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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
6
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
32
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
days into GRACE mission
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
days into GRACE mission
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
days into GRACE mission
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
days into GRACE mission
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
days into GRACE mission
deca
y ra
te (
km/d
ay)
Actual Daily Flux 13-month Averaged Flux
Figure 3.8: Orbital Decay Rate (Epoch: 1/1/89)
0 100 200 300 400 500 600100
150
200
250
300
350
days into GRACE mission
F10
.7
Actual Daily Flux 13-month Averaged Flux
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
days into GRACE mission
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 3.10: Orbital Decay (Epoch: 3/1/92)
0 100 200 300 400 500 60010
-2
10-1
100
days into GRACE mission
deca
y ra
te (
km/d
ay)
Actual Daily Flux 13-month Averaged Flux
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
days into GRACE mission
F10
.7
Actual Daily Flux 13-month Averaged Flux
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
44
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
56
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
(Initial Eccentricity, eo = 0.001)
Initial Omega (deg) [Phase w.r.t. Sun]
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
Initial Omega (deg) [Phase w.r.t. Sun]
Life
time
(yea
rs)
(Initial Eccentricity, eo = 0.005)
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
(Initial Eccentricity, eo = 0.005)
Initial Omega (deg) [Phase w.r.t. Sun]
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
Initial Omega (deg) [Phase w.r.t. Sun]
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
Initial Omega (deg) [Phase w.r.t. Sun]
Life
time
(yea
rs)
(Initial Altitude, ho = 475 km)
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
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 5.19: 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
Initial Omega (deg) [Phase w.r.t. Sun]
Life
time
(yea
rs)
(Initial Eccentricity, eo = 0.005)
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
Initial Omega (deg) [Phase w.r.t. Sun]
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 5.21: 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
Initial Omega (deg) [Phase w.r.t. Sun]
Life
time
(yea
rs)
(Initial Altitude, ho = 475 km)
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
Initial Omega (deg) [Phase w.r.t. Sun]
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
(Initial Eccentricity, eo = 0.001)
Initial Omega (deg) [Phase w.r.t. Sun]
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
Initial Omega (deg) [Phase w.r.t. Sun]
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
(Initial Eccentricity, eo = 0.005)
Initial Omega (deg) [Phase w.r.t. Sun]
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)
(Initial Altitude, ho = 450 km)
0 50 100 150 200
2
2.5
3
3.5
Initial Omega (deg)
Life
time
(yea
rs)
(Initial Altitude, ho = 455 km)
METDTM
0 50 100 150 200
2.5
3
3.5
4
Initial Omega (deg)
Life
time
(yea
rs)
(Initial Altitude, ho = 460 km)
0 50 100 150 2002.5
3
3.5
4
4.5
5
Initial Omega (deg)
Life
time
(yea
rs)
(Initial Altitude, ho = 465 km)
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)
(Initial Eccentricity, eo = 0.003)
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
82
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
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103
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