MASTER's THESIS Behavior and Relative Velocity of Debris Near

2010:048 MASTER'S THESIS

Behavior and Relative Velocity of Debris near Geostationary Orbit

Lin Gao

Luleå University of Technology Master Thesis, Continuation Courses Space Science and Technology Department of Space Science, Kiruna

2010:048 - ISSN: 1653-0187 - ISRN: LTU-PB-EX--10/048--SE

CRANFIELD UNIVERSITY

SCHOOL OF ENGINEERING

MSc THESIS

Academic Year 2009-10

Lin Gao

Behavior and Relative Velocity of Debris near Geostationary Orbit

Supervisor: Dr. S.E.Hobbs

May 2010

This thesis is submitted in partial (45%) fulﬁllment of the requirements for the degree of Master of Science

Abstract

A general model is developed describing third-body gravity perturbation to debris’ orbit. Applying this model to debris released from geostationary orbit tells their motion in both short and long term. Without considering the Moon’s precession around the solar pole, the relative velocity between GEO debris can be calculated. This is an important coeﬃcient for simulating the GEO debris environment and can serve as an input to break up models. ii iii

Acknowledgements

To my parents and dear friend M.Z. for her support. Sincerely thank you to Dr. S.E.Hobbs for his supervision. iv CONTENTS v

Contents

Contents v

List of ﬁgures viii

Abbreviations x

1 Introduction 1

1.1Background...... 1

1.2AimoftheThesis...... 1

1.3DocumentStructure...... 2

2 Literature Review 3

2.1GeosynchronousRegion...... 3

2.2SpaceDebrisModels...... 4

2.3PerturbationSource...... 5

2.3.1 NonhomogeneityandOblatenessoftheEarth...... 6

2.3.2 AtmosphereDrag...... 6

2.3.3 GravitationalPerturbation...... 7

2.3.4 SolarRadiationandSolarWind...... 7

2.4MotionoftheMoon...... 8

2.4.1 Cassini’sLaws...... 8

2.4.2 Relative Motion between the Moon, Sun, and Earth ...... 8

2.4.3 LunarStandstill...... 9 vi CONTENTS

3 Gravity Perturbation Model Development 11

3.1GeneralAnalysis...... 11

3.2GravityPerturbationModel...... 12

3.2.1 AnalyticalApproach...... 12

3.2.2 QuantitativeApproach...... 13

3.2.3 Validation...... 17

3.2.4 AmendmentofTheModel...... 17

4 Relative Velocity Between GEO Debris 23

4.1VeriﬁcationoftheModel...... 23

4.1.1 ComparisonBetweenOtherModels...... 23

4.1.2 Application of the Model to the Precession of Lunar Orbit . . 26

4.2RelativeVelocityoftheGEODebris...... 28

4.2.1 Motion of GEO Debris with Sun’s Perturbation ...... 28

4.2.2 CombinationofLunarandSolarGravityPerturbation.... 33

4.3TheRelativeVelocity...... 35

5 Discussion of the Results 39

5.1AboutTheModel...... 39

5.1.1 TheValidityoftheModel...... 39

5.1.2 TheMeaningofDevelopingsuchaModel...... 40

5.2AboutTheResult...... 40

6 Conclusions 42

References 43

A Calculation of iR 46

B Matlab code 48

B.1Numericallysolvethegroupofdiﬀerentialfunctions...... 48 Contents vii

B.2PlotthePrecessionofLunarOrbitalPole...... 50

B.3 Plot Inclination and the Precession of GEO Debris’ Orbital Pole . . . 51

B.4PlotoftheRevolvingAscendingNode...... 53

B.5SimulationoftheDistributionofDebris...... 54

B.6SimulationoftheDistributionofRelativeVelocities...... 54

C Application of Matlab cftool toolbox 57 viii LIST OF FIGURES

List of Figures

2.1DistributionofGEODebris[1]...... 3

2.2PayloadsandupperstageslaunchedintoGEO[2]...... 4

2.3GEOregion...... 5

2.4ApparentpathsoftheSunandMoononthecelestialsphere...... 9

2.5ApparentmotionoftheSunandtheMoon.[3]...... 9

2.6Parametersofthelunarorbit...... 10

3.1MaximumrelativevelocitybetweenGEOdebris...... 12

3.2TypicalpositionbetweenSunandEarth...... 13

3.3MomentappliedbySuntoGEOdebrisatSolstice...... 13

3.4GeneralsituationforSolstice...... 14

3.5 Position of M and m in (x,y,z)and(x,y,z) ...... 14

3.6 iR is in the direction of OA when the inclination of orbital plane i =0 18

4.1 The New Deﬁned i ...... 24

4.2ComparisonofCoordinateSystems...... 25

4.3ThePrecessionofLunarOrbitalPole...... 27

4.4Moon’sOrbitalInclinationtoEclipticPlane...... 27

4.5TrackoftheOrbitalPoleofGEODebris...... 29

4.6TheInclinationoftheOrbit...... 29

4.7Debris’RelativeMotiontotheEarthSurface...... 30 LIST OF FIGURES ix

4.8 The Ascending Node is perpendicular to the Projection of the Orbital Pole...... 31

4.9 The Ascending Node Evolves with the Same Period as ω ...... 31

4.10 Curves have diﬀerent amplitude and phase. They are all moving westwardsbecauseoftheEarthrotation...... 33

4.11DistributionofGEOdebris...... 33

4.12 The simulation of debris distribution is quite similar to the observation. 34

4.13TheDistributionofRelativeVelocities...... 37

A.1 Direction of iR changswithL ...... 46

C.1 Curve Fitting Toolbox Interface. Thetaz vs Time is plotted as blue andcurveﬁttingresultistheredline...... 57

C.2CurveFittingResult...... 58 x Abbreviations

Acronyms and deﬁnitions

GEO Geostationary Earth Orbit IADC Inter-Agency Space Debris Coordination Committee SHM Simple-Harmonic-Motion Lunar Orbital Pole the pole perpendicular to the lunar orbit and it points to the north Ecliptic Pole the pole perpendicular to the ecliptic plane and it points to the north Introduction 1

Chapter 1

Introduction

1.1 Background

Geostationary Earth Orbit (GEO) is widely used because of its unique characteristics. Its orbital period is exactly one sidereal day with the altitude of 35,786 km. Satellites using this orbit can thus stay above a ﬁxed point relative to the Earth, which is very important for some satellites like communication satellites.

But after their lifetimes these satellites are no longer actively controlled and become orbital debris. Break-ups and explosions would further contribute to the debris population. Since there is no nature removal mechanism in GEO region, these debris would have lifetimes exceeding a million years. Their orbit would drift from 15 degrees north to 15 degrees south of the equatorial plane and 52 km above and below the geosynchronous arc because of the gravity perturbation from the Sun and the Moon [4]. The nonhomogeneity and oblateness of the Earth cause migration west and east around the Earth. Net eﬀect of these motions leads to a torus around the Earth, where only 32% of the 1,124 known objects (2004) are under active control [1].

Because of the threat placed by these debris, risk and damage assessment are in- dispensable in spacecrafts design. In breakup models the initial condition, e.g., the relative velocity, is of extreme importance in the case of collision (rather than explo- sion or rupture) [5]. And the distribution of relative velocity between GEO debris is exactly what this thesis is pursuing.

1.2 Aim of the Thesis

The speed of any GEO objects is around 3.07 km/s with no more than 0.48% variation. So the relative velocity is decided mainly by the direction, i.e., the orbit inclination. This places the necessity of a thorough understanding of third-body 2 Introduction gravity perturbation on debris’ orbit. After developing a model of gravity perturbation, the motion of GEO debris can be studied and relative velocity would be a direct result.

1.3 Document Structure

The key to know the motion of GEO debris is to understand the mechanism how the Sun and the Moon apply gravity perturbation to them. Most of the theories describing third-body gravity perturbation were done under the assumption that the orbital inclination i ≈ 0. Chapter 3 is the development of a perturbation model without this assumption.

In Section 4.1, solution of the model was compared to the work by Alby.F. It was further applied to the lunar orbit and results were quite optimistic. These prove that my model is reasonable and correct.

The motion of GEO debris in short and long term considering only the Sun is discussed in Section 4.2.1. SHM for GEO debris in short term is assumed to be a good approximation because of the similarity between the actual and computed result from the model considering only the Sun. In Section 4.2.2 detailed discussion about the approximation is provided. Calculation of the relative velocity between debris is then listed in Section 4.3. Literature Review 3

Chapter 2

Literature Review

Geosynchronous region is fairly crowded with retired satellites and fragments that were generated by break-ups, explosions and collisions. And the number of launches each year is generally increasing (Figure 2.2). In all the cataloged GEO objects only 31% are under control (Figure 2.1 (a)). Break-ups of spacecrafts contribute about 43% of catalogued objects, and 85% of all space debris larger than 5 cm in diameter [6]. Explosions and collisions happen less frequently but contribute about 50 percent of all tracked objects [1]. Figure 2.1 (b) is a plot of the GEO debris torus.

(a) cataloged GEO objects (b) debris torus around GEO

Figure 2.1: Distribution of GEO Debris [1].

2.1 Geosynchronous Region

Because of the uniqueness of this orbit, [7] deﬁnes the protected geosynchronous region as a segment of the spherical shell with:

• lower altitude = geostationary altitude minus 200 km • upper altitude = geostationary altitude plus 200 km 4 Literature Review

Figure 2.2: Payloads and upper stages launched into GEO [2]

• -15 degrees ≤ latitude ≤ +15 degrees

• geostationary altitude (ZGEO) = 35,786 km (the altitude of the geostationary Earth orbit)

This region is where this thesis focuses at. [7] requires that spacecrafts be pushed away from geostationary orbit at the end of the disposal to avoid interference with active spacecrafts. According to their guide, the satellites must meet the following two conditions:

1. A minimum increase in perigee altitude of: 235km+(1000·CR · A/m)

2. An eccentricity less than or equal to 0.003.

Here CR is the solar radiation pressure coeﬃcient. A/m is the aspect area to dry mass ratio (m2kg−1).

2.2 Space Debris Models

NASA’s breakup model of EVOLVE 4.0, also called the NASA Standard Breakup Model, is quite helpful in designing satellites by providing results of fragmentations Literature Review 5

Figure 2.3: GEO region in the debris environment. Initial conditions of the breakup such as total mass of the parent object and the collision velocity are basic inputs of the model. [5]

There are some well developed models about debris environment, such as EVOLVE (NASA) and MASTER-2005 (ESA). They both emphasize at estimating the future evolution of the debris environment. Take EVOLVE as an example, typical time frame is 100 years and its premise is to predict future debris environment according to recent launch and debris degradation rates [8]. MASTER (Meteoroid and Space Debris Terrestrial Environment Reference) is a software used to analyze space debris ﬂux and spatial densities [9].

2.3 Perturbation Source

Under the ideal case that GEO debris experience no other force than Earth gravity, they will continuously move in the geostationary orbit with the same speed and relatively still to neighboring objects. Then there is no need to worry about them. Unfortunately that does not exist in practice. In practice perturbing accelerations acting on space objects result from nonhomogeneity and oblateness of the Earth, atmosphere drag, third-body gravitational perturbation, solar radiation and solar wind. 6 Literature Review

2.3.1 Nonhomogeneity and Oblateness of the Earth

The Earth is oblate and its mass is not distributed homogeneously. This results in the diﬀerence between reality and the ideal model of the Earth gravity potential function as:

μ U(r)=− (2.1) r

[10] gives a simpliﬁed form of U: ∞ μ Re n U ≈− 1 − ( ) JnPn(sφ) r n=2 r μ = [U0 + UJ2 + UJ3 + ···] (2.2) r

− Re 2 1 2 − here U0 = 1, UJ2 =( r ) J2 2 (3 sin φ 1). J2 is at least 400 times larger than any other Jn coeﬃcients. As a result the orbit of GEO debris can be viewed as getting perturbation from UJ2. The Lagrange’s planetary equations can be used: da 2 ∂U = , (2.3) dt na ∂M √ − 2 − 2 de 1 e ∂U − 1 e ∂U = 2 2 , (2.4) dt na e ∂M nae ∂ω di −1 ∂U ∂U = √ +cos(i) , (2.5) dt na2 1 − e2 sin(i) ∂Ω ∂ω dΩ 1 ∂U = √ , (2.6) dt 2 − 2 ∂i √na 1 e sin(i) dω 1 − e2 ∂U cos(i) ∂U = − √ , (2.7) dt na2e ∂e na2 1 − e2 sin(i) ∂i dM 2 ∂U 1 − e2 ∂U = n − − (2.8) dt na ∂a na2e ∂e

Substitute UJ2 into Eq. 2.3, Eq. 2.4 and Eq. 2.5: da de di = = = 0 (2.9) dt dt dt

This means the change of da, de and di is zero for each period.

2.3.2 Atmosphere Drag

GEO orbit locates in the exosphere of the Earth atmosphere. Here particles are so far away thus hardly collide with each other. The atmosphere drag can be ignored. Literature Review 7

2.3.3 Gravitational Perturbation

Theoretically all planets and the Sun apply gravitational perturbation on GEO debris. According to [10] its magnitude is determined by Kp :

Gmp Kp = 3 (2.10) nrp where G is Gravitational Constant, mp is the mass of the perturbing body, rp is its distance from Earth center, n is the mean motion of the GEO object. And

−3 Kmoon =5.844 × 10 deg/day (2.11) −3 Ksun =2.69 × 10 deg/day (2.12)

Even the closet planet to the Earth, Venus, its distance from Earth is at least (depends on its relative position with the Earth) 108 times that of the Moon while itsmassisonlyabout4.49 times the Moon’s mass. The planet with highest mass in the solar system is Jupiter, which is 2.58 × 104 times heavier but also 1.64 × 103 times further than Moon from the center of Earth. As a result, none of these planets will apply inﬂuence to the orbit of GEO debris by more than 10−5 times of Moon’s inﬂuence. And only gravitational perturbation from the Sun and the Moon need to be considered.

2.3.4 Solar Radiation and Solar Wind

The Sun radiates electromagnetic waves from X-rays to radio waves, which cause pressure on any objects in the GEO orbit and change the eccentricity of the their orbit. Solar wind contains emitted ionized nuclei and electrons from the Sun. The momentum ﬂux caused by solar wind is much less than that caused by the solar radiation by a factor of 100 to 1,000 [10]. As a result in this thesis only solar radiation pressure is considered.

The solar radiation pressure |FR| is the product of the momentum ﬂux P , the cross- sectional area A of the debris perpendicular to the sun line and the absorption characteristic coeﬃcient Cp: |FR| = PACp (2.13)

Cp range from 0 to 2; Cp = 1 for a black body and Cp = 2 for a body reﬂecting all light.

In the orbits around the Earth, P is:

452.7 2 P = × 10−8(N/m ) (2.14) 1.0004 + 0.334 cos(D) 8 Literature Review

D is the “phase” of the year starting on July 4, the day of earth aphelion [11].

From Eq. 2.13 and Eq. 2.14:

452.7ACp −8 |FR| = × 10 N (2.15) 1.0004 + 0.334 cos(D)

Eq. 2.15 shows that the solar radiation pressure relates directly to the “phase” of the year.

2.4 Motion of the Moon

2.4.1 Cassini’s Laws

The Cassini’s laws describe the motion of the rotation of the Moon based on observation. [12]

1. The Moon rotates counterclockwise with an uniform motion around an axis ﬁxed with respect to its surface; the period of rotation is the same as the Moon’s sidereal period of revolution around the Earth.

2. The angle between the rotation axis and the ecliptic pole of the Moon is constant ≈ 1◦35.

3. The ecliptic pole of the Moon, the axis normal to the Moon’s orbit and its rotation axis are always in the same plane.

2.4.2 Relative Motion between the Moon, Sun, and Earth

The Moon revolves around the Earth in a month (27.3 days), the Earth revolves around the Sun in a year (365.25 days). The Sun’s track in the celestial sphere lies in a ﬁxed plane tilted 23.44◦ to the equatorial plane, while the lunar orbital inclination varies from 23.736◦ +5.133◦ =28.869◦ to 23.736◦ − 5.133◦ =18.603◦ (Figure 2.4). This is caused by the precession of the lunar pole around the ecliptic pole with an cone angle of 5.13◦ (Figure 2.5).

Orbital parameters of the Moon is deﬁned in Figure 2.6: ism, im and is are the inclination between the ecliptic orbit and the lunar orbit, the lunar orbit and the equatorial orbit, the ecliptic orbit the equatorial orbit respectively. Ω is the right ascension angle of the lunar orbit in the ecliptic orbit from the vernal equinox axis. [10] gives an approximate expression of im as:

im[deg] = 23.736 + 5.133 sin [ωm(t − 2, 001.7433)] (2.16) Literature Review 9

Figure 2.4: Apparent paths of the Sun and Moon on the celestial sphere.

Figure 2.5: Apparent motion of the Sun and the Moon.[3]

here ωm is the frequency of the precession of lunar pole with a period of 18.6 Years, t is time expressed in years.

2.4.3 Lunar Standstill

Due to the gravity of the Sun, the Moon’s orbit is not ﬁxed relative to the Earth but precessing around the solar pole with period of 18.6 Years. With constant obliquity of the ecliptic of 5.133◦, the inclination of lunar orbit to the equatorial plane varies from 18.603◦ to 28.869◦. Moving in the celestial sphere, the orbit inclination is the extreme declination that the Moon reaches in a month. When the inclination is 10 Literature Review

Figure 2.6: Parameters of the lunar orbit.

18.603◦ (called minor standstill) or 28.869◦ (called major standstill), the change of inclination is very slow. The most recent lunar major standstill happened in 2006 with consequences be able to be observed for more than 3 years. Gravity Perturbation Model Development 11

Chapter 3

Gravity Perturbation Model Development

3.1 General Analysis

According to the deﬁnition of GEO region [7], the radius of concerned orbits range from (rGEO-200) km to (rGEO+200) km, rGEO = 42164 (km) is the GEO orbital radius. So the minimum velocity of GEO objects is: μ vmin = rGEO + 200

μ is the geocentric gravitational constant, μ = 398600.4415 ± 0.0008 km3s−2 [13].

398600.44 vmin = =3.0673(km/s) (3.1) 42164 + 200

Similarly, 398600.44 vmax = =3.0820(km/s) (3.2) 42164 − 200

And the variation of velocity is:

vmax − 1=0.48% (3.3) vmin

That’s to say the speed of all objects in GEO region relative to the geocentric inertial system have the same magnitude around 3.07 km/s. The debris’s velocity vector is drifted 15 degrees north to 15 degrees south of the equatorial plane (Figure 3.1). So the maximum relative velocity between GEO debris is: 12 Gravity Perturbation Model Development

Figure 3.1: Maximum relative velocity between GEO debris.

3.07 × sin(15◦) × 2=1.59(km/s) (3.4)

As a conclusion, the relative velocity of GEO debris must be a distribution between 0 and 1.59 km/s.

3.2 Gravity Perturbation Model

Gravity perturbations of the GEO orbit come mainly from the Sun and Moon. They are quite similar so a general model of their gravity inﬂuence can be built up.

3.2.1 Analytical Approach

Before building up the model, an analytical approach is made to show main characters of the perturbation.

Take the Sun for an example. Typical relative positions between Sun and Earth happen at Solstice. The Sun is to the north of GEO orbit when it’s Summer Solstice and in the south at Winter Solstice.

Sun stays to the north of an equatorial orbit when it’s Summer Solstice (point A in Figure 3.3 (a)). As a result the average force applied by the Sun to a small fragment of GEO debris m is (f + δf) during the day, which means m moves on the +x side of y axis. During the night Sun is a little bit further from m thus exerts (f − δf) on it. Gravity Perturbation Model Development 13

Figure 3.2: Typical position between Sun and Earth

(a) Summer Solstice (point A) (b) Winter Solstice (point B)

Figure 3.3: Moment applied by Sun to GEO debris at Solstice

The situation at Winter Solstice (point B) is similar. The Sun stays to the south of the equatorial plane and applies (f + δf) during the day and (f − δf)atnight (Figure 3.3 (b)) to m. The net eﬀect for both Summer and Winter Solstice can be equivalent to applying δf only, as in Figure 3.4:

δf leads to a clockwise moment Mf around the y axis (−y direction) at both A and B point. Its direction changes while Sun moves between A and B, but a reasonable guess is that the average torque for a year is in (−y) direction.

3.2.2 Quantitative Approach

A general model is built up to study the gravitational inﬂuence of a distant mass M on the orbit of GEO debris. M can be the Moon or Sun. The Moon’s orbit is an ellipse with semi-major axis of 3.844 × 105 km. It is large compared with the GEO orbit radius (4.216 × 104) km. The distance between Sun and Earth is 1.496 × 108 km, which is 3.548 × 103 times the GEO orbit radius. One of base assumption of this model is that the ratio r between GEO orbit radius r and orbit radius of M R0 satisﬁes: 14 Gravity Perturbation Model Development

Figure 3.4: General situation for Solstice

r r = << 1 (3.5) R0

To start, Moon’s orbit is viewed as a circle. This general model for both Moon and Sun will later be speciﬁed considering their detailed orbit characters.

Figure 3.5: Position of M and m in (x,y,z)and(x,y,z) Gravity Perturbation Model Development 15

There are two sets of coordinates in Figure 3.5. (x,y,z) is an Earth-ﬁxed-inertial(ECI) system with x,y axis in the equatorial plane. (x,y,z) is a inertial coordinate system rotated from (x,y,z)abouty(y)axisforθ until M is on x axis.

M is moving in the plane (x ,y ) and its orbit radius is R0. Write its position in (x ,y ,z ) and use rotation matrix from (x ,y ,z )to(x,y,z), R 0 in (x,y,z)is:

⎛ ⎞ cos θ cos α ⎜ ⎟ R 0 = R0 ⎝ sin α ⎠ (3.6) sin θ cos α

α = w1t − α0 (3.7)

2π w1 = (3.8) T

T is the period of M’s motion. For Sun Ts = 365.242190402 (days) [14] and for Moon Tm =27.321582 (days) [15].

Position of a small fragment of GEO debris with mass m is deﬁned by φ (Figure 3.5):

r = r(cos φ, sin φ, 0) (3.9)

φ = w2t − φ0 (3.10)

2π w2 = (3.11) TG

TG is the period of GEO orbit. TG = 1 sidereal day = 86164.09054s .

Force applied by M to m is:

GMm f = R (3.12) |R|3

⎛ ⎞ R0 cos θ cos α − r cos φ ⎜ ⎟ R = R 0 − r = ⎝ R0 sin α − r sin φ ⎠ (3.13) R0 sin θ cos α 16 Gravity Perturbation Model Development

And

1 |R| = R0(1 + Δ) 2 (3.14)

Here

r 2 r Δ= − 2 (cos θ cos α cos φ +sinα sin φ) R0 R0 = r2 − 2r(cos θ cos α cos φ +sinα sin φ) (3.15)

Because of Eq. 3.5: Δ << 1 (3.16)

From Eq. 3.5, Eq. 3.12, Eq. 3.13, Eq. 3.14:

GMm − 3 2 f = 2 (1 + Δ) · K (3.17) R0

Here ⎛ ⎞ cos θ cos α − r cos φ ⎜ ⎟ K = ⎝ sin α − r sin φ ⎠ (3.18) sin θ cos α

Considering that the Moon’s orbit radius is no more than 10r, inequality Eq. 3.5 − 3 may not be satisﬁed to some extent. Expand (1 + Δ) 2 around 1 up to the second order as: − 3 3 15 2 (1 + Δ) 2 =1− Δ+ Δ + ··· (3.19) 2 8

The torque applied to m is: ⎛ ⎞ cos φ GMmr 3 15 2 ⎜ ⎟ τ = r × f = 2 (1 − Δ+ Δ ) ⎝ sin φ ⎠ × K (3.20) 0 2 8 R 0

Integrate τ from 0s to TG: T 2π G GMmr 3 15 2 δt · τ = 2 δφ(1 − Δ+ Δ ) (3.21) 0 0 ⎛w2R0 2 8 ⎞ sin θ cos α sin φ ⎜ ⎟ · ⎝ − sin θ cos α cos φ ⎠ sin α cos φ − cos θ cos α sin φ

Put in Eq. 3.15 the integration result is: ⎛ ⎞ 1 TG 4 sin θ sin 2α · GMmr − 15 3 ⎜ − 1 2 ⎟ δt τ = 2 (3r r ) ⎝ 4 sin 2θ cos α ⎠ (3.22) 0 2 0 2 w R 0 Gravity Perturbation Model Development 17

3.2.3 Validation

TG ( 0 δt · τ) is the accumulated torque applied by M during one day. Consider M as Sun now. α equals to zero at point A and π at point B (Figure 3.4). With that value the only non-zero component in Eq. 3.22 is in (−y) direction. This result is the same as for the analytical approach.

Furthermore, the x component of this accumulated torque is a function of sin 2α (α indicates the position of Sun). Average torque in x axis is zero if integration is done for period TS:

⎛ ⎞ 0 GMmr 15 3 ⎜ ⎟ τ = − 2 (3r − r ) ⎝ sin 2θ ⎠ (3.23) 0 2 8R 0

This result is compatible with the guess that average moment in a year applies only in (−y) direction.

In summary, the model works ﬁne as long as the two conditions it is based on are satisﬁed:

1. r << 1;

2. r = r(cos φ, sin φ, 0), or r stays in (x,y) plane.

Expanding Δ to the second order takes care of the difference between reality and condition 1. But the angular velocity of GEO debris w will move towards (−y) axis because of the average torque applied by Sun Eq. 3.23 or Moon. As a result in longer time scale condition 2 is no longer satisfied and this model needs to be modified.

3.2.4 Amendment of The Model

Dealing With Moving Coordinate System

Since GEO debris no longer stay in the (x, y) plane, a new coordinate system ﬁxed to the orbital plane can be built up to study the perturbation. The torque equation would stay the same and so does the result. But only the movement relative to Earth is of concern. The rotation matrix from this new coordinate system to (x,y,z) is changing with time; thus makes the problem complicated.

Goes back to the physics meaning, the applied moment τy in (−y) direction coupled to the angular momentum L about z tends to rotate the angular momentum vector about x axis. The L vector is moving towards (−y) (Figure 3.6). 18 Gravity Perturbation Model Development

Figure 3.6: iR is in the direction of OA when the inclination of orbital plane i =0

According to its magnitude Eq. 3.23 and its direction,

GMmr 15 3 τ = 2 sin θ cos θ(3r − r ) · (ix ×iz) (3.24) 4R0 2

ix andiz are unit vectors in x and z axis. θ is the angle betweeniR and the equatorial plane.

We eliminate these coordinate-dependent vectors by replacing ix ×iz with parameters relate directly to the orbit to avoid the complexity introduced by the moving coordinate. Angular momentum vector L is:

⎛ ⎞ Lx ⎜ ⎟ L = ⎝ Ly ⎠ (3.25) Lz

Another one can beiR. It is the unit vector from the center of the Earth to M and is perpendicular to the ascending node. It is deﬁned to the north of the orbital plane. In the case that orbital inclination i =0,iR is in the direction of OA (Eq. 3.6). When the orbital inclination i is changed from 0 the direction of iR is also changed.

Relationship between ix, iz, L and iR is:

1 L ix ×iz = ·iR × (3.26) cos θ L Gravity Perturbation Model Development 19

From Eq. 3.24 and Eq. 3.26,

GMmr 15 3 L τ = 2 sin θ(3r − r ) · (iR × ) (3.27) 4R0 2 L

Refer to appendix 1 for the deduction of iR, the result is:

⎛ ⎞ cos θ ⎜ Ly ⎟ R 1 · ⎝ ⎠ i = c Lx cos θ+Lz sin θ (3.28) sin θ

Here C1 is a normalization constant. Considering that:

dL τ = (3.29) dt

Eq. 3.27 become a group of diﬀerential equations about Lx, Ly, Lz

Diﬀerential Equations

From Eq. A.1, Eq. 3.27 and Eq. 3.29: ⎛ ⎞ ⎛ ⎞ dLx Lz−Lx tan θ dt L +L tan θ cos θLy ⎜ ⎟ GMmr 15 Lx cos θ + Lz sin θ ⎜ x z ⎟ ⎝ dLy ⎠ − 3 ⎝ − ⎠ dt = 2 (3r r ) 2 2 2 Lx sin θ cos θLz (3.30) dL 4R0 2 Lx + Ly + Lz Lz−Lx tan θ z y dt Lx+Lz tan θ sin θL

This is a group of diﬀerential equations about Lx, Ly and Lz.

GMmr 15 3 C2 = 2 (3r − r ) (3.31) 4R0 2

Rewrite Eq. 3.30: − dLx · Lx cos θ + Lz sin θ · Lz Lx tan θ = C2 2 2 2 cos θLy (3.32) dt Lx + Ly + Lz Lx + Lz tan θ

dLy · Lx cos θ + Lz sin θ · − = C2 2 2 2 (Lx sin θ cos θLz) (3.33) dt Lx + Ly + Lz 20 Gravity Perturbation Model Development

− dLz · Lx cos θ + Lz sin θ · Lz Lx tan θ = C2 2 2 2 sin θLy (3.34) dt Lx + Ly + Lz Lx + Lz tan θ

From Eq. 3.32 and Eq. 3.34,

dLz dLx =tanθ · (3.35) dt dt

Apply boundary condition:

2π 2 x z when i=0, L =0,L = TG mr

The solution of Eq. 3.35 is:

2π 2 Lz − tan θLx = mr (3.36) TG

Substitute Lz with Lx, Eq. 3.32 and Eq. 3.33 is simpliﬁed to be:

2 4 2π C2mr cos θ · · Ly dLx TG = π 2 2 2 2π 2 2 2 (3.37) x y dt (L + TG sin 2θmr ) +(L cos θ) +(TG mr cos θ)

2 2 2π π 2 − 2 · · x · dLy C mr cos θ TG (L + TG sin 2θ mr ) = π 2 2 2 2π 2 2 2 (3.38) dt x y (L + TG sin 2θmr ) +(L cos θ) +(TG mr cos θ)

Set new variables:

π 2 x x = L + TG mr sin 2θ (3.39) y =cosθLy (3.40)

Eq. 3.37 and Eq. 3.38 become:

dx 2π 3 2 y = cos θmr C2 2 2 2 (3.41) dt TG x +y +J dy 2π 3 2 x = − cos θmr C2 2 2 2 (3.42) dt TG x +y +J

2π 2 2 Where J = TG mr cos θ. Gravity Perturbation Model Development 21

Deﬁne the inclination vector i as an unit vector aligned with the L :

L TG i = (3.43) mr2 2π

Redeﬁne x, y and J as:

x = ix +sinθ cos θ (3.44)

y =cosθiy (3.45)

J =cos2 θ (3.46)

Result of substituting i into Eq. 3.41 and Eq. 3.42 using new deﬁned x, y and J is:

dx y dt = C3 x2+y2+J 2 dy − x (3.47) dt = C3 x2+y2+J 2

here new constant C3:

GMTG 3 15 2 C3 = 3 cos θ(3 − r ) (3.48) 8πR0 2

At t =0,Lx = Ly =0,ix = iy = 0, from Eq. 3.44 and Eq. 3.45: x(0) = sin θ cos θ (3.49) y(0) = 0

According to the conclusion in the analytical approach (Chapter 3.2.1), at t =0,

dLy < 0 dt

Thus dy diy =cosθ < 0 dt dt 22 Gravity Perturbation Model Development

Solution

Solution of Eq. 3.47 is not straightforward, so I pursued numerical solution using Matlab code in appendix B.1. According to the result, ix and iy change sinusoidally. With this clue and the boundary conditions, solution of Eq. 3.47 is:

1 x = 2 sin(2θ)cos(wt) − 1 (3.50) y = 2 sin(2θ) sin(wt)

And GMTG 15 2 w = 3 cos θ(3 − r ) (3.51) 8πR0 2

Substitute Eq. 3.44 and Eq. 3.45 into Eq. 3.50, 2 wt ix = − sin(2θ)sin ( ) 2 (3.52) iy = − sin(θ) sin(wt)

And according to Eq. 3.36 and Eq. 3.43,

2 2 wt 2 2 iz = −2sin θ sin ( )+1=cos θ +sin θ cos(wt) (3.53) 2 Relative Velocity Between GEO Debris 23

Chapter 4

Relative Velocity Between GEO Debris

4.1 Veriﬁcation of the Model

The calculation done by Alby.F. [16] is ﬁrst used to check my model. His result can be applied only to geostationary objects. The approximation he made, such as null orbital inclination and the perturbation body is inﬁnitely far away, limit its usage. His model can be viewed as a special case in my model when debris are just released from geostationary orbit. As a simple check, I considered only the solar gravity perturbation in the comparison.

As a nature satellite of the Earth, the Moon’s orbit receives only the Sun’s gravity perturbation. Mechanical properties of its orbit are well studied. So I further applied my model to calculate the precession period of the lunar orbital pole around the ecliptic pole to verify the model. [12] gives the precession period as 18.6 (years).

4.1.1 Comparison Between Other Models

di dix y Alby. F. gives the time derivative of the inclination vector dt and dt for satellites on the Geostationary orbit [16]:

dix 3 = − Kmoon sin(Ωm)sin(2im) + 0 (4.1) dt 8

diy 3 3 = Kmoon cos(Ωm)sin(2im)+ Ksun sin(2is) (4.2) dt 8 8 24 Relative Velocity Between GEO Debris

Here GTGmp Kp = 3 2πrp mp is the mass of the perturbing body and rp is its distance from the center of the Earth. In his work, inclination vector i is deﬁned as a vector aligned with the node line whose norm equals to the value of i (Figure 4.1).

Figure 4.1: The New Deﬁned i

But the deﬁnition of i used in the last chapter is in Eq. 3.43:

L TG i = mr2 2π

Coordinate systems we used are not the same neither(Figure 4.2). In Alby’s work the vernal equinox axis is in (+Xγ) direction while I used it as (−y) direction. With my coordinate system (x − y), deﬁnition of i: ix = −ixy cos(α) (4.3) iy = −ixy sin(α)

ixy is the value of projection of the inclination vector on (x,y) plane.

Using Alby’s coordinate system and the deﬁnition of components of inclination vector ix and iy: ix = i cos(α) (4.4) iy = i sin(α) Relative Velocity Between GEO Debris 25

Figure 4.2: Comparison of Coordinate Systems

Since according to their deﬁnition the magnitude of ixy isthesameasi .Compare Eq. 4.3 and Eq. 4.4,

ix = −ix (4.5) iy = −iy

Replace ix and iy with ix and iy in Alby’s result and consider only the Sun’s gravity perturbation, which is the second parts on the right side of Eq. 4.1 and Eq. 4.2. Use M as the mass of the Sun and θ as the solar orbital inclination:

dix = 0 (4.6) dt

diy −3 −3GMTG = Ksun sin(2is)= 3 sin(2θ) (4.7) dt 8 16πrp

Alby’s result can be viewed as a special case in my model when t = 0, i.e., the debris is just released from the geostationary orbit. Apply the model with only the Sun’s inﬂuence, according to Eq. 3.52, 26 Relative Velocity Between GEO Debris

dix w |t=0 = − sin(2θ) sin(wt) = 0 (4.8) dt 2

diy |t=0 = −w sin(θ)cos(wt)=w sin(θ) (4.9) dt GMTG 15 2 = − 3 sin(2θ)(3 − r ) (4.10) 16πR0 2

For Sun r << 1, the results of my model and Alby’s are the same. This proves the validity of my model.

4.1.2 Application of the Model to the Precession of Lunar Orbit

According the the result from Eq. 3.52 and Eq. 3.51, ⎧ ⎪ − 2 wt ⎨ ix = sin(2θ)sin ( 2 ) iy = − sin(θ) sin(wt) ⎩⎪ 2 2 iz =cos θ +sin θ cos(wt)

GMTG 15 2 w = 3 cos θ(3 − r ) 8πR0 2

So the precession period is:

16π2R3 1 T = 0 3GMT cos(θ) 1 − 2.5r2 16π2R3 ≈ 0 (1 + 2.5r2) (4.11) 3GMTm cos(θ)

8 30 Substitute R0 =1.496 × 10 (km), M =1.9891 × 10 (kg), Tm =27.32 (days), θ =5.133◦ into Eq. 4.11: T =17.9(years) (4.12)

This value is very close to observation result of 18.6 Years [12]. The diﬀerence may result from:

• The oblateness of the Earth and the nonohomogeneity;

• The solar radiation and solar wind; Relative Velocity Between GEO Debris 27

• The Moon’s orbit is not circular;

• The Sun’s trajectory is not circular.

Plot i using Matlab code in appendix B.2. It precesses around the ecliptic pole (Figure 4.3). In Figure 4.4 , thetaz = arccos(iz) is the Moon’s orbital inclination to the ecliptic plane.

1.4 Solar Orbital Pole 1.2

0.8 z i 0.6

0.4

0.2

0 0.1 0 0 −0.1 −0.05 −0.1 −0.2 −0.15 i i y x

Figure 4.3: The Precession of Lunar Orbital Pole

1.01

1 z i

0.99

0.98 0 10 20 30 40 50 time/year

10 z theta 5

0 0 10 20 30 40 50 time/year

Figure 4.4: Moon’s Orbital Inclination to Ecliptic Plane

The minimum value of iz according to Eq. 3.53 happens when: 28 Relative Velocity Between GEO Debris

cos(wt)=−1 (4.13)

So,

2 2 izmin =cos θ − sin θ =cos(2θ) (4.14)

That means the diﬀerence between maximum and minimum value of thetaz is 2θ.For this case 2θ =2× 5.133◦ =10.266◦. The lunar orbital inclination to the equatorial plane reach its maximum of 28.869◦ at lunar major standstill and minimum of 18.603◦ at minor standstill, diﬀerence of them is exactly 10.266◦. From Eq. 3.53 and Eq. 4.13, at both positions:

diz = − sin2 θw sin(wt) = 0 (4.15) dt

Thus thetaz = arccos(iz) is changing with velocity of zero too. This matches the fact that the lunar orbital inclination changes very slowly at lunar standstill. And this is exactly what still means in the word standstill.

In conclusion, my model is correct and can explain the precession of the lunar orbit quite well.

4.2 Relative Velocity of the GEO Debris

Because of the different orbital period and inclination of the Moon and the Sun, an comprehensive solution of GEO debris would be extremely complex. Instead of pursuing a general solution, analysis with gravity perturbation from only the Sun is first applied. The resulting distribution of GEO debris is very similar to observation in Figure 4.12. With similar influence to the GEO debris, count in the Moon’s gravity will change the period and amplitude but not other characteristics of debris’ short term motion. Assume the actual motion of GEO debris is similar to the motion considering only the Sun. The relative velocity of the GEO debris can be calculated.

4.2.1 Motion of GEO Debris with Sun’s Perturbation

Results from the Model

Apply the model to the GEO debris under perturbation from the Sun. Plot the evolving track of i in Figure 4.5 (code refer to appendix B.3). Similar to the lunar orbit, the orbit of GEO debris precesses around the ecliptic pole with constant cone Relative Velocity Between GEO Debris 29

1.4

1.2 Solar Orbital Pole

0.8 z i 0.6

0.4

0.2

0 0.2 0 0 −0.2 −0.4 −0.2 −0.4 −0.8 −0.6 i i y x

Figure 4.5: Track of the Orbital Pole of GEO Debris

1 z i 0.8

0.6

0 200 400 600 800 1000 time/year

40 z theta 20

0 0 200 400 600 800 1000 time/year

Figure 4.6: The Inclination of the Orbit

◦ angle of 23.44 . thetaz = arccos(iz) is the inclination of the orbit to the equatorial plane.

From Eq. 4.11, the precession period with solar gravity perturbation is:

2 3 ≈ 16π Rs TGs ◦ = 530(Years) (4.16) 3GMsTG cos(23.44 ) 30 Relative Velocity Between GEO Debris

(a) GEO Debris on a Drifted Orbit. (b) A Snapshot of Debris in the same orbit. The longitude of the ascending node is initially assumed to be 0◦.

Figure 4.7: Debris’ Relative Motion to the Earth Surface

Relative Motion between GEO Debris and the Earth Surface

Figure 4.7 (a) is a sketch about the motion of a piece of GEO debris on the celestial sphere. It crosses the equatorial plane at point A. At this moment point A on the earth surface is at the same position of its projection. A rotates with the earth surface so is moved to D2 after a while. At the same time the debris is moved to D1. D2 corresponds to D2 on the celestial sphere. Because the debris stay in the GEO region with orbital period almost equal to one sidereal day, angular velocity of the movement of debris is almost the same with the earth’s rotation. D1andD2stayin the same longitude. D1 moves relative to D2 in the direction perpendicular to the equatorial plane. Figure 4.7 (b) is a snapshot of distributed debris’ projections on the Earth surface. These debris are in the same orbit. Points on the curve stand for individual debris and they move vertically in Figure (b). The longitude of its track depends on where it crosses the equatorial plane (point A in Figure (a)). Deﬁne the longitude of vernal equinox axis to be (−90◦)onthe(x,y) plane in the Earth-ﬁxed inertial system. This curve moves westwards because of the rotation of the Earth. It is like shaking one end of a rope on a horizontal plane and causes a moving wave on it: the wave moves on the rope horizontally and mass points on the rope oscillate vertically.

The longitude of the ascending node OA is determined by the orbital pole (Figure 4.8). This is because the orbital pole OP is perpendicular to the orbit plane and OA is inside the plane, OA is perpendicular to OP. The line between P and its projection on the equatorial plane H is perpendicular to the equatorial plane thus also perpendicular to OA,soOA is perpendicular to OH.

Plot the position of H and the resulting revolving ascending node dynamically using data from the model by Matlab (code refer to B.4). Figure 4.9 (a) is a snapshot of the result. Deﬁne the orientation of ascending node as φ. φ is the angle from +x direction to the upper half of the ascending node. φ changes in the interval of Relative Velocity Between GEO Debris 31

Figure 4.8: The Ascending Node is perpendicular to the Projection of the Orbital Pole.

(0, 180◦). That results in the rapid changes in the end of each period (Figure 4.9).

1 180

0.8 Ascending Node 160 0.6 140 0.4 120 0.2 100

y 0 O

/degrees 80 −0.2 φ H 60 −0.4 40 −0.6

−0.8 20

−1 0 −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 x T/period

(a) A Snapshot the Ascending Node, plotted (b) φ Evolves with Constant Speed as the red line.

Figure 4.9: The Ascending Node Evolves with the Same Period as ω

After a piece of GEO debris is released from geostationary orbit, its orbital pole precesses around the ecliptic pole. The resulting OH on the (x,y) plane revolves with period time same to the precession period P . Meanwhile, φ decreases from 180◦ to 0◦ (Figure 4.10). Together with the curve ﬁtting result in Appendix C, in time interval of (0, P ) the drift speed of ascending node is:

π W = (4.17) P

Any debris released at the same longitude oscillates at this longitude, but the am- 32 Relative Velocity Between GEO Debris plitude and phase of their oscillation depends on when it was released. With Eq. 4.17, the amplitude is decided by:

A = θ sin(WT) (4.18)

Here θ is the long term maximum orbital inclination, T is the period since when the debris was released from geostationary orbit. In the case of solar perturbation on GEO debris,

θ =2× 23.44◦ =46.88◦

The phase of oscillation is:

wt + WT + i0 + longi (4.19)

2π t is the time during the day, w = 1Day , longi is the longitude of the debris. i0 is the initial phase of the oscillation, which is decided by the time when the measurement starts. WT result from the shift of ascending node. WT and wt stands for long term and short term motion of debris respectively.

From Eq. 4.18 and Eq. 4.19, latitude of any particular piece of GEO debris is:

lati(t, T, longi)=θ sin(WT) sin(wt + WT + i0 + longi) (4.20)

If assume there is not preferable longitude in GEO, take a snapshot of all debris near GEO it should look like sum of curves similar to Figure 4.7 (b) with diﬀerent phase and amplitude (Figure 4.10). If assume that the launch activities have been kept at the same level since the 1950s, weight of each of these curves are the same.

The result of adding all these curve together is in Figure 4.11 (a) (matlab code refer to appendix B.5). An observation is provided by [4], which is for the situation in 1995 (Figure 4.11 (b)). These two plots are similar, but in (b) dots are more concentrated close to the equatorial plane.

Considering that the number of launches before 1967 were fairly limited (Figure 2.2), the density of GEO debris on diﬀerent orbits is in fact not the same. Longitude of debris’ ascending nodes is distributed only in the interval between 180◦ and [180◦ − (1995 − 1967)/54 × 180◦] ≈ 90◦. Change the interval of ascending nodes to be (90◦, 180◦), the result of adding these curves is in Figure 4.12 (b).

This is closer to the observation result in Figure 4.12 (a). The only diﬀerence is the amplitude. That’s because no lunar perturbation is considered up to now. But Figure 4.12 indicates the similarity between the solar perturbation and the real case comprise both the Sun and the Moon. Relative Velocity Between GEO Debris 33

Figure 4.10: Curves have diﬀerent amplitude and phase. They are all moving westwards because of the Earth rotation.

−20 latitude(degrees)

−40

−60

−80

0 50 100 150 200 250 300 350 longitude(degrees)

(a) Add all the Curves with Ascending Node (b) Observation from 0◦ to 180◦

Figure 4.11: Distribution of GEO debris

4.2.2 Combination of Lunar and Solar Gravity Perturbation

As mentioned before, the combination of lunar and solar gravity perturbation is extremely complex because the Sun and the Moon move with diﬀerent period and orbital inclination. Debris’ motion with only solar gravity was discussed in Chapter 4.2.1. Similarity between Figure 4.12 (a) and (b) reveals the similarity between the solar perturbation and the real perturbation with both the Sun and the Moon.

From Eq. 4.20, GEO debris follow simple-harmonic-motion (SHM) in one day applying solar gravity. The motion with only the lunar gravity perturbation should also be SHM. The real motion is the combination of these two. Write the force 34 Relative Velocity Between GEO Debris

−20 latitude(degrees)

−40

−60

−80

0 50 100 150 200 250 300 350 longitude(degrees)

(a) Observation (b) Simulation

Figure 4.12: The simulation of debris distribution is quite similar to the observation. applied to SHM particles in the form of:

Fsun = −ks(x − 23.44) (4.21)

Fmoon = −km(x − 23.736 − 5.133 sin wmT ) (4.22)

Here Fsun and Fmoon are apparent force applied to GEO debris. x is the latitude. wm is the velocity of the Moon’s 18.6 years precession period. Eq. 4.22 is simpliﬁed from Eq. 2.16 assuming T counts from 2001. For SHM, the oscillation velocity is: k w = (4.23) m

For the Sun and the Moon : ks ws = (4.24) m

km wm = (4.25) m

From Eq. 4.21 and Eq. 4.22, for the combination of Fsun and Fmoon:

Fsum = −(ks + km)x +23.44ks +23.736km +5.133km sin(wmT ) (4.26) ks and km are constants. The motion with FSum is still SHM if ignore 5.133km sin(wmT ). Angular velocity of this motion is:

ks + km 2 2 wsum = = w + w (4.27) m s m Relative Velocity Between GEO Debris 35

The coeﬃcient Kp in Eq. 2.10 given by [10] determines ws and wm (refer to Chapter 2.3.3 for details): Gmp Kp = 3 nrp For the Moon and the Sun,

−3 Kmoon =5.844 × 10 deg/day

−3 Ksun =2.69 × 10 deg/day

These can be viewed as the maximum velocity driven by Fmoon and Fsun.And

Kp = wA (4.28)

Here A is the amplitude of SHM. Assume the amplitude the motion is known to be 15◦, from Eq. 4.27 and Eq. 4.28,

2 2 Kmoon + Ksun −4 wsum = =4.289 × 10 /day = 0.1566/year (4.29) 15◦

The period of the precession is: 2π T = =40.1years (4.30) w

This result is close to the 54 years period of GEO debris’ precession. Diﬀerence is because the Moon’s 18.6 years precession period is not considered in Eq. 4.26. With Eq. 4.23 and Eq. 4.28, Kp k = m( )2 (4.31) A

The ecliptic plane is tilted to the equatorial plane by a ﬁxed angle so As is constant. ◦ ◦ But for the lunar orbit 37.206

4.3 The Relative Velocity

Assume that the motion applying solar and lunar perturbation together is similar with the case considering only the Sun’s gravity. All debris are oscillating at a ﬁxed longitude with period of one day. Their motion can be described by Eq. 4.20: 36 Relative Velocity Between GEO Debris

lati(t, T )=θ sin(WT) sin(wt + WT + i0 + longi)

Take the maximum inclination of 15◦ and precession period of 54 years as already known. Since there is no preferable longitude in GEO, the relative velocity of debris at longitude=0◦ is studied. By changing the start time of measurement, we can use 0 as value of i0. Simply Eq. 4.20 to be: 1 lati(t, T )= θ [cos(wt) − cos(wt +2WT)] (4.32) 2

Two pieces of debris with the same longitude and latitude at the same t may be released from geostationary orbit at diﬀerent time, thus have diﬀerent values of T : 1 lati1 = θ [cos(wt) − cos(wt +2WT1)] 2 1 = lati2 = θ [cos(wt) − cos(wt +2WT2)] (4.33) 2

As a result: cos(wt +2WT1)=cos(wt +2WT2) (4.34)

The maximum speed of GEO debris in the latitudinal direction when crossing the equatorial plane is:

v0 sin θ (4.35)

Here v0 =3.07 (km/s) is the speed of GEO objects. From Eq. 4.32 and Eq. 4.35, velocity of debris is:

sin θv0 v(t, T )= [− sin(wt)+sin(wt +2WT)] (4.36) 2

Then the relative velocity between two debris at the same point and time is:

sin θv0 v1 − v2 = [sin(wt +2WT1) − sin(wt +2WT2)] (4.37) 2

From Eq. 4.34, sin(wt +2WT1)=± sin(wt +2WT2) (4.38)

Ignore the solution sin(wt +2WT1)=sin(wt +2WT2) because in that case v1 = v2, the relative velocity between debris is zero and it’s meaningless. Using sin(wt + 2WT1)=− sin(wt +2WT2), Eq. 4.37 becomes:

v1 − v2 =sinθv0 sin(wt +2WT1) (4.39) Relative Velocity Between GEO Debris 37

Calculate diﬀerent values of relative velocity when t and T varies using Matlab (code refer to Appendix B.6). And the result is Figure 4.13. This simulation is done for now (2010) under three assumptions:

• The launch activity is maintained at the same level since 1967;

• There is no preferable longitude of GEO satellites;

• The orbital inclination change with time sinusoidally.

0.1

0.09

0.08

0.07

0.06

0.05

probability 0.04

0.03

0.02

0.01

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 velocity(km/s)

Figure 4.13: The Distribution of Relative Velocities

By changing interval value of T ime inthecodethesimulationcanbedoneforany time, e.g., now, 10 years later or 1000 years later. The result is almost identical. But special attention need to be paid to the case when T ime < 27. Because only when

π T1 − T2 = = 27(Years) (4.40) 2W the solution:

sin(wt +2WT1)=− sin(wt +2WT2) (4.41) for Eq. 4.34 exists. That’s to say any two piece of debris with T1 − T2 =27(Years) at the same longitude will for sure meet, and relative velocity between them must satisﬁes the distribution in Figure 4.13. If T ime < 27, then T1 − T2 < 27 and this solution of Eq. 4.41 do not exist thus these two pieces of debris won’t meet at any time. That’s easy to understand. Consider that in the early stage of space industry, 38 Relative Velocity Between GEO Debris the GEO debris are still close to the geostationary orbit thus have no chance to collide with each other.

The relative velocity distributes between 0 and 0.79 km/s. The probability increases rapidly its value get close to 0.79 km/s. This means the relative velocity between GEO debris have higher chance to be in the interval from 0.6 km/s to 0.79 km/s. This meets the fact that the relative velocity between debris in GEO is low (< 1.5 km/s) according to [17].

Recall the conclusion in Chapter 3.1, the relative velocity between GEO debris should maximum value of 1.59 km/s, which is 2 sin θv0. But here the maximum relative velocity is sin θv0. Reason is that according to Eq. 4.36:

sin θv0 v(t, T )= [− sin(wt)+sin(wt +2WT)] 2 v(t, T ) can be any value between (-sin θv0,sinθv0). But with the same t, Eq. 4.37 gives:

sin θv0 v1 − v2 = [sin(wt +2WT1) − sin(wt +2WT2)] 2

So the magnitude of relative velocity can not exceed sin θv0. Discussion of the Results 39

Chapter 5

Discussion of the Results

5.1 About The Model

5.1.1 The Validity of the Model

The result in Chapter 4.1 applying my model to the lunar orbit shows that the lunar orbital pole precesses around the ecliptic pole with a constant cone angle of 5.133◦. This is the same with observation. The precession period according to the model is 17.9 Years, which is very close to the actual 18.6 Years precession period. A possible reason for that diﬀerence is the solar radiation pressure. [11] gives the solar radiation pressure:

4.527ACp −6 |FR| = × 10 N (5.1) 1.0004 + 0.334 cos(D)

D is the “phase” of the year starting on July 4, the day of earth aphelion. According to Eq. 3.23, third-body gravity perturbation is:

2 GMmr 15 2 τy = − 3 (3 − r ) sin 2θ 8R0 2

Deﬁne the characteristic force FG. It is diﬀerent from the actual force applied by the third-body but only for the use of estimating its magnitude.

GMmr FG = 3 (5.2) R0

−6 −6 For the Sun FGs =1.7m × 10 (N), for the Moon FGm =3.7m × 10 (N). Assume space debris are in solid state thus have density in the order of 103 (kg/m3). View debris as ball-shaped with radius of r (m). Its mass is πr3 (kg) and its cross area is 40 Discussion of the Results

2 2 πr (m ). Compare FGs, FGm with |FR| in Eq.5.1, if r is in the order of 1 m, FG is 3 about 10 times of FR. FG and FR are comparable for debris have radius as small as 10−3 m. Thus for the Moon, solar radiation pressure can be ignored.

Another possible reason of the diﬀerence is the Earth oblateness and nonhomogenity. According to the discussion in Chapter 2.3.1, the resulting average change of da, de and di is null for a day. That’s quite obvious because when the GEO debris move around the Earth for one circle, they move relative to the Earth surface latitudinally and the inﬂuence of Earth oblateness and nonhomogenity average out.

The only possible reason of the diﬀerence is the fact that the moon’s orbit is not circular. Its orbit eccentricity is e =0.0549 [15]. The precession period in Eq. 4.11:

16π2R3 T = 0 (1 + 2.5r2)=17.9(years) 3GMTm cos(θ)

The perturbed body inﬂuent the result by term Tm. Guess the change in the model introduced by the non-circularity of perturbed body has the same magnitude as e. The possible range of T is: T ×(1−0.0549) = 16.9(years)toT ×(1+0.0549) = 18.9 (years). This seems to be a possible explanation of the real period of 18.6 (Years).

The orbital period of geosynchronous debris is not perfectly 1 sidereal day. But as discussed in section 3.1, the diﬀerence won’t be more than 0.48% thus can be ignored.

5.1.2 The Meaning of Developing such a Model

Most of the mature work about the dynamic of GEO objects apply only to the Geostationary orbit. That’s to say by making the approximation of very small inclination and eccentricity, Eq. 2.5 can be simpliﬁed to:

di 1 ∂Up = − (5.3) dt ina2 ∂Ω

This makes it possible to analytically solve ix and iy. My work includes no such approximation and more appropriate in dealing with GEO debris’s motion since their maximum inclination is up to 15◦.

5.2 About The Result

The model is developed with the assumption that the perturbing body is far away from the Earth compared with the perturbed body. The orbital radius rGEO of GEO is 42164 km. The Sun-Earth distance is 1.496 × 108 km and that’s more than 3 2 × 10 times of rGEO. The Moon is much closer to the Earth compared with the Discussion of the Results 41

Sun. Semi-major axis of the lunar orbit is 3.84399 × 105 km [15], about 9 times of rGEO. This makes it necessary to expand Δ to the second order in Eq. 3.19. The model was developed only for perturbing bodies with orbital eccentricity being null. The Moon’s elliptical orbit requires modiﬁcation for the lunar gravity perturbation.

The orbit of the perturbed body is assumed to be circular in the model, while actually it’s not true for GEO debris. According to [4], the solar and lunar gravity cause GEO debris to migrate 52 km above and below the geostationary orbit. This 2×52 leads to eccentricity no more than e = 42164 =0.07. According to the discussion in Chapter 5.1.1, the resulting change in precession period won’t be more than 7%.

To simulate the relative velocity, I assumed SHM for debris’ motion. It’s the premise to simplify Eq. 4.20 to Eq. 4.32 in Chapter 4.3. But it’s not accurate according to Chapter 4.2.2 because the Moon’s orbit is precessing around the ecliptic pole with period of 18.6 years. Actual motion should be SHM with modulated amplitude. Future work can be done numerically and will lead to more actuate result of relative velocity.

In the calculation of relative velocity, I assumed that:

• The launch activity is maintained at the same level since 1967;

• There is no preferable longitude in GEO.

This is the ideal case. Actually in the early stage of space industry most of the launches were done by the US or the Soviet Union. They must have some preferable longitude for their GEO satellites, e.g., above their country or countries they are interested in. Thus satellites aiming at using particular longitude should care more about debris released from that longitude. Eq. 4.39 needs to be modified for this reason. Generally space activies have been increasing so the weight of each curves in Figure 4.10 will be different. In Eq. 4.39, this means different probability of T . The distribution of relative velocities will be changed because of the change in Eq. 4.39. 42 Conclusions

Chapter 6

Conclusions

The development of a gravity perturbation model in Chapter 3 was really time consuming. Luckily according to the result in Chapter 4.1, it describes the behavior of lunar orbit quite well. This proves the model is reasonable and correct.

The model was applied to the GEO debris considering only the Sun. The movement of debris during one day (short term) and the precession of their orbit (long term) was combined to reveal the behavior of GEO debris. The resulting distribution of debris is very similar to the observation in Figure 4.12. This indicates the similarity between the solar perturbation and the real perturbation applied by both the Sun and the Moon. With only the Sun debris follow SHM in short term and the dynamic functions were studied. Another SHM due to lunar gravity needs to be added. Result shows that if not consider the Moon’s 18.6 (years) percession period, combination of these two SHMs is still SHM. With assumption of SHM and based on the fact that the maximum inclination of GEO debris is 15◦ and precession period is 54 (years), the relative velocity between GEO debris was simulated.

The gravity perturbation model is useful in studying the behavior of satellites. With- out approximation of null orbital inclination, it’s more appropriate in describing motion of GEO debris. The resulting distribution of relative velocity can be used as input to break up models.

Further work can be done by taking 5.133km sin(wmT ) into account in Eq. 4.26. Numerical result would be easier in that case. The accurate rate of launches into GEO and preference of any particular longitude can be considered to improve the accuracy of relative velocity. REFERENCES 43

References

[1] European Space Operations Centre(ESOC) space debris oﬃce. http://www. esa.int/SPECIALS/ESOC/SEMN2VM5NDF_mg_1.html. Accessed on May 31th, 2010.

[2] Scientiﬁc and Technical Subcommittee of the United Nations Committee on the Peaceful uses of Outer Space. Technical report on space debris. Technical report, United Nations, 1999. ISBN 92-1-100813-1.

[3] B.N. Agrawal. Design of geosynchronous spacecraft. Prentice-Hall, Inc. Upper Saddle River, NJ, USA, 1986.

[4] Oﬃce of Science and Technology Policy (OSTP). Interagency report on space debris 1995. Technical report, The National Science and Technology Council and Committee on Transportation Research and Development, US, 1995.

[5] NL Johnson, PH Krisko, J.C. Liou, and PD Anz-Meador. NASA’s new breakup model of EVOLVE 4.0. Advances in Space Research, 28(9):1377–1384, 2001.

[6] Working Group 4, IADC. Support document to the iadc space debris mitigation guidelines (ai 20.3). Technical report, Inter-Agency Space Debris Coordination Committee, 2004.

[7] Steering Group, IADC. IADC space debris mitigation guidelines. Technical report, Inter-Agency Space Debris Coordination Committee, 2002.

[8] P.H.Krisko. The predicted growth of the low earth orbit space debris environment - an assessment of future risk for spacecraft. Proceedings of the Institution of Mechanical Engineers, 221(6), 2007.

[9] The Institute of Aerospace Systems at the Technische Universit¨at Braun- schweig. http://www.master-2005.net/index.html. accessed on May 20th, 2010.

[10] M.J. Sidi. Spacecraft dynamics and control: a practical engineering approach. Cambridge Univ Pr, 2000.

[11] J.R. Wertz. Spacecraft attitude determination and control. Kluwer Academic Pub, 1978. 44 References

[12] VV Beletsky and FH Lutze. Essays on the motion of celestial bodies. Applied Mechanics Reviews, 56:B79, 2003.

[13] EM Standish. Report of the IAU WGAS Sub-group on Numerical Standards. Highlights of astronomy, 12:180–184, 1995.

[14] JL Simon, P. Bretagnon, J. Chapront, M. Chapront-Touz´e, G. Francou, and J. Laskar. Numerical expressions for precession formulae and mean elements for the Moon and the planets. Astronomy and Astrophysics (ISSN 0004-6361), 282(2), 1994.

[15] M.A. Wieczorek, B.L. Jolliﬀ, A. Khan, M.E. Pritchard, B.P. Weiss, J.G. Williams, L.L. Hood, K. Righter, C.R. Neal, C.K. Shearer, et al. The con- stitution and structure of the lunar interior. Reviews in Mineralogy and Geo- chemistry, 60(1):221, 2006.

[16] F. Alby. The Motion of the Satellite, Lectures and Exercises on Space Mechan- ics, 1983.

[17] NL Johnson, E Stansbery, DO Whitlock, KJ Abercromby, and D Shoots. His- tory of on-orbit satellite fragmentations. NASA STI/Recon Technical Report N, 84:34461, 2008. References 45 46 Calculation of iR

Appendix A

Calculation of iR

iR is the unit vector from the center of the Earth to M and is perpendicular to the ascending node. It is north of the orbital plane. In the case that orbital inclination i =0,iR is in the direction of OA (Figure 3.6). When the angular momentum L is changed to be L , the direction of iR is also changed to be iR (Figure A.1).

Figure A.1: Direction of iR changs with L

iR is decided by the position of M on its orbit (point A in Figure A.1), which means it’s a function of α (3.6). According to its deﬁnition, point A is also the point in the orbit of M when the distance between A and orbital plane is maximum.

The orbital plane contains the point (0,0,0) and is perpendicular to L . It’s function is:

Lx · x + Ly · y + Lz · z =0

The distance between M and this plane is maximum when the following is maximum: Calculation of iR 47

Lx · Rx + Ly · Ry + Lz · Rz

Rx,Ry and Rz are corresponding components in R 0, R 0 is the position vector of M. Put in their values from (3.6):

R0(Lx cos θ cos α + Ly sin α + Lz sin θ cos α)

At its maximum its ﬁrst order derivative equals to zero:

d R0 [(Lx cos θ + Lz sin θ)cosα + Ly sin α]=0 dα

The solution is: Ly tan α = Lx cos θ + Lz sin θ

So:

⎛ ⎞ cos θ ⎜ Ly ⎟ R 1 · ⎝ ⎠ i = c Lx cos θ+Lz sin θ (A.1) sin θ

Where C1 is a normalization constant which is independent of α.

As a simple check, put in the initial value i =0,

Lx = Ly =0

So, ⎛ ⎞ cos θ ⎜ ⎟ iR = C1 · ⎝ 0 ⎠ ,C1 =1 sin θ

This means that initially iR exists in the x-z plane and its angle with the x axis is θ. This is compatible with the real case. 48 Matlab code

Appendix B

Matlab code

B.1 Numerically solve the group of diﬀerential functions

%NR2005.m

%This code is to numerically solve the group of differential equations

%This code is modified for the case of the Sun's gravity perturbation on %the lunar orbit

%Lin Gao, 18:55, 31 May 2010 function [T,C]=NR2005 clc; clear all; %Run the program with memory cleared.

%theta is the angle between the ecliptic plane the lunar orbital plane theta=(5.133)/180*pi;

%TM is the orbital period of the Moon. TM=86400*27.21222;

%Function ode45 can solve initial value problems for ordinary differential %equations with medium accuracy. Maxstep of iteration is set to limit %error. The interval of integration is set to be 100 years. options=odeset('Maxstep',TM); [T,C]=ode45(@per,[0,TM*12*100],[sin(theta)*cos(theta),0],options);

%Record the ix, iy and iz and assign them to the base workspace for further %use. ix=C(:,1)−sin(theta)*cos(theta); iy=C(:,2)/cos(theta); iz=(C(:,1)−sin(theta)*cos(theta))*tan(theta)+1; assignin('base','ix',ix); Matlab code 49 assignin('base','iy',iy); assignin('base','iz',iz); assignin('base','T',T/86400/365.24);

%i is a unit vector according to its definition. Accuracy of integration %can be tracked by comparing the integrated magnitude of i with 1. goodness=(ix.ˆ2+iy.ˆ2+iz.ˆ2).ˆ0.5;

%Plot out the change of orbital inclination with regards to time. subplot(4,1,1) hold on; plot(T/86400/365.24,ix); hold off; ylabel('i x') subplot(4,1,2) hold on; plot(T/86400/365.24,iy); hold off; ylabel('i y') subplot(4,1,3) hold on; plot(T/86400/365.24,iz) hold off; ylabel('i z')

%Supervise the accuracy of integration. subplot(4,1,4) hold on; plot(T/86400/365.24,goodness); hold off; ylabel('i') xlabel('time/year')

%Plot the 3−D diagram of the revolving lunar pole in a new figure. figure;

%Define the pole. x=zeros(100:100); y=zeros(100:100); z=zeros(100:100); for i=1:100 %When i increases the lunar pole revolves. for j=1:100 %Define 100 points on each pole for plotting. x(i,j)=j/100*ix(i*10); y(i,j)=j/100*iy(i*10); z(i,j)=j/100*iz(i*10); end end

%Plot the surface of the track of the lunar pole. surf(x,y,z); xlabel('i x'); ylabel('i y'); zlabel('i z');; 50 Matlab code

%plot the ecliptic pole hold on; plot3([0,−1.3*sin(theta)],[0,0],[0,1.3*cos(theta)]);

%Define the differential equations to be solved. function dc=per(t,c) dc=zeros(2,1); theta=(5.133)/180*pi; m sun=1.9891e30; %mass of the Sun G = 6.67428e−11; %gravitational constant d sun=1.496e11; %Earth−Sun distance TM=86400*27.21222; r=384399000; %orbit radius C3=G*m sun*cos(theta)ˆ3*(3−7.5*rˆ2/d sunˆ2)*TM/8/pi/(d sun)ˆ3; assignin('base','C3',C3); J2=cos(theta)ˆ2; assignin('base','J2',J2); dc(1)=C3*c(2)/(c(1)ˆ2+c(2)ˆ2+J2ˆ2); dc(2)=−C3*c(1)/(c(1)ˆ2+c(2)ˆ2+J2ˆ2);

B.2 Plot the Precession of Lunar Orbital Pole

%PrecessionSM2905.m

%This code is to plot the precessing lunar orbital pole around the solar %orbital pole.

%Lin Gao, 23:52, May 29, 2010 clc; clear; %run the code with memory cleared theta=(5.133)/180*pi; %The maximum inclination of lunar orbital pole regards to the ecliptic %plane is 5.133 degrees. m sun=1.9891e30; %mass of the Sun G = 6.67428e−11; %gravitational constant d sun=1.496e11; %Earth−Sun distance TM=86400*27.321661; %the orbital period of the Moon r=384399000; %the Earth−Moon distance w=G*m sun*TM*cos(theta)*(3−2.5*rˆ2/(d sun)ˆ2)/8/pi/(d sun)ˆ3; %the precession speed

DUR=20; %plot made for 20 years ΔT=86400*365.242190402*DUR/100; %time interval Matlab code 51

T=1:100; %100 discrete time series ix=−sin(2*theta).*(sin(w.*T.*ΔT./2)).ˆ2; iy=sin(theta).*sin(w.*T.*ΔT); iz=−2*(sin(theta))ˆ2.*(sin(w.*T.*ΔT./2)).ˆ2+1; %create ix, iy and iz at the end of each time interval x=zeros(100:100); y=zeros(100:100); z=zeros(100:100); %initialize the points on the cone for i=1:100 for j=1:100 x(i,j)=j/100*ix(i); y(i,j)=j/100*iy(i); z(i,j)=j/100*iz(i); end %generate 100 discrete points on the orbital pole at the end of each %time interval end

%Plot the surface of the track of the lunar pole. surf(x,y,z); xlabel('i x'); ylabel('i y'); zlabel('i z');

%plot the ecliptic pole hold on; plot3([0,−1.3*sin(theta)],[0,0],[0,1.3*cos(theta)]);

B.3 Plot Inclination and the Precession of GEO Debris’ Orbital Pole

%PrecessionSGEO2905.m

%This code is to plot the inclination and precessing GEO debris' orbital %pole around the ecliptic pole.

%Lin Gao, 18:12, May 30, 2010 clc; clear; %run the code with memory cleared theta=(23.44)/180*pi; %The maximum inclination of GEO orbital pole regards to the ecliptic %plane is 23.44 degrees. m sun=1.9891e30; %mass of the Sun G = 6.67428e−11; %gravitational constant d sun=1.496e11; %Earth−Sun distance 52 Matlab code

TG=86164; %the orbital period of the GEO debris r=42164170; %GEO orbital radius w=G*m sun*TG*cos(theta)*(3−2.5*rˆ2/(d sun)ˆ2)/8/pi/(d sun)ˆ3; %the precession speed

DUR=1000; %plot made for 1000 years ΔT=86400*365.242190402*DUR/1000; %time interval

T=1:1000; %1000 discrete time series ix=−sin(2*theta).*(sin(w.*T.*ΔT./2)).ˆ2; iy=sin(theta).*sin(w.*T.*ΔT); iz=−2*(sin(theta))ˆ2.*(sin(w.*T.*ΔT./2)).ˆ2+1; %create ix, iy and iz at the end of each time interval x=zeros(1000:100); y=zeros(1000:100); z=zeros(1000:100); %initialize the points on the cone for i=1:1000 for j=1:100 x(i,j)=j/100*ix(i); y(i,j)=j/100*iy(i); z(i,j)=j/100*iz(i); end %generate 100 discrete points on the orbital pole at the end of each %time interval end

%Plot the surface of the track of the lunar pole. surf(x,y,z); xlabel('i x'); ylabel('i y'); zlabel('i z'); %axis([−0.2 0 −0.1 0.1 0 1]);

%plot the ecliptic pole hold on; plot3([0,−1.3*sin(theta)],[0,0],[0,1.3*cos(theta)]);

%plot the inclination in a new figure figure;

Time=DUR.*T./1000; subplot(2,1,1) plot(Time,iz); xlabel('time/year'); ylabel('i z'); subplot(2,1,2) theta z=acos(iz)./pi.*180; Matlab code 53 plot(Time,theta z); xlabel('time/year'); ylabel('theta z');

B.4 Plot of the Revolving Ascending Node

%ASNODE3105.m

%This code is to dynamically show the revolve of the ascending node of the %debris plane.

%Lin Gao, 05:17, 01 June 2010 clc; clear; %Run the code with cleared memory. time(1,300)=0; %Assume the angular velocity is 2*pi. Dynamically plot the ascending node %for 3 periods. Refresh it 100 times in each period. x(1,300)=0; y(1,300)=0; %Define and initialize x, y and time. (x,y) is the position of H. for t=1:300 time(t)=0.01*t; w=2*pi; x(t)=0.5*sin(5.133*2/180*pi)*(cos(w*time(t))−1); y(t)=−sin(5.133/180*pi)*sin(w*time(t));

plot([0,x(t)],[0,y(t)]); %plot OH hold on;

norm=(x(t)ˆ2+y(t)ˆ2)ˆ0.5/2; %normalization constant plot([y(t)/norm,−y(t)/norm],[−x(t)/norm,x(t)/norm],'r'); %plot OA in red

grid on; xlabel('x'); ylabel('y'); zlabel('z'); axis([−0.8 0.8 −0.8 0.8]); %set labels and axis range properly

pause(0.01); %pause for visualization

m(t)=getframe(gcf); %grab frames to make a movie of the precession

hold off; end movie2avi(m, 'ascendingnode.avi', 'compression', 'Cinepak'); %Export the result as an avi file.

%Open a new window for plotting the orientation change of the ascending 54 Matlab code

%node. figure; plot(time, atan(y./x)./pi.*180+90) xlabel('T/period'); ylabel('\phi /degrees');

B.5 Simulation of the Distribution of Debris

%SIMU2605.m

%This code is to simulate the distribution of debris.

%Lin Gao, 09:41, 26 May 2010

%Simulation is done by adding all the curves with different amplitudes and %phases. clc; clear; %run the code with memory cleared

N=140; %N decides how dense are the curves in the figure. It's set for better %visualization. n=0.5; %n decides when the simulation is done for. If assume 1967 is the beginning %of space industry, the simulation is done for 1967+n*54. In the case of %1995, n=0.5 . for i=1:360 %Plot is done in the interval between 0 and 360 degrees. for t=1:(N*n) %t stands for time since released from geostationary orbit hold on; plot(i,46.88*sin(t/N*pi)*sin(i/180*pi+t/N*pi)); end end axis([0 360 −90 90]); grid on; xlabel('longitude(degrees)'); ylabel('latitude(degrees)');

B.6 Simulation of the Distribution of Relative Ve- locities

%DISTR2805.m

%This code is to simulate the distribution of relative velocity between Matlab code 55

%debris

%Lin Gao, 22:27, 27 May 2010

%Separate the range of relative velocity to be 50 parts and record the %number of data falling into each interval (densi). Probability of %different values of velocity is (prob), which is calculated by normalizing %(densi).

N=50; %Separate the range of relative velocity by 50 parts.

Time=43; %Set 1967 as the beginning of space industry and it's %now 43 years from then. Change of Time will not %influence the result. val(1:1000)=0; %the velocity intervals prob(1:1000)=0; %the probability series densi(1:1000)=0; %the density series vrela=0; %the relative velocity w1=0.01; %w1 is the angular velocity for the movement in one day w2=0.01; %w2 is the angular velocity for the movement of ascending node

for i=1:N

val(i)=3.07*sin(15/180*pi)/N*(i−1); %the value of velocity

for T=1:(3.14/w2/54*Time) %T stands for time after released from geostationary orbit. for t=1:(6.28/w1) %t stands for time during a day

vrela=3.07*sin(15/180*pi)*sin(w1*t+w2*T); %Maximum velocity is 3.07*sin(15/180*pi). vrela is a function %of t and T.

if ((abs(vrela)≥3.07*sin(15/180*pi)/N*(i−1)) && ... (abs(vrela)<3.07*sin(15/180*pi)/N*(i))) densi(i)=densi(i)+1; end; %count the number of data falling into each interval

end end;

prob(i)=densi(i)/(3.14/w2/54*Time*6.28/w1); %normalize densi

hold on; plot(val(i),prob(i),'+'); %plot the value of velocity and its probability

end axis([0 1 0 0.1]); 56 Matlab code xlabel('velocity(km/s)'); ylabel('probability'); Application of Matlab cftool toolbox 57

Appendix C

Application of Matlab cftool toolbox

Figure C.1: Curve Fitting Toolbox Interface. Thetaz vs Time is plotted as blue and curve ﬁtting result is the red line. thetaz is the orbital inclination and

thetaz = arccos(iz)(C.1)

With (3.53), 2 2 thetaz = arccos(cos θ +sin θ cos(wt)) (C.2) 58 Application of Matlab cftool toolbox

And analytical solution of thetaz is very complex. According to Figure 4.6 it changes periodically. So I applied the cftool in Matlab to ﬁnd numerical expression of thetaz. During one period of time, it is more or less sinusoidal. Choose ‘Type of ﬁt’ to be ‘Sum of Sin Functions’.

The result is shown in Figure C.2.

Figure C.2: Curve Fitting Result

With R-squared and Adjusted R-squared value very close to 1, the model can explain thetaz quite well. In conclusion, thetaz can be expressed with a sinusoidal function during period [0,T], where T is the precession period. From Figure C.1, the phase change of this sinusoidal function in T is π so the angular velocity of change is:

π w = (C.3) T