Preliminary Mission Analysis and Design for a Small Satellite Swarm

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...... 46 4.7 Area projection. [10]...... 47 4.8 Pitch torque profile for true anomaly, ν = 270◦ ...... 48 4.9 Pitch torque profile for true anomaly, ν = 0◦ ...... 48

vi List of Figures List of Figures

4.10 Pitch torque of separate components, for true anomaly, ν = 270◦ ...... 49 4.11 Pitch torque of separate components, for true anomaly, ν = 0◦ ...... 50 4.12 Yaw torque profile...... 50 4.13 Roll torque profile...... 51 4.14 Yaw torque of separate components...... 51 ◦ 4.15 Euler angles φi for different ωi,t=t0 , and panel angle of θ = −45 ...... 53 ◦ 4.16 Euler angles φi for different ωi,t=t0 , and panel angle of θ = −60 ...... 53

5.1 Orbit Trajectory...... 55 5.2 Attitude angles for diﬀerent panel angle...... 56

6.1 Reference coordinate frames. [11]...... 59 6.2 Relative position and velocity, in a circular orbit...... 62 6.3 Relative position and velocity, in a elliptical orbit...... 63

vii List of Tables

2.1 Orbital Parameters. [2]...... 6 2.2 Classical Orbital Elements...... 6 2.3 The ﬁrst six Earth zonal harmonics.[11]...... 8

3.1 Atmospheric model input parameters...... 19 3.2 Initial orbital decay conditions...... 34

4.1 Satellite dimensions and mass properties...... 39 4.2 Location of the satellite center of mass...... 40 4.3 The principal moments of inertia of the satellite...... 40 4.4 Location of the radius of the shaded panel length...... 47 4.5 Initial orbit simulation conditions...... 52 4.6 Initial conditions for pitch dynamics for diﬀerent angular velocities with a panel angle, θ = −45◦...... 52 4.7 Initial conditions for pitch dynamics for diﬀerent angular velocities with a panel angle, θ = −60◦...... 53

5.1 Orbit radius and velocity at perigee...... 56 5.2 Orbit radius and velocity at perigee...... 56 5.3 Initial condition for diﬀerent panel angles...... 56

viii

Chapter 1

Introduction

1.1 Background

The development of small satellite technology has shown to reduce cost and shorten development time. This has led to an aﬀordable way for universities to send experiments and instruments into space. The CubeSat (10 × 10 × 10 cm3 with mass ≤ 1 kg) was jointly standardized by Standford University and California Polytechnic University. The satellite allows up to two or three cubes to be connected together to construct a larger CubeSats (2-U or 3-U). Figure 1.1 shows CAD drawings for the skeleton of a 2-U and 3-U cube satellites [1]. As cube satellites are designed with low cost access to space in mind, substantial constraints are put on the satellite’s mass, volume and power subsystem. In order to minimize power consumption, passive attitude control methods, such as gravity gradient and aerodynamic controls are ideal for such platforms.

Currently, Kungliga Tekniska Högskolan (KTH ) has been involved in CubeSat projects, such as, the SWIM Project. On October 2011, the Swedish National Space Board (SNSB), called for ideas of innovative low-cost scientific satellite missions. A scientific mission to study the structures and dynamics of magnetic field-aligned currents, Alfvénwaves, density cavities and accelerated particles, which are associated with the aurora, was proposed by the Space and Plasma Physics department at KTH, Stockholm, Sweden. The mission objective is to make multi-scale in situ observations between the altitudes of 6000 km and 200 km, where these altitudes represent the auroral acceleration region down to the auroral ionosphere. The expected duration of the mission is approximately 3 months.

Figure 1.1: Illustration of 2-U and 3-U Cube Satellite. [1]

2 Chapter 1. Introduction 1.2. Problem Statement

1.2 Problem Statement

The mission requires multiple measuring points which lead to the concept of using a closely spaced 3-U CubeSat formation to make an in situ observation of the aurora. The formation is made of 10 or more 3-U CubeSat released from a single launcher. For a CubeSat, magnetorquers, gravity gradient torques, and aerodynamic torques are desired for the formation maintenance and attitude stabilization.

1.3 Scope

The MATLAB program developed previously for the SWIM Project is expanded to suit this project [12]. The thesis aims to understand the orbit trajectory and satellite geometry that would meet the mission requirements. A suitable insertion orbit for the mission is investigated, by analyzing the inﬂuence of the upper atmosphere. Additionally, passive methods of attitude control using aerodynamic torque and gravity gradient torque are also examined. A brief account of a leader-follower satellite formation is investigated as an attempt to understand the complexity of formation ﬂying.

1.4 Thesis Overview

The following chapters of this thesis are Theoretical Background, Orbital Decay, Attitude Simulations, Passively Stabilized Perigee Passage, Satellite Formation Design, and Conclusion.

Theoretical Background covers the theory in the subjects of orbital mechanics, attitude kinematics and attitude dynamics.

The Orbital Decay chapter ﬁrst introduces the atmospheric model that is used, and discuss the ad- vantages and disadvantages between the Jacchia J70 and NRLMSISE-00 atmospheric model. Then in the next section, the orbital decay simulation results are discussed.

Attitude Simulations studies the feasibility of using aerodynamic drag panels as a mean of decelerating the satellite as well as attitude control during the perigee passage. This chapter explores the use of partial accommodation coeﬃcient on the satellite surfaces, and satellite self-shadow. The analysis in this chapter aims to provide an understanding of aerodynamic torque on the principal axis.

The Passively Stabilized Perigee Passage chapter explores the stability of the satellite during the perigee passage. Simulations are done for each panel angle, in order to obtain the initial conditions at the entrance of the perigee trajectory.

Satellite Formation Design touches on the subject of formation ﬂying of two satellites in a leader-follower formation. Results from the simulations on the relative translation between the leader and follower satellite formation are discussed.

3 Chapter 2

Theoretical Background

2.1 Orbital Mechanics 2.1.1 The Two-Body Problem The equations of motion of two mass particles interaction is described by Newton’s law of gravitation. It states that two particles attract each other with a force, acting along the line between them, is inversely proportional to the square of the distance between them and proportional to the product of their masses. The interaction can be described analytically and show the motion of a two particles system whose masses are m1 and m2. Let the position vector (r1, r2) and velocity vectors of the (v1, v2) be expressed with respect to an Earth-Centered coordinate system as

r1 = x1 · ix + y1 · iy + z1 · iz

r2 = x2 · ix + y2 · iy + z1 · iz (2.1) dr dx dy dz v = 1 = 1 · i + 1 · i + 1 · i 1 dt dt x dt y dt z dr dx dy dz v = 2 = 2 · i + 2 · i + 2 · i 2 dt dt x dt y dt z Then let, p r12 = |r2 − r1| = (r2 − r1) · (r2 − r1) (2.2) represent the distance between mass m1 and m2, in order for the magnitude of the force attracting the Gm1m2 two particles to be 2 . G here is the universal graitational constant. Furthermore, the directions of r12 force can be expressed in unit vector terms, where the force acting on m1 due to m2 has the direction (r2−r1) . The force on m2 due to m1 is then the opposite direction. Therefore, total force f1 acting on r12 m1 is and f2 acting on m2 are written as,

m1m2 f1 = G 3 (r2 − r1) r12 (2.3) m2m1 f2 = G 3 (r1 − r2) r21 Substituting Newton’s second law of motion, d2r dv f = m 1 ≡ m 1 1 1 dt2 1 dt (2.4) d2r dv f = m 2 ≡ m 2 2 2 dt2 2 dt

4 Chapter 2. Theoretical Background 2.1. Orbital Mechanics

into equation (2.4) yields

2 d r1 m2 2 = G 3 (r2 − r1) dt r12 (2.5) 2 d r2 m1 2 = G 3 (r1 − r2) dt r21

2.1.2 Equation of Relative Motion The equation of relative motion of two mass particles, can be obtained from equations (2.4) and (2.5). Then, the equation of motion of two bodies can be described by the pair of non-linear diﬀerential equations [13]. The pair of equations in (2.5), are combined to give the form,

d2r µ + r = 0 (2.6) dt2 r3 where r = r2 − r1 and µ = G(m1 + m2).

Equation (2.6) is the fundamental diﬀerential equation of the two-body problem. The satellite’s orbital position and velocity are determined using the two-body problem. It is important to note that the assumptions used in the derivation of equation (2.6) are, gravity is the only force, the Earth is spherically symmetric, and the Earth and the satellite are the only two bodies in the system. [2]

Figure 2.1: Geometry of an Ellipse and Orbital Parameters. [2]

5 Chapter 2. Theoretical Background 2.1. Orbital Mechanics

r: position vector of the satellite relative to Earth’s center. V: velocity vector of the satellite relative to Earth’s center. φ: ﬂight-path-angle, the angle between the velocity vector and a line perpendicular to the position vector. a: semi-major axis of the ellipse. b: semi-minor axis of the ellipse. c: the distance from the center of the orbit to one of the focii. ν: the polar angle of the ellipse, also called true anomaly, measured in the direction of motion from the direction of perigee to the position vector. rA: radius of apogee, the distance from Earth’s center to the farthest point on the ellipse. rP : radius of perigee, the distance from Earth’s center to the nearest approach to the Earth.

Table 2.1: Orbital Parameters. [2]

Figure 2.1 shows the parameters of an elliptical orbit and table 2.1 deﬁnes the variables. For a satellite orbiting Earth, the polar equation of a conic section is a solution to the two-body equation of motion. The solution yields the magnitude of the position vector in terms of the location in the orbit is,

a(1 − e2) r = (2.7) 1 + e cos(ν) where a is the semi-major axis, e is the eccentricity, and ν is the polar angle or true anomaly. The ratio c a in ﬁgure 2.1, is equivalent to the eccentricity e for the ellipse.

2.1.3 Classical Orbital Elements The solution of the two-body equations of motion is solved with six integration (initial conditions) constants. The results from the six integration constants give the three components of position and velocity of the orbit at any time. As an alternative, the orbit can be described with ﬁve constants and one quantity varying with time. These six parameters, called classical orbital elements, are deﬁned in table 2.2.

a: semimajor axis e: eccentricity i: inclination Ω: right ascension of the ascending node ω: argument of perigee ν: true anomaly

Table 2.2: Classical Orbital Elements

The semi-major axis a is the distance from the center of the ellipse to the farthest edge of the ellipse. All conic sections can be deﬁned in terms of eccentricity e, where an ellipse has an eccentricity of 0 < e < 1 and the eccentricity of a circle is e = 0. The inclination of the orbital plane i is the angle between the angular momentum vector and the unit vector in the Z-direction. The right ascension of the ascending node Ω is deﬁned as the angle from the vernal equinox to the ascending node. The ascending node is the point where the satellite passes through the equatorial plane moving from south to north. Right ascension is measured as a right-handed rotation about the pole, Z axis. The argument of perigee ω is

6 Chapter 2. Theoretical Background 2.1. Orbital Mechanics the angle from the ascending node to the eccentricity vector measured in the direction of the satellite’s motion. The eccentricity vector points from the center of the Earth to perigee with a magnitude equal to the eccentricity of the orbit. And lastly, the true anomaly ν is the angle from the eccentricity vector to the satellite position vector, which is measured in the direction of the satellite’s motion. Instead of using true anomaly, the time since perigee passage T could also be used [2]. Figure 2.2 shows all six classical orbital elements.

Figure 2.2: Classical Orbital Elements. [3]

2.1.4 Special Perturbation: Cowell’s Method There are two main methods that are used to calculate perturbations: Cowell’s method and Encke’s method. In this work, Cowell’s Method is used due to its straight forward step-by-step integration of the two-body equation. From equation (2.6), the equation of motion may be given to include the perturbation accelerations:

d2r µ + r = a (2.8) dt2 r3 p Encke’s method computes the dominant trajectory by using the closed-form Keplerian solution. Then it numerically solves a second order differential equation for the deviations δ from the two-body solution. The difference between the primary acceleration and all perturbing accelerations is integrated. The reference orbit is called the osculating orbit. The osculating orbit is the resultant orbit if there were no perturbing accelerations at a particular time. Figure 2.3 shows that at an initial time the osculating and true orbits are in contact. Any particular osculating orbit is fine until the true orbit deviates too far from it. Then a rectification process is done to continue the integration. The rectification means that a new time and a starting point will be chosen to coincide with the true orbital path. After that the true radius and velocity vector is used to calculate the new osculating orbit.

Figure 2.3: Illustration of Encke’s method. [4]

7 Chapter 2. Theoretical Background 2.1. Orbital Mechanics

For the numerical integration of Cowell’s method, equation (2.8) would be reduced to a first-order ordinary differential equation, where ap is the vector sum of all perturbing accelerations to be included in the integration [6]. Perturbation accelerations that are commonly included are gravitational potential, atmospheric drag, third body attraction, solar pressure, and magnetic field. The gravitational potential and atmospheric drag perturbations are the main concern of this work, since other perturbations are very small and therefore could be ignored.

2.1.5 Orbit Perturbation: Aerodynamic Drag The aerodynamics drag is one of the inﬂuences that aﬀect orbit trajectory when a satellite is in low-Earth orbit. At altitudes below 1500 km, the satellite experiences air molecules in the direction of motion. Thus, the resultant force on the surface of the satellite, from the change of momentum of air molecules, is known as atmospheric drag. The atmospheric drag force re-written in the form of acceleration,

−1 C A a = ρV 2 D i (2.9) D 2 m v where ρ is the atmospheric density, V is the velocity, CD is the coeﬃcient of drag, m is the mass of the satellite, A is the eﬀective projection area of the satellite, and iv is the unit vector of the satellite velocity relative to the atmosphere.

The acceleration due to drag is a function of local density of the atmosphere, and the cross-sectional area of the satellite in the direction of motion. The value of coeﬃcient of drag is addressed later in sections 3.2.2 and 4.2.2.

2.1.6 Orbit Perturbation: Non-spherical Earth A spherically symmetric mass distribution Earth is assumed when developing the two body equations of motion. However, in reality the Earth has a bulge at the equator and ﬂattening at the poles. The satellite’s acceleration can be found by taking the gradient of the gravitational potential function, Φ. A widely used gravity potential function is

" ∞ # µ X RE Φ = − 1 − ( )nJ P (sinL) (2.10) r r n n n=2 where µ = GM is the Earth’s gravitational constant, Pn are Legendre polynomials, RE Earth radius, Jk are the dimensionless geopotential coefficients, L is a geocentric latitude. The Earth’s geopotential coefficients are given by the first six zonal harmonics in table 2.3.

J2 = 1082e-6

J3 = -2.52e-6

J4 = -1.61e-6

J5 = -0.15e-6

J6 = 0.57e-6

Table 2.3: The ﬁrst six Earth zonal harmonics.[11]

The J2 harmonic represents the oblateness perturbation. As seen in table 2.3, the J2 is the dominant harmonic which causes noticeable precession at Low-Earth orbits. Knowing the harmonics, the gravitational perturbation function for J2 is then given by,

8 Chapter 2. Theoretical Background 2.2. Attitude Kinematics

J µ R 2 R(r) = − 2 e 3 sin2φ − 1 (2.11) 2 r r where Re is Earth radius.

Then by computing the gradient of R(r) and substituting z/r into sinφ, the perturbation acceleration aJ2 due to J2 is [11],

 2  1 − 5 z x 2 r r 3 µ R  2  a = − J e  1 − 5 z y  (2.12) J2 2 2 r2 r  r r   z 2 z  3 − 5 r r where aJ2 is now given in terms of inertial Cartesian Coordinates.

In a simulation that requires higher precision, higher order of aJi perturbation accelerations can be added.

2.2 Attitude Kinematics 2.2.1 Coordinate System and Rotation Matrix The orientation of a satellite is described by the attitude kinematics which involves a body-ﬁxed reference frame. The satellite’s orbital position and velocity are described by the orbital reference frame. The inertial reference frame that is used is the Earth-Centered Inertial (ECI). These are three diﬀerent coordinate frames used in this work.

The body-ﬁxed frame B = (b1, b2, b3), is an orthogonal principal axis where b1 is the roll axis, b2 is the pitch axis, and b3 is the yaw axis. The orientation of the body is shown between the body-ﬁxed frame B and the orbital frame A.

The local vertical local horizon reference frame A = (a1, a2, a3), is a rotating frame where the a1 vector points along the satellite’s orbital path, a3 vector points to the center of the Earth and a2 vector is orthogonal to the other two vectors.

The Earth-Centered Inertial frame N = (n1, n2, n3), has the vector n1 pointing to the vernal equinox, the n3 vector points out of the Earth’s North pole, and the n2 vector is orthogonal to the other two vectors.

The coordinate frames, along with the body-ﬁxed frame above are shown in ﬁgure 2.4.

Figure 2.4: Shows ECI as N, A and B reference frame. [5]

9 Chapter 2. Theoretical Background 2.2. Attitude Kinematics

2.2.2 Direction Cosine Matrix The displacements of the body-ﬁxed referenced frames are used to describe the satellite orientations. Consider a right-hand set of three orthogonal unit vectors {a1, a2, a3} to be reference frame A and another right-hand set of three orthogonal unit vectors {b1, b2, b3} to be reference frame B. The rotation matrix that rotates the vectors A to the vectors B is in the form,     b1 · a1 b1 · a2 b1 · a3 b1 B/A R = b2 · a1 b2 · a2 b2 · a3 = b2 · a1 a2 a3 (2.13) b3 · a1 b3 · a2 b3 · a3 b3

B/A where R is simply called direction cosine matrix. The scalar product bj ·ai is the cosine angle between T T bj and ai. Generally, a square matrix A can be called orthogonal if AA (note that AA = I where I is the identity matrix) is a diagonal matrix and it is called an orthonormal matrix if AAT is an identity matrix. For matrix A to be orthonormal it has to have A−1 = AT and |A| = ±1. In addition, the rotation of the ﬁrst, second and third axes of the reference frame A are described by the following three elementary rotation matrices:

1 0 0  R1(θ1) = 0 cos θ1 sin θ1  (2.14) 0 − sin θ1 cos θ1

  cos θ2 0 − sin θ2 R2(θ2) =  0 1 0  (2.15) sin θ2 0 cos θ2

  cos θ3 sin θ3 0 R3(θ3) = − sin θ3 cos θ3 0 (2.16) 0 0 1

th where Ri(θi) is the direction cosine matrix C of an elementary rotation about the i axis of A with an angle θi.

2.2.3 Euler Angles Euler Angles are the most commonly used sets of attitude parameters to describe the reference frame B relative to an arbitrary reference frame N, not to be confused with Earth-centered inertial frame. This is done through a consecutive rotation angles (θ1, θ2, θ3) about the displaced body-fixed axes b. It is good to note that the order of consecutive rotation of the three angles euler angles is important. For rotations about the third, second, and first body axis shown as (3-2-1), does not yield the same orientation as a (1-2-3) rotation. The Euler angles (3-2-1) describe the satellite’s yaw, pitch, and roll (ψ, θ, φ). The three rotations are represented by a set of direction cosine matrices introduced previously in equations (2.14), (2.15), and (2.16). To transform components from vector N frame into B frame done through a sequence of Euler angle rotations, the reference axes are first rotated about the n3 axis by the yaw angle ψ, then about the n2 axis by the pitch angle θ, and finally rotated about the n1 axis by the roll angle φ. The rotation sequence (3-2-1) is shown in figure 2.5.

10 Chapter 2. Theoretical Background 2.2. Attitude Kinematics

Figure 2.5: Euler angle rotation sequence (3-2-1). [6]

The Euler angle sequence for a 3-2-1 rotation from vector N frame into B frame is deﬁned as:

B/N R = R(θ1)R(θ2)R(θ3) (2.17) Therefore the resultant direction cosine matrix in terms of (3-2-1) Euler angles   cos θ3 cos θ2 sin θ3 cos θ1 + cos θ3 sin θ2 sin θ1 sin θ3 sin θ1 − cos θ3 sin θ2 cos θ1 B/N R = − sin θ3 cos θ2 cos θ3 cos θ1 − sin θ3 sin θ2 sin θ1 cos θ3 sin θ1 + sin θ3 sin θ2 cos θ1 (2.18) sin θ2 − cos θ2 sin θ1 cos θ2 cos θ1

2.2.4 Euler’s Principal Rotation The Euler’s principal rotation theorem states that, a rigid body or coordinate frame of reference can be brought from its initial orientation to an arbitrary ﬁnal orientation by a single rotation through a principal angle Φ about the principal axis eˆ [6]. The Euler’s principal rotation is also known as Euler axis or eigenaxis. The ﬁgure 2.6, visualizes this theorem.

Figure 2.6: Illustration of Euler’s principal rotation theorem. [6]

Let unit vector eˆ be represented in B and N frame components as ˆ ˆ ˆ eˆ = eb1 b1 + eb2 b2 + eb3 b3 (2.19)

eˆ = en1 nˆ1 + en2 nˆ2 + en3 nˆ3 (2.20) According to the Euler’s principal rotation theorem, the components in frame B as well as in frame N has the same vector components ine ˆ, ie. ebi = eni = ei [6]. The Euler principal rotation of N frame to B frame is then described by a rotation matrix [RB/N ] as seen in equation (2.21).

11 Chapter 2. Theoretical Background 2.2. Attitude Kinematics

    e1 e1 B/N e2 = [R ] e2 (2.21) e3 e3 This leads to the parametrization of the direction cosine matrix

 2  e1(1 − cosΦ) + cosΦ e1e2(1 − cosΦ) + e3sinΦ e1e3(1 − cosΦ) − e2sinΦ B/N 2 R = e2e1(1 − cosΦ) − e3sinΦ e2(1 − cosΦ) + cosΦ e2e3(1 − cosΦ) + e1sinΦ (2.22) 2 e3e1(1 − cosΦ) + e2sinΦ e3e2(1 − cosΦ) − e1sinΦ e3(1 − cosΦ) + cosΦ

B/N The direction cosine matrix [R ] depends on four scalar quantities e1, e2, e3, Φ. It is good to note 2 2 2 that the vector component ei must abide by the unit constraint e1 + e2 + e3 = 1. [6]

2.2.5 Euler Parameters (Quaternions) The Euler parameters or quaternions, are deﬁned from Euler principal rotation as such: Φ q = e sin( ) 1 1 2 Φ q2 = e2sin( ) 2 (2.23) Φ q = e sin( ) 3 3 2 Φ q = cos( ) 4 2 where the e = (e1, e2, e3) is the eigenaxis and Φ is the rotation angle. Since the constraints on the 2 2 2 2 2 2 2 eigenaxis is e1 + e2 + e3 = 1, the quaternions also must satisfy a similar constraint q1 + q2 + q3 + q4 = 1. It is good to note that this constraint geometrically describes a four-dimensional unit sphere. For a given attitude, there are two sets of quaternions that will describe the same orientation. This is due to non-uniqueness of the eigenaxis rotations themselves. The sets of (eˆ, Φ) and (−eˆ, Φ) will result in the same quaternion vector ~q.[6]

From quaternions, the direction cosine matrix that relates the inertial coordinate to the body-ﬁxed coordinate can be written as:

 2 2  1 − 2(q2 + q3) 2(q1q2 + q3q4) 2(q1q3 − q2q4) B/N 2 2 R = 2(q2q1 − q3q4) 1 − 2(q1 + q3) 2(q2q3 + q1q4) (2.24) 2 2 2(q3q1 + q2q4) 2(q3q2 − q1q4) 1 − 2(q1 + q2) Unlike the previous section, the superscript and subscript denoted as N is the inertial coordinate frame.The inverse transformation from rotation matrix R to the eigenaxis can be found through in- spection of equation (2.24) to be

R23 − R32 q1 = 4q4 R − R q = 31 13 2 4q 4 (2.25) R12 − R21 q3 = 4q4 1p q = ± R + R + R + 1 4 2 11 22 33 A very important composite rotation property of the quaternion is that they allow an overall combine rotation consisting of two sequential rotations. Let the quaternion vector q0 describe the ﬁrst, q00 describe

12 Chapter 2. Theoretical Background 2.2. Attitude Kinematics the second, and q the composite rotation. Equation (2.26) describes the rotation from N frame to F frame through the B frame.

[FN(q)] = [FB(q00)][BN(q0) (2.26) Using equation (2.24) and substituting it into equation (2.26) and equating corresponding elements yields the following transformation       q1 q400 q300 −q200 q100 q10 q2 −q300 q400 q100 q200 q20   =     (2.27) q3  q200 −q100 q400 q300 q30 q4 −q100 −q200 −q300 q400 q40 Equation (2.27), shows the quaternion multiplication rule in matrix form. The 4 by 4 matrix is an orthonormal matrix and is called the quaternion matrix.

2.2.6 Kinematic Diﬀerential Equations In previous sections, the orientation of a reference frame is described in terms of direction cosine matrix, Euler angles, and quaternions. In this section the relative orientation between two reference frames as time dependent. Kinematic diﬀerential equation represents the time-dependent relationship between two reference frames.

Consider two reference frames A = (a1, a2, a3) and B = (b1, b2, b3). Reference frame B is rotating with respect to reference frame A with an angular velocity of ω ≡ ωB/A. The angular velocity expressed in terms of vectors of B as,

ω = ω1b1 + ω2b2 + ω3b3 (2.28) The relationship between reference frame A to reference frame B and frame B to A is written as,         a1 b1 b1 a1 T a2 = R b2 ←→ b2 = R a2 (2.29) a3 b3 b3 a3 where R is the direction cosine matrix RB/A.

Since reference frame A and B are rotating relative to each other, the direction cosine matrix R is a function of time. Then taking the time derivative of of equation (2.29) yields,

     ˙  0 b1 b1 ˙ T T ˙ 0 = R b2 + R b2 (2.30a) 0 b3 b˙3     b1 ω × b1 ˙ T T = R b2 + R ω × b2 (2.30b) b3 ω × b3       b1 0 −ω3 ω2 b1 ˙ T T = R b2 − R  ω3 0 −ω1 b2 (2.30c) b3 −ω2 ω1 0 b3

Here Ω is deﬁned as the skew symmetric matrix from equation (2.30),   0 −ω3 ω2 Ω =  ω3 0 −ω1 (2.31) −ω2 ω1 0 then the equation (2.30) is rewritten as,

13 Chapter 2. Theoretical Background 2.3. Attitude Dynamics

    b1 0 h ˙ T T i R − R Ω b2 = 0 (2.32) b3 0 which then simpliﬁed as,

R˙ T − RT Ω = 0 (2.33) Then knowing that the transpose of a skew symmetric matrix is ΩT = −Ω, equation (2.33) becomes,

R˙ T + ΩR = 0 (2.34) where the equation is called the kinematic diﬀerential equation for the direction cosine matrix R. In this thesis, equation (2.34) is rewritten in terms of quaternions. The quaternion is a non-singular representa- tion of the Euler angles, hence, the kinematic diﬀerential equation for quaternions is as,       q˙1 0 ω3 −ω2 ω1 q1 q˙2 1 −ω3 0 ω1 ω2 q2   =     (2.35) q˙3 2  ω2 −ω1 0 ω3 q3 q˙4 −ω1 −ω2 −ω3 0 q4 Then from equation (2.35), the equation is rewritten in the form, 1 q˙ = (q ω − ω × q) (2.36) 2 4 1 q˙ = − ωT q (2.37) 4 2 where q = [q1, q2, q3].

2.3 Attitude Dynamics 2.3.1 Rotational Dynamics Attitude and determination control analysis describes the rotational behavior of a satellite body frame subjected to the forces imposed upon it. The analysis involves the use of Newton’s law of motion; therefore, in order to obtain the rotational behavior of the satellite, the velocity and acceleration needs to be described in the inertial frame. From ﬁgure 2.7, ρ is a vector given in the body frame of reference with an angular velocity ω. The time diﬀerential equation for a rotating frame is written as,

dρ dρ = + ω × ρ (2.38) dt i dt b

Figure 2.7: Time diﬀerentiation in rotating frame. [7]

The body frame ρ then is written in the inertial frame as,

14 Chapter 2. Theoretical Background 2.3. Attitude Dynamics

r = R + ρ (2.39) Then diﬀerentiate equation (2.39) and substitute equation (2.38) into (2.39) to get the velocity,

dr dR dρ v = = + + ω × ρ (2.40) dt i dt dt b and diﬀerentiating once more for the acceleration d2ρ d2R d2ρ dρ dω a = = 2 + 2 + 2ω × + × ρ + ω × (ω × ρ) (2.41) dt i dt dt b dt b dt The third term on the right in equation (2.41) is the Coriolis force, and the last term on the right-hand side is called the centrifugal force.

In rotational dynamics, the quantities of interest are moment of inertia, angular momentum, and rotational kinetic energy. The angular momentum of a mass is the moment of its linear momentum about the origin. As seen in ﬁgure 2.7, in the inertial frame, the angular momentum of mass mi about the origin is written as,

H = ri × mivi (2.42) then the total angular momentum is, X Ht = ri × mivi (2.43) Applying equations (2.39) and (2.40) with V = dR/dt, and assuming that the origin of the rotating P frame lies on the body’s center of mass ( mjρj = 0), and the position vectors ρj are ﬁxed in the body frame ( dρ/dt = 0), the equation for the total angular momentum becomes,

X X dρi H = m R × V + m ρ × = H + H (2.44) t i i i dt orb b The ﬁrst term describes the angular momentum of the rigid body due to its translational velocity V in the inertial frame. The second term describes the body angular momentum due to its rotational velocity about the body’s center of mass. Then by assuming a rigid body, equation (2.38) becomes, dρ j = ω × ρ (2.45) dt i From equation (2.44), the body angular momentum is rewritten as,

X dρi X H = m ρ × = m ρ × (ω × ρ ) = Iω (2.46) i i dt i i i where I is the inertia matrix. The components of the matrix are,

X 2 2 Ixx = mi ρi2 + ρi3 (2.47a) X 2 2 Iyy = mi ρi1 + ρi3 (2.47b) X 2 2 Izz = mi ρi1 + ρi2 (2.47c) X Ixy = Iyx = − miρi1ρi2 (2.47d) X Ixz = Izx = − miρi1ρi3 (2.47e) X Iyz = Izy = − miρi2ρi3 (2.47f)

The diagonal components of the matrix are (Ixx,Iyy,Izz), which are the principal moments of inertia. Here the rotational equations for a rigid body are derived. Consider the cross product of force (Fi) and a body position (ρi),

15 Chapter 2. Theoretical Background 2.3. Attitude Dynamics

Ti = ρi × Fi (2.48) The cross product is the torque, which can be summed to be the net torque from all such forces is then,

X X d2r T = ρ × F = ρ × m (2.49) i i i i dt2 2 dri The result of the expansion of the dt2 term is, dH dH T = = + ω × H (2.50) dt dt body Here the body is considered to be a lumped mass, where the total kinetic energy is given by,

2 2 1 X dri 1 X dRi dρi E = = m + (2.51) 2 dt 2 i dt dt

Then, by choosing the center-of-mass for the ρi, equation (2.51) can be separated into translational and rotational components

2 2 1 X dRi 1 X dρi E = m + m = E + E (2.52) 2 i dt 2 i dt trans rot After expanding equation (2.52) with a rigid body assumption such that H = Iω, the rotational component becomes, 1 E = ωT Iω (2.53) rot 2 Assuming a body-ﬁxed center-of-mass system in which H,T and ω are expressed, equation (2.50) is rearranged as, dH = T − ω × Iω (2.54) dt body then expanding into components

H˙1 = Ixxω˙ 1 = T1 + (Iyy − Izz) ω2ω3 (2.55a)

H˙2 = Iyyω˙ 2 = T2 + (Izz − Ixx) ω3ω1 (2.55b)

H˙3 = Izzω˙ 3 = T3 + (Ixx − Iyy) ω1ω2 (2.55c)

Equation (2.55), the Euler equations for the motion of a rigid body with inﬂuence of an external torque. The external torques that are considered in this work are the Gravity Gradient Torque, Magnetic Torque, and Aerodynamic Torque. These torques are subsumed into the three torque components (T1,T2,T3) in equation (2.55). Thus, the net external torque on the system can be summed together as such, Tnet = Tgrav + Tmag + Taero.

2.3.2 Gravity Gradient Torque The gravitational field of Earth decreases as the distance r from the center of Earth increases, stated by Newton’s 1/r2 law, as long as higher order harmonics in section 2.1.6 are neglected. As a result, a satellite at lower orbit is going to experience greater gravitational attraction than a satellite at higher orbits. A rigid body with unequal principal moments of inertia are affected by the differential attraction causing a torque to rotate the satellite to align its minimum inertial axis with the local vertical. Figure 2.8 shows a satellite in low-Earth orbit, where it displays the gravitational acceleration vector. If the satellite is perturbed from its equilibrium, a restoring torque is produced towards the stable vertical position, where it causes a periodic oscillatory motion (libration motion). However, satellite’s dissipated

16 Chapter 2. Theoretical Background 2.3. Attitude Dynamics energy will damp this motion [7].

The gravity gradient torque for a satellite is given by, Z Tgrav = r × agdm (2.56) where the gravitational acceleration, ag, is R + r a = −µ (2.57) g |R + r|3 Here µ is the Earth’s gravitational parameter (3.98601 × 105 km3/s2), R is the center of mass of the satellite with respect to Earth.

Figure 2.8: Gravity gradient torques on a low-Earth orbit satellite. [8]

To subsume the torque Tgrav in Euler’s equation, the integral (2.56) is evaluated in the principal-axis body frame. Here let the components R = [R1,R2,R3] be in the body frame and the components r = [r1, r2, r3]. Then develop equation (2.56), as in Wiesel, W [8], the gravity gradient torque becomes, 3µ T = R × IR (2.58) grav R5 I is the principal moments of inertia. The torque components are then separately written in the body-ﬁxed frame,

3µ T = R R (I − I ) (2.59a) grav1 R5 2 3 zz yy 3µ T = R R (I − I ) (2.59b) grav2 R5 1 3 xx zz 3µ T = R R (I − I ) (2.59c) grav3 R5 1 2 yy xx

Observing equations (2.59), it is good to note that only an asymmetrical body will experience gravitational torque due to the diﬀerence in principal moment of inertia.

2.3.3 Magnetic Torque The Earth’s magnetic field is another torque that is exerted on to the satellite in low-Earth orbits. This torque is given by, Tmag = M × B (2.60) where M is the magnetic dipole moment due to current loops and residual magnetization in the satellite. B is the Earth magnetic field vector that is expressed in body-fixed coordinates. The magnitude is proportional to 1/r3, where r is the radius vector from the center of Earth to the satellite.

17 Chapter 2. Theoretical Background 2.3. Attitude Dynamics

2.3.4 Aerodynamic Torque The eﬀect of the upper atmosphere on the satellite decay was discussed in section 2.1.5. The same drag force also produces a disturbance torque on the satellite due to any oﬀsets between the aerodynamic center of pressure and the center of mass. The aerodynamic torque is written as,

Taero = rcp × Faero (2.61) where rcp is the center-of-pressure (CP) vector in body-ﬁxed coordinates and Faero is the aerodynamic force applied. Here, the aerodynamic force is the mass of the satellite multiplied by the same drag acceleration in section 2.1.5. Further expansion of aerodynamic torque will be discussed later in section 4.2.2, where partial accommodation of incoming particles are considered.

18 Chapter 3

Orbital Decay

In order to investigate the orbital decay, the Earth atmosphere must ﬁrst be examined since aerodynamic drag is the main cause of decay. Satellites in low earth orbits still encounter drag, lift, heating and experience orbit decay as a consequence of passing through the upper atmosphere. At higher altitudes such as 600 km, the perturbations in the orbital period are so minute that the drag can simply be taken into consideration without accurate knowledge of the atmosphere [2]. At intermediate altitudes, to be able to predict variations in the atmospheric density and generate orbit perturbations, atmospheric models need to be examined.

3.1 Comparing Atmospheric Models

The two empirical models used to predict the densities and temperatures of the Earth’s thermosphere (90 to about 600 km altitude) are the Jacchia J70 model and the US Naval Research Laboratory Mass Spectrometer Incoherent Scatter Radar Exosphere 2000 (NRLMSISE-00). Both models discussed in this section use similar input parameters. These parameters include time of day, day of year, altitude and geographic location of the satellite. The density variations are also expressed terms of solar and geomagnetic activity indices, such as the 10.7 cm solar ﬂux (F10.7 index) and the Ap for geomagnetic activity. The inputs used in both atmospheric models are shown in table 3.1.

F10.7 index (Maximum) = 225 F10.7 index (Minimum) = 75 Average Ap index (Maximum) = 20 Average Ap index (Minimum) = 8

Table 3.1: Atmospheric model input parameters.

3.1.1 Jacchia J70 Model The Jacchia models are based on satellite drag data collected from ground-based tracking of selected satellites [14]. The altitude of the model ranges from 90 to 2500 km, and the exospheric temperatures range from 600 to 2000 K. Any altitude below 90 km uses the standard atmosphere to estimate the density and temperature. It is important to note that during the time when the model was made, good observational data did not exist above 1100 km. Hence, all the output above the height are considered to be unconfirmed extrapolation [14]. The model’s prediction method is derived from the static diffusion model, where it defines temperature and chemical composition. The diffusion model provides densities that correspond with satellite drag observations, with rocket probe measurements [15].

In order to determine the variability of the Jacchia J70 model, the model is tested with diﬀerent seasons,

19 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models latitudes and solar activities. Figure 3.1 shows density maps comparing several months during high and low solar activity. As discussed above while it is important to see the density variation between diﬀer- ent solar activity conditions, it is also signiﬁcant to understand the variations between local solar time. Figure 3.1, shows the variations from night to day. The density variation on the daytime can be 3 times higher than the nighttime, during solar maximum. However, the dramatic variations in the density are lower during solar minimum, with only the daytime density being about one and a half times higher than the nighttime.

20 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

(a) Density map: September with high solar activity. (b) Density map: September with low solar activity.

(e) Density map: May with high solar activity. (f) Density map: May with low solar activity.

Figure 3.1: J70: Density maps at 300 km altitude of 3 months showing high and solar activity conditions.

Figure 3.2 provides a density ratio map by dividing the solar maximum over the solar minimum for various altitudes and latitudes of diﬀerent seasons (September, January, May). For the same seasons, ﬁgure 3.3 shows the atmospheric density as a function of altitude for solar maximum and solar minimum

21 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models conditions. During the summer months (September and May) a higher variability with solar activity is observed. At altitudes of approximately 150 km, the solar activity does not strongly affect the density as seen in figure 3.3. However, at higher altitudes such as 500-800 km, the density varies approximately 2 to 3 orders of magnitude between solar minimum and solar maximum. The order of magnitude is significant because the density variations would affect the satellite drag, which leads to a faster decay during the period of solar maximum.

(a) September (b) January

Figure 3.2: J70: Density map of the solar maximum-minimum ratio for diﬀerent altitudes and latitudes of diﬀerent months.

22 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

(a) September (b) January

Figure 3.3: J70: Log10 of density as a function of altitude for solar maximum and solar minimum at diﬀerent months.

Figure 3.4 shows the variation of density proﬁle normalized to the equatorial value. The densities are up to three times the equatorial value in the polar regions. However, the densities at the polar regions are lower during solar minimum, compared to the density ratio during solar maximum, for the months September and May. This shows that the density is dependent on latitude and season. At altitude of 200 km, density in the polar regions are 15% greater than the equatorial region during the summer season. On the contrary, at the same altitude there is a 10% decrease in the polar region density during the winter. These observations of the model are in agreement with the statement in Newton and Pelz, that a weak dependency of density on latitude and season exists such that in summer at 200 km height the polar region density is 15% higher than lower altitudes, whereas in winter the density decreased by 10% [16]. In addition, large density ratios (> 3) are observed only near the polar regions, both during solar maximum and solar minimum.

23 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

(a) Density ratio during September with high solar activity. (b) Density ratio during September with low solar activity.

(e) Density ratio during May with high solar activity. (f) Density ratio during May with low solar activity.

Figure 3.4: J70: Ratio of density values (normalized to the equator value) of high and low solar activity at longitude of −90◦.

24 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

3.1.2 NRLMSISE-00 Model The NRLMSISE-00 (Mass Spectrometer and Incoherent Scatter Radar Extended) model is an empirical density model, developed by the US Naval Research Laboratory, based on the MSISE-90 model. The NRLMSISE-00 model ﬁnds the total density by taking into account the contributions of N2, O, O2, He, Ar and H. The satellite total mass density data are from satellite accelerometers and from orbit determination (including Jacchia data), where it is valid from 0 to 1000 km. The incoherent scatter data provide the temperature and the molecular number oxygen density is from solar ultraviolet occultation aboard the Solar Maximum Mission [14]. The major diﬀerences between the NRLMSISE-00 model and the MSISE-90 model are: (1) the total mass density uses the drag and accelerometer data extensively, including Jacchia and Barlier data sets. (2) at altitudes above 500 km, the total mass density accounts + for large contributions from O and hot oxygen. (3) the SSM UV occultation data on O2 are included. (4) the temperature data from incoherent scatter data covers 1981-1997 [17][18].

The density variations as the satellite passes from the nighttime to daytime are represented in ﬁgure 3.5. During solar maximum, the daytime density is 3 times greater than the nighttime, which is similar to the Jacchia J70 model in ﬁgure 3.1. As for the variation during solar minimum, the daytime-nighttime variations are less, with the exception of the variation in the month of September. In September the density is a magnitude less than the months of January and May. During the months of January and May, the daytime density is 0.8 times greater than the nighttime. The NRLMSISE-00 model’s nighttime- daytime density variations are unevenly distributed with respect to the latitudes compared to the Jacchia J70 model. A reason for this might be due to the NRLMSISE-00 model’s inclusion of numerous satellite drag and accelerometer data [18].

25 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

(a) Density map: September with high solar activity. (b) Density map: September with low solar activity.

(e) Density map: May with high solar activity. (f) Density map: May with low solar activity.

Figure 3.5: NRLMSISE-00: Density maps at 300 km altitude of 4 months showing high and low solar activity conditions.

The density ratio map of the NRLMSISE-00 are shown in ﬁgure 3.6, where the density at solar maximum is divided by density at solar minimum conditions. These values are plotted for diﬀerent altitudes,

26 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models latitudes and seasons. In figure 3.6 the NRLMSISE-00 model shows an agreement with the Jacchia J70 model, that there is a greater variability with solar activity in the North Pole than the South Pole during the summer. Further, figure 3.7 shows the variability of the density during high and low solar activity. The density between solar maximum and solar minimum diverges at altitudes greater than 150 km. It is observed that at the altitude of 600 km, there is a density difference of 3 orders of magnitude between the solar maximum and solar minimum.

(a) September (b) January

Figure 3.6: NRLMSISE-00: Density map of the solar maximum-minimum ratio for diﬀerent altitudes and latitudes of diﬀerent months.

27 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

(a) September (b) January

Figure 3.7: NRLMSISE-00: Log10 of density as a function of altitude for solar maximum and solar minimum at diﬀerent months.

The density ratios of latitude to the equatorial density are presented in ﬁgures 3.8. These ﬁgures represent the density ratio of altitudes 100 to 1000 km and all latitudes. In summer months (September and May), the North Pole density is about two and half times greater than the equator. Similarly during the winter month (January), the South Pole density is approximately 4 to 5 times greater than the equator.

28 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

(a) September ratio of density values (normalized to the equa- (b) September ratio of density values (normalized to the equator value) maps of high solar activity tor value) maps of low solar activity

(c) January ratio of density values (normalized to the equator (d) January ratio of density values (normalized to the equator value) maps of high solar activity value) maps of low solar activity

(e) May ratio of density values (normalized to the equator (f) May ratio of density values (normalized to the equator value) maps of high solar activity value) maps of low solar activity

Figure 3.8: NRLMSISE-00: Ratio density map of diﬀerent months and solar activity

In order to compare the NRLMSISE-00 model to the Jacchia J70 model directly, the density ratio of the

29 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models two models are plotted in figure 3.9. The regions that the NRLMSISE-00 show significant differences are the North and South Poles. At solar minimum, the NRLMSISE-00 density is two times greater than the J70 model, at approximately 600 km altitude. However, at solar maximum, the Jacchia J70 model’s North Pole density is greater than the NRLMSISE-00 at altitudes of 1000 km. This is observed in figure 3.9.

30 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

(a) Density ratio during September with high solar activity. (b) Density ratio during September with low solar activity.

(e) Density ratio during May with high solar activity. (f) Density ratio during May with low solar activity.

Figure 3.9: Density ratio between NRLMSISE-00 over J70.

31 Chapter 3. Orbital Decay 3.1. Comparing Atmospheric Models

3.1.3 Summary of Atmospheric Models Jacchia J70 Model: The Jacchia J70 model’s altitude ranges from 0 km to 2,500 km, where the standard atmospheric model is used at altitudes below 90 km. The density is found to be dependent on latitude, season and solar activity.

1. Density variation due to local time at 300 km altitude. (a) The daytime density is 3 times greater than the night time density during solar maximum. (b) During solar minimum, the daytime density is 1.5 times greater than the nighttime density.

2. Density variation due to solar activity. (a) Solar activity does not aﬀect the density below the altitude of 150 km. (b) A density diﬀerence of 2 to 3 orders of magnitude are noticed at altitudes of 500-800 km.

3. Density variation due to latitude. (a) During the summer months, the density in the polar regions are 15% greater than the equatorial region. (b) During the winter month, the density in the polar regions decrease by approximately 10% compared to the equatorial regions.

NRLMSISE-00: The NRLMSISE-00 model’s altitude ranges from 0 km to 1,000 km. The density in this model is also dependent on latitude, season and solar activity.

1. Density variation due to local time at 300 km altitude. (a) During solar maximum, daytime density is 3 times greater than the nighttime. (b) For September during solar minimum, the daytime density is a magnitude lower than the daytime density of January and May. (c) For January and May, during solar minimum, the daytime density is approximately 0.8 times greater than the nighttime density. 2. Density variation due to solar activity.

(a) The solar activity does not affect the density at altitudes lower than 150 km. (b) Solar maximum and minimum affects the density at altitudes above 600 km, where there are density differences up to 3 orders of magnitude. 3. Density variation due to latitude.

(a) The North Pole density during solar maximum is approximately 2.5 times greather than the equatorial value. (b) The South Pole density during solar maximum is approximately 4 to 5 times greater than the equatorial value. (c) During solar minimum, the density variations happen both in the North Pole as well as the South Pole.

Even though the range of altitude covered by the Jacchia J70 model is greater, the detail of the density variations in the NRLMSISE-00 model is greater. Additionally, the NRLMSISE-00 also includes numerous satellite drag and accelerometer data on total mass density. Hence, the NRLMSISE-00 is more suitable for this work since the analysis of the satellite extensively uses atmospheric density.

32 Chapter 3. Orbital Decay 3.2. Decay Lifetime Simulation

3.2 Decay Lifetime Simulation

Low-Earth orbit satellites have a finite lifetime due to the effects of atmospheric drag. The effects of drag are cumulative and eventually become significant; therefore, it is necessary to understand these effects in order to determine the lifetime of the satellite. Here, simulations are run using different ballistic coefficients, M CBallistic = (3.1) Cd · A where M is the mass of the satellite, Cd is the coefficient of drag, and A is the ram-facing area. First, effects of atmospheric drag is analyzed, then the effect of Earth oblateness is added. As a result, by performing the decay lifetime simulations, an appropriate altitude for orbit insertion can be determined.

3.2.1 Methodology The simulations are done in MATLAB, where the equations of motion with perturbations (Cowell’s Method), in section 2.1.4, are solved using numerical integration Runge-Kutta(4-5) scheme. The built-in MATLAB Runge-Kutta(4-5) function ODE45 is used to solve the initial value problem, with both relative and absolute tolerance of 1 × 10−10.

Initial Condition Output r = ODE45 Integrator 0 u = [x, y, z, x,˙ y,˙ z˙] [x, y, z, x,˙ y,˙ z˙] Cowell’s Method t = t0 (time)

Figure 3.10: Block diagram represents the orbit propagation simulation process.

Here from equation (2.8), consider that r = [x, y, z, x,˙ y,˙ z˙] be the state-variable in the Cartesian coordinate form:

x =x ˙ y =y ˙ z =z ˙ µ x˙ = − x + a (3.2) 3 px px2 + y2 + z2 µ y˙ = − y + a 3 py px2 + y2 + z2 µ z˙ = − z + a 3 pz px2 + y2 + z2

After the integration the returned state vector u = [x, y, z, x,˙ y,˙ z˙] correspond to the orbit propagation. The ﬁrst three vectors are the trajectory and the remaining are the velocity.

3.2.2 Eﬀect of Atmospheric Drag Orbital decay is caused by drag in low-Earth orbits. The density obtained from the NRLMSISE-00 atmospheric model, is used to calculate the aerodynamic drag deceleration, using equation (2.9), in section 2.1.5. The coeﬃcient of drag, CD, is assumed to be a constant value of 2.2 [19]. This assumption is near the lowest limitation value of CD; however, at altitudes above 250 km (K. Moe et M.M. Moe) found that the MSISE-90 model overestimates the density by as much as 15% [20].

33 Chapter 3. Orbital Decay 3.2. Decay Lifetime Simulation

The inﬂuence of the upper atmosphere on the satellite is simulated for diﬀerent perigees. These simulations are done at orbits with apogee of 6000 km and perigees of 190 km, 170 km, and 150 km. The acceleration due to drag is added to the equations of motion in section 2.1.4. Table 3.2 presents the initial conditions for the orbital decay simulation.

Apogee a = 6000 km Perigee p = 150 km, 170 km, 190 km Inclination i = 90◦ True anomaly θ = 0◦ Right ascension of the ascending node Ω = 0 ◦ Argument of perigee ω = 0 ◦ Mass m = 3 kg Ram facing area A = 0.1 m2

Table 3.2: Initial orbital decay conditions.

Simulations are run for orbital decays with a lifetime of 3 months. As expected, the lower the initial perigee the faster the decay time. In ﬁgure 3.11, the simulation of Perigee = 150 km, de-orbits before the 3 months. The simulations that start at Perigee = 170 km and Perigee = 190 km, does not de-orbit in 3 months. This shows the inﬂuence of atmospheric density on the rate of orbit decay.

Figure 3.11: Orbital Decay due to aerodynamic drag.

3.2.3 Eﬀect of J2 Perturbation Atmospheric drag is not the only perturbation that aﬀects the satellite in low-Earth orbits. As mentioned earlier in section 2.1.6, Earth oblateness is a crucial perturbation to subsume into the equations of motion. Simulations are run for the same initial conditions as in section 3.2.2, for initial perigees of 170 km and 190 km.

34 Chapter 3. Orbital Decay 3.2. Decay Lifetime Simulation

Figure 3.12: Orbital Decay due to drag and Earth oblateness.

The effects of the Earth oblateness are the oscillations in the decay. This is evident in figure 3.12, where it shows both decay with and without Earth oblateness perturbation. Additionally, the orbit decay time is also affected by the Earth oblateness. Figure 3.13, compares different inclinations to show the cause of the oscillations. The orbit decay trajectory with zero inclination does not have any oscillations in contrast to the orbit trajectories with 30◦, and 90◦ inclination. Then from the comparison, it is clear the oscillations are prominent due to the 90◦ inclination of the orbit.

Figure 3.13: Earth oblateness eﬀect on diﬀerent orbit inclinations.

The appropriate range of insertion altitudes are found using an iterative method. The range of altitudes are narrowed down from 150-190 km to approximately 160-170 km, by observing the decay time of each perigee insertion. The iterative results are shown in ﬁgure 3.14, where the appropriate perigee insertion of the satellite should be between 165-170 km.

35 Chapter 3. Orbital Decay 3.2. Decay Lifetime Simulation

Figure 3.14: Altitude versus time in ﬁnal decay.

Taking a value between 165 km and 170 km, a simulation is done at a perigee altitude of 168 km. The result showed in ﬁgure 3.14 suggested that an orbit with a perigee of 168 km and an apogee of 6000 km can be an option for insertion of the satellite. Additionally, the insertion orbit altitude can be varied by changing the ballistic coeﬃcient of the satellite.

3.2.4 Effect of different ram-facing areas In the previous section, the insertion altitude has been discussed, where the effect of changing the ballistic is investigated to examine the decay time of the satellite. The different ram-facing areas of the satellite, yield different decay times. Figure 3.15 shows the decay time dependency on the ram-facing areas. The simulations are done at an altitude of 168 km, with areas of 0.01 m2, 0.03 m2, 0.05 m2, 0.07 m2, and 0.1 m2.

Figure 3.15: Range of ram-facing areas.

In ﬁgure 3.15, with the ram-facing area of 0.1 m2, the satellite decayed in 90 days. However, setting the ram-facing areas to be less than 0.1 m2 showed that the satellite did not decay in 90 days. This means that if the satellite is able to keep a ram-facing area of 0.1 m2 in an orbit with perigee of 168 km and apogee of 6000 km, the satellite will de-orbit in 3 months.

3.2.5 Summary of Simulations The results from the decay simulation of the satellite are summarized here. The simulation investigated the eﬀects of aerodynamic drag and Earth-oblateness perturbation on satellite decay, as well as, estab-

36 Chapter 3. Orbital Decay 3.2. Decay Lifetime Simulation lished a possible insertion orbit. In addition, the eﬀect of changing the ballistic coeﬃcients by means of varying the ram-facing area is also examined.

The results indicated that an appropriate insertion orbit for the satellite is around perigee of 168 km and apogee of 6000 km, to be able to de-orbit in 90 days. However, this is quite ideal since the satellite would have to keep a constant ram-facing area of 0.1 m2 as showed later in section 3.2.4. In addition, due to the high inclination orbit required for the mission, the eﬀect of Earth oblateness perturbation inﬂuences the satellite’s trajectory as seen in section 3.2.3.

37 Chapter 4

Satellite Attitude Simulation

In low-Earth-orbits, satellites are subjected to many different forces that affect its attitude and stability. This chapter studies the feasibility of using aerodynamic drag panels as a mean of decelerating the satellite during the perigee pass. As seen in section 3.2.4, having a certain ballistic coefficient, the satellite is able to decelerate and reduce its apogee. Existing literature proposes the combined use of aerodynamic and magnetic torquing, to achieve three axis attitude control [21]. In addition, building on previous work that used gravity gradient stabilization and magnetic torquing [12], a drag panel is added to the satellite configuration. The analysis in this chapter aims to provide an understanding of aerodynamic torque on each principal axis.

4.1 Satellite Geometry

According to section 3.2.4, a constant ram-surface area of 0.1 m2 allows the satellite to decay in three months. In order to adhere to the compact requirements of a CubeSat, a deployable panel is positioned at a certain angle once the satellite is in orbit. The CubeSat conﬁguration in this section is an attempt to understand the behavior of using aerodynamic drag as means of orbit, attitude control, and formation maintenance. A speciﬁc geometry of a 3U-CubeSat is established consisting of a control panel placed to the rear of the center of gravity. An idea taken from a badminton shuttlecock, the panel provides passive stability about the pitch axis. Additionally, a mass constraint of ≤ 4 kg is put on the satellite.

Figure 4.1: Satellite geometry isometric view.

38 Chapter 4. Satellite Attitude Simulation 4.1. Satellite Geometry

Figure 4.2: Satellite geometry top and side views.

The angle of attack φ of the satellite is defined as the angle between body axis xb and the velocity vector va,1, where velocity vector va,1 is defined in the local vertical local horizon orbital frame. The flight path angle γ is defined as the angle between velocity vector va,1 and the local horizon.

4.1.1 Satellite Dimensions and Mass Properties The satellite dimensions are listed in table 4.1, which includes the dimensions of the bus, boom and panel. The satellite dimensions follow the guideline of the 3U-CubeSat requirements. The dimension and mass property of the boom are kept from previous work done on gravity gradient [12]. For the control panel, the thickness of the panel is neglected, so that thin plate is assumed. In addition, the mass of each part of the satellite is also deﬁned here.

Satellite Bus L = 0.15 m w = 0.05 m d = 0.05 m mbus = 3 kg Satellite Boom Lboom = 0.9 m dboom = 0.015 m dtip = 0.025 m Boom density per length = 0.08342 kg/m Boom tip mass = 100 g Satellite Panel Lpanel = 0.15 m Panel density = 15.8 kg/m2 2 Areapanel = 0.03 m mpanel = 0.47 kg θ = 0,-15, -30, -45, -60 degrees

Table 4.1: Satellite dimensions and mass properties.

39 Chapter 4. Satellite Attitude Simulation 4.1. Satellite Geometry

4.1.2 Center of Gravity The knowledge of a rigid body’s center of mass is vital, since the attitude control takes place around the center of mass. The center of mass is calculated by,

1 X R = m r (4.1) c.m. M i i where Rc.m. is the center of mass, M is M is the total mass of the system of particles, ri is the average position of particle i, and mi is the mass of particle i.

The center of mass of each components of the satellite are listed in Table 4.2, where it is good to note that the center of mass changes as the control panel angle changes.

Element Mass Rc.m,1 Rc.m,2 Rc.m,3 Bus mbus 0 0 0 Boom mboom l 0 Lboom/2 Tip mtip l 0 L Panel mpanel − (d + Lpanelcosθ) 0 − (w + wpanelsinθ)

Table 4.2: Location of the satellite center of mass.

4.1.3 Principal Moments of Inertia The principle moments of inertia stated in equation are used in the calculation of angular momentum in section section:RotDynamics. The moment of inertia dictates rotation around the body axes, which are computed based on the knowledge of component mass and location. The equation for the moment of inertia is stated as,

X 2 Ii = ri mi (4.2) where Ii is the is the moment of inertia, ri is the position vector and mi is the mass. Since the moment of inertia matrix is symmetric, the vectors ωi corresponding to the three Ii are perpendicular. There- fore, there are only three possible axes in the rigid body about which angular momentum H and ω are parallel. The three special Ii are the diagonal terms of the moment of inertia matrix, which are termed the principal moments of inertia. While the corresponding ωi are called the principal moments of inertia [8]. The principal moments of inertia of each satellite component are shown in Table 4.3.

Element Mass I1 I2 I3 mbus 2 2 mbus 2 2 mbus 2 2 Bus mbus 12 4d + 4w 12 4d + 4L 12 4w + 4L mboom 2 mboom 2 Boom mboom 12 Lboom 12 Lboom 0 Tip mtip 0 0 0 mpanel 2 2 Panel mpanel 0 0 12 4wpanel + 4Lpanel

Table 4.3: The principal moments of inertia of the satellite.

As the origin of the body frame changes, it is important to know how the moment of inertia matrix changes. Hence the parallel axis theorem gives the moment of inertia matrix about a new origin. The new principal moments of inertia are given by,

I = I + md2 (4.3)

40 Chapter 4. Satellite Attitude Simulation 4.2. Attitude Simulation where I is the principal moments of inertia for a parallel axis, I is the moment of inertia at the center of mass, m is the mass of the element and d is the distance of the parallel axis. The ﬁnal moments of inertia used in the calculations are stated,

m m I = bus 4d2 + 4w2 + m d2 + boom L2 + m d2 + m d2 1 12 bus 1,bus 12 boom boom 1,boom tip tip,1 2 + mpaneld3,panel mbus 2 2 2 mboom 2 2 2 I2 = 4d + 4L + mbusd2,bus + Lboom + mboomd2,boom + mtipdtip,2 12 12 (4.4) 2 + mpaneld3,panel m m I = bus 4w2 + 4L2 + m d2 + m d2 + m d2 + panel 4w2 + 4L2 3 12 bus 3,bus boom 3,boom tip tip,3 12 panel panel 2 + mpaneld3,panel

4.2 Attitude Simulation

The attitude simulation in this section combines orbital mechanics with satellite dynamics and kinematics. The simulation includes external torques; therefore, it is important to calculate the aerodynamic torque profiles over different attitude angles for roll, pitch and yaw axes. The torque profiles are plotted for attitude angles during the perigee passage, where they are established for attitude angles of -90 to 90 degrees, and all principle axes. Then the torque profiles are calculated for true anomalies, ν = 0◦ and ν = 270◦, which represents the perigee passage of the satellite. In addition, different panel settings have also been taken into consideration. Since different panel angles produce different torques, the panel is tested for angles 0, -15, -30, -45 and -60 degrees. Furthermore, the aerodynamic shadowing by the ram surface reduces the drag effects on the control panel. Hence, the aerodynamic shadowing is modeled by setting the length of the shadowed panel to zero.

4.2.1 Attitude Propagation In order to estimate the satellite’s behavior due to external torques, a simulation is done with the same equations in section 3.2 for the position and velocity are considered. Taking r to be r = (x, y, z) then the state variables are in the form,

x =x ˙ y =y ˙ z =z ˙ µ x˙ = − x + a 3 px px2 + y2 + z2 (4.5) µ y˙ = − y + a 3 py px2 + y2 + z2 µ z˙ = − z + a 3 pz px2 + y2 + z2

Likewise, the Euler equations of motion for a rigid body from section 2.3 are rewritten in state-variable form,

41 Chapter 4. Satellite Attitude Simulation 4.2. Attitude Simulation

(Iyy − Izz) ω2ω3 ω˙ 1 = + T1 Ixx (Izz − Ixx) ω3ω1 ω˙ 2 = + T2 (4.6) Iyy

(Ixx − Iyy) ω1ω2 ω˙ 3 = + T3 Izz where T1,T2, andT3 are the sum of all external torques, in this case the Gravity Gradient Torque and Aerodynamic Torque. Furthermore, the kinematic diﬀerential equations from section 2.2.6 are expanded as the following,

1 q˙ = (ω q − ω q + ω q ) 1 2 3 2 2 3 1 4 1 q˙2 = (−ω1q3 + ω2q4 + ω3q1) 2 (4.7) 1 q˙ = (ω q − ω q + ω q ) 3 2 2 1 1 2 3 4 1 q˙ = (−ω q − ω q − ω q ) 4 2 1 1 2 2 3 3 As a result, equations (4.5), (4.6) and (4.7) constitutes 13 non-linear coupled ordinary diﬀerential equations, which are solved using Runge-Kutta(4-5) numerical method. The initial state variables in this case are given in the vector form,

T r0 = [x0, y0, z0, x˙0, y˙0, z˙0, ω1, ω2, ω3, q1, q2, q3, q4] (4.8)

4.2.2 Aerodynamic Torque with Partial Accomodation Coefficient Satellites in Low-Earth orbits are placed in an atmospheric environment where an assumption of free- molecular flow is made. Free-molecular flow model treats the incoming molecules to the surface and the outgoing from the surface separately. The aerodynamic torque calculations take into account the transfer of momentum of gas molecules in the atmosphere when it impacts the surface of the satellite. The momentum transfer happens when molecules arrive at the surface and leaves the surface, then by the assumption of free-molecular flow, the two cases are added together. There are two extreme cases of momentum transfer when a particle impacts any physical surface. First is the specular reflection and the other is diffuse reflection. In the specular reflection case, each reflected particle from the surface has no change in energy, therefore; the angle of reflection and outgoing velocity equals to the angle of incidence and the incoming velocity. On the contrary, in the case of diffuse reflection, the surface completely accommodates the incoming particles. As a result, the particle leaves with a probabilistic kinetic energy characteristic of the surface temperature and probabilistic direction governed by a cosine distribution. [22]

Figure 4.3: Specular and diﬀuse molecular reﬂection. [9]

42 Chapter 4. Satellite Attitude Simulation 4.2. Attitude Simulation

Figure 4.3, represents the two concepts of specular and diffuse molecular reflection. In regards of the two cases of specular and diffuse molecular reflection, a reasonable assumption lies somewhere between the two. The particle at least becomes partially accommodated by the surface but not totally accommodated. The analysis of aerodynamic torque in this work assumes partial surface accommodation of the incoming particles.

Based on partial surface accommodation, the force and torque expressions are derived to include the momentum of molecules leaving the surface. Figure 4.4, shows the force acting on the satellite surface (dA).

Figure 4.4: Molecules incident on an element of the satellite surface. [9]

The force acting on the surface (dA) by the local atmosphere are decomposed into inward facing normal component (n) and a tangential component (t). As a result, the force components follow the same decomposition, having a normal component (dFn) and a tangential component (dFt). Trivially, summing the normal and tangential components, the force acting on (dA) is written as such:

dF = dFn + dFt (4.9) Through the assumption that the mean thermal motion of the atmosphere is much smaller than the speed of the satellite through the atmosphere, the incoming atmosphere is viewed as a parallel molecular beam VR of density (ρ) and velocity VR. As seen in ﬁgure 4.4, let V = and n be the unit inward normal to |VR| the surface of (dA)[22].

cos α = VR · n (4.10) where the velocity vector VR is in the body frame of the satellite moving in Earth’s atmosphere, and n is the vector normal to the plane. The velocity vector is described by,

VR = Vi · RBN (4.11) where Vi is the velocity vector in the inertial frame and RBN is the transformation matrix from the inertial frame to the body frame.

The angle of incidence (α) is the angle which the particle impacts the surface of the satellite, and is calculated by equation (4.10). The projected area acted upon by local atmosphere is (dA cos α) in the normal direction and (dA sin α) in the tangential direction. The molecular momentum ﬂux through (dA cos α) is a force imparted to (dA)[22][9]. Hence, the force is written as:

2 2 dF = ρ VR cosα VR dA + ρ VR sinα VR dA (4.12) where ρ is the local atmospheric density.

43 Chapter 4. Satellite Attitude Simulation 4.2. Attitude Simulation

When the satellite travels through space, it is good to note that not all surfaces will impact the upper atmosphere. Thus, if cosα ≥ 0 then that element is exposed to the ﬂow. On the contrary, if cosα < 0 then that element is shadowed from the ﬂow. Consequently, using the Heaviside function, H(·), the conditions above are then accounted for in the force and torque calculations [9]. For example, if cosα ≥ 0, then H(cosα) = 1, and if cosα < 0, H(cosα) = 0.

Incorporating the Heaviside function into the equation (4.12), the force expressions for normal and tangential components for diﬀuse reﬂection becomes,

Diffuse dfn = H(cosα)ρVRcosα[VRcosα + Vb]ndA (4.13)

Diffuse 2 dft = H(cosα)ρVRsinαcosαtdA (4.14) where Vs is the molecular exit velocity.

The particle velocity, Vs, is described by the kinetic theory of gases, which relates Vs to the surface temperature Ts.

πRT 1/2 V = s (4.15) s 2m where R is the universal gas constant (R = 8.314 × 103J/kg · mole · deg C) and m is the molecular weight of the gas. A typical estimation of Vs for the altitudes in the upper atmosphere is 5% of VR.

On the other extreme, the force expressions for normal and tangential components for specular reﬂection case are written as,

Specular 2 2 dFn = 2H(cosα)ρVRcos αndA (4.16)

Specular dFt = 0 (4.17) In order to approximate the gas-surface interaction to be as close to reality as possible, a factor is introduced. This factor is called accommodation coeﬃcient, σ, where it is also separated into the tangential (σt) and normal (σn) components. Therefore, the elemental normal and tangential force becomes,

Diffuse Specular dFn = σndFn + (1 − σn)dFn (4.18)

Diffuse Specular dFt = σtdFt + (1 − σt)dFt (4.19) From equation (4.18) and (4.19),the partial accommodation elemental force equation is derived by substituting equations (4.13),(4.14), (4.16), and (4.17) into the above equations (equations (4.18) and (4.19)). Now, seeing that ~tsinα = VˆR − ncosαˆ , the force equation is rewritten as, 2 Vs dF = H(cosα)ρVRcosα (2 − σn − σt)cosα + σn n + σtVR dA (4.20) VR In order to ﬁnd the total force acting on the satellite by the local atmosphere, the equation (4.20), is integrated over the entire satellite which yields. 2 Vs F = ρVR σtApVR + σn Ap + (2 − σn − σt) App (4.21) VR where Ap, Ap, and App are deﬁned as

Ap , { H(cosα)(cosα)dA (4.22)

Ap , { H(cosα)(cosα)ndA (4.23)

44 Chapter 4. Satellite Attitude Simulation 4.3. Torque Proﬁles

2 App , { H(cosα)(cos α)ndA (4.24)

Average values for σt and σn are used, these coeﬃcients usually have values somewhere in the range of 0.8 < σ < 0.9. From the total force derived in equation (4.21), the expression for the aerodynamic torque is then,

Taero = { r × dF (4.25)

In equation (4.25), the aerodynamic torque (Taero) is calculated by crossing the distance from the center- of-mass to the surface element (r) and the force vector acting on that surface element (dF ), then integrating over the entire satellite.

4.3 Torque Proﬁles

The collective torque aﬀecting the satellite at a given attitude is acquired by summing the individual torques of diﬀerent satellite components.

Figure 4.5: Vectors in torque calculations.

The torque proﬁle results included self-shadowing of the satellite bus on the control panel. This depends on the angle of the control panel and the pitch angle to the velocity vector. Figure 4.6 illustrated how the length of the shaded panel were approximated.

45 Chapter 4. Satellite Attitude Simulation 4.3. Torque Proﬁles

(a)

(b)

Figure 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.

The length of the shaded panel on the topside was calculated using area projection from one plane to another by relating the angle between them.

46 Chapter 4. Satellite Attitude Simulation 4.3. Torque Proﬁles

Figure 4.7: Area projection. [10]

The new area, A0 is calculated by,

A0 = Acosθ (4.26) Similarly, by using the pitch angle and the angle between the velocity vector and the normal vector of the panel, as in ﬁgure 4.6, the shadowed area of the panel by the satellite bus can be calculated. Therefore, since the width of the panel and satellite bus are the same, the shaded lengths are calculated by,

Lshadowed,f = 2Lsin(α)cos(φ) (4.27)

Lshadowed,r = 2Lsin(β)cos(φ) (4.28) ◦ where β = 90 − α, Lshadow,f is the length of the shaded top side and Lshadow,r is the length of the shaded bottom side of the panel.

In the algorithm for calculating the shaded length of the panel, Lshadow,f and Lshadow,r, are both set equal to zero when φ ≤ 0. In addition, for conditions when φ > 0, the shaded top side of the panel Lshadow,f is set to zero if the shaded length is greater than the length of the panel (Lshadow,f > Lpanel). Similarly, the shaded bottom side of the panel Lshadow,r is set to zero if the shaded length is greater than the length of the panel (Lshadow,f > Lpanel). Otherwise in other conditions the shaded length in equations (4.27) and (4.28) are used to calculate the radius from the center of the shaded length to the center of gravity of the satellite. The calculation of the the radius of the shadowed panel is shown in table 4.4.

Side rp,xb rp,yb rp,zb Lshadowed,f Lshadowed,f rshadowed,f (Top Side) - 2 cos θ 0 2 sin θ Lshadowed,r Lshadowed,r rshadowed,r (Bottom Side) - 2 cos θ 0 2 sin θ

Table 4.4: Location of the radius of the shaded panel length.

rp,f = rshadowed,f − CGposition (4.29)

rp,r = rshadowed,r − CGposition (4.30)

Once the radius rp from the shaded region of the panel to the center of gravity are known, the forces and moments are then calculated using the same method as in section 4.2.2. Furthermore, to calculate the actual force and moment acting on the panel,

Factual = Funshadowed − Fshadowed (4.31)

Mactual = Munshadowed − Mshadowed (4.32)

47 Chapter 4. Satellite Attitude Simulation 4.3. Torque Proﬁles

4.3.1 Pitch Torque The pitch torque profiles are established for different true anomalies and different panel angles. Since the control surface torques are dominant in the pitch axis due to gravity gradient boom and the aerodynamic control panel, the magnitude of different external torques are determined.

Figure 4.8: Pitch torque proﬁle for true anomaly, ν = 270◦

Figure 4.9: Pitch torque proﬁle for true anomaly, ν = 0◦

The torque as a function of angle to velocity vector, in ﬁgures 4.8 and 4.9, illustrates the torque depen-

48 Chapter 4. Satellite Attitude Simulation 4.3. Torque Proﬁles dency to the pitch angle. When the control panel is shadowed, the torque curve is determined by the gravity gradient boom. However, at true anomaly of ν = 270◦, the gravity gradient torque is dominant due to the low atmospheric density. On the contrary, at true anomaly, ν = 0◦, the aerodynamic torques dominate. The equilibria are determined by zero crossing of the torque curve. The characterization of the zero crossing of the torque curve depends on which external torques are dominant. At the perigee passage (ν = 0), the zero crossing of the torque curve is characterized by the aerodynamic torque due to higher atmosphere density. At ν = 270◦, gravity gradient torque characterizes the torque curve due to lower atmospheric density.

Figure 4.10: Pitch torque of separate components, for true anomaly, ν = 270◦

49 Chapter 4. Satellite Attitude Simulation 4.3. Torque Proﬁles

Figure 4.11: Pitch torque of separate components, for true anomaly, ν = 0◦

Figures 4.10 and 4.11, show the aerodynamic torque proﬁles for the satellite bus, boom and control panel. The shadowing of the control panel can also be seen in both ﬁgures.

4.3.2 Yaw and Roll Torque Proﬁles

Figure 4.12: Yaw torque proﬁle.

50 Chapter 4. Satellite Attitude Simulation 4.3. Torque Proﬁles

Figure 4.13: Roll torque proﬁle

The yaw torque profile plotted in figure 4.12 is dominated by the aerodynamic torque. Since gravity gradient torque does not function in the yaw axis, the aerodynamic panel characterizes the torque curve. The equilibria are determined to be at yaw angles, −90◦, 0◦, and 90◦ due to the zero torque. The roll torque profile in figure 4.13 is dominated by the gravity gradient torque, since the gravity gradient torque applies only to the pitch and roll axes. Consequently, there is no panel aerodynamic torque in the roll axis due to the assumption that the thickness of the panel is zero.

Figure 4.14: Yaw torque of separate components.

51 Chapter 4. Satellite Attitude Simulation 4.4. Pitch Dynamics

There are torque profiles on the satellite bus, in figure 4.14, due to the center of mass shifting as the panel angle changes. It is interesting that the yaw torque has the same magnitude as the pitch torque. For instance the maximum pitch torque is 4×10−4 and the maximum yaw torque is 0.5×10−4, this could be due to the fact that the Euler equations of motion are coupled. Hence, the pitch torque also contributes to the yaw torque. Additionally, there are no yaw torque profiles for the boom because here the boom is assumed to always be aligned with the center of mass.

4.4 Pitch Dynamics

To find a suitable attitude upon the satellite’s entrance into the perigee passage, the attitude dynamics are investigated. The previous section explored the torque profiles of the satellite. From the torque profiles, the equilibria are determined by the zero crossing of the curve. Hence in this section multiple simulations are done with different panel angles θ, angular rates ω, and attitude angles φ, in order to find the combination that keeps the satellite’s attitude stable.

4.4.1 Initial Orbit Conditions for Pitch Dynamics The initial orbit simulation conditions are presented in table 4.5. These values are decided based on decay simulation results in section 3.2.4, where it was decided that an orbit with perigee of 168 km was appropriate for the insertion orbit. The true anomaly is set at ν = 270◦, to indicate the perigee entrance of the satellite. The run time of the simulation is t = 2800 s, which is approximately where the satellite exits the perigee passage.

Apogee a = 6000 km Perigee p = 168 km Inclination i = 90◦ True anomaly θ = 270◦ Right ascension of the ascending node Ω = 0 ◦ Argument of perigee ω = 0 ◦ Eccentricity e = 0.3082 Orbital period T = 9160 s Mean motion n = 6.86×10−4

Table 4.5: Initial orbit simulation conditions.

4.4.2 Simulation Results In the previous section, torque proﬁles of panel angles at θ = −45◦ and θ = −60◦ had the equilibria ◦ ◦ crossings between φ2 = 0 and φ2 = −20 . In order to keep the satellite stable, it is good to keep the ◦ initial angle deviation as close to φ2 = 0 as possible. The pitch behavior has been primarily investigated due to the satellite having control surfaces in the pitch axis, where these values are presented in tables 4.6 and 4.7.

ω0 [rad/s] θ [deg] Roll φ1 [deg] Pitch φ2 [deg] Yaw φ3 [deg] (0, -n, 0) −45◦ 0◦ −10◦ 0◦ (0, -1.5n, 0) −45◦ 0◦ −10◦ 0◦ (0, -2n, 0) −45◦ 0◦ −10◦ 0◦

Table 4.6: Initial conditions for pitch dynamics for diﬀerent angular velocities with a panel angle, θ = −45◦.

52 Chapter 4. Satellite Attitude Simulation 4.4. Pitch Dynamics

◦ Figure 4.15: Euler angles φi for diﬀerent ωi,t=t0 , and panel angle of θ = −45 .

The simulation results in ﬁgure 4.15, have shown that the satellite appears to be stabilized throughout the ◦ ◦ perigee passage, with a pitch angle φ2 = −10 , angular rate ω0 = (0, −2n, 0), and panel angle θ = −45 . In addition, some yaw and roll coupling can be seen, however the magnitude is very small and can be ignored.

ω0 [rad/s] θ [deg] Roll φ1 [deg] Pitch φ2 [deg] Yaw φ3 [deg] (0, -n, 0) −60◦ 0◦ −3◦ 0◦ (0, -1.5n, 0) −60◦ 0◦ −3◦ 0◦ (0, -1.78n, 0) −60◦ 0◦ −3◦ 0◦

Table 4.7: Initial conditions for pitch dynamics for diﬀerent angular velocities with a panel angle, θ = −60◦.

◦ Figure 4.16: Euler angles φi for diﬀerent ωi,t=t0 , and panel angle of θ = −60 .

The simulation results in ﬁgure 4.15, have shown that the satellite appears to be stabilized throughout ◦ the perigee passage, with a pitch angle φ2 = −3 , angular rate ω0 = (0, −1.78n, 0), and panel angle θ = −60◦. There is also some yaw and roll coupling that occurred, but again the magnitude is very small and it can be ignored.

The two results demonstrated that the satellite can be stabilized in the perigee passage by with the combination of aerodynamic drag panel and gravity gradient boom. The satellite with a deployable con- ◦ ◦ trol panel angle of θ = −45 is stable when entering at a pitch angle of φ2 = −10 and an initial angular ◦ velocity of ω2,t=t0 = −2n [rad/s]. Similarly, the deployable control panel angle of θ = −60 is stable ◦ when entering at a pitch angle of φ2 = −3 and an initial angular velocity of ω2,t=t0 = −1.78n [rad/s]. Additionally, it is good to note that although the yaw and roll coupling is minor, the oscillations seem to increase with time. Therefore, the yaw and roll oscillations could render the satellite unstable after many orbits, if it is not corrected.

53 Chapter 4. Satellite Attitude Simulation 4.5. Summary

4.5 Summary

In this section, the torque proﬁles were established to better understand the behavior of the satellite during the perigee passage. The torque proﬁles done with multiple panel angles yielded equilibria zero crossings, where it could be possible to stabilize the satellite. The panel angles that provided the zero crossing of the torque with minimum deviation from the zero pitch angle were θ = −45◦ and −60◦. Then simulations were done to investigate the attitude and angular rate of the satellite during the perigee passage. It was found that at control panel angle of θ = −45◦, the satellite was stable given the initial ◦ conditions of pitch angle and angular rate to be, φ2 = −10 and ω2,t=t0 = −2n [rad/s]. Similarly, the control panel angle of θ = −60◦ was stable given the initial conditions of pitch angle and angular rate to ◦ be, φ2 = −3 and ω2,t=t0 = −1.78 [rad/s].

54 Chapter 5

Passively Stabilized Perigee Passage

In order for a cube satellite to minimize power consumption, passive attitude control methods are analyzed. In this chapter the stability of the satellite during the perigee passage are simulated for each panel angle. The initial conditions at the perigee entrance are found through backward integration of the equations of motion knowing the conditions at the perigee. Figure 5.1 shows a single satellite orbit, where the entrance and exit of the perigee passage are indicated.

Figure 5.1: Orbit Trajectory.

5.1 Initial Conditions for Perigee Passage

In section 4.4, it was shown that for two selected panel angles, θ = −45◦ & − 60◦, the satellite was able to obtain a stable perigee passage. The term stable here means that the satellite does not tumble during the passage. It was also clear that the initial conditions governing one perigee passage of the satellite are the pitch angular velocity ω2 and pitch angle φ. Therefore, for each panel angle, a set of initial conditions at the entrance of perigee are found by backward integrating the equations of motion. This is done using the pitch angle at perigee that yielded zero torque for each panel angle and the orbital angular velocity.

The orbital angular velocity at perigee (True anomaly ν = 0) is obtained from,

55 Chapter 5. Passively Stabilized Perigee Passage 5.2. Simulation Results

r × v ω = (5.1) 2,ν=0 |r|2 where r is the radius and v is the orbital velocity. The radius and velocity values used to calculate equation (5.1) are presented in table 5.1.

Variables x y z Radius, r [km] 6552.089 0 -2377.168 Velocity, v [km/s] 0.2571354 0 8.917172

Table 5.1: Orbit radius and velocity at perigee.

Table 5.2, shows the pitch angle that yields zero torque at the perigee for diﬀerent panel angles. These conditions are used for backward integration of equations of motion during the perigee passage, in order to get the initial condition at the entrance of the perigee trajectory.

◦ ◦ Torques [N·m] Panel Angle θ [ ] Zero-torque Pitch Angle φ2,ν=0 [ ] 0 0 -55 0 -15 -44 0 -30 -32 0 -45 -18 0 -60 -3

Table 5.2: Orbit radius and velocity at perigee.

5.2 Simulation Results

◦ ◦ Panel Angle θ [ ] Initial Pitch Angle φν=270 [ ] Pitch Angular Velocity ω2,ν=270 [rad/s] -60 -4.332 -0.0012622 -45 -13.68 -0.0014083 -30 -24.39 -0.0014728 -15 -30.64 -0.0015326 0 -36.88 -0.0015925

Table 5.3: Initial condition for diﬀerent panel angles.

Figure 5.2: Attitude angles for diﬀerent panel angle.

56 Chapter 5. Passively Stabilized Perigee Passage 5.3. Summary

The initial conditions at the entrance of the perigee passage are shown in table 5.3, where it ensures that the satellite does not tumble. For each panel angle, there is a corresponding initial pitch angle φν=270 and a pitch angular velocity ω2,ν=270. In ﬁgure 5.2 it is seen that for all panel angles simulated there are some disturbances at the perigee, this could be due to the same orbital angular velocity ω2,ν=0 used to backward integrate the initial conditions.

5.3 Summary

This section investigates the initial conditions for each panel setting to be able to remain stable through the perigee passage. A stable pitch attitude depends on the on the initial pitch angular velocity and pitch angle, where they were obtain by backward integrating the pitch angular velocity and pitch angle conditions at the perigee. The the initial pitch angular velocity and pitch angle that yields a stable perigee passage were found by using this method.

57 Chapter 6

Satellite Formation Design

In order to make multiple in situ measurements of the Aurora, a formation of satellites need to be examined. As the satellite industry realizes the potential of cost reduction replacing conventional satellites with a multiple Cube Satellite formation, formation flying has become an increasingly popular subject of study. There are a few different types of satellite formations, such as orbit tracking, leader-follower, virtual structure and swarming. The type of formation flying that is investigated in this thesis, is the leader-follower satellite formation in order to simplify the simulation to two satellites. This chapter includes the definitions of coordinate reference frames and coordinate frame transformations used in the modeling of the leader-follower formation. A derivation of elliptic Earth orbits of the relative translation between the leader and follower satellite formation is shown [23].

6.1 Coordinate Frame

The Earth-Centered-Inertia (ECI) coordinate reference frame described in section 2.2.1 is reiterated here to help clarify the satellite formation reference frames. In addition, the leader orbit reference frame and follower orbit reference frame are also deﬁned.

Earth-Centered-Inertia (ECI), is denoted as frame, i = (ix, iy, iz), has the vector ix pointing to the vernal equinox, the iz vector points out of the Earth’s north pole, and the iy vector is orthogonal to the other two vectors, and the location of its origin is at the center of the Earth.

Leader orbit reference frame is denoted as Fl, where its origin is at the center of mass of the leader satellite.

58 Chapter 6. Satellite Formation Design 6.2. Formation Dynamics

Figure 6.1: Reference coordinate frames. [11]

The er axis in the frame is parallel to the rl which points from the center of the Earth to the satellite. The eh axis is parallel to the orbit momentum vector, where it points in the orbit normal direction. Lastly, the eθ axis completes the right-handed orthogonal frame. These basis vectors are deﬁned as,

rl er = rl eθ = eh × er (6.1) h e = h h where h = rl × r˙l is the angular momentum vector of the orbit and h = |h|.

Follower orbital frame is denoted as Ff , where it has its origin at the center of mass of the follower satellite. The vector rf is the vector pointing from the center of the Earth to the center of the follower orbit frame. In ﬁgure 6.1, the origin of the follower orbit reference frame, expressed in Ff is located by a relative orbit position vector p = [x y z]T . The frame unit vectors align with the basis vector Fl. The relative orbit position vector is deﬁned as,

p = rf − rl = xer + yeθ + zeh (6.2) Body reference frame is previously deﬁned in section 2.2.1. The body frames in this section are denoted as Fl and Ff , which the subscript l is the body reference frame of the leader satellite. Similarly the subscript f denotes the follower body reference frame.

6.2 Formation Dynamics

The general equation of relative motion of two bodies, equation (2.6) from section 2.1.1, is adapted to express the dynamics of the leader and follower satellite as,

µ fdl µ fdf r¨l = − 3 rl + r¨f = − 3 rf + (6.3) rl ml rf mf where fdl and fdf are the orbital perturbations terms. Then the second order derivative of the relative position is expressed as,

59 Chapter 6. Satellite Formation Design 6.2. Formation Dynamics

p¨ = r¨f − r¨l

µ fdf µ fdl = − 3 rf + + 3 rl − (6.4) rf mf rl ml then it is rewritten as,

! rl + p rl mf mf p¨ = −mf µ 3 − + fdf − fdl (6.5) (rl + p) rl ml

From equation (6.2), the dynamics of the follower satellite relative to the leader satellite is expressed in the orbit frame Fl as,

rf = rl + p (6.6) = (rl + x) er + yeθ + zeh then diﬀerentiating twice with respect to time gives,

r˙f = (¨rl +x ¨) er + 2 (r ˙l +x ˙) e˙r + (rl + x) e¨r +y ¨eθ + 2y ˙e˙θ + ye¨θ +z ¨eh + 2z ˙e˙h + ze¨h (6.7) Furthermore, using true anomaly ν for the leader satellite the relationship between the true anomaly and the follower orbit reference frame as [23],

e˙r =ν ˙eθ (6.8)

e˙θ = −ν˙er (6.9) 2 e¨r =ν ¨eθ − ν¨ er (6.10) 2 e¨θ = −ν¨er − ν¨ eθ (6.11)

The ideal case of the satellite is when there are no out of plane motion, therefore, e˙h = e¨ = 0. Then inserting equations (6.8)-(6.11) into equation (6.7) yields,

2 2 r¨f = r¨l +x ¨ − 2y ˙ν˙ − ν˙ (rl + x) − yν¨ er + y¨ + 2ν ˙ (r ˙l +x ˙) +ν ¨ (rl + x) − yν˙ eθ +z ¨eh (6.12)

Additionally, the position of the leader satellite is expressed as rl = rler where it is diﬀerentiate twice with respect to time and inserting to equations (6.8)-(6.11) to become,

r¨l =r ¨ler + 2r ˙le˙r + rle¨r (6.13) 2 = r¨l − rlν˙ er + (2r ˙lν˙ + rlν¨) eθ

By taking the diﬀerence between equation (6.12) and (6.13), the result is the acceleration vector in the Fl reference frame as,

p¨ = r¨f + r¨l (6.14) 2 2 = x¨ − 2ν ˙y˙ − ν˙ x − νy¨ er + y¨ + 2ν ˙x˙ +νx ¨ − ν˙ y eθ +z ¨eh

Substituting (6.14) into (6.5) and setting v = p˙, produces the nonlinear relative equations of motion in the form,

p˙ = v (6.15)

60 Chapter 6. Satellite Formation Design 6.3. Satellite Formation Simulation

mf v˙ = Fd − Ct (ν ˙) v − Dt (ν, ˙ nu,¨ rf ) p − nt (rl, rf ) (6.16) where 0 −1 0 Ct (ν ˙) = 2mf ν˙ 1 0 0 (6.17) 0 0 0

 µ 2  3 − ν˙ −ν¨ 0 rf  µ 2  D (ν, ˙ nu,¨ r ) p = m ν¨ 3 − ν˙ 0 p (6.18) t f f  rf   µ  0 0 3 rf

rl 1  3 − 2  rf rl nt (rl, rf ) = mf µ  0  (6.19) 0

Here, Ct (ν ˙) is a skew symmetric Coriolis like matrix and Dt (ν, ˙ nu,¨ rf ) can be seen as a time varying potential force. The force vector Fd includes the perturbation force and are given as,

mf Fd = fdf − fdl (6.20) ml The rate of true anomaly of the leader satellite is given by,

2 nl (1 + el cos ν (t)) ν˙ = 3 (6.21) 2 2 (1 − el ) p 3 where nl = µ/al is the mean motion of the leader, al is the semi-major axis of the leader and el is the orbit’s eccentricity. Then by diﬀerentiating equation (6.21) the acceleration of true anomaly is,

2 3 −2nl el (1 + el cos ν (t)) sin ν (t) ν¨ (t) = 3 (6.22) 2 2 (1 − el ) 6.3 Satellite Formation Simulation

In order to illustrate the impact of perturbing forces on a satellite formation, simulations are done for two satellites in a leader-follower formation. The effects of atmospheric drag and Earth oblateness were included in the simulations. The first simulation illustrates the circular orbit with perturbation and second simulation examines the effect of elliptical trajectory on the satellite formation.

The ﬁrst simulation the leader satellite is assumed to have a circular orbit with an inclination of 22.5◦, altitude of 250 km and a constant altitude relative to the leader orbital frame. The follower satellite is located initially at 10 m behind the leader in the along-track direction (eθ) with the same initial orbit velocity and altitude as the leader.

61 Chapter 6. Satellite Formation Design 6.3. Satellite Formation Simulation

Figure 6.2: Relative position and velocity, in a circular orbit.

Figure 6.2 shows the relative position and velocity between two satellites in formation for a circular orbit. If there were no perturbations the relative translation would be constant. Hence it is seen in the figure that the perturbation forces cause rather large oscillations in the relative translation. The Earth oblateness causes the follower satellite to experience the variation in gravity field, where as the leader satellite only experiences the ideal gravity force as if the Earth was a single point. The result is a stable cyclic oscillations in the radial direction (er). Similar oscillations are also seen in the cross-track direction (eh), which are due to the same variation in the gravity field that causes the oscillations in the radial directions. Since Earth is not a point mass, the follower satellite is then drawn towards the side with the largest gravitational pull. It also seems that the cross-track motion is less stable than the radial motion. As for the along-track distance (eθ) between the leader and follower satellites, is not only oscillating but also de- caying. This behavior is due to the contribution of the atmospheric drag that are affecting both satellites.

The second simulation the leader satellite is assumed to have an elliptical orbit with an inclination of 90◦, an apogee altitude of 6000 km and perigee altitude of 168 km. The follower satellite is initially located 10 m behind the leader in the along-track direction (eθ) as well.

62 Chapter 6. Satellite Formation Design 6.4. Summary

Figure 6.3: Relative position and velocity, in a elliptical orbit.

Figure 6.3 shows the relative position and velocity between two satellites in formation for an elliptical orbit. The stable axis seems to be the cross-track direction (eh). It is clear from the ﬁgure that the two other axes are unstable due to the increasing distance in both the radial (er) and cross-track (eθ) direction. This indicates the the follower satellite’s distances in the radial and cross-track increases dramatically with time having a spiral motion in the radial-cross-track plane.

6.4 Summary

In this section the satellite leader-follower formation was simulated for circular and elliptical orbit. The leader satellite’s initial conditions in a circular orbit are an altitude of 250 km, and an inclination of 22.5◦ and the follower satellite is initially located 10 m behind the leader. The leader satellite’s initial conditions for the elliptical orbit are inclination of 90◦, an apogee altitude of 6000 km and perigee altitude of 168 km. In the circular orbit simulation, the relative translation was found to oscillate in all directions, whereas in the elliptical orbit, only the cross-track direction (eh) exhibits the oscillation behavior. This shows that in the elliptical orbit, the motion spirals out of the radial-cross-track plane. This condition is known as out-of-plane drift, which is caused by the mismatch period of the leader and follower satellite. This could be corrected in the future work by selecting initial conditions for obtaining a no-drift solution in eccentric orbits through energy matching method.

63 Chapter 7

Conclusion

The CubeSat configuration has been proven to reduce cost and development time. This has enabled an affordable way for universities to perform experiments and test instruments in space. The Space and Plasma Physics department at Kungliga Tekniska Högskolan has proposed a scientific mission to study the aurora in response to the Swedish National Space Board’s appeal for ideas of innovative low-cost scientific satellite missions. The mission requires a closely spaced 3-U CubeSat formation in order to make multiple in situ measurements of the aurora. The apogee and perigee of the mission are 6000 km and 200 km. In addition, the thesis also explores the use of passive control methods for orbit and formation control. This was done through gravity gradient and aerodynamic torques. The thesis is a preliminary mission analysis and design of a small satellite swarm. Further expansion of previous simulation code developed for the SWIM project was done in the program MATLAB.

In the nature of a highly eccentric orbit, a more accurate atmospheric model was needed to approximate orbital decay. The NRLMSISE-00 atmospheric model was chosen not only because of the extensive satellite drag data and accelerometer data on the total mass density, but also the greater detail of the density variations. From the results of the orbital decay simulation, the insertion orbit was found to be at a perigee of 168 km and an apogee of 6000 km, where it was possible for the satellite with a ram-facing area of atleast 0.1 m2 to decay in 90 days.

The satellite’s combination of gravity gradient and aerodynamic torques were analyzed to establish the behavior of the satellite during the perigee passage. The satellite was configured with a gravity gradient boom and an aerodynamic drag panel. The pitch torque profiles were found for multiple panel angles that supplied zero-torque. The panel angles -45◦ and -60◦ were then simulated to establish the initial conditions of pitch angle and angular rate at the entrance of the perigee passage that returned a non- tumbling trajectory. Furthermore, a method was developed to find initial conditions for all panel angles by utilizing pitch angle that yielded zero-torque at the perigee. Knowing this the initial conditions for all panel angles that returned a non-tumbling perigee trajectory were found by backwards integration of the pitch angle and angular rate conditions at perigee. The simulations indicate that for atleast one perigee passage at a perigee altitude of 168 km, the satellite was able to maintain its attitude and not tumble.

Since a swarm of satellites would eventually be used to make an in situ observation, a brief simulation of a satellite formation was done on a leader-follower conﬁguration. A circular and elliptical orbits of the relative translation were simulated. It was found 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. In addition, the simulation has shown that a leader-follower formation with a elliptical reference orbit, the relative translation would spiral out of the radial-cross-track plane.

The preliminary study done in this thesis have found that a satellite with a constant ram-faced area of 0.1 m2 would be able to achieve an orbit decay of 90 days. However, further investigations are needed to establish a more linear orbit decay. Although a stable pitch attitudes with the combination of gravity gradient and aerodynamic drag panel were found, they were only simulated for a single perigee passage

64 Chapter 7. Conclusion and perigee altitude. Further analyses are needed in order to obtain other initial conditions at the entrance of the perigee passage, which would ensure a non-tumbling passage as the satellite decays. Additionally, further studies are needed to examine if the satellite yaw axis can be controlled, by adding additional panels on the side. Lastly, simulations of the leader-follower satellite yielded an out of plane behavior for an elliptical orbit. This could be corrected in future works by determining the initial conditions that matches the period of the leader and follower satellite.

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