Neutral Density Estimation from Multiple Equivalent Platforms

NEUTRAL DENSITY ESTIMATION FROM MULTIPLE EQUIVALENT PLATFORMS

ADISSERTATION SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Alan Li May 2016

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/rb805sm7205

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Sigrid Close, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Simone D'Amico

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Stephen Rock

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

Accurate modeling of atmospheric neutral density variations has been a challenge to the atmospheric science and space community for the past 50 years. The study of this topic gives insights to the dynamical processes active within our atmosphere, as well as enabling accurate prediction of the motion of objects within this region. Atmospheric density itself is a complex phenomenon that varies spatially and temporally, and is inherently linked to the behavior of the sun. Although multiple methods and models have been utilized to measure and predict neutral densities in the past, the lower thermosphere is particularly dicult to probe. This is due to the fact in this region, satellites that can provide direct measurements of density quickly deorbit and in situ instrumentation missions are infrequent due to cost and operational issues. In addition, models usually exhibit a 15% error in their estimated densities, which can increase to beyond 50% during periods of high solar activity and active geomagnetic conditions.

With the drastic increase of small satellite constellations and abundance of meteoroid observations in recent years, new opportunities have arisen for atmospheric science, unprecedented in coverage and scope. This thesis presents a new methodology for estimating neutral densities using large quantities of measurements that are becoming increasingly available. The focus is on the concept of equivalent platforms, and approaches the problem from a stochastic viewpoint. By utilizing order statistics in combination with physical laws, the probability distribution of the variations between platforms can be inferred. The method does not depend upon prior models of the

iv atmosphere, and is a novel way to derive neutral densities. It also is able to provide a new framework in which uncertainty across platforms may be combined with uncertainty inherent in physical models.

The neutral density estimation methodology was applied to two particular scenarios: a constellation of low Earth orbit CubeSats and meteoroid observations as measured by a high power large aperture radar. Results show that this estimation scheme is capable of predicting trends as seen by accepted models, but is also able to derive densities not otherwise predicted. This is due to the neutral density estimates being directly data- based, where models will often make predictions based solely upon a few preselected parameters. In the case for meteoroids, a new partitioned approach is able to predict densities per a speciﬁc layer of the atmosphere. Estimated standard deviations can be decreased to less than 5% and 12% for satellite and meteoroid derived densities under idealized scenarios, respectively. In the event that the measurements are noisy, the standard deviations will increase, to approximately 10% and 16%, respectively. Moreover, the method is able to observe trends not otherwise reported by ocial models. As increasing numbers of satellite constellations are launched and highly sensitive radars are built in the future, the topics covered in this thesis will aid neutral density estimation within the least explored region of the atmosphere.

v Acknowledgements

As with any serious body of work, this thesis would not have been possible if not for the support and contributions of many di↵erent people. In no manner is the list presented here exhaustive, as there have been too many positive inﬂuences throughout my PhD career that have shaped and matured my perspectives.

First and foremost, I would like to thank my advisor, Prof. Sigrid Close for her invaluable guidance, advice and insight throughout the entirety of my time at Stanford. She has been a patient, kind, and wonderful mentor as well as friend throughout the often dicult passages associated with academic research. In particular, when I proposed to explore subject material closer aligned with my own interests, her unwavering support was truly encouraging.

Along the same lines, I would like to thank the faculty and sta↵at Stanford, particularly within the Aeronautics and Astronautics department. This is especially directed at my committee, Prof. Stephen Rock, Prof. Simone D’Amico, Prof. An- drew Barrows, and Prof. Elizabeth Pate-Cornell for their time in reviewing my work. Iwouldalsoliketothankthevarioussupportsta↵suchasPatrickFerguson,Sherann Ellsworth, and Dana Parga for their assistance with the often overlooked but very important administrative duties, allowing us PhD students to focus on our research free of worry.

Iwouldalsoliketoextendmythankstothevariousorganizationsthathavehelpedme in my work. Without Prof. Scott Hubbard and the Federal Aviation Administration (FAA) Center of Excellence - Commercial Space Transportation (CoE-CST) program,

vi this work deﬁnitely would not have been possible. A special thanks to Planet Labs, in particular Cyrus Foster and James Mason, for providing much of the open source satellite data analyzed within this thesis. My time spent working there, pondering the e↵ects of drag on constellations of satellites, was instrumental in providing me inspiration to continue research into this topic.

IwouldliketoalsothankmyfriendsandlabmateswithintheSpaceEnvironment and Satellite Systems (SESS) Lab that I have interacted with during my past 7 years at Stanford. There are too many to name here, so I will try my best (in no particular order): Yichen Jin, Gordon Cheung, Ashish Goel, Alejandro Campos, Ozhan Turgit, Siddharth Krishnamoorthy, Ved Chirayath, Jonathan Yee, Alex Fletcher, Theresa Johnson, Andrew Nuttall, Glenn Sugar, Monica Hew, Travis Swenson, Ana Tarano, Paul Tarantino, and Lorenzo Limonta, to name a few. I would also like to give a very noteworthy mention here to Nicolas Lee, for providing invaluable suggestions and comments in regards to this thesis, as well as for his unparalleled dedication to SESS.

Finally, a very special thanks is dedicated to my parents, Zhu Ju Hua and Li Shen Sheng, for their constant encouragement and support throughout not only my academic career, but also throughout my life. Without their constant and unmatched belief in me, all that I have thus far achieved would have been impossible.

vii Nomenclature

A Partial derivative matrix of observations relative to initial state

F Force (Subscripts: Fg Gravity; FD Drag; FL Lift; FSRP SRP; F3body

3rd body; Fu Unmodeled) Pˆ Predicted covariance

R Position vector from Earth (Subscripts: Rsun Distance from sun)

U Random molecular thermal velocity, consisting of (U1,U2,U3)

V Inertial velocity (Subscripts: Vrel Relative velocity to atmosphere; Va Velocity of atmosphere)

W Weighting matrix

X State (Subscripts: X0 Initial state; Xi State at ti)

Xˆ Predicted state (Subscripts: Xˆ i Predicted state at ti) a Acceleration (Subscripts: ag Gravity; aD Drag; aSRP SRP) b˜ Residual (Subscripts: b˜i Residual at ti)

ˆe d Unit vector of desired direction of force for free molecular ﬂow

ˆe v Unit vector of inertial velocity (Subscripts: ˆe v,rel Unit vector of relative velocity) p Linear momentum

viii pos Position of meteoroid, consisting of (posx,posy,posz) u Molecular velocity, consisting of (u1,u2,u3)(Subscripts:ui Incident ve-

locity; ur Reﬂected velocity) A Frontal cross sectional area

Ap Geomagnetic planetary A-index

Aref Reference area

Az Azimuth angle of radar (Subscripts: Azm Monopulse azimuth angle with

respect to radar boresight; Azt Total azimuth deﬁned as Az+Azm) B Ballistic factor

B¯ Mean ballistic factor

C Non-dimensional momentum ﬂux coecient (see CD and CA)

CA Axial drag coecient

CD Drag coecient

CH Heat transfer coecient

CR Reﬂective coecient

D Ablation parameter D = 1 CH 6 CDH⇤

Ei Incident energy ﬂux

El Elevation angle of radar (Subscripts: Elm Monopulse elevation angle

with respect to radar boresight; Elt Total elevation deﬁned as El+Elm)

Er Reﬂected energy ﬂux

Ew Energy ﬂux if molecules reemitted at wall temperature G Gravitational constant

H Scale height

H⇤ Heat of enthalpy or vaporization I Indicator notation

ix th K Kparameter(Subscripts: Kj Kparameterofj platform; Kmk Mini-

mum K amongst all platforms at time tk) Kn Knudsen number

Kp Geomagnetic planetary K-index

Lref Characteristic length or dimension

Mi Molecular mass of gas constituent i N Total number, usually referring to the number of platforms

NA Avogadro’s number

Ngas Number density of gas

Ni Number density of gas constituent i P Pressure

R Radius (of meteoroid)

RR Range rate

S Molecular speed ratio

T Temperature (Subscripts: Ti Incident; Tr Reﬂected; Tw Spacecraft surface temperature)

T Exospheric temperature 1 Tr Traverse angle

W Wind factor

Xi Random variable X indexed by i

th X(i) i order statistic of X a Semi-major parameter aM Mean semi-major parameter ap 3-hour interval geomagnetic a-index b Ballistic coecient

x c Speed of light e Eccentricity erf Error function g Gravity gr Ratio of geometric series h Altitude incl Inclination k Boltzmann constant m Mass mgas Mean mass of gas mi Mass of gas constituent i n0 Mean motion nM Mean mean motion p Normal momentum transfer (Subscripts: pi Incident; pr Reﬂected; pw Reﬂected at surface temperature) ps Semiparameter r Distance, generally from the center of the Earth (Subscripts: rsun Dis- tance from sun) rng Radar range t Time (Subscripts: ti Initial time; tf Final time) v Speed or magnitude of inertial velocity vector (Subscripts: vrel Magni-

tude of relative velocity; vgas Bulk speed of gas; vth Thermal speed) x Property of body that is prone to variation, such as B for satellites or

R⇢m for meteoroids (Subscripts: xmk Minimum order statistic of x at

time tk)

xi y Variable used to derive the quotient distribution (y = xmk) z Variable used to derive the quotient distribution (z = x ) xmk

Scripts and Greek Symbols

↵ Accommodation coecient

Meteoroid approach angle measured from surface horizontal

C Cumulative distribution function (CDF)

M Correctional term to transform osculating mean motion to mean mean motion f Maxwellian velocity distribution function

F Complement to CDF: F =1 C Used in free molecular ﬂow = S cos ✓

1 Used in meteoroid literature = 2 CD H Heaviside function

 Tuning parameter for smoothness

Mean free path

µE Earth’s gravitational parameter N Normal distribution (Gaussian)

⌫ True anomaly

⌦ Longitude of Ascending Node or Rising Ascension of Ascending Node (RAAN)

! Argument of periapsis or perigee

⌦E Angular velocity of Earth’s rotation P Probability density function (PDF)

xii ⇢a Neutral mass density of atmosphere (Subscripts: ⇢a,est Estimated density)

⇢¯a Averaged neutral mass density of atmosphere

⇢m Bulk density (of meteoroid)

A Collisional area of the particles within a gas

K Standard deviation of K

N Normal momentum accommodation coecient

T Tangential momentum accommodation coecient

⌧ Tangential momentum transfer (Subscripts: ⌧i Incident; ⌧r Reﬂected;

⌧w Reﬂected at surface temperature) ✓ Rotation angle of satellite

U Aspherical potential function

Abbreviations

ADACS Attitude Determination and Control System

ALTAIR ARPA Long-range Tracking and Instrumentation Radar

ARPA Advanced Research Projects Agency

ASAT Anti-Satellite

AU Astronomical unit

CDF Cumulative distribution function

CHAMP Challenging Mini-Satellite Payload

CIRA COSPAR International Reference Atmosphere

CME Coronal Mass Ejection

COSPAR Committee on Space Research

xiii COTS Commercial o↵–the–shelf dBsm Decibels relative to a square meter

DCA Dynamic Calibration of the Atmosphere doy Day of year

DSMC Direct Simulation Monte Carlo

Dst Disturbance storm time

DTM Drag Temperature Model

EGM Earth Gravitational Model

ESR European Incoherent Scatter Svalbard Radar

EUV Extreme Ultra–Violet

FUV Far Ultra-Violet

GAMDM Global Average Mass Density Model

GOCE Gravity Field and Steady-State Ocean Circulation Explorer

GOES Geostationary Operational Environmental Satellite

GOST Government Standard (of the USSR)

GPS Global Positioning System

GRACE Gravity Recovery and Climate Experiment

GUVI Global Ultraviolet Imager

HASDM High-Accuracy Satellite Drag Model

HPLA High Power Large Aperture

HWM Horizontal Wind Model

IID Independent and identically distributed

IR Infrared

ISR Incoherent Scatter Radar

xiv ISS International Space Station

JB Jacchia-Bowman empirical thermosphere model

JSpOC Joint Space Operations Center

L1 Lagrange Point 1

LC Left-circular

LDEF Long Duration Exposure Facility

LEO Low Earth Orbit

MEO Medium Earth Orbit

Mg II Magnesium II Core-to-Wing ratio

MLT Mesosphere and Lower Thermosphere

MSIS Mass Spectrometer and Incoherent Scatter

NMS Neutral Mass Spectrometer

NORAD North American Aerospace Defense Command

NRL Naval Research Laboratory

NRLMSISE Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Extended empirical thermosphere model

PDF Probability density function

PRF Pulse–repetition frequency

RAAN Rising Ascension of Ascending Node ⌦

RC Right-circular

RCS Radar Cross Section

RMS (Weighted) Root mean squared error

RSSI Received signal strength indication

RTI Range-time-intensity

RTP Ray-Tracing Panel

xv SFU Solar Flux Units

SGP Simpliﬁed General Perturbations

SLR Satellite Laser Ranging

SNR Signal to Noise Ratio

SOHO Solar and Heliospheric Observatory

SRP Solar Radiation Pressure

SSN Space Surveillance Network

STK Systems Tool Kit

TIMED Thermosphere Ionosphere Mesosphere Energetics and Dynamics

TLE Two Line Element

UHF Ultra High Frequency

UTC Coordinated Universal Time

UV Ultraviolet

VHF Very High Frequency

xvi Contents

Abstract iv

Acknowledgements vi

Nomenclature viii

Contents xvii

List of Tables xxi

List of Figures xxii

1 Introduction 1 1.1 Overview ...... 2 1.2 Motivation ...... 4 1.2.1 Solar and Terrestrial Sciences ...... 4 1.2.2 Engineering and Satellite Applications ...... 7 1.3 Summary of Contributions ...... 11 1.4 Reader’s Guide ...... 13

2 Background 15 2.1 Nature of the Atmosphere ...... 15 2.1.1 Energy Sources and Indices ...... 16 2.1.2 Structure and Composition ...... 19 2.1.3 Neutral Density Variations ...... 27

xvii 2.2 Density Measurements ...... 33 2.2.1 In Situ Instrumentation ...... 33 2.2.2 Satellite Orbital Decay ...... 34 2.2.3 Remote Sensing ...... 35 2.2.4 Relation to Thesis: Measurements from Equivalent Platforms 39 2.3 Atmospheric Density Modeling ...... 40 2.3.1 Empirical Models ...... 40 2.3.2 Physics Based Models ...... 43 2.4 Summary ...... 44

3 Dynamics 45 3.1 Force Modeling ...... 46 3.1.1 Gravity ...... 47 3.1.2 Drag ...... 48 3.1.3 Solar Radiation Pressure (SRP) ...... 51 3.1.4 Other Perturbing Forces ...... 52 3.2 Free Molecular Flow ...... 53 3.2.1 Surface-Flow Interactions ...... 54

3.2.2 Uncertainty in CD ...... 60 3.2.3 Numerical Methods ...... 63 3.3 Satellites ...... 64 3.3.1 Orbital Dynamics ...... 64 3.3.2 K Parameter ...... 69 3.4 Meteoroids ...... 75 3.4.1 Meteoroid Characteristics ...... 75 3.4.2 Equations of Motion ...... 76 3.5 Summary ...... 79

4 Methodology 80 4.1 Statistical Methodology ...... 81 4.1.1 Motivation ...... 81 4.1.2 Major Assumptions ...... 83

xviii 4.1.3 Order Statistics ...... 87 4.1.4 Ratio Statistics ...... 89 4.1.5 Numerical Formulation ...... 93 4.1.6 Errors and Uncertainty ...... 95

4.1.7 Calculating ⇢a ...... 99 4.2 Satellites ...... 101 4.3 Meteoroids ...... 105 4.3.1 Successive Layer Approximation ...... 106 4.3.2 Calculating Density ...... 108 4.4 Summary ...... 111

5 Application and Results 112 5.1 Satellites ...... 112 5.1.1 Simulation ...... 113 5.1.2 Constellation Data ...... 117 5.2 Meteoroids ...... 127 5.2.1 Radar Characteristics ...... 128 5.2.2 Meteoroid Data ...... 130 5.3 Summary ...... 139

6 Conclusions 141 6.1 Review of Contributions ...... 141 6.1.1 Density Estimation Method ...... 142 6.1.2 Applications ...... 143 6.2 Discussion of Methodology ...... 143 6.3 Future Work ...... 145 6.3.1 Advanced Modeling ...... 145 6.3.2 Recursive Estimation ...... 145 6.3.3 Data Collection and Analysis ...... 146 6.3.4 Orbital Debris ...... 147 6.4 Concluding Remarks ...... 148

xix A Planet Labs CubeSat Data 149

BALTAIRMeteorData 155

Bibliography 164

xx List of Tables

3.1 Comparison of orbital elements due to various forces after 1 day ... 68 3.2 List of variables for meteoroid ablation ...... 77

dv 1 ⇢a,ref 5.1 Statistics of 2 over multiple bins at 18:00 UTC, doy 312, 2007 137 dt v ⇢a

5.2 F10.7 index for solar irradiance and Kp index for geomagnetic activity on analyzed days ...... 137

A.1 Planet Labs Flock 1B Cubesats catalogue identiﬁcation and launch dates/times ...... 151 A.2 Planet Labs ground station locations and altitudes ...... 153 A.3 Sample of Planet Labs ranging data ...... 154

xxi List of Figures

1.1 Gr¨un model of meteoroid ﬂux at Earth’s orbit or 1 astronomical unit (AU) ...... 6 1.2 Sporadic meteoroid sources relative to the Earth’s velocity vector and the Sun ...... 7 1.3 Tracked space objects over 10 cm in diameter as of April 2016. .... 10

2.1 Layers of the atmosphere...... 21 2.2 Temperature of the atmosphere at speciﬁc altitudes as modeled by NRLMSISE ...... 23 2.3 Number density of atmospheric constituents in the thermosphere as modeled by NRLMSISE ...... 25 2.4 Neutral density of the atmosphere under high and low solar activity as modeled by NRLMSISE ...... 26 2.5 Neutral density of the atmosphere at 400 km worldwide under moderate solar conditions as modeled by NRLMSISE ...... 27 2.6 Comparison of 27 day variations in density, EUV, and MG II ..... 30 2.7 Intra–annual density variations of neutral density at 400 km as modeled by NRLMSISE ...... 31 2.8 History of sunspot number ...... 32 2.9 Meteor head and trail echoes as observed by the Jicamarca Incoherent Scatter Radar ...... 38

3.1 Major forces acting upon satellites and meteoroids ...... 46 3.2 Relative contributions of forces in LEO ...... 52

xxii 3.3 Hyperthermal flow vs hypothermal flow ...... 55 3.4 Specular vs di↵use fluxes ...... 57

3.5 Flat plate with incident particle approaching at angle ✓i ...... 59

3.6 CD and CA evaluated at various angles of attack for a ﬂat plate ... 60

3.7 Variation of CD versus ↵, S,andTw for a ﬂat plate ...... 62 3.8 A 2D elliptical orbit ...... 65 3.9 A 3D orbit as deﬁned by Keplerian elements ...... 66 3.10 E↵ects of perturbations on orbital elements ...... 67 3.11 Representative two-line element set ...... 70

4.1 The methodology ﬂowchart ...... 81 4.2 The concept of equivalent platforms ...... 84 4.3 The probability distribution of the minimum order statistic applied on the normal distribution for varying N ...... 88 4.4 Estimates on a Gamma Distribution using the geometric series numerical method ...... 95 4.5 The e↵ects of error in K on the ratio distribution ...... 96 4.6 CubeSat and its orientation relative to its velocity vector ...... 103

4.7 E↵ect of ↵ on CD,giventhatCD is distributed according to a half normal distribution and ↵ N(2.2, 0.1) ...... 104 ⇠

5.1 Probability distributions for drag coecient CD used in STK simulations113 5.2 STK simulation of 10 satellites evenly spaced along-track in an orbit . 114 5.3 Comparison between ratio distribution K between theoretical and Kmk simulation data...... 115

5.4 Estimated distribution of CD and CD,min compared to the original .. 116 5.5 Comparison of mean vs ordered estimates of neutral density given the amount of error in K...... 117 5.6 Comparison of errors in density estimates for mean vs minimum (ordered) estimates given varying amounts of error in K ...... 118 5.7 Schematic of Planet Labs CubeSat ...... 120

xxiii 5.8 Measure of residuals from least squares technique on ranging measurements from ground stations ...... 121 5.9 Evolution of mean motion of satellites ...... 122 5.10 K parameter as measured from Planet Labs TLEs, color-coded for di↵erent satellites ...... 123 5.11 B¯ CDF ﬁtted to a combination of a gamma distribution with a 4th B¯mk order polynomial ...... 124

5.12 Numerical CDF estimate of B¯ and B¯mk given Bmin↵ =0.022 ..... 125

5.13 Estimate of B¯mk given↵ ¯ =0.95 and ↵ =0.02 ...... 126 5.14 Estimate and comparison of neutral densities to existing models. ... 127 5.15 Minimum and mean neutral density estimates compared to existing models ...... 128 5.16 Fitting of velocity data vs altitude as a sum of two exponentials for one particular meteoroid detection ...... 130 5.17 38 meteoroid detections with 3D speeds of over 40 km/s as detected by ALTAIR ...... 132 dv 1 5.18 dt v2 of meteoroids traveling at speeds over 40 km/s as detected by ALTAIR ...... 133

5.19 Density ratios calculated from minimization procedure with ⇢a,110km as reference over 38 meteoroids ...... 134 5.20 Calculated density ratios on two separate days over 9 bins across 20 minutes ...... 135 1 1 5.21 Variation of meteoroids (R ⇢m )ontwoseparatedays ...... 136 5.22 Estimated densities on two separate days over 9 bins across 20 minutes 136 5.23 Histogram of estimated ablation parameters ...... 138

A.1 Launch direction of Planet Labs CubeSats ...... 150 A.2 Planet Labs ground station locations on world map ...... 152 A.3 Sample of Planet Labs generated TLEs ...... 152

B.1 Radar and relative-to-radar meteoroid angles ...... 158

xxiv B.2 Sample calculated elevation and traverse angles of a meteoroid detection derived from ALTAIR data with respect to altitude ...... 159 B.3 Sample range rate and 3D speed of a meteoroid as detected by ALTAIR160 B.4 Sample range rate and 3D deceleration of a meteoroid as detected by ALTAIR, calculated using ﬁnite di↵erencing ...... 161 B.5 LC and RC polarized RCS returns as detected by ALTAIR with respect to altitude for a sample meteoroid ...... 162 B.6 Fitted azimuth and elevation of a sample meteoroid as detected by ALTAIR, color coded by LC RCS ...... 163

xxv Chapter 1

Introduction

Fifty years ago, the launch of Sputnik heralded the beginning of the space age. Ever since then, increasing numbers of artiﬁcial satellites have been launched into space, with the majority of them orbiting the Earth in order to provide essential daily services for humanity. Predicting and controlling the motion of these satellites is thus essential to their continued operation and functionality. Most of these satellites reside in Low Earth Orbit (LEO), which extends to an altitude of 2000 km and lie within the Earth’s atmosphere. Here, the dominant force acting on these satellites besides gravity is atmospheric drag, which is by deﬁnition the force acting opposite to the relative motion of any object moving with respect to the surrounding medium. Understanding and modeling atmospheric neutral density is thus integral to satellite drag modeling [1, 2].

This thesis will focus on the thermosphere, a layer of the Earth’s atmosphere that extends from approximately 85 km to 690 km, primarily consisting of the gases N2,

O2, O, He and H. It is in this region that the International Space Station (ISS) orbits the Earth along with hundreds of other LEO satellites, while in the lower reaches of the same region meteoroids ablate as they enter the atmosphere. The goal of this thesis is to utilize the increasing number of observed satellites and meteoroids to

1 CHAPTER 1. INTRODUCTION 2

measure the thermospheric neutral density for improved satellite drag modeling. The concept can be applied to give real-time updates of neutral density, if the frequency and quality of data can be maintained. In particular, the focus will be on the use of equivalent platforms, which we deﬁne as objects that share general characteristics but are variable across certain quantiﬁable properties.

1.1 Overview

Although there is no firm boundary as to where “space” begins, it is convenient to set it at the Kármán line, which lies at an altitude of 100 km above Earth’s sea level. Here our instinctive preconceptions of pressure, temperature and density become blurred, as the individual molecules in the atmosphere drift further and further apart. The thermosphere begins at approximately 85 km, and is where most of the Sun’s X-ray and ultraviolet (UV) radiation is absorbed by molecules, causing excitation, dissociation and ionization. Below the thermosphere is the mesosphere, where the mixing of the various constituents of the atmosphere still occur due to turbulence. The exosphere, the region beyond the thermosphere where gas particles have become so di↵use that they rarely interact at all, begins somewhere in between 500 to 1000 km in altitude, and marks where the void of space has begun.

Although the atmosphere within the thermosphere is very thin, there is still enough air here to a↵ect the motion of objects traveling within. The concern in this region is mostly centered about the neutral particles, as they are far more abundant than their charged counterparts [3]. The atmospheric mass density, ⇢a,isdeﬁnedasthemassper unit volume of the atmosphere’s constituents, which generally decreases exponentially with increasing altitude. Within this thesis, the use of the term “density” will refer to the atmospheric mass density, as opposed to number density, unless otherwise stated.

Neutral density is a temporally and spatially varying phenomenon. The Sun is the driving factor for much of these variations, as the abundance or lack of UV irradiance CHAPTER 1. INTRODUCTION 3

causes the atmosphere to either heat up and expand or cool down and contract. During periods of high solar activity, radiation from the Sun ﬂoods the atmosphere with highly energetic particles, which causes further increases in density. It is then of no surprise that the most common temporal variations are experienced diurnally (day/night cycle), monthly (Sun’s rotational period), seasonally (varying distance from the Earth to the Sun as well as the changing position of the sub-solar point) and every 11 years (solar cycle).

To drive the modeling and prediction of neutral densities, there is a need for accurate and precise measurements of the thermosphere. However, it is often dicult to obtain these measurements in the lower thermosphere, as sounding rocket missions are infrequent and satellites quickly deorbit in this region. Commonly used models such as the Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Extended (NRLMSISE) empirical thermosphere model [4]canprovideglobalesti- mates of neutral mass densities, but tend to possess standard deviations and errors of roughly 10-20% [5, 6, 2]. During more turbulent times, this can reach as high as 20%-50% [7, 8]. These errors can have profound consequences on satellite operations, as changes in density can a↵ect their predicted trajectories and orbital motion.

This thesis provides a novel method to address these challenges by analyzing the drag across many short-lived nanosatellites at low altitudes, as well as by analyzing the abundance of meteoroids ablating within the atmosphere. As the number of satellites in LEO continually increases due to the commercialization of space, it is our hope that a system where constant observation of these satellites can be realized, such that real-time estimates of neutral densities within the lower thermosphere can be made. This notion can be further extended to lower altitudes by observing the billions of meteoroids ablating within our atmosphere daily, especially since direct measurements regarding this region are relatively scarce. The approach described herein comes from a stochastic viewpoint, whereby instead of focusing on singular measurements, we make observations across a large family of similar objects traversing the medium. The goal is to calculate the statistical measures of the variation across said objects in CHAPTER 1. INTRODUCTION 4

addition to estimating the background neutral density, with standard deviations that are lower than the oft quoted 15% average. Ultimately, this would lead to a better understanding and prediction of the dynamic processes and physics that govern our atmosphere.

1.2 Motivation

The motivation to study thermospheric neutral densities stems not only from a scien- tiﬁc perspective, but also from its importance in day to day satellite operations and maintenance. This section will give an overview of the associated ﬁelds of application that are a↵ected by our ability to model and predict neutral density behavior.

1.2.1 Solar and Terrestrial Sciences

Solar-Atmospheric Interaction

The thermosphere is inherently linked with the conditions of our Sun, as solar UV irradiance is the primary driver in determining how our atmosphere behaves. This behavior is also the result of complex interactions between solar emissions, the neutral atmosphere, ionosphere, and magnetosphere [9]. By studying the e↵ects on the atmosphere due to the Sun, we can better model the complex dynamics present in such a system and anticipate future behaviors as well.

Earth Science

Many applications to Earth science research depend upon accurate localization or geo-referencing measurements from satellites. Since drag is a direct result of neutral CHAPTER 1. INTRODUCTION 5

particles interacting with the surface of a satellite, it becomes necessary to model these e↵ects. Two notable missions include the Challenging Mini-Satellite Payload (CHAMP) [10]andtheGravityRecoveryandClimateExperiment(GRACE)[11] missions in the 2000s, which carried high precision accelerometers to determine the gravity ﬁeld of the Earth [12]. Non-conservative forces such as drag interfere with these measurements, and hence must be ﬁltered out to provide accurate data.

For other missions, Satellite Laser Ranging (SLR) is used to accurately determine satellite positions to within meters [13, 14]. By analyzing the change over time of satellite orbits and altitudes, it is possible to perform a variety of scientiﬁc missions. These include measuring the movement of the Earth’s surface, changing sea levels, ice volumes, and mass distribution of the Earth, all of which are aimed at long– term research linked with climate change [15, 16]. All these studies require accurate modeling of the thermosphere such that we are able to remove the unwanted e↵ects due to drag.

Meteor Science

Meteoroids are classiﬁed as small bodies moving within the solar system that are much smaller than asteroids. We will focus on the smallest subclass of meteoroids, often termed as “micrometeoroids” (µm to mm in diameter). When a meteoroid ablates in the Earth’s atmosphere, it forms a plasma, which is created from the high energy interactions between the meteoroid and the atmosphere. This plasma is commonly referred to as a meteor.

Meteoroid sources can be classiﬁed into two categories: “streams”, which is the result of the Earth passing through the wake of another space object (e.g. comets or asteroids), and “sporadics”, where the meteoroid cannot be associated with its original parent body. Often streams are linked with meteor showers, while sporadics are considered to be always present in the background and are the dominant source CHAPTER 1. INTRODUCTION 6

Figure 1.1: Grün model of meteoroid flux at Earth’s orbit or 1 astronomical unit (AU). for particles smaller than 1 µg. The commonly used Grün interplanetary flux model predicts that the cumulative number of meteoroids increases according to the power law as the mass decreases, as shown in Figure 1.1 [17, 18].

Ground-based radar surveys have attempted to determine the source directions of the incoming sporadic meteoroids [19]. Most predictably originate from the Apex (approximately 33% of all sporadics), which is the direction of the Earth’s velocity vector, as Earth “sweeps” through its orbit. The next two largest sources are the Helion and the Anti-Helion (roughly 22% each), referring to the sunward and anti- sunward directions respectively. The remaining and smallest contributions come from the North and South Toroidal sources, which are located 60 north and south of the equator. Figure 1.2 shows the respective direction of these sources relative to Earth’s velocity vector and the Sun.

The study of meteoroids provides insight regarding the formation of the solar system, since they are some of the oldest interplanetary particles still in existence. They also represent a signiﬁcant source of upper atmospheric mass deposition that is critical to the study of aeronomy of the upper mesosphere and lower thermosphere [21, 22]. As they enter the Earth’s atmosphere, they ablate and thus their material is mixed into CHAPTER 1. INTRODUCTION 7

Figure 1.2: Sporadic meteoroid sources relative to the Earth’s velocity vector and the Sun [20]. the surrounding medium. Measurements from the Long Duration Exposure Facility (LDEF) satellite have estimated that the mass deposited by the smallest particles (< 1 mm in diameter) is comparable to or greater than that deposited by larger bodies over 1 cm in diameter [21]. The accurate estimation of neutral atmospheric densities thus aids in the calculation of meteoroid composition and masses through ablation and ionization modeling. As will be detailed in this thesis, the reverse scenario of estimating neutral densities from meteoroid observations can also be accomplished, if the population and characteristics of the meteoroids are well known.

1.2.2 Engineering and Satellite Applications

There are currently over 1000 on-orbit satellites that are active, providing essential imagery, communications and scientiﬁc data [23]. What is more troubling, however, is the millions of untracked space debris that orbit alongside our space assets. Accurate force modeling is thus critical to the day-to-day operation of satellites and in the area of on-orbit collision mitigation. CHAPTER 1. INTRODUCTION 8

Satellite Orbit Determination

In LEO, the most prevalent forces are central gravity and atmospheric drag. The decay of orbits under drag is a well known phenomenon, in which elliptical orbits slowly circularize over time, and henceforth decrease in altitude. As the neutral density increases, this process accelerates. Although drag mostly a↵ects a satellite’s in-track motion, lift can also cause forces normal to the orbit plane to manifest, giving rise to further perturbations in the satellite’s motion. Combined with gravitational perturbations, there exist drag-gravity coupling e↵ects as well, albeit small. To truly predict the motion of these satellites, all these e↵ects must be well understood and modeled.

If a satellite is to perform an on-orbit maneuver, such as for altitude raising purposes or for collision avoidance, then we need to ascertain the required force necessary to move the satellite in relation to the background density. It might also be necessary to maneuver the satellite into a more favorable orbit particular to its objective, such as maintaining a certain ground track. Because orbital dynamics dictate these processes, alargerorsmallerthanpredictedbackgroundneutraldensitywouldrequiremoreor less e↵ort in maneuvering the satellite, respectively. The consequences are that we might overshoot or undershoot a particular desired orbit, and this directly impacts the fuel consumption and time spent on such a maneuver.

Finally, over a longer time frame, we may wish to predict how long a satellite remains useful; because drag constantly causes an orbit to decay, it will eventually reenter the (lower) atmosphere [24]. Usually the time to orbital decay inﬂuences the planning of mission extensions or spacecraft decommissioning, although this aspect is also considered during mission design. CHAPTER 1. INTRODUCTION 9

Space Object Catalogue and Tracking

The Space Object Catalogue is a database of all known (unclassiﬁed) satellites that have orbited the Earth since the dawn of the space age. The updating and maintenance of this catalogue is performed by the North American Aerospace Defense Com- mand (NORAD) through the Joint Space Operations Center (JSpOC), who tasks the US Space Surveillance Network (SSN), a globally distributed network of interferom- eter, radar and optical tracking systems [25], to observe space objects. As it is not possible to track every single object constantly, we require good dynamics models to predict their motion. This includes the need for neutral density models, as drag is a key factor in modeling the motion of LEO satellites. We are therefore able to identify new objects as well as update the orbits of existing objects in the catalogue by using force modeling and correlating predictions with detections.

Figure 1.3 shows the number of on-orbit space objects greater than 10 cm in diameter as of April 2016. We see that due to primarily two events, the Chinese anti-satellite (ASAT) test in 2007 [26]andtheIridium-Kosmoscollisionin2008[27], we have more than doubled the population of tracked objects greater than 10 cm in diameter (primarily due to the increase of space debris) in orbit around the Earth. Many more objects are too small to be detected and hence go untracked (the current threshold is approximately 1 cm or greater in diameter [28]), where their numbers may reach into the millions.

Satellite Conjunction Analysis

Because of the high number of space objects in orbit, there exists the very likely scenario of a high velocity collision between two objects, causing fragmentation and creating more space debris. In fact, an estimate of 10 to 30 objects larger than 10 cm per day come within a tolerated distance (a “warning box” approximately 25 km along-track, and 5 km in the other directions) from the ISS, and approximately 10 CHAPTER 1. INTRODUCTION 10

Figure 1.3: Tracked space objects over 10 cm in diameter as of January 2016. Source: NASA: Orbital Debris Quarterly News Volume 20, Issue 1 & 2, April 2016 maneuvers per year must be made due to a close proximity object [29]. As can be seen from Figure 1.3 where there exists two dramatic increases in space object population in 2007 and 2009, catastrophic collisions have enormous consequences on the future and safety of the space environment.

The current method of assessing collision risk consists of predicting an error ellipsoid for each space object such that if another object is projected to be within 1 km, a warning is issued. These error ellipsoids can only be predicted a few days in advance and are prone to sudden changes under ﬂuctuating and uncertain solar conditions. If better models were in place to predict such ﬂuctuations in the neutral densities, or if the e↵ects of drag were better understood, then our predictive capabilities would be enhanced in determining these conjunction events. CHAPTER 1. INTRODUCTION 11

Atmospheric Reentry

Given enough time, atmospheric drag causes all LEO objects to decay in orbit and eventually reenter the atmosphere. If the object is particularly large or massive, it might not entirely disintegrate in the atmosphere and hence reach the Earth’s surface. Although not a primary concern since these objects usually crash into the ocean or in remote locations, there is the possibility that these objects might contain hazardous materials and come into proximity with a highly populated area [30]. The trajectories become increasingly dicult to predict as the object’s altitude decreases, due to a multitude of e↵ects including heating and ablation, highly varying densities, temperature, and winds. The modeling of the background density ﬂuctuations could bring us one step closer to estimating a reentering object’s trajectory during its initial stages within the lower thermosphere and upper mesosphere.

Another aspect of reentry not regularly addressed, at least in orbit determination, is in the area of meteoroids. As these interplanetary particles enter the Earth’s atmosphere, most of them ablate and disintegrate similarly to small satellites, but they enter the Earth at rates of billions per day. Larger meteoroids are also able to reach the Earth’s surface and release tons of energy due to their higher velocities and mass, as evidenced by the Chelyabinsk event of 2013 [31]. The composition of the atmosphere determines how these objects interact with it, and likewise we can infer the makeup of the lower thermosphere by observing their trajectories.

1.3 Summary of Contributions

The general scope of this work is to make measurements from multiple equivalent platforms over one region to arrive at neutral density estimates. A brief explanation is warranted here as to the deﬁnition of equivalent platforms, which is key in understanding this topic. Simply put, we seek to analyze the motions of a set of objects CHAPTER 1. INTRODUCTION 12

that are similar in some way (either by construction, such as satellites, or in shape, such as spherical meteoroids) but varying in others (attitude for satellites, composition for meteoroids). A simpliﬁcation that we make here is that we only consider one main attribute across all platforms to drive a certain variation, although more can be appended if necessary, at the cost of increased uncertainty in the density estimates. In contrast, previous work in this ﬁeld has used far fewer or highly variable platforms over multiple regions to achieve estimated density.

The method introduced here is based upon order statistics, and the basic concept is that if all platforms travel through a certain medium, they must all encounter the same background density. Therefore, we can make observations as to the statistical likelihood of observing the minimum (or for that matter, any order) measurement amongst all measurements arising from the innate variation across all platforms. We then are able to estimate the variation itself as well as the background density with associated standard deviations. In short, we estimate the mean neutral density by observing multiple equivalent platforms passing through similar regions of space over similar time scales.

In summary, this dissertation presents the following major contributions:

Development of a new estimation technique based upon order statistics to de- • termine the neutral atmospheric density for di↵erent regions of the atmosphere. This is accomplished by analyzing the motion of multiple similar objects traversing these regions, calculating the e↵ects of drag, and negating its e↵ects through the ratio distribution of the minimum order statistic. Two types of estimates for neutral density can be made: the mean or the ordered estimate. The former performs well in the case of large errors in the measurement method coupled with small variations across the platforms, while the latter excels in situations where the measurement method is highly accurate but the platforms di↵er from one another considerably. The method is independent of prior density models, and o↵ers a new perspective for the estimation of neutral densities. CHAPTER 1. INTRODUCTION 13

Development of the statistical framework necessary for this data driven method. • By numerically calculating the variation across all platforms across all time periods, we can calculate the standard deviation of our neutral density estimates. This quantity decreases drastically if we employ the ordered estimate, especially if we can guarantee the quality of observation. For idealized scenarios, standard deviations of lower than 5% for satellites and 12% for meteoroids have been shown. For practical data, standard deviations of approximately 10% for satellites and 16% for meteoroids are realizable. The method depends upon correctly identifying the underlying physics of the problem or possessing accurate historical measurements, although additional error can be injected into the estimate if we are able to model this uncertainty.

The ﬁrst application of the new estimation scheme to the following two equiv- • alent platforms:

– Satellites: a satellite constellation (i.e. CubeSats) where all satellites are similar in mass distribution, material composition, and controlled behavior, all within the same orbit. Simulation and on–orbit data are analyzed for a circular LEO orbit of 400 km.

– Meteoroids: by observing the deceleration and inferred ablation during entry into the Earth’s atmosphere, a new method was developed to estimate altitude speciﬁc neutral densities. Data from radar detections of meteoroid plasma provides the bulk of the analysis.

1.4 Reader’s Guide

The motivation for studying the atmosphere from scientiﬁc, operational and risk assessment perspectives have been described in this chapter. The following chapters are organized as follows: CHAPTER 1. INTRODUCTION 14

Chapter 2 will provide information on the characteristics and behavior of the • atmosphere, how it reacts to external forcing functions, and how we currently model and measure its properties.

Chapter 3 will further the discussion by exploring the dynamics of how objects • interact with the atmosphere as they move through it, with particular emphasis on free molecular ﬂow theory. As with much of the following chapters contained within this thesis, two separate sections (Section 3.3 and Section 3.4)provide the details of the theory as applied to satellites and meteoroids, respectively.

Chapter 4 will detail the main contributions of this thesis, including the method- • ology of order statistics, ratio distributions and estimation of the variability across multiple platforms with the goal of estimating neutral density. Similar to the previous chapter, the application of the theory to both satellites and meteoroids are separately documented in Sections 4.2 and 4.3,respectively.

Chapter 5 will move from the theoretical to the practical, by applying sim- • ulation, on–orbit and radar data to produce empirical estimates. Particular applications for satellites and meteoroids are again separated into their respective sections (Sections 5.1 and 5.2).

Chapter 6 concludes this thesis by discussing the results of the methodology • and what the future of this research might hold.

Appendix A gives details regarding the satellites from where we obtain our • on–orbit data.

Appendix B details the radar data pertaining to meteoroid plasma detections. • Chapter 2

Background

This chapter will provide the context necessary to understand our atmosphere and how it is modeled. The chapter is organized as follows: Section 2.1 describes the behavior and structure of the atmosphere, particularly the thermosphere, its small and large scale physics and how the Sun drives many of its interactions. Section 2.2 describes the many methods of measurement that we have historically applied in determining the neutral density of the thermosphere. Finally, Section 2.3 details the history and applications of atmospheric modeling, including present–day models available to us as well as their particular implementations.

2.1 Nature of the Atmosphere

With respect to the size of the Earth, our atmosphere is almost negligibly thin in comparison. To put it into perspective, with the mean Earth’s radius being 6371 km, commercial airliners ﬂy at around 10 km, the record altitude of high-altitude balloons is 53 km, meteoroids ablate at approximately 100 km, and the ISS orbits the Earth

15 CHAPTER 2. BACKGROUND 16

at about 400 km. As one ascends in altitude with the atmosphere becoming increasingly di↵use, solar radiation becomes more prominent in interacting with individual particles as well as being inﬂuenced by Earth’s magnetic ﬁeld. It is of no surprise then that much of the behavior of the atmosphere is inherently tied to that of the Sun.

2.1.1 Energy Sources and Indices

We begin our discussion by identifying the main source of energy acting upon our atmosphere: the Sun. Unsurprisingly, changes in solar radiation output determine most of the atmospheric variations that we observe. This section will explore the major forms of energy and their interactions with the atmosphere, and how we characterize and measure them in the form of indices.

EUV

The Sun’s radiation spectrum can be approximated as a blackbody. Although it emits less electromagnetic radiation at the lower and higher frequencies compared to that in the visible spectrum, it is the shortest wavelength emissions (170 nm and less) that interact with the Earth’s thermosphere. Extreme ultra–violet (EUV) radiation, which spans wavelengths from 10 nm to 124 nm, is particularly important as the atmosphere absorbs much of its energy. The 11 year solar cycle causes ﬂuctuations in the amount of solar radiation emitted (roughly 3% across the entire spectrum), with periods of high and low activity termed solar maxima and solar minima,respectively. Low wavelength emissions come from the Sun’s chromosphere and corona, and have been to known to vary by a factor of 2 for EUV radiation up to extremes of 1000 for X-rays [32].

Absorption of EUV radiation by O2 and N2 in the upper atmosphere is the primary CHAPTER 2. BACKGROUND 17

contributor to atmospheric heating, although other species such as H2O, Na, NO, N, and O play a factor as well. During this process, these molecular compounds undergo dissociation and ionization, in which electrons and ions are produced, and in turn recombine [33, 34]. The deposition of energy raises the temperature of these particles, leading to an increase in the neutral density in the upper atmosphere.

Since the EUV radiation is absorbed by the atmosphere, it cannot be measured on the ground. However, it has been determined that the incoming radiation at a wavelength of 10.7 cm originates from the same layers of the Sun as the EUV. Hence the often cited F10.7 index is used as a proxy to EUV, measured in SFU (Solar Flux Units):

22 Watts 1 SFU = 1 10 (2.1) ⇥ m2Hz

Typical values for F10.7 range from less than 70 to more than 300 SFU [35]. In addition, the moving centered average F¯10.7 spanning over 81 days (approximately 3 solar rotations) is used in many models, since it captures the long term e↵ects and slowly varying components of solar radiation. It has been found that a time lag of 1 day is appropriate when utilizing the F10.7 index [36].

In more recent years, there has been an emergence of other indices (along with their 81 day averages) for the purposes of thermospheric density modeling [37, 38]. One recog- nized example is the Mg II core-to-wing ratio ﬁrst proposed by Heath and Schlesinger

[39], and later adapted into the M10.7 index [36, 37]. The original index is computed by taking the ratio between the variable chromospheric Mg II h and k lines at 279.56 and 280.27 nm, respectively, with the non-varying wings about the 280 nm wavelength. This ratio has been shown to be a good proxy for solar far ultra-violet (FUV) and EUV radiation [40].

There are a few additional solar indices in use today by the newest models [37,

38]. The ﬁrst is the SEUV ,orS10.7 index, which takes X-ray emissions at 26–34 nm observed by the Solar and Heliospheric Observatory (SOHO) research satellite located at the Lagrange Point 1 (L1) in the Earth-Sun system [41]. The second is CHAPTER 2. BACKGROUND 18

the composite Y10.7 index, which combines measurements at 0.1–0.8 nm from the Geostationary Operational Environmental Satellite (GOES) System at solar maxima with the Lyman-↵ emissions at 121 nm during solar minima [42].

Geomagnetic

The Sun constantly emits a stream of electrons and ions into space at approximately 400 km/s, known as the solar wind. While most of these particles are deflected by the Earth’s magnetic field, some become trapped within the magnetosphere, where they move along field lines and sometimes enter the atmosphere near the polar regions where the field lines are open. The entire system is quite complex, comprising of interactions between the solar wind, the magnetosphere, and the ionosphere along with fields induced by moving charged particles [43]. Geomagnetic storms are the result of large disturbances in these processes. Thus, while solar irradiance can be seen as almost predictable, solar wind is much less so, even during geomagnetically quiet periods.

Of interest to us is the mechanism by which geomagnetic activity a↵ects the thermosphere. The two primary methods in which energy is dissipated into the polar upper atmosphere is by Joule heating and by particle precipitation [44]. The former refers to the transformation of electric current energy into thermal energy through collisions between charged particles accelerated by an electric ﬁeld [45]. The latter is related to the bombardment of highly energetic particles generally into the auroral regions, resulting in ionization and dissociation of atmospheric constituents. During geomagnetic storm events, the energy deposited by these two mechanisms can exceed that of EUV by a factor of two [46].

In measuring the magnitude of geomagnetic activity, the number of geomagnetic indices is just as varied as the number of solar indices. The most commonly used index is the Kp index (subscript p for planetary), based upon 3 hour measurements CHAPTER 2. BACKGROUND 19

of the K indices, which are quasi-logarithmic indices measuring the magnitude of the local magnetic ﬁeld at 12 separate stations [47]. The Kp index ranges from 0 to 9, with denominations in thirds (e.g. 0+, 1-, 1o, 1+), with larger numbers indicating greater amounts of geomagnetic activity.

Just as commonly used as the Kp index is the ap index. This index is simply a linear equivalent to the Kp,andassuchitisalsomeasuredin3hourintervals,whichcan be averaged to provide the daily planetary amplitude Ap.TherangeofvaluesforAp is from 0 to 400 (in units of nT), with average values of 10-20 being quite common. Aconversionprocessandcodecanbefoundvia[1].

Finally, newer versions of density models utilize the disturbance storm time (Dst), which is used to assess the severity of geomagnetic storms [38]. It is expressed in nT and is the average value of the horizontal component of the Earth’s magnetic ﬁeld measured hourly from four near-equatorial geomagnetic observatories. Other choices include the an and am,whichareconsideredtobeimprovedversionsoftheap.Sofar, the “best” index to utilize is inconclusive, as each can perform well or poorly during di↵erent periods [48].

2.1.2 Structure and Composition

We continue our discussion by deﬁning the basic properties of the atmosphere, including its composition and primary layers. In particular, the core concept of hydrostatic equilibrium, as well as the relationship between pressure, temperature, and atmospheric constituents is explained. CHAPTER 2. BACKGROUND 20

Hydrostatic Equilibrium

The basic model of the atmosphere comes from the balance of pressure and gravity, as deﬁned by: dP = g⇢ dh (2.2) dh a where P is the pressure, h the altitude, g the gravitational acceleration, and ⇢a the density of the atmosphere. If we assume the ideal gas law:

P kT = (2.3) ⇢a mgas where k is the Boltzmann constant, T the temperature, and mgas the mean molecular mass of the gas, we can combine Eqns. (2.2)and(2.3)into:

dP m g 1 = gas dh = dh (2.4) P kT H where we often make the substitution H =(kT/mgasg). The height H is known as the isothermal scale height of the atmosphere, and describes the exponential rate of decrease in pressure and density over altitude. For the Earth’s atmosphere, this amounts to an altitude of roughly 8 km.

In the case of isothermal equilibrium, where T is uniform over the entire control volume (seen as a column of atmosphere), we obtain:

h P = P exp 0 H ✓ ◆ h ⇢ = ⇢ exp (2.5) a a0 H ✓ ◆ where P0 and ⇢a0 are the pressure and density of the atmosphere at sea level. Even if we assume non-thermal equilibrium and horizontal or vertical motion (such as winds), it can be shown that the variation of density is predominantly determined by gravity, CHAPTER 2. BACKGROUND 21

which is one order of magnitude greater than the variation due to temperature. Vari- ation due to winds is negligible in this regard, and can be regarded as second order compared to the variations due to gravity [33].

It should be noted that di↵erent species possess di↵ering scale heights as determined by their molecular weights. This is not an issue within the troposphere, where the gases are thoroughly mixed, but within the thermosphere, molecular di↵usion can separate the various species apart from one another. This results in separate treatment for each atmospheric constituent when analyzing these low density regions.

Atmospheric Layers

Figure 2.1: Layers of the atmosphere. Source: http://www.dreamstime.com/ stock-images-earth-s-atmosphere-layers-image22603834

The atmosphere can be primarily divided into 5 regions, as seen in Figure 2.1: CHAPTER 2. BACKGROUND 22

The troposphere, in which the bulk of all human activities takes place and where • all weather occurs on Earth (0 km – 20 km)

The stratosphere, where the ﬁrst temperature inversion (hotter regions higher • in altitude than colder regions) takes place above the hydrostatic equilibrium point (20 km – 50 km)

The mesosphere, where the majority of meteoroids disintegrate completely and • where the coldest naturally occurring place on Earth is located (50 km – 85 km)

The thermosphere, where solar radiation causes photoionization/photodissociation • of the molecules present and molecular di↵usion starts to occur (85 km – 690 km)

The exosphere, where virtually no interactions between molecules take place • and the beginning of interplanetary space (690 km and beyond)

The thermosphere and exosphere comprise the ionosphere, in which we observe the presence of increasing amounts of charged particles due to ionization from solar radiation. However, the neutral particles far outnumber their charged counterparts at altitudes lower than 500 km by at least 2 orders of magnitude [3]. For the purposes of this thesis, we will focus mainly upon the thermosphere, where the ablation of meteoroids starts to occur at its lower altitudes and where the majority of LEO satellites orbit the Earth.

Temperature

The separation of regions of the atmosphere as described previously is mainly due to the proﬁle of temperature, as can be seen in Figure 2.2.Theseparationboundariesare known as the tropopause, stratopause, and mesopause, with the preﬁx of each pause referring to the atmospheric layer directly below it. Temperature inversion is the CHAPTER 2. BACKGROUND 23

Figure 2.2: Temperature of the atmosphere at speciﬁc altitudes as modeled by the Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Extended (NRLMSISE) empirical thermosphere model [4]. Separation of speciﬁc layers of the atmosphere is shown as well. primary factor in determining where particular regions begin and end, which in turn is determined by a multitude of factors such as geographic location, time of year, and solar conditions. The air near the surface is warmed by the Earth itself, but it initially plummets with increasing altitude due to the drop in pressure and initial thinning of the troposphere. However, when one reaches the stratosphere, the concentration of ozone present absorbs UV radiation from the Sun and the temperature begins to rise. The next layer, the mesosphere, contains less ozone and hence we see the temperature drop again. The thermosphere sees another temperature inversion since the majority of the Sun’s EUV and X-ray radiation is absorbed by the particles present here. Finally, the temperature decreases in the exosphere as interplanetary space begins.

Temperature is the starting point for many models of the atmosphere. By combining temperature with the integration of the di↵usion equation, one can derive the number density of individual species and calculate the neutral density. However, one should CHAPTER 2. BACKGROUND 24

note the many, often empirical, corrections necessary to determine the temperature itself. Physically, temperature is determined by the amount of energy absorbed by individual particles due to external sources, primarily from the Sun at higher altitudes. However, the complex nature of all sources and interactions make the entire system very dicult to model.

Composition

The most abundant species within the thermosphere are N2, O, He and H. This is shown Figure 2.3 for both high and low solar activity. Because of the di↵use nature of the upper atmosphere, these various species do not mix homogeneously as in the trosposphere. Thus we need to turn to other methods to characterize their mutual interactions, which leads to the concept of the mean free path, ,deﬁnedas:

1 = (2.6) NgasA where Ngas is the number density of the gas, and A the collisional area of the particles within the gas. The mean free path is therefore the average distance a gas molecule would travel before it collides with another, and this distance increases with altitude. Above 100 km, the mean free path is so large that individual species are able to di↵use faster than turbulence can mix them. This molecular di↵usion is what causes the individual species to behave independently regardless of one another, according to Eq. (2.5). Since the lighter particles possess larger scale heights, they are more abundant within the thermosphere than their heavier counterparts. CHAPTER 2. BACKGROUND 25

(a) Number density of atmospheric constituents during high solar activity: (F10.7) = 60, Ap =5

(b) Number density of atmospheric constituents during low solar activity: F10.7 = 200, Ap = 15 Figure 2.3: Number density of atmospheric constituents in the thermosphere under high and low solar activity as modeled by NRLMSISE. Constituents include helium (He), atomic oxygen (O), molecular nitrogen (N2), molecular oxygen (O2), argon (Ar), hydrogen (H), and nitrogen (N). CHAPTER 2. BACKGROUND 26

Figure 2.4: Neutral density of the atmosphere under high (F10.7 =200,Ap =15)and low (F10.7 =60,Ap = 5) solar activity as modeled by NRLMSISE at midnight, 0 longitude, 0 latitude.

Density

Density is calculated as the total sum of all existing species within a volume:

M N ⇢ = i i (2.7) a N i A X where Mi is the molecular mass of each constituent, Ni the number density of each constituent, and NA the Avogadro’s number.

Generally, neutral density decreases exponentially with increasing altitude, as described by Eq. (2.5). Figure 2.4 shows this trend during both high and low solar conditions. Density also varies depending upon the geographic location, as can be CHAPTER 2. BACKGROUND 27

Figure 2.5: Neutral density of the atmosphere under moderate (F10.7 =130,Ap =10) solar conditions as modeled by NRLMSISE at 400 km over the entire Earth. The 0th degree in both longitude and latitude (prime meridian and equator) have been labeled. seen in Figure 2.5, although this is primarily due to the day/night cycle. Other factors include interactions between di↵erent layers of the atmosphere, such as semi-diurnal tides at 100 km altitude, or additional sources of energy such as Joule heating and particle precipitation close to the poles [45, 49, 48], which become dominant during periods of high solar and geomagnetic activity.

2.1.3 Neutral Density Variations

As a temporal and spatially varying phenomenon, thermospheric neutral density follows certain predictable patterns in its behavior. This section will explore these variations. CHAPTER 2. BACKGROUND 28

Geographic

Spatial variations in density are the easiest to visualize: for example, a satellite passing over the equator would see a di↵erent density compared to when it passes over the poles. This type of variation is classiﬁed as static,sinceweassumeallatmospheric parameters as constant. The equatorial bulge also e↵ectively changes the actual altitude (relative to sea level) of a passing spacecraft, and hence, the atmospheric density that inﬂuences its motion is di↵erent as well [1]. It should be noted, however, that this does not mean that spatial and temporal variations are independent of one another; for example, longitudinal variations arise mainly due to the day/night cycle, and seasonal-latitudinal variations are mainly due to the changing sub-solar point.

The most evident spatial variation are latitudinal variations.Thisisprimarilydue to the di↵ering amounts of sunlight the Earth receives over di↵erent latitudes, but also due to the fact that the auroral regions receive additional sources of atmospheric heating from the geomagnetic ﬁeld. This e↵ect results in an increase in the average densities at latitudes greater than 55o by at least 50% compared to equatorial regions, with the e↵ect being more pronounced during nighttime [50, 51].

The longitudinal variations are harder to characterize, since they are strongly coupled with time-varying models. It has been shown, however, that the amount of longitudinal variation is higher in the winter hemispheres than in the summer hemispheres (approximately 11% to 6%) [52]. This is due to the fact that Joule heating and particle heating in the auroral regions are much weaker than the solar EUV radiation experienced in the summer for geomagnetically quiet conditions [46]. Furthermore, the asymmetry of the pole locations on the Earth contributes to di↵erent amounts of heating when comparing northern and southern hemispheres [52]. Finally, to a lesser extent and only considering up to the mesosphere, di↵erent geographic features on the Earth (such as oceans vs. mountains) play a role in the wind, density, and temperature experienced by the atmospheric layers above these areas [1]. CHAPTER 2. BACKGROUND 29

Diurnal

The most prominent density variation that is easily observed is diurnal, due to the uneven heating by the Sun on the Earth as seen in Figure 2.5.Thisphenomenonis subject to seasonal variations as well, as the Sun moves from the northern to southern hemisphere. The result is a pronounced atmospheric bulge that lags in the general direction of the Sun at local times 2:00–2:30 P.M. (and a corresponding minimum at 4:00 A.M.). The ratio between the day–night densities can easily exceed 3, and can reach a factor of 5 at altitudes of 500 km and higher [7].

Solar Rotation

The 27 day solar rotation cycle corresponds to the Sun’s rotational period, which can cause densities to vary in excess of 15% (peak to peak variation of 30%) [54]. ± This e↵ect can be seen in Figure 2.6, where increases in EUV radiation of 15% and increases in the Mg II index (see section 2.1.1)of2%correspondtodensityvariations of up to 50% [53]. The often unpredictable growth and decay of active solar regions as they rotate with the Sun are the main drivers of this phenomenon.

Intra–annual

The intra–annual variations refer to both semiannual and annual oscillations in densities, as ﬁrst reported by Paetzold and Zsch¨orner [55]. This type of variation is shown in Figure 2.7, where a 6 month periodicity occurs with maximums in April and Oc- tober, and minimums in January and July. The e↵ect is worldwide, can change from year to year by 60% during solar minima to over 250% during solar maxima, and is highly correlated with solar activity [56, 57]. Initially thought to be the result of the eccentricity of Earth’s orbit about the Sun, it has been shown that eccentricity CHAPTER 2. BACKGROUND 30

Figure 2.6: Comparison of ﬁltered 27 day variations in density, EUV, and Mg II during 2003 in terms of percent variation about the mean. Reproduced from [53]. accounts for less than half of the observed e↵ects [58]. Many theories have been proposed to account for this anomaly, such as uneven heating of the hemispheres [58], global transport [59], gravity waves [60], and dissipating tides [61]. However, because of the many interdependent phenomena that could give rise to these variations, the exact cause is still not fully explained [62].

Solar cycle

The 11 year solar cycle is the strongest driver of the amount of solar radiation that reaches the Earth. Sunspots, which are regions of the sun that appear dark due to CHAPTER 2. BACKGROUND 31

Figure 2.7: The intra–annual density variations at 400 km as modeled by NRLMSISE under moderate solar conditions (F10.7 =140,Ap =10). their cooler temperatures and stronger magnetic ﬁelds compared to its surroundings, increase in number and strength during high solar activity and decrease during low solar activity. As can be seen in Figure 2.8,thesunspotnumberishighlycorrelated with the F10.7 index, with the peaks varying from cycle to cycle. The solar cycle also a↵ects shorter time scale variations, such that the amplitudes of semiannual or monthly variations generally increase during more turbulent solar periods.

Geomagnetic Activity

As discussed in Section 2.1.1,manyenergeticparticlesoriginatingfromthesolarwind become trapped within the Earth’s magnetic ﬁeld. Geomagnetic activity is the result of these particles interacting with the ionosphere, depositing energy particularly in the auroral regions. The dissipation of solar wind energy also a↵ects the constituents CHAPTER 2. BACKGROUND 32

st Figure 2.8: History of sunspot number versus F10.7 index, retrieved on Dec. 1 ,2015 from http://omniweb.gsfc.nasa.gov of the atmosphere, such as an increase in heavier gas particles, altitude dependent behavior of atomic oxygen, and a signiﬁcant decrease of helium (demonstratively this might also be classiﬁed as a geographic latitude variation, which is another indication that spatial and temporal variations are inherently connected) [34, 44].

During certain periods, particularly during solar maxima, the number of charged particles and their speeds may increase drastically. These sudden outbursts are known as coronal mass ejections (CMEs) and solar ﬂares, which cause geomagnetic storms on the Earth. Although temporary, the mass density of the thermosphere tends to increase about 90 minutes after a major solar storm [63], with the magnitude of the increase dependent upon the altitude and strength of the geomagnetic disturbance [48]. CHAPTER 2. BACKGROUND 33

2.2 Density Measurements

There have been many methods employed in measuring the properties of the atmosphere, including its pressure, temperature, and respective constituents. Veriﬁcation and calibration of atmospheric density models depend upon this collected data, and is often not limited or restricted to just one methodology. For our purposes, this section will focus primarily on the measurement of neutral density within the thermosphere.

2.2.1 In Situ Instrumentation

The use of instrumentation aboard sounding rockets and satellites is the most direct method to probe the atmosphere. The earliest experiments in the 1960s and 1970s used pressure gauges, which in recent years have been updated with the use of ionization gauges, with typical errors of roughly 2% [49, 64]. Their region of application is within the mesosphere and lower thermosphere (MLT), around 60 km–300 km in altitude.

The bulk of our early knowledge of thermospheric composition, however, comes from neutral mass spectrometers (NMS) launched during the 1960s, 1970s and early 1980s aboard the Atmospheric Explorer and Dynamics Explorer series of satellites. These instruments are capable of determining the number densities of the atmosphere’s key constituents directly, particularly He, O, N2 and Ar [65, 66]. With the termination of the Dynamics Explorer 2 mission in 1983, there has been a lack of new data of this type in recent years.

In situ methods of atmospheric density determination are typically performed with the use of accelerometers. They are able to measure the non-conservative forces on asatellite,andhencederivetheneutraldensitythroughthee↵ectsofdrag(e.g.the CHAMP and GRACE missions) [67, 68, 69, 70]. The Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) [71]andtheSwarmmission[72]aremissions CHAPTER 2. BACKGROUND 34

continuing with the theme of making measurements using high precision accelerometers.

The beneﬁts of these methods are readily obvious: they provide direct measurements of the atmosphere, often with very good resolution and accuracy. Much of the validation of our models comes from such data. However, calibration of these sensors has been a constant issue, as they often disturb the medium in which they take measurements. Deriving atmospheric densities from these instruments also require careful separation of drag and solar radiation pressure (SRP), which is the momentum imparted upon the satellite by impacting photons, is necessary and sometimes dicult. Another drawback is one of scale; because of expensive instrumentation and the spe- ciﬁc nature of these experiments, the frequency of these missions is sparse. Thus, the results tend towards point measurements that do not give us a full temporal or spatial resolution of Earth’s entire atmosphere due to inadequate coverage.

2.2.2 Satellite Orbital Decay

Under orbital decay, elliptical orbits tend to circularize, and then proceed to shrink as momentum is imparted upon the satellite surfaces by collisions with the particles in the surrounding medium. By measuring how an orbit changes over time, the neutral density can be deduced. The dynamics of this process are further detailed in Chapter 3.

The ﬁrst source of data relating to this method of density estimation comes from two line element (TLE) sets. This format, maintained by NORAD and JSpOC, contains continuously updated information regarding all unclassiﬁed and tracked satellites and their orbits. Through judiciously analyzing particular satellites’ TLEs, reliable estimations can be made regarding the neutral density over a certain period [73, 74]. Global averages of density can also be calculated by analyzing large numbers of satellites over longer time frames [75]. CHAPTER 2. BACKGROUND 35

A more accurate version of orbit-derived atmospheric mass density, based upon similar principles of observing changing orbits, has been developed recently. Instead of utilizing TLEs, they instead rely upon higher quality datasets gathered from the Global Positioning System (GPS) [76]orfromSLR[13]. These tracking techniques are able to give position measurements on the order of centimeters, instead of kilometers in the case of TLEs.

Although it might seem prudent to use datasets from the latter methods, it has to be noted that this is a more recent development and not all satellites are equipped to provide these measurements. Furthermore, most missions carrying such sensitive instruments do so with the primary purpose of not measuring drag, but rather on missions related to geodesy. As such, they su↵er from the same problems as previously discussed for in situ instrumentation: the temporal and spatial sparsity of data. In contrast, the advantages of TLE datasets lie mainly in its availability, abundance and history. From this, we are able to gain a global perspective on changing thermospheric conditions across longer time periods. However, the drawback here is that resolution is lost, since TLEs are mean elements (further discussed in Chapter 3)thatdonot capture the short–periodic nature of density variations. The method of orbit–derived density also shares a complication discussed previously for accelerometers: the separation of other perturbing forces from drag is still relevant and must be addressed. Furthermore, drag in itself is a complicated phenomenon that is heavily dependent upon a satellite’s shape, orientation, velocity, and the surrounding medium.

2.2.3 Remote Sensing

Remote sensing is a broad term used to refer to the science of obtaining information from a source without physical contact. For our purposes, this usually implies inferring neutral densities through observation of phenomena that interact with the atmosphere. CHAPTER 2. BACKGROUND 36

Remote sensing o↵ers new ways to verify the neutral density and compare them with traditional methods. Because of their nature in observation rather than direct measurement, they can often be accomplished on the ground and are less constrained by mission timelines. However, they do depend upon phenomena that may occur only at certain points in time and location, and not globally. As well, many complex models on how the atmosphere a↵ects or reacts to these phenomena must be simpliﬁed to deduce neutral density itself.

Radar Measurements

The use of radar in aerospace applications has traditionally been for the tracking of missiles, aircraft, and satellites. They operate by transmitting a pulse of electromagnetic energy and receive the reflected signal. The power of the reflection is dependent upon the transmit power, the antenna gain, the distance and the radar cross section (RCS) of the object. The RCS is particularly important since it is a measure of the detectability of the object to the radar (e.g. a commercial jetliner would be readily detectable, while a stealth fighter would be virtually undetectable). By measuring the time it takes for the signal to return, the distance to the object can be measured. By either di↵erentiating the range rate or through Doppler techniques, the velocity can be obtained as well.

A special subset of particularly sensitive radars known as High–Power Large–Aperture (HPLA) radars are classiﬁed based upon their megawatt–class transmitters, high gain, and large sizes. Examples include the European Incoherent Scatter Scientiﬁc Associ- ation (EISCAT) radars in northern Scandinavia, Millstone Hill near Boston, and the Advanced Research Projects Agency (ARPA) Long-range Tracking and Instrumen- tation Radar (ALTAIR) located on the Kwajalein Atoll. The latter two radars are also utilized within the SSN to track satellites and orbital debris. Most transmit at frequencies greater than 30 MHz but less than 3 GHz, in the Very High Frequency (VHF) and Ultra High Frequency (UHF) ranges. Recently however, they have been CHAPTER 2. BACKGROUND 37

re–purposed for other applications such as probing ionospheric phenomena, or for detecting ablating meteoroids in the lower thermosphere [77].

The use of incoherent scatter radar (ISR) to deduce atmospheric properties comes from measuring the ion and electron temperatures, velocities and number densities. During geomagnetically quiet periods, the ion–neutral collision frequency can be obtained through plasma parameters inferred from radar data [78]. If we assume that most of the upper thermosphere is dominated by atomic oxygen, the number density of O can be calculated. This has recently been accomplished at the European Incoherent Scatter Svalbard Radar (ESR) at altitudes of 300–400 km [78, 57].

The use of radars to collect meteoroid data has been well established since the late 1940s, but more recently HPLA radars have yielded signiﬁcant insights into the ﬁeld of meteor physics [79, 80, 81, 82, 83]. What a radar actually detects when a meteoroid passes through its beam is the meteor. These detections are split into two categories: head echoes and trail echoes. The head echo represents the scattering from the dense ball of plasma traveling with the meteoroid at velocities between 11 to 72 km/s [84, 85], while the trail echo consists of the plasma di↵using in the meteoroid’s wake. Figure 2.9 shows a meteor in a range–time–intensity (RTI) plot, where we plot the range vs. time with the data color–coded based upon the Signal to Noise Ratio (SNR). The head echo, which is our primary interest, can easily be seen as a streak outlined in green preceding all other detections. The meteoroid’s speed can then be determined by observing the head echo’s speed through the change in altitude, or through the Doppler shift. If interferometry or monopulse data are available, angular measurements can be made such that the full 3D velocity and acceleration can be calculated. Using these measurements, the masses and bulk densities of meteoroids can be derived [85, 86]. Conversely, and shown later in this thesis, we can utilize knowledge regarding the distribution of meteoroid properties to deduce atmospheric density [87]. CHAPTER 2. BACKGROUND 38

Figure 2.9: Examples of meteor head and trail echoes as observed by the Jicamarca Incoherent Scatter Radar on a RTI plot. The head echo is outlined in green, the trail echo is outlined in red, and a ﬂare event is outlined in yellow. Reproduced from [77].

Dayside Airglow

Another method of deducing atmospheric properties comes from the observation of the dayside airglow caused by photoelectron excitation of N2,O,andO2 due to FUV radiation. The Thermosphere, Ionosphere, Mesosphere Energetics and Dynamics (TIMED) satellite carried onboard the Global Ultraviolet Imager (GUVI) instrument to perform these measurements at an altitude range of 60-180 km [88]. Infrared (IR) airglow observations can also give anomalous oxygen quantities at altitudes below 110 km [89].

Atmospheric Occultation

Recent developments in atmospheric occultation have been quite promising. By measuring the attenuation of IR, UV and X-ray sources as they traverse the atmosphere, we can determine the composition as well as the total mass density of the MLT region [90, 91]. This method focuses mostly on altitudes of 70–150 km, and sources include CHAPTER 2. BACKGROUND 39

either the Sun, celestial bodies, or satellites positioned in Medium Earth orbit (MEO). Compared to NRLMSISE, deviations of predicted density can vary up to 50% [91].

Another similar area is the use of radio occultation from GPS signals received by satellite constellations to determine the makeup of the atmosphere and properties of the ionosphere [92, 93]. Relying upon the refraction and bending of the signal due to the atmosphere, this new method has been shown to be capable of scanning successive layers of the atmosphere, aiding in weather forecasting and monitoring climate change.

2.2.4 Relation to Thesis: Measurements from Equivalent Plat- forms

This section describes how density estimation as covered by this thesis is related to the measurement methods as described previously. The main goal of this thesis is to provide a new framework in which we can leverage measurements from similar platforms over similar regions of space to deduce neutral densities within those regions. In particular, we focus on CubeSats, which are miniaturized satellites composed of multiples of 10 10 11.35 cm cubic units. They possess a mass ranging from ⇥ ⇥ 1–10 kg and are usually built using commercial o↵–the–shelf (COTS) components. Data collected from a constellation of these satellites and from meteoroid ablation observations were used to determine neutral densities.

Due to the recent advances of CubeSats and increased launch opportunities, there has been an upward trend in a↵ordable small satellites. As a result, we are beginning to see the emergence of LEO constellations, which presents us with a unique opportunity to measure neutral atmospheric density en masse through similar satellites through orbit–derived mass density. The short lifespan of satellites within this region becomes almost a non–issue, as replacement from the ISS or other forms of delivery are frequent and cheap enough to provide a steady supply of new spacecraft [94]. Although we CHAPTER 2. BACKGROUND 40

have chosen TLE data for analysis within this thesis due to availability, the general concept can be applied to any density measurement method, such as GPS or SLR.

The second application focuses on the MLT region at approximately 100 km in altitude. Because incoming meteoroids interact with the atmosphere, they have been used as probes for atmospheric science before the advent of in situ monitoring [95]. In the context of this thesis, we use HPLA radars to observe vast quantities of incoming meteoroids as they ablate in Earth’s atmosphere. Coupled with a few physical laws (drag and ablation), we take a data–driven, remote sensing approach to determine the neutral densities in a region where measurements are relatively scarce. However, the underlying method still relies upon drag and the concept of equivalent platforms, similar to that of satellite constellation tracking.

2.3 Atmospheric Density Modeling

Neutral atmospheric density modeling has been an ongoing endeavor for the past 50 years. The two main types of models in use today are either semi–empirical or “physical” in nature [48]. This section will describe brieﬂy their nature, uses, and provide examples from each.

2.3.1 Empirical Models

Empirical models are derived based upon data collected from various satellites and instruments to ﬁt a parameterized mathematical formulation of the atmosphere. They are probably the most widely known models, and are used for many satellite applications since they are fast, readily available, and possess a long history. As such, they are often used for benchmarking and parameter retrieval purposes for new models or instruments. Because of the wealth of data already collected and “coded” into these CHAPTER 2. BACKGROUND 41

models, they also serve to provide useful estimates or forecasts of the environment based upon only a few parameters when measurements are lacking.

All empirical models at their starting point use the exospheric temperature, T as its 1 defining parameter. This is usually a function of indices such as F10.7 and Kp.The temperatures across an entire altitude range (usually from the temperature inflection point at 85 km to the exospheric boundary) are calculated with considerations for geographic location, local solar time, and periodic variations, all of which can be quite complex. The parameterization of how the temperature profile varies with altitude also depends upon the specific model. The di↵usive equilibrium condition is employed to calculate species specific number densities and hence the total neutral density.

All models do fairly well under stable conditions and low solar activity, with a standard deviation of roughly 10-20% [5, 6, 2]. It has been shown that errors can reach as high as 20-50% during turbulent periods of high solar activity [7, 8]. Also, model assumptions and simpliﬁcations, which are always inherent, do not capture the full dynamics present in such a complex system.

Jacchia Model

The first empirical atmospheric model to be realized was by Jacchia [96, 97], and was adopted as the Committee on Space Research (COSPAR) International Reference Atmosphere in 1972, henceforth known as CIRA–72. The model was based upon the first observations of orbital changes in satellites under the influence of drag. The latest version of this model is JB2008 [38], where the di↵usive equilibrium condition has been relaxed at all altitudes to better match intra–annual variations and the addition of solar and geomagnetic indices. CHAPTER 2. BACKGROUND 42

Mass Spectrometer and Incoherent Scatter (MSIS)

The MSIS model, created by Hedin in 1977, was originally based solely on mass spectrometer measurements and incoherent scatter radar observations [98]. They were considered advantageous to drag–derived densities since they directly measured the atmosphere’s constituents via in situ methods and were veriﬁed through radar observations. MSIS–86 replaced Jacchia’s model as the COSPAR International Ref- erence Atmosphere in 1986, henceforth known as CIRA–86 [66]. Eventually, MSIS incorporated UV occultation, accelerometer and orbit–derived density data as well. The newest version of MSIS is known as NRLMSISE-00, developed by the US Naval Research Laboratory (NRL) with the ‘00’ indicating the year of 2000 [4].

Drag Temperature Model (DTM)

The original DTM model was based upon orbit–derived density data similar to Jac- chia’s [99], but has recently been modiﬁed to include accelerometer, mass spectrometer, ISR, and optical airglow data as well. It is quite similar to the MSIS model, except for its treatment of temporal and spatial variations. The newest iteration is DTM–2013 [100].

Satellite Based Models

Besides the three main models mentioned above, the Russian space surveillance system uses the Russian Government Standard (GOST) model, which derives its methods from observations from the Russian Cosmos satellites [101]. In terms of accuracy, it possesses an error equivalent to the older 1971 Jacchia model [1]. Recently, the Global Average Mass Density Model (GAMDM) [102] has also been developed, based upon orbit–derived density as a function of altitude, solar ﬂux, geomagnetic activity, and day of year using 5000 Earth–orbiting objects. CHAPTER 2. BACKGROUND 43

Of particular interest also are Dynamic Calibration of the Atmosphere (DCA) methods, which are used to improve or correct atmospheric density. They do so by selecting from a set of “calibration” satellites, whose drag coecients are well known or estimated beforehand. By analyzing the orbits of these particular satellites, density corrections are made every 3 hours, which may be factored into existing models such as MSIS. The US Air Force Space Command’s High–Accuracy Satellite Drag Model (HASDM) accomplishes this through the tracking of 75–80 calibration satellites [103].

2.3.2 Physics Based Models

Physics based or “physical” models numerically solve the fluid equations from first principles, while attempting to incorporate all the intricacies inherent in the atmosphere (e.g. heating and cooling processes, species specific reaction rates, gravity waves). Recent versions usually are part of larger coupled thermosphere–ionosphere– magnetosphere models. These models can simulate new geophysical conditions lacking in historical data, but are higher in complexity and still depend mostly on external drivers. Boundary conditions are imposed when the fluid equations become invalid beyond 650 km in the upper case, but lower bounds may vary.

Because of their complexity, physical models are rarely seen in engineering applications, and are used mostly to identify, analyze and interpret physical processes that occur climatologically or during speciﬁc events [48, 104]. In terms of accuracy, they are on par with empirical models [105]. The topic of physical models will remain outside the scope of this thesis. CHAPTER 2. BACKGROUND 44

2.4 Summary

This chapter has covered the fundamentals of the atmosphere: its structure, properties, and the sources of energy that drive its behavior. The neutral density of the thermosphere, deﬁned as the aggregate mass of all existing species within a volume, was the primary focus. Its spatial and temporal variations were explored, as well as current and historical methods of measurement and modeling. Finally, we have set the stage regarding our approach to neutral density estimation, in particular from the observation of satellite constellations in LEO and from meteoroid ablation in the MLT region. Chapter 3

Dynamics

This chapter will set the mathematical and physical groundwork for the equivalent platform density estimation methodology developed in the later chapters. To understand how objects move within the atmosphere, Section 3.1 will review the concepts of force modeling within this context. Section 3.2 will go further in detail into the nature of drag forces under free molecular ﬂow theory, which is applicable to the regions we are interested in. A simpliﬁed derivation of drag is presented here, with the complete derivation available within Sentman’s analysis [113].

In Sections 3.3 and 3.4,twoparticularapplicationsrelevanttothisthesiswillbe highlighted: how both satellites and meteoroids behave under atmospheric forces and how we extract required drag-related variables by observing their motion. In particular, changes in the mean motion drives much of our satellite analysis, while velocity and acceleration measurements are required for meteoroid data. We deﬁne the important K parameter, which combines the e↵ect of the neutral density with the properties of the platform, which forms the basis for much of the material covered later in Chapter 4.

45 CHAPTER 3. DYNAMICS 46

3.1 Force Modeling

The study of the motion of objects is deﬁned by Newton’s second law:

dp dV F = = m (3.1) dt dt where F is the total resultant force on an object, p the linear momentum, m the object mass, a the inertial acceleration of the object, and V the inertial velocity. Within the context of this thesis, we adopt the bold emphasis to denote vector quantities.

Figure 3.1: Major forces acting upon satellites (left) and meteoroids (right). Def- initions are as follows: Fg gravity, FD drag, FL lift, Fsrp SRP, F3body 3rd body perturbations, and Fu unmodeled forces (e.g. solid tides, electromagnetic, albedo). Note that the direction for Fu is arbitrary, and ultimately depends on the net sum of all unmodeled forces.

Figure 3.1 shows the major forces acting upon satellites and meteoroids as they traverse the atmosphere. These include central-body gravity Fg,dragFD,liftFL,SRP

Fsrp,3rdbodyperturbationsF3body,andunmodeledforcesFu,suchassolidtides, electromagnetic forces, and albedo e↵ects (note that the direction chosen for Fu in the ﬁgure is arbitrary, and ultimately depends on the net sum of all unmodeled forces). At altitudes lower than 600 km, atmospheric e↵ects such as drag and lift dominate, depending on the object’s mass, orientation and shape. However, the number of atmospheric molecules that interact with the object decreases with increasing altitude, and hence SRP becomes more relevant at higher altitudes. Beyond LEO, e↵ects such as 3rd body perturbations from the Sun and moon as well as the Lorentz force due CHAPTER 3. DYNAMICS 47

to the charge di↵erential between the object and its surrounding plasma can become relevant as well. The primary focus of this thesis will be on drag as it is the dominant force in the regions of interest to us, with the other forces (aside from gravity) at least an order of magnitude smaller.

Due to the dichotomy of the subject material as it relates to both satellites and meteoroids, treatment of the speciﬁc methodology will be divided into two independent sections, 3.3 and 3.4 respectively. This section will give a general overview of the physics models we employ to model all relevant forces.

3.1.1 Gravity

Earth’s gravity is the dominant force experienced by all objects within its sphere of inﬂuence (approximately 106 km in radius). The gravitational force is deﬁned as:

Gm m R F = 1 2 (3.2) g R 3 | | where G is the gravitational constant, m1 and m2 are the masses of the two objects in question, and R is the vector from the secondary body to the attracting body.

This deﬁnition is exact for point masses, but Earth is not in reality a point mass, or perfectly spherical for that matter. Thus we turn to the mathematical formulation described by an aspherical-potential function, U. Note that because gravity is a conservative force (work done in moving a particle between two points is independent of the path taken), we can formulate gravitational acceleration as:

a = U (3.3) g r

Conventionally we express U using spherical harmonics, which consist of a series of Legendre polynomials that describe a function deﬁned on a sphere’s surface. In CHAPTER 3. DYNAMICS 48

the same way that a Fourier series is able to model any periodic function through an infinite series of trigonometric functions, spherical harmonics can model deformations beyond that of a perfectly spherical surface through the use of an infinite series of Legendre polynomials. Just as a Fourier series requires constants to define a function’s shape, spherical harmonics requires constants for its associated Legendre polynomials. These constants are labeled by specific indices known as the degree and order, with higher order terms capturing higher frequency features. Therefore, the accuracy and resolution of spherical gravity models are specified by the maximum of these indices. Current implementations of gravity models can reach maximum degree and order values of 2159, such as the Earth Gravitational Model 2008 (EGM2008) [106], although for common engineering applications, typical values of 40 are seen as sucient. The derivation of U can be found in [1], as it is beyond the scope of this thesis.

3.1.2 Drag

The deceleration due to drag derived from the conservation of linear momentum between a moving object and the surrounding gas molecules is described by:

1 A V a = C ⇢ V2 rel (3.4) D 2 D m a rel V | rel| where aD is the (negative) acceleration due to drag, CD the drag coecient, A the object’s frontal area, ⇢a the atmospheric density, and Vrel the relative velocity of the object with respect to the atmosphere. This is the most commonly used formulation by aerodynamicists for objects moving through a ﬂuid, with the drag force simply deﬁned as:

FD = maD (3.5)

Note that drag always acts in the opposite direction to the relative velocity. The factor 1 1 2 of 2 comes from the deﬁnition of dynamic pressure as 2 ⇢aVrel from low velocity aerodynamics, and is thus retained to preserve consistent notation. In meteoroid CHAPTER 3. DYNAMICS 49

literature, this factor is often dropped or replaced by :

1 = C (3.6) 2 D

Ballistic Factor, B

As can be seen from Eq. (3.4), the drag force is a function of many parameters. To ease notation, we often combine various parameters that are properties of the body into the ballistic factor, B,deﬁnedas:

1 CDA B = b = (3.7) m where b is the ballistic coecient. Thus the formulation combines an object’s drag coecient, frontal area and mass into one variable that deﬁnes dynamically similar conﬁgurations.

The drag coecient, CD,arisesfromthefactthatindividualparticlesofaﬂuid continuously strike the surface of a moving object, imparting their momentum upon it. This interaction is di↵erent for di↵erent shapes, and thus gives rise to di↵erent values of CD as a result. Treatment of how CD is evaluated under free molecular ﬂow (i.e. the mean free path is larger than the size of the object), which is relevant to objects in the thermosphere, is described in Section 3.2.

The ballistic factor is also dependent upon the area and mass of the object. The frontal area, A,isdeﬁnedastheareaoftheobjectasprojectedontotheplaneor- thogonal to the velocity vector. The ratio between A and m determines how sensitive the object is to drag. Thus it is quite common for smaller objects to experience larger perturbations due to drag than larger objects of similar construction because of the square–cube law in area–to–mass ratios. CHAPTER 3. DYNAMICS 50

Background Density ⇢a

The background neutral density, ⇢a (described in Section 2.1.2)isthemainparameter we wish to solve for in our equations. A meteoroid or a satellite will traverse di↵erent regions of the atmosphere, either in altitude or latitude/longitude, and thus the atmospheric density can change (drastically) along the object’s trajectory. Often, when we integrate over the time period as the object traverses its path, what we actually end up solving for is the mean neutral density,⇢ ¯a over that period.

Relative Velocity, Vrel and Wind Factor, W

Vrel is deﬁned as the velocity of an object relative to the background atmosphere. It is important to make this distinction because the velocity of the atmosphere is mostly driven by co-rotation with the Earth, but also includes contributions from background winds. Thus we deﬁne:

Vrel = V + Va (3.8)

where V is the inertial velocity of the object and Va is the velocity of the background atmosphere (including co-rotation and winds). To simplify calculations, King-Hele [107]deﬁnesthewindfactorW as:

v2 r⌦ W = rel (ê ê ) 1 E cos(incl) (3.9) v2 v,rel · v ⇡ v ✓ ◆ where vrel is the magnitude of the relative velocity, v the magnitude of the inertial velocity, ê v,rel the unit vector of the relative velocity, ê v the unit vector of the inertial velocity, r the distance of the object from the center of the earth, ⌦E the angular velocity of Earth’s rotation, and incl the inclination of the orbit. This approximation is shown to be accurate [73] to within 3% of the Horizontal Wind Model (HWM-93) [108]. CHAPTER 3. DYNAMICS 51

Combined with Eqs.(3.4)and(3.7), we arrive at the formulation:

1 A 1 a ˆe = C ⇢ Wv2 = B⇢ Wv2 (3.10) D · v 2 D m a 2 a

This equation will be used in Section 3.3 when we isolate the e↵ects of drag.

3.1.3 Solar Radiation Pressure (SRP)

Solar radiation pressure is the e↵ect of photons from the Sun impacting upon an object’s surface, and thereby transferring their momentum onto the object. The solar radiation ﬂux at a distance of 1 AU from the Sun is approximately 1361 W/m2.The acceleration experienced by an object is thus calculated as:

SolarF lux A R a = C sun (3.11) SRP cr2 m R R sun | sun| where Rsun is the vector from the Sun to the object, rsun the distance between the Sun and the object, c the speed of light, and CR the reflective coecient. CR varies from 0 to 2, where 0 indicates that the object is translucent and no momentum transfer occurs, 1 indicates that the object fully absorbs all solar radiation, and 2 indicates that the object fully reflects all incoming radiation towards its source (the Sun). The treatment of A,thearea,requirescarefulconsiderationaswell,sinceitisdependent upon the complexity of the surfaces and the angle at which they are oriented in relation to the incoming solar flux. The entire process becomes quite complicated if we consider eclipsing bodies, albedo from the Earth, and varying Sun-Earth distances.

The e↵ects of SRP are felt more keenly in the upper regions of the thermosphere beyond an altitude of 600 km, where it becomes equivalent to that of drag (refer to Figure 3.2). It can cause periodic orbital changes as well as torque e↵ects, which become more prominent with increasing area–to–mass ratios. For the purposes of this thesis, where the analysis of drag is the primary concern for altitudes below 500 km, we will consider the e↵ects of SRP negligible (it contributes to less than 1% of the total CHAPTER 3. DYNAMICS 52

error in neutral density estimates). Note that our methodology of calculating neutral densities (described in Chapter 4)canstillbeappliedifweareabletoe↵ectively recognize and account for the e↵ects of SRP on our system beforehand.

3.1.4 Other Perturbing Forces

Figure 3.2: Relative contributions of forces in LEO. Note that force contributions other than central gravity are additional perturbing forces. An area–to–mass ratio of 0.2 m2/kg is adopted for drag and SRP.

The remaining perturbing forces in the lower thermosphere are considered orders of magnitude smaller than drag (see Figure 3.2), and so are generally ignored for short term predictions. These include 3rd body perturbations (lunar or solar), tides, and electromagnetic forces. We also ignore thrust from propulsion since we are focusing mostly on satellites and meteoroids. Figure 3.2 shows the relative contributions of all major forces on objects in LEO (Table 3.1 in Section 3.3.1 also gives the changes CHAPTER 3. DYNAMICS 53

in orbital parameters resulting from these forces) given an area–to–mass ratio of 0.2 m2/kg for drag and SRP. We see that for objects below 400 km in altitude, drag is at least 2 orders of magnitude larger than all other forces except for gravity. The 2 2 ⇥ and 8 8(degree/orderof2and8)gravityperturbationsarealsoshown,although ⇥ their e↵ects are conservative. Models for many of these perturbing accelerations can be found in [109]and[1].

3.2 Free Molecular Flow

The region of space we are interested in is dominated by rareﬁed gas dynamics. The concept is centered about the mean free path , introduced earlier in Section 2.1.2 and Eq. (2.6). Here we also introduce the Knudsen number, Kn,deﬁnedas:

Kn = (3.12) Lref where Lref is the characteristic length of a particular object. Normally, a value of Kn 10 indicates that the flow should be treated as particulate in nature, while values of Kn 1 indicate that the flow is best described by the Navier-Stokes equations  for continuum flow.

Another important parameter in free molecular ﬂow is the molecular speed ratio, S:

v S = gas (3.13) vth

Here, vgas is treated as the bulk speed of the gas relative to the moving object, and vth the thermal speed, which is deﬁned as:

2kT v = (3.14) th m r i where mi is the mass of a speciﬁc molecule. Like the Knudsen number, the speed ratio CHAPTER 3. DYNAMICS 54

also separates two types of flows, known as hyperthermal and hypothermal flows. The former is assumed for S 5, and describes the condition where the bulk velocity is much greater than that of the thermal velocity, where the particles of the gas act as a collimated beam. Hypothermal flow, assumed for S<5, indicates that the random Maxwellian velocities of the individual particles are still important, and hence must be considered in drag calculations. Generally the hypothermal condition becomes increasingly important for parallel or near-parallel surfaces to the flow, analogous to skin friction. The di↵erence between the two types of flows is illustrated in Figure 3.3.

Analytical solutions of free molecular ﬂow for simple shapes have had an extensive history [111, 112, 113, 114]. More recent methods have made use of increased computing power to implement numerical methods, one of the more popular being Direct Simulation Monte Carlo (DSMC) [115]. A good overview of the available methods can be found in [110]. The following sections will detail the classical treatment of free molecular ﬂow on a particular surface.

3.2.1 Surface-Flow Interactions

Drag force is the result of constant impingement of ﬂuid molecules upon the surface of the object in question. Thus we must start by analyzing these interactions, chieﬂy the transfer of momentum between the molecules and the object surface.

Momentum Flux

As an object moves through a fluid (or vice versa, i.e. the fluid flows around an object), the transfer of momentum between the fluid molecules and the object surface gives rise to drag. The momentum flux is thus defined as the momentum flow per unit area. To continue our analysis, we need to define vector quantities using a CHAPTER 3. DYNAMICS 55

Figure 3.3: The di↵erence between hyperthermal flow (top) and hypothermal flow (bottom). The fluid particles act as a collimated beam for hyperthermal flow, while random motions of the particles still contribute for hypothermal flow. The latter scenario becomes important when analyzing oblong shapes traveling parallel to the flow. Reproduced from [110] . CHAPTER 3. DYNAMICS 56

Cartesian coordinate system attached to a surface element, with the x–axis normal to the surface. If we let u =(u1,u2,u3)bethevelocityofanincomingmoleculeonto a surface element, and ê d be the unit vector in the desired force direction, then the element will experience an incident momentum flux along direction ê d:

+ + + dFincident 1 1 1 = miNi (ˆe d u)u1f(u, U)du (3.15) dA 0 · Z Z1 Z1 where f is the Maxwellian velocity distribution function deﬁned as:

3 m 2 m f(u,U)= i exp i (u U )2 +(u U )2 +(u U )2 (3.16) 2⇡kT 2kT 1 1 2 2 3 3 ⇣ ⌘ ⇣ ⇥ ⇤⌘ Here, U=(U1,U2,U3)indicatestherandomthermalvelocityofthemolecule.

Likewise the same concept can be applied to Eq. (3.15)forthereflectedflux,andso the total momentum flux over an area element in the ê d direction is simply:

dF dF dF total = incident reflected (3.17) dA dA dA

Note here that the reflected flux is defined positive in the same direction as the incident flux, and hence it will be negative since it acts in the opposite direction. Due to conservation laws, the incident number of molecules per unit time must also be equal to the reflected number of molecules per unit time.

Reﬂected Flux

Much of the uncertainty related to the modeling of drag comes from the nature of the molecules after they have been reﬂected from a surface. First, we deﬁne the accommodation coecient, ↵,as:

E E ↵ = i r (3.18) E E i w CHAPTER 3. DYNAMICS 57

where Ei is the incident energy flux, Er the reflected energy flux, and Ew the energy flux if all molecules were di↵usely reemitted at the same thermal temperature as the spacecraft surface. Thus ↵ can be thought of as a ratio of the transfer in energy between the actual scenario versus an idealized case. In the event that all molecules retain their energy after impact, ↵ =0,whereasiftheyacquirethesametempera- ture as the surface, ↵ =1.Sinceenergyisassociatedwiththetemperatureofthe molecules, we can define: T = T (1 ↵)+↵T (3.19) r i w where the subscripts i, r and w indicates the incident, reflected and surface temperatures respectively.

Equally important is the concept of di↵use versus specular reflections, as seen in Figure 3.4.Theformerdescribesthecasewhereparticlescanbereemittedinany particular direction, while the latter fixes the angle of incidence, ✓i,tobeequaltothe angle of reflection, ✓r (sometimes this is modified to specify the Knudsen cosine law distribution). The simplest form of its usage postulates that for every unit area of surface, a certain percentage of molecules are reflected di↵usely, while the remainder are reflected specularly (known as the Maxwellian model).

Figure 3.4: A particle approaches the surface at angle ✓i and reflects o↵at angle ✓r. For specular reflections, ✓i = ✓r,whiledi↵usereflectionsareunpredictableandexit at random angles. CHAPTER 3. DYNAMICS 58

The accommodation coecient can be further separated into two terms [111]:

pi pr N = (3.20) pi pw ⌧ ⌧ ⌧ ⌧ = i r = i r T ⌧ ⌧ ⌧ i w i where N is the normal momentum accommodation coecient and T is the tangential momentum accommodation coecient. Similar to Eq. (3.18), p is the momentum transfer in the normal direction and ⌧ is the momentum transfer in the tangential direction, while the subscripts i, r and w indicates the incident, reflected and surface accommodation values respectively. By definition of di↵use reflections, ⌧w =0since the speed distribution is symmetrical about the surface normal. A few other idealized cases include the following: N = T = 0 for no energy exchange, N = T =1for perfect accommodation to surface temperatures, and T =0fordi↵usereflections.

Drag Coecient

The drag coecient is the non-dimensionalized version of the total momentum ﬂux. Non-dimensional numbers are useful since they provide information regarding the physics of a problem in a general way. In the case of drag, knowing the drag coecient allows us to infer the amount of drag the object encounters without directly knowing its shape. Following Eq.(3.17), the calculation of the non-dimensional coecient C is quite straightforward:

1 dC 1 dF = ⇢ v2A total (3.21) dA 2 a ref dA ✓ ◆

We aggregate all the molecules within the gas into ⇢a to obtain mean mass density. Also note that the integration performed in Eq. (3.15) is not trivial and depends upon how molecules are reﬂected o↵of the surface. Generally, when ˆe d is in the direction of the velocity vector, we get the drag coecient CD,andwhenitisalong the direction perpendicular to that of the velocity vector, we get the axial coecient CHAPTER 3. DYNAMICS 59

CA.

Figure 3.5: Flat plate with incident particle approaching at angle ✓i.

Referring to Figure 3.5,ifwelet✓i be the angle of incidence at which particles strike the surface of a ﬂat plate, the drag coecient becomes:

A 1 2 CD = (2 N )cos✓i cos ✓i (1 + erf()) + e (3.22) A Sp⇡ ref " ✓ ◆

2 N N Tr p⇡ 1 S2 + (1 + erf()) + (1 + erf(S)) + e 2S2 2 T S S2 r i ✓ ◆#

where Aref is the reference area, = S cos ✓i,anderfistheerrorfunctiondeﬁnedas:

x 2 y2 erf(x)= e dy (3.23) p⇡ Z0 Note here that Eq. 3.22 depends on a variety of parameters such as the momentum accommodation coecients, the speed ratio, as well as the surface and incident molecular temperatures.

Figure 3.6 shows C and C at various angles of attack (90 ✓ )underconditions D A i where S =7and Tr =0.425. When the plate is perpendicular to the ﬂow, we see Ti q CHAPTER 3. DYNAMICS 60

Figure 3.6: CD and CA evaluated at various angles of attack for a ﬂat plate. Speed ratio S =7andtemperatureratio Tr =0.425. Reproduced from [113, 116]. Ti q the often quoted value of CD =2.2amongstsatelliteoperatorsandengineersappear.

3.2.2 Uncertainty in CD

The atmosphere through which objects ﬂy through is composed of many constituents, and its exact makeup is uncertain at any point in space and time. Attempting to solve for the mean atmospheric mass density directly translates into attempting to solve for the mean molecular mass. Unfortunately, this does not enable us to ascertain the exact composition and percentages of each species present. This a↵ects calculations CHAPTER 3. DYNAMICS 61

of CD since we are unable to account for how the spacecraft surface might interact with certain atmospheric constituents (particularly O, which determines specular vs. di↵use reﬂections).

The atmospheric and spacecraft surface temperatures also a↵ect the calculation of the speed ratio S and of CD, but are generally not known. Higher temperatures and lighter atmospheric species such as hydrogen or helium both lead to increased force coecients.

However, more important is the choice of the accommodation coecient. It is dependent on a multitude of factors, such as the atmospheric composition, temperature, and surface material of the object [117, 118]. Recent contributions have seen a possible relationship between Langmuir isotherms and accommodation coecients as well [119, 120]. Because the Sun’s behavior directly inﬂuences many of these parameters, there is a tendency to separately evaluate periods of high vs. low solar activity.

The surfaces of objects in the lower thermosphere usually become coated with the abundant adsorbed atomic oxygen [118], which causes reﬂecting molecules to be highly accommodated (↵ = 1) and di↵use in nature. This is due to the fact that incoming molecules tend to bounce about or stick to the uneven contaminated surface, and hence lose a signiﬁcant portion of their kinetic energy and eventually escape in random directions. During periods of high solar activity, the region where ↵ =1caneasily extend up to 500 km [121, 122]. If we extend further up in altitude or analyze periods of low solar activity, ↵ tends to decrease as there are lower quantities of atomic oxygen, and hence the di↵usely reemitted particle assumption tends to break down.

Figure 3.7 shows the variation of CD with respect to ↵, S and Tw.Byfarthe most sensitive parameter is the accommodation coecient, although the speed ratio does a↵ect the drag coecient slightly. Speciﬁcally for satellites, their velocities and atmospheric temperatures can be considered to be fairly constant over a short period of time, while their surface temperatures does very little in a↵ecting CD. CHAPTER 3. DYNAMICS 62

(a) Variation of CD vs ↵ using S =7.5236, Ti = 1000 K, Tw = 300 K

(b) Variation of CD vs S using ↵ =0.95, Ti = 1000 K, Tw = 300 K

Figure 3.7: Variation of CD versus ↵, S,andTw for a ﬂat plate. Note that changes in ↵ have the greatest e↵ect on the drag coecient CD. CHAPTER 3. DYNAMICS 63

3.2.3 Numerical Methods

With recent increases in computing power, numerical methods have been applied to free molecular ﬂows with great success. These are generally used for analyzing the aerodynamics of complex body shapes.

The simplest method, known as panel methods, divides the body into a number of discrete panels that are each modeled as a flat plate with one side exposed to the flow. By combining and summing up the contributions of each individual panel, the total body force coecient can be computed. One can increase the complexity to include other types of basic shapes as well, such as cylinders or spheres, where necessary. This method is only applicable for convex shapes, since regions of concavity shielded from the flow cannot be modeled realistically.

Further increasing the complexity results in Ray-Tracing Panel (RTP) methods, in which shadowing e↵ects from other parts of the spacecraft a↵ect the flow upon certain surfaces. By utilizing ray-tracing techniques, the shadowed panels can be identified and removed from the overall calculation. This tends to be valid only under hyperthermal conditions, as random thermal motions within hypothermal flows are not considered.

Finally the most complex methods are Monte Carlo in nature, with the DSMC method directly simulating molecules [115]. The molecules are allowed to collide with one another and with the surface, which is typically represented as a mesh. Macroscopic quantities such as temperature and pressure can be calculated at each time step and averaged. This method has shown promise in complex conditions such as transition regimes between free molecular and continuum ﬂows, such as during spacecraft reentry [123]. However, with increasing number of particles within the simulation, the computational cost becomes prohibitively expensive. CHAPTER 3. DYNAMICS 64

3.3 Satellites

The many di↵erent forces exerted upon satellites slowly alter their orbits over time. By analyzing their motion, we are able to estimate these forces, given appropriate dynamical models. This section will focus on the change of orbital characteristics and parameters of satellites in LEO as they predominantly undergo drag.

3.3.1 Orbital Dynamics

Orbital Elements

To define any specific orbit requires the use of 6 independent parameters, such as 3D position and velocity, or the Keplerian orbital elements. These elements arise from the two–body solution as defined by Eq. (3.2), which forms the basis of all orbital motion. In its simplest definition, the motion of objects about a larger body follows the equations for conic sections (the elliptical case is shown in Figure 3.8), as described by: p r = s (3.24) 1+e cos ⌫ · where ps is the semilatus rectum, e the eccentricity, and ⌫ the true anomaly (from perigee). Not shown in Eq.(3.24) is the semi-major parameter a,whichisthedistance from the center of the ellipse to its furthest point (in the case of an elliptical orbit).

It can also be related to ps as: p = a(1 e2)(3.25) s If we transform everything to the 3D case as seen in Figure 3.9,thenweintroduce incl,theinclination(weuseincl here instead of i as the latter will be used frequently for indexing), ⌦the longitude of the ascending node (also known as the Rising Ascen- sion of Ascending Node (RAAN)), and ! the argument of perigee. The 6 Keplerian elements are thus [a e incl ⌦ !⌫]. CHAPTER 3. DYNAMICS 65

Figure 3.8: A 2D elliptical orbit as deﬁned by the semi-major parameter a,eccentricity e,semilatusrectumps,distancefromcentralbodyr,andtrueanomaly⌫.

Another important relation is the period of the orbit, which is the time required for an orbiting object to make one revolution about the central body. However, we will deﬁne another parameter known as the mean motion, n0,whichwillbeusedmuch more frequently: µ n = E (3.26) 0 a3 r where µE is Earth’s gravitational parameter, deﬁned as:

14 3 2 µ = GM 3.986 10 m s (3.27) E Earth ⇡ ⇥

Detailed derivation of Kepler’s laws can be found in many astrodynamics textbooks [124, 1].

Mean Elements and the Osculating Orbit

Because of perturbational e↵ects, no satellite orbit actually follows a perfectly elliptical orbit. As seen from Figure 3.10,orbitalelementstendtoﬂuctuateoveran extended period of time. These cyclical variations are divided according to the speed CHAPTER 3. DYNAMICS 66

Figure 3.9: A 3D orbit as deﬁned by Keplerian elements: semi-major parameter a, RAAN ⌦, inclination incl, argument of perigee !,andtrue anomaly ⌫. To view the orbit in the orbital plane, refer to Figure 3.8. Reproduced from http://what-when-how.com/space-science-and-technology/ precision-orbit-determination-for-earth-observation-systems/ of their variations. The fastest variations are termed short-periodic e↵ects, and typically are on the order of a satellite’s period or less, such as the diurnal cycle. On a longer time scale are long-periodic e↵ects, which take place over days or weeks, such as from higher order gravitational harmonics or coupling. Finally, not included in periodic e↵ects are secular changes, which is the steady increase or decrease of an orbital parameter.

Here we deﬁne the concept of osculating elements,whicharesimplytheorbitalele- ments describing an orbit at a speciﬁc point in time. This is related to the osculating orbit,whichdescribesanorbitasifitweresimplyatwo-bodyproblemwithoutper- turbations. Hence the osculating orbit changes at every point in time, yet it is tangent to the actual orbit at those points (the term comes from the Latin word meaning “to CHAPTER 3. DYNAMICS 67

Figure 3.10: E↵ects of perturbations on orbital elements. The short-periodic, long- periodic and secular variations due to perturbations, as well as their combined e↵ects, are shown. Reproduced from [1] kiss”).

The mean elements are the averaged values of the osculating elements over a time period. They vary smoothly since they do not exhibit the short-period behavior, and may or may not follow the long-periodic perturbations. Note that the averaging process is performed over an arbitrary time frame relevant to the speciﬁc orbit, but will generally give information regarding a satellite’s long term behavior. Averaging over a few orbits masks out the short-periodic features, while over weeks masks out the long-periodic e↵ects. The frequently utilized Kozai or Brouwer means are examples of orbit-averaged values that take into account higher order gravitational terms and asimpliﬁeddragmodel[125, 126].

Inﬂuence of Drag

There are many methods that model the inﬂuence of drag on an orbit. Using elementary principles of energy and angular momentum, Burns arrived at the classical perturbation equations [127]. Vallado showed these derivations through variation of parameters for both Lagrangian and Gaussian cases (conservative and nonconservative, respectively) [1]. The main points can be summarized as follows: CHAPTER 3. DYNAMICS 68

Secular changes occur in a, e,andincl,witha and e decreasing over time •

Periodic changes occur in all orbital elements •

Drag tends to ﬁrst circularize an orbit, then cause it to shrink •

Perigee height remains relatively constant •

The diurnal bulge of the atmosphere also gives rise to the advance of perigee ! • for circular and low eccentricity orbits [128, 129]

Table 3.1 shows the relative contributions to changes in key orbital parameters (a, e and n0) from all notable forces. Note that for satellites in LEO, the e↵ects due to drag lowered the semi-major parameter a by 5 km whereas e↵ects from other forces were negligible in comparison (less than 0.1 km change).

rad Forces a [km] e n0 [ s ] 4 3 Fg 6776.849 1.9007 10 1.1317 10 ⇥ 4 ⇥ 3 Fg & FD 6771.227 1.8738 10 1.1331 10 ⇥ 4 ⇥ 3 Fg & Fsrp 6776.851 1.9375 10 1.1317 10 ⇥ 4 ⇥ 3 Fg, FD & Fsrp 6771.229 1.9082 10 1.1331 10 ⇥ 4 ⇥ 3 F & F 6776.847 1.8718 10 1.1317 10 g 3body ⇥ ⇥ Table 3.1: Comparison of orbital elements due to various forces for an equatorial 400 km circular orbit after 1 day, using only Fg as the baseline. An area–to–mass ratio of 0.2 m2/kg is adopted for drag and SRP. Note that the e↵ects due to drag dominates over all other forces at this altitude.

We are more concerned with the secular changes in an orbit due to drag of time scales on the order of a few orbits to a few days. To determine the e↵ect of drag on orbital parameters, there are two prevailing methods: general perturbations and special perturbations. General perturbational methods estimate Keplerian elements of the orbit from analytical equations that take into account a perturbed two-body solution and a simpliﬁed exponential atmospheric drag model. Alternatively, special perturbational methods seek to directly integrate Newton’s di↵erential equations CHAPTER 3. DYNAMICS 69

with all modeled forces. With this method, a state vector of the position and velocity of the satellite is required in order to ﬁt over a time span to correspond with ranging measurements, through nonlinear least squares or more advanced ﬁltering algorithms. Both algorithms are compatible with the proposed methodology outlined in this thesis.

3.3.2 K Parameter

Revisiting Eq. (3.10) for drag acceleration, we see that it is a function of B, ⇢a, W ,andv. While the velocity can be measured and the wind factor modeled, the two former terms are often inseparable. In other words, any observed change in the motion of the satellite can be the attributed to the neutral density and/or the ballistic factor, and either one can compensate for the other. Thus, we deﬁne K,animportant parameter used throughout the rest of this thesis, as:

K =¯⇢aB¯ (3.28) where⇢ ¯ is the mean neutral density and B¯ is the mean ballistic factor. An important fact here is that we are using orbit-averaged quantities, denoted by the bar over the variables, since we will be deriving K from observations integrated over time. Note that in literature, K is also known as B⇤,andistheaggregateofdrage↵ectsfrom both the atmosphere and from a satellite’s properties.

TLEs

The original derivation for neutral density estimates from TLE data as well as from special perturbations orbit determination was accomplished by Picone et al. [73]. A brief overview of the process will be outlined here, although the neutral density estimation technique presented in Chapter 4 can be utilized in conjunction with any CHAPTER 3. DYNAMICS 70

Figure 3.11: Representative two-line element set. Retrieved from http:// spaceflight.nasa.gov/ particular method.

In general, TLEs contain mean orbital elements at a given point in time (formally known as an epoch)estimatedusingtheSimpliﬁedGeneralPerturbations(SGP4) propagator [130]. A sample of the TLE format is shown in Figure 3.11.There is an abundant number of propagators in existence accounting for di↵erent levels of perturbation from Earth’s aspherical gravity, lunar/solar gravity, and nonconservative e↵ects. However, since TLEs are a standardized form that is regularly reported by JSpOC, we ﬁrst restrict our approach to use only this data format.

We begin with the change of the semi-major parameter over time t due to drag, which is given by [107]: da 2a2v = ~a D eˆv (3.29) dt µE · Substituting Eq. (3.4)intoEq.(3.29)yields:

da a2⇢ Bv3W = a (3.30) dt µE where W is the wind factor deﬁned in Eq. (3.9). CHAPTER 3. DYNAMICS 71

We can then write the full change of the semi-major parameter as:

da a2 ⇢ Bv3W da da da = M a + + + (3.31) dt µ dt dt dt G D G SRP where the ﬁrst term is the aforementioned drag term, and the following terms arise respectively from gravity, drag-gravity coupling and solar radiation pressure (SRP).

Notice that we have replaced in the ﬁrst term a with aM ,themeanvalueofthe semi-major parameter. This is possible since the short-periodic variation in a is attributable primarily to gravitational perturbations, and drag can be viewed as a secular change of the mean value. We will include only the drag term going forwards since 1) the SGP4 propagator inherently accounts for gravity perturbations and 2) the drag-gravity coupling and SRP terms will be small. This last assumption is usually justiﬁed for orbits with perigees in the thermosphere (< 600 km), but must be relaxed for higher orbits.

Referring back to the mean motion deﬁned in Eq. (3.26), we make an additional correction to arrive at the mean mean motion, nM [131]:

µ n = E (1 + )(3.32) M a3 M r M

Conveniently, this additional term M is inherently included in the mean motion reported from TLEs, which transforms the osculating mean motion into a mean value as afunctionofeccentricity,semi-majorparameter,inclinationandlowerordergravita- 3 tional perturbation terms of order O(10 ). We are more concerned with the change of mean motion: dnM 3 1 2 2 = (n ) 3 µ 3 ⇢ Bv3W (1 + ) 3 (3.33) dt 2 M E a M

Note that in Eq. (3.33) we have ignored SRP forces and assumed that M is small [1].

We integrate Eq. (3.33)betweentwotimeperiods,ti and tf (initial and ﬁnal times, CHAPTER 3. DYNAMICS 72

respectively) while neglecting the correction term:

t 3 2 f 1 n (t ) n (t )= µ 3 (n ) 3 ⇢ Bv3Wdt (3.34) M f M i 2 E M a Zti

It can be seen then that if we assume a mean density,⇢ ¯a,wecanremoveitfromthe integral, and arrive at: 2 2µ 3 (n (t ) n (t )) ⇢¯ = E M f M i (3.35) a tf 1 3 3 (nM ) 3 Bv Wdt Zti Similarly, if we assume a mean B factor, B¯,weget:

2 2µ 3 (n (t ) n (t )) ⇢¯ B¯ = E M f M i = K (3.36) a tf 1 3 3 (nM ) 3 v Wdt Zti We have abbreviated the right hand side of Eq. (3.36)assimplytheK we desire, calculated from the TLEs. It should be noted here that removing⇢ ¯ and B¯ from the

tf integral essentially assumes that ⇢Bdt =¯⇢aB¯.Thisisnotunreasonableifweare ti concerned with the mean valuesR in our estimations [132]. Note that there are other methods (e.g. analysis of the eccentricity or advancement of the argument of perigee) that can be employed to calculate K, but ultimately the equations still depend upon

⇢¯a and B¯.

Di↵erential Correction and Batch Least Squares

Di↵erential correction is a method to determine the orbit through a series of observations. It relies upon least squares, which itself is based upon a 2-norm minimization scheme that seeks to minimize the errors associated with the residual between the observed and expected values of the state. Because orbital motion is inherently nonlinear, we must linearize the equations of motion as a consequence. Thus the method becomes nonlinear least squares, in which we need to supply an initial guess that CHAPTER 3. DYNAMICS 73

approximates the initial orbit. The process can be found in many orbital textbooks [109, 1]. A brief overview will be provided herein.

We define our state as X,suchthatthenominalstateisX0 and the state at each ti is Xi (note that subscript i does not represent the initial state). We will let Xˆ be the predicted state, such that Xˆ i is the predicted state at ti.Thegeneralalgorithmfor state estimation is described in Algorithm 1.Afewotherdefinitionsarenecessary as well: b˜ is the residual calculated from the observations minus what would have been observed if we propagate Xˆ 0 accordingly, A is the partial derivative of the observations relative to the initial state X0, W is the weighting matrix, and RMS is the weighted root-mean-squared error. Often we make use of the definition:

1 Pˆ = ATWA (3.37) ⇣ ⌘ where Pˆ is the predicted covariance matrix.

The general idea is to iterate on the RMS until we converge (according to a preset convergence tolerance), and the direction that we move in for each correction step is determined by the partial derivative matrix A. Note that the propagation step can be as complicated as we desire, but if we introduce too many model uncertainties or if the model itself is erroneous, we may converge on an incorrect answer or diverge completely. If we do not converge, we simply update the initial state Xˆ 0 and repeat the process. As expected, if our initial state is too far from the actual state, then the nonlinearity of the problem does not guarantee that we will move in the correct direction during minimization. The solution does not guaranteed a global minimum, as the problem itself can be highly nonlinear.

The batch least squares method is an extension to least squares: we now take new incoming observations in conjunction with what we have already calculated to develop a better estimate for X0.ItreliesonBayesTheorem,andwemakethefollowing CHAPTER 3. DYNAMICS 74

Algorithm 1 Di↵erential Correction while RMS not converged do for #ofobservationsdo

Xˆ i =Propagate(X0) . Can be complicated function of X

b˜i =Obs-Obs(Xˆ i) . Calculate residuals Find A = @(obs) @Xˆ . Either through integration or ﬁnite di↵erencing @Xˆ @Xˆ 0 Accumulate ATWA and ATWb˜ end for 1 ˆx = ATWA ATWb˜ RMS = b˜TWb˜ Npoints if RMS Convergedq then . Check convergence of RMS break else . Update initial X

Xˆ 0 = Xˆ 0 + x end if end while adjustments to the earlier matrices:

A W 0 b˜ A = old , W = old , b˜ = old (3.38) ˜ "Anew# " 0 Wnew# "bnew# where the old subscript indicates values already calculated, and the new subscript represents new incoming data. Thus if we have already processed data from our previous observations, we do not have to reprocess everything again. Note that the estimates of the state and covariance are referenced to the initial epoch time.

At this point, we are able to use observations or pre-processed TLEs to form the variable K, which is the product of the mean neutral density and the mean ballistic factor. This is based upon the e↵ects of drag on a satellite orbit and how we expect it to change over time. Treatment of the separation between ⇢a and B will be described CHAPTER 3. DYNAMICS 75

in Chapter 4.

3.4 Meteoroids

This section will focus on the inﬂuence of the atmosphere upon incoming meteoroids. Analogous to the satellite case, we will estimate the K factor, under a di↵erent set of assumptions. Hence K will be a function of di↵erent variables compared to that of satellites.

3.4.1 Meteoroid Characteristics

Most meteoroids originate in interplanetary space and follow heliocentric orbits (barring e↵ects from major planets and moons), with less than 5% classiﬁed as interstellar [133]. Their velocity detected at the Earth in general lies between 11 km/s and 73 km/s. The lower limit corresponds to Earth’s escape velocity, while the upper limit is bound by the sum between the heliocentric escape velocity and Earth’s orbital velocity.

The composition of meteoroids also tend to vary depending upon the makeup of the meteoroid itself. Iron-rich meteoroids tend to possess densities ranging from 4 to 5.5 g/cm3, while stony meteoroids possess lower but still large densities at approximately 3g/cm3. However, icy cometary meteoroids are the most frequent, and thus densities close to 1 g/cm3 are fairly common. Within the lower limits are meteoroids composed of very porous carbonaceous materials, which correspond to densities as low as 0.1 g/cm3 [134, 135].

When a meteoroid ﬁrst enters the Earth’s atmosphere, it is subject to a number of physical processes as it collides with molecules within Earth’s atmosphere. Chief amongst these interactions is the meteoroid’s deceleration and its ablation, which CHAPTER 3. DYNAMICS 76

is deﬁned as the removal of the meteoroid mass via phase transformations as its temperature increases. The ablation process is a complicated phenomenon and has been studied and modeled extensively [136, 137, 138, 139, 140, 141]. It is dependent upon many factors such as the meteoroid’s composition, mass and velocity as well as atmospheric conditions. Often, meteoroids do not have sucient mass to reach Earth’s surface, with the least massive meteoroids vaporizing completely in the lower thermosphere. As a meteoroid ablates, the head echo plasma moves with the velocity of the meteoroid [142, 143, 144]; these are easily detected using HPLA radars.

3.4.2 Equations of Motion

When a meteoroid collides with air molecules, the conservation of linear momentum (neglecting external forces) can be described similarly to the satellite case by Eq. (3.4). In meteoroid literature, we often simplify Eq. (3.4)to:

dv A = ⇢ v2 (3.39) dt M a where we take the frame tangent to the trajectory to relieve the vector notation and substitute in from Eq. (3.6). Note that in this formulation we neglect gravity and the relative velocity of the atmosphere due to their negligible contributions, as well as higher order terms that result from the momentum equation. We also assume that the meteoroid travels linearly along its trajectory during the entire timespan of detection.

The ablation equation characterizes the mass loss of the meteoroid:

dm 1 CH 3 = ⇢aAv (3.40) dt 2 H⇤ where CH is the heat transfer coecient and H⇤ the heat of enthalpy (otherwise known as latent heat of vaporization, sublimation or fusion). Many subtleties are contained within Eqns. (3.39)and(3.40), a major one being that the variables CD, CH and H⇤ CHAPTER 3. DYNAMICS 77

are not truly constant over the meteoroid’s lifetime. Moreover, di↵erential ablation may contribute to di↵erent rates of sublimation across the meteoroid’s surface and alter its assumed spherical shape [145]. However for our purposes, we will take these variables as constant over short observed time spans of at most a few seconds, which is valid in a free molecular ﬂow regime [146, 141]. Table 3.2 shows the relevant variables within this thesis and the assumptions of each.

Variable Comments Solve for? V -Velocity Observed from head echoes 7 Taken from statistics to solve for M -Mass 7 actual densities Taken from statistics to solve for ⇢ -Meteoroidbulkdensity 7 m actual densities A -Frontalarea Related to radius 7 R -Radius Related to mass and bulk density 7 Often assumed = 2, combined into D, C -Dragcoecient ? D held constant Often assumed = 1, combined into D, C - Heat transfer coecient ? H held constant H⇤ - Heat of enthalpy Combined into D,heldconstant ? D - Ablation parameter D = 1 CH ,heldconstant 3 6 CDH⇤ ⇢a -Atmosphericdensity Main to-solve-for variable 3

Table 3.2: List of variables for meteoroid ablation and brief comments regarding each. Note that ‘?’ indicates that the variable can either take an assumed value (generally measured or calculated from physics), or can be treated as an unknown that we need not explicitly solve for. D,theablationparameterwhichisacombinationofCD, CH , and H⇤,canhoweverbesolvedfor.

By dividing (3.40)by(3.39), we can form the di↵erential equation:

1 dm C dv = H v (3.41) m dt CDH⇤ dt

Given the initial mass and velocity of the meteoroid at the start of detection, and assuming CH , CD and H⇤ as constant, we arrive at the following relation:

CH 2 2 2C H (v v0 ) m = m0e D ⇤ (3.42) CHAPTER 3. DYNAMICS 78

If we assume a spherical shape, Eq. (3.42)becomes:

CH 2 2 6C H (v v0 ) R = R0e D ⇤ (3.43) where R is the radius of the meteoroid. Since we assume isothermal heating for small meteoroids, the constant spherical shape factor assumption is valid [147].

The same spherical assumption can be applied to Equation (3.39):

1 dv 3 CD⇢a 2 = (3.44) v dt 8 ⇢mR where ⇢m is the bulk density of the meteoroid. Furthermore, we can relate velocity to altitude with the following identity:

dh = v sin (3.45) dt where h is the altitude and the approach angle at which meteoroid’s trajectory makes relative to the Earth’s surface.

Following Eq. (3.44), velocity and acceleration may be measured, while we need to estimate ⇢a,theneutraldensity.ThusanalogoustothesatellitedeﬁnitionofK,we deﬁne it for the meteoroid case as:

R⇢ K = m (3.46) ⇢a

We have neglected to include CD in the formulation since we have assumed all meteoroids to be spherical, and hence possess equivalent drag coecients. Compared to K from Eq. (3.36), we see that in the case for satellites the main source of variation was from the ballistic factor, while in the meteoroid case it is due to the radius combined with the composition. CHAPTER 3. DYNAMICS 79

3.5 Summary

Both satellites and meteoroids experience a multitude of forces as they traverse the atmosphere. In both cases, drag is the dominant force, altering the orbit of satellites and causing meteoroids to decelerate. On a molecular level, drag force is the result from the transfer of momentum between the fluid flow and the object traveling through it. Free molecular flow physics dominate in these particular regions where the atmosphere is fairly thin and the mean free path is significantly larger than that of the characteristic length of the object (Kn 10). By measuring the changes in the orbit of satellites and the deceleration of meteoroids, we can estimate the neutral density necessary for such changes. To accomplish this, we have defined the K parameter in Eqs. (3.36)and(3.46), which encompasses the e↵ects of drag as a function of the neutral density and some property of the object in question. Chapter 4

Methodology

The central methodology of the estimation of atmospheric neutral densities from multiple equivalent platforms will be presented in this chapter. Although the use of order and ratio statistics exists within current literature in other fields [148, 149], the combination and application of its theory to satellite constellations and meteoroids is a first. Section 4.1 will set the groundwork and theory from a probabilistic and order statistics perspective. The general format is to first remove the e↵ect of the atmosphere from our measurements, then estimate the variation of the platform, followed by calculating the neutral density. Specifically, there are two types of estimates that result: the mean and the ordered estimate. The mean estimate works well when there exists large errors in measurement, while the ordered estimate works well when there is considerable variation across the platforms. Section 4.2 and 4.3 will tailor the method for specific applications to satellites and meteoroids, respectively.

80 CHAPTER 4. METHODOLOGY 81

Figure 4.1: The methodology ﬂowchart. Given the data, we ﬁrst remove the bias due to background density, estimate the variation across the platforms, then calculate the density. In the case of error in the measurements, we must recurse to reestimate the variation.

4.1 Statistical Methodology

In this section, we detail the mathematical formulation for the estimation process. We ﬁrst motivate the need to use a stochastic approach, and present the assumptions necessary for the method to succeed. In broad terms, the approach can be thought of as a three stage process (shown in Figure 4.1):

1. Remove the bias in the data

2. Estimate the variation across multiple platforms

3. Calculate the neutral atmospheric density

In the case of error in the measurements, we must recurse on the variation estimate, and calculate the density until convergence.

4.1.1 Motivation

Before delving into the mathematics, we must first address what we are trying to achieve and our approach. Specifically, we turn to order statistics, defined as the branch of statistics dealing with a random sample that has been arranged usually in CHAPTER 4. METHODOLOGY 82

ascending order, to help us. The idea is that given enough sample sets, the ordered value per set (e.g. minimum, maximum, median) must asymptotically approach a certain value, usually derived from physics or historical data. Instead of calculating unique values for the parameters describing each platform, we can assign a probabilistic distribution to the variable parameters across all platforms. More detail regarding the mathematics is presented in Section 4.1.3.

Let us ﬁrst consider Eqs. (3.36)and(3.46), in particular the variable K.Itcanbe seen that K is the direct result of the atmospheric density, ⇢a,multipliedbysome property of the platform (B for satellites, R⇢m for meteoroids). This is what can be observed, but it is not actually possible to separate out the two contributing factors to K.Inotherwords,thevariablesthatmakeupK are not observable. To estimate them will require additional information.

Since we are unable to estimate the characteristics of each platform speciﬁcally, we represent it instead as a probabilistic distribution. If we were restricted to just one platform, then this probability would be dicult to estimate as well, since we would never be certain if the contribution to changes in K is associated with the background atmospheric density or with the platform characteristics. However, when we introduce increasing numbers of platforms, then we can observe not only changes across time, but also the variation across all present platforms. This is because at a certain point in time, all platforms within the system are expected to encounter the same neutral density background, especially if they travel through similar regions of space. However, over time, they would all be a↵ected by the temporal variations of density, but maintain their probabilistic distributions for platform characteristics.

Another advantage in approaching mean density estimation from a multiple platform perspective is less sensitivity to errors and extreme cases. For example, in the case for one satellite, any onboard instrument failure would be catastrophic. However, in aconstellation,suchaneventwouldnotendangerthemissionasawhole.Inour case, the estimation procedure can still continue despite certain platforms exhibit- ing anomalous characteristics. We can even devise statistical tests for determining CHAPTER 4. METHODOLOGY 83

the likelihood that a set of observations is erroneous by examining its probabilistic distribution.

4.1.2 Major Assumptions

The estimation procedure outlined in this chapter is not without its assumptions. To reiterate Section 1.3,thefourmainpointsthatwillbecoveredinclude:

1. Estimation of mean quantities

2. Equivalence amongst platforms

3. Regional similarity

4. Time scale similarity

We will show in this section under what conditions our treatment of the problem is valid.

Mean estimation

The keyword mean is used frequently in this thesis, and is closely related to the concept of regionality and time scales to be detailed later. Because we often integrate changes over time and over a traversed path, the parameters utilized within the process will contain the accumulation of all e↵ects encountered over that time period and path. Thus the ﬁnal estimated density is the mean density over the entirety of the path traversed. As a consequence, the estimate su↵ers over longer time periods, where an object may traverse highly di↵erent regions and undergo extensive variations (all of which are averaged into the ﬁnal product). CHAPTER 4. METHODOLOGY 84

Figure 4.2: The concept of equivalent platforms. Left: all platforms are identical, but the variation in orientation a↵ects individual measurements. Right: all platforms are spherical, but the variation in sizes a↵ect individual measurements.

Equivalence

The concept of equivalence refers to how similar the platforms are versus the main source of variation across platforms. An illustration is shown in Figure 4.2,wherewe see that all objects in each scenario are similar in a certain way, but a main source of variation drives the di↵erences across measurements. In the case of satellites, they might all be composed of the same materials and possess the same mass and area, but their orientation relative to the incoming stream of particles may di↵er from one another. In the case of meteoroids, their shapes may all be spherical, but their sizes and radii may vary.

The purpose is therefore to characterize these variations. In the case of multiple contributing variations, we will seek higher dimensional representations. We will assume for our purposes that the variations across platforms are independent and identically distributed (IID). That is, the variations experienced by every platform is distributed according to the same, albeit unknown, distribution [132]. This is valid if the platforms all originate from the same source and experience similar forces.

There are two ways that this assumption may be violated. First, if the incorrect source of variation is assumed, the mathematical basis (speciﬁcally the constant of integration, see Section 4.1.6)forthemethodologywouldbecomeerroneous.For CHAPTER 4. METHODOLOGY 85

example, if for meteoroids we assume that the variation arises mainly due to form factor di↵erences, then the contributions of varying meteoroid bulk densities and radii will be ignored. In addition, if the statistics of the variation are desired, then any derived value from the calculations will be attributed to the incorrect source. Thus it is important to identify and characterize the main source of variation.

The second type of violation is that of the IID assumption. This is more problematic, as the mathematics developed later in this section are only valid under this condition. As a result, the probability calculation itself will be ﬂawed if the IID assumption is violated, which would drastically a↵ect the estimated background density. Fortunately, it is not likely that one platform’s variation will be directly correlated with another’s. What is possible, however, is the existence of hidden variables contributing to correlated variations across a particular time frame (e.g. radiation spikes a↵ecting all attitude control systems, or sudden appearance of shower meteoroids vs. sporadics). We will assume that there are no hidden variables, or equivalently that 1) their e↵ects are small or 2) occur infrequently as to not overly a↵ect the statistics.

Regionality

Regionality deals with the issue of comparing bodies that travel through di↵erent regions of space. As can be expected, since density is a spatially varying phenomenon, K values will be biased according to what speciﬁc regions are traversed. We will assume in the simplest case that all platforms travel through similar regions of space, and hence encounter the same background density. Moreover, this assumption is strengthened if similar regions are traversed by each platform over a short period of time. Violation of this assumption would lead to biased results for atmospheric density, or an average of the di↵erent regions involved.

To remedy this issue, we can either introduce another parameter that characterizes the di↵erent regions (such as an exponential function across di↵ering altitudes) or to CHAPTER 4. METHODOLOGY 86

partition the regions accordingly. The ﬁrst method assumes a certain parameterization, with an additional parameter that needs to be estimated. The second method does not assume any particular characterization, but instead splits the problem into regions that each contain its own averaged density estimate. The tradeo↵is such that the ﬁrst solution can be applied when there is relatively limited data across multiple regions, while the second method requires many more measurements to be made.

Time Scales

The time scale of the problem refers to the period of time that we integrate over that corresponds to the measurements. The choice of integration time is mostly dependent upon how long we expect the e↵ects due to neutral density to become evident or measurable. This varies depending upon the method of observation. For orbit-derived density, the observed semi-major parameter takes over 1 day to change noticeably (depending upon altitude), while for GPS it might take as little as a few orbits (i.e. several hours). For radar-detected meteoroids, it is near-instantaneous, as velocity measurements are made over very short time spans (8.7 ms) as the meteoroid rapidly descends in altitude. Thus the integrated result for neutral density for the meteoroid case can be regarded as almost an instantaneous measure, while for satellites it is representative of the mean density encountered by the satellite over the entirety of a day or more.

However, a hidden consequence of time scales is that they are closely related to earlier assumptions of platform equivalence and regionality. Over longer periods of time, an object may undergo many variations or traverse many di↵erent regions, leading to the averaging of platform characteristics and neutral densities. Since neutral density is a temporally (and spatially) varying phenomenon, an increase in time scale means that we discard our ability to perceive shorter term trends. The ideal time scale over which to integrate is relative to the system dynamics involved, which varies from case to case. CHAPTER 4. METHODOLOGY 87

The definition of the time scale will ultimately a↵ect how we interpret the final estimate. Since we are estimating mean quantities, a longer time scale will entail that the final result is an averaged estimate over longer periods. In the case of shorter time scales, we can view the estimate to be more accurate, but on longer time scales, only general trends associated with the background density will be deducible. We will for our purposes within the thesis assume that the neutral density does not vary drastically across a certain time scale.

4.1.3 Order Statistics

Order statistics is the branch of statistics dealing with ordered values of a sample of observations. It is often used in areas where statistics of the extremes of a distribution are important, such as in instrumentation failure rates or medical/clinical trials [148,

149]. In mathematical terms, we let Xi for i =1, 2...N be a set of N real valued random variables. If we order the values of X such that X X ... X , i (1)  (2)   (N) th then X(i) is known as the i order statistic of Xi.Moreover,ifXi is IID on some cumulative distribution function (CDF) C(x), then the CDF of the rth order statistic can be expressed as:

N N i N i C (x)= C (x)[1 C(x)] (4.1) (r) i i=r X ✓ ◆ In other words, Eq. (4.1)expressesthelikelihoodofx given N independent draws. In particular, the probability of the minimum or 1st order statistic can be expressed as:

N 1 P (x)=NP(x)[1 C(x)] (4.2) (1) C (x)=1 [1 C(x)]N (1) where P is the probability density function (PDF). For a distribution possessing hard bounds on the tails, the extremes are more likely to be achieved with increasing N. CHAPTER 4. METHODOLOGY 88

Figure 4.3: The probability distribution of the minimum order statistic applied on the normal distribution for varying N.TheredcurvedescribesthePDFoftheminimum value in a set of ﬁve draws from the PDF described by the blue curve. Note that with increasing N, the distribution for the minimum becomes increasingly narrow.

Figure 4.3 shows this e↵ect on the normal distribution for N =5andN =10.

In our case of density estimation, we let x be the set of random variables that represent the main source of variability between platforms. For satellites, this distribution has ahardminimumboundduetothephysicsofthedragcoecientCD.Formeteoroids it is more ambiguous, but we can set the minimum bound as the smallest meteoroid aradarcandetect.ThelargeroursamplesizeN,themorelikelythataplatform exists that will be at its minimum (or rth order statistic) conﬁguration. All these characteristics of the problem lends itself well to the application of order statistics. CHAPTER 4. METHODOLOGY 89

4.1.4 Ratio Statistics

Referring to Figure 4.1, we see that the ﬁrst step is to the remove the bias. In our context, bias is the e↵ect of atmospheric density on a set of measurements within atimescale.BecauseK is the product of ⇢a with the property of the body, every observation made at a certain time will have contributions due to the immediate background density at that time. This bias must be removed if we are to compare measurements made at di↵erent times.

For every time period tk, we deﬁne the minimum K value amongst N platforms as:

Kmk(tk) = min Kj(tk)(4.3) j

We will, without loss of generality, continue to use x to represent the main source of variability between platforms. Recall that for this thesis, x = B¯ for satellites, and x = R¯⇢m for meteoroids. Note the mean quantities implied in both cases. Analogous to Eq. (4.3), we let:

xmk(tk) = min xj(tk)(4.4) j To remove the bias (assuming slowly varying densities over time, such that all bodies see the same⇢ ¯a), dividing K by Kmk equates to:

K (t ) x (t ) j k = j k (4.5) Kmk(tk) xmk(tk)

We proceed to amalgamate measurements across all tk to arrive at a constructed CDF x of . The result is a quotient distribution between variables x and xmk.Ingeneral, xmk x if one has two independent variables x and y,thequotientdistributionofz= y takes the following form: + 1 P(z)= y P (zy,y)dy (4.6) | | x,y Z1 where Px,y is the joint probability distribution of independent variables x and y. CHAPTER 4. METHODOLOGY 90

However, since variable zy is dependent on y,weexpandEq.(4.6)to:

+ 1 P(z)= y P (zy y)P (y)dy (4.7) | | x | y Z1 The conditional probability P (zy y)canbeexpressedas: x | 1 P (zy y)= H (x y) P (zy)(4.8) x | 1 C (y) x x where H is the Heaviside function. Since y is the minimum and is positive in nature, we can remove the Heaviside function and absolute symbol from our equations.

As well, we recognize that Py(y)issimplytheprobabilityoftheminimumoverN observations: N 1 P (y)=NP (y)(1 C (y)) (4.9) y x x This is exactly the probability of the 1st order statistic expressed in Eq. (4.2).

Eq. (4.7)thenbecomes:

+ 1 N 2 P (z)= NyP (zy) P (y)(1 C (y)) dy (4.10) z x x x Z1 AsubtlepointisthatEq.(4.7)removesallentrieswherez = 1. However, working with the CDF is much more preferable than the PDF, since it can be generated directly from data. Thus we must integrate Eq. (4.10), beginning with the following transformation:

xmax N 1 N @C (zy) d 1 (1 Cx(y)) P (z)= x dy (4.11) z N 1 @z · dy Zxmin ⇥ ⇤

We have also replaced the limits of integration to reﬂect physical cases of xmin and xmax. We then take the integral with respect to the dummy variable ⌧ (not to be confused with the tangential momentum accommodation coecient from Section CHAPTER 4. METHODOLOGY 91

3.2.1):

z Cz(z)= Pz(⌧)d⌧ Z1 z xmax N 1 N @C (⌧y) d 1 (1 Cx(y)) = x dyd⌧ (4.12) N 1 @⌧ · dy Z1 Zxmin ⇥ ⇤ Since only the first term inside the integral in Eq. (4.12) contains the variable ⌧,we can move the outer integral with respect to ⌧ inside and only apply it to the first term: z @C (⌧y) x d⌧ = C (zy) C (y)(4.13) @⌧ x x Z1 We make the definition for the complement of C here:

F (x)=1 C (x)(4.14) x x

Combining Eqs (4.12), (4.13)and(4.14)thenyields:

xmax N 1 N 1 N d 1 F (y) d 1 F (y) C (z)= F (zy) x + F (y) x dy (4.15) z N 1 x dy x dy Zxmin ⇥ ⇤ ⇥ ⇤ The ﬁrst term inside the integral of Eq. (4.15)canbefurthersimpliﬁed,suchthat:

N 1 N 1 d 1 Fx (y) d Fx (y) Fx(zy) dy = Fx(zy) dy (4.16) ⇥ dy ⇤ dy CHAPTER 4. METHODOLOGY 92

The second term can be simpliﬁed through integration by parts:

⇠⇠⇠: 0 xmax N 1 ⇠⇠⇠ xmax d 1 Fx (y) N⇠⇠1 F (y) dy = F (y) 1 F⇠ (y) x x ⇠⇠⇠ x xmin dy ⇠⇠ Z ⇥ ⇤ ⇠⇠ xmin n xmax o dFB¯ (y) N 1 1 Fx (y) dy dy Zxmin xmax 1 d FN (y) dF (y) = x x dy N dy dy xmin Z xmax 1 N = Fx (y) Fx(y) N x n o min 1 N 1 = +1= (4.17) N N

The derivation assumes that the limits of Fx approaches 1 at xmax and 0 at xmin. Finally, the result of integration on Eq. (4.10)isgivenas:

xmax N 1 N d F (y) C (z)= F (zy) x dy +1 (4.18) z N 1 x · dy Zxmin

The left side Cz is derived from the data we collect in the form of a CDF. Moreover, we can amalgamate data from any number of time sources as long as we can assure the IID assumption is upheld. This is because the division by the minimum order statistic yields an unbiased result independent of atmospheric density. It is also preferable to make an equal number of measurements across each time frame such they can be combined easily. This condition is easily met for satellites (since the same number of satellites are within a constellation at every point in time), but might be ambiguous in the case of meteoroids. The distribution Fx,whichisthestatisticalmeasureofthe variation across all platforms, is what we will solve for. CHAPTER 4. METHODOLOGY 93

4.1.5 Numerical Formulation

If we refer to Figure 4.1,wearenowreadytosolveforthevariationacrossplatforms.

To solve Eq. (4.18)forFx given the constructed Cz(z)fromdata,weturntoa numerical formulation of Eq. (4.18). Let us discretize the problem and deﬁne the points of evaluation as a geometric series:

i 1 let zi = gr i 1 yi = xmingr where i =1, 2,...,n (4.19)

and gr is the ratio of the geometric series, chosen such that the problem is tenable

(approximately 200 points). Usually n is determined at a point such that C(zn) u 0, although there is no consequence for letting n exceed this value. We then approximate Eq. (4.18)asasummation:

n N N 1 N 1 C (z ) 1= F (z y ) F (y ) F (y ) (4.20) z j N 1 x j i · x i+1 x i i=1 X ⇥ ⇤ Since we have assumed a geometric series for both y and z,wecanrewriteEq.(4.20) as: n N 1 N 1 N 1 (C(zj) 1) = Fx,i Fx,ij+2 Fx,ij+1 (4.21) N · i=j X ⇥ ⇤ Note that Eq. (4.21)takesthefunctionFx and simply evaluates it at discrete points interspersed at geometric intervals, labeled by indices i and j.Thisinturnisrelated to the cumulative distribution at points zj.CarefulmanipulationofEq.(4.21) CHAPTER 4. METHODOLOGY 94

transforms it into matrix form:

N 1 N 1 C(z ) F 0 0 F F n x,n ··· x,2 x,1 02 3 1 2 3 2 N 1 N 1 3 C(zn 1) Fx,n 1 Fx,n 0 Fx,3 Fx,2 N 1 ··· B6 . 7 1C = 6 . . . . 7 6 . 7 (4.22) N B6 . 7 C 6 . . .. . 7 6 . 7 B6 7 C 6 7 6 7 B6 7 C 6 7 6 N 1 N 1 7 B6 C(z ) 7 C 6 F F F 7 6F F 7 B6 2 7 C 6 x,2 x,3 x,n7 6 x,n x,n 17 B6 7 C 6 ··· 7 6 7 @4 5 A 4 5 4 5 The above Eq. (4.22)issolvedwithMatlab’sfmincon procedure, not as an absolute equality, but by minimizing the 2-norm error between the resulting vectors on opposite sides of the equality sign. Mathematically, it can be described as:

dC Minimize: min (LHS RHS)2 +  max x (4.23) ⇤ dz ✓X ✓ ◆◆ Subject to: 0 = F > F > > F =1 x,1 x,2 ··· x,n where LHS and RHS refer to the left hand side and right hand side of Eq. (4.22)

dCz respectively and  is the magnitude assigned to the tuning parameter dz ,which determines the smoothness of the solution. While solving for Fx,itisusefulto include the inequalities F > F > > F and the equalities F =1and x,1 x,2 ··· x,n x,1 Fx,n = 0 as constraints passed to the solver. This stems from the nature of Fx being related to the CDF. Eq. (4.22) is inherently nonlinear, so an initial guess must be provided; our tests have shown that a simple constant can still converge to the correct solution (shown in Figure 4.4).

Although we have a solution for the distribution of x,thereisanadditionalissuewe must address: solving for a function inside an integral inevitably leads to a loss of information, namely the constant of integration. In this matter, it is possible to shift and stretch the curve of Cx along the x axis to achieve a similar quotient distribution as measured. This is where a good knowledge of the location of the rth ratio statistic (we will term this as an anchor)isrequired,suchastheminimumormedian.Inthe case of satellites, this comes in the form of the minimum ballistic coecient, while in the case of meteoroids, we require the median of R⇢m.Boththesecaseswillbe CHAPTER 4. METHODOLOGY 95

(b) CDF Estimate of x (a) CDF Estimate of x xmin Figure 4.4: Estimates on a Gamma Distribution using the geometric series numerical method. The estimated CDF and consequent ratio distribution closely matches that of the actual, even though the initial guess was erroneous. covered in sections 4.2 and 4.3 respectively.

4.1.6 Errors and Uncertainty

Error plays a large role in any estimation scheme, arising from imprecise measurements, uncertainty in constants, and sensor noise. Thus quantifying and ﬁltering the error is essential to deriving the neutral density. There are mainly two methods of uncertainty to be addressed within our methodology: the errors in the estimation of K,anduncertaintyintheintegrationconstantoranchor.

Error in K

The ﬁrst type of uncertainty mainly arises from imprecise observations or measurements (e.g. orbital parameter changes, meteoroid velocities), a↵ecting the value of K. Hence the observed K can be expressed as the sum of K and K,whereK is CHAPTER 4. METHODOLOGY 96

the error associated with the estimation process. This directly has an e↵ect on x:

K + K = ⇢a (x + x)(4.24)

We construct the minimum ratio statistic with this error included, resulting in:

(x + x) C (z)=C (4.25) z (x + x) ✓ mk ◆ The e↵ect is shown in Figure 4.5,whereweseeadrasticchangeoftheratiodistri- bution from errors in K. However, if we follow the algorithm as outlined in Section 4.1.4,theresultwillbex + x rather than x.

Figure 4.5: The e↵ects of error in K on the ratio distribution. Shown are the theoretical, simulations from data, and 10% error in K ratio distributions.

We do not know the error characteristics on x,butwemayinferitfromK.This information may come from the variability in the elements used to calculate K itself, or from the least squares algorithm in the form of the covariance matrix. We do have to make an assumption that K behaves as a Gaussian (or at least according to a distribution in which we can model) independent of the background density. CHAPTER 4. METHODOLOGY 97

In actuality, this is not entirely the case, as K is dependent upon ⇢a,althoughit can be approximated as a Gaussian for small variabilities in ⇢a (within 1 order of magnitude). This variability does not manifest itself in Eq. (4.25), since ⇢a has been entirely removed.

We can thus estimate x as:

x N 0, K (4.26) ⇠ ⇢ ✓ a,est ◆ where N is the normal distribution with mean 0 and standard deviation K ,where ⇢a,est

K is the standard deviation of K and ⇢a,est is the estimated density. We expect x to follow this distribution regardless of time, and ideally K and ⇢a,est will need to be calculated for every individual time period. However, if the variation in ⇢a is not excessive, then we can simply use the mean quantities of K and ⇢a,est across all time frames. This relaxation is applicable especially within the ﬁrst iteration, when we are still unsure of the error in K and thus the estimate of ⇢a,est is still questionable. Upon convergence, we can switch to a time frame by time frame calculation such the distribution of x is more accurately portrayed.

In the context of error, the CDF that we solve for can be expressed as:

Cx+x =[Cx ? Px](4.27)

where the ? symbol represents convolution. Cx+x is what we will have solved for from Section 4.1.5,andifwehavetheerrorstatisticofx,thenwecandeconvolveit to attain Cx. It is then possible to calculate a new ⇢a,est based upon this Cx,feedit back into the formulation of x in Eq. (4.26), and proceed with the recursion process until we have converged upon a suitable x and ⇢a,est. This is the case when the new

⇢a,est is only within a few percent of the previous estimate. CHAPTER 4. METHODOLOGY 98

Uncertainty in Integration

The second type of uncertainty that we address is the uncertainty in the integration constant that must be used to anchor the CDF Cx.Thistypeofuncertaintycanbe caused by erroneous physics, incorrectly identifying the variations, or through faulty historic data. Although the speciﬁcs of calculating this integration constant will be unique for di↵erent applications (covered in Section 4.2 and 4.3), we will characterize its uncertainty with a PDF. Usually, this PDF will rely upon historical data for its formulation.

The probability of observing x in this context can be expressed as:

+ 1 P (x)= P (x x ) P(x )dx (4.28) x x | 0 0 0 Z1 where x0 is the integration constant previously mentioned. In other words, because at every time period we are uncertain whether a speciﬁc ith ratio statistic has been th achieved, we let P(x0)describethisuncertainty.Thisi ratio statistic can depend upon other parameters, such as the accommodation coecient.

Eq. (4.28) can be reformulated for the case of a CDF:

+ 1 C (x)= C (x x ) P(x )dx (4.29) x x | 0 0 0 Z1 AveryimportantnotehereisthatwhatwehavecalculatedpreviouslyfromSection 4.1.5 is actually C (x x ). In that case, we were uncertain as to what integration x | 0 constant to apply, so now in Eq. (4.29)wehaveappliedallpossibleintegration constants with a probability assigned to each. Moreover, applying it to the minimum CHAPTER 4. METHODOLOGY 99

order statistic (recall y = xmk)yields:

+ 1 P (y)= P (y x ) P(x )dx (4.30) y y | 0 0 0 1 Z + 1 C (y)= C (y x ) P(x )dx y y | 0 0 0 Z1 Usually in terms of limits, we do not take it from to + ,butratherfromthe 1 1 absolute minimum x can physically be, to where the probability of x0 approaches 0. In the limit that we have N platforms and N ,webecomeincreasinglycertain !1 that the minimum statistic can be achieved, and hence P P(x ). This ambiguity y ! 0 is the direct result of not precisely knowing the integration constant. It can also be seen that if we incorrectly identify the source of variation, or that if our underlying physics is faulty, the ﬁnal estimated density will possess a bias.

4.1.7 Calculating ⇢a

We are now ready to calculate the neutral density ⇢a from our dataset consisting of multiple but equivalent platforms. In reaching this point, we have removed the bias due to atmospheric density in K per observation period and accounted for the errors in measurements as well as physical uncertainties. We now possess the CDF for the variation of the platform in the form of Cx, or the variation of the order statistic

Cy.Fromthesefunctions,wecancalculatethemeanandvarianceoftheplatform variation. Additionally, there are two methods in which we may calculate the neutral density itself: the mean estimate and the ordered estimate. CHAPTER 4. METHODOLOGY 100

Mean Estimate

The mean estimate can be expressed as:

K¯k ⇢a,k = (4.31) xµ

Here, we calculate the density as K¯ ,themeanvalueofallK of all platforms at time tk,dividedbythemeanvalueofx calculated from Cx. This formulation is e↵ective if we perceive the error in K to be large, as averaging the Ksfromallplatformswill decrease the error by the central limit theorem. However, the tradeo↵is that the error in x is large in comparison, as we take the mean and standard deviation of x according to the full probability distribution Px.

Ordered Estimate

The ordered estimate can be expressed as:

K ⇢ = mk (4.32) a,k y

Here, we calculate density as the ordered statistic Kmk (in this particular example it is the 1st order statistic) divided by the associated order statistic y calculated from

Cy.Thisformulationise↵ectiveifweperceivetheerrorinK to be small, as we are certain of the measurements we make. However, we are quite uncertain regarding the variation of the platform, and hence taking the order statistic drastically decreases the size of the probabilistic distribution in that regard. The tradeo↵here is that although the error in y is small, large errors in K will produce erroneous estimates, as we have taken a particular K to work with rather than averaging them. CHAPTER 4. METHODOLOGY 101

Discussion

The type of estimate to use is dependent upon how well we know the error in our measurements. If we expect errors in K to be large in comparison to the variation of the platform, then it might be suitable to use the mean estimate. If we expect the errors in K to be small but expect large variations across the platforms, then it would be suitable to use the ordered estimate. In other words, we must choose which error to minimize depending upon the nature of the problem.

The ⇢a calculated either by the mean or ordered estimate can be fed back into the recursion process as outlined in Section 4.1.6.Inthisway,weareabletoarriveat better estimates of Px and in turn, density itself. The standard deviation in the ﬁnal estimate of ⇢ can be calculated from K and in the case of the mean estimate, a pN x or K and y in the case of the ordered estimate.

We must emphasize here that the entire estimation scheme is an o✏ine algorithm. That is, it takes a batch of data already collected to perform the analysis rather than sequentially updating an estimate with each new measurement. However, with increasing number of measurements across larger time spans, we can be increasingly certain of the Cz approaching the actual ratio distribution, and hence the calculation of Cx will also improve over time. Ultimately, the method is able to attenuate the e↵ect of large errors in K on the estimation of x,thevariationoftheplatform.

4.2 Satellites

This section will focus on the methodology as covered in Section 4.1 as applied to a constellation of satellites. This scenario lends itself well to our methodology since a constellation will often possess identical satellites and are usually launched together CHAPTER 4. METHODOLOGY 102

from one source, and thus will exhibit similar initial orbital characteristics. Further- more, satellites within a constellation are usually tasked with one particular goal, so they tend to share similar programming and operational states. CubeSat constellations designed for Earth observation in particular are relevant in our case, since they adhere to all our previous assumptions and are easily replaced in regions where measurements are scarce (400 km in altitude or lower). These satellites are usually constrained to be Earth pointing, but combined with large solar panels and low mass, small changes in attitude noticeably a↵ect their drag proﬁles.

We start our analysis for satellites by observing their changes in orbit. From Section 3.3,weshowedthattheK parameter can either be calculated from TLEs or from the batch least squares methods, and the time scale in which we perform these estimations of K relates to the mean density that we will calculate. We also stipulated that the satellites must at least traverse one orbit, such that we actually calculate the orbit averaged mean density. Although this algorithm can work on smaller spatial and temporal scales of less than one orbit, we would require very precise instrumentation that measures the e↵ects of drag within a very speciﬁc geographic location and to ensure that each satellite passes this region within a short time frame. Unfortunately, TLE data, although plentiful, does not possess this type of accuracy or precision.

We attribute the variation of the platform in the case of satellites entirely to its orientation. We assume all satellites to be of the same mass, form factor, and general make, including material, components, and even software. This arises from our speciﬁc assumption of equivalent platforms, and limiting the satellites we analyze to constellations manufactured by one particular entity. We deﬁne ✓ as the rotation angle of the satellite about its spin axis (shown in Figure 4.6)relativetoitsvelocity vector, and that it is IID across all satellites. We also assume that all satellites possess the same type of Attitude Determination and Control Systems (ADACS) onboard that only allows slight variations in ✓.Inreality,theremightbeslightvariationsin angles in other axes as well, but they contribute little to the overall change in frontal area. For comparison, a 5 angular change in orientation in ✓ causes the frontal area CHAPTER 4. METHODOLOGY 103

to increase by 50% for spacecraft with extended solar panels, but only 2% along other axes.

Figure 4.6: CubeSat and its orientation relative to its velocity vector, denoted by ✓. Amin and Amax represent the minimum and maximum values of the satellite frontal area, aligned with either the side of the spacecraft or the solar panel surface, respectively.

The ballistic factor B represents the variation of the platform, which was symbolized by x in Section 4.1. B is directly related to ✓ as described by:

C (✓)A(✓) B(✓)= D (4.33) m

Thus B is IID with some unknown distribution. We deﬁne Bmin as the condition when ✓ =0: C A B = D,min min (4.34) min m

Under this condition, CD and A both take on their minimum values. Recall that CD can be calculated from free molecular theory as covered in Section 3.2,whileA is known beforehand from satellite speciﬁcations. We let this minimum condition on

B represent the integration constant when solving for Cx in Section 4.1.6.Wethen CHAPTER 4. METHODOLOGY 104

rewrite Eq. (4.18)intoanappropriateformintermsofB:

Bmax N 1 B N B d F (Bmk) C = F B B dB +1 (4.35) B N 1 B B · mk · dB mk ✓ mk ◆ ZBmin ✓ mk ◆ mk

Figure 4.7: E↵ect of ↵ on CD,giventhatCD is distributed according to a half normal distribution and ↵ N(2.2, 0.1) ⇠

However, recall that there is still an uncertainty regarding the values that Bmin can take. This can be traced directly to the accommodation coecient ↵ (see Section 3.2.1), which governs the characteristics of particles reﬂecting o↵of the satellite’s surface. We need to set the probability of ↵ from prior research [150, 121, 122, 120]undervaryingsolarconditions.Therearefewsourcesthatgivetheprobability density of ↵,althoughonecandrawfromadistributiongivenenoughhistoricaldata. For example, ↵ tends to be larger during solar maximums at equivalent altitudes compared to solar minimums due to atmospheric expansion. The probability of seeing CHAPTER 4. METHODOLOGY 105

a particular ballistic parameter with a certain ↵ is then:

P(B,↵)=P(B B )P(B ↵)P(↵)(4.36) | min min|

Because Bmin is a function of ↵,wecanreduceEq.(4.36)totheformofEqs.(4.29) and (4.30):

Bmax C (B)= C (B B ) P(B )dB (4.37) B B | min min min ZBmin,↵=1 Bmax C (B )= C (B B ) P(B )dB Bmk mk Bmk mk| min min min ZBmin,↵=1

The e↵ect of uncertainty in ↵ on CD is illustrated in Figure 4.7.Itcanbeseenthat this uncertainty essentially stretches the original distribution. Note that this type of error is di↵erent from that existing in K;whereaswemustaccountfortheerrorinK to solve for CB,wemustaddinerrordueto↵ to CB to achieve the ﬁnal distribution in B.ThisisultimatelythedistributionutilizedinSection4.1.7 to estimate ⇢a.

4.3 Meteoroids

Analogous to Section 4.2, this section will apply the methodology to meteoroids. However, before we do so, we must address the issue of meteoroids traversing multiple altitudes during their descent into the atmosphere. It is not appropriate to use the mean approximation here, as the densities vary by orders of magnitude over the range of altitudes, and the radius of the meteoroids decreases rapidly as the entire body ablates. Thus we will adopt the successive layer approximation before we apply the methodology of estimating Cx. CHAPTER 4. METHODOLOGY 106

4.3.1 Successive Layer Approximation

Since meteoroids traverse through multiple regions where the neutral density varies by orders of magnitude, applying the previous methodology directly would not be advisable. Instead, we use the successive layer approximation to specify constant regions of density per speciﬁc altitude, and thus estimating it for each layer. Given enough data points, this method is advantageous over a parameterized formulation of the atmospheric density since we do not assume a priori a density proﬁle.

We begin with the equations of motion for meteoroids developed in Section 3.4.2.If we take the logarithm of the ratio of Eq. (3.44)betweentwodistinctpointsintime

(t1 and t2)foracertainmeteoroid,weget:

1 dv 1 dv ⇢ ln 2 ln 1 = D v2 v2 +ln a2 (4.38) v2 dt v2 dt 1 2 ⇢ ✓ 2 ◆ ✓ 1 ◆ ✓ a1 ◆ 1 CH where we abbreviate D = .Thespeedsforthemeteoroidattimest1 and 6 CDH⇤ t2 are represented as v1 and v2,respectively.Similarly,⇢a1 and ⇢a2 represent the neutral densities encountered by the meteoroid at times t1 and t2.Eq.(4.38)relates the observations of the velocity and deceleration of the meteoroid with the unknown constants as well as a density ratio parameter. Again, we have taken an approach where the atmosphere is divided into successive layers, with each layer approximated to have a constant neutral density, with no horizontal density variation.

In general, we can write Eq. (4.38)fortheith meteoroid at the jth altitude as:

L = D v2 v2 +ln(⇢ ) (4.39) i,j i i,j i,j+1 rj where 1 dv 1 dv L =ln i,j+1 ln i,j i,j v2 dt v2 dt ✓ i,j+1 ◆ ✓ i,j ◆ ⇢a,j+1 ⇢rj = ⇢a,j CHAPTER 4. METHODOLOGY 107

Here we let Li,j represent the left hand side of Eq. (4.38), while Di is the unknown but constant ablation parameter per meteoroid. Eq. (4.39)takessuccessiveratiosbetween the jth layer of the atmosphere and the next (j +1)th layer, but this formulation can easily be altered such that the ratio is between one layer of the atmosphere and aparticularreferencelayer.Theadvantageoftheformerformulationisthatwe can incorporate more measurements within our minimization scheme, since not all meteoroids are observed throughout all layers, while the latter formulation requires all the analyzed meteoroids to pass through at least the reference layer. However, with the latter method the results are more consistent since we ensure that there is an equal number of meteoroid measurements across all altitudes and avoid areas where data might be lacking.

We focus on the successive ratio formulation to leverage the large amounts of data we possess across multiple altitudes. Eq. (4.39)canbemanipulatedintomatrixform for the ith meteoroid:

ln(⇢ ) 2 2 r1 Fi,1 Ii,j=1 0 0 vi,1 vi,2 ··· 2 ln(⇢r2) 3 2 F 3 2 0 I 0 v2 v2 3 i,2 i,j=2 i,2 i,3 . . = . . ···. . . 6 . 7 (4.40) 6 . 7 6 ...... 7 6 7 6 7 6 7 6ln(⇢ )7 6 7 6 2 2 7 6 rm 7 6Fi,m7 6 00 Ii,j=m vi,m vi,m+17 6 7 6 7 6 ··· 7 6 Di 7 4 5 4 5 6 7 4 5 Here we use I as the indicator function, such that Ii,j=k =1ifthereisdataforthe th th i meteoroid at the k altitude, and Ii,j=k = 0 otherwise. Note that by this logic, there are instances where Li,k =0andvi,k =0ifthereisnodataavailableforthe ith meteoroid at the kth altitude. In these cases, rows consisting entirely of zeros can be neglected. In essence, we split the atmosphere into m +1layersandmake observations from a total of N meteoroids. Ideally if all meteoroids travel through all m +1altitudes,thenwewouldestimatem + N parameters from m N equations. ⇥ CHAPTER 4. METHODOLOGY 108

We estimate the unknowns ⇢rj and Di by solving:

2 2 minimize Li,j Di(vi,j vi,j+1) ⇢rj (4.41) i,j P subject to Di > 0

Note that Eq. (4.41)isconvexandhencepossessesaglobalminimum.Wechoose to minimize the 1-norm over the 2-norm to reject outliers (meteoroids that do not conform to our ablation model, such as fragmentation events) and impose the positive restriction Di > 0 such that the results are physical. It is interesting to note here that although what we seek to solve for are the density ratios ⇢rj,aconsequenceof Eq. (4.41) is that we can solve for each meteoroid’s ablation constant D.Thiscan provide insight into a meteoroid’s characteristics, such as its composition or how it interacts with the atmosphere.

4.3.2 Calculating Density

If we take the ratio of Eq. (3.44) again between two arbitrary meteoroids (1 and 2) at the start of their ablation processes, we get:

dv2 1 dt v2 R ⇢ ⇢ 2 = 1 m1 a2 (4.42) dv1 1 2 R2⇢m2⇢a1 dt v1

Since the meteoroids might initially be detected at di↵erent altitudes, the density ratio ⇢a2 is appended to the end of the equation. However, having solved for the ⇢a1 density ratios in the previous section, we can compare all densities to a reference density, ⇢a,ref : dv2 1 ⇢a,ref dt v2 ⇢ R ⇢ K 2 a2 = 1 m1 = 1 (4.43) dv1 1 ⇢a,ref 2 R2⇢m2 K2 dt v1 ⇢a1 Note that we can calculate K from the observed velocity, deceleration and atmospheric density ratios. This in turn relates directly to the radius and bulk density of the CHAPTER 4. METHODOLOGY 109

meteoroid. Because errors in the measurements directly translate into erroneous ratios across K,thisimpactsonthesupposeddistributionofR⇢m. As covered in Section 4.1.6,thise↵ectcanbeaccountedforifweknowtheerrorassociatedwith the measurements and if it is assumed to be Gaussian.

Much of the methodology as covered in Section 4.1 can be applied now on K,if we deﬁne K2 to be the minimum statistic Kmk. However, the major IID assumption must still hold, such that all meteoroids must all be distributed according to the same distribution across all time periods. This assumption is valid if we make measurements across smaller time scales such as an hour (with relatively stable solar and geomagnetic conditions), or across similar local times spanning over multiple days if we expect the same sporadic background of meteoroids to be present. Care must be taken while combining measurements to ensure that the IID assumption is maintained.

We again solve for the CDF Cx,nowsuchthatx = R⇢m.Inthecaseoftheunknown integration constant, we choose the median statistic to anchor our predictions for

R⇢m. The reason for this choice is because there are no hard bounds for the minimum or maximum statistic, as meteoroids can be arbitrarily large or small. Also, the characteristics of the radar highly inﬂuence the detectable meteoroid mass thresholds. The median does not vary greatly from sample to sample, and hence inferring its mean and variance from historical data is preferable. Thus we use the probability of the median in conjunction with Eqs. (4.29)and(4.30)toarriveatourﬁnalestimatefor

CR⇢m .

We proceed to calculate the neutral density at the reference altitude:

8 (R⇢m) dv ⇢a,ref ⇢a,ref = 2 (4.44) 3 CDv dt ⇢a

Note that we have all the elements necessary to calculate ⇢a,ref ,withCD being inferred from free molecular ﬂow theory. The densities at other altitudes can simply be calculated by multiplying Eq. (4.44) with the density ratios derived in Section 4.3.1.

Since we also possess the variational characteristics of (R⇢m) from its CDF, we can CHAPTER 4. METHODOLOGY 110

calculate the standard deviation relevant to the estimated ⇢a,ref .Ifweincreasethe thickness of the atmospheric layers, the assumption of a constant density per layer becomes increasing invalid, to the point that each layer di↵ers from the previous by large margins. Conversely, thinning the layers is limited by the velocity data available to us, and thus we introduce additional error from needing to interpolate an increasing number of points. Additional errors in velocity and more importantly, deceleration also propagate into the calculation of neutral density.

There are a few embedded assumptions within the neutral density estimation process as applied to meteoroids, which we will now address. Besides the IID assumption, we have assumed that the ablation process starts at the ﬁrst detection point of the meteoroid. The statistics of the meteoroids’ sizes and bulk densities pertains to their exospheric characteristics, but a meteoroid might have begun ablating earlier only to cross the radar beam afterwards. This can be accounted for by selecting meteoroids where the deceleration begins at approximately zero with a low RCS to indicate relatively little to no starting ablation. Care must be taken to accurately remove the beam pattern from the RCS data, since the RCS or SNR variation with range/time is primarily due to the particle traversing the radar beam (see Section 5.2).

Another assumption is the constant properties (CH , H⇤ and CD) of the ablating meteoroid, which holds true for short duration observations over speciﬁc portions of the ablating meteoroid’s lifetime (most head echo plasmas were observed for < 1sec). Although this implies that not all meteoroids traverse all layers of the atmosphere, recall that this is unnecessary in estimating the densities, since all we require is that the cumulative number of all meteoroids traverses all the appropriate altitudes.

Finally, erroneous or noisy measurements of the head plasma’s velocity and deceleration, which we assume to be traveling with the meteoroid itself, can lead to incorrect estimates of the neutral densities and its ratios. This inconsistency can be checked if we are able to divide the data gathered over a particular time frame into separate bins, and evaluate each bin independently. The normalized error of Eq. (4.41)canbe calculated to determine how well the estimated densities match observational data. CHAPTER 4. METHODOLOGY 111

4.4 Summary

This chapter has introduced the methodology of estimating the mean neutral atmospheric density through multiple equivalent platforms. Key assumptions made are:

1. All platforms possess similarities but quantiﬁable di↵erences that are IID according to some unknown distribution

2. All platforms pass through similar regions of space

3. All platforms are observed over similar time scales

The mathematics of the estimation scheme are based upon on the ratio distribution of the minimum order statistic. The distribution of the variation is calculated numerically, given knowledge of an integration constant (referred to as an anchor), which is estimated either through physics or historical data. The neutral density estimate can either be an ordered estimate, which fares well in the case of large errors in the variation of the platform, or a mean estimate, which performs better in the case of noisy measurements.

The methodology was adapted to the cases of both satellites and meteoroids. The former saw the main source of variation as the attitudes of the satellites, while for the latter the variation arose from the physical properties of various meteoroids (size and bulk density). The mean neutral density and its associated standard deviation can thus be accurately estimated across a speciﬁc orbit, or across multiple layers of the atmosphere through the successive layer approximation. Chapter 5

Application and Results

This chapter will utilize the methodology developed in Chapter 4 to determine the mean neutral density using simulated and real data sources. Section 5.1 will detail a space environment where we analyze a constellation of satellites within one particular orbit plane. Section 5.2 will use radar data of ablating meteoroids to determine the neutral densities within the lower thermosphere. Satellite simulation results show that the density can be estimated to within less than 5% of its actual value, while standard deviations for on-orbit and radar data estimates are approximately 10% and 16%, respectively.

5.1 Satellites

The method of neutral density estimation is applied to a constellation of satellites.

Section 5.1.1 will detail a simulation environment with the drag coecient CD de- ﬁned according to a distribution, while Section 5.1.2 will look at orbital data from a CubeSat constellation.

112 CHAPTER 5. APPLICATION AND RESULTS 113

5.1.1 Simulation

The simulation environment was implemented with AGI’s Systems ToolKit (STK) in conjunction with Mathworks MATLAB (See Figure 5.2). 10 reference satellites were created, each with a mass of 4.25 kg and a frontal area of 0.04 m2,andwere spaced evenly along-track in a speciﬁc orbit and propagated over time. The reference atmosphere used is Jacchia-Roberts [97], and TLEs are generated from the orbits over each simulated day. The initial orbital parameters are: semi-major parameter a =

6778.137 km, eccentricity e =0,inclinationincl =28.5, and RAAN ⌦= 0.A4 4 ⇥ gravity model was utilized such that we capture the large spherical harmonic e↵ects, alongside perturbational SRP and lunar gravitational forces. The total simulation period was 50 days.

(a) PDF of simulated CD (b) CDF of simulated CD

Figure 5.1: Probability distributions for drag coecient CD used in STK simulations. Ahalf–normaldistributionwasusedtorepresentalargeprobabilityofthesatellite in its nominal (minimum–drag) orientation. Slight deviations in its attitude from nominal increases the value of CD.ThelimitingcaseforCD =2.2comesfromfree molecular ﬂow theory for a ﬂat plate.

The CD value for every satellite was allowed to vary randomly according to a half– normal distribution, but was held constant for the duration of the ﬁt span (in this case, 1 day). In reality, this would be analogous to C¯D,whichrepresentsthemean

CD value as experienced by a satellite over a time span. A half–normal distribution was chosen since the satellite’s attitude proﬁle is expected to be symmetric about its optimal mode of operation (✓ = 0), shown in Figure 5.1. CHAPTER 5. APPLICATION AND RESULTS 114

Figure 5.2: STK simulation of 10 satellites evenly spaced along-track in an 400 km circular orbit. Other parameters include the Jacchia–Roberts reference atmosphere, a4 4gravityﬁeld,andSRP. ⇥

We can calculate the K values per satellite per time period accordingly and construct the CDF of the ratio distribution by amalgamating all measurements from all time periods together. Error in the measurements was injected directly into the estimated K values in the form of a Gaussian distribution. This is depicted in Figure 4.5,where we see that, without error, our data points very closely resemble the theoretical distribution, while with 10% error in K, our distribution can become quite di↵erent. Mostly the di↵erence manifests itself by stretching the ratio distribution along the K axis by a factor of at least 50%. This is also highlighted in Figure 5.3,where Kmin analytical and simulation data points for the ratio distributions are compared. Note that although we have given a prior distribution to CD,thevariationthatweestimate is for B, the ballistic factor. The fundamental parameter that drives this variation is ✓,theorientationofthesatellitewithrespecttotheincomingﬂow,whichdetermines the value of A and CD,andconsequentlyB.

Given that we know the standard deviation of K, we can apply the iterative method to calculate the CDF of CD. The ﬁnal result of this estimation is shown in Figure 5.4. CHAPTER 5. APPLICATION AND RESULTS 115

(a) Ratio distribution with no error in K. (b) Ratio distribution with 10% error in K.

Figure 5.3: Comparison between ratio distribution K between theoretical and sim- Kmk ulation data. A 10% error in K extends the distribution by at least 50% along the K axis. This e↵ect must be taken into account during the estimation of neutral Kmk density, as measurements made will always exhibit some error distribution.

Here we have assumed that we know the minimum value of CD =2.2 beforehand, so we can be certain that the distribution begins with that value. This would not be true in the real case, so we would add an additional uncertainty to this minimum

CD,andhencestretchthedistributionslightlyfromabout2.1onwards(thise↵ectis shown in Figure 4.7). We see that the estimate also does not perfectly match that of the theoretical, but the minimum order estimate matches quite well (this is usually the case since the bounds on the ordered estimates are much tighter). Because we estimate our CDF from data, it will never perfectly match that of the theoretical. Moreover, we often have to employ smoothing algorithms for the CDF such that we can perform the minimization function of Eq. (4.23) appropriately on discretized points located between data points. One example that was used was bootstrapping, where we take a set of measurements from random days, form the ratio distribution for each day, and combine the values into one sorted vector. This is repeated n times, where we eventually take the mean value across all n vectors to form the x-axis of the ratio distribution PDF or CDF.

We can now proceed to calculate the mean or ordered (minimum) estimate for neutral densities, shown in Figure 5.5. We see that given small errors in K, the minimum estimate does better and generally has tighter bounds. This is due to the fact that CHAPTER 5. APPLICATION AND RESULTS 116

Figure 5.4: Estimated distribution of CD and CD,min compared to the original. The estimate has accounted for errors in K and approximates the original distribution. We actually overestimate the error, and thus the estimated neutral density standard deviations will be larger than they need to be. Note that the distribution for the minimum CD is very localized and hence we are quite certain of what the minimum value should be.

because K has virtually no error, we can shrink our bounds on CD,mk and produce very accurate estimations. However, the ordered estimate can become quite inaccurate in the case of large errors in K, as seen in Figure 5.5b. Again, this is due to the fact that when choosing Kmk,itiscorruptedbylargeamountsoferror,andwecaneven see bias creeping into the estimate (error on a half–normal distribution tends to bias K high). The mean estimate decreases this error since the central limit theorem is applied. If the relative variation in CD is small, the mean estimate fares much better than the ordered estimate.

The errors in the estimates for neutral density are depicted in Figure 5.6. Note that although the mean and minimum estimates look similar in Figure 5.6a,larger variations in CD do allow minimum estimates to perform better in general. The CHAPTER 5. APPLICATION AND RESULTS 117

(a) Density estimates for minimal error in K. (b) Density estimates for large errors in K.

Figure 5.5: Comparison of mean vs ordered estimates of neutral density given the amount of error in K.Dashedlinesrepresent1 values. Note that the minimum (or ordered) estimate performs better in the case of minimal error in K,whilethemean estimate performs better when errors in K are large. standard deviation of the estimates are not shown here, which would be smaller for the ordered estimate in the case where K is small. We can see that for the case of large K values, the minimum estimate does quite poorly (although the mean estimate also performs worse than in the case where K is small, by virtue of larger errors in K).

5.1.2 Constellation Data

The on-orbit data that we analyzed was obtained from CubeSats launched by Planet Labs in August of 2014. The satellites were released from the ISS and attitude stabilized within a few days of their deployment. During their lifetimes, the satellites were subject to a number of attitude maneuvers making use of di↵erential drag to separate them along-track. The ocial SpaceTrack satellite IDs are 40122–40127, and 40131–40134. More information regarding these satellites can be found in Appendix A. CHAPTER 5. APPLICATION AND RESULTS 118

(a) Standard deviation of neutral density es- (b) Standard deviation of neutral density estimates for minimal errors (approximately timates for large errors (approximately 10%) 1%) in K. in K.

Figure 5.6: Comparison of errors in density estimates for mean vs minimum (ordered) estimates given varying amounts of error in K.ForsmallerrorsinK,thestandard deviation of the estimate is lower for the minimum estimate, but for large errors in K,thestandarddeviationislowerforthemeanestimate.

Although the SSN tracks these satellites and the information is publicly available in the form of TLEs, Planet Labs was able to perform ranging measurements from their own ground stations worldwide. These measurements were used to update and correct the TLEs using a batch least squares method, updated daily. The ﬁtting process accounts for ionospheric delay during transmission, drag, and higher order gravity terms. The residuals of these measurements were on the order of 1 km. The orbital parameters estimated by Planet Labs are more accurate than those reported by JSpOC due to the fact that downlinks are performed frequently (at least once per day) and are of higher quality than observations made by the SSN. This is evidenced by changes in orbital parameters in the JSpOC TLEs that are unphysical, such as decreases in the mean motion at certain points in time [151]. We have used the corrected Planet Labs TLEs for our analysis.

The geometry of these satellites can be approximated as a rectangular prism with a large ﬂat array of solar panels attached to one side. A small cylinder which houses the optical instruments is positioned on the end of the rectangular prism. Because CHAPTER 5. APPLICATION AND RESULTS 119

these satellites in question are identical and possess similar ADACS, we can assume that their orientation with respect to their forward–facing direction to be IID. The same can be assumed for the drag coecient CD since it depends directly upon the angle of orientation ✓. Figure 5.7 shows a schematic of the CubeSat where Amax and

Amin are the respective maximum and minimum surface areas of the CubeSat. Since this particular CubeSat has deployed solar panels, we use Amin and Amax to deﬁne the surface areas of a side, and of the solar panels, respectively. If the CubeSat did not have deployed solar panels, then all four sides would possess the same area. We have ignored the top and bottom surfaces of the satellite because their rotation is mainly about the longitudinal axis. Small rotations about the other axes contribute little to the overall frontal area. The CubeSats themselves were designed for Earth observation and hence must be nadir pointing at all times. Planet Labs’ quoted distribution of ✓ was estimated to possess a 3 range of roughly 10.

We can perform the least squares estimation technique detailed in Section 3.3.2 on the ranging data to determine the orbit of these satellites. An illustration is shown in Figure 5.8 for the ﬁrst and last iteration of the process. The ﬁnal error characteristics consists of an RMS error of approximately 0.6 km on the residuals and the estimation of K produces an error of approximately 0.038K.Moredetailsregardingorbit K ⇡ determination for Planet Labs CubeSats can be found in [151].

Figure 5.9 shows the evolution of the mean motion of 10 satellites over a period of 40 days according to ranging measurements obtained from Planet Labs. There are also times when the mean motion increases drastically faster for some satellites compared to others, a trait indicative of controlled attitude changes to make use of di↵erential drag. Attempts were made to remove measurements made during these maneuvers since they are detrimental to the IID assumption.

Figure 5.10 shows the K parameters from Eq. (3.36)onadailybasisforall10 satellites. Most follow a general behavior but there are usually at least a few satellites that display large values of K,indicativeofattitudemaneuversthatalignthesolar panels normal to the velocity vector. Since the satellites at most are a few kilometers CHAPTER 5. APPLICATION AND RESULTS 120

Figure 5.7: Schematic of Planet Labs CubeSat. The frontal proﬁle can be approximated as a rectangular ﬂat plate (Amin), with one side consisting of a large array of solar panels (Amax). A small cylinder rests atop the satellite which houses the optical instruments. During minimum drag operation, the large solar panel array is oriented parallel to the velocity vector. apart in altitude and lie in the same orbital plane, they must all encounter roughly the same neutral density.

Amalgamating all measurements from all time steps (except those during obvious attitude maneuvers), we achieve the resulting curve shown in Figure 5.11.Thesmooth fit was a combination of a gamma distribution with a 4th order polynomial. The former was used to describe the initial characteristics of the curve, which exhibits a small but noticeable increase in slope up to approximately B¯ =1.05. The polynomial B¯mk then describes the remainder of the curve. One can see that final fit closely matches the distribution recovered from data. This smoothing process aids in the numerical method in recovering the distribution of B. CHAPTER 5. APPLICATION AND RESULTS 121

(a) Ranging residuals at ﬁrst step of least (b) Ranging residuals at last step of least squares algorithm, given initial estimates for squares algorithm, with RMS converged to position and velocity. within 0.1% of previous estimate.

Figure 5.8: Measure of residuals from least squares technique on ranging measurements from ground stations. The left figure shows the initial iteration given initial estimates of position and velocity, while the right figure shows the converged final solution. Note that the RMS was approximately 0.6 km and K 0.038K (assuming Gaussian) for the final estimate. ⇡

B¯ Estimation of the distribution of B¯ given B¯mk and uses the method outlined in B¯mk Section 4.1,inparticularEq.(4.22). The result is shown in Figure 5.12.Recallingthe constant of integration ambiguity, we have allowed it in this case be Bmin↵ =0.022. It should be noted that using the numerical method through fmincon occasionally gives results that are nonsmooth or contain points where the PDF goes to 0 during intermediate parts of the distribution. This is generally treated as unphysical and disregarded. The CDF of B¯mk is then calculated as per Eq. (4.2).

We are now left with the estimation of Bmin↵.Theaccommodationcoecient↵ was reported to be as high as 0.96 at altitudes of 400 km during solar maximums [122, 121]. The period of August 2014 during which our measurements were made was a fairly mild solar maximum. Other estimation procedures are possible based upon

Langmuir isotherms [119, 120]orthroughhistoricaldata.ThedragcoecientCD of the satellite was calculated using Sentman’s analysis [113]. General mean temperature of the satellite surface was taken to be 300 K and the mean atmospheric temperature CHAPTER 5. APPLICATION AND RESULTS 122

Figure 5.9: Evolution of mean motion of satellites, color-coded for di↵erent satellites. The mean motion always increases as a result of drag. The satellites begin to separate in altitude near the end as evidenced by the separation of the curves. to be 1000 K, although these do not signiﬁcantly impact the resulting ballistic factor (less than 1%, as evidence by Figure 3.7). The atmosphere was assumed to consist of atomic oxygen (NRLMSISE shows that the region to be 96% atomic oxygen by volume), and so it reasonable to believe that the satellite would be coated with the element, further enhancing di↵use reﬂections.

The drag coecient CD of the satellite was calculated from theoretical models of a ﬂat plate and cylinder. We have chosen to ignore the e↵ect of the side solar panel for the minimum achievable B¯ on the assumption of hyperthermal ﬂow. Including this e↵ect will raise CD by roughly 0.1 and B¯ by 0.001.

We have set the accommodation coecient ↵ normally distributed about a mean value of 0.95 with a standard deviation of ↵ =0.02. This is again based upon information from previous solar maxima and NRLMSISE parameters showing the atmosphere in the region to consist mostly of atomic oxygen. The resulting estimate of B¯ is shown in Figure 5.13.InthelimitofN satellites, this distribution mk !1 would become exactly the distribution dictated by ↵. However, with fewer satellites, CHAPTER 5. APPLICATION AND RESULTS 123

Figure 5.10: K parameter as measured from Planet Labs TLEs, color-coded for dif- ferent satellites. All satellite roughly follow the same patterns, with large variations mostly due to di↵erential drag maneuvers. The small di↵erences between the K parameters are attributed to minute variations in attitude. this distribution is a combination of uncertainty in ↵ and uncertainty in the inferred minimum B¯.

Figure 5.14 gives the estimated neutral densities with comparisons with NRLMSISE- 00 and JB2006 models. The standard deviation of the estimated densities is roughly 9%–10%, since it includes errors that may reside in K (barring errors in K,thiscan be reduced to approximately 3%, shown in simulations). This is much better than taking a single random measurement of a random satellite to estimate the density, because with more satellites we become increasingly certain that the minimum B¯ approaches the theoretical limit.

We emphasize here that this method was used on mean elements, and thus the result we obtain is the mean density encountered by the satellite over the entirety of one day. With greater time resolution on updated orbital elements, we can use this method to achieve mean densities averaged over a shorter time span. However, we cannot CHAPTER 5. APPLICATION AND RESULTS 124

Figure 5.11: B¯ CDF ﬁtted to a combination of a gamma distribution with a 4th B¯mk order polynomial. The former was used to describe the initial characteristics of the curve, which exhibits a small but noticeable increase in slope up to approximately B¯ =1.05. The polynomial then describes the remainder of the curve. B¯mk achieve estimates at a rate faster than one orbit, since the satellites must all traverse the same path at least once per time period to ensure equivalent averaged densities. Another consequence of using mean elements is that B¯ is unlikely to take the actual value of Bmin↵, since it is unlikely that the satellite is perfectly at ✓ = 0 attitude throughout the entire integration period.

Referring again to Figure 5.14,itcanbeseenthatthereisacertainvariabilitybetween our estimates and predicted values from NRLMSISE. It is interesting, however, to note that although the measurements do not often coincide with the predicted, they share much of the same patterns. There are points where there are large di↵erences in trends, such as on day 4, where measurements indicate a relative large drop in density as opposed to that predicted by JB2006. However, if we look closely at Figure 5.10 we see that almost all satellites exhibit this behavior, which becomes quite dicult CHAPTER 5. APPLICATION AND RESULTS 125

Figure 5.12: Numerical CDF estimate of B¯ and B¯mk given Bmin↵ =0.022, with the initial guess as simply a constant value. Note that the estimated B¯mk is very close to the actual lower bound (we have not yet placed any variation on the accommodation coecient ↵). The smoothness of the estimate is attributed to the earlier smooth ﬁt of the data in Figure 5.11. This distribution is later used to calculate the neutral density with associated standard deviations. to dismiss as simply coincidental. In most cases, almost all the satellites follow very similar trends exhibited by the minimum, which make the trends themselves dicult to dismiss as erroneous. There are large discrepancies from days 28 to 33 between measured and predicted, and this is attributed to the fact that JB2006 has been known to overestimate the neutral densities during periods of high solar activity [122]

(in this case the F10.7 index did markedly increase during that time).

Another curious artifact appears on days 17 and 18, where the estimate drastically decreases. However, the minimum estimate on those days displayed unusually increased error compared to estimates on other days. We can perform a chi–squared distribution hypothesis test on the ratio distribution for degree equal to the number of satellites to check if observations on particular days conform to the amalgamated

CDF Cz. The chi–squared test reject the measurements made on days 17 and 18 CHAPTER 5. APPLICATION AND RESULTS 126

Figure 5.13: Estimate of B¯mk given↵ ¯ =0.95 and ↵ =0.02. The estimate is a mix of uncertainty in ↵ and uncertainty of the inferred minimum B¯. due to their data points being far from the expected probabilistic distribution. This check, however, depends on the fact that in general,thedistributioniscorrect.

Figure 5.15 depicts the mean estimate in addition to the minimum estimate. We see here that the densities also follow the same general trends, but the e↵ects of outliers are much more pronounced. Because one or two satellites may not conform to the IID assumption at all times (recall that some satellites were performing di↵erential drag maneuvers), this drastically a↵ects the mean estimate, while having less e↵ect on the ordered estimate.

It should be noted that the satellites used in question had a form factor where one side consisted of a large array of solar panels. This leads to e↵ects where the solar panels may shadow the body of the CubeSat. The estimation would fare much better in the case of something similar in shape to a simple rectangular prism, which is symmetric about all 3 axes. CHAPTER 5. APPLICATION AND RESULTS 127

Figure 5.14: Estimate and comparison of neutral densities to existing models. The ordered (minimum) estimate is depicted here with its standard deviation. Major di↵erences include larger than usual densities from days 11–16, and lower than usual densities from days 27–32. However, because observations across all satellites exhibit similar behavior, it is dicult to dismiss this as simply coincidental. On days 17 and 18, the lower than usual density estimates are certainly erroneous, as a chi-square hypothesis test will show that the distribution of K on those days do not agree with the usual ratio distribution.

5.2 Meteoroids

Because of the abundance of meteoroids ablating in the atmosphere, we are able to apply our method of neutral density estimation to head plasma detections made by HPLA radars. Instead of attributing the variation of the platform (i.e. the meteoroids) to the ballistic factor, we now attribute it to meteoroid properties, such as size and bulk density. This section will begin by describing the radar with which the observations are made with, and continue with the estimation of neutral density according to the successive layer approximation detailed previously in Section 4.3.1. CHAPTER 5. APPLICATION AND RESULTS 128

Figure 5.15: Minimum and mean neutral density estimates compared to existing models. Because di↵erential drag maneuvers are mixed with the observations, the mean estimate is sensitive to these outliers, while the ordered estimate is relatively una↵ected.

5.2.1 Radar Characteristics

The radar used in this study is the ARPA Long-range Tracking and Instrumentation Radar (ALTAIR), which is used routinely to track satellites and rockets in orbit as part of the SSN. ALTAIR is a 46-m diameter HPLA radar that is located on the Kwajalein Atoll at 9 Nand167 E. It transmits at a peak power of 6 MW simultaneously at two frequencies with right-circularly (RC) polarized signal energy in a half-power beamwidth of 2.8 and 1.1 at VHF and UHF (typical operating frequencies are 160 MHz and 422 MHz), respectively. ALTAIR receives both right- circular and left-circular (LC) energy and has four additional receiving horns for the purpose of angle measurement, which gives the position, velocity and deceleration of an object in three dimensions.

Radar meteor data were collected simultaneously at VHF and UHF at ALTAIR in CHAPTER 5. APPLICATION AND RESULTS 129

2007 and 2008 and include approximately 30 hours of data. In particular, amplitude and phase data from meteoroid plasma (head and trail echoes) were observed in each frequency and four receiving channels for altitudes spanning 80-140 km. The two ALTAIR waveforms used to collect the data were a 100 µs VHF chirped pulse (“V7100”) and a 150 µs UHF chirped pulse (“U7100”), both with 15-m range spacing. These waveforms were chosen for their high sensitivity and range resolution. A 115 Hz pulse-repetition frequency (PRF) was utilized due to ALTAIR system requirements associated with these two waveforms. With this, ALTAIR can detect a target as small as -50 decibels-relative-to-a-square-meter (dBsm) at VHF and -65 dBsm at UHF at arangeof100km[152](roughlyequivalenttoaRCSof10µm2 and 0.316 µm2, respectively).

Delaunay triangulation was used to automatically detect head echoes within the 30 hours of data [86]. The head echo range rates and 3D speeds were computed by applying a phase-derived matching technique, described in [153], which reduces the range rate error to the order of 1 m/s. The errors in the monopulse can be assumed to be on average 11.2 mdeg in azimuth and elevation [154], which at a range of 100 km gives an average velocity error of 2.3 km/s in the worst case for meteoroids traveling completely perpendicular to the radar boresight. This corresponds to an error of less than 6% for meteoroids traveling over 40 km/s. The acceleration error is more dicult to quantify, as the interpulse timing of 8.7 msec makes finite di↵erencing inaccurate. In the best case scenario for meteoroids traveling “down the beam”, finite di↵erencing leads to an acceleration error of approximately 4%. We thus model the velocity vs altitude from a meteoroid streak as the sum of two exponential functions, which possesses an average R2 0.99 and RMS error of 0.023. Figure 5.16 shows an ⇡ example of this fit for one meteoroid. Acceleration is taken as the product between dv the derivative of the function with respective to altitude ( dh )withthechangein dh altitude ( dt ), which is closely related to the range rate. CHAPTER 5. APPLICATION AND RESULTS 130

Figure 5.16: Fitting of velocity data vs altitude as a sum of two exponentials for dv dh one particular meteoroid detection. The acceleration is calculated as dt dt .Errorsin range rate are considered small. Adjusted R2 0.9997 and RMS error 0.0088 for this ﬁt. ⇡ ⇡

5.2.2 Meteoroid Data

The majority of the data analyzed consists of two continuous sets of observations on two separate days. Each data set spans 20 minutes and was collected at 18:00 Coordinated Universal Time (UTC), which corresponds to 6:00 AM locally. The ﬁrst collection occurred on day 312 in 2007, and the second took place on day 15 in 2008. The datasets on these two days were chosen for their continuity and quantity of observations, as well as the local time being favorable for the detection of North apex meteoroids traveling down the radar beam. Preliminary analysis was performed on data collected at 16:00 UTC on day 131 in 2007. This was meant to serve as a preliminary test to the methodology, which unfortunately due to the lack of further data on this day made cross-validation dicult. More information regarding radar data can be found in Appendix B. CHAPTER 5. APPLICATION AND RESULTS 131

The methodology presented in Section 4 is applied to the ALTAIR dataset. We restrict our observations to VHF detections since they tend to possess higher RCS values and longer durations due to a wider beam. We ﬁlter our results such that we discount very short duration observations (<0.05 seconds) and low velocity meteoroids (< 40 km/s). Longer duration observations have the following advantages:

1. These observations reﬂect the idea that the meteoroid is traveling nearly parallel to the radar boresight (as opposed to traversing the beam), and hence the monopulse error is minimized. Since ALTAIR is located close to the equator, it is able to detect Apex sporadics in this fashion.

2. They tend to possess higher SNR since the ablating meteoroid contains more material. Shorter duration observations tend to be small/weak with a low SNR, are noisy, and the meteoroid path cannot be accurately reconstructed with only afewdatapoints.

3. They are correlated to meteoroids that traverse through multiple layers of the atmosphere (as opposed to “earth-skimming” meteoroids)

4. They are associated with a lower likelihood of anomalous physics that we are not modeling (such as ﬂares or breakups) [147, 145].

Another beneﬁt of choosing higher velocity meteoroids is that the deceleration becomes more evident. However, there exists a mass-velocity selection e↵ect associated with any plasma detection such that it is insensitive to both low-mass, low-velocity meteoroids (due to low SNR) and low-mass, high-velocity meteoroids due to high velocity meteoroids ablating at higher altitudes where the radius of the plasma is large 7 (i.e. almost no meteoroids below 1 10 ginmassandover70km/sareobserved) ⇥ [155].

Figure 5.17 shows the 3D speeds (>40 km/s) of 38 meteoroids detected over approx- 1 imately 3 minutes. Figure 5.18 shows the deceleration multiplied by v2 of the same CHAPTER 5. APPLICATION AND RESULTS 132

Figure 5.17: 38 meteoroid detections with 3D speeds of over 40 km/s as detected by ALTAIR. Each color represents a separate meteoroid event (note that colors are reused). Data taken at 16:00 UTC, day of year (doy) 131, 2007 spanning over approximately 3 minutes. meteoroids (refer to Equation (3.44)). In both cases, the deceleration is quite evident as the meteoroid traverses lower in altitude where the neutral density exponentially increases.

We now apply the minimization of Eq. (4.41)tothedecelerationdata.There- sulting density ratio ⇢a is shown in Figure 5.19.Comparisonsaremadetothe ⇢a,110km NRLMSISE-00 and exponential (8 km scale height, hydrostatic equilibrium) models. However, our data–drive estimation can be corrupted by noisy or incomplete data, so the issue of consistency amongst all estimates must be addressed. Furthermore, if we desire actual density values instead of ratios, we must start amalgamating ratio statistics while keeping with the IID assumption of incoming meteoroids. Both of these problems can be remedied by binning data over a longer period into multiple sets, and evaluating each set independently. CHAPTER 5. APPLICATION AND RESULTS 133

dv 1 Figure 5.18: dt v2 of meteoroids traveling at speeds over 40 km/s as detected by ALTAIR. The traces are color coded similarly to that of Figure 5.17 (i.e. a green curve here and a green plot in Figure 5.17 corresponds to the exact same meteoroid). Data taken at 16:00 UTC, doy 131, 2007. Note an outlier (black line) in the data at 105 km; the methodology is able to ignore such erroneous measurements.

The result of taking data over 20 minutes and binning it into 9 equal intervals is shown in Figure 5.20.Becausewetakedatafromashorttimeinterval,wecanbe fairly certain that the observed meteoroids follow the same distribution and that the background neutral density remains fairly constant. This is a good assumption given that geomagnetic indices are only updated every 3 hours. This actually negates the need to amalgamate ratio statistics since the method is primarily designed to counter the e↵ects of changing atmospheric densities. We can see from Figure 5.20 that estimates with high mean squared error generally appear as outliers and hence can be considered as poor results. Note that the method is not without false positives and false negatives. An example of the former can be found in Figure 5.20b where a result possessing large error is located close to the estimated mean, and an example of the latter can be seen in Figure 5.20a where a result possessing low error is located very close to an erroneous estimate. These anomalies exist since our method depends upon CHAPTER 5. APPLICATION AND RESULTS 134

Figure 5.19: Density ratios calculated from minimization procedure with ⇢a,110km as reference over 38 meteoroids. The reference altitude begins at 110 km and ends at 97 km, due to the lower quantities of meteoroid data beyond these altitudes. Comparisons are made with NRLMSISE and hydrostatic exponential models. Data taken at 16:00 UTC, doy 131, 2007. incoming data that could at instances be highly variable, especially if an insucient amount of data is fed into the minimization process. Overall, the estimates tend to cluster together in such a way that we can calculate a meaningful average and standard deviation. As expected, the aforementioned average closely agrees with the result calculated from all data over the 20 minute period.

If we desire the actual density values, we require statistics on the masses and bulk densities of the incoming meteoroids. If we refer to Eq. (4.44), we see that measuring dv 1 ⇢a,ref 1 the variation across 2 is equivalent to measuring the variation across . dt v ⇢a R⇢m

This is due to the fact that we take CD to be constant across all meteoroids and we scale the measurements based upon one reference density. Figure 5.21 shows this variation for two separate days. We scale the measurements relative to the median CHAPTER 5. APPLICATION AND RESULTS 135

(a) Density ratios as calculated from 45 me- (b) Density ratios as calculated from 30 meteoroids per bin at 18:00 UTC, doy 312, 2007. teoroids per bin at 18:00 UTC, doy 015, 2008.

Figure 5.20: Calculated density ratios on two separate days over 9 bins across 20 minutes. The mean estimate was calculated from all binned estimates possessing RMS error less than 0.15. The error bars indicate the standard deviation of these estimates (not including high error estimates, but including false positive estimates). since it is less sensitive to outliers, especially since masses and bulk densities vary across 2 orders of magnitude between meteoroids. Based upon previous ALTAIR 5 measurements, their respective median values are approximately m =1.36 10 g ⇥ 3 and ⇢m =0.45 g/cm [155, 86]. Figure 5.22 shows the resulting estimated densities.

It should be noted that although we have set the median to a speciﬁc value, there is still an uncertainty related to this value. This uncertainty can be minimized if we increase our sample size, but there is a theoretical limit as the median was derived from other parameters that are also uncertain (i.e. the masses and bulk densities were derived from an analytical scattering theory with its own inherent assumptions [85]). In particular, detections made in VHF at ALTAIR are only sensitive to meteoroids 6 4 of a certain mass range (6 10 gto3 10 g[155]) and so our median must come ⇥ ⇥ from this particular source. We can also look at the median statistics per bin to gain a rough measure of this uncertainty, shown in Table 5.1. Note that median is not the standard deviation of measurements about the median value, but is instead the standard deviation of the median itself. From this we would expect roughly a 12% standard deviation on our density estimates, given that we know the statistics CHAPTER 5. APPLICATION AND RESULTS 136

1 1 1 1 (a) Variation of meteoroids (R ⇢m ) scaled (b) Variation of meteoroids (R ⇢m ) scaled to median on 18:00 UTC, doy 312, 2007. to median on 18:00 UTC, doy 015, 2008.

1 1 Figure 5.21: Variation of meteoroids (R ⇢m ) on two separate days. Note that on the latter day there are less detected meteoroids as compared to day 312. The data reﬂects the total number of meteoroids across all bins on that particular day.

(a) Densities as calculated from 45 mete- (b) Densities as calculated from 30 meteoroids per bin at 1800 UTC, doy 312, 2007. oroids per bin at 1800 UTC, doy 015, 2008.

Figure 5.22: Estimated densities on two separate days over 9 bins across 20 minutes. The mean estimate was calculated from all binned estimates possessing RMS error less than 0.15. The error bars indicate the standard deviation of these estimates (not including high error estimates, but including false positive estimates). CHAPTER 5. APPLICATION AND RESULTS 137

Bin Median median median as % 4 5 1 9.882 10 13.678 10 13.8 ⇥ 4 ⇥ 5 2 8.639 10 6.084 10 7.0 ⇥ 4 ⇥ 5 3 8.607 10 9.184 10 10.7 ⇥ 4 ⇥ 4 4 8.677 10 11.709 10 13.5 ⇥ 4 ⇥ 4 5 6.750 10 7.752 10 11.5 ⇥ 4 ⇥ 4 6 8.206 10 10.063 10 12.3 ⇥ 4 ⇥ 4 7 10.011 10 15.641 10 15.6 ⇥ 4 ⇥ 4 8 5.802 10 5.501 10 9.5 ⇥ 4 ⇥ 5 9 7.628 10 10.420 10 13.7 ⇥ ⇥ dv 1 ⇢a,ref Table 5.1: Statistics of 2 over multiple bins at 18:00 UTC, doy 312, 2007. dt v ⇢a The standard deviation here corresponds to the uncertainty in the median meteoroid properties per bin. The last column indicates this as a percentage, calculated as median Median 100%. Red rows indicate bins where the mean squared errors of the estimate were large.⇥ Units in 1/km.

Day F10.7 81 day average F¯10.7 Kp 312 70 72 1- 015 74 75 2+

Table 5.2: F10.7 index for solar irradiance and Kp index for geomagnetic activity on analyzed days. of the median as well as the velocities and accelerations perfectly. In comparison, we calculate from Figure 5.22 that the standard deviation is roughly 10%. Both these uncertainties are associated with the methodology, as we assume that most meteoroids follow the given ablation and drag model. Additional error from inaccurate velocities and deceleration also increases the standard deviation (the best case scenario being an additional 4% in the case of the ordered estimate).

Comparing the neutral densities between the two days, we observe that it is signiﬁ- cantly lower on the ﬁrst day. However, NRLMSISE predicts that the densities across both days to be roughly equivalent. If we refer to the solar and geophysical parameters

(shown in Table 5.2), we see that on the second day the F10.7 and Kp indices are both greater compared to the ﬁrst, possibly explaining the increase in neutral densities not reﬂected by NRLMSISE. Although these indices are included within NRLMSISE (in CHAPTER 5. APPLICATION AND RESULTS 138

fact, NRLMSISE only exhibits a slight dependence on the F10.7 index and practically no dependence upon geomagnetic indices at these altitudes), the model does tend to predict the average behavior of the neutral densities and not phenomena on a shorter time scale (less than 1 day) [4]. Furthermore, there could be other factors in the complex dynamics of the thermosphere that go unmodeled.

Figure 5.23: Histogram of estimated ablation parameters: log H⇤ is estimated 10 CH from the minimization procedure over approximately 500 meteoroids.⇣ ⌘ Data taken at 16:00 UTC, doy 132, 2007.

Another interesting analysis we can perform is on the values of D from the minimization process in Equation (4.41). Recall that D is the ablation parameter, which is acombinationofconstants(CD, CH and H⇤) related to the ablation process. Since meteoroids ablate in the free molecular ﬂow regime (Kn > 10) at these altitudes, H we can safely assume that CD =2[139]. The histogram of ⇤ is shown in Figure CH CHAPTER 5. APPLICATION AND RESULTS 139

5.23. We infer here that CH is variable across meteoroids, since if we take the often assumed CH = 1 for free molecular ﬂow, the corresponding H⇤ values tend to be too large in comparison to known materials. Approximately 10% of the estimates made for H⇤ are far too large, citing the possibility that these meteoroids do not follow our model and other physics is at work (e.g. variable ablation parameters, di↵erential ablation, sputtering, fragmentation [136, 147, 145]).

Potentially, the proposed method can be applied at other HPLA sites if certain conditions are met. However, given the relative scarcity of these facilities and their narrow observation ranges, the method can only provide a local measure of dynamically changing neutral densities. The measurements can be used to further improve atmospheric models locally, or can serve as a benchmark for global atmospheric models at specific locations. Regarding the observations, there is no particular need for “down the beam” echoes, although these types of echoes do tend to last longer and possess better noise characteristics. It is however a necessity for accurate acceleration and velocity measurements in 3D, particularly at ALTAIR, since the radar attempts to align itself with the meteoroid streak for monopulse detection purposes. More importantly, the sensitivity of the radar to specific frequencies and hence size of meteoroids must be known beforehand, such that the median R⇢m can be accurately identified. This has no bearing upon the calculation of density ratios, but is crucial for estimating actual densities. Finally, the beam pattern correction for the SNR is not critical, but may provide additional information such as plasma properties, what stage the ablation process is at, and for filtering meteoroids outside the main beam.

5.3 Summary

In this chapter, the neutral density estimation methodology was first tested on a satellite constellation simulation, then applied to on-orbit data and meteoroid plasma measurements from an HPLA radar. The goal was to first define the main source of CHAPTER 5. APPLICATION AND RESULTS 140

variation (B in the case of satellites, R⇢m in the case of meteoroids), estimate its distribution, and then calculate the neutral density with associated standard deviations.

For the case of satellites, it was shown that the mean estimate does indeed perform better for large errors in the measurements given small variations of the platform. However, the ordered estimate performs better in the opposing case of small measurement errors but large platform variation. Analyzing the on-orbit data, the neutral density can be estimated with a standard deviation of less than 10%. Comparing the results to NRLMSISE, we observe periods where the densities diverges from the models. However, since we directly observe these changes across all satellites for a speciﬁc time span, it adds credence to the idea that we are observing actual density changes, unless if an anomaly or disturbance a↵ected all satellites at once. Estimates at a certain time period can also be validated via hypothesis testing to verify if a certain distribution of satellite measurements matches the calculated distribution.

For the case of meteoroids, we predominantly utilized the ordered median estimate, as the large variation in sizes and bulk densities of the meteoroids does not lend itself well to the mean estimate. By employing the successive layer approximation to velocity and acceleration data, we can initially calculate the density ratio between each layer to a reference layer without requiring knowledge regarding meteoroid properties. However, to calculate the actual density values, we need to know a priori the mass, radius, and bulk density distributions of the sporadics (only knowledge of two of these properties is necessary). We cross–validated our estimates by separating our measurements into separate bins, and performing our method on each independently. The results agree well, with a standard deviation of approximately 12% on neutral density estimates. Additional error in velocity and acceleration can compound this uncertainty, estimated approximately at 4%. Chapter 6

Conclusions

This thesis provides a new methodology for estimating the neutral density of the atmosphere through the observation of many similar platforms. The methodology was applied to two separate platforms, including satellites and meteoroids, which occur in two di↵erent regimes and under di↵erent conditions, proving its adaptability to changing conditions of the problem itself. Section 6.1 discusses the contributions made to the area of density estimation. Section 6.2 addresses certain aspects and caveats of the methodology. Section 6.3 discusses future work, including proposed new research that can stem from the ideas presented within this dissertation.

6.1 Review of Contributions

This section will review the contributions made by this thesis on the topic of neutral density estimation.

141 CHAPTER 6. CONCLUSIONS 142

6.1.1 Density Estimation Method

A new density estimation method was developed that took measurements from multiple equivalent platforms to formulate a statistical approach to determining neutral densities. The platforms were allowed to vary in an IID manner, and the measurements were amalgamated across all platforms across all time periods. The unknown distribution was solved numerically through the use of the minimum ratio distribution. Treatment of error in the measurements was handled by adding a recursive component to the methodology, and uncertainty for the integration constant was derived from either basic physical models or from historical data.

We created a statistical framework for the methodology to estimate the standard deviations associated with this data driven scheme. The density estimate is based upon the initial measurement K and the amount of variation within the system, with the mean estimate performing well when the errors in K were large in comparison, and the ordered estimate performing well when the variation across the platforms were large. Given the error characteristics of K and the distribution in the variation, we can then calculate the appropriate bounds on the density estimate. Current estimates put the standard deviation at approximately 10% for satellites, and 16% for meteoroids.

These results show that we can e↵ectively reduce the standard deviation of our measurements to well below the 15% of most models. For the lower thermosphere, our neutral density estimations derived from meteoroid ablation provide a technique for capturing the time–varying behavior, unlike current atmospheric models, which mostly yield a static estimate. Our methodology is thus able to make use of the abundant data in these di↵erent regions that are historically dicult to probe. CHAPTER 6. CONCLUSIONS 143

6.1.2 Applications

Satellites

The density estimation method was applied to the ﬁrst constellation of CubeSats in LEO launched from the ISS at an altitude of approximately 400 km. As such they followed roughly the same orbital path and would have observed similar densities. Comparisons were made to known models and show good agreement during certain periods, but were di↵erent during others. However, as we observe similar behaviors across all satellites during these periods, it is more likely to be indicative of neutral density variations.

Meteoroids

The density estimation method was applied to ablating meteoroids as observed by ALTAIR at an altitude of approximately 100 km. A new successive layered method was developed in which the atmosphere was divided into layers, and the ratio of densities was calculated between one layer relative to the previous. Combined with physical laws and historical data of meteoroid sizes and bulk densities, the neutral density was calculated across multiple altitudes.

6.2 Discussion of Methodology

This section will discuss the application of the methodology, and specifically where it is meant to be applied as well as its limitations. Firstly, the method is meant to be applied on platforms in which the major source of variation can be readily identified and modeled. This has ramifications on the integration constant, as it is dependent either on historical data or physical laws, both of which relies upon the correctly CHAPTER 6. CONCLUSIONS 144

deﬁned variation parameter. Given that the variation source is well known, the method is able to estimate the variation of the platform by decoupling the density contribution and platform variation that are inherent within the observations. In addition, the platform variation can be estimated in the form of a probability function, which in turn gives us mean and standard deviation values to estimate the neutral density.

There are, however, caveats to using this method. First and most importantly is the IID assumption, which states that all the platforms analyzed must adhere to one distribution (albeit unknown) and be independent of one another. Violation of this assumption will lead to incorrect estimates of the estimated CDF and hence neutral density as well. The method also inherently gives mean estimates of the neutral density, so the time scales of integration (particularly for satellites) and the regions traversed are important. Furthermore, because of the innate dependence upon existing models, the method o↵ers precision rather than accuracy, as biases that occur across all platforms would be dicult to detect.

The method itself is an o✏ine one, meaning that it does not iteratively improve upon previous states of estimation. In this sense, it is closer to a least squares method than an iterative filter, as it defines a constant standard deviation based on the total data set available. Of course, with more measurements and over longer time periods, the method can be applied to a larger data set, resulting in a smaller standard deviation. Finally, although the method is mostly data driven, it still depends upon physical models, such as the drag equation, free molecular flow, and ablation relations. These fundamental equations also a↵ect the constant of integration that must be calculated beforehand to anchor the variation distribution itself. To account for the uncertainties within the physics and historical data regarding the integration constant, we can attach additional probabilistic elements to the anchor, at the cost of larger standard deviations on our density estimates. CHAPTER 6. CONCLUSIONS 145

6.3 Future Work

Neutral density estimation is a crucial component to our operations in space and to our general understanding of the atmosphere. As we continue to populate LEO with additional satellites, collision avoidance and risk management becomes an increasingly pressing issue, one that relies heavily upon accurate orbital prediction. As the Earth’s climate changes, accurate measurements throughout our entire atmosphere will help us understand and give us further insight to its behavior. The methodology developed in this thesis becomes increasingly relevant when we are able to leverage an increasing number of similar objects within the atmosphere, particularly in regions where measurements are still notoriously dicult to obtain.

6.3.1 Advanced Modeling

The results presented within this thesis show good agreement with current models, and as such the trends they observe can be considered to be relevant and truthful. However, there are still a few outstanding issues that can be further explored in future work, mainly dealing with the underlying models inherent to the methodology. These include more advanced techniques for determining the drag coecient CD,sensitivity to underlying assumptions such as the quadratic drag law, and di↵erent methods for deducing the K parameter.

6.3.2 Recursive Estimation

As mentioned previously, the method presented here is an o✏ine one. However, it can potentially be adapted into a iterative/recursive estimation technique. If every consecutive estimate of the probability distribution can be corrected given a new set of data, and a measure of the variance of the probability distribution can be given, CHAPTER 6. CONCLUSIONS 146

then we can eventually converge onto the true distribution after enough iterations. Filtering in this sense can be powerful, especially if we want to bring this methodology onto platforms with limited computing power or if the application requires fast response times.

This method would still rely upon similar assumptions of IID data. Hypothesis tests can be employed to determine whether a certain data set matches a known distribution, and by how much it di↵ers. We can thus reject estimates that do not match our previously calculated variation distribution.

6.3.3 Data Collection and Analysis

The future of this methodology particularly relies upon a wealth of data. In particular, if we have enough similar satellites in enough constellations, then we can achieve ahighspatialandtemporalresolutionofneutraldensitieswithinLEOthrougha tomographic approach. We can thus partition the atmosphere, and by observing all the orbital changes of all satellites through all regions, we can estimate region- speciﬁc neutral densities. The similarity between satellites is still required so that we can adequately compare measurements while accounting for the variation between platforms. This is something that traditional methods such as HASDM su↵er from, since they rely upon a small number of calibrated satellites. With the increased commercialization of space, companies such as Spire, Planet Labs, Terra Bella, and OneWeb might able to provide a wealth of data for atmospheric density estimation in addition to the other services they provide for humanity.

In the area of meteoroids, the data analyzed within this thesis have been collected on days of sporadic meteoroid activity. The major reason that the estimates for density derived from meteoroids have larger standard deviations is because meteoroids are vastly di↵erent from one another in terms of composition and size. If sucient data could be gathered during meteor shower events (which are expected to have less CHAPTER 6. CONCLUSIONS 147

variation in meteoroid composition than sporadics), then we can drastically improve our density estimates as well. Although traditionally HPLA radars are more sensitive to sporadics, shower events have been observed previously [81, 156], and it is the hope that continued improvements in this area could lead to the use of our methodology for those data sets. Likewise, orbit analysis techniques for meteors could be employed such that we ﬁlter meteoroids from speciﬁc sources [143, 157]. This would lead to less uncertainty regarding our statistics and better estimates overall.

We can also take a more complex view regarding meteoroid ablation and introduce variable parameters. However, doing so might potentially increase the degrees of freedom within our to-solve-for parameters, particularly if we require a model that can ﬁt every observation. Ultimately, what we require is a model that ﬁts most incoming meteoroids without being a↵ected by outlying cases, such that the statistics of large numbers dictate the derivation of neutral densities.

6.3.4 Orbital Debris

Orbital debris is deﬁned as man-made objects in space that no longer serve any useful purpose. This can range from small objects such as paint ﬂakes, to large objects such as defunct rocket bodies. The amount of debris in orbit numbers in the millions, and daily observations are made for the larger pieces (greater than 10 cm in diameter). This data source can potentially be used with our methodology to not only measure neutral densities, but also to assess the variability between debris.

However, due to the large di↵erences in size, shape, and composition, the density estimates derived here might possess too high of an error to be of value. One solution is to classify debris into categories so they can analyzed separately. Analysis of debris data still poses a major challenge simply because of the fact that they are dicult to track and are highly variable. CHAPTER 6. CONCLUSIONS 148

6.4 Concluding Remarks

In the coming decades when commercial space access will become more a↵ordable, we might see an explosion of businesses and ventures that take advantage of our LEO regions. Many companies have already started in on this trend, with CubeSats providing daily imagery, and high altitude balloons providing key services such as Internet access. By utilizing this untapped resource of new observations and measurements, we might be able to derive additional science, simply from the wealth of data that will be available.

Neutral density estimation will always be relevant for our endeavors in space, whether it be to plan satellite operations or to study and model the behavior of our atmosphere. As we move towards placing more and more of our assets beyond the surface of the Earth, our understanding and prediction of the processes that govern these regions will surely need to be improved if we are to truly succeed in our aspirations. Appendix A

Planet Labs CubeSat Data

Planet Labs deployed its 1B ﬂock of CubeSats from the ISS in mid-August, 2014. The ocial SpaceTrack satellite IDs are 40122–40127, and 40131–40134. The Nanoracks CubeSat Deployer ideally launches the CubeSats in a direction opposite of the ISS velocity vector 45 towards nadir pointing, shown in Figure A.1.Thisminimizes the collision risk between the ISS and the deployed CubeSats and ensures that they will reenter the atmosphere within a span of a few months to a few years. The CubeSats initially exhibit similar orbital parameters as the ISS, with a circular orbit approximately 400 km in altitude at an inclination of 51.65.Deploymentinformation can be found in Table A.1.

The CubeSats themselves are of similar model and make, each with a mass of 4.3 kg in a standard 3U CubeSat form factor. They are not equipped with a propulsion system, have deployable solar arrays, and a limited 3-axis attitude control capability. Their frontal area can vary from 0.04 m2 when the solar panels are aligned with the velocity vector, to approximately 0.24 m2 when the solar panels are perpendicular to the velocity vector (refer to Figure 5.7 for the schematic). Usually a few days after deployment, the satellites go through commissioning, stabilize their attitude, and spend their remaining lifetimes in as close to a minimum drag state as possible.

149 APPENDIX A. PLANET LABS CUBESAT DATA 150

Figure A.1: Launch direction of Planet Labs CubeSats, in the direction opposite to the ISS velocity vector and nadir pointing

Exceptions occur when the satellites perform di↵erential drag maneuvers to space out the constellation evenly along-track.

Planet Labs has ground stations located worldwide as shown in Figure A.2 and listed in Table A.2.Theycangenerallyestablishcommunicationwithasatellitemoments after it has breached the horizon, and a half cone angle of 60 is considered within the region of good linkage. Data intensive imagery is downloaded from the satellites via a high-bandwidth X-band link, while regular communication is established over UHF using a low speed transceiver. Unlike the X-band link which requires pointing accuracy within 0.2,thetransceivercantolerategroundstationpointingerrorsin excess of 10.

The method in which ranging measurements are made is through two-way radio ranging using the UHF transceiver. It is not a coherent transponder and thus the data is noisy, with a standard deviation of approximately 0.8 km. The data itself exhibits a maximum range bias of 200 meters, a time bias of approximately 17 ms, and occasional outliers caused by glitches within the system (which are rare). A sample of the data is shown in Table A.3,displayingthetimeoftransmission,distanceas APPENDIX A. PLANET LABS CUBESAT DATA 151

SpaceTrack ID Planet Labs ID Deploy date Deploy time UTC 40122 Flock 1B–24 2014–08–19 18:25:02 40123 Flock 1B–23 2014–08–19 18:25:02 40124 Flock 1B–26 2014–08–20 02:26:13 40125 Flock 1B–25 2014–08–20 02:26:13 40126 Flock 1B–15 2014–08–20 09:50:02 40127 Flock 1B–16 2014–08–20 09:50:02 40131 Flock 1B–1 2014–08–21 13:37:00 40132 Flock 1B–2 2014–08–21 13:37:00 40133 Flock 1B–8 2014–08–23 19:43:57 40134 Flock 1B–7 2014–08–23 19:43:57

Table A.1: Planet Labs Flock 1B Cubesats catalogue identiﬁcation and launch dates/times. measured by the light-time, valid data ﬂag, received signal strength indication (RSSI) for uplink and downlink, and the ID of the CubeSat.

The orbit can thus be estimated from ranging data spanning 0.5 to 1 day over multiple ground stations. Usually, more than a few thousand data points will be incorporated in the orbit determination process. Planet Labs publishes their own TLEs, which have been veriﬁed to be much more accurate than thse published by JSpOC [151]. A sample is shown in Figure A.3. APPENDIX A. PLANET LABS CUBESAT DATA 152

Figure A.2: Planet Labs ground station locations on world map

Figure A.3: Sample of Planet Labs generated TLEs APPENDIX A. PLANET LABS CUBESAT DATA 153

Ground Station Latitude (deg) Longitude (deg) Altitude (m) Atlantis 51.142830 -1.437020 88.0 Berna 37.384998 -122.413002 22.0 Buran 38.191670 -83.438890 222.0 Calabrese 48.145600 -119.701300 380.0 Caracara 50.051950 -5.181780 115.0 Clanor 38.191670 -83.438890 222.0 Clementine 48.146376 -119.700943 380.0 Gardner 50.051950 -5.181780 115.0 Hamlin 48.146400 -119.701100 400.0 Joppa 50.051950 -5.181780 115.0 Joppahsd 50.051950 -5.181780 115.0 Macetera 46.910326 -96.795151 255.0 Mandarin 37.384998 -122.413002 22.0 Marrs 64.793610 -147.536300 162.0 Nmsu 32.280000 -106.750000 1520.0 Premier -46.529070 168.381540 8.0 Premierhsd -46.529070 168.381540 8.0 Satsuma 43.736000 -96.622600 458.0 Seleta 37.782990 -122.394240 22.0 Tangerine 45.502153 13.739822 0.0 Tarocco 37.384998 -122.413002 22.0 Valencia 50.331110 8.473889 386.0 Verna 48.145909 -119.699913 378.0

Table A.2: Planet Labs ground station locations and altitudes. APPENDIX A. PLANET LABS CUBESAT DATA 154

Unix Time One-way Valid RSSI RSSI ID Light-time (km) Data Downlink Uplink 1416746228.894 521.683 1 -81.5 -74.0 0812 1416746228.956 519.818 1 -80.5 -75.0 0812 1416746229.017 519.107 1 -81.0 -74.5 0812 1416746229.079 520.706 1 -80.0 -73.5 0812 1416746229.140 520.973 1 -81.0 -73.5 0812 1416746229.202 520.351 1 -80.5 -73.5 0812 1416746229.264 518.397 1 -80.5 -73.5 0812 1416746229.328 518.841 1 -80.5 -73.5 0812 1416746229.393 518.930 1 -80.5 -73.5 0812 1416746229.455 518.663 1 -80.5 -74.5 0812 1416746229.517 519.196 1 -79.5 -73.5 0812 1416746229.582 518.219 1 -79.0 -74.0 0812 ......

Table A.3: Sample of Planet Labs ranging data. Appendix B

ALTAIR Meteor Data

The original ALTAIR data consists of in-phase and quadrature components of the reﬂected signal (both LC and RC polarizations) from the illuminated target. Also, the four additional receive horns collect LC signal returns in the form of two additional channels, which consists of the di↵erenced energies in the form of the LC azimuth di↵erence and the LC elevation di↵erence. These two di↵erences are combined in a process known as amplitude comparison monopulse, a form of phase interferometry, to measure the angle within the radar boresight that the object is located at. The range measurement is derived from the time delay of the target return, and so ALTAIR combines all 3 measurements into the position of the object in 3 dimensions. The range rate is calculated using the interpulse Doppler method [153]withhighaccuracy, and combined with the azimuth and elevation di↵erence data gives 3D velocities by ﬁnite di↵erencing.

The traverse (can be thought of as the azimuth in the plane of the elevation angle) and elevation angle o↵sets with respect to the radar boresight are calculated from the raw data. Conversion from the traverse to azimuth angle is accomplished by:

Az =asin(sin(Tr) sec(El)) (B.1) m ·

155 APPENDIX B. ALTAIR METEOR DATA 156

where Azm is monopulse azimuth angle with respect to the radar boresight, Tr is the traverse angle, and El is the elevation angle of the radar boresight (see Figure B.1). Alinearﬁttingisperformedontheﬁnaltraverseandelevationestimates,shownin Figure B.2.

The position of the meteoroid is calculated as:

pos rng cos(El + El )sin(Az + Az ) x · m m pos = 2pos 3 = 2rng cos(El + El )cos(Az + Az )3 (B.2) y · m m 6posz 7 6 rng sin(El + Elm) 7 6 7 6 · 7 4 5 4 5 where pos is the position vector of the meteoroid (consisting of (posx, posy, posz) as the respective x, y and z coordinates), rng is the range of the detection, Az is the azimuth of the radar boresight, and Elm is the monopulse elevation angle with respect to the radar boresight.

The velocity of the meteoroid is calculated as

d(posx) RR c s rng s s Elt rng c c Azt dt · E A · A E dt · E A dt 2 d(posy) 3 = 2 RR c c rng c s Elt rng c s Azt 3 dt · E A · A E dt · E A dt d(posz) Elt 6 7 6 RR sE + rng cE 7 6 dt 7 6 · · dt 7 4 5 4 5 where Azt = Az + Azm

Elt = El + Elm

sE =sin(Elt)

cE =cos(Elt)

sA =sin(Azt)

cA =cos(Azt)(B.3) where RR is the range rate phase velocity calculated using the Doppler technique. The

Elt Azt changes in elevation and azimuth, dt and dt , are calculated using ﬁnite di↵erencing. The range rate and 3D speed of a sample meteoroid detection is shown in Figure B.3. APPENDIX B. ALTAIR METEOR DATA 157

The deceleration can be further derived by ﬁnite di↵erencing, but as the results are noisy (refer to Figure B.4), we instead utilize the ﬁtting technique detailed in Section 5.2.

The RCS of the return can give additional information regarding the detection. Figure B.5 shows the LC and RC polarized returns with respect to altitude for a sample meteoroid as detected by ALTAIR. Figure B.6 shows the same meteoroid as a function of the ﬁtted azimuth and elevation, color coded by LC RCS. Both ﬁgures show an increase in the RCS during the middle of the detection. This can give indication that the meteoroid itself has indeed passed through the center of the radar beam. By analyzing the RCS information, one can also deduce the strength of the plasma created and the amount of ablation taking place. Further application of plasma scattering theory can lead to estimates of meteoroid bulk densities and composition. APPENDIX B. ALTAIR METEOR DATA 158

Figure B.1: Radar and relative-to-radar meteoroid angles. Shown here are azimuth Az,monopulseazimuthAzm, elevation El,andtraverseTr angles. The monopulse elevation angle Elm is not shown, as the meteoroid is located as the same elevation angle as that of the radar boresight. Note that the traverse angle is the angle between the two vectors in the plane tilted by the radar boresight elevation angle. APPENDIX B. ALTAIR METEOR DATA 159

Figure B.2: Sample calculated elevation and traverse angles of a meteoroid detection derived from ALTAIR data with respect to altitude. A linear ﬁt is then performed on both quantities. APPENDIX B. ALTAIR METEOR DATA 160

Figure B.3: Sample range rate and 3D speed of a meteoroid as detected by ALTAIR. Note that the 3D speed is considerably larger than the range rate as the former does not take into account the meteoroid’s direction of travel perpendicular to the radar boresight. APPENDIX B. ALTAIR METEOR DATA 161

Figure B.4: Sample range rate and 3D deceleration of a meteoroid as detected by ALTAIR, calculated using ﬁnite di↵erencing. Note the noisy characteristics of the estimate at the start of detection, an artifact that arises due to ﬁnite di↵erencing. APPENDIX B. ALTAIR METEOR DATA 162

Figure B.5: LC and RC polarized RCS returns as detected by ALTAIR with respect to altitude for a sample meteoroid. The radar beam pattern can be clearly seen as an increase in the middle of the detection due to the plasma located close to the radar boresight. APPENDIX B. ALTAIR METEOR DATA 163

Figure B.6: Fitted azimuth and elevation of a sample meteoroid as detected by AL- TAIR, color coded by LC RCS. The radar beam pattern can be clearly seen as an increase in the middle of the detection due to the plasma located close to the radar boresight. Bibliography

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