fout,'%11.3e',drdt); fprintf(fout,'%11.3e',y(1)); fprintf(fout,'%11.3e',r); fprintf(fout,'%11.3e',y(2)); fprintf(fout,'%11.3e',dthetadt); fprintf(fout,'%11.3e',y(3)); fprintf(fout,'%11.3e',y(4)); fprintf(fout,'%11.3e',m); fprintf(fout,'%11.3e',V); fprintf(fout,'\r\n'); yrow=y(1:end); ii=0; end % %check escape velocity c3=2*((V^2)/2-(mu/r)); % if c3 >= 0 % disp(V) % disp(xp) % disp(r) % disp(y(2)) % disp(y(1)) % disp(drdt) % disp(y(4)) % disp(m) % break % else %...march the solution forward by interval h x=x+h; for i=1:n y(i)=yout(i); end drhodt(jn+1)=y(1); rho(jn+1)=y(2); dtheta(jn+1)=y(3); theta(jn+1)=y(4); b(jn+1)=nu; % end end %...plot the sprial orbit in polar coordinates %rho=rho+.523; %to plot Mars orbit %rho=rho+4.2; %to plot Jupiter orbit polarplot(theta,rho,'k'); hold on %polarplot(theta(jn),rho(jn),'o--b'); pax=gca; pax.FontSize=18; pax.FontName='Cambria Math'; 167 %legend('Transit to Mars','Mars Orbit','Transit to Earth','Earth Orbit'); %legend('Transit to Jupiter','Jupiter Orbit','Transit to Earth','Earth Orbit'); %rlim([0 4]); %rticks([1 2 4 6 8 10 12 14]); %pax.RLim=[0 1.5]; hold on %for k=1:jn+1 %if b(k)==0 %polarplot(theta(k),rho(k),'x--b'); %break %else %end %end disp(V) %disp(xp) disp(r) %disp(y(1)) %rfinal=y(2)*r0/(1.49597871*(10^8)); %disp(y(2)) %disp(drdt) %disp(y(3)) disp(y(4)) disp(m) %disp(c3) %disp(dthetadt) status=fclose('all'); 168 BIBLIOGRAPHY [1] H. Curtis, Orbital Mechanics for Engineering Students, Oxford: Elsevier Ltd., 2009, pp. 70-82. [2] D. Greenwood, Principles of Dynamics, Upper Sadler River, NJ: Prentice-Hall Inc., 1988. [3] J. Irving, Space Technology, New York: Wiley, 1959. [4] "Dawn," 25 April 2019. [Online]. Available: https://solarsystem.nasa.gov/missions/dawn/overview/. [Accessed 22 September 2019]. [5] R. McNutt, J. Horsewood and D. Fiehler, "Human Missions Throughout the Outer Solar System: Requirements and Implementations," Johns Hopkins APL Technical Digest, vol. 28, no. 4, pp. 373-388, 2010. [6] J. Gilland, D. Fiehler and V. Lyons, "Electric Propulsion Concepts Enabled by High Power Systems for Space Exploration," in Second International Energy Conversion Engineering Conference, Providence, 2004. [7] J. Fisher, "NEXT-C Flight Ion Propulsion System Development Status," in 35th International Electric Propulsion Conference, Atlanta, 2017. [8] R. Shastry, "Current Status of NASA's NEXT-C Ion Propulsion System Developmental Project," in Joint Propulsion Conference, Cleveland, 2014. [9] "Technology Readiness Level," October 2020. [Online]. Available: https://www.nasa.gov/directorates/heo/scan/engineering/technology/txt_accordion1.htm l. [Accessed January 2021]. [10] S. Borowski, D. McCurdy and T. Packard, "Nuclear Thermal Propulsion (NTP): A Proven Growth Technology for Human NEO/Mars Exploration Missions," in IEEE Aerospace Conference, 2021, Big Sky. [11] S. Borowski and D. McCurdy, "Conventional and Bimodal Nuclear Thermal Rocket (NTR) Artificial Gravity Mars Transfer Vehicle Concepts," in Joint Propulsion Conference, Cleveland, 2014. [12] "Nuclear Fusion Power," July 2019. [Online]. Available: https://www.world- nuclear.org/information-library/current-and-future-generation/nuclear-fusion- power.aspx. [Accessed November 2019]. [13] "Physics of Uranium and Nuclear Energy," February 2018. [Online]. Available: https://www.world-nuclear.org/information-library/nuclear-fuel- cycle/introduction/physics-of-nuclear-energy.aspx. [Accessed November 2019]. [14] M. LaPointe, R. Adams, J. Cassibry, M. Zweiner and J. Gilland, "Preliminary Analysis of the Gradient Field Imploding Liner Fusion Propulsion Concept," in Joint Propulsion Conference, Cincinnati, 2018. 169 [15] S. Farokhi, Aircraft Propulsion, John Wiley & Sons Inc., 2009. [16] Aerojet Rocketdyne, "Powering Deep Space Exploration," [Online]. Available: https://www.rocket.com/space/liquid-engines/rs-25-engine. [Accessed 19 February 2021]. 170