GRACE ORBIT ANALYSIS TOOL AND PARAMETRIC ANALYSIS · 2001. 1. 9. · GRACE Orbit Analysis Tool and...

GRACE ORBIT ANALYSIS TOOL AND PARAMETRIC ANALYSIS

by

Philip Curell

Center for Space Research

The University of Texas at Austin

December 1998

CSR-TM-98-05

This work was supported by NASA Contract NAS5-97213

Center for Space ResearchThe University of Texas at Austin

Austin, Texas 78712

Principal Investigator:Byron D. Tapley

Supervised by:R. Steven Nerem

GRACE Orbit Analysis Tool and Parametric Analysis

by

Philip Claude Curell, M.S.E.

The Center for Space Research of the University of Texas at Austin (UTCSR)

is spearheading a satellite mission to produce a new model of the gravity field

with unprecedented accuracy every 30 days throughout a 5-year mission. This

mission, called the Gravity Recovery and Climate Experiment (GRACE), will

employ a pair of identical satellites orbiting in the same orbit plane at an

altitude between 300 and 500 kilometers. As one satellite “chases” the other

satellite approximately 200 kilometers ahead, a microwave link will measure

the range change between them. In addition to providing global high reso-

lution estimates of the Earth’s mean gravity field, the GRACE mission will

also determine the time variability of the gravity field over the duration of

its mission. This thesis presents a quick-look analysis tool called “GOAT”

(GRACE Orbit Analysis Tool) which can be used as a mission design tool for

GRACE. For greater computational speed, GOAT uses a semi-analytic per-

turbation technique for the propagation of the mean orbital elements. The

theories and concepts that were incorporated in the development of GOAT

are presented in addition to the guidelines for using GOAT. Following the de-

scription of GOAT is a presentation of an analysis of the lifetime variability

for given GRACE mission design parameters.

Table of Contents

Abstract vi

List of Tables xii

List of Figures xiv

List of Constants xviii

List of Abbreviations xx

1 Introduction 1

1.1 GRACE Mission Background . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 GRACE Orbit Analysis Tool (GOAT) . . . . . . . . . 3

1.2.2 GRACE Mission Parametric Analysis . . . . . . . . . . 5

2 Development of GOAT 7

2.1 Propagation of the Orbital Elements . . . . . . . . . . . . . . 11

2.1.1 Background of Orbit Propagation . . . . . . . . . . . . 11

2.1.2 Semi-Analytic Liu Theory . . . . . . . . . . . . . . . . 11

2.2 Calculation of Spacecraft Parameters . . . . . . . . . . . . . . 20

2.2.1 Interpretation of AMA/LaRC Data Files . . . . . . . . 20

2.2.2 Definition of Cone and Clock Angles . . . . . . . . . . 22

2.2.3 Transformation to the Satellite Body-Fixed Frame . . . 24

2.2.4 Determination of the Drag Accelerations . . . . . . . . 25

2.2.5 Determination of the SRP Accelerations . . . . . . . . 27

2.3 Atmospheric Density Models . . . . . . . . . . . . . . . . . . . 28

2.3.1 Exponential Atmospheric Model . . . . . . . . . . . . . 28

2.3.2 Drag Temperature Model . . . . . . . . . . . . . . . . 30

2.3.3 Marshall Engineering Thermosphere Model . . . . . . . 30

3 Validation of GOAT 33

3.1 Validating GOAT with MSODP . . . . . . . . . . . . . . . . . 33

3.2 Using 13-month Averaged Flux Predictions . . . . . . . . . . . 38

3.3 Conclusions of the GOAT Validation . . . . . . . . . . . . . . 43

4 Guidelines for Using GOAT 45

4.1 MATLAB Version Guidelines . . . . . . . . . . . . . . . . . . 45

4.1.1 Starting GOAT . . . . . . . . . . . . . . . . . . . . . . 45

4.1.2 Running a GOAT simulation . . . . . . . . . . . . . . . 46

4.1.3 After a GOAT simulation . . . . . . . . . . . . . . . . 50

4.2 FORTRAN Version Guidelines . . . . . . . . . . . . . . . . . . 54

4.2.1 Editing the Inputs . . . . . . . . . . . . . . . . . . . . 54

4.2.2 Compiling and Running GOAT . . . . . . . . . . . . . 54

4.2.3 After a GOAT Simulation . . . . . . . . . . . . . . . . 55

5 GRACE Mission Parametric Analysis 57

5.1 Description of the Analysis . . . . . . . . . . . . . . . . . . . . 57

5.1.1 Parameters of Interest . . . . . . . . . . . . . . . . . . 57

5.1.2 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 Initial Results: Scatter Plots . . . . . . . . . . . . . . . 61

5.2.2 Further Analysis . . . . . . . . . . . . . . . . . . . . . 68

5.3 Conclusions of the Analysis . . . . . . . . . . . . . . . . . . . 79

5.4 Recommendations for the GRACE mission . . . . . . . . . . . 81

A Various Supporting Algorithms 83

A.1 Drag Temperature Model (DTM) . . . . . . . . . . . . . . . . 83

A.2 Modeling SRP Perturbations with the Cylindrical Shadow Model 90

A.3 Conversion between Orbital Elements and Inertial (ECI) State

Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A.3.1 Orbital Elements to Inertial (ECI) State Vector . . . . 92

A.3.2 Inertial (ECI) State Vector to Orbital Elements . . . . 94

A.4 Conversion of Orbit Epoch to Greenwich Sidereal Time and

Julian Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

A.5 Conversion of Inertial (ECI) Vector to RTN Vector . . . . . . 97

A.6 Determination of Sun Vector . . . . . . . . . . . . . . . . . . . 98

A.7 Conversion of Inertial (ECI) Position to Earth-Fixed (ECEF)

Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Bibliography 101

List of Tables

2.1 Sample Points and Weights for a 13-point Gauss-Legendre Quadra-

ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Exponential Atmospheric Model . . . . . . . . . . . . . . . . . 29

3.1 Test Case Profile for GOAT Validation . . . . . . . . . . . . . 34

3.2 Test Case Profiles for Flux Averaging Comparison . . . . . . . 39

5.1 Parameters of Interest for the Parametric Analysis . . . . . . . 58

5.2 Run Profile for the Parametric Analysis . . . . . . . . . . . . . 59

A.1 Spherical Harmonics Expansion Coefficients, Ai, for the Thermo-

pause Temperature and the Atmospheric Constituents . . . . . 85

A.2 Thermal Diffusion Factors and Molecular Weights of the At-

mospheric Constituents . . . . . . . . . . . . . . . . . . . . . . 89

List of Figures

1.1 Illustration of the GRACE Mission Satellite Pair (UTCSR) . . 2

2.1 GOAT Simulation Flow Diagram . . . . . . . . . . . . . . . . 10

2.2 Zonal Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Normalized Forces Data Structure . . . . . . . . . . . . . . . . 21

2.4 Cone Angle α and Clock Angle β . . . . . . . . . . . . . . . . 23

2.5 Transformation from the RTN (Trajectory-Fixed) Frame to the

Satellite (Body-Fixed) Frame . . . . . . . . . . . . . . . . . . 26

3.1 GOAT Run Output . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 MSODP Run Output . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Element-by-Element Differences Between GOAT and MSODP

Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Percent Difference in Semi-major Axis Decay . . . . . . . . . . 37

3.5 MSFC Flux Predictions . . . . . . . . . . . . . . . . . . . . . 39

3.6 Flux Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7 Orbital Decay (Epoch: 1/1/89) . . . . . . . . . . . . . . . . . 41

3.8 Orbital Decay Rate (Epoch: 1/1/89) . . . . . . . . . . . . . . 41

3.9 13-month Averaged Flux and Daily Flux (Epoch: 1/1/89) . . 41

3.10 Orbital Decay (Epoch: 3/1/92) . . . . . . . . . . . . . . . . . 42

3.11 Orbital Decay Rate (Epoch: 3/1/92) . . . . . . . . . . . . . . 42

3.12 13-month Averaged Flux and Daily Flux (Epoch: 3/1/92) . . 42

4.1 GOAT Command Window . . . . . . . . . . . . . . . . . . . . 46

4.2 Various Popup Warnings . . . . . . . . . . . . . . . . . . . . . 47

4.3 Choose Input Type Window . . . . . . . . . . . . . . . . . . . 47

4.4 Initialization with Orbital Elements Window . . . . . . . . . . 48

4.5 Initialization with State Vector Window . . . . . . . . . . . . 48

4.6 Set Options Window . . . . . . . . . . . . . . . . . . . . . . . 49

4.7 MATLAB Command Window . . . . . . . . . . . . . . . . . . 51

4.8 Sim Complete Popup . . . . . . . . . . . . . . . . . . . . . . . 52

4.9 Load Workspace Window . . . . . . . . . . . . . . . . . . . . . 52

4.10 Save Workspace Window . . . . . . . . . . . . . . . . . . . . . 52

4.11 Plot Long Term Data Window . . . . . . . . . . . . . . . . . . 53

5.1 Scatter Plot of Lifetime vs. Initial Altitude . . . . . . . . . . . 63

5.2 Grouping of Initial Altitude Data Points . . . . . . . . . . . . 63

5.3 Scatter Plot of Lifetime vs. Atmosphere Model . . . . . . . . . 64

5.4 Grouping of Atmosphere Model Data Points . . . . . . . . . . 64

5.5 Scatter Plot of Lifetime vs. Flux Profile . . . . . . . . . . . . 65

5.6 Grouping of Flux Profile Data Points . . . . . . . . . . . . . . 65

5.7 Scatter Plot of Lifetime vs. Initial Eccentricity . . . . . . . . . 66

5.8 Grouping of Initial Eccentricity Data Points . . . . . . . . . . 66

5.9 Scatter Plot of Lifetime vs. Initial Phase w.r.t. Sun . . . . . . 67

5.10 Grouping of Initial Phase w.r.t. Sun Data Points . . . . . . . 67

5.11 Atmopheric Bulge and Several Orbit Trajectories . . . . . . . 68

5.12 Lifetime vs. Initial Altitude (Exponential Model) . . . . . . . 69

5.13 Lifetime vs. Phase (eo = 0.001, DTM, +2σ flux) . . . . . . . . 71

5.14 Close-up of Variation (ho = 475 km, eo = 0.001, DTM, +2σ flux) 71

5.15 Lifetime vs. Phase (eo = 0.005, DTM, +2σ flux) . . . . . . . . 72

5.16 Close-up of Variation (ho = 475 km, eo = 0.005, DTM, +2σ flux) 72

5.17 Lifetime vs. Phase (ho = 450 km, DTM, +2σ flux) . . . . . . 73

5.18 Lifetime vs. Phase (ho = 475 km, DTM, +2σ flux) . . . . . . 73

5.19 Lifetime vs. Phase (eo = 0.001, DTM, +2σ daily flux) . . . . 74

5.20 Lifetime vs. Phase (eo = 0.005, DTM, +2σ daily flux) . . . . 74

5.21 Lifetime vs. Phase (ho = 450 km, DTM, +2σ daily flux) . . . 75

5.22 Lifetime vs. Phase (ho = 475 km, DTM, +2σ daily flux) . . . 75

5.23 Lifetime vs. Phase (eo = 0.001, MET, +2σ flux) . . . . . . . . 77

5.24 Close-up of Variation (ho = 460 km, eo = 0.001, MET, +2σ flux) 77

5.25 Lifetime vs. Phase (eo = 0.005, MET, +2σ flux) . . . . . . . . 78

5.26 Close-up of Variation (ho = 465 km, eo = 0.005, MET, +2σ flux) 78

5.27 DTM vs. MET – Four Plots with Lifetime vs. Phase (eo = 0.003) 80

5.28 DTM vs. MET – Average Lifetime vs. Initial Altitude (eo =

0.003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.1 Flux Data Structure . . . . . . . . . . . . . . . . . . . . . . . 84

A.2 Cylindrical Shadow Model . . . . . . . . . . . . . . . . . . . . 91

A.3 Classical Orbital Elements . . . . . . . . . . . . . . . . . . . . 92

List of Constants

ae, Re Earth mean equatorial radius 6378136.3 (m)

be Earth semi-minor (polar) radius 6356751.6 (m)

fe Earth flattening factor 0.00335

g120 gravitational acceleration at 120 km 9.446626 (ms2

)

J2 2nd Earth zonal harmonic coefficient 0.0010826269

J3 3rd Earth zonal harmonic coefficient −0.0000025323

J4 4th Earth zonal harmonic coefficient −0.0000016204

k Boltzmann’s constant 8.31432 ( kg m2

mole s2 deg)

NA Avogadro’s number 1.660421× 10−24

P solar radiation pressure constant 4.61× 10−6 ( Nm2 )

µ Earth gravitational parameter 3.986004415× 1014 (m3

s2)

ωe Earth rotation rate 7.2921158553× 10−5 ( rads

)

2π/24 ( radhr

)

Ωe Earth orbital rate (around Sun) 2π/365 ( radday

)

List of Abbreviations

AMA Analytical Mechanics Associates, Inc.

AN Ascending Node

APL Applied Physics Laboratory

CHAMP Challenging Mini-Satellite Payload

CPU Computer Processing Unit

DASA Daimler-Benz Aerospace

DLR Deutsches Zentrum fur Luft- und Raumfahrt

DN Descending Node

DSS Dornier Satellitensysteme

DSST Draper Semianalytical Satellite Theory

DTM Drag Temperature Model

ECEF Earth-Centered Earth-Fixed

ECI Earth-Centered Inertial

ESSP Earth System Science Pathfinders

FORTRAN Formula Translation (computer language)

FTP File Transfer Protocal

GFZ GeoForschungsZentrum Potsdam

GOAT GRACE Orbit Analysis Tool

GPS Global Positioning System

GRACE Gravity Recovery and Climate Experiment

GSOC German Space Operations Center

GST Greenwich Sidereal Time

GUI Graphical User Interface

GVOP Gaussian Variation of Parameters

HANDE Hoots Analytic Dynamic Ephemeris Theory

JD Julian Date

JPL Jet Propulsion Laboratory

LaRC Langley Research Center

LEO Low Earth Orbit

MJD Modified Julian Date

MSFC Marshall Spaceflight Center

MSODP Multi-Satellite Orbit Determination Program

MTPE Mission to Planet Earth

NASA National Aeronautics and Space Administration

ONERA Office National d’Etudes et de Recherches Aerospatiales

P Perigee Point

PI Principal Investigator

RTN Radial-Transverse-Normal

SALT Semi-Analytic Liu Theory

SRP Solar Radiation Pressure

SST Satellite-to-Satellite Tracking

UT University of Texas at Austin

UTCSR Center for Space Research at the University of Texas at Austin

VOP Variation of Parameters

Chapter 1

Introduction

1.1 GRACE Mission Background

The Center for Space Research of the University of Texas at Austin (UTCSR)

is spearheading efforts to produce a new model of the gravity field with un-

precedented accuracy approximately every 30 days throughout a 5-year mis-

sion. This mission is called the Gravity Recovery and Climate Experiment

or, more simply, GRACE. GRACE will employ a pair of identical satellites

(Figure 1.1) to make global measurements of the gravity field. With a launch

from the Russian launch site Plesetzk on June 23, 2001, the GRACE satellites

will be placed in the same orbit plane. The near polar inclination (87 to

92) will ensure the global coverage and the low altitude (300 km to 500 km)

will ensure sensitivity for a high resolution gravity field. The pair of satellites

will be separated by 100 to 500 km along track (nominally: 200 km ±50 km).

As the satellites fly over the different mass concentrations in the Earth, their

separation distance will vary allowing the determination of the Earth’s gravity

field. In addition to determining the mean gravity field, the GRACE mission

will also determine the time variability of the gravity field. These results will

address a wide range of issues in Earth system science modeling, including

oceanography, hydrology, glaciology, geodesy, and the solid Earth sciences.

Measuring the Earth’s gravity field from space can be accomplished in a

variety of ways. The GRACE mission will employ two methods of measure-

1

Figure 1.1: Illustration of the GRACE Mission Satellite Pair (UTCSR)

ment, satellite-to-satellite “low-low” tracking (SST) and Global Positioning

System (GPS) “high-low” tracking. SST utilizes microwave tracking to accu-

rately measure the separation between the two GRACE satellites flying in low

Earth orbit (LEO). As one satellite “chases” the other satellite approximately

230 kilometers ahead (tandem formation), the microwave link will measure the

exact separation between them. Since the gravitational accelerations will be

different for each satellite, their separation distance will vary. The microwave

tracking measurements will be used with GPS double-differenced phase mea-

surements to determine the gravity field. GPS receivers onboard each satellite

will obtain accurate position and velocity data from the GPS satellite con-

stellation. To correct the SST and GPS measurements for nongravitational

effects such as atmospheric drag and solar radiation pressure, an electrostatic

accelerometer is carried onboard each satellite.

Science resulting from the GRACE mission will further investigations in a

wide range of disciplines which includes oceanography, hydrology, glaciology,

geodesy, as well as the solid Earth sciences. By producing a new model of the

Earth’s gravity field every 30 days, the GRACE Mission will enable the direct

observation of mass movement within the combined solid Earth, ocean and

atmosphere system. Observations of the Earth’s time-varying global gravita-

tional field allows the investigation of the changing mass distribution of the

Earth and the processes involved [7]. The continually changing distribution

2

of ice, snow, and groundwater, the hydromagnetic motion in the Earth’s core,

ocean circulation and sea level, and post-glacial rebound are among the many

processes that can become better understood with satellite gravity measure-

ments.

Further information regarding the GRACE mission and scientific objectives

can be found in the GRACE Science and Mission Requirements Document [16].

1.2 Thesis Overview

This thesis is divided into two major parts: the first covering the development

of the GRACE Orbit Analysis Tool (GOAT) and the second covering a para-

metric analysis of the GRACE mission. The goal of this thesis is to address

certain GRACE mission design issues such as lifetime prediction and orbit

selection. GOAT is a tool that was developed for long term orbit analysis. Af-

ter the testing and validation of GOAT, it was used to perform a parametric

analysis of the lifetime of the GRACE mission.

1.2.1 GRACE Orbit Analysis Tool (GOAT)

This part of the thesis includes Chapters 2, 3, and 4. Chapter 2 presents the

theories and concepts that were incorporated in the development of GOAT.

This includes an introduction to the semi-analytic propagation technique called

SALT (Semi-Analytic Liu Theory), an explanation of the use of parametric

data files generated by Analytical Mechanics Associates (AMA) and NASA’s

Langley Research Center (LaRC), and a discussion of the various atmospheric

density models used. Each section of Chapter 2 addresses a particular moti-

vation behind the development of GOAT. Currently, the Multi-Satellite Orbit

Determination Program (MSODP) developed by UTCSR is used for GRACE

mission analaysis. These simulations are usually time-consuming and expen-

sive. To achieve greater computational speeds, GOAT implements a modified

SALT algorithm for the propagation of the mean orbital elements. Given

the availability of parametric data files from the aerothermal design work

3

performed by AMA and LaRC, the accurate calculation of spacecraft non-

gravitational parameters (such as the ballistic coefficient and reflectivity) is

possible with GOAT. The flexibility of SALT allows GOAT to incorporate

three different atmospheric density models which were incorporated primarily

for the parametric analysis. An additional motivation for the development of

GOAT was the fact that the source code for existing commercial orbit analysis

software is unavailable. With GOAT, modifications to the code is always an

option.

Next, Chapter 3 presents the testing and validation of GOAT. GOAT simu-

lation runs are compared to similar runs executed by UTCSR’s Multi-Satellite

Orbit Determination Program (MSODP). Also, since only 13-month averaged

flux predictions are available for the GRACE mission time-frame, the use of

these type of predictions is compared with the use of acutal daily flux mea-

surements.

Chapter 4 presents guidelines for using GOAT. GOAT can be used to run

mission lifetime simulations given a different set of GRACE mission param-

eters, environment models and initial conditions. GOAT was written in the

MATLAB programming environment as well as in FORTRAN. Several reasons

can be given by the author for this choice, the most important being his fa-

miliarity with MATLAB and FORTRAN programming, as well as the ability

to convert the MATLAB code to C++ with a code conversion utility. The

MATLAB programming environment provided the author with the ability to

use graphical user interfaces (GUIs). Once “goat” is typed in the MATLAB

Command Window, the GOAT user need only “point and click.” The vari-

ous GUI windows that are encountered by the user are explained in detail.

Although the FORTRAN version of GOAT is somewhat less user-friendly, it

does provide added computational speed. (Unlike MATLAB, compilation and

linking of the FORTRAN program is performed before program execution.)

4

1.2.2 GRACE Mission Parametric Analysis

This part of the thesis includes Chapter 5. This chapter first presents the

objectives of the parametric analysis. It also briefly describes the GOAT sim-

ulation runs which provide the input to the parametric analysis.

Chapter 5 then presents the results of the parametric analysis. Under-

standing the parameters that affect the lifetime of the GRACE mission is of

great importance to GRACE mission designers. In this analysis, the design

parameters of interest are the initial altitude, phase with respect to the Sun

(initial longitude of the ascending node), and initial eccentricity. The envi-

ronment model parameters of interest are the atmospheric density model and

flux profile. GOAT was used to generate the data for the parametric analysis

which will provide additional insight to the factors that govern the GRACE

mission.

5

Chapter 2

Development of GOAT

The GRACE Orbit Analysis Tool (GOAT) was developed to provide GRACE

mission planners with a quick-look analysis tool, primarily to assist in the para-

metric analysis of the GRACE mission. The specific goals in the development

of GOAT are:

1. Incorporation of an efficient orbit propagation algorithm.

2. Accurate calculation of spacecraft aerodynamic parameters.

3. Incorporation of various atmospheric models (some which model the

dynamics of the atmosphere using solar activity measurements).

The first goal is addressed with the incorporation of a unique combination

of general and special perturbation techniques that were originally developed

by Liu and Alford [13]. (These combination techniques are commonly classified

as “numeric-analytical” or “semi-analytical.”) Liu and Alford give a semi-

analytic solution for the motion of a satellite under the combined influences of

gravity and atmospheric drag. Their theory, referred to as the Semi-Analytic

Liu Theory (SALT), uses the gravitational effects of the zonal harmonics J2, J3,

and J4, along with the drag effects of an arbitrary atmospheric drag model to

predict satellite motion. Their theory uses the method of averaging to obtain a

set of averaged variational equations which account for the perturbation effects

of gravity and drag. In GOAT, this theory is slightly modified. To account

for the perturbations of drag and solar radiation pressure (SRP), the Gaussian

7

Variation of Parameters (GVOP) is used instead of the drag equations given

by Liu and Alford. A detailed discussion of the modified SALT algorithm is

given in Section 2.1.

Due to their availability, parametric data files that were generated by An-

alytical Mechanics Associates (AMA) and NASA’s Langley Research Center

(LaRC) are incorporated to address the second goal. These data files, which

are based on AMA/LaRC aerodynamic and thermal model calculations using

FREEMOL simulations [15], contain normalized aerodynamic and solar radia-

tion forces as a function of the cone and clock angles. These angles are defined

in Section 2.2 where a description of the AMA/LaRC data files is also given.

To address the third goal, three different atmospheric density models are

incorporated. The GOAT user is given the option of implementing the Drag

Temperature Model (DTM) [1], Marshall Engineering Thermosphere (MET)

Model [10], or the more simple, yet computationally speedy, Exponential At-

mospheric Model [18]. Section 2.3 presents a brief discussion of these models.

Their concepts are presented with greater detail in Appendix A.

Since AMA/LaRC has also provided parametric data files for SRP pertur-

bation modeling, the solar radiation pressure (SRP) on the satellite can be

modeled for further simulation accuracy. Since SALT does not include the

effects of SRP, the Gaussian Variation of Parameters (GVOP) method [9], a

general perturbation technique, can be used with a cylindrical Earth shadow

model to simulate the perturbation effects of SRP. It was found that these

perturbations did not have a significant effect on the simulation, therefore, the

current versions of GOAT do not implement this modeling. For completeness,

the concept is briefly discussed in Appendix A.

GOAT was primarily developed in the MATLAB programming environ-

ment. All references to MATLAB-related code will carry the suffix “.m” and

may be referred to as an “m-file”. As increased computational speed was de-

sired later in the development process, a FORTRAN version of GOAT was

coded. References to FORTRAN-related code will not carry a suffix and will

usually contain all capital letters. (Guidelines for using both versions are pro-

8

vided in Chapter 4.)

The simulation flow diagram shown in Figure 2.1 illustrates how the GOAT

simulation was constructed. This construction is used for each simulation run

for the parametric analysis of the GRACE mission lifetime variability. No-

tice that the “Execute Directive” block shows a propagation scheme for the

two GRACE satellites without orbit restoration maneuvers (reboosts) or sep-

aration maintenance. Although GOAT was coded in MATLAB and in FOR-

TRAN, the illustration describes the general algorithm used in both versions.

9

Initialize Run

Set Run Options

Input Initial Mean Orbital Elements

Timing Parameters: • Input Duration of Simulation• Input Simulation Step Size

Execute Directive

Start Simulation

yes

n > 13 ?

yes

no

Determine relative velocity, v rel

Determine density, ρ

Evaluate GVOP function.

Add result to summation.

Determine altitude, h

no

Obtain aerodynamic force fromAMA/LaRC data files.Compute the averaged drag perturbation terms

(numerical portion of SALT algorithm)CompleteQuadraturecomputation.

Simulation complete?or

Below min. altitude?

Input Simulation Epoch

Spacecraft Parameters:• Input Initial Spacecraft Mass• Select Source for Ballistic Coefficients (use LaRC or use specified parameters)

Propagation: • Select a Type of Propagation (maintenance, lifetime, etc.)• Select an Integrator• Select an Atmospheric Drag Model

Quit Simulation ACall Integrator (for both satellites)

A Compute the timederivatives of the meanorbital elements (withSALT) for the integrator.

Compute the total averaged perturbation effect(then return the derivatives to the integrator)

Evaluate the averaged variationalequations due to J2,J3, and J4

Compute the averaged gravity perturbation terms(analytical portion of SALT algorithm)

Advance simulation time

Compute range between satellites

Approximate theintegrals with a 13-ptGauss-LegendreQuadrature.

n > 13 ?

yes

no

Determine Sun-to-satellitevector and Earth shadowbinary number.

Evaluate GVOP function.

Add result to summation.

Determine Sun position vector

Obtain SRP force fromAMA/LaRC data files.Compute the averaged SRP perturbation terms

(numerical portion of SALT algorithm)CompleteQuadraturecomputation.

Approximate theintegrals with a 13-ptGauss-LegendreQuadrature.

Figure 2.1: GOAT Simulation Flow Diagram

10

2.1 Propagation of the Orbital Elements

This section presents the modified Semi-Analytic Liu Theory (SALT) which is

used for the propagation of the mean orbital elements. This modified theory

accounts for the gravity, atmospheric drag, and SRP perturbation effects.

2.1.1 Background of Orbit Propagation

The two main techniques for modeling the motion of a satellite are special and

general perturbation techniques. Special perturbation techniques numerically

integrate the equations of motion including all relevant perturbations. Gen-

eral perturbation techniques replace the equations of motion with analytical

expressions that approximate the motion of the satellite. The Gaussion Varia-

tion of Parameters method is an example of a general perturbation technique.

Combinations of the two techniques have been developed to use the best at-

tributes of each and are often referred to as semi-analytic techniques. These

semi-analytic techniques typically use the speed from the general techniques

and the accuracy from the special techniques in some optimal fashion. Ana-

lytical expressions are often used to model the gravitational effects, whereas

the drag effects are usually evaluated numerically to maintain the accuracy of

the atmospheric density models.

A semi-analytic technique developed by Liu and Alford [13] was imple-

mented in GOAT and is briefly explained in the following section. A vari-

ety of alternative semi-analytic methods have been developed including those

by Cefola et al (with the Draper Semi-analytical Satellite Theory, DSST) [5]

and by Hoots et al (with the Hoots Analytic Dynamic Ephemeris Theory,

HANDE) [11].

2.1.2 Semi-Analytic Liu Theory

Liu and Alford give a semi-analytic solution for the motion of a satellite un-

der the combined influences of gravity and atmospheric drag. Their theory,

referred to as SALT, uses the gravitational effects of the zonal harmonics J2,

11

J3, and J4, along with the drag effects of an arbitrary atmospheric drag model

to predict satellite motion. SALT basically extends a system of first-order

ordinary differential equations for a set of well-defined mean orbital elements

to include the drag effect due to a rotating atmosphere [14]. Notice that al-

though J22 is a second-order effect, it is included since it is of the same order

as J3 and J4. The implementation of SALT is found in the funciton Deriv.m

(MATLAB) and DERIV (FORTRAN) subroutines.

To remove the dependence on the fast variable, the mean anomaly (M),

the

method of averaging [12] is applied. After the method of averaging is applied,

the equations of motion for a satellite are:

am = 〈ad〉+ 〈as〉+ 〈ag〉em = 〈ed〉+ 〈es〉+ 〈eg〉im = 〈id〉+ 〈is〉+ 〈ig〉

Ωm = 〈Ωd〉+ 〈Ωs〉+ 〈Ωg〉ωm = 〈ωd〉+ 〈ωs〉+ 〈ωg〉Mm = 〈Md〉+ 〈Ms〉+ 〈Mg〉 (2.1)

where a is the semi-major axis, e is the eccentricity, i is the orbit inclination,

Ω is the longitude of the ascending node, ω is the argument of perigee, and M

is the mean anomaly. The 〈x〉 refers to the time rate of change of the given

orbital element x over one cycle of mean anomaly. The subscripts of “d”, “s”

and “g” refer to drag, SRP and gravity, respectively.

Computation of the Averaged Gravity Effects

The 2nd, 3rd, and 4th zonal harmonics of the oblateness of the Earth form

the potential for the gravity perturbation effects which are modeled by SALT.

Figure 2.2 depicts the gravitational departure from a perfect sphere due to

12

J2

J3

J4

Figure 2.2: Zonal Harmonics

J2, J3, and J4. The bands of latitude show zones in which the potential is

alternately increasing and decreasing. J2 is by far the strongest perturbation

and is considered to be a first order gravity perturbation. The second order

effects of J3, J4, and J22 are also considered in the gravity modeling.

The averaged gravity effects, as given by Liu and Alford [13], are:

〈ag〉 = 0 (2.2)

〈eg〉 = − 3

32nJ2

2

(Re

p

)4

sin2i(14−15 sin2i

)e(1−e2

)sin2ω

− 3

8nJ3

(Re

p

)3

sini(4−5 sin2i

) (1−e2

)cosω

− 15

32nJ4

(Re

p

)4

sin2i(6−7 sin2i

)e(1−e2

)sin2ω (2.3)

〈ig〉 = − 3

64nJ2

2

(Re

p

)4

sin 2i(14−15 sin2i

)e2 sin2ω

− 3

8nJ3

(Re

p

)3

cosi(4−5 sin2i

)e cosω

− 15

64nJ4

(Re

p

)4

sin 2i(6−7 sin2i

)e2 sin2ω (2.4)

〈Ωg〉 = − 3

2nJ2

(Re

p

)2

cosi

13

− 3

2nJ2

2

(Re

p

)4

cosi[

94+ 3

2

√1−e2 − sin2i

(52

+ 94

√1−e2

)+ e2

4

(1+ 5

4sin2i

)+ e2

8

(7−15 sin2i

)cos2ω

]− 3

8nJ3

(Re

p

)3(15 sin2i−4

)e coti sinω

+15

16nJ4

(Re

p

)4

cosi[(

4−7 sin2i) (

1+ 32e2)

−(3−7 sin2i

)e2 cos2ω

](2.5)

〈ωg〉 =3

4nJ2

(Re

p

)2(4−5 sin2i

)

+3

16nJ2

2

(Re

p

)448−103 sin2i+ 215

4sin4i+

(7− 9

2sin2i

− 458

sin4i)e2 + 6

(1− 3

2sin2i

) (4−5 sin2i

)√1−e2

− 14

[2(14−15 sin2i

)sin2i−

(28− 158 sin2 i

+ 135 sin4i)e2]

cos2ω

+3

8nJ3

(Re

p

)3 [(4−5 sin2i

)sin2i−e2cos2i

e sini+ 2 sini (13

−15 sin2i)e]

sinω

− 15

32nJ4

(Re

p

)416− 62 sin2i+ 49 sin4i+ 3

4

(24−84 sin2i

+ 63 sin4i)e2 +

[sin2i

(6−7 sin2i

)− 1

2(12

− 70 sin2i+ 63 sin4i)e2]

cos2ω

(2.6)

〈Mg〉 = n+3

2nJ2

(Re

p

)2(1− 3

2sin2i

)√1−e2

14

+3

2nJ2

2

(Re

p

)4(1− 3

2sin2i

)2(1−e2) +

[54

(1− 5

2sin2i

+ 138

sin4i)

+ 58

(1−sin2i− 5

8sin4 i

)e2

+ 116

sin2i(14−15 sin2i

) (1− 5

2e2)

cos2ω]√

1−e2

+3

8nJ2

2

(Re

p

)4

1√1−e2

3[3− 15

2sin2i+ 47

8sin4i+

(32−5 sin2i

+ 11716

sin4i)e2 − 1

8

(1+5 sin2i− 101

8sin4i

)e4]

+ e2

8sin2i

[70−123 sin2i+

(56−66 sin2i

)e2]

cos2ω

+ 27128e4 sin4i cos4ω

− 3

8nJ3

(Re

p

)3

sini(4−5 sin2i

) 1−4e2

e

√1−e2 sinω

− 45

128nJ4

(Re

p

)4(8−40 sin2i+35 sin4i

)e2√

1−e2

+15

64nJ4

(Re

p

)4

sin2i(6−7 sin2i

) (2−5e2

)√1−e2 cos2ω (2.7)

where the mean motion n=√µ/a3, Re is the Earth’s equatorial radius, and p

is the semi-parameter (semi-latus rectum) of the orbit and is given by:

p = a(1− e2

)(2.8)

Computation of the Averaged Drag and SRP Effects

The modification of SALT is introduced in the modeling of the drag and SRP

perturbations. Instead of using the expressions given by Liu and Alford for

drag, the Gaussian Variation of Parameters (GVOP) is used to model the

perturbations of drag and SRP.

15

Lagrange and Gauss both developed variation of parameter (VOP) meth-

ods to analyze perturbations on orbital motion. The Gaussian VOP, a version

of the Lagrange VOP or the Lagrange planetary equations, is used to express

the rates of change of the orbital elements. Convienently, Gauss developed

his VOP equations in the RTN trajectory-fixed frame. (Notice that the x-axis

for this frame is along the radius vector, the y-axis is along the direction of

satellite motion in the orbit plane, and the z-axis is normal to the orbit plane.)

A disturbance in this frame can be defined by:

~f = fRR + fT T + fNN (2.9)

where ~f is used to indicate an acceleration, or a specific force. The Gaussian

VOP equations can then be defined:

da

dt=

2

n√

1−e2

[e sinνfR+

p

rfT

]de

dt=

√1−e2

na

[sinνfR+

(cosν+

e+cosν

1+e cosν

)fT

]di

dt=

r cos(ω+ν)

na2√

1−e2fN

dΩ

dt=

r sin(ω+ν)

na2√

1−e2 sinifN

dω

dt=

√1−e2

nae

[− cosνfR+sinν

(1+

r

p

)fT

]− r coti sin(ω+ν)

na2√

1−e2fN

dM

dt=

1

na2e[(p cosν − 2er)fR − (p+ r)sinνfT ] (2.10)

where dxdt

is the rate of change in an orbital element x over time. With the

GVOP implementation in GOAT, the disturbance ~f is either the drag or SRP

perturbation. Given the GVOP equations above, the method of averaging is

applied in the following fashion:

16

〈x〉 =1

2π

∫ 2π

0

dx

dtdM (2.11)

where the transformation from mean anomaly M to true anomaly ν is:

dM =(r/a)2√(1− e2)

dν (2.12)

Determination of Relative Velocity

To account for the drag perturbation, the relative velocity of the satellite with

respect to the atmosphere must be determined. V , the velocity of the satellite

relative to the atmosphere, is given by:

V =

[µ

p

(1+e2+2e cosν

)] 12

1− (1−e2)32

1+e2+2e cosν

ωen

cosi

(2.13)

Determination of Radial Position and Altitude

The determination of the radial position r and altitude z of the satellite

is required for the computation of the density by a given atmospheric den-

sity model. The function Quadrature.m (MATLAB), DRAG QUADRATURE

(FORTRAN) subroutines use the following equations to solve for r and z.

Given the mean orbital elements, the orbit radius is given by:

rtwo−body =p

1 + e cosν(2.14)

and the short period variation to the radial position is:

17

δr = J2R2e

p

1

4sin2i cos 2 (ω+ν)

−[1

2− 3

4sin2i

] [1+

e cosν

1+√

1−e2+

2√1−e2

r

a

](2.15)

The resulting radial position is:

r = rtwo−body + δr (2.16)

To approximate the altitude, the Earth’s surface directly below the satellite

(the reference ellipsoid) must first be determined. The Earth’s surface is given

by:

rsurface = Re

1−fe [sini sin(ω+ν)]2

(2.17)

where the Earth flattening factor fe=0.00335. The altitude z is simply found

by:

z = r − rsurface (2.18)

Integral Evaluation with the Gauss-Legendre Quadrature

The Gauss-Legendre quadrature [6] is used to approximate the integration

which gives the averaged drag perturbation effects on the mean orbital ele-

ments (see the GVOP equations). The Gauss-Legendre quadrature is imple-

mented with the function Quadrature.m (MATLAB) and DRAG QUADRATURE

(FORTRAN) subroutines.

18

i sample point, xi weight, γi

1 4.9690399098278e-02 0.04048400476532 2.5887226364506e-01 0.09212149983783 6.2336081246353e-01 0.13887351021984 1.1235926877659e+00 0.17814598076195 1.7326111217452e+00 0.20781604753696 2.4175865012266e+00 0.22628318026297 3.1415926535898e+00 0.23255155323098 3.8655988059530e+00 0.22628318026299 4.5505741854343e+00 0.207816047536910 5.1595926194137e+00 0.178145980761911 5.6598244947161e+00 0.138873510219812 6.0243130435345e+00 0.092121499837813 6.2334949080813e+00 0.0404840047653

Table 2.1: Sample Points and Weights for a 13-point Gauss-Legendre Quadra-ture

For a given function, f(x), over the closed interval [a, b], the n-point Gauss-

Legendre quadrature approximates the integration of f(x) by:

∫ b

af (x) dx ≈ b− a

2

n∑i=1

γif (xi) (2.19)

where γi is the tabulated Gaussian weight associated with the tabulated Gaus-

sian sample point xi in [a, b]. A 13-point Gauss-Legendre quadrature is suffi-

cient for the integration of the drag equations used in the SALT algorithm [11].

The table of Gaussian sample points and associated weights for the 13-point

quadrature is given in Table 2.1. For SALT, the sample points are the true

anomaly ν over the closed interval [0, 2π].

19

2.2 Calculation of Spacecraft Parameters

Using the FREEMOL simulation [15], Analytical Mechanics Associates (AMA)

and NASA’s Langley Research Center (LaRC) generated drag and SRP para-

metric data files (so-called “AMA/LaRC data files”) that can be used to model

spacecraft non-gravitational parameters such as the ballistic coefficient (B) and

reflectivity (η). These data files contain the normalized aerodynamic forces and

solar radiation forces as a function of the cone and clock angle. The follow-

ing sections present how the AMA/LaRC data files are used to determine the

effects of drag and SRP perturbations. The discussions include: the interpre-

tation of the AMA/LaRC data files, the definition of cone and clock angles,

the transformation of a given unit vector to the satellite body-fixed frame,

and lastly, the determination of the drag and SRP accelerations. Notice that

the transformation mentioned is necessary for the correct determination of the

cone and clock angles.

2.2.1 Interpretation of AMA/LaRC Data Files

This section provides an explanation of how the AMA/LaRC data files are

converted to GOAT data structures (specifically for the MATLAB version

of GOAT). Notice that the FORTRAN version of GOAT directly uses the

AMA/LaRC FORTRAN subroutine lib.f that accompanied the data files.

The MATLAB algorithm which was used to convert the drag and SRP

data files to two-dimensional data structures was adapted from the lib.f sub-

routine. This algorithm, found in the init LaRC data.m (MATLAB) subrou-

tine, is used to operate on the two AMA/LaRC data files and construct the

six two-dimensional data structures used by the MATLAB version of GOAT:

fdx , fdy , fdz , fSRPx , fSRPy , and fSRPz . The resulting data structure for each of

the normalized aerodynamic and solar radiation forces is shown in Figure 2.3.

Notice that the clock angle β for each structure varies from 0 to 360 in

increments of 5. The increments of cone angle α between 0-10 and 170-180

are only 1. Elsewhere, the cone angle α incrementation is 5.

20

α

β

normalized forces

Figure 2.3: Normalized Forces Data Structure

21

2.2.2 Definition of Cone and Clock Angles

In order to use the AMA/LaRC data files, two angles must be determined:

the cone angle α and the clock angle β. The determination of α and β for

the aerodynamics is found in the function invBfromLaRC.m (MATLAB) and

GET BCOEFF (FORTRAN) subroutines where the ballistic coefficient B is

determined and then used in SALT. (The determination of α and β for the

SRP accelerations is not currently coded.)

The cone and clock angles are measured from the satellite body-fixed frame

as shown in Figure 2.4. In this frame, the x-axis is the long axis of symmetry

of the satellite, pointing in the direction of the Ku/Ka (microwave) horn, the

y-axis is the vertical axis of symmetry, pointing towards the radiator, and the

z-axis completes the right-handed coordinate system. The purpose of showing

both satellites in Figure 2.4 is due to the fact that the x-axes of their body-

fixed frames are pointed toward one another, thus making the visualization of

the cone and clock angle slightly different for each.

In the case of aerodynamic drag, the cone and clock angles are measured

from the body-fixed frame to the wind unit vector,vwind. This case is shown in

Figure 2.4 for both GRACE satellites. (The wind unit vector,vwind, points in

the opposite direction of the velocity unit vector,vsatellite.) Notice that these

angles are greatly exaggerated for illustration purposes. (In the case of SRP,

the cone and clock angles are measured from the body-fixed frame to the sun-

to-satellite unit vector, vsun−to−sat.) The following trigonometric definitions

apply to the motion through the atmosphere (in the case of drag) as well as

motion through solar radiation (in the case of SRP). The cone angle, α, is

defined as the angle between the −x axis and the wind unit vector (or the

sun-to-satellite unit vector) and is given by:

cosα = −x ⇒ α = cos−1 (−x) (2.20)

The clock angle, β, is defined as the rotation angle about the −x axis and is

22

α

β

z

y

xvwind

ClockAnglePlane

Cone AnglePlane

α

z

yx

vwind Cone AnglePlane

β

Clock AnglePlane

LeadingSatellite

LaggingSatellite

Figure 2.4: Cone Angle α and Clock Angle β

23

given by:

tan β =sin β

cos β=

y

−z ⇒ β = tan−1(y

−z

)(2.21)

Notice that quadrant verification is necessary with the clock angle definition.

(In the code, the atan2 function is used to accomplish this check.)

Once these two angles are determined, the normalized aerodynamic and

solar radiation forces (which are a function of α and β) can be found. Given

these two angles as input, GOAT uses an interpolation algorithm to “look-up”

the particular normalized force in the drag or SRP data structures (depending

on the perturbation being modeled). Notice that the interpolation algorithms

are found in the function INTERP lin2.m (MATLAB) and LATTICE (FOR-

TRAN) subroutines.

2.2.3 Transformation to the Satellite Body-Fixed Frame

Since the GRACE satellites must be facing each other to maintain the mi-

crowave link between them, their orientation, specifically their cone and clock

angles, will be quite different. Given this difference, the transformation from

the RTN (trajectory-fixed) frame to the satellite body-fixed frame is different

as well. This transformation is illustrated in Figure 2.5. At the top, notice how

the transverse direction (T for the RTN trajectory-fixed frame) is in the same

direction for each satellite. At the bottom, in order to maintain a direct line

of sight for the microwave link, notice how the x-direction (x for the satellite

body-fixed frame) for each satellite is directed toward the other satellite. Given

this different orientation with respect to the atmosphere and solar radiation,

the cone and clock angles will be quite different and therefore yield different

drag and SRP accelerations. This is due to the fact that different surfaces on

each satellite are being exposed to the relative wind and solar radiation.

For the leading satellite, to convert a given RTN vector to the intermediate

satellite frame, the transformation is simply:

24

x′ = −Ty′ = N

z′ = −R (2.22)

and for the lagging satellite, the transformation is:

x′ = T

y′ = −Nz′ = −R (2.23)

For the transformation from the intermediate satellite frame to the actual

satellite frame, a rotation matrix used. For a −1 pitch about the y′-axis, the

rotation matrix is:

R2(−1) =

cos(−1) 0 − sin(−1)0 1 0

sin(−1) 0 cos(−1)

(2.24)

2.2.4 Determination of the Drag Accelerations

The force of drag acting on a satellite is given by:

~Fd = msat~ad = CdA1

2ρv2ev

= CdAqev

= ~fdq (2.25)

25

NR

T

y'

x'

z'

y

x

z

Leading SatelliteLagging Satellite

NR

T

y'

x'

z'y

x

z

180˚ rotation

180˚ rotation

-1˚ rotation

-1˚ rotation

NOTE: Scales are exaggerated for illustration.

Figure 2.5: Transformation from the RTN (Trajectory-Fixed) Frame to theSatellite (Body-Fixed) Frame

26

where Cd is the satellite’s coefficient of drag, A is the characteristic area, ρ

is the atmospheric density, v is the velocity of the satellite relative to the co-

rotating atmosphere, and ev is the unit vector in the direction of the velocity

v. With the dynamic pressure q = 12ρv2, the remaining variables form the

normalized force ~fd = CdAev that is given in the AMA/LaRC data files where

fd is given for each satellite body-fixed axis, x, y, and z.

Therefore, to “un-normalize” the normalized force fd found in the AMA/LaRC

data files, fd need only be scaled by the dynamic pressure, q.

2.2.5 Determination of the SRP Accelerations

It was shown that SRP perturbations do not have a significant effect on the

simulation, therefore they are not modeled in the current versions of GOAT.

The discussion that follows has been included for completeness.

The force of SRP acting on a satellite is given by:

~FSRP = P ε (1 + η)Aesun−to−sat

=P

2ε ~fSRP (2.26)

where P is the solar radiation pressure constant, ε is the Earth shadow binary

(0 for shadow and 1 for sunlight), η is the satellite’s surface reflectivity, A

is the characteristic area, and esun−to−sat is the unit vector in the direction

of the solar radiation (from the sun to the satellite). The normalized force~fSRP = 2 (1 + η)Aesun−to−sat is given in the AMA/LaRC data files where

fSRP is given for each satellite body-fixed axis, x, y, and z.

Therefore, to “un-normalize” the normalized force fd found in the AMA/LaRC

data files, fd need only be scaled by half of the solar radiation pressure con-

stant, P/2, and the Earth shadow binary, ε.

27

2.3 Atmospheric Density Models

The drag on a satellite’s motion is directly caused by atmospheric density. The

atmospheric density ρ is related to the drag acceleration ~ad by the following

equation:

~ad = −1

2

CdA

msat

ρ|~vrel|2vrel (2.27)

where Cd is the coefficient of drag, A is the characteristic area, msat is the

satellite mass, and ~vrel is the satellite velocity relative to the atmosphere.

The ballistic coefficient B= CdAmsat

and the atmospheric density ρ are the most

difficult parameters in the equation to determine. The option of using aerody-

namic data files from NASA’s Langley Research Center simplifies the process

of accurately determining the ballistic coefficient. To find atmospheric density,

an atmospheric density model must be selected.

Atmospheric density models range widely from simple and efficient to

complex and computationally intensive. Three different atmospheric density

models are used in GOAT. The Marshall Engineering Thermosphere (MET)

Model [10] was used since it is the adopted atmosphere model for GRACE

mission design. The Drag Temperature Model (DTM) [1] is also used, al-

though primarily for comparison purposes. Both DTM and MET model the

dynamics of the atmosphere using solar flux measurements. For greater speed

(at the expense of less accuracy), the GOAT user can choose to implement the

Exponential Atmospheric Model [18] which is a static, exponential decaying

model of the atmosphere.

2.3.1 Exponential Atmospheric Model

This static model of the atmosphere assumes that the density decays exponen-

tially from the surface of the Earth. Using Table 2.2, the atmospheric density,

ρ, is found by:

28

Actual Altitude Reference Altitude Reference Density Scale Heighth (km) h0 (km) ρ0 (kg/m3) H (km)

0-25 0 1.225 7.24925-30 25 3.899e-02 6.34930-40 30 1.774e-02 6.68240-50 40 3.972e-03 7.55450-60 50 1.057e-03 8.38260-70 60 3.206e-04 7.71470-80 70 8.770e-05 6.54980-90 80 1.905e-05 5.79990-100 90 3.396e-06 5.382100-110 100 5.297e-07 5.877110-120 110 9.661e-08 7.263120-130 120 2.438e-08 9.473130-140 130 8.484e-09 12.636140-150 140 3.845e-09 16.149150-180 150 2.070e-09 22.523180-200 180 5.464e-10 29.740200-250 200 2.789e-10 37.105250-300 250 7.248e-11 45.546300-350 300 2.418e-11 53.628350-400 350 9.158e-12 53.298400-450 400 3.725e-12 58.515450-500 450 1.585e-12 60.828500-600 500 6.967e-13 63.822

Table 2.2: Exponential Atmospheric Model

ρ = ρ0e−h−h0

H (2.28)

where the reference altitude h0, reference density ρ0, and scale height H are

used with the actual altitude h. This model uses the U.S. Standard Atmosphere

(1976) for 0 km, CIRA-72 for 25-500 km, and CIRA-72 with T∞=1000K for

above 500 km.

The Exponential Atmospheric Model is implemented with the function Density-

Exp.m (MATLAB) and EXP DENSITY (FORTRAN) subroutines.

29

2.3.2 Drag Temperature Model

With the DTM model, various constituent densities contributing to the total

density are expanded in terms of spherical harmonics. The atmospheric con-

stituents that are of greatest significance are molecular nitrogen (N2), atomic

oxygen (O), molecular oxygen (O2), helium (He), and molecular hydrogen

(H2). Each constituent has a density that can be represented by:

ρj(z) = A1jeGj(L)−1fi(z) (2.29)

where A1j is the first coefficient for the spherical harmonics expansion for

constituent j, Gj(L) is a function which represents a spherical harmonic ex-

pansion using the remaining 35 coefficients (A2j→A36j). The function fj(z)

results from the integration of a diffusive equilibrium distribution. The 36

coefficients for the thermopause temperature and the atmospheric constituent

densities are found in Table A.1 of the Appendix. Once these functions are

determined, the total density is simply found by:

ρtotal(z) =5∑j=1

ρj(z) (2.30)

For much greater detail on the implementation of the DTM atmospheric

density model, see Appendix A.1. DTM was is implemented with the func-

tion Density-DTM.m and function GDELRB.m subroutines for the MATLAB

version of GOAT and with the DTM DENSITY and GDELRB subroutines for

the FORTRAN version of GOAT.

2.3.3 Marshall Engineering Thermosphere Model

The Marshall Engineering Thermosphere Model (MET) is a modified Jacchia

1970 model that includes some of the spatial and temporal variation patterns

30

found in the Jacchia 1971 model. MET was developed at NASA’s Marshall

Spaceflight Center (MSFC) primarily for engineering applications. The MET

code is retrievable from the National Space Science Data Center’s anonymous

FTP site:

ftp://nssdc.gsfc.nasa.gov/pub/models/atmospheric/met/

MET was is implemented only with the MET DENSITY subroutine for

the FORTRAN version of GOAT.

31

Chapter 3

Validation of GOAT

The next few sections will present how GOAT was tested and validated. In this

process, the UTCSR’s Multi-Satellite Orbit Determination Program (MSODP)

was used as the validation tool. Also, since only 13-month averaged flux

predictions are available for the GRACE mission time-frame, the use of these

type of predictions are compared with the use of actual daily flux.

3.1 Validating GOAT with MSODP

The objective in the testing and validation of GOAT was to demonstrate

reasonable simulation results with GOAT as compared to results obtained with

MSODP for a given test case. The GRACE mission parameters, environment

models and initial conditions for the test case are given in Table 3.1.

The plots of the mean orbital elements which resulted from running the

test case with GOAT and MSODP are shown in Figures 3.1 and 3.2, respec-

tively. Figure 3.3 shows a plot of the element-by-element differences between

the MSODP and GOAT runs. The difference in the given element x is given

by: ∆x = x(MSODP ) − x(GOAT ). It was expected that there will be some dif-

ferences since MSODP numerically integrates the equations of motion thereby

propagating the initial state and GOAT uses a semi-analytic propagation tech-

nique to propagate the mean orbital elements. Also, in order to be able to

make comparisons to the GOAT simulation results, the MSODP inertial states

33

Parameter

Spacecraft Mass 400 kgInitial Altitude 465 kmInitial Orbital Elementsa0 6843.111 kme0 0.00121i0 87

Ω0 0

ω0 0

M0 0

Orbit Epoch 1/10/1991Atmospheric Density Model DTMFlux Profile 2σ

Table 3.1: Test Case Profile for GOAT Validation

go through a conversion algorithm to obtain the mean orbital elements.

As shown by the decay in the semi-major axis a, the GOAT simulation

predicts a lifetime of 680 days, whereas the MSODP simulation predicts a

lifetime of 595 days. Although this may appear to imply that the accuracy

of GOAT is rather poor, further analysis shows that the decay predicted by

GOAT is within 10% of the decay predicted by MSODP until the 400th day

of the simulation is reached. This is shown in Figure 3.4.

The period of the eccentricity variations is quite similar, however the ampli-

tude is somewhat different. The largest difference in the inclination i is 0.015

which makes this difference almost insignificant. The rate of the longitude of

ascending node Ω for the GOAT simulation is very comparable to the MSODP

simulation, both approximately equal to -0.41/day. Although the argument

of perigee ω for each run appears to vary greatly from one another, when the

eccentricity is not approaching zero, the values for the argument of perigee

are quite comparable. Notice that for each run, the rate of the argument of

perigee ω is approximately equal to -3.83/day when the eccentricity is above

0.0005.

This confirms the inherent limitation in the use of the classical orbital ele-

34

ments. As the eccentricity approaches zero, the argument of perigee becomes

undefined. Evidence of this limitation is shown in the plots. Therefore, the

use of GOAT is limited to simulation runs where there are no occurences of

small eccentricity (below 0.0005).

Another noteworthy comparison is the simulation time. Compilation, link-

ing and execution of the FORTRAN version of GOAT took no more than 10

seconds. On the other hand, a similar MSODP run took nearly 8 CPU hours

on a Cray J90.

35

100 200 300 400 500 6006600

6700

6800

a (k

m)

100 200 300 400 500 600

0.51

1.52

2.5x 10

-3e

100 200 300 400 500 600

86.97

86.98

86.99

i (de

g)

100 200 300 400 500 600100

200

300

Om

ega

(deg

)

100 200 300 400 500 600

100

200

300

days into GRACE mission

omeg

a (d

eg)

Figure 3.1: GOAT Run Output

0 100 200 300 400 500

6600

6700

6800

a (k

m)

0 100 200 300 400 500

0.51

1.52

x 10-3

e

0 100 200 300 400 50086.96

86.98

i (de

g)

0 100 200 300 400 500

150200250300350

Oem

ga (

deg)

0 100 200 300 400 5000

200


omeg

a (d

eg)

Figure 3.2: MSODP Run Output

36

50 100 150 200 250 300 350 400 450 500 550

-150

-100

-50

0

del a

(km

)

50 100 150 200 250 300 350 400 450 500 550-10

-5

0

5

x 10-4

del e

50 100 150 200 250 300 350 400 450 500 550-15-10

-50

x 10-3

del i

(de

g)

50 100 150 200 250 300 350 400 450 500 550-2

-1

del O

meg

a (d

eg)

50 100 150 200 250 300 350 400 450 500 550-300-200-100

0


del o

meg

a (d

eg)

Figure 3.3: Element-by-Element Differences Between GOAT and MSODPRuns

50 100 150 200 250 300 350 400 450 500 550

10

20

30

40

50

60

70

80

90

100


Per

cent

Diff

eren

ce in

SM

A D

ecay

(%

)

Figure 3.4: Percent Difference in Semi-major Axis Decay

37

3.2 Using 13-month Averaged Flux Predictions

The orbit epoch for the parametric analysis is June 23, 2001, which is near the

maximum for solar cycle 23. Since only 13-month averaged flux predictions

are available for this time-frame, the reliability of using 13-month averaged

flux versus using daily flux must be evaluated.

For the parametric analysis found in Chapter 5, 13-month averaged flux

predictions from NASA’s Marshall Spaceflight Center (MSFC) were used. Fig-

ure 3.5 shows the MSFC flux predictions for solar cycles 23 and 24. Given these

predictions, there is a 95% probability that the flux will be no greater than

the value shown by the 95% curve. The same follows for the 50% and 5%

curves shown. Figure 3.6 illustrates how the acutal flux data (solar cycle 22)

is “smoothed” by taking midpoint averages of the daily flux over a given pe-

riod of time. Midpoint averages of the flux over 6 solar rotation periods (164

days) and 13 months (395 days) are shown to illustrate the effect on the flux

data.

To realize the consequences of using the 13-month averaged flux predic-

tions, two test cases were considered. The first test case began in 1989 and

the second began in 1992. The run profile for each test case is shown in Ta-

ble 3.2. For each test case, two identical simulation runs were performed with

MSODP – one using actual daily flux and the other using 13-month averages of

the daily flux. (A 13-month midpoint averaging algorithm was used to obtain

the additional MSODP flux data file.)

Figures 3.7, 3.8, and 3.9 show the run results for the first test case. The

most notable consequence of using the 13-month averaged flux is the longer

expected lifetime of the satellite as shown in Figure 3.7. The cause for such

a difference in decay is more clearly shown in Figures 3.8 and 3.9, where the

orbital decay rates and the flux profiles are plotted.

Figures 3.10, 3.11, and 3.12 show the run results for the second test case. In

this case, the difference in orbital decay is much less apparent. The decay rates

shown in Figure 3.11 are a little more comparable. Notice that between 60

and 240 days into the mission, the decay rate for the averaged flux is actually

38

1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 201860

80

100

120

140

160

180

200

220

240

260

year

13-m

onth

ave

rage

d flu

x

cycle 23 cycle 24

95%50%5%

Figure 3.5: MSFC Flux Predictions

Parameter Test Case #1 Test Case #2

Spacecraft Mass 400 kg 400 kgInitial Altitude 465 km 465 kmInitial Orbital Elementsa0 6843.111 km 6843.111 kme0 0.00121 0.00121i0 87 87

Ω0 0 0

ω0 0 0

M0 0 0

Orbit Epoch 1/1/1989 3/1/1992Atmospheric Density Model DTM DTMFlux Profile 2σ 2σ

Table 3.2: Test Case Profiles for Flux Averaging Comparison

39

89.5 90 90.5 91 91.5 92 92.5 93

100

150

200

250

300

350

year

F10

.7

Daily Flux 164-day averaged Flux 13-month averaged Flux

Figure 3.6: Flux Smoothing

greater. This is due to the 13-month averaged flux shown in Figure 3.12 being

greater than actual daily flux (between 60 and 240 days).

Therefore, by using the averaged flux predictions, it is very difficult to say

what kind of orbital lifetime will result. Given these two test cases, it appears

that, if anything, the lifetime predictions will be greater when using 13-month

averaged flux.

40

0 100 200 300 400 500 600200

250

300

350

400

450

500


aliti

tude

(km

)

Simulation Epoch = 1/1/89Initial Altitude = 465 kmSpacecraft Mass = 400 kgDTM Atmospheric Density Model(2-sigma Flux Profile)

Actual Daily Flux 13-month Averaged Flux

Figure 3.7: Orbital Decay (Epoch: 1/1/89)

0 100 200 300 400 500 60010

-2

10-1

100

101


deca

y ra

te (

km/d

ay)


Figure 3.8: Orbital Decay Rate (Epoch: 1/1/89)

0 100 200 300 400 500 600100

150

200

250

300

350


F10

.7


Figure 3.9: 13-month Averaged Flux and Daily Flux (Epoch: 1/1/89)

41

0 100 200 300 400 500 600200

250

300

350

400

450

500


aliti

tude

(km

)

Simulation Epoch = 1/1/89Initial Altitude = 465 kmSpacecraft Mass = 400 kgDTM Atmospheric Density Model(2-sigma Flux Profile)


Figure 3.10: Orbital Decay (Epoch: 3/1/92)

0 100 200 300 400 500 60010

-2

10-1

100


deca

y ra

te (

km/d

ay)


Figure 3.11: Orbital Decay Rate (Epoch: 3/1/92)

0 100 200 300 400 500 60080

100

120

140

160

180

200

220

240


F10

.7


Figure 3.12: 13-month Averaged Flux and Daily Flux (Epoch: 3/1/92)

42

3.3 Conclusions of the GOAT Validation

Two issues were addressed in this chapter: GOAT validation with MSODP,

and understanding the consequences of using 13-month averaged flux predic-

tions. In comparing the GOAT simulation run with the MSODP simulation

run, it should be noted that 1), running a simulation with GOAT is much faster

and 2), there is an inherent limitation when using GOAT which uses SALT

to propagate the mean orbital elements. Although, up to a point, the orbital

decay predicted by GOAT was within 10% of the decay predicted by MSODP,

eccentricities below 0.0005 appeared to cause problems in the propagation,

especially with the argument of perigee.

Comparing the use of 13-month averaged flux with the use of actual daily

flux demonstrated that the averaged flux will likely lead to greater lifetime

predictions. Nevertheless, the reliability of using the MSFC flux predictions

(which are in the form of 13-month averages) has been evaluated.

43

Chapter 4

Guidelines for Using GOAT

This chapter will present the guidelines for running the GRACE Orbit Anal-

ysis Tool (GOAT). The MATLAB version of GOAT was written in the MAT-

LAB 5.1 programming environment and, the FORTRAN version of GOAT was

written for use with a FORTRAN-77 compiler. These guidelines assume that

the GOAT user has a basic understanding of the MATLAB and FORTRAN

programming environments.

4.1 MATLAB Version Guidelines

The next few sections will present the guidelines for running the MATLAB

version of the GRACE Orbit Analysis Tool.

4.1.1 Starting GOAT

Start the MATLAB application from the GOAT directory (which contains

the goat.m and supporting files). Once the MATLAB application is running,

simply type “goat” in the MATLAB Command Window to start GOAT. The

GOAT Command Window shown in Figure 4.1 will appear once GOAT begins.

In addition to the seven buttons shown in the GOAT Command Window,

the user can also select a command from the pull-down menu structure labeled

“GOAT”. When starting GOAT, the user has two immediate options: initial-

izing a new run (with the Initialize Run button) or loading the workspace of

45

Figure 4.1: GOAT Command Window

a previous run (with the Load Workspace button). Given that the procedures

for using GOAT are dependent on the data that is initially available, pressing

certain buttons may produce a popup warning. Pressing the Set Options but-

ton without first initializing, pressing the Execute Run button without first

initializing and setting options, and pressing the Plot Data button without

first producing simulation run data will all produce a popup warning. The

various popup warnings are shown in Figure 4.2. (Without run data, pressing

the Save Workspace button will simply save an empty binary file.)

4.1.2 Running a GOAT simulation

The first step in running a GOAT simulation is initializing the run with ei-

ther the mean orbital elements or the state vectors (see Figure 4.3). Choosing

to initialize with the orbital elements will produce the window shown in Fig-

ure 4.4. Choosing to initialize with the state vectors will produce the window

shown in Figure 4.5. Initialization also requires entry of the orbit epoch. No-

tice that with the DTM atmospheric density model, the flux data file limits

GOAT simulation between January 1, 2001 and December 31, 2006.

46

Figure 4.2: Various Popup Warnings

Figure 4.3: Choose Input Type Window

47

Figure 4.4: Initialization with Orbital Elements Window

Figure 4.5: Initialization with State Vector Window

48

Figure 4.6: Set Options Window

The next step in running a GOAT simulation is setting up the options

for the run. Once the Set Options button is pressed, the window shown in

Figure 4.6 will appear.

The timing parameters given are the simulation duration and step size.

The simulation duration is entered in units of days. The step size is entered

in units of seconds. Below this frame, the user can enter text for simulation

run identification.

The types of propagation intended for GOAT include normal two-satellite

propagation with and without maintenance, and lifetime one-satellite propa-

gation. Maintenance rules for the given propagation schemes are entered in

the next two text boxes shown. The user then can select a type of integrator

and relative tolerance for the integration. The last pull-down selector provides

the user with the selection of an atmospheric density model. The choices for

the MATLAB version of GOAT are the DTM and Exponential Atmospheric

Model.

The final step in running a GOAT simulation is executing the run. This is

49

simply accomplished by pressing the Execute Run button. Verification of the

simulation directive is given in the MATLAB Command Window. As shown

in Figure 4.7, many of the GOAT commands and statuses can be verified in

this window. Upon completion of the simulation, the Simulation Complete

popup will appear (Figure 4.8).

4.1.3 After a GOAT simulation

GOAT simulation data can be obtained by running a simulation (as explained

in the previous section) or by loading the variable workspace from a previous

GOAT run. Press the Load Workspace button to load data from a previous

run. A choice can then be made with the window shown in Figure 4.9.

After a GOAT simulation, the user can save the variable workspace and

plot the simulation data. Press the Save Workspace button to save data from

a current simulation run. A choice can then be made with the window shown

in Figure 4.10. To plot the data, simply press the Plot Data button and

the window in Figure 4.11 appears. From this window, various plots can be

generated, including a custom plot where the user selects the x-axis and y-axis

variables to plot.

Between simulation runs or loading, a good precaution is to clear the vari-

able workspace. This is accomplished by typing “clear all” in the MATLAB

Command Window.

To quit GOAT, simply press the Quit GOAT button.

50

Figure 4.7: MATLAB Command Window

51

Figure 4.8: Sim Complete Popup

Figure 4.9: Load Workspace Window

Figure 4.10: Save Workspace Window

52

Figure 4.11: Plot Long Term Data Window

53

4.2 FORTRAN Version Guidelines

The next few sections will present the guidelines for running the FORTRAN

version of the GRACE Orbit Analysis Tool. Notice that the required files for

creating a new GOAT executable are: FLUX.DAT, FAERO.DAT, INTEG.F,

LIB.F, and MET.F. FLUX.DAT contains the flux data for the simulation

time-frame. FAERO.DAT was obtained from AMA/LaRC and is used with

AMA/LaRC’s LIB.F library of subroutines to obtain the normalized aerdy-

namic forces. INTEG.F contains a library of subroutines that is used for the

integration of the SALT equations. MET.F was obtained from one of the Na-

tional Space Science Data Center’s FTP sites and contains the code for the

MET atmospheric model.

4.2.1 Editing the Inputs

The first step in running a simulation with the FORTRAN version of GOAT

is to edit the inputs. These are found directly in goat.f. Near the beginning,

5 input decks can be found. The first input deck is where the user enters the

initial mean orbital elements for each satellite. Notice that the length unit is

meters and the angular unit is radians. (Notice that the degrees-to-radians

conversion variable D2R can be used.) The second input deck is where the

spacecraft mass is entered in kilograms. The third input deck is where the orbit

epoch is entered in years, months, days, hours, minutes and seconds. Notice

that these values should be integers. The fourth input deck is where timing

parameters are entered. The initial time, final time, and step size are all in

units of seconds. The method of integration is also specified. The fifth input

deck is where the user selects the atmospheric density model. The user can

select from the Exponential (1), DTM (2), and MET (3) atmospheric models.

4.2.2 Compiling and Running GOAT

Once the inputs have been entered, the next step is to compile. The makegoat

script can be used or the user can simply type: f77 goat.f integ.f lib.f met.f

54

-o rungoat If no errors are encountered, the user need only type rungoat to

execute GOAT.

4.2.3 After a GOAT Simulation

Three output files are generated by running GOAT: SAT1.OUT, SAT2.OUT,

and RANGE.OUT. For the parametric analysis found in this thesis, the author

transferred these output files to a MacIntosh Computer where MATLAB was

used to generate the plots.

55

Chapter 5

GRACE Mission ParametricAnalysis

5.1 Description of the Analysis

This chapter will present a description of the parametric analysis of the life-

time of the GRACE mission. The goal of this analysis is to present some of

the parameters that affect the lifetime of the current design of the GRACE

satellites. (As of the writing of this thesis, the current design is Configuration

D.)

5.1.1 Parameters of Interest

The parameters of interest, and how they are varied from run to run, are found

in Table 5.1. The initial altitude, initial longitude of ascending node (phase

with respect to the Sun), and initial eccentricity are the mission design pa-

rameters. The atmospheric density model and flux profile are the environment

model parameters. Notice that the flux profiles only apply to the DTM and

MET atmospheric density models. The nominal flux profile refers to the 50%

flux predictions (from MSFC) shown in Figure 3.5. The +2σ flux profile refers

to the 95% flux predictions shown in Figure 3.5.

The remaining mission parameters and initial conditions for these runs that

are not shown in Table 5.1 are given in Table 5.2. Notice that the orbit epoch

57

Initial Altitude Phase w.r.t. Sun Initial Density Model Flux Profile(km) (Initial Ω, deg) Eccentricity450 0 0.001 Exponential nominal455 20 0.002 DTM +2σ460 40 0.003 MET465 60 0.004 (for DTM & MET)

470 80 0.005475 100480 120485 140490 160495 180500505510515520525530535540545550

Table 5.1: Parameters of Interest for the Parametric Analysis

58

Parameter

Spacecraft Mass 400 kgInitial Altitude variable (see Table 5.1)Initial Orbital Elementsa0 variable (see Initial Altitude)e0 variable (see Table 5.1)i0 87

Ω0 variable (see Table 5.1)ω0 0

M0 0

Orbit Epoch 6/23/2001Atmospheric Density Model variable (see Table 5.1)Flux Profile variable (see Table 5.1)

Table 5.2: Run Profile for the Parametric Analysis

is June 23, 2001, which is the approximate time-frame for the beginning of the

GRACE mission.

5.1.2 Strategy

A total of 5,250 simulation runs were generated by varying the parameters of

interest as shown in Table 5.1. All simulation runs were allowed to run for

5 years or until the minimum altitude of 120 kilometers was reached – which

ever came first. As a result, some of the lifetime predictions of 5 years may

well have lasted longer if these simulation limits were not applied.

The FORTRAN version of GOAT was used to process the simulation runs

for the parametric analysis. A driver program was created which looped

through all the parameters of interest, varying them according to Table 5.1.

From within the driver program, a modified version of GOAT was called and

ran the particular simulation. After each call to GOAT, the run identification

number, the particular parameters of interest, and the predicted lifetime result

was written to an output file.

Once the 5,250 simulation runs were completed, MATLAB was used to

process the data and produce plots. The initial results are in the form of

59

five scatter plots. Each scatter plot has the predicted lifetime plotted for a

particular parameter of interest. The initial results and the further analysis is

found the next section.

60

5.2 Results of the Analysis

This chapter will present the results of the parametric analysis of the GRACE

mission. These results are based on the GOAT simulation runs that were

explained in the previous section. Section 5.2.1 contains the initial results of

the analysis, primarily in the form of scatter plots. Section 5.2.2 goes into a

more detailed analysis of some of the parameters.

5.2.1 Initial Results: Scatter Plots

Each scatter plot contains 5,250 data points of the dependent variable which

is the GRACE mission lifetime. These data points are plotted against the

independent variables which are the 5 parameters of interest: initial altitiude,

initial longitude of the ascending node (phase with respect to the Sun), initial

eccentricity, atmospheric density model, and flux profile. (Recall, these are

the parameters found listed in Table 5.1.)

Lifetime vs. Initial Altitude

In Figure 5.1, the lifetime was plotted for different initial altitudes. Three

bands of data points can be identified in this scatter plot. As the data was

examined further, the bands where identified as runs using the Exponential

Atmospheric Model, MET Atmospheric Model, and DTM Atmospheric Model

as shown in Figure 5.2. Notice how the Exponential Model predicted the

least decay (longer lifetime predictions) and DTM predicted the greatest de-

cay (shorter lifetime predictions). The data points across the top of the scatter

plot resulted from runs which were initialized with parameters that, as shown,

yielded a lifetime of 5 years or greater. Further analysis of using DTM will

focus on the altitudes ranging from 450 km to 475 km. Further MET anal-

ysis will focus on the altitudes ranging from 450 km to 465 km, whereas the

Exponential Model will only focus on the 450 to 460 km range.

61

Lifetime vs. Atmosphere Model

The scatter plot for the use of different atmospheric models is shown in Fig-

ure 5.3. In this plot, a value of ‘1’ indicates use of the Exponential Model, a

value of ‘2’ indicates use of DTM, and a value of ‘3’ indicates use of MET. The

distribution of the data points belonging to each model is found in Figure 5.4.

The grouping shown by this plot agrees with what was shown in the scatter

plot for different initial altitudes.

Lifetime vs. Flux Profile

The scatter plot of lifetime versus the flux profile used is shown in Figure 5.5.

In this plot, a value of ‘1’ indicates use of the nominal (50%) flux profile,

a value of ‘2’ indicates use of the +2σ (95%) flux profile, and a value of

‘0’ indicates that the flux profile is not applicable to the atmospheric density

model (namely, the Exponential Model). Figure 5.6 shows how the data points

can be grouped. Notice that using the nominal flux profile yielded predicted

lifetimes of 5 years or greater. Since the greatest spread of the data points is

shown by using the +2σ flux profile, further analysis of this data will focus on

the use of the +2σ flux predictions.

Lifetime vs. Initial Eccentricity

Figure 5.7 shows the scatter plot of lifetime versus the initial eccentricity. In

Figure 5.8, notice the downward trend for lifetime as the initial eccentricity

is increased. Further analysis will vary initial eccentricity through the entire

range from 0.001 to 0.005.

Lifetime vs. Phase w.r.t. Sun

Finally, in Figure 5.9, the scatter plot of lifetime versus the initial longitude of

ascending node. Variation of this parameter is used to analyze how the phase

with respect to the Sun affects the lifetime. Figure 5.10 shows how the data

points tend to “wave” with the lifetime variations of the phase w.r.t. the Sun.

Further analysis will vary the phase through the entire range from 0 to 180.

62

450 460 470 480 490 500 510 520 530 540 550

2

2.5

3

3.5

4

4.5

5

Initial Altitude (km)

Life

time

(yea

rs)

Figure 5.1: Scatter Plot of Lifetime vs. Initial Altitude

Figure 5.2: Grouping of Initial Altitude Data Points

63

0.5 1 1.5 2 2.5 3 3.5

2

2.5

3

3.5

4

4.5

5

Atmosphere Model (1=Exp,2=DTM,3=MET)

Life

time

(yea

rs)

Figure 5.3: Scatter Plot of Lifetime vs. Atmosphere Model

Figure 5.4: Grouping of Atmosphere Model Data Points

64

-0.5 0 0.5 1 1.5 2 2.5

2

2.5

3

3.5

4

4.5

5

Flux Profile (1=nominal,2=2sigma)

Life

time

(yea

rs)

Figure 5.5: Scatter Plot of Lifetime vs. Flux Profile

σ

Figure 5.6: Grouping of Flux Profile Data Points

65

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

x 10-3

2

2.5

3

3.5

4

4.5

5

Initial Eccentricity

Life

time

(yea

rs)

Figure 5.7: Scatter Plot of Lifetime vs. Initial Eccentricity

Figure 5.8: Grouping of Initial Eccentricity Data Points

66

0 20 40 60 80 100 120 140 160 180

2

2.5

3

3.5

4

4.5

5

Initial Omega (Phase wrt Sun) (deg)

Life

time

(yea

rs)

Figure 5.9: Scatter Plot of Lifetime vs. Initial Phase w.r.t. Sun

Figure 5.10: Grouping of Initial Phase w.r.t. Sun Data Points

67

5.2.2 Further Analysis

The scatter plots in the previous section helped provide some insight as to

what required further analysis. Basically, this analysis takes the different en-

vironment model parameters (atmosphere model and flux profile) and applies

them to the different design parameters (initial altitude, phase w.r.t. Sun, and

initial eccentricity). As a result, the analysis is divided into four areas: use of

the Exponential Model, DTM, and MET, and fourthly, a comparison of DTM

and MET use.

The Exponential Model is a static model of the atmosphere, whereas DTM

and MET are dynamic models of the atmosphere. Notice that an important

feature of the dynamic atmosphere is the diurnal temperature variation. The

part of Earth’s atmosphere that is exposed to the Sun experiences higher

temperatures which cause the atmosphere to swell, or bulge. Figure 5.11

illustrates the atmospheric bulge and several orbit trajectories. It will be

shown that changing the three design parameters for a given atmosphere model

affects the lifetime prediction.

Orbit Trajectories

NOTE: Scales areexagerrated for illustration.

Atmosphere

Earth

(diurnal bulge)

Sun

Figure 5.11: Atmopheric Bulge and Several Orbit Trajectories

68

Use of the Exponential Model

Using the Exponential Model, a static representation of the atmosphere, does

not yield the fluctuations that appear when using the dynamic atmosphere

models. Therefore, as shown in Figure 5.12, the lifetime is plotted for the

initial altitudes. Notice how the curves for the various initial eccentricities

eventually converge on the 5-year lifetime level. It would be expected that if

the simulations where allowed to run beyond the 5-year mark, the 5 eccen-

tricity curves would have continued in parallel (for the most part), with the

greatest initial eccentricity yielding the greatest orbital decay (shortest life-

time prediction). It might be that the usefulness of the Exponential Model is

only found in its simplicity.

450 451 452 453 454 455 456 457 458 459 4604.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

5.1

Life

time

(yea

rs)

Initial Altitude, ho (km)

eo = 0.001

eo = 0.002

eo = 0.003

eo = 0.004

eo = 0.005

Figure 5.12: Lifetime vs. Initial Altitude (Exponential Model)

Use of DTM

All data points that are found in the following plots pertain to simulation runs

that have used the DTM atmospheric density model with a +2σ flux profile. To

69

illustrate how the lifetime depends on initial altitude , the initial eccentricity

is fixed. In Figure 5.13, the initial eccentricity is fixed at 0.001 and then the

lifetime is plotted for changes in the phase w.r.t. the Sun. Notice the lower

altitudes have shorter lifetime predictions with smaller lifetime variations. The

larger variations in the higher altitudes are due to the effects of the passing in

and out of the atmospheric bulge caused by exposure to the Sun. Figure 5.14

provides a close-up of the lifetime variation found for an initial altitude of 475

km.

In Figure 5.15, the initial eccentricity is fixed at 0.005 and the lifetime is

plotted for changes in the phase w.r.t. the Sun. Notice that, with this greater

eccentricity, the lifetime predictions are shorter. Also, since the shape of the

orbit is more elliptical, the lifetime variations are less symmetrical. This is

shown in Figure 5.16 which is a close-up with the initial altitude of 475 km.

Fixing the initial altitude, the next two plots illustrate how the lifetime

depends on initial eccentricity, and how the initial eccentricity plays a part in

the symmetry of the lifetime variations. Figures 5.17 and 5.18 show where the

initial altitudes have been fixed to 450 km and 475 km, respectively.

ASIDE: Use of DTM with Daily Flux

It was demonstrated in Chapter 3 that using 13-month averaged flux instead

of daily flux will likely lead to greater lifetime predictions. To confirm this

expectation, new lifetime predictions using DTM with +2σ daily flux were

generated. Note that the predicted daily flux profile used in generating these

new lifetime predictions was obtained from UTCSR. (Recall that the 13-month

averaged flux profile was obtained from MSFC.)

Figures 5.19 - 5.22 are the resulting plots. Notice how Figure 5.19 is com-

pared with Figure 5.13, Figure 5.20 is compared with Figure 5.15, Figure 5.21 is

compared with Figure 5.17, and Figure 5.22 is compared with Figure 5.18. As

expected, the lifetime predictions using the daily flux are consistently shorter

than the lifetime predictions using the 13-month averaged flux.

70

0 20 40 60 80 100 120 140 160 180 200 220

2

2.5

3

3.5

4

4.5

Initial Omega (deg) [Phase w.r.t. Sun]

Life

time

(yea

rs)

(Initial Eccentricity, eo = 0.001)

ho = 450 km

ho = 455 km

ho = 460 km

ho = 465 km

ho = 470 km

ho = 475 km

Figure 5.13: Lifetime vs. Phase (eo = 0.001, DTM, +2σ flux)

0 20 40 60 80 100 120 140 160 180 200 2204.3

4.32

4.34

4.36

4.38

4.4

4.42

4.44



Life

time

(yea

rs)

ho = 475 km

Figure 5.14: Close-up of Variation (ho = 475 km, eo = 0.001, DTM, +2σ flux)

71

0 20 40 60 80 100 120 140 160 180 200 220

2

2.5

3

3.5

4


Life

time

(yea

rs)


ho = 450 km

ho = 455 km

ho = 460 km

ho = 465 km

ho = 470 km

ho = 475 km

Figure 5.15: Lifetime vs. Phase (eo = 0.005, DTM, +2σ flux)

0 20 40 60 80 100 120 140 160 180 200 2203.87

3.88

3.89

3.9

3.91

3.92

3.93

3.94

3.95

3.96



Life

time

(yea

rs)

ho = 475 km

Figure 5.16: Close-up of Variation (ho = 475 km, eo = 0.005, DTM, +2σ flux)

72

0 20 40 60 80 100 120 140 160 180 200 2201.72

1.74

1.76

1.78

1.8

1.82

1.84

1.86


Life

time

(yea

rs)

(Initial Altitude, ho = 450 km)

eo = 0.001

eo = 0.002

eo = 0.003

eo = 0.004

eo = 0.005

Figure 5.17: Lifetime vs. Phase (ho = 450 km, DTM, +2σ flux)

0 20 40 60 80 100 120 140 160 180 200 220

3.9

4

4.1

4.2

4.3

4.4


Life

time

(yea

rs)


eo = 0.001

eo = 0.002

eo = 0.003

eo = 0.004

eo = 0.005

Figure 5.18: Lifetime vs. Phase (ho = 475 km, DTM, +2σ flux)

73

0 20 40 60 80 100 120 140 160 180 200 2201

1.5

2

2.5

3

3.5

4


Life

time

(yea

rs)


ho = 450 km

ho = 455 km

ho = 460 km

ho = 465 km

ho = 470 km

ho = 475 km

Figure 5.19: Lifetime vs. Phase (eo = 0.001, DTM, +2σ daily flux)

0 20 40 60 80 100 120 140 160 180 200 2201

1.5

2

2.5

3

3.5


Life

time

(yea

rs)


ho = 450 km

ho = 455 km

ho = 460 km

ho = 465 km

ho = 470 km

ho = 475 km

Figure 5.20: Lifetime vs. Phase (eo = 0.005, DTM, +2σ daily flux)

74

0 20 40 60 80 100 120 140 160 180 200 2201.18

1.2

1.22

1.24

1.26

1.28

1.3


Life

time

(yea

rs)


eo = 0.001

eo = 0.002

eo = 0.003

eo = 0.004

eo = 0.005

Figure 5.21: Lifetime vs. Phase (ho = 450 km, DTM, +2σ daily flux)

0 20 40 60 80 100 120 140 160 180 200 220

3

3.1

3.2

3.3

3.4

3.5

3.6


Life

time

(yea

rs)


eo = 0.001

eo = 0.002

eo = 0.003

eo = 0.004

eo = 0.005

Figure 5.22: Lifetime vs. Phase (ho = 475 km, DTM, +2σ daily flux)

75

Use of MET

All data points that are found in the following plots pertain to simulation

runs that have used the MET atmospheric density model with a +2σ flux

profile. Once again, to illustrate the dependence of lifetime on initial altitude,

the initial eccentricity is fixed. In Figure 5.23, the initial eccentricity is fixed

at 0.001 and then the lifetime is plotted for changes in the phase w.r.t. the

Sun. Notice the lower altitudes have shorter lifetime predictions as expected.

However, the lifetime variations do not appear as they do in the use of DTM.

This is more apparent in a close-up of the curve for the initial altitude of 460

km is shown in Figure 5.24.

As the initial eccentricity is fixed at 0.005 and the lifetime is plotted for

changes in the phase w.r.t. the Sun, the result is similar, except with shorter

lifetime predictions (see Figure 5.25). Again, notice the difference in the life-

time variation shown in Figure 5.26.

76

0 20 40 60 80 100 120 140 160 180 200 2203

3.5

4

4.5

5


Life

time

(yea

rs)

(Initial Eccentricity, eo = 0.001) h

o = 450 km

ho = 455 km

ho = 460 km

ho = 465 km

Figure 5.23: Lifetime vs. Phase (eo = 0.001, MET, +2σ flux)

0 20 40 60 80 100 120 140 160 180 200 2204.14

4.16

4.18

4.2

4.22

4.24

4.26

4.28

4.3

4.32



Life

time

(yea

rs)

ho = 460 km

Figure 5.24: Close-up of Variation (ho = 460 km, eo = 0.001, MET, +2σ flux)

77

0 20 40 60 80 100 120 140 160 180 200 2202.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6


Life

time

(yea

rs)

(Initial Eccentricity, eo = 0.005) h

o = 450 km

ho = 455 km

ho = 460 km

ho = 465 km

Figure 5.25: Lifetime vs. Phase (eo = 0.005, MET, +2σ flux)

0 20 40 60 80 100 120 140 160 180 200 2204.61

4.62

4.63

4.64

4.65

4.66

4.67



Life

time

(yea

rs)

ho = 465 km

Figure 5.26: Close-up of Variation (ho = 465 km, eo = 0.005, MET, +2σ flux)

78

Comparison of DTM and MET

To make the comparison between DTM and MET, the initial eccentricity was

fixed at 0.003. Figure 5.27 contains four plots which show how the models

differ in their prediction of lifetime. Figure 5.28 shows the result of taking the

averaged lifetime predictions and plotting them against the initial altitudes.

The expected curves result. Analysis of the entire data set suggests that

DTM will consistently predict shorter lifetimes than MET. Also notice the

plots from the two previous sections indicate that the atmospheric modeling

is quite different in the two atmospheric models as evident in their lifetime

variation differences.

5.3 Conclusions of the Analysis

First, notice that the conclusions of this parametric analysis are preliminary

and further analysis is suggested. This analysis has shown how the parameters

of interest affect the lifetime of the GRACE mission. Although some of the

results were expected, some still deserve further analysis.

The Exponential Model, in its simplicity, provides lifetime predictions that

are not correlated with the uncertainty that is in the flux predictions used

by the dynamic atmosphere models. When the dynamic models implement

the nominal flux profile, no significant orbital decay occurred. All lifetimes

predicted were 5 years or greater. Use of the +2σ flux profile did yield inter-

esting results. The phase with respect to the Sun was clearly an issue given

DTM’s prediction of lifetime. With MET, the randomness of the lifetime pre-

dictions made it difficult to ascertain how the phase with respect to the Sun

affects lifetime. Nevertheless, with both dynamic atmosphere models, the de-

pendence of lifetime on the phase with respect to the Sun was shown to be

relatively weak. The greatest lifetime variation for a given initial altitude and

eccentricity was approximately 35 days. The dependence of lifetime on the

eccentricty was alittle stronger at higher initial altitudes. For example, at an

initial altitude of 475 km, the difference in the lifetime prediction between an

79

0 50 100 150 200

2

2.5

3

Initial Omega (deg)

Life

time

(yea

rs)


0 50 100 150 200

2

2.5

3

3.5

Initial Omega (deg)

Life

time

(yea

rs)


METDTM

0 50 100 150 200

2.5

3

3.5

4

Initial Omega (deg)

Life

time

(yea

rs)


0 50 100 150 2002.5

3

3.5

4

4.5

5

Initial Omega (deg)

Life

time

(yea

rs)


Figure 5.27: DTM vs. MET – Four Plots with Lifetime vs. Phase (eo = 0.003)

450 452 454 456 458 460 462 464 466 468 4701.5

2

2.5

3

3.5

4

4.5

5

5.5

Initial Altitude (km)

Ave

rage

Life

time

(yea

rs)


METDTM

Figure 5.28: DTM vs. MET – Average Lifetime vs. Initial Altitude (eo =0.003)

80

initial eccentricity of 0.001 and 0.005 was approximately 164 days (0.45 years).

On the other hand, at an initial altitude of 450 km, this difference was only

36 days. Of all the design parameters, the initial altitude had the greatest

affect on the predicted lifetime. With MET, a change in the initial altitude of

+10 km resulted in a lifetime increase of approximately one year. With DTM,

a change in the initial altitude of +10 km resulted in a lifetime increase of

approximately 0.6 to 1.5 years. A comparison of the two dynamic models did

show that DTM will consistently predict shorter lifetimes than MET given the

same initial altitude, phase with respect to the Sun, and initial eccentricity.

5.4 Recommendations for the GRACE mis-

sion

The recommendations discussed in this section are based on the parametric

analysis of the GRACE mission found in this chapter. To be conservative, the

environment model parameters that produced the shortest lifetime predictions

are considered. These are 1), the use of the DTM atmospheric density model

and 2), a +2σ flux profile.

The minimum altitude recommended is 480 km. The initial altitudes lower

than 480 km clearly had lifetime predictions of less than 5 years regardless

of the initial eccentricity and initial longitude of the asending node. The

maximum eccentricity recommended is 0.001. It was clear from all the plots

that as the initial eccentricity was increased, the predicted lifetime decreased.

The initial longitude of the ascending node recommended, which yields the

“best” phase with respect to the Sun, is 40. Each lifetime variation achieved

a maximum near that initial longitude. (Notice that it was shown that the

lifetime predictions were affected very little by changes in the initial longitude

of the ascending node.)

81

Appendix A

Various Supporting Algorithms

This appendix will present the various supporting algorithms used with GOAT.

Notice that some of these algorithms are not complete subroutines and instead

are found as code within a particular subroutine.

A.1 Drag Temperature Model (DTM)

With the DTM model, various densities contributing to the total density are

expanded in terms of spherical harmonics. The atmospheric constituents that

are of greatest significance are molecular nitrogen (N2), atomic oxygen (O),

molecular oxygen (O2), helium (He), and molecular hydrogen (H2). Each

constituent has a density that can be represented by:

ρj(z) = A1jeGj(L)−1fi(z) (A.1)

where A1j is the first coefficient for the spherical harmonics expansion for

constituent j, Gj(L) is a function which represents a spherical harmonic ex-

pansion using the remaining 35 coefficients (A2j→A36j). The function fj(z)

results from the integration of a diffusive equilibrium distribution. The 36

coefficients for the thermopause temperature and the atmospheric constituent

densities are found in Table A.1 of the Appendix. Once these functions are

determined, the total density is simply found by:

83

MJDsmoothed solar fluxsolar flux geomagnetic index

Figure A.1: Flux Data Structure

ρtotal(z) =5∑j=1

ρj(z) (A.2)

(Note that to obtain the units of kg/m3, one would also have to multiply ρtotal

by Avogadro’s Number NA.)

The first step is to obtain the solar flux F10.7, the smoothed solar flux

F10.7, and the planetary geomagnetic index Kp given the current Modified

Julian Date (MJD). GOAT “looks-up” these values in a solar flux data file.

The flux data structure is shown in Figure A.1. (Solar flux is measured in

units of 10−22 Wm2Hz

.)

The next step is, given the geodetic latitude φ, to compute the following

Legendre polynomials (Pm,n):

P0,1 = sinφ

P0,2 =1

2

(3 sin2φ− 1

)P0,3 =

1

2

(5 sin2φ− 3

)sinφ

P0,4 =1

8

(35 sin4φ− 30 sin2φ+ 3

)P0,5 =

1

8

(63 sin4φ− 70 sin2φ+ 15

)sinφ

84

i T∞ N2 O He H2

1 0.99980E+03 3.84200E+17 0.93000E+17 3.00000E+13 1.76100E+112 -0.36357E-02 0.28076E-01 -0.16598E-02 0.12000E+00 -1.33700E-013 0.24593E-01 0.48462E-01 -0.99095E-01 -0.15000E+00 04 0.13259E-02 -0.81017E-03 0.78453E-03 0.23799E-02 -1.24600E-025 -0.56234E-05 0.20983E-04 -0.23733E-04 -0.31008E-04 06 0.25361E-02 0.29998E-02 0.80001E-02 0.56980E-02 -1.93000E-027 0.17656E-01 -0.10000E-01 0.70000E-01 0.17103E-01 -6.00000E-028 0.33677E-01 0.80000E-01 -0.18000E+00 -0.17997E+00 -0.20000E-029 -0.37643E-02 0.53709E-01 0.14597E+00 -0.13251E+00 5.87800E-0210 0.17452E-01 -0.13732E+00 0.10517E+00 -0.64239E-01 011 -3.63831E+00 1.48687E+00 0.64263E-01 3.80793E+00 1.58727E+0012 -0.27270E-02 0.19930E-01 0.24620E+00 0.24859E+00 013 0.27465E-01 -0.84711E-01 -0.50845E-01 -0.17732E+00 014 -1.63795E+00 1.53685E+01 1.85356E+00 1.81331E+00 015 -0.13373E+00 -0.49083E-01 0.39103E+00 -0.11071E+01 3.30100E-0116 -0.27321E-01 0.91420E-02 0.96719E-01 -0.36255E-01 1.04500E-0117 -0.96732E-02 -0.16362E-01 0.12624E+00 -0.10180E+00 018 -2.50880E-01 8.46944E-01 -0.28570E+00 -3.36273E+00 -0.25408E+0019 -0.27469E-01 -0.46712E-01 -0.14463E+00 0.11711E+00 -9.06500E-0220 -2.99288E+00 9.07841E-01 1.88607E+00 -3.70403E+00 -1.23857E+0021 -0.66567E-01 0 -0.20686E+00 -0.31594E+00 2.09400E-0122 -0.59604E-02 0 0.82922E-02 0.52452E-01 2.83000E-0223 0.67446E-02 0 -0.30261E-01 -0.31686E-01 024 -0.26620E-01 0 0.14237E+00 -0.13975E+00 8.57100E-0225 0.14691E-01 0 -0.28977E-01 0.83399E-01 -2.47500E-0226 -0.10971E+00 0 0.22409E+00 0.21382E+00 3.83000E-0127 0.88700E-02 0 -0.79313E-01 -0.61816E-01 2.94100E-0228 0.36918E-02 0 -0.16385E-01 -0.15026E-01 029 0.12219E-01 0 -0.10113E+00 0.10574E+00 -3.97400E-0330 -0.76358E-02 0 0.65531E-01 -0.97446E-01 4.35600E-0231 -0.44894E-02 0 0.53655E-01 0.22606E-01 032 0.23646E-02 0 -0.23722E-02 0.12125E-01 033 0.50569E-02 0 0.18910E-01 -0.22391E-01 034 0.10792E-02 0 -0.26522E-02 -0.24648E-02 035 -0.71610E-03 0 0.83050E-02 0.32432E-02 036 0.96385E-03 0 -0.38860E-02 -0.57766E-02 0

Table A.1: Spherical Harmonics Expansion Coefficients, Ai, for the Thermo-pause Temperature and the Atmospheric Constituents

85

P1,1 = cosφ

P1,2 = 3 sinφ cosφ

P1,3 =3

2

(5 sin2φ− 1

)cosφ

P1,5 =1

8

(315 sin4φ− 210 sin2φ+ 15

)cosφ

P2,2 = 3 cos2φ

P2,3 = 15 sinφ cos2φ

P3,3 = 15 cos3φ (A.3)

Next, the function G(L), which represents further expansion of the spher-

ical harmonics, is determined by:

G(L) = ZL+ SF +GM

+ ANeven + ANodd + SANeven + SANodd

+D + SD + TD (A.4)

The term “ZL” refers to the dependence of G(L) on zonal latitude and is

expressed as:

ZL = 1 + A2P0,2 + A3P0,4 (A.5)

The term “SF” refers to the dependence of G(L) on direct solar flux and is

expressed as:

SF = A4

(F10.7 − F10.7

)+ A5

(F10.7 − F10.7

)2+ A6

(F10.7 − 150

)(A.6)

86

The term “GM” refers to the dependence of G(L) on planetary geomagnetic

effect and is expressed as:

GM = (A7 + A8P0,2)Kp (A.7)

The term “ANeven” refers to the dependence of G(L) on the even-latitude

annual variations and is expressed as:

ANeven = (A9 + A10P0,2) cos[Ωe (d− A11)] (A.8)

where Ωe = 2π/365 (“per day”) and d is the day count in the year. The

term “SANeven” refers to the dependence of G(L) on even-latitude semi-annual

variations and is expressed as:

SANeven = (A12 + A13P0,2) cos[2Ωe (d− A14)] (A.9)

The term “ANodd” refers to the dependence of G(L) on the odd-latitude annual

variations and is expressed as:

ANodd = (A15P0,1 + A16P0,3 + A17P0,5) cos[Ωe (d− A18)] (A.10)

The term “SANodd” refers to the dependence of G(L) on odd-latitude semi-

annual variations and is expressed as:

SANodd = A19P0,1 cos[2Ωe (d− A20)] (A.11)

87

The term “D” refers to the dependence of G(L) on diurnal variations and is

expressed as:

D = A21P1,1 + A22P1,3 + A23P1,5

+ (A24P1,1 + A25P1,2) cos[Ωe (d− A18)] cosωet

+ A26P1,1 + A27P1,3 + A28P1,5

+ (A29P1,1 + A30P1,2) cos[Ωe (d− A18)] sinωet (A.12)

where ωe = 2π/24 (“per hour”) and t is the local solar time. The term “SD”

refers to the dependence of G(L) on semidiurnal variations and is expressed

as:

SD = A31P2,2 + A32P2,3 cos[Ωe (d− A18)] cosωet

+ A33P2,2 + A34P2,3 cos[Ωe (d− A18)] sinωet (A.13)

The term “TD” refers to the dependence of G(L) on terdiurnal variations and

is expressed as:

TD = A35P3,3 cos3ωet+ A36P3,3 sin3ωet (A.14)

The last step is to evaluate the function fj(z) which results from the inte-

gration of a diffusive equilibrium distribution. For a given constituent j, the

function is given by:

fj(z) =(

1− a1− aeσζ

)1+αj+γj

e−σγjζ (A.15)

88

j γj mj (kg/mole)

H2 -0.40 2.0e-03He -0.38 4.0e-03O 0 16.0e-03N2 0 28.0e-03O2 0 32.0e-03

Table A.2: Thermal Diffusion Factors and Molecular Weights of the Atmos-pheric Constituents

where a=(T∞−T120)/T∞ and αj is the thermal diffusion factor (see Table A.2).

The thermopause temperature T∞ is found using the G(L) funciton and is

simply T∞=A1G(L). The geopotential altitude ζ is given by:

ζ =(z − 120)(R + 120)

R + z(A.16)

with R = 6356.77 km and z being the current altitude. The dimensionless

parameter γj is defined as:

γj =mjg120

σkT∞(A.17)

where mj is the molecular weight of constituent j (see Table A.2), g120 is

the gravitational acceleration at an altitude of 120 km, and k is Boltzmann’s

constant. Finally, the quantity σ is related to the temperature gradient tgrad

by:

σ = tgrad +1

(R + 120)(A.18)

89

A.2 Modeling SRP Perturbations with the Cylin-

drical Shadow Model

The pressure on the surface of a satellite caused by solar radiation can change

the motion of a satellite. The acceleration due to solar radiation pressure

(SRP) is shown by:

~aSRP = P (1 + η)A

msat

ε vsun−to−sat (A.19)

where P is the solar radiation pressure constant, η is the reflectivity of the

satellite’s surface, A is the area of this surface, msat is the satellite mass, ε is

the Earth shadow binary (0 for shadow, 1 for sunlight), and vsun−to−sat is the

sun-to-satellite unit vector. In this equation, the combined term, (1 + η)A, is

the most difficult to determine. The option of using aerodynamic data files

from LaRC simplifies the process of accurately determining ~aSRP provided that

P , ε, and vsun−to−sat are known.

To make the determination of whether the satellite is orbiting in the shadow

of the Earth or in the stream of solar radiation, GOAT uses the cylindrical

shadow model. The cylindrical shadow model, which is the simple alternative

to the conical shadow model, is illustrated in Figure A.2. In Figure A.2, the

distance D is the projection of ~rsat onto ~rsun and is given by:

D =~rsat · ~rsun|~rsun|

(A.20)

and distance H is the “height” of the satellite above ~rsun and is given by:

H =√|~rsat|2 −D2 (A.21)

90

Earth Sun

D

H

position A

position C

position B

NOTE: Scales are exaggerated for illustration.

rsun

rsat

Figure A.2: Cylindrical Shadow Model

via the Pythagorean theorem. If distance D is greater than zero, the satellite

is in sunlight as shown by position A and ε is equal to 1. If D is not greater

than zero, the satellite may or may not be in shadow. In this case, the distance

H must be determined. If H is greater than the Earth’s polar radius be, the

satellite is above the Earth’s shadow as shown by position B and ε is equal to

1. If H is not greater than the Earth’s polar radius, the satellite is within the

Earth’s shadow as shown by position C and ε is equal to 0. Notice with ε=0,

the SRP perturbation is essentially “turned off”.

A.3 Conversion between Orbital Elements and

Inertial (ECI) State Vector

As the mean orbital elements are propagated with SALT, a conversion to the

6-element (position and velocity) inertial state vector is performed with each

new set of orbital elements. Occassionally there is also a need to convert the

inertial state vector to the orbital elements. Figure A.3 provides an illustration

of the six classical orbital elements where a is the semi-major axis, e is the

eccentricity, i is the orbit inclination, Ω is the longitude of the ascending node,

ω is the argument of perigee, and ν is the true anomaly. Also illustrated, are

the position vector ~r and velocity vector ~v which combine to form the interial

state vector. (In the figure, AN refers to the “ascending node”, DN refers to

91

h

e

n

I

J

K

rv

ν

ω

Ω

i

AN

DNP

NOTE: Scales are exaggeratedfor illustration.

Figure A.3: Classical Orbital Elements

the “descending node”, and P refers to the point of “perigee”.)

A.3.1 Orbital Elements to Inertial (ECI) State Vector

To convert from orbital elements to inertial state vector, the perifocal coordi-

nate frame must be defined. The x-axis for the perifocal frame points toward

the perigee (along ~e in Figure A.3) and the z-axis is normal to the orbital

plane (along ~h). (The y-axis is simply in the direction of ~h× ~e.) This coordi-

nate frame is easily defined if the orbital elements are known. The perifocal

position and velocity of the satellite is given as:

92

~rperifocal =

p cosν

(1+e cosν)p sinν

(1+e cosν)

0

(A.22)

~vperifocal =

−√

µp

sinν√µp(e+ cosν)

0

(A.23)

where µ is the Earth’s gravitational parameter, and the semi-parameter p is

defined by Equation 2.8. The next step is to transform the perifocal position

and velocity to the inertial position and velocity. This is accomplished with

the following rotations defined by:

R3(−Ω) =

cos(−Ω) sin(−Ω) 0− sin(−Ω) cos(−Ω) 0

0 0 1

(A.24)

R1(−i) =

1 0 00 cos(−i) sin(−i)0 − sin(−i) cos(−i)

(A.25)

R3(−ω) =

cos(−ω) sin(−ω) 0− sin(−ω) cos(−ω) 0

0 0 1

(A.26)

and the resulting transformation matrix for perifocal to inertial is:

Rinertialperifocal = R3(−Ω)R1(−i)R3(−ω) (A.27)

The final step would be simply:

~rinertial = Rinertialperifocal ~rperifocal

~vinertial = Rinertialperifocal ~vperifocal (A.28)

93

A.3.2 Inertial (ECI) State Vector to Orbital Elements

To convert from inertial state vector to orbital elements, the inertial position

and velocity vectors are first used to find the angular momentum vector ~h, the

ascending node pointing vector ~n, the eccentricity vector ~e, and the specific

energy of the orbit ξ. The angular momentum vector, which is perpendicular

to the orbit plane, is given by the following cross product of the inertial position

and velocity vectors:

~h = ~r × ~v (A.29)

With the unit vector along the inertial z-axis defined by k = [0, 0, 1], the

ascending node pointing vector is given by this cross product:

~n = k × ~h (A.30)

The eccentricity vector, which points towards perigee, is defined by:

~e =~v × ~hµ− ~r

|~r| (A.31)

The specific energy of the orbit is defined by:

ξ =|~v|22− µ

|~r| (A.32)

Finally, the 6 orbital elements can be defined:

a = − µ

2ξ

94

e = |~e|

i = cos−1

(hk

|~h|

)

Ω = cos−1

(ni|~n|

)

ω = cos−1

(~n · ~e|~n||~e|

)

ν = cos−1

(~e · ~r|~e||~r|

)(A.33)

A.4 Conversion of Orbit Epoch to Greenwich

Sidereal Time and Julian Date

At the beginning of each simulation run, the epoch that is entered with the

associated initial conditions must be converted to an initial Greenwich sidereal

time GST0 and Julian date JD0. The following equations demonstrate how

GST0 and JD0 are derived from the epoch [17]. Let the input epoch be:

EPOCH =

YEARMONTHDAYHOURMINSEC

(A.34)

The Julian date at the epoch’s start of day is given by:

JD = 367 · YEAR

+ integer

7[YEAR + integer

(MONTH+9

12

)]4

+ integer

(275 ·MONTH

9

)

95

+DAY

+ 1721013.5 (A.35)

Given JD, the number of Julian centuries that have elapsed since the J2000

epoch is given by:

TUT1 =JD − 2451545.0

36525(A.36)

Given TUT1, Greenwich sidereal time at the epoch’s start of day is given by:

GST = 100.4606184

+ 36000.77005361 · TUT1

+ 0.00038793 · T 2UT1

− 2.6× 10−8 · T 3UT1 (A.37)

To reduce GST to range between 0 and 360, the following conversion may

be necessary:

GST = GST − integer(GST

360

)360 (A.38)

Finally, to obtain JD and GST at the epoch, consider the time of the day:

JD0 = JD +

[(SEC60

+MIN)60

+HOUR]

24(A.39)

GST0 = GST + ωe (HOUR · 3600 +MIN · 60 + SEC) (A.40)

96

A.5 Conversion of Inertial (ECI) Vector to RTN

Vector

Although vwind and vsun−to−sat are in the satellite frame, it is necessary to

convert the velocity unit vector from the inertial frame to the RTN frame before

performing the relevant conversions shown in Equations 2.22 and 2.23. This

subroutine computes the tranformation matrix that is necessary to convert an

inertial vector to an RTN vector. This matrix is defined by:

RRTNinertial =

Rx Ry Rz

Tx Ty TzNx Ny Nz

(A.41)

The inertial state vector (which consists of the inertial position and velocity

of the satellite), is required to compute the transformation matrix. For the

following explanation, the inertial position vector is defined by ~r and the in-

ertial velocity vector defined by ~v. The 1st row in the transformation matrix

is essentially the radial unit vector R and is found by:

R =rx|~r| x+

ry|~r| y +

rz|~r| z (A.42)

The 3rd row in the transformation matrix is the normal unit vector N . The

normal unit vector is found by:

N =hx

|~h|x+

hy

|~h|y +

hz

|~h|z (A.43)

where the angular momentum vector ~h is given as:

~h = ~r × ~v (A.44)

97

The 2nd row in the transformation matrix is the transverse unit vector T which

completes the orthogonal system with:

T = N × R (A.45)

A.6 Determination of Sun Vector

The position of the Sun is required for use in the cylindrical shadow model.

This subroutine computes the Sun position vector ~rsun with the following equa-

tions [17]. With JD defined as the current Julian Date, the number of Julian

centuries that have elapsed since the J2000 epoch is given by Equation A.36.

Given TUT1, the mean longitude of the Sun is:

λMsun = 280.4606184 + 36000.77005361 · TUT1 (A.46)

the mean anomaly of the Sun is:

Msun = 357.5277233 + 35999.05034 · TUT1 (A.47)

and the obliquity of the ecliptic is:

ε = 23.439291 − 0.0130042 · TUT1 (A.48)

Given λMsun and Msun, the ecliptic latitude of the Sun is:

98

λecliptic = λMsun + 1.914666471 sinMsun + 0.019994643 sin 2Msun (A.49)

and the Sun position magnitude is:

rsun = 1.000140612− 0.016708617 cosMsun − 0.000139589 cos 2Msun (A.50)

Finally the Sun position vector ~rsun can be found:

~rsun =

rsun cosλeclipticrsun cos ε sinλeclipticrsun sin ε sinλecliptic

(A.51)

A.7 Conversion of Inertial (ECI) Position to

Earth-Fixed (ECEF) Position

Since a co-rotating atmosphere is assumed, the position of the satellite and

the Sun in the Earth-Centered Earth-Fixed (ECEF) frame is required for the

DTM atmospheric density model. This subroutine is rather simple and is

described with the following equations. Given the current Greenwich Sidereal

Time GST , the rotation about the z-axis of the inertial frame is:

R3(GST ) =

cos(GST ) sin(GST ) 0− sin(GST ) cos(GST ) 0

0 0 1

(A.52)

and then the conversion of an inertial position vector to an ECEF position

vector is:

99

~rECEF = R3(GST ) ~rinertial (A.53)

100

Bibliography

[1] Barlier, F., Berger, C., Falin, J. L., Kockarts, G., and Thuillier, G., “A

Thermospheric Model Based on Satellite Drag Data,” Ann. Geophys., T.

34, Fasc. 1, 1978, p. 9-24.

[2] Bate, R. R., Mueller, D. D., and White, J. E., Fundamentals of Astrody-

namics, Dover Publications Inc., New York, NY, 1971.

[3] Battin, D. A., An Introduction to the Mathematics and Methods of As-

trodynamics, AIAA Inc., New York, NY, 1987.

[4] Canale, R. P., and Chapra, S. C., Numerical Methods for Engineers, 2nd

ed., McGraw-Hill, Inc., New York, NY, 1988.

[5] Cefola, P. J., Long, A. C., and Holloway, G., “The Long-Term Predic-

tion of Artificial Satellite Orbits,” AIAA Paper 74-170, AIAA Aerospace

Sciences Meeting, Washington, D.C., 1974.

[6] Davis, P. J., and Polonsky, I., “Numerical Interpolation, Differentia-

tion, and Integration,” Handbook of Mathematical Functions with Formu-

las, Graphs, and Mathematical Tables, U.S. Government Printing Office,

Washington D.C., 1967.

[7] Dickey, J. O., Bentley, C. R., Bilham, R., Carton, R., Eanes, R. J., Her-

ring, T. A., Kaula, W. M., Lagerloef, G. S. E., Rojstaczer, S., Smith,

W. H. F., Van den Dool, H. M., Wahr, J. M., and Zuber, M. T., “Con-

tributions of Satellite Gravity Measurements to Earth Science, Global

101

Change, and Natural Hazards Research,” Spring Meeting of the Ameri-

can Geophysical Union, Baltimore, Maryland, May 27-30, 1997.

[8] Euler, H., “Solar Activity Inputs for Upper Atmospheric Models Used

in Programs to Estimate Spacecraft Orbital Lifetime,” Memorandom

from EL23/Chief, Electromagnetics and Aerospace Environments Branch,

MSFC, Alabama, May 12, 1998. Mathematical Tables, U.S. Government

Printing Office, Washington D.C., 1967.

[9] Gauss, K. F., Theory of the Motion of the Heavenly Bodies Moving about

the Sun, 1809, Translation and Reprint in 1963, Dover Publications, New

York, NY.

[10] Hickey, M. P., “The NASA Engineering Thermosphere Model,” NASA

CR-179359, Washington, D.C., 1988.

[11] Hoots, F. R., and France, R. G., “An Analytical Satellite Theory Using

Gravity and a Dynamic Atmosphere,” Celestial Mechanics, Vol. 40, 1987,

p. 1-18.

[12] Kyner, W. T., “Averaging Method in Celestial Mechanics,” The Theory of

Orbits in the Solar System and in Stellar System, edited by Contopoulous,

G., Academic Press, London, 1966.

[13] Liu, J. J. F., and Alford, R. L., “Semianalytic Theory for a Close Earth

Artificial Satellite,” Journal of Guidance and Control, Vol. 3, No. 4, 1980,

p. 304-311; also AIAA Paper 79-0123, Jan. 1979.

[14] Liu, J. J. F., “Advances in Orbit Theory for an Artificial Satellite with

Drag,” Journal of Astronautical Sciences, Vol. 31, 1983, p. 165.

[15] Mazanek, D. and Kumar, R. R., private communication, NASA Langley

Research Center and Analytical Mechanics Associates, Inc., Hampton,

VA, 1998.

102

[16] Stanton, R., Bettadpur, S., Dunn, C., Renner, K. P., and Watkins,

M., GRACE Science and Mission Requirements Document, Jet Propul-

sion Laboratory, University of Texas Center for Space Research, Geo-

ForschungsZentrum Potsdam, 327-200, baselined May 15, 1998.

[17] Vallado, D. A., Fundamentals of Astrodynamics and Applications,

McGraw-Hill, Inc., New York, NY, 1997.

[18] Wertz, J. A., Spacecraft Attitude Determination and Control, Reidel Pub-

lishing Company, Dordrecht, Holland, 1978.

103

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