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Delta-V to Near-Earth Asteroids: an Examination of the Shoemaker-Helin Equations

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Delta-V to Near-Earth Asteroids: an Examination of the Shoemaker-Helin Equations DELTA-V TO NEAR-EARTH ASTEROIDS: AN EXAMINATION OF THE SHOEMAKER-HELIN EQUATIONS By Max Murphy A thesis presented to the Department of Astronomy in partial fulfillment of the requirements for the degree of Bachelor of Arts Advisors: Martin Elvis José Luis Galache Sukrit Ranjan April 11, 2015 Harvard College 2 Delta-V to Near-Earth Asteroids Abstract Near-Earth asteroids present promising opportunities for future human spaceflight missions. Many of these objects have orbits that come within close proximity to the Earth, allowing a significantly lower cost in comparison to missions to Mars or even the Moon. This cost is directly related to the change in velocity (Delta-V) necessary to move a vehicle from low-Earth orbit to the orbit of a near-Earth asteroid. This paper will discuss the merits of asteroid rendezvous missions in the near future and assess the validity of Delta-V approximations for ~1200 known asteroids candidates. A 1979 paper by Shoemaker and Helin suggested a class-specific set of equations for approximating the Delta-V for these objects. Modern numerical calculations provide accurate measurements for these values and will serve as a test to the strength of the Shoemaker-Helin equations. Max Murphy Contents 1 Equations and Constants 2 Introduction 2.1 History 2.2 NEA Classes 2.3 Asteroid Naming 2.4 Commercial Interest and the Space Economy 2.5 Spaceflight Mission Considerations 2.6 Delta-V of Asteroids 3 Delta-V Approximation 3.1 Orbit Measurements 3.2 Trajectory Calculations 3.3 Shoemaker-Helin Equations 3.4 Near-Earth Object Human Space Flight Accessible Targets Study (NHATS) 4 Methods 4.1 Acquiring Data 4.2 Comparing SH and NHATS Delta-V 4.3 Modifying the Shoemaker-Helin Equations 5 Results and Conclusions 6 Figures and Tables 4 Delta-V to Near-Earth Asteroids 1 Equations and Constants I will use the following constants and equations in my calculations and analysis. Equations 1. Escape velocity: �! = 2�� � ∆! ! ! ! !!"" ! 2. Tsiolkovsky's "Rocket Equation: ∆� = �!""�� and � = !! !! !!"" 3. Specific Impulse: �!" = !! 4. Shoemaker & Helin: a. � = �! + �! ! ! b. �! = �! + � − �! c. �! = 3 − ! − 2 2� cos (� ) ! !!! � + 1 2 ! � ! d. �! = �! − 2�!�! cos 2 + �! e. �! = ! − ! − ! 2 cos (� ) !,!" ! !!! ! � + 1 2 f. �! = ! − ! − ! � 1 − �! !,!" ! ! ! � g. �! = ! − ! − ! ! !,!" ! !!! ! !!! h. �! = ! − ! − ! � 1 − �! cos (� ) !,!" ! ! ! � 2 i. �! = ! − 1 − 2(!) 2 − � !,!" ! ! j. ∆� = 30� + 0.5 Max Murphy 5. Semi-minor axis: � = �(1 − �) 6. Semi-major axis: � = �(1 + �) 7. Hyperbolic excess velocity: �! = � −� 8. Radius of ellipse: � = ! !!!(!"#$) 1.2 Constants 1. Gravitational constant (G): 6.67384 × 10-11 m3 kg-1 s-2 2. Mass of Earth (M): 5.972 × 10-24 kg 3. Effective exhaust velocity (�!"") a. Solid rocket: 2.50 km/s b. Bipropellant liquid rocket: 4.40 km/s c. Ion thruster: 29.0 km/s -2 4. Standard acceleration due to gravity (�!): 9.806 m s 5. Mean orbital velocity of Earth (v0): 29.78 km/s 6. Orbital velocity at LEO – 6,680 km (U0): 7.73 km/s 7. Escape velocity from Earth (S): 11.2 km/s 6 Delta-V to Near-Earth Asteroids 2 Introduction Asteroids are remnants of the early Solar System as it formed 4.5 billion years ago in the Orion–Cygnus Arm of the Milky Way. A group of planetesimals – precursors to planet/moon sized objects – not captured by the gravitational pull of neighboring protoplanetary disks became the Main Belt. In comparison to the icier comets that formed beyond Saturn, these rocky objects formed from dust and ice grains in the warm region between Mars and Jupiter. The significant size of Jupiter caused gravitational perturbations that altered the fate of Main Belt planetesimals. In combination with high orbital velocities and population density, these perturbations prevented the formation of a protoplanet (or protoplanets) in the region and triggered destructive collisions that split apart planetesimals into smaller objects known today as asteroids. Continued resonances between Jupiter and Main Belt asteroids scattered many of the objects into orbits that allow close passes with Earth every few decades. Those that fall within 1.3 AU of the Sun are known as Near-Earth Asteroids (NEAs). While NEAs are considered to be dynamically young – their orbits have been altered significantly on hundred million year timescales – they accurately represent the populations of asteroids found in the Main Belt (Cheng, 1997). As the leftover building blocks that created our Solar System, asteroids can provide evidence for the ingredients of planet formation and allow us to test models of planetary-system evolution. However, asteroids are not all alike in their composition and are divided into three main groups. Carbonaceous asteroids (C-type) have a low albedo (~0.03) and are similar in composition to the Sun – carbon rich with concentrations of Nitrogen and Oxygen – but depleted in Hydrogen and Helium. C-type Max Murphy asteroids form 75% of the known asteroid population and are typically found in the outer regions of the Main Belt. Silicaeous asteroids (S-type) are brighter than C-types with an albedo of ~0.15 and are predominately metallic iron mixed with silicates. The remaining 8% are mostly M-type, metallic, asteroids that have a composition dominated by metallic iron and heavier metals including Nickel, Gold, and Aluminum. In addition, the 3µm absorption line found in infrared spectra of S-type NEAs 443 Eros and 1036 Ganymed provide evidence for water ice and Hydrogen Peroxide present on the surface of asteroids (Emery, et al., 2015). Meteorites found on Earth can also give insight into the composition of NEAs. Rocky objects that are smaller than an asteroid – less than a meter across – are referred to as meteoroids and are often scattered parts of other space rocks. Once they enter the Earth's atmosphere most meteoroids burn up, but any part that reaches the surface is called a meteorite. Chemical analysis of meteorites thought to have once been part of asteroids have shown deposits of precious metals including Gold, Platinum, Palladium, and Rhenium – used in the construction of jet engines – and the semiconductor Germanium (Ross, 2001). Beyond improving our understanding of a young Solar System's formative years, the presence of rare materials in asteroids offers another valuable opportunity. An unmanned mining vehicle deployed on an NEA could excavate metals used in the fabrication of spacecraft and surface-mission vehicles as well as Hydrogen and Oxygen, the critical elements of rocket fuel. Using resources acquired from NEAs at space stations orbiting Earth may be cheaper than either returning the spacecraft to a surface facility or shipping materials mined on Earth, both of which require more frequent trips 8 Delta-V to Near-Earth Asteroids out of Earth's gravitational well. SpaceX lists the launch cost of the Falcon 9 v1.1 rocket as $61.2 million (or about $2111 per pound for the 28,991 lb payload necessary to get to LEO), and NASA's shuttle program saw launch costs exceeding $300 million (SpaceX Capabilities and Services, 2014). Asteroid mining may help eliminate some of these expenditures and make future human spaceflight a more efficient enterprise. Determining a suitable candidate for an asteroid mining mission would require knowledge of both the object's composition and orbit. NASA's NEOWISE survey estimates a population of 19,500 NEAs smaller than 1 kilometer but larger than 100 meters, and 981 larger than 1 kilometer (Mainzer et al. 2014). One of the first filters a mining entrepreneur may apply to this lengthy list is finding the objects that require the least amount of travel time and fuel cost. As I will show in the following sections, these parameters are fundamentally connected to the acceleration – change in speed and direction - necessary to move a spacecraft from LEO to the orbit of an NEA. The goal of this project is to test the Shoemaker-Helin model for calculating the change in velocity (Delta-V) of NEAs and to discuss additional considerations that may improve the statistical accuracy of Delta-V estimates. 2.1 History The tale of asteroid study begins at the turn of the 19th century (quite literally January 1st 1801) when Giuseppe Piazzi became the first scientist to observe an asteroid. Initially thought to be another planet orbiting near 2.8 AU, as other such objects were found it eventually became clear that they were something new - minor planets. In 1944, O. J. Schmidt proposed that these objects were remnants of a failed Max Murphy planet formation due to influence from Jupiter's gravitational pull, a theory that is widely accepted today. Tom Gehrels initiated the first formal discussion of a potential asteroid mission in 1971, and detailed proposals from Stuhlinger et al. (1972) and by Whipple et al. (1973) soon followed. A few unusual ideas regarding the exploitation of asteroids also became topics of lively debate, including a forced impact to create another Panama Canal (Herrick 1979). Astronomers quickly realized that any visit to an asteroid would require extensive observations of NEAs and a method of quantifying the difficulty of reaching each object. Dr. Eugene Shoemaker, appointed chief scientist at the USGS Center for Astrogeology in 1965, and Eleanor Helin, the principal investigator of the Near Earth Asteroid Tracking (NEAT) program run by NASA's Jet Propulsion Laboratory (JPL), spearheaded the first systematic search for NEAs at the 0.46-meter Schmidt Telescope on Palomar Mountain. One of Dr. Shoemaker and Helin's papers, written in 1979, outlined a method (see Section 3.3) for determining the Delta-V of the asteroids they observed in order to gain a better understanding of flight trajectories between Earth and NEAs (Shoemaker & Helin, 1979). The development of this method was one of the important first steps toward an unmanned rendezvous mission with an NEA. In the past three and a half decades, many more papers have advanced our understanding of asteroid orbits in and out of the Main Belt.
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