A comparison of nuclear thermal rockets with traditional chemical rockets for space transport

The Solar System has multiple destinations that private and governmental space agencies are planning to explore. Missions within the Solar System are both exorbitantly expensive and time intensive projects that involve high risks for the organizations involved. A mission that is currently being developed by multiple agencies is a human mission to Mars. In planning a mission to Mars and other similar targets, a critical design parameter to consider is the most effective propulsion system to transport the crew and payload to the destination in a given time frame. To understand the most effective propulsion system for space exploration, this analysis will use current parameters for a mission to Mars to compare the advantages and disadvantages of the chemical rocket with nuclear thermal propulsion. This assessment compares the two rockets due to their similar thrust capabilities and years of testing and development. Through numerical, graphical, and simulation data, this study validates that nuclear thermal propulsion, perating under an initial thrust-to-weight ratio of 0.11, can complete a mission from low Earth orbit to low Mars orbit within the same ten-month time frame as a chemical rocket. The numerical data in this analysis also shows that the nuclear thermal rocket is able to deliver 127,727.11 kilograms of payload to low Mars orbit, which is an extra 50,807 kilograms of payload when compared to a chemical rocket of the same size. By utilizing the extra payload carried by a nuclear thermal rocket, and an estimated budget of $23 billion for NASA to develop a chemical rocket as part of the Space Launch System to be used for a crewed mission beyond the moon's orbit [1], this paper shows that a nuclear thermal rocket would decrease the overall payload cost per kilogram by $430,000. This decrease would result in a total payload cost difference of $9.18 billion when considering the single mission. Once future program budgets are developed showing the total life-cycle of the Space Launch System program, a new payload cost difference can be calculated to compare the costs of the rockets' payloads over multiple missions. Due to the nuclear thermal rocket's superior payload capacities, which results in a large difference in the cost per kilogram of payload, this analysis hopes to provide further evidence that nuclear thermal rockets should be seriously considered for space exploration missions. It is the recommendation of this study that an in -depth analysis of the total cost to design, build and launch a nuclear thermal rocket be performed. If the nuclear thermal rocket can provide future savings in space exploration, the government along with other private organizations need to devote more resources to designing a safe and efficient nuclear thermal rocket that will allow for better access to the Solar System.

A COMPARISON OF NUCLEAR THERMAL ROCKETS WITH TRADITIONAL CHEMICAL ROCKETS FOR SPACE TRANSPORT by Joshua T. Hanes A Senior Honors Thesis Submitted to the Faculty of The University of Utah In Partial Fulfillment of the Requirements for the Honors Degree in Bachelor of Science In The Department of Physics and Astronomy Approved: ______________________________ Clayton Williams Thesis Faculty Supervisor _____________________________ Carleton DeTar Chair, Department of Physics & Astronomy _______________________________ Anil Chandra Seth Honors Faculty Advisor _____________________________ Sylvia D. Torti, PhD Dean, Honors College April 2016 ABSTRACT The Solar System has multiple destinations that private and governmental space agencies are planning to explore. Missions within the Solar System are both exorbitantly expensive and time intensive projects that involve high risks for the organizations involved. A mission that is currently being developed by multiple agencies is a human mission to Mars. In planning a mission to Mars and other similar targets, a critical design parameter to consider is the most effective propulsion system to transport the crew and payload to the destination in a given time frame. To understand the most effective propulsion system for space exploration, this analysis will use current parameters for a mission to Mars to compare the advantages and disadvantages of the chemical rocket with nuclear thermal propulsion. This assessment compares the two rockets due to their similar thrust capabilities and years of testing and development. Through numerical, graphical, and simulation data, this study validates that nuclear thermal propulsion, operating under an initial thrust-to-weight ratio of 0.11, can complete a mission from low Earth orbit to low Mars orbit within the same ten-month time frame as a chemical rocket. The numerical data in this analysis also shows that the nuclear thermal rocket is able to deliver 127,727.11 kilograms of payload to low Mars orbit, which is an extra 50,807 kilograms of payload when compared to a chemical rocket of the same size. By utilizing the extra payload carried by a nuclear thermal rocket, and an estimated budget of $23 billion for NASA to develop a chemical rocket as part of the Space Launch System to be used for a crewed mission beyond the moon’s orbit [1], this paper shows that a nuclear thermal rocket would decrease the overall payload cost per kilogram by $430,000. This ii decrease would result in a total payload cost difference of $9.18 billion when considering the single mission. Once future program budgets are developed showing the total lifecycle of the Space Launch System program, a new payload cost difference can be calculated to compare the costs of the rockets’ payloads over multiple missions. Due to the nuclear thermal rocket’s superior payload capacities, which results in a large difference in the cost per kilogram of payload, this analysis hopes to provide further evidence that nuclear thermal rockets should be seriously considered for space exploration missions. It is the recommendation of this study that an in-depth analysis of the total cost to design, build and launch a nuclear thermal rocket be performed. If the nuclear thermal rocket can provide future savings in space exploration, the government along with other private organizations need to devote more resources to designing a safe and efficient nuclear thermal rocket that will allow for better access to the Solar System. iii NOMENCLATURE LEO IMLEO mi mf mgross mpl mprop mFT mOT MR λ (lambda) Isp MXR T W T/W Ε (epsilon) Δv ϕ (phi) ve fTSW fFT fOT ρ FT/OT TCM MOI NTP NTR Lbf kg km m S = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Low Earth Orbit Initial Mass in Low Earth Orbit Initial Mass Final Mass Gross mass or IMLEO Payload Mass Propellant Mass Fuel Tank Mass Oxidizer Tank Mass Mass Ratio Payload Ratio Specific Impulse Mixture Ratio Thrust Weight Thrust-to-Weight Ratio Structural Coefficient Delta Velocity Gross Mass Coefficient Exit Velocity Ratio Factor Between Thrust and Gross Weight Fuel Tank Factor Oxidizer Tank Factor Density Fuel Tank / Oxidizer Tank Transfer Correction Maneuver Mars Orbit Insertion Nuclear Thermal Propulsion Nuclear Thermal Rocket Pounds force Kilogram Kilometer Meter Second iv v TABLE OF CONTENTS ABSTRACT ........................................................................................................................... ii NOMENCLATURE ................................................................................................................. iv 1 INTRODUCTION ................................................................................................................. 1 1.1 History of Chemical Rockets .................................................................................... 1 1.2 History of Nuclear Thermal Rockets ........................................................................ 3 2 METHODS ......................................................................................................................... 5 2.1 Derivations & GMAT Parameters ............................................................................ 6 3 RESULTS AND ANALYSIS................................................................................................. 14 3.1 Mission Simulation ................................................................................................. 14 3.2 Mission Analysis ..................................................................................................... 18 3.3 Payload Cost Difference ......................................................................................... 25 4 CONCLUSION................................................................................................................... 29 5. REFERENCES .................................................................................................................. 31 APPENDIX A....................................................................................................................... 33 APPENDIX B ....................................................................................................................... 52 vi 1 1. INTRODUCTION Within the last century, propulsion systems have seen great advances, reaching a pinnacle in the Apollo 11 mission that rocketed humans to take their first steps on the moon. Humankind is now working towards its next leap in space exploration as multiple agencies including NASA, SpaceX, and Mars One plan manned missions to Mars. Understandably, these mission will cost billions of dollars and require decades of testing. Due to this large risk and investment, it is of the highest importance to consider what the most efficient, safe, and cost effective means will be to make the trip to Mars. There are currently many advanced propulsion systems that exist, but only the NTR is comparable to the chemical rocket in relation to engine testing and thrust ratios. The mission analysis used in this document will only evaluate the chemical rocket and NTR due to these similarities. The comparison will take place from low earth orbit (LEO) to low Mars orbit (LMO) because the NTR engine selected for this study performs poorly for vehicle thrust-to-weight (T/W) ratios greater than 0.6, which are required to leave the Earth’s surface. If the NTR’s engine were able to achieve engine T/W ratios greater than 6, this study could be used as a baseline analysis for the viability of NTR launches from the Earth’s surface, but because the only proven and tested NTR engines have achieved engine T/W ratios of approximately 3, those engines and ratios will be used for this analysis. 1.1. History of Chemical Rockets The first chemical rockets were launched by the ancient Chinese in the form of solid propellant fireworks during the 12th century. The liquid chemical rockets used today were not developed until 1926, when Robert Goddard, the father of modern rocket propulsion, developed and successfully tested the first rocket using liquid fuel [4]. Chemical liquid rockets have a great advantage over solid propellants because the liquid rockets have a higher specific impulse (Isp), can be throttled, shut-off, and restarted. Isp is an important efficiency parameter in rocketry because it expresses the 2 amount of impulse or thrust provided per unit weight of propellant. Isp is defined as the total impulse of the rocket engine divided by the total weight of the propellant, which is in units of seconds. Rocket engines with higher Isp need less fuel in order to provide the same amount of thrust, which allows the vehicle to carry more payload. Chemical rockets saw great advancements during the 20th century partially due to research done by countries building weapon propulsion systems, along with the race to send humans to the moon. By the end of the 20th century, chemical liquid rockets became the main rocket engine for space flight and have now been proven as an efficient means to send Figure 1. RS-25 rocket engines powering the main stage of the space shuttle. Picture provided by NASA [20] payloads to orbit and beyond. This document defines a chemical rocket as a rocket that uses liquid propellant. One of the most well-known rocket engines, the RS-25, used for the US Space Shuttle’s main engines, has an Isp of 452 seconds, thrust of 512,300 force pounds (lbf) in vacuum, and an engine weight of 7,775 pounds (see Figure 1). The RS-25 exemplifies the advantage of being able to generate large amounts of thrust per engine mass. This is partially due to the small weight of the engines, and also due to the amount of thrust created during combustion. The example above has a T/W ratio of 66:1, which shows the ability of the chemical rocket to provide a great amount of thrust when compared to the total weight of the engine (Wengine). The chemical rocket engine that will be used in this study is the RL10C-1 developed by Aerojet Rocketdyne [5], which is part of the RL10 class of rocket engines. These engines were used to launch spacecraft such as Voyager 1 and New Horizons to different locations throughout the Solar System and into interstellar space. The RL10C-1 has an Isp of 449.7 s, weight of 420 lbf, and thrust of 22,890 lbf, allowing the rocket to operate with an engine T/W ratio of 54.5. With an Isp of 449.7 s, the RL10-C-1 can generate 449.7 Newton-Seconds of thrust per Newton of propellant consumed. 3 1.2. History of Nuclear Thermal Rockets In the 1960’s and 70’s, NASA and the Atomic Energy Commission created the Rover and Nuclear Engine for Rocket Vehicle Applications (NERVA) programs to develop nuclear-powered propulsion that would help get man to the moon and beyond. Throughout the program, scientists performed test firings on over 20 different nuclear thermal rocket (NTR) engines and advanced the technology enough to become a competitor with the chemical rockets being used for the missions to the moon and space station [6]. See Figure 2 for reference of the size and type of engines tested during the Rover and NERVA programs. Though the programs officially ended in 1972, research has continued through other government programs, and NERVA-derived engines have now achieved a specific impulse up to 925 seconds, almost double the best chemical rockets [7]. One such engine, named “Pewee”, was one of the later generations of NTR engines developed at the NERVA Program, and has become the central design for future NTR systems due to its high efficiency and relatively small mass. The “Pewee” engine is shown on the right in Figure 2. NASA’s current “Nuclear Cryogenic Propulsion Stage Project” at the Marshall Space Flight Center is using a Pewee-class NTR engine, and has based its research and modeling around this engine [8]. This study will also use the Pewee-class engine specifications for the rocket comparison in this study. Figure 2. Main Series of Nuclear Thermal Rockets during the Project Rover [6]. 4 Nuclear rocket engines used for space propulsion are based on the use of nuclear fission to create heat energy. Fission involves the splitting of large nuclei into smaller nuclei, which releases a great amount of energy in the process. The NTR then pumps liquid hydrogen from a propellant tank through the solid core reactor where it is heated to high temperatures. Due to the reactor’s extreme operating temperatures, which can reach up to 3000 degrees Kelvin, the hydrogen undergoes rapid expansion and is expelled from the rocket at high velocities. Through the high exhaust velocity of the hydrogen, the NTR is able to achieve an Isp in the range of 900 seconds depending on the engine selected. A publication from Los Alamos Laboratory explains the potential advantage of nuclear propulsion in the following statement: A nuclear reaction typically releases ten million times the energy of a chemical reaction. Thus, it would seem that the weight of the fuel necessary for a nuclear rocket to deliver a certain payload would be significantly less than the weight of the fuel a chemical rocket would need to heave the same weight the same distance. So, a larger fraction of a nuclear rocket’s total weight— including the weights of the rocket engine(s), the airframe, the fuel, and the payload—could go into the payload weight [7]. Fission is one of the most energetic processes in the universe and allows nuclear reactors to harness tremendous amounts of energy per fuel volume, which in the case of the NTR is usually highly enriched uranium-235. Though nuclear propulsion has a high energy density, the question that this study will answer is whether the NTR has limiting factors such as a large engine weight and a massive rocket structure that will physically stop the rocket from completing missions within the Solar System. Other important factors must be considered—such as the NTR’s commercial viability due to legal, safety, and political barriers. There are many variables associated with each barrier mentioned, however this study will only focus on the physical and economic viability of the rockets. This will be accomplished by understanding the efficiency and payload cost differences found during a mission to Mars for a chemical and a nuclear thermal rocket. 5 2. METHODS To compare the advantages and disadvantages of a chemical rocket with the NTR, this paper seeks to evaluate the amount of payload each system can deliver to low Mars orbit within a ten-month mission window. To ensure each rocket can complete the mission within ten months, each rocket’s trajectory and orbit paths will be simulated numerically using specific mission parameters. Once the rockets’ trajectory simulations confirm that each rocket completes the mission, the rockets’ efficiencies can be compared by calculating various rocket parameters such as Isp, Wengine, payload mass, propellant mass, and structure mass. Using these calculated parameters, it will then be possible to create a graphical analysis to understand the advantages and disadvantages of each rocket design. By knowing how the rockets compare during the mission to Mars, along with the current budgets for developing the different rocket systems, this study will create a breakdown of the cost per kilogram of payload. In order to simulate the rockets’ trajectory paths, NASA’s mission simulation software, “General Mission Analysis Tool” (GMAT), can be used to plot each rocket’s orbit and trajectory in time. GMAT is open source software maintained by NASA, which is used for “real-world engineering studies” and “to fly operational spacecraft” [9]. GMAT can be used to calculate mission flight parameters such as mission duration, required propellant mass, and flight trajectories. By using GMAT to simulate the mission to Mars, this study is able to validate that the rockets are able to complete the mission within ten months under the various mission conditions. The GMAT simulation is also used to calculate the required delta velocity (v), or change in velocity of the rocket during each maneuver. 6 Using specifications set by NASA’s Nuclear Thermal Propulsion project for a mission to Mars [10], this paper sets the mass constraint for the total initial mass in low Earth orbit (IMLEO) for both spacecraft to 300,000 kg. By using 300,000 kg for IMLEO and a v of 5.6 km/s, each rocket’s required total thrust, number of engines, and propellant mass can be derived to create a simulation of the rocket from LEO to LMO. The following section defines the equations used to calculate the rocket parameters needed for GMAT and the rocket comparison. 2.1. Derivations & GMAT Parameters To generate a GMAT simulation from LEO to LMO, GMAT requires multiple parameters from the rocket structure, fuel tank, engines, initial position, and launch date. This section will be used to specify and derive each equation and parameter necessary for both the simulation and the rocket comparison analysis. To fully describe the rocket’s structure in GMAT, GMAT requires the rocket’s dry mass and surface area. The dry mass is the mass of the rocket and payload without the propellant. GMAT uses both the dry mass and propellant mass independently throughout the simulation, which is why the two terms are separated. The surface area is incorporated into GMAT to allow calculations for the radiation pressure experienced during the transit from Earth to Mars. As mentioned earlier, the advantage of setting the vehicle’s gross mass to 300,000 kg is that the propellant mass fraction (PMF) can then be used to solve for the total propellant mass – which is a required GMAT input to run the program. PMF is the ratio of the amount of propellant at LEO to the gross mass. Once the propellant mass (mprop) is found, the dry mass can be calculated by subtracting mprop from IMLEO. 7 A simple method of finding the amount of propellant needed with relation to the gross mass of the vehicle is through the Mass Ratio (MR). MR is the initial mass divided by the final mass of the rocket, and can be calculated using the ideal rocket equation [11]. The ideal rocket equation is based on the principle of conservation of momentum. This is the basic principle of rocketry and states that the forward momentum of the rocket will change by the same amount as the reverse momentum of the expelled propellant, which is represented by the following equation. 𝑚𝑟𝑜𝑐𝑘𝑒𝑡 ∗ 𝑑𝑣 = −𝑣𝑒𝑥ℎ𝑎𝑢𝑠𝑡 ∗ 𝑑𝑚𝑝𝑟𝑜𝑝𝑒𝑙𝑙𝑎𝑛𝑡 (1) Integrating and solving for Δv gives, Δ𝜐 = 𝑣𝑒𝑥ℎ𝑎𝑢𝑠𝑡 ∗ ln⁡( 𝑚𝑖 𝑚𝑓 ) (2) Equation 2 can be easily rearranged to give the value of MR (mi/mf). In order to solve for PMF using MR, the assumption must be made that the only change in the vehicle’s mass is due to the expulsion of propellant through the rocket nozzle. This assumption is shown in the following equation. 𝑀𝑅 = 𝑚𝑖 𝑚𝑓 = 𝑚𝑔𝑟𝑜𝑠 𝑚𝑔𝑟𝑜𝑠𝑠 −𝑚𝑝𝑟𝑜𝑝 (3) By using the ideal rocket equation and setting ve=Isp*g0, MR can be written as, Δ𝑣 𝑀𝑅 = 𝑒 𝐼𝑠𝑝∗𝑔0 (4) From Equation 4, the MR ratio can be solved since the other parameters are already known. The Isp is specified by the engine selected for the rocket. The chemical rocket engine has an Isp of 449.7 s, and the NTR engine has an Isp of 900 s. The g0 term is 9.81 m/s^2 and acts as a conversion factor between mass and weight. Δ𝑣 is solved by GMAT and found to be 5.6 km/s from LEO to LMO. By re-arranging equation 3, PMF can now be solved in terms of MR [12]. 8 The PMF is defined as follows, 𝑃𝑀𝐹 = 𝑚𝑝𝑟𝑜𝑝 𝑚𝑔𝑟𝑜𝑠𝑠 = 𝑀𝑅−1 𝑀𝑅 =1− 1 𝑀𝑅 (5) With the PMF now defined in terms of MR, mprop can be found by multiplying the PMF by the gross mass as shown below. 𝑚𝑝𝑟𝑜𝑝 = 𝑃𝑀𝐹 ∗ 𝑚𝑔𝑟𝑜𝑠𝑠 (6) As mentioned earlier, the GMAT input for dry mass can be calculated by subtracting mprop from IMLEO. With the dry mass of the rocket structure now specified in GMAT, the surface area of the rocket must be calculated to find the forces acting on the vehicle during flight such as drag and radiation pressure. Atmospheric drag can be neglected due to the mission only operating in vacuum, but the surface area is still included due to radiation pressure from the Sun. The approximate surface area is calculated by assuming that the propellant tank’s geometry for the rockets are cylindrical-elliptical for liquid hydrogen (LH2) and spherical for liquid oxygen (LOX). It is important to note that the chemical rocket has a mixture ratio (MXR) of 5.5:1 for fuel LH2 and oxidizer LOX respectively. The NTR only uses LH2 and thus has a MXR of zero. The volume of each tank is calculated by dividing the propellant mass by the density of each fuel type. Once the volumes of the tanks are known, it is a matter of calculating the surface area by using each tank’s geometry. In GMAT, after the vehicle parameters have been set, the initial launch coordinates and date must be specified. For this analysis, the time of launch matches the launch of the MAVEN satellite to Mars in November of 2013. Using this launch window ensures reliability, and an optimal window for the comparison. Both rockets start at the same location in LEO and begin their respective maneuvers from that location. 9 Once the coordinates have been set in GMAT, the propellant tank variables need to be defined. The required variables for the propellant tank are propellant mass, propellant density, and tank volume. For both rockets, the GMAT propellant mass input can be calculated by Equation 6. In GMAT, there is only one input for propellant density, so the density for the propellant of the chemical rocket is the average density of LOX and LH2, while the propellant density for the NTR is the density of LH2 alone because the NTR does not undergo combustion. The last field for the propellant tank is the volume of the tank, which is found during the surface area calculation. After the propellant tank parameters have been set in GMAT, the final input parameters that must be included to perform the trajectory simulation are the rocket’s thrust and Isp. As mentioned earlier, the thrust and Isp are constants specific to the rocket’s engine. In this case, the R10C-1 has an Isp of 449.7 s and a thrust of 101,819.76 N in vacuum, while the Pewee has an Isp of 900 s, with a thrust of 111,205.5 N. The following chart shows all of the previously mentioned variables calculated for implementation into GMAT. For other specifications needed to run the GMAT simulation, see appendix A, which provides a description of GMAT along with the script for the NTR simulation. With the calculations in Table 1, along with the other specifications shown in appendix A, GMAT is able to create a simulation of the mission for each rocket type. The mission summary included in GMAT describes the total mission time, trajectory plots, fuel consumption, mission duration, engine burn times, and locations for each rocket. See the Results section for the plots and data for each rocket’s simulation. 10 Table 1. The calculated parameters that are require before running the GMAT simulation IMLEO Dry Mass Propellant Mass Payload Mass Total Thrust Engine Isp Propellant Vol. Surface Area Chemical Rocket 300,000.00 84,300.63 215,699.37 76,920.48 323,730.00 449.70 645.07 637.16 NTR 300,000.00 159,096.20 140,903.80 127,727.11 323,730.00 900.00 2,038.82 1,133.95 Units kg kg kg kg N sec m^3 m^2 Along with the mission duration, the other goal of this analysis is to understand the payload delivered to Mars by each rocket. This can be calculated through the payload ratio (λ). The payload ratio is equal to the mass of the payload divided by IMLEO. To derive an equation for λ, the different structural and propellant mass terms which affect λ must first be found. The following equation derivations are provided by Kirk Sorensen in his “Rocket Design Theory” [13]. This study utilizes the assumption that the structure has parts dependent on the propellant mass (mpl) and the gross mass. The propellant massdependent term is referred to as the structural coefficient (ε), which in this study relates to the propellant tank. This is apparent when considering that for each kilogram of propellant added to the tank, the volume, and consequently the mass of the tank, must increase by a proportional amount. The amount of mass by which the tank increases is found through the mass estimating relationship tables (MERs). MERs are tables which include mass estimating ratios for different tank designs. In the table, each tank’s corresponding ratio is the tank’s mass divided by the propellant mass carried within the tank. These ratios are used to find the total tank mass required once the total propellant mass is known. The gross mass-dependent term can be found by using the gross mass coefficient (ϕ). For the simplified rocket designs used in this study, ϕ only relates to the thrust structure mass and the total mass of the engines. The thrust structure is the frame of the rocket, 11 which includes all of the parts which provide support during propulsive maneuvers. The assumption that ϕ includes the thrust structure is made because the thrust structure must be proportional to the amount of thrust the rocket experiences in order to keep the rocket from breaking apart. This allows the definition of the gross mass to include the ϕ ratio. With the ε and ϕ terms defined, the mgross in units of kilograms can now be defined as, 𝑚𝑔𝑟𝑜𝑠𝑠 = 𝑚𝑝𝑙 + 𝜙 ∗ 𝑚𝑔𝑟𝑜𝑠𝑠 + 𝑚𝑝𝑟𝑜𝑝 + 𝜉 ∗ 𝑚𝑝𝑟𝑜𝑝 = 𝑚𝑝𝑙 + 𝜙 ∗ 𝑚𝑔𝑟𝑜𝑠𝑠 + 𝑚𝑝𝑟𝑜𝑝 (1 + 𝜉) (7) With the mgross in terms of payload, propellant, and structure masses, the payload ratio can be derived [13]. Using equation 3 to solve for mprop in terms of MR and mgross, the payload ratio is found to be, 𝜆= 1−𝜉(𝑀𝑅−1)−𝜙𝑀𝑅 𝑀𝑅 (8) Using equation 5, λ can be written in terms of PMF, ε, and ϕ. 𝜆 = 1 − 𝑃𝑀𝐹 − 𝜉(𝑃𝑀𝐹) − 𝜙 (9) From equation 9, the last terms that need to be derived are ε and ϕ. As explained earlier, ε is the structural coefficient, or the mass of the structure dependent upon the propellant mass. This structure-dependent mass can be defined as the mass of the fuel tank (mFT) and oxidizer tank (mOT) if we assume the propellant only contains LH2 and LOX. 𝜉= mFT +mOT mpropellant ( 10 ) where mFT and mOT are the mass of the fuel and oxidizer tanks respectively. With the use of an MER table [14], both tank masses can be converted into volume ratios. By including an ullage factor (fullage), which is the percentage of space in a propellant tank left empty in case of propellant expansion, ε will also account for the mass of the propellant not included to allow for the ullage space. 𝜉= 𝑀𝐸𝑅𝐹𝑇 ∗𝑉𝐹𝑇 +𝑀𝐸𝑅𝑂𝑇 ∗𝑉𝑂𝑇 𝑚𝑝𝑟𝑜𝑝 (1−𝑓𝑢𝑙𝑙𝑎𝑔𝑒 ) ( 11 ) 12 By assuming the volume of the propellant tanks is equal to the volume of the propellant plus the ullage volume, the volume terms in Equation 11 are re-written in terms of propellant mass, density, and the ullage factor. 𝑀𝐸𝑅𝐹𝑇 ∗( 𝜉= 𝑚𝑓𝑢𝑒𝑙 𝑚 )+𝑀𝐸𝑅𝑂𝑇 ∗( 𝑂𝑇 ) 𝜌𝑓𝑢𝑒𝑙 𝜌𝑂𝑇 𝑚 ( 12 ) 𝑝𝑟𝑜𝑝(1−𝑓𝑢𝑙𝑙𝑎𝑔𝑒 ) The advantage of equation 12 is that the structure coefficient ε is now in terms of the variables that are dependent on the type of rocket engine selected for the mission. By selecting the RL10C-1 and assuming an ullage factor of 3%, ε is then found by specifying the type of tanks used, referencing a MER table, and entering the propellant masses and densities, which were derived from GMAT. The advantage of calculating ε during a rocket’s design is that it allows one to quickly understand the propellant-sensitive masses when mgross of the rocket is specified in the mission design. In this analysis, because mgross is specified in the mission parameters obtained by NASA [10], the propellant-sensitive masses can quickly be calculated. The last variable that must be found in order to solve for the payload ratio is the gross mass-dependent term ϕ. For a mission in vacuum, and assuming a simplified rocket system, the only gross mass-dependent objects are the rocket engines and thrust structure, because they are proportional to the maximum amount of force the rocket will experience during the mission. With this proportionality assumption, ϕ is written as, 𝜙= 𝑚𝑒𝑛𝑔𝑖𝑛𝑒𝑠 +𝑚𝑡ℎ𝑟𝑢𝑠𝑡−𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝑚𝑔𝑟𝑜𝑠𝑠 ( 13 ) By using the weight conversion factor of 9.81 (m/s^2), each mass term in Equation 13 is converted into a weight term. By converting the above terms to weight instead of mass, the thrust and weight of the engines, which are defined by the engine selected during the 13 rocket design, can be used. The conversion of Equation 13 into weight terms is shown below, 𝜙= 𝑇 ) 𝑊 𝑒𝑛𝑔𝑖𝑛𝑒−𝑣𝑎𝑐𝑢𝑢𝑚 𝑛∗𝑊𝑒𝑛𝑔𝑖𝑛𝑒 (1+𝑓𝑇𝑆𝑊 ∗( ) 𝑊𝑔𝑟𝑜𝑠𝑠 ( 14 ) where n is the number of engines on the rocket. The Wgross is then re-written in terms of the total initial thrust-to-weight ratio (T/W)initial defined in the mission specifications [15]. Depending on the specific (T/W)initial set, the gross weight is, 𝑊𝑔𝑟𝑜𝑠𝑠 = 𝑛∗𝑡ℎ𝑟𝑢𝑠𝑡𝑒𝑛𝑔𝑖𝑛𝑒 ( 15 ) 𝑇 𝑊 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ( ) The derivation for the Wgross in Equation 15 is then substituted back into equation 14 to yield, 𝑇 𝜙= (𝑊) 𝑇 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 (1+𝑓𝑇𝑆𝑊∗(𝑊) 𝑒𝑛𝑔𝑖𝑛𝑒−𝑣𝑎𝑐𝑢𝑢𝑚 𝑇 𝑊𝑒𝑛𝑔𝑖𝑛𝑒−𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ⁡⁡) ( 16 ) For vacuum, the (T/W)engine-vacuum and (T/W)engine-initial are equal. Equation 16, emphasizes the importance of a smaller engine weight through the (T/W)Engine term. The advantage of calculating the ϕ factor is that it allows one to understand how each rocket’s structural masses compare just by knowing the (T/W)initial and (T/W)Engine. The ϕ term can be calculated for each rocket design without requiring a mission simulation. Now that ϕ and ε are defined, it is possible to calculate λ simply by specifying the rocket engine type and the mission (T/W)initial. This assumes that the design of the rocket is accounted for through the MER chart, ullage factor, and thrust structure factor used in the derivations. With the previous derivations, it is now possible to understand the mission duration through the GMAT simulation, and how much payload will be delivered to Mars through the payload ratio. The next section describes the results of each rocket’s mission and compares each rocket through the parameters derived in this section. 14 3. RESULTS AND ANALYSIS From the parameters previously defined, GMAT is able to run a simulation for both the chemical rocket and the NTR. The mission summaries for both rockets are provided in appendix B, while the results of the mission are described below in the Mission Simulation section. After the results of the mission have been explained, the Mission Analysis section examines the different characteristics of the rockets and describes graphically the corresponding advantages and disadvantages. 3.1. Mission Simulation After running the GMAT script, the following trajectory plots are created. Figure 3 shows the trajectory plots for the NTR and chemical rockets. These plots are important because they show the successful maneuvers and orbit captures of each rocket. A maneuver is the time during which the rocket fires its engines to change its orbit trajectory. An orbit capture is a specific maneuver in which the rocket fires its engines in order to provide the required thrust to change trajectory and begin orbiting an object. Figure 3 shows that the NTR and chemical rocket began orbiting Mars on September 19th, 2014 and September 11th, 2014 respectively, which means each rocket completed the mission within Figure 3. Trajectory plot of the NTR from LEO to LMO from November 16th, 2013 to September 19th, 2014. The red line plots the path of the rocket, the green line indicates Earth’s orbit, while the yellow line plots the orbit of Mars around the sun. 15 the ten-month timeframe. The plots also show that both missions complete a successful “Hohmann Transfer” maneuver from Earth to Mars. A Hohmann Transfer is utilized because it allows the rocket to perform propulsive maneuvers at the perigee of the different rocket orbits. This allows the rocket to minimize the required amount of energy used for each maneuver. In this example, the perigee is defined at the location of the rocket’s orbit that is closest to the body being orbited. The next set of plots obtained by GMAT, shown in Figure 4, show the initial maneuvers performed by each rocket in order to provide the rocket the escape velocity required to overcome Earth’s gravitational pull. Notice that instead of one propulsive maneuver, each rocket utilizes six propulsive maneuvers at their respective orbit perigee in order to take advantage of the “Oberth Effect” [16]. For a manned mission, this could be changed to a single maneuver to limit the amount of times the vehicle passes through the Earth’s Van Allen belts. The Oberth Effect is defined as the increase in a rocket’s efficiency due to the rocket thrusting at the point of maximum velocity, which is when the rocket reaches the perigee of the orbit. This increase in efficiency is a result of a greater amount of mechanical energy generated if the change in velocity occurs when the object is already traveling at Figure 4. Trajectory plots for the first six propulsive maneuvers around the earth. The red line indicated the rockets path towards apogee, while the blue line indicates the rockets path back towards perigee. The yellow line close to the earth indicates the duration of the path during which the rocket is thrusting. This thrust period is known as burn time. Each burn time in shown in a chart below. 16 high velocities. The increase in the amount of mechanical energy generated from rockets traveling at higher velocities is due to the velocity term in the kinetic energy equation shown below. 1 1 2 2 𝐾𝐸 = ∗ 𝑚 ∗ (𝑣 𝑖 + Δ𝑣)2 = ∗ 𝑚 ∗ (𝑣𝑖2 + 2𝑣𝑖 Δ𝑣 + Δ𝑣 2 ) ( 17) As equation 17 demonstrates, the change in KE will be greater for maneuvers that have greater initial velocities due to the 2*vi*Δv term. The final propulsive maneuver in Figure 4 is performed to provide the rocket with an eccentricity greater than one, which allows the rocket to propagate to the required transfer correction maneuver (TCM) location. The burn times for each rocket are indicated by the yellow section of the trajectory plot. Due to the difference in each rocket’s Isp, the rockets expel a different amount of propellant every second. While a higher Isp allows the NTR to use less propellant mass during the mission, it also means that the chemical rocket will weigh less during each maneuver because it is using more fuel. The different rate at which propellant is expelled means that the rockets will have different weights and consequently different burn times for each maneuver. The difference in burn times for each maneuver is described in Table 2. It is important to consider the burn time for each maneuver. As the burn time increases, the rocket becomes less efficient relative to the Oberth Effect because the rocket begins to slow down from gravitational pull. Table 2. This table shows the total burn time for each maneuver. Notice that the chemical rocket’s burn times are less because it uses much more fuel for each maneuver, and consequently has less mass to propel. Maneuver 1: Raise apogee to 10,000km Maneuver 2: Raise apogee to 15,000km Maneuver 3: Raise apogee to 25,000km Maneuver 4: Raise apogee to 40,000km Maneuver 5: Raise apogee to 100,000km Maneuver 6: Achieve ECC of 1.22 Maneuver 7: TCM Maneuver 8: MOI NTR Burn Time Chemical Rocket Burn Time Units 413.44 537.24 498.78 312.38 342.34 612.77 38.35 987.94 402.77 491.54 424.25 250.55 260.59 429.26 56.52 573.69 sec sec sec sec sec sec sec sec 17 While Table 2 shows that the NTR has burn times that are significantly higher than the chemical rocket, the GMAT simulation and trajectory plots confirm that the burn times for the NTR are relatively small compared to the orbit period, which allows for minimal loss in efficiency. It is important to note that for missions with longer burn times, the rocket will require greater T/W ratios in order to minimize the burn time and energy losses due to gravity, and to maximize the efficiencies gained from the Oberth Effect. The final plot created in GMAT is the Mars orbit insertion (MOI) maneuver. As shown in Table 2, the MOI maneuver requires the greatest burn time. The dramatic increase in burn time is due to the required velocity change the rocket must provide during the Mars orbit capture. Unlike the first six maneuvers around Earth, the MOI can only have one propulsive maneuver, otherwise the rocket will miss the planet. Each rocket’s burn time with relation to the total orbit path around Mars is shown by the yellow line in Figure 5. Figure 5. Plot of the MOI maneuver and orbit around Mars for each rocket. Notice the burn time plotted in yellow for each rocket, while the red line indicates the rocket trajectory and orbit path around Mars. 18 3.2. Mission Analysis With the trajectory, mission durations, and final fuel masses now known, the next step is to compare the two rockets using the previously defined terms in the “Methods” section. The main terms to focus on are the specific impulse, payload ratio, structure coefficient, gross mass coefficient, engine weight, and propellant mass. Table 3 uses each equation to compare the results of the rockets side-by-side. Table 3. Numeric comparison of the chemical rocket with an NTR for a mission from LEO to LMO. Many of the terms come from Kirk Sorensen “Rocket Design” derivation [13], and NASA’s rocket pages [11]. LH2 Density ρfuel LOX Density ρox LH2 Tank Factor (MERFT) LOX Tank Factor (MEROT) Ullage Factor (fullage) Thrust Structure Factor (fTSF) Stage Radius Dome Factor Delta-V (Δv) Initial T/W Rocket (T/W)initial Gross Mass (mgross) Specific Impulse (Isp) Mixture Ratio (MXR) Engine Thrust Engine Weight (WEngine) Engine Vacuum T/W (T/W)Engine-Vacuum Structural Coefficient (ε) Mass Ratio (MR) Propellant Mass Fraction (PMF) Gross Mass Coefficient (ϕ) Payload Ratio (λ) Propellant Mass (mprop) Payload Mass (mpl) Propellant Sensitive Mass (Tanks) Fuel Mass (LH2) Oxidizer Mass (LOX) Fuel Tank Volume (VFT) Oxidizer Tank Volume (VOT) Fuel Tank Mass (mFT) Oxidizer Tank Mass (mOX) Gross Sensitive Mass (Engines / mTS) Required Thrust (Thrust total) Number of Engines (n) Total Engine Weight (n*WEngine) Thrust Structure Weight (WTS) Chemical Rocket 71.25 1,141.00 9.48 12.88 0.03 0.003 4.20 .71 5.60 0.11 300,000.00 449.70 5.50 101,819.76 1,868.25 54.50 .0309 3.56 0.72 0.0023 0.2564 215,699.37 76,920.48 6,675.65 33,184.52 182,514.85 480.17 164.91 4,551.83 2,123.82 704.5 323,730.00 3.18 5,940.00 971.19 NTR 71.25 n/a 9.48 n/a 0.03 0.003 4.20 .71 5.60 0.11 300,000.00 900.00 0 111,205.50 40,245.33 2.76 .1372 1.89 0.47 0.0401 0.426 140,903.80 127,727.11 19,327.38 140,903.80 n/a 2,038.82 n/a 19,327.38 n/a 12,041.72 323,730.00 2.91 117,158.07 971.19 Units kg/m3 kg/m3 kg/m3 kg/m3 m km/s kg sec N N kg kg kg kg kg m3 m3 kg kg kg N N N 19 From Table 3, it becomes clear that for a mission to Mars from LEO to LMO, the NTR is able to deliver nearly double the payload mass to LMO when the (T/W)initial for both rockets is set as 0.11. The table also highlights the advantages and disadvantages of each rocket, which are explained below. It is important to note that this table includes all of the same parameters as the mission validated by the GMAT simulation. The key factor that allows the NTR to deliver almost double the payload to Mars when compared to a chemical rocket is the Pewee engine’s Isp of 900 s, which is double that of a chemical rocket engine’s Isp of 449.70 s. With a higher Isp, the NTR requires less propellant because each kilogram of propellant provides more impulse, as defined by Isp. As mentioned in the earlier, the Isp is the total impulse divided by the weight of the propellant. This can be expressed in terms of the exhaust velocity to provide a more intuitive understanding of why the NTR requires less fuel. The following derivation defines Isp in terms of exhaust velocity (vexh). 𝐼𝑠𝑝 = 𝑇𝑜𝑡𝑎𝑙⁡𝐼𝑚𝑝𝑢𝑙𝑠𝑒 𝑊𝑒𝑖𝑔ℎ𝑡 = 𝐼 𝑚∗𝑔0 = 𝑚∗𝑣𝑒𝑥ℎ 𝑚∗𝑔0 = 𝑣𝑒𝑥ℎ 𝑔0 ( 18 ) Using this derivation to solve for vexh, the NTR has an exhaust velocity that is twice that of the chemical rocket. With a greater vexh, and the total thrust of both rockets equal, the NTR requires a smaller mass flow rate (mdot) to achieve the same amount of thrust. This is shown by the derivation of the thrust equation in vacuum, with F1 equal to F2, which are the force thrust for the NTR and chemical rocket respectively. 𝑡ℎ𝑟𝑢𝑠𝑡 = 𝐹 = 𝑚𝑑𝑜𝑡 ∗ 𝑣𝑒𝑥ℎ ( 19 ) 𝐹1 = 𝐹2 → 𝑚𝑑𝑜𝑡1 ∗ 𝑣𝑒𝑥ℎ1 = 𝑚𝑑𝑜𝑡2 ∗ 𝑣𝑒𝑥ℎ2 ( 20 ) 𝑣𝑒𝑥ℎ1 > 𝑣𝑒𝑥ℎ2 → 𝑚𝑑𝑜𝑡1 < 𝑚𝑑𝑜𝑡2 ( 21 ) 20 As Equation 20 illustrates, the mdot of the NTR must be smaller than the mdot of the chemical rocket. The exact value for each rocket’s mass flow rate can be solved for with the following derivation. 𝑚𝑑𝑜𝑡 = 𝑑𝑚 𝑑𝑡 = 𝑡ℎ𝑟𝑢𝑠𝑡𝑒𝑛𝑔𝑖𝑛𝑒 𝐼𝑠𝑝 ∗𝑔0 ∗𝑛= 𝑡ℎ𝑟𝑢𝑠𝑡𝑡𝑜𝑡𝑎𝑙 𝐼𝑠𝑝 ∗𝑔0 ( 22 ) Using Equation 20, the mass flow rate for the NTR is 36.67 kg/s, while the chemical rocket is 73.38 kg/s for the same amount of total thrust. Multiplying these terms by the total burn time of each rocket shows that the total propellant mass used during the mission will be greater for the chemical rocket than the NTR. The difference in the total amount of propellant carried during the mission is shown in Table 3 in the mprop column. Table 3 shows that the total propellant mass for the chemical rocket is 215,699.37 kg, while the total propellant mass for the NTR is only 140,903.80 kg. GMAT confirms this calculation showing that both propellant masses have been expelled by the end of the mission. It is important to note that the (T/W)initial ratio is set to 0.11 for this mission. This is an important consideration because as (T/W)initial increases, the total thrust required will also increase, which will require the number of Pewee engines on the NTR to increase. For a (T/W)initial of 0.11, the NTR only requires approximately three engines as shown in Table 3. If the (T/W)initial requirement were to increase, the NTR would become less efficient because more of the payload mass is replaced by the mass of the engines. This conclusion will be shown graphically later in the analysis. While the NTR has shown that it is more efficient than a chemical rocket from LEO to LMO under certain mission specifications such as a (T/W)initial of 0.11, Table 3 shows that the NTR also has many disadvantages. The first disadvantage that can be seen in Table 3 is the engine weight. The weight of the NTR engine is 21.54 times greater than that of the 21 chemical rocket. This disadvantage can also be understood by the gross mass-sensitive term ϕ. For this analysis, the ϕ term for the NTR is .0401, which is greater than the chemical rocket’s ϕ factor by an order of magnitude. This means that more of the payload mass for an NTR is replaced by engine mass and thrust structure than is the case for the chemical rocket. Though the NTR has greater engine and thrust structure masses, it is still able to carry more payload because it requires much less total propellant mass. The other factor that can be evaluated to understand the disadvantages of the NTR compared to the chemical rocket is the structural coefficient ε. Similar to ϕ, the epsilon factor for the NTR is almost an order of magnitude greater than the chemical rocket’s ε. By analyzing the derivation for ε in Equation 12, it becomes evident that the NTR is disadvantaged by the low density of LH2. The Pewee engine only utilizes LH2 because it provides propulsion through the expansion of LH2, not combustion of a fuel and oxidant utilized in chemical rockets. The fact that the mox term in epsilon is zero, means that the structure coefficient is only dependent upon the mass and density of LH2. The impact of only using LH2 as the propellant is that it requires massive tanks to store the low density substance, which is shown by a higher epsilon factor. This is seen in the comparative total masses of the propellant tanks for each rocket. The chemical rocket has a total tank mass of 6,676.65 kg, while the NTR has a tank mass of 19,327.38 kg. The derivation that summarizes these advantages and disadvantages of the NTR is the payload ratio (λ) in Equation 9. This derivation shows that λ is a function of PMF, ε, and ϕ. It is possible to create a graphical representation of the effects of the (T/W)initial ratio on the payload ratio by utilizing Equation 9, and a delta-velocity requirement of 5.6 km/s 22 Figure 6. This chart demonstrates the effects of PMF, ε, and ϕ on the payload ratio as the initial thrust-toweight ratio is varied from .01 to 1.0. Notice that because the NTR has a much smaller PMF ratio, lambda is greater for low T/W ratios, but as the T/W ratio requirements increase towards 1.0, the effects of epsilon and phi dramatically decrease the NTR’s payload ratio, and the chemical rocket become more efficient than the NTR. Also notice how the slope of the chemical rocket only varies slightly. The small slope of the chemical rocket in comparison to the NTR is due to the light weight engine size of the rocket, along with the use of the denser LOX propellant. Figure 6 includes the effects of PMF, ε, and ϕ on the payload ratio (λ) as (T/W)initial is varied from .01 to 1. This data is obtained by varying the (T/W)initial field in Table 3, and plotting the data. There are three very intriguing aspects that Figure 6 illustrates in relation to the rocket’s comparison from Table 3. The first important aspect illustrated by Figure 6 is the NTR’s superior payload fractions for lower T/W ratios. The NTR’s better payload ratio is to be expected because nuclear reactions provide a great amount of additional energy per unit mass, resulting in a higher vexh and Isp when compared with combustion energy. The NTR’s negative slope can be understood by the ε and ϕ terms in the payload ratio 𝜆 = 1 − 𝑃𝑀𝐹 − 𝜀(𝑃𝑀𝐹) − 𝜙. The ε term has a relativity small effect on λ because it is multiplied by PMF, while the ϕ term can have a large effect on λ because it is not decreased by PMF. The large effect of ϕ on 23 λ is directly seen in Figure 6. As the (T/W)initial increases, it causes a large increase in ϕ, and the λ ratio of the NTR drops. The requirement for an NTR engine to have a low (T/W)Engine ratio is one of its main limiting factors. Thus the only missions for which an NTR would be a viable system in comparison to a chemical rocket are missions that require an overall T/W ratio less than 0.6. This (T/W)initial requirement of less than 0.6 guarantees that with current NTR technology, the NTR will never be a viable system for launches from the Earth’s surface. On the other hand, this same T/W requirement ensures that the NTR design is viable for missions similar to the mission from LEO to LMO which have (T/W)initial requirements less than 0.6. The second important aspect illustrated in Figure 6 is the slope of the lines for the chemical rocket and NTR. The small slope of the chemical rocket is important because it shows that for any changes in the (T/W)initial, the overall payload is only affected slightly. The chemical rocket’s relatively constant slope leads to an advantage at or above a T/W ratio of 0.6 because the chemical rocket is able to carry more payload than the NTR. The advantage is related to the chemical rocket’s engine which has a small mass and provides a relatively large amount of thrust for its size. The chemical rocket’s small engine mass means that when the design requires adding engines to the rocket to increase its thrust, there will not be a dramatic change in the payload fraction of the rocket. Thus for missions that require large Δv and high (T/W)initial, a chemical rocket is a better option because it is not disadvantaged by large engines or tank masses. The steep slope of the NTR is important because it demonstrates that the NTR will benefit greatly from any advancements in the system’s technology. The NTR could become much more efficient with advances in engine designs leading to higher Isp and lower ε and ϕ factors. Advances 24 in the types of propellant used, the weight of the engines and nuclear reactors, and the maximum temperature of the engines could make the NTR a more viable alternative for space transport. On the other hand, improvements in the chemical rocket ε or ϕ factor cannot greatly affect the payload ratio because the slope of the line in Figure 6 is nearly horizontal. The only major improvement to a chemical rocket would come from an increase in Isp, but this would not likely change the payload ratio in any significant way, because the chemical rocket’s research and development is already relatively mature. The third important aspect illustrated in Figure 6 is the (T/W)initial point of 0.6 at which the NTR is able to carry a greater payload than the chemical rocket. It is at this point that the NTR option for space transport begins to be intriguing, but other considerations such as the costs to design the rocket still need to be considered. In Figure 7, the 0.6 point can be used as the starting point to demonstrate at what (T/W)initial the NTR will carry enough extra payload to offset any disadvantages it may face in costs, legal, and safety issues— such as using highly enriched uranium for the rocket’s nuclear reactor. Many of the barriers Figure 7. Shows the extra payload carried by an NTR for a mission from LEO to LMO with a delta velocity of 5.6(km/s) at different (T/W) ratios. 25 associated with the rocket’s use of highly enriched uranium require extra development, transportation, and launch procedures needed in order to ensure the safety of workers and communities near the testing areas. This study selected the 0.11 T/W ratio point because as Figure 7 shows, the NTR carries an extra 50,807 kg of payload when compared to the chemical rocket. It is at this point that the study assumes that the extra payload carried can offset the cost of barriers of a legal, safety, and economic nature. In order to assess if the 0.11 T/W ratio point allows the NTR to become more economically viable than the chemical rocket, a payload cost difference analysis can be performed using current and past budgets set for similar missions of the two rocket types. 3.3. Payload Cost Difference By comparing the Space Launch System (SLS), NASA’s proposed “$23 billion” chemical rocket, and the NTR, a basic payload cost analysis can be developed to determine the economic viability of the NTR. NASA states that the SLS will be “the world’s most powerful rocket”, and will launch astronauts to “an asteroid and eventually Mars” [17]. NASA is planning to build multiple configurations of the SLS. The initial configuration, which will be used in this comparison, will deliver 70 metric tons (mton) to LEO for a Lunar flyby mission. The U.S. Government Accountability Office (GAO) states that this configuration is “estimated to cost nearly $23 billion to demonstrate initial capabilities” by the year 2022 [1]. This cost includes designing, building, and launching the boosters, core stage, Orion spacecraft, interim cryogenic propulsion stage (ICPS), the flight systems, as well as all of the infrastructure needed for the agency to complete the SLS launch by 2022. It is important to note that the ICPS is the stage of the SLS that is launched once the SLS reaches LEO. The NTR is similar to the ICPS in that it is launched from LEO, and requires 26 many of the same flight systems. The only major design differences between an NTR and the ICPS are the nuclear thermal rocket engines and larger propellant tanks. Similar to the ICPS, the NTR would also attach to the Orion spacecraft in order to transport the crew during the mission. For the purpose of creating a cost analysis of the NTR by using the SLS mission parameters mentioned above, assume the NTR must cost less than or equal to the $23 billion SLS budget. An advantage of using the NTR is that it does not require many of the costly stages and infrastructures needed for the SLS. For example, the extra costs experienced by the SLS include the launch pad, mobile launcher, boosters, core stage, and the transporter needed to move the SLS. As mentioned above, because the NTR requires a similar design to the ICPS, it will consequently have many similar costs. However, the NTR has additional costs in comparison to the ICPS due to use of nuclear thermal rocket engines and larger propellant tanks. Another added cost for the NTR is the launching of the rocket and payload into LEO. In order to incorporate that launch cost, this study will use prices provided by the private company SpaceX. SpaceX quotes the price of sending 6,400 kg of payload into LEO as $90 million dollars [18]. At this price, the cost of sending the 70 mton NTR into LEO would be approximately $1 billion. This leaves $22 billion still in the budget for developing an NTR with a similar design to the ICPS that incorporates NTR engines and larger propellant tanks. The Nuclear Engine for Rocket Vehicle Application (NERVA) project, which took place between 1962-1972, designed and tested twenty different NTR engines including the Pewee engine used in this study. The cost of the NERVA program, including inflation, is 27 $8.79 billion. With the cost to develop, build, and test 20 different nuclear thermal rocket engines estimated at $9 billion, it is more than reasonable to assume that a new NTR rocket could be developed and built for much less than the previously mentioned $22 billion budget. With these assumptions defined, a required Δv for the SLS Lunar mission must be less than 5.6 km/s with a (T/W)initial of 0.11 in order to use the calculations derived for the Mars mission studied in this paper. With this requirement, a total payload cost difference between the two rockets can be developed. By using the calculations in Table 3, and knowing the IMLEO for the mission is 70 mton, the cost per kilogram of payload for the SLS is approximately $1.2 million, while the cost per kilogram of payload for the NTR is approximately $770 thousand, which is $430 thousand less than the chemical rocket. Table 4 demonstrates that even when assuming the NTR is equal to the cost of the SLS program, which requires a more complex infrastructure, there is a large cost difference between the SLS and NTR payload. The table shows that the NTR provides a payload cost difference of $9.18 billion to send a single payload of 17,948 kg for the planned SLS Lunar mission. This is not an exhaustive cost analysis, but it can be used to provoke thought, and provide insight on the large cost savings an NTR could provide after multiple missions. Table 4. This table provides the numerical analysis of the payload cost difference between the SLS and an NTR for the lunar flyby mission planned for year 2022. The costs are in billions (B) USD, except for the “Payload Cost Per Kilogram” given in millions (M) USD. The comparison assumes the worst case, where the cost of developing, building, and launching the NTR into LEO is equal to the $23 billion SLS budget. SLS NTR IMLE O(kg) Cost to Send to LEO (B) Project Cost (B) Total Cost (B) Payload (kg) Payload Cost per kilogram (M) 70,000 70,000 n/a $.98 $23 $22 $23 $23 17,948 29,803 $1.2 $.77 Cost Difference: Cost of 17,948(kg) Payload (B) $23 $13.82 $9.18 28 Currently NASA uses commercial companies to send payloads to LEO. If NASA were to continue outsourcing its LEO launches to commercial companies, the NTR would become a viable option due to the greater amount of payload carried, which would lead to a significantly cheaper payload cost per kilogram as shown in Table 4. Though this specific example uses figures from NASA and the GOA, the same principle could be applied to all exploration missions throughout the Solar System with (T/W)initial and Δv requirements similar to the Mars mission simulated in this study. As mentioned earlier, 0.11 is not necessarily the optimal (T/W)initial, but it is a point on Figure 6 which provides the NTR with a significantly better payload ratio. The optimal (T/W)initial would need to be calculated for each mission depending on the mission’s specific parameters. With the NTR and chemical rocket missions validated through the GMAT simulation, and the graphical and numeric cost analysis for each rocket’s payloads, it is evident that the NTR has great potential for advancing exploration within the Solar System. Though the NTR has shown that it is superior for certain types of missions, there is still much development, safety considerations, and economic analysis that must be done to definitively conclude that the NTR is a viable option and should become an integral vehicle for space transport. The final section explores the future research and development that must take place in order to use NTR’s for future missions. 29 4. CONCLUSION Though the NTR still requires many years of design and testing, the system offers great potential for opening new possibilities in space exploration. The analysis has theoretically shown that the NTR system launching from LEO can have payload ratios 40% greater than that of a chemical rocket for missions with low initial thrust-to-weight ratio requirements. The significant factor impacting the efficiency of an NTR system is the higher exhaust velocity, which allows the NTR to obtain an Isp that is double the chemical rocket’s Isp. The limiting factor for the NTR is its large engine mass. This causes a dramatic decrease in the NTR’s payload capacity for large T/W ratios. Though the NTR’s efficiency is hindered by the engine mass, it is still a viable option for space exploration because these missions require smaller T/W ratios. During missions similar to the mission analyzed in this study, the NTR can provide large payload cost differences when compared to the chemical rocket, assuming development costs are similar. The payload cost difference for the mission analysis in this study is approximately $9.18 billion dollars, based upon previously selected assumptions. This cost difference, along with the numeric and graphic analysis, help show that at low T/W ratios, the NTR’s potential to provide economic and efficiency advantages over the chemical rocket in space exploration is significant. There are currently multiple organizations such as the Center for Space Nuclear Research in Idaho [19] and the “Cryogenic Nuclear Propulsion Stage” project being developed by NASA which are working towards improving the efficiency and cost of the NTR [3]. These organizations are focusing on creating nuclear rocket engines that can be applied commercially by using commercial grade uranium instead of the highly enriched uranium used in the Pewee engine. NASA is also testing new research techniques in order 30 to decrease the cost of developing and testing future NTR technology. These decreases in costs are emphasized by a quote from the manager of the Marshall testing facility when he said, “The cost savings is remarkable. Whereas it costs tens of millions of dollars to perform full-scale testing of nuclear rocket fuel elements in specially designed nuclear reactors, our research costs just tens of thousands – and no radiation protection is required!” [3]. This quote highlights the fact that from current advancements in technology, and new testing procedures, the NTR shows great potential for improvements in efficiency, safety, and cost. This is not true for the chemical rocket, as it is based on combustion engine technology, which has likely reached a relative maximum point due to the years of development, testing, and use the engine has undergone. It is the recommendation of this study that an in-depth analysis of the total cost to design, build and launch an NTR be performed. With an in-depth cost analysis, along with the data in this study, companies and governments will have the ability to understand how NTR technology can be applied to provide economic benefits for space exploration. If a cost analysis is performed, and the NTR provides overall savings when compared to a chemical rocket, it should be seriously considered for the main propulsion system used in future space exploration missions throughout the Solar System. 31 5. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] GAO, "Space Launch System Management Tools Should Better Track to Cost and Schedule Commitment to Adequately Monitor Increasing Risk," 2015. Universities Space Research Association, "Center For Space Nuclear Research," [Online]. Available: http://csnr.usra.edu/public/default.cfm?content=347. [Accessed April 2016]. NASA, "NASA's Marshall Center's 'NTREES' Facility Tests 'Ticket to Mars' Technologies," [Online]. Available: http://www.nasa.gov/centers/marshall/news/news/releases/2014/ntrees.html. NASA, "Dr. Robert H. Goddard, American Rocketry Pioneer," 18 April 2015. [Online]. Available: https://www.nasa.gov/centers/goddard/about/history/dr_goddard.html. [Accessed 18 April 2015]. Aerojet Rocketdyne, "RL10 Propulsion System," February 2016. [Online]. Available: https://rocket.com/files/aerojet/documents/Capabilities/PDFs/RL10%20data%20s heet%20Feb%202016.pdf. [Accessed 6 April 2016]. National Aeronautics and Space Administration, "Overview of Rover Engine Program Tests Final Report," George C. Marshall Space Flight Center, Huntsville, 1991. Los Alamos National Labroratory, "A Half Century of Los Alamos in Space," National Security Science, no. 1 2011 LALP-11-015, 2011. M. G. Houts, "Space Nuclear Power and Propulsion," in AIAA, 2014. NASA, "GMAT," [Online]. Available: https://gmat.gsfc.nasa.gov/. [Accessed April 2016]. T. K. W. J. E. R. R. H. J. W. B. H. P. G. G. E. D. R. B. A. Michael G Houts, "M132984_Paper.txt," 2013. [Online]. Available: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20140003221.pdf. [Accessed 6 April 2016]. NASA, "Ideal Rocekt Equation," 2014. [Online]. Available: https://spaceflightsystems.grc.nasa.gov/education/rocket/rktpow.html. [Accessed April 2016]. NASA, "Mass Ratios," October 2014. [Online]. Available: https://spaceflightsystems.grc.nasa.gov/education/rocket/rktwtp.html. [Accessed April 2016]. K. Sorensen, "Selenian Boondocks," 2010. [Online]. Available: http://selenianboondocks.com/category/rocket-design-theory/. [Accessed April 2016]. NASA Langley Research Center, "tankMERs," 2010. [Online]. Available: http://selenianboondocks.com/2010/02/calculating-propellant-mass-sensitiveterm/tankmers/. [Accessed April 2016]. 32 [15] K. Sorensen, "Calculating Gross-Mass-Sensitive Term," 2010. [Online]. Available: http://selenianboondocks.com/2010/02/calculating-gross-mass-sensitive-term/. [Accessed April 2016]. [16] Wikipedia, "Oberth Effect," [Online]. Available: https://en.wikipedia.org/wiki/Oberth_effect. [Accessed April 2016]. [17] NASA, "Space Launch System (SLS) Overview," 10 February 2016. [Online]. Available: https://www.nasa.gov/exploration/systems/sls/overview.html. [Accessed April 2016]. [18] SpaceX, "Capabilities and Services," [Online]. Available: http://www.SpaceX.com/about/capabilities. [Accessed 2016]. [19] Center For Space Nuclear Research, [Online]. Available: http://csnr.usra.edu/public/default.cfm?content=347. [Accessed April 2016]. [20] NASA, "Name of Image: Orbiter Atlantis (STS-110) Launch with New Block II Engines," [Online]. Available: https://mix.msfc.nasa.gov/abstracts.php?p=2388. [Accessed 6 April 2016]. [21] B. W. Samm Wagner, "Low-Thrust Trajectory Optimization for Asteroid Exploration, Redirect, and Deflection Missions," AAS, 2011. [22] S. D. Howe, "Assessment of the Advantages and Feasibility of a Nuclear Rocket for a Manned Mars Mission," 1985. 33 APPENDIX A The following GMAT script code can be copied into the “General Mission Analysis Tool” (GMAT) and run to validate mission parameters and convergence from low Earth orbit to Mars orbit. For further information about GMAT, NASA provides web-sites and documentation. The following was obtained from the GMAT home website page, “GMAT is developed by a team of NASA, private industry, and public and private contributors. It is used for real-world engineering studies, as a tool for education and public engagement, and to ply operational spacecraft” [9]. %General Mission Analysis Tool(GMAT) Script for NTR %---------------------------------------%---------- Spacecraft %---------------------------------------Create Spacecraft MarsSC; GMAT MarsSC.DateFormat = UTCGregorian; GMAT MarsSC.Epoch = '16 Nov 2013 21:55:24.315'; GMAT MarsSC.CoordinateSystem = EarthMJ2000Eq; GMAT MarsSC.DisplayStateType = Cartesian; GMAT MarsSC.X = 7100; GMAT MarsSC.Y = 100; GMAT MarsSC.Z = -2000; GMAT MarsSC.VX = 0; GMAT MarsSC.VY = 7.35; GMAT MarsSC.VZ = 1; GMAT MarsSC.DryMass = 159096.2; GMAT MarsSC.Cd = 2.2; 34 GMAT MarsSC.Cr = 1.8; GMAT MarsSC.DragArea = 1133.95; GMAT MarsSC.SRPArea = 996.0876273; GMAT MarsSC.Tanks = {MainTank}; GMAT MarsSC.Thrusters = {Thruster, ThrusterMOI}; GMAT MarsSC.NAIFId = -123456789; GMAT MarsSC.NAIFIdReferenceFrame = -123456789; GMAT MarsSC.OrbitColor = Red; GMAT MarsSC.TargetColor = Teal; GMAT MarsSC.Id = 'SatId'; GMAT MarsSC.Attitude = CoordinateSystemFixed; GMAT MarsSC.SPADSRPScaleFactor = 1; GMAT MarsSC.ModelFile = '../data/vehicle/models/aura.3ds'; GMAT MarsSC.ModelOffsetX = 0; GMAT MarsSC.ModelOffsetY = 0; GMAT MarsSC.ModelOffsetZ = 0; GMAT MarsSC.ModelRotationX = 0; GMAT MarsSC.ModelRotationY = 0; GMAT MarsSC.ModelRotationZ = 0; GMAT MarsSC.ModelScale = 1; GMAT MarsSC.AttitudeDisplayStateType = 'Quaternion'; GMAT MarsSC.AttitudeRateDisplayStateType = 'AngularVelocity'; GMAT MarsSC.AttitudeCoordinateSystem = EarthMJ2000Eq; GMAT MarsSC.EulerAngleSequence = '321'; %---------------------------------------%---------- Hardware Components %---------------------------------------Create ChemicalTank MainTank; 35 GMAT MainTank.AllowNegativeFuelMass = true; GMAT MainTank.FuelMass = 140903.8; GMAT MainTank.Pressure = 5000; GMAT MainTank.Temperature = 20; GMAT MainTank.RefTemperature = 20; GMAT MainTank.Volume = 2038.82; GMAT MainTank.FuelDensity = 71.25; GMAT MainTank.PressureModel = PressureRegulated; Create ChemicalThruster Thruster; GMAT Thruster.CoordinateSystem = Local; GMAT Thruster.Origin = Earth; GMAT Thruster.Axes = VNB; GMAT Thruster.ThrustDirection1 = 1; GMAT Thruster.ThrustDirection2 = 0; GMAT Thruster.ThrustDirection3 = 0; GMAT Thruster.DutyCycle = 1; GMAT Thruster.ThrustScaleFactor = 1; GMAT Thruster.DecrementMass = true; GMAT Thruster.Tank = {MainTank}; GMAT Thruster.GravitationalAccel = 9.800000000000001; GMAT Thruster.C1 = 323730; GMAT Thruster.C2 = 0; GMAT Thruster.C3 = 0; GMAT Thruster.C4 = 0; GMAT Thruster.C5 = 0; GMAT Thruster.C6 = 0; GMAT Thruster.C7 = 0; GMAT Thruster.C8 = 0; 36 GMAT Thruster.C9 = 0; GMAT Thruster.C10 = 0; GMAT Thruster.C11 = 0; GMAT Thruster.C12 = 0; GMAT Thruster.C13 = 0; GMAT Thruster.C14 = 0; GMAT Thruster.C15 = 0; GMAT Thruster.C16 = 0; GMAT Thruster.K1 = 900; GMAT Thruster.K2 = 0; GMAT Thruster.K3 = 0; GMAT Thruster.K4 = 0; GMAT Thruster.K5 = 0; GMAT Thruster.K6 = 0; GMAT Thruster.K7 = 0; GMAT Thruster.K8 = 0; GMAT Thruster.K9 = 0; GMAT Thruster.K10 = 0; GMAT Thruster.K11 = 0; GMAT Thruster.K12 = 0; GMAT Thruster.K13 = 0; GMAT Thruster.K14 = 0; GMAT Thruster.K15 = 0; GMAT Thruster.K16 = 0; Create ChemicalThruster ThrusterMOI; GMAT ThrusterMOI.CoordinateSystem = Local; GMAT ThrusterMOI.Origin = Mars; 37 GMAT ThrusterMOI.Axes = VNB; GMAT ThrusterMOI.ThrustDirection1 = -1; GMAT ThrusterMOI.ThrustDirection2 = 0; GMAT ThrusterMOI.ThrustDirection3 = 0; GMAT ThrusterMOI.DutyCycle = 1; GMAT ThrusterMOI.ThrustScaleFactor = 1; GMAT ThrusterMOI.DecrementMass = true; GMAT ThrusterMOI.Tank = {MainTank}; GMAT ThrusterMOI.GravitationalAccel = 9.800000000000001; GMAT ThrusterMOI.C1 = 323730; GMAT ThrusterMOI.C2 = 0; GMAT ThrusterMOI.C3 = 0; GMAT ThrusterMOI.C4 = 0; GMAT ThrusterMOI.C5 = 0; GMAT ThrusterMOI.C6 = 0; GMAT ThrusterMOI.C7 = 0; GMAT ThrusterMOI.C8 = 0; GMAT ThrusterMOI.C9 = 0; GMAT ThrusterMOI.C10 = 0; GMAT ThrusterMOI.C11 = 0; GMAT ThrusterMOI.C12 = 0; GMAT ThrusterMOI.C13 = 0; GMAT ThrusterMOI.C14 = 0; GMAT ThrusterMOI.C15 = 0; GMAT ThrusterMOI.C16 = 0; GMAT ThrusterMOI.K1 = 900; GMAT ThrusterMOI.K2 = 0; GMAT ThrusterMOI.K3 = 0; 38 GMAT ThrusterMOI.K4 = 0; GMAT ThrusterMOI.K5 = 0; GMAT ThrusterMOI.K6 = 0; GMAT ThrusterMOI.K7 = 0; GMAT ThrusterMOI.K8 = 0; GMAT ThrusterMOI.K9 = 0; GMAT ThrusterMOI.K10 = 0; GMAT ThrusterMOI.K11 = 0; GMAT ThrusterMOI.K12 = 0; GMAT ThrusterMOI.K13 = 0; GMAT ThrusterMOI.K14 = 0; GMAT ThrusterMOI.K15 = 0; GMAT ThrusterMOI.K16 = 0; %---------------------------------------%---------- ForceModels and Propagators %---------------------------------------Create ForceModel DefaultProp_ForceModel; GMAT DefaultProp_ForceModel.CeNTRalBody = Earth; GMAT DefaultProp_ForceModel.PointMasses = {Earth}; GMAT DefaultProp_ForceModel.Drag = None; GMAT DefaultProp_ForceModel.SRP = Off; GMAT DefaultProp_ForceModel.RelativisticCorrection = Off; GMAT DefaultProp_ForceModel.ErrorCoNTRol = RSSStep; Create ForceModel DeepSpace_ForceModel; GMAT DeepSpace_ForceModel.CeNTRalBody = Sun; GMAT DeepSpace_ForceModel.PointMasses = {Earth, Jupiter, Luna, Mars, Neptune, Saturn, Sun, Uranus, Venus}; GMAT DeepSpace_ForceModel.Drag = None; 39 GMAT DeepSpace_ForceModel.SRP = On; GMAT DeepSpace_ForceModel.RelativisticCorrection = Off; GMAT DeepSpace_ForceModel.ErrorCoNTRol = RSSStep; GMAT DeepSpace_ForceModel.SRP.Flux = 1367; GMAT DeepSpace_ForceModel.SRP.SRPModel = Spherical; GMAT DeepSpace_ForceModel.SRP.Nominal_Sun = 149597870.691; Create ForceModel NearEarth_ForceModel; GMAT NearEarth_ForceModel.CeNTRalBody = Earth; GMAT NearEarth_ForceModel.PrimaryBodies = {Earth}; GMAT NearEarth_ForceModel.PointMasses = {Luna, Sun}; GMAT NearEarth_ForceModel.Drag = None; GMAT NearEarth_ForceModel.SRP = On; GMAT NearEarth_ForceModel.RelativisticCorrection = Off; GMAT NearEarth_ForceModel.ErrorCoNTRol = RSSStep; GMAT NearEarth_ForceModel.GravityField.Earth.Degree = 8; GMAT NearEarth_ForceModel.GravityField.Earth.Order = 8; GMAT NearEarth_ForceModel.GravityField.Earth.PotentialFile = 'JGM2.cof'; GMAT NearEarth_ForceModel.GravityField.Earth.EarthTideModel = 'None'; GMAT NearEarth_ForceModel.SRP.Flux = 1367; GMAT NearEarth_ForceModel.SRP.SRPModel = Spherical; GMAT NearEarth_ForceModel.SRP.Nominal_Sun = 149597870.691; Create ForceModel NearMars_ForceModel; GMAT NearMars_ForceModel.CeNTRalBody = Mars; GMAT NearMars_ForceModel.PrimaryBodies = {Mars}; GMAT NearMars_ForceModel.PointMasses = {Sun}; GMAT NearMars_ForceModel.Drag = None; GMAT NearMars_ForceModel.SRP = On; 40 GMAT NearMars_ForceModel.RelativisticCorrection = Off; GMAT NearMars_ForceModel.ErrorCoNTRol = RSSStep; GMAT NearMars_ForceModel.GravityField.Mars.Degree = 8; GMAT NearMars_ForceModel.GravityField.Mars.Order = 8; GMAT NearMars_ForceModel.GravityField.Mars.PotentialFile = 'Mars50c.cof'; GMAT NearMars_ForceModel.SRP.Flux = 1367; GMAT NearMars_ForceModel.SRP.SRPModel = Spherical; GMAT NearMars_ForceModel.SRP.Nominal_Sun = 149597870.691; %---------------------------------------%---------- Propagators %---------------------------------------Create Propagator DeepSpace; GMAT DeepSpace.FM = DeepSpace_ForceModel; GMAT DeepSpace.Type = PrinceDormand78; GMAT DeepSpace.InitialStepSize = 600; GMAT DeepSpace.Accuracy = 1e-012; GMAT DeepSpace.MinStep = 0; GMAT DeepSpace.MaxStep = 864000; GMAT DeepSpace.MaxStepAttempts = 50; GMAT DeepSpace.StopIfAccuracyIsViolated = true; Create Propagator NearEarth; GMAT NearEarth.FM = NearEarth_ForceModel; GMAT NearEarth.Type = RungeKutta89; GMAT NearEarth.InitialStepSize = 600; GMAT NearEarth.Accuracy = 1e-013; GMAT NearEarth.MinStep = 0; GMAT NearEarth.MaxStep = 600; GMAT NearEarth.MaxStepAttempts = 50; 41 GMAT NearEarth.StopIfAccuracyIsViolated = true; Create Propagator NearMars; GMAT NearMars.FM = NearMars_ForceModel; GMAT NearMars.Type = PrinceDormand78; GMAT NearMars.InitialStepSize = 60; GMAT NearMars.Accuracy = 1e-012; GMAT NearMars.MinStep = 0; GMAT NearMars.MaxStep = 86400; GMAT NearMars.MaxStepAttempts = 50; GMAT NearMars.StopIfAccuracyIsViolated = true; %---------------------------------------%---------- Burns %---------------------------------------Create FiniteBurn FiniteTOI_1; GMAT FiniteTOI_1.Thrusters = {Thruster}; GMAT FiniteTOI_1.ThrottleLogicAlgorithm = 'MaxNumberOfThrusters'; Create FiniteBurn Finite_2; GMAT Finite_2.Thrusters = {Thruster}; GMAT Finite_2.ThrottleLogicAlgorithm = 'MaxNumberOfThrusters'; Create FiniteBurn Finite_3; GMAT Finite_3.Thrusters = {Thruster}; GMAT Finite_3.ThrottleLogicAlgorithm = 'MaxNumberOfThrusters'; Create FiniteBurn Finite_4; GMAT Finite_4.Thrusters = {Thruster}; GMAT Finite_4.ThrottleLogicAlgorithm = 'MaxNumberOfThrusters'; Create FiniteBurn Finite_5; GMAT Finite_5.Thrusters = {Thruster}; GMAT Finite_5.ThrottleLogicAlgorithm = 'MaxNumberOfThrusters'; 42 Create FiniteBurn Finite_ECC; GMAT Finite_ECC.Thrusters = {Thruster}; GMAT Finite_ECC.ThrottleLogicAlgorithm = 'MaxNumberOfThrusters'; Create FiniteBurn FiniteTCM; GMAT FiniteTCM.Thrusters = {Thruster}; GMAT FiniteTCM.ThrottleLogicAlgorithm = 'MaxNumberOfThrusters'; Create FiniteBurn FiniteMOI; GMAT FiniteMOI.Thrusters = {ThrusterMOI}; GMAT FiniteMOI.ThrottleLogicAlgorithm = 'MaxNumberOfThrusters'; %---------------------------------------%---------- Coordinate Systems %---------------------------------------Create CoordinateSystem SunEcliptic; GMAT SunEcliptic.Origin = Sun; GMAT SunEcliptic.Axes = MJ2000Ec; Create CoordinateSystem MarsInertial; GMAT MarsInertial.Origin = Mars; GMAT MarsInertial.Axes = BodyInertial; %---------------------------------------%---------- Solvers %---------------------------------------Create DifferentialCorrector DC1; GMAT DC1.ShowProgress = true; GMAT DC1.ReportStyle = Normal; GMAT DC1.ReportFile = 'DifferentialCorrectorDC1.data'; GMAT DC1.MaximumIterations = 50; GMAT DC1.DerivativeMethod = ForwardDifference; GMAT DC1.Algorithm = NewtonRaphson; 43 %---------------------------------------%---------- Plots/Reports %---------------------------------------Create OrbitView EarthView; GMAT EarthView.SolverIterations = Current; GMAT EarthView.UpperLeft = [ 0.002305741295826608 0.2737889273356401 ]; GMAT EarthView.Size = [ 0.3652294212589348 0.3460207612456747 ]; GMAT EarthView.RelativeZOrder = 3993; GMAT EarthView.Maximized = false; GMAT EarthView.Add = {Earth, MarsSC}; GMAT EarthView.CoordinateSystem = EarthMJ2000Eq; GMAT EarthView.DrawObject = [ true true ]; GMAT EarthView.DataCollectFrequency = 1; GMAT EarthView.UpdatePlotFrequency = 50; GMAT EarthView.NumPointsToRedraw = 0; GMAT EarthView.ShowPlot = true; GMAT EarthView.ShowLabels = true; GMAT EarthView.ViewPoiNTReference = Earth; GMAT EarthView.ViewPointVector = [ 0 0 30000 ]; GMAT EarthView.ViewDirection = Earth; GMAT EarthView.ViewScaleFactor = 4; GMAT EarthView.ViewUpCoordinateSystem = EarthMJ2000Eq; GMAT EarthView.ViewUpAxis = Z; GMAT EarthView.EclipticPlane = Off; GMAT EarthView.XYPlane = On; GMAT EarthView.WireFrame = Off; GMAT EarthView.Axes = On; GMAT EarthView.Grid = Off; 44 GMAT EarthView.SunLine = Off; GMAT EarthView.UseInitialView = On; GMAT EarthView.StarCount = 7000; GMAT EarthView.EnableStars = On; GMAT EarthView.EnableConstellations = On; Create OrbitView SolarSystemView; GMAT SolarSystemView.SolverIterations = Current; GMAT SolarSystemView.UpperLeft = [ 0.3329490431173622 0.008650519031141869 ]; GMAT SolarSystemView.Size = [ 0.6665898086234725 0.9619377162629758 ]; GMAT SolarSystemView.RelativeZOrder = 4010; GMAT SolarSystemView.Maximized = false; GMAT SolarSystemView.Add = {MarsSC, Earth, Mars, Sun}; GMAT SolarSystemView.CoordinateSystem = SunEcliptic; GMAT SolarSystemView.DrawObject = [ true true true true ]; GMAT SolarSystemView.DataCollectFrequency = 1; GMAT SolarSystemView.UpdatePlotFrequency = 50; GMAT SolarSystemView.NumPointsToRedraw = 0; GMAT SolarSystemView.ShowPlot = true; GMAT SolarSystemView.ShowLabels = true; GMAT SolarSystemView.ViewPoiNTReference = Sun; GMAT SolarSystemView.ViewPointVector = [ 0 0 600000000 ]; GMAT SolarSystemView.ViewDirection = Earth; GMAT SolarSystemView.ViewScaleFactor = 1; GMAT SolarSystemView.ViewUpCoordinateSystem = SunEcliptic; GMAT SolarSystemView.ViewUpAxis = Z; GMAT SolarSystemView.EclipticPlane = Off; GMAT SolarSystemView.XYPlane = On; GMAT SolarSystemView.WireFrame = Off; 45 GMAT SolarSystemView.Axes = On; GMAT SolarSystemView.Grid = Off; GMAT SolarSystemView.SunLine = Off; GMAT SolarSystemView.UseInitialView = On; GMAT SolarSystemView.StarCount = 7000; GMAT SolarSystemView.EnableStars = On; GMAT SolarSystemView.EnableConstellations = On; Create OrbitView MarsView; GMAT MarsView.SolverIterations = Current; GMAT MarsView.UpperLeft = [ 0 0.6219723183391004 ]; GMAT MarsView.Size = [ 0.36914918146184 0.3512110726643599 ]; GMAT MarsView.RelativeZOrder = 3998; GMAT MarsView.Maximized = false; GMAT MarsView.Add = {MarsSC, Mars, Earth}; GMAT MarsView.CoordinateSystem = MarsInertial; GMAT MarsView.DrawObject = [ true true true ]; GMAT MarsView.DataCollectFrequency = 1; GMAT MarsView.UpdatePlotFrequency = 50; GMAT MarsView.NumPointsToRedraw = 0; GMAT MarsView.ShowPlot = true; GMAT MarsView.ShowLabels = true; GMAT MarsView.ViewPoiNTReference = Mars; GMAT MarsView.ViewPointVector = [ 22000 22000 0 ]; GMAT MarsView.ViewDirection = Mars; GMAT MarsView.ViewScaleFactor = 1; GMAT MarsView.ViewUpCoordinateSystem = MarsInertial; GMAT MarsView.ViewUpAxis = Z; GMAT MarsView.EclipticPlane = Off; 46 GMAT MarsView.XYPlane = On; GMAT MarsView.WireFrame = Off; GMAT MarsView.Axes = On; GMAT MarsView.Grid = Off; GMAT MarsView.SunLine = Off; GMAT MarsView.UseInitialView = On; GMAT MarsView.StarCount = 7000; GMAT MarsView.EnableStars = On; GMAT MarsView.EnableConstellations = On; %---------------------------------------%---------- Arrays, Variables, Strings %---------------------------------------Create Variable BurnDuration I ForLoopBT ForLoopBR EpochTCM; GMAT BurnDuration = 0; GMAT ForLoopBT = 0; GMAT ForLoopBR = 0; GMAT EpochTCM = 0; %---------------------------------------%---------- Mission Sequence %---------------------------------------- BeginMissionSequence; %Propagate 'Prop To Periapsis' NearEarth(MarsSC) {MarsSC.Earth.Apoapsis, OrbitColor = [0 0 255]}; Propagate 'Prop 1.6 sec' NearEarth(MarsSC) {MarsSC.ElapsedSecs = 243, OrbitColor = [255 0 255]}; Target 'Raise Apogee 10,000' DC1 ShowProgressWindow = true}; {SolveMode = Solve, ExitMode = SaveAndContinue, 47 Vary 'Vary Burn Duration' DC1(BurnDuration = 413.4406047956213, {Perturbation = 0.0001, Lower = 0.0, Upper = 10000, MaxStep = 20, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); BeginFiniteBurn 'Turn Thruster On' FiniteTOI_1(MarsSC); Propagate 'Prop BurnDuration' NearEarth(MarsSC) {MarsSC.ElapsedSecs = BurnDuration, OrbitColor = [255 255 0]}; EndFiniteBurn 'Turn Thruster Off' FiniteTOI_1(MarsSC); Propagate 'Prop To Apoapsis' NearEarth(MarsSC) {MarsSC.Earth.Apoapsis, OrbitColor = [255 0 0]}; Achieve 'Achieve Apogee Radius = 10,000' DC1(MarsSC.Earth.RMAG = 10000, {Tolerance = 0.005}); EndTarget; % For targeter DC1 Propagate 'Prop To Perigee' NearEarth(MarsSC) {MarsSC.Earth.Periapsis, OrbitColor = [0 0 255]}; Target 'Raise Apogee 15,000' DC1 {SolveMode = Solve, ExitMode = SaveAndContinue, ShowProgressWindow = true}; Vary 'Vary Burn Duration' DC1(BurnDuration = 537.2427453217656, {Perturbation = 0.0001, Lower = 0.0, Upper = 10000, MaxStep = 20, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); BeginFiniteBurn 'Turn Thruster On' Finite_2(MarsSC); Propagate 'Prop BurnDuration' NearEarth(MarsSC) {MarsSC.ElapsedSecs = BurnDuration, OrbitColor = [255 255 0]}; EndFiniteBurn 'Turn Thruster Off' Finite_2(MarsSC); Propagate 'Prop To Apogee' NearEarth(MarsSC) {MarsSC.Earth.Apoapsis}; Achieve 'Achieve Apogee Radius = 15,000' DC1(MarsSC.Earth.RMAG = 15000, {Tolerance = 0.005}); EndTarget; % For targeter DC1 Propagate 'Prop To Perigee' NearEarth(MarsSC) {MarsSC.Earth.Periapsis, OrbitColor = [0 0 255]}; Target 'Raise Apogee 25,000' DC1 {SolveMode = Solve, ExitMode = SaveAndContinue, ShowProgressWindow = true}; Vary 'Vary Burn Duration' DC1(BurnDuration = 498.7815096809487, {Perturbation = 0.0001, Lower = 0.0, Upper = 10000, MaxStep = 50, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); 48 BeginFiniteBurn 'Turn Thruster On' Finite_3(MarsSC); Propagate 'Prop BurnDuration' NearEarth(MarsSC) {MarsSC.ElapsedSecs = BurnDuration, OrbitColor = [255 255 0]}; EndFiniteBurn 'Turn Thruster Off' Finite_3(MarsSC); Propagate 'Prop To Apogee' NearEarth(MarsSC) {MarsSC.Earth.Apoapsis}; Achieve 'Achieve Apogee Radius = 25,000' DC1(MarsSC.Earth.RMAG = 25000, {Tolerance = 0.005}); EndTarget; % For targeter DC1 Propagate 'Prop To Perigee' NearEarth(MarsSC) {MarsSC.Earth.Periapsis, OrbitColor = [0 0 255]}; Target 'Raise Apogee 40,000' DC1 {SolveMode = Solve, ExitMode = SaveAndContinue, ShowProgressWindow = true}; Vary 'Vary Burn Duration' DC1(BurnDuration = 312.3842207244039, {Perturbation = 0.0001, Lower = 0.0, Upper = 10000, MaxStep = 50, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); BeginFiniteBurn 'Turn Thruster On' Finite_4(MarsSC); Propagate 'Prop BurnDuration' NearEarth(MarsSC) {MarsSC.ElapsedSecs = BurnDuration, OrbitColor = [255 255 0]}; EndFiniteBurn 'Turn Thruster Off' Finite_4(MarsSC); Propagate 'Prop To Apogee' NearEarth(MarsSC) {MarsSC.Earth.Apoapsis}; Achieve 'Achieve Apogee Radius = 40,000' DC1(MarsSC.Earth.RMAG = 40000, {Tolerance = 0.005}); EndTarget; % For targeter DC1 Propagate 'Prop To Perigee' NearEarth(MarsSC) {MarsSC.Earth.Periapsis, OrbitColor = [0 0 255]}; Target 'Raise Apogee 100,000' DC1 {SolveMode = Solve, ExitMode = SaveAndContinue, ShowProgressWindow = true}; Vary 'Vary Burn Duration' DC1(BurnDuration = 342.338889155234, {Perturbation = 0.0001, Lower = 0.0, Upper = 10000, MaxStep = 20, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); BeginFiniteBurn 'Turn Thruster On' Finite_5(MarsSC); 49 Propagate 'Prop BurnDuration' NearEarth(MarsSC) {MarsSC.ElapsedSecs = BurnDuration, OrbitColor = [255 255 0]}; EndFiniteBurn 'Turn Thruster Off' Finite_5(MarsSC); Propagate 'Prop To Apogee' NearEarth(MarsSC) {MarsSC.Earth.Apoapsis}; Achieve 'Achieve Apogee Radius = 100,000' DC1(MarsSC.Earth.RMAG = 100000, {Tolerance = 0.005}); EndTarget; % For targeter DC1 Propagate 'Prop To Perigee' NearEarth(MarsSC) {MarsSC.Earth.Periapsis, OrbitColor = [0 0 255]}; Target 'Change ECC to 1.13' DC1 {SolveMode = Solve, ExitMode = SaveAndContinue, ShowProgressWindow = true}; Vary 'Vary Burn Duration' DC1(BurnDuration = 612.7705999929157, {Perturbation = 0.0001, Lower = 0.0, Upper = 10000, MaxStep = 100, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); BeginFiniteBurn 'Turn Thruster On' Finite_ECC(MarsSC); Propagate 'Prop BurnDuration' NearEarth(MarsSC) {MarsSC.ElapsedSecs = BurnDuration, OrbitColor = [255 255 0]}; EndFiniteBurn 'Turn Thruster Off' Finite_ECC(MarsSC); Achieve 'Achieve ECC 1.13' DC1(MarsSC.Earth.ECC = 1.22, {Tolerance = 0.005}); EndTarget; % For targeter DC1 Propagate NearEarth(MarsSC) {MarsSC.ElapsedSecs = 200000}; Target 'TCM' DC1 {SolveMode = Solve, ExitMode = DiscardAndContinue, ShowProgressWindow = true}; Vary 'Vary Burn Duration' DC1(BurnDuration = 38.34673676108495, {Perturbation = .001, Lower = 10000, Upper = 10000, MaxStep = 5, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); Vary 'Vary.V' DC1(MarsSC.Thruster.ThrustDirection1 = 6.119425356259391, {Perturbation = 0.0001, Lower = -10, Upper = 10, MaxStep = .2, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); Vary 'Vary.B' DC1(MarsSC.Thruster.ThrustDirection2 = 8.104575805963847, {Perturbation = 0.0001, Lower = -10, Upper = 10, MaxStep = .2, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); 50 Vary 'Vary.N' DC1(MarsSC.Thruster.ThrustDirection3 = -0.6426540445722402, {Perturbation = 0.0001, Lower = -10, Upper = 10, MaxStep = .2, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); BeginFiniteBurn 'Turn Thruster On' FiniteTCM(MarsSC); Propagate 'Prop BurnDuration' NearEarth(MarsSC) {MarsSC.ElapsedSecs = BurnDuration, OrbitColor = [255 255 0]}; EndFiniteBurn 'Turn Thruster Off' FiniteTCM(MarsSC); Propagate 'Prop 280 Days' DeepSpace(MarsSC) {MarsSC.ElapsedDays = 280}; Propagate 'Prop To Periapsis' NearMars(MarsSC) {MarsSC.Mars.Periapsis}; Achieve 'Achieve BdotT' DC1(MarsSC.MarsInertial.BdotT = 0, {Tolerance = 0.01}); Achieve 'Achieve BdotR' DC1(MarsSC.MarsInertial.BdotR = -8000, {Tolerance = 0.01}); EndTarget; % For targeter DC1 Target 'MOI' DC1 {SolveMode = Solve, ExitMode = SaveAndContinue, ShowProgressWindow = true}; Vary 'Vary Burn Duration' DC1(BurnDuration = 987.9466021007054, {Perturbation = .001, Lower = 1, Upper = 10000, MaxStep = 50, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); %Vary 'Vary.V' DC1(MarsSC.ThrusterMOI.ThrustDirection1 = -1, {Perturbation = 0.0001, Lower = -30, Upper = 0, MaxStep = .2, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); %Vary 'Vary.B' DC1(MarsSC.ThrusterMOI.ThrustDirection2 = 0, {Perturbation = 0.0001, Lower = -30, Upper = 30, MaxStep = .2, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); %Vary 'Vary.N' DC1(MarsSC.ThrusterMOI.ThrustDirection3 = 0, {Perturbation = 0.0001, Lower = -30, Upper = 30, MaxStep = .2, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0}); BeginFiniteBurn 'Turn Thruster On' FiniteMOI(MarsSC); Propagate 'Prop BurnDuration' NearMars(MarsSC) {MarsSC.ElapsedSecs = BurnDuration, OrbitColor = [255 255 0]}; EndFiniteBurn 'Turn Thruster Off' FiniteMOI(MarsSC); Propagate 'Prop To Periapsis' NearMars(MarsSC) {MarsSC.Mars.Apoapsis}; Achieve 'Achieve RMAG' DC1(MarsSC.Mars.RMAG = 13000, {Tolerance = 0.1}); EndTarget; % For targeter DC1 51 Propagate NearMars(MarsSC) {MarsSC.ElapsedDays = 1}; %End of GMAT Code 52 APPENDIX B Figure 8. NTR Mission Summary once in orbit around Mars. Mars is set as the inertial frame of reference. 53 Figure 9. Chemical rocket Mission Summary once in orbit around Mars. Mars is set as the inertial frame of reference.