Trade Study Of Earth To Pluto Trajectories Utilizing A Jovian Gravitational Assist

Transcription

1 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 2009, Orlando, Florida AIAA Trade Study Of Earth To Pluto Trajectories Utilizing A Jovian Gravitational Assist Alan R. Campbell, Charles M. Cimet, and Nathan T. Depenbusch Pennsylvania State University, University Park, PA, 16802, U.S.A. V T F T h R Due to Pluto s extreme distance from our region of the solar system, very few scientific studies have been conducted on this solar body. Because of this distance, it is advantageous to employ a gravitational assist so as to reduce the necessary transfer time and propellant consumption. It is the objective of this paper to analyze the launch window from January 1, 2009 to December 31, 2019 for feasible trajectories to Pluto using a gravitational fly-by of Jupiter. Although this general problem has been studied and solved previously, the specific considerations in this paper appear to be unique. Trajectories were calculated beginning in a Low-Earth parking orbit with a single maneuver into a Lambert s transfer targeting Jupiter. At Jupiter, the geometry of a dark side swing-by was determined to enter a second Lambert s transfer targeting Pluto. A second burn was then calculated for an orbit capture at Pluto. All calculations were performed in MATLAB by varying the launch dates and both transfer times over small intervals. Orbital position assumptions were made using a linearly varying ephemerides model corresponding to the years C.E C.E Gravitational effects for other solar bodies, particularly the moons of Jupiter, were assumed to be inconsequential. Finally, since the Lambert s transfers were calculated independently, the approach and departure velocities at Jupiter were not set to be equal, however, only those that closely matched were considered. As expected, the results indicate a yearly cyclical pattern of viable launch dates with perturbations resulting from the orbital procession of Jupiter and Pluto. An inverse relationship between the necessary propellant usage and the total time of flight is also evident. Specifically, the geometry appears to be most favorable as soon as possible, particularly the window from June 18, 2009 to October 11, Although there are many viable options for an Earth to Pluto mission in the next dozen years, the most beneficial launch windows require mission planning to begin in the very near future. Impulsive change in velocity, km/s Time of flight, years Date, years after J2000 altitude, km radius, km Subscript E Earth J Jupiter P Pluto t Turning T OT Total Nomenclature Undergraduate Student, Department of Aerospace Engineering, 229 Hammond Building, Univeristy Park, PA 16802, Student Member, AIAA. 1 of 8 Copyright 2009 by the American Institute of Aeronautics and American Astronautics, Institute Inc. All of rights Aeronautics reserved. and Astronautics

2 I. Introduction Despite its recent demotion from planethood, Pluto is the most familiar solar object not yet visited. This is soon to change with the 2006 launch of the New Horizons mission scheduled to perform a fly-by of Pluto in July of More can be learned about the body, however, by putting a scientific satellite into its orbit. This paper focuses on finding feasible trajectories and launch dates to accomplish this mission. Due to Pluto s distance and eccentricity about the Sun, it is nearing a phase characterized by the freezing of its atmosphere, therefore making the effectiveness of an observationary probe drop significantly. 1 In order to reach Pluto before the atmosphere freezes, we must consider launches within the upcoming decade, focusing on faster trajectories. One way of doing this is to enlist the help of another celestial body in a gravitational assist. Studies have shown that a Jovian swing-by is an ideal candidate for a direct route to Pluto. 2 It is a result of the similarity of this mission to the current New Horizons mission that much of the numerical analysis is roughly filtered by the capabilities of New Horizons. A. Background The en route New Horizons mission, designed and built by the Johns Hopkins Applied Physics Laboratory (APL) for the National Aeronautics and Space Administration (NASA) was launched January 19, 2006 and flew by Jupiter on February 28, It is scheduled to reach Pluto in July The scientific goals of the New Horizons mission are primarily observatory in regards to Pluto and its moon, Charon. The mission is based on several scientific studies to be performed on Pluto and Charon as the probe flys by. 3 Pluto s atmosphere is also currently escaping out to space. There are no other celestial bodies presently acting in such a way, and it is thought that Earth s early atmosphere of hydrogen and helium was lost in this manner. By studying this phenomenon, we may be able to gain insight into the early atmospheric formation of our planet. In addition, Pluto s composition is known to be similar to that of an asteroid in that it contains large amounts of water and carbon compounds. It is these ingredients that may have sparked life on Earth, transported to our planet by an asteroid impact. These studies, along with the others, are limited by the extreme relative speed of the probe with respect to the Pluto-Charon system. As a result, New Horizons has only a seven month study window. B. Overview In an attempt to lengthen the available window, an orbital capture maneuver will be performed at Pluto to allow prolonged study of its environment. Some simplifying assumptions are made. The probe is taken to already be in parked orbit around Earth, and will undergo a pair of transfer ellipses obtained from the solution of Lambert s problem to the vicinity of Pluto. An initial burn will set us onto the first transfer ellipse heading toward the Jovian swing-by. The swing-by will then send the probe on course to Pluto, where it shall manuever into orbit around the body. Minor correctional manuevers may be required to keep the spacecraft on its interplanetary paths. These will not be considered as significant in this paper. The gravitational effects of all other solar objects (such as other planets and moons) will also be ignored. This is a reasonable assumption throughout most of the journey, with the possible exception of the Jupiter swing-by where any of the numerous moons of Jupiter could have an effect. II. Model The solution of trajectories to fit the mission involved a two-part process: calculating possible trajectories and determining which of these solutions are feasible. Calculation was done by iterating launch date (T E ), total time of flight (T F EP ), and time of flight from Earth to Jupiter (T F EJ ). Trajectories were analyzed by limiting particular characteristics to those allowed by physics, mission requirements, and/or human capability. A. Calculative Procedure Determining the most efficient trajectories to Pluto required solving several separate smaller problems. In order to do so, a MATLAB program was compiled to take all of these scenarios into effect. An entirely temporal constraint of the iteration was used. 2 of 8

3 The first part of the problem was solved by writing an ephemerides function to compute the approximate time-varying positions of the three important solar bodies in this mission. Linear approximation functions valid for the time period C.E were obtained from a Caltech Solar System Dynamics Group paper. 5 These functions were evaluated over a relevant time period for this mission. Planetary position and velocity vectors were calculated in the solar inertial reference frame. Another major part of the solution was the creation of a functional solution to Lambert s Problem. This was done based on an algorithm developed by R.L. Anderson (Univ. of Colorado). 6 Requiring an initial and final position vector, as well as desired time of flight, this algorithm implements a change of variables to solve the problem, then iterates over a differential value to achieve time of flight convergence. At that point, the Stumpff functions are calculated. From here, the initial and final velocity vectors are obtained and used in the main function. Trajectory calculation was performed by decoupling the Earth to Jupiter Lambert transfer from the Jupiter to Pluto transfer. Since the trajectories are decoupled at this point, the velocities with respect to Jupiter do not necessarily match. For this to be physically viable, the velocity magnitude change with respect to the planet in the Jovian sphere of influence had to be limited to small values able to be carried on board. Values were set at a V of 0.5 km/s for calculation. One further constraint was placed inside Jupiter s sphere of influence. The minimum turning radius, (R t ) was set at one Jupiter radius (R J ) to prevent a collision with the planet. The turning radius was calculated geometrically with a dark-side swing-by assumption. 7 These results were implemented together with other less complex functions to allow for the complete calculation of trajectory characteristics. Since the main focus of this study was to find trajectories from Earth to Pluto, the analysis was not concerned with the details of ground to Earth orbit launch; it was therefore assumed that the spacecraft would start in an Earth parking orbit (h 350 km). A parking orbit at Pluto was also assumed (h 50 km). B. Analytic Procedure Upon review of the data calculated by the code, analysis of the prescribed trajectories commenced by discarding what was considered unfeasible. For a trajectory to be considered feasible it had to meet several constraints. These constraints were set with regards to a number of factors, including human technological capability, physical representation, and radiative effects. First, all trajectories with a total V greater than 32 km/s were rejected. This value was chosen as roughly double the V of the New Horizons mission and used as a conservative upper limit. The next constraint was to limit the necessary V t to a value less than km/s, a number based on the amount of corrective propellant carried by New Horizons. Another limitation imposed was a higher limit on the minimum turning radius at Jupiter. Research has shown that fly-bys of a radius less than 5R J can induce potentially catastrophic radiative effects on the spacecraft. Radiation is a concern up to a distance of 14R J, but depending on the hardiness of the craft, such trajectories are possible. III. Results After applying constraints, the remaining data points fall into three distinct launch windows. Each window is identified by the group of data points falling in a small range of launch dates. The first window of launch dates fall between June 18, 2009 and October 11, 2009 (132 days), the second between March 18, 2018 and June 9, 2018 (112 days), and the third between March 16, 2019 and July 6, 2019 (149 days). The first window contains 127 feasible trajectories, the second contains 87, and the third 86 (see figure 1). There are also several notable patterns within the data. As expected, the V T OT vs. T E has a sinusoidal pattern (see figure 2), based upon revolution of Earth around the Sun, skewed by the slower revolutions of Jupiter and Pluto about the Sun. The data curve is incomplete due to aforementioned calculation constraints applied to the trajectory determination. The V T OT vs. T F EP exhibits an inverse relationship (see figure 3), which was expected under the assumption that it takes more propellant to achieve a higher speed. In this plot, each launch window is grouped in a different region of the plot. Windows one, two, and three all have groupings in the upper middle and upper right regions; the upper middle region corresponds to high total V s and moderate times of flight while the upper right region corresponds to high total V s and high times of flight, respectively. Window one also has values starting at approximately a V of 29km/s and time of flight of 8 years, following 3 of 8

4 an inverse trend to V s of approximately 14 km/s with a time of flight of 17.5 years. This range of values also contains trajectories with low V s and moderate times of flight; specific results include a V of 18.00km/s and a time of flight of years. Overall, the data indicates a larger range of possible trajectories earlier in the time period studied. This can be noted in the R t vs. T E plot (see figure 4). This graph also indicates a much higher average R t early in the time period. Together, these observations indicate a much more favorable geometry in earlier dates. A large gap in resulting trajectories was observed between T s of 12 and 15 (correlating to the years of 2012, 2013, and 2014). This gap, when read in conjunction with the above R t graph can be explained by poor planetary alignment. For example, in this range, the trajectory would require the probe to travel through Jupiter in order for the spacecraft to reach the correct Lambert s transfer ellipse. Based on the R t vs. T F EP graph (see figure 5), there is a Gaussian-like trend in the turning radius for viable trajectories in window 1, which peaks for a 10.5 year time of flight and subsequently decreases for all greater times of flight. This trend does not appear in launch windows two and three, which show a slightly inverse correlation between turning radius and total time of flight. Given that a larger turning radius results in less energy being transfered from Jupiter to the spacecraft, this graph also shows that the spacecraft requires a lower energy transfer for low time of flight trajectories in launch window one. IV. Conclusion It is evident that planetary geometry is most favorable in the first launch window. Trajectories calculated in this period had both lower V s and times of flight than those in windows two and three. Given that window one is in 2009, and the less favorable windows two and three occur in 2018 and 2019, respectively, it can be deduced that planetary geometry will become less favorable as the next decade progresses. Since Pluto s atomosphere should completely freeze by 2020, the second and third launch windows will yield arrival dates after this process has ended. This loss of atmosphere is unique, and in order to sufficiently study this phenomena the initial launch window must be used. In order to reach Pluto in time to be of any scientific benefit, shorter flight times from the first launch window need to be utilized. This naturally brings about an increase in necessary V. For missions constrained by fuel, the first launch window also appears advantageous because trajectories with lower V s can be attained. An additional benefit of the planetary geometry in the first launch window is a result of the higher necessary turning radii. This would allow the hardiness of the spacecraft to be of less concern. A specific trajectory can be chosen in the initial launch window to meet specific mission needs. Trade-offs involving total V, on-board propellant, arrival date, and turning radius should be considered in depth in choosing a definite trajectory. 4 of 8

5 Appendix Figure 1. Total time of flight as a function of launch date. 5 of 8

6 Figure 2. Total V as a function of launch date. Figure 3. Total V as a function of total time of flight. 6 of 8

7 Figure 4. Turning radius as a function of launch date. Figure 5. Turning radius as a function of total time of flight. 7 of 8

8 Acknowledgments The authors would like to collectively thank Dr. David B. Spencer, Associate Professor of Aerospace Engineering, Pennsylvania State University, for both the inspiration and for divulging a good deal of the theory behind our study. References 1 Guo, Y. and Farquhar, R.W., Baseline design of newhorizons mission to Pluto and the Kuiper belt, Acta Astronautica, Vol. 58, April 2006, pp Minovitch, M.A., Fast Missions to Pluto Using Jupiter Gravity-Assist and Small Launch Vehicles Journal of Spacecraft and Rockets, Vol. 31, No. 6, November-December 1994 pp Kusnierkiewicz, D.Y., et al., A description of the Pluto-bound New Horizons spacecraft Acta Astronautica, Vol. 57, May 2005, pp Beisser, K., Why Go to Pluto?, New Horizons: NASAs Pluto-Kuiper Belt Mission, [cited 8 December 2007]. 5 Standish, E.M., Keplerian Elements for Approximate Positions of the Major Planets Solar System Dynamics Group, Jet Propulsion Laboratory, California Institute of Technology. 6 Anderson, R.L., Solution to the Lambert Problem using Universal Variables University of Colorado; rla/lambert.pdf, [cited 23 November 2007]. 7 Wiesel, W.E. Interplanetary Trajectories Spaceflight Dynamics 2nd Ed., Chapter 11, McGraw-Hill, of 8