EFFECTS OF GRAVITY PERTURBATIONS ON OPTIMAL LOW-THRUST TRANSFER SOLUTIONS FOR LOW EARTH ORBITS By ALEXA HORN AN HONORS THESIS PRESENTED TO THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE MAGNA CUM LAUDE UNIVERSITY OF FLORIDA 2019 ⃝c 2019 Alexa Horn This thesis is dedicated to my friends and family who have supported me throughout my academic journey. I would not be submitting this project without them. ACKNOWLEDGMENTS This thesis would not have been possible without the mentorship and guidance of my thesis chair, Dr. Anil Rao. Thank you as well to Dr. William Hager and Dr. Richard Lind for serving on my thesis committee. Finally, thank you to the University of Florida and the friends and family who have supported me throughout my education. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 4 LIST OF FIGURES .................................... 6 ABSTRACT ........................................ 7 CHAPTER 1 INTRODUCTION .................................. 8 2 Background ...................................... 9 2.1 Modified Equinoctial Elements ......................... 9 2.2 Optimization Problem Definition ....................... 9 2.3 Initial Orbit and Other Parameters ...................... 12 2.4 Survey of Earth-Centered Perturbations ................... 12 2.4.1 Zonal Harmonics ............................ 13 2.4.2 Analysis ................................. 13 2.4.3 Perturbation Selection ......................... 14 3 Methodology ..................................... 17 3.1 GPOPS-II .................................... 17 3.2 Comparing Solutions With and Without J2 .................. 17 3.3 Variations with Inclination ........................... 18 3.4 Variations with Eccentricity .......................... 18 3.5 Final Orbit Selection .............................. 18 4 Results ......................................... 19 4.1 Near-Equatorial Orbits ............................. 19 4.1.1 Eccentricity of 0.1 ............................ 19 4.1.2 Eccentricity of 0.2 ............................ 20 4.1.3 Eccentricity of 0.3 ............................ 20 4.2 Polar Orbits ................................... 25 4.2.1 Eccentricity of 0.1 ............................ 25 4.2.2 Eccentricity of 0.2 ............................ 25 4.2.3 Eccentricity of 0.3 ............................ 26 4.3 Tundra Orbit .................................. 31 5 Conclusions ...................................... 33 BIOGRAPHICAL SKETCH ................................ 36 5 LIST OF FIGURES Figure page 2-1 Longitude of the ascending node versus time in hours for a final orbit with an eccentricity of 0. ................................... 15 2-2 Argument of periapsis versus time in hours for a final orbit with an eccentricity of 0. .......................................... 15 2-3 Longitude of the ascending node versus time in hours for a final orbit with an eccentricity of 0.737. ................................. 16 2-4 Argument of periapsis versus time in hours for a final orbit with an eccentricity of 0.737. ........................................ 16 4-1 Maximum deviations in COE's for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.1. ........... 22 4-2 Maximum deviations in COE's for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.2. ........... 23 4-3 Maximum deviations in COE's for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.3. ........... 24 4-4 Maximum deviations in COE's for J2 and no-perturbation solutions as a function of inclination for a Polar orbit with eccentricity 0.1. ................ 28 4-5 Maximum deviations in COE's for J2 and no-perturbation solutions as a function of inclination for a Polar orbit with eccentricity 0.2. ................ 29 4-6 Maximum deviations in COE's for J2 and no-perturbation solutions as a function of inclination for a Polar orbit with eccentricity 0.3. ................ 30 4-7 Maximum deviations in COE's for J2 and no-perturbation solutions as a function of inclination for a Tundra orbit. .......................... 32 6 Abstract of Honors Thesis Presented to the University of Florida in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science Magna Cum Laude EFFECTS OF GRAVITY PERTURBATIONS ON OPTIMAL LOW-THRUST TRANSFER SOLUTIONS FOR LOW EARTH ORBITS By Alexa Horn May 2019 Chair: Dr. Anil V. Rao Major: Aerospace Engineering This thesis explores the effects of gravity perturbations on the optimal solutions for transfers in Low Earth Orbit. The focus is on very low-thrust maneuvers, which are not time dependent, but are capable of minimizing fuel consumption. Using optimal control software, GPOPS-II, transfer orbits are calculated from a space shuttle parking orbit to various final orbits. The final orbits, Near-Equatorial, Polar, and Tundra, are tested over a range of inclinations. Solutions are found and compared that include and exclude the zonal harmonic, J2. Calculating the numerical differences in the transfer orbit's orbital parameters for both cases leads to the conclusion that J2 can, at certain final orbit inclinations, have a significant effect on the optimal transfer solution. It is, in some cases, imperative to consider J2 in spacecraft trajectory optimization in order to realistically model earth-centric maneuvers. It is important to note that this thesis only seeks to demonstrate the influence of J2. Additional, in-depth study is required to ascertain why J2 has the effects it does on these select orbits. 7 CHAPTER 1 INTRODUCTION Low-thrust trajectories have, historically, been appealing to missions that depend on minimizing fuel consumption. As long as the duration of the trajectory isn't an issue, low-thrust propulsion methods such as electric thrusters and solar sails produce more delta-V with the same amount of fuel than chemical engines. One caveat of low-thrust maneuvers, however, is that forces due to aerodynamic drag and perturbations can have an outsized impact on the very low thrust [1]. The focus of this project is an optimization problem of an earth-centered, low-thrust orbit transfer with a goal of minimizing fuel consumption. The thrust model that is used simulates the low thrust that is produced by electric propulsion. The objective is to demonstrate just how significant the impact of perturbations is on optimal low-thrust trajectories and to explore how that impact can vary with orbital parameters like inclination. The project finds that gravity perturbations can have a noticeable, and sometimes egregious, effect on low-thrust orbit transfers, depending on final orbit inclination. 8 CHAPTER 2 BACKGROUND The low-thrust optimization problem utilized by John Betts in Ref. [2] serves as the basis for the optimization of various low-thrust transfers in this thesis. In addition to the problem definition, the parameters for the initial parking orbit are also borrowed. Just as in Betts' problems, Modified Equinoctial Elements (MEE) are used here because of their greater simplicity in calculating transfer orbit solutions than Classical Orbital Elements (COE). 2.1 Modified Equinoctial Elements The use of MEE (p, f , g, h, k, L) eliminates the singularities that arise in COE (a, e, i, Ω, !, ν) when e=0 and i=0, 90◦ [1]. While it is easier to calculate solutions with MEE, it is easier to visualize and input orbits in their COE form. The following equations are used to transfer between the two sets of elements. p = a(1 − e2) a = p=(1 − f 2 − g2) p f = ecos(! + Ω) e = f 2 + g2 p 2 h2+k2 g = esin(! + Ω) i = arctan( 1−h2−k2 ) k h = tan(i=2)cos(Ω) Ω = arctan( h ) gh−fk k = tan(i=2)sin(Ω) ! = arctan( fh+gk ) L = Ω + ! + ν ν = L − Ω − ! 2.2 Optimization Problem Definition The dynamics are defined by the state and the control variables given below. [yT , w] = [p, f , g, h, k, L, w] T u = [ur , uθ, uh] The equations of motion used for a vehicle with varying thrust are: _y=A(y)∆+b _w = −T [1 + 0.01τ]=Isp 9 0 =ku∥−1 τL ≤ τ ≤ 0 where w is weight, T is the maximum thrust, τ is the throttle factor and Isp is the engine's specific impulse. The equinoctial dynamics are defined by: 2 q 3 2p p 6 0 q 0 7 6q q µ q 7 6 7 6 p sinL p 1 [(q + 1)cosL + f ] − p g [hsinL − kcosL]7 6 µ µ q µ q 7 6q q q 7 6 p p 1 − p g − 7 6 µ cosL µ q [(q + 1)sinL + g] µ q [hsinL kcosL]7 A = 6 q 7 6 2 7 6 0 0 p s cosL 7 6 µ 2q 7 6 q 7 6 p s2sinL 7 6 0 0 7 4 q µ 2q 5 p 1 − 0 0 µ q [hsinL kcosL] p T q 2 b = [0 0 0 0 0 µp( p ) ] where q = 1 + fcosL + gsinL p r = q α2 = h2 − k 2 p χ = h2 + k 2 s2 = 1 + χ2. The following expressions relate the equinoctial coordinates to Cartesian coordinates. 2 3 r 2 6 2 (cosL + α cosL + 2hksinL)7 6 s 7 6 7 r = 6 r (sinL − α2sinL + 2hkcosL) 7 4 s2 5 2r − s2 (hsinL kcosL) 10 2 q 3 − 1 µ (sinL + 2sinL − 2hkcosL + g − 2fhk + 2g) 6 s2 p α α 7 6 q 7 6 1 µ 2 2 7 v = 6− 2 (−cosL + α cosL − 2hksinL − f − 2ghk + α f )7 4 s p q 5 2 µ s2 p (hcosL + ksinL + fh + gk) The disturbing acceleration is defined by ∆ = ∆g + ∆T where ∆g is due to oblate earth effects and ∆T is from thrust. The coordinate axes of the disturbing acceleration are given as[ ] r (r × v) × r r × v Q = [i i i ] = r r θ h jj r jj jj r × v jjjj r jj jj r × v jj in a rotating radial frame. A local horizontal reference frame for oblate gravity models is δg = δgnin − δgr ir where e − (eT i )i i = n n r r . n jj T jj en(en ir )ir Ignoring tesseral harmonics and focusing solely on the first four zonal harmonics, gravitational perturbations can be calculated with µ cos ϕ X4 Re δg = − ( )k P0 J n r 2 r k k k=2 µ X4 Re δg = − (k + 1)( )k P J r r 2 r k k k=2 where ϕ is the geocentric latitude, Re is the radius of the earth, Pk is the k-th order Legendre polynomial and Jk are the zonal harmonic coefficients.