Analytical Low-Thrust Trajectory Design using the Simplified General Perturbations model J. G. P. de Jong November 2018 - Technische Universiteit Delft - Master Thesis Analytical Low-Thrust Trajectory Design using the Simplified General Perturbations model by J. G. P. de Jong to obtain the degree of Master of Science at the Delft University of Technology. to be defended publicly on Thursday December 20, 2018 at 13:00. Student number: 4001532 Project duration: November 29, 2017 - November 26, 2018 Supervisor: Ir. R. Noomen Thesis committee: Dr. Ir. E.J.O. Schrama TU Delft Ir. R. Noomen TU Delft Dr. S. Speretta TU Delft November 26, 2018 An electronic version of this thesis is available at http://repository.tudelft.nl/. Frontpage picture: NASA. Preface Ever since I was a little girl, I knew I was going to study in Delft. Which study exactly remained unknown until the day before my high school graduation. There I was, reading a flyer about aerospace engineering and suddenly I realized: I was going to study aerospace engineering. During the bachelor it soon became clear that space is the best part of the word aerospace and thus the space flight master was chosen. Looking back this should have been clear already years ago: all those books about space I have read when growing up... After quite some time I have come to the end of my studies. Especially the last years were not an easy journey, but I pulled through and made it to the end. This was not possible without a lot of people and I would like this opportunity to thank them here. First of all, my supervisor Ron Noomen. Thanks for your input and com- ments. But mostly, thank you for your understanding, patience and kind words when I needed them the most. The graduation room on the 9th floor was the perfect place for me to work on my thesis. Some students graduated before me and left the room as friends: Laurens, Tim and Aleix. And of course Evelyne, my friend throughout the entire studies, during good and bad times. Our trip to Greece was the perfect break during my thesis. The current residents of the graduation room: thank you for your company and support. Amanthla, Corné, Datta, Maaike, Max, Raam, Sunayna and Rosalie: I am really going to miss the cakes on Wednesday afternoon. My bouldermattie and hockeyteamie Laura, thank you for proofreading parts of this thesis and listening to me during our boulder sessions. I still think that sometimes we actually talked more than we were bouldering. Lastly, I would like to thank my mother and brother. Thank you for being there for me. For you too, the last years were not easy but both of you always helped me out when I needed it the most. J. G. P.de Jong, Delft, November 2018 iii Abstract It is expected that the use of low-thrust propulsion for spacecraft in orbit around Earth will increase in the upcoming years. Compared to conventional chemical propulsion, low-thrust propulsion has a higher energy density per unit of mass, resulting in a significant reduction of propellant mass required for a given velocity increment to be gained. Due to the small acceleration force of low-thrust propulsion and the long travel time, perturbations have a large influence on a low-thrust transfer orbit. Numerical methods are required to ob- tain a feasible and precise solution of a trajectory problem, but these methods can be computationally time consuming. An analytical first-guess approximation, which takes perturbations into account, is thus desired. This can then be used as a base-line for the numerical optimization. The perturbations acting on a spacecraft in an orbit around Earth are currently modelled analytically in the Simplified General Perturbations Model 4 (SGP4) model, which is used to propagate spacecraft forward in time. In this thesis research, the SGP4 method is combined with an existing analytical method for a low-thrust trajectory. This is the solution suggested by Edelbaum and improved by (Kechichian, 1997). This resulted in the SGP4-LT tool. An iterative SGP4 version version was required. In the development of this tool, various modifications had to be made. In order to successfully implement this low-thrust solution, an iterative ver- sion of SGP4 was developed. To convert osculating orbital elements in mean orbital elements, an existing procedure was corrected and improved. Originally, this solution had poor convergence rates, but after the correction and modifications, a convergence rate of 99.5 % was reached for 17542 objects of the satellite cat- alogue. Convergence was not reached in case of combinations for inclinations smaller than 0.05° and eccen- 6 tricities lower than 4 10¡ . It is expected that this is caused by floating-point errors in the conversion process. ¢ It was shown that SGP4-LT combines perturbations and low-thrust acceleration correctly. Due to the per- turbations, the transfer time will increase slightly and the evaluation of the semi-major axis and inclination showed similar behaviour for a pure Edelbaum solution. When looking at the orientation of the orbit, the right ascension of the ascending node changed with a rate as expected by SGP4 at its corresponding altitude. Normally, a time-consuming numerical integration is required to obtain data to plot the trajectory. SGP4-LT requires only 5 seconds to produce the data required to plot trajectories with flight times of 180 days. The versatility of the SGP4-LT tool is shown by applying the tool to three different applications. Orbit rais- ing, non-coplanar orbit changes and altitude maintenance in low Earth orbits were considered. SGP4-LT obtained expected results for the orbit raising and non-coplanar orbit change. The altitude maintenance is based on the GOCE spacecraft, which used an ion engine to fly drag-free in a low Earth orbit. It was found that for the lowest thrust acceleration possible, 27.18 kg xenon propellant was required to maintain the altitude during the mission life time. Using historical TLE GOCE data, this was found to be a reasonable first approx- imation of the propellant mass. Although too low, the values obtained were in the range of the Gravity field and steady-state Ocean Circulation Explorer (GOCE)’s thruster and similar results to GOCE’s mission can be obtained when a higher thrust acceleration is used. In conclusion, a correct functioning tool has been developed which obtains a first approximation solution for transfer trajectories performing whilst orbiting around Earth. SGP4-LT is fast, robust and capable of cal- culating ¢V budgets for maintaining the altitude in low Earth orbits. v Acronyms CATO Computer Algorithm for Trajectory Optimization DFAC Drag-Free Attitude Control DITAN Direct Interplanetary Trajectory Analysis DoD Department of Defense ECI Earth-Centered Inertial EOR Electric Orbit Raising ESA European Space Agency GEO Geostationary Earth Orbit GOCE Gravity field and steady-state Ocean Circulation Explorer GPS Global Positioning System GSFC Goddard Space Flight Center GTO Geostationary Transfer Orbit HEO Highly Elliptical Orbit IPS Ion Propulsion System JAXA Japan Aerospace Exploration Agency JD Julian Date LEO Low-Earth Orbit LOTNAV Low-Thrust Interplanetary Navigation Tool MATLAB MATrix LABoratory MEO Medium Earth Orbit MJD Modified Julian Date MMO Mercury Magnetospheric Orbiter MPO Mercury Planetary Orbiter MTM Mercury Transfer Module NAIF Navigation and Ancillary Information Facility NASA National Aeronautics and Space Administration NGA US National Geospatial-Intelligence Agency NEP Nuclear Electric Propulsion NORAD North American Aerospace Defence Command NSSCC National Space Surveillance Control Center vii viii 0. Abstract RAAN Right Ascension of the Ascending Node REP Radioisotope Electric Propulsion RIC Radial, In-track, Cross-track RK Runge-Kutta RPS Radioisotope Power Systems SATCAT Satellite Catalogue SDP4 Simplified Deep Space Perturbations SEP Solar Electric Propulsion SEPTOP Solar Electric Propulsion Trajectory Optimization Program SGP Simplified General Perturbations SGP4 Simplified General Perturbations Model 4 SSN Space Surveillance Network STK Systems Tool Kit TEME True Equator and Mean Equinox TLE Two Line Elements TUDAT TU Delft Astrodynamics Toolbox USSTRATCOM United States Strategic Command XIPS Xenon Ion Propulsion System XIPSTOP Xenon Ion Propulsion System Trajectory Optimization Program Nomenclature Greek Symbols Symbol Description ® Mean angular position rad ¯ Thrust yaw angle rad ± Angle of velocity vector rad ²0 Electric constant 8.854e-12 F/m ´ Engine efficiency - γ Flight path angle rad ¹ Gravitational parameter m3/s2 Á Latitude rad ½ Density of atmosphere kg/m3 σ Stefan-Boltzmann constant 5.670367e-8 J/(sm2K4) θ Longitude rad θ True anomaly rad ! Argument of pericenter rad ­ Right ascension of ascending node rad Roman Symbols Symbol Description a Semi-major axis m 2 aT Thrust acceleration m/s c Speed of light 299792458 m/s CD Drag coefficient - CR Reflectivity coefficient - e Eccentricity - e Unit vector - E Eccentric anomaly rad E Power density W/m E Total specific energy m2/s2 f Perturbing forces acceleration vector m/s2 ix x Nomenclature fmee Modified equinoctial element - g Local gravitational acceleration m/s2 g1,0 First Gauss geomagnetic coefficient -2.96e-5 T gmee Modified equinoctial element - G Universal gravitational constant 6.670e-11 m3/(kgs2) hee Equinoctial element - hmee Modified equinoctial element - i Inclination angle rad Isp Specific impulse s J Model parameter gravity field potential - kee Equinoctial element - kmee Modified equinoctial element - L True longitude rad m Mass kg m˙ Mass flow kg/s M Mass kg M Mean anomaly rad n Mean angular motion rad/s p Semi-latus rectum m P Input power W P(x) Legendre polynomial - 2 P0 Solar constant N/m q Electric charge C qee Equinoctial element r Radial distance m r Position vector m R Perturbing potential m2/s2 R Radius of the Earth 6378.135e3 m S Effective cross-sectional area m2 t Time s T Thrust vector N T Temperature K T Thrust N U Gravitational potential m2/s2 Nomenclature xi U ¤ Electrical potential difference V V Velocity m/s Ve Exit velocity m/s W Power density W/m2 x Coordinate in x-direction m X State vector m,m/s y Coordinate in y-direction m z Coordinate in z-direction m Superscripts Symbol Description 0 First derivative w.r.t.