Space and Plasma Physics Department KTH, Kungliga Tekniska H¨ogskolan SE-100 44 Stockholm Sweden PRELIMINARY MISSION ANALYSIS AND DESIGN FOR A SMALL SATELLITE SWARM September 5, 2012 Author: Supervisor: Noravidhya Tanapura Nickolay Ivchenko Master of Science Thesis Stockholm, Sweden 2012 Abstract The thesis is a preliminary mission analysis and design of a small satellite swarm. The concept of the mission is to probe altitudes between 200 km and 6000 km to study the structures and dynamics of the magnetic field aligned currents. The mission lifetime is about 3 months. Aerodynamic drag at low altitudes is used for orbit and formation control. During the perigee passage, the satellite would decelerate due to drag, therefore, reducing its apogee. In addition, the attitude control of the spacecraft during the perigee passage could be used for formation control by changing its cross-sectional area. The simulations indicated that an appropriate insertion orbit should be at the perigee of 168 km and an apogee of 6000 km. Moreover, from the orbital decay simulations, it was found that by maintaining a constant ram-facing area of 0.1 m2, it is possible for the satellite to decay in 90 days. The attitude simulations show that for at least one perigee passage at a perigee altitude of 168 km, the satellite is able to maintain its attitude and not tumble throughout the trajectory. In addition, investigation of the leader-follower satellite formation yielded that the relative translation of a circular orbit oscillates in all relative directions whereas in an elliptical orbit it only oscillates in the cross-track direction. Furthermore, the simulation has also shown that the relative translation of a leader-follower formation with a elliptical reference orbit, would spiral out of the radial-cross-track plane. Acknowledgements It would not have been possible for me to complete an undertaking as challenging as this project on my own. I would like to give special thanks to my supervisor Dr. Nickolay Ivchenko, for the opportunity to work on such a project and my gratitude for his patience and guidance that nudged me in the right direction when I needed it. I would like to thank my parents for their endless source of inspiration and motivation throughout my journey, for I could not have imagined coming this far without their support and vision. Finally, I would like to thank all my friends here at KTH and all those who have helped me along the way, for your invaluable support and feedback during the course of my studies. Noravidhya Tanapura I know that I am mortal and ephemeral. But when I search for the close-knit encompassing convolutions of the stars, my feet no longer touch the earth, but in the presence of Zeus himself I take my fill of ambrosia which the gods produce. -Kepler’s epigram ascribed to Ptolemy Contents 1 Introduction 2 1.1 Background............................................2 1.2 Problem Statement........................................3 1.3 Scope...............................................3 1.4 Thesis Overview.........................................3 2 Theoretical Background4 2.1 Orbital Mechanics........................................4 2.1.1 The Two-Body Problem.................................4 2.1.2 Equation of Relative Motion..............................5 2.1.3 Classical Orbital Elements................................6 2.1.4 Special Perturbation: Cowell’s Method.........................7 2.1.5 Orbit Perturbation: Aerodynamic Drag........................8 2.1.6 Orbit Perturbation: Non-spherical Earth........................8 2.2 Attitude Kinematics.......................................9 2.2.1 Coordinate System and Rotation Matrix........................9 2.2.2 Direction Cosine Matrix................................. 10 2.2.3 Euler Angles....................................... 10 2.2.4 Euler’s Principal Rotation................................ 11 2.2.5 Euler Parameters (Quaternions)............................ 12 2.2.6 Kinematic Differential Equations............................ 13 2.3 Attitude Dynamics........................................ 14 2.3.1 Rotational Dynamics................................... 14 2.3.2 Gravity Gradient Torque................................ 16 2.3.3 Magnetic Torque..................................... 17 2.3.4 Aerodynamic Torque................................... 18 3 Orbital Decay 19 3.1 Comparing Atmospheric Models................................ 19 3.1.1 Jacchia J70 Model.................................... 19 3.1.2 NRLMSISE-00 Model.................................. 25 3.1.3 Summary of Atmospheric Models............................ 32 3.2 Decay Lifetime Simulation.................................... 33 3.2.1 Methodology....................................... 33 3.2.2 Effect of Atmospheric Drag............................... 33 3.2.3 Effect of J2 Perturbation................................ 34 3.2.4 Effect of different ram-facing areas........................... 36 3.2.5 Summary of Simulations................................. 36 4 Satellite Attitude Simulation 38 4.1 Satellite Geometry........................................ 38 4.1.1 Satellite Dimensions and Mass Properties....................... 39 4.1.2 Center of Gravity..................................... 40 4.1.3 Principal Moments of Inertia.............................. 40 iv Contents Contents 4.2 Attitude Simulation....................................... 41 4.2.1 Attitude Propagation.................................. 41 4.2.2 Aerodynamic Torque with Partial Accomodation Coefficient............. 42 4.3 Torque Profiles.......................................... 45 4.3.1 Pitch Torque....................................... 48 4.3.2 Yaw and Roll Torque Profiles.............................. 50 4.4 Pitch Dynamics.......................................... 52 4.4.1 Initial Orbit Conditions for Pitch Dynamics...................... 52 4.4.2 Simulation Results.................................... 52 4.5 Summary............................................. 54 5 Passively Stabilized Perigee Passage 55 5.1 Initial Conditions for Perigee Passage............................. 55 5.2 Simulation Results........................................ 56 5.3 Summary............................................. 57 6 Satellite Formation Design 58 6.1 Coordinate Frame........................................ 58 6.2 Formation Dynamics....................................... 59 6.3 Satellite Formation Simulation................................. 61 6.4 Summary............................................. 63 7 Conclusion 64 v List of Figures 1.1 Illustration of 2-U and 3-U Cube Satellite. [1].........................2 2.1 Geometry of an Ellipse and Orbital Parameters. [2]......................5 2.2 Classical Orbital Elements. [3].................................7 2.3 Illustration of Encke’s method. [4]...............................7 2.4 Shows ECI as N, A and B reference frame. [5].........................9 2.5 Euler angle rotation sequence (3-2-1). [6]........................... 11 2.6 Illustration of Euler’s principal rotation theorem. [6]..................... 11 2.7 Time differentiation in rotating frame. [7]........................... 14 2.8 Gravity gradient torques on a low-Earth orbit satellite. [8].................. 17 3.1 J70: Density maps at 300 km altitude of 3 months showing high and solar activity conditions. 21 3.2 J70: Density map of the solar maximum-minimum ratio for different altitudes and latitudes of different months......................................... 22 3.3 J70: Log10 of density as a function of altitude for solar maximum and solar minimum at different months.......................................... 23 3.4 J70: Ratio of density values (normalized to the equator value) of high and low solar activity at longitude of −90◦........................................ 24 3.5 NRLMSISE-00: Density maps at 300 km altitude of 4 months showing high and low solar activity conditions......................................... 26 3.6 NRLMSISE-00: Density map of the solar maximum-minimum ratio for different altitudes and latitudes of different months................................. 27 3.7 NRLMSISE-00: Log10 of density as a function of altitude for solar maximum and solar minimum at different months................................... 28 3.8 NRLMSISE-00: Ratio density map of different months and solar activity.......... 29 3.9 Density ratio between NRLMSISE-00 over J70......................... 31 3.10 Block diagram represents the orbit propagation simulation process.............. 33 3.11 Orbital Decay due to aerodynamic drag............................. 34 3.12 Orbital Decay due to drag and Earth oblateness........................ 35 3.13 Earth oblateness effect on different orbit inclinations...................... 35 3.14 Altitude versus time in final decay................................ 36 3.15 Range of ram-facing areas..................................... 36 4.1 Satellite geometry isometric view................................. 38 4.2 Satellite geometry top and side views.............................. 39 4.3 Specular and diffuse molecular reflection. [9].......................... 42 4.4 Molecules incident on an element of the satellite surface. [9]................. 43 4.5 Vectors in torque calculations................................... 45 4.6 Satellite Geometry showing vectors used in self-shadowing calculations: a)Shows the angles and vectors for topside control panel shadow. b)Shows the angles and vectors for bottomside control panel shadow. ......................................